LMFDB - Embedded newform 304.6.i.d.49.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,6,Mod(49,304)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(304, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("304.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 304 = 2^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	304.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$48.7566812231$
Analytic rank:	$0$
Dimension:	$18$
Relative dimension:	$9$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 5 x^{17} - 3057 x^{16} + 14564 x^{15} + 3829838 x^{14} - 15907074 x^{13} + \cdots + 66\!\cdots\!83 $ x^18 - 5x^17 - 3057x^16 + 14564x^15 + 3829838x^14 - 15907074x^13 - 2546775754x^12 + 7879525640x^11 + 976140188391x^10 - 1502012839499x^9 - 219575028272723x^8 - 104014033904304x^7 + 28115925509664606x^6 + 76771209356887914x^5 - 1810997176122162810x^4 - 9301294316540690604x^3 + 37701980111568799233x^2 + 354684213311311368351*x + 668276771042404734183
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{16}\cdot 3^{3}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 76)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			49.3
Root			$17.4003 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	304.49
Dual form			304.6.i.d.273.3

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-7.95017 + 13.7701i) q^{3} +(-15.4645 + 26.7852i) q^{5} -132.225 q^{7} +(-4.91049 - 8.50521i) q^{9} +O(q^{10})$	\(q+(-7.95017 + 13.7701i) q^{3} +(-15.4645 + 26.7852i) q^{5} -132.225 q^{7} +(-4.91049 - 8.50521i) q^{9} -670.612 q^{11} +(-411.330 - 712.445i) q^{13} +(-245.890 - 425.895i) q^{15} +(731.251 - 1266.56i) q^{17} +(1573.26 + 30.7850i) q^{19} +(1051.21 - 1820.76i) q^{21} +(1145.60 + 1984.23i) q^{23} +(1084.20 + 1877.89i) q^{25} -3707.63 q^{27} +(-1381.18 - 2392.27i) q^{29} -10591.8 q^{31} +(5331.48 - 9234.40i) q^{33} +(2044.79 - 3541.69i) q^{35} +4815.22 q^{37} +13080.6 q^{39} +(-7285.18 + 12618.3i) q^{41} +(5150.23 - 8920.46i) q^{43} +303.752 q^{45} +(2555.09 + 4425.55i) q^{47} +676.549 q^{49} +(11627.1 + 20138.8i) q^{51} +(-3724.23 - 6450.55i) q^{53} +(10370.7 - 17962.5i) q^{55} +(-12931.6 + 21419.2i) q^{57} +(17150.1 - 29704.9i) q^{59} +(20608.6 + 35695.1i) q^{61} +(649.291 + 1124.60i) q^{63} +25444.0 q^{65} +(19625.6 + 33992.5i) q^{67} -36430.7 q^{69} +(-5848.69 + 10130.2i) q^{71} +(-13696.8 + 23723.6i) q^{73} -34478.3 q^{75} +88671.9 q^{77} +(2583.48 - 4474.71i) q^{79} +(30669.5 - 53121.2i) q^{81} +86152.6 q^{83} +(22616.8 + 39173.5i) q^{85} +43922.4 q^{87} +(25384.0 + 43966.4i) q^{89} +(54388.3 + 94203.3i) q^{91} +(84206.9 - 145851. i) q^{93} +(-25154.2 + 41664.1i) q^{95} +(-13304.7 + 23044.5i) q^{97} +(3293.03 + 5703.70i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 11 q^{3} + 11 q^{5} - 336 q^{7} - 902 q^{9}+O(q^{10}) $ 18 * q + 11 * q^3 + 11 * q^5 - 336 * q^7 - 902 * q^9	$ 18 q + 11 q^{3} + 11 q^{5} - 336 q^{7} - 902 q^{9} + 320 q^{11} + 227 q^{13} + 101 q^{15} + 179 q^{17} + 868 q^{19} - 5700 q^{21} + 3425 q^{23} - 7054 q^{25} - 14722 q^{27} - 7349 q^{29} + 9960 q^{31} - 2998 q^{33} - 15888 q^{35} + 26444 q^{37} + 30246 q^{39} - 7311 q^{41} + 8283 q^{43} - 62164 q^{45} - 37603 q^{47} + 124738 q^{49} - 47227 q^{51} - 20337 q^{53} - 716 q^{55} - 57555 q^{57} + 74455 q^{59} - 7569 q^{61} + 52544 q^{63} + 188998 q^{65} + 26177 q^{67} + 116282 q^{69} + 53463 q^{71} - 14103 q^{73} - 120912 q^{75} - 31960 q^{77} - 31825 q^{79} - 21137 q^{81} - 82600 q^{83} - 50787 q^{85} + 339766 q^{87} - 155197 q^{89} + 2800 q^{91} - 46460 q^{93} - 49315 q^{95} + 111241 q^{97} + 193544 q^{99}+O(q^{100}) $ 18 * q + 11 * q^3 + 11 * q^5 - 336 * q^7 - 902 * q^9 + 320 * q^11 + 227 * q^13 + 101 * q^15 + 179 * q^17 + 868 * q^19 - 5700 * q^21 + 3425 * q^23 - 7054 * q^25 - 14722 * q^27 - 7349 * q^29 + 9960 * q^31 - 2998 * q^33 - 15888 * q^35 + 26444 * q^37 + 30246 * q^39 - 7311 * q^41 + 8283 * q^43 - 62164 * q^45 - 37603 * q^47 + 124738 * q^49 - 47227 * q^51 - 20337 * q^53 - 716 * q^55 - 57555 * q^57 + 74455 * q^59 - 7569 * q^61 + 52544 * q^63 + 188998 * q^65 + 26177 * q^67 + 116282 * q^69 + 53463 * q^71 - 14103 * q^73 - 120912 * q^75 - 31960 * q^77 - 31825 * q^79 - 21137 * q^81 - 82600 * q^83 - 50787 * q^85 + 339766 * q^87 - 155197 * q^89 + 2800 * q^91 - 46460 * q^93 - 49315 * q^95 + 111241 * q^97 + 193544 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/304\mathbb{Z}\right)^\times$.

$n$	$97$	$191$	$229$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−7.95017	+	13.7701i	−0.510004	+	0.883352i	0.489929	+	0.871762i	$0.337023\pi$
$3$	−7.95017	+	13.7701i	−0.510004	+	0.883352i	−0.999933	+	0.0115902i	$0.996311\pi$
$4$		0			0
$5$	−15.4645	+	26.7852i	−0.276637	+	0.479149i	−0.970547	−	0.240913i	$-0.922553\pi$
$5$	−15.4645	+	26.7852i	−0.276637	+	0.479149i	0.693910	+	0.720062i	$0.255887\pi$
$6$		0			0
$7$	−132.225			−1.01993			−0.509964	−	0.860196i	$-0.670341\pi$
$7$	−132.225			−1.01993			−0.509964	+	0.860196i	$0.670341\pi$
$8$		0			0
$9$	−4.91049	−	8.50521i	−0.0202078	−	0.0350009i
$10$		0			0
$11$	−670.612			−1.67105			−0.835525	−	0.549452i	$-0.814837\pi$
$11$	−670.612			−1.67105			−0.835525	+	0.549452i	$0.814837\pi$
$12$		0			0
$13$	−411.330	−	712.445i	−0.675044	−	1.16921i	−0.976456	−	0.215717i	$-0.930791\pi$
$13$	−411.330	−	712.445i	−0.675044	−	1.16921i	0.301412	−	0.953494i	$-0.402542\pi$
$14$		0			0
$15$	−245.890	−	425.895i	−0.282172	−	0.488736i
$16$		0			0
$17$	731.251	−	1266.56i	0.613683	−	1.06293i	−0.376931	−	0.926241i	$-0.623021\pi$
$17$	731.251	−	1266.56i	0.613683	−	1.06293i	0.990614	−	0.136689i	$-0.0436461\pi$
$18$		0			0
$19$	1573.26	+	30.7850i	0.999809	+	0.0195639i
$19$	1573.26	+	30.7850i	0.999809	+	0.0195639i
$20$		0			0
$21$	1051.21	−	1820.76i	0.520167	−	0.900956i
$22$		0			0
$23$	1145.60	+	1984.23i	0.451556	+	0.782118i	0.998483	−	0.0550625i	$-0.0175358\pi$
$23$	1145.60	+	1984.23i	0.451556	+	0.782118i	−0.546927	+	0.837180i	$0.684202\pi$
$24$		0			0
$25$	1084.20	+	1877.89i	0.346944	+	0.600925i
$26$		0			0
$27$	−3707.63			−0.978783
$28$		0			0
$29$	−1381.18	−	2392.27i	−0.304968	−	0.528221i	0.672286	−	0.740292i	$-0.265313\pi$
$29$	−1381.18	−	2392.27i	−0.304968	−	0.528221i	−0.977254	+	0.212071i	$0.931979\pi$
$30$		0			0
$31$	−10591.8			−1.97955			−0.989776	−	0.142628i	$-0.954445\pi$
$31$	−10591.8			−1.97955			−0.989776	+	0.142628i	$0.954445\pi$
$32$		0			0
$33$	5331.48	−	9234.40i	0.852242	−	1.47613i
$34$		0			0
$35$	2044.79	−	3541.69i	0.282150	−	0.488698i
$36$		0			0
$37$	4815.22			0.578245			0.289122	−	0.957292i	$-0.406637\pi$
$37$	4815.22			0.578245			0.289122	+	0.957292i	$0.406637\pi$
$38$		0			0
$39$	13080.6			1.37710
$40$		0			0
$41$	−7285.18	+	12618.3i	−0.676832	+	1.17231i	0.299098	+	0.954222i	$0.403314\pi$
$41$	−7285.18	+	12618.3i	−0.676832	+	1.17231i	−0.975930	+	0.218085i	$0.930019\pi$
$42$		0			0
$43$	5150.23	−	8920.46i	0.424772	−	0.735726i	−0.571628	−	0.820513i	$-0.693688\pi$
$43$	5150.23	−	8920.46i	0.424772	−	0.735726i	0.996399	+	0.0847874i	$0.0270211\pi$
$44$		0			0
$45$	303.752			0.0223608
$46$		0			0
$47$	2555.09	+	4425.55i	0.168718	+	0.292228i	0.937969	−	0.346718i	$-0.112704\pi$
$47$	2555.09	+	4425.55i	0.168718	+	0.292228i	−0.769251	+	0.638946i	$0.779371\pi$
$48$		0			0
$49$	676.549			0.0402540
$50$		0			0
$51$	11627.1	+	20138.8i	0.625961	+	1.08420i
$52$		0			0
$53$	−3724.23	−	6450.55i	−0.182115	−	0.315433i	0.760485	−	0.649355i	$-0.224961\pi$
$53$	−3724.23	−	6450.55i	−0.182115	−	0.315433i	−0.942601	+	0.333922i	$0.891628\pi$
$54$		0			0
$55$	10370.7	−	17962.5i	0.462274	−	0.800682i
$56$		0			0
$57$	−12931.6	+	21419.2i	−0.527188	+	0.873206i
$58$		0			0
$59$	17150.1	−	29704.9i	0.641412	−	1.11096i	−0.343705	−	0.939077i	$-0.611682\pi$
$59$	17150.1	−	29704.9i	0.641412	−	1.11096i	0.985118	−	0.171881i	$-0.0549845\pi$
$60$		0			0
$61$	20608.6	+	35695.1i	0.709126	+	1.22824i	0.965182	+	0.261580i	$0.0842434\pi$
$61$	20608.6	+	35695.1i	0.709126	+	1.22824i	−0.256056	+	0.966662i	$0.582423\pi$
$62$		0			0
$63$	649.291	+	1124.60i	0.0206105	+	0.0356984i
$64$		0			0
$65$	25444.0			0.746968
$66$		0			0
$67$	19625.6	+	33992.5i	0.534116	+	0.925116i	0.999206	+	0.0398521i	$0.0126887\pi$
$67$	19625.6	+	33992.5i	0.534116	+	0.925116i	−0.465090	+	0.885263i	$0.653978\pi$
$68$		0			0
$69$	−36430.7			−0.921181
$70$		0			0
$71$	−5848.69	+	10130.2i	−0.137693	+	0.238492i	−0.926623	−	0.375992i	$-0.877302\pi$
$71$	−5848.69	+	10130.2i	−0.137693	+	0.238492i	0.788930	+	0.614483i	$0.210635\pi$
$72$		0			0
$73$	−13696.8	+	23723.6i	−0.300824	+	0.521043i	−0.976323	−	0.216318i	$-0.930595\pi$
$73$	−13696.8	+	23723.6i	−0.300824	+	0.521043i	0.675498	+	0.737361i	$0.263928\pi$
$74$		0			0
$75$	−34478.3			−0.707771
$76$		0			0
$77$	88671.9			1.70435
$78$		0			0
$79$	2583.48	−	4474.71i	0.0465733	−	0.0806673i	−0.841799	−	0.539791i	$-0.818503\pi$
$79$	2583.48	−	4474.71i	0.0465733	−	0.0806673i	0.888372	+	0.459124i	$0.151837\pi$
$80$		0			0
$81$	30669.5	−	53121.2i	0.519391	−	0.899612i
$82$		0			0
$83$	86152.6			1.37269			0.686346	−	0.727275i	$-0.259214\pi$
$83$	86152.6			1.37269			0.686346	+	0.727275i	$0.259214\pi$
$84$		0			0
$85$	22616.8	+	39173.5i	0.339535	+	0.588091i
$86$		0			0
$87$	43922.4			0.622140
$88$		0			0
$89$	25384.0	+	43966.4i	0.339692	+	0.588364i	0.984375	−	0.176087i	$-0.0563438\pi$
$89$	25384.0	+	43966.4i	0.339692	+	0.588364i	−0.644683	+	0.764450i	$0.723011\pi$
$90$		0			0
$91$	54388.3	+	94203.3i	0.688497	+	1.19251i
$92$		0			0
$93$	84206.9	−	145851.i	1.00958	−	1.74864i
$94$		0			0
$95$	−25154.2	+	41664.1i	−0.285958	+	0.473645i
$96$		0			0
$97$	−13304.7	+	23044.5i	−0.143574	+	0.248678i	−0.928840	−	0.370481i	$-0.879193\pi$
$97$	−13304.7	+	23044.5i	−0.143574	+	0.248678i	0.785266	+	0.619159i	$0.212526\pi$
$98$		0			0
$99$	3293.03	+	5703.70i	0.0337682	+	0.0584882i
$100$		0			0
$101$	32085.7	+	55574.0i	0.312974	+	0.542086i	0.979005	−	0.203838i	$-0.0653416\pi$
$101$	32085.7	+	55574.0i	0.312974	+	0.542086i	−0.666031	+	0.745924i	$0.732008\pi$
$102$		0			0
$103$	−66441.0			−0.617082			−0.308541	−	0.951211i	$-0.599841\pi$
$103$	−66441.0			−0.617082			−0.308541	+	0.951211i	$0.599841\pi$
$104$		0			0
$105$	32512.9	+	56314.1i	0.287795	+	0.498475i
$106$		0			0
$107$	187958.			1.58709			0.793546	−	0.608510i	$-0.208232\pi$
$107$	187958.			1.58709			0.793546	+	0.608510i	$0.208232\pi$
$108$		0			0
$109$	89626.3	−	155237.i	0.722552	−	1.25150i	−0.237422	−	0.971407i	$-0.576302\pi$
$109$	89626.3	−	155237.i	0.722552	−	1.25150i	0.959974	−	0.280090i	$-0.0903643\pi$
$110$		0			0
$111$	−38281.8	+	66306.0i	−0.294907	+	0.510794i
$112$		0			0
$113$	−59332.4			−0.437115			−0.218557	−	0.975824i	$-0.570135\pi$
$113$	−59332.4			−0.437115			−0.218557	+	0.975824i	$0.570135\pi$
$114$		0			0
$115$	−70864.1			−0.499668
$116$		0			0
$117$	−4039.66	+	6996.90i	−0.0272823	+	0.0472542i
$118$		0			0
$119$	−96689.9	+	167472.i	−0.625913	+	1.08411i
$120$		0			0
$121$	288669.			1.79241
$122$		0			0
$123$	−115837.	−	200635.i	−0.690374	−	1.19576i
$124$		0			0
$125$	−163719.			−0.937184
$126$		0			0
$127$	33405.1	+	57859.3i	0.183782	+	0.318320i	0.943165	−	0.332324i	$-0.107833\pi$
$127$	33405.1	+	57859.3i	0.183782	+	0.318320i	−0.759383	+	0.650643i	$0.774499\pi$
$128$		0			0
$129$	81890.4	+	141838.i	0.433270	+	0.750446i
$130$		0			0
$131$	175425.	−	303846.i	0.893129	−	1.54695i	0.0570264	−	0.998373i	$-0.481838\pi$
$131$	175425.	−	303846.i	0.893129	−	1.54695i	0.836103	−	0.548573i	$-0.184829\pi$
$132$		0			0
$133$	−208025.	−	4070.56i	−1.01973	−	0.0199538i
$134$		0			0
$135$	57336.5	−	99309.7i	0.270767	−	0.468983i
$136$		0			0
$137$	−124390.	−	215450.i	−0.566219	−	0.980719i	−0.996935	−	0.0782321i	$-0.975072\pi$
$137$	−124390.	−	215450.i	−0.566219	−	0.980719i	0.430717	−	0.902487i	$-0.358261\pi$
$138$		0			0
$139$	−221518.	−	383680.i	−0.972459	−	1.68435i	−0.688078	−	0.725637i	$-0.741545\pi$
$139$	−221518.	−	383680.i	−0.972459	−	1.68435i	−0.284381	−	0.958711i	$-0.591788\pi$
$140$		0			0
$141$	−81253.7			−0.344188
$142$		0			0
$143$	275843.	+	477774.i	1.12803	+	1.95381i
$144$		0			0
$145$	85436.8			0.337462
$146$		0			0
$147$	−5378.68	+	9316.15i	−0.0205297	+	0.0355585i
$148$		0			0
$149$	−120138.	+	208085.i	−0.443317	+	0.767847i	−0.997933	−	0.0642591i	$-0.979532\pi$
$149$	−120138.	+	208085.i	−0.443317	+	0.767847i	0.554617	+	0.832106i	$0.312865\pi$
$150$		0			0
$151$	166073.			0.592730			0.296365	−	0.955075i	$-0.404226\pi$
$151$	166073.			0.592730			0.296365	+	0.955075i	$0.404226\pi$
$152$		0			0
$153$	−14363.2			−0.0496046
$154$		0			0
$155$	163797.	−	283705.i	0.547617	−	0.948501i
$156$		0			0
$157$	−50926.7	+	88207.7i	−0.164891	+	0.285599i	−0.936617	−	0.350356i	$-0.886061\pi$
$157$	−50926.7	+	88207.7i	−0.164891	+	0.285599i	0.771726	+	0.635956i	$0.219394\pi$
$158$		0			0
$159$	118433.			0.371518
$160$		0			0
$161$	−151477.	−	262365.i	−0.460555	−	0.797704i
$162$		0			0
$163$	256024.			0.754764			0.377382	−	0.926058i	$-0.376824\pi$
$163$	256024.			0.754764			0.377382	+	0.926058i	$0.376824\pi$
$164$		0			0
$165$	164897.	+	285610.i	0.471523	+	0.816702i
$166$		0			0
$167$	−53946.9	−	93438.8i	−0.149684	−	0.259260i	0.781427	−	0.623997i	$-0.214492\pi$
$167$	−53946.9	−	93438.8i	−0.149684	−	0.259260i	−0.931111	+	0.364737i	$0.881159\pi$
$168$		0			0
$169$	−152738.	+	264551.i	−0.411369	+	0.712512i
$170$		0			0
$171$	−7463.64	−	13532.1i	−0.0195191	−	0.0353895i
$172$		0			0
$173$	197251.	−	341649.i	0.501077	−	0.867890i	−0.498922	−	0.866647i	$-0.666271\pi$
$173$	197251.	−	341649.i	0.501077	−	0.867890i	0.999999	−	0.00124378i	$-0.000395908\pi$
$174$		0			0
$175$	−143359.	−	248305.i	−0.353858	−	0.612900i
$176$		0			0
$177$	272693.	+	472318.i	0.654245	+	1.13319i
$178$		0			0
$179$	229285.			0.534863			0.267432	−	0.963577i	$-0.413825\pi$
$179$	229285.			0.534863			0.267432	+	0.963577i	$0.413825\pi$
$180$		0			0
$181$	−269882.	−	467449.i	−0.612318	−	1.06057i	−0.990849	−	0.134977i	$-0.956904\pi$
$181$	−269882.	−	467449.i	−0.612318	−	1.06057i	0.378531	−	0.925589i	$-0.376429\pi$
$182$		0			0
$183$	−655367.			−1.44663
$184$		0			0
$185$	−74464.8	+	128977.i	−0.159964	+	0.277065i
$186$		0			0
$187$	−490386.	+	849373.i	−1.02550	+	1.77621i
$188$		0			0
$189$	490242.			0.998289
$190$		0			0
$191$	−448848.			−0.890259			−0.445129	−	0.895466i	$-0.646842\pi$
$191$	−448848.			−0.890259			−0.445129	+	0.895466i	$0.646842\pi$
$192$		0			0
$193$	23904.7	−	41404.2i	0.0461945	−	0.0800113i	−0.842004	−	0.539472i	$-0.818624\pi$
$193$	23904.7	−	41404.2i	0.0461945	−	0.0800113i	0.888198	+	0.459461i	$0.151957\pi$
$194$		0			0
$195$	−202284.	+	350366.i	−0.380957	+	0.659836i
$196$		0			0
$197$	43802.6			0.0804145			0.0402072	−	0.999191i	$-0.487198\pi$
$197$	43802.6			0.0804145			0.0402072	+	0.999191i	$0.487198\pi$
$198$		0			0
$199$	−300067.	−	519730.i	−0.537137	−	0.930348i	−0.999057	−	0.0434264i	$-0.986173\pi$
$199$	−300067.	−	519730.i	−0.537137	−	0.930348i	0.461920	−	0.886922i	$-0.347161\pi$
$200$		0			0
$201$	−624107.			−1.08960
$202$		0			0
$203$	182627.	+	316319.i	0.311046	+	0.538747i
$204$		0			0
$205$	−225323.	−	390271.i	−0.374473	−	0.648607i
$206$		0			0
$207$	11250.9	−	19487.1i	0.0182499	−	0.0316097i
$208$		0			0
$209$	−1.05505e6	−	20644.8i	−1.67073	−	0.0326923i
$210$		0			0
$211$	−239624.	+	415040.i	−0.370530	+	0.641777i	−0.989647	−	0.143522i	$-0.954157\pi$
$211$	−239624.	+	415040.i	−0.370530	+	0.641777i	0.619117	+	0.785299i	$0.287491\pi$
$212$		0			0
$213$	−92996.2	−	161074.i	−0.140448	−	0.243263i
$214$		0			0
$215$	159291.	+	275900.i	0.235015	+	0.407058i
$216$		0			0
$217$	1.40051e6			2.01900
$218$		0			0
$219$	−217784.	−	377214.i	−0.306843	−	0.531468i
$220$		0			0
$221$	−1.20314e6			−1.65705
$222$		0			0
$223$	93000.5	−	161082.i	0.125234	−	0.216912i	−0.796590	−	0.604520i	$-0.793365\pi$
$223$	93000.5	−	161082.i	0.125234	−	0.216912i	0.921824	+	0.387608i	$0.126698\pi$
$224$		0			0
$225$	10647.9	−	18442.7i	0.0140219	−	0.0242867i
$226$		0			0
$227$	−131418.			−0.169274			−0.0846371	−	0.996412i	$-0.526973\pi$
$227$	−131418.			−0.169274			−0.0846371	+	0.996412i	$0.526973\pi$
$228$		0			0
$229$	105774.			0.133288			0.0666440	−	0.997777i	$-0.478771\pi$
$229$	105774.			0.133288			0.0666440	+	0.997777i	$0.478771\pi$
$230$		0			0
$231$	−704957.	+	1.22102e6i	−0.869226	+	1.50554i
$232$		0			0
$233$	504870.	−	874460.i	0.609241	−	1.05524i	−0.382124	−	0.924111i	$-0.624807\pi$
$233$	504870.	−	874460.i	0.609241	−	1.05524i	0.991366	−	0.131126i	$-0.0418593\pi$
$234$		0			0
$235$	−158053.			−0.186695
$236$		0			0
$237$	41078.2	+	71149.5i	0.0475051	+	0.0822813i
$238$		0			0
$239$	538983.			0.610351			0.305176	−	0.952296i	$-0.401285\pi$
$239$	538983.			0.610351			0.305176	+	0.952296i	$0.401285\pi$
$240$		0			0
$241$	618134.	+	1.07064e6i	0.685551	+	1.18741i	0.973263	+	0.229692i	$0.0737720\pi$
$241$	618134.	+	1.07064e6i	0.685551	+	1.18741i	−0.287712	+	0.957717i	$0.592895\pi$
$242$		0			0
$243$	37179.3	+	64396.4i	0.0403911	+	0.0699594i
$244$		0			0
$245$	−10462.5	+	18121.5i	−0.0111357	+	0.0192877i
$246$		0			0
$247$	−625197.	−	1.13352e6i	−0.652040	−	1.18219i
$248$		0			0
$249$	−684928.	+	1.18633e6i	−0.700078	+	1.21257i
$250$		0			0
$251$	665047.	+	1.15190e6i	0.666298	+	1.15406i	0.978932	+	0.204188i	$0.0654553\pi$
$251$	665047.	+	1.15190e6i	0.666298	+	1.15406i	−0.312634	+	0.949874i	$0.601211\pi$
$252$		0			0
$253$	−768250.	−	1.33065e6i	−0.754573	−	1.30696i
$254$		0			0
$255$	−719230.			−0.692656
$256$		0			0
$257$	471458.	+	816590.i	0.445257	+	0.771207i	0.998070	−	0.0620979i	$-0.0197791\pi$
$257$	471458.	+	816590.i	0.445257	+	0.771207i	−0.552813	+	0.833305i	$0.686446\pi$
$258$		0			0
$259$	−636694.			−0.589768
$260$		0			0
$261$	−13564.5	+	23494.4i	−0.0123255	+	0.0213483i
$262$		0			0
$263$	848058.	−	1.46888e6i	0.756025	−	1.30947i	−0.188838	−	0.982008i	$-0.560472\pi$
$263$	848058.	−	1.46888e6i	0.756025	−	1.30947i	0.944863	−	0.327465i	$-0.106194\pi$
$264$		0			0
$265$	230373.			0.201519
$266$		0			0
$267$	−807229.			−0.692977
$268$		0			0
$269$	403755.	−	699325.i	0.340203	−	0.589248i	−0.644268	−	0.764800i	$-0.722838\pi$
$269$	403755.	−	699325.i	0.340203	−	0.589248i	0.984470	+	0.175552i	$0.0561710\pi$
$270$		0			0
$271$	1.02987e6	−	1.78379e6i	0.851845	−	1.47544i	−0.0276959	−	0.999616i	$-0.508817\pi$
$271$	1.02987e6	−	1.78379e6i	0.851845	−	1.47544i	0.879541	−	0.475823i	$-0.157850\pi$
$272$		0			0
$273$	−1.72958e6			−1.40454
$274$		0			0
$275$	−727078.	−	1.25934e6i	−0.579761	−	1.00418i
$276$		0			0
$277$	−1.06729e6			−0.835761			−0.417881	−	0.908502i	$-0.637227\pi$
$277$	−1.06729e6			−0.835761			−0.417881	+	0.908502i	$0.637227\pi$
$278$		0			0
$279$	52011.1	+	90085.8i	0.0400023	+	0.0692861i
$280$		0			0
$281$	−352862.	−	611176.i	−0.266587	−	0.461743i	0.701391	−	0.712777i	$-0.252563\pi$
$281$	−352862.	−	611176.i	−0.266587	−	0.461743i	−0.967978	+	0.251034i	$0.919229\pi$
$282$		0			0
$283$	−190924.	+	330690.i	−0.141708	+	0.245446i	−0.928140	−	0.372231i	$-0.878593\pi$
$283$	−190924.	+	330690.i	−0.141708	+	0.245446i	0.786432	+	0.617677i	$0.211926\pi$
$284$		0			0
$285$	−373739.	−	677613.i	−0.272556	−	0.494162i
$286$		0			0
$287$	963286.	−	1.66846e6i	0.690320	−	1.19567i
$288$		0			0
$289$	−359527.	−	622720.i	−0.253214	−	0.438579i
$290$		0			0
$291$	−211550.	−	366415.i	−0.146447	−	0.253653i
$292$		0			0
$293$	−2.08651e6			−1.41988			−0.709941	−	0.704261i	$-0.751278\pi$
$293$	−2.08651e6			−1.41988			−0.709941	+	0.704261i	$0.751278\pi$
$294$		0			0
$295$	530435.	+	918740.i	0.354876	+	0.614664i
$296$		0			0
$297$	2.48638e6			1.63560
$298$		0			0
$299$	942435.	−	1.63235e6i	0.609640	−	1.05593i
$300$		0			0
$301$	−680991.	+	1.17951e6i	−0.433237	+	0.750388i
$302$		0			0
$303$	−1.02035e6			−0.638471
$304$		0			0
$305$	−1.27480e6			−0.784681
$306$		0			0
$307$	−805874.	+	1.39582e6i	−0.488002	+	0.845244i	−0.999905	−	0.0137995i	$-0.995607\pi$
$307$	−805874.	+	1.39582e6i	−0.488002	+	0.845244i	0.511903	+	0.859043i	$0.328941\pi$
$308$		0			0
$309$	528217.	−	914899.i	0.314714	−	0.545101i
$310$		0			0
$311$	−785721.			−0.460646			−0.230323	−	0.973114i	$-0.573978\pi$
$311$	−785721.			−0.460646			−0.230323	+	0.973114i	$0.573978\pi$
$312$		0			0
$313$	1.34106e6	+	2.32278e6i	0.773724	+	1.34013i	0.935509	+	0.353303i	$0.114942\pi$
$313$	1.34106e6	+	2.32278e6i	0.773724	+	1.34013i	−0.161785	+	0.986826i	$0.551725\pi$
$314$		0			0
$315$	−40163.7			−0.0228065
$316$		0			0
$317$	−1.30544e6	−	2.26109e6i	−0.729642	−	1.26378i	−0.957034	−	0.289974i	$-0.906353\pi$
$317$	−1.30544e6	−	2.26109e6i	−0.729642	−	1.26378i	0.227392	−	0.973803i	$-0.426980\pi$
$318$		0			0
$319$	926235.	+	1.60429e6i	0.509618	+	0.882684i
$320$		0			0
$321$	−1.49430e6	+	2.58821e6i	−0.809423	+	1.40196i
$322$		0			0
$323$	1.18944e6	−	1.97012e6i	0.634361	−	1.05072i
$324$		0			0
$325$	891929.	−	1.54487e6i	0.468405	−	0.811302i
$326$		0			0
$327$	1.42509e6	+	2.46833e6i	0.737008	+	1.27654i
$328$		0			0
$329$	−337848.	−	585170.i	−0.172080	−	0.298052i
$330$		0			0
$331$	−2.09316e6			−1.05010			−0.525052	−	0.851070i	$-0.675954\pi$
$331$	−2.09316e6			−1.05010			−0.525052	+	0.851070i	$0.675954\pi$
$332$		0			0
$333$	−23645.1	−	40954.4i	−0.0116850	−	0.0202391i
$334$		0			0
$335$	−1.21400e6			−0.591024
$336$		0			0
$337$	−1.63109e6	+	2.82513e6i	−0.782353	+	1.35508i	0.148214	+	0.988955i	$0.452647\pi$
$337$	−1.63109e6	+	2.82513e6i	−0.782353	+	1.35508i	−0.930568	+	0.366120i	$0.880686\pi$
$338$		0			0
$339$	471703.	−	817013.i	0.222930	−	0.386127i
$340$		0			0
$341$	7.10301e6			3.30793
$342$		0			0
$343$	2.13285e6			0.978872
$344$		0			0
$345$	563382.	−	975805.i	0.254833	−	0.441383i
$346$		0			0
$347$	−602372.	+	1.04334e6i	−0.268560	+	0.465159i	−0.968490	−	0.249052i	$-0.919881\pi$
$347$	−602372.	+	1.04334e6i	−0.268560	+	0.465159i	0.699930	+	0.714211i	$0.253214\pi$
$348$		0			0
$349$	2.97327e6			1.30668			0.653342	−	0.757063i	$-0.273366\pi$
$349$	2.97327e6			1.30668			0.653342	+	0.757063i	$0.273366\pi$
$350$		0			0
$351$	1.52506e6	+	2.64148e6i	0.660722	+	1.14440i
$352$		0			0
$353$	2.66290e6			1.13741			0.568706	−	0.822541i	$-0.307444\pi$
$353$	2.66290e6			1.13741			0.568706	+	0.822541i	$0.307444\pi$
$354$		0			0
$355$	−180894.	−	313317.i	−0.0761820	−	0.131951i
$356$		0			0
$357$	−1.53740e6	−	2.66286e6i	−0.638436	−	1.10580i
$358$		0			0
$359$	−415704.	+	720020.i	−0.170235	+	0.294855i	−0.938502	−	0.345274i	$-0.887786\pi$
$359$	−415704.	+	720020.i	−0.170235	+	0.294855i	0.768267	+	0.640129i	$0.221119\pi$
$360$		0			0
$361$	2.47420e6	+	96865.8i	0.999235	+	0.0391203i
$362$		0			0
$363$	−2.29497e6	+	3.97501e6i	−0.914136	+	1.58333i
$364$		0			0
$365$	−423629.	−	733746.i	−0.166438	−	0.288279i
$366$		0			0
$367$	−543635.	−	941604.i	−0.210689	−	0.364925i	0.741241	−	0.671239i	$-0.234237\pi$
$367$	−543635.	−	941604.i	−0.210689	−	0.364925i	−0.951930	+	0.306314i	$0.900904\pi$
$368$		0			0
$369$	143095.			0.0547090
$370$		0			0
$371$	492438.	+	852927.i	0.185745	+	0.321719i
$372$		0			0
$373$	3.00979e6			1.12012			0.560059	−	0.828452i	$-0.310778\pi$
$373$	3.00979e6			1.12012			0.560059	+	0.828452i	$0.310778\pi$
$374$		0			0
$375$	1.30160e6	−	2.25443e6i	0.477967	−	0.827863i
$376$		0			0
$377$	−1.13624e6	+	1.96803e6i	−0.411734	+	0.713145i
$378$		0			0
$379$	1.58924e6			0.568317			0.284158	−	0.958777i	$-0.408286\pi$
$379$	1.58924e6			0.568317			0.284158	+	0.958777i	$0.408286\pi$
$380$		0			0
$381$	−1.06230e6			−0.374918
$382$		0			0
$383$	−1.14784e6	+	1.98811e6i	−0.399837	+	0.692538i	−0.993705	−	0.112024i	$-0.964267\pi$
$383$	−1.14784e6	+	1.98811e6i	−0.399837	+	0.692538i	0.593869	+	0.804562i	$0.297600\pi$
$384$		0			0
$385$	−1.37126e6	+	2.37510e6i	−0.471486	+	0.816638i
$386$		0			0
$387$	−101160.			−0.0343347
$388$		0			0
$389$	−2.12902e6	−	3.68757e6i	−0.713355	−	1.23557i	−0.963590	−	0.267383i	$-0.913841\pi$
$389$	−2.12902e6	−	3.68757e6i	−0.713355	−	1.23557i	0.250235	−	0.968185i	$-0.419492\pi$
$390$		0			0
$391$	3.35087e6			1.10845
$392$		0			0
$393$	2.78933e6	+	4.83125e6i	0.910999	+	1.57790i
$394$		0			0
$395$	79904.2	+	138398.i	0.0257678	+	0.0446311i
$396$		0			0
$397$	−173073.	+	299772.i	−0.0551130	+	0.0954585i	−0.892266	−	0.451511i	$-0.850885\pi$
$397$	−173073.	+	299772.i	−0.0551130	+	0.0954585i	0.837153	+	0.546969i	$0.184219\pi$
$398$		0			0
$399$	1.70989e6	−	2.83216e6i	0.537694	−	0.890607i
$400$		0			0
$401$	777399.	−	1.34649e6i	0.241425	−	0.418161i	−0.719695	−	0.694290i	$-0.755718\pi$
$401$	777399.	−	1.34649e6i	0.241425	−	0.418161i	0.961121	+	0.276129i	$0.0890517\pi$
$402$		0			0
$403$	4.35674e6	+	7.54610e6i	1.33629	+	2.31451i
$404$		0			0
$405$	948576.	+	1.64298e6i	0.287365	+	0.497731i
$406$		0			0
$407$	−3.22914e6			−0.966276
$408$		0			0
$409$	−796042.	−	1.37878e6i	−0.235303	−	0.407557i	0.724058	−	0.689739i	$-0.242275\pi$
$409$	−796042.	−	1.37878e6i	−0.235303	−	0.407557i	−0.959361	+	0.282183i	$0.908942\pi$
$410$		0			0
$411$	3.95569e6			1.15509
$412$		0			0
$413$	−2.26768e6	+	3.92774e6i	−0.654195	+	1.13310i
$414$		0			0
$415$	−1.33230e6	+	2.30762e6i	−0.379737	+	0.657724i
$416$		0			0
$417$	7.04441e6			1.98383
$418$		0			0
$419$	4.11283e6			1.14447			0.572236	−	0.820089i	$-0.306076\pi$
$419$	4.11283e6			1.14447			0.572236	+	0.820089i	$0.306076\pi$
$420$		0			0
$421$	1.57964e6	−	2.73602e6i	0.434364	−	0.752341i	−0.562879	−	0.826539i	$-0.690306\pi$
$421$	1.57964e6	−	2.73602e6i	0.434364	−	0.752341i	0.997244	+	0.0741981i	$0.0236397\pi$
$422$		0			0
$423$	25093.5	−	43463.2i	0.00681883	−	0.0118106i
$424$		0			0
$425$	3.17129e6			0.851655
$426$		0			0
$427$	−2.72498e6	−	4.71980e6i	−0.723257	−	1.25272i
$428$		0			0
$429$	−8.77199e6			−2.30120
$430$		0			0
$431$	−508299.	−	880400.i	−0.131803	−	0.228290i	0.792568	−	0.609783i	$-0.208743\pi$
$431$	−508299.	−	880400.i	−0.131803	−	0.228290i	−0.924372	+	0.381493i	$0.875410\pi$
$432$		0			0
$433$	−1.49339e6	−	2.58663e6i	−0.382784	−	0.663001i	0.608675	−	0.793419i	$-0.291701\pi$
$433$	−1.49339e6	−	2.58663e6i	−0.382784	−	0.663001i	−0.991459	+	0.130419i	$0.958368\pi$
$434$		0			0
$435$	−679237.	+	1.17647e6i	−0.172107	+	0.298098i
$436$		0			0
$437$	1.74124e6	+	3.15698e6i	0.436168	+	0.790802i
$438$		0			0
$439$	3.05381e6	−	5.28936e6i	0.756277	−	1.30991i	−0.188459	−	0.982081i	$-0.560349\pi$
$439$	3.05381e6	−	5.28936e6i	0.756277	−	1.30991i	0.944737	−	0.327830i	$-0.106317\pi$
$440$		0			0
$441$	−3322.18	−	5754.19i	−0.000813443	−	0.00140892i
$442$		0			0
$443$	−2.29170e6	−	3.96935e6i	−0.554816	−	0.960970i	−0.997918	−	0.0644983i	$-0.979455\pi$
$443$	−2.29170e6	−	3.96935e6i	−0.554816	−	0.960970i	0.443102	−	0.896471i	$-0.353878\pi$
$444$		0			0
$445$	−1.57020e6			−0.375885
$446$		0			0
$447$	−1.91023e6	−	3.30862e6i	−0.452186	−	0.783210i
$448$		0			0
$449$	−2.65968e6			−0.622607			−0.311304	−	0.950310i	$-0.600766\pi$
$449$	−2.65968e6			−0.622607			−0.311304	+	0.950310i	$0.600766\pi$
$450$		0			0
$451$	4.88553e6	−	8.46199e6i	1.13102	−	1.95899i
$452$		0			0
$453$	−1.32031e6	+	2.28684e6i	−0.302294	+	0.523589i
$454$		0			0
$455$	−3.36434e6			−0.761854
$456$		0			0
$457$	5.09945e6			1.14217			0.571087	−	0.820889i	$-0.306522\pi$
$457$	5.09945e6			1.14217			0.571087	+	0.820889i	$0.306522\pi$
$458$		0			0
$459$	−2.71121e6	+	4.69595e6i	−0.600663	+	1.04038i
$460$		0			0
$461$	3.18373e6	−	5.51438e6i	0.697725	−	1.20849i	−0.271529	−	0.962430i	$-0.587529\pi$
$461$	3.18373e6	−	5.51438e6i	0.697725	−	1.20849i	0.969254	−	0.246064i	$-0.0791374\pi$
$462$		0			0
$463$	5.07803e6			1.10089			0.550443	−	0.834873i	$-0.314459\pi$
$463$	5.07803e6			1.10089			0.550443	+	0.834873i	$0.314459\pi$
$464$		0			0
$465$	2.60443e6	+	4.51101e6i	0.558574	+	0.967478i
$466$		0			0
$467$	−3.58423e6			−0.760507			−0.380254	−	0.924882i	$-0.624163\pi$
$467$	−3.58423e6			−0.760507			−0.380254	+	0.924882i	$0.624163\pi$
$468$		0			0
$469$	−2.59500e6	−	4.49467e6i	−0.544760	−	0.943552i
$470$		0			0
$471$	−809753.	−	1.40253e6i	−0.168190	−	0.291314i
$472$		0			0
$473$	−3.45381e6	+	5.98217e6i	−0.709815	+	1.22944i
$474$		0			0
$475$	1.64792e6	+	2.98779e6i	0.335121	+	0.607598i
$476$		0			0
$477$	−36575.5	+	63350.7i	−0.00736029	+	0.0127484i
$478$		0			0
$479$	3.65416e6	+	6.32919e6i	0.727694	+	1.26040i	0.957855	+	0.287251i	$0.0927414\pi$
$479$	3.65416e6	+	6.32919e6i	0.727694	+	1.26040i	−0.230161	+	0.973153i	$0.573925\pi$
$480$		0			0
$481$	−1.98064e6	−	3.43058e6i	−0.390341	−	0.676090i
$482$		0			0
$483$	4.81706e6			0.939539
$484$		0			0
$485$	−411501.	−	712740.i	−0.0794358	−	0.137587i
$486$		0			0
$487$	−5.63917e6			−1.07744			−0.538719	−	0.842485i	$-0.681092\pi$
$487$	−5.63917e6			−1.07744			−0.538719	+	0.842485i	$0.681092\pi$
$488$		0			0
$489$	−2.03543e6	+	3.52547e6i	−0.384932	+	0.666723i
$490$		0			0
$491$	3.98866e6	−	6.90857e6i	0.746661	−	1.29325i	−0.202754	−	0.979230i	$-0.564989\pi$
$491$	3.98866e6	−	6.90857e6i	0.746661	−	1.29325i	0.949415	−	0.314025i	$-0.101678\pi$
$492$		0			0
$493$	−4.03995e6			−0.748616
$494$		0			0
$495$	−203700.			−0.0373661
$496$		0			0
$497$	773345.	−	1.33947e6i	0.140437	−	0.243244i
$498$		0			0
$499$	−2.71915e6	+	4.70971e6i	−0.488858	+	0.846727i	−0.999918	−	0.0128184i	$-0.995920\pi$
$499$	−2.71915e6	+	4.70971e6i	−0.488858	+	0.846727i	0.511060	+	0.859545i	$0.329253\pi$
$500$		0			0
$501$	1.71555e6			0.305358
$502$		0			0
$503$	−4.08679e6	−	7.07853e6i	−0.720216	−	1.24745i	−0.960913	−	0.276850i	$-0.910709\pi$
$503$	−4.08679e6	−	7.07853e6i	−0.720216	−	1.24745i	0.240697	−	0.970600i	$-0.422624\pi$
$504$		0			0
$505$	−1.98475e6			−0.346320
$506$		0			0
$507$	−2.42859e6	−	4.20645e6i	−0.419599	−	0.726767i
$508$		0			0
$509$	−5.63560e6	−	9.76115e6i	−0.964153	−	1.66996i	−0.711873	−	0.702308i	$-0.752153\pi$
$509$	−5.63560e6	−	9.76115e6i	−0.964153	−	1.66996i	−0.252280	−	0.967654i	$-0.581180\pi$
$510$		0			0
$511$	1.81107e6	−	3.13686e6i	0.306819	−	0.531427i
$512$		0			0
$513$	−5.83307e6	−	114139.i	−0.978596	−	0.0191488i
$514$		0			0
$515$	1.02747e6	−	1.77964e6i	0.170708	−	0.295674i
$516$		0			0
$517$	−1.71348e6	−	2.96783e6i	−0.281937	−	0.488329i
$518$		0			0
$519$	3.13636e6	+	5.43234e6i	0.511102	+	0.885255i
$520$		0			0
$521$	6.41682e6			1.03568			0.517840	−	0.855477i	$-0.326736\pi$
$521$	6.41682e6			1.03568			0.517840	+	0.855477i	$0.326736\pi$
$522$		0			0
$523$	1.77525e6	+	3.07483e6i	0.283796	+	0.491549i	0.972317	−	0.233668i	$-0.0750728\pi$
$523$	1.77525e6	+	3.07483e6i	0.283796	+	0.491549i	−0.688520	+	0.725217i	$0.741739\pi$
$524$		0			0
$525$	4.55891e6			0.721876
$526$		0			0
$527$	−7.74529e6	+	1.34152e7i	−1.21482	+	2.10413i
$528$		0			0
$529$	593395.	−	1.02779e6i	0.0921945	−	0.159686i
$530$		0			0
$531$	−336862.			−0.0518460
$532$		0			0
$533$	1.19865e7			1.82757
$534$		0			0
$535$	−2.90668e6	+	5.03451e6i	−0.439048	+	0.760454i
$536$		0			0
$537$	−1.82285e6	+	3.15727e6i	−0.272782	+	0.472473i
$538$		0			0
$539$	−453702.			−0.0672665
$540$		0			0
$541$	2.87812e6	+	4.98505e6i	0.422781	+	0.732279i	0.996210	−	0.0869765i	$-0.0277205\pi$
$541$	2.87812e6	+	4.98505e6i	0.422781	+	0.732279i	−0.573429	+	0.819255i	$0.694387\pi$
$542$		0			0
$543$	8.58242e6			1.24914
$544$		0			0
$545$	2.77204e6	+	4.80132e6i	0.399769	+	0.692420i
$546$		0			0
$547$	−561663.	−	972828.i	−0.0802615	−	0.139017i	0.823101	−	0.567895i	$-0.192242\pi$
$547$	−561663.	−	972828.i	−0.0802615	−	0.139017i	−0.903362	+	0.428878i	$0.858909\pi$
$548$		0			0
$549$	202396.	−	350560.i	0.0286597	−	0.0496400i
$550$		0			0
$551$	−2.09931e6	−	3.80619e6i	−0.294576	−	0.534086i
$552$		0			0
$553$	−341601.	+	591671.i	−0.0475014	+	0.0822749i
$554$		0			0
$555$	−1.18402e6	−	2.05077e6i	−0.163164	−	0.282609i
$556$		0			0
$557$	2.85326e6	+	4.94199e6i	0.389676	+	0.674938i	0.992406	−	0.123007i	$-0.0392537\pi$
$557$	2.85326e6	+	4.94199e6i	0.389676	+	0.674938i	−0.602730	+	0.797945i	$0.705920\pi$
$558$		0			0
$559$	−8.47378e6			−1.14696
$560$		0			0
$561$	−7.79730e6	−	1.35053e7i	−1.04601	−	1.81175i
$562$		0			0
$563$	−1.30560e7			−1.73596			−0.867980	−	0.496599i	$-0.834582\pi$
$563$	−1.30560e7			−1.73596			−0.867980	+	0.496599i	$0.834582\pi$
$564$		0			0
$565$	917543.	−	1.58923e6i	0.120922	−	0.209443i
$566$		0			0
$567$	−4.05529e6	+	7.02397e6i	−0.529742	+	0.917540i
$568$		0			0
$569$	8.04953e6			1.04229			0.521147	−	0.853467i	$-0.325504\pi$
$569$	8.04953e6			1.04229			0.521147	+	0.853467i	$0.325504\pi$
$570$		0			0
$571$	604331.			0.0775683			0.0387842	−	0.999248i	$-0.487652\pi$
$571$	604331.			0.0775683			0.0387842	+	0.999248i	$0.487652\pi$
$572$		0			0
$573$	3.56842e6	−	6.18069e6i	0.454035	−	0.786412i
$574$		0			0
$575$	−2.48411e6	+	4.30260e6i	−0.313329	+	0.542702i
$576$		0			0
$577$	3.91150e6			0.489107			0.244554	−	0.969636i	$-0.421359\pi$
$577$	3.91150e6			0.489107			0.244554	+	0.969636i	$0.421359\pi$
$578$		0			0
$579$	380093.	+	658341.i	0.0471188	+	0.0816121i
$580$		0			0
$581$	−1.13916e7			−1.40005
$582$		0			0
$583$	2.49751e6	+	4.32582e6i	0.304324	+	0.527105i
$584$		0			0
$585$	−124942.	−	216407.i	−0.0150945	−	0.0261445i
$586$		0			0
$587$	720645.	−	1.24819e6i	0.0863229	−	0.149516i	−0.819631	−	0.572891i	$-0.805822\pi$
$587$	720645.	−	1.24819e6i	0.0863229	−	0.149516i	0.905954	+	0.423376i	$0.139155\pi$
$588$		0			0
$589$	−1.66637e7	−	326070.i	−1.97917	−	0.0387278i
$590$		0			0
$591$	−348238.	+	603166.i	−0.0410117	+	0.0710343i
$592$		0			0
$593$	−1.01304e6	−	1.75464e6i	−0.118302	−	0.204904i	0.800793	−	0.598941i	$-0.204412\pi$
$593$	−1.01304e6	−	1.75464e6i	−0.118302	−	0.204904i	−0.919095	+	0.394037i	$0.871078\pi$
$594$		0			0
$595$	−2.99052e6	−	5.17973e6i	−0.346301	−	0.599811i
$596$		0			0
$597$	9.54232e6			1.09577
$598$		0			0
$599$	−697546.	−	1.20819e6i	−0.0794339	−	0.137584i	0.823572	−	0.567212i	$-0.191978\pi$
$599$	−697546.	−	1.20819e6i	−0.0794339	−	0.137584i	−0.903006	+	0.429628i	$0.858645\pi$
$600$		0			0
$601$	−1.12873e7			−1.27469			−0.637346	−	0.770578i	$-0.719968\pi$
$601$	−1.12873e7			−1.27469			−0.637346	+	0.770578i	$0.719968\pi$
$602$		0			0
$603$	192742.	−	333839.i	0.0215866	−	0.0373890i
$604$		0			0
$605$	−4.46412e6	+	7.73208e6i	−0.495847	+	0.858832i
$606$		0			0
$607$	−1.45297e7			−1.60061			−0.800306	−	0.599591i	$-0.795330\pi$
$607$	−1.45297e7			−1.60061			−0.800306	+	0.599591i	$0.795330\pi$
$608$		0			0
$609$	−5.80766e6			−0.634538
$610$		0			0
$611$	2.10197e6	−	3.64072e6i	0.227784	−	0.394534i
$612$		0			0
$613$	−7.68601e6	+	1.33126e7i	−0.826133	+	1.43090i	0.0749172	+	0.997190i	$0.476131\pi$
$613$	−7.68601e6	+	1.33126e7i	−0.826133	+	1.43090i	−0.901050	+	0.433715i	$0.857203\pi$
$614$		0			0
$615$	7.16543e6			0.763931
$616$		0			0
$617$	1.39641e6	+	2.41866e6i	0.147673	+	0.255777i	0.930367	−	0.366630i	$-0.119488\pi$
$617$	1.39641e6	+	2.41866e6i	0.147673	+	0.255777i	−0.782694	+	0.622407i	$0.786155\pi$
$618$		0			0
$619$	8.26283e6			0.866766			0.433383	−	0.901210i	$-0.357320\pi$
$619$	8.26283e6			0.866766			0.433383	+	0.901210i	$0.357320\pi$
$620$		0			0
$621$	−4.24744e6	−	7.35678e6i	−0.441975	−	0.765524i
$622$		0			0
$623$	−3.35641e6	−	5.81347e6i	−0.346461	−	0.600089i
$624$		0			0
$625$	−856296.	+	1.48315e6i	−0.0876848	+	0.151874i
$626$		0			0
$627$	8.67209e6	−	1.43640e7i	0.880958	−	1.45917i
$628$		0			0
$629$	3.52113e6	−	6.09878e6i	0.354859	−	0.614634i
$630$		0			0
$631$	329136.	+	570081.i	0.0329081	+	0.0569985i	0.882010	−	0.471230i	$-0.156190\pi$
$631$	329136.	+	570081.i	0.0329081	+	0.0569985i	−0.849102	+	0.528228i	$0.822856\pi$
$632$		0			0
$633$	−3.81010e6	−	6.59929e6i	−0.377944	−	0.654618i
$634$		0			0
$635$	−2.06637e6			−0.203364
$636$		0			0
$637$	−278285.	−	482004.i	−0.0271732	−	0.0470654i
$638$		0			0
$639$	114880.			0.0111299
$640$		0			0
$641$	−626336.	+	1.08485e6i	−0.0602091	+	0.104285i	−0.894559	−	0.446950i	$-0.852510\pi$
$641$	−626336.	+	1.08485e6i	−0.0602091	+	0.104285i	0.834350	+	0.551236i	$0.185843\pi$
$642$		0			0
$643$	−7.69468e6	+	1.33276e7i	−0.733945	+	1.27123i	0.221240	+	0.975219i	$0.428990\pi$
$643$	−7.69468e6	+	1.33276e7i	−0.733945	+	1.27123i	−0.955185	+	0.296010i	$0.904344\pi$
$644$		0			0
$645$	−5.06557e6			−0.479434
$646$		0			0
$647$	−1.31905e6			−0.123880			−0.0619398	−	0.998080i	$-0.519729\pi$
$647$	−1.31905e6			−0.123880			−0.0619398	+	0.998080i	$0.519729\pi$
$648$		0			0
$649$	−1.15011e7	+	1.99205e7i	−1.07183	+	1.85647i
$650$		0			0
$651$	−1.11343e7	+	1.92852e7i	−1.02970	+	1.78349i
$652$		0			0
$653$	−2.35430e6			−0.216062			−0.108031	−	0.994148i	$-0.534455\pi$
$653$	−2.35430e6			−0.216062			−0.108031	+	0.994148i	$0.534455\pi$
$654$		0			0
$655$	5.42572e6	+	9.39762e6i	0.494145	+	0.855884i
$656$		0			0
$657$	269032.			0.0243160
$658$		0			0
$659$	−1.16918e6	−	2.02508e6i	−0.104874	−	0.181647i	0.808813	−	0.588066i	$-0.200111\pi$
$659$	−1.16918e6	−	2.02508e6i	−0.104874	−	0.181647i	−0.913687	+	0.406419i	$0.866777\pi$
$660$		0			0
$661$	4.27183e6	+	7.39902e6i	0.380286	+	0.658674i	0.991103	−	0.133098i	$-0.0424924\pi$
$661$	4.27183e6	+	7.39902e6i	0.380286	+	0.658674i	−0.610817	+	0.791772i	$0.709159\pi$
$662$		0			0
$663$	9.56519e6	−	1.65674e7i	0.845103	−	1.46376i
$664$		0			0
$665$	3.32603e6	−	5.50905e6i	0.291657	−	0.483084i
$666$		0			0
$667$	3.16454e6	−	5.48115e6i	0.275421	−	0.477042i
$668$		0			0
$669$	1.47874e6	+	2.56125e6i	0.127740	+	0.221252i
$670$		0			0
$671$	−1.38204e7	−	2.39376e7i	−1.18498	−	2.05245i
$672$		0			0
$673$	7.81154e6			0.664813			0.332406	−	0.943136i	$-0.392139\pi$
$673$	7.81154e6			0.664813			0.332406	+	0.943136i	$0.392139\pi$
$674$		0			0
$675$	−4.01981e6	−	6.96252e6i	−0.339583	−	0.588175i
$676$		0			0
$677$	−9.38338e6			−0.786842			−0.393421	−	0.919358i	$-0.628709\pi$
$677$	−9.38338e6			−0.786842			−0.393421	+	0.919358i	$0.628709\pi$
$678$		0			0
$679$	1.75922e6	−	3.04706e6i	0.146435	−	0.253634i
$680$		0			0
$681$	1.04480e6	−	1.80964e6i	0.0863305	−	0.149529i
$682$		0			0
$683$	1.64103e7			1.34606			0.673031	−	0.739614i	$-0.264992\pi$
$683$	1.64103e7			1.34606			0.673031	+	0.739614i	$0.264992\pi$
$684$		0			0
$685$	7.69450e6			0.626547
$686$		0			0
$687$	−840923.	+	1.45652e6i	−0.0679774	+	0.117740i
$688$		0			0
$689$	−3.06377e6	+	5.30661e6i	−0.245872	+	0.425862i
$690$		0			0
$691$	1.44148e7			1.14846			0.574228	−	0.818695i	$-0.305302\pi$
$691$	1.44148e7			1.14846			0.574228	+	0.818695i	$0.305302\pi$
$692$		0			0
$693$	−435422.	−	754173.i	−0.0344411	−	0.0596538i
$694$		0			0
$695$	1.37026e7			1.07607
$696$		0			0
$697$	1.06546e7	+	1.84543e7i	0.830721	+	1.43885i
$698$		0			0
$699$	8.02760e6	+	1.39042e7i	0.621431	+	1.07635i
$700$		0			0
$701$	9.22639e6	−	1.59806e7i	0.709148	−	1.22828i	−0.256025	−	0.966670i	$-0.582413\pi$
$701$	9.22639e6	−	1.59806e7i	0.709148	−	1.22828i	0.965174	−	0.261610i	$-0.0842536\pi$
$702$		0			0
$703$	7.57559e6	+	148237.i	0.578134	+	0.0113127i
$704$		0			0
$705$	1.25654e6	−	2.17640e6i	0.0952150	−	0.164917i
$706$		0			0
$707$	−4.24254e6	−	7.34829e6i	−0.319211	−	0.552889i
$708$		0			0
$709$	2.36759e6	+	4.10078e6i	0.176885	+	0.306373i	0.940812	−	0.338929i	$-0.110065\pi$
$709$	2.36759e6	+	4.10078e6i	0.176885	+	0.306373i	−0.763927	+	0.645302i	$0.776731\pi$
$710$		0			0
$711$	−50744.5			−0.00376457
$712$		0			0
$713$	−1.21340e7	−	2.10166e7i	−0.893879	−	1.54824i
$714$		0			0
$715$	−1.70631e7			−1.24822
$716$		0			0
$717$	−4.28500e6	+	7.42185e6i	−0.311282	+	0.539155i
$718$		0			0
$719$	1.63001e6	−	2.82326e6i	0.117589	−	0.203671i	−0.801222	−	0.598367i	$-0.795817\pi$
$719$	1.63001e6	−	2.82326e6i	0.117589	−	0.203671i	0.918812	+	0.394696i	$0.129150\pi$
$720$		0			0
$721$	8.78518e6			0.629379
$722$		0			0
$723$	−1.96571e7			−1.39853
$724$		0			0
$725$	2.99495e6	−	5.18740e6i	0.211614	−	0.366526i
$726$		0			0
$727$	−1.20435e7	+	2.08600e7i	−0.845117	+	1.46379i	0.0404018	+	0.999184i	$0.487136\pi$
$727$	−1.20435e7	+	2.08600e7i	−0.845117	+	1.46379i	−0.885519	+	0.464603i	$0.846197\pi$
$728$		0			0
$729$	1.37231e7			0.956384
$730$		0			0
$731$	−7.53222e6	−	1.30462e7i	−0.521350	−	0.903005i
$732$		0			0
$733$	8.14935e6			0.560225			0.280113	−	0.959967i	$-0.409628\pi$
$733$	8.14935e6			0.560225			0.280113	+	0.959967i	$0.409628\pi$
$734$		0			0
$735$	−166357.	−	288138.i	−0.0113585	−	0.0196736i
$736$		0			0
$737$	−1.31611e7	−	2.27958e7i	−0.892535	−	1.54592i
$738$		0			0
$739$	3.07531e6	−	5.32659e6i	0.207146	−	0.358788i	−0.743668	−	0.668549i	$-0.766916\pi$
$739$	3.07531e6	−	5.32659e6i	0.207146	−	0.358788i	0.950814	+	0.309761i	$0.100249\pi$
$740$		0			0
$741$	2.05792e7	+	402686.i	1.37684	+	0.0269415i
$742$		0			0
$743$	6.28020e6	−	1.08776e7i	0.417351	−	0.722873i	−0.578321	−	0.815809i	$-0.696292\pi$
$743$	6.28020e6	−	1.08776e7i	0.417351	−	0.722873i	0.995672	+	0.0929364i	$0.0296253\pi$
$744$		0			0
$745$	−3.71573e6	−	6.43584e6i	−0.245275	−	0.424829i
$746$		0			0
$747$	−423051.	−	732746.i	−0.0277390	−	0.0480454i
$748$		0			0
$749$	−2.48529e7			−1.61872
$750$		0			0
$751$	6.87642e6	+	1.19103e7i	0.444900	+	0.770590i	0.998045	−	0.0624953i	$-0.0199058\pi$
$751$	6.87642e6	+	1.19103e7i	0.444900	+	0.770590i	−0.553145	+	0.833085i	$0.686572\pi$
$752$		0			0
$753$	−2.11490e7			−1.35926
$754$		0			0
$755$	−2.56823e6	+	4.44831e6i	−0.163971	+	0.284006i
$756$		0			0
$757$	−3.85263e6	+	6.67296e6i	−0.244353	+	0.423232i	−0.961950	−	0.273227i	$-0.911909\pi$
$757$	−3.85263e6	+	6.67296e6i	−0.244353	+	0.423232i	0.717596	+	0.696459i	$0.245242\pi$
$758$		0			0
$759$	2.44309e7			1.53934
$760$		0			0
$761$	−2.30446e7			−1.44247			−0.721236	−	0.692689i	$-0.756426\pi$
$761$	−2.30446e7			−1.44247			−0.721236	+	0.692689i	$0.756426\pi$
$762$		0			0
$763$	−1.18509e7	+	2.05263e7i	−0.736951	+	1.27644i
$764$		0			0
$765$	222119.	−	384721.i	0.0137225	−	0.0237680i
$766$		0			0
$767$	−2.82174e7			−1.73193
$768$		0			0
$769$	8.54029e6	+	1.47922e7i	0.520783	+	0.902023i	0.999708	+	0.0241667i	$0.00769325\pi$
$769$	8.54029e6	+	1.47922e7i	0.520783	+	0.902023i	−0.478925	+	0.877856i	$0.658973\pi$
$770$		0			0
$771$	−1.49927e7			−0.908330
$772$		0			0
$773$	−7.20473e6	−	1.24790e7i	−0.433680	−	0.751155i	0.563507	−	0.826111i	$-0.309452\pi$
$773$	−7.20473e6	−	1.24790e7i	−0.433680	−	0.751155i	−0.997187	+	0.0749561i	$0.976118\pi$
$774$		0			0
$775$	−1.14837e7	−	1.98903e7i	−0.686794	−	1.18956i
$776$		0			0
$777$	5.06183e6	−	8.76734e6i	0.300784	−	0.520973i
$778$		0			0
$779$	−1.18500e7	+	1.96276e7i	−0.699637	+	1.15884i
$780$		0			0
$781$	3.92220e6	−	6.79345e6i	0.230092	−	0.398532i
$782$		0			0
$783$	5.12090e6	+	8.86965e6i	0.298498	+	0.517014i
$784$		0			0
$785$	−1.57511e6	−	2.72817e6i	−0.0912298	−	0.158015i
$786$		0			0
$787$	2.96282e7			1.70518			0.852588	−	0.522584i	$-0.175032\pi$
$787$	2.96282e7			1.70518			0.852588	+	0.522584i	$0.175032\pi$
$788$		0			0
$789$	1.34844e7	+	2.33557e7i	0.771151	+	1.33567i
$790$		0			0
$791$	7.84525e6			0.445826
$792$		0			0
$793$	1.69538e7	−	2.93649e7i	0.957382	−	1.65823i
$794$		0			0
$795$	−1.83150e6	+	3.17226e6i	−0.102776	+	0.178013i
$796$		0			0
$797$	2.29223e7			1.27824			0.639121	−	0.769107i	$-0.279298\pi$
$797$	2.29223e7			1.27824			0.639121	+	0.769107i	$0.279298\pi$
$798$		0			0
$799$	7.47365e6			0.414158
$800$		0			0
$801$	249296.	−	431793.i	0.0137288	−	0.0237790i
$802$		0			0
$803$	9.18526e6	−	1.59093e7i	0.502693	−	0.870690i
$804$		0			0
$805$	9.37003e6			0.509625
$806$		0			0
$807$	6.41985e6	+	1.11195e7i	0.347009	+	0.601038i
$808$		0			0
$809$	4.65984e6			0.250322			0.125161	−	0.992136i	$-0.460055\pi$
$809$	4.65984e6			0.250322			0.125161	+	0.992136i	$0.460055\pi$
$810$		0			0
$811$	−848246.	−	1.46921e6i	−0.0452866	−	0.0784387i	0.842494	−	0.538706i	$-0.181087\pi$
$811$	−848246.	−	1.46921e6i	−0.0452866	−	0.0784387i	−0.887780	+	0.460268i	$0.847753\pi$
$812$		0			0
$813$	1.63753e7	+	2.83629e7i	0.868889	+	1.50496i
$814$		0			0
$815$	−3.95927e6	+	6.85766e6i	−0.208795	+	0.361644i
$816$		0			0
$817$	8.37727e6	−	1.38757e7i	0.439084	−	0.727275i
$818$		0			0
$819$	534146.	−	925167.i	0.0278259	−	0.0481959i
$820$		0			0
$821$	469873.	+	813845.i	0.0243289	+	0.0421389i	0.877934	−	0.478783i	$-0.158922\pi$
$821$	469873.	+	813845.i	0.0243289	+	0.0421389i	−0.853605	+	0.520921i	$0.825588\pi$
$822$		0			0
$823$	1.57070e7	+	2.72053e7i	0.808338	+	1.40008i	0.914014	+	0.405682i	$0.132966\pi$
$823$	1.57070e7	+	2.72053e7i	0.808338	+	1.40008i	−0.105676	+	0.994401i	$0.533701\pi$
$824$		0			0
$825$	2.31216e7			1.18272
$826$		0			0
$827$	−4.21728e6	−	7.30454e6i	−0.214421	−	0.371389i	0.738672	−	0.674065i	$-0.235453\pi$
$827$	−4.21728e6	−	7.30454e6i	−0.214421	−	0.371389i	−0.953093	+	0.302676i	$0.902120\pi$
$828$		0			0
$829$	−9.81671e6			−0.496112			−0.248056	−	0.968746i	$-0.579792\pi$
$829$	−9.81671e6			−0.496112			−0.248056	+	0.968746i	$0.579792\pi$
$830$		0			0
$831$	8.48513e6	−	1.46967e7i	0.426242	−	0.738272i
$832$		0			0
$833$	494727.	−	856892.i	0.0247032	−	0.0427872i
$834$		0			0
$835$	3.33704e6			0.165632
$836$		0			0
$837$	3.92706e7			1.93755
$838$		0			0
$839$	1.20733e7	−	2.09116e7i	0.592135	−	1.02561i	−0.401809	−	0.915723i	$-0.631618\pi$
$839$	1.20733e7	−	2.09116e7i	0.592135	−	1.02561i	0.993944	−	0.109885i	$-0.0350482\pi$
$840$		0			0
$841$	6.44027e6	−	1.11549e7i	0.313989	−	0.543844i
$842$		0			0
$843$	1.12213e7			0.543842
$844$		0			0
$845$	−4.72403e6	−	8.18227e6i	−0.227599	−	0.394214i
$846$		0			0
$847$	−3.81694e7			−1.82813
$848$		0			0
$849$	−3.03576e6	−	5.25809e6i	−0.144543	−	0.250356i
$850$		0			0
$851$	5.51629e6	+	9.55449e6i	0.261110	+	0.452255i
$852$		0			0
$853$	−1.09389e7	+	1.89467e7i	−0.514754	+	0.891580i	0.485099	+	0.874459i	$0.338783\pi$
$853$	−1.09389e7	+	1.89467e7i	−0.514754	+	0.891580i	−0.999853	+	0.0171210i	$0.994550\pi$
$854$		0			0
$855$	477881.	+	9351.02i	0.0223566	+	0.000437465i
$856$		0			0
$857$	664571.	−	1.15107e6i	0.0309093	−	0.0535365i	−0.850157	−	0.526529i	$-0.823493\pi$
$857$	664571.	−	1.15107e6i	0.0309093	−	0.0535365i	0.881066	+	0.472993i	$0.156826\pi$
$858$		0			0
$859$	8.07444e6	+	1.39853e7i	0.373362	+	0.646681i	0.990080	−	0.140502i	$-0.0448717\pi$
$859$	8.07444e6	+	1.39853e7i	0.373362	+	0.646681i	−0.616719	+	0.787184i	$0.711538\pi$
$860$		0			0
$861$	1.53166e7	+	2.65291e7i	0.704132	+	1.21959i
$862$		0			0
$863$	1.37431e7			0.628141			0.314071	−	0.949400i	$-0.398307\pi$
$863$	1.37431e7			0.628141			0.314071	+	0.949400i	$0.398307\pi$
$864$		0			0
$865$	6.10077e6	+	1.05668e7i	0.277233	+	0.480181i
$866$		0			0
$867$	1.14332e7			0.516560
$868$		0			0
$869$	−1.73251e6	+	3.00080e6i	−0.0778263	+	0.134799i
$870$		0			0
$871$	1.61452e7	−	2.79643e7i	0.721103	−	1.24899i
$872$		0			0
$873$	261331.			0.0116053
$874$		0			0
$875$	2.16478e7			0.955860
$876$		0			0
$877$	−1.51904e7	+	2.63106e7i	−0.666916	+	1.15513i	0.311846	+	0.950133i	$0.399053\pi$
$877$	−1.51904e7	+	2.63106e7i	−0.666916	+	1.15513i	−0.978762	+	0.205000i	$0.934281\pi$
$878$		0			0
$879$	1.65881e7	−	2.87315e7i	0.724145	−	1.25426i
$880$		0			0
$881$	−1.96130e7			−0.851341			−0.425671	−	0.904878i	$-0.639962\pi$
$881$	−1.96130e7			−0.851341			−0.425671	+	0.904878i	$0.639962\pi$
$882$		0			0
$883$	9.76660e6	+	1.69162e7i	0.421543	+	0.730133i	0.996091	−	0.0883376i	$-0.0281554\pi$
$883$	9.76660e6	+	1.69162e7i	0.421543	+	0.730133i	−0.574548	+	0.818471i	$0.694822\pi$
$884$		0			0
$885$	−1.68682e7			−0.723953
$886$		0			0
$887$	4.00975e6	+	6.94508e6i	0.171123	+	0.296393i	0.938813	−	0.344428i	$-0.111927\pi$
$887$	4.00975e6	+	6.94508e6i	0.171123	+	0.296393i	−0.767690	+	0.640822i	$0.778594\pi$
$888$		0			0
$889$	−4.41700e6	−	7.65047e6i	−0.187445	−	0.324663i
$890$		0			0
$891$	−2.05674e7	+	3.56237e7i	−0.867929	+	1.50330i
$892$		0			0
$893$	3.88359e6	+	7.04120e6i	0.162969	+	0.295473i
$894$		0			0
$895$	−3.54577e6	+	6.14145e6i	−0.147963	+	0.256279i
$896$		0			0
$897$	1.49850e7	+	2.59549e7i	0.621838	+	1.07705i
$898$		0			0
$899$	1.46292e7	+	2.53386e7i	0.603701	+	1.04564i
$900$		0			0
$901$	−1.08934e7			−0.447045
$902$		0			0
$903$	−1.08280e7	−	1.87546e7i	−0.441905	−	0.765401i
$904$		0			0
$905$	1.66943e7			0.677558
$906$		0			0
$907$	1.11897e6	−	1.93811e6i	0.0451647	−	0.0782276i	−0.842559	−	0.538604i	$-0.818952\pi$
$907$	1.11897e6	−	1.93811e6i	0.0451647	−	0.0782276i	0.887724	+	0.460376i	$0.152285\pi$
$908$		0			0
$909$	315112.	−	545791.i	0.0126490	−	0.0219087i
$910$		0			0
$911$	3.98221e7			1.58975			0.794874	−	0.606774i	$-0.207537\pi$
$911$	3.98221e7			1.58975			0.794874	+	0.606774i	$0.207537\pi$
$912$		0			0
$913$	−5.77750e7			−2.29384
$914$		0			0
$915$	1.01349e7	−	1.75542e7i	0.400190	−	0.693150i
$916$		0			0
$917$	−2.31957e7	+	4.01761e7i	−0.910928	+	1.57777i
$918$		0			0
$919$	−8.32391e6			−0.325116			−0.162558	−	0.986699i	$-0.551975\pi$
$919$	−8.32391e6			−0.325116			−0.162558	+	0.986699i	$0.551975\pi$
$920$		0			0
$921$	−1.28137e7	−	2.21939e7i	−0.497765	−	0.862155i
$922$		0			0
$923$	9.62297e6			0.371796
$924$		0			0
$925$	5.22066e6	+	9.04245e6i	0.200619	+	0.347482i
$926$		0			0
$927$	326257.	+	565094.i	0.0124698	+	0.0215984i
$928$		0			0
$929$	4.49343e6	−	7.78285e6i	0.170820	−	0.295869i	−0.767887	−	0.640586i	$-0.778692\pi$
$929$	4.49343e6	−	7.78285e6i	0.170820	−	0.295869i	0.938707	+	0.344717i	$0.112025\pi$
$930$		0			0
$931$	1.06439e6	+	20827.6i	0.0402463	+	0.000787526i
$932$		0			0
$933$	6.24662e6	−	1.08195e7i	0.234931	−	0.406913i
$934$		0			0
$935$	−1.51671e7	−	2.62702e7i	−0.567380	−	0.982730i
$936$		0			0
$937$	−623801.	−	1.08045e6i	−0.0232112	−	0.0402029i	0.854187	−	0.519967i	$-0.174056\pi$
$937$	−623801.	−	1.08045e6i	−0.0232112	−	0.0402029i	−0.877398	+	0.479764i	$0.840722\pi$
$938$		0			0
$939$	−4.26465e7			−1.57841
$940$		0			0
$941$	−1.30000e6	−	2.25167e6i	−0.0478596	−	0.0828953i	0.841103	−	0.540875i	$-0.181907\pi$
$941$	−1.30000e6	−	2.25167e6i	−0.0478596	−	0.0828953i	−0.888963	+	0.457979i	$0.848573\pi$
$942$		0			0
$943$	−3.33835e7			−1.22251
$944$		0			0
$945$	−7.58134e6	+	1.31313e7i	−0.276163	+	0.478329i
$946$		0			0
$947$	−1.03456e7	+	1.79192e7i	−0.374871	+	0.649296i	−0.990308	−	0.138891i	$-0.955646\pi$
$947$	−1.03456e7	+	1.79192e7i	−0.374871	+	0.649296i	0.615437	+	0.788186i	$0.288980\pi$
$948$		0			0
$949$	2.25357e7			0.812279
$950$		0			0
$951$	4.15140e7			1.48848
$952$		0			0
$953$	1.50162e7	−	2.60089e7i	0.535586	−	0.927662i	−0.463549	−	0.886071i	$-0.653424\pi$
$953$	1.50162e7	−	2.60089e7i	0.535586	−	0.927662i	0.999135	−	0.0415907i	$-0.0132426\pi$
$954$		0			0
$955$	6.94120e6	−	1.20225e7i	0.246278	−	0.426567i
$956$		0			0
$957$	−2.94549e7			−1.03963
$958$		0			0
$959$	1.64475e7	+	2.84879e7i	0.577502	+	1.00026i
$960$		0			0
$961$	8.35579e7			2.91863
$962$		0			0
$963$	−922967.	−	1.59863e6i	−0.0320716	−	0.0555496i
$964$		0			0
$965$	739348.	+	1.28059e6i	0.0255582	+	0.0442681i
$966$		0			0
$967$	−7.38839e6	+	1.27971e7i	−0.254088	+	0.440093i	−0.964647	−	0.263544i	$-0.915109\pi$
$967$	−7.38839e6	+	1.27971e7i	−0.254088	+	0.440093i	0.710560	+	0.703637i	$0.248442\pi$
$968$		0			0
$969$	1.76726e7	+	3.20415e7i	0.604630	+	1.09624i
$970$		0			0
$971$	5.30678e6	−	9.19161e6i	0.180627	−	0.312855i	−0.761467	−	0.648203i	$-0.775521\pi$
$971$	5.30678e6	−	9.19161e6i	0.180627	−	0.312855i	0.942094	+	0.335348i	$0.108854\pi$
$972$		0			0
$973$	2.92903e7	+	5.07322e7i	0.991839	+	1.71791i
$974$		0			0
$975$	1.41820e7	+	2.45639e7i	0.477777	+	0.827534i
$976$		0			0
$977$	−3.66481e7			−1.22833			−0.614165	−	0.789177i	$-0.710507\pi$
$977$	−3.66481e7			−1.22833			−0.614165	+	0.789177i	$0.710507\pi$
$978$		0			0
$979$	−1.70228e7	−	2.94844e7i	−0.567642	−	0.983186i
$980$		0			0
$981$	−1.76043e6			−0.0584046
$982$		0			0
$983$	6.80511e6	−	1.17868e7i	0.224621	−	0.389056i	−0.731584	−	0.681751i	$-0.761219\pi$
$983$	6.80511e6	−	1.17868e7i	0.224621	−	0.389056i	0.956206	+	0.292695i	$0.0945521\pi$
$984$		0			0
$985$	−677384.	+	1.17326e6i	−0.0222456	+	0.0385305i
$986$		0			0
$987$	1.07438e7			0.351047
$988$		0			0
$989$	2.36003e7			0.767232
$990$		0			0
$991$	−1.59467e6	+	2.76205e6i	−0.0515806	+	0.0893402i	−0.890663	−	0.454664i	$-0.849759\pi$
$991$	−1.59467e6	+	2.76205e6i	−0.0515806	+	0.0893402i	0.839082	+	0.544005i	$0.183093\pi$
$992$		0			0
$993$	1.66410e7	−	2.88230e7i	0.535557	−	0.927612i
$994$		0			0
$995$	1.85615e7			0.594367
$996$		0			0
$997$	−1.71729e7	−	2.97444e7i	−0.547150	−	0.947692i	−0.998468	−	0.0553288i	$-0.982379\pi$
$997$	−1.71729e7	−	2.97444e7i	−0.547150	−	0.947692i	0.451318	−	0.892363i	$-0.350954\pi$
$998$		0			0
$999$	−1.78530e7			−0.565976

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.6.i.d.49.3		18
4.3	odd	2		76.6.e.a.49.7	yes	18
12.11	even	2		684.6.k.f.505.5		18
19.7	even	3	inner	304.6.i.d.273.3		18
76.7	odd	6		76.6.e.a.45.7	✓	18
228.83	even	6		684.6.k.f.577.5		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.6.e.a.45.7	✓	18	76.7	odd	6
76.6.e.a.49.7	yes	18	4.3	odd	2
304.6.i.d.49.3		18	1.1	even	1	trivial
304.6.i.d.273.3		18	19.7	even	3	inner
684.6.k.f.505.5		18	12.11	even	2
684.6.k.f.577.5		18	228.83	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	304.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-7.95017 + 13.7701i) q^{3} +(-15.4645 + 26.7852i) q^{5} -132.225 q^{7} +(-4.91049 - 8.50521i) q^{9} +O(q^{10})\)	\(q+(-7.95017 + 13.7701i) q^{3} +(-15.4645 + 26.7852i) q^{5} -132.225 q^{7} +(-4.91049 - 8.50521i) q^{9} -670.612 q^{11} +(-411.330 - 712.445i) q^{13} +(-245.890 - 425.895i) q^{15} +(731.251 - 1266.56i) q^{17} +(1573.26 + 30.7850i) q^{19} +(1051.21 - 1820.76i) q^{21} +(1145.60 + 1984.23i) q^{23} +(1084.20 + 1877.89i) q^{25} -3707.63 q^{27} +(-1381.18 - 2392.27i) q^{29} -10591.8 q^{31} +(5331.48 - 9234.40i) q^{33} +(2044.79 - 3541.69i) q^{35} +4815.22 q^{37} +13080.6 q^{39} +(-7285.18 + 12618.3i) q^{41} +(5150.23 - 8920.46i) q^{43} +303.752 q^{45} +(2555.09 + 4425.55i) q^{47} +676.549 q^{49} +(11627.1 + 20138.8i) q^{51} +(-3724.23 - 6450.55i) q^{53} +(10370.7 - 17962.5i) q^{55} +(-12931.6 + 21419.2i) q^{57} +(17150.1 - 29704.9i) q^{59} +(20608.6 + 35695.1i) q^{61} +(649.291 + 1124.60i) q^{63} +25444.0 q^{65} +(19625.6 + 33992.5i) q^{67} -36430.7 q^{69} +(-5848.69 + 10130.2i) q^{71} +(-13696.8 + 23723.6i) q^{73} -34478.3 q^{75} +88671.9 q^{77} +(2583.48 - 4474.71i) q^{79} +(30669.5 - 53121.2i) q^{81} +86152.6 q^{83} +(22616.8 + 39173.5i) q^{85} +43922.4 q^{87} +(25384.0 + 43966.4i) q^{89} +(54388.3 + 94203.3i) q^{91} +(84206.9 - 145851. i) q^{93} +(-25154.2 + 41664.1i) q^{95} +(-13304.7 + 23044.5i) q^{97} +(3293.03 + 5703.70i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 11 q^{3} + 11 q^{5} - 336 q^{7} - 902 q^{9}+O(q^{10}) \) 18 * q + 11 * q^3 + 11 * q^5 - 336 * q^7 - 902 * q^9	\( 18 q + 11 q^{3} + 11 q^{5} - 336 q^{7} - 902 q^{9} + 320 q^{11} + 227 q^{13} + 101 q^{15} + 179 q^{17} + 868 q^{19} - 5700 q^{21} + 3425 q^{23} - 7054 q^{25} - 14722 q^{27} - 7349 q^{29} + 9960 q^{31} - 2998 q^{33} - 15888 q^{35} + 26444 q^{37} + 30246 q^{39} - 7311 q^{41} + 8283 q^{43} - 62164 q^{45} - 37603 q^{47} + 124738 q^{49} - 47227 q^{51} - 20337 q^{53} - 716 q^{55} - 57555 q^{57} + 74455 q^{59} - 7569 q^{61} + 52544 q^{63} + 188998 q^{65} + 26177 q^{67} + 116282 q^{69} + 53463 q^{71} - 14103 q^{73} - 120912 q^{75} - 31960 q^{77} - 31825 q^{79} - 21137 q^{81} - 82600 q^{83} - 50787 q^{85} + 339766 q^{87} - 155197 q^{89} + 2800 q^{91} - 46460 q^{93} - 49315 q^{95} + 111241 q^{97} + 193544 q^{99}+O(q^{100}) \) 18 * q + 11 * q^3 + 11 * q^5 - 336 * q^7 - 902 * q^9 + 320 * q^11 + 227 * q^13 + 101 * q^15 + 179 * q^17 + 868 * q^19 - 5700 * q^21 + 3425 * q^23 - 7054 * q^25 - 14722 * q^27 - 7349 * q^29 + 9960 * q^31 - 2998 * q^33 - 15888 * q^35 + 26444 * q^37 + 30246 * q^39 - 7311 * q^41 + 8283 * q^43 - 62164 * q^45 - 37603 * q^47 + 124738 * q^49 - 47227 * q^51 - 20337 * q^53 - 716 * q^55 - 57555 * q^57 + 74455 * q^59 - 7569 * q^61 + 52544 * q^63 + 188998 * q^65 + 26177 * q^67 + 116282 * q^69 + 53463 * q^71 - 14103 * q^73 - 120912 * q^75 - 31960 * q^77 - 31825 * q^79 - 21137 * q^81 - 82600 * q^83 - 50787 * q^85 + 339766 * q^87 - 155197 * q^89 + 2800 * q^91 - 46460 * q^93 - 49315 * q^95 + 111241 * q^97 + 193544 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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