LMFDB - Embedded newform 684.6.k.f.577.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,6,Mod(505,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, 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("684.505");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	684.k (of order $3$, degree $2$, 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:	$109.702532752$
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} - 7 x^{17} + 17616 x^{16} - 456301 x^{15} + 216301789 x^{14} - 6076762674 x^{13} + \cdots + 55\!\cdots\!41 $ x^18 - 7x^17 + 17616x^16 - 456301x^15 + 216301789x^14 - 6076762674x^13 + 1365504992251x^12 - 54625271848675x^11 + 6304911261795867x^10 - 242577082836320134x^9 + 18704695568091019759x^8 - 704026968823840581411x^7 + 38243816152807979345695x^6 - 1065755622605542033556002x^5 + 37300806349205873742869889x^4 - 655224799021744228503297133x^3 + 19168823714556940699533734824x^2 - 236484931518202417328772433599*x + 5519876026771720332419776049541
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{16}\cdot 3^{7}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 76)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			577.5
Root			$16.4645 - 26.7852i$ of defining polynomial
Character	$\chi$	$=$	684.577
Dual form			684.6.k.f.505.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(15.4645 + 26.7852i) q^{5} +132.225 q^{7} +O(q^{10})$	$q+(15.4645 + 26.7852i) q^{5} +132.225 q^{7} -670.612 q^{11} +(-411.330 + 712.445i) q^{13} +(-731.251 - 1266.56i) q^{17} +(-1573.26 + 30.7850i) q^{19} +(1145.60 - 1984.23i) q^{23} +(1084.20 - 1877.89i) q^{25} +(1381.18 - 2392.27i) q^{29} +10591.8 q^{31} +(2044.79 + 3541.69i) q^{35} +4815.22 q^{37} +(7285.18 + 12618.3i) q^{41} +(-5150.23 - 8920.46i) q^{43} +(2555.09 - 4425.55i) q^{47} +676.549 q^{49} +(3724.23 - 6450.55i) q^{53} +(-10370.7 - 17962.5i) q^{55} +(17150.1 + 29704.9i) q^{59} +(20608.6 - 35695.1i) q^{61} -25444.0 q^{65} +(-19625.6 + 33992.5i) q^{67} +(-5848.69 - 10130.2i) q^{71} +(-13696.8 - 23723.6i) q^{73} -88671.9 q^{77} +(-2583.48 - 4474.71i) q^{79} +86152.6 q^{83} +(22616.8 - 39173.5i) q^{85} +(-25384.0 + 43966.4i) q^{89} +(-54388.3 + 94203.3i) q^{91} +(-25154.2 - 41664.1i) q^{95} +(-13304.7 - 23044.5i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 11 q^{5} + 336 q^{7}+O(q^{10}) $ 18 * q - 11 * q^5 + 336 * q^7	$ 18 q - 11 q^{5} + 336 q^{7} + 320 q^{11} + 227 q^{13} - 179 q^{17} - 868 q^{19} + 3425 q^{23} - 7054 q^{25} + 7349 q^{29} - 9960 q^{31} - 15888 q^{35} + 26444 q^{37} + 7311 q^{41} - 8283 q^{43} - 37603 q^{47} + 124738 q^{49} + 20337 q^{53} + 716 q^{55} + 74455 q^{59} - 7569 q^{61} - 188998 q^{65} - 26177 q^{67} + 53463 q^{71} - 14103 q^{73} + 31960 q^{77} + 31825 q^{79} - 82600 q^{83} - 50787 q^{85} + 155197 q^{89} - 2800 q^{91} - 49315 q^{95} + 111241 q^{97}+O(q^{100}) $ 18 * q - 11 * q^5 + 336 * q^7 + 320 * q^11 + 227 * q^13 - 179 * q^17 - 868 * q^19 + 3425 * q^23 - 7054 * q^25 + 7349 * q^29 - 9960 * q^31 - 15888 * q^35 + 26444 * q^37 + 7311 * q^41 - 8283 * q^43 - 37603 * q^47 + 124738 * q^49 + 20337 * q^53 + 716 * q^55 + 74455 * q^59 - 7569 * q^61 - 188998 * q^65 - 26177 * q^67 + 53463 * q^71 - 14103 * q^73 + 31960 * q^77 + 31825 * q^79 - 82600 * q^83 - 50787 * q^85 + 155197 * q^89 - 2800 * q^91 - 49315 * q^95 + 111241 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$.

$n$	$325$	$343$	$533$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	15.4645	+	26.7852i	0.276637	+	0.479149i	0.970547	−	0.240913i	$-0.0774468\pi$
$5$	15.4645	+	26.7852i	0.276637	+	0.479149i	−0.693910	+	0.720062i	$0.744113\pi$
$6$		0			0
$7$	132.225			1.01993			0.509964	−	0.860196i	$-0.329659\pi$
$7$	132.225			1.01993			0.509964	+	0.860196i	$0.329659\pi$
$8$		0			0
$9$		0			0
$10$		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	0.301412	+	0.953494i	$0.402542\pi$
$13$	−411.330	+	712.445i	−0.675044	+	1.16921i	−0.976456	+	0.215717i	$0.930791\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−731.251	−	1266.56i	−0.613683	−	1.06293i	−0.990614	−	0.136689i	$-0.956354\pi$
$17$	−731.251	−	1266.56i	−0.613683	−	1.06293i	0.376931	−	0.926241i	$-0.376979\pi$
$18$		0			0
$19$	−1573.26	+	30.7850i	−0.999809	+	0.0195639i
$19$	−1573.26	+	30.7850i	−0.999809	+	0.0195639i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1145.60	−	1984.23i	0.451556	−	0.782118i	−0.546927	−	0.837180i	$-0.684202\pi$
$23$	1145.60	−	1984.23i	0.451556	−	0.782118i	0.998483	+	0.0550625i	$0.0175358\pi$
$24$		0			0
$25$	1084.20	−	1877.89i	0.346944	−	0.600925i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1381.18	−	2392.27i	0.304968	−	0.528221i	−0.672286	−	0.740292i	$-0.734687\pi$
$29$	1381.18	−	2392.27i	0.304968	−	0.528221i	0.977254	+	0.212071i	$0.0680208\pi$
$30$		0			0
$31$	10591.8			1.97955			0.989776	−	0.142628i	$-0.0455552\pi$
$31$	10591.8			1.97955			0.989776	+	0.142628i	$0.0455552\pi$
$32$		0			0
$33$		0			0
$34$		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
$39$		0			0
$40$		0			0
$41$	7285.18	+	12618.3i	0.676832	+	1.17231i	0.975930	+	0.218085i	$0.0699810\pi$
$41$	7285.18	+	12618.3i	0.676832	+	1.17231i	−0.299098	+	0.954222i	$0.596686\pi$
$42$		0			0
$43$	−5150.23	−	8920.46i	−0.424772	−	0.735726i	0.571628	−	0.820513i	$-0.306312\pi$
$43$	−5150.23	−	8920.46i	−0.424772	−	0.735726i	−0.996399	+	0.0847874i	$0.972979\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2555.09	−	4425.55i	0.168718	−	0.292228i	−0.769251	−	0.638946i	$-0.779371\pi$
$47$	2555.09	−	4425.55i	0.168718	−	0.292228i	0.937969	+	0.346718i	$0.112704\pi$
$48$		0			0
$49$	676.549			0.0402540
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3724.23	−	6450.55i	0.182115	−	0.315433i	−0.760485	−	0.649355i	$-0.775039\pi$
$53$	3724.23	−	6450.55i	0.182115	−	0.315433i	0.942601	+	0.333922i	$0.108372\pi$
$54$		0			0
$55$	−10370.7	−	17962.5i	−0.462274	−	0.800682i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	17150.1	+	29704.9i	0.641412	+	1.11096i	0.985118	+	0.171881i	$0.0549845\pi$
$59$	17150.1	+	29704.9i	0.641412	+	1.11096i	−0.343705	+	0.939077i	$0.611682\pi$
$60$		0			0
$61$	20608.6	−	35695.1i	0.709126	−	1.22824i	−0.256056	−	0.966662i	$-0.582423\pi$
$61$	20608.6	−	35695.1i	0.709126	−	1.22824i	0.965182	−	0.261580i	$-0.0842434\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−25444.0			−0.746968
$66$		0			0
$67$	−19625.6	+	33992.5i	−0.534116	+	0.925116i	0.465090	+	0.885263i	$0.346022\pi$
$67$	−19625.6	+	33992.5i	−0.534116	+	0.925116i	−0.999206	+	0.0398521i	$0.987311\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−5848.69	−	10130.2i	−0.137693	−	0.238492i	0.788930	−	0.614483i	$-0.210635\pi$
$71$	−5848.69	−	10130.2i	−0.137693	−	0.238492i	−0.926623	+	0.375992i	$0.877302\pi$
$72$		0			0
$73$	−13696.8	−	23723.6i	−0.300824	−	0.521043i	0.675498	−	0.737361i	$-0.263928\pi$
$73$	−13696.8	−	23723.6i	−0.300824	−	0.521043i	−0.976323	+	0.216318i	$0.930595\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−88671.9			−1.70435
$78$		0			0
$79$	−2583.48	−	4474.71i	−0.0465733	−	0.0806673i	0.841799	−	0.539791i	$-0.181497\pi$
$79$	−2583.48	−	4474.71i	−0.0465733	−	0.0806673i	−0.888372	+	0.459124i	$0.848163\pi$
$80$		0			0
$81$		0			0
$82$		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
$87$		0			0
$88$		0			0
$89$	−25384.0	+	43966.4i	−0.339692	+	0.588364i	−0.984375	−	0.176087i	$-0.943656\pi$
$89$	−25384.0	+	43966.4i	−0.339692	+	0.588364i	0.644683	+	0.764450i	$0.276989\pi$
$90$		0			0
$91$	−54388.3	+	94203.3i	−0.688497	+	1.19251i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−25154.2	−	41664.1i	−0.285958	−	0.473645i
$96$		0			0
$97$	−13304.7	−	23044.5i	−0.143574	−	0.248678i	0.785266	−	0.619159i	$-0.212526\pi$
$97$	−13304.7	−	23044.5i	−0.143574	−	0.248678i	−0.928840	+	0.370481i	$0.879193\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−32085.7	+	55574.0i	−0.312974	+	0.542086i	−0.979005	−	0.203838i	$-0.934658\pi$
$101$	−32085.7	+	55574.0i	−0.312974	+	0.542086i	0.666031	+	0.745924i	$0.267992\pi$
$102$		0			0
$103$	66441.0			0.617082			0.308541	−	0.951211i	$-0.400159\pi$
$103$	66441.0			0.617082			0.308541	+	0.951211i	$0.400159\pi$
$104$		0			0
$105$		0			0
$106$		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	0.959974	+	0.280090i	$0.0903643\pi$
$109$	89626.3	+	155237.i	0.722552	+	1.25150i	−0.237422	+	0.971407i	$0.576302\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	59332.4			0.437115			0.218557	−	0.975824i	$-0.429865\pi$
$113$	59332.4			0.437115			0.218557	+	0.975824i	$0.429865\pi$
$114$		0			0
$115$	70864.1			0.499668
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−96689.9	−	167472.i	−0.625913	−	1.08411i
$120$		0			0
$121$	288669.			1.79241
$122$		0			0
$123$		0			0
$124$		0			0
$125$	163719.			0.937184
$126$		0			0
$127$	−33405.1	+	57859.3i	−0.183782	+	0.318320i	−0.943165	−	0.332324i	$-0.892167\pi$
$127$	−33405.1	+	57859.3i	−0.183782	+	0.318320i	0.759383	+	0.650643i	$0.225501\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	175425.	+	303846.i	0.893129	+	1.54695i	0.836103	+	0.548573i	$0.184829\pi$
$131$	175425.	+	303846.i	0.893129	+	1.54695i	0.0570264	+	0.998373i	$0.481838\pi$
$132$		0			0
$133$	−208025.	+	4070.56i	−1.01973	+	0.0199538i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	124390.	−	215450.i	0.566219	−	0.980719i	−0.430717	−	0.902487i	$-0.641739\pi$
$137$	124390.	−	215450.i	0.566219	−	0.980719i	0.996935	−	0.0782321i	$-0.0249275\pi$
$138$		0			0
$139$	221518.	−	383680.i	0.972459	−	1.68435i	0.284381	−	0.958711i	$-0.408212\pi$
$139$	221518.	−	383680.i	0.972459	−	1.68435i	0.688078	−	0.725637i	$-0.258455\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	275843.	−	477774.i	1.12803	−	1.95381i
$144$		0			0
$145$	85436.8			0.337462
$146$		0			0
$147$		0			0
$148$		0			0
$149$	120138.	+	208085.i	0.443317	+	0.767847i	0.997933	−	0.0642591i	$-0.0204684\pi$
$149$	120138.	+	208085.i	0.443317	+	0.767847i	−0.554617	+	0.832106i	$0.687135\pi$
$150$		0			0
$151$	−166073.			−0.592730			−0.296365	−	0.955075i	$-0.595774\pi$
$151$	−166073.			−0.592730			−0.296365	+	0.955075i	$0.595774\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	163797.	+	283705.i	0.547617	+	0.948501i
$156$		0			0
$157$	−50926.7	−	88207.7i	−0.164891	−	0.285599i	0.771726	−	0.635956i	$-0.219394\pi$
$157$	−50926.7	−	88207.7i	−0.164891	−	0.285599i	−0.936617	+	0.350356i	$0.886061\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	151477.	−	262365.i	0.460555	−	0.797704i
$162$		0			0
$163$	−256024.			−0.754764			−0.377382	−	0.926058i	$-0.623176\pi$
$163$	−256024.			−0.754764			−0.377382	+	0.926058i	$0.623176\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−53946.9	+	93438.8i	−0.149684	+	0.259260i	−0.931111	−	0.364737i	$-0.881159\pi$
$167$	−53946.9	+	93438.8i	−0.149684	+	0.259260i	0.781427	+	0.623997i	$0.214492\pi$
$168$		0			0
$169$	−152738.	−	264551.i	−0.411369	−	0.712512i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−197251.	−	341649.i	−0.501077	−	0.867890i	−0.999999	−	0.00124378i	$-0.999604\pi$
$173$	−197251.	−	341649.i	−0.501077	−	0.867890i	0.498922	−	0.866647i	$-0.333729\pi$
$174$		0			0
$175$	143359.	−	248305.i	0.353858	−	0.612900i
$176$		0			0
$177$		0			0
$178$		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	0.378531	+	0.925589i	$0.376429\pi$
$181$	−269882.	+	467449.i	−0.612318	+	1.06057i	−0.990849	+	0.134977i	$0.956904\pi$
$182$		0			0
$183$		0			0
$184$		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
$189$		0			0
$190$		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	0.888198	−	0.459461i	$-0.151957\pi$
$193$	23904.7	+	41404.2i	0.0461945	+	0.0800113i	−0.842004	+	0.539472i	$0.818624\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−43802.6			−0.0804145			−0.0402072	−	0.999191i	$-0.512802\pi$
$197$	−43802.6			−0.0804145			−0.0402072	+	0.999191i	$0.512802\pi$
$198$		0			0
$199$	300067.	−	519730.i	0.537137	−	0.930348i	−0.461920	−	0.886922i	$-0.652839\pi$
$199$	300067.	−	519730.i	0.537137	−	0.930348i	0.999057	−	0.0434264i	$-0.0138274\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	182627.	−	316319.i	0.311046	−	0.538747i
$204$		0			0
$205$	−225323.	+	390271.i	−0.374473	+	0.648607i
$206$		0			0
$207$		0			0
$208$		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	$-0.0458427\pi$
$211$	239624.	+	415040.i	0.370530	+	0.641777i	−0.619117	+	0.785299i	$0.712509\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	159291.	−	275900.i	0.235015	−	0.407058i
$216$		0			0
$217$	1.40051e6			2.01900
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.20314e6			1.65705
$222$		0			0
$223$	−93000.5	−	161082.i	−0.125234	−	0.216912i	0.796590	−	0.604520i	$-0.206635\pi$
$223$	−93000.5	−	161082.i	−0.125234	−	0.216912i	−0.921824	+	0.387608i	$0.873302\pi$
$224$		0			0
$225$		0			0
$226$		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
$231$		0			0
$232$		0			0
$233$	−504870.	−	874460.i	−0.609241	−	1.05524i	−0.991366	−	0.131126i	$-0.958141\pi$
$233$	−504870.	−	874460.i	−0.609241	−	1.05524i	0.382124	−	0.924111i	$-0.375193\pi$
$234$		0			0
$235$	158053.			0.186695
$236$		0			0
$237$		0			0
$238$		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	−0.287712	−	0.957717i	$-0.592895\pi$
$241$	618134.	−	1.07064e6i	0.685551	−	1.18741i	0.973263	−	0.229692i	$-0.0737720\pi$
$242$		0			0
$243$		0			0
$244$		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
$249$		0			0
$250$		0			0
$251$	665047.	−	1.15190e6i	0.666298	−	1.15406i	−0.312634	−	0.949874i	$-0.601211\pi$
$251$	665047.	−	1.15190e6i	0.666298	−	1.15406i	0.978932	−	0.204188i	$-0.0654553\pi$
$252$		0			0
$253$	−768250.	+	1.33065e6i	−0.754573	+	1.30696i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−471458.	+	816590.i	−0.445257	+	0.771207i	−0.998070	−	0.0620979i	$-0.980221\pi$
$257$	−471458.	+	816590.i	−0.445257	+	0.771207i	0.552813	+	0.833305i	$0.313554\pi$
$258$		0			0
$259$	636694.			0.589768
$260$		0			0
$261$		0			0
$262$		0			0
$263$	848058.	+	1.46888e6i	0.756025	+	1.30947i	0.944863	+	0.327465i	$0.106194\pi$
$263$	848058.	+	1.46888e6i	0.756025	+	1.30947i	−0.188838	+	0.982008i	$0.560472\pi$
$264$		0			0
$265$	230373.			0.201519
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−403755.	−	699325.i	−0.340203	−	0.589248i	0.644268	−	0.764800i	$-0.277162\pi$
$269$	−403755.	−	699325.i	−0.340203	−	0.589248i	−0.984470	+	0.175552i	$0.943829\pi$
$270$		0			0
$271$	−1.02987e6	−	1.78379e6i	−0.851845	−	1.47544i	−0.879541	−	0.475823i	$-0.842150\pi$
$271$	−1.02987e6	−	1.78379e6i	−0.851845	−	1.47544i	0.0276959	−	0.999616i	$-0.491183\pi$
$272$		0			0
$273$		0			0
$274$		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
$279$		0			0
$280$		0			0
$281$	352862.	−	611176.i	0.266587	−	0.461743i	−0.701391	−	0.712777i	$-0.747437\pi$
$281$	352862.	−	611176.i	0.266587	−	0.461743i	0.967978	+	0.251034i	$0.0807706\pi$
$282$		0			0
$283$	190924.	+	330690.i	0.141708	+	0.245446i	0.928140	−	0.372231i	$-0.121407\pi$
$283$	190924.	+	330690.i	0.141708	+	0.245446i	−0.786432	+	0.617677i	$0.788074\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	963286.	+	1.66846e6i	0.690320	+	1.19567i
$288$		0			0
$289$	−359527.	+	622720.i	−0.253214	+	0.438579i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2.08651e6			1.41988			0.709941	−	0.704261i	$-0.248722\pi$
$293$	2.08651e6			1.41988			0.709941	+	0.704261i	$0.248722\pi$
$294$		0			0
$295$	−530435.	+	918740.i	−0.354876	+	0.614664i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	942435.	+	1.63235e6i	0.609640	+	1.05593i
$300$		0			0
$301$	−680991.	−	1.17951e6i	−0.433237	−	0.750388i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.27480e6			0.784681
$306$		0			0
$307$	805874.	+	1.39582e6i	0.488002	+	0.845244i	0.999905	−	0.0137995i	$-0.00439266\pi$
$307$	805874.	+	1.39582e6i	0.488002	+	0.845244i	−0.511903	+	0.859043i	$0.671059\pi$
$308$		0			0
$309$		0			0
$310$		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	−0.161785	−	0.986826i	$-0.551725\pi$
$313$	1.34106e6	−	2.32278e6i	0.773724	−	1.34013i	0.935509	−	0.353303i	$-0.114942\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.30544e6	−	2.26109e6i	0.729642	−	1.26378i	−0.227392	−	0.973803i	$-0.573020\pi$
$317$	1.30544e6	−	2.26109e6i	0.729642	−	1.26378i	0.957034	−	0.289974i	$-0.0936467\pi$
$318$		0			0
$319$	−926235.	+	1.60429e6i	−0.509618	+	0.882684i
$320$		0			0
$321$		0			0
$322$		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
$327$		0			0
$328$		0			0
$329$	337848.	−	585170.i	0.172080	−	0.298052i
$330$		0			0
$331$	2.09316e6			1.05010			0.525052	−	0.851070i	$-0.324046\pi$
$331$	2.09316e6			1.05010			0.525052	+	0.851070i	$0.324046\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.21400e6			−0.591024
$336$		0			0
$337$	−1.63109e6	−	2.82513e6i	−0.782353	−	1.35508i	−0.930568	−	0.366120i	$-0.880686\pi$
$337$	−1.63109e6	−	2.82513e6i	−0.782353	−	1.35508i	0.148214	−	0.988955i	$-0.452647\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−7.10301e6			−3.30793
$342$		0			0
$343$	−2.13285e6			−0.978872
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−602372.	−	1.04334e6i	−0.268560	−	0.465159i	0.699930	−	0.714211i	$-0.253214\pi$
$347$	−602372.	−	1.04334e6i	−0.268560	−	0.465159i	−0.968490	+	0.249052i	$0.919881\pi$
$348$		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
$351$		0			0
$352$		0			0
$353$	−2.66290e6			−1.13741			−0.568706	−	0.822541i	$-0.692556\pi$
$353$	−2.66290e6			−1.13741			−0.568706	+	0.822541i	$0.692556\pi$
$354$		0			0
$355$	180894.	−	313317.i	0.0761820	−	0.131951i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−415704.	−	720020.i	−0.170235	−	0.294855i	0.768267	−	0.640129i	$-0.221119\pi$
$359$	−415704.	−	720020.i	−0.170235	−	0.294855i	−0.938502	+	0.345274i	$0.887786\pi$
$360$		0			0
$361$	2.47420e6	−	96865.8i	0.999235	−	0.0391203i
$362$		0			0
$363$		0			0
$364$		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	$-0.765763\pi$
$367$	543635.	−	941604.i	0.210689	−	0.364925i	0.951930	+	0.306314i	$0.0990958\pi$
$368$		0			0
$369$		0			0
$370$		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
$375$		0			0
$376$		0			0
$377$	1.13624e6	+	1.96803e6i	0.411734	+	0.713145i
$378$		0			0
$379$	−1.58924e6			−0.568317			−0.284158	−	0.958777i	$-0.591714\pi$
$379$	−1.58924e6			−0.568317			−0.284158	+	0.958777i	$0.591714\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.14784e6	−	1.98811e6i	−0.399837	−	0.692538i	0.593869	−	0.804562i	$-0.297600\pi$
$383$	−1.14784e6	−	1.98811e6i	−0.399837	−	0.692538i	−0.993705	+	0.112024i	$0.964267\pi$
$384$		0			0
$385$	−1.37126e6	−	2.37510e6i	−0.471486	−	0.816638i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2.12902e6	−	3.68757e6i	0.713355	−	1.23557i	−0.250235	−	0.968185i	$-0.580508\pi$
$389$	2.12902e6	−	3.68757e6i	0.713355	−	1.23557i	0.963590	−	0.267383i	$-0.0861588\pi$
$390$		0			0
$391$	−3.35087e6			−1.10845
$392$		0			0
$393$		0			0
$394$		0			0
$395$	79904.2	−	138398.i	0.0257678	−	0.0446311i
$396$		0			0
$397$	−173073.	−	299772.i	−0.0551130	−	0.0954585i	0.837153	−	0.546969i	$-0.184219\pi$
$397$	−173073.	−	299772.i	−0.0551130	−	0.0954585i	−0.892266	+	0.451511i	$0.850885\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−777399.	−	1.34649e6i	−0.241425	−	0.418161i	0.719695	−	0.694290i	$-0.244282\pi$
$401$	−777399.	−	1.34649e6i	−0.241425	−	0.418161i	−0.961121	+	0.276129i	$0.910948\pi$
$402$		0			0
$403$	−4.35674e6	+	7.54610e6i	−1.33629	+	2.31451i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3.22914e6			−0.966276
$408$		0			0
$409$	−796042.	+	1.37878e6i	−0.235303	+	0.407557i	−0.959361	−	0.282183i	$-0.908942\pi$
$409$	−796042.	+	1.37878e6i	−0.235303	+	0.407557i	0.724058	+	0.689739i	$0.242275\pi$
$410$		0			0
$411$		0			0
$412$		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
$417$		0			0
$418$		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	0.997244	−	0.0741981i	$-0.0236397\pi$
$421$	1.57964e6	+	2.73602e6i	0.434364	+	0.752341i	−0.562879	+	0.826539i	$0.690306\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.17129e6			−0.851655
$426$		0			0
$427$	2.72498e6	−	4.71980e6i	0.723257	−	1.25272i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−508299.	+	880400.i	−0.131803	+	0.228290i	−0.924372	−	0.381493i	$-0.875410\pi$
$431$	−508299.	+	880400.i	−0.131803	+	0.228290i	0.792568	+	0.609783i	$0.208743\pi$
$432$		0			0
$433$	−1.49339e6	+	2.58663e6i	−0.382784	+	0.663001i	−0.991459	−	0.130419i	$-0.958368\pi$
$433$	−1.49339e6	+	2.58663e6i	−0.382784	+	0.663001i	0.608675	+	0.793419i	$0.291701\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.74124e6	+	3.15698e6i	−0.436168	+	0.790802i
$438$		0			0
$439$	−3.05381e6	−	5.28936e6i	−0.756277	−	1.30991i	−0.944737	−	0.327830i	$-0.893683\pi$
$439$	−3.05381e6	−	5.28936e6i	−0.756277	−	1.30991i	0.188459	−	0.982081i	$-0.439651\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2.29170e6	+	3.96935e6i	−0.554816	+	0.960970i	0.443102	+	0.896471i	$0.353878\pi$
$443$	−2.29170e6	+	3.96935e6i	−0.554816	+	0.960970i	−0.997918	+	0.0644983i	$0.979455\pi$
$444$		0			0
$445$	−1.57020e6			−0.375885
$446$		0			0
$447$		0			0
$448$		0			0
$449$	2.65968e6			0.622607			0.311304	−	0.950310i	$-0.399234\pi$
$449$	2.65968e6			0.622607			0.311304	+	0.950310i	$0.399234\pi$
$450$		0			0
$451$	−4.88553e6	−	8.46199e6i	−1.13102	−	1.95899i
$452$		0			0
$453$		0			0
$454$		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
$459$		0			0
$460$		0			0
$461$	−3.18373e6	−	5.51438e6i	−0.697725	−	1.20849i	−0.969254	−	0.246064i	$-0.920863\pi$
$461$	−3.18373e6	−	5.51438e6i	−0.697725	−	1.20849i	0.271529	−	0.962430i	$-0.412471\pi$
$462$		0			0
$463$	−5.07803e6			−1.10089			−0.550443	−	0.834873i	$-0.685541\pi$
$463$	−5.07803e6			−1.10089			−0.550443	+	0.834873i	$0.685541\pi$
$464$		0			0
$465$		0			0
$466$		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
$471$		0			0
$472$		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
$477$		0			0
$478$		0			0
$479$	3.65416e6	−	6.32919e6i	0.727694	−	1.26040i	−0.230161	−	0.973153i	$-0.573925\pi$
$479$	3.65416e6	−	6.32919e6i	0.727694	−	1.26040i	0.957855	−	0.287251i	$-0.0927414\pi$
$480$		0			0
$481$	−1.98064e6	+	3.43058e6i	−0.390341	+	0.676090i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	411501.	−	712740.i	0.0794358	−	0.137587i
$486$		0			0
$487$	5.63917e6			1.07744			0.538719	−	0.842485i	$-0.318908\pi$
$487$	5.63917e6			1.07744			0.538719	+	0.842485i	$0.318908\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	3.98866e6	+	6.90857e6i	0.746661	+	1.29325i	0.949415	+	0.314025i	$0.101678\pi$
$491$	3.98866e6	+	6.90857e6i	0.746661	+	1.29325i	−0.202754	+	0.979230i	$0.564989\pi$
$492$		0			0
$493$	−4.03995e6			−0.748616
$494$		0			0
$495$		0			0
$496$		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	$-0.00408034\pi$
$499$	2.71915e6	+	4.70971e6i	0.488858	+	0.846727i	−0.511060	+	0.859545i	$0.670747\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−4.08679e6	+	7.07853e6i	−0.720216	+	1.24745i	0.240697	+	0.970600i	$0.422624\pi$
$503$	−4.08679e6	+	7.07853e6i	−0.720216	+	1.24745i	−0.960913	+	0.276850i	$0.910709\pi$
$504$		0			0
$505$	−1.98475e6			−0.346320
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5.63560e6	−	9.76115e6i	0.964153	−	1.66996i	0.252280	−	0.967654i	$-0.418820\pi$
$509$	5.63560e6	−	9.76115e6i	0.964153	−	1.66996i	0.711873	−	0.702308i	$-0.247847\pi$
$510$		0			0
$511$	−1.81107e6	−	3.13686e6i	−0.306819	−	0.531427i
$512$		0			0
$513$		0			0
$514$		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
$519$		0			0
$520$		0			0
$521$	−6.41682e6			−1.03568			−0.517840	−	0.855477i	$-0.673264\pi$
$521$	−6.41682e6			−1.03568			−0.517840	+	0.855477i	$0.673264\pi$
$522$		0			0
$523$	−1.77525e6	+	3.07483e6i	−0.283796	+	0.491549i	−0.972317	−	0.233668i	$-0.924927\pi$
$523$	−1.77525e6	+	3.07483e6i	−0.283796	+	0.491549i	0.688520	+	0.725217i	$0.258261\pi$
$524$		0			0
$525$		0			0
$526$		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
$531$		0			0
$532$		0			0
$533$	−1.19865e7			−1.82757
$534$		0			0
$535$	2.90668e6	+	5.03451e6i	0.439048	+	0.760454i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−453702.			−0.0672665
$540$		0			0
$541$	2.87812e6	−	4.98505e6i	0.422781	−	0.732279i	−0.573429	−	0.819255i	$-0.694387\pi$
$541$	2.87812e6	−	4.98505e6i	0.422781	−	0.732279i	0.996210	+	0.0869765i	$0.0277205\pi$
$542$		0			0
$543$		0			0
$544$		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	$-0.807758\pi$
$547$	561663.	−	972828.i	0.0802615	−	0.139017i	0.903362	+	0.428878i	$0.141091\pi$
$548$		0			0
$549$		0			0
$550$		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
$555$		0			0
$556$		0			0
$557$	−2.85326e6	+	4.94199e6i	−0.389676	+	0.674938i	−0.992406	−	0.123007i	$-0.960746\pi$
$557$	−2.85326e6	+	4.94199e6i	−0.389676	+	0.674938i	0.602730	+	0.797945i	$0.294080\pi$
$558$		0			0
$559$	8.47378e6			1.14696
$560$		0			0
$561$		0			0
$562$		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
$567$		0			0
$568$		0			0
$569$	−8.04953e6			−1.04229			−0.521147	−	0.853467i	$-0.674496\pi$
$569$	−8.04953e6			−1.04229			−0.521147	+	0.853467i	$0.674496\pi$
$570$		0			0
$571$	−604331.			−0.0775683			−0.0387842	−	0.999248i	$-0.512348\pi$
$571$	−604331.			−0.0775683			−0.0387842	+	0.999248i	$0.512348\pi$
$572$		0			0
$573$		0			0
$574$		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
$579$		0			0
$580$		0			0
$581$	1.13916e7			1.40005
$582$		0			0
$583$	−2.49751e6	+	4.32582e6i	−0.304324	+	0.527105i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	720645.	+	1.24819e6i	0.0863229	+	0.149516i	0.905954	−	0.423376i	$-0.139155\pi$
$587$	720645.	+	1.24819e6i	0.0863229	+	0.149516i	−0.819631	+	0.572891i	$0.805822\pi$
$588$		0			0
$589$	−1.66637e7	+	326070.i	−1.97917	+	0.0387278i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1.01304e6	−	1.75464e6i	0.118302	−	0.204904i	−0.800793	−	0.598941i	$-0.795588\pi$
$593$	1.01304e6	−	1.75464e6i	0.118302	−	0.204904i	0.919095	+	0.394037i	$0.128922\pi$
$594$		0			0
$595$	2.99052e6	−	5.17973e6i	0.346301	−	0.599811i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−697546.	+	1.20819e6i	−0.0794339	+	0.137584i	−0.903006	−	0.429628i	$-0.858645\pi$
$599$	−697546.	+	1.20819e6i	−0.0794339	+	0.137584i	0.823572	+	0.567212i	$0.191978\pi$
$600$		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
$603$		0			0
$604$		0			0
$605$	4.46412e6	+	7.73208e6i	0.495847	+	0.858832i
$606$		0			0
$607$	1.45297e7			1.60061			0.800306	−	0.599591i	$-0.204670\pi$
$607$	1.45297e7			1.60061			0.800306	+	0.599591i	$0.204670\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2.10197e6	+	3.64072e6i	0.227784	+	0.394534i
$612$		0			0
$613$	−7.68601e6	−	1.33126e7i	−0.826133	−	1.43090i	−0.901050	−	0.433715i	$-0.857203\pi$
$613$	−7.68601e6	−	1.33126e7i	−0.826133	−	1.43090i	0.0749172	−	0.997190i	$-0.476131\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.39641e6	+	2.41866e6i	−0.147673	+	0.255777i	−0.930367	−	0.366630i	$-0.880512\pi$
$617$	−1.39641e6	+	2.41866e6i	−0.147673	+	0.255777i	0.782694	+	0.622407i	$0.213845\pi$
$618$		0			0
$619$	−8.26283e6			−0.866766			−0.433383	−	0.901210i	$-0.642680\pi$
$619$	−8.26283e6			−0.866766			−0.433383	+	0.901210i	$0.642680\pi$
$620$		0			0
$621$		0			0
$622$		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
$627$		0			0
$628$		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	$-0.843810\pi$
$631$	−329136.	+	570081.i	−0.0329081	+	0.0569985i	0.849102	+	0.528228i	$0.177144\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2.06637e6			−0.203364
$636$		0			0
$637$	−278285.	+	482004.i	−0.0271732	+	0.0470654i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	626336.	+	1.08485e6i	0.0602091	+	0.104285i	0.894559	−	0.446950i	$-0.147490\pi$
$641$	626336.	+	1.08485e6i	0.0602091	+	0.104285i	−0.834350	+	0.551236i	$0.814157\pi$
$642$		0			0
$643$	7.69468e6	+	1.33276e7i	0.733945	+	1.27123i	0.955185	+	0.296010i	$0.0956562\pi$
$643$	7.69468e6	+	1.33276e7i	0.733945	+	1.27123i	−0.221240	+	0.975219i	$0.571010\pi$
$644$		0			0
$645$		0			0
$646$		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
$651$		0			0
$652$		0			0
$653$	2.35430e6			0.216062			0.108031	−	0.994148i	$-0.465545\pi$
$653$	2.35430e6			0.216062			0.108031	+	0.994148i	$0.465545\pi$
$654$		0			0
$655$	−5.42572e6	+	9.39762e6i	−0.494145	+	0.855884i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.16918e6	+	2.02508e6i	−0.104874	+	0.181647i	−0.913687	−	0.406419i	$-0.866777\pi$
$659$	−1.16918e6	+	2.02508e6i	−0.104874	+	0.181647i	0.808813	+	0.588066i	$0.200111\pi$
$660$		0			0
$661$	4.27183e6	−	7.39902e6i	0.380286	−	0.658674i	−0.610817	−	0.791772i	$-0.709159\pi$
$661$	4.27183e6	−	7.39902e6i	0.380286	−	0.658674i	0.991103	+	0.133098i	$0.0424924\pi$
$662$		0			0
$663$		0			0
$664$		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
$669$		0			0
$670$		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
$675$		0			0
$676$		0			0
$677$	9.38338e6			0.786842			0.393421	−	0.919358i	$-0.371291\pi$
$677$	9.38338e6			0.786842			0.393421	+	0.919358i	$0.371291\pi$
$678$		0			0
$679$	−1.75922e6	−	3.04706e6i	−0.146435	−	0.253634i
$680$		0			0
$681$		0			0
$682$		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
$687$		0			0
$688$		0			0
$689$	3.06377e6	+	5.30661e6i	0.245872	+	0.425862i
$690$		0			0
$691$	−1.44148e7			−1.14846			−0.574228	−	0.818695i	$-0.694698\pi$
$691$	−1.44148e7			−1.14846			−0.574228	+	0.818695i	$0.694698\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.37026e7			1.07607
$696$		0			0
$697$	1.06546e7	−	1.84543e7i	0.830721	−	1.43885i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−9.22639e6	−	1.59806e7i	−0.709148	−	1.22828i	−0.965174	−	0.261610i	$-0.915746\pi$
$701$	−9.22639e6	−	1.59806e7i	−0.709148	−	1.22828i	0.256025	−	0.966670i	$-0.417587\pi$
$702$		0			0
$703$	−7.57559e6	+	148237.i	−0.578134	+	0.0113127i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−4.24254e6	+	7.34829e6i	−0.319211	+	0.552889i
$708$		0			0
$709$	2.36759e6	−	4.10078e6i	0.176885	−	0.306373i	−0.763927	−	0.645302i	$-0.776731\pi$
$709$	2.36759e6	−	4.10078e6i	0.176885	−	0.306373i	0.940812	+	0.338929i	$0.110065\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.21340e7	−	2.10166e7i	0.893879	−	1.54824i
$714$		0			0
$715$	1.70631e7			1.24822
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1.63001e6	+	2.82326e6i	0.117589	+	0.203671i	0.918812	−	0.394696i	$-0.129150\pi$
$719$	1.63001e6	+	2.82326e6i	0.117589	+	0.203671i	−0.801222	+	0.598367i	$0.795817\pi$
$720$		0			0
$721$	8.78518e6			0.629379
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2.99495e6	−	5.18740e6i	−0.211614	−	0.366526i
$726$		0			0
$727$	1.20435e7	+	2.08600e7i	0.845117	+	1.46379i	0.885519	+	0.464603i	$0.153803\pi$
$727$	1.20435e7	+	2.08600e7i	0.845117	+	1.46379i	−0.0404018	+	0.999184i	$0.512864\pi$
$728$		0			0
$729$		0			0
$730$		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
$735$		0			0
$736$		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	$-0.233084\pi$
$739$	−3.07531e6	−	5.32659e6i	−0.207146	−	0.358788i	−0.950814	+	0.309761i	$0.899751\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	6.28020e6	+	1.08776e7i	0.417351	+	0.722873i	0.995672	−	0.0929364i	$-0.0296253\pi$
$743$	6.28020e6	+	1.08776e7i	0.417351	+	0.722873i	−0.578321	+	0.815809i	$0.696292\pi$
$744$		0			0
$745$	−3.71573e6	+	6.43584e6i	−0.245275	+	0.424829i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	2.48529e7			1.61872
$750$		0			0
$751$	−6.87642e6	+	1.19103e7i	−0.444900	+	0.770590i	−0.998045	−	0.0624953i	$-0.980094\pi$
$751$	−6.87642e6	+	1.19103e7i	−0.444900	+	0.770590i	0.553145	+	0.833085i	$0.313428\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−2.56823e6	−	4.44831e6i	−0.163971	−	0.284006i
$756$		0			0
$757$	−3.85263e6	−	6.67296e6i	−0.244353	−	0.423232i	0.717596	−	0.696459i	$-0.245242\pi$
$757$	−3.85263e6	−	6.67296e6i	−0.244353	−	0.423232i	−0.961950	+	0.273227i	$0.911909\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	2.30446e7			1.44247			0.721236	−	0.692689i	$-0.243574\pi$
$761$	2.30446e7			1.44247			0.721236	+	0.692689i	$0.243574\pi$
$762$		0			0
$763$	1.18509e7	+	2.05263e7i	0.736951	+	1.27644i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.82174e7			−1.73193
$768$		0			0
$769$	8.54029e6	−	1.47922e7i	0.520783	−	0.902023i	−0.478925	−	0.877856i	$-0.658973\pi$
$769$	8.54029e6	−	1.47922e7i	0.520783	−	0.902023i	0.999708	−	0.0241667i	$-0.00769325\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	7.20473e6	−	1.24790e7i	0.433680	−	0.751155i	−0.563507	−	0.826111i	$-0.690548\pi$
$773$	7.20473e6	−	1.24790e7i	0.433680	−	0.751155i	0.997187	+	0.0749561i	$0.0238817\pi$
$774$		0			0
$775$	1.14837e7	−	1.98903e7i	0.686794	−	1.18956i
$776$		0			0
$777$		0			0
$778$		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
$783$		0			0
$784$		0			0
$785$	1.57511e6	−	2.72817e6i	0.0912298	−	0.158015i
$786$		0			0
$787$	−2.96282e7			−1.70518			−0.852588	−	0.522584i	$-0.824968\pi$
$787$	−2.96282e7			−1.70518			−0.852588	+	0.522584i	$0.824968\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	7.84525e6			0.445826
$792$		0			0
$793$	1.69538e7	+	2.93649e7i	0.957382	+	1.65823i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−2.29223e7			−1.27824			−0.639121	−	0.769107i	$-0.720702\pi$
$797$	−2.29223e7			−1.27824			−0.639121	+	0.769107i	$0.720702\pi$
$798$		0			0
$799$	−7.47365e6			−0.414158
$800$		0			0
$801$		0			0
$802$		0			0
$803$	9.18526e6	+	1.59093e7i	0.502693	+	0.870690i
$804$		0			0
$805$	9.37003e6			0.509625
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−4.65984e6			−0.250322			−0.125161	−	0.992136i	$-0.539945\pi$
$809$	−4.65984e6			−0.250322			−0.125161	+	0.992136i	$0.539945\pi$
$810$		0			0
$811$	848246.	−	1.46921e6i	0.0452866	−	0.0784387i	−0.842494	−	0.538706i	$-0.818913\pi$
$811$	848246.	−	1.46921e6i	0.0452866	−	0.0784387i	0.887780	+	0.460268i	$0.152247\pi$
$812$		0			0
$813$		0			0
$814$		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
$819$		0			0
$820$		0			0
$821$	−469873.	+	813845.i	−0.0243289	+	0.0421389i	−0.877934	−	0.478783i	$-0.841078\pi$
$821$	−469873.	+	813845.i	−0.0243289	+	0.0421389i	0.853605	+	0.520921i	$0.174412\pi$
$822$		0			0
$823$	−1.57070e7	+	2.72053e7i	−0.808338	+	1.40008i	0.105676	+	0.994401i	$0.466299\pi$
$823$	−1.57070e7	+	2.72053e7i	−0.808338	+	1.40008i	−0.914014	+	0.405682i	$0.867034\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−4.21728e6	+	7.30454e6i	−0.214421	+	0.371389i	−0.953093	−	0.302676i	$-0.902120\pi$
$827$	−4.21728e6	+	7.30454e6i	−0.214421	+	0.371389i	0.738672	+	0.674065i	$0.235453\pi$
$828$		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
$831$		0			0
$832$		0			0
$833$	−494727.	−	856892.i	−0.0247032	−	0.0427872i
$834$		0			0
$835$	−3.33704e6			−0.165632
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1.20733e7	+	2.09116e7i	0.592135	+	1.02561i	0.993944	+	0.109885i	$0.0350482\pi$
$839$	1.20733e7	+	2.09116e7i	0.592135	+	1.02561i	−0.401809	+	0.915723i	$0.631618\pi$
$840$		0			0
$841$	6.44027e6	+	1.11549e7i	0.313989	+	0.543844i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	4.72403e6	−	8.18227e6i	0.227599	−	0.394214i
$846$		0			0
$847$	3.81694e7			1.82813
$848$		0			0
$849$		0			0
$850$		0			0
$851$	5.51629e6	−	9.55449e6i	0.261110	−	0.452255i
$852$		0			0
$853$	−1.09389e7	−	1.89467e7i	−0.514754	−	0.891580i	−0.999853	−	0.0171210i	$-0.994550\pi$
$853$	−1.09389e7	−	1.89467e7i	−0.514754	−	0.891580i	0.485099	−	0.874459i	$-0.338783\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−664571.	−	1.15107e6i	−0.0309093	−	0.0535365i	0.850157	−	0.526529i	$-0.176507\pi$
$857$	−664571.	−	1.15107e6i	−0.0309093	−	0.0535365i	−0.881066	+	0.472993i	$0.843174\pi$
$858$		0			0
$859$	−8.07444e6	+	1.39853e7i	−0.373362	+	0.646681i	−0.990080	−	0.140502i	$-0.955128\pi$
$859$	−8.07444e6	+	1.39853e7i	−0.373362	+	0.646681i	0.616719	+	0.787184i	$0.288462\pi$
$860$		0			0
$861$		0			0
$862$		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
$867$		0			0
$868$		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
$873$		0			0
$874$		0			0
$875$	2.16478e7			0.955860
$876$		0			0
$877$	−1.51904e7	−	2.63106e7i	−0.666916	−	1.15513i	−0.978762	−	0.205000i	$-0.934281\pi$
$877$	−1.51904e7	−	2.63106e7i	−0.666916	−	1.15513i	0.311846	−	0.950133i	$-0.399053\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1.96130e7			0.851341			0.425671	−	0.904878i	$-0.360038\pi$
$881$	1.96130e7			0.851341			0.425671	+	0.904878i	$0.360038\pi$
$882$		0			0
$883$	−9.76660e6	+	1.69162e7i	−0.421543	+	0.730133i	−0.996091	−	0.0883376i	$-0.971845\pi$
$883$	−9.76660e6	+	1.69162e7i	−0.421543	+	0.730133i	0.574548	+	0.818471i	$0.305178\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	4.00975e6	−	6.94508e6i	0.171123	−	0.296393i	−0.767690	−	0.640822i	$-0.778594\pi$
$887$	4.00975e6	−	6.94508e6i	0.171123	−	0.296393i	0.938813	+	0.344428i	$0.111927\pi$
$888$		0			0
$889$	−4.41700e6	+	7.65047e6i	−0.187445	+	0.324663i
$890$		0			0
$891$		0			0
$892$		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
$897$		0			0
$898$		0			0
$899$	1.46292e7	−	2.53386e7i	0.603701	−	1.04564i
$900$		0			0
$901$	−1.08934e7			−0.447045
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1.66943e7			−0.677558
$906$		0			0
$907$	−1.11897e6	−	1.93811e6i	−0.0451647	−	0.0782276i	0.842559	−	0.538604i	$-0.181048\pi$
$907$	−1.11897e6	−	1.93811e6i	−0.0451647	−	0.0782276i	−0.887724	+	0.460376i	$0.847715\pi$
$908$		0			0
$909$		0			0
$910$		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
$915$		0			0
$916$		0			0
$917$	2.31957e7	+	4.01761e7i	0.910928	+	1.57777i
$918$		0			0
$919$	8.32391e6			0.325116			0.162558	−	0.986699i	$-0.448025\pi$
$919$	8.32391e6			0.325116			0.162558	+	0.986699i	$0.448025\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	9.62297e6			0.371796
$924$		0			0
$925$	5.22066e6	−	9.04245e6i	0.200619	−	0.347482i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−4.49343e6	−	7.78285e6i	−0.170820	−	0.295869i	0.767887	−	0.640586i	$-0.221308\pi$
$929$	−4.49343e6	−	7.78285e6i	−0.170820	−	0.295869i	−0.938707	+	0.344717i	$0.887975\pi$
$930$		0			0
$931$	−1.06439e6	+	20827.6i	−0.0402463	+	0.000787526i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.51671e7	+	2.62702e7i	−0.567380	+	0.982730i
$936$		0			0
$937$	−623801.	+	1.08045e6i	−0.0232112	+	0.0402029i	−0.877398	−	0.479764i	$-0.840722\pi$
$937$	−623801.	+	1.08045e6i	−0.0232112	+	0.0402029i	0.854187	+	0.519967i	$0.174056\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.30000e6	−	2.25167e6i	0.0478596	−	0.0828953i	−0.841103	−	0.540875i	$-0.818093\pi$
$941$	1.30000e6	−	2.25167e6i	0.0478596	−	0.0828953i	0.888963	+	0.457979i	$0.151427\pi$
$942$		0			0
$943$	3.33835e7			1.22251
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1.03456e7	−	1.79192e7i	−0.374871	−	0.649296i	0.615437	−	0.788186i	$-0.288980\pi$
$947$	−1.03456e7	−	1.79192e7i	−0.374871	−	0.649296i	−0.990308	+	0.138891i	$0.955646\pi$
$948$		0			0
$949$	2.25357e7			0.812279
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.50162e7	−	2.60089e7i	−0.535586	−	0.927662i	−0.999135	−	0.0415907i	$-0.986757\pi$
$953$	−1.50162e7	−	2.60089e7i	−0.535586	−	0.927662i	0.463549	−	0.886071i	$-0.346576\pi$
$954$		0			0
$955$	−6.94120e6	−	1.20225e7i	−0.246278	−	0.426567i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1.64475e7	−	2.84879e7i	0.577502	−	1.00026i
$960$		0			0
$961$	8.35579e7			2.91863
$962$		0			0
$963$		0			0
$964$		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	$-0.0848915\pi$
$967$	7.38839e6	+	1.27971e7i	0.254088	+	0.440093i	−0.710560	+	0.703637i	$0.751558\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	5.30678e6	+	9.19161e6i	0.180627	+	0.312855i	0.942094	−	0.335348i	$-0.108854\pi$
$971$	5.30678e6	+	9.19161e6i	0.180627	+	0.312855i	−0.761467	+	0.648203i	$0.775521\pi$
$972$		0			0
$973$	2.92903e7	−	5.07322e7i	0.991839	−	1.71791i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	3.66481e7			1.22833			0.614165	−	0.789177i	$-0.289493\pi$
$977$	3.66481e7			1.22833			0.614165	+	0.789177i	$0.289493\pi$
$978$		0			0
$979$	1.70228e7	−	2.94844e7i	0.567642	−	0.983186i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	6.80511e6	+	1.17868e7i	0.224621	+	0.389056i	0.956206	−	0.292695i	$-0.0945521\pi$
$983$	6.80511e6	+	1.17868e7i	0.224621	+	0.389056i	−0.731584	+	0.681751i	$0.761219\pi$
$984$		0			0
$985$	−677384.	−	1.17326e6i	−0.0222456	−	0.0385305i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2.36003e7			−0.767232
$990$		0			0
$991$	1.59467e6	+	2.76205e6i	0.0515806	+	0.0893402i	0.890663	−	0.454664i	$-0.150241\pi$
$991$	1.59467e6	+	2.76205e6i	0.0515806	+	0.0893402i	−0.839082	+	0.544005i	$0.816907\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1.85615e7			0.594367
$996$		0			0
$997$	−1.71729e7	+	2.97444e7i	−0.547150	+	0.947692i	0.451318	+	0.892363i	$0.350954\pi$
$997$	−1.71729e7	+	2.97444e7i	−0.547150	+	0.947692i	−0.998468	+	0.0553288i	$0.982379\pi$
$998$		0			0
$999$		0			0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.6.e.a.45.7	✓	18	3.2	odd	2
76.6.e.a.49.7	yes	18	57.11	odd	6
304.6.i.d.49.3		18	228.11	even	6
304.6.i.d.273.3		18	12.11	even	2
684.6.k.f.505.5		18	19.11	even	3	inner
684.6.k.f.577.5		18	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	684.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(15.4645 + 26.7852i) q^{5} +132.225 q^{7} +O(q^{10})\)	\(q+(15.4645 + 26.7852i) q^{5} +132.225 q^{7} -670.612 q^{11} +(-411.330 + 712.445i) q^{13} +(-731.251 - 1266.56i) q^{17} +(-1573.26 + 30.7850i) q^{19} +(1145.60 - 1984.23i) q^{23} +(1084.20 - 1877.89i) q^{25} +(1381.18 - 2392.27i) q^{29} +10591.8 q^{31} +(2044.79 + 3541.69i) q^{35} +4815.22 q^{37} +(7285.18 + 12618.3i) q^{41} +(-5150.23 - 8920.46i) q^{43} +(2555.09 - 4425.55i) q^{47} +676.549 q^{49} +(3724.23 - 6450.55i) q^{53} +(-10370.7 - 17962.5i) q^{55} +(17150.1 + 29704.9i) q^{59} +(20608.6 - 35695.1i) q^{61} -25444.0 q^{65} +(-19625.6 + 33992.5i) q^{67} +(-5848.69 - 10130.2i) q^{71} +(-13696.8 - 23723.6i) q^{73} -88671.9 q^{77} +(-2583.48 - 4474.71i) q^{79} +86152.6 q^{83} +(22616.8 - 39173.5i) q^{85} +(-25384.0 + 43966.4i) q^{89} +(-54388.3 + 94203.3i) q^{91} +(-25154.2 - 41664.1i) q^{95} +(-13304.7 - 23044.5i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 11 q^{5} + 336 q^{7}+O(q^{10}) \) 18 * q - 11 * q^5 + 336 * q^7	\( 18 q - 11 q^{5} + 336 q^{7} + 320 q^{11} + 227 q^{13} - 179 q^{17} - 868 q^{19} + 3425 q^{23} - 7054 q^{25} + 7349 q^{29} - 9960 q^{31} - 15888 q^{35} + 26444 q^{37} + 7311 q^{41} - 8283 q^{43} - 37603 q^{47} + 124738 q^{49} + 20337 q^{53} + 716 q^{55} + 74455 q^{59} - 7569 q^{61} - 188998 q^{65} - 26177 q^{67} + 53463 q^{71} - 14103 q^{73} + 31960 q^{77} + 31825 q^{79} - 82600 q^{83} - 50787 q^{85} + 155197 q^{89} - 2800 q^{91} - 49315 q^{95} + 111241 q^{97}+O(q^{100}) \) 18 * q - 11 * q^5 + 336 * q^7 + 320 * q^11 + 227 * q^13 - 179 * q^17 - 868 * q^19 + 3425 * q^23 - 7054 * q^25 + 7349 * q^29 - 9960 * q^31 - 15888 * q^35 + 26444 * q^37 + 7311 * q^41 - 8283 * q^43 - 37603 * q^47 + 124738 * q^49 + 20337 * q^53 + 716 * q^55 + 74455 * q^59 - 7569 * q^61 - 188998 * q^65 - 26177 * q^67 + 53463 * q^71 - 14103 * q^73 + 31960 * q^77 + 31825 * q^79 - 82600 * q^83 - 50787 * q^85 + 155197 * q^89 - 2800 * q^91 - 49315 * q^95 + 111241 * q^97

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