LMFDB - Embedded newform 84.6.i.c.37.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,6,Mod(25,84)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(84, 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("84.25");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	84.i (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:	$13.4722408643$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 703x^{6} + 2770x^{5} + 427565x^{4} + 718170x^{3} + 42175732x^{2} - 40929504x + 3559792896 $ x^8 - 2x^7 + 703x^6 + 2770x^5 + 427565x^4 + 718170x^3 + 42175732x^2 - 40929504*x + 3559792896
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{3}\cdot 7 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			37.2
Root			$-5.49618 - 9.51967i$ of defining polynomial
Character	$\chi$	$=$	84.37
Dual form			84.6.i.c.25.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.50000 + 7.79423i) q^{3} +(-23.0577 + 39.9371i) q^{5} +(112.271 - 64.8240i) q^{7} +(-40.5000 + 70.1481i) q^{9} +O(q^{10})$	\(q+(4.50000 + 7.79423i) q^{3} +(-23.0577 + 39.9371i) q^{5} +(112.271 - 64.8240i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(315.582 + 546.605i) q^{11} -1079.22 q^{13} -415.038 q^{15} +(-80.5778 - 139.565i) q^{17} +(-588.428 + 1019.19i) q^{19} +(1010.47 + 583.360i) q^{21} +(-1081.73 + 1873.61i) q^{23} +(499.186 + 864.615i) q^{25} -729.000 q^{27} -4492.01 q^{29} +(-159.130 - 275.621i) q^{31} +(-2840.24 + 4919.44i) q^{33} +(0.166415 + 5978.48i) q^{35} +(-7593.41 + 13152.2i) q^{37} +(-4856.51 - 8411.73i) q^{39} +20587.2 q^{41} -455.118 q^{43} +(-1867.67 - 3234.90i) q^{45} +(10381.4 - 17981.1i) q^{47} +(8402.69 - 14555.8i) q^{49} +(725.200 - 1256.08i) q^{51} +(9650.03 + 16714.3i) q^{53} -29106.4 q^{55} -10591.7 q^{57} +(-3184.15 - 5515.11i) q^{59} +(24572.6 - 42560.9i) q^{61} +(0.292303 + 10501.0i) q^{63} +(24884.4 - 43101.1i) q^{65} +(-17027.0 - 29491.7i) q^{67} -19471.1 q^{69} +62962.4 q^{71} +(4433.88 + 7679.70i) q^{73} +(-4492.67 + 7781.54i) q^{75} +(70864.0 + 40910.7i) q^{77} +(-17206.6 + 29802.7i) q^{79} +(-3280.50 - 5681.99i) q^{81} -7041.42 q^{83} +7431.75 q^{85} +(-20214.1 - 35011.8i) q^{87} +(10121.4 - 17530.7i) q^{89} +(-121166. + 69959.7i) q^{91} +(1432.17 - 2480.59i) q^{93} +(-27135.6 - 47000.2i) q^{95} +54066.6 q^{97} -51124.4 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 36 q^{3} - 42 q^{7} - 324 q^{9}+O(q^{10}) $ 8 * q + 36 * q^3 - 42 * q^7 - 324 * q^9	$ 8 q + 36 q^{3} - 42 q^{7} - 324 q^{9} - 462 q^{11} - 1204 q^{13} + 228 q^{17} + 358 q^{19} + 1404 q^{21} - 2148 q^{23} - 5454 q^{25} - 5832 q^{27} - 11064 q^{29} + 830 q^{31} + 4158 q^{33} + 7692 q^{35} - 3914 q^{37} - 5418 q^{39} - 16632 q^{41} - 29036 q^{43} + 41700 q^{47} + 41876 q^{49} - 2052 q^{51} + 22164 q^{53} + 7784 q^{55} + 6444 q^{57} + 32886 q^{59} + 83732 q^{61} + 16038 q^{63} - 93192 q^{65} - 80034 q^{67} - 38664 q^{69} + 89544 q^{71} - 22470 q^{73} + 49086 q^{75} + 138732 q^{77} - 75286 q^{79} - 26244 q^{81} - 34836 q^{83} + 278504 q^{85} - 49788 q^{87} + 28944 q^{89} - 12058 q^{91} - 7470 q^{93} - 144120 q^{95} - 433356 q^{97} + 74844 q^{99}+O(q^{100}) $ 8 * q + 36 * q^3 - 42 * q^7 - 324 * q^9 - 462 * q^11 - 1204 * q^13 + 228 * q^17 + 358 * q^19 + 1404 * q^21 - 2148 * q^23 - 5454 * q^25 - 5832 * q^27 - 11064 * q^29 + 830 * q^31 + 4158 * q^33 + 7692 * q^35 - 3914 * q^37 - 5418 * q^39 - 16632 * q^41 - 29036 * q^43 + 41700 * q^47 + 41876 * q^49 - 2052 * q^51 + 22164 * q^53 + 7784 * q^55 + 6444 * q^57 + 32886 * q^59 + 83732 * q^61 + 16038 * q^63 - 93192 * q^65 - 80034 * q^67 - 38664 * q^69 + 89544 * q^71 - 22470 * q^73 + 49086 * q^75 + 138732 * q^77 - 75286 * q^79 - 26244 * q^81 - 34836 * q^83 + 278504 * q^85 - 49788 * q^87 + 28944 * q^89 - 12058 * q^91 - 7470 * q^93 - 144120 * q^95 - 433356 * q^97 + 74844 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$.

$n$	$29$	$43$	$73$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$3$	4.50000	+	7.79423i	0.288675	+	0.500000i
$3$	4.50000	+	7.79423i	0.288675	+	0.500000i
$4$		0			0
$5$	−23.0577	+	39.9371i	−0.412469	+	0.714416i	−0.995159	−	0.0982777i	$-0.968667\pi$
$5$	−23.0577	+	39.9371i	−0.412469	+	0.714416i	0.582691	+	0.812694i	$0.302000\pi$
$6$		0			0
$7$	112.271	−	64.8240i	0.866011	−	0.500024i
$7$	112.271	−	64.8240i	0.866011	−	0.500024i
$8$		0			0
$9$	−40.5000	+	70.1481i	−0.166667	+	0.288675i
$10$		0			0
$11$	315.582	+	546.605i	0.786378	+	1.36205i	0.928173	+	0.372150i	$0.121379\pi$
$11$	315.582	+	546.605i	0.786378	+	1.36205i	−0.141795	+	0.989896i	$0.545287\pi$
$12$		0			0
$13$	−1079.22			−1.77114			−0.885571	−	0.464503i	$-0.846233\pi$
$13$	−1079.22			−1.77114			−0.885571	+	0.464503i	$0.846233\pi$
$14$		0			0
$15$	−415.038			−0.476278
$16$		0			0
$17$	−80.5778	−	139.565i	−0.0676228	−	0.117126i	0.830232	−	0.557419i	$-0.188208\pi$
$17$	−80.5778	−	139.565i	−0.0676228	−	0.117126i	−0.897854	+	0.440292i	$0.854875\pi$
$18$		0			0
$19$	−588.428	+	1019.19i	−0.373946	+	0.647694i	−0.990169	−	0.139878i	$-0.955329\pi$
$19$	−588.428	+	1019.19i	−0.373946	+	0.647694i	0.616222	+	0.787572i	$0.288662\pi$
$20$		0			0
$21$	1010.47	+	583.360i	0.500008	+	0.288661i
$22$		0			0
$23$	−1081.73	+	1873.61i	−0.426382	+	0.738516i	−0.996548	−	0.0830136i	$-0.973546\pi$
$23$	−1081.73	+	1873.61i	−0.426382	+	0.738516i	0.570166	+	0.821529i	$0.306879\pi$
$24$		0			0
$25$	499.186	+	864.615i	0.159739	+	0.276677i
$26$		0			0
$27$	−729.000			−0.192450
$28$		0			0
$29$	−4492.01			−0.991850			−0.495925	−	0.868365i	$-0.665171\pi$
$29$	−4492.01			−0.991850			−0.495925	+	0.868365i	$0.665171\pi$
$30$		0			0
$31$	−159.130	−	275.621i	−0.0297405	−	0.0515120i	0.850772	−	0.525535i	$-0.176135\pi$
$31$	−159.130	−	275.621i	−0.0297405	−	0.0515120i	−0.880513	+	0.474023i	$0.842801\pi$
$32$		0			0
$33$	−2840.24	+	4919.44i	−0.454015	+	0.786378i
$34$		0			0
$35$	0.166415	+	5978.48i	2.29627e−5	+	0.824937i
$36$		0			0
$37$	−7593.41	+	13152.2i	−0.911869	+	1.57940i	−0.100447	+	0.994942i	$0.532027\pi$
$37$	−7593.41	+	13152.2i	−0.911869	+	1.57940i	−0.811422	+	0.584461i	$0.801306\pi$
$38$		0			0
$39$	−4856.51	−	8411.73i	−0.511285	−	0.885571i
$40$		0			0
$41$	20587.2			1.91266			0.956330	−	0.292289i	$-0.0944169\pi$
$41$	20587.2			1.91266			0.956330	+	0.292289i	$0.0944169\pi$
$42$		0			0
$43$	−455.118			−0.0375364			−0.0187682	−	0.999824i	$-0.505974\pi$
$43$	−455.118			−0.0375364			−0.0187682	+	0.999824i	$0.505974\pi$
$44$		0			0
$45$	−1867.67	−	3234.90i	−0.137490	−	0.238139i
$46$		0			0
$47$	10381.4	−	17981.1i	0.685504	−	1.18733i	−0.287774	−	0.957698i	$-0.592915\pi$
$47$	10381.4	−	17981.1i	0.685504	−	1.18733i	0.973278	−	0.229630i	$-0.0737516\pi$
$48$		0			0
$49$	8402.69	−	14555.8i	0.499952	−	0.866053i
$50$		0			0
$51$	725.200	−	1256.08i	0.0390420	−	0.0676228i
$52$		0			0
$53$	9650.03	+	16714.3i	0.471888	+	0.817334i	0.999483	−	0.0321622i	$-0.0102393\pi$
$53$	9650.03	+	16714.3i	0.471888	+	0.817334i	−0.527595	+	0.849496i	$0.676906\pi$
$54$		0			0
$55$	−29106.4			−1.29742
$56$		0			0
$57$	−10591.7			−0.431796
$58$		0			0
$59$	−3184.15	−	5515.11i	−0.119087	−	0.206264i	0.800319	−	0.599574i	$-0.204663\pi$
$59$	−3184.15	−	5515.11i	−0.119087	−	0.206264i	−0.919406	+	0.393310i	$0.871330\pi$
$60$		0			0
$61$	24572.6	−	42560.9i	0.845524	−	1.46449i	−0.0396416	−	0.999214i	$-0.512622\pi$
$61$	24572.6	−	42560.9i	0.845524	−	1.46449i	0.885165	−	0.465276i	$-0.154045\pi$
$62$		0			0
$63$	0.292303	+	10501.0i	9.27857e−6	+	0.333333i
$64$		0			0
$65$	24884.4	−	43101.1i	0.730541	−	1.26533i
$66$		0			0
$67$	−17027.0	−	29491.7i	−0.463395	−	0.802624i	0.535732	−	0.844388i	$-0.320036\pi$
$67$	−17027.0	−	29491.7i	−0.463395	−	0.802624i	−0.999128	+	0.0417639i	$0.986702\pi$
$68$		0			0
$69$	−19471.1			−0.492344
$70$		0			0
$71$	62962.4			1.48230			0.741149	−	0.671341i	$-0.234281\pi$
$71$	62962.4			1.48230			0.741149	+	0.671341i	$0.234281\pi$
$72$		0			0
$73$	4433.88	+	7679.70i	0.0973815	+	0.168670i	0.910600	−	0.413289i	$-0.135620\pi$
$73$	4433.88	+	7679.70i	0.0973815	+	0.168670i	−0.813219	+	0.581958i	$0.802287\pi$
$74$		0			0
$75$	−4492.67	+	7781.54i	−0.0922256	+	0.159739i
$76$		0			0
$77$	70864.0	+	40910.7i	1.36207	+	0.786340i
$78$		0			0
$79$	−17206.6	+	29802.7i	−0.310190	+	0.537265i	−0.978403	−	0.206705i	$-0.933726\pi$
$79$	−17206.6	+	29802.7i	−0.310190	+	0.537265i	0.668213	+	0.743970i	$0.267059\pi$
$80$		0			0
$81$	−3280.50	−	5681.99i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	−7041.42			−0.112193			−0.0560964	−	0.998425i	$-0.517865\pi$
$83$	−7041.42			−0.112193			−0.0560964	+	0.998425i	$0.517865\pi$
$84$		0			0
$85$	7431.75			0.111569
$86$		0			0
$87$	−20214.1	−	35011.8i	−0.286323	−	0.495925i
$88$		0			0
$89$	10121.4	−	17530.7i	0.135445	−	0.234598i	−0.790322	−	0.612692i	$-0.790087\pi$
$89$	10121.4	−	17530.7i	0.135445	−	0.234598i	0.925768	+	0.378093i	$0.123420\pi$
$90$		0			0
$91$	−121166.	+	69959.7i	−1.53383	+	0.885614i
$92$		0			0
$93$	1432.17	−	2480.59i	0.0171707	−	0.0297405i
$94$		0			0
$95$	−27135.6	−	47000.2i	−0.308482	−	0.534307i
$96$		0			0
$97$	54066.6			0.583444			0.291722	−	0.956503i	$-0.405772\pi$
$97$	54066.6			0.583444			0.291722	+	0.956503i	$0.405772\pi$
$98$		0			0
$99$	−51124.4			−0.524252
$100$		0			0
$101$	45743.2	+	79229.5i	0.446193	+	0.772829i	0.998134	−	0.0610540i	$-0.0194462\pi$
$101$	45743.2	+	79229.5i	0.446193	+	0.772829i	−0.551942	+	0.833883i	$0.686113\pi$
$102$		0			0
$103$	−37690.7	+	65282.2i	−0.350059	+	0.606320i	−0.986259	−	0.165204i	$-0.947172\pi$
$103$	−37690.7	+	65282.2i	−0.350059	+	0.606320i	0.636201	+	0.771524i	$0.280505\pi$
$104$		0			0
$105$	−46596.9	+	26904.5i	−0.412462	+	0.238150i
$106$		0			0
$107$	−76066.3	+	131751.i	−0.642293	+	1.11248i	0.342627	+	0.939472i	$0.388683\pi$
$107$	−76066.3	+	131751.i	−0.642293	+	1.11248i	−0.984920	+	0.173012i	$0.944650\pi$
$108$		0			0
$109$	−38336.0	−	66399.8i	−0.309058	−	0.535304i	0.669099	−	0.743174i	$-0.266680\pi$
$109$	−38336.0	−	66399.8i	−0.309058	−	0.535304i	−0.978157	+	0.207869i	$0.933347\pi$
$110$		0			0
$111$	−136681.			−1.05294
$112$		0			0
$113$	228515.			1.68352			0.841760	−	0.539852i	$-0.181520\pi$
$113$	228515.			1.68352			0.841760	+	0.539852i	$0.181520\pi$
$114$		0			0
$115$	−49884.4	−	86402.3i	−0.351739	−	0.609229i
$116$		0			0
$117$	43708.6	−	75705.5i	0.295190	−	0.511285i
$118$		0			0
$119$	−18093.7	−	10445.7i	−0.117128	−	0.0676195i
$120$		0			0
$121$	−118659.	+	205524.i	−0.736780	+	1.27614i
$122$		0			0
$123$	92642.4	+	160461.i	0.552137	+	0.956330i
$124$		0			0
$125$	−190151.			−1.08849
$126$		0			0
$127$	122111.			0.671809			0.335905	−	0.941896i	$-0.390958\pi$
$127$	122111.			0.671809			0.335905	+	0.941896i	$0.390958\pi$
$128$		0			0
$129$	−2048.03	−	3547.30i	−0.0108358	−	0.0187682i
$130$		0			0
$131$	−37897.7	+	65640.7i	−0.192945	+	0.334191i	−0.946225	−	0.323509i	$-0.895137\pi$
$131$	−37897.7	+	65640.7i	−0.192945	+	0.334191i	0.753280	+	0.657700i	$0.228471\pi$
$132$		0			0
$133$	4.24689	+	152570.i	2.08181e−5	+	0.747892i
$134$		0			0
$135$	16809.1	−	29114.1i	0.0793796	−	0.137490i
$136$		0			0
$137$	120806.	+	209242.i	0.549903	+	0.952460i	0.998281	+	0.0586154i	$0.0186686\pi$
$137$	120806.	+	209242.i	0.549903	+	0.952460i	−0.448378	+	0.893844i	$0.647998\pi$
$138$		0			0
$139$	125657.			0.551634			0.275817	−	0.961210i	$-0.411052\pi$
$139$	125657.			0.551634			0.275817	+	0.961210i	$0.411052\pi$
$140$		0			0
$141$	186865.			0.791552
$142$		0			0
$143$	−340584.	−	589910.i	−1.39279	−	2.41238i
$144$		0			0
$145$	103575.	−	179398.i	0.409107	−	0.708594i
$146$		0			0
$147$	151263.	−	8.42103i	0.577350	−	3.21419e-5i
$148$		0			0
$149$	−18023.3	+	31217.2i	−0.0665071	+	0.115194i	−0.897362	−	0.441296i	$-0.854519\pi$
$149$	−18023.3	+	31217.2i	−0.0665071	+	0.115194i	0.830854	+	0.556490i	$0.187852\pi$
$150$		0			0
$151$	−75334.0	−	130482.i	−0.268874	−	0.465703i	0.699697	−	0.714439i	$-0.253318\pi$
$151$	−75334.0	−	130482.i	−0.268874	−	0.465703i	−0.968571	+	0.248736i	$0.919985\pi$
$152$		0			0
$153$	13053.6			0.0450819
$154$		0			0
$155$	14676.7			0.0490680
$156$		0			0
$157$	−213207.	−	369285.i	−0.690323	−	1.19567i	−0.971732	−	0.236086i	$-0.924135\pi$
$157$	−213207.	−	369285.i	−0.690323	−	1.19567i	0.281409	−	0.959588i	$-0.409198\pi$
$158$		0			0
$159$	−86850.3	+	150429.i	−0.272445	+	0.471888i
$160$		0			0
$161$	7.80722	+	280475.i	2.37373e−5	+	0.852765i
$162$		0			0
$163$	96255.0	−	166718.i	0.283762	−	0.491490i	−0.688546	−	0.725192i	$-0.741751\pi$
$163$	96255.0	−	166718.i	0.283762	−	0.491490i	0.972308	+	0.233702i	$0.0750841\pi$
$164$		0			0
$165$	−130979.	−	226862.i	−0.374534	−	0.648712i
$166$		0			0
$167$	164987.			0.457782			0.228891	−	0.973452i	$-0.426490\pi$
$167$	164987.			0.457782			0.228891	+	0.973452i	$0.426490\pi$
$168$		0			0
$169$	793433.			2.13695
$170$		0			0
$171$	−47662.6	−	82554.1i	−0.124649	−	0.215898i
$172$		0			0
$173$	164749.	−	285353.i	0.418511	−	0.724883i	−0.577279	−	0.816547i	$-0.695885\pi$
$173$	164749.	−	285353.i	0.418511	−	0.724883i	0.995790	+	0.0916644i	$0.0292187\pi$
$174$		0			0
$175$	112092.	+	64712.2i	0.276681	+	0.159732i
$176$		0			0
$177$	28657.4	−	49636.0i	0.0687548	−	0.119087i
$178$		0			0
$179$	−184174.	−	318999.i	−0.429631	−	0.744143i	0.567209	−	0.823574i	$-0.308023\pi$
$179$	−184174.	−	318999.i	−0.429631	−	0.744143i	−0.996840	+	0.0794308i	$0.974690\pi$
$180$		0			0
$181$	−79607.3			−0.180616			−0.0903080	−	0.995914i	$-0.528785\pi$
$181$	−79607.3			−0.180616			−0.0903080	+	0.995914i	$0.528785\pi$
$182$		0			0
$183$	442306.			0.976327
$184$		0			0
$185$	−350173.	−	606517.i	−0.752234	−	1.30291i
$186$		0			0
$187$	50857.9	−	88088.4i	0.106354	−	0.184211i
$188$		0			0
$189$	−81845.8	+	47256.7i	−0.166664	+	0.0962297i
$190$		0			0
$191$	−282957.	+	490096.i	−0.561225	+	0.972070i	0.436165	+	0.899867i	$0.356336\pi$
$191$	−282957.	+	490096.i	−0.561225	+	0.972070i	−0.997390	+	0.0722036i	$0.976997\pi$
$192$		0			0
$193$	19332.9	+	33485.6i	0.0373597	+	0.0647089i	0.884101	−	0.467296i	$-0.154772\pi$
$193$	19332.9	+	33485.6i	0.0373597	+	0.0647089i	−0.846741	+	0.532005i	$0.821439\pi$
$194$		0			0
$195$	447920.			0.843556
$196$		0			0
$197$	−334957.			−0.614927			−0.307463	−	0.951560i	$-0.599480\pi$
$197$	−334957.			−0.614927			−0.307463	+	0.951560i	$0.599480\pi$
$198$		0			0
$199$	300123.	+	519828.i	0.537237	+	0.930522i	0.999051	+	0.0435454i	$0.0138653\pi$
$199$	300123.	+	519828.i	0.537237	+	0.930522i	−0.461814	+	0.886977i	$0.652801\pi$
$200$		0			0
$201$	153243.	−	265425.i	0.267541	−	0.463395i
$202$		0			0
$203$	−504324.	+	291191.i	−0.858954	+	0.495949i
$204$		0			0
$205$	−474693.	+	822193.i	−0.788912	+	1.36644i
$206$		0			0
$207$	−87620.1	−	151762.i	−0.142127	−	0.246172i
$208$		0			0
$209$	−742790.			−1.17625
$210$		0			0
$211$	1.06504e6			1.64687			0.823433	−	0.567414i	$-0.192056\pi$
$211$	1.06504e6			1.64687			0.823433	+	0.567414i	$0.192056\pi$
$212$		0			0
$213$	283331.	+	490743.i	0.427903	+	0.741149i
$214$		0			0
$215$	10494.0	−	18176.1i	0.0154826	−	0.0268166i
$216$		0			0
$217$	−35732.6	−	20628.9i	−0.0515128	−	0.0297390i
$218$		0			0
$219$	−39904.9	+	69117.3i	−0.0562232	+	0.0973815i
$220$		0			0
$221$	86961.6	+	150622.i	0.119770	+	0.207447i
$222$		0			0
$223$	−1.37380e6			−1.84995			−0.924976	−	0.380027i	$-0.875915\pi$
$223$	−1.37380e6			−1.84995			−0.924976	+	0.380027i	$0.875915\pi$
$224$		0			0
$225$	−80868.1			−0.106493
$226$		0			0
$227$	−162440.	−	281354.i	−0.209232	−	0.362400i	0.742241	−	0.670133i	$-0.233763\pi$
$227$	−162440.	−	281354.i	−0.209232	−	0.362400i	−0.951473	+	0.307733i	$0.900430\pi$
$228$		0			0
$229$	411196.	−	712212.i	0.518155	−	0.897472i	−0.481622	−	0.876379i	$-0.659952\pi$
$229$	411196.	−	712212.i	0.518155	−	0.897472i	0.999778	−	0.0210926i	$-0.00671448\pi$
$230$		0			0
$231$	20.4990	+	736428.i	2.52757e−5	+	0.908031i
$232$		0			0
$233$	−569070.	+	985657.i	−0.686713	+	1.18942i	0.286182	+	0.958175i	$0.407614\pi$
$233$	−569070.	+	985657.i	−0.686713	+	1.18942i	−0.972895	+	0.231247i	$0.925719\pi$
$234$		0			0
$235$	478741.	+	829204.i	0.565498	+	0.979471i
$236$		0			0
$237$	−309719.			−0.358176
$238$		0			0
$239$	483125.			0.547097			0.273549	−	0.961858i	$-0.411803\pi$
$239$	483125.			0.547097			0.273549	+	0.961858i	$0.411803\pi$
$240$		0			0
$241$	−507406.	−	878853.i	−0.562747	−	0.974706i	−0.997255	−	0.0740384i	$-0.976411\pi$
$241$	−507406.	−	878853.i	−0.562747	−	0.974706i	0.434509	−	0.900668i	$-0.356922\pi$
$242$		0			0
$243$	29524.5	−	51137.9i	0.0320750	−	0.0555556i
$244$		0			0
$245$	387568.	+	671201.i	0.412508	+	0.714393i
$246$		0			0
$247$	635046.	−	1.09993e6i	0.662312	−	1.14716i
$248$		0			0
$249$	−31686.4	−	54882.4i	−0.0323873	−	0.0560964i
$250$		0			0
$251$	415812.			0.416593			0.208297	−	0.978066i	$-0.433208\pi$
$251$	415812.			0.416593			0.208297	+	0.978066i	$0.433208\pi$
$252$		0			0
$253$	−1.36550e6			−1.34119
$254$		0			0
$255$	33442.9	+	57924.8i	0.0322072	+	0.0557845i
$256$		0			0
$257$	−505998.	+	876414.i	−0.477876	+	0.827706i	−0.999678	−	0.0253604i	$-0.991927\pi$
$257$	−505998.	+	876414.i	−0.477876	+	0.827706i	0.521802	+	0.853067i	$0.325260\pi$
$258$		0			0
$259$	54.8043	+	1.96885e6i	5.07651e−5	+	1.82374i
$260$		0			0
$261$	181927.	−	315106.i	0.165308	−	0.286323i
$262$		0			0
$263$	−1.02885e6	−	1.78202e6i	−0.917195	−	1.58863i	−0.803656	−	0.595094i	$-0.797115\pi$
$263$	−1.02885e6	−	1.78202e6i	−0.917195	−	1.58863i	−0.113539	−	0.993534i	$-0.536219\pi$
$264$		0			0
$265$	−890030.			−0.778556
$266$		0			0
$267$	182185.			0.156399
$268$		0			0
$269$	403998.	+	699745.i	0.340407	+	0.589602i	0.984508	−	0.175338i	$-0.0561018\pi$
$269$	403998.	+	699745.i	0.340407	+	0.589602i	−0.644101	+	0.764940i	$0.722768\pi$
$270$		0			0
$271$	−98011.4	+	169761.i	−0.0810687	+	0.140415i	−0.903709	−	0.428147i	$-0.859167\pi$
$271$	−98011.4	+	169761.i	−0.0810687	+	0.140415i	0.822641	+	0.568562i	$0.192500\pi$
$272$		0			0
$273$	−1.09053e6	−	629577.i	−0.885586	−	0.511260i
$274$		0			0
$275$	−315069.	+	545715.i	−0.251231	+	0.435145i
$276$		0			0
$277$	−151292.	−	262046.i	−0.118472	−	0.205200i	0.800690	−	0.599079i	$-0.204466\pi$
$277$	−151292.	−	262046.i	−0.118472	−	0.205200i	−0.919162	+	0.393879i	$0.871133\pi$
$278$		0			0
$279$	25779.0			0.0198270
$280$		0			0
$281$	646014.			0.488063			0.244032	−	0.969767i	$-0.421530\pi$
$281$	646014.			0.488063			0.244032	+	0.969767i	$0.421530\pi$
$282$		0			0
$283$	553748.	+	959119.i	0.411004	+	0.711879i	0.995000	−	0.0998771i	$-0.0318450\pi$
$283$	553748.	+	959119.i	0.411004	+	0.711879i	−0.583996	+	0.811757i	$0.698512\pi$
$284$		0			0
$285$	244220.	−	423002.i	0.178102	−	0.308482i
$286$		0			0
$287$	2.31135e6	−	1.33455e6i	1.65639	−	0.956376i
$288$		0			0
$289$	696943.	−	1.20714e6i	0.490854	−	0.850185i
$290$		0			0
$291$	243300.	+	421407.i	0.168426	+	0.291722i
$292$		0			0
$293$	1.89396e6			1.28885			0.644425	−	0.764668i	$-0.277097\pi$
$293$	1.89396e6			1.28885			0.644425	+	0.764668i	$0.277097\pi$
$294$		0			0
$295$	293677.			0.196478
$296$		0			0
$297$	−230060.	−	398475.i	−0.151338	−	0.262126i
$298$		0			0
$299$	1.16743e6	−	2.02205e6i	0.755184	−	1.30802i
$300$		0			0
$301$	−51096.7	+	29502.6i	−0.0325070	+	0.0187691i
$302$		0			0
$303$	−411688.	+	713065.i	−0.257610	+	0.446193i
$304$		0			0
$305$	1.13317e6	+	1.96271e6i	0.697504	+	1.20811i
$306$		0			0
$307$	1.97803e6			1.19781			0.598905	−	0.800820i	$-0.295603\pi$
$307$	1.97803e6			1.19781			0.598905	+	0.800820i	$0.295603\pi$
$308$		0			0
$309$	−678432.			−0.404213
$310$		0			0
$311$	−1.29393e6	−	2.24116e6i	−0.758596	−	1.31393i	−0.943566	−	0.331183i	$-0.892552\pi$
$311$	−1.29393e6	−	2.24116e6i	−0.758596	−	1.31393i	0.184970	−	0.982744i	$-0.440781\pi$
$312$		0			0
$313$	70678.5	−	122419.i	0.0407781	−	0.0706297i	−0.844916	−	0.534899i	$-0.820350\pi$
$313$	70678.5	−	122419.i	0.0407781	−	0.0706297i	0.885694	+	0.464269i	$0.153683\pi$
$314$		0			0
$315$	−419386.	−	242117.i	−0.238143	−	0.137483i
$316$		0			0
$317$	−586423.	+	1.01571e6i	−0.327765	+	0.567706i	−0.982068	−	0.188527i	$-0.939629\pi$
$317$	−586423.	+	1.01571e6i	−0.327765	+	0.567706i	0.654303	+	0.756233i	$0.272962\pi$
$318$		0			0
$319$	−1.41760e6	−	2.45536e6i	−0.779969	−	1.35095i
$320$		0			0
$321$	−1.36919e6			−0.741656
$322$		0			0
$323$	189657.			0.101149
$324$		0			0
$325$	−538734.	−	933114.i	−0.282921	−	0.490034i
$326$		0			0
$327$	345024.	−	597598.i	0.178435	−	0.309058i
$328$		0			0
$329$	−74.9260	−	2.69172e6i	−3.81630e−5	−	1.37101i
$330$		0			0
$331$	−653305.	+	1.13156e6i	−0.327752	+	0.567684i	−0.982066	−	0.188540i	$-0.939624\pi$
$331$	−653305.	+	1.13156e6i	−0.327752	+	0.567684i	0.654313	+	0.756224i	$0.272958\pi$
$332$		0			0
$333$	−615066.	−	1.06533e6i	−0.303956	−	0.526468i
$334$		0			0
$335$	1.57041e6			0.764544
$336$		0			0
$337$	−265059.			−0.127136			−0.0635679	−	0.997978i	$-0.520248\pi$
$337$	−265059.			−0.127136			−0.0635679	+	0.997978i	$0.520248\pi$
$338$		0			0
$339$	1.02832e6	+	1.78110e6i	0.485991	+	0.841760i
$340$		0			0
$341$	100437.	−	173962.i	0.0467745	−	0.0810157i
$342$		0			0
$343$	−181.953	−	2.17889e6i	−8.35072e−5	−	1.00000i
$344$		0			0
$345$	448959.	−	777621.i	0.203076	−	0.351739i
$346$		0			0
$347$	1.52791e6	+	2.64642e6i	0.681200	+	1.17987i	0.974615	+	0.223888i	$0.0718751\pi$
$347$	1.52791e6	+	2.64642e6i	0.681200	+	1.17987i	−0.293414	+	0.955985i	$0.594792\pi$
$348$		0			0
$349$	−1.47164e6			−0.646753			−0.323377	−	0.946270i	$-0.604818\pi$
$349$	−1.47164e6			−0.646753			−0.323377	+	0.946270i	$0.604818\pi$
$350$		0			0
$351$	786755.			0.340857
$352$		0			0
$353$	192592.	+	333578.i	0.0822623	+	0.142482i	0.904221	−	0.427064i	$-0.140452\pi$
$353$	192592.	+	333578.i	0.0822623	+	0.142482i	−0.821959	+	0.569547i	$0.807119\pi$
$354$		0			0
$355$	−1.45177e6	+	2.51454e6i	−0.611401	+	1.05898i
$356$		0			0
$357$	−5.23402	−	188033.i	−2.17353e−6	−	0.0780841i
$358$		0			0
$359$	820939.	−	1.42191e6i	0.336182	−	0.582285i	−0.647529	−	0.762041i	$-0.724198\pi$
$359$	820939.	−	1.42191e6i	0.336182	−	0.582285i	0.983711	+	0.179756i	$0.0575309\pi$
$360$		0			0
$361$	545555.	+	944929.i	0.220328	+	0.381620i
$362$		0			0
$363$	−2.13586e6			−0.850760
$364$		0			0
$365$	−408940.			−0.160667
$366$		0			0
$367$	561616.	+	972747.i	0.217658	+	0.376994i	0.954091	−	0.299516i	$-0.0968250\pi$
$367$	561616.	+	972747.i	0.217658	+	0.376994i	−0.736434	+	0.676510i	$0.763492\pi$
$368$		0			0
$369$	−833782.	+	1.44415e6i	−0.318777	+	0.552137i
$370$		0			0
$371$	2.16691e6	+	1.25099e6i	0.817347	+	0.471865i
$372$		0			0
$373$	2.11052e6	−	3.65554e6i	0.785450	−	1.36044i	−0.143280	−	0.989682i	$-0.545765\pi$
$373$	2.11052e6	−	3.65554e6i	0.785450	−	1.36044i	0.928730	−	0.370757i	$-0.120902\pi$
$374$		0			0
$375$	−855679.	−	1.48208e6i	−0.314219	−	0.544244i
$376$		0			0
$377$	4.84789e6			1.75671
$378$		0			0
$379$	−4.49923e6			−1.60894			−0.804471	−	0.593993i	$-0.797551\pi$
$379$	−4.49923e6			−1.60894			−0.804471	+	0.593993i	$0.797551\pi$
$380$		0			0
$381$	549500.	+	951762.i	0.193935	+	0.335905i
$382$		0			0
$383$	−2.21557e6	+	3.83748e6i	−0.771771	+	1.33675i	0.164820	+	0.986324i	$0.447296\pi$
$383$	−2.21557e6	+	3.83748e6i	−0.771771	+	1.33675i	−0.936591	+	0.350423i	$0.886038\pi$
$384$		0			0
$385$	−3.26781e6	+	1.88680e6i	−1.12358	+	0.648743i
$386$		0			0
$387$	18432.3	−	31925.7i	0.00625607	−	0.0108358i
$388$		0			0
$389$	2.72111e6	+	4.71311e6i	0.911744	+	1.57919i	0.811600	+	0.584214i	$0.198597\pi$
$389$	2.72111e6	+	4.71311e6i	0.911744	+	1.57919i	0.100144	+	0.994973i	$0.468070\pi$
$390$		0			0
$391$	348654.			0.115333
$392$		0			0
$393$	−682158.			−0.222794
$394$		0			0
$395$	−793489.	−	1.37436e6i	−0.255887	−	0.443209i
$396$		0			0
$397$	−1.34397e6	+	2.32782e6i	−0.427969	+	0.741264i	−0.996693	−	0.0812644i	$-0.974104\pi$
$397$	−1.34397e6	+	2.32782e6i	−0.427969	+	0.741264i	0.568723	+	0.822529i	$0.307438\pi$
$398$		0			0
$399$	−1.18914e6	+	686597.i	−0.373940	+	0.215908i
$400$		0			0
$401$	−221647.	+	383904.i	−0.0688337	+	0.119223i	−0.898388	−	0.439202i	$-0.855261\pi$
$401$	−221647.	+	383904.i	−0.0688337	+	0.119223i	0.829554	+	0.558426i	$0.188594\pi$
$402$		0			0
$403$	171737.	+	297457.i	0.0526746	+	0.0912351i
$404$		0			0
$405$	302563.			0.0916597
$406$		0			0
$407$	−9.58539e6			−2.86829
$408$		0			0
$409$	2.44789e6	+	4.23987e6i	0.723575	+	1.25327i	0.959558	+	0.281511i	$0.0908355\pi$
$409$	2.44789e6	+	4.23987e6i	0.723575	+	1.25327i	−0.235983	+	0.971757i	$0.575831\pi$
$410$		0			0
$411$	−1.08725e6	+	1.88317e6i	−0.317487	+	0.549903i
$412$		0			0
$413$	−715001.	−	412779.i	−0.206268	−	0.119081i
$414$		0			0
$415$	162359.	−	281214.i	0.0462760	−	0.0801523i
$416$		0			0
$417$	565458.	+	979402.i	0.159243	+	0.275817i
$418$		0			0
$419$	−2.23186e6			−0.621058			−0.310529	−	0.950564i	$-0.600506\pi$
$419$	−2.23186e6			−0.621058			−0.310529	+	0.950564i	$0.600506\pi$
$420$		0			0
$421$	5.48208e6			1.50744			0.753721	−	0.657195i	$-0.228257\pi$
$421$	5.48208e6			1.50744			0.753721	+	0.657195i	$0.228257\pi$
$422$		0			0
$423$	840891.	+	1.45647e6i	0.228501	+	0.395776i
$424$		0			0
$425$	80446.6	−	139338.i	0.0216041	−	0.0374193i
$426$		0			0
$427$	−177.349	−	6.37126e6i	−4.70715e−5	−	1.69105i
$428$		0			0
$429$	3.06526e6	−	5.30919e6i	0.804126	−	1.39279i
$430$		0			0
$431$	−2.31183e6	−	4.00421e6i	−0.599463	−	1.03830i	−0.992900	−	0.118949i	$-0.962047\pi$
$431$	−2.31183e6	−	4.00421e6i	−0.599463	−	1.03830i	0.393437	−	0.919352i	$-0.371286\pi$
$432$		0			0
$433$	−3.08314e6			−0.790267			−0.395134	−	0.918624i	$-0.629302\pi$
$433$	−3.08314e6			−0.790267			−0.395134	+	0.918624i	$0.629302\pi$
$434$		0			0
$435$	1.86436e6			0.472396
$436$		0			0
$437$	−1.27304e6	−	2.20497e6i	−0.318888	−	0.552330i
$438$		0			0
$439$	−217235.	+	376262.i	−0.0537983	+	0.0931815i	−0.891670	−	0.452685i	$-0.850466\pi$
$439$	−217235.	+	376262.i	−0.0537983	+	0.0931815i	0.837872	+	0.545867i	$0.183799\pi$
$440$		0			0
$441$	680749.	+	1.17894e6i	0.166683	+	0.288666i
$442$		0			0
$443$	−860317.	+	1.49011e6i	−0.208281	+	0.360753i	−0.951173	−	0.308658i	$-0.900120\pi$
$443$	−860317.	+	1.49011e6i	−0.208281	+	0.360753i	0.742892	+	0.669411i	$0.233453\pi$
$444$		0			0
$445$	466751.	+	808437.i	0.111734	+	0.193529i
$446$		0			0
$447$	−324419.			−0.0767958
$448$		0			0
$449$	−4.06508e6			−0.951598			−0.475799	−	0.879554i	$-0.657841\pi$
$449$	−4.06508e6			−0.951598			−0.475799	+	0.879554i	$0.657841\pi$
$450$		0			0
$451$	6.49696e6	+	1.12531e7i	1.50407	+	2.60513i
$452$		0			0
$453$	678006.	−	1.17434e6i	0.155234	−	0.268874i
$454$		0			0
$455$	−179.600	−	6.45213e6i	−4.06703e−5	−	1.46108i
$456$		0			0
$457$	−1.13092e6	+	1.95881e6i	−0.253303	+	0.438734i	−0.964433	−	0.264326i	$-0.914850\pi$
$457$	−1.13092e6	+	1.95881e6i	−0.253303	+	0.438734i	0.711130	+	0.703061i	$0.248184\pi$
$458$		0			0
$459$	58741.2	+	101743.i	0.0130140	+	0.0225409i
$460$		0			0
$461$	7.80980e6			1.71154			0.855771	−	0.517355i	$-0.173083\pi$
$461$	7.80980e6			1.71154			0.855771	+	0.517355i	$0.173083\pi$
$462$		0			0
$463$	−525518.			−0.113929			−0.0569647	−	0.998376i	$-0.518142\pi$
$463$	−525518.			−0.113929			−0.0569647	+	0.998376i	$0.518142\pi$
$464$		0			0
$465$	66045.0	+	114393.i	0.0141647	+	0.0245340i
$466$		0			0
$467$	−2.79688e6	+	4.84434e6i	−0.593447	+	1.02788i	0.400318	+	0.916376i	$0.368900\pi$
$467$	−2.79688e6	+	4.84434e6i	−0.593447	+	1.02788i	−0.993764	+	0.111503i	$0.964434\pi$
$468$		0			0
$469$	−3.82341e6	−	2.20731e6i	−0.802637	−	0.463373i
$470$		0			0
$471$	1.91886e6	−	3.32357e6i	0.398558	−	0.690323i
$472$		0			0
$473$	−143627.	−	248770.i	−0.0295178	−	0.0511264i
$474$		0			0
$475$	−1.17494e6			−0.238936
$476$		0			0
$477$	−1.56330e6			−0.314592
$478$		0			0
$479$	−906232.	−	1.56964e6i	−0.180468	−	0.312580i	0.761572	−	0.648080i	$-0.224428\pi$
$479$	−906232.	−	1.56964e6i	−0.180468	−	0.312580i	−0.942040	+	0.335500i	$0.891095\pi$
$480$		0			0
$481$	8.19499e6	−	1.41941e7i	1.61505	−	2.79735i
$482$		0			0
$483$	−2.18605e6	+	1.26220e6i	−0.426375	+	0.246184i
$484$		0			0
$485$	−1.24665e6	+	2.15926e6i	−0.240652	+	0.416822i
$486$		0			0
$487$	−720340.	−	1.24767e6i	−0.137631	−	0.238383i	0.788969	−	0.614433i	$-0.210615\pi$
$487$	−720340.	−	1.24767e6i	−0.137631	−	0.238383i	−0.926599	+	0.376050i	$0.877282\pi$
$488$		0			0
$489$	1.73259e6			0.327660
$490$		0			0
$491$	5.25026e6			0.982827			0.491413	−	0.870926i	$-0.336480\pi$
$491$	5.25026e6			0.982827			0.491413	+	0.870926i	$0.336480\pi$
$492$		0			0
$493$	361957.	+	626927.i	0.0670717	+	0.116172i
$494$		0			0
$495$	1.17881e6	−	2.04176e6i	0.216237	−	0.374534i
$496$		0			0
$497$	7.06887e6	−	4.08148e6i	1.28369	−	0.741185i
$498$		0			0
$499$	4.07080e6	−	7.05083e6i	0.731861	−	1.26762i	−0.224227	−	0.974537i	$-0.571986\pi$
$499$	4.07080e6	−	7.05083e6i	0.731861	−	1.26762i	0.956087	−	0.293083i	$-0.0946811\pi$
$500$		0			0
$501$	742442.	+	1.28595e6i	0.132150	+	0.228891i
$502$		0			0
$503$	−7.98029e6			−1.40637			−0.703183	−	0.711009i	$-0.748239\pi$
$503$	−7.98029e6			−1.40637			−0.703183	+	0.711009i	$0.748239\pi$
$504$		0			0
$505$	−4.21893e6			−0.736162
$506$		0			0
$507$	3.57045e6	+	6.18420e6i	0.616883	+	1.06847i
$508$		0			0
$509$	−4.07141e6	+	7.05189e6i	−0.696547	+	1.20646i	0.273109	+	0.961983i	$0.411948\pi$
$509$	−4.07141e6	+	7.05189e6i	−0.696547	+	1.20646i	−0.969656	+	0.244472i	$0.921385\pi$
$510$		0			0
$511$	995626.	+	574788.i	0.168672	+	0.0973768i
$512$		0			0
$513$	428964.	−	742987.i	0.0719660	−	0.124649i
$514$		0			0
$515$	−1.73812e6	−	3.01051e6i	−0.288776	−	0.500176i
$516$		0			0
$517$	1.31047e7			2.15626
$518$		0			0
$519$	2.96548e6			0.483255
$520$		0			0
$521$	−2.55984e6	−	4.43377e6i	−0.413160	−	0.715613i	0.582074	−	0.813136i	$-0.302241\pi$
$521$	−2.55984e6	−	4.43377e6i	−0.413160	−	0.715613i	−0.995233	+	0.0975227i	$0.968908\pi$
$522$		0			0
$523$	−2.36736e6	+	4.10040e6i	−0.378452	+	0.655498i	−0.990837	−	0.135061i	$-0.956877\pi$
$523$	−2.36736e6	+	4.10040e6i	−0.378452	+	0.655498i	0.612385	+	0.790560i	$0.290210\pi$
$524$		0			0
$525$	32.4252	+	1.16488e6i	5.13433e−6	+	0.184451i
$526$		0			0
$527$	−25644.7	+	44417.9i	−0.00402226	+	0.00696677i
$528$		0			0
$529$	877893.	+	1.52055e6i	0.136396	+	0.236245i
$530$		0			0
$531$	515833.			0.0793912
$532$		0			0
$533$	−2.22182e7			−3.38759
$534$		0			0
$535$	−3.50783e6	−	6.07574e6i	−0.529851	−	0.917729i
$536$		0			0
$537$	1.65757e6	−	2.87099e6i	0.248048	−	0.429631i
$538$		0			0
$539$	1.06080e7	−	590.562i	1.57276	−	8.75576e-5i
$540$		0			0
$541$	4.10226e6	−	7.10533e6i	0.602602	−	1.04374i	−0.389824	−	0.920890i	$-0.627464\pi$
$541$	4.10226e6	−	7.10533e6i	0.602602	−	1.04374i	0.992426	−	0.122848i	$-0.0392026\pi$
$542$		0			0
$543$	−358233.	−	620477.i	−0.0521394	−	0.0903080i
$544$		0			0
$545$	3.53575e6			0.509907
$546$		0			0
$547$	−1.57733e6			−0.225400			−0.112700	−	0.993629i	$-0.535950\pi$
$547$	−1.57733e6			−0.225400			−0.112700	+	0.993629i	$0.535950\pi$
$548$		0			0
$549$	1.99038e6	+	3.44744e6i	0.281841	+	0.488163i
$550$		0			0
$551$	2.64323e6	−	4.57820e6i	0.370899	−	0.642415i
$552$		0			0
$553$	124.186	+	4.46139e6i	1.72687e−5	+	0.620380i
$554$		0			0
$555$	3.15156e6	−	5.45865e6i	0.434303	−	0.752234i
$556$		0			0
$557$	1.05205e6	+	1.82221e6i	0.143681	+	0.248863i	0.928880	−	0.370380i	$-0.120773\pi$
$557$	1.05205e6	+	1.82221e6i	0.143681	+	0.248863i	−0.785199	+	0.619244i	$0.787439\pi$
$558$		0			0
$559$	491175.			0.0664824
$560$		0			0
$561$	915442.			0.122807
$562$		0			0
$563$	−3.84583e6	−	6.66118e6i	−0.511351	−	0.885687i	−0.999913	−	0.0131574i	$-0.995812\pi$
$563$	−3.84583e6	−	6.66118e6i	−0.511351	−	0.885687i	0.488562	−	0.872529i	$-0.337522\pi$
$564$		0			0
$565$	−5.26903e6	+	9.12622e6i	−0.694399	+	1.20273i
$566$		0			0
$567$	−736636.	−	425269.i	−0.0962266	−	0.0555529i
$568$		0			0
$569$	−3.65070e6	+	6.32319e6i	−0.472710	+	0.818758i	−0.999512	−	0.0312298i	$-0.990058\pi$
$569$	−3.65070e6	+	6.32319e6i	−0.472710	+	0.818758i	0.526802	+	0.849988i	$0.323391\pi$
$570$		0			0
$571$	−1.70967e6	−	2.96124e6i	−0.219443	−	0.380087i	0.735195	−	0.677856i	$-0.237091\pi$
$571$	−1.70967e6	−	2.96124e6i	−0.219443	−	0.380087i	−0.954638	+	0.297769i	$0.903757\pi$
$572$		0			0
$573$	−5.09323e6			−0.648047
$574$		0			0
$575$	−2.15994e6			−0.272440
$576$		0			0
$577$	5.46447e6	+	9.46473e6i	0.683295	+	1.18350i	0.973969	+	0.226680i	$0.0727870\pi$
$577$	5.46447e6	+	9.46473e6i	0.683295	+	1.18350i	−0.290674	+	0.956822i	$0.593880\pi$
$578$		0			0
$579$	−173996.	+	301370.i	−0.0215696	+	0.0373597i
$580$		0			0
$581$	−790549.	+	456453.i	−0.0971602	+	0.0560991i
$582$		0			0
$583$	−6.09076e6	+	1.05495e7i	−0.742164	+	1.28547i
$584$		0			0
$585$	2.01564e6	+	3.49119e6i	0.243514	+	0.421778i
$586$		0			0
$587$	2.25268e6			0.269839			0.134919	−	0.990857i	$-0.456922\pi$
$587$	2.25268e6			0.269839			0.134919	+	0.990857i	$0.456922\pi$
$588$		0			0
$589$	374546.			0.0444853
$590$		0			0
$591$	−1.50731e6	−	2.61073e6i	−0.177514	−	0.307463i
$592$		0			0
$593$	2.84120e6	−	4.92111e6i	0.331792	−	0.574680i	−0.651072	−	0.759016i	$-0.725680\pi$
$593$	2.84120e6	−	4.92111e6i	0.331792	−	0.574680i	0.982863	+	0.184336i	$0.0590136\pi$
$594$		0			0
$595$	834372.	−	481756.i	0.0966201	−	0.0557872i
$596$		0			0
$597$	−2.70110e6	+	4.67845e6i	−0.310174	+	0.537237i
$598$		0			0
$599$	6.40132e6	+	1.10874e7i	0.728958	+	1.26259i	0.957324	+	0.289017i	$0.0933283\pi$
$599$	6.40132e6	+	1.10874e7i	0.728958	+	1.26259i	−0.228366	+	0.973575i	$0.573338\pi$
$600$		0			0
$601$	−9.25870e6			−1.04560			−0.522798	−	0.852457i	$-0.675112\pi$
$601$	−9.25870e6			−1.04560			−0.522798	+	0.852457i	$0.675112\pi$
$602$		0			0
$603$	2.75838e6			0.308930
$604$		0			0
$605$	−5.47201e6	−	9.47780e6i	−0.607797	−	1.05274i
$606$		0			0
$607$	200660.	−	347553.i	0.0221049	−	0.0382868i	−0.854761	−	0.519021i	$-0.826297\pi$
$607$	200660.	−	347553.i	0.0221049	−	0.0382868i	0.876866	+	0.480734i	$0.159630\pi$
$608$		0			0
$609$	−4.53906e6	−	2.62046e6i	−0.495933	−	0.286309i
$610$		0			0
$611$	−1.12038e7	+	1.94056e7i	−1.21413	+	2.10293i
$612$		0			0
$613$	−3.41198e6	−	5.90973e6i	−0.366738	−	0.635208i	0.622316	−	0.782766i	$-0.286192\pi$
$613$	−3.41198e6	−	5.90973e6i	−0.366738	−	0.635208i	−0.989053	+	0.147558i	$0.952859\pi$
$614$		0			0
$615$	−8.54448e6			−0.910957
$616$		0			0
$617$	−336274.			−0.0355615			−0.0177808	−	0.999842i	$-0.505660\pi$
$617$	−336274.			−0.0355615			−0.0177808	+	0.999842i	$0.505660\pi$
$618$		0			0
$619$	4.80575e6	+	8.32381e6i	0.504121	+	0.873163i	0.999989	+	0.00476520i	$0.00151681\pi$
$619$	4.80575e6	+	8.32381e6i	0.504121	+	0.873163i	−0.495868	+	0.868398i	$0.665150\pi$
$620$		0			0
$621$	788581.	−	1.36586e6i	0.0820573	−	0.142127i
$622$		0			0
$623$	−73.0495	−	2.62431e6i	−7.54045e−6	−	0.270891i
$624$		0			0
$625$	2.82448e6	−	4.89215e6i	0.289227	−	0.500956i
$626$		0			0
$627$	−3.34255e6	−	5.78947e6i	−0.339555	−	0.588126i
$628$		0			0
$629$	2.44744e6			0.246652
$630$		0			0
$631$	1.28813e7			1.28791			0.643957	−	0.765062i	$-0.277292\pi$
$631$	1.28813e7			1.28791			0.643957	+	0.765062i	$0.277292\pi$
$632$		0			0
$633$	4.79266e6	+	8.30113e6i	0.475409	+	0.823433i
$634$		0			0
$635$	−2.81560e6	+	4.87676e6i	−0.277100	+	0.479951i
$636$		0			0
$637$	−9.06839e6	+	1.57089e7i	−0.885486	+	1.53390i
$638$		0			0
$639$	−2.54998e6	+	4.41669e6i	−0.247050	+	0.427903i
$640$		0			0
$641$	7.47479e6	+	1.29467e7i	0.718545	+	1.24456i	0.961576	+	0.274538i	$0.0885250\pi$
$641$	7.47479e6	+	1.29467e7i	0.718545	+	1.24456i	−0.243031	+	0.970018i	$0.578142\pi$
$642$		0			0
$643$	−8.03892e6			−0.766780			−0.383390	−	0.923587i	$-0.625243\pi$
$643$	−8.03892e6			−0.766780			−0.383390	+	0.923587i	$0.625243\pi$
$644$		0			0
$645$	188892.			0.0178778
$646$		0			0
$647$	3.34388e6	+	5.79176e6i	0.314043	+	0.543939i	0.979234	−	0.202735i	$-0.0649830\pi$
$647$	3.34388e6	+	5.79176e6i	0.314043	+	0.543939i	−0.665190	+	0.746674i	$0.731650\pi$
$648$		0			0
$649$	2.00973e6	−	3.48095e6i	0.187294	−	0.324404i
$650$		0			0
$651$	−10.3365	−	371338.i	−9.55915e−7	−	0.0343413i
$652$		0			0
$653$	522213.	−	904499.i	0.0479253	−	0.0830090i	−0.841068	−	0.540930i	$-0.818072\pi$
$653$	522213.	−	904499.i	0.0479253	−	0.0830090i	0.888993	+	0.457921i	$0.151406\pi$
$654$		0			0
$655$	−1.74766e6	−	3.02704e6i	−0.159168	−	0.275687i
$656$		0			0
$657$	−718288.			−0.0649210
$658$		0			0
$659$	2.10237e7			1.88580			0.942902	−	0.333071i	$-0.108085\pi$
$659$	2.10237e7			1.88580			0.942902	+	0.333071i	$0.108085\pi$
$660$		0			0
$661$	−4.73340e6	−	8.19849e6i	−0.421376	−	0.729845i	0.574698	−	0.818365i	$-0.305119\pi$
$661$	−4.73340e6	−	8.19849e6i	−0.421376	−	0.729845i	−0.996074	+	0.0885207i	$0.971786\pi$
$662$		0			0
$663$	−782654.	+	1.35560e6i	−0.0691490	+	0.119770i
$664$		0			0
$665$	−6.09329e6	−	3.51773e6i	−0.534315	−	0.308467i
$666$		0			0
$667$	4.85915e6	−	8.41629e6i	0.422908	−	0.732497i
$668$		0			0
$669$	−6.18208e6	−	1.07077e7i	−0.534035	−	0.924976i
$670$		0			0
$671$	3.10187e7			2.65960
$672$		0			0
$673$	−1.95188e7			−1.66117			−0.830587	−	0.556889i	$-0.811995\pi$
$673$	−1.95188e7			−1.66117			−0.830587	+	0.556889i	$0.811995\pi$
$674$		0			0
$675$	−363906.	−	630304.i	−0.0307419	−	0.0532465i
$676$		0			0
$677$	8.52722e6	−	1.47696e7i	0.715049	−	1.23850i	−0.247892	−	0.968788i	$-0.579738\pi$
$677$	8.52722e6	−	1.47696e7i	0.715049	−	1.23850i	0.962941	−	0.269713i	$-0.0869289\pi$
$678$		0			0
$679$	6.07012e6	−	3.50481e6i	0.505270	−	0.291736i
$680$		0			0
$681$	1.46196e6	−	2.53219e6i	0.120800	−	0.209232i
$682$		0			0
$683$	−5.47375e6	−	9.48080e6i	−0.448986	−	0.777667i	0.549334	−	0.835603i	$-0.314881\pi$
$683$	−5.47375e6	−	9.48080e6i	−0.448986	−	0.777667i	−0.998320	+	0.0579361i	$0.981548\pi$
$684$		0			0
$685$	−1.11420e7			−0.907270
$686$		0			0
$687$	7.40153e6			0.598314
$688$		0			0
$689$	−1.04146e7	−	1.80385e7i	−0.835781	−	1.44762i
$690$		0			0
$691$	−5.22690e6	+	9.05326e6i	−0.416437	+	0.721290i	−0.995578	−	0.0939372i	$-0.970055\pi$
$691$	−5.22690e6	+	9.05326e6i	−0.416437	+	0.721290i	0.579141	+	0.815227i	$0.303388\pi$
$692$		0			0
$693$	−5.73980e6	+	3.31409e6i	−0.454008	+	0.262139i
$694$		0			0
$695$	−2.89737e6	+	5.01839e6i	−0.227532	+	0.394096i
$696$		0			0
$697$	−1.65887e6	−	2.87325e6i	−0.129339	−	0.224022i
$698$		0			0
$699$	−1.02433e7			−0.792948
$700$		0			0
$701$	8.71564e6			0.669891			0.334946	−	0.942237i	$-0.391282\pi$
$701$	8.71564e6			0.669891			0.334946	+	0.942237i	$0.391282\pi$
$702$		0			0
$703$	−8.93634e6	−	1.54782e7i	−0.681980	−	1.18122i
$704$		0			0
$705$	−4.30867e6	+	7.46284e6i	−0.326490	+	0.565498i
$706$		0			0
$707$	1.02716e7	+	5.92994e6i	0.772841	+	0.446171i
$708$		0			0
$709$	469220.	−	812712.i	0.0350559	−	0.0607185i	−0.847965	−	0.530052i	$-0.822172\pi$
$709$	469220.	−	812712.i	0.0350559	−	0.0607185i	0.883021	+	0.469333i	$0.155506\pi$
$710$		0			0
$711$	−1.39374e6	−	2.41402e6i	−0.103397	−	0.179088i
$712$		0			0
$713$	688542.			0.0507232
$714$		0			0
$715$	3.14124e7			2.29792
$716$		0			0
$717$	2.17406e6	+	3.76559e6i	0.157933	+	0.273549i
$718$		0			0
$719$	−3.29537e6	+	5.70774e6i	−0.237729	+	0.411758i	−0.960062	−	0.279787i	$-0.909736\pi$
$719$	−3.29537e6	+	5.70774e6i	−0.237729	+	0.411758i	0.722334	+	0.691545i	$0.243070\pi$
$720$		0			0
$721$	272.027	+	9.77258e6i	1.94883e−5	+	0.700118i
$722$		0			0
$723$	4.56665e6	−	7.90968e6i	0.324902	−	0.562747i
$724$		0			0
$725$	−2.24235e6	−	3.88386e6i	−0.158438	−	0.274422i
$726$		0			0
$727$	−2.32586e7			−1.63210			−0.816052	−	0.577979i	$-0.803842\pi$
$727$	−2.32586e7			−1.63210			−0.816052	+	0.577979i	$0.803842\pi$
$728$		0			0
$729$	531441.			0.0370370
$730$		0			0
$731$	36672.4	+	63518.5i	0.00253832	+	0.00439650i
$732$		0			0
$733$	1.01427e7	−	1.75677e7i	0.697259	−	1.20769i	−0.272154	−	0.962254i	$-0.587736\pi$
$733$	1.01427e7	−	1.75677e7i	0.697259	−	1.20769i	0.969413	−	0.245434i	$-0.0789305\pi$
$734$		0			0
$735$	−3.48744e6	+	6.04120e6i	−0.238116	+	0.412482i
$736$		0			0
$737$	1.07469e7	−	1.86141e7i	0.728807	−	1.26233i
$738$		0			0
$739$	−296714.	−	513923.i	−0.0199860	−	0.0346168i	0.855859	−	0.517209i	$-0.173029\pi$
$739$	−296714.	−	513923.i	−0.0199860	−	0.0346168i	−0.875845	+	0.482592i	$0.839696\pi$
$740$		0			0
$741$	1.14308e7			0.764772
$742$		0			0
$743$	7.70228e6			0.511855			0.255928	−	0.966696i	$-0.417619\pi$
$743$	7.70228e6			0.511855			0.255928	+	0.966696i	$0.417619\pi$
$744$		0			0
$745$	−831151.	−	1.43959e6i	−0.0548642	−	0.0950276i
$746$		0			0
$747$	285177.	−	493942.i	0.0186988	−	0.0323873i
$748$		0			0
$749$	548.997	+	1.97228e7i	3.57574e−5	+	1.28459i
$750$		0			0
$751$	1.34166e7	−	2.32383e7i	0.868048	−	1.50350i	0.00405860	−	0.999992i	$-0.498708\pi$
$751$	1.34166e7	−	2.32383e7i	0.868048	−	1.50350i	0.863989	−	0.503511i	$-0.167959\pi$
$752$		0			0
$753$	1.87115e6	+	3.24093e6i	0.120260	+	0.208297i
$754$		0			0
$755$	6.94811e6			0.443608
$756$		0			0
$757$	2.34943e7			1.49013			0.745063	−	0.666994i	$-0.232419\pi$
$757$	2.34943e7			1.49013			0.745063	+	0.666994i	$0.232419\pi$
$758$		0			0
$759$	−6.14475e6	−	1.06430e7i	−0.387168	−	0.670595i
$760$		0			0
$761$	6.30959e6	−	1.09285e7i	0.394947	−	0.684069i	−0.598147	−	0.801386i	$-0.704096\pi$
$761$	6.30959e6	−	1.09285e7i	0.394947	−	0.684069i	0.993094	+	0.117317i	$0.0374295\pi$
$762$		0			0
$763$	−8.60833e6	−	4.96970e6i	−0.535313	−	0.309043i
$764$		0			0
$765$	−300986.	+	521323.i	−0.0185948	+	0.0322072i
$766$		0			0
$767$	3.43642e6	+	5.95205e6i	0.210920	+	0.365324i
$768$		0			0
$769$	2.73674e6			0.166885			0.0834425	−	0.996513i	$-0.473408\pi$
$769$	2.73674e6			0.166885			0.0834425	+	0.996513i	$0.473408\pi$
$770$		0			0
$771$	−9.10796e6			−0.551804
$772$		0			0
$773$	−1.07752e7	−	1.86633e7i	−0.648602	−	1.12341i	−0.983457	−	0.181141i	$-0.942021\pi$
$773$	−1.07752e7	−	1.86633e7i	−0.648602	−	1.12341i	0.334856	−	0.942269i	$-0.391312\pi$
$774$		0			0
$775$	158871.	−	275172.i	0.00950145	−	0.0164570i
$776$		0			0
$777$	−1.53454e7	+	8.86023e6i	−0.911854	+	0.526493i
$778$		0			0
$779$	−1.21141e7	+	2.09822e7i	−0.715232	+	1.23882i
$780$		0			0
$781$	1.98698e7	+	3.44156e7i	1.16565	+	2.01896i
$782$		0			0

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:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	84.i (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(4.50000 + 7.79423i) q^{3} +(-23.0577 + 39.9371i) q^{5} +(112.271 - 64.8240i) q^{7} +(-40.5000 + 70.1481i) q^{9} +O(q^{10})\)	\(q+(4.50000 + 7.79423i) q^{3} +(-23.0577 + 39.9371i) q^{5} +(112.271 - 64.8240i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(315.582 + 546.605i) q^{11} -1079.22 q^{13} -415.038 q^{15} +(-80.5778 - 139.565i) q^{17} +(-588.428 + 1019.19i) q^{19} +(1010.47 + 583.360i) q^{21} +(-1081.73 + 1873.61i) q^{23} +(499.186 + 864.615i) q^{25} -729.000 q^{27} -4492.01 q^{29} +(-159.130 - 275.621i) q^{31} +(-2840.24 + 4919.44i) q^{33} +(0.166415 + 5978.48i) q^{35} +(-7593.41 + 13152.2i) q^{37} +(-4856.51 - 8411.73i) q^{39} +20587.2 q^{41} -455.118 q^{43} +(-1867.67 - 3234.90i) q^{45} +(10381.4 - 17981.1i) q^{47} +(8402.69 - 14555.8i) q^{49} +(725.200 - 1256.08i) q^{51} +(9650.03 + 16714.3i) q^{53} -29106.4 q^{55} -10591.7 q^{57} +(-3184.15 - 5515.11i) q^{59} +(24572.6 - 42560.9i) q^{61} +(0.292303 + 10501.0i) q^{63} +(24884.4 - 43101.1i) q^{65} +(-17027.0 - 29491.7i) q^{67} -19471.1 q^{69} +62962.4 q^{71} +(4433.88 + 7679.70i) q^{73} +(-4492.67 + 7781.54i) q^{75} +(70864.0 + 40910.7i) q^{77} +(-17206.6 + 29802.7i) q^{79} +(-3280.50 - 5681.99i) q^{81} -7041.42 q^{83} +7431.75 q^{85} +(-20214.1 - 35011.8i) q^{87} +(10121.4 - 17530.7i) q^{89} +(-121166. + 69959.7i) q^{91} +(1432.17 - 2480.59i) q^{93} +(-27135.6 - 47000.2i) q^{95} +54066.6 q^{97} -51124.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 36 q^{3} - 42 q^{7} - 324 q^{9}+O(q^{10}) \) 8 * q + 36 * q^3 - 42 * q^7 - 324 * q^9	\( 8 q + 36 q^{3} - 42 q^{7} - 324 q^{9} - 462 q^{11} - 1204 q^{13} + 228 q^{17} + 358 q^{19} + 1404 q^{21} - 2148 q^{23} - 5454 q^{25} - 5832 q^{27} - 11064 q^{29} + 830 q^{31} + 4158 q^{33} + 7692 q^{35} - 3914 q^{37} - 5418 q^{39} - 16632 q^{41} - 29036 q^{43} + 41700 q^{47} + 41876 q^{49} - 2052 q^{51} + 22164 q^{53} + 7784 q^{55} + 6444 q^{57} + 32886 q^{59} + 83732 q^{61} + 16038 q^{63} - 93192 q^{65} - 80034 q^{67} - 38664 q^{69} + 89544 q^{71} - 22470 q^{73} + 49086 q^{75} + 138732 q^{77} - 75286 q^{79} - 26244 q^{81} - 34836 q^{83} + 278504 q^{85} - 49788 q^{87} + 28944 q^{89} - 12058 q^{91} - 7470 q^{93} - 144120 q^{95} - 433356 q^{97} + 74844 q^{99}+O(q^{100}) \) 8 * q + 36 * q^3 - 42 * q^7 - 324 * q^9 - 462 * q^11 - 1204 * q^13 + 228 * q^17 + 358 * q^19 + 1404 * q^21 - 2148 * q^23 - 5454 * q^25 - 5832 * q^27 - 11064 * q^29 + 830 * q^31 + 4158 * q^33 + 7692 * q^35 - 3914 * q^37 - 5418 * q^39 - 16632 * q^41 - 29036 * q^43 + 41700 * q^47 + 41876 * q^49 - 2052 * q^51 + 22164 * q^53 + 7784 * q^55 + 6444 * q^57 + 32886 * q^59 + 83732 * q^61 + 16038 * q^63 - 93192 * q^65 - 80034 * q^67 - 38664 * q^69 + 89544 * q^71 - 22470 * q^73 + 49086 * q^75 + 138732 * q^77 - 75286 * q^79 - 26244 * q^81 - 34836 * q^83 + 278504 * q^85 - 49788 * q^87 + 28944 * q^89 - 12058 * q^91 - 7470 * q^93 - 144120 * q^95 - 433356 * q^97 + 74844 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more