LMFDB - Embedded newform 3468.2.j.b.3217.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3468,2,Mod(829,3468)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3468, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3468.829");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3468 = 2^{2} \cdot 3 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3468.j (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$27.6921194210$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 204)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			3217.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	3468.3217
Dual form			3468.2.j.b.829.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 + 0.707107i) q^{3} +(0.707107 - 0.707107i) q^{5} +(-1.41421 - 1.41421i) q^{7} -1.00000i q^{9} +O(q^{10})$	\(q+(-0.707107 + 0.707107i) q^{3} +(0.707107 - 0.707107i) q^{5} +(-1.41421 - 1.41421i) q^{7} -1.00000i q^{9} +(2.12132 + 2.12132i) q^{11} -3.00000 q^{13} +1.00000i q^{15} -3.00000i q^{19} +2.00000 q^{21} +(2.12132 + 2.12132i) q^{23} +4.00000i q^{25} +(0.707107 + 0.707107i) q^{27} +(-4.24264 + 4.24264i) q^{29} +(1.41421 - 1.41421i) q^{31} -3.00000 q^{33} -2.00000 q^{35} +(7.07107 - 7.07107i) q^{37} +(2.12132 - 2.12132i) q^{39} +(2.12132 + 2.12132i) q^{41} +1.00000i q^{43} +(-0.707107 - 0.707107i) q^{45} -2.00000 q^{47} -3.00000i q^{49} +6.00000i q^{53} +3.00000 q^{55} +(2.12132 + 2.12132i) q^{57} +(7.07107 + 7.07107i) q^{61} +(-1.41421 + 1.41421i) q^{63} +(-2.12132 + 2.12132i) q^{65} -4.00000 q^{67} -3.00000 q^{69} +(5.65685 - 5.65685i) q^{71} +(9.89949 - 9.89949i) q^{73} +(-2.82843 - 2.82843i) q^{75} -6.00000i q^{77} +(8.48528 + 8.48528i) q^{79} -1.00000 q^{81} -14.0000i q^{83} -6.00000i q^{87} +2.00000 q^{89} +(4.24264 + 4.24264i) q^{91} +2.00000i q^{93} +(-2.12132 - 2.12132i) q^{95} +(2.12132 - 2.12132i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 12 q^{13} + 8 q^{21} - 12 q^{33} - 8 q^{35} - 8 q^{47} + 12 q^{55} - 16 q^{67} - 12 q^{69} - 4 q^{81} + 8 q^{89}+O(q^{100}) $ 4 * q - 12 * q^13 + 8 * q^21 - 12 * q^33 - 8 * q^35 - 8 * q^47 + 12 * q^55 - 16 * q^67 - 12 * q^69 - 4 * q^81 + 8 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3468\mathbb{Z}\right)^\times$.

$n$	$1157$	$1735$	$2893$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$

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
$3$	−0.707107	+	0.707107i	−0.408248	+	0.408248i
$3$	−0.707107	+	0.707107i	−0.408248	+	0.408248i
$4$		0			0
$5$	0.707107	−	0.707107i	0.316228	−	0.316228i	−0.531089	−	0.847316i	$-0.678217\pi$
$5$	0.707107	−	0.707107i	0.316228	−	0.316228i	0.847316	+	0.531089i	$0.178217\pi$
$6$		0			0
$7$	−1.41421	−	1.41421i	−0.534522	−	0.534522i	0.387392	−	0.921915i	$-0.373376\pi$
$7$	−1.41421	−	1.41421i	−0.534522	−	0.534522i	−0.921915	+	0.387392i	$0.873376\pi$
$8$		0			0
$9$		−	1.00000i		−	0.333333i
$10$		0			0
$11$	2.12132	+	2.12132i	0.639602	+	0.639602i	0.950457	−	0.310855i	$-0.100615\pi$
$11$	2.12132	+	2.12132i	0.639602	+	0.639602i	−0.310855	+	0.950457i	$0.600615\pi$
$12$		0			0
$13$	−3.00000			−0.832050			−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000			−0.832050			−0.416025	+	0.909353i	$0.636577\pi$
$14$		0			0
$15$			1.00000i			0.258199i
$16$		0			0
$17$		0			0
$17$		0			0
$18$		0			0
$19$		−	3.00000i		−	0.688247i	−0.938924	−	0.344124i	$-0.888176\pi$
$19$		−	3.00000i		−	0.688247i	0.938924	−	0.344124i	$-0.111824\pi$
$20$		0			0
$21$	2.00000			0.436436
$22$		0			0
$23$	2.12132	+	2.12132i	0.442326	+	0.442326i	0.892793	−	0.450467i	$-0.148743\pi$
$23$	2.12132	+	2.12132i	0.442326	+	0.442326i	−0.450467	+	0.892793i	$0.648743\pi$
$24$		0			0
$25$			4.00000i			0.800000i
$26$		0			0
$27$	0.707107	+	0.707107i	0.136083	+	0.136083i
$28$		0			0
$29$	−4.24264	+	4.24264i	−0.787839	+	0.787839i	−0.981140	−	0.193301i	$-0.938081\pi$
$29$	−4.24264	+	4.24264i	−0.787839	+	0.787839i	0.193301	+	0.981140i	$0.438081\pi$
$30$		0			0
$31$	1.41421	−	1.41421i	0.254000	−	0.254000i	−0.568608	−	0.822608i	$-0.692518\pi$
$31$	1.41421	−	1.41421i	0.254000	−	0.254000i	0.822608	+	0.568608i	$0.192518\pi$
$32$		0			0
$33$	−3.00000			−0.522233
$34$		0			0
$35$	−2.00000			−0.338062
$36$		0			0
$37$	7.07107	−	7.07107i	1.16248	−	1.16248i	0.178545	−	0.983932i	$-0.442861\pi$
$37$	7.07107	−	7.07107i	1.16248	−	1.16248i	0.983932	−	0.178545i	$-0.0571389\pi$
$38$		0			0
$39$	2.12132	−	2.12132i	0.339683	−	0.339683i
$40$		0			0
$41$	2.12132	+	2.12132i	0.331295	+	0.331295i	0.853078	−	0.521783i	$-0.174733\pi$
$41$	2.12132	+	2.12132i	0.331295	+	0.331295i	−0.521783	+	0.853078i	$0.674733\pi$
$42$		0			0
$43$			1.00000i			0.152499i	0.997089	+	0.0762493i	$0.0242945\pi$
$43$			1.00000i			0.152499i	−0.997089	+	0.0762493i	$0.975706\pi$
$44$		0			0
$45$	−0.707107	−	0.707107i	−0.105409	−	0.105409i
$46$		0			0
$47$	−2.00000			−0.291730			−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000			−0.291730			−0.145865	+	0.989305i	$0.546597\pi$
$48$		0			0
$49$		−	3.00000i		−	0.428571i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			6.00000i			0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$			6.00000i			0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$		0			0
$55$	3.00000			0.404520
$56$		0			0
$57$	2.12132	+	2.12132i	0.280976	+	0.280976i
$58$		0			0
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$	7.07107	+	7.07107i	0.905357	+	0.905357i	0.995893	−	0.0905357i	$-0.0288579\pi$
$61$	7.07107	+	7.07107i	0.905357	+	0.905357i	−0.0905357	+	0.995893i	$0.528858\pi$
$62$		0			0
$63$	−1.41421	+	1.41421i	−0.178174	+	0.178174i
$64$		0			0
$65$	−2.12132	+	2.12132i	−0.263117	+	0.263117i
$66$		0			0
$67$	−4.00000			−0.488678			−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000			−0.488678			−0.244339	+	0.969690i	$0.578571\pi$
$68$		0			0
$69$	−3.00000			−0.361158
$70$		0			0
$71$	5.65685	−	5.65685i	0.671345	−	0.671345i	−0.286681	−	0.958026i	$-0.592552\pi$
$71$	5.65685	−	5.65685i	0.671345	−	0.671345i	0.958026	+	0.286681i	$0.0925520\pi$
$72$		0			0
$73$	9.89949	−	9.89949i	1.15865	−	1.15865i	0.173882	−	0.984767i	$-0.444369\pi$
$73$	9.89949	−	9.89949i	1.15865	−	1.15865i	0.984767	−	0.173882i	$-0.0556310\pi$
$74$		0			0
$75$	−2.82843	−	2.82843i	−0.326599	−	0.326599i
$76$		0			0
$77$		−	6.00000i		−	0.683763i
$78$		0			0
$79$	8.48528	+	8.48528i	0.954669	+	0.954669i	0.999016	−	0.0443474i	$-0.0141209\pi$
$79$	8.48528	+	8.48528i	0.954669	+	0.954669i	−0.0443474	+	0.999016i	$0.514121\pi$
$80$		0			0
$81$	−1.00000			−0.111111
$82$		0			0
$83$		−	14.0000i		−	1.53670i	−0.640030	−	0.768350i	$-0.721078\pi$
$83$		−	14.0000i		−	1.53670i	0.640030	−	0.768350i	$-0.278922\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		−	6.00000i		−	0.643268i
$88$		0			0
$89$	2.00000			0.212000			0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000			0.212000			0.106000	+	0.994366i	$0.466196\pi$
$90$		0			0
$91$	4.24264	+	4.24264i	0.444750	+	0.444750i
$92$		0			0
$93$			2.00000i			0.207390i
$94$		0			0
$95$	−2.12132	−	2.12132i	−0.217643	−	0.217643i
$96$		0			0
$97$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$97$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$98$		0			0
$99$	2.12132	−	2.12132i	0.213201	−	0.213201i
$100$		0			0
$101$	12.0000			1.19404			0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000			1.19404			0.597022	+	0.802225i	$0.296350\pi$
$102$		0			0
$103$	−13.0000			−1.28093			−0.640464	−	0.767988i	$-0.721258\pi$
$103$	−13.0000			−1.28093			−0.640464	+	0.767988i	$0.721258\pi$
$104$		0			0
$105$	1.41421	−	1.41421i	0.138013	−	0.138013i
$106$		0			0
$107$	9.19239	−	9.19239i	0.888662	−	0.888662i	−0.105733	−	0.994395i	$-0.533719\pi$
$107$	9.19239	−	9.19239i	0.888662	−	0.888662i	0.994395	+	0.105733i	$0.0337188\pi$
$108$		0			0
$109$	−2.82843	−	2.82843i	−0.270914	−	0.270914i	0.558554	−	0.829468i	$-0.311356\pi$
$109$	−2.82843	−	2.82843i	−0.270914	−	0.270914i	−0.829468	+	0.558554i	$0.811356\pi$
$110$		0			0
$111$			10.0000i			0.949158i
$112$		0			0
$113$	13.4350	+	13.4350i	1.26386	+	1.26386i	0.949206	+	0.314655i	$0.101889\pi$
$113$	13.4350	+	13.4350i	1.26386	+	1.26386i	0.314655	+	0.949206i	$0.398111\pi$
$114$		0			0
$115$	3.00000			0.279751
$116$		0			0
$117$			3.00000i			0.277350i
$118$		0			0
$119$		0			0
$120$		0			0
$121$		−	2.00000i		−	0.181818i
$122$		0			0
$123$	−3.00000			−0.270501
$124$		0			0
$125$	6.36396	+	6.36396i	0.569210	+	0.569210i
$126$		0			0
$127$		−	13.0000i		−	1.15356i	−0.816898	−	0.576782i	$-0.804308\pi$
$127$		−	13.0000i		−	1.15356i	0.816898	−	0.576782i	$-0.195692\pi$
$128$		0			0
$129$	−0.707107	−	0.707107i	−0.0622573	−	0.0622573i
$130$		0			0
$131$	−14.8492	+	14.8492i	−1.29738	+	1.29738i	−0.367270	+	0.930114i	$0.619707\pi$
$131$	−14.8492	+	14.8492i	−1.29738	+	1.29738i	−0.930114	+	0.367270i	$0.880293\pi$
$132$		0			0
$133$	−4.24264	+	4.24264i	−0.367884	+	0.367884i
$134$		0			0
$135$	1.00000			0.0860663
$136$		0			0
$137$	16.0000			1.36697			0.683486	−	0.729964i	$-0.260463\pi$
$137$	16.0000			1.36697			0.683486	+	0.729964i	$0.260463\pi$
$138$		0			0
$139$	5.65685	−	5.65685i	0.479808	−	0.479808i	−0.425262	−	0.905070i	$-0.639818\pi$
$139$	5.65685	−	5.65685i	0.479808	−	0.479808i	0.905070	+	0.425262i	$0.139818\pi$
$140$		0			0
$141$	1.41421	−	1.41421i	0.119098	−	0.119098i
$142$		0			0
$143$	−6.36396	−	6.36396i	−0.532181	−	0.532181i
$144$		0			0
$145$			6.00000i			0.498273i
$146$		0			0
$147$	2.12132	+	2.12132i	0.174964	+	0.174964i
$148$		0			0
$149$	8.00000			0.655386			0.327693	−	0.944784i	$-0.393729\pi$
$149$	8.00000			0.655386			0.327693	+	0.944784i	$0.393729\pi$
$150$		0			0
$151$			20.0000i			1.62758i	0.581161	+	0.813788i	$0.302599\pi$
$151$			20.0000i			1.62758i	−0.581161	+	0.813788i	$0.697401\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	2.00000i		−	0.160644i
$156$		0			0
$157$	5.00000			0.399043			0.199522	−	0.979893i	$-0.436061\pi$
$157$	5.00000			0.399043			0.199522	+	0.979893i	$0.436061\pi$
$158$		0			0
$159$	−4.24264	−	4.24264i	−0.336463	−	0.336463i
$160$		0			0
$161$		−	6.00000i		−	0.472866i
$162$		0			0
$163$	1.41421	+	1.41421i	0.110770	+	0.110770i	0.760319	−	0.649550i	$-0.225042\pi$
$163$	1.41421	+	1.41421i	0.110770	+	0.110770i	−0.649550	+	0.760319i	$0.725042\pi$
$164$		0			0
$165$	−2.12132	+	2.12132i	−0.165145	+	0.165145i
$166$		0			0
$167$	10.6066	−	10.6066i	0.820763	−	0.820763i	−0.165454	−	0.986218i	$-0.552909\pi$
$167$	10.6066	−	10.6066i	0.820763	−	0.820763i	0.986218	+	0.165454i	$0.0529089\pi$
$168$		0			0
$169$	−4.00000			−0.307692
$170$		0			0
$171$	−3.00000			−0.229416
$172$		0			0
$173$	13.4350	−	13.4350i	1.02145	−	1.02145i	0.0216814	−	0.999765i	$-0.493098\pi$
$173$	13.4350	−	13.4350i	1.02145	−	1.02145i	0.999765	−	0.0216814i	$-0.00690194\pi$
$174$		0			0
$175$	5.65685	−	5.65685i	0.427618	−	0.427618i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	26.0000i		−	1.94333i	−0.236360	−	0.971666i	$-0.575954\pi$
$179$		−	26.0000i		−	1.94333i	0.236360	−	0.971666i	$-0.424046\pi$
$180$		0			0
$181$	−2.82843	−	2.82843i	−0.210235	−	0.210235i	0.594132	−	0.804367i	$-0.297496\pi$
$181$	−2.82843	−	2.82843i	−0.210235	−	0.210235i	−0.804367	+	0.594132i	$0.797496\pi$
$182$		0			0
$183$	−10.0000			−0.739221
$184$		0			0
$185$		−	10.0000i		−	0.735215i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		−	2.00000i		−	0.145479i
$190$		0			0
$191$	−16.0000			−1.15772			−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000			−1.15772			−0.578860	+	0.815427i	$0.696502\pi$
$192$		0			0
$193$	15.5563	+	15.5563i	1.11977	+	1.11977i	0.991775	+	0.127996i	$0.0408544\pi$
$193$	15.5563	+	15.5563i	1.11977	+	1.11977i	0.127996	+	0.991775i	$0.459146\pi$
$194$		0			0
$195$		−	3.00000i		−	0.214834i
$196$		0			0
$197$	14.8492	+	14.8492i	1.05796	+	1.05796i	0.998213	+	0.0597514i	$0.0190308\pi$
$197$	14.8492	+	14.8492i	1.05796	+	1.05796i	0.0597514	+	0.998213i	$0.480969\pi$
$198$		0			0
$199$	−16.9706	+	16.9706i	−1.20301	+	1.20301i	−0.229765	+	0.973246i	$0.573796\pi$
$199$	−16.9706	+	16.9706i	−1.20301	+	1.20301i	−0.973246	+	0.229765i	$0.926204\pi$
$200$		0			0
$201$	2.82843	−	2.82843i	0.199502	−	0.199502i
$202$		0			0
$203$	12.0000			0.842235
$204$		0			0
$205$	3.00000			0.209529
$206$		0			0
$207$	2.12132	−	2.12132i	0.147442	−	0.147442i
$208$		0			0
$209$	6.36396	−	6.36396i	0.440204	−	0.440204i
$210$		0			0
$211$	4.24264	+	4.24264i	0.292075	+	0.292075i	0.837900	−	0.545824i	$-0.183783\pi$
$211$	4.24264	+	4.24264i	0.292075	+	0.292075i	−0.545824	+	0.837900i	$0.683783\pi$
$212$		0			0
$213$			8.00000i			0.548151i
$214$		0			0
$215$	0.707107	+	0.707107i	0.0482243	+	0.0482243i
$216$		0			0
$217$	−4.00000			−0.271538
$218$		0			0
$219$			14.0000i			0.946032i
$220$		0			0
$221$		0			0
$222$		0			0
$223$			1.00000i			0.0669650i	0.999439	+	0.0334825i	$0.0106598\pi$
$223$			1.00000i			0.0669650i	−0.999439	+	0.0334825i	$0.989340\pi$
$224$		0			0
$225$	4.00000			0.266667
$226$		0			0
$227$	−2.12132	−	2.12132i	−0.140797	−	0.140797i	0.633195	−	0.773992i	$-0.281743\pi$
$227$	−2.12132	−	2.12132i	−0.140797	−	0.140797i	−0.773992	+	0.633195i	$0.781743\pi$
$228$		0			0
$229$			10.0000i			0.660819i	0.943838	+	0.330409i	$0.107187\pi$
$229$			10.0000i			0.660819i	−0.943838	+	0.330409i	$0.892813\pi$
$230$		0			0
$231$	4.24264	+	4.24264i	0.279145	+	0.279145i
$232$		0			0
$233$	13.4350	−	13.4350i	0.880158	−	0.880158i	−0.113392	−	0.993550i	$-0.536172\pi$
$233$	13.4350	−	13.4350i	0.880158	−	0.880158i	0.993550	+	0.113392i	$0.0361717\pi$
$234$		0			0
$235$	−1.41421	+	1.41421i	−0.0922531	+	0.0922531i
$236$		0			0
$237$	−12.0000			−0.779484
$238$		0			0
$239$	12.0000			0.776215			0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000			0.776215			0.388108	+	0.921614i	$0.373129\pi$
$240$		0			0
$241$	9.89949	−	9.89949i	0.637683	−	0.637683i	−0.312301	−	0.949983i	$-0.601100\pi$
$241$	9.89949	−	9.89949i	0.637683	−	0.637683i	0.949983	+	0.312301i	$0.101100\pi$
$242$		0			0
$243$	0.707107	−	0.707107i	0.0453609	−	0.0453609i
$244$		0			0
$245$	−2.12132	−	2.12132i	−0.135526	−	0.135526i
$246$		0			0
$247$			9.00000i			0.572656i
$248$		0			0
$249$	9.89949	+	9.89949i	0.627355	+	0.627355i
$250$		0			0
$251$	30.0000			1.89358			0.946792	−	0.321847i	$-0.104304\pi$
$251$	30.0000			1.89358			0.946792	+	0.321847i	$0.104304\pi$
$252$		0			0
$253$			9.00000i			0.565825i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			26.0000i			1.62184i	0.585160	+	0.810918i	$0.301032\pi$
$257$			26.0000i			1.62184i	−0.585160	+	0.810918i	$0.698968\pi$
$258$		0			0
$259$	−20.0000			−1.24274
$260$		0			0
$261$	4.24264	+	4.24264i	0.262613	+	0.262613i
$262$		0			0
$263$			14.0000i			0.863277i	0.902047	+	0.431638i	$0.142064\pi$
$263$			14.0000i			0.863277i	−0.902047	+	0.431638i	$0.857936\pi$
$264$		0			0
$265$	4.24264	+	4.24264i	0.260623	+	0.260623i
$266$		0			0
$267$	−1.41421	+	1.41421i	−0.0865485	+	0.0865485i
$268$		0			0
$269$	−10.6066	+	10.6066i	−0.646696	+	0.646696i	−0.952193	−	0.305497i	$-0.901177\pi$
$269$	−10.6066	+	10.6066i	−0.646696	+	0.646696i	0.305497	+	0.952193i	$0.401177\pi$
$270$		0			0
$271$	−7.00000			−0.425220			−0.212610	−	0.977137i	$-0.568196\pi$
$271$	−7.00000			−0.425220			−0.212610	+	0.977137i	$0.568196\pi$
$272$		0			0
$273$	−6.00000			−0.363137
$274$		0			0
$275$	−8.48528	+	8.48528i	−0.511682	+	0.511682i
$276$		0			0
$277$	7.07107	−	7.07107i	0.424859	−	0.424859i	−0.462014	−	0.886873i	$-0.652873\pi$
$277$	7.07107	−	7.07107i	0.424859	−	0.424859i	0.886873	+	0.462014i	$0.152873\pi$
$278$		0			0
$279$	−1.41421	−	1.41421i	−0.0846668	−	0.0846668i
$280$		0			0
$281$		−	2.00000i		−	0.119310i	−0.998219	−	0.0596550i	$-0.981000\pi$
$281$		−	2.00000i		−	0.119310i	0.998219	−	0.0596550i	$-0.0190001\pi$
$282$		0			0
$283$	14.1421	+	14.1421i	0.840663	+	0.840663i	0.988945	−	0.148282i	$-0.0473744\pi$
$283$	14.1421	+	14.1421i	0.840663	+	0.840663i	−0.148282	+	0.988945i	$0.547374\pi$
$284$		0			0
$285$	3.00000			0.177705
$286$		0			0
$287$		−	6.00000i		−	0.354169i
$288$		0			0
$289$		0			0
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−6.00000			−0.350524			−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000			−0.350524			−0.175262	+	0.984522i	$0.556077\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$			3.00000i			0.174078i
$298$		0			0
$299$	−6.36396	−	6.36396i	−0.368037	−	0.368037i
$300$		0			0
$301$	1.41421	−	1.41421i	0.0815139	−	0.0815139i
$302$		0			0
$303$	−8.48528	+	8.48528i	−0.487467	+	0.487467i
$304$		0			0
$305$	10.0000			0.572598
$306$		0			0
$307$	12.0000			0.684876			0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000			0.684876			0.342438	+	0.939540i	$0.388747\pi$
$308$		0			0
$309$	9.19239	−	9.19239i	0.522937	−	0.522937i
$310$		0			0
$311$	5.65685	−	5.65685i	0.320771	−	0.320771i	−0.528292	−	0.849063i	$-0.677167\pi$
$311$	5.65685	−	5.65685i	0.320771	−	0.320771i	0.849063	+	0.528292i	$0.177167\pi$
$312$		0			0
$313$	5.65685	+	5.65685i	0.319744	+	0.319744i	0.848669	−	0.528925i	$-0.177405\pi$
$313$	5.65685	+	5.65685i	0.319744	+	0.319744i	−0.528925	+	0.848669i	$0.677405\pi$
$314$		0			0
$315$			2.00000i			0.112687i
$316$		0			0
$317$	−4.24264	−	4.24264i	−0.238290	−	0.238290i	0.577851	−	0.816142i	$-0.303891\pi$
$317$	−4.24264	−	4.24264i	−0.238290	−	0.238290i	−0.816142	+	0.577851i	$0.803891\pi$
$318$		0			0
$319$	−18.0000			−1.00781
$320$		0			0
$321$			13.0000i			0.725589i
$322$		0			0
$323$		0			0
$324$		0			0
$325$		−	12.0000i		−	0.665640i
$326$		0			0
$327$	4.00000			0.221201
$328$		0			0
$329$	2.82843	+	2.82843i	0.155936	+	0.155936i
$330$		0			0
$331$			15.0000i			0.824475i	0.911077	+	0.412237i	$0.135253\pi$
$331$			15.0000i			0.824475i	−0.911077	+	0.412237i	$0.864747\pi$
$332$		0			0
$333$	−7.07107	−	7.07107i	−0.387492	−	0.387492i
$334$		0			0
$335$	−2.82843	+	2.82843i	−0.154533	+	0.154533i
$336$		0			0
$337$	5.65685	−	5.65685i	0.308148	−	0.308148i	−0.536043	−	0.844191i	$-0.680081\pi$
$337$	5.65685	−	5.65685i	0.308148	−	0.308148i	0.844191	+	0.536043i	$0.180081\pi$
$338$		0			0
$339$	−19.0000			−1.03194
$340$		0			0
$341$	6.00000			0.324918
$342$		0			0
$343$	−14.1421	+	14.1421i	−0.763604	+	0.763604i
$344$		0			0
$345$	−2.12132	+	2.12132i	−0.114208	+	0.114208i
$346$		0			0
$347$	−16.9706	−	16.9706i	−0.911028	−	0.911028i	0.0853256	−	0.996353i	$-0.472807\pi$
$347$	−16.9706	−	16.9706i	−0.911028	−	0.911028i	−0.996353	+	0.0853256i	$0.972807\pi$
$348$		0			0
$349$			23.0000i			1.23116i	0.788074	+	0.615581i	$0.211079\pi$
$349$			23.0000i			1.23116i	−0.788074	+	0.615581i	$0.788921\pi$
$350$		0			0
$351$	−2.12132	−	2.12132i	−0.113228	−	0.113228i
$352$		0			0
$353$	−8.00000			−0.425797			−0.212899	−	0.977074i	$-0.568290\pi$
$353$	−8.00000			−0.425797			−0.212899	+	0.977074i	$0.568290\pi$
$354$		0			0
$355$		−	8.00000i		−	0.424596i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			30.0000i			1.58334i	0.610949	+	0.791670i	$0.290788\pi$
$359$			30.0000i			1.58334i	−0.610949	+	0.791670i	$0.709212\pi$
$360$		0			0
$361$	10.0000			0.526316
$362$		0			0
$363$	1.41421	+	1.41421i	0.0742270	+	0.0742270i
$364$		0			0
$365$		−	14.0000i		−	0.732793i
$366$		0			0
$367$	−15.5563	−	15.5563i	−0.812035	−	0.812035i	0.172904	−	0.984939i	$-0.444685\pi$
$367$	−15.5563	−	15.5563i	−0.812035	−	0.812035i	−0.984939	+	0.172904i	$0.944685\pi$
$368$		0			0
$369$	2.12132	−	2.12132i	0.110432	−	0.110432i
$370$		0			0
$371$	8.48528	−	8.48528i	0.440534	−	0.440534i
$372$		0			0
$373$	−18.0000			−0.932005			−0.466002	−	0.884783i	$-0.654306\pi$
$373$	−18.0000			−0.932005			−0.466002	+	0.884783i	$0.654306\pi$
$374$		0			0
$375$	−9.00000			−0.464758
$376$		0			0
$377$	12.7279	−	12.7279i	0.655521	−	0.655521i
$378$		0			0
$379$	−7.07107	+	7.07107i	−0.363216	+	0.363216i	−0.864996	−	0.501779i	$-0.832679\pi$
$379$	−7.07107	+	7.07107i	−0.363216	+	0.363216i	0.501779	+	0.864996i	$0.332679\pi$
$380$		0			0
$381$	9.19239	+	9.19239i	0.470940	+	0.470940i
$382$		0			0
$383$		−	24.0000i		−	1.22634i	−0.789950	−	0.613171i	$-0.789894\pi$
$383$		−	24.0000i		−	1.22634i	0.789950	−	0.613171i	$-0.210106\pi$
$384$		0			0
$385$	−4.24264	−	4.24264i	−0.216225	−	0.216225i
$386$		0			0
$387$	1.00000			0.0508329
$388$		0			0
$389$		−	26.0000i		−	1.31825i	−0.752032	−	0.659126i	$-0.770926\pi$
$389$		−	26.0000i		−	1.31825i	0.752032	−	0.659126i	$-0.229074\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		−	21.0000i		−	1.05931i
$394$		0			0
$395$	12.0000			0.603786
$396$		0			0
$397$	−22.6274	−	22.6274i	−1.13564	−	1.13564i	−0.989223	−	0.146414i	$-0.953227\pi$
$397$	−22.6274	−	22.6274i	−1.13564	−	1.13564i	−0.146414	−	0.989223i	$-0.546773\pi$
$398$		0			0
$399$		−	6.00000i		−	0.300376i
$400$		0			0
$401$	10.6066	+	10.6066i	0.529668	+	0.529668i	0.920473	−	0.390805i	$-0.127803\pi$
$401$	10.6066	+	10.6066i	0.529668	+	0.529668i	−0.390805	+	0.920473i	$0.627803\pi$
$402$		0			0
$403$	−4.24264	+	4.24264i	−0.211341	+	0.211341i
$404$		0			0
$405$	−0.707107	+	0.707107i	−0.0351364	+	0.0351364i
$406$		0			0
$407$	30.0000			1.48704
$408$		0			0
$409$	−7.00000			−0.346128			−0.173064	−	0.984911i	$-0.555367\pi$
$409$	−7.00000			−0.346128			−0.173064	+	0.984911i	$0.555367\pi$
$410$		0			0
$411$	−11.3137	+	11.3137i	−0.558064	+	0.558064i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−9.89949	−	9.89949i	−0.485947	−	0.485947i
$416$		0			0
$417$			8.00000i			0.391762i
$418$		0			0
$419$	−11.3137	−	11.3137i	−0.552711	−	0.552711i	0.374511	−	0.927222i	$-0.377810\pi$
$419$	−11.3137	−	11.3137i	−0.552711	−	0.552711i	−0.927222	+	0.374511i	$0.877810\pi$
$420$		0			0
$421$	25.0000			1.21843			0.609213	−	0.793007i	$-0.291486\pi$
$421$	25.0000			1.21843			0.609213	+	0.793007i	$0.291486\pi$
$422$		0			0
$423$			2.00000i			0.0972433i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	20.0000i		−	0.967868i
$428$		0			0
$429$	9.00000			0.434524
$430$		0			0
$431$	2.82843	+	2.82843i	0.136241	+	0.136241i	0.771938	−	0.635698i	$-0.219287\pi$
$431$	2.82843	+	2.82843i	0.136241	+	0.136241i	−0.635698	+	0.771938i	$0.719287\pi$
$432$		0			0
$433$		−	11.0000i		−	0.528626i	−0.964437	−	0.264313i	$-0.914855\pi$
$433$		−	11.0000i		−	0.528626i	0.964437	−	0.264313i	$-0.0851452\pi$
$434$		0			0
$435$	−4.24264	−	4.24264i	−0.203419	−	0.203419i
$436$		0			0
$437$	6.36396	−	6.36396i	0.304430	−	0.304430i
$438$		0			0
$439$	16.9706	−	16.9706i	0.809961	−	0.809961i	−0.174667	−	0.984628i	$-0.555885\pi$
$439$	16.9706	−	16.9706i	0.809961	−	0.809961i	0.984628	+	0.174667i	$0.0558848\pi$
$440$		0			0
$441$	−3.00000			−0.142857
$442$		0			0
$443$	22.0000			1.04525			0.522626	−	0.852562i	$-0.324953\pi$
$443$	22.0000			1.04525			0.522626	+	0.852562i	$0.324953\pi$
$444$		0			0
$445$	1.41421	−	1.41421i	0.0670402	−	0.0670402i
$446$		0			0
$447$	−5.65685	+	5.65685i	−0.267560	+	0.267560i
$448$		0			0
$449$	21.2132	+	21.2132i	1.00111	+	1.00111i	0.999999	+	0.00111359i	$0.000354466\pi$
$449$	21.2132	+	21.2132i	1.00111	+	1.00111i	0.00111359	+	0.999999i	$0.499646\pi$
$450$		0			0
$451$			9.00000i			0.423793i
$452$		0			0
$453$	−14.1421	−	14.1421i	−0.664455	−	0.664455i
$454$		0			0
$455$	6.00000			0.281284
$456$		0			0
$457$			25.0000i			1.16945i	0.811231	+	0.584725i	$0.198798\pi$
$457$			25.0000i			1.16945i	−0.811231	+	0.584725i	$0.801202\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			16.0000i			0.745194i	0.927993	+	0.372597i	$0.121533\pi$
$461$			16.0000i			0.745194i	−0.927993	+	0.372597i	$0.878467\pi$
$462$		0			0
$463$	24.0000			1.11537			0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000			1.11537			0.557687	+	0.830051i	$0.311689\pi$
$464$		0			0
$465$	1.41421	+	1.41421i	0.0655826	+	0.0655826i
$466$		0			0
$467$		−	20.0000i		−	0.925490i	−0.886492	−	0.462745i	$-0.846865\pi$
$467$		−	20.0000i		−	0.925490i	0.886492	−	0.462745i	$-0.153135\pi$
$468$		0			0
$469$	5.65685	+	5.65685i	0.261209	+	0.261209i
$470$		0			0
$471$	−3.53553	+	3.53553i	−0.162909	+	0.162909i
$472$		0			0
$473$	−2.12132	+	2.12132i	−0.0975384	+	0.0975384i
$474$		0			0
$475$	12.0000			0.550598
$476$		0			0
$477$	6.00000			0.274721
$478$		0			0
$479$	−17.6777	+	17.6777i	−0.807713	+	0.807713i	−0.984287	−	0.176574i	$-0.943499\pi$
$479$	−17.6777	+	17.6777i	−0.807713	+	0.807713i	0.176574	+	0.984287i	$0.443499\pi$
$480$		0			0
$481$	−21.2132	+	21.2132i	−0.967239	+	0.967239i
$482$		0			0
$483$	4.24264	+	4.24264i	0.193047	+	0.193047i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$487$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$488$		0			0
$489$	−2.00000			−0.0904431
$490$		0			0
$491$		−	30.0000i		−	1.35388i	−0.736038	−	0.676941i	$-0.763305\pi$
$491$		−	30.0000i		−	1.35388i	0.736038	−	0.676941i	$-0.236695\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		−	3.00000i		−	0.134840i
$496$		0			0
$497$	−16.0000			−0.717698
$498$		0			0
$499$	−12.7279	−	12.7279i	−0.569780	−	0.569780i	0.362287	−	0.932067i	$-0.381996\pi$
$499$	−12.7279	−	12.7279i	−0.569780	−	0.569780i	−0.932067	+	0.362287i	$0.881996\pi$
$500$		0			0
$501$			15.0000i			0.670151i
$502$		0			0
$503$	−9.19239	−	9.19239i	−0.409868	−	0.409868i	0.471824	−	0.881693i	$-0.343596\pi$
$503$	−9.19239	−	9.19239i	−0.409868	−	0.409868i	−0.881693	+	0.471824i	$0.843596\pi$
$504$		0			0
$505$	8.48528	−	8.48528i	0.377590	−	0.377590i
$506$		0			0
$507$	2.82843	−	2.82843i	0.125615	−	0.125615i
$508$		0			0
$509$	16.0000			0.709188			0.354594	−	0.935020i	$-0.384619\pi$
$509$	16.0000			0.709188			0.354594	+	0.935020i	$0.384619\pi$
$510$		0			0
$511$	−28.0000			−1.23865
$512$		0			0
$513$	2.12132	−	2.12132i	0.0936586	−	0.0936586i
$514$		0			0
$515$	−9.19239	+	9.19239i	−0.405065	+	0.405065i
$516$		0			0
$517$	−4.24264	−	4.24264i	−0.186591	−	0.186591i
$518$		0			0
$519$			19.0000i			0.834007i
$520$		0			0
$521$	−10.6066	−	10.6066i	−0.464684	−	0.464684i	0.435503	−	0.900187i	$-0.356570\pi$
$521$	−10.6066	−	10.6066i	−0.464684	−	0.464684i	−0.900187	+	0.435503i	$0.856570\pi$
$522$		0			0
$523$	−20.0000			−0.874539			−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000			−0.874539			−0.437269	+	0.899331i	$0.644054\pi$
$524$		0			0
$525$			8.00000i			0.349149i
$526$		0			0
$527$		0			0
$528$		0			0
$529$		−	14.0000i		−	0.608696i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−6.36396	−	6.36396i	−0.275654	−	0.275654i
$534$		0			0
$535$		−	13.0000i		−	0.562039i
$536$		0			0
$537$	18.3848	+	18.3848i	0.793362	+	0.793362i
$538$		0			0
$539$	6.36396	−	6.36396i	0.274115	−	0.274115i
$540$		0			0
$541$	−32.5269	+	32.5269i	−1.39844	+	1.39844i	−0.593909	+	0.804532i	$0.702416\pi$
$541$	−32.5269	+	32.5269i	−1.39844	+	1.39844i	−0.804532	+	0.593909i	$0.797584\pi$
$542$		0			0
$543$	4.00000			0.171656
$544$		0			0
$545$	−4.00000			−0.171341
$546$		0			0
$547$	−15.5563	+	15.5563i	−0.665141	+	0.665141i	−0.956587	−	0.291446i	$-0.905864\pi$
$547$	−15.5563	+	15.5563i	−0.665141	+	0.665141i	0.291446	+	0.956587i	$0.405864\pi$
$548$		0			0
$549$	7.07107	−	7.07107i	0.301786	−	0.301786i
$550$		0			0
$551$	12.7279	+	12.7279i	0.542228	+	0.542228i
$552$		0			0
$553$		−	24.0000i		−	1.02058i
$554$		0			0
$555$	7.07107	+	7.07107i	0.300150	+	0.300150i
$556$		0			0
$557$	−20.0000			−0.847427			−0.423714	−	0.905796i	$-0.639274\pi$
$557$	−20.0000			−0.847427			−0.423714	+	0.905796i	$0.639274\pi$
$558$		0			0
$559$		−	3.00000i		−	0.126886i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	18.0000i		−	0.758610i	−0.925272	−	0.379305i	$-0.876163\pi$
$563$		−	18.0000i		−	0.758610i	0.925272	−	0.379305i	$-0.123837\pi$
$564$		0			0
$565$	19.0000			0.799336
$566$		0			0
$567$	1.41421	+	1.41421i	0.0593914	+	0.0593914i
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	−19.7990	−	19.7990i	−0.828562	−	0.828562i	0.158756	−	0.987318i	$-0.449252\pi$
$571$	−19.7990	−	19.7990i	−0.828562	−	0.828562i	−0.987318	+	0.158756i	$0.949252\pi$
$572$		0			0
$573$	11.3137	−	11.3137i	0.472637	−	0.472637i
$574$		0			0
$575$	−8.48528	+	8.48528i	−0.353861	+	0.353861i
$576$		0			0
$577$	−35.0000			−1.45707			−0.728535	−	0.685009i	$-0.759798\pi$
$577$	−35.0000			−1.45707			−0.728535	+	0.685009i	$0.759798\pi$
$578$		0			0
$579$	−22.0000			−0.914289
$580$		0			0
$581$	−19.7990	+	19.7990i	−0.821401	+	0.821401i
$582$		0			0
$583$	−12.7279	+	12.7279i	−0.527137	+	0.527137i
$584$		0			0
$585$	2.12132	+	2.12132i	0.0877058	+	0.0877058i
$586$		0			0
$587$		−	36.0000i		−	1.48588i	−0.669359	−	0.742940i	$-0.733431\pi$
$587$		−	36.0000i		−	1.48588i	0.669359	−	0.742940i	$-0.266569\pi$
$588$		0			0
$589$	−4.24264	−	4.24264i	−0.174815	−	0.174815i
$590$		0			0
$591$	−21.0000			−0.863825
$592$		0			0
$593$		−	38.0000i		−	1.56047i	−0.625485	−	0.780236i	$-0.715099\pi$
$593$		−	38.0000i		−	1.56047i	0.625485	−	0.780236i	$-0.284901\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	24.0000i		−	0.982255i
$598$		0			0
$599$	4.00000			0.163436			0.0817178	−	0.996656i	$-0.473959\pi$
$599$	4.00000			0.163436			0.0817178	+	0.996656i	$0.473959\pi$
$600$		0			0
$601$	15.5563	+	15.5563i	0.634557	+	0.634557i	0.949208	−	0.314651i	$-0.101887\pi$
$601$	15.5563	+	15.5563i	0.634557	+	0.634557i	−0.314651	+	0.949208i	$0.601887\pi$
$602$		0			0
$603$			4.00000i			0.162893i
$604$		0			0
$605$	−1.41421	−	1.41421i	−0.0574960	−	0.0574960i
$606$		0			0
$607$	19.7990	−	19.7990i	0.803616	−	0.803616i	−0.180043	−	0.983659i	$-0.557624\pi$
$607$	19.7990	−	19.7990i	0.803616	−	0.803616i	0.983659	+	0.180043i	$0.0576236\pi$
$608$		0			0
$609$	−8.48528	+	8.48528i	−0.343841	+	0.343841i
$610$		0			0
$611$	6.00000			0.242734
$612$		0			0
$613$	−25.0000			−1.00974			−0.504870	−	0.863195i	$-0.668460\pi$
$613$	−25.0000			−1.00974			−0.504870	+	0.863195i	$0.668460\pi$
$614$		0			0
$615$	−2.12132	+	2.12132i	−0.0855399	+	0.0855399i
$616$		0			0
$617$	−29.6985	+	29.6985i	−1.19562	+	1.19562i	−0.220150	+	0.975466i	$0.570655\pi$
$617$	−29.6985	+	29.6985i	−1.19562	+	1.19562i	−0.975466	+	0.220150i	$0.929345\pi$
$618$		0			0
$619$	−14.1421	−	14.1421i	−0.568420	−	0.568420i	0.363265	−	0.931686i	$-0.381662\pi$
$619$	−14.1421	−	14.1421i	−0.568420	−	0.568420i	−0.931686	+	0.363265i	$0.881662\pi$
$620$		0			0
$621$			3.00000i			0.120386i
$622$		0			0
$623$	−2.82843	−	2.82843i	−0.113319	−	0.113319i
$624$		0			0
$625$	−11.0000			−0.440000
$626$		0			0
$627$			9.00000i			0.359425i
$628$		0			0
$629$		0			0
$630$		0			0
$631$			29.0000i			1.15447i	0.816577	+	0.577236i	$0.195869\pi$
$631$			29.0000i			1.15447i	−0.816577	+	0.577236i	$0.804131\pi$
$632$		0			0
$633$	−6.00000			−0.238479
$634$		0			0
$635$	−9.19239	−	9.19239i	−0.364789	−	0.364789i
$636$		0			0
$637$			9.00000i			0.356593i
$638$		0			0
$639$	−5.65685	−	5.65685i	−0.223782	−	0.223782i
$640$		0			0
$641$	26.1630	−	26.1630i	1.03337	−	1.03337i	0.0339509	−	0.999424i	$-0.489191\pi$
$641$	26.1630	−	26.1630i	1.03337	−	1.03337i	0.999424	−	0.0339509i	$-0.0108090\pi$
$642$		0			0
$643$	25.4558	−	25.4558i	1.00388	−	1.00388i	0.00388805	−	0.999992i	$-0.498762\pi$
$643$	25.4558	−	25.4558i	1.00388	−	1.00388i	0.999992	−	0.00388805i	$-0.00123761\pi$
$644$		0			0
$645$	−1.00000			−0.0393750
$646$		0			0
$647$	16.0000			0.629025			0.314512	−	0.949253i	$-0.398159\pi$
$647$	16.0000			0.629025			0.314512	+	0.949253i	$0.398159\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	2.82843	−	2.82843i	0.110855	−	0.110855i
$652$		0			0
$653$	−4.94975	−	4.94975i	−0.193699	−	0.193699i	0.603594	−	0.797292i	$-0.293735\pi$
$653$	−4.94975	−	4.94975i	−0.193699	−	0.193699i	−0.797292	+	0.603594i	$0.793735\pi$
$654$		0			0
$655$			21.0000i			0.820538i
$656$		0			0
$657$	−9.89949	−	9.89949i	−0.386216	−	0.386216i
$658$		0			0
$659$	28.0000			1.09073			0.545363	−	0.838200i	$-0.316392\pi$
$659$	28.0000			1.09073			0.545363	+	0.838200i	$0.316392\pi$
$660$		0			0
$661$			17.0000i			0.661223i	0.943767	+	0.330612i	$0.107255\pi$
$661$			17.0000i			0.661223i	−0.943767	+	0.330612i	$0.892745\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			6.00000i			0.232670i
$666$		0			0
$667$	−18.0000			−0.696963
$668$		0			0
$669$	−0.707107	−	0.707107i	−0.0273383	−	0.0273383i
$670$		0			0
$671$			30.0000i			1.15814i
$672$		0			0
$673$	−4.24264	−	4.24264i	−0.163542	−	0.163542i	0.620592	−	0.784134i	$-0.286892\pi$
$673$	−4.24264	−	4.24264i	−0.163542	−	0.163542i	−0.784134	+	0.620592i	$0.786892\pi$
$674$		0			0
$675$	−2.82843	+	2.82843i	−0.108866	+	0.108866i
$676$		0			0
$677$	−10.6066	+	10.6066i	−0.407645	+	0.407645i	−0.880917	−	0.473272i	$-0.843073\pi$
$677$	−10.6066	+	10.6066i	−0.407645	+	0.407645i	0.473272	+	0.880917i	$0.343073\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	3.00000			0.114960
$682$		0			0
$683$	−7.77817	+	7.77817i	−0.297624	+	0.297624i	−0.840082	−	0.542459i	$-0.817493\pi$
$683$	−7.77817	+	7.77817i	−0.297624	+	0.297624i	0.542459	+	0.840082i	$0.317493\pi$
$684$		0			0
$685$	11.3137	−	11.3137i	0.432275	−	0.432275i
$686$		0			0
$687$	−7.07107	−	7.07107i	−0.269778	−	0.269778i
$688$		0			0
$689$		−	18.0000i		−	0.685745i
$690$		0			0
$691$	−31.1127	−	31.1127i	−1.18358	−	1.18358i	−0.978809	−	0.204773i	$-0.934354\pi$
$691$	−31.1127	−	31.1127i	−1.18358	−	1.18358i	−0.204773	−	0.978809i	$-0.565646\pi$
$692$		0			0
$693$	−6.00000			−0.227921
$694$		0			0
$695$		−	8.00000i		−	0.303457i
$696$		0			0
$697$		0			0
$698$		0			0
$699$			19.0000i			0.718646i
$700$		0			0
$701$	−40.0000			−1.51078			−0.755390	−	0.655276i	$-0.772552\pi$
$701$	−40.0000			−1.51078			−0.755390	+	0.655276i	$0.772552\pi$
$702$		0			0
$703$	−21.2132	−	21.2132i	−0.800071	−	0.800071i
$704$		0			0
$705$		−	2.00000i		−	0.0753244i
$706$		0			0
$707$	−16.9706	−	16.9706i	−0.638244	−	0.638244i
$708$		0			0
$709$	12.7279	−	12.7279i	0.478007	−	0.478007i	−0.426487	−	0.904494i	$-0.640249\pi$
$709$	12.7279	−	12.7279i	0.478007	−	0.478007i	0.904494	+	0.426487i	$0.140249\pi$
$710$		0			0
$711$	8.48528	−	8.48528i	0.318223	−	0.318223i
$712$		0			0
$713$	6.00000			0.224702
$714$		0			0
$715$	−9.00000			−0.336581
$716$		0			0
$717$	−8.48528	+	8.48528i	−0.316889	+	0.316889i
$718$		0			0
$719$	−26.1630	+	26.1630i	−0.975713	+	0.975713i	−0.999712	−	0.0239986i	$-0.992360\pi$
$719$	−26.1630	+	26.1630i	−0.975713	+	0.975713i	0.0239986	+	0.999712i	$0.492360\pi$
$720$		0			0
$721$	18.3848	+	18.3848i	0.684685	+	0.684685i
$722$		0			0
$723$			14.0000i			0.520666i
$724$		0			0
$725$	−16.9706	−	16.9706i	−0.630271	−	0.630271i
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$			1.00000i			0.0370370i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		−	14.0000i		−	0.517102i	−0.965998	−	0.258551i	$-0.916755\pi$
$733$		−	14.0000i		−	0.517102i	0.965998	−	0.258551i	$-0.0832450\pi$
$734$		0			0
$735$	3.00000			0.110657
$736$		0			0
$737$	−8.48528	−	8.48528i	−0.312559	−	0.312559i
$738$		0			0
$739$			11.0000i			0.404642i	0.979319	+	0.202321i	$0.0648484\pi$
$739$			11.0000i			0.404642i	−0.979319	+	0.202321i	$0.935152\pi$
$740$		0			0
$741$	−6.36396	−	6.36396i	−0.233786	−	0.233786i
$742$		0			0
$743$	−22.6274	+	22.6274i	−0.830119	+	0.830119i	−0.987533	−	0.157413i	$-0.949684\pi$
$743$	−22.6274	+	22.6274i	−0.830119	+	0.830119i	0.157413	+	0.987533i	$0.449684\pi$
$744$		0			0
$745$	5.65685	−	5.65685i	0.207251	−	0.207251i
$746$		0			0
$747$	−14.0000			−0.512233
$748$		0			0
$749$	−26.0000			−0.950019
$750$		0			0
$751$	−35.3553	+	35.3553i	−1.29013	+	1.29013i	−0.355433	+	0.934702i	$0.615667\pi$
$751$	−35.3553	+	35.3553i	−1.29013	+	1.29013i	−0.934702	+	0.355433i	$0.884333\pi$
$752$		0			0
$753$	−21.2132	+	21.2132i	−0.773052	+	0.773052i
$754$		0			0
$755$	14.1421	+	14.1421i	0.514685	+	0.514685i
$756$		0			0
$757$		−	25.0000i		−	0.908640i	−0.890838	−	0.454320i	$-0.849882\pi$
$757$		−	25.0000i		−	0.908640i	0.890838	−	0.454320i	$-0.150118\pi$
$758$		0			0
$759$	−6.36396	−	6.36396i	−0.230997	−	0.230997i
$760$		0			0
$761$	6.00000			0.217500			0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000			0.217500			0.108750	+	0.994069i	$0.465315\pi$
$762$		0			0
$763$			8.00000i			0.289619i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−21.0000			−0.757279			−0.378640	−	0.925544i	$-0.623608\pi$
$769$	−21.0000			−0.757279			−0.378640	+	0.925544i	$0.623608\pi$
$770$		0			0
$771$	−18.3848	−	18.3848i	−0.662112	−	0.662112i
$772$		0			0
$773$			20.0000i			0.719350i	0.933078	+	0.359675i	$0.117112\pi$
$773$			20.0000i			0.719350i	−0.933078	+	0.359675i	$0.882888\pi$
$774$		0			0
$775$	5.65685	+	5.65685i	0.203200	+	0.203200i
$776$		0			0
$777$	14.1421	−	14.1421i	0.507346	−	0.507346i
$778$		0			0
$779$	6.36396	−	6.36396i	0.228013	−	0.228013i
$780$		0			0
$781$	24.0000			0.858788
$782$		0			0
$783$	−6.00000			−0.214423
$784$		0			0
$785$	3.53553	−	3.53553i	0.126189	−	0.126189i
$786$		0			0
$787$	−19.7990	+	19.7990i	−0.705758	+	0.705758i	−0.965640	−	0.259882i	$-0.916316\pi$
$787$	−19.7990	+	19.7990i	−0.705758	+	0.705758i	0.259882	+	0.965640i	$0.416316\pi$
$788$		0			0
$789$	−9.89949	−	9.89949i	−0.352431	−	0.352431i
$790$		0			0
$791$		−	38.0000i		−	1.35112i
$792$		0			0
$793$	−21.2132	−	21.2132i	−0.753303	−	0.753303i
$794$		0			0
$795$	−6.00000			−0.212798
$796$		0			0
$797$			12.0000i			0.425062i	0.977154	+	0.212531i	$0.0681706\pi$
$797$			12.0000i			0.425062i	−0.977154	+	0.212531i	$0.931829\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		−	2.00000i		−	0.0706665i
$802$		0			0
$803$	42.0000			1.48215
$804$		0			0
$805$	−4.24264	−	4.24264i	−0.149533	−	0.149533i
$806$		0			0
$807$		−	15.0000i		−	0.528025i
$808$		0			0
$809$	−14.8492	−	14.8492i	−0.522072	−	0.522072i	0.396125	−	0.918197i	$-0.370355\pi$
$809$	−14.8492	−	14.8492i	−0.522072	−	0.522072i	−0.918197	+	0.396125i	$0.870355\pi$
$810$		0			0
$811$	−7.07107	+	7.07107i	−0.248299	+	0.248299i	−0.820272	−	0.571973i	$-0.806178\pi$
$811$	−7.07107	+	7.07107i	−0.248299	+	0.248299i	0.571973	+	0.820272i	$0.306178\pi$
$812$		0			0
$813$	4.94975	−	4.94975i	0.173595	−	0.173595i
$814$		0			0
$815$	2.00000			0.0700569
$816$		0			0
$817$	3.00000			0.104957
$818$		0			0
$819$	4.24264	−	4.24264i	0.148250	−	0.148250i
$820$		0			0
$821$	9.19239	−	9.19239i	0.320817	−	0.320817i	−0.528264	−	0.849080i	$-0.677157\pi$
$821$	9.19239	−	9.19239i	0.320817	−	0.320817i	0.849080	+	0.528264i	$0.177157\pi$
$822$		0			0
$823$	15.5563	+	15.5563i	0.542260	+	0.542260i	0.924191	−	0.381931i	$-0.124741\pi$
$823$	15.5563	+	15.5563i	0.542260	+	0.542260i	−0.381931	+	0.924191i	$0.624741\pi$
$824$		0			0
$825$		−	12.0000i		−	0.417786i
$826$		0			0
$827$	20.5061	+	20.5061i	0.713067	+	0.713067i	0.967176	−	0.254109i	$-0.0817821\pi$
$827$	20.5061	+	20.5061i	0.713067	+	0.713067i	−0.254109	+	0.967176i	$0.581782\pi$
$828$		0			0
$829$	−30.0000			−1.04194			−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000			−1.04194			−0.520972	+	0.853574i	$0.674430\pi$
$830$		0			0
$831$			10.0000i			0.346896i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		−	15.0000i		−	0.519096i
$836$		0			0
$837$	2.00000			0.0691301
$838$		0			0
$839$	−30.4056	−	30.4056i	−1.04972	−	1.04972i	−0.998698	−	0.0510200i	$-0.983753\pi$
$839$	−30.4056	−	30.4056i	−1.04972	−	1.04972i	−0.0510200	−	0.998698i	$-0.516247\pi$
$840$		0			0
$841$		−	7.00000i		−	0.241379i
$842$		0			0
$843$	1.41421	+	1.41421i	0.0487081	+	0.0487081i
$844$		0			0
$845$	−2.82843	+	2.82843i	−0.0973009	+	0.0973009i
$846$		0			0
$847$	−2.82843	+	2.82843i	−0.0971859	+	0.0971859i
$848$		0			0
$849$	−20.0000			−0.686398
$850$		0			0
$851$	30.0000			1.02839
$852$		0			0
$853$	−14.1421	+	14.1421i	−0.484218	+	0.484218i	−0.906476	−	0.422258i	$-0.861238\pi$
$853$	−14.1421	+	14.1421i	−0.484218	+	0.484218i	0.422258	+	0.906476i	$0.361238\pi$
$854$		0			0
$855$	−2.12132	+	2.12132i	−0.0725476	+	0.0725476i
$856$		0			0
$857$	12.7279	+	12.7279i	0.434778	+	0.434778i	0.890250	−	0.455472i	$-0.150530\pi$
$857$	12.7279	+	12.7279i	0.434778	+	0.434778i	−0.455472	+	0.890250i	$0.650530\pi$
$858$		0			0
$859$			44.0000i			1.50126i	0.660722	+	0.750630i	$0.270250\pi$
$859$			44.0000i			1.50126i	−0.660722	+	0.750630i	$0.729750\pi$
$860$		0			0
$861$	4.24264	+	4.24264i	0.144589	+	0.144589i
$862$		0			0
$863$	10.0000			0.340404			0.170202	−	0.985409i	$-0.445558\pi$
$863$	10.0000			0.340404			0.170202	+	0.985409i	$0.445558\pi$
$864$		0			0
$865$		−	19.0000i		−	0.646019i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			36.0000i			1.22122i
$870$		0			0
$871$	12.0000			0.406604
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	18.0000i		−	0.608511i
$876$		0			0
$877$	5.65685	+	5.65685i	0.191018	+	0.191018i	0.796136	−	0.605118i	$-0.206874\pi$
$877$	5.65685	+	5.65685i	0.191018	+	0.191018i	−0.605118	+	0.796136i	$0.706874\pi$
$878$		0			0
$879$	4.24264	−	4.24264i	0.143101	−	0.143101i
$880$		0			0
$881$	1.41421	−	1.41421i	0.0476461	−	0.0476461i	−0.682882	−	0.730528i	$-0.739274\pi$
$881$	1.41421	−	1.41421i	0.0476461	−	0.0476461i	0.730528	+	0.682882i	$0.239274\pi$
$882$		0			0
$883$	−43.0000			−1.44707			−0.723533	−	0.690290i	$-0.757483\pi$
$883$	−43.0000			−1.44707			−0.723533	+	0.690290i	$0.757483\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−10.6066	+	10.6066i	−0.356135	+	0.356135i	−0.862386	−	0.506251i	$-0.831031\pi$
$887$	−10.6066	+	10.6066i	−0.356135	+	0.356135i	0.506251	+	0.862386i	$0.331031\pi$
$888$		0			0
$889$	−18.3848	+	18.3848i	−0.616606	+	0.616606i
$890$		0			0
$891$	−2.12132	−	2.12132i	−0.0710669	−	0.0710669i
$892$		0			0
$893$			6.00000i			0.200782i
$894$		0			0
$895$	−18.3848	−	18.3848i	−0.614535	−	0.614535i
$896$		0			0
$897$	9.00000			0.300501
$898$		0			0
$899$			12.0000i			0.400222i
$900$		0			0
$901$		0			0
$902$		0			0
$903$			2.00000i			0.0665558i
$904$		0			0
$905$	−4.00000			−0.132964
$906$		0			0
$907$	−5.65685	−	5.65685i	−0.187833	−	0.187833i	0.606926	−	0.794759i	$-0.292403\pi$
$907$	−5.65685	−	5.65685i	−0.187833	−	0.187833i	−0.794759	+	0.606926i	$0.792403\pi$
$908$		0			0
$909$		−	12.0000i		−	0.398015i
$910$		0			0
$911$	−21.9203	−	21.9203i	−0.726252	−	0.726252i	0.243619	−	0.969871i	$-0.421665\pi$
$911$	−21.9203	−	21.9203i	−0.726252	−	0.726252i	−0.969871	+	0.243619i	$0.921665\pi$
$912$		0			0
$913$	29.6985	−	29.6985i	0.982876	−	0.982876i
$914$		0			0
$915$	−7.07107	+	7.07107i	−0.233762	+	0.233762i
$916$		0			0
$917$	42.0000			1.38696
$918$		0			0
$919$	45.0000			1.48441			0.742207	−	0.670171i	$-0.233779\pi$
$919$	45.0000			1.48441			0.742207	+	0.670171i	$0.233779\pi$
$920$		0			0
$921$	−8.48528	+	8.48528i	−0.279600	+	0.279600i
$922$		0			0
$923$	−16.9706	+	16.9706i	−0.558593	+	0.558593i
$924$		0			0
$925$	28.2843	+	28.2843i	0.929981	+	0.929981i
$926$		0			0
$927$			13.0000i			0.426976i
$928$		0			0
$929$	−3.53553	−	3.53553i	−0.115997	−	0.115997i	0.646726	−	0.762723i	$-0.276138\pi$
$929$	−3.53553	−	3.53553i	−0.115997	−	0.115997i	−0.762723	+	0.646726i	$0.776138\pi$
$930$		0			0
$931$	−9.00000			−0.294963
$932$		0			0
$933$			8.00000i			0.261908i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	22.0000i		−	0.718709i	−0.933201	−	0.359354i	$-0.882997\pi$
$937$		−	22.0000i		−	0.718709i	0.933201	−	0.359354i	$-0.117003\pi$
$938$		0			0
$939$	−8.00000			−0.261070
$940$		0			0
$941$	21.2132	+	21.2132i	0.691531	+	0.691531i	0.962569	−	0.271038i	$-0.0873669\pi$
$941$	21.2132	+	21.2132i	0.691531	+	0.691531i	−0.271038	+	0.962569i	$0.587367\pi$
$942$		0			0
$943$			9.00000i			0.293080i
$944$		0			0
$945$	−1.41421	−	1.41421i	−0.0460044	−	0.0460044i
$946$		0			0
$947$	11.3137	−	11.3137i	0.367646	−	0.367646i	−0.498972	−	0.866618i	$-0.666289\pi$
$947$	11.3137	−	11.3137i	0.367646	−	0.367646i	0.866618	+	0.498972i	$0.166289\pi$
$948$		0			0
$949$	−29.6985	+	29.6985i	−0.964054	+	0.964054i
$950$		0			0
$951$	6.00000			0.194563
$952$		0			0
$953$	12.0000			0.388718			0.194359	−	0.980930i	$-0.437737\pi$
$953$	12.0000			0.388718			0.194359	+	0.980930i	$0.437737\pi$
$954$		0			0
$955$	−11.3137	+	11.3137i	−0.366103	+	0.366103i
$956$		0			0
$957$	12.7279	−	12.7279i	0.411435	−	0.411435i
$958$		0			0
$959$	−22.6274	−	22.6274i	−0.730677	−	0.730677i
$960$		0			0
$961$			27.0000i			0.870968i
$962$		0			0
$963$	−9.19239	−	9.19239i	−0.296221	−	0.296221i
$964$		0			0
$965$	22.0000			0.708205
$966$		0			0
$967$			19.0000i			0.610999i	0.952192	+	0.305499i	$0.0988234\pi$
$967$			19.0000i			0.610999i	−0.952192	+	0.305499i	$0.901177\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	38.0000i		−	1.21948i	−0.792602	−	0.609739i	$-0.791274\pi$
$971$		−	38.0000i		−	1.21948i	0.792602	−	0.609739i	$-0.208726\pi$
$972$		0			0
$973$	−16.0000			−0.512936
$974$		0			0
$975$	8.48528	+	8.48528i	0.271746	+	0.271746i
$976$		0			0
$977$		−	40.0000i		−	1.27971i	−0.768494	−	0.639857i	$-0.778994\pi$
$977$		−	40.0000i		−	1.27971i	0.768494	−	0.639857i	$-0.221006\pi$
$978$		0			0
$979$	4.24264	+	4.24264i	0.135595	+	0.135595i
$980$		0			0
$981$	−2.82843	+	2.82843i	−0.0903047	+	0.0903047i
$982$		0			0
$983$	0.707107	−	0.707107i	0.0225532	−	0.0225532i	−0.695740	−	0.718293i	$-0.744924\pi$
$983$	0.707107	−	0.707107i	0.0225532	−	0.0225532i	0.718293	+	0.695740i	$0.244924\pi$
$984$		0			0
$985$	21.0000			0.669116
$986$		0			0
$987$	−4.00000			−0.127321
$988$		0			0
$989$	−2.12132	+	2.12132i	−0.0674541	+	0.0674541i
$990$		0			0
$991$	5.65685	−	5.65685i	0.179696	−	0.179696i	−0.611527	−	0.791223i	$-0.709445\pi$
$991$	5.65685	−	5.65685i	0.179696	−	0.179696i	0.791223	+	0.611527i	$0.209445\pi$
$992$		0			0
$993$	−10.6066	−	10.6066i	−0.336590	−	0.336590i
$994$		0			0
$995$			24.0000i			0.760851i
$996$		0			0
$997$	8.48528	+	8.48528i	0.268732	+	0.268732i	0.828589	−	0.559857i	$-0.189144\pi$
$997$	8.48528	+	8.48528i	0.268732	+	0.268732i	−0.559857	+	0.828589i	$0.689144\pi$
$998$		0			0
$999$	10.0000			0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3468.2.j.b.3217.1		4
17.2	even	8		204.2.b.a.169.2	yes	2
17.4	even	4	inner	3468.2.j.b.829.2		4
17.8	even	8		3468.2.a.h.1.1		1
17.9	even	8		3468.2.a.b.1.1		1
17.13	even	4	inner	3468.2.j.b.829.1		4
17.15	even	8		204.2.b.a.169.1	✓	2
17.16	even	2	inner	3468.2.j.b.3217.2		4
51.2	odd	8		612.2.b.b.577.1		2
51.32	odd	8		612.2.b.b.577.2		2
68.15	odd	8		816.2.c.d.577.2		2
68.19	odd	8		816.2.c.d.577.1		2
85.2	odd	8		5100.2.k.d.4249.2		2
85.19	even	8		5100.2.e.b.1801.1		2
85.32	odd	8		5100.2.k.c.4249.1		2
85.49	even	8		5100.2.e.b.1801.2		2
85.53	odd	8		5100.2.k.c.4249.2		2
85.83	odd	8		5100.2.k.d.4249.1		2
136.19	odd	8		3264.2.c.b.577.2		2
136.53	even	8		3264.2.c.a.577.1		2
136.83	odd	8		3264.2.c.b.577.1		2
136.117	even	8		3264.2.c.a.577.2		2
204.83	even	8		2448.2.c.o.577.2		2
204.155	even	8		2448.2.c.o.577.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
204.2.b.a.169.1	✓	2	17.15	even	8
204.2.b.a.169.2	yes	2	17.2	even	8
612.2.b.b.577.1		2	51.2	odd	8
612.2.b.b.577.2		2	51.32	odd	8
816.2.c.d.577.1		2	68.19	odd	8
816.2.c.d.577.2		2	68.15	odd	8
2448.2.c.o.577.1		2	204.155	even	8
2448.2.c.o.577.2		2	204.83	even	8
3264.2.c.a.577.1		2	136.53	even	8
3264.2.c.a.577.2		2	136.117	even	8
3264.2.c.b.577.1		2	136.83	odd	8
3264.2.c.b.577.2		2	136.19	odd	8
3468.2.a.b.1.1		1	17.9	even	8
3468.2.a.h.1.1		1	17.8	even	8
3468.2.j.b.829.1		4	17.13	even	4	inner
3468.2.j.b.829.2		4	17.4	even	4	inner
3468.2.j.b.3217.1		4	1.1	even	1	trivial
3468.2.j.b.3217.2		4	17.16	even	2	inner
5100.2.e.b.1801.1		2	85.19	even	8
5100.2.e.b.1801.2		2	85.49	even	8
5100.2.k.c.4249.1		2	85.32	odd	8
5100.2.k.c.4249.2		2	85.53	odd	8
5100.2.k.d.4249.1		2	85.83	odd	8
5100.2.k.d.4249.2		2	85.2	odd	8

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:	\( N \)	\(=\)	\( 3468 = 2^{2} \cdot 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3468.j (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 + 0.707107i) q^{3} +(0.707107 - 0.707107i) q^{5} +(-1.41421 - 1.41421i) q^{7} -1.00000i q^{9} +O(q^{10})\)	\(q+(-0.707107 + 0.707107i) q^{3} +(0.707107 - 0.707107i) q^{5} +(-1.41421 - 1.41421i) q^{7} -1.00000i q^{9} +(2.12132 + 2.12132i) q^{11} -3.00000 q^{13} +1.00000i q^{15} -3.00000i q^{19} +2.00000 q^{21} +(2.12132 + 2.12132i) q^{23} +4.00000i q^{25} +(0.707107 + 0.707107i) q^{27} +(-4.24264 + 4.24264i) q^{29} +(1.41421 - 1.41421i) q^{31} -3.00000 q^{33} -2.00000 q^{35} +(7.07107 - 7.07107i) q^{37} +(2.12132 - 2.12132i) q^{39} +(2.12132 + 2.12132i) q^{41} +1.00000i q^{43} +(-0.707107 - 0.707107i) q^{45} -2.00000 q^{47} -3.00000i q^{49} +6.00000i q^{53} +3.00000 q^{55} +(2.12132 + 2.12132i) q^{57} +(7.07107 + 7.07107i) q^{61} +(-1.41421 + 1.41421i) q^{63} +(-2.12132 + 2.12132i) q^{65} -4.00000 q^{67} -3.00000 q^{69} +(5.65685 - 5.65685i) q^{71} +(9.89949 - 9.89949i) q^{73} +(-2.82843 - 2.82843i) q^{75} -6.00000i q^{77} +(8.48528 + 8.48528i) q^{79} -1.00000 q^{81} -14.0000i q^{83} -6.00000i q^{87} +2.00000 q^{89} +(4.24264 + 4.24264i) q^{91} +2.00000i q^{93} +(-2.12132 - 2.12132i) q^{95} +(2.12132 - 2.12132i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 12 q^{13} + 8 q^{21} - 12 q^{33} - 8 q^{35} - 8 q^{47} + 12 q^{55} - 16 q^{67} - 12 q^{69} - 4 q^{81} + 8 q^{89}+O(q^{100}) \) 4 * q - 12 * q^13 + 8 * q^21 - 12 * q^33 - 8 * q^35 - 8 * q^47 + 12 * q^55 - 16 * q^67 - 12 * q^69 - 4 * q^81 + 8 * q^89

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