LMFDB - Embedded newform 1156.2.b.a.577.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1156,2,Mod(577,1156)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1156, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1156.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1156 = 2^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1156.b (of order $2$, degree $1$, 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:	$9.23070647366$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - x^{2} + 2x + 27 $ x^4 - 2x^3 - x^2 + 2x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 68)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.4
Root			$2.30278 + 1.41421i$ of defining polynomial
Character	$\chi$	$=$	1156.577
Dual form			1156.2.b.a.577.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.25662i q^{3} -1.41421i q^{5} -3.25662i q^{7} -7.60555 q^{9} +O(q^{10})$	$q+3.25662i q^{3} -1.41421i q^{5} -3.25662i q^{7} -7.60555 q^{9} -0.428189i q^{11} -2.60555 q^{13} +4.60555 q^{15} -0.605551 q^{19} +10.6056 q^{21} -6.08504i q^{23} +3.00000 q^{25} -14.9985i q^{27} -2.27059i q^{29} -6.08504i q^{31} +1.39445 q^{33} -4.60555 q^{35} -4.24264i q^{37} -8.48528i q^{39} -1.41421i q^{41} -3.39445 q^{43} +10.7559i q^{45} +4.00000 q^{47} -3.60555 q^{49} +5.21110 q^{53} -0.605551 q^{55} -1.97205i q^{57} -8.60555 q^{59} +8.78383i q^{61} +24.7684i q^{63} +3.68481i q^{65} +9.21110 q^{67} +19.8167 q^{69} -4.11300i q^{71} -9.89949i q^{73} +9.76985i q^{75} -1.39445 q^{77} -0.428189i q^{79} +26.0278 q^{81} -17.8167 q^{83} +7.39445 q^{87} -7.81665 q^{89} +8.48528i q^{91} +19.8167 q^{93} +0.856379i q^{95} -10.7559i q^{97} +3.25662i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{9}+O(q^{10}) $ 4 * q - 16 * q^9	$ 4 q - 16 q^{9} + 4 q^{13} + 4 q^{15} + 12 q^{19} + 28 q^{21} + 12 q^{25} + 20 q^{33} - 4 q^{35} - 28 q^{43} + 16 q^{47} - 8 q^{53} + 12 q^{55} - 20 q^{59} + 8 q^{67} + 36 q^{69} - 20 q^{77} + 32 q^{81} - 28 q^{83} + 44 q^{87} + 12 q^{89} + 36 q^{93}+O(q^{100}) $ 4 * q - 16 * q^9 + 4 * q^13 + 4 * q^15 + 12 * q^19 + 28 * q^21 + 12 * q^25 + 20 * q^33 - 4 * q^35 - 28 * q^43 + 16 * q^47 - 8 * q^53 + 12 * q^55 - 20 * q^59 + 8 * q^67 + 36 * q^69 - 20 * q^77 + 32 * q^81 - 28 * q^83 + 44 * q^87 + 12 * q^89 + 36 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1156\mathbb{Z}\right)^\times$.

$n$	$579$	$581$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	3.25662i	1.88021i	0.340887	+	0.940104i	$0.389273\pi$
$3$	3.25662i	1.88021i	−0.340887	+	0.940104i	$0.610727\pi$
$4$	0	0
$5$	− 1.41421i	− 0.632456i	−0.948683	−	0.316228i	$-0.897584\pi$
$5$	− 1.41421i	− 0.632456i	0.948683	−	0.316228i	$-0.102416\pi$
$6$	0	0
$7$	− 3.25662i	− 1.23089i	−0.788182	−	0.615443i	$-0.788977\pi$
$7$	− 3.25662i	− 1.23089i	0.788182	−	0.615443i	$-0.211023\pi$
$8$	0	0
$9$	−7.60555	−2.53518
$10$	0	0
$11$	− 0.428189i	− 0.129104i	−0.997914	−	0.0645520i	$-0.979438\pi$
$11$	− 0.428189i	− 0.129104i	0.997914	−	0.0645520i	$-0.0205618\pi$
$12$	0	0
$13$	−2.60555	−0.722650	−0.361325	−	0.932440i	$-0.617675\pi$
$13$	−2.60555	−0.722650	−0.361325	+	0.932440i	$0.617675\pi$
$14$	0	0
$15$	4.60555	1.18915
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−0.605551	−0.138923	−0.0694615	−	0.997585i	$-0.522128\pi$
$19$	−0.605551	−0.138923	−0.0694615	+	0.997585i	$0.522128\pi$
$20$	0	0
$21$	10.6056	2.31432
$22$	0	0
$23$	− 6.08504i	− 1.26882i	−0.772997	−	0.634410i	$-0.781243\pi$
$23$	− 6.08504i	− 1.26882i	0.772997	−	0.634410i	$-0.218757\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	− 14.9985i	− 2.88647i
$28$	0	0
$29$	− 2.27059i	− 0.421638i	−0.977525	−	0.210819i	$-0.932387\pi$
$29$	− 2.27059i	− 0.421638i	0.977525	−	0.210819i	$-0.0676131\pi$
$30$	0	0
$31$	− 6.08504i	− 1.09291i	−0.837490	−	0.546453i	$-0.815978\pi$
$31$	− 6.08504i	− 1.09291i	0.837490	−	0.546453i	$-0.184022\pi$
$32$	0	0
$33$	1.39445	0.242742
$34$	0	0
$35$	−4.60555	−0.778480
$36$	0	0
$37$	− 4.24264i	− 0.697486i	−0.937218	−	0.348743i	$-0.886609\pi$
$37$	− 4.24264i	− 0.697486i	0.937218	−	0.348743i	$-0.113391\pi$
$38$	0	0
$39$	− 8.48528i	− 1.35873i
$40$	0	0
$41$	− 1.41421i	− 0.220863i	−0.993884	−	0.110432i	$-0.964777\pi$
$41$	− 1.41421i	− 0.220863i	0.993884	−	0.110432i	$-0.0352233\pi$
$42$	0	0
$43$	−3.39445	−0.517649	−0.258824	−	0.965924i	$-0.583335\pi$
$43$	−3.39445	−0.517649	−0.258824	+	0.965924i	$0.583335\pi$
$44$	0	0
$45$	10.7559i	1.60339i
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	−3.60555	−0.515079
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5.21110	0.715800	0.357900	−	0.933760i	$-0.383493\pi$
$53$	5.21110	0.715800	0.357900	+	0.933760i	$0.383493\pi$
$54$	0	0
$55$	−0.605551	−0.0816525
$56$	0	0
$57$	− 1.97205i	− 0.261204i
$58$	0	0
$59$	−8.60555	−1.12035	−0.560174	−	0.828375i	$-0.689266\pi$
$59$	−8.60555	−1.12035	−0.560174	+	0.828375i	$0.689266\pi$
$60$	0	0
$61$	8.78383i	1.12465i	0.826915	+	0.562327i	$0.190094\pi$
$61$	8.78383i	1.12465i	−0.826915	+	0.562327i	$0.809906\pi$
$62$	0	0
$63$	24.7684i	3.12052i
$64$	0	0
$65$	3.68481i	0.457044i
$66$	0	0
$67$	9.21110	1.12532	0.562658	−	0.826690i	$-0.309779\pi$
$67$	9.21110	1.12532	0.562658	+	0.826690i	$0.309779\pi$
$68$	0	0
$69$	19.8167	2.38564
$70$	0	0
$71$	− 4.11300i	− 0.488123i	−0.969760	−	0.244061i	$-0.921520\pi$
$71$	− 4.11300i	− 0.488123i	0.969760	−	0.244061i	$-0.0784798\pi$
$72$	0	0
$73$	− 9.89949i	− 1.15865i	−0.815097	−	0.579324i	$-0.803317\pi$
$73$	− 9.89949i	− 1.15865i	0.815097	−	0.579324i	$-0.196683\pi$
$74$	0	0
$75$	9.76985i	1.12813i
$76$	0	0
$77$	−1.39445	−0.158912
$78$	0	0
$79$	− 0.428189i	− 0.0481751i	−0.999710	−	0.0240875i	$-0.992332\pi$
$79$	− 0.428189i	− 0.0481751i	0.999710	−	0.0240875i	$-0.00766804\pi$
$80$	0	0
$81$	26.0278	2.89197
$82$	0	0
$83$	−17.8167	−1.95563	−0.977816	−	0.209466i	$-0.932827\pi$
$83$	−17.8167	−1.95563	−0.977816	+	0.209466i	$0.932827\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	7.39445	0.792768
$88$	0	0
$89$	−7.81665	−0.828564	−0.414282	−	0.910149i	$-0.635967\pi$
$89$	−7.81665	−0.828564	−0.414282	+	0.910149i	$0.635967\pi$
$90$	0	0
$91$	8.48528i	0.889499i
$92$	0	0
$93$	19.8167	2.05489
$94$	0	0
$95$	0.856379i	0.0878626i
$96$	0	0
$97$	− 10.7559i	− 1.09209i	−0.837755	−	0.546047i	$-0.816132\pi$
$97$	− 10.7559i	− 1.09209i	0.837755	−	0.546047i	$-0.183868\pi$
$98$	0	0
$99$	3.25662i	0.327302i
$100$	0	0
$101$	10.6056	1.05529	0.527646	−	0.849464i	$-0.323075\pi$
$101$	10.6056	1.05529	0.527646	+	0.849464i	$0.323075\pi$
$102$	0	0
$103$	−13.2111	−1.30173	−0.650864	−	0.759194i	$-0.725593\pi$
$103$	−13.2111	−1.30173	−0.650864	+	0.759194i	$0.725593\pi$
$104$	0	0
$105$	− 14.9985i	− 1.46371i
$106$	0	0
$107$	8.05709i	0.778908i	0.921046	+	0.389454i	$0.127336\pi$
$107$	8.05709i	0.778908i	−0.921046	+	0.389454i	$0.872664\pi$
$108$	0	0
$109$	8.78383i	0.841338i	0.907214	+	0.420669i	$0.138205\pi$
$109$	8.78383i	0.841338i	−0.907214	+	0.420669i	$0.861795\pi$
$110$	0	0
$111$	13.8167	1.31142
$112$	0	0
$113$	13.5843i	1.27790i	0.769247	+	0.638952i	$0.220632\pi$
$113$	13.5843i	1.27790i	−0.769247	+	0.638952i	$0.779368\pi$
$114$	0	0
$115$	−8.60555	−0.802472
$116$	0	0
$117$	19.8167	1.83205
$118$	0	0
$119$	0	0
$120$	0	0
$121$	10.8167	0.983332
$122$	0	0
$123$	4.60555	0.415269
$124$	0	0
$125$	− 11.3137i	− 1.01193i
$126$	0	0
$127$	−0.605551	−0.0537340	−0.0268670	−	0.999639i	$-0.508553\pi$
$127$	−0.605551	−0.0537340	−0.0268670	+	0.999639i	$0.508553\pi$
$128$	0	0
$129$	− 11.0544i	− 0.973287i
$130$	0	0
$131$	− 8.91347i	− 0.778774i	−0.921074	−	0.389387i	$-0.872687\pi$
$131$	− 8.91347i	− 0.778774i	0.921074	−	0.389387i	$-0.127313\pi$
$132$	0	0
$133$	1.97205i	0.170998i
$134$	0	0
$135$	−21.2111	−1.82556
$136$	0	0
$137$	−2.60555	−0.222607	−0.111304	−	0.993786i	$-0.535503\pi$
$137$	−2.60555	−0.222607	−0.111304	+	0.993786i	$0.535503\pi$
$138$	0	0
$139$	− 0.428189i	− 0.0363186i	−0.999835	−	0.0181593i	$-0.994219\pi$
$139$	− 0.428189i	− 0.0363186i	0.999835	−	0.0181593i	$-0.00578059\pi$
$140$	0	0
$141$	13.0265i	1.09703i
$142$	0	0
$143$	1.11567i	0.0932970i
$144$	0	0
$145$	−3.21110	−0.266668
$146$	0	0
$147$	− 11.7419i	− 0.968455i
$148$	0	0
$149$	−8.42221	−0.689974	−0.344987	−	0.938607i	$-0.612117\pi$
$149$	−8.42221	−0.689974	−0.344987	+	0.938607i	$0.612117\pi$
$150$	0	0
$151$	−4.60555	−0.374794	−0.187397	−	0.982284i	$-0.560005\pi$
$151$	−4.60555	−0.374794	−0.187397	+	0.982284i	$0.560005\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−8.60555	−0.691215
$156$	0	0
$157$	8.42221	0.672165	0.336083	−	0.941833i	$-0.390898\pi$
$157$	8.42221	0.672165	0.336083	+	0.941833i	$0.390898\pi$
$158$	0	0
$159$	16.9706i	1.34585i
$160$	0	0
$161$	−19.8167	−1.56177
$162$	0	0
$163$	6.08504i	0.476617i	0.971189	+	0.238309i	$0.0765930\pi$
$163$	6.08504i	0.476617i	−0.971189	+	0.238309i	$0.923407\pi$
$164$	0	0
$165$	− 1.97205i	− 0.153524i
$166$	0	0
$167$	− 21.0836i	− 1.63149i	−0.578408	−	0.815747i	$-0.696326\pi$
$167$	− 21.0836i	− 1.63149i	0.578408	−	0.815747i	$-0.303674\pi$
$168$	0	0
$169$	−6.21110	−0.477777
$170$	0	0
$171$	4.60555	0.352195
$172$	0	0
$173$	20.9539i	1.59310i	0.604575	+	0.796548i	$0.293343\pi$
$173$	20.9539i	1.59310i	−0.604575	+	0.796548i	$0.706657\pi$
$174$	0	0
$175$	− 9.76985i	− 0.738531i
$176$	0	0
$177$	− 28.0250i	− 2.10649i
$178$	0	0
$179$	−15.0278	−1.12323	−0.561614	−	0.827400i	$-0.689819\pi$
$179$	−15.0278	−1.12323	−0.561614	+	0.827400i	$0.689819\pi$
$180$	0	0
$181$	− 18.3848i	− 1.36653i	−0.730171	−	0.683265i	$-0.760559\pi$
$181$	− 18.3848i	− 1.36653i	0.730171	−	0.683265i	$-0.239441\pi$
$182$	0	0
$183$	−28.6056	−2.11458
$184$	0	0
$185$	−6.00000	−0.441129
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−48.8444	−3.55291
$190$	0	0
$191$	−6.78890	−0.491227	−0.245614	−	0.969368i	$-0.578989\pi$
$191$	−6.78890	−0.491227	−0.245614	+	0.969368i	$0.578989\pi$
$192$	0	0
$193$	− 11.6123i	− 0.835868i	−0.908477	−	0.417934i	$-0.862754\pi$
$193$	− 11.6123i	− 0.835868i	0.908477	−	0.417934i	$-0.137246\pi$
$194$	0	0
$195$	−12.0000	−0.859338
$196$	0	0
$197$	24.8980i	1.77391i	0.461856	+	0.886955i	$0.347184\pi$
$197$	24.8980i	1.77391i	−0.461856	+	0.886955i	$0.652816\pi$
$198$	0	0
$199$	− 14.3110i	− 1.01448i	−0.861804	−	0.507241i	$-0.830665\pi$
$199$	− 14.3110i	− 1.01448i	0.861804	−	0.507241i	$-0.169335\pi$
$200$	0	0
$201$	29.9970i	2.11583i
$202$	0	0
$203$	−7.39445	−0.518989
$204$	0	0
$205$	−2.00000	−0.139686
$206$	0	0
$207$	46.2801i	3.21669i
$208$	0	0
$209$	0.259291i	0.0179355i
$210$	0	0
$211$	− 21.9399i	− 1.51041i	−0.655490	−	0.755204i	$-0.727538\pi$
$211$	− 21.9399i	− 1.51041i	0.655490	−	0.755204i	$-0.272462\pi$
$212$	0	0
$213$	13.3944	0.917773
$214$	0	0
$215$	4.80048i	0.327390i
$216$	0	0
$217$	−19.8167	−1.34524
$218$	0	0
$219$	32.2389	2.17850
$220$	0	0
$221$	0	0
$222$	0	0
$223$	17.8167	1.19309	0.596546	−	0.802579i	$-0.296539\pi$
$223$	17.8167	1.19309	0.596546	+	0.802579i	$0.296539\pi$
$224$	0	0
$225$	−22.8167	−1.52111
$226$	0	0
$227$	− 0.428189i	− 0.0284199i	−0.999899	−	0.0142100i	$-0.995477\pi$
$227$	− 0.428189i	− 0.0284199i	0.999899	−	0.0142100i	$-0.00452332\pi$
$228$	0	0
$229$	4.60555	0.304343	0.152172	−	0.988354i	$-0.451373\pi$
$229$	4.60555	0.304343	0.152172	+	0.988354i	$0.451373\pi$
$230$	0	0
$231$	− 4.54118i	− 0.298788i
$232$	0	0
$233$	7.92745i	0.519344i	0.965697	+	0.259672i	$0.0836145\pi$
$233$	7.92745i	0.519344i	−0.965697	+	0.259672i	$0.916386\pi$
$234$	0	0
$235$	− 5.65685i	− 0.369012i
$236$	0	0
$237$	1.39445	0.0905792
$238$	0	0
$239$	−14.7889	−0.956614	−0.478307	−	0.878193i	$-0.658749\pi$
$239$	−14.7889	−0.956614	−0.478307	+	0.878193i	$0.658749\pi$
$240$	0	0
$241$	− 12.7279i	− 0.819878i	−0.912113	−	0.409939i	$-0.865550\pi$
$241$	− 12.7279i	− 0.819878i	0.912113	−	0.409939i	$-0.134450\pi$
$242$	0	0
$243$	39.7669i	2.55105i
$244$	0	0
$245$	5.09902i	0.325764i
$246$	0	0
$247$	1.57779	0.100393
$248$	0	0
$249$	− 58.0220i	− 3.67700i
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	−2.60555	−0.163810
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−19.3944	−1.20979	−0.604896	−	0.796304i	$-0.706785\pi$
$257$	−19.3944	−1.20979	−0.604896	+	0.796304i	$0.706785\pi$
$258$	0	0
$259$	−13.8167	−0.858525
$260$	0	0
$261$	17.2691i	1.06893i
$262$	0	0
$263$	23.0278	1.41995	0.709976	−	0.704226i	$-0.248706\pi$
$263$	23.0278	1.41995	0.709976	+	0.704226i	$0.248706\pi$
$264$	0	0
$265$	− 7.36961i	− 0.452712i
$266$	0	0
$267$	− 25.4558i	− 1.55787i
$268$	0	0
$269$	− 24.6387i	− 1.50225i	−0.660160	−	0.751125i	$-0.729512\pi$
$269$	− 24.6387i	− 1.50225i	0.660160	−	0.751125i	$-0.270488\pi$
$270$	0	0
$271$	1.21110	0.0735692	0.0367846	−	0.999323i	$-0.488288\pi$
$271$	1.21110	0.0735692	0.0367846	+	0.999323i	$0.488288\pi$
$272$	0	0
$273$	−27.6333	−1.67244
$274$	0	0
$275$	− 1.28457i	− 0.0774624i
$276$	0	0
$277$	5.09902i	0.306370i	0.988197	+	0.153185i	$0.0489531\pi$
$277$	5.09902i	0.306370i	−0.988197	+	0.153185i	$0.951047\pi$
$278$	0	0
$279$	46.2801i	2.77072i
$280$	0	0
$281$	18.4222	1.09898	0.549488	−	0.835501i	$-0.314823\pi$
$281$	18.4222	1.09898	0.549488	+	0.835501i	$0.314823\pi$
$282$	0	0
$283$	2.99733i	0.178173i	0.996024	+	0.0890863i	$0.0283947\pi$
$283$	2.99733i	0.178173i	−0.996024	+	0.0890863i	$0.971605\pi$
$284$	0	0
$285$	−2.78890	−0.165200
$286$	0	0
$287$	−4.60555	−0.271857
$288$	0	0
$289$	0	0
$290$	0	0
$291$	35.0278	2.05336
$292$	0	0
$293$	21.6333	1.26383	0.631916	−	0.775037i	$-0.282269\pi$
$293$	21.6333	1.26383	0.631916	+	0.775037i	$0.282269\pi$
$294$	0	0
$295$	12.1701i	0.708570i
$296$	0	0
$297$	−6.42221	−0.372654
$298$	0	0
$299$	15.8549i	0.916912i
$300$	0	0
$301$	11.0544i	0.637166i
$302$	0	0
$303$	34.5382i	1.98417i
$304$	0	0
$305$	12.4222	0.711293
$306$	0	0
$307$	21.2111	1.21058	0.605291	−	0.796004i	$-0.293057\pi$
$307$	21.2111	1.21058	0.605291	+	0.796004i	$0.293057\pi$
$308$	0	0
$309$	− 43.0235i	− 2.44752i
$310$	0	0
$311$	26.7404i	1.51631i	0.652075	+	0.758155i	$0.273899\pi$
$311$	26.7404i	1.51631i	−0.652075	+	0.758155i	$0.726101\pi$
$312$	0	0
$313$	− 8.78383i	− 0.496491i	−0.968697	−	0.248246i	$-0.920146\pi$
$313$	− 8.78383i	− 0.496491i	0.968697	−	0.248246i	$-0.0798540\pi$
$314$	0	0
$315$	35.0278	1.97359
$316$	0	0
$317$	27.7264i	1.55727i	0.627476	+	0.778636i	$0.284088\pi$
$317$	27.7264i	1.55727i	−0.627476	+	0.778636i	$0.715912\pi$
$318$	0	0
$319$	−0.972244	−0.0544352
$320$	0	0
$321$	−26.2389	−1.46451
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−7.81665	−0.433590
$326$	0	0
$327$	−28.6056	−1.58189
$328$	0	0
$329$	− 13.0265i	− 0.718172i
$330$	0	0
$331$	−3.39445	−0.186576	−0.0932879	−	0.995639i	$-0.529738\pi$
$331$	−3.39445	−0.186576	−0.0932879	+	0.995639i	$0.529738\pi$
$332$	0	0
$333$	32.2676i	1.76825i
$334$	0	0
$335$	− 13.0265i	− 0.711712i
$336$	0	0
$337$	22.9260i	1.24886i	0.781082	+	0.624428i	$0.214668\pi$
$337$	22.9260i	1.24886i	−0.781082	+	0.624428i	$0.785332\pi$
$338$	0	0
$339$	−44.2389	−2.40273
$340$	0	0
$341$	−2.60555	−0.141099
$342$	0	0
$343$	− 11.0544i	− 0.596882i
$344$	0	0
$345$	− 28.0250i	− 1.50881i
$346$	0	0
$347$	7.79780i	0.418608i	0.977851	+	0.209304i	$0.0671198\pi$
$347$	7.79780i	0.418608i	−0.977851	+	0.209304i	$0.932880\pi$
$348$	0	0
$349$	23.6333	1.26506	0.632531	−	0.774535i	$-0.282016\pi$
$349$	23.6333	1.26506	0.632531	+	0.774535i	$0.282016\pi$
$350$	0	0
$351$	39.0794i	2.08590i
$352$	0	0
$353$	31.2111	1.66120	0.830600	−	0.556870i	$-0.187998\pi$
$353$	31.2111	1.66120	0.830600	+	0.556870i	$0.187998\pi$
$354$	0	0
$355$	−5.81665	−0.308716
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−16.6056	−0.876407	−0.438204	−	0.898876i	$-0.644385\pi$
$359$	−16.6056	−0.876407	−0.438204	+	0.898876i	$0.644385\pi$
$360$	0	0
$361$	−18.6333	−0.980700
$362$	0	0
$363$	35.2257i	1.84887i
$364$	0	0
$365$	−14.0000	−0.732793
$366$	0	0
$367$	9.76985i	0.509982i	0.966943	+	0.254991i	$0.0820725\pi$
$367$	9.76985i	0.509982i	−0.966943	+	0.254991i	$0.917927\pi$
$368$	0	0
$369$	10.7559i	0.559928i
$370$	0	0
$371$	− 16.9706i	− 0.881068i
$372$	0	0
$373$	21.0278	1.08878	0.544388	−	0.838834i	$-0.316762\pi$
$373$	21.0278	1.08878	0.544388	+	0.838834i	$0.316762\pi$
$374$	0	0
$375$	36.8444	1.90264
$376$	0	0
$377$	5.91614i	0.304697i
$378$	0	0
$379$	− 29.3095i	− 1.50553i	−0.658289	−	0.752765i	$-0.728720\pi$
$379$	− 29.3095i	− 1.50553i	0.658289	−	0.752765i	$-0.271280\pi$
$380$	0	0
$381$	− 1.97205i	− 0.101031i
$382$	0	0
$383$	−19.3944	−0.991010	−0.495505	−	0.868605i	$-0.665017\pi$
$383$	−19.3944	−0.991010	−0.495505	+	0.868605i	$0.665017\pi$
$384$	0	0
$385$	1.97205i	0.100505i
$386$	0	0
$387$	25.8167	1.31233
$388$	0	0
$389$	−10.1833	−0.516316	−0.258158	−	0.966103i	$-0.583116\pi$
$389$	−10.1833	−0.516316	−0.258158	+	0.966103i	$0.583116\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	29.0278	1.46426
$394$	0	0
$395$	−0.605551	−0.0304686
$396$	0	0
$397$	− 12.7279i	− 0.638796i	−0.947621	−	0.319398i	$-0.896519\pi$
$397$	− 12.7279i	− 0.638796i	0.947621	−	0.319398i	$-0.103481\pi$
$398$	0	0
$399$	−6.42221	−0.321512
$400$	0	0
$401$	9.89949i	0.494357i	0.968970	+	0.247179i	$0.0795034\pi$
$401$	9.89949i	0.494357i	−0.968970	+	0.247179i	$0.920497\pi$
$402$	0	0
$403$	15.8549i	0.789788i
$404$	0	0
$405$	− 36.8088i	− 1.82904i
$406$	0	0
$407$	−1.81665	−0.0900482
$408$	0	0
$409$	23.2111	1.14772	0.573858	−	0.818955i	$-0.305446\pi$
$409$	23.2111	1.14772	0.573858	+	0.818955i	$0.305446\pi$
$410$	0	0
$411$	− 8.48528i	− 0.418548i
$412$	0	0
$413$	28.0250i	1.37902i
$414$	0	0
$415$	25.1966i	1.23685i
$416$	0	0
$417$	1.39445	0.0682864
$418$	0	0
$419$	3.25662i	0.159096i	0.996831	+	0.0795481i	$0.0253477\pi$
$419$	3.25662i	0.159096i	−0.996831	+	0.0795481i	$0.974652\pi$
$420$	0	0
$421$	−18.6056	−0.906779	−0.453390	−	0.891312i	$-0.649785\pi$
$421$	−18.6056	−0.906779	−0.453390	+	0.891312i	$0.649785\pi$
$422$	0	0
$423$	−30.4222	−1.47918
$424$	0	0
$425$	0	0
$426$	0	0
$427$	28.6056	1.38432
$428$	0	0
$429$	−3.63331	−0.175418
$430$	0	0
$431$	− 2.65953i	− 0.128105i	−0.997947	−	0.0640525i	$-0.979597\pi$
$431$	− 2.65953i	− 0.128105i	0.997947	−	0.0640525i	$-0.0204025\pi$
$432$	0	0
$433$	−15.0278	−0.722188	−0.361094	−	0.932529i	$-0.617597\pi$
$433$	−15.0278	−0.722188	−0.361094	+	0.932529i	$0.617597\pi$
$434$	0	0
$435$	− 10.4573i	− 0.501391i
$436$	0	0
$437$	3.68481i	0.176268i
$438$	0	0
$439$	37.1977i	1.77535i	0.460469	+	0.887676i	$0.347681\pi$
$439$	37.1977i	1.77535i	−0.460469	+	0.887676i	$0.652319\pi$
$440$	0	0
$441$	27.4222	1.30582
$442$	0	0
$443$	−17.2111	−0.817724	−0.408862	−	0.912596i	$-0.634074\pi$
$443$	−17.2111	−0.817724	−0.408862	+	0.912596i	$0.634074\pi$
$444$	0	0
$445$	11.0544i	0.524030i
$446$	0	0
$447$	− 27.4279i	− 1.29729i
$448$	0	0
$449$	23.7823i	1.12236i	0.827694	+	0.561179i	$0.189652\pi$
$449$	23.7823i	1.12236i	−0.827694	+	0.561179i	$0.810348\pi$
$450$	0	0
$451$	−0.605551	−0.0285143
$452$	0	0
$453$	− 14.9985i	− 0.704692i
$454$	0	0
$455$	12.0000	0.562569
$456$	0	0
$457$	36.6056	1.71234	0.856168	−	0.516698i	$-0.172839\pi$
$457$	36.6056	1.71234	0.856168	+	0.516698i	$0.172839\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−29.2111	−1.36050	−0.680248	−	0.732982i	$-0.738128\pi$
$461$	−29.2111	−1.36050	−0.680248	+	0.732982i	$0.738128\pi$
$462$	0	0
$463$	39.6333	1.84192	0.920958	−	0.389662i	$-0.127408\pi$
$463$	39.6333	1.84192	0.920958	+	0.389662i	$0.127408\pi$
$464$	0	0
$465$	− 28.0250i	− 1.29963i
$466$	0	0
$467$	16.2389	0.751445	0.375722	−	0.926732i	$-0.377395\pi$
$467$	16.2389	0.751445	0.375722	+	0.926732i	$0.377395\pi$
$468$	0	0
$469$	− 29.9970i	− 1.38513i
$470$	0	0
$471$	27.4279i	1.26381i
$472$	0	0
$473$	1.45347i	0.0668305i
$474$	0	0
$475$	−1.81665	−0.0833538
$476$	0	0
$477$	−39.6333	−1.81468
$478$	0	0
$479$	− 17.3988i	− 0.794969i	−0.917609	−	0.397485i	$-0.869883\pi$
$479$	− 17.3988i	− 0.794969i	0.917609	−	0.397485i	$-0.130117\pi$
$480$	0	0
$481$	11.0544i	0.504038i
$482$	0	0
$483$	− 64.5352i	− 2.93646i
$484$	0	0
$485$	−15.2111	−0.690701
$486$	0	0
$487$	− 24.5091i	− 1.11061i	−0.831646	−	0.555306i	$-0.812601\pi$
$487$	− 24.5091i	− 1.11061i	0.831646	−	0.555306i	$-0.187399\pi$
$488$	0	0
$489$	−19.8167	−0.896140
$490$	0	0
$491$	9.81665	0.443019	0.221510	−	0.975158i	$-0.428902\pi$
$491$	9.81665	0.443019	0.221510	+	0.975158i	$0.428902\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	4.60555	0.207004
$496$	0	0
$497$	−13.3944	−0.600823
$498$	0	0
$499$	14.5703i	0.652257i	0.945325	+	0.326129i	$0.105744\pi$
$499$	14.5703i	0.652257i	−0.945325	+	0.326129i	$0.894256\pi$
$500$	0	0
$501$	68.6611	3.06755
$502$	0	0
$503$	− 38.9105i	− 1.73493i	−0.497495	−	0.867467i	$-0.665747\pi$
$503$	− 38.9105i	− 1.73493i	0.497495	−	0.867467i	$-0.334253\pi$
$504$	0	0
$505$	− 14.9985i	− 0.667425i
$506$	0	0
$507$	− 20.2272i	− 0.898321i
$508$	0	0
$509$	−3.21110	−0.142330	−0.0711648	−	0.997465i	$-0.522672\pi$
$509$	−3.21110	−0.142330	−0.0711648	+	0.997465i	$0.522672\pi$
$510$	0	0
$511$	−32.2389	−1.42616
$512$	0	0
$513$	9.08237i	0.400996i
$514$	0	0
$515$	18.6833i	0.823285i
$516$	0	0
$517$	− 1.71276i	− 0.0753270i
$518$	0	0
$519$	−68.2389	−2.99535
$520$	0	0
$521$	18.3848i	0.805452i	0.915321	+	0.402726i	$0.131937\pi$
$521$	18.3848i	0.805452i	−0.915321	+	0.402726i	$0.868063\pi$
$522$	0	0
$523$	12.0000	0.524723	0.262362	−	0.964970i	$-0.415499\pi$
$523$	12.0000	0.524723	0.262362	+	0.964970i	$0.415499\pi$
$524$	0	0
$525$	31.8167	1.38859
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−14.0278	−0.609902
$530$	0	0
$531$	65.4500	2.84029
$532$	0	0
$533$	3.68481i	0.159607i
$534$	0	0
$535$	11.3944	0.492625
$536$	0	0
$537$	− 48.9396i	− 2.11190i
$538$	0	0
$539$	1.54386i	0.0664987i
$540$	0	0
$541$	− 4.24264i	− 0.182405i	−0.995832	−	0.0912027i	$-0.970929\pi$
$541$	− 4.24264i	− 0.182405i	0.995832	−	0.0912027i	$-0.0290711\pi$
$542$	0	0
$543$	59.8722	2.56936
$544$	0	0
$545$	12.4222	0.532109
$546$	0	0
$547$	16.2831i	0.696214i	0.937455	+	0.348107i	$0.113175\pi$
$547$	16.2831i	0.696214i	−0.937455	+	0.348107i	$0.886825\pi$
$548$	0	0
$549$	− 66.8058i	− 2.85120i
$550$	0	0
$551$	1.37496i	0.0585753i
$552$	0	0
$553$	−1.39445	−0.0592980
$554$	0	0
$555$	− 19.5397i	− 0.829414i
$556$	0	0
$557$	21.3944	0.906512	0.453256	−	0.891380i	$-0.350262\pi$
$557$	21.3944	0.906512	0.453256	+	0.891380i	$0.350262\pi$
$558$	0	0
$559$	8.84441	0.374079
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−20.6056	−0.868420	−0.434210	−	0.900812i	$-0.642972\pi$
$563$	−20.6056	−0.868420	−0.434210	+	0.900812i	$0.642972\pi$
$564$	0	0
$565$	19.2111	0.808217
$566$	0	0
$567$	− 84.7624i	− 3.55969i
$568$	0	0
$569$	−30.4222	−1.27537	−0.637683	−	0.770299i	$-0.720107\pi$
$569$	−30.4222	−1.27537	−0.637683	+	0.770299i	$0.720107\pi$
$570$	0	0
$571$	− 3.25662i	− 0.136285i	−0.997676	−	0.0681426i	$-0.978293\pi$
$571$	− 3.25662i	− 0.136285i	0.997676	−	0.0681426i	$-0.0217073\pi$
$572$	0	0
$573$	− 22.1088i	− 0.923610i
$574$	0	0
$575$	− 18.2551i	− 0.761292i
$576$	0	0
$577$	−31.4500	−1.30928	−0.654640	−	0.755941i	$-0.727180\pi$
$577$	−31.4500	−1.30928	−0.654640	+	0.755941i	$0.727180\pi$
$578$	0	0
$579$	37.8167	1.57161
$580$	0	0
$581$	58.0220i	2.40716i
$582$	0	0
$583$	− 2.23134i	− 0.0924126i
$584$	0	0
$585$	− 28.0250i	− 1.15869i
$586$	0	0
$587$	−4.60555	−0.190091	−0.0950457	−	0.995473i	$-0.530300\pi$
$587$	−4.60555	−0.190091	−0.0950457	+	0.995473i	$0.530300\pi$
$588$	0	0
$589$	3.68481i	0.151830i
$590$	0	0
$591$	−81.0833	−3.33532
$592$	0	0
$593$	−1.57779	−0.0647923	−0.0323961	−	0.999475i	$-0.510314\pi$
$593$	−1.57779	−0.0647923	−0.0323961	+	0.999475i	$0.510314\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	46.6056	1.90744
$598$	0	0
$599$	−26.0555	−1.06460	−0.532300	−	0.846556i	$-0.678672\pi$
$599$	−26.0555	−1.06460	−0.532300	+	0.846556i	$0.678672\pi$
$600$	0	0
$601$	− 5.95540i	− 0.242926i	−0.992596	−	0.121463i	$-0.961241\pi$
$601$	− 5.95540i	− 0.242926i	0.992596	−	0.121463i	$-0.0387585\pi$
$602$	0	0
$603$	−70.0555	−2.85288
$604$	0	0
$605$	− 15.2971i	− 0.621914i
$606$	0	0
$607$	13.4547i	0.546108i	0.961999	+	0.273054i	$0.0880337\pi$
$607$	13.4547i	0.546108i	−0.961999	+	0.273054i	$0.911966\pi$
$608$	0	0
$609$	− 24.0809i	− 0.975807i
$610$	0	0
$611$	−10.4222	−0.421637
$612$	0	0
$613$	40.4222	1.63264	0.816319	−	0.577602i	$-0.196011\pi$
$613$	40.4222	1.63264	0.816319	+	0.577602i	$0.196011\pi$
$614$	0	0
$615$	− 6.51323i	− 0.262639i
$616$	0	0
$617$	2.27059i	0.0914106i	0.998955	+	0.0457053i	$0.0145535\pi$
$617$	2.27059i	0.0914106i	−0.998955	+	0.0457053i	$0.985446\pi$
$618$	0	0
$619$	− 20.4865i	− 0.823421i	−0.911315	−	0.411710i	$-0.864932\pi$
$619$	− 20.4865i	− 0.823421i	0.911315	−	0.411710i	$-0.135068\pi$
$620$	0	0
$621$	−91.2666	−3.66240
$622$	0	0
$623$	25.4558i	1.01987i
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	−0.844410	−0.0337225
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−24.2389	−0.964934	−0.482467	−	0.875914i	$-0.660259\pi$
$631$	−24.2389	−0.964934	−0.482467	+	0.875914i	$0.660259\pi$
$632$	0	0
$633$	71.4500	2.83988
$634$	0	0
$635$	0.856379i	0.0339844i
$636$	0	0
$637$	9.39445	0.372222
$638$	0	0
$639$	31.2816i	1.23748i
$640$	0	0
$641$	43.5813i	1.72136i	0.509147	+	0.860680i	$0.329961\pi$
$641$	43.5813i	1.72136i	−0.509147	+	0.860680i	$0.670039\pi$
$642$	0	0
$643$	− 5.82575i	− 0.229745i	−0.993380	−	0.114873i	$-0.963354\pi$
$643$	− 5.82575i	− 0.229745i	0.993380	−	0.114873i	$-0.0366460\pi$
$644$	0	0
$645$	−15.6333	−0.615561
$646$	0	0
$647$	−17.2111	−0.676638	−0.338319	−	0.941031i	$-0.609858\pi$
$647$	−17.2111	−0.676638	−0.338319	+	0.941031i	$0.609858\pi$
$648$	0	0
$649$	3.68481i	0.144641i
$650$	0	0
$651$	− 64.5352i	− 2.52934i
$652$	0	0
$653$	− 35.0960i	− 1.37341i	−0.726934	−	0.686707i	$-0.759055\pi$
$653$	− 35.0960i	− 1.37341i	0.726934	−	0.686707i	$-0.240945\pi$
$654$	0	0
$655$	−12.6056	−0.492540
$656$	0	0
$657$	75.2911i	2.93739i
$658$	0	0
$659$	32.0000	1.24654	0.623272	−	0.782006i	$-0.285803\pi$
$659$	32.0000	1.24654	0.623272	+	0.782006i	$0.285803\pi$
$660$	0	0
$661$	1.21110	0.0471064	0.0235532	−	0.999723i	$-0.492502\pi$
$661$	1.21110	0.0471064	0.0235532	+	0.999723i	$0.492502\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.78890	0.108149
$666$	0	0
$667$	−13.8167	−0.534983
$668$	0	0
$669$	58.0220i	2.24326i
$670$	0	0
$671$	3.76114	0.145197
$672$	0	0
$673$	− 7.92745i	− 0.305581i	−0.988259	−	0.152790i	$-0.951174\pi$
$673$	− 7.92745i	− 0.305581i	0.988259	−	0.152790i	$-0.0488259\pi$
$674$	0	0
$675$	− 44.9955i	− 1.73188i
$676$	0	0
$677$	22.6667i	0.871151i	0.900152	+	0.435575i	$0.143455\pi$
$677$	22.6667i	0.871151i	−0.900152	+	0.435575i	$0.856545\pi$
$678$	0	0
$679$	−35.0278	−1.34424
$680$	0	0
$681$	1.39445	0.0534354
$682$	0	0
$683$	1.02528i	0.0392312i	0.999808	+	0.0196156i	$0.00624423\pi$
$683$	1.02528i	0.0392312i	−0.999808	+	0.0196156i	$0.993756\pi$
$684$	0	0
$685$	3.68481i	0.140789i
$686$	0	0
$687$	14.9985i	0.572229i
$688$	0	0
$689$	−13.5778	−0.517273
$690$	0	0
$691$	− 7.79780i	− 0.296642i	−0.988939	−	0.148321i	$-0.952613\pi$
$691$	− 7.79780i	− 0.296642i	0.988939	−	0.148321i	$-0.0473869\pi$
$692$	0	0
$693$	10.6056	0.402872
$694$	0	0
$695$	−0.605551	−0.0229699
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−25.8167	−0.976476
$700$	0	0
$701$	7.81665	0.295231	0.147615	−	0.989045i	$-0.452840\pi$
$701$	7.81665	0.295231	0.147615	+	0.989045i	$0.452840\pi$
$702$	0	0
$703$	2.56914i	0.0968968i
$704$	0	0
$705$	18.4222	0.693820
$706$	0	0
$707$	− 34.5382i	− 1.29894i
$708$	0	0
$709$	20.3568i	0.764517i	0.924056	+	0.382258i	$0.124854\pi$
$709$	20.3568i	0.764517i	−0.924056	+	0.382258i	$0.875146\pi$
$710$	0	0
$711$	3.25662i	0.122133i
$712$	0	0
$713$	−37.0278	−1.38670
$714$	0	0
$715$	1.57779	0.0590062
$716$	0	0
$717$	− 48.1618i	− 1.79863i
$718$	0	0
$719$	− 6.68213i	− 0.249201i	−0.992207	−	0.124601i	$-0.960235\pi$
$719$	− 6.68213i	− 0.249201i	0.992207	−	0.124601i	$-0.0397650\pi$
$720$	0	0
$721$	43.0235i	1.60228i
$722$	0	0
$723$	41.4500	1.54154
$724$	0	0
$725$	− 6.81178i	− 0.252983i
$726$	0	0
$727$	6.42221	0.238186	0.119093	−	0.992883i	$-0.462001\pi$
$727$	6.42221	0.238186	0.119093	+	0.992883i	$0.462001\pi$
$728$	0	0
$729$	−51.4222	−1.90453
$730$	0	0
$731$	0	0
$732$	0	0
$733$	46.0555	1.70110	0.850550	−	0.525895i	$-0.176269\pi$
$733$	46.0555	1.70110	0.850550	+	0.525895i	$0.176269\pi$
$734$	0	0
$735$	−16.6056	−0.612505
$736$	0	0
$737$	− 3.94410i	− 0.145283i
$738$	0	0
$739$	34.6611	1.27503	0.637514	−	0.770439i	$-0.279963\pi$
$739$	34.6611	1.27503	0.637514	+	0.770439i	$0.279963\pi$
$740$	0	0
$741$	5.13827i	0.188759i
$742$	0	0
$743$	20.8243i	0.763968i	0.924169	+	0.381984i	$0.124759\pi$
$743$	20.8243i	0.763968i	−0.924169	+	0.381984i	$0.875241\pi$
$744$	0	0
$745$	11.9108i	0.436378i
$746$	0	0
$747$	135.505	4.95789
$748$	0	0
$749$	26.2389	0.958747
$750$	0	0
$751$	− 2.40024i	− 0.0875859i	−0.999041	−	0.0437930i	$-0.986056\pi$
$751$	− 2.40024i	− 0.0875859i	0.999041	−	0.0437930i	$-0.0139442\pi$
$752$	0	0
$753$	39.0794i	1.42413i
$754$	0	0
$755$	6.51323i	0.237041i
$756$	0	0
$757$	39.0278	1.41849	0.709244	−	0.704963i	$-0.249036\pi$
$757$	39.0278	1.41849	0.709244	+	0.704963i	$0.249036\pi$
$758$	0	0
$759$	− 8.48528i	− 0.307996i
$760$	0	0
$761$	31.4500	1.14006	0.570030	−	0.821624i	$-0.306932\pi$
$761$	31.4500	1.14006	0.570030	+	0.821624i	$0.306932\pi$
$762$	0	0
$763$	28.6056	1.03559
$764$	0	0
$765$	0	0
$766$	0	0
$767$	22.4222	0.809619
$768$	0	0
$769$	18.6056	0.670933	0.335467	−	0.942052i	$-0.391106\pi$
$769$	18.6056	0.670933	0.335467	+	0.942052i	$0.391106\pi$
$770$	0	0
$771$	− 63.1603i	− 2.27466i
$772$	0	0
$773$	−28.6056	−1.02887	−0.514435	−	0.857529i	$-0.671998\pi$
$773$	−28.6056	−1.02887	−0.514435	+	0.857529i	$0.671998\pi$
$774$	0	0
$775$	− 18.2551i	− 0.655744i
$776$	0	0
$777$	− 44.9955i	− 1.61421i
$778$	0	0
$779$	0.856379i	0.0306830i
$780$	0	0
$781$	−1.76114	−0.0630186
$782$	0	0
$783$	−34.0555	−1.21704
$784$	0	0
$785$	− 11.9108i	− 0.425115i
$786$	0	0
$787$	20.8243i	0.742305i	0.928572	+	0.371152i	$0.121037\pi$
$787$	20.8243i	0.742305i	−0.928572	+	0.371152i	$0.878963\pi$
$788$	0	0
$789$	74.9926i	2.66981i
$790$	0	0
$791$	44.2389	1.57295
$792$	0	0
$793$	− 22.8867i	− 0.812731i
$794$	0	0
$795$	24.0000	0.851192
$796$	0	0
$797$	−3.63331	−0.128698	−0.0643492	−	0.997927i	$-0.520497\pi$
$797$	−3.63331	−0.128698	−0.0643492	+	0.997927i	$0.520497\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	59.4500	2.10056
$802$	0	0
$803$	−4.23886	−0.149586
$804$	0	0
$805$	28.0250i	0.987751i
$806$	0	0
$807$	80.2389	2.82454
$808$	0	0
$809$	29.4392i	1.03503i	0.855675	+	0.517513i	$0.173142\pi$
$809$	29.4392i	1.03503i	−0.855675	+	0.517513i	$0.826858\pi$
$810$	0	0
$811$	15.1674i	0.532600i	0.963890	+	0.266300i	$0.0858012\pi$
$811$	15.1674i	0.532600i	−0.963890	+	0.266300i	$0.914199\pi$
$812$	0	0
$813$	3.94410i	0.138326i
$814$	0	0
$815$	8.60555	0.301439
$816$	0	0
$817$	2.05551	0.0719133
$818$	0	0
$819$	− 64.5352i	− 2.25504i
$820$	0	0
$821$	18.1255i	0.632584i	0.948662	+	0.316292i	$0.102438\pi$
$821$	18.1255i	0.632584i	−0.948662	+	0.316292i	$0.897562\pi$
$822$	0	0
$823$	− 18.2551i	− 0.636334i	−0.948035	−	0.318167i	$-0.896933\pi$
$823$	− 18.2551i	− 0.636334i	0.948035	−	0.318167i	$-0.103067\pi$
$824$	0	0
$825$	4.18335	0.145645
$826$	0	0
$827$	8.91347i	0.309952i	0.987918	+	0.154976i	$0.0495300\pi$
$827$	8.91347i	0.309952i	−0.987918	+	0.154976i	$0.950470\pi$
$828$	0	0
$829$	18.8444	0.654493	0.327247	−	0.944939i	$-0.393879\pi$
$829$	18.8444	0.654493	0.327247	+	0.944939i	$0.393879\pi$
$830$	0	0
$831$	−16.6056	−0.576040
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−29.8167	−1.03185
$836$	0	0
$837$	−91.2666	−3.15464
$838$	0	0
$839$	− 21.0836i	− 0.727885i	−0.931421	−	0.363943i	$-0.881430\pi$
$839$	− 21.0836i	− 0.727885i	0.931421	−	0.363943i	$-0.118570\pi$
$840$	0	0
$841$	23.8444	0.822221
$842$	0	0
$843$	59.9941i	2.06631i
$844$	0	0
$845$	8.78383i	0.302173i
$846$	0	0
$847$	− 35.2257i	− 1.21037i
$848$	0	0
$849$	−9.76114	−0.335001
$850$	0	0
$851$	−25.8167	−0.884983
$852$	0	0
$853$	− 41.6093i	− 1.42467i	−0.701837	−	0.712337i	$-0.747637\pi$
$853$	− 41.6093i	− 1.42467i	0.701837	−	0.712337i	$-0.252363\pi$
$854$	0	0
$855$	− 6.51323i	− 0.222748i
$856$	0	0
$857$	26.3515i	0.900149i	0.892991	+	0.450075i	$0.148603\pi$
$857$	26.3515i	0.900149i	−0.892991	+	0.450075i	$0.851397\pi$
$858$	0	0
$859$	1.81665	0.0619834	0.0309917	−	0.999520i	$-0.490133\pi$
$859$	1.81665	0.0619834	0.0309917	+	0.999520i	$0.490133\pi$
$860$	0	0
$861$	− 14.9985i	− 0.511148i
$862$	0	0
$863$	52.4777	1.78636	0.893181	−	0.449697i	$-0.148468\pi$
$863$	52.4777	1.78636	0.893181	+	0.449697i	$0.148468\pi$
$864$	0	0
$865$	29.6333	1.00756
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−0.183346	−0.00621959
$870$	0	0
$871$	−24.0000	−0.813209
$872$	0	0
$873$	81.8043i	2.76866i
$874$	0	0
$875$	−36.8444	−1.24557
$876$	0	0
$877$	− 2.52988i	− 0.0854281i	−0.999087	−	0.0427140i	$-0.986400\pi$
$877$	− 2.52988i	− 0.0854281i	0.999087	−	0.0427140i	$-0.0136004\pi$
$878$	0	0
$879$	70.4514i	2.37627i
$880$	0	0
$881$	34.2397i	1.15356i	0.816898	+	0.576782i	$0.195692\pi$
$881$	34.2397i	1.15356i	−0.816898	+	0.576782i	$0.804308\pi$
$882$	0	0
$883$	25.5778	0.860761	0.430381	−	0.902647i	$-0.358379\pi$
$883$	25.5778	0.860761	0.430381	+	0.902647i	$0.358379\pi$
$884$	0	0
$885$	−39.6333	−1.33226
$886$	0	0
$887$	16.5424i	0.555439i	0.960662	+	0.277719i	$0.0895785\pi$
$887$	16.5424i	0.555439i	−0.960662	+	0.277719i	$0.910421\pi$
$888$	0	0
$889$	1.97205i	0.0661404i
$890$	0	0
$891$	− 11.1448i	− 0.373365i
$892$	0	0
$893$	−2.42221	−0.0810560
$894$	0	0
$895$	21.2525i	0.710391i
$896$	0	0
$897$	−51.6333	−1.72399
$898$	0	0
$899$	−13.8167	−0.460811
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−36.0000	−1.19800
$904$	0	0
$905$	−26.0000	−0.864269
$906$	0	0
$907$	34.7071i	1.15243i	0.817298	+	0.576215i	$0.195471\pi$
$907$	34.7071i	1.15243i	−0.817298	+	0.576215i	$0.804529\pi$
$908$	0	0
$909$	−80.6611	−2.67536
$910$	0	0
$911$	− 6.68213i	− 0.221389i	−0.993854	−	0.110694i	$-0.964693\pi$
$911$	− 6.68213i	− 0.221389i	0.993854	−	0.110694i	$-0.0353075\pi$
$912$	0	0
$913$	7.62890i	0.252480i
$914$	0	0
$915$	40.4544i	1.33738i
$916$	0	0
$917$	−29.0278	−0.958581
$918$	0	0
$919$	40.8444	1.34733	0.673666	−	0.739036i	$-0.264718\pi$
$919$	40.8444	1.34733	0.673666	+	0.739036i	$0.264718\pi$
$920$	0	0
$921$	69.0764i	2.27615i
$922$	0	0
$923$	10.7166i	0.352742i
$924$	0	0
$925$	− 12.7279i	− 0.418491i
$926$	0	0
$927$	100.478	3.30012
$928$	0	0
$929$	1.41421i	0.0463988i	0.999731	+	0.0231994i	$0.00738527\pi$
$929$	1.41421i	0.0463988i	−0.999731	+	0.0231994i	$0.992615\pi$
$930$	0	0
$931$	2.18335	0.0715563
$932$	0	0
$933$	−87.0833	−2.85098
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−6.42221	−0.209804	−0.104902	−	0.994483i	$-0.533453\pi$
$937$	−6.42221	−0.209804	−0.104902	+	0.994483i	$0.533453\pi$
$938$	0	0
$939$	28.6056	0.933507
$940$	0	0
$941$	− 22.6667i	− 0.738912i	−0.929248	−	0.369456i	$-0.879544\pi$
$941$	− 22.6667i	− 0.738912i	0.929248	−	0.369456i	$-0.120456\pi$
$942$	0	0
$943$	−8.60555	−0.280235
$944$	0	0
$945$	69.0764i	2.24706i
$946$	0	0
$947$	− 13.1169i	− 0.426241i	−0.977026	−	0.213120i	$-0.931637\pi$
$947$	− 13.1169i	− 0.426241i	0.977026	−	0.213120i	$-0.0683626\pi$
$948$	0	0
$949$	25.7936i	0.837297i
$950$	0	0
$951$	−90.2944	−2.92800
$952$	0	0
$953$	50.2389	1.62740	0.813698	−	0.581288i	$-0.197451\pi$
$953$	50.2389	1.62740	0.813698	+	0.581288i	$0.197451\pi$
$954$	0	0
$955$	9.60095i	0.310679i
$956$	0	0
$957$	− 3.16622i	− 0.102350i
$958$	0	0
$959$	8.48528i	0.274004i
$960$	0	0
$961$	−6.02776	−0.194444
$962$	0	0
$963$	− 61.2786i	− 1.97468i
$964$	0	0
$965$	−16.4222	−0.528649
$966$	0	0
$967$	−21.8167	−0.701576	−0.350788	−	0.936455i	$-0.614086\pi$
$967$	−21.8167	−0.701576	−0.350788	+	0.936455i	$0.614086\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−55.0278	−1.76592	−0.882962	−	0.469444i	$-0.844454\pi$
$971$	−55.0278	−1.76592	−0.882962	+	0.469444i	$0.844454\pi$
$972$	0	0
$973$	−1.39445	−0.0447040
$974$	0	0
$975$	− 25.4558i	− 0.815239i
$976$	0	0
$977$	16.8444	0.538900	0.269450	−	0.963014i	$-0.413158\pi$
$977$	16.8444	0.538900	0.269450	+	0.963014i	$0.413158\pi$
$978$	0	0
$979$	3.34701i	0.106971i
$980$	0	0
$981$	− 66.8058i	− 2.13295i
$982$	0	0
$983$	− 11.4826i	− 0.366238i	−0.983091	−	0.183119i	$-0.941381\pi$
$983$	− 11.4826i	− 0.366238i	0.983091	−	0.183119i	$-0.0586194\pi$
$984$	0	0
$985$	35.2111	1.12192
$986$	0	0
$987$	42.4222	1.35031
$988$	0	0
$989$	20.6554i	0.656803i
$990$	0	0
$991$	− 33.2536i	− 1.05634i	−0.849140	−	0.528168i	$-0.822879\pi$
$991$	− 33.2536i	− 1.05634i	0.849140	−	0.528168i	$-0.177121\pi$
$992$	0	0
$993$	− 11.0544i	− 0.350801i
$994$	0	0
$995$	−20.2389	−0.641615
$996$	0	0
$997$	55.4921i	1.75745i	0.477325	+	0.878727i	$0.341606\pi$
$997$	55.4921i	1.75745i	−0.477325	+	0.878727i	$0.658394\pi$
$998$	0	0
$999$	−63.6333	−2.01327

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1156.2.b.a.577.4		4
17.2	even	8		1156.2.e.c.829.2		4
17.3	odd	16		1156.2.h.e.977.1		16
17.4	even	4		1156.2.a.h.1.4		4
17.5	odd	16		1156.2.h.e.757.1		16
17.6	odd	16		1156.2.h.e.1001.4		16
17.7	odd	16		1156.2.h.e.733.4		16
17.8	even	8		1156.2.e.c.905.2		4
17.9	even	8		68.2.e.a.21.1	yes	4
17.10	odd	16		1156.2.h.e.733.1		16
17.11	odd	16		1156.2.h.e.1001.1		16
17.12	odd	16		1156.2.h.e.757.4		16
17.13	even	4		1156.2.a.h.1.1		4
17.14	odd	16		1156.2.h.e.977.4		16
17.15	even	8		68.2.e.a.13.1	✓	4
17.16	even	2	inner	1156.2.b.a.577.1		4
51.26	odd	8		612.2.k.e.361.2		4
51.32	odd	8		612.2.k.e.217.2		4
68.15	odd	8		272.2.o.g.81.2		4
68.43	odd	8		272.2.o.g.225.2		4
68.47	odd	4		4624.2.a.bq.1.4		4
68.55	odd	4		4624.2.a.bq.1.1		4
85.9	even	8		1700.2.o.c.701.2		4
85.32	odd	8		1700.2.m.a.149.1		4
85.43	odd	8		1700.2.m.a.1449.1		4
85.49	even	8		1700.2.o.c.1101.2		4
85.77	odd	8		1700.2.m.b.1449.2		4
85.83	odd	8		1700.2.m.b.149.2		4
136.43	odd	8		1088.2.o.s.769.1		4
136.77	even	8		1088.2.o.t.769.2		4
136.83	odd	8		1088.2.o.s.897.1		4
136.117	even	8		1088.2.o.t.897.2		4
204.83	even	8		2448.2.be.u.1441.1		4
204.179	even	8		2448.2.be.u.1585.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
68.2.e.a.13.1	✓	4	17.15	even	8
68.2.e.a.21.1	yes	4	17.9	even	8
272.2.o.g.81.2		4	68.15	odd	8
272.2.o.g.225.2		4	68.43	odd	8
612.2.k.e.217.2		4	51.32	odd	8
612.2.k.e.361.2		4	51.26	odd	8
1088.2.o.s.769.1		4	136.43	odd	8
1088.2.o.s.897.1		4	136.83	odd	8
1088.2.o.t.769.2		4	136.77	even	8
1088.2.o.t.897.2		4	136.117	even	8
1156.2.a.h.1.1		4	17.13	even	4
1156.2.a.h.1.4		4	17.4	even	4
1156.2.b.a.577.1		4	17.16	even	2	inner
1156.2.b.a.577.4		4	1.1	even	1	trivial
1156.2.e.c.829.2		4	17.2	even	8
1156.2.e.c.905.2		4	17.8	even	8
1156.2.h.e.733.1		16	17.10	odd	16
1156.2.h.e.733.4		16	17.7	odd	16
1156.2.h.e.757.1		16	17.5	odd	16
1156.2.h.e.757.4		16	17.12	odd	16
1156.2.h.e.977.1		16	17.3	odd	16
1156.2.h.e.977.4		16	17.14	odd	16
1156.2.h.e.1001.1		16	17.11	odd	16
1156.2.h.e.1001.4		16	17.6	odd	16
1700.2.m.a.149.1		4	85.32	odd	8
1700.2.m.a.1449.1		4	85.43	odd	8
1700.2.m.b.149.2		4	85.83	odd	8
1700.2.m.b.1449.2		4	85.77	odd	8
1700.2.o.c.701.2		4	85.9	even	8
1700.2.o.c.1101.2		4	85.49	even	8
2448.2.be.u.1441.1		4	204.83	even	8
2448.2.be.u.1585.1		4	204.179	even	8
4624.2.a.bq.1.1		4	68.55	odd	4
4624.2.a.bq.1.4		4	68.47	odd	4

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:	\( N \)	\(=\)	\( 1156 = 2^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1156.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.25662i q^{3} -1.41421i q^{5} -3.25662i q^{7} -7.60555 q^{9} +O(q^{10})\)	\(q+3.25662i q^{3} -1.41421i q^{5} -3.25662i q^{7} -7.60555 q^{9} -0.428189i q^{11} -2.60555 q^{13} +4.60555 q^{15} -0.605551 q^{19} +10.6056 q^{21} -6.08504i q^{23} +3.00000 q^{25} -14.9985i q^{27} -2.27059i q^{29} -6.08504i q^{31} +1.39445 q^{33} -4.60555 q^{35} -4.24264i q^{37} -8.48528i q^{39} -1.41421i q^{41} -3.39445 q^{43} +10.7559i q^{45} +4.00000 q^{47} -3.60555 q^{49} +5.21110 q^{53} -0.605551 q^{55} -1.97205i q^{57} -8.60555 q^{59} +8.78383i q^{61} +24.7684i q^{63} +3.68481i q^{65} +9.21110 q^{67} +19.8167 q^{69} -4.11300i q^{71} -9.89949i q^{73} +9.76985i q^{75} -1.39445 q^{77} -0.428189i q^{79} +26.0278 q^{81} -17.8167 q^{83} +7.39445 q^{87} -7.81665 q^{89} +8.48528i q^{91} +19.8167 q^{93} +0.856379i q^{95} -10.7559i q^{97} +3.25662i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{9}+O(q^{10}) \) 4 * q - 16 * q^9	\( 4 q - 16 q^{9} + 4 q^{13} + 4 q^{15} + 12 q^{19} + 28 q^{21} + 12 q^{25} + 20 q^{33} - 4 q^{35} - 28 q^{43} + 16 q^{47} - 8 q^{53} + 12 q^{55} - 20 q^{59} + 8 q^{67} + 36 q^{69} - 20 q^{77} + 32 q^{81} - 28 q^{83} + 44 q^{87} + 12 q^{89} + 36 q^{93}+O(q^{100}) \) 4 * q - 16 * q^9 + 4 * q^13 + 4 * q^15 + 12 * q^19 + 28 * q^21 + 12 * q^25 + 20 * q^33 - 4 * q^35 - 28 * q^43 + 16 * q^47 - 8 * q^53 + 12 * q^55 - 20 * q^59 + 8 * q^67 + 36 * q^69 - 20 * q^77 + 32 * q^81 - 28 * q^83 + 44 * q^87 + 12 * q^89 + 36 * q^93

Introduction

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