LMFDB - Embedded newform 1352.2.f.d.337.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1352,2,Mod(337,1352)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1352, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1352.337"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1352 = 2^{3} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1352.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,2,0,0,0,0,0,6,0,0,0,0,0,0,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$10.7957743533$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.3
Root			$-2.56155i$ of defining polynomial
Character	$\chi$	$=$	1352.337
Dual form			1352.2.f.d.337.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155 q^{3} -3.56155i q^{5} +2.56155i q^{7} +3.56155 q^{9} -2.56155i q^{11} -9.12311i q^{15} +5.00000 q^{17} -2.56155i q^{19} +6.56155i q^{21} +3.68466 q^{23} -7.68466 q^{25} +1.43845 q^{27} -5.00000 q^{29} -8.00000i q^{31} -6.56155i q^{33} +9.12311 q^{35} +1.00000i q^{37} +9.24621i q^{41} +6.56155 q^{43} -12.6847i q^{45} -4.00000i q^{47} +0.438447 q^{49} +12.8078 q^{51} +4.43845 q^{53} -9.12311 q^{55} -6.56155i q^{57} +2.56155i q^{59} -7.24621 q^{61} +9.12311i q^{63} -9.43845i q^{67} +9.43845 q^{69} +7.68466i q^{71} +1.31534i q^{73} -19.6847 q^{75} +6.56155 q^{77} -4.00000 q^{79} -7.00000 q^{81} +2.24621i q^{83} -17.8078i q^{85} -12.8078 q^{87} +9.68466i q^{89} -20.4924i q^{93} -9.12311 q^{95} +2.80776i q^{97} -9.12311i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 6 q^{9} + 20 q^{17} - 10 q^{23} - 6 q^{25} + 14 q^{27} - 20 q^{29} + 20 q^{35} + 18 q^{43} + 10 q^{49} + 10 q^{51} + 26 q^{53} - 20 q^{55} + 4 q^{61} + 46 q^{69} - 54 q^{75} + 18 q^{77}+ \cdots - 20 q^{95}+O(q^{100}) $ 4 * q + 2 * q^3 + 6 * q^9 + 20 * q^17 - 10 * q^23 - 6 * q^25 + 14 * q^27 - 20 * q^29 + 20 * q^35 + 18 * q^43 + 10 * q^49 + 10 * q^51 + 26 * q^53 - 20 * q^55 + 4 * q^61 + 46 * q^69 - 54 * q^75 + 18 * q^77 - 16 * q^79 - 28 * q^81 - 10 * q^87 - 20 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1352\mathbb{Z}\right)^\times$.

$n$	$677$	$1015$	$1185$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	2.56155	1.47891	0.739457	−	0.673204i	$-0.235083\pi$
$3$	2.56155	1.47891	0.739457	+	0.673204i	$0.235083\pi$
$4$	0	0
$5$	− 3.56155i	− 1.59277i	−0.604787	−	0.796387i	$-0.706742\pi$
$5$	− 3.56155i	− 1.59277i	0.604787	−	0.796387i	$-0.293258\pi$
$6$	0	0
$7$	2.56155i	0.968176i	0.875019	+	0.484088i	$0.160849\pi$
$7$	2.56155i	0.968176i	−0.875019	+	0.484088i	$0.839151\pi$
$8$	0	0
$9$	3.56155	1.18718
$10$	0	0
$11$	− 2.56155i	− 0.772337i	−0.922428	−	0.386169i	$-0.873798\pi$
$11$	− 2.56155i	− 0.772337i	0.922428	−	0.386169i	$-0.126202\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	− 9.12311i	− 2.35558i
$16$	0	0
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	0	0
$19$	− 2.56155i	− 0.587661i	−0.955858	−	0.293830i	$-0.905070\pi$
$19$	− 2.56155i	− 0.587661i	0.955858	−	0.293830i	$-0.0949300\pi$
$20$	0	0
$21$	6.56155i	1.43185i
$22$	0	0
$23$	3.68466	0.768304	0.384152	−	0.923270i	$-0.374494\pi$
$23$	3.68466	0.768304	0.384152	+	0.923270i	$0.374494\pi$
$24$	0	0
$25$	−7.68466	−1.53693
$26$	0	0
$27$	1.43845	0.276829
$28$	0	0
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	− 8.00000i	− 1.43684i	−0.695608	−	0.718421i	$-0.744865\pi$
$31$	− 8.00000i	− 1.43684i	0.695608	−	0.718421i	$-0.255135\pi$
$32$	0	0
$33$	− 6.56155i	− 1.14222i
$34$	0	0
$35$	9.12311	1.54209
$36$	0	0
$37$	1.00000i	0.164399i	0.996616	+	0.0821995i	$0.0261945\pi$
$37$	1.00000i	0.164399i	−0.996616	+	0.0821995i	$0.973806\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.24621i	1.44402i	0.691885	+	0.722008i	$0.256781\pi$
$41$	9.24621i	1.44402i	−0.691885	+	0.722008i	$0.743219\pi$
$42$	0	0
$43$	6.56155	1.00063	0.500314	−	0.865844i	$-0.333218\pi$
$43$	6.56155	1.00063	0.500314	+	0.865844i	$0.333218\pi$
$44$	0	0
$45$	− 12.6847i	− 1.89092i
$46$	0	0
$47$	− 4.00000i	− 0.583460i	−0.956501	−	0.291730i	$-0.905769\pi$
$47$	− 4.00000i	− 0.583460i	0.956501	−	0.291730i	$-0.0942309\pi$
$48$	0	0
$49$	0.438447	0.0626353
$50$	0	0
$51$	12.8078	1.79345
$52$	0	0
$53$	4.43845	0.609668	0.304834	−	0.952406i	$-0.401399\pi$
$53$	4.43845	0.609668	0.304834	+	0.952406i	$0.401399\pi$
$54$	0	0
$55$	−9.12311	−1.23016
$56$	0	0
$57$	− 6.56155i	− 0.869099i
$58$	0	0
$59$	2.56155i	0.333486i	0.986000	+	0.166743i	$0.0533250\pi$
$59$	2.56155i	0.333486i	−0.986000	+	0.166743i	$0.946675\pi$
$60$	0	0
$61$	−7.24621	−0.927782	−0.463891	−	0.885892i	$-0.653547\pi$
$61$	−7.24621	−0.927782	−0.463891	+	0.885892i	$0.653547\pi$
$62$	0	0
$63$	9.12311i	1.14940i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 9.43845i	− 1.15309i	−0.817065	−	0.576545i	$-0.804401\pi$
$67$	− 9.43845i	− 1.15309i	0.817065	−	0.576545i	$-0.195599\pi$
$68$	0	0
$69$	9.43845	1.13626
$70$	0	0
$71$	7.68466i	0.912001i	0.889979	+	0.456001i	$0.150719\pi$
$71$	7.68466i	0.912001i	−0.889979	+	0.456001i	$0.849281\pi$
$72$	0	0
$73$	1.31534i	0.153949i	0.997033	+	0.0769745i	$0.0245260\pi$
$73$	1.31534i	0.153949i	−0.997033	+	0.0769745i	$0.975474\pi$
$74$	0	0
$75$	−19.6847	−2.27299
$76$	0	0
$77$	6.56155	0.747758
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	2.24621i	0.246554i	0.992372	+	0.123277i	$0.0393403\pi$
$83$	2.24621i	0.246554i	−0.992372	+	0.123277i	$0.960660\pi$
$84$	0	0
$85$	− 17.8078i	− 1.93152i
$86$	0	0
$87$	−12.8078	−1.37314
$88$	0	0
$89$	9.68466i	1.02657i	0.858218	+	0.513286i	$0.171572\pi$
$89$	9.68466i	1.02657i	−0.858218	+	0.513286i	$0.828428\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 20.4924i	− 2.12497i
$94$	0	0
$95$	−9.12311	−0.936011
$96$	0	0
$97$	2.80776i	0.285085i	0.989789	+	0.142543i	$0.0455278\pi$
$97$	2.80776i	0.285085i	−0.989789	+	0.142543i	$0.954472\pi$
$98$	0	0
$99$	− 9.12311i	− 0.916907i
$100$	0	0
$101$	−5.24621	−0.522018	−0.261009	−	0.965336i	$-0.584055\pi$
$101$	−5.24621	−0.522018	−0.261009	+	0.965336i	$0.584055\pi$
$102$	0	0
$103$	−14.2462	−1.40372	−0.701860	−	0.712314i	$-0.747647\pi$
$103$	−14.2462	−1.40372	−0.701860	+	0.712314i	$0.747647\pi$
$104$	0	0
$105$	23.3693	2.28061
$106$	0	0
$107$	−3.68466	−0.356209	−0.178105	−	0.984012i	$-0.556997\pi$
$107$	−3.68466	−0.356209	−0.178105	+	0.984012i	$0.556997\pi$
$108$	0	0
$109$	8.24621i	0.789844i	0.918715	+	0.394922i	$0.129228\pi$
$109$	8.24621i	0.789844i	−0.918715	+	0.394922i	$0.870772\pi$
$110$	0	0
$111$	2.56155i	0.243132i
$112$	0	0
$113$	9.24621	0.869810	0.434905	−	0.900476i	$-0.356782\pi$
$113$	9.24621	0.869810	0.434905	+	0.900476i	$0.356782\pi$
$114$	0	0
$115$	− 13.1231i	− 1.22374i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	12.8078i	1.17409i
$120$	0	0
$121$	4.43845	0.403495
$122$	0	0
$123$	23.6847i	2.13557i
$124$	0	0
$125$	9.56155i	0.855211i
$126$	0	0
$127$	−12.8078	−1.13651	−0.568253	−	0.822854i	$-0.692380\pi$
$127$	−12.8078	−1.13651	−0.568253	+	0.822854i	$0.692380\pi$
$128$	0	0
$129$	16.8078	1.47984
$130$	0	0
$131$	18.2462	1.59418	0.797089	−	0.603861i	$-0.206372\pi$
$131$	18.2462	1.59418	0.797089	+	0.603861i	$0.206372\pi$
$132$	0	0
$133$	6.56155	0.568959
$134$	0	0
$135$	− 5.12311i	− 0.440927i
$136$	0	0
$137$	21.4924i	1.83622i	0.396324	+	0.918111i	$0.370286\pi$
$137$	21.4924i	1.83622i	−0.396324	+	0.918111i	$0.629714\pi$
$138$	0	0
$139$	19.6847	1.66963	0.834815	−	0.550530i	$-0.185574\pi$
$139$	19.6847	1.66963	0.834815	+	0.550530i	$0.185574\pi$
$140$	0	0
$141$	− 10.2462i	− 0.862887i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	17.8078i	1.47885i
$146$	0	0
$147$	1.12311	0.0926322
$148$	0	0
$149$	13.2462i	1.08517i	0.840000	+	0.542586i	$0.182555\pi$
$149$	13.2462i	1.08517i	−0.840000	+	0.542586i	$0.817445\pi$
$150$	0	0
$151$	12.4924i	1.01662i	0.861174	+	0.508309i	$0.169729\pi$
$151$	12.4924i	1.01662i	−0.861174	+	0.508309i	$0.830271\pi$
$152$	0	0
$153$	17.8078	1.43967
$154$	0	0
$155$	−28.4924	−2.28857
$156$	0	0
$157$	−17.8078	−1.42121	−0.710607	−	0.703589i	$-0.751580\pi$
$157$	−17.8078	−1.42121	−0.710607	+	0.703589i	$0.751580\pi$
$158$	0	0
$159$	11.3693	0.901645
$160$	0	0
$161$	9.43845i	0.743854i
$162$	0	0
$163$	− 5.93087i	− 0.464542i	−0.972651	−	0.232271i	$-0.925384\pi$
$163$	− 5.93087i	− 0.464542i	0.972651	−	0.232271i	$-0.0746156\pi$
$164$	0	0
$165$	−23.3693	−1.81930
$166$	0	0
$167$	5.43845i	0.420840i	0.977611	+	0.210420i	$0.0674831\pi$
$167$	5.43845i	0.420840i	−0.977611	+	0.210420i	$0.932517\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	− 9.12311i	− 0.697661i
$172$	0	0
$173$	15.9309	1.21120	0.605601	−	0.795769i	$-0.292933\pi$
$173$	15.9309	1.21120	0.605601	+	0.795769i	$0.292933\pi$
$174$	0	0
$175$	− 19.6847i	− 1.48802i
$176$	0	0
$177$	6.56155i	0.493197i
$178$	0	0
$179$	−4.31534	−0.322544	−0.161272	−	0.986910i	$-0.551560\pi$
$179$	−4.31534	−0.322544	−0.161272	+	0.986910i	$0.551560\pi$
$180$	0	0
$181$	9.80776	0.729005	0.364503	−	0.931202i	$-0.381239\pi$
$181$	9.80776	0.729005	0.364503	+	0.931202i	$0.381239\pi$
$182$	0	0
$183$	−18.5616	−1.37211
$184$	0	0
$185$	3.56155	0.261851
$186$	0	0
$187$	− 12.8078i	− 0.936596i
$188$	0	0
$189$	3.68466i	0.268019i
$190$	0	0
$191$	−4.31534	−0.312247	−0.156124	−	0.987738i	$-0.549900\pi$
$191$	−4.31534	−0.312247	−0.156124	+	0.987738i	$0.549900\pi$
$192$	0	0
$193$	− 3.00000i	− 0.215945i	−0.994154	−	0.107972i	$-0.965564\pi$
$193$	− 3.00000i	− 0.215945i	0.994154	−	0.107972i	$-0.0344358\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.31534i	0.164961i	0.996593	+	0.0824806i	$0.0262843\pi$
$197$	2.31534i	0.164961i	−0.996593	+	0.0824806i	$0.973716\pi$
$198$	0	0
$199$	−9.43845	−0.669074	−0.334537	−	0.942383i	$-0.608580\pi$
$199$	−9.43845	−0.669074	−0.334537	+	0.942383i	$0.608580\pi$
$200$	0	0
$201$	− 24.1771i	− 1.70532i
$202$	0	0
$203$	− 12.8078i	− 0.898929i
$204$	0	0
$205$	32.9309	2.29999
$206$	0	0
$207$	13.1231	0.912119
$208$	0	0
$209$	−6.56155	−0.453872
$210$	0	0
$211$	12.3153	0.847823	0.423912	−	0.905704i	$-0.360657\pi$
$211$	12.3153	0.847823	0.423912	+	0.905704i	$0.360657\pi$
$212$	0	0
$213$	19.6847i	1.34877i
$214$	0	0
$215$	− 23.3693i	− 1.59377i
$216$	0	0
$217$	20.4924	1.39112
$218$	0	0
$219$	3.36932i	0.227677i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 3.68466i	− 0.246743i	−0.992361	−	0.123371i	$-0.960629\pi$
$223$	− 3.68466i	− 0.246743i	0.992361	−	0.123371i	$-0.0393707\pi$
$224$	0	0
$225$	−27.3693	−1.82462
$226$	0	0
$227$	− 7.05398i	− 0.468189i	−0.972214	−	0.234094i	$-0.924788\pi$
$227$	− 7.05398i	− 0.468189i	0.972214	−	0.234094i	$-0.0752125\pi$
$228$	0	0
$229$	4.24621i	0.280598i	0.990109	+	0.140299i	$0.0448063\pi$
$229$	4.24621i	0.280598i	−0.990109	+	0.140299i	$0.955194\pi$
$230$	0	0
$231$	16.8078	1.10587
$232$	0	0
$233$	−4.24621	−0.278179	−0.139089	−	0.990280i	$-0.544417\pi$
$233$	−4.24621	−0.278179	−0.139089	+	0.990280i	$0.544417\pi$
$234$	0	0
$235$	−14.2462	−0.929320
$236$	0	0
$237$	−10.2462	−0.665563
$238$	0	0
$239$	1.75379i	0.113443i	0.998390	+	0.0567216i	$0.0180647\pi$
$239$	1.75379i	0.113443i	−0.998390	+	0.0567216i	$0.981935\pi$
$240$	0	0
$241$	25.4924i	1.64211i	0.570848	+	0.821056i	$0.306614\pi$
$241$	25.4924i	1.64211i	−0.570848	+	0.821056i	$0.693386\pi$
$242$	0	0
$243$	−22.2462	−1.42710
$244$	0	0
$245$	− 1.56155i	− 0.0997639i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	5.75379i	0.364632i
$250$	0	0
$251$	−29.9309	−1.88922	−0.944610	−	0.328195i	$-0.893560\pi$
$251$	−29.9309	−1.88922	−0.944610	+	0.328195i	$0.893560\pi$
$252$	0	0
$253$	− 9.43845i	− 0.593390i
$254$	0	0
$255$	− 45.6155i	− 2.85656i
$256$	0	0
$257$	11.2462	0.701519	0.350760	−	0.936466i	$-0.385923\pi$
$257$	11.2462	0.701519	0.350760	+	0.936466i	$0.385923\pi$
$258$	0	0
$259$	−2.56155	−0.159167
$260$	0	0
$261$	−17.8078	−1.10227
$262$	0	0
$263$	8.80776	0.543110	0.271555	−	0.962423i	$-0.412462\pi$
$263$	8.80776	0.543110	0.271555	+	0.962423i	$0.412462\pi$
$264$	0	0
$265$	− 15.8078i	− 0.971063i
$266$	0	0
$267$	24.8078i	1.51821i
$268$	0	0
$269$	14.8078	0.902845	0.451423	−	0.892310i	$-0.350917\pi$
$269$	14.8078	0.902845	0.451423	+	0.892310i	$0.350917\pi$
$270$	0	0
$271$	5.43845i	0.330362i	0.986263	+	0.165181i	$0.0528209\pi$
$271$	5.43845i	0.330362i	−0.986263	+	0.165181i	$0.947179\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	19.6847i	1.18703i
$276$	0	0
$277$	−11.0000	−0.660926	−0.330463	−	0.943819i	$-0.607205\pi$
$277$	−11.0000	−0.660926	−0.330463	+	0.943819i	$0.607205\pi$
$278$	0	0
$279$	− 28.4924i	− 1.70580i
$280$	0	0
$281$	− 14.6847i	− 0.876013i	−0.898972	−	0.438007i	$-0.855685\pi$
$281$	− 14.6847i	− 0.876013i	0.898972	−	0.438007i	$-0.144315\pi$
$282$	0	0
$283$	21.4384	1.27438	0.637192	−	0.770705i	$-0.280096\pi$
$283$	21.4384	1.27438	0.637192	+	0.770705i	$0.280096\pi$
$284$	0	0
$285$	−23.3693	−1.38428
$286$	0	0
$287$	−23.6847	−1.39806
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	7.19224i	0.421616i
$292$	0	0
$293$	− 11.4924i	− 0.671394i	−0.941970	−	0.335697i	$-0.891028\pi$
$293$	− 11.4924i	− 0.671394i	0.941970	−	0.335697i	$-0.108972\pi$
$294$	0	0
$295$	9.12311	0.531168
$296$	0	0
$297$	− 3.68466i	− 0.213806i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	16.8078i	0.968783i
$302$	0	0
$303$	−13.4384	−0.772019
$304$	0	0
$305$	25.8078i	1.47775i
$306$	0	0
$307$	26.2462i	1.49795i	0.662598	+	0.748975i	$0.269454\pi$
$307$	26.2462i	1.49795i	−0.662598	+	0.748975i	$0.730546\pi$
$308$	0	0
$309$	−36.4924	−2.07598
$310$	0	0
$311$	14.2462	0.807829	0.403914	−	0.914797i	$-0.367649\pi$
$311$	14.2462	0.807829	0.403914	+	0.914797i	$0.367649\pi$
$312$	0	0
$313$	−32.2462	−1.82266	−0.911332	−	0.411673i	$-0.864945\pi$
$313$	−32.2462	−1.82266	−0.911332	+	0.411673i	$0.864945\pi$
$314$	0	0
$315$	32.4924	1.83074
$316$	0	0
$317$	16.4384i	0.923275i	0.887069	+	0.461638i	$0.152738\pi$
$317$	16.4384i	0.923275i	−0.887069	+	0.461638i	$0.847262\pi$
$318$	0	0
$319$	12.8078i	0.717097i
$320$	0	0
$321$	−9.43845	−0.526803
$322$	0	0
$323$	− 12.8078i	− 0.712643i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	21.1231i	1.16811i
$328$	0	0
$329$	10.2462	0.564892
$330$	0	0
$331$	21.9309i	1.20543i	0.797957	+	0.602715i	$0.205914\pi$
$331$	21.9309i	1.20543i	−0.797957	+	0.602715i	$0.794086\pi$
$332$	0	0
$333$	3.56155i	0.195172i
$334$	0	0
$335$	−33.6155	−1.83661
$336$	0	0
$337$	−16.4384	−0.895459	−0.447730	−	0.894169i	$-0.647767\pi$
$337$	−16.4384	−0.895459	−0.447730	+	0.894169i	$0.647767\pi$
$338$	0	0
$339$	23.6847	1.28637
$340$	0	0
$341$	−20.4924	−1.10973
$342$	0	0
$343$	19.0540i	1.02882i
$344$	0	0
$345$	− 33.6155i	− 1.80980i
$346$	0	0
$347$	−35.0540	−1.88180	−0.940898	−	0.338690i	$-0.890016\pi$
$347$	−35.0540	−1.88180	−0.940898	+	0.338690i	$0.890016\pi$
$348$	0	0
$349$	− 20.5616i	− 1.10063i	−0.834956	−	0.550317i	$-0.814507\pi$
$349$	− 20.5616i	− 1.10063i	0.834956	−	0.550317i	$-0.185493\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 9.00000i	− 0.479022i	−0.970894	−	0.239511i	$-0.923013\pi$
$353$	− 9.00000i	− 0.479022i	0.970894	−	0.239511i	$-0.0769871\pi$
$354$	0	0
$355$	27.3693	1.45261
$356$	0	0
$357$	32.8078i	1.73637i
$358$	0	0
$359$	4.49242i	0.237101i	0.992948	+	0.118550i	$0.0378247\pi$
$359$	4.49242i	0.237101i	−0.992948	+	0.118550i	$0.962175\pi$
$360$	0	0
$361$	12.4384	0.654655
$362$	0	0
$363$	11.3693	0.596734
$364$	0	0
$365$	4.68466	0.245206
$366$	0	0
$367$	−32.1771	−1.67963	−0.839815	−	0.542873i	$-0.817337\pi$
$367$	−32.1771	−1.67963	−0.839815	+	0.542873i	$0.817337\pi$
$368$	0	0
$369$	32.9309i	1.71431i
$370$	0	0
$371$	11.3693i	0.590266i
$372$	0	0
$373$	−11.2462	−0.582307	−0.291153	−	0.956676i	$-0.594039\pi$
$373$	−11.2462	−0.582307	−0.291153	+	0.956676i	$0.594039\pi$
$374$	0	0
$375$	24.4924i	1.26478i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 25.9309i	− 1.33198i	−0.745961	−	0.665990i	$-0.768009\pi$
$379$	− 25.9309i	− 1.33198i	0.745961	−	0.665990i	$-0.231991\pi$
$380$	0	0
$381$	−32.8078	−1.68079
$382$	0	0
$383$	16.1771i	0.826610i	0.910593	+	0.413305i	$0.135626\pi$
$383$	16.1771i	0.826610i	−0.910593	+	0.413305i	$0.864374\pi$
$384$	0	0
$385$	− 23.3693i	− 1.19101i
$386$	0	0
$387$	23.3693	1.18793
$388$	0	0
$389$	17.3153	0.877923	0.438961	−	0.898506i	$-0.355347\pi$
$389$	17.3153	0.877923	0.438961	+	0.898506i	$0.355347\pi$
$390$	0	0
$391$	18.4233	0.931706
$392$	0	0
$393$	46.7386	2.35765
$394$	0	0
$395$	14.2462i	0.716805i
$396$	0	0
$397$	21.6847i	1.08832i	0.838981	+	0.544161i	$0.183152\pi$
$397$	21.6847i	1.08832i	−0.838981	+	0.544161i	$0.816848\pi$
$398$	0	0
$399$	16.8078	0.841441
$400$	0	0
$401$	9.49242i	0.474029i	0.971506	+	0.237014i	$0.0761689\pi$
$401$	9.49242i	0.474029i	−0.971506	+	0.237014i	$0.923831\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	24.9309i	1.23882i
$406$	0	0
$407$	2.56155	0.126971
$408$	0	0
$409$	1.24621i	0.0616212i	0.999525	+	0.0308106i	$0.00980887\pi$
$409$	1.24621i	0.0616212i	−0.999525	+	0.0308106i	$0.990191\pi$
$410$	0	0
$411$	55.0540i	2.71561i
$412$	0	0
$413$	−6.56155	−0.322873
$414$	0	0
$415$	8.00000	0.392705
$416$	0	0
$417$	50.4233	2.46924
$418$	0	0
$419$	−13.9309	−0.680568	−0.340284	−	0.940323i	$-0.610523\pi$
$419$	−13.9309	−0.680568	−0.340284	+	0.940323i	$0.610523\pi$
$420$	0	0
$421$	− 21.3153i	− 1.03885i	−0.854517	−	0.519423i	$-0.826147\pi$
$421$	− 21.3153i	− 1.03885i	0.854517	−	0.519423i	$-0.173853\pi$
$422$	0	0
$423$	− 14.2462i	− 0.692674i
$424$	0	0
$425$	−38.4233	−1.86380
$426$	0	0
$427$	− 18.5616i	− 0.898256i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2.06913i	0.0996665i	0.998758	+	0.0498333i	$0.0158690\pi$
$431$	2.06913i	0.0996665i	−0.998758	+	0.0498333i	$0.984131\pi$
$432$	0	0
$433$	1.49242	0.0717212	0.0358606	−	0.999357i	$-0.488583\pi$
$433$	1.49242	0.0717212	0.0358606	+	0.999357i	$0.488583\pi$
$434$	0	0
$435$	45.6155i	2.18710i
$436$	0	0
$437$	− 9.43845i	− 0.451502i
$438$	0	0
$439$	37.9309	1.81034	0.905171	−	0.425048i	$-0.139743\pi$
$439$	37.9309	1.81034	0.905171	+	0.425048i	$0.139743\pi$
$440$	0	0
$441$	1.56155	0.0743597
$442$	0	0
$443$	21.7538	1.03355	0.516777	−	0.856120i	$-0.327132\pi$
$443$	21.7538	1.03355	0.516777	+	0.856120i	$0.327132\pi$
$444$	0	0
$445$	34.4924	1.63510
$446$	0	0
$447$	33.9309i	1.60488i
$448$	0	0
$449$	− 13.0540i	− 0.616055i	−0.951377	−	0.308028i	$-0.900331\pi$
$449$	− 13.0540i	− 0.616055i	0.951377	−	0.308028i	$-0.0996689\pi$
$450$	0	0
$451$	23.6847	1.11527
$452$	0	0
$453$	32.0000i	1.50349i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 9.00000i	− 0.421002i	−0.977594	−	0.210501i	$-0.932490\pi$
$457$	− 9.00000i	− 0.421002i	0.977594	−	0.210501i	$-0.0675096\pi$
$458$	0	0
$459$	7.19224	0.335705
$460$	0	0
$461$	− 9.49242i	− 0.442106i	−0.975262	−	0.221053i	$-0.929051\pi$
$461$	− 9.49242i	− 0.442106i	0.975262	−	0.221053i	$-0.0709494\pi$
$462$	0	0
$463$	− 32.9848i	− 1.53294i	−0.642283	−	0.766468i	$-0.722012\pi$
$463$	− 32.9848i	− 1.53294i	0.642283	−	0.766468i	$-0.277988\pi$
$464$	0	0
$465$	−72.9848	−3.38459
$466$	0	0
$467$	−9.75379	−0.451352	−0.225676	−	0.974202i	$-0.572459\pi$
$467$	−9.75379	−0.451352	−0.225676	+	0.974202i	$0.572459\pi$
$468$	0	0
$469$	24.1771	1.11639
$470$	0	0
$471$	−45.6155	−2.10185
$472$	0	0
$473$	− 16.8078i	− 0.772822i
$474$	0	0
$475$	19.6847i	0.903194i
$476$	0	0
$477$	15.8078	0.723788
$478$	0	0
$479$	− 33.3002i	− 1.52152i	−0.649031	−	0.760762i	$-0.724825\pi$
$479$	− 33.3002i	− 1.52152i	0.649031	−	0.760762i	$-0.275175\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	24.1771i	1.10010i
$484$	0	0
$485$	10.0000	0.454077
$486$	0	0
$487$	− 38.5616i	− 1.74739i	−0.486473	−	0.873695i	$-0.661717\pi$
$487$	− 38.5616i	− 1.74739i	0.486473	−	0.873695i	$-0.338283\pi$
$488$	0	0
$489$	− 15.1922i	− 0.687017i
$490$	0	0
$491$	−35.0540	−1.58196	−0.790982	−	0.611840i	$-0.790430\pi$
$491$	−35.0540	−1.58196	−0.790982	+	0.611840i	$0.790430\pi$
$492$	0	0
$493$	−25.0000	−1.12594
$494$	0	0
$495$	−32.4924	−1.46043
$496$	0	0
$497$	−19.6847	−0.882978
$498$	0	0
$499$	16.4924i	0.738302i	0.929369	+	0.369151i	$0.120352\pi$
$499$	16.4924i	0.738302i	−0.929369	+	0.369151i	$0.879648\pi$
$500$	0	0
$501$	13.9309i	0.622385i
$502$	0	0
$503$	−20.3153	−0.905816	−0.452908	−	0.891557i	$-0.649613\pi$
$503$	−20.3153	−0.905816	−0.452908	+	0.891557i	$0.649613\pi$
$504$	0	0
$505$	18.6847i	0.831456i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 33.0000i	− 1.46270i	−0.682003	−	0.731350i	$-0.738891\pi$
$509$	− 33.0000i	− 1.46270i	0.682003	−	0.731350i	$-0.261109\pi$
$510$	0	0
$511$	−3.36932	−0.149050
$512$	0	0
$513$	− 3.68466i	− 0.162682i
$514$	0	0
$515$	50.7386i	2.23581i
$516$	0	0
$517$	−10.2462	−0.450628
$518$	0	0
$519$	40.8078	1.79126
$520$	0	0
$521$	−28.0540	−1.22907	−0.614533	−	0.788891i	$-0.710656\pi$
$521$	−28.0540	−1.22907	−0.614533	+	0.788891i	$0.710656\pi$
$522$	0	0
$523$	41.3002	1.80593	0.902966	−	0.429713i	$-0.141385\pi$
$523$	41.3002	1.80593	0.902966	+	0.429713i	$0.141385\pi$
$524$	0	0
$525$	− 50.4233i	− 2.20065i
$526$	0	0
$527$	− 40.0000i	− 1.74243i
$528$	0	0
$529$	−9.42329	−0.409708
$530$	0	0
$531$	9.12311i	0.395909i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	13.1231i	0.567361i
$536$	0	0
$537$	−11.0540	−0.477014
$538$	0	0
$539$	− 1.12311i	− 0.0483756i
$540$	0	0
$541$	− 6.68466i	− 0.287396i	−0.989622	−	0.143698i	$-0.954101\pi$
$541$	− 6.68466i	− 0.287396i	0.989622	−	0.143698i	$-0.0458994\pi$
$542$	0	0
$543$	25.1231	1.07814
$544$	0	0
$545$	29.3693	1.25804
$546$	0	0
$547$	−16.4924	−0.705165	−0.352583	−	0.935781i	$-0.614696\pi$
$547$	−16.4924	−0.705165	−0.352583	+	0.935781i	$0.614696\pi$
$548$	0	0
$549$	−25.8078	−1.10145
$550$	0	0
$551$	12.8078i	0.545629i
$552$	0	0
$553$	− 10.2462i	− 0.435713i
$554$	0	0
$555$	9.12311	0.387254
$556$	0	0
$557$	1.49242i	0.0632360i	0.999500	+	0.0316180i	$0.0100660\pi$
$557$	1.49242i	0.0632360i	−0.999500	+	0.0316180i	$0.989934\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	− 32.8078i	− 1.38514i
$562$	0	0
$563$	31.0540	1.30877	0.654385	−	0.756162i	$-0.272928\pi$
$563$	31.0540	1.30877	0.654385	+	0.756162i	$0.272928\pi$
$564$	0	0
$565$	− 32.9309i	− 1.38541i
$566$	0	0
$567$	− 17.9309i	− 0.753026i
$568$	0	0
$569$	−6.31534	−0.264753	−0.132376	−	0.991200i	$-0.542261\pi$
$569$	−6.31534	−0.264753	−0.132376	+	0.991200i	$0.542261\pi$
$570$	0	0
$571$	−24.4924	−1.02498	−0.512488	−	0.858694i	$-0.671276\pi$
$571$	−24.4924	−1.02498	−0.512488	+	0.858694i	$0.671276\pi$
$572$	0	0
$573$	−11.0540	−0.461786
$574$	0	0
$575$	−28.3153	−1.18083
$576$	0	0
$577$	− 31.5616i	− 1.31392i	−0.753923	−	0.656962i	$-0.771841\pi$
$577$	− 31.5616i	− 1.31392i	0.753923	−	0.656962i	$-0.228159\pi$
$578$	0	0
$579$	− 7.68466i	− 0.319364i
$580$	0	0
$581$	−5.75379	−0.238707
$582$	0	0
$583$	− 11.3693i	− 0.470869i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 13.9309i	− 0.574989i	−0.957782	−	0.287494i	$-0.907178\pi$
$587$	− 13.9309i	− 0.574989i	0.957782	−	0.287494i	$-0.0928222\pi$
$588$	0	0
$589$	−20.4924	−0.844376
$590$	0	0
$591$	5.93087i	0.243963i
$592$	0	0
$593$	23.5616i	0.967557i	0.875190	+	0.483779i	$0.160736\pi$
$593$	23.5616i	0.967557i	−0.875190	+	0.483779i	$0.839264\pi$
$594$	0	0
$595$	45.6155	1.87005
$596$	0	0
$597$	−24.1771	−0.989502
$598$	0	0
$599$	−22.7386	−0.929075	−0.464538	−	0.885553i	$-0.653779\pi$
$599$	−22.7386	−0.929075	−0.464538	+	0.885553i	$0.653779\pi$
$600$	0	0
$601$	−37.9848	−1.54943	−0.774717	−	0.632308i	$-0.782108\pi$
$601$	−37.9848	−1.54943	−0.774717	+	0.632308i	$0.782108\pi$
$602$	0	0
$603$	− 33.6155i	− 1.36893i
$604$	0	0
$605$	− 15.8078i	− 0.642677i
$606$	0	0
$607$	31.6847	1.28604	0.643020	−	0.765849i	$-0.277681\pi$
$607$	31.6847	1.28604	0.643020	+	0.765849i	$0.277681\pi$
$608$	0	0
$609$	− 32.8078i	− 1.32944i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 9.00000i	− 0.363507i	−0.983344	−	0.181753i	$-0.941823\pi$
$613$	− 9.00000i	− 0.363507i	0.983344	−	0.181753i	$-0.0581772\pi$
$614$	0	0
$615$	84.3542	3.40149
$616$	0	0
$617$	8.75379i	0.352414i	0.984353	+	0.176207i	$0.0563829\pi$
$617$	8.75379i	0.352414i	−0.984353	+	0.176207i	$0.943617\pi$
$618$	0	0
$619$	− 36.9848i	− 1.48655i	−0.668988	−	0.743273i	$-0.733272\pi$
$619$	− 36.9848i	− 1.48655i	0.668988	−	0.743273i	$-0.266728\pi$
$620$	0	0
$621$	5.30019	0.212689
$622$	0	0
$623$	−24.8078	−0.993902
$624$	0	0
$625$	−4.36932	−0.174773
$626$	0	0
$627$	−16.8078	−0.671237
$628$	0	0
$629$	5.00000i	0.199363i
$630$	0	0
$631$	27.0540i	1.07700i	0.842625	+	0.538501i	$0.181009\pi$
$631$	27.0540i	1.07700i	−0.842625	+	0.538501i	$0.818991\pi$
$632$	0	0
$633$	31.5464	1.25386
$634$	0	0
$635$	45.6155i	1.81020i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	27.3693i	1.08271i
$640$	0	0
$641$	−31.4924	−1.24388	−0.621938	−	0.783067i	$-0.713654\pi$
$641$	−31.4924	−1.24388	−0.621938	+	0.783067i	$0.713654\pi$
$642$	0	0
$643$	− 33.3002i	− 1.31323i	−0.754225	−	0.656616i	$-0.771987\pi$
$643$	− 33.3002i	− 1.31323i	0.754225	−	0.656616i	$-0.228013\pi$
$644$	0	0
$645$	− 59.8617i	− 2.35705i
$646$	0	0
$647$	5.93087	0.233167	0.116583	−	0.993181i	$-0.462806\pi$
$647$	5.93087	0.233167	0.116583	+	0.993181i	$0.462806\pi$
$648$	0	0
$649$	6.56155	0.257563
$650$	0	0
$651$	52.4924	2.05734
$652$	0	0
$653$	45.5464	1.78237	0.891184	−	0.453642i	$-0.149875\pi$
$653$	45.5464	1.78237	0.891184	+	0.453642i	$0.149875\pi$
$654$	0	0
$655$	− 64.9848i	− 2.53917i
$656$	0	0
$657$	4.68466i	0.182766i
$658$	0	0
$659$	17.4384	0.679305	0.339653	−	0.940551i	$-0.389690\pi$
$659$	17.4384	0.679305	0.339653	+	0.940551i	$0.389690\pi$
$660$	0	0
$661$	43.7386i	1.70124i	0.525784	+	0.850618i	$0.323772\pi$
$661$	43.7386i	1.70124i	−0.525784	+	0.850618i	$0.676228\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 23.3693i	− 0.906223i
$666$	0	0
$667$	−18.4233	−0.713353
$668$	0	0
$669$	− 9.43845i	− 0.364911i
$670$	0	0
$671$	18.5616i	0.716561i
$672$	0	0
$673$	−0.753789	−0.0290564	−0.0145282	−	0.999894i	$-0.504625\pi$
$673$	−0.753789	−0.0290564	−0.0145282	+	0.999894i	$0.504625\pi$
$674$	0	0
$675$	−11.0540	−0.425468
$676$	0	0
$677$	20.7386	0.797050	0.398525	−	0.917157i	$-0.369522\pi$
$677$	20.7386	0.797050	0.398525	+	0.917157i	$0.369522\pi$
$678$	0	0
$679$	−7.19224	−0.276013
$680$	0	0
$681$	− 18.0691i	− 0.692411i
$682$	0	0
$683$	− 19.6847i	− 0.753213i	−0.926373	−	0.376606i	$-0.877091\pi$
$683$	− 19.6847i	− 0.753213i	0.926373	−	0.376606i	$-0.122909\pi$
$684$	0	0
$685$	76.5464	2.92469
$686$	0	0
$687$	10.8769i	0.414979i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 21.4384i	− 0.815557i	−0.913081	−	0.407778i	$-0.866304\pi$
$691$	− 21.4384i	− 0.815557i	0.913081	−	0.407778i	$-0.133696\pi$
$692$	0	0
$693$	23.3693	0.887727
$694$	0	0
$695$	− 70.1080i	− 2.65935i
$696$	0	0
$697$	46.2311i	1.75113i
$698$	0	0
$699$	−10.8769	−0.411402
$700$	0	0
$701$	16.7386	0.632209	0.316105	−	0.948724i	$-0.397625\pi$
$701$	16.7386	0.632209	0.316105	+	0.948724i	$0.397625\pi$
$702$	0	0
$703$	2.56155	0.0966108
$704$	0	0
$705$	−36.4924	−1.37438
$706$	0	0
$707$	− 13.4384i	− 0.505405i
$708$	0	0
$709$	15.7386i	0.591077i	0.955331	+	0.295538i	$0.0954990\pi$
$709$	15.7386i	0.591077i	−0.955331	+	0.295538i	$0.904501\pi$
$710$	0	0
$711$	−14.2462	−0.534275
$712$	0	0
$713$	− 29.4773i	− 1.10393i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	4.49242i	0.167773i
$718$	0	0
$719$	14.0691	0.524690	0.262345	−	0.964974i	$-0.415504\pi$
$719$	14.0691	0.524690	0.262345	+	0.964974i	$0.415504\pi$
$720$	0	0
$721$	− 36.4924i	− 1.35905i
$722$	0	0
$723$	65.3002i	2.42854i
$724$	0	0
$725$	38.4233	1.42701
$726$	0	0
$727$	42.7386	1.58509	0.792544	−	0.609815i	$-0.208756\pi$
$727$	42.7386	1.58509	0.792544	+	0.609815i	$0.208756\pi$
$728$	0	0
$729$	−35.9848	−1.33277
$730$	0	0
$731$	32.8078	1.21344
$732$	0	0
$733$	− 21.8078i	− 0.805488i	−0.915313	−	0.402744i	$-0.868056\pi$
$733$	− 21.8078i	− 0.805488i	0.915313	−	0.402744i	$-0.131944\pi$
$734$	0	0
$735$	− 4.00000i	− 0.147542i
$736$	0	0
$737$	−24.1771	−0.890574
$738$	0	0
$739$	− 18.0691i	− 0.664684i	−0.943159	−	0.332342i	$-0.892161\pi$
$739$	− 18.0691i	− 0.664684i	0.943159	−	0.332342i	$-0.107839\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 37.9309i	− 1.39155i	−0.718260	−	0.695774i	$-0.755061\pi$
$743$	− 37.9309i	− 1.39155i	0.718260	−	0.695774i	$-0.244939\pi$
$744$	0	0
$745$	47.1771	1.72843
$746$	0	0
$747$	8.00000i	0.292705i
$748$	0	0
$749$	− 9.43845i	− 0.344873i
$750$	0	0
$751$	−5.30019	−0.193407	−0.0967033	−	0.995313i	$-0.530830\pi$
$751$	−5.30019	−0.193407	−0.0967033	+	0.995313i	$0.530830\pi$
$752$	0	0
$753$	−76.6695	−2.79399
$754$	0	0
$755$	44.4924	1.61925
$756$	0	0
$757$	53.5464	1.94618	0.973088	−	0.230432i	$-0.0740139\pi$
$757$	53.5464	1.94618	0.973088	+	0.230432i	$0.0740139\pi$
$758$	0	0
$759$	− 24.1771i	− 0.877572i
$760$	0	0
$761$	51.9309i	1.88249i	0.337721	+	0.941246i	$0.390344\pi$
$761$	51.9309i	1.88249i	−0.337721	+	0.941246i	$0.609656\pi$
$762$	0	0
$763$	−21.1231	−0.764708
$764$	0	0
$765$	− 63.4233i	− 2.29307i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 27.9309i	− 1.00721i	−0.863933	−	0.503606i	$-0.832006\pi$
$769$	− 27.9309i	− 1.00721i	0.863933	−	0.503606i	$-0.167994\pi$
$770$	0	0
$771$	28.8078	1.03749
$772$	0	0
$773$	54.8078i	1.97130i	0.168807	+	0.985649i	$0.446009\pi$
$773$	54.8078i	1.97130i	−0.168807	+	0.985649i	$0.553991\pi$
$774$	0	0
$775$	61.4773i	2.20833i
$776$	0	0
$777$	−6.56155	−0.235394
$778$	0	0
$779$	23.6847	0.848591
$780$	0	0
$781$	19.6847	0.704372
$782$	0	0
$783$	−7.19224	−0.257030
$784$	0	0
$785$	63.4233i	2.26367i
$786$	0	0
$787$	− 34.4233i	− 1.22706i	−0.789672	−	0.613529i	$-0.789749\pi$
$787$	− 34.4233i	− 1.22706i	0.789672	−	0.613529i	$-0.210251\pi$
$788$	0	0
$789$	22.5616	0.803213
$790$	0	0
$791$	23.6847i	0.842130i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 40.4924i	− 1.43612i
$796$	0	0
$797$	15.9309	0.564300	0.282150	−	0.959370i	$-0.408952\pi$
$797$	15.9309	0.564300	0.282150	+	0.959370i	$0.408952\pi$
$798$	0	0
$799$	− 20.0000i	− 0.707549i
$800$	0	0
$801$	34.4924i	1.21873i
$802$	0	0
$803$	3.36932	0.118901
$804$	0	0
$805$	33.6155	1.18479
$806$	0	0
$807$	37.9309	1.33523
$808$	0	0
$809$	−6.75379	−0.237451	−0.118725	−	0.992927i	$-0.537881\pi$
$809$	−6.75379	−0.237451	−0.118725	+	0.992927i	$0.537881\pi$
$810$	0	0
$811$	8.00000i	0.280918i	0.990086	+	0.140459i	$0.0448578\pi$
$811$	8.00000i	0.280918i	−0.990086	+	0.140459i	$0.955142\pi$
$812$	0	0
$813$	13.9309i	0.488577i
$814$	0	0
$815$	−21.1231	−0.739910
$816$	0	0
$817$	− 16.8078i	− 0.588029i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	6.80776i	0.237593i	0.992919	+	0.118796i	$0.0379035\pi$
$821$	6.80776i	0.237593i	−0.992919	+	0.118796i	$0.962096\pi$
$822$	0	0
$823$	−44.1771	−1.53992	−0.769958	−	0.638094i	$-0.779723\pi$
$823$	−44.1771	−1.53992	−0.769958	+	0.638094i	$0.779723\pi$
$824$	0	0
$825$	50.4233i	1.75551i
$826$	0	0
$827$	− 28.9848i	− 1.00790i	−0.863732	−	0.503951i	$-0.831879\pi$
$827$	− 28.9848i	− 1.00790i	0.863732	−	0.503951i	$-0.168121\pi$
$828$	0	0
$829$	−39.4924	−1.37163	−0.685814	−	0.727777i	$-0.740554\pi$
$829$	−39.4924	−1.37163	−0.685814	+	0.727777i	$0.740554\pi$
$830$	0	0
$831$	−28.1771	−0.977452
$832$	0	0
$833$	2.19224	0.0759565
$834$	0	0
$835$	19.3693	0.670303
$836$	0	0
$837$	− 11.5076i	− 0.397760i
$838$	0	0
$839$	− 39.6847i	− 1.37007i	−0.728512	−	0.685033i	$-0.759788\pi$
$839$	− 39.6847i	− 1.37007i	0.728512	−	0.685033i	$-0.240212\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	− 37.6155i	− 1.29555i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	11.3693i	0.390654i
$848$	0	0
$849$	54.9157	1.88470
$850$	0	0
$851$	3.68466i	0.126308i
$852$	0	0
$853$	− 13.4233i	− 0.459605i	−0.973237	−	0.229802i	$-0.926192\pi$
$853$	− 13.4233i	− 0.459605i	0.973237	−	0.229802i	$-0.0738080\pi$
$854$	0	0
$855$	−32.4924	−1.11122
$856$	0	0
$857$	52.0540	1.77813	0.889065	−	0.457781i	$-0.151356\pi$
$857$	52.0540	1.77813	0.889065	+	0.457781i	$0.151356\pi$
$858$	0	0
$859$	−5.75379	−0.196317	−0.0981584	−	0.995171i	$-0.531295\pi$
$859$	−5.75379	−0.196317	−0.0981584	+	0.995171i	$0.531295\pi$
$860$	0	0
$861$	−60.6695	−2.06761
$862$	0	0
$863$	− 38.2462i	− 1.30192i	−0.759114	−	0.650958i	$-0.774367\pi$
$863$	− 38.2462i	− 1.30192i	0.759114	−	0.650958i	$-0.225633\pi$
$864$	0	0
$865$	− 56.7386i	− 1.92917i
$866$	0	0
$867$	20.4924	0.695959
$868$	0	0
$869$	10.2462i	0.347579i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	10.0000i	0.338449i
$874$	0	0
$875$	−24.4924	−0.827995
$876$	0	0
$877$	9.73863i	0.328850i	0.986390	+	0.164425i	$0.0525769\pi$
$877$	9.73863i	0.328850i	−0.986390	+	0.164425i	$0.947423\pi$
$878$	0	0
$879$	− 29.4384i	− 0.992934i
$880$	0	0
$881$	−1.24621	−0.0419859	−0.0209930	−	0.999780i	$-0.506683\pi$
$881$	−1.24621	−0.0419859	−0.0209930	+	0.999780i	$0.506683\pi$
$882$	0	0
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	0	0
$885$	23.3693	0.785551
$886$	0	0
$887$	8.31534	0.279202	0.139601	−	0.990208i	$-0.455418\pi$
$887$	8.31534	0.279202	0.139601	+	0.990208i	$0.455418\pi$
$888$	0	0
$889$	− 32.8078i	− 1.10034i
$890$	0	0
$891$	17.9309i	0.600707i
$892$	0	0
$893$	−10.2462	−0.342876
$894$	0	0
$895$	15.3693i	0.513740i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	40.0000i	1.33407i
$900$	0	0
$901$	22.1922	0.739331
$902$	0	0
$903$	43.0540i	1.43275i
$904$	0	0
$905$	− 34.9309i	− 1.16114i
$906$	0	0
$907$	−13.3002	−0.441625	−0.220813	−	0.975316i	$-0.570871\pi$
$907$	−13.3002	−0.441625	−0.220813	+	0.975316i	$0.570871\pi$
$908$	0	0
$909$	−18.6847	−0.619731
$910$	0	0
$911$	18.7386	0.620839	0.310419	−	0.950600i	$-0.399531\pi$
$911$	18.7386	0.620839	0.310419	+	0.950600i	$0.399531\pi$
$912$	0	0
$913$	5.75379	0.190423
$914$	0	0
$915$	66.1080i	2.18546i
$916$	0	0
$917$	46.7386i	1.54345i
$918$	0	0
$919$	−11.5464	−0.380881	−0.190440	−	0.981699i	$-0.560992\pi$
$919$	−11.5464	−0.380881	−0.190440	+	0.981699i	$0.560992\pi$
$920$	0	0
$921$	67.2311i	2.21534i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 7.68466i	− 0.252670i
$926$	0	0
$927$	−50.7386	−1.66648
$928$	0	0
$929$	27.9848i	0.918153i	0.888397	+	0.459076i	$0.151820\pi$
$929$	27.9848i	0.918153i	−0.888397	+	0.459076i	$0.848180\pi$
$930$	0	0
$931$	− 1.12311i	− 0.0368083i
$932$	0	0
$933$	36.4924	1.19471
$934$	0	0
$935$	−45.6155	−1.49179
$936$	0	0
$937$	−1.31534	−0.0429703	−0.0214852	−	0.999769i	$-0.506839\pi$
$937$	−1.31534	−0.0429703	−0.0214852	+	0.999769i	$0.506839\pi$
$938$	0	0
$939$	−82.6004	−2.69556
$940$	0	0
$941$	14.0000i	0.456387i	0.973616	+	0.228193i	$0.0732819\pi$
$941$	14.0000i	0.456387i	−0.973616	+	0.228193i	$0.926718\pi$
$942$	0	0
$943$	34.0691i	1.10944i
$944$	0	0
$945$	13.1231	0.426895
$946$	0	0
$947$	46.4233i	1.50855i	0.656556	+	0.754277i	$0.272012\pi$
$947$	46.4233i	1.50855i	−0.656556	+	0.754277i	$0.727988\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	42.1080i	1.36544i
$952$	0	0
$953$	5.19224	0.168193	0.0840965	−	0.996458i	$-0.473200\pi$
$953$	5.19224	0.168193	0.0840965	+	0.996458i	$0.473200\pi$
$954$	0	0
$955$	15.3693i	0.497339i
$956$	0	0
$957$	32.8078i	1.06052i
$958$	0	0
$959$	−55.0540	−1.77779
$960$	0	0
$961$	−33.0000	−1.06452
$962$	0	0
$963$	−13.1231	−0.422886
$964$	0	0
$965$	−10.6847	−0.343951
$966$	0	0
$967$	22.2462i	0.715390i	0.933838	+	0.357695i	$0.116437\pi$
$967$	22.2462i	0.715390i	−0.933838	+	0.357695i	$0.883563\pi$
$968$	0	0
$969$	− 32.8078i	− 1.05394i
$970$	0	0
$971$	31.6847	1.01681	0.508405	−	0.861118i	$-0.330235\pi$
$971$	31.6847	1.01681	0.508405	+	0.861118i	$0.330235\pi$
$972$	0	0
$973$	50.4233i	1.61650i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 15.2462i	− 0.487770i	−0.969804	−	0.243885i	$-0.921578\pi$
$977$	− 15.2462i	− 0.487770i	0.969804	−	0.243885i	$-0.0784219\pi$
$978$	0	0
$979$	24.8078	0.792860
$980$	0	0
$981$	29.3693i	0.937690i
$982$	0	0
$983$	− 40.9848i	− 1.30721i	−0.756834	−	0.653607i	$-0.773255\pi$
$983$	− 40.9848i	− 1.30721i	0.756834	−	0.653607i	$-0.226745\pi$
$984$	0	0
$985$	8.24621	0.262746
$986$	0	0
$987$	26.2462	0.835426
$988$	0	0
$989$	24.1771	0.768786
$990$	0	0
$991$	57.3002	1.82020	0.910100	−	0.414388i	$-0.136004\pi$
$991$	57.3002	1.82020	0.910100	+	0.414388i	$0.136004\pi$
$992$	0	0
$993$	56.1771i	1.78273i
$994$	0	0
$995$	33.6155i	1.06568i
$996$	0	0
$997$	−26.7538	−0.847301	−0.423650	−	0.905826i	$-0.639252\pi$
$997$	−26.7538	−0.847301	−0.423650	+	0.905826i	$0.639252\pi$
$998$	0	0
$999$	1.43845i	0.0455105i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1352.2.f.d.337.3		4
4.3	odd	2		2704.2.f.l.337.1		4
13.2	odd	12		104.2.i.b.9.1	✓	4
13.3	even	3		1352.2.o.c.1161.1		8
13.4	even	6		1352.2.o.c.361.2		8
13.5	odd	4		1352.2.a.f.1.2		2
13.6	odd	12		104.2.i.b.81.1	yes	4
13.7	odd	12		1352.2.i.e.1329.1		4
13.8	odd	4		1352.2.a.h.1.2		2
13.9	even	3		1352.2.o.c.361.1		8
13.10	even	6		1352.2.o.c.1161.2		8
13.11	odd	12		1352.2.i.e.529.1		4
13.12	even	2	inner	1352.2.f.d.337.4		4
39.2	even	12		936.2.t.f.217.2		4
39.32	even	12		936.2.t.f.289.2		4
52.15	even	12		208.2.i.e.113.2		4
52.19	even	12		208.2.i.e.81.2		4
52.31	even	4		2704.2.a.q.1.1		2
52.47	even	4		2704.2.a.r.1.1		2
52.51	odd	2		2704.2.f.l.337.2		4
104.19	even	12		832.2.i.l.705.1		4
104.45	odd	12		832.2.i.o.705.2		4
104.67	even	12		832.2.i.l.321.1		4
104.93	odd	12		832.2.i.o.321.2		4
156.71	odd	12		1872.2.t.s.289.2		4
156.119	odd	12		1872.2.t.s.1153.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.i.b.9.1	✓	4	13.2	odd	12
104.2.i.b.81.1	yes	4	13.6	odd	12
208.2.i.e.81.2		4	52.19	even	12
208.2.i.e.113.2		4	52.15	even	12
832.2.i.l.321.1		4	104.67	even	12
832.2.i.l.705.1		4	104.19	even	12
832.2.i.o.321.2		4	104.93	odd	12
832.2.i.o.705.2		4	104.45	odd	12
936.2.t.f.217.2		4	39.2	even	12
936.2.t.f.289.2		4	39.32	even	12
1352.2.a.f.1.2		2	13.5	odd	4
1352.2.a.h.1.2		2	13.8	odd	4
1352.2.f.d.337.3		4	1.1	even	1	trivial
1352.2.f.d.337.4		4	13.12	even	2	inner
1352.2.i.e.529.1		4	13.11	odd	12
1352.2.i.e.1329.1		4	13.7	odd	12
1352.2.o.c.361.1		8	13.9	even	3
1352.2.o.c.361.2		8	13.4	even	6
1352.2.o.c.1161.1		8	13.3	even	3
1352.2.o.c.1161.2		8	13.10	even	6
1872.2.t.s.289.2		4	156.71	odd	12
1872.2.t.s.1153.2		4	156.119	odd	12
2704.2.a.q.1.1		2	52.31	even	4
2704.2.a.r.1.1		2	52.47	even	4
2704.2.f.l.337.1		4	4.3	odd	2
2704.2.f.l.337.2		4	52.51	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1352 = 2^{3} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1352.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.56155 q^{3} -3.56155i q^{5} +2.56155i q^{7} +3.56155 q^{9} -2.56155i q^{11} -9.12311i q^{15} +5.00000 q^{17} -2.56155i q^{19} +6.56155i q^{21} +3.68466 q^{23} -7.68466 q^{25} +1.43845 q^{27} -5.00000 q^{29} -8.00000i q^{31} -6.56155i q^{33} +9.12311 q^{35} +1.00000i q^{37} +9.24621i q^{41} +6.56155 q^{43} -12.6847i q^{45} -4.00000i q^{47} +0.438447 q^{49} +12.8078 q^{51} +4.43845 q^{53} -9.12311 q^{55} -6.56155i q^{57} +2.56155i q^{59} -7.24621 q^{61} +9.12311i q^{63} -9.43845i q^{67} +9.43845 q^{69} +7.68466i q^{71} +1.31534i q^{73} -19.6847 q^{75} +6.56155 q^{77} -4.00000 q^{79} -7.00000 q^{81} +2.24621i q^{83} -17.8078i q^{85} -12.8078 q^{87} +9.68466i q^{89} -20.4924i q^{93} -9.12311 q^{95} +2.80776i q^{97} -9.12311i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 6 q^{9} + 20 q^{17} - 10 q^{23} - 6 q^{25} + 14 q^{27} - 20 q^{29} + 20 q^{35} + 18 q^{43} + 10 q^{49} + 10 q^{51} + 26 q^{53} - 20 q^{55} + 4 q^{61} + 46 q^{69} - 54 q^{75} + 18 q^{77}+ \cdots - 20 q^{95}+O(q^{100}) \) 4 * q + 2 * q^3 + 6 * q^9 + 20 * q^17 - 10 * q^23 - 6 * q^25 + 14 * q^27 - 20 * q^29 + 20 * q^35 + 18 * q^43 + 10 * q^49 + 10 * q^51 + 26 * q^53 - 20 * q^55 + 4 * q^61 + 46 * q^69 - 54 * q^75 + 18 * q^77 - 16 * q^79 - 28 * q^81 - 10 * q^87 - 20 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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