LMFDB - Embedded newform 7232.2.a.bn.1.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7232,2,Mod(1,7232)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7232.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7232, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7232 = 2^{6} \cdot 113 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7232.a (trivial)

Newform invariants

comment:select newform

sage:traces = [18,0,0,0,-10,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$57.7478107418$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 43 x^{16} + 760 x^{14} - 7095 x^{12} + 37240 x^{10} - 107142 x^{8} + 149152 x^{6} - 72200 x^{4} + \cdots - 800 $ x^18 - 43x^16 + 760x^14 - 7095x^12 + 37240x^10 - 107142x^8 + 149152x^6 - 72200x^4 + 13504x^2 - 800
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{10}\cdot 5 $
Twist minimal:	no (minimal twist has level 3616)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$1.83905$ of defining polynomial
Character	$\chi$	$=$	7232.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.83905 q^{3} +3.63208 q^{5} +2.00576 q^{7} +0.382087 q^{9} -0.615471 q^{11} +6.26971 q^{13} +6.67957 q^{15} +2.75179 q^{17} -0.214330 q^{19} +3.68868 q^{21} -4.90647 q^{23} +8.19203 q^{25} -4.81446 q^{27} +5.27427 q^{29} -9.69962 q^{31} -1.13188 q^{33} +7.28507 q^{35} -0.0565910 q^{37} +11.5303 q^{39} +1.14920 q^{41} +9.36491 q^{43} +1.38777 q^{45} +0.589828 q^{47} -2.97694 q^{49} +5.06068 q^{51} +2.44243 q^{53} -2.23544 q^{55} -0.394162 q^{57} +13.9532 q^{59} -6.55332 q^{61} +0.766373 q^{63} +22.7721 q^{65} -0.763726 q^{67} -9.02321 q^{69} +2.42663 q^{71} -3.33086 q^{73} +15.0655 q^{75} -1.23448 q^{77} -9.46737 q^{79} -10.0003 q^{81} -11.7285 q^{83} +9.99475 q^{85} +9.69962 q^{87} +17.8784 q^{89} +12.5755 q^{91} -17.8380 q^{93} -0.778464 q^{95} +2.74266 q^{97} -0.235163 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 10 q^{5} + 32 q^{9} + 12 q^{13} + 16 q^{17} - 14 q^{21} + 44 q^{25} - 22 q^{29} + 24 q^{33} + 4 q^{37} + 50 q^{41} - 32 q^{45} + 58 q^{49} - 2 q^{53} + 46 q^{57} - 10 q^{61} + 40 q^{65} - 22 q^{69}+ \cdots + 48 q^{97}+O(q^{100}) $ 18 * q - 10 * q^5 + 32 * q^9 + 12 * q^13 + 16 * q^17 - 14 * q^21 + 44 * q^25 - 22 * q^29 + 24 * q^33 + 4 * q^37 + 50 * q^41 - 32 * q^45 + 58 * q^49 - 2 * q^53 + 46 * q^57 - 10 * q^61 + 40 * q^65 - 22 * q^69 + 62 * q^73 - 14 * q^77 + 46 * q^81 - 12 * q^85 + 52 * q^89 - 16 * q^93 + 48 * q^97

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
$3$	1.83905	1.06177	0.530887	−	0.847443i	$-0.321859\pi$
$3$	1.83905	1.06177	0.530887	+	0.847443i	$0.321859\pi$
$4$	0	0
$5$	3.63208	1.62432	0.812159	−	0.583437i	$-0.198292\pi$
$5$	3.63208	1.62432	0.812159	+	0.583437i	$0.198292\pi$
$6$	0	0
$7$	2.00576	0.758104	0.379052	−	0.925375i	$-0.376250\pi$
$7$	2.00576	0.758104	0.379052	+	0.925375i	$0.376250\pi$
$8$	0	0
$9$	0.382087	0.127362
$10$	0	0
$11$	−0.615471	−0.185571	−0.0927857	−	0.995686i	$-0.529577\pi$
$11$	−0.615471	−0.185571	−0.0927857	+	0.995686i	$0.529577\pi$
$12$	0	0
$13$	6.26971	1.73890	0.869452	−	0.494018i	$-0.164472\pi$
$13$	6.26971	1.73890	0.869452	+	0.494018i	$0.164472\pi$
$14$	0	0
$15$	6.67957	1.72466
$16$	0	0
$17$	2.75179	0.667408	0.333704	−	0.942678i	$-0.391701\pi$
$17$	2.75179	0.667408	0.333704	+	0.942678i	$0.391701\pi$
$18$	0	0
$19$	−0.214330	−0.0491706	−0.0245853	−	0.999698i	$-0.507827\pi$
$19$	−0.214330	−0.0491706	−0.0245853	+	0.999698i	$0.507827\pi$
$20$	0	0
$21$	3.68868	0.804935
$22$	0	0
$23$	−4.90647	−1.02307	−0.511534	−	0.859263i	$-0.670923\pi$
$23$	−4.90647	−1.02307	−0.511534	+	0.859263i	$0.670923\pi$
$24$	0	0
$25$	8.19203	1.63841
$26$	0	0
$27$	−4.81446	−0.926543
$28$	0	0
$29$	5.27427	0.979408	0.489704	−	0.871889i	$-0.337105\pi$
$29$	5.27427	0.979408	0.489704	+	0.871889i	$0.337105\pi$
$30$	0	0
$31$	−9.69962	−1.74210	−0.871052	−	0.491191i	$-0.836562\pi$
$31$	−9.69962	−1.74210	−0.871052	+	0.491191i	$0.836562\pi$
$32$	0	0
$33$	−1.13188	−0.197035
$34$	0	0
$35$	7.28507	1.23140
$36$	0	0
$37$	−0.0565910	−0.00930351	−0.00465176	−	0.999989i	$-0.501481\pi$
$37$	−0.0565910	−0.00930351	−0.00465176	+	0.999989i	$0.501481\pi$
$38$	0	0
$39$	11.5303	1.84632
$40$	0	0
$41$	1.14920	0.179476	0.0897378	−	0.995965i	$-0.471397\pi$
$41$	1.14920	0.179476	0.0897378	+	0.995965i	$0.471397\pi$
$42$	0	0
$43$	9.36491	1.42814	0.714068	−	0.700076i	$-0.246851\pi$
$43$	9.36491	1.42814	0.714068	+	0.700076i	$0.246851\pi$
$44$	0	0
$45$	1.38777	0.206877
$46$	0	0
$47$	0.589828	0.0860353	0.0430176	−	0.999074i	$-0.486303\pi$
$47$	0.589828	0.0860353	0.0430176	+	0.999074i	$0.486303\pi$
$48$	0	0
$49$	−2.97694	−0.425278
$50$	0	0
$51$	5.06068	0.708636
$52$	0	0
$53$	2.44243	0.335494	0.167747	−	0.985830i	$-0.446351\pi$
$53$	2.44243	0.335494	0.167747	+	0.985830i	$0.446351\pi$
$54$	0	0
$55$	−2.23544	−0.301427
$56$	0	0
$57$	−0.394162	−0.0522081
$58$	0	0
$59$	13.9532	1.81655	0.908277	−	0.418369i	$-0.137398\pi$
$59$	13.9532	1.81655	0.908277	+	0.418369i	$0.137398\pi$
$60$	0	0
$61$	−6.55332	−0.839066	−0.419533	−	0.907740i	$-0.637806\pi$
$61$	−6.55332	−0.839066	−0.419533	+	0.907740i	$0.637806\pi$
$62$	0	0
$63$	0.766373	0.0965539
$64$	0	0
$65$	22.7721	2.82453
$66$	0	0
$67$	−0.763726	−0.0933040	−0.0466520	−	0.998911i	$-0.514855\pi$
$67$	−0.763726	−0.0933040	−0.0466520	+	0.998911i	$0.514855\pi$
$68$	0	0
$69$	−9.02321	−1.08627
$70$	0	0
$71$	2.42663	0.287988	0.143994	−	0.989579i	$-0.454005\pi$
$71$	2.42663	0.287988	0.143994	+	0.989579i	$0.454005\pi$
$72$	0	0
$73$	−3.33086	−0.389848	−0.194924	−	0.980818i	$-0.562446\pi$
$73$	−3.33086	−0.389848	−0.194924	+	0.980818i	$0.562446\pi$
$74$	0	0
$75$	15.0655	1.73962
$76$	0	0
$77$	−1.23448	−0.140682
$78$	0	0
$79$	−9.46737	−1.06516	−0.532581	−	0.846379i	$-0.678778\pi$
$79$	−9.46737	−1.06516	−0.532581	+	0.846379i	$0.678778\pi$
$80$	0	0
$81$	−10.0003	−1.11114
$82$	0	0
$83$	−11.7285	−1.28737	−0.643683	−	0.765292i	$-0.722594\pi$
$83$	−11.7285	−1.28737	−0.643683	+	0.765292i	$0.722594\pi$
$84$	0	0
$85$	9.99475	1.08408
$86$	0	0
$87$	9.69962	1.03991
$88$	0	0
$89$	17.8784	1.89511	0.947555	−	0.319594i	$-0.103547\pi$
$89$	17.8784	1.89511	0.947555	+	0.319594i	$0.103547\pi$
$90$	0	0
$91$	12.5755	1.31827
$92$	0	0
$93$	−17.8380	−1.84972
$94$	0	0
$95$	−0.778464	−0.0798687
$96$	0	0
$97$	2.74266	0.278475	0.139238	−	0.990259i	$-0.455535\pi$
$97$	2.74266	0.278475	0.139238	+	0.990259i	$0.455535\pi$
$98$	0	0
$99$	−0.235163	−0.0236348
$100$	0	0
$101$	−13.0583	−1.29935	−0.649677	−	0.760211i	$-0.725096\pi$
$101$	−13.0583	−1.29935	−0.649677	+	0.760211i	$0.725096\pi$
$102$	0	0
$103$	−0.974823	−0.0960521	−0.0480261	−	0.998846i	$-0.515293\pi$
$103$	−0.974823	−0.0960521	−0.0480261	+	0.998846i	$0.515293\pi$
$104$	0	0
$105$	13.3976	1.30747
$106$	0	0
$107$	7.12237	0.688545	0.344273	−	0.938870i	$-0.388126\pi$
$107$	7.12237	0.688545	0.344273	+	0.938870i	$0.388126\pi$
$108$	0	0
$109$	4.91579	0.470847	0.235424	−	0.971893i	$-0.424352\pi$
$109$	4.91579	0.470847	0.235424	+	0.971893i	$0.424352\pi$
$110$	0	0
$111$	−0.104073	−0.00987822
$112$	0	0
$113$	−1.00000	−0.0940721
$113$	−1.00000	−0.0940721
$114$	0	0
$115$	−17.8207	−1.66179
$116$	0	0
$117$	2.39557	0.221471
$118$	0	0
$119$	5.51943	0.505965
$120$	0	0
$121$	−10.6212	−0.965563
$122$	0	0
$123$	2.11344	0.190562
$124$	0	0
$125$	11.5937	1.03698
$126$	0	0
$127$	12.9296	1.14731	0.573657	−	0.819096i	$-0.305524\pi$
$127$	12.9296	1.14731	0.573657	+	0.819096i	$0.305524\pi$
$128$	0	0
$129$	17.2225	1.51636
$130$	0	0
$131$	−7.12461	−0.622480	−0.311240	−	0.950331i	$-0.600744\pi$
$131$	−7.12461	−0.622480	−0.311240	+	0.950331i	$0.600744\pi$
$132$	0	0
$133$	−0.429893	−0.0372765
$134$	0	0
$135$	−17.4865	−1.50500
$136$	0	0
$137$	−5.49173	−0.469190	−0.234595	−	0.972093i	$-0.575376\pi$
$137$	−5.49173	−0.469190	−0.234595	+	0.972093i	$0.575376\pi$
$138$	0	0
$139$	−10.5690	−0.896451	−0.448226	−	0.893921i	$-0.647944\pi$
$139$	−10.5690	−0.896451	−0.448226	+	0.893921i	$0.647944\pi$
$140$	0	0
$141$	1.08472	0.0913499
$142$	0	0
$143$	−3.85882	−0.322691
$144$	0	0
$145$	19.1566	1.59087
$146$	0	0
$147$	−5.47474	−0.451549
$148$	0	0
$149$	−6.83760	−0.560158	−0.280079	−	0.959977i	$-0.590361\pi$
$149$	−6.83760	−0.560158	−0.280079	+	0.959977i	$0.590361\pi$
$150$	0	0
$151$	−17.1643	−1.39681	−0.698404	−	0.715704i	$-0.746106\pi$
$151$	−17.1643	−1.39681	−0.698404	+	0.715704i	$0.746106\pi$
$152$	0	0
$153$	1.05143	0.0850027
$154$	0	0
$155$	−35.2299	−2.82973
$156$	0	0
$157$	2.32795	0.185791	0.0928953	−	0.995676i	$-0.470388\pi$
$157$	2.32795	0.185791	0.0928953	+	0.995676i	$0.470388\pi$
$158$	0	0
$159$	4.49174	0.356219
$160$	0	0
$161$	−9.84117	−0.775593
$162$	0	0
$163$	21.5911	1.69114	0.845572	−	0.533862i	$-0.179260\pi$
$163$	21.5911	1.69114	0.845572	+	0.533862i	$0.179260\pi$
$164$	0	0
$165$	−4.11108	−0.320047
$166$	0	0
$167$	−1.25184	−0.0968702	−0.0484351	−	0.998826i	$-0.515423\pi$
$167$	−1.25184	−0.0968702	−0.0484351	+	0.998826i	$0.515423\pi$
$168$	0	0
$169$	26.3092	2.02379
$170$	0	0
$171$	−0.0818926	−0.00626249
$172$	0	0
$173$	−11.0577	−0.840699	−0.420350	−	0.907362i	$-0.638093\pi$
$173$	−11.0577	−0.840699	−0.420350	+	0.907362i	$0.638093\pi$
$174$	0	0
$175$	16.4312	1.24208
$176$	0	0
$177$	25.6606	1.92877
$178$	0	0
$179$	−7.75968	−0.579986	−0.289993	−	0.957029i	$-0.593653\pi$
$179$	−7.75968	−0.579986	−0.289993	+	0.957029i	$0.593653\pi$
$180$	0	0
$181$	9.64945	0.717238	0.358619	−	0.933484i	$-0.383248\pi$
$181$	9.64945	0.717238	0.358619	+	0.933484i	$0.383248\pi$
$182$	0	0
$183$	−12.0518	−0.890898
$184$	0	0
$185$	−0.205543	−0.0151119
$186$	0	0
$187$	−1.69365	−0.123852
$188$	0	0
$189$	−9.65663	−0.702417
$190$	0	0
$191$	−13.1689	−0.952870	−0.476435	−	0.879210i	$-0.658071\pi$
$191$	−13.1689	−0.952870	−0.476435	+	0.879210i	$0.658071\pi$
$192$	0	0
$193$	19.4942	1.40322	0.701612	−	0.712559i	$-0.252464\pi$
$193$	19.4942	1.40322	0.701612	+	0.712559i	$0.252464\pi$
$194$	0	0
$195$	41.8789	2.99901
$196$	0	0
$197$	−8.76754	−0.624662	−0.312331	−	0.949973i	$-0.601110\pi$
$197$	−8.76754	−0.624662	−0.312331	+	0.949973i	$0.601110\pi$
$198$	0	0
$199$	24.7557	1.75488	0.877441	−	0.479684i	$-0.159249\pi$
$199$	24.7557	1.75488	0.877441	+	0.479684i	$0.159249\pi$
$200$	0	0
$201$	−1.40453	−0.0990677
$202$	0	0
$203$	10.5789	0.742493
$204$	0	0
$205$	4.17401	0.291525
$206$	0	0
$207$	−1.87470	−0.130300
$208$	0	0
$209$	0.131914	0.00912466
$210$	0	0
$211$	−20.1288	−1.38572	−0.692861	−	0.721071i	$-0.743650\pi$
$211$	−20.1288	−1.38572	−0.692861	+	0.721071i	$0.743650\pi$
$212$	0	0
$213$	4.46268	0.305778
$214$	0	0
$215$	34.0141	2.31975
$216$	0	0
$217$	−19.4551	−1.32070
$218$	0	0
$219$	−6.12561	−0.413930
$220$	0	0
$221$	17.2529	1.16056
$222$	0	0
$223$	−10.6749	−0.714845	−0.357422	−	0.933943i	$-0.616344\pi$
$223$	−10.6749	−0.714845	−0.357422	+	0.933943i	$0.616344\pi$
$224$	0	0
$225$	3.13007	0.208671
$226$	0	0
$227$	−4.43422	−0.294309	−0.147155	−	0.989113i	$-0.547012\pi$
$227$	−4.43422	−0.294309	−0.147155	+	0.989113i	$0.547012\pi$
$228$	0	0
$229$	−4.75387	−0.314145	−0.157072	−	0.987587i	$-0.550206\pi$
$229$	−4.75387	−0.314145	−0.157072	+	0.987587i	$0.550206\pi$
$230$	0	0
$231$	−2.27027	−0.149373
$232$	0	0
$233$	−12.3974	−0.812178	−0.406089	−	0.913833i	$-0.633108\pi$
$233$	−12.3974	−0.812178	−0.406089	+	0.913833i	$0.633108\pi$
$234$	0	0
$235$	2.14231	0.139749
$236$	0	0
$237$	−17.4109	−1.13096
$238$	0	0
$239$	1.43869	0.0930614	0.0465307	−	0.998917i	$-0.485183\pi$
$239$	1.43869	0.0930614	0.0465307	+	0.998917i	$0.485183\pi$
$240$	0	0
$241$	27.3237	1.76008	0.880038	−	0.474904i	$-0.157517\pi$
$241$	27.3237	1.76008	0.880038	+	0.474904i	$0.157517\pi$
$242$	0	0
$243$	−3.94757	−0.253237
$244$	0	0
$245$	−10.8125	−0.690786
$246$	0	0
$247$	−1.34379	−0.0855030
$248$	0	0
$249$	−21.5692	−1.36689
$250$	0	0
$251$	−7.89513	−0.498336	−0.249168	−	0.968460i	$-0.580157\pi$
$251$	−7.89513	−0.498336	−0.249168	+	0.968460i	$0.580157\pi$
$252$	0	0
$253$	3.01979	0.189852
$254$	0	0
$255$	18.3808	1.15105
$256$	0	0
$257$	20.1250	1.25536	0.627682	−	0.778470i	$-0.284004\pi$
$257$	20.1250	1.25536	0.627682	+	0.778470i	$0.284004\pi$
$258$	0	0
$259$	−0.113508	−0.00705303
$260$	0	0
$261$	2.01523	0.124740
$262$	0	0
$263$	6.34551	0.391281	0.195641	−	0.980676i	$-0.437321\pi$
$263$	6.34551	0.391281	0.195641	+	0.980676i	$0.437321\pi$
$264$	0	0
$265$	8.87112	0.544949
$266$	0	0
$267$	32.8792	2.01218
$268$	0	0
$269$	−29.7248	−1.81235	−0.906175	−	0.422902i	$-0.861011\pi$
$269$	−29.7248	−1.81235	−0.906175	+	0.422902i	$0.861011\pi$
$270$	0	0
$271$	19.8818	1.20773	0.603867	−	0.797085i	$-0.293626\pi$
$271$	19.8818	1.20773	0.603867	+	0.797085i	$0.293626\pi$
$272$	0	0
$273$	23.1269	1.39970
$274$	0	0
$275$	−5.04196	−0.304041
$276$	0	0
$277$	−17.0909	−1.02689	−0.513446	−	0.858122i	$-0.671631\pi$
$277$	−17.0909	−1.02689	−0.513446	+	0.858122i	$0.671631\pi$
$278$	0	0
$279$	−3.70610	−0.221878
$280$	0	0
$281$	23.7803	1.41862	0.709308	−	0.704899i	$-0.249008\pi$
$281$	23.7803	1.41862	0.709308	+	0.704899i	$0.249008\pi$
$282$	0	0
$283$	18.0362	1.07214	0.536070	−	0.844173i	$-0.319908\pi$
$283$	18.0362	1.07214	0.536070	+	0.844173i	$0.319908\pi$
$284$	0	0
$285$	−1.43163	−0.0848025
$286$	0	0
$287$	2.30502	0.136061
$288$	0	0
$289$	−9.42762	−0.554566
$290$	0	0
$291$	5.04388	0.295678
$292$	0	0
$293$	−0.00534046	−0.000311993 0	−0.000155997	−	1.00000i	$-0.500050\pi$
$293$	−0.00534046	−0.000311993 0	−0.000155997		1.00000i	$0.500050\pi$
$294$	0	0
$295$	50.6793	2.95066
$296$	0	0
$297$	2.96316	0.171940
$298$	0	0
$299$	−30.7621	−1.77902
$300$	0	0
$301$	18.7837	1.08268
$302$	0	0
$303$	−24.0149	−1.37962
$304$	0	0
$305$	−23.8022	−1.36291
$306$	0	0
$307$	−28.9922	−1.65467	−0.827335	−	0.561708i	$-0.810144\pi$
$307$	−28.9922	−1.65467	−0.827335	+	0.561708i	$0.810144\pi$
$308$	0	0
$309$	−1.79274	−0.101986
$310$	0	0
$311$	−9.06808	−0.514203	−0.257102	−	0.966384i	$-0.582768\pi$
$311$	−9.06808	−0.514203	−0.257102	+	0.966384i	$0.582768\pi$
$312$	0	0
$313$	−34.3485	−1.94149	−0.970746	−	0.240110i	$-0.922816\pi$
$313$	−34.3485	−1.94149	−0.970746	+	0.240110i	$0.922816\pi$
$314$	0	0
$315$	2.78353	0.156834
$316$	0	0
$317$	16.6587	0.935647	0.467824	−	0.883822i	$-0.345038\pi$
$317$	16.6587	0.935647	0.467824	+	0.883822i	$0.345038\pi$
$318$	0	0
$319$	−3.24616	−0.181750
$320$	0	0
$321$	13.0984	0.731079
$322$	0	0
$323$	−0.589792	−0.0328169
$324$	0	0
$325$	51.3616	2.84903
$326$	0	0
$327$	9.04036	0.499933
$328$	0	0
$329$	1.18305	0.0652237
$330$	0	0
$331$	3.16595	0.174016	0.0870082	−	0.996208i	$-0.472269\pi$
$331$	3.16595	0.174016	0.0870082	+	0.996208i	$0.472269\pi$
$332$	0	0
$333$	−0.0216227	−0.00118492
$334$	0	0
$335$	−2.77392	−0.151555
$336$	0	0
$337$	−15.1682	−0.826263	−0.413131	−	0.910671i	$-0.635565\pi$
$337$	−15.1682	−0.826263	−0.413131	+	0.910671i	$0.635565\pi$
$338$	0	0
$339$	−1.83905	−0.0998832
$340$	0	0
$341$	5.96983	0.323285
$342$	0	0
$343$	−20.0113	−1.08051
$344$	0	0
$345$	−32.7731	−1.76444
$346$	0	0
$347$	−24.1063	−1.29410	−0.647048	−	0.762450i	$-0.723996\pi$
$347$	−24.1063	−1.29410	−0.647048	+	0.762450i	$0.723996\pi$
$348$	0	0
$349$	−14.6137	−0.782253	−0.391126	−	0.920337i	$-0.627914\pi$
$349$	−14.6137	−0.782253	−0.391126	+	0.920337i	$0.627914\pi$
$350$	0	0
$351$	−30.1853	−1.61117
$352$	0	0
$353$	2.89176	0.153913	0.0769564	−	0.997034i	$-0.475480\pi$
$353$	2.89176	0.153913	0.0769564	+	0.997034i	$0.475480\pi$
$354$	0	0
$355$	8.81372	0.467784
$356$	0	0
$357$	10.1505	0.537220
$358$	0	0
$359$	−1.49402	−0.0788513	−0.0394257	−	0.999223i	$-0.512553\pi$
$359$	−1.49402	−0.0788513	−0.0394257	+	0.999223i	$0.512553\pi$
$360$	0	0
$361$	−18.9541	−0.997582
$362$	0	0
$363$	−19.5329	−1.02521
$364$	0	0
$365$	−12.0980	−0.633237
$366$	0	0
$367$	6.10176	0.318509	0.159255	−	0.987238i	$-0.449091\pi$
$367$	6.10176	0.318509	0.159255	+	0.987238i	$0.449091\pi$
$368$	0	0
$369$	0.439096	0.0228584
$370$	0	0
$371$	4.89892	0.254339
$372$	0	0
$373$	21.2606	1.10083	0.550417	−	0.834890i	$-0.314469\pi$
$373$	21.2606	1.10083	0.550417	+	0.834890i	$0.314469\pi$
$374$	0	0
$375$	21.3214	1.10103
$376$	0	0
$377$	33.0681	1.70310
$378$	0	0
$379$	7.92412	0.407035	0.203517	−	0.979071i	$-0.434763\pi$
$379$	7.92412	0.407035	0.203517	+	0.979071i	$0.434763\pi$
$380$	0	0
$381$	23.7781	1.21819
$382$	0	0
$383$	−10.8179	−0.552768	−0.276384	−	0.961047i	$-0.589136\pi$
$383$	−10.8179	−0.552768	−0.276384	+	0.961047i	$0.589136\pi$
$384$	0	0
$385$	−4.48375	−0.228513
$386$	0	0
$387$	3.57821	0.181891
$388$	0	0
$389$	1.71313	0.0868593	0.0434296	−	0.999056i	$-0.486172\pi$
$389$	1.71313	0.0868593	0.0434296	+	0.999056i	$0.486172\pi$
$390$	0	0
$391$	−13.5016	−0.682805
$392$	0	0
$393$	−13.1025	−0.660933
$394$	0	0
$395$	−34.3863	−1.73016
$396$	0	0
$397$	10.6575	0.534884	0.267442	−	0.963574i	$-0.413822\pi$
$397$	10.6575	0.534884	0.267442	+	0.963574i	$0.413822\pi$
$398$	0	0
$399$	−0.790593	−0.0395792
$400$	0	0
$401$	39.2440	1.95975	0.979875	−	0.199611i	$-0.0639680\pi$
$401$	39.2440	1.95975	0.979875	+	0.199611i	$0.0639680\pi$
$402$	0	0
$403$	−60.8138	−3.02935
$404$	0	0
$405$	−36.3218	−1.80485
$406$	0	0
$407$	0.0348301	0.00172647
$408$	0	0
$409$	−24.5923	−1.21601	−0.608006	−	0.793933i	$-0.708030\pi$
$409$	−24.5923	−1.21601	−0.608006	+	0.793933i	$0.708030\pi$
$410$	0	0
$411$	−10.0995	−0.498174
$412$	0	0
$413$	27.9868	1.37714
$414$	0	0
$415$	−42.5987	−2.09109
$416$	0	0
$417$	−19.4369	−0.951828
$418$	0	0
$419$	8.68346	0.424215	0.212107	−	0.977246i	$-0.431967\pi$
$419$	8.68346	0.424215	0.212107	+	0.977246i	$0.431967\pi$
$420$	0	0
$421$	−5.93496	−0.289252	−0.144626	−	0.989486i	$-0.546198\pi$
$421$	−5.93496	−0.289252	−0.144626	+	0.989486i	$0.546198\pi$
$422$	0	0
$423$	0.225366	0.0109577
$424$	0	0
$425$	22.5428	1.09349
$426$	0	0
$427$	−13.1444	−0.636100
$428$	0	0
$429$	−7.09654	−0.342624
$430$	0	0
$431$	−21.8054	−1.05033	−0.525165	−	0.851000i	$-0.675997\pi$
$431$	−21.8054	−1.05033	−0.525165	+	0.851000i	$0.675997\pi$
$432$	0	0
$433$	13.6960	0.658190	0.329095	−	0.944297i	$-0.393256\pi$
$433$	13.6960	0.658190	0.329095	+	0.944297i	$0.393256\pi$
$434$	0	0
$435$	35.2299	1.68914
$436$	0	0
$437$	1.05160	0.0503049
$438$	0	0
$439$	25.5350	1.21872	0.609358	−	0.792895i	$-0.291427\pi$
$439$	25.5350	1.21872	0.609358	+	0.792895i	$0.291427\pi$
$440$	0	0
$441$	−1.13745	−0.0541644
$442$	0	0
$443$	−24.5003	−1.16405	−0.582023	−	0.813172i	$-0.697739\pi$
$443$	−24.5003	−1.16405	−0.582023	+	0.813172i	$0.697739\pi$
$444$	0	0
$445$	64.9359	3.07826
$446$	0	0
$447$	−12.5747	−0.594761
$448$	0	0
$449$	−5.95389	−0.280981	−0.140491	−	0.990082i	$-0.544868\pi$
$449$	−5.95389	−0.280981	−0.140491	+	0.990082i	$0.544868\pi$
$450$	0	0
$451$	−0.707302	−0.0333055
$452$	0	0
$453$	−31.5659	−1.48309
$454$	0	0
$455$	45.6753	2.14129
$456$	0	0
$457$	−10.7028	−0.500657	−0.250328	−	0.968161i	$-0.580539\pi$
$457$	−10.7028	−0.500657	−0.250328	+	0.968161i	$0.580539\pi$
$458$	0	0
$459$	−13.2484	−0.618383
$460$	0	0
$461$	11.9029	0.554374	0.277187	−	0.960816i	$-0.410598\pi$
$461$	11.9029	0.554374	0.277187	+	0.960816i	$0.410598\pi$
$462$	0	0
$463$	−34.3933	−1.59839	−0.799195	−	0.601072i	$-0.794741\pi$
$463$	−34.3933	−1.59839	−0.799195	+	0.601072i	$0.794741\pi$
$464$	0	0
$465$	−64.7893	−3.00453
$466$	0	0
$467$	33.1760	1.53520	0.767601	−	0.640928i	$-0.221450\pi$
$467$	33.1760	1.53520	0.767601	+	0.640928i	$0.221450\pi$
$468$	0	0
$469$	−1.53185	−0.0707342
$470$	0	0
$471$	4.28120	0.197267
$472$	0	0
$473$	−5.76383	−0.265021
$474$	0	0
$475$	−1.75580	−0.0805615
$476$	0	0
$477$	0.933222	0.0427293
$478$	0	0
$479$	−39.2357	−1.79273	−0.896363	−	0.443321i	$-0.853800\pi$
$479$	−39.2357	−1.79273	−0.896363	+	0.443321i	$0.853800\pi$
$480$	0	0
$481$	−0.354809	−0.0161779
$482$	0	0
$483$	−18.0984	−0.823504
$484$	0	0
$485$	9.96158	0.452332
$486$	0	0
$487$	−38.7135	−1.75428	−0.877139	−	0.480237i	$-0.840551\pi$
$487$	−38.7135	−1.75428	−0.877139	+	0.480237i	$0.840551\pi$
$488$	0	0
$489$	39.7070	1.79561
$490$	0	0
$491$	23.2910	1.05111	0.525553	−	0.850761i	$-0.323858\pi$
$491$	23.2910	1.05111	0.525553	+	0.850761i	$0.323858\pi$
$492$	0	0
$493$	14.5137	0.653665
$494$	0	0
$495$	−0.854133	−0.0383904
$496$	0	0
$497$	4.86723	0.218325
$498$	0	0
$499$	15.3428	0.686839	0.343419	−	0.939182i	$-0.388415\pi$
$499$	15.3428	0.686839	0.343419	+	0.939182i	$0.388415\pi$
$500$	0	0
$501$	−2.30219	−0.102854
$502$	0	0
$503$	−9.78313	−0.436208	−0.218104	−	0.975926i	$-0.569987\pi$
$503$	−9.78313	−0.436208	−0.218104	+	0.975926i	$0.569987\pi$
$504$	0	0
$505$	−47.4290	−2.11056
$506$	0	0
$507$	48.3838	2.14880
$508$	0	0
$509$	−17.2667	−0.765333	−0.382666	−	0.923887i	$-0.624994\pi$
$509$	−17.2667	−0.765333	−0.382666	+	0.923887i	$0.624994\pi$
$510$	0	0
$511$	−6.68090	−0.295546
$512$	0	0
$513$	1.03188	0.0455587
$514$	0	0
$515$	−3.54064	−0.156019
$516$	0	0
$517$	−0.363022	−0.0159657
$518$	0	0
$519$	−20.3356	−0.892632
$520$	0	0
$521$	22.0112	0.964329	0.482165	−	0.876081i	$-0.339851\pi$
$521$	22.0112	0.964329	0.482165	+	0.876081i	$0.339851\pi$
$522$	0	0
$523$	43.8921	1.91927	0.959633	−	0.281256i	$-0.0907509\pi$
$523$	43.8921	1.91927	0.959633	+	0.281256i	$0.0907509\pi$
$524$	0	0
$525$	30.2178	1.31881
$526$	0	0
$527$	−26.6914	−1.16269
$528$	0	0
$529$	1.07341	0.0466698
$530$	0	0
$531$	5.33134	0.231361
$532$	0	0
$533$	7.20518	0.312091
$534$	0	0
$535$	25.8690	1.11842
$536$	0	0
$537$	−14.2704	−0.615813
$538$	0	0
$539$	1.83222	0.0789194
$540$	0	0
$541$	−9.76862	−0.419986	−0.209993	−	0.977703i	$-0.567344\pi$
$541$	−9.76862	−0.419986	−0.209993	+	0.977703i	$0.567344\pi$
$542$	0	0
$543$	17.7458	0.761544
$544$	0	0
$545$	17.8546	0.764805
$546$	0	0
$547$	−16.2117	−0.693162	−0.346581	−	0.938020i	$-0.612657\pi$
$547$	−16.2117	−0.693162	−0.346581	+	0.938020i	$0.612657\pi$
$548$	0	0
$549$	−2.50394	−0.106865
$550$	0	0
$551$	−1.13043	−0.0481581
$552$	0	0
$553$	−18.9892	−0.807504
$554$	0	0
$555$	−0.378004	−0.0160454
$556$	0	0
$557$	13.7696	0.583436	0.291718	−	0.956504i	$-0.405773\pi$
$557$	13.7696	0.583436	0.291718	+	0.956504i	$0.405773\pi$
$558$	0	0
$559$	58.7153	2.48339
$560$	0	0
$561$	−3.11470	−0.131503
$562$	0	0
$563$	−23.2868	−0.981421	−0.490711	−	0.871323i	$-0.663263\pi$
$563$	−23.2868	−0.981421	−0.490711	+	0.871323i	$0.663263\pi$
$564$	0	0
$565$	−3.63208	−0.152803
$566$	0	0
$567$	−20.0581	−0.842361
$568$	0	0
$569$	14.8249	0.621494	0.310747	−	0.950493i	$-0.399421\pi$
$569$	14.8249	0.621494	0.310747	+	0.950493i	$0.399421\pi$
$570$	0	0
$571$	15.8111	0.661674	0.330837	−	0.943688i	$-0.392669\pi$
$571$	15.8111	0.661674	0.330837	+	0.943688i	$0.392669\pi$
$572$	0	0
$573$	−24.2183	−1.01173
$574$	0	0
$575$	−40.1939	−1.67620
$576$	0	0
$577$	38.3816	1.59785	0.798923	−	0.601433i	$-0.205403\pi$
$577$	38.3816	1.59785	0.798923	+	0.601433i	$0.205403\pi$
$578$	0	0
$579$	35.8507	1.48991
$580$	0	0
$581$	−23.5244	−0.975957
$582$	0	0
$583$	−1.50325	−0.0622581
$584$	0	0
$585$	8.70092	0.359739
$586$	0	0
$587$	26.7495	1.10407	0.552036	−	0.833820i	$-0.313851\pi$
$587$	26.7495	1.10407	0.552036	+	0.833820i	$0.313851\pi$
$588$	0	0
$589$	2.07892	0.0856604
$590$	0	0
$591$	−16.1239	−0.663249
$592$	0	0
$593$	26.8844	1.10401	0.552005	−	0.833841i	$-0.313863\pi$
$593$	26.8844	1.10401	0.552005	+	0.833841i	$0.313863\pi$
$594$	0	0
$595$	20.0470	0.821848
$596$	0	0
$597$	45.5268	1.86329
$598$	0	0
$599$	0.374571	0.0153046	0.00765228	−	0.999971i	$-0.497564\pi$
$599$	0.374571	0.0153046	0.00765228	+	0.999971i	$0.497564\pi$
$600$	0	0
$601$	−25.2617	−1.03045	−0.515223	−	0.857056i	$-0.672291\pi$
$601$	−25.2617	−1.03045	−0.515223	+	0.857056i	$0.672291\pi$
$602$	0	0
$603$	−0.291810	−0.0118834
$604$	0	0
$605$	−38.5771	−1.56838
$606$	0	0
$607$	35.1534	1.42683	0.713417	−	0.700740i	$-0.247147\pi$
$607$	35.1534	1.42683	0.713417	+	0.700740i	$0.247147\pi$
$608$	0	0
$609$	19.4551	0.788359
$610$	0	0
$611$	3.69805	0.149607
$612$	0	0
$613$	2.42638	0.0980006	0.0490003	−	0.998799i	$-0.484396\pi$
$613$	2.42638	0.0980006	0.0490003	+	0.998799i	$0.484396\pi$
$614$	0	0
$615$	7.67619	0.309534
$616$	0	0
$617$	9.50523	0.382666	0.191333	−	0.981525i	$-0.438719\pi$
$617$	9.50523	0.382666	0.191333	+	0.981525i	$0.438719\pi$
$618$	0	0
$619$	28.1754	1.13247	0.566233	−	0.824245i	$-0.308400\pi$
$619$	28.1754	1.13247	0.566233	+	0.824245i	$0.308400\pi$
$620$	0	0
$621$	23.6220	0.947918
$622$	0	0
$623$	35.8597	1.43669
$624$	0	0
$625$	1.14925	0.0459700
$626$	0	0
$627$	0.242595	0.00968832
$628$	0	0
$629$	−0.155727	−0.00620924
$630$	0	0
$631$	13.6784	0.544530	0.272265	−	0.962222i	$-0.412227\pi$
$631$	13.6784	0.544530	0.272265	+	0.962222i	$0.412227\pi$
$632$	0	0
$633$	−37.0177	−1.47132
$634$	0	0
$635$	46.9613	1.86360
$636$	0	0
$637$	−18.6646	−0.739517
$638$	0	0
$639$	0.927184	0.0366788
$640$	0	0
$641$	29.1539	1.15151	0.575754	−	0.817623i	$-0.304708\pi$
$641$	29.1539	1.15151	0.575754	+	0.817623i	$0.304708\pi$
$642$	0	0
$643$	−47.0346	−1.85486	−0.927432	−	0.373993i	$-0.877988\pi$
$643$	−47.0346	−1.85486	−0.927432	+	0.373993i	$0.877988\pi$
$644$	0	0
$645$	62.5536	2.46304
$646$	0	0
$647$	−40.1235	−1.57742	−0.788710	−	0.614766i	$-0.789251\pi$
$647$	−40.1235	−1.57742	−0.788710	+	0.614766i	$0.789251\pi$
$648$	0	0
$649$	−8.58780	−0.337101
$650$	0	0
$651$	−35.7788	−1.40228
$652$	0	0
$653$	−34.7319	−1.35917	−0.679583	−	0.733599i	$-0.737839\pi$
$653$	−34.7319	−1.35917	−0.679583	+	0.733599i	$0.737839\pi$
$654$	0	0
$655$	−25.8772	−1.01111
$656$	0	0
$657$	−1.27268	−0.0496520
$658$	0	0
$659$	−9.29415	−0.362049	−0.181024	−	0.983479i	$-0.557941\pi$
$659$	−9.29415	−0.362049	−0.181024	+	0.983479i	$0.557941\pi$
$660$	0	0
$661$	−3.07255	−0.119508	−0.0597542	−	0.998213i	$-0.519032\pi$
$661$	−3.07255	−0.119508	−0.0597542	+	0.998213i	$0.519032\pi$
$662$	0	0
$663$	31.7289	1.23225
$664$	0	0
$665$	−1.56141	−0.0605488
$666$	0	0
$667$	−25.8780	−1.00200
$668$	0	0
$669$	−19.6316	−0.759003
$670$	0	0
$671$	4.03337	0.155707
$672$	0	0
$673$	16.7063	0.643981	0.321991	−	0.946743i	$-0.395648\pi$
$673$	16.7063	0.643981	0.321991	+	0.946743i	$0.395648\pi$
$674$	0	0
$675$	−39.4402	−1.51805
$676$	0	0
$677$	−13.6036	−0.522829	−0.261415	−	0.965227i	$-0.584189\pi$
$677$	−13.6036	−0.522829	−0.261415	+	0.965227i	$0.584189\pi$
$678$	0	0
$679$	5.50111	0.211113
$680$	0	0
$681$	−8.15473	−0.312490
$682$	0	0
$683$	−23.1106	−0.884302	−0.442151	−	0.896941i	$-0.645785\pi$
$683$	−23.1106	−0.884302	−0.442151	+	0.896941i	$0.645785\pi$
$684$	0	0
$685$	−19.9464	−0.762114
$686$	0	0
$687$	−8.74259	−0.333551
$688$	0	0
$689$	15.3133	0.583392
$690$	0	0
$691$	16.2173	0.616933	0.308467	−	0.951235i	$-0.400184\pi$
$691$	16.2173	0.616933	0.308467	+	0.951235i	$0.400184\pi$
$692$	0	0
$693$	−0.471680	−0.0179176
$694$	0	0
$695$	−38.3875	−1.45612
$696$	0	0
$697$	3.16238	0.119784
$698$	0	0
$699$	−22.7993	−0.862349
$700$	0	0
$701$	−48.1277	−1.81776	−0.908878	−	0.417061i	$-0.863060\pi$
$701$	−48.1277	−1.81776	−0.908878	+	0.417061i	$0.863060\pi$
$702$	0	0
$703$	0.0121291	0.000457460 0
$704$	0	0
$705$	3.93980	0.148381
$706$	0	0
$707$	−26.1918	−0.985046
$708$	0	0
$709$	19.6844	0.739262	0.369631	−	0.929179i	$-0.379484\pi$
$709$	19.6844	0.739262	0.369631	+	0.929179i	$0.379484\pi$
$710$	0	0
$711$	−3.61736	−0.135662
$712$	0	0
$713$	47.5909	1.78229
$714$	0	0
$715$	−14.0156	−0.524152
$716$	0	0
$717$	2.64582	0.0988101
$718$	0	0
$719$	16.7596	0.625026	0.312513	−	0.949913i	$-0.398829\pi$
$719$	16.7596	0.625026	0.312513	+	0.949913i	$0.398829\pi$
$720$	0	0
$721$	−1.95526	−0.0728175
$722$	0	0
$723$	50.2496	1.86880
$724$	0	0
$725$	43.2070	1.60467
$726$	0	0
$727$	−9.86131	−0.365736	−0.182868	−	0.983138i	$-0.558538\pi$
$727$	−9.86131	−0.365736	−0.182868	+	0.983138i	$0.558538\pi$
$728$	0	0
$729$	22.7411	0.842261
$730$	0	0
$731$	25.7703	0.953150
$732$	0	0
$733$	38.7802	1.43238	0.716190	−	0.697906i	$-0.245885\pi$
$733$	38.7802	1.43238	0.716190	+	0.697906i	$0.245885\pi$
$734$	0	0
$735$	−19.8847	−0.733458
$736$	0	0
$737$	0.470051	0.0173146
$738$	0	0
$739$	−35.9799	−1.32354	−0.661772	−	0.749705i	$-0.730195\pi$
$739$	−35.9799	−1.32354	−0.661772	+	0.749705i	$0.730195\pi$
$740$	0	0
$741$	−2.47128	−0.0907848
$742$	0	0
$743$	45.7849	1.67968	0.839842	−	0.542830i	$-0.182647\pi$
$743$	45.7849	1.67968	0.839842	+	0.542830i	$0.182647\pi$
$744$	0	0
$745$	−24.8347	−0.909874
$746$	0	0
$747$	−4.48129	−0.163962
$748$	0	0
$749$	14.2857	0.521989
$750$	0	0
$751$	46.8047	1.70793	0.853963	−	0.520333i	$-0.174192\pi$
$751$	46.8047	1.70793	0.853963	+	0.520333i	$0.174192\pi$
$752$	0	0
$753$	−14.5195	−0.529120
$754$	0	0
$755$	−62.3420	−2.26886
$756$	0	0
$757$	24.2150	0.880109	0.440055	−	0.897971i	$-0.354959\pi$
$757$	24.2150	0.880109	0.440055	+	0.897971i	$0.354959\pi$
$758$	0	0
$759$	5.55352	0.201580
$760$	0	0
$761$	21.5962	0.782862	0.391431	−	0.920207i	$-0.371980\pi$
$761$	21.5962	0.782862	0.391431	+	0.920207i	$0.371980\pi$
$762$	0	0
$763$	9.85987	0.356951
$764$	0	0
$765$	3.81886	0.138071
$766$	0	0
$767$	87.4826	3.15881
$768$	0	0
$769$	24.9845	0.900963	0.450482	−	0.892786i	$-0.351252\pi$
$769$	24.9845	0.900963	0.450482	+	0.892786i	$0.351252\pi$
$770$	0	0
$771$	37.0108	1.33291
$772$	0	0
$773$	25.7255	0.925280	0.462640	−	0.886546i	$-0.346902\pi$
$773$	25.7255	0.925280	0.462640	+	0.886546i	$0.346902\pi$
$774$	0	0
$775$	−79.4597	−2.85428
$776$	0	0
$777$	−0.208746	−0.00748872
$778$	0	0
$779$	−0.246309	−0.00882493
$780$	0	0
$781$	−1.49352	−0.0534423
$782$	0	0
$783$	−25.3928	−0.907464
$784$	0	0
$785$	8.45531	0.301783
$786$	0	0
$787$	−16.4695	−0.587074	−0.293537	−	0.955948i	$-0.594832\pi$
$787$	−16.4695	−0.587074	−0.293537	+	0.955948i	$0.594832\pi$
$788$	0	0
$789$	11.6697	0.415452
$790$	0	0
$791$	−2.00576	−0.0713165
$792$	0	0
$793$	−41.0874	−1.45906
$794$	0	0
$795$	16.3144	0.578612
$796$	0	0
$797$	−28.7592	−1.01870	−0.509351	−	0.860559i	$-0.670115\pi$
$797$	−28.7592	−1.01870	−0.509351	+	0.860559i	$0.670115\pi$
$798$	0	0
$799$	1.62309	0.0574207
$800$	0	0
$801$	6.83111	0.241366
$802$	0	0
$803$	2.05005	0.0723446
$804$	0	0
$805$	−35.7440	−1.25981
$806$	0	0
$807$	−54.6652	−1.92431
$808$	0	0
$809$	−32.9165	−1.15728	−0.578641	−	0.815582i	$-0.696417\pi$
$809$	−32.9165	−1.15728	−0.578641	+	0.815582i	$0.696417\pi$
$810$	0	0
$811$	52.8067	1.85430	0.927148	−	0.374696i	$-0.122253\pi$
$811$	52.8067	1.85430	0.927148	+	0.374696i	$0.122253\pi$
$812$	0	0
$813$	36.5635	1.28234
$814$	0	0
$815$	78.4206	2.74695
$816$	0	0
$817$	−2.00718	−0.0702223
$818$	0	0
$819$	4.80493	0.167898
$820$	0	0
$821$	26.9972	0.942210	0.471105	−	0.882077i	$-0.343855\pi$
$821$	26.9972	0.942210	0.471105	+	0.882077i	$0.343855\pi$
$822$	0	0
$823$	−52.5234	−1.83085	−0.915425	−	0.402488i	$-0.868146\pi$
$823$	−52.5234	−1.83085	−0.915425	+	0.402488i	$0.868146\pi$
$824$	0	0
$825$	−9.27239	−0.322823
$826$	0	0
$827$	9.30750	0.323653	0.161827	−	0.986819i	$-0.448261\pi$
$827$	9.30750	0.323653	0.161827	+	0.986819i	$0.448261\pi$
$828$	0	0
$829$	28.0309	0.973554	0.486777	−	0.873526i	$-0.338172\pi$
$829$	28.0309	0.973554	0.486777	+	0.873526i	$0.338172\pi$
$830$	0	0
$831$	−31.4309	−1.09033
$832$	0	0
$833$	−8.19194	−0.283834
$834$	0	0
$835$	−4.54678	−0.157348
$836$	0	0
$837$	46.6985	1.61413
$838$	0	0
$839$	17.1213	0.591092	0.295546	−	0.955328i	$-0.404498\pi$
$839$	17.1213	0.591092	0.295546	+	0.955328i	$0.404498\pi$
$840$	0	0
$841$	−1.18205	−0.0407604
$842$	0	0
$843$	43.7331	1.50625
$844$	0	0
$845$	95.5573	3.28727
$846$	0	0
$847$	−21.3035	−0.731998
$848$	0	0
$849$	33.1694	1.13837
$850$	0	0
$851$	0.277662	0.00951813
$852$	0	0
$853$	−18.3512	−0.628332	−0.314166	−	0.949368i	$-0.601725\pi$
$853$	−18.3512	−0.628332	−0.314166	+	0.949368i	$0.601725\pi$
$854$	0	0
$855$	−0.297441	−0.0101723
$856$	0	0
$857$	−11.1645	−0.381374	−0.190687	−	0.981651i	$-0.561071\pi$
$857$	−11.1645	−0.381374	−0.190687	+	0.981651i	$0.561071\pi$
$858$	0	0
$859$	46.8419	1.59822	0.799112	−	0.601182i	$-0.205303\pi$
$859$	46.8419	1.59822	0.799112	+	0.601182i	$0.205303\pi$
$860$	0	0
$861$	4.23904	0.144466
$862$	0	0
$863$	−22.8801	−0.778847	−0.389424	−	0.921059i	$-0.627326\pi$
$863$	−22.8801	−0.778847	−0.389424	+	0.921059i	$0.627326\pi$
$864$	0	0
$865$	−40.1624	−1.36556
$866$	0	0
$867$	−17.3378	−0.588823
$868$	0	0
$869$	5.82689	0.197664
$870$	0	0
$871$	−4.78834	−0.162247
$872$	0	0
$873$	1.04794	0.0354673
$874$	0	0
$875$	23.2542	0.786135
$876$	0	0
$877$	34.9776	1.18111	0.590555	−	0.806998i	$-0.298909\pi$
$877$	34.9776	1.18111	0.590555	+	0.806998i	$0.298909\pi$
$878$	0	0
$879$	−0.00982135	−0.000331266 0
$880$	0	0
$881$	−3.81577	−0.128557	−0.0642783	−	0.997932i	$-0.520475\pi$
$881$	−3.81577	−0.128557	−0.0642783	+	0.997932i	$0.520475\pi$
$882$	0	0
$883$	10.0990	0.339859	0.169930	−	0.985456i	$-0.445646\pi$
$883$	10.0990	0.339859	0.169930	+	0.985456i	$0.445646\pi$
$884$	0	0
$885$	93.2015	3.13293
$886$	0	0
$887$	−40.9816	−1.37603	−0.688014	−	0.725697i	$-0.741517\pi$
$887$	−40.9816	−1.37603	−0.688014	+	0.725697i	$0.741517\pi$
$888$	0	0
$889$	25.9336	0.869784
$890$	0	0
$891$	6.15487	0.206196
$892$	0	0
$893$	−0.126418	−0.00423041
$894$	0	0
$895$	−28.1838	−0.942081
$896$	0	0
$897$	−56.5729	−1.88891
$898$	0	0
$899$	−51.1585	−1.70623
$900$	0	0
$901$	6.72107	0.223911
$902$	0	0
$903$	34.5441	1.14956
$904$	0	0
$905$	35.0476	1.16502
$906$	0	0
$907$	−38.7156	−1.28553	−0.642765	−	0.766064i	$-0.722213\pi$
$907$	−38.7156	−1.28553	−0.642765	+	0.766064i	$0.722213\pi$
$908$	0	0
$909$	−4.98942	−0.165489
$910$	0	0
$911$	19.3561	0.641296	0.320648	−	0.947198i	$-0.396099\pi$
$911$	19.3561	0.641296	0.320648	+	0.947198i	$0.396099\pi$
$912$	0	0
$913$	7.21852	0.238898
$914$	0	0
$915$	−43.7733	−1.44710
$916$	0	0
$917$	−14.2902	−0.471905
$918$	0	0
$919$	46.6129	1.53762	0.768809	−	0.639478i	$-0.220850\pi$
$919$	46.6129	1.53762	0.768809	+	0.639478i	$0.220850\pi$
$920$	0	0
$921$	−53.3179	−1.75689
$922$	0	0
$923$	15.2143	0.500783
$924$	0	0
$925$	−0.463596	−0.0152429
$926$	0	0
$927$	−0.372467	−0.0122334
$928$	0	0
$929$	−16.8552	−0.553003	−0.276501	−	0.961014i	$-0.589175\pi$
$929$	−16.8552	−0.553003	−0.276501	+	0.961014i	$0.589175\pi$
$930$	0	0
$931$	0.638048	0.0209112
$932$	0	0
$933$	−16.6766	−0.545967
$934$	0	0
$935$	−6.15148	−0.201175
$936$	0	0
$937$	6.54747	0.213897	0.106948	−	0.994265i	$-0.465892\pi$
$937$	6.54747	0.213897	0.106948	+	0.994265i	$0.465892\pi$
$938$	0	0
$939$	−63.1684	−2.06142
$940$	0	0
$941$	−41.3776	−1.34887	−0.674436	−	0.738333i	$-0.735613\pi$
$941$	−41.3776	−1.34887	−0.674436	+	0.738333i	$0.735613\pi$
$942$	0	0
$943$	−5.63853	−0.183616
$944$	0	0
$945$	−35.0737	−1.14095
$946$	0	0
$947$	25.1244	0.816432	0.408216	−	0.912885i	$-0.366151\pi$
$947$	25.1244	0.816432	0.408216	+	0.912885i	$0.366151\pi$
$948$	0	0
$949$	−20.8835	−0.677908
$950$	0	0
$951$	30.6361	0.993445
$952$	0	0
$953$	−8.65605	−0.280397	−0.140198	−	0.990123i	$-0.544774\pi$
$953$	−8.65605	−0.280397	−0.140198	+	0.990123i	$0.544774\pi$
$954$	0	0
$955$	−47.8306	−1.54776
$956$	0	0
$957$	−5.96983	−0.192977
$958$	0	0
$959$	−11.0151	−0.355695
$960$	0	0
$961$	63.0827	2.03493
$962$	0	0
$963$	2.72136	0.0876947
$964$	0	0
$965$	70.8046	2.27928
$966$	0	0
$967$	−12.7235	−0.409160	−0.204580	−	0.978850i	$-0.565583\pi$
$967$	−12.7235	−0.409160	−0.204580	+	0.978850i	$0.565583\pi$
$968$	0	0
$969$	−1.08465	−0.0348441
$970$	0	0
$971$	12.1496	0.389899	0.194950	−	0.980813i	$-0.437546\pi$
$971$	12.1496	0.389899	0.194950	+	0.980813i	$0.437546\pi$
$972$	0	0
$973$	−21.1988	−0.679603
$974$	0	0
$975$	94.4564	3.02503
$976$	0	0
$977$	−51.5461	−1.64911	−0.824554	−	0.565784i	$-0.808574\pi$
$977$	−51.5461	−1.64911	−0.824554	+	0.565784i	$0.808574\pi$
$978$	0	0
$979$	−11.0036	−0.351678
$980$	0	0
$981$	1.87826	0.0599682
$982$	0	0
$983$	−10.6683	−0.340265	−0.170132	−	0.985421i	$-0.554420\pi$
$983$	−10.6683	−0.340265	−0.170132	+	0.985421i	$0.554420\pi$
$984$	0	0
$985$	−31.8445	−1.01465
$986$	0	0
$987$	2.17568	0.0692528
$988$	0	0
$989$	−45.9486	−1.46108
$990$	0	0
$991$	48.1671	1.53008	0.765039	−	0.643984i	$-0.222720\pi$
$991$	48.1671	1.53008	0.765039	+	0.643984i	$0.222720\pi$
$992$	0	0
$993$	5.82233	0.184766
$994$	0	0
$995$	89.9147	2.85049
$996$	0	0
$997$	21.6045	0.684222	0.342111	−	0.939660i	$-0.388858\pi$
$997$	21.6045	0.684222	0.342111	+	0.939660i	$0.388858\pi$
$998$	0	0
$999$	0.272455	0.00862011

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7232.2.a.bn.1.13		18
4.3	odd	2	inner	7232.2.a.bn.1.6		18
8.3	odd	2		3616.2.a.j.1.13	yes	18
8.5	even	2		3616.2.a.j.1.6	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3616.2.a.j.1.6	✓	18	8.5	even	2
3616.2.a.j.1.13	yes	18	8.3	odd	2
7232.2.a.bn.1.6		18	4.3	odd	2	inner
7232.2.a.bn.1.13		18	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7232 = 2^{6} \cdot 113 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7232.a (trivial)

\(f(q)\)	\(=\)	\(q+1.83905 q^{3} +3.63208 q^{5} +2.00576 q^{7} +0.382087 q^{9} -0.615471 q^{11} +6.26971 q^{13} +6.67957 q^{15} +2.75179 q^{17} -0.214330 q^{19} +3.68868 q^{21} -4.90647 q^{23} +8.19203 q^{25} -4.81446 q^{27} +5.27427 q^{29} -9.69962 q^{31} -1.13188 q^{33} +7.28507 q^{35} -0.0565910 q^{37} +11.5303 q^{39} +1.14920 q^{41} +9.36491 q^{43} +1.38777 q^{45} +0.589828 q^{47} -2.97694 q^{49} +5.06068 q^{51} +2.44243 q^{53} -2.23544 q^{55} -0.394162 q^{57} +13.9532 q^{59} -6.55332 q^{61} +0.766373 q^{63} +22.7721 q^{65} -0.763726 q^{67} -9.02321 q^{69} +2.42663 q^{71} -3.33086 q^{73} +15.0655 q^{75} -1.23448 q^{77} -9.46737 q^{79} -10.0003 q^{81} -11.7285 q^{83} +9.99475 q^{85} +9.69962 q^{87} +17.8784 q^{89} +12.5755 q^{91} -17.8380 q^{93} -0.778464 q^{95} +2.74266 q^{97} -0.235163 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 10 q^{5} + 32 q^{9} + 12 q^{13} + 16 q^{17} - 14 q^{21} + 44 q^{25} - 22 q^{29} + 24 q^{33} + 4 q^{37} + 50 q^{41} - 32 q^{45} + 58 q^{49} - 2 q^{53} + 46 q^{57} - 10 q^{61} + 40 q^{65} - 22 q^{69}+ \cdots + 48 q^{97}+O(q^{100}) \) 18 * q - 10 * q^5 + 32 * q^9 + 12 * q^13 + 16 * q^17 - 14 * q^21 + 44 * q^25 - 22 * q^29 + 24 * q^33 + 4 * q^37 + 50 * q^41 - 32 * q^45 + 58 * q^49 - 2 * q^53 + 46 * q^57 - 10 * q^61 + 40 * q^65 - 22 * q^69 + 62 * q^73 - 14 * q^77 + 46 * q^81 - 12 * q^85 + 52 * q^89 - 16 * q^93 + 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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