LMFDB - Embedded newform 7744.2.a.dw.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7744,2,Mod(1,7744)] 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("7744.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7744, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7744 = 2^{6} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7744.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,5,0,0,0,2,0,7,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$61.8361513253$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.19898000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 10x^{4} + 7x^{3} + 24x^{2} - 15x - 5 $ x^6 - x^5 - 10x^4 + 7x^3 + 24x^2 - 15x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 352)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-2.18328$ of defining polynomial
Character	$\chi$	$=$	7744.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.18328 q^{3} +0.0919008 q^{5} +3.89322 q^{7} +7.13330 q^{9} -2.60679 q^{13} +0.292546 q^{15} +0.530665 q^{17} +5.03067 q^{19} +12.3932 q^{21} -3.05227 q^{23} -4.99155 q^{25} +13.1575 q^{27} +6.60717 q^{29} -4.76590 q^{31} +0.357790 q^{35} -8.24453 q^{37} -8.29815 q^{39} +2.90115 q^{41} +6.01518 q^{43} +0.655556 q^{45} +4.36536 q^{47} +8.15716 q^{49} +1.68926 q^{51} -10.8598 q^{53} +16.0141 q^{57} +11.2054 q^{59} +11.4546 q^{61} +27.7715 q^{63} -0.239566 q^{65} +7.33468 q^{67} -9.71623 q^{69} +11.2163 q^{71} +2.52675 q^{73} -15.8895 q^{75} -8.63148 q^{79} +20.4841 q^{81} +1.53368 q^{83} +0.0487685 q^{85} +21.0325 q^{87} -3.93257 q^{89} -10.1488 q^{91} -15.1712 q^{93} +0.462323 q^{95} -4.80184 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 5 q^{3} + 2 q^{7} + 7 q^{9} + 4 q^{13} - 8 q^{15} + 9 q^{17} - 5 q^{19} + 12 q^{21} - 6 q^{23} + 4 q^{25} + 26 q^{27} + 10 q^{29} - 12 q^{31} + 26 q^{35} + 12 q^{37} + 2 q^{39} + 17 q^{41} - 11 q^{43}+ \cdots - 21 q^{97}+O(q^{100}) $ 6 * q + 5 * q^3 + 2 * q^7 + 7 * q^9 + 4 * q^13 - 8 * q^15 + 9 * q^17 - 5 * q^19 + 12 * q^21 - 6 * q^23 + 4 * q^25 + 26 * q^27 + 10 * q^29 - 12 * q^31 + 26 * q^35 + 12 * q^37 + 2 * q^39 + 17 * q^41 - 11 * q^43 - 14 * q^45 - 22 * q^47 + 6 * q^51 - 18 * q^53 + 6 * q^57 + 7 * q^59 + 18 * q^61 + 26 * q^63 + 26 * q^65 + 31 * q^67 - 16 * q^69 - 22 * q^71 - q^73 + 11 * q^75 - 28 * q^79 + 34 * q^81 + 13 * q^83 + 2 * q^85 + 38 * q^87 - q^89 + 26 * q^91 - 34 * q^93 + 2 * q^95 - 21 * 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
$3$	3.18328	1.83787	0.918935	−	0.394409i	$-0.129051\pi$
$3$	3.18328	1.83787	0.918935	+	0.394409i	$0.129051\pi$
$4$	0	0
$5$	0.0919008	0.0410993	0.0205496	−	0.999789i	$-0.493458\pi$
$5$	0.0919008	0.0410993	0.0205496	+	0.999789i	$0.493458\pi$
$6$	0	0
$7$	3.89322	1.47150	0.735749	−	0.677254i	$-0.236830\pi$
$7$	3.89322	1.47150	0.735749	+	0.677254i	$0.236830\pi$
$8$	0	0
$9$	7.13330	2.37777
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−2.60679	−0.722993	−0.361496	−	0.932374i	$-0.617734\pi$
$13$	−2.60679	−0.722993	−0.361496	+	0.932374i	$0.617734\pi$
$14$	0	0
$15$	0.292546	0.0755352
$16$	0	0
$17$	0.530665	0.128705	0.0643525	−	0.997927i	$-0.479502\pi$
$17$	0.530665	0.128705	0.0643525	+	0.997927i	$0.479502\pi$
$18$	0	0
$19$	5.03067	1.15412	0.577058	−	0.816703i	$-0.304201\pi$
$19$	5.03067	1.15412	0.577058	+	0.816703i	$0.304201\pi$
$20$	0	0
$21$	12.3932	2.70442
$22$	0	0
$23$	−3.05227	−0.636442	−0.318221	−	0.948017i	$-0.603085\pi$
$23$	−3.05227	−0.636442	−0.318221	+	0.948017i	$0.603085\pi$
$24$	0	0
$25$	−4.99155	−0.998311
$26$	0	0
$27$	13.1575	2.53216
$28$	0	0
$29$	6.60717	1.22692	0.613460	−	0.789726i	$-0.289777\pi$
$29$	6.60717	1.22692	0.613460	+	0.789726i	$0.289777\pi$
$30$	0	0
$31$	−4.76590	−0.855980	−0.427990	−	0.903783i	$-0.640778\pi$
$31$	−4.76590	−0.855980	−0.427990	+	0.903783i	$0.640778\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.357790	0.0604775
$36$	0	0
$37$	−8.24453	−1.35539	−0.677696	−	0.735342i	$-0.737021\pi$
$37$	−8.24453	−1.35539	−0.677696	+	0.735342i	$0.737021\pi$
$38$	0	0
$39$	−8.29815	−1.32877
$40$	0	0
$41$	2.90115	0.453083	0.226542	−	0.974001i	$-0.427258\pi$
$41$	2.90115	0.453083	0.226542	+	0.974001i	$0.427258\pi$
$42$	0	0
$43$	6.01518	0.917306	0.458653	−	0.888616i	$-0.348332\pi$
$43$	6.01518	0.917306	0.458653	+	0.888616i	$0.348332\pi$
$44$	0	0
$45$	0.655556	0.0977245
$46$	0	0
$47$	4.36536	0.636753	0.318376	−	0.947964i	$-0.396862\pi$
$47$	4.36536	0.636753	0.318376	+	0.947964i	$0.396862\pi$
$48$	0	0
$49$	8.15716	1.16531
$50$	0	0
$51$	1.68926	0.236543
$52$	0	0
$53$	−10.8598	−1.49170	−0.745851	−	0.666112i	$-0.767957\pi$
$53$	−10.8598	−1.49170	−0.745851	+	0.666112i	$0.767957\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	16.0141	2.12111
$58$	0	0
$59$	11.2054	1.45882	0.729410	−	0.684077i	$-0.239795\pi$
$59$	11.2054	1.45882	0.729410	+	0.684077i	$0.239795\pi$
$60$	0	0
$61$	11.4546	1.46661	0.733303	−	0.679902i	$-0.237978\pi$
$61$	11.4546	1.46661	0.733303	+	0.679902i	$0.237978\pi$
$62$	0	0
$63$	27.7715	3.49888
$64$	0	0
$65$	−0.239566	−0.0297145
$66$	0	0
$67$	7.33468	0.896074	0.448037	−	0.894015i	$-0.352123\pi$
$67$	7.33468	0.896074	0.448037	+	0.894015i	$0.352123\pi$
$68$	0	0
$69$	−9.71623	−1.16970
$70$	0	0
$71$	11.2163	1.33113	0.665565	−	0.746340i	$-0.268191\pi$
$71$	11.2163	1.33113	0.665565	+	0.746340i	$0.268191\pi$
$72$	0	0
$73$	2.52675	0.295734	0.147867	−	0.989007i	$-0.452759\pi$
$73$	2.52675	0.295734	0.147867	+	0.989007i	$0.452759\pi$
$74$	0	0
$75$	−15.8895	−1.83477
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−8.63148	−0.971117	−0.485559	−	0.874204i	$-0.661384\pi$
$79$	−8.63148	−0.971117	−0.485559	+	0.874204i	$0.661384\pi$
$80$	0	0
$81$	20.4841	2.27601
$82$	0	0
$83$	1.53368	0.168344	0.0841719	−	0.996451i	$-0.473176\pi$
$83$	1.53368	0.168344	0.0841719	+	0.996451i	$0.473176\pi$
$84$	0	0
$85$	0.0487685	0.00528969
$86$	0	0
$87$	21.0325	2.25492
$88$	0	0
$89$	−3.93257	−0.416852	−0.208426	−	0.978038i	$-0.566834\pi$
$89$	−3.93257	−0.416852	−0.208426	+	0.978038i	$0.566834\pi$
$90$	0	0
$91$	−10.1488	−1.06388
$92$	0	0
$93$	−15.1712	−1.57318
$94$	0	0
$95$	0.462323	0.0474333
$96$	0	0
$97$	−4.80184	−0.487553	−0.243776	−	0.969831i	$-0.578386\pi$
$97$	−4.80184	−0.487553	−0.243776	+	0.969831i	$0.578386\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.82504	−0.480109	−0.240055	−	0.970759i	$-0.577165\pi$
$101$	−4.82504	−0.480109	−0.240055	+	0.970759i	$0.577165\pi$
$102$	0	0
$103$	2.07599	0.204553	0.102277	−	0.994756i	$-0.467387\pi$
$103$	2.07599	0.204553	0.102277	+	0.994756i	$0.467387\pi$
$104$	0	0
$105$	1.13895	0.111150
$106$	0	0
$107$	−1.10608	−0.106929	−0.0534645	−	0.998570i	$-0.517026\pi$
$107$	−1.10608	−0.106929	−0.0534645	+	0.998570i	$0.517026\pi$
$108$	0	0
$109$	1.18772	0.113763	0.0568813	−	0.998381i	$-0.481884\pi$
$109$	1.18772	0.113763	0.0568813	+	0.998381i	$0.481884\pi$
$110$	0	0
$111$	−26.2447	−2.49103
$112$	0	0
$113$	−4.22459	−0.397416	−0.198708	−	0.980059i	$-0.563674\pi$
$113$	−4.22459	−0.397416	−0.198708	+	0.980059i	$0.563674\pi$
$114$	0	0
$115$	−0.280506	−0.0261573
$116$	0	0
$117$	−18.5950	−1.71911
$118$	0	0
$119$	2.06599	0.189389
$120$	0	0
$121$	0	0
$122$	0	0
$123$	9.23518	0.832708
$124$	0	0
$125$	−0.918232	−0.0821291
$126$	0	0
$127$	−9.22897	−0.818939	−0.409469	−	0.912324i	$-0.634286\pi$
$127$	−9.22897	−0.818939	−0.409469	+	0.912324i	$0.634286\pi$
$128$	0	0
$129$	19.1480	1.68589
$130$	0	0
$131$	2.96831	0.259343	0.129671	−	0.991557i	$-0.458608\pi$
$131$	2.96831	0.259343	0.129671	+	0.991557i	$0.458608\pi$
$132$	0	0
$133$	19.5855	1.69828
$134$	0	0
$135$	1.20918	0.104070
$136$	0	0
$137$	−16.0863	−1.37434	−0.687171	−	0.726496i	$-0.741148\pi$
$137$	−16.0863	−1.37434	−0.687171	+	0.726496i	$0.741148\pi$
$138$	0	0
$139$	13.9490	1.18314	0.591568	−	0.806255i	$-0.298509\pi$
$139$	13.9490	1.18314	0.591568	+	0.806255i	$0.298509\pi$
$140$	0	0
$141$	13.8962	1.17027
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0.607204	0.0504256
$146$	0	0
$147$	25.9666	2.14169
$148$	0	0
$149$	−15.9323	−1.30522	−0.652612	−	0.757692i	$-0.726327\pi$
$149$	−15.9323	−1.30522	−0.652612	+	0.757692i	$0.726327\pi$
$150$	0	0
$151$	14.3868	1.17078	0.585391	−	0.810751i	$-0.300941\pi$
$151$	14.3868	1.17078	0.585391	+	0.810751i	$0.300941\pi$
$152$	0	0
$153$	3.78539	0.306031
$154$	0	0
$155$	−0.437990	−0.0351802
$156$	0	0
$157$	20.7821	1.65859	0.829296	−	0.558809i	$-0.188742\pi$
$157$	20.7821	1.65859	0.829296	+	0.558809i	$0.188742\pi$
$158$	0	0
$159$	−34.5697	−2.74156
$160$	0	0
$161$	−11.8831	−0.936523
$162$	0	0
$163$	1.94881	0.152642	0.0763211	−	0.997083i	$-0.475683\pi$
$163$	1.94881	0.152642	0.0763211	+	0.997083i	$0.475683\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.80062	−0.526248	−0.263124	−	0.964762i	$-0.584753\pi$
$167$	−6.80062	−0.526248	−0.263124	+	0.964762i	$0.584753\pi$
$168$	0	0
$169$	−6.20466	−0.477281
$170$	0	0
$171$	35.8853	2.74422
$172$	0	0
$173$	−6.89550	−0.524255	−0.262128	−	0.965033i	$-0.584424\pi$
$173$	−6.89550	−0.524255	−0.262128	+	0.965033i	$0.584424\pi$
$174$	0	0
$175$	−19.4332	−1.46901
$176$	0	0
$177$	35.6700	2.68112
$178$	0	0
$179$	7.98643	0.596934	0.298467	−	0.954420i	$-0.403525\pi$
$179$	7.98643	0.596934	0.298467	+	0.954420i	$0.403525\pi$
$180$	0	0
$181$	−8.85237	−0.657992	−0.328996	−	0.944331i	$-0.606710\pi$
$181$	−8.85237	−0.657992	−0.328996	+	0.944331i	$0.606710\pi$
$182$	0	0
$183$	36.4631	2.69543
$184$	0	0
$185$	−0.757679	−0.0557056
$186$	0	0
$187$	0	0
$188$	0	0
$189$	51.2250	3.72607
$190$	0	0
$191$	19.7629	1.43000	0.714998	−	0.699127i	$-0.246428\pi$
$191$	19.7629	1.43000	0.714998	+	0.699127i	$0.246428\pi$
$192$	0	0
$193$	−24.7216	−1.77950	−0.889750	−	0.456448i	$-0.849122\pi$
$193$	−24.7216	−1.77950	−0.889750	+	0.456448i	$0.849122\pi$
$194$	0	0
$195$	−0.762606	−0.0546114
$196$	0	0
$197$	−9.40207	−0.669870	−0.334935	−	0.942241i	$-0.608714\pi$
$197$	−9.40207	−0.669870	−0.334935	+	0.942241i	$0.608714\pi$
$198$	0	0
$199$	−6.96116	−0.493464	−0.246732	−	0.969084i	$-0.579357\pi$
$199$	−6.96116	−0.493464	−0.246732	+	0.969084i	$0.579357\pi$
$200$	0	0
$201$	23.3484	1.64687
$202$	0	0
$203$	25.7232	1.80541
$204$	0	0
$205$	0.266618	0.0186214
$206$	0	0
$207$	−21.7727	−1.51331
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−10.6821	−0.735389	−0.367694	−	0.929947i	$-0.619853\pi$
$211$	−10.6821	−0.735389	−0.367694	+	0.929947i	$0.619853\pi$
$212$	0	0
$213$	35.7047	2.44644
$214$	0	0
$215$	0.552799	0.0377006
$216$	0	0
$217$	−18.5547	−1.25957
$218$	0	0
$219$	8.04337	0.543520
$220$	0	0
$221$	−1.38333	−0.0930528
$222$	0	0
$223$	−20.9248	−1.40123	−0.700614	−	0.713541i	$-0.747090\pi$
$223$	−20.9248	−1.40123	−0.700614	+	0.713541i	$0.747090\pi$
$224$	0	0
$225$	−35.6063	−2.37375
$226$	0	0
$227$	−12.1965	−0.809512	−0.404756	−	0.914425i	$-0.632644\pi$
$227$	−12.1965	−0.809512	−0.404756	+	0.914425i	$0.632644\pi$
$228$	0	0
$229$	−11.4257	−0.755032	−0.377516	−	0.926003i	$-0.623222\pi$
$229$	−11.4257	−0.755032	−0.377516	+	0.926003i	$0.623222\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.42684	0.421036	0.210518	−	0.977590i	$-0.432485\pi$
$233$	6.42684	0.421036	0.210518	+	0.977590i	$0.432485\pi$
$234$	0	0
$235$	0.401180	0.0261701
$236$	0	0
$237$	−27.4765	−1.78479
$238$	0	0
$239$	−13.8013	−0.892731	−0.446365	−	0.894851i	$-0.647282\pi$
$239$	−13.8013	−0.892731	−0.446365	+	0.894851i	$0.647282\pi$
$240$	0	0
$241$	21.0631	1.35680	0.678398	−	0.734694i	$-0.262674\pi$
$241$	21.0631	1.35680	0.678398	+	0.734694i	$0.262674\pi$
$242$	0	0
$243$	25.7343	1.65085
$244$	0	0
$245$	0.749649	0.0478933
$246$	0	0
$247$	−13.1139	−0.834417
$248$	0	0
$249$	4.88215	0.309394
$250$	0	0
$251$	−12.0570	−0.761028	−0.380514	−	0.924775i	$-0.624253\pi$
$251$	−12.0570	−0.761028	−0.380514	+	0.924775i	$0.624253\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0.155244	0.00972176
$256$	0	0
$257$	−24.8884	−1.55249	−0.776247	−	0.630428i	$-0.782879\pi$
$257$	−24.8884	−1.55249	−0.776247	+	0.630428i	$0.782879\pi$
$258$	0	0
$259$	−32.0978	−1.99446
$260$	0	0
$261$	47.1309	2.91733
$262$	0	0
$263$	22.1354	1.36492	0.682462	−	0.730921i	$-0.260909\pi$
$263$	22.1354	1.36492	0.682462	+	0.730921i	$0.260909\pi$
$264$	0	0
$265$	−0.998021	−0.0613079
$266$	0	0
$267$	−12.5185	−0.766120
$268$	0	0
$269$	−1.32824	−0.0809842	−0.0404921	−	0.999180i	$-0.512893\pi$
$269$	−1.32824	−0.0809842	−0.0404921	+	0.999180i	$0.512893\pi$
$270$	0	0
$271$	−0.673010	−0.0408824	−0.0204412	−	0.999791i	$-0.506507\pi$
$271$	−0.673010	−0.0408824	−0.0204412	+	0.999791i	$0.506507\pi$
$272$	0	0
$273$	−32.3065	−1.95528
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.54134	0.152695	0.0763473	−	0.997081i	$-0.475674\pi$
$277$	2.54134	0.152695	0.0763473	+	0.997081i	$0.475674\pi$
$278$	0	0
$279$	−33.9966	−2.03532
$280$	0	0
$281$	20.2813	1.20988	0.604939	−	0.796272i	$-0.293197\pi$
$281$	20.2813	1.20988	0.604939	+	0.796272i	$0.293197\pi$
$282$	0	0
$283$	−11.9672	−0.711378	−0.355689	−	0.934604i	$-0.615754\pi$
$283$	−11.9672	−0.711378	−0.355689	+	0.934604i	$0.615754\pi$
$284$	0	0
$285$	1.47170	0.0871763
$286$	0	0
$287$	11.2948	0.666711
$288$	0	0
$289$	−16.7184	−0.983435
$290$	0	0
$291$	−15.2856	−0.896058
$292$	0	0
$293$	18.4143	1.07578	0.537888	−	0.843017i	$-0.319222\pi$
$293$	18.4143	1.07578	0.537888	+	0.843017i	$0.319222\pi$
$294$	0	0
$295$	1.02979	0.0599565
$296$	0	0
$297$	0	0
$298$	0	0
$299$	7.95661	0.460143
$300$	0	0
$301$	23.4184	1.34981
$302$	0	0
$303$	−15.3595	−0.882379
$304$	0	0
$305$	1.05268	0.0602764
$306$	0	0
$307$	−21.3297	−1.21735	−0.608675	−	0.793420i	$-0.708298\pi$
$307$	−21.3297	−1.21735	−0.608675	+	0.793420i	$0.708298\pi$
$308$	0	0
$309$	6.60846	0.375942
$310$	0	0
$311$	−10.2512	−0.581292	−0.290646	−	0.956831i	$-0.593870\pi$
$311$	−10.2512	−0.581292	−0.290646	+	0.956831i	$0.593870\pi$
$312$	0	0
$313$	−22.7735	−1.28724	−0.643618	−	0.765347i	$-0.722568\pi$
$313$	−22.7735	−1.28724	−0.643618	+	0.765347i	$0.722568\pi$
$314$	0	0
$315$	2.55222	0.143802
$316$	0	0
$317$	21.8610	1.22783	0.613917	−	0.789370i	$-0.289593\pi$
$317$	21.8610	1.22783	0.613917	+	0.789370i	$0.289593\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−3.52098	−0.196522
$322$	0	0
$323$	2.66960	0.148540
$324$	0	0
$325$	13.0119	0.721772
$326$	0	0
$327$	3.78084	0.209081
$328$	0	0
$329$	16.9953	0.936981
$330$	0	0
$331$	3.18234	0.174917	0.0874587	−	0.996168i	$-0.472125\pi$
$331$	3.18234	0.174917	0.0874587	+	0.996168i	$0.472125\pi$
$332$	0	0
$333$	−58.8107	−3.22281
$334$	0	0
$335$	0.674063	0.0368280
$336$	0	0
$337$	−15.3438	−0.835830	−0.417915	−	0.908486i	$-0.637239\pi$
$337$	−15.3438	−0.835830	−0.417915	+	0.908486i	$0.637239\pi$
$338$	0	0
$339$	−13.4481	−0.730398
$340$	0	0
$341$	0	0
$342$	0	0
$343$	4.50507	0.243251
$344$	0	0
$345$	−0.892930	−0.0480737
$346$	0	0
$347$	20.1222	1.08022	0.540109	−	0.841595i	$-0.318383\pi$
$347$	20.1222	1.08022	0.540109	+	0.841595i	$0.318383\pi$
$348$	0	0
$349$	27.0712	1.44909	0.724544	−	0.689228i	$-0.242050\pi$
$349$	27.0712	1.44909	0.724544	+	0.689228i	$0.242050\pi$
$350$	0	0
$351$	−34.2988	−1.83073
$352$	0	0
$353$	−24.9212	−1.32642	−0.663210	−	0.748433i	$-0.730807\pi$
$353$	−24.9212	−1.32642	−0.663210	+	0.748433i	$0.730807\pi$
$354$	0	0
$355$	1.03079	0.0547085
$356$	0	0
$357$	6.57665	0.348073
$358$	0	0
$359$	−10.8174	−0.570922	−0.285461	−	0.958390i	$-0.592147\pi$
$359$	−10.8174	−0.570922	−0.285461	+	0.958390i	$0.592147\pi$
$360$	0	0
$361$	6.30766	0.331982
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0.232210	0.0121544
$366$	0	0
$367$	−12.8331	−0.669884	−0.334942	−	0.942239i	$-0.608717\pi$
$367$	−12.8331	−0.669884	−0.334942	+	0.942239i	$0.608717\pi$
$368$	0	0
$369$	20.6948	1.07733
$370$	0	0
$371$	−42.2794	−2.19504
$372$	0	0
$373$	6.70036	0.346932	0.173466	−	0.984840i	$-0.444503\pi$
$373$	6.70036	0.346932	0.173466	+	0.984840i	$0.444503\pi$
$374$	0	0
$375$	−2.92299	−0.150943
$376$	0	0
$377$	−17.2235	−0.887055
$378$	0	0
$379$	−3.93510	−0.202132	−0.101066	−	0.994880i	$-0.532225\pi$
$379$	−3.93510	−0.202132	−0.101066	+	0.994880i	$0.532225\pi$
$380$	0	0
$381$	−29.3784	−1.50510
$382$	0	0
$383$	11.5765	0.591532	0.295766	−	0.955260i	$-0.404425\pi$
$383$	11.5765	0.591532	0.295766	+	0.955260i	$0.404425\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	42.9081	2.18114
$388$	0	0
$389$	23.5070	1.19185	0.595927	−	0.803039i	$-0.296785\pi$
$389$	23.5070	1.19185	0.595927	+	0.803039i	$0.296785\pi$
$390$	0	0
$391$	−1.61973	−0.0819133
$392$	0	0
$393$	9.44899	0.476639
$394$	0	0
$395$	−0.793240	−0.0399122
$396$	0	0
$397$	4.30776	0.216200	0.108100	−	0.994140i	$-0.465523\pi$
$397$	4.30776	0.216200	0.108100	+	0.994140i	$0.465523\pi$
$398$	0	0
$399$	62.3463	3.12122
$400$	0	0
$401$	23.2378	1.16044	0.580220	−	0.814460i	$-0.302967\pi$
$401$	23.2378	1.16044	0.580220	+	0.814460i	$0.302967\pi$
$402$	0	0
$403$	12.4237	0.618868
$404$	0	0
$405$	1.88250	0.0935424
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−2.06671	−0.102192	−0.0510962	−	0.998694i	$-0.516272\pi$
$409$	−2.06671	−0.102192	−0.0510962	+	0.998694i	$0.516272\pi$
$410$	0	0
$411$	−51.2071	−2.52586
$412$	0	0
$413$	43.6251	2.14665
$414$	0	0
$415$	0.140947	0.00691881
$416$	0	0
$417$	44.4035	2.17445
$418$	0	0
$419$	−1.19093	−0.0581809	−0.0290904	−	0.999577i	$-0.509261\pi$
$419$	−1.19093	−0.0581809	−0.0290904	+	0.999577i	$0.509261\pi$
$420$	0	0
$421$	8.51037	0.414770	0.207385	−	0.978259i	$-0.433505\pi$
$421$	8.51037	0.414770	0.207385	+	0.978259i	$0.433505\pi$
$422$	0	0
$423$	31.1394	1.51405
$424$	0	0
$425$	−2.64884	−0.128488
$426$	0	0
$427$	44.5951	2.15811
$428$	0	0
$429$	0	0
$430$	0	0
$431$	24.9531	1.20195	0.600974	−	0.799269i	$-0.294780\pi$
$431$	24.9531	1.20195	0.600974	+	0.799269i	$0.294780\pi$
$432$	0	0
$433$	2.96861	0.142662	0.0713311	−	0.997453i	$-0.477275\pi$
$433$	2.96861	0.142662	0.0713311	+	0.997453i	$0.477275\pi$
$434$	0	0
$435$	1.93290	0.0926756
$436$	0	0
$437$	−15.3550	−0.734527
$438$	0	0
$439$	7.11680	0.339666	0.169833	−	0.985473i	$-0.445677\pi$
$439$	7.11680	0.339666	0.169833	+	0.985473i	$0.445677\pi$
$440$	0	0
$441$	58.1875	2.77083
$442$	0	0
$443$	−34.1530	−1.62266	−0.811328	−	0.584591i	$-0.801255\pi$
$443$	−34.1530	−1.62266	−0.811328	+	0.584591i	$0.801255\pi$
$444$	0	0
$445$	−0.361407	−0.0171323
$446$	0	0
$447$	−50.7170	−2.39883
$448$	0	0
$449$	1.14282	0.0539329	0.0269665	−	0.999636i	$-0.491415\pi$
$449$	1.14282	0.0539329	0.0269665	+	0.999636i	$0.491415\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	45.7973	2.15175
$454$	0	0
$455$	−0.932683	−0.0437248
$456$	0	0
$457$	−4.23764	−0.198228	−0.0991142	−	0.995076i	$-0.531601\pi$
$457$	−4.23764	−0.198228	−0.0991142	+	0.995076i	$0.531601\pi$
$458$	0	0
$459$	6.98221	0.325902
$460$	0	0
$461$	22.1743	1.03276	0.516380	−	0.856360i	$-0.327279\pi$
$461$	22.1743	1.03276	0.516380	+	0.856360i	$0.327279\pi$
$462$	0	0
$463$	−14.1466	−0.657450	−0.328725	−	0.944426i	$-0.606619\pi$
$463$	−14.1466	−0.657450	−0.328725	+	0.944426i	$0.606619\pi$
$464$	0	0
$465$	−1.39425	−0.0646566
$466$	0	0
$467$	−13.2099	−0.611283	−0.305642	−	0.952147i	$-0.598871\pi$
$467$	−13.2099	−0.611283	−0.305642	+	0.952147i	$0.598871\pi$
$468$	0	0
$469$	28.5555	1.31857
$470$	0	0
$471$	66.1554	3.04828
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−25.1109	−1.15217
$476$	0	0
$477$	−77.4660	−3.54692
$478$	0	0
$479$	−14.8348	−0.677819	−0.338910	−	0.940819i	$-0.610058\pi$
$479$	−14.8348	−0.677819	−0.338910	+	0.940819i	$0.610058\pi$
$480$	0	0
$481$	21.4917	0.979939
$482$	0	0
$483$	−37.8274	−1.72121
$484$	0	0
$485$	−0.441293	−0.0200381
$486$	0	0
$487$	−37.3255	−1.69138	−0.845690	−	0.533674i	$-0.820811\pi$
$487$	−37.3255	−1.69138	−0.845690	+	0.533674i	$0.820811\pi$
$488$	0	0
$489$	6.20360	0.280537
$490$	0	0
$491$	−14.5067	−0.654676	−0.327338	−	0.944907i	$-0.606152\pi$
$491$	−14.5067	−0.654676	−0.327338	+	0.944907i	$0.606152\pi$
$492$	0	0
$493$	3.50619	0.157911
$494$	0	0
$495$	0	0
$496$	0	0
$497$	43.6675	1.95876
$498$	0	0
$499$	−16.7216	−0.748561	−0.374281	−	0.927316i	$-0.622110\pi$
$499$	−16.7216	−0.748561	−0.374281	+	0.927316i	$0.622110\pi$
$500$	0	0
$501$	−21.6483	−0.967175
$502$	0	0
$503$	−40.1781	−1.79145	−0.895727	−	0.444605i	$-0.853344\pi$
$503$	−40.1781	−1.79145	−0.895727	+	0.444605i	$0.853344\pi$
$504$	0	0
$505$	−0.443425	−0.0197322
$506$	0	0
$507$	−19.7512	−0.877181
$508$	0	0
$509$	30.5738	1.35516	0.677580	−	0.735450i	$-0.263029\pi$
$509$	30.5738	1.35516	0.677580	+	0.735450i	$0.263029\pi$
$510$	0	0
$511$	9.83719	0.435172
$512$	0	0
$513$	66.1910	2.92240
$514$	0	0
$515$	0.190785	0.00840699
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−21.9504	−0.963513
$520$	0	0
$521$	9.11390	0.399287	0.199644	−	0.979869i	$-0.436022\pi$
$521$	9.11390	0.399287	0.199644	+	0.979869i	$0.436022\pi$
$522$	0	0
$523$	−18.4292	−0.805854	−0.402927	−	0.915232i	$-0.632007\pi$
$523$	−18.4292	−0.805854	−0.402927	+	0.915232i	$0.632007\pi$
$524$	0	0
$525$	−61.8615	−2.69986
$526$	0	0
$527$	−2.52909	−0.110169
$528$	0	0
$529$	−13.6837	−0.594942
$530$	0	0
$531$	79.9316	3.46874
$532$	0	0
$533$	−7.56268	−0.327576
$534$	0	0
$535$	−0.101650	−0.00439471
$536$	0	0
$537$	25.4231	1.09709
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−31.1327	−1.33850	−0.669250	−	0.743037i	$-0.733385\pi$
$541$	−31.1327	−1.33850	−0.669250	+	0.743037i	$0.733385\pi$
$542$	0	0
$543$	−28.1796	−1.20930
$544$	0	0
$545$	0.109152	0.00467556
$546$	0	0
$547$	10.1201	0.432704	0.216352	−	0.976315i	$-0.430584\pi$
$547$	10.1201	0.432704	0.216352	+	0.976315i	$0.430584\pi$
$548$	0	0
$549$	81.7088	3.48725
$550$	0	0
$551$	33.2385	1.41601
$552$	0	0
$553$	−33.6042	−1.42900
$554$	0	0
$555$	−2.41191	−0.102380
$556$	0	0
$557$	−13.0959	−0.554892	−0.277446	−	0.960741i	$-0.589488\pi$
$557$	−13.0959	−0.554892	−0.277446	+	0.960741i	$0.589488\pi$
$558$	0	0
$559$	−15.6803	−0.663205
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−21.7813	−0.917971	−0.458985	−	0.888444i	$-0.651787\pi$
$563$	−21.7813	−0.917971	−0.458985	+	0.888444i	$0.651787\pi$
$564$	0	0
$565$	−0.388243	−0.0163335
$566$	0	0
$567$	79.7491	3.34915
$568$	0	0
$569$	8.26297	0.346402	0.173201	−	0.984887i	$-0.444589\pi$
$569$	8.26297	0.346402	0.173201	+	0.984887i	$0.444589\pi$
$570$	0	0
$571$	41.2789	1.72747	0.863733	−	0.503949i	$-0.168120\pi$
$571$	41.2789	1.72747	0.863733	+	0.503949i	$0.168120\pi$
$572$	0	0
$573$	62.9110	2.62815
$574$	0	0
$575$	15.2356	0.635366
$576$	0	0
$577$	16.7087	0.695592	0.347796	−	0.937570i	$-0.386930\pi$
$577$	16.7087	0.695592	0.347796	+	0.937570i	$0.386930\pi$
$578$	0	0
$579$	−78.6959	−3.27049
$580$	0	0
$581$	5.97097	0.247718
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−1.70890	−0.0706541
$586$	0	0
$587$	28.4066	1.17247	0.586233	−	0.810142i	$-0.300610\pi$
$587$	28.4066	1.17247	0.586233	+	0.810142i	$0.300610\pi$
$588$	0	0
$589$	−23.9757	−0.987900
$590$	0	0
$591$	−29.9295	−1.23113
$592$	0	0
$593$	−8.51973	−0.349864	−0.174932	−	0.984581i	$-0.555970\pi$
$593$	−8.51973	−0.349864	−0.174932	+	0.984581i	$0.555970\pi$
$594$	0	0
$595$	0.189866	0.00778377
$596$	0	0
$597$	−22.1594	−0.906922
$598$	0	0
$599$	−10.0123	−0.409090	−0.204545	−	0.978857i	$-0.565571\pi$
$599$	−10.0123	−0.409090	−0.204545	+	0.978857i	$0.565571\pi$
$600$	0	0
$601$	−43.8325	−1.78796	−0.893982	−	0.448104i	$-0.852100\pi$
$601$	−43.8325	−1.78796	−0.893982	+	0.448104i	$0.852100\pi$
$602$	0	0
$603$	52.3205	2.13066
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−17.9372	−0.728049	−0.364025	−	0.931389i	$-0.618598\pi$
$607$	−17.9372	−0.728049	−0.364025	+	0.931389i	$0.618598\pi$
$608$	0	0
$609$	81.8841	3.31811
$610$	0	0
$611$	−11.3796	−0.460368
$612$	0	0
$613$	−10.8880	−0.439761	−0.219880	−	0.975527i	$-0.570567\pi$
$613$	−10.8880	−0.439761	−0.219880	+	0.975527i	$0.570567\pi$
$614$	0	0
$615$	0.848721	0.0342237
$616$	0	0
$617$	11.9150	0.479680	0.239840	−	0.970812i	$-0.422905\pi$
$617$	11.9150	0.479680	0.239840	+	0.970812i	$0.422905\pi$
$618$	0	0
$619$	7.66159	0.307945	0.153973	−	0.988075i	$-0.450793\pi$
$619$	7.66159	0.307945	0.153973	+	0.988075i	$0.450793\pi$
$620$	0	0
$621$	−40.1601	−1.61157
$622$	0	0
$623$	−15.3104	−0.613397
$624$	0	0
$625$	24.8734	0.994935
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4.37508	−0.174446
$630$	0	0
$631$	22.1860	0.883209	0.441605	−	0.897210i	$-0.354409\pi$
$631$	22.1860	0.883209	0.441605	+	0.897210i	$0.354409\pi$
$632$	0	0
$633$	−34.0043	−1.35155
$634$	0	0
$635$	−0.848150	−0.0336578
$636$	0	0
$637$	−21.2640	−0.842510
$638$	0	0
$639$	80.0093	3.16512
$640$	0	0
$641$	12.8536	0.507685	0.253843	−	0.967246i	$-0.418305\pi$
$641$	12.8536	0.507685	0.253843	+	0.967246i	$0.418305\pi$
$642$	0	0
$643$	−2.63803	−0.104034	−0.0520169	−	0.998646i	$-0.516565\pi$
$643$	−2.63803	−0.104034	−0.0520169	+	0.998646i	$0.516565\pi$
$644$	0	0
$645$	1.75972	0.0692888
$646$	0	0
$647$	−4.97696	−0.195664	−0.0978322	−	0.995203i	$-0.531191\pi$
$647$	−4.97696	−0.195664	−0.0978322	+	0.995203i	$0.531191\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−59.0648	−2.31493
$652$	0	0
$653$	24.8958	0.974249	0.487125	−	0.873333i	$-0.338046\pi$
$653$	24.8958	0.974249	0.487125	+	0.873333i	$0.338046\pi$
$654$	0	0
$655$	0.272790	0.0106588
$656$	0	0
$657$	18.0241	0.703186
$658$	0	0
$659$	−11.7528	−0.457823	−0.228911	−	0.973447i	$-0.573517\pi$
$659$	−11.7528	−0.457823	−0.228911	+	0.973447i	$0.573517\pi$
$660$	0	0
$661$	−20.7879	−0.808554	−0.404277	−	0.914637i	$-0.632477\pi$
$661$	−20.7879	−0.808554	−0.404277	+	0.914637i	$0.632477\pi$
$662$	0	0
$663$	−4.40353	−0.171019
$664$	0	0
$665$	1.79992	0.0697981
$666$	0	0
$667$	−20.1668	−0.780863
$668$	0	0
$669$	−66.6096	−2.57527
$670$	0	0
$671$	0	0
$672$	0	0
$673$	28.6730	1.10526	0.552631	−	0.833426i	$-0.313624\pi$
$673$	28.6730	1.10526	0.552631	+	0.833426i	$0.313624\pi$
$674$	0	0
$675$	−65.6763	−2.52788
$676$	0	0
$677$	47.7481	1.83511	0.917553	−	0.397613i	$-0.130161\pi$
$677$	47.7481	1.83511	0.917553	+	0.397613i	$0.130161\pi$
$678$	0	0
$679$	−18.6946	−0.717433
$680$	0	0
$681$	−38.8250	−1.48778
$682$	0	0
$683$	8.67362	0.331887	0.165943	−	0.986135i	$-0.446933\pi$
$683$	8.67362	0.331887	0.165943	+	0.986135i	$0.446933\pi$
$684$	0	0
$685$	−1.47834	−0.0564845
$686$	0	0
$687$	−36.3713	−1.38765
$688$	0	0
$689$	28.3091	1.07849
$690$	0	0
$691$	28.4269	1.08141	0.540706	−	0.841212i	$-0.318157\pi$
$691$	28.4269	1.08141	0.540706	+	0.841212i	$0.318157\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.28192	0.0486260
$696$	0	0
$697$	1.53954	0.0583141
$698$	0	0
$699$	20.4585	0.773810
$700$	0	0
$701$	40.8040	1.54115	0.770573	−	0.637352i	$-0.219970\pi$
$701$	40.8040	1.54115	0.770573	+	0.637352i	$0.219970\pi$
$702$	0	0
$703$	−41.4755	−1.56428
$704$	0	0
$705$	1.27707	0.0480972
$706$	0	0
$707$	−18.7849	−0.706480
$708$	0	0
$709$	−7.23483	−0.271710	−0.135855	−	0.990729i	$-0.543378\pi$
$709$	−7.23483	−0.271710	−0.135855	+	0.990729i	$0.543378\pi$
$710$	0	0
$711$	−61.5710	−2.30909
$712$	0	0
$713$	14.5468	0.544781
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−43.9334	−1.64072
$718$	0	0
$719$	28.6751	1.06940	0.534701	−	0.845041i	$-0.320424\pi$
$719$	28.6751	1.06940	0.534701	+	0.845041i	$0.320424\pi$
$720$	0	0
$721$	8.08228	0.301000
$722$	0	0
$723$	67.0500	2.49362
$724$	0	0
$725$	−32.9800	−1.22485
$726$	0	0
$727$	−44.2322	−1.64048	−0.820241	−	0.572018i	$-0.806161\pi$
$727$	−44.2322	−1.64048	−0.820241	+	0.572018i	$0.806161\pi$
$728$	0	0
$729$	20.4672	0.758046
$730$	0	0
$731$	3.19204	0.118062
$732$	0	0
$733$	−30.2981	−1.11909	−0.559543	−	0.828801i	$-0.689023\pi$
$733$	−30.2981	−1.11909	−0.559543	+	0.828801i	$0.689023\pi$
$734$	0	0
$735$	2.38635	0.0880218
$736$	0	0
$737$	0	0
$738$	0	0
$739$	16.7014	0.614373	0.307186	−	0.951649i	$-0.400613\pi$
$739$	16.7014	0.614373	0.307186	+	0.951649i	$0.400613\pi$
$740$	0	0
$741$	−41.7453	−1.53355
$742$	0	0
$743$	3.75599	0.137794	0.0688970	−	0.997624i	$-0.478052\pi$
$743$	3.75599	0.137794	0.0688970	+	0.997624i	$0.478052\pi$
$744$	0	0
$745$	−1.46419	−0.0536438
$746$	0	0
$747$	10.9402	0.400282
$748$	0	0
$749$	−4.30622	−0.157346
$750$	0	0
$751$	12.3624	0.451110	0.225555	−	0.974230i	$-0.427581\pi$
$751$	12.3624	0.451110	0.225555	+	0.974230i	$0.427581\pi$
$752$	0	0
$753$	−38.3807	−1.39867
$754$	0	0
$755$	1.32216	0.0481183
$756$	0	0
$757$	−15.8860	−0.577388	−0.288694	−	0.957421i	$-0.593221\pi$
$757$	−15.8860	−0.577388	−0.288694	+	0.957421i	$0.593221\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	48.3362	1.75218	0.876092	−	0.482143i	$-0.160142\pi$
$761$	48.3362	1.75218	0.876092	+	0.482143i	$0.160142\pi$
$762$	0	0
$763$	4.62404	0.167401
$764$	0	0
$765$	0.347880	0.0125776
$766$	0	0
$767$	−29.2101	−1.05472
$768$	0	0
$769$	−44.3792	−1.60035	−0.800177	−	0.599763i	$-0.795261\pi$
$769$	−44.3792	−1.60035	−0.800177	+	0.599763i	$0.795261\pi$
$770$	0	0
$771$	−79.2268	−2.85328
$772$	0	0
$773$	18.3181	0.658857	0.329429	−	0.944180i	$-0.393144\pi$
$773$	18.3181	0.658857	0.329429	+	0.944180i	$0.393144\pi$
$774$	0	0
$775$	23.7892	0.854534
$776$	0	0
$777$	−102.176	−3.66555
$778$	0	0
$779$	14.5947	0.522910
$780$	0	0
$781$	0	0
$782$	0	0
$783$	86.9337	3.10676
$784$	0	0
$785$	1.90989	0.0681670
$786$	0	0
$787$	−40.7153	−1.45135	−0.725673	−	0.688040i	$-0.758471\pi$
$787$	−40.7153	−1.45135	−0.725673	+	0.688040i	$0.758471\pi$
$788$	0	0
$789$	70.4631	2.50855
$790$	0	0
$791$	−16.4472	−0.584797
$792$	0	0
$793$	−29.8596	−1.06035
$794$	0	0
$795$	−3.17698	−0.112676
$796$	0	0
$797$	−40.3900	−1.43069	−0.715343	−	0.698773i	$-0.753730\pi$
$797$	−40.3900	−1.43069	−0.715343	+	0.698773i	$0.753730\pi$
$798$	0	0
$799$	2.31654	0.0819533
$800$	0	0
$801$	−28.0522	−0.991177
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−1.09207	−0.0384904
$806$	0	0
$807$	−4.22816	−0.148838
$808$	0	0
$809$	−38.2759	−1.34571	−0.672855	−	0.739775i	$-0.734932\pi$
$809$	−38.2759	−1.34571	−0.672855	+	0.739775i	$0.734932\pi$
$810$	0	0
$811$	−33.8925	−1.19013	−0.595063	−	0.803679i	$-0.702873\pi$
$811$	−33.8925	−1.19013	−0.595063	+	0.803679i	$0.702873\pi$
$812$	0	0
$813$	−2.14238	−0.0751366
$814$	0	0
$815$	0.179097	0.00627349
$816$	0	0
$817$	30.2604	1.05868
$818$	0	0
$819$	−72.3944	−2.52967
$820$	0	0
$821$	49.5643	1.72980	0.864902	−	0.501940i	$-0.167380\pi$
$821$	49.5643	1.72980	0.864902	+	0.501940i	$0.167380\pi$
$822$	0	0
$823$	21.4293	0.746977	0.373488	−	0.927635i	$-0.378162\pi$
$823$	21.4293	0.746977	0.373488	+	0.927635i	$0.378162\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	41.0128	1.42615	0.713077	−	0.701086i	$-0.247301\pi$
$827$	41.0128	1.42615	0.713077	+	0.701086i	$0.247301\pi$
$828$	0	0
$829$	13.4849	0.468350	0.234175	−	0.972194i	$-0.424761\pi$
$829$	13.4849	0.468350	0.234175	+	0.972194i	$0.424761\pi$
$830$	0	0
$831$	8.08982	0.280633
$832$	0	0
$833$	4.32872	0.149981
$834$	0	0
$835$	−0.624982	−0.0216284
$836$	0	0
$837$	−62.7072	−2.16748
$838$	0	0
$839$	−35.5305	−1.22665	−0.613324	−	0.789832i	$-0.710168\pi$
$839$	−35.5305	−1.22665	−0.613324	+	0.789832i	$0.710168\pi$
$840$	0	0
$841$	14.6547	0.505334
$842$	0	0
$843$	64.5610	2.22360
$844$	0	0
$845$	−0.570213	−0.0196159
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−38.0951	−1.30742
$850$	0	0
$851$	25.1645	0.862628
$852$	0	0
$853$	−35.2623	−1.20736	−0.603680	−	0.797227i	$-0.706299\pi$
$853$	−35.2623	−1.20736	−0.603680	+	0.797227i	$0.706299\pi$
$854$	0	0
$855$	3.29789	0.112785
$856$	0	0
$857$	−50.3640	−1.72040	−0.860201	−	0.509955i	$-0.829662\pi$
$857$	−50.3640	−1.72040	−0.860201	+	0.509955i	$0.829662\pi$
$858$	0	0
$859$	10.0493	0.342876	0.171438	−	0.985195i	$-0.445159\pi$
$859$	10.0493	0.342876	0.171438	+	0.985195i	$0.445159\pi$
$860$	0	0
$861$	35.9546	1.22533
$862$	0	0
$863$	37.9846	1.29301	0.646505	−	0.762909i	$-0.276230\pi$
$863$	37.9846	1.29301	0.646505	+	0.762909i	$0.276230\pi$
$864$	0	0
$865$	−0.633702	−0.0215465
$866$	0	0
$867$	−53.2194	−1.80743
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−19.1200	−0.647855
$872$	0	0
$873$	−34.2529	−1.15929
$874$	0	0
$875$	−3.57488	−0.120853
$876$	0	0
$877$	−47.9508	−1.61918	−0.809592	−	0.586993i	$-0.800311\pi$
$877$	−47.9508	−1.61918	−0.809592	+	0.586993i	$0.800311\pi$
$878$	0	0
$879$	58.6180	1.97714
$880$	0	0
$881$	−26.0141	−0.876439	−0.438219	−	0.898868i	$-0.644391\pi$
$881$	−26.0141	−0.876439	−0.438219	+	0.898868i	$0.644391\pi$
$882$	0	0
$883$	5.74740	0.193415	0.0967077	−	0.995313i	$-0.469169\pi$
$883$	5.74740	0.193415	0.0967077	+	0.995313i	$0.469169\pi$
$884$	0	0
$885$	3.27810	0.110192
$886$	0	0
$887$	42.5731	1.42946	0.714732	−	0.699398i	$-0.246549\pi$
$887$	42.5731	1.42946	0.714732	+	0.699398i	$0.246549\pi$
$888$	0	0
$889$	−35.9304	−1.20507
$890$	0	0
$891$	0	0
$892$	0	0
$893$	21.9607	0.734886
$894$	0	0
$895$	0.733960	0.0245336
$896$	0	0
$897$	25.3282	0.845683
$898$	0	0
$899$	−31.4891	−1.05022
$900$	0	0
$901$	−5.76289	−0.191990
$902$	0	0
$903$	74.5474	2.48078
$904$	0	0
$905$	−0.813540	−0.0270430
$906$	0	0
$907$	−51.0644	−1.69557	−0.847783	−	0.530344i	$-0.822063\pi$
$907$	−51.0644	−1.69557	−0.847783	+	0.530344i	$0.822063\pi$
$908$	0	0
$909$	−34.4185	−1.14159
$910$	0	0
$911$	48.8976	1.62005	0.810026	−	0.586395i	$-0.199453\pi$
$911$	48.8976	1.62005	0.810026	+	0.586395i	$0.199453\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	3.35099	0.110780
$916$	0	0
$917$	11.5563	0.381623
$918$	0	0
$919$	59.1482	1.95112	0.975559	−	0.219738i	$-0.0705203\pi$
$919$	59.1482	1.95112	0.975559	+	0.219738i	$0.0705203\pi$
$920$	0	0
$921$	−67.8984	−2.23733
$922$	0	0
$923$	−29.2385	−0.962398
$924$	0	0
$925$	41.1530	1.35310
$926$	0	0
$927$	14.8086	0.486380
$928$	0	0
$929$	−0.781411	−0.0256373	−0.0128186	−	0.999918i	$-0.504080\pi$
$929$	−0.781411	−0.0256373	−0.0128186	+	0.999918i	$0.504080\pi$
$930$	0	0
$931$	41.0360	1.34490
$932$	0	0
$933$	−32.6325	−1.06834
$934$	0	0
$935$	0	0
$936$	0	0
$937$	5.79935	0.189457	0.0947283	−	0.995503i	$-0.469802\pi$
$937$	5.79935	0.189457	0.0947283	+	0.995503i	$0.469802\pi$
$938$	0	0
$939$	−72.4947	−2.36577
$940$	0	0
$941$	−43.8198	−1.42848	−0.714242	−	0.699898i	$-0.753229\pi$
$941$	−43.8198	−1.42848	−0.714242	+	0.699898i	$0.753229\pi$
$942$	0	0
$943$	−8.85508	−0.288361
$944$	0	0
$945$	4.70761	0.153139
$946$	0	0
$947$	−12.2117	−0.396828	−0.198414	−	0.980118i	$-0.563579\pi$
$947$	−12.2117	−0.396828	−0.198414	+	0.980118i	$0.563579\pi$
$948$	0	0
$949$	−6.58670	−0.213813
$950$	0	0
$951$	69.5897	2.25660
$952$	0	0
$953$	14.5487	0.471278	0.235639	−	0.971841i	$-0.424282\pi$
$953$	14.5487	0.471278	0.235639	+	0.971841i	$0.424282\pi$
$954$	0	0
$955$	1.81623	0.0587718
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−62.6273	−2.02234
$960$	0	0
$961$	−8.28623	−0.267298
$962$	0	0
$963$	−7.89002	−0.254253
$964$	0	0
$965$	−2.27194	−0.0731362
$966$	0	0
$967$	10.3647	0.333306	0.166653	−	0.986016i	$-0.446704\pi$
$967$	10.3647	0.333306	0.166653	+	0.986016i	$0.446704\pi$
$968$	0	0
$969$	8.49810	0.272998
$970$	0	0
$971$	−24.4508	−0.784663	−0.392332	−	0.919824i	$-0.628331\pi$
$971$	−24.4508	−0.784663	−0.392332	+	0.919824i	$0.628331\pi$
$972$	0	0
$973$	54.3064	1.74098
$974$	0	0
$975$	41.4207	1.32652
$976$	0	0
$977$	52.1861	1.66958	0.834790	−	0.550569i	$-0.185589\pi$
$977$	52.1861	1.66958	0.834790	+	0.550569i	$0.185589\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	8.47234	0.270501
$982$	0	0
$983$	−6.89154	−0.219806	−0.109903	−	0.993942i	$-0.535054\pi$
$983$	−6.89154	−0.219806	−0.109903	+	0.993942i	$0.535054\pi$
$984$	0	0
$985$	−0.864058	−0.0275312
$986$	0	0
$987$	54.1008	1.72205
$988$	0	0
$989$	−18.3599	−0.583811
$990$	0	0
$991$	−43.2270	−1.37315	−0.686576	−	0.727058i	$-0.740887\pi$
$991$	−43.2270	−1.37315	−0.686576	+	0.727058i	$0.740887\pi$
$992$	0	0
$993$	10.1303	0.321475
$994$	0	0
$995$	−0.639736	−0.0202810
$996$	0	0
$997$	9.56907	0.303056	0.151528	−	0.988453i	$-0.451581\pi$
$997$	9.56907	0.303056	0.151528	+	0.988453i	$0.451581\pi$
$998$	0	0
$999$	−108.477	−3.43207

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7744.2.a.dw.1.6		6
4.3	odd	2		7744.2.a.dt.1.1		6
8.3	odd	2		3872.2.a.bp.1.6		6
8.5	even	2		3872.2.a.bo.1.1		6
11.7	odd	10		704.2.m.n.577.1		12
11.8	odd	10		704.2.m.n.449.1		12
11.10	odd	2		7744.2.a.dv.1.6		6
44.7	even	10		704.2.m.m.577.3		12
44.19	even	10		704.2.m.m.449.3		12
44.43	even	2		7744.2.a.du.1.1		6
88.19	even	10		352.2.m.e.97.1	✓	12
88.21	odd	2		3872.2.a.bn.1.1		6
88.29	odd	10		352.2.m.f.225.3	yes	12
88.43	even	2		3872.2.a.bq.1.6		6
88.51	even	10		352.2.m.e.225.1	yes	12
88.85	odd	10		352.2.m.f.97.3	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
352.2.m.e.97.1	✓	12	88.19	even	10
352.2.m.e.225.1	yes	12	88.51	even	10
352.2.m.f.97.3	yes	12	88.85	odd	10
352.2.m.f.225.3	yes	12	88.29	odd	10
704.2.m.m.449.3		12	44.19	even	10
704.2.m.m.577.3		12	44.7	even	10
704.2.m.n.449.1		12	11.8	odd	10
704.2.m.n.577.1		12	11.7	odd	10
3872.2.a.bn.1.1		6	88.21	odd	2
3872.2.a.bo.1.1		6	8.5	even	2
3872.2.a.bp.1.6		6	8.3	odd	2
3872.2.a.bq.1.6		6	88.43	even	2
7744.2.a.dt.1.1		6	4.3	odd	2
7744.2.a.du.1.1		6	44.43	even	2
7744.2.a.dv.1.6		6	11.10	odd	2
7744.2.a.dw.1.6		6	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:	\( N \)	\(=\)	\( 7744 = 2^{6} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7744.a (trivial)

\(f(q)\)	\(=\)	\(q+3.18328 q^{3} +0.0919008 q^{5} +3.89322 q^{7} +7.13330 q^{9} -2.60679 q^{13} +0.292546 q^{15} +0.530665 q^{17} +5.03067 q^{19} +12.3932 q^{21} -3.05227 q^{23} -4.99155 q^{25} +13.1575 q^{27} +6.60717 q^{29} -4.76590 q^{31} +0.357790 q^{35} -8.24453 q^{37} -8.29815 q^{39} +2.90115 q^{41} +6.01518 q^{43} +0.655556 q^{45} +4.36536 q^{47} +8.15716 q^{49} +1.68926 q^{51} -10.8598 q^{53} +16.0141 q^{57} +11.2054 q^{59} +11.4546 q^{61} +27.7715 q^{63} -0.239566 q^{65} +7.33468 q^{67} -9.71623 q^{69} +11.2163 q^{71} +2.52675 q^{73} -15.8895 q^{75} -8.63148 q^{79} +20.4841 q^{81} +1.53368 q^{83} +0.0487685 q^{85} +21.0325 q^{87} -3.93257 q^{89} -10.1488 q^{91} -15.1712 q^{93} +0.462323 q^{95} -4.80184 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 5 q^{3} + 2 q^{7} + 7 q^{9} + 4 q^{13} - 8 q^{15} + 9 q^{17} - 5 q^{19} + 12 q^{21} - 6 q^{23} + 4 q^{25} + 26 q^{27} + 10 q^{29} - 12 q^{31} + 26 q^{35} + 12 q^{37} + 2 q^{39} + 17 q^{41} - 11 q^{43}+ \cdots - 21 q^{97}+O(q^{100}) \) 6 * q + 5 * q^3 + 2 * q^7 + 7 * q^9 + 4 * q^13 - 8 * q^15 + 9 * q^17 - 5 * q^19 + 12 * q^21 - 6 * q^23 + 4 * q^25 + 26 * q^27 + 10 * q^29 - 12 * q^31 + 26 * q^35 + 12 * q^37 + 2 * q^39 + 17 * q^41 - 11 * q^43 - 14 * q^45 - 22 * q^47 + 6 * q^51 - 18 * q^53 + 6 * q^57 + 7 * q^59 + 18 * q^61 + 26 * q^63 + 26 * q^65 + 31 * q^67 - 16 * q^69 - 22 * q^71 - q^73 + 11 * q^75 - 28 * q^79 + 34 * q^81 + 13 * q^83 + 2 * q^85 + 38 * q^87 - q^89 + 26 * q^91 - 34 * q^93 + 2 * q^95 - 21 * q^97

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