LMFDB - Embedded newform 6724.2.a.j.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6724,2,Mod(1,6724)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6724, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6724.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6724 = 2^{2} \cdot 41^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6724.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$53.6914103191$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 39x^{14} + 594x^{12} - 4428x^{10} + 16529x^{8} - 28236x^{6} + 17856x^{4} - 4032x^{2} + 256 $ x^16 - 39x^14 + 594x^12 - 4428x^10 + 16529x^8 - 28236x^6 + 17856x^4 - 4032*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 164)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$0.322385$ of defining polynomial
Character	$\chi$	$=$	6724.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.322385 q^{3} +2.66810 q^{5} -3.97926 q^{7} -2.89607 q^{9} +O(q^{10})$	$q+0.322385 q^{3} +2.66810 q^{5} -3.97926 q^{7} -2.89607 q^{9} -1.32070 q^{11} +0.557202 q^{13} +0.860158 q^{15} +1.80538 q^{17} -5.17178 q^{19} -1.28285 q^{21} +7.59512 q^{23} +2.11878 q^{25} -1.90081 q^{27} +5.27995 q^{29} -7.69901 q^{31} -0.425773 q^{33} -10.6171 q^{35} +11.4713 q^{37} +0.179634 q^{39} +0.0780374 q^{43} -7.72701 q^{45} -3.35841 q^{47} +8.83447 q^{49} +0.582027 q^{51} +9.05323 q^{53} -3.52375 q^{55} -1.66731 q^{57} +2.08768 q^{59} -6.32449 q^{61} +11.5242 q^{63} +1.48667 q^{65} -8.71817 q^{67} +2.44856 q^{69} -10.1054 q^{71} +9.63398 q^{73} +0.683065 q^{75} +5.25538 q^{77} +10.8632 q^{79} +8.07541 q^{81} +0.977095 q^{83} +4.81693 q^{85} +1.70218 q^{87} -13.1167 q^{89} -2.21725 q^{91} -2.48205 q^{93} -13.7988 q^{95} +9.66916 q^{97} +3.82482 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 6 q^{5} + 30 q^{9}+O(q^{10}) $ 16 * q + 6 * q^5 + 30 * q^9	$ 16 q + 6 q^{5} + 30 q^{9} + 46 q^{21} - 8 q^{23} + 38 q^{25} - 2 q^{31} + 20 q^{33} + 12 q^{37} + 8 q^{39} - 48 q^{43} + 92 q^{45} - 12 q^{49} + 42 q^{51} + 22 q^{57} - 32 q^{59} + 6 q^{61} - 34 q^{73} + 92 q^{77} + 108 q^{81} + 12 q^{83} - 34 q^{87}+O(q^{100}) $ 16 * q + 6 * q^5 + 30 * q^9 + 46 * q^21 - 8 * q^23 + 38 * q^25 - 2 * q^31 + 20 * q^33 + 12 * q^37 + 8 * q^39 - 48 * q^43 + 92 * q^45 - 12 * q^49 + 42 * q^51 + 22 * q^57 - 32 * q^59 + 6 * q^61 - 34 * q^73 + 92 * q^77 + 108 * q^81 + 12 * q^83 - 34 * q^87

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0.322385	0.186129	0.0930647	−	0.995660i	$-0.470334\pi$
$3$	0.322385	0.186129	0.0930647	+	0.995660i	$0.470334\pi$
$4$	0	0
$5$	2.66810	1.19321	0.596606	−	0.802534i	$-0.296515\pi$
$5$	2.66810	1.19321	0.596606	+	0.802534i	$0.296515\pi$
$6$	0	0
$7$	−3.97926	−1.50402	−0.752009	−	0.659153i	$-0.770915\pi$
$7$	−3.97926	−1.50402	−0.752009	+	0.659153i	$0.770915\pi$
$8$	0	0
$9$	−2.89607	−0.965356
$10$	0	0
$11$	−1.32070	−0.398205	−0.199102	−	0.979979i	$-0.563803\pi$
$11$	−1.32070	−0.398205	−0.199102	+	0.979979i	$0.563803\pi$
$12$	0	0
$13$	0.557202	0.154540	0.0772700	−	0.997010i	$-0.475380\pi$
$13$	0.557202	0.154540	0.0772700	+	0.997010i	$0.475380\pi$
$14$	0	0
$15$	0.860158	0.222092
$16$	0	0
$17$	1.80538	0.437868	0.218934	−	0.975740i	$-0.429742\pi$
$17$	1.80538	0.437868	0.218934	+	0.975740i	$0.429742\pi$
$18$	0	0
$19$	−5.17178	−1.18649	−0.593243	−	0.805023i	$-0.702153\pi$
$19$	−5.17178	−1.18649	−0.593243	+	0.805023i	$0.702153\pi$
$20$	0	0
$21$	−1.28285	−0.279942
$22$	0	0
$23$	7.59512	1.58369	0.791846	−	0.610721i	$-0.209120\pi$
$23$	7.59512	1.58369	0.791846	+	0.610721i	$0.209120\pi$
$24$	0	0
$25$	2.11878	0.423757
$26$	0	0
$27$	−1.90081	−0.365810
$28$	0	0
$29$	5.27995	0.980462	0.490231	−	0.871593i	$-0.336912\pi$
$29$	5.27995	0.980462	0.490231	+	0.871593i	$0.336912\pi$
$30$	0	0
$31$	−7.69901	−1.38278	−0.691392	−	0.722480i	$-0.743002\pi$
$31$	−7.69901	−1.38278	−0.691392	+	0.722480i	$0.743002\pi$
$32$	0	0
$33$	−0.425773	−0.0741175
$34$	0	0
$35$	−10.6171	−1.79461
$36$	0	0
$37$	11.4713	1.88587	0.942937	−	0.332971i	$-0.108051\pi$
$37$	11.4713	1.88587	0.942937	+	0.332971i	$0.108051\pi$
$38$	0	0
$39$	0.179634	0.0287644
$40$	0	0
$41$	0	0
$41$	0	0
$42$	0	0
$43$	0.0780374	0.0119006	0.00595029	−	0.999982i	$-0.498106\pi$
$43$	0.0780374	0.0119006	0.00595029	+	0.999982i	$0.498106\pi$
$44$	0	0
$45$	−7.72701	−1.15187
$46$	0	0
$47$	−3.35841	−0.489874	−0.244937	−	0.969539i	$-0.578767\pi$
$47$	−3.35841	−0.489874	−0.244937	+	0.969539i	$0.578767\pi$
$48$	0	0
$49$	8.83447	1.26207
$50$	0	0
$51$	0.582027	0.0815000
$52$	0	0
$53$	9.05323	1.24356	0.621779	−	0.783193i	$-0.286410\pi$
$53$	9.05323	1.24356	0.621779	+	0.783193i	$0.286410\pi$
$54$	0	0
$55$	−3.52375	−0.475143
$56$	0	0
$57$	−1.66731	−0.220840
$58$	0	0
$59$	2.08768	0.271793	0.135896	−	0.990723i	$-0.456609\pi$
$59$	2.08768	0.271793	0.135896	+	0.990723i	$0.456609\pi$
$60$	0	0
$61$	−6.32449	−0.809768	−0.404884	−	0.914368i	$-0.632688\pi$
$61$	−6.32449	−0.809768	−0.404884	+	0.914368i	$0.632688\pi$
$62$	0	0
$63$	11.5242	1.45191
$64$	0	0
$65$	1.48667	0.184399
$66$	0	0
$67$	−8.71817	−1.06509	−0.532547	−	0.846401i	$-0.678765\pi$
$67$	−8.71817	−1.06509	−0.532547	+	0.846401i	$0.678765\pi$
$68$	0	0
$69$	2.44856	0.294771
$70$	0	0
$71$	−10.1054	−1.19929	−0.599643	−	0.800267i	$-0.704691\pi$
$71$	−10.1054	−1.19929	−0.599643	+	0.800267i	$0.704691\pi$
$72$	0	0
$73$	9.63398	1.12757	0.563786	−	0.825921i	$-0.309344\pi$
$73$	9.63398	1.12757	0.563786	+	0.825921i	$0.309344\pi$
$74$	0	0
$75$	0.683065	0.0788736
$76$	0	0
$77$	5.25538	0.598906
$78$	0	0
$79$	10.8632	1.22220	0.611101	−	0.791553i	$-0.290727\pi$
$79$	10.8632	1.22220	0.611101	+	0.791553i	$0.290727\pi$
$80$	0	0
$81$	8.07541	0.897268
$82$	0	0
$83$	0.977095	0.107250	0.0536251	−	0.998561i	$-0.482922\pi$
$83$	0.977095	0.107250	0.0536251	+	0.998561i	$0.482922\pi$
$84$	0	0
$85$	4.81693	0.522469
$86$	0	0
$87$	1.70218	0.182493
$88$	0	0
$89$	−13.1167	−1.39037	−0.695186	−	0.718830i	$-0.744678\pi$
$89$	−13.1167	−1.39037	−0.695186	+	0.718830i	$0.744678\pi$
$90$	0	0
$91$	−2.21725	−0.232431
$92$	0	0
$93$	−2.48205	−0.257377
$94$	0	0
$95$	−13.7988	−1.41573
$96$	0	0
$97$	9.66916	0.981754	0.490877	−	0.871229i	$-0.336676\pi$
$97$	9.66916	0.981754	0.490877	+	0.871229i	$0.336676\pi$
$98$	0	0
$99$	3.82482	0.384409
$100$	0	0
$101$	−10.5982	−1.05456	−0.527280	−	0.849692i	$-0.676788\pi$
$101$	−10.5982	−1.05456	−0.527280	+	0.849692i	$0.676788\pi$
$102$	0	0
$103$	7.32885	0.722133	0.361066	−	0.932540i	$-0.382413\pi$
$103$	7.32885	0.722133	0.361066	+	0.932540i	$0.382413\pi$
$104$	0	0
$105$	−3.42279	−0.334030
$106$	0	0
$107$	6.61132	0.639141	0.319570	−	0.947563i	$-0.396461\pi$
$107$	6.61132	0.639141	0.319570	+	0.947563i	$0.396461\pi$
$108$	0	0
$109$	3.56001	0.340987	0.170494	−	0.985359i	$-0.445464\pi$
$109$	3.56001	0.340987	0.170494	+	0.985359i	$0.445464\pi$
$110$	0	0
$111$	3.69819	0.351017
$112$	0	0
$113$	4.99639	0.470021	0.235010	−	0.971993i	$-0.424488\pi$
$113$	4.99639	0.470021	0.235010	+	0.971993i	$0.424488\pi$
$114$	0	0
$115$	20.2646	1.88968
$116$	0	0
$117$	−1.61369	−0.149186
$118$	0	0
$119$	−7.18405	−0.658561
$120$	0	0
$121$	−9.25576	−0.841433
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−7.68739	−0.687581
$126$	0	0
$127$	17.2992	1.53506	0.767529	−	0.641014i	$-0.221486\pi$
$127$	17.2992	1.53506	0.767529	+	0.641014i	$0.221486\pi$
$128$	0	0
$129$	0.0251581	0.00221505
$130$	0	0
$131$	8.52501	0.744833	0.372417	−	0.928066i	$-0.378529\pi$
$131$	8.52501	0.744833	0.372417	+	0.928066i	$0.378529\pi$
$132$	0	0
$133$	20.5798	1.78450
$134$	0	0
$135$	−5.07155	−0.436490
$136$	0	0
$137$	6.33985	0.541650	0.270825	−	0.962629i	$-0.412704\pi$
$137$	6.33985	0.541650	0.270825	+	0.962629i	$0.412704\pi$
$138$	0	0
$139$	4.91982	0.417294	0.208647	−	0.977991i	$-0.433094\pi$
$139$	4.91982	0.417294	0.208647	+	0.977991i	$0.433094\pi$
$140$	0	0
$141$	−1.08270	−0.0911799
$142$	0	0
$143$	−0.735894	−0.0615385
$144$	0	0
$145$	14.0875	1.16990
$146$	0	0
$147$	2.84811	0.234908
$148$	0	0
$149$	23.1721	1.89833	0.949166	−	0.314775i	$-0.101929\pi$
$149$	23.1721	1.89833	0.949166	+	0.314775i	$0.101929\pi$
$150$	0	0
$151$	10.3952	0.845948	0.422974	−	0.906142i	$-0.360986\pi$
$151$	10.3952	0.845948	0.422974	+	0.906142i	$0.360986\pi$
$152$	0	0
$153$	−5.22849	−0.422698
$154$	0	0
$155$	−20.5418	−1.64995
$156$	0	0
$157$	16.3674	1.30626	0.653130	−	0.757246i	$-0.273456\pi$
$157$	16.3674	1.30626	0.653130	+	0.757246i	$0.273456\pi$
$158$	0	0
$159$	2.91863	0.231463
$160$	0	0
$161$	−30.2229	−2.38190
$162$	0	0
$163$	15.6925	1.22913	0.614564	−	0.788867i	$-0.289332\pi$
$163$	15.6925	1.22913	0.614564	+	0.788867i	$0.289332\pi$
$164$	0	0
$165$	−1.13601	−0.0884380
$166$	0	0
$167$	11.2182	0.868094	0.434047	−	0.900890i	$-0.357085\pi$
$167$	11.2182	0.868094	0.434047	+	0.900890i	$0.357085\pi$
$168$	0	0
$169$	−12.6895	−0.976117
$170$	0	0
$171$	14.9778	1.14538
$172$	0	0
$173$	25.4129	1.93211	0.966053	−	0.258343i	$-0.0831766\pi$
$173$	25.4129	1.93211	0.966053	+	0.258343i	$0.0831766\pi$
$174$	0	0
$175$	−8.43118	−0.637337
$176$	0	0
$177$	0.673037	0.0505886
$178$	0	0
$179$	10.6717	0.797639	0.398819	−	0.917030i	$-0.369420\pi$
$179$	10.6717	0.797639	0.398819	+	0.917030i	$0.369420\pi$
$180$	0	0
$181$	−2.13074	−0.158377	−0.0791884	−	0.996860i	$-0.525233\pi$
$181$	−2.13074	−0.158377	−0.0791884	+	0.996860i	$0.525233\pi$
$182$	0	0
$183$	−2.03892	−0.150722
$184$	0	0
$185$	30.6067	2.25025
$186$	0	0
$187$	−2.38435	−0.174361
$188$	0	0
$189$	7.56379	0.550185
$190$	0	0
$191$	−17.2709	−1.24968	−0.624839	−	0.780754i	$-0.714835\pi$
$191$	−17.2709	−1.24968	−0.624839	+	0.780754i	$0.714835\pi$
$192$	0	0
$193$	−19.1561	−1.37889	−0.689444	−	0.724339i	$-0.742145\pi$
$193$	−19.1561	−1.37889	−0.689444	+	0.724339i	$0.742145\pi$
$194$	0	0
$195$	0.479282	0.0343221
$196$	0	0
$197$	4.38654	0.312528	0.156264	−	0.987715i	$-0.450055\pi$
$197$	4.38654	0.312528	0.156264	+	0.987715i	$0.450055\pi$
$198$	0	0
$199$	20.1231	1.42649	0.713245	−	0.700915i	$-0.247225\pi$
$199$	20.1231	1.42649	0.713245	+	0.700915i	$0.247225\pi$
$200$	0	0
$201$	−2.81061	−0.198245
$202$	0	0
$203$	−21.0103	−1.47463
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−21.9960	−1.52883
$208$	0	0
$209$	6.83034	0.472464
$210$	0	0
$211$	4.41134	0.303689	0.151845	−	0.988404i	$-0.451479\pi$
$211$	4.41134	0.303689	0.151845	+	0.988404i	$0.451479\pi$
$212$	0	0
$213$	−3.25782	−0.223222
$214$	0	0
$215$	0.208212	0.0141999
$216$	0	0
$217$	30.6363	2.07973
$218$	0	0
$219$	3.10586	0.209874
$220$	0	0
$221$	1.00596	0.0676681
$222$	0	0
$223$	7.83960	0.524978	0.262489	−	0.964935i	$-0.415457\pi$
$223$	7.83960	0.524978	0.262489	+	0.964935i	$0.415457\pi$
$224$	0	0
$225$	−6.13614	−0.409076
$226$	0	0
$227$	1.25770	0.0834764	0.0417382	−	0.999129i	$-0.486710\pi$
$227$	1.25770	0.0834764	0.0417382	+	0.999129i	$0.486710\pi$
$228$	0	0
$229$	9.21705	0.609080	0.304540	−	0.952500i	$-0.401497\pi$
$229$	9.21705	0.609080	0.304540	+	0.952500i	$0.401497\pi$
$230$	0	0
$231$	1.69426	0.111474
$232$	0	0
$233$	−10.9231	−0.715596	−0.357798	−	0.933799i	$-0.616472\pi$
$233$	−10.9231	−0.715596	−0.357798	+	0.933799i	$0.616472\pi$
$234$	0	0
$235$	−8.96058	−0.584524
$236$	0	0
$237$	3.50213	0.227488
$238$	0	0
$239$	24.1067	1.55933	0.779666	−	0.626195i	$-0.215389\pi$
$239$	24.1067	1.55933	0.779666	+	0.626195i	$0.215389\pi$
$240$	0	0
$241$	7.21028	0.464455	0.232228	−	0.972661i	$-0.425399\pi$
$241$	7.21028	0.464455	0.232228	+	0.972661i	$0.425399\pi$
$242$	0	0
$243$	8.30581	0.532818
$244$	0	0
$245$	23.5713	1.50591
$246$	0	0
$247$	−2.88172	−0.183360
$248$	0	0
$249$	0.315001	0.0199624
$250$	0	0
$251$	−15.0699	−0.951206	−0.475603	−	0.879660i	$-0.657770\pi$
$251$	−15.0699	−0.951206	−0.475603	+	0.879660i	$0.657770\pi$
$252$	0	0
$253$	−10.0308	−0.630633
$254$	0	0
$255$	1.55291	0.0972469
$256$	0	0
$257$	2.97232	0.185408	0.0927042	−	0.995694i	$-0.470449\pi$
$257$	2.97232	0.185408	0.0927042	+	0.995694i	$0.470449\pi$
$258$	0	0
$259$	−45.6473	−2.83639
$260$	0	0
$261$	−15.2911	−0.946495
$262$	0	0
$263$	−11.9733	−0.738304	−0.369152	−	0.929369i	$-0.620352\pi$
$263$	−11.9733	−0.738304	−0.369152	+	0.929369i	$0.620352\pi$
$264$	0	0
$265$	24.1550	1.48383
$266$	0	0
$267$	−4.22865	−0.258789
$268$	0	0
$269$	−14.4142	−0.878850	−0.439425	−	0.898279i	$-0.644818\pi$
$269$	−14.4142	−0.878850	−0.439425	+	0.898279i	$0.644818\pi$
$270$	0	0
$271$	21.3659	1.29788	0.648942	−	0.760838i	$-0.275212\pi$
$271$	21.3659	1.29788	0.648942	+	0.760838i	$0.275212\pi$
$272$	0	0
$273$	−0.714809	−0.0432622
$274$	0	0
$275$	−2.79827	−0.168742
$276$	0	0
$277$	25.2905	1.51956	0.759779	−	0.650181i	$-0.225307\pi$
$277$	25.2905	1.51956	0.759779	+	0.650181i	$0.225307\pi$
$278$	0	0
$279$	22.2969	1.33488
$280$	0	0
$281$	−14.6434	−0.873552	−0.436776	−	0.899570i	$-0.643880\pi$
$281$	−14.6434	−0.873552	−0.436776	+	0.899570i	$0.643880\pi$
$282$	0	0
$283$	−17.4187	−1.03544	−0.517718	−	0.855552i	$-0.673218\pi$
$283$	−17.4187	−1.03544	−0.517718	+	0.855552i	$0.673218\pi$
$284$	0	0
$285$	−4.44855	−0.263509
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.7406	−0.808272
$290$	0	0
$291$	3.11720	0.182733
$292$	0	0
$293$	19.9102	1.16317	0.581583	−	0.813487i	$-0.302433\pi$
$293$	19.9102	1.16317	0.581583	+	0.813487i	$0.302433\pi$
$294$	0	0
$295$	5.57014	0.324306
$296$	0	0
$297$	2.51039	0.145667
$298$	0	0
$299$	4.23201	0.244744
$300$	0	0
$301$	−0.310531	−0.0178987
$302$	0	0
$303$	−3.41670	−0.196285
$304$	0	0
$305$	−16.8744	−0.966225
$306$	0	0
$307$	−31.0281	−1.77087	−0.885433	−	0.464767i	$-0.846138\pi$
$307$	−31.0281	−1.77087	−0.885433	+	0.464767i	$0.846138\pi$
$308$	0	0
$309$	2.36271	0.134410
$310$	0	0
$311$	18.4589	1.04671	0.523353	−	0.852116i	$-0.324681\pi$
$311$	18.4589	1.04671	0.523353	+	0.852116i	$0.324681\pi$
$312$	0	0
$313$	−1.71723	−0.0970636	−0.0485318	−	0.998822i	$-0.515454\pi$
$313$	−1.71723	−0.0970636	−0.0485318	+	0.998822i	$0.515454\pi$
$314$	0	0
$315$	30.7478	1.73244
$316$	0	0
$317$	−11.3547	−0.637741	−0.318870	−	0.947798i	$-0.603303\pi$
$317$	−11.3547	−0.637741	−0.318870	+	0.947798i	$0.603303\pi$
$318$	0	0
$319$	−6.97320	−0.390424
$320$	0	0
$321$	2.13139	0.118963
$322$	0	0
$323$	−9.33699	−0.519524
$324$	0	0
$325$	1.18059	0.0654874
$326$	0	0
$327$	1.14770	0.0634678
$328$	0	0
$329$	13.3640	0.736779
$330$	0	0
$331$	9.45566	0.519730	0.259865	−	0.965645i	$-0.416322\pi$
$331$	9.45566	0.519730	0.259865	+	0.965645i	$0.416322\pi$
$332$	0	0
$333$	−33.2217	−1.82054
$334$	0	0
$335$	−23.2610	−1.27088
$336$	0	0
$337$	−13.7436	−0.748660	−0.374330	−	0.927295i	$-0.622127\pi$
$337$	−13.7436	−0.748660	−0.374330	+	0.927295i	$0.622127\pi$
$338$	0	0
$339$	1.61076	0.0874846
$340$	0	0
$341$	10.1680	0.550631
$342$	0	0
$343$	−7.29983	−0.394154
$344$	0	0
$345$	6.53300	0.351725
$346$	0	0
$347$	−14.7192	−0.790167	−0.395083	−	0.918645i	$-0.629284\pi$
$347$	−14.7192	−0.790167	−0.395083	+	0.918645i	$0.629284\pi$
$348$	0	0
$349$	−21.4731	−1.14943	−0.574713	−	0.818355i	$-0.694886\pi$
$349$	−21.4731	−1.14943	−0.574713	+	0.818355i	$0.694886\pi$
$350$	0	0
$351$	−1.05913	−0.0565323
$352$	0	0
$353$	36.2052	1.92701	0.963503	−	0.267696i	$-0.0862624\pi$
$353$	36.2052	1.92701	0.963503	+	0.267696i	$0.0862624\pi$
$354$	0	0
$355$	−26.9622	−1.43100
$356$	0	0
$357$	−2.31603	−0.122577
$358$	0	0
$359$	−36.1445	−1.90763	−0.953816	−	0.300390i	$-0.902883\pi$
$359$	−36.1445	−1.90763	−0.953816	+	0.300390i	$0.902883\pi$
$360$	0	0
$361$	7.74726	0.407750
$362$	0	0
$363$	−2.98392	−0.156615
$364$	0	0
$365$	25.7045	1.34543
$366$	0	0
$367$	23.2411	1.21318	0.606588	−	0.795016i	$-0.292538\pi$
$367$	23.2411	1.21318	0.606588	+	0.795016i	$0.292538\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−36.0251	−1.87033
$372$	0	0
$373$	−1.79300	−0.0928380	−0.0464190	−	0.998922i	$-0.514781\pi$
$373$	−1.79300	−0.0928380	−0.0464190	+	0.998922i	$0.514781\pi$
$374$	0	0
$375$	−2.47830	−0.127979
$376$	0	0
$377$	2.94200	0.151521
$378$	0	0
$379$	7.13926	0.366719	0.183359	−	0.983046i	$-0.441303\pi$
$379$	7.13926	0.366719	0.183359	+	0.983046i	$0.441303\pi$
$380$	0	0
$381$	5.57702	0.285719
$382$	0	0
$383$	−18.9386	−0.967715	−0.483858	−	0.875147i	$-0.660765\pi$
$383$	−18.9386	−0.967715	−0.483858	+	0.875147i	$0.660765\pi$
$384$	0	0
$385$	14.0219	0.714623
$386$	0	0
$387$	−0.226002	−0.0114883
$388$	0	0
$389$	15.5030	0.786034	0.393017	−	0.919531i	$-0.371431\pi$
$389$	15.5030	0.786034	0.393017	+	0.919531i	$0.371431\pi$
$390$	0	0
$391$	13.7120	0.693448
$392$	0	0
$393$	2.74834	0.138635
$394$	0	0
$395$	28.9841	1.45835
$396$	0	0
$397$	3.86127	0.193792	0.0968958	−	0.995295i	$-0.469109\pi$
$397$	3.86127	0.193792	0.0968958	+	0.995295i	$0.469109\pi$
$398$	0	0
$399$	6.63463	0.332147
$400$	0	0
$401$	10.2685	0.512786	0.256393	−	0.966573i	$-0.417466\pi$
$401$	10.2685	0.512786	0.256393	+	0.966573i	$0.417466\pi$
$402$	0	0
$403$	−4.28990	−0.213695
$404$	0	0
$405$	21.5460	1.07063
$406$	0	0
$407$	−15.1501	−0.750964
$408$	0	0
$409$	26.2789	1.29941	0.649705	−	0.760187i	$-0.274892\pi$
$409$	26.2789	1.29941	0.649705	+	0.760187i	$0.274892\pi$
$410$	0	0
$411$	2.04388	0.100817
$412$	0	0
$413$	−8.30740	−0.408781
$414$	0	0
$415$	2.60699	0.127972
$416$	0	0
$417$	1.58608	0.0776707
$418$	0	0
$419$	−19.8559	−0.970023	−0.485012	−	0.874508i	$-0.661185\pi$
$419$	−19.8559	−0.970023	−0.485012	+	0.874508i	$0.661185\pi$
$420$	0	0
$421$	−35.6499	−1.73747	−0.868735	−	0.495278i	$-0.835066\pi$
$421$	−35.6499	−1.73747	−0.868735	+	0.495278i	$0.835066\pi$
$422$	0	0
$423$	9.72617	0.472903
$424$	0	0
$425$	3.82520	0.185549
$426$	0	0
$427$	25.1668	1.21790
$428$	0	0
$429$	−0.237241	−0.0114541
$430$	0	0
$431$	14.1320	0.680713	0.340356	−	0.940297i	$-0.389452\pi$
$431$	14.1320	0.680713	0.340356	+	0.940297i	$0.389452\pi$
$432$	0	0
$433$	11.5266	0.553931	0.276966	−	0.960880i	$-0.410671\pi$
$433$	11.5266	0.553931	0.276966	+	0.960880i	$0.410671\pi$
$434$	0	0
$435$	4.54159	0.217753
$436$	0	0
$437$	−39.2802	−1.87903
$438$	0	0
$439$	−6.25709	−0.298635	−0.149317	−	0.988789i	$-0.547708\pi$
$439$	−6.25709	−0.298635	−0.149317	+	0.988789i	$0.547708\pi$
$440$	0	0
$441$	−25.5852	−1.21834
$442$	0	0
$443$	−14.5668	−0.692089	−0.346045	−	0.938218i	$-0.612475\pi$
$443$	−14.5668	−0.692089	−0.346045	+	0.938218i	$0.612475\pi$
$444$	0	0
$445$	−34.9968	−1.65901
$446$	0	0
$447$	7.47035	0.353335
$448$	0	0
$449$	−14.4605	−0.682435	−0.341218	−	0.939984i	$-0.610839\pi$
$449$	−14.4605	−0.682435	−0.341218	+	0.939984i	$0.610839\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	3.35126	0.157456
$454$	0	0
$455$	−5.91585	−0.277339
$456$	0	0
$457$	11.7284	0.548633	0.274316	−	0.961639i	$-0.411548\pi$
$457$	11.7284	0.548633	0.274316	+	0.961639i	$0.411548\pi$
$458$	0	0
$459$	−3.43167	−0.160177
$460$	0	0
$461$	29.3866	1.36867	0.684335	−	0.729168i	$-0.260093\pi$
$461$	29.3866	1.36867	0.684335	+	0.729168i	$0.260093\pi$
$462$	0	0
$463$	30.6866	1.42613	0.713064	−	0.701099i	$-0.247307\pi$
$463$	30.6866	1.42613	0.713064	+	0.701099i	$0.247307\pi$
$464$	0	0
$465$	−6.62237	−0.307105
$466$	0	0
$467$	15.6288	0.723216	0.361608	−	0.932330i	$-0.382228\pi$
$467$	15.6288	0.723216	0.361608	+	0.932330i	$0.382228\pi$
$468$	0	0
$469$	34.6918	1.60192
$470$	0	0
$471$	5.27661	0.243133
$472$	0	0
$473$	−0.103064	−0.00473887
$474$	0	0
$475$	−10.9579	−0.502782
$476$	0	0
$477$	−26.2188	−1.20048
$478$	0	0
$479$	−15.2926	−0.698736	−0.349368	−	0.936986i	$-0.613604\pi$
$479$	−15.2926	−0.698736	−0.349368	+	0.936986i	$0.613604\pi$
$480$	0	0
$481$	6.39184	0.291443
$482$	0	0
$483$	−9.74343	−0.443341
$484$	0	0
$485$	25.7983	1.17144
$486$	0	0
$487$	−19.5991	−0.888121	−0.444060	−	0.895997i	$-0.646462\pi$
$487$	−19.5991	−0.888121	−0.444060	+	0.895997i	$0.646462\pi$
$488$	0	0
$489$	5.05902	0.228777
$490$	0	0
$491$	−34.0426	−1.53632	−0.768160	−	0.640258i	$-0.778828\pi$
$491$	−34.0426	−1.53632	−0.768160	+	0.640258i	$0.778828\pi$
$492$	0	0
$493$	9.53229	0.429313
$494$	0	0
$495$	10.2050	0.458682
$496$	0	0
$497$	40.2119	1.80375
$498$	0	0
$499$	−44.1702	−1.97733	−0.988665	−	0.150137i	$-0.952028\pi$
$499$	−44.1702	−1.97733	−0.988665	+	0.150137i	$0.952028\pi$
$500$	0	0
$501$	3.61660	0.161578
$502$	0	0
$503$	−7.46217	−0.332722	−0.166361	−	0.986065i	$-0.553202\pi$
$503$	−7.46217	−0.332722	−0.166361	+	0.986065i	$0.553202\pi$
$504$	0	0
$505$	−28.2771	−1.25831
$506$	0	0
$507$	−4.09092	−0.181684
$508$	0	0
$509$	−38.6546	−1.71334	−0.856668	−	0.515869i	$-0.827469\pi$
$509$	−38.6546	−1.71334	−0.856668	+	0.515869i	$0.827469\pi$
$510$	0	0
$511$	−38.3361	−1.69589
$512$	0	0
$513$	9.83054	0.434029
$514$	0	0
$515$	19.5541	0.861658
$516$	0	0
$517$	4.43543	0.195070
$518$	0	0
$519$	8.19275	0.359622
$520$	0	0
$521$	22.1855	0.971963	0.485981	−	0.873969i	$-0.338462\pi$
$521$	22.1855	0.971963	0.485981	+	0.873969i	$0.338462\pi$
$522$	0	0
$523$	20.1698	0.881962	0.440981	−	0.897516i	$-0.354631\pi$
$523$	20.1698	0.881962	0.440981	+	0.897516i	$0.354631\pi$
$524$	0	0
$525$	−2.71809	−0.118627
$526$	0	0
$527$	−13.8996	−0.605476
$528$	0	0
$529$	34.6858	1.50808
$530$	0	0
$531$	−6.04606	−0.262377
$532$	0	0
$533$	0	0
$534$	0	0
$535$	17.6397	0.762631
$536$	0	0
$537$	3.44039	0.148464
$538$	0	0
$539$	−11.6676	−0.502561
$540$	0	0
$541$	−14.8144	−0.636920	−0.318460	−	0.947936i	$-0.603166\pi$
$541$	−14.8144	−0.636920	−0.318460	+	0.947936i	$0.603166\pi$
$542$	0	0
$543$	−0.686920	−0.0294786
$544$	0	0
$545$	9.49849	0.406871
$546$	0	0
$547$	17.1232	0.732137	0.366069	−	0.930588i	$-0.380704\pi$
$547$	17.1232	0.732137	0.366069	+	0.930588i	$0.380704\pi$
$548$	0	0
$549$	18.3161	0.781714
$550$	0	0
$551$	−27.3067	−1.16330
$552$	0	0
$553$	−43.2273	−1.83821
$554$	0	0
$555$	9.86715	0.418837
$556$	0	0
$557$	−6.76858	−0.286794	−0.143397	−	0.989665i	$-0.545803\pi$
$557$	−6.76858	−0.286794	−0.143397	+	0.989665i	$0.545803\pi$
$558$	0	0
$559$	0.0434826	0.00183912
$560$	0	0
$561$	−0.768680	−0.0324537
$562$	0	0
$563$	−46.5483	−1.96178	−0.980889	−	0.194566i	$-0.937670\pi$
$563$	−46.5483	−1.96178	−0.980889	+	0.194566i	$0.937670\pi$
$564$	0	0
$565$	13.3309	0.560835
$566$	0	0
$567$	−32.1341	−1.34951
$568$	0	0
$569$	6.21085	0.260373	0.130186	−	0.991490i	$-0.458442\pi$
$569$	6.21085	0.260373	0.130186	+	0.991490i	$0.458442\pi$
$570$	0	0
$571$	36.8209	1.54091	0.770454	−	0.637495i	$-0.220029\pi$
$571$	36.8209	1.54091	0.770454	+	0.637495i	$0.220029\pi$
$572$	0	0
$573$	−5.56788	−0.232602
$574$	0	0
$575$	16.0924	0.671100
$576$	0	0
$577$	1.20820	0.0502982	0.0251491	−	0.999684i	$-0.491994\pi$
$577$	1.20820	0.0502982	0.0251491	+	0.999684i	$0.491994\pi$
$578$	0	0
$579$	−6.17565	−0.256651
$580$	0	0
$581$	−3.88811	−0.161306
$582$	0	0
$583$	−11.9566	−0.495190
$584$	0	0
$585$	−4.30551	−0.178011
$586$	0	0
$587$	−7.47929	−0.308703	−0.154352	−	0.988016i	$-0.549329\pi$
$587$	−7.47929	−0.308703	−0.154352	+	0.988016i	$0.549329\pi$
$588$	0	0
$589$	39.8176	1.64065
$590$	0	0
$591$	1.41416	0.0581706
$592$	0	0
$593$	−22.1727	−0.910524	−0.455262	−	0.890358i	$-0.650454\pi$
$593$	−22.1727	−0.910524	−0.455262	+	0.890358i	$0.650454\pi$
$594$	0	0
$595$	−19.1678	−0.785803
$596$	0	0
$597$	6.48740	0.265512
$598$	0	0
$599$	−35.6021	−1.45466	−0.727331	−	0.686287i	$-0.759239\pi$
$599$	−35.6021	−1.45466	−0.727331	+	0.686287i	$0.759239\pi$
$600$	0	0
$601$	9.57919	0.390743	0.195372	−	0.980729i	$-0.437409\pi$
$601$	9.57919	0.390743	0.195372	+	0.980729i	$0.437409\pi$
$602$	0	0
$603$	25.2484	1.02819
$604$	0	0
$605$	−24.6954	−1.00401
$606$	0	0
$607$	24.6236	0.999443	0.499721	−	0.866186i	$-0.333436\pi$
$607$	24.6236	0.999443	0.499721	+	0.866186i	$0.333436\pi$
$608$	0	0
$609$	−6.77340	−0.274472
$610$	0	0
$611$	−1.87131	−0.0757051
$612$	0	0
$613$	−1.01671	−0.0410646	−0.0205323	−	0.999789i	$-0.506536\pi$
$613$	−1.01671	−0.0410646	−0.0205323	+	0.999789i	$0.506536\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	17.1500	0.690432	0.345216	−	0.938523i	$-0.387806\pi$
$617$	17.1500	0.690432	0.345216	+	0.938523i	$0.387806\pi$
$618$	0	0
$619$	−22.5414	−0.906016	−0.453008	−	0.891506i	$-0.649649\pi$
$619$	−22.5414	−0.906016	−0.453008	+	0.891506i	$0.649649\pi$
$620$	0	0
$621$	−14.4369	−0.579331
$622$	0	0
$623$	52.1949	2.09114
$624$	0	0
$625$	−31.1047	−1.24419
$626$	0	0
$627$	2.20200	0.0879395
$628$	0	0
$629$	20.7100	0.825763
$630$	0	0
$631$	−1.02704	−0.0408860	−0.0204430	−	0.999791i	$-0.506508\pi$
$631$	−1.02704	−0.0408860	−0.0204430	+	0.999791i	$0.506508\pi$
$632$	0	0
$633$	1.42215	0.0565255
$634$	0	0
$635$	46.1562	1.83165
$636$	0	0
$637$	4.92258	0.195040
$638$	0	0
$639$	29.2658	1.15774
$640$	0	0
$641$	5.15819	0.203736	0.101868	−	0.994798i	$-0.467518\pi$
$641$	5.15819	0.203736	0.101868	+	0.994798i	$0.467518\pi$
$642$	0	0
$643$	33.9530	1.33898	0.669489	−	0.742822i	$-0.266513\pi$
$643$	33.9530	1.33898	0.669489	+	0.742822i	$0.266513\pi$
$644$	0	0
$645$	0.0671245	0.00264302
$646$	0	0
$647$	33.2144	1.30579	0.652896	−	0.757447i	$-0.273554\pi$
$647$	33.2144	1.30579	0.652896	+	0.757447i	$0.273554\pi$
$648$	0	0
$649$	−2.75719	−0.108229
$650$	0	0
$651$	9.87671	0.387099
$652$	0	0
$653$	−12.8872	−0.504314	−0.252157	−	0.967686i	$-0.581140\pi$
$653$	−12.8872	−0.504314	−0.252157	+	0.967686i	$0.581140\pi$
$654$	0	0
$655$	22.7456	0.888745
$656$	0	0
$657$	−27.9007	−1.08851
$658$	0	0
$659$	−11.9650	−0.466088	−0.233044	−	0.972466i	$-0.574869\pi$
$659$	−11.9650	−0.466088	−0.233044	+	0.972466i	$0.574869\pi$
$660$	0	0
$661$	15.4571	0.601211	0.300605	−	0.953749i	$-0.402811\pi$
$661$	15.4571	0.601211	0.300605	+	0.953749i	$0.402811\pi$
$662$	0	0
$663$	0.324306	0.0125950
$664$	0	0
$665$	54.9091	2.12928
$666$	0	0
$667$	40.1018	1.55275
$668$	0	0
$669$	2.52737	0.0977139
$670$	0	0
$671$	8.35272	0.322453
$672$	0	0
$673$	−5.40431	−0.208321	−0.104160	−	0.994561i	$-0.533216\pi$
$673$	−5.40431	−0.208321	−0.104160	+	0.994561i	$0.533216\pi$
$674$	0	0
$675$	−4.02740	−0.155015
$676$	0	0
$677$	29.0829	1.11775	0.558873	−	0.829253i	$-0.311234\pi$
$677$	29.0829	1.11775	0.558873	+	0.829253i	$0.311234\pi$
$678$	0	0
$679$	−38.4760	−1.47658
$680$	0	0
$681$	0.405464	0.0155374
$682$	0	0
$683$	36.6328	1.40172	0.700858	−	0.713301i	$-0.252801\pi$
$683$	36.6328	1.40172	0.700858	+	0.713301i	$0.252801\pi$
$684$	0	0
$685$	16.9154	0.646304
$686$	0	0
$687$	2.97144	0.113368
$688$	0	0
$689$	5.04448	0.192179
$690$	0	0
$691$	22.9988	0.874914	0.437457	−	0.899239i	$-0.355879\pi$
$691$	22.9988	0.874914	0.437457	+	0.899239i	$0.355879\pi$
$692$	0	0
$693$	−15.2199	−0.578158
$694$	0	0
$695$	13.1266	0.497921
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−3.52145	−0.133193
$700$	0	0
$701$	−15.8741	−0.599555	−0.299778	−	0.954009i	$-0.596912\pi$
$701$	−15.8741	−0.599555	−0.299778	+	0.954009i	$0.596912\pi$
$702$	0	0
$703$	−59.3271	−2.23756
$704$	0	0
$705$	−2.88876	−0.108797
$706$	0	0
$707$	42.1729	1.58608
$708$	0	0
$709$	−26.9703	−1.01289	−0.506445	−	0.862272i	$-0.669041\pi$
$709$	−26.9703	−1.01289	−0.506445	+	0.862272i	$0.669041\pi$
$710$	0	0
$711$	−31.4605	−1.17986
$712$	0	0
$713$	−58.4749	−2.18990
$714$	0	0
$715$	−1.96344	−0.0734286
$716$	0	0
$717$	7.77165	0.290238
$718$	0	0
$719$	13.0984	0.488490	0.244245	−	0.969714i	$-0.421460\pi$
$719$	13.0984	0.488490	0.244245	+	0.969714i	$0.421460\pi$
$720$	0	0
$721$	−29.1634	−1.08610
$722$	0	0
$723$	2.32449	0.0864487
$724$	0	0
$725$	11.1871	0.415477
$726$	0	0
$727$	−2.94375	−0.109178	−0.0545889	−	0.998509i	$-0.517385\pi$
$727$	−2.94375	−0.109178	−0.0545889	+	0.998509i	$0.517385\pi$
$728$	0	0
$729$	−21.5486	−0.798095
$730$	0	0
$731$	0.140887	0.00521088
$732$	0	0
$733$	−26.5067	−0.979047	−0.489524	−	0.871990i	$-0.662829\pi$
$733$	−26.5067	−0.979047	−0.489524	+	0.871990i	$0.662829\pi$
$734$	0	0
$735$	7.59904	0.280295
$736$	0	0
$737$	11.5140	0.424125
$738$	0	0
$739$	−9.05773	−0.333194	−0.166597	−	0.986025i	$-0.553278\pi$
$739$	−9.05773	−0.333194	−0.166597	+	0.986025i	$0.553278\pi$
$740$	0	0
$741$	−0.929026	−0.0341286
$742$	0	0
$743$	−1.45053	−0.0532147	−0.0266074	−	0.999646i	$-0.508470\pi$
$743$	−1.45053	−0.0532147	−0.0266074	+	0.999646i	$0.508470\pi$
$744$	0	0
$745$	61.8256	2.26511
$746$	0	0
$747$	−2.82973	−0.103535
$748$	0	0
$749$	−26.3081	−0.961278
$750$	0	0
$751$	29.7674	1.08623	0.543115	−	0.839659i	$-0.317245\pi$
$751$	29.7674	1.08623	0.543115	+	0.839659i	$0.317245\pi$
$752$	0	0
$753$	−4.85833	−0.177047
$754$	0	0
$755$	27.7355	1.00940
$756$	0	0
$757$	−4.90465	−0.178263	−0.0891313	−	0.996020i	$-0.528409\pi$
$757$	−4.90465	−0.178263	−0.0891313	+	0.996020i	$0.528409\pi$
$758$	0	0
$759$	−3.23380	−0.117379
$760$	0	0
$761$	33.7673	1.22406	0.612032	−	0.790833i	$-0.290353\pi$
$761$	33.7673	1.22406	0.612032	+	0.790833i	$0.290353\pi$
$762$	0	0
$763$	−14.1662	−0.512851
$764$	0	0
$765$	−13.9502	−0.504369
$766$	0	0
$767$	1.16326	0.0420028
$768$	0	0
$769$	14.1117	0.508879	0.254440	−	0.967089i	$-0.418109\pi$
$769$	14.1117	0.508879	0.254440	+	0.967089i	$0.418109\pi$
$770$	0	0
$771$	0.958233	0.0345099
$772$	0	0
$773$	−5.22407	−0.187897	−0.0939484	−	0.995577i	$-0.529949\pi$
$773$	−5.22407	−0.187897	−0.0939484	+	0.995577i	$0.529949\pi$
$774$	0	0
$775$	−16.3125	−0.585964
$776$	0	0
$777$	−14.7160	−0.527935
$778$	0	0
$779$	0	0
$780$	0	0
$781$	13.3461	0.477561
$782$	0	0
$783$	−10.0362	−0.358663
$784$	0	0
$785$	43.6699	1.55865
$786$	0	0
$787$	13.7350	0.489600	0.244800	−	0.969574i	$-0.421278\pi$
$787$	13.7350	0.489600	0.244800	+	0.969574i	$0.421278\pi$
$788$	0	0
$789$	−3.86001	−0.137420
$790$	0	0
$791$	−19.8819	−0.706919
$792$	0	0
$793$	−3.52402	−0.125142
$794$	0	0
$795$	7.78721	0.276184
$796$	0	0
$797$	35.2914	1.25009	0.625044	−	0.780590i	$-0.285081\pi$
$797$	35.2914	1.25009	0.625044	+	0.780590i	$0.285081\pi$
$798$	0	0
$799$	−6.06318	−0.214500
$800$	0	0
$801$	37.9870	1.34220
$802$	0	0
$803$	−12.7235	−0.449004
$804$	0	0
$805$	−80.6379	−2.84211
$806$	0	0
$807$	−4.64693	−0.163580
$808$	0	0
$809$	48.5369	1.70647	0.853233	−	0.521530i	$-0.174639\pi$
$809$	48.5369	1.70647	0.853233	+	0.521530i	$0.174639\pi$
$810$	0	0
$811$	−20.6721	−0.725897	−0.362948	−	0.931809i	$-0.618230\pi$
$811$	−20.6721	−0.725897	−0.362948	+	0.931809i	$0.618230\pi$
$812$	0	0
$813$	6.88805	0.241574
$814$	0	0
$815$	41.8691	1.46661
$816$	0	0
$817$	−0.403592	−0.0141199
$818$	0	0
$819$	6.42130	0.224378
$820$	0	0
$821$	−27.8597	−0.972312	−0.486156	−	0.873872i	$-0.661601\pi$
$821$	−27.8597	−0.972312	−0.486156	+	0.873872i	$0.661601\pi$
$822$	0	0
$823$	26.3402	0.918162	0.459081	−	0.888395i	$-0.348179\pi$
$823$	26.3402	0.918162	0.459081	+	0.888395i	$0.348179\pi$
$824$	0	0
$825$	−0.902121	−0.0314078
$826$	0	0
$827$	−18.5792	−0.646061	−0.323030	−	0.946389i	$-0.604702\pi$
$827$	−18.5792	−0.646061	−0.323030	+	0.946389i	$0.604702\pi$
$828$	0	0
$829$	13.3990	0.465368	0.232684	−	0.972552i	$-0.425249\pi$
$829$	13.3990	0.465368	0.232684	+	0.972552i	$0.425249\pi$
$830$	0	0
$831$	8.15329	0.282835
$832$	0	0
$833$	15.9495	0.552619
$834$	0	0
$835$	29.9314	1.03582
$836$	0	0
$837$	14.6343	0.505836
$838$	0	0
$839$	47.5522	1.64169	0.820843	−	0.571154i	$-0.193504\pi$
$839$	47.5522	1.64169	0.820843	+	0.571154i	$0.193504\pi$
$840$	0	0
$841$	−1.12214	−0.0386944
$842$	0	0
$843$	−4.72082	−0.162594
$844$	0	0
$845$	−33.8570	−1.16472
$846$	0	0
$847$	36.8310	1.26553
$848$	0	0
$849$	−5.61554	−0.192725
$850$	0	0
$851$	87.1261	2.98664
$852$	0	0
$853$	−1.89326	−0.0648241	−0.0324120	−	0.999475i	$-0.510319\pi$
$853$	−1.89326	−0.0648241	−0.0324120	+	0.999475i	$0.510319\pi$
$854$	0	0
$855$	39.9624	1.36668
$856$	0	0
$857$	0.0176156	0.000601739 0	0.000300869	−	1.00000i	$-0.499904\pi$
$857$	0.0176156	0.000601739 0	0.000300869		1.00000i	$0.499904\pi$
$858$	0	0
$859$	1.38392	0.0472189	0.0236095	−	0.999721i	$-0.492484\pi$
$859$	1.38392	0.0472189	0.0236095	+	0.999721i	$0.492484\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	6.05654	0.206167	0.103084	−	0.994673i	$-0.467129\pi$
$863$	6.05654	0.206167	0.103084	+	0.994673i	$0.467129\pi$
$864$	0	0
$865$	67.8043	2.30541
$866$	0	0
$867$	−4.42978	−0.150443
$868$	0	0
$869$	−14.3469	−0.486686
$870$	0	0
$871$	−4.85778	−0.164600
$872$	0	0
$873$	−28.0025	−0.947742
$874$	0	0
$875$	30.5901	1.03413
$876$	0	0
$877$	14.0802	0.475455	0.237727	−	0.971332i	$-0.423598\pi$
$877$	14.0802	0.475455	0.237727	+	0.971332i	$0.423598\pi$
$878$	0	0
$879$	6.41876	0.216499
$880$	0	0
$881$	−4.19149	−0.141215	−0.0706074	−	0.997504i	$-0.522494\pi$
$881$	−4.19149	−0.141215	−0.0706074	+	0.997504i	$0.522494\pi$
$882$	0	0
$883$	−50.5539	−1.70127	−0.850636	−	0.525755i	$-0.823783\pi$
$883$	−50.5539	−1.70127	−0.850636	+	0.525755i	$0.823783\pi$
$884$	0	0
$885$	1.79573	0.0603629
$886$	0	0
$887$	49.1329	1.64972	0.824861	−	0.565336i	$-0.191253\pi$
$887$	49.1329	1.64972	0.824861	+	0.565336i	$0.191253\pi$
$888$	0	0
$889$	−68.8381	−2.30875
$890$	0	0
$891$	−10.6652	−0.357296
$892$	0	0
$893$	17.3689	0.581229
$894$	0	0
$895$	28.4732	0.951753
$896$	0	0
$897$	1.36434	0.0455540
$898$	0	0
$899$	−40.6504	−1.35577
$900$	0	0
$901$	16.3445	0.544514
$902$	0	0
$903$	−0.100111	−0.00333147
$904$	0	0
$905$	−5.68504	−0.188977
$906$	0	0
$907$	3.93829	0.130769	0.0653844	−	0.997860i	$-0.479173\pi$
$907$	3.93829	0.130769	0.0653844	+	0.997860i	$0.479173\pi$
$908$	0	0
$909$	30.6931	1.01803
$910$	0	0
$911$	−23.6950	−0.785049	−0.392524	−	0.919742i	$-0.628398\pi$
$911$	−23.6950	−0.785049	−0.392524	+	0.919742i	$0.628398\pi$
$912$	0	0
$913$	−1.29044	−0.0427075
$914$	0	0
$915$	−5.44006	−0.179843
$916$	0	0
$917$	−33.9232	−1.12024
$918$	0	0
$919$	−1.41363	−0.0466314	−0.0233157	−	0.999728i	$-0.507422\pi$
$919$	−1.41363	−0.0466314	−0.0233157	+	0.999728i	$0.507422\pi$
$920$	0	0
$921$	−10.0030	−0.329610
$922$	0	0
$923$	−5.63073	−0.185338
$924$	0	0
$925$	24.3053	0.799152
$926$	0	0
$927$	−21.2248	−0.697115
$928$	0	0
$929$	52.2797	1.71524	0.857621	−	0.514282i	$-0.171942\pi$
$929$	52.2797	1.71524	0.857621	+	0.514282i	$0.171942\pi$
$930$	0	0
$931$	−45.6899	−1.49743
$932$	0	0
$933$	5.95087	0.194823
$934$	0	0
$935$	−6.36170	−0.208050
$936$	0	0
$937$	27.5653	0.900519	0.450259	−	0.892898i	$-0.351332\pi$
$937$	27.5653	0.900519	0.450259	+	0.892898i	$0.351332\pi$
$938$	0	0
$939$	−0.553610	−0.0180664
$940$	0	0
$941$	−1.49909	−0.0488691	−0.0244345	−	0.999701i	$-0.507779\pi$
$941$	−1.49909	−0.0488691	−0.0244345	+	0.999701i	$0.507779\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	20.1810	0.656488
$946$	0	0
$947$	−19.9995	−0.649896	−0.324948	−	0.945732i	$-0.605347\pi$
$947$	−19.9995	−0.649896	−0.324948	+	0.945732i	$0.605347\pi$
$948$	0	0
$949$	5.36807	0.174255
$950$	0	0
$951$	−3.66057	−0.118702
$952$	0	0
$953$	−16.4273	−0.532133	−0.266066	−	0.963955i	$-0.585724\pi$
$953$	−16.4273	−0.532133	−0.266066	+	0.963955i	$0.585724\pi$
$954$	0	0
$955$	−46.0805	−1.49113
$956$	0	0
$957$	−2.24806	−0.0726694
$958$	0	0
$959$	−25.2279	−0.814651
$960$	0	0
$961$	28.2748	0.912089
$962$	0	0
$963$	−19.1468	−0.616998
$964$	0	0
$965$	−51.1105	−1.64531
$966$	0	0
$967$	−17.0530	−0.548388	−0.274194	−	0.961674i	$-0.588411\pi$
$967$	−17.0530	−0.548388	−0.274194	+	0.961674i	$0.588411\pi$
$968$	0	0
$969$	−3.01011	−0.0966987
$970$	0	0
$971$	−7.05651	−0.226454	−0.113227	−	0.993569i	$-0.536119\pi$
$971$	−7.05651	−0.226454	−0.113227	+	0.993569i	$0.536119\pi$
$972$	0	0
$973$	−19.5772	−0.627617
$974$	0	0
$975$	0.380605	0.0121891
$976$	0	0
$977$	−9.19350	−0.294126	−0.147063	−	0.989127i	$-0.546982\pi$
$977$	−9.19350	−0.294126	−0.147063	+	0.989127i	$0.546982\pi$
$978$	0	0
$979$	17.3232	0.553652
$980$	0	0
$981$	−10.3100	−0.329174
$982$	0	0
$983$	−41.4917	−1.32338	−0.661690	−	0.749777i	$-0.730161\pi$
$983$	−41.4917	−1.32338	−0.661690	+	0.749777i	$0.730161\pi$
$984$	0	0
$985$	11.7038	0.372912
$986$	0	0
$987$	4.30835	0.137136
$988$	0	0
$989$	0.592703	0.0188469
$990$	0	0
$991$	−39.0959	−1.24192	−0.620961	−	0.783842i	$-0.713257\pi$
$991$	−39.0959	−1.24192	−0.620961	+	0.783842i	$0.713257\pi$
$992$	0	0
$993$	3.04837	0.0967371
$994$	0	0
$995$	53.6906	1.70211
$996$	0	0
$997$	49.9130	1.58076	0.790380	−	0.612617i	$-0.209883\pi$
$997$	49.9130	1.58076	0.790380	+	0.612617i	$0.209883\pi$
$998$	0	0
$999$	−21.8048	−0.689872

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6724.2.a.j.1.9		16
41.20	even	20		164.2.k.a.113.3	yes	16
41.39	even	20		164.2.k.a.45.2	✓	16
41.40	even	2	inner	6724.2.a.j.1.8		16
123.20	odd	20		1476.2.bb.b.1261.4		16
123.80	odd	20		1476.2.bb.b.865.4		16
164.39	odd	20		656.2.be.e.209.3		16
164.143	odd	20		656.2.be.e.113.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.k.a.45.2	✓	16	41.39	even	20
164.2.k.a.113.3	yes	16	41.20	even	20
656.2.be.e.113.2		16	164.143	odd	20
656.2.be.e.209.3		16	164.39	odd	20
1476.2.bb.b.865.4		16	123.80	odd	20
1476.2.bb.b.1261.4		16	123.20	odd	20
6724.2.a.j.1.8		16	41.40	even	2	inner
6724.2.a.j.1.9		16	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 6724 = 2^{2} \cdot 41^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6724.a (trivial)

\(f(q)\)	\(=\)	\(q+0.322385 q^{3} +2.66810 q^{5} -3.97926 q^{7} -2.89607 q^{9} +O(q^{10})\)	\(q+0.322385 q^{3} +2.66810 q^{5} -3.97926 q^{7} -2.89607 q^{9} -1.32070 q^{11} +0.557202 q^{13} +0.860158 q^{15} +1.80538 q^{17} -5.17178 q^{19} -1.28285 q^{21} +7.59512 q^{23} +2.11878 q^{25} -1.90081 q^{27} +5.27995 q^{29} -7.69901 q^{31} -0.425773 q^{33} -10.6171 q^{35} +11.4713 q^{37} +0.179634 q^{39} +0.0780374 q^{43} -7.72701 q^{45} -3.35841 q^{47} +8.83447 q^{49} +0.582027 q^{51} +9.05323 q^{53} -3.52375 q^{55} -1.66731 q^{57} +2.08768 q^{59} -6.32449 q^{61} +11.5242 q^{63} +1.48667 q^{65} -8.71817 q^{67} +2.44856 q^{69} -10.1054 q^{71} +9.63398 q^{73} +0.683065 q^{75} +5.25538 q^{77} +10.8632 q^{79} +8.07541 q^{81} +0.977095 q^{83} +4.81693 q^{85} +1.70218 q^{87} -13.1167 q^{89} -2.21725 q^{91} -2.48205 q^{93} -13.7988 q^{95} +9.66916 q^{97} +3.82482 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 6 q^{5} + 30 q^{9}+O(q^{10}) \) 16 * q + 6 * q^5 + 30 * q^9	\( 16 q + 6 q^{5} + 30 q^{9} + 46 q^{21} - 8 q^{23} + 38 q^{25} - 2 q^{31} + 20 q^{33} + 12 q^{37} + 8 q^{39} - 48 q^{43} + 92 q^{45} - 12 q^{49} + 42 q^{51} + 22 q^{57} - 32 q^{59} + 6 q^{61} - 34 q^{73} + 92 q^{77} + 108 q^{81} + 12 q^{83} - 34 q^{87}+O(q^{100}) \) 16 * q + 6 * q^5 + 30 * q^9 + 46 * q^21 - 8 * q^23 + 38 * q^25 - 2 * q^31 + 20 * q^33 + 12 * q^37 + 8 * q^39 - 48 * q^43 + 92 * q^45 - 12 * q^49 + 42 * q^51 + 22 * q^57 - 32 * q^59 + 6 * q^61 - 34 * q^73 + 92 * q^77 + 108 * q^81 + 12 * q^83 - 34 * q^87

Introduction

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