LMFDB - Embedded newform 3328.2.a.bl.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3328,2,Mod(1,3328)] 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("3328.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3328, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3328 = 2^{8} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3328.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,2,0,0,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$26.5742137927$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.13448.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 2 $ x^4 - 7*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1664)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.546295$ of defining polynomial
Character	$\chi$	$=$	3328.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.11473 q^{3} +3.70156 q^{5} -0.546295 q^{7} +6.70156 q^{9} +3.66103 q^{11} +1.00000 q^{13} -11.5294 q^{15} +5.70156 q^{17} -2.56844 q^{19} +1.70156 q^{21} +6.22947 q^{23} +8.70156 q^{25} -11.5294 q^{27} +2.00000 q^{29} +4.75362 q^{31} -11.4031 q^{33} -2.02214 q^{35} -0.298438 q^{37} -3.11473 q^{39} -1.40312 q^{41} +12.6220 q^{43} +24.8062 q^{45} +8.96094 q^{47} -6.70156 q^{49} -17.7588 q^{51} -11.4031 q^{53} +13.5515 q^{55} +8.00000 q^{57} -9.89049 q^{59} -3.66103 q^{63} +3.70156 q^{65} -13.1683 q^{67} -19.4031 q^{69} -6.77576 q^{71} -5.40312 q^{73} -27.1030 q^{75} -2.00000 q^{77} +13.5515 q^{79} +15.8062 q^{81} -1.47585 q^{83} +21.1047 q^{85} -6.22947 q^{87} -1.40312 q^{89} -0.546295 q^{91} -14.8062 q^{93} -9.50723 q^{95} +10.0000 q^{97} +24.5346 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{5} + 14 q^{9} + 4 q^{13} + 10 q^{17} - 6 q^{21} + 22 q^{25} + 8 q^{29} - 20 q^{33} - 14 q^{37} + 20 q^{41} + 48 q^{45} - 14 q^{49} - 20 q^{53} + 32 q^{57} + 2 q^{65} - 52 q^{69} + 4 q^{73} - 8 q^{77}+ \cdots + 40 q^{97}+O(q^{100}) $ 4 * q + 2 * q^5 + 14 * q^9 + 4 * q^13 + 10 * q^17 - 6 * q^21 + 22 * q^25 + 8 * q^29 - 20 * q^33 - 14 * q^37 + 20 * q^41 + 48 * q^45 - 14 * q^49 - 20 * q^53 + 32 * q^57 + 2 * q^65 - 52 * q^69 + 4 * q^73 - 8 * q^77 + 12 * q^81 + 46 * q^85 + 20 * q^89 - 8 * q^93 + 40 * 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
$3$	−3.11473	−1.79829	−0.899146	−	0.437649i	$-0.855811\pi$
$3$	−3.11473	−1.79829	−0.899146	+	0.437649i	$0.855811\pi$
$4$	0	0
$5$	3.70156	1.65539	0.827694	−	0.561179i	$-0.189652\pi$
$5$	3.70156	1.65539	0.827694	+	0.561179i	$0.189652\pi$
$6$	0	0
$7$	−0.546295	−0.206480	−0.103240	−	0.994656i	$-0.532921\pi$
$7$	−0.546295	−0.206480	−0.103240	+	0.994656i	$0.532921\pi$
$8$	0	0
$9$	6.70156	2.23385
$10$	0	0
$11$	3.66103	1.10384	0.551921	−	0.833897i	$-0.313895\pi$
$11$	3.66103	1.10384	0.551921	+	0.833897i	$0.313895\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−11.5294	−2.97687
$16$	0	0
$17$	5.70156	1.38283	0.691416	−	0.722457i	$-0.256987\pi$
$17$	5.70156	1.38283	0.691416	+	0.722457i	$0.256987\pi$
$18$	0	0
$19$	−2.56844	−0.589240	−0.294620	−	0.955614i	$-0.595193\pi$
$19$	−2.56844	−0.589240	−0.294620	+	0.955614i	$0.595193\pi$
$20$	0	0
$21$	1.70156	0.371311
$22$	0	0
$23$	6.22947	1.29893	0.649467	−	0.760390i	$-0.274992\pi$
$23$	6.22947	1.29893	0.649467	+	0.760390i	$0.274992\pi$
$24$	0	0
$25$	8.70156	1.74031
$26$	0	0
$27$	−11.5294	−2.21883
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	4.75362	0.853775	0.426887	−	0.904305i	$-0.359610\pi$
$31$	4.75362	0.853775	0.426887	+	0.904305i	$0.359610\pi$
$32$	0	0
$33$	−11.4031	−1.98503
$34$	0	0
$35$	−2.02214	−0.341805
$36$	0	0
$37$	−0.298438	−0.0490629	−0.0245314	−	0.999699i	$-0.507809\pi$
$37$	−0.298438	−0.0490629	−0.0245314	+	0.999699i	$0.507809\pi$
$38$	0	0
$39$	−3.11473	−0.498756
$40$	0	0
$41$	−1.40312	−0.219131	−0.109566	−	0.993980i	$-0.534946\pi$
$41$	−1.40312	−0.219131	−0.109566	+	0.993980i	$0.534946\pi$
$42$	0	0
$43$	12.6220	1.92483	0.962416	−	0.271580i	$-0.0875460\pi$
$43$	12.6220	1.92483	0.962416	+	0.271580i	$0.0875460\pi$
$44$	0	0
$45$	24.8062	3.69790
$46$	0	0
$47$	8.96094	1.30709	0.653544	−	0.756889i	$-0.273282\pi$
$47$	8.96094	1.30709	0.653544	+	0.756889i	$0.273282\pi$
$48$	0	0
$49$	−6.70156	−0.957366
$50$	0	0
$51$	−17.7588	−2.48674
$52$	0	0
$53$	−11.4031	−1.56634	−0.783170	−	0.621808i	$-0.786398\pi$
$53$	−11.4031	−1.56634	−0.783170	+	0.621808i	$0.786398\pi$
$54$	0	0
$55$	13.5515	1.82729
$56$	0	0
$57$	8.00000	1.05963
$58$	0	0
$59$	−9.89049	−1.28763	−0.643816	−	0.765180i	$-0.722650\pi$
$59$	−9.89049	−1.28763	−0.643816	+	0.765180i	$0.722650\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	−3.66103	−0.461246
$64$	0	0
$65$	3.70156	0.459122
$66$	0	0
$67$	−13.1683	−1.60876	−0.804380	−	0.594116i	$-0.797502\pi$
$67$	−13.1683	−1.60876	−0.804380	+	0.594116i	$0.797502\pi$
$68$	0	0
$69$	−19.4031	−2.33586
$70$	0	0
$71$	−6.77576	−0.804135	−0.402067	−	0.915610i	$-0.631708\pi$
$71$	−6.77576	−0.804135	−0.402067	+	0.915610i	$0.631708\pi$
$72$	0	0
$73$	−5.40312	−0.632388	−0.316194	−	0.948695i	$-0.602405\pi$
$73$	−5.40312	−0.632388	−0.316194	+	0.948695i	$0.602405\pi$
$74$	0	0
$75$	−27.1030	−3.12959
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	13.5515	1.52467	0.762333	−	0.647185i	$-0.224054\pi$
$79$	13.5515	1.52467	0.762333	+	0.647185i	$0.224054\pi$
$80$	0	0
$81$	15.8062	1.75625
$82$	0	0
$83$	−1.47585	−0.161995	−0.0809977	−	0.996714i	$-0.525811\pi$
$83$	−1.47585	−0.161995	−0.0809977	+	0.996714i	$0.525811\pi$
$84$	0	0
$85$	21.1047	2.28912
$86$	0	0
$87$	−6.22947	−0.667869
$88$	0	0
$89$	−1.40312	−0.148731	−0.0743654	−	0.997231i	$-0.523693\pi$
$89$	−1.40312	−0.148731	−0.0743654	+	0.997231i	$0.523693\pi$
$90$	0	0
$91$	−0.546295	−0.0572672
$92$	0	0
$93$	−14.8062	−1.53534
$94$	0	0
$95$	−9.50723	−0.975422
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	24.5346	2.46582
$100$	0	0
$101$	−18.8062	−1.87129	−0.935646	−	0.352940i	$-0.885182\pi$
$101$	−18.8062	−1.87129	−0.935646	+	0.352940i	$0.885182\pi$
$102$	0	0
$103$	−4.04429	−0.398495	−0.199248	−	0.979949i	$-0.563850\pi$
$103$	−4.04429	−0.398495	−0.199248	+	0.979949i	$0.563850\pi$
$104$	0	0
$105$	6.29844	0.614665
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	7.70156	0.737676	0.368838	−	0.929494i	$-0.379756\pi$
$109$	7.70156	0.737676	0.368838	+	0.929494i	$0.379756\pi$
$110$	0	0
$111$	0.929554	0.0882294
$112$	0	0
$113$	11.4031	1.07272	0.536358	−	0.843991i	$-0.319800\pi$
$113$	11.4031	1.07272	0.536358	+	0.843991i	$0.319800\pi$
$114$	0	0
$115$	23.0588	2.15024
$116$	0	0
$117$	6.70156	0.619560
$118$	0	0
$119$	−3.11473	−0.285527
$120$	0	0
$121$	2.40312	0.218466
$122$	0	0
$123$	4.37036	0.394062
$124$	0	0
$125$	13.7016	1.22550
$126$	0	0
$127$	−18.6884	−1.65833	−0.829164	−	0.559006i	$-0.811183\pi$
$127$	−18.6884	−1.65833	−0.829164	+	0.559006i	$0.811183\pi$
$128$	0	0
$129$	−39.3141	−3.46141
$130$	0	0
$131$	−11.5294	−1.00733	−0.503663	−	0.863900i	$-0.668015\pi$
$131$	−11.5294	−1.00733	−0.503663	+	0.863900i	$0.668015\pi$
$132$	0	0
$133$	1.40312	0.121666
$134$	0	0
$135$	−42.6767	−3.67303
$136$	0	0
$137$	14.0000	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$137$	14.0000	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$138$	0	0
$139$	0.929554	0.0788438	0.0394219	−	0.999223i	$-0.487448\pi$
$139$	0.929554	0.0788438	0.0394219	+	0.999223i	$0.487448\pi$
$140$	0	0
$141$	−27.9109	−2.35052
$142$	0	0
$143$	3.66103	0.306151
$144$	0	0
$145$	7.40312	0.614796
$146$	0	0
$147$	20.8736	1.72162
$148$	0	0
$149$	−12.8062	−1.04913	−0.524564	−	0.851371i	$-0.675772\pi$
$149$	−12.8062	−1.04913	−0.524564	+	0.851371i	$0.675772\pi$
$150$	0	0
$151$	15.9569	1.29856	0.649278	−	0.760551i	$-0.275071\pi$
$151$	15.9569	1.29856	0.649278	+	0.760551i	$0.275071\pi$
$152$	0	0
$153$	38.2094	3.08904
$154$	0	0
$155$	17.5958	1.41333
$156$	0	0
$157$	−17.4031	−1.38892	−0.694460	−	0.719531i	$-0.744357\pi$
$157$	−17.4031	−1.38892	−0.694460	+	0.719531i	$0.744357\pi$
$158$	0	0
$159$	35.5177	2.81674
$160$	0	0
$161$	−3.40312	−0.268204
$162$	0	0
$163$	13.9348	1.09146	0.545728	−	0.837962i	$-0.316253\pi$
$163$	13.9348	1.09146	0.545728	+	0.837962i	$0.316253\pi$
$164$	0	0
$165$	−42.2094	−3.28600
$166$	0	0
$167$	−0.383260	−0.0296575	−0.0148288	−	0.999890i	$-0.504720\pi$
$167$	−0.383260	−0.0296575	−0.0148288	+	0.999890i	$0.504720\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−17.2125	−1.31628
$172$	0	0
$173$	8.59688	0.653608	0.326804	−	0.945092i	$-0.394028\pi$
$173$	8.59688	0.653608	0.326804	+	0.945092i	$0.394028\pi$
$174$	0	0
$175$	−4.75362	−0.359340
$176$	0	0
$177$	30.8062	2.31554
$178$	0	0
$179$	−8.57768	−0.641126	−0.320563	−	0.947227i	$-0.603872\pi$
$179$	−8.57768	−0.641126	−0.320563	+	0.947227i	$0.603872\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.10469	−0.0812182
$186$	0	0
$187$	20.8736	1.52643
$188$	0	0
$189$	6.29844	0.458144
$190$	0	0
$191$	1.09259	0.0790570	0.0395285	−	0.999218i	$-0.487414\pi$
$191$	1.09259	0.0790570	0.0395285	+	0.999218i	$0.487414\pi$
$192$	0	0
$193$	−4.80625	−0.345961	−0.172981	−	0.984925i	$-0.555340\pi$
$193$	−4.80625	−0.345961	−0.172981	+	0.984925i	$0.555340\pi$
$194$	0	0
$195$	−11.5294	−0.825636
$196$	0	0
$197$	19.1047	1.36115	0.680576	−	0.732677i	$-0.261729\pi$
$197$	19.1047	1.36115	0.680576	+	0.732677i	$0.261729\pi$
$198$	0	0
$199$	8.41464	0.596498	0.298249	−	0.954488i	$-0.403597\pi$
$199$	8.41464	0.596498	0.298249	+	0.954488i	$0.403597\pi$
$200$	0	0
$201$	41.0156	2.89302
$202$	0	0
$203$	−1.09259	−0.0766847
$204$	0	0
$205$	−5.19375	−0.362747
$206$	0	0
$207$	41.7472	2.90163
$208$	0	0
$209$	−9.40312	−0.650428
$210$	0	0
$211$	5.29991	0.364861	0.182430	−	0.983219i	$-0.441604\pi$
$211$	5.29991	0.364861	0.182430	+	0.983219i	$0.441604\pi$
$212$	0	0
$213$	21.1047	1.44607
$214$	0	0
$215$	46.7210	3.18635
$216$	0	0
$217$	−2.59688	−0.176287
$218$	0	0
$219$	16.8293	1.13722
$220$	0	0
$221$	5.70156	0.383529
$222$	0	0
$223$	−9.72746	−0.651399	−0.325699	−	0.945473i	$-0.605600\pi$
$223$	−9.72746	−0.651399	−0.325699	+	0.945473i	$0.605600\pi$
$224$	0	0
$225$	58.3141	3.88760
$226$	0	0
$227$	24.5346	1.62842	0.814209	−	0.580571i	$-0.197171\pi$
$227$	24.5346	1.62842	0.814209	+	0.580571i	$0.197171\pi$
$228$	0	0
$229$	21.9109	1.44792	0.723958	−	0.689844i	$-0.242321\pi$
$229$	21.9109	1.44792	0.723958	+	0.689844i	$0.242321\pi$
$230$	0	0
$231$	6.22947	0.409869
$232$	0	0
$233$	−10.5078	−0.688390	−0.344195	−	0.938898i	$-0.611848\pi$
$233$	−10.5078	−0.688390	−0.344195	+	0.938898i	$0.611848\pi$
$234$	0	0
$235$	33.1695	2.16374
$236$	0	0
$237$	−42.2094	−2.74179
$238$	0	0
$239$	−26.8828	−1.73890	−0.869452	−	0.494017i	$-0.835528\pi$
$239$	−26.8828	−1.73890	−0.869452	+	0.494017i	$0.835528\pi$
$240$	0	0
$241$	2.59688	0.167279	0.0836397	−	0.996496i	$-0.473346\pi$
$241$	2.59688	0.167279	0.0836397	+	0.996496i	$0.473346\pi$
$242$	0	0
$243$	−14.6441	−0.939420
$244$	0	0
$245$	−24.8062	−1.58481
$246$	0	0
$247$	−2.56844	−0.163426
$248$	0	0
$249$	4.59688	0.291315
$250$	0	0
$251$	15.4106	0.972710	0.486355	−	0.873761i	$-0.338326\pi$
$251$	15.4106	0.972710	0.486355	+	0.873761i	$0.338326\pi$
$252$	0	0
$253$	22.8062	1.43382
$254$	0	0
$255$	−65.7355	−4.11651
$256$	0	0
$257$	4.29844	0.268129	0.134065	−	0.990973i	$-0.457197\pi$
$257$	4.29844	0.268129	0.134065	+	0.990973i	$0.457197\pi$
$258$	0	0
$259$	0.163035	0.0101305
$260$	0	0
$261$	13.4031	0.829633
$262$	0	0
$263$	2.95170	0.182009	0.0910047	−	0.995850i	$-0.470992\pi$
$263$	2.95170	0.182009	0.0910047	+	0.995850i	$0.470992\pi$
$264$	0	0
$265$	−42.2094	−2.59290
$266$	0	0
$267$	4.37036	0.267462
$268$	0	0
$269$	−14.5969	−0.889987	−0.444994	−	0.895534i	$-0.646794\pi$
$269$	−14.5969	−0.889987	−0.444994	+	0.895534i	$0.646794\pi$
$270$	0	0
$271$	−15.9569	−0.969314	−0.484657	−	0.874704i	$-0.661056\pi$
$271$	−15.9569	−0.969314	−0.484657	+	0.874704i	$0.661056\pi$
$272$	0	0
$273$	1.70156	0.102983
$274$	0	0
$275$	31.8567	1.92103
$276$	0	0
$277$	12.0000	0.721010	0.360505	−	0.932757i	$-0.382604\pi$
$277$	12.0000	0.721010	0.360505	+	0.932757i	$0.382604\pi$
$278$	0	0
$279$	31.8567	1.90721
$280$	0	0
$281$	1.40312	0.0837034	0.0418517	−	0.999124i	$-0.486674\pi$
$281$	1.40312	0.0837034	0.0418517	+	0.999124i	$0.486674\pi$
$282$	0	0
$283$	−7.32206	−0.435251	−0.217626	−	0.976032i	$-0.569831\pi$
$283$	−7.32206	−0.435251	−0.217626	+	0.976032i	$0.569831\pi$
$284$	0	0
$285$	29.6125	1.75409
$286$	0	0
$287$	0.766519	0.0452462
$288$	0	0
$289$	15.5078	0.912224
$290$	0	0
$291$	−31.1473	−1.82589
$292$	0	0
$293$	−11.1047	−0.648743	−0.324371	−	0.945930i	$-0.605153\pi$
$293$	−11.1047	−0.648743	−0.324371	+	0.945930i	$0.605153\pi$
$294$	0	0
$295$	−36.6103	−2.13153
$296$	0	0
$297$	−42.2094	−2.44924
$298$	0	0
$299$	6.22947	0.360259
$300$	0	0
$301$	−6.89531	−0.397439
$302$	0	0
$303$	58.5764	3.36513
$304$	0	0
$305$	0	0
$306$	0	0
$307$	5.52014	0.315051	0.157525	−	0.987515i	$-0.449648\pi$
$307$	5.52014	0.315051	0.157525	+	0.987515i	$0.449648\pi$
$308$	0	0
$309$	12.5969	0.716611
$310$	0	0
$311$	−19.0145	−1.07821	−0.539106	−	0.842238i	$-0.681238\pi$
$311$	−19.0145	−1.07821	−0.539106	+	0.842238i	$0.681238\pi$
$312$	0	0
$313$	13.1047	0.740721	0.370360	−	0.928888i	$-0.379234\pi$
$313$	13.1047	0.740721	0.370360	+	0.928888i	$0.379234\pi$
$314$	0	0
$315$	−13.5515	−0.763542
$316$	0	0
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	7.32206	0.409956
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−14.6441	−0.814820
$324$	0	0
$325$	8.70156	0.482676
$326$	0	0
$327$	−23.9883	−1.32656
$328$	0	0
$329$	−4.89531	−0.269887
$330$	0	0
$331$	6.61273	0.363468	0.181734	−	0.983348i	$-0.441829\pi$
$331$	6.61273	0.363468	0.181734	+	0.983348i	$0.441829\pi$
$332$	0	0
$333$	−2.00000	−0.109599
$334$	0	0
$335$	−48.7431	−2.66312
$336$	0	0
$337$	−4.50781	−0.245556	−0.122778	−	0.992434i	$-0.539180\pi$
$337$	−4.50781	−0.245556	−0.122778	+	0.992434i	$0.539180\pi$
$338$	0	0
$339$	−35.5177	−1.92906
$340$	0	0
$341$	17.4031	0.942432
$342$	0	0
$343$	7.48509	0.404157
$344$	0	0
$345$	−71.8219	−3.86676
$346$	0	0
$347$	28.0326	1.50487	0.752434	−	0.658667i	$-0.228879\pi$
$347$	28.0326	1.50487	0.752434	+	0.658667i	$0.228879\pi$
$348$	0	0
$349$	3.70156	0.198140	0.0990700	−	0.995080i	$-0.468413\pi$
$349$	3.70156	0.198140	0.0990700	+	0.995080i	$0.468413\pi$
$350$	0	0
$351$	−11.5294	−0.615393
$352$	0	0
$353$	0.806248	0.0429123	0.0214561	−	0.999770i	$-0.493170\pi$
$353$	0.806248	0.0429123	0.0214561	+	0.999770i	$0.493170\pi$
$354$	0	0
$355$	−25.0809	−1.33116
$356$	0	0
$357$	9.70156	0.513461
$358$	0	0
$359$	4.75362	0.250886	0.125443	−	0.992101i	$-0.459965\pi$
$359$	4.75362	0.250886	0.125443	+	0.992101i	$0.459965\pi$
$360$	0	0
$361$	−12.4031	−0.652796
$362$	0	0
$363$	−7.48509	−0.392865
$364$	0	0
$365$	−20.0000	−1.04685
$366$	0	0
$367$	−7.64813	−0.399229	−0.199614	−	0.979875i	$-0.563969\pi$
$367$	−7.64813	−0.399229	−0.199614	+	0.979875i	$0.563969\pi$
$368$	0	0
$369$	−9.40312	−0.489507
$370$	0	0
$371$	6.22947	0.323418
$372$	0	0
$373$	9.40312	0.486875	0.243438	−	0.969917i	$-0.421725\pi$
$373$	9.40312	0.486875	0.243438	+	0.969917i	$0.421725\pi$
$374$	0	0
$375$	−42.6767	−2.20382
$376$	0	0
$377$	2.00000	0.103005
$378$	0	0
$379$	−18.3051	−0.940272	−0.470136	−	0.882594i	$-0.655795\pi$
$379$	−18.3051	−0.940272	−0.470136	+	0.882594i	$0.655795\pi$
$380$	0	0
$381$	58.2094	2.98216
$382$	0	0
$383$	17.3756	0.887851	0.443925	−	0.896064i	$-0.353586\pi$
$383$	17.3756	0.887851	0.443925	+	0.896064i	$0.353586\pi$
$384$	0	0
$385$	−7.40312	−0.377298
$386$	0	0
$387$	84.5869	4.29979
$388$	0	0
$389$	−6.59688	−0.334475	−0.167237	−	0.985917i	$-0.553485\pi$
$389$	−6.59688	−0.334475	−0.167237	+	0.985917i	$0.553485\pi$
$390$	0	0
$391$	35.5177	1.79621
$392$	0	0
$393$	35.9109	1.81147
$394$	0	0
$395$	50.1618	2.52391
$396$	0	0
$397$	12.8062	0.642727	0.321364	−	0.946956i	$-0.395859\pi$
$397$	12.8062	0.642727	0.321364	+	0.946956i	$0.395859\pi$
$398$	0	0
$399$	−4.37036	−0.218792
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	4.75362	0.236795
$404$	0	0
$405$	58.5078	2.90728
$406$	0	0
$407$	−1.09259	−0.0541576
$408$	0	0
$409$	−11.1938	−0.553495	−0.276748	−	0.960943i	$-0.589257\pi$
$409$	−11.1938	−0.553495	−0.276748	+	0.960943i	$0.589257\pi$
$410$	0	0
$411$	−43.6063	−2.15094
$412$	0	0
$413$	5.40312	0.265870
$414$	0	0
$415$	−5.46295	−0.268166
$416$	0	0
$417$	−2.89531	−0.141784
$418$	0	0
$419$	−13.3885	−0.654070	−0.327035	−	0.945012i	$-0.606050\pi$
$419$	−13.3885	−0.654070	−0.327035	+	0.945012i	$0.606050\pi$
$420$	0	0
$421$	−15.1047	−0.736157	−0.368079	−	0.929795i	$-0.619984\pi$
$421$	−15.1047	−0.736157	−0.368079	+	0.929795i	$0.619984\pi$
$422$	0	0
$423$	60.0523	2.91984
$424$	0	0
$425$	49.6125	2.40656
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−11.4031	−0.550548
$430$	0	0
$431$	−10.8200	−0.521183	−0.260592	−	0.965449i	$-0.583918\pi$
$431$	−10.8200	−0.521183	−0.260592	+	0.965449i	$0.583918\pi$
$432$	0	0
$433$	−21.9109	−1.05297	−0.526486	−	0.850184i	$-0.676491\pi$
$433$	−21.9109	−1.05297	−0.526486	+	0.850184i	$0.676491\pi$
$434$	0	0
$435$	−23.0588	−1.10558
$436$	0	0
$437$	−16.0000	−0.765384
$438$	0	0
$439$	−10.2738	−0.490340	−0.245170	−	0.969480i	$-0.578844\pi$
$439$	−10.2738	−0.490340	−0.245170	+	0.969480i	$0.578844\pi$
$440$	0	0
$441$	−44.9109	−2.13862
$442$	0	0
$443$	1.25562	0.0596565	0.0298283	−	0.999555i	$-0.490504\pi$
$443$	1.25562	0.0596565	0.0298283	+	0.999555i	$0.490504\pi$
$444$	0	0
$445$	−5.19375	−0.246207
$446$	0	0
$447$	39.8880	1.88664
$448$	0	0
$449$	−23.6125	−1.11434	−0.557171	−	0.830398i	$-0.688113\pi$
$449$	−23.6125	−1.11434	−0.557171	+	0.830398i	$0.688113\pi$
$450$	0	0
$451$	−5.13688	−0.241886
$452$	0	0
$453$	−49.7016	−2.33518
$454$	0	0
$455$	−2.02214	−0.0947996
$456$	0	0
$457$	1.40312	0.0656354	0.0328177	−	0.999461i	$-0.489552\pi$
$457$	1.40312	0.0656354	0.0328177	+	0.999461i	$0.489552\pi$
$458$	0	0
$459$	−65.7355	−3.06827
$460$	0	0
$461$	−33.9109	−1.57939	−0.789695	−	0.613500i	$-0.789761\pi$
$461$	−33.9109	−1.57939	−0.789695	+	0.613500i	$0.789761\pi$
$462$	0	0
$463$	−0.383260	−0.0178116	−0.00890579	−	0.999960i	$-0.502835\pi$
$463$	−0.383260	−0.0178116	−0.00890579	+	0.999960i	$0.502835\pi$
$464$	0	0
$465$	−54.8062	−2.54158
$466$	0	0
$467$	34.4251	1.59300	0.796502	−	0.604636i	$-0.206681\pi$
$467$	34.4251	1.59300	0.796502	+	0.604636i	$0.206681\pi$
$468$	0	0
$469$	7.19375	0.332177
$470$	0	0
$471$	54.2061	2.49768
$472$	0	0
$473$	46.2094	2.12471
$474$	0	0
$475$	−22.3494	−1.02546
$476$	0	0
$477$	−76.4187	−3.49897
$478$	0	0
$479$	3.82406	0.174726	0.0873629	−	0.996177i	$-0.472156\pi$
$479$	3.82406	0.174726	0.0873629	+	0.996177i	$0.472156\pi$
$480$	0	0
$481$	−0.298438	−0.0136076
$482$	0	0
$483$	10.5998	0.482309
$484$	0	0
$485$	37.0156	1.68079
$486$	0	0
$487$	−34.0418	−1.54258	−0.771291	−	0.636482i	$-0.780389\pi$
$487$	−34.0418	−1.54258	−0.771291	+	0.636482i	$0.780389\pi$
$488$	0	0
$489$	−43.4031	−1.96276
$490$	0	0
$491$	2.34821	0.105973	0.0529867	−	0.998595i	$-0.483126\pi$
$491$	2.34821	0.105973	0.0529867	+	0.998595i	$0.483126\pi$
$492$	0	0
$493$	11.4031	0.513571
$494$	0	0
$495$	90.8164	4.08189
$496$	0	0
$497$	3.70156	0.166038
$498$	0	0
$499$	−0.383260	−0.0171571	−0.00857853	−	0.999963i	$-0.502731\pi$
$499$	−0.383260	−0.0171571	−0.00857853	+	0.999963i	$0.502731\pi$
$500$	0	0
$501$	1.19375	0.0533329
$502$	0	0
$503$	−30.0547	−1.34007	−0.670037	−	0.742327i	$-0.733722\pi$
$503$	−30.0547	−1.34007	−0.670037	+	0.742327i	$0.733722\pi$
$504$	0	0
$505$	−69.6125	−3.09772
$506$	0	0
$507$	−3.11473	−0.138330
$508$	0	0
$509$	−15.6125	−0.692012	−0.346006	−	0.938232i	$-0.612462\pi$
$509$	−15.6125	−0.692012	−0.346006	+	0.938232i	$0.612462\pi$
$510$	0	0
$511$	2.95170	0.130575
$512$	0	0
$513$	29.6125	1.30742
$514$	0	0
$515$	−14.9702	−0.659665
$516$	0	0
$517$	32.8062	1.44282
$518$	0	0
$519$	−26.7770	−1.17538
$520$	0	0
$521$	13.1047	0.574127	0.287063	−	0.957912i	$-0.407321\pi$
$521$	13.1047	0.574127	0.287063	+	0.957912i	$0.407321\pi$
$522$	0	0
$523$	8.74071	0.382205	0.191102	−	0.981570i	$-0.438794\pi$
$523$	8.74071	0.382205	0.191102	+	0.981570i	$0.438794\pi$
$524$	0	0
$525$	14.8062	0.646198
$526$	0	0
$527$	27.1030	1.18063
$528$	0	0
$529$	15.8062	0.687228
$530$	0	0
$531$	−66.2818	−2.87638
$532$	0	0
$533$	−1.40312	−0.0607761
$534$	0	0
$535$	0	0
$536$	0	0
$537$	26.7172	1.15293
$538$	0	0
$539$	−24.5346	−1.05678
$540$	0	0
$541$	−31.7016	−1.36296	−0.681478	−	0.731838i	$-0.738663\pi$
$541$	−31.7016	−1.36296	−0.681478	+	0.731838i	$0.738663\pi$
$542$	0	0
$543$	−43.6063	−1.87132
$544$	0	0
$545$	28.5078	1.22114
$546$	0	0
$547$	21.8031	0.932235	0.466117	−	0.884723i	$-0.345652\pi$
$547$	21.8031	0.932235	0.466117	+	0.884723i	$0.345652\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−5.13688	−0.218838
$552$	0	0
$553$	−7.40312	−0.314813
$554$	0	0
$555$	3.44080	0.146054
$556$	0	0
$557$	32.2984	1.36853	0.684264	−	0.729234i	$-0.260123\pi$
$557$	32.2984	1.36853	0.684264	+	0.729234i	$0.260123\pi$
$558$	0	0
$559$	12.6220	0.533852
$560$	0	0
$561$	−65.0156	−2.74496
$562$	0	0
$563$	−35.6807	−1.50376	−0.751882	−	0.659298i	$-0.770854\pi$
$563$	−35.6807	−1.50376	−0.751882	+	0.659298i	$0.770854\pi$
$564$	0	0
$565$	42.2094	1.77576
$566$	0	0
$567$	−8.63487	−0.362630
$568$	0	0
$569$	8.89531	0.372911	0.186455	−	0.982463i	$-0.440300\pi$
$569$	8.89531	0.372911	0.186455	+	0.982463i	$0.440300\pi$
$570$	0	0
$571$	31.3104	1.31030	0.655149	−	0.755500i	$-0.272606\pi$
$571$	31.3104	1.31030	0.655149	+	0.755500i	$0.272606\pi$
$572$	0	0
$573$	−3.40312	−0.142168
$574$	0	0
$575$	54.2061	2.26055
$576$	0	0
$577$	40.2094	1.67394	0.836969	−	0.547250i	$-0.184325\pi$
$577$	40.2094	1.67394	0.836969	+	0.547250i	$0.184325\pi$
$578$	0	0
$579$	14.9702	0.622139
$580$	0	0
$581$	0.806248	0.0334488
$582$	0	0
$583$	−41.7472	−1.72899
$584$	0	0
$585$	24.8062	1.02561
$586$	0	0
$587$	2.89451	0.119469	0.0597346	−	0.998214i	$-0.480975\pi$
$587$	2.89451	0.119469	0.0597346	+	0.998214i	$0.480975\pi$
$588$	0	0
$589$	−12.2094	−0.503078
$590$	0	0
$591$	−59.5060	−2.44775
$592$	0	0
$593$	14.0000	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$594$	0	0
$595$	−11.5294	−0.472658
$596$	0	0
$597$	−26.2094	−1.07268
$598$	0	0
$599$	−26.3365	−1.07608	−0.538041	−	0.842919i	$-0.680835\pi$
$599$	−26.3365	−1.07608	−0.538041	+	0.842919i	$0.680835\pi$
$600$	0	0
$601$	23.3141	0.951000	0.475500	−	0.879716i	$-0.342267\pi$
$601$	23.3141	0.951000	0.475500	+	0.879716i	$0.342267\pi$
$602$	0	0
$603$	−88.2479	−3.59373
$604$	0	0
$605$	8.89531	0.361646
$606$	0	0
$607$	27.8696	1.13119	0.565595	−	0.824683i	$-0.308646\pi$
$607$	27.8696	1.13119	0.565595	+	0.824683i	$0.308646\pi$
$608$	0	0
$609$	3.40312	0.137902
$610$	0	0
$611$	8.96094	0.362521
$612$	0	0
$613$	−30.4187	−1.22860	−0.614301	−	0.789072i	$-0.710562\pi$
$613$	−30.4187	−1.22860	−0.614301	+	0.789072i	$0.710562\pi$
$614$	0	0
$615$	16.1771	0.652326
$616$	0	0
$617$	40.2094	1.61877	0.809384	−	0.587280i	$-0.199801\pi$
$617$	40.2094	1.61877	0.809384	+	0.587280i	$0.199801\pi$
$618$	0	0
$619$	−48.6860	−1.95685	−0.978427	−	0.206594i	$-0.933762\pi$
$619$	−48.6860	−1.95685	−0.978427	+	0.206594i	$0.933762\pi$
$620$	0	0
$621$	−71.8219	−2.88211
$622$	0	0
$623$	0.766519	0.0307099
$624$	0	0
$625$	7.20937	0.288375
$626$	0	0
$627$	29.2882	1.16966
$628$	0	0
$629$	−1.70156	−0.0678457
$630$	0	0
$631$	−0.872365	−0.0347283	−0.0173641	−	0.999849i	$-0.505527\pi$
$631$	−0.872365	−0.0347283	−0.0173641	+	0.999849i	$0.505527\pi$
$632$	0	0
$633$	−16.5078	−0.656127
$634$	0	0
$635$	−69.1763	−2.74518
$636$	0	0
$637$	−6.70156	−0.265526
$638$	0	0
$639$	−45.4082	−1.79632
$640$	0	0
$641$	30.2094	1.19320	0.596599	−	0.802539i	$-0.296518\pi$
$641$	30.2094	1.19320	0.596599	+	0.802539i	$0.296518\pi$
$642$	0	0
$643$	0.709330	0.0279732	0.0139866	−	0.999902i	$-0.495548\pi$
$643$	0.709330	0.0279732	0.0139866	+	0.999902i	$0.495548\pi$
$644$	0	0
$645$	−145.523	−5.72998
$646$	0	0
$647$	−24.9179	−0.979622	−0.489811	−	0.871829i	$-0.662934\pi$
$647$	−24.9179	−0.979622	−0.489811	+	0.871829i	$0.662934\pi$
$648$	0	0
$649$	−36.2094	−1.42134
$650$	0	0
$651$	8.08857	0.317016
$652$	0	0
$653$	−10.5969	−0.414688	−0.207344	−	0.978268i	$-0.566482\pi$
$653$	−10.5969	−0.414688	−0.207344	+	0.978268i	$0.566482\pi$
$654$	0	0
$655$	−42.6767	−1.66752
$656$	0	0
$657$	−36.2094	−1.41266
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	0	0
$663$	−17.7588	−0.689696
$664$	0	0
$665$	5.19375	0.201405
$666$	0	0
$667$	12.4589	0.482412
$668$	0	0
$669$	30.2984	1.17141
$670$	0	0
$671$	0	0
$672$	0	0
$673$	11.3141	0.436125	0.218062	−	0.975935i	$-0.430026\pi$
$673$	11.3141	0.436125	0.218062	+	0.975935i	$0.430026\pi$
$674$	0	0
$675$	−100.324	−3.86146
$676$	0	0
$677$	−49.0156	−1.88382	−0.941912	−	0.335861i	$-0.890973\pi$
$677$	−49.0156	−1.88382	−0.941912	+	0.335861i	$0.890973\pi$
$678$	0	0
$679$	−5.46295	−0.209649
$680$	0	0
$681$	−76.4187	−2.92837
$682$	0	0
$683$	−34.3679	−1.31505	−0.657526	−	0.753432i	$-0.728397\pi$
$683$	−34.3679	−1.31505	−0.657526	+	0.753432i	$0.728397\pi$
$684$	0	0
$685$	51.8219	1.98001
$686$	0	0
$687$	−68.2467	−2.60377
$688$	0	0
$689$	−11.4031	−0.434424
$690$	0	0
$691$	−17.2125	−0.654796	−0.327398	−	0.944886i	$-0.606172\pi$
$691$	−17.2125	−0.654796	−0.327398	+	0.944886i	$0.606172\pi$
$692$	0	0
$693$	−13.4031	−0.509143
$694$	0	0
$695$	3.44080	0.130517
$696$	0	0
$697$	−8.00000	−0.303022
$698$	0	0
$699$	32.7290	1.23793
$700$	0	0
$701$	−16.8062	−0.634763	−0.317382	−	0.948298i	$-0.602804\pi$
$701$	−16.8062	−0.634763	−0.317382	+	0.948298i	$0.602804\pi$
$702$	0	0
$703$	0.766519	0.0289098
$704$	0	0
$705$	−103.314	−3.89103
$706$	0	0
$707$	10.2738	0.386384
$708$	0	0
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	0	0
$711$	90.8164	3.40588
$712$	0	0
$713$	29.6125	1.10900
$714$	0	0
$715$	13.5515	0.506798
$716$	0	0
$717$	83.7328	3.12706
$718$	0	0
$719$	−50.1618	−1.87072	−0.935360	−	0.353698i	$-0.884924\pi$
$719$	−50.1618	−1.87072	−0.935360	+	0.353698i	$0.884924\pi$
$720$	0	0
$721$	2.20937	0.0822813
$722$	0	0
$723$	−8.08857	−0.300817
$724$	0	0
$725$	17.4031	0.646336
$726$	0	0
$727$	45.7914	1.69831	0.849155	−	0.528143i	$-0.177112\pi$
$727$	45.7914	1.69831	0.849155	+	0.528143i	$0.177112\pi$
$728$	0	0
$729$	−1.80625	−0.0668981
$730$	0	0
$731$	71.9649	2.66172
$732$	0	0
$733$	−44.1203	−1.62962	−0.814810	−	0.579728i	$-0.803159\pi$
$733$	−44.1203	−1.62962	−0.814810	+	0.579728i	$0.803159\pi$
$734$	0	0
$735$	77.2648	2.84996
$736$	0	0
$737$	−48.2094	−1.77582
$738$	0	0
$739$	2.56844	0.0944815	0.0472408	−	0.998884i	$-0.484957\pi$
$739$	2.56844	0.0944815	0.0472408	+	0.998884i	$0.484957\pi$
$740$	0	0
$741$	8.00000	0.293887
$742$	0	0
$743$	18.4682	0.677532	0.338766	−	0.940871i	$-0.389991\pi$
$743$	18.4682	0.677532	0.338766	+	0.940871i	$0.389991\pi$
$744$	0	0
$745$	−47.4031	−1.73672
$746$	0	0
$747$	−9.89049	−0.361874
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−10.2738	−0.374895	−0.187447	−	0.982275i	$-0.560021\pi$
$751$	−10.2738	−0.374895	−0.187447	+	0.982275i	$0.560021\pi$
$752$	0	0
$753$	−48.0000	−1.74922
$754$	0	0
$755$	59.0655	2.14962
$756$	0	0
$757$	−10.2094	−0.371066	−0.185533	−	0.982638i	$-0.559401\pi$
$757$	−10.2094	−0.371066	−0.185533	+	0.982638i	$0.559401\pi$
$758$	0	0
$759$	−71.0354	−2.57842
$760$	0	0
$761$	1.40312	0.0508632	0.0254316	−	0.999677i	$-0.491904\pi$
$761$	1.40312	0.0508632	0.0254316	+	0.999677i	$0.491904\pi$
$762$	0	0
$763$	−4.20732	−0.152315
$764$	0	0
$765$	141.434	5.11357
$766$	0	0
$767$	−9.89049	−0.357125
$768$	0	0
$769$	−14.5969	−0.526377	−0.263188	−	0.964744i	$-0.584774\pi$
$769$	−14.5969	−0.526377	−0.263188	+	0.964744i	$0.584774\pi$
$770$	0	0
$771$	−13.3885	−0.482175
$772$	0	0
$773$	−36.2984	−1.30556	−0.652782	−	0.757546i	$-0.726398\pi$
$773$	−36.2984	−1.30556	−0.652782	+	0.757546i	$0.726398\pi$
$774$	0	0
$775$	41.3639	1.48583
$776$	0	0
$777$	−0.507811	−0.0182176
$778$	0	0
$779$	3.60384	0.129121
$780$	0	0
$781$	−24.8062	−0.887637
$782$	0	0
$783$	−23.0588	−0.824053
$784$	0	0
$785$	−64.4187	−2.29920
$786$	0	0
$787$	−9.12397	−0.325235	−0.162617	−	0.986689i	$-0.551994\pi$
$787$	−9.12397	−0.325235	−0.162617	+	0.986689i	$0.551994\pi$
$788$	0	0
$789$	−9.19375	−0.327306
$790$	0	0
$791$	−6.22947	−0.221494
$792$	0	0
$793$	0	0
$794$	0	0
$795$	131.471	4.66279
$796$	0	0
$797$	7.79063	0.275958	0.137979	−	0.990435i	$-0.455939\pi$
$797$	7.79063	0.275958	0.137979	+	0.990435i	$0.455939\pi$
$798$	0	0
$799$	51.0914	1.80748
$800$	0	0
$801$	−9.40312	−0.332243
$802$	0	0
$803$	−19.7810	−0.698056
$804$	0	0
$805$	−12.5969	−0.443982
$806$	0	0
$807$	45.4654	1.60046
$808$	0	0
$809$	−0.507811	−0.0178537	−0.00892683	−	0.999960i	$-0.502842\pi$
$809$	−0.507811	−0.0178537	−0.00892683	+	0.999960i	$0.502842\pi$
$810$	0	0
$811$	30.4380	1.06882	0.534411	−	0.845225i	$-0.320533\pi$
$811$	30.4380	1.06882	0.534411	+	0.845225i	$0.320533\pi$
$812$	0	0
$813$	49.7016	1.74311
$814$	0	0
$815$	51.5805	1.80678
$816$	0	0
$817$	−32.4187	−1.13419
$818$	0	0
$819$	−3.66103	−0.127927
$820$	0	0
$821$	50.5078	1.76273	0.881367	−	0.472432i	$-0.156624\pi$
$821$	50.5078	1.76273	0.881367	+	0.472432i	$0.156624\pi$
$822$	0	0
$823$	21.1996	0.738973	0.369487	−	0.929236i	$-0.379534\pi$
$823$	21.1996	0.738973	0.369487	+	0.929236i	$0.379534\pi$
$824$	0	0
$825$	−99.2250	−3.45457
$826$	0	0
$827$	−9.89049	−0.343926	−0.171963	−	0.985103i	$-0.555011\pi$
$827$	−9.89049	−0.343926	−0.171963	+	0.985103i	$0.555011\pi$
$828$	0	0
$829$	−24.5969	−0.854285	−0.427142	−	0.904184i	$-0.640480\pi$
$829$	−24.5969	−0.854285	−0.427142	+	0.904184i	$0.640480\pi$
$830$	0	0
$831$	−37.3768	−1.29659
$832$	0	0
$833$	−38.2094	−1.32388
$834$	0	0
$835$	−1.41866	−0.0490947
$836$	0	0
$837$	−54.8062	−1.89438
$838$	0	0
$839$	6.28666	0.217039	0.108520	−	0.994094i	$-0.465389\pi$
$839$	6.28666	0.217039	0.108520	+	0.994094i	$0.465389\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−4.37036	−0.150523
$844$	0	0
$845$	3.70156	0.127338
$846$	0	0
$847$	−1.31281	−0.0451088
$848$	0	0
$849$	22.8062	0.782708
$850$	0	0
$851$	−1.85911	−0.0637294
$852$	0	0
$853$	−27.7016	−0.948483	−0.474242	−	0.880395i	$-0.657278\pi$
$853$	−27.7016	−0.948483	−0.474242	+	0.880395i	$0.657278\pi$
$854$	0	0
$855$	−63.7133	−2.17895
$856$	0	0
$857$	−26.4187	−0.902447	−0.451224	−	0.892411i	$-0.649012\pi$
$857$	−26.4187	−0.902447	−0.451224	+	0.892411i	$0.649012\pi$
$858$	0	0
$859$	−5.90340	−0.201421	−0.100711	−	0.994916i	$-0.532112\pi$
$859$	−5.90340	−0.201421	−0.100711	+	0.994916i	$0.532112\pi$
$860$	0	0
$861$	−2.38750	−0.0813659
$862$	0	0
$863$	18.4682	0.628664	0.314332	−	0.949313i	$-0.398219\pi$
$863$	18.4682	0.628664	0.314332	+	0.949313i	$0.398219\pi$
$864$	0	0
$865$	31.8219	1.08198
$866$	0	0
$867$	−48.3027	−1.64045
$868$	0	0
$869$	49.6125	1.68299
$870$	0	0
$871$	−13.1683	−0.446190
$872$	0	0
$873$	67.0156	2.26814
$874$	0	0
$875$	−7.48509	−0.253042
$876$	0	0
$877$	−9.31406	−0.314513	−0.157257	−	0.987558i	$-0.550265\pi$
$877$	−9.31406	−0.314513	−0.157257	+	0.987558i	$0.550265\pi$
$878$	0	0
$879$	34.5881	1.16663
$880$	0	0
$881$	5.91093	0.199144	0.0995722	−	0.995030i	$-0.468253\pi$
$881$	5.91093	0.199144	0.0995722	+	0.995030i	$0.468253\pi$
$882$	0	0
$883$	29.1252	0.980141	0.490070	−	0.871683i	$-0.336971\pi$
$883$	29.1252	0.980141	0.490070	+	0.871683i	$0.336971\pi$
$884$	0	0
$885$	114.031	3.83312
$886$	0	0
$887$	16.0628	0.539335	0.269668	−	0.962953i	$-0.413086\pi$
$887$	16.0628	0.539335	0.269668	+	0.962953i	$0.413086\pi$
$888$	0	0
$889$	10.2094	0.342411
$890$	0	0
$891$	57.8671	1.93862
$892$	0	0
$893$	−23.0156	−0.770188
$894$	0	0
$895$	−31.7508	−1.06131
$896$	0	0
$897$	−19.4031	−0.647851
$898$	0	0
$899$	9.50723	0.317084
$900$	0	0
$901$	−65.0156	−2.16598
$902$	0	0
$903$	21.4771	0.714712
$904$	0	0
$905$	51.8219	1.72262
$906$	0	0
$907$	1.69607	0.0563172	0.0281586	−	0.999603i	$-0.491036\pi$
$907$	1.69607	0.0563172	0.0281586	+	0.999603i	$0.491036\pi$
$908$	0	0
$909$	−126.031	−4.18019
$910$	0	0
$911$	−23.8253	−0.789367	−0.394683	−	0.918817i	$-0.629146\pi$
$911$	−23.8253	−0.789367	−0.394683	+	0.918817i	$0.629146\pi$
$912$	0	0
$913$	−5.40312	−0.178817
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.29844	0.207993
$918$	0	0
$919$	40.9806	1.35183	0.675913	−	0.736981i	$-0.263749\pi$
$919$	40.9806	1.35183	0.675913	+	0.736981i	$0.263749\pi$
$920$	0	0
$921$	−17.1938	−0.566553
$922$	0	0
$923$	−6.77576	−0.223027
$924$	0	0
$925$	−2.59688	−0.0853847
$926$	0	0
$927$	−27.1030	−0.890181
$928$	0	0
$929$	33.4031	1.09592	0.547960	−	0.836504i	$-0.315404\pi$
$929$	33.4031	1.09592	0.547960	+	0.836504i	$0.315404\pi$
$930$	0	0
$931$	17.2125	0.564119
$932$	0	0
$933$	59.2250	1.93894
$934$	0	0
$935$	77.2648	2.52683
$936$	0	0
$937$	−47.6125	−1.55543	−0.777716	−	0.628616i	$-0.783622\pi$
$937$	−47.6125	−1.55543	−0.777716	+	0.628616i	$0.783622\pi$
$938$	0	0
$939$	−40.8176	−1.33203
$940$	0	0
$941$	43.5234	1.41882	0.709412	−	0.704794i	$-0.248961\pi$
$941$	43.5234	1.41882	0.709412	+	0.704794i	$0.248961\pi$
$942$	0	0
$943$	−8.74071	−0.284637
$944$	0	0
$945$	23.3141	0.758406
$946$	0	0
$947$	58.6336	1.90534	0.952669	−	0.304011i	$-0.0983259\pi$
$947$	58.6336	1.90534	0.952669	+	0.304011i	$0.0983259\pi$
$948$	0	0
$949$	−5.40312	−0.175393
$950$	0	0
$951$	−18.6884	−0.606013
$952$	0	0
$953$	19.7016	0.638196	0.319098	−	0.947722i	$-0.396620\pi$
$953$	19.7016	0.638196	0.319098	+	0.947722i	$0.396620\pi$
$954$	0	0
$955$	4.04429	0.130870
$956$	0	0
$957$	−22.8062	−0.737221
$958$	0	0
$959$	−7.64813	−0.246971
$960$	0	0
$961$	−8.40312	−0.271069
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−17.7906	−0.572701
$966$	0	0
$967$	−21.4199	−0.688817	−0.344408	−	0.938820i	$-0.611920\pi$
$967$	−21.4199	−0.688817	−0.344408	+	0.938820i	$0.611920\pi$
$968$	0	0
$969$	45.6125	1.46528
$970$	0	0
$971$	−3.11473	−0.0999565	−0.0499783	−	0.998750i	$-0.515915\pi$
$971$	−3.11473	−0.0999565	−0.0499783	+	0.998750i	$0.515915\pi$
$972$	0	0
$973$	−0.507811	−0.0162797
$974$	0	0
$975$	−27.1030	−0.867992
$976$	0	0
$977$	32.8062	1.04956	0.524782	−	0.851236i	$-0.324147\pi$
$977$	32.8062	1.04956	0.524782	+	0.851236i	$0.324147\pi$
$978$	0	0
$979$	−5.13688	−0.164175
$980$	0	0
$981$	51.6125	1.64786
$982$	0	0
$983$	−49.6155	−1.58249	−0.791244	−	0.611500i	$-0.790566\pi$
$983$	−49.6155	−1.58249	−0.791244	+	0.611500i	$0.790566\pi$
$984$	0	0
$985$	70.7172	2.25324
$986$	0	0
$987$	15.2476	0.485336
$988$	0	0
$989$	78.6281	2.50023
$990$	0	0
$991$	40.3285	1.28108	0.640538	−	0.767926i	$-0.278711\pi$
$991$	40.3285	1.28108	0.640538	+	0.767926i	$0.278711\pi$
$992$	0	0
$993$	−20.5969	−0.653622
$994$	0	0
$995$	31.1473	0.987437
$996$	0	0
$997$	26.2094	0.830059	0.415030	−	0.909808i	$-0.363771\pi$
$997$	26.2094	0.830059	0.415030	+	0.909808i	$0.363771\pi$
$998$	0	0
$999$	3.44080	0.108862

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3328.2.a.bl.1.1		4
4.3	odd	2	inner	3328.2.a.bl.1.4		4
8.3	odd	2		3328.2.a.bk.1.1		4
8.5	even	2		3328.2.a.bk.1.4		4
16.3	odd	4		1664.2.b.j.833.7	yes	8
16.5	even	4		1664.2.b.j.833.8	yes	8
16.11	odd	4		1664.2.b.j.833.2	yes	8
16.13	even	4		1664.2.b.j.833.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1664.2.b.j.833.1	✓	8	16.13	even	4
1664.2.b.j.833.2	yes	8	16.11	odd	4
1664.2.b.j.833.7	yes	8	16.3	odd	4
1664.2.b.j.833.8	yes	8	16.5	even	4
3328.2.a.bk.1.1		4	8.3	odd	2
3328.2.a.bk.1.4		4	8.5	even	2
3328.2.a.bl.1.1		4	1.1	even	1	trivial
3328.2.a.bl.1.4		4	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 3328 = 2^{8} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3328.a (trivial)

\(f(q)\)	\(=\)	\(q-3.11473 q^{3} +3.70156 q^{5} -0.546295 q^{7} +6.70156 q^{9} +3.66103 q^{11} +1.00000 q^{13} -11.5294 q^{15} +5.70156 q^{17} -2.56844 q^{19} +1.70156 q^{21} +6.22947 q^{23} +8.70156 q^{25} -11.5294 q^{27} +2.00000 q^{29} +4.75362 q^{31} -11.4031 q^{33} -2.02214 q^{35} -0.298438 q^{37} -3.11473 q^{39} -1.40312 q^{41} +12.6220 q^{43} +24.8062 q^{45} +8.96094 q^{47} -6.70156 q^{49} -17.7588 q^{51} -11.4031 q^{53} +13.5515 q^{55} +8.00000 q^{57} -9.89049 q^{59} -3.66103 q^{63} +3.70156 q^{65} -13.1683 q^{67} -19.4031 q^{69} -6.77576 q^{71} -5.40312 q^{73} -27.1030 q^{75} -2.00000 q^{77} +13.5515 q^{79} +15.8062 q^{81} -1.47585 q^{83} +21.1047 q^{85} -6.22947 q^{87} -1.40312 q^{89} -0.546295 q^{91} -14.8062 q^{93} -9.50723 q^{95} +10.0000 q^{97} +24.5346 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{5} + 14 q^{9} + 4 q^{13} + 10 q^{17} - 6 q^{21} + 22 q^{25} + 8 q^{29} - 20 q^{33} - 14 q^{37} + 20 q^{41} + 48 q^{45} - 14 q^{49} - 20 q^{53} + 32 q^{57} + 2 q^{65} - 52 q^{69} + 4 q^{73} - 8 q^{77}+ \cdots + 40 q^{97}+O(q^{100}) \) 4 * q + 2 * q^5 + 14 * q^9 + 4 * q^13 + 10 * q^17 - 6 * q^21 + 22 * q^25 + 8 * q^29 - 20 * q^33 - 14 * q^37 + 20 * q^41 + 48 * q^45 - 14 * q^49 - 20 * q^53 + 32 * q^57 + 2 * q^65 - 52 * q^69 + 4 * q^73 - 8 * q^77 + 12 * q^81 + 46 * q^85 + 20 * q^89 - 8 * q^93 + 40 * q^97

Introduction

L-functions

Modular forms

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