LMFDB - Embedded newform 1248.2.g.a.625.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1248,2,Mod(625,1248)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1248, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1248.625"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1248 = 2^{5} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1248.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$9.96533017226$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{20})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 312)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			625.3
Root			$-0.951057 - 0.309017i$ of defining polynomial
Character	$\chi$	$=$	1248.625
Dual form			1248.2.g.a.625.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +1.28408i q^{5} -0.442463 q^{7} -1.00000 q^{9} -0.841616i q^{11} +1.00000i q^{13} +1.28408 q^{15} +0.351141 q^{17} +3.67853i q^{19} +0.442463i q^{21} +7.04029 q^{23} +3.35114 q^{25} +1.00000i q^{27} +0.217017i q^{29} +5.14475 q^{31} -0.841616 q^{33} -0.568158i q^{35} -5.60845i q^{37} +1.00000 q^{39} +4.49890 q^{41} +10.4317i q^{43} -1.28408i q^{45} +6.90096 q^{47} -6.80423 q^{49} -0.351141i q^{51} +3.35706i q^{53} +1.08070 q^{55} +3.67853 q^{57} -4.74559i q^{59} +7.23607i q^{61} +0.442463 q^{63} -1.28408 q^{65} +7.67853i q^{67} -7.04029i q^{69} +7.78589 q^{71} +4.23015 q^{73} -3.35114i q^{75} +0.372384i q^{77} +0.230146 q^{79} +1.00000 q^{81} -7.86067i q^{83} +0.450893i q^{85} +0.217017 q^{87} +5.81787 q^{89} -0.442463i q^{91} -5.14475i q^{93} -4.72353 q^{95} +9.66780 q^{97} +0.841616i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{7} - 8 q^{9} + 4 q^{15} - 16 q^{17} + 8 q^{23} + 8 q^{25} + 4 q^{31} + 8 q^{39} + 36 q^{41} - 24 q^{47} - 24 q^{49} + 40 q^{55} + 12 q^{57} + 4 q^{63} - 4 q^{65} - 16 q^{71} + 32 q^{73} + 8 q^{81}+ \cdots - 56 q^{97}+O(q^{100}) $ 8 * q - 4 * q^7 - 8 * q^9 + 4 * q^15 - 16 * q^17 + 8 * q^23 + 8 * q^25 + 4 * q^31 + 8 * q^39 + 36 * q^41 - 24 * q^47 - 24 * q^49 + 40 * q^55 + 12 * q^57 + 4 * q^63 - 4 * q^65 - 16 * q^71 + 32 * q^73 + 8 * q^81 + 8 * q^87 + 60 * q^89 + 24 * q^95 - 56 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1248\mathbb{Z}\right)^\times$.

$n$	$703$	$769$	$833$	$1093$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	1.28408i	0.574258i	0.957892	+	0.287129i	$0.0927008\pi$
$5$	1.28408i	0.574258i	−0.957892	+	0.287129i	$0.907299\pi$
$6$	0	0
$7$	−0.442463	−0.167235	−0.0836177	−	0.996498i	$-0.526647\pi$
$7$	−0.442463	−0.167235	−0.0836177	+	0.996498i	$0.526647\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 0.841616i	− 0.253757i	−0.991918	−	0.126878i	$-0.959504\pi$
$11$	− 0.841616i	− 0.253757i	0.991918	−	0.126878i	$-0.0404958\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	0	0
$15$	1.28408	0.331548
$16$	0	0
$17$	0.351141	0.0851642	0.0425821	−	0.999093i	$-0.486442\pi$
$17$	0.351141	0.0851642	0.0425821	+	0.999093i	$0.486442\pi$
$18$	0	0
$19$	3.67853i	0.843913i	0.906616	+	0.421956i	$0.138657\pi$
$19$	3.67853i	0.843913i	−0.906616	+	0.421956i	$0.861343\pi$
$20$	0	0
$21$	0.442463i	0.0965534i
$22$	0	0
$23$	7.04029	1.46800	0.734001	−	0.679148i	$-0.237651\pi$
$23$	7.04029	1.46800	0.734001	+	0.679148i	$0.237651\pi$
$24$	0	0
$25$	3.35114	0.670228
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	0.217017i	0.0402991i	0.999797	+	0.0201495i	$0.00641423\pi$
$29$	0.217017i	0.0402991i	−0.999797	+	0.0201495i	$0.993586\pi$
$30$	0	0
$31$	5.14475	0.924024	0.462012	−	0.886874i	$-0.347128\pi$
$31$	5.14475	0.924024	0.462012	+	0.886874i	$0.347128\pi$
$32$	0	0
$33$	−0.841616	−0.146506
$34$	0	0
$35$	− 0.568158i	− 0.0960362i
$36$	0	0
$37$	− 5.60845i	− 0.922024i	−0.887394	−	0.461012i	$-0.847487\pi$
$37$	− 5.60845i	− 0.922024i	0.887394	−	0.461012i	$-0.152513\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	4.49890	0.702611	0.351305	−	0.936261i	$-0.385738\pi$
$41$	4.49890	0.702611	0.351305	+	0.936261i	$0.385738\pi$
$42$	0	0
$43$	10.4317i	1.59082i	0.606069	+	0.795412i	$0.292745\pi$
$43$	10.4317i	1.59082i	−0.606069	+	0.795412i	$0.707255\pi$
$44$	0	0
$45$	− 1.28408i	− 0.191419i
$46$	0	0
$47$	6.90096	1.00661	0.503304	−	0.864109i	$-0.332117\pi$
$47$	6.90096	1.00661	0.503304	+	0.864109i	$0.332117\pi$
$48$	0	0
$49$	−6.80423	−0.972032
$50$	0	0
$51$	− 0.351141i	− 0.0491696i
$52$	0	0
$53$	3.35706i	0.461128i	0.973057	+	0.230564i	$0.0740571\pi$
$53$	3.35706i	0.461128i	−0.973057	+	0.230564i	$0.925943\pi$
$54$	0	0
$55$	1.08070	0.145722
$56$	0	0
$57$	3.67853	0.487233
$58$	0	0
$59$	− 4.74559i	− 0.617824i	−0.951091	−	0.308912i	$-0.900035\pi$
$59$	− 4.74559i	− 0.617824i	0.951091	−	0.308912i	$-0.0999648\pi$
$60$	0	0
$61$	7.23607i	0.926484i	0.886232	+	0.463242i	$0.153314\pi$
$61$	7.23607i	0.926484i	−0.886232	+	0.463242i	$0.846686\pi$
$62$	0	0
$63$	0.442463	0.0557452
$64$	0	0
$65$	−1.28408	−0.159270
$66$	0	0
$67$	7.67853i	0.938082i	0.883176	+	0.469041i	$0.155400\pi$
$67$	7.67853i	0.938082i	−0.883176	+	0.469041i	$0.844600\pi$
$68$	0	0
$69$	− 7.04029i	− 0.847552i
$70$	0	0
$71$	7.78589	0.924015	0.462007	−	0.886876i	$-0.347129\pi$
$71$	7.78589	0.924015	0.462007	+	0.886876i	$0.347129\pi$
$72$	0	0
$73$	4.23015	0.495101	0.247551	−	0.968875i	$-0.420374\pi$
$73$	4.23015	0.495101	0.247551	+	0.968875i	$0.420374\pi$
$74$	0	0
$75$	− 3.35114i	− 0.386956i
$76$	0	0
$77$	0.372384i	0.0424371i
$78$	0	0
$79$	0.230146	0.0258935	0.0129467	−	0.999916i	$-0.495879\pi$
$79$	0.230146	0.0258935	0.0129467	+	0.999916i	$0.495879\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 7.86067i	− 0.862820i	−0.902156	−	0.431410i	$-0.858016\pi$
$83$	− 7.86067i	− 0.862820i	0.902156	−	0.431410i	$-0.141984\pi$
$84$	0	0
$85$	0.450893i	0.0489062i
$86$	0	0
$87$	0.217017	0.0232667
$88$	0	0
$89$	5.81787	0.616692	0.308346	−	0.951274i	$-0.400224\pi$
$89$	5.81787	0.616692	0.308346	+	0.951274i	$0.400224\pi$
$90$	0	0
$91$	− 0.442463i	− 0.0463828i
$92$	0	0
$93$	− 5.14475i	− 0.533485i
$94$	0	0
$95$	−4.72353	−0.484623
$96$	0	0
$97$	9.66780	0.981616	0.490808	−	0.871268i	$-0.336702\pi$
$97$	9.66780	0.981616	0.490808	+	0.871268i	$0.336702\pi$
$98$	0	0
$99$	0.841616i	0.0845855i
$100$	0	0
$101$	5.92522i	0.589582i	0.955562	+	0.294791i	$0.0952500\pi$
$101$	5.92522i	0.589582i	−0.955562	+	0.294791i	$0.904750\pi$
$102$	0	0
$103$	−0.608565	−0.0599637	−0.0299818	−	0.999550i	$-0.509545\pi$
$103$	−0.608565	−0.0599637	−0.0299818	+	0.999550i	$0.509545\pi$
$104$	0	0
$105$	−0.568158	−0.0554466
$106$	0	0
$107$	− 18.3701i	− 1.77590i	−0.459938	−	0.887951i	$-0.652128\pi$
$107$	− 18.3701i	− 1.77590i	0.459938	−	0.887951i	$-0.347872\pi$
$108$	0	0
$109$	− 2.54691i	− 0.243950i	−0.992533	−	0.121975i	$-0.961077\pi$
$109$	− 2.54691i	− 0.243950i	0.992533	−	0.121975i	$-0.0389228\pi$
$110$	0	0
$111$	−5.60845	−0.532331
$112$	0	0
$113$	−10.7273	−1.00914	−0.504568	−	0.863372i	$-0.668348\pi$
$113$	−10.7273	−1.00914	−0.504568	+	0.863372i	$0.668348\pi$
$114$	0	0
$115$	9.04029i	0.843012i
$116$	0	0
$117$	− 1.00000i	− 0.0924500i
$118$	0	0
$119$	−0.155367	−0.0142425
$120$	0	0
$121$	10.2917	0.935608
$122$	0	0
$123$	− 4.49890i	− 0.405653i
$124$	0	0
$125$	10.7235i	0.959141i
$126$	0	0
$127$	−8.92303	−0.791791	−0.395895	−	0.918296i	$-0.629566\pi$
$127$	−8.92303	−0.791791	−0.395895	+	0.918296i	$0.629566\pi$
$128$	0	0
$129$	10.4317	0.918463
$130$	0	0
$131$	− 7.19566i	− 0.628688i	−0.949309	−	0.314344i	$-0.898215\pi$
$131$	− 7.19566i	− 0.628688i	0.949309	−	0.314344i	$-0.101785\pi$
$132$	0	0
$133$	− 1.62762i	− 0.141132i
$134$	0	0
$135$	−1.28408	−0.110516
$136$	0	0
$137$	7.99448	0.683014	0.341507	−	0.939879i	$-0.389063\pi$
$137$	7.99448	0.683014	0.341507	+	0.939879i	$0.389063\pi$
$138$	0	0
$139$	− 9.27855i	− 0.786997i	−0.919325	−	0.393498i	$-0.871265\pi$
$139$	− 9.27855i	− 0.786997i	0.919325	−	0.393498i	$-0.128735\pi$
$140$	0	0
$141$	− 6.90096i	− 0.581166i
$142$	0	0
$143$	0.841616	0.0703794
$144$	0	0
$145$	−0.278667	−0.0231420
$146$	0	0
$147$	6.80423i	0.561203i
$148$	0	0
$149$	− 17.3161i	− 1.41859i	−0.704910	−	0.709297i	$-0.749013\pi$
$149$	− 17.3161i	− 1.41859i	0.704910	−	0.709297i	$-0.250987\pi$
$150$	0	0
$151$	0.555344	0.0451933	0.0225966	−	0.999745i	$-0.492807\pi$
$151$	0.555344	0.0451933	0.0225966	+	0.999745i	$0.492807\pi$
$152$	0	0
$153$	−0.351141	−0.0283881
$154$	0	0
$155$	6.60626i	0.530628i
$156$	0	0
$157$	22.8826i	1.82623i	0.407700	+	0.913116i	$0.366331\pi$
$157$	22.8826i	1.82623i	−0.407700	+	0.913116i	$0.633669\pi$
$158$	0	0
$159$	3.35706	0.266232
$160$	0	0
$161$	−3.11507	−0.245502
$162$	0	0
$163$	20.0853i	1.57320i	0.617462	+	0.786601i	$0.288161\pi$
$163$	20.0853i	1.57320i	−0.617462	+	0.786601i	$0.711839\pi$
$164$	0	0
$165$	− 1.08070i	− 0.0841325i
$166$	0	0
$167$	−17.0776	−1.32150	−0.660751	−	0.750605i	$-0.729762\pi$
$167$	−17.0776	−1.32150	−0.660751	+	0.750605i	$0.729762\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	− 3.67853i	− 0.281304i
$172$	0	0
$173$	− 17.1421i	− 1.30329i	−0.758523	−	0.651646i	$-0.774079\pi$
$173$	− 17.1421i	− 1.30329i	0.758523	−	0.651646i	$-0.225921\pi$
$174$	0	0
$175$	−1.48276	−0.112086
$176$	0	0
$177$	−4.74559	−0.356701
$178$	0	0
$179$	− 5.24418i	− 0.391969i	−0.980607	−	0.195984i	$-0.937210\pi$
$179$	− 5.24418i	− 0.391969i	0.980607	−	0.195984i	$-0.0627902\pi$
$180$	0	0
$181$	16.4105i	1.21978i	0.792485	+	0.609891i	$0.208787\pi$
$181$	16.4105i	1.21978i	−0.792485	+	0.609891i	$0.791213\pi$
$182$	0	0
$183$	7.23607	0.534906
$184$	0	0
$185$	7.20170	0.529479
$186$	0	0
$187$	− 0.295526i	− 0.0216110i
$188$	0	0
$189$	− 0.442463i	− 0.0321845i
$190$	0	0
$191$	−14.9537	−1.08201	−0.541005	−	0.841019i	$-0.681956\pi$
$191$	−14.9537	−1.08201	−0.541005	+	0.841019i	$0.681956\pi$
$192$	0	0
$193$	−20.6868	−1.48907	−0.744536	−	0.667582i	$-0.767329\pi$
$193$	−20.6868	−1.48907	−0.744536	+	0.667582i	$0.767329\pi$
$194$	0	0
$195$	1.28408i	0.0919548i
$196$	0	0
$197$	8.12048i	0.578561i	0.957244	+	0.289280i	$0.0934160\pi$
$197$	8.12048i	0.578561i	−0.957244	+	0.289280i	$0.906584\pi$
$198$	0	0
$199$	−25.9098	−1.83670	−0.918348	−	0.395774i	$-0.870476\pi$
$199$	−25.9098	−1.83670	−0.918348	+	0.395774i	$0.870476\pi$
$200$	0	0
$201$	7.67853	0.541602
$202$	0	0
$203$	− 0.0960221i	− 0.00673943i
$204$	0	0
$205$	5.77695i	0.403480i
$206$	0	0
$207$	−7.04029	−0.489334
$208$	0	0
$209$	3.09591	0.214149
$210$	0	0
$211$	− 19.4589i	− 1.33961i	−0.742539	−	0.669803i	$-0.766379\pi$
$211$	− 19.4589i	− 1.33961i	0.742539	−	0.669803i	$-0.233621\pi$
$212$	0	0
$213$	− 7.78589i	− 0.533480i
$214$	0	0
$215$	−13.3952	−0.913543
$216$	0	0
$217$	−2.27636	−0.154530
$218$	0	0
$219$	− 4.23015i	− 0.285847i
$220$	0	0
$221$	0.351141i	0.0236203i
$222$	0	0
$223$	14.5325	0.973165	0.486582	−	0.873635i	$-0.338243\pi$
$223$	14.5325	0.973165	0.486582	+	0.873635i	$0.338243\pi$
$224$	0	0
$225$	−3.35114	−0.223409
$226$	0	0
$227$	8.94345i	0.593597i	0.954940	+	0.296799i	$0.0959190\pi$
$227$	8.94345i	0.593597i	−0.954940	+	0.296799i	$0.904081\pi$
$228$	0	0
$229$	21.0343i	1.38998i	0.719018	+	0.694991i	$0.244592\pi$
$229$	21.0343i	1.38998i	−0.719018	+	0.694991i	$0.755408\pi$
$230$	0	0
$231$	0.372384	0.0245011
$232$	0	0
$233$	−9.94646	−0.651615	−0.325807	−	0.945436i	$-0.605636\pi$
$233$	−9.94646	−0.651615	−0.325807	+	0.945436i	$0.605636\pi$
$234$	0	0
$235$	8.86138i	0.578053i
$236$	0	0
$237$	− 0.230146i	− 0.0149496i
$238$	0	0
$239$	1.83441	0.118658	0.0593290	−	0.998238i	$-0.481104\pi$
$239$	1.83441	0.118658	0.0593290	+	0.998238i	$0.481104\pi$
$240$	0	0
$241$	−8.11727	−0.522879	−0.261440	−	0.965220i	$-0.584197\pi$
$241$	−8.11727	−0.522879	−0.261440	+	0.965220i	$0.584197\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	− 8.73716i	− 0.558197i
$246$	0	0
$247$	−3.67853	−0.234059
$248$	0	0
$249$	−7.86067	−0.498149
$250$	0	0
$251$	31.3723i	1.98020i	0.140363	+	0.990100i	$0.455173\pi$
$251$	31.3723i	1.98020i	−0.140363	+	0.990100i	$0.544827\pi$
$252$	0	0
$253$	− 5.92522i	− 0.372515i
$254$	0	0
$255$	0.450893	0.0282360
$256$	0	0
$257$	−20.8789	−1.30239	−0.651195	−	0.758911i	$-0.725732\pi$
$257$	−20.8789	−1.30239	−0.651195	+	0.758911i	$0.725732\pi$
$258$	0	0
$259$	2.48154i	0.154195i
$260$	0	0
$261$	− 0.217017i	− 0.0134330i
$262$	0	0
$263$	4.83139	0.297916	0.148958	−	0.988843i	$-0.452408\pi$
$263$	4.83139	0.297916	0.148958	+	0.988843i	$0.452408\pi$
$264$	0	0
$265$	−4.31073	−0.264806
$266$	0	0
$267$	− 5.81787i	− 0.356048i
$268$	0	0
$269$	11.7022i	0.713494i	0.934201	+	0.356747i	$0.116114\pi$
$269$	11.7022i	0.713494i	−0.934201	+	0.356747i	$0.883886\pi$
$270$	0	0
$271$	9.27385	0.563347	0.281673	−	0.959510i	$-0.409111\pi$
$271$	9.27385	0.563347	0.281673	+	0.959510i	$0.409111\pi$
$272$	0	0
$273$	−0.442463	−0.0267791
$274$	0	0
$275$	− 2.82037i	− 0.170075i
$276$	0	0
$277$	− 0.676024i	− 0.0406183i	−0.999794	−	0.0203092i	$-0.993535\pi$
$277$	− 0.676024i	− 0.0406183i	0.999794	−	0.0203092i	$-0.00646506\pi$
$278$	0	0
$279$	−5.14475	−0.308008
$280$	0	0
$281$	−1.07891	−0.0643621	−0.0321811	−	0.999482i	$-0.510245\pi$
$281$	−1.07891	−0.0643621	−0.0321811	+	0.999482i	$0.510245\pi$
$282$	0	0
$283$	− 6.72353i	− 0.399672i	−0.979829	−	0.199836i	$-0.935959\pi$
$283$	− 6.72353i	− 0.399672i	0.979829	−	0.199836i	$-0.0640410\pi$
$284$	0	0
$285$	4.72353i	0.279797i
$286$	0	0
$287$	−1.99060	−0.117501
$288$	0	0
$289$	−16.8767	−0.992747
$290$	0	0
$291$	− 9.66780i	− 0.566736i
$292$	0	0
$293$	− 32.9152i	− 1.92293i	−0.274934	−	0.961463i	$-0.588656\pi$
$293$	− 32.9152i	− 1.92293i	0.274934	−	0.961463i	$-0.411344\pi$
$294$	0	0
$295$	6.09372	0.354790
$296$	0	0
$297$	0.841616	0.0488355
$298$	0	0
$299$	7.04029i	0.407151i
$300$	0	0
$301$	− 4.61566i	− 0.266042i
$302$	0	0
$303$	5.92522	0.340395
$304$	0	0
$305$	−9.29168	−0.532040
$306$	0	0
$307$	10.8105i	0.616986i	0.951227	+	0.308493i	$0.0998246\pi$
$307$	10.8105i	0.616986i	−0.951227	+	0.308493i	$0.900175\pi$
$308$	0	0
$309$	0.608565i	0.0346201i
$310$	0	0
$311$	23.7426	1.34632	0.673159	−	0.739498i	$-0.264937\pi$
$311$	23.7426	1.34632	0.673159	+	0.739498i	$0.264937\pi$
$312$	0	0
$313$	−12.4398	−0.703142	−0.351571	−	0.936161i	$-0.614352\pi$
$313$	−12.4398	−0.703142	−0.351571	+	0.936161i	$0.614352\pi$
$314$	0	0
$315$	0.568158i	0.0320121i
$316$	0	0
$317$	− 28.4643i	− 1.59871i	−0.600856	−	0.799357i	$-0.705173\pi$
$317$	− 28.4643i	− 1.59871i	0.600856	−	0.799357i	$-0.294827\pi$
$318$	0	0
$319$	0.182645	0.0102262
$320$	0	0
$321$	−18.3701	−1.02532
$322$	0	0
$323$	1.29168i	0.0718712i
$324$	0	0
$325$	3.35114i	0.185888i
$326$	0	0
$327$	−2.54691	−0.140845
$328$	0	0
$329$	−3.05342	−0.168341
$330$	0	0
$331$	− 12.1021i	− 0.665194i	−0.943069	−	0.332597i	$-0.892075\pi$
$331$	− 12.1021i	− 0.665194i	0.943069	−	0.332597i	$-0.107925\pi$
$332$	0	0
$333$	5.60845i	0.307341i
$334$	0	0
$335$	−9.85984	−0.538701
$336$	0	0
$337$	−34.9882	−1.90593	−0.952964	−	0.303084i	$-0.901984\pi$
$337$	−34.9882	−1.90593	−0.952964	+	0.303084i	$0.901984\pi$
$338$	0	0
$339$	10.7273i	0.582625i
$340$	0	0
$341$	− 4.32990i	− 0.234477i
$342$	0	0
$343$	6.10787	0.329794
$344$	0	0
$345$	9.04029	0.486713
$346$	0	0
$347$	− 14.9561i	− 0.802886i	−0.915884	−	0.401443i	$-0.868509\pi$
$347$	− 14.9561i	− 0.802886i	0.915884	−	0.401443i	$-0.131491\pi$
$348$	0	0
$349$	− 21.1802i	− 1.13375i	−0.823803	−	0.566876i	$-0.808152\pi$
$349$	− 21.1802i	− 1.13375i	0.823803	−	0.566876i	$-0.191848\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−5.64462	−0.300433	−0.150216	−	0.988653i	$-0.547997\pi$
$353$	−5.64462	−0.300433	−0.150216	+	0.988653i	$0.547997\pi$
$354$	0	0
$355$	9.99769i	0.530623i
$356$	0	0
$357$	0.155367i	0.00822290i
$358$	0	0
$359$	−1.54993	−0.0818023	−0.0409011	−	0.999163i	$-0.513023\pi$
$359$	−1.54993	−0.0818023	−0.0409011	+	0.999163i	$0.513023\pi$
$360$	0	0
$361$	5.46841	0.287811
$362$	0	0
$363$	− 10.2917i	− 0.540173i
$364$	0	0
$365$	5.43184i	0.284316i
$366$	0	0
$367$	−16.7367	−0.873646	−0.436823	−	0.899547i	$-0.643896\pi$
$367$	−16.7367	−0.873646	−0.436823	+	0.899547i	$0.643896\pi$
$368$	0	0
$369$	−4.49890	−0.234204
$370$	0	0
$371$	− 1.48538i	− 0.0771170i
$372$	0	0
$373$	22.3532i	1.15741i	0.815538	+	0.578703i	$0.196441\pi$
$373$	22.3532i	1.15741i	−0.815538	+	0.578703i	$0.803559\pi$
$374$	0	0
$375$	10.7235	0.553760
$376$	0	0
$377$	−0.217017	−0.0111769
$378$	0	0
$379$	− 4.22545i	− 0.217047i	−0.994094	−	0.108523i	$-0.965388\pi$
$379$	− 4.22545i	− 0.217047i	0.994094	−	0.108523i	$-0.0346122\pi$
$380$	0	0
$381$	8.92303i	0.457141i
$382$	0	0
$383$	25.2602	1.29074	0.645368	−	0.763872i	$-0.276704\pi$
$383$	25.2602	1.29074	0.645368	+	0.763872i	$0.276704\pi$
$384$	0	0
$385$	−0.478171	−0.0243698
$386$	0	0
$387$	− 10.4317i	− 0.530275i
$388$	0	0
$389$	12.3130i	0.624296i	0.950034	+	0.312148i	$0.101048\pi$
$389$	12.3130i	0.624296i	−0.950034	+	0.312148i	$0.898952\pi$
$390$	0	0
$391$	2.47214	0.125021
$392$	0	0
$393$	−7.19566	−0.362973
$394$	0	0
$395$	0.295526i	0.0148695i
$396$	0	0
$397$	− 24.0271i	− 1.20588i	−0.797785	−	0.602942i	$-0.793995\pi$
$397$	− 24.0271i	− 1.20588i	0.797785	−	0.602942i	$-0.206005\pi$
$398$	0	0
$399$	−1.62762	−0.0814827
$400$	0	0
$401$	36.4832	1.82189	0.910943	−	0.412532i	$-0.135355\pi$
$401$	36.4832	1.82189	0.910943	+	0.412532i	$0.135355\pi$
$402$	0	0
$403$	5.14475i	0.256278i
$404$	0	0
$405$	1.28408i	0.0638064i
$406$	0	0
$407$	−4.72016	−0.233970
$408$	0	0
$409$	35.6796	1.76424	0.882122	−	0.471020i	$-0.156114\pi$
$409$	35.6796	1.76424	0.882122	+	0.471020i	$0.156114\pi$
$410$	0	0
$411$	− 7.99448i	− 0.394338i
$412$	0	0
$413$	2.09975i	0.103322i
$414$	0	0
$415$	10.0937	0.495481
$416$	0	0
$417$	−9.27855	−0.454373
$418$	0	0
$419$	− 38.6383i	− 1.88761i	−0.330509	−	0.943803i	$-0.607220\pi$
$419$	− 38.6383i	− 1.88761i	0.330509	−	0.943803i	$-0.392780\pi$
$420$	0	0
$421$	− 21.9454i	− 1.06956i	−0.844993	−	0.534778i	$-0.820395\pi$
$421$	− 21.9454i	− 1.06956i	0.844993	−	0.534778i	$-0.179605\pi$
$422$	0	0
$423$	−6.90096	−0.335536
$424$	0	0
$425$	1.17672	0.0570795
$426$	0	0
$427$	− 3.20170i	− 0.154941i
$428$	0	0
$429$	− 0.841616i	− 0.0406336i
$430$	0	0
$431$	−18.9340	−0.912021	−0.456010	−	0.889974i	$-0.650722\pi$
$431$	−18.9340	−0.912021	−0.456010	+	0.889974i	$0.650722\pi$
$432$	0	0
$433$	−13.8790	−0.666982	−0.333491	−	0.942753i	$-0.608227\pi$
$433$	−13.8790	−0.666982	−0.333491	+	0.942753i	$0.608227\pi$
$434$	0	0
$435$	0.278667i	0.0133611i
$436$	0	0
$437$	25.8979i	1.23887i
$438$	0	0
$439$	−2.56585	−0.122461	−0.0612307	−	0.998124i	$-0.519503\pi$
$439$	−2.56585	−0.122461	−0.0612307	+	0.998124i	$0.519503\pi$
$440$	0	0
$441$	6.80423	0.324011
$442$	0	0
$443$	31.6297i	1.50277i	0.659863	+	0.751386i	$0.270614\pi$
$443$	31.6297i	1.50277i	−0.659863	+	0.751386i	$0.729386\pi$
$444$	0	0
$445$	7.47060i	0.354140i
$446$	0	0
$447$	−17.3161	−0.819026
$448$	0	0
$449$	−20.9329	−0.987886	−0.493943	−	0.869494i	$-0.664445\pi$
$449$	−20.9329	−0.987886	−0.493943	+	0.869494i	$0.664445\pi$
$450$	0	0
$451$	− 3.78635i	− 0.178292i
$452$	0	0
$453$	− 0.555344i	− 0.0260923i
$454$	0	0
$455$	0.568158	0.0266357
$456$	0	0
$457$	32.5740	1.52375	0.761873	−	0.647726i	$-0.224280\pi$
$457$	32.5740	1.52375	0.761873	+	0.647726i	$0.224280\pi$
$458$	0	0
$459$	0.351141i	0.0163899i
$460$	0	0
$461$	− 38.3837i	− 1.78771i	−0.448358	−	0.893854i	$-0.647991\pi$
$461$	− 38.3837i	− 1.78771i	0.448358	−	0.893854i	$-0.352009\pi$
$462$	0	0
$463$	−8.83401	−0.410551	−0.205276	−	0.978704i	$-0.565809\pi$
$463$	−8.83401	−0.410551	−0.205276	+	0.978704i	$0.565809\pi$
$464$	0	0
$465$	6.60626	0.306358
$466$	0	0
$467$	2.58502i	0.119620i	0.998210	+	0.0598102i	$0.0190495\pi$
$467$	2.58502i	0.119620i	−0.998210	+	0.0598102i	$0.980950\pi$
$468$	0	0
$469$	− 3.39747i	− 0.156881i
$470$	0	0
$471$	22.8826	1.05438
$472$	0	0
$473$	8.77951	0.403682
$474$	0	0
$475$	12.3273i	0.565614i
$476$	0	0
$477$	− 3.35706i	− 0.153709i
$478$	0	0
$479$	−12.5138	−0.571770	−0.285885	−	0.958264i	$-0.592288\pi$
$479$	−12.5138	−0.571770	−0.285885	+	0.958264i	$0.592288\pi$
$480$	0	0
$481$	5.60845	0.255723
$482$	0	0
$483$	3.11507i	0.141741i
$484$	0	0
$485$	12.4142i	0.563700i
$486$	0	0
$487$	−36.2215	−1.64135	−0.820676	−	0.571394i	$-0.806403\pi$
$487$	−36.2215	−1.64135	−0.820676	+	0.571394i	$0.806403\pi$
$488$	0	0
$489$	20.0853	0.908288
$490$	0	0
$491$	23.7638i	1.07245i	0.844076	+	0.536223i	$0.180149\pi$
$491$	23.7638i	1.07245i	−0.844076	+	0.536223i	$0.819851\pi$
$492$	0	0
$493$	0.0762036i	0.00343204i
$494$	0	0
$495$	−1.08070	−0.0485739
$496$	0	0
$497$	−3.44497	−0.154528
$498$	0	0
$499$	− 30.6430i	− 1.37177i	−0.727710	−	0.685885i	$-0.759415\pi$
$499$	− 30.6430i	− 1.37177i	0.727710	−	0.685885i	$-0.240585\pi$
$500$	0	0
$501$	17.0776i	0.762970i
$502$	0	0
$503$	10.0748	0.449212	0.224606	−	0.974450i	$-0.427890\pi$
$503$	10.0748	0.449212	0.224606	+	0.974450i	$0.427890\pi$
$504$	0	0
$505$	−7.60845	−0.338572
$506$	0	0
$507$	1.00000i	0.0444116i
$508$	0	0
$509$	− 0.833186i	− 0.0369303i	−0.999830	−	0.0184652i	$-0.994122\pi$
$509$	− 0.833186i	− 0.0369303i	0.999830	−	0.0184652i	$-0.00587798\pi$
$510$	0	0
$511$	−1.87169	−0.0827985
$512$	0	0
$513$	−3.67853	−0.162411
$514$	0	0
$515$	− 0.781445i	− 0.0344346i
$516$	0	0
$517$	− 5.80796i	− 0.255434i
$518$	0	0
$519$	−17.1421	−0.752456
$520$	0	0
$521$	44.9453	1.96909	0.984545	−	0.175130i	$-0.0560345\pi$
$521$	44.9453	1.96909	0.984545	+	0.175130i	$0.0560345\pi$
$522$	0	0
$523$	− 29.3048i	− 1.28141i	−0.767787	−	0.640705i	$-0.778642\pi$
$523$	− 29.3048i	− 1.28141i	0.767787	−	0.640705i	$-0.221358\pi$
$524$	0	0
$525$	1.48276i	0.0647128i
$526$	0	0
$527$	1.80653	0.0786937
$528$	0	0
$529$	26.5657	1.15503
$530$	0	0
$531$	4.74559i	0.205941i
$532$	0	0
$533$	4.49890i	0.194869i
$534$	0	0
$535$	23.5886	1.01983
$536$	0	0
$537$	−5.24418	−0.226303
$538$	0	0
$539$	5.72654i	0.246660i
$540$	0	0
$541$	21.7177i	0.933719i	0.884332	+	0.466859i	$0.154615\pi$
$541$	21.7177i	0.933719i	−0.884332	+	0.466859i	$0.845385\pi$
$542$	0	0
$543$	16.4105	0.704241
$544$	0	0
$545$	3.27044	0.140090
$546$	0	0
$547$	37.4660i	1.60193i	0.598711	+	0.800965i	$0.295680\pi$
$547$	37.4660i	1.60193i	−0.598711	+	0.800965i	$0.704320\pi$
$548$	0	0
$549$	− 7.23607i	− 0.308828i
$550$	0	0
$551$	−0.798304	−0.0340089
$552$	0	0
$553$	−0.101831	−0.00433030
$554$	0	0
$555$	− 7.20170i	− 0.305695i
$556$	0	0
$557$	39.0824i	1.65597i	0.560747	+	0.827987i	$0.310514\pi$
$557$	39.0824i	1.65597i	−0.560747	+	0.827987i	$0.689486\pi$
$558$	0	0
$559$	−10.4317	−0.441215
$560$	0	0
$561$	−0.295526	−0.0124771
$562$	0	0
$563$	− 2.72956i	− 0.115037i	−0.998344	−	0.0575186i	$-0.981681\pi$
$563$	− 2.72956i	− 0.115037i	0.998344	−	0.0575186i	$-0.0183189\pi$
$564$	0	0
$565$	− 13.7746i	− 0.579504i
$566$	0	0
$567$	−0.442463	−0.0185817
$568$	0	0
$569$	34.4068	1.44241	0.721203	−	0.692724i	$-0.243589\pi$
$569$	34.4068	1.44241	0.721203	+	0.692724i	$0.243589\pi$
$570$	0	0
$571$	0.642937i	0.0269061i	0.999910	+	0.0134530i	$0.00428236\pi$
$571$	0.642937i	0.0269061i	−0.999910	+	0.0134530i	$0.995718\pi$
$572$	0	0
$573$	14.9537i	0.624699i
$574$	0	0
$575$	23.5930	0.983897
$576$	0	0
$577$	38.0210	1.58284	0.791418	−	0.611276i	$-0.209343\pi$
$577$	38.0210	1.58284	0.791418	+	0.611276i	$0.209343\pi$
$578$	0	0
$579$	20.6868i	0.859716i
$580$	0	0
$581$	3.47806i	0.144294i
$582$	0	0
$583$	2.82536	0.117014
$584$	0	0
$585$	1.28408	0.0530901
$586$	0	0
$587$	25.8546i	1.06713i	0.845758	+	0.533567i	$0.179149\pi$
$587$	25.8546i	1.06713i	−0.845758	+	0.533567i	$0.820851\pi$
$588$	0	0
$589$	18.9251i	0.779795i
$590$	0	0
$591$	8.12048	0.334032
$592$	0	0
$593$	−45.7853	−1.88018	−0.940089	−	0.340928i	$-0.889259\pi$
$593$	−45.7853	−1.88018	−0.940089	+	0.340928i	$0.889259\pi$
$594$	0	0
$595$	− 0.199504i	− 0.00817885i
$596$	0	0
$597$	25.9098i	1.06042i
$598$	0	0
$599$	−20.6319	−0.842996	−0.421498	−	0.906829i	$-0.638496\pi$
$599$	−20.6319	−0.842996	−0.421498	+	0.906829i	$0.638496\pi$
$600$	0	0
$601$	15.6688	0.639144	0.319572	−	0.947562i	$-0.396461\pi$
$601$	15.6688	0.639144	0.319572	+	0.947562i	$0.396461\pi$
$602$	0	0
$603$	− 7.67853i	− 0.312694i
$604$	0	0
$605$	13.2153i	0.537280i
$606$	0	0
$607$	22.1224	0.897921	0.448961	−	0.893552i	$-0.351794\pi$
$607$	22.1224	0.897921	0.448961	+	0.893552i	$0.351794\pi$
$608$	0	0
$609$	−0.0960221	−0.00389101
$610$	0	0
$611$	6.90096i	0.279183i
$612$	0	0
$613$	− 2.50278i	− 0.101086i	−0.998722	−	0.0505431i	$-0.983905\pi$
$613$	− 2.50278i	− 0.101086i	0.998722	−	0.0505431i	$-0.0160952\pi$
$614$	0	0
$615$	5.77695	0.232949
$616$	0	0
$617$	−15.8712	−0.638949	−0.319475	−	0.947595i	$-0.603506\pi$
$617$	−15.8712	−0.638949	−0.319475	+	0.947595i	$0.603506\pi$
$618$	0	0
$619$	2.24088i	0.0900686i	0.998985	+	0.0450343i	$0.0143397\pi$
$619$	2.24088i	0.0900686i	−0.998985	+	0.0450343i	$0.985660\pi$
$620$	0	0
$621$	7.04029i	0.282517i
$622$	0	0
$623$	−2.57419	−0.103133
$624$	0	0
$625$	2.98585	0.119434
$626$	0	0
$627$	− 3.09591i	− 0.123639i
$628$	0	0
$629$	− 1.96936i	− 0.0785234i
$630$	0	0
$631$	35.9870	1.43262	0.716309	−	0.697783i	$-0.245830\pi$
$631$	35.9870	1.43262	0.716309	+	0.697783i	$0.245830\pi$
$632$	0	0
$633$	−19.4589	−0.773422
$634$	0	0
$635$	− 11.4579i	− 0.454692i
$636$	0	0
$637$	− 6.80423i	− 0.269593i
$638$	0	0
$639$	−7.78589	−0.308005
$640$	0	0
$641$	11.5943	0.457947	0.228974	−	0.973433i	$-0.426463\pi$
$641$	11.5943	0.457947	0.228974	+	0.973433i	$0.426463\pi$
$642$	0	0
$643$	− 38.8146i	− 1.53070i	−0.643615	−	0.765350i	$-0.722566\pi$
$643$	− 38.8146i	− 1.53070i	0.643615	−	0.765350i	$-0.277434\pi$
$644$	0	0
$645$	13.3952i	0.527434i
$646$	0	0
$647$	−2.24637	−0.0883141	−0.0441570	−	0.999025i	$-0.514060\pi$
$647$	−2.24637	−0.0883141	−0.0441570	+	0.999025i	$0.514060\pi$
$648$	0	0
$649$	−3.99397	−0.156777
$650$	0	0
$651$	2.27636i	0.0892177i
$652$	0	0
$653$	− 49.2254i	− 1.92634i	−0.268894	−	0.963170i	$-0.586658\pi$
$653$	− 49.2254i	− 1.92634i	0.268894	−	0.963170i	$-0.413342\pi$
$654$	0	0
$655$	9.23980	0.361029
$656$	0	0
$657$	−4.23015	−0.165034
$658$	0	0
$659$	− 13.3783i	− 0.521145i	−0.965454	−	0.260572i	$-0.916089\pi$
$659$	− 13.3783i	− 0.521145i	0.965454	−	0.260572i	$-0.0839113\pi$
$660$	0	0
$661$	− 16.8637i	− 0.655921i	−0.944691	−	0.327961i	$-0.893639\pi$
$661$	− 16.8637i	− 0.655921i	0.944691	−	0.327961i	$-0.106361\pi$
$662$	0	0
$663$	0.351141	0.0136372
$664$	0	0
$665$	2.08999	0.0810462
$666$	0	0
$667$	1.52786i	0.0591591i
$668$	0	0
$669$	− 14.5325i	− 0.561857i
$670$	0	0
$671$	6.08999	0.235101
$672$	0	0
$673$	−13.1707	−0.507693	−0.253846	−	0.967245i	$-0.581696\pi$
$673$	−13.1707	−0.507693	−0.253846	+	0.967245i	$0.581696\pi$
$674$	0	0
$675$	3.35114i	0.128985i
$676$	0	0
$677$	9.70590i	0.373028i	0.982452	+	0.186514i	$0.0597190\pi$
$677$	9.70590i	0.373028i	−0.982452	+	0.186514i	$0.940281\pi$
$678$	0	0
$679$	−4.27765	−0.164161
$680$	0	0
$681$	8.94345	0.342714
$682$	0	0
$683$	17.1371i	0.655734i	0.944724	+	0.327867i	$0.106330\pi$
$683$	17.1371i	0.655734i	−0.944724	+	0.327867i	$0.893670\pi$
$684$	0	0
$685$	10.2655i	0.392226i
$686$	0	0
$687$	21.0343	0.802507
$688$	0	0
$689$	−3.35706	−0.127894
$690$	0	0
$691$	18.1669i	0.691101i	0.938400	+	0.345551i	$0.112308\pi$
$691$	18.1669i	0.691101i	−0.938400	+	0.345551i	$0.887692\pi$
$692$	0	0
$693$	− 0.372384i	− 0.0141457i
$694$	0	0
$695$	11.9144	0.451939
$696$	0	0
$697$	1.57975	0.0598373
$698$	0	0
$699$	9.94646i	0.376210i
$700$	0	0
$701$	23.6870i	0.894644i	0.894373	+	0.447322i	$0.147622\pi$
$701$	23.6870i	0.894644i	−0.894373	+	0.447322i	$0.852378\pi$
$702$	0	0
$703$	20.6309	0.778108
$704$	0	0
$705$	8.86138	0.333739
$706$	0	0
$707$	− 2.62169i	− 0.0985989i
$708$	0	0
$709$	− 22.8135i	− 0.856779i	−0.903594	−	0.428390i	$-0.859081\pi$
$709$	− 22.8135i	− 0.856779i	0.903594	−	0.428390i	$-0.140919\pi$
$710$	0	0
$711$	−0.230146	−0.00863115
$712$	0	0
$713$	36.2205	1.35647
$714$	0	0
$715$	1.08070i	0.0404159i
$716$	0	0
$717$	− 1.83441i	− 0.0685072i
$718$	0	0
$719$	29.6237	1.10478	0.552388	−	0.833587i	$-0.313717\pi$
$719$	29.6237	1.10478	0.552388	+	0.833587i	$0.313717\pi$
$720$	0	0
$721$	0.269268	0.0100281
$722$	0	0
$723$	8.11727i	0.301884i
$724$	0	0
$725$	0.727255i	0.0270096i
$726$	0	0
$727$	−25.1105	−0.931296	−0.465648	−	0.884970i	$-0.654179\pi$
$727$	−25.1105	−0.931296	−0.465648	+	0.884970i	$0.654179\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	3.66301i	0.135481i
$732$	0	0
$733$	− 20.5850i	− 0.760325i	−0.924920	−	0.380162i	$-0.875868\pi$
$733$	− 20.5850i	− 0.760325i	0.924920	−	0.380162i	$-0.124132\pi$
$734$	0	0
$735$	−8.73716	−0.322275
$736$	0	0
$737$	6.46237	0.238045
$738$	0	0
$739$	− 34.0773i	− 1.25355i	−0.779199	−	0.626777i	$-0.784374\pi$
$739$	− 34.0773i	− 1.25355i	0.779199	−	0.626777i	$-0.215626\pi$
$740$	0	0
$741$	3.67853i	0.135134i
$742$	0	0
$743$	−18.6165	−0.682973	−0.341486	−	0.939887i	$-0.610930\pi$
$743$	−18.6165	−0.682973	−0.341486	+	0.939887i	$0.610930\pi$
$744$	0	0
$745$	22.2353	0.814638
$746$	0	0
$747$	7.86067i	0.287607i
$748$	0	0
$749$	8.12809i	0.296994i
$750$	0	0
$751$	−2.57046	−0.0937975	−0.0468988	−	0.998900i	$-0.514934\pi$
$751$	−2.57046	−0.0937975	−0.0468988	+	0.998900i	$0.514934\pi$
$752$	0	0
$753$	31.3723	1.14327
$754$	0	0
$755$	0.713106i	0.0259526i
$756$	0	0
$757$	− 30.9632i	− 1.12538i	−0.826669	−	0.562689i	$-0.809767\pi$
$757$	− 30.9632i	− 1.12538i	0.826669	−	0.562689i	$-0.190233\pi$
$758$	0	0
$759$	−5.92522	−0.215072
$760$	0	0
$761$	20.0113	0.725410	0.362705	−	0.931904i	$-0.381853\pi$
$761$	20.0113	0.725410	0.362705	+	0.931904i	$0.381853\pi$
$762$	0	0
$763$	1.12692i	0.0407971i
$764$	0	0
$765$	− 0.450893i	− 0.0163021i
$766$	0	0
$767$	4.74559	0.171353
$768$	0	0
$769$	−9.24418	−0.333354	−0.166677	−	0.986012i	$-0.553304\pi$
$769$	−9.24418	−0.333354	−0.166677	+	0.986012i	$0.553304\pi$
$770$	0	0
$771$	20.8789i	0.751935i
$772$	0	0
$773$	18.2225i	0.655419i	0.944778	+	0.327710i	$0.106277\pi$
$773$	18.2225i	0.655419i	−0.944778	+	0.327710i	$0.893723\pi$
$774$	0	0
$775$	17.2408	0.619307
$776$	0	0
$777$	2.48154	0.0890246
$778$	0	0
$779$	16.5494i	0.592942i
$780$	0	0
$781$	− 6.55272i	− 0.234475i
$782$	0	0
$783$	−0.217017	−0.00775556
$784$	0	0
$785$	−29.3831	−1.04873
$786$	0	0
$787$	− 14.3323i	− 0.510891i	−0.966823	−	0.255446i	$-0.917778\pi$
$787$	− 14.3323i	− 0.510891i	0.966823	−	0.255446i	$-0.0822221\pi$
$788$	0	0
$789$	− 4.83139i	− 0.172002i
$790$	0	0
$791$	4.74642	0.168763
$792$	0	0
$793$	−7.23607	−0.256960
$794$	0	0
$795$	4.31073i	0.152886i
$796$	0	0
$797$	22.7044i	0.804230i	0.915589	+	0.402115i	$0.131725\pi$
$797$	22.7044i	0.804230i	−0.915589	+	0.402115i	$0.868275\pi$
$798$	0	0
$799$	2.42321	0.0857270
$800$	0	0
$801$	−5.81787	−0.205564
$802$	0	0
$803$	− 3.56016i	− 0.125635i
$804$	0	0
$805$	− 4.00000i	− 0.140981i
$806$	0	0
$807$	11.7022	0.411936
$808$	0	0
$809$	3.11134	0.109389	0.0546945	−	0.998503i	$-0.482582\pi$
$809$	3.11134	0.109389	0.0546945	+	0.998503i	$0.482582\pi$
$810$	0	0
$811$	− 38.9979i	− 1.36940i	−0.728824	−	0.684701i	$-0.759933\pi$
$811$	− 38.9979i	− 1.36940i	0.728824	−	0.684701i	$-0.240067\pi$
$812$	0	0
$813$	− 9.27385i	− 0.325248i
$814$	0	0
$815$	−25.7911	−0.903423
$816$	0	0
$817$	−38.3734	−1.34252
$818$	0	0
$819$	0.442463i	0.0154609i
$820$	0	0
$821$	− 18.3837i	− 0.641596i	−0.947148	−	0.320798i	$-0.896049\pi$
$821$	− 18.3837i	− 0.641596i	0.947148	−	0.320798i	$-0.103951\pi$
$822$	0	0
$823$	23.1232	0.806024	0.403012	−	0.915195i	$-0.367963\pi$
$823$	23.1232	0.806024	0.403012	+	0.915195i	$0.367963\pi$
$824$	0	0
$825$	−2.82037	−0.0981928
$826$	0	0
$827$	40.9499i	1.42397i	0.702196	+	0.711984i	$0.252203\pi$
$827$	40.9499i	1.42397i	−0.702196	+	0.711984i	$0.747797\pi$
$828$	0	0
$829$	− 21.8956i	− 0.760467i	−0.924891	−	0.380233i	$-0.875844\pi$
$829$	− 21.8956i	− 0.760467i	0.924891	−	0.380233i	$-0.124156\pi$
$830$	0	0
$831$	−0.676024	−0.0234510
$832$	0	0
$833$	−2.38924	−0.0827824
$834$	0	0
$835$	− 21.9290i	− 0.758883i
$836$	0	0
$837$	5.14475i	0.177828i
$838$	0	0
$839$	−16.8477	−0.581645	−0.290823	−	0.956777i	$-0.593929\pi$
$839$	−16.8477	−0.581645	−0.290823	+	0.956777i	$0.593929\pi$
$840$	0	0
$841$	28.9529	0.998376
$842$	0	0
$843$	1.07891i	0.0371595i
$844$	0	0
$845$	− 1.28408i	− 0.0441737i
$846$	0	0
$847$	−4.55369	−0.156467
$848$	0	0
$849$	−6.72353	−0.230751
$850$	0	0
$851$	− 39.4852i	− 1.35353i
$852$	0	0
$853$	30.5527i	1.04611i	0.852300	+	0.523053i	$0.175207\pi$
$853$	30.5527i	1.04611i	−0.852300	+	0.523053i	$0.824793\pi$
$854$	0	0
$855$	4.72353	0.161541
$856$	0	0
$857$	−13.5549	−0.463027	−0.231514	−	0.972832i	$-0.574368\pi$
$857$	−13.5549	−0.463027	−0.231514	+	0.972832i	$0.574368\pi$
$858$	0	0
$859$	− 0.192044i	− 0.00655247i	−0.999995	−	0.00327623i	$-0.998957\pi$
$859$	− 0.192044i	− 0.00655247i	0.999995	−	0.00327623i	$-0.00104286\pi$
$860$	0	0
$861$	1.99060i	0.0678395i
$862$	0	0
$863$	2.78227	0.0947096	0.0473548	−	0.998878i	$-0.484921\pi$
$863$	2.78227	0.0947096	0.0473548	+	0.998878i	$0.484921\pi$
$864$	0	0
$865$	22.0118	0.748425
$866$	0	0
$867$	16.8767i	0.573163i
$868$	0	0
$869$	− 0.193695i	− 0.00657064i
$870$	0	0
$871$	−7.67853	−0.260177
$872$	0	0
$873$	−9.66780	−0.327205
$874$	0	0
$875$	− 4.74477i	− 0.160402i
$876$	0	0
$877$	− 12.8805i	− 0.434945i	−0.976066	−	0.217473i	$-0.930219\pi$
$877$	− 12.8805i	− 0.434945i	0.976066	−	0.217473i	$-0.0697812\pi$
$878$	0	0
$879$	−32.9152	−1.11020
$880$	0	0
$881$	23.9300	0.806223	0.403111	−	0.915151i	$-0.367929\pi$
$881$	23.9300	0.806223	0.403111	+	0.915151i	$0.367929\pi$
$882$	0	0
$883$	39.6941i	1.33581i	0.744246	+	0.667906i	$0.232809\pi$
$883$	39.6941i	1.33581i	−0.744246	+	0.667906i	$0.767191\pi$
$884$	0	0
$885$	− 6.09372i	− 0.204838i
$886$	0	0
$887$	34.5433	1.15985	0.579926	−	0.814669i	$-0.303082\pi$
$887$	34.5433	1.15985	0.579926	+	0.814669i	$0.303082\pi$
$888$	0	0
$889$	3.94811	0.132416
$890$	0	0
$891$	− 0.841616i	− 0.0281952i
$892$	0	0
$893$	25.3854i	0.849490i
$894$	0	0
$895$	6.73394	0.225091
$896$	0	0
$897$	7.04029	0.235069
$898$	0	0
$899$	1.11650i	0.0372373i
$900$	0	0
$901$	1.17880i	0.0392716i
$902$	0	0
$903$	−4.61566	−0.153600
$904$	0	0
$905$	−21.0724	−0.700469
$906$	0	0
$907$	− 13.0082i	− 0.431931i	−0.976401	−	0.215966i	$-0.930710\pi$
$907$	− 13.0082i	− 0.431931i	0.976401	−	0.215966i	$-0.0692899\pi$
$908$	0	0
$909$	− 5.92522i	− 0.196527i
$910$	0	0
$911$	−13.4420	−0.445355	−0.222677	−	0.974892i	$-0.571480\pi$
$911$	−13.4420	−0.445355	−0.222677	+	0.974892i	$0.571480\pi$
$912$	0	0
$913$	−6.61566	−0.218946
$914$	0	0
$915$	9.29168i	0.307174i
$916$	0	0
$917$	3.18382i	0.105139i
$918$	0	0
$919$	49.5539	1.63463	0.817316	−	0.576190i	$-0.195461\pi$
$919$	49.5539	1.63463	0.817316	+	0.576190i	$0.195461\pi$
$920$	0	0
$921$	10.8105	0.356217
$922$	0	0
$923$	7.78589i	0.256276i
$924$	0	0
$925$	− 18.7947i	− 0.617966i
$926$	0	0
$927$	0.608565	0.0199879
$928$	0	0
$929$	−30.8996	−1.01378	−0.506892	−	0.862010i	$-0.669206\pi$
$929$	−30.8996	−1.01378	−0.506892	+	0.862010i	$0.669206\pi$
$930$	0	0
$931$	− 25.0296i	− 0.820311i
$932$	0	0
$933$	− 23.7426i	− 0.777297i
$934$	0	0
$935$	0.379478	0.0124103
$936$	0	0
$937$	−38.4580	−1.25637	−0.628184	−	0.778065i	$-0.716201\pi$
$937$	−38.4580	−1.25637	−0.628184	+	0.778065i	$0.716201\pi$
$938$	0	0
$939$	12.4398i	0.405959i
$940$	0	0
$941$	− 8.97232i	− 0.292489i	−0.989248	−	0.146245i	$-0.953281\pi$
$941$	− 8.97232i	− 0.292489i	0.989248	−	0.146245i	$-0.0467187\pi$
$942$	0	0
$943$	31.6736	1.03143
$944$	0	0
$945$	0.568158	0.0184822
$946$	0	0
$947$	− 48.9032i	− 1.58914i	−0.607173	−	0.794570i	$-0.707696\pi$
$947$	− 48.9032i	− 1.58914i	0.607173	−	0.794570i	$-0.292304\pi$
$948$	0	0
$949$	4.23015i	0.137316i
$950$	0	0
$951$	−28.4643	−0.923018
$952$	0	0
$953$	−36.2690	−1.17487	−0.587435	−	0.809272i	$-0.699862\pi$
$953$	−36.2690	−1.17487	−0.587435	+	0.809272i	$0.699862\pi$
$954$	0	0
$955$	− 19.2017i	− 0.621352i
$956$	0	0
$957$	− 0.182645i	− 0.00590407i
$958$	0	0
$959$	−3.53726	−0.114224
$960$	0	0
$961$	−4.53159	−0.146180
$962$	0	0
$963$	18.3701i	0.591968i
$964$	0	0
$965$	− 26.5635i	− 0.855111i
$966$	0	0
$967$	−1.19828	−0.0385341	−0.0192671	−	0.999814i	$-0.506133\pi$
$967$	−1.19828	−0.0385341	−0.0192671	+	0.999814i	$0.506133\pi$
$968$	0	0
$969$	1.29168	0.0414948
$970$	0	0
$971$	43.8767i	1.40807i	0.710165	+	0.704035i	$0.248620\pi$
$971$	43.8767i	1.40807i	−0.710165	+	0.704035i	$0.751380\pi$
$972$	0	0
$973$	4.10542i	0.131614i
$974$	0	0
$975$	3.35114	0.107322
$976$	0	0
$977$	−22.5003	−0.719849	−0.359925	−	0.932981i	$-0.617198\pi$
$977$	−22.5003	−0.719849	−0.359925	+	0.932981i	$0.617198\pi$
$978$	0	0
$979$	− 4.89641i	− 0.156490i
$980$	0	0
$981$	2.54691i	0.0813167i
$982$	0	0
$983$	51.6457	1.64724	0.823621	−	0.567140i	$-0.191950\pi$
$983$	51.6457	1.64724	0.823621	+	0.567140i	$0.191950\pi$
$984$	0	0
$985$	−10.4273	−0.332243
$986$	0	0
$987$	3.05342i	0.0971915i
$988$	0	0
$989$	73.4424i	2.33533i
$990$	0	0
$991$	−49.2348	−1.56400	−0.781998	−	0.623281i	$-0.785799\pi$
$991$	−49.2348	−1.56400	−0.781998	+	0.623281i	$0.785799\pi$
$992$	0	0
$993$	−12.1021	−0.384050
$994$	0	0
$995$	− 33.2702i	− 1.05474i
$996$	0	0
$997$	− 33.9073i	− 1.07386i	−0.843628	−	0.536928i	$-0.819585\pi$
$997$	− 33.9073i	− 1.07386i	0.843628	−	0.536928i	$-0.180415\pi$
$998$	0	0
$999$	5.60845	0.177444

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1248.2.g.a.625.3		8
3.2	odd	2		3744.2.g.d.1873.3		8
4.3	odd	2		312.2.g.a.157.7	✓	8
8.3	odd	2		312.2.g.a.157.8	yes	8
8.5	even	2	inner	1248.2.g.a.625.6		8
12.11	even	2		936.2.g.d.469.2		8
16.3	odd	4		9984.2.a.y.1.3		4
16.5	even	4		9984.2.a.s.1.2		4
16.11	odd	4		9984.2.a.bb.1.2		4
16.13	even	4		9984.2.a.bh.1.3		4
24.5	odd	2		3744.2.g.d.1873.6		8
24.11	even	2		936.2.g.d.469.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.g.a.157.7	✓	8	4.3	odd	2
312.2.g.a.157.8	yes	8	8.3	odd	2
936.2.g.d.469.1		8	24.11	even	2
936.2.g.d.469.2		8	12.11	even	2
1248.2.g.a.625.3		8	1.1	even	1	trivial
1248.2.g.a.625.6		8	8.5	even	2	inner
3744.2.g.d.1873.3		8	3.2	odd	2
3744.2.g.d.1873.6		8	24.5	odd	2
9984.2.a.s.1.2		4	16.5	even	4
9984.2.a.y.1.3		4	16.3	odd	4
9984.2.a.bb.1.2		4	16.11	odd	4
9984.2.a.bh.1.3		4	16.13	even	4

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:	\( N \)	\(=\)	\( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1248.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +1.28408i q^{5} -0.442463 q^{7} -1.00000 q^{9} -0.841616i q^{11} +1.00000i q^{13} +1.28408 q^{15} +0.351141 q^{17} +3.67853i q^{19} +0.442463i q^{21} +7.04029 q^{23} +3.35114 q^{25} +1.00000i q^{27} +0.217017i q^{29} +5.14475 q^{31} -0.841616 q^{33} -0.568158i q^{35} -5.60845i q^{37} +1.00000 q^{39} +4.49890 q^{41} +10.4317i q^{43} -1.28408i q^{45} +6.90096 q^{47} -6.80423 q^{49} -0.351141i q^{51} +3.35706i q^{53} +1.08070 q^{55} +3.67853 q^{57} -4.74559i q^{59} +7.23607i q^{61} +0.442463 q^{63} -1.28408 q^{65} +7.67853i q^{67} -7.04029i q^{69} +7.78589 q^{71} +4.23015 q^{73} -3.35114i q^{75} +0.372384i q^{77} +0.230146 q^{79} +1.00000 q^{81} -7.86067i q^{83} +0.450893i q^{85} +0.217017 q^{87} +5.81787 q^{89} -0.442463i q^{91} -5.14475i q^{93} -4.72353 q^{95} +9.66780 q^{97} +0.841616i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{7} - 8 q^{9} + 4 q^{15} - 16 q^{17} + 8 q^{23} + 8 q^{25} + 4 q^{31} + 8 q^{39} + 36 q^{41} - 24 q^{47} - 24 q^{49} + 40 q^{55} + 12 q^{57} + 4 q^{63} - 4 q^{65} - 16 q^{71} + 32 q^{73} + 8 q^{81}+ \cdots - 56 q^{97}+O(q^{100}) \) 8 * q - 4 * q^7 - 8 * q^9 + 4 * q^15 - 16 * q^17 + 8 * q^23 + 8 * q^25 + 4 * q^31 + 8 * q^39 + 36 * q^41 - 24 * q^47 - 24 * q^49 + 40 * q^55 + 12 * q^57 + 4 * q^63 - 4 * q^65 - 16 * q^71 + 32 * q^73 + 8 * q^81 + 8 * q^87 + 60 * q^89 + 24 * q^95 - 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

Properties

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