LMFDB - Embedded newform 832.2.b.c.417.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [832,2,Mod(417,832)] 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("832.417"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(832, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 832 = 2^{6} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	832.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,-12,0,0,0,0,0,-36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	no
Analytic conductor:	$6.64355344817$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.195105024.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 5x^{6} + 4x^{5} - 20x^{4} + 12x^{3} + 45x^{2} - 108x + 81 $ x^8 - 4x^7 + 5x^6 + 4x^5 - 20x^4 + 12x^3 + 45x^2 - 108*x + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			417.5
Root			$1.72124 - 0.193255i$ of defining polynomial
Character	$\chi$	$=$	832.417
Dual form			832.2.b.c.417.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.386509i q^{3} +0.386509i q^{5} -4.17452 q^{7} +2.85061 q^{9} -3.78801i q^{11} -1.00000i q^{13} -0.149389 q^{15} -4.49843 q^{17} -5.25211i q^{19} -1.61349i q^{21} +6.11192 q^{23} +4.85061 q^{25} +2.26131i q^{27} -8.88494i q^{29} -4.44911 q^{31} +1.46410 q^{33} -1.61349i q^{35} +3.96254i q^{37} +0.386509 q^{39} +8.11192 q^{41} -12.7356i q^{43} +1.10179i q^{45} -7.40150 q^{47} +10.4266 q^{49} -1.73869i q^{51} -9.04013i q^{53} +1.46410 q^{55} +2.02999 q^{57} +4.56103i q^{59} +1.33891i q^{61} -11.8999 q^{63} +0.386509 q^{65} +6.67296i q^{67} +2.36231i q^{69} +4.59850 q^{71} -11.6880 q^{73} +1.87481i q^{75} +15.8131i q^{77} -2.57916 q^{79} +7.67781 q^{81} -2.67296i q^{83} -1.73869i q^{85} +3.43411 q^{87} -10.3490 q^{89} +4.17452i q^{91} -1.71962i q^{93} +2.02999 q^{95} +0.237120 q^{97} -10.7982i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{9} - 36 q^{15} - 4 q^{17} + 24 q^{23} + 4 q^{25} - 20 q^{31} - 16 q^{33} - 4 q^{39} + 40 q^{41} - 40 q^{47} - 4 q^{49} - 16 q^{55} - 8 q^{57} - 44 q^{63} - 4 q^{65} + 56 q^{71} - 16 q^{73} - 32 q^{79}+ \cdots - 40 q^{97}+O(q^{100}) $ 8 * q - 12 * q^9 - 36 * q^15 - 4 * q^17 + 24 * q^23 + 4 * q^25 - 20 * q^31 - 16 * q^33 - 4 * q^39 + 40 * q^41 - 40 * q^47 - 4 * q^49 - 16 * q^55 - 8 * q^57 - 44 * q^63 - 4 * q^65 + 56 * q^71 - 16 * q^73 - 32 * q^79 + 40 * q^81 + 24 * q^87 - 16 * q^89 - 8 * q^95 - 40 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/832\mathbb{Z}\right)^\times$.

$n$	$261$	$703$	$769$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	0.386509i	0.223151i	0.993756	+	0.111576i	$0.0355897\pi$
$3$	0.386509i	0.223151i	−0.993756	+	0.111576i	$0.964410\pi$
$4$	0	0
$5$	0.386509i	0.172852i	0.996258	+	0.0864261i	$0.0275446\pi$
$5$	0.386509i	0.172852i	−0.996258	+	0.0864261i	$0.972455\pi$
$6$	0	0
$7$	−4.17452	−1.57782	−0.788911	−	0.614508i	$-0.789355\pi$
$7$	−4.17452	−1.57782	−0.788911	+	0.614508i	$0.789355\pi$
$8$	0	0
$9$	2.85061	0.950204
$10$	0	0
$11$	− 3.78801i	− 1.14213i	−0.820905	−	0.571064i	$-0.806531\pi$
$11$	− 3.78801i	− 1.14213i	0.820905	−	0.571064i	$-0.193469\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	−0.149389	−0.0385721
$16$	0	0
$17$	−4.49843	−1.09103	−0.545515	−	0.838101i	$-0.683666\pi$
$17$	−4.49843	−1.09103	−0.545515	+	0.838101i	$0.683666\pi$
$18$	0	0
$19$	− 5.25211i	− 1.20492i	−0.798150	−	0.602459i	$-0.794188\pi$
$19$	− 5.25211i	− 1.20492i	0.798150	−	0.602459i	$-0.205812\pi$
$20$	0	0
$21$	− 1.61349i	− 0.352093i
$22$	0	0
$23$	6.11192	1.27442	0.637212	−	0.770688i	$-0.280087\pi$
$23$	6.11192	1.27442	0.637212	+	0.770688i	$0.280087\pi$
$24$	0	0
$25$	4.85061	0.970122
$26$	0	0
$27$	2.26131i	0.435190i
$28$	0	0
$29$	− 8.88494i	− 1.64989i	−0.565211	−	0.824946i	$-0.691205\pi$
$29$	− 8.88494i	− 1.64989i	0.565211	−	0.824946i	$-0.308795\pi$
$30$	0	0
$31$	−4.44911	−0.799083	−0.399542	−	0.916715i	$-0.630831\pi$
$31$	−4.44911	−0.799083	−0.399542	+	0.916715i	$0.630831\pi$
$32$	0	0
$33$	1.46410	0.254867
$34$	0	0
$35$	− 1.61349i	− 0.272730i
$36$	0	0
$37$	3.96254i	0.651437i	0.945467	+	0.325718i	$0.105606\pi$
$37$	3.96254i	0.651437i	−0.945467	+	0.325718i	$0.894394\pi$
$38$	0	0
$39$	0.386509	0.0618910
$40$	0	0
$41$	8.11192	1.26687	0.633435	−	0.773796i	$-0.281644\pi$
$41$	8.11192	1.26687	0.633435	+	0.773796i	$0.281644\pi$
$42$	0	0
$43$	− 12.7356i	− 1.94215i	−0.238767	−	0.971077i	$-0.576743\pi$
$43$	− 12.7356i	− 1.94215i	0.238767	−	0.971077i	$-0.423257\pi$
$44$	0	0
$45$	1.10179i	0.164245i
$46$	0	0
$47$	−7.40150	−1.07962	−0.539810	−	0.841787i	$-0.681504\pi$
$47$	−7.40150	−1.07962	−0.539810	+	0.841787i	$0.681504\pi$
$48$	0	0
$49$	10.4266	1.48952
$50$	0	0
$51$	− 1.73869i	− 0.243465i
$52$	0	0
$53$	− 9.04013i	− 1.24176i	−0.783907	−	0.620879i	$-0.786776\pi$
$53$	− 9.04013i	− 1.24176i	0.783907	−	0.620879i	$-0.213224\pi$
$54$	0	0
$55$	1.46410	0.197419
$56$	0	0
$57$	2.02999	0.268879
$58$	0	0
$59$	4.56103i	0.593796i	0.954909	+	0.296898i	$0.0959521\pi$
$59$	4.56103i	0.593796i	−0.954909	+	0.296898i	$0.904048\pi$
$60$	0	0
$61$	1.33891i	0.171429i	0.996320	+	0.0857147i	$0.0273174\pi$
$61$	1.33891i	0.171429i	−0.996320	+	0.0857147i	$0.972683\pi$
$62$	0	0
$63$	−11.8999	−1.49925
$64$	0	0
$65$	0.386509	0.0479406
$66$	0	0
$67$	6.67296i	0.815231i	0.913154	+	0.407616i	$0.133640\pi$
$67$	6.67296i	0.815231i	−0.913154	+	0.407616i	$0.866360\pi$
$68$	0	0
$69$	2.36231i	0.284389i
$70$	0	0
$71$	4.59850	0.545741	0.272871	−	0.962051i	$-0.412027\pi$
$71$	4.59850	0.545741	0.272871	+	0.962051i	$0.412027\pi$
$72$	0	0
$73$	−11.6880	−1.36797	−0.683986	−	0.729495i	$-0.739755\pi$
$73$	−11.6880	−1.36797	−0.683986	+	0.729495i	$0.739755\pi$
$74$	0	0
$75$	1.87481i	0.216484i
$76$	0	0
$77$	15.8131i	1.80208i
$78$	0	0
$79$	−2.57916	−0.290178	−0.145089	−	0.989419i	$-0.546347\pi$
$79$	−2.57916	−0.290178	−0.145089	+	0.989419i	$0.546347\pi$
$80$	0	0
$81$	7.67781	0.853090
$82$	0	0
$83$	− 2.67296i	− 0.293395i	−0.989181	−	0.146698i	$-0.953136\pi$
$83$	− 2.67296i	− 0.293395i	0.989181	−	0.146698i	$-0.0468644\pi$
$84$	0	0
$85$	− 1.73869i	− 0.188587i
$86$	0	0
$87$	3.43411	0.368175
$88$	0	0
$89$	−10.3490	−1.09700	−0.548498	−	0.836152i	$-0.684800\pi$
$89$	−10.3490	−1.09700	−0.548498	+	0.836152i	$0.684800\pi$
$90$	0	0
$91$	4.17452i	0.437609i
$92$	0	0
$93$	− 1.71962i	− 0.178316i
$94$	0	0
$95$	2.02999	0.208273
$96$	0	0
$97$	0.237120	0.0240759	0.0120379	−	0.999928i	$-0.496168\pi$
$97$	0.237120	0.0240759	0.0120379	+	0.999928i	$0.496168\pi$
$98$	0	0
$99$	− 10.7982i	− 1.08526i
$100$	0	0
$101$	− 7.15205i	− 0.711656i	−0.934551	−	0.355828i	$-0.884199\pi$
$101$	− 7.15205i	− 0.711656i	0.934551	−	0.355828i	$-0.115801\pi$
$102$	0	0
$103$	10.1552	1.00062	0.500310	−	0.865846i	$-0.333219\pi$
$103$	10.1552	1.00062	0.500310	+	0.865846i	$0.333219\pi$
$104$	0	0
$105$	0.623629	0.0608600
$106$	0	0
$107$	2.12519i	0.205450i	0.994710	+	0.102725i	$0.0327562\pi$
$107$	2.12519i	0.205450i	−0.994710	+	0.102725i	$0.967244\pi$
$108$	0	0
$109$	− 19.1146i	− 1.83085i	−0.402494	−	0.915423i	$-0.631856\pi$
$109$	− 19.1146i	− 1.83085i	0.402494	−	0.915423i	$-0.368144\pi$
$110$	0	0
$111$	−1.53156	−0.145369
$112$	0	0
$113$	−9.73121	−0.915435	−0.457718	−	0.889098i	$-0.651333\pi$
$113$	−9.73121	−0.915435	−0.457718	+	0.889098i	$0.651333\pi$
$114$	0	0
$115$	2.36231i	0.220287i
$116$	0	0
$117$	− 2.85061i	− 0.263539i
$118$	0	0
$119$	18.7788	1.72145
$120$	0	0
$121$	−3.34904	−0.304459
$122$	0	0
$123$	3.13533i	0.282703i
$124$	0	0
$125$	3.80735i	0.340540i
$126$	0	0
$127$	−3.43411	−0.304728	−0.152364	−	0.988324i	$-0.548689\pi$
$127$	−3.43411	−0.304728	−0.152364	+	0.988324i	$0.548689\pi$
$128$	0	0
$129$	4.92241	0.433394
$130$	0	0
$131$	13.4699i	1.17687i	0.808544	+	0.588435i	$0.200256\pi$
$131$	13.4699i	1.17687i	−0.808544	+	0.588435i	$0.799744\pi$
$132$	0	0
$133$	21.9251i	1.90114i
$134$	0	0
$135$	−0.874019	−0.0752235
$136$	0	0
$137$	−0.497224	−0.0424807	−0.0212403	−	0.999774i	$-0.506762\pi$
$137$	−0.497224	−0.0424807	−0.0212403	+	0.999774i	$0.506762\pi$
$138$	0	0
$139$	1.45831i	0.123692i	0.998086	+	0.0618459i	$0.0196987\pi$
$139$	1.45831i	0.123692i	−0.998086	+	0.0618459i	$0.980301\pi$
$140$	0	0
$141$	− 2.86075i	− 0.240919i
$142$	0	0
$143$	−3.78801	−0.316770
$144$	0	0
$145$	3.43411	0.285187
$146$	0	0
$147$	4.02999i	0.332388i
$148$	0	0
$149$	9.15205i	0.749765i	0.927072	+	0.374883i	$0.122317\pi$
$149$	9.15205i	0.749765i	−0.927072	+	0.374883i	$0.877683\pi$
$150$	0	0
$151$	−2.32971	−0.189589	−0.0947945	−	0.995497i	$-0.530219\pi$
$151$	−2.32971	−0.189589	−0.0947945	+	0.995497i	$0.530219\pi$
$152$	0	0
$153$	−12.8233	−1.03670
$154$	0	0
$155$	− 1.71962i	− 0.138123i
$156$	0	0
$157$	3.20400i	0.255707i	0.991793	+	0.127853i	$0.0408087\pi$
$157$	3.20400i	0.255707i	−0.991793	+	0.127853i	$0.959191\pi$
$158$	0	0
$159$	3.49409	0.277100
$160$	0	0
$161$	−25.5144	−2.01081
$162$	0	0
$163$	0.635960i	0.0498122i	0.999690	+	0.0249061i	$0.00792868\pi$
$163$	0.635960i	0.0498122i	−0.999690	+	0.0249061i	$0.992071\pi$
$164$	0	0
$165$	0.565889i	0.0440544i
$166$	0	0
$167$	3.32704	0.257454	0.128727	−	0.991680i	$-0.458911\pi$
$167$	3.32704	0.257454	0.128727	+	0.991680i	$0.458911\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	− 14.9717i	− 1.14492i
$172$	0	0
$173$	25.2473i	1.91951i	0.280833	+	0.959757i	$0.409389\pi$
$173$	25.2473i	1.91951i	−0.280833	+	0.959757i	$0.590611\pi$
$174$	0	0
$175$	−20.2490	−1.53068
$176$	0	0
$177$	−1.76288	−0.132506
$178$	0	0
$179$	14.1177i	1.05521i	0.849490	+	0.527604i	$0.176909\pi$
$179$	14.1177i	1.05521i	−0.849490	+	0.527604i	$0.823091\pi$
$180$	0	0
$181$	− 4.84168i	− 0.359879i	−0.983678	−	0.179940i	$-0.942410\pi$
$181$	− 4.84168i	− 0.359879i	0.983678	−	0.179940i	$-0.0575903\pi$
$182$	0	0
$183$	−0.517500	−0.0382547
$184$	0	0
$185$	−1.53156	−0.112602
$186$	0	0
$187$	17.0401i	1.24610i
$188$	0	0
$189$	− 9.43991i	− 0.686652i
$190$	0	0
$191$	−6.75975	−0.489118	−0.244559	−	0.969634i	$-0.578643\pi$
$191$	−6.75975	−0.489118	−0.244559	+	0.969634i	$0.578643\pi$
$192$	0	0
$193$	8.66810	0.623943	0.311972	−	0.950091i	$-0.399011\pi$
$193$	8.66810	0.623943	0.311972	+	0.950091i	$0.399011\pi$
$194$	0	0
$195$	0.149389i	0.0106980i
$196$	0	0
$197$	24.5717i	1.75066i	0.483526	+	0.875330i	$0.339356\pi$
$197$	24.5717i	1.75066i	−0.483526	+	0.875330i	$0.660644\pi$
$198$	0	0
$199$	22.8533	1.62003	0.810013	−	0.586412i	$-0.199460\pi$
$199$	22.8533	1.62003	0.810013	+	0.586412i	$0.199460\pi$
$200$	0	0
$201$	−2.57916	−0.181920
$202$	0	0
$203$	37.0904i	2.60324i
$204$	0	0
$205$	3.13533i	0.218981i
$206$	0	0
$207$	17.4227	1.21096
$208$	0	0
$209$	−19.8951	−1.37617
$210$	0	0
$211$	− 8.38651i	− 0.577351i	−0.957427	−	0.288676i	$-0.906785\pi$
$211$	− 8.38651i	− 0.577351i	0.957427	−	0.288676i	$-0.0932149\pi$
$212$	0	0
$213$	1.77736i	0.121783i
$214$	0	0
$215$	4.92241	0.335705
$216$	0	0
$217$	18.5729	1.26081
$218$	0	0
$219$	− 4.51750i	− 0.305264i
$220$	0	0
$221$	4.49843i	0.302597i
$222$	0	0
$223$	13.6819	0.916207	0.458103	−	0.888899i	$-0.348529\pi$
$223$	13.6819	0.916207	0.458103	+	0.888899i	$0.348529\pi$
$224$	0	0
$225$	13.8272	0.921814
$226$	0	0
$227$	15.4892i	1.02806i	0.857773	+	0.514028i	$0.171847\pi$
$227$	15.4892i	1.02806i	−0.857773	+	0.514028i	$0.828153\pi$
$228$	0	0
$229$	− 24.0128i	− 1.58681i	−0.608694	−	0.793405i	$-0.708306\pi$
$229$	− 24.0128i	− 1.58681i	0.608694	−	0.793405i	$-0.291694\pi$
$230$	0	0
$231$	−6.11192	−0.402135
$232$	0	0
$233$	−13.0027	−0.851833	−0.425916	−	0.904763i	$-0.640048\pi$
$233$	−13.0027	−0.851833	−0.425916	+	0.904763i	$0.640048\pi$
$234$	0	0
$235$	− 2.86075i	− 0.186615i
$236$	0	0
$237$	− 0.996868i	− 0.0647535i
$238$	0	0
$239$	−13.0341	−0.843103	−0.421552	−	0.906804i	$-0.638514\pi$
$239$	−13.0341	−0.843103	−0.421552	+	0.906804i	$0.638514\pi$
$240$	0	0
$241$	−24.3693	−1.56977	−0.784883	−	0.619644i	$-0.787277\pi$
$241$	−24.3693	−1.56977	−0.784883	+	0.619644i	$0.787277\pi$
$242$	0	0
$243$	9.75149i	0.625558i
$244$	0	0
$245$	4.02999i	0.257467i
$246$	0	0
$247$	−5.25211	−0.334184
$248$	0	0
$249$	1.03312	0.0654714
$250$	0	0
$251$	4.37904i	0.276402i	0.990404	+	0.138201i	$0.0441320\pi$
$251$	4.37904i	0.276402i	−0.990404	+	0.138201i	$0.955868\pi$
$252$	0	0
$253$	− 23.1521i	− 1.45556i
$254$	0	0
$255$	0.672018	0.0420834
$256$	0	0
$257$	−12.0058	−0.748901	−0.374450	−	0.927247i	$-0.622169\pi$
$257$	−12.0058	−0.748901	−0.374450	+	0.927247i	$0.622169\pi$
$258$	0	0
$259$	− 16.5417i	− 1.02785i
$260$	0	0
$261$	− 25.3275i	− 1.56773i
$262$	0	0
$263$	19.3089	1.19064	0.595319	−	0.803489i	$-0.297026\pi$
$263$	19.3089	1.19064	0.595319	+	0.803489i	$0.297026\pi$
$264$	0	0
$265$	3.49409	0.214640
$266$	0	0
$267$	− 4.00000i	− 0.244796i
$268$	0	0
$269$	16.5862i	1.01128i	0.862746	+	0.505638i	$0.168743\pi$
$269$	16.5862i	1.01128i	−0.862746	+	0.505638i	$0.831257\pi$
$270$	0	0
$271$	32.6125	1.98107	0.990534	−	0.137265i	$-0.0438311\pi$
$271$	32.6125	1.98107	0.990534	+	0.137265i	$0.0438311\pi$
$272$	0	0
$273$	−1.61349	−0.0976529
$274$	0	0
$275$	− 18.3742i	− 1.10800i
$276$	0	0
$277$	− 9.72663i	− 0.584416i	−0.956355	−	0.292208i	$-0.905610\pi$
$277$	− 9.72663i	− 0.584416i	0.956355	−	0.292208i	$-0.0943900\pi$
$278$	0	0
$279$	−12.6827	−0.759292
$280$	0	0
$281$	18.7967	1.12132	0.560660	−	0.828046i	$-0.310548\pi$
$281$	18.7967	1.12132	0.560660	+	0.828046i	$0.310548\pi$
$282$	0	0
$283$	− 19.2207i	− 1.14255i	−0.820758	−	0.571277i	$-0.806448\pi$
$283$	− 19.2207i	− 1.14255i	0.820758	−	0.571277i	$-0.193552\pi$
$284$	0	0
$285$	0.784610i	0.0464763i
$286$	0	0
$287$	−33.8634	−1.99889
$288$	0	0
$289$	3.23591	0.190348
$290$	0	0
$291$	0.0916490i	0.00537256i
$292$	0	0
$293$	− 11.6638i	− 0.681404i	−0.940171	−	0.340702i	$-0.889335\pi$
$293$	− 11.6638i	− 0.681404i	0.940171	−	0.340702i	$-0.110665\pi$
$294$	0	0
$295$	−1.76288	−0.102639
$296$	0	0
$297$	8.56589	0.497043
$298$	0	0
$299$	− 6.11192i	− 0.353462i
$300$	0	0
$301$	53.1649i	3.06437i
$302$	0	0
$303$	2.76433	0.158807
$304$	0	0
$305$	−0.517500	−0.0296319
$306$	0	0
$307$	− 9.29537i	− 0.530515i	−0.964178	−	0.265258i	$-0.914543\pi$
$307$	− 9.29537i	− 0.530515i	0.964178	−	0.265258i	$-0.0854570\pi$
$308$	0	0
$309$	3.92507i	0.223290i
$310$	0	0
$311$	14.7643	0.837209	0.418604	−	0.908169i	$-0.362519\pi$
$311$	14.7643	0.837209	0.418604	+	0.908169i	$0.362519\pi$
$312$	0	0
$313$	−13.8806	−0.784578	−0.392289	−	0.919842i	$-0.628317\pi$
$313$	−13.8806	−0.784578	−0.392289	+	0.919842i	$0.628317\pi$
$314$	0	0
$315$	− 4.59943i	− 0.259149i
$316$	0	0
$317$	− 30.3606i	− 1.70522i	−0.522545	−	0.852612i	$-0.675017\pi$
$317$	− 30.3606i	− 1.70522i	0.522545	−	0.852612i	$-0.324983\pi$
$318$	0	0
$319$	−33.6563	−1.88439
$320$	0	0
$321$	−0.821407	−0.0458465
$322$	0	0
$323$	23.6263i	1.31460i
$324$	0	0
$325$	− 4.85061i	− 0.269063i
$326$	0	0
$327$	7.38796	0.408555
$328$	0	0
$329$	30.8977	1.70345
$330$	0	0
$331$	9.35077i	0.513965i	0.966416	+	0.256982i	$0.0827282\pi$
$331$	9.35077i	0.513965i	−0.966416	+	0.256982i	$0.917272\pi$
$332$	0	0
$333$	11.2956i	0.618998i
$334$	0	0
$335$	−2.57916	−0.140914
$336$	0	0
$337$	17.8806	0.974018	0.487009	−	0.873397i	$-0.338088\pi$
$337$	17.8806	0.974018	0.487009	+	0.873397i	$0.338088\pi$
$338$	0	0
$339$	− 3.76120i	− 0.204280i
$340$	0	0
$341$	16.8533i	0.912656i
$342$	0	0
$343$	−14.3046	−0.772374
$344$	0	0
$345$	−0.913056	−0.0491573
$346$	0	0
$347$	2.84047i	0.152485i	0.997089	+	0.0762423i	$0.0242922\pi$
$347$	2.84047i	0.152485i	−0.997089	+	0.0762423i	$0.975708\pi$
$348$	0	0
$349$	− 19.3147i	− 1.03389i	−0.856018	−	0.516946i	$-0.827069\pi$
$349$	− 19.3147i	− 1.03389i	0.856018	−	0.516946i	$-0.172931\pi$
$350$	0	0
$351$	2.26131	0.120700
$352$	0	0
$353$	8.61783	0.458681	0.229340	−	0.973346i	$-0.426343\pi$
$353$	8.61783	0.458681	0.229340	+	0.973346i	$0.426343\pi$
$354$	0	0
$355$	1.77736i	0.0943325i
$356$	0	0
$357$	7.25818i	0.384144i
$358$	0	0
$359$	0.672957	0.0355173	0.0177586	−	0.999842i	$-0.494347\pi$
$359$	0.672957	0.0355173	0.0177586	+	0.999842i	$0.494347\pi$
$360$	0	0
$361$	−8.58471	−0.451827
$362$	0	0
$363$	− 1.29444i	− 0.0679403i
$364$	0	0
$365$	− 4.51750i	− 0.236457i
$366$	0	0
$367$	−14.1189	−0.737002	−0.368501	−	0.929627i	$-0.620129\pi$
$367$	−14.1189	−0.737002	−0.368501	+	0.929627i	$0.620129\pi$
$368$	0	0
$369$	23.1239	1.20378
$370$	0	0
$371$	37.7382i	1.95927i
$372$	0	0
$373$	− 5.22698i	− 0.270643i	−0.990802	−	0.135321i	$-0.956793\pi$
$373$	− 5.22698i	− 0.270643i	0.990802	−	0.135321i	$-0.0432067\pi$
$374$	0	0
$375$	−1.47158	−0.0759918
$376$	0	0
$377$	−8.88494	−0.457598
$378$	0	0
$379$	− 34.5426i	− 1.77434i	−0.461447	−	0.887168i	$-0.652669\pi$
$379$	− 34.5426i	− 1.77434i	0.461447	−	0.887168i	$-0.347331\pi$
$380$	0	0
$381$	− 1.32732i	− 0.0680004i
$382$	0	0
$383$	22.1296	1.13077	0.565384	−	0.824828i	$-0.308728\pi$
$383$	22.1296	1.13077	0.565384	+	0.824828i	$0.308728\pi$
$384$	0	0
$385$	−6.11192	−0.311493
$386$	0	0
$387$	− 36.3041i	− 1.84544i
$388$	0	0
$389$	34.4426i	1.74631i	0.487445	+	0.873154i	$0.337929\pi$
$389$	34.4426i	1.74631i	−0.487445	+	0.873154i	$0.662071\pi$
$390$	0	0
$391$	−27.4941	−1.39044
$392$	0	0
$393$	−5.20624	−0.262620
$394$	0	0
$395$	− 0.996868i	− 0.0501579i
$396$	0	0
$397$	− 16.5042i	− 0.828324i	−0.910203	−	0.414162i	$-0.864075\pi$
$397$	− 16.5042i	− 0.828324i	0.910203	−	0.414162i	$-0.135925\pi$
$398$	0	0
$399$	−8.47424	−0.424243
$400$	0	0
$401$	19.5530	0.976432	0.488216	−	0.872723i	$-0.337648\pi$
$401$	19.5530	0.976432	0.488216	+	0.872723i	$0.337648\pi$
$402$	0	0
$403$	4.44911i	0.221626i
$404$	0	0
$405$	2.96754i	0.147458i
$406$	0	0
$407$	15.0101	0.744025
$408$	0	0
$409$	3.36545	0.166411	0.0832053	−	0.996532i	$-0.473484\pi$
$409$	3.36545	0.166411	0.0832053	+	0.996532i	$0.473484\pi$
$410$	0	0
$411$	− 0.192181i	− 0.00947961i
$412$	0	0
$413$	− 19.0401i	− 0.936903i
$414$	0	0
$415$	1.03312	0.0507140
$416$	0	0
$417$	−0.563648	−0.0276020
$418$	0	0
$419$	− 28.0877i	− 1.37218i	−0.727519	−	0.686088i	$-0.759327\pi$
$419$	− 28.0877i	− 1.37218i	0.727519	−	0.686088i	$-0.240673\pi$
$420$	0	0
$421$	− 2.31784i	− 0.112965i	−0.998404	−	0.0564824i	$-0.982012\pi$
$421$	− 2.31784i	− 0.112965i	0.998404	−	0.0564824i	$-0.0179885\pi$
$422$	0	0
$423$	−21.0988	−1.02586
$424$	0	0
$425$	−21.8202	−1.05843
$426$	0	0
$427$	− 5.58930i	− 0.270485i
$428$	0	0
$429$	− 1.46410i	− 0.0706875i
$430$	0	0
$431$	28.9591	1.39491	0.697456	−	0.716627i	$-0.254315\pi$
$431$	28.9591	1.39491	0.697456	+	0.716627i	$0.254315\pi$
$432$	0	0
$433$	27.1578	1.30512	0.652561	−	0.757736i	$-0.273694\pi$
$433$	27.1578	1.30512	0.652561	+	0.757736i	$0.273694\pi$
$434$	0	0
$435$	1.32732i	0.0636399i
$436$	0	0
$437$	− 32.1005i	− 1.53558i
$438$	0	0
$439$	−19.3592	−0.923963	−0.461982	−	0.886889i	$-0.652861\pi$
$439$	−19.3592	−0.923963	−0.461982	+	0.886889i	$0.652861\pi$
$440$	0	0
$441$	29.7223	1.41535
$442$	0	0
$443$	− 6.21638i	− 0.295349i	−0.989036	−	0.147674i	$-0.952821\pi$
$443$	− 6.21638i	− 0.295349i	0.989036	−	0.147674i	$-0.0471788\pi$
$444$	0	0
$445$	− 4.00000i	− 0.189618i
$446$	0	0
$447$	−3.53735	−0.167311
$448$	0	0
$449$	16.2971	0.769108	0.384554	−	0.923103i	$-0.374355\pi$
$449$	16.2971	0.769108	0.384554	+	0.923103i	$0.374355\pi$
$450$	0	0
$451$	− 30.7281i	− 1.44693i
$452$	0	0
$453$	− 0.900453i	− 0.0423070i
$454$	0	0
$455$	−1.61349	−0.0756416
$456$	0	0
$457$	24.1552	1.12993	0.564966	−	0.825114i	$-0.308889\pi$
$457$	24.1552	1.12993	0.564966	+	0.825114i	$0.308889\pi$
$458$	0	0
$459$	− 10.1724i	− 0.474806i
$460$	0	0
$461$	− 16.4102i	− 0.764301i	−0.924100	−	0.382150i	$-0.875184\pi$
$461$	− 16.4102i	− 0.764301i	0.924100	−	0.382150i	$-0.124816\pi$
$462$	0	0
$463$	−18.7029	−0.869200	−0.434600	−	0.900624i	$-0.643110\pi$
$463$	−18.7029	−0.869200	−0.434600	+	0.900624i	$0.643110\pi$
$464$	0	0
$465$	0.664649	0.0308224
$466$	0	0
$467$	− 41.2410i	− 1.90841i	−0.299160	−	0.954203i	$-0.596707\pi$
$467$	− 41.2410i	− 1.90841i	0.299160	−	0.954203i	$-0.403293\pi$
$468$	0	0
$469$	− 27.8564i	− 1.28629i
$470$	0	0
$471$	−1.23837	−0.0570613
$472$	0	0
$473$	−48.2424	−2.21819
$474$	0	0
$475$	− 25.4760i	− 1.16892i
$476$	0	0
$477$	− 25.7699i	− 1.17992i
$478$	0	0
$479$	−28.7474	−1.31350	−0.656752	−	0.754107i	$-0.728070\pi$
$479$	−28.7474	−1.31350	−0.656752	+	0.754107i	$0.728070\pi$
$480$	0	0
$481$	3.96254	0.180676
$482$	0	0
$483$	− 9.86154i	− 0.448715i
$484$	0	0
$485$	0.0916490i	0.00416157i
$486$	0	0
$487$	12.1625	0.551134	0.275567	−	0.961282i	$-0.411134\pi$
$487$	12.1625	0.551134	0.275567	+	0.961282i	$0.411134\pi$
$488$	0	0
$489$	−0.245804	−0.0111157
$490$	0	0
$491$	32.4102i	1.46265i	0.682027	+	0.731327i	$0.261099\pi$
$491$	32.4102i	1.46265i	−0.682027	+	0.731327i	$0.738901\pi$
$492$	0	0
$493$	39.9683i	1.80008i
$494$	0	0
$495$	4.17358	0.187589
$496$	0	0
$497$	−19.1965	−0.861082
$498$	0	0
$499$	21.1726i	0.947816i	0.880574	+	0.473908i	$0.157157\pi$
$499$	21.1726i	0.947816i	−0.880574	+	0.473908i	$0.842843\pi$
$500$	0	0
$501$	1.28593i	0.0574512i
$502$	0	0
$503$	15.5760	0.694501	0.347250	−	0.937772i	$-0.387115\pi$
$503$	15.5760	0.694501	0.347250	+	0.937772i	$0.387115\pi$
$504$	0	0
$505$	2.76433	0.123011
$506$	0	0
$507$	− 0.386509i	− 0.0171655i
$508$	0	0
$509$	− 4.35773i	− 0.193153i	−0.995326	−	0.0965765i	$-0.969211\pi$
$509$	− 4.35773i	− 0.193153i	0.995326	−	0.0965765i	$-0.0307892\pi$
$510$	0	0
$511$	48.7916	2.15841
$512$	0	0
$513$	11.8767	0.524368
$514$	0	0
$515$	3.92507i	0.172959i
$516$	0	0
$517$	28.0370i	1.23307i
$518$	0	0
$519$	−9.75829	−0.428342
$520$	0	0
$521$	−17.6568	−0.773556	−0.386778	−	0.922173i	$-0.626412\pi$
$521$	−17.6568	−0.773556	−0.386778	+	0.922173i	$0.626412\pi$
$522$	0	0
$523$	10.3490i	0.452532i	0.974066	+	0.226266i	$0.0726519\pi$
$523$	10.3490i	0.452532i	−0.974066	+	0.226266i	$0.927348\pi$
$524$	0	0
$525$	− 7.82642i	− 0.341573i
$526$	0	0
$527$	20.0140	0.871824
$528$	0	0
$529$	14.3556	0.624158
$530$	0	0
$531$	13.0017i	0.564227i
$532$	0	0
$533$	− 8.11192i	− 0.351366i
$534$	0	0
$535$	−0.821407	−0.0355125
$536$	0	0
$537$	−5.45663	−0.235471
$538$	0	0
$539$	− 39.4962i	− 1.70122i
$540$	0	0
$541$	− 20.0375i	− 0.861478i	−0.902477	−	0.430739i	$-0.858253\pi$
$541$	− 20.0375i	− 0.861478i	0.902477	−	0.430739i	$-0.141747\pi$
$542$	0	0
$543$	1.87135	0.0803075
$544$	0	0
$545$	7.38796	0.316466
$546$	0	0
$547$	29.1446i	1.24613i	0.782169	+	0.623066i	$0.214113\pi$
$547$	29.1446i	1.24613i	−0.782169	+	0.623066i	$0.785887\pi$
$548$	0	0
$549$	3.81670i	0.162893i
$550$	0	0
$551$	−46.6647	−1.98798
$552$	0	0
$553$	10.7668	0.457849
$554$	0	0
$555$	− 0.591960i	− 0.0251273i
$556$	0	0
$557$	23.1411i	0.980521i	0.871576	+	0.490261i	$0.163098\pi$
$557$	23.1411i	0.980521i	−0.871576	+	0.490261i	$0.836902\pi$
$558$	0	0
$559$	−12.7356	−0.538657
$560$	0	0
$561$	−6.58616	−0.278068
$562$	0	0
$563$	1.19578i	0.0503962i	0.999682	+	0.0251981i	$0.00802165\pi$
$563$	1.19578i	0.0503962i	−0.999682	+	0.0251981i	$0.991978\pi$
$564$	0	0
$565$	− 3.76120i	− 0.158235i
$566$	0	0
$567$	−32.0512	−1.34602
$568$	0	0
$569$	34.6087	1.45087	0.725436	−	0.688290i	$-0.241638\pi$
$569$	34.6087	1.45087	0.725436	+	0.688290i	$0.241638\pi$
$570$	0	0
$571$	27.6372i	1.15658i	0.815831	+	0.578291i	$0.196280\pi$
$571$	27.6372i	1.15658i	−0.815831	+	0.578291i	$0.803720\pi$
$572$	0	0
$573$	− 2.61270i	− 0.109147i
$574$	0	0
$575$	29.6466	1.23635
$576$	0	0
$577$	−2.65241	−0.110421	−0.0552106	−	0.998475i	$-0.517583\pi$
$577$	−2.65241	−0.110421	−0.0552106	+	0.998475i	$0.517583\pi$
$578$	0	0
$579$	3.35030i	0.139234i
$580$	0	0
$581$	11.1583i	0.462925i
$582$	0	0
$583$	−34.2441	−1.41825
$584$	0	0
$585$	1.10179	0.0455533
$586$	0	0
$587$	18.7302i	0.773079i	0.922273	+	0.386540i	$0.126330\pi$
$587$	18.7302i	0.773079i	−0.922273	+	0.386540i	$0.873670\pi$
$588$	0	0
$589$	23.3672i	0.962829i
$590$	0	0
$591$	−9.49718	−0.390662
$592$	0	0
$593$	−4.74761	−0.194961	−0.0974806	−	0.995237i	$-0.531078\pi$
$593$	−4.74761	−0.194961	−0.0974806	+	0.995237i	$0.531078\pi$
$594$	0	0
$595$	7.25818i	0.297556i
$596$	0	0
$597$	8.83300i	0.361511i
$598$	0	0
$599$	−19.0217	−0.777207	−0.388603	−	0.921405i	$-0.627042\pi$
$599$	−19.0217	−0.777207	−0.388603	+	0.921405i	$0.627042\pi$
$600$	0	0
$601$	34.0585	1.38927	0.694637	−	0.719360i	$-0.255565\pi$
$601$	34.0585	1.38927	0.694637	+	0.719360i	$0.255565\pi$
$602$	0	0
$603$	19.0220i	0.774636i
$604$	0	0
$605$	− 1.29444i	− 0.0526263i
$606$	0	0
$607$	40.2908	1.63535	0.817677	−	0.575677i	$-0.195261\pi$
$607$	40.2908	1.63535	0.817677	+	0.575677i	$0.195261\pi$
$608$	0	0
$609$	−14.3358	−0.580915
$610$	0	0
$611$	7.40150i	0.299433i
$612$	0	0
$613$	1.01472i	0.0409843i	0.999790	+	0.0204921i	$0.00652331\pi$
$613$	1.01472i	0.0409843i	−0.999790	+	0.0204921i	$0.993477\pi$
$614$	0	0
$615$	−1.21183	−0.0488659
$616$	0	0
$617$	−8.74303	−0.351981	−0.175991	−	0.984392i	$-0.556313\pi$
$617$	−8.74303	−0.351981	−0.175991	+	0.984392i	$0.556313\pi$
$618$	0	0
$619$	14.1433i	0.568468i	0.958755	+	0.284234i	$0.0917393\pi$
$619$	14.1433i	0.568468i	−0.958755	+	0.284234i	$0.908261\pi$
$620$	0	0
$621$	13.8210i	0.554617i
$622$	0	0
$623$	43.2023	1.73086
$624$	0	0
$625$	22.7815	0.911259
$626$	0	0
$627$	− 7.68963i	− 0.307094i
$628$	0	0
$629$	− 17.8252i	− 0.710737i
$630$	0	0
$631$	−22.2697	−0.886544	−0.443272	−	0.896387i	$-0.646182\pi$
$631$	−22.2697	−0.886544	−0.443272	+	0.896387i	$0.646182\pi$
$632$	0	0
$633$	3.24146	0.128837
$634$	0	0
$635$	− 1.32732i	− 0.0526729i
$636$	0	0
$637$	− 10.4266i	− 0.413118i
$638$	0	0
$639$	13.1085	0.518565
$640$	0	0
$641$	−41.8317	−1.65225	−0.826127	−	0.563484i	$-0.809461\pi$
$641$	−41.8317	−1.65225	−0.826127	+	0.563484i	$0.809461\pi$
$642$	0	0
$643$	10.7692i	0.424695i	0.977194	+	0.212348i	$0.0681109\pi$
$643$	10.7692i	0.424695i	−0.977194	+	0.212348i	$0.931889\pi$
$644$	0	0
$645$	1.90256i	0.0749130i
$646$	0	0
$647$	28.6548	1.12654	0.563269	−	0.826274i	$-0.309544\pi$
$647$	28.6548	1.12654	0.563269	+	0.826274i	$0.309544\pi$
$648$	0	0
$649$	17.2772	0.678191
$650$	0	0
$651$	7.17859i	0.281351i
$652$	0	0
$653$	11.9048i	0.465871i	0.972492	+	0.232935i	$0.0748330\pi$
$653$	11.9048i	0.465871i	−0.972492	+	0.232935i	$0.925167\pi$
$654$	0	0
$655$	−5.20624	−0.203425
$656$	0	0
$657$	−33.3178	−1.29985
$658$	0	0
$659$	− 33.6915i	− 1.31243i	−0.754572	−	0.656217i	$-0.772155\pi$
$659$	− 33.6915i	− 1.31243i	0.754572	−	0.656217i	$-0.227845\pi$
$660$	0	0
$661$	− 11.5963i	− 0.451044i	−0.974238	−	0.225522i	$-0.927591\pi$
$661$	− 11.5963i	− 0.451044i	0.974238	−	0.225522i	$-0.0724087\pi$
$662$	0	0
$663$	−1.73869	−0.0675250
$664$	0	0
$665$	−8.47424	−0.328617
$666$	0	0
$667$	− 54.3041i	− 2.10266i
$668$	0	0
$669$	5.28817i	0.204453i
$670$	0	0
$671$	5.07180	0.195795
$672$	0	0
$673$	−18.4414	−0.710862	−0.355431	−	0.934702i	$-0.615666\pi$
$673$	−18.4414	−0.710862	−0.355431	+	0.934702i	$0.615666\pi$
$674$	0	0
$675$	10.9688i	0.422188i
$676$	0	0
$677$	9.47111i	0.364004i	0.983298	+	0.182002i	$0.0582578\pi$
$677$	9.47111i	0.364004i	−0.983298	+	0.182002i	$0.941742\pi$
$678$	0	0
$679$	−0.989862	−0.0379874
$680$	0	0
$681$	−5.98673	−0.229412
$682$	0	0
$683$	5.61087i	0.214694i	0.994222	+	0.107347i	$0.0342356\pi$
$683$	5.61087i	0.214694i	−0.994222	+	0.107347i	$0.965764\pi$
$684$	0	0
$685$	− 0.192181i	− 0.00734288i
$686$	0	0
$687$	9.28117	0.354099
$688$	0	0
$689$	−9.04013	−0.344401
$690$	0	0
$691$	3.96618i	0.150881i	0.997150	+	0.0754403i	$0.0240362\pi$
$691$	3.96618i	0.150881i	−0.997150	+	0.0754403i	$0.975964\pi$
$692$	0	0
$693$	45.0771i	1.71234i
$694$	0	0
$695$	−0.563648	−0.0213804
$696$	0	0
$697$	−36.4910	−1.38219
$698$	0	0
$699$	− 5.02565i	− 0.190087i
$700$	0	0
$701$	15.5893i	0.588800i	0.955682	+	0.294400i	$0.0951197\pi$
$701$	15.5893i	0.588800i	−0.955682	+	0.294400i	$0.904880\pi$
$702$	0	0
$703$	20.8117	0.784928
$704$	0	0
$705$	1.10571	0.0416433
$706$	0	0
$707$	29.8564i	1.12287i
$708$	0	0
$709$	− 1.82642i	− 0.0685925i	−0.999412	−	0.0342963i	$-0.989081\pi$
$709$	− 1.82642i	− 0.0685925i	0.999412	−	0.0342963i	$-0.0109190\pi$
$710$	0	0
$711$	−7.35218	−0.275728
$712$	0	0
$713$	−27.1926	−1.01837
$714$	0	0
$715$	− 1.46410i	− 0.0547543i
$716$	0	0
$717$	− 5.03778i	− 0.188139i
$718$	0	0
$719$	−43.9991	−1.64089	−0.820444	−	0.571727i	$-0.806274\pi$
$719$	−43.9991	−1.64089	−0.820444	+	0.571727i	$0.806274\pi$
$720$	0	0
$721$	−42.3930	−1.57880
$722$	0	0
$723$	− 9.41896i	− 0.350295i
$724$	0	0
$725$	− 43.0974i	− 1.60060i
$726$	0	0
$727$	33.4878	1.24199	0.620997	−	0.783813i	$-0.286728\pi$
$727$	33.4878	1.24199	0.620997	+	0.783813i	$0.286728\pi$
$728$	0	0
$729$	19.2644	0.713496
$730$	0	0
$731$	57.2901i	2.11895i
$732$	0	0
$733$	− 10.4817i	− 0.387151i	−0.981085	−	0.193575i	$-0.937992\pi$
$733$	− 10.4817i	− 0.387151i	0.981085	−	0.193575i	$-0.0620084\pi$
$734$	0	0
$735$	−1.55763	−0.0574540
$736$	0	0
$737$	25.2772	0.931099
$738$	0	0
$739$	− 36.8854i	− 1.35685i	−0.734669	−	0.678426i	$-0.762662\pi$
$739$	− 36.8854i	− 1.35685i	0.734669	−	0.678426i	$-0.237338\pi$
$740$	0	0
$741$	− 2.02999i	− 0.0745736i
$742$	0	0
$743$	30.3800	1.11453	0.557267	−	0.830334i	$-0.311850\pi$
$743$	30.3800	1.11453	0.557267	+	0.830334i	$0.311850\pi$
$744$	0	0
$745$	−3.53735	−0.129599
$746$	0	0
$747$	− 7.61956i	− 0.278785i
$748$	0	0
$749$	− 8.87167i	− 0.324164i
$750$	0	0
$751$	10.0230	0.365744	0.182872	−	0.983137i	$-0.441461\pi$
$751$	10.0230	0.365744	0.182872	+	0.983137i	$0.441461\pi$
$752$	0	0
$753$	−1.69254	−0.0616795
$754$	0	0
$755$	− 0.900453i	− 0.0327708i
$756$	0	0
$757$	− 15.8218i	− 0.575054i	−0.957772	−	0.287527i	$-0.907167\pi$
$757$	− 15.8218i	− 0.575054i	0.957772	−	0.287527i	$-0.0928331\pi$
$758$	0	0
$759$	8.94848	0.324809
$760$	0	0
$761$	−18.2787	−0.662602	−0.331301	−	0.943525i	$-0.607488\pi$
$761$	−18.2787	−0.662602	−0.331301	+	0.943525i	$0.607488\pi$
$762$	0	0
$763$	79.7943i	2.88875i
$764$	0	0
$765$	− 4.95632i	− 0.179196i
$766$	0	0
$767$	4.56103	0.164689
$768$	0	0
$769$	45.0674	1.62517	0.812586	−	0.582841i	$-0.198059\pi$
$769$	45.0674	1.62517	0.812586	+	0.582841i	$0.198059\pi$
$770$	0	0
$771$	− 4.64035i	− 0.167118i
$772$	0	0
$773$	24.8873i	0.895134i	0.894251	+	0.447567i	$0.147709\pi$
$773$	24.8873i	0.895134i	−0.894251	+	0.447567i	$0.852291\pi$
$774$	0	0
$775$	−21.5809	−0.775208
$776$	0	0
$777$	6.39352	0.229366
$778$	0	0
$779$	− 42.6048i	− 1.52647i
$780$	0	0
$781$	− 17.4192i	− 0.623307i
$782$	0	0
$783$	20.0916	0.718017
$784$	0	0
$785$	−1.23837	−0.0441995
$786$	0	0
$787$	41.2857i	1.47167i	0.677158	+	0.735837i	$0.263211\pi$
$787$	41.2857i	1.47167i	−0.677158	+	0.735837i	$0.736789\pi$
$788$	0	0
$789$	7.46307i	0.265692i
$790$	0	0
$791$	40.6232	1.44439
$792$	0	0
$793$	1.33891	0.0475460
$794$	0	0
$795$	1.35050i	0.0478972i
$796$	0	0
$797$	7.69496i	0.272569i	0.990670	+	0.136285i	$0.0435162\pi$
$797$	7.69496i	0.272569i	−0.990670	+	0.136285i	$0.956484\pi$
$798$	0	0
$799$	33.2952	1.17790
$800$	0	0
$801$	−29.5011	−1.04237
$802$	0	0
$803$	44.2741i	1.56240i
$804$	0	0
$805$	− 9.86154i	− 0.347573i
$806$	0	0
$807$	−6.41070	−0.225667
$808$	0	0
$809$	21.7457	0.764538	0.382269	−	0.924051i	$-0.375143\pi$
$809$	21.7457	0.764538	0.382269	+	0.924051i	$0.375143\pi$
$810$	0	0
$811$	− 37.7501i	− 1.32558i	−0.748803	−	0.662792i	$-0.769371\pi$
$811$	− 37.7501i	− 1.32558i	0.748803	−	0.662792i	$-0.230629\pi$
$812$	0	0
$813$	12.6050i	0.442078i
$814$	0	0
$815$	−0.245804	−0.00861015
$816$	0	0
$817$	−66.8886	−2.34014
$818$	0	0
$819$	11.8999i	0.415817i
$820$	0	0
$821$	41.7827i	1.45823i	0.684393	+	0.729113i	$0.260067\pi$
$821$	41.7827i	1.45823i	−0.684393	+	0.729113i	$0.739933\pi$
$822$	0	0
$823$	−26.8990	−0.937639	−0.468819	−	0.883294i	$-0.655320\pi$
$823$	−26.8990	−0.937639	−0.468819	+	0.883294i	$0.655320\pi$
$824$	0	0
$825$	7.10179	0.247252
$826$	0	0
$827$	28.2939i	0.983876i	0.870630	+	0.491938i	$0.163711\pi$
$827$	28.2939i	0.983876i	−0.870630	+	0.491938i	$0.836289\pi$
$828$	0	0
$829$	11.9251i	0.414175i	0.978322	+	0.207087i	$0.0663985\pi$
$829$	11.9251i	0.414175i	−0.978322	+	0.207087i	$0.933602\pi$
$830$	0	0
$831$	3.75943	0.130413
$832$	0	0
$833$	−46.9035	−1.62511
$834$	0	0
$835$	1.28593i	0.0445015i
$836$	0	0
$837$	− 10.0608i	− 0.347753i
$838$	0	0
$839$	−12.8465	−0.443512	−0.221756	−	0.975102i	$-0.571179\pi$
$839$	−12.8465	−0.443512	−0.221756	+	0.975102i	$0.571179\pi$
$840$	0	0
$841$	−49.9422	−1.72215
$842$	0	0
$843$	7.26511i	0.250224i
$844$	0	0
$845$	− 0.386509i	− 0.0132963i
$846$	0	0
$847$	13.9807	0.480381
$848$	0	0
$849$	7.42898	0.254962
$850$	0	0
$851$	24.2187i	0.830207i
$852$	0	0
$853$	37.8311i	1.29531i	0.761933	+	0.647656i	$0.224250\pi$
$853$	37.8311i	1.29531i	−0.761933	+	0.647656i	$0.775750\pi$
$854$	0	0
$855$	5.78671	0.197901
$856$	0	0
$857$	−16.3041	−0.556938	−0.278469	−	0.960445i	$-0.589827\pi$
$857$	−16.3041	−0.556938	−0.278469	+	0.960445i	$0.589827\pi$
$858$	0	0
$859$	45.3512i	1.54736i	0.633574	+	0.773682i	$0.281587\pi$
$859$	45.3512i	1.54736i	−0.633574	+	0.773682i	$0.718413\pi$
$860$	0	0
$861$	− 13.0885i	− 0.446055i
$862$	0	0
$863$	−49.2816	−1.67757	−0.838783	−	0.544466i	$-0.816733\pi$
$863$	−49.2816	−1.67757	−0.838783	+	0.544466i	$0.816733\pi$
$864$	0	0
$865$	−9.75829	−0.331792
$866$	0	0
$867$	1.25071i	0.0424763i
$868$	0	0
$869$	9.76989i	0.331421i
$870$	0	0
$871$	6.67296	0.226105
$872$	0	0
$873$	0.675936	0.0228770
$874$	0	0
$875$	− 15.8939i	− 0.537311i
$876$	0	0
$877$	39.2687i	1.32601i	0.748614	+	0.663006i	$0.230719\pi$
$877$	39.2687i	1.32601i	−0.748614	+	0.663006i	$0.769281\pi$
$878$	0	0
$879$	4.50815	0.152056
$880$	0	0
$881$	46.5627	1.56874	0.784369	−	0.620295i	$-0.212987\pi$
$881$	46.5627	1.56874	0.784369	+	0.620295i	$0.212987\pi$
$882$	0	0
$883$	− 5.74737i	− 0.193414i	−0.995313	−	0.0967072i	$-0.969169\pi$
$883$	− 5.74737i	− 0.193414i	0.995313	−	0.0967072i	$-0.0308310\pi$
$884$	0	0
$885$	− 0.681369i	− 0.0229040i
$886$	0	0
$887$	−30.0007	−1.00733	−0.503663	−	0.863900i	$-0.668015\pi$
$887$	−30.0007	−1.00733	−0.503663	+	0.863900i	$0.668015\pi$
$888$	0	0
$889$	14.3358	0.480806
$890$	0	0
$891$	− 29.0837i	− 0.974339i
$892$	0	0
$893$	38.8736i	1.30085i
$894$	0	0
$895$	−5.45663	−0.182395
$896$	0	0
$897$	2.36231	0.0788754
$898$	0	0
$899$	39.5301i	1.31840i
$900$	0	0
$901$	40.6664i	1.35479i
$902$	0	0
$903$	−20.5487	−0.683818
$904$	0	0
$905$	1.87135	0.0622059
$906$	0	0
$907$	1.37124i	0.0455313i	0.999741	+	0.0227657i	$0.00724717\pi$
$907$	1.37124i	0.0455313i	−0.999741	+	0.0227657i	$0.992753\pi$
$908$	0	0
$909$	− 20.3877i	− 0.676218i
$910$	0	0
$911$	−18.5959	−0.616109	−0.308054	−	0.951369i	$-0.599678\pi$
$911$	−18.5959	−0.616109	−0.308054	+	0.951369i	$0.599678\pi$
$912$	0	0
$913$	−10.1252	−0.335095
$914$	0	0
$915$	− 0.200018i	− 0.00661240i
$916$	0	0
$917$	− 56.2304i	− 1.85689i
$918$	0	0
$919$	−51.3927	−1.69529	−0.847645	−	0.530564i	$-0.821980\pi$
$919$	−51.3927	−1.69529	−0.847645	+	0.530564i	$0.821980\pi$
$920$	0	0
$921$	3.59275	0.118385
$922$	0	0
$923$	− 4.59850i	− 0.151361i
$924$	0	0
$925$	19.2207i	0.631973i
$926$	0	0
$927$	28.9485	0.950793
$928$	0	0
$929$	18.4090	0.603981	0.301990	−	0.953311i	$-0.402349\pi$
$929$	18.4090	0.603981	0.301990	+	0.953311i	$0.402349\pi$
$930$	0	0
$931$	− 54.7619i	− 1.79475i
$932$	0	0
$933$	5.70655i	0.186824i
$934$	0	0
$935$	−6.58616	−0.215391
$936$	0	0
$937$	14.4143	0.470893	0.235447	−	0.971887i	$-0.424345\pi$
$937$	14.4143	0.470893	0.235447	+	0.971887i	$0.424345\pi$
$938$	0	0
$939$	− 5.36498i	− 0.175079i
$940$	0	0
$941$	45.4458i	1.48149i	0.671786	+	0.740745i	$0.265527\pi$
$941$	45.4458i	1.48149i	−0.671786	+	0.740745i	$0.734473\pi$
$942$	0	0
$943$	49.5795	1.61453
$944$	0	0
$945$	3.64861	0.118689
$946$	0	0
$947$	− 53.9688i	− 1.75375i	−0.480718	−	0.876875i	$-0.659624\pi$
$947$	− 53.9688i	− 1.75375i	0.480718	−	0.876875i	$-0.340376\pi$
$948$	0	0
$949$	11.6880i	0.379407i
$950$	0	0
$951$	11.7347	0.380522
$952$	0	0
$953$	−18.3732	−0.595168	−0.297584	−	0.954696i	$-0.596181\pi$
$953$	−18.3732	−0.595168	−0.297584	+	0.954696i	$0.596181\pi$
$954$	0	0
$955$	− 2.61270i	− 0.0845451i
$956$	0	0
$957$	− 13.0085i	− 0.420504i
$958$	0	0
$959$	2.07567	0.0670269
$960$	0	0
$961$	−11.2055	−0.361466
$962$	0	0
$963$	6.05810i	0.195220i
$964$	0	0
$965$	3.35030i	0.107850i
$966$	0	0
$967$	21.2966	0.684852	0.342426	−	0.939545i	$-0.388751\pi$
$967$	21.2966	0.684852	0.342426	+	0.939545i	$0.388751\pi$
$968$	0	0
$969$	−9.13178	−0.293355
$970$	0	0
$971$	− 9.81949i	− 0.315122i	−0.987509	−	0.157561i	$-0.949637\pi$
$971$	− 9.81949i	− 0.315122i	0.987509	−	0.157561i	$-0.0503631\pi$
$972$	0	0
$973$	− 6.08773i	− 0.195164i
$974$	0	0
$975$	1.87481	0.0600418
$976$	0	0
$977$	55.6703	1.78105	0.890525	−	0.454934i	$-0.150337\pi$
$977$	55.6703	1.78105	0.890525	+	0.454934i	$0.150337\pi$
$978$	0	0
$979$	39.2023i	1.25291i
$980$	0	0
$981$	− 54.4883i	− 1.73968i
$982$	0	0
$983$	18.2697	0.582714	0.291357	−	0.956614i	$-0.405893\pi$
$983$	18.2697	0.582714	0.291357	+	0.956614i	$0.405893\pi$
$984$	0	0
$985$	−9.49718	−0.302605
$986$	0	0
$987$	11.9423i	0.380126i
$988$	0	0
$989$	− 77.8388i	− 2.47513i
$990$	0	0
$991$	28.0503	0.891046	0.445523	−	0.895270i	$-0.353018\pi$
$991$	28.0503	0.891046	0.445523	+	0.895270i	$0.353018\pi$
$992$	0	0
$993$	−3.61416	−0.114692
$994$	0	0
$995$	8.83300i	0.280025i
$996$	0	0
$997$	− 22.9739i	− 0.727590i	−0.931479	−	0.363795i	$-0.881481\pi$
$997$	− 22.9739i	− 0.727590i	0.931479	−	0.363795i	$-0.118519\pi$
$998$	0	0
$999$	−8.96054	−0.283499

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	832.2.b.c.417.5	yes	8
4.3	odd	2		832.2.b.d.417.4	yes	8
8.3	odd	2		832.2.b.d.417.5	yes	8
8.5	even	2	inner	832.2.b.c.417.4	✓	8
16.3	odd	4		3328.2.a.bi.1.3		4
16.5	even	4		3328.2.a.bj.1.3		4
16.11	odd	4		3328.2.a.bn.1.2		4
16.13	even	4		3328.2.a.bm.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
832.2.b.c.417.4	✓	8	8.5	even	2	inner
832.2.b.c.417.5	yes	8	1.1	even	1	trivial
832.2.b.d.417.4	yes	8	4.3	odd	2
832.2.b.d.417.5	yes	8	8.3	odd	2
3328.2.a.bi.1.3		4	16.3	odd	4
3328.2.a.bj.1.3		4	16.5	even	4
3328.2.a.bm.1.2		4	16.13	even	4
3328.2.a.bn.1.2		4	16.11	odd	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:	\( N \)	\(=\)	\( 832 = 2^{6} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	832.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.386509i q^{3} +0.386509i q^{5} -4.17452 q^{7} +2.85061 q^{9} -3.78801i q^{11} -1.00000i q^{13} -0.149389 q^{15} -4.49843 q^{17} -5.25211i q^{19} -1.61349i q^{21} +6.11192 q^{23} +4.85061 q^{25} +2.26131i q^{27} -8.88494i q^{29} -4.44911 q^{31} +1.46410 q^{33} -1.61349i q^{35} +3.96254i q^{37} +0.386509 q^{39} +8.11192 q^{41} -12.7356i q^{43} +1.10179i q^{45} -7.40150 q^{47} +10.4266 q^{49} -1.73869i q^{51} -9.04013i q^{53} +1.46410 q^{55} +2.02999 q^{57} +4.56103i q^{59} +1.33891i q^{61} -11.8999 q^{63} +0.386509 q^{65} +6.67296i q^{67} +2.36231i q^{69} +4.59850 q^{71} -11.6880 q^{73} +1.87481i q^{75} +15.8131i q^{77} -2.57916 q^{79} +7.67781 q^{81} -2.67296i q^{83} -1.73869i q^{85} +3.43411 q^{87} -10.3490 q^{89} +4.17452i q^{91} -1.71962i q^{93} +2.02999 q^{95} +0.237120 q^{97} -10.7982i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{9} - 36 q^{15} - 4 q^{17} + 24 q^{23} + 4 q^{25} - 20 q^{31} - 16 q^{33} - 4 q^{39} + 40 q^{41} - 40 q^{47} - 4 q^{49} - 16 q^{55} - 8 q^{57} - 44 q^{63} - 4 q^{65} + 56 q^{71} - 16 q^{73} - 32 q^{79}+ \cdots - 40 q^{97}+O(q^{100}) \) 8 * q - 12 * q^9 - 36 * q^15 - 4 * q^17 + 24 * q^23 + 4 * q^25 - 20 * q^31 - 16 * q^33 - 4 * q^39 + 40 * q^41 - 40 * q^47 - 4 * q^49 - 16 * q^55 - 8 * q^57 - 44 * q^63 - 4 * q^65 + 56 * q^71 - 16 * q^73 - 32 * q^79 + 40 * q^81 + 24 * q^87 - 16 * q^89 - 8 * q^95 - 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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