LMFDB - Embedded newform 405.3.c.a.161.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [405,3,Mod(161,405)] 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("405.161"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	405.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,-32,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$11.0354507066$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 48x^{14} + 912x^{12} + 8704x^{10} + 43602x^{8} + 109032x^{6} + 117844x^{4} + 36000x^{2} + 81 $ x^16 + 48x^14 + 912x^12 + 8704x^10 + 43602x^8 + 109032x^6 + 117844x^4 + 36000*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{14} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.12
Root			$1.83391i$ of defining polynomial
Character	$\chi$	$=$	405.161
Dual form			405.3.c.a.161.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.83391i q^{2} +0.636762 q^{4} +2.23607i q^{5} -6.46796 q^{7} +8.50342i q^{8} -4.10076 q^{10} +10.6775i q^{11} +11.0675 q^{13} -11.8617i q^{14} -13.0475 q^{16} -12.6328i q^{17} -32.1403 q^{19} +1.42384i q^{20} -19.5817 q^{22} +11.3422i q^{23} -5.00000 q^{25} +20.2968i q^{26} -4.11855 q^{28} +40.6724i q^{29} -31.8858 q^{31} +10.0857i q^{32} +23.1674 q^{34} -14.4628i q^{35} +45.4499 q^{37} -58.9425i q^{38} -19.0142 q^{40} +29.4196i q^{41} -37.6185 q^{43} +6.79904i q^{44} -20.8007 q^{46} -57.5178i q^{47} -7.16547 q^{49} -9.16957i q^{50} +7.04735 q^{52} +33.3940i q^{53} -23.8757 q^{55} -54.9998i q^{56} -74.5897 q^{58} -4.09814i q^{59} -66.2390 q^{61} -58.4757i q^{62} -70.6863 q^{64} +24.7477i q^{65} +70.9780 q^{67} -8.04405i q^{68} +26.5235 q^{70} -13.1123i q^{71} +109.273 q^{73} +83.3511i q^{74} -20.4657 q^{76} -69.0618i q^{77} +99.7782 q^{79} -29.1751i q^{80} -53.9530 q^{82} +85.7333i q^{83} +28.2477 q^{85} -68.9891i q^{86} -90.7955 q^{88} -63.6372i q^{89} -71.5841 q^{91} +7.22230i q^{92} +105.483 q^{94} -71.8679i q^{95} +86.4129 q^{97} -13.1409i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{4} - 4 q^{7} + 20 q^{13} + 64 q^{16} - 52 q^{19} + 48 q^{22} - 80 q^{25} + 32 q^{28} - 64 q^{31} - 108 q^{34} + 44 q^{37} + 60 q^{40} + 248 q^{43} - 108 q^{46} + 108 q^{49} - 124 q^{52} - 180 q^{58}+ \cdots + 284 q^{97}+O(q^{100}) $ 16 * q - 32 * q^4 - 4 * q^7 + 20 * q^13 + 64 * q^16 - 52 * q^19 + 48 * q^22 - 80 * q^25 + 32 * q^28 - 64 * q^31 - 108 * q^34 + 44 * q^37 + 60 * q^40 + 248 * q^43 - 108 * q^46 + 108 * q^49 - 124 * q^52 - 180 * q^58 - 124 * q^61 + 256 * q^64 - 28 * q^67 + 120 * q^70 - 268 * q^73 + 212 * q^76 + 80 * q^79 - 204 * q^82 - 60 * q^85 - 288 * q^88 + 136 * q^91 + 300 * q^94 + 284 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.83391i	0.916957i	0.888706	+	0.458478i	$0.151605\pi$
$2$	1.83391i	0.916957i	−0.888706	+	0.458478i	$0.848395\pi$
$3$	0	0
$3$	0	0
$4$	0.636762	0.159190
$5$	2.23607i	0.447214i
$5$	2.23607i	0.447214i
$6$	0	0
$7$	−6.46796	−0.923995	−0.461997	−	0.886881i	$-0.652867\pi$
$7$	−6.46796	−0.923995	−0.461997	+	0.886881i	$0.652867\pi$
$8$	8.50342i	1.06293i
$9$	0	0
$10$	−4.10076	−0.410076
$11$	10.6775i	0.970684i	0.874324	+	0.485342i	$0.161305\pi$
$11$	10.6775i	0.970684i	−0.874324	+	0.485342i	$0.838695\pi$
$12$	0	0
$13$	11.0675	0.851345	0.425673	−	0.904877i	$-0.360038\pi$
$13$	11.0675	0.851345	0.425673	+	0.904877i	$0.360038\pi$
$14$	− 11.8617i	− 0.847263i
$15$	0	0
$16$	−13.0475	−0.815468
$17$	− 12.6328i	− 0.743103i	−0.928412	−	0.371552i	$-0.878826\pi$
$17$	− 12.6328i	− 0.743103i	0.928412	−	0.371552i	$-0.121174\pi$
$18$	0	0
$19$	−32.1403	−1.69159	−0.845797	−	0.533505i	$-0.820875\pi$
$19$	−32.1403	−1.69159	−0.845797	+	0.533505i	$0.820875\pi$
$20$	1.42384i	0.0711921i
$21$	0	0
$22$	−19.5817	−0.890075
$23$	11.3422i	0.493140i	0.969125	+	0.246570i	$0.0793036\pi$
$23$	11.3422i	0.493140i	−0.969125	+	0.246570i	$0.920696\pi$
$24$	0	0
$25$	−5.00000	−0.200000
$26$	20.2968i	0.780647i
$27$	0	0
$28$	−4.11855	−0.147091
$29$	40.6724i	1.40250i	0.712916	+	0.701249i	$0.247374\pi$
$29$	40.6724i	1.40250i	−0.712916	+	0.701249i	$0.752626\pi$
$30$	0	0
$31$	−31.8858	−1.02857	−0.514286	−	0.857618i	$-0.671943\pi$
$31$	−31.8858	−1.02857	−0.514286	+	0.857618i	$0.671943\pi$
$32$	10.0857i	0.315179i
$33$	0	0
$34$	23.1674	0.681393
$35$	− 14.4628i	− 0.413223i
$36$	0	0
$37$	45.4499	1.22837	0.614187	−	0.789160i	$-0.289484\pi$
$37$	45.4499	1.22837	0.614187	+	0.789160i	$0.289484\pi$
$38$	− 58.9425i	− 1.55112i
$39$	0	0
$40$	−19.0142	−0.475356
$41$	29.4196i	0.717552i	0.933424	+	0.358776i	$0.116806\pi$
$41$	29.4196i	0.717552i	−0.933424	+	0.358776i	$0.883194\pi$
$42$	0	0
$43$	−37.6185	−0.874849	−0.437424	−	0.899255i	$-0.644109\pi$
$43$	−37.6185	−0.874849	−0.437424	+	0.899255i	$0.644109\pi$
$44$	6.79904i	0.154524i
$45$	0	0
$46$	−20.8007	−0.452188
$47$	− 57.5178i	− 1.22378i	−0.790941	−	0.611892i	$-0.790409\pi$
$47$	− 57.5178i	− 1.22378i	0.790941	−	0.611892i	$-0.209591\pi$
$48$	0	0
$49$	−7.16547	−0.146234
$50$	− 9.16957i	− 0.183391i
$51$	0	0
$52$	7.04735	0.135526
$53$	33.3940i	0.630075i	0.949079	+	0.315038i	$0.102017\pi$
$53$	33.3940i	0.630075i	−0.949079	+	0.315038i	$0.897983\pi$
$54$	0	0
$55$	−23.8757	−0.434103
$56$	− 54.9998i	− 0.982139i
$57$	0	0
$58$	−74.5897	−1.28603
$59$	− 4.09814i	− 0.0694600i	−0.999397	−	0.0347300i	$-0.988943\pi$
$59$	− 4.09814i	− 0.0694600i	0.999397	−	0.0347300i	$-0.0110571\pi$
$60$	0	0
$61$	−66.2390	−1.08589	−0.542943	−	0.839770i	$-0.682690\pi$
$61$	−66.2390	−1.08589	−0.542943	+	0.839770i	$0.682690\pi$
$62$	− 58.4757i	− 0.943157i
$63$	0	0
$64$	−70.6863	−1.10447
$65$	24.7477i	0.380733i
$66$	0	0
$67$	70.9780	1.05937	0.529686	−	0.848194i	$-0.322310\pi$
$67$	70.9780	1.05937	0.529686	+	0.848194i	$0.322310\pi$
$68$	− 8.04405i	− 0.118295i
$69$	0	0
$70$	26.5235	0.378908
$71$	− 13.1123i	− 0.184680i	−0.995728	−	0.0923400i	$-0.970565\pi$
$71$	− 13.1123i	− 0.184680i	0.995728	−	0.0923400i	$-0.0294347\pi$
$72$	0	0
$73$	109.273	1.49688	0.748442	−	0.663200i	$-0.230802\pi$
$73$	109.273	1.49688	0.748442	+	0.663200i	$0.230802\pi$
$74$	83.3511i	1.12637i
$75$	0	0
$76$	−20.4657	−0.269286
$77$	− 69.0618i	− 0.896907i
$78$	0	0
$79$	99.7782	1.26301	0.631507	−	0.775370i	$-0.282437\pi$
$79$	99.7782	1.26301	0.631507	+	0.775370i	$0.282437\pi$
$80$	− 29.1751i	− 0.364688i
$81$	0	0
$82$	−53.9530	−0.657964
$83$	85.7333i	1.03293i	0.856308	+	0.516466i	$0.172753\pi$
$83$	85.7333i	1.03293i	−0.856308	+	0.516466i	$0.827247\pi$
$84$	0	0
$85$	28.2477	0.332326
$86$	− 68.9891i	− 0.802199i
$87$	0	0
$88$	−90.7955	−1.03177
$89$	− 63.6372i	− 0.715025i	−0.933908	−	0.357513i	$-0.883625\pi$
$89$	− 63.6372i	− 0.715025i	0.933908	−	0.357513i	$-0.116375\pi$
$90$	0	0
$91$	−71.5841	−0.786638
$92$	7.22230i	0.0785032i
$93$	0	0
$94$	105.483	1.12216
$95$	− 71.8679i	− 0.756504i
$96$	0	0
$97$	86.4129	0.890854	0.445427	−	0.895318i	$-0.353052\pi$
$97$	86.4129	0.890854	0.445427	+	0.895318i	$0.353052\pi$
$98$	− 13.1409i	− 0.134090i
$99$	0	0
$100$	−3.18381	−0.0318381
$101$	− 44.0860i	− 0.436495i	−0.975893	−	0.218247i	$-0.929966\pi$
$101$	− 44.0860i	− 0.436495i	0.975893	−	0.218247i	$-0.0700339\pi$
$102$	0	0
$103$	93.8716	0.911375	0.455688	−	0.890140i	$-0.349393\pi$
$103$	93.8716	0.911375	0.455688	+	0.890140i	$0.349393\pi$
$104$	94.1115i	0.904918i
$105$	0	0
$106$	−61.2417	−0.577752
$107$	138.518i	1.29456i	0.762252	+	0.647280i	$0.224094\pi$
$107$	138.518i	1.29456i	−0.762252	+	0.647280i	$0.775906\pi$
$108$	0	0
$109$	176.733	1.62140	0.810700	−	0.585461i	$-0.199087\pi$
$109$	176.733	1.62140	0.810700	+	0.585461i	$0.199087\pi$
$110$	− 43.7859i	− 0.398054i
$111$	0	0
$112$	84.3907	0.753488
$113$	88.7306i	0.785227i	0.919704	+	0.392613i	$0.128429\pi$
$113$	88.7306i	0.785227i	−0.919704	+	0.392613i	$0.871571\pi$
$114$	0	0
$115$	−25.3620	−0.220539
$116$	25.8986i	0.223264i
$117$	0	0
$118$	7.51564	0.0636918
$119$	81.7082i	0.686623i
$120$	0	0
$121$	6.99045	0.0577723
$122$	− 121.477i	− 0.995710i
$123$	0	0
$124$	−20.3036	−0.163739
$125$	− 11.1803i	− 0.0894427i
$126$	0	0
$127$	92.0203	0.724569	0.362285	−	0.932068i	$-0.381997\pi$
$127$	92.0203	0.724569	0.362285	+	0.932068i	$0.381997\pi$
$128$	− 89.2897i	− 0.697576i
$129$	0	0
$130$	−45.3851	−0.349116
$131$	160.480i	1.22504i	0.790457	+	0.612518i	$0.209843\pi$
$131$	160.480i	1.22504i	−0.790457	+	0.612518i	$0.790157\pi$
$132$	0	0
$133$	207.882	1.56302
$134$	130.167i	0.971399i
$135$	0	0
$136$	107.422	0.789865
$137$	− 90.8642i	− 0.663243i	−0.943413	−	0.331621i	$-0.892404\pi$
$137$	− 90.8642i	− 0.663243i	0.943413	−	0.331621i	$-0.107596\pi$
$138$	0	0
$139$	−64.9730	−0.467431	−0.233716	−	0.972305i	$-0.575088\pi$
$139$	−64.9730	−0.467431	−0.233716	+	0.972305i	$0.575088\pi$
$140$	− 9.20936i	− 0.0657811i
$141$	0	0
$142$	24.0468	0.169344
$143$	118.173i	0.826387i
$144$	0	0
$145$	−90.9464	−0.627216
$146$	200.396i	1.37258i
$147$	0	0
$148$	28.9407	0.195545
$149$	144.297i	0.968433i	0.874948	+	0.484217i	$0.160895\pi$
$149$	144.297i	0.968433i	−0.874948	+	0.484217i	$0.839105\pi$
$150$	0	0
$151$	136.775	0.905796	0.452898	−	0.891562i	$-0.350390\pi$
$151$	136.775	0.905796	0.452898	+	0.891562i	$0.350390\pi$
$152$	− 273.302i	− 1.79804i
$153$	0	0
$154$	126.653	0.822425
$155$	− 71.2987i	− 0.459992i
$156$	0	0
$157$	−52.1929	−0.332439	−0.166220	−	0.986089i	$-0.553156\pi$
$157$	−52.1929	−0.332439	−0.166220	+	0.986089i	$0.553156\pi$
$158$	182.985i	1.15813i
$159$	0	0
$160$	−22.5523	−0.140952
$161$	− 73.3611i	− 0.455659i
$162$	0	0
$163$	−103.226	−0.633287	−0.316643	−	0.948545i	$-0.602556\pi$
$163$	−103.226	−0.633287	−0.316643	+	0.948545i	$0.602556\pi$
$164$	18.7333i	0.114227i
$165$	0	0
$166$	−157.228	−0.947154
$167$	− 112.819i	− 0.675565i	−0.941224	−	0.337782i	$-0.890323\pi$
$167$	− 112.819i	− 0.675565i	0.941224	−	0.337782i	$-0.109677\pi$
$168$	0	0
$169$	−46.5107	−0.275211
$170$	51.8038i	0.304728i
$171$	0	0
$172$	−23.9540	−0.139268
$173$	131.316i	0.759052i	0.925181	+	0.379526i	$0.123913\pi$
$173$	131.316i	0.759052i	−0.925181	+	0.379526i	$0.876087\pi$
$174$	0	0
$175$	32.3398	0.184799
$176$	− 139.315i	− 0.791562i
$177$	0	0
$178$	116.705	0.655647
$179$	297.076i	1.65964i	0.558028	+	0.829822i	$0.311558\pi$
$179$	297.076i	1.65964i	−0.558028	+	0.829822i	$0.688442\pi$
$180$	0	0
$181$	28.1168	0.155342	0.0776708	−	0.996979i	$-0.475252\pi$
$181$	28.1168	0.155342	0.0776708	+	0.996979i	$0.475252\pi$
$182$	− 131.279i	− 0.721313i
$183$	0	0
$184$	−96.4477	−0.524173
$185$	101.629i	0.549346i
$186$	0	0
$187$	134.887	0.721318
$188$	− 36.6252i	− 0.194815i
$189$	0	0
$190$	131.799	0.693681
$191$	9.17035i	0.0480123i	0.999712	+	0.0240062i	$0.00764213\pi$
$191$	9.17035i	0.0480123i	−0.999712	+	0.0240062i	$0.992358\pi$
$192$	0	0
$193$	94.7659	0.491015	0.245508	−	0.969395i	$-0.421045\pi$
$193$	94.7659	0.491015	0.245508	+	0.969395i	$0.421045\pi$
$194$	158.474i	0.816875i
$195$	0	0
$196$	−4.56270	−0.0232791
$197$	− 267.330i	− 1.35701i	−0.734597	−	0.678504i	$-0.762629\pi$
$197$	− 267.330i	− 1.35701i	0.734597	−	0.678504i	$-0.237371\pi$
$198$	0	0
$199$	−158.327	−0.795615	−0.397808	−	0.917469i	$-0.630229\pi$
$199$	−158.327	−0.795615	−0.397808	+	0.917469i	$0.630229\pi$
$200$	− 42.5171i	− 0.212585i
$201$	0	0
$202$	80.8498	0.400247
$203$	− 263.068i	− 1.29590i
$204$	0	0
$205$	−65.7843	−0.320899
$206$	172.152i	0.835692i
$207$	0	0
$208$	−144.403	−0.694245
$209$	− 343.179i	− 1.64200i
$210$	0	0
$211$	317.573	1.50508	0.752542	−	0.658545i	$-0.228828\pi$
$211$	317.573	1.50508	0.752542	+	0.658545i	$0.228828\pi$
$212$	21.2640i	0.100302i
$213$	0	0
$214$	−254.030	−1.18706
$215$	− 84.1175i	− 0.391244i
$216$	0	0
$217$	206.236	0.950396
$218$	324.112i	1.48675i
$219$	0	0
$220$	−15.2031	−0.0691050
$221$	− 139.813i	− 0.632637i
$222$	0	0
$223$	−123.991	−0.556013	−0.278007	−	0.960579i	$-0.589674\pi$
$223$	−123.991	−0.556013	−0.278007	+	0.960579i	$0.589674\pi$
$224$	− 65.2340i	− 0.291223i
$225$	0	0
$226$	−162.724	−0.720019
$227$	− 217.477i	− 0.958049i	−0.877801	−	0.479025i	$-0.840990\pi$
$227$	− 217.477i	− 0.958049i	0.877801	−	0.479025i	$-0.159010\pi$
$228$	0	0
$229$	74.9546	0.327313	0.163656	−	0.986517i	$-0.447671\pi$
$229$	74.9546	0.327313	0.163656	+	0.986517i	$0.447671\pi$
$230$	− 46.5117i	− 0.202225i
$231$	0	0
$232$	−345.855	−1.49075
$233$	− 142.781i	− 0.612792i	−0.951904	−	0.306396i	$-0.900877\pi$
$233$	− 142.781i	− 0.612792i	0.951904	−	0.306396i	$-0.0991232\pi$
$234$	0	0
$235$	128.614	0.547293
$236$	− 2.60954i	− 0.0110574i
$237$	0	0
$238$	−149.846	−0.629604
$239$	− 200.181i	− 0.837577i	−0.908084	−	0.418789i	$-0.862455\pi$
$239$	− 200.181i	− 0.837577i	0.908084	−	0.418789i	$-0.137545\pi$
$240$	0	0
$241$	193.920	0.804647	0.402324	−	0.915498i	$-0.368203\pi$
$241$	193.920	0.804647	0.402324	+	0.915498i	$0.368203\pi$
$242$	12.8199i	0.0529747i
$243$	0	0
$244$	−42.1784	−0.172862
$245$	− 16.0225i	− 0.0653979i
$246$	0	0
$247$	−355.712	−1.44013
$248$	− 271.138i	− 1.09330i
$249$	0	0
$250$	20.5038	0.0820151
$251$	− 301.288i	− 1.20035i	−0.799868	−	0.600176i	$-0.795097\pi$
$251$	− 301.288i	− 1.20035i	0.799868	−	0.600176i	$-0.204903\pi$
$252$	0	0
$253$	−121.107	−0.478684
$254$	168.757i	0.664398i
$255$	0	0
$256$	−118.996	−0.464827
$257$	− 210.010i	− 0.817158i	−0.912723	−	0.408579i	$-0.866025\pi$
$257$	− 210.010i	− 0.817158i	0.912723	−	0.408579i	$-0.133975\pi$
$258$	0	0
$259$	−293.968	−1.13501
$260$	15.7584i	0.0606091i
$261$	0	0
$262$	−294.306	−1.12330
$263$	151.418i	0.575735i	0.957670	+	0.287868i	$0.0929463\pi$
$263$	151.418i	0.575735i	−0.957670	+	0.287868i	$0.907054\pi$
$264$	0	0
$265$	−74.6712	−0.281778
$266$	381.238i	1.43323i
$267$	0	0
$268$	45.1960	0.168642
$269$	411.264i	1.52886i	0.644706	+	0.764431i	$0.276980\pi$
$269$	411.264i	1.52886i	−0.644706	+	0.764431i	$0.723020\pi$
$270$	0	0
$271$	−255.156	−0.941534	−0.470767	−	0.882257i	$-0.656023\pi$
$271$	−255.156	−0.941534	−0.470767	+	0.882257i	$0.656023\pi$
$272$	164.826i	0.605977i
$273$	0	0
$274$	166.637	0.608165
$275$	− 53.3876i	− 0.194137i
$276$	0	0
$277$	−185.560	−0.669891	−0.334945	−	0.942238i	$-0.608718\pi$
$277$	−185.560	−0.669891	−0.334945	+	0.942238i	$0.608718\pi$
$278$	− 119.155i	− 0.428614i
$279$	0	0
$280$	122.983	0.439226
$281$	104.142i	0.370610i	0.982681	+	0.185305i	$0.0593274\pi$
$281$	104.142i	0.370610i	−0.982681	+	0.185305i	$0.940673\pi$
$282$	0	0
$283$	−51.8563	−0.183238	−0.0916189	−	0.995794i	$-0.529204\pi$
$283$	−51.8563	−0.183238	−0.0916189	+	0.995794i	$0.529204\pi$
$284$	− 8.34940i	− 0.0293993i
$285$	0	0
$286$	−216.720	−0.757762
$287$	− 190.285i	− 0.663014i
$288$	0	0
$289$	129.414	0.447798
$290$	− 166.788i	− 0.575130i
$291$	0	0
$292$	69.5806	0.238290
$293$	331.670i	1.13198i	0.824412	+	0.565990i	$0.191506\pi$
$293$	331.670i	1.13198i	−0.824412	+	0.565990i	$0.808494\pi$
$294$	0	0
$295$	9.16372	0.0310635
$296$	386.479i	1.30567i
$297$	0	0
$298$	−264.627	−0.888012
$299$	125.530i	0.419833i
$300$	0	0
$301$	243.315	0.808356
$302$	250.834i	0.830576i
$303$	0	0
$304$	419.350	1.37944
$305$	− 148.115i	− 0.485623i
$306$	0	0
$307$	−292.274	−0.952031	−0.476016	−	0.879437i	$-0.657919\pi$
$307$	−292.274	−0.952031	−0.476016	+	0.879437i	$0.657919\pi$
$308$	− 43.9759i	− 0.142779i
$309$	0	0
$310$	130.756	0.421793
$311$	− 197.791i	− 0.635985i	−0.948093	−	0.317992i	$-0.896991\pi$
$311$	− 197.791i	− 0.635985i	0.948093	−	0.317992i	$-0.103009\pi$
$312$	0	0
$313$	215.698	0.689130	0.344565	−	0.938762i	$-0.388026\pi$
$313$	215.698	0.689130	0.344565	+	0.938762i	$0.388026\pi$
$314$	− 95.7173i	− 0.304832i
$315$	0	0
$316$	63.5349	0.201060
$317$	− 203.636i	− 0.642384i	−0.947014	−	0.321192i	$-0.895916\pi$
$317$	− 203.636i	− 0.642384i	0.947014	−	0.321192i	$-0.104084\pi$
$318$	0	0
$319$	−434.281	−1.36138
$320$	− 158.059i	− 0.493935i
$321$	0	0
$322$	134.538	0.417820
$323$	406.020i	1.25703i
$324$	0	0
$325$	−55.3374	−0.170269
$326$	− 189.307i	− 0.580697i
$327$	0	0
$328$	−250.167	−0.762705
$329$	372.023i	1.13077i
$330$	0	0
$331$	94.0231	0.284058	0.142029	−	0.989863i	$-0.454637\pi$
$331$	94.0231	0.284058	0.142029	+	0.989863i	$0.454637\pi$
$332$	54.5917i	0.164433i
$333$	0	0
$334$	206.901	0.619464
$335$	158.712i	0.473766i
$336$	0	0
$337$	−336.137	−0.997438	−0.498719	−	0.866764i	$-0.666196\pi$
$337$	−336.137	−0.997438	−0.498719	+	0.866764i	$0.666196\pi$
$338$	− 85.2965i	− 0.252357i
$339$	0	0
$340$	17.9870	0.0529031
$341$	− 340.461i	− 0.998419i
$342$	0	0
$343$	363.276	1.05911
$344$	− 319.886i	− 0.929901i
$345$	0	0
$346$	−240.822	−0.696018
$347$	− 62.5234i	− 0.180183i	−0.995934	−	0.0900914i	$-0.971284\pi$
$347$	− 62.5234i	− 0.180183i	0.995934	−	0.0900914i	$-0.0287159\pi$
$348$	0	0
$349$	−320.951	−0.919629	−0.459815	−	0.888015i	$-0.652084\pi$
$349$	−320.951	−0.919629	−0.459815	+	0.888015i	$0.652084\pi$
$350$	59.3084i	0.169453i
$351$	0	0
$352$	−107.690	−0.305939
$353$	− 166.213i	− 0.470859i	−0.971891	−	0.235430i	$-0.924350\pi$
$353$	− 166.213i	− 0.470859i	0.971891	−	0.235430i	$-0.0756497\pi$
$354$	0	0
$355$	29.3200	0.0825914
$356$	− 40.5217i	− 0.113825i
$357$	0	0
$358$	−544.812	−1.52182
$359$	− 199.670i	− 0.556183i	−0.960555	−	0.278091i	$-0.910298\pi$
$359$	− 199.670i	− 0.556183i	0.960555	−	0.278091i	$-0.0897018\pi$
$360$	0	0
$361$	671.998	1.86149
$362$	51.5638i	0.142442i
$363$	0	0
$364$	−45.5820	−0.125225
$365$	244.341i	0.669427i
$366$	0	0
$367$	−160.642	−0.437716	−0.218858	−	0.975757i	$-0.570233\pi$
$367$	−160.642	−0.437716	−0.218858	+	0.975757i	$0.570233\pi$
$368$	− 147.988i	− 0.402140i
$369$	0	0
$370$	−186.379	−0.503726
$371$	− 215.991i	− 0.582186i
$372$	0	0
$373$	−171.238	−0.459082	−0.229541	−	0.973299i	$-0.573723\pi$
$373$	−171.238	−0.459082	−0.229541	+	0.973299i	$0.573723\pi$
$374$	247.370i	0.661418i
$375$	0	0
$376$	489.098	1.30079
$377$	450.142i	1.19401i
$378$	0	0
$379$	−297.486	−0.784922	−0.392461	−	0.919769i	$-0.628376\pi$
$379$	−297.486	−0.784922	−0.392461	+	0.919769i	$0.628376\pi$
$380$	− 45.7627i	− 0.120428i
$381$	0	0
$382$	−16.8176	−0.0440252
$383$	480.748i	1.25522i	0.778530	+	0.627608i	$0.215966\pi$
$383$	480.748i	1.25522i	−0.778530	+	0.627608i	$0.784034\pi$
$384$	0	0
$385$	154.427	0.401109
$386$	173.793i	0.450240i
$387$	0	0
$388$	55.0244	0.141815
$389$	594.441i	1.52812i	0.645142	+	0.764062i	$0.276798\pi$
$389$	594.441i	1.52812i	−0.645142	+	0.764062i	$0.723202\pi$
$390$	0	0
$391$	143.284	0.366454
$392$	− 60.9310i	− 0.155436i
$393$	0	0
$394$	490.261	1.24432
$395$	223.111i	0.564837i
$396$	0	0
$397$	−106.303	−0.267765	−0.133883	−	0.990997i	$-0.542745\pi$
$397$	−106.303	−0.267765	−0.133883	+	0.990997i	$0.542745\pi$
$398$	− 290.359i	− 0.729545i
$399$	0	0
$400$	65.2374	0.163094
$401$	30.3925i	0.0757918i	0.999282	+	0.0378959i	$0.0120655\pi$
$401$	30.3925i	0.0757918i	−0.999282	+	0.0378959i	$0.987934\pi$
$402$	0	0
$403$	−352.895	−0.875671
$404$	− 28.0722i	− 0.0694857i
$405$	0	0
$406$	482.444	1.18828
$407$	485.292i	1.19236i
$408$	0	0
$409$	73.4390	0.179557	0.0897787	−	0.995962i	$-0.471384\pi$
$409$	73.4390	0.179557	0.0897787	+	0.995962i	$0.471384\pi$
$410$	− 120.643i	− 0.294250i
$411$	0	0
$412$	59.7738	0.145082
$413$	26.5066i	0.0641807i
$414$	0	0
$415$	−191.706	−0.461941
$416$	111.624i	0.268326i
$417$	0	0
$418$	629.360	1.50565
$419$	− 166.914i	− 0.398362i	−0.979963	−	0.199181i	$-0.936172\pi$
$419$	− 166.914i	− 0.398362i	0.979963	−	0.199181i	$-0.0638282\pi$
$420$	0	0
$421$	−305.491	−0.725632	−0.362816	−	0.931861i	$-0.618185\pi$
$421$	−305.491	−0.725632	−0.362816	+	0.931861i	$0.618185\pi$
$422$	582.401i	1.38010i
$423$	0	0
$424$	−283.963	−0.669724
$425$	63.1638i	0.148621i
$426$	0	0
$427$	428.431	1.00335
$428$	88.2029i	0.206081i
$429$	0	0
$430$	154.264	0.358754
$431$	− 221.026i	− 0.512820i	−0.966568	−	0.256410i	$-0.917460\pi$
$431$	− 221.026i	− 0.512820i	0.966568	−	0.256410i	$-0.0825398\pi$
$432$	0	0
$433$	194.055	0.448164	0.224082	−	0.974570i	$-0.428062\pi$
$433$	194.055	0.448164	0.224082	+	0.974570i	$0.428062\pi$
$434$	378.219i	0.871472i
$435$	0	0
$436$	112.537	0.258111
$437$	− 364.543i	− 0.834193i
$438$	0	0
$439$	−154.690	−0.352369	−0.176184	−	0.984357i	$-0.556376\pi$
$439$	−154.690	−0.352369	−0.176184	+	0.984357i	$0.556376\pi$
$440$	− 203.025i	− 0.461420i
$441$	0	0
$442$	256.405	0.580101
$443$	− 483.147i	− 1.09063i	−0.838233	−	0.545313i	$-0.816411\pi$
$443$	− 483.147i	− 1.09063i	0.838233	−	0.545313i	$-0.183589\pi$
$444$	0	0
$445$	142.297	0.319769
$446$	− 227.389i	− 0.509840i
$447$	0	0
$448$	457.196	1.02053
$449$	− 679.146i	− 1.51257i	−0.654240	−	0.756287i	$-0.727011\pi$
$449$	− 679.146i	− 1.51257i	0.654240	−	0.756287i	$-0.272989\pi$
$450$	0	0
$451$	−314.129	−0.696516
$452$	56.5002i	0.125001i
$453$	0	0
$454$	398.834	0.878490
$455$	− 160.067i	− 0.351795i
$456$	0	0
$457$	−80.5139	−0.176179	−0.0880896	−	0.996113i	$-0.528076\pi$
$457$	−80.5139	−0.176179	−0.0880896	+	0.996113i	$0.528076\pi$
$458$	137.460i	0.300131i
$459$	0	0
$460$	−16.1495	−0.0351077
$461$	618.742i	1.34217i	0.741379	+	0.671087i	$0.234172\pi$
$461$	618.742i	1.34217i	−0.741379	+	0.671087i	$0.765828\pi$
$462$	0	0
$463$	329.457	0.711570	0.355785	−	0.934568i	$-0.384213\pi$
$463$	329.457	0.711570	0.355785	+	0.934568i	$0.384213\pi$
$464$	− 530.673i	− 1.14369i
$465$	0	0
$466$	261.847	0.561904
$467$	− 216.349i	− 0.463275i	−0.972802	−	0.231637i	$-0.925592\pi$
$467$	− 216.349i	− 0.463275i	0.972802	−	0.231637i	$-0.0744083\pi$
$468$	0	0
$469$	−459.083	−0.978855
$470$	235.867i	0.501844i
$471$	0	0
$472$	34.8482	0.0738310
$473$	− 401.673i	− 0.849202i
$474$	0	0
$475$	160.701	0.338319
$476$	52.0286i	0.109304i
$477$	0	0
$478$	367.114	0.768022
$479$	− 138.523i	− 0.289193i	−0.989491	−	0.144596i	$-0.953812\pi$
$479$	− 138.523i	− 0.289193i	0.989491	−	0.144596i	$-0.0461884\pi$
$480$	0	0
$481$	503.016	1.04577
$482$	355.632i	0.737827i
$483$	0	0
$484$	4.45125	0.00919679
$485$	193.225i	0.398402i
$486$	0	0
$487$	93.0290	0.191025	0.0955123	−	0.995428i	$-0.469551\pi$
$487$	93.0290	0.191025	0.0955123	+	0.995428i	$0.469551\pi$
$488$	− 563.258i	− 1.15422i
$489$	0	0
$490$	29.3838	0.0599670
$491$	234.598i	0.477796i	0.971045	+	0.238898i	$0.0767861\pi$
$491$	234.598i	0.477796i	−0.971045	+	0.238898i	$0.923214\pi$
$492$	0	0
$493$	513.805	1.04220
$494$	− 652.346i	− 1.32054i
$495$	0	0
$496$	416.029	0.838768
$497$	84.8097i	0.170643i
$498$	0	0
$499$	−661.151	−1.32495	−0.662476	−	0.749083i	$-0.730494\pi$
$499$	−661.151	−1.32495	−0.662476	+	0.749083i	$0.730494\pi$
$500$	− 7.11921i	− 0.0142384i
$501$	0	0
$502$	552.537	1.10067
$503$	− 184.352i	− 0.366504i	−0.983066	−	0.183252i	$-0.941338\pi$
$503$	− 184.352i	− 0.366504i	0.983066	−	0.183252i	$-0.0586624\pi$
$504$	0	0
$505$	98.5792	0.195206
$506$	− 222.100i	− 0.438932i
$507$	0	0
$508$	58.5950	0.115344
$509$	− 810.916i	− 1.59316i	−0.604536	−	0.796578i	$-0.706642\pi$
$509$	− 810.916i	− 1.59316i	0.604536	−	0.796578i	$-0.293358\pi$
$510$	0	0
$511$	−706.771	−1.38311
$512$	− 575.386i	− 1.12380i
$513$	0	0
$514$	385.139	0.749298
$515$	209.903i	0.407579i
$516$	0	0
$517$	614.148	1.18791
$518$	− 539.112i	− 1.04076i
$519$	0	0
$520$	−210.440	−0.404692
$521$	787.925i	1.51233i	0.654379	+	0.756167i	$0.272930\pi$
$521$	787.925i	1.51233i	−0.654379	+	0.756167i	$0.727070\pi$
$522$	0	0
$523$	495.806	0.948004	0.474002	−	0.880524i	$-0.342809\pi$
$523$	495.806	0.948004	0.474002	+	0.880524i	$0.342809\pi$
$524$	102.187i	0.195014i
$525$	0	0
$526$	−277.688	−0.527924
$527$	402.805i	0.764336i
$528$	0	0
$529$	400.354	0.756812
$530$	− 136.941i	− 0.258378i
$531$	0	0
$532$	132.371	0.248818
$533$	325.601i	0.610884i
$534$	0	0
$535$	−309.735	−0.578945
$536$	603.556i	1.12604i
$537$	0	0
$538$	−754.222	−1.40190
$539$	− 76.5095i	− 0.141947i
$540$	0	0
$541$	−395.636	−0.731305	−0.365652	−	0.930751i	$-0.619154\pi$
$541$	−395.636	−0.731305	−0.365652	+	0.930751i	$0.619154\pi$
$542$	− 467.934i	− 0.863346i
$543$	0	0
$544$	127.410	0.234210
$545$	395.186i	0.725112i
$546$	0	0
$547$	161.798	0.295792	0.147896	−	0.989003i	$-0.452750\pi$
$547$	161.798	0.295792	0.147896	+	0.989003i	$0.452750\pi$
$548$	− 57.8589i	− 0.105582i
$549$	0	0
$550$	97.9083	0.178015
$551$	− 1307.22i	− 2.37246i
$552$	0	0
$553$	−645.361	−1.16702
$554$	− 340.300i	− 0.614261i
$555$	0	0
$556$	−41.3723	−0.0744106
$557$	199.042i	0.357347i	0.983908	+	0.178673i	$0.0571806\pi$
$557$	199.042i	0.357347i	−0.983908	+	0.178673i	$0.942819\pi$
$558$	0	0
$559$	−416.342	−0.744799
$560$	188.703i	0.336970i
$561$	0	0
$562$	−190.987	−0.339834
$563$	907.583i	1.61205i	0.591882	+	0.806024i	$0.298385\pi$
$563$	907.583i	1.61205i	−0.591882	+	0.806024i	$0.701615\pi$
$564$	0	0
$565$	−198.408	−0.351164
$566$	− 95.1000i	− 0.168021i
$567$	0	0
$568$	111.499	0.196301
$569$	− 896.509i	− 1.57559i	−0.615940	−	0.787793i	$-0.711224\pi$
$569$	− 896.509i	− 1.57559i	0.615940	−	0.787793i	$-0.288776\pi$
$570$	0	0
$571$	−609.617	−1.06763	−0.533815	−	0.845601i	$-0.679242\pi$
$571$	−609.617	−1.06763	−0.533815	+	0.845601i	$0.679242\pi$
$572$	75.2483i	0.131553i
$573$	0	0
$574$	348.966	0.607955
$575$	− 56.7112i	− 0.0986281i
$576$	0	0
$577$	477.854	0.828170	0.414085	−	0.910238i	$-0.364102\pi$
$577$	477.854	0.828170	0.414085	+	0.910238i	$0.364102\pi$
$578$	237.333i	0.410611i
$579$	0	0
$580$	−57.9111	−0.0998468
$581$	− 554.520i	− 0.954423i
$582$	0	0
$583$	−356.565	−0.611604
$584$	929.191i	1.59108i
$585$	0	0
$586$	−608.254	−1.03798
$587$	− 254.840i	− 0.434139i	−0.976156	−	0.217070i	$-0.930350\pi$
$587$	− 254.840i	− 0.434139i	0.976156	−	0.217070i	$-0.0696499\pi$
$588$	0	0
$589$	1024.82	1.73993
$590$	16.8055i	0.0284839i
$591$	0	0
$592$	−593.007	−1.00170
$593$	604.184i	1.01886i	0.860512	+	0.509430i	$0.170144\pi$
$593$	604.184i	1.01886i	−0.860512	+	0.509430i	$0.829856\pi$
$594$	0	0
$595$	−182.705	−0.307067
$596$	91.8825i	0.154165i
$597$	0	0
$598$	−230.211	−0.384969
$599$	190.331i	0.317749i	0.987299	+	0.158874i	$0.0507865\pi$
$599$	190.331i	0.317749i	−0.987299	+	0.158874i	$0.949214\pi$
$600$	0	0
$601$	937.324	1.55961	0.779804	−	0.626024i	$-0.215319\pi$
$601$	937.324	1.55961	0.779804	+	0.626024i	$0.215319\pi$
$602$	446.219i	0.741227i
$603$	0	0
$604$	87.0932	0.144194
$605$	15.6311i	0.0258366i
$606$	0	0
$607$	−967.995	−1.59472	−0.797360	−	0.603504i	$-0.793771\pi$
$607$	−967.995	−1.59472	−0.797360	+	0.603504i	$0.793771\pi$
$608$	− 324.158i	− 0.533154i
$609$	0	0
$610$	271.630	0.445295
$611$	− 636.578i	− 1.04186i
$612$	0	0
$613$	14.3244	0.0233677	0.0116839	−	0.999932i	$-0.496281\pi$
$613$	14.3244	0.0233677	0.0116839	+	0.999932i	$0.496281\pi$
$614$	− 536.005i	− 0.872972i
$615$	0	0
$616$	587.262	0.953347
$617$	1067.61i	1.73033i	0.501488	+	0.865165i	$0.332786\pi$
$617$	1067.61i	1.73033i	−0.501488	+	0.865165i	$0.667214\pi$
$618$	0	0
$619$	1164.33	1.88099	0.940494	−	0.339810i	$-0.110363\pi$
$619$	1164.33	1.88099	0.940494	+	0.339810i	$0.110363\pi$
$620$	− 45.4003i	− 0.0732263i
$621$	0	0
$622$	362.732	0.583170
$623$	411.603i	0.660679i
$624$	0	0
$625$	25.0000	0.0400000
$626$	395.571i	0.631902i
$627$	0	0
$628$	−33.2344	−0.0529211
$629$	− 574.157i	− 0.912809i
$630$	0	0
$631$	−88.9538	−0.140973	−0.0704864	−	0.997513i	$-0.522455\pi$
$631$	−88.9538	−0.140973	−0.0704864	+	0.997513i	$0.522455\pi$
$632$	848.456i	1.34249i
$633$	0	0
$634$	373.451	0.589039
$635$	205.764i	0.324037i
$636$	0	0
$637$	−79.3038	−0.124496
$638$	− 796.434i	− 1.24833i
$639$	0	0
$640$	199.658	0.311965
$641$	133.208i	0.207813i	0.994587	+	0.103906i	$0.0331343\pi$
$641$	133.208i	0.207813i	−0.994587	+	0.103906i	$0.966866\pi$
$642$	0	0
$643$	−483.457	−0.751877	−0.375939	−	0.926645i	$-0.622680\pi$
$643$	−483.457	−0.751877	−0.375939	+	0.926645i	$0.622680\pi$
$644$	− 46.7135i	− 0.0725365i
$645$	0	0
$646$	−744.606	−1.15264
$647$	347.759i	0.537495i	0.963211	+	0.268748i	$0.0866097\pi$
$647$	347.759i	0.537495i	−0.963211	+	0.268748i	$0.913390\pi$
$648$	0	0
$649$	43.7580	0.0674237
$650$	− 101.484i	− 0.156129i
$651$	0	0
$652$	−65.7302	−0.100813
$653$	− 867.744i	− 1.32886i	−0.747352	−	0.664428i	$-0.768675\pi$
$653$	− 867.744i	− 1.32886i	0.747352	−	0.664428i	$-0.231325\pi$
$654$	0	0
$655$	−358.843	−0.547853
$656$	− 383.852i	− 0.585140i
$657$	0	0
$658$	−682.258	−1.03687
$659$	929.076i	1.40983i	0.709294	+	0.704913i	$0.249014\pi$
$659$	929.076i	1.40983i	−0.709294	+	0.704913i	$0.750986\pi$
$660$	0	0
$661$	−927.123	−1.40261	−0.701303	−	0.712863i	$-0.747398\pi$
$661$	−927.123	−1.40261	−0.701303	+	0.712863i	$0.747398\pi$
$662$	172.430i	0.260469i
$663$	0	0
$664$	−729.027	−1.09793
$665$	464.839i	0.699005i
$666$	0	0
$667$	−461.316	−0.691629
$668$	− 71.8390i	− 0.107543i
$669$	0	0
$670$	−291.063	−0.434423
$671$	− 707.269i	− 1.05405i
$672$	0	0
$673$	945.992	1.40563	0.702817	−	0.711370i	$-0.251925\pi$
$673$	945.992	1.40563	0.702817	+	0.711370i	$0.251925\pi$
$674$	− 616.446i	− 0.914608i
$675$	0	0
$676$	−29.6162	−0.0438110
$677$	614.584i	0.907804i	0.891051	+	0.453902i	$0.149968\pi$
$677$	614.584i	0.907804i	−0.891051	+	0.453902i	$0.850032\pi$
$678$	0	0
$679$	−558.915	−0.823145
$680$	240.202i	0.353238i
$681$	0	0
$682$	624.376	0.915507
$683$	258.893i	0.379053i	0.981876	+	0.189527i	$0.0606954\pi$
$683$	258.893i	0.379053i	−0.981876	+	0.189527i	$0.939305\pi$
$684$	0	0
$685$	203.179	0.296611
$686$	666.217i	0.971162i
$687$	0	0
$688$	490.827	0.713411
$689$	369.588i	0.536412i
$690$	0	0
$691$	−970.974	−1.40517	−0.702586	−	0.711599i	$-0.747971\pi$
$691$	−970.974	−1.40517	−0.702586	+	0.711599i	$0.747971\pi$
$692$	83.6170i	0.120834i
$693$	0	0
$694$	114.663	0.165220
$695$	− 145.284i	− 0.209042i
$696$	0	0
$697$	371.651	0.533215
$698$	− 588.596i	− 0.843260i
$699$	0	0
$700$	20.5927	0.0294182
$701$	718.418i	1.02485i	0.858733	+	0.512424i	$0.171252\pi$
$701$	718.418i	1.02485i	−0.858733	+	0.512424i	$0.828748\pi$
$702$	0	0
$703$	−1460.77	−2.07791
$704$	− 754.755i	− 1.07209i
$705$	0	0
$706$	304.821	0.431758
$707$	285.146i	0.403319i
$708$	0	0
$709$	−988.445	−1.39414	−0.697070	−	0.717003i	$-0.745513\pi$
$709$	−988.445	−1.39414	−0.697070	+	0.717003i	$0.745513\pi$
$710$	53.7703i	0.0757328i
$711$	0	0
$712$	541.134	0.760020
$713$	− 361.656i	− 0.507231i
$714$	0	0
$715$	−264.244	−0.369572
$716$	189.167i	0.264199i
$717$	0	0
$718$	366.177	0.509995
$719$	1280.53i	1.78099i	0.454994	+	0.890495i	$0.349642\pi$
$719$	1280.53i	1.78099i	−0.454994	+	0.890495i	$0.650358\pi$
$720$	0	0
$721$	−607.158	−0.842106
$722$	1232.39i	1.70691i
$723$	0	0
$724$	17.9037	0.0247289
$725$	− 203.362i	− 0.280500i
$726$	0	0
$727$	−649.647	−0.893600	−0.446800	−	0.894634i	$-0.647436\pi$
$727$	−649.647	−0.893600	−0.446800	+	0.894634i	$0.647436\pi$
$728$	− 608.710i	− 0.836140i
$729$	0	0
$730$	−448.100	−0.613836
$731$	475.225i	0.650103i
$732$	0	0
$733$	1170.09	1.59630	0.798150	−	0.602459i	$-0.205812\pi$
$733$	1170.09	1.59630	0.798150	+	0.602459i	$0.205812\pi$
$734$	− 294.603i	− 0.401367i
$735$	0	0
$736$	−114.394	−0.155427
$737$	757.869i	1.02832i
$738$	0	0
$739$	1175.25	1.59033	0.795164	−	0.606394i	$-0.207385\pi$
$739$	1175.25	1.59033	0.795164	+	0.606394i	$0.207385\pi$
$740$	64.7134i	0.0874506i
$741$	0	0
$742$	396.109	0.533840
$743$	1017.43i	1.36936i	0.728844	+	0.684680i	$0.240058\pi$
$743$	1017.43i	1.36936i	−0.728844	+	0.684680i	$0.759942\pi$
$744$	0	0
$745$	−322.657	−0.433097
$746$	− 314.035i	− 0.420958i
$747$	0	0
$748$	85.8906	0.114827
$749$	− 895.928i	− 1.19617i
$750$	0	0
$751$	−994.458	−1.32418	−0.662089	−	0.749425i	$-0.730330\pi$
$751$	−994.458	−1.32418	−0.662089	+	0.749425i	$0.730330\pi$
$752$	750.463i	0.997957i
$753$	0	0
$754$	−825.521	−1.09486
$755$	305.839i	0.405084i
$756$	0	0
$757$	−659.088	−0.870658	−0.435329	−	0.900271i	$-0.643368\pi$
$757$	−659.088	−0.870658	−0.435329	+	0.900271i	$0.643368\pi$
$758$	− 545.563i	− 0.719740i
$759$	0	0
$760$	611.123	0.804109
$761$	− 646.260i	− 0.849224i	−0.905375	−	0.424612i	$-0.860410\pi$
$761$	− 646.260i	− 0.849224i	0.905375	−	0.424612i	$-0.139590\pi$
$762$	0	0
$763$	−1143.10	−1.49817
$764$	5.83933i	0.00764310i
$765$	0	0
$766$	−881.649	−1.15098
$767$	− 45.3561i	− 0.0591345i
$768$	0	0
$769$	−246.857	−0.321010	−0.160505	−	0.987035i	$-0.551312\pi$
$769$	−246.857	−0.321010	−0.160505	+	0.987035i	$0.551312\pi$
$770$	283.206i	0.367800i
$771$	0	0
$772$	60.3433	0.0781649
$773$	− 263.071i	− 0.340325i	−0.985416	−	0.170162i	$-0.945571\pi$
$773$	− 263.071i	− 0.340325i	0.985416	−	0.170162i	$-0.0544292\pi$
$774$	0	0
$775$	159.429	0.205715
$776$	734.805i	0.946914i
$777$	0	0
$778$	−1090.15	−1.40122
$779$	− 945.555i	− 1.21381i
$780$	0	0
$781$	140.007	0.179266
$782$	262.770i	0.336023i
$783$	0	0
$784$	93.4914	0.119249
$785$	− 116.707i	− 0.148671i
$786$	0	0
$787$	1076.78	1.36820	0.684102	−	0.729387i	$-0.260194\pi$
$787$	1076.78	1.36820	0.684102	+	0.729387i	$0.260194\pi$
$788$	− 170.226i	− 0.216022i
$789$	0	0
$790$	−409.166	−0.517931
$791$	− 573.906i	− 0.725545i
$792$	0	0
$793$	−733.100	−0.924463
$794$	− 194.950i	− 0.245529i
$795$	0	0
$796$	−100.817	−0.126654
$797$	− 913.930i	− 1.14671i	−0.819306	−	0.573356i	$-0.805641\pi$
$797$	− 913.930i	− 1.14671i	0.819306	−	0.573356i	$-0.194359\pi$
$798$	0	0
$799$	−726.609	−0.909398
$800$	− 50.4286i	− 0.0630357i
$801$	0	0
$802$	−55.7373	−0.0694978
$803$	1166.76i	1.45300i
$804$	0	0
$805$	164.040	0.203777
$806$	− 647.179i	− 0.802952i
$807$	0	0
$808$	374.881	0.463962
$809$	− 1008.67i	− 1.24681i	−0.781897	−	0.623407i	$-0.785748\pi$
$809$	− 1008.67i	− 1.24681i	0.781897	−	0.623407i	$-0.214252\pi$
$810$	0	0
$811$	952.468	1.17444	0.587218	−	0.809429i	$-0.300223\pi$
$811$	952.468	1.17444	0.587218	+	0.809429i	$0.300223\pi$
$812$	− 167.511i	− 0.206295i
$813$	0	0
$814$	−889.984	−1.09335
$815$	− 230.820i	− 0.283215i
$816$	0	0
$817$	1209.07	1.47989
$818$	134.681i	0.164646i
$819$	0	0
$820$	−41.8889	−0.0510840
$821$	− 1411.45i	− 1.71918i	−0.510984	−	0.859590i	$-0.670719\pi$
$821$	− 1411.45i	− 1.71918i	0.510984	−	0.859590i	$-0.329281\pi$
$822$	0	0
$823$	761.733	0.925557	0.462778	−	0.886474i	$-0.346853\pi$
$823$	761.733	0.925557	0.462778	+	0.886474i	$0.346853\pi$
$824$	798.230i	0.968726i
$825$	0	0
$826$	−48.6108	−0.0588509
$827$	− 718.144i	− 0.868373i	−0.900823	−	0.434186i	$-0.857036\pi$
$827$	− 718.144i	− 0.868373i	0.900823	−	0.434186i	$-0.142964\pi$
$828$	0	0
$829$	400.411	0.483005	0.241503	−	0.970400i	$-0.422360\pi$
$829$	400.411	0.483005	0.241503	+	0.970400i	$0.422360\pi$
$830$	− 351.571i	− 0.423580i
$831$	0	0
$832$	−782.320	−0.940288
$833$	90.5196i	0.108667i
$834$	0	0
$835$	252.272	0.302122
$836$	− 218.523i	− 0.261391i
$837$	0	0
$838$	306.105	0.365281
$839$	− 709.335i	− 0.845453i	−0.906257	−	0.422726i	$-0.861073\pi$
$839$	− 709.335i	− 0.845453i	0.906257	−	0.422726i	$-0.138927\pi$
$840$	0	0
$841$	−813.248	−0.967001
$842$	− 560.244i	− 0.665373i
$843$	0	0
$844$	202.218	0.239595
$845$	− 104.001i	− 0.123078i
$846$	0	0
$847$	−45.2139	−0.0533813
$848$	− 435.708i	− 0.513806i
$849$	0	0
$850$	−115.837	−0.136279
$851$	515.503i	0.605761i
$852$	0	0
$853$	886.227	1.03895	0.519476	−	0.854485i	$-0.326127\pi$
$853$	886.227	1.03895	0.519476	+	0.854485i	$0.326127\pi$
$854$	785.706i	0.920030i
$855$	0	0
$856$	−1177.88	−1.37602
$857$	327.514i	0.382163i	0.981574	+	0.191081i	$0.0611995\pi$
$857$	327.514i	0.382163i	−0.981574	+	0.191081i	$0.938801\pi$
$858$	0	0
$859$	−70.1670	−0.0816845	−0.0408423	−	0.999166i	$-0.513004\pi$
$859$	−70.1670	−0.0816845	−0.0408423	+	0.999166i	$0.513004\pi$
$860$	− 53.5628i	− 0.0622823i
$861$	0	0
$862$	405.342	0.470234
$863$	− 982.709i	− 1.13871i	−0.822091	−	0.569356i	$-0.807193\pi$
$863$	− 982.709i	− 1.13871i	0.822091	−	0.569356i	$-0.192807\pi$
$864$	0	0
$865$	−293.632	−0.339459
$866$	355.880i	0.410947i
$867$	0	0
$868$	131.323	0.151294
$869$	1065.38i	1.22599i
$870$	0	0
$871$	785.548	0.901892
$872$	1502.83i	1.72343i
$873$	0	0
$874$	668.539	0.764919
$875$	72.3140i	0.0826446i
$876$	0	0
$877$	926.895	1.05689	0.528446	−	0.848967i	$-0.322775\pi$
$877$	926.895	1.05689	0.528446	+	0.848967i	$0.322775\pi$
$878$	− 283.688i	− 0.323107i
$879$	0	0
$880$	311.518	0.353997
$881$	− 766.920i	− 0.870510i	−0.900307	−	0.435255i	$-0.856658\pi$
$881$	− 766.920i	− 0.870510i	0.900307	−	0.435255i	$-0.143342\pi$
$882$	0	0
$883$	1206.08	1.36589	0.682946	−	0.730469i	$-0.260699\pi$
$883$	1206.08	1.36589	0.682946	+	0.730469i	$0.260699\pi$
$884$	− 89.0275i	− 0.100710i
$885$	0	0
$886$	886.050	1.00006
$887$	− 752.566i	− 0.848440i	−0.905559	−	0.424220i	$-0.860548\pi$
$887$	− 752.566i	− 0.848440i	0.905559	−	0.424220i	$-0.139452\pi$
$888$	0	0
$889$	−595.184	−0.669498
$890$	260.961i	0.293214i
$891$	0	0
$892$	−78.9527	−0.0885120
$893$	1848.64i	2.07015i
$894$	0	0
$895$	−664.283	−0.742216
$896$	577.522i	0.644556i
$897$	0	0
$898$	1245.49	1.38696
$899$	− 1296.87i	− 1.44257i
$900$	0	0
$901$	421.858	0.468211
$902$	− 576.085i	− 0.638675i
$903$	0	0
$904$	−754.514	−0.834639
$905$	62.8711i	0.0694709i
$906$	0	0
$907$	−354.739	−0.391113	−0.195556	−	0.980692i	$-0.562651\pi$
$907$	−354.739	−0.391113	−0.195556	+	0.980692i	$0.562651\pi$
$908$	− 138.481i	− 0.152512i
$909$	0	0
$910$	293.549	0.322581
$911$	324.402i	0.356094i	0.984022	+	0.178047i	$0.0569780\pi$
$911$	324.402i	0.356094i	−0.984022	+	0.178047i	$0.943022\pi$
$912$	0	0
$913$	−915.420	−1.00265
$914$	− 147.655i	− 0.161549i
$915$	0	0
$916$	47.7282	0.0521050
$917$	− 1037.98i	− 1.13193i
$918$	0	0
$919$	−604.200	−0.657453	−0.328727	−	0.944425i	$-0.606619\pi$
$919$	−604.200	−0.657453	−0.328727	+	0.944425i	$0.606619\pi$
$920$	− 215.664i	− 0.234417i
$921$	0	0
$922$	−1134.72	−1.23071
$923$	− 145.120i	− 0.157227i
$924$	0	0
$925$	−227.249	−0.245675
$926$	604.196i	0.652479i
$927$	0	0
$928$	−410.211	−0.442037
$929$	− 1636.77i	− 1.76186i	−0.473246	−	0.880930i	$-0.656918\pi$
$929$	− 1636.77i	− 1.76186i	0.473246	−	0.880930i	$-0.343082\pi$
$930$	0	0
$931$	230.300	0.247369
$932$	− 90.9172i	− 0.0975506i
$933$	0	0
$934$	396.766	0.424803
$935$	301.615i	0.322583i
$936$	0	0
$937$	−266.212	−0.284111	−0.142055	−	0.989859i	$-0.545371\pi$
$937$	−266.212	−0.284111	−0.142055	+	0.989859i	$0.545371\pi$
$938$	− 841.918i	− 0.897567i
$939$	0	0
$940$	81.8963	0.0871238
$941$	− 162.090i	− 0.172253i	−0.996284	−	0.0861264i	$-0.972551\pi$
$941$	− 162.090i	− 0.172253i	0.996284	−	0.0861264i	$-0.0274489\pi$
$942$	0	0
$943$	−333.684	−0.353854
$944$	53.4704i	0.0566424i
$945$	0	0
$946$	736.633	0.778681
$947$	− 797.434i	− 0.842063i	−0.907046	−	0.421032i	$-0.861668\pi$
$947$	− 797.434i	− 0.842063i	0.907046	−	0.421032i	$-0.138332\pi$
$948$	0	0
$949$	1209.37	1.27437
$950$	294.713i	0.310224i
$951$	0	0
$952$	−694.799	−0.729831
$953$	1382.90i	1.45110i	0.688168	+	0.725551i	$0.258415\pi$
$953$	1382.90i	1.45110i	−0.688168	+	0.725551i	$0.741585\pi$
$954$	0	0
$955$	−20.5055	−0.0214718
$956$	− 127.468i	− 0.133334i
$957$	0	0
$958$	254.040	0.265177
$959$	587.706i	0.612833i
$960$	0	0
$961$	55.7017	0.0579622
$962$	922.488i	0.958927i
$963$	0	0
$964$	123.481	0.128092
$965$	211.903i	0.219589i
$966$	0	0
$967$	369.511	0.382121	0.191061	−	0.981578i	$-0.438807\pi$
$967$	369.511	0.382121	0.191061	+	0.981578i	$0.438807\pi$
$968$	59.4427i	0.0614077i
$969$	0	0
$970$	−354.358	−0.365318
$971$	961.450i	0.990165i	0.868846	+	0.495082i	$0.164862\pi$
$971$	961.450i	0.990165i	−0.868846	+	0.495082i	$0.835138\pi$
$972$	0	0
$973$	420.243	0.431904
$974$	170.607i	0.175161i
$975$	0	0
$976$	864.253	0.885505
$977$	1718.15i	1.75860i	0.476268	+	0.879300i	$0.341989\pi$
$977$	1718.15i	1.75860i	−0.476268	+	0.879300i	$0.658011\pi$
$978$	0	0
$979$	679.488	0.694063
$980$	− 10.2025i	− 0.0104107i
$981$	0	0
$982$	−430.232	−0.438118
$983$	− 1281.67i	− 1.30383i	−0.758292	−	0.651915i	$-0.773966\pi$
$983$	− 1281.67i	− 1.30383i	0.758292	−	0.651915i	$-0.226034\pi$
$984$	0	0
$985$	597.769	0.606872
$986$	942.274i	0.955653i
$987$	0	0
$988$	−226.504	−0.229255
$989$	− 426.678i	− 0.431423i
$990$	0	0
$991$	952.926	0.961581	0.480790	−	0.876836i	$-0.340350\pi$
$991$	952.926	0.961581	0.480790	+	0.876836i	$0.340350\pi$
$992$	− 321.591i	− 0.324184i
$993$	0	0
$994$	−155.534	−0.156473
$995$	− 354.031i	− 0.355810i
$996$	0	0
$997$	1769.67	1.77499	0.887495	−	0.460817i	$-0.152444\pi$
$997$	1769.67	1.77499	0.887495	+	0.460817i	$0.152444\pi$
$998$	− 1212.49i	− 1.21492i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.3.c.a.161.12		16
3.2	odd	2	inner	405.3.c.a.161.5		16
9.2	odd	6		135.3.i.a.71.3		16
9.4	even	3		135.3.i.a.116.3		16
9.5	odd	6		45.3.i.a.11.6	✓	16
9.7	even	3		45.3.i.a.41.6	yes	16
36.7	odd	6		720.3.bs.c.401.4		16
36.11	even	6		2160.3.bs.c.881.6		16
36.23	even	6		720.3.bs.c.641.4		16
36.31	odd	6		2160.3.bs.c.1601.6		16
45.2	even	12		675.3.i.c.449.5		32
45.4	even	6		675.3.j.b.251.6		16
45.7	odd	12		225.3.i.b.149.12		32
45.13	odd	12		675.3.i.c.224.5		32
45.14	odd	6		225.3.j.b.101.3		16
45.22	odd	12		675.3.i.c.224.12		32
45.23	even	12		225.3.i.b.74.12		32
45.29	odd	6		675.3.j.b.476.6		16
45.32	even	12		225.3.i.b.74.5		32
45.34	even	6		225.3.j.b.176.3		16
45.38	even	12		675.3.i.c.449.12		32
45.43	odd	12		225.3.i.b.149.5		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.3.i.a.11.6	✓	16	9.5	odd	6
45.3.i.a.41.6	yes	16	9.7	even	3
135.3.i.a.71.3		16	9.2	odd	6
135.3.i.a.116.3		16	9.4	even	3
225.3.i.b.74.5		32	45.32	even	12
225.3.i.b.74.12		32	45.23	even	12
225.3.i.b.149.5		32	45.43	odd	12
225.3.i.b.149.12		32	45.7	odd	12
225.3.j.b.101.3		16	45.14	odd	6
225.3.j.b.176.3		16	45.34	even	6
405.3.c.a.161.5		16	3.2	odd	2	inner
405.3.c.a.161.12		16	1.1	even	1	trivial
675.3.i.c.224.5		32	45.13	odd	12
675.3.i.c.224.12		32	45.22	odd	12
675.3.i.c.449.5		32	45.2	even	12
675.3.i.c.449.12		32	45.38	even	12
675.3.j.b.251.6		16	45.4	even	6
675.3.j.b.476.6		16	45.29	odd	6
720.3.bs.c.401.4		16	36.7	odd	6
720.3.bs.c.641.4		16	36.23	even	6
2160.3.bs.c.881.6		16	36.11	even	6
2160.3.bs.c.1601.6		16	36.31	odd	6

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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	405.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.83391i q^{2} +0.636762 q^{4} +2.23607i q^{5} -6.46796 q^{7} +8.50342i q^{8} -4.10076 q^{10} +10.6775i q^{11} +11.0675 q^{13} -11.8617i q^{14} -13.0475 q^{16} -12.6328i q^{17} -32.1403 q^{19} +1.42384i q^{20} -19.5817 q^{22} +11.3422i q^{23} -5.00000 q^{25} +20.2968i q^{26} -4.11855 q^{28} +40.6724i q^{29} -31.8858 q^{31} +10.0857i q^{32} +23.1674 q^{34} -14.4628i q^{35} +45.4499 q^{37} -58.9425i q^{38} -19.0142 q^{40} +29.4196i q^{41} -37.6185 q^{43} +6.79904i q^{44} -20.8007 q^{46} -57.5178i q^{47} -7.16547 q^{49} -9.16957i q^{50} +7.04735 q^{52} +33.3940i q^{53} -23.8757 q^{55} -54.9998i q^{56} -74.5897 q^{58} -4.09814i q^{59} -66.2390 q^{61} -58.4757i q^{62} -70.6863 q^{64} +24.7477i q^{65} +70.9780 q^{67} -8.04405i q^{68} +26.5235 q^{70} -13.1123i q^{71} +109.273 q^{73} +83.3511i q^{74} -20.4657 q^{76} -69.0618i q^{77} +99.7782 q^{79} -29.1751i q^{80} -53.9530 q^{82} +85.7333i q^{83} +28.2477 q^{85} -68.9891i q^{86} -90.7955 q^{88} -63.6372i q^{89} -71.5841 q^{91} +7.22230i q^{92} +105.483 q^{94} -71.8679i q^{95} +86.4129 q^{97} -13.1409i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{4} - 4 q^{7} + 20 q^{13} + 64 q^{16} - 52 q^{19} + 48 q^{22} - 80 q^{25} + 32 q^{28} - 64 q^{31} - 108 q^{34} + 44 q^{37} + 60 q^{40} + 248 q^{43} - 108 q^{46} + 108 q^{49} - 124 q^{52} - 180 q^{58}+ \cdots + 284 q^{97}+O(q^{100}) \) 16 * q - 32 * q^4 - 4 * q^7 + 20 * q^13 + 64 * q^16 - 52 * q^19 + 48 * q^22 - 80 * q^25 + 32 * q^28 - 64 * q^31 - 108 * q^34 + 44 * q^37 + 60 * q^40 + 248 * q^43 - 108 * q^46 + 108 * q^49 - 124 * q^52 - 180 * q^58 - 124 * q^61 + 256 * q^64 - 28 * q^67 + 120 * q^70 - 268 * q^73 + 212 * q^76 + 80 * q^79 - 204 * q^82 - 60 * q^85 - 288 * q^88 + 136 * q^91 + 300 * q^94 + 284 * q^97

Introduction

L-functions

Modular forms

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Database

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