LMFDB - Embedded newform 480.3.g.a.271.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [480,3,Mod(271,480)] 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("480.271"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(480, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 480 = 2^{5} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	480.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$13.0790526893$
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} - 4 x^{15} + x^{14} + 24 x^{13} - 44 x^{12} - 32 x^{11} + 180 x^{10} - 64 x^{9} - 352 x^{8} + \cdots + 65536 $ x^16 - 4x^15 + x^14 + 24x^13 - 44x^12 - 32x^11 + 180x^10 - 64x^9 - 352x^8 - 256x^7 + 2880x^6 - 2048x^5 - 11264x^4 + 24576x^3 + 4096x^2 - 65536x + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{32} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			271.8
Root			$1.13200 - 1.64881i$ of defining polynomial
Character	$\chi$	$=$	480.271
Dual form			480.3.g.a.271.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{3} +2.23607i q^{5} +11.6935i q^{7} +3.00000 q^{9} -19.3398 q^{11} -12.9092i q^{13} -3.87298i q^{15} +3.08243 q^{17} +10.1462 q^{19} -20.2537i q^{21} -12.5038i q^{23} -5.00000 q^{25} -5.19615 q^{27} -0.924932i q^{29} -32.5955i q^{31} +33.4975 q^{33} -26.1474 q^{35} +3.15307i q^{37} +22.3595i q^{39} -68.9305 q^{41} -69.7882 q^{43} +6.70820i q^{45} -46.2730i q^{47} -87.7376 q^{49} -5.33892 q^{51} -16.7503i q^{53} -43.2451i q^{55} -17.5737 q^{57} -72.0926 q^{59} +37.8730i q^{61} +35.0805i q^{63} +28.8660 q^{65} +121.224 q^{67} +21.6572i q^{69} -67.4107i q^{71} -14.7754 q^{73} +8.66025 q^{75} -226.149i q^{77} +132.433i q^{79} +9.00000 q^{81} +33.3714 q^{83} +6.89251i q^{85} +1.60203i q^{87} -60.3607 q^{89} +150.954 q^{91} +56.4571i q^{93} +22.6876i q^{95} -13.2188 q^{97} -58.0193 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 48 q^{9} - 64 q^{11} + 32 q^{19} - 80 q^{25} + 192 q^{43} - 80 q^{49} - 96 q^{51} + 96 q^{57} - 128 q^{59} + 64 q^{67} - 160 q^{73} + 144 q^{81} + 192 q^{91} - 224 q^{97} - 192 q^{99}+O(q^{100}) $ 16 * q + 48 * q^9 - 64 * q^11 + 32 * q^19 - 80 * q^25 + 192 * q^43 - 80 * q^49 - 96 * q^51 + 96 * q^57 - 128 * q^59 + 64 * q^67 - 160 * q^73 + 144 * q^81 + 192 * q^91 - 224 * q^97 - 192 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/480\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$421$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$3$	−1.73205	−0.577350
$3$	−1.73205	−0.577350
$4$	0	0
$5$	2.23607i	0.447214i
$5$	2.23607i	0.447214i
$6$	0	0
$7$	11.6935i	1.67050i	0.549872	+	0.835249i	$0.314676\pi$
$7$	11.6935i	1.67050i	−0.549872	+	0.835249i	$0.685324\pi$
$8$	0	0
$9$	3.00000	0.333333
$10$	0	0
$11$	−19.3398	−1.75816	−0.879081	−	0.476673i	$-0.841843\pi$
$11$	−19.3398	−1.75816	−0.879081	+	0.476673i	$0.841843\pi$
$12$	0	0
$13$	− 12.9092i	− 0.993019i	−0.868031	−	0.496510i	$-0.834615\pi$
$13$	− 12.9092i	− 0.993019i	0.868031	−	0.496510i	$-0.165385\pi$
$14$	0	0
$15$	− 3.87298i	− 0.258199i
$16$	0	0
$17$	3.08243	0.181319	0.0906596	−	0.995882i	$-0.471102\pi$
$17$	3.08243	0.181319	0.0906596	+	0.995882i	$0.471102\pi$
$18$	0	0
$19$	10.1462	0.534010	0.267005	−	0.963695i	$-0.413966\pi$
$19$	10.1462	0.534010	0.267005	+	0.963695i	$0.413966\pi$
$20$	0	0
$21$	− 20.2537i	− 0.964462i
$22$	0	0
$23$	− 12.5038i	− 0.543643i	−0.962348	−	0.271821i	$-0.912374\pi$
$23$	− 12.5038i	− 0.543643i	0.962348	−	0.271821i	$-0.0876260\pi$
$24$	0	0
$25$	−5.00000	−0.200000
$26$	0	0
$27$	−5.19615	−0.192450
$28$	0	0
$29$	− 0.924932i	− 0.0318942i	−0.999873	−	0.0159471i	$-0.994924\pi$
$29$	− 0.924932i	− 0.0318942i	0.999873	−	0.0159471i	$-0.00507633\pi$
$30$	0	0
$31$	− 32.5955i	− 1.05147i	−0.850649	−	0.525734i	$-0.823791\pi$
$31$	− 32.5955i	− 1.05147i	0.850649	−	0.525734i	$-0.176209\pi$
$32$	0	0
$33$	33.4975	1.01508
$34$	0	0
$35$	−26.1474	−0.747069
$36$	0	0
$37$	3.15307i	0.0852181i	0.999092	+	0.0426090i	$0.0135670\pi$
$37$	3.15307i	0.0852181i	−0.999092	+	0.0426090i	$0.986433\pi$
$38$	0	0
$39$	22.3595i	0.573320i
$40$	0	0
$41$	−68.9305	−1.68123	−0.840616	−	0.541631i	$-0.817807\pi$
$41$	−68.9305	−1.68123	−0.840616	+	0.541631i	$0.817807\pi$
$42$	0	0
$43$	−69.7882	−1.62298	−0.811491	−	0.584365i	$-0.801344\pi$
$43$	−69.7882	−1.62298	−0.811491	+	0.584365i	$0.801344\pi$
$44$	0	0
$45$	6.70820i	0.149071i
$46$	0	0
$47$	− 46.2730i	− 0.984531i	−0.870445	−	0.492265i	$-0.836169\pi$
$47$	− 46.2730i	− 0.984531i	0.870445	−	0.492265i	$-0.163831\pi$
$48$	0	0
$49$	−87.7376	−1.79056
$50$	0	0
$51$	−5.33892	−0.104685
$52$	0	0
$53$	− 16.7503i	− 0.316043i	−0.987436	−	0.158021i	$-0.949488\pi$
$53$	− 16.7503i	− 0.316043i	0.987436	−	0.158021i	$-0.0505115\pi$
$54$	0	0
$55$	− 43.2451i	− 0.786274i
$56$	0	0
$57$	−17.5737	−0.308311
$58$	0	0
$59$	−72.0926	−1.22191	−0.610954	−	0.791666i	$-0.709214\pi$
$59$	−72.0926	−1.22191	−0.610954	+	0.791666i	$0.709214\pi$
$60$	0	0
$61$	37.8730i	0.620869i	0.950595	+	0.310434i	$0.100474\pi$
$61$	37.8730i	0.620869i	−0.950595	+	0.310434i	$0.899526\pi$
$62$	0	0
$63$	35.0805i	0.556833i
$64$	0	0
$65$	28.8660	0.444092
$66$	0	0
$67$	121.224	1.80931	0.904654	−	0.426148i	$-0.140130\pi$
$67$	121.224	1.80931	0.904654	+	0.426148i	$0.140130\pi$
$68$	0	0
$69$	21.6572i	0.313872i
$70$	0	0
$71$	− 67.4107i	− 0.949447i	−0.880135	−	0.474724i	$-0.842548\pi$
$71$	− 67.4107i	− 0.949447i	0.880135	−	0.474724i	$-0.157452\pi$
$72$	0	0
$73$	−14.7754	−0.202402	−0.101201	−	0.994866i	$-0.532269\pi$
$73$	−14.7754	−0.202402	−0.101201	+	0.994866i	$0.532269\pi$
$74$	0	0
$75$	8.66025	0.115470
$76$	0	0
$77$	− 226.149i	− 2.93701i
$78$	0	0
$79$	132.433i	1.67637i	0.545390	+	0.838183i	$0.316382\pi$
$79$	132.433i	1.67637i	−0.545390	+	0.838183i	$0.683618\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	33.3714	0.402065	0.201033	−	0.979585i	$-0.435570\pi$
$83$	33.3714	0.402065	0.201033	+	0.979585i	$0.435570\pi$
$84$	0	0
$85$	6.89251i	0.0810884i
$86$	0	0
$87$	1.60203i	0.0184141i
$88$	0	0
$89$	−60.3607	−0.678210	−0.339105	−	0.940749i	$-0.610124\pi$
$89$	−60.3607	−0.678210	−0.339105	+	0.940749i	$0.610124\pi$
$90$	0	0
$91$	150.954	1.65884
$92$	0	0
$93$	56.4571i	0.607065i
$94$	0	0
$95$	22.6876i	0.238817i
$96$	0	0
$97$	−13.2188	−0.136277	−0.0681383	−	0.997676i	$-0.521706\pi$
$97$	−13.2188	−0.136277	−0.0681383	+	0.997676i	$0.521706\pi$
$98$	0	0
$99$	−58.0193	−0.586054
$100$	0	0
$101$	148.736i	1.47263i	0.676636	+	0.736317i	$0.263437\pi$
$101$	148.736i	1.47263i	−0.676636	+	0.736317i	$0.736563\pi$
$102$	0	0
$103$	− 133.780i	− 1.29883i	−0.760434	−	0.649416i	$-0.775014\pi$
$103$	− 133.780i	− 1.29883i	0.760434	−	0.649416i	$-0.224986\pi$
$104$	0	0
$105$	45.2887	0.431321
$106$	0	0
$107$	−55.2036	−0.515921	−0.257961	−	0.966155i	$-0.583051\pi$
$107$	−55.2036	−0.515921	−0.257961	+	0.966155i	$0.583051\pi$
$108$	0	0
$109$	77.6680i	0.712550i	0.934381	+	0.356275i	$0.115953\pi$
$109$	77.6680i	0.712550i	−0.934381	+	0.356275i	$0.884047\pi$
$110$	0	0
$111$	− 5.46128i	− 0.0492007i
$112$	0	0
$113$	−90.7924	−0.803472	−0.401736	−	0.915755i	$-0.631593\pi$
$113$	−90.7924	−0.803472	−0.401736	+	0.915755i	$0.631593\pi$
$114$	0	0
$115$	27.9593	0.243124
$116$	0	0
$117$	− 38.7277i	− 0.331006i
$118$	0	0
$119$	36.0443i	0.302893i
$120$	0	0
$121$	253.027	2.09113
$122$	0	0
$123$	119.391	0.970660
$124$	0	0
$125$	− 11.1803i	− 0.0894427i
$126$	0	0
$127$	45.7755i	0.360437i	0.983627	+	0.180219i	$0.0576805\pi$
$127$	45.7755i	0.360437i	−0.983627	+	0.180219i	$0.942319\pi$
$128$	0	0
$129$	120.877	0.937029
$130$	0	0
$131$	−186.279	−1.42197	−0.710987	−	0.703205i	$-0.751752\pi$
$131$	−186.279	−1.42197	−0.710987	+	0.703205i	$0.751752\pi$
$132$	0	0
$133$	118.644i	0.892063i
$134$	0	0
$135$	− 11.6190i	− 0.0860663i
$136$	0	0
$137$	32.9523	0.240528	0.120264	−	0.992742i	$-0.461626\pi$
$137$	32.9523	0.240528	0.120264	+	0.992742i	$0.461626\pi$
$138$	0	0
$139$	83.8571	0.603289	0.301644	−	0.953421i	$-0.402464\pi$
$139$	83.8571	0.603289	0.301644	+	0.953421i	$0.402464\pi$
$140$	0	0
$141$	80.1471i	0.568419i
$142$	0	0
$143$	249.662i	1.74589i
$144$	0	0
$145$	2.06821	0.0142635
$146$	0	0
$147$	151.966	1.03378
$148$	0	0
$149$	81.4421i	0.546591i	0.961930	+	0.273296i	$0.0881137\pi$
$149$	81.4421i	0.546591i	−0.961930	+	0.273296i	$0.911886\pi$
$150$	0	0
$151$	84.7219i	0.561072i	0.959843	+	0.280536i	$0.0905123\pi$
$151$	84.7219i	0.561072i	−0.959843	+	0.280536i	$0.909488\pi$
$152$	0	0
$153$	9.24728	0.0604397
$154$	0	0
$155$	72.8858	0.470231
$156$	0	0
$157$	− 167.846i	− 1.06908i	−0.845143	−	0.534540i	$-0.820485\pi$
$157$	− 167.846i	− 1.06908i	0.845143	−	0.534540i	$-0.179515\pi$
$158$	0	0
$159$	29.0123i	0.182467i
$160$	0	0
$161$	146.213	0.908154
$162$	0	0
$163$	−82.8713	−0.508413	−0.254206	−	0.967150i	$-0.581814\pi$
$163$	−82.8713	−0.508413	−0.254206	+	0.967150i	$0.581814\pi$
$164$	0	0
$165$	74.9026i	0.453955i
$166$	0	0
$167$	− 37.2973i	− 0.223337i	−0.993746	−	0.111669i	$-0.964380\pi$
$167$	− 37.2973i	− 0.223337i	0.993746	−	0.111669i	$-0.0356195\pi$
$168$	0	0
$169$	2.35134	0.0139132
$170$	0	0
$171$	30.4386	0.178003
$172$	0	0
$173$	− 211.828i	− 1.22444i	−0.790689	−	0.612218i	$-0.790277\pi$
$173$	− 211.828i	− 1.22444i	0.790689	−	0.612218i	$-0.209723\pi$
$174$	0	0
$175$	− 58.4674i	− 0.334100i
$176$	0	0
$177$	124.868	0.705469
$178$	0	0
$179$	−86.4715	−0.483081	−0.241540	−	0.970391i	$-0.577653\pi$
$179$	−86.4715	−0.483081	−0.241540	+	0.970391i	$0.577653\pi$
$180$	0	0
$181$	79.4036i	0.438694i	0.975647	+	0.219347i	$0.0703926\pi$
$181$	79.4036i	0.438694i	−0.975647	+	0.219347i	$0.929607\pi$
$182$	0	0
$183$	− 65.5979i	− 0.358459i
$184$	0	0
$185$	−7.05048	−0.0381107
$186$	0	0
$187$	−59.6134	−0.318788
$188$	0	0
$189$	− 60.7611i	− 0.321487i
$190$	0	0
$191$	− 290.341i	− 1.52011i	−0.649858	−	0.760056i	$-0.725172\pi$
$191$	− 290.341i	− 1.52011i	0.649858	−	0.760056i	$-0.274828\pi$
$192$	0	0
$193$	182.850	0.947408	0.473704	−	0.880684i	$-0.342917\pi$
$193$	182.850	0.947408	0.473704	+	0.880684i	$0.342917\pi$
$194$	0	0
$195$	−49.9973	−0.256396
$196$	0	0
$197$	98.1640i	0.498295i	0.968466	+	0.249147i	$0.0801503\pi$
$197$	98.1640i	0.498295i	−0.968466	+	0.249147i	$0.919850\pi$
$198$	0	0
$199$	225.355i	1.13244i	0.824255	+	0.566218i	$0.191594\pi$
$199$	225.355i	1.13244i	−0.824255	+	0.566218i	$0.808406\pi$
$200$	0	0
$201$	−209.965	−1.04460
$202$	0	0
$203$	10.8157	0.0532792
$204$	0	0
$205$	− 154.133i	− 0.751870i
$206$	0	0
$207$	− 37.5113i	− 0.181214i
$208$	0	0
$209$	−196.225	−0.938877
$210$	0	0
$211$	−302.658	−1.43440	−0.717199	−	0.696869i	$-0.754576\pi$
$211$	−302.658	−1.43440	−0.717199	+	0.696869i	$0.754576\pi$
$212$	0	0
$213$	116.759i	0.548164i
$214$	0	0
$215$	− 156.051i	− 0.725820i
$216$	0	0
$217$	381.155	1.75647
$218$	0	0
$219$	25.5917	0.116857
$220$	0	0
$221$	− 39.7918i	− 0.180053i
$222$	0	0
$223$	432.537i	1.93963i	0.243842	+	0.969815i	$0.421592\pi$
$223$	432.537i	1.93963i	−0.243842	+	0.969815i	$0.578408\pi$
$224$	0	0
$225$	−15.0000	−0.0666667
$226$	0	0
$227$	359.320	1.58291	0.791453	−	0.611230i	$-0.209325\pi$
$227$	359.320	1.58291	0.791453	+	0.611230i	$0.209325\pi$
$228$	0	0
$229$	212.212i	0.926688i	0.886179	+	0.463344i	$0.153351\pi$
$229$	212.212i	0.926688i	−0.886179	+	0.463344i	$0.846649\pi$
$230$	0	0
$231$	391.702i	1.69568i
$232$	0	0
$233$	−276.656	−1.18736	−0.593682	−	0.804699i	$-0.702326\pi$
$233$	−276.656	−1.18736	−0.593682	+	0.804699i	$0.702326\pi$
$234$	0	0
$235$	103.469	0.440296
$236$	0	0
$237$	− 229.380i	− 0.967850i
$238$	0	0
$239$	− 27.9106i	− 0.116781i	−0.998294	−	0.0583904i	$-0.981403\pi$
$239$	− 27.9106i	− 0.116781i	0.998294	−	0.0583904i	$-0.0185968\pi$
$240$	0	0
$241$	−115.420	−0.478920	−0.239460	−	0.970906i	$-0.576970\pi$
$241$	−115.420	−0.478920	−0.239460	+	0.970906i	$0.576970\pi$
$242$	0	0
$243$	−15.5885	−0.0641500
$244$	0	0
$245$	− 196.187i	− 0.800764i
$246$	0	0
$247$	− 130.980i	− 0.530282i
$248$	0	0
$249$	−57.8010	−0.232132
$250$	0	0
$251$	26.6696	0.106253	0.0531267	−	0.998588i	$-0.483081\pi$
$251$	26.6696	0.106253	0.0531267	+	0.998588i	$0.483081\pi$
$252$	0	0
$253$	241.820i	0.955812i
$254$	0	0
$255$	− 11.9382i	− 0.0468164i
$256$	0	0
$257$	170.155	0.662082	0.331041	−	0.943616i	$-0.392600\pi$
$257$	170.155	0.662082	0.331041	+	0.943616i	$0.392600\pi$
$258$	0	0
$259$	−36.8704	−0.142357
$260$	0	0
$261$	− 2.77479i	− 0.0106314i
$262$	0	0
$263$	105.658i	0.401741i	0.979618	+	0.200870i	$0.0643770\pi$
$263$	105.658i	0.401741i	−0.979618	+	0.200870i	$0.935623\pi$
$264$	0	0
$265$	37.4547	0.141339
$266$	0	0
$267$	104.548	0.391565
$268$	0	0
$269$	206.889i	0.769103i	0.923104	+	0.384551i	$0.125644\pi$
$269$	206.889i	0.769103i	−0.923104	+	0.384551i	$0.874356\pi$
$270$	0	0
$271$	− 247.759i	− 0.914241i	−0.889405	−	0.457120i	$-0.848881\pi$
$271$	− 247.759i	− 0.914241i	0.889405	−	0.457120i	$-0.151119\pi$
$272$	0	0
$273$	−261.460	−0.957729
$274$	0	0
$275$	96.6989	0.351632
$276$	0	0
$277$	− 26.9487i	− 0.0972879i	−0.998816	−	0.0486439i	$-0.984510\pi$
$277$	− 26.9487i	− 0.0972879i	0.998816	−	0.0486439i	$-0.0154900\pi$
$278$	0	0
$279$	− 97.7865i	− 0.350489i
$280$	0	0
$281$	−289.489	−1.03021	−0.515106	−	0.857127i	$-0.672247\pi$
$281$	−289.489	−1.03021	−0.515106	+	0.857127i	$0.672247\pi$
$282$	0	0
$283$	−3.01272	−0.0106457	−0.00532283	−	0.999986i	$-0.501694\pi$
$283$	−3.01272	−0.0106457	−0.00532283	+	0.999986i	$0.501694\pi$
$284$	0	0
$285$	− 39.2961i	− 0.137881i
$286$	0	0
$287$	− 806.038i	− 2.80849i
$288$	0	0
$289$	−279.499	−0.967123
$290$	0	0
$291$	22.8957	0.0786794
$292$	0	0
$293$	95.1952i	0.324898i	0.986717	+	0.162449i	$0.0519393\pi$
$293$	95.1952i	0.324898i	−0.986717	+	0.162449i	$0.948061\pi$
$294$	0	0
$295$	− 161.204i	− 0.546454i
$296$	0	0
$297$	100.492	0.338358
$298$	0	0
$299$	−161.414	−0.539848
$300$	0	0
$301$	− 816.068i	− 2.71119i
$302$	0	0
$303$	− 257.618i	− 0.850226i
$304$	0	0
$305$	−84.6866	−0.277661
$306$	0	0
$307$	−167.348	−0.545108	−0.272554	−	0.962140i	$-0.587868\pi$
$307$	−167.348	−0.545108	−0.272554	+	0.962140i	$0.587868\pi$
$308$	0	0
$309$	231.713i	0.749881i
$310$	0	0
$311$	299.478i	0.962951i	0.876460	+	0.481475i	$0.159899\pi$
$311$	299.478i	0.962951i	−0.876460	+	0.481475i	$0.840101\pi$
$312$	0	0
$313$	−123.623	−0.394963	−0.197481	−	0.980307i	$-0.563276\pi$
$313$	−123.623	−0.394963	−0.197481	+	0.980307i	$0.563276\pi$
$314$	0	0
$315$	−78.4423	−0.249023
$316$	0	0
$317$	135.949i	0.428862i	0.976739	+	0.214431i	$0.0687896\pi$
$317$	135.949i	0.428862i	−0.976739	+	0.214431i	$0.931210\pi$
$318$	0	0
$319$	17.8880i	0.0560752i
$320$	0	0
$321$	95.6154	0.297867
$322$	0	0
$323$	31.2749	0.0968263
$324$	0	0
$325$	64.5462i	0.198604i
$326$	0	0
$327$	− 134.525i	− 0.411391i
$328$	0	0
$329$	541.092	1.64466
$330$	0	0
$331$	41.1886	0.124437	0.0622185	−	0.998063i	$-0.480182\pi$
$331$	41.1886	0.124437	0.0622185	+	0.998063i	$0.480182\pi$
$332$	0	0
$333$	9.45921i	0.0284060i
$334$	0	0
$335$	271.064i	0.809147i
$336$	0	0
$337$	−565.599	−1.67834	−0.839168	−	0.543873i	$-0.816957\pi$
$337$	−565.599	−1.67834	−0.839168	+	0.543873i	$0.816957\pi$
$338$	0	0
$339$	157.257	0.463885
$340$	0	0
$341$	630.390i	1.84865i
$342$	0	0
$343$	− 452.977i	− 1.32063i
$344$	0	0
$345$	−48.4269	−0.140368
$346$	0	0
$347$	−134.668	−0.388092	−0.194046	−	0.980992i	$-0.562161\pi$
$347$	−134.668	−0.388092	−0.194046	+	0.980992i	$0.562161\pi$
$348$	0	0
$349$	− 461.309i	− 1.32180i	−0.750473	−	0.660901i	$-0.770174\pi$
$349$	− 461.309i	− 1.32180i	0.750473	−	0.660901i	$-0.229826\pi$
$350$	0	0
$351$	67.0784i	0.191107i
$352$	0	0
$353$	−429.790	−1.21754	−0.608768	−	0.793349i	$-0.708336\pi$
$353$	−429.790	−1.21754	−0.608768	+	0.793349i	$0.708336\pi$
$354$	0	0
$355$	150.735	0.424606
$356$	0	0
$357$	− 62.4305i	− 0.174875i
$358$	0	0
$359$	312.402i	0.870200i	0.900382	+	0.435100i	$0.143287\pi$
$359$	312.402i	0.870200i	−0.900382	+	0.435100i	$0.856713\pi$
$360$	0	0
$361$	−258.055	−0.714833
$362$	0	0
$363$	−438.256	−1.20732
$364$	0	0
$365$	− 33.0387i	− 0.0905170i
$366$	0	0
$367$	138.931i	0.378558i	0.981923	+	0.189279i	$0.0606151\pi$
$367$	138.931i	0.378558i	−0.981923	+	0.189279i	$0.939385\pi$
$368$	0	0
$369$	−206.792	−0.560411
$370$	0	0
$371$	195.869	0.527949
$372$	0	0
$373$	422.559i	1.13287i	0.824108	+	0.566433i	$0.191677\pi$
$373$	422.559i	1.13287i	−0.824108	+	0.566433i	$0.808323\pi$
$374$	0	0
$375$	19.3649i	0.0516398i
$376$	0	0
$377$	−11.9402	−0.0316715
$378$	0	0
$379$	−419.245	−1.10619	−0.553094	−	0.833119i	$-0.686553\pi$
$379$	−419.245	−1.10619	−0.553094	+	0.833119i	$0.686553\pi$
$380$	0	0
$381$	− 79.2856i	− 0.208099i
$382$	0	0
$383$	306.762i	0.800944i	0.916309	+	0.400472i	$0.131154\pi$
$383$	306.762i	0.800944i	−0.916309	+	0.400472i	$0.868846\pi$
$384$	0	0
$385$	505.685	1.31347
$386$	0	0
$387$	−209.365	−0.540994
$388$	0	0
$389$	− 441.866i	− 1.13590i	−0.823062	−	0.567951i	$-0.807736\pi$
$389$	− 441.866i	− 1.13590i	0.823062	−	0.567951i	$-0.192264\pi$
$390$	0	0
$391$	− 38.5420i	− 0.0985728i
$392$	0	0
$393$	322.644	0.820978
$394$	0	0
$395$	−296.129	−0.749693
$396$	0	0
$397$	160.391i	0.404007i	0.979385	+	0.202004i	$0.0647453\pi$
$397$	160.391i	0.404007i	−0.979385	+	0.202004i	$0.935255\pi$
$398$	0	0
$399$	− 205.498i	− 0.515033i
$400$	0	0
$401$	193.791	0.483270	0.241635	−	0.970367i	$-0.422316\pi$
$401$	193.791	0.483270	0.241635	+	0.970367i	$0.422316\pi$
$402$	0	0
$403$	−420.783	−1.04413
$404$	0	0
$405$	20.1246i	0.0496904i
$406$	0	0
$407$	− 60.9797i	− 0.149827i
$408$	0	0
$409$	−459.509	−1.12349	−0.561747	−	0.827309i	$-0.689870\pi$
$409$	−459.509	−1.12349	−0.561747	+	0.827309i	$0.689870\pi$
$410$	0	0
$411$	−57.0751	−0.138869
$412$	0	0
$413$	− 843.013i	− 2.04119i
$414$	0	0
$415$	74.6207i	0.179809i
$416$	0	0
$417$	−145.245	−0.348309
$418$	0	0
$419$	−223.442	−0.533274	−0.266637	−	0.963797i	$-0.585912\pi$
$419$	−223.442	−0.533274	−0.266637	+	0.963797i	$0.585912\pi$
$420$	0	0
$421$	− 753.492i	− 1.78977i	−0.446299	−	0.894884i	$-0.647258\pi$
$421$	− 753.492i	− 1.78977i	0.446299	−	0.894884i	$-0.352742\pi$
$422$	0	0
$423$	− 138.819i	− 0.328177i
$424$	0	0
$425$	−15.4121	−0.0362638
$426$	0	0
$427$	−442.867	−1.03716
$428$	0	0
$429$	− 432.427i	− 1.00799i
$430$	0	0
$431$	801.595i	1.85985i	0.367752	+	0.929924i	$0.380128\pi$
$431$	801.595i	1.85985i	−0.367752	+	0.929924i	$0.619872\pi$
$432$	0	0
$433$	725.058	1.67450	0.837249	−	0.546822i	$-0.184162\pi$
$433$	725.058	1.67450	0.837249	+	0.546822i	$0.184162\pi$
$434$	0	0
$435$	−3.58224	−0.00823505
$436$	0	0
$437$	− 126.866i	− 0.290311i
$438$	0	0
$439$	− 551.061i	− 1.25526i	−0.778510	−	0.627632i	$-0.784024\pi$
$439$	− 551.061i	− 1.25526i	0.778510	−	0.627632i	$-0.215976\pi$
$440$	0	0
$441$	−263.213	−0.596854
$442$	0	0
$443$	−36.7779	−0.0830201	−0.0415100	−	0.999138i	$-0.513217\pi$
$443$	−36.7779	−0.0830201	−0.0415100	+	0.999138i	$0.513217\pi$
$444$	0	0
$445$	− 134.971i	− 0.303305i
$446$	0	0
$447$	− 141.062i	− 0.315574i
$448$	0	0
$449$	332.641	0.740849	0.370424	−	0.928863i	$-0.379212\pi$
$449$	332.641	0.740849	0.370424	+	0.928863i	$0.379212\pi$
$450$	0	0
$451$	1333.10	2.95588
$452$	0	0
$453$	− 146.743i	− 0.323935i
$454$	0	0
$455$	337.544i	0.741854i
$456$	0	0
$457$	−343.279	−0.751158	−0.375579	−	0.926790i	$-0.622556\pi$
$457$	−343.279	−0.751158	−0.375579	+	0.926790i	$0.622556\pi$
$458$	0	0
$459$	−16.0168	−0.0348949
$460$	0	0
$461$	208.768i	0.452859i	0.974028	+	0.226429i	$0.0727053\pi$
$461$	208.768i	0.452859i	−0.974028	+	0.226429i	$0.927295\pi$
$462$	0	0
$463$	− 239.500i	− 0.517278i	−0.965974	−	0.258639i	$-0.916726\pi$
$463$	− 239.500i	− 0.517278i	0.965974	−	0.258639i	$-0.0832740\pi$
$464$	0	0
$465$	−126.242	−0.271488
$466$	0	0
$467$	295.094	0.631892	0.315946	−	0.948777i	$-0.397678\pi$
$467$	295.094	0.631892	0.315946	+	0.948777i	$0.397678\pi$
$468$	0	0
$469$	1417.53i	3.02244i
$470$	0	0
$471$	290.717i	0.617234i
$472$	0	0
$473$	1349.69	2.85346
$474$	0	0
$475$	−50.7310	−0.106802
$476$	0	0
$477$	− 50.2508i	− 0.105348i
$478$	0	0
$479$	− 533.361i	− 1.11349i	−0.830684	−	0.556745i	$-0.812050\pi$
$479$	− 533.361i	− 1.11349i	0.830684	−	0.556745i	$-0.187950\pi$
$480$	0	0
$481$	40.7037	0.0846232
$482$	0	0
$483$	−253.248	−0.524323
$484$	0	0
$485$	− 29.5582i	− 0.0609448i
$486$	0	0
$487$	285.675i	0.586601i	0.956020	+	0.293301i	$0.0947537\pi$
$487$	285.675i	0.586601i	−0.956020	+	0.293301i	$0.905246\pi$
$488$	0	0
$489$	143.537	0.293532
$490$	0	0
$491$	−188.244	−0.383388	−0.191694	−	0.981455i	$-0.561398\pi$
$491$	−188.244	−0.383388	−0.191694	+	0.981455i	$0.561398\pi$
$492$	0	0
$493$	− 2.85103i	− 0.00578303i
$494$	0	0
$495$	− 129.735i	− 0.262091i
$496$	0	0
$497$	788.266	1.58605
$498$	0	0
$499$	−625.534	−1.25357	−0.626787	−	0.779190i	$-0.715631\pi$
$499$	−625.534	−1.25357	−0.626787	+	0.779190i	$0.715631\pi$
$500$	0	0
$501$	64.6008i	0.128944i
$502$	0	0
$503$	232.863i	0.462949i	0.972841	+	0.231474i	$0.0743549\pi$
$503$	232.863i	0.462949i	−0.972841	+	0.231474i	$0.925645\pi$
$504$	0	0
$505$	−332.584	−0.658582
$506$	0	0
$507$	−4.07263	−0.00803280
$508$	0	0
$509$	615.343i	1.20893i	0.796633	+	0.604463i	$0.206612\pi$
$509$	615.343i	1.20893i	−0.796633	+	0.604463i	$0.793388\pi$
$510$	0	0
$511$	− 172.775i	− 0.338112i
$512$	0	0
$513$	−52.7212	−0.102770
$514$	0	0
$515$	299.140	0.580855
$516$	0	0
$517$	894.909i	1.73096i
$518$	0	0
$519$	366.896i	0.706929i
$520$	0	0
$521$	132.731	0.254761	0.127381	−	0.991854i	$-0.459343\pi$
$521$	132.731	0.254761	0.127381	+	0.991854i	$0.459343\pi$
$522$	0	0
$523$	911.886	1.74357	0.871784	−	0.489890i	$-0.162963\pi$
$523$	911.886	1.74357	0.871784	+	0.489890i	$0.162963\pi$
$524$	0	0
$525$	101.269i	0.192892i
$526$	0	0
$527$	− 100.473i	− 0.190651i
$528$	0	0
$529$	372.655	0.704453
$530$	0	0
$531$	−216.278	−0.407303
$532$	0	0
$533$	889.841i	1.66950i
$534$	0	0
$535$	− 123.439i	− 0.230727i
$536$	0	0
$537$	149.773	0.278907
$538$	0	0
$539$	1696.83	3.14810
$540$	0	0
$541$	− 373.215i	− 0.689861i	−0.938628	−	0.344931i	$-0.887902\pi$
$541$	− 373.215i	− 0.689861i	0.938628	−	0.344931i	$-0.112098\pi$
$542$	0	0
$543$	− 137.531i	− 0.253280i
$544$	0	0
$545$	−173.671	−0.318662
$546$	0	0
$547$	175.297	0.320469	0.160235	−	0.987079i	$-0.448775\pi$
$547$	175.297	0.320469	0.160235	+	0.987079i	$0.448775\pi$
$548$	0	0
$549$	113.619i	0.206956i
$550$	0	0
$551$	− 9.38454i	− 0.0170318i
$552$	0	0
$553$	−1548.60	−2.80036
$554$	0	0
$555$	12.2118	0.0220032
$556$	0	0
$557$	− 403.527i	− 0.724466i	−0.932088	−	0.362233i	$-0.882015\pi$
$557$	− 403.527i	− 0.724466i	0.932088	−	0.362233i	$-0.117985\pi$
$558$	0	0
$559$	900.913i	1.61165i
$560$	0	0
$561$	103.253	0.184053
$562$	0	0
$563$	236.345	0.419796	0.209898	−	0.977723i	$-0.432687\pi$
$563$	236.345	0.419796	0.209898	+	0.977723i	$0.432687\pi$
$564$	0	0
$565$	− 203.018i	− 0.359324i
$566$	0	0
$567$	105.241i	0.185611i
$568$	0	0
$569$	−757.837	−1.33187	−0.665937	−	0.746008i	$-0.731968\pi$
$569$	−757.837	−1.33187	−0.665937	+	0.746008i	$0.731968\pi$
$570$	0	0
$571$	−198.009	−0.346775	−0.173388	−	0.984854i	$-0.555471\pi$
$571$	−198.009	−0.346775	−0.173388	+	0.984854i	$0.555471\pi$
$572$	0	0
$573$	502.886i	0.877637i
$574$	0	0
$575$	62.5189i	0.108729i
$576$	0	0
$577$	−479.109	−0.830346	−0.415173	−	0.909743i	$-0.636279\pi$
$577$	−479.109	−0.830346	−0.415173	+	0.909743i	$0.636279\pi$
$578$	0	0
$579$	−316.705	−0.546986
$580$	0	0
$581$	390.228i	0.671649i
$582$	0	0
$583$	323.947i	0.555654i
$584$	0	0
$585$	86.5979	0.148031
$586$	0	0
$587$	658.065	1.12106	0.560532	−	0.828133i	$-0.310597\pi$
$587$	658.065	1.12106	0.560532	+	0.828133i	$0.310597\pi$
$588$	0	0
$589$	− 330.720i	− 0.561495i
$590$	0	0
$591$	− 170.025i	− 0.287691i
$592$	0	0
$593$	202.971	0.342279	0.171139	−	0.985247i	$-0.445255\pi$
$593$	202.971	0.342279	0.171139	+	0.985247i	$0.445255\pi$
$594$	0	0
$595$	−80.5975	−0.135458
$596$	0	0
$597$	− 390.326i	− 0.653813i
$598$	0	0
$599$	11.7767i	0.0196605i	0.999952	+	0.00983027i	$0.00312912\pi$
$599$	11.7767i	0.0196605i	−0.999952	+	0.00983027i	$0.996871\pi$
$600$	0	0
$601$	206.863	0.344198	0.172099	−	0.985080i	$-0.444945\pi$
$601$	206.863	0.344198	0.172099	+	0.985080i	$0.444945\pi$
$602$	0	0
$603$	363.671	0.603102
$604$	0	0
$605$	565.786i	0.935183i
$606$	0	0
$607$	885.649i	1.45906i	0.683950	+	0.729529i	$0.260261\pi$
$607$	885.649i	1.45906i	−0.683950	+	0.729529i	$0.739739\pi$
$608$	0	0
$609$	−18.7333	−0.0307607
$610$	0	0
$611$	−597.349	−0.977658
$612$	0	0
$613$	− 979.091i	− 1.59721i	−0.601855	−	0.798606i	$-0.705571\pi$
$613$	− 979.091i	− 1.59721i	0.601855	−	0.798606i	$-0.294429\pi$
$614$	0	0
$615$	266.967i	0.434092i
$616$	0	0
$617$	1064.97	1.72604	0.863019	−	0.505171i	$-0.168571\pi$
$617$	1064.97	1.72604	0.863019	+	0.505171i	$0.168571\pi$
$618$	0	0
$619$	658.875	1.06442	0.532209	−	0.846613i	$-0.321362\pi$
$619$	658.875	1.06442	0.532209	+	0.846613i	$0.321362\pi$
$620$	0	0
$621$	64.9716i	0.104624i
$622$	0	0
$623$	− 705.827i	− 1.13295i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	339.872	0.542061
$628$	0	0
$629$	9.71910i	0.0154517i
$630$	0	0
$631$	− 1074.30i	− 1.70253i	−0.524736	−	0.851265i	$-0.675836\pi$
$631$	− 1074.30i	− 1.70253i	0.524736	−	0.851265i	$-0.324164\pi$
$632$	0	0
$633$	524.219	0.828150
$634$	0	0
$635$	−102.357	−0.161192
$636$	0	0
$637$	1132.63i	1.77806i
$638$	0	0
$639$	− 202.232i	− 0.316482i
$640$	0	0
$641$	170.402	0.265838	0.132919	−	0.991127i	$-0.457565\pi$
$641$	170.402	0.265838	0.132919	+	0.991127i	$0.457565\pi$
$642$	0	0
$643$	387.322	0.602367	0.301183	−	0.953566i	$-0.402618\pi$
$643$	387.322	0.602367	0.301183	+	0.953566i	$0.402618\pi$
$644$	0	0
$645$	270.289i	0.419052i
$646$	0	0
$647$	1098.39i	1.69766i	0.528665	+	0.848830i	$0.322693\pi$
$647$	1098.39i	1.69766i	−0.528665	+	0.848830i	$0.677307\pi$
$648$	0	0
$649$	1394.25	2.14831
$650$	0	0
$651$	−660.180	−1.01410
$652$	0	0
$653$	− 801.196i	− 1.22695i	−0.789716	−	0.613473i	$-0.789772\pi$
$653$	− 801.196i	− 1.22695i	0.789716	−	0.613473i	$-0.210228\pi$
$654$	0	0
$655$	− 416.532i	− 0.635927i
$656$	0	0
$657$	−44.3261	−0.0674674
$658$	0	0
$659$	204.225	0.309901	0.154950	−	0.987922i	$-0.450478\pi$
$659$	204.225	0.309901	0.154950	+	0.987922i	$0.450478\pi$
$660$	0	0
$661$	− 283.821i	− 0.429382i	−0.976682	−	0.214691i	$-0.931126\pi$
$661$	− 283.821i	− 0.429382i	0.976682	−	0.214691i	$-0.0688744\pi$
$662$	0	0
$663$	68.9214i	0.103954i
$664$	0	0
$665$	−265.297	−0.398943
$666$	0	0
$667$	−11.5651	−0.0173390
$668$	0	0
$669$	− 749.177i	− 1.11985i
$670$	0	0
$671$	− 732.455i	− 1.09159i
$672$	0	0
$673$	−371.454	−0.551937	−0.275969	−	0.961167i	$-0.588999\pi$
$673$	−371.454	−0.551937	−0.275969	+	0.961167i	$0.588999\pi$
$674$	0	0
$675$	25.9808	0.0384900
$676$	0	0
$677$	1084.06i	1.60127i	0.599150	+	0.800637i	$0.295505\pi$
$677$	1084.06i	1.60127i	−0.599150	+	0.800637i	$0.704495\pi$
$678$	0	0
$679$	− 154.574i	− 0.227650i
$680$	0	0
$681$	−622.360	−0.913891
$682$	0	0
$683$	580.301	0.849636	0.424818	−	0.905279i	$-0.360338\pi$
$683$	580.301	0.849636	0.424818	+	0.905279i	$0.360338\pi$
$684$	0	0
$685$	73.6836i	0.107567i
$686$	0	0
$687$	− 367.561i	− 0.535023i
$688$	0	0
$689$	−216.233	−0.313837
$690$	0	0
$691$	438.911	0.635183	0.317592	−	0.948228i	$-0.397126\pi$
$691$	438.911	0.635183	0.317592	+	0.948228i	$0.397126\pi$
$692$	0	0
$693$	− 678.448i	− 0.979002i
$694$	0	0
$695$	187.510i	0.269799i
$696$	0	0
$697$	−212.473	−0.304840
$698$	0	0
$699$	479.182	0.685525
$700$	0	0
$701$	− 173.658i	− 0.247730i	−0.992299	−	0.123865i	$-0.960471\pi$
$701$	− 173.658i	− 0.247730i	0.992299	−	0.123865i	$-0.0395289\pi$
$702$	0	0
$703$	31.9917i	0.0455073i
$704$	0	0
$705$	−179.214	−0.254205
$706$	0	0
$707$	−1739.24	−2.46003
$708$	0	0
$709$	− 1231.35i	− 1.73674i	−0.495913	−	0.868372i	$-0.665166\pi$
$709$	− 1231.35i	− 1.73674i	0.495913	−	0.868372i	$-0.334834\pi$
$710$	0	0
$711$	397.299i	0.558788i
$712$	0	0
$713$	−407.567	−0.571623
$714$	0	0
$715$	−558.261	−0.780785
$716$	0	0
$717$	48.3426i	0.0674234i
$718$	0	0
$719$	− 327.782i	− 0.455885i	−0.973675	−	0.227943i	$-0.926800\pi$
$719$	− 327.782i	− 0.455885i	0.973675	−	0.227943i	$-0.0731999\pi$
$720$	0	0
$721$	1564.35	2.16970
$722$	0	0
$723$	199.913	0.276505
$724$	0	0
$725$	4.62466i	0.00637884i
$726$	0	0
$727$	− 217.872i	− 0.299686i	−0.988710	−	0.149843i	$-0.952123\pi$
$727$	− 217.872i	− 0.299686i	0.988710	−	0.149843i	$-0.0478768\pi$
$728$	0	0
$729$	27.0000	0.0370370
$730$	0	0
$731$	−215.117	−0.294278
$732$	0	0
$733$	− 154.099i	− 0.210231i	−0.994460	−	0.105115i	$-0.966479\pi$
$733$	− 154.099i	− 0.210231i	0.994460	−	0.105115i	$-0.0335212\pi$
$734$	0	0
$735$	339.806i	0.462321i
$736$	0	0
$737$	−2344.44	−3.18106
$738$	0	0
$739$	−497.686	−0.673459	−0.336729	−	0.941601i	$-0.609321\pi$
$739$	−497.686	−0.673459	−0.336729	+	0.941601i	$0.609321\pi$
$740$	0	0
$741$	226.864i	0.306159i
$742$	0	0
$743$	657.468i	0.884883i	0.896797	+	0.442441i	$0.145888\pi$
$743$	657.468i	0.884883i	−0.896797	+	0.442441i	$0.854112\pi$
$744$	0	0
$745$	−182.110	−0.244443
$746$	0	0
$747$	100.114	0.134022
$748$	0	0
$749$	− 645.522i	− 0.861846i
$750$	0	0
$751$	379.552i	0.505396i	0.967545	+	0.252698i	$0.0813179\pi$
$751$	379.552i	0.505396i	−0.967545	+	0.252698i	$0.918682\pi$
$752$	0	0
$753$	−46.1931	−0.0613455
$754$	0	0
$755$	−189.444	−0.250919
$756$	0	0
$757$	− 195.966i	− 0.258872i	−0.991588	−	0.129436i	$-0.958683\pi$
$757$	− 195.966i	− 0.258872i	0.991588	−	0.129436i	$-0.0413166\pi$
$758$	0	0
$759$	− 418.845i	− 0.551838i
$760$	0	0
$761$	−963.030	−1.26548	−0.632740	−	0.774364i	$-0.718070\pi$
$761$	−963.030	−1.26548	−0.632740	+	0.774364i	$0.718070\pi$
$762$	0	0
$763$	−908.209	−1.19031
$764$	0	0
$765$	20.6775i	0.0270295i
$766$	0	0
$767$	930.661i	1.21338i
$768$	0	0
$769$	81.3670	0.105809	0.0529044	−	0.998600i	$-0.483152\pi$
$769$	81.3670	0.105809	0.0529044	+	0.998600i	$0.483152\pi$
$770$	0	0
$771$	−294.717	−0.382253
$772$	0	0
$773$	781.463i	1.01095i	0.862842	+	0.505474i	$0.168682\pi$
$773$	781.463i	1.01095i	−0.862842	+	0.505474i	$0.831318\pi$
$774$	0	0
$775$	162.978i	0.210294i
$776$	0	0
$777$	63.8613	0.0821896
$778$	0	0
$779$	−699.383	−0.897795
$780$	0	0
$781$	1303.71i	1.66928i
$782$	0	0
$783$	4.80609i	0.00613804i
$784$	0	0
$785$	375.314	0.478107
$786$	0	0
$787$	−1030.41	−1.30929	−0.654644	−	0.755937i	$-0.727182\pi$
$787$	−1030.41	−1.30929	−0.654644	+	0.755937i	$0.727182\pi$
$788$	0	0
$789$	− 183.005i	− 0.231945i
$790$	0	0
$791$	− 1061.68i	− 1.34220i
$792$	0	0
$793$	488.912	0.616534
$794$	0	0
$795$	−64.8735	−0.0816019
$796$	0	0
$797$	85.4710i	0.107241i	0.998561	+	0.0536205i	$0.0170761\pi$
$797$	85.4710i	0.107241i	−0.998561	+	0.0536205i	$0.982924\pi$
$798$	0	0
$799$	− 142.633i	− 0.178514i
$800$	0	0
$801$	−181.082	−0.226070
$802$	0	0
$803$	285.752	0.355856
$804$	0	0
$805$	326.942i	0.406139i
$806$	0	0
$807$	− 358.342i	− 0.444042i
$808$	0	0
$809$	749.874	0.926915	0.463458	−	0.886119i	$-0.346609\pi$
$809$	749.874	0.926915	0.463458	+	0.886119i	$0.346609\pi$
$810$	0	0
$811$	454.606	0.560550	0.280275	−	0.959920i	$-0.409574\pi$
$811$	454.606	0.560550	0.280275	+	0.959920i	$0.409574\pi$
$812$	0	0
$813$	429.132i	0.527837i
$814$	0	0
$815$	− 185.306i	− 0.227369i
$816$	0	0
$817$	−708.085	−0.866689
$818$	0	0
$819$	452.862	0.552945
$820$	0	0
$821$	− 40.2949i	− 0.0490802i	−0.999699	−	0.0245401i	$-0.992188\pi$
$821$	− 40.2949i	− 0.0490802i	0.999699	−	0.0245401i	$-0.00781214\pi$
$822$	0	0
$823$	− 1109.53i	− 1.34815i	−0.738662	−	0.674076i	$-0.764542\pi$
$823$	− 1109.53i	− 1.34815i	0.738662	−	0.674076i	$-0.235458\pi$
$824$	0	0
$825$	−167.487	−0.203015
$826$	0	0
$827$	−750.168	−0.907096	−0.453548	−	0.891232i	$-0.649842\pi$
$827$	−750.168	−0.907096	−0.453548	+	0.891232i	$0.649842\pi$
$828$	0	0
$829$	− 182.949i	− 0.220687i	−0.993894	−	0.110343i	$-0.964805\pi$
$829$	− 182.949i	− 0.220687i	0.993894	−	0.110343i	$-0.0351950\pi$
$830$	0	0
$831$	46.6766i	0.0561692i
$832$	0	0
$833$	−270.445	−0.324663
$834$	0	0
$835$	83.3993	0.0998794
$836$	0	0
$837$	169.371i	0.202355i
$838$	0	0
$839$	− 719.345i	− 0.857384i	−0.903451	−	0.428692i	$-0.858975\pi$
$839$	− 719.345i	− 0.857384i	0.903451	−	0.428692i	$-0.141025\pi$
$840$	0	0
$841$	840.145	0.998983
$842$	0	0
$843$	501.410	0.594793
$844$	0	0
$845$	5.25774i	0.00622218i
$846$	0	0
$847$	2958.77i	3.49323i
$848$	0	0
$849$	5.21819	0.00614628
$850$	0	0
$851$	39.4253	0.0463282
$852$	0	0
$853$	− 593.155i	− 0.695375i	−0.937610	−	0.347688i	$-0.886967\pi$
$853$	− 593.155i	− 0.695375i	0.937610	−	0.347688i	$-0.113033\pi$
$854$	0	0
$855$	68.0628i	0.0796056i
$856$	0	0
$857$	−180.069	−0.210116	−0.105058	−	0.994466i	$-0.533503\pi$
$857$	−180.069	−0.210116	−0.105058	+	0.994466i	$0.533503\pi$
$858$	0	0
$859$	−509.587	−0.593232	−0.296616	−	0.954997i	$-0.595858\pi$
$859$	−509.587	−0.593232	−0.296616	+	0.954997i	$0.595858\pi$
$860$	0	0
$861$	1396.10i	1.62149i
$862$	0	0
$863$	− 1430.93i	− 1.65809i	−0.559181	−	0.829046i	$-0.688884\pi$
$863$	− 1430.93i	− 1.65809i	0.559181	−	0.829046i	$-0.311116\pi$
$864$	0	0
$865$	473.661	0.547585
$866$	0	0
$867$	484.106	0.558369
$868$	0	0
$869$	− 2561.22i	− 2.94732i
$870$	0	0
$871$	− 1564.91i	− 1.79668i
$872$	0	0
$873$	−39.6565	−0.0454256
$874$	0	0
$875$	130.737	0.149414
$876$	0	0
$877$	− 905.426i	− 1.03241i	−0.856464	−	0.516207i	$-0.827344\pi$
$877$	− 905.426i	− 1.03241i	0.856464	−	0.516207i	$-0.172656\pi$
$878$	0	0
$879$	− 164.883i	− 0.187580i
$880$	0	0
$881$	1479.51	1.67935	0.839677	−	0.543086i	$-0.182744\pi$
$881$	1479.51	1.67935	0.839677	+	0.543086i	$0.182744\pi$
$882$	0	0
$883$	−360.404	−0.408158	−0.204079	−	0.978954i	$-0.565420\pi$
$883$	−360.404	−0.408158	−0.204079	+	0.978954i	$0.565420\pi$
$884$	0	0
$885$	279.213i	0.315495i
$886$	0	0
$887$	− 1106.42i	− 1.24738i	−0.781674	−	0.623688i	$-0.785634\pi$
$887$	− 1106.42i	− 1.24738i	0.781674	−	0.623688i	$-0.214366\pi$
$888$	0	0
$889$	−535.276	−0.602110
$890$	0	0
$891$	−174.058	−0.195351
$892$	0	0
$893$	− 469.494i	− 0.525750i
$894$	0	0
$895$	− 193.356i	− 0.216040i
$896$	0	0
$897$	279.578	0.311681
$898$	0	0
$899$	−30.1486	−0.0335357
$900$	0	0
$901$	− 51.6315i	− 0.0573046i
$902$	0	0
$903$	1413.47i	1.56530i
$904$	0	0
$905$	−177.552	−0.196190
$906$	0	0
$907$	−1467.30	−1.61775	−0.808876	−	0.587979i	$-0.799924\pi$
$907$	−1467.30	−1.61775	−0.808876	+	0.587979i	$0.799924\pi$
$908$	0	0
$909$	446.208i	0.490878i
$910$	0	0
$911$	1188.75i	1.30488i	0.757840	+	0.652441i	$0.226255\pi$
$911$	1188.75i	1.30488i	−0.757840	+	0.652441i	$0.773745\pi$
$912$	0	0
$913$	−645.396	−0.706896
$914$	0	0
$915$	146.681	0.160308
$916$	0	0
$917$	− 2178.25i	− 2.37541i
$918$	0	0
$919$	− 917.844i	− 0.998743i	−0.866388	−	0.499371i	$-0.833564\pi$
$919$	− 917.844i	− 0.998743i	0.866388	−	0.499371i	$-0.166436\pi$
$920$	0	0
$921$	289.856	0.314718
$922$	0	0
$923$	−870.222	−0.942819
$924$	0	0
$925$	− 15.7653i	− 0.0170436i
$926$	0	0
$927$	− 401.339i	− 0.432944i
$928$	0	0
$929$	−326.140	−0.351066	−0.175533	−	0.984474i	$-0.556165\pi$
$929$	−326.140	−0.351066	−0.175533	+	0.984474i	$0.556165\pi$
$930$	0	0
$931$	−890.203	−0.956179
$932$	0	0
$933$	− 518.710i	− 0.555960i
$934$	0	0
$935$	− 133.300i	− 0.142566i
$936$	0	0
$937$	803.356	0.857370	0.428685	−	0.903454i	$-0.358977\pi$
$937$	803.356	0.857370	0.428685	+	0.903454i	$0.358977\pi$
$938$	0	0
$939$	214.122	0.228032
$940$	0	0
$941$	− 617.150i	− 0.655845i	−0.944705	−	0.327922i	$-0.893652\pi$
$941$	− 617.150i	− 0.655845i	0.944705	−	0.327922i	$-0.106348\pi$
$942$	0	0
$943$	861.892i	0.913990i
$944$	0	0
$945$	135.866	0.143774
$946$	0	0
$947$	1372.28	1.44908	0.724542	−	0.689231i	$-0.242052\pi$
$947$	1372.28	1.44908	0.724542	+	0.689231i	$0.242052\pi$
$948$	0	0
$949$	190.739i	0.200989i
$950$	0	0
$951$	− 235.471i	− 0.247603i
$952$	0	0
$953$	−313.823	−0.329300	−0.164650	−	0.986352i	$-0.552649\pi$
$953$	−313.823	−0.329300	−0.164650	+	0.986352i	$0.552649\pi$
$954$	0	0
$955$	649.223	0.679815
$956$	0	0
$957$	− 30.9829i	− 0.0323750i
$958$	0	0
$959$	385.327i	0.401801i
$960$	0	0
$961$	−101.467	−0.105585
$962$	0	0
$963$	−165.611	−0.171974
$964$	0	0
$965$	408.865i	0.423694i
$966$	0	0
$967$	191.090i	0.197611i	0.995107	+	0.0988055i	$0.0315022\pi$
$967$	191.090i	0.197611i	−0.995107	+	0.0988055i	$0.968498\pi$
$968$	0	0
$969$	−54.1697	−0.0559027
$970$	0	0
$971$	−88.4914	−0.0911343	−0.0455671	−	0.998961i	$-0.514509\pi$
$971$	−88.4914	−0.0911343	−0.0455671	+	0.998961i	$0.514509\pi$
$972$	0	0
$973$	980.582i	1.00779i
$974$	0	0
$975$	− 111.797i	− 0.114664i
$976$	0	0
$977$	−1367.80	−1.40000	−0.700001	−	0.714142i	$-0.746817\pi$
$977$	−1367.80	−1.40000	−0.700001	+	0.714142i	$0.746817\pi$
$978$	0	0
$979$	1167.36	1.19240
$980$	0	0
$981$	233.004i	0.237517i
$982$	0	0
$983$	− 266.437i	− 0.271045i	−0.990774	−	0.135522i	$-0.956729\pi$
$983$	− 266.437i	− 0.271045i	0.990774	−	0.135522i	$-0.0432713\pi$
$984$	0	0
$985$	−219.501	−0.222844
$986$	0	0
$987$	−937.199	−0.949543
$988$	0	0
$989$	872.617i	0.882322i
$990$	0	0
$991$	842.673i	0.850326i	0.905117	+	0.425163i	$0.139783\pi$
$991$	842.673i	0.850326i	−0.905117	+	0.425163i	$0.860217\pi$
$992$	0	0
$993$	−71.3408	−0.0718437
$994$	0	0
$995$	−503.909	−0.506441
$996$	0	0
$997$	1590.97i	1.59576i	0.602817	+	0.797879i	$0.294045\pi$
$997$	1590.97i	1.59576i	−0.602817	+	0.797879i	$0.705955\pi$
$998$	0	0
$999$	− 16.3838i	− 0.0164002i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	480.3.g.a.271.8		16
3.2	odd	2		1440.3.g.c.271.8		16
4.3	odd	2		120.3.g.a.91.7	✓	16
5.2	odd	4		2400.3.p.b.1999.22		32
5.3	odd	4		2400.3.p.b.1999.9		32
5.4	even	2		2400.3.g.b.751.9		16
8.3	odd	2	inner	480.3.g.a.271.1		16
8.5	even	2		120.3.g.a.91.8	yes	16
12.11	even	2		360.3.g.c.91.10		16
20.3	even	4		600.3.p.b.499.6		32
20.7	even	4		600.3.p.b.499.27		32
20.19	odd	2		600.3.g.d.451.10		16
24.5	odd	2		360.3.g.c.91.9		16
24.11	even	2		1440.3.g.c.271.9		16
40.3	even	4		2400.3.p.b.1999.21		32
40.13	odd	4		600.3.p.b.499.28		32
40.19	odd	2		2400.3.g.b.751.16		16
40.27	even	4		2400.3.p.b.1999.10		32
40.29	even	2		600.3.g.d.451.9		16
40.37	odd	4		600.3.p.b.499.5		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.3.g.a.91.7	✓	16	4.3	odd	2
120.3.g.a.91.8	yes	16	8.5	even	2
360.3.g.c.91.9		16	24.5	odd	2
360.3.g.c.91.10		16	12.11	even	2
480.3.g.a.271.1		16	8.3	odd	2	inner
480.3.g.a.271.8		16	1.1	even	1	trivial
600.3.g.d.451.9		16	40.29	even	2
600.3.g.d.451.10		16	20.19	odd	2
600.3.p.b.499.5		32	40.37	odd	4
600.3.p.b.499.6		32	20.3	even	4
600.3.p.b.499.27		32	20.7	even	4
600.3.p.b.499.28		32	40.13	odd	4
1440.3.g.c.271.8		16	3.2	odd	2
1440.3.g.c.271.9		16	24.11	even	2
2400.3.g.b.751.9		16	5.4	even	2
2400.3.g.b.751.16		16	40.19	odd	2
2400.3.p.b.1999.9		32	5.3	odd	4
2400.3.p.b.1999.10		32	40.27	even	4
2400.3.p.b.1999.21		32	40.3	even	4
2400.3.p.b.1999.22		32	5.2	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:	\( N \)	\(=\)	\( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	480.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} +2.23607i q^{5} +11.6935i q^{7} +3.00000 q^{9} -19.3398 q^{11} -12.9092i q^{13} -3.87298i q^{15} +3.08243 q^{17} +10.1462 q^{19} -20.2537i q^{21} -12.5038i q^{23} -5.00000 q^{25} -5.19615 q^{27} -0.924932i q^{29} -32.5955i q^{31} +33.4975 q^{33} -26.1474 q^{35} +3.15307i q^{37} +22.3595i q^{39} -68.9305 q^{41} -69.7882 q^{43} +6.70820i q^{45} -46.2730i q^{47} -87.7376 q^{49} -5.33892 q^{51} -16.7503i q^{53} -43.2451i q^{55} -17.5737 q^{57} -72.0926 q^{59} +37.8730i q^{61} +35.0805i q^{63} +28.8660 q^{65} +121.224 q^{67} +21.6572i q^{69} -67.4107i q^{71} -14.7754 q^{73} +8.66025 q^{75} -226.149i q^{77} +132.433i q^{79} +9.00000 q^{81} +33.3714 q^{83} +6.89251i q^{85} +1.60203i q^{87} -60.3607 q^{89} +150.954 q^{91} +56.4571i q^{93} +22.6876i q^{95} -13.2188 q^{97} -58.0193 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 48 q^{9} - 64 q^{11} + 32 q^{19} - 80 q^{25} + 192 q^{43} - 80 q^{49} - 96 q^{51} + 96 q^{57} - 128 q^{59} + 64 q^{67} - 160 q^{73} + 144 q^{81} + 192 q^{91} - 224 q^{97} - 192 q^{99}+O(q^{100}) \) 16 * q + 48 * q^9 - 64 * q^11 + 32 * q^19 - 80 * q^25 + 192 * q^43 - 80 * q^49 - 96 * q^51 + 96 * q^57 - 128 * q^59 + 64 * q^67 - 160 * q^73 + 144 * q^81 + 192 * q^91 - 224 * q^97 - 192 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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