LMFDB - Embedded newform 1024.3.d.k.511.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1024,3,Mod(511,1024)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1024.511");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1024 = 2^{10} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1024.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$27.9019790705$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.653473922154496.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4x^{10} + 13x^{8} - 28x^{6} + 52x^{4} - 64x^{2} + 64 $ x^12 - 4x^10 + 13x^8 - 28x^6 + 52x^4 - 64*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{30} $
Twist minimal:	no (minimal twist has level 16)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			511.5
Root			$1.35489 - 0.405301i$ of defining polynomial
Character	$\chi$	$=$	1024.511
Dual form			1024.3.d.k.511.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.206992 q^{3} -5.21257i q^{5} +9.66442i q^{7} -8.95715 q^{9} -7.80371 q^{11} -8.86897i q^{13} +1.07896i q^{15} +6.78623 q^{17} +19.1174 q^{19} -2.00046i q^{21} +17.0790i q^{23} -2.17092 q^{25} +3.71699 q^{27} -6.86851i q^{29} -5.25662i q^{31} +1.61531 q^{33} +50.3765 q^{35} +25.7183i q^{37} +1.83581i q^{39} +48.2302 q^{41} +77.0907 q^{43} +46.6898i q^{45} -40.4015i q^{47} -44.4011 q^{49} -1.40470 q^{51} +15.4144i q^{53} +40.6774i q^{55} -3.95715 q^{57} +71.9690 q^{59} +24.0624i q^{61} -86.5657i q^{63} -46.2302 q^{65} -32.4126 q^{67} -3.53521i q^{69} +51.6047i q^{71} -78.5032 q^{73} +0.449364 q^{75} -75.4184i q^{77} -108.512i q^{79} +79.8450 q^{81} +81.0589 q^{83} -35.3737i q^{85} +1.42173i q^{87} +44.1276 q^{89} +85.7135 q^{91} +1.08808i q^{93} -99.6510i q^{95} +112.700 q^{97} +69.8991 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{9} + 8 q^{17} + 20 q^{25} - 8 q^{33} + 92 q^{49} + 72 q^{57} + 24 q^{65} + 96 q^{73} - 172 q^{81} - 160 q^{89} - 8 q^{97}+O(q^{100}) $ 12 * q + 12 * q^9 + 8 * q^17 + 20 * q^25 - 8 * q^33 + 92 * q^49 + 72 * q^57 + 24 * q^65 + 96 * q^73 - 172 * q^81 - 160 * q^89 - 8 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times$.

$n$	$5$	$1023$
$\chi(n)$	$-1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	−0.206992	−0.0689974	−0.0344987	−	0.999405i	$-0.510983\pi$
$3$	−0.206992	−0.0689974	−0.0344987	+	0.999405i	$0.510983\pi$
$4$	0	0
$5$	− 5.21257i	− 1.04251i	−0.853400	−	0.521257i	$-0.825463\pi$
$5$	− 5.21257i	− 1.04251i	0.853400	−	0.521257i	$-0.174537\pi$
$6$	0	0
$7$	9.66442i	1.38063i	0.723508	+	0.690316i	$0.242528\pi$
$7$	9.66442i	1.38063i	−0.723508	+	0.690316i	$0.757472\pi$
$8$	0	0
$9$	−8.95715	−0.995239
$10$	0	0
$11$	−7.80371	−0.709428	−0.354714	−	0.934975i	$-0.615422\pi$
$11$	−7.80371	−0.709428	−0.354714	+	0.934975i	$0.615422\pi$
$12$	0	0
$13$	− 8.86897i	− 0.682228i	−0.940022	−	0.341114i	$-0.889196\pi$
$13$	− 8.86897i	− 0.682228i	0.940022	−	0.341114i	$-0.110804\pi$
$14$	0	0
$15$	1.07896i	0.0719308i
$16$	0	0
$17$	6.78623	0.399190	0.199595	−	0.979878i	$-0.436037\pi$
$17$	6.78623	0.399190	0.199595	+	0.979878i	$0.436037\pi$
$18$	0	0
$19$	19.1174	1.00618	0.503090	−	0.864234i	$-0.332196\pi$
$19$	19.1174	1.00618	0.503090	+	0.864234i	$0.332196\pi$
$20$	0	0
$21$	− 2.00046i	− 0.0952599i
$22$	0	0
$23$	17.0790i	0.742564i	0.928520	+	0.371282i	$0.121082\pi$
$23$	17.0790i	0.742564i	−0.928520	+	0.371282i	$0.878918\pi$
$24$	0	0
$25$	−2.17092	−0.0868370
$26$	0	0
$27$	3.71699	0.137666
$28$	0	0
$29$	− 6.86851i	− 0.236845i	−0.992963	−	0.118423i	$-0.962216\pi$
$29$	− 6.86851i	− 0.236845i	0.992963	−	0.118423i	$-0.0377837\pi$
$30$	0	0
$31$	− 5.25662i	− 0.169568i	−0.996399	−	0.0847841i	$-0.972980\pi$
$31$	− 5.25662i	− 0.169568i	0.996399	−	0.0847841i	$-0.0270201\pi$
$32$	0	0
$33$	1.61531	0.0489487
$34$	0	0
$35$	50.3765	1.43933
$36$	0	0
$37$	25.7183i	0.695090i	0.937663	+	0.347545i	$0.112985\pi$
$37$	25.7183i	0.695090i	−0.937663	+	0.347545i	$0.887015\pi$
$38$	0	0
$39$	1.83581i	0.0470720i
$40$	0	0
$41$	48.2302	1.17635	0.588173	−	0.808735i	$-0.299848\pi$
$41$	48.2302	1.17635	0.588173	+	0.808735i	$0.299848\pi$
$42$	0	0
$43$	77.0907	1.79281	0.896403	−	0.443240i	$-0.146171\pi$
$43$	77.0907	1.79281	0.896403	+	0.443240i	$0.146171\pi$
$44$	0	0
$45$	46.6898i	1.03755i
$46$	0	0
$47$	− 40.4015i	− 0.859607i	−0.902922	−	0.429804i	$-0.858583\pi$
$47$	− 40.4015i	− 0.859607i	0.902922	−	0.429804i	$-0.141417\pi$
$48$	0	0
$49$	−44.4011	−0.906144
$50$	0	0
$51$	−1.40470	−0.0275431
$52$	0	0
$53$	15.4144i	0.290837i	0.989370	+	0.145419i	$0.0464529\pi$
$53$	15.4144i	0.290837i	−0.989370	+	0.145419i	$0.953547\pi$
$54$	0	0
$55$	40.6774i	0.739590i
$56$	0	0
$57$	−3.95715	−0.0694238
$58$	0	0
$59$	71.9690	1.21981	0.609907	−	0.792473i	$-0.291207\pi$
$59$	71.9690	1.21981	0.609907	+	0.792473i	$0.291207\pi$
$60$	0	0
$61$	24.0624i	0.394466i	0.980357	+	0.197233i	$0.0631955\pi$
$61$	24.0624i	0.394466i	−0.980357	+	0.197233i	$0.936804\pi$
$62$	0	0
$63$	− 86.5657i	− 1.37406i
$64$	0	0
$65$	−46.2302	−0.711233
$66$	0	0
$67$	−32.4126	−0.483769	−0.241885	−	0.970305i	$-0.577766\pi$
$67$	−32.4126	−0.483769	−0.241885	+	0.970305i	$0.577766\pi$
$68$	0	0
$69$	− 3.53521i	− 0.0512349i
$70$	0	0
$71$	51.6047i	0.726827i	0.931628	+	0.363414i	$0.118389\pi$
$71$	51.6047i	0.726827i	−0.931628	+	0.363414i	$0.881611\pi$
$72$	0	0
$73$	−78.5032	−1.07539	−0.537693	−	0.843141i	$-0.680704\pi$
$73$	−78.5032	−1.07539	−0.537693	+	0.843141i	$0.680704\pi$
$74$	0	0
$75$	0.449364	0.00599152
$76$	0	0
$77$	− 75.4184i	− 0.979459i
$78$	0	0
$79$	− 108.512i	− 1.37357i	−0.726859	−	0.686787i	$-0.759021\pi$
$79$	− 108.512i	− 1.37357i	0.726859	−	0.686787i	$-0.240979\pi$
$80$	0	0
$81$	79.8450	0.985741
$82$	0	0
$83$	81.0589	0.976613	0.488307	−	0.872672i	$-0.337615\pi$
$83$	81.0589	0.976613	0.488307	+	0.872672i	$0.337615\pi$
$84$	0	0
$85$	− 35.3737i	− 0.416161i
$86$	0	0
$87$	1.42173i	0.0163417i
$88$	0	0
$89$	44.1276	0.495816	0.247908	−	0.968784i	$-0.420257\pi$
$89$	44.1276	0.495816	0.247908	+	0.968784i	$0.420257\pi$
$90$	0	0
$91$	85.7135	0.941906
$92$	0	0
$93$	1.08808i	0.0116998i
$94$	0	0
$95$	− 99.6510i	− 1.04896i
$96$	0	0
$97$	112.700	1.16185	0.580926	−	0.813956i	$-0.302691\pi$
$97$	112.700	1.16185	0.580926	+	0.813956i	$0.302691\pi$
$98$	0	0
$99$	69.8991	0.706051
$100$	0	0
$101$	− 137.724i	− 1.36361i	−0.731534	−	0.681804i	$-0.761195\pi$
$101$	− 137.724i	− 1.36361i	0.731534	−	0.681804i	$-0.238805\pi$
$102$	0	0
$103$	138.698i	1.34658i	0.739379	+	0.673290i	$0.235119\pi$
$103$	138.698i	1.34658i	−0.739379	+	0.673290i	$0.764881\pi$
$104$	0	0
$105$	−10.4275	−0.0993099
$106$	0	0
$107$	44.8851	0.419487	0.209743	−	0.977756i	$-0.432737\pi$
$107$	44.8851	0.419487	0.209743	+	0.977756i	$0.432737\pi$
$108$	0	0
$109$	1.00735i	0.00924179i	0.999989	+	0.00462089i	$0.00147088\pi$
$109$	1.00735i	0.00924179i	−0.999989	+	0.00462089i	$0.998529\pi$
$110$	0	0
$111$	− 5.32349i	− 0.0479594i
$112$	0	0
$113$	−14.8888	−0.131759	−0.0658795	−	0.997828i	$-0.520985\pi$
$113$	−14.8888	−0.131759	−0.0658795	+	0.997828i	$0.520985\pi$
$114$	0	0
$115$	89.0253	0.774133
$116$	0	0
$117$	79.4407i	0.678981i
$118$	0	0
$119$	65.5850i	0.551134i
$120$	0	0
$121$	−60.1021	−0.496711
$122$	0	0
$123$	−9.98326	−0.0811647
$124$	0	0
$125$	− 118.998i	− 0.951986i
$126$	0	0
$127$	106.861i	0.841425i	0.907194	+	0.420712i	$0.138220\pi$
$127$	106.861i	0.841425i	−0.907194	+	0.420712i	$0.861780\pi$
$128$	0	0
$129$	−15.9572	−0.123699
$130$	0	0
$131$	216.655	1.65386	0.826928	−	0.562307i	$-0.190086\pi$
$131$	216.655	1.65386	0.826928	+	0.562307i	$0.190086\pi$
$132$	0	0
$133$	184.759i	1.38916i
$134$	0	0
$135$	− 19.3751i	− 0.143519i
$136$	0	0
$137$	75.1700	0.548686	0.274343	−	0.961632i	$-0.411540\pi$
$137$	75.1700	0.548686	0.274343	+	0.961632i	$0.411540\pi$
$138$	0	0
$139$	151.922	1.09297	0.546483	−	0.837471i	$-0.315966\pi$
$139$	151.922	1.09297	0.546483	+	0.837471i	$0.315966\pi$
$140$	0	0
$141$	8.36280i	0.0593106i
$142$	0	0
$143$	69.2109i	0.483992i
$144$	0	0
$145$	−35.8026	−0.246915
$146$	0	0
$147$	9.19067	0.0625216
$148$	0	0
$149$	− 206.519i	− 1.38603i	−0.720921	−	0.693017i	$-0.756281\pi$
$149$	− 206.519i	− 1.38603i	0.720921	−	0.693017i	$-0.243719\pi$
$150$	0	0
$151$	220.513i	1.46035i	0.683260	+	0.730175i	$0.260561\pi$
$151$	220.513i	1.46035i	−0.683260	+	0.730175i	$0.739439\pi$
$152$	0	0
$153$	−60.7853	−0.397290
$154$	0	0
$155$	−27.4005	−0.176777
$156$	0	0
$157$	154.942i	0.986893i	0.869776	+	0.493447i	$0.164263\pi$
$157$	154.942i	0.986893i	−0.869776	+	0.493447i	$0.835737\pi$
$158$	0	0
$159$	− 3.19066i	− 0.0200670i
$160$	0	0
$161$	−165.058	−1.02521
$162$	0	0
$163$	−80.2963	−0.492615	−0.246308	−	0.969192i	$-0.579217\pi$
$163$	−80.2963	−0.492615	−0.246308	+	0.969192i	$0.579217\pi$
$164$	0	0
$165$	− 8.41990i	− 0.0510297i
$166$	0	0
$167$	− 106.677i	− 0.638781i	−0.947623	−	0.319391i	$-0.896522\pi$
$167$	− 106.677i	− 0.638781i	0.947623	−	0.319391i	$-0.103478\pi$
$168$	0	0
$169$	90.3414	0.534564
$170$	0	0
$171$	−171.238	−1.00139
$172$	0	0
$173$	252.240i	1.45803i	0.684496	+	0.729016i	$0.260022\pi$
$173$	252.240i	1.45803i	−0.684496	+	0.729016i	$0.739978\pi$
$174$	0	0
$175$	− 20.9807i	− 0.119890i
$176$	0	0
$177$	−14.8970	−0.0841639
$178$	0	0
$179$	85.5065	0.477690	0.238845	−	0.971058i	$-0.423231\pi$
$179$	85.5065	0.477690	0.238845	+	0.971058i	$0.423231\pi$
$180$	0	0
$181$	208.049i	1.14944i	0.818349	+	0.574721i	$0.194890\pi$
$181$	208.049i	1.14944i	−0.818349	+	0.574721i	$0.805110\pi$
$182$	0	0
$183$	− 4.98073i	− 0.0272171i
$184$	0	0
$185$	134.059	0.724642
$186$	0	0
$187$	−52.9578	−0.283197
$188$	0	0
$189$	35.9225i	0.190066i
$190$	0	0
$191$	− 106.861i	− 0.559481i	−0.960076	−	0.279741i	$-0.909752\pi$
$191$	− 106.861i	− 0.559481i	0.960076	−	0.279741i	$-0.0902485\pi$
$192$	0	0
$193$	68.1873	0.353302	0.176651	−	0.984274i	$-0.443474\pi$
$193$	68.1873	0.353302	0.176651	+	0.984274i	$0.443474\pi$
$194$	0	0
$195$	9.56927	0.0490732
$196$	0	0
$197$	87.4732i	0.444027i	0.975044	+	0.222013i	$0.0712628\pi$
$197$	87.4732i	0.444027i	−0.975044	+	0.222013i	$0.928737\pi$
$198$	0	0
$199$	158.466i	0.796310i	0.917318	+	0.398155i	$0.130349\pi$
$199$	158.466i	0.796310i	−0.917318	+	0.398155i	$0.869651\pi$
$200$	0	0
$201$	6.70914	0.0333788
$202$	0	0
$203$	66.3802	0.326996
$204$	0	0
$205$	− 251.403i	− 1.22636i
$206$	0	0
$207$	− 152.979i	− 0.739028i
$208$	0	0
$209$	−149.187	−0.713813
$210$	0	0
$211$	−278.644	−1.32059	−0.660295	−	0.751007i	$-0.729569\pi$
$211$	−278.644	−1.32059	−0.660295	+	0.751007i	$0.729569\pi$
$212$	0	0
$213$	− 10.6818i	− 0.0501492i
$214$	0	0
$215$	− 401.841i	− 1.86903i
$216$	0	0
$217$	50.8022	0.234111
$218$	0	0
$219$	16.2495	0.0741988
$220$	0	0
$221$	− 60.1869i	− 0.272339i
$222$	0	0
$223$	15.7698i	0.0707168i	0.999375	+	0.0353584i	$0.0112573\pi$
$223$	15.7698i	0.0707168i	−0.999375	+	0.0353584i	$0.988743\pi$
$224$	0	0
$225$	19.4453	0.0864236
$226$	0	0
$227$	−281.838	−1.24158	−0.620788	−	0.783978i	$-0.713187\pi$
$227$	−281.838	−1.24158	−0.620788	+	0.783978i	$0.713187\pi$
$228$	0	0
$229$	− 326.007i	− 1.42361i	−0.702377	−	0.711805i	$-0.747878\pi$
$229$	− 326.007i	− 1.42361i	0.702377	−	0.711805i	$-0.252122\pi$
$230$	0	0
$231$	15.6110i	0.0675801i
$232$	0	0
$233$	−344.791	−1.47979	−0.739895	−	0.672722i	$-0.765125\pi$
$233$	−344.791	−1.47979	−0.739895	+	0.672722i	$0.765125\pi$
$234$	0	0
$235$	−210.596	−0.896153
$236$	0	0
$237$	22.4612i	0.0947729i
$238$	0	0
$239$	77.1978i	0.323004i	0.986872	+	0.161502i	$0.0516337\pi$
$239$	77.1978i	0.323004i	−0.986872	+	0.161502i	$0.948366\pi$
$240$	0	0
$241$	293.483	1.21777	0.608885	−	0.793259i	$-0.291617\pi$
$241$	293.483	1.21777	0.608885	+	0.793259i	$0.291617\pi$
$242$	0	0
$243$	−49.9802	−0.205680
$244$	0	0
$245$	231.444i	0.944669i
$246$	0	0
$247$	− 169.552i	− 0.686445i
$248$	0	0
$249$	−16.7785	−0.0673837
$250$	0	0
$251$	112.617	0.448673	0.224337	−	0.974512i	$-0.427979\pi$
$251$	112.617	0.448673	0.224337	+	0.974512i	$0.427979\pi$
$252$	0	0
$253$	− 133.279i	− 0.526796i
$254$	0	0
$255$	7.32208i	0.0287140i
$256$	0	0
$257$	221.860	0.863270	0.431635	−	0.902048i	$-0.357937\pi$
$257$	221.860	0.863270	0.431635	+	0.902048i	$0.357937\pi$
$258$	0	0
$259$	−248.553	−0.959664
$260$	0	0
$261$	61.5223i	0.235718i
$262$	0	0
$263$	− 374.223i	− 1.42290i	−0.702736	−	0.711451i	$-0.748039\pi$
$263$	− 374.223i	− 1.42290i	0.702736	−	0.711451i	$-0.251961\pi$
$264$	0	0
$265$	80.3486	0.303202
$266$	0	0
$267$	−9.13407	−0.0342100
$268$	0	0
$269$	− 506.247i	− 1.88196i	−0.338465	−	0.940979i	$-0.609908\pi$
$269$	− 506.247i	− 1.88196i	0.338465	−	0.940979i	$-0.390092\pi$
$270$	0	0
$271$	359.030i	1.32484i	0.749135	+	0.662418i	$0.230470\pi$
$271$	359.030i	1.32484i	−0.749135	+	0.662418i	$0.769530\pi$
$272$	0	0
$273$	−17.7420	−0.0649890
$274$	0	0
$275$	16.9413	0.0616046
$276$	0	0
$277$	− 497.471i	− 1.79592i	−0.440074	−	0.897961i	$-0.645048\pi$
$277$	− 497.471i	− 1.79592i	0.440074	−	0.897961i	$-0.354952\pi$
$278$	0	0
$279$	47.0843i	0.168761i
$280$	0	0
$281$	−191.390	−0.681103	−0.340552	−	0.940226i	$-0.610614\pi$
$281$	−191.390	−0.681103	−0.340552	+	0.940226i	$0.610614\pi$
$282$	0	0
$283$	−44.2818	−0.156473	−0.0782363	−	0.996935i	$-0.524929\pi$
$283$	−44.2818	−0.156473	−0.0782363	+	0.996935i	$0.524929\pi$
$284$	0	0
$285$	20.6270i	0.0723753i
$286$	0	0
$287$	466.117i	1.62410i
$288$	0	0
$289$	−242.947	−0.840647
$290$	0	0
$291$	−23.3279	−0.0801647
$292$	0	0
$293$	130.233i	0.444483i	0.974992	+	0.222241i	$0.0713373\pi$
$293$	130.233i	0.444483i	−0.974992	+	0.222241i	$0.928663\pi$
$294$	0	0
$295$	− 375.144i	− 1.27167i
$296$	0	0
$297$	−29.0063	−0.0976643
$298$	0	0
$299$	151.473	0.506598
$300$	0	0
$301$	745.037i	2.47521i
$302$	0	0
$303$	28.5079i	0.0940854i
$304$	0	0
$305$	125.427	0.411236
$306$	0	0
$307$	−364.254	−1.18649	−0.593247	−	0.805020i	$-0.702154\pi$
$307$	−364.254	−1.18649	−0.593247	+	0.805020i	$0.702154\pi$
$308$	0	0
$309$	− 28.7093i	− 0.0929104i
$310$	0	0
$311$	− 130.914i	− 0.420946i	−0.977600	−	0.210473i	$-0.932500\pi$
$311$	− 130.914i	− 0.420946i	0.977600	−	0.210473i	$-0.0675004\pi$
$312$	0	0
$313$	−51.8354	−0.165608	−0.0828041	−	0.996566i	$-0.526388\pi$
$313$	−51.8354	−0.165608	−0.0828041	+	0.996566i	$0.526388\pi$
$314$	0	0
$315$	−451.230	−1.43248
$316$	0	0
$317$	155.049i	0.489114i	0.969635	+	0.244557i	$0.0786425\pi$
$317$	155.049i	0.489114i	−0.969635	+	0.244557i	$0.921357\pi$
$318$	0	0
$319$	53.5999i	0.168025i
$320$	0	0
$321$	−9.29086	−0.0289435
$322$	0	0
$323$	129.735	0.401657
$324$	0	0
$325$	19.2539i	0.0592426i
$326$	0	0
$327$	− 0.208514i	0 0.000637659i
$328$	0	0
$329$	390.458	1.18680
$330$	0	0
$331$	−457.111	−1.38100	−0.690500	−	0.723332i	$-0.742610\pi$
$331$	−457.111	−1.38100	−0.690500	+	0.723332i	$0.742610\pi$
$332$	0	0
$333$	− 230.363i	− 0.691781i
$334$	0	0
$335$	168.953i	0.504337i
$336$	0	0
$337$	−315.159	−0.935191	−0.467596	−	0.883943i	$-0.654880\pi$
$337$	−315.159	−0.935191	−0.467596	+	0.883943i	$0.654880\pi$
$338$	0	0
$339$	3.08186	0.00909102
$340$	0	0
$341$	41.0211i	0.120297i
$342$	0	0
$343$	44.4459i	0.129580i
$344$	0	0
$345$	−18.4275	−0.0534132
$346$	0	0
$347$	−434.967	−1.25351	−0.626753	−	0.779218i	$-0.715616\pi$
$347$	−434.967	−1.25351	−0.626753	+	0.779218i	$0.715616\pi$
$348$	0	0
$349$	− 241.068i	− 0.690740i	−0.938467	−	0.345370i	$-0.887753\pi$
$349$	− 241.068i	− 0.690740i	0.938467	−	0.345370i	$-0.112247\pi$
$350$	0	0
$351$	− 32.9659i	− 0.0939198i
$352$	0	0
$353$	238.136	0.674606	0.337303	−	0.941396i	$-0.390485\pi$
$353$	238.136	0.674606	0.337303	+	0.941396i	$0.390485\pi$
$354$	0	0
$355$	268.993	0.757728
$356$	0	0
$357$	− 13.5756i	− 0.0380268i
$358$	0	0
$359$	− 33.6470i	− 0.0937241i	−0.998901	−	0.0468620i	$-0.985078\pi$
$359$	− 33.6470i	− 0.0937241i	0.998901	−	0.0468620i	$-0.0149221\pi$
$360$	0	0
$361$	4.47577	0.0123983
$362$	0	0
$363$	12.4407	0.0342718
$364$	0	0
$365$	409.203i	1.12111i
$366$	0	0
$367$	240.758i	0.656016i	0.944675	+	0.328008i	$0.106377\pi$
$367$	240.758i	0.656016i	−0.944675	+	0.328008i	$0.893623\pi$
$368$	0	0
$369$	−432.005	−1.17075
$370$	0	0
$371$	−148.971	−0.401539
$372$	0	0
$373$	611.653i	1.63982i	0.572492	+	0.819910i	$0.305977\pi$
$373$	611.653i	1.63982i	−0.572492	+	0.819910i	$0.694023\pi$
$374$	0	0
$375$	24.6317i	0.0656845i
$376$	0	0
$377$	−60.9166	−0.161583
$378$	0	0
$379$	−247.086	−0.651943	−0.325971	−	0.945380i	$-0.605691\pi$
$379$	−247.086	−0.651943	−0.325971	+	0.945380i	$0.605691\pi$
$380$	0	0
$381$	− 22.1194i	− 0.0580561i
$382$	0	0
$383$	673.381i	1.75817i	0.476661	+	0.879087i	$0.341847\pi$
$383$	673.381i	1.75817i	−0.476661	+	0.879087i	$0.658153\pi$
$384$	0	0
$385$	−393.124	−1.02110
$386$	0	0
$387$	−690.513	−1.78427
$388$	0	0
$389$	388.408i	0.998478i	0.866464	+	0.499239i	$0.166387\pi$
$389$	388.408i	0.998478i	−0.866464	+	0.499239i	$0.833613\pi$
$390$	0	0
$391$	115.902i	0.296424i
$392$	0	0
$393$	−44.8459	−0.114112
$394$	0	0
$395$	−565.628	−1.43197
$396$	0	0
$397$	− 383.611i	− 0.966275i	−0.875545	−	0.483137i	$-0.839497\pi$
$397$	− 383.611i	− 0.966275i	0.875545	−	0.483137i	$-0.160503\pi$
$398$	0	0
$399$	− 38.2436i	− 0.0958487i
$400$	0	0
$401$	415.193	1.03539	0.517697	−	0.855564i	$-0.326790\pi$
$401$	415.193	1.03539	0.517697	+	0.855564i	$0.326790\pi$
$402$	0	0
$403$	−46.6208	−0.115684
$404$	0	0
$405$	− 416.198i	− 1.02765i
$406$	0	0
$407$	− 200.699i	− 0.493117i
$408$	0	0
$409$	634.686	1.55180	0.775900	−	0.630856i	$-0.217296\pi$
$409$	634.686	1.55180	0.775900	+	0.630856i	$0.217296\pi$
$410$	0	0
$411$	−15.5596	−0.0378579
$412$	0	0
$413$	695.539i	1.68411i
$414$	0	0
$415$	− 422.525i	− 1.01813i
$416$	0	0
$417$	−31.4467	−0.0754117
$418$	0	0
$419$	−27.2500	−0.0650358	−0.0325179	−	0.999471i	$-0.510353\pi$
$419$	−27.2500	−0.0650358	−0.0325179	+	0.999471i	$0.510353\pi$
$420$	0	0
$421$	− 345.783i	− 0.821337i	−0.911785	−	0.410668i	$-0.865295\pi$
$421$	− 345.783i	− 0.821337i	0.911785	−	0.410668i	$-0.134705\pi$
$422$	0	0
$423$	361.883i	0.855515i
$424$	0	0
$425$	−14.7324	−0.0346644
$426$	0	0
$427$	−232.549	−0.544612
$428$	0	0
$429$	− 14.3261i	− 0.0333942i
$430$	0	0
$431$	337.331i	0.782670i	0.920248	+	0.391335i	$0.127987\pi$
$431$	337.331i	0.782670i	−0.920248	+	0.391335i	$0.872013\pi$
$432$	0	0
$433$	424.560	0.980508	0.490254	−	0.871580i	$-0.336904\pi$
$433$	424.560	0.980508	0.490254	+	0.871580i	$0.336904\pi$
$434$	0	0
$435$	7.41086	0.0170365
$436$	0	0
$437$	326.506i	0.747153i
$438$	0	0
$439$	− 162.004i	− 0.369029i	−0.982830	−	0.184514i	$-0.940929\pi$
$439$	− 162.004i	− 0.369029i	0.982830	−	0.184514i	$-0.0590712\pi$
$440$	0	0
$441$	397.707	0.901831
$442$	0	0
$443$	696.061	1.57124	0.785622	−	0.618707i	$-0.212343\pi$
$443$	696.061	1.57124	0.785622	+	0.618707i	$0.212343\pi$
$444$	0	0
$445$	− 230.018i	− 0.516895i
$446$	0	0
$447$	42.7478i	0.0956327i
$448$	0	0
$449$	195.434	0.435266	0.217633	−	0.976031i	$-0.430166\pi$
$449$	195.434	0.435266	0.217633	+	0.976031i	$0.430166\pi$
$450$	0	0
$451$	−376.374	−0.834533
$452$	0	0
$453$	− 45.6444i	− 0.100760i
$454$	0	0
$455$	− 446.788i	− 0.981951i
$456$	0	0
$457$	−386.874	−0.846552	−0.423276	−	0.906001i	$-0.639120\pi$
$457$	−386.874	−0.846552	−0.423276	+	0.906001i	$0.639120\pi$
$458$	0	0
$459$	25.2243	0.0549550
$460$	0	0
$461$	− 246.640i	− 0.535011i	−0.963556	−	0.267505i	$-0.913801\pi$
$461$	− 246.640i	− 0.535011i	0.963556	−	0.267505i	$-0.0861993\pi$
$462$	0	0
$463$	− 60.5295i	− 0.130733i	−0.997861	−	0.0653666i	$-0.979178\pi$
$463$	− 60.5295i	− 0.130733i	0.997861	−	0.0653666i	$-0.0208217\pi$
$464$	0	0
$465$	5.67168	0.0121972
$466$	0	0
$467$	433.431	0.928118	0.464059	−	0.885804i	$-0.346393\pi$
$467$	433.431	0.928118	0.464059	+	0.885804i	$0.346393\pi$
$468$	0	0
$469$	− 313.249i	− 0.667908i
$470$	0	0
$471$	− 32.0718i	− 0.0680930i
$472$	0	0
$473$	−601.593	−1.27187
$474$	0	0
$475$	−41.5025	−0.0873736
$476$	0	0
$477$	− 138.069i	− 0.289453i
$478$	0	0
$479$	376.452i	0.785912i	0.919557	+	0.392956i	$0.128547\pi$
$479$	376.452i	0.785912i	−0.919557	+	0.392956i	$0.871453\pi$
$480$	0	0
$481$	228.095	0.474210
$482$	0	0
$483$	34.1658	0.0707366
$484$	0	0
$485$	− 587.455i	− 1.21125i
$486$	0	0
$487$	− 77.2033i	− 0.158528i	−0.996854	−	0.0792641i	$-0.974743\pi$
$487$	− 77.2033i	− 0.158528i	0.996854	−	0.0792641i	$-0.0252571\pi$
$488$	0	0
$489$	16.6207	0.0339892
$490$	0	0
$491$	822.277	1.67470	0.837350	−	0.546668i	$-0.184104\pi$
$491$	822.277	1.67470	0.837350	+	0.546668i	$0.184104\pi$
$492$	0	0
$493$	− 46.6113i	− 0.0945462i
$494$	0	0
$495$	− 364.354i	− 0.736069i
$496$	0	0
$497$	−498.730	−1.00348
$498$	0	0
$499$	−246.081	−0.493149	−0.246574	−	0.969124i	$-0.579305\pi$
$499$	−246.081	−0.493149	−0.246574	+	0.969124i	$0.579305\pi$
$500$	0	0
$501$	22.0812i	0.0440742i
$502$	0	0
$503$	− 355.262i	− 0.706286i	−0.935569	−	0.353143i	$-0.885113\pi$
$503$	− 355.262i	− 0.706286i	0.935569	−	0.353143i	$-0.114887\pi$
$504$	0	0
$505$	−717.899	−1.42158
$506$	0	0
$507$	−18.7000	−0.0368835
$508$	0	0
$509$	− 395.509i	− 0.777032i	−0.921442	−	0.388516i	$-0.872988\pi$
$509$	− 395.509i	− 0.777032i	0.921442	−	0.388516i	$-0.127012\pi$
$510$	0	0
$511$	− 758.688i	− 1.48471i
$512$	0	0
$513$	71.0592	0.138517
$514$	0	0
$515$	722.972	1.40383
$516$	0	0
$517$	315.282i	0.609830i
$518$	0	0
$519$	− 52.2116i	− 0.100600i
$520$	0	0
$521$	−705.745	−1.35460	−0.677299	−	0.735708i	$-0.736849\pi$
$521$	−705.745	−1.35460	−0.677299	+	0.735708i	$0.736849\pi$
$522$	0	0
$523$	264.122	0.505013	0.252506	−	0.967595i	$-0.418745\pi$
$523$	264.122	0.505013	0.252506	+	0.967595i	$0.418745\pi$
$524$	0	0
$525$	4.34284i	0.00827208i
$526$	0	0
$527$	− 35.6726i	− 0.0676899i
$528$	0	0
$529$	237.309	0.448599
$530$	0	0
$531$	−644.637	−1.21401
$532$	0	0
$533$	− 427.752i	− 0.802536i
$534$	0	0
$535$	− 233.967i	− 0.437321i
$536$	0	0
$537$	−17.6992	−0.0329593
$538$	0	0
$539$	346.493	0.642845
$540$	0	0
$541$	− 168.679i	− 0.311792i	−0.987774	−	0.155896i	$-0.950174\pi$
$541$	− 168.679i	− 0.311792i	0.987774	−	0.155896i	$-0.0498264\pi$
$542$	0	0
$543$	− 43.0645i	− 0.0793085i
$544$	0	0
$545$	5.25091	0.00963470
$546$	0	0
$547$	200.072	0.365762	0.182881	−	0.983135i	$-0.441458\pi$
$547$	200.072	0.365762	0.182881	+	0.983135i	$0.441458\pi$
$548$	0	0
$549$	− 215.531i	− 0.392588i
$550$	0	0
$551$	− 131.308i	− 0.238309i
$552$	0	0
$553$	1048.71	1.89640
$554$	0	0
$555$	−27.7491	−0.0499984
$556$	0	0
$557$	531.576i	0.954355i	0.878807	+	0.477178i	$0.158340\pi$
$557$	531.576i	0.954355i	−0.878807	+	0.477178i	$0.841660\pi$
$558$	0	0
$559$	− 683.715i	− 1.22310i
$560$	0	0
$561$	10.9618	0.0195398
$562$	0	0
$563$	431.828	0.767012	0.383506	−	0.923538i	$-0.374716\pi$
$563$	431.828	0.767012	0.383506	+	0.923538i	$0.374716\pi$
$564$	0	0
$565$	77.6088i	0.137361i
$566$	0	0
$567$	771.656i	1.36095i
$568$	0	0
$569$	296.778	0.521578	0.260789	−	0.965396i	$-0.416017\pi$
$569$	296.778	0.521578	0.260789	+	0.965396i	$0.416017\pi$
$570$	0	0
$571$	−491.745	−0.861201	−0.430600	−	0.902543i	$-0.641698\pi$
$571$	−491.745	−0.861201	−0.430600	+	0.902543i	$0.641698\pi$
$572$	0	0
$573$	22.1194i	0.0386027i
$574$	0	0
$575$	− 37.0771i	− 0.0644820i
$576$	0	0
$577$	−189.382	−0.328218	−0.164109	−	0.986442i	$-0.552475\pi$
$577$	−189.382	−0.328218	−0.164109	+	0.986442i	$0.552475\pi$
$578$	0	0
$579$	−14.1142	−0.0243769
$580$	0	0
$581$	783.387i	1.34834i
$582$	0	0
$583$	− 120.289i	− 0.206328i
$584$	0	0
$585$	414.091	0.707847
$586$	0	0
$587$	906.775	1.54476	0.772381	−	0.635160i	$-0.219066\pi$
$587$	906.775	1.54476	0.772381	+	0.635160i	$0.219066\pi$
$588$	0	0
$589$	− 100.493i	− 0.170616i
$590$	0	0
$591$	− 18.1063i	− 0.0306367i
$592$	0	0
$593$	127.909	0.215697	0.107849	−	0.994167i	$-0.465604\pi$
$593$	127.909	0.215697	0.107849	+	0.994167i	$0.465604\pi$
$594$	0	0
$595$	341.867	0.574566
$596$	0	0
$597$	− 32.8011i	− 0.0549433i
$598$	0	0
$599$	794.804i	1.32688i	0.748227	+	0.663442i	$0.230905\pi$
$599$	794.804i	1.32688i	−0.748227	+	0.663442i	$0.769095\pi$
$600$	0	0
$601$	−89.2746	−0.148543	−0.0742717	−	0.997238i	$-0.523663\pi$
$601$	−89.2746	−0.148543	−0.0742717	+	0.997238i	$0.523663\pi$
$602$	0	0
$603$	290.324	0.481466
$604$	0	0
$605$	313.286i	0.517829i
$606$	0	0
$607$	− 316.002i	− 0.520596i	−0.965528	−	0.260298i	$-0.916179\pi$
$607$	− 316.002i	− 0.520596i	0.965528	−	0.260298i	$-0.0838208\pi$
$608$	0	0
$609$	−13.7402	−0.0225619
$610$	0	0
$611$	−358.320	−0.586448
$612$	0	0
$613$	271.534i	0.442959i	0.975165	+	0.221479i	$0.0710885\pi$
$613$	271.534i	0.442959i	−0.975165	+	0.221479i	$0.928911\pi$
$614$	0	0
$615$	52.0385i	0.0846154i
$616$	0	0
$617$	105.762	0.171413	0.0857066	−	0.996320i	$-0.472685\pi$
$617$	105.762	0.171413	0.0857066	+	0.996320i	$0.472685\pi$
$618$	0	0
$619$	−783.218	−1.26530	−0.632648	−	0.774440i	$-0.718032\pi$
$619$	−783.218	−1.26530	−0.632648	+	0.774440i	$0.718032\pi$
$620$	0	0
$621$	63.4823i	0.102226i
$622$	0	0
$623$	426.468i	0.684539i
$624$	0	0
$625$	−674.560	−1.07930
$626$	0	0
$627$	30.8805	0.0492512
$628$	0	0
$629$	174.531i	0.277473i
$630$	0	0
$631$	762.907i	1.20904i	0.796589	+	0.604522i	$0.206636\pi$
$631$	762.907i	1.20904i	−0.796589	+	0.604522i	$0.793364\pi$
$632$	0	0
$633$	57.6772	0.0911172
$634$	0	0
$635$	557.020	0.877198
$636$	0	0
$637$	393.792i	0.618197i
$638$	0	0
$639$	− 462.232i	− 0.723367i
$640$	0	0
$641$	−412.834	−0.644046	−0.322023	−	0.946732i	$-0.604363\pi$
$641$	−412.834	−0.644046	−0.322023	+	0.946732i	$0.604363\pi$
$642$	0	0
$643$	526.815	0.819308	0.409654	−	0.912241i	$-0.365649\pi$
$643$	526.815	0.819308	0.409654	+	0.912241i	$0.365649\pi$
$644$	0	0
$645$	83.1778i	0.128958i
$646$	0	0
$647$	1170.94i	1.80980i	0.425627	+	0.904899i	$0.360054\pi$
$647$	1170.94i	1.80980i	−0.425627	+	0.904899i	$0.639946\pi$
$648$	0	0
$649$	−561.625	−0.865370
$650$	0	0
$651$	−10.5156	−0.0161531
$652$	0	0
$653$	19.4487i	0.0297836i	0.999889	+	0.0148918i	$0.00474038\pi$
$653$	19.4487i	0.0297836i	−0.999889	+	0.0148918i	$0.995260\pi$
$654$	0	0
$655$	− 1129.33i	− 1.72417i
$656$	0	0
$657$	703.165	1.07027
$658$	0	0
$659$	400.433	0.607637	0.303818	−	0.952730i	$-0.401738\pi$
$659$	400.433	0.607637	0.303818	+	0.952730i	$0.401738\pi$
$660$	0	0
$661$	406.636i	0.615182i	0.951519	+	0.307591i	$0.0995229\pi$
$661$	406.636i	0.615182i	−0.951519	+	0.307591i	$0.900477\pi$
$662$	0	0
$663$	12.4582i	0.0187907i
$664$	0	0
$665$	963.069	1.44822
$666$	0	0
$667$	117.307	0.175873
$668$	0	0
$669$	− 3.26423i	− 0.00487927i
$670$	0	0
$671$	− 187.776i	− 0.279845i
$672$	0	0
$673$	−45.5265	−0.0676471	−0.0338236	−	0.999428i	$-0.510768\pi$
$673$	−45.5265	−0.0676471	−0.0338236	+	0.999428i	$0.510768\pi$
$674$	0	0
$675$	−8.06930	−0.0119545
$676$	0	0
$677$	− 294.639i	− 0.435213i	−0.976037	−	0.217606i	$-0.930175\pi$
$677$	− 294.639i	− 0.435213i	0.976037	−	0.217606i	$-0.0698249\pi$
$678$	0	0
$679$	1089.18i	1.60409i
$680$	0	0
$681$	58.3382	0.0856655
$682$	0	0
$683$	310.626	0.454796	0.227398	−	0.973802i	$-0.426978\pi$
$683$	310.626	0.454796	0.227398	+	0.973802i	$0.426978\pi$
$684$	0	0
$685$	− 391.829i	− 0.572013i
$686$	0	0
$687$	67.4808i	0.0982254i
$688$	0	0
$689$	136.710	0.198418
$690$	0	0
$691$	980.037	1.41829	0.709144	−	0.705063i	$-0.249081\pi$
$691$	980.037	1.41829	0.709144	+	0.705063i	$0.249081\pi$
$692$	0	0
$693$	675.534i	0.974797i
$694$	0	0
$695$	− 791.905i	− 1.13943i
$696$	0	0
$697$	327.301	0.469585
$698$	0	0
$699$	71.3690	0.102102
$700$	0	0
$701$	− 276.305i	− 0.394158i	−0.980388	−	0.197079i	$-0.936854\pi$
$701$	− 276.305i	− 0.394158i	0.980388	−	0.197079i	$-0.0631456\pi$
$702$	0	0
$703$	491.668i	0.699386i
$704$	0	0
$705$	43.5917	0.0618322
$706$	0	0
$707$	1331.03	1.88264
$708$	0	0
$709$	− 449.838i	− 0.634468i	−0.948347	−	0.317234i	$-0.897246\pi$
$709$	− 449.838i	− 0.634468i	0.948347	−	0.317234i	$-0.102754\pi$
$710$	0	0
$711$	971.962i	1.36703i
$712$	0	0
$713$	89.7775	0.125915
$714$	0	0
$715$	360.767	0.504569
$716$	0	0
$717$	− 15.9793i	− 0.0222864i
$718$	0	0
$719$	1122.38i	1.56103i	0.625139	+	0.780514i	$0.285042\pi$
$719$	1122.38i	1.56103i	−0.625139	+	0.780514i	$0.714958\pi$
$720$	0	0
$721$	−1340.43	−1.85913
$722$	0	0
$723$	−60.7486	−0.0840229
$724$	0	0
$725$	14.9110i	0.0205669i
$726$	0	0
$727$	529.192i	0.727911i	0.931416	+	0.363956i	$0.118574\pi$
$727$	529.192i	0.727911i	−0.931416	+	0.363956i	$0.881426\pi$
$728$	0	0
$729$	−708.260	−0.971549
$730$	0	0
$731$	523.155	0.715670
$732$	0	0
$733$	372.110i	0.507653i	0.967250	+	0.253826i	$0.0816892\pi$
$733$	372.110i	0.507653i	−0.967250	+	0.253826i	$0.918311\pi$
$734$	0	0
$735$	− 47.9070i	− 0.0651797i
$736$	0	0
$737$	252.938	0.343200
$738$	0	0
$739$	62.9975	0.0852469	0.0426235	−	0.999091i	$-0.486428\pi$
$739$	62.9975	0.0852469	0.0426235	+	0.999091i	$0.486428\pi$
$740$	0	0
$741$	35.0959i	0.0473629i
$742$	0	0
$743$	− 762.894i	− 1.02678i	−0.858157	−	0.513388i	$-0.828390\pi$
$743$	− 762.894i	− 1.02678i	0.858157	−	0.513388i	$-0.171610\pi$
$744$	0	0
$745$	−1076.50	−1.44496
$746$	0	0
$747$	−726.057	−0.971964
$748$	0	0
$749$	433.789i	0.579157i
$750$	0	0
$751$	− 1342.93i	− 1.78819i	−0.447876	−	0.894095i	$-0.647820\pi$
$751$	− 1342.93i	− 1.78819i	0.447876	−	0.894095i	$-0.352180\pi$
$752$	0	0
$753$	−23.3108	−0.0309573
$754$	0	0
$755$	1149.44	1.52244
$756$	0	0
$757$	− 558.375i	− 0.737615i	−0.929506	−	0.368808i	$-0.879766\pi$
$757$	− 558.375i	− 0.737615i	0.929506	−	0.368808i	$-0.120234\pi$
$758$	0	0
$759$	27.5878i	0.0363475i
$760$	0	0
$761$	−480.213	−0.631029	−0.315514	−	0.948921i	$-0.602177\pi$
$761$	−480.213	−0.631029	−0.315514	+	0.948921i	$0.602177\pi$
$762$	0	0
$763$	−9.73550	−0.0127595
$764$	0	0
$765$	316.848i	0.414180i
$766$	0	0
$767$	− 638.291i	− 0.832191i
$768$	0	0
$769$	472.763	0.614777	0.307388	−	0.951584i	$-0.400545\pi$
$769$	472.763	0.614777	0.307388	+	0.951584i	$0.400545\pi$
$770$	0	0
$771$	−45.9234	−0.0595634
$772$	0	0
$773$	− 1213.02i	− 1.56924i	−0.619979	−	0.784619i	$-0.712859\pi$
$773$	− 1213.02i	− 1.56924i	0.619979	−	0.784619i	$-0.287141\pi$
$774$	0	0
$775$	11.4117i	0.0147248i
$776$	0	0
$777$	51.4485	0.0662143
$778$	0	0
$779$	922.036	1.18362
$780$	0	0
$781$	− 402.708i	− 0.515632i
$782$	0	0
$783$	− 25.5302i	− 0.0326056i
$784$	0	0
$785$	807.648	1.02885
$786$	0	0
$787$	240.919	0.306123	0.153062	−	0.988217i	$-0.451087\pi$
$787$	240.919	0.306123	0.153062	+	0.988217i	$0.451087\pi$
$788$	0	0
$789$	77.4612i	0.0981764i
$790$	0	0
$791$	− 143.891i	− 0.181911i
$792$	0	0
$793$	213.409	0.269116
$794$	0	0
$795$	−16.6315	−0.0209202
$796$	0	0
$797$	− 1181.68i	− 1.48266i	−0.671143	−	0.741328i	$-0.734197\pi$
$797$	− 1181.68i	− 1.48266i	0.671143	−	0.741328i	$-0.265803\pi$
$798$	0	0
$799$	− 274.174i	− 0.343147i
$800$	0	0
$801$	−395.258	−0.493456
$802$	0	0
$803$	612.616	0.762909
$804$	0	0
$805$	860.379i	1.06879i
$806$	0	0
$807$	104.789i	0.129850i
$808$	0	0
$809$	371.926	0.459735	0.229868	−	0.973222i	$-0.426171\pi$
$809$	371.926	0.459735	0.229868	+	0.973222i	$0.426171\pi$
$810$	0	0
$811$	389.798	0.480639	0.240320	−	0.970694i	$-0.422748\pi$
$811$	389.798	0.480639	0.240320	+	0.970694i	$0.422748\pi$
$812$	0	0
$813$	− 74.3165i	− 0.0914102i
$814$	0	0
$815$	418.550i	0.513559i
$816$	0	0
$817$	1473.77	1.80389
$818$	0	0
$819$	−767.749	−0.937422
$820$	0	0
$821$	− 1279.76i	− 1.55878i	−0.626542	−	0.779388i	$-0.715530\pi$
$821$	− 1279.76i	− 1.55878i	0.626542	−	0.779388i	$-0.284470\pi$
$822$	0	0
$823$	523.237i	0.635768i	0.948130	+	0.317884i	$0.102972\pi$
$823$	523.237i	0.635768i	−0.948130	+	0.317884i	$0.897028\pi$
$824$	0	0
$825$	−3.50671	−0.00425055
$826$	0	0
$827$	1022.20	1.23603	0.618017	−	0.786165i	$-0.287936\pi$
$827$	1022.20	1.23603	0.618017	+	0.786165i	$0.287936\pi$
$828$	0	0
$829$	405.003i	0.488544i	0.969707	+	0.244272i	$0.0785489\pi$
$829$	405.003i	0.488544i	−0.969707	+	0.244272i	$0.921451\pi$
$830$	0	0
$831$	102.972i	0.123914i
$832$	0	0
$833$	−301.316	−0.361724
$834$	0	0
$835$	−556.059	−0.665939
$836$	0	0
$837$	− 19.5388i	− 0.0233438i
$838$	0	0
$839$	− 1353.58i	− 1.61333i	−0.591008	−	0.806666i	$-0.701270\pi$
$839$	− 1353.58i	− 1.61333i	0.591008	−	0.806666i	$-0.298730\pi$
$840$	0	0
$841$	793.824	0.943904
$842$	0	0
$843$	39.6162	0.0469943
$844$	0	0
$845$	− 470.911i	− 0.557291i
$846$	0	0
$847$	− 580.852i	− 0.685776i
$848$	0	0
$849$	9.16597	0.0107962
$850$	0	0
$851$	−439.243	−0.516149
$852$	0	0
$853$	− 945.053i	− 1.10792i	−0.832544	−	0.553958i	$-0.813117\pi$
$853$	− 945.053i	− 1.10792i	0.832544	−	0.553958i	$-0.186883\pi$
$854$	0	0
$855$	892.589i	1.04396i
$856$	0	0
$857$	−488.688	−0.570230	−0.285115	−	0.958493i	$-0.592032\pi$
$857$	−488.688	−0.570230	−0.285115	+	0.958493i	$0.592032\pi$
$858$	0	0
$859$	−380.166	−0.442568	−0.221284	−	0.975209i	$-0.571025\pi$
$859$	−380.166	−0.442568	−0.221284	+	0.975209i	$0.571025\pi$
$860$	0	0
$861$	− 96.4824i	− 0.112059i
$862$	0	0
$863$	152.667i	0.176903i	0.996080	+	0.0884514i	$0.0281918\pi$
$863$	152.667i	0.176903i	−0.996080	+	0.0884514i	$0.971808\pi$
$864$	0	0
$865$	1314.82	1.52002
$866$	0	0
$867$	50.2881	0.0580024
$868$	0	0
$869$	846.799i	0.974452i
$870$	0	0
$871$	287.466i	0.330041i
$872$	0	0
$873$	−1009.47	−1.15632
$874$	0	0
$875$	1150.05	1.31434
$876$	0	0
$877$	230.004i	0.262262i	0.991365	+	0.131131i	$0.0418609\pi$
$877$	230.004i	0.262262i	−0.991365	+	0.131131i	$0.958139\pi$
$878$	0	0
$879$	− 26.9573i	− 0.0306681i
$880$	0	0
$881$	−873.243	−0.991196	−0.495598	−	0.868552i	$-0.665051\pi$
$881$	−873.243	−0.991196	−0.495598	+	0.868552i	$0.665051\pi$
$882$	0	0
$883$	−325.304	−0.368408	−0.184204	−	0.982888i	$-0.558971\pi$
$883$	−325.304	−0.368408	−0.184204	+	0.982888i	$0.558971\pi$
$884$	0	0
$885$	77.6517i	0.0877421i
$886$	0	0
$887$	430.685i	0.485552i	0.970082	+	0.242776i	$0.0780580\pi$
$887$	430.685i	0.485552i	−0.970082	+	0.242776i	$0.921942\pi$
$888$	0	0
$889$	−1032.75	−1.16170
$890$	0	0
$891$	−623.087	−0.699312
$892$	0	0
$893$	− 772.373i	− 0.864920i
$894$	0	0
$895$	− 445.709i	− 0.497999i
$896$	0	0
$897$	−31.3537	−0.0349539
$898$	0	0
$899$	−36.1051	−0.0401614
$900$	0	0
$901$	104.606i	0.116099i
$902$	0	0
$903$	− 154.217i	− 0.170783i
$904$	0	0
$905$	1084.47	1.19831
$906$	0	0
$907$	−31.4326	−0.0346556	−0.0173278	−	0.999850i	$-0.505516\pi$
$907$	−31.4326	−0.0346556	−0.0173278	+	0.999850i	$0.505516\pi$
$908$	0	0
$909$	1233.62i	1.35712i
$910$	0	0
$911$	− 1399.85i	− 1.53661i	−0.640083	−	0.768306i	$-0.721100\pi$
$911$	− 1399.85i	− 1.53661i	0.640083	−	0.768306i	$-0.278900\pi$
$912$	0	0
$913$	−632.560	−0.692837
$914$	0	0
$915$	−25.9624	−0.0283742
$916$	0	0
$917$	2093.85i	2.28337i
$918$	0	0
$919$	− 806.944i	− 0.878068i	−0.898470	−	0.439034i	$-0.855321\pi$
$919$	− 806.944i	− 0.878068i	0.898470	−	0.439034i	$-0.144679\pi$
$920$	0	0
$921$	75.3976	0.0818650
$922$	0	0
$923$	457.681	0.495862
$924$	0	0
$925$	− 55.8326i	− 0.0603595i
$926$	0	0
$927$	− 1242.34i	− 1.34017i
$928$	0	0
$929$	−1620.69	−1.74455	−0.872276	−	0.489013i	$-0.837357\pi$
$929$	−1620.69	−1.74455	−0.872276	+	0.489013i	$0.837357\pi$
$930$	0	0
$931$	−848.834	−0.911744
$932$	0	0
$933$	27.0982i	0.0290442i
$934$	0	0
$935$	276.046i	0.295237i
$936$	0	0
$937$	598.181	0.638400	0.319200	−	0.947687i	$-0.396586\pi$
$937$	598.181	0.638400	0.319200	+	0.947687i	$0.396586\pi$
$938$	0	0
$939$	10.7295	0.0114265
$940$	0	0
$941$	− 1382.88i	− 1.46958i	−0.678294	−	0.734791i	$-0.737280\pi$
$941$	− 1382.88i	− 1.46958i	0.678294	−	0.734791i	$-0.262720\pi$
$942$	0	0
$943$	823.721i	0.873511i
$944$	0	0
$945$	187.249	0.198147
$946$	0	0
$947$	−1170.80	−1.23633	−0.618163	−	0.786050i	$-0.712123\pi$
$947$	−1170.80	−1.23633	−0.618163	+	0.786050i	$0.712123\pi$
$948$	0	0
$949$	696.242i	0.733659i
$950$	0	0
$951$	− 32.0939i	− 0.0337476i
$952$	0	0
$953$	1846.78	1.93786	0.968930	−	0.247333i	$-0.0795543\pi$
$953$	1846.78	1.93786	0.968930	+	0.247333i	$0.0795543\pi$
$954$	0	0
$955$	−557.020	−0.583267
$956$	0	0
$957$	− 11.0947i	− 0.0115933i
$958$	0	0
$959$	726.475i	0.757534i
$960$	0	0
$961$	933.368	0.971247
$962$	0	0
$963$	−402.043	−0.417490
$964$	0	0
$965$	− 355.431i	− 0.368323i
$966$	0	0
$967$	363.922i	0.376341i	0.982136	+	0.188170i	$0.0602557\pi$
$967$	363.922i	0.376341i	−0.982136	+	0.188170i	$0.939744\pi$
$968$	0	0
$969$	−26.8542	−0.0277133
$970$	0	0
$971$	−1642.32	−1.69137	−0.845685	−	0.533683i	$-0.820808\pi$
$971$	−1642.32	−1.69137	−0.845685	+	0.533683i	$0.820808\pi$
$972$	0	0
$973$	1468.24i	1.50898i
$974$	0	0
$975$	− 3.98540i	− 0.00408759i
$976$	0	0
$977$	−1159.63	−1.18693	−0.593467	−	0.804858i	$-0.702241\pi$
$977$	−1159.63	−1.18693	−0.593467	+	0.804858i	$0.702241\pi$
$978$	0	0
$979$	−344.359	−0.351746
$980$	0	0
$981$	− 9.02303i	− 0.00919779i
$982$	0	0
$983$	− 1780.51i	− 1.81131i	−0.424020	−	0.905653i	$-0.639382\pi$
$983$	− 1780.51i	− 1.81131i	0.424020	−	0.905653i	$-0.360618\pi$
$984$	0	0
$985$	455.961	0.462904
$986$	0	0
$987$	−80.8216	−0.0818861
$988$	0	0
$989$	1316.63i	1.33127i
$990$	0	0
$991$	675.783i	0.681920i	0.940078	+	0.340960i	$0.110752\pi$
$991$	675.783i	0.681920i	−0.940078	+	0.340960i	$0.889248\pi$
$992$	0	0
$993$	94.6184	0.0952854
$994$	0	0
$995$	826.014	0.830165
$996$	0	0
$997$	− 13.3638i	− 0.0134040i	−0.999978	−	0.00670200i	$-0.997867\pi$
$997$	− 13.3638i	− 0.0134040i	0.999978	−	0.00670200i	$-0.00213333\pi$
$998$	0	0
$999$	95.5948i	0.0956905i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1024.3.d.k.511.5		12
4.3	odd	2	inner	1024.3.d.k.511.7		12
8.3	odd	2	inner	1024.3.d.k.511.6		12
8.5	even	2	inner	1024.3.d.k.511.8		12
16.3	odd	4		1024.3.c.j.1023.7		12
16.5	even	4		1024.3.c.j.1023.8		12
16.11	odd	4		1024.3.c.j.1023.6		12
16.13	even	4		1024.3.c.j.1023.5		12
32.3	odd	8		16.3.f.a.11.2	yes	6
32.5	even	8		16.3.f.a.3.2	✓	6
32.11	odd	8		128.3.f.a.95.2		6
32.13	even	8		128.3.f.a.31.2		6
32.19	odd	8		128.3.f.b.31.2		6
32.21	even	8		128.3.f.b.95.2		6
32.27	odd	8		64.3.f.a.47.2		6
32.29	even	8		64.3.f.a.15.2		6
96.5	odd	8		144.3.m.a.19.2		6
96.11	even	8		1152.3.m.b.991.3		6
96.29	odd	8		576.3.m.a.271.1		6
96.35	even	8		144.3.m.a.91.2		6
96.53	odd	8		1152.3.m.a.991.3		6
96.59	even	8		576.3.m.a.559.1		6
96.77	odd	8		1152.3.m.b.415.3		6
96.83	even	8		1152.3.m.a.415.3		6
160.3	even	8		400.3.k.d.299.1		6
160.37	odd	8		400.3.k.d.99.1		6
160.67	even	8		400.3.k.c.299.3		6
160.69	even	8		400.3.r.c.51.2		6
160.99	odd	8		400.3.r.c.251.2		6
160.133	odd	8		400.3.k.c.99.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.3.f.a.3.2	✓	6	32.5	even	8
16.3.f.a.11.2	yes	6	32.3	odd	8
64.3.f.a.15.2		6	32.29	even	8
64.3.f.a.47.2		6	32.27	odd	8
128.3.f.a.31.2		6	32.13	even	8
128.3.f.a.95.2		6	32.11	odd	8
128.3.f.b.31.2		6	32.19	odd	8
128.3.f.b.95.2		6	32.21	even	8
144.3.m.a.19.2		6	96.5	odd	8
144.3.m.a.91.2		6	96.35	even	8
400.3.k.c.99.3		6	160.133	odd	8
400.3.k.c.299.3		6	160.67	even	8
400.3.k.d.99.1		6	160.37	odd	8
400.3.k.d.299.1		6	160.3	even	8
400.3.r.c.51.2		6	160.69	even	8
400.3.r.c.251.2		6	160.99	odd	8
576.3.m.a.271.1		6	96.29	odd	8
576.3.m.a.559.1		6	96.59	even	8
1024.3.c.j.1023.5		12	16.13	even	4
1024.3.c.j.1023.6		12	16.11	odd	4
1024.3.c.j.1023.7		12	16.3	odd	4
1024.3.c.j.1023.8		12	16.5	even	4
1024.3.d.k.511.5		12	1.1	even	1	trivial
1024.3.d.k.511.6		12	8.3	odd	2	inner
1024.3.d.k.511.7		12	4.3	odd	2	inner
1024.3.d.k.511.8		12	8.5	even	2	inner
1152.3.m.a.415.3		6	96.83	even	8
1152.3.m.a.991.3		6	96.53	odd	8
1152.3.m.b.415.3		6	96.77	odd	8
1152.3.m.b.991.3		6	96.11	even	8

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1024 = 2^{10} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1024.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.206992 q^{3} -5.21257i q^{5} +9.66442i q^{7} -8.95715 q^{9} -7.80371 q^{11} -8.86897i q^{13} +1.07896i q^{15} +6.78623 q^{17} +19.1174 q^{19} -2.00046i q^{21} +17.0790i q^{23} -2.17092 q^{25} +3.71699 q^{27} -6.86851i q^{29} -5.25662i q^{31} +1.61531 q^{33} +50.3765 q^{35} +25.7183i q^{37} +1.83581i q^{39} +48.2302 q^{41} +77.0907 q^{43} +46.6898i q^{45} -40.4015i q^{47} -44.4011 q^{49} -1.40470 q^{51} +15.4144i q^{53} +40.6774i q^{55} -3.95715 q^{57} +71.9690 q^{59} +24.0624i q^{61} -86.5657i q^{63} -46.2302 q^{65} -32.4126 q^{67} -3.53521i q^{69} +51.6047i q^{71} -78.5032 q^{73} +0.449364 q^{75} -75.4184i q^{77} -108.512i q^{79} +79.8450 q^{81} +81.0589 q^{83} -35.3737i q^{85} +1.42173i q^{87} +44.1276 q^{89} +85.7135 q^{91} +1.08808i q^{93} -99.6510i q^{95} +112.700 q^{97} +69.8991 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{9} + 8 q^{17} + 20 q^{25} - 8 q^{33} + 92 q^{49} + 72 q^{57} + 24 q^{65} + 96 q^{73} - 172 q^{81} - 160 q^{89} - 8 q^{97}+O(q^{100}) \) 12 * q + 12 * q^9 + 8 * q^17 + 20 * q^25 - 8 * q^33 + 92 * q^49 + 72 * q^57 + 24 * q^65 + 96 * q^73 - 172 * q^81 - 160 * q^89 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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