LMFDB - Embedded newform 1024.3.c.j.1023.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1024,3,Mod(1023,1024)]

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, 0]))

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

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

chi := DirichletCharacter("1024.1023");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1024 = 2^{10} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1024.c (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			1023.7
Root			$-1.35489 + 0.405301i$ of defining polynomial
Character	$\chi$	$=$	1024.1023
Dual form			1024.3.c.j.1023.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.206992i q^{3} -5.21257 q^{5} +9.66442i q^{7} +8.95715 q^{9} +O(q^{10})$	$q+0.206992i q^{3} -5.21257 q^{5} +9.66442i q^{7} +8.95715 q^{9} -7.80371i q^{11} +8.86897 q^{13} -1.07896i q^{15} +6.78623 q^{17} -19.1174i q^{19} -2.00046 q^{21} +17.0790i q^{23} +2.17092 q^{25} +3.71699i q^{27} +6.86851 q^{29} +5.25662i q^{31} +1.61531 q^{33} -50.3765i q^{35} +25.7183 q^{37} +1.83581i q^{39} -48.2302 q^{41} +77.0907i q^{43} -46.6898 q^{45} +40.4015i q^{47} -44.4011 q^{49} +1.40470i q^{51} +15.4144 q^{53} +40.6774i q^{55} +3.95715 q^{57} +71.9690i q^{59} -24.0624 q^{61} +86.5657i q^{63} -46.2302 q^{65} +32.4126i q^{67} -3.53521 q^{69} +51.6047i q^{71} +78.5032 q^{73} +0.449364i q^{75} +75.4184 q^{77} +108.512i q^{79} +79.8450 q^{81} -81.0589i q^{83} -35.3737 q^{85} +1.42173i q^{87} -44.1276 q^{89} +85.7135i q^{91} -1.08808 q^{93} +99.6510i q^{95} +112.700 q^{97} -69.8991i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^9	$ 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

Display 100 coefficients 10 coefficients

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
$3$	0.206992i	0.0689974i	0.999405	+	0.0344987i	$0.0109834\pi$
$3$	0.206992i	0.0689974i	−0.999405	+	0.0344987i	$0.989017\pi$
$4$	0	0
$5$	−5.21257	−1.04251	−0.521257	−	0.853400i	$-0.674537\pi$
$5$	−5.21257	−1.04251	−0.521257	+	0.853400i	$0.674537\pi$
$6$	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
$11$	− 7.80371i	− 0.709428i	−0.934975	−	0.354714i	$-0.884578\pi$
$11$	− 7.80371i	− 0.709428i	0.934975	−	0.354714i	$-0.115422\pi$
$12$	0	0
$13$	8.86897	0.682228	0.341114	−	0.940022i	$-0.389196\pi$
$13$	8.86897	0.682228	0.341114	+	0.940022i	$0.389196\pi$
$14$	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
$19$	− 19.1174i	− 1.00618i	−0.864234	−	0.503090i	$-0.832196\pi$
$19$	− 19.1174i	− 1.00618i	0.864234	−	0.503090i	$-0.167804\pi$
$20$	0	0
$21$	−2.00046	−0.0952599
$22$	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
$27$	3.71699i	0.137666i
$28$	0	0
$29$	6.86851	0.236845	0.118423	−	0.992963i	$-0.462216\pi$
$29$	6.86851	0.236845	0.118423	+	0.992963i	$0.462216\pi$
$30$	0	0
$31$	5.25662i	0.169568i	0.996399	+	0.0847841i	$0.0270201\pi$
$31$	5.25662i	0.169568i	−0.996399	+	0.0847841i	$0.972980\pi$
$32$	0	0
$33$	1.61531	0.0489487
$34$	0	0
$35$	− 50.3765i	− 1.43933i
$36$	0	0
$37$	25.7183	0.695090	0.347545	−	0.937663i	$-0.387015\pi$
$37$	25.7183	0.695090	0.347545	+	0.937663i	$0.387015\pi$
$38$	0	0
$39$	1.83581i	0.0470720i
$40$	0	0
$41$	−48.2302	−1.17635	−0.588173	−	0.808735i	$-0.700152\pi$
$41$	−48.2302	−1.17635	−0.588173	+	0.808735i	$0.700152\pi$
$42$	0	0
$43$	77.0907i	1.79281i	0.443240	+	0.896403i	$0.353829\pi$
$43$	77.0907i	1.79281i	−0.443240	+	0.896403i	$0.646171\pi$
$44$	0	0
$45$	−46.6898	−1.03755
$46$	0	0
$47$	40.4015i	0.859607i	0.902922	+	0.429804i	$0.141417\pi$
$47$	40.4015i	0.859607i	−0.902922	+	0.429804i	$0.858583\pi$
$48$	0	0
$49$	−44.4011	−0.906144
$50$	0	0
$51$	1.40470i	0.0275431i
$52$	0	0
$53$	15.4144	0.290837	0.145419	−	0.989370i	$-0.453547\pi$
$53$	15.4144	0.290837	0.145419	+	0.989370i	$0.453547\pi$
$54$	0	0
$55$	40.6774i	0.739590i
$56$	0	0
$57$	3.95715	0.0694238
$58$	0	0
$59$	71.9690i	1.21981i	0.792473	+	0.609907i	$0.208793\pi$
$59$	71.9690i	1.21981i	−0.792473	+	0.609907i	$0.791207\pi$
$60$	0	0
$61$	−24.0624	−0.394466	−0.197233	−	0.980357i	$-0.563196\pi$
$61$	−24.0624	−0.394466	−0.197233	+	0.980357i	$0.563196\pi$
$62$	0	0
$63$	86.5657i	1.37406i
$64$	0	0
$65$	−46.2302	−0.711233
$66$	0	0
$67$	32.4126i	0.483769i	0.970305	+	0.241885i	$0.0777656\pi$
$67$	32.4126i	0.483769i	−0.970305	+	0.241885i	$0.922234\pi$
$68$	0	0
$69$	−3.53521	−0.0512349
$70$	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	$-0.319296\pi$
$73$	78.5032	1.07539	0.537693	+	0.843141i	$0.319296\pi$
$74$	0	0
$75$	0.449364i	0.00599152i
$76$	0	0
$77$	75.4184	0.979459
$78$	0	0
$79$	108.512i	1.37357i	0.726859	+	0.686787i	$0.240979\pi$
$79$	108.512i	1.37357i	−0.726859	+	0.686787i	$0.759021\pi$
$80$	0	0
$81$	79.8450	0.985741
$82$	0	0
$83$	− 81.0589i	− 0.976613i	−0.872672	−	0.488307i	$-0.837615\pi$
$83$	− 81.0589i	− 0.976613i	0.872672	−	0.488307i	$-0.162385\pi$
$84$	0	0
$85$	−35.3737	−0.416161
$86$	0	0
$87$	1.42173i	0.0163417i
$88$	0	0
$89$	−44.1276	−0.495816	−0.247908	−	0.968784i	$-0.579743\pi$
$89$	−44.1276	−0.495816	−0.247908	+	0.968784i	$0.579743\pi$
$90$	0	0
$91$	85.7135i	0.941906i
$92$	0	0
$93$	−1.08808	−0.0116998
$94$	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
$99$	− 69.8991i	− 0.706051i
$100$	0	0
$101$	−137.724	−1.36361	−0.681804	−	0.731534i	$-0.738805\pi$
$101$	−137.724	−1.36361	−0.681804	+	0.731534i	$0.738805\pi$
$102$	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
$107$	44.8851i	0.419487i	0.977756	+	0.209743i	$0.0672629\pi$
$107$	44.8851i	0.419487i	−0.977756	+	0.209743i	$0.932737\pi$
$108$	0	0
$109$	−1.00735	−0.00924179	−0.00462089	−	0.999989i	$-0.501471\pi$
$109$	−1.00735	−0.00924179	−0.00462089	+	0.999989i	$0.501471\pi$
$110$	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
$115$	− 89.0253i	− 0.774133i
$116$	0	0
$117$	79.4407	0.678981
$118$	0	0
$119$	65.5850i	0.551134i
$120$	0	0
$121$	60.1021	0.496711
$122$	0	0
$123$	− 9.98326i	− 0.0811647i
$124$	0	0
$125$	118.998	0.951986
$126$	0	0
$127$	− 106.861i	− 0.841425i	−0.907194	−	0.420712i	$-0.861780\pi$
$127$	− 106.861i	− 0.841425i	0.907194	−	0.420712i	$-0.138220\pi$
$128$	0	0
$129$	−15.9572	−0.123699
$130$	0	0
$131$	− 216.655i	− 1.65386i	−0.562307	−	0.826928i	$-0.690086\pi$
$131$	− 216.655i	− 1.65386i	0.562307	−	0.826928i	$-0.309914\pi$
$132$	0	0
$133$	184.759	1.38916
$134$	0	0
$135$	− 19.3751i	− 0.143519i
$136$	0	0
$137$	−75.1700	−0.548686	−0.274343	−	0.961632i	$-0.588460\pi$
$137$	−75.1700	−0.548686	−0.274343	+	0.961632i	$0.588460\pi$
$138$	0	0
$139$	151.922i	1.09297i	0.837471	+	0.546483i	$0.184034\pi$
$139$	151.922i	1.09297i	−0.837471	+	0.546483i	$0.815966\pi$
$140$	0	0
$141$	−8.36280	−0.0593106
$142$	0	0
$143$	− 69.2109i	− 0.483992i
$144$	0	0
$145$	−35.8026	−0.246915
$146$	0	0
$147$	− 9.19067i	− 0.0625216i
$148$	0	0
$149$	−206.519	−1.38603	−0.693017	−	0.720921i	$-0.743719\pi$
$149$	−206.519	−1.38603	−0.693017	+	0.720921i	$0.743719\pi$
$150$	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
$155$	− 27.4005i	− 0.176777i
$156$	0	0
$157$	−154.942	−0.986893	−0.493447	−	0.869776i	$-0.664263\pi$
$157$	−154.942	−0.986893	−0.493447	+	0.869776i	$0.664263\pi$
$158$	0	0
$159$	3.19066i	0.0200670i
$160$	0	0
$161$	−165.058	−1.02521
$162$	0	0
$163$	80.2963i	0.492615i	0.969192	+	0.246308i	$0.0792174\pi$
$163$	80.2963i	0.492615i	−0.969192	+	0.246308i	$0.920783\pi$
$164$	0	0
$165$	−8.41990	−0.0510297
$166$	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
$171$	− 171.238i	− 1.00139i
$172$	0	0
$173$	−252.240	−1.45803	−0.729016	−	0.684496i	$-0.760022\pi$
$173$	−252.240	−1.45803	−0.729016	+	0.684496i	$0.760022\pi$
$174$	0	0
$175$	20.9807i	0.119890i
$176$	0	0
$177$	−14.8970	−0.0841639
$178$	0	0
$179$	− 85.5065i	− 0.477690i	−0.971058	−	0.238845i	$-0.923231\pi$
$179$	− 85.5065i	− 0.477690i	0.971058	−	0.238845i	$-0.0767688\pi$
$180$	0	0
$181$	208.049	1.14944	0.574721	−	0.818349i	$-0.305110\pi$
$181$	208.049	1.14944	0.574721	+	0.818349i	$0.305110\pi$
$182$	0	0
$183$	− 4.98073i	− 0.0272171i
$184$	0	0
$185$	−134.059	−0.724642
$186$	0	0
$187$	− 52.9578i	− 0.283197i
$188$	0	0
$189$	−35.9225	−0.190066
$190$	0	0
$191$	106.861i	0.559481i	0.960076	+	0.279741i	$0.0902485\pi$
$191$	106.861i	0.559481i	−0.960076	+	0.279741i	$0.909752\pi$
$192$	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
$195$	− 9.56927i	− 0.0490732i
$196$	0	0
$197$	87.4732	0.444027	0.222013	−	0.975044i	$-0.428737\pi$
$197$	87.4732	0.444027	0.222013	+	0.975044i	$0.428737\pi$
$198$	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
$203$	66.3802i	0.326996i
$204$	0	0
$205$	251.403	1.22636
$206$	0	0
$207$	152.979i	0.739028i
$208$	0	0
$209$	−149.187	−0.713813
$210$	0	0
$211$	278.644i	1.32059i	0.751007	+	0.660295i	$0.229569\pi$
$211$	278.644i	1.32059i	−0.751007	+	0.660295i	$0.770431\pi$
$212$	0	0
$213$	−10.6818	−0.0501492
$214$	0	0
$215$	− 401.841i	− 1.86903i
$216$	0	0
$217$	−50.8022	−0.234111
$218$	0	0
$219$	16.2495i	0.0741988i
$220$	0	0
$221$	60.1869	0.272339
$222$	0	0
$223$	− 15.7698i	− 0.0707168i	−0.999375	−	0.0353584i	$-0.988743\pi$
$223$	− 15.7698i	− 0.0707168i	0.999375	−	0.0353584i	$-0.0112573\pi$
$224$	0	0
$225$	19.4453	0.0864236
$226$	0	0
$227$	281.838i	1.24158i	0.783978	+	0.620788i	$0.213187\pi$
$227$	281.838i	1.24158i	−0.783978	+	0.620788i	$0.786813\pi$
$228$	0	0
$229$	−326.007	−1.42361	−0.711805	−	0.702377i	$-0.752122\pi$
$229$	−326.007	−1.42361	−0.711805	+	0.702377i	$0.752122\pi$
$230$	0	0
$231$	15.6110i	0.0675801i
$232$	0	0
$233$	344.791	1.47979	0.739895	−	0.672722i	$-0.234875\pi$
$233$	344.791	1.47979	0.739895	+	0.672722i	$0.234875\pi$
$234$	0	0
$235$	− 210.596i	− 0.896153i
$236$	0	0
$237$	−22.4612	−0.0947729
$238$	0	0
$239$	− 77.1978i	− 0.323004i	−0.986872	−	0.161502i	$-0.948366\pi$
$239$	− 77.1978i	− 0.323004i	0.986872	−	0.161502i	$-0.0516337\pi$
$240$	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
$243$	49.9802i	0.205680i
$244$	0	0
$245$	231.444	0.944669
$246$	0	0
$247$	− 169.552i	− 0.686445i
$248$	0	0
$249$	16.7785	0.0673837
$250$	0	0
$251$	112.617i	0.448673i	0.974512	+	0.224337i	$0.0720215\pi$
$251$	112.617i	0.448673i	−0.974512	+	0.224337i	$0.927979\pi$
$252$	0	0
$253$	133.279	0.526796
$254$	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
$259$	248.553i	0.959664i
$260$	0	0
$261$	61.5223	0.235718
$262$	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
$267$	− 9.13407i	− 0.0342100i
$268$	0	0
$269$	506.247	1.88196	0.940979	−	0.338465i	$-0.109908\pi$
$269$	506.247	1.88196	0.940979	+	0.338465i	$0.109908\pi$
$270$	0	0
$271$	− 359.030i	− 1.32484i	−0.749135	−	0.662418i	$-0.769530\pi$
$271$	− 359.030i	− 1.32484i	0.749135	−	0.662418i	$-0.230470\pi$
$272$	0	0
$273$	−17.7420	−0.0649890
$274$	0	0
$275$	− 16.9413i	− 0.0616046i
$276$	0	0
$277$	−497.471	−1.79592	−0.897961	−	0.440074i	$-0.854952\pi$
$277$	−497.471	−1.79592	−0.897961	+	0.440074i	$0.854952\pi$
$278$	0	0
$279$	47.0843i	0.168761i
$280$	0	0
$281$	191.390	0.681103	0.340552	−	0.940226i	$-0.389386\pi$
$281$	191.390	0.681103	0.340552	+	0.940226i	$0.389386\pi$
$282$	0	0
$283$	− 44.2818i	− 0.156473i	−0.996935	−	0.0782363i	$-0.975071\pi$
$283$	− 44.2818i	− 0.156473i	0.996935	−	0.0782363i	$-0.0249289\pi$
$284$	0	0
$285$	−20.6270	−0.0723753
$286$	0	0
$287$	− 466.117i	− 1.62410i
$288$	0	0
$289$	−242.947	−0.840647
$290$	0	0
$291$	23.3279i	0.0801647i
$292$	0	0
$293$	130.233	0.444483	0.222241	−	0.974992i	$-0.428663\pi$
$293$	130.233	0.444483	0.222241	+	0.974992i	$0.428663\pi$
$294$	0	0
$295$	− 375.144i	− 1.27167i
$296$	0	0
$297$	29.0063	0.0976643
$298$	0	0
$299$	151.473i	0.506598i
$300$	0	0
$301$	−745.037	−2.47521
$302$	0	0
$303$	− 28.5079i	− 0.0940854i
$304$	0	0
$305$	125.427	0.411236
$306$	0	0
$307$	364.254i	1.18649i	0.805020	+	0.593247i	$0.202154\pi$
$307$	364.254i	1.18649i	−0.805020	+	0.593247i	$0.797846\pi$
$308$	0	0
$309$	−28.7093	−0.0929104
$310$	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	$-0.473612\pi$
$313$	51.8354	0.165608	0.0828041	+	0.996566i	$0.473612\pi$
$314$	0	0
$315$	− 451.230i	− 1.43248i
$316$	0	0
$317$	−155.049	−0.489114	−0.244557	−	0.969635i	$-0.578643\pi$
$317$	−155.049	−0.489114	−0.244557	+	0.969635i	$0.578643\pi$
$318$	0	0
$319$	− 53.5999i	− 0.168025i
$320$	0	0
$321$	−9.29086	−0.0289435
$322$	0	0
$323$	− 129.735i	− 0.401657i
$324$	0	0
$325$	19.2539	0.0592426
$326$	0	0
$327$	− 0.208514i	0 0.000637659i
$328$	0	0
$329$	−390.458	−1.18680
$330$	0	0
$331$	− 457.111i	− 1.38100i	−0.723332	−	0.690500i	$-0.757390\pi$
$331$	− 457.111i	− 1.38100i	0.723332	−	0.690500i	$-0.242610\pi$
$332$	0	0
$333$	230.363	0.691781
$334$	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
$339$	− 3.08186i	− 0.00909102i
$340$	0	0
$341$	41.0211	0.120297
$342$	0	0
$343$	44.4459i	0.129580i
$344$	0	0
$345$	18.4275	0.0534132
$346$	0	0
$347$	− 434.967i	− 1.25351i	−0.779218	−	0.626753i	$-0.784384\pi$
$347$	− 434.967i	− 1.25351i	0.779218	−	0.626753i	$-0.215616\pi$
$348$	0	0
$349$	241.068	0.690740	0.345370	−	0.938467i	$-0.387753\pi$
$349$	241.068	0.690740	0.345370	+	0.938467i	$0.387753\pi$
$350$	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
$355$	− 268.993i	− 0.757728i
$356$	0	0
$357$	−13.5756	−0.0380268
$358$	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
$363$	12.4407i	0.0342718i
$364$	0	0
$365$	−409.203	−1.12111
$366$	0	0
$367$	− 240.758i	− 0.656016i	−0.944675	−	0.328008i	$-0.893623\pi$
$367$	− 240.758i	− 0.656016i	0.944675	−	0.328008i	$-0.106377\pi$
$368$	0	0
$369$	−432.005	−1.17075
$370$	0	0
$371$	148.971i	0.401539i
$372$	0	0
$373$	611.653	1.63982	0.819910	−	0.572492i	$-0.194023\pi$
$373$	611.653	1.63982	0.819910	+	0.572492i	$0.194023\pi$
$374$	0	0
$375$	24.6317i	0.0656845i
$376$	0	0
$377$	60.9166	0.161583
$378$	0	0
$379$	− 247.086i	− 0.651943i	−0.945380	−	0.325971i	$-0.894309\pi$
$379$	− 247.086i	− 0.651943i	0.945380	−	0.325971i	$-0.105691\pi$
$380$	0	0
$381$	22.1194	0.0580561
$382$	0	0
$383$	− 673.381i	− 1.75817i	−0.476661	−	0.879087i	$-0.658153\pi$
$383$	− 673.381i	− 1.75817i	0.476661	−	0.879087i	$-0.341847\pi$
$384$	0	0
$385$	−393.124	−1.02110
$386$	0	0
$387$	690.513i	1.78427i
$388$	0	0
$389$	388.408	0.998478	0.499239	−	0.866464i	$-0.333613\pi$
$389$	388.408	0.998478	0.499239	+	0.866464i	$0.333613\pi$
$390$	0	0
$391$	115.902i	0.296424i
$392$	0	0
$393$	44.8459	0.114112
$394$	0	0
$395$	− 565.628i	− 1.43197i
$396$	0	0
$397$	383.611	0.966275	0.483137	−	0.875545i	$-0.339497\pi$
$397$	383.611	0.966275	0.483137	+	0.875545i	$0.339497\pi$
$398$	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
$403$	46.6208i	0.115684i
$404$	0	0
$405$	−416.198	−1.02765
$406$	0	0
$407$	− 200.699i	− 0.493117i
$408$	0	0
$409$	−634.686	−1.55180	−0.775900	−	0.630856i	$-0.782704\pi$
$409$	−634.686	−1.55180	−0.775900	+	0.630856i	$0.782704\pi$
$410$	0	0
$411$	− 15.5596i	− 0.0378579i
$412$	0	0
$413$	−695.539	−1.68411
$414$	0	0
$415$	422.525i	1.01813i
$416$	0	0
$417$	−31.4467	−0.0754117
$418$	0	0
$419$	27.2500i	0.0650358i	0.999471	+	0.0325179i	$0.0103526\pi$
$419$	27.2500i	0.0650358i	−0.999471	+	0.0325179i	$0.989647\pi$
$420$	0	0
$421$	−345.783	−0.821337	−0.410668	−	0.911785i	$-0.634705\pi$
$421$	−345.783	−0.821337	−0.410668	+	0.911785i	$0.634705\pi$
$422$	0	0
$423$	361.883i	0.855515i
$424$	0	0
$425$	14.7324	0.0346644
$426$	0	0
$427$	− 232.549i	− 0.544612i
$428$	0	0
$429$	14.3261	0.0333942
$430$	0	0
$431$	− 337.331i	− 0.782670i	−0.920248	−	0.391335i	$-0.872013\pi$
$431$	− 337.331i	− 0.782670i	0.920248	−	0.391335i	$-0.127987\pi$
$432$	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
$435$	− 7.41086i	− 0.0170365i
$436$	0	0
$437$	326.506	0.747153
$438$	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
$443$	696.061i	1.57124i	0.618707	+	0.785622i	$0.287657\pi$
$443$	696.061i	1.57124i	−0.618707	+	0.785622i	$0.712343\pi$
$444$	0	0
$445$	230.018	0.516895
$446$	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
$451$	376.374i	0.834533i
$452$	0	0
$453$	−45.6444	−0.100760
$454$	0	0
$455$	− 446.788i	− 0.981951i
$456$	0	0
$457$	386.874	0.846552	0.423276	−	0.906001i	$-0.360880\pi$
$457$	386.874	0.846552	0.423276	+	0.906001i	$0.360880\pi$
$458$	0	0
$459$	25.2243i	0.0549550i
$460$	0	0
$461$	246.640	0.535011	0.267505	−	0.963556i	$-0.413801\pi$
$461$	246.640	0.535011	0.267505	+	0.963556i	$0.413801\pi$
$462$	0	0
$463$	60.5295i	0.130733i	0.997861	+	0.0653666i	$0.0208217\pi$
$463$	60.5295i	0.130733i	−0.997861	+	0.0653666i	$0.979178\pi$
$464$	0	0
$465$	5.67168	0.0121972
$466$	0	0
$467$	− 433.431i	− 0.928118i	−0.885804	−	0.464059i	$-0.846393\pi$
$467$	− 433.431i	− 0.928118i	0.885804	−	0.464059i	$-0.153607\pi$
$468$	0	0
$469$	−313.249	−0.667908
$470$	0	0
$471$	− 32.0718i	− 0.0680930i
$472$	0	0
$473$	601.593	1.27187
$474$	0	0
$475$	− 41.5025i	− 0.0873736i
$476$	0	0
$477$	138.069	0.289453
$478$	0	0
$479$	− 376.452i	− 0.785912i	−0.919557	−	0.392956i	$-0.871453\pi$
$479$	− 376.452i	− 0.785912i	0.919557	−	0.392956i	$-0.128547\pi$
$480$	0	0
$481$	228.095	0.474210
$482$	0	0
$483$	− 34.1658i	− 0.0707366i
$484$	0	0
$485$	−587.455	−1.21125
$486$	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
$491$	822.277i	1.67470i	0.546668	+	0.837350i	$0.315896\pi$
$491$	822.277i	1.67470i	−0.546668	+	0.837350i	$0.684104\pi$
$492$	0	0
$493$	46.6113	0.0945462
$494$	0	0
$495$	364.354i	0.736069i
$496$	0	0
$497$	−498.730	−1.00348
$498$	0	0
$499$	246.081i	0.493149i	0.969124	+	0.246574i	$0.0793050\pi$
$499$	246.081i	0.493149i	−0.969124	+	0.246574i	$0.920695\pi$
$500$	0	0
$501$	22.0812	0.0440742
$502$	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
$507$	− 18.7000i	− 0.0368835i
$508$	0	0
$509$	395.509	0.777032	0.388516	−	0.921442i	$-0.372988\pi$
$509$	395.509	0.777032	0.388516	+	0.921442i	$0.372988\pi$
$510$	0	0
$511$	758.688i	1.48471i
$512$	0	0
$513$	71.0592	0.138517
$514$	0	0
$515$	− 722.972i	− 1.40383i
$516$	0	0
$517$	315.282	0.609830
$518$	0	0
$519$	− 52.2116i	− 0.100600i
$520$	0	0
$521$	705.745	1.35460	0.677299	−	0.735708i	$-0.263151\pi$
$521$	705.745	1.35460	0.677299	+	0.735708i	$0.263151\pi$
$522$	0	0
$523$	264.122i	0.505013i	0.967595	+	0.252506i	$0.0812549\pi$
$523$	264.122i	0.505013i	−0.967595	+	0.252506i	$0.918745\pi$
$524$	0	0
$525$	−4.34284	−0.00827208
$526$	0	0
$527$	35.6726i	0.0676899i
$528$	0	0
$529$	237.309	0.448599
$530$	0	0
$531$	644.637i	1.21401i
$532$	0	0
$533$	−427.752	−0.802536
$534$	0	0
$535$	− 233.967i	− 0.437321i
$536$	0	0
$537$	17.6992	0.0329593
$538$	0	0
$539$	346.493i	0.642845i
$540$	0	0
$541$	168.679	0.311792	0.155896	−	0.987774i	$-0.450174\pi$
$541$	168.679	0.311792	0.155896	+	0.987774i	$0.450174\pi$
$542$	0	0
$543$	43.0645i	0.0793085i
$544$	0	0
$545$	5.25091	0.00963470
$546$	0	0
$547$	− 200.072i	− 0.365762i	−0.983135	−	0.182881i	$-0.941458\pi$
$547$	− 200.072i	− 0.365762i	0.983135	−	0.182881i	$-0.0585422\pi$
$548$	0	0
$549$	−215.531	−0.392588
$550$	0	0
$551$	− 131.308i	− 0.238309i
$552$	0	0
$553$	−1048.71	−1.89640
$554$	0	0
$555$	− 27.7491i	− 0.0499984i
$556$	0	0
$557$	−531.576	−0.954355	−0.477178	−	0.878807i	$-0.658340\pi$
$557$	−531.576	−0.954355	−0.477178	+	0.878807i	$0.658340\pi$
$558$	0	0
$559$	683.715i	1.22310i
$560$	0	0
$561$	10.9618	0.0195398
$562$	0	0
$563$	− 431.828i	− 0.767012i	−0.923538	−	0.383506i	$-0.874716\pi$
$563$	− 431.828i	− 0.767012i	0.923538	−	0.383506i	$-0.125284\pi$
$564$	0	0
$565$	77.6088	0.137361
$566$	0	0
$567$	771.656i	1.36095i
$568$	0	0
$569$	−296.778	−0.521578	−0.260789	−	0.965396i	$-0.583983\pi$
$569$	−296.778	−0.521578	−0.260789	+	0.965396i	$0.583983\pi$
$570$	0	0
$571$	− 491.745i	− 0.861201i	−0.902543	−	0.430600i	$-0.858302\pi$
$571$	− 491.745i	− 0.861201i	0.902543	−	0.430600i	$-0.141698\pi$
$572$	0	0
$573$	−22.1194	−0.0386027
$574$	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
$579$	14.1142i	0.0243769i
$580$	0	0
$581$	783.387	1.34834
$582$	0	0
$583$	− 120.289i	− 0.206328i
$584$	0	0
$585$	−414.091	−0.707847
$586$	0	0
$587$	906.775i	1.54476i	0.635160	+	0.772381i	$0.280934\pi$
$587$	906.775i	1.54476i	−0.635160	+	0.772381i	$0.719066\pi$
$588$	0	0
$589$	100.493	0.170616
$590$	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
$595$	− 341.867i	− 0.574566i
$596$	0	0
$597$	−32.8011	−0.0549433
$598$	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	$-0.476337\pi$
$601$	89.2746	0.148543	0.0742717	+	0.997238i	$0.476337\pi$
$602$	0	0
$603$	290.324i	0.481466i
$604$	0	0
$605$	−313.286	−0.517829
$606$	0	0
$607$	316.002i	0.520596i	0.965528	+	0.260298i	$0.0838208\pi$
$607$	316.002i	0.520596i	−0.965528	+	0.260298i	$0.916179\pi$
$608$	0	0
$609$	−13.7402	−0.0225619
$610$	0	0
$611$	358.320i	0.586448i
$612$	0	0
$613$	271.534	0.442959	0.221479	−	0.975165i	$-0.428911\pi$
$613$	271.534	0.442959	0.221479	+	0.975165i	$0.428911\pi$
$614$	0	0
$615$	52.0385i	0.0846154i
$616$	0	0
$617$	−105.762	−0.171413	−0.0857066	−	0.996320i	$-0.527315\pi$
$617$	−105.762	−0.171413	−0.0857066	+	0.996320i	$0.527315\pi$
$618$	0	0
$619$	− 783.218i	− 1.26530i	−0.774440	−	0.632648i	$-0.781968\pi$
$619$	− 783.218i	− 1.26530i	0.774440	−	0.632648i	$-0.218032\pi$
$620$	0	0
$621$	−63.4823	−0.102226
$622$	0	0
$623$	− 426.468i	− 0.684539i
$624$	0	0
$625$	−674.560	−1.07930
$626$	0	0
$627$	− 30.8805i	− 0.0492512i
$628$	0	0
$629$	174.531	0.277473
$630$	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
$635$	557.020i	0.877198i
$636$	0	0
$637$	−393.792	−0.618197
$638$	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
$643$	− 526.815i	− 0.819308i	−0.912241	−	0.409654i	$-0.865649\pi$
$643$	− 526.815i	− 0.819308i	0.912241	−	0.409654i	$-0.134351\pi$
$644$	0	0
$645$	83.1778	0.128958
$646$	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
$651$	− 10.5156i	− 0.0161531i
$652$	0	0
$653$	−19.4487	−0.0297836	−0.0148918	−	0.999889i	$-0.504740\pi$
$653$	−19.4487	−0.0297836	−0.0148918	+	0.999889i	$0.504740\pi$
$654$	0	0
$655$	1129.33i	1.72417i
$656$	0	0
$657$	703.165	1.07027
$658$	0	0
$659$	− 400.433i	− 0.607637i	−0.952730	−	0.303818i	$-0.901738\pi$
$659$	− 400.433i	− 0.607637i	0.952730	−	0.303818i	$-0.0982616\pi$
$660$	0	0
$661$	406.636	0.615182	0.307591	−	0.951519i	$-0.400477\pi$
$661$	406.636	0.615182	0.307591	+	0.951519i	$0.400477\pi$
$662$	0	0
$663$	12.4582i	0.0187907i
$664$	0	0
$665$	−963.069	−1.44822
$666$	0	0
$667$	117.307i	0.175873i
$668$	0	0
$669$	3.26423	0.00487927
$670$	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
$675$	8.06930i	0.0119545i
$676$	0	0
$677$	−294.639	−0.435213	−0.217606	−	0.976037i	$-0.569825\pi$
$677$	−294.639	−0.435213	−0.217606	+	0.976037i	$0.569825\pi$
$678$	0	0
$679$	1089.18i	1.60409i
$680$	0	0
$681$	−58.3382	−0.0856655
$682$	0	0
$683$	310.626i	0.454796i	0.973802	+	0.227398i	$0.0730218\pi$
$683$	310.626i	0.454796i	−0.973802	+	0.227398i	$0.926978\pi$
$684$	0	0
$685$	391.829	0.572013
$686$	0	0
$687$	− 67.4808i	− 0.0982254i
$688$	0	0
$689$	136.710	0.198418
$690$	0	0
$691$	− 980.037i	− 1.41829i	−0.705063	−	0.709144i	$-0.749081\pi$
$691$	− 980.037i	− 1.41829i	0.705063	−	0.709144i	$-0.250919\pi$
$692$	0	0
$693$	675.534	0.974797
$694$	0	0
$695$	− 791.905i	− 1.13943i
$696$	0	0
$697$	−327.301	−0.469585
$698$	0	0
$699$	71.3690i	0.102102i
$700$	0	0
$701$	276.305	0.394158	0.197079	−	0.980388i	$-0.436854\pi$
$701$	276.305	0.394158	0.197079	+	0.980388i	$0.436854\pi$
$702$	0	0
$703$	− 491.668i	− 0.699386i
$704$	0	0
$705$	43.5917	0.0618322
$706$	0	0
$707$	− 1331.03i	− 1.88264i
$708$	0	0
$709$	−449.838	−0.634468	−0.317234	−	0.948347i	$-0.602754\pi$
$709$	−449.838	−0.634468	−0.317234	+	0.948347i	$0.602754\pi$
$710$	0	0
$711$	971.962i	1.36703i
$712$	0	0
$713$	−89.7775	−0.125915
$714$	0	0
$715$	360.767i	0.504569i
$716$	0	0
$717$	15.9793	0.0222864
$718$	0	0
$719$	− 1122.38i	− 1.56103i	−0.625139	−	0.780514i	$-0.714958\pi$
$719$	− 1122.38i	− 1.56103i	0.625139	−	0.780514i	$-0.285042\pi$
$720$	0	0
$721$	−1340.43	−1.85913
$722$	0	0
$723$	60.7486i	0.0840229i
$724$	0	0
$725$	14.9110	0.0205669
$726$	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
$731$	523.155i	0.715670i
$732$	0	0
$733$	−372.110	−0.507653	−0.253826	−	0.967250i	$-0.581689\pi$
$733$	−372.110	−0.507653	−0.253826	+	0.967250i	$0.581689\pi$
$734$	0	0
$735$	47.9070i	0.0651797i
$736$	0	0
$737$	252.938	0.343200
$738$	0	0
$739$	− 62.9975i	− 0.0852469i	−0.999091	−	0.0426235i	$-0.986428\pi$
$739$	− 62.9975i	− 0.0852469i	0.999091	−	0.0426235i	$-0.0135716\pi$
$740$	0	0
$741$	35.0959	0.0473629
$742$	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
$747$	− 726.057i	− 0.971964i
$748$	0	0
$749$	−433.789	−0.579157
$750$	0	0
$751$	1342.93i	1.78819i	0.447876	+	0.894095i	$0.352180\pi$
$751$	1342.93i	1.78819i	−0.447876	+	0.894095i	$0.647820\pi$
$752$	0	0
$753$	−23.3108	−0.0309573
$754$	0	0
$755$	− 1149.44i	− 1.52244i
$756$	0	0
$757$	−558.375	−0.737615	−0.368808	−	0.929506i	$-0.620234\pi$
$757$	−558.375	−0.737615	−0.368808	+	0.929506i	$0.620234\pi$
$758$	0	0
$759$	27.5878i	0.0363475i
$760$	0	0
$761$	480.213	0.631029	0.315514	−	0.948921i	$-0.397823\pi$
$761$	480.213	0.631029	0.315514	+	0.948921i	$0.397823\pi$
$762$	0	0
$763$	− 9.73550i	− 0.0127595i
$764$	0	0
$765$	−316.848	−0.414180
$766$	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
$771$	45.9234i	0.0595634i
$772$	0	0
$773$	−1213.02	−1.56924	−0.784619	−	0.619979i	$-0.787141\pi$
$773$	−1213.02	−1.56924	−0.784619	+	0.619979i	$0.787141\pi$
$774$	0	0
$775$	11.4117i	0.0147248i
$776$	0	0
$777$	−51.4485	−0.0662143
$778$	0	0
$779$	922.036i	1.18362i
$780$	0	0
$781$	402.708	0.515632
$782$	0	0
$783$	25.5302i	0.0326056i
$784$	0	0
$785$	807.648	1.02885
$786$	0	0
$787$	− 240.919i	− 0.306123i	−0.988217	−	0.153062i	$-0.951087\pi$
$787$	− 240.919i	− 0.306123i	0.988217	−	0.153062i	$-0.0489133\pi$
$788$	0	0
$789$	77.4612	0.0981764
$790$	0	0
$791$	− 143.891i	− 0.181911i
$792$	0	0
$793$	−213.409	−0.269116
$794$	0	0
$795$	− 16.6315i	− 0.0209202i
$796$	0	0
$797$	1181.68	1.48266	0.741328	−	0.671143i	$-0.234197\pi$
$797$	1181.68	1.48266	0.741328	+	0.671143i	$0.234197\pi$
$798$	0	0
$799$	274.174i	0.343147i
$800$	0	0
$801$	−395.258	−0.493456
$802$	0	0
$803$	− 612.616i	− 0.762909i
$804$	0	0
$805$	860.379	1.06879
$806$	0	0
$807$	104.789i	0.129850i
$808$	0	0
$809$	−371.926	−0.459735	−0.229868	−	0.973222i	$-0.573829\pi$
$809$	−371.926	−0.459735	−0.229868	+	0.973222i	$0.573829\pi$
$810$	0	0
$811$	389.798i	0.480639i	0.970694	+	0.240320i	$0.0772522\pi$
$811$	389.798i	0.480639i	−0.970694	+	0.240320i	$0.922748\pi$
$812$	0	0
$813$	74.3165	0.0914102
$814$	0	0
$815$	− 418.550i	− 0.513559i
$816$	0	0
$817$	1473.77	1.80389
$818$	0	0
$819$	767.749i	0.937422i
$820$	0	0
$821$	−1279.76	−1.55878	−0.779388	−	0.626542i	$-0.784470\pi$
$821$	−1279.76	−1.55878	−0.779388	+	0.626542i	$0.784470\pi$
$822$	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
$827$	1022.20i	1.23603i	0.786165	+	0.618017i	$0.212064\pi$
$827$	1022.20i	1.23603i	−0.786165	+	0.618017i	$0.787936\pi$
$828$	0	0
$829$	−405.003	−0.488544	−0.244272	−	0.969707i	$-0.578549\pi$
$829$	−405.003	−0.488544	−0.244272	+	0.969707i	$0.578549\pi$
$830$	0	0
$831$	− 102.972i	− 0.123914i
$832$	0	0
$833$	−301.316	−0.361724
$834$	0	0
$835$	556.059i	0.665939i
$836$	0	0
$837$	−19.5388	−0.0233438
$838$	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
$843$	39.6162i	0.0469943i
$844$	0	0
$845$	470.911	0.557291
$846$	0	0
$847$	580.852i	0.685776i
$848$	0	0
$849$	9.16597	0.0107962
$850$	0	0
$851$	439.243i	0.516149i
$852$	0	0
$853$	−945.053	−1.10792	−0.553958	−	0.832544i	$-0.686883\pi$
$853$	−945.053	−1.10792	−0.553958	+	0.832544i	$0.686883\pi$
$854$	0	0
$855$	892.589i	1.04396i
$856$	0	0
$857$	488.688	0.570230	0.285115	−	0.958493i	$-0.407968\pi$
$857$	488.688	0.570230	0.285115	+	0.958493i	$0.407968\pi$
$858$	0	0
$859$	− 380.166i	− 0.442568i	−0.975209	−	0.221284i	$-0.928975\pi$
$859$	− 380.166i	− 0.442568i	0.975209	−	0.221284i	$-0.0710247\pi$
$860$	0	0
$861$	96.4824	0.112059
$862$	0	0
$863$	− 152.667i	− 0.176903i	−0.996080	−	0.0884514i	$-0.971808\pi$
$863$	− 152.667i	− 0.176903i	0.996080	−	0.0884514i	$-0.0281918\pi$
$864$	0	0
$865$	1314.82	1.52002
$866$	0	0
$867$	− 50.2881i	− 0.0580024i
$868$	0	0
$869$	846.799	0.974452
$870$	0	0
$871$	287.466i	0.330041i
$872$	0	0
$873$	1009.47	1.15632
$874$	0	0
$875$	1150.05i	1.31434i
$876$	0	0
$877$	−230.004	−0.262262	−0.131131	−	0.991365i	$-0.541861\pi$
$877$	−230.004	−0.262262	−0.131131	+	0.991365i	$0.541861\pi$
$878$	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
$883$	325.304i	0.368408i	0.982888	+	0.184204i	$0.0589707\pi$
$883$	325.304i	0.368408i	−0.982888	+	0.184204i	$0.941029\pi$
$884$	0	0
$885$	77.6517	0.0877421
$886$	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
$891$	− 623.087i	− 0.699312i
$892$	0	0
$893$	772.373	0.864920
$894$	0	0
$895$	445.709i	0.497999i
$896$	0	0
$897$	−31.3537	−0.0349539
$898$	0	0
$899$	36.1051i	0.0401614i
$900$	0	0
$901$	104.606	0.116099
$902$	0	0
$903$	− 154.217i	− 0.170783i
$904$	0	0
$905$	−1084.47	−1.19831
$906$	0	0
$907$	− 31.4326i	− 0.0346556i	−0.999850	−	0.0173278i	$-0.994484\pi$
$907$	− 31.4326i	− 0.0346556i	0.999850	−	0.0173278i	$-0.00551588\pi$
$908$	0	0
$909$	−1233.62	−1.35712
$910$	0	0
$911$	1399.85i	1.53661i	0.640083	+	0.768306i	$0.278900\pi$
$911$	1399.85i	1.53661i	−0.640083	+	0.768306i	$0.721100\pi$
$912$	0	0
$913$	−632.560	−0.692837
$914$	0	0
$915$	25.9624i	0.0283742i
$916$	0	0
$917$	2093.85	2.28337
$918$	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
$923$	457.681i	0.495862i
$924$	0	0
$925$	55.8326	0.0603595
$926$	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
$931$	848.834i	0.911744i
$932$	0	0
$933$	27.0982	0.0290442
$934$	0	0
$935$	276.046i	0.295237i
$936$	0	0
$937$	−598.181	−0.638400	−0.319200	−	0.947687i	$-0.603414\pi$
$937$	−598.181	−0.638400	−0.319200	+	0.947687i	$0.603414\pi$
$938$	0	0
$939$	10.7295i	0.0114265i
$940$	0	0
$941$	1382.88	1.46958	0.734791	−	0.678294i	$-0.237280\pi$
$941$	1382.88	1.46958	0.734791	+	0.678294i	$0.237280\pi$
$942$	0	0
$943$	− 823.721i	− 0.873511i
$944$	0	0
$945$	187.249	0.198147
$946$	0	0
$947$	1170.80i	1.23633i	0.786050	+	0.618163i	$0.212123\pi$
$947$	1170.80i	1.23633i	−0.786050	+	0.618163i	$0.787877\pi$
$948$	0	0
$949$	696.242	0.733659
$950$	0	0
$951$	− 32.0939i	− 0.0337476i
$952$	0	0
$953$	−1846.78	−1.93786	−0.968930	−	0.247333i	$-0.920446\pi$
$953$	−1846.78	−1.93786	−0.968930	+	0.247333i	$0.920446\pi$
$954$	0	0
$955$	− 557.020i	− 0.583267i
$956$	0	0
$957$	11.0947	0.0115933
$958$	0	0
$959$	− 726.475i	− 0.757534i
$960$	0	0
$961$	933.368	0.971247
$962$	0	0
$963$	402.043i	0.417490i
$964$	0	0
$965$	−355.431	−0.368323
$966$	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
$971$	− 1642.32i	− 1.69137i	−0.533683	−	0.845685i	$-0.679192\pi$
$971$	− 1642.32i	− 1.69137i	0.533683	−	0.845685i	$-0.320808\pi$
$972$	0	0
$973$	−1468.24	−1.50898
$974$	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
$979$	344.359i	0.351746i
$980$	0	0
$981$	−9.02303	−0.00919779
$982$	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
$987$	− 80.8216i	− 0.0818861i
$988$	0	0
$989$	−1316.63	−1.33127
$990$	0	0
$991$	− 675.783i	− 0.681920i	−0.940078	−	0.340960i	$-0.889248\pi$
$991$	− 675.783i	− 0.681920i	0.940078	−	0.340960i	$-0.110752\pi$
$992$	0	0
$993$	94.6184	0.0952854
$994$	0	0
$995$	− 826.014i	− 0.830165i
$996$	0	0
$997$	−13.3638	−0.0134040	−0.00670200	−	0.999978i	$-0.502133\pi$
$997$	−13.3638	−0.0134040	−0.00670200	+	0.999978i	$0.502133\pi$
$998$	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.c.j.1023.7		12
4.3	odd	2	inner	1024.3.c.j.1023.5		12
8.3	odd	2	inner	1024.3.c.j.1023.8		12
8.5	even	2	inner	1024.3.c.j.1023.6		12
16.3	odd	4		1024.3.d.k.511.8		12
16.5	even	4		1024.3.d.k.511.7		12
16.11	odd	4		1024.3.d.k.511.5		12
16.13	even	4		1024.3.d.k.511.6		12
32.3	odd	8		16.3.f.a.3.2	✓	6
32.5	even	8		128.3.f.b.31.2		6
32.11	odd	8		64.3.f.a.15.2		6
32.13	even	8		128.3.f.a.95.2		6
32.19	odd	8		128.3.f.b.95.2		6
32.21	even	8		16.3.f.a.11.2	yes	6
32.27	odd	8		128.3.f.a.31.2		6
32.29	even	8		64.3.f.a.47.2		6
96.5	odd	8		1152.3.m.a.415.3		6
96.11	even	8		576.3.m.a.271.1		6
96.29	odd	8		576.3.m.a.559.1		6
96.35	even	8		144.3.m.a.19.2		6
96.53	odd	8		144.3.m.a.91.2		6
96.59	even	8		1152.3.m.b.415.3		6
96.77	odd	8		1152.3.m.b.991.3		6
96.83	even	8		1152.3.m.a.991.3		6
160.3	even	8		400.3.k.c.99.3		6
160.53	odd	8		400.3.k.d.299.1		6
160.67	even	8		400.3.k.d.99.1		6
160.99	odd	8		400.3.r.c.51.2		6
160.117	odd	8		400.3.k.c.299.3		6
160.149	even	8		400.3.r.c.251.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.3.f.a.3.2	✓	6	32.3	odd	8
16.3.f.a.11.2	yes	6	32.21	even	8
64.3.f.a.15.2		6	32.11	odd	8
64.3.f.a.47.2		6	32.29	even	8
128.3.f.a.31.2		6	32.27	odd	8
128.3.f.a.95.2		6	32.13	even	8
128.3.f.b.31.2		6	32.5	even	8
128.3.f.b.95.2		6	32.19	odd	8
144.3.m.a.19.2		6	96.35	even	8
144.3.m.a.91.2		6	96.53	odd	8
400.3.k.c.99.3		6	160.3	even	8
400.3.k.c.299.3		6	160.117	odd	8
400.3.k.d.99.1		6	160.67	even	8
400.3.k.d.299.1		6	160.53	odd	8
400.3.r.c.51.2		6	160.99	odd	8
400.3.r.c.251.2		6	160.149	even	8
576.3.m.a.271.1		6	96.11	even	8
576.3.m.a.559.1		6	96.29	odd	8
1024.3.c.j.1023.5		12	4.3	odd	2	inner
1024.3.c.j.1023.6		12	8.5	even	2	inner
1024.3.c.j.1023.7		12	1.1	even	1	trivial
1024.3.c.j.1023.8		12	8.3	odd	2	inner
1024.3.d.k.511.5		12	16.11	odd	4
1024.3.d.k.511.6		12	16.13	even	4
1024.3.d.k.511.7		12	16.5	even	4
1024.3.d.k.511.8		12	16.3	odd	4
1152.3.m.a.415.3		6	96.5	odd	8
1152.3.m.a.991.3		6	96.83	even	8
1152.3.m.b.415.3		6	96.59	even	8
1152.3.m.b.991.3		6	96.77	odd	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.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.206992i q^{3} -5.21257 q^{5} +9.66442i q^{7} +8.95715 q^{9} +O(q^{10})\)	\(q+0.206992i q^{3} -5.21257 q^{5} +9.66442i q^{7} +8.95715 q^{9} -7.80371i q^{11} +8.86897 q^{13} -1.07896i q^{15} +6.78623 q^{17} -19.1174i q^{19} -2.00046 q^{21} +17.0790i q^{23} +2.17092 q^{25} +3.71699i q^{27} +6.86851 q^{29} +5.25662i q^{31} +1.61531 q^{33} -50.3765i q^{35} +25.7183 q^{37} +1.83581i q^{39} -48.2302 q^{41} +77.0907i q^{43} -46.6898 q^{45} +40.4015i q^{47} -44.4011 q^{49} +1.40470i q^{51} +15.4144 q^{53} +40.6774i q^{55} +3.95715 q^{57} +71.9690i q^{59} -24.0624 q^{61} +86.5657i q^{63} -46.2302 q^{65} +32.4126i q^{67} -3.53521 q^{69} +51.6047i q^{71} +78.5032 q^{73} +0.449364i q^{75} +75.4184 q^{77} +108.512i q^{79} +79.8450 q^{81} -81.0589i q^{83} -35.3737 q^{85} +1.42173i q^{87} -44.1276 q^{89} +85.7135i q^{91} -1.08808 q^{93} +99.6510i q^{95} +112.700 q^{97} -69.8991i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^9	\( 12 q - 12 q^{9} + 8 q^{17} - 20 q^{25} - 8 q^{33} + 92 q^{49} - 72 q^{57} + 24 q^{65} - 96 q^{73} - 172 q^{81} + 160 q^{89} - 8 q^{97}+O(q^{100}) \) 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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