LMFDB - Embedded newform 1444.3.c.d.721.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1444,3,Mod(721,1444)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1444, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("1444.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1444 = 2^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1444.c (of order $2$, degree $1$, 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:	$39.3461501736$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			721.2
Character	$\chi$	$=$	1444.721
Dual form			1444.3.c.d.721.23

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.26351i q^{3} +0.413588 q^{5} +11.9368 q^{7} -18.7045 q^{9} +O(q^{10})$	$q-5.26351i q^{3} +0.413588 q^{5} +11.9368 q^{7} -18.7045 q^{9} -2.31281 q^{11} +15.3811i q^{13} -2.17693i q^{15} +28.1989 q^{17} -62.8293i q^{21} +23.3288 q^{23} -24.8289 q^{25} +51.0800i q^{27} -25.3152i q^{29} -22.4453i q^{31} +12.1735i q^{33} +4.93690 q^{35} -0.679928i q^{37} +80.9587 q^{39} -77.1676i q^{41} -40.3183 q^{43} -7.73598 q^{45} +54.2466 q^{47} +93.4862 q^{49} -148.425i q^{51} -62.2490i q^{53} -0.956551 q^{55} -34.3802i q^{59} +91.7505 q^{61} -223.272 q^{63} +6.36145i q^{65} +32.8505i q^{67} -122.791i q^{69} +53.6912i q^{71} +15.8025 q^{73} +130.687i q^{75} -27.6074 q^{77} +27.6923i q^{79} +100.519 q^{81} -27.3740 q^{83} +11.6627 q^{85} -133.247 q^{87} +67.4693i q^{89} +183.601i q^{91} -118.141 q^{93} -115.959i q^{97} +43.2601 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 12 q^{5} + 40 q^{7} - 104 q^{9}+O(q^{10}) $ 24 * q - 12 * q^5 + 40 * q^7 - 104 * q^9	$ 24 q - 12 q^{5} + 40 q^{7} - 104 q^{9} - 64 q^{11} - 60 q^{17} + 100 q^{23} + 24 q^{25} - 144 q^{35} + 68 q^{39} - 16 q^{43} + 248 q^{45} + 180 q^{47} - 208 q^{49} - 532 q^{55} + 104 q^{61} - 1216 q^{63} - 484 q^{73} + 136 q^{77} + 460 q^{81} + 744 q^{83} + 32 q^{85} - 1296 q^{87} - 256 q^{93} + 804 q^{99}+O(q^{100}) $ 24 * q - 12 * q^5 + 40 * q^7 - 104 * q^9 - 64 * q^11 - 60 * q^17 + 100 * q^23 + 24 * q^25 - 144 * q^35 + 68 * q^39 - 16 * q^43 + 248 * q^45 + 180 * q^47 - 208 * q^49 - 532 * q^55 + 104 * q^61 - 1216 * q^63 - 484 * q^73 + 136 * q^77 + 460 * q^81 + 744 * q^83 + 32 * q^85 - 1296 * q^87 - 256 * q^93 + 804 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1444\mathbb{Z}\right)^\times$.

$n$	$723$	$1085$
$\chi(n)$	$1$	$-1$

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
$3$	− 5.26351i	− 1.75450i	−0.480031	−	0.877252i	$-0.659374\pi$
$3$	− 5.26351i	− 1.75450i	0.480031	−	0.877252i	$-0.340626\pi$
$4$	0	0
$5$	0.413588	0.0827176	0.0413588	−	0.999144i	$-0.486831\pi$
$5$	0.413588	0.0827176	0.0413588	+	0.999144i	$0.486831\pi$
$6$	0	0
$7$	11.9368	1.70525	0.852626	−	0.522522i	$-0.175009\pi$
$7$	11.9368	1.70525	0.852626	+	0.522522i	$0.175009\pi$
$8$	0	0
$9$	−18.7045	−2.07828
$10$	0	0
$11$	−2.31281	−0.210255	−0.105128	−	0.994459i	$-0.533525\pi$
$11$	−2.31281	−0.210255	−0.105128	+	0.994459i	$0.533525\pi$
$12$	0	0
$13$	15.3811i	1.18316i	0.806245	+	0.591582i	$0.201496\pi$
$13$	15.3811i	1.18316i	−0.806245	+	0.591582i	$0.798504\pi$
$14$	0	0
$15$	− 2.17693i	− 0.145128i
$16$	0	0
$17$	28.1989	1.65876	0.829380	−	0.558685i	$-0.188694\pi$
$17$	28.1989	1.65876	0.829380	+	0.558685i	$0.188694\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	− 62.8293i	− 2.99187i
$22$	0	0
$23$	23.3288	1.01430	0.507148	−	0.861859i	$-0.330700\pi$
$23$	23.3288	1.01430	0.507148	+	0.861859i	$0.330700\pi$
$24$	0	0
$25$	−24.8289	−0.993158
$26$	0	0
$27$	51.0800i	1.89185i
$28$	0	0
$29$	− 25.3152i	− 0.872937i	−0.899720	−	0.436468i	$-0.856229\pi$
$29$	− 25.3152i	− 0.872937i	0.899720	−	0.436468i	$-0.143771\pi$
$30$	0	0
$31$	− 22.4453i	− 0.724041i	−0.932170	−	0.362021i	$-0.882087\pi$
$31$	− 22.4453i	− 0.724041i	0.932170	−	0.362021i	$-0.117913\pi$
$32$	0	0
$33$	12.1735i	0.368894i
$34$	0	0
$35$	4.93690	0.141054
$36$	0	0
$37$	− 0.679928i	− 0.0183764i	−0.999958	−	0.00918822i	$-0.997075\pi$
$37$	− 0.679928i	− 0.0183764i	0.999958	−	0.00918822i	$-0.00292474\pi$
$38$	0	0
$39$	80.9587	2.07586
$40$	0	0
$41$	− 77.1676i	− 1.88214i	−0.338216	−	0.941069i	$-0.609823\pi$
$41$	− 77.1676i	− 1.88214i	0.338216	−	0.941069i	$-0.390177\pi$
$42$	0	0
$43$	−40.3183	−0.937635	−0.468817	−	0.883295i	$-0.655320\pi$
$43$	−40.3183	−0.937635	−0.468817	+	0.883295i	$0.655320\pi$
$44$	0	0
$45$	−7.73598	−0.171911
$46$	0	0
$47$	54.2466	1.15418	0.577092	−	0.816679i	$-0.304187\pi$
$47$	54.2466	1.15418	0.577092	+	0.816679i	$0.304187\pi$
$48$	0	0
$49$	93.4862	1.90788
$50$	0	0
$51$	− 148.425i	− 2.91030i
$52$	0	0
$53$	− 62.2490i	− 1.17451i	−0.809402	−	0.587255i	$-0.800209\pi$
$53$	− 62.2490i	− 1.17451i	0.809402	−	0.587255i	$-0.199791\pi$
$54$	0	0
$55$	−0.956551	−0.0173918
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 34.3802i	− 0.582714i	−0.956614	−	0.291357i	$-0.905893\pi$
$59$	− 34.3802i	− 0.582714i	0.956614	−	0.291357i	$-0.0941068\pi$
$60$	0	0
$61$	91.7505	1.50411	0.752053	−	0.659103i	$-0.229064\pi$
$61$	91.7505	1.50411	0.752053	+	0.659103i	$0.229064\pi$
$62$	0	0
$63$	−223.272	−3.54399
$64$	0	0
$65$	6.36145i	0.0978685i
$66$	0	0
$67$	32.8505i	0.490306i	0.969484	+	0.245153i	$0.0788383\pi$
$67$	32.8505i	0.490306i	−0.969484	+	0.245153i	$0.921162\pi$
$68$	0	0
$69$	− 122.791i	− 1.77958i
$70$	0	0
$71$	53.6912i	0.756214i	0.925762	+	0.378107i	$0.123425\pi$
$71$	53.6912i	0.756214i	−0.925762	+	0.378107i	$0.876575\pi$
$72$	0	0
$73$	15.8025	0.216473	0.108236	−	0.994125i	$-0.465480\pi$
$73$	15.8025	0.216473	0.108236	+	0.994125i	$0.465480\pi$
$74$	0	0
$75$	130.687i	1.74250i
$76$	0	0
$77$	−27.6074	−0.358538
$78$	0	0
$79$	27.6923i	0.350536i	0.984521	+	0.175268i	$0.0560792\pi$
$79$	27.6923i	0.350536i	−0.984521	+	0.175268i	$0.943921\pi$
$80$	0	0
$81$	100.519	1.24098
$82$	0	0
$83$	−27.3740	−0.329808	−0.164904	−	0.986310i	$-0.552731\pi$
$83$	−27.3740	−0.329808	−0.164904	+	0.986310i	$0.552731\pi$
$84$	0	0
$85$	11.6627	0.137209
$86$	0	0
$87$	−133.247	−1.53157
$88$	0	0
$89$	67.4693i	0.758082i	0.925380	+	0.379041i	$0.123746\pi$
$89$	67.4693i	0.758082i	−0.925380	+	0.379041i	$0.876254\pi$
$90$	0	0
$91$	183.601i	2.01759i
$92$	0	0
$93$	−118.141	−1.27033
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 115.959i	− 1.19546i	−0.801699	−	0.597728i	$-0.796070\pi$
$97$	− 115.959i	− 1.19546i	0.801699	−	0.597728i	$-0.203930\pi$
$98$	0	0
$99$	43.2601	0.436970
$100$	0	0
$101$	42.5365	0.421154	0.210577	−	0.977577i	$-0.432466\pi$
$101$	42.5365	0.421154	0.210577	+	0.977577i	$0.432466\pi$
$102$	0	0
$103$	− 59.3166i	− 0.575890i	−0.957647	−	0.287945i	$-0.907028\pi$
$103$	− 59.3166i	− 0.575890i	0.957647	−	0.287945i	$-0.0929720\pi$
$104$	0	0
$105$	− 25.9854i	− 0.247480i
$106$	0	0
$107$	153.719i	1.43663i	0.695719	+	0.718314i	$0.255086\pi$
$107$	153.719i	1.43663i	−0.695719	+	0.718314i	$0.744914\pi$
$108$	0	0
$109$	− 125.686i	− 1.15308i	−0.817069	−	0.576539i	$-0.804403\pi$
$109$	− 125.686i	− 1.15308i	0.817069	−	0.576539i	$-0.195597\pi$
$110$	0	0
$111$	−3.57881	−0.0322415
$112$	0	0
$113$	101.999i	0.902643i	0.892361	+	0.451321i	$0.149047\pi$
$113$	101.999i	0.902643i	−0.892361	+	0.451321i	$0.850953\pi$
$114$	0	0
$115$	9.64851	0.0839001
$116$	0	0
$117$	− 287.697i	− 2.45895i
$118$	0	0
$119$	336.604	2.82860
$120$	0	0
$121$	−115.651	−0.955793
$122$	0	0
$123$	−406.173	−3.30222
$124$	0	0
$125$	−20.6087	−0.164869
$126$	0	0
$127$	− 89.3326i	− 0.703406i	−0.936112	−	0.351703i	$-0.885603\pi$
$127$	− 89.3326i	− 0.703406i	0.936112	−	0.351703i	$-0.114397\pi$
$128$	0	0
$129$	212.216i	1.64508i
$130$	0	0
$131$	15.5243	0.118506	0.0592530	−	0.998243i	$-0.481128\pi$
$131$	15.5243	0.118506	0.0592530	+	0.998243i	$0.481128\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	21.1261i	0.156489i
$136$	0	0
$137$	89.4187	0.652692	0.326346	−	0.945250i	$-0.394183\pi$
$137$	89.4187	0.652692	0.326346	+	0.945250i	$0.394183\pi$
$138$	0	0
$139$	80.5416	0.579436	0.289718	−	0.957112i	$-0.406438\pi$
$139$	80.5416	0.579436	0.289718	+	0.957112i	$0.406438\pi$
$140$	0	0
$141$	− 285.528i	− 2.02502i
$142$	0	0
$143$	− 35.5736i	− 0.248767i
$144$	0	0
$145$	− 10.4701i	− 0.0722073i
$146$	0	0
$147$	− 492.066i	− 3.34738i
$148$	0	0
$149$	−174.416	−1.17058	−0.585289	−	0.810825i	$-0.699019\pi$
$149$	−174.416	−1.17058	−0.585289	+	0.810825i	$0.699019\pi$
$150$	0	0
$151$	− 249.140i	− 1.64993i	−0.565183	−	0.824966i	$-0.691194\pi$
$151$	− 249.140i	− 1.64993i	0.565183	−	0.824966i	$-0.308806\pi$
$152$	0	0
$153$	−527.448	−3.44737
$154$	0	0
$155$	− 9.28310i	− 0.0598910i
$156$	0	0
$157$	46.1555	0.293984	0.146992	−	0.989138i	$-0.453041\pi$
$157$	46.1555	0.293984	0.146992	+	0.989138i	$0.453041\pi$
$158$	0	0
$159$	−327.648	−2.06068
$160$	0	0
$161$	278.470	1.72963
$162$	0	0
$163$	−241.830	−1.48362	−0.741811	−	0.670609i	$-0.766033\pi$
$163$	−241.830	−1.48362	−0.741811	+	0.670609i	$0.766033\pi$
$164$	0	0
$165$	5.03482i	0.0305140i
$166$	0	0
$167$	90.4040i	0.541342i	0.962672	+	0.270671i	$0.0872455\pi$
$167$	90.4040i	0.541342i	−0.962672	+	0.270671i	$0.912755\pi$
$168$	0	0
$169$	−67.5790	−0.399876
$170$	0	0
$171$	0	0
$172$	0	0
$173$	213.804i	1.23586i	0.786232	+	0.617931i	$0.212029\pi$
$173$	213.804i	1.23586i	−0.786232	+	0.617931i	$0.787971\pi$
$174$	0	0
$175$	−296.377	−1.69358
$176$	0	0
$177$	−180.960	−1.02237
$178$	0	0
$179$	250.297i	1.39831i	0.714973	+	0.699153i	$0.246439\pi$
$179$	250.297i	1.39831i	−0.714973	+	0.699153i	$0.753561\pi$
$180$	0	0
$181$	− 80.4569i	− 0.444513i	−0.974988	−	0.222257i	$-0.928658\pi$
$181$	− 80.4569i	− 0.444513i	0.974988	−	0.222257i	$-0.0713423\pi$
$182$	0	0
$183$	− 482.930i	− 2.63896i
$184$	0	0
$185$	− 0.281210i	− 0.00152006i
$186$	0	0
$187$	−65.2187	−0.348763
$188$	0	0
$189$	609.729i	3.22608i
$190$	0	0
$191$	−44.9849	−0.235523	−0.117762	−	0.993042i	$-0.537572\pi$
$191$	−44.9849	−0.235523	−0.117762	+	0.993042i	$0.537572\pi$
$192$	0	0
$193$	− 63.9232i	− 0.331209i	−0.986192	−	0.165604i	$-0.947043\pi$
$193$	− 63.9232i	− 0.331209i	0.986192	−	0.165604i	$-0.0529574\pi$
$194$	0	0
$195$	33.4836	0.171711
$196$	0	0
$197$	−293.074	−1.48769	−0.743843	−	0.668354i	$-0.766999\pi$
$197$	−293.074	−1.48769	−0.743843	+	0.668354i	$0.766999\pi$
$198$	0	0
$199$	−107.812	−0.541768	−0.270884	−	0.962612i	$-0.587316\pi$
$199$	−107.812	−0.541768	−0.270884	+	0.962612i	$0.587316\pi$
$200$	0	0
$201$	172.909	0.860244
$202$	0	0
$203$	− 302.181i	− 1.48858i
$204$	0	0
$205$	− 31.9156i	− 0.155686i
$206$	0	0
$207$	−436.354	−2.10799
$208$	0	0
$209$	0	0
$210$	0	0
$211$	67.2498i	0.318719i	0.987221	+	0.159360i	$0.0509429\pi$
$211$	67.2498i	0.318719i	−0.987221	+	0.159360i	$0.949057\pi$
$212$	0	0
$213$	282.604	1.32678
$214$	0	0
$215$	−16.6752	−0.0775589
$216$	0	0
$217$	− 267.924i	− 1.23467i
$218$	0	0
$219$	− 83.1766i	− 0.379802i
$220$	0	0
$221$	433.731i	1.96258i
$222$	0	0
$223$	47.0616i	0.211039i	0.994417	+	0.105519i	$0.0336505\pi$
$223$	47.0616i	0.211039i	−0.994417	+	0.105519i	$0.966349\pi$
$224$	0	0
$225$	464.414	2.06406
$226$	0	0
$227$	− 218.041i	− 0.960535i	−0.877122	−	0.480267i	$-0.840540\pi$
$227$	− 218.041i	− 0.960535i	0.877122	−	0.480267i	$-0.159460\pi$
$228$	0	0
$229$	343.739	1.50104	0.750522	−	0.660845i	$-0.229802\pi$
$229$	343.739	1.50104	0.750522	+	0.660845i	$0.229802\pi$
$230$	0	0
$231$	145.312i	0.629057i
$232$	0	0
$233$	163.366	0.701142	0.350571	−	0.936536i	$-0.385987\pi$
$233$	163.366	0.701142	0.350571	+	0.936536i	$0.385987\pi$
$234$	0	0
$235$	22.4358	0.0954713
$236$	0	0
$237$	145.759	0.615017
$238$	0	0
$239$	172.340	0.721089	0.360545	−	0.932742i	$-0.382591\pi$
$239$	172.340	0.721089	0.360545	+	0.932742i	$0.382591\pi$
$240$	0	0
$241$	− 192.395i	− 0.798321i	−0.916881	−	0.399160i	$-0.869302\pi$
$241$	− 192.395i	− 0.798321i	0.916881	−	0.399160i	$-0.130698\pi$
$242$	0	0
$243$	− 69.3636i	− 0.285447i
$244$	0	0
$245$	38.6648	0.157815
$246$	0	0
$247$	0	0
$248$	0	0
$249$	144.084i	0.578649i
$250$	0	0
$251$	32.7510	0.130482	0.0652411	−	0.997870i	$-0.479218\pi$
$251$	32.7510	0.130482	0.0652411	+	0.997870i	$0.479218\pi$
$252$	0	0
$253$	−53.9550	−0.213261
$254$	0	0
$255$	− 61.3869i	− 0.240733i
$256$	0	0
$257$	432.950i	1.68463i	0.538984	+	0.842316i	$0.318808\pi$
$257$	432.950i	1.68463i	−0.538984	+	0.842316i	$0.681192\pi$
$258$	0	0
$259$	− 8.11614i	− 0.0313365i
$260$	0	0
$261$	473.509i	1.81421i
$262$	0	0
$263$	−183.159	−0.696422	−0.348211	−	0.937416i	$-0.613211\pi$
$263$	−183.159	−0.696422	−0.348211	+	0.937416i	$0.613211\pi$
$264$	0	0
$265$	− 25.7454i	− 0.0971526i
$266$	0	0
$267$	355.125	1.33006
$268$	0	0
$269$	32.5512i	0.121008i	0.998168	+	0.0605040i	$0.0192708\pi$
$269$	32.5512i	0.121008i	−0.998168	+	0.0605040i	$0.980729\pi$
$270$	0	0
$271$	−153.982	−0.568200	−0.284100	−	0.958795i	$-0.591695\pi$
$271$	−153.982	−0.568200	−0.284100	+	0.958795i	$0.591695\pi$
$272$	0	0
$273$	966.385	3.53987
$274$	0	0
$275$	57.4246	0.208817
$276$	0	0
$277$	−207.479	−0.749020	−0.374510	−	0.927223i	$-0.622189\pi$
$277$	−207.479	−0.749020	−0.374510	+	0.927223i	$0.622189\pi$
$278$	0	0
$279$	419.829i	1.50476i
$280$	0	0
$281$	217.122i	0.772676i	0.922357	+	0.386338i	$0.126260\pi$
$281$	217.122i	0.772676i	−0.922357	+	0.386338i	$0.873740\pi$
$282$	0	0
$283$	346.880	1.22572	0.612862	−	0.790190i	$-0.290018\pi$
$283$	346.880	1.22572	0.612862	+	0.790190i	$0.290018\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 921.131i	− 3.20952i
$288$	0	0
$289$	506.179	1.75148
$290$	0	0
$291$	−610.352	−2.09743
$292$	0	0
$293$	271.073i	0.925165i	0.886576	+	0.462583i	$0.153077\pi$
$293$	271.073i	0.925165i	−0.886576	+	0.462583i	$0.846923\pi$
$294$	0	0
$295$	− 14.2192i	− 0.0482008i
$296$	0	0
$297$	− 118.138i	− 0.397772i
$298$	0	0
$299$	358.823i	1.20008i
$300$	0	0
$301$	−481.270	−1.59890
$302$	0	0
$303$	− 223.891i	− 0.738916i
$304$	0	0
$305$	37.9469	0.124416
$306$	0	0
$307$	− 40.5198i	− 0.131986i	−0.997820	−	0.0659932i	$-0.978978\pi$
$307$	− 40.5198i	− 0.131986i	0.997820	−	0.0659932i	$-0.0210216\pi$
$308$	0	0
$309$	−312.214	−1.01040
$310$	0	0
$311$	62.4872	0.200923	0.100462	−	0.994941i	$-0.467968\pi$
$311$	62.4872	0.200923	0.100462	+	0.994941i	$0.467968\pi$
$312$	0	0
$313$	246.361	0.787096	0.393548	−	0.919304i	$-0.371248\pi$
$313$	246.361	0.787096	0.393548	+	0.919304i	$0.371248\pi$
$314$	0	0
$315$	−92.3425	−0.293151
$316$	0	0
$317$	356.541i	1.12474i	0.826887	+	0.562368i	$0.190109\pi$
$317$	356.541i	1.12474i	−0.826887	+	0.562368i	$0.809891\pi$
$318$	0	0
$319$	58.5492i	0.183540i
$320$	0	0
$321$	809.103	2.52057
$322$	0	0
$323$	0	0
$324$	0	0
$325$	− 381.897i	− 1.17507i
$326$	0	0
$327$	−661.547	−2.02308
$328$	0	0
$329$	647.529	1.96817
$330$	0	0
$331$	− 46.1175i	− 0.139328i	−0.997571	−	0.0696639i	$-0.977807\pi$
$331$	− 46.1175i	− 0.139328i	0.997571	−	0.0696639i	$-0.0221927\pi$
$332$	0	0
$333$	12.7178i	0.0381914i
$334$	0	0
$335$	13.5866i	0.0405570i
$336$	0	0
$337$	− 11.8320i	− 0.0351099i	−0.999846	−	0.0175549i	$-0.994412\pi$
$337$	− 11.8320i	− 0.0351099i	0.999846	−	0.0175549i	$-0.00558820\pi$
$338$	0	0
$339$	536.871	1.58369
$340$	0	0
$341$	51.9117i	0.152234i
$342$	0	0
$343$	531.021	1.54817
$344$	0	0
$345$	− 50.7850i	− 0.147203i
$346$	0	0
$347$	−595.687	−1.71668	−0.858338	−	0.513084i	$-0.828503\pi$
$347$	−595.687	−1.71668	−0.858338	+	0.513084i	$0.828503\pi$
$348$	0	0
$349$	−79.6574	−0.228245	−0.114122	−	0.993467i	$-0.536406\pi$
$349$	−79.6574	−0.228245	−0.114122	+	0.993467i	$0.536406\pi$
$350$	0	0
$351$	−785.668	−2.23837
$352$	0	0
$353$	535.410	1.51674	0.758371	−	0.651824i	$-0.225996\pi$
$353$	535.410	1.51674	0.758371	+	0.651824i	$0.225996\pi$
$354$	0	0
$355$	22.2061i	0.0625523i
$356$	0	0
$357$	− 1771.72i	− 4.96279i
$358$	0	0
$359$	−141.427	−0.393947	−0.196974	−	0.980409i	$-0.563111\pi$
$359$	−141.427	−0.393947	−0.196974	+	0.980409i	$0.563111\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	608.730i	1.67694i
$364$	0	0
$365$	6.53573	0.0179061
$366$	0	0
$367$	336.226	0.916146	0.458073	−	0.888915i	$-0.348540\pi$
$367$	336.226	0.916146	0.458073	+	0.888915i	$0.348540\pi$
$368$	0	0
$369$	1443.39i	3.91161i
$370$	0	0
$371$	− 743.051i	− 2.00283i
$372$	0	0
$373$	289.856i	0.777094i	0.921429	+	0.388547i	$0.127023\pi$
$373$	289.856i	0.777094i	−0.921429	+	0.388547i	$0.872977\pi$
$374$	0	0
$375$	108.474i	0.289264i
$376$	0	0
$377$	389.376	1.03283
$378$	0	0
$379$	− 410.636i	− 1.08347i	−0.840549	−	0.541736i	$-0.817767\pi$
$379$	− 410.636i	− 1.08347i	0.840549	−	0.541736i	$-0.182233\pi$
$380$	0	0
$381$	−470.203	−1.23413
$382$	0	0
$383$	114.831i	0.299820i	0.988700	+	0.149910i	$0.0478984\pi$
$383$	114.831i	0.299820i	−0.988700	+	0.149910i	$0.952102\pi$
$384$	0	0
$385$	−11.4181	−0.0296574
$386$	0	0
$387$	754.135	1.94867
$388$	0	0
$389$	−0.353826	−0.000909577 0	−0.000454789	−	1.00000i	$-0.500145\pi$
$389$	−0.353826	−0.000909577 0	−0.000454789		1.00000i	$0.500145\pi$
$390$	0	0
$391$	657.846	1.68247
$392$	0	0
$393$	− 81.7123i	− 0.207919i
$394$	0	0
$395$	11.4532i	0.0289955i
$396$	0	0
$397$	268.164	0.675477	0.337738	−	0.941240i	$-0.390338\pi$
$397$	268.164	0.675477	0.337738	+	0.941240i	$0.390338\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 207.050i	− 0.516334i	−0.966100	−	0.258167i	$-0.916881\pi$
$401$	− 207.050i	− 0.516334i	0.966100	−	0.258167i	$-0.0831185\pi$
$402$	0	0
$403$	345.234	0.856659
$404$	0	0
$405$	41.5735	0.102651
$406$	0	0
$407$	1.57254i	0.00386375i
$408$	0	0
$409$	− 287.676i	− 0.703363i	−0.936120	−	0.351682i	$-0.885610\pi$
$409$	− 287.676i	− 0.703363i	0.936120	−	0.351682i	$-0.114390\pi$
$410$	0	0
$411$	− 470.656i	− 1.14515i
$412$	0	0
$413$	− 410.388i	− 0.993674i
$414$	0	0
$415$	−11.3216	−0.0272809
$416$	0	0
$417$	− 423.932i	− 1.01662i
$418$	0	0
$419$	340.158	0.811832	0.405916	−	0.913910i	$-0.366952\pi$
$419$	340.158	0.811832	0.405916	+	0.913910i	$0.366952\pi$
$420$	0	0
$421$	− 247.625i	− 0.588182i	−0.955777	−	0.294091i	$-0.904983\pi$
$421$	− 247.625i	− 0.588182i	0.955777	−	0.294091i	$-0.0950169\pi$
$422$	0	0
$423$	−1014.66	−2.39872
$424$	0	0
$425$	−700.149	−1.64741
$426$	0	0
$427$	1095.20	2.56488
$428$	0	0
$429$	−187.242	−0.436462
$430$	0	0
$431$	82.4353i	0.191265i	0.995417	+	0.0956326i	$0.0304874\pi$
$431$	82.4353i	0.191265i	−0.995417	+	0.0956326i	$0.969513\pi$
$432$	0	0
$433$	370.542i	0.855755i	0.903837	+	0.427878i	$0.140739\pi$
$433$	370.542i	0.855755i	−0.903837	+	0.427878i	$0.859261\pi$
$434$	0	0
$435$	−55.1092	−0.126688
$436$	0	0
$437$	0	0
$438$	0	0
$439$	545.103i	1.24169i	0.783932	+	0.620846i	$0.213211\pi$
$439$	545.103i	1.24169i	−0.783932	+	0.620846i	$0.786789\pi$
$440$	0	0
$441$	−1748.62	−3.96512
$442$	0	0
$443$	511.589	1.15483	0.577415	−	0.816451i	$-0.304062\pi$
$443$	511.589	1.15483	0.577415	+	0.816451i	$0.304062\pi$
$444$	0	0
$445$	27.9045i	0.0627067i
$446$	0	0
$447$	918.041i	2.05378i
$448$	0	0
$449$	98.0419i	0.218356i	0.994022	+	0.109178i	$0.0348219\pi$
$449$	98.0419i	0.218356i	−0.994022	+	0.109178i	$0.965178\pi$
$450$	0	0
$451$	178.474i	0.395730i
$452$	0	0
$453$	−1311.35	−2.89481
$454$	0	0
$455$	75.9351i	0.166890i
$456$	0	0
$457$	−720.640	−1.57689	−0.788446	−	0.615104i	$-0.789114\pi$
$457$	−720.640	−1.57689	−0.788446	+	0.615104i	$0.789114\pi$
$458$	0	0
$459$	1440.40i	3.13813i
$460$	0	0
$461$	46.9650	0.101876	0.0509382	−	0.998702i	$-0.483779\pi$
$461$	46.9650	0.101876	0.0509382	+	0.998702i	$0.483779\pi$
$462$	0	0
$463$	129.373	0.279423	0.139711	−	0.990192i	$-0.455383\pi$
$463$	129.373	0.279423	0.139711	+	0.990192i	$0.455383\pi$
$464$	0	0
$465$	−48.8617	−0.105079
$466$	0	0
$467$	−621.361	−1.33054	−0.665268	−	0.746604i	$-0.731683\pi$
$467$	−621.361	−1.33054	−0.665268	+	0.746604i	$0.731683\pi$
$468$	0	0
$469$	392.129i	0.836096i
$470$	0	0
$471$	− 242.940i	− 0.515797i
$472$	0	0
$473$	93.2485	0.197143
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1164.34i	2.44096i
$478$	0	0
$479$	781.639	1.63181	0.815907	−	0.578183i	$-0.196238\pi$
$479$	781.639	1.63181	0.815907	+	0.578183i	$0.196238\pi$
$480$	0	0
$481$	10.4581	0.0217423
$482$	0	0
$483$	− 1465.73i	− 3.03464i
$484$	0	0
$485$	− 47.9593i	− 0.0988853i
$486$	0	0
$487$	96.0339i	0.197195i	0.995127	+	0.0985975i	$0.0314356\pi$
$487$	96.0339i	0.197195i	−0.995127	+	0.0985975i	$0.968564\pi$
$488$	0	0
$489$	1272.88i	2.60302i
$490$	0	0
$491$	665.003	1.35438	0.677192	−	0.735806i	$-0.263197\pi$
$491$	665.003	1.35438	0.677192	+	0.735806i	$0.263197\pi$
$492$	0	0
$493$	− 713.860i	− 1.44799i
$494$	0	0
$495$	17.8918	0.0361451
$496$	0	0
$497$	640.899i	1.28954i
$498$	0	0
$499$	50.2694	0.100740	0.0503701	−	0.998731i	$-0.483960\pi$
$499$	50.2694	0.100740	0.0503701	+	0.998731i	$0.483960\pi$
$500$	0	0
$501$	475.843	0.949786
$502$	0	0
$503$	554.459	1.10230	0.551152	−	0.834405i	$-0.314188\pi$
$503$	554.459	1.10230	0.551152	+	0.834405i	$0.314188\pi$
$504$	0	0
$505$	17.5926	0.0348368
$506$	0	0
$507$	355.703i	0.701583i
$508$	0	0
$509$	450.372i	0.884817i	0.896814	+	0.442409i	$0.145876\pi$
$509$	450.372i	0.884817i	−0.896814	+	0.442409i	$0.854124\pi$
$510$	0	0
$511$	188.631	0.369140
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 24.5327i	− 0.0476362i
$516$	0	0
$517$	−125.462	−0.242673
$518$	0	0
$519$	1125.36	2.16832
$520$	0	0
$521$	− 793.092i	− 1.52225i	−0.648605	−	0.761125i	$-0.724647\pi$
$521$	− 793.092i	− 1.52225i	0.648605	−	0.761125i	$-0.275353\pi$
$522$	0	0
$523$	802.623i	1.53465i	0.641258	+	0.767326i	$0.278413\pi$
$523$	802.623i	1.53465i	−0.641258	+	0.767326i	$0.721587\pi$
$524$	0	0
$525$	1559.98i	2.97140i
$526$	0	0
$527$	− 632.932i	− 1.20101i
$528$	0	0
$529$	15.2324	0.0287947
$530$	0	0
$531$	643.065i	1.21105i
$532$	0	0
$533$	1186.92	2.22688
$534$	0	0
$535$	63.5765i	0.118835i
$536$	0	0
$537$	1317.44	2.45333
$538$	0	0
$539$	−216.216	−0.401142
$540$	0	0
$541$	−541.587	−1.00109	−0.500543	−	0.865712i	$-0.666866\pi$
$541$	−541.587	−1.00109	−0.500543	+	0.865712i	$0.666866\pi$
$542$	0	0
$543$	−423.486	−0.779900
$544$	0	0
$545$	− 51.9821i	− 0.0953799i
$546$	0	0
$547$	528.303i	0.965818i	0.875670	+	0.482909i	$0.160420\pi$
$547$	528.303i	0.965818i	−0.875670	+	0.482909i	$0.839580\pi$
$548$	0	0
$549$	−1716.15	−3.12596
$550$	0	0
$551$	0	0
$552$	0	0
$553$	330.557i	0.597752i
$554$	0	0
$555$	−1.48015	−0.00266694
$556$	0	0
$557$	523.780	0.940360	0.470180	−	0.882571i	$-0.344189\pi$
$557$	523.780	0.940360	0.470180	+	0.882571i	$0.344189\pi$
$558$	0	0
$559$	− 620.141i	− 1.10938i
$560$	0	0
$561$	343.279i	0.611906i
$562$	0	0
$563$	334.527i	0.594186i	0.954848	+	0.297093i	$0.0960172\pi$
$563$	334.527i	0.594186i	−0.954848	+	0.297093i	$0.903983\pi$
$564$	0	0
$565$	42.1854i	0.0746645i
$566$	0	0
$567$	1199.87	2.11618
$568$	0	0
$569$	− 257.524i	− 0.452590i	−0.974059	−	0.226295i	$-0.927339\pi$
$569$	− 257.524i	− 0.452590i	0.974059	−	0.226295i	$-0.0726613\pi$
$570$	0	0
$571$	−957.626	−1.67710	−0.838552	−	0.544822i	$-0.816597\pi$
$571$	−957.626	−1.67710	−0.838552	+	0.544822i	$0.816597\pi$
$572$	0	0
$573$	236.779i	0.413226i
$574$	0	0
$575$	−579.229	−1.00736
$576$	0	0
$577$	756.954	1.31188	0.655940	−	0.754813i	$-0.272273\pi$
$577$	756.954	1.31188	0.655940	+	0.754813i	$0.272273\pi$
$578$	0	0
$579$	−336.461	−0.581107
$580$	0	0
$581$	−326.757	−0.562405
$582$	0	0
$583$	143.970i	0.246947i
$584$	0	0
$585$	− 118.988i	− 0.203398i
$586$	0	0
$587$	−1139.29	−1.94086	−0.970431	−	0.241379i	$-0.922400\pi$
$587$	−1139.29	−1.94086	−0.970431	+	0.241379i	$0.922400\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	1542.60i	2.61015i
$592$	0	0
$593$	−1034.88	−1.74516	−0.872580	−	0.488471i	$-0.837555\pi$
$593$	−1034.88	−1.74516	−0.872580	+	0.488471i	$0.837555\pi$
$594$	0	0
$595$	139.215	0.233975
$596$	0	0
$597$	567.468i	0.950533i
$598$	0	0
$599$	− 1132.76i	− 1.89109i	−0.325496	−	0.945543i	$-0.605531\pi$
$599$	− 1132.76i	− 1.89109i	0.325496	−	0.945543i	$-0.394469\pi$
$600$	0	0
$601$	430.032i	0.715528i	0.933812	+	0.357764i	$0.116461\pi$
$601$	430.032i	0.715528i	−0.933812	+	0.357764i	$0.883539\pi$
$602$	0	0
$603$	− 614.454i	− 1.01900i
$604$	0	0
$605$	−47.8319	−0.0790609
$606$	0	0
$607$	400.109i	0.659158i	0.944128	+	0.329579i	$0.106907\pi$
$607$	400.109i	0.659158i	−0.944128	+	0.329579i	$0.893093\pi$
$608$	0	0
$609$	−1590.53	−2.61171
$610$	0	0
$611$	834.374i	1.36559i
$612$	0	0
$613$	595.824	0.971981	0.485990	−	0.873964i	$-0.338459\pi$
$613$	595.824	0.971981	0.485990	+	0.873964i	$0.338459\pi$
$614$	0	0
$615$	−167.988	−0.273152
$616$	0	0
$617$	49.3880	0.0800454	0.0400227	−	0.999199i	$-0.487257\pi$
$617$	49.3880	0.0800454	0.0400227	+	0.999199i	$0.487257\pi$
$618$	0	0
$619$	111.633	0.180345	0.0901724	−	0.995926i	$-0.471258\pi$
$619$	111.633	0.180345	0.0901724	+	0.995926i	$0.471258\pi$
$620$	0	0
$621$	1191.63i	1.91890i
$622$	0	0
$623$	805.364i	1.29272i
$624$	0	0
$625$	612.200	0.979520
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 19.1732i	− 0.0304821i
$630$	0	0
$631$	−757.785	−1.20093	−0.600464	−	0.799652i	$-0.705017\pi$
$631$	−757.785	−1.20093	−0.600464	+	0.799652i	$0.705017\pi$
$632$	0	0
$633$	353.970	0.559194
$634$	0	0
$635$	− 36.9469i	− 0.0581841i
$636$	0	0
$637$	1437.92i	2.25734i
$638$	0	0
$639$	− 1004.27i	− 1.57163i
$640$	0	0
$641$	682.346i	1.06450i	0.846586	+	0.532251i	$0.178654\pi$
$641$	682.346i	1.06450i	−0.846586	+	0.532251i	$0.821346\pi$
$642$	0	0
$643$	−1165.75	−1.81298	−0.906491	−	0.422224i	$-0.861249\pi$
$643$	−1165.75	−1.81298	−0.906491	+	0.422224i	$0.861249\pi$
$644$	0	0
$645$	87.7699i	0.136077i
$646$	0	0
$647$	162.102	0.250543	0.125272	−	0.992122i	$-0.460020\pi$
$647$	162.102	0.250543	0.125272	+	0.992122i	$0.460020\pi$
$648$	0	0
$649$	79.5148i	0.122519i
$650$	0	0
$651$	−1410.22	−2.16624
$652$	0	0
$653$	−1281.73	−1.96283	−0.981413	−	0.191906i	$-0.938533\pi$
$653$	−1281.73	−1.96283	−0.981413	+	0.191906i	$0.938533\pi$
$654$	0	0
$655$	6.42066	0.00980254
$656$	0	0
$657$	−295.579	−0.449891
$658$	0	0
$659$	325.333i	0.493676i	0.969057	+	0.246838i	$0.0793916\pi$
$659$	325.333i	0.493676i	−0.969057	+	0.246838i	$0.920608\pi$
$660$	0	0
$661$	− 1090.15i	− 1.64924i	−0.565684	−	0.824622i	$-0.691388\pi$
$661$	− 1090.15i	− 1.64924i	0.565684	−	0.824622i	$-0.308612\pi$
$662$	0	0
$663$	2282.95	3.44336
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 590.572i	− 0.885416i
$668$	0	0
$669$	247.709	0.370268
$670$	0	0
$671$	−212.201	−0.316246
$672$	0	0
$673$	494.770i	0.735172i	0.929990	+	0.367586i	$0.119816\pi$
$673$	494.770i	0.735172i	−0.929990	+	0.367586i	$0.880184\pi$
$674$	0	0
$675$	− 1268.26i	− 1.87891i
$676$	0	0
$677$	− 1098.17i	− 1.62211i	−0.584970	−	0.811055i	$-0.698894\pi$
$677$	− 1098.17i	− 1.62211i	0.584970	−	0.811055i	$-0.301106\pi$
$678$	0	0
$679$	− 1384.18i	− 2.03855i
$680$	0	0
$681$	−1147.66	−1.68526
$682$	0	0
$683$	− 824.731i	− 1.20751i	−0.797169	−	0.603756i	$-0.793670\pi$
$683$	− 824.731i	− 1.20751i	0.797169	−	0.603756i	$-0.206330\pi$
$684$	0	0
$685$	36.9825	0.0539891
$686$	0	0
$687$	− 1809.28i	− 2.63359i
$688$	0	0
$689$	957.459	1.38964
$690$	0	0
$691$	−1008.51	−1.45949	−0.729746	−	0.683718i	$-0.760362\pi$
$691$	−1008.51	−1.45949	−0.729746	+	0.683718i	$0.760362\pi$
$692$	0	0
$693$	516.385	0.745144
$694$	0	0
$695$	33.3110	0.0479296
$696$	0	0
$697$	− 2176.04i	− 3.12201i
$698$	0	0
$699$	− 859.880i	− 1.23016i
$700$	0	0
$701$	−112.875	−0.161021	−0.0805103	−	0.996754i	$-0.525655\pi$
$701$	−112.875	−0.161021	−0.0805103	+	0.996754i	$0.525655\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	− 118.091i	− 0.167505i
$706$	0	0
$707$	507.748	0.718173
$708$	0	0
$709$	−423.968	−0.597980	−0.298990	−	0.954256i	$-0.596650\pi$
$709$	−423.968	−0.597980	−0.298990	+	0.954256i	$0.596650\pi$
$710$	0	0
$711$	− 517.973i	− 0.728513i
$712$	0	0
$713$	− 523.621i	− 0.734391i
$714$	0	0
$715$	− 14.7128i	− 0.0205774i
$716$	0	0
$717$	− 907.115i	− 1.26515i
$718$	0	0
$719$	1126.83	1.56722	0.783608	−	0.621255i	$-0.213377\pi$
$719$	1126.83	1.56722	0.783608	+	0.621255i	$0.213377\pi$
$720$	0	0
$721$	− 708.048i	− 0.982036i
$722$	0	0
$723$	−1012.68	−1.40066
$724$	0	0
$725$	628.549i	0.866964i
$726$	0	0
$727$	−532.434	−0.732371	−0.366186	−	0.930542i	$-0.619336\pi$
$727$	−532.434	−0.732371	−0.366186	+	0.930542i	$0.619336\pi$
$728$	0	0
$729$	539.576	0.740159
$730$	0	0
$731$	−1136.93	−1.55531
$732$	0	0
$733$	256.314	0.349678	0.174839	−	0.984597i	$-0.444060\pi$
$733$	256.314	0.349678	0.174839	+	0.984597i	$0.444060\pi$
$734$	0	0
$735$	− 203.513i	− 0.276888i
$736$	0	0
$737$	− 75.9770i	− 0.103090i
$738$	0	0
$739$	−510.850	−0.691272	−0.345636	−	0.938369i	$-0.612337\pi$
$739$	−510.850	−0.691272	−0.345636	+	0.938369i	$0.612337\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 78.9836i	− 0.106304i	−0.998586	−	0.0531518i	$-0.983073\pi$
$743$	− 78.9836i	− 0.106304i	0.998586	−	0.0531518i	$-0.0169267\pi$
$744$	0	0
$745$	−72.1364	−0.0968274
$746$	0	0
$747$	512.019	0.685434
$748$	0	0
$749$	1834.91i	2.44981i
$750$	0	0
$751$	− 1039.20i	− 1.38375i	−0.722018	−	0.691874i	$-0.756785\pi$
$751$	− 1039.20i	− 1.38375i	0.722018	−	0.691874i	$-0.243215\pi$
$752$	0	0
$753$	− 172.385i	− 0.228931i
$754$	0	0
$755$	− 103.041i	− 0.136478i
$756$	0	0
$757$	−696.674	−0.920309	−0.460154	−	0.887839i	$-0.652206\pi$
$757$	−696.674	−0.920309	−0.460154	+	0.887839i	$0.652206\pi$
$758$	0	0
$759$	283.993i	0.374167i
$760$	0	0
$761$	1286.07	1.68997	0.844986	−	0.534789i	$-0.179609\pi$
$761$	1286.07	1.68997	0.844986	+	0.534789i	$0.179609\pi$
$762$	0	0
$763$	− 1500.28i	− 1.96629i
$764$	0	0
$765$	−218.146	−0.285158
$766$	0	0
$767$	528.805	0.689446
$768$	0	0
$769$	581.307	0.755927	0.377963	−	0.925821i	$-0.376625\pi$
$769$	581.307	0.755927	0.377963	+	0.925821i	$0.376625\pi$
$770$	0	0
$771$	2278.84	2.95569
$772$	0	0
$773$	− 28.0560i	− 0.0362950i	−0.999835	−	0.0181475i	$-0.994223\pi$
$773$	− 28.0560i	− 0.0362950i	0.999835	−	0.0181475i	$-0.00577684\pi$
$774$	0	0
$775$	557.293i	0.719087i
$776$	0	0
$777$	−42.7194	−0.0549799
$778$	0	0
$779$	0	0
$780$	0	0
$781$	− 124.178i	− 0.158998i
$782$	0	0
$783$	1293.10	1.65147
$784$	0	0
$785$	19.0894	0.0243177
$786$	0	0
$787$	592.049i	0.752286i	0.926562	+	0.376143i	$0.122750\pi$
$787$	592.049i	0.752286i	−0.926562	+	0.376143i	$0.877250\pi$
$788$	0	0
$789$	964.059i	1.22188i
$790$	0	0
$791$	1217.53i	1.53923i
$792$	0	0
$793$	1411.23i	1.77960i
$794$	0	0
$795$	−135.511	−0.170455
$796$	0	0
$797$	577.304i	0.724346i	0.932111	+	0.362173i	$0.117965\pi$
$797$	577.304i	0.724346i	−0.932111	+	0.362173i	$0.882035\pi$
$798$	0	0
$799$	1529.70	1.91451
$800$	0	0
$801$	− 1261.98i	− 1.57551i
$802$	0	0
$803$	−36.5482	−0.0455145
$804$	0	0
$805$	115.172	0.143071
$806$	0	0
$807$	171.333	0.212309
$808$	0	0
$809$	−207.068	−0.255955	−0.127978	−	0.991777i	$-0.540849\pi$
$809$	−207.068	−0.255955	−0.127978	+	0.991777i	$0.540849\pi$
$810$	0	0
$811$	1093.74i	1.34863i	0.738445	+	0.674314i	$0.235561\pi$
$811$	1093.74i	1.34863i	−0.738445	+	0.674314i	$0.764439\pi$
$812$	0	0
$813$	810.487i	0.996909i
$814$	0	0
$815$	−100.018	−0.122722
$816$	0	0
$817$	0	0
$818$	0	0
$819$	− 3434.17i	− 4.19312i
$820$	0	0
$821$	510.459	0.621753	0.310876	−	0.950450i	$-0.399377\pi$
$821$	510.459	0.621753	0.310876	+	0.950450i	$0.399377\pi$
$822$	0	0
$823$	−814.509	−0.989683	−0.494842	−	0.868983i	$-0.664774\pi$
$823$	−814.509	−0.989683	−0.494842	+	0.868983i	$0.664774\pi$
$824$	0	0
$825$	− 302.255i	− 0.366370i
$826$	0	0
$827$	− 370.958i	− 0.448558i	−0.974525	−	0.224279i	$-0.927997\pi$
$827$	− 370.958i	− 0.448558i	0.974525	−	0.224279i	$-0.0720027\pi$
$828$	0	0
$829$	593.243i	0.715613i	0.933796	+	0.357806i	$0.116475\pi$
$829$	593.243i	0.715613i	−0.933796	+	0.357806i	$0.883525\pi$
$830$	0	0
$831$	1092.07i	1.31416i
$832$	0	0
$833$	2636.21	3.16472
$834$	0	0
$835$	37.3900i	0.0447785i
$836$	0	0
$837$	1146.50	1.36978
$838$	0	0
$839$	734.720i	0.875710i	0.899046	+	0.437855i	$0.144262\pi$
$839$	734.720i	0.875710i	−0.899046	+	0.437855i	$0.855738\pi$
$840$	0	0
$841$	200.142	0.237981
$842$	0	0
$843$	1142.82	1.35566
$844$	0	0
$845$	−27.9499	−0.0330768
$846$	0	0
$847$	−1380.50	−1.62987
$848$	0	0
$849$	− 1825.81i	− 2.15054i
$850$	0	0
$851$	− 15.8619i	− 0.0186391i
$852$	0	0
$853$	758.649	0.889389	0.444694	−	0.895682i	$-0.353312\pi$
$853$	758.649	0.889389	0.444694	+	0.895682i	$0.353312\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1128.37i	− 1.31665i	−0.752732	−	0.658327i	$-0.771264\pi$
$857$	− 1128.37i	− 1.31665i	0.752732	−	0.658327i	$-0.228736\pi$
$858$	0	0
$859$	165.223	0.192344	0.0961718	−	0.995365i	$-0.469340\pi$
$859$	165.223	0.192344	0.0961718	+	0.995365i	$0.469340\pi$
$860$	0	0
$861$	−4848.38	−5.63111
$862$	0	0
$863$	954.045i	1.10550i	0.833348	+	0.552749i	$0.186421\pi$
$863$	954.045i	1.10550i	−0.833348	+	0.552749i	$0.813579\pi$
$864$	0	0
$865$	88.4269i	0.102228i
$866$	0	0
$867$	− 2664.28i	− 3.07298i
$868$	0	0
$869$	− 64.0471i	− 0.0737021i
$870$	0	0
$871$	−505.278	−0.580113
$872$	0	0
$873$	2168.96i	2.48449i
$874$	0	0
$875$	−246.001	−0.281144
$876$	0	0
$877$	786.755i	0.897098i	0.893758	+	0.448549i	$0.148059\pi$
$877$	786.755i	0.897098i	−0.893758	+	0.448549i	$0.851941\pi$
$878$	0	0
$879$	1426.80	1.62321
$880$	0	0
$881$	−843.198	−0.957092	−0.478546	−	0.878063i	$-0.658836\pi$
$881$	−843.198	−0.957092	−0.478546	+	0.878063i	$0.658836\pi$
$882$	0	0
$883$	−747.809	−0.846896	−0.423448	−	0.905920i	$-0.639180\pi$
$883$	−747.809	−0.846896	−0.423448	+	0.905920i	$0.639180\pi$
$884$	0	0
$885$	−74.8430	−0.0845684
$886$	0	0
$887$	1387.31i	1.56405i	0.623246	+	0.782026i	$0.285813\pi$
$887$	1387.31i	1.56405i	−0.623246	+	0.782026i	$0.714187\pi$
$888$	0	0
$889$	− 1066.34i	− 1.19948i
$890$	0	0
$891$	−232.482	−0.260922
$892$	0	0
$893$	0	0
$894$	0	0
$895$	103.520i	0.115664i
$896$	0	0
$897$	1888.67	2.10554
$898$	0	0
$899$	−568.206	−0.632042
$900$	0	0
$901$	− 1755.35i	− 1.94823i
$902$	0	0
$903$	2533.17i	2.80528i
$904$	0	0
$905$	− 33.2760i	− 0.0367691i
$906$	0	0
$907$	1460.66i	1.61043i	0.592983	+	0.805215i	$0.297950\pi$
$907$	1460.66i	1.61043i	−0.592983	+	0.805215i	$0.702050\pi$
$908$	0	0
$909$	−795.626	−0.875277
$910$	0	0
$911$	1467.89i	1.61129i	0.592399	+	0.805645i	$0.298181\pi$
$911$	1467.89i	1.61129i	−0.592399	+	0.805645i	$0.701819\pi$
$912$	0	0
$913$	63.3110	0.0693439
$914$	0	0
$915$	− 199.734i	− 0.218289i
$916$	0	0
$917$	185.310	0.202083
$918$	0	0
$919$	1506.37	1.63914	0.819572	−	0.572976i	$-0.194211\pi$
$919$	1506.37	1.63914	0.819572	+	0.572976i	$0.194211\pi$
$920$	0	0
$921$	−213.276	−0.231571
$922$	0	0
$923$	−825.831	−0.894725
$924$	0	0
$925$	16.8819i	0.0182507i
$926$	0	0
$927$	1109.49i	1.19686i
$928$	0	0
$929$	576.585	0.620652	0.310326	−	0.950630i	$-0.399562\pi$
$929$	576.585	0.620652	0.310326	+	0.950630i	$0.399562\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 328.902i	− 0.352521i
$934$	0	0
$935$	−26.9737	−0.0288489
$936$	0	0
$937$	869.123	0.927559	0.463780	−	0.885951i	$-0.346493\pi$
$937$	869.123	0.927559	0.463780	+	0.885951i	$0.346493\pi$
$938$	0	0
$939$	− 1296.72i	− 1.38096i
$940$	0	0
$941$	− 597.560i	− 0.635026i	−0.948254	−	0.317513i	$-0.897152\pi$
$941$	− 597.560i	− 0.635026i	0.948254	−	0.317513i	$-0.102848\pi$
$942$	0	0
$943$	− 1800.23i	− 1.90904i
$944$	0	0
$945$	252.177i	0.266854i
$946$	0	0
$947$	−154.426	−0.163068	−0.0815341	−	0.996671i	$-0.525982\pi$
$947$	−154.426	−0.163068	−0.0815341	+	0.996671i	$0.525982\pi$
$948$	0	0
$949$	243.060i	0.256122i
$950$	0	0
$951$	1876.66	1.97335
$952$	0	0
$953$	− 1099.24i	− 1.15345i	−0.816937	−	0.576727i	$-0.804330\pi$
$953$	− 1099.24i	− 1.15345i	0.816937	−	0.576727i	$-0.195670\pi$
$954$	0	0
$955$	−18.6052	−0.0194819
$956$	0	0
$957$	308.174	0.322021
$958$	0	0
$959$	1067.37	1.11300
$960$	0	0
$961$	457.210	0.475764
$962$	0	0
$963$	− 2875.25i	− 2.98572i
$964$	0	0
$965$	− 26.4379i	− 0.0273968i
$966$	0	0
$967$	−1259.27	−1.30224	−0.651122	−	0.758973i	$-0.725701\pi$
$967$	−1259.27	−1.30224	−0.651122	+	0.758973i	$0.725701\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1268.23i	− 1.30611i	−0.757312	−	0.653053i	$-0.773488\pi$
$971$	− 1268.23i	− 1.30611i	0.757312	−	0.653053i	$-0.226512\pi$
$972$	0	0
$973$	961.405	0.988084
$974$	0	0
$975$	−2010.12	−2.06166
$976$	0	0
$977$	1449.00i	1.48312i	0.670889	+	0.741558i	$0.265913\pi$
$977$	1449.00i	1.48312i	−0.670889	+	0.741558i	$0.734087\pi$
$978$	0	0
$979$	− 156.044i	− 0.159391i
$980$	0	0
$981$	2350.89i	2.39642i
$982$	0	0
$983$	− 322.999i	− 0.328585i	−0.986412	−	0.164293i	$-0.947466\pi$
$983$	− 322.999i	− 0.328585i	0.986412	−	0.164293i	$-0.0525341\pi$
$984$	0	0
$985$	−121.212	−0.123058
$986$	0	0
$987$	− 3408.28i	− 3.45317i
$988$	0	0
$989$	−940.577	−0.951038
$990$	0	0
$991$	− 1012.46i	− 1.02165i	−0.859685	−	0.510825i	$-0.829340\pi$
$991$	− 1012.46i	− 1.02165i	0.859685	−	0.510825i	$-0.170660\pi$
$992$	0	0
$993$	−242.740	−0.244451
$994$	0	0
$995$	−44.5897	−0.0448137
$996$	0	0
$997$	−659.056	−0.661039	−0.330519	−	0.943799i	$-0.607224\pi$
$997$	−659.056	−0.661039	−0.330519	+	0.943799i	$0.607224\pi$
$998$	0	0
$999$	34.7307	0.0347655

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1444.3.c.d.721.2	✓	24
19.18	odd	2	inner	1444.3.c.d.721.23	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1444.3.c.d.721.2	✓	24	1.1	even	1	trivial
1444.3.c.d.721.23	yes	24	19.18	odd	2	inner

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:	\( N \)	\(=\)	\( 1444 = 2^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1444.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.26351i q^{3} +0.413588 q^{5} +11.9368 q^{7} -18.7045 q^{9} +O(q^{10})\)	\(q-5.26351i q^{3} +0.413588 q^{5} +11.9368 q^{7} -18.7045 q^{9} -2.31281 q^{11} +15.3811i q^{13} -2.17693i q^{15} +28.1989 q^{17} -62.8293i q^{21} +23.3288 q^{23} -24.8289 q^{25} +51.0800i q^{27} -25.3152i q^{29} -22.4453i q^{31} +12.1735i q^{33} +4.93690 q^{35} -0.679928i q^{37} +80.9587 q^{39} -77.1676i q^{41} -40.3183 q^{43} -7.73598 q^{45} +54.2466 q^{47} +93.4862 q^{49} -148.425i q^{51} -62.2490i q^{53} -0.956551 q^{55} -34.3802i q^{59} +91.7505 q^{61} -223.272 q^{63} +6.36145i q^{65} +32.8505i q^{67} -122.791i q^{69} +53.6912i q^{71} +15.8025 q^{73} +130.687i q^{75} -27.6074 q^{77} +27.6923i q^{79} +100.519 q^{81} -27.3740 q^{83} +11.6627 q^{85} -133.247 q^{87} +67.4693i q^{89} +183.601i q^{91} -118.141 q^{93} -115.959i q^{97} +43.2601 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 12 q^{5} + 40 q^{7} - 104 q^{9}+O(q^{10}) \) 24 * q - 12 * q^5 + 40 * q^7 - 104 * q^9	\( 24 q - 12 q^{5} + 40 q^{7} - 104 q^{9} - 64 q^{11} - 60 q^{17} + 100 q^{23} + 24 q^{25} - 144 q^{35} + 68 q^{39} - 16 q^{43} + 248 q^{45} + 180 q^{47} - 208 q^{49} - 532 q^{55} + 104 q^{61} - 1216 q^{63} - 484 q^{73} + 136 q^{77} + 460 q^{81} + 744 q^{83} + 32 q^{85} - 1296 q^{87} - 256 q^{93} + 804 q^{99}+O(q^{100}) \) 24 * q - 12 * q^5 + 40 * q^7 - 104 * q^9 - 64 * q^11 - 60 * q^17 + 100 * q^23 + 24 * q^25 - 144 * q^35 + 68 * q^39 - 16 * q^43 + 248 * q^45 + 180 * q^47 - 208 * q^49 - 532 * q^55 + 104 * q^61 - 1216 * q^63 - 484 * q^73 + 136 * q^77 + 460 * q^81 + 744 * q^83 + 32 * q^85 - 1296 * q^87 - 256 * q^93 + 804 * q^99

Introduction

L-functions

Modular forms

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Database

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