LMFDB - Embedded newform 1900.2.c.e.1749.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,2,Mod(1749,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1900.1749");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1900.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:	$15.1715763840$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1749.2
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1900.1749
Dual form			1900.2.c.e.1749.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.732051i q^{3} -2.00000i q^{7} +2.46410 q^{9} +O(q^{10})$	$q-0.732051i q^{3} -2.00000i q^{7} +2.46410 q^{9} +3.46410 q^{11} +0.732051i q^{13} -3.46410i q^{17} -1.00000 q^{19} -1.46410 q^{21} +3.46410i q^{23} -4.00000i q^{27} +3.46410 q^{29} +5.46410 q^{31} -2.53590i q^{33} -3.26795i q^{37} +0.535898 q^{39} -6.00000 q^{41} +8.92820i q^{43} +0.928203i q^{47} +3.00000 q^{49} -2.53590 q^{51} -7.26795i q^{53} +0.732051i q^{57} +6.92820 q^{59} -8.39230 q^{61} -4.92820i q^{63} -3.26795i q^{67} +2.53590 q^{69} -9.46410 q^{71} -7.46410i q^{73} -6.92820i q^{77} +10.9282 q^{79} +4.46410 q^{81} -3.46410i q^{83} -2.53590i q^{87} +8.53590 q^{89} +1.46410 q^{91} -4.00000i q^{93} -14.5885i q^{97} +8.53590 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 4 q - 4 q^{9} - 4 q^{19} + 8 q^{21} + 8 q^{31} + 16 q^{39} - 24 q^{41} + 12 q^{49} - 24 q^{51} + 8 q^{61} + 24 q^{69} - 24 q^{71} + 16 q^{79} + 4 q^{81} + 48 q^{89} - 8 q^{91} + 48 q^{99}+O(q^{100}) $ 4 * q - 4 * q^9 - 4 * q^19 + 8 * q^21 + 8 * q^31 + 16 * q^39 - 24 * q^41 + 12 * q^49 - 24 * q^51 + 8 * q^61 + 24 * q^69 - 24 * q^71 + 16 * q^79 + 4 * q^81 + 48 * q^89 - 8 * q^91 + 48 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$.

$n$	$77$	$401$	$951$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 0.732051i	− 0.422650i	−0.977416	−	0.211325i	$-0.932222\pi$
$3$	− 0.732051i	− 0.422650i	0.977416	−	0.211325i	$-0.0677778\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	0	0
$9$	2.46410	0.821367
$10$	0	0
$11$	3.46410	1.04447	0.522233	−	0.852803i	$-0.325099\pi$
$11$	3.46410	1.04447	0.522233	+	0.852803i	$0.325099\pi$
$12$	0	0
$13$	0.732051i	0.203034i	0.994834	+	0.101517i	$0.0323697\pi$
$13$	0.732051i	0.203034i	−0.994834	+	0.101517i	$0.967630\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 3.46410i	− 0.840168i	−0.907485	−	0.420084i	$-0.862001\pi$
$17$	− 3.46410i	− 0.840168i	0.907485	−	0.420084i	$-0.137999\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.46410	−0.319493
$22$	0	0
$23$	3.46410i	0.722315i	0.932505	+	0.361158i	$0.117618\pi$
$23$	3.46410i	0.722315i	−0.932505	+	0.361158i	$0.882382\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 4.00000i	− 0.769800i
$28$	0	0
$29$	3.46410	0.643268	0.321634	−	0.946864i	$-0.395768\pi$
$29$	3.46410	0.643268	0.321634	+	0.946864i	$0.395768\pi$
$30$	0	0
$31$	5.46410	0.981382	0.490691	−	0.871334i	$-0.336744\pi$
$31$	5.46410	0.981382	0.490691	+	0.871334i	$0.336744\pi$
$32$	0	0
$33$	− 2.53590i	− 0.441443i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 3.26795i	− 0.537248i	−0.963245	−	0.268624i	$-0.913431\pi$
$37$	− 3.26795i	− 0.537248i	0.963245	−	0.268624i	$-0.0865688\pi$
$38$	0	0
$39$	0.535898	0.0858124
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	8.92820i	1.36154i	0.732498	+	0.680769i	$0.238354\pi$
$43$	8.92820i	1.36154i	−0.732498	+	0.680769i	$0.761646\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.928203i	0.135392i	0.997706	+	0.0676962i	$0.0215649\pi$
$47$	0.928203i	0.135392i	−0.997706	+	0.0676962i	$0.978435\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	−2.53590	−0.355097
$52$	0	0
$53$	− 7.26795i	− 0.998330i	−0.866507	−	0.499165i	$-0.833640\pi$
$53$	− 7.26795i	− 0.998330i	0.866507	−	0.499165i	$-0.166360\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.732051i	0.0969625i
$58$	0	0
$59$	6.92820	0.901975	0.450988	−	0.892530i	$-0.351072\pi$
$59$	6.92820	0.901975	0.450988	+	0.892530i	$0.351072\pi$
$60$	0	0
$61$	−8.39230	−1.07452	−0.537262	−	0.843415i	$-0.680541\pi$
$61$	−8.39230	−1.07452	−0.537262	+	0.843415i	$0.680541\pi$
$62$	0	0
$63$	− 4.92820i	− 0.620895i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 3.26795i	− 0.399244i	−0.979873	−	0.199622i	$-0.936029\pi$
$67$	− 3.26795i	− 0.399244i	0.979873	−	0.199622i	$-0.0639713\pi$
$68$	0	0
$69$	2.53590	0.305286
$70$	0	0
$71$	−9.46410	−1.12318	−0.561591	−	0.827415i	$-0.689811\pi$
$71$	−9.46410	−1.12318	−0.561591	+	0.827415i	$0.689811\pi$
$72$	0	0
$73$	− 7.46410i	− 0.873607i	−0.899557	−	0.436804i	$-0.856111\pi$
$73$	− 7.46410i	− 0.873607i	0.899557	−	0.436804i	$-0.143889\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 6.92820i	− 0.789542i
$78$	0	0
$79$	10.9282	1.22952	0.614759	−	0.788715i	$-0.289253\pi$
$79$	10.9282	1.22952	0.614759	+	0.788715i	$0.289253\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	− 3.46410i	− 0.380235i	−0.981761	−	0.190117i	$-0.939113\pi$
$83$	− 3.46410i	− 0.380235i	0.981761	−	0.190117i	$-0.0608868\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 2.53590i	− 0.271877i
$88$	0	0
$89$	8.53590	0.904803	0.452402	−	0.891814i	$-0.350567\pi$
$89$	8.53590	0.904803	0.452402	+	0.891814i	$0.350567\pi$
$90$	0	0
$91$	1.46410	0.153480
$92$	0	0
$93$	− 4.00000i	− 0.414781i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 14.5885i	− 1.48123i	−0.671928	−	0.740617i	$-0.734533\pi$
$97$	− 14.5885i	− 1.48123i	0.671928	−	0.740617i	$-0.265467\pi$
$98$	0	0
$99$	8.53590	0.857890
$100$	0	0
$101$	4.39230	0.437051	0.218525	−	0.975831i	$-0.429875\pi$
$101$	4.39230	0.437051	0.218525	+	0.975831i	$0.429875\pi$
$102$	0	0
$103$	− 3.66025i	− 0.360656i	−0.983607	−	0.180328i	$-0.942284\pi$
$103$	− 3.66025i	− 0.360656i	0.983607	−	0.180328i	$-0.0577158\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 11.6603i	− 1.12724i	−0.826034	−	0.563620i	$-0.809408\pi$
$107$	− 11.6603i	− 1.12724i	0.826034	−	0.563620i	$-0.190592\pi$
$108$	0	0
$109$	−6.39230	−0.612272	−0.306136	−	0.951988i	$-0.599036\pi$
$109$	−6.39230	−0.612272	−0.306136	+	0.951988i	$0.599036\pi$
$110$	0	0
$111$	−2.39230	−0.227068
$112$	0	0
$113$	− 18.5885i	− 1.74865i	−0.485336	−	0.874327i	$-0.661303\pi$
$113$	− 18.5885i	− 1.74865i	0.485336	−	0.874327i	$-0.338697\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.80385i	0.166766i
$118$	0	0
$119$	−6.92820	−0.635107
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	4.39230i	0.396041i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	10.5885i	0.939574i	0.882780	+	0.469787i	$0.155669\pi$
$127$	10.5885i	0.939574i	−0.882780	+	0.469787i	$0.844331\pi$
$128$	0	0
$129$	6.53590	0.575454
$130$	0	0
$131$	−5.07180	−0.443125	−0.221562	−	0.975146i	$-0.571116\pi$
$131$	−5.07180	−0.443125	−0.221562	+	0.975146i	$0.571116\pi$
$132$	0	0
$133$	2.00000i	0.173422i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	17.3205i	1.47979i	0.672722	+	0.739895i	$0.265125\pi$
$137$	17.3205i	1.47979i	−0.672722	+	0.739895i	$0.734875\pi$
$138$	0	0
$139$	−4.53590	−0.384730	−0.192365	−	0.981323i	$-0.561616\pi$
$139$	−4.53590	−0.384730	−0.192365	+	0.981323i	$0.561616\pi$
$140$	0	0
$141$	0.679492	0.0572235
$142$	0	0
$143$	2.53590i	0.212062i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 2.19615i	− 0.181136i
$148$	0	0
$149$	−16.3923	−1.34291	−0.671455	−	0.741045i	$-0.734330\pi$
$149$	−16.3923	−1.34291	−0.671455	+	0.741045i	$0.734330\pi$
$150$	0	0
$151$	12.3923	1.00847	0.504236	−	0.863566i	$-0.331774\pi$
$151$	12.3923	1.00847	0.504236	+	0.863566i	$0.331774\pi$
$152$	0	0
$153$	− 8.53590i	− 0.690086i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 16.5359i	− 1.31971i	−0.751394	−	0.659854i	$-0.770618\pi$
$157$	− 16.5359i	− 1.31971i	0.751394	−	0.659854i	$-0.229382\pi$
$158$	0	0
$159$	−5.32051	−0.421944
$160$	0	0
$161$	6.92820	0.546019
$162$	0	0
$163$	20.9282i	1.63922i	0.572919	+	0.819612i	$0.305811\pi$
$163$	20.9282i	1.63922i	−0.572919	+	0.819612i	$0.694189\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.26795i	0.562411i	0.959648	+	0.281205i	$0.0907342\pi$
$167$	7.26795i	0.562411i	−0.959648	+	0.281205i	$0.909266\pi$
$168$	0	0
$169$	12.4641	0.958777
$170$	0	0
$171$	−2.46410	−0.188435
$172$	0	0
$173$	14.1962i	1.07931i	0.841885	+	0.539657i	$0.181446\pi$
$173$	14.1962i	1.07931i	−0.841885	+	0.539657i	$0.818554\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 5.07180i	− 0.381220i
$178$	0	0
$179$	20.7846	1.55351	0.776757	−	0.629800i	$-0.216863\pi$
$179$	20.7846	1.55351	0.776757	+	0.629800i	$0.216863\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	6.14359i	0.454148i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 12.0000i	− 0.877527i
$188$	0	0
$189$	−8.00000	−0.581914
$190$	0	0
$191$	−6.92820	−0.501307	−0.250654	−	0.968077i	$-0.580646\pi$
$191$	−6.92820	−0.501307	−0.250654	+	0.968077i	$0.580646\pi$
$192$	0	0
$193$	− 18.1962i	− 1.30979i	−0.755721	−	0.654894i	$-0.772713\pi$
$193$	− 18.1962i	− 1.30979i	0.755721	−	0.654894i	$-0.227287\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.9282i	0.921096i	0.887635	+	0.460548i	$0.152347\pi$
$197$	12.9282i	0.921096i	−0.887635	+	0.460548i	$0.847653\pi$
$198$	0	0
$199$	−13.0718	−0.926635	−0.463318	−	0.886192i	$-0.653341\pi$
$199$	−13.0718	−0.926635	−0.463318	+	0.886192i	$0.653341\pi$
$200$	0	0
$201$	−2.39230	−0.168740
$202$	0	0
$203$	− 6.92820i	− 0.486265i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	8.53590i	0.593286i
$208$	0	0
$209$	−3.46410	−0.239617
$210$	0	0
$211$	−15.3205	−1.05471	−0.527354	−	0.849646i	$-0.676816\pi$
$211$	−15.3205	−1.05471	−0.527354	+	0.849646i	$0.676816\pi$
$212$	0	0
$213$	6.92820i	0.474713i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 10.9282i	− 0.741855i
$218$	0	0
$219$	−5.46410	−0.369230
$220$	0	0
$221$	2.53590	0.170583
$222$	0	0
$223$	7.66025i	0.512969i	0.966548	+	0.256484i	$0.0825642\pi$
$223$	7.66025i	0.512969i	−0.966548	+	0.256484i	$0.917436\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.19615i	0.145764i	0.997341	+	0.0728819i	$0.0232196\pi$
$227$	2.19615i	0.145764i	−0.997341	+	0.0728819i	$0.976780\pi$
$228$	0	0
$229$	−10.5359	−0.696232	−0.348116	−	0.937452i	$-0.613178\pi$
$229$	−10.5359	−0.696232	−0.348116	+	0.937452i	$0.613178\pi$
$230$	0	0
$231$	−5.07180	−0.333700
$232$	0	0
$233$	12.9282i	0.846955i	0.905907	+	0.423477i	$0.139191\pi$
$233$	12.9282i	0.846955i	−0.905907	+	0.423477i	$0.860809\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 8.00000i	− 0.519656i
$238$	0	0
$239$	−13.8564	−0.896296	−0.448148	−	0.893959i	$-0.647916\pi$
$239$	−13.8564	−0.896296	−0.448148	+	0.893959i	$0.647916\pi$
$240$	0	0
$241$	15.8564	1.02140	0.510700	−	0.859759i	$-0.329386\pi$
$241$	15.8564	1.02140	0.510700	+	0.859759i	$0.329386\pi$
$242$	0	0
$243$	− 15.2679i	− 0.979439i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 0.732051i	− 0.0465793i
$248$	0	0
$249$	−2.53590	−0.160706
$250$	0	0
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	0	0
$253$	12.0000i	0.754434i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 9.12436i	− 0.569162i	−0.958652	−	0.284581i	$-0.908146\pi$
$257$	− 9.12436i	− 0.569162i	0.958652	−	0.284581i	$-0.0918544\pi$
$258$	0	0
$259$	−6.53590	−0.406121
$260$	0	0
$261$	8.53590	0.528359
$262$	0	0
$263$	15.4641i	0.953557i	0.879023	+	0.476779i	$0.158196\pi$
$263$	15.4641i	0.953557i	−0.879023	+	0.476779i	$0.841804\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 6.24871i	− 0.382415i
$268$	0	0
$269$	10.3923	0.633630	0.316815	−	0.948487i	$-0.397387\pi$
$269$	10.3923	0.633630	0.316815	+	0.948487i	$0.397387\pi$
$270$	0	0
$271$	18.3923	1.11725	0.558626	−	0.829419i	$-0.311329\pi$
$271$	18.3923	1.11725	0.558626	+	0.829419i	$0.311329\pi$
$272$	0	0
$273$	− 1.07180i	− 0.0648681i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.39230i	0.143740i	0.997414	+	0.0718698i	$0.0228966\pi$
$277$	2.39230i	0.143740i	−0.997414	+	0.0718698i	$0.977103\pi$
$278$	0	0
$279$	13.4641	0.806075
$280$	0	0
$281$	−12.9282	−0.771232	−0.385616	−	0.922659i	$-0.626011\pi$
$281$	−12.9282	−0.771232	−0.385616	+	0.922659i	$0.626011\pi$
$282$	0	0
$283$	− 2.39230i	− 0.142208i	−0.997469	−	0.0711039i	$-0.977348\pi$
$283$	− 2.39230i	− 0.142208i	0.997469	−	0.0711039i	$-0.0226522\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12.0000i	0.708338i
$288$	0	0
$289$	5.00000	0.294118
$290$	0	0
$291$	−10.6795	−0.626043
$292$	0	0
$293$	11.6603i	0.681199i	0.940208	+	0.340600i	$0.110630\pi$
$293$	11.6603i	0.681199i	−0.940208	+	0.340600i	$0.889370\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 13.8564i	− 0.804030i
$298$	0	0
$299$	−2.53590	−0.146655
$300$	0	0
$301$	17.8564	1.02923
$302$	0	0
$303$	− 3.21539i	− 0.184719i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	18.1962i	1.03851i	0.854620	+	0.519255i	$0.173790\pi$
$307$	18.1962i	1.03851i	−0.854620	+	0.519255i	$0.826210\pi$
$308$	0	0
$309$	−2.67949	−0.152431
$310$	0	0
$311$	−22.3923	−1.26975	−0.634876	−	0.772614i	$-0.718949\pi$
$311$	−22.3923	−1.26975	−0.634876	+	0.772614i	$0.718949\pi$
$312$	0	0
$313$	− 30.7846i	− 1.74005i	−0.493008	−	0.870025i	$-0.664103\pi$
$313$	− 30.7846i	− 1.74005i	0.493008	−	0.870025i	$-0.335897\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.1244i	1.18646i	0.805032	+	0.593231i	$0.202148\pi$
$317$	21.1244i	1.18646i	−0.805032	+	0.593231i	$0.797852\pi$
$318$	0	0
$319$	12.0000	0.671871
$320$	0	0
$321$	−8.53590	−0.476427
$322$	0	0
$323$	3.46410i	0.192748i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4.67949i	0.258776i
$328$	0	0
$329$	1.85641	0.102347
$330$	0	0
$331$	26.2487	1.44276	0.721380	−	0.692540i	$-0.243508\pi$
$331$	26.2487	1.44276	0.721380	+	0.692540i	$0.243508\pi$
$332$	0	0
$333$	− 8.05256i	− 0.441278i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	20.7321i	1.12935i	0.825314	+	0.564673i	$0.190998\pi$
$337$	20.7321i	1.12935i	−0.825314	+	0.564673i	$0.809002\pi$
$338$	0	0
$339$	−13.6077	−0.739069
$340$	0	0
$341$	18.9282	1.02502
$342$	0	0
$343$	− 20.0000i	− 1.07990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	31.1769i	1.67366i	0.547459	+	0.836832i	$0.315595\pi$
$347$	31.1769i	1.67366i	−0.547459	+	0.836832i	$0.684405\pi$
$348$	0	0
$349$	−7.07180	−0.378545	−0.189272	−	0.981925i	$-0.560613\pi$
$349$	−7.07180	−0.378545	−0.189272	+	0.981925i	$0.560613\pi$
$350$	0	0
$351$	2.92820	0.156296
$352$	0	0
$353$	22.3923i	1.19182i	0.803050	+	0.595911i	$0.203209\pi$
$353$	22.3923i	1.19182i	−0.803050	+	0.595911i	$0.796791\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	5.07180i	0.268428i
$358$	0	0
$359$	15.4641	0.816164	0.408082	−	0.912945i	$-0.366198\pi$
$359$	15.4641	0.816164	0.408082	+	0.912945i	$0.366198\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	− 0.732051i	− 0.0384227i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	23.8564i	1.24529i	0.782503	+	0.622647i	$0.213943\pi$
$367$	23.8564i	1.24529i	−0.782503	+	0.622647i	$0.786057\pi$
$368$	0	0
$369$	−14.7846	−0.769656
$370$	0	0
$371$	−14.5359	−0.754666
$372$	0	0
$373$	14.5885i	0.755362i	0.925936	+	0.377681i	$0.123278\pi$
$373$	14.5885i	0.755362i	−0.925936	+	0.377681i	$0.876722\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.53590i	0.130605i
$378$	0	0
$379$	−1.07180	−0.0550545	−0.0275273	−	0.999621i	$-0.508763\pi$
$379$	−1.07180	−0.0550545	−0.0275273	+	0.999621i	$0.508763\pi$
$380$	0	0
$381$	7.75129	0.397111
$382$	0	0
$383$	23.6603i	1.20898i	0.796612	+	0.604491i	$0.206624\pi$
$383$	23.6603i	1.20898i	−0.796612	+	0.604491i	$0.793376\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	22.0000i	1.11832i
$388$	0	0
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	3.71281i	0.187287i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	12.5359i	0.629159i	0.949231	+	0.314579i	$0.101863\pi$
$397$	12.5359i	0.629159i	−0.949231	+	0.314579i	$0.898137\pi$
$398$	0	0
$399$	1.46410	0.0732968
$400$	0	0
$401$	−2.78461	−0.139057	−0.0695284	−	0.997580i	$-0.522149\pi$
$401$	−2.78461	−0.139057	−0.0695284	+	0.997580i	$0.522149\pi$
$402$	0	0
$403$	4.00000i	0.199254i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 11.3205i	− 0.561137i
$408$	0	0
$409$	−6.39230	−0.316079	−0.158040	−	0.987433i	$-0.550517\pi$
$409$	−6.39230	−0.316079	−0.158040	+	0.987433i	$0.550517\pi$
$410$	0	0
$411$	12.6795	0.625433
$412$	0	0
$413$	− 13.8564i	− 0.681829i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	3.32051i	0.162606i
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−34.0000	−1.65706	−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000	−1.65706	−0.828529	+	0.559946i	$0.810822\pi$
$422$	0	0
$423$	2.28719i	0.111207i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	16.7846i	0.812264i
$428$	0	0
$429$	1.85641	0.0896281
$430$	0	0
$431$	−26.5359	−1.27819	−0.639095	−	0.769128i	$-0.720691\pi$
$431$	−26.5359	−1.27819	−0.639095	+	0.769128i	$0.720691\pi$
$432$	0	0
$433$	− 39.6603i	− 1.90595i	−0.303049	−	0.952975i	$-0.598004\pi$
$433$	− 39.6603i	− 1.90595i	0.303049	−	0.952975i	$-0.401996\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 3.46410i	− 0.165710i
$438$	0	0
$439$	−23.7128	−1.13175	−0.565875	−	0.824491i	$-0.691462\pi$
$439$	−23.7128	−1.13175	−0.565875	+	0.824491i	$0.691462\pi$
$440$	0	0
$441$	7.39230	0.352015
$442$	0	0
$443$	34.3923i	1.63403i	0.576618	+	0.817014i	$0.304372\pi$
$443$	34.3923i	1.63403i	−0.576618	+	0.817014i	$0.695628\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	12.0000i	0.567581i
$448$	0	0
$449$	−3.46410	−0.163481	−0.0817405	−	0.996654i	$-0.526048\pi$
$449$	−3.46410	−0.163481	−0.0817405	+	0.996654i	$0.526048\pi$
$450$	0	0
$451$	−20.7846	−0.978709
$452$	0	0
$453$	− 9.07180i	− 0.426230i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.78461i	0.317371i	0.987329	+	0.158685i	$0.0507255\pi$
$457$	6.78461i	0.317371i	−0.987329	+	0.158685i	$0.949274\pi$
$458$	0	0
$459$	−13.8564	−0.646762
$460$	0	0
$461$	33.7128	1.57016	0.785081	−	0.619393i	$-0.212621\pi$
$461$	33.7128	1.57016	0.785081	+	0.619393i	$0.212621\pi$
$462$	0	0
$463$	11.4641i	0.532782i	0.963865	+	0.266391i	$0.0858312\pi$
$463$	11.4641i	0.532782i	−0.963865	+	0.266391i	$0.914169\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 5.32051i	− 0.246204i	−0.992394	−	0.123102i	$-0.960716\pi$
$467$	− 5.32051i	− 0.246204i	0.992394	−	0.123102i	$-0.0392842\pi$
$468$	0	0
$469$	−6.53590	−0.301800
$470$	0	0
$471$	−12.1051	−0.557774
$472$	0	0
$473$	30.9282i	1.42208i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 17.9090i	− 0.819995i
$478$	0	0
$479$	1.60770	0.0734575	0.0367287	−	0.999325i	$-0.488306\pi$
$479$	1.60770	0.0734575	0.0367287	+	0.999325i	$0.488306\pi$
$480$	0	0
$481$	2.39230	0.109080
$482$	0	0
$483$	− 5.07180i	− 0.230775i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	1.80385i	0.0817401i	0.999164	+	0.0408701i	$0.0130130\pi$
$487$	1.80385i	0.0817401i	−0.999164	+	0.0408701i	$0.986987\pi$
$488$	0	0
$489$	15.3205	0.692817
$490$	0	0
$491$	5.07180	0.228887	0.114443	−	0.993430i	$-0.463492\pi$
$491$	5.07180	0.228887	0.114443	+	0.993430i	$0.463492\pi$
$492$	0	0
$493$	− 12.0000i	− 0.540453i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	18.9282i	0.849046i
$498$	0	0
$499$	−16.5359	−0.740248	−0.370124	−	0.928982i	$-0.620685\pi$
$499$	−16.5359	−0.740248	−0.370124	+	0.928982i	$0.620685\pi$
$500$	0	0
$501$	5.32051	0.237703
$502$	0	0
$503$	8.53590i	0.380597i	0.981726	+	0.190298i	$0.0609456\pi$
$503$	8.53590i	0.380597i	−0.981726	+	0.190298i	$0.939054\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 9.12436i	− 0.405227i
$508$	0	0
$509$	−29.3205	−1.29961	−0.649804	−	0.760102i	$-0.725149\pi$
$509$	−29.3205	−1.29961	−0.649804	+	0.760102i	$0.725149\pi$
$510$	0	0
$511$	−14.9282	−0.660385
$512$	0	0
$513$	4.00000i	0.176604i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	3.21539i	0.141413i
$518$	0	0
$519$	10.3923	0.456172
$520$	0	0
$521$	26.7846	1.17346	0.586728	−	0.809784i	$-0.300416\pi$
$521$	26.7846	1.17346	0.586728	+	0.809784i	$0.300416\pi$
$522$	0	0
$523$	35.3731i	1.54676i	0.633945	+	0.773378i	$0.281435\pi$
$523$	35.3731i	1.54676i	−0.633945	+	0.773378i	$0.718565\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 18.9282i	− 0.824525i
$528$	0	0
$529$	11.0000	0.478261
$530$	0	0
$531$	17.0718	0.740853
$532$	0	0
$533$	− 4.39230i	− 0.190252i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 15.2154i	− 0.656593i
$538$	0	0
$539$	10.3923	0.447628
$540$	0	0
$541$	33.1769	1.42639	0.713193	−	0.700967i	$-0.247248\pi$
$541$	33.1769	1.42639	0.713193	+	0.700967i	$0.247248\pi$
$542$	0	0
$543$	− 10.2487i	− 0.439814i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 3.26795i	− 0.139727i	−0.997557	−	0.0698637i	$-0.977744\pi$
$547$	− 3.26795i	− 0.139727i	0.997557	−	0.0698637i	$-0.0222564\pi$
$548$	0	0
$549$	−20.6795	−0.882579
$550$	0	0
$551$	−3.46410	−0.147576
$552$	0	0
$553$	− 21.8564i	− 0.929429i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	29.3205i	1.24235i	0.783672	+	0.621175i	$0.213344\pi$
$557$	29.3205i	1.24235i	−0.783672	+	0.621175i	$0.786656\pi$
$558$	0	0
$559$	−6.53590	−0.276439
$560$	0	0
$561$	−8.78461	−0.370887
$562$	0	0
$563$	33.8038i	1.42466i	0.701844	+	0.712331i	$0.252361\pi$
$563$	33.8038i	1.42466i	−0.701844	+	0.712331i	$0.747639\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 8.92820i	− 0.374949i
$568$	0	0
$569$	−5.32051	−0.223047	−0.111524	−	0.993762i	$-0.535573\pi$
$569$	−5.32051	−0.223047	−0.111524	+	0.993762i	$0.535573\pi$
$570$	0	0
$571$	−2.39230	−0.100115	−0.0500574	−	0.998746i	$-0.515940\pi$
$571$	−2.39230	−0.100115	−0.0500574	+	0.998746i	$0.515940\pi$
$572$	0	0
$573$	5.07180i	0.211877i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 34.7846i	− 1.44810i	−0.689746	−	0.724051i	$-0.742278\pi$
$577$	− 34.7846i	− 1.44810i	0.689746	−	0.724051i	$-0.257722\pi$
$578$	0	0
$579$	−13.3205	−0.553581
$580$	0	0
$581$	−6.92820	−0.287430
$582$	0	0
$583$	− 25.1769i	− 1.04272i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19.1769i	0.791516i	0.918355	+	0.395758i	$0.129518\pi$
$587$	19.1769i	0.791516i	−0.918355	+	0.395758i	$0.870482\pi$
$588$	0	0
$589$	−5.46410	−0.225144
$590$	0	0
$591$	9.46410	0.389301
$592$	0	0
$593$	4.14359i	0.170157i	0.996374	+	0.0850785i	$0.0271141\pi$
$593$	4.14359i	0.170157i	−0.996374	+	0.0850785i	$0.972886\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9.56922i	0.391642i
$598$	0	0
$599$	−20.7846	−0.849236	−0.424618	−	0.905373i	$-0.639592\pi$
$599$	−20.7846	−0.849236	−0.424618	+	0.905373i	$0.639592\pi$
$600$	0	0
$601$	−44.6410	−1.82095	−0.910473	−	0.413570i	$-0.864282\pi$
$601$	−44.6410	−1.82095	−0.910473	+	0.413570i	$0.864282\pi$
$602$	0	0
$603$	− 8.05256i	− 0.327926i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 25.9090i	− 1.05161i	−0.850604	−	0.525806i	$-0.823764\pi$
$607$	− 25.9090i	− 1.05161i	0.850604	−	0.525806i	$-0.176236\pi$
$608$	0	0
$609$	−5.07180	−0.205520
$610$	0	0
$611$	−0.679492	−0.0274893
$612$	0	0
$613$	− 14.3923i	− 0.581300i	−0.956829	−	0.290650i	$-0.906129\pi$
$613$	− 14.3923i	− 0.581300i	0.956829	−	0.290650i	$-0.0938715\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 22.3923i	− 0.901480i	−0.892655	−	0.450740i	$-0.851160\pi$
$617$	− 22.3923i	− 0.901480i	0.892655	−	0.450740i	$-0.148840\pi$
$618$	0	0
$619$	−18.3923	−0.739249	−0.369625	−	0.929181i	$-0.620514\pi$
$619$	−18.3923	−0.739249	−0.369625	+	0.929181i	$0.620514\pi$
$620$	0	0
$621$	13.8564	0.556038
$622$	0	0
$623$	− 17.0718i	− 0.683967i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.53590i	0.101274i
$628$	0	0
$629$	−11.3205	−0.451378
$630$	0	0
$631$	4.53590	0.180571	0.0902856	−	0.995916i	$-0.471222\pi$
$631$	4.53590	0.180571	0.0902856	+	0.995916i	$0.471222\pi$
$632$	0	0
$633$	11.2154i	0.445772i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.19615i	0.0870147i
$638$	0	0
$639$	−23.3205	−0.922545
$640$	0	0
$641$	−11.0718	−0.437310	−0.218655	−	0.975802i	$-0.570167\pi$
$641$	−11.0718	−0.437310	−0.218655	+	0.975802i	$0.570167\pi$
$642$	0	0
$643$	6.39230i	0.252088i	0.992025	+	0.126044i	$0.0402280\pi$
$643$	6.39230i	0.252088i	−0.992025	+	0.126044i	$0.959772\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 8.53590i	− 0.335581i	−0.985823	−	0.167790i	$-0.946337\pi$
$647$	− 8.53590i	− 0.335581i	0.985823	−	0.167790i	$-0.0536632\pi$
$648$	0	0
$649$	24.0000	0.942082
$650$	0	0
$651$	−8.00000	−0.313545
$652$	0	0
$653$	20.5359i	0.803632i	0.915720	+	0.401816i	$0.131621\pi$
$653$	20.5359i	0.803632i	−0.915720	+	0.401816i	$0.868379\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 18.3923i	− 0.717552i
$658$	0	0
$659$	32.7846	1.27711	0.638554	−	0.769577i	$-0.279533\pi$
$659$	32.7846	1.27711	0.638554	+	0.769577i	$0.279533\pi$
$660$	0	0
$661$	46.7846	1.81971	0.909855	−	0.414926i	$-0.136193\pi$
$661$	46.7846	1.81971	0.909855	+	0.414926i	$0.136193\pi$
$662$	0	0
$663$	− 1.85641i	− 0.0720969i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	12.0000i	0.464642i
$668$	0	0
$669$	5.60770	0.216806
$670$	0	0
$671$	−29.0718	−1.12230
$672$	0	0
$673$	− 47.2679i	− 1.82205i	−0.412356	−	0.911023i	$-0.635294\pi$
$673$	− 47.2679i	− 1.82205i	0.412356	−	0.911023i	$-0.364706\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	34.9808i	1.34442i	0.740361	+	0.672210i	$0.234655\pi$
$677$	34.9808i	1.34442i	−0.740361	+	0.672210i	$0.765345\pi$
$678$	0	0
$679$	−29.1769	−1.11971
$680$	0	0
$681$	1.60770	0.0616070
$682$	0	0
$683$	− 0.339746i	− 0.0130000i	−0.999979	−	0.00650001i	$-0.997931\pi$
$683$	− 0.339746i	− 0.0130000i	0.999979	−	0.00650001i	$-0.00206903\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	7.71281i	0.294262i
$688$	0	0
$689$	5.32051	0.202695
$690$	0	0
$691$	−11.1769	−0.425190	−0.212595	−	0.977140i	$-0.568191\pi$
$691$	−11.1769	−0.425190	−0.212595	+	0.977140i	$0.568191\pi$
$692$	0	0
$693$	− 17.0718i	− 0.648504i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	20.7846i	0.787273i
$698$	0	0
$699$	9.46410	0.357965
$700$	0	0
$701$	−33.4641	−1.26392	−0.631961	−	0.775000i	$-0.717750\pi$
$701$	−33.4641	−1.26392	−0.631961	+	0.775000i	$0.717750\pi$
$702$	0	0
$703$	3.26795i	0.123253i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 8.78461i	− 0.330379i
$708$	0	0
$709$	−10.7846	−0.405025	−0.202512	−	0.979280i	$-0.564911\pi$
$709$	−10.7846	−0.405025	−0.202512	+	0.979280i	$0.564911\pi$
$710$	0	0
$711$	26.9282	1.00989
$712$	0	0
$713$	18.9282i	0.708867i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	10.1436i	0.378819i
$718$	0	0
$719$	17.3205	0.645946	0.322973	−	0.946408i	$-0.395318\pi$
$719$	17.3205	0.645946	0.322973	+	0.946408i	$0.395318\pi$
$720$	0	0
$721$	−7.32051	−0.272630
$722$	0	0
$723$	− 11.6077i	− 0.431695i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 27.8564i	− 1.03314i	−0.856246	−	0.516568i	$-0.827209\pi$
$727$	− 27.8564i	− 1.03314i	0.856246	−	0.516568i	$-0.172791\pi$
$728$	0	0
$729$	2.21539	0.0820515
$730$	0	0
$731$	30.9282	1.14392
$732$	0	0
$733$	− 30.7846i	− 1.13706i	−0.822664	−	0.568528i	$-0.807513\pi$
$733$	− 30.7846i	− 1.13706i	0.822664	−	0.568528i	$-0.192487\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 11.3205i	− 0.416996i
$738$	0	0
$739$	28.0000	1.03000	0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000	1.03000	0.514998	+	0.857191i	$0.327793\pi$
$740$	0	0
$741$	−0.535898	−0.0196867
$742$	0	0
$743$	45.1244i	1.65545i	0.561132	+	0.827726i	$0.310366\pi$
$743$	45.1244i	1.65545i	−0.561132	+	0.827726i	$0.689634\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 8.53590i	− 0.312312i
$748$	0	0
$749$	−23.3205	−0.852113
$750$	0	0
$751$	−18.5359	−0.676385	−0.338192	−	0.941077i	$-0.609815\pi$
$751$	−18.5359	−0.676385	−0.338192	+	0.941077i	$0.609815\pi$
$752$	0	0
$753$	17.5692i	0.640258i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	10.0000i	0.363456i	0.983349	+	0.181728i	$0.0581691\pi$
$757$	10.0000i	0.363456i	−0.983349	+	0.181728i	$0.941831\pi$
$758$	0	0
$759$	8.78461	0.318861
$760$	0	0
$761$	40.3923	1.46422	0.732110	−	0.681186i	$-0.238536\pi$
$761$	40.3923	1.46422	0.732110	+	0.681186i	$0.238536\pi$
$762$	0	0
$763$	12.7846i	0.462834i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5.07180i	0.183132i
$768$	0	0
$769$	37.4641	1.35099	0.675495	−	0.737365i	$-0.263930\pi$
$769$	37.4641	1.35099	0.675495	+	0.737365i	$0.263930\pi$
$770$	0	0
$771$	−6.67949	−0.240556
$772$	0	0
$773$	− 16.0526i	− 0.577370i	−0.957424	−	0.288685i	$-0.906782\pi$
$773$	− 16.0526i	− 0.577370i	0.957424	−	0.288685i	$-0.0932181\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	4.78461i	0.171647i
$778$	0	0
$779$	6.00000	0.214972
$780$	0	0
$781$	−32.7846	−1.17313
$782$	0	0
$783$	− 13.8564i	− 0.495188i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.80385i	0.0643002i	0.999483	+	0.0321501i	$0.0102355\pi$
$787$	1.80385i	0.0643002i	−0.999483	+	0.0321501i	$0.989765\pi$
$788$	0	0
$789$	11.3205	0.403021
$790$	0	0
$791$	−37.1769	−1.32186
$792$	0	0
$793$	− 6.14359i	− 0.218165i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 11.6603i	− 0.413027i	−0.978444	−	0.206514i	$-0.933788\pi$
$797$	− 11.6603i	− 0.413027i	0.978444	−	0.206514i	$-0.0662118\pi$
$798$	0	0
$799$	3.21539	0.113752
$800$	0	0
$801$	21.0333	0.743176
$802$	0	0
$803$	− 25.8564i	− 0.912453i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 7.60770i	− 0.267804i
$808$	0	0
$809$	45.7128	1.60718	0.803588	−	0.595185i	$-0.202921\pi$
$809$	45.7128	1.60718	0.803588	+	0.595185i	$0.202921\pi$
$810$	0	0
$811$	36.3923	1.27791	0.638953	−	0.769246i	$-0.279368\pi$
$811$	36.3923	1.27791	0.638953	+	0.769246i	$0.279368\pi$
$812$	0	0
$813$	− 13.4641i	− 0.472207i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 8.92820i	− 0.312358i
$818$	0	0
$819$	3.60770	0.126063
$820$	0	0
$821$	−43.8564	−1.53060	−0.765300	−	0.643674i	$-0.777409\pi$
$821$	−43.8564	−1.53060	−0.765300	+	0.643674i	$0.777409\pi$
$822$	0	0
$823$	43.5692i	1.51873i	0.650666	+	0.759364i	$0.274490\pi$
$823$	43.5692i	1.51873i	−0.650666	+	0.759364i	$0.725510\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.3397i	0.429095i	0.976714	+	0.214548i	$0.0688277\pi$
$827$	12.3397i	0.429095i	−0.976714	+	0.214548i	$0.931172\pi$
$828$	0	0
$829$	40.2487	1.39790	0.698948	−	0.715173i	$-0.253652\pi$
$829$	40.2487	1.39790	0.698948	+	0.715173i	$0.253652\pi$
$830$	0	0
$831$	1.75129	0.0607515
$832$	0	0
$833$	− 10.3923i	− 0.360072i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 21.8564i	− 0.755468i
$838$	0	0
$839$	−5.07180	−0.175098	−0.0875489	−	0.996160i	$-0.527903\pi$
$839$	−5.07180	−0.175098	−0.0875489	+	0.996160i	$0.527903\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	0	0
$843$	9.46410i	0.325961i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 2.00000i	− 0.0687208i
$848$	0	0
$849$	−1.75129	−0.0601041
$850$	0	0
$851$	11.3205	0.388062
$852$	0	0
$853$	24.6410i	0.843692i	0.906667	+	0.421846i	$0.138618\pi$
$853$	24.6410i	0.843692i	−0.906667	+	0.421846i	$0.861382\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 14.1962i	− 0.484931i	−0.970160	−	0.242466i	$-0.922044\pi$
$857$	− 14.1962i	− 0.484931i	0.970160	−	0.242466i	$-0.0779562\pi$
$858$	0	0
$859$	7.71281	0.263158	0.131579	−	0.991306i	$-0.457995\pi$
$859$	7.71281	0.263158	0.131579	+	0.991306i	$0.457995\pi$
$860$	0	0
$861$	8.78461	0.299379
$862$	0	0
$863$	− 40.7321i	− 1.38654i	−0.720680	−	0.693268i	$-0.756170\pi$
$863$	− 40.7321i	− 1.38654i	0.720680	−	0.693268i	$-0.243830\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 3.66025i	− 0.124309i
$868$	0	0
$869$	37.8564	1.28419
$870$	0	0
$871$	2.39230	0.0810602
$872$	0	0
$873$	− 35.9474i	− 1.21664i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	50.9808i	1.72150i	0.509030	+	0.860749i	$0.330004\pi$
$877$	50.9808i	1.72150i	−0.509030	+	0.860749i	$0.669996\pi$
$878$	0	0
$879$	8.53590	0.287909
$880$	0	0
$881$	−35.3205	−1.18998	−0.594989	−	0.803734i	$-0.702844\pi$
$881$	−35.3205	−1.18998	−0.594989	+	0.803734i	$0.702844\pi$
$882$	0	0
$883$	− 22.0000i	− 0.740359i	−0.928960	−	0.370179i	$-0.879296\pi$
$883$	− 22.0000i	− 0.740359i	0.928960	−	0.370179i	$-0.120704\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 54.5885i	− 1.83290i	−0.400148	−	0.916451i	$-0.631041\pi$
$887$	− 54.5885i	− 1.83290i	0.400148	−	0.916451i	$-0.368959\pi$
$888$	0	0
$889$	21.1769	0.710251
$890$	0	0
$891$	15.4641	0.518067
$892$	0	0
$893$	− 0.928203i	− 0.0310611i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.85641i	0.0619836i
$898$	0	0
$899$	18.9282	0.631291
$900$	0	0
$901$	−25.1769	−0.838765
$902$	0	0
$903$	− 13.0718i	− 0.435002i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 38.5885i	− 1.28131i	−0.767829	−	0.640654i	$-0.778663\pi$
$907$	− 38.5885i	− 1.28131i	0.767829	−	0.640654i	$-0.221337\pi$
$908$	0	0
$909$	10.8231	0.358979
$910$	0	0
$911$	21.4641	0.711137	0.355569	−	0.934650i	$-0.384287\pi$
$911$	21.4641	0.711137	0.355569	+	0.934650i	$0.384287\pi$
$912$	0	0
$913$	− 12.0000i	− 0.397142i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	10.1436i	0.334971i
$918$	0	0
$919$	52.4974	1.73173	0.865865	−	0.500278i	$-0.166769\pi$
$919$	52.4974	1.73173	0.865865	+	0.500278i	$0.166769\pi$
$920$	0	0
$921$	13.3205	0.438926
$922$	0	0
$923$	− 6.92820i	− 0.228045i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 9.01924i	− 0.296231i
$928$	0	0
$929$	−35.5692	−1.16699	−0.583494	−	0.812117i	$-0.698315\pi$
$929$	−35.5692	−1.16699	−0.583494	+	0.812117i	$0.698315\pi$
$930$	0	0
$931$	−3.00000	−0.0983210
$932$	0	0
$933$	16.3923i	0.536660i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	11.1769i	0.365134i	0.983193	+	0.182567i	$0.0584406\pi$
$937$	11.1769i	0.365134i	−0.983193	+	0.182567i	$0.941559\pi$
$938$	0	0
$939$	−22.5359	−0.735431
$940$	0	0
$941$	−16.6410	−0.542482	−0.271241	−	0.962512i	$-0.587434\pi$
$941$	−16.6410	−0.542482	−0.271241	+	0.962512i	$0.587434\pi$
$942$	0	0
$943$	− 20.7846i	− 0.676840i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 41.3205i	− 1.34274i	−0.741124	−	0.671368i	$-0.765707\pi$
$947$	− 41.3205i	− 1.34274i	0.741124	−	0.671368i	$-0.234293\pi$
$948$	0	0
$949$	5.46410	0.177372
$950$	0	0
$951$	15.4641	0.501458
$952$	0	0
$953$	29.4115i	0.952733i	0.879247	+	0.476367i	$0.158047\pi$
$953$	29.4115i	0.952733i	−0.879247	+	0.476367i	$0.841953\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 8.78461i	− 0.283966i
$958$	0	0
$959$	34.6410	1.11862
$960$	0	0
$961$	−1.14359	−0.0368901
$962$	0	0
$963$	− 28.7321i	− 0.925877i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	10.6795i	0.343429i	0.985147	+	0.171715i	$0.0549307\pi$
$967$	10.6795i	0.343429i	−0.985147	+	0.171715i	$0.945069\pi$
$968$	0	0
$969$	2.53590	0.0814648
$970$	0	0
$971$	−21.4641	−0.688816	−0.344408	−	0.938820i	$-0.611920\pi$
$971$	−21.4641	−0.688816	−0.344408	+	0.938820i	$0.611920\pi$
$972$	0	0
$973$	9.07180i	0.290828i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 21.8038i	− 0.697567i	−0.937203	−	0.348783i	$-0.886595\pi$
$977$	− 21.8038i	− 0.697567i	0.937203	−	0.348783i	$-0.113405\pi$
$978$	0	0
$979$	29.5692	0.945036
$980$	0	0
$981$	−15.7513	−0.502900
$982$	0	0
$983$	56.4449i	1.80031i	0.435568	+	0.900156i	$0.356548\pi$
$983$	56.4449i	1.80031i	−0.435568	+	0.900156i	$0.643452\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 1.35898i	− 0.0432569i
$988$	0	0
$989$	−30.9282	−0.983460
$990$	0	0
$991$	12.3923	0.393655	0.196827	−	0.980438i	$-0.436936\pi$
$991$	12.3923	0.393655	0.196827	+	0.980438i	$0.436936\pi$
$992$	0	0
$993$	− 19.2154i	− 0.609782i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 9.60770i	− 0.304279i	−0.988359	−	0.152139i	$-0.951384\pi$
$997$	− 9.60770i	− 0.304279i	0.988359	−	0.152139i	$-0.0486163\pi$
$998$	0	0
$999$	−13.0718	−0.413573

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.2.c.e.1749.2		4
5.2	odd	4		380.2.a.d.1.1	✓	2
5.3	odd	4		1900.2.a.d.1.2		2
5.4	even	2	inner	1900.2.c.e.1749.3		4
15.2	even	4		3420.2.a.h.1.1		2
20.3	even	4		7600.2.a.bf.1.1		2
20.7	even	4		1520.2.a.l.1.2		2
40.27	even	4		6080.2.a.bj.1.1		2
40.37	odd	4		6080.2.a.z.1.2		2
95.37	even	4		7220.2.a.h.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.a.d.1.1	✓	2	5.2	odd	4
1520.2.a.l.1.2		2	20.7	even	4
1900.2.a.d.1.2		2	5.3	odd	4
1900.2.c.e.1749.2		4	1.1	even	1	trivial
1900.2.c.e.1749.3		4	5.4	even	2	inner
3420.2.a.h.1.1		2	15.2	even	4
6080.2.a.z.1.2		2	40.37	odd	4
6080.2.a.bj.1.1		2	40.27	even	4
7220.2.a.h.1.2		2	95.37	even	4
7600.2.a.bf.1.1		2	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1900.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.732051i q^{3} -2.00000i q^{7} +2.46410 q^{9} +O(q^{10})\)	\(q-0.732051i q^{3} -2.00000i q^{7} +2.46410 q^{9} +3.46410 q^{11} +0.732051i q^{13} -3.46410i q^{17} -1.00000 q^{19} -1.46410 q^{21} +3.46410i q^{23} -4.00000i q^{27} +3.46410 q^{29} +5.46410 q^{31} -2.53590i q^{33} -3.26795i q^{37} +0.535898 q^{39} -6.00000 q^{41} +8.92820i q^{43} +0.928203i q^{47} +3.00000 q^{49} -2.53590 q^{51} -7.26795i q^{53} +0.732051i q^{57} +6.92820 q^{59} -8.39230 q^{61} -4.92820i q^{63} -3.26795i q^{67} +2.53590 q^{69} -9.46410 q^{71} -7.46410i q^{73} -6.92820i q^{77} +10.9282 q^{79} +4.46410 q^{81} -3.46410i q^{83} -2.53590i q^{87} +8.53590 q^{89} +1.46410 q^{91} -4.00000i q^{93} -14.5885i q^{97} +8.53590 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} - 4 q^{19} + 8 q^{21} + 8 q^{31} + 16 q^{39} - 24 q^{41} + 12 q^{49} - 24 q^{51} + 8 q^{61} + 24 q^{69} - 24 q^{71} + 16 q^{79} + 4 q^{81} + 48 q^{89} - 8 q^{91} + 48 q^{99}+O(q^{100}) \) 4 * q - 4 * q^9 - 4 * q^19 + 8 * q^21 + 8 * q^31 + 16 * q^39 - 24 * q^41 + 12 * q^49 - 24 * q^51 + 8 * q^61 + 24 * q^69 - 24 * q^71 + 16 * q^79 + 4 * q^81 + 48 * q^89 - 8 * q^91 + 48 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

Properties

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