LMFDB - Embedded newform 1900.3.e.f.1101.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,3,Mod(1101,1900)]

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, 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("1900.1101");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1900.e (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:	$51.7712502285$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 62x^{10} + 1445x^{8} + 15924x^{6} + 83244x^{4} + 170640x^{2} + 55600 $ x^12 + 62x^10 + 1445x^8 + 15924x^6 + 83244x^4 + 170640*x^2 + 55600
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{12}\cdot 5 $
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1101.12
Root			$4.83157i$ of defining polynomial
Character	$\chi$	$=$	1900.1101
Dual form			1900.3.e.f.1101.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.83157i q^{3} +3.72856 q^{7} -14.3441 q^{9} +O(q^{10})$	$q+4.83157i q^{3} +3.72856 q^{7} -14.3441 q^{9} -13.8938 q^{11} -4.10699i q^{13} +15.3087 q^{17} +(-16.4588 - 9.49250i) q^{19} +18.0148i q^{21} -17.4452 q^{23} -25.8203i q^{27} -16.3924i q^{29} -15.9121i q^{31} -67.1291i q^{33} -5.14304i q^{37} +19.8432 q^{39} +42.2785i q^{41} -1.56664 q^{43} +54.5821 q^{47} -35.0979 q^{49} +73.9649i q^{51} -55.9822i q^{53} +(45.8637 - 79.5219i) q^{57} -101.822i q^{59} +80.5719 q^{61} -53.4827 q^{63} -24.8838i q^{67} -84.2876i q^{69} -121.245i q^{71} +33.9798 q^{73} -51.8040 q^{77} -102.623i q^{79} -4.34417 q^{81} -80.3324 q^{83} +79.2009 q^{87} +12.0884i q^{89} -15.3131i q^{91} +76.8802 q^{93} +161.837i q^{97} +199.294 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{7} - 16 q^{9}+O(q^{10}) $ 12 * q + 12 * q^7 - 16 * q^9	$ 12 q + 12 q^{7} - 16 q^{9} - 32 q^{11} + 12 q^{17} + 24 q^{19} - 4 q^{23} + 124 q^{39} + 176 q^{43} + 72 q^{47} - 24 q^{49} - 140 q^{57} + 152 q^{61} - 48 q^{63} + 148 q^{73} - 376 q^{77} - 468 q^{81} + 208 q^{83} + 84 q^{87} + 184 q^{93} + 392 q^{99}+O(q^{100}) $ 12 * q + 12 * q^7 - 16 * q^9 - 32 * q^11 + 12 * q^17 + 24 * q^19 - 4 * q^23 + 124 * q^39 + 176 * q^43 + 72 * q^47 - 24 * q^49 - 140 * q^57 + 152 * q^61 - 48 * q^63 + 148 * q^73 - 376 * q^77 - 468 * q^81 + 208 * q^83 + 84 * q^87 + 184 * q^93 + 392 * 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
$3$			4.83157i			1.61052i	0.592919	+	0.805262i	$0.297975\pi$
$3$			4.83157i			1.61052i	−0.592919	+	0.805262i	$0.702025\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	3.72856			0.532651			0.266326	−	0.963883i	$-0.414190\pi$
$7$	3.72856			0.532651			0.266326	+	0.963883i	$0.414190\pi$
$8$		0			0
$9$	−14.3441			−1.59379
$10$		0			0
$11$	−13.8938			−1.26308			−0.631539	−	0.775344i	$-0.717576\pi$
$11$	−13.8938			−1.26308			−0.631539	+	0.775344i	$0.717576\pi$
$12$		0			0
$13$		−	4.10699i		−	0.315922i	−0.987445	−	0.157961i	$-0.949508\pi$
$13$		−	4.10699i		−	0.315922i	0.987445	−	0.157961i	$-0.0504920\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	15.3087			0.900510			0.450255	−	0.892900i	$-0.351333\pi$
$17$	15.3087			0.900510			0.450255	+	0.892900i	$0.351333\pi$
$18$		0			0
$19$	−16.4588	−	9.49250i	−0.866253	−	0.499605i
$19$	−16.4588	−	9.49250i	−0.866253	−	0.499605i
$20$		0			0
$21$			18.0148i			0.857847i
$22$		0			0
$23$	−17.4452			−0.758486			−0.379243	−	0.925297i	$-0.623816\pi$
$23$	−17.4452			−0.758486			−0.379243	+	0.925297i	$0.623816\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		−	25.8203i		−	0.956307i
$28$		0			0
$29$		−	16.3924i		−	0.565254i	−0.959230	−	0.282627i	$-0.908794\pi$
$29$		−	16.3924i		−	0.565254i	0.959230	−	0.282627i	$-0.0912059\pi$
$30$		0			0
$31$		−	15.9121i		−	0.513292i	−0.966505	−	0.256646i	$-0.917383\pi$
$31$		−	15.9121i		−	0.513292i	0.966505	−	0.256646i	$-0.0826175\pi$
$32$		0			0
$33$		−	67.1291i		−	2.03422i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	5.14304i		−	0.139001i	−0.997582	−	0.0695005i	$-0.977859\pi$
$37$		−	5.14304i		−	0.139001i	0.997582	−	0.0695005i	$-0.0221405\pi$
$38$		0			0
$39$	19.8432			0.508800
$40$		0			0
$41$			42.2785i			1.03118i	0.856835	+	0.515591i	$0.172428\pi$
$41$			42.2785i			1.03118i	−0.856835	+	0.515591i	$0.827572\pi$
$42$		0			0
$43$	−1.56664			−0.0364334			−0.0182167	−	0.999834i	$-0.505799\pi$
$43$	−1.56664			−0.0364334			−0.0182167	+	0.999834i	$0.505799\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	54.5821			1.16132			0.580660	−	0.814146i	$-0.302794\pi$
$47$	54.5821			1.16132			0.580660	+	0.814146i	$0.302794\pi$
$48$		0			0
$49$	−35.0979			−0.716283
$50$		0			0
$51$			73.9649i			1.45029i
$52$		0			0
$53$		−	55.9822i		−	1.05627i	−0.849161	−	0.528134i	$-0.822892\pi$
$53$		−	55.9822i		−	1.05627i	0.849161	−	0.528134i	$-0.177108\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	45.8637	−	79.5219i	0.804626	−	1.39512i
$58$		0			0
$59$		−	101.822i		−	1.72580i	−0.505376	−	0.862899i	$-0.668646\pi$
$59$		−	101.822i		−	1.72580i	0.505376	−	0.862899i	$-0.331354\pi$
$60$		0			0
$61$	80.5719			1.32085			0.660426	−	0.750892i	$-0.270376\pi$
$61$	80.5719			1.32085			0.660426	+	0.750892i	$0.270376\pi$
$62$		0			0
$63$	−53.4827			−0.848932
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	24.8838i		−	0.371400i	−0.982606	−	0.185700i	$-0.940545\pi$
$67$		−	24.8838i		−	0.371400i	0.982606	−	0.185700i	$-0.0594552\pi$
$68$		0			0
$69$		−	84.2876i		−	1.22156i
$70$		0			0
$71$		−	121.245i		−	1.70767i	−0.520545	−	0.853835i	$-0.674271\pi$
$71$		−	121.245i		−	1.70767i	0.520545	−	0.853835i	$-0.325729\pi$
$72$		0			0
$73$	33.9798			0.465476			0.232738	−	0.972539i	$-0.425231\pi$
$73$	33.9798			0.465476			0.232738	+	0.972539i	$0.425231\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−51.8040			−0.672779
$78$		0			0
$79$		−	102.623i		−	1.29902i	−0.760351	−	0.649512i	$-0.774973\pi$
$79$		−	102.623i		−	1.29902i	0.760351	−	0.649512i	$-0.225027\pi$
$80$		0			0
$81$	−4.34417			−0.0536317
$82$		0			0
$83$	−80.3324			−0.967860			−0.483930	−	0.875107i	$-0.660791\pi$
$83$	−80.3324			−0.967860			−0.483930	+	0.875107i	$0.660791\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	79.2009			0.910355
$88$		0			0
$89$			12.0884i			0.135824i	0.997691	+	0.0679121i	$0.0216338\pi$
$89$			12.0884i			0.135824i	−0.997691	+	0.0679121i	$0.978366\pi$
$90$		0			0
$91$		−	15.3131i		−	0.168276i
$92$		0			0
$93$	76.8802			0.826669
$94$		0			0
$95$		0			0
$96$		0			0
$97$			161.837i			1.66842i	0.551447	+	0.834210i	$0.314076\pi$
$97$			161.837i			1.66842i	−0.551447	+	0.834210i	$0.685924\pi$
$98$		0			0
$99$	199.294			2.01308
$100$		0			0
$101$	−6.07921			−0.0601901			−0.0300951	−	0.999547i	$-0.509581\pi$
$101$	−6.07921			−0.0601901			−0.0300951	+	0.999547i	$0.509581\pi$
$102$		0			0
$103$		−	27.8929i		−	0.270805i	−0.990791	−	0.135402i	$-0.956767\pi$
$103$		−	27.8929i		−	0.270805i	0.990791	−	0.135402i	$-0.0432327\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	61.4652i		−	0.574441i	−0.957865	−	0.287220i	$-0.907269\pi$
$107$		−	61.4652i		−	0.574441i	0.957865	−	0.287220i	$-0.0927312\pi$
$108$		0			0
$109$			30.2310i			0.277348i	0.990338	+	0.138674i	$0.0442841\pi$
$109$			30.2310i			0.277348i	−0.990338	+	0.138674i	$0.955716\pi$
$110$		0			0
$111$	24.8489			0.223864
$112$		0			0
$113$			178.939i			1.58353i	0.610826	+	0.791765i	$0.290837\pi$
$113$			178.939i			1.58353i	−0.610826	+	0.791765i	$0.709163\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$			58.9109i			0.503512i
$118$		0			0
$119$	57.0793			0.479658
$120$		0			0
$121$	72.0390			0.595364
$122$		0			0
$123$	−204.272			−1.66074
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	53.4071i		−	0.420528i	−0.977645	−	0.210264i	$-0.932568\pi$
$127$		−	53.4071i		−	0.420528i	0.977645	−	0.210264i	$-0.0674324\pi$
$128$		0			0
$129$		−	7.56932i		−	0.0586769i
$130$		0			0
$131$	−109.035			−0.832327			−0.416163	−	0.909290i	$-0.636626\pi$
$131$	−109.035			−0.832327			−0.416163	+	0.909290i	$0.636626\pi$
$132$		0			0
$133$	−61.3676	−	35.3933i	−0.461411	−	0.266115i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	24.9987			0.182472			0.0912362	−	0.995829i	$-0.470918\pi$
$137$	24.9987			0.182472			0.0912362	+	0.995829i	$0.470918\pi$
$138$		0			0
$139$	115.083			0.827936			0.413968	−	0.910291i	$-0.364143\pi$
$139$	115.083			0.827936			0.413968	+	0.910291i	$0.364143\pi$
$140$		0			0
$141$			263.717i			1.87033i
$142$		0			0
$143$			57.0619i			0.399034i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		−	169.578i		−	1.15359i
$148$		0			0
$149$	−141.364			−0.948750			−0.474375	−	0.880323i	$-0.657326\pi$
$149$	−141.364			−0.948750			−0.474375	+	0.880323i	$0.657326\pi$
$150$		0			0
$151$		−	250.107i		−	1.65634i	−0.560478	−	0.828169i	$-0.689383\pi$
$151$		−	250.107i		−	1.65634i	0.560478	−	0.828169i	$-0.310617\pi$
$152$		0			0
$153$	−219.589			−1.43522
$154$		0			0
$155$		0			0
$156$		0			0
$157$	14.5947			0.0929601			0.0464801	−	0.998919i	$-0.485200\pi$
$157$	14.5947			0.0929601			0.0464801	+	0.998919i	$0.485200\pi$
$158$		0			0
$159$	270.482			1.70115
$160$		0			0
$161$	−65.0454			−0.404009
$162$		0			0
$163$	9.57924			0.0587684			0.0293842	−	0.999568i	$-0.490645\pi$
$163$	9.57924			0.0587684			0.0293842	+	0.999568i	$0.490645\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	302.815i		−	1.81326i	−0.421923	−	0.906631i	$-0.638645\pi$
$167$		−	302.815i		−	1.81326i	0.421923	−	0.906631i	$-0.361355\pi$
$168$		0			0
$169$	152.133			0.900193
$170$		0			0
$171$	236.086	+	136.161i	1.38062	+	0.796264i
$172$		0			0
$173$			111.084i			0.642102i	0.947062	+	0.321051i	$0.104036\pi$
$173$			111.084i			0.642102i	−0.947062	+	0.321051i	$0.895964\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	491.961			2.77944
$178$		0			0
$179$			247.724i			1.38393i	0.721929	+	0.691967i	$0.243256\pi$
$179$			247.724i			1.38393i	−0.721929	+	0.691967i	$0.756744\pi$
$180$		0			0
$181$		−	234.560i		−	1.29591i	−0.761678	−	0.647956i	$-0.775624\pi$
$181$		−	234.560i		−	1.29591i	0.761678	−	0.647956i	$-0.224376\pi$
$182$		0			0
$183$			389.289i			2.12726i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−212.696			−1.13741
$188$		0			0
$189$		−	96.2724i		−	0.509378i
$190$		0			0
$191$	−250.384			−1.31091			−0.655455	−	0.755235i	$-0.727523\pi$
$191$	−250.384			−1.31091			−0.655455	+	0.755235i	$0.727523\pi$
$192$		0			0
$193$			122.208i			0.633204i	0.948558	+	0.316602i	$0.102542\pi$
$193$			122.208i			0.633204i	−0.948558	+	0.316602i	$0.897458\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	343.177			1.74201			0.871007	−	0.491271i	$-0.163467\pi$
$197$	343.177			1.74201			0.871007	+	0.491271i	$0.163467\pi$
$198$		0			0
$199$	297.583			1.49539			0.747696	−	0.664042i	$-0.231160\pi$
$199$	297.583			1.49539			0.747696	+	0.664042i	$0.231160\pi$
$200$		0			0
$201$	120.228			0.598148
$202$		0			0
$203$		−	61.1199i		−	0.301083i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	250.235			1.20886
$208$		0			0
$209$	228.676	+	131.887i	1.09414	+	0.631040i
$210$		0			0
$211$		−	237.792i		−	1.12698i	−0.826124	−	0.563488i	$-0.809459\pi$
$211$		−	237.792i		−	1.12698i	0.826124	−	0.563488i	$-0.190541\pi$
$212$		0			0
$213$	585.801			2.75024
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	59.3290i		−	0.273406i
$218$		0			0
$219$			164.176i			0.749661i
$220$		0			0
$221$		−	62.8725i		−	0.284491i
$222$		0			0
$223$			152.390i			0.683365i	0.939815	+	0.341682i	$0.110997\pi$
$223$			152.390i			0.683365i	−0.939815	+	0.341682i	$0.889003\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			205.814i			0.906670i	0.891340	+	0.453335i	$0.149766\pi$
$227$			205.814i			0.906670i	−0.891340	+	0.453335i	$0.850234\pi$
$228$		0			0
$229$	135.060			0.589783			0.294892	−	0.955531i	$-0.404716\pi$
$229$	135.060			0.589783			0.294892	+	0.955531i	$0.404716\pi$
$230$		0			0
$231$		−	250.295i		−	1.08353i
$232$		0			0
$233$	−100.593			−0.431728			−0.215864	−	0.976423i	$-0.569257\pi$
$233$	−100.593			−0.431728			−0.215864	+	0.976423i	$0.569257\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	495.830			2.09211
$238$		0			0
$239$	−328.565			−1.37475			−0.687373	−	0.726304i	$-0.741236\pi$
$239$	−328.565			−1.37475			−0.687373	+	0.726304i	$0.741236\pi$
$240$		0			0
$241$			144.249i			0.598543i	0.954168	+	0.299272i	$0.0967437\pi$
$241$			144.249i			0.598543i	−0.954168	+	0.299272i	$0.903256\pi$
$242$		0			0
$243$		−	253.372i		−	1.04268i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−38.9856	+	67.5961i	−0.157836	+	0.273669i
$248$		0			0
$249$		−	388.132i		−	1.55876i
$250$		0			0
$251$	66.0565			0.263173			0.131587	−	0.991305i	$-0.457993\pi$
$251$	66.0565			0.263173			0.131587	+	0.991305i	$0.457993\pi$
$252$		0			0
$253$	242.381			0.958027
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	268.255i		−	1.04379i	−0.853009	−	0.521896i	$-0.825225\pi$
$257$		−	268.255i		−	1.04379i	0.853009	−	0.521896i	$-0.174775\pi$
$258$		0			0
$259$		−	19.1761i		−	0.0740390i
$260$		0			0
$261$			235.133i			0.900895i
$262$		0			0
$263$	241.131			0.916848			0.458424	−	0.888734i	$-0.348414\pi$
$263$	241.131			0.916848			0.458424	+	0.888734i	$0.348414\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−58.4058			−0.218748
$268$		0			0
$269$			270.274i			1.00473i	0.864654	+	0.502367i	$0.167538\pi$
$269$			270.274i			1.00473i	−0.864654	+	0.502367i	$0.832462\pi$
$270$		0			0
$271$	159.872			0.589934			0.294967	−	0.955507i	$-0.404691\pi$
$271$	159.872			0.589934			0.294967	+	0.955507i	$0.404691\pi$
$272$		0			0
$273$	73.9865			0.271013
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−480.862			−1.73596			−0.867981	−	0.496597i	$-0.834583\pi$
$277$	−480.862			−1.73596			−0.867981	+	0.496597i	$0.834583\pi$
$278$		0			0
$279$			228.244i			0.818078i
$280$		0			0
$281$		−	125.506i		−	0.446639i	−0.974745	−	0.223320i	$-0.928311\pi$
$281$		−	125.506i		−	0.446639i	0.974745	−	0.223320i	$-0.0716894\pi$
$282$		0			0
$283$	169.824			0.600086			0.300043	−	0.953926i	$-0.402999\pi$
$283$	169.824			0.600086			0.300043	+	0.953926i	$0.402999\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			157.638i			0.549261i
$288$		0			0
$289$	−54.6445			−0.189081
$290$		0			0
$291$	−781.925			−2.68703
$292$		0			0
$293$		−	103.213i		−	0.352263i	−0.984367	−	0.176131i	$-0.943642\pi$
$293$		−	103.213i		−	0.352263i	0.984367	−	0.176131i	$-0.0563583\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$			358.743i			1.20789i
$298$		0			0
$299$			71.6472i			0.239623i
$300$		0			0
$301$	−5.84130			−0.0194063
$302$		0			0
$303$		−	29.3721i		−	0.0969377i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	470.447i		−	1.53240i	−0.642602	−	0.766200i	$-0.722145\pi$
$307$		−	470.447i		−	1.53240i	0.642602	−	0.766200i	$-0.277855\pi$
$308$		0			0
$309$	134.767			0.436138
$310$		0			0
$311$	−104.966			−0.337511			−0.168756	−	0.985658i	$-0.553975\pi$
$311$	−104.966			−0.337511			−0.168756	+	0.985658i	$0.553975\pi$
$312$		0			0
$313$	−358.845			−1.14647			−0.573234	−	0.819392i	$-0.694311\pi$
$313$	−358.845			−1.14647			−0.573234	+	0.819392i	$0.694311\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	367.285i		−	1.15863i	−0.815104	−	0.579314i	$-0.803320\pi$
$317$		−	367.285i		−	1.15863i	0.815104	−	0.579314i	$-0.196680\pi$
$318$		0			0
$319$			227.753i			0.713960i
$320$		0			0
$321$	296.973			0.925150
$322$		0			0
$323$	−251.963	−	145.318i	−0.780070	−	0.449900i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−146.063			−0.446676
$328$		0			0
$329$	203.512			0.618579
$330$		0			0
$331$			496.428i			1.49978i	0.661561	+	0.749892i	$0.269894\pi$
$331$			496.428i			1.49978i	−0.661561	+	0.749892i	$0.730106\pi$
$332$		0			0
$333$			73.7721i			0.221538i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	52.5036i		−	0.155797i	−0.996961	−	0.0778986i	$-0.975179\pi$
$337$		−	52.5036i		−	0.155797i	0.996961	−	0.0778986i	$-0.0248210\pi$
$338$		0			0
$339$	−864.556			−2.55031
$340$		0			0
$341$			221.080i			0.648328i
$342$		0			0
$343$	−313.564			−0.914180
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−409.556			−1.18028			−0.590138	−	0.807302i	$-0.700927\pi$
$347$	−409.556			−1.18028			−0.590138	+	0.807302i	$0.700927\pi$
$348$		0			0
$349$	562.063			1.61050			0.805248	−	0.592938i	$-0.202032\pi$
$349$	562.063			1.61050			0.805248	+	0.592938i	$0.202032\pi$
$350$		0			0
$351$	−106.044			−0.302118
$352$		0			0
$353$	−252.453			−0.715165			−0.357583	−	0.933881i	$-0.616399\pi$
$353$	−252.453			−0.715165			−0.357583	+	0.933881i	$0.616399\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$			275.783i			0.772500i
$358$		0			0
$359$	−119.344			−0.332433			−0.166217	−	0.986089i	$-0.553155\pi$
$359$	−119.344			−0.332433			−0.166217	+	0.986089i	$0.553155\pi$
$360$		0			0
$361$	180.785	+	312.470i	0.500789	+	0.865569i
$362$		0			0
$363$			348.062i			0.958848i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	690.635			1.88184			0.940920	−	0.338629i	$-0.109963\pi$
$367$	690.635			1.88184			0.940920	+	0.338629i	$0.109963\pi$
$368$		0			0
$369$		−	606.446i		−	1.64349i
$370$		0			0
$371$		−	208.733i		−	0.562623i
$372$		0			0
$373$		−	491.052i		−	1.31649i	−0.752803	−	0.658246i	$-0.771299\pi$
$373$		−	491.052i		−	1.31649i	0.752803	−	0.658246i	$-0.228701\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−67.3233			−0.178576
$378$		0			0
$379$		−	691.775i		−	1.82526i	−0.408782	−	0.912632i	$-0.634046\pi$
$379$		−	691.775i		−	1.82526i	0.408782	−	0.912632i	$-0.365954\pi$
$380$		0			0
$381$	258.040			0.677271
$382$		0			0
$383$			252.889i			0.660285i	0.943931	+	0.330142i	$0.107097\pi$
$383$			252.889i			0.660285i	−0.943931	+	0.330142i	$0.892903\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	22.4720			0.0580671
$388$		0			0
$389$	−504.764			−1.29759			−0.648797	−	0.760961i	$-0.724728\pi$
$389$	−504.764			−1.29759			−0.648797	+	0.760961i	$0.724728\pi$
$390$		0			0
$391$	−267.063			−0.683025
$392$		0			0
$393$		−	526.809i		−	1.34048i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−313.581			−0.789876			−0.394938	−	0.918708i	$-0.629234\pi$
$397$	−313.581			−0.789876			−0.394938	+	0.918708i	$0.629234\pi$
$398$		0			0
$399$	171.005	−	296.502i	0.428585	−	0.743113i
$400$		0			0
$401$			561.838i			1.40109i	0.713607	+	0.700546i	$0.247060\pi$
$401$			561.838i			1.40109i	−0.713607	+	0.700546i	$0.752940\pi$
$402$		0			0
$403$	−65.3506			−0.162160
$404$		0			0
$405$		0			0
$406$		0			0
$407$			71.4566i			0.175569i
$408$		0			0
$409$		−	165.613i		−	0.404921i	−0.979290	−	0.202460i	$-0.935106\pi$
$409$		−	165.613i		−	0.404921i	0.979290	−	0.202460i	$-0.0648937\pi$
$410$		0			0
$411$			120.783i			0.293876i
$412$		0			0
$413$		−	379.650i		−	0.919249i
$414$		0			0
$415$		0			0
$416$		0			0
$417$			556.032i			1.33341i
$418$		0			0
$419$	−440.399			−1.05107			−0.525536	−	0.850771i	$-0.676135\pi$
$419$	−440.399			−1.05107			−0.525536	+	0.850771i	$0.676135\pi$
$420$		0			0
$421$		−	786.831i		−	1.86896i	−0.356018	−	0.934479i	$-0.615866\pi$
$421$		−	786.831i		−	1.86896i	0.356018	−	0.934479i	$-0.384134\pi$
$422$		0			0
$423$	−782.929			−1.85090
$424$		0			0
$425$		0			0
$426$		0			0
$427$	300.417			0.703553
$428$		0			0
$429$	−275.698			−0.642654
$430$		0			0
$431$		−	219.539i		−	0.509372i	−0.967024	−	0.254686i	$-0.918028\pi$
$431$		−	219.539i		−	0.509372i	0.967024	−	0.254686i	$-0.0819721\pi$
$432$		0			0
$433$			450.900i			1.04134i	0.853758	+	0.520670i	$0.174318\pi$
$433$			450.900i			1.04134i	−0.853758	+	0.520670i	$0.825682\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	287.127	+	165.598i	0.657041	+	0.378944i
$438$		0			0
$439$			319.069i			0.726808i	0.931632	+	0.363404i	$0.118385\pi$
$439$			319.069i			0.726808i	−0.931632	+	0.363404i	$0.881615\pi$
$440$		0			0
$441$	503.446			1.14160
$442$		0			0
$443$	−780.776			−1.76247			−0.881237	−	0.472674i	$-0.843289\pi$
$443$	−780.776			−1.76247			−0.881237	+	0.472674i	$0.843289\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	683.009i		−	1.52798i
$448$		0			0
$449$		−	523.244i		−	1.16535i	−0.812704	−	0.582677i	$-0.802005\pi$
$449$		−	523.244i		−	1.16535i	0.812704	−	0.582677i	$-0.197995\pi$
$450$		0			0
$451$		−	587.411i		−	1.30246i
$452$		0			0
$453$	1208.41			2.66757
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−874.643			−1.91388			−0.956940	−	0.290285i	$-0.906250\pi$
$457$	−874.643			−1.91388			−0.956940	+	0.290285i	$0.906250\pi$
$458$		0			0
$459$		−	395.274i		−	0.861164i
$460$		0			0
$461$	120.368			0.261102			0.130551	−	0.991442i	$-0.458325\pi$
$461$	120.368			0.261102			0.130551	+	0.991442i	$0.458325\pi$
$462$		0			0
$463$	−256.911			−0.554883			−0.277441	−	0.960743i	$-0.589486\pi$
$463$	−256.911			−0.554883			−0.277441	+	0.960743i	$0.589486\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−490.911			−1.05120			−0.525601	−	0.850731i	$-0.676160\pi$
$467$	−490.911			−1.05120			−0.525601	+	0.850731i	$0.676160\pi$
$468$		0			0
$469$		−	92.7807i		−	0.197827i
$470$		0			0
$471$			70.5155i			0.149714i
$472$		0			0
$473$	21.7666			0.0460182
$474$		0			0
$475$		0			0
$476$		0			0
$477$			803.014i			1.68347i
$478$		0			0
$479$	545.752			1.13936			0.569679	−	0.821868i	$-0.307068\pi$
$479$	545.752			1.13936			0.569679	+	0.821868i	$0.307068\pi$
$480$		0			0
$481$	−21.1224			−0.0439135
$482$		0			0
$483$		−	314.271i		−	0.650665i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			243.032i			0.499039i	0.968370	+	0.249519i	$0.0802726\pi$
$487$			243.032i			0.499039i	−0.968370	+	0.249519i	$0.919727\pi$
$488$		0			0
$489$			46.2828i			0.0946479i
$490$		0			0
$491$	−555.332			−1.13102			−0.565511	−	0.824740i	$-0.691321\pi$
$491$	−555.332			−1.13102			−0.565511	+	0.824740i	$0.691321\pi$
$492$		0			0
$493$		−	250.946i		−	0.509017i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	452.067i		−	0.909592i
$498$		0			0
$499$	−231.235			−0.463397			−0.231698	−	0.972788i	$-0.574428\pi$
$499$	−231.235			−0.463397			−0.231698	+	0.972788i	$0.574428\pi$
$500$		0			0
$501$	1463.07			2.92030
$502$		0			0
$503$	206.048			0.409638			0.204819	−	0.978800i	$-0.434340\pi$
$503$	206.048			0.409638			0.204819	+	0.978800i	$0.434340\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			735.040i			1.44978i
$508$		0			0
$509$			398.832i			0.783560i	0.920059	+	0.391780i	$0.128141\pi$
$509$			398.832i			0.783560i	−0.920059	+	0.391780i	$0.871859\pi$
$510$		0			0
$511$	126.696			0.247936
$512$		0			0
$513$	−245.099	+	424.971i	−0.477776	+	0.828404i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−758.355			−1.46684
$518$		0			0
$519$	−536.708			−1.03412
$520$		0			0
$521$		−	262.617i		−	0.504064i	−0.967719	−	0.252032i	$-0.918901\pi$
$521$		−	262.617i		−	0.504064i	0.967719	−	0.252032i	$-0.0810988\pi$
$522$		0			0
$523$		−	814.031i		−	1.55646i	−0.627977	−	0.778232i	$-0.716117\pi$
$523$		−	814.031i		−	1.55646i	0.627977	−	0.778232i	$-0.283883\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	243.592i		−	0.462225i
$528$		0			0
$529$	−224.666			−0.424699
$530$		0			0
$531$			1460.54i			2.75055i
$532$		0			0
$533$	173.637			0.325773
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−1196.90			−2.22886
$538$		0			0
$539$	487.644			0.904720
$540$		0			0
$541$	−196.753			−0.363684			−0.181842	−	0.983328i	$-0.558206\pi$
$541$	−196.753			−0.363684			−0.181842	+	0.983328i	$0.558206\pi$
$542$		0			0
$543$	1133.29			2.08710
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	759.052i		−	1.38766i	−0.720137	−	0.693832i	$-0.755921\pi$
$547$		−	759.052i		−	1.38766i	0.720137	−	0.693832i	$-0.244079\pi$
$548$		0			0
$549$	−1155.73			−2.10515
$550$		0			0
$551$	−155.605	+	269.799i	−0.282404	+	0.489653i
$552$		0			0
$553$		−	382.636i		−	0.691927i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−705.968			−1.26745			−0.633723	−	0.773560i	$-0.718474\pi$
$557$	−705.968			−1.26745			−0.633723	+	0.773560i	$0.718474\pi$
$558$		0			0
$559$			6.43416i			0.0115101i
$560$		0			0
$561$		−	1027.66i		−	1.83183i
$562$		0			0
$563$		−	357.744i		−	0.635424i	−0.948187	−	0.317712i	$-0.897086\pi$
$563$		−	357.744i		−	0.635424i	0.948187	−	0.317712i	$-0.102914\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−16.1975			−0.0285670
$568$		0			0
$569$			448.411i			0.788069i	0.919096	+	0.394034i	$0.128921\pi$
$569$			448.411i			0.788069i	−0.919096	+	0.394034i	$0.871079\pi$
$570$		0			0
$571$	203.672			0.356694			0.178347	−	0.983968i	$-0.442925\pi$
$571$	203.672			0.356694			0.178347	+	0.983968i	$0.442925\pi$
$572$		0			0
$573$		−	1209.75i		−	2.11125i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−232.447			−0.402854			−0.201427	−	0.979504i	$-0.564558\pi$
$577$	−232.447			−0.402854			−0.201427	+	0.979504i	$0.564558\pi$
$578$		0			0
$579$	−590.458			−1.01979
$580$		0			0
$581$	−299.524			−0.515532
$582$		0			0
$583$			777.809i			1.33415i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1101.80			1.87701			0.938503	−	0.345271i	$-0.112213\pi$
$587$	1101.80			1.87701			0.938503	+	0.345271i	$0.112213\pi$
$588$		0			0
$589$	−151.045	+	261.894i	−0.256443	+	0.444641i
$590$		0			0
$591$			1658.08i			2.80555i
$592$		0			0
$593$	387.409			0.653303			0.326651	−	0.945145i	$-0.394080\pi$
$593$	387.409			0.653303			0.326651	+	0.945145i	$0.394080\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			1437.79i			2.40836i
$598$		0			0
$599$			649.288i			1.08395i	0.840393	+	0.541977i	$0.182324\pi$
$599$			649.288i			1.08395i	−0.840393	+	0.541977i	$0.817676\pi$
$600$		0			0
$601$		−	564.191i		−	0.938754i	−0.882998	−	0.469377i	$-0.844479\pi$
$601$		−	564.191i		−	0.938754i	0.882998	−	0.469377i	$-0.155521\pi$
$602$		0			0
$603$			356.935i			0.591932i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	431.505i		−	0.710882i	−0.934699	−	0.355441i	$-0.884331\pi$
$607$		−	431.505i		−	0.710882i	0.934699	−	0.355441i	$-0.115669\pi$
$608$		0			0
$609$	295.305			0.484902
$610$		0			0
$611$		−	224.168i		−	0.366887i
$612$		0			0
$613$	−360.161			−0.587539			−0.293769	−	0.955876i	$-0.594910\pi$
$613$	−360.161			−0.587539			−0.293769	+	0.955876i	$0.594910\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	255.900			0.414749			0.207375	−	0.978262i	$-0.433508\pi$
$617$	255.900			0.414749			0.207375	+	0.978262i	$0.433508\pi$
$618$		0			0
$619$	−231.220			−0.373538			−0.186769	−	0.982404i	$-0.559802\pi$
$619$	−231.220			−0.373538			−0.186769	+	0.982404i	$0.559802\pi$
$620$		0			0
$621$			450.440i			0.725346i
$622$		0			0
$623$			45.0722i			0.0723470i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−637.223	+	1104.87i	−1.01630	+	1.76215i
$628$		0			0
$629$		−	78.7331i		−	0.125172i
$630$		0			0
$631$	55.6688			0.0882232			0.0441116	−	0.999027i	$-0.485954\pi$
$631$	55.6688			0.0882232			0.0441116	+	0.999027i	$0.485954\pi$
$632$		0			0
$633$	1148.91			1.81502
$634$		0			0
$635$		0			0
$636$		0			0
$637$			144.146i			0.226290i
$638$		0			0
$639$			1739.14i			2.72166i
$640$		0			0
$641$		−	1086.58i		−	1.69513i	−0.530692	−	0.847565i	$-0.678068\pi$
$641$		−	1086.58i		−	1.69513i	0.530692	−	0.847565i	$-0.321932\pi$
$642$		0			0
$643$	−132.021			−0.205320			−0.102660	−	0.994717i	$-0.532735\pi$
$643$	−132.021			−0.205320			−0.102660	+	0.994717i	$0.532735\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−139.001			−0.214840			−0.107420	−	0.994214i	$-0.534259\pi$
$647$	−139.001			−0.214840			−0.107420	+	0.994214i	$0.534259\pi$
$648$		0			0
$649$			1414.70i			2.17982i
$650$		0			0
$651$	286.652			0.440326
$652$		0			0
$653$	2.61433			0.00400356			0.00200178	−	0.999998i	$-0.499363\pi$
$653$	2.61433			0.00400356			0.00200178	+	0.999998i	$0.499363\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−487.408			−0.741870
$658$		0			0
$659$		−	1115.25i		−	1.69234i	−0.532911	−	0.846171i	$-0.678902\pi$
$659$		−	1115.25i		−	1.69234i	0.532911	−	0.846171i	$-0.321098\pi$
$660$		0			0
$661$		−	1032.56i		−	1.56212i	−0.624453	−	0.781062i	$-0.714678\pi$
$661$		−	1032.56i		−	1.56212i	0.624453	−	0.781062i	$-0.285322\pi$
$662$		0			0
$663$	303.773			0.458180
$664$		0			0
$665$		0			0
$666$		0			0
$667$			285.968i			0.428738i
$668$		0			0
$669$	−736.285			−1.10058
$670$		0			0
$671$	−1119.45			−1.66834
$672$		0			0
$673$			923.804i			1.37267i	0.727288	+	0.686333i	$0.240781\pi$
$673$			923.804i			1.37267i	−0.727288	+	0.686333i	$0.759219\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	1144.25i		−	1.69017i	−0.534631	−	0.845086i	$-0.679549\pi$
$677$		−	1144.25i		−	1.69017i	0.534631	−	0.845086i	$-0.320451\pi$
$678$		0			0
$679$			603.417i			0.888686i
$680$		0			0
$681$	−994.406			−1.46021
$682$		0			0
$683$			686.997i			1.00585i	0.864330	+	0.502926i	$0.167743\pi$
$683$			686.997i			1.00585i	−0.864330	+	0.502926i	$0.832257\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$			652.554i			0.949860i
$688$		0			0
$689$	−229.918			−0.333699
$690$		0			0
$691$	−1216.77			−1.76088			−0.880442	−	0.474154i	$-0.842754\pi$
$691$	−1216.77			−1.76088			−0.880442	+	0.474154i	$0.842754\pi$
$692$		0			0
$693$	743.081			1.07227
$694$		0			0
$695$		0			0
$696$		0			0
$697$			647.228i			0.928591i
$698$		0			0
$699$		−	486.020i		−	0.695308i
$700$		0			0
$701$	−1082.00			−1.54351			−0.771753	−	0.635922i	$-0.780620\pi$
$701$	−1082.00			−1.54351			−0.771753	+	0.635922i	$0.780620\pi$
$702$		0			0
$703$	−48.8203	+	84.6482i	−0.0694456	+	0.120410i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−22.6667			−0.0320604
$708$		0			0
$709$	−639.509			−0.901987			−0.450994	−	0.892527i	$-0.648930\pi$
$709$	−639.509			−0.901987			−0.450994	+	0.892527i	$0.648930\pi$
$710$		0			0
$711$			1472.03i			2.07037i
$712$		0			0
$713$			277.589i			0.389325i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		−	1587.48i		−	2.21406i
$718$		0			0
$719$	−95.9878			−0.133502			−0.0667509	−	0.997770i	$-0.521263\pi$
$719$	−95.9878			−0.133502			−0.0667509	+	0.997770i	$0.521263\pi$
$720$		0			0
$721$		−	104.000i		−	0.144245i
$722$		0			0
$723$	−696.949			−0.963968
$724$		0			0
$725$		0			0
$726$		0			0
$727$	368.875			0.507394			0.253697	−	0.967284i	$-0.418353\pi$
$727$	368.875			0.507394			0.253697	+	0.967284i	$0.418353\pi$
$728$		0			0
$729$	1185.09			1.62563
$730$		0			0
$731$	−23.9831			−0.0328087
$732$		0			0
$733$	865.251			1.18042			0.590212	−	0.807248i	$-0.299044\pi$
$733$	865.251			1.18042			0.590212	+	0.807248i	$0.299044\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			345.732i			0.469107i
$738$		0			0
$739$	−934.331			−1.26432			−0.632159	−	0.774839i	$-0.717831\pi$
$739$	−934.331			−1.26432			−0.632159	+	0.774839i	$0.717831\pi$
$740$		0			0
$741$	−326.595	−	188.362i	−0.440750	−	0.254199i
$742$		0			0
$743$			235.551i			0.317027i	0.987357	+	0.158514i	$0.0506702\pi$
$743$			235.551i			0.317027i	−0.987357	+	0.158514i	$0.949330\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	1152.29			1.54256
$748$		0			0
$749$		−	229.176i		−	0.305976i
$750$		0			0
$751$		−	971.042i		−	1.29300i	−0.762915	−	0.646499i	$-0.776232\pi$
$751$		−	971.042i		−	1.29300i	0.762915	−	0.646499i	$-0.223768\pi$
$752$		0			0
$753$			319.156i			0.423847i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	177.074			0.233916			0.116958	−	0.993137i	$-0.462686\pi$
$757$	177.074			0.233916			0.116958	+	0.993137i	$0.462686\pi$
$758$		0			0
$759$			1171.08i			1.54292i
$760$		0			0
$761$	−159.521			−0.209621			−0.104810	−	0.994492i	$-0.533424\pi$
$761$	−159.521			−0.209621			−0.104810	+	0.994492i	$0.533424\pi$
$762$		0			0
$763$			112.718i			0.147730i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−418.182			−0.545218
$768$		0			0
$769$	707.134			0.919551			0.459775	−	0.888035i	$-0.347930\pi$
$769$	707.134			0.919551			0.459775	+	0.888035i	$0.347930\pi$
$770$		0			0
$771$	1296.09			1.68105
$772$		0			0
$773$			799.572i			1.03438i	0.855872	+	0.517188i	$0.173021\pi$
$773$			799.572i			1.03438i	−0.855872	+	0.517188i	$0.826979\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	92.6507			0.119242
$778$		0			0
$779$	401.329	−	695.854i	0.515184	−	0.893265i
$780$		0			0
$781$			1684.55i			2.15692i
$782$		0			0
$783$	−423.256			−0.540557
$784$		0			0
$785$		0			0
$786$		0			0
$787$			1366.97i			1.73693i	0.495749	+	0.868466i	$0.334893\pi$
$787$			1366.97i			1.73693i	−0.495749	+	0.868466i	$0.665107\pi$
$788$		0			0
$789$			1165.04i			1.47661i
$790$		0			0
$791$			667.184i			0.843469i
$792$		0			0
$793$		−	330.908i		−	0.417286i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			121.258i			0.152143i	0.997102	+	0.0760717i	$0.0242378\pi$
$797$			121.258i			0.152143i	−0.997102	+	0.0760717i	$0.975762\pi$
$798$		0			0
$799$	835.579			1.04578
$800$		0			0
$801$		−	173.396i		−	0.216475i
$802$		0			0
$803$	−472.110			−0.587933
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−1305.85			−1.61815
$808$		0			0
$809$	820.488			1.01420			0.507100	−	0.861887i	$-0.330717\pi$
$809$	820.488			1.01420			0.507100	+	0.861887i	$0.330717\pi$
$810$		0			0
$811$			537.722i			0.663036i	0.943449	+	0.331518i	$0.107561\pi$
$811$			537.722i			0.663036i	−0.943449	+	0.331518i	$0.892439\pi$
$812$		0			0
$813$			772.434i			0.950103i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	25.7850	+	14.8713i	0.0315606	+	0.0182023i
$818$		0			0
$819$			219.653i			0.268196i
$820$		0			0
$821$	−54.0424			−0.0658250			−0.0329125	−	0.999458i	$-0.510478\pi$
$821$	−54.0424			−0.0658250			−0.0329125	+	0.999458i	$0.510478\pi$
$822$		0			0
$823$	599.052			0.727889			0.363944	−	0.931421i	$-0.381430\pi$
$823$	599.052			0.727889			0.363944	+	0.931421i	$0.381430\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	620.339i		−	0.750108i	−0.927003	−	0.375054i	$-0.877624\pi$
$827$		−	620.339i		−	0.750108i	0.927003	−	0.375054i	$-0.122376\pi$
$828$		0			0
$829$			36.8107i			0.0444038i	0.999754	+	0.0222019i	$0.00706766\pi$
$829$			36.8107i			0.0444038i	−0.999754	+	0.0222019i	$0.992932\pi$
$830$		0			0
$831$		−	2323.32i		−	2.79581i
$832$		0			0
$833$	−537.302			−0.645020
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−410.854			−0.490865
$838$		0			0
$839$			531.549i			0.633551i	0.948501	+	0.316776i	$0.102600\pi$
$839$			531.549i			0.633551i	−0.948501	+	0.316776i	$0.897400\pi$
$840$		0			0
$841$	572.290			0.680488
$842$		0			0
$843$	606.389			0.719323
$844$		0			0
$845$		0			0
$846$		0			0
$847$	268.602			0.317121
$848$		0			0
$849$			820.519i			0.966453i
$850$		0			0
$851$			89.7212i			0.105430i
$852$		0			0
$853$	−16.6518			−0.0195214			−0.00976072	−	0.999952i	$-0.503107\pi$
$853$	−16.6518			−0.0195214			−0.00976072	+	0.999952i	$0.503107\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	656.701i		−	0.766279i	−0.923691	−	0.383139i	$-0.874843\pi$
$857$		−	656.701i		−	0.766279i	0.923691	−	0.383139i	$-0.125157\pi$
$858$		0			0
$859$	−7.72122			−0.00898862			−0.00449431	−	0.999990i	$-0.501431\pi$
$859$	−7.72122			−0.00898862			−0.00449431	+	0.999990i	$0.501431\pi$
$860$		0			0
$861$	−761.638			−0.884597
$862$		0			0
$863$			1132.71i			1.31253i	0.754530	+	0.656266i	$0.227865\pi$
$863$			1132.71i			1.31253i	−0.754530	+	0.656266i	$0.772135\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	264.019i		−	0.304520i
$868$		0			0
$869$			1425.83i			1.64077i
$870$		0			0
$871$	−102.197			−0.117333
$872$		0			0
$873$		−	2321.40i		−	2.65910i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	826.040i		−	0.941893i	−0.882162	−	0.470947i	$-0.843912\pi$
$877$		−	826.040i		−	0.941893i	0.882162	−	0.470947i	$-0.156088\pi$
$878$		0			0
$879$	498.681			0.567327
$880$		0			0
$881$	691.162			0.784520			0.392260	−	0.919854i	$-0.371693\pi$
$881$	691.162			0.784520			0.392260	+	0.919854i	$0.371693\pi$
$882$		0			0
$883$	1339.08			1.51651			0.758255	−	0.651958i	$-0.226052\pi$
$883$	1339.08			1.51651			0.758255	+	0.651958i	$0.226052\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			1263.16i			1.42408i	0.702139	+	0.712040i	$0.252228\pi$
$887$			1263.16i			1.42408i	−0.702139	+	0.712040i	$0.747772\pi$
$888$		0			0
$889$		−	199.131i		−	0.223995i
$890$		0			0
$891$	60.3572			0.0677409
$892$		0			0
$893$	−898.356	−	518.120i	−1.00600	−	0.580202i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−346.168			−0.385918
$898$		0			0
$899$	−260.836			−0.290141
$900$		0			0
$901$		−	857.014i		−	0.951181i
$902$		0			0
$903$		−	28.2227i		−	0.0312543i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	531.490i		−	0.585987i	−0.956114	−	0.292994i	$-0.905349\pi$
$907$		−	531.490i		−	0.585987i	0.956114	−	0.292994i	$-0.0946515\pi$
$908$		0			0
$909$	87.2006			0.0959302
$910$		0			0
$911$		−	214.421i		−	0.235368i	−0.993051	−	0.117684i	$-0.962453\pi$
$911$		−	214.421i		−	0.235368i	0.993051	−	0.117684i	$-0.0375471\pi$
$912$		0			0
$913$	1116.13			1.22248
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−406.542			−0.443340
$918$		0			0
$919$	−1523.80			−1.65811			−0.829055	−	0.559167i	$-0.811121\pi$
$919$	−1523.80			−1.65811			−0.829055	+	0.559167i	$0.811121\pi$
$920$		0			0
$921$	2273.00			2.46797
$922$		0			0
$923$	−497.950			−0.539490
$924$		0			0
$925$		0			0
$926$		0			0
$927$			400.098i			0.431605i
$928$		0			0
$929$	−269.533			−0.290132			−0.145066	−	0.989422i	$-0.546339\pi$
$929$	−269.533			−0.290132			−0.145066	+	0.989422i	$0.546339\pi$
$930$		0			0
$931$	577.669	+	333.166i	0.620482	+	0.357859i
$932$		0			0
$933$		−	507.151i		−	0.543570i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	492.223			0.525318			0.262659	−	0.964889i	$-0.415401\pi$
$937$	492.223			0.525318			0.262659	+	0.964889i	$0.415401\pi$
$938$		0			0
$939$		−	1733.78i		−	1.84641i
$940$		0			0
$941$		−	1364.05i		−	1.44958i	−0.688972	−	0.724788i	$-0.741938\pi$
$941$		−	1364.05i		−	1.44958i	0.688972	−	0.724788i	$-0.258062\pi$
$942$		0			0
$943$		−	737.556i		−	0.782138i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1082.40			1.14298			0.571490	−	0.820609i	$-0.306366\pi$
$947$	1082.40			1.14298			0.571490	+	0.820609i	$0.306366\pi$
$948$		0			0
$949$		−	139.554i		−	0.147054i
$950$		0			0
$951$	1774.56			1.86600
$952$		0			0
$953$		−	196.828i		−	0.206535i	−0.994654	−	0.103268i	$-0.967070\pi$
$953$		−	196.828i		−	0.206535i	0.994654	−	0.103268i	$-0.0329298\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−1100.41			−1.14985
$958$		0			0
$959$	93.2092			0.0971941
$960$		0			0
$961$	707.806			0.736531
$962$		0			0
$963$			881.661i			0.915536i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1381.14			−1.42828			−0.714138	−	0.700005i	$-0.753181\pi$
$967$	−1381.14			−1.42828			−0.714138	+	0.700005i	$0.753181\pi$
$968$		0			0
$969$	702.112	−	1217.37i	0.724574	−	1.25632i
$970$		0			0
$971$			1116.21i			1.14954i	0.818314	+	0.574771i	$0.194909\pi$
$971$			1116.21i			1.14954i	−0.818314	+	0.574771i	$0.805091\pi$
$972$		0			0
$973$	429.094			0.441001
$974$		0			0
$975$		0			0
$976$		0			0
$977$			1784.43i			1.82644i	0.407471	+	0.913218i	$0.366411\pi$
$977$			1784.43i			1.82644i	−0.407471	+	0.913218i	$0.633589\pi$
$978$		0			0
$979$		−	167.954i		−	0.171557i
$980$		0			0
$981$		−	433.635i		−	0.442034i
$982$		0			0
$983$			600.203i			0.610583i	0.952259	+	0.305291i	$0.0987538\pi$
$983$			600.203i			0.610583i	−0.952259	+	0.305291i	$0.901246\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$			983.285i			0.996236i
$988$		0			0
$989$	27.3303			0.0276343
$990$		0			0
$991$			14.0179i			0.0141452i	0.999975	+	0.00707259i	$0.00225129\pi$
$991$			14.0179i			0.0141452i	−0.999975	+	0.00707259i	$0.997749\pi$
$992$		0			0
$993$	−2398.53			−2.41544
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−137.727			−0.138142			−0.0690708	−	0.997612i	$-0.522003\pi$
$997$	−137.727			−0.138142			−0.0690708	+	0.997612i	$0.522003\pi$
$998$		0			0
$999$	−132.795			−0.132928

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.3.e.f.1101.12		12
5.2	odd	4		1900.3.g.c.949.23		24
5.3	odd	4		1900.3.g.c.949.2		24
5.4	even	2		380.3.e.a.341.1	✓	12
15.14	odd	2		3420.3.o.a.721.4		12
19.18	odd	2	inner	1900.3.e.f.1101.1		12
20.19	odd	2		1520.3.h.b.721.12		12
95.18	even	4		1900.3.g.c.949.24		24
95.37	even	4		1900.3.g.c.949.1		24
95.94	odd	2		380.3.e.a.341.12	yes	12
285.284	even	2		3420.3.o.a.721.3		12
380.379	even	2		1520.3.h.b.721.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.3.e.a.341.1	✓	12	5.4	even	2
380.3.e.a.341.12	yes	12	95.94	odd	2
1520.3.h.b.721.1		12	380.379	even	2
1520.3.h.b.721.12		12	20.19	odd	2
1900.3.e.f.1101.1		12	19.18	odd	2	inner
1900.3.e.f.1101.12		12	1.1	even	1	trivial
1900.3.g.c.949.1		24	95.37	even	4
1900.3.g.c.949.2		24	5.3	odd	4
1900.3.g.c.949.23		24	5.2	odd	4
1900.3.g.c.949.24		24	95.18	even	4
3420.3.o.a.721.3		12	285.284	even	2
3420.3.o.a.721.4		12	15.14	odd	2

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1900.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.83157i q^{3} +3.72856 q^{7} -14.3441 q^{9} +O(q^{10})\)	\(q+4.83157i q^{3} +3.72856 q^{7} -14.3441 q^{9} -13.8938 q^{11} -4.10699i q^{13} +15.3087 q^{17} +(-16.4588 - 9.49250i) q^{19} +18.0148i q^{21} -17.4452 q^{23} -25.8203i q^{27} -16.3924i q^{29} -15.9121i q^{31} -67.1291i q^{33} -5.14304i q^{37} +19.8432 q^{39} +42.2785i q^{41} -1.56664 q^{43} +54.5821 q^{47} -35.0979 q^{49} +73.9649i q^{51} -55.9822i q^{53} +(45.8637 - 79.5219i) q^{57} -101.822i q^{59} +80.5719 q^{61} -53.4827 q^{63} -24.8838i q^{67} -84.2876i q^{69} -121.245i q^{71} +33.9798 q^{73} -51.8040 q^{77} -102.623i q^{79} -4.34417 q^{81} -80.3324 q^{83} +79.2009 q^{87} +12.0884i q^{89} -15.3131i q^{91} +76.8802 q^{93} +161.837i q^{97} +199.294 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{7} - 16 q^{9}+O(q^{10}) \) 12 * q + 12 * q^7 - 16 * q^9	\( 12 q + 12 q^{7} - 16 q^{9} - 32 q^{11} + 12 q^{17} + 24 q^{19} - 4 q^{23} + 124 q^{39} + 176 q^{43} + 72 q^{47} - 24 q^{49} - 140 q^{57} + 152 q^{61} - 48 q^{63} + 148 q^{73} - 376 q^{77} - 468 q^{81} + 208 q^{83} + 84 q^{87} + 184 q^{93} + 392 q^{99}+O(q^{100}) \) 12 * q + 12 * q^7 - 16 * q^9 - 32 * q^11 + 12 * q^17 + 24 * q^19 - 4 * q^23 + 124 * q^39 + 176 * q^43 + 72 * q^47 - 24 * q^49 - 140 * q^57 + 152 * q^61 - 48 * q^63 + 148 * q^73 - 376 * q^77 - 468 * q^81 + 208 * q^83 + 84 * q^87 + 184 * q^93 + 392 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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