LMFDB - Embedded newform 2100.4.k.a.1849.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2100,4,Mod(1849,2100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2100, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2100.1849");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2100.k (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:	$123.904011012$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 420)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	2100.1849
Dual form			2100.4.k.a.1849.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} -7.00000i q^{7} -9.00000 q^{9} -44.0000 q^{11} +42.0000i q^{13} -94.0000i q^{17} +36.0000 q^{19} -21.0000 q^{21} -24.0000i q^{23} +27.0000i q^{27} -54.0000 q^{29} -112.000 q^{31} +132.000i q^{33} -322.000i q^{37} +126.000 q^{39} -22.0000 q^{41} -292.000i q^{43} +272.000i q^{47} -49.0000 q^{49} -282.000 q^{51} +578.000i q^{53} -108.000i q^{57} +44.0000 q^{59} -26.0000 q^{61} +63.0000i q^{63} +12.0000i q^{67} -72.0000 q^{69} -280.000 q^{71} -410.000i q^{73} +308.000i q^{77} +320.000 q^{79} +81.0000 q^{81} +1252.00i q^{83} +162.000i q^{87} +38.0000 q^{89} +294.000 q^{91} +336.000i q^{93} +1250.00i q^{97} +396.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{9} - 88 q^{11} + 72 q^{19} - 42 q^{21} - 108 q^{29} - 224 q^{31} + 252 q^{39} - 44 q^{41} - 98 q^{49} - 564 q^{51} + 88 q^{59} - 52 q^{61} - 144 q^{69} - 560 q^{71} + 640 q^{79} + 162 q^{81}+ \cdots + 792 q^{99}+O(q^{100}) $ 2 * q - 18 * q^9 - 88 * q^11 + 72 * q^19 - 42 * q^21 - 108 * q^29 - 224 * q^31 + 252 * q^39 - 44 * q^41 - 98 * q^49 - 564 * q^51 + 88 * q^59 - 52 * q^61 - 144 * q^69 - 560 * q^71 + 640 * q^79 + 162 * q^81 + 76 * q^89 + 588 * q^91 + 792 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$.

$n$	$701$	$1051$	$1177$	$1501$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 7.00000i	− 0.377964i
$7$	− 7.00000i	− 0.377964i
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	−44.0000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−44.0000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	42.0000i	0.896054i	0.894020	+	0.448027i	$0.147873\pi$
$13$	42.0000i	0.896054i	−0.894020	+	0.448027i	$0.852127\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 94.0000i	− 1.34108i	−0.741874	−	0.670540i	$-0.766063\pi$
$17$	− 94.0000i	− 1.34108i	0.741874	−	0.670540i	$-0.233937\pi$
$18$	0	0
$19$	36.0000	0.434682	0.217341	−	0.976096i	$-0.430262\pi$
$19$	36.0000	0.434682	0.217341	+	0.976096i	$0.430262\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	− 24.0000i	− 0.217580i	−0.994065	−	0.108790i	$-0.965302\pi$
$23$	− 24.0000i	− 0.217580i	0.994065	−	0.108790i	$-0.0346976\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	−54.0000	−0.345778	−0.172889	−	0.984941i	$-0.555310\pi$
$29$	−54.0000	−0.345778	−0.172889	+	0.984941i	$0.555310\pi$
$30$	0	0
$31$	−112.000	−0.648897	−0.324448	−	0.945903i	$-0.605179\pi$
$31$	−112.000	−0.648897	−0.324448	+	0.945903i	$0.605179\pi$
$32$	0	0
$33$	132.000i	0.696311i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 322.000i	− 1.43072i	−0.698758	−	0.715358i	$-0.746264\pi$
$37$	− 322.000i	− 1.43072i	0.698758	−	0.715358i	$-0.253736\pi$
$38$	0	0
$39$	126.000	0.517337
$40$	0	0
$41$	−22.0000	−0.0838006	−0.0419003	−	0.999122i	$-0.513341\pi$
$41$	−22.0000	−0.0838006	−0.0419003	+	0.999122i	$0.513341\pi$
$42$	0	0
$43$	− 292.000i	− 1.03557i	−0.855510	−	0.517786i	$-0.826756\pi$
$43$	− 292.000i	− 1.03557i	0.855510	−	0.517786i	$-0.173244\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	272.000i	0.844155i	0.906560	+	0.422077i	$0.138699\pi$
$47$	272.000i	0.844155i	−0.906560	+	0.422077i	$0.861301\pi$
$48$	0	0
$49$	−49.0000	−0.142857
$50$	0	0
$51$	−282.000	−0.774273
$52$	0	0
$53$	578.000i	1.49801i	0.662566	+	0.749004i	$0.269468\pi$
$53$	578.000i	1.49801i	−0.662566	+	0.749004i	$0.730532\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 108.000i	− 0.250964i
$58$	0	0
$59$	44.0000	0.0970900	0.0485450	−	0.998821i	$-0.484542\pi$
$59$	44.0000	0.0970900	0.0485450	+	0.998821i	$0.484542\pi$
$60$	0	0
$61$	−26.0000	−0.0545731	−0.0272865	−	0.999628i	$-0.508687\pi$
$61$	−26.0000	−0.0545731	−0.0272865	+	0.999628i	$0.508687\pi$
$62$	0	0
$63$	63.0000i	0.125988i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.0000i	0.0218811i	0.999940	+	0.0109405i	$0.00348255\pi$
$67$	12.0000i	0.0218811i	−0.999940	+	0.0109405i	$0.996517\pi$
$68$	0	0
$69$	−72.0000	−0.125620
$70$	0	0
$71$	−280.000	−0.468027	−0.234013	−	0.972233i	$-0.575186\pi$
$71$	−280.000	−0.468027	−0.234013	+	0.972233i	$0.575186\pi$
$72$	0	0
$73$	− 410.000i	− 0.657354i	−0.944442	−	0.328677i	$-0.893397\pi$
$73$	− 410.000i	− 0.657354i	0.944442	−	0.328677i	$-0.106603\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	308.000i	0.455842i
$78$	0	0
$79$	320.000	0.455732	0.227866	−	0.973693i	$-0.426825\pi$
$79$	320.000	0.455732	0.227866	+	0.973693i	$0.426825\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1252.00i	1.65572i	0.560934	+	0.827861i	$0.310442\pi$
$83$	1252.00i	1.65572i	−0.560934	+	0.827861i	$0.689558\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	162.000i	0.199635i
$88$	0	0
$89$	38.0000	0.0452583	0.0226292	−	0.999744i	$-0.492796\pi$
$89$	38.0000	0.0452583	0.0226292	+	0.999744i	$0.492796\pi$
$90$	0	0
$91$	294.000	0.338677
$92$	0	0
$93$	336.000i	0.374641i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1250.00i	1.30844i	0.756306	+	0.654218i	$0.227002\pi$
$97$	1250.00i	1.30844i	−0.756306	+	0.654218i	$0.772998\pi$
$98$	0	0
$99$	396.000	0.402015
$100$	0	0
$101$	−834.000	−0.821645	−0.410822	−	0.911715i	$-0.634758\pi$
$101$	−834.000	−0.821645	−0.410822	+	0.911715i	$0.634758\pi$
$102$	0	0
$103$	728.000i	0.696427i	0.937415	+	0.348213i	$0.113212\pi$
$103$	728.000i	0.696427i	−0.937415	+	0.348213i	$0.886788\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 108.000i	− 0.0975771i	−0.998809	−	0.0487886i	$-0.984464\pi$
$107$	− 108.000i	− 0.0975771i	0.998809	−	0.0487886i	$-0.0155361\pi$
$108$	0	0
$109$	−534.000	−0.469247	−0.234624	−	0.972086i	$-0.575386\pi$
$109$	−534.000	−0.469247	−0.234624	+	0.972086i	$0.575386\pi$
$110$	0	0
$111$	−966.000	−0.826024
$112$	0	0
$113$	1774.00i	1.47685i	0.674336	+	0.738424i	$0.264430\pi$
$113$	1774.00i	1.47685i	−0.674336	+	0.738424i	$0.735570\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 378.000i	− 0.298685i
$118$	0	0
$119$	−658.000	−0.506880
$120$	0	0
$121$	605.000	0.454545
$122$	0	0
$123$	66.0000i	0.0483823i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2512.00i	1.75515i	0.479440	+	0.877575i	$0.340840\pi$
$127$	2512.00i	1.75515i	−0.479440	+	0.877575i	$0.659160\pi$
$128$	0	0
$129$	−876.000	−0.597888
$130$	0	0
$131$	1388.00	0.925726	0.462863	−	0.886430i	$-0.346822\pi$
$131$	1388.00	0.925726	0.462863	+	0.886430i	$0.346822\pi$
$132$	0	0
$133$	− 252.000i	− 0.164295i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2406.00i	− 1.50043i	−0.661196	−	0.750213i	$-0.729951\pi$
$137$	− 2406.00i	− 1.50043i	0.661196	−	0.750213i	$-0.270049\pi$
$138$	0	0
$139$	860.000	0.524779	0.262389	−	0.964962i	$-0.415490\pi$
$139$	860.000	0.524779	0.262389	+	0.964962i	$0.415490\pi$
$140$	0	0
$141$	816.000	0.487373
$142$	0	0
$143$	− 1848.00i	− 1.08068i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	147.000i	0.0824786i
$148$	0	0
$149$	−798.000	−0.438756	−0.219378	−	0.975640i	$-0.570403\pi$
$149$	−798.000	−0.438756	−0.219378	+	0.975640i	$0.570403\pi$
$150$	0	0
$151$	3416.00	1.84099	0.920497	−	0.390749i	$-0.127784\pi$
$151$	3416.00	1.84099	0.920497	+	0.390749i	$0.127784\pi$
$152$	0	0
$153$	846.000i	0.447026i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	70.0000i	0.0355835i	0.999842	+	0.0177917i	$0.00566359\pi$
$157$	70.0000i	0.0355835i	−0.999842	+	0.0177917i	$0.994336\pi$
$158$	0	0
$159$	1734.00	0.864875
$160$	0	0
$161$	−168.000	−0.0822376
$162$	0	0
$163$	− 2444.00i	− 1.17441i	−0.809438	−	0.587205i	$-0.800228\pi$
$163$	− 2444.00i	− 1.17441i	0.809438	−	0.587205i	$-0.199772\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 648.000i	− 0.300262i	−0.988666	−	0.150131i	$-0.952030\pi$
$167$	− 648.000i	− 0.300262i	0.988666	−	0.150131i	$-0.0479695\pi$
$168$	0	0
$169$	433.000	0.197087
$170$	0	0
$171$	−324.000	−0.144894
$172$	0	0
$173$	− 1526.00i	− 0.670634i	−0.942105	−	0.335317i	$-0.891157\pi$
$173$	− 1526.00i	− 0.670634i	0.942105	−	0.335317i	$-0.108843\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 132.000i	− 0.0560550i
$178$	0	0
$179$	84.0000	0.0350752	0.0175376	−	0.999846i	$-0.494417\pi$
$179$	84.0000	0.0350752	0.0175376	+	0.999846i	$0.494417\pi$
$180$	0	0
$181$	−1970.00	−0.809000	−0.404500	−	0.914538i	$-0.632554\pi$
$181$	−1970.00	−0.809000	−0.404500	+	0.914538i	$0.632554\pi$
$182$	0	0
$183$	78.0000i	0.0315078i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4136.00i	1.61740i
$188$	0	0
$189$	189.000	0.0727393
$190$	0	0
$191$	720.000	0.272761	0.136381	−	0.990657i	$-0.456453\pi$
$191$	720.000	0.272761	0.136381	+	0.990657i	$0.456453\pi$
$192$	0	0
$193$	1534.00i	0.572123i	0.958211	+	0.286061i	$0.0923462\pi$
$193$	1534.00i	0.572123i	−0.958211	+	0.286061i	$0.907654\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3950.00i	1.42856i	0.699861	+	0.714279i	$0.253245\pi$
$197$	3950.00i	1.42856i	−0.699861	+	0.714279i	$0.746755\pi$
$198$	0	0
$199$	−3752.00	−1.33654	−0.668272	−	0.743917i	$-0.732966\pi$
$199$	−3752.00	−1.33654	−0.668272	+	0.743917i	$0.732966\pi$
$200$	0	0
$201$	36.0000	0.0126331
$202$	0	0
$203$	378.000i	0.130692i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	216.000i	0.0725268i
$208$	0	0
$209$	−1584.00	−0.524247
$210$	0	0
$211$	5596.00	1.82580	0.912902	−	0.408179i	$-0.133836\pi$
$211$	5596.00	1.82580	0.912902	+	0.408179i	$0.133836\pi$
$212$	0	0
$213$	840.000i	0.270215i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	784.000i	0.245260i
$218$	0	0
$219$	−1230.00	−0.379524
$220$	0	0
$221$	3948.00	1.20168
$222$	0	0
$223$	2672.00i	0.802378i	0.915995	+	0.401189i	$0.131403\pi$
$223$	2672.00i	0.802378i	−0.915995	+	0.401189i	$0.868597\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	620.000i	0.181281i	0.995884	+	0.0906406i	$0.0288915\pi$
$227$	620.000i	0.181281i	−0.995884	+	0.0906406i	$0.971109\pi$
$228$	0	0
$229$	−4318.00	−1.24603	−0.623016	−	0.782209i	$-0.714093\pi$
$229$	−4318.00	−1.24603	−0.623016	+	0.782209i	$0.714093\pi$
$230$	0	0
$231$	924.000	0.263181
$232$	0	0
$233$	4902.00i	1.37829i	0.724625	+	0.689143i	$0.242013\pi$
$233$	4902.00i	1.37829i	−0.724625	+	0.689143i	$0.757987\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 960.000i	− 0.263117i
$238$	0	0
$239$	−2176.00	−0.588928	−0.294464	−	0.955663i	$-0.595141\pi$
$239$	−2176.00	−0.588928	−0.294464	+	0.955663i	$0.595141\pi$
$240$	0	0
$241$	1330.00	0.355489	0.177744	−	0.984077i	$-0.443120\pi$
$241$	1330.00	0.355489	0.177744	+	0.984077i	$0.443120\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1512.00i	0.389499i
$248$	0	0
$249$	3756.00	0.955931
$250$	0	0
$251$	−972.000	−0.244431	−0.122215	−	0.992504i	$-0.539000\pi$
$251$	−972.000	−0.244431	−0.122215	+	0.992504i	$0.539000\pi$
$252$	0	0
$253$	1056.00i	0.262412i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1102.00i	− 0.267474i	−0.991017	−	0.133737i	$-0.957302\pi$
$257$	− 1102.00i	− 0.267474i	0.991017	−	0.133737i	$-0.0426978\pi$
$258$	0	0
$259$	−2254.00	−0.540760
$260$	0	0
$261$	486.000	0.115259
$262$	0	0
$263$	4600.00i	1.07851i	0.842143	+	0.539255i	$0.181294\pi$
$263$	4600.00i	1.07851i	−0.842143	+	0.539255i	$0.818706\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 114.000i	− 0.0261299i
$268$	0	0
$269$	3498.00	0.792851	0.396425	−	0.918067i	$-0.370251\pi$
$269$	3498.00	0.792851	0.396425	+	0.918067i	$0.370251\pi$
$270$	0	0
$271$	8256.00	1.85061	0.925307	−	0.379219i	$-0.123807\pi$
$271$	8256.00	1.85061	0.925307	+	0.379219i	$0.123807\pi$
$272$	0	0
$273$	− 882.000i	− 0.195535i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 594.000i	− 0.128845i	−0.997923	−	0.0644224i	$-0.979480\pi$
$277$	− 594.000i	− 0.128845i	0.997923	−	0.0644224i	$-0.0205205\pi$
$278$	0	0
$279$	1008.00	0.216299
$280$	0	0
$281$	−2326.00	−0.493799	−0.246900	−	0.969041i	$-0.579412\pi$
$281$	−2326.00	−0.493799	−0.246900	+	0.969041i	$0.579412\pi$
$282$	0	0
$283$	5164.00i	1.08469i	0.840155	+	0.542346i	$0.182464\pi$
$283$	5164.00i	1.08469i	−0.840155	+	0.542346i	$0.817536\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	154.000i	0.0316736i
$288$	0	0
$289$	−3923.00	−0.798494
$290$	0	0
$291$	3750.00	0.755426
$292$	0	0
$293$	9602.00i	1.91452i	0.289226	+	0.957261i	$0.406602\pi$
$293$	9602.00i	1.91452i	−0.289226	+	0.957261i	$0.593398\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1188.00i	− 0.232104i
$298$	0	0
$299$	1008.00	0.194964
$300$	0	0
$301$	−2044.00	−0.391409
$302$	0	0
$303$	2502.00i	0.474377i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 836.000i	− 0.155417i	−0.996976	−	0.0777085i	$-0.975240\pi$
$307$	− 836.000i	− 0.155417i	0.996976	−	0.0777085i	$-0.0247604\pi$
$308$	0	0
$309$	2184.00	0.402082
$310$	0	0
$311$	1512.00	0.275684	0.137842	−	0.990454i	$-0.455983\pi$
$311$	1512.00	0.275684	0.137842	+	0.990454i	$0.455983\pi$
$312$	0	0
$313$	9142.00i	1.65091i	0.564464	+	0.825457i	$0.309083\pi$
$313$	9142.00i	1.65091i	−0.564464	+	0.825457i	$0.690917\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	22.0000i	0.00389793i	0.999998	+	0.00194896i	$0.000620375\pi$
$317$	22.0000i	0.00389793i	−0.999998	+	0.00194896i	$0.999380\pi$
$318$	0	0
$319$	2376.00	0.417023
$320$	0	0
$321$	−324.000	−0.0563362
$322$	0	0
$323$	− 3384.00i	− 0.582944i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1602.00i	0.270920i
$328$	0	0
$329$	1904.00	0.319061
$330$	0	0
$331$	8932.00	1.48322	0.741612	−	0.670829i	$-0.234062\pi$
$331$	8932.00	1.48322	0.741612	+	0.670829i	$0.234062\pi$
$332$	0	0
$333$	2898.00i	0.476905i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 3086.00i	− 0.498828i	−0.968397	−	0.249414i	$-0.919762\pi$
$337$	− 3086.00i	− 0.498828i	0.968397	−	0.249414i	$-0.0802380\pi$
$338$	0	0
$339$	5322.00	0.852659
$340$	0	0
$341$	4928.00	0.782599
$342$	0	0
$343$	343.000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 3356.00i	− 0.519192i	−0.965717	−	0.259596i	$-0.916411\pi$
$347$	− 3356.00i	− 0.519192i	0.965717	−	0.259596i	$-0.0835893\pi$
$348$	0	0
$349$	9722.00	1.49114	0.745568	−	0.666429i	$-0.232178\pi$
$349$	9722.00	1.49114	0.745568	+	0.666429i	$0.232178\pi$
$350$	0	0
$351$	−1134.00	−0.172446
$352$	0	0
$353$	10254.0i	1.54608i	0.634359	+	0.773039i	$0.281264\pi$
$353$	10254.0i	1.54608i	−0.634359	+	0.773039i	$0.718736\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1974.00i	0.292648i
$358$	0	0
$359$	−9800.00	−1.44074	−0.720368	−	0.693592i	$-0.756027\pi$
$359$	−9800.00	−1.44074	−0.720368	+	0.693592i	$0.756027\pi$
$360$	0	0
$361$	−5563.00	−0.811051
$362$	0	0
$363$	− 1815.00i	− 0.262432i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 7328.00i	− 1.04228i	−0.853470	−	0.521142i	$-0.825506\pi$
$367$	− 7328.00i	− 1.04228i	0.853470	−	0.521142i	$-0.174494\pi$
$368$	0	0
$369$	198.000	0.0279335
$370$	0	0
$371$	4046.00	0.566194
$372$	0	0
$373$	− 9902.00i	− 1.37455i	−0.726399	−	0.687274i	$-0.758807\pi$
$373$	− 9902.00i	− 1.37455i	0.726399	−	0.687274i	$-0.241193\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2268.00i	− 0.309835i
$378$	0	0
$379$	5292.00	0.717234	0.358617	−	0.933485i	$-0.383248\pi$
$379$	5292.00	0.717234	0.358617	+	0.933485i	$0.383248\pi$
$380$	0	0
$381$	7536.00	1.01334
$382$	0	0
$383$	6144.00i	0.819696i	0.912154	+	0.409848i	$0.134418\pi$
$383$	6144.00i	0.819696i	−0.912154	+	0.409848i	$0.865582\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2628.00i	0.345191i
$388$	0	0
$389$	4402.00	0.573754	0.286877	−	0.957967i	$-0.407383\pi$
$389$	4402.00	0.573754	0.286877	+	0.957967i	$0.407383\pi$
$390$	0	0
$391$	−2256.00	−0.291792
$392$	0	0
$393$	− 4164.00i	− 0.534468i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 7850.00i	− 0.992393i	−0.868210	−	0.496197i	$-0.834729\pi$
$397$	− 7850.00i	− 0.992393i	0.868210	−	0.496197i	$-0.165271\pi$
$398$	0	0
$399$	−756.000	−0.0948555
$400$	0	0
$401$	11122.0	1.38505	0.692526	−	0.721393i	$-0.256498\pi$
$401$	11122.0	1.38505	0.692526	+	0.721393i	$0.256498\pi$
$402$	0	0
$403$	− 4704.00i	− 0.581446i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	14168.0i	1.72551i
$408$	0	0
$409$	3030.00	0.366317	0.183159	−	0.983083i	$-0.441368\pi$
$409$	3030.00	0.366317	0.183159	+	0.983083i	$0.441368\pi$
$410$	0	0
$411$	−7218.00	−0.866272
$412$	0	0
$413$	− 308.000i	− 0.0366966i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 2580.00i	− 0.302981i
$418$	0	0
$419$	13620.0	1.58802	0.794010	−	0.607904i	$-0.207990\pi$
$419$	13620.0	1.58802	0.794010	+	0.607904i	$0.207990\pi$
$420$	0	0
$421$	−1282.00	−0.148411	−0.0742053	−	0.997243i	$-0.523642\pi$
$421$	−1282.00	−0.148411	−0.0742053	+	0.997243i	$0.523642\pi$
$422$	0	0
$423$	− 2448.00i	− 0.281385i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	182.000i	0.0206267i
$428$	0	0
$429$	−5544.00	−0.623932
$430$	0	0
$431$	−13056.0	−1.45913	−0.729565	−	0.683911i	$-0.760278\pi$
$431$	−13056.0	−1.45913	−0.729565	+	0.683911i	$0.760278\pi$
$432$	0	0
$433$	9550.00i	1.05992i	0.848024	+	0.529958i	$0.177792\pi$
$433$	9550.00i	1.05992i	−0.848024	+	0.529958i	$0.822208\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 864.000i	− 0.0945783i
$438$	0	0
$439$	−3640.00	−0.395735	−0.197868	−	0.980229i	$-0.563402\pi$
$439$	−3640.00	−0.395735	−0.197868	+	0.980229i	$0.563402\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	10940.0i	1.17331i	0.809838	+	0.586654i	$0.199555\pi$
$443$	10940.0i	1.17331i	−0.809838	+	0.586654i	$0.800445\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2394.00i	0.253316i
$448$	0	0
$449$	−13602.0	−1.42966	−0.714831	−	0.699297i	$-0.753496\pi$
$449$	−13602.0	−1.42966	−0.714831	+	0.699297i	$0.753496\pi$
$450$	0	0
$451$	968.000	0.101067
$452$	0	0
$453$	− 10248.0i	− 1.06290i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	15130.0i	1.54869i	0.632763	+	0.774345i	$0.281921\pi$
$457$	15130.0i	1.54869i	−0.632763	+	0.774345i	$0.718079\pi$
$458$	0	0
$459$	2538.00	0.258091
$460$	0	0
$461$	−9770.00	−0.987059	−0.493530	−	0.869729i	$-0.664294\pi$
$461$	−9770.00	−0.987059	−0.493530	+	0.869729i	$0.664294\pi$
$462$	0	0
$463$	1408.00i	0.141329i	0.997500	+	0.0706645i	$0.0225120\pi$
$463$	1408.00i	0.141329i	−0.997500	+	0.0706645i	$0.977488\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 8644.00i	− 0.856524i	−0.903655	−	0.428262i	$-0.859126\pi$
$467$	− 8644.00i	− 0.856524i	0.903655	−	0.428262i	$-0.140874\pi$
$468$	0	0
$469$	84.0000	0.00827028
$470$	0	0
$471$	210.000	0.0205441
$472$	0	0
$473$	12848.0i	1.24895i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 5202.00i	− 0.499336i
$478$	0	0
$479$	−17184.0	−1.63916	−0.819579	−	0.572966i	$-0.805793\pi$
$479$	−17184.0	−1.63916	−0.819579	+	0.572966i	$0.805793\pi$
$480$	0	0
$481$	13524.0	1.28200
$482$	0	0
$483$	504.000i	0.0474799i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	8744.00i	0.813611i	0.913515	+	0.406805i	$0.133357\pi$
$487$	8744.00i	0.813611i	−0.913515	+	0.406805i	$0.866643\pi$
$488$	0	0
$489$	−7332.00	−0.678046
$490$	0	0
$491$	9012.00	0.828322	0.414161	−	0.910204i	$-0.364075\pi$
$491$	9012.00	0.828322	0.414161	+	0.910204i	$0.364075\pi$
$492$	0	0
$493$	5076.00i	0.463715i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1960.00i	0.176897i
$498$	0	0
$499$	18564.0	1.66541	0.832704	−	0.553718i	$-0.186791\pi$
$499$	18564.0	1.66541	0.832704	+	0.553718i	$0.186791\pi$
$500$	0	0
$501$	−1944.00	−0.173356
$502$	0	0
$503$	− 552.000i	− 0.0489313i	−0.999701	−	0.0244657i	$-0.992212\pi$
$503$	− 552.000i	− 0.0489313i	0.999701	−	0.0244657i	$-0.00778844\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 1299.00i	− 0.113788i
$508$	0	0
$509$	−19558.0	−1.70313	−0.851565	−	0.524249i	$-0.824346\pi$
$509$	−19558.0	−1.70313	−0.851565	+	0.524249i	$0.824346\pi$
$510$	0	0
$511$	−2870.00	−0.248457
$512$	0	0
$513$	972.000i	0.0836547i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 11968.0i	− 1.01809i
$518$	0	0
$519$	−4578.00	−0.387191
$520$	0	0
$521$	10346.0	0.869993	0.434997	−	0.900432i	$-0.356750\pi$
$521$	10346.0	0.869993	0.434997	+	0.900432i	$0.356750\pi$
$522$	0	0
$523$	3740.00i	0.312694i	0.987702	+	0.156347i	$0.0499718\pi$
$523$	3740.00i	0.312694i	−0.987702	+	0.156347i	$0.950028\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10528.0i	0.870222i
$528$	0	0
$529$	11591.0	0.952659
$530$	0	0
$531$	−396.000	−0.0323633
$532$	0	0
$533$	− 924.000i	− 0.0750898i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 252.000i	− 0.0202507i
$538$	0	0
$539$	2156.00	0.172292
$540$	0	0
$541$	−15450.0	−1.22781	−0.613907	−	0.789378i	$-0.710403\pi$
$541$	−15450.0	−1.22781	−0.613907	+	0.789378i	$0.710403\pi$
$542$	0	0
$543$	5910.00i	0.467076i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	13260.0i	1.03648i	0.855234	+	0.518242i	$0.173413\pi$
$547$	13260.0i	1.03648i	−0.855234	+	0.518242i	$0.826587\pi$
$548$	0	0
$549$	234.000	0.0181910
$550$	0	0
$551$	−1944.00	−0.150303
$552$	0	0
$553$	− 2240.00i	− 0.172250i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	15238.0i	1.15916i	0.814914	+	0.579582i	$0.196784\pi$
$557$	15238.0i	1.15916i	−0.814914	+	0.579582i	$0.803216\pi$
$558$	0	0
$559$	12264.0	0.927928
$560$	0	0
$561$	12408.0	0.933808
$562$	0	0
$563$	11620.0i	0.869848i	0.900467	+	0.434924i	$0.143225\pi$
$563$	11620.0i	0.869848i	−0.900467	+	0.434924i	$0.856775\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 567.000i	− 0.0419961i
$568$	0	0
$569$	−25770.0	−1.89865	−0.949327	−	0.314289i	$-0.898234\pi$
$569$	−25770.0	−1.89865	−0.949327	+	0.314289i	$0.898234\pi$
$570$	0	0
$571$	−6828.00	−0.500425	−0.250213	−	0.968191i	$-0.580501\pi$
$571$	−6828.00	−0.500425	−0.250213	+	0.968191i	$0.580501\pi$
$572$	0	0
$573$	− 2160.00i	− 0.157479i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 25886.0i	− 1.86767i	−0.357698	−	0.933837i	$-0.616438\pi$
$577$	− 25886.0i	− 1.86767i	0.357698	−	0.933837i	$-0.383562\pi$
$578$	0	0
$579$	4602.00	0.330315
$580$	0	0
$581$	8764.00	0.625804
$582$	0	0
$583$	− 25432.0i	− 1.80667i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1180.00i	− 0.0829707i	−0.999139	−	0.0414854i	$-0.986791\pi$
$587$	− 1180.00i	− 0.0829707i	0.999139	−	0.0414854i	$-0.0132090\pi$
$588$	0	0
$589$	−4032.00	−0.282064
$590$	0	0
$591$	11850.0	0.824778
$592$	0	0
$593$	− 16386.0i	− 1.13473i	−0.823468	−	0.567363i	$-0.807964\pi$
$593$	− 16386.0i	− 1.13473i	0.823468	−	0.567363i	$-0.192036\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	11256.0i	0.771654i
$598$	0	0
$599$	1320.00	0.0900396	0.0450198	−	0.998986i	$-0.485665\pi$
$599$	1320.00	0.0900396	0.0450198	+	0.998986i	$0.485665\pi$
$600$	0	0
$601$	1866.00	0.126648	0.0633242	−	0.997993i	$-0.479830\pi$
$601$	1866.00	0.126648	0.0633242	+	0.997993i	$0.479830\pi$
$602$	0	0
$603$	− 108.000i	− 0.00729370i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	28240.0i	1.88835i	0.329450	+	0.944173i	$0.393137\pi$
$607$	28240.0i	1.88835i	−0.329450	+	0.944173i	$0.606863\pi$
$608$	0	0
$609$	1134.00	0.0754548
$610$	0	0
$611$	−11424.0	−0.756408
$612$	0	0
$613$	− 1278.00i	− 0.0842054i	−0.999113	−	0.0421027i	$-0.986594\pi$
$613$	− 1278.00i	− 0.0842054i	0.999113	−	0.0421027i	$-0.0134057\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 2662.00i	− 0.173692i	−0.996222	−	0.0868461i	$-0.972321\pi$
$617$	− 2662.00i	− 0.173692i	0.996222	−	0.0868461i	$-0.0276788\pi$
$618$	0	0
$619$	15676.0	1.01789	0.508943	−	0.860800i	$-0.330036\pi$
$619$	15676.0	1.01789	0.508943	+	0.860800i	$0.330036\pi$
$620$	0	0
$621$	648.000	0.0418733
$622$	0	0
$623$	− 266.000i	− 0.0171060i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4752.00i	0.302674i
$628$	0	0
$629$	−30268.0	−1.91870
$630$	0	0
$631$	23768.0	1.49951	0.749754	−	0.661717i	$-0.230172\pi$
$631$	23768.0	1.49951	0.749754	+	0.661717i	$0.230172\pi$
$632$	0	0
$633$	− 16788.0i	− 1.05413i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 2058.00i	− 0.128008i
$638$	0	0
$639$	2520.00	0.156009
$640$	0	0
$641$	−5758.00	−0.354801	−0.177400	−	0.984139i	$-0.556769\pi$
$641$	−5758.00	−0.354801	−0.177400	+	0.984139i	$0.556769\pi$
$642$	0	0
$643$	7604.00i	0.466365i	0.972433	+	0.233182i	$0.0749139\pi$
$643$	7604.00i	0.466365i	−0.972433	+	0.233182i	$0.925086\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 8072.00i	− 0.490484i	−0.969462	−	0.245242i	$-0.921133\pi$
$647$	− 8072.00i	− 0.490484i	0.969462	−	0.245242i	$-0.0788674\pi$
$648$	0	0
$649$	−1936.00	−0.117095
$650$	0	0
$651$	2352.00	0.141601
$652$	0	0
$653$	− 15302.0i	− 0.917019i	−0.888689	−	0.458510i	$-0.848383\pi$
$653$	− 15302.0i	− 0.917019i	0.888689	−	0.458510i	$-0.151617\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3690.00i	0.219118i
$658$	0	0
$659$	4244.00	0.250869	0.125435	−	0.992102i	$-0.459967\pi$
$659$	4244.00	0.250869	0.125435	+	0.992102i	$0.459967\pi$
$660$	0	0
$661$	−10706.0	−0.629978	−0.314989	−	0.949095i	$-0.602001\pi$
$661$	−10706.0	−0.629978	−0.314989	+	0.949095i	$0.602001\pi$
$662$	0	0
$663$	− 11844.0i	− 0.693790i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1296.00i	0.0752344i
$668$	0	0
$669$	8016.00	0.463253
$670$	0	0
$671$	1144.00	0.0658176
$672$	0	0
$673$	10526.0i	0.602894i	0.953483	+	0.301447i	$0.0974696\pi$
$673$	10526.0i	0.602894i	−0.953483	+	0.301447i	$0.902530\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 10146.0i	− 0.575986i	−0.957633	−	0.287993i	$-0.907012\pi$
$677$	− 10146.0i	− 0.575986i	0.957633	−	0.287993i	$-0.0929880\pi$
$678$	0	0
$679$	8750.00	0.494542
$680$	0	0
$681$	1860.00	0.104663
$682$	0	0
$683$	10348.0i	0.579729i	0.957068	+	0.289865i	$0.0936103\pi$
$683$	10348.0i	0.579729i	−0.957068	+	0.289865i	$0.906390\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	12954.0i	0.719397i
$688$	0	0
$689$	−24276.0	−1.34230
$690$	0	0
$691$	−17252.0	−0.949779	−0.474889	−	0.880046i	$-0.657512\pi$
$691$	−17252.0	−0.949779	−0.474889	+	0.880046i	$0.657512\pi$
$692$	0	0
$693$	− 2772.00i	− 0.151947i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2068.00i	0.112383i
$698$	0	0
$699$	14706.0	0.795754
$700$	0	0
$701$	−35882.0	−1.93330	−0.966651	−	0.256098i	$-0.917563\pi$
$701$	−35882.0	−1.93330	−0.966651	+	0.256098i	$0.917563\pi$
$702$	0	0
$703$	− 11592.0i	− 0.621907i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5838.00i	0.310552i
$708$	0	0
$709$	−21406.0	−1.13388	−0.566939	−	0.823760i	$-0.691872\pi$
$709$	−21406.0	−1.13388	−0.566939	+	0.823760i	$0.691872\pi$
$710$	0	0
$711$	−2880.00	−0.151911
$712$	0	0
$713$	2688.00i	0.141187i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6528.00i	0.340018i
$718$	0	0
$719$	11184.0	0.580101	0.290051	−	0.957011i	$-0.406328\pi$
$719$	11184.0	0.580101	0.290051	+	0.957011i	$0.406328\pi$
$720$	0	0
$721$	5096.00	0.263225
$722$	0	0
$723$	− 3990.00i	− 0.205242i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 2280.00i	− 0.116314i	−0.998307	−	0.0581572i	$-0.981478\pi$
$727$	− 2280.00i	− 0.116314i	0.998307	−	0.0581572i	$-0.0185225\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−27448.0	−1.38878
$732$	0	0
$733$	− 18118.0i	− 0.912965i	−0.889732	−	0.456483i	$-0.849109\pi$
$733$	− 18118.0i	− 0.912965i	0.889732	−	0.456483i	$-0.150891\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 528.000i	− 0.0263896i
$738$	0	0
$739$	−30124.0	−1.49950	−0.749749	−	0.661722i	$-0.769826\pi$
$739$	−30124.0	−1.49950	−0.749749	+	0.661722i	$0.769826\pi$
$740$	0	0
$741$	4536.00	0.224877
$742$	0	0
$743$	3768.00i	0.186049i	0.995664	+	0.0930246i	$0.0296535\pi$
$743$	3768.00i	0.186049i	−0.995664	+	0.0930246i	$0.970346\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 11268.0i	− 0.551907i
$748$	0	0
$749$	−756.000	−0.0368807
$750$	0	0
$751$	−18880.0	−0.917365	−0.458682	−	0.888600i	$-0.651678\pi$
$751$	−18880.0	−0.917365	−0.458682	+	0.888600i	$0.651678\pi$
$752$	0	0
$753$	2916.00i	0.141122i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 9266.00i	− 0.444886i	−0.974946	−	0.222443i	$-0.928597\pi$
$757$	− 9266.00i	− 0.444886i	0.974946	−	0.222443i	$-0.0714031\pi$
$758$	0	0
$759$	3168.00	0.151503
$760$	0	0
$761$	−4134.00	−0.196922	−0.0984608	−	0.995141i	$-0.531392\pi$
$761$	−4134.00	−0.196922	−0.0984608	+	0.995141i	$0.531392\pi$
$762$	0	0
$763$	3738.00i	0.177359i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1848.00i	0.0869979i
$768$	0	0
$769$	18110.0	0.849237	0.424619	−	0.905372i	$-0.360408\pi$
$769$	18110.0	0.849237	0.424619	+	0.905372i	$0.360408\pi$
$770$	0	0
$771$	−3306.00	−0.154426
$772$	0	0
$773$	− 35358.0i	− 1.64520i	−0.568621	−	0.822600i	$-0.692523\pi$
$773$	− 35358.0i	− 1.64520i	0.568621	−	0.822600i	$-0.307477\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	6762.00i	0.312208i
$778$	0	0
$779$	−792.000	−0.0364266
$780$	0	0
$781$	12320.0	0.564461
$782$	0	0
$783$	− 1458.00i	− 0.0665449i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 31652.0i	− 1.43364i	−0.697260	−	0.716818i	$-0.745598\pi$
$787$	− 31652.0i	− 1.43364i	0.697260	−	0.716818i	$-0.254402\pi$
$788$	0	0
$789$	13800.0	0.622678
$790$	0	0
$791$	12418.0	0.558196
$792$	0	0
$793$	− 1092.00i	− 0.0489005i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 8442.00i	− 0.375196i	−0.982246	−	0.187598i	$-0.939930\pi$
$797$	− 8442.00i	− 0.375196i	0.982246	−	0.187598i	$-0.0600702\pi$
$798$	0	0
$799$	25568.0	1.13208
$800$	0	0
$801$	−342.000	−0.0150861
$802$	0	0
$803$	18040.0i	0.792799i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 10494.0i	− 0.457753i
$808$	0	0
$809$	−14298.0	−0.621373	−0.310687	−	0.950512i	$-0.600559\pi$
$809$	−14298.0	−0.621373	−0.310687	+	0.950512i	$0.600559\pi$
$810$	0	0
$811$	−44956.0	−1.94651	−0.973254	−	0.229730i	$-0.926216\pi$
$811$	−44956.0	−1.94651	−0.973254	+	0.229730i	$0.926216\pi$
$812$	0	0
$813$	− 24768.0i	− 1.06845i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 10512.0i	− 0.450145i
$818$	0	0
$819$	−2646.00	−0.112892
$820$	0	0
$821$	−45090.0	−1.91675	−0.958375	−	0.285512i	$-0.907836\pi$
$821$	−45090.0	−1.91675	−0.958375	+	0.285512i	$0.907836\pi$
$822$	0	0
$823$	− 35128.0i	− 1.48783i	−0.668274	−	0.743916i	$-0.732966\pi$
$823$	− 35128.0i	− 1.48783i	0.668274	−	0.743916i	$-0.267034\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	11204.0i	0.471102i	0.971862	+	0.235551i	$0.0756894\pi$
$827$	11204.0i	0.471102i	−0.971862	+	0.235551i	$0.924311\pi$
$828$	0	0
$829$	2266.00	0.0949354	0.0474677	−	0.998873i	$-0.484885\pi$
$829$	2266.00	0.0949354	0.0474677	+	0.998873i	$0.484885\pi$
$830$	0	0
$831$	−1782.00	−0.0743886
$832$	0	0
$833$	4606.00i	0.191583i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 3024.00i	− 0.124880i
$838$	0	0
$839$	22408.0	0.922062	0.461031	−	0.887384i	$-0.347480\pi$
$839$	22408.0	0.922062	0.461031	+	0.887384i	$0.347480\pi$
$840$	0	0
$841$	−21473.0	−0.880438
$842$	0	0
$843$	6978.00i	0.285095i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 4235.00i	− 0.171802i
$848$	0	0
$849$	15492.0	0.626247
$850$	0	0
$851$	−7728.00	−0.311295
$852$	0	0
$853$	21394.0i	0.858753i	0.903126	+	0.429377i	$0.141267\pi$
$853$	21394.0i	0.858753i	−0.903126	+	0.429377i	$0.858733\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 26022.0i	− 1.03722i	−0.855012	−	0.518608i	$-0.826450\pi$
$857$	− 26022.0i	− 1.03722i	0.855012	−	0.518608i	$-0.173550\pi$
$858$	0	0
$859$	−12756.0	−0.506670	−0.253335	−	0.967379i	$-0.581527\pi$
$859$	−12756.0	−0.506670	−0.253335	+	0.967379i	$0.581527\pi$
$860$	0	0
$861$	462.000	0.0182868
$862$	0	0
$863$	2384.00i	0.0940351i	0.998894	+	0.0470176i	$0.0149717\pi$
$863$	2384.00i	0.0940351i	−0.998894	+	0.0470176i	$0.985028\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	11769.0i	0.461011i
$868$	0	0
$869$	−14080.0	−0.549633
$870$	0	0
$871$	−504.000	−0.0196066
$872$	0	0
$873$	− 11250.0i	− 0.436145i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 29482.0i	− 1.13516i	−0.823318	−	0.567581i	$-0.807880\pi$
$877$	− 29482.0i	− 1.13516i	0.823318	−	0.567581i	$-0.192120\pi$
$878$	0	0
$879$	28806.0	1.10535
$880$	0	0
$881$	3266.00	0.124897	0.0624485	−	0.998048i	$-0.480109\pi$
$881$	3266.00	0.124897	0.0624485	+	0.998048i	$0.480109\pi$
$882$	0	0
$883$	4068.00i	0.155039i	0.996991	+	0.0775193i	$0.0246999\pi$
$883$	4068.00i	0.155039i	−0.996991	+	0.0775193i	$0.975300\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 27544.0i	− 1.04266i	−0.853356	−	0.521329i	$-0.825437\pi$
$887$	− 27544.0i	− 1.04266i	0.853356	−	0.521329i	$-0.174563\pi$
$888$	0	0
$889$	17584.0	0.663384
$890$	0	0
$891$	−3564.00	−0.134005
$892$	0	0
$893$	9792.00i	0.366939i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 3024.00i	− 0.112562i
$898$	0	0
$899$	6048.00	0.224374
$900$	0	0
$901$	54332.0	2.00895
$902$	0	0
$903$	6132.00i	0.225980i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	9092.00i	0.332850i	0.986054	+	0.166425i	$0.0532224\pi$
$907$	9092.00i	0.332850i	−0.986054	+	0.166425i	$0.946778\pi$
$908$	0	0
$909$	7506.00	0.273882
$910$	0	0
$911$	−22464.0	−0.816976	−0.408488	−	0.912764i	$-0.633944\pi$
$911$	−22464.0	−0.816976	−0.408488	+	0.912764i	$0.633944\pi$
$912$	0	0
$913$	− 55088.0i	− 1.99687i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 9716.00i	− 0.349892i
$918$	0	0
$919$	−1304.00	−0.0468063	−0.0234032	−	0.999726i	$-0.507450\pi$
$919$	−1304.00	−0.0468063	−0.0234032	+	0.999726i	$0.507450\pi$
$920$	0	0
$921$	−2508.00	−0.0897301
$922$	0	0
$923$	− 11760.0i	− 0.419377i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 6552.00i	− 0.232142i
$928$	0	0
$929$	10094.0	0.356484	0.178242	−	0.983987i	$-0.442959\pi$
$929$	10094.0	0.356484	0.178242	+	0.983987i	$0.442959\pi$
$930$	0	0
$931$	−1764.00	−0.0620975
$932$	0	0
$933$	− 4536.00i	− 0.159166i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	21338.0i	0.743951i	0.928243	+	0.371975i	$0.121319\pi$
$937$	21338.0i	0.743951i	−0.928243	+	0.371975i	$0.878681\pi$
$938$	0	0
$939$	27426.0	0.953156
$940$	0	0
$941$	46934.0	1.62593	0.812967	−	0.582309i	$-0.197851\pi$
$941$	46934.0	1.62593	0.812967	+	0.582309i	$0.197851\pi$
$942$	0	0
$943$	528.000i	0.0182333i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 9652.00i	− 0.331202i	−0.986193	−	0.165601i	$-0.947044\pi$
$947$	− 9652.00i	− 0.331202i	0.986193	−	0.165601i	$-0.0529563\pi$
$948$	0	0
$949$	17220.0	0.589025
$950$	0	0
$951$	66.0000	0.00225047
$952$	0	0
$953$	34998.0i	1.18961i	0.803871	+	0.594804i	$0.202770\pi$
$953$	34998.0i	1.18961i	−0.803871	+	0.594804i	$0.797230\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 7128.00i	− 0.240769i
$958$	0	0
$959$	−16842.0	−0.567108
$960$	0	0
$961$	−17247.0	−0.578933
$962$	0	0
$963$	972.000i	0.0325257i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	9160.00i	0.304618i	0.988333	+	0.152309i	$0.0486709\pi$
$967$	9160.00i	0.304618i	−0.988333	+	0.152309i	$0.951329\pi$
$968$	0	0
$969$	−10152.0	−0.336563
$970$	0	0
$971$	−6204.00	−0.205042	−0.102521	−	0.994731i	$-0.532691\pi$
$971$	−6204.00	−0.205042	−0.102521	+	0.994731i	$0.532691\pi$
$972$	0	0
$973$	− 6020.00i	− 0.198348i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 49614.0i	− 1.62466i	−0.583198	−	0.812330i	$-0.698199\pi$
$977$	− 49614.0i	− 1.62466i	0.583198	−	0.812330i	$-0.301801\pi$
$978$	0	0
$979$	−1672.00	−0.0545836
$980$	0	0
$981$	4806.00	0.156416
$982$	0	0
$983$	25816.0i	0.837642i	0.908069	+	0.418821i	$0.137557\pi$
$983$	25816.0i	0.837642i	−0.908069	+	0.418821i	$0.862443\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 5712.00i	− 0.184210i
$988$	0	0
$989$	−7008.00	−0.225320
$990$	0	0
$991$	48592.0	1.55759	0.778797	−	0.627276i	$-0.215830\pi$
$991$	48592.0	1.55759	0.778797	+	0.627276i	$0.215830\pi$
$992$	0	0
$993$	− 26796.0i	− 0.856340i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 9634.00i	− 0.306030i	−0.988224	−	0.153015i	$-0.951102\pi$
$997$	− 9634.00i	− 0.306030i	0.988224	−	0.153015i	$-0.0488983\pi$
$998$	0	0
$999$	8694.00	0.275341

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2100.4.k.a.1849.1		2
5.2	odd	4		2100.4.a.d.1.1		1
5.3	odd	4		420.4.a.d.1.1	✓	1
5.4	even	2	inner	2100.4.k.a.1849.2		2
15.8	even	4		1260.4.a.c.1.1		1
20.3	even	4		1680.4.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.4.a.d.1.1	✓	1	5.3	odd	4
1260.4.a.c.1.1		1	15.8	even	4
1680.4.a.k.1.1		1	20.3	even	4
2100.4.a.d.1.1		1	5.2	odd	4
2100.4.k.a.1849.1		2	1.1	even	1	trivial
2100.4.k.a.1849.2		2	5.4	even	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:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2100.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} -7.00000i q^{7} -9.00000 q^{9} -44.0000 q^{11} +42.0000i q^{13} -94.0000i q^{17} +36.0000 q^{19} -21.0000 q^{21} -24.0000i q^{23} +27.0000i q^{27} -54.0000 q^{29} -112.000 q^{31} +132.000i q^{33} -322.000i q^{37} +126.000 q^{39} -22.0000 q^{41} -292.000i q^{43} +272.000i q^{47} -49.0000 q^{49} -282.000 q^{51} +578.000i q^{53} -108.000i q^{57} +44.0000 q^{59} -26.0000 q^{61} +63.0000i q^{63} +12.0000i q^{67} -72.0000 q^{69} -280.000 q^{71} -410.000i q^{73} +308.000i q^{77} +320.000 q^{79} +81.0000 q^{81} +1252.00i q^{83} +162.000i q^{87} +38.0000 q^{89} +294.000 q^{91} +336.000i q^{93} +1250.00i q^{97} +396.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{9} - 88 q^{11} + 72 q^{19} - 42 q^{21} - 108 q^{29} - 224 q^{31} + 252 q^{39} - 44 q^{41} - 98 q^{49} - 564 q^{51} + 88 q^{59} - 52 q^{61} - 144 q^{69} - 560 q^{71} + 640 q^{79} + 162 q^{81}+ \cdots + 792 q^{99}+O(q^{100}) \) 2 * q - 18 * q^9 - 88 * q^11 + 72 * q^19 - 42 * q^21 - 108 * q^29 - 224 * q^31 + 252 * q^39 - 44 * q^41 - 98 * q^49 - 564 * q^51 + 88 * q^59 - 52 * q^61 - 144 * q^69 - 560 * q^71 + 640 * q^79 + 162 * q^81 + 76 * q^89 + 588 * q^91 + 792 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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