LMFDB - Embedded newform 207.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,4,Mod(1,207)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(207, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("207.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	207.a (trivial)

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:	yes
Analytic conductor:	$12.2133953712$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 23)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	207.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} -4.00000 q^{4} +6.00000 q^{5} -8.00000 q^{7} -24.0000 q^{8} +O(q^{10})$

\(q+2.00000 q^{2} -4.00000 q^{4} +6.00000 q^{5} -8.00000 q^{7} -24.0000 q^{8} +12.0000 q^{10} -34.0000 q^{11} -57.0000 q^{13} -16.0000 q^{14} -16.0000 q^{16} +80.0000 q^{17} -70.0000 q^{19} -24.0000 q^{20} -68.0000 q^{22} -23.0000 q^{23} -89.0000 q^{25} -114.000 q^{26} +32.0000 q^{28} -245.000 q^{29} +103.000 q^{31} +160.000 q^{32} +160.000 q^{34} -48.0000 q^{35} -298.000 q^{37} -140.000 q^{38} -144.000 q^{40} -95.0000 q^{41} +88.0000 q^{43} +136.000 q^{44} -46.0000 q^{46} +357.000 q^{47} -279.000 q^{49} -178.000 q^{50} +228.000 q^{52} +414.000 q^{53} -204.000 q^{55} +192.000 q^{56} -490.000 q^{58} +408.000 q^{59} +822.000 q^{61} +206.000 q^{62} +448.000 q^{64} -342.000 q^{65} +926.000 q^{67} -320.000 q^{68} -96.0000 q^{70} -335.000 q^{71} -899.000 q^{73} -596.000 q^{74} +280.000 q^{76} +272.000 q^{77} -1322.00 q^{79} -96.0000 q^{80} -190.000 q^{82} +36.0000 q^{83} +480.000 q^{85} +176.000 q^{86} +816.000 q^{88} +460.000 q^{89} +456.000 q^{91} +92.0000 q^{92} +714.000 q^{94} -420.000 q^{95} -964.000 q^{97} -558.000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 $

$2$	2.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	2.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	0	0
$3$	0	0
$4$	−4.00000	−0.500000
$5$	6.00000	0.536656	0.268328	−	0.963328i	$-0.413529\pi$
$5$	6.00000	0.536656	0.268328	+	0.963328i	$0.413529\pi$
$6$	0	0
$7$	−8.00000	−0.431959	−0.215980	−	0.976398i	$-0.569295\pi$
$7$	−8.00000	−0.431959	−0.215980	+	0.976398i	$0.569295\pi$
$8$	−24.0000	−1.06066
$9$	0	0
$10$	12.0000	0.379473
$11$	−34.0000	−0.931944	−0.465972	−	0.884799i	$-0.654295\pi$
$11$	−34.0000	−0.931944	−0.465972	+	0.884799i	$0.654295\pi$
$12$	0	0
$13$	−57.0000	−1.21607	−0.608037	−	0.793909i	$-0.708043\pi$
$13$	−57.0000	−1.21607	−0.608037	+	0.793909i	$0.708043\pi$
$14$	−16.0000	−0.305441
$15$	0	0
$16$	−16.0000	−0.250000
$17$	80.0000	1.14134	0.570672	−	0.821178i	$-0.306683\pi$
$17$	80.0000	1.14134	0.570672	+	0.821178i	$0.306683\pi$
$18$	0	0
$19$	−70.0000	−0.845216	−0.422608	−	0.906313i	$-0.638885\pi$
$19$	−70.0000	−0.845216	−0.422608	+	0.906313i	$0.638885\pi$
$20$	−24.0000	−0.268328
$21$	0	0
$22$	−68.0000	−0.658984
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	−89.0000	−0.712000
$26$	−114.000	−0.859894
$27$	0	0
$28$	32.0000	0.215980
$29$	−245.000	−1.56881	−0.784403	−	0.620252i	$-0.787030\pi$
$29$	−245.000	−1.56881	−0.784403	+	0.620252i	$0.787030\pi$
$30$	0	0
$31$	103.000	0.596753	0.298377	−	0.954448i	$-0.403555\pi$
$31$	103.000	0.596753	0.298377	+	0.954448i	$0.403555\pi$
$32$	160.000	0.883883
$33$	0	0
$34$	160.000	0.807052
$35$	−48.0000	−0.231814
$36$	0	0
$37$	−298.000	−1.32408	−0.662039	−	0.749469i	$-0.730309\pi$
$37$	−298.000	−1.32408	−0.662039	+	0.749469i	$0.730309\pi$
$38$	−140.000	−0.597658
$39$	0	0
$40$	−144.000	−0.569210
$41$	−95.0000	−0.361866	−0.180933	−	0.983495i	$-0.557912\pi$
$41$	−95.0000	−0.361866	−0.180933	+	0.983495i	$0.557912\pi$
$42$	0	0
$43$	88.0000	0.312090	0.156045	−	0.987750i	$-0.450125\pi$
$43$	88.0000	0.312090	0.156045	+	0.987750i	$0.450125\pi$
$44$	136.000	0.465972
$45$	0	0
$46$	−46.0000	−0.147442
$47$	357.000	1.10795	0.553977	−	0.832532i	$-0.313110\pi$
$47$	357.000	1.10795	0.553977	+	0.832532i	$0.313110\pi$
$48$	0	0
$49$	−279.000	−0.813411
$50$	−178.000	−0.503460
$51$	0	0
$52$	228.000	0.608037
$53$	414.000	1.07297	0.536484	−	0.843911i	$-0.319752\pi$
$53$	414.000	1.07297	0.536484	+	0.843911i	$0.319752\pi$
$54$	0	0
$55$	−204.000	−0.500134
$56$	192.000	0.458162
$57$	0	0
$58$	−490.000	−1.10931
$59$	408.000	0.900289	0.450145	−	0.892956i	$-0.351372\pi$
$59$	408.000	0.900289	0.450145	+	0.892956i	$0.351372\pi$
$60$	0	0
$61$	822.000	1.72535	0.862675	−	0.505759i	$-0.168788\pi$
$61$	822.000	1.72535	0.862675	+	0.505759i	$0.168788\pi$
$62$	206.000	0.421968
$63$	0	0
$64$	448.000	0.875000
$65$	−342.000	−0.652614
$66$	0	0
$67$	926.000	1.68849	0.844246	−	0.535957i	$-0.180049\pi$
$67$	926.000	1.68849	0.844246	+	0.535957i	$0.180049\pi$
$68$	−320.000	−0.570672
$69$	0	0
$70$	−96.0000	−0.163917
$71$	−335.000	−0.559960	−0.279980	−	0.960006i	$-0.590328\pi$
$71$	−335.000	−0.559960	−0.279980	+	0.960006i	$0.590328\pi$
$72$	0	0
$73$	−899.000	−1.44137	−0.720685	−	0.693263i	$-0.756173\pi$
$73$	−899.000	−1.44137	−0.720685	+	0.693263i	$0.756173\pi$
$74$	−596.000	−0.936265
$75$	0	0
$76$	280.000	0.422608
$77$	272.000	0.402562
$78$	0	0
$79$	−1322.00	−1.88274	−0.941371	−	0.337373i	$-0.890462\pi$
$79$	−1322.00	−1.88274	−0.941371	+	0.337373i	$0.890462\pi$
$80$	−96.0000	−0.134164
$81$	0	0
$82$	−190.000	−0.255878
$83$	36.0000	0.0476086	0.0238043	−	0.999717i	$-0.492422\pi$
$83$	36.0000	0.0476086	0.0238043	+	0.999717i	$0.492422\pi$
$84$	0	0
$85$	480.000	0.612510
$86$	176.000	0.220681
$87$	0	0
$88$	816.000	0.988476
$89$	460.000	0.547864	0.273932	−	0.961749i	$-0.411676\pi$
$89$	460.000	0.547864	0.273932	+	0.961749i	$0.411676\pi$
$90$	0	0
$91$	456.000	0.525294
$92$	92.0000	0.104257
$93$	0	0
$94$	714.000	0.783441
$95$	−420.000	−0.453590
$96$	0	0
$97$	−964.000	−1.00907	−0.504533	−	0.863393i	$-0.668335\pi$
$97$	−964.000	−1.00907	−0.504533	+	0.863393i	$0.668335\pi$
$98$	−558.000	−0.575168
$99$	0	0
$100$	356.000	0.356000
$101$	310.000	0.305407	0.152704	−	0.988272i	$-0.451202\pi$
$101$	310.000	0.305407	0.152704	+	0.988272i	$0.451202\pi$
$102$	0	0
$103$	1044.00	0.998722	0.499361	−	0.866394i	$-0.333568\pi$
$103$	1044.00	0.998722	0.499361	+	0.866394i	$0.333568\pi$
$104$	1368.00	1.28984
$105$	0	0
$106$	828.000	0.758703
$107$	−414.000	−0.374046	−0.187023	−	0.982356i	$-0.559884\pi$
$107$	−414.000	−0.374046	−0.187023	+	0.982356i	$0.559884\pi$
$108$	0	0
$109$	704.000	0.618633	0.309316	−	0.950959i	$-0.399900\pi$
$109$	704.000	0.618633	0.309316	+	0.950959i	$0.399900\pi$
$110$	−408.000	−0.353648
$111$	0	0
$112$	128.000	0.107990
$113$	−952.000	−0.792537	−0.396268	−	0.918135i	$-0.629695\pi$
$113$	−952.000	−0.792537	−0.396268	+	0.918135i	$0.629695\pi$
$114$	0	0
$115$	−138.000	−0.111901
$116$	980.000	0.784403
$117$	0	0
$118$	816.000	0.636601
$119$	−640.000	−0.493014
$120$	0	0
$121$	−175.000	−0.131480
$122$	1644.00	1.22001
$123$	0	0
$124$	−412.000	−0.298377
$125$	−1284.00	−0.918756
$126$	0	0
$127$	261.000	0.182362	0.0911811	−	0.995834i	$-0.470936\pi$
$127$	261.000	0.182362	0.0911811	+	0.995834i	$0.470936\pi$
$128$	−384.000	−0.265165
$129$	0	0
$130$	−684.000	−0.461467
$131$	1441.00	0.961074	0.480537	−	0.876974i	$-0.340442\pi$
$131$	1441.00	0.961074	0.480537	+	0.876974i	$0.340442\pi$
$132$	0	0
$133$	560.000	0.365099
$134$	1852.00	1.19394
$135$	0	0
$136$	−1920.00	−1.21058
$137$	−1556.00	−0.970351	−0.485175	−	0.874417i	$-0.661244\pi$
$137$	−1556.00	−0.970351	−0.485175	+	0.874417i	$0.661244\pi$
$138$	0	0
$139$	25.0000	0.0152552	0.00762760	−	0.999971i	$-0.497572\pi$
$139$	25.0000	0.0152552	0.00762760	+	0.999971i	$0.497572\pi$
$140$	192.000	0.115907
$141$	0	0
$142$	−670.000	−0.395952
$143$	1938.00	1.13331
$144$	0	0
$145$	−1470.00	−0.841909
$146$	−1798.00	−1.01920
$147$	0	0
$148$	1192.00	0.662039
$149$	−822.000	−0.451952	−0.225976	−	0.974133i	$-0.572557\pi$
$149$	−822.000	−0.451952	−0.225976	+	0.974133i	$0.572557\pi$
$150$	0	0
$151$	−1489.00	−0.802471	−0.401235	−	0.915975i	$-0.631419\pi$
$151$	−1489.00	−0.802471	−0.401235	+	0.915975i	$0.631419\pi$
$152$	1680.00	0.896487
$153$	0	0
$154$	544.000	0.284654
$155$	618.000	0.320251
$156$	0	0
$157$	−632.000	−0.321268	−0.160634	−	0.987014i	$-0.551354\pi$
$157$	−632.000	−0.321268	−0.160634	+	0.987014i	$0.551354\pi$
$158$	−2644.00	−1.33130
$159$	0	0
$160$	960.000	0.474342
$161$	184.000	0.0900698
$162$	0	0
$163$	−3043.00	−1.46225	−0.731123	−	0.682245i	$-0.761004\pi$
$163$	−3043.00	−1.46225	−0.731123	+	0.682245i	$0.761004\pi$
$164$	380.000	0.180933
$165$	0	0
$166$	72.0000	0.0336644
$167$	2224.00	1.03053	0.515264	−	0.857031i	$-0.327694\pi$
$167$	2224.00	1.03053	0.515264	+	0.857031i	$0.327694\pi$
$168$	0	0
$169$	1052.00	0.478835
$170$	960.000	0.433110
$171$	0	0
$172$	−352.000	−0.156045
$173$	−3230.00	−1.41949	−0.709747	−	0.704457i	$-0.751191\pi$
$173$	−3230.00	−1.41949	−0.709747	+	0.704457i	$0.751191\pi$
$174$	0	0
$175$	712.000	0.307555
$176$	544.000	0.232986
$177$	0	0
$178$	920.000	0.387398
$179$	−369.000	−0.154080	−0.0770401	−	0.997028i	$-0.524547\pi$
$179$	−369.000	−0.154080	−0.0770401	+	0.997028i	$0.524547\pi$
$180$	0	0
$181$	−1370.00	−0.562604	−0.281302	−	0.959619i	$-0.590766\pi$
$181$	−1370.00	−0.562604	−0.281302	+	0.959619i	$0.590766\pi$
$182$	912.000	0.371439
$183$	0	0
$184$	552.000	0.221163
$185$	−1788.00	−0.710575
$186$	0	0
$187$	−2720.00	−1.06367
$188$	−1428.00	−0.553977
$189$	0	0
$190$	−840.000	−0.320737
$191$	−4410.00	−1.67066	−0.835331	−	0.549747i	$-0.814724\pi$
$191$	−4410.00	−1.67066	−0.835331	+	0.549747i	$0.814724\pi$
$192$	0	0
$193$	−135.000	−0.0503498	−0.0251749	−	0.999683i	$-0.508014\pi$
$193$	−135.000	−0.0503498	−0.0251749	+	0.999683i	$0.508014\pi$
$194$	−1928.00	−0.713517
$195$	0	0
$196$	1116.00	0.406706
$197$	−1221.00	−0.441587	−0.220794	−	0.975321i	$-0.570865\pi$
$197$	−1221.00	−0.441587	−0.220794	+	0.975321i	$0.570865\pi$
$198$	0	0
$199$	−1098.00	−0.391131	−0.195566	−	0.980691i	$-0.562654\pi$
$199$	−1098.00	−0.391131	−0.195566	+	0.980691i	$0.562654\pi$
$200$	2136.00	0.755190
$201$	0	0
$202$	620.000	0.215956
$203$	1960.00	0.677660
$204$	0	0
$205$	−570.000	−0.194198
$206$	2088.00	0.706203
$207$	0	0
$208$	912.000	0.304018
$209$	2380.00	0.787694
$210$	0	0
$211$	−3676.00	−1.19937	−0.599683	−	0.800238i	$-0.704707\pi$
$211$	−3676.00	−1.19937	−0.599683	+	0.800238i	$0.704707\pi$
$212$	−1656.00	−0.536484
$213$	0	0
$214$	−828.000	−0.264490
$215$	528.000	0.167485
$216$	0	0
$217$	−824.000	−0.257773
$218$	1408.00	0.437439
$219$	0	0
$220$	816.000	0.250067
$221$	−4560.00	−1.38796
$222$	0	0
$223$	1656.00	0.497282	0.248641	−	0.968596i	$-0.420016\pi$
$223$	1656.00	0.497282	0.248641	+	0.968596i	$0.420016\pi$
$224$	−1280.00	−0.381802
$225$	0	0
$226$	−1904.00	−0.560408
$227$	−2940.00	−0.859624	−0.429812	−	0.902918i	$-0.641420\pi$
$227$	−2940.00	−0.859624	−0.429812	+	0.902918i	$0.641420\pi$
$228$	0	0
$229$	3612.00	1.04230	0.521152	−	0.853464i	$-0.325502\pi$
$229$	3612.00	1.04230	0.521152	+	0.853464i	$0.325502\pi$
$230$	−276.000	−0.0791257
$231$	0	0
$232$	5880.00	1.66397
$233$	4325.00	1.21605	0.608026	−	0.793917i	$-0.291962\pi$
$233$	4325.00	1.21605	0.608026	+	0.793917i	$0.291962\pi$
$234$	0	0
$235$	2142.00	0.594590
$236$	−1632.00	−0.450145
$237$	0	0
$238$	−1280.00	−0.348614
$239$	−2735.00	−0.740219	−0.370110	−	0.928988i	$-0.620680\pi$
$239$	−2735.00	−0.740219	−0.370110	+	0.928988i	$0.620680\pi$
$240$	0	0
$241$	−6710.00	−1.79348	−0.896741	−	0.442556i	$-0.854072\pi$
$241$	−6710.00	−1.79348	−0.896741	+	0.442556i	$0.854072\pi$
$242$	−350.000	−0.0929705
$243$	0	0
$244$	−3288.00	−0.862675
$245$	−1674.00	−0.436522
$246$	0	0
$247$	3990.00	1.02784
$248$	−2472.00	−0.632952
$249$	0	0
$250$	−2568.00	−0.649658
$251$	6948.00	1.74723	0.873613	−	0.486621i	$-0.161771\pi$
$251$	6948.00	1.74723	0.873613	+	0.486621i	$0.161771\pi$
$252$	0	0
$253$	782.000	0.194324
$254$	522.000	0.128950
$255$	0	0
$256$	−4352.00	−1.06250
$257$	4929.00	1.19635	0.598176	−	0.801365i	$-0.295892\pi$
$257$	4929.00	1.19635	0.598176	+	0.801365i	$0.295892\pi$
$258$	0	0
$259$	2384.00	0.571948
$260$	1368.00	0.326307
$261$	0	0
$262$	2882.00	0.679582
$263$	−6138.00	−1.43911	−0.719554	−	0.694437i	$-0.755654\pi$
$263$	−6138.00	−1.43911	−0.719554	+	0.694437i	$0.755654\pi$
$264$	0	0
$265$	2484.00	0.575815
$266$	1120.00	0.258164
$267$	0	0
$268$	−3704.00	−0.844246
$269$	2063.00	0.467596	0.233798	−	0.972285i	$-0.424885\pi$
$269$	2063.00	0.467596	0.233798	+	0.972285i	$0.424885\pi$
$270$	0	0
$271$	−1064.00	−0.238500	−0.119250	−	0.992864i	$-0.538049\pi$
$271$	−1064.00	−0.238500	−0.119250	+	0.992864i	$0.538049\pi$
$272$	−1280.00	−0.285336
$273$	0	0
$274$	−3112.00	−0.686142
$275$	3026.00	0.663544
$276$	0	0
$277$	5729.00	1.24268	0.621340	−	0.783541i	$-0.286589\pi$
$277$	5729.00	1.24268	0.621340	+	0.783541i	$0.286589\pi$
$278$	50.0000	0.0107871
$279$	0	0
$280$	1152.00	0.245876
$281$	960.000	0.203804	0.101902	−	0.994794i	$-0.467507\pi$
$281$	960.000	0.203804	0.101902	+	0.994794i	$0.467507\pi$
$282$	0	0
$283$	−114.000	−0.0239456	−0.0119728	−	0.999928i	$-0.503811\pi$
$283$	−114.000	−0.0239456	−0.0119728	+	0.999928i	$0.503811\pi$
$284$	1340.00	0.279980
$285$	0	0
$286$	3876.00	0.801373
$287$	760.000	0.156311
$288$	0	0
$289$	1487.00	0.302666
$290$	−2940.00	−0.595320
$291$	0	0
$292$	3596.00	0.720685
$293$	7048.00	1.40529	0.702643	−	0.711543i	$-0.252003\pi$
$293$	7048.00	1.40529	0.702643	+	0.711543i	$0.252003\pi$
$294$	0	0
$295$	2448.00	0.483146
$296$	7152.00	1.40440
$297$	0	0
$298$	−1644.00	−0.319578
$299$	1311.00	0.253569
$300$	0	0
$301$	−704.000	−0.134810
$302$	−2978.00	−0.567433
$303$	0	0
$304$	1120.00	0.211304
$305$	4932.00	0.925920
$306$	0	0
$307$	3872.00	0.719826	0.359913	−	0.932986i	$-0.382806\pi$
$307$	3872.00	0.719826	0.359913	+	0.932986i	$0.382806\pi$
$308$	−1088.00	−0.201281
$309$	0	0
$310$	1236.00	0.226452
$311$	4977.00	0.907459	0.453730	−	0.891139i	$-0.350093\pi$
$311$	4977.00	0.907459	0.453730	+	0.891139i	$0.350093\pi$
$312$	0	0
$313$	−2536.00	−0.457965	−0.228983	−	0.973430i	$-0.573540\pi$
$313$	−2536.00	−0.457965	−0.228983	+	0.973430i	$0.573540\pi$
$314$	−1264.00	−0.227171
$315$	0	0
$316$	5288.00	0.941371
$317$	−1434.00	−0.254074	−0.127037	−	0.991898i	$-0.540547\pi$
$317$	−1434.00	−0.254074	−0.127037	+	0.991898i	$0.540547\pi$
$318$	0	0
$319$	8330.00	1.46204
$320$	2688.00	0.469574
$321$	0	0
$322$	368.000	0.0636889
$323$	−5600.00	−0.964682
$324$	0	0
$325$	5073.00	0.865844
$326$	−6086.00	−1.03396
$327$	0	0
$328$	2280.00	0.383817
$329$	−2856.00	−0.478591
$330$	0	0
$331$	5469.00	0.908167	0.454084	−	0.890959i	$-0.349967\pi$
$331$	5469.00	0.908167	0.454084	+	0.890959i	$0.349967\pi$
$332$	−144.000	−0.0238043
$333$	0	0
$334$	4448.00	0.728694
$335$	5556.00	0.906139
$336$	0	0
$337$	−7796.00	−1.26016	−0.630082	−	0.776529i	$-0.716979\pi$
$337$	−7796.00	−1.26016	−0.630082	+	0.776529i	$0.716979\pi$
$338$	2104.00	0.338587
$339$	0	0
$340$	−1920.00	−0.306255
$341$	−3502.00	−0.556141
$342$	0	0
$343$	4976.00	0.783320
$344$	−2112.00	−0.331022
$345$	0	0
$346$	−6460.00	−1.00373
$347$	10068.0	1.55758	0.778788	−	0.627288i	$-0.215835\pi$
$347$	10068.0	1.55758	0.778788	+	0.627288i	$0.215835\pi$
$348$	0	0
$349$	−7495.00	−1.14956	−0.574782	−	0.818306i	$-0.694913\pi$
$349$	−7495.00	−1.14956	−0.574782	+	0.818306i	$0.694913\pi$
$350$	1424.00	0.217474
$351$	0	0
$352$	−5440.00	−0.823730
$353$	−10617.0	−1.60081	−0.800405	−	0.599460i	$-0.795382\pi$
$353$	−10617.0	−1.60081	−0.800405	+	0.599460i	$0.795382\pi$
$354$	0	0
$355$	−2010.00	−0.300506
$356$	−1840.00	−0.273932
$357$	0	0
$358$	−738.000	−0.108951
$359$	−2522.00	−0.370769	−0.185384	−	0.982666i	$-0.559353\pi$
$359$	−2522.00	−0.370769	−0.185384	+	0.982666i	$0.559353\pi$
$360$	0	0
$361$	−1959.00	−0.285610
$362$	−2740.00	−0.397821
$363$	0	0
$364$	−1824.00	−0.262647
$365$	−5394.00	−0.773520
$366$	0	0
$367$	7204.00	1.02465	0.512324	−	0.858792i	$-0.328785\pi$
$367$	7204.00	1.02465	0.512324	+	0.858792i	$0.328785\pi$
$368$	368.000	0.0521286
$369$	0	0
$370$	−3576.00	−0.502452
$371$	−3312.00	−0.463478
$372$	0	0
$373$	−13310.0	−1.84763	−0.923815	−	0.382840i	$-0.874946\pi$
$373$	−13310.0	−1.84763	−0.923815	+	0.382840i	$0.874946\pi$
$374$	−5440.00	−0.752128
$375$	0	0
$376$	−8568.00	−1.17516
$377$	13965.0	1.90778
$378$	0	0
$379$	12952.0	1.75541	0.877704	−	0.479203i	$-0.159074\pi$
$379$	12952.0	1.75541	0.877704	+	0.479203i	$0.159074\pi$
$380$	1680.00	0.226795
$381$	0	0
$382$	−8820.00	−1.18134
$383$	2812.00	0.375161	0.187580	−	0.982249i	$-0.439936\pi$
$383$	2812.00	0.375161	0.187580	+	0.982249i	$0.439936\pi$
$384$	0	0
$385$	1632.00	0.216037
$386$	−270.000	−0.0356027
$387$	0	0
$388$	3856.00	0.504533
$389$	−1264.00	−0.164749	−0.0823745	−	0.996601i	$-0.526250\pi$
$389$	−1264.00	−0.164749	−0.0823745	+	0.996601i	$0.526250\pi$
$390$	0	0
$391$	−1840.00	−0.237987
$392$	6696.00	0.862753
$393$	0	0
$394$	−2442.00	−0.312249
$395$	−7932.00	−1.01039
$396$	0	0
$397$	7119.00	0.899981	0.449990	−	0.893033i	$-0.351427\pi$
$397$	7119.00	0.899981	0.449990	+	0.893033i	$0.351427\pi$
$398$	−2196.00	−0.276572
$399$	0	0
$400$	1424.00	0.178000
$401$	−4262.00	−0.530758	−0.265379	−	0.964144i	$-0.585497\pi$
$401$	−4262.00	−0.530758	−0.265379	+	0.964144i	$0.585497\pi$
$402$	0	0
$403$	−5871.00	−0.725696
$404$	−1240.00	−0.152704
$405$	0	0
$406$	3920.00	0.479178
$407$	10132.0	1.23397
$408$	0	0
$409$	229.000	0.0276854	0.0138427	−	0.999904i	$-0.495594\pi$
$409$	229.000	0.0276854	0.0138427	+	0.999904i	$0.495594\pi$
$410$	−1140.00	−0.137319
$411$	0	0
$412$	−4176.00	−0.499361
$413$	−3264.00	−0.388888
$414$	0	0
$415$	216.000	0.0255495
$416$	−9120.00	−1.07487
$417$	0	0
$418$	4760.00	0.556984
$419$	−15776.0	−1.83940	−0.919699	−	0.392623i	$-0.871568\pi$
$419$	−15776.0	−1.83940	−0.919699	+	0.392623i	$0.871568\pi$
$420$	0	0
$421$	−8728.00	−1.01040	−0.505198	−	0.863003i	$-0.668581\pi$
$421$	−8728.00	−1.01040	−0.505198	+	0.863003i	$0.668581\pi$
$422$	−7352.00	−0.848080
$423$	0	0
$424$	−9936.00	−1.13805
$425$	−7120.00	−0.812637
$426$	0	0
$427$	−6576.00	−0.745281
$428$	1656.00	0.187023
$429$	0	0
$430$	1056.00	0.118430
$431$	2928.00	0.327232	0.163616	−	0.986524i	$-0.447684\pi$
$431$	2928.00	0.327232	0.163616	+	0.986524i	$0.447684\pi$
$432$	0	0
$433$	−5314.00	−0.589780	−0.294890	−	0.955531i	$-0.595283\pi$
$433$	−5314.00	−0.589780	−0.294890	+	0.955531i	$0.595283\pi$
$434$	−1648.00	−0.182273
$435$	0	0
$436$	−2816.00	−0.309316
$437$	1610.00	0.176240
$438$	0	0
$439$	2585.00	0.281037	0.140519	−	0.990078i	$-0.455123\pi$
$439$	2585.00	0.281037	0.140519	+	0.990078i	$0.455123\pi$
$440$	4896.00	0.530472
$441$	0	0
$442$	−9120.00	−0.981435
$443$	2997.00	0.321426	0.160713	−	0.987001i	$-0.448621\pi$
$443$	2997.00	0.321426	0.160713	+	0.987001i	$0.448621\pi$
$444$	0	0
$445$	2760.00	0.294015
$446$	3312.00	0.351632
$447$	0	0
$448$	−3584.00	−0.377964
$449$	16562.0	1.74078	0.870389	−	0.492365i	$-0.163868\pi$
$449$	16562.0	1.74078	0.870389	+	0.492365i	$0.163868\pi$
$450$	0	0
$451$	3230.00	0.337239
$452$	3808.00	0.396268
$453$	0	0
$454$	−5880.00	−0.607846
$455$	2736.00	0.281903
$456$	0	0
$457$	3924.00	0.401656	0.200828	−	0.979626i	$-0.435637\pi$
$457$	3924.00	0.401656	0.200828	+	0.979626i	$0.435637\pi$
$458$	7224.00	0.737020
$459$	0	0
$460$	552.000	0.0559503
$461$	4543.00	0.458977	0.229489	−	0.973311i	$-0.426295\pi$
$461$	4543.00	0.458977	0.229489	+	0.973311i	$0.426295\pi$
$462$	0	0
$463$	9616.00	0.965213	0.482606	−	0.875837i	$-0.339690\pi$
$463$	9616.00	0.965213	0.482606	+	0.875837i	$0.339690\pi$
$464$	3920.00	0.392201
$465$	0	0
$466$	8650.00	0.859879
$467$	−7826.00	−0.775469	−0.387735	−	0.921771i	$-0.626742\pi$
$467$	−7826.00	−0.775469	−0.387735	+	0.921771i	$0.626742\pi$
$468$	0	0
$469$	−7408.00	−0.729360
$470$	4284.00	0.420439
$471$	0	0
$472$	−9792.00	−0.954901
$473$	−2992.00	−0.290851
$474$	0	0
$475$	6230.00	0.601794
$476$	2560.00	0.246507
$477$	0	0
$478$	−5470.00	−0.523414
$479$	−11404.0	−1.08781	−0.543906	−	0.839146i	$-0.683055\pi$
$479$	−11404.0	−1.08781	−0.543906	+	0.839146i	$0.683055\pi$
$480$	0	0
$481$	16986.0	1.61018
$482$	−13420.0	−1.26818
$483$	0	0
$484$	700.000	0.0657400
$485$	−5784.00	−0.541521
$486$	0	0
$487$	−9267.00	−0.862275	−0.431137	−	0.902286i	$-0.641888\pi$
$487$	−9267.00	−0.862275	−0.431137	+	0.902286i	$0.641888\pi$
$488$	−19728.0	−1.83001
$489$	0	0
$490$	−3348.00	−0.308668
$491$	18191.0	1.67199	0.835996	−	0.548735i	$-0.184890\pi$
$491$	18191.0	1.67199	0.835996	+	0.548735i	$0.184890\pi$
$492$	0	0
$493$	−19600.0	−1.79055
$494$	7980.00	0.726796
$495$	0	0
$496$	−1648.00	−0.149188
$497$	2680.00	0.241880
$498$	0	0
$499$	19315.0	1.73278	0.866391	−	0.499366i	$-0.166434\pi$
$499$	19315.0	1.73278	0.866391	+	0.499366i	$0.166434\pi$
$500$	5136.00	0.459378
$501$	0	0
$502$	13896.0	1.23548
$503$	−8422.00	−0.746557	−0.373279	−	0.927719i	$-0.621766\pi$
$503$	−8422.00	−0.746557	−0.373279	+	0.927719i	$0.621766\pi$
$504$	0	0
$505$	1860.00	0.163899
$506$	1564.00	0.137408
$507$	0	0
$508$	−1044.00	−0.0911811
$509$	863.000	0.0751509	0.0375754	−	0.999294i	$-0.488037\pi$
$509$	863.000	0.0751509	0.0375754	+	0.999294i	$0.488037\pi$
$510$	0	0
$511$	7192.00	0.622613
$512$	−5632.00	−0.486136
$513$	0	0
$514$	9858.00	0.845949
$515$	6264.00	0.535971
$516$	0	0
$517$	−12138.0	−1.03255
$518$	4768.00	0.404428
$519$	0	0
$520$	8208.00	0.692201
$521$	−19260.0	−1.61957	−0.809785	−	0.586727i	$-0.800416\pi$
$521$	−19260.0	−1.61957	−0.809785	+	0.586727i	$0.800416\pi$
$522$	0	0
$523$	−11740.0	−0.981557	−0.490779	−	0.871284i	$-0.663288\pi$
$523$	−11740.0	−0.981557	−0.490779	+	0.871284i	$0.663288\pi$
$524$	−5764.00	−0.480537
$525$	0	0
$526$	−12276.0	−1.01760
$527$	8240.00	0.681101
$528$	0	0
$529$	529.000	0.0434783
$530$	4968.00	0.407163
$531$	0	0
$532$	−2240.00	−0.182549
$533$	5415.00	0.440056
$534$	0	0
$535$	−2484.00	−0.200734
$536$	−22224.0	−1.79092
$537$	0	0
$538$	4126.00	0.330640
$539$	9486.00	0.758054
$540$	0	0
$541$	17741.0	1.40988	0.704940	−	0.709267i	$-0.250974\pi$
$541$	17741.0	1.40988	0.704940	+	0.709267i	$0.250974\pi$
$542$	−2128.00	−0.168645
$543$	0	0
$544$	12800.0	1.00882
$545$	4224.00	0.331993
$546$	0	0
$547$	−6571.00	−0.513630	−0.256815	−	0.966461i	$-0.582673\pi$
$547$	−6571.00	−0.513630	−0.256815	+	0.966461i	$0.582673\pi$
$548$	6224.00	0.485175
$549$	0	0
$550$	6052.00	0.469197
$551$	17150.0	1.32598
$552$	0	0
$553$	10576.0	0.813268
$554$	11458.0	0.878707
$555$	0	0
$556$	−100.000	−0.00762760
$557$	1372.00	0.104369	0.0521845	−	0.998637i	$-0.483382\pi$
$557$	1372.00	0.104369	0.0521845	+	0.998637i	$0.483382\pi$
$558$	0	0
$559$	−5016.00	−0.379524
$560$	768.000	0.0579534
$561$	0	0
$562$	1920.00	0.144111
$563$	−4332.00	−0.324284	−0.162142	−	0.986767i	$-0.551840\pi$
$563$	−4332.00	−0.324284	−0.162142	+	0.986767i	$0.551840\pi$
$564$	0	0
$565$	−5712.00	−0.425320
$566$	−228.000	−0.0169321
$567$	0	0
$568$	8040.00	0.593928
$569$	3546.00	0.261258	0.130629	−	0.991431i	$-0.458300\pi$
$569$	3546.00	0.261258	0.130629	+	0.991431i	$0.458300\pi$
$570$	0	0
$571$	−6160.00	−0.451468	−0.225734	−	0.974189i	$-0.572478\pi$
$571$	−6160.00	−0.451468	−0.225734	+	0.974189i	$0.572478\pi$
$572$	−7752.00	−0.566656
$573$	0	0
$574$	1520.00	0.110529
$575$	2047.00	0.148462
$576$	0	0
$577$	2953.00	0.213059	0.106529	−	0.994310i	$-0.466026\pi$
$577$	2953.00	0.213059	0.106529	+	0.994310i	$0.466026\pi$
$578$	2974.00	0.214017
$579$	0	0
$580$	5880.00	0.420955
$581$	−288.000	−0.0205650
$582$	0	0
$583$	−14076.0	−0.999946
$584$	21576.0	1.52880
$585$	0	0
$586$	14096.0	0.993687
$587$	2949.00	0.207356	0.103678	−	0.994611i	$-0.466939\pi$
$587$	2949.00	0.207356	0.103678	+	0.994611i	$0.466939\pi$
$588$	0	0
$589$	−7210.00	−0.504385
$590$	4896.00	0.341636
$591$	0	0
$592$	4768.00	0.331020
$593$	−16390.0	−1.13500	−0.567501	−	0.823372i	$-0.692090\pi$
$593$	−16390.0	−1.13500	−0.567501	+	0.823372i	$0.692090\pi$
$594$	0	0
$595$	−3840.00	−0.264579
$596$	3288.00	0.225976
$597$	0	0
$598$	2622.00	0.179300
$599$	12920.0	0.881297	0.440648	−	0.897680i	$-0.354749\pi$
$599$	12920.0	0.881297	0.440648	+	0.897680i	$0.354749\pi$
$600$	0	0
$601$	−13835.0	−0.939004	−0.469502	−	0.882931i	$-0.655567\pi$
$601$	−13835.0	−0.939004	−0.469502	+	0.882931i	$0.655567\pi$
$602$	−1408.00	−0.0953252
$603$	0	0
$604$	5956.00	0.401235
$605$	−1050.00	−0.0705596
$606$	0	0
$607$	6004.00	0.401474	0.200737	−	0.979645i	$-0.435666\pi$
$607$	6004.00	0.401474	0.200737	+	0.979645i	$0.435666\pi$
$608$	−11200.0	−0.747072
$609$	0	0
$610$	9864.00	0.654724
$611$	−20349.0	−1.34735
$612$	0	0
$613$	−16416.0	−1.08162	−0.540812	−	0.841143i	$-0.681883\pi$
$613$	−16416.0	−1.08162	−0.540812	+	0.841143i	$0.681883\pi$
$614$	7744.00	0.508994
$615$	0	0
$616$	−6528.00	−0.426982
$617$	−3786.00	−0.247032	−0.123516	−	0.992343i	$-0.539417\pi$
$617$	−3786.00	−0.247032	−0.123516	+	0.992343i	$0.539417\pi$
$618$	0	0
$619$	15824.0	1.02750	0.513748	−	0.857941i	$-0.328257\pi$
$619$	15824.0	1.02750	0.513748	+	0.857941i	$0.328257\pi$
$620$	−2472.00	−0.160126
$621$	0	0
$622$	9954.00	0.641670
$623$	−3680.00	−0.236655
$624$	0	0
$625$	3421.00	0.218944
$626$	−5072.00	−0.323830
$627$	0	0
$628$	2528.00	0.160634
$629$	−23840.0	−1.51123
$630$	0	0
$631$	17852.0	1.12627	0.563135	−	0.826365i	$-0.309595\pi$
$631$	17852.0	1.12627	0.563135	+	0.826365i	$0.309595\pi$
$632$	31728.0	1.99695
$633$	0	0
$634$	−2868.00	−0.179657
$635$	1566.00	0.0978658
$636$	0	0
$637$	15903.0	0.989168
$638$	16660.0	1.03382
$639$	0	0
$640$	−2304.00	−0.142302
$641$	−10324.0	−0.636152	−0.318076	−	0.948065i	$-0.603037\pi$
$641$	−10324.0	−0.636152	−0.318076	+	0.948065i	$0.603037\pi$
$642$	0	0
$643$	−14702.0	−0.901696	−0.450848	−	0.892601i	$-0.648878\pi$
$643$	−14702.0	−0.901696	−0.450848	+	0.892601i	$0.648878\pi$
$644$	−736.000	−0.0450349
$645$	0	0
$646$	−11200.0	−0.682133
$647$	−11939.0	−0.725457	−0.362728	−	0.931895i	$-0.618155\pi$
$647$	−11939.0	−0.725457	−0.362728	+	0.931895i	$0.618155\pi$
$648$	0	0
$649$	−13872.0	−0.839019
$650$	10146.0	0.612244
$651$	0	0
$652$	12172.0	0.731123
$653$	−6159.00	−0.369097	−0.184548	−	0.982823i	$-0.559082\pi$
$653$	−6159.00	−0.369097	−0.184548	+	0.982823i	$0.559082\pi$
$654$	0	0
$655$	8646.00	0.515767
$656$	1520.00	0.0904665
$657$	0	0
$658$	−5712.00	−0.338415
$659$	21692.0	1.28225	0.641123	−	0.767438i	$-0.278469\pi$
$659$	21692.0	1.28225	0.641123	+	0.767438i	$0.278469\pi$
$660$	0	0
$661$	16502.0	0.971034	0.485517	−	0.874227i	$-0.338631\pi$
$661$	16502.0	0.971034	0.485517	+	0.874227i	$0.338631\pi$
$662$	10938.0	0.642171
$663$	0	0
$664$	−864.000	−0.0504965
$665$	3360.00	0.195933
$666$	0	0
$667$	5635.00	0.327119
$668$	−8896.00	−0.515264
$669$	0	0
$670$	11112.0	0.640737
$671$	−27948.0	−1.60793
$672$	0	0
$673$	−27733.0	−1.58845	−0.794226	−	0.607622i	$-0.792124\pi$
$673$	−27733.0	−1.58845	−0.794226	+	0.607622i	$0.792124\pi$
$674$	−15592.0	−0.891070
$675$	0	0
$676$	−4208.00	−0.239417
$677$	8814.00	0.500369	0.250184	−	0.968198i	$-0.419509\pi$
$677$	8814.00	0.500369	0.250184	+	0.968198i	$0.419509\pi$
$678$	0	0
$679$	7712.00	0.435875
$680$	−11520.0	−0.649664
$681$	0	0
$682$	−7004.00	−0.393251
$683$	22999.0	1.28848	0.644240	−	0.764823i	$-0.277174\pi$
$683$	22999.0	1.28848	0.644240	+	0.764823i	$0.277174\pi$
$684$	0	0
$685$	−9336.00	−0.520745
$686$	9952.00	0.553891
$687$	0	0
$688$	−1408.00	−0.0780225
$689$	−23598.0	−1.30481
$690$	0	0
$691$	−12140.0	−0.668346	−0.334173	−	0.942512i	$-0.608457\pi$
$691$	−12140.0	−0.668346	−0.334173	+	0.942512i	$0.608457\pi$
$692$	12920.0	0.709747
$693$	0	0
$694$	20136.0	1.10137
$695$	150.000	0.00818680
$696$	0	0
$697$	−7600.00	−0.413014
$698$	−14990.0	−0.812865
$699$	0	0
$700$	−2848.00	−0.153778
$701$	20024.0	1.07888	0.539441	−	0.842024i	$-0.318636\pi$
$701$	20024.0	1.07888	0.539441	+	0.842024i	$0.318636\pi$
$702$	0	0
$703$	20860.0	1.11913
$704$	−15232.0	−0.815451
$705$	0	0
$706$	−21234.0	−1.13194
$707$	−2480.00	−0.131924
$708$	0	0
$709$	−4956.00	−0.262520	−0.131260	−	0.991348i	$-0.541902\pi$
$709$	−4956.00	−0.262520	−0.131260	+	0.991348i	$0.541902\pi$
$710$	−4020.00	−0.212490
$711$	0	0
$712$	−11040.0	−0.581098
$713$	−2369.00	−0.124432
$714$	0	0
$715$	11628.0	0.608199
$716$	1476.00	0.0770401
$717$	0	0
$718$	−5044.00	−0.262173
$719$	−2760.00	−0.143158	−0.0715790	−	0.997435i	$-0.522804\pi$
$719$	−2760.00	−0.143158	−0.0715790	+	0.997435i	$0.522804\pi$
$720$	0	0
$721$	−8352.00	−0.431407
$722$	−3918.00	−0.201957
$723$	0	0
$724$	5480.00	0.281302
$725$	21805.0	1.11699
$726$	0	0
$727$	7746.00	0.395163	0.197581	−	0.980287i	$-0.436691\pi$
$727$	7746.00	0.395163	0.197581	+	0.980287i	$0.436691\pi$
$728$	−10944.0	−0.557159
$729$	0	0
$730$	−10788.0	−0.546961
$731$	7040.00	0.356202
$732$	0	0
$733$	−11976.0	−0.603470	−0.301735	−	0.953392i	$-0.597566\pi$
$733$	−11976.0	−0.603470	−0.301735	+	0.953392i	$0.597566\pi$
$734$	14408.0	0.724535
$735$	0	0
$736$	−3680.00	−0.184302
$737$	−31484.0	−1.57358
$738$	0	0
$739$	15057.0	0.749500	0.374750	−	0.927126i	$-0.377728\pi$
$739$	15057.0	0.749500	0.374750	+	0.927126i	$0.377728\pi$
$740$	7152.00	0.355287
$741$	0	0
$742$	−6624.00	−0.327729
$743$	−18532.0	−0.915038	−0.457519	−	0.889200i	$-0.651262\pi$
$743$	−18532.0	−0.915038	−0.457519	+	0.889200i	$0.651262\pi$
$744$	0	0
$745$	−4932.00	−0.242543
$746$	−26620.0	−1.30647
$747$	0	0
$748$	10880.0	0.531834
$749$	3312.00	0.161573
$750$	0	0
$751$	−192.000	−0.00932913	−0.00466457	−	0.999989i	$-0.501485\pi$
$751$	−192.000	−0.00932913	−0.00466457	+	0.999989i	$0.501485\pi$
$752$	−5712.00	−0.276988
$753$	0	0
$754$	27930.0	1.34901
$755$	−8934.00	−0.430651
$756$	0	0
$757$	−9830.00	−0.471965	−0.235982	−	0.971757i	$-0.575831\pi$
$757$	−9830.00	−0.471965	−0.235982	+	0.971757i	$0.575831\pi$
$758$	25904.0	1.24126
$759$	0	0
$760$	10080.0	0.481105
$761$	30219.0	1.43947	0.719736	−	0.694248i	$-0.244263\pi$
$761$	30219.0	1.43947	0.719736	+	0.694248i	$0.244263\pi$
$762$	0	0
$763$	−5632.00	−0.267224
$764$	17640.0	0.835331
$765$	0	0
$766$	5624.00	0.265279
$767$	−23256.0	−1.09482
$768$	0	0
$769$	1122.00	0.0526142	0.0263071	−	0.999654i	$-0.491625\pi$
$769$	1122.00	0.0526142	0.0263071	+	0.999654i	$0.491625\pi$
$770$	3264.00	0.152762
$771$	0	0
$772$	540.000	0.0251749
$773$	−19300.0	−0.898024	−0.449012	−	0.893526i	$-0.648224\pi$
$773$	−19300.0	−0.898024	−0.449012	+	0.893526i	$0.648224\pi$
$774$	0	0
$775$	−9167.00	−0.424888
$776$	23136.0	1.07028
$777$	0	0
$778$	−2528.00	−0.116495
$779$	6650.00	0.305855
$780$	0	0
$781$	11390.0	0.521852
$782$	−3680.00	−0.168282
$783$	0	0
$784$	4464.00	0.203353
$785$	−3792.00	−0.172411
$786$	0	0
$787$	−19396.0	−0.878517	−0.439258	−	0.898361i	$-0.644759\pi$
$787$	−19396.0	−0.878517	−0.439258	+	0.898361i	$0.644759\pi$
$788$	4884.00	0.220794
$789$	0	0
$790$	−15864.0	−0.714450
$791$	7616.00	0.342344
$792$	0	0
$793$	−46854.0	−2.09815
$794$	14238.0	0.636383
$795$	0	0
$796$	4392.00	0.195566
$797$	39034.0	1.73482	0.867412	−	0.497590i	$-0.165782\pi$
$797$	39034.0	1.73482	0.867412	+	0.497590i	$0.165782\pi$
$798$	0	0
$799$	28560.0	1.26456
$800$	−14240.0	−0.629325
$801$	0	0
$802$	−8524.00	−0.375303
$803$	30566.0	1.34328
$804$	0	0
$805$	1104.00	0.0483365
$806$	−11742.0	−0.513144
$807$	0	0
$808$	−7440.00	−0.323934
$809$	10310.0	0.448060	0.224030	−	0.974582i	$-0.428079\pi$
$809$	10310.0	0.448060	0.224030	+	0.974582i	$0.428079\pi$
$810$	0	0
$811$	−40693.0	−1.76193	−0.880965	−	0.473182i	$-0.843105\pi$
$811$	−40693.0	−1.76193	−0.880965	+	0.473182i	$0.843105\pi$
$812$	−7840.00	−0.338830
$813$	0	0
$814$	20264.0	0.872546
$815$	−18258.0	−0.784724
$816$	0	0
$817$	−6160.00	−0.263784
$818$	458.000	0.0195765
$819$	0	0
$820$	2280.00	0.0970988
$821$	13934.0	0.592326	0.296163	−	0.955137i	$-0.404293\pi$
$821$	13934.0	0.592326	0.296163	+	0.955137i	$0.404293\pi$
$822$	0	0
$823$	6175.00	0.261539	0.130770	−	0.991413i	$-0.458255\pi$
$823$	6175.00	0.261539	0.130770	+	0.991413i	$0.458255\pi$
$824$	−25056.0	−1.05930
$825$	0	0
$826$	−6528.00	−0.274986
$827$	−28664.0	−1.20525	−0.602627	−	0.798023i	$-0.705879\pi$
$827$	−28664.0	−1.20525	−0.602627	+	0.798023i	$0.705879\pi$
$828$	0	0
$829$	−39590.0	−1.65865	−0.829323	−	0.558770i	$-0.811274\pi$
$829$	−39590.0	−1.65865	−0.829323	+	0.558770i	$0.811274\pi$
$830$	432.000	0.0180662
$831$	0	0
$832$	−25536.0	−1.06406
$833$	−22320.0	−0.928382
$834$	0	0
$835$	13344.0	0.553040
$836$	−9520.00	−0.393847
$837$	0	0
$838$	−31552.0	−1.30065
$839$	14316.0	0.589086	0.294543	−	0.955638i	$-0.404833\pi$
$839$	14316.0	0.589086	0.294543	+	0.955638i	$0.404833\pi$
$840$	0	0
$841$	35636.0	1.46115
$842$	−17456.0	−0.714458
$843$	0	0
$844$	14704.0	0.599683
$845$	6312.00	0.256970
$846$	0	0
$847$	1400.00	0.0567941
$848$	−6624.00	−0.268242
$849$	0	0
$850$	−14240.0	−0.574621
$851$	6854.00	0.276089
$852$	0	0
$853$	28366.0	1.13861	0.569304	−	0.822127i	$-0.307213\pi$
$853$	28366.0	1.13861	0.569304	+	0.822127i	$0.307213\pi$
$854$	−13152.0	−0.526993
$855$	0	0
$856$	9936.00	0.396735
$857$	−19283.0	−0.768605	−0.384303	−	0.923207i	$-0.625558\pi$
$857$	−19283.0	−0.768605	−0.384303	+	0.923207i	$0.625558\pi$
$858$	0	0
$859$	−26101.0	−1.03673	−0.518367	−	0.855158i	$-0.673460\pi$
$859$	−26101.0	−1.03673	−0.518367	+	0.855158i	$0.673460\pi$
$860$	−2112.00	−0.0837426
$861$	0	0
$862$	5856.00	0.231388
$863$	−973.000	−0.0383793	−0.0191896	−	0.999816i	$-0.506109\pi$
$863$	−973.000	−0.0383793	−0.0191896	+	0.999816i	$0.506109\pi$
$864$	0	0
$865$	−19380.0	−0.761780
$866$	−10628.0	−0.417037
$867$	0	0
$868$	3296.00	0.128887
$869$	44948.0	1.75461
$870$	0	0
$871$	−52782.0	−2.05333
$872$	−16896.0	−0.656159
$873$	0	0
$874$	3220.00	0.124620
$875$	10272.0	0.396865
$876$	0	0
$877$	5694.00	0.219239	0.109620	−	0.993974i	$-0.465037\pi$
$877$	5694.00	0.219239	0.109620	+	0.993974i	$0.465037\pi$
$878$	5170.00	0.198723
$879$	0	0
$880$	3264.00	0.125033
$881$	−45960.0	−1.75758	−0.878792	−	0.477205i	$-0.841650\pi$
$881$	−45960.0	−1.75758	−0.878792	+	0.477205i	$0.841650\pi$
$882$	0	0
$883$	17188.0	0.655065	0.327532	−	0.944840i	$-0.393783\pi$
$883$	17188.0	0.655065	0.327532	+	0.944840i	$0.393783\pi$
$884$	18240.0	0.693979
$885$	0	0
$886$	5994.00	0.227283
$887$	−8451.00	−0.319906	−0.159953	−	0.987125i	$-0.551134\pi$
$887$	−8451.00	−0.319906	−0.159953	+	0.987125i	$0.551134\pi$
$888$	0	0
$889$	−2088.00	−0.0787731
$890$	5520.00	0.207900
$891$	0	0
$892$	−6624.00	−0.248641
$893$	−24990.0	−0.936460
$894$	0	0
$895$	−2214.00	−0.0826881
$896$	3072.00	0.114541
$897$	0	0
$898$	33124.0	1.23092
$899$	−25235.0	−0.936190
$900$	0	0
$901$	33120.0	1.22463
$902$	6460.00	0.238464
$903$	0	0
$904$	22848.0	0.840612
$905$	−8220.00	−0.301925
$906$	0	0
$907$	32774.0	1.19983	0.599913	−	0.800065i	$-0.295202\pi$
$907$	32774.0	1.19983	0.599913	+	0.800065i	$0.295202\pi$
$908$	11760.0	0.429812
$909$	0	0
$910$	5472.00	0.199335
$911$	23690.0	0.861564	0.430782	−	0.902456i	$-0.358238\pi$
$911$	23690.0	0.861564	0.430782	+	0.902456i	$0.358238\pi$
$912$	0	0
$913$	−1224.00	−0.0443686
$914$	7848.00	0.284014
$915$	0	0
$916$	−14448.0	−0.521152
$917$	−11528.0	−0.415145
$918$	0	0
$919$	−30044.0	−1.07841	−0.539206	−	0.842174i	$-0.681275\pi$
$919$	−30044.0	−1.07841	−0.539206	+	0.842174i	$0.681275\pi$
$920$	3312.00	0.118688
$921$	0	0
$922$	9086.00	0.324546
$923$	19095.0	0.680953
$924$	0	0
$925$	26522.0	0.942744
$926$	19232.0	0.682508
$927$	0	0
$928$	−39200.0	−1.38664
$929$	39705.0	1.40224	0.701119	−	0.713044i	$-0.252684\pi$
$929$	39705.0	1.40224	0.701119	+	0.713044i	$0.252684\pi$
$930$	0	0
$931$	19530.0	0.687508
$932$	−17300.0	−0.608026
$933$	0	0
$934$	−15652.0	−0.548339
$935$	−16320.0	−0.570825
$936$	0	0
$937$	17422.0	0.607419	0.303710	−	0.952765i	$-0.401775\pi$
$937$	17422.0	0.607419	0.303710	+	0.952765i	$0.401775\pi$
$938$	−14816.0	−0.515735
$939$	0	0
$940$	−8568.00	−0.297295
$941$	25292.0	0.876191	0.438095	−	0.898928i	$-0.355653\pi$
$941$	25292.0	0.876191	0.438095	+	0.898928i	$0.355653\pi$
$942$	0	0
$943$	2185.00	0.0754543
$944$	−6528.00	−0.225072
$945$	0	0
$946$	−5984.00	−0.205662
$947$	−33211.0	−1.13961	−0.569806	−	0.821779i	$-0.692982\pi$
$947$	−33211.0	−1.13961	−0.569806	+	0.821779i	$0.692982\pi$
$948$	0	0
$949$	51243.0	1.75281
$950$	12460.0	0.425532
$951$	0	0
$952$	15360.0	0.522921
$953$	14154.0	0.481105	0.240552	−	0.970636i	$-0.422671\pi$
$953$	14154.0	0.481105	0.240552	+	0.970636i	$0.422671\pi$
$954$	0	0
$955$	−26460.0	−0.896571
$956$	10940.0	0.370110
$957$	0	0
$958$	−22808.0	−0.769199
$959$	12448.0	0.419152
$960$	0	0
$961$	−19182.0	−0.643886
$962$	33972.0	1.13857
$963$	0	0
$964$	26840.0	0.896741
$965$	−810.000	−0.0270205
$966$	0	0
$967$	−46343.0	−1.54115	−0.770574	−	0.637350i	$-0.780030\pi$
$967$	−46343.0	−1.54115	−0.770574	+	0.637350i	$0.780030\pi$
$968$	4200.00	0.139456
$969$	0	0
$970$	−11568.0	−0.382914
$971$	−11710.0	−0.387015	−0.193508	−	0.981099i	$-0.561986\pi$
$971$	−11710.0	−0.387015	−0.193508	+	0.981099i	$0.561986\pi$
$972$	0	0
$973$	−200.000	−0.00658963
$974$	−18534.0	−0.609720
$975$	0	0
$976$	−13152.0	−0.431337
$977$	−47854.0	−1.56703	−0.783513	−	0.621375i	$-0.786574\pi$
$977$	−47854.0	−1.56703	−0.783513	+	0.621375i	$0.786574\pi$
$978$	0	0
$979$	−15640.0	−0.510579
$980$	6696.00	0.218261
$981$	0	0
$982$	36382.0	1.18228
$983$	22078.0	0.716357	0.358178	−	0.933653i	$-0.383398\pi$
$983$	22078.0	0.716357	0.358178	+	0.933653i	$0.383398\pi$
$984$	0	0
$985$	−7326.00	−0.236980
$986$	−39200.0	−1.26611
$987$	0	0
$988$	−15960.0	−0.513922
$989$	−2024.00	−0.0650753
$990$	0	0
$991$	−4288.00	−0.137450	−0.0687249	−	0.997636i	$-0.521893\pi$
$991$	−4288.00	−0.137450	−0.0687249	+	0.997636i	$0.521893\pi$
$992$	16480.0	0.527460
$993$	0	0
$994$	5360.00	0.171035
$995$	−6588.00	−0.209903
$996$	0	0
$997$	28966.0	0.920123	0.460061	−	0.887887i	$-0.347827\pi$
$997$	28966.0	0.920123	0.460061	+	0.887887i	$0.347827\pi$
$998$	38630.0	1.22526
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.4.a.a.1.1		1
3.2	odd	2		23.4.a.a.1.1	✓	1
12.11	even	2		368.4.a.d.1.1		1
15.2	even	4		575.4.b.b.24.1		2
15.8	even	4		575.4.b.b.24.2		2
15.14	odd	2		575.4.a.g.1.1		1
21.20	even	2		1127.4.a.a.1.1		1
24.5	odd	2		1472.4.a.h.1.1		1
24.11	even	2		1472.4.a.c.1.1		1
69.68	even	2		529.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.4.a.a.1.1	✓	1	3.2	odd	2
207.4.a.a.1.1		1	1.1	even	1	trivial
368.4.a.d.1.1		1	12.11	even	2
529.4.a.a.1.1		1	69.68	even	2
575.4.a.g.1.1		1	15.14	odd	2
575.4.b.b.24.1		2	15.2	even	4
575.4.b.b.24.2		2	15.8	even	4
1127.4.a.a.1.1		1	21.20	even	2
1472.4.a.c.1.1		1	24.11	even	2
1472.4.a.h.1.1		1	24.5	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:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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