LMFDB - Embedded newform 1170.4.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1170,4,Mod(1,1170)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1170, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1170.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1170.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:	$69.0322347067$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1170.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} -24.0000 q^{7} +8.00000 q^{8} +O(q^{10})$

\(q+2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} -24.0000 q^{7} +8.00000 q^{8} +10.0000 q^{10} +32.0000 q^{11} +13.0000 q^{13} -48.0000 q^{14} +16.0000 q^{16} -78.0000 q^{17} -32.0000 q^{19} +20.0000 q^{20} +64.0000 q^{22} -158.000 q^{23} +25.0000 q^{25} +26.0000 q^{26} -96.0000 q^{28} -126.000 q^{29} +250.000 q^{31} +32.0000 q^{32} -156.000 q^{34} -120.000 q^{35} +38.0000 q^{37} -64.0000 q^{38} +40.0000 q^{40} -90.0000 q^{41} -428.000 q^{43} +128.000 q^{44} -316.000 q^{46} +84.0000 q^{47} +233.000 q^{49} +50.0000 q^{50} +52.0000 q^{52} -236.000 q^{53} +160.000 q^{55} -192.000 q^{56} -252.000 q^{58} +188.000 q^{59} -170.000 q^{61} +500.000 q^{62} +64.0000 q^{64} +65.0000 q^{65} -78.0000 q^{67} -312.000 q^{68} -240.000 q^{70} -16.0000 q^{71} -1012.00 q^{73} +76.0000 q^{74} -128.000 q^{76} -768.000 q^{77} -1264.00 q^{79} +80.0000 q^{80} -180.000 q^{82} +60.0000 q^{83} -390.000 q^{85} -856.000 q^{86} +256.000 q^{88} +1010.00 q^{89} -312.000 q^{91} -632.000 q^{92} +168.000 q^{94} -160.000 q^{95} +372.000 q^{97} +466.000 q^{98} +O(q^{100})\)

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
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−24.0000	−1.29588	−0.647939	−	0.761692i	$-0.724369\pi$
$7$	−24.0000	−1.29588	−0.647939	+	0.761692i	$0.724369\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	10.0000	0.316228
$11$	32.0000	0.877124	0.438562	−	0.898701i	$-0.355488\pi$
$11$	32.0000	0.877124	0.438562	+	0.898701i	$0.355488\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	−48.0000	−0.916324
$15$	0	0
$16$	16.0000	0.250000
$17$	−78.0000	−1.11281	−0.556405	−	0.830911i	$-0.687820\pi$
$17$	−78.0000	−1.11281	−0.556405	+	0.830911i	$0.687820\pi$
$18$	0	0
$19$	−32.0000	−0.386384	−0.193192	−	0.981161i	$-0.561884\pi$
$19$	−32.0000	−0.386384	−0.193192	+	0.981161i	$0.561884\pi$
$20$	20.0000	0.223607
$21$	0	0
$22$	64.0000	0.620220
$23$	−158.000	−1.43240	−0.716202	−	0.697893i	$-0.754121\pi$
$23$	−158.000	−1.43240	−0.716202	+	0.697893i	$0.754121\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	26.0000	0.196116
$27$	0	0
$28$	−96.0000	−0.647939
$29$	−126.000	−0.806814	−0.403407	−	0.915021i	$-0.632174\pi$
$29$	−126.000	−0.806814	−0.403407	+	0.915021i	$0.632174\pi$
$30$	0	0
$31$	250.000	1.44843	0.724215	−	0.689574i	$-0.242202\pi$
$31$	250.000	1.44843	0.724215	+	0.689574i	$0.242202\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	−156.000	−0.786876
$35$	−120.000	−0.579534
$36$	0	0
$37$	38.0000	0.168842	0.0844211	−	0.996430i	$-0.473096\pi$
$37$	38.0000	0.168842	0.0844211	+	0.996430i	$0.473096\pi$
$38$	−64.0000	−0.273215
$39$	0	0
$40$	40.0000	0.158114
$41$	−90.0000	−0.342820	−0.171410	−	0.985200i	$-0.554832\pi$
$41$	−90.0000	−0.342820	−0.171410	+	0.985200i	$0.554832\pi$
$42$	0	0
$43$	−428.000	−1.51789	−0.758946	−	0.651153i	$-0.774286\pi$
$43$	−428.000	−1.51789	−0.758946	+	0.651153i	$0.774286\pi$
$44$	128.000	0.438562
$45$	0	0
$46$	−316.000	−1.01286
$47$	84.0000	0.260695	0.130347	−	0.991468i	$-0.458391\pi$
$47$	84.0000	0.260695	0.130347	+	0.991468i	$0.458391\pi$
$48$	0	0
$49$	233.000	0.679300
$50$	50.0000	0.141421
$51$	0	0
$52$	52.0000	0.138675
$53$	−236.000	−0.611643	−0.305822	−	0.952089i	$-0.598931\pi$
$53$	−236.000	−0.611643	−0.305822	+	0.952089i	$0.598931\pi$
$54$	0	0
$55$	160.000	0.392262
$56$	−192.000	−0.458162
$57$	0	0
$58$	−252.000	−0.570504
$59$	188.000	0.414839	0.207420	−	0.978252i	$-0.433493\pi$
$59$	188.000	0.414839	0.207420	+	0.978252i	$0.433493\pi$
$60$	0	0
$61$	−170.000	−0.356824	−0.178412	−	0.983956i	$-0.557096\pi$
$61$	−170.000	−0.356824	−0.178412	+	0.983956i	$0.557096\pi$
$62$	500.000	1.02419
$63$	0	0
$64$	64.0000	0.125000
$65$	65.0000	0.124035
$66$	0	0
$67$	−78.0000	−0.142227	−0.0711136	−	0.997468i	$-0.522655\pi$
$67$	−78.0000	−0.142227	−0.0711136	+	0.997468i	$0.522655\pi$
$68$	−312.000	−0.556405
$69$	0	0
$70$	−240.000	−0.409793
$71$	−16.0000	−0.0267444	−0.0133722	−	0.999911i	$-0.504257\pi$
$71$	−16.0000	−0.0267444	−0.0133722	+	0.999911i	$0.504257\pi$
$72$	0	0
$73$	−1012.00	−1.62254	−0.811272	−	0.584670i	$-0.801224\pi$
$73$	−1012.00	−1.62254	−0.811272	+	0.584670i	$0.801224\pi$
$74$	76.0000	0.119389
$75$	0	0
$76$	−128.000	−0.193192
$77$	−768.000	−1.13665
$78$	0	0
$79$	−1264.00	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−1264.00	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	80.0000	0.111803
$81$	0	0
$82$	−180.000	−0.242411
$83$	60.0000	0.0793477	0.0396738	−	0.999213i	$-0.487368\pi$
$83$	60.0000	0.0793477	0.0396738	+	0.999213i	$0.487368\pi$
$84$	0	0
$85$	−390.000	−0.497664
$86$	−856.000	−1.07331
$87$	0	0
$88$	256.000	0.310110
$89$	1010.00	1.20292	0.601459	−	0.798903i	$-0.294586\pi$
$89$	1010.00	1.20292	0.601459	+	0.798903i	$0.294586\pi$
$90$	0	0
$91$	−312.000	−0.359412
$92$	−632.000	−0.716202
$93$	0	0
$94$	168.000	0.184339
$95$	−160.000	−0.172796
$96$	0	0
$97$	372.000	0.389390	0.194695	−	0.980864i	$-0.437628\pi$
$97$	372.000	0.389390	0.194695	+	0.980864i	$0.437628\pi$
$98$	466.000	0.480338
$99$	0	0
$100$	100.000	0.100000
$101$	−850.000	−0.837408	−0.418704	−	0.908123i	$-0.637515\pi$
$101$	−850.000	−0.837408	−0.418704	+	0.908123i	$0.637515\pi$
$102$	0	0
$103$	−144.000	−0.137755	−0.0688774	−	0.997625i	$-0.521942\pi$
$103$	−144.000	−0.137755	−0.0688774	+	0.997625i	$0.521942\pi$
$104$	104.000	0.0980581
$105$	0	0
$106$	−472.000	−0.432497
$107$	−1380.00	−1.24682	−0.623410	−	0.781896i	$-0.714253\pi$
$107$	−1380.00	−1.24682	−0.623410	+	0.781896i	$0.714253\pi$
$108$	0	0
$109$	−328.000	−0.288227	−0.144113	−	0.989561i	$-0.546033\pi$
$109$	−328.000	−0.288227	−0.144113	+	0.989561i	$0.546033\pi$
$110$	320.000	0.277371
$111$	0	0
$112$	−384.000	−0.323970
$113$	−2134.00	−1.77655	−0.888274	−	0.459315i	$-0.848095\pi$
$113$	−2134.00	−1.77655	−0.888274	+	0.459315i	$0.848095\pi$
$114$	0	0
$115$	−790.000	−0.640590
$116$	−504.000	−0.403407
$117$	0	0
$118$	376.000	0.293336
$119$	1872.00	1.44207
$120$	0	0
$121$	−307.000	−0.230654
$122$	−340.000	−0.252313
$123$	0	0
$124$	1000.00	0.724215
$125$	125.000	0.0894427
$126$	0	0
$127$	32.0000	0.0223586	0.0111793	−	0.999938i	$-0.496441\pi$
$127$	32.0000	0.0223586	0.0111793	+	0.999938i	$0.496441\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	130.000	0.0877058
$131$	1458.00	0.972413	0.486206	−	0.873844i	$-0.338380\pi$
$131$	1458.00	0.972413	0.486206	+	0.873844i	$0.338380\pi$
$132$	0	0
$133$	768.000	0.500707
$134$	−156.000	−0.100570
$135$	0	0
$136$	−624.000	−0.393438
$137$	722.000	0.450253	0.225126	−	0.974330i	$-0.427721\pi$
$137$	722.000	0.450253	0.225126	+	0.974330i	$0.427721\pi$
$138$	0	0
$139$	−852.000	−0.519897	−0.259949	−	0.965622i	$-0.583706\pi$
$139$	−852.000	−0.519897	−0.259949	+	0.965622i	$0.583706\pi$
$140$	−480.000	−0.289767
$141$	0	0
$142$	−32.0000	−0.0189111
$143$	416.000	0.243270
$144$	0	0
$145$	−630.000	−0.360818
$146$	−2024.00	−1.14731
$147$	0	0
$148$	152.000	0.0844211
$149$	114.000	0.0626795	0.0313397	−	0.999509i	$-0.490023\pi$
$149$	114.000	0.0626795	0.0313397	+	0.999509i	$0.490023\pi$
$150$	0	0
$151$	−114.000	−0.0614383	−0.0307192	−	0.999528i	$-0.509780\pi$
$151$	−114.000	−0.0614383	−0.0307192	+	0.999528i	$0.509780\pi$
$152$	−256.000	−0.136608
$153$	0	0
$154$	−1536.00	−0.803730
$155$	1250.00	0.647758
$156$	0	0
$157$	2650.00	1.34709	0.673545	−	0.739147i	$-0.264771\pi$
$157$	2650.00	1.34709	0.673545	+	0.739147i	$0.264771\pi$
$158$	−2528.00	−1.27289
$159$	0	0
$160$	160.000	0.0790569
$161$	3792.00	1.85622
$162$	0	0
$163$	502.000	0.241225	0.120612	−	0.992700i	$-0.461514\pi$
$163$	502.000	0.241225	0.120612	+	0.992700i	$0.461514\pi$
$164$	−360.000	−0.171410
$165$	0	0
$166$	120.000	0.0561073
$167$	2008.00	0.930441	0.465221	−	0.885195i	$-0.345975\pi$
$167$	2008.00	0.930441	0.465221	+	0.885195i	$0.345975\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	−780.000	−0.351902
$171$	0	0
$172$	−1712.00	−0.758946
$173$	−3072.00	−1.35006	−0.675028	−	0.737792i	$-0.735869\pi$
$173$	−3072.00	−1.35006	−0.675028	+	0.737792i	$0.735869\pi$
$174$	0	0
$175$	−600.000	−0.259176
$176$	512.000	0.219281
$177$	0	0
$178$	2020.00	0.850592
$179$	4110.00	1.71618	0.858089	−	0.513501i	$-0.171652\pi$
$179$	4110.00	1.71618	0.858089	+	0.513501i	$0.171652\pi$
$180$	0	0
$181$	−270.000	−0.110878	−0.0554391	−	0.998462i	$-0.517656\pi$
$181$	−270.000	−0.110878	−0.0554391	+	0.998462i	$0.517656\pi$
$182$	−624.000	−0.254143
$183$	0	0
$184$	−1264.00	−0.506431
$185$	190.000	0.0755085
$186$	0	0
$187$	−2496.00	−0.976073
$188$	336.000	0.130347
$189$	0	0
$190$	−320.000	−0.122185
$191$	1044.00	0.395504	0.197752	−	0.980252i	$-0.436636\pi$
$191$	1044.00	0.395504	0.197752	+	0.980252i	$0.436636\pi$
$192$	0	0
$193$	−3452.00	−1.28746	−0.643732	−	0.765251i	$-0.722615\pi$
$193$	−3452.00	−1.28746	−0.643732	+	0.765251i	$0.722615\pi$
$194$	744.000	0.275341
$195$	0	0
$196$	932.000	0.339650
$197$	−1866.00	−0.674858	−0.337429	−	0.941351i	$-0.609557\pi$
$197$	−1866.00	−0.674858	−0.337429	+	0.941351i	$0.609557\pi$
$198$	0	0
$199$	−5132.00	−1.82813	−0.914065	−	0.405568i	$-0.867074\pi$
$199$	−5132.00	−1.82813	−0.914065	+	0.405568i	$0.867074\pi$
$200$	200.000	0.0707107
$201$	0	0
$202$	−1700.00	−0.592137
$203$	3024.00	1.04553
$204$	0	0
$205$	−450.000	−0.153314
$206$	−288.000	−0.0974073
$207$	0	0
$208$	208.000	0.0693375
$209$	−1024.00	−0.338907
$210$	0	0
$211$	2580.00	0.841775	0.420887	−	0.907113i	$-0.361719\pi$
$211$	2580.00	0.841775	0.420887	+	0.907113i	$0.361719\pi$
$212$	−944.000	−0.305822
$213$	0	0
$214$	−2760.00	−0.881634
$215$	−2140.00	−0.678822
$216$	0	0
$217$	−6000.00	−1.87699
$218$	−656.000	−0.203807
$219$	0	0
$220$	640.000	0.196131
$221$	−1014.00	−0.308638
$222$	0	0
$223$	1460.00	0.438425	0.219213	−	0.975677i	$-0.429651\pi$
$223$	1460.00	0.438425	0.219213	+	0.975677i	$0.429651\pi$
$224$	−768.000	−0.229081
$225$	0	0
$226$	−4268.00	−1.25621
$227$	−1780.00	−0.520453	−0.260226	−	0.965548i	$-0.583797\pi$
$227$	−1780.00	−0.520453	−0.260226	+	0.965548i	$0.583797\pi$
$228$	0	0
$229$	−376.000	−0.108501	−0.0542506	−	0.998527i	$-0.517277\pi$
$229$	−376.000	−0.108501	−0.0542506	+	0.998527i	$0.517277\pi$
$230$	−1580.00	−0.452966
$231$	0	0
$232$	−1008.00	−0.285252
$233$	−970.000	−0.272733	−0.136367	−	0.990658i	$-0.543542\pi$
$233$	−970.000	−0.272733	−0.136367	+	0.990658i	$0.543542\pi$
$234$	0	0
$235$	420.000	0.116586
$236$	752.000	0.207420
$237$	0	0
$238$	3744.00	1.01970
$239$	776.000	0.210022	0.105011	−	0.994471i	$-0.466512\pi$
$239$	776.000	0.210022	0.105011	+	0.994471i	$0.466512\pi$
$240$	0	0
$241$	−2202.00	−0.588561	−0.294281	−	0.955719i	$-0.595080\pi$
$241$	−2202.00	−0.588561	−0.294281	+	0.955719i	$0.595080\pi$
$242$	−614.000	−0.163097
$243$	0	0
$244$	−680.000	−0.178412
$245$	1165.00	0.303792
$246$	0	0
$247$	−416.000	−0.107164
$248$	2000.00	0.512097
$249$	0	0
$250$	250.000	0.0632456
$251$	−1790.00	−0.450135	−0.225067	−	0.974343i	$-0.572260\pi$
$251$	−1790.00	−0.450135	−0.225067	+	0.974343i	$0.572260\pi$
$252$	0	0
$253$	−5056.00	−1.25640
$254$	64.0000	0.0158099
$255$	0	0
$256$	256.000	0.0625000
$257$	2322.00	0.563589	0.281795	−	0.959475i	$-0.409070\pi$
$257$	2322.00	0.563589	0.281795	+	0.959475i	$0.409070\pi$
$258$	0	0
$259$	−912.000	−0.218799
$260$	260.000	0.0620174
$261$	0	0
$262$	2916.00	0.687600
$263$	5802.00	1.36033	0.680165	−	0.733059i	$-0.261908\pi$
$263$	5802.00	1.36033	0.680165	+	0.733059i	$0.261908\pi$
$264$	0	0
$265$	−1180.00	−0.273535
$266$	1536.00	0.354053
$267$	0	0
$268$	−312.000	−0.0711136
$269$	38.0000	0.00861301	0.00430651	−	0.999991i	$-0.498629\pi$
$269$	38.0000	0.00861301	0.00430651	+	0.999991i	$0.498629\pi$
$270$	0	0
$271$	−6942.00	−1.55608	−0.778038	−	0.628218i	$-0.783785\pi$
$271$	−6942.00	−1.55608	−0.778038	+	0.628218i	$0.783785\pi$
$272$	−1248.00	−0.278203
$273$	0	0
$274$	1444.00	0.318377
$275$	800.000	0.175425
$276$	0	0
$277$	−8686.00	−1.88408	−0.942042	−	0.335496i	$-0.891096\pi$
$277$	−8686.00	−1.88408	−0.942042	+	0.335496i	$0.891096\pi$
$278$	−1704.00	−0.367623
$279$	0	0
$280$	−960.000	−0.204896
$281$	7502.00	1.59264	0.796320	−	0.604876i	$-0.206777\pi$
$281$	7502.00	1.59264	0.796320	+	0.604876i	$0.206777\pi$
$282$	0	0
$283$	3416.00	0.717527	0.358763	−	0.933429i	$-0.383198\pi$
$283$	3416.00	0.717527	0.358763	+	0.933429i	$0.383198\pi$
$284$	−64.0000	−0.0133722
$285$	0	0
$286$	832.000	0.172018
$287$	2160.00	0.444254
$288$	0	0
$289$	1171.00	0.238347
$290$	−1260.00	−0.255137
$291$	0	0
$292$	−4048.00	−0.811272
$293$	4274.00	0.852183	0.426092	−	0.904680i	$-0.359890\pi$
$293$	4274.00	0.852183	0.426092	+	0.904680i	$0.359890\pi$
$294$	0	0
$295$	940.000	0.185522
$296$	304.000	0.0596947
$297$	0	0
$298$	228.000	0.0443211
$299$	−2054.00	−0.397277
$300$	0	0
$301$	10272.0	1.96700
$302$	−228.000	−0.0434435
$303$	0	0
$304$	−512.000	−0.0965961
$305$	−850.000	−0.159577
$306$	0	0
$307$	1446.00	0.268819	0.134410	−	0.990926i	$-0.457086\pi$
$307$	1446.00	0.268819	0.134410	+	0.990926i	$0.457086\pi$
$308$	−3072.00	−0.568323
$309$	0	0
$310$	2500.00	0.458034
$311$	924.000	0.168473	0.0842367	−	0.996446i	$-0.473155\pi$
$311$	924.000	0.168473	0.0842367	+	0.996446i	$0.473155\pi$
$312$	0	0
$313$	−3538.00	−0.638912	−0.319456	−	0.947601i	$-0.603500\pi$
$313$	−3538.00	−0.638912	−0.319456	+	0.947601i	$0.603500\pi$
$314$	5300.00	0.952536
$315$	0	0
$316$	−5056.00	−0.900070
$317$	5246.00	0.929479	0.464739	−	0.885448i	$-0.346148\pi$
$317$	5246.00	0.929479	0.464739	+	0.885448i	$0.346148\pi$
$318$	0	0
$319$	−4032.00	−0.707676
$320$	320.000	0.0559017
$321$	0	0
$322$	7584.00	1.31255
$323$	2496.00	0.429973
$324$	0	0
$325$	325.000	0.0554700
$326$	1004.00	0.170572
$327$	0	0
$328$	−720.000	−0.121205
$329$	−2016.00	−0.337829
$330$	0	0
$331$	7924.00	1.31584	0.657919	−	0.753089i	$-0.271437\pi$
$331$	7924.00	1.31584	0.657919	+	0.753089i	$0.271437\pi$
$332$	240.000	0.0396738
$333$	0	0
$334$	4016.00	0.657921
$335$	−390.000	−0.0636059
$336$	0	0
$337$	4862.00	0.785905	0.392953	−	0.919559i	$-0.371454\pi$
$337$	4862.00	0.785905	0.392953	+	0.919559i	$0.371454\pi$
$338$	338.000	0.0543928
$339$	0	0
$340$	−1560.00	−0.248832
$341$	8000.00	1.27045
$342$	0	0
$343$	2640.00	0.415588
$344$	−3424.00	−0.536656
$345$	0	0
$346$	−6144.00	−0.954634
$347$	7084.00	1.09593	0.547967	−	0.836500i	$-0.315402\pi$
$347$	7084.00	1.09593	0.547967	+	0.836500i	$0.315402\pi$
$348$	0	0
$349$	−8124.00	−1.24604	−0.623020	−	0.782206i	$-0.714094\pi$
$349$	−8124.00	−1.24604	−0.623020	+	0.782206i	$0.714094\pi$
$350$	−1200.00	−0.183265
$351$	0	0
$352$	1024.00	0.155055
$353$	−8790.00	−1.32534	−0.662669	−	0.748912i	$-0.730576\pi$
$353$	−8790.00	−1.32534	−0.662669	+	0.748912i	$0.730576\pi$
$354$	0	0
$355$	−80.0000	−0.0119604
$356$	4040.00	0.601459
$357$	0	0
$358$	8220.00	1.21352
$359$	−5776.00	−0.849152	−0.424576	−	0.905392i	$-0.639577\pi$
$359$	−5776.00	−0.849152	−0.424576	+	0.905392i	$0.639577\pi$
$360$	0	0
$361$	−5835.00	−0.850707
$362$	−540.000	−0.0784027
$363$	0	0
$364$	−1248.00	−0.179706
$365$	−5060.00	−0.725623
$366$	0	0
$367$	11496.0	1.63511	0.817556	−	0.575849i	$-0.195328\pi$
$367$	11496.0	1.63511	0.817556	+	0.575849i	$0.195328\pi$
$368$	−2528.00	−0.358101
$369$	0	0
$370$	380.000	0.0533926
$371$	5664.00	0.792615
$372$	0	0
$373$	7894.00	1.09581	0.547903	−	0.836542i	$-0.315426\pi$
$373$	7894.00	1.09581	0.547903	+	0.836542i	$0.315426\pi$
$374$	−4992.00	−0.690188
$375$	0	0
$376$	672.000	0.0921696
$377$	−1638.00	−0.223770
$378$	0	0
$379$	9188.00	1.24527	0.622633	−	0.782514i	$-0.286063\pi$
$379$	9188.00	1.24527	0.622633	+	0.782514i	$0.286063\pi$
$380$	−640.000	−0.0863982
$381$	0	0
$382$	2088.00	0.279663
$383$	4048.00	0.540060	0.270030	−	0.962852i	$-0.412966\pi$
$383$	4048.00	0.540060	0.270030	+	0.962852i	$0.412966\pi$
$384$	0	0
$385$	−3840.00	−0.508323
$386$	−6904.00	−0.910374
$387$	0	0
$388$	1488.00	0.194695
$389$	10906.0	1.42148	0.710741	−	0.703454i	$-0.248360\pi$
$389$	10906.0	1.42148	0.710741	+	0.703454i	$0.248360\pi$
$390$	0	0
$391$	12324.0	1.59399
$392$	1864.00	0.240169
$393$	0	0
$394$	−3732.00	−0.477197
$395$	−6320.00	−0.805047
$396$	0	0
$397$	10326.0	1.30541	0.652704	−	0.757613i	$-0.273634\pi$
$397$	10326.0	1.30541	0.652704	+	0.757613i	$0.273634\pi$
$398$	−10264.0	−1.29268
$399$	0	0
$400$	400.000	0.0500000
$401$	−10414.0	−1.29688	−0.648442	−	0.761264i	$-0.724579\pi$
$401$	−10414.0	−1.29688	−0.648442	+	0.761264i	$0.724579\pi$
$402$	0	0
$403$	3250.00	0.401722
$404$	−3400.00	−0.418704
$405$	0	0
$406$	6048.00	0.739303
$407$	1216.00	0.148096
$408$	0	0
$409$	−3694.00	−0.446593	−0.223297	−	0.974751i	$-0.571682\pi$
$409$	−3694.00	−0.446593	−0.223297	+	0.974751i	$0.571682\pi$
$410$	−900.000	−0.108409
$411$	0	0
$412$	−576.000	−0.0688774
$413$	−4512.00	−0.537581
$414$	0	0
$415$	300.000	0.0354854
$416$	416.000	0.0490290
$417$	0	0
$418$	−2048.00	−0.239643
$419$	10906.0	1.27158	0.635791	−	0.771861i	$-0.280674\pi$
$419$	10906.0	1.27158	0.635791	+	0.771861i	$0.280674\pi$
$420$	0	0
$421$	12740.0	1.47484	0.737422	−	0.675432i	$-0.236043\pi$
$421$	12740.0	1.47484	0.737422	+	0.675432i	$0.236043\pi$
$422$	5160.00	0.595225
$423$	0	0
$424$	−1888.00	−0.216249
$425$	−1950.00	−0.222562
$426$	0	0
$427$	4080.00	0.462401
$428$	−5520.00	−0.623410
$429$	0	0
$430$	−4280.00	−0.480000
$431$	14688.0	1.64152	0.820761	−	0.571272i	$-0.193550\pi$
$431$	14688.0	1.64152	0.820761	+	0.571272i	$0.193550\pi$
$432$	0	0
$433$	1102.00	0.122307	0.0611533	−	0.998128i	$-0.480522\pi$
$433$	1102.00	0.122307	0.0611533	+	0.998128i	$0.480522\pi$
$434$	−12000.0	−1.32723
$435$	0	0
$436$	−1312.00	−0.144113
$437$	5056.00	0.553458
$438$	0	0
$439$	−8356.00	−0.908451	−0.454226	−	0.890887i	$-0.650084\pi$
$439$	−8356.00	−0.908451	−0.454226	+	0.890887i	$0.650084\pi$
$440$	1280.00	0.138685
$441$	0	0
$442$	−2028.00	−0.218240
$443$	8712.00	0.934356	0.467178	−	0.884163i	$-0.345271\pi$
$443$	8712.00	0.934356	0.467178	+	0.884163i	$0.345271\pi$
$444$	0	0
$445$	5050.00	0.537962
$446$	2920.00	0.310013
$447$	0	0
$448$	−1536.00	−0.161985
$449$	−7758.00	−0.815418	−0.407709	−	0.913112i	$-0.633672\pi$
$449$	−7758.00	−0.815418	−0.407709	+	0.913112i	$0.633672\pi$
$450$	0	0
$451$	−2880.00	−0.300696
$452$	−8536.00	−0.888274
$453$	0	0
$454$	−3560.00	−0.368016
$455$	−1560.00	−0.160734
$456$	0	0
$457$	−5948.00	−0.608831	−0.304415	−	0.952539i	$-0.598461\pi$
$457$	−5948.00	−0.608831	−0.304415	+	0.952539i	$0.598461\pi$
$458$	−752.000	−0.0767219
$459$	0	0
$460$	−3160.00	−0.320295
$461$	8730.00	0.881988	0.440994	−	0.897510i	$-0.354626\pi$
$461$	8730.00	0.881988	0.440994	+	0.897510i	$0.354626\pi$
$462$	0	0
$463$	−9028.00	−0.906192	−0.453096	−	0.891462i	$-0.649680\pi$
$463$	−9028.00	−0.906192	−0.453096	+	0.891462i	$0.649680\pi$
$464$	−2016.00	−0.201704
$465$	0	0
$466$	−1940.00	−0.192851
$467$	−600.000	−0.0594533	−0.0297266	−	0.999558i	$-0.509464\pi$
$467$	−600.000	−0.0594533	−0.0297266	+	0.999558i	$0.509464\pi$
$468$	0	0
$469$	1872.00	0.184309
$470$	840.000	0.0824390
$471$	0	0
$472$	1504.00	0.146668
$473$	−13696.0	−1.33138
$474$	0	0
$475$	−800.000	−0.0772769
$476$	7488.00	0.721033
$477$	0	0
$478$	1552.00	0.148508
$479$	−10192.0	−0.972201	−0.486101	−	0.873903i	$-0.661581\pi$
$479$	−10192.0	−0.972201	−0.486101	+	0.873903i	$0.661581\pi$
$480$	0	0
$481$	494.000	0.0468284
$482$	−4404.00	−0.416176
$483$	0	0
$484$	−1228.00	−0.115327
$485$	1860.00	0.174141
$486$	0	0
$487$	8240.00	0.766715	0.383357	−	0.923600i	$-0.374768\pi$
$487$	8240.00	0.766715	0.383357	+	0.923600i	$0.374768\pi$
$488$	−1360.00	−0.126156
$489$	0	0
$490$	2330.00	0.214814
$491$	−994.000	−0.0913617	−0.0456808	−	0.998956i	$-0.514546\pi$
$491$	−994.000	−0.0913617	−0.0456808	+	0.998956i	$0.514546\pi$
$492$	0	0
$493$	9828.00	0.897831
$494$	−832.000	−0.0757762
$495$	0	0
$496$	4000.00	0.362107
$497$	384.000	0.0346575
$498$	0	0
$499$	−13384.0	−1.20070	−0.600351	−	0.799737i	$-0.704972\pi$
$499$	−13384.0	−1.20070	−0.600351	+	0.799737i	$0.704972\pi$
$500$	500.000	0.0447214
$501$	0	0
$502$	−3580.00	−0.318293
$503$	6442.00	0.571043	0.285521	−	0.958372i	$-0.407833\pi$
$503$	6442.00	0.571043	0.285521	+	0.958372i	$0.407833\pi$
$504$	0	0
$505$	−4250.00	−0.374500
$506$	−10112.0	−0.888406
$507$	0	0
$508$	128.000	0.0111793
$509$	4458.00	0.388207	0.194104	−	0.980981i	$-0.437820\pi$
$509$	4458.00	0.388207	0.194104	+	0.980981i	$0.437820\pi$
$510$	0	0
$511$	24288.0	2.10262
$512$	512.000	0.0441942
$513$	0	0
$514$	4644.00	0.398518
$515$	−720.000	−0.0616058
$516$	0	0
$517$	2688.00	0.228662
$518$	−1824.00	−0.154714
$519$	0	0
$520$	520.000	0.0438529
$521$	−2160.00	−0.181634	−0.0908170	−	0.995868i	$-0.528948\pi$
$521$	−2160.00	−0.181634	−0.0908170	+	0.995868i	$0.528948\pi$
$522$	0	0
$523$	8244.00	0.689264	0.344632	−	0.938738i	$-0.388004\pi$
$523$	8244.00	0.689264	0.344632	+	0.938738i	$0.388004\pi$
$524$	5832.00	0.486206
$525$	0	0
$526$	11604.0	0.961898
$527$	−19500.0	−1.61183
$528$	0	0
$529$	12797.0	1.05178
$530$	−2360.00	−0.193419
$531$	0	0
$532$	3072.00	0.250354
$533$	−1170.00	−0.0950813
$534$	0	0
$535$	−6900.00	−0.557594
$536$	−624.000	−0.0502849
$537$	0	0
$538$	76.0000	0.00609032
$539$	7456.00	0.595831
$540$	0	0
$541$	−2588.00	−0.205669	−0.102834	−	0.994698i	$-0.532791\pi$
$541$	−2588.00	−0.205669	−0.102834	+	0.994698i	$0.532791\pi$
$542$	−13884.0	−1.10031
$543$	0	0
$544$	−2496.00	−0.196719
$545$	−1640.00	−0.128899
$546$	0	0
$547$	−4676.00	−0.365505	−0.182753	−	0.983159i	$-0.558501\pi$
$547$	−4676.00	−0.365505	−0.182753	+	0.983159i	$0.558501\pi$
$548$	2888.00	0.225126
$549$	0	0
$550$	1600.00	0.124044
$551$	4032.00	0.311740
$552$	0	0
$553$	30336.0	2.33276
$554$	−17372.0	−1.33225
$555$	0	0
$556$	−3408.00	−0.259949
$557$	21858.0	1.66275	0.831376	−	0.555710i	$-0.187553\pi$
$557$	21858.0	1.66275	0.831376	+	0.555710i	$0.187553\pi$
$558$	0	0
$559$	−5564.00	−0.420988
$560$	−1920.00	−0.144884
$561$	0	0
$562$	15004.0	1.12617
$563$	−22128.0	−1.65645	−0.828227	−	0.560392i	$-0.810650\pi$
$563$	−22128.0	−1.65645	−0.828227	+	0.560392i	$0.810650\pi$
$564$	0	0
$565$	−10670.0	−0.794496
$566$	6832.00	0.507368
$567$	0	0
$568$	−128.000	−0.00945556
$569$	−9544.00	−0.703173	−0.351586	−	0.936155i	$-0.614358\pi$
$569$	−9544.00	−0.703173	−0.351586	+	0.936155i	$0.614358\pi$
$570$	0	0
$571$	11812.0	0.865704	0.432852	−	0.901465i	$-0.357507\pi$
$571$	11812.0	0.865704	0.432852	+	0.901465i	$0.357507\pi$
$572$	1664.00	0.121635
$573$	0	0
$574$	4320.00	0.314135
$575$	−3950.00	−0.286481
$576$	0	0
$577$	−23364.0	−1.68571	−0.842856	−	0.538139i	$-0.819128\pi$
$577$	−23364.0	−1.68571	−0.842856	+	0.538139i	$0.819128\pi$
$578$	2342.00	0.168537
$579$	0	0
$580$	−2520.00	−0.180409
$581$	−1440.00	−0.102825
$582$	0	0
$583$	−7552.00	−0.536487
$584$	−8096.00	−0.573656
$585$	0	0
$586$	8548.00	0.602585
$587$	852.000	0.0599077	0.0299538	−	0.999551i	$-0.490464\pi$
$587$	852.000	0.0599077	0.0299538	+	0.999551i	$0.490464\pi$
$588$	0	0
$589$	−8000.00	−0.559651
$590$	1880.00	0.131184
$591$	0	0
$592$	608.000	0.0422106
$593$	−9466.00	−0.655518	−0.327759	−	0.944761i	$-0.606293\pi$
$593$	−9466.00	−0.655518	−0.327759	+	0.944761i	$0.606293\pi$
$594$	0	0
$595$	9360.00	0.644912
$596$	456.000	0.0313397
$597$	0	0
$598$	−4108.00	−0.280917
$599$	14088.0	0.960968	0.480484	−	0.877003i	$-0.340461\pi$
$599$	14088.0	0.960968	0.480484	+	0.877003i	$0.340461\pi$
$600$	0	0
$601$	1398.00	0.0948845	0.0474423	−	0.998874i	$-0.484893\pi$
$601$	1398.00	0.0948845	0.0474423	+	0.998874i	$0.484893\pi$
$602$	20544.0	1.39088
$603$	0	0
$604$	−456.000	−0.0307192
$605$	−1535.00	−0.103151
$606$	0	0
$607$	−22976.0	−1.53635	−0.768177	−	0.640237i	$-0.778836\pi$
$607$	−22976.0	−1.53635	−0.768177	+	0.640237i	$0.778836\pi$
$608$	−1024.00	−0.0683038
$609$	0	0
$610$	−1700.00	−0.112838
$611$	1092.00	0.0723038
$612$	0	0
$613$	25766.0	1.69768	0.848841	−	0.528648i	$-0.177301\pi$
$613$	25766.0	1.69768	0.848841	+	0.528648i	$0.177301\pi$
$614$	2892.00	0.190084
$615$	0	0
$616$	−6144.00	−0.401865
$617$	26790.0	1.74801	0.874007	−	0.485913i	$-0.161513\pi$
$617$	26790.0	1.74801	0.874007	+	0.485913i	$0.161513\pi$
$618$	0	0
$619$	23852.0	1.54878	0.774388	−	0.632711i	$-0.218058\pi$
$619$	23852.0	1.54878	0.774388	+	0.632711i	$0.218058\pi$
$620$	5000.00	0.323879
$621$	0	0
$622$	1848.00	0.119129
$623$	−24240.0	−1.55884
$624$	0	0
$625$	625.000	0.0400000
$626$	−7076.00	−0.451779
$627$	0	0
$628$	10600.0	0.673545
$629$	−2964.00	−0.187889
$630$	0	0
$631$	−22390.0	−1.41257	−0.706285	−	0.707927i	$-0.749630\pi$
$631$	−22390.0	−1.41257	−0.706285	+	0.707927i	$0.749630\pi$
$632$	−10112.0	−0.636446
$633$	0	0
$634$	10492.0	0.657241
$635$	160.000	0.00999907
$636$	0	0
$637$	3029.00	0.188404
$638$	−8064.00	−0.500403
$639$	0	0
$640$	640.000	0.0395285
$641$	15504.0	0.955337	0.477669	−	0.878540i	$-0.341482\pi$
$641$	15504.0	0.955337	0.477669	+	0.878540i	$0.341482\pi$
$642$	0	0
$643$	−7454.00	−0.457165	−0.228582	−	0.973525i	$-0.573409\pi$
$643$	−7454.00	−0.457165	−0.228582	+	0.973525i	$0.573409\pi$
$644$	15168.0	0.928110
$645$	0	0
$646$	4992.00	0.304037
$647$	19794.0	1.20275	0.601377	−	0.798965i	$-0.294619\pi$
$647$	19794.0	1.20275	0.601377	+	0.798965i	$0.294619\pi$
$648$	0	0
$649$	6016.00	0.363865
$650$	650.000	0.0392232
$651$	0	0
$652$	2008.00	0.120612
$653$	−23516.0	−1.40927	−0.704634	−	0.709571i	$-0.748889\pi$
$653$	−23516.0	−1.40927	−0.704634	+	0.709571i	$0.748889\pi$
$654$	0	0
$655$	7290.00	0.434876
$656$	−1440.00	−0.0857051
$657$	0	0
$658$	−4032.00	−0.238881
$659$	−13978.0	−0.826260	−0.413130	−	0.910672i	$-0.635565\pi$
$659$	−13978.0	−0.826260	−0.413130	+	0.910672i	$0.635565\pi$
$660$	0	0
$661$	9416.00	0.554070	0.277035	−	0.960860i	$-0.410648\pi$
$661$	9416.00	0.554070	0.277035	+	0.960860i	$0.410648\pi$
$662$	15848.0	0.930438
$663$	0	0
$664$	480.000	0.0280536
$665$	3840.00	0.223923
$666$	0	0
$667$	19908.0	1.15568
$668$	8032.00	0.465221
$669$	0	0
$670$	−780.000	−0.0449762
$671$	−5440.00	−0.312979
$672$	0	0
$673$	21226.0	1.21575	0.607877	−	0.794031i	$-0.292021\pi$
$673$	21226.0	1.21575	0.607877	+	0.794031i	$0.292021\pi$
$674$	9724.00	0.555719
$675$	0	0
$676$	676.000	0.0384615
$677$	5576.00	0.316548	0.158274	−	0.987395i	$-0.449407\pi$
$677$	5576.00	0.316548	0.158274	+	0.987395i	$0.449407\pi$
$678$	0	0
$679$	−8928.00	−0.504603
$680$	−3120.00	−0.175951
$681$	0	0
$682$	16000.0	0.898346
$683$	−32164.0	−1.80193	−0.900967	−	0.433887i	$-0.857142\pi$
$683$	−32164.0	−1.80193	−0.900967	+	0.433887i	$0.857142\pi$
$684$	0	0
$685$	3610.00	0.201359
$686$	5280.00	0.293865
$687$	0	0
$688$	−6848.00	−0.379473
$689$	−3068.00	−0.169639
$690$	0	0
$691$	−9132.00	−0.502746	−0.251373	−	0.967890i	$-0.580882\pi$
$691$	−9132.00	−0.502746	−0.251373	+	0.967890i	$0.580882\pi$
$692$	−12288.0	−0.675028
$693$	0	0
$694$	14168.0	0.774942
$695$	−4260.00	−0.232505
$696$	0	0
$697$	7020.00	0.381494
$698$	−16248.0	−0.881083
$699$	0	0
$700$	−2400.00	−0.129588
$701$	14762.0	0.795368	0.397684	−	0.917522i	$-0.369814\pi$
$701$	14762.0	0.795368	0.397684	+	0.917522i	$0.369814\pi$
$702$	0	0
$703$	−1216.00	−0.0652380
$704$	2048.00	0.109640
$705$	0	0
$706$	−17580.0	−0.937156
$707$	20400.0	1.08518
$708$	0	0
$709$	6712.00	0.355535	0.177768	−	0.984072i	$-0.443112\pi$
$709$	6712.00	0.355535	0.177768	+	0.984072i	$0.443112\pi$
$710$	−160.000	−0.00845731
$711$	0	0
$712$	8080.00	0.425296
$713$	−39500.0	−2.07474
$714$	0	0
$715$	2080.00	0.108794
$716$	16440.0	0.858089
$717$	0	0
$718$	−11552.0	−0.600441
$719$	−3144.00	−0.163076	−0.0815378	−	0.996670i	$-0.525983\pi$
$719$	−3144.00	−0.163076	−0.0815378	+	0.996670i	$0.525983\pi$
$720$	0	0
$721$	3456.00	0.178513
$722$	−11670.0	−0.601541
$723$	0	0
$724$	−1080.00	−0.0554391
$725$	−3150.00	−0.161363
$726$	0	0
$727$	10736.0	0.547698	0.273849	−	0.961773i	$-0.411703\pi$
$727$	10736.0	0.547698	0.273849	+	0.961773i	$0.411703\pi$
$728$	−2496.00	−0.127071
$729$	0	0
$730$	−10120.0	−0.513093
$731$	33384.0	1.68913
$732$	0	0
$733$	−24806.0	−1.24997	−0.624987	−	0.780635i	$-0.714896\pi$
$733$	−24806.0	−1.24997	−0.624987	+	0.780635i	$0.714896\pi$
$734$	22992.0	1.15620
$735$	0	0
$736$	−5056.00	−0.253216
$737$	−2496.00	−0.124751
$738$	0	0
$739$	−17308.0	−0.861549	−0.430775	−	0.902459i	$-0.641760\pi$
$739$	−17308.0	−0.861549	−0.430775	+	0.902459i	$0.641760\pi$
$740$	760.000	0.0377543
$741$	0	0
$742$	11328.0	0.560464
$743$	8940.00	0.441422	0.220711	−	0.975339i	$-0.429162\pi$
$743$	8940.00	0.441422	0.220711	+	0.975339i	$0.429162\pi$
$744$	0	0
$745$	570.000	0.0280311
$746$	15788.0	0.774852
$747$	0	0
$748$	−9984.00	−0.488036
$749$	33120.0	1.61573
$750$	0	0
$751$	−37036.0	−1.79955	−0.899776	−	0.436353i	$-0.856270\pi$
$751$	−37036.0	−1.79955	−0.899776	+	0.436353i	$0.856270\pi$
$752$	1344.00	0.0651737
$753$	0	0
$754$	−3276.00	−0.158229
$755$	−570.000	−0.0274761
$756$	0	0
$757$	−15726.0	−0.755048	−0.377524	−	0.926000i	$-0.623224\pi$
$757$	−15726.0	−0.755048	−0.377524	+	0.926000i	$0.623224\pi$
$758$	18376.0	0.880536
$759$	0	0
$760$	−1280.00	−0.0610927
$761$	−37626.0	−1.79230	−0.896151	−	0.443750i	$-0.853648\pi$
$761$	−37626.0	−1.79230	−0.896151	+	0.443750i	$0.853648\pi$
$762$	0	0
$763$	7872.00	0.373507
$764$	4176.00	0.197752
$765$	0	0
$766$	8096.00	0.381880
$767$	2444.00	0.115056
$768$	0	0
$769$	7234.00	0.339226	0.169613	−	0.985511i	$-0.445748\pi$
$769$	7234.00	0.339226	0.169613	+	0.985511i	$0.445748\pi$
$770$	−7680.00	−0.359439
$771$	0	0
$772$	−13808.0	−0.643732
$773$	2938.00	0.136704	0.0683522	−	0.997661i	$-0.478226\pi$
$773$	2938.00	0.136704	0.0683522	+	0.997661i	$0.478226\pi$
$774$	0	0
$775$	6250.00	0.289686
$776$	2976.00	0.137670
$777$	0	0
$778$	21812.0	1.00514
$779$	2880.00	0.132460
$780$	0	0
$781$	−512.000	−0.0234581
$782$	24648.0	1.12712
$783$	0	0
$784$	3728.00	0.169825
$785$	13250.0	0.602437
$786$	0	0
$787$	−10550.0	−0.477849	−0.238924	−	0.971038i	$-0.576795\pi$
$787$	−10550.0	−0.477849	−0.238924	+	0.971038i	$0.576795\pi$
$788$	−7464.00	−0.337429
$789$	0	0
$790$	−12640.0	−0.569254
$791$	51216.0	2.30219
$792$	0	0
$793$	−2210.00	−0.0989652
$794$	20652.0	0.923063
$795$	0	0
$796$	−20528.0	−0.914065
$797$	−35956.0	−1.59803	−0.799013	−	0.601314i	$-0.794644\pi$
$797$	−35956.0	−1.59803	−0.799013	+	0.601314i	$0.794644\pi$
$798$	0	0
$799$	−6552.00	−0.290104
$800$	800.000	0.0353553
$801$	0	0
$802$	−20828.0	−0.917035
$803$	−32384.0	−1.42317
$804$	0	0
$805$	18960.0	0.830127
$806$	6500.00	0.284060
$807$	0	0
$808$	−6800.00	−0.296068
$809$	29004.0	1.26048	0.630239	−	0.776401i	$-0.282957\pi$
$809$	29004.0	1.26048	0.630239	+	0.776401i	$0.282957\pi$
$810$	0	0
$811$	4392.00	0.190165	0.0950826	−	0.995469i	$-0.469688\pi$
$811$	4392.00	0.190165	0.0950826	+	0.995469i	$0.469688\pi$
$812$	12096.0	0.522766
$813$	0	0
$814$	2432.00	0.104719
$815$	2510.00	0.107879
$816$	0	0
$817$	13696.0	0.586490
$818$	−7388.00	−0.315789
$819$	0	0
$820$	−1800.00	−0.0766570
$821$	13842.0	0.588416	0.294208	−	0.955741i	$-0.404944\pi$
$821$	13842.0	0.588416	0.294208	+	0.955741i	$0.404944\pi$
$822$	0	0
$823$	42328.0	1.79278	0.896392	−	0.443262i	$-0.146179\pi$
$823$	42328.0	1.79278	0.896392	+	0.443262i	$0.146179\pi$
$824$	−1152.00	−0.0487037
$825$	0	0
$826$	−9024.00	−0.380127
$827$	31620.0	1.32955	0.664773	−	0.747045i	$-0.268528\pi$
$827$	31620.0	1.32955	0.664773	+	0.747045i	$0.268528\pi$
$828$	0	0
$829$	−30866.0	−1.29315	−0.646574	−	0.762851i	$-0.723799\pi$
$829$	−30866.0	−1.29315	−0.646574	+	0.762851i	$0.723799\pi$
$830$	600.000	0.0250919
$831$	0	0
$832$	832.000	0.0346688
$833$	−18174.0	−0.755933
$834$	0	0
$835$	10040.0	0.416106
$836$	−4096.00	−0.169453
$837$	0	0
$838$	21812.0	0.899144
$839$	25800.0	1.06164	0.530819	−	0.847485i	$-0.321884\pi$
$839$	25800.0	1.06164	0.530819	+	0.847485i	$0.321884\pi$
$840$	0	0
$841$	−8513.00	−0.349051
$842$	25480.0	1.04287
$843$	0	0
$844$	10320.0	0.420887
$845$	845.000	0.0344010
$846$	0	0
$847$	7368.00	0.298899
$848$	−3776.00	−0.152911
$849$	0	0
$850$	−3900.00	−0.157375
$851$	−6004.00	−0.241850
$852$	0	0
$853$	−35018.0	−1.40562	−0.702810	−	0.711378i	$-0.748071\pi$
$853$	−35018.0	−1.40562	−0.702810	+	0.711378i	$0.748071\pi$
$854$	8160.00	0.326967
$855$	0	0
$856$	−11040.0	−0.440817
$857$	−23262.0	−0.927205	−0.463603	−	0.886043i	$-0.653443\pi$
$857$	−23262.0	−0.927205	−0.463603	+	0.886043i	$0.653443\pi$
$858$	0	0
$859$	−38228.0	−1.51842	−0.759210	−	0.650846i	$-0.774414\pi$
$859$	−38228.0	−1.51842	−0.759210	+	0.650846i	$0.774414\pi$
$860$	−8560.00	−0.339411
$861$	0	0
$862$	29376.0	1.16073
$863$	−7920.00	−0.312399	−0.156199	−	0.987726i	$-0.549924\pi$
$863$	−7920.00	−0.312399	−0.156199	+	0.987726i	$0.549924\pi$
$864$	0	0
$865$	−15360.0	−0.603764
$866$	2204.00	0.0864838
$867$	0	0
$868$	−24000.0	−0.938494
$869$	−40448.0	−1.57895
$870$	0	0
$871$	−1014.00	−0.0394467
$872$	−2624.00	−0.101904
$873$	0	0
$874$	10112.0	0.391354
$875$	−3000.00	−0.115907
$876$	0	0
$877$	−30190.0	−1.16242	−0.581211	−	0.813753i	$-0.697421\pi$
$877$	−30190.0	−1.16242	−0.581211	+	0.813753i	$0.697421\pi$
$878$	−16712.0	−0.642372
$879$	0	0
$880$	2560.00	0.0980654
$881$	−6896.00	−0.263714	−0.131857	−	0.991269i	$-0.542094\pi$
$881$	−6896.00	−0.263714	−0.131857	+	0.991269i	$0.542094\pi$
$882$	0	0
$883$	−41316.0	−1.57463	−0.787313	−	0.616554i	$-0.788528\pi$
$883$	−41316.0	−1.57463	−0.787313	+	0.616554i	$0.788528\pi$
$884$	−4056.00	−0.154319
$885$	0	0
$886$	17424.0	0.660689
$887$	−7022.00	−0.265812	−0.132906	−	0.991129i	$-0.542431\pi$
$887$	−7022.00	−0.265812	−0.132906	+	0.991129i	$0.542431\pi$
$888$	0	0
$889$	−768.000	−0.0289740
$890$	10100.0	0.380396
$891$	0	0
$892$	5840.00	0.219213
$893$	−2688.00	−0.100728
$894$	0	0
$895$	20550.0	0.767498
$896$	−3072.00	−0.114541
$897$	0	0
$898$	−15516.0	−0.576588
$899$	−31500.0	−1.16861
$900$	0	0
$901$	18408.0	0.680643
$902$	−5760.00	−0.212624
$903$	0	0
$904$	−17072.0	−0.628104
$905$	−1350.00	−0.0495862
$906$	0	0
$907$	−3120.00	−0.114220	−0.0571102	−	0.998368i	$-0.518189\pi$
$907$	−3120.00	−0.114220	−0.0571102	+	0.998368i	$0.518189\pi$
$908$	−7120.00	−0.260226
$909$	0	0
$910$	−3120.00	−0.113656
$911$	22800.0	0.829196	0.414598	−	0.910005i	$-0.363922\pi$
$911$	22800.0	0.829196	0.414598	+	0.910005i	$0.363922\pi$
$912$	0	0
$913$	1920.00	0.0695977
$914$	−11896.0	−0.430508
$915$	0	0
$916$	−1504.00	−0.0542506
$917$	−34992.0	−1.26013
$918$	0	0
$919$	14740.0	0.529083	0.264542	−	0.964374i	$-0.414779\pi$
$919$	14740.0	0.529083	0.264542	+	0.964374i	$0.414779\pi$
$920$	−6320.00	−0.226483
$921$	0	0
$922$	17460.0	0.623660
$923$	−208.000	−0.00741756
$924$	0	0
$925$	950.000	0.0337684
$926$	−18056.0	−0.640774
$927$	0	0
$928$	−4032.00	−0.142626
$929$	−25306.0	−0.893717	−0.446858	−	0.894605i	$-0.647457\pi$
$929$	−25306.0	−0.893717	−0.446858	+	0.894605i	$0.647457\pi$
$930$	0	0
$931$	−7456.00	−0.262471
$932$	−3880.00	−0.136367
$933$	0	0
$934$	−1200.00	−0.0420398
$935$	−12480.0	−0.436513
$936$	0	0
$937$	−46414.0	−1.61823	−0.809114	−	0.587652i	$-0.800052\pi$
$937$	−46414.0	−1.61823	−0.809114	+	0.587652i	$0.800052\pi$
$938$	3744.00	0.130326
$939$	0	0
$940$	1680.00	0.0582931
$941$	−13122.0	−0.454586	−0.227293	−	0.973826i	$-0.572987\pi$
$941$	−13122.0	−0.454586	−0.227293	+	0.973826i	$0.572987\pi$
$942$	0	0
$943$	14220.0	0.491057
$944$	3008.00	0.103710
$945$	0	0
$946$	−27392.0	−0.941428
$947$	47564.0	1.63212	0.816062	−	0.577964i	$-0.196152\pi$
$947$	47564.0	1.63212	0.816062	+	0.577964i	$0.196152\pi$
$948$	0	0
$949$	−13156.0	−0.450012
$950$	−1600.00	−0.0546430
$951$	0	0
$952$	14976.0	0.509848
$953$	−37594.0	−1.27785	−0.638924	−	0.769270i	$-0.720620\pi$
$953$	−37594.0	−1.27785	−0.638924	+	0.769270i	$0.720620\pi$
$954$	0	0
$955$	5220.00	0.176875
$956$	3104.00	0.105011
$957$	0	0
$958$	−20384.0	−0.687450
$959$	−17328.0	−0.583473
$960$	0	0
$961$	32709.0	1.09795
$962$	988.000	0.0331127
$963$	0	0
$964$	−8808.00	−0.294281
$965$	−17260.0	−0.575771
$966$	0	0
$967$	−21176.0	−0.704213	−0.352107	−	0.935960i	$-0.614535\pi$
$967$	−21176.0	−0.704213	−0.352107	+	0.935960i	$0.614535\pi$
$968$	−2456.00	−0.0815484
$969$	0	0
$970$	3720.00	0.123136
$971$	54954.0	1.81623	0.908114	−	0.418723i	$-0.137522\pi$
$971$	54954.0	1.81623	0.908114	+	0.418723i	$0.137522\pi$
$972$	0	0
$973$	20448.0	0.673723
$974$	16480.0	0.542149
$975$	0	0
$976$	−2720.00	−0.0892060
$977$	−12090.0	−0.395899	−0.197950	−	0.980212i	$-0.563428\pi$
$977$	−12090.0	−0.395899	−0.197950	+	0.980212i	$0.563428\pi$
$978$	0	0
$979$	32320.0	1.05511
$980$	4660.00	0.151896
$981$	0	0
$982$	−1988.00	−0.0646025
$983$	31972.0	1.03738	0.518692	−	0.854961i	$-0.326419\pi$
$983$	31972.0	1.03738	0.518692	+	0.854961i	$0.326419\pi$
$984$	0	0
$985$	−9330.00	−0.301806
$986$	19656.0	0.634863
$987$	0	0
$988$	−1664.00	−0.0535819
$989$	67624.0	2.17423
$990$	0	0
$991$	7292.00	0.233742	0.116871	−	0.993147i	$-0.462714\pi$
$991$	7292.00	0.233742	0.116871	+	0.993147i	$0.462714\pi$
$992$	8000.00	0.256049
$993$	0	0
$994$	768.000	0.0245065
$995$	−25660.0	−0.817565
$996$	0	0
$997$	−31282.0	−0.993692	−0.496846	−	0.867839i	$-0.665509\pi$
$997$	−31282.0	−0.993692	−0.496846	+	0.867839i	$0.665509\pi$
$998$	−26768.0	−0.849024
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1170.4.a.l.1.1	yes	1
3.2	odd	2		1170.4.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1170.4.a.a.1.1	✓	1	3.2	odd	2
1170.4.a.l.1.1	yes	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1170.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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