LMFDB - Embedded newform 361.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [361,4,Mod(1,361)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(361, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("361.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 361 = 19^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	361.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:	$21.2996895121$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 19)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	361.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{2} +5.00000 q^{3} +1.00000 q^{4} -12.0000 q^{5} +15.0000 q^{6} +11.0000 q^{7} -21.0000 q^{8} -2.00000 q^{9} +O(q^{10})$

\(q+3.00000 q^{2} +5.00000 q^{3} +1.00000 q^{4} -12.0000 q^{5} +15.0000 q^{6} +11.0000 q^{7} -21.0000 q^{8} -2.00000 q^{9} -36.0000 q^{10} -54.0000 q^{11} +5.00000 q^{12} -11.0000 q^{13} +33.0000 q^{14} -60.0000 q^{15} -71.0000 q^{16} -93.0000 q^{17} -6.00000 q^{18} -12.0000 q^{20} +55.0000 q^{21} -162.000 q^{22} +183.000 q^{23} -105.000 q^{24} +19.0000 q^{25} -33.0000 q^{26} -145.000 q^{27} +11.0000 q^{28} +249.000 q^{29} -180.000 q^{30} -56.0000 q^{31} -45.0000 q^{32} -270.000 q^{33} -279.000 q^{34} -132.000 q^{35} -2.00000 q^{36} +250.000 q^{37} -55.0000 q^{39} +252.000 q^{40} -240.000 q^{41} +165.000 q^{42} -196.000 q^{43} -54.0000 q^{44} +24.0000 q^{45} +549.000 q^{46} -168.000 q^{47} -355.000 q^{48} -222.000 q^{49} +57.0000 q^{50} -465.000 q^{51} -11.0000 q^{52} -435.000 q^{53} -435.000 q^{54} +648.000 q^{55} -231.000 q^{56} +747.000 q^{58} -195.000 q^{59} -60.0000 q^{60} -358.000 q^{61} -168.000 q^{62} -22.0000 q^{63} +433.000 q^{64} +132.000 q^{65} -810.000 q^{66} +961.000 q^{67} -93.0000 q^{68} +915.000 q^{69} -396.000 q^{70} +246.000 q^{71} +42.0000 q^{72} +353.000 q^{73} +750.000 q^{74} +95.0000 q^{75} -594.000 q^{77} -165.000 q^{78} +34.0000 q^{79} +852.000 q^{80} -671.000 q^{81} -720.000 q^{82} +234.000 q^{83} +55.0000 q^{84} +1116.00 q^{85} -588.000 q^{86} +1245.00 q^{87} +1134.00 q^{88} +168.000 q^{89} +72.0000 q^{90} -121.000 q^{91} +183.000 q^{92} -280.000 q^{93} -504.000 q^{94} -225.000 q^{96} -758.000 q^{97} -666.000 q^{98} +108.000 q^{99} +O(q^{100})\)

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 $

$2$	3.00000	1.06066	0.530330	−	0.847791i	$-0.322068\pi$
$2$	3.00000	1.06066	0.530330	+	0.847791i	$0.322068\pi$
$3$	5.00000	0.962250	0.481125	−	0.876652i	$-0.340228\pi$
$3$	5.00000	0.962250	0.481125	+	0.876652i	$0.340228\pi$
$4$	1.00000	0.125000
$5$	−12.0000	−1.07331	−0.536656	−	0.843801i	$-0.680313\pi$
$5$	−12.0000	−1.07331	−0.536656	+	0.843801i	$0.680313\pi$
$6$	15.0000	1.02062
$7$	11.0000	0.593944	0.296972	−	0.954886i	$-0.404023\pi$
$7$	11.0000	0.593944	0.296972	+	0.954886i	$0.404023\pi$
$8$	−21.0000	−0.928078
$9$	−2.00000	−0.0740741
$10$	−36.0000	−1.13842
$11$	−54.0000	−1.48015	−0.740073	−	0.672526i	$-0.765209\pi$
$11$	−54.0000	−1.48015	−0.740073	+	0.672526i	$0.765209\pi$
$12$	5.00000	0.120281
$13$	−11.0000	−0.234681	−0.117340	−	0.993092i	$-0.537437\pi$
$13$	−11.0000	−0.234681	−0.117340	+	0.993092i	$0.537437\pi$
$14$	33.0000	0.629973
$15$	−60.0000	−1.03280
$16$	−71.0000	−1.10938
$17$	−93.0000	−1.32681	−0.663406	−	0.748259i	$-0.730890\pi$
$17$	−93.0000	−1.32681	−0.663406	+	0.748259i	$0.730890\pi$
$18$	−6.00000	−0.0785674
$19$	0	0
$19$	0	0
$20$	−12.0000	−0.134164
$21$	55.0000	0.571523
$22$	−162.000	−1.56993
$23$	183.000	1.65905	0.829525	−	0.558470i	$-0.188611\pi$
$23$	183.000	1.65905	0.829525	+	0.558470i	$0.188611\pi$
$24$	−105.000	−0.893043
$25$	19.0000	0.152000
$26$	−33.0000	−0.248917
$27$	−145.000	−1.03353
$28$	11.0000	0.0742430
$29$	249.000	1.59442	0.797209	−	0.603703i	$-0.206309\pi$
$29$	249.000	1.59442	0.797209	+	0.603703i	$0.206309\pi$
$30$	−180.000	−1.09545
$31$	−56.0000	−0.324448	−0.162224	−	0.986754i	$-0.551867\pi$
$31$	−56.0000	−0.324448	−0.162224	+	0.986754i	$0.551867\pi$
$32$	−45.0000	−0.248592
$33$	−270.000	−1.42427
$34$	−279.000	−1.40730
$35$	−132.000	−0.637488
$36$	−2.00000	−0.00925926
$37$	250.000	1.11080	0.555402	−	0.831582i	$-0.312564\pi$
$37$	250.000	1.11080	0.555402	+	0.831582i	$0.312564\pi$
$38$	0	0
$39$	−55.0000	−0.225822
$40$	252.000	0.996117
$41$	−240.000	−0.914188	−0.457094	−	0.889418i	$-0.651110\pi$
$41$	−240.000	−0.914188	−0.457094	+	0.889418i	$0.651110\pi$
$42$	165.000	0.606192
$43$	−196.000	−0.695110	−0.347555	−	0.937660i	$-0.612988\pi$
$43$	−196.000	−0.695110	−0.347555	+	0.937660i	$0.612988\pi$
$44$	−54.0000	−0.185018
$45$	24.0000	0.0795046
$46$	549.000	1.75969
$47$	−168.000	−0.521390	−0.260695	−	0.965421i	$-0.583952\pi$
$47$	−168.000	−0.521390	−0.260695	+	0.965421i	$0.583952\pi$
$48$	−355.000	−1.06750
$49$	−222.000	−0.647230
$50$	57.0000	0.161220
$51$	−465.000	−1.27673
$52$	−11.0000	−0.0293351
$53$	−435.000	−1.12739	−0.563697	−	0.825982i	$-0.690621\pi$
$53$	−435.000	−1.12739	−0.563697	+	0.825982i	$0.690621\pi$
$54$	−435.000	−1.09622
$55$	648.000	1.58866
$56$	−231.000	−0.551226
$57$	0	0
$58$	747.000	1.69114
$59$	−195.000	−0.430285	−0.215143	−	0.976583i	$-0.569022\pi$
$59$	−195.000	−0.430285	−0.215143	+	0.976583i	$0.569022\pi$
$60$	−60.0000	−0.129099
$61$	−358.000	−0.751430	−0.375715	−	0.926735i	$-0.622603\pi$
$61$	−358.000	−0.751430	−0.375715	+	0.926735i	$0.622603\pi$
$62$	−168.000	−0.344129
$63$	−22.0000	−0.0439959
$64$	433.000	0.845703
$65$	132.000	0.251886
$66$	−810.000	−1.51067
$67$	961.000	1.75231	0.876155	−	0.482029i	$-0.160100\pi$
$67$	961.000	1.75231	0.876155	+	0.482029i	$0.160100\pi$
$68$	−93.0000	−0.165852
$69$	915.000	1.59642
$70$	−396.000	−0.676158
$71$	246.000	0.411195	0.205597	−	0.978637i	$-0.434086\pi$
$71$	246.000	0.411195	0.205597	+	0.978637i	$0.434086\pi$
$72$	42.0000	0.0687465
$73$	353.000	0.565966	0.282983	−	0.959125i	$-0.408676\pi$
$73$	353.000	0.565966	0.282983	+	0.959125i	$0.408676\pi$
$74$	750.000	1.17819
$75$	95.0000	0.146262
$76$	0	0
$77$	−594.000	−0.879124
$78$	−165.000	−0.239520
$79$	34.0000	0.0484215	0.0242108	−	0.999707i	$-0.492293\pi$
$79$	34.0000	0.0484215	0.0242108	+	0.999707i	$0.492293\pi$
$80$	852.000	1.19071
$81$	−671.000	−0.920439
$82$	−720.000	−0.969643
$83$	234.000	0.309456	0.154728	−	0.987957i	$-0.450550\pi$
$83$	234.000	0.309456	0.154728	+	0.987957i	$0.450550\pi$
$84$	55.0000	0.0714404
$85$	1116.00	1.42408
$86$	−588.000	−0.737275
$87$	1245.00	1.53423
$88$	1134.00	1.37369
$89$	168.000	0.200089	0.100045	−	0.994983i	$-0.468101\pi$
$89$	168.000	0.200089	0.100045	+	0.994983i	$0.468101\pi$
$90$	72.0000	0.0843274
$91$	−121.000	−0.139387
$92$	183.000	0.207381
$93$	−280.000	−0.312201
$94$	−504.000	−0.553017
$95$	0	0
$96$	−225.000	−0.239208
$97$	−758.000	−0.793435	−0.396718	−	0.917941i	$-0.629851\pi$
$97$	−758.000	−0.793435	−0.396718	+	0.917941i	$0.629851\pi$
$98$	−666.000	−0.686491
$99$	108.000	0.109640
$100$	19.0000	0.0190000
$101$	−726.000	−0.715245	−0.357622	−	0.933866i	$-0.616412\pi$
$101$	−726.000	−0.715245	−0.357622	+	0.933866i	$0.616412\pi$
$102$	−1395.00	−1.35417
$103$	−2.00000	−0.00191326	−0.000956630	−	1.00000i	$-0.500305\pi$
$103$	−2.00000	−0.00191326	−0.000956630		1.00000i	$0.500305\pi$
$104$	231.000	0.217802
$105$	−660.000	−0.613423
$106$	−1305.00	−1.19578
$107$	−1413.00	−1.27663	−0.638317	−	0.769773i	$-0.720369\pi$
$107$	−1413.00	−1.27663	−0.638317	+	0.769773i	$0.720369\pi$
$108$	−145.000	−0.129191
$109$	−389.000	−0.341830	−0.170915	−	0.985286i	$-0.554672\pi$
$109$	−389.000	−0.341830	−0.170915	+	0.985286i	$0.554672\pi$
$110$	1944.00	1.68503
$111$	1250.00	1.06887
$112$	−781.000	−0.658907
$113$	−342.000	−0.284714	−0.142357	−	0.989815i	$-0.545468\pi$
$113$	−342.000	−0.284714	−0.142357	+	0.989815i	$0.545468\pi$
$114$	0	0
$115$	−2196.00	−1.78068
$116$	249.000	0.199302
$117$	22.0000	0.0173838
$118$	−585.000	−0.456387
$119$	−1023.00	−0.788053
$120$	1260.00	0.958514
$121$	1585.00	1.19083
$122$	−1074.00	−0.797011
$123$	−1200.00	−0.879678
$124$	−56.0000	−0.0405560
$125$	1272.00	0.910169
$126$	−66.0000	−0.0466647
$127$	1150.00	0.803512	0.401756	−	0.915747i	$-0.368400\pi$
$127$	1150.00	0.803512	0.401756	+	0.915747i	$0.368400\pi$
$128$	1659.00	1.14560
$129$	−980.000	−0.668870
$130$	396.000	0.267165
$131$	−1452.00	−0.968411	−0.484205	−	0.874954i	$-0.660891\pi$
$131$	−1452.00	−0.968411	−0.484205	+	0.874954i	$0.660891\pi$
$132$	−270.000	−0.178034
$133$	0	0
$134$	2883.00	1.85861
$135$	1740.00	1.10930
$136$	1953.00	1.23139
$137$	−1689.00	−1.05329	−0.526646	−	0.850085i	$-0.676551\pi$
$137$	−1689.00	−1.05329	−0.526646	+	0.850085i	$0.676551\pi$
$138$	2745.00	1.69326
$139$	2144.00	1.30829	0.654143	−	0.756371i	$-0.273030\pi$
$139$	2144.00	1.30829	0.654143	+	0.756371i	$0.273030\pi$
$140$	−132.000	−0.0796860
$141$	−840.000	−0.501708
$142$	738.000	0.436138
$143$	594.000	0.347362
$144$	142.000	0.0821759
$145$	−2988.00	−1.71131
$146$	1059.00	0.600298
$147$	−1110.00	−0.622798
$148$	250.000	0.138850
$149$	−3000.00	−1.64946	−0.824730	−	0.565527i	$-0.808673\pi$
$149$	−3000.00	−1.64946	−0.824730	+	0.565527i	$0.808673\pi$
$150$	285.000	0.155134
$151$	1006.00	0.542166	0.271083	−	0.962556i	$-0.412618\pi$
$151$	1006.00	0.542166	0.271083	+	0.962556i	$0.412618\pi$
$152$	0	0
$153$	186.000	0.0982824
$154$	−1782.00	−0.932452
$155$	672.000	0.348234
$156$	−55.0000	−0.0282277
$157$	2846.00	1.44672	0.723362	−	0.690469i	$-0.242596\pi$
$157$	2846.00	1.44672	0.723362	+	0.690469i	$0.242596\pi$
$158$	102.000	0.0513588
$159$	−2175.00	−1.08483
$160$	540.000	0.266817
$161$	2013.00	0.985383
$162$	−2013.00	−0.976273
$163$	−1600.00	−0.768845	−0.384422	−	0.923157i	$-0.625599\pi$
$163$	−1600.00	−0.768845	−0.384422	+	0.923157i	$0.625599\pi$
$164$	−240.000	−0.114273
$165$	3240.00	1.52869
$166$	702.000	0.328228
$167$	2004.00	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	2004.00	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	−1155.00	−0.530418
$169$	−2076.00	−0.944925
$170$	3348.00	1.51047
$171$	0	0
$172$	−196.000	−0.0868887
$173$	462.000	0.203036	0.101518	−	0.994834i	$-0.467630\pi$
$173$	462.000	0.203036	0.101518	+	0.994834i	$0.467630\pi$
$174$	3735.00	1.62730
$175$	209.000	0.0902795
$176$	3834.00	1.64204
$177$	−975.000	−0.414042
$178$	504.000	0.212227
$179$	−720.000	−0.300644	−0.150322	−	0.988637i	$-0.548031\pi$
$179$	−720.000	−0.300644	−0.150322	+	0.988637i	$0.548031\pi$
$180$	24.0000	0.00993808
$181$	2338.00	0.960122	0.480061	−	0.877235i	$-0.340614\pi$
$181$	2338.00	0.960122	0.480061	+	0.877235i	$0.340614\pi$
$182$	−363.000	−0.147843
$183$	−1790.00	−0.723063
$184$	−3843.00	−1.53973
$185$	−3000.00	−1.19224
$186$	−840.000	−0.331139
$187$	5022.00	1.96388
$188$	−168.000	−0.0651737
$189$	−1595.00	−0.613858
$190$	0	0
$191$	2871.00	1.08763	0.543817	−	0.839204i	$-0.316978\pi$
$191$	2871.00	1.08763	0.543817	+	0.839204i	$0.316978\pi$
$192$	2165.00	0.813778
$193$	−1658.00	−0.618370	−0.309185	−	0.951002i	$-0.600056\pi$
$193$	−1658.00	−0.618370	−0.309185	+	0.951002i	$0.600056\pi$
$194$	−2274.00	−0.841565
$195$	660.000	0.242377
$196$	−222.000	−0.0809038
$197$	−4176.00	−1.51029	−0.755146	−	0.655556i	$-0.772434\pi$
$197$	−4176.00	−1.51029	−0.755146	+	0.655556i	$0.772434\pi$
$198$	324.000	0.116291
$199$	−241.000	−0.0858494	−0.0429247	−	0.999078i	$-0.513668\pi$
$199$	−241.000	−0.0858494	−0.0429247	+	0.999078i	$0.513668\pi$
$200$	−399.000	−0.141068
$201$	4805.00	1.68616
$202$	−2178.00	−0.758631
$203$	2739.00	0.946996
$204$	−465.000	−0.159591
$205$	2880.00	0.981209
$206$	−6.00000	−0.00202932
$207$	−366.000	−0.122893
$208$	781.000	0.260349
$209$	0	0
$210$	−1980.00	−0.650633
$211$	745.000	0.243071	0.121535	−	0.992587i	$-0.461218\pi$
$211$	745.000	0.243071	0.121535	+	0.992587i	$0.461218\pi$
$212$	−435.000	−0.140924
$213$	1230.00	0.395672
$214$	−4239.00	−1.35408
$215$	2352.00	0.746070
$216$	3045.00	0.959194
$217$	−616.000	−0.192704
$218$	−1167.00	−0.362565
$219$	1765.00	0.544601
$220$	648.000	0.198583
$221$	1023.00	0.311377
$222$	3750.00	1.13371
$223$	1978.00	0.593976	0.296988	−	0.954881i	$-0.404018\pi$
$223$	1978.00	0.593976	0.296988	+	0.954881i	$0.404018\pi$
$224$	−495.000	−0.147650
$225$	−38.0000	−0.0112593
$226$	−1026.00	−0.301985
$227$	−5355.00	−1.56574	−0.782872	−	0.622183i	$-0.786246\pi$
$227$	−5355.00	−1.56574	−0.782872	+	0.622183i	$0.786246\pi$
$228$	0	0
$229$	−6370.00	−1.83817	−0.919086	−	0.394057i	$-0.871071\pi$
$229$	−6370.00	−1.83817	−0.919086	+	0.394057i	$0.871071\pi$
$230$	−6588.00	−1.88870
$231$	−2970.00	−0.845938
$232$	−5229.00	−1.47974
$233$	−2838.00	−0.797955	−0.398978	−	0.916961i	$-0.630635\pi$
$233$	−2838.00	−0.797955	−0.398978	+	0.916961i	$0.630635\pi$
$234$	66.0000	0.0184383
$235$	2016.00	0.559614
$236$	−195.000	−0.0537857
$237$	170.000	0.0465936
$238$	−3069.00	−0.835856
$239$	−369.000	−0.0998687	−0.0499344	−	0.998753i	$-0.515901\pi$
$239$	−369.000	−0.0998687	−0.0499344	+	0.998753i	$0.515901\pi$
$240$	4260.00	1.14576
$241$	−6608.00	−1.76622	−0.883109	−	0.469167i	$-0.844554\pi$
$241$	−6608.00	−1.76622	−0.883109	+	0.469167i	$0.844554\pi$
$242$	4755.00	1.26307
$243$	560.000	0.147835
$244$	−358.000	−0.0939287
$245$	2664.00	0.694680
$246$	−3600.00	−0.933039
$247$	0	0
$248$	1176.00	0.301113
$249$	1170.00	0.297774
$250$	3816.00	0.965380
$251$	4674.00	1.17538	0.587690	−	0.809086i	$-0.300038\pi$
$251$	4674.00	1.17538	0.587690	+	0.809086i	$0.300038\pi$
$252$	−22.0000	−0.00549948
$253$	−9882.00	−2.45564
$254$	3450.00	0.852253
$255$	5580.00	1.37033
$256$	1513.00	0.369385
$257$	−4512.00	−1.09514	−0.547570	−	0.836760i	$-0.684447\pi$
$257$	−4512.00	−1.09514	−0.547570	+	0.836760i	$0.684447\pi$
$258$	−2940.00	−0.709443
$259$	2750.00	0.659756
$260$	132.000	0.0314857
$261$	−498.000	−0.118105
$262$	−4356.00	−1.02715
$263$	3768.00	0.883440	0.441720	−	0.897153i	$-0.354368\pi$
$263$	3768.00	0.883440	0.441720	+	0.897153i	$0.354368\pi$
$264$	5670.00	1.32183
$265$	5220.00	1.21005
$266$	0	0
$267$	840.000	0.192536
$268$	961.000	0.219039
$269$	−4758.00	−1.07844	−0.539220	−	0.842165i	$-0.681281\pi$
$269$	−4758.00	−1.07844	−0.539220	+	0.842165i	$0.681281\pi$
$270$	5220.00	1.17659
$271$	−2041.00	−0.457498	−0.228749	−	0.973485i	$-0.573463\pi$
$271$	−2041.00	−0.457498	−0.228749	+	0.973485i	$0.573463\pi$
$272$	6603.00	1.47193
$273$	−605.000	−0.134126
$274$	−5067.00	−1.11718
$275$	−1026.00	−0.224982
$276$	915.000	0.199553
$277$	1964.00	0.426012	0.213006	−	0.977051i	$-0.431675\pi$
$277$	1964.00	0.426012	0.213006	+	0.977051i	$0.431675\pi$
$278$	6432.00	1.38765
$279$	112.000	0.0240332
$280$	2772.00	0.591638
$281$	5496.00	1.16678	0.583388	−	0.812194i	$-0.301727\pi$
$281$	5496.00	1.16678	0.583388	+	0.812194i	$0.301727\pi$
$282$	−2520.00	−0.532141
$283$	3098.00	0.650731	0.325366	−	0.945588i	$-0.394513\pi$
$283$	3098.00	0.650731	0.325366	+	0.945588i	$0.394513\pi$
$284$	246.000	0.0513993
$285$	0	0
$286$	1782.00	0.368433
$287$	−2640.00	−0.542977
$288$	90.0000	0.0184142
$289$	3736.00	0.760432
$290$	−8964.00	−1.81512
$291$	−3790.00	−0.763484
$292$	353.000	0.0707458
$293$	−117.000	−0.0233284	−0.0116642	−	0.999932i	$-0.503713\pi$
$293$	−117.000	−0.0233284	−0.0116642	+	0.999932i	$0.503713\pi$
$294$	−3330.00	−0.660577
$295$	2340.00	0.461831
$296$	−5250.00	−1.03091
$297$	7830.00	1.52977
$298$	−9000.00	−1.74952
$299$	−2013.00	−0.389347
$300$	95.0000	0.0182828
$301$	−2156.00	−0.412856
$302$	3018.00	0.575054
$303$	−3630.00	−0.688244
$304$	0	0
$305$	4296.00	0.806519
$306$	558.000	0.104244
$307$	1420.00	0.263986	0.131993	−	0.991251i	$-0.457862\pi$
$307$	1420.00	0.263986	0.131993	+	0.991251i	$0.457862\pi$
$308$	−594.000	−0.109891
$309$	−10.0000	−0.00184104
$310$	2016.00	0.369358
$311$	−6561.00	−1.19627	−0.598135	−	0.801395i	$-0.704091\pi$
$311$	−6561.00	−1.19627	−0.598135	+	0.801395i	$0.704091\pi$
$312$	1155.00	0.209580
$313$	−1483.00	−0.267809	−0.133904	−	0.990994i	$-0.542751\pi$
$313$	−1483.00	−0.267809	−0.133904	+	0.990994i	$0.542751\pi$
$314$	8538.00	1.53448
$315$	264.000	0.0472213
$316$	34.0000	0.00605269
$317$	1239.00	0.219524	0.109762	−	0.993958i	$-0.464991\pi$
$317$	1239.00	0.219524	0.109762	+	0.993958i	$0.464991\pi$
$318$	−6525.00	−1.15064
$319$	−13446.0	−2.35997
$320$	−5196.00	−0.907704
$321$	−7065.00	−1.22844
$322$	6039.00	1.04516
$323$	0	0
$324$	−671.000	−0.115055
$325$	−209.000	−0.0356715
$326$	−4800.00	−0.815483
$327$	−1945.00	−0.328926
$328$	5040.00	0.848437
$329$	−1848.00	−0.309676
$330$	9720.00	1.62142
$331$	8899.00	1.47774	0.738872	−	0.673846i	$-0.235359\pi$
$331$	8899.00	1.47774	0.738872	+	0.673846i	$0.235359\pi$
$332$	234.000	0.0386820
$333$	−500.000	−0.0822818
$334$	6012.00	0.984916
$335$	−11532.0	−1.88078
$336$	−3905.00	−0.634033
$337$	−5816.00	−0.940112	−0.470056	−	0.882637i	$-0.655766\pi$
$337$	−5816.00	−0.940112	−0.470056	+	0.882637i	$0.655766\pi$
$338$	−6228.00	−1.00224
$339$	−1710.00	−0.273966
$340$	1116.00	0.178011
$341$	3024.00	0.480231
$342$	0	0
$343$	−6215.00	−0.978363
$344$	4116.00	0.645116
$345$	−10980.0	−1.71346
$346$	1386.00	0.215352
$347$	−1578.00	−0.244125	−0.122063	−	0.992522i	$-0.538951\pi$
$347$	−1578.00	−0.244125	−0.122063	+	0.992522i	$0.538951\pi$
$348$	1245.00	0.191779
$349$	1658.00	0.254300	0.127150	−	0.991883i	$-0.459417\pi$
$349$	1658.00	0.254300	0.127150	+	0.991883i	$0.459417\pi$
$350$	627.000	0.0957559
$351$	1595.00	0.242549
$352$	2430.00	0.367953
$353$	−11367.0	−1.71389	−0.856947	−	0.515405i	$-0.827641\pi$
$353$	−11367.0	−1.71389	−0.856947	+	0.515405i	$0.827641\pi$
$354$	−2925.00	−0.439158
$355$	−2952.00	−0.441341
$356$	168.000	0.0250112
$357$	−5115.00	−0.758304
$358$	−2160.00	−0.318881
$359$	2553.00	0.375326	0.187663	−	0.982233i	$-0.439909\pi$
$359$	2553.00	0.375326	0.187663	+	0.982233i	$0.439909\pi$
$360$	−504.000	−0.0737865
$361$	0	0
$362$	7014.00	1.01836
$363$	7925.00	1.14588
$364$	−121.000	−0.0174234
$365$	−4236.00	−0.607459
$366$	−5370.00	−0.766925
$367$	−196.000	−0.0278777	−0.0139389	−	0.999903i	$-0.504437\pi$
$367$	−196.000	−0.0278777	−0.0139389	+	0.999903i	$0.504437\pi$
$368$	−12993.0	−1.84051
$369$	480.000	0.0677176
$370$	−9000.00	−1.26456
$371$	−4785.00	−0.669609
$372$	−280.000	−0.0390251
$373$	−9353.00	−1.29834	−0.649169	−	0.760644i	$-0.724883\pi$
$373$	−9353.00	−1.29834	−0.649169	+	0.760644i	$0.724883\pi$
$374$	15066.0	2.08301
$375$	6360.00	0.875811
$376$	3528.00	0.483890
$377$	−2739.00	−0.374180
$378$	−4785.00	−0.651095
$379$	−3827.00	−0.518680	−0.259340	−	0.965786i	$-0.583505\pi$
$379$	−3827.00	−0.518680	−0.259340	+	0.965786i	$0.583505\pi$
$380$	0	0
$381$	5750.00	0.773180
$382$	8613.00	1.15361
$383$	−5694.00	−0.759660	−0.379830	−	0.925056i	$-0.624018\pi$
$383$	−5694.00	−0.759660	−0.379830	+	0.925056i	$0.624018\pi$
$384$	8295.00	1.10235
$385$	7128.00	0.943575
$386$	−4974.00	−0.655881
$387$	392.000	0.0514896
$388$	−758.000	−0.0991794
$389$	1290.00	0.168138	0.0840689	−	0.996460i	$-0.473208\pi$
$389$	1290.00	0.168138	0.0840689	+	0.996460i	$0.473208\pi$
$390$	1980.00	0.257080
$391$	−17019.0	−2.20125
$392$	4662.00	0.600680
$393$	−7260.00	−0.931854
$394$	−12528.0	−1.60191
$395$	−408.000	−0.0519714
$396$	108.000	0.0137051
$397$	6536.00	0.826278	0.413139	−	0.910668i	$-0.364432\pi$
$397$	6536.00	0.826278	0.413139	+	0.910668i	$0.364432\pi$
$398$	−723.000	−0.0910571
$399$	0	0
$400$	−1349.00	−0.168625
$401$	−2328.00	−0.289912	−0.144956	−	0.989438i	$-0.546304\pi$
$401$	−2328.00	−0.289912	−0.144956	+	0.989438i	$0.546304\pi$
$402$	14415.0	1.78844
$403$	616.000	0.0761418
$404$	−726.000	−0.0894056
$405$	8052.00	0.987919
$406$	8217.00	1.00444
$407$	−13500.0	−1.64415
$408$	9765.00	1.18490
$409$	6676.00	0.807107	0.403554	−	0.914956i	$-0.367775\pi$
$409$	6676.00	0.807107	0.403554	+	0.914956i	$0.367775\pi$
$410$	8640.00	1.04073
$411$	−8445.00	−1.01353
$412$	−2.00000	−0.000239158 0
$413$	−2145.00	−0.255565
$414$	−1098.00	−0.130347
$415$	−2808.00	−0.332143
$416$	495.000	0.0583398
$417$	10720.0	1.25890
$418$	0	0
$419$	−8136.00	−0.948615	−0.474307	−	0.880359i	$-0.657301\pi$
$419$	−8136.00	−0.948615	−0.474307	+	0.880359i	$0.657301\pi$
$420$	−660.000	−0.0766779
$421$	8665.00	1.00310	0.501551	−	0.865128i	$-0.332763\pi$
$421$	8665.00	1.00310	0.501551	+	0.865128i	$0.332763\pi$
$422$	2235.00	0.257815
$423$	336.000	0.0386215
$424$	9135.00	1.04631
$425$	−1767.00	−0.201676
$426$	3690.00	0.419674
$427$	−3938.00	−0.446307
$428$	−1413.00	−0.159579
$429$	2970.00	0.334249
$430$	7056.00	0.791327
$431$	−750.000	−0.0838196	−0.0419098	−	0.999121i	$-0.513344\pi$
$431$	−750.000	−0.0838196	−0.0419098	+	0.999121i	$0.513344\pi$
$432$	10295.0	1.14657
$433$	4858.00	0.539170	0.269585	−	0.962977i	$-0.413113\pi$
$433$	4858.00	0.539170	0.269585	+	0.962977i	$0.413113\pi$
$434$	−1848.00	−0.204394
$435$	−14940.0	−1.64671
$436$	−389.000	−0.0427287
$437$	0	0
$438$	5295.00	0.577637
$439$	−6500.00	−0.706670	−0.353335	−	0.935497i	$-0.614952\pi$
$439$	−6500.00	−0.706670	−0.353335	+	0.935497i	$0.614952\pi$
$440$	−13608.0	−1.47440
$441$	444.000	0.0479430
$442$	3069.00	0.330266
$443$	3486.00	0.373871	0.186936	−	0.982372i	$-0.440144\pi$
$443$	3486.00	0.373871	0.186936	+	0.982372i	$0.440144\pi$
$444$	1250.00	0.133609
$445$	−2016.00	−0.214759
$446$	5934.00	0.630007
$447$	−15000.0	−1.58719
$448$	4763.00	0.502300
$449$	15030.0	1.57975	0.789877	−	0.613265i	$-0.210144\pi$
$449$	15030.0	1.57975	0.789877	+	0.613265i	$0.210144\pi$
$450$	−114.000	−0.0119422
$451$	12960.0	1.35313
$452$	−342.000	−0.0355892
$453$	5030.00	0.521700
$454$	−16065.0	−1.66072
$455$	1452.00	0.149606
$456$	0	0
$457$	−2959.00	−0.302880	−0.151440	−	0.988466i	$-0.548391\pi$
$457$	−2959.00	−0.302880	−0.151440	+	0.988466i	$0.548391\pi$
$458$	−19110.0	−1.94968
$459$	13485.0	1.37130
$460$	−2196.00	−0.222585
$461$	−156.000	−0.0157606	−0.00788031	−	0.999969i	$-0.502508\pi$
$461$	−156.000	−0.0157606	−0.00788031	+	0.999969i	$0.502508\pi$
$462$	−8910.00	−0.897253
$463$	4484.00	0.450085	0.225042	−	0.974349i	$-0.427748\pi$
$463$	4484.00	0.450085	0.225042	+	0.974349i	$0.427748\pi$
$464$	−17679.0	−1.76881
$465$	3360.00	0.335089
$466$	−8514.00	−0.846359
$467$	8766.00	0.868613	0.434306	−	0.900765i	$-0.356994\pi$
$467$	8766.00	0.868613	0.434306	+	0.900765i	$0.356994\pi$
$468$	22.0000	0.00217297
$469$	10571.0	1.04077
$470$	6048.00	0.593561
$471$	14230.0	1.39211
$472$	4095.00	0.399338
$473$	10584.0	1.02886
$474$	510.000	0.0494200
$475$	0	0
$476$	−1023.00	−0.0985066
$477$	870.000	0.0835106
$478$	−1107.00	−0.105927
$479$	−18996.0	−1.81200	−0.906001	−	0.423275i	$-0.860881\pi$
$479$	−18996.0	−1.81200	−0.906001	+	0.423275i	$0.860881\pi$
$480$	2700.00	0.256745
$481$	−2750.00	−0.260684
$482$	−19824.0	−1.87336
$483$	10065.0	0.948185
$484$	1585.00	0.148854
$485$	9096.00	0.851604
$486$	1680.00	0.156803
$487$	7450.00	0.693207	0.346603	−	0.938012i	$-0.387335\pi$
$487$	7450.00	0.693207	0.346603	+	0.938012i	$0.387335\pi$
$488$	7518.00	0.697385
$489$	−8000.00	−0.739821
$490$	7992.00	0.736820
$491$	6180.00	0.568023	0.284012	−	0.958821i	$-0.408335\pi$
$491$	6180.00	0.568023	0.284012	+	0.958821i	$0.408335\pi$
$492$	−1200.00	−0.109960
$493$	−23157.0	−2.11549
$494$	0	0
$495$	−1296.00	−0.117679
$496$	3976.00	0.359935
$497$	2706.00	0.244227
$498$	3510.00	0.315837
$499$	2576.00	0.231097	0.115549	−	0.993302i	$-0.463137\pi$
$499$	2576.00	0.231097	0.115549	+	0.993302i	$0.463137\pi$
$500$	1272.00	0.113771
$501$	10020.0	0.893534
$502$	14022.0	1.24668
$503$	−10545.0	−0.934748	−0.467374	−	0.884060i	$-0.654800\pi$
$503$	−10545.0	−0.934748	−0.467374	+	0.884060i	$0.654800\pi$
$504$	462.000	0.0408316
$505$	8712.00	0.767681
$506$	−29646.0	−2.60460
$507$	−10380.0	−0.909254
$508$	1150.00	0.100439
$509$	14694.0	1.27957	0.639784	−	0.768555i	$-0.279024\pi$
$509$	14694.0	1.27957	0.639784	+	0.768555i	$0.279024\pi$
$510$	16740.0	1.45345
$511$	3883.00	0.336152
$512$	−8733.00	−0.753804
$513$	0	0
$514$	−13536.0	−1.16157
$515$	24.0000	0.00205353
$516$	−980.000	−0.0836087
$517$	9072.00	0.771733
$518$	8250.00	0.699776
$519$	2310.00	0.195371
$520$	−2772.00	−0.233770
$521$	−10332.0	−0.868816	−0.434408	−	0.900716i	$-0.643042\pi$
$521$	−10332.0	−0.868816	−0.434408	+	0.900716i	$0.643042\pi$
$522$	−1494.00	−0.125269
$523$	−10937.0	−0.914420	−0.457210	−	0.889359i	$-0.651151\pi$
$523$	−10937.0	−0.914420	−0.457210	+	0.889359i	$0.651151\pi$
$524$	−1452.00	−0.121051
$525$	1045.00	0.0868715
$526$	11304.0	0.937030
$527$	5208.00	0.430482
$528$	19170.0	1.58005
$529$	21322.0	1.75245
$530$	15660.0	1.28345
$531$	390.000	0.0318730
$532$	0	0
$533$	2640.00	0.214542
$534$	2520.00	0.204215
$535$	16956.0	1.37023
$536$	−20181.0	−1.62628
$537$	−3600.00	−0.289295
$538$	−14274.0	−1.14386
$539$	11988.0	0.957996
$540$	1740.00	0.138662
$541$	18578.0	1.47640	0.738198	−	0.674584i	$-0.235677\pi$
$541$	18578.0	1.47640	0.738198	+	0.674584i	$0.235677\pi$
$542$	−6123.00	−0.485250
$543$	11690.0	0.923878
$544$	4185.00	0.329835
$545$	4668.00	0.366890
$546$	−1815.00	−0.142262
$547$	−21404.0	−1.67307	−0.836535	−	0.547914i	$-0.815422\pi$
$547$	−21404.0	−1.67307	−0.836535	+	0.547914i	$0.815422\pi$
$548$	−1689.00	−0.131662
$549$	716.000	0.0556614
$550$	−3078.00	−0.238630
$551$	0	0
$552$	−19215.0	−1.48160
$553$	374.000	0.0287597
$554$	5892.00	0.451854
$555$	−15000.0	−1.14723
$556$	2144.00	0.163536
$557$	−3948.00	−0.300327	−0.150163	−	0.988661i	$-0.547980\pi$
$557$	−3948.00	−0.300327	−0.150163	+	0.988661i	$0.547980\pi$
$558$	336.000	0.0254911
$559$	2156.00	0.163129
$560$	9372.00	0.707213
$561$	25110.0	1.88974
$562$	16488.0	1.23755
$563$	−5724.00	−0.428486	−0.214243	−	0.976780i	$-0.568729\pi$
$563$	−5724.00	−0.428486	−0.214243	+	0.976780i	$0.568729\pi$
$564$	−840.000	−0.0627134
$565$	4104.00	0.305587
$566$	9294.00	0.690205
$567$	−7381.00	−0.546689
$568$	−5166.00	−0.381621
$569$	20592.0	1.51716	0.758578	−	0.651582i	$-0.225895\pi$
$569$	20592.0	1.51716	0.758578	+	0.651582i	$0.225895\pi$
$570$	0	0
$571$	20684.0	1.51593	0.757967	−	0.652293i	$-0.226193\pi$
$571$	20684.0	1.51593	0.757967	+	0.652293i	$0.226193\pi$
$572$	594.000	0.0434203
$573$	14355.0	1.04658
$574$	−7920.00	−0.575914
$575$	3477.00	0.252176
$576$	−866.000	−0.0626447
$577$	−19573.0	−1.41219	−0.706096	−	0.708116i	$-0.749545\pi$
$577$	−19573.0	−1.41219	−0.706096	+	0.708116i	$0.749545\pi$
$578$	11208.0	0.806559
$579$	−8290.00	−0.595027
$580$	−2988.00	−0.213914
$581$	2574.00	0.183800
$582$	−11370.0	−0.809797
$583$	23490.0	1.66871
$584$	−7413.00	−0.525260
$585$	−264.000	−0.0186582
$586$	−351.000	−0.0247435
$587$	13524.0	0.950929	0.475464	−	0.879735i	$-0.342280\pi$
$587$	13524.0	0.950929	0.475464	+	0.879735i	$0.342280\pi$
$588$	−1110.00	−0.0778497
$589$	0	0
$590$	7020.00	0.489845
$591$	−20880.0	−1.45328
$592$	−17750.0	−1.23230
$593$	8994.00	0.622832	0.311416	−	0.950274i	$-0.399197\pi$
$593$	8994.00	0.622832	0.311416	+	0.950274i	$0.399197\pi$
$594$	23490.0	1.62257
$595$	12276.0	0.845827
$596$	−3000.00	−0.206183
$597$	−1205.00	−0.0826087
$598$	−6039.00	−0.412965
$599$	−10128.0	−0.690850	−0.345425	−	0.938446i	$-0.612265\pi$
$599$	−10128.0	−0.690850	−0.345425	+	0.938446i	$0.612265\pi$
$600$	−1995.00	−0.135743
$601$	22696.0	1.54041	0.770207	−	0.637794i	$-0.220153\pi$
$601$	22696.0	1.54041	0.770207	+	0.637794i	$0.220153\pi$
$602$	−6468.00	−0.437900
$603$	−1922.00	−0.129801
$604$	1006.00	0.0677708
$605$	−19020.0	−1.27814
$606$	−10890.0	−0.729993
$607$	5182.00	0.346509	0.173254	−	0.984877i	$-0.444572\pi$
$607$	5182.00	0.346509	0.173254	+	0.984877i	$0.444572\pi$
$608$	0	0
$609$	13695.0	0.911247
$610$	12888.0	0.855442
$611$	1848.00	0.122360
$612$	186.000	0.0122853
$613$	10082.0	0.664287	0.332144	−	0.943229i	$-0.392228\pi$
$613$	10082.0	0.664287	0.332144	+	0.943229i	$0.392228\pi$
$614$	4260.00	0.279999
$615$	14400.0	0.944169
$616$	12474.0	0.815896
$617$	−12174.0	−0.794338	−0.397169	−	0.917745i	$-0.630007\pi$
$617$	−12174.0	−0.794338	−0.397169	+	0.917745i	$0.630007\pi$
$618$	−30.0000	−0.00195271
$619$	7490.00	0.486347	0.243173	−	0.969983i	$-0.421812\pi$
$619$	7490.00	0.486347	0.243173	+	0.969983i	$0.421812\pi$
$620$	672.000	0.0435293
$621$	−26535.0	−1.71467
$622$	−19683.0	−1.26884
$623$	1848.00	0.118842
$624$	3905.00	0.250521
$625$	−17639.0	−1.12890
$626$	−4449.00	−0.284054
$627$	0	0
$628$	2846.00	0.180840
$629$	−23250.0	−1.47383
$630$	792.000	0.0500858
$631$	11072.0	0.698525	0.349263	−	0.937025i	$-0.386432\pi$
$631$	11072.0	0.698525	0.349263	+	0.937025i	$0.386432\pi$
$632$	−714.000	−0.0449389
$633$	3725.00	0.233895
$634$	3717.00	0.232841
$635$	−13800.0	−0.862419
$636$	−2175.00	−0.135604
$637$	2442.00	0.151893
$638$	−40338.0	−2.50313
$639$	−492.000	−0.0304589
$640$	−19908.0	−1.22958
$641$	18894.0	1.16422	0.582112	−	0.813108i	$-0.302226\pi$
$641$	18894.0	1.16422	0.582112	+	0.813108i	$0.302226\pi$
$642$	−21195.0	−1.30296
$643$	−19834.0	−1.21645	−0.608224	−	0.793765i	$-0.708118\pi$
$643$	−19834.0	−1.21645	−0.608224	+	0.793765i	$0.708118\pi$
$644$	2013.00	0.123173
$645$	11760.0	0.717906
$646$	0	0
$647$	3375.00	0.205077	0.102539	−	0.994729i	$-0.467303\pi$
$647$	3375.00	0.205077	0.102539	+	0.994729i	$0.467303\pi$
$648$	14091.0	0.854239
$649$	10530.0	0.636885
$650$	−627.000	−0.0378353
$651$	−3080.00	−0.185430
$652$	−1600.00	−0.0961056
$653$	−24948.0	−1.49509	−0.747543	−	0.664214i	$-0.768766\pi$
$653$	−24948.0	−1.49509	−0.747543	+	0.664214i	$0.768766\pi$
$654$	−5835.00	−0.348879
$655$	17424.0	1.03941
$656$	17040.0	1.01418
$657$	−706.000	−0.0419234
$658$	−5544.00	−0.328461
$659$	9879.00	0.583962	0.291981	−	0.956424i	$-0.405686\pi$
$659$	9879.00	0.583962	0.291981	+	0.956424i	$0.405686\pi$
$660$	3240.00	0.191086
$661$	14155.0	0.832928	0.416464	−	0.909152i	$-0.363269\pi$
$661$	14155.0	0.832928	0.416464	+	0.909152i	$0.363269\pi$
$662$	26697.0	1.56738
$663$	5115.00	0.299623
$664$	−4914.00	−0.287199
$665$	0	0
$666$	−1500.00	−0.0872730
$667$	45567.0	2.64522
$668$	2004.00	0.116073
$669$	9890.00	0.571554
$670$	−34596.0	−1.99487
$671$	19332.0	1.11223
$672$	−2475.00	−0.142076
$673$	−8948.00	−0.512511	−0.256256	−	0.966609i	$-0.582489\pi$
$673$	−8948.00	−0.512511	−0.256256	+	0.966609i	$0.582489\pi$
$674$	−17448.0	−0.997139
$675$	−2755.00	−0.157096
$676$	−2076.00	−0.118116
$677$	11511.0	0.653477	0.326738	−	0.945115i	$-0.394050\pi$
$677$	11511.0	0.653477	0.326738	+	0.945115i	$0.394050\pi$
$678$	−5130.00	−0.290585
$679$	−8338.00	−0.471256
$680$	−23436.0	−1.32166
$681$	−26775.0	−1.50664
$682$	9072.00	0.509362
$683$	10476.0	0.586900	0.293450	−	0.955974i	$-0.405197\pi$
$683$	10476.0	0.586900	0.293450	+	0.955974i	$0.405197\pi$
$684$	0	0
$685$	20268.0	1.13051
$686$	−18645.0	−1.03771
$687$	−31850.0	−1.76878
$688$	13916.0	0.771137
$689$	4785.00	0.264578
$690$	−32940.0	−1.81740
$691$	30098.0	1.65699	0.828496	−	0.559995i	$-0.189197\pi$
$691$	30098.0	1.65699	0.828496	+	0.559995i	$0.189197\pi$
$692$	462.000	0.0253795
$693$	1188.00	0.0651203
$694$	−4734.00	−0.258934
$695$	−25728.0	−1.40420
$696$	−26145.0	−1.42388
$697$	22320.0	1.21296
$698$	4974.00	0.269726
$699$	−14190.0	−0.767833
$700$	209.000	0.0112849
$701$	−14700.0	−0.792028	−0.396014	−	0.918245i	$-0.629607\pi$
$701$	−14700.0	−0.792028	−0.396014	+	0.918245i	$0.629607\pi$
$702$	4785.00	0.257262
$703$	0	0
$704$	−23382.0	−1.25176
$705$	10080.0	0.538489
$706$	−34101.0	−1.81786
$707$	−7986.00	−0.424815
$708$	−975.000	−0.0517553
$709$	31178.0	1.65150	0.825751	−	0.564035i	$-0.190752\pi$
$709$	31178.0	1.65150	0.825751	+	0.564035i	$0.190752\pi$
$710$	−8856.00	−0.468112
$711$	−68.0000	−0.00358678
$712$	−3528.00	−0.185699
$713$	−10248.0	−0.538276
$714$	−15345.0	−0.804303
$715$	−7128.00	−0.372828
$716$	−720.000	−0.0375805
$717$	−1845.00	−0.0960987
$718$	7659.00	0.398094
$719$	−33285.0	−1.72645	−0.863227	−	0.504815i	$-0.831561\pi$
$719$	−33285.0	−1.72645	−0.863227	+	0.504815i	$0.831561\pi$
$720$	−1704.00	−0.0882005
$721$	−22.0000	−0.00113637
$722$	0	0
$723$	−33040.0	−1.69954
$724$	2338.00	0.120015
$725$	4731.00	0.242352
$726$	23775.0	1.21539
$727$	−34729.0	−1.77170	−0.885851	−	0.463970i	$-0.846425\pi$
$727$	−34729.0	−1.77170	−0.885851	+	0.463970i	$0.846425\pi$
$728$	2541.00	0.129362
$729$	20917.0	1.06269
$730$	−12708.0	−0.644307
$731$	18228.0	0.922280
$732$	−1790.00	−0.0903829
$733$	4196.00	0.211436	0.105718	−	0.994396i	$-0.466286\pi$
$733$	4196.00	0.211436	0.105718	+	0.994396i	$0.466286\pi$
$734$	−588.000	−0.0295688
$735$	13320.0	0.668457
$736$	−8235.00	−0.412427
$737$	−51894.0	−2.59368
$738$	1440.00	0.0718254
$739$	−10744.0	−0.534810	−0.267405	−	0.963584i	$-0.586166\pi$
$739$	−10744.0	−0.534810	−0.267405	+	0.963584i	$0.586166\pi$
$740$	−3000.00	−0.149030
$741$	0	0
$742$	−14355.0	−0.710227
$743$	2208.00	0.109022	0.0545112	−	0.998513i	$-0.482640\pi$
$743$	2208.00	0.109022	0.0545112	+	0.998513i	$0.482640\pi$
$744$	5880.00	0.289746
$745$	36000.0	1.77039
$746$	−28059.0	−1.37710
$747$	−468.000	−0.0229227
$748$	5022.00	0.245485
$749$	−15543.0	−0.758249
$750$	19080.0	0.928937
$751$	−13160.0	−0.639434	−0.319717	−	0.947513i	$-0.603588\pi$
$751$	−13160.0	−0.639434	−0.319717	+	0.947513i	$0.603588\pi$
$752$	11928.0	0.578417
$753$	23370.0	1.13101
$754$	−8217.00	−0.396877
$755$	−12072.0	−0.581914
$756$	−1595.00	−0.0767323
$757$	758.000	0.0363936	0.0181968	−	0.999834i	$-0.494207\pi$
$757$	758.000	0.0363936	0.0181968	+	0.999834i	$0.494207\pi$
$758$	−11481.0	−0.550143
$759$	−49410.0	−2.36294
$760$	0	0
$761$	4851.00	0.231076	0.115538	−	0.993303i	$-0.463141\pi$
$761$	4851.00	0.231076	0.115538	+	0.993303i	$0.463141\pi$
$762$	17250.0	0.820081
$763$	−4279.00	−0.203028
$764$	2871.00	0.135954
$765$	−2232.00	−0.105488
$766$	−17082.0	−0.805741
$767$	2145.00	0.100980
$768$	7565.00	0.355441
$769$	−33091.0	−1.55175	−0.775873	−	0.630890i	$-0.782690\pi$
$769$	−33091.0	−1.55175	−0.775873	+	0.630890i	$0.782690\pi$
$770$	21384.0	1.00081
$771$	−22560.0	−1.05380
$772$	−1658.00	−0.0772963
$773$	−42357.0	−1.97086	−0.985430	−	0.170079i	$-0.945598\pi$
$773$	−42357.0	−1.97086	−0.985430	+	0.170079i	$0.945598\pi$
$774$	1176.00	0.0546130
$775$	−1064.00	−0.0493161
$776$	15918.0	0.736370
$777$	13750.0	0.634850
$778$	3870.00	0.178337
$779$	0	0
$780$	660.000	0.0302972
$781$	−13284.0	−0.608629
$782$	−51057.0	−2.33478
$783$	−36105.0	−1.64788
$784$	15762.0	0.718021
$785$	−34152.0	−1.55279
$786$	−21780.0	−0.988380
$787$	39877.0	1.80618	0.903089	−	0.429454i	$-0.141294\pi$
$787$	39877.0	1.80618	0.903089	+	0.429454i	$0.141294\pi$
$788$	−4176.00	−0.188787
$789$	18840.0	0.850091
$790$	−1224.00	−0.0551240
$791$	−3762.00	−0.169104
$792$	−2268.00	−0.101755
$793$	3938.00	0.176346
$794$	19608.0	0.876400
$795$	26100.0	1.16437
$796$	−241.000	−0.0107312
$797$	30033.0	1.33478	0.667392	−	0.744706i	$-0.267410\pi$
$797$	30033.0	1.33478	0.667392	+	0.744706i	$0.267410\pi$
$798$	0	0
$799$	15624.0	0.691786
$800$	−855.000	−0.0377860
$801$	−336.000	−0.0148214
$802$	−6984.00	−0.307498
$803$	−19062.0	−0.837713
$804$	4805.00	0.210770
$805$	−24156.0	−1.05762
$806$	1848.00	0.0807606
$807$	−23790.0	−1.03773
$808$	15246.0	0.663802
$809$	585.000	0.0254234	0.0127117	−	0.999919i	$-0.495954\pi$
$809$	585.000	0.0254234	0.0127117	+	0.999919i	$0.495954\pi$
$810$	24156.0	1.04785
$811$	−28361.0	−1.22798	−0.613989	−	0.789315i	$-0.710436\pi$
$811$	−28361.0	−1.22798	−0.613989	+	0.789315i	$0.710436\pi$
$812$	2739.00	0.118374
$813$	−10205.0	−0.440228
$814$	−40500.0	−1.74389
$815$	19200.0	0.825211
$816$	33015.0	1.41637
$817$	0	0
$818$	20028.0	0.856067
$819$	242.000	0.0103250
$820$	2880.00	0.122651
$821$	25068.0	1.06563	0.532813	−	0.846233i	$-0.321135\pi$
$821$	25068.0	1.06563	0.532813	+	0.846233i	$0.321135\pi$
$822$	−25335.0	−1.07501
$823$	10901.0	0.461707	0.230854	−	0.972989i	$-0.425848\pi$
$823$	10901.0	0.461707	0.230854	+	0.972989i	$0.425848\pi$
$824$	42.0000	0.00177565
$825$	−5130.00	−0.216489
$826$	−6435.00	−0.271068
$827$	−12027.0	−0.505707	−0.252854	−	0.967505i	$-0.581369\pi$
$827$	−12027.0	−0.505707	−0.252854	+	0.967505i	$0.581369\pi$
$828$	−366.000	−0.0153616
$829$	19339.0	0.810219	0.405109	−	0.914268i	$-0.367233\pi$
$829$	19339.0	0.810219	0.405109	+	0.914268i	$0.367233\pi$
$830$	−8424.00	−0.352291
$831$	9820.00	0.409930
$832$	−4763.00	−0.198470
$833$	20646.0	0.858753
$834$	32160.0	1.33526
$835$	−24048.0	−0.996665
$836$	0	0
$837$	8120.00	0.335326
$838$	−24408.0	−1.00616
$839$	13188.0	0.542670	0.271335	−	0.962485i	$-0.412535\pi$
$839$	13188.0	0.542670	0.271335	+	0.962485i	$0.412535\pi$
$840$	13860.0	0.569304
$841$	37612.0	1.54217
$842$	25995.0	1.06395
$843$	27480.0	1.12273
$844$	745.000	0.0303838
$845$	24912.0	1.01420
$846$	1008.00	0.0409642
$847$	17435.0	0.707289
$848$	30885.0	1.25070
$849$	15490.0	0.626167
$850$	−5301.00	−0.213909
$851$	45750.0	1.84288
$852$	1230.00	0.0494590
$853$	−4678.00	−0.187775	−0.0938873	−	0.995583i	$-0.529929\pi$
$853$	−4678.00	−0.187775	−0.0938873	+	0.995583i	$0.529929\pi$
$854$	−11814.0	−0.473380
$855$	0	0
$856$	29673.0	1.18482
$857$	−15252.0	−0.607933	−0.303966	−	0.952683i	$-0.598311\pi$
$857$	−15252.0	−0.607933	−0.303966	+	0.952683i	$0.598311\pi$
$858$	8910.00	0.354525
$859$	−610.000	−0.0242293	−0.0121146	−	0.999927i	$-0.503856\pi$
$859$	−610.000	−0.0242293	−0.0121146	+	0.999927i	$0.503856\pi$
$860$	2352.00	0.0932588
$861$	−13200.0	−0.522479
$862$	−2250.00	−0.0889041
$863$	−774.000	−0.0305299	−0.0152649	−	0.999883i	$-0.504859\pi$
$863$	−774.000	−0.0305299	−0.0152649	+	0.999883i	$0.504859\pi$
$864$	6525.00	0.256927
$865$	−5544.00	−0.217921
$866$	14574.0	0.571876
$867$	18680.0	0.731726
$868$	−616.000	−0.0240880
$869$	−1836.00	−0.0716709
$870$	−44820.0	−1.74660
$871$	−10571.0	−0.411234
$872$	8169.00	0.317245
$873$	1516.00	0.0587730
$874$	0	0
$875$	13992.0	0.540590
$876$	1765.00	0.0680751
$877$	31039.0	1.19511	0.597556	−	0.801827i	$-0.296139\pi$
$877$	31039.0	1.19511	0.597556	+	0.801827i	$0.296139\pi$
$878$	−19500.0	−0.749537
$879$	−585.000	−0.0224477
$880$	−46008.0	−1.76242
$881$	33678.0	1.28790	0.643950	−	0.765067i	$-0.277294\pi$
$881$	33678.0	1.28790	0.643950	+	0.765067i	$0.277294\pi$
$882$	1332.00	0.0508512
$883$	−42982.0	−1.63812	−0.819060	−	0.573708i	$-0.805504\pi$
$883$	−42982.0	−1.63812	−0.819060	+	0.573708i	$0.805504\pi$
$884$	1023.00	0.0389222
$885$	11700.0	0.444397
$886$	10458.0	0.396550
$887$	−4494.00	−0.170117	−0.0850585	−	0.996376i	$-0.527108\pi$
$887$	−4494.00	−0.170117	−0.0850585	+	0.996376i	$0.527108\pi$
$888$	−26250.0	−0.991996
$889$	12650.0	0.477241
$890$	−6048.00	−0.227786
$891$	36234.0	1.36238
$892$	1978.00	0.0742470
$893$	0	0
$894$	−45000.0	−1.68347
$895$	8640.00	0.322685
$896$	18249.0	0.680420
$897$	−10065.0	−0.374649
$898$	45090.0	1.67558
$899$	−13944.0	−0.517306
$900$	−38.0000	−0.00140741
$901$	40455.0	1.49584
$902$	38880.0	1.43521
$903$	−10780.0	−0.397271
$904$	7182.00	0.264236
$905$	−28056.0	−1.03051
$906$	15090.0	0.553346
$907$	23839.0	0.872724	0.436362	−	0.899771i	$-0.356267\pi$
$907$	23839.0	0.872724	0.436362	+	0.899771i	$0.356267\pi$
$908$	−5355.00	−0.195718
$909$	1452.00	0.0529811
$910$	4356.00	0.158681
$911$	10332.0	0.375757	0.187878	−	0.982192i	$-0.439839\pi$
$911$	10332.0	0.375757	0.187878	+	0.982192i	$0.439839\pi$
$912$	0	0
$913$	−12636.0	−0.458040
$914$	−8877.00	−0.321253
$915$	21480.0	0.776073
$916$	−6370.00	−0.229772
$917$	−15972.0	−0.575182
$918$	40455.0	1.45448
$919$	−14371.0	−0.515838	−0.257919	−	0.966166i	$-0.583037\pi$
$919$	−14371.0	−0.515838	−0.257919	+	0.966166i	$0.583037\pi$
$920$	46116.0	1.65261
$921$	7100.00	0.254021
$922$	−468.000	−0.0167167
$923$	−2706.00	−0.0964995
$924$	−2970.00	−0.105742
$925$	4750.00	0.168842
$926$	13452.0	0.477387
$927$	4.00000	0.000141723 0
$928$	−11205.0	−0.396360
$929$	26889.0	0.949623	0.474811	−	0.880088i	$-0.342516\pi$
$929$	26889.0	0.949623	0.474811	+	0.880088i	$0.342516\pi$
$930$	10080.0	0.355415
$931$	0	0
$932$	−2838.00	−0.0997444
$933$	−32805.0	−1.15111
$934$	26298.0	0.921303
$935$	−60264.0	−2.10785
$936$	−462.000	−0.0161335
$937$	785.000	0.0273691	0.0136845	−	0.999906i	$-0.495644\pi$
$937$	785.000	0.0273691	0.0136845	+	0.999906i	$0.495644\pi$
$938$	31713.0	1.10391
$939$	−7415.00	−0.257699
$940$	2016.00	0.0699518
$941$	18141.0	0.628459	0.314229	−	0.949347i	$-0.398254\pi$
$941$	18141.0	0.628459	0.314229	+	0.949347i	$0.398254\pi$
$942$	42690.0	1.47656
$943$	−43920.0	−1.51668
$944$	13845.0	0.477348
$945$	19140.0	0.658862
$946$	31752.0	1.09128
$947$	23100.0	0.792660	0.396330	−	0.918108i	$-0.370284\pi$
$947$	23100.0	0.792660	0.396330	+	0.918108i	$0.370284\pi$
$948$	170.000	0.00582420
$949$	−3883.00	−0.132821
$950$	0	0
$951$	6195.00	0.211237
$952$	21483.0	0.731374
$953$	−45690.0	−1.55304	−0.776519	−	0.630094i	$-0.783016\pi$
$953$	−45690.0	−1.55304	−0.776519	+	0.630094i	$0.783016\pi$
$954$	2610.00	0.0885764
$955$	−34452.0	−1.16737
$956$	−369.000	−0.0124836
$957$	−67230.0	−2.27089
$958$	−56988.0	−1.92192
$959$	−18579.0	−0.625597
$960$	−25980.0	−0.873438
$961$	−26655.0	−0.894733
$962$	−8250.00	−0.276498
$963$	2826.00	0.0945655
$964$	−6608.00	−0.220777
$965$	19896.0	0.663705
$966$	30195.0	1.00570
$967$	21584.0	0.717781	0.358891	−	0.933380i	$-0.383155\pi$
$967$	21584.0	0.717781	0.358891	+	0.933380i	$0.383155\pi$
$968$	−33285.0	−1.10519
$969$	0	0
$970$	27288.0	0.903263
$971$	50556.0	1.67087	0.835437	−	0.549586i	$-0.185214\pi$
$971$	50556.0	1.67087	0.835437	+	0.549586i	$0.185214\pi$
$972$	560.000	0.0184794
$973$	23584.0	0.777049
$974$	22350.0	0.735257
$975$	−1045.00	−0.0343249
$976$	25418.0	0.833617
$977$	−8568.00	−0.280568	−0.140284	−	0.990111i	$-0.544802\pi$
$977$	−8568.00	−0.280568	−0.140284	+	0.990111i	$0.544802\pi$
$978$	−24000.0	−0.784699
$979$	−9072.00	−0.296162
$980$	2664.00	0.0868351
$981$	778.000	0.0253207
$982$	18540.0	0.602480
$983$	−29706.0	−0.963860	−0.481930	−	0.876210i	$-0.660064\pi$
$983$	−29706.0	−0.963860	−0.481930	+	0.876210i	$0.660064\pi$
$984$	25200.0	0.816409
$985$	50112.0	1.62102
$986$	−69471.0	−2.24382
$987$	−9240.00	−0.297986
$988$	0	0
$989$	−35868.0	−1.15322
$990$	−3888.00	−0.124817
$991$	−30512.0	−0.978048	−0.489024	−	0.872270i	$-0.662647\pi$
$991$	−30512.0	−0.978048	−0.489024	+	0.872270i	$0.662647\pi$
$992$	2520.00	0.0806553
$993$	44495.0	1.42196
$994$	8118.00	0.259042
$995$	2892.00	0.0921433
$996$	1170.00	0.0372218
$997$	47756.0	1.51700	0.758499	−	0.651674i	$-0.225933\pi$
$997$	47756.0	1.51700	0.758499	+	0.651674i	$0.225933\pi$
$998$	7728.00	0.245116
$999$	−36250.0	−1.14805

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	361.4.a.b.1.1		1
19.18	odd	2		19.4.a.a.1.1	✓	1
57.56	even	2		171.4.a.d.1.1		1
76.75	even	2		304.4.a.b.1.1		1
95.18	even	4		475.4.b.c.324.2		2
95.37	even	4		475.4.b.c.324.1		2
95.94	odd	2		475.4.a.e.1.1		1
133.132	even	2		931.4.a.a.1.1		1
152.37	odd	2		1216.4.a.f.1.1		1
152.75	even	2		1216.4.a.a.1.1		1
209.208	even	2		2299.4.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.4.a.a.1.1	✓	1	19.18	odd	2
171.4.a.d.1.1		1	57.56	even	2
304.4.a.b.1.1		1	76.75	even	2
361.4.a.b.1.1		1	1.1	even	1	trivial
475.4.a.e.1.1		1	95.94	odd	2
475.4.b.c.324.1		2	95.37	even	4
475.4.b.c.324.2		2	95.18	even	4
931.4.a.a.1.1		1	133.132	even	2
1216.4.a.a.1.1		1	152.75	even	2
1216.4.a.f.1.1		1	152.37	odd	2
2299.4.a.b.1.1		1	209.208	even	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:	\( N \)	\(=\)	\( 361 = 19^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	361.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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