LMFDB - Embedded newform 475.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,4,Mod(1,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, 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("475.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	475.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:	$28.0259072527$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q-5.00000 q^{2} -4.00000 q^{3} +17.0000 q^{4} +20.0000 q^{6} +32.0000 q^{7} -45.0000 q^{8} -11.0000 q^{9} -12.0000 q^{11} -68.0000 q^{12} +42.0000 q^{13} -160.000 q^{14} +89.0000 q^{16} -114.000 q^{17} +55.0000 q^{18} +19.0000 q^{19} -128.000 q^{21} +60.0000 q^{22} -160.000 q^{23} +180.000 q^{24} -210.000 q^{26} +152.000 q^{27} +544.000 q^{28} +214.000 q^{29} -144.000 q^{31} -85.0000 q^{32} +48.0000 q^{33} +570.000 q^{34} -187.000 q^{36} -94.0000 q^{37} -95.0000 q^{38} -168.000 q^{39} -6.00000 q^{41} +640.000 q^{42} +308.000 q^{43} -204.000 q^{44} +800.000 q^{46} -184.000 q^{47} -356.000 q^{48} +681.000 q^{49} +456.000 q^{51} +714.000 q^{52} +274.000 q^{53} -760.000 q^{54} -1440.00 q^{56} -76.0000 q^{57} -1070.00 q^{58} +276.000 q^{59} -826.000 q^{61} +720.000 q^{62} -352.000 q^{63} -287.000 q^{64} -240.000 q^{66} -52.0000 q^{67} -1938.00 q^{68} +640.000 q^{69} -344.000 q^{71} +495.000 q^{72} +166.000 q^{73} +470.000 q^{74} +323.000 q^{76} -384.000 q^{77} +840.000 q^{78} -688.000 q^{79} -311.000 q^{81} +30.0000 q^{82} -996.000 q^{83} -2176.00 q^{84} -1540.00 q^{86} -856.000 q^{87} +540.000 q^{88} +1578.00 q^{89} +1344.00 q^{91} -2720.00 q^{92} +576.000 q^{93} +920.000 q^{94} +340.000 q^{96} -786.000 q^{97} -3405.00 q^{98} +132.000 q^{99} +O(q^{100})\)

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 $

$2$	−5.00000	−1.76777	−0.883883	−	0.467707i	$-0.845080\pi$
$2$	−5.00000	−1.76777	−0.883883	+	0.467707i	$0.845080\pi$
$3$	−4.00000	−0.769800	−0.384900	−	0.922958i	$-0.625764\pi$
$3$	−4.00000	−0.769800	−0.384900	+	0.922958i	$0.625764\pi$
$4$	17.0000	2.12500
$5$	0	0
$5$	0	0
$6$	20.0000	1.36083
$7$	32.0000	1.72784	0.863919	−	0.503631i	$-0.168003\pi$
$7$	32.0000	1.72784	0.863919	+	0.503631i	$0.168003\pi$
$8$	−45.0000	−1.98874
$9$	−11.0000	−0.407407
$10$	0	0
$11$	−12.0000	−0.328921	−0.164461	−	0.986384i	$-0.552588\pi$
$11$	−12.0000	−0.328921	−0.164461	+	0.986384i	$0.552588\pi$
$12$	−68.0000	−1.63583
$13$	42.0000	0.896054	0.448027	−	0.894020i	$-0.352127\pi$
$13$	42.0000	0.896054	0.448027	+	0.894020i	$0.352127\pi$
$14$	−160.000	−3.05441
$15$	0	0
$16$	89.0000	1.39062
$17$	−114.000	−1.62642	−0.813208	−	0.581974i	$-0.802281\pi$
$17$	−114.000	−1.62642	−0.813208	+	0.581974i	$0.802281\pi$
$18$	55.0000	0.720201
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	−128.000	−1.33009
$22$	60.0000	0.581456
$23$	−160.000	−1.45054	−0.725268	−	0.688467i	$-0.758284\pi$
$23$	−160.000	−1.45054	−0.725268	+	0.688467i	$0.758284\pi$
$24$	180.000	1.53093
$25$	0	0
$26$	−210.000	−1.58401
$27$	152.000	1.08342
$28$	544.000	3.67165
$29$	214.000	1.37030	0.685152	−	0.728400i	$-0.259736\pi$
$29$	214.000	1.37030	0.685152	+	0.728400i	$0.259736\pi$
$30$	0	0
$31$	−144.000	−0.834296	−0.417148	−	0.908839i	$-0.636970\pi$
$31$	−144.000	−0.834296	−0.417148	+	0.908839i	$0.636970\pi$
$32$	−85.0000	−0.469563
$33$	48.0000	0.253204
$34$	570.000	2.87512
$35$	0	0
$36$	−187.000	−0.865741
$37$	−94.0000	−0.417662	−0.208831	−	0.977952i	$-0.566966\pi$
$37$	−94.0000	−0.417662	−0.208831	+	0.977952i	$0.566966\pi$
$38$	−95.0000	−0.405554
$39$	−168.000	−0.689783
$40$	0	0
$41$	−6.00000	−0.0228547	−0.0114273	−	0.999935i	$-0.503638\pi$
$41$	−6.00000	−0.0228547	−0.0114273	+	0.999935i	$0.503638\pi$
$42$	640.000	2.35129
$43$	308.000	1.09232	0.546158	−	0.837682i	$-0.316090\pi$
$43$	308.000	1.09232	0.546158	+	0.837682i	$0.316090\pi$
$44$	−204.000	−0.698958
$45$	0	0
$46$	800.000	2.56421
$47$	−184.000	−0.571046	−0.285523	−	0.958372i	$-0.592167\pi$
$47$	−184.000	−0.571046	−0.285523	+	0.958372i	$0.592167\pi$
$48$	−356.000	−1.07050
$49$	681.000	1.98542
$50$	0	0
$51$	456.000	1.25202
$52$	714.000	1.90412
$53$	274.000	0.710128	0.355064	−	0.934842i	$-0.384459\pi$
$53$	274.000	0.710128	0.355064	+	0.934842i	$0.384459\pi$
$54$	−760.000	−1.91524
$55$	0	0
$56$	−1440.00	−3.43622
$57$	−76.0000	−0.176604
$58$	−1070.00	−2.42238
$59$	276.000	0.609019	0.304510	−	0.952509i	$-0.401507\pi$
$59$	276.000	0.609019	0.304510	+	0.952509i	$0.401507\pi$
$60$	0	0
$61$	−826.000	−1.73375	−0.866873	−	0.498530i	$-0.833873\pi$
$61$	−826.000	−1.73375	−0.866873	+	0.498530i	$0.833873\pi$
$62$	720.000	1.47484
$63$	−352.000	−0.703934
$64$	−287.000	−0.560547
$65$	0	0
$66$	−240.000	−0.447605
$67$	−52.0000	−0.0948181	−0.0474090	−	0.998876i	$-0.515096\pi$
$67$	−52.0000	−0.0948181	−0.0474090	+	0.998876i	$0.515096\pi$
$68$	−1938.00	−3.45613
$69$	640.000	1.11662
$70$	0	0
$71$	−344.000	−0.575004	−0.287502	−	0.957780i	$-0.592825\pi$
$71$	−344.000	−0.575004	−0.287502	+	0.957780i	$0.592825\pi$
$72$	495.000	0.810227
$73$	166.000	0.266148	0.133074	−	0.991106i	$-0.457515\pi$
$73$	166.000	0.266148	0.133074	+	0.991106i	$0.457515\pi$
$74$	470.000	0.738330
$75$	0	0
$76$	323.000	0.487508
$77$	−384.000	−0.568323
$78$	840.000	1.21938
$79$	−688.000	−0.979823	−0.489912	−	0.871772i	$-0.662971\pi$
$79$	−688.000	−0.979823	−0.489912	+	0.871772i	$0.662971\pi$
$80$	0	0
$81$	−311.000	−0.426612
$82$	30.0000	0.0404018
$83$	−996.000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−996.000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	−2176.00	−2.82644
$85$	0	0
$86$	−1540.00	−1.93096
$87$	−856.000	−1.05486
$88$	540.000	0.654139
$89$	1578.00	1.87941	0.939706	−	0.341983i	$-0.111099\pi$
$89$	1578.00	1.87941	0.939706	+	0.341983i	$0.111099\pi$
$90$	0	0
$91$	1344.00	1.54824
$92$	−2720.00	−3.08239
$93$	576.000	0.642241
$94$	920.000	1.00948
$95$	0	0
$96$	340.000	0.361470
$97$	−786.000	−0.822744	−0.411372	−	0.911467i	$-0.634950\pi$
$97$	−786.000	−0.822744	−0.411372	+	0.911467i	$0.634950\pi$
$98$	−3405.00	−3.50976
$99$	132.000	0.134005
$100$	0	0
$101$	126.000	0.124133	0.0620667	−	0.998072i	$-0.480231\pi$
$101$	126.000	0.124133	0.0620667	+	0.998072i	$0.480231\pi$
$102$	−2280.00	−2.21327
$103$	368.000	0.352040	0.176020	−	0.984387i	$-0.443678\pi$
$103$	368.000	0.352040	0.176020	+	0.984387i	$0.443678\pi$
$104$	−1890.00	−1.78202
$105$	0	0
$106$	−1370.00	−1.25534
$107$	−204.000	−0.184312	−0.0921562	−	0.995745i	$-0.529376\pi$
$107$	−204.000	−0.184312	−0.0921562	+	0.995745i	$0.529376\pi$
$108$	2584.00	2.30227
$109$	−1434.00	−1.26011	−0.630056	−	0.776549i	$-0.716968\pi$
$109$	−1434.00	−1.26011	−0.630056	+	0.776549i	$0.716968\pi$
$110$	0	0
$111$	376.000	0.321517
$112$	2848.00	2.40277
$113$	−1218.00	−1.01398	−0.506990	−	0.861952i	$-0.669242\pi$
$113$	−1218.00	−1.01398	−0.506990	+	0.861952i	$0.669242\pi$
$114$	380.000	0.312195
$115$	0	0
$116$	3638.00	2.91190
$117$	−462.000	−0.365059
$118$	−1380.00	−1.07660
$119$	−3648.00	−2.81018
$120$	0	0
$121$	−1187.00	−0.891811
$122$	4130.00	3.06486
$123$	24.0000	0.0175936
$124$	−2448.00	−1.77288
$125$	0	0
$126$	1760.00	1.24439
$127$	−904.000	−0.631630	−0.315815	−	0.948821i	$-0.602278\pi$
$127$	−904.000	−0.631630	−0.315815	+	0.948821i	$0.602278\pi$
$128$	2115.00	1.46048
$129$	−1232.00	−0.840865
$130$	0	0
$131$	−2180.00	−1.45395	−0.726975	−	0.686664i	$-0.759074\pi$
$131$	−2180.00	−1.45395	−0.726975	+	0.686664i	$0.759074\pi$
$132$	816.000	0.538058
$133$	608.000	0.396393
$134$	260.000	0.167616
$135$	0	0
$136$	5130.00	3.23451
$137$	2566.00	1.60021	0.800103	−	0.599863i	$-0.204778\pi$
$137$	2566.00	1.60021	0.800103	+	0.599863i	$0.204778\pi$
$138$	−3200.00	−1.97393
$139$	1988.00	1.21309	0.606547	−	0.795048i	$-0.292554\pi$
$139$	1988.00	1.21309	0.606547	+	0.795048i	$0.292554\pi$
$140$	0	0
$141$	736.000	0.439591
$142$	1720.00	1.01647
$143$	−504.000	−0.294731
$144$	−979.000	−0.566551
$145$	0	0
$146$	−830.000	−0.470488
$147$	−2724.00	−1.52838
$148$	−1598.00	−0.887532
$149$	1134.00	0.623496	0.311748	−	0.950165i	$-0.399086\pi$
$149$	1134.00	0.623496	0.311748	+	0.950165i	$0.399086\pi$
$150$	0	0
$151$	−2440.00	−1.31500	−0.657498	−	0.753456i	$-0.728385\pi$
$151$	−2440.00	−1.31500	−0.657498	+	0.753456i	$0.728385\pi$
$152$	−855.000	−0.456248
$153$	1254.00	0.662614
$154$	1920.00	1.00466
$155$	0	0
$156$	−2856.00	−1.46579
$157$	−3238.00	−1.64599	−0.822995	−	0.568048i	$-0.807699\pi$
$157$	−3238.00	−1.64599	−0.822995	+	0.568048i	$0.807699\pi$
$158$	3440.00	1.73210
$159$	−1096.00	−0.546657
$160$	0	0
$161$	−5120.00	−2.50629
$162$	1555.00	0.754150
$163$	1420.00	0.682350	0.341175	−	0.940000i	$-0.389175\pi$
$163$	1420.00	0.682350	0.341175	+	0.940000i	$0.389175\pi$
$164$	−102.000	−0.0485662
$165$	0	0
$166$	4980.00	2.32845
$167$	2336.00	1.08243	0.541213	−	0.840886i	$-0.317965\pi$
$167$	2336.00	1.08243	0.541213	+	0.840886i	$0.317965\pi$
$168$	5760.00	2.64520
$169$	−433.000	−0.197087
$170$	0	0
$171$	−209.000	−0.0934657
$172$	5236.00	2.32117
$173$	−1206.00	−0.530003	−0.265001	−	0.964248i	$-0.585372\pi$
$173$	−1206.00	−0.530003	−0.265001	+	0.964248i	$0.585372\pi$
$174$	4280.00	1.86475
$175$	0	0
$176$	−1068.00	−0.457406
$177$	−1104.00	−0.468823
$178$	−7890.00	−3.32236
$179$	−1412.00	−0.589597	−0.294798	−	0.955559i	$-0.595253\pi$
$179$	−1412.00	−0.589597	−0.294798	+	0.955559i	$0.595253\pi$
$180$	0	0
$181$	3742.00	1.53669	0.768344	−	0.640037i	$-0.221081\pi$
$181$	3742.00	1.53669	0.768344	+	0.640037i	$0.221081\pi$
$182$	−6720.00	−2.73692
$183$	3304.00	1.33464
$184$	7200.00	2.88473
$185$	0	0
$186$	−2880.00	−1.13533
$187$	1368.00	0.534963
$188$	−3128.00	−1.21347
$189$	4864.00	1.87198
$190$	0	0
$191$	−1472.00	−0.557645	−0.278822	−	0.960343i	$-0.589944\pi$
$191$	−1472.00	−0.557645	−0.278822	+	0.960343i	$0.589944\pi$
$192$	1148.00	0.431509
$193$	−1458.00	−0.543778	−0.271889	−	0.962329i	$-0.587648\pi$
$193$	−1458.00	−0.543778	−0.271889	+	0.962329i	$0.587648\pi$
$194$	3930.00	1.45442
$195$	0	0
$196$	11577.0	4.21902
$197$	−2046.00	−0.739957	−0.369978	−	0.929040i	$-0.620635\pi$
$197$	−2046.00	−0.739957	−0.369978	+	0.929040i	$0.620635\pi$
$198$	−660.000	−0.236890
$199$	−1496.00	−0.532908	−0.266454	−	0.963848i	$-0.585852\pi$
$199$	−1496.00	−0.532908	−0.266454	+	0.963848i	$0.585852\pi$
$200$	0	0
$201$	208.000	0.0729910
$202$	−630.000	−0.219439
$203$	6848.00	2.36766
$204$	7752.00	2.66053
$205$	0	0
$206$	−1840.00	−0.622325
$207$	1760.00	0.590959
$208$	3738.00	1.24608
$209$	−228.000	−0.0754598
$210$	0	0
$211$	844.000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	844.000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	4658.00	1.50902
$213$	1376.00	0.442638
$214$	1020.00	0.325821
$215$	0	0
$216$	−6840.00	−2.15464
$217$	−4608.00	−1.44153
$218$	7170.00	2.22759
$219$	−664.000	−0.204881
$220$	0	0
$221$	−4788.00	−1.45736
$222$	−1880.00	−0.568366
$223$	3400.00	1.02099	0.510495	−	0.859881i	$-0.329462\pi$
$223$	3400.00	1.02099	0.510495	+	0.859881i	$0.329462\pi$
$224$	−2720.00	−0.811329
$225$	0	0
$226$	6090.00	1.79248
$227$	−1844.00	−0.539166	−0.269583	−	0.962977i	$-0.586886\pi$
$227$	−1844.00	−0.539166	−0.269583	+	0.962977i	$0.586886\pi$
$228$	−1292.00	−0.375284
$229$	−1090.00	−0.314538	−0.157269	−	0.987556i	$-0.550269\pi$
$229$	−1090.00	−0.314538	−0.157269	+	0.987556i	$0.550269\pi$
$230$	0	0
$231$	1536.00	0.437495
$232$	−9630.00	−2.72517
$233$	−2842.00	−0.799080	−0.399540	−	0.916716i	$-0.630830\pi$
$233$	−2842.00	−0.799080	−0.399540	+	0.916716i	$0.630830\pi$
$234$	2310.00	0.645339
$235$	0	0
$236$	4692.00	1.29417
$237$	2752.00	0.754268
$238$	18240.0	4.96775
$239$	−2400.00	−0.649553	−0.324776	−	0.945791i	$-0.605289\pi$
$239$	−2400.00	−0.649553	−0.324776	+	0.945791i	$0.605289\pi$
$240$	0	0
$241$	2130.00	0.569317	0.284658	−	0.958629i	$-0.408120\pi$
$241$	2130.00	0.569317	0.284658	+	0.958629i	$0.408120\pi$
$242$	5935.00	1.57651
$243$	−2860.00	−0.755017
$244$	−14042.0	−3.68421
$245$	0	0
$246$	−120.000	−0.0311013
$247$	798.000	0.205569
$248$	6480.00	1.65920
$249$	3984.00	1.01396
$250$	0	0
$251$	−2364.00	−0.594480	−0.297240	−	0.954803i	$-0.596066\pi$
$251$	−2364.00	−0.594480	−0.297240	+	0.954803i	$0.596066\pi$
$252$	−5984.00	−1.49586
$253$	1920.00	0.477112
$254$	4520.00	1.11657
$255$	0	0
$256$	−8279.00	−2.02124
$257$	−6290.00	−1.52669	−0.763345	−	0.645991i	$-0.776444\pi$
$257$	−6290.00	−1.52669	−0.763345	+	0.645991i	$0.776444\pi$
$258$	6160.00	1.48645
$259$	−3008.00	−0.721653
$260$	0	0
$261$	−2354.00	−0.558272
$262$	10900.0	2.57025
$263$	−8112.00	−1.90193	−0.950965	−	0.309300i	$-0.899905\pi$
$263$	−8112.00	−1.90193	−0.950965	+	0.309300i	$0.899905\pi$
$264$	−2160.00	−0.503556
$265$	0	0
$266$	−3040.00	−0.700731
$267$	−6312.00	−1.44677
$268$	−884.000	−0.201488
$269$	−4794.00	−1.08660	−0.543300	−	0.839539i	$-0.682825\pi$
$269$	−4794.00	−1.08660	−0.543300	+	0.839539i	$0.682825\pi$
$270$	0	0
$271$	304.000	0.0681427	0.0340714	−	0.999419i	$-0.489153\pi$
$271$	304.000	0.0681427	0.0340714	+	0.999419i	$0.489153\pi$
$272$	−10146.0	−2.26173
$273$	−5376.00	−1.19183
$274$	−12830.0	−2.82879
$275$	0	0
$276$	10880.0	2.37282
$277$	−2062.00	−0.447269	−0.223635	−	0.974673i	$-0.571792\pi$
$277$	−2062.00	−0.447269	−0.223635	+	0.974673i	$0.571792\pi$
$278$	−9940.00	−2.14447
$279$	1584.00	0.339898
$280$	0	0
$281$	−4054.00	−0.860645	−0.430323	−	0.902675i	$-0.641600\pi$
$281$	−4054.00	−0.860645	−0.430323	+	0.902675i	$0.641600\pi$
$282$	−3680.00	−0.777095
$283$	−7996.00	−1.67955	−0.839775	−	0.542934i	$-0.817313\pi$
$283$	−7996.00	−1.67955	−0.839775	+	0.542934i	$0.817313\pi$
$284$	−5848.00	−1.22188
$285$	0	0
$286$	2520.00	0.521017
$287$	−192.000	−0.0394892
$288$	935.000	0.191303
$289$	8083.00	1.64523
$290$	0	0
$291$	3144.00	0.633349
$292$	2822.00	0.565565
$293$	3906.00	0.778809	0.389404	−	0.921067i	$-0.372681\pi$
$293$	3906.00	0.778809	0.389404	+	0.921067i	$0.372681\pi$
$294$	13620.0	2.70182
$295$	0	0
$296$	4230.00	0.830621
$297$	−1824.00	−0.356361
$298$	−5670.00	−1.10220
$299$	−6720.00	−1.29976
$300$	0	0
$301$	9856.00	1.88734
$302$	12200.0	2.32461
$303$	−504.000	−0.0955579
$304$	1691.00	0.319031
$305$	0	0
$306$	−6270.00	−1.17135
$307$	1820.00	0.338348	0.169174	−	0.985586i	$-0.445890\pi$
$307$	1820.00	0.338348	0.169174	+	0.985586i	$0.445890\pi$
$308$	−6528.00	−1.20769
$309$	−1472.00	−0.271000
$310$	0	0
$311$	712.000	0.129819	0.0649097	−	0.997891i	$-0.479324\pi$
$311$	712.000	0.129819	0.0649097	+	0.997891i	$0.479324\pi$
$312$	7560.00	1.37180
$313$	−9130.00	−1.64875	−0.824374	−	0.566046i	$-0.808473\pi$
$313$	−9130.00	−1.64875	−0.824374	+	0.566046i	$0.808473\pi$
$314$	16190.0	2.90973
$315$	0	0
$316$	−11696.0	−2.08212
$317$	−6342.00	−1.12367	−0.561833	−	0.827251i	$-0.689904\pi$
$317$	−6342.00	−1.12367	−0.561833	+	0.827251i	$0.689904\pi$
$318$	5480.00	0.966362
$319$	−2568.00	−0.450722
$320$	0	0
$321$	816.000	0.141884
$322$	25600.0	4.43053
$323$	−2166.00	−0.373125
$324$	−5287.00	−0.906550
$325$	0	0
$326$	−7100.00	−1.20624
$327$	5736.00	0.970035
$328$	270.000	0.0454520
$329$	−5888.00	−0.986675
$330$	0	0
$331$	−4748.00	−0.788440	−0.394220	−	0.919016i	$-0.628985\pi$
$331$	−4748.00	−0.788440	−0.394220	+	0.919016i	$0.628985\pi$
$332$	−16932.0	−2.79899
$333$	1034.00	0.170159
$334$	−11680.0	−1.91348
$335$	0	0
$336$	−11392.0	−1.84966
$337$	−5154.00	−0.833105	−0.416552	−	0.909112i	$-0.636762\pi$
$337$	−5154.00	−0.833105	−0.416552	+	0.909112i	$0.636762\pi$
$338$	2165.00	0.348404
$339$	4872.00	0.780563
$340$	0	0
$341$	1728.00	0.274418
$342$	1045.00	0.165226
$343$	10816.0	1.70265
$344$	−13860.0	−2.17233
$345$	0	0
$346$	6030.00	0.936921
$347$	12148.0	1.87936	0.939681	−	0.342051i	$-0.111122\pi$
$347$	12148.0	1.87936	0.939681	+	0.342051i	$0.111122\pi$
$348$	−14552.0	−2.24158
$349$	−602.000	−0.0923333	−0.0461666	−	0.998934i	$-0.514701\pi$
$349$	−602.000	−0.0923333	−0.0461666	+	0.998934i	$0.514701\pi$
$350$	0	0
$351$	6384.00	0.970805
$352$	1020.00	0.154449
$353$	−2.00000	−0.000301556 0	−0.000150778	−	1.00000i	$-0.500048\pi$
$353$	−2.00000	−0.000301556 0	−0.000150778		1.00000i	$0.500048\pi$
$354$	5520.00	0.828770
$355$	0	0
$356$	26826.0	3.99375
$357$	14592.0	2.16328
$358$	7060.00	1.04227
$359$	1960.00	0.288147	0.144074	−	0.989567i	$-0.453980\pi$
$359$	1960.00	0.288147	0.144074	+	0.989567i	$0.453980\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	−18710.0	−2.71651
$363$	4748.00	0.686516
$364$	22848.0	3.29000
$365$	0	0
$366$	−16520.0	−2.35933
$367$	5864.00	0.834055	0.417028	−	0.908894i	$-0.363072\pi$
$367$	5864.00	0.834055	0.417028	+	0.908894i	$0.363072\pi$
$368$	−14240.0	−2.01715
$369$	66.0000	0.00931117
$370$	0	0
$371$	8768.00	1.22699
$372$	9792.00	1.36476
$373$	−558.000	−0.0774588	−0.0387294	−	0.999250i	$-0.512331\pi$
$373$	−558.000	−0.0774588	−0.0387294	+	0.999250i	$0.512331\pi$
$374$	−6840.00	−0.945690
$375$	0	0
$376$	8280.00	1.13566
$377$	8988.00	1.22787
$378$	−24320.0	−3.30922
$379$	−4876.00	−0.660853	−0.330427	−	0.943832i	$-0.607193\pi$
$379$	−4876.00	−0.660853	−0.330427	+	0.943832i	$0.607193\pi$
$380$	0	0
$381$	3616.00	0.486229
$382$	7360.00	0.985786
$383$	424.000	0.0565676	0.0282838	−	0.999600i	$-0.490996\pi$
$383$	424.000	0.0565676	0.0282838	+	0.999600i	$0.490996\pi$
$384$	−8460.00	−1.12428
$385$	0	0
$386$	7290.00	0.961273
$387$	−3388.00	−0.445017
$388$	−13362.0	−1.74833
$389$	−9890.00	−1.28906	−0.644528	−	0.764581i	$-0.722946\pi$
$389$	−9890.00	−1.28906	−0.644528	+	0.764581i	$0.722946\pi$
$390$	0	0
$391$	18240.0	2.35917
$392$	−30645.0	−3.94849
$393$	8720.00	1.11925
$394$	10230.0	1.30807
$395$	0	0
$396$	2244.00	0.284761
$397$	234.000	0.0295822	0.0147911	−	0.999891i	$-0.495292\pi$
$397$	234.000	0.0295822	0.0147911	+	0.999891i	$0.495292\pi$
$398$	7480.00	0.942057
$399$	−2432.00	−0.305144
$400$	0	0
$401$	11602.0	1.44483	0.722414	−	0.691461i	$-0.243032\pi$
$401$	11602.0	1.44483	0.722414	+	0.691461i	$0.243032\pi$
$402$	−1040.00	−0.129031
$403$	−6048.00	−0.747574
$404$	2142.00	0.263783
$405$	0	0
$406$	−34240.0	−4.18547
$407$	1128.00	0.137378
$408$	−20520.0	−2.48993
$409$	−14806.0	−1.79000	−0.894999	−	0.446067i	$-0.852824\pi$
$409$	−14806.0	−1.79000	−0.894999	+	0.446067i	$0.852824\pi$
$410$	0	0
$411$	−10264.0	−1.23184
$412$	6256.00	0.748085
$413$	8832.00	1.05229
$414$	−8800.00	−1.04468
$415$	0	0
$416$	−3570.00	−0.420754
$417$	−7952.00	−0.933840
$418$	1140.00	0.133395
$419$	6252.00	0.728950	0.364475	−	0.931213i	$-0.381248\pi$
$419$	6252.00	0.728950	0.364475	+	0.931213i	$0.381248\pi$
$420$	0	0
$421$	−10482.0	−1.21345	−0.606724	−	0.794913i	$-0.707517\pi$
$421$	−10482.0	−1.21345	−0.606724	+	0.794913i	$0.707517\pi$
$422$	−4220.00	−0.486792
$423$	2024.00	0.232648
$424$	−12330.0	−1.41226
$425$	0	0
$426$	−6880.00	−0.782481
$427$	−26432.0	−2.99563
$428$	−3468.00	−0.391664
$429$	2016.00	0.226884
$430$	0	0
$431$	3936.00	0.439885	0.219943	−	0.975513i	$-0.429413\pi$
$431$	3936.00	0.439885	0.219943	+	0.975513i	$0.429413\pi$
$432$	13528.0	1.50663
$433$	−10946.0	−1.21485	−0.607426	−	0.794376i	$-0.707798\pi$
$433$	−10946.0	−1.21485	−0.607426	+	0.794376i	$0.707798\pi$
$434$	23040.0	2.54828
$435$	0	0
$436$	−24378.0	−2.67774
$437$	−3040.00	−0.332776
$438$	3320.00	0.362182
$439$	−7800.00	−0.848004	−0.424002	−	0.905661i	$-0.639375\pi$
$439$	−7800.00	−0.848004	−0.424002	+	0.905661i	$0.639375\pi$
$440$	0	0
$441$	−7491.00	−0.808876
$442$	23940.0	2.57627
$443$	11364.0	1.21878	0.609390	−	0.792870i	$-0.291414\pi$
$443$	11364.0	1.21878	0.609390	+	0.792870i	$0.291414\pi$
$444$	6392.00	0.683223
$445$	0	0
$446$	−17000.0	−1.80487
$447$	−4536.00	−0.479967
$448$	−9184.00	−0.968534
$449$	7330.00	0.770432	0.385216	−	0.922826i	$-0.374127\pi$
$449$	7330.00	0.770432	0.385216	+	0.922826i	$0.374127\pi$
$450$	0	0
$451$	72.0000	0.00751740
$452$	−20706.0	−2.15471
$453$	9760.00	1.01228
$454$	9220.00	0.953119
$455$	0	0
$456$	3420.00	0.351220
$457$	12774.0	1.30753	0.653766	−	0.756696i	$-0.273188\pi$
$457$	12774.0	1.30753	0.653766	+	0.756696i	$0.273188\pi$
$458$	5450.00	0.556030
$459$	−17328.0	−1.76210
$460$	0	0
$461$	−3786.00	−0.382498	−0.191249	−	0.981542i	$-0.561254\pi$
$461$	−3786.00	−0.382498	−0.191249	+	0.981542i	$0.561254\pi$
$462$	−7680.00	−0.773389
$463$	19448.0	1.95211	0.976053	−	0.217532i	$-0.0698007\pi$
$463$	19448.0	1.95211	0.976053	+	0.217532i	$0.0698007\pi$
$464$	19046.0	1.90558
$465$	0	0
$466$	14210.0	1.41259
$467$	−4596.00	−0.455412	−0.227706	−	0.973730i	$-0.573123\pi$
$467$	−4596.00	−0.455412	−0.227706	+	0.973730i	$0.573123\pi$
$468$	−7854.00	−0.775751
$469$	−1664.00	−0.163830
$470$	0	0
$471$	12952.0	1.26708
$472$	−12420.0	−1.21118
$473$	−3696.00	−0.359286
$474$	−13760.0	−1.33337
$475$	0	0
$476$	−62016.0	−5.97164
$477$	−3014.00	−0.289311
$478$	12000.0	1.14826
$479$	12432.0	1.18587	0.592936	−	0.805250i	$-0.297969\pi$
$479$	12432.0	1.18587	0.592936	+	0.805250i	$0.297969\pi$
$480$	0	0
$481$	−3948.00	−0.374248
$482$	−10650.0	−1.00642
$483$	20480.0	1.92934
$484$	−20179.0	−1.89510
$485$	0	0
$486$	14300.0	1.33469
$487$	18016.0	1.67635	0.838175	−	0.545401i	$-0.183622\pi$
$487$	18016.0	1.67635	0.838175	+	0.545401i	$0.183622\pi$
$488$	37170.0	3.44796
$489$	−5680.00	−0.525273
$490$	0	0
$491$	1972.00	0.181253	0.0906264	−	0.995885i	$-0.471113\pi$
$491$	1972.00	0.181253	0.0906264	+	0.995885i	$0.471113\pi$
$492$	408.000	0.0373863
$493$	−24396.0	−2.22868
$494$	−3990.00	−0.363398
$495$	0	0
$496$	−12816.0	−1.16019
$497$	−11008.0	−0.993514
$498$	−19920.0	−1.79244
$499$	18780.0	1.68479	0.842393	−	0.538864i	$-0.181146\pi$
$499$	18780.0	1.68479	0.842393	+	0.538864i	$0.181146\pi$
$500$	0	0
$501$	−9344.00	−0.833252
$502$	11820.0	1.05090
$503$	−8256.00	−0.731843	−0.365921	−	0.930646i	$-0.619246\pi$
$503$	−8256.00	−0.731843	−0.365921	+	0.930646i	$0.619246\pi$
$504$	15840.0	1.39994
$505$	0	0
$506$	−9600.00	−0.843423
$507$	1732.00	0.151718
$508$	−15368.0	−1.34221
$509$	−13290.0	−1.15731	−0.578653	−	0.815574i	$-0.696421\pi$
$509$	−13290.0	−1.15731	−0.578653	+	0.815574i	$0.696421\pi$
$510$	0	0
$511$	5312.00	0.459861
$512$	24475.0	2.11260
$513$	2888.00	0.248554
$514$	31450.0	2.69883
$515$	0	0
$516$	−20944.0	−1.78684
$517$	2208.00	0.187829
$518$	15040.0	1.27571
$519$	4824.00	0.407996
$520$	0	0
$521$	7610.00	0.639924	0.319962	−	0.947430i	$-0.396330\pi$
$521$	7610.00	0.639924	0.319962	+	0.947430i	$0.396330\pi$
$522$	11770.0	0.986894
$523$	13636.0	1.14008	0.570039	−	0.821618i	$-0.306928\pi$
$523$	13636.0	1.14008	0.570039	+	0.821618i	$0.306928\pi$
$524$	−37060.0	−3.08964
$525$	0	0
$526$	40560.0	3.36217
$527$	16416.0	1.35691
$528$	4272.00	0.352112
$529$	13433.0	1.10405
$530$	0	0
$531$	−3036.00	−0.248119
$532$	10336.0	0.842335
$533$	−252.000	−0.0204790
$534$	31560.0	2.55756
$535$	0	0
$536$	2340.00	0.188568
$537$	5648.00	0.453872
$538$	23970.0	1.92086
$539$	−8172.00	−0.653048
$540$	0	0
$541$	1350.00	0.107285	0.0536424	−	0.998560i	$-0.482917\pi$
$541$	1350.00	0.107285	0.0536424	+	0.998560i	$0.482917\pi$
$542$	−1520.00	−0.120460
$543$	−14968.0	−1.18294
$544$	9690.00	0.763705
$545$	0	0
$546$	26880.0	2.10688
$547$	20396.0	1.59428	0.797139	−	0.603796i	$-0.206346\pi$
$547$	20396.0	1.59428	0.797139	+	0.603796i	$0.206346\pi$
$548$	43622.0	3.40044
$549$	9086.00	0.706341
$550$	0	0
$551$	4066.00	0.314369
$552$	−28800.0	−2.22067
$553$	−22016.0	−1.69298
$554$	10310.0	0.790668
$555$	0	0
$556$	33796.0	2.57782
$557$	4458.00	0.339123	0.169562	−	0.985520i	$-0.445765\pi$
$557$	4458.00	0.339123	0.169562	+	0.985520i	$0.445765\pi$
$558$	−7920.00	−0.600861
$559$	12936.0	0.978774
$560$	0	0
$561$	−5472.00	−0.411815
$562$	20270.0	1.52142
$563$	−18228.0	−1.36451	−0.682255	−	0.731115i	$-0.739000\pi$
$563$	−18228.0	−1.36451	−0.682255	+	0.731115i	$0.739000\pi$
$564$	12512.0	0.934132
$565$	0	0
$566$	39980.0	2.96905
$567$	−9952.00	−0.737116
$568$	15480.0	1.14353
$569$	−8246.00	−0.607540	−0.303770	−	0.952745i	$-0.598245\pi$
$569$	−8246.00	−0.607540	−0.303770	+	0.952745i	$0.598245\pi$
$570$	0	0
$571$	−924.000	−0.0677201	−0.0338601	−	0.999427i	$-0.510780\pi$
$571$	−924.000	−0.0677201	−0.0338601	+	0.999427i	$0.510780\pi$
$572$	−8568.00	−0.626304
$573$	5888.00	0.429275
$574$	960.000	0.0698077
$575$	0	0
$576$	3157.00	0.228371
$577$	−16322.0	−1.17763	−0.588816	−	0.808267i	$-0.700406\pi$
$577$	−16322.0	−1.17763	−0.588816	+	0.808267i	$0.700406\pi$
$578$	−40415.0	−2.90838
$579$	5832.00	0.418600
$580$	0	0
$581$	−31872.0	−2.27586
$582$	−15720.0	−1.11961
$583$	−3288.00	−0.233576
$584$	−7470.00	−0.529299
$585$	0	0
$586$	−19530.0	−1.37675
$587$	−17500.0	−1.23050	−0.615249	−	0.788333i	$-0.710945\pi$
$587$	−17500.0	−1.23050	−0.615249	+	0.788333i	$0.710945\pi$
$588$	−46308.0	−3.24781
$589$	−2736.00	−0.191401
$590$	0	0
$591$	8184.00	0.569619
$592$	−8366.00	−0.580812
$593$	1486.00	0.102905	0.0514525	−	0.998675i	$-0.483615\pi$
$593$	1486.00	0.102905	0.0514525	+	0.998675i	$0.483615\pi$
$594$	9120.00	0.629963
$595$	0	0
$596$	19278.0	1.32493
$597$	5984.00	0.410233
$598$	33600.0	2.29767
$599$	24616.0	1.67910	0.839551	−	0.543280i	$-0.182818\pi$
$599$	24616.0	1.67910	0.839551	+	0.543280i	$0.182818\pi$
$600$	0	0
$601$	−13334.0	−0.905000	−0.452500	−	0.891764i	$-0.649468\pi$
$601$	−13334.0	−0.905000	−0.452500	+	0.891764i	$0.649468\pi$
$602$	−49280.0	−3.33638
$603$	572.000	0.0386296
$604$	−41480.0	−2.79437
$605$	0	0
$606$	2520.00	0.168924
$607$	12056.0	0.806158	0.403079	−	0.915165i	$-0.367940\pi$
$607$	12056.0	0.806158	0.403079	+	0.915165i	$0.367940\pi$
$608$	−1615.00	−0.107725
$609$	−27392.0	−1.82263
$610$	0	0
$611$	−7728.00	−0.511688
$612$	21318.0	1.40805
$613$	98.0000	0.00645707	0.00322853	−	0.999995i	$-0.498972\pi$
$613$	98.0000	0.00645707	0.00322853	+	0.999995i	$0.498972\pi$
$614$	−9100.00	−0.598121
$615$	0	0
$616$	17280.0	1.13025
$617$	−2938.00	−0.191701	−0.0958504	−	0.995396i	$-0.530557\pi$
$617$	−2938.00	−0.191701	−0.0958504	+	0.995396i	$0.530557\pi$
$618$	7360.00	0.479066
$619$	25316.0	1.64384	0.821919	−	0.569604i	$-0.192903\pi$
$619$	25316.0	1.64384	0.821919	+	0.569604i	$0.192903\pi$
$620$	0	0
$621$	−24320.0	−1.57154
$622$	−3560.00	−0.229490
$623$	50496.0	3.24732
$624$	−14952.0	−0.959229
$625$	0	0
$626$	45650.0	2.91460
$627$	912.000	0.0580890
$628$	−55046.0	−3.49773
$629$	10716.0	0.679292
$630$	0	0
$631$	1256.00	0.0792402	0.0396201	−	0.999215i	$-0.487385\pi$
$631$	1256.00	0.0792402	0.0396201	+	0.999215i	$0.487385\pi$
$632$	30960.0	1.94861
$633$	−3376.00	−0.211981
$634$	31710.0	1.98638
$635$	0	0
$636$	−18632.0	−1.16165
$637$	28602.0	1.77905
$638$	12840.0	0.796772
$639$	3784.00	0.234261
$640$	0	0
$641$	20290.0	1.25024	0.625122	−	0.780527i	$-0.285049\pi$
$641$	20290.0	1.25024	0.625122	+	0.780527i	$0.285049\pi$
$642$	−4080.00	−0.250817
$643$	−10676.0	−0.654775	−0.327388	−	0.944890i	$-0.606168\pi$
$643$	−10676.0	−0.654775	−0.327388	+	0.944890i	$0.606168\pi$
$644$	−87040.0	−5.32586
$645$	0	0
$646$	10830.0	0.659599
$647$	−11264.0	−0.684441	−0.342221	−	0.939620i	$-0.611179\pi$
$647$	−11264.0	−0.684441	−0.342221	+	0.939620i	$0.611179\pi$
$648$	13995.0	0.848419
$649$	−3312.00	−0.200320
$650$	0	0
$651$	18432.0	1.10969
$652$	24140.0	1.44999
$653$	−25878.0	−1.55082	−0.775409	−	0.631459i	$-0.782456\pi$
$653$	−25878.0	−1.55082	−0.775409	+	0.631459i	$0.782456\pi$
$654$	−28680.0	−1.71480
$655$	0	0
$656$	−534.000	−0.0317823
$657$	−1826.00	−0.108431
$658$	29440.0	1.74421
$659$	1500.00	0.0886672	0.0443336	−	0.999017i	$-0.485884\pi$
$659$	1500.00	0.0886672	0.0443336	+	0.999017i	$0.485884\pi$
$660$	0	0
$661$	−7618.00	−0.448269	−0.224135	−	0.974558i	$-0.571956\pi$
$661$	−7618.00	−0.448269	−0.224135	+	0.974558i	$0.571956\pi$
$662$	23740.0	1.39378
$663$	19152.0	1.12187
$664$	44820.0	2.61951
$665$	0	0
$666$	−5170.00	−0.300801
$667$	−34240.0	−1.98767
$668$	39712.0	2.30015
$669$	−13600.0	−0.785959
$670$	0	0
$671$	9912.00	0.570266
$672$	10880.0	0.624561
$673$	4110.00	0.235407	0.117703	−	0.993049i	$-0.462447\pi$
$673$	4110.00	0.235407	0.117703	+	0.993049i	$0.462447\pi$
$674$	25770.0	1.47273
$675$	0	0
$676$	−7361.00	−0.418810
$677$	21474.0	1.21907	0.609537	−	0.792758i	$-0.291355\pi$
$677$	21474.0	1.21907	0.609537	+	0.792758i	$0.291355\pi$
$678$	−24360.0	−1.37985
$679$	−25152.0	−1.42157
$680$	0	0
$681$	7376.00	0.415050
$682$	−8640.00	−0.485107
$683$	−4668.00	−0.261517	−0.130758	−	0.991414i	$-0.541741\pi$
$683$	−4668.00	−0.261517	−0.130758	+	0.991414i	$0.541741\pi$
$684$	−3553.00	−0.198615
$685$	0	0
$686$	−54080.0	−3.00989
$687$	4360.00	0.242132
$688$	27412.0	1.51900
$689$	11508.0	0.636313
$690$	0	0
$691$	13292.0	0.731768	0.365884	−	0.930661i	$-0.380767\pi$
$691$	13292.0	0.731768	0.365884	+	0.930661i	$0.380767\pi$
$692$	−20502.0	−1.12626
$693$	4224.00	0.231539
$694$	−60740.0	−3.32228
$695$	0	0
$696$	38520.0	2.09784
$697$	684.000	0.0371712
$698$	3010.00	0.163224
$699$	11368.0	0.615132
$700$	0	0
$701$	7206.00	0.388255	0.194128	−	0.980976i	$-0.437812\pi$
$701$	7206.00	0.388255	0.194128	+	0.980976i	$0.437812\pi$
$702$	−31920.0	−1.71616
$703$	−1786.00	−0.0958183
$704$	3444.00	0.184376
$705$	0	0
$706$	10.0000	0.000533081 0
$707$	4032.00	0.214482
$708$	−18768.0	−0.996249
$709$	−5666.00	−0.300128	−0.150064	−	0.988676i	$-0.547948\pi$
$709$	−5666.00	−0.300128	−0.150064	+	0.988676i	$0.547948\pi$
$710$	0	0
$711$	7568.00	0.399187
$712$	−71010.0	−3.73766
$713$	23040.0	1.21018
$714$	−72960.0	−3.82417
$715$	0	0
$716$	−24004.0	−1.25289
$717$	9600.00	0.500026
$718$	−9800.00	−0.509377
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	11776.0	0.608268
$722$	−1805.00	−0.0930404
$723$	−8520.00	−0.438260
$724$	63614.0	3.26546
$725$	0	0
$726$	−23740.0	−1.21360
$727$	4048.00	0.206509	0.103254	−	0.994655i	$-0.467074\pi$
$727$	4048.00	0.206509	0.103254	+	0.994655i	$0.467074\pi$
$728$	−60480.0	−3.07904
$729$	19837.0	1.00782
$730$	0	0
$731$	−35112.0	−1.77656
$732$	56168.0	2.83611
$733$	−12134.0	−0.611432	−0.305716	−	0.952123i	$-0.598896\pi$
$733$	−12134.0	−0.611432	−0.305716	+	0.952123i	$0.598896\pi$
$734$	−29320.0	−1.47442
$735$	0	0
$736$	13600.0	0.681118
$737$	624.000	0.0311877
$738$	−330.000	−0.0164600
$739$	−13556.0	−0.674784	−0.337392	−	0.941364i	$-0.609545\pi$
$739$	−13556.0	−0.674784	−0.337392	+	0.941364i	$0.609545\pi$
$740$	0	0
$741$	−3192.00	−0.158247
$742$	−43840.0	−2.16903
$743$	−4368.00	−0.215675	−0.107837	−	0.994169i	$-0.534393\pi$
$743$	−4368.00	−0.215675	−0.107837	+	0.994169i	$0.534393\pi$
$744$	−25920.0	−1.27725
$745$	0	0
$746$	2790.00	0.136929
$747$	10956.0	0.536625
$748$	23256.0	1.13680
$749$	−6528.00	−0.318462
$750$	0	0
$751$	3872.00	0.188138	0.0940688	−	0.995566i	$-0.470013\pi$
$751$	3872.00	0.188138	0.0940688	+	0.995566i	$0.470013\pi$
$752$	−16376.0	−0.794111
$753$	9456.00	0.457631
$754$	−44940.0	−2.17058
$755$	0	0
$756$	82688.0	3.97795
$757$	−6190.00	−0.297199	−0.148599	−	0.988897i	$-0.547476\pi$
$757$	−6190.00	−0.297199	−0.148599	+	0.988897i	$0.547476\pi$
$758$	24380.0	1.16823
$759$	−7680.00	−0.367281
$760$	0	0
$761$	−7062.00	−0.336396	−0.168198	−	0.985753i	$-0.553795\pi$
$761$	−7062.00	−0.336396	−0.168198	+	0.985753i	$0.553795\pi$
$762$	−18080.0	−0.859540
$763$	−45888.0	−2.17727
$764$	−25024.0	−1.18500
$765$	0	0
$766$	−2120.00	−0.0999983
$767$	11592.0	0.545714
$768$	33116.0	1.55595
$769$	−5438.00	−0.255006	−0.127503	−	0.991838i	$-0.540696\pi$
$769$	−5438.00	−0.255006	−0.127503	+	0.991838i	$0.540696\pi$
$770$	0	0
$771$	25160.0	1.17525
$772$	−24786.0	−1.15553
$773$	−1182.00	−0.0549982	−0.0274991	−	0.999622i	$-0.508754\pi$
$773$	−1182.00	−0.0549982	−0.0274991	+	0.999622i	$0.508754\pi$
$774$	16940.0	0.786687
$775$	0	0
$776$	35370.0	1.63622
$777$	12032.0	0.555528
$778$	49450.0	2.27875
$779$	−114.000	−0.00524323
$780$	0	0
$781$	4128.00	0.189131
$782$	−91200.0	−4.17047
$783$	32528.0	1.48462
$784$	60609.0	2.76098
$785$	0	0
$786$	−43600.0	−1.97858
$787$	−12452.0	−0.563997	−0.281999	−	0.959415i	$-0.590997\pi$
$787$	−12452.0	−0.563997	−0.281999	+	0.959415i	$0.590997\pi$
$788$	−34782.0	−1.57241
$789$	32448.0	1.46411
$790$	0	0
$791$	−38976.0	−1.75199
$792$	−5940.00	−0.266501
$793$	−34692.0	−1.55353
$794$	−1170.00	−0.0522944
$795$	0	0
$796$	−25432.0	−1.13243
$797$	−15526.0	−0.690037	−0.345018	−	0.938596i	$-0.612127\pi$
$797$	−15526.0	−0.690037	−0.345018	+	0.938596i	$0.612127\pi$
$798$	12160.0	0.539423
$799$	20976.0	0.928758
$800$	0	0
$801$	−17358.0	−0.765686
$802$	−58010.0	−2.55412
$803$	−1992.00	−0.0875419
$804$	3536.00	0.155106
$805$	0	0
$806$	30240.0	1.32154
$807$	19176.0	0.836465
$808$	−5670.00	−0.246869
$809$	31034.0	1.34870	0.674349	−	0.738412i	$-0.264424\pi$
$809$	31034.0	1.34870	0.674349	+	0.738412i	$0.264424\pi$
$810$	0	0
$811$	−34636.0	−1.49967	−0.749836	−	0.661623i	$-0.769868\pi$
$811$	−34636.0	−1.49967	−0.749836	+	0.661623i	$0.769868\pi$
$812$	116416.	5.03128
$813$	−1216.00	−0.0524563
$814$	−5640.00	−0.242852
$815$	0	0
$816$	40584.0	1.74108
$817$	5852.00	0.250594
$818$	74030.0	3.16430
$819$	−14784.0	−0.630763
$820$	0	0
$821$	−20082.0	−0.853674	−0.426837	−	0.904328i	$-0.640372\pi$
$821$	−20082.0	−0.853674	−0.426837	+	0.904328i	$0.640372\pi$
$822$	51320.0	2.17760
$823$	−33568.0	−1.42176	−0.710879	−	0.703314i	$-0.751703\pi$
$823$	−33568.0	−1.42176	−0.710879	+	0.703314i	$0.751703\pi$
$824$	−16560.0	−0.700115
$825$	0	0
$826$	−44160.0	−1.86020
$827$	−19644.0	−0.825984	−0.412992	−	0.910735i	$-0.635516\pi$
$827$	−19644.0	−0.825984	−0.412992	+	0.910735i	$0.635516\pi$
$828$	29920.0	1.25579
$829$	726.000	0.0304162	0.0152081	−	0.999884i	$-0.495159\pi$
$829$	726.000	0.0304162	0.0152081	+	0.999884i	$0.495159\pi$
$830$	0	0
$831$	8248.00	0.344308
$832$	−12054.0	−0.502280
$833$	−77634.0	−3.22912
$834$	39760.0	1.65081
$835$	0	0
$836$	−3876.00	−0.160352
$837$	−21888.0	−0.903895
$838$	−31260.0	−1.28861
$839$	3512.00	0.144515	0.0722573	−	0.997386i	$-0.476980\pi$
$839$	3512.00	0.144515	0.0722573	+	0.997386i	$0.476980\pi$
$840$	0	0
$841$	21407.0	0.877732
$842$	52410.0	2.14509
$843$	16216.0	0.662525
$844$	14348.0	0.585164
$845$	0	0
$846$	−10120.0	−0.411268
$847$	−37984.0	−1.54090
$848$	24386.0	0.987522
$849$	31984.0	1.29292
$850$	0	0
$851$	15040.0	0.605834
$852$	23392.0	0.940606
$853$	39442.0	1.58320	0.791599	−	0.611041i	$-0.209249\pi$
$853$	39442.0	1.58320	0.791599	+	0.611041i	$0.209249\pi$
$854$	132160.	5.29558
$855$	0	0
$856$	9180.00	0.366549
$857$	40454.0	1.61246	0.806232	−	0.591599i	$-0.201503\pi$
$857$	40454.0	1.61246	0.806232	+	0.591599i	$0.201503\pi$
$858$	−10080.0	−0.401079
$859$	−3436.00	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−3436.00	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	768.000	0.0303988
$862$	−19680.0	−0.777614
$863$	10056.0	0.396651	0.198326	−	0.980136i	$-0.436450\pi$
$863$	10056.0	0.396651	0.198326	+	0.980136i	$0.436450\pi$
$864$	−12920.0	−0.508735
$865$	0	0
$866$	54730.0	2.14758
$867$	−32332.0	−1.26650
$868$	−78336.0	−3.06325
$869$	8256.00	0.322285
$870$	0	0
$871$	−2184.00	−0.0849621
$872$	64530.0	2.50603
$873$	8646.00	0.335192
$874$	15200.0	0.588270
$875$	0	0
$876$	−11288.0	−0.435372
$877$	44394.0	1.70933	0.854663	−	0.519183i	$-0.173764\pi$
$877$	44394.0	1.70933	0.854663	+	0.519183i	$0.173764\pi$
$878$	39000.0	1.49907
$879$	−15624.0	−0.599527
$880$	0	0
$881$	−18222.0	−0.696839	−0.348419	−	0.937339i	$-0.613281\pi$
$881$	−18222.0	−0.696839	−0.348419	+	0.937339i	$0.613281\pi$
$882$	37455.0	1.42990
$883$	29404.0	1.12064	0.560319	−	0.828277i	$-0.310679\pi$
$883$	29404.0	1.12064	0.560319	+	0.828277i	$0.310679\pi$
$884$	−81396.0	−3.09688
$885$	0	0
$886$	−56820.0	−2.15452
$887$	−18576.0	−0.703180	−0.351590	−	0.936154i	$-0.614359\pi$
$887$	−18576.0	−0.703180	−0.351590	+	0.936154i	$0.614359\pi$
$888$	−16920.0	−0.639412
$889$	−28928.0	−1.09135
$890$	0	0
$891$	3732.00	0.140322
$892$	57800.0	2.16960
$893$	−3496.00	−0.131007
$894$	22680.0	0.848471
$895$	0	0
$896$	67680.0	2.52347
$897$	26880.0	1.00055
$898$	−36650.0	−1.36194
$899$	−30816.0	−1.14324
$900$	0	0
$901$	−31236.0	−1.15496
$902$	−360.000	−0.0132890
$903$	−39424.0	−1.45288
$904$	54810.0	2.01654
$905$	0	0
$906$	−48800.0	−1.78948
$907$	2644.00	0.0967945	0.0483972	−	0.998828i	$-0.484589\pi$
$907$	2644.00	0.0967945	0.0483972	+	0.998828i	$0.484589\pi$
$908$	−31348.0	−1.14573
$909$	−1386.00	−0.0505728
$910$	0	0
$911$	39744.0	1.44542	0.722710	−	0.691151i	$-0.242896\pi$
$911$	39744.0	1.44542	0.722710	+	0.691151i	$0.242896\pi$
$912$	−6764.00	−0.245590
$913$	11952.0	0.433246
$914$	−63870.0	−2.31141
$915$	0	0
$916$	−18530.0	−0.668393
$917$	−69760.0	−2.51219
$918$	86640.0	3.11497
$919$	−19720.0	−0.707838	−0.353919	−	0.935276i	$-0.615151\pi$
$919$	−19720.0	−0.707838	−0.353919	+	0.935276i	$0.615151\pi$
$920$	0	0
$921$	−7280.00	−0.260461
$922$	18930.0	0.676167
$923$	−14448.0	−0.515235
$924$	26112.0	0.929677
$925$	0	0
$926$	−97240.0	−3.45087
$927$	−4048.00	−0.143424
$928$	−18190.0	−0.643444
$929$	36258.0	1.28050	0.640251	−	0.768166i	$-0.278830\pi$
$929$	36258.0	1.28050	0.640251	+	0.768166i	$0.278830\pi$
$930$	0	0
$931$	12939.0	0.455487
$932$	−48314.0	−1.69804
$933$	−2848.00	−0.0999350
$934$	22980.0	0.805063
$935$	0	0
$936$	20790.0	0.726007
$937$	−14586.0	−0.508542	−0.254271	−	0.967133i	$-0.581836\pi$
$937$	−14586.0	−0.508542	−0.254271	+	0.967133i	$0.581836\pi$
$938$	8320.00	0.289614
$939$	36520.0	1.26921
$940$	0	0
$941$	30182.0	1.04560	0.522798	−	0.852457i	$-0.324888\pi$
$941$	30182.0	1.04560	0.522798	+	0.852457i	$0.324888\pi$
$942$	−64760.0	−2.23991
$943$	960.000	0.0331515
$944$	24564.0	0.846917
$945$	0	0
$946$	18480.0	0.635134
$947$	−15828.0	−0.543127	−0.271563	−	0.962421i	$-0.587541\pi$
$947$	−15828.0	−0.543127	−0.271563	+	0.962421i	$0.587541\pi$
$948$	46784.0	1.60282
$949$	6972.00	0.238483
$950$	0	0
$951$	25368.0	0.864999
$952$	164160.	5.58871
$953$	−22746.0	−0.773153	−0.386577	−	0.922257i	$-0.626343\pi$
$953$	−22746.0	−0.773153	−0.386577	+	0.922257i	$0.626343\pi$
$954$	15070.0	0.511435
$955$	0	0
$956$	−40800.0	−1.38030
$957$	10272.0	0.346966
$958$	−62160.0	−2.09634
$959$	82112.0	2.76490
$960$	0	0
$961$	−9055.00	−0.303951
$962$	19740.0	0.661583
$963$	2244.00	0.0750902
$964$	36210.0	1.20980
$965$	0	0
$966$	−102400.	−3.41063
$967$	−16480.0	−0.548047	−0.274023	−	0.961723i	$-0.588355\pi$
$967$	−16480.0	−0.548047	−0.274023	+	0.961723i	$0.588355\pi$
$968$	53415.0	1.77358
$969$	8664.00	0.287232
$970$	0	0
$971$	−16940.0	−0.559867	−0.279933	−	0.960019i	$-0.590312\pi$
$971$	−16940.0	−0.559867	−0.279933	+	0.960019i	$0.590312\pi$
$972$	−48620.0	−1.60441
$973$	63616.0	2.09603
$974$	−90080.0	−2.96340
$975$	0	0
$976$	−73514.0	−2.41099
$977$	40062.0	1.31187	0.655935	−	0.754817i	$-0.272275\pi$
$977$	40062.0	1.31187	0.655935	+	0.754817i	$0.272275\pi$
$978$	28400.0	0.928560
$979$	−18936.0	−0.618179
$980$	0	0
$981$	15774.0	0.513379
$982$	−9860.00	−0.320413
$983$	−4832.00	−0.156782	−0.0783911	−	0.996923i	$-0.524978\pi$
$983$	−4832.00	−0.156782	−0.0783911	+	0.996923i	$0.524978\pi$
$984$	−1080.00	−0.0349890
$985$	0	0
$986$	121980.	3.93979
$987$	23552.0	0.759542
$988$	13566.0	0.436834
$989$	−49280.0	−1.58444
$990$	0	0
$991$	−4144.00	−0.132834	−0.0664170	−	0.997792i	$-0.521157\pi$
$991$	−4144.00	−0.132834	−0.0664170	+	0.997792i	$0.521157\pi$
$992$	12240.0	0.391754
$993$	18992.0	0.606941
$994$	55040.0	1.75630
$995$	0	0
$996$	67728.0	2.15466
$997$	−35294.0	−1.12114	−0.560568	−	0.828109i	$-0.689417\pi$
$997$	−35294.0	−1.12114	−0.560568	+	0.828109i	$0.689417\pi$
$998$	−93900.0	−2.97831
$999$	−14288.0	−0.452505

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.4.a.a.1.1		1
5.2	odd	4		475.4.b.a.324.1		2
5.3	odd	4		475.4.b.a.324.2		2
5.4	even	2		95.4.a.d.1.1	✓	1
15.14	odd	2		855.4.a.a.1.1		1
20.19	odd	2		1520.4.a.c.1.1		1
95.94	odd	2		1805.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.4.a.d.1.1	✓	1	5.4	even	2
475.4.a.a.1.1		1	1.1	even	1	trivial
475.4.b.a.324.1		2	5.2	odd	4
475.4.b.a.324.2		2	5.3	odd	4
855.4.a.a.1.1		1	15.14	odd	2
1520.4.a.c.1.1		1	20.19	odd	2
1805.4.a.a.1.1		1	95.94	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:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	475.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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