LMFDB - Embedded newform 475.6.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,6,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, 6, names="a")

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

chi := DirichletCharacter("475.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
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:	$76.1823144112$
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	$\chi$	$=$	475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +1.00000 q^{3} -28.0000 q^{4} +2.00000 q^{6} +167.000 q^{7} -120.000 q^{8} -242.000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} +1.00000 q^{3} -28.0000 q^{4} +2.00000 q^{6} +167.000 q^{7} -120.000 q^{8} -242.000 q^{9} +262.000 q^{11} -28.0000 q^{12} -749.000 q^{13} +334.000 q^{14} +656.000 q^{16} +1597.00 q^{17} -484.000 q^{18} -361.000 q^{19} +167.000 q^{21} +524.000 q^{22} +2011.00 q^{23} -120.000 q^{24} -1498.00 q^{26} -485.000 q^{27} -4676.00 q^{28} -1055.00 q^{29} -1548.00 q^{31} +5152.00 q^{32} +262.000 q^{33} +3194.00 q^{34} +6776.00 q^{36} -9378.00 q^{37} -722.000 q^{38} -749.000 q^{39} -10248.0 q^{41} +334.000 q^{42} -10544.0 q^{43} -7336.00 q^{44} +4022.00 q^{46} +6912.00 q^{47} +656.000 q^{48} +11082.0 q^{49} +1597.00 q^{51} +20972.0 q^{52} +35291.0 q^{53} -970.000 q^{54} -20040.0 q^{56} -361.000 q^{57} -2110.00 q^{58} +33655.0 q^{59} -26218.0 q^{61} -3096.00 q^{62} -40414.0 q^{63} -10688.0 q^{64} +524.000 q^{66} -45083.0 q^{67} -44716.0 q^{68} +2011.00 q^{69} +30942.0 q^{71} +29040.0 q^{72} -46969.0 q^{73} -18756.0 q^{74} +10108.0 q^{76} +43754.0 q^{77} -1498.00 q^{78} -64430.0 q^{79} +58321.0 q^{81} -20496.0 q^{82} +13986.0 q^{83} -4676.00 q^{84} -21088.0 q^{86} -1055.00 q^{87} -31440.0 q^{88} -137700. q^{89} -125083. q^{91} -56308.0 q^{92} -1548.00 q^{93} +13824.0 q^{94} +5152.00 q^{96} +22162.0 q^{97} +22164.0 q^{98} -63404.0 q^{99} +O(q^{100})\)

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 $

$2$	2.00000	0.353553	0.176777	−	0.984251i	$-0.443433\pi$
$2$	2.00000	0.353553	0.176777	+	0.984251i	$0.443433\pi$
$3$	1.00000	0.0641500	0.0320750	−	0.999485i	$-0.489788\pi$
$3$	1.00000	0.0641500	0.0320750	+	0.999485i	$0.489788\pi$
$4$	−28.0000	−0.875000
$5$	0	0
$5$	0	0
$6$	2.00000	0.0226805
$7$	167.000	1.28816	0.644082	−	0.764956i	$-0.277239\pi$
$7$	167.000	1.28816	0.644082	+	0.764956i	$0.277239\pi$
$8$	−120.000	−0.662913
$9$	−242.000	−0.995885
$10$	0	0
$11$	262.000	0.652859	0.326430	−	0.945222i	$-0.394154\pi$
$11$	262.000	0.652859	0.326430	+	0.945222i	$0.394154\pi$
$12$	−28.0000	−0.0561313
$13$	−749.000	−1.22920	−0.614601	−	0.788838i	$-0.710683\pi$
$13$	−749.000	−1.22920	−0.614601	+	0.788838i	$0.710683\pi$
$14$	334.000	0.455435
$15$	0	0
$16$	656.000	0.640625
$17$	1597.00	1.34024	0.670120	−	0.742253i	$-0.266243\pi$
$17$	1597.00	1.34024	0.670120	+	0.742253i	$0.266243\pi$
$18$	−484.000	−0.352098
$19$	−361.000	−0.229416
$19$	−361.000	−0.229416
$20$	0	0
$21$	167.000	0.0826358
$22$	524.000	0.230821
$23$	2011.00	0.792670	0.396335	−	0.918106i	$-0.370282\pi$
$23$	2011.00	0.792670	0.396335	+	0.918106i	$0.370282\pi$
$24$	−120.000	−0.0425259
$25$	0	0
$26$	−1498.00	−0.434589
$27$	−485.000	−0.128036
$28$	−4676.00	−1.12714
$29$	−1055.00	−0.232947	−0.116474	−	0.993194i	$-0.537159\pi$
$29$	−1055.00	−0.232947	−0.116474	+	0.993194i	$0.537159\pi$
$30$	0	0
$31$	−1548.00	−0.289312	−0.144656	−	0.989482i	$-0.546208\pi$
$31$	−1548.00	−0.289312	−0.144656	+	0.989482i	$0.546208\pi$
$32$	5152.00	0.889408
$33$	262.000	0.0418809
$34$	3194.00	0.473846
$35$	0	0
$36$	6776.00	0.871399
$37$	−9378.00	−1.12618	−0.563088	−	0.826397i	$-0.690387\pi$
$37$	−9378.00	−1.12618	−0.563088	+	0.826397i	$0.690387\pi$
$38$	−722.000	−0.0811107
$39$	−749.000	−0.0788534
$40$	0	0
$41$	−10248.0	−0.952093	−0.476047	−	0.879420i	$-0.657931\pi$
$41$	−10248.0	−0.952093	−0.476047	+	0.879420i	$0.657931\pi$
$42$	334.000	0.0292162
$43$	−10544.0	−0.869629	−0.434815	−	0.900520i	$-0.643186\pi$
$43$	−10544.0	−0.869629	−0.434815	+	0.900520i	$0.643186\pi$
$44$	−7336.00	−0.571252
$45$	0	0
$46$	4022.00	0.280251
$47$	6912.00	0.456414	0.228207	−	0.973613i	$-0.426714\pi$
$47$	6912.00	0.456414	0.228207	+	0.973613i	$0.426714\pi$
$48$	656.000	0.0410961
$49$	11082.0	0.659368
$50$	0	0
$51$	1597.00	0.0859764
$52$	20972.0	1.07555
$53$	35291.0	1.72574	0.862868	−	0.505430i	$-0.168666\pi$
$53$	35291.0	1.72574	0.862868	+	0.505430i	$0.168666\pi$
$54$	−970.000	−0.0452676
$55$	0	0
$56$	−20040.0	−0.853941
$57$	−361.000	−0.0147170
$58$	−2110.00	−0.0823593
$59$	33655.0	1.25869	0.629346	−	0.777125i	$-0.283323\pi$
$59$	33655.0	1.25869	0.629346	+	0.777125i	$0.283323\pi$
$60$	0	0
$61$	−26218.0	−0.902142	−0.451071	−	0.892488i	$-0.648958\pi$
$61$	−26218.0	−0.902142	−0.451071	+	0.892488i	$0.648958\pi$
$62$	−3096.00	−0.102287
$63$	−40414.0	−1.28286
$64$	−10688.0	−0.326172
$65$	0	0
$66$	524.000	0.0148071
$67$	−45083.0	−1.22695	−0.613473	−	0.789715i	$-0.710228\pi$
$67$	−45083.0	−1.22695	−0.613473	+	0.789715i	$0.710228\pi$
$68$	−44716.0	−1.17271
$69$	2011.00	0.0508498
$70$	0	0
$71$	30942.0	0.728455	0.364227	−	0.931310i	$-0.381333\pi$
$71$	30942.0	0.728455	0.364227	+	0.931310i	$0.381333\pi$
$72$	29040.0	0.660185
$73$	−46969.0	−1.03158	−0.515791	−	0.856714i	$-0.672502\pi$
$73$	−46969.0	−1.03158	−0.515791	+	0.856714i	$0.672502\pi$
$74$	−18756.0	−0.398163
$75$	0	0
$76$	10108.0	0.200739
$77$	43754.0	0.840990
$78$	−1498.00	−0.0278789
$79$	−64430.0	−1.16150	−0.580752	−	0.814081i	$-0.697241\pi$
$79$	−64430.0	−1.16150	−0.580752	+	0.814081i	$0.697241\pi$
$80$	0	0
$81$	58321.0	0.987671
$82$	−20496.0	−0.336616
$83$	13986.0	0.222843	0.111421	−	0.993773i	$-0.464460\pi$
$83$	13986.0	0.222843	0.111421	+	0.993773i	$0.464460\pi$
$84$	−4676.00	−0.0723063
$85$	0	0
$86$	−21088.0	−0.307460
$87$	−1055.00	−0.0149436
$88$	−31440.0	−0.432789
$89$	−137700.	−1.84272	−0.921359	−	0.388713i	$-0.872920\pi$
$89$	−137700.	−1.84272	−0.921359	+	0.388713i	$0.872920\pi$
$90$	0	0
$91$	−125083.	−1.58342
$92$	−56308.0	−0.693586
$93$	−1548.00	−0.0185594
$94$	13824.0	0.161367
$95$	0	0
$96$	5152.00	0.0570555
$97$	22162.0	0.239155	0.119578	−	0.992825i	$-0.461846\pi$
$97$	22162.0	0.239155	0.119578	+	0.992825i	$0.461846\pi$
$98$	22164.0	0.233122
$99$	−63404.0	−0.650173
$100$	0	0
$101$	−93598.0	−0.912984	−0.456492	−	0.889728i	$-0.650894\pi$
$101$	−93598.0	−0.912984	−0.456492	+	0.889728i	$0.650894\pi$
$102$	3194.00	0.0303973
$103$	−148474.	−1.37898	−0.689489	−	0.724296i	$-0.742165\pi$
$103$	−148474.	−1.37898	−0.689489	+	0.724296i	$0.742165\pi$
$104$	89880.0	0.814854
$105$	0	0
$106$	70582.0	0.610140
$107$	−224313.	−1.89407	−0.947033	−	0.321137i	$-0.895935\pi$
$107$	−224313.	−1.89407	−0.947033	+	0.321137i	$0.895935\pi$
$108$	13580.0	0.112032
$109$	−191405.	−1.54307	−0.771537	−	0.636184i	$-0.780512\pi$
$109$	−191405.	−1.54307	−0.771537	+	0.636184i	$0.780512\pi$
$110$	0	0
$111$	−9378.00	−0.0722442
$112$	109552.	0.825230
$113$	−112814.	−0.831126	−0.415563	−	0.909564i	$-0.636415\pi$
$113$	−112814.	−0.831126	−0.415563	+	0.909564i	$0.636415\pi$
$114$	−722.000	−0.00520325
$115$	0	0
$116$	29540.0	0.203829
$117$	181258.	1.22414
$118$	67310.0	0.445015
$119$	266699.	1.72645
$120$	0	0
$121$	−92407.0	−0.573775
$122$	−52436.0	−0.318955
$123$	−10248.0	−0.0610768
$124$	43344.0	0.253148
$125$	0	0
$126$	−80828.0	−0.453561
$127$	86882.0	0.477992	0.238996	−	0.971021i	$-0.423182\pi$
$127$	86882.0	0.477992	0.238996	+	0.971021i	$0.423182\pi$
$128$	−186240.	−1.00473
$129$	−10544.0	−0.0557868
$130$	0	0
$131$	−309368.	−1.57506	−0.787530	−	0.616276i	$-0.788640\pi$
$131$	−309368.	−1.57506	−0.787530	+	0.616276i	$0.788640\pi$
$132$	−7336.00	−0.0366458
$133$	−60287.0	−0.295525
$134$	−90166.0	−0.433791
$135$	0	0
$136$	−191640.	−0.888462
$137$	301977.	1.37459	0.687294	−	0.726379i	$-0.258798\pi$
$137$	301977.	1.37459	0.687294	+	0.726379i	$0.258798\pi$
$138$	4022.00	0.0179781
$139$	266420.	1.16958	0.584790	−	0.811185i	$-0.301177\pi$
$139$	266420.	1.16958	0.584790	+	0.811185i	$0.301177\pi$
$140$	0	0
$141$	6912.00	0.0292790
$142$	61884.0	0.257548
$143$	−196238.	−0.802496
$144$	−158752.	−0.637989
$145$	0	0
$146$	−93938.0	−0.364720
$147$	11082.0	0.0422985
$148$	262584.	0.985403
$149$	185400.	0.684139	0.342069	−	0.939675i	$-0.388872\pi$
$149$	185400.	0.684139	0.342069	+	0.939675i	$0.388872\pi$
$150$	0	0
$151$	−235638.	−0.841013	−0.420507	−	0.907289i	$-0.638148\pi$
$151$	−235638.	−0.841013	−0.420507	+	0.907289i	$0.638148\pi$
$152$	43320.0	0.152083
$153$	−386474.	−1.33472
$154$	87508.0	0.297335
$155$	0	0
$156$	20972.0	0.0689967
$157$	−12018.0	−0.0389120	−0.0194560	−	0.999811i	$-0.506193\pi$
$157$	−12018.0	−0.0389120	−0.0194560	+	0.999811i	$0.506193\pi$
$158$	−128860.	−0.410653
$159$	35291.0	0.110706
$160$	0	0
$161$	335837.	1.02109
$162$	116642.	0.349195
$163$	−10764.0	−0.0317325	−0.0158663	−	0.999874i	$-0.505051\pi$
$163$	−10764.0	−0.0317325	−0.0158663	+	0.999874i	$0.505051\pi$
$164$	286944.	0.833082
$165$	0	0
$166$	27972.0	0.0787868
$167$	171332.	0.475387	0.237694	−	0.971340i	$-0.423609\pi$
$167$	171332.	0.475387	0.237694	+	0.971340i	$0.423609\pi$
$168$	−20040.0	−0.0547803
$169$	189708.	0.510939
$170$	0	0
$171$	87362.0	0.228472
$172$	295232.	0.760926
$173$	121866.	0.309576	0.154788	−	0.987948i	$-0.450531\pi$
$173$	121866.	0.309576	0.154788	+	0.987948i	$0.450531\pi$
$174$	−2110.00	−0.00528335
$175$	0	0
$176$	171872.	0.418238
$177$	33655.0	0.0807451
$178$	−275400.	−0.651499
$179$	−438960.	−1.02398	−0.511991	−	0.858991i	$-0.671092\pi$
$179$	−438960.	−1.02398	−0.511991	+	0.858991i	$0.671092\pi$
$180$	0	0
$181$	107842.	0.244676	0.122338	−	0.992488i	$-0.460961\pi$
$181$	107842.	0.244676	0.122338	+	0.992488i	$0.460961\pi$
$182$	−250166.	−0.559822
$183$	−26218.0	−0.0578724
$184$	−241320.	−0.525471
$185$	0	0
$186$	−3096.00	−0.00656173
$187$	418414.	0.874988
$188$	−193536.	−0.399362
$189$	−80995.0	−0.164932
$190$	0	0
$191$	−702123.	−1.39261	−0.696305	−	0.717746i	$-0.745174\pi$
$191$	−702123.	−1.39261	−0.696305	+	0.717746i	$0.745174\pi$
$192$	−10688.0	−0.0209239
$193$	−43774.0	−0.0845908	−0.0422954	−	0.999105i	$-0.513467\pi$
$193$	−43774.0	−0.0845908	−0.0422954	+	0.999105i	$0.513467\pi$
$194$	44324.0	0.0845541
$195$	0	0
$196$	−310296.	−0.576947
$197$	146312.	0.268605	0.134303	−	0.990940i	$-0.457121\pi$
$197$	146312.	0.268605	0.134303	+	0.990940i	$0.457121\pi$
$198$	−126808.	−0.229871
$199$	−677475.	−1.21272	−0.606360	−	0.795190i	$-0.707371\pi$
$199$	−677475.	−1.21272	−0.606360	+	0.795190i	$0.707371\pi$
$200$	0	0
$201$	−45083.0	−0.0787087
$202$	−187196.	−0.322789
$203$	−176185.	−0.300074
$204$	−44716.0	−0.0752294
$205$	0	0
$206$	−296948.	−0.487542
$207$	−486662.	−0.789408
$208$	−491344.	−0.787458
$209$	−94582.0	−0.149776
$210$	0	0
$211$	221307.	0.342207	0.171104	−	0.985253i	$-0.445267\pi$
$211$	221307.	0.342207	0.171104	+	0.985253i	$0.445267\pi$
$212$	−988148.	−1.51002
$213$	30942.0	0.0467304
$214$	−448626.	−0.669653
$215$	0	0
$216$	58200.0	0.0848767
$217$	−258516.	−0.372682
$218$	−382810.	−0.545559
$219$	−46969.0	−0.0661761
$220$	0	0
$221$	−1.19615e6	−1.64743
$222$	−18756.0	−0.0255422
$223$	−316974.	−0.426836	−0.213418	−	0.976961i	$-0.568460\pi$
$223$	−316974.	−0.426836	−0.213418	+	0.976961i	$0.568460\pi$
$224$	860384.	1.14570
$225$	0	0
$226$	−225628.	−0.293847
$227$	152097.	0.195910	0.0979549	−	0.995191i	$-0.468770\pi$
$227$	152097.	0.195910	0.0979549	+	0.995191i	$0.468770\pi$
$228$	10108.0	0.0128774
$229$	1.35001e6	1.70117	0.850586	−	0.525836i	$-0.176247\pi$
$229$	1.35001e6	1.70117	0.850586	+	0.525836i	$0.176247\pi$
$230$	0	0
$231$	43754.0	0.0539495
$232$	126600.	0.154424
$233$	859686.	1.03741	0.518705	−	0.854954i	$-0.326414\pi$
$233$	859686.	1.03741	0.518705	+	0.854954i	$0.326414\pi$
$234$	362516.	0.432800
$235$	0	0
$236$	−942340.	−1.10136
$237$	−64430.0	−0.0745105
$238$	533398.	0.610392
$239$	−1.07780e6	−1.22051	−0.610255	−	0.792205i	$-0.708933\pi$
$239$	−1.07780e6	−1.22051	−0.610255	+	0.792205i	$0.708933\pi$
$240$	0	0
$241$	1.70211e6	1.88775	0.943877	−	0.330296i	$-0.107149\pi$
$241$	1.70211e6	1.88775	0.943877	+	0.330296i	$0.107149\pi$
$242$	−184814.	−0.202860
$243$	176176.	0.191395
$244$	734104.	0.789374
$245$	0	0
$246$	−20496.0	−0.0215939
$247$	270389.	0.281998
$248$	185760.	0.191789
$249$	13986.0	0.0142954
$250$	0	0
$251$	665602.	0.666853	0.333427	−	0.942776i	$-0.391795\pi$
$251$	665602.	0.666853	0.333427	+	0.942776i	$0.391795\pi$
$252$	1.13159e6	1.12251
$253$	526882.	0.517502
$254$	173764.	0.168996
$255$	0	0
$256$	−30464.0	−0.0290527
$257$	−809428.	−0.764444	−0.382222	−	0.924071i	$-0.624841\pi$
$257$	−809428.	−0.764444	−0.382222	+	0.924071i	$0.624841\pi$
$258$	−21088.0	−0.0197236
$259$	−1.56613e6	−1.45070
$260$	0	0
$261$	255310.	0.231989
$262$	−618736.	−0.556868
$263$	1.78862e6	1.59451	0.797256	−	0.603641i	$-0.206284\pi$
$263$	1.78862e6	1.59451	0.797256	+	0.603641i	$0.206284\pi$
$264$	−31440.0	−0.0277634
$265$	0	0
$266$	−120574.	−0.104484
$267$	−137700.	−0.118210
$268$	1.26232e6	1.07358
$269$	−321350.	−0.270768	−0.135384	−	0.990793i	$-0.543227\pi$
$269$	−321350.	−0.270768	−0.135384	+	0.990793i	$0.543227\pi$
$270$	0	0
$271$	−275523.	−0.227895	−0.113947	−	0.993487i	$-0.536350\pi$
$271$	−275523.	−0.227895	−0.113947	+	0.993487i	$0.536350\pi$
$272$	1.04763e6	0.858591
$273$	−125083.	−0.101576
$274$	603954.	0.485990
$275$	0	0
$276$	−56308.0	−0.0444936
$277$	1.46919e6	1.15048	0.575240	−	0.817985i	$-0.304909\pi$
$277$	1.46919e6	1.15048	0.575240	+	0.817985i	$0.304909\pi$
$278$	532840.	0.413509
$279$	374616.	0.288122
$280$	0	0
$281$	2.54219e6	1.92062	0.960312	−	0.278927i	$-0.0899786\pi$
$281$	2.54219e6	1.92062	0.960312	+	0.278927i	$0.0899786\pi$
$282$	13824.0	0.0103517
$283$	−2.43841e6	−1.80984	−0.904922	−	0.425577i	$-0.860071\pi$
$283$	−2.43841e6	−1.80984	−0.904922	+	0.425577i	$0.860071\pi$
$284$	−866376.	−0.637398
$285$	0	0
$286$	−392476.	−0.283725
$287$	−1.71142e6	−1.22645
$288$	−1.24678e6	−0.885748
$289$	1.13055e6	0.796244
$290$	0	0
$291$	22162.0	0.0153418
$292$	1.31513e6	0.902635
$293$	−172739.	−0.117550	−0.0587748	−	0.998271i	$-0.518719\pi$
$293$	−172739.	−0.117550	−0.0587748	+	0.998271i	$0.518719\pi$
$294$	22164.0	0.0149548
$295$	0	0
$296$	1.12536e6	0.746556
$297$	−127070.	−0.0835895
$298$	370800.	0.241880
$299$	−1.50624e6	−0.974352
$300$	0	0
$301$	−1.76085e6	−1.12023
$302$	−471276.	−0.297343
$303$	−93598.0	−0.0585679
$304$	−236816.	−0.146969
$305$	0	0
$306$	−772948.	−0.471896
$307$	−2.35091e6	−1.42361	−0.711803	−	0.702379i	$-0.752121\pi$
$307$	−2.35091e6	−1.42361	−0.711803	+	0.702379i	$0.752121\pi$
$308$	−1.22511e6	−0.735866
$309$	−148474.	−0.0884615
$310$	0	0
$311$	3.31824e6	1.94539	0.972694	−	0.232089i	$-0.0745562\pi$
$311$	3.31824e6	1.94539	0.972694	+	0.232089i	$0.0745562\pi$
$312$	89880.0	0.0522729
$313$	1.51021e6	0.871318	0.435659	−	0.900112i	$-0.356515\pi$
$313$	1.51021e6	0.871318	0.435659	+	0.900112i	$0.356515\pi$
$314$	−24036.0	−0.0137575
$315$	0	0
$316$	1.80404e6	1.01632
$317$	−348143.	−0.194585	−0.0972925	−	0.995256i	$-0.531018\pi$
$317$	−348143.	−0.194585	−0.0972925	+	0.995256i	$0.531018\pi$
$318$	70582.0	0.0391405
$319$	−276410.	−0.152082
$320$	0	0
$321$	−224313.	−0.121504
$322$	671674.	0.361010
$323$	−576517.	−0.307472
$324$	−1.63299e6	−0.864212
$325$	0	0
$326$	−21528.0	−0.0112191
$327$	−191405.	−0.0989883
$328$	1.22976e6	0.631155
$329$	1.15430e6	0.587937
$330$	0	0
$331$	−2.85406e6	−1.43184	−0.715918	−	0.698184i	$-0.753992\pi$
$331$	−2.85406e6	−1.43184	−0.715918	+	0.698184i	$0.753992\pi$
$332$	−391608.	−0.194987
$333$	2.26948e6	1.12154
$334$	342664.	0.168075
$335$	0	0
$336$	109552.	0.0529386
$337$	−355588.	−0.170558	−0.0852790	−	0.996357i	$-0.527178\pi$
$337$	−355588.	−0.170558	−0.0852790	+	0.996357i	$0.527178\pi$
$338$	379416.	0.180644
$339$	−112814.	−0.0533168
$340$	0	0
$341$	−405576.	−0.188880
$342$	174724.	0.0807769
$343$	−956075.	−0.438790
$344$	1.26528e6	0.576488
$345$	0	0
$346$	243732.	0.109452
$347$	219202.	0.0977284	0.0488642	−	0.998805i	$-0.484440\pi$
$347$	219202.	0.0977284	0.0488642	+	0.998805i	$0.484440\pi$
$348$	29540.0	0.0130756
$349$	−905730.	−0.398048	−0.199024	−	0.979995i	$-0.563777\pi$
$349$	−905730.	−0.398048	−0.199024	+	0.979995i	$0.563777\pi$
$350$	0	0
$351$	363265.	0.157382
$352$	1.34982e6	0.580658
$353$	1.91611e6	0.818434	0.409217	−	0.912437i	$-0.365802\pi$
$353$	1.91611e6	0.818434	0.409217	+	0.912437i	$0.365802\pi$
$354$	67310.0	0.0285477
$355$	0	0
$356$	3.85560e6	1.61238
$357$	266699.	0.110752
$358$	−877920.	−0.362032
$359$	−718685.	−0.294308	−0.147154	−	0.989114i	$-0.547011\pi$
$359$	−718685.	−0.294308	−0.147154	+	0.989114i	$0.547011\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	215684.	0.0865061
$363$	−92407.0	−0.0368077
$364$	3.50232e6	1.38549
$365$	0	0
$366$	−52436.0	−0.0204610
$367$	−3.03483e6	−1.17617	−0.588083	−	0.808800i	$-0.700117\pi$
$367$	−3.03483e6	−1.17617	−0.588083	+	0.808800i	$0.700117\pi$
$368$	1.31922e6	0.507804
$369$	2.48002e6	0.948175
$370$	0	0
$371$	5.89360e6	2.22303
$372$	43344.0	0.0162395
$373$	−920239.	−0.342475	−0.171237	−	0.985230i	$-0.554777\pi$
$373$	−920239.	−0.342475	−0.171237	+	0.985230i	$0.554777\pi$
$374$	836828.	0.309355
$375$	0	0
$376$	−829440.	−0.302563
$377$	790195.	0.286339
$378$	−161990.	−0.0583121
$379$	−3.49650e6	−1.25036	−0.625182	−	0.780479i	$-0.714975\pi$
$379$	−3.49650e6	−1.25036	−0.625182	+	0.780479i	$0.714975\pi$
$380$	0	0
$381$	86882.0	0.0306632
$382$	−1.40425e6	−0.492362
$383$	5.38811e6	1.87689	0.938446	−	0.345426i	$-0.112266\pi$
$383$	5.38811e6	1.87689	0.938446	+	0.345426i	$0.112266\pi$
$384$	−186240.	−0.0644533
$385$	0	0
$386$	−87548.0	−0.0299074
$387$	2.55165e6	0.866051
$388$	−620536.	−0.209261
$389$	−576510.	−0.193167	−0.0965835	−	0.995325i	$-0.530791\pi$
$389$	−576510.	−0.193167	−0.0965835	+	0.995325i	$0.530791\pi$
$390$	0	0
$391$	3.21157e6	1.06237
$392$	−1.32984e6	−0.437103
$393$	−309368.	−0.101040
$394$	292624.	0.0949663
$395$	0	0
$396$	1.77531e6	0.568901
$397$	704252.	0.224260	0.112130	−	0.993694i	$-0.464233\pi$
$397$	704252.	0.224260	0.112130	+	0.993694i	$0.464233\pi$
$398$	−1.35495e6	−0.428761
$399$	−60287.0	−0.0189580
$400$	0	0
$401$	−2.80013e6	−0.869595	−0.434797	−	0.900528i	$-0.643180\pi$
$401$	−2.80013e6	−0.869595	−0.434797	+	0.900528i	$0.643180\pi$
$402$	−90166.0	−0.0278277
$403$	1.15945e6	0.355623
$404$	2.62074e6	0.798861
$405$	0	0
$406$	−352370.	−0.106092
$407$	−2.45704e6	−0.735234
$408$	−191640.	−0.0569949
$409$	2.68312e6	0.793107	0.396554	−	0.918012i	$-0.370206\pi$
$409$	2.68312e6	0.793107	0.396554	+	0.918012i	$0.370206\pi$
$410$	0	0
$411$	301977.	0.0881798
$412$	4.15727e6	1.20661
$413$	5.62038e6	1.62140
$414$	−973324.	−0.279098
$415$	0	0
$416$	−3.85885e6	−1.09326
$417$	266420.	0.0750286
$418$	−189164.	−0.0529539
$419$	2.74162e6	0.762908	0.381454	−	0.924388i	$-0.375423\pi$
$419$	2.74162e6	0.762908	0.381454	+	0.924388i	$0.375423\pi$
$420$	0	0
$421$	−6.58570e6	−1.81091	−0.905455	−	0.424442i	$-0.860470\pi$
$421$	−6.58570e6	−1.81091	−0.905455	+	0.424442i	$0.860470\pi$
$422$	442614.	0.120988
$423$	−1.67270e6	−0.454536
$424$	−4.23492e6	−1.14401
$425$	0	0
$426$	61884.0	0.0165217
$427$	−4.37841e6	−1.16211
$428$	6.28076e6	1.65731
$429$	−196238.	−0.0514802
$430$	0	0
$431$	2.03796e6	0.528449	0.264224	−	0.964461i	$-0.414884\pi$
$431$	2.03796e6	0.528449	0.264224	+	0.964461i	$0.414884\pi$
$432$	−318160.	−0.0820231
$433$	3.55081e6	0.910138	0.455069	−	0.890456i	$-0.349615\pi$
$433$	3.55081e6	0.910138	0.455069	+	0.890456i	$0.349615\pi$
$434$	−517032.	−0.131763
$435$	0	0
$436$	5.35934e6	1.35019
$437$	−725971.	−0.181851
$438$	−93938.0	−0.0233968
$439$	2.16110e6	0.535197	0.267598	−	0.963531i	$-0.413770\pi$
$439$	2.16110e6	0.535197	0.267598	+	0.963531i	$0.413770\pi$
$440$	0	0
$441$	−2.68184e6	−0.656655
$442$	−2.39231e6	−0.582453
$443$	−844114.	−0.204358	−0.102179	−	0.994766i	$-0.532581\pi$
$443$	−844114.	−0.204358	−0.102179	+	0.994766i	$0.532581\pi$
$444$	262584.	0.0632136
$445$	0	0
$446$	−633948.	−0.150909
$447$	185400.	0.0438875
$448$	−1.78490e6	−0.420163
$449$	208350.	0.0487728	0.0243864	−	0.999703i	$-0.492237\pi$
$449$	208350.	0.0487728	0.0243864	+	0.999703i	$0.492237\pi$
$450$	0	0
$451$	−2.68498e6	−0.621583
$452$	3.15879e6	0.727235
$453$	−235638.	−0.0539510
$454$	304194.	0.0692645
$455$	0	0
$456$	43320.0	0.00975610
$457$	−4.74043e6	−1.06176	−0.530881	−	0.847446i	$-0.678139\pi$
$457$	−4.74043e6	−1.06176	−0.530881	+	0.847446i	$0.678139\pi$
$458$	2.70002e6	0.601455
$459$	−774545.	−0.171599
$460$	0	0
$461$	4.33213e6	0.949400	0.474700	−	0.880148i	$-0.342557\pi$
$461$	4.33213e6	0.949400	0.474700	+	0.880148i	$0.342557\pi$
$462$	87508.0	0.0190740
$463$	1.78328e6	0.386604	0.193302	−	0.981139i	$-0.438080\pi$
$463$	1.78328e6	0.386604	0.193302	+	0.981139i	$0.438080\pi$
$464$	−692080.	−0.149232
$465$	0	0
$466$	1.71937e6	0.366780
$467$	−3.11964e6	−0.661930	−0.330965	−	0.943643i	$-0.607374\pi$
$467$	−3.11964e6	−0.661930	−0.330965	+	0.943643i	$0.607374\pi$
$468$	−5.07522e6	−1.07113
$469$	−7.52886e6	−1.58051
$470$	0	0
$471$	−12018.0	−0.00249620
$472$	−4.03860e6	−0.834403
$473$	−2.76253e6	−0.567746
$474$	−128860.	−0.0263434
$475$	0	0
$476$	−7.46757e6	−1.51064
$477$	−8.54042e6	−1.71863
$478$	−2.15559e6	−0.431516
$479$	−6.33442e6	−1.26144	−0.630722	−	0.776009i	$-0.717241\pi$
$479$	−6.33442e6	−1.26144	−0.630722	+	0.776009i	$0.717241\pi$
$480$	0	0
$481$	7.02412e6	1.38430
$482$	3.40422e6	0.667422
$483$	335837.	0.0655029
$484$	2.58740e6	0.502053
$485$	0	0
$486$	352352.	0.0676684
$487$	5.13068e6	0.980286	0.490143	−	0.871642i	$-0.336945\pi$
$487$	5.13068e6	0.980286	0.490143	+	0.871642i	$0.336945\pi$
$488$	3.14616e6	0.598041
$489$	−10764.0	−0.00203564
$490$	0	0
$491$	6.35173e6	1.18902	0.594509	−	0.804089i	$-0.297346\pi$
$491$	6.35173e6	1.18902	0.594509	+	0.804089i	$0.297346\pi$
$492$	286944.	0.0534422
$493$	−1.68484e6	−0.312205
$494$	540778.	0.0997015
$495$	0	0
$496$	−1.01549e6	−0.185341
$497$	5.16731e6	0.938369
$498$	27972.0	0.00505417
$499$	1.09526e6	0.196909	0.0984546	−	0.995142i	$-0.468610\pi$
$499$	1.09526e6	0.196909	0.0984546	+	0.995142i	$0.468610\pi$
$500$	0	0
$501$	171332.	0.0304961
$502$	1.33120e6	0.235768
$503$	−874909.	−0.154185	−0.0770926	−	0.997024i	$-0.524564\pi$
$503$	−874909.	−0.154185	−0.0770926	+	0.997024i	$0.524564\pi$
$504$	4.84968e6	0.850426
$505$	0	0
$506$	1.05376e6	0.182965
$507$	189708.	0.0327767
$508$	−2.43270e6	−0.418243
$509$	−1.02477e7	−1.75321	−0.876604	−	0.481213i	$-0.840196\pi$
$509$	−1.02477e7	−1.75321	−0.876604	+	0.481213i	$0.840196\pi$
$510$	0	0
$511$	−7.84382e6	−1.32885
$512$	5.89875e6	0.994455
$513$	175085.	0.0293735
$514$	−1.61886e6	−0.270272
$515$	0	0
$516$	295232.	0.0488134
$517$	1.81094e6	0.297974
$518$	−3.13225e6	−0.512900
$519$	121866.	0.0198593
$520$	0	0
$521$	−6.01595e6	−0.970979	−0.485489	−	0.874243i	$-0.661359\pi$
$521$	−6.01595e6	−0.970979	−0.485489	+	0.874243i	$0.661359\pi$
$522$	510620.	0.0820203
$523$	−3.19743e6	−0.511148	−0.255574	−	0.966789i	$-0.582264\pi$
$523$	−3.19743e6	−0.511148	−0.255574	+	0.966789i	$0.582264\pi$
$524$	8.66230e6	1.37818
$525$	0	0
$526$	3.57723e6	0.563745
$527$	−2.47216e6	−0.387748
$528$	171872.	0.0268300
$529$	−2.39222e6	−0.371674
$530$	0	0
$531$	−8.14451e6	−1.25351
$532$	1.68804e6	0.258585
$533$	7.67575e6	1.17032
$534$	−275400.	−0.0417937
$535$	0	0
$536$	5.40996e6	0.813359
$537$	−438960.	−0.0656885
$538$	−642700.	−0.0957310
$539$	2.90348e6	0.430475
$540$	0	0
$541$	−4.86884e6	−0.715208	−0.357604	−	0.933873i	$-0.616406\pi$
$541$	−4.86884e6	−0.715208	−0.357604	+	0.933873i	$0.616406\pi$
$542$	−551046.	−0.0805730
$543$	107842.	0.0156960
$544$	8.22774e6	1.19202
$545$	0	0
$546$	−250166.	−0.0359126
$547$	7.54713e6	1.07848	0.539242	−	0.842151i	$-0.318711\pi$
$547$	7.54713e6	1.07848	0.539242	+	0.842151i	$0.318711\pi$
$548$	−8.45536e6	−1.20276
$549$	6.34476e6	0.898430
$550$	0	0
$551$	380855.	0.0534417
$552$	−241320.	−0.0337090
$553$	−1.07598e7	−1.49621
$554$	2.93838e6	0.406756
$555$	0	0
$556$	−7.45976e6	−1.02338
$557$	−5.86653e6	−0.801204	−0.400602	−	0.916252i	$-0.631199\pi$
$557$	−5.86653e6	−0.801204	−0.400602	+	0.916252i	$0.631199\pi$
$558$	749232.	0.101866
$559$	7.89746e6	1.06895
$560$	0	0
$561$	418414.	0.0561305
$562$	5.08438e6	0.679043
$563$	−4.29068e6	−0.570500	−0.285250	−	0.958453i	$-0.592077\pi$
$563$	−4.29068e6	−0.570500	−0.285250	+	0.958453i	$0.592077\pi$
$564$	−193536.	−0.0256191
$565$	0	0
$566$	−4.87683e6	−0.639877
$567$	9.73961e6	1.27228
$568$	−3.71304e6	−0.482902
$569$	2.80536e6	0.363252	0.181626	−	0.983368i	$-0.441864\pi$
$569$	2.80536e6	0.363252	0.181626	+	0.983368i	$0.441864\pi$
$570$	0	0
$571$	5.85289e6	0.751243	0.375621	−	0.926773i	$-0.377429\pi$
$571$	5.85289e6	0.751243	0.375621	+	0.926773i	$0.377429\pi$
$572$	5.49466e6	0.702184
$573$	−702123.	−0.0893360
$574$	−3.42283e6	−0.433617
$575$	0	0
$576$	2.58650e6	0.324830
$577$	−4.59012e6	−0.573964	−0.286982	−	0.957936i	$-0.592652\pi$
$577$	−4.59012e6	−0.573964	−0.286982	+	0.957936i	$0.592652\pi$
$578$	2.26110e6	0.281515
$579$	−43774.0	−0.00542650
$580$	0	0
$581$	2.33566e6	0.287058
$582$	44324.0	0.00542415
$583$	9.24624e6	1.12666
$584$	5.63628e6	0.683849
$585$	0	0
$586$	−345478.	−0.0415601
$587$	−477148.	−0.0571555	−0.0285777	−	0.999592i	$-0.509098\pi$
$587$	−477148.	−0.0571555	−0.0285777	+	0.999592i	$0.509098\pi$
$588$	−310296.	−0.0370112
$589$	558828.	0.0663728
$590$	0	0
$591$	146312.	0.0172310
$592$	−6.15197e6	−0.721456
$593$	−8.60867e6	−1.00531	−0.502654	−	0.864487i	$-0.667643\pi$
$593$	−8.60867e6	−1.00531	−0.502654	+	0.864487i	$0.667643\pi$
$594$	−254140.	−0.0295534
$595$	0	0
$596$	−5.19120e6	−0.598621
$597$	−677475.	−0.0777960
$598$	−3.01248e6	−0.344485
$599$	−1.35640e7	−1.54461	−0.772306	−	0.635251i	$-0.780897\pi$
$599$	−1.35640e7	−1.54461	−0.772306	+	0.635251i	$0.780897\pi$
$600$	0	0
$601$	−1.19487e7	−1.34939	−0.674693	−	0.738098i	$-0.735724\pi$
$601$	−1.19487e7	−1.34939	−0.674693	+	0.738098i	$0.735724\pi$
$602$	−3.52170e6	−0.396060
$603$	1.09101e7	1.22190
$604$	6.59786e6	0.735887
$605$	0	0
$606$	−187196.	−0.0207069
$607$	−5.47192e6	−0.602793	−0.301396	−	0.953499i	$-0.597453\pi$
$607$	−5.47192e6	−0.602793	−0.301396	+	0.953499i	$0.597453\pi$
$608$	−1.85987e6	−0.204044
$609$	−176185.	−0.0192498
$610$	0	0
$611$	−5.17709e6	−0.561025
$612$	1.08213e7	1.16788
$613$	−4.20527e6	−0.452005	−0.226002	−	0.974127i	$-0.572566\pi$
$613$	−4.20527e6	−0.452005	−0.226002	+	0.974127i	$0.572566\pi$
$614$	−4.70182e6	−0.503321
$615$	0	0
$616$	−5.25048e6	−0.557503
$617$	8.20450e6	0.867639	0.433820	−	0.901000i	$-0.357165\pi$
$617$	8.20450e6	0.867639	0.433820	+	0.901000i	$0.357165\pi$
$618$	−296948.	−0.0312759
$619$	−1.02115e7	−1.07118	−0.535591	−	0.844477i	$-0.679911\pi$
$619$	−1.02115e7	−1.07118	−0.535591	+	0.844477i	$0.679911\pi$
$620$	0	0
$621$	−975335.	−0.101490
$622$	6.63647e6	0.687799
$623$	−2.29959e7	−2.37372
$624$	−491344.	−0.0505154
$625$	0	0
$626$	3.02042e6	0.308058
$627$	−94582.0	−0.00960815
$628$	336504.	0.0340480
$629$	−1.49767e7	−1.50935
$630$	0	0
$631$	5.36515e6	0.536425	0.268212	−	0.963360i	$-0.413567\pi$
$631$	5.36515e6	0.536425	0.268212	+	0.963360i	$0.413567\pi$
$632$	7.73160e6	0.769975
$633$	221307.	0.0219526
$634$	−696286.	−0.0687962
$635$	0	0
$636$	−988148.	−0.0968677
$637$	−8.30042e6	−0.810497
$638$	−552820.	−0.0537690
$639$	−7.48796e6	−0.725457
$640$	0	0
$641$	1.38079e7	1.32734	0.663671	−	0.748025i	$-0.268998\pi$
$641$	1.38079e7	1.32734	0.663671	+	0.748025i	$0.268998\pi$
$642$	−448626.	−0.0429583
$643$	1.84380e7	1.75868	0.879338	−	0.476197i	$-0.157985\pi$
$643$	1.84380e7	1.75868	0.879338	+	0.476197i	$0.157985\pi$
$644$	−9.40344e6	−0.893453
$645$	0	0
$646$	−1.15303e6	−0.108708
$647$	−5.30937e6	−0.498635	−0.249317	−	0.968422i	$-0.580206\pi$
$647$	−5.30937e6	−0.498635	−0.249317	+	0.968422i	$0.580206\pi$
$648$	−6.99852e6	−0.654740
$649$	8.81761e6	0.821749
$650$	0	0
$651$	−258516.	−0.0239075
$652$	301392.	0.0277660
$653$	−1.83492e7	−1.68397	−0.841986	−	0.539500i	$-0.818613\pi$
$653$	−1.83492e7	−1.68397	−0.841986	+	0.539500i	$0.818613\pi$
$654$	−382810.	−0.0349977
$655$	0	0
$656$	−6.72269e6	−0.609935
$657$	1.13665e7	1.02734
$658$	2.30861e6	0.207867
$659$	1.64541e7	1.47591	0.737957	−	0.674847i	$-0.235791\pi$
$659$	1.64541e7	1.47591	0.737957	+	0.674847i	$0.235791\pi$
$660$	0	0
$661$	4.86195e6	0.432819	0.216410	−	0.976303i	$-0.430565\pi$
$661$	4.86195e6	0.432819	0.216410	+	0.976303i	$0.430565\pi$
$662$	−5.70813e6	−0.506231
$663$	−1.19615e6	−0.105682
$664$	−1.67832e6	−0.147725
$665$	0	0
$666$	4.53895e6	0.396524
$667$	−2.12160e6	−0.184650
$668$	−4.79730e6	−0.415964
$669$	−316974.	−0.0273816
$670$	0	0
$671$	−6.86912e6	−0.588972
$672$	860384.	0.0734969
$673$	1.29160e6	0.109923	0.0549616	−	0.998488i	$-0.482496\pi$
$673$	1.29160e6	0.109923	0.0549616	+	0.998488i	$0.482496\pi$
$674$	−711176.	−0.0603014
$675$	0	0
$676$	−5.31182e6	−0.447071
$677$	−7.75570e6	−0.650354	−0.325177	−	0.945653i	$-0.605424\pi$
$677$	−7.75570e6	−0.650354	−0.325177	+	0.945653i	$0.605424\pi$
$678$	−225628.	−0.0188503
$679$	3.70105e6	0.308071
$680$	0	0
$681$	152097.	0.0125676
$682$	−811152.	−0.0667792
$683$	−2.37260e6	−0.194614	−0.0973069	−	0.995254i	$-0.531023\pi$
$683$	−2.37260e6	−0.194614	−0.0973069	+	0.995254i	$0.531023\pi$
$684$	−2.44614e6	−0.199913
$685$	0	0
$686$	−1.91215e6	−0.155136
$687$	1.35001e6	0.109130
$688$	−6.91686e6	−0.557106
$689$	−2.64330e7	−2.12128
$690$	0	0
$691$	−1.82118e7	−1.45097	−0.725484	−	0.688239i	$-0.758384\pi$
$691$	−1.82118e7	−1.45097	−0.725484	+	0.688239i	$0.758384\pi$
$692$	−3.41225e6	−0.270879
$693$	−1.05885e7	−0.837529
$694$	438404.	0.0345522
$695$	0	0
$696$	126600.	0.00990628
$697$	−1.63661e7	−1.27603
$698$	−1.81146e6	−0.140731
$699$	859686.	0.0665498
$700$	0	0
$701$	−5.71599e6	−0.439335	−0.219668	−	0.975575i	$-0.570497\pi$
$701$	−5.71599e6	−0.439335	−0.219668	+	0.975575i	$0.570497\pi$
$702$	726530.	0.0556430
$703$	3.38546e6	0.258362
$704$	−2.80026e6	−0.212944
$705$	0	0
$706$	3.83222e6	0.289360
$707$	−1.56309e7	−1.17607
$708$	−942340.	−0.0706520
$709$	1.26458e7	0.944776	0.472388	−	0.881391i	$-0.343392\pi$
$709$	1.26458e7	0.944776	0.472388	+	0.881391i	$0.343392\pi$
$710$	0	0
$711$	1.55921e7	1.15672
$712$	1.65240e7	1.22156
$713$	−3.11303e6	−0.229329
$714$	533398.	0.0391567
$715$	0	0
$716$	1.22909e7	0.895984
$717$	−1.07780e6	−0.0782958
$718$	−1.43737e6	−0.104054
$719$	7.84702e6	0.566087	0.283043	−	0.959107i	$-0.408656\pi$
$719$	7.84702e6	0.566087	0.283043	+	0.959107i	$0.408656\pi$
$720$	0	0
$721$	−2.47952e7	−1.77635
$722$	260642.	0.0186081
$723$	1.70211e6	0.121100
$724$	−3.01958e6	−0.214092
$725$	0	0
$726$	−184814.	−0.0130135
$727$	1.55340e7	1.09005	0.545026	−	0.838419i	$-0.316520\pi$
$727$	1.55340e7	1.09005	0.545026	+	0.838419i	$0.316520\pi$
$728$	1.50100e7	1.04967
$729$	−1.39958e7	−0.975393
$730$	0	0
$731$	−1.68388e7	−1.16551
$732$	734104.	0.0506384
$733$	−4.45074e6	−0.305966	−0.152983	−	0.988229i	$-0.548888\pi$
$733$	−4.45074e6	−0.305966	−0.152983	+	0.988229i	$0.548888\pi$
$734$	−6.06966e6	−0.415838
$735$	0	0
$736$	1.03607e7	0.705007
$737$	−1.18117e7	−0.801024
$738$	4.96003e6	0.335231
$739$	8.62912e6	0.581240	0.290620	−	0.956839i	$-0.406138\pi$
$739$	8.62912e6	0.581240	0.290620	+	0.956839i	$0.406138\pi$
$740$	0	0
$741$	270389.	0.0180902
$742$	1.17872e7	0.785960
$743$	2.03501e7	1.35237	0.676183	−	0.736734i	$-0.263633\pi$
$743$	2.03501e7	1.35237	0.676183	+	0.736734i	$0.263633\pi$
$744$	185760.	0.0123033
$745$	0	0
$746$	−1.84048e6	−0.121083
$747$	−3.38461e6	−0.221926
$748$	−1.17156e7	−0.765615
$749$	−3.74603e7	−2.43987
$750$	0	0
$751$	−5.97197e6	−0.386383	−0.193191	−	0.981161i	$-0.561884\pi$
$751$	−5.97197e6	−0.386383	−0.193191	+	0.981161i	$0.561884\pi$
$752$	4.53427e6	0.292390
$753$	665602.	0.0427787
$754$	1.58039e6	0.101236
$755$	0	0
$756$	2.26786e6	0.144315
$757$	1.95462e6	0.123972	0.0619859	−	0.998077i	$-0.480257\pi$
$757$	1.95462e6	0.123972	0.0619859	+	0.998077i	$0.480257\pi$
$758$	−6.99301e6	−0.442070
$759$	526882.	0.0331978
$760$	0	0
$761$	−9.04197e6	−0.565981	−0.282990	−	0.959123i	$-0.591326\pi$
$761$	−9.04197e6	−0.565981	−0.282990	+	0.959123i	$0.591326\pi$
$762$	173764.	0.0108411
$763$	−3.19646e7	−1.98773
$764$	1.96594e7	1.21853
$765$	0	0
$766$	1.07762e7	0.663581
$767$	−2.52076e7	−1.54719
$768$	−30464.0	−0.00186373
$769$	−2.53152e6	−0.154371	−0.0771853	−	0.997017i	$-0.524593\pi$
$769$	−2.53152e6	−0.154371	−0.0771853	+	0.997017i	$0.524593\pi$
$770$	0	0
$771$	−809428.	−0.0490391
$772$	1.22567e6	0.0740169
$773$	1.23803e7	0.745219	0.372609	−	0.927988i	$-0.378463\pi$
$773$	1.23803e7	0.745219	0.372609	+	0.927988i	$0.378463\pi$
$774$	5.10330e6	0.306195
$775$	0	0
$776$	−2.65944e6	−0.158539
$777$	−1.56613e6	−0.0930624
$778$	−1.15302e6	−0.0682948
$779$	3.69953e6	0.218425
$780$	0	0
$781$	8.10680e6	0.475578
$782$	6.42313e6	0.375604
$783$	511675.	0.0298256
$784$	7.26979e6	0.422408
$785$	0	0
$786$	−618736.	−0.0357231
$787$	5.63282e6	0.324182	0.162091	−	0.986776i	$-0.448176\pi$
$787$	5.63282e6	0.324182	0.162091	+	0.986776i	$0.448176\pi$
$788$	−4.09674e6	−0.235030
$789$	1.78862e6	0.102288
$790$	0	0
$791$	−1.88399e7	−1.07063
$792$	7.60848e6	0.431008
$793$	1.96373e7	1.10892
$794$	1.40850e6	0.0792879
$795$	0	0
$796$	1.89693e7	1.06113
$797$	−6.81883e6	−0.380246	−0.190123	−	0.981760i	$-0.560889\pi$
$797$	−6.81883e6	−0.380246	−0.190123	+	0.981760i	$0.560889\pi$
$798$	−120574.	−0.00670265
$799$	1.10385e7	0.611705
$800$	0	0
$801$	3.33234e7	1.83513
$802$	−5.60026e6	−0.307448
$803$	−1.23059e7	−0.673479
$804$	1.26232e6	0.0688701
$805$	0	0
$806$	2.31890e6	0.125732
$807$	−321350.	−0.0173698
$808$	1.12318e7	0.605229
$809$	1.26018e6	0.0676960	0.0338480	−	0.999427i	$-0.489224\pi$
$809$	1.26018e6	0.0676960	0.0338480	+	0.999427i	$0.489224\pi$
$810$	0	0
$811$	5.18260e6	0.276691	0.138346	−	0.990384i	$-0.455822\pi$
$811$	5.18260e6	0.276691	0.138346	+	0.990384i	$0.455822\pi$
$812$	4.93318e6	0.262565
$813$	−275523.	−0.0146195
$814$	−4.91407e6	−0.259944
$815$	0	0
$816$	1.04763e6	0.0550787
$817$	3.80638e6	0.199507
$818$	5.36624e6	0.280406
$819$	3.02701e7	1.57690
$820$	0	0
$821$	−1.43552e7	−0.743276	−0.371638	−	0.928378i	$-0.621204\pi$
$821$	−1.43552e7	−0.743276	−0.371638	+	0.928378i	$0.621204\pi$
$822$	603954.	0.0311763
$823$	2.14492e7	1.10385	0.551927	−	0.833893i	$-0.313893\pi$
$823$	2.14492e7	1.10385	0.551927	+	0.833893i	$0.313893\pi$
$824$	1.78169e7	0.914142
$825$	0	0
$826$	1.12408e7	0.573252
$827$	3.97290e6	0.201996	0.100998	−	0.994887i	$-0.467796\pi$
$827$	3.97290e6	0.201996	0.100998	+	0.994887i	$0.467796\pi$
$828$	1.36265e7	0.690732
$829$	2.67452e7	1.35164	0.675819	−	0.737068i	$-0.263790\pi$
$829$	2.67452e7	1.35164	0.675819	+	0.737068i	$0.263790\pi$
$830$	0	0
$831$	1.46919e6	0.0738033
$832$	8.00531e6	0.400931
$833$	1.76980e7	0.883712
$834$	532840.	0.0265266
$835$	0	0
$836$	2.64830e6	0.131054
$837$	750780.	0.0370424
$838$	5.48324e6	0.269729
$839$	−2.50564e7	−1.22889	−0.614445	−	0.788960i	$-0.710620\pi$
$839$	−2.50564e7	−1.22889	−0.614445	+	0.788960i	$0.710620\pi$
$840$	0	0
$841$	−1.93981e7	−0.945736
$842$	−1.31714e7	−0.640253
$843$	2.54219e6	0.123208
$844$	−6.19660e6	−0.299431
$845$	0	0
$846$	−3.34541e6	−0.160703
$847$	−1.54320e7	−0.739116
$848$	2.31509e7	1.10555
$849$	−2.43841e6	−0.116102
$850$	0	0
$851$	−1.88592e7	−0.892685
$852$	−866376.	−0.0408891
$853$	−1.15098e7	−0.541622	−0.270811	−	0.962633i	$-0.587292\pi$
$853$	−1.15098e7	−0.541622	−0.270811	+	0.962633i	$0.587292\pi$
$854$	−8.75681e6	−0.410867
$855$	0	0
$856$	2.69176e7	1.25560
$857$	3.32557e7	1.54673	0.773364	−	0.633962i	$-0.218573\pi$
$857$	3.32557e7	1.54673	0.773364	+	0.633962i	$0.218573\pi$
$858$	−392476.	−0.0182010
$859$	4.41153e6	0.203989	0.101994	−	0.994785i	$-0.467478\pi$
$859$	4.41153e6	0.203989	0.101994	+	0.994785i	$0.467478\pi$
$860$	0	0
$861$	−1.71142e6	−0.0786770
$862$	4.07592e6	0.186835
$863$	1.51528e7	0.692576	0.346288	−	0.938128i	$-0.387442\pi$
$863$	1.51528e7	0.692576	0.346288	+	0.938128i	$0.387442\pi$
$864$	−2.49872e6	−0.113876
$865$	0	0
$866$	7.10161e6	0.321782
$867$	1.13055e6	0.0510790
$868$	7.23845e6	0.326097
$869$	−1.68807e7	−0.758298
$870$	0	0
$871$	3.37672e7	1.50817
$872$	2.29686e7	1.02292
$873$	−5.36320e6	−0.238171
$874$	−1.45194e6	−0.0642940
$875$	0	0
$876$	1.31513e6	0.0579041
$877$	2.21611e7	0.972952	0.486476	−	0.873694i	$-0.338282\pi$
$877$	2.21611e7	0.972952	0.486476	+	0.873694i	$0.338282\pi$
$878$	4.32220e6	0.189221
$879$	−172739.	−0.00754081
$880$	0	0
$881$	1.83390e6	0.0796043	0.0398021	−	0.999208i	$-0.487327\pi$
$881$	1.83390e6	0.0796043	0.0398021	+	0.999208i	$0.487327\pi$
$882$	−5.36369e6	−0.232162
$883$	7.90709e6	0.341283	0.170642	−	0.985333i	$-0.445416\pi$
$883$	7.90709e6	0.341283	0.170642	+	0.985333i	$0.445416\pi$
$884$	3.34923e7	1.44150
$885$	0	0
$886$	−1.68823e6	−0.0722515
$887$	−3.73237e7	−1.59285	−0.796427	−	0.604734i	$-0.793279\pi$
$887$	−3.73237e7	−1.59285	−0.796427	+	0.604734i	$0.793279\pi$
$888$	1.12536e6	0.0478916
$889$	1.45093e7	0.615732
$890$	0	0
$891$	1.52801e7	0.644810
$892$	8.87527e6	0.373482
$893$	−2.49523e6	−0.104709
$894$	370800.	0.0155166
$895$	0	0
$896$	−3.11021e7	−1.29425
$897$	−1.50624e6	−0.0625047
$898$	416700.	0.0172438
$899$	1.63314e6	0.0673945
$900$	0	0
$901$	5.63597e7	2.31290
$902$	−5.36995e6	−0.219763
$903$	−1.76085e6	−0.0718625
$904$	1.35377e7	0.550964
$905$	0	0
$906$	−471276.	−0.0190746
$907$	−4.65709e7	−1.87974	−0.939868	−	0.341539i	$-0.889052\pi$
$907$	−4.65709e7	−1.87974	−0.939868	+	0.341539i	$0.889052\pi$
$908$	−4.25872e6	−0.171421
$909$	2.26507e7	0.909227
$910$	0	0
$911$	6.77047e6	0.270286	0.135143	−	0.990826i	$-0.456851\pi$
$911$	6.77047e6	0.270286	0.135143	+	0.990826i	$0.456851\pi$
$912$	−236816.	−0.00942809
$913$	3.66433e6	0.145485
$914$	−9.48087e6	−0.375390
$915$	0	0
$916$	−3.78003e7	−1.48853
$917$	−5.16645e7	−2.02894
$918$	−1.54909e6	−0.0606694
$919$	−3.40525e7	−1.33003	−0.665014	−	0.746831i	$-0.731574\pi$
$919$	−3.40525e7	−1.33003	−0.665014	+	0.746831i	$0.731574\pi$
$920$	0	0
$921$	−2.35091e6	−0.0913243
$922$	8.66426e6	0.335664
$923$	−2.31756e7	−0.895418
$924$	−1.22511e6	−0.0472059
$925$	0	0
$926$	3.56655e6	0.136685
$927$	3.59307e7	1.37330
$928$	−5.43536e6	−0.207185
$929$	−3.31153e7	−1.25889	−0.629447	−	0.777043i	$-0.716719\pi$
$929$	−3.31153e7	−1.25889	−0.629447	+	0.777043i	$0.716719\pi$
$930$	0	0
$931$	−4.00060e6	−0.151269
$932$	−2.40712e7	−0.907733
$933$	3.31824e6	0.124797
$934$	−6.23928e6	−0.234028
$935$	0	0
$936$	−2.17510e7	−0.811501
$937$	2.86378e7	1.06559	0.532797	−	0.846243i	$-0.321141\pi$
$937$	2.86378e7	1.06559	0.532797	+	0.846243i	$0.321141\pi$
$938$	−1.50577e7	−0.558795
$939$	1.51021e6	0.0558951
$940$	0	0
$941$	3.01462e7	1.10983	0.554917	−	0.831906i	$-0.312750\pi$
$941$	3.01462e7	1.10983	0.554917	+	0.831906i	$0.312750\pi$
$942$	−24036.0	−0.000882541 0
$943$	−2.06087e7	−0.754696
$944$	2.20777e7	0.806350
$945$	0	0
$946$	−5.52506e6	−0.200728
$947$	−2.09887e7	−0.760521	−0.380260	−	0.924879i	$-0.624166\pi$
$947$	−2.09887e7	−0.760521	−0.380260	+	0.924879i	$0.624166\pi$
$948$	1.80404e6	0.0651967
$949$	3.51798e7	1.26802
$950$	0	0
$951$	−348143.	−0.0124826
$952$	−3.20039e7	−1.14449
$953$	2.52874e7	0.901928	0.450964	−	0.892542i	$-0.351080\pi$
$953$	2.52874e7	0.901928	0.450964	+	0.892542i	$0.351080\pi$
$954$	−1.70808e7	−0.607629
$955$	0	0
$956$	3.01783e7	1.06795
$957$	−276410.	−0.00975605
$958$	−1.26688e7	−0.445988
$959$	5.04302e7	1.77070
$960$	0	0
$961$	−2.62328e7	−0.916298
$962$	1.40482e7	0.489423
$963$	5.42837e7	1.88627
$964$	−4.76591e7	−1.65179
$965$	0	0
$966$	671674.	0.0231588
$967$	3.13246e7	1.07726	0.538628	−	0.842543i	$-0.318943\pi$
$967$	3.13246e7	1.07726	0.538628	+	0.842543i	$0.318943\pi$
$968$	1.10888e7	0.380363
$969$	−576517.	−0.0197243
$970$	0	0
$971$	1.38582e7	0.471693	0.235846	−	0.971790i	$-0.424214\pi$
$971$	1.38582e7	0.471693	0.235846	+	0.971790i	$0.424214\pi$
$972$	−4.93293e6	−0.167471
$973$	4.44921e7	1.50661
$974$	1.02614e7	0.346583
$975$	0	0
$976$	−1.71990e7	−0.577935
$977$	5.10509e7	1.71107	0.855533	−	0.517748i	$-0.173229\pi$
$977$	5.10509e7	1.71107	0.855533	+	0.517748i	$0.173229\pi$
$978$	−21528.0	−0.000719708 0
$979$	−3.60774e7	−1.20304
$980$	0	0
$981$	4.63200e7	1.53672
$982$	1.27035e7	0.420381
$983$	−2.44065e6	−0.0805605	−0.0402803	−	0.999188i	$-0.512825\pi$
$983$	−2.44065e6	−0.0805605	−0.0402803	+	0.999188i	$0.512825\pi$
$984$	1.22976e6	0.0404886
$985$	0	0
$986$	−3.36967e6	−0.110381
$987$	1.15430e6	0.0377161
$988$	−7.57089e6	−0.246749
$989$	−2.12040e7	−0.689329
$990$	0	0
$991$	2.61437e7	0.845633	0.422817	−	0.906215i	$-0.361041\pi$
$991$	2.61437e7	0.845633	0.422817	+	0.906215i	$0.361041\pi$
$992$	−7.97530e6	−0.257316
$993$	−2.85406e6	−0.0918524
$994$	1.03346e7	0.331764
$995$	0	0
$996$	−391608.	−0.0125084
$997$	−2.77522e7	−0.884220	−0.442110	−	0.896961i	$-0.645770\pi$
$997$	−2.77522e7	−0.884220	−0.442110	+	0.896961i	$0.645770\pi$
$998$	2.19052e6	0.0696179
$999$	4.54833e6	0.144191

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.6.a.a.1.1		1
5.4	even	2		19.6.a.b.1.1	✓	1
15.14	odd	2		171.6.a.b.1.1		1
20.19	odd	2		304.6.a.b.1.1		1
35.34	odd	2		931.6.a.b.1.1		1
95.94	odd	2		361.6.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.6.a.b.1.1	✓	1	5.4	even	2
171.6.a.b.1.1		1	15.14	odd	2
304.6.a.b.1.1		1	20.19	odd	2
361.6.a.b.1.1		1	95.94	odd	2
475.6.a.a.1.1		1	1.1	even	1	trivial
931.6.a.b.1.1		1	35.34	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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	475.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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