LMFDB - Embedded newform 845.6.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [845,6,Mod(1,845)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(845, 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("845.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 845 = 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	845.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:	$135.524327742$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	845.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q-5.00000 q^{2} +6.00000 q^{3} -7.00000 q^{4} +25.0000 q^{5} -30.0000 q^{6} +244.000 q^{7} +195.000 q^{8} -207.000 q^{9} -125.000 q^{10} -794.000 q^{11} -42.0000 q^{12} -1220.00 q^{14} +150.000 q^{15} -751.000 q^{16} -1534.00 q^{17} +1035.00 q^{18} -2706.00 q^{19} -175.000 q^{20} +1464.00 q^{21} +3970.00 q^{22} -702.000 q^{23} +1170.00 q^{24} +625.000 q^{25} -2700.00 q^{27} -1708.00 q^{28} -5038.00 q^{29} -750.000 q^{30} +3634.00 q^{31} -2485.00 q^{32} -4764.00 q^{33} +7670.00 q^{34} +6100.00 q^{35} +1449.00 q^{36} +7058.00 q^{37} +13530.0 q^{38} +4875.00 q^{40} +294.000 q^{41} -7320.00 q^{42} +7618.00 q^{43} +5558.00 q^{44} -5175.00 q^{45} +3510.00 q^{46} +3020.00 q^{47} -4506.00 q^{48} +42729.0 q^{49} -3125.00 q^{50} -9204.00 q^{51} +626.000 q^{53} +13500.0 q^{54} -19850.0 q^{55} +47580.0 q^{56} -16236.0 q^{57} +25190.0 q^{58} +30066.0 q^{59} -1050.00 q^{60} -5806.00 q^{61} -18170.0 q^{62} -50508.0 q^{63} +36457.0 q^{64} +23820.0 q^{66} +12436.0 q^{67} +10738.0 q^{68} -4212.00 q^{69} -30500.0 q^{70} -4734.00 q^{71} -40365.0 q^{72} +14694.0 q^{73} -35290.0 q^{74} +3750.00 q^{75} +18942.0 q^{76} -193736. q^{77} -39804.0 q^{79} -18775.0 q^{80} +34101.0 q^{81} -1470.00 q^{82} +41776.0 q^{83} -10248.0 q^{84} -38350.0 q^{85} -38090.0 q^{86} -30228.0 q^{87} -154830. q^{88} -7970.00 q^{89} +25875.0 q^{90} +4914.00 q^{92} +21804.0 q^{93} -15100.0 q^{94} -67650.0 q^{95} -14910.0 q^{96} +78050.0 q^{97} -213645. q^{98} +164358. q^{99} +O(q^{100})\)

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	−0.883883	−0.441942	−	0.897044i	$-0.645710\pi$
$2$	−5.00000	−0.883883	−0.441942	+	0.897044i	$0.645710\pi$
$3$	6.00000	0.384900	0.192450	−	0.981307i	$-0.438357\pi$
$3$	6.00000	0.384900	0.192450	+	0.981307i	$0.438357\pi$
$4$	−7.00000	−0.218750
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	−30.0000	−0.340207
$7$	244.000	1.88211	0.941054	−	0.338255i	$-0.109837\pi$
$7$	244.000	1.88211	0.941054	+	0.338255i	$0.109837\pi$
$8$	195.000	1.07723
$9$	−207.000	−0.851852
$10$	−125.000	−0.395285
$11$	−794.000	−1.97851	−0.989256	−	0.146192i	$-0.953298\pi$
$11$	−794.000	−1.97851	−0.989256	+	0.146192i	$0.953298\pi$
$12$	−42.0000	−0.0841969
$13$	0	0
$13$	0	0
$14$	−1220.00	−1.66356
$15$	150.000	0.172133
$16$	−751.000	−0.733398
$17$	−1534.00	−1.28737	−0.643685	−	0.765291i	$-0.722595\pi$
$17$	−1534.00	−1.28737	−0.643685	+	0.765291i	$0.722595\pi$
$18$	1035.00	0.752938
$19$	−2706.00	−1.71966	−0.859832	−	0.510576i	$-0.829432\pi$
$19$	−2706.00	−1.71966	−0.859832	+	0.510576i	$0.829432\pi$
$20$	−175.000	−0.0978280
$21$	1464.00	0.724424
$22$	3970.00	1.74877
$23$	−702.000	−0.276705	−0.138353	−	0.990383i	$-0.544181\pi$
$23$	−702.000	−0.276705	−0.138353	+	0.990383i	$0.544181\pi$
$24$	1170.00	0.414627
$25$	625.000	0.200000
$26$	0	0
$27$	−2700.00	−0.712778
$28$	−1708.00	−0.411711
$29$	−5038.00	−1.11241	−0.556203	−	0.831047i	$-0.687742\pi$
$29$	−5038.00	−1.11241	−0.556203	+	0.831047i	$0.687742\pi$
$30$	−750.000	−0.152145
$31$	3634.00	0.679173	0.339587	−	0.940575i	$-0.389713\pi$
$31$	3634.00	0.679173	0.339587	+	0.940575i	$0.389713\pi$
$32$	−2485.00	−0.428994
$33$	−4764.00	−0.761530
$34$	7670.00	1.13788
$35$	6100.00	0.841705
$36$	1449.00	0.186343
$37$	7058.00	0.847573	0.423787	−	0.905762i	$-0.360701\pi$
$37$	7058.00	0.847573	0.423787	+	0.905762i	$0.360701\pi$
$38$	13530.0	1.51998
$39$	0	0
$40$	4875.00	0.481753
$41$	294.000	0.0273141	0.0136571	−	0.999907i	$-0.495653\pi$
$41$	294.000	0.0273141	0.0136571	+	0.999907i	$0.495653\pi$
$42$	−7320.00	−0.640306
$43$	7618.00	0.628304	0.314152	−	0.949373i	$-0.398280\pi$
$43$	7618.00	0.628304	0.314152	+	0.949373i	$0.398280\pi$
$44$	5558.00	0.432800
$45$	−5175.00	−0.380960
$46$	3510.00	0.244575
$47$	3020.00	0.199417	0.0997085	−	0.995017i	$-0.468209\pi$
$47$	3020.00	0.199417	0.0997085	+	0.995017i	$0.468209\pi$
$48$	−4506.00	−0.282285
$49$	42729.0	2.54233
$50$	−3125.00	−0.176777
$51$	−9204.00	−0.495509
$52$	0	0
$53$	626.000	0.0306115	0.0153058	−	0.999883i	$-0.495128\pi$
$53$	626.000	0.0306115	0.0153058	+	0.999883i	$0.495128\pi$
$54$	13500.0	0.630013
$55$	−19850.0	−0.884818
$56$	47580.0	2.02747
$57$	−16236.0	−0.661899
$58$	25190.0	0.983237
$59$	30066.0	1.12446	0.562232	−	0.826979i	$-0.309943\pi$
$59$	30066.0	1.12446	0.562232	+	0.826979i	$0.309943\pi$
$60$	−1050.00	−0.0376540
$61$	−5806.00	−0.199780	−0.0998901	−	0.994998i	$-0.531849\pi$
$61$	−5806.00	−0.199780	−0.0998901	+	0.994998i	$0.531849\pi$
$62$	−18170.0	−0.600310
$63$	−50508.0	−1.60328
$64$	36457.0	1.11258
$65$	0	0
$66$	23820.0	0.673104
$67$	12436.0	0.338449	0.169225	−	0.985577i	$-0.445874\pi$
$67$	12436.0	0.338449	0.169225	+	0.985577i	$0.445874\pi$
$68$	10738.0	0.281612
$69$	−4212.00	−0.106504
$70$	−30500.0	−0.743969
$71$	−4734.00	−0.111451	−0.0557253	−	0.998446i	$-0.517747\pi$
$71$	−4734.00	−0.111451	−0.0557253	+	0.998446i	$0.517747\pi$
$72$	−40365.0	−0.917643
$73$	14694.0	0.322725	0.161363	−	0.986895i	$-0.448411\pi$
$73$	14694.0	0.322725	0.161363	+	0.986895i	$0.448411\pi$
$74$	−35290.0	−0.749156
$75$	3750.00	0.0769800
$76$	18942.0	0.376177
$77$	−193736.	−3.72378
$78$	0	0
$79$	−39804.0	−0.717561	−0.358781	−	0.933422i	$-0.616807\pi$
$79$	−39804.0	−0.717561	−0.358781	+	0.933422i	$0.616807\pi$
$80$	−18775.0	−0.327986
$81$	34101.0	0.577503
$82$	−1470.00	−0.0241425
$83$	41776.0	0.665628	0.332814	−	0.942992i	$-0.392002\pi$
$83$	41776.0	0.665628	0.332814	+	0.942992i	$0.392002\pi$
$84$	−10248.0	−0.158468
$85$	−38350.0	−0.575729
$86$	−38090.0	−0.555348
$87$	−30228.0	−0.428165
$88$	−154830.	−2.13132
$89$	−7970.00	−0.106656	−0.0533278	−	0.998577i	$-0.516983\pi$
$89$	−7970.00	−0.106656	−0.0533278	+	0.998577i	$0.516983\pi$
$90$	25875.0	0.336724
$91$	0	0
$92$	4914.00	0.0605293
$93$	21804.0	0.261414
$94$	−15100.0	−0.176261
$95$	−67650.0	−0.769057
$96$	−14910.0	−0.165120
$97$	78050.0	0.842255	0.421127	−	0.907001i	$-0.361634\pi$
$97$	78050.0	0.842255	0.421127	+	0.907001i	$0.361634\pi$
$98$	−213645.	−2.24713
$99$	164358.	1.68540
$100$	−4375.00	−0.0437500
$101$	−23010.0	−0.224447	−0.112223	−	0.993683i	$-0.535797\pi$
$101$	−23010.0	−0.224447	−0.112223	+	0.993683i	$0.535797\pi$
$102$	46020.0	0.437972
$103$	121706.	1.13037	0.565183	−	0.824966i	$-0.308806\pi$
$103$	121706.	1.13037	0.565183	+	0.824966i	$0.308806\pi$
$104$	0	0
$105$	36600.0	0.323972
$106$	−3130.00	−0.0270570
$107$	−70142.0	−0.592269	−0.296134	−	0.955146i	$-0.595698\pi$
$107$	−70142.0	−0.592269	−0.296134	+	0.955146i	$0.595698\pi$
$108$	18900.0	0.155920
$109$	195878.	1.57914	0.789568	−	0.613663i	$-0.210305\pi$
$109$	195878.	1.57914	0.789568	+	0.613663i	$0.210305\pi$
$110$	99250.0	0.782076
$111$	42348.0	0.326231
$112$	−183244.	−1.38034
$113$	−100238.	−0.738476	−0.369238	−	0.929335i	$-0.620381\pi$
$113$	−100238.	−0.738476	−0.369238	+	0.929335i	$0.620381\pi$
$114$	81180.0	0.585042
$115$	−17550.0	−0.123746
$116$	35266.0	0.243339
$117$	0	0
$118$	−150330.	−0.993895
$119$	−374296.	−2.42297
$120$	29250.0	0.185427
$121$	469385.	2.91451
$122$	29030.0	0.176582
$123$	1764.00	0.0105132
$124$	−25438.0	−0.148569
$125$	15625.0	0.0894427
$126$	252540.	1.41711
$127$	39286.0	0.216137	0.108068	−	0.994143i	$-0.465533\pi$
$127$	39286.0	0.216137	0.108068	+	0.994143i	$0.465533\pi$
$128$	−102765.	−0.554396
$129$	45708.0	0.241834
$130$	0	0
$131$	211460.	1.07659	0.538295	−	0.842757i	$-0.319069\pi$
$131$	211460.	1.07659	0.538295	+	0.842757i	$0.319069\pi$
$132$	33348.0	0.166585
$133$	−660264.	−3.23660
$134$	−62180.0	−0.299150
$135$	−67500.0	−0.318764
$136$	−299130.	−1.38680
$137$	−26302.0	−0.119726	−0.0598628	−	0.998207i	$-0.519066\pi$
$137$	−26302.0	−0.119726	−0.0598628	+	0.998207i	$0.519066\pi$
$138$	21060.0	0.0941371
$139$	1344.00	0.00590014	0.00295007	−	0.999996i	$-0.499061\pi$
$139$	1344.00	0.00590014	0.00295007	+	0.999996i	$0.499061\pi$
$140$	−42700.0	−0.184123
$141$	18120.0	0.0767557
$142$	23670.0	0.0985093
$143$	0	0
$144$	155457.	0.624747
$145$	−125950.	−0.497483
$146$	−73470.0	−0.285251
$147$	256374.	0.978545
$148$	−49406.0	−0.185407
$149$	49086.0	0.181131	0.0905653	−	0.995891i	$-0.471133\pi$
$149$	49086.0	0.181131	0.0905653	+	0.995891i	$0.471133\pi$
$150$	−18750.0	−0.0680414
$151$	357998.	1.27773	0.638864	−	0.769320i	$-0.279405\pi$
$151$	357998.	1.27773	0.638864	+	0.769320i	$0.279405\pi$
$152$	−527670.	−1.85248
$153$	317538.	1.09665
$154$	968680.	3.29138
$155$	90850.0	0.303736
$156$	0	0
$157$	45450.0	0.147158	0.0735791	−	0.997289i	$-0.476558\pi$
$157$	45450.0	0.147158	0.0735791	+	0.997289i	$0.476558\pi$
$158$	199020.	0.634241
$159$	3756.00	0.0117824
$160$	−62125.0	−0.191852
$161$	−171288.	−0.520790
$162$	−170505.	−0.510446
$163$	−5892.00	−0.0173698	−0.00868488	−	0.999962i	$-0.502765\pi$
$163$	−5892.00	−0.0173698	−0.00868488	+	0.999962i	$0.502765\pi$
$164$	−2058.00	−0.00597497
$165$	−119100.	−0.340566
$166$	−208880.	−0.588338
$167$	−212772.	−0.590369	−0.295184	−	0.955440i	$-0.595381\pi$
$167$	−212772.	−0.590369	−0.295184	+	0.955440i	$0.595381\pi$
$168$	285480.	0.780373
$169$	0	0
$170$	191750.	0.508877
$171$	560142.	1.46490
$172$	−53326.0	−0.137442
$173$	503178.	1.27822	0.639111	−	0.769114i	$-0.279302\pi$
$173$	503178.	1.27822	0.639111	+	0.769114i	$0.279302\pi$
$174$	151140.	0.378448
$175$	152500.	0.376422
$176$	596294.	1.45104
$177$	180396.	0.432806
$178$	39850.0	0.0942710
$179$	581724.	1.35701	0.678507	−	0.734594i	$-0.262627\pi$
$179$	581724.	1.35701	0.678507	+	0.734594i	$0.262627\pi$
$180$	36225.0	0.0833349
$181$	202202.	0.458764	0.229382	−	0.973337i	$-0.426330\pi$
$181$	202202.	0.458764	0.229382	+	0.973337i	$0.426330\pi$
$182$	0	0
$183$	−34836.0	−0.0768954
$184$	−136890.	−0.298076
$185$	176450.	0.379046
$186$	−109020.	−0.231059
$187$	1.21800e6	2.54708
$188$	−21140.0	−0.0436225
$189$	−658800.	−1.34153
$190$	338250.	0.679757
$191$	−340608.	−0.675572	−0.337786	−	0.941223i	$-0.609678\pi$
$191$	−340608.	−0.675572	−0.337786	+	0.941223i	$0.609678\pi$
$192$	218742.	0.428232
$193$	−275614.	−0.532608	−0.266304	−	0.963889i	$-0.585803\pi$
$193$	−275614.	−0.532608	−0.266304	+	0.963889i	$0.585803\pi$
$194$	−390250.	−0.744455
$195$	0	0
$196$	−299103.	−0.556135
$197$	−538218.	−0.988081	−0.494041	−	0.869439i	$-0.664481\pi$
$197$	−538218.	−0.988081	−0.494041	+	0.869439i	$0.664481\pi$
$198$	−821790.	−1.48970
$199$	−853840.	−1.52842	−0.764212	−	0.644965i	$-0.776872\pi$
$199$	−853840.	−1.52842	−0.764212	+	0.644965i	$0.776872\pi$
$200$	121875.	0.215447
$201$	74616.0	0.130269
$202$	115050.	0.198385
$203$	−1.22927e6	−2.09367
$204$	64428.0	0.108392
$205$	7350.00	0.0122153
$206$	−608530.	−0.999111
$207$	145314.	0.235712
$208$	0	0
$209$	2.14856e6	3.40238
$210$	−183000.	−0.286354
$211$	−1.00112e6	−1.54804	−0.774019	−	0.633162i	$-0.781757\pi$
$211$	−1.00112e6	−1.54804	−0.774019	+	0.633162i	$0.781757\pi$
$212$	−4382.00	−0.00669627
$213$	−28404.0	−0.0428974
$214$	350710.	0.523496
$215$	190450.	0.280986
$216$	−526500.	−0.767828
$217$	886696.	1.27828
$218$	−979390.	−1.39577
$219$	88164.0	0.124217
$220$	138950.	0.193554
$221$	0	0
$222$	−211740.	−0.288350
$223$	−21364.0	−0.0287687	−0.0143844	−	0.999897i	$-0.504579\pi$
$223$	−21364.0	−0.0287687	−0.0143844	+	0.999897i	$0.504579\pi$
$224$	−606340.	−0.807414
$225$	−129375.	−0.170370
$226$	501190.	0.652727
$227$	880748.	1.13445	0.567227	−	0.823561i	$-0.308016\pi$
$227$	880748.	1.13445	0.567227	+	0.823561i	$0.308016\pi$
$228$	113652.	0.144790
$229$	13030.0	0.0164193	0.00820967	−	0.999966i	$-0.497387\pi$
$229$	13030.0	0.0164193	0.00820967	+	0.999966i	$0.497387\pi$
$230$	87750.0	0.109377
$231$	−1.16242e6	−1.43328
$232$	−982410.	−1.19832
$233$	−1.20700e6	−1.45652	−0.728260	−	0.685300i	$-0.759671\pi$
$233$	−1.20700e6	−1.45652	−0.728260	+	0.685300i	$0.759671\pi$
$234$	0	0
$235$	75500.0	0.0891820
$236$	−210462.	−0.245977
$237$	−238824.	−0.276189
$238$	1.87148e6	2.14162
$239$	187038.	0.211804	0.105902	−	0.994377i	$-0.466227\pi$
$239$	187038.	0.211804	0.105902	+	0.994377i	$0.466227\pi$
$240$	−112650.	−0.126242
$241$	−271690.	−0.301322	−0.150661	−	0.988585i	$-0.548140\pi$
$241$	−271690.	−0.301322	−0.150661	+	0.988585i	$0.548140\pi$
$242$	−2.34692e6	−2.57609
$243$	860706.	0.935059
$244$	40642.0	0.0437019
$245$	1.06822e6	1.13697
$246$	−8820.00	−0.00929246
$247$	0	0
$248$	708630.	0.731628
$249$	250656.	0.256200
$250$	−78125.0	−0.0790569
$251$	102648.	0.102841	0.0514205	−	0.998677i	$-0.483625\pi$
$251$	102648.	0.102841	0.0514205	+	0.998677i	$0.483625\pi$
$252$	353556.	0.350717
$253$	557388.	0.547465
$254$	−196430.	−0.191040
$255$	−230100.	−0.221598
$256$	−652799.	−0.622558
$257$	−221182.	−0.208890	−0.104445	−	0.994531i	$-0.533307\pi$
$257$	−221182.	−0.208890	−0.104445	+	0.994531i	$0.533307\pi$
$258$	−228540.	−0.213753
$259$	1.72215e6	1.59523
$260$	0	0
$261$	1.04287e6	0.947605
$262$	−1.05730e6	−0.951579
$263$	1.40317e6	1.25090	0.625449	−	0.780265i	$-0.284916\pi$
$263$	1.40317e6	1.25090	0.625449	+	0.780265i	$0.284916\pi$
$264$	−928980.	−0.820345
$265$	15650.0	0.0136899
$266$	3.30132e6	2.86077
$267$	−47820.0	−0.0410517
$268$	−87052.0	−0.0740358
$269$	−582954.	−0.491195	−0.245597	−	0.969372i	$-0.578984\pi$
$269$	−582954.	−0.491195	−0.245597	+	0.969372i	$0.578984\pi$
$270$	337500.	0.281750
$271$	1.04690e6	0.865930	0.432965	−	0.901411i	$-0.357467\pi$
$271$	1.04690e6	0.865930	0.432965	+	0.901411i	$0.357467\pi$
$272$	1.15203e6	0.944154
$273$	0	0
$274$	131510.	0.105824
$275$	−496250.	−0.395702
$276$	29484.0	0.0232977
$277$	1.10461e6	0.864987	0.432493	−	0.901637i	$-0.357634\pi$
$277$	1.10461e6	0.864987	0.432493	+	0.901637i	$0.357634\pi$
$278$	−6720.00	−0.00521504
$279$	−752238.	−0.578555
$280$	1.18950e6	0.906712
$281$	−908826.	−0.686618	−0.343309	−	0.939223i	$-0.611548\pi$
$281$	−908826.	−0.686618	−0.343309	+	0.939223i	$0.611548\pi$
$282$	−90600.0	−0.0678431
$283$	−449254.	−0.333446	−0.166723	−	0.986004i	$-0.553319\pi$
$283$	−449254.	−0.333446	−0.166723	+	0.986004i	$0.553319\pi$
$284$	33138.0	0.0243798
$285$	−405900.	−0.296010
$286$	0	0
$287$	71736.0	0.0514082
$288$	514395.	0.365440
$289$	933299.	0.657319
$290$	629750.	0.439717
$291$	468300.	0.324184
$292$	−102858.	−0.0705961
$293$	1.96083e6	1.33435	0.667175	−	0.744901i	$-0.267503\pi$
$293$	1.96083e6	1.33435	0.667175	+	0.744901i	$0.267503\pi$
$294$	−1.28187e6	−0.864919
$295$	751650.	0.502876
$296$	1.37631e6	0.913034
$297$	2.14380e6	1.41024
$298$	−245430.	−0.160098
$299$	0	0
$300$	−26250.0	−0.0168394
$301$	1.85879e6	1.18254
$302$	−1.78999e6	−1.12936
$303$	−138060.	−0.0863896
$304$	2.03221e6	1.26120
$305$	−145150.	−0.0893444
$306$	−1.58769e6	−0.969309
$307$	1.79385e6	1.08627	0.543137	−	0.839644i	$-0.317236\pi$
$307$	1.79385e6	1.08627	0.543137	+	0.839644i	$0.317236\pi$
$308$	1.35615e6	0.814576
$309$	730236.	0.435078
$310$	−454250.	−0.268467
$311$	2.41233e6	1.41428	0.707141	−	0.707072i	$-0.249985\pi$
$311$	2.41233e6	1.41428	0.707141	+	0.707072i	$0.249985\pi$
$312$	0	0
$313$	−2.15436e6	−1.24296	−0.621480	−	0.783430i	$-0.713468\pi$
$313$	−2.15436e6	−1.24296	−0.621480	+	0.783430i	$0.713468\pi$
$314$	−227250.	−0.130071
$315$	−1.26270e6	−0.717008
$316$	278628.	0.156967
$317$	−2.59616e6	−1.45105	−0.725526	−	0.688195i	$-0.758403\pi$
$317$	−2.59616e6	−1.45105	−0.725526	+	0.688195i	$0.758403\pi$
$318$	−18780.0	−0.0104142
$319$	4.00017e6	2.20091
$320$	911425.	0.497561
$321$	−420852.	−0.227964
$322$	856440.	0.460317
$323$	4.15100e6	2.21384
$324$	−238707.	−0.126329
$325$	0	0
$326$	29460.0	0.0153528
$327$	1.17527e6	0.607810
$328$	57330.0	0.0294237
$329$	736880.	0.375325
$330$	595500.	0.301021
$331$	917226.	0.460157	0.230079	−	0.973172i	$-0.426102\pi$
$331$	917226.	0.460157	0.230079	+	0.973172i	$0.426102\pi$
$332$	−292432.	−0.145606
$333$	−1.46101e6	−0.722007
$334$	1.06386e6	0.521817
$335$	310900.	0.151359
$336$	−1.09946e6	−0.531291
$337$	2.23894e6	1.07391	0.536954	−	0.843611i	$-0.319575\pi$
$337$	2.23894e6	1.07391	0.536954	+	0.843611i	$0.319575\pi$
$338$	0	0
$339$	−601428.	−0.284239
$340$	268450.	0.125941
$341$	−2.88540e6	−1.34375
$342$	−2.80071e6	−1.29480
$343$	6.32497e6	2.90284
$344$	1.48551e6	0.676830
$345$	−105300.	−0.0476300
$346$	−2.51589e6	−1.12980
$347$	3.41808e6	1.52391	0.761954	−	0.647631i	$-0.224240\pi$
$347$	3.41808e6	1.52391	0.761954	+	0.647631i	$0.224240\pi$
$348$	211596.	0.0936611
$349$	−2.35691e6	−1.03581	−0.517905	−	0.855438i	$-0.673288\pi$
$349$	−2.35691e6	−1.03581	−0.517905	+	0.855438i	$0.673288\pi$
$350$	−762500.	−0.332713
$351$	0	0
$352$	1.97309e6	0.848770
$353$	3.76395e6	1.60771	0.803854	−	0.594827i	$-0.202779\pi$
$353$	3.76395e6	1.60771	0.803854	+	0.594827i	$0.202779\pi$
$354$	−901980.	−0.382550
$355$	−118350.	−0.0498422
$356$	55790.0	0.0233309
$357$	−2.24578e6	−0.932601
$358$	−2.90862e6	−1.19944
$359$	−3.28216e6	−1.34407	−0.672037	−	0.740517i	$-0.734581\pi$
$359$	−3.28216e6	−1.34407	−0.672037	+	0.740517i	$0.734581\pi$
$360$	−1.00913e6	−0.410382
$361$	4.84634e6	1.95725
$362$	−1.01101e6	−0.405494
$363$	2.81631e6	1.12180
$364$	0	0
$365$	367350.	0.144327
$366$	174180.	0.0679666
$367$	−2.42605e6	−0.940233	−0.470116	−	0.882604i	$-0.655788\pi$
$367$	−2.42605e6	−0.940233	−0.470116	+	0.882604i	$0.655788\pi$
$368$	527202.	0.202935
$369$	−60858.0	−0.0232676
$370$	−882250.	−0.335033
$371$	152744.	0.0576142
$372$	−152628.	−0.0571843
$373$	2.80635e6	1.04441	0.522204	−	0.852820i	$-0.325110\pi$
$373$	2.80635e6	1.04441	0.522204	+	0.852820i	$0.325110\pi$
$374$	−6.08998e6	−2.25132
$375$	93750.0	0.0344265
$376$	588900.	0.214819
$377$	0	0
$378$	3.29400e6	1.18575
$379$	−3.15392e6	−1.12785	−0.563927	−	0.825825i	$-0.690710\pi$
$379$	−3.15392e6	−1.12785	−0.563927	+	0.825825i	$0.690710\pi$
$380$	473550.	0.168231
$381$	235716.	0.0831911
$382$	1.70304e6	0.597127
$383$	475044.	0.165477	0.0827384	−	0.996571i	$-0.473633\pi$
$383$	475044.	0.165477	0.0827384	+	0.996571i	$0.473633\pi$
$384$	−616590.	−0.213387
$385$	−4.84340e6	−1.66532
$386$	1.37807e6	0.470764
$387$	−1.57693e6	−0.535222
$388$	−546350.	−0.184243
$389$	150566.	0.0504490	0.0252245	−	0.999682i	$-0.491970\pi$
$389$	150566.	0.0504490	0.0252245	+	0.999682i	$0.491970\pi$
$390$	0	0
$391$	1.07687e6	0.356222
$392$	8.33216e6	2.73869
$393$	1.26876e6	0.414379
$394$	2.69109e6	0.873349
$395$	−995100.	−0.320903
$396$	−1.15051e6	−0.368681
$397$	−241686.	−0.0769618	−0.0384809	−	0.999259i	$-0.512252\pi$
$397$	−241686.	−0.0769618	−0.0384809	+	0.999259i	$0.512252\pi$
$398$	4.26920e6	1.35095
$399$	−3.96158e6	−1.24577
$400$	−469375.	−0.146680
$401$	3.19679e6	0.992780	0.496390	−	0.868100i	$-0.334659\pi$
$401$	3.19679e6	0.992780	0.496390	+	0.868100i	$0.334659\pi$
$402$	−373080.	−0.115143
$403$	0	0
$404$	161070.	0.0490977
$405$	852525.	0.258267
$406$	6.14636e6	1.85056
$407$	−5.60405e6	−1.67693
$408$	−1.79478e6	−0.533778
$409$	−423282.	−0.125119	−0.0625593	−	0.998041i	$-0.519926\pi$
$409$	−423282.	−0.125119	−0.0625593	+	0.998041i	$0.519926\pi$
$410$	−36750.0	−0.0107969
$411$	−157812.	−0.0460824
$412$	−851942.	−0.247267
$413$	7.33610e6	2.11636
$414$	−726570.	−0.208342
$415$	1.04440e6	0.297678
$416$	0	0
$417$	8064.00	0.00227096
$418$	−1.07428e7	−3.00731
$419$	−1.13159e6	−0.314887	−0.157444	−	0.987528i	$-0.550325\pi$
$419$	−1.13159e6	−0.314887	−0.157444	+	0.987528i	$0.550325\pi$
$420$	−256200.	−0.0708689
$421$	−3.47699e6	−0.956088	−0.478044	−	0.878336i	$-0.658654\pi$
$421$	−3.47699e6	−0.956088	−0.478044	+	0.878336i	$0.658654\pi$
$422$	5.00562e6	1.36829
$423$	−625140.	−0.169874
$424$	122070.	0.0329757
$425$	−958750.	−0.257474
$426$	142020.	0.0379163
$427$	−1.41666e6	−0.376008
$428$	490994.	0.129559
$429$	0	0
$430$	−952250.	−0.248359
$431$	−3.41044e6	−0.884335	−0.442168	−	0.896932i	$-0.645790\pi$
$431$	−3.41044e6	−0.884335	−0.442168	+	0.896932i	$0.645790\pi$
$432$	2.02770e6	0.522750
$433$	−3.40722e6	−0.873335	−0.436667	−	0.899623i	$-0.643841\pi$
$433$	−3.40722e6	−0.873335	−0.436667	+	0.899623i	$0.643841\pi$
$434$	−4.43348e6	−1.12985
$435$	−755700.	−0.191481
$436$	−1.37115e6	−0.345436
$437$	1.89961e6	0.475840
$438$	−440820.	−0.109793
$439$	−7.09114e6	−1.75612	−0.878061	−	0.478549i	$-0.841163\pi$
$439$	−7.09114e6	−1.75612	−0.878061	+	0.478549i	$0.841163\pi$
$440$	−3.87075e6	−0.953155
$441$	−8.84490e6	−2.16569
$442$	0	0
$443$	−8.23508e6	−1.99369	−0.996847	−	0.0793445i	$-0.974717\pi$
$443$	−8.23508e6	−1.99369	−0.996847	+	0.0793445i	$0.974717\pi$
$444$	−296436.	−0.0713631
$445$	−199250.	−0.0476978
$446$	106820.	0.0254282
$447$	294516.	0.0697172
$448$	8.89551e6	2.09400
$449$	1.29601e6	0.303383	0.151691	−	0.988428i	$-0.451528\pi$
$449$	1.29601e6	0.303383	0.151691	+	0.988428i	$0.451528\pi$
$450$	646875.	0.150588
$451$	−233436.	−0.0540414
$452$	701666.	0.161542
$453$	2.14799e6	0.491798
$454$	−4.40374e6	−1.00273
$455$	0	0
$456$	−3.16602e6	−0.713020
$457$	−1.68196e6	−0.376725	−0.188363	−	0.982100i	$-0.560318\pi$
$457$	−1.68196e6	−0.376725	−0.188363	+	0.982100i	$0.560318\pi$
$458$	−65150.0	−0.0145128
$459$	4.14180e6	0.917608
$460$	122850.	0.0270695
$461$	3.20663e6	0.702743	0.351372	−	0.936236i	$-0.385715\pi$
$461$	3.20663e6	0.702743	0.351372	+	0.936236i	$0.385715\pi$
$462$	5.81208e6	1.26685
$463$	5.26370e6	1.14114	0.570570	−	0.821249i	$-0.306722\pi$
$463$	5.26370e6	1.14114	0.570570	+	0.821249i	$0.306722\pi$
$464$	3.78354e6	0.815837
$465$	545100.	0.116908
$466$	6.03499e6	1.28739
$467$	−8.26813e6	−1.75435	−0.877173	−	0.480175i	$-0.840573\pi$
$467$	−8.26813e6	−1.75435	−0.877173	+	0.480175i	$0.840573\pi$
$468$	0	0
$469$	3.03438e6	0.636999
$470$	−377500.	−0.0788265
$471$	272700.	0.0566413
$472$	5.86287e6	1.21131
$473$	−6.04869e6	−1.24311
$474$	1.19412e6	0.244119
$475$	−1.69125e6	−0.343933
$476$	2.62007e6	0.530024
$477$	−129582.	−0.0260765
$478$	−935190.	−0.187210
$479$	−3.65468e6	−0.727797	−0.363899	−	0.931439i	$-0.618555\pi$
$479$	−3.65468e6	−0.727797	−0.363899	+	0.931439i	$0.618555\pi$
$480$	−372750.	−0.0738439
$481$	0	0
$482$	1.35845e6	0.266334
$483$	−1.02773e6	−0.200452
$484$	−3.28570e6	−0.637549
$485$	1.95125e6	0.376668
$486$	−4.30353e6	−0.826483
$487$	−7.13084e6	−1.36244	−0.681221	−	0.732077i	$-0.738551\pi$
$487$	−7.13084e6	−1.36244	−0.681221	+	0.732077i	$0.738551\pi$
$488$	−1.13217e6	−0.215210
$489$	−35352.0	−0.00668562
$490$	−5.34113e6	−1.00495
$491$	5.72551e6	1.07179	0.535896	−	0.844284i	$-0.319974\pi$
$491$	5.72551e6	1.07179	0.535896	+	0.844284i	$0.319974\pi$
$492$	−12348.0	−0.00229977
$493$	7.72829e6	1.43208
$494$	0	0
$495$	4.10895e6	0.753734
$496$	−2.72913e6	−0.498105
$497$	−1.15510e6	−0.209762
$498$	−1.25328e6	−0.226451
$499$	7.17251e6	1.28950	0.644748	−	0.764395i	$-0.276962\pi$
$499$	7.17251e6	1.28950	0.644748	+	0.764395i	$0.276962\pi$
$500$	−109375.	−0.0195656
$501$	−1.27663e6	−0.227233
$502$	−513240.	−0.0908994
$503$	−2.90611e6	−0.512143	−0.256072	−	0.966658i	$-0.582428\pi$
$503$	−2.90611e6	−0.512143	−0.256072	+	0.966658i	$0.582428\pi$
$504$	−9.84906e6	−1.72710
$505$	−575250.	−0.100376
$506$	−2.78694e6	−0.483895
$507$	0	0
$508$	−275002.	−0.0472799
$509$	8.37125e6	1.43217	0.716087	−	0.698011i	$-0.245931\pi$
$509$	8.37125e6	1.43217	0.716087	+	0.698011i	$0.245931\pi$
$510$	1.15050e6	0.195867
$511$	3.58534e6	0.607404
$512$	6.55248e6	1.10466
$513$	7.30620e6	1.22574
$514$	1.10591e6	0.184634
$515$	3.04265e6	0.505515
$516$	−319956.	−0.0529013
$517$	−2.39788e6	−0.394549
$518$	−8.61076e6	−1.40999
$519$	3.01907e6	0.491988
$520$	0	0
$521$	5.37332e6	0.867258	0.433629	−	0.901092i	$-0.357233\pi$
$521$	5.37332e6	0.867258	0.433629	+	0.901092i	$0.357233\pi$
$522$	−5.21433e6	−0.837572
$523$	5.26875e6	0.842274	0.421137	−	0.906997i	$-0.361631\pi$
$523$	5.26875e6	0.842274	0.421137	+	0.906997i	$0.361631\pi$
$524$	−1.48022e6	−0.235504
$525$	915000.	0.144885
$526$	−7.01587e6	−1.10565
$527$	−5.57456e6	−0.874347
$528$	3.57776e6	0.558505
$529$	−5.94354e6	−0.923434
$530$	−78250.0	−0.0121003
$531$	−6.22366e6	−0.957877
$532$	4.62185e6	0.708005
$533$	0	0
$534$	239100.	0.0362849
$535$	−1.75355e6	−0.264871
$536$	2.42502e6	0.364589
$537$	3.49034e6	0.522315
$538$	2.91477e6	0.434159
$539$	−3.39268e7	−5.03004
$540$	472500.	0.0697296
$541$	6.07956e6	0.893056	0.446528	−	0.894770i	$-0.352660\pi$
$541$	6.07956e6	0.893056	0.446528	+	0.894770i	$0.352660\pi$
$542$	−5.23451e6	−0.765381
$543$	1.21321e6	0.176578
$544$	3.81199e6	0.552274
$545$	4.89695e6	0.706211
$546$	0	0
$547$	−7.88715e6	−1.12707	−0.563536	−	0.826091i	$-0.690559\pi$
$547$	−7.88715e6	−1.12707	−0.563536	+	0.826091i	$0.690559\pi$
$548$	184114.	0.0261900
$549$	1.20184e6	0.170183
$550$	2.48125e6	0.349755
$551$	1.36328e7	1.91296
$552$	−821340.	−0.114730
$553$	−9.71218e6	−1.35053
$554$	−5.52305e6	−0.764548
$555$	1.05870e6	0.145895
$556$	−9408.00	−0.00129066
$557$	5.88545e6	0.803788	0.401894	−	0.915686i	$-0.368352\pi$
$557$	5.88545e6	0.803788	0.401894	+	0.915686i	$0.368352\pi$
$558$	3.76119e6	0.511375
$559$	0	0
$560$	−4.58110e6	−0.617305
$561$	7.30798e6	0.980370
$562$	4.54413e6	0.606890
$563$	−3.91526e6	−0.520583	−0.260291	−	0.965530i	$-0.583819\pi$
$563$	−3.91526e6	−0.520583	−0.260291	+	0.965530i	$0.583819\pi$
$564$	−126840.	−0.0167903
$565$	−2.50595e6	−0.330256
$566$	2.24627e6	0.294728
$567$	8.32064e6	1.08692
$568$	−923130.	−0.120058
$569$	9.78180e6	1.26660	0.633298	−	0.773908i	$-0.281701\pi$
$569$	9.78180e6	1.26660	0.633298	+	0.773908i	$0.281701\pi$
$570$	2.02950e6	0.261639
$571$	−1.08198e7	−1.38877	−0.694386	−	0.719603i	$-0.744324\pi$
$571$	−1.08198e7	−1.38877	−0.694386	+	0.719603i	$0.744324\pi$
$572$	0	0
$573$	−2.04365e6	−0.260028
$574$	−358680.	−0.0454389
$575$	−438750.	−0.0553411
$576$	−7.54660e6	−0.947753
$577$	−1.48792e7	−1.86055	−0.930274	−	0.366865i	$-0.880431\pi$
$577$	−1.48792e7	−1.86055	−0.930274	+	0.366865i	$0.880431\pi$
$578$	−4.66650e6	−0.580993
$579$	−1.65368e6	−0.205001
$580$	881650.	0.108824
$581$	1.01933e7	1.25278
$582$	−2.34150e6	−0.286541
$583$	−497044.	−0.0605652
$584$	2.86533e6	0.347650
$585$	0	0
$586$	−9.80413e6	−1.17941
$587$	−1.22649e7	−1.46916	−0.734578	−	0.678525i	$-0.762620\pi$
$587$	−1.22649e7	−1.46916	−0.734578	+	0.678525i	$0.762620\pi$
$588$	−1.79462e6	−0.214057
$589$	−9.83360e6	−1.16795
$590$	−3.75825e6	−0.444483
$591$	−3.22931e6	−0.380313
$592$	−5.30056e6	−0.621609
$593$	1.54878e7	1.80864	0.904320	−	0.426856i	$-0.140379\pi$
$593$	1.54878e7	1.80864	0.904320	+	0.426856i	$0.140379\pi$
$594$	−1.07190e7	−1.24649
$595$	−9.35740e6	−1.08358
$596$	−343602.	−0.0396223
$597$	−5.12304e6	−0.588291
$598$	0	0
$599$	9.75710e6	1.11110	0.555551	−	0.831483i	$-0.312507\pi$
$599$	9.75710e6	1.11110	0.555551	+	0.831483i	$0.312507\pi$
$600$	731250.	0.0829254
$601$	−7.57967e6	−0.855981	−0.427990	−	0.903783i	$-0.640778\pi$
$601$	−7.57967e6	−0.855981	−0.427990	+	0.903783i	$0.640778\pi$
$602$	−9.29396e6	−1.04522
$603$	−2.57425e6	−0.288309
$604$	−2.50599e6	−0.279503
$605$	1.17346e7	1.30341
$606$	690300.	0.0763583
$607$	1.36231e7	1.50073	0.750367	−	0.661022i	$-0.229877\pi$
$607$	1.36231e7	1.50073	0.750367	+	0.661022i	$0.229877\pi$
$608$	6.72441e6	0.737726
$609$	−7.37563e6	−0.805853
$610$	725750.	0.0789701
$611$	0	0
$612$	−2.22277e6	−0.239892
$613$	1.20366e7	1.29376	0.646880	−	0.762592i	$-0.276073\pi$
$613$	1.20366e7	1.29376	0.646880	+	0.762592i	$0.276073\pi$
$614$	−8.96924e6	−0.960140
$615$	44100.0	0.00470166
$616$	−3.77785e7	−4.01137
$617$	−8.55509e6	−0.904715	−0.452358	−	0.891837i	$-0.649417\pi$
$617$	−8.55509e6	−0.904715	−0.452358	+	0.891837i	$0.649417\pi$
$618$	−3.65118e6	−0.384558
$619$	1.33018e7	1.39535	0.697675	−	0.716414i	$-0.254218\pi$
$619$	1.33018e7	1.39535	0.697675	+	0.716414i	$0.254218\pi$
$620$	−635950.	−0.0664422
$621$	1.89540e6	0.197230
$622$	−1.20617e7	−1.25006
$623$	−1.94468e6	−0.200737
$624$	0	0
$625$	390625.	0.0400000
$626$	1.07718e7	1.09863
$627$	1.28914e7	1.30958
$628$	−318150.	−0.0321909
$629$	−1.08270e7	−1.09114
$630$	6.31350e6	0.633751
$631$	−9.16681e6	−0.916526	−0.458263	−	0.888817i	$-0.651528\pi$
$631$	−9.16681e6	−0.916526	−0.458263	+	0.888817i	$0.651528\pi$
$632$	−7.76178e6	−0.772981
$633$	−6.00674e6	−0.595840
$634$	1.29808e7	1.28256
$635$	982150.	0.0966593
$636$	−26292.0	−0.00257739
$637$	0	0
$638$	−2.00009e7	−1.94535
$639$	979938.	0.0949394
$640$	−2.56912e6	−0.247934
$641$	9.96437e6	0.957866	0.478933	−	0.877851i	$-0.341024\pi$
$641$	9.96437e6	0.957866	0.478933	+	0.877851i	$0.341024\pi$
$642$	2.10426e6	0.201494
$643$	−6.64194e6	−0.633530	−0.316765	−	0.948504i	$-0.602597\pi$
$643$	−6.64194e6	−0.633530	−0.316765	+	0.948504i	$0.602597\pi$
$644$	1.19902e6	0.113923
$645$	1.14270e6	0.108152
$646$	−2.07550e7	−1.95678
$647$	844766.	0.0793370	0.0396685	−	0.999213i	$-0.487370\pi$
$647$	844766.	0.0793370	0.0396685	+	0.999213i	$0.487370\pi$
$648$	6.64969e6	0.622106
$649$	−2.38724e7	−2.22477
$650$	0	0
$651$	5.32018e6	0.492010
$652$	41244.0	0.00379963
$653$	5.79681e6	0.531993	0.265997	−	0.963974i	$-0.414299\pi$
$653$	5.79681e6	0.531993	0.265997	+	0.963974i	$0.414299\pi$
$654$	−5.87634e6	−0.537233
$655$	5.28650e6	0.481465
$656$	−220794.	−0.0200322
$657$	−3.04166e6	−0.274914
$658$	−3.68440e6	−0.331743
$659$	−1.12406e7	−1.00827	−0.504136	−	0.863624i	$-0.668189\pi$
$659$	−1.12406e7	−1.00827	−0.504136	+	0.863624i	$0.668189\pi$
$660$	833700.	0.0744989
$661$	1.54928e7	1.37920	0.689599	−	0.724191i	$-0.257787\pi$
$661$	1.54928e7	1.37920	0.689599	+	0.724191i	$0.257787\pi$
$662$	−4.58613e6	−0.406725
$663$	0	0
$664$	8.14632e6	0.717037
$665$	−1.65066e7	−1.44745
$666$	7.30503e6	0.638170
$667$	3.53668e6	0.307809
$668$	1.48940e6	0.129143
$669$	−128184.	−0.0110731
$670$	−1.55450e6	−0.133784
$671$	4.60996e6	0.395268
$672$	−3.63804e6	−0.310774
$673$	−723294.	−0.0615570	−0.0307785	−	0.999526i	$-0.509799\pi$
$673$	−723294.	−0.0615570	−0.0307785	+	0.999526i	$0.509799\pi$
$674$	−1.11947e7	−0.949210
$675$	−1.68750e6	−0.142556
$676$	0	0
$677$	−7.57359e6	−0.635082	−0.317541	−	0.948244i	$-0.602857\pi$
$677$	−7.57359e6	−0.635082	−0.317541	+	0.948244i	$0.602857\pi$
$678$	3.00714e6	0.251235
$679$	1.90442e7	1.58522
$680$	−7.47825e6	−0.620194
$681$	5.28449e6	0.436652
$682$	1.44270e7	1.18772
$683$	1.65552e7	1.35794	0.678972	−	0.734164i	$-0.262426\pi$
$683$	1.65552e7	1.35794	0.678972	+	0.734164i	$0.262426\pi$
$684$	−3.92099e6	−0.320447
$685$	−657550.	−0.0535430
$686$	−3.16248e7	−2.56577
$687$	78180.0	0.00631981
$688$	−5.72112e6	−0.460797
$689$	0	0
$690$	526500.	0.0420994
$691$	2.04593e7	1.63003	0.815016	−	0.579438i	$-0.196728\pi$
$691$	2.04593e7	1.63003	0.815016	+	0.579438i	$0.196728\pi$
$692$	−3.52225e6	−0.279611
$693$	4.01034e7	3.17211
$694$	−1.70904e7	−1.34696
$695$	33600.0	0.00263862
$696$	−5.89446e6	−0.461234
$697$	−450996.	−0.0351634
$698$	1.17846e7	0.915535
$699$	−7.24199e6	−0.560615
$700$	−1.06750e6	−0.0823423
$701$	1.52050e7	1.16867	0.584334	−	0.811514i	$-0.301356\pi$
$701$	1.52050e7	1.16867	0.584334	+	0.811514i	$0.301356\pi$
$702$	0	0
$703$	−1.90989e7	−1.45754
$704$	−2.89469e7	−2.20125
$705$	453000.	0.0343262
$706$	−1.88198e7	−1.42103
$707$	−5.61444e6	−0.422433
$708$	−1.26277e6	−0.0946764
$709$	1.80833e7	1.35102	0.675509	−	0.737351i	$-0.263924\pi$
$709$	1.80833e7	1.35102	0.675509	+	0.737351i	$0.263924\pi$
$710$	591750.	0.0440547
$711$	8.23943e6	0.611256
$712$	−1.55415e6	−0.114893
$713$	−2.55107e6	−0.187931
$714$	1.12289e7	0.824311
$715$	0	0
$716$	−4.07207e6	−0.296847
$717$	1.12223e6	0.0815236
$718$	1.64108e7	1.18801
$719$	−2.08096e7	−1.50121	−0.750604	−	0.660752i	$-0.770237\pi$
$719$	−2.08096e7	−1.50121	−0.750604	+	0.660752i	$0.770237\pi$
$720$	3.88643e6	0.279395
$721$	2.96963e7	2.12747
$722$	−2.42317e7	−1.72998
$723$	−1.63014e6	−0.115979
$724$	−1.41541e6	−0.100355
$725$	−3.14875e6	−0.222481
$726$	−1.40816e7	−0.991537
$727$	−2.59006e7	−1.81750	−0.908749	−	0.417344i	$-0.862961\pi$
$727$	−2.59006e7	−1.81750	−0.908749	+	0.417344i	$0.862961\pi$
$728$	0	0
$729$	−3.12231e6	−0.217599
$730$	−1.83675e6	−0.127568
$731$	−1.16860e7	−0.808859
$732$	243852.	0.0168209
$733$	1.96307e7	1.34951	0.674754	−	0.738043i	$-0.264250\pi$
$733$	1.96307e7	1.34951	0.674754	+	0.738043i	$0.264250\pi$
$734$	1.21303e7	0.831056
$735$	6.40935e6	0.437618
$736$	1.74447e6	0.118705
$737$	−9.87418e6	−0.669626
$738$	304290.	0.0205659
$739$	1.67436e7	1.12781	0.563906	−	0.825839i	$-0.309298\pi$
$739$	1.67436e7	1.12781	0.563906	+	0.825839i	$0.309298\pi$
$740$	−1.23515e6	−0.0829164
$741$	0	0
$742$	−763720.	−0.0509242
$743$	−5.57725e6	−0.370637	−0.185318	−	0.982679i	$-0.559332\pi$
$743$	−5.57725e6	−0.370637	−0.185318	+	0.982679i	$0.559332\pi$
$744$	4.25178e6	0.281604
$745$	1.22715e6	0.0810041
$746$	−1.40318e7	−0.923135
$747$	−8.64763e6	−0.567016
$748$	−8.52597e6	−0.557173
$749$	−1.71146e7	−1.11471
$750$	−468750.	−0.0304290
$751$	1.24035e7	0.802499	0.401250	−	0.915969i	$-0.368576\pi$
$751$	1.24035e7	0.802499	0.401250	+	0.915969i	$0.368576\pi$
$752$	−2.26802e6	−0.146252
$753$	615888.	0.0395835
$754$	0	0
$755$	8.94995e6	0.571417
$756$	4.61160e6	0.293459
$757$	4.37170e6	0.277275	0.138637	−	0.990343i	$-0.455728\pi$
$757$	4.37170e6	0.277275	0.138637	+	0.990343i	$0.455728\pi$
$758$	1.57696e7	0.996892
$759$	3.34433e6	0.210719
$760$	−1.31918e7	−0.828454
$761$	2.10490e7	1.31756	0.658780	−	0.752335i	$-0.271073\pi$
$761$	2.10490e7	1.31756	0.658780	+	0.752335i	$0.271073\pi$
$762$	−1.17858e6	−0.0735312
$763$	4.77942e7	2.97210
$764$	2.38426e6	0.147781
$765$	7.93845e6	0.490436
$766$	−2.37522e6	−0.146262
$767$	0	0
$768$	−3.91679e6	−0.239623
$769$	−2.26551e7	−1.38150	−0.690748	−	0.723096i	$-0.742718\pi$
$769$	−2.26551e7	−1.38150	−0.690748	+	0.723096i	$0.742718\pi$
$770$	2.42170e7	1.47195
$771$	−1.32709e6	−0.0804017
$772$	1.92930e6	0.116508
$773$	−1.15053e7	−0.692545	−0.346272	−	0.938134i	$-0.612553\pi$
$773$	−1.15053e7	−0.692545	−0.346272	+	0.938134i	$0.612553\pi$
$774$	7.88463e6	0.473074
$775$	2.27125e6	0.135835
$776$	1.52198e7	0.907305
$777$	1.03329e7	0.614003
$778$	−752830.	−0.0445911
$779$	−795564.	−0.0469712
$780$	0	0
$781$	3.75880e6	0.220506
$782$	−5.38434e6	−0.314859
$783$	1.36026e7	0.792898
$784$	−3.20895e7	−1.86454
$785$	1.13625e6	0.0658112
$786$	−6.34380e6	−0.366263
$787$	−967112.	−0.0556596	−0.0278298	−	0.999613i	$-0.508860\pi$
$787$	−967112.	−0.0556596	−0.0278298	+	0.999613i	$0.508860\pi$
$788$	3.76753e6	0.216143
$789$	8.41904e6	0.481471
$790$	4.97550e6	0.283641
$791$	−2.44581e7	−1.38989
$792$	3.20498e7	1.81557
$793$	0	0
$794$	1.20843e6	0.0680253
$795$	93900.0	0.00526924
$796$	5.97688e6	0.334343
$797$	−2.85072e7	−1.58968	−0.794838	−	0.606821i	$-0.792444\pi$
$797$	−2.85072e7	−1.58968	−0.794838	+	0.606821i	$0.792444\pi$
$798$	1.98079e7	1.10111
$799$	−4.63268e6	−0.256723
$800$	−1.55313e6	−0.0857988
$801$	1.64979e6	0.0908547
$802$	−1.59840e7	−0.877502
$803$	−1.16670e7	−0.638516
$804$	−522312.	−0.0284964
$805$	−4.28220e6	−0.232904
$806$	0	0
$807$	−3.49772e6	−0.189061
$808$	−4.48695e6	−0.241781
$809$	1.08912e7	0.585065	0.292533	−	0.956256i	$-0.405502\pi$
$809$	1.08912e7	0.585065	0.292533	+	0.956256i	$0.405502\pi$
$810$	−4.26262e6	−0.228278
$811$	−1.28535e7	−0.686228	−0.343114	−	0.939294i	$-0.611482\pi$
$811$	−1.28535e7	−0.686228	−0.343114	+	0.939294i	$0.611482\pi$
$812$	8.60490e6	0.457990
$813$	6.28141e6	0.333297
$814$	2.80203e7	1.48221
$815$	−147300.	−0.00776799
$816$	6.91220e6	0.363405
$817$	−2.06143e7	−1.08047
$818$	2.11641e6	0.110590
$819$	0	0
$820$	−51450.0	−0.00267209
$821$	9.60605e6	0.497378	0.248689	−	0.968583i	$-0.420000\pi$
$821$	9.60605e6	0.497378	0.248689	+	0.968583i	$0.420000\pi$
$822$	789060.	0.0407315
$823$	1.42909e7	0.735460	0.367730	−	0.929933i	$-0.380135\pi$
$823$	1.42909e7	0.735460	0.367730	+	0.929933i	$0.380135\pi$
$824$	2.37327e7	1.21767
$825$	−2.97750e6	−0.152306
$826$	−3.66805e7	−1.87062
$827$	−2.40317e7	−1.22186	−0.610930	−	0.791685i	$-0.709204\pi$
$827$	−2.40317e7	−1.22186	−0.610930	+	0.791685i	$0.709204\pi$
$828$	−1.01720e6	−0.0515620
$829$	1.10830e7	0.560107	0.280053	−	0.959984i	$-0.409648\pi$
$829$	1.10830e7	0.560107	0.280053	+	0.959984i	$0.409648\pi$
$830$	−5.22200e6	−0.263113
$831$	6.62766e6	0.332934
$832$	0	0
$833$	−6.55463e7	−3.27292
$834$	−40320.0	−0.00200727
$835$	−5.31930e6	−0.264021
$836$	−1.50399e7	−0.744270
$837$	−9.81180e6	−0.484100
$838$	5.65796e6	0.278323
$839$	6.89303e6	0.338069	0.169034	−	0.985610i	$-0.445935\pi$
$839$	6.89303e6	0.338069	0.169034	+	0.985610i	$0.445935\pi$
$840$	7.13700e6	0.348994
$841$	4.87030e6	0.237446
$842$	1.73849e7	0.845070
$843$	−5.45296e6	−0.264279
$844$	7.00787e6	0.338633
$845$	0	0
$846$	3.12570e6	0.150149
$847$	1.14530e8	5.48543
$848$	−470126.	−0.0224504
$849$	−2.69552e6	−0.128344
$850$	4.79375e6	0.227577
$851$	−4.95472e6	−0.234528
$852$	198828.	0.00938380
$853$	683466.	0.0321621	0.0160810	−	0.999871i	$-0.494881\pi$
$853$	683466.	0.0321621	0.0160810	+	0.999871i	$0.494881\pi$
$854$	7.08332e6	0.332347
$855$	1.40035e7	0.655123
$856$	−1.36777e7	−0.638011
$857$	−7.89742e6	−0.367310	−0.183655	−	0.982991i	$-0.558793\pi$
$857$	−7.89742e6	−0.367310	−0.183655	+	0.982991i	$0.558793\pi$
$858$	0	0
$859$	3.52556e7	1.63021	0.815107	−	0.579310i	$-0.196678\pi$
$859$	3.52556e7	1.63021	0.815107	+	0.579310i	$0.196678\pi$
$860$	−1.33315e6	−0.0614657
$861$	430416.	0.0197870
$862$	1.70522e7	0.781649
$863$	−1.76565e7	−0.807007	−0.403503	−	0.914978i	$-0.632208\pi$
$863$	−1.76565e7	−0.807007	−0.403503	+	0.914978i	$0.632208\pi$
$864$	6.70950e6	0.305778
$865$	1.25794e7	0.571638
$866$	1.70361e7	0.771926
$867$	5.59979e6	0.253002
$868$	−6.20687e6	−0.279623
$869$	3.16044e7	1.41970
$870$	3.77850e6	0.169247
$871$	0	0
$872$	3.81962e7	1.70110
$873$	−1.61564e7	−0.717476
$874$	−9.49806e6	−0.420587
$875$	3.81250e6	0.168341
$876$	−617148.	−0.0271725
$877$	6.40016e6	0.280991	0.140495	−	0.990081i	$-0.455131\pi$
$877$	6.40016e6	0.280991	0.140495	+	0.990081i	$0.455131\pi$
$878$	3.54557e7	1.55221
$879$	1.17650e7	0.513592
$880$	1.49074e7	0.648924
$881$	−1.14571e7	−0.497318	−0.248659	−	0.968591i	$-0.579990\pi$
$881$	−1.14571e7	−0.497318	−0.248659	+	0.968591i	$0.579990\pi$
$882$	4.42245e7	1.91422
$883$	2.42296e7	1.04579	0.522896	−	0.852397i	$-0.324852\pi$
$883$	2.42296e7	1.04579	0.522896	+	0.852397i	$0.324852\pi$
$884$	0	0
$885$	4.50990e6	0.193557
$886$	4.11754e7	1.76219
$887$	8.66087e6	0.369617	0.184809	−	0.982775i	$-0.440833\pi$
$887$	8.66087e6	0.369617	0.184809	+	0.982775i	$0.440833\pi$
$888$	8.25786e6	0.351427
$889$	9.58578e6	0.406793
$890$	996250.	0.0421593
$891$	−2.70762e7	−1.14260
$892$	149548.	0.00629316
$893$	−8.17212e6	−0.342930
$894$	−1.47258e6	−0.0616219
$895$	1.45431e7	0.606875
$896$	−2.50747e7	−1.04343
$897$	0	0
$898$	−6.48003e6	−0.268155
$899$	−1.83081e7	−0.755516
$900$	905625.	0.0372685
$901$	−960284.	−0.0394083
$902$	1.16718e6	0.0477663
$903$	1.11528e7	0.455159
$904$	−1.95464e7	−0.795511
$905$	5.05505e6	0.205165
$906$	−1.07399e7	−0.434692
$907$	−7.84287e6	−0.316561	−0.158280	−	0.987394i	$-0.550595\pi$
$907$	−7.84287e6	−0.316561	−0.158280	+	0.987394i	$0.550595\pi$
$908$	−6.16524e6	−0.248162
$909$	4.76307e6	0.191195
$910$	0	0
$911$	−942576.	−0.0376288	−0.0188144	−	0.999823i	$-0.505989\pi$
$911$	−942576.	−0.0376288	−0.0188144	+	0.999823i	$0.505989\pi$
$912$	1.21932e7	0.485436
$913$	−3.31701e7	−1.31695
$914$	8.40979e6	0.332981
$915$	−870900.	−0.0343887
$916$	−91210.0	−0.00359173
$917$	5.15962e7	2.02626
$918$	−2.07090e7	−0.811059
$919$	−2.00734e7	−0.784030	−0.392015	−	0.919959i	$-0.628222\pi$
$919$	−2.00734e7	−0.784030	−0.392015	+	0.919959i	$0.628222\pi$
$920$	−3.42225e6	−0.133304
$921$	1.07631e7	0.418107
$922$	−1.60332e7	−0.621143
$923$	0	0
$924$	8.13691e6	0.313530
$925$	4.41125e6	0.169515
$926$	−2.63185e7	−1.00863
$927$	−2.51931e7	−0.962904
$928$	1.25194e7	0.477216
$929$	−1.10181e7	−0.418858	−0.209429	−	0.977824i	$-0.567160\pi$
$929$	−1.10181e7	−0.418858	−0.209429	+	0.977824i	$0.567160\pi$
$930$	−2.72550e6	−0.103333
$931$	−1.15625e8	−4.37196
$932$	8.44899e6	0.318614
$933$	1.44740e7	0.544358
$934$	4.13406e7	1.55064
$935$	3.04499e7	1.13909
$936$	0	0
$937$	3.59532e7	1.33779	0.668896	−	0.743356i	$-0.266767\pi$
$937$	3.59532e7	1.33779	0.668896	+	0.743356i	$0.266767\pi$
$938$	−1.51719e7	−0.563032
$939$	−1.29261e7	−0.478415
$940$	−528500.	−0.0195086
$941$	−1.28845e7	−0.474345	−0.237172	−	0.971468i	$-0.576221\pi$
$941$	−1.28845e7	−0.474345	−0.237172	+	0.971468i	$0.576221\pi$
$942$	−1.36350e6	−0.0500643
$943$	−206388.	−0.00755797
$944$	−2.25796e7	−0.824680
$945$	−1.64700e7	−0.599949
$946$	3.02435e7	1.09876
$947$	1.18911e7	0.430871	0.215436	−	0.976518i	$-0.430883\pi$
$947$	1.18911e7	0.430871	0.215436	+	0.976518i	$0.430883\pi$
$948$	1.67177e6	0.0604164
$949$	0	0
$950$	8.45625e6	0.303997
$951$	−1.55769e7	−0.558510
$952$	−7.29877e7	−2.61010
$953$	4.40094e7	1.56969	0.784844	−	0.619694i	$-0.212743\pi$
$953$	4.40094e7	1.56969	0.784844	+	0.619694i	$0.212743\pi$
$954$	647910.	0.0230486
$955$	−8.51520e6	−0.302125
$956$	−1.30927e6	−0.0463322
$957$	2.40010e7	0.847130
$958$	1.82734e7	0.643288
$959$	−6.41769e6	−0.225337
$960$	5.46855e6	0.191511
$961$	−1.54232e7	−0.538723
$962$	0	0
$963$	1.45194e7	0.504525
$964$	1.90183e6	0.0659142
$965$	−6.89035e6	−0.238190
$966$	5.13864e6	0.177176
$967$	−2.11144e7	−0.726128	−0.363064	−	0.931764i	$-0.618269\pi$
$967$	−2.11144e7	−0.726128	−0.363064	+	0.931764i	$0.618269\pi$
$968$	9.15301e7	3.13961
$969$	2.49060e7	0.852109
$970$	−9.75625e6	−0.332931
$971$	2.44293e7	0.831502	0.415751	−	0.909478i	$-0.363519\pi$
$971$	2.44293e7	0.831502	0.415751	+	0.909478i	$0.363519\pi$
$972$	−6.02494e6	−0.204544
$973$	327936.	0.0111047
$974$	3.56542e7	1.20424
$975$	0	0
$976$	4.36031e6	0.146518
$977$	−5.15549e7	−1.72796	−0.863980	−	0.503527i	$-0.832036\pi$
$977$	−5.15549e7	−1.72796	−0.863980	+	0.503527i	$0.832036\pi$
$978$	176760.	0.00590931
$979$	6.32818e6	0.211019
$980$	−7.47758e6	−0.248711
$981$	−4.05467e7	−1.34519
$982$	−2.86276e7	−0.947339
$983$	1.38938e7	0.458604	0.229302	−	0.973355i	$-0.426356\pi$
$983$	1.38938e7	0.458604	0.229302	+	0.973355i	$0.426356\pi$
$984$	343980.	0.0113252
$985$	−1.34554e7	−0.441883
$986$	−3.86415e7	−1.26579
$987$	4.42128e6	0.144463
$988$	0	0
$989$	−5.34784e6	−0.173855
$990$	−2.05448e7	−0.666213
$991$	3.31496e7	1.07225	0.536123	−	0.844140i	$-0.319888\pi$
$991$	3.31496e7	1.07225	0.536123	+	0.844140i	$0.319888\pi$
$992$	−9.03049e6	−0.291361
$993$	5.50336e6	0.177115
$994$	5.77548e6	0.185405
$995$	−2.13460e7	−0.683532
$996$	−1.75459e6	−0.0560438
$997$	9.45871e6	0.301366	0.150683	−	0.988582i	$-0.451853\pi$
$997$	9.45871e6	0.301366	0.150683	+	0.988582i	$0.451853\pi$
$998$	−3.58626e7	−1.13976
$999$	−1.90566e7	−0.604132

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.6.a.a.1.1		1
13.12	even	2		65.6.a.a.1.1	✓	1
39.38	odd	2		585.6.a.a.1.1		1
52.51	odd	2		1040.6.a.a.1.1		1
65.12	odd	4		325.6.b.a.274.2		2
65.38	odd	4		325.6.b.a.274.1		2
65.64	even	2		325.6.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.6.a.a.1.1	✓	1	13.12	even	2
325.6.a.a.1.1		1	65.64	even	2
325.6.b.a.274.1		2	65.38	odd	4
325.6.b.a.274.2		2	65.12	odd	4
585.6.a.a.1.1		1	39.38	odd	2
845.6.a.a.1.1		1	1.1	even	1	trivial
1040.6.a.a.1.1		1	52.51	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:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	845.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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