LMFDB - Embedded newform 115.6.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [115,6,Mod(1,115)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(115, 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("115.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$115 = 5 \cdot 23$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	115.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:	$18.4441392785$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{10} - 2 x^{9} - 220 x^{8} + 541 x^{7} + 15887 x^{6} - 50180 x^{5} - 417450 x^{4} + 1703213 x^{3} + \cdots + 15136200$ x^10 - 2x^9 - 220x^8 + 541x^7 + 15887x^6 - 50180x^5 - 417450x^4 + 1703213x^3 + 2235790x^2 - 15551780*x + 15136200
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$2\cdot 3$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$7.17227$ of defining polynomial
Character	$\chi$	$=$	115.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.17227 q^{2} +13.4003 q^{3} +34.7860 q^{4} -25.0000 q^{5} -109.510 q^{6} +85.3046 q^{7} -22.7676 q^{8} -63.4333 q^{9} +204.307 q^{10} -551.213 q^{11} +466.141 q^{12} +400.829 q^{13} -697.132 q^{14} -335.006 q^{15} -927.088 q^{16} -333.547 q^{17} +518.394 q^{18} +2668.99 q^{19} -869.649 q^{20} +1143.10 q^{21} +4504.66 q^{22} -529.000 q^{23} -305.092 q^{24} +625.000 q^{25} -3275.68 q^{26} -4106.28 q^{27} +2967.40 q^{28} -1249.92 q^{29} +2737.76 q^{30} -2604.42 q^{31} +8304.97 q^{32} -7386.39 q^{33} +2725.83 q^{34} -2132.62 q^{35} -2206.59 q^{36} -16493.7 q^{37} -21811.7 q^{38} +5371.21 q^{39} +569.191 q^{40} -7787.12 q^{41} -9341.75 q^{42} -19679.5 q^{43} -19174.5 q^{44} +1585.83 q^{45} +4323.13 q^{46} +7018.58 q^{47} -12423.2 q^{48} -9530.12 q^{49} -5107.67 q^{50} -4469.61 q^{51} +13943.2 q^{52} -23203.4 q^{53} +33557.6 q^{54} +13780.3 q^{55} -1942.19 q^{56} +35765.2 q^{57} +10214.7 q^{58} -36537.3 q^{59} -11653.5 q^{60} +55063.4 q^{61} +21284.0 q^{62} -5411.15 q^{63} -38203.7 q^{64} -10020.7 q^{65} +60363.6 q^{66} +1882.41 q^{67} -11602.7 q^{68} -7088.73 q^{69} +17428.3 q^{70} -20720.9 q^{71} +1444.23 q^{72} +4194.38 q^{73} +134791. q^{74} +8375.16 q^{75} +92843.5 q^{76} -47021.0 q^{77} -43894.9 q^{78} -2776.34 q^{79} +23177.2 q^{80} -39610.9 q^{81} +63638.4 q^{82} -30026.9 q^{83} +39763.9 q^{84} +8338.67 q^{85} +160827. q^{86} -16749.2 q^{87} +12549.8 q^{88} -125647. q^{89} -12959.8 q^{90} +34192.5 q^{91} -18401.8 q^{92} -34899.8 q^{93} -57357.7 q^{94} -66724.8 q^{95} +111289. q^{96} +104126. q^{97} +77882.7 q^{98} +34965.2 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$10 q - 12 q^{2} - 18 q^{3} + 138 q^{4} - 250 q^{5} + 78 q^{6} - 15 q^{7} - 285 q^{8} + 788 q^{9} + 300 q^{10} - 594 q^{11} - 1698 q^{12} - 1048 q^{13} + 639 q^{14} + 450 q^{15} + 234 q^{16} - 851 q^{17}+ \cdots - 244126 q^{99}+O(q^{100})$ 10 * q - 12 * q^2 - 18 * q^3 + 138 * q^4 - 250 * q^5 + 78 * q^6 - 15 * q^7 - 285 * q^8 + 788 * q^9 + 300 * q^10 - 594 * q^11 - 1698 * q^12 - 1048 * q^13 + 639 * q^14 + 450 * q^15 + 234 * q^16 - 851 * q^17 - 6534 * q^18 - 178 * q^19 - 3450 * q^20 - 3116 * q^21 - 6103 * q^22 - 5290 * q^23 + 1689 * q^24 + 6250 * q^25 - 8914 * q^26 + 1800 * q^27 - 10177 * q^28 - 5527 * q^29 - 1950 * q^30 - 14999 * q^31 - 44832 * q^32 - 27368 * q^33 - 14369 * q^34 + 375 * q^35 - 34128 * q^36 - 25503 * q^37 - 53451 * q^38 - 41640 * q^39 + 7125 * q^40 - 7147 * q^41 - 52736 * q^42 + 5652 * q^43 - 3135 * q^44 - 19700 * q^45 + 6348 * q^46 - 57752 * q^47 - 51470 * q^48 + 19617 * q^49 - 7500 * q^50 + 13956 * q^51 + 46680 * q^52 - 74635 * q^53 - 5901 * q^54 + 14850 * q^55 + 6825 * q^56 - 55844 * q^57 + 46373 * q^58 - 58843 * q^59 + 42450 * q^60 + 43344 * q^61 + 40430 * q^62 - 55165 * q^63 + 223597 * q^64 + 26200 * q^65 + 273973 * q^66 + 68051 * q^67 + 53021 * q^68 + 9522 * q^69 - 15975 * q^70 - 19237 * q^71 + 253050 * q^72 - 9160 * q^73 + 92210 * q^74 - 11250 * q^75 + 309393 * q^76 - 238146 * q^77 + 184189 * q^78 - 61112 * q^79 - 5850 * q^80 + 107738 * q^81 - 57922 * q^82 - 106785 * q^83 + 235538 * q^84 + 21275 * q^85 + 175458 * q^86 - 149624 * q^87 + 20709 * q^88 - 172774 * q^89 + 163350 * q^90 + 314634 * q^91 - 73002 * q^92 - 82480 * q^93 + 328193 * q^94 + 4450 * q^95 + 724596 * q^96 + 120712 * q^97 + 515589 * q^98 - 244126 * q^99

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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	−8.17227	−1.44467	−0.722333	−	0.691545i	$-0.756930\pi$
$2$	−8.17227	−1.44467	−0.722333	+	0.691545i	$0.756930\pi$
$3$	13.4003	0.859626	0.429813	−	0.902918i	$-0.358579\pi$
$3$	13.4003	0.859626	0.429813	+	0.902918i	$0.358579\pi$
$4$	34.7860	1.08706
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	−109.510	−1.24187
$7$	85.3046	0.658002	0.329001	−	0.944330i	$-0.393288\pi$
$7$	85.3046	0.658002	0.329001	+	0.944330i	$0.393288\pi$
$8$	−22.7676	−0.125775
$9$	−63.4333	−0.261042
$10$	204.307	0.646075
$11$	−551.213	−1.37353	−0.686764	−	0.726880i	$-0.740969\pi$
$11$	−551.213	−1.37353	−0.686764	+	0.726880i	$0.740969\pi$
$12$	466.141	0.934467
$13$	400.829	0.657810	0.328905	−	0.944363i	$-0.393320\pi$
$13$	400.829	0.657810	0.328905	+	0.944363i	$0.393320\pi$
$14$	−697.132	−0.950594
$15$	−335.006	−0.384437
$16$	−927.088	−0.905359
$17$	−333.547	−0.279920	−0.139960	−	0.990157i	$-0.544697\pi$
$17$	−333.547	−0.279920	−0.139960	+	0.990157i	$0.544697\pi$
$18$	518.394	0.377119
$19$	2668.99	1.69615	0.848073	−	0.529879i	$-0.177763\pi$
$19$	2668.99	1.69615	0.848073	+	0.529879i	$0.177763\pi$
$20$	−869.649	−0.486149
$21$	1143.10	0.565636
$22$	4504.66	1.98429
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	−305.092	−0.108119
$25$	625.000	0.200000
$26$	−3275.68	−0.950316
$27$	−4106.28	−1.08403
$28$	2967.40	0.715289
$29$	−1249.92	−0.275985	−0.137993	−	0.990433i	$-0.544065\pi$
$29$	−1249.92	−0.275985	−0.137993	+	0.990433i	$0.544065\pi$
$30$	2737.76	0.555383
$31$	−2604.42	−0.486750	−0.243375	−	0.969932i	$-0.578255\pi$
$31$	−2604.42	−0.486750	−0.243375	+	0.969932i	$0.578255\pi$
$32$	8304.97	1.43372
$33$	−7386.39	−1.18072
$34$	2725.83	0.404391
$35$	−2132.62	−0.294268
$36$	−2206.59	−0.283769
$37$	−16493.7	−1.98068	−0.990340	−	0.138657i	$-0.955721\pi$
$37$	−16493.7	−1.98068	−0.990340	+	0.138657i	$0.955721\pi$
$38$	−21811.7	−2.45037
$39$	5371.21	0.565471
$40$	569.191	0.0562481
$41$	−7787.12	−0.723464	−0.361732	−	0.932282i	$-0.617815\pi$
$41$	−7787.12	−0.723464	−0.361732	+	0.932282i	$0.617815\pi$
$42$	−9341.75	−0.817156
$43$	−19679.5	−1.62310	−0.811548	−	0.584286i	$-0.801375\pi$
$43$	−19679.5	−1.62310	−0.811548	+	0.584286i	$0.801375\pi$
$44$	−19174.5	−1.49311
$45$	1585.83	0.116742
$46$	4323.13	0.301234
$47$	7018.58	0.463452	0.231726	−	0.972781i	$-0.425563\pi$
$47$	7018.58	0.463452	0.231726	+	0.972781i	$0.425563\pi$
$48$	−12423.2	−0.778270
$49$	−9530.12	−0.567033
$50$	−5107.67	−0.288933
$51$	−4469.61	−0.240627
$52$	13943.2	0.715080
$53$	−23203.4	−1.13465	−0.567324	−	0.823495i	$-0.692021\pi$
$53$	−23203.4	−1.13465	−0.567324	+	0.823495i	$0.692021\pi$
$54$	33557.6	1.56606
$55$	13780.3	0.614261
$56$	−1942.19	−0.0827600
$57$	35765.2	1.45805
$58$	10214.7	0.398707
$59$	−36537.3	−1.36649	−0.683245	−	0.730190i	$-0.739432\pi$
$59$	−36537.3	−1.36649	−0.683245	+	0.730190i	$0.739432\pi$
$60$	−11653.5	−0.417906
$61$	55063.4	1.89469	0.947345	−	0.320214i	$-0.103755\pi$
$61$	55063.4	1.89469	0.947345	+	0.320214i	$0.103755\pi$
$62$	21284.0	0.703192
$63$	−5411.15	−0.171766
$64$	−38203.7	−1.16588
$65$	−10020.7	−0.294182
$66$	60363.6	1.70575
$67$	1882.41	0.0512304	0.0256152	−	0.999672i	$-0.491846\pi$
$67$	1882.41	0.0512304	0.0256152	+	0.999672i	$0.491846\pi$
$68$	−11602.7	−0.304290
$69$	−7088.73	−0.179245
$70$	17428.3	0.425119
$71$	−20720.9	−0.487823	−0.243912	−	0.969797i	$-0.578431\pi$
$71$	−20720.9	−0.487823	−0.243912	+	0.969797i	$0.578431\pi$
$72$	1444.23	0.0328325
$73$	4194.38	0.0921214	0.0460607	−	0.998939i	$-0.485333\pi$
$73$	4194.38	0.0921214	0.0460607	+	0.998939i	$0.485333\pi$
$74$	134791.	2.86142
$75$	8375.16	0.171925
$76$	92843.5	1.84382
$77$	−47021.0	−0.903785
$78$	−43894.9	−0.816917
$79$	−2776.34	−0.0500501	−0.0250251	−	0.999687i	$-0.507967\pi$
$79$	−2776.34	−0.0500501	−0.0250251	+	0.999687i	$0.507967\pi$
$80$	23177.2	0.404889
$81$	−39610.9	−0.670815
$82$	63638.4	1.04516
$83$	−30026.9	−0.478427	−0.239213	−	0.970967i	$-0.576890\pi$
$83$	−30026.9	−0.478427	−0.239213	+	0.970967i	$0.576890\pi$
$84$	39763.9	0.614881
$85$	8338.67	0.125184
$86$	160827.	2.34483
$87$	−16749.2	−0.237244
$88$	12549.8	0.172755
$89$	−125647.	−1.68142	−0.840712	−	0.541482i	$-0.817863\pi$
$89$	−125647.	−1.68142	−0.840712	+	0.541482i	$0.817863\pi$
$90$	−12959.8	−0.168653
$91$	34192.5	0.432841
$92$	−18401.8	−0.226668
$93$	−34899.8	−0.418423
$94$	−57357.7	−0.669533
$95$	−66724.8	−0.758540
$96$	111289.	1.23246
$97$	104126.	1.12365	0.561825	−	0.827256i	$-0.310099\pi$
$97$	104126.	1.12365	0.561825	+	0.827256i	$0.310099\pi$
$98$	77882.7	0.819174
$99$	34965.2	0.358549
$100$	21741.2	0.217412
$101$	110179.	1.07472	0.537358	−	0.843354i	$-0.319422\pi$
$101$	110179.	1.07472	0.537358	+	0.843354i	$0.319422\pi$
$102$	36526.8	0.347626
$103$	−157717.	−1.46482	−0.732411	−	0.680863i	$-0.761605\pi$
$103$	−157717.	−1.46482	−0.732411	+	0.680863i	$0.761605\pi$
$104$	−9125.93	−0.0827358
$105$	−28577.6	−0.252960
$106$	189624.	1.63919
$107$	−3468.74	−0.0292896	−0.0146448	−	0.999893i	$-0.504662\pi$
$107$	−3468.74	−0.0292896	−0.0146448	+	0.999893i	$0.504662\pi$
$108$	−142841.	−1.17840
$109$	101306.	0.816712	0.408356	−	0.912823i	$-0.366102\pi$
$109$	101306.	0.816712	0.408356	+	0.912823i	$0.366102\pi$
$110$	−112616.	−0.887402
$111$	−221020.	−1.70265
$112$	−79084.8	−0.595728
$113$	6017.65	0.0443334	0.0221667	−	0.999754i	$-0.492944\pi$
$113$	6017.65	0.0443334	0.0221667	+	0.999754i	$0.492944\pi$
$114$	−292283.	−2.10640
$115$	13225.0	0.0932505
$116$	−43479.6	−0.300013
$117$	−25425.9	−0.171716
$118$	298592.	1.97412
$119$	−28453.1	−0.184188
$120$	7627.30	0.0483524
$121$	142785.	0.886580
$122$	−449993.	−2.73720
$123$	−104349.	−0.621909
$124$	−90597.1	−0.529127
$125$	−15625.0	−0.0894427
$126$	44221.4	0.248145
$127$	−27459.5	−0.151072	−0.0755359	−	0.997143i	$-0.524067\pi$
$127$	−27459.5	−0.151072	−0.0755359	+	0.997143i	$0.524067\pi$
$128$	46451.4	0.250596
$129$	−263711.	−1.39526
$130$	81892.0	0.424994
$131$	−297127.	−1.51274	−0.756368	−	0.654146i	$-0.773028\pi$
$131$	−297127.	−1.51274	−0.756368	+	0.654146i	$0.773028\pi$
$132$	−256943.	−1.28352
$133$	227677.	1.11607
$134$	−15383.6	−0.0740109
$135$	102657.	0.484791
$136$	7594.07	0.0352069
$137$	294915.	1.34244	0.671220	−	0.741258i	$-0.265771\pi$
$137$	294915.	1.34244	0.671220	+	0.741258i	$0.265771\pi$
$138$	57931.0	0.258949
$139$	335549.	1.47306	0.736528	−	0.676407i	$-0.236464\pi$
$139$	335549.	1.47306	0.736528	+	0.676407i	$0.236464\pi$
$140$	−74185.1	−0.319887
$141$	94050.7	0.398396
$142$	169337.	0.704742
$143$	−220942.	−0.903521
$144$	58808.2	0.236337
$145$	31247.9	0.123424
$146$	−34277.6	−0.133085
$147$	−127706.	−0.487437
$148$	−573750.	−2.15312
$149$	488334.	1.80199	0.900993	−	0.433834i	$-0.142839\pi$
$149$	488334.	1.80199	0.900993	+	0.433834i	$0.142839\pi$
$150$	−68444.0	−0.248375
$151$	354250.	1.26435	0.632176	−	0.774825i	$-0.282162\pi$
$151$	354250.	1.26435	0.632176	+	0.774825i	$0.282162\pi$
$152$	−60766.7	−0.213332
$153$	21158.0	0.0730710
$154$	384268.	1.30567
$155$	65110.4	0.217681
$156$	186843.	0.614702
$157$	−308639.	−0.999314	−0.499657	−	0.866223i	$-0.666541\pi$
$157$	−308639.	−0.999314	−0.499657	+	0.866223i	$0.666541\pi$
$158$	22689.0	0.0723057
$159$	−310931.	−0.975374
$160$	−207624.	−0.641177
$161$	−45126.1	−0.137203
$162$	323711.	0.969103
$163$	−299295.	−0.882329	−0.441165	−	0.897426i	$-0.645435\pi$
$163$	−299295.	−0.882329	−0.441165	+	0.897426i	$0.645435\pi$
$164$	−270882.	−0.786450
$165$	184660.	0.528035
$166$	245388.	0.691167
$167$	−116466.	−0.323152	−0.161576	−	0.986860i	$-0.551658\pi$
$167$	−116466.	−0.323152	−0.161576	+	0.986860i	$0.551658\pi$
$168$	−26025.8	−0.0711427
$169$	−210629.	−0.567286
$170$	−68145.8	−0.180849
$171$	−169303.	−0.442766
$172$	−684572.	−1.76440
$173$	−331800.	−0.842871	−0.421435	−	0.906858i	$-0.638474\pi$
$173$	−331800.	−0.842871	−0.421435	+	0.906858i	$0.638474\pi$
$174$	136879.	0.342739
$175$	53315.4	0.131600
$176$	511023.	1.24354
$177$	−489609.	−1.17467
$178$	1.02682e6	2.42910
$179$	280820.	0.655082	0.327541	−	0.944837i	$-0.393780\pi$
$179$	280820.	0.655082	0.327541	+	0.944837i	$0.393780\pi$
$180$	55164.7	0.126905
$181$	−220.792	−0.000500941 0	−0.000250470	−	1.00000i	$-0.500080\pi$
$181$	−220.792	−0.000500941 0	−0.000250470		1.00000i	$0.500080\pi$
$182$	−279431.	−0.625310
$183$	737863.	1.62873
$184$	12044.1	0.0262258
$185$	412343.	0.885787
$186$	285211.	0.604482
$187$	183855.	0.384478
$188$	244148.	0.503801
$189$	−350285.	−0.713291
$190$	545293.	1.09584
$191$	−166609.	−0.330457	−0.165229	−	0.986255i	$-0.552836\pi$
$191$	−166609.	−0.330457	−0.165229	+	0.986255i	$0.552836\pi$
$192$	−511939.	−1.00222
$193$	20214.5	0.0390634	0.0195317	−	0.999809i	$-0.493782\pi$
$193$	20214.5	0.0390634	0.0195317	+	0.999809i	$0.493782\pi$
$194$	−850949.	−1.62330
$195$	−134280.	−0.252886
$196$	−331515.	−0.616400
$197$	−588783.	−1.08091	−0.540456	−	0.841373i	$-0.681748\pi$
$197$	−588783.	−1.08091	−0.540456	+	0.841373i	$0.681748\pi$
$198$	−285745.	−0.517984
$199$	740471.	1.32549	0.662743	−	0.748847i	$-0.269392\pi$
$199$	740471.	1.32549	0.662743	+	0.748847i	$0.269392\pi$
$200$	−14229.8	−0.0251549
$201$	25224.8	0.0440390
$202$	−900409.	−1.55261
$203$	−106624.	−0.181599
$204$	−155480.	−0.261576
$205$	194678.	0.323543
$206$	1.28890e6	2.11618
$207$	33556.2	0.0544311
$208$	−371603.	−0.595554
$209$	−1.47118e6	−2.32971
$210$	233544.	0.365443
$211$	−435057.	−0.672729	−0.336365	−	0.941732i	$-0.609197\pi$
$211$	−435057.	−0.672729	−0.336365	+	0.941732i	$0.609197\pi$
$212$	−807151.	−1.23343
$213$	−277665.	−0.419346
$214$	28347.5	0.0423136
$215$	491989.	0.725870
$216$	93490.4	0.136343
$217$	−222169.	−0.320283
$218$	−827900.	−1.17988
$219$	56205.7	0.0791900
$220$	479362.	0.667739
$221$	−133695.	−0.184134
$222$	1.80624e6	2.45976
$223$	288757.	0.388839	0.194420	−	0.980918i	$-0.437718\pi$
$223$	288757.	0.388839	0.194420	+	0.980918i	$0.437718\pi$
$224$	708452.	0.943389
$225$	−39645.8	−0.0522085
$226$	−49177.9	−0.0640470
$227$	533584.	0.687287	0.343644	−	0.939100i	$-0.388339\pi$
$227$	533584.	0.687287	0.343644	+	0.939100i	$0.388339\pi$
$228$	1.24413e6	1.58499
$229$	209610.	0.264133	0.132067	−	0.991241i	$-0.457839\pi$
$229$	209610.	0.264133	0.132067	+	0.991241i	$0.457839\pi$
$230$	−108078.	−0.134716
$231$	−630093.	−0.776917
$232$	28457.7	0.0347120
$233$	824744.	0.995243	0.497622	−	0.867394i	$-0.334207\pi$
$233$	824744.	0.995243	0.497622	+	0.867394i	$0.334207\pi$
$234$	207787.	0.248073
$235$	−175465.	−0.207262
$236$	−1.27098e6	−1.48546
$237$	−37203.7	−0.0430244
$238$	232526.	0.266090
$239$	−670029.	−0.758751	−0.379375	−	0.925243i	$-0.623861\pi$
$239$	−670029.	−0.758751	−0.379375	+	0.925243i	$0.623861\pi$
$240$	310580.	0.348053
$241$	−157659.	−0.174855	−0.0874273	−	0.996171i	$-0.527865\pi$
$241$	−157659.	−0.174855	−0.0874273	+	0.996171i	$0.527865\pi$
$242$	−1.16687e6	−1.28081
$243$	467030.	0.507375
$244$	1.91543e6	2.05964
$245$	238253.	0.253585
$246$	852771.	0.898451
$247$	1.06981e6	1.11574
$248$	59296.4	0.0612208
$249$	−402368.	−0.411268
$250$	127692.	0.129215
$251$	−1.30181e6	−1.30426	−0.652128	−	0.758109i	$-0.726123\pi$
$251$	−1.30181e6	−1.30426	−0.652128	+	0.758109i	$0.726123\pi$
$252$	−188232.	−0.186721
$253$	291592.	0.286400
$254$	224407.	0.218248
$255$	111740.	0.107612
$256$	842904.	0.803856
$257$	1.08611e6	1.02575	0.512874	−	0.858464i	$-0.328581\pi$
$257$	1.08611e6	1.02575	0.512874	+	0.858464i	$0.328581\pi$
$258$	2.15512e6	2.01568
$259$	−1.40699e6	−1.30329
$260$	−348580.	−0.319793
$261$	79286.4	0.0720439
$262$	2.42820e6	2.18540
$263$	1.74183e6	1.55281	0.776404	−	0.630236i	$-0.217042\pi$
$263$	1.74183e6	1.55281	0.776404	+	0.630236i	$0.217042\pi$
$264$	168171.	0.148505
$265$	580084.	0.507430
$266$	−1.86064e6	−1.61235
$267$	−1.68370e6	−1.44540
$268$	65481.5	0.0556906
$269$	954865.	0.804566	0.402283	−	0.915515i	$-0.368217\pi$
$269$	954865.	0.804566	0.402283	+	0.915515i	$0.368217\pi$
$270$	−838941.	−0.700361
$271$	211939.	0.175302	0.0876512	−	0.996151i	$-0.472064\pi$
$271$	211939.	0.175302	0.0876512	+	0.996151i	$0.472064\pi$
$272$	309227.	0.253428
$273$	458189.	0.372081
$274$	−2.41012e6	−1.93938
$275$	−344508.	−0.274706
$276$	−246588.	−0.194850
$277$	1.35738e6	1.06293	0.531463	−	0.847081i	$-0.321642\pi$
$277$	1.35738e6	1.06293	0.531463	+	0.847081i	$0.321642\pi$
$278$	−2.74220e6	−2.12807
$279$	165207.	0.127062
$280$	48554.6	0.0370114
$281$	1.68620e6	1.27393	0.636963	−	0.770894i	$-0.280190\pi$
$281$	1.68620e6	1.27393	0.636963	+	0.770894i	$0.280190\pi$
$282$	−768608.	−0.575549
$283$	540012.	0.400809	0.200404	−	0.979713i	$-0.435774\pi$
$283$	540012.	0.400809	0.200404	+	0.979713i	$0.435774\pi$
$284$	−720796.	−0.530294
$285$	−894129.	−0.652061
$286$	1.80560e6	1.30529
$287$	−664277.	−0.476041
$288$	−526812.	−0.374261
$289$	−1.30860e6	−0.921645
$290$	−255366.	−0.178307
$291$	1.39532e6	0.965920
$292$	145905.	0.100142
$293$	59213.8	0.0402953	0.0201476	−	0.999797i	$-0.493586\pi$
$293$	59213.8	0.0402953	0.0201476	+	0.999797i	$0.493586\pi$
$294$	1.04365e6	0.704183
$295$	913432.	0.611113
$296$	375523.	0.249119
$297$	2.26344e6	1.48894
$298$	−3.99080e6	−2.60327
$299$	−212038.	−0.137163
$300$	291338.	0.186893
$301$	−1.67876e6	−1.06800
$302$	−2.89503e6	−1.82657
$303$	1.47642e6	0.923854
$304$	−2.47439e6	−1.53562
$305$	−1.37658e6	−0.847331
$306$	−172909.	−0.105563
$307$	−25932.8	−0.0157037	−0.00785187	−	0.999969i	$-0.502499\pi$
$307$	−25932.8	−0.0157037	−0.00785187	+	0.999969i	$0.502499\pi$
$308$	−1.63567e6	−0.982470
$309$	−2.11344e6	−1.25920
$310$	−532100.	−0.314477
$311$	−1.61509e6	−0.946879	−0.473440	−	0.880826i	$-0.656988\pi$
$311$	−1.61509e6	−0.946879	−0.473440	+	0.880826i	$0.656988\pi$
$312$	−122290.	−0.0711219
$313$	−2.57528e6	−1.48581	−0.742906	−	0.669396i	$-0.766553\pi$
$313$	−2.57528e6	−1.48581	−0.742906	+	0.669396i	$0.766553\pi$
$314$	2.52228e6	1.44368
$315$	135279.	0.0768163
$316$	−96577.7	−0.0544075
$317$	−1.12868e6	−0.630847	−0.315423	−	0.948951i	$-0.602147\pi$
$317$	−1.12868e6	−0.630847	−0.315423	+	0.948951i	$0.602147\pi$
$318$	2.54101e6	1.40909
$319$	688970.	0.379074
$320$	955091.	0.521399
$321$	−46482.0	−0.0251781
$322$	368783.	0.198213
$323$	−890234.	−0.474786
$324$	−1.37790e6	−0.729217
$325$	250518.	0.131562
$326$	2.44592e6	1.27467
$327$	1.35753e6	0.702067
$328$	177294.	0.0909935
$329$	598717.	0.304952
$330$	−1.50909e6	−0.762834
$331$	1.59828e6	0.801829	0.400915	−	0.916115i	$-0.368692\pi$
$331$	1.59828e6	0.801829	0.400915	+	0.916115i	$0.368692\pi$
$332$	−1.04451e6	−0.520079
$333$	1.04625e6	0.517042
$334$	951789.	0.466847
$335$	−47060.3	−0.0229109
$336$	−1.05976e6	−0.512104
$337$	65142.4	0.0312456	0.0156228	−	0.999878i	$-0.495027\pi$
$337$	65142.4	0.0312456	0.0156228	+	0.999878i	$0.495027\pi$
$338$	1.72132e6	0.819539
$339$	80638.1	0.0381102
$340$	290069.	0.136083
$341$	1.43559e6	0.668565
$342$	1.38359e6	0.639649
$343$	−2.24668e6	−1.03111
$344$	448057.	0.204144
$345$	177218.	0.0801606
$346$	2.71156e6	1.21767
$347$	−196819.	−0.0877492	−0.0438746	−	0.999037i	$-0.513970\pi$
$347$	−196819.	−0.0877492	−0.0438746	+	0.999037i	$0.513970\pi$
$348$	−582637.	−0.257899
$349$	−1.44649e6	−0.635702	−0.317851	−	0.948141i	$-0.602961\pi$
$349$	−1.44649e6	−0.635702	−0.317851	+	0.948141i	$0.602961\pi$
$350$	−435708.	−0.190119
$351$	−1.64592e6	−0.713083
$352$	−4.57781e6	−1.96925
$353$	−3.49462e6	−1.49267	−0.746334	−	0.665572i	$-0.768188\pi$
$353$	−3.49462e6	−1.49267	−0.746334	+	0.665572i	$0.768188\pi$
$354$	4.00121e6	1.69701
$355$	518022.	0.218161
$356$	−4.37076e6	−1.82781
$357$	−381278.	−0.158333
$358$	−2.29494e6	−0.946375
$359$	−1.31793e6	−0.539704	−0.269852	−	0.962902i	$-0.586975\pi$
$359$	−1.31793e6	−0.539704	−0.269852	+	0.962902i	$0.586975\pi$
$360$	−36105.7	−0.0146831
$361$	4.64742e6	1.87691
$362$	1804.37	0.000723692 0
$363$	1.91335e6	0.762128
$364$	1.18942e6	0.470524
$365$	−104859.	−0.0411979
$366$	−6.03001e6	−2.35297
$367$	3.94840e6	1.53023	0.765114	−	0.643895i	$-0.222683\pi$
$367$	3.94840e6	1.53023	0.765114	+	0.643895i	$0.222683\pi$
$368$	490429.	0.188780
$369$	493963.	0.188855
$370$	−3.36978e6	−1.27967
$371$	−1.97935e6	−0.746601
$372$	−1.21402e6	−0.454852
$373$	−4.06998e6	−1.51468	−0.757339	−	0.653022i	$-0.773501\pi$
$373$	−4.06998e6	−1.51468	−0.757339	+	0.653022i	$0.773501\pi$
$374$	−1.50251e6	−0.555443
$375$	−209379.	−0.0768873
$376$	−159797.	−0.0582905
$377$	−501003.	−0.181546
$378$	2.86262e6	1.03047
$379$	−3.08289e6	−1.10245	−0.551226	−	0.834356i	$-0.685840\pi$
$379$	−3.08289e6	−1.10245	−0.551226	+	0.834356i	$0.685840\pi$
$380$	−2.32109e6	−0.824579
$381$	−367964.	−0.129865
$382$	1.36157e6	0.477400
$383$	−413860.	−0.144164	−0.0720820	−	0.997399i	$-0.522964\pi$
$383$	−413860.	−0.144164	−0.0720820	+	0.997399i	$0.522964\pi$
$384$	622461.	0.215419
$385$	1.17552e6	0.404185
$386$	−165198.	−0.0564336
$387$	1.24834e6	0.423697
$388$	3.62214e6	1.22148
$389$	5.93753e6	1.98944	0.994722	−	0.102605i	$-0.0327178\pi$
$389$	5.93753e6	1.98944	0.994722	+	0.102605i	$0.0327178\pi$
$390$	1.09737e6	0.365336
$391$	176446.	0.0583674
$392$	216978.	0.0713184
$393$	−3.98157e6	−1.30039
$394$	4.81170e6	1.56156
$395$	69408.5	0.0223831
$396$	1.21630e6	0.389765
$397$	−3.92824e6	−1.25090	−0.625449	−	0.780265i	$-0.715084\pi$
$397$	−3.92824e6	−1.25090	−0.625449	+	0.780265i	$0.715084\pi$
$398$	−6.05133e6	−1.91489
$399$	3.05093e6	0.959402
$400$	−579430.	−0.181072
$401$	−4.27894e6	−1.32885	−0.664424	−	0.747356i	$-0.731323\pi$
$401$	−4.27894e6	−1.32885	−0.664424	+	0.747356i	$0.731323\pi$
$402$	−206144.	−0.0636217
$403$	−1.04392e6	−0.320189
$404$	3.83267e6	1.16828
$405$	990273.	0.299997
$406$	871357.	0.262350
$407$	9.09155e6	2.72052
$408$	101762.	0.0302648
$409$	4.02846e6	1.19078	0.595389	−	0.803437i	$-0.296998\pi$
$409$	4.02846e6	1.19078	0.595389	+	0.803437i	$0.296998\pi$
$410$	−1.59096e6	−0.467412
$411$	3.95193e6	1.15400
$412$	−5.48633e6	−1.59235
$413$	−3.11680e6	−0.899153
$414$	−274230.	−0.0786348
$415$	750673.	0.213959
$416$	3.32887e6	0.943113
$417$	4.49644e6	1.26628
$418$	1.20229e7	3.36565
$419$	6.85980e6	1.90887	0.954435	−	0.298419i	$-0.0964591\pi$
$419$	6.85980e6	1.90887	0.954435	+	0.298419i	$0.0964591\pi$
$420$	−994099.	−0.274983
$421$	−2.74935e6	−0.756005	−0.378003	−	0.925805i	$-0.623389\pi$
$421$	−2.74935e6	−0.756005	−0.378003	+	0.925805i	$0.623389\pi$
$422$	3.55541e6	0.971870
$423$	−445212.	−0.120981
$424$	528286.	0.142710
$425$	−208467.	−0.0559840
$426$	2.26915e6	0.605815
$427$	4.69716e6	1.24671
$428$	−120664.	−0.0318395
$429$	−2.96068e6	−0.776690
$430$	−4.02066e6	−1.04864
$431$	1.30645e6	0.338765	0.169382	−	0.985550i	$-0.445823\pi$
$431$	1.30645e6	0.338765	0.169382	+	0.985550i	$0.445823\pi$
$432$	3.80688e6	0.981432
$433$	−2.98729e6	−0.765697	−0.382849	−	0.923811i	$-0.625057\pi$
$433$	−2.98729e6	−0.765697	−0.382849	+	0.923811i	$0.625057\pi$
$434$	1.81562e6	0.462702
$435$	418730.	0.106099
$436$	3.52403e6	0.887816
$437$	−1.41190e6	−0.353671
$438$	−459328.	−0.114403
$439$	−3.23159e6	−0.800305	−0.400153	−	0.916449i	$-0.631043\pi$
$439$	−3.23159e6	−0.800305	−0.400153	+	0.916449i	$0.631043\pi$
$440$	−313745.	−0.0772584
$441$	604527.	0.148020
$442$	1.09259e6	0.266013
$443$	1.88453e6	0.456241	0.228121	−	0.973633i	$-0.426742\pi$
$443$	1.88453e6	0.456241	0.228121	+	0.973633i	$0.426742\pi$
$444$	−7.68840e6	−1.85088
$445$	3.14118e6	0.751956
$446$	−2.35980e6	−0.561743
$447$	6.54380e6	1.54903
$448$	−3.25895e6	−0.767154
$449$	5.28563e6	1.23732	0.618659	−	0.785660i	$-0.287676\pi$
$449$	5.28563e6	1.23732	0.618659	+	0.785660i	$0.287676\pi$
$450$	323996.	0.0754238
$451$	4.29236e6	0.993699
$452$	209330.	0.0481931
$453$	4.74704e6	1.08687
$454$	−4.36059e6	−0.992901
$455$	−854814.	−0.193572
$456$	−814289.	−0.183386
$457$	−6.09884e6	−1.36602	−0.683010	−	0.730409i	$-0.739329\pi$
$457$	−6.09884e6	−1.36602	−0.683010	+	0.730409i	$0.739329\pi$
$458$	−1.71299e6	−0.381584
$459$	1.36964e6	0.303441
$460$	460044.	0.101369
$461$	−5.03999e6	−1.10453	−0.552265	−	0.833669i	$-0.686236\pi$
$461$	−5.03999e6	−1.10453	−0.552265	+	0.833669i	$0.686236\pi$
$462$	5.14929e6	1.12239
$463$	−4.33452e6	−0.939698	−0.469849	−	0.882747i	$-0.655692\pi$
$463$	−4.33452e6	−0.939698	−0.469849	+	0.882747i	$0.655692\pi$
$464$	1.15878e6	0.249866
$465$	872495.	0.187125
$466$	−6.74003e6	−1.43779
$467$	2.73020e6	0.579299	0.289649	−	0.957133i	$-0.406461\pi$
$467$	2.73020e6	0.579299	0.289649	+	0.957133i	$0.406461\pi$
$468$	−884464.	−0.186666
$469$	160579.	0.0337097
$470$	1.43394e6	0.299424
$471$	−4.13584e6	−0.859037
$472$	831868.	0.171870
$473$	1.08476e7	2.22937
$474$	304038.	0.0621559
$475$	1.66812e6	0.339229
$476$	−989767.	−0.200224
$477$	1.47187e6	0.296191
$478$	5.47566e6	1.09614
$479$	3.85339e6	0.767369	0.383684	−	0.923464i	$-0.374655\pi$
$479$	3.85339e6	0.767369	0.383684	+	0.923464i	$0.374655\pi$
$480$	−2.78222e6	−0.551173
$481$	−6.61116e6	−1.30291
$482$	1.28843e6	0.252607
$483$	−604702.	−0.117943
$484$	4.96690e6	0.963767
$485$	−2.60316e6	−0.502512
$486$	−3.81670e6	−0.732988
$487$	−4.77493e6	−0.912315	−0.456158	−	0.889899i	$-0.650775\pi$
$487$	−4.77493e6	−0.912315	−0.456158	+	0.889899i	$0.650775\pi$
$488$	−1.25366e6	−0.238304
$489$	−4.01063e6	−0.758474
$490$	−1.94707e6	−0.366346
$491$	−501195.	−0.0938216	−0.0469108	−	0.998899i	$-0.514938\pi$
$491$	−501195.	−0.0938216	−0.0469108	+	0.998899i	$0.514938\pi$
$492$	−3.62989e6	−0.676053
$493$	416906.	0.0772539
$494$	−8.74277e6	−1.61188
$495$	−874131.	−0.160348
$496$	2.41452e6	0.440684
$497$	−1.76759e6	−0.320989
$498$	3.28826e6	0.594145
$499$	2.32923e6	0.418757	0.209378	−	0.977835i	$-0.432856\pi$
$499$	2.32923e6	0.418757	0.209378	+	0.977835i	$0.432856\pi$
$500$	−543531.	−0.0972297
$501$	−1.56067e6	−0.277790
$502$	1.06387e7	1.88421
$503$	−9.10450e6	−1.60449	−0.802243	−	0.596997i	$-0.796360\pi$
$503$	−9.10450e6	−1.60449	−0.802243	+	0.596997i	$0.796360\pi$
$504$	123199.	0.0216039
$505$	−2.75446e6	−0.480628
$506$	−2.38296e6	−0.413753
$507$	−2.82249e6	−0.487654
$508$	−955206.	−0.164224
$509$	−3.50476e6	−0.599602	−0.299801	−	0.954002i	$-0.596920\pi$
$509$	−3.50476e6	−0.599602	−0.299801	+	0.954002i	$0.596920\pi$
$510$	−913171.	−0.155463
$511$	357800.	0.0606161
$512$	−8.37488e6	−1.41190
$513$	−1.09596e7	−1.83867
$514$	−8.87598e6	−1.48186
$515$	3.94292e6	0.655088
$516$	−9.17344e6	−1.51673
$517$	−3.86873e6	−0.636564
$518$	1.14983e7	1.88282
$519$	−4.44620e6	−0.724554
$520$	228148.	0.0370006
$521$	5.26368e6	0.849562	0.424781	−	0.905296i	$-0.360351\pi$
$521$	5.26368e6	0.849562	0.424781	+	0.905296i	$0.360351\pi$
$522$	−647949.	−0.104079
$523$	−2.84855e6	−0.455375	−0.227688	−	0.973734i	$-0.573117\pi$
$523$	−2.84855e6	−0.455375	−0.227688	+	0.973734i	$0.573117\pi$
$524$	−1.03358e7	−1.64444
$525$	714439.	0.113127
$526$	−1.42347e7	−2.24329
$527$	868694.	0.136251
$528$	6.84783e6	1.06898
$529$	279841.	0.0434783
$530$	−4.74060e6	−0.733067
$531$	2.31768e6	0.356712
$532$	7.91998e6	1.21323
$533$	−3.12130e6	−0.475902
$534$	1.37597e7	2.08812
$535$	86718.6	0.0130987
$536$	−42858.1	−0.00644349
$537$	3.76306e6	0.563126
$538$	−7.80342e6	−1.16233
$539$	5.25313e6	0.778836
$540$	3.57103e6	0.526997
$541$	−8.01867e6	−1.17790	−0.588951	−	0.808169i	$-0.700459\pi$
$541$	−8.01867e6	−1.17790	−0.588951	+	0.808169i	$0.700459\pi$
$542$	−1.73202e6	−0.253254
$543$	−2958.66	−0.000430622 0
$544$	−2.77010e6	−0.401326
$545$	−2.53265e6	−0.365245
$546$	−3.74444e6	−0.537533
$547$	7.93649e6	1.13412	0.567061	−	0.823676i	$-0.308080\pi$
$547$	7.93649e6	1.13412	0.567061	+	0.823676i	$0.308080\pi$
$548$	1.02589e7	1.45931
$549$	−3.49285e6	−0.494594
$550$	2.81541e6	0.396858
$551$	−3.33602e6	−0.468112
$552$	161394.	0.0225444
$553$	−236835.	−0.0329331
$554$	−1.10929e7	−1.53557
$555$	5.52550e6	0.761446
$556$	1.16724e7	1.60130
$557$	2.43838e6	0.333015	0.166508	−	0.986040i	$-0.446751\pi$
$557$	2.43838e6	0.333015	0.166508	+	0.986040i	$0.446751\pi$
$558$	−1.35011e6	−0.183563
$559$	−7.88813e6	−1.06769
$560$	1.97712e6	0.266418
$561$	2.46371e6	0.330508
$562$	−1.37801e7	−1.84040
$563$	1.32223e7	1.75807	0.879034	−	0.476759i	$-0.158188\pi$
$563$	1.32223e7	1.75807	0.879034	+	0.476759i	$0.158188\pi$
$564$	3.27165e6	0.433080
$565$	−150441.	−0.0198265
$566$	−4.41312e6	−0.579035
$567$	−3.37899e6	−0.441397
$568$	471766.	0.0613558
$569$	−5.12347e6	−0.663412	−0.331706	−	0.943383i	$-0.607624\pi$
$569$	−5.12347e6	−0.663412	−0.331706	+	0.943383i	$0.607624\pi$
$570$	7.30706e6	0.942011
$571$	1.28462e7	1.64887	0.824434	−	0.565958i	$-0.191494\pi$
$571$	1.28462e7	1.64887	0.824434	+	0.565958i	$0.191494\pi$
$572$	−7.68568e6	−0.982182
$573$	−2.23260e6	−0.284070
$574$	5.42865e6	0.687721
$575$	−330625.	−0.0417029
$576$	2.42338e6	0.304345
$577$	−7.09107e6	−0.886691	−0.443345	−	0.896351i	$-0.646209\pi$
$577$	−7.09107e6	−0.886691	−0.443345	+	0.896351i	$0.646209\pi$
$578$	1.06943e7	1.33147
$579$	270879.	0.0335799
$580$	1.08699e6	0.134170
$581$	−2.56143e6	−0.314806
$582$	−1.14029e7	−1.39543
$583$	1.27900e7	1.55847
$584$	−95496.1	−0.0115865
$585$	635647.	0.0767939
$586$	−483911.	−0.0582132
$587$	−6.16460e6	−0.738431	−0.369215	−	0.929344i	$-0.620374\pi$
$587$	−6.16460e6	−0.738431	−0.369215	+	0.929344i	$0.620374\pi$
$588$	−4.44238e6	−0.529873
$589$	−6.95117e6	−0.825599
$590$	−7.46481e6	−0.882854
$591$	−7.88985e6	−0.929180
$592$	1.52911e7	1.79323
$593$	458736.	0.0535706	0.0267853	−	0.999641i	$-0.491473\pi$
$593$	458736.	0.0535706	0.0267853	+	0.999641i	$0.491473\pi$
$594$	−1.84974e7	−2.15102
$595$	711327.	0.0823714
$596$	1.69872e7	1.95887
$597$	9.92249e6	1.13942
$598$	1.73283e6	0.198155
$599$	1.12961e7	1.28636	0.643181	−	0.765714i	$-0.277614\pi$
$599$	1.12961e7	1.28636	0.643181	+	0.765714i	$0.277614\pi$
$600$	−190683.	−0.0216238
$601$	−1.36855e6	−0.154552	−0.0772760	−	0.997010i	$-0.524622\pi$
$601$	−1.36855e6	−0.154552	−0.0772760	+	0.997010i	$0.524622\pi$
$602$	1.37192e7	1.54290
$603$	−119408.	−0.0133733
$604$	1.23229e7	1.37443
$605$	−3.56961e6	−0.396491
$606$	−1.20657e7	−1.33466
$607$	−3.18822e6	−0.351218	−0.175609	−	0.984460i	$-0.556189\pi$
$607$	−3.18822e6	−0.351218	−0.175609	+	0.984460i	$0.556189\pi$
$608$	2.21659e7	2.43179
$609$	−1.42878e6	−0.156107
$610$	1.12498e7	1.22411
$611$	2.81325e6	0.304863
$612$	736000.	0.0794327
$613$	−4.67188e6	−0.502158	−0.251079	−	0.967967i	$-0.580785\pi$
$613$	−4.67188e6	−0.502158	−0.251079	+	0.967967i	$0.580785\pi$
$614$	211930.	0.0226867
$615$	2.60873e6	0.278126
$616$	1.07056e6	0.113673
$617$	−4.58446e6	−0.484815	−0.242407	−	0.970175i	$-0.577937\pi$
$617$	−4.58446e6	−0.484815	−0.242407	+	0.970175i	$0.577937\pi$
$618$	1.72716e7	1.81912
$619$	−6.87650e6	−0.721341	−0.360670	−	0.932693i	$-0.617452\pi$
$619$	−6.87650e6	−0.721341	−0.360670	+	0.932693i	$0.617452\pi$
$620$	2.26493e6	0.236633
$621$	2.17222e6	0.226035
$622$	1.31989e7	1.36792
$623$	−1.07183e7	−1.10638
$624$	−4.97958e6	−0.511954
$625$	390625.	0.0400000
$626$	2.10459e7	2.14650
$627$	−1.97142e7	−2.00268
$628$	−1.07363e7	−1.08632
$629$	5.50143e6	0.554433
$630$	−1.10553e6	−0.110974
$631$	−582950.	−0.0582851	−0.0291426	−	0.999575i	$-0.509278\pi$
$631$	−582950.	−0.0582851	−0.0291426	+	0.999575i	$0.509278\pi$
$632$	63210.7	0.00629503
$633$	−5.82988e6	−0.578296
$634$	9.22390e6	0.911363
$635$	686488.	0.0675614
$636$	−1.08160e7	−1.06029
$637$	−3.81995e6	−0.373000
$638$	−5.63045e6	−0.547635
$639$	1.31439e6	0.127343
$640$	−1.16129e6	−0.112070
$641$	−2.49002e6	−0.239363	−0.119682	−	0.992812i	$-0.538187\pi$
$641$	−2.49002e6	−0.239363	−0.119682	+	0.992812i	$0.538187\pi$
$642$	379864.	0.0363739
$643$	4.28616e6	0.408828	0.204414	−	0.978885i	$-0.434471\pi$
$643$	4.28616e6	0.408828	0.204414	+	0.978885i	$0.434471\pi$
$644$	−1.56976e6	−0.149148
$645$	6.59277e6	0.623977
$646$	7.27523e6	0.685907
$647$	−4.08917e6	−0.384038	−0.192019	−	0.981391i	$-0.561504\pi$
$647$	−4.08917e6	−0.384038	−0.192019	+	0.981391i	$0.561504\pi$
$648$	901847.	0.0843715
$649$	2.01398e7	1.87691
$650$	−2.04730e6	−0.190063
$651$	−2.97712e6	−0.275323
$652$	−1.04113e7	−0.959146
$653$	−1.72639e7	−1.58437	−0.792185	−	0.610281i	$-0.791056\pi$
$653$	−1.72639e7	−1.58437	−0.792185	+	0.610281i	$0.791056\pi$
$654$	−1.10941e7	−1.01425
$655$	7.42817e6	0.676517
$656$	7.21934e6	0.654995
$657$	−266063.	−0.0240476
$658$	−4.89288e6	−0.440555
$659$	5.70243e6	0.511501	0.255751	−	0.966743i	$-0.417677\pi$
$659$	5.70243e6	0.511501	0.255751	+	0.966743i	$0.417677\pi$
$660$	6.42357e6	0.574006
$661$	−1.27038e7	−1.13091	−0.565457	−	0.824778i	$-0.691300\pi$
$661$	−1.27038e7	−1.13091	−0.565457	+	0.824778i	$0.691300\pi$
$662$	−1.30615e7	−1.15838
$663$	−1.79155e6	−0.158287
$664$	683642.	0.0601739
$665$	−5.69193e6	−0.499121
$666$	−8.55025e6	−0.746953
$667$	661206.	0.0575470
$668$	−4.05137e6	−0.351286
$669$	3.86942e6	0.334257
$670$	384590.	0.0330987
$671$	−3.03516e7	−2.60241
$672$	9.49344e6	0.810962
$673$	4.04898e6	0.344594	0.172297	−	0.985045i	$-0.444881\pi$
$673$	4.04898e6	0.344594	0.172297	+	0.985045i	$0.444881\pi$
$674$	−532361.	−0.0451395
$675$	−2.56643e6	−0.216805
$676$	−7.32694e6	−0.616675
$677$	−1.53240e7	−1.28500	−0.642498	−	0.766287i	$-0.722102\pi$
$677$	−1.53240e7	−1.28500	−0.642498	+	0.766287i	$0.722102\pi$
$678$	−658996.	−0.0550565
$679$	8.88246e6	0.739365
$680$	−189852.	−0.0157450
$681$	7.15016e6	0.590811
$682$	−1.17320e7	−0.965853
$683$	9.07115e6	0.744065	0.372032	−	0.928220i	$-0.378661\pi$
$683$	9.07115e6	0.744065	0.372032	+	0.928220i	$0.378661\pi$
$684$	−5.88937e6	−0.481314
$685$	−7.37286e6	−0.600357
$686$	1.83605e7	1.48961
$687$	2.80882e6	0.227056
$688$	1.82447e7	1.46948
$689$	−9.30058e6	−0.746383
$690$	−1.44828e6	−0.115805
$691$	5.99487e6	0.477623	0.238811	−	0.971066i	$-0.423242\pi$
$691$	5.99487e6	0.477623	0.238811	+	0.971066i	$0.423242\pi$
$692$	−1.15420e7	−0.916252
$693$	2.98270e6	0.235926
$694$	1.60846e6	0.126768
$695$	−8.38873e6	−0.658770
$696$	381340.	0.0298393
$697$	2.59737e6	0.202512
$698$	1.18211e7	0.918377
$699$	1.10518e7	0.855538
$700$	1.85463e6	0.143058
$701$	4.32960e6	0.332776	0.166388	−	0.986060i	$-0.446790\pi$
$701$	4.32960e6	0.332776	0.166388	+	0.986060i	$0.446790\pi$
$702$	1.34509e7	1.03017
$703$	−4.40216e7	−3.35953
$704$	2.10583e7	1.60137
$705$	−2.35127e6	−0.178168
$706$	2.85590e7	2.15641
$707$	9.39874e6	0.707165
$708$	−1.70315e7	−1.27694
$709$	5.08191e6	0.379674	0.189837	−	0.981816i	$-0.439204\pi$
$709$	5.08191e6	0.379674	0.189837	+	0.981816i	$0.439204\pi$
$710$	−4.23342e6	−0.315170
$711$	176112.	0.0130652
$712$	2.86069e6	0.211481
$713$	1.37774e6	0.101494
$714$	3.11591e6	0.228738
$715$	5.52355e6	0.404067
$716$	9.76860e6	0.712114
$717$	−8.97856e6	−0.652242
$718$	1.07705e7	0.779693
$719$	2.03791e7	1.47015	0.735075	−	0.677986i	$-0.237147\pi$
$719$	2.03791e7	1.47015	0.735075	+	0.677986i	$0.237147\pi$
$720$	−1.47021e6	−0.105693
$721$	−1.34540e7	−0.963856
$722$	−3.79800e7	−2.71151
$723$	−2.11267e6	−0.150310
$724$	−7680.45	−0.000544553 0
$725$	−781198.	−0.0551971
$726$	−1.56364e7	−1.10102
$727$	2.60136e7	1.82543	0.912713	−	0.408601i	$-0.133983\pi$
$727$	2.60136e7	1.82543	0.912713	+	0.408601i	$0.133983\pi$
$728$	−778484.	−0.0544404
$729$	1.58838e7	1.10697
$730$	856939.	0.0595173
$731$	6.56405e6	0.454337
$732$	2.56673e7	1.77052
$733$	1.74641e6	0.120057	0.0600283	−	0.998197i	$-0.480881\pi$
$733$	1.74641e6	0.120057	0.0600283	+	0.998197i	$0.480881\pi$
$734$	−3.22674e7	−2.21067
$735$	3.19265e6	0.217988
$736$	−4.39333e6	−0.298951
$737$	−1.03761e6	−0.0703664
$738$	−4.03680e6	−0.272832
$739$	−2.22491e7	−1.49865	−0.749326	−	0.662201i	$-0.769622\pi$
$739$	−2.22491e7	−1.49865	−0.749326	+	0.662201i	$0.769622\pi$
$740$	1.43438e7	0.962905
$741$	1.43357e7	0.959122
$742$	1.61758e7	1.07859
$743$	−1.10893e7	−0.736938	−0.368469	−	0.929640i	$-0.620118\pi$
$743$	−1.10893e7	−0.736938	−0.368469	+	0.929640i	$0.620118\pi$
$744$	794587.	0.0526270
$745$	−1.22084e7	−0.805873
$746$	3.32610e7	2.18821
$747$	1.90471e6	0.124890
$748$	6.39558e6	0.417952
$749$	−295900.	−0.0192726
$750$	1.71110e6	0.111077
$751$	2.22405e7	1.43895	0.719474	−	0.694519i	$-0.244383\pi$
$751$	2.22405e7	1.43895	0.719474	+	0.694519i	$0.244383\pi$
$752$	−6.50684e6	−0.419590
$753$	−1.74445e7	−1.12117
$754$	4.09433e6	0.262273
$755$	−8.85626e6	−0.565435
$756$	−1.21850e7	−0.775391
$757$	8.33656e6	0.528746	0.264373	−	0.964421i	$-0.414835\pi$
$757$	8.33656e6	0.528746	0.264373	+	0.964421i	$0.414835\pi$
$758$	2.51942e7	1.59267
$759$	3.90740e6	0.246197
$760$	1.51917e6	0.0954051
$761$	1.85004e7	1.15803	0.579014	−	0.815317i	$-0.303438\pi$
$761$	1.85004e7	1.15803	0.579014	+	0.815317i	$0.303438\pi$
$762$	3.00710e6	0.187612
$763$	8.64187e6	0.537398
$764$	−5.79566e6	−0.359227
$765$	−528949.	−0.0326784
$766$	3.38218e6	0.208269
$767$	−1.46452e7	−0.898890
$768$	1.12951e7	0.691015
$769$	−2.80478e6	−0.171034	−0.0855172	−	0.996337i	$-0.527254\pi$
$769$	−2.80478e6	−0.171034	−0.0855172	+	0.996337i	$0.527254\pi$
$770$	−9.60671e6	−0.583912
$771$	1.45541e7	0.881760
$772$	703181.	0.0424643
$773$	−8.78805e6	−0.528986	−0.264493	−	0.964388i	$-0.585205\pi$
$773$	−8.78805e6	−0.528986	−0.264493	+	0.964388i	$0.585205\pi$
$774$	−1.02018e7	−0.612100
$775$	−1.62776e6	−0.0973500
$776$	−2.37071e6	−0.141327
$777$	−1.88540e7	−1.12034
$778$	−4.85231e7	−2.87408
$779$	−2.07838e7	−1.22710
$780$	−4.67106e6	−0.274903
$781$	1.14216e7	0.670039
$782$	−1.44197e6	−0.0843214
$783$	5.13251e6	0.299175
$784$	8.83526e6	0.513368
$785$	7.71598e6	0.446907
$786$	3.25385e7	1.87863
$787$	−2.68037e7	−1.54262	−0.771309	−	0.636461i	$-0.780398\pi$
$787$	−2.68037e7	−1.54262	−0.771309	+	0.636461i	$0.780398\pi$
$788$	−2.04814e7	−1.17502
$789$	2.33410e7	1.33483
$790$	−567225.	−0.0323361
$791$	513334.	0.0291715
$792$	−796076.	−0.0450964
$793$	2.20710e7	1.24635
$794$	3.21026e7	1.80713
$795$	7.77327e6	0.436200
$796$	2.57580e7	1.44089
$797$	−2.76614e7	−1.54251	−0.771257	−	0.636524i	$-0.780372\pi$
$797$	−2.76614e7	−1.54251	−0.771257	+	0.636524i	$0.780372\pi$
$798$	−2.49331e7	−1.38602
$799$	−2.34102e6	−0.129730
$800$	5.19061e6	0.286743
$801$	7.97021e6	0.438923
$802$	3.49687e7	1.91974
$803$	−2.31199e6	−0.126531
$804$	877469.	0.0478731
$805$	1.12815e6	0.0613590
$806$	8.53123e6	0.462566
$807$	1.27954e7	0.691626
$808$	−2.50851e6	−0.135172
$809$	2.90367e7	1.55983	0.779913	−	0.625888i	$-0.215263\pi$
$809$	2.90367e7	1.55983	0.779913	+	0.625888i	$0.215263\pi$
$810$	−8.09278e6	−0.433396
$811$	−1.48264e7	−0.791561	−0.395781	−	0.918345i	$-0.629526\pi$
$811$	−1.48264e7	−0.791561	−0.395781	+	0.918345i	$0.629526\pi$
$812$	−3.70901e6	−0.197409
$813$	2.84004e6	0.150695
$814$	−7.42986e7	−3.93025
$815$	7.48238e6	0.394590
$816$	4.14372e6	0.217854
$817$	−5.25246e7	−2.75301
$818$	−3.29217e7	−1.72028
$819$	−2.16895e6	−0.112990
$820$	6.77206e6	0.351711
$821$	−1.57559e7	−0.815804	−0.407902	−	0.913026i	$-0.633740\pi$
$821$	−1.57559e7	−0.815804	−0.407902	+	0.913026i	$0.633740\pi$
$822$	−3.22962e7	−1.66714
$823$	−184301.	−0.00948481	−0.00474241	−	0.999989i	$-0.501510\pi$
$823$	−184301.	−0.00948481	−0.00474241	+	0.999989i	$0.501510\pi$
$824$	3.59084e6	0.184237
$825$	−4.61649e6	−0.236144
$826$	2.54713e7	1.29898
$827$	3.34618e7	1.70132	0.850659	−	0.525718i	$-0.176203\pi$
$827$	3.34618e7	1.70132	0.850659	+	0.525718i	$0.176203\pi$
$828$	1.16729e6	0.0591699
$829$	4.68688e6	0.236863	0.118432	−	0.992962i	$-0.462213\pi$
$829$	4.68688e6	0.236863	0.118432	+	0.992962i	$0.462213\pi$
$830$	−6.13470e6	−0.309099
$831$	1.81893e7	0.913720
$832$	−1.53131e7	−0.766930
$833$	3.17874e6	0.158724
$834$	−3.67461e7	−1.82935
$835$	2.91164e6	0.144518
$836$	−5.11765e7	−2.53253
$837$	1.06945e7	0.527649
$838$	−5.60601e7	−2.75768
$839$	1.02280e6	0.0501631	0.0250816	−	0.999685i	$-0.492015\pi$
$839$	1.02280e6	0.0501631	0.0250816	+	0.999685i	$0.492015\pi$
$840$	650644.	0.0318160
$841$	−1.89489e7	−0.923832
$842$	2.24684e7	1.09218
$843$	2.25956e7	1.09510
$844$	−1.51339e7	−0.731298
$845$	5.26573e6	0.253698
$846$	3.63839e6	0.174777
$847$	1.21802e7	0.583372
$848$	2.15116e7	1.02726
$849$	7.23630e6	0.344546
$850$	1.70365e6	0.0808783
$851$	8.72518e6	0.413001
$852$	−9.65885e6	−0.455854
$853$	5.30980e6	0.249865	0.124932	−	0.992165i	$-0.460129\pi$
$853$	5.30980e6	0.249865	0.124932	+	0.992165i	$0.460129\pi$
$854$	−3.83864e7	−1.80108
$855$	4.23258e6	0.198011
$856$	78975.1	0.00368388
$857$	1.95248e7	0.908099	0.454050	−	0.890976i	$-0.349979\pi$
$857$	1.95248e7	0.908099	0.454050	+	0.890976i	$0.349979\pi$
$858$	2.41955e7	1.12206
$859$	1.11237e6	0.0514358	0.0257179	−	0.999669i	$-0.491813\pi$
$859$	1.11237e6	0.0514358	0.0257179	+	0.999669i	$0.491813\pi$
$860$	1.71143e7	0.789065
$861$	−8.90148e6	−0.409218
$862$	−1.06766e7	−0.489402
$863$	3.79774e7	1.73579	0.867896	−	0.496746i	$-0.165472\pi$
$863$	3.79774e7	1.73579	0.867896	+	0.496746i	$0.165472\pi$
$864$	−3.41026e7	−1.55418
$865$	8.29500e6	0.376943
$866$	2.44129e7	1.10618
$867$	−1.75356e7	−0.792270
$868$	−7.72835e6	−0.348167
$869$	1.53035e6	0.0687452
$870$	−3.42197e6	−0.153278
$871$	754525.	0.0336999
$872$	−2.30650e6	−0.102722
$873$	−6.60508e6	−0.293320
$874$	1.15384e7	0.510937
$875$	−1.33288e6	−0.0588535
$876$	1.95517e6	0.0860843
$877$	2.95488e6	0.129730	0.0648650	−	0.997894i	$-0.479338\pi$
$877$	2.95488e6	0.129730	0.0648650	+	0.997894i	$0.479338\pi$
$878$	2.64095e7	1.15617
$879$	793480.	0.0346389
$880$	−1.27756e7	−0.556126
$881$	2.35223e7	1.02103	0.510516	−	0.859868i	$-0.329454\pi$
$881$	2.35223e7	1.02103	0.510516	+	0.859868i	$0.329454\pi$
$882$	−4.94036e6	−0.213839
$883$	−2.63551e7	−1.13753	−0.568766	−	0.822499i	$-0.692579\pi$
$883$	−2.63551e7	−1.13753	−0.568766	+	0.822499i	$0.692579\pi$
$884$	−4.65071e6	−0.200165
$885$	1.22402e7	0.525329
$886$	−1.54009e7	−0.659116
$887$	−2.54991e7	−1.08822	−0.544108	−	0.839015i	$-0.683132\pi$
$887$	−2.54991e7	−1.08822	−0.544108	+	0.839015i	$0.683132\pi$
$888$	5.03211e6	0.214150
$889$	−2.34242e6	−0.0994056
$890$	−2.56705e7	−1.08633
$891$	2.18341e7	0.921383
$892$	1.00447e7	0.422692
$893$	1.87325e7	0.786082
$894$	−5.34777e7	−2.23784
$895$	−7.02050e6	−0.292962
$896$	3.96252e6	0.164893
$897$	−2.84137e6	−0.117909
$898$	−4.31956e7	−1.78751
$899$	3.25530e6	0.134336
$900$	−1.37912e6	−0.0567538
$901$	7.73941e6	0.317611
$902$	−3.50783e7	−1.43556
$903$	−2.24958e7	−0.918081
$904$	−137008.	−0.00557602
$905$	5519.79	0.000224027 0
$906$	−3.87941e7	−1.57017
$907$	2.91884e7	1.17813	0.589063	−	0.808087i	$-0.299497\pi$
$907$	2.91884e7	1.17813	0.589063	+	0.808087i	$0.299497\pi$
$908$	1.85612e7	0.747124
$909$	−6.98899e6	−0.280546
$910$	6.98577e6	0.279647
$911$	2.24208e7	0.895064	0.447532	−	0.894268i	$-0.352303\pi$
$911$	2.24208e7	0.895064	0.447532	+	0.894268i	$0.352303\pi$
$912$	−3.31574e7	−1.32006
$913$	1.65512e7	0.657133
$914$	4.98414e7	1.97344
$915$	−1.84466e7	−0.728388
$916$	7.29148e6	0.287129
$917$	−2.53463e7	−0.995384
$918$	−1.11930e7	−0.438371
$919$	−9.78554e6	−0.382205	−0.191102	−	0.981570i	$-0.561206\pi$
$919$	−9.78554e6	−0.382205	−0.191102	+	0.981570i	$0.561206\pi$
$920$	−301102.	−0.0117285
$921$	−347506.	−0.0134994
$922$	4.11881e7	1.59568
$923$	−8.30553e6	−0.320895
$924$	−2.19184e7	−0.844557
$925$	−1.03086e7	−0.396136
$926$	3.54228e7	1.35755
$927$	1.00045e7	0.382380
$928$	−1.03805e7	−0.395685
$929$	8.39431e6	0.319114	0.159557	−	0.987189i	$-0.448993\pi$
$929$	8.39431e6	0.319114	0.159557	+	0.987189i	$0.448993\pi$
$930$	−7.13027e6	−0.270333
$931$	−2.54358e7	−0.961771
$932$	2.86895e7	1.08189
$933$	−2.16426e7	−0.813963
$934$	−2.23119e7	−0.836894
$935$	−4.59638e6	−0.171944
$936$	578888.	0.0215976
$937$	2.34833e7	0.873795	0.436897	−	0.899511i	$-0.356077\pi$
$937$	2.34833e7	0.873795	0.436897	+	0.899511i	$0.356077\pi$
$938$	−1.31229e6	−0.0486993
$939$	−3.45094e7	−1.27724
$940$	−6.10370e6	−0.225307
$941$	3.37480e7	1.24244	0.621218	−	0.783638i	$-0.286638\pi$
$941$	3.37480e7	1.24244	0.621218	+	0.783638i	$0.286638\pi$
$942$	3.37992e7	1.24102
$943$	4.11939e6	0.150853
$944$	3.38733e7	1.23716
$945$	8.75712e6	0.318994
$946$	−8.86497e7	−3.22069
$947$	2.58607e7	0.937056	0.468528	−	0.883449i	$-0.344784\pi$
$947$	2.58607e7	0.937056	0.468528	+	0.883449i	$0.344784\pi$
$948$	−1.29417e6	−0.0467702
$949$	1.68123e6	0.0605984
$950$	−1.36323e7	−0.490073
$951$	−1.51246e7	−0.542293
$952$	647809.	0.0231662
$953$	1.58198e7	0.564245	0.282122	−	0.959378i	$-0.408962\pi$
$953$	1.58198e7	0.564245	0.282122	+	0.959378i	$0.408962\pi$
$954$	−1.20285e7	−0.427898
$955$	4.16523e6	0.147785
$956$	−2.33076e7	−0.824808
$957$	9.23238e6	0.325862
$958$	−3.14909e7	−1.10859
$959$	2.51576e7	0.883328
$960$	1.27985e7	0.448208
$961$	−2.18462e7	−0.763074
$962$	5.40282e7	1.88227
$963$	220034.	0.00764582
$964$	−5.48433e6	−0.190078
$965$	−505363.	−0.0174697
$966$	4.94178e6	0.170389
$967$	4.67474e7	1.60765	0.803825	−	0.594865i	$-0.202795\pi$
$967$	4.67474e7	1.60765	0.803825	+	0.594865i	$0.202795\pi$
$968$	−3.25087e6	−0.111509
$969$	−1.19294e7	−0.408138
$970$	2.12737e7	0.725962
$971$	2.86862e7	0.976393	0.488196	−	0.872734i	$-0.337655\pi$
$971$	2.86862e7	0.976393	0.488196	+	0.872734i	$0.337655\pi$
$972$	1.62461e7	0.551548
$973$	2.86239e7	0.969274
$974$	3.90220e7	1.31799
$975$	3.35700e6	0.113094
$976$	−5.10486e7	−1.71537
$977$	−4.19859e6	−0.140724	−0.0703618	−	0.997522i	$-0.522415\pi$
$977$	−4.19859e6	−0.140724	−0.0703618	+	0.997522i	$0.522415\pi$
$978$	3.27759e7	1.09574
$979$	6.92583e7	2.30948
$980$	8.28786e6	0.275662
$981$	−6.42617e6	−0.213196
$982$	4.09590e6	0.135541
$983$	−5.60414e7	−1.84980	−0.924900	−	0.380209i	$-0.875852\pi$
$983$	−5.60414e7	−1.84980	−0.924900	+	0.380209i	$0.875852\pi$
$984$	2.37579e6	0.0782204
$985$	1.47196e7	0.483398
$986$	−3.40707e6	−0.111606
$987$	8.02296e6	0.262145
$988$	3.72143e7	1.21288
$989$	1.04105e7	0.338439
$990$	7.14363e6	0.231649
$991$	−4.50808e7	−1.45817	−0.729083	−	0.684425i	$-0.760053\pi$
$991$	−4.50808e7	−1.45817	−0.729083	+	0.684425i	$0.760053\pi$
$992$	−2.16296e7	−0.697862
$993$	2.14173e7	0.689274
$994$	1.44452e7	0.463722
$995$	−1.85118e7	−0.592776
$996$	−1.39968e7	−0.447074
$997$	1.84395e7	0.587503	0.293752	−	0.955882i	$-0.405096\pi$
$997$	1.84395e7	0.587503	0.293752	+	0.955882i	$0.405096\pi$
$998$	−1.90351e7	−0.604964
$999$	6.77279e7	2.14711

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.6.a.d.1.2	✓	10
3.2	odd	2		1035.6.a.j.1.9		10
5.4	even	2		575.6.a.f.1.9		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.6.a.d.1.2	✓	10	1.1	even	1	trivial
575.6.a.f.1.9		10	5.4	even	2
1035.6.a.j.1.9		10	3.2	odd	2

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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Varieties

Fields

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Groups

Database

Properties

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Newspace parameters

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Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	115.6.a.d.1.2

Level	$115$
Weight	$6$
Character	115.1
Self dual	yes
Analytic conductor	$18.444$
Analytic rank	$1$
Dimension	$10$
CM	no
Inner twists	$1$