LMFDB - Embedded newform 65.6.a.d.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [65,6,Mod(1,65)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(65, 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("65.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 65 = 5 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	65.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:	$10.4249482878$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 163x^{4} - 8x^{3} + 6120x^{2} + 6624x - 19440 $ x^6 - 163x^4 - 8x^3 + 6120x^2 + 6624x - 19440
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$8.51599$ of defining polynomial
Character	$\chi$	$=$	65.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+8.51599 q^{2} +11.2297 q^{3} +40.5221 q^{4} -25.0000 q^{5} +95.6316 q^{6} +229.647 q^{7} +72.5738 q^{8} -116.895 q^{9} -212.900 q^{10} +502.772 q^{11} +455.049 q^{12} +169.000 q^{13} +1955.67 q^{14} -280.741 q^{15} -678.668 q^{16} -453.600 q^{17} -995.476 q^{18} -1894.12 q^{19} -1013.05 q^{20} +2578.86 q^{21} +4281.60 q^{22} -1313.22 q^{23} +814.979 q^{24} +625.000 q^{25} +1439.20 q^{26} -4041.49 q^{27} +9305.78 q^{28} -4570.28 q^{29} -2390.79 q^{30} +3205.38 q^{31} -8101.89 q^{32} +5645.95 q^{33} -3862.86 q^{34} -5741.18 q^{35} -4736.82 q^{36} +7686.60 q^{37} -16130.3 q^{38} +1897.81 q^{39} -1814.35 q^{40} +16247.4 q^{41} +21961.5 q^{42} -15615.8 q^{43} +20373.4 q^{44} +2922.37 q^{45} -11183.4 q^{46} -24741.7 q^{47} -7621.21 q^{48} +35930.9 q^{49} +5322.49 q^{50} -5093.78 q^{51} +6848.23 q^{52} -19027.8 q^{53} -34417.3 q^{54} -12569.3 q^{55} +16666.4 q^{56} -21270.4 q^{57} -38920.5 q^{58} +24196.1 q^{59} -11376.2 q^{60} -22730.3 q^{61} +27297.0 q^{62} -26844.6 q^{63} -47278.3 q^{64} -4225.00 q^{65} +48080.9 q^{66} +65647.6 q^{67} -18380.8 q^{68} -14747.0 q^{69} -48891.8 q^{70} -24448.0 q^{71} -8483.51 q^{72} -17510.1 q^{73} +65459.0 q^{74} +7018.53 q^{75} -76753.8 q^{76} +115460. q^{77} +16161.7 q^{78} +53681.2 q^{79} +16966.7 q^{80} -16979.1 q^{81} +138363. q^{82} +27706.4 q^{83} +104501. q^{84} +11340.0 q^{85} -132984. q^{86} -51322.7 q^{87} +36488.1 q^{88} +35466.0 q^{89} +24886.9 q^{90} +38810.4 q^{91} -53214.4 q^{92} +35995.3 q^{93} -210700. q^{94} +47353.1 q^{95} -90981.4 q^{96} +58497.8 q^{97} +305987. q^{98} -58771.5 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 38 q^{3} + 134 q^{4} - 150 q^{5} + 318 q^{6} + 220 q^{7} + 24 q^{8} + 518 q^{9} - 170 q^{11} + 2238 q^{12} + 1014 q^{13} - 1440 q^{14} - 950 q^{15} + 3506 q^{16} + 728 q^{17} + 7788 q^{18} + 1218 q^{19}+ \cdots - 32270 q^{99}+O(q^{100}) $ 6 * q + 38 * q^3 + 134 * q^4 - 150 * q^5 + 318 * q^6 + 220 * q^7 + 24 * q^8 + 518 * q^9 - 170 * q^11 + 2238 * q^12 + 1014 * q^13 - 1440 * q^14 - 950 * q^15 + 3506 * q^16 + 728 * q^17 + 7788 * q^18 + 1218 * q^19 - 3350 * q^20 - 396 * q^21 + 5154 * q^22 + 8954 * q^23 + 12618 * q^24 + 3750 * q^25 + 13112 * q^27 + 13212 * q^28 + 8364 * q^29 - 7950 * q^30 + 2862 * q^31 - 29672 * q^32 - 3772 * q^33 - 21484 * q^34 - 5500 * q^35 - 2138 * q^36 + 13840 * q^37 - 27706 * q^38 + 6422 * q^39 - 600 * q^40 + 2248 * q^41 - 47028 * q^42 + 3822 * q^43 - 73482 * q^44 - 12950 * q^45 - 39886 * q^46 - 10860 * q^47 + 13962 * q^48 + 85902 * q^49 - 55028 * q^51 + 22646 * q^52 + 6584 * q^53 + 45504 * q^54 + 4250 * q^55 - 188800 * q^56 + 11692 * q^57 - 88500 * q^58 - 28874 * q^59 - 55950 * q^60 + 972 * q^61 + 123466 * q^62 - 125236 * q^63 + 14022 * q^64 - 25350 * q^65 - 158040 * q^66 + 60040 * q^67 + 3504 * q^68 + 153088 * q^69 + 36000 * q^70 - 74174 * q^71 + 298668 * q^72 + 48960 * q^73 + 226112 * q^74 + 23750 * q^75 - 93422 * q^76 + 231588 * q^77 + 53742 * q^78 + 214508 * q^79 - 87650 * q^80 + 116006 * q^81 + 58732 * q^82 + 123260 * q^83 - 235268 * q^84 - 18200 * q^85 - 304666 * q^86 + 228144 * q^87 + 127622 * q^88 + 94028 * q^89 - 194700 * q^90 + 37180 * q^91 + 430298 * q^92 + 192892 * q^93 - 125920 * q^94 - 30450 * q^95 - 385858 * q^96 + 246284 * q^97 - 4736 * q^98 - 32270 * 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 $

$2$	8.51599	1.50543	0.752714	−	0.658347i	$-0.228744\pi$
$2$	8.51599	1.50543	0.752714	+	0.658347i	$0.228744\pi$
$3$	11.2297	0.720382	0.360191	−	0.932878i	$-0.382711\pi$
$3$	11.2297	0.720382	0.360191	+	0.932878i	$0.382711\pi$
$4$	40.5221	1.26631
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	95.6316	1.08448
$7$	229.647	1.77140	0.885699	−	0.464260i	$-0.153680\pi$
$7$	229.647	1.77140	0.885699	+	0.464260i	$0.153680\pi$
$8$	72.5738	0.400918
$9$	−116.895	−0.481049
$10$	−212.900	−0.673248
$11$	502.772	1.25282	0.626411	−	0.779493i	$-0.284523\pi$
$11$	502.772	1.25282	0.626411	+	0.779493i	$0.284523\pi$
$12$	455.049	0.912231
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	1955.67	2.66671
$15$	−280.741	−0.322165
$16$	−678.668	−0.662762
$17$	−453.600	−0.380672	−0.190336	−	0.981719i	$-0.560958\pi$
$17$	−453.600	−0.380672	−0.190336	+	0.981719i	$0.560958\pi$
$18$	−995.476	−0.724185
$19$	−1894.12	−1.20372	−0.601859	−	0.798603i	$-0.705573\pi$
$19$	−1894.12	−1.20372	−0.601859	+	0.798603i	$0.705573\pi$
$20$	−1013.05	−0.566313
$21$	2578.86	1.27608
$22$	4281.60	1.88603
$23$	−1313.22	−0.517628	−0.258814	−	0.965927i	$-0.583332\pi$
$23$	−1313.22	−0.517628	−0.258814	+	0.965927i	$0.583332\pi$
$24$	814.979	0.288814
$25$	625.000	0.200000
$26$	1439.20	0.417531
$27$	−4041.49	−1.06692
$28$	9305.78	2.24315
$29$	−4570.28	−1.00913	−0.504566	−	0.863373i	$-0.668348\pi$
$29$	−4570.28	−1.00913	−0.504566	+	0.863373i	$0.668348\pi$
$30$	−2390.79	−0.484996
$31$	3205.38	0.599067	0.299534	−	0.954086i	$-0.403169\pi$
$31$	3205.38	0.599067	0.299534	+	0.954086i	$0.403169\pi$
$32$	−8101.89	−1.39866
$33$	5645.95	0.902511
$34$	−3862.86	−0.573075
$35$	−5741.18	−0.792193
$36$	−4736.82	−0.609160
$37$	7686.60	0.923060	0.461530	−	0.887125i	$-0.347301\pi$
$37$	7686.60	0.923060	0.461530	+	0.887125i	$0.347301\pi$
$38$	−16130.3	−1.81211
$39$	1897.81	0.199798
$40$	−1814.35	−0.179296
$41$	16247.4	1.50947	0.754737	−	0.656028i	$-0.227765\pi$
$41$	16247.4	1.50947	0.754737	+	0.656028i	$0.227765\pi$
$42$	21961.5	1.92105
$43$	−15615.8	−1.28793	−0.643965	−	0.765055i	$-0.722712\pi$
$43$	−15615.8	−1.28793	−0.643965	+	0.765055i	$0.722712\pi$
$44$	20373.4	1.58647
$45$	2922.37	0.215132
$46$	−11183.4	−0.779252
$47$	−24741.7	−1.63375	−0.816875	−	0.576815i	$-0.804295\pi$
$47$	−24741.7	−1.63375	−0.816875	+	0.576815i	$0.804295\pi$
$48$	−7621.21	−0.477442
$49$	35930.9	2.13785
$50$	5322.49	0.301086
$51$	−5093.78	−0.274230
$52$	6848.23	0.351212
$53$	−19027.8	−0.930460	−0.465230	−	0.885190i	$-0.654028\pi$
$53$	−19027.8	−0.930460	−0.465230	+	0.885190i	$0.654028\pi$
$54$	−34417.3	−1.60617
$55$	−12569.3	−0.560279
$56$	16666.4	0.710185
$57$	−21270.4	−0.867137
$58$	−38920.5	−1.51918
$59$	24196.1	0.904932	0.452466	−	0.891782i	$-0.350544\pi$
$59$	24196.1	0.904932	0.452466	+	0.891782i	$0.350544\pi$
$60$	−11376.2	−0.407962
$61$	−22730.3	−0.782133	−0.391067	−	0.920362i	$-0.627894\pi$
$61$	−22730.3	−0.782133	−0.391067	+	0.920362i	$0.627894\pi$
$62$	27297.0	0.901853
$63$	−26844.6	−0.852130
$64$	−47278.3	−1.44282
$65$	−4225.00	−0.124035
$66$	48080.9	1.35867
$67$	65647.6	1.78662	0.893309	−	0.449442i	$-0.148377\pi$
$67$	65647.6	1.78662	0.893309	+	0.449442i	$0.148377\pi$
$68$	−18380.8	−0.482051
$69$	−14747.0	−0.372890
$70$	−48891.8	−1.19259
$71$	−24448.0	−0.575568	−0.287784	−	0.957695i	$-0.592919\pi$
$71$	−24448.0	−0.575568	−0.287784	+	0.957695i	$0.592919\pi$
$72$	−8483.51	−0.192861
$73$	−17510.1	−0.384576	−0.192288	−	0.981339i	$-0.561591\pi$
$73$	−17510.1	−0.384576	−0.192288	+	0.981339i	$0.561591\pi$
$74$	65459.0	1.38960
$75$	7018.53	0.144076
$76$	−76753.8	−1.52429
$77$	115460.	2.21925
$78$	16161.7	0.300782
$79$	53681.2	0.967730	0.483865	−	0.875143i	$-0.339233\pi$
$79$	53681.2	0.967730	0.483865	+	0.875143i	$0.339233\pi$
$80$	16966.7	0.296396
$81$	−16979.1	−0.287543
$82$	138363.	2.27240
$83$	27706.4	0.441454	0.220727	−	0.975336i	$-0.429157\pi$
$83$	27706.4	0.441454	0.220727	+	0.975336i	$0.429157\pi$
$84$	104501.	1.61592
$85$	11340.0	0.170242
$86$	−132984.	−1.93889
$87$	−51322.7	−0.726961
$88$	36488.1	0.502278
$89$	35466.0	0.474610	0.237305	−	0.971435i	$-0.423736\pi$
$89$	35466.0	0.474610	0.237305	+	0.971435i	$0.423736\pi$
$90$	24886.9	0.323865
$91$	38810.4	0.491298
$92$	−53214.4	−0.655480
$93$	35995.3	0.431558
$94$	−210700.	−2.45949
$95$	47353.1	0.538319
$96$	−90981.4	−1.00757
$97$	58497.8	0.631263	0.315632	−	0.948882i	$-0.397784\pi$
$97$	58497.8	0.631263	0.315632	+	0.948882i	$0.397784\pi$
$98$	305987.	3.21838
$99$	−58771.5	−0.602669
$100$	25326.3	0.253263
$101$	119105.	1.16179	0.580894	−	0.813979i	$-0.302703\pi$
$101$	119105.	1.16179	0.580894	+	0.813979i	$0.302703\pi$
$102$	−43378.5	−0.412833
$103$	140545.	1.30533	0.652667	−	0.757645i	$-0.273650\pi$
$103$	140545.	1.30533	0.652667	+	0.757645i	$0.273650\pi$
$104$	12265.0	0.111195
$105$	−64471.5	−0.570682
$106$	−162040.	−1.40074
$107$	155974.	1.31703	0.658513	−	0.752570i	$-0.271186\pi$
$107$	155974.	1.31703	0.658513	+	0.752570i	$0.271186\pi$
$108$	−163770.	−1.35106
$109$	78243.4	0.630785	0.315392	−	0.948961i	$-0.397864\pi$
$109$	78243.4	0.630785	0.315392	+	0.948961i	$0.397864\pi$
$110$	−107040.	−0.843460
$111$	86317.8	0.664956
$112$	−155854.	−1.17402
$113$	−28676.6	−0.211267	−0.105634	−	0.994405i	$-0.533687\pi$
$113$	−28676.6	−0.211267	−0.105634	+	0.994405i	$0.533687\pi$
$114$	−181138.	−1.30541
$115$	32830.5	0.231490
$116$	−185197.	−1.27788
$117$	−19755.2	−0.133419
$118$	206054.	1.36231
$119$	−104168.	−0.674322
$120$	−20374.5	−0.129162
$121$	91728.6	0.569562
$122$	−193571.	−1.17745
$123$	182453.	1.08740
$124$	129889.	0.758608
$125$	−15625.0	−0.0894427
$126$	−228608.	−1.28282
$127$	−198070.	−1.08970	−0.544852	−	0.838532i	$-0.683414\pi$
$127$	−198070.	−1.08970	−0.544852	+	0.838532i	$0.683414\pi$
$128$	−143360.	−0.773401
$129$	−175360.	−0.927803
$130$	−35980.1	−0.186725
$131$	−376991.	−1.91934	−0.959671	−	0.281126i	$-0.909292\pi$
$131$	−376991.	−1.91934	−0.959671	+	0.281126i	$0.909292\pi$
$132$	228786.	1.14286
$133$	−434981.	−2.13226
$134$	559054.	2.68963
$135$	101037.	0.477142
$136$	−32919.5	−0.152618
$137$	115360.	0.525114	0.262557	−	0.964916i	$-0.415434\pi$
$137$	115360.	0.525114	0.262557	+	0.964916i	$0.415434\pi$
$138$	−125585.	−0.561360
$139$	−349891.	−1.53602	−0.768008	−	0.640440i	$-0.778752\pi$
$139$	−349891.	−1.53602	−0.768008	+	0.640440i	$0.778752\pi$
$140$	−232645.	−1.00317
$141$	−277841.	−1.17692
$142$	−208198.	−0.866476
$143$	84968.5	0.347470
$144$	79332.9	0.318821
$145$	114257.	0.451298
$146$	−149116.	−0.578952
$147$	403491.	1.54007
$148$	311477.	1.16888
$149$	−161771.	−0.596947	−0.298473	−	0.954418i	$-0.596477\pi$
$149$	−161771.	−0.596947	−0.298473	+	0.954418i	$0.596477\pi$
$150$	59769.7	0.216897
$151$	258115.	0.921237	0.460618	−	0.887598i	$-0.347628\pi$
$151$	258115.	0.921237	0.460618	+	0.887598i	$0.347628\pi$
$152$	−137464.	−0.482591
$153$	53023.6	0.183122
$154$	983258.	3.34092
$155$	−80134.6	−0.267911
$156$	76903.2	0.253007
$157$	444441.	1.43901	0.719507	−	0.694486i	$-0.244368\pi$
$157$	444441.	1.43901	0.719507	+	0.694486i	$0.244368\pi$
$158$	457148.	1.45685
$159$	−213675.	−0.670287
$160$	202547.	0.625499
$161$	−301577.	−0.916926
$162$	−144594.	−0.432875
$163$	−140828.	−0.415166	−0.207583	−	0.978217i	$-0.566560\pi$
$163$	−140828.	−0.415166	−0.207583	+	0.978217i	$0.566560\pi$
$164$	658380.	1.91147
$165$	−141149.	−0.403615
$166$	235948.	0.664577
$167$	−78814.3	−0.218682	−0.109341	−	0.994004i	$-0.534874\pi$
$167$	−78814.3	−0.218682	−0.109341	+	0.994004i	$0.534874\pi$
$168$	187158.	0.511605
$169$	28561.0	0.0769231
$170$	96571.4	0.256287
$171$	221414.	0.579047
$172$	−632784.	−1.63093
$173$	282076.	0.716558	0.358279	−	0.933615i	$-0.383364\pi$
$173$	282076.	0.716558	0.358279	+	0.933615i	$0.383364\pi$
$174$	−437063.	−1.09439
$175$	143530.	0.354280
$176$	−341215.	−0.830323
$177$	271714.	0.651897
$178$	302028.	0.714492
$179$	−112083.	−0.261460	−0.130730	−	0.991418i	$-0.541732\pi$
$179$	−112083.	−0.261460	−0.130730	+	0.991418i	$0.541732\pi$
$180$	118421.	0.272424
$181$	151962.	0.344777	0.172389	−	0.985029i	$-0.444852\pi$
$181$	151962.	0.344777	0.172389	+	0.985029i	$0.444852\pi$
$182$	330509.	0.739613
$183$	−255254.	−0.563435
$184$	−95305.4	−0.207526
$185$	−192165.	−0.412805
$186$	306536.	0.649679
$187$	−228058.	−0.476914
$188$	−1.00259e6	−2.06884
$189$	−928118.	−1.88994
$190$	403259.	0.810400
$191$	−616584.	−1.22295	−0.611475	−	0.791264i	$-0.709423\pi$
$191$	−616584.	−1.22295	−0.611475	+	0.791264i	$0.709423\pi$
$192$	−530918.	−1.03938
$193$	−590265.	−1.14065	−0.570327	−	0.821418i	$-0.693183\pi$
$193$	−590265.	−1.14065	−0.570327	+	0.821418i	$0.693183\pi$
$194$	498167.	0.950321
$195$	−47445.3	−0.0893524
$196$	1.45599e6	2.70719
$197$	−585138.	−1.07422	−0.537109	−	0.843513i	$-0.680484\pi$
$197$	−585138.	−1.07422	−0.537109	+	0.843513i	$0.680484\pi$
$198$	−500497.	−0.907275
$199$	455610.	0.815570	0.407785	−	0.913078i	$-0.366301\pi$
$199$	455610.	0.815570	0.407785	+	0.913078i	$0.366301\pi$
$200$	45358.6	0.0801835
$201$	737200.	1.28705
$202$	1.01430e6	1.74899
$203$	−1.04955e6	−1.78758
$204$	−206410.	−0.347261
$205$	−406186.	−0.675057
$206$	1.19688e6	1.96509
$207$	153509.	0.249005
$208$	−114695.	−0.183817
$209$	−952313.	−1.50804
$210$	−549038.	−0.859121
$211$	477952.	0.739057	0.369529	−	0.929219i	$-0.379519\pi$
$211$	477952.	0.739057	0.369529	+	0.929219i	$0.379519\pi$
$212$	−771044.	−1.17826
$213$	−274542.	−0.414629
$214$	1.32828e6	1.98269
$215$	390394.	0.575980
$216$	−293307.	−0.427748
$217$	736108.	1.06119
$218$	666320.	0.949601
$219$	−196633.	−0.277042
$220$	−509334.	−0.709489
$221$	−76658.5	−0.105579
$222$	735082.	1.00104
$223$	1.38878e6	1.87013	0.935065	−	0.354478i	$-0.115341\pi$
$223$	1.38878e6	1.87013	0.935065	+	0.354478i	$0.115341\pi$
$224$	−1.86058e6	−2.47758
$225$	−73059.3	−0.0962098
$226$	−244210.	−0.318048
$227$	−159073.	−0.204896	−0.102448	−	0.994738i	$-0.532667\pi$
$227$	−159073.	−0.204896	−0.102448	+	0.994738i	$0.532667\pi$
$228$	−861919.	−1.09807
$229$	−333919.	−0.420777	−0.210389	−	0.977618i	$-0.567473\pi$
$229$	−333919.	−0.420777	−0.210389	+	0.977618i	$0.567473\pi$
$230$	279584.	0.348492
$231$	1.29658e6	1.59871
$232$	−331683.	−0.404579
$233$	−453757.	−0.547562	−0.273781	−	0.961792i	$-0.588274\pi$
$233$	−453757.	−0.547562	−0.273781	+	0.961792i	$0.588274\pi$
$234$	−168235.	−0.200853
$235$	618543.	0.730635
$236$	980477.	1.14593
$237$	602821.	0.697136
$238$	−887095.	−1.01514
$239$	−752595.	−0.852249	−0.426124	−	0.904665i	$-0.640121\pi$
$239$	−752595.	−0.852249	−0.426124	+	0.904665i	$0.640121\pi$
$240$	190530.	0.213519
$241$	219818.	0.243792	0.121896	−	0.992543i	$-0.461103\pi$
$241$	219818.	0.243792	0.121896	+	0.992543i	$0.461103\pi$
$242$	781160.	0.857435
$243$	791414.	0.859781
$244$	−921079.	−0.990427
$245$	−898272.	−0.956077
$246$	1.55377e6	1.63700
$247$	−320107.	−0.333851
$248$	232627.	0.240177
$249$	311133.	0.318015
$250$	−133062.	−0.134650
$251$	−673696.	−0.674963	−0.337481	−	0.941332i	$-0.609575\pi$
$251$	−673696.	−0.674963	−0.337481	+	0.941332i	$0.609575\pi$
$252$	−1.08780e6	−1.07906
$253$	−660250.	−0.648496
$254$	−1.68676e6	−1.64047
$255$	127344.	0.122639
$256$	292048.	0.278518
$257$	−95463.7	−0.0901582	−0.0450791	−	0.998983i	$-0.514354\pi$
$257$	−95463.7	−0.0901582	−0.0450791	+	0.998983i	$0.514354\pi$
$258$	−1.49336e6	−1.39674
$259$	1.76521e6	1.63511
$260$	−171206.	−0.157067
$261$	534243.	0.485442
$262$	−3.21045e6	−2.88943
$263$	−140198.	−0.124983	−0.0624916	−	0.998045i	$-0.519905\pi$
$263$	−140198.	−0.124983	−0.0624916	+	0.998045i	$0.519905\pi$
$264$	409748.	0.361832
$265$	475694.	0.416114
$266$	−3.70429e6	−3.20997
$267$	398271.	0.341901
$268$	2.66018e6	2.26242
$269$	987335.	0.831924	0.415962	−	0.909382i	$-0.363445\pi$
$269$	987335.	0.831924	0.415962	+	0.909382i	$0.363445\pi$
$270$	860433.	0.718303
$271$	−1.09742e6	−0.907713	−0.453857	−	0.891075i	$-0.649952\pi$
$271$	−1.09742e6	−0.907713	−0.453857	+	0.891075i	$0.649952\pi$
$272$	307844.	0.252295
$273$	435827.	0.353922
$274$	982404.	0.790522
$275$	314232.	0.250564
$276$	−597579.	−0.472196
$277$	−1.75035e6	−1.37065	−0.685325	−	0.728238i	$-0.740340\pi$
$277$	−1.75035e6	−1.37065	−0.685325	+	0.728238i	$0.740340\pi$
$278$	−2.97967e6	−2.31236
$279$	−374693.	−0.288181
$280$	−416660.	−0.317604
$281$	783692.	0.592079	0.296039	−	0.955176i	$-0.404334\pi$
$281$	783692.	0.592079	0.296039	+	0.955176i	$0.404334\pi$
$282$	−2.36609e6	−1.77178
$283$	1.32705e6	0.984963	0.492482	−	0.870323i	$-0.336090\pi$
$283$	1.32705e6	0.984963	0.492482	+	0.870323i	$0.336090\pi$
$284$	−990682.	−0.728850
$285$	531759.	0.387795
$286$	723590.	0.523092
$287$	3.73118e6	2.67388
$288$	947070.	0.672823
$289$	−1.21410e6	−0.855089
$290$	973012.	0.679396
$291$	656910.	0.454751
$292$	−709547.	−0.486995
$293$	495312.	0.337062	0.168531	−	0.985696i	$-0.446098\pi$
$293$	495312.	0.337062	0.168531	+	0.985696i	$0.446098\pi$
$294$	3.43613e6	2.31847
$295$	−604903.	−0.404698
$296$	557846.	0.370071
$297$	−2.03195e6	−1.33666
$298$	−1.37764e6	−0.898660
$299$	−221934.	−0.143564
$300$	284405.	0.182446
$301$	−3.58612e6	−2.28144
$302$	2.19811e6	1.38686
$303$	1.33751e6	0.836932
$304$	1.28548e6	0.797778
$305$	568258.	0.349781
$306$	451548.	0.275677
$307$	−691003.	−0.418441	−0.209221	−	0.977868i	$-0.567093\pi$
$307$	−691003.	−0.418441	−0.209221	+	0.977868i	$0.567093\pi$
$308$	4.67869e6	2.81026
$309$	1.57827e6	0.940340
$310$	−682425.	−0.403321
$311$	1.51561e6	0.888560	0.444280	−	0.895888i	$-0.353460\pi$
$311$	1.51561e6	0.888560	0.444280	+	0.895888i	$0.353460\pi$
$312$	137731.	0.0801026
$313$	829838.	0.478776	0.239388	−	0.970924i	$-0.423053\pi$
$313$	829838.	0.478776	0.239388	+	0.970924i	$0.423053\pi$
$314$	3.78485e6	2.16633
$315$	671115.	0.381084
$316$	2.17527e6	1.22545
$317$	−2.54241e6	−1.42101	−0.710507	−	0.703691i	$-0.751534\pi$
$317$	−2.54241e6	−1.42101	−0.710507	+	0.703691i	$0.751534\pi$
$318$	−1.81965e6	−1.00907
$319$	−2.29781e6	−1.26426
$320$	1.18196e6	0.645248
$321$	1.75154e6	0.948762
$322$	−2.56823e6	−1.38037
$323$	859176.	0.458222
$324$	−688028.	−0.364119
$325$	105625.	0.0554700
$326$	−1.19929e6	−0.625002
$327$	878646.	0.454406
$328$	1.17914e6	0.605174
$329$	−5.68187e6	−2.89402
$330$	−1.20202e6	−0.607614
$331$	−2.87653e6	−1.44311	−0.721553	−	0.692359i	$-0.756572\pi$
$331$	−2.87653e6	−1.44311	−0.721553	+	0.692359i	$0.756572\pi$
$332$	1.12272e6	0.559019
$333$	−898525.	−0.444037
$334$	−671182.	−0.329211
$335$	−1.64119e6	−0.799000
$336$	−1.75019e6	−0.845740
$337$	1.34564e6	0.645437	0.322719	−	0.946495i	$-0.395403\pi$
$337$	1.34564e6	0.645437	0.322719	+	0.946495i	$0.395403\pi$
$338$	243225.	0.115802
$339$	−322029.	−0.152193
$340$	459521.	0.215580
$341$	1.61158e6	0.750525
$342$	1.88556e6	0.871714
$343$	4.39175e6	2.01559
$344$	−1.13330e6	−0.516354
$345$	368675.	0.166762
$346$	2.40216e6	1.07873
$347$	3.90267e6	1.73995	0.869977	−	0.493092i	$-0.164133\pi$
$347$	3.90267e6	1.73995	0.869977	+	0.493092i	$0.164133\pi$
$348$	−2.07970e6	−0.920562
$349$	4.23422e6	1.86084	0.930422	−	0.366489i	$-0.119440\pi$
$349$	4.23422e6	1.86084	0.930422	+	0.366489i	$0.119440\pi$
$350$	1.22230e6	0.533343
$351$	−683013.	−0.295911
$352$	−4.07340e6	−1.75227
$353$	−3.24104e6	−1.38436	−0.692178	−	0.721727i	$-0.743349\pi$
$353$	−3.24104e6	−1.38436	−0.692178	+	0.721727i	$0.743349\pi$
$354$	2.31391e6	0.981384
$355$	611199.	0.257402
$356$	1.43716e6	0.601006
$357$	−1.16977e6	−0.485770
$358$	−954495.	−0.393610
$359$	2.89904e6	1.18718	0.593591	−	0.804767i	$-0.297710\pi$
$359$	2.89904e6	1.18718	0.593591	+	0.804767i	$0.297710\pi$
$360$	212088.	0.0862501
$361$	1.11161e6	0.448936
$362$	1.29411e6	0.519038
$363$	1.03008e6	0.410303
$364$	1.57268e6	0.622137
$365$	437753.	0.171988
$366$	−2.17374e6	−0.848211
$367$	−463452.	−0.179614	−0.0898068	−	0.995959i	$-0.528625\pi$
$367$	−463452.	−0.179614	−0.0898068	+	0.995959i	$0.528625\pi$
$368$	891241.	0.343064
$369$	−1.89924e6	−0.726131
$370$	−1.63648e6	−0.621448
$371$	−4.36967e6	−1.64822
$372$	1.45861e6	0.546488
$373$	1.91392e6	0.712283	0.356142	−	0.934432i	$-0.384092\pi$
$373$	1.91392e6	0.712283	0.356142	+	0.934432i	$0.384092\pi$
$374$	−1.94214e6	−0.717961
$375$	−175463.	−0.0644330
$376$	−1.79560e6	−0.654999
$377$	−772378.	−0.279883
$378$	−7.90385e6	−2.84517
$379$	−815893.	−0.291766	−0.145883	−	0.989302i	$-0.546602\pi$
$379$	−815893.	−0.291766	−0.145883	+	0.989302i	$0.546602\pi$
$380$	1.91885e6	0.681681
$381$	−2.22425e6	−0.785004
$382$	−5.25082e6	−1.84106
$383$	2.97503e6	1.03632	0.518160	−	0.855283i	$-0.326617\pi$
$383$	2.97503e6	1.03632	0.518160	+	0.855283i	$0.326617\pi$
$384$	−1.60989e6	−0.557144
$385$	−2.88651e6	−0.992477
$386$	−5.02669e6	−1.71717
$387$	1.82541e6	0.619558
$388$	2.37045e6	0.799378
$389$	−4.54774e6	−1.52378	−0.761889	−	0.647707i	$-0.775728\pi$
$389$	−4.54774e6	−1.52378	−0.761889	+	0.647707i	$0.775728\pi$
$390$	−404043.	−0.134514
$391$	595677.	0.197047
$392$	2.60764e6	0.857102
$393$	−4.23347e6	−1.38266
$394$	−4.98303e6	−1.61716
$395$	−1.34203e6	−0.432782
$396$	−2.38154e6	−0.763168
$397$	1.81318e6	0.577385	0.288693	−	0.957422i	$-0.406779\pi$
$397$	1.81318e6	0.577385	0.288693	+	0.957422i	$0.406779\pi$
$398$	3.87997e6	1.22778
$399$	−4.88468e6	−1.53604
$400$	−424168.	−0.132552
$401$	2.43351e6	0.755740	0.377870	−	0.925859i	$-0.376657\pi$
$401$	2.43351e6	0.755740	0.377870	+	0.925859i	$0.376657\pi$
$402$	6.27798e6	1.93756
$403$	541710.	0.166151
$404$	4.82639e6	1.47119
$405$	424477.	0.128593
$406$	−8.93798e6	−2.69107
$407$	3.86461e6	1.15643
$408$	−369675.	−0.109943
$409$	1.67145e6	0.494066	0.247033	−	0.969007i	$-0.420544\pi$
$409$	1.67145e6	0.494066	0.247033	+	0.969007i	$0.420544\pi$
$410$	−3.45908e6	−1.01625
$411$	1.29545e6	0.378283
$412$	5.69517e6	1.65296
$413$	5.55658e6	1.60299
$414$	1.30728e6	0.374859
$415$	−692660.	−0.197424
$416$	−1.36922e6	−0.387918
$417$	−3.92915e6	−1.10652
$418$	−8.10988e6	−2.27025
$419$	−6.10400e6	−1.69855	−0.849277	−	0.527948i	$-0.822962\pi$
$419$	−6.10400e6	−1.69855	−0.849277	+	0.527948i	$0.822962\pi$
$420$	−2.61252e6	−0.722663
$421$	2.83501e6	0.779559	0.389780	−	0.920908i	$-0.372551\pi$
$421$	2.83501e6	0.779559	0.389780	+	0.920908i	$0.372551\pi$
$422$	4.07023e6	1.11260
$423$	2.89218e6	0.785914
$424$	−1.38092e6	−0.373038
$425$	−283500.	−0.0761344
$426$	−2.33800e6	−0.624194
$427$	−5.21996e6	−1.38547
$428$	6.32041e6	1.66777
$429$	954166.	0.250311
$430$	3.32459e6	0.867097
$431$	−666945.	−0.172941	−0.0864703	−	0.996254i	$-0.527559\pi$
$431$	−666945.	−0.172941	−0.0864703	+	0.996254i	$0.527559\pi$
$432$	2.74283e6	0.707115
$433$	6.17798e6	1.58353	0.791766	−	0.610825i	$-0.209162\pi$
$433$	6.17798e6	1.58353	0.791766	+	0.610825i	$0.209162\pi$
$434$	6.26868e6	1.59754
$435$	1.28307e6	0.325107
$436$	3.17058e6	0.798772
$437$	2.48740e6	0.623078
$438$	−1.67452e6	−0.417067
$439$	6.33242e6	1.56823	0.784113	−	0.620618i	$-0.213118\pi$
$439$	6.33242e6	1.56823	0.784113	+	0.620618i	$0.213118\pi$
$440$	−912202.	−0.224626
$441$	−4.20014e6	−1.02841
$442$	−652823.	−0.158942
$443$	−5.49568e6	−1.33049	−0.665246	−	0.746624i	$-0.731673\pi$
$443$	−5.49568e6	−1.33049	−0.665246	+	0.746624i	$0.731673\pi$
$444$	3.49778e6	0.842044
$445$	−886650.	−0.212252
$446$	1.18268e7	2.81535
$447$	−1.81663e6	−0.430030
$448$	−1.08573e7	−2.55581
$449$	−2.72617e6	−0.638170	−0.319085	−	0.947726i	$-0.603376\pi$
$449$	−2.72617e6	−0.638170	−0.319085	+	0.947726i	$0.603376\pi$
$450$	−622173.	−0.144837
$451$	8.16876e6	1.89110
$452$	−1.16204e6	−0.267531
$453$	2.89854e6	0.663643
$454$	−1.35467e6	−0.308456
$455$	−970260.	−0.219715
$456$	−1.54367e6	−0.347650
$457$	−2.69725e6	−0.604131	−0.302065	−	0.953287i	$-0.597676\pi$
$457$	−2.69725e6	−0.604131	−0.302065	+	0.953287i	$0.597676\pi$
$458$	−2.84365e6	−0.633450
$459$	1.83322e6	0.406148
$460$	1.33036e6	0.293140
$461$	6.41429e6	1.40571	0.702856	−	0.711332i	$-0.251908\pi$
$461$	6.41429e6	1.40571	0.702856	+	0.711332i	$0.251908\pi$
$462$	1.10416e7	2.40674
$463$	−6.06138e6	−1.31407	−0.657035	−	0.753860i	$-0.728190\pi$
$463$	−6.06138e6	−1.31407	−0.657035	+	0.753860i	$0.728190\pi$
$464$	3.10171e6	0.668815
$465$	−899883.	−0.192998
$466$	−3.86419e6	−0.824315
$467$	−4.16010e6	−0.882697	−0.441348	−	0.897336i	$-0.645500\pi$
$467$	−4.16010e6	−0.882697	−0.441348	+	0.897336i	$0.645500\pi$
$468$	−800523.	−0.168950
$469$	1.50758e7	3.16481
$470$	5.26751e6	1.09992
$471$	4.99092e6	1.03664
$472$	1.75601e6	0.362803
$473$	−7.85117e6	−1.61355
$474$	5.13362e6	1.04949
$475$	−1.18383e6	−0.240743
$476$	−4.22111e6	−0.853904
$477$	2.22425e6	0.447597
$478$	−6.40909e6	−1.28300
$479$	−4.84248e6	−0.964338	−0.482169	−	0.876078i	$-0.660151\pi$
$479$	−4.84248e6	−0.964338	−0.482169	+	0.876078i	$0.660151\pi$
$480$	2.27454e6	0.450598
$481$	1.29904e6	0.256011
$482$	1.87196e6	0.367012
$483$	−3.38661e6	−0.660537
$484$	3.71703e6	0.721245
$485$	−1.46245e6	−0.282309
$486$	6.73967e6	1.29434
$487$	5.38041e6	1.02800	0.514000	−	0.857790i	$-0.328163\pi$
$487$	5.38041e6	1.02800	0.514000	+	0.857790i	$0.328163\pi$
$488$	−1.64963e6	−0.313571
$489$	−1.58145e6	−0.299078
$490$	−7.64967e6	−1.43930
$491$	7.11622e6	1.33213	0.666063	−	0.745895i	$-0.267978\pi$
$491$	7.11622e6	1.33213	0.666063	+	0.745895i	$0.267978\pi$
$492$	7.39338e6	1.37699
$493$	2.07308e6	0.384149
$494$	−2.72603e6	−0.502589
$495$	1.46929e6	0.269522
$496$	−2.17539e6	−0.397039
$497$	−5.61441e6	−1.01956
$498$	2.64961e6	0.478749
$499$	−4.84522e6	−0.871088	−0.435544	−	0.900167i	$-0.643444\pi$
$499$	−4.84522e6	−0.871088	−0.435544	+	0.900167i	$0.643444\pi$
$500$	−633157.	−0.113263
$501$	−885057.	−0.157535
$502$	−5.73719e6	−1.01611
$503$	−9.07263e6	−1.59887	−0.799435	−	0.600752i	$-0.794868\pi$
$503$	−9.07263e6	−1.59887	−0.799435	+	0.600752i	$0.794868\pi$
$504$	−1.94822e6	−0.341634
$505$	−2.97763e6	−0.519567
$506$	−5.62268e6	−0.976264
$507$	320730.	0.0554140
$508$	−8.02619e6	−1.37991
$509$	4.27353e6	0.731126	0.365563	−	0.930787i	$-0.380877\pi$
$509$	4.27353e6	0.731126	0.365563	+	0.930787i	$0.380877\pi$
$510$	1.08446e6	0.184625
$511$	−4.02116e6	−0.681238
$512$	7.07461e6	1.19269
$513$	7.65509e6	1.28427
$514$	−812968.	−0.135727
$515$	−3.51362e6	−0.583763
$516$	−7.10594e6	−1.17489
$517$	−1.24394e7	−2.04680
$518$	1.50325e7	2.46154
$519$	3.16762e6	0.516196
$520$	−306624.	−0.0497277
$521$	−1.08227e7	−1.74679	−0.873396	−	0.487010i	$-0.838087\pi$
$521$	−1.08227e7	−1.74679	−0.873396	+	0.487010i	$0.838087\pi$
$522$	4.54961e6	0.730799
$523$	8.74432e6	1.39789	0.698943	−	0.715178i	$-0.253654\pi$
$523$	8.74432e6	1.39789	0.698943	+	0.715178i	$0.253654\pi$
$524$	−1.52764e7	−2.43049
$525$	1.61179e6	0.255217
$526$	−1.19392e6	−0.188153
$527$	−1.45396e6	−0.228048
$528$	−3.83173e6	−0.598150
$529$	−4.71180e6	−0.732061
$530$	4.05100e6	0.626431
$531$	−2.82841e6	−0.435317
$532$	−1.76263e7	−2.70012
$533$	2.74582e6	0.418653
$534$	3.39167e6	0.514708
$535$	−3.89936e6	−0.588992
$536$	4.76430e6	0.716287
$537$	−1.25865e6	−0.188352
$538$	8.40813e6	1.25240
$539$	1.80650e7	2.67835
$540$	4.09424e6	0.604212
$541$	−1.21182e7	−1.78011	−0.890054	−	0.455856i	$-0.849333\pi$
$541$	−1.21182e7	−1.78011	−0.890054	+	0.455856i	$0.849333\pi$
$542$	−9.34560e6	−1.36650
$543$	1.70648e6	0.248372
$544$	3.67502e6	0.532430
$545$	−1.95608e6	−0.282096
$546$	3.71150e6	0.532804
$547$	8.62953e6	1.23316	0.616579	−	0.787293i	$-0.288518\pi$
$547$	8.62953e6	1.23316	0.616579	+	0.787293i	$0.288518\pi$
$548$	4.67463e6	0.664960
$549$	2.65706e6	0.376245
$550$	2.67600e6	0.377207
$551$	8.65669e6	1.21471
$552$	−1.07025e6	−0.149498
$553$	1.23277e7	1.71424
$554$	−1.49060e7	−2.06341
$555$	−2.15795e6	−0.297378
$556$	−1.41783e7	−1.94508
$557$	−5.42393e6	−0.740758	−0.370379	−	0.928881i	$-0.620772\pi$
$557$	−5.42393e6	−0.740758	−0.370379	+	0.928881i	$0.620772\pi$
$558$	−3.19088e6	−0.433836
$559$	−2.63907e6	−0.357208
$560$	3.89636e6	0.525036
$561$	−2.56101e6	−0.343561
$562$	6.67391e6	0.891332
$563$	4.98711e6	0.663099	0.331549	−	0.943438i	$-0.392429\pi$
$563$	4.98711e6	0.663099	0.331549	+	0.943438i	$0.392429\pi$
$564$	−1.12587e7	−1.49036
$565$	716916.	0.0944816
$566$	1.13011e7	1.48279
$567$	−3.89920e6	−0.509352
$568$	−1.77428e6	−0.230755
$569$	−7.02975e6	−0.910246	−0.455123	−	0.890429i	$-0.650405\pi$
$569$	−7.02975e6	−0.910246	−0.455123	+	0.890429i	$0.650405\pi$
$570$	4.52845e6	0.583798
$571$	−4.95071e6	−0.635443	−0.317722	−	0.948184i	$-0.602918\pi$
$571$	−4.95071e6	−0.635443	−0.317722	+	0.948184i	$0.602918\pi$
$572$	3.44310e6	0.440007
$573$	−6.92402e6	−0.880992
$574$	3.17747e7	4.02533
$575$	−820763.	−0.103526
$576$	5.52659e6	0.694066
$577$	860779.	0.107635	0.0538173	−	0.998551i	$-0.482861\pi$
$577$	860779.	0.107635	0.0538173	+	0.998551i	$0.482861\pi$
$578$	−1.03393e7	−1.28727
$579$	−6.62847e6	−0.821707
$580$	4.62993e6	0.571485
$581$	6.36270e6	0.781990
$582$	5.59424e6	0.684595
$583$	−9.56662e6	−1.16570
$584$	−1.27078e6	−0.154183
$585$	493881.	0.0596668
$586$	4.21807e6	0.507423
$587$	−6.96051e6	−0.833769	−0.416884	−	0.908959i	$-0.636878\pi$
$587$	−6.96051e6	−0.833769	−0.416884	+	0.908959i	$0.636878\pi$
$588$	1.63503e7	1.95021
$589$	−6.07140e6	−0.721108
$590$	−5.15135e6	−0.609244
$591$	−6.57089e6	−0.773848
$592$	−5.21665e6	−0.611769
$593$	3.89146e6	0.454439	0.227220	−	0.973844i	$-0.427036\pi$
$593$	3.89146e6	0.454439	0.227220	+	0.973844i	$0.427036\pi$
$594$	−1.73041e7	−2.01225
$595$	2.60420e6	0.301566
$596$	−6.55530e6	−0.755922
$597$	5.11635e6	0.587522
$598$	−1.88999e6	−0.216126
$599$	−2.95768e6	−0.336809	−0.168405	−	0.985718i	$-0.553862\pi$
$599$	−2.95768e6	−0.336809	−0.168405	+	0.985718i	$0.553862\pi$
$600$	509362.	0.0577628
$601$	−3.86928e6	−0.436962	−0.218481	−	0.975841i	$-0.570110\pi$
$601$	−3.86928e6	−0.436962	−0.218481	+	0.975841i	$0.570110\pi$
$602$	−3.05394e7	−3.43454
$603$	−7.67387e6	−0.859452
$604$	1.04594e7	1.16658
$605$	−2.29321e6	−0.254716
$606$	1.13902e7	1.25994
$607$	3.31072e6	0.364713	0.182356	−	0.983233i	$-0.441628\pi$
$607$	3.31072e6	0.364713	0.182356	+	0.983233i	$0.441628\pi$
$608$	1.53460e7	1.68359
$609$	−1.17861e7	−1.28774
$610$	4.83928e6	0.526570
$611$	−4.18135e6	−0.453121
$612$	2.14863e6	0.231890
$613$	1.00301e7	1.07809	0.539046	−	0.842276i	$-0.318785\pi$
$613$	1.00301e7	1.07809	0.539046	+	0.842276i	$0.318785\pi$
$614$	−5.88458e6	−0.629933
$615$	−4.56133e6	−0.486299
$616$	8.37939e6	0.889735
$617$	−1.03851e6	−0.109824	−0.0549121	−	0.998491i	$-0.517488\pi$
$617$	−1.03851e6	−0.109824	−0.0549121	+	0.998491i	$0.517488\pi$
$618$	1.34405e7	1.41561
$619$	−3.76934e6	−0.395402	−0.197701	−	0.980262i	$-0.563348\pi$
$619$	−3.76934e6	−0.395402	−0.197701	+	0.980262i	$0.563348\pi$
$620$	−3.24722e6	−0.339260
$621$	5.30737e6	0.552269
$622$	1.29069e7	1.33766
$623$	8.14467e6	0.840724
$624$	−1.28798e6	−0.132419
$625$	390625.	0.0400000
$626$	7.06689e6	0.720763
$627$	−1.06941e7	−1.08637
$628$	1.80097e7	1.82224
$629$	−3.48665e6	−0.351383
$630$	5.71521e6	0.573695
$631$	1.12994e7	1.12975	0.564874	−	0.825177i	$-0.308925\pi$
$631$	1.12994e7	1.12975	0.564874	+	0.825177i	$0.308925\pi$
$632$	3.89585e6	0.387980
$633$	5.36723e6	0.532404
$634$	−2.16512e7	−2.13923
$635$	4.95174e6	0.487331
$636$	−8.65855e6	−0.848794
$637$	6.07232e6	0.592933
$638$	−1.95681e7	−1.90326
$639$	2.85784e6	0.276876
$640$	3.58401e6	0.345875
$641$	7.88738e6	0.758207	0.379104	−	0.925354i	$-0.376232\pi$
$641$	7.88738e6	0.758207	0.379104	+	0.925354i	$0.376232\pi$
$642$	1.49161e7	1.42829
$643$	−1.12400e7	−1.07211	−0.536056	−	0.844183i	$-0.680086\pi$
$643$	−1.12400e7	−1.07211	−0.536056	+	0.844183i	$0.680086\pi$
$644$	−1.22205e7	−1.16112
$645$	4.38399e6	0.414926
$646$	7.31673e6	0.689820
$647$	3.37493e6	0.316960	0.158480	−	0.987362i	$-0.449341\pi$
$647$	3.37493e6	0.316960	0.158480	+	0.987362i	$0.449341\pi$
$648$	−1.23224e6	−0.115281
$649$	1.21651e7	1.13372
$650$	899501.	0.0835061
$651$	8.26623e6	0.764461
$652$	−5.70666e6	−0.525730
$653$	−9.31325e6	−0.854709	−0.427355	−	0.904084i	$-0.640554\pi$
$653$	−9.31325e6	−0.854709	−0.427355	+	0.904084i	$0.640554\pi$
$654$	7.48254e6	0.684076
$655$	9.42476e6	0.858356
$656$	−1.10266e7	−1.00042
$657$	2.04685e6	0.185000
$658$	−4.83868e7	−4.35674
$659$	−1.79159e7	−1.60704	−0.803519	−	0.595279i	$-0.797041\pi$
$659$	−1.79159e7	−1.60704	−0.803519	+	0.595279i	$0.797041\pi$
$660$	−5.71964e6	−0.511104
$661$	6.41296e6	0.570893	0.285447	−	0.958395i	$-0.407858\pi$
$661$	6.41296e6	0.570893	0.285447	+	0.958395i	$0.407858\pi$
$662$	−2.44965e7	−2.17249
$663$	−860848.	−0.0760576
$664$	2.01076e6	0.176987
$665$	1.08745e7	0.953577
$666$	−7.65183e6	−0.668466
$667$	6.00179e6	0.522355
$668$	−3.19372e6	−0.276921
$669$	1.55955e7	1.34721
$670$	−1.39764e7	−1.20284
$671$	−1.14282e7	−0.979874
$672$	−2.08936e7	−1.78481
$673$	1.79299e7	1.52595	0.762977	−	0.646426i	$-0.223737\pi$
$673$	1.79299e7	1.52595	0.762977	+	0.646426i	$0.223737\pi$
$674$	1.14594e7	0.971659
$675$	−2.52593e6	−0.213384
$676$	1.15735e6	0.0974088
$677$	−1.90216e7	−1.59505	−0.797526	−	0.603284i	$-0.793858\pi$
$677$	−1.90216e7	−1.59505	−0.797526	+	0.603284i	$0.793858\pi$
$678$	−2.74239e6	−0.229116
$679$	1.34339e7	1.11822
$680$	822988.	0.0682529
$681$	−1.78634e6	−0.147603
$682$	1.37242e7	1.12986
$683$	−7.06814e6	−0.579767	−0.289884	−	0.957062i	$-0.593617\pi$
$683$	−7.06814e6	−0.579767	−0.289884	+	0.957062i	$0.593617\pi$
$684$	8.97214e6	0.733256
$685$	−2.88400e6	−0.234838
$686$	3.74001e7	3.03433
$687$	−3.74979e6	−0.303121
$688$	1.05979e7	0.853591
$689$	−3.21569e6	−0.258063
$690$	3.13963e6	0.251048
$691$	−1.15060e6	−0.0916703	−0.0458352	−	0.998949i	$-0.514595\pi$
$691$	−1.15060e6	−0.0916703	−0.0458352	+	0.998949i	$0.514595\pi$
$692$	1.14303e7	0.907388
$693$	−1.34967e7	−1.06757
$694$	3.32351e7	2.61938
$695$	8.74728e6	0.686928
$696$	−3.72468e6	−0.291452
$697$	−7.36985e6	−0.574615
$698$	3.60586e7	2.80137
$699$	−5.09553e6	−0.394454
$700$	5.81611e6	0.448630
$701$	−2.35577e6	−0.181066	−0.0905331	−	0.995893i	$-0.528857\pi$
$701$	−2.35577e6	−0.181066	−0.0905331	+	0.995893i	$0.528857\pi$
$702$	−5.81653e6	−0.445473
$703$	−1.45594e7	−1.11110
$704$	−2.37702e7	−1.80759
$705$	6.94603e6	0.526337
$706$	−2.76007e7	−2.08405
$707$	2.73522e7	2.05799
$708$	1.10104e7	0.825507
$709$	−2.42423e6	−0.181117	−0.0905585	−	0.995891i	$-0.528865\pi$
$709$	−2.42423e6	−0.181117	−0.0905585	+	0.995891i	$0.528865\pi$
$710$	5.20496e6	0.387500
$711$	−6.27506e6	−0.465526
$712$	2.57390e6	0.190280
$713$	−4.20937e6	−0.310094
$714$	−9.96176e6	−0.731292
$715$	−2.12421e6	−0.155393
$716$	−4.54182e6	−0.331091
$717$	−8.45138e6	−0.613945
$718$	2.46882e7	1.78722
$719$	1.50460e7	1.08542	0.542712	−	0.839919i	$-0.317398\pi$
$719$	1.50460e7	1.08542	0.542712	+	0.839919i	$0.317398\pi$
$720$	−1.98332e6	−0.142581
$721$	3.22757e7	2.31227
$722$	9.46645e6	0.675841
$723$	2.46847e6	0.175624
$724$	6.15782e6	0.436597
$725$	−2.85643e6	−0.201826
$726$	8.77215e6	0.617681
$727$	1.82431e7	1.28016	0.640078	−	0.768310i	$-0.278902\pi$
$727$	1.82431e7	1.28016	0.640078	+	0.768310i	$0.278902\pi$
$728$	2.81662e6	0.196970
$729$	1.30132e7	0.906914
$730$	3.72790e6	0.258915
$731$	7.08332e6	0.490279
$732$	−1.03434e7	−0.713486
$733$	−3.52470e6	−0.242305	−0.121152	−	0.992634i	$-0.538659\pi$
$733$	−3.52470e6	−0.242305	−0.121152	+	0.992634i	$0.538659\pi$
$734$	−3.94675e6	−0.270395
$735$	−1.00873e7	−0.688741
$736$	1.06396e7	0.723985
$737$	3.30058e7	2.23832
$738$	−1.61739e7	−1.09314
$739$	−1.26675e7	−0.853255	−0.426627	−	0.904427i	$-0.640298\pi$
$739$	−1.26675e7	−0.853255	−0.426627	+	0.904427i	$0.640298\pi$
$740$	−7.78692e6	−0.522741
$741$	−3.59469e6	−0.240501
$742$	−3.72121e7	−2.48127
$743$	−2.58128e7	−1.71539	−0.857694	−	0.514160i	$-0.828104\pi$
$743$	−2.58128e7	−1.71539	−0.857694	+	0.514160i	$0.828104\pi$
$744$	2.61232e6	0.173019
$745$	4.04428e6	0.266963
$746$	1.62990e7	1.07229
$747$	−3.23874e6	−0.212361
$748$	−9.24136e6	−0.603924
$749$	3.58191e7	2.33298
$750$	−1.49424e6	−0.0969992
$751$	−4.98464e6	−0.322503	−0.161251	−	0.986913i	$-0.551553\pi$
$751$	−4.98464e6	−0.322503	−0.161251	+	0.986913i	$0.551553\pi$
$752$	1.67914e7	1.08279
$753$	−7.56537e6	−0.486231
$754$	−6.57756e6	−0.421344
$755$	−6.45288e6	−0.411990
$756$	−3.76093e7	−2.39326
$757$	−2.60215e7	−1.65041	−0.825206	−	0.564831i	$-0.808941\pi$
$757$	−2.60215e7	−1.65041	−0.825206	+	0.564831i	$0.808941\pi$
$758$	−6.94813e6	−0.439233
$759$	−7.41438e6	−0.467165
$760$	3.43660e6	0.215821
$761$	1.00530e7	0.629264	0.314632	−	0.949214i	$-0.398119\pi$
$761$	1.00530e7	0.629264	0.314632	+	0.949214i	$0.398119\pi$
$762$	−1.89417e7	−1.18177
$763$	1.79684e7	1.11737
$764$	−2.49852e7	−1.54864
$765$	−1.32559e6	−0.0818947
$766$	2.53353e7	1.56011
$767$	4.08915e6	0.250983
$768$	3.27959e6	0.200640
$769$	−1.26849e7	−0.773522	−0.386761	−	0.922180i	$-0.626406\pi$
$769$	−1.26849e7	−0.773522	−0.386761	+	0.922180i	$0.626406\pi$
$770$	−2.45814e7	−1.49410
$771$	−1.07202e6	−0.0649484
$772$	−2.39188e7	−1.44443
$773$	744727.	0.0448279	0.0224139	−	0.999749i	$-0.492865\pi$
$773$	744727.	0.0448279	0.0224139	+	0.999749i	$0.492865\pi$
$774$	1.55451e7	0.932700
$775$	2.00336e6	0.119813
$776$	4.24541e6	0.253084
$777$	1.98227e7	1.17790
$778$	−3.87285e7	−2.29394
$779$	−3.07747e7	−1.81698
$780$	−1.92258e6	−0.113148
$781$	−1.22917e7	−0.721084
$782$	5.07278e6	0.296640
$783$	1.84708e7	1.07667
$784$	−2.43851e7	−1.41689
$785$	−1.11110e7	−0.643546
$786$	−3.60522e7	−2.08150
$787$	−2.69272e7	−1.54972	−0.774861	−	0.632132i	$-0.782180\pi$
$787$	−2.69272e7	−1.54972	−0.774861	+	0.632132i	$0.782180\pi$
$788$	−2.37110e7	−1.36030
$789$	−1.57437e6	−0.0900357
$790$	−1.14287e7	−0.651522
$791$	−6.58551e6	−0.374238
$792$	−4.26527e6	−0.241621
$793$	−3.84142e6	−0.216925
$794$	1.54411e7	0.869212
$795$	5.34188e6	0.299762
$796$	1.84623e7	1.03277
$797$	2.60489e7	1.45259	0.726296	−	0.687382i	$-0.241240\pi$
$797$	2.60489e7	1.45259	0.726296	+	0.687382i	$0.241240\pi$
$798$	−4.15979e7	−2.31241
$799$	1.12229e7	0.621923
$800$	−5.06368e6	−0.279732
$801$	−4.14580e6	−0.228311
$802$	2.07238e7	1.13771
$803$	−8.80361e6	−0.481805
$804$	2.98729e7	1.62981
$805$	7.53944e6	0.410062
$806$	4.61319e6	0.250129
$807$	1.10874e7	0.599304
$808$	8.64391e6	0.465781
$809$	−9.26480e6	−0.497697	−0.248848	−	0.968542i	$-0.580052\pi$
$809$	−9.26480e6	−0.497697	−0.248848	+	0.968542i	$0.580052\pi$
$810$	3.61485e6	0.193587
$811$	−7.02749e6	−0.375187	−0.187594	−	0.982247i	$-0.560069\pi$
$811$	−7.02749e6	−0.375187	−0.187594	+	0.982247i	$0.560069\pi$
$812$	−4.25301e7	−2.26363
$813$	−1.23236e7	−0.653901
$814$	3.29110e7	1.74092
$815$	3.52071e6	0.185668
$816$	3.45698e6	0.181749
$817$	2.95782e7	1.55030
$818$	1.42340e7	0.743781
$819$	−4.53674e6	−0.236338
$820$	−1.64595e7	−0.854835
$821$	−5.97329e6	−0.309283	−0.154641	−	0.987971i	$-0.549422\pi$
$821$	−5.97329e6	−0.309283	−0.154641	+	0.987971i	$0.549422\pi$
$822$	1.10321e7	0.569478
$823$	3.83061e7	1.97137	0.985684	−	0.168601i	$-0.0539249\pi$
$823$	3.83061e7	1.97137	0.985684	+	0.168601i	$0.0539249\pi$
$824$	1.01999e7	0.523331
$825$	3.52872e6	0.180502
$826$	4.73197e7	2.41319
$827$	2.55362e7	1.29835	0.649177	−	0.760637i	$-0.275113\pi$
$827$	2.55362e7	1.29835	0.649177	+	0.760637i	$0.275113\pi$
$828$	6.22049e6	0.315318
$829$	−3.30695e7	−1.67125	−0.835625	−	0.549301i	$-0.814894\pi$
$829$	−3.30695e7	−1.67125	−0.835625	+	0.549301i	$0.814894\pi$
$830$	−5.89869e6	−0.297208
$831$	−1.96559e7	−0.987392
$832$	−7.99002e6	−0.400166
$833$	−1.62983e7	−0.813821
$834$	−3.34606e7	−1.66579
$835$	1.97036e6	0.0977978
$836$	−3.85897e7	−1.90966
$837$	−1.29545e7	−0.639158
$838$	−5.19816e7	−2.55705
$839$	−5.58580e6	−0.273956	−0.136978	−	0.990574i	$-0.543739\pi$
$839$	−5.58580e6	−0.273956	−0.136978	+	0.990574i	$0.543739\pi$
$840$	−4.67894e6	−0.228797
$841$	376341.	0.0183481
$842$	2.41429e7	1.17357
$843$	8.80059e6	0.426523
$844$	1.93676e7	0.935879
$845$	−714025.	−0.0344010
$846$	2.46298e7	1.18314
$847$	2.10652e7	1.00892
$848$	1.29135e7	0.616674
$849$	1.49023e7	0.709550
$850$	−2.41429e6	−0.114615
$851$	−1.00942e7	−0.477802
$852$	−1.11250e7	−0.525051
$853$	1.67371e7	0.787604	0.393802	−	0.919195i	$-0.371160\pi$
$853$	1.67371e7	0.787604	0.393802	+	0.919195i	$0.371160\pi$
$854$	−4.44531e7	−2.08573
$855$	−5.53534e6	−0.258958
$856$	1.13197e7	0.528019
$857$	2.22774e7	1.03613	0.518063	−	0.855343i	$-0.326653\pi$
$857$	2.22774e7	1.03613	0.518063	+	0.855343i	$0.326653\pi$
$858$	8.12567e6	0.376826
$859$	2.58125e7	1.19357	0.596785	−	0.802401i	$-0.296445\pi$
$859$	2.58125e7	1.19357	0.596785	+	0.802401i	$0.296445\pi$
$860$	1.58196e7	0.729372
$861$	4.18999e7	1.92622
$862$	−5.67970e6	−0.260350
$863$	1.79739e7	0.821517	0.410758	−	0.911744i	$-0.365264\pi$
$863$	1.79739e7	0.821517	0.410758	+	0.911744i	$0.365264\pi$
$864$	3.27438e7	1.49226
$865$	−7.05191e6	−0.320454
$866$	5.26116e7	2.38389
$867$	−1.36340e7	−0.615991
$868$	2.98286e7	1.34380
$869$	2.69894e7	1.21239
$870$	1.09266e7	0.489425
$871$	1.10944e7	0.495519
$872$	5.67842e6	0.252893
$873$	−6.83810e6	−0.303669
$874$	2.11827e7	0.937999
$875$	−3.58824e6	−0.158439
$876$	−7.96797e6	−0.350822
$877$	−1.87373e7	−0.822637	−0.411318	−	0.911492i	$-0.634932\pi$
$877$	−1.87373e7	−0.822637	−0.411318	+	0.911492i	$0.634932\pi$
$878$	5.39268e7	2.36085
$879$	5.56218e6	0.242813
$880$	8.53038e6	0.371332
$881$	5.88375e6	0.255396	0.127698	−	0.991813i	$-0.459241\pi$
$881$	5.88375e6	0.255396	0.127698	+	0.991813i	$0.459241\pi$
$882$	−3.57683e7	−1.54820
$883$	−2.17863e7	−0.940333	−0.470166	−	0.882578i	$-0.655806\pi$
$883$	−2.17863e7	−0.940333	−0.470166	+	0.882578i	$0.655806\pi$
$884$	−3.10636e6	−0.133697
$885$	−6.79285e6	−0.291537
$886$	−4.68012e7	−2.00296
$887$	3.27848e6	0.139915	0.0699574	−	0.997550i	$-0.477714\pi$
$887$	3.27848e6	0.139915	0.0699574	+	0.997550i	$0.477714\pi$
$888$	6.26442e6	0.266593
$889$	−4.54862e7	−1.93030
$890$	−7.55070e6	−0.319531
$891$	−8.53661e6	−0.360240
$892$	5.62763e7	2.36817
$893$	4.68639e7	1.96657
$894$	−1.54704e7	−0.647379
$895$	2.80207e6	0.116929
$896$	−3.29223e7	−1.37000
$897$	−2.49224e6	−0.103421
$898$	−2.32160e7	−0.960719
$899$	−1.46495e7	−0.604538
$900$	−2.96052e6	−0.121832
$901$	8.63100e6	0.354200
$902$	6.95651e7	2.84692
$903$	−4.02709e7	−1.64351
$904$	−2.08117e6	−0.0847007
$905$	−3.79905e6	−0.154189
$906$	2.46840e7	0.999067
$907$	−1.09733e7	−0.442914	−0.221457	−	0.975170i	$-0.571081\pi$
$907$	−1.09733e7	−0.442914	−0.221457	+	0.975170i	$0.571081\pi$
$908$	−6.44598e6	−0.259462
$909$	−1.39228e7	−0.558877
$910$	−8.26272e6	−0.330765
$911$	−3.55315e7	−1.41846	−0.709232	−	0.704975i	$-0.750958\pi$
$911$	−3.55315e7	−1.41846	−0.709232	+	0.704975i	$0.750958\pi$
$912$	1.44355e7	0.574705
$913$	1.39300e7	0.553063
$914$	−2.29698e7	−0.909476
$915$	6.38134e6	0.251976
$916$	−1.35311e7	−0.532837
$917$	−8.65749e7	−3.39992
$918$	1.56117e7	0.611426
$919$	2.84725e7	1.11208	0.556042	−	0.831154i	$-0.312319\pi$
$919$	2.84725e7	1.11208	0.556042	+	0.831154i	$0.312319\pi$
$920$	2.38264e6	0.0928086
$921$	−7.75973e6	−0.301438
$922$	5.46240e7	2.11620
$923$	−4.13170e6	−0.159634
$924$	5.25400e7	2.02446
$925$	4.80413e6	0.184612
$926$	−5.16186e7	−1.97824
$927$	−1.64290e7	−0.627930
$928$	3.70279e7	1.41143
$929$	8.20156e6	0.311786	0.155893	−	0.987774i	$-0.450174\pi$
$929$	8.20156e6	0.311786	0.155893	+	0.987774i	$0.450174\pi$
$930$	−7.66340e6	−0.290545
$931$	−6.80576e7	−2.57337
$932$	−1.83872e7	−0.693386
$933$	1.70198e7	0.640103
$934$	−3.54274e7	−1.32884
$935$	5.70144e6	0.213283
$936$	−1.43371e6	−0.0534900
$937$	2.94945e7	1.09747	0.548734	−	0.835997i	$-0.315110\pi$
$937$	2.94945e7	1.09747	0.548734	+	0.835997i	$0.315110\pi$
$938$	1.28385e8	4.76440
$939$	9.31879e6	0.344902
$940$	2.50647e7	0.925214
$941$	2.45596e7	0.904165	0.452083	−	0.891976i	$-0.350681\pi$
$941$	2.45596e7	0.904165	0.452083	+	0.891976i	$0.350681\pi$
$942$	4.25026e7	1.56059
$943$	−2.13365e7	−0.781346
$944$	−1.64211e7	−0.599754
$945$	2.32030e7	0.845208
$946$	−6.68605e7	−2.42908
$947$	8.92873e6	0.323530	0.161765	−	0.986829i	$-0.448281\pi$
$947$	8.92873e6	0.323530	0.161765	+	0.986829i	$0.448281\pi$
$948$	2.44275e7	0.882793
$949$	−2.95921e6	−0.106662
$950$	−1.00815e7	−0.362422
$951$	−2.85504e7	−1.02367
$952$	−7.55988e6	−0.270348
$953$	−3.05698e7	−1.09033	−0.545167	−	0.838327i	$-0.683534\pi$
$953$	−3.05698e7	−1.09033	−0.545167	+	0.838327i	$0.683534\pi$
$954$	1.89417e7	0.673825
$955$	1.54146e7	0.546920
$956$	−3.04967e7	−1.07922
$957$	−2.58036e7	−0.910753
$958$	−4.12385e7	−1.45174
$959$	2.64921e7	0.930186
$960$	1.32730e7	0.464825
$961$	−1.83547e7	−0.641118
$962$	1.10626e7	0.385406
$963$	−1.82326e7	−0.633554
$964$	8.90746e6	0.308718
$965$	1.47566e7	0.510116
$966$	−2.88403e7	−0.994391
$967$	−1.99903e7	−0.687469	−0.343734	−	0.939067i	$-0.611692\pi$
$967$	−1.99903e7	−0.687469	−0.343734	+	0.939067i	$0.611692\pi$
$968$	6.65709e6	0.228348
$969$	9.64825e6	0.330095
$970$	−1.24542e7	−0.424997
$971$	−5.02538e7	−1.71049	−0.855245	−	0.518224i	$-0.826593\pi$
$971$	−5.02538e7	−1.71049	−0.855245	+	0.518224i	$0.826593\pi$
$972$	3.20697e7	1.08875
$973$	−8.03515e7	−2.72090
$974$	4.58195e7	1.54758
$975$	1.18613e6	0.0399596
$976$	1.54263e7	0.518368
$977$	2.08198e6	0.0697816	0.0348908	−	0.999391i	$-0.488892\pi$
$977$	2.08198e6	0.0697816	0.0348908	+	0.999391i	$0.488892\pi$
$978$	−1.34677e7	−0.450241
$979$	1.78313e7	0.594602
$980$	−3.63998e7	−1.21069
$981$	−9.14625e6	−0.303439
$982$	6.06016e7	2.00542
$983$	3.36402e7	1.11039	0.555194	−	0.831721i	$-0.312644\pi$
$983$	3.36402e7	1.11039	0.555194	+	0.831721i	$0.312644\pi$
$984$	1.32413e7	0.435957
$985$	1.46284e7	0.480405
$986$	1.76543e7	0.578308
$987$	−6.38054e7	−2.08480
$988$	−1.29714e7	−0.422761
$989$	2.05069e7	0.666669
$990$	1.25124e7	0.405746
$991$	−3.75872e6	−0.121578	−0.0607891	−	0.998151i	$-0.519362\pi$
$991$	−3.75872e6	−0.121578	−0.0607891	+	0.998151i	$0.519362\pi$
$992$	−2.59697e7	−0.837891
$993$	−3.23024e7	−1.03959
$994$	−4.78122e7	−1.53487
$995$	−1.13903e7	−0.364734
$996$	1.26078e7	0.402708
$997$	1.49889e6	0.0477563	0.0238782	−	0.999715i	$-0.492399\pi$
$997$	1.49889e6	0.0477563	0.0238782	+	0.999715i	$0.492399\pi$
$998$	−4.12619e7	−1.31136
$999$	−3.10654e7	−0.984833

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	65.6.a.d.1.5	✓	6
3.2	odd	2		585.6.a.m.1.2		6
4.3	odd	2		1040.6.a.q.1.3		6
5.2	odd	4		325.6.b.g.274.10		12
5.3	odd	4		325.6.b.g.274.3		12
5.4	even	2		325.6.a.g.1.2		6
13.12	even	2		845.6.a.h.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.6.a.d.1.5	✓	6	1.1	even	1	trivial
325.6.a.g.1.2		6	5.4	even	2
325.6.b.g.274.3		12	5.3	odd	4
325.6.b.g.274.10		12	5.2	odd	4
585.6.a.m.1.2		6	3.2	odd	2
845.6.a.h.1.2		6	13.12	even	2
1040.6.a.q.1.3		6	4.3	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:	\( N \)	\(=\)	\( 65 = 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	65.a (trivial)

\(f(q)\)	\(=\)	\(q+8.51599 q^{2} +11.2297 q^{3} +40.5221 q^{4} -25.0000 q^{5} +95.6316 q^{6} +229.647 q^{7} +72.5738 q^{8} -116.895 q^{9} -212.900 q^{10} +502.772 q^{11} +455.049 q^{12} +169.000 q^{13} +1955.67 q^{14} -280.741 q^{15} -678.668 q^{16} -453.600 q^{17} -995.476 q^{18} -1894.12 q^{19} -1013.05 q^{20} +2578.86 q^{21} +4281.60 q^{22} -1313.22 q^{23} +814.979 q^{24} +625.000 q^{25} +1439.20 q^{26} -4041.49 q^{27} +9305.78 q^{28} -4570.28 q^{29} -2390.79 q^{30} +3205.38 q^{31} -8101.89 q^{32} +5645.95 q^{33} -3862.86 q^{34} -5741.18 q^{35} -4736.82 q^{36} +7686.60 q^{37} -16130.3 q^{38} +1897.81 q^{39} -1814.35 q^{40} +16247.4 q^{41} +21961.5 q^{42} -15615.8 q^{43} +20373.4 q^{44} +2922.37 q^{45} -11183.4 q^{46} -24741.7 q^{47} -7621.21 q^{48} +35930.9 q^{49} +5322.49 q^{50} -5093.78 q^{51} +6848.23 q^{52} -19027.8 q^{53} -34417.3 q^{54} -12569.3 q^{55} +16666.4 q^{56} -21270.4 q^{57} -38920.5 q^{58} +24196.1 q^{59} -11376.2 q^{60} -22730.3 q^{61} +27297.0 q^{62} -26844.6 q^{63} -47278.3 q^{64} -4225.00 q^{65} +48080.9 q^{66} +65647.6 q^{67} -18380.8 q^{68} -14747.0 q^{69} -48891.8 q^{70} -24448.0 q^{71} -8483.51 q^{72} -17510.1 q^{73} +65459.0 q^{74} +7018.53 q^{75} -76753.8 q^{76} +115460. q^{77} +16161.7 q^{78} +53681.2 q^{79} +16966.7 q^{80} -16979.1 q^{81} +138363. q^{82} +27706.4 q^{83} +104501. q^{84} +11340.0 q^{85} -132984. q^{86} -51322.7 q^{87} +36488.1 q^{88} +35466.0 q^{89} +24886.9 q^{90} +38810.4 q^{91} -53214.4 q^{92} +35995.3 q^{93} -210700. q^{94} +47353.1 q^{95} -90981.4 q^{96} +58497.8 q^{97} +305987. q^{98} -58771.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 38 q^{3} + 134 q^{4} - 150 q^{5} + 318 q^{6} + 220 q^{7} + 24 q^{8} + 518 q^{9} - 170 q^{11} + 2238 q^{12} + 1014 q^{13} - 1440 q^{14} - 950 q^{15} + 3506 q^{16} + 728 q^{17} + 7788 q^{18} + 1218 q^{19}+ \cdots - 32270 q^{99}+O(q^{100}) \) 6 * q + 38 * q^3 + 134 * q^4 - 150 * q^5 + 318 * q^6 + 220 * q^7 + 24 * q^8 + 518 * q^9 - 170 * q^11 + 2238 * q^12 + 1014 * q^13 - 1440 * q^14 - 950 * q^15 + 3506 * q^16 + 728 * q^17 + 7788 * q^18 + 1218 * q^19 - 3350 * q^20 - 396 * q^21 + 5154 * q^22 + 8954 * q^23 + 12618 * q^24 + 3750 * q^25 + 13112 * q^27 + 13212 * q^28 + 8364 * q^29 - 7950 * q^30 + 2862 * q^31 - 29672 * q^32 - 3772 * q^33 - 21484 * q^34 - 5500 * q^35 - 2138 * q^36 + 13840 * q^37 - 27706 * q^38 + 6422 * q^39 - 600 * q^40 + 2248 * q^41 - 47028 * q^42 + 3822 * q^43 - 73482 * q^44 - 12950 * q^45 - 39886 * q^46 - 10860 * q^47 + 13962 * q^48 + 85902 * q^49 - 55028 * q^51 + 22646 * q^52 + 6584 * q^53 + 45504 * q^54 + 4250 * q^55 - 188800 * q^56 + 11692 * q^57 - 88500 * q^58 - 28874 * q^59 - 55950 * q^60 + 972 * q^61 + 123466 * q^62 - 125236 * q^63 + 14022 * q^64 - 25350 * q^65 - 158040 * q^66 + 60040 * q^67 + 3504 * q^68 + 153088 * q^69 + 36000 * q^70 - 74174 * q^71 + 298668 * q^72 + 48960 * q^73 + 226112 * q^74 + 23750 * q^75 - 93422 * q^76 + 231588 * q^77 + 53742 * q^78 + 214508 * q^79 - 87650 * q^80 + 116006 * q^81 + 58732 * q^82 + 123260 * q^83 - 235268 * q^84 - 18200 * q^85 - 304666 * q^86 + 228144 * q^87 + 127622 * q^88 + 94028 * q^89 - 194700 * q^90 + 37180 * q^91 + 430298 * q^92 + 192892 * q^93 - 125920 * q^94 - 30450 * q^95 - 385858 * q^96 + 246284 * q^97 - 4736 * q^98 - 32270 * q^99

Introduction

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