LMFDB - Embedded newform 230.6.a.i.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [230,6,Mod(1,230)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("230.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 230 = 2 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	230.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,24,15] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$36.8882785570$
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} - 3x^{5} - 1156x^{4} + 593x^{3} + 338133x^{2} + 408388x - 13033476 $ x^6 - 3x^5 - 1156x^4 + 593x^3 + 338133x^2 + 408388*x - 13033476
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$5.93160$ of defining polynomial
Character	$\chi$	$=$	230.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.00000 q^{2} +7.93160 q^{3} +16.0000 q^{4} -25.0000 q^{5} +31.7264 q^{6} +221.199 q^{7} +64.0000 q^{8} -180.090 q^{9} -100.000 q^{10} +508.302 q^{11} +126.906 q^{12} +148.017 q^{13} +884.795 q^{14} -198.290 q^{15} +256.000 q^{16} -1625.64 q^{17} -720.359 q^{18} +1657.06 q^{19} -400.000 q^{20} +1754.46 q^{21} +2033.21 q^{22} -529.000 q^{23} +507.622 q^{24} +625.000 q^{25} +592.070 q^{26} -3355.78 q^{27} +3539.18 q^{28} +6957.76 q^{29} -793.160 q^{30} -813.687 q^{31} +1024.00 q^{32} +4031.65 q^{33} -6502.54 q^{34} -5529.97 q^{35} -2881.44 q^{36} +13541.4 q^{37} +6628.24 q^{38} +1174.01 q^{39} -1600.00 q^{40} -7502.30 q^{41} +7017.84 q^{42} +8204.16 q^{43} +8132.83 q^{44} +4502.24 q^{45} -2116.00 q^{46} +4348.32 q^{47} +2030.49 q^{48} +32121.9 q^{49} +2500.00 q^{50} -12893.9 q^{51} +2368.28 q^{52} +37777.3 q^{53} -13423.1 q^{54} -12707.5 q^{55} +14156.7 q^{56} +13143.1 q^{57} +27831.0 q^{58} -33536.0 q^{59} -3172.64 q^{60} -26658.3 q^{61} -3254.75 q^{62} -39835.6 q^{63} +4096.00 q^{64} -3700.44 q^{65} +16126.6 q^{66} -1721.91 q^{67} -26010.2 q^{68} -4195.81 q^{69} -22119.9 q^{70} -39497.2 q^{71} -11525.7 q^{72} -51003.5 q^{73} +54165.7 q^{74} +4957.25 q^{75} +26513.0 q^{76} +112436. q^{77} +4696.06 q^{78} +60468.8 q^{79} -6400.00 q^{80} +17145.1 q^{81} -30009.2 q^{82} +42121.0 q^{83} +28071.4 q^{84} +40640.9 q^{85} +32816.7 q^{86} +55186.1 q^{87} +32531.3 q^{88} -86689.6 q^{89} +18009.0 q^{90} +32741.3 q^{91} -8464.00 q^{92} -6453.84 q^{93} +17393.3 q^{94} -41426.5 q^{95} +8121.95 q^{96} +19131.8 q^{97} +128488. q^{98} -91540.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 24 q^{2} + 15 q^{3} + 96 q^{4} - 150 q^{5} + 60 q^{6} + 106 q^{7} + 384 q^{8} + 899 q^{9} - 600 q^{10} + 321 q^{11} + 240 q^{12} + 527 q^{13} + 424 q^{14} - 375 q^{15} + 1536 q^{16} - 660 q^{17} + 3596 q^{18}+ \cdots - 2973 q^{99}+O(q^{100}) $ 6 * q + 24 * q^2 + 15 * q^3 + 96 * q^4 - 150 * q^5 + 60 * q^6 + 106 * q^7 + 384 * q^8 + 899 * q^9 - 600 * q^10 + 321 * q^11 + 240 * q^12 + 527 * q^13 + 424 * q^14 - 375 * q^15 + 1536 * q^16 - 660 * q^17 + 3596 * q^18 + 2749 * q^19 - 2400 * q^20 + 6002 * q^21 + 1284 * q^22 - 3174 * q^23 + 960 * q^24 + 3750 * q^25 + 2108 * q^26 + 15372 * q^27 + 1696 * q^28 + 3337 * q^29 - 1500 * q^30 + 31094 * q^31 + 6144 * q^32 + 15087 * q^33 - 2640 * q^34 - 2650 * q^35 + 14384 * q^36 + 27037 * q^37 + 10996 * q^38 + 38528 * q^39 - 9600 * q^40 + 33608 * q^41 + 24008 * q^42 + 17024 * q^43 + 5136 * q^44 - 22475 * q^45 - 12696 * q^46 + 16864 * q^47 + 3840 * q^48 + 6002 * q^49 + 15000 * q^50 + 5719 * q^51 + 8432 * q^52 - 8475 * q^53 + 61488 * q^54 - 8025 * q^55 + 6784 * q^56 + 9566 * q^57 + 13348 * q^58 + 7899 * q^59 - 6000 * q^60 + 25437 * q^61 + 124376 * q^62 - 13333 * q^63 + 24576 * q^64 - 13175 * q^65 + 60348 * q^66 - 25517 * q^67 - 10560 * q^68 - 7935 * q^69 - 10600 * q^70 + 17204 * q^71 + 57536 * q^72 + 760 * q^73 + 108148 * q^74 + 9375 * q^75 + 43984 * q^76 + 102330 * q^77 + 154112 * q^78 + 66972 * q^79 - 38400 * q^80 + 115874 * q^81 + 134432 * q^82 + 58523 * q^83 + 96032 * q^84 + 16500 * q^85 + 68096 * q^86 - 70854 * q^87 + 20544 * q^88 + 38406 * q^89 - 89900 * q^90 + 25111 * q^91 - 50784 * q^92 + 130338 * q^93 + 67456 * q^94 - 68725 * q^95 + 15360 * q^96 + 82861 * q^97 + 24008 * q^98 - 2973 * 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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	7.93160	0.508812	0.254406	−	0.967097i	$-0.418120\pi$
$3$	7.93160	0.508812	0.254406	+	0.967097i	$0.418120\pi$
$4$	16.0000	0.500000
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	31.7264	0.359785
$7$	221.199	1.70623	0.853115	−	0.521722i	$-0.174710\pi$
$7$	221.199	1.70623	0.853115	+	0.521722i	$0.174710\pi$
$8$	64.0000	0.353553
$9$	−180.090	−0.741110
$10$	−100.000	−0.316228
$11$	508.302	1.26660	0.633301	−	0.773906i	$-0.281700\pi$
$11$	508.302	1.26660	0.633301	+	0.773906i	$0.281700\pi$
$12$	126.906	0.254406
$13$	148.017	0.242915	0.121458	−	0.992597i	$-0.461243\pi$
$13$	148.017	0.242915	0.121458	+	0.992597i	$0.461243\pi$
$14$	884.795	1.20649
$15$	−198.290	−0.227548
$16$	256.000	0.250000
$17$	−1625.64	−1.36427	−0.682136	−	0.731225i	$-0.738949\pi$
$17$	−1625.64	−1.36427	−0.682136	+	0.731225i	$0.738949\pi$
$18$	−720.359	−0.524044
$19$	1657.06	1.05306	0.526531	−	0.850156i	$-0.323492\pi$
$19$	1657.06	1.05306	0.526531	+	0.850156i	$0.323492\pi$
$20$	−400.000	−0.223607
$21$	1754.46	0.868151
$22$	2033.21	0.895622
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	507.622	0.179892
$25$	625.000	0.200000
$26$	592.070	0.171767
$27$	−3355.78	−0.885898
$28$	3539.18	0.853115
$29$	6957.76	1.53629	0.768147	−	0.640273i	$-0.221179\pi$
$29$	6957.76	1.53629	0.768147	+	0.640273i	$0.221179\pi$
$30$	−793.160	−0.160901
$31$	−813.687	−0.152073	−0.0760367	−	0.997105i	$-0.524227\pi$
$31$	−813.687	−0.152073	−0.0760367	+	0.997105i	$0.524227\pi$
$32$	1024.00	0.176777
$33$	4031.65	0.644462
$34$	−6502.54	−0.964686
$35$	−5529.97	−0.763049
$36$	−2881.44	−0.370555
$37$	13541.4	1.62615	0.813074	−	0.582161i	$-0.197793\pi$
$37$	13541.4	1.62615	0.813074	+	0.582161i	$0.197793\pi$
$38$	6628.24	0.744628
$39$	1174.01	0.123598
$40$	−1600.00	−0.158114
$41$	−7502.30	−0.697003	−0.348502	−	0.937308i	$-0.613309\pi$
$41$	−7502.30	−0.697003	−0.348502	+	0.937308i	$0.613309\pi$
$42$	7017.84	0.613875
$43$	8204.16	0.676649	0.338324	−	0.941030i	$-0.390140\pi$
$43$	8204.16	0.676649	0.338324	+	0.941030i	$0.390140\pi$
$44$	8132.83	0.633301
$45$	4502.24	0.331435
$46$	−2116.00	−0.147442
$47$	4348.32	0.287129	0.143564	−	0.989641i	$-0.454144\pi$
$47$	4348.32	0.287129	0.143564	+	0.989641i	$0.454144\pi$
$48$	2030.49	0.127203
$49$	32121.9	1.91122
$50$	2500.00	0.141421
$51$	−12893.9	−0.694158
$52$	2368.28	0.121458
$53$	37777.3	1.84731	0.923657	−	0.383219i	$-0.125185\pi$
$53$	37777.3	1.84731	0.923657	+	0.383219i	$0.125185\pi$
$54$	−13423.1	−0.626424
$55$	−12707.5	−0.566441
$56$	14156.7	0.603244
$57$	13143.1	0.535811
$58$	27831.0	1.08632
$59$	−33536.0	−1.25424	−0.627122	−	0.778921i	$-0.715767\pi$
$59$	−33536.0	−1.25424	−0.627122	+	0.778921i	$0.715767\pi$
$60$	−3172.64	−0.113774
$61$	−26658.3	−0.917292	−0.458646	−	0.888619i	$-0.651665\pi$
$61$	−26658.3	−0.917292	−0.458646	+	0.888619i	$0.651665\pi$
$62$	−3254.75	−0.107532
$63$	−39835.6	−1.26450
$64$	4096.00	0.125000
$65$	−3700.44	−0.108635
$66$	16126.6	0.455704
$67$	−1721.91	−0.0468623	−0.0234311	−	0.999725i	$-0.507459\pi$
$67$	−1721.91	−0.0468623	−0.0234311	+	0.999725i	$0.507459\pi$
$68$	−26010.2	−0.682136
$69$	−4195.81	−0.106095
$70$	−22119.9	−0.539557
$71$	−39497.2	−0.929867	−0.464934	−	0.885346i	$-0.653922\pi$
$71$	−39497.2	−0.929867	−0.464934	+	0.885346i	$0.653922\pi$
$72$	−11525.7	−0.262022
$73$	−51003.5	−1.12019	−0.560096	−	0.828428i	$-0.689236\pi$
$73$	−51003.5	−1.12019	−0.560096	+	0.828428i	$0.689236\pi$
$74$	54165.7	1.14986
$75$	4957.25	0.101762
$76$	26513.0	0.526531
$77$	112436.	2.16111
$78$	4696.06	0.0873971
$79$	60468.8	1.09009	0.545046	−	0.838406i	$-0.316512\pi$
$79$	60468.8	1.09009	0.545046	+	0.838406i	$0.316512\pi$
$80$	−6400.00	−0.111803
$81$	17145.1	0.290355
$82$	−30009.2	−0.492856
$83$	42121.0	0.671126	0.335563	−	0.942018i	$-0.391073\pi$
$83$	42121.0	0.671126	0.335563	+	0.942018i	$0.391073\pi$
$84$	28071.4	0.434075
$85$	40640.9	0.610121
$86$	32816.7	0.478463
$87$	55186.1	0.781685
$88$	32531.3	0.447811
$89$	−86689.6	−1.16009	−0.580045	−	0.814584i	$-0.696965\pi$
$89$	−86689.6	−1.16009	−0.580045	+	0.814584i	$0.696965\pi$
$90$	18009.0	0.234360
$91$	32741.3	0.414469
$92$	−8464.00	−0.104257
$93$	−6453.84	−0.0773768
$94$	17393.3	0.203031
$95$	−41426.5	−0.470944
$96$	8121.95	0.0899461
$97$	19131.8	0.206456	0.103228	−	0.994658i	$-0.467083\pi$
$97$	19131.8	0.206456	0.103228	+	0.994658i	$0.467083\pi$
$98$	128488.	1.35144
$99$	−91540.0	−0.938691
$100$	10000.0	0.100000
$101$	67653.5	0.659914	0.329957	−	0.943996i	$-0.392966\pi$
$101$	67653.5	0.659914	0.329957	+	0.943996i	$0.392966\pi$
$102$	−51575.6	−0.490844
$103$	−84138.0	−0.781446	−0.390723	−	0.920508i	$-0.627775\pi$
$103$	−84138.0	−0.781446	−0.390723	+	0.920508i	$0.627775\pi$
$104$	9473.12	0.0858835
$105$	−43861.5	−0.388249
$106$	151109.	1.30625
$107$	1260.97	0.0106475	0.00532373	−	0.999986i	$-0.498305\pi$
$107$	1260.97	0.0106475	0.00532373	+	0.999986i	$0.498305\pi$
$108$	−53692.4	−0.442949
$109$	114660.	0.924368	0.462184	−	0.886784i	$-0.347066\pi$
$109$	114660.	0.924368	0.462184	+	0.886784i	$0.347066\pi$
$110$	−50830.2	−0.400535
$111$	107405.	0.827403
$112$	56626.9	0.426558
$113$	−177717.	−1.30928	−0.654640	−	0.755941i	$-0.727180\pi$
$113$	−177717.	−1.30928	−0.654640	+	0.755941i	$0.727180\pi$
$114$	52572.5	0.378876
$115$	13225.0	0.0932505
$116$	111324.	0.768147
$117$	−26656.4	−0.180027
$118$	−134144.	−0.886884
$119$	−359589.	−2.32776
$120$	−12690.6	−0.0804503
$121$	97319.8	0.604279
$122$	−106633.	−0.648623
$123$	−59505.2	−0.354644
$124$	−13019.0	−0.0760367
$125$	−15625.0	−0.0894427
$126$	−159343.	−0.894140
$127$	−195975.	−1.07818	−0.539089	−	0.842249i	$-0.681231\pi$
$127$	−195975.	−1.07818	−0.539089	+	0.842249i	$0.681231\pi$
$128$	16384.0	0.0883883
$129$	65072.1	0.344287
$130$	−14801.7	−0.0768165
$131$	−327925.	−1.66954	−0.834769	−	0.550600i	$-0.814399\pi$
$131$	−327925.	−1.66954	−0.834769	+	0.550600i	$0.814399\pi$
$132$	64506.3	0.322231
$133$	366540.	1.79677
$134$	−6887.64	−0.0331366
$135$	83894.4	0.396186
$136$	−104041.	−0.482343
$137$	−97406.4	−0.443390	−0.221695	−	0.975116i	$-0.571159\pi$
$137$	−97406.4	−0.443390	−0.221695	+	0.975116i	$0.571159\pi$
$138$	−16783.3	−0.0750203
$139$	420401.	1.84555	0.922776	−	0.385337i	$-0.125915\pi$
$139$	420401.	1.84555	0.922776	+	0.385337i	$0.125915\pi$
$140$	−88479.5	−0.381525
$141$	34489.1	0.146095
$142$	−157989.	−0.657515
$143$	75237.6	0.307677
$144$	−46103.0	−0.185278
$145$	−173944.	−0.687052
$146$	−204014.	−0.792095
$147$	254778.	0.972453
$148$	216663.	0.813074
$149$	−474317.	−1.75026	−0.875131	−	0.483887i	$-0.839225\pi$
$149$	−474317.	−1.75026	−0.875131	+	0.483887i	$0.839225\pi$
$150$	19829.0	0.0719569
$151$	−198394.	−0.708085	−0.354043	−	0.935229i	$-0.615193\pi$
$151$	−198394.	−0.708085	−0.354043	+	0.935229i	$0.615193\pi$
$152$	106052.	0.372314
$153$	292760.	1.01108
$154$	449743.	1.52814
$155$	20342.2	0.0680093
$156$	18784.2	0.0617991
$157$	35380.4	0.114555	0.0572774	−	0.998358i	$-0.481758\pi$
$157$	35380.4	0.114555	0.0572774	+	0.998358i	$0.481758\pi$
$158$	241875.	0.770812
$159$	299634.	0.939936
$160$	−25600.0	−0.0790569
$161$	−117014.	−0.355774
$162$	68580.6	0.205312
$163$	109172.	0.321840	0.160920	−	0.986967i	$-0.448554\pi$
$163$	109172.	0.321840	0.160920	+	0.986967i	$0.448554\pi$
$164$	−120037.	−0.348502
$165$	−100791.	−0.288212
$166$	168484.	0.474558
$167$	−348050.	−0.965717	−0.482859	−	0.875698i	$-0.660402\pi$
$167$	−348050.	−0.965717	−0.482859	+	0.875698i	$0.660402\pi$
$168$	112285.	0.306938
$169$	−349384.	−0.940992
$170$	162564.	0.431421
$171$	−298420.	−0.780436
$172$	131267.	0.338324
$173$	−312017.	−0.792617	−0.396308	−	0.918117i	$-0.629709\pi$
$173$	−312017.	−0.792617	−0.396308	+	0.918117i	$0.629709\pi$
$174$	220745.	0.552735
$175$	138249.	0.341246
$176$	130125.	0.316650
$177$	−265994.	−0.638174
$178$	−346758.	−0.820308
$179$	379189.	0.884552	0.442276	−	0.896879i	$-0.354171\pi$
$179$	379189.	0.884552	0.442276	+	0.896879i	$0.354171\pi$
$180$	72035.9	0.165717
$181$	−372084.	−0.844199	−0.422099	−	0.906550i	$-0.638707\pi$
$181$	−372084.	−0.844199	−0.422099	+	0.906550i	$0.638707\pi$
$182$	130965.	0.293074
$183$	−211443.	−0.466729
$184$	−33856.0	−0.0737210
$185$	−338535.	−0.727235
$186$	−25815.4	−0.0547137
$187$	−826314.	−1.72799
$188$	69573.1	0.143564
$189$	−742294.	−1.51155
$190$	−165706.	−0.333008
$191$	−260701.	−0.517082	−0.258541	−	0.966000i	$-0.583242\pi$
$191$	−260701.	−0.517082	−0.258541	+	0.966000i	$0.583242\pi$
$192$	32487.8	0.0636015
$193$	−681976.	−1.31788	−0.658940	−	0.752195i	$-0.728995\pi$
$193$	−681976.	−1.31788	−0.658940	+	0.752195i	$0.728995\pi$
$194$	76527.4	0.145986
$195$	−29350.4	−0.0552748
$196$	513951.	0.955611
$197$	20599.0	0.0378165	0.0189083	−	0.999821i	$-0.493981\pi$
$197$	20599.0	0.0378165	0.0189083	+	0.999821i	$0.493981\pi$
$198$	−366160.	−0.663755
$199$	−592413.	−1.06045	−0.530227	−	0.847856i	$-0.677893\pi$
$199$	−592413.	−1.06045	−0.530227	+	0.847856i	$0.677893\pi$
$200$	40000.0	0.0707107
$201$	−13657.5	−0.0238441
$202$	270614.	0.466629
$203$	1.53905e6	2.62127
$204$	−206302.	−0.347079
$205$	187558.	0.311709
$206$	−336552.	−0.552566
$207$	95267.5	0.154532
$208$	37892.5	0.0607288
$209$	842287.	1.33381
$210$	−175446.	−0.274533
$211$	−1.16771e6	−1.80563	−0.902817	−	0.430025i	$-0.858505\pi$
$211$	−1.16771e6	−1.80563	−0.902817	+	0.430025i	$0.858505\pi$
$212$	604436.	0.923657
$213$	−313276.	−0.473128
$214$	5043.89	0.00752889
$215$	−205104.	−0.302606
$216$	−214770.	−0.313212
$217$	−179987.	−0.259472
$218$	458639.	0.653627
$219$	−404539.	−0.569967
$220$	−203321.	−0.283221
$221$	−240623.	−0.331402
$222$	429620.	0.585063
$223$	222295.	0.299342	0.149671	−	0.988736i	$-0.452179\pi$
$223$	222295.	0.299342	0.149671	+	0.988736i	$0.452179\pi$
$224$	226508.	0.301622
$225$	−112556.	−0.148222
$226$	−710868.	−0.925801
$227$	−1.06302e6	−1.36923	−0.684613	−	0.728907i	$-0.740029\pi$
$227$	−1.06302e6	−1.36923	−0.684613	+	0.728907i	$0.740029\pi$
$228$	210290.	0.267906
$229$	1.29480e6	1.63160	0.815799	−	0.578336i	$-0.196298\pi$
$229$	1.29480e6	1.63160	0.815799	+	0.578336i	$0.196298\pi$
$230$	52900.0	0.0659380
$231$	891795.	1.09960
$232$	445297.	0.543162
$233$	−48701.5	−0.0587696	−0.0293848	−	0.999568i	$-0.509355\pi$
$233$	−48701.5	−0.0587696	−0.0293848	+	0.999568i	$0.509355\pi$
$234$	−106626.	−0.127298
$235$	−108708.	−0.128408
$236$	−536577.	−0.627122
$237$	479614.	0.554652
$238$	−1.43835e6	−1.64598
$239$	1.20769e6	1.36761	0.683803	−	0.729667i	$-0.260325\pi$
$239$	1.20769e6	1.36761	0.683803	+	0.729667i	$0.260325\pi$
$240$	−50762.2	−0.0568869
$241$	−709454.	−0.786831	−0.393415	−	0.919361i	$-0.628707\pi$
$241$	−709454.	−0.786831	−0.393415	+	0.919361i	$0.628707\pi$
$242$	389279.	0.427290
$243$	951442.	1.03363
$244$	−426532.	−0.458646
$245$	−803048.	−0.854725
$246$	−238021.	−0.250771
$247$	245274.	0.255805
$248$	−52076.0	−0.0537661
$249$	334087.	0.341477
$250$	−62500.0	−0.0632456
$251$	−772995.	−0.774448	−0.387224	−	0.921986i	$-0.626566\pi$
$251$	−772995.	−0.774448	−0.387224	+	0.921986i	$0.626566\pi$
$252$	−637370.	−0.632252
$253$	−268892.	−0.264105
$254$	−783898.	−0.762387
$255$	322347.	0.310437
$256$	65536.0	0.0625000
$257$	−35070.4	−0.0331213	−0.0165607	−	0.999863i	$-0.505272\pi$
$257$	−35070.4	−0.0331213	−0.0165607	+	0.999863i	$0.505272\pi$
$258$	260288.	0.243448
$259$	2.99535e6	2.77458
$260$	−59207.0	−0.0543175
$261$	−1.25302e6	−1.13856
$262$	−1.31170e6	−1.18054
$263$	318972.	0.284357	0.142178	−	0.989841i	$-0.454589\pi$
$263$	318972.	0.284357	0.142178	+	0.989841i	$0.454589\pi$
$264$	258025.	0.227852
$265$	−944432.	−0.826144
$266$	1.46616e6	1.27051
$267$	−687587.	−0.590268
$268$	−27550.6	−0.0234311
$269$	2.19198e6	1.84695	0.923477	−	0.383654i	$-0.125334\pi$
$269$	2.19198e6	1.84695	0.923477	+	0.383654i	$0.125334\pi$
$270$	335578.	0.280146
$271$	−94605.3	−0.0782515	−0.0391257	−	0.999234i	$-0.512457\pi$
$271$	−94605.3	−0.0782515	−0.0391257	+	0.999234i	$0.512457\pi$
$272$	−416163.	−0.341068
$273$	259691.	0.210887
$274$	−389626.	−0.313524
$275$	317689.	0.253320
$276$	−67133.0	−0.0530473
$277$	2.20154e6	1.72396	0.861980	−	0.506942i	$-0.169224\pi$
$277$	2.20154e6	1.72396	0.861980	+	0.506942i	$0.169224\pi$
$278$	1.68160e6	1.30500
$279$	146537.	0.112703
$280$	−353918.	−0.269779
$281$	2.32212e6	1.75436	0.877178	−	0.480165i	$-0.159423\pi$
$281$	2.32212e6	1.75436	0.877178	+	0.480165i	$0.159423\pi$
$282$	137956.	0.103304
$283$	−1.14696e6	−0.851297	−0.425649	−	0.904889i	$-0.639954\pi$
$283$	−1.14696e6	−0.851297	−0.425649	+	0.904889i	$0.639954\pi$
$284$	−631956.	−0.464934
$285$	−328578.	−0.239622
$286$	300950.	0.217560
$287$	−1.65950e6	−1.18925
$288$	−184412.	−0.131011
$289$	1.22284e6	0.861238
$290$	−695776.	−0.485819
$291$	151746.	0.105047
$292$	−816055.	−0.560096
$293$	869490.	0.591692	0.295846	−	0.955236i	$-0.404398\pi$
$293$	869490.	0.591692	0.295846	+	0.955236i	$0.404398\pi$
$294$	1.01911e6	0.687628
$295$	838401.	0.560915
$296$	866651.	0.574930
$297$	−1.70575e6	−1.12208
$298$	−1.89727e6	−1.23762
$299$	−78301.2	−0.0506513
$300$	79316.0	0.0508812
$301$	1.81475e6	1.15452
$302$	−793575.	−0.500692
$303$	536601.	0.335772
$304$	424207.	0.263266
$305$	666457.	0.410225
$306$	1.17104e6	0.714939
$307$	−1.63549e6	−0.990380	−0.495190	−	0.868785i	$-0.664902\pi$
$307$	−1.63549e6	−0.990380	−0.495190	+	0.868785i	$0.664902\pi$
$308$	1.79897e6	1.08056
$309$	−667349.	−0.397609
$310$	81368.7	0.0480898
$311$	677317.	0.397092	0.198546	−	0.980092i	$-0.436378\pi$
$311$	677317.	0.397092	0.198546	+	0.980092i	$0.436378\pi$
$312$	75137.0	0.0436986
$313$	1.76300e6	1.01716	0.508581	−	0.861014i	$-0.330170\pi$
$313$	1.76300e6	1.01716	0.508581	+	0.861014i	$0.330170\pi$
$314$	141521.	0.0810025
$315$	995891.	0.565504
$316$	967500.	0.545046
$317$	−2.33802e6	−1.30678	−0.653388	−	0.757023i	$-0.726653\pi$
$317$	−2.33802e6	−1.30678	−0.653388	+	0.757023i	$0.726653\pi$
$318$	1.19854e6	0.664635
$319$	3.53664e6	1.94587
$320$	−102400.	−0.0559017
$321$	10001.5	0.00541756
$322$	−468057.	−0.251570
$323$	−2.69378e6	−1.43666
$324$	274322.	0.145177
$325$	92510.9	0.0485830
$326$	436686.	0.227576
$327$	909436.	0.470330
$328$	−480147.	−0.246428
$329$	961843.	0.489908
$330$	−403165.	−0.203797
$331$	1.71176e6	0.858763	0.429382	−	0.903123i	$-0.358732\pi$
$331$	1.71176e6	0.858763	0.429382	+	0.903123i	$0.358732\pi$
$332$	673937.	0.335563
$333$	−2.43867e6	−1.20515
$334$	−1.39220e6	−0.682865
$335$	43047.8	0.0209575
$336$	449142.	0.217038
$337$	−1.87213e6	−0.897969	−0.448984	−	0.893540i	$-0.648214\pi$
$337$	−1.87213e6	−0.897969	−0.448984	+	0.893540i	$0.648214\pi$
$338$	−1.39754e6	−0.665382
$339$	−1.40958e6	−0.666178
$340$	650254.	0.305061
$341$	−413599.	−0.192616
$342$	−1.19368e6	−0.551851
$343$	3.38764e6	1.55476
$344$	525066.	0.239231
$345$	104895.	0.0474470
$346$	−1.24807e6	−0.560465
$347$	−720471.	−0.321213	−0.160606	−	0.987019i	$-0.551345\pi$
$347$	−720471.	−0.321213	−0.160606	+	0.987019i	$0.551345\pi$
$348$	882978.	0.390843
$349$	2.15337e6	0.946356	0.473178	−	0.880967i	$-0.343107\pi$
$349$	2.15337e6	0.946356	0.473178	+	0.880967i	$0.343107\pi$
$350$	552997.	0.241297
$351$	−496714.	−0.215198
$352$	520501.	0.223906
$353$	4.07233e6	1.73943	0.869713	−	0.493559i	$-0.164304\pi$
$353$	4.07233e6	1.73943	0.869713	+	0.493559i	$0.164304\pi$
$354$	−1.06398e6	−0.451257
$355$	987431.	0.415849
$356$	−1.38703e6	−0.580045
$357$	−2.85211e6	−1.18439
$358$	1.51676e6	0.625473
$359$	1.89142e6	0.774555	0.387277	−	0.921963i	$-0.373416\pi$
$359$	1.89142e6	0.774555	0.387277	+	0.921963i	$0.373416\pi$
$360$	288144.	0.117180
$361$	269749.	0.108941
$362$	−1.48834e6	−0.596939
$363$	771901.	0.307465
$364$	523861.	0.207235
$365$	1.27509e6	0.500965
$366$	−845771.	−0.330027
$367$	660710.	0.256062	0.128031	−	0.991770i	$-0.459134\pi$
$367$	660710.	0.256062	0.128031	+	0.991770i	$0.459134\pi$
$368$	−135424.	−0.0521286
$369$	1.35109e6	0.516556
$370$	−1.35414e6	−0.514233
$371$	8.35629e6	3.15194
$372$	−103261.	−0.0386884
$373$	228810.	0.0851535	0.0425767	−	0.999093i	$-0.486443\pi$
$373$	228810.	0.0851535	0.0425767	+	0.999093i	$0.486443\pi$
$374$	−3.30526e6	−1.22187
$375$	−123931.	−0.0455095
$376$	278292.	0.101515
$377$	1.02987e6	0.373189
$378$	−2.96918e6	−1.06882
$379$	−879111.	−0.314373	−0.157187	−	0.987569i	$-0.550242\pi$
$379$	−879111.	−0.314373	−0.157187	+	0.987569i	$0.550242\pi$
$380$	−662824.	−0.235472
$381$	−1.55439e6	−0.548590
$382$	−1.04280e6	−0.365632
$383$	1.46746e6	0.511175	0.255588	−	0.966786i	$-0.417731\pi$
$383$	1.46746e6	0.511175	0.255588	+	0.966786i	$0.417731\pi$
$384$	129951.	0.0449731
$385$	−2.81089e6	−0.966480
$386$	−2.72791e6	−0.931882
$387$	−1.47749e6	−0.501471
$388$	306110.	0.103228
$389$	−4.15891e6	−1.39349	−0.696747	−	0.717317i	$-0.745370\pi$
$389$	−4.15891e6	−1.39349	−0.696747	+	0.717317i	$0.745370\pi$
$390$	−117401.	−0.0390852
$391$	859961.	0.284470
$392$	2.05580e6	0.675719
$393$	−2.60097e6	−0.849481
$394$	82396.2	0.0267403
$395$	−1.51172e6	−0.487504
$396$	−1.46464e6	−0.469346
$397$	382116.	0.121680	0.0608399	−	0.998148i	$-0.480622\pi$
$397$	382116.	0.121680	0.0608399	+	0.998148i	$0.480622\pi$
$398$	−2.36965e6	−0.749854
$399$	2.90725e6	0.914217
$400$	160000.	0.0500000
$401$	−1.62999e6	−0.506202	−0.253101	−	0.967440i	$-0.581451\pi$
$401$	−1.62999e6	−0.506202	−0.253101	+	0.967440i	$0.581451\pi$
$402$	−54630.0	−0.0168603
$403$	−120440.	−0.0369409
$404$	1.08246e6	0.329957
$405$	−428629.	−0.129851
$406$	6.15619e6	1.85352
$407$	6.88313e6	2.05968
$408$	−825209.	−0.245422
$409$	2.54388e6	0.751949	0.375975	−	0.926630i	$-0.377308\pi$
$409$	2.54388e6	0.751949	0.375975	+	0.926630i	$0.377308\pi$
$410$	750230.	0.220412
$411$	−772588.	−0.225602
$412$	−1.34621e6	−0.390723
$413$	−7.41813e6	−2.14003
$414$	381070.	0.109271
$415$	−1.05303e6	−0.300137
$416$	151570.	0.0429417
$417$	3.33445e6	0.939039
$418$	3.36915e6	0.943147
$419$	−5.19902e6	−1.44673	−0.723363	−	0.690468i	$-0.757405\pi$
$419$	−5.19902e6	−1.44673	−0.723363	+	0.690468i	$0.757405\pi$
$420$	−701784.	−0.194124
$421$	1.48220e6	0.407569	0.203784	−	0.979016i	$-0.434676\pi$
$421$	1.48220e6	0.407569	0.203784	+	0.979016i	$0.434676\pi$
$422$	−4.67085e6	−1.27678
$423$	−783087.	−0.212794
$424$	2.41775e6	0.653124
$425$	−1.01602e6	−0.272854
$426$	−1.25310e6	−0.334552
$427$	−5.89678e6	−1.56511
$428$	20175.6	0.00532373
$429$	596754.	0.156550
$430$	−820416.	−0.213975
$431$	−1.88531e6	−0.488865	−0.244432	−	0.969666i	$-0.578602\pi$
$431$	−1.88531e6	−0.488865	−0.244432	+	0.969666i	$0.578602\pi$
$432$	−859079.	−0.221475
$433$	−4.29778e6	−1.10160	−0.550800	−	0.834637i	$-0.685677\pi$
$433$	−4.29778e6	−1.10160	−0.550800	+	0.834637i	$0.685677\pi$
$434$	−719947.	−0.183475
$435$	−1.37965e6	−0.349580
$436$	1.83456e6	0.462184
$437$	−876585.	−0.219579
$438$	−1.61816e6	−0.403028
$439$	−3.24244e6	−0.802991	−0.401496	−	0.915861i	$-0.631510\pi$
$439$	−3.24244e6	−0.802991	−0.401496	+	0.915861i	$0.631510\pi$
$440$	−813283.	−0.200267
$441$	−5.78483e6	−1.41643
$442$	−962490.	−0.234337
$443$	5.95768e6	1.44234	0.721170	−	0.692758i	$-0.243605\pi$
$443$	5.95768e6	1.44234	0.721170	+	0.692758i	$0.243605\pi$
$444$	1.71848e6	0.413702
$445$	2.16724e6	0.518808
$446$	889180.	0.211667
$447$	−3.76209e6	−0.890554
$448$	906030.	0.213279
$449$	−5.17530e6	−1.21149	−0.605745	−	0.795659i	$-0.707125\pi$
$449$	−5.17530e6	−1.21149	−0.605745	+	0.795659i	$0.707125\pi$
$450$	−450224.	−0.104809
$451$	−3.81343e6	−0.882826
$452$	−2.84347e6	−0.654640
$453$	−1.57358e6	−0.360282
$454$	−4.25206e6	−0.968189
$455$	−818532.	−0.185356
$456$	841160.	0.189438
$457$	−207637.	−0.0465067	−0.0232533	−	0.999730i	$-0.507402\pi$
$457$	−207637.	−0.0465067	−0.0232533	+	0.999730i	$0.507402\pi$
$458$	5.17919e6	1.15371
$459$	5.45527e6	1.20861
$460$	211600.	0.0466252
$461$	5.35749e6	1.17411	0.587056	−	0.809547i	$-0.300287\pi$
$461$	5.35749e6	1.17411	0.587056	+	0.809547i	$0.300287\pi$
$462$	3.56718e6	0.777535
$463$	497554.	0.107867	0.0539334	−	0.998545i	$-0.482824\pi$
$463$	497554.	0.107867	0.0539334	+	0.998545i	$0.482824\pi$
$464$	1.78119e6	0.384074
$465$	161346.	0.0346040
$466$	−194806.	−0.0415564
$467$	−2.16518e6	−0.459412	−0.229706	−	0.973260i	$-0.573776\pi$
$467$	−2.16518e6	−0.459412	−0.229706	+	0.973260i	$0.573776\pi$
$468$	−426503.	−0.0900135
$469$	−380885.	−0.0799579
$470$	−434832.	−0.0907981
$471$	280623.	0.0582869
$472$	−2.14631e6	−0.443442
$473$	4.17019e6	0.857044
$474$	1.91846e6	0.392198
$475$	1.03566e6	0.210613
$476$	−5.75342e6	−1.16388
$477$	−6.80330e6	−1.36906
$478$	4.83076e6	0.967043
$479$	−2.48498e6	−0.494863	−0.247431	−	0.968905i	$-0.579586\pi$
$479$	−2.48498e6	−0.494863	−0.247431	+	0.968905i	$0.579586\pi$
$480$	−203049.	−0.0402251
$481$	2.00437e6	0.395016
$482$	−2.83781e6	−0.556374
$483$	−928109.	−0.181022
$484$	1.55712e6	0.302140
$485$	−478296.	−0.0923299
$486$	3.80577e6	0.730890
$487$	3.99273e6	0.762864	0.381432	−	0.924397i	$-0.375431\pi$
$487$	3.99273e6	0.762864	0.381432	+	0.924397i	$0.375431\pi$
$488$	−1.70613e6	−0.324312
$489$	865905.	0.163756
$490$	−3.21219e6	−0.604382
$491$	−3.97858e6	−0.744773	−0.372387	−	0.928078i	$-0.621460\pi$
$491$	−3.97858e6	−0.744773	−0.372387	+	0.928078i	$0.621460\pi$
$492$	−952084.	−0.177322
$493$	−1.13108e7	−2.09592
$494$	981096.	0.180881
$495$	2.28850e6	0.419795
$496$	−208304.	−0.0380184
$497$	−8.73674e6	−1.58657
$498$	1.33635e6	0.241461
$499$	7.33673e6	1.31902	0.659509	−	0.751696i	$-0.270764\pi$
$499$	7.33673e6	1.31902	0.659509	+	0.751696i	$0.270764\pi$
$500$	−250000.	−0.0447214
$501$	−2.76059e6	−0.491369
$502$	−3.09198e6	−0.547617
$503$	1.23312e6	0.217312	0.108656	−	0.994079i	$-0.465345\pi$
$503$	1.23312e6	0.217312	0.108656	+	0.994079i	$0.465345\pi$
$504$	−2.54948e6	−0.447070
$505$	−1.69134e6	−0.295122
$506$	−1.07557e6	−0.186750
$507$	−2.77117e6	−0.478788
$508$	−3.13559e6	−0.539089
$509$	−3.69590e6	−0.632304	−0.316152	−	0.948709i	$-0.602391\pi$
$509$	−3.69590e6	−0.632304	−0.316152	+	0.948709i	$0.602391\pi$
$510$	1.28939e6	0.219512
$511$	−1.12819e7	−1.91131
$512$	262144.	0.0441942
$513$	−5.56072e6	−0.932906
$514$	−140282.	−0.0234203
$515$	2.10345e6	0.349473
$516$	1.04115e6	0.172144
$517$	2.21026e6	0.363678
$518$	1.19814e7	1.96193
$519$	−2.47479e6	−0.403293
$520$	−236828.	−0.0384083
$521$	−1.80338e6	−0.291067	−0.145533	−	0.989353i	$-0.546490\pi$
$521$	−1.80338e6	−0.291067	−0.145533	+	0.989353i	$0.546490\pi$
$522$	−5.01209e6	−0.805086
$523$	455239.	0.0727755	0.0363878	−	0.999338i	$-0.488415\pi$
$523$	455239.	0.0727755	0.0363878	+	0.999338i	$0.488415\pi$
$524$	−5.24680e6	−0.834769
$525$	1.09654e6	0.173630
$526$	1.27589e6	0.201070
$527$	1.32276e6	0.207470
$528$	1.03210e6	0.161116
$529$	279841.	0.0434783
$530$	−3.77773e6	−0.584172
$531$	6.03950e6	0.929532
$532$	5.86464e6	0.898384
$533$	−1.11047e6	−0.169313
$534$	−2.75035e6	−0.417383
$535$	−31524.3	−0.00476169
$536$	−110202.	−0.0165683
$537$	3.00758e6	0.450071
$538$	8.76792e6	1.30599
$539$	1.63276e7	2.42076
$540$	1.34231e6	0.198093
$541$	1.32966e7	1.95321	0.976603	−	0.215048i	$-0.0689909\pi$
$541$	1.32966e7	1.95321	0.976603	+	0.215048i	$0.0689909\pi$
$542$	−378421.	−0.0553321
$543$	−2.95122e6	−0.429539
$544$	−1.66465e6	−0.241172
$545$	−2.86650e6	−0.413390
$546$	1.03876e6	0.149120
$547$	4.51679e6	0.645449	0.322724	−	0.946493i	$-0.395401\pi$
$547$	4.51679e6	0.645449	0.322724	+	0.946493i	$0.395401\pi$
$548$	−1.55850e6	−0.221695
$549$	4.80088e6	0.679814
$550$	1.27075e6	0.179124
$551$	1.15294e7	1.61781
$552$	−268532.	−0.0375101
$553$	1.33756e7	1.85995
$554$	8.80616e6	1.21902
$555$	−2.68513e6	−0.370026
$556$	6.72641e6	0.922776
$557$	−1.43003e7	−1.95303	−0.976514	−	0.215456i	$-0.930876\pi$
$557$	−1.43003e7	−1.95303	−0.976514	+	0.215456i	$0.930876\pi$
$558$	586147.	0.0796932
$559$	1.21436e6	0.164368
$560$	−1.41567e6	−0.190762
$561$	−6.55399e6	−0.879222
$562$	9.28846e6	1.24052
$563$	−362252.	−0.0481659	−0.0240830	−	0.999710i	$-0.507667\pi$
$563$	−362252.	−0.0481659	−0.0240830	+	0.999710i	$0.507667\pi$
$564$	551826.	0.0730473
$565$	4.44292e6	0.585528
$566$	−4.58783e6	−0.601958
$567$	3.79249e6	0.495412
$568$	−2.52782e6	−0.328758
$569$	1.12892e7	1.46178	0.730889	−	0.682497i	$-0.239106\pi$
$569$	1.12892e7	1.46178	0.730889	+	0.682497i	$0.239106\pi$
$570$	−1.31431e6	−0.169438
$571$	−7.90939e6	−1.01520	−0.507602	−	0.861592i	$-0.669468\pi$
$571$	−7.90939e6	−1.01520	−0.507602	+	0.861592i	$0.669468\pi$
$572$	1.20380e6	0.153838
$573$	−2.06778e6	−0.263098
$574$	−6.63800e6	−0.840926
$575$	−330625.	−0.0417029
$576$	−737648.	−0.0926388
$577$	−1.20265e7	−1.50384	−0.751919	−	0.659256i	$-0.770871\pi$
$577$	−1.20265e7	−1.50384	−0.751919	+	0.659256i	$0.770871\pi$
$578$	4.89134e6	0.608987
$579$	−5.40916e6	−0.670554
$580$	−2.78310e6	−0.343526
$581$	9.31713e6	1.14510
$582$	606984.	0.0742797
$583$	1.92023e7	2.33981
$584$	−3.26422e6	−0.396048
$585$	666411.	0.0805105
$586$	3.47796e6	0.418389
$587$	1.73287e6	0.207573	0.103787	−	0.994600i	$-0.466904\pi$
$587$	1.73287e6	0.207573	0.103787	+	0.994600i	$0.466904\pi$
$588$	4.07645e6	0.486227
$589$	−1.34833e6	−0.160143
$590$	3.35360e6	0.396627
$591$	163383.	0.0192415
$592$	3.46660e6	0.406537
$593$	−511607.	−0.0597448	−0.0298724	−	0.999554i	$-0.509510\pi$
$593$	−511607.	−0.0597448	−0.0298724	+	0.999554i	$0.509510\pi$
$594$	−6.82299e6	−0.793430
$595$	8.98972e6	1.04101
$596$	−7.58907e6	−0.875131
$597$	−4.69878e6	−0.539572
$598$	−313205.	−0.0358159
$599$	−1.10348e7	−1.25660	−0.628299	−	0.777972i	$-0.716249\pi$
$599$	−1.10348e7	−1.25660	−0.628299	+	0.777972i	$0.716249\pi$
$600$	317264.	0.0359785
$601$	5.47736e6	0.618564	0.309282	−	0.950970i	$-0.399911\pi$
$601$	5.47736e6	0.618564	0.309282	+	0.950970i	$0.399911\pi$
$602$	7.25901e6	0.816368
$603$	310098.	0.0347301
$604$	−3.17430e6	−0.354043
$605$	−2.43299e6	−0.270242
$606$	2.14640e6	0.237427
$607$	1.50745e7	1.66063	0.830313	−	0.557297i	$-0.188162\pi$
$607$	1.50745e7	1.66063	0.830313	+	0.557297i	$0.188162\pi$
$608$	1.69683e6	0.186157
$609$	1.22071e7	1.33374
$610$	2.66583e6	0.290073
$611$	643627.	0.0697479
$612$	4.68417e6	0.505538
$613$	−1.60324e7	−1.72325	−0.861624	−	0.507548i	$-0.830552\pi$
$613$	−1.60324e7	−1.72325	−0.861624	+	0.507548i	$0.830552\pi$
$614$	−6.54196e6	−0.700304
$615$	1.48763e6	0.158602
$616$	7.19589e6	0.764069
$617$	−1.81041e7	−1.91454	−0.957269	−	0.289199i	$-0.906611\pi$
$617$	−1.81041e7	−1.91454	−0.957269	+	0.289199i	$0.906611\pi$
$618$	−2.66939e6	−0.281152
$619$	1.07745e7	1.13024	0.565118	−	0.825010i	$-0.308831\pi$
$619$	1.07745e7	1.13024	0.565118	+	0.825010i	$0.308831\pi$
$620$	325475.	0.0340046
$621$	1.77521e6	0.184723
$622$	2.70927e6	0.280786
$623$	−1.91756e7	−1.97938
$624$	300548.	0.0308995
$625$	390625.	0.0400000
$626$	7.05198e6	0.719243
$627$	6.68068e6	0.678659
$628$	566086.	0.0572774
$629$	−2.20134e7	−2.21851
$630$	3.98356e6	0.399872
$631$	−4926.90	−0.000492607 0	−0.000246304	−	1.00000i	$-0.500078\pi$
$631$	−4926.90	−0.000492607 0	−0.000246304		1.00000i	$0.500078\pi$
$632$	3.87000e6	0.385406
$633$	−9.26182e6	−0.918728
$634$	−9.35210e6	−0.924030
$635$	4.89936e6	0.482176
$636$	4.79415e6	0.469968
$637$	4.75461e6	0.464265
$638$	1.41466e7	1.37594
$639$	7.11305e6	0.689134
$640$	−409600.	−0.0395285
$641$	1.18363e7	1.13781	0.568906	−	0.822403i	$-0.307367\pi$
$641$	1.18363e7	1.13781	0.568906	+	0.822403i	$0.307367\pi$
$642$	40006.1	0.00383079
$643$	1.28400e6	0.122472	0.0612362	−	0.998123i	$-0.480496\pi$
$643$	1.28400e6	0.122472	0.0612362	+	0.998123i	$0.480496\pi$
$644$	−1.87223e6	−0.177887
$645$	−1.62680e6	−0.153970
$646$	−1.07751e7	−1.01587
$647$	−9.98021e6	−0.937301	−0.468650	−	0.883384i	$-0.655260\pi$
$647$	−9.98021e6	−0.937301	−0.468650	+	0.883384i	$0.655260\pi$
$648$	1.09729e6	0.102656
$649$	−1.70464e7	−1.58863
$650$	370044.	0.0343534
$651$	−1.42758e6	−0.132023
$652$	1.74675e6	0.160920
$653$	1.09561e7	1.00548	0.502740	−	0.864438i	$-0.332325\pi$
$653$	1.09561e7	1.00548	0.502740	+	0.864438i	$0.332325\pi$
$654$	3.63774e6	0.332573
$655$	8.19813e6	0.746640
$656$	−1.92059e6	−0.174251
$657$	9.18520e6	0.830186
$658$	3.84737e6	0.346417
$659$	1.98759e7	1.78284	0.891420	−	0.453177i	$-0.149710\pi$
$659$	1.98759e7	1.78284	0.891420	+	0.453177i	$0.149710\pi$
$660$	−1.61266e6	−0.144106
$661$	−1.50432e7	−1.33917	−0.669585	−	0.742736i	$-0.733528\pi$
$661$	−1.50432e7	−1.33917	−0.669585	+	0.742736i	$0.733528\pi$
$662$	6.84705e6	0.607237
$663$	−1.90852e6	−0.168622
$664$	2.69575e6	0.237279
$665$	−9.16349e6	−0.803539
$666$	−9.75468e6	−0.852173
$667$	−3.68066e6	−0.320340
$668$	−5.56879e6	−0.482859
$669$	1.76315e6	0.152309
$670$	172191.	0.0148192
$671$	−1.35504e7	−1.16184
$672$	1.79657e6	0.153469
$673$	−1.72807e6	−0.147070	−0.0735351	−	0.997293i	$-0.523428\pi$
$673$	−1.72807e6	−0.147070	−0.0735351	+	0.997293i	$0.523428\pi$
$674$	−7.48852e6	−0.634960
$675$	−2.09736e6	−0.177180
$676$	−5.59014e6	−0.470496
$677$	−3.83766e6	−0.321807	−0.160903	−	0.986970i	$-0.551441\pi$
$677$	−3.83766e6	−0.321807	−0.160903	+	0.986970i	$0.551441\pi$
$678$	−5.63832e6	−0.471059
$679$	4.23194e6	0.352262
$680$	2.60102e6	0.215710
$681$	−8.43141e6	−0.696678
$682$	−1.65440e6	−0.136200
$683$	−1.96704e7	−1.61347	−0.806737	−	0.590910i	$-0.798769\pi$
$683$	−1.96704e7	−1.61347	−0.806737	+	0.590910i	$0.798769\pi$
$684$	−4.77471e6	−0.390218
$685$	2.43516e6	0.198290
$686$	1.35506e7	1.09938
$687$	1.02698e7	0.830177
$688$	2.10027e6	0.169162
$689$	5.59170e6	0.448741
$690$	419581.	0.0335501
$691$	−2.06835e6	−0.164790	−0.0823948	−	0.996600i	$-0.526257\pi$
$691$	−2.06835e6	−0.164790	−0.0823948	+	0.996600i	$0.526257\pi$
$692$	−4.99228e6	−0.396308
$693$	−2.02485e7	−1.60162
$694$	−2.88189e6	−0.227132
$695$	−1.05100e7	−0.825356
$696$	3.53191e6	0.276367
$697$	1.21960e7	0.950902
$698$	8.61347e6	0.669174
$699$	−386281.	−0.0299027
$700$	2.21199e6	0.170623
$701$	−2.26675e6	−0.174225	−0.0871123	−	0.996198i	$-0.527764\pi$
$701$	−2.26675e6	−0.174225	−0.0871123	+	0.996198i	$0.527764\pi$
$702$	−1.98685e6	−0.152168
$703$	2.24389e7	1.71244
$704$	2.08200e6	0.158325
$705$	−862227.	−0.0653355
$706$	1.62893e7	1.22996
$707$	1.49649e7	1.12596
$708$	−4.25591e6	−0.319087
$709$	4.88196e6	0.364736	0.182368	−	0.983230i	$-0.441624\pi$
$709$	4.88196e6	0.364736	0.182368	+	0.983230i	$0.441624\pi$
$710$	3.94972e6	0.294050
$711$	−1.08898e7	−0.807879
$712$	−5.54813e6	−0.410154
$713$	430441.	0.0317095
$714$	−1.14085e7	−0.837493
$715$	−1.88094e6	−0.137597
$716$	6.06703e6	0.442276
$717$	9.57891e6	0.695854
$718$	7.56568e6	0.547693
$719$	−9.79754e6	−0.706797	−0.353399	−	0.935473i	$-0.614974\pi$
$719$	−9.79754e6	−0.706797	−0.353399	+	0.935473i	$0.614974\pi$
$720$	1.15257e6	0.0828586
$721$	−1.86112e7	−1.33333
$722$	1.07900e6	0.0770331
$723$	−5.62710e6	−0.400349
$724$	−5.95335e6	−0.422099
$725$	4.34860e6	0.307259
$726$	3.08760e6	0.217410
$727$	−1.45372e6	−0.102010	−0.0510052	−	0.998698i	$-0.516243\pi$
$727$	−1.45372e6	−0.102010	−0.0510052	+	0.998698i	$0.516243\pi$
$728$	2.09544e6	0.146537
$729$	3.38019e6	0.235571
$730$	5.10035e6	0.354236
$731$	−1.33370e7	−0.923133
$732$	−3.38308e6	−0.233365
$733$	1.90099e7	1.30683	0.653417	−	0.756999i	$-0.273335\pi$
$733$	1.90099e7	1.30683	0.653417	+	0.756999i	$0.273335\pi$
$734$	2.64284e6	0.181063
$735$	−6.36945e6	−0.434894
$736$	−541696.	−0.0368605
$737$	−875250.	−0.0593558
$738$	5.40435e6	0.365261
$739$	1.67293e7	1.12685	0.563425	−	0.826167i	$-0.309483\pi$
$739$	1.67293e7	1.12685	0.563425	+	0.826167i	$0.309483\pi$
$740$	−5.41657e6	−0.363618
$741$	1.94541e6	0.130157
$742$	3.34252e7	2.22876
$743$	−3.02797e6	−0.201224	−0.100612	−	0.994926i	$-0.532080\pi$
$743$	−3.02797e6	−0.201224	−0.100612	+	0.994926i	$0.532080\pi$
$744$	−413046.	−0.0273568
$745$	1.18579e7	0.782741
$746$	915239.	0.0602126
$747$	−7.58557e6	−0.497378
$748$	−1.32210e7	−0.863994
$749$	278926.	0.0181670
$750$	−495725.	−0.0321801
$751$	−3.45504e6	−0.223539	−0.111770	−	0.993734i	$-0.535652\pi$
$751$	−3.45504e6	−0.223539	−0.111770	+	0.993734i	$0.535652\pi$
$752$	1.11317e6	0.0717822
$753$	−6.13108e6	−0.394049
$754$	4.11948e6	0.263885
$755$	4.95984e6	0.316665
$756$	−1.18767e7	−0.755773
$757$	1.99821e7	1.26736	0.633681	−	0.773594i	$-0.281543\pi$
$757$	1.99821e7	1.26736	0.633681	+	0.773594i	$0.281543\pi$
$758$	−3.51644e6	−0.222295
$759$	−2.13274e6	−0.134380
$760$	−2.65130e6	−0.166504
$761$	1.21696e7	0.761755	0.380877	−	0.924626i	$-0.375622\pi$
$761$	1.21696e7	0.761755	0.380877	+	0.924626i	$0.375622\pi$
$762$	−6.21756e6	−0.387912
$763$	2.53626e7	1.57719
$764$	−4.17122e6	−0.258541
$765$	−7.31901e6	−0.452167
$766$	5.86985e6	0.361455
$767$	−4.96392e6	−0.304675
$768$	519805.	0.0318008
$769$	3.01762e7	1.84013	0.920066	−	0.391764i	$-0.128135\pi$
$769$	3.01762e7	1.84013	0.920066	+	0.391764i	$0.128135\pi$
$770$	−1.12436e7	−0.683404
$771$	−278164.	−0.0168525
$772$	−1.09116e7	−0.658940
$773$	1.85630e6	0.111737	0.0558687	−	0.998438i	$-0.482207\pi$
$773$	1.85630e6	0.111737	0.0558687	+	0.998438i	$0.482207\pi$
$774$	−5.90994e6	−0.354594
$775$	−508555.	−0.0304147
$776$	1.22444e6	0.0729932
$777$	2.37579e7	1.41174
$778$	−1.66356e7	−0.985350
$779$	−1.24318e7	−0.733988
$780$	−469606.	−0.0276374
$781$	−2.00765e7	−1.17777
$782$	3.43985e6	0.201151
$783$	−2.33487e7	−1.36100
$784$	8.22321e6	0.477806
$785$	−884509.	−0.0512305
$786$	−1.04039e7	−0.600674
$787$	9.60964e6	0.553058	0.276529	−	0.961006i	$-0.410816\pi$
$787$	9.60964e6	0.553058	0.276529	+	0.961006i	$0.410816\pi$
$788$	329585.	0.0189083
$789$	2.52996e6	0.144684
$790$	−6.04688e6	−0.344718
$791$	−3.93108e7	−2.23393
$792$	−5.85856e6	−0.331877
$793$	−3.94589e6	−0.222824
$794$	1.52846e6	0.0860406
$795$	−7.49085e6	−0.420352
$796$	−9.47861e6	−0.530227
$797$	−2.16173e7	−1.20547	−0.602734	−	0.797942i	$-0.705922\pi$
$797$	−2.16173e7	−1.20547	−0.602734	+	0.797942i	$0.705922\pi$
$798$	1.16290e7	0.646449
$799$	−7.06878e6	−0.391722
$800$	640000.	0.0353553
$801$	1.56119e7	0.859755
$802$	−6.51996e6	−0.357939
$803$	−2.59252e7	−1.41884
$804$	−218520.	−0.0119221
$805$	2.92535e6	0.159107
$806$	−481760.	−0.0261212
$807$	1.73859e7	0.939753
$808$	4.32983e6	0.233315
$809$	7.92785e6	0.425877	0.212938	−	0.977066i	$-0.431697\pi$
$809$	7.92785e6	0.425877	0.212938	+	0.977066i	$0.431697\pi$
$810$	−1.71451e6	−0.0918182
$811$	3.63619e7	1.94131	0.970654	−	0.240480i	$-0.0773047\pi$
$811$	3.63619e7	1.94131	0.970654	+	0.240480i	$0.0773047\pi$
$812$	2.46248e7	1.31064
$813$	−750371.	−0.0398153
$814$	2.75325e7	1.45641
$815$	−2.72929e6	−0.143931
$816$	−3.30084e6	−0.173540
$817$	1.35948e7	0.712553
$818$	1.01755e7	0.531708
$819$	−5.89637e6	−0.307167
$820$	3.00092e6	0.155855
$821$	−3.29493e7	−1.70604	−0.853018	−	0.521881i	$-0.825230\pi$
$821$	−3.29493e7	−1.70604	−0.853018	+	0.521881i	$0.825230\pi$
$822$	−3.09035e6	−0.159525
$823$	−3.14482e7	−1.61844	−0.809218	−	0.587508i	$-0.800109\pi$
$823$	−3.14482e7	−1.61844	−0.809218	+	0.587508i	$0.800109\pi$
$824$	−5.38483e6	−0.276283
$825$	2.51978e6	0.128892
$826$	−2.96725e7	−1.51323
$827$	3.47551e7	1.76707	0.883537	−	0.468361i	$-0.155155\pi$
$827$	3.47551e7	1.76707	0.883537	+	0.468361i	$0.155155\pi$
$828$	1.52428e6	0.0772661
$829$	−2.66691e6	−0.134779	−0.0673895	−	0.997727i	$-0.521467\pi$
$829$	−2.66691e6	−0.134779	−0.0673895	+	0.997727i	$0.521467\pi$
$830$	−4.21210e6	−0.212229
$831$	1.74617e7	0.877172
$832$	606280.	0.0303644
$833$	−5.22185e7	−2.60743
$834$	1.33378e7	0.664001
$835$	8.70124e6	0.431882
$836$	1.34766e7	0.666905
$837$	2.73055e6	0.134722
$838$	−2.07961e7	−1.02299
$839$	−2.87306e7	−1.40909	−0.704546	−	0.709658i	$-0.748849\pi$
$839$	−2.87306e7	−1.40909	−0.704546	+	0.709658i	$0.748849\pi$
$840$	−2.80714e6	−0.137267
$841$	2.78993e7	1.36020
$842$	5.92879e6	0.288195
$843$	1.84181e7	0.892638
$844$	−1.86834e7	−0.902817
$845$	8.73460e6	0.420825
$846$	−3.13235e6	−0.150468
$847$	2.15270e7	1.03104
$848$	9.67098e6	0.461829
$849$	−9.09720e6	−0.433150
$850$	−4.06409e6	−0.192937
$851$	−7.16341e6	−0.339075
$852$	−5.01242e6	−0.236564
$853$	−1.79041e7	−0.842521	−0.421260	−	0.906940i	$-0.638412\pi$
$853$	−1.79041e7	−0.842521	−0.421260	+	0.906940i	$0.638412\pi$
$854$	−2.35871e7	−1.10670
$855$	7.46049e6	0.349021
$856$	80702.2	0.00376445
$857$	1.08685e7	0.505494	0.252747	−	0.967532i	$-0.418666\pi$
$857$	1.08685e7	0.505494	0.252747	+	0.967532i	$0.418666\pi$
$858$	2.38702e6	0.110697
$859$	2.96355e6	0.137034	0.0685172	−	0.997650i	$-0.478173\pi$
$859$	2.96355e6	0.137034	0.0685172	+	0.997650i	$0.478173\pi$
$860$	−3.28167e6	−0.151303
$861$	−1.31625e7	−0.605104
$862$	−7.54122e6	−0.345680
$863$	−3.16541e7	−1.44678	−0.723391	−	0.690438i	$-0.757418\pi$
$863$	−3.16541e7	−1.44678	−0.723391	+	0.690438i	$0.757418\pi$
$864$	−3.43632e6	−0.156606
$865$	7.80043e6	0.354469
$866$	−1.71911e7	−0.778949
$867$	9.69903e6	0.438208
$868$	−2.87979e6	−0.129736
$869$	3.07364e7	1.38071
$870$	−5.51861e6	−0.247191
$871$	−254873.	−0.0113836
$872$	7.33823e6	0.326814
$873$	−3.44545e6	−0.153007
$874$	−3.50634e6	−0.155266
$875$	−3.45623e6	−0.152610
$876$	−6.47262e6	−0.284984
$877$	3.92674e7	1.72398	0.861992	−	0.506921i	$-0.169217\pi$
$877$	3.92674e7	1.72398	0.861992	+	0.506921i	$0.169217\pi$
$878$	−1.29698e7	−0.567801
$879$	6.89645e6	0.301060
$880$	−3.25313e6	−0.141610
$881$	1.81431e6	0.0787540	0.0393770	−	0.999224i	$-0.487463\pi$
$881$	1.81431e6	0.0787540	0.0393770	+	0.999224i	$0.487463\pi$
$882$	−2.31393e7	−1.00156
$883$	1.44217e7	0.622465	0.311233	−	0.950334i	$-0.399258\pi$
$883$	1.44217e7	0.622465	0.311233	+	0.950334i	$0.399258\pi$
$884$	−3.84996e6	−0.165701
$885$	6.64986e6	0.285400
$886$	2.38307e7	1.01989
$887$	1.99385e7	0.850910	0.425455	−	0.904980i	$-0.360114\pi$
$887$	1.99385e7	0.850910	0.425455	+	0.904980i	$0.360114\pi$
$888$	6.87392e6	0.292531
$889$	−4.33493e7	−1.83962
$890$	8.66896e6	0.366853
$891$	8.71491e6	0.367763
$892$	3.55672e6	0.149671
$893$	7.20542e6	0.302365
$894$	−1.50484e7	−0.629717
$895$	−9.47973e6	−0.395584
$896$	3.62412e6	0.150811
$897$	−621054.	−0.0257720
$898$	−2.07012e7	−0.856652
$899$	−5.66144e6	−0.233630
$900$	−1.80090e6	−0.0741110
$901$	−6.14121e7	−2.52024
$902$	−1.52537e7	−0.624252
$903$	1.43939e7	0.587433
$904$	−1.13739e7	−0.462901
$905$	9.30210e6	0.377537
$906$	−6.29431e6	−0.254758
$907$	1.79214e7	0.723361	0.361680	−	0.932302i	$-0.382203\pi$
$907$	1.79214e7	0.723361	0.361680	+	0.932302i	$0.382203\pi$
$908$	−1.70082e7	−0.684613
$909$	−1.21837e7	−0.489069
$910$	−3.27413e6	−0.131067
$911$	3.84312e6	0.153422	0.0767111	−	0.997053i	$-0.475558\pi$
$911$	3.84312e6	0.153422	0.0767111	+	0.997053i	$0.475558\pi$
$912$	3.36464e6	0.133953
$913$	2.14102e7	0.850049
$914$	−830550.	−0.0328852
$915$	5.28607e6	0.208728
$916$	2.07168e7	0.815799
$917$	−7.25367e7	−2.84862
$918$	2.18211e7	0.854613
$919$	−1.88160e7	−0.734917	−0.367459	−	0.930040i	$-0.619772\pi$
$919$	−1.88160e7	−0.734917	−0.367459	+	0.930040i	$0.619772\pi$
$920$	846400.	0.0329690
$921$	−1.29720e7	−0.503917
$922$	2.14300e7	0.830222
$923$	−5.84628e6	−0.225879
$924$	1.42687e7	0.549801
$925$	8.46339e6	0.325229
$926$	1.99022e6	0.0762734
$927$	1.51524e7	0.579138
$928$	7.12475e6	0.271581
$929$	2.15714e7	0.820048	0.410024	−	0.912075i	$-0.365520\pi$
$929$	2.15714e7	0.820048	0.410024	+	0.912075i	$0.365520\pi$
$930$	645384.	0.0244687
$931$	5.32280e7	2.01264
$932$	−779224.	−0.0293848
$933$	5.37221e6	0.202045
$934$	−8.66073e6	−0.324853
$935$	2.06578e7	0.772780
$936$	−1.70601e6	−0.0636491
$937$	−5.12515e7	−1.90703	−0.953515	−	0.301344i	$-0.902565\pi$
$937$	−5.12515e7	−1.90703	−0.953515	+	0.301344i	$0.902565\pi$
$938$	−1.52354e6	−0.0565387
$939$	1.39834e7	0.517545
$940$	−1.73933e6	−0.0642039
$941$	3.93093e7	1.44717	0.723587	−	0.690233i	$-0.242492\pi$
$941$	3.93093e7	1.44717	0.723587	+	0.690233i	$0.242492\pi$
$942$	1.12249e6	0.0412150
$943$	3.96872e6	0.145335
$944$	−8.58523e6	−0.313561
$945$	1.85573e7	0.675984
$946$	1.66808e7	0.606022
$947$	−4.72717e7	−1.71288	−0.856439	−	0.516249i	$-0.827328\pi$
$947$	−4.72717e7	−1.71288	−0.856439	+	0.516249i	$0.827328\pi$
$948$	7.67382e6	0.277326
$949$	−7.54940e6	−0.272112
$950$	4.14265e6	0.148926
$951$	−1.85443e7	−0.664903
$952$	−2.30137e7	−0.822988
$953$	−5.19223e7	−1.85192	−0.925960	−	0.377623i	$-0.876742\pi$
$953$	−5.19223e7	−1.85192	−0.925960	+	0.377623i	$0.876742\pi$
$954$	−2.72132e7	−0.968074
$955$	6.51753e6	0.231246
$956$	1.93230e7	0.683803
$957$	2.80512e7	0.990084
$958$	−9.93993e6	−0.349921
$959$	−2.15462e7	−0.756526
$960$	−812195.	−0.0284435
$961$	−2.79671e7	−0.976874
$962$	8.01747e6	0.279318
$963$	−227088.	−0.00789094
$964$	−1.13513e7	−0.393415
$965$	1.70494e7	0.589374
$966$	−3.71244e6	−0.128002
$967$	−3.73911e7	−1.28588	−0.642942	−	0.765915i	$-0.722287\pi$
$967$	−3.73911e7	−1.28588	−0.642942	+	0.765915i	$0.722287\pi$
$968$	6.22847e6	0.213645
$969$	−2.13659e7	−0.730992
$970$	−1.91318e6	−0.0652871
$971$	−2.44905e7	−0.833585	−0.416793	−	0.909002i	$-0.636846\pi$
$971$	−2.44905e7	−0.833585	−0.416793	+	0.909002i	$0.636846\pi$
$972$	1.52231e7	0.516817
$973$	9.29921e7	3.14894
$974$	1.59709e7	0.539426
$975$	733759.	0.0247196
$976$	−6.82452e6	−0.229323
$977$	−6.82012e6	−0.228589	−0.114295	−	0.993447i	$-0.536461\pi$
$977$	−6.82012e6	−0.228589	−0.114295	+	0.993447i	$0.536461\pi$
$978$	3.46362e6	0.115793
$979$	−4.40645e7	−1.46937
$980$	−1.28488e7	−0.427362
$981$	−2.06491e7	−0.685059
$982$	−1.59143e7	−0.526634
$983$	1.76180e7	0.581532	0.290766	−	0.956794i	$-0.406090\pi$
$983$	1.76180e7	0.581532	0.290766	+	0.956794i	$0.406090\pi$
$984$	−3.80833e6	−0.125386
$985$	−514976.	−0.0169121
$986$	−4.52431e7	−1.48204
$987$	7.62895e6	0.249271
$988$	3.92438e6	0.127902
$989$	−4.34000e6	−0.141091
$990$	9.15400e6	0.296840
$991$	1.22781e7	0.397144	0.198572	−	0.980086i	$-0.436370\pi$
$991$	1.22781e7	0.397144	0.198572	+	0.980086i	$0.436370\pi$
$992$	−833216.	−0.0268830
$993$	1.35770e7	0.436949
$994$	−3.49470e7	−1.12187
$995$	1.48103e7	0.474249
$996$	5.34539e6	0.170738
$997$	−4.31192e7	−1.37383	−0.686914	−	0.726738i	$-0.741035\pi$
$997$	−4.31192e7	−1.37383	−0.686914	+	0.726738i	$0.741035\pi$
$998$	2.93469e7	0.932687
$999$	−4.54420e7	−1.44060

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	230.6.a.i.1.4	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.6.a.i.1.4	✓	6	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 230 = 2 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	230.a (trivial)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} +7.93160 q^{3} +16.0000 q^{4} -25.0000 q^{5} +31.7264 q^{6} +221.199 q^{7} +64.0000 q^{8} -180.090 q^{9} -100.000 q^{10} +508.302 q^{11} +126.906 q^{12} +148.017 q^{13} +884.795 q^{14} -198.290 q^{15} +256.000 q^{16} -1625.64 q^{17} -720.359 q^{18} +1657.06 q^{19} -400.000 q^{20} +1754.46 q^{21} +2033.21 q^{22} -529.000 q^{23} +507.622 q^{24} +625.000 q^{25} +592.070 q^{26} -3355.78 q^{27} +3539.18 q^{28} +6957.76 q^{29} -793.160 q^{30} -813.687 q^{31} +1024.00 q^{32} +4031.65 q^{33} -6502.54 q^{34} -5529.97 q^{35} -2881.44 q^{36} +13541.4 q^{37} +6628.24 q^{38} +1174.01 q^{39} -1600.00 q^{40} -7502.30 q^{41} +7017.84 q^{42} +8204.16 q^{43} +8132.83 q^{44} +4502.24 q^{45} -2116.00 q^{46} +4348.32 q^{47} +2030.49 q^{48} +32121.9 q^{49} +2500.00 q^{50} -12893.9 q^{51} +2368.28 q^{52} +37777.3 q^{53} -13423.1 q^{54} -12707.5 q^{55} +14156.7 q^{56} +13143.1 q^{57} +27831.0 q^{58} -33536.0 q^{59} -3172.64 q^{60} -26658.3 q^{61} -3254.75 q^{62} -39835.6 q^{63} +4096.00 q^{64} -3700.44 q^{65} +16126.6 q^{66} -1721.91 q^{67} -26010.2 q^{68} -4195.81 q^{69} -22119.9 q^{70} -39497.2 q^{71} -11525.7 q^{72} -51003.5 q^{73} +54165.7 q^{74} +4957.25 q^{75} +26513.0 q^{76} +112436. q^{77} +4696.06 q^{78} +60468.8 q^{79} -6400.00 q^{80} +17145.1 q^{81} -30009.2 q^{82} +42121.0 q^{83} +28071.4 q^{84} +40640.9 q^{85} +32816.7 q^{86} +55186.1 q^{87} +32531.3 q^{88} -86689.6 q^{89} +18009.0 q^{90} +32741.3 q^{91} -8464.00 q^{92} -6453.84 q^{93} +17393.3 q^{94} -41426.5 q^{95} +8121.95 q^{96} +19131.8 q^{97} +128488. q^{98} -91540.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 24 q^{2} + 15 q^{3} + 96 q^{4} - 150 q^{5} + 60 q^{6} + 106 q^{7} + 384 q^{8} + 899 q^{9} - 600 q^{10} + 321 q^{11} + 240 q^{12} + 527 q^{13} + 424 q^{14} - 375 q^{15} + 1536 q^{16} - 660 q^{17} + 3596 q^{18}+ \cdots - 2973 q^{99}+O(q^{100}) \) 6 * q + 24 * q^2 + 15 * q^3 + 96 * q^4 - 150 * q^5 + 60 * q^6 + 106 * q^7 + 384 * q^8 + 899 * q^9 - 600 * q^10 + 321 * q^11 + 240 * q^12 + 527 * q^13 + 424 * q^14 - 375 * q^15 + 1536 * q^16 - 660 * q^17 + 3596 * q^18 + 2749 * q^19 - 2400 * q^20 + 6002 * q^21 + 1284 * q^22 - 3174 * q^23 + 960 * q^24 + 3750 * q^25 + 2108 * q^26 + 15372 * q^27 + 1696 * q^28 + 3337 * q^29 - 1500 * q^30 + 31094 * q^31 + 6144 * q^32 + 15087 * q^33 - 2640 * q^34 - 2650 * q^35 + 14384 * q^36 + 27037 * q^37 + 10996 * q^38 + 38528 * q^39 - 9600 * q^40 + 33608 * q^41 + 24008 * q^42 + 17024 * q^43 + 5136 * q^44 - 22475 * q^45 - 12696 * q^46 + 16864 * q^47 + 3840 * q^48 + 6002 * q^49 + 15000 * q^50 + 5719 * q^51 + 8432 * q^52 - 8475 * q^53 + 61488 * q^54 - 8025 * q^55 + 6784 * q^56 + 9566 * q^57 + 13348 * q^58 + 7899 * q^59 - 6000 * q^60 + 25437 * q^61 + 124376 * q^62 - 13333 * q^63 + 24576 * q^64 - 13175 * q^65 + 60348 * q^66 - 25517 * q^67 - 10560 * q^68 - 7935 * q^69 - 10600 * q^70 + 17204 * q^71 + 57536 * q^72 + 760 * q^73 + 108148 * q^74 + 9375 * q^75 + 43984 * q^76 + 102330 * q^77 + 154112 * q^78 + 66972 * q^79 - 38400 * q^80 + 115874 * q^81 + 134432 * q^82 + 58523 * q^83 + 96032 * q^84 + 16500 * q^85 + 68096 * q^86 - 70854 * q^87 + 20544 * q^88 + 38406 * q^89 - 89900 * q^90 + 25111 * q^91 - 50784 * q^92 + 130338 * q^93 + 67456 * q^94 - 68725 * q^95 + 15360 * q^96 + 82861 * q^97 + 24008 * q^98 - 2973 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more