LMFDB - Embedded newform 1815.4.a.s.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1815,4,Mod(1,1815)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1815, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1815.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1815 = 3 \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1815.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:	$107.088466660$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.23612.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 20x + 26 $ x^3 - x^2 - 20*x + 26
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$4.26150$ of defining polynomial
Character	$\chi$	$=$	1815.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+5.26150 q^{2} -3.00000 q^{3} +19.6833 q^{4} -5.00000 q^{5} -15.7845 q^{6} +10.3207 q^{7} +61.4719 q^{8} +9.00000 q^{9} -26.3075 q^{10} -59.0500 q^{12} -63.9817 q^{13} +54.3024 q^{14} +15.0000 q^{15} +165.967 q^{16} -17.1461 q^{17} +47.3535 q^{18} -90.2104 q^{19} -98.4167 q^{20} -30.9621 q^{21} -212.605 q^{23} -184.416 q^{24} +25.0000 q^{25} -336.639 q^{26} -27.0000 q^{27} +203.146 q^{28} -57.5461 q^{29} +78.9224 q^{30} -141.704 q^{31} +381.462 q^{32} -90.2140 q^{34} -51.6035 q^{35} +177.150 q^{36} -257.963 q^{37} -474.642 q^{38} +191.945 q^{39} -307.359 q^{40} +225.914 q^{41} -162.907 q^{42} +347.445 q^{43} -45.0000 q^{45} -1118.62 q^{46} +404.364 q^{47} -497.902 q^{48} -236.483 q^{49} +131.537 q^{50} +51.4382 q^{51} -1259.37 q^{52} +259.568 q^{53} -142.060 q^{54} +634.433 q^{56} +270.631 q^{57} -302.779 q^{58} -853.067 q^{59} +295.250 q^{60} +203.699 q^{61} -745.573 q^{62} +92.8864 q^{63} +679.320 q^{64} +319.908 q^{65} +266.890 q^{67} -337.492 q^{68} +637.814 q^{69} -271.512 q^{70} +92.4460 q^{71} +553.247 q^{72} +242.026 q^{73} -1357.27 q^{74} -75.0000 q^{75} -1775.64 q^{76} +1009.92 q^{78} +1021.60 q^{79} -829.837 q^{80} +81.0000 q^{81} +1188.65 q^{82} -706.415 q^{83} -609.438 q^{84} +85.7303 q^{85} +1828.08 q^{86} +172.638 q^{87} -440.218 q^{89} -236.767 q^{90} -660.336 q^{91} -4184.77 q^{92} +425.111 q^{93} +2127.56 q^{94} +451.052 q^{95} -1144.38 q^{96} -197.761 q^{97} -1244.25 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 4 q^{2} - 9 q^{3} + 22 q^{4} - 15 q^{5} - 12 q^{6} + 4 q^{7} + 48 q^{8} + 27 q^{9} - 20 q^{10} - 66 q^{12} - 56 q^{14} + 45 q^{15} + 50 q^{16} + 218 q^{17} + 36 q^{18} - 146 q^{19} - 110 q^{20} - 12 q^{21}+ \cdots - 572 q^{98}+O(q^{100}) $ 3 * q + 4 * q^2 - 9 * q^3 + 22 * q^4 - 15 * q^5 - 12 * q^6 + 4 * q^7 + 48 * q^8 + 27 * q^9 - 20 * q^10 - 66 * q^12 - 56 * q^14 + 45 * q^15 + 50 * q^16 + 218 * q^17 + 36 * q^18 - 146 * q^19 - 110 * q^20 - 12 * q^21 - 200 * q^23 - 144 * q^24 + 75 * q^25 - 508 * q^26 - 81 * q^27 + 340 * q^28 - 68 * q^29 + 60 * q^30 - 68 * q^31 + 688 * q^32 - 176 * q^34 - 20 * q^35 + 198 * q^36 - 390 * q^37 - 316 * q^38 - 240 * q^40 + 196 * q^41 + 168 * q^42 + 524 * q^43 - 135 * q^45 - 1160 * q^46 - 60 * q^47 - 150 * q^48 - 157 * q^49 + 100 * q^50 - 654 * q^51 - 1020 * q^52 - 158 * q^53 - 108 * q^54 + 1368 * q^56 + 438 * q^57 + 1092 * q^58 - 1044 * q^59 + 330 * q^60 - 642 * q^61 - 88 * q^62 + 36 * q^63 + 1166 * q^64 - 236 * q^67 - 144 * q^68 + 600 * q^69 + 280 * q^70 - 544 * q^71 + 432 * q^72 - 900 * q^73 - 1536 * q^74 - 225 * q^75 - 1996 * q^76 + 1524 * q^78 + 1586 * q^79 - 250 * q^80 + 243 * q^81 + 380 * q^82 + 1582 * q^83 - 1020 * q^84 - 1090 * q^85 + 3568 * q^86 + 204 * q^87 - 2122 * q^89 - 180 * q^90 - 8 * q^91 - 4128 * q^92 + 204 * q^93 + 2152 * q^94 + 730 * q^95 - 2064 * q^96 + 618 * q^97 - 572 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	5.26150	1.86022	0.930110	−	0.367281i	$-0.119711\pi$
$2$	5.26150	1.86022	0.930110	+	0.367281i	$0.119711\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	19.6833	2.46042
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−15.7845	−1.07400
$7$	10.3207	0.557266	0.278633	−	0.960398i	$-0.410119\pi$
$7$	10.3207	0.557266	0.278633	+	0.960398i	$0.410119\pi$
$8$	61.4719	2.71670
$9$	9.00000	0.333333
$10$	−26.3075	−0.831916
$11$	0	0
$11$	0	0
$12$	−59.0500	−1.42052
$13$	−63.9817	−1.36502	−0.682512	−	0.730874i	$-0.739113\pi$
$13$	−63.9817	−1.36502	−0.682512	+	0.730874i	$0.739113\pi$
$14$	54.3024	1.03664
$15$	15.0000	0.258199
$16$	165.967	2.59324
$17$	−17.1461	−0.244620	−0.122310	−	0.992492i	$-0.539030\pi$
$17$	−17.1461	−0.244620	−0.122310	+	0.992492i	$0.539030\pi$
$18$	47.3535	0.620073
$19$	−90.2104	−1.08925	−0.544623	−	0.838681i	$-0.683327\pi$
$19$	−90.2104	−1.08925	−0.544623	+	0.838681i	$0.683327\pi$
$20$	−98.4167	−1.10033
$21$	−30.9621	−0.321738
$22$	0	0
$23$	−212.605	−1.92744	−0.963721	−	0.266913i	$-0.913996\pi$
$23$	−212.605	−1.92744	−0.963721	+	0.266913i	$0.913996\pi$
$24$	−184.416	−1.56849
$25$	25.0000	0.200000
$26$	−336.639	−2.53925
$27$	−27.0000	−0.192450
$28$	203.146	1.37111
$29$	−57.5461	−0.368484	−0.184242	−	0.982881i	$-0.558983\pi$
$29$	−57.5461	−0.368484	−0.184242	+	0.982881i	$0.558983\pi$
$30$	78.9224	0.480307
$31$	−141.704	−0.820991	−0.410496	−	0.911863i	$-0.634644\pi$
$31$	−141.704	−0.820991	−0.410496	+	0.911863i	$0.634644\pi$
$32$	381.462	2.10730
$33$	0	0
$34$	−90.2140	−0.455046
$35$	−51.6035	−0.249217
$36$	177.150	0.820139
$37$	−257.963	−1.14619	−0.573093	−	0.819490i	$-0.694257\pi$
$37$	−257.963	−1.14619	−0.573093	+	0.819490i	$0.694257\pi$
$38$	−474.642	−2.02624
$39$	191.945	0.788097
$40$	−307.359	−1.21494
$41$	225.914	0.860533	0.430266	−	0.902702i	$-0.358420\pi$
$41$	225.914	0.860533	0.430266	+	0.902702i	$0.358420\pi$
$42$	−162.907	−0.598503
$43$	347.445	1.23221	0.616103	−	0.787666i	$-0.288711\pi$
$43$	347.445	1.23221	0.616103	+	0.787666i	$0.288711\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	−1118.62	−3.58547
$47$	404.364	1.25495	0.627473	−	0.778638i	$-0.284089\pi$
$47$	404.364	1.25495	0.627473	+	0.778638i	$0.284089\pi$
$48$	−497.902	−1.49721
$49$	−236.483	−0.689455
$50$	131.537	0.372044
$51$	51.4382	0.141231
$52$	−1259.37	−3.35853
$53$	259.568	0.672726	0.336363	−	0.941732i	$-0.390803\pi$
$53$	259.568	0.672726	0.336363	+	0.941732i	$0.390803\pi$
$54$	−142.060	−0.358000
$55$	0	0
$56$	634.433	1.51392
$57$	270.631	0.628877
$58$	−302.779	−0.685462
$59$	−853.067	−1.88237	−0.941185	−	0.337891i	$-0.890287\pi$
$59$	−853.067	−1.88237	−0.941185	+	0.337891i	$0.890287\pi$
$60$	295.250	0.635277
$61$	203.699	0.427558	0.213779	−	0.976882i	$-0.431423\pi$
$61$	203.699	0.427558	0.213779	+	0.976882i	$0.431423\pi$
$62$	−745.573	−1.52722
$63$	92.8864	0.185755
$64$	679.320	1.32680
$65$	319.908	0.610458
$66$	0	0
$67$	266.890	0.486653	0.243327	−	0.969944i	$-0.421761\pi$
$67$	266.890	0.486653	0.243327	+	0.969944i	$0.421761\pi$
$68$	−337.492	−0.601866
$69$	637.814	1.11281
$70$	−271.512	−0.463598
$71$	92.4460	0.154526	0.0772629	−	0.997011i	$-0.475382\pi$
$71$	92.4460	0.154526	0.0772629	+	0.997011i	$0.475382\pi$
$72$	553.247	0.905566
$73$	242.026	0.388040	0.194020	−	0.980998i	$-0.437847\pi$
$73$	242.026	0.388040	0.194020	+	0.980998i	$0.437847\pi$
$74$	−1357.27	−2.13216
$75$	−75.0000	−0.115470
$76$	−1775.64	−2.68000
$77$	0	0
$78$	1009.92	1.46603
$79$	1021.60	1.45492	0.727460	−	0.686150i	$-0.240701\pi$
$79$	1021.60	1.45492	0.727460	+	0.686150i	$0.240701\pi$
$80$	−829.837	−1.15973
$81$	81.0000	0.111111
$82$	1188.65	1.60078
$83$	−706.415	−0.934206	−0.467103	−	0.884203i	$-0.654702\pi$
$83$	−706.415	−0.934206	−0.467103	+	0.884203i	$0.654702\pi$
$84$	−609.438	−0.791609
$85$	85.7303	0.109397
$86$	1828.08	2.29217
$87$	172.638	0.212745
$88$	0	0
$89$	−440.218	−0.524304	−0.262152	−	0.965027i	$-0.584432\pi$
$89$	−440.218	−0.524304	−0.262152	+	0.965027i	$0.584432\pi$
$90$	−236.767	−0.277305
$91$	−660.336	−0.760682
$92$	−4184.77	−4.74231
$93$	425.111	0.473999
$94$	2127.56	2.33448
$95$	451.052	0.487126
$96$	−1144.38	−1.21665
$97$	−197.761	−0.207006	−0.103503	−	0.994629i	$-0.533005\pi$
$97$	−197.761	−0.207006	−0.103503	+	0.994629i	$0.533005\pi$
$98$	−1244.25	−1.28254
$99$	0	0
$100$	492.084	0.492084
$101$	−1400.62	−1.37987	−0.689937	−	0.723870i	$-0.742362\pi$
$101$	−1400.62	−1.37987	−0.689937	+	0.723870i	$0.742362\pi$
$102$	270.642	0.262721
$103$	−1345.70	−1.28734	−0.643669	−	0.765304i	$-0.722589\pi$
$103$	−1345.70	−1.28734	−0.643669	+	0.765304i	$0.722589\pi$
$104$	−3933.07	−3.70836
$105$	154.811	0.143885
$106$	1365.72	1.25142
$107$	889.178	0.803366	0.401683	−	0.915779i	$-0.368425\pi$
$107$	889.178	0.803366	0.401683	+	0.915779i	$0.368425\pi$
$108$	−531.450	−0.473508
$109$	−1256.29	−1.10395	−0.551974	−	0.833861i	$-0.686125\pi$
$109$	−1256.29	−1.10395	−0.551974	+	0.833861i	$0.686125\pi$
$110$	0	0
$111$	773.890	0.661751
$112$	1712.90	1.44512
$113$	−2394.01	−1.99301	−0.996504	−	0.0835448i	$-0.973376\pi$
$113$	−2394.01	−1.99301	−0.996504	+	0.0835448i	$0.973376\pi$
$114$	1423.93	1.16985
$115$	1063.02	0.861978
$116$	−1132.70	−0.906626
$117$	−575.835	−0.455008
$118$	−4488.41	−3.50162
$119$	−176.960	−0.136318
$120$	922.078	0.701449
$121$	0	0
$122$	1071.76	0.795352
$123$	−677.742	−0.496829
$124$	−2789.20	−2.01998
$125$	−125.000	−0.0894427
$126$	488.721	0.345546
$127$	−2065.57	−1.44322	−0.721612	−	0.692298i	$-0.756598\pi$
$127$	−2065.57	−1.44322	−0.721612	+	0.692298i	$0.756598\pi$
$128$	522.548	0.360837
$129$	−1042.33	−0.711414
$130$	1683.20	1.13559
$131$	−785.526	−0.523907	−0.261953	−	0.965081i	$-0.584367\pi$
$131$	−785.526	−0.523907	−0.261953	+	0.965081i	$0.584367\pi$
$132$	0	0
$133$	−931.035	−0.607000
$134$	1404.24	0.905282
$135$	135.000	0.0860663
$136$	−1054.00	−0.664558
$137$	1276.24	0.795885	0.397942	−	0.917410i	$-0.369724\pi$
$137$	1276.24	0.795885	0.397942	+	0.917410i	$0.369724\pi$
$138$	3355.86	2.07007
$139$	2703.21	1.64952	0.824760	−	0.565482i	$-0.191310\pi$
$139$	2703.21	1.64952	0.824760	+	0.565482i	$0.191310\pi$
$140$	−1015.73	−0.613178
$141$	−1213.09	−0.724544
$142$	486.405	0.287452
$143$	0	0
$144$	1493.71	0.864413
$145$	287.731	0.164791
$146$	1273.42	0.721840
$147$	709.449	0.398057
$148$	−5077.58	−2.82010
$149$	2400.99	1.32011	0.660056	−	0.751217i	$-0.270533\pi$
$149$	2400.99	1.32011	0.660056	+	0.751217i	$0.270533\pi$
$150$	−394.612	−0.214800
$151$	2517.30	1.35665	0.678326	−	0.734761i	$-0.262706\pi$
$151$	2517.30	1.35665	0.678326	+	0.734761i	$0.262706\pi$
$152$	−5545.40	−2.95916
$153$	−154.315	−0.0815399
$154$	0	0
$155$	708.518	0.367158
$156$	3778.12	1.93905
$157$	1391.42	0.707310	0.353655	−	0.935376i	$-0.384939\pi$
$157$	1391.42	0.707310	0.353655	+	0.935376i	$0.384939\pi$
$158$	5375.13	2.70647
$159$	−778.705	−0.388398
$160$	−1907.31	−0.942412
$161$	−2194.23	−1.07410
$162$	426.181	0.206691
$163$	−2720.53	−1.30729	−0.653644	−	0.756802i	$-0.726761\pi$
$163$	−2720.53	−1.30729	−0.653644	+	0.756802i	$0.726761\pi$
$164$	4446.74	2.11727
$165$	0	0
$166$	−3716.80	−1.73783
$167$	−2950.25	−1.36705	−0.683525	−	0.729927i	$-0.739554\pi$
$167$	−2950.25	−1.36705	−0.683525	+	0.729927i	$0.739554\pi$
$168$	−1903.30	−0.874064
$169$	1896.65	0.863292
$170$	451.070	0.203503
$171$	−811.894	−0.363082
$172$	6838.88	3.03174
$173$	−537.049	−0.236018	−0.118009	−	0.993013i	$-0.537651\pi$
$173$	−537.049	−0.236018	−0.118009	+	0.993013i	$0.537651\pi$
$174$	908.336	0.395752
$175$	258.018	0.111453
$176$	0	0
$177$	2559.20	1.08679
$178$	−2316.21	−0.975320
$179$	2891.25	1.20728	0.603638	−	0.797259i	$-0.293717\pi$
$179$	2891.25	1.20728	0.603638	+	0.797259i	$0.293717\pi$
$180$	−885.751	−0.366778
$181$	435.209	0.178723	0.0893615	−	0.995999i	$-0.471517\pi$
$181$	435.209	0.178723	0.0893615	+	0.995999i	$0.471517\pi$
$182$	−3474.36	−1.41504
$183$	−611.098	−0.246851
$184$	−13069.2	−5.23628
$185$	1289.82	0.512590
$186$	2236.72	0.881743
$187$	0	0
$188$	7959.23	3.08769
$189$	−278.659	−0.107246
$190$	2373.21	0.906161
$191$	−3779.49	−1.43180	−0.715901	−	0.698202i	$-0.753984\pi$
$191$	−3779.49	−1.43180	−0.715901	+	0.698202i	$0.753984\pi$
$192$	−2037.96	−0.766027
$193$	−3751.91	−1.39932	−0.699660	−	0.714476i	$-0.746665\pi$
$193$	−3751.91	−1.39932	−0.699660	+	0.714476i	$0.746665\pi$
$194$	−1040.52	−0.385076
$195$	−959.725	−0.352448
$196$	−4654.78	−1.69635
$197$	3920.73	1.41797	0.708986	−	0.705223i	$-0.249153\pi$
$197$	3920.73	1.41797	0.708986	+	0.705223i	$0.249153\pi$
$198$	0	0
$199$	−597.084	−0.212694	−0.106347	−	0.994329i	$-0.533915\pi$
$199$	−597.084	−0.212694	−0.106347	+	0.994329i	$0.533915\pi$
$200$	1536.80	0.543340
$201$	−800.669	−0.280969
$202$	−7369.37	−2.56687
$203$	−593.917	−0.205344
$204$	1012.48	0.347488
$205$	−1129.57	−0.384842
$206$	−7080.40	−2.39473
$207$	−1913.44	−0.642480
$208$	−10618.9	−3.53984
$209$	0	0
$210$	814.536	0.267659
$211$	4384.55	1.43054	0.715272	−	0.698846i	$-0.246303\pi$
$211$	4384.55	1.43054	0.715272	+	0.698846i	$0.246303\pi$
$212$	5109.17	1.65519
$213$	−277.338	−0.0892155
$214$	4678.41	1.49444
$215$	−1737.22	−0.551059
$216$	−1659.74	−0.522829
$217$	−1462.48	−0.457510
$218$	−6609.95	−2.05359
$219$	−726.077	−0.224035
$220$	0	0
$221$	1097.03	0.333912
$222$	4071.82	1.23100
$223$	−2333.03	−0.700587	−0.350294	−	0.936640i	$-0.613918\pi$
$223$	−2333.03	−0.700587	−0.350294	+	0.936640i	$0.613918\pi$
$224$	3936.95	1.17433
$225$	225.000	0.0666667
$226$	−12596.1	−3.70743
$227$	2120.00	0.619864	0.309932	−	0.950759i	$-0.399694\pi$
$227$	2120.00	0.619864	0.309932	+	0.950759i	$0.399694\pi$
$228$	5326.93	1.54730
$229$	2347.12	0.677301	0.338651	−	0.940912i	$-0.390030\pi$
$229$	2347.12	0.677301	0.338651	+	0.940912i	$0.390030\pi$
$230$	5593.10	1.60347
$231$	0	0
$232$	−3537.47	−1.00106
$233$	375.499	0.105578	0.0527891	−	0.998606i	$-0.483189\pi$
$233$	375.499	0.105578	0.0527891	+	0.998606i	$0.483189\pi$
$234$	−3029.75	−0.846415
$235$	−2021.82	−0.561229
$236$	−16791.2	−4.63142
$237$	−3064.79	−0.839998
$238$	−931.072	−0.253582
$239$	1428.15	0.386524	0.193262	−	0.981147i	$-0.438093\pi$
$239$	1428.15	0.386524	0.193262	+	0.981147i	$0.438093\pi$
$240$	2489.51	0.669572
$241$	−190.819	−0.0510032	−0.0255016	−	0.999675i	$-0.508118\pi$
$241$	−190.819	−0.0510032	−0.0255016	+	0.999675i	$0.508118\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	4009.49	1.05197
$245$	1182.41	0.308334
$246$	−3565.94	−0.924211
$247$	5771.81	1.48685
$248$	−8710.79	−2.23039
$249$	2119.24	0.539364
$250$	−657.687	−0.166383
$251$	−6294.80	−1.58297	−0.791483	−	0.611191i	$-0.790691\pi$
$251$	−6294.80	−1.58297	−0.791483	+	0.611191i	$0.790691\pi$
$252$	1828.31	0.457036
$253$	0	0
$254$	−10868.0	−2.68471
$255$	−257.191	−0.0631605
$256$	−2685.18	−0.655561
$257$	4459.44	1.08238	0.541191	−	0.840900i	$-0.317974\pi$
$257$	4459.44	1.08238	0.541191	+	0.840900i	$0.317974\pi$
$258$	−5484.24	−1.32339
$259$	−2662.36	−0.638731
$260$	6296.87	1.50198
$261$	−517.915	−0.122828
$262$	−4133.04	−0.974581
$263$	4416.65	1.03552	0.517761	−	0.855525i	$-0.326766\pi$
$263$	4416.65	1.03552	0.517761	+	0.855525i	$0.326766\pi$
$264$	0	0
$265$	−1297.84	−0.300852
$266$	−4898.64	−1.12915
$267$	1320.65	0.302707
$268$	5253.28	1.19737
$269$	1914.86	0.434020	0.217010	−	0.976169i	$-0.430370\pi$
$269$	1914.86	0.434020	0.217010	+	0.976169i	$0.430370\pi$
$270$	710.302	0.160102
$271$	−6088.34	−1.36472	−0.682362	−	0.731014i	$-0.739047\pi$
$271$	−6088.34	−1.36472	−0.682362	+	0.731014i	$0.739047\pi$
$272$	−2845.69	−0.634357
$273$	1981.01	0.439180
$274$	6714.91	1.48052
$275$	0	0
$276$	12554.3	2.73798
$277$	832.321	0.180539	0.0902696	−	0.995917i	$-0.471227\pi$
$277$	832.321	0.180539	0.0902696	+	0.995917i	$0.471227\pi$
$278$	14222.9	3.06847
$279$	−1275.33	−0.273664
$280$	−3172.17	−0.677047
$281$	2545.32	0.540360	0.270180	−	0.962810i	$-0.412917\pi$
$281$	2545.32	0.540360	0.270180	+	0.962810i	$0.412917\pi$
$282$	−6382.67	−1.34781
$283$	5911.71	1.24175	0.620874	−	0.783911i	$-0.286778\pi$
$283$	5911.71	1.24175	0.620874	+	0.783911i	$0.286778\pi$
$284$	1819.65	0.380198
$285$	−1353.16	−0.281242
$286$	0	0
$287$	2331.59	0.479545
$288$	3433.15	0.702433
$289$	−4619.01	−0.940161
$290$	1513.89	0.306548
$291$	593.282	0.119515
$292$	4763.87	0.954742
$293$	6871.03	1.37000	0.685000	−	0.728543i	$-0.259802\pi$
$293$	6871.03	1.37000	0.685000	+	0.728543i	$0.259802\pi$
$294$	3732.76	0.740473
$295$	4265.34	0.841822
$296$	−15857.5	−3.11384
$297$	0	0
$298$	12632.8	2.45570
$299$	13602.8	2.63100
$300$	−1476.25	−0.284105
$301$	3585.88	0.686666
$302$	13244.7	2.52367
$303$	4201.87	0.796670
$304$	−14972.0	−2.82468
$305$	−1018.50	−0.191210
$306$	−811.926	−0.151682
$307$	−200.179	−0.0372144	−0.0186072	−	0.999827i	$-0.505923\pi$
$307$	−200.179	−0.0372144	−0.0186072	+	0.999827i	$0.505923\pi$
$308$	0	0
$309$	4037.10	0.743245
$310$	3727.87	0.682995
$311$	5734.93	1.04565	0.522827	−	0.852439i	$-0.324878\pi$
$311$	5734.93	1.04565	0.522827	+	0.852439i	$0.324878\pi$
$312$	11799.2	2.14102
$313$	−3077.36	−0.555727	−0.277864	−	0.960621i	$-0.589626\pi$
$313$	−3077.36	−0.555727	−0.277864	+	0.960621i	$0.589626\pi$
$314$	7320.97	1.31575
$315$	−464.432	−0.0830723
$316$	20108.5	3.57971
$317$	2142.38	0.379584	0.189792	−	0.981824i	$-0.439219\pi$
$317$	2142.38	0.379584	0.189792	+	0.981824i	$0.439219\pi$
$318$	−4097.15	−0.722506
$319$	0	0
$320$	−3396.60	−0.593362
$321$	−2667.54	−0.463823
$322$	−11544.9	−1.99806
$323$	1546.75	0.266451
$324$	1594.35	0.273380
$325$	−1599.54	−0.273005
$326$	−14314.0	−2.43184
$327$	3768.86	0.637365
$328$	13887.4	2.33781
$329$	4173.32	0.699339
$330$	0	0
$331$	−1618.23	−0.268719	−0.134359	−	0.990933i	$-0.542898\pi$
$331$	−1618.23	−0.268719	−0.134359	+	0.990933i	$0.542898\pi$
$332$	−13904.6	−2.29854
$333$	−2321.67	−0.382062
$334$	−15522.7	−2.54301
$335$	−1334.45	−0.217638
$336$	−5138.70	−0.834343
$337$	−2406.47	−0.388988	−0.194494	−	0.980904i	$-0.562306\pi$
$337$	−2406.47	−0.388988	−0.194494	+	0.980904i	$0.562306\pi$
$338$	9979.23	1.60591
$339$	7182.04	1.15066
$340$	1687.46	0.269163
$341$	0	0
$342$	−4271.78	−0.675413
$343$	−5980.67	−0.941475
$344$	21358.1	3.34753
$345$	−3189.07	−0.497663
$346$	−2825.68	−0.439045
$347$	6612.89	1.02305	0.511525	−	0.859268i	$-0.329081\pi$
$347$	6612.89	1.02305	0.511525	+	0.859268i	$0.329081\pi$
$348$	3398.10	0.523441
$349$	−349.871	−0.0536623	−0.0268311	−	0.999640i	$-0.508542\pi$
$349$	−349.871	−0.0536623	−0.0268311	+	0.999640i	$0.508542\pi$
$350$	1357.56	0.207327
$351$	1727.50	0.262699
$352$	0	0
$353$	1723.29	0.259835	0.129917	−	0.991525i	$-0.458529\pi$
$353$	1723.29	0.259835	0.129917	+	0.991525i	$0.458529\pi$
$354$	13465.2	2.02166
$355$	−462.230	−0.0691060
$356$	−8664.97	−1.29001
$357$	530.879	0.0787033
$358$	15212.3	2.24580
$359$	−5875.74	−0.863816	−0.431908	−	0.901918i	$-0.642159\pi$
$359$	−5875.74	−0.863816	−0.431908	+	0.901918i	$0.642159\pi$
$360$	−2766.24	−0.404982
$361$	1278.92	0.186458
$362$	2289.85	0.332464
$363$	0	0
$364$	−12997.6	−1.87159
$365$	−1210.13	−0.173537
$366$	−3215.29	−0.459197
$367$	−5368.28	−0.763548	−0.381774	−	0.924256i	$-0.624687\pi$
$367$	−5368.28	−0.763548	−0.381774	+	0.924256i	$0.624687\pi$
$368$	−35285.5	−4.99832
$369$	2033.23	0.286844
$370$	6786.37	0.953531
$371$	2678.93	0.374887
$372$	8367.60	1.16624
$373$	−10393.9	−1.44282	−0.721412	−	0.692506i	$-0.756507\pi$
$373$	−10393.9	−1.44282	−0.721412	+	0.692506i	$0.756507\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	24857.0	3.40931
$377$	3681.90	0.502990
$378$	−1466.16	−0.199501
$379$	10918.9	1.47986	0.739928	−	0.672686i	$-0.234860\pi$
$379$	10918.9	1.47986	0.739928	+	0.672686i	$0.234860\pi$
$380$	8878.21	1.19853
$381$	6196.70	0.833246
$382$	−19885.8	−2.66347
$383$	−11663.6	−1.55609	−0.778044	−	0.628210i	$-0.783788\pi$
$383$	−11663.6	−1.55609	−0.778044	+	0.628210i	$0.783788\pi$
$384$	−1567.64	−0.208329
$385$	0	0
$386$	−19740.7	−2.60304
$387$	3127.00	0.410735
$388$	−3892.59	−0.509321
$389$	−5827.00	−0.759487	−0.379744	−	0.925092i	$-0.623988\pi$
$389$	−5827.00	−0.759487	−0.379744	+	0.925092i	$0.623988\pi$
$390$	−5049.59	−0.655631
$391$	3645.34	0.471490
$392$	−14537.1	−1.87304
$393$	2356.58	0.302478
$394$	20628.9	2.63774
$395$	−5107.99	−0.650660
$396$	0	0
$397$	7366.99	0.931332	0.465666	−	0.884961i	$-0.345815\pi$
$397$	7366.99	0.931332	0.465666	+	0.884961i	$0.345815\pi$
$398$	−3141.55	−0.395658
$399$	2793.11	0.350452
$400$	4149.18	0.518648
$401$	14604.2	1.81870	0.909349	−	0.416035i	$-0.136581\pi$
$401$	14604.2	1.81870	0.909349	+	0.416035i	$0.136581\pi$
$402$	−4212.72	−0.522665
$403$	9066.43	1.12067
$404$	−27569.0	−3.39507
$405$	−405.000	−0.0496904
$406$	−3124.89	−0.381985
$407$	0	0
$408$	3162.00	0.383683
$409$	−12581.2	−1.52103	−0.760515	−	0.649320i	$-0.775054\pi$
$409$	−12581.2	−1.52103	−0.760515	+	0.649320i	$0.775054\pi$
$410$	−5943.23	−0.715891
$411$	−3828.71	−0.459504
$412$	−26487.9	−3.16739
$413$	−8804.26	−1.04898
$414$	−10067.6	−1.19516
$415$	3532.07	0.417790
$416$	−24406.6	−2.87651
$417$	−8109.63	−0.952351
$418$	0	0
$419$	−3776.01	−0.440263	−0.220131	−	0.975470i	$-0.570649\pi$
$419$	−3776.01	−0.440263	−0.220131	+	0.975470i	$0.570649\pi$
$420$	3047.19	0.354018
$421$	12683.4	1.46830	0.734148	−	0.678989i	$-0.237582\pi$
$421$	12683.4	1.46830	0.734148	+	0.678989i	$0.237582\pi$
$422$	23069.3	2.66113
$423$	3639.27	0.418316
$424$	15956.2	1.82759
$425$	−428.652	−0.0489239
$426$	−1459.21	−0.165960
$427$	2102.32	0.238263
$428$	17502.0	1.97662
$429$	0	0
$430$	−9140.40	−1.02509
$431$	−14152.6	−1.58168	−0.790841	−	0.612022i	$-0.790356\pi$
$431$	−14152.6	−1.58168	−0.790841	+	0.612022i	$0.790356\pi$
$432$	−4481.12	−0.499069
$433$	−10950.2	−1.21532	−0.607661	−	0.794197i	$-0.707892\pi$
$433$	−10950.2	−1.21532	−0.607661	+	0.794197i	$0.707892\pi$
$434$	−7694.84	−0.851070
$435$	−863.192	−0.0951423
$436$	−24727.9	−2.71618
$437$	19179.2	2.09946
$438$	−3820.25	−0.416755
$439$	11221.0	1.21993	0.609964	−	0.792429i	$-0.291184\pi$
$439$	11221.0	1.21993	0.609964	+	0.792429i	$0.291184\pi$
$440$	0	0
$441$	−2128.35	−0.229818
$442$	5772.04	0.621149
$443$	9647.11	1.03465	0.517323	−	0.855790i	$-0.326929\pi$
$443$	9647.11	1.03465	0.517323	+	0.855790i	$0.326929\pi$
$444$	15232.7	1.62818
$445$	2201.09	0.234476
$446$	−12275.2	−1.30325
$447$	−7202.96	−0.762167
$448$	7011.07	0.739379
$449$	6482.03	0.681305	0.340652	−	0.940189i	$-0.389352\pi$
$449$	6482.03	0.681305	0.340652	+	0.940189i	$0.389352\pi$
$450$	1183.84	0.124015
$451$	0	0
$452$	−47122.2	−4.90363
$453$	−7551.89	−0.783264
$454$	11154.4	1.15308
$455$	3301.68	0.340187
$456$	16636.2	1.70847
$457$	−11319.8	−1.15868	−0.579342	−	0.815085i	$-0.696690\pi$
$457$	−11319.8	−1.15868	−0.579342	+	0.815085i	$0.696690\pi$
$458$	12349.4	1.25993
$459$	462.944	0.0470771
$460$	20923.9	2.12083
$461$	8406.73	0.849329	0.424664	−	0.905351i	$-0.360392\pi$
$461$	8406.73	0.849329	0.424664	+	0.905351i	$0.360392\pi$
$462$	0	0
$463$	9758.56	0.979523	0.489761	−	0.871857i	$-0.337084\pi$
$463$	9758.56	0.979523	0.489761	+	0.871857i	$0.337084\pi$
$464$	−9550.78	−0.955568
$465$	−2125.55	−0.211979
$466$	1975.68	0.196399
$467$	−16388.7	−1.62394	−0.811969	−	0.583701i	$-0.801604\pi$
$467$	−16388.7	−1.62394	−0.811969	+	0.583701i	$0.801604\pi$
$468$	−11334.4	−1.11951
$469$	2754.49	0.271195
$470$	−10637.8	−1.04401
$471$	−4174.27	−0.408365
$472$	−52439.6	−5.11384
$473$	0	0
$474$	−16125.4	−1.56258
$475$	−2255.26	−0.217849
$476$	−3483.16	−0.335400
$477$	2336.11	0.224242
$478$	7514.20	0.719020
$479$	−13829.0	−1.31913	−0.659567	−	0.751646i	$-0.729260\pi$
$479$	−13829.0	−1.31913	−0.659567	+	0.751646i	$0.729260\pi$
$480$	5721.92	0.544102
$481$	16504.9	1.56457
$482$	−1004.00	−0.0948771
$483$	6582.69	0.620130
$484$	0	0
$485$	988.804	0.0925758
$486$	−1278.54	−0.119333
$487$	13264.4	1.23423	0.617113	−	0.786875i	$-0.288302\pi$
$487$	13264.4	1.23423	0.617113	+	0.786875i	$0.288302\pi$
$488$	12521.8	1.16155
$489$	8161.58	0.754763
$490$	6221.27	0.573568
$491$	7468.22	0.686428	0.343214	−	0.939257i	$-0.388484\pi$
$491$	7468.22	0.686428	0.343214	+	0.939257i	$0.388484\pi$
$492$	−13340.2	−1.22241
$493$	986.690	0.0901385
$494$	30368.4	2.76586
$495$	0	0
$496$	−23518.2	−2.12903
$497$	954.109	0.0861119
$498$	11150.4	1.00334
$499$	−5276.64	−0.473377	−0.236688	−	0.971586i	$-0.576062\pi$
$499$	−5276.64	−0.473377	−0.236688	+	0.971586i	$0.576062\pi$
$500$	−2460.42	−0.220067
$501$	8850.76	0.789267
$502$	−33120.1	−2.94466
$503$	10956.8	0.971253	0.485626	−	0.874166i	$-0.338592\pi$
$503$	10956.8	0.971253	0.485626	+	0.874166i	$0.338592\pi$
$504$	5709.90	0.504641
$505$	7003.12	0.617098
$506$	0	0
$507$	−5689.96	−0.498422
$508$	−40657.3	−3.55093
$509$	12734.4	1.10892	0.554462	−	0.832209i	$-0.312924\pi$
$509$	12734.4	1.10892	0.554462	+	0.832209i	$0.312924\pi$
$510$	−1353.21	−0.117492
$511$	2497.87	0.216242
$512$	−18308.4	−1.58032
$513$	2435.68	0.209626
$514$	23463.3	2.01347
$515$	6728.51	0.575715
$516$	−20516.6	−1.75038
$517$	0	0
$518$	−14008.0	−1.18818
$519$	1611.15	0.136265
$520$	19665.4	1.65843
$521$	−5650.70	−0.475166	−0.237583	−	0.971367i	$-0.576355\pi$
$521$	−5650.70	−0.475166	−0.237583	+	0.971367i	$0.576355\pi$
$522$	−2725.01	−0.228487
$523$	−14103.2	−1.17914	−0.589572	−	0.807716i	$-0.700703\pi$
$523$	−14103.2	−1.17914	−0.589572	+	0.807716i	$0.700703\pi$
$524$	−15461.8	−1.28903
$525$	−774.053	−0.0643475
$526$	23238.2	1.92630
$527$	2429.66	0.200830
$528$	0	0
$529$	33033.8	2.71503
$530$	−6828.59	−0.559651
$531$	−7677.60	−0.627457
$532$	−18325.9	−1.49347
$533$	−14454.4	−1.17465
$534$	6948.62	0.563102
$535$	−4445.89	−0.359276
$536$	16406.2	1.32209
$537$	−8673.75	−0.697021
$538$	10075.0	0.807372
$539$	0	0
$540$	2657.25	0.211759
$541$	6391.90	0.507965	0.253983	−	0.967209i	$-0.418259\pi$
$541$	6391.90	0.507965	0.253983	+	0.967209i	$0.418259\pi$
$542$	−32033.8	−2.53869
$543$	−1305.63	−0.103186
$544$	−6540.57	−0.515486
$545$	6281.43	0.493701
$546$	10423.1	0.816971
$547$	−20786.7	−1.62482	−0.812409	−	0.583088i	$-0.801844\pi$
$547$	−20786.7	−1.62482	−0.812409	+	0.583088i	$0.801844\pi$
$548$	25120.6	1.95821
$549$	1833.29	0.142519
$550$	0	0
$551$	5191.26	0.401370
$552$	39207.6	3.02317
$553$	10543.6	0.810777
$554$	4379.26	0.335843
$555$	−3869.45	−0.295944
$556$	53208.2	4.05851
$557$	15125.9	1.15064	0.575320	−	0.817928i	$-0.304877\pi$
$557$	15125.9	1.15064	0.575320	+	0.817928i	$0.304877\pi$
$558$	−6710.16	−0.509075
$559$	−22230.1	−1.68199
$560$	−8564.50	−0.646279
$561$	0	0
$562$	13392.2	1.00519
$563$	−8706.42	−0.651744	−0.325872	−	0.945414i	$-0.605658\pi$
$563$	−8706.42	−0.651744	−0.325872	+	0.945414i	$0.605658\pi$
$564$	−23877.7	−1.78268
$565$	11970.1	0.891300
$566$	31104.4	2.30992
$567$	835.977	0.0619184
$568$	5682.83	0.419800
$569$	−7067.76	−0.520731	−0.260366	−	0.965510i	$-0.583843\pi$
$569$	−7067.76	−0.520731	−0.260366	+	0.965510i	$0.583843\pi$
$570$	−7119.63	−0.523172
$571$	3326.42	0.243794	0.121897	−	0.992543i	$-0.461102\pi$
$571$	3326.42	0.243794	0.121897	+	0.992543i	$0.461102\pi$
$572$	0	0
$573$	11338.5	0.826651
$574$	12267.7	0.892060
$575$	−5315.12	−0.385488
$576$	6113.88	0.442266
$577$	−3308.06	−0.238676	−0.119338	−	0.992854i	$-0.538077\pi$
$577$	−3308.06	−0.238676	−0.119338	+	0.992854i	$0.538077\pi$
$578$	−24302.9	−1.74891
$579$	11255.7	0.807897
$580$	5663.50	0.405455
$581$	−7290.70	−0.520601
$582$	3121.55	0.222324
$583$	0	0
$584$	14877.8	1.05419
$585$	2879.17	0.203486
$586$	36151.9	2.54850
$587$	−5694.88	−0.400431	−0.200215	−	0.979752i	$-0.564164\pi$
$587$	−5694.88	−0.400431	−0.200215	+	0.979752i	$0.564164\pi$
$588$	13964.3	0.979387
$589$	12783.1	0.894262
$590$	22442.0	1.56597
$591$	−11762.2	−0.818666
$592$	−42813.5	−2.97234
$593$	3907.69	0.270606	0.135303	−	0.990804i	$-0.456799\pi$
$593$	3907.69	0.270606	0.135303	+	0.990804i	$0.456799\pi$
$594$	0	0
$595$	884.798	0.0609633
$596$	47259.5	3.24803
$597$	1791.25	0.122799
$598$	71571.1	4.89425
$599$	10727.2	0.731720	0.365860	−	0.930670i	$-0.380775\pi$
$599$	10727.2	0.731720	0.365860	+	0.930670i	$0.380775\pi$
$600$	−4610.39	−0.313697
$601$	−3348.98	−0.227301	−0.113650	−	0.993521i	$-0.536254\pi$
$601$	−3348.98	−0.227301	−0.113650	+	0.993521i	$0.536254\pi$
$602$	18867.1	1.27735
$603$	2402.01	0.162218
$604$	49548.8	3.33793
$605$	0	0
$606$	22108.1	1.48198
$607$	21539.9	1.44032	0.720162	−	0.693806i	$-0.244068\pi$
$607$	21539.9	1.44032	0.720162	+	0.693806i	$0.244068\pi$
$608$	−34411.8	−2.29537
$609$	1781.75	0.118555
$610$	−5358.82	−0.355692
$611$	−25871.9	−1.71303
$612$	−3037.43	−0.200622
$613$	9284.33	0.611730	0.305865	−	0.952075i	$-0.401054\pi$
$613$	9284.33	0.611730	0.305865	+	0.952075i	$0.401054\pi$
$614$	−1053.24	−0.0692270
$615$	3388.71	0.222189
$616$	0	0
$617$	−20711.3	−1.35139	−0.675695	−	0.737181i	$-0.736156\pi$
$617$	−20711.3	−1.35139	−0.675695	+	0.737181i	$0.736156\pi$
$618$	21241.2	1.38260
$619$	−13282.9	−0.862496	−0.431248	−	0.902233i	$-0.641927\pi$
$619$	−13282.9	−0.862496	−0.431248	+	0.902233i	$0.641927\pi$
$620$	13946.0	0.903363
$621$	5740.33	0.370936
$622$	30174.3	1.94515
$623$	−4543.36	−0.292177
$624$	31856.6	2.04373
$625$	625.000	0.0400000
$626$	−16191.5	−1.03378
$627$	0	0
$628$	27387.9	1.74028
$629$	4423.06	0.280380
$630$	−2443.61	−0.154533
$631$	−14789.9	−0.933086	−0.466543	−	0.884498i	$-0.654501\pi$
$631$	−14789.9	−0.933086	−0.466543	+	0.884498i	$0.654501\pi$
$632$	62799.5	3.95258
$633$	−13153.6	−0.825925
$634$	11272.1	0.706109
$635$	10327.8	0.645429
$636$	−15327.5	−0.955622
$637$	15130.6	0.941123
$638$	0	0
$639$	832.014	0.0515086
$640$	−2612.74	−0.161371
$641$	−5808.52	−0.357914	−0.178957	−	0.983857i	$-0.557272\pi$
$641$	−5808.52	−0.357914	−0.178957	+	0.983857i	$0.557272\pi$
$642$	−14035.2	−0.862813
$643$	−18891.2	−1.15862	−0.579311	−	0.815106i	$-0.696678\pi$
$643$	−18891.2	−1.15862	−0.579311	+	0.815106i	$0.696678\pi$
$644$	−43189.8	−2.64273
$645$	5211.67	0.318154
$646$	8138.24	0.495657
$647$	−243.046	−0.0147684	−0.00738418	−	0.999973i	$-0.502350\pi$
$647$	−243.046	−0.0147684	−0.00738418	+	0.999973i	$0.502350\pi$
$648$	4979.22	0.301855
$649$	0	0
$650$	−8415.98	−0.507849
$651$	4387.44	0.264144
$652$	−53549.0	−3.21648
$653$	−15920.4	−0.954081	−0.477041	−	0.878881i	$-0.658291\pi$
$653$	−15920.4	−0.954081	−0.477041	+	0.878881i	$0.658291\pi$
$654$	19829.8	1.18564
$655$	3927.63	0.234298
$656$	37494.4	2.23157
$657$	2178.23	0.129347
$658$	21957.9	1.30092
$659$	7476.38	0.441940	0.220970	−	0.975281i	$-0.429078\pi$
$659$	7476.38	0.441940	0.220970	+	0.975281i	$0.429078\pi$
$660$	0	0
$661$	−14920.1	−0.877948	−0.438974	−	0.898500i	$-0.644658\pi$
$661$	−14920.1	−0.877948	−0.438974	+	0.898500i	$0.644658\pi$
$662$	−8514.30	−0.499876
$663$	−3291.10	−0.192784
$664$	−43424.7	−2.53796
$665$	4655.18	0.271459
$666$	−12215.5	−0.710720
$667$	12234.6	0.710232
$668$	−58070.8	−3.36351
$669$	6999.08	0.404484
$670$	−7021.20	−0.404854
$671$	0	0
$672$	−11810.9	−0.677997
$673$	−563.692	−0.0322864	−0.0161432	−	0.999870i	$-0.505139\pi$
$673$	−563.692	−0.0322864	−0.0161432	+	0.999870i	$0.505139\pi$
$674$	−12661.6	−0.723602
$675$	−675.000	−0.0384900
$676$	37332.5	2.12406
$677$	−13280.2	−0.753914	−0.376957	−	0.926231i	$-0.623030\pi$
$677$	−13280.2	−0.753914	−0.376957	+	0.926231i	$0.623030\pi$
$678$	37788.3	2.14049
$679$	−2041.03	−0.115357
$680$	5270.01	0.297199
$681$	−6359.99	−0.357879
$682$	0	0
$683$	6856.80	0.384141	0.192070	−	0.981381i	$-0.438480\pi$
$683$	6856.80	0.384141	0.192070	+	0.981381i	$0.438480\pi$
$684$	−15980.8	−0.893334
$685$	−6381.18	−0.355930
$686$	−31467.3	−1.75135
$687$	−7041.36	−0.391040
$688$	57664.5	3.19541
$689$	−16607.6	−0.918287
$690$	−16779.3	−0.925763
$691$	−28374.6	−1.56211	−0.781057	−	0.624460i	$-0.785319\pi$
$691$	−28374.6	−1.56211	−0.781057	+	0.624460i	$0.785319\pi$
$692$	−10570.9	−0.580702
$693$	0	0
$694$	34793.7	1.90310
$695$	−13516.1	−0.737688
$696$	10612.4	0.577963
$697$	−3873.54	−0.210503
$698$	−1840.84	−0.0998237
$699$	−1126.50	−0.0609556
$700$	5078.65	0.274221
$701$	−667.753	−0.0359781	−0.0179891	−	0.999838i	$-0.505726\pi$
$701$	−667.753	−0.0359781	−0.0179891	+	0.999838i	$0.505726\pi$
$702$	9089.26	0.488678
$703$	23271.0	1.24848
$704$	0	0
$705$	6065.45	0.324026
$706$	9067.10	0.483349
$707$	−14455.4	−0.768956
$708$	50373.6	2.67395
$709$	23667.8	1.25368	0.626842	−	0.779147i	$-0.284347\pi$
$709$	23667.8	1.25368	0.626842	+	0.779147i	$0.284347\pi$
$710$	−2432.02	−0.128552
$711$	9194.37	0.484973
$712$	−27061.0	−1.42438
$713$	30126.9	1.58241
$714$	2793.22	0.146405
$715$	0	0
$716$	56909.5	2.97040
$717$	−4284.45	−0.223160
$718$	−30915.2	−1.60689
$719$	11835.5	0.613896	0.306948	−	0.951726i	$-0.400692\pi$
$719$	11835.5	0.613896	0.306948	+	0.951726i	$0.400692\pi$
$720$	−7468.53	−0.386577
$721$	−13888.6	−0.717390
$722$	6729.01	0.346853
$723$	572.458	0.0294467
$724$	8566.38	0.439733
$725$	−1438.65	−0.0736969
$726$	0	0
$727$	15633.2	0.797530	0.398765	−	0.917053i	$-0.369439\pi$
$727$	15633.2	0.797530	0.398765	+	0.917053i	$0.369439\pi$
$728$	−40592.1	−2.06654
$729$	729.000	0.0370370
$730$	−6367.08	−0.322817
$731$	−5957.31	−0.301422
$732$	−12028.5	−0.607356
$733$	14870.7	0.749335	0.374668	−	0.927159i	$-0.377757\pi$
$733$	14870.7	0.749335	0.374668	+	0.927159i	$0.377757\pi$
$734$	−28245.2	−1.42037
$735$	−3547.24	−0.178016
$736$	−81100.6	−4.06169
$737$	0	0
$738$	10697.8	0.533593
$739$	−20850.3	−1.03788	−0.518939	−	0.854812i	$-0.673673\pi$
$739$	−20850.3	−1.03788	−0.518939	+	0.854812i	$0.673673\pi$
$740$	25387.9	1.26119
$741$	−17315.4	−0.858432
$742$	14095.2	0.697372
$743$	29254.7	1.44448	0.722242	−	0.691641i	$-0.243112\pi$
$743$	29254.7	1.44448	0.722242	+	0.691641i	$0.243112\pi$
$744$	26132.4	1.28771
$745$	−12004.9	−0.590372
$746$	−54687.2	−2.68397
$747$	−6357.73	−0.311402
$748$	0	0
$749$	9176.95	0.447688
$750$	1973.06	0.0960613
$751$	10936.8	0.531411	0.265705	−	0.964054i	$-0.414395\pi$
$751$	10936.8	0.531411	0.265705	+	0.964054i	$0.414395\pi$
$752$	67111.2	3.25438
$753$	18884.4	0.913926
$754$	19372.3	0.935672
$755$	−12586.5	−0.606714
$756$	−5484.94	−0.263870
$757$	−8476.34	−0.406972	−0.203486	−	0.979078i	$-0.565227\pi$
$757$	−8476.34	−0.406972	−0.203486	+	0.979078i	$0.565227\pi$
$758$	57449.6	2.75286
$759$	0	0
$760$	27727.0	1.32337
$761$	−913.964	−0.0435364	−0.0217682	−	0.999763i	$-0.506930\pi$
$761$	−913.964	−0.0435364	−0.0217682	+	0.999763i	$0.506930\pi$
$762$	32603.9	1.55002
$763$	−12965.8	−0.615193
$764$	−74393.0	−3.52283
$765$	771.573	0.0364657
$766$	−61367.9	−2.89466
$767$	54580.6	2.56948
$768$	8055.53	0.378488
$769$	−32215.2	−1.51067	−0.755337	−	0.655337i	$-0.772527\pi$
$769$	−32215.2	−1.51067	−0.755337	+	0.655337i	$0.772527\pi$
$770$	0	0
$771$	−13378.3	−0.624914
$772$	−73850.2	−3.44291
$773$	72.6900	0.00338225	0.00169112	−	0.999999i	$-0.499462\pi$
$773$	72.6900	0.00338225	0.00169112	+	0.999999i	$0.499462\pi$
$774$	16452.7	0.764058
$775$	−3542.59	−0.164198
$776$	−12156.7	−0.562373
$777$	7987.09	0.368771
$778$	−30658.7	−1.41281
$779$	−20379.8	−0.937332
$780$	−18890.6	−0.867169
$781$	0	0
$782$	19179.9	0.877075
$783$	1553.75	0.0709148
$784$	−39248.5	−1.78792
$785$	−6957.11	−0.316319
$786$	12399.1	0.562675
$787$	−487.318	−0.0220724	−0.0110362	−	0.999939i	$-0.503513\pi$
$787$	−487.318	−0.0220724	−0.0110362	+	0.999939i	$0.503513\pi$
$788$	77173.1	3.48880
$789$	−13249.9	−0.597859
$790$	−26875.7	−1.21037
$791$	−24707.9	−1.11064
$792$	0	0
$793$	−13033.0	−0.583627
$794$	38761.4	1.73248
$795$	3893.52	0.173697
$796$	−11752.6	−0.523317
$797$	31379.9	1.39465	0.697324	−	0.716756i	$-0.254374\pi$
$797$	31379.9	1.39465	0.697324	+	0.716756i	$0.254374\pi$
$798$	14695.9	0.651917
$799$	−6933.25	−0.306985
$800$	9536.54	0.421460
$801$	−3961.96	−0.174768
$802$	76839.8	3.38318
$803$	0	0
$804$	−15759.8	−0.691302
$805$	10971.2	0.480351
$806$	47703.0	2.08470
$807$	−5744.59	−0.250581
$808$	−86099.0	−3.74870
$809$	−1824.26	−0.0792802	−0.0396401	−	0.999214i	$-0.512621\pi$
$809$	−1824.26	−0.0792802	−0.0396401	+	0.999214i	$0.512621\pi$
$810$	−2130.91	−0.0924351
$811$	4364.52	0.188976	0.0944878	−	0.995526i	$-0.469879\pi$
$811$	4364.52	0.188976	0.0944878	+	0.995526i	$0.469879\pi$
$812$	−11690.3	−0.505232
$813$	18265.0	0.787924
$814$	0	0
$815$	13602.6	0.584637
$816$	8537.06	0.366246
$817$	−31343.1	−1.34218
$818$	−66196.1	−2.82945
$819$	−5943.02	−0.253561
$820$	−22233.7	−0.946872
$821$	−3306.16	−0.140543	−0.0702714	−	0.997528i	$-0.522387\pi$
$821$	−3306.16	−0.140543	−0.0702714	+	0.997528i	$0.522387\pi$
$822$	−20144.7	−0.854779
$823$	−19183.7	−0.812519	−0.406259	−	0.913758i	$-0.633167\pi$
$823$	−19183.7	−0.812519	−0.406259	+	0.913758i	$0.633167\pi$
$824$	−82722.8	−3.49731
$825$	0	0
$826$	−46323.6	−1.95134
$827$	−20646.4	−0.868131	−0.434066	−	0.900881i	$-0.642921\pi$
$827$	−20646.4	−0.868131	−0.434066	+	0.900881i	$0.642921\pi$
$828$	−37663.0	−1.58077
$829$	5345.44	0.223950	0.111975	−	0.993711i	$-0.464282\pi$
$829$	5345.44	0.223950	0.111975	+	0.993711i	$0.464282\pi$
$830$	18584.0	0.777181
$831$	−2496.96	−0.104234
$832$	−43464.0	−1.81111
$833$	4054.75	0.168654
$834$	−42668.8	−1.77158
$835$	14751.3	0.611363
$836$	0	0
$837$	3826.00	0.158000
$838$	−19867.5	−0.818985
$839$	29284.9	1.20504	0.602519	−	0.798105i	$-0.294164\pi$
$839$	29284.9	1.20504	0.602519	+	0.798105i	$0.294164\pi$
$840$	9516.50	0.390893
$841$	−21077.4	−0.864219
$842$	66733.8	2.73135
$843$	−7635.97	−0.311977
$844$	86302.6	3.51974
$845$	−9483.26	−0.386076
$846$	19148.0	0.778159
$847$	0	0
$848$	43079.9	1.74454
$849$	−17735.1	−0.716923
$850$	−2255.35	−0.0910092
$851$	54844.2	2.20921
$852$	−5458.94	−0.219507
$853$	8070.62	0.323954	0.161977	−	0.986795i	$-0.448213\pi$
$853$	8070.62	0.323954	0.161977	+	0.986795i	$0.448213\pi$
$854$	11061.4	0.443222
$855$	4059.47	0.162375
$856$	54659.5	2.18250
$857$	11344.2	0.452169	0.226085	−	0.974108i	$-0.427407\pi$
$857$	11344.2	0.452169	0.226085	+	0.974108i	$0.427407\pi$
$858$	0	0
$859$	25470.6	1.01170	0.505848	−	0.862623i	$-0.331180\pi$
$859$	25470.6	1.01170	0.505848	+	0.862623i	$0.331180\pi$
$860$	−34194.4	−1.35584
$861$	−6994.78	−0.276866
$862$	−74463.6	−2.94228
$863$	−14558.4	−0.574243	−0.287122	−	0.957894i	$-0.592698\pi$
$863$	−14558.4	−0.574243	−0.287122	+	0.957894i	$0.592698\pi$
$864$	−10299.5	−0.405550
$865$	2685.24	0.105550
$866$	−57614.6	−2.26077
$867$	13857.0	0.542802
$868$	−28786.5	−1.12567
$869$	0	0
$870$	−4541.68	−0.176986
$871$	−17076.0	−0.664294
$872$	−77226.3	−2.99910
$873$	−1779.85	−0.0690019
$874$	100911.	3.90546
$875$	−1290.09	−0.0498434
$876$	−14291.6	−0.551220
$877$	−185.528	−0.00714350	−0.00357175	−	0.999994i	$-0.501137\pi$
$877$	−185.528	−0.00714350	−0.00357175	+	0.999994i	$0.501137\pi$
$878$	59039.1	2.26933
$879$	−20613.1	−0.790969
$880$	0	0
$881$	18950.1	0.724681	0.362340	−	0.932046i	$-0.381978\pi$
$881$	18950.1	0.724681	0.362340	+	0.932046i	$0.381978\pi$
$882$	−11198.3	−0.427513
$883$	21258.3	0.810189	0.405095	−	0.914275i	$-0.367239\pi$
$883$	21258.3	0.810189	0.405095	+	0.914275i	$0.367239\pi$
$884$	21593.3	0.821563
$885$	−12796.0	−0.486026
$886$	50758.2	1.92467
$887$	−35707.4	−1.35168	−0.675839	−	0.737049i	$-0.736219\pi$
$887$	−35707.4	−1.35168	−0.675839	+	0.737049i	$0.736219\pi$
$888$	47572.5	1.79778
$889$	−21318.1	−0.804259
$890$	11581.0	0.436177
$891$	0	0
$892$	−45921.8	−1.72374
$893$	−36477.8	−1.36695
$894$	−37898.4	−1.41780
$895$	−14456.3	−0.539910
$896$	5393.06	0.201082
$897$	−40808.4	−1.51901
$898$	34105.2	1.26738
$899$	8154.49	0.302522
$900$	4428.75	0.164028
$901$	−4450.58	−0.164562
$902$	0	0
$903$	−10757.6	−0.396447
$904$	−147165.	−5.41440
$905$	−2176.05	−0.0799274
$906$	−39734.2	−1.45704
$907$	−6542.52	−0.239516	−0.119758	−	0.992803i	$-0.538212\pi$
$907$	−6542.52	−0.239516	−0.119758	+	0.992803i	$0.538212\pi$
$908$	41728.6	1.52512
$909$	−12605.6	−0.459958
$910$	17371.8	0.632823
$911$	31171.2	1.13364	0.566821	−	0.823841i	$-0.308173\pi$
$911$	31171.2	1.13364	0.566821	+	0.823841i	$0.308173\pi$
$912$	44915.9	1.63083
$913$	0	0
$914$	−59559.2	−2.15541
$915$	3055.49	0.110395
$916$	46199.2	1.66644
$917$	−8107.19	−0.291955
$918$	2435.78	0.0875737
$919$	−12031.9	−0.431877	−0.215938	−	0.976407i	$-0.569281\pi$
$919$	−12031.9	−0.431877	−0.215938	+	0.976407i	$0.569281\pi$
$920$	65346.1	2.34174
$921$	600.537	0.0214857
$922$	44232.0	1.57994
$923$	−5914.85	−0.210931
$924$	0	0
$925$	−6449.08	−0.229237
$926$	51344.7	1.82213
$927$	−12111.3	−0.429113
$928$	−21951.6	−0.776506
$929$	−12546.2	−0.443085	−0.221542	−	0.975151i	$-0.571109\pi$
$929$	−12546.2	−0.443085	−0.221542	+	0.975151i	$0.571109\pi$
$930$	−11183.6	−0.394328
$931$	21333.2	0.750986
$932$	7391.07	0.259767
$933$	−17204.8	−0.603708
$934$	−86229.1	−3.02088
$935$	0	0
$936$	−35397.7	−1.23612
$937$	−17909.8	−0.624427	−0.312214	−	0.950012i	$-0.601070\pi$
$937$	−17909.8	−0.624427	−0.312214	+	0.950012i	$0.601070\pi$
$938$	14492.7	0.504483
$939$	9232.08	0.320849
$940$	−39796.2	−1.38086
$941$	−829.893	−0.0287500	−0.0143750	−	0.999897i	$-0.504576\pi$
$941$	−829.893	−0.0287500	−0.0143750	+	0.999897i	$0.504576\pi$
$942$	−21962.9	−0.759650
$943$	−48030.4	−1.65863
$944$	−141581.	−4.88144
$945$	1393.30	0.0479618
$946$	0	0
$947$	−17654.9	−0.605814	−0.302907	−	0.953020i	$-0.597957\pi$
$947$	−17654.9	−0.605814	−0.302907	+	0.953020i	$0.597957\pi$
$948$	−60325.4	−2.06675
$949$	−15485.2	−0.529685
$950$	−11866.0	−0.405248
$951$	−6427.14	−0.219153
$952$	−10878.0	−0.370335
$953$	−30736.6	−1.04476	−0.522380	−	0.852713i	$-0.674956\pi$
$953$	−30736.6	−1.04476	−0.522380	+	0.852713i	$0.674956\pi$
$954$	12291.5	0.417139
$955$	18897.4	0.640321
$956$	28110.8	0.951012
$957$	0	0
$958$	−72761.5	−2.45388
$959$	13171.7	0.443519
$960$	10189.8	0.342578
$961$	−9711.08	−0.325974
$962$	86840.6	2.91045
$963$	8002.61	0.267789
$964$	−3755.97	−0.125489
$965$	18759.6	0.625794
$966$	34634.8	1.15358
$967$	−23645.2	−0.786327	−0.393163	−	0.919469i	$-0.628619\pi$
$967$	−23645.2	−0.786327	−0.393163	+	0.919469i	$0.628619\pi$
$968$	0	0
$969$	−4640.26	−0.153836
$970$	5202.59	0.172211
$971$	27402.0	0.905635	0.452818	−	0.891603i	$-0.350419\pi$
$971$	27402.0	0.905635	0.452818	+	0.891603i	$0.350419\pi$
$972$	−4783.05	−0.157836
$973$	27899.1	0.919222
$974$	69790.6	2.29593
$975$	4798.62	0.157619
$976$	33807.5	1.10876
$977$	−49118.7	−1.60844	−0.804220	−	0.594332i	$-0.797416\pi$
$977$	−49118.7	−1.60844	−0.804220	+	0.594332i	$0.797416\pi$
$978$	42942.1	1.40403
$979$	0	0
$980$	23273.9	0.758630
$981$	−11306.6	−0.367983
$982$	39294.0	1.27691
$983$	52630.5	1.70768	0.853842	−	0.520533i	$-0.174267\pi$
$983$	52630.5	1.70768	0.853842	+	0.520533i	$0.174267\pi$
$984$	−41662.1	−1.34973
$985$	−19603.7	−0.634136
$986$	5191.47	0.167677
$987$	−12520.0	−0.403764
$988$	113609.	3.65827
$989$	−73868.4	−2.37500
$990$	0	0
$991$	45472.7	1.45761	0.728803	−	0.684724i	$-0.240077\pi$
$991$	45472.7	1.45761	0.728803	+	0.684724i	$0.240077\pi$
$992$	−54054.5	−1.73007
$993$	4854.68	0.155145
$994$	5020.04	0.160187
$995$	2985.42	0.0951197
$996$	41713.8	1.32706
$997$	−55068.1	−1.74927	−0.874637	−	0.484779i	$-0.838900\pi$
$997$	−55068.1	−1.74927	−0.874637	+	0.484779i	$0.838900\pi$
$998$	−27763.0	−0.880585
$999$	6965.01	0.220584

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1815.4.a.s.1.3		3
11.10	odd	2		165.4.a.d.1.1	✓	3
33.32	even	2		495.4.a.l.1.3		3
55.32	even	4		825.4.c.l.199.1		6
55.43	even	4		825.4.c.l.199.6		6
55.54	odd	2		825.4.a.s.1.3		3
165.164	even	2		2475.4.a.s.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.a.d.1.1	✓	3	11.10	odd	2
495.4.a.l.1.3		3	33.32	even	2
825.4.a.s.1.3		3	55.54	odd	2
825.4.c.l.199.1		6	55.32	even	4
825.4.c.l.199.6		6	55.43	even	4
1815.4.a.s.1.3		3	1.1	even	1	trivial
2475.4.a.s.1.1		3	165.164	even	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:	\( N \)	\(=\)	\( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1815.a (trivial)

\(f(q)\)	\(=\)	\(q+5.26150 q^{2} -3.00000 q^{3} +19.6833 q^{4} -5.00000 q^{5} -15.7845 q^{6} +10.3207 q^{7} +61.4719 q^{8} +9.00000 q^{9} -26.3075 q^{10} -59.0500 q^{12} -63.9817 q^{13} +54.3024 q^{14} +15.0000 q^{15} +165.967 q^{16} -17.1461 q^{17} +47.3535 q^{18} -90.2104 q^{19} -98.4167 q^{20} -30.9621 q^{21} -212.605 q^{23} -184.416 q^{24} +25.0000 q^{25} -336.639 q^{26} -27.0000 q^{27} +203.146 q^{28} -57.5461 q^{29} +78.9224 q^{30} -141.704 q^{31} +381.462 q^{32} -90.2140 q^{34} -51.6035 q^{35} +177.150 q^{36} -257.963 q^{37} -474.642 q^{38} +191.945 q^{39} -307.359 q^{40} +225.914 q^{41} -162.907 q^{42} +347.445 q^{43} -45.0000 q^{45} -1118.62 q^{46} +404.364 q^{47} -497.902 q^{48} -236.483 q^{49} +131.537 q^{50} +51.4382 q^{51} -1259.37 q^{52} +259.568 q^{53} -142.060 q^{54} +634.433 q^{56} +270.631 q^{57} -302.779 q^{58} -853.067 q^{59} +295.250 q^{60} +203.699 q^{61} -745.573 q^{62} +92.8864 q^{63} +679.320 q^{64} +319.908 q^{65} +266.890 q^{67} -337.492 q^{68} +637.814 q^{69} -271.512 q^{70} +92.4460 q^{71} +553.247 q^{72} +242.026 q^{73} -1357.27 q^{74} -75.0000 q^{75} -1775.64 q^{76} +1009.92 q^{78} +1021.60 q^{79} -829.837 q^{80} +81.0000 q^{81} +1188.65 q^{82} -706.415 q^{83} -609.438 q^{84} +85.7303 q^{85} +1828.08 q^{86} +172.638 q^{87} -440.218 q^{89} -236.767 q^{90} -660.336 q^{91} -4184.77 q^{92} +425.111 q^{93} +2127.56 q^{94} +451.052 q^{95} -1144.38 q^{96} -197.761 q^{97} -1244.25 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 4 q^{2} - 9 q^{3} + 22 q^{4} - 15 q^{5} - 12 q^{6} + 4 q^{7} + 48 q^{8} + 27 q^{9} - 20 q^{10} - 66 q^{12} - 56 q^{14} + 45 q^{15} + 50 q^{16} + 218 q^{17} + 36 q^{18} - 146 q^{19} - 110 q^{20} - 12 q^{21}+ \cdots - 572 q^{98}+O(q^{100}) \) 3 * q + 4 * q^2 - 9 * q^3 + 22 * q^4 - 15 * q^5 - 12 * q^6 + 4 * q^7 + 48 * q^8 + 27 * q^9 - 20 * q^10 - 66 * q^12 - 56 * q^14 + 45 * q^15 + 50 * q^16 + 218 * q^17 + 36 * q^18 - 146 * q^19 - 110 * q^20 - 12 * q^21 - 200 * q^23 - 144 * q^24 + 75 * q^25 - 508 * q^26 - 81 * q^27 + 340 * q^28 - 68 * q^29 + 60 * q^30 - 68 * q^31 + 688 * q^32 - 176 * q^34 - 20 * q^35 + 198 * q^36 - 390 * q^37 - 316 * q^38 - 240 * q^40 + 196 * q^41 + 168 * q^42 + 524 * q^43 - 135 * q^45 - 1160 * q^46 - 60 * q^47 - 150 * q^48 - 157 * q^49 + 100 * q^50 - 654 * q^51 - 1020 * q^52 - 158 * q^53 - 108 * q^54 + 1368 * q^56 + 438 * q^57 + 1092 * q^58 - 1044 * q^59 + 330 * q^60 - 642 * q^61 - 88 * q^62 + 36 * q^63 + 1166 * q^64 - 236 * q^67 - 144 * q^68 + 600 * q^69 + 280 * q^70 - 544 * q^71 + 432 * q^72 - 900 * q^73 - 1536 * q^74 - 225 * q^75 - 1996 * q^76 + 1524 * q^78 + 1586 * q^79 - 250 * q^80 + 243 * q^81 + 380 * q^82 + 1582 * q^83 - 1020 * q^84 - 1090 * q^85 + 3568 * q^86 + 204 * q^87 - 2122 * q^89 - 180 * q^90 - 8 * q^91 - 4128 * q^92 + 204 * q^93 + 2152 * q^94 + 730 * q^95 - 2064 * q^96 + 618 * q^97 - 572 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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