LMFDB - Embedded newform 2366.4.a.bi.1.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2366,4,Mod(1,2366)] 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("2366.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2366, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2366 = 2 \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2366.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,-30,-5,60,26] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$139.598519074$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 5 x^{14} - 272 x^{13} + 1126 x^{12} + 29249 x^{11} - 95770 x^{10} - 1588299 x^{9} + \cdots - 13037372712 $ x^15 - 5x^14 - 272x^13 + 1126x^12 + 29249x^11 - 95770x^10 - 1588299x^9 + 3793496x^8 + 46836812x^7 - 66634081x^6 - 752083205x^5 + 321098572x^4 + 5907626255x^3 + 2610199536x^2 - 15008729652x - 13037372712
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 7\cdot 13^{4} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-2.45772$ of defining polynomial
Character	$\chi$	$=$	2366.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} +2.45772 q^{3} +4.00000 q^{4} +21.4663 q^{5} -4.91545 q^{6} +7.00000 q^{7} -8.00000 q^{8} -20.9596 q^{9} -42.9327 q^{10} -42.1847 q^{11} +9.83090 q^{12} -14.0000 q^{14} +52.7583 q^{15} +16.0000 q^{16} +45.1263 q^{17} +41.9192 q^{18} -16.3522 q^{19} +85.8653 q^{20} +17.2041 q^{21} +84.3694 q^{22} +98.1670 q^{23} -19.6618 q^{24} +335.803 q^{25} -117.871 q^{27} +28.0000 q^{28} -43.2528 q^{29} -105.517 q^{30} +92.9766 q^{31} -32.0000 q^{32} -103.678 q^{33} -90.2526 q^{34} +150.264 q^{35} -83.8384 q^{36} -77.6622 q^{37} +32.7043 q^{38} -171.731 q^{40} -453.446 q^{41} -34.4081 q^{42} -43.8052 q^{43} -168.739 q^{44} -449.925 q^{45} -196.334 q^{46} +487.388 q^{47} +39.3236 q^{48} +49.0000 q^{49} -671.606 q^{50} +110.908 q^{51} -428.977 q^{53} +235.743 q^{54} -905.551 q^{55} -56.0000 q^{56} -40.1891 q^{57} +86.5057 q^{58} +514.832 q^{59} +211.033 q^{60} +623.552 q^{61} -185.953 q^{62} -146.717 q^{63} +64.0000 q^{64} +207.357 q^{66} +775.921 q^{67} +180.505 q^{68} +241.267 q^{69} -300.529 q^{70} +37.4567 q^{71} +167.677 q^{72} +426.048 q^{73} +155.324 q^{74} +825.312 q^{75} -65.4086 q^{76} -295.293 q^{77} +656.199 q^{79} +343.461 q^{80} +276.213 q^{81} +906.891 q^{82} +1430.27 q^{83} +68.8163 q^{84} +968.696 q^{85} +87.6103 q^{86} -106.304 q^{87} +337.478 q^{88} -527.502 q^{89} +899.851 q^{90} +392.668 q^{92} +228.511 q^{93} -974.776 q^{94} -351.021 q^{95} -78.6472 q^{96} -531.618 q^{97} -98.0000 q^{98} +884.175 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 30 q^{2} - 5 q^{3} + 60 q^{4} + 26 q^{5} + 10 q^{6} + 105 q^{7} - 120 q^{8} + 164 q^{9} - 52 q^{10} + 7 q^{11} - 20 q^{12} - 210 q^{14} + 86 q^{15} + 240 q^{16} + 16 q^{17} - 328 q^{18} + 511 q^{19}+ \cdots + 2790 q^{99}+O(q^{100}) $ 15 * q - 30 * q^2 - 5 * q^3 + 60 * q^4 + 26 * q^5 + 10 * q^6 + 105 * q^7 - 120 * q^8 + 164 * q^9 - 52 * q^10 + 7 * q^11 - 20 * q^12 - 210 * q^14 + 86 * q^15 + 240 * q^16 + 16 * q^17 - 328 * q^18 + 511 * q^19 + 104 * q^20 - 35 * q^21 - 14 * q^22 - 268 * q^23 + 40 * q^24 + 639 * q^25 - 557 * q^27 + 420 * q^28 - 283 * q^29 - 172 * q^30 + 127 * q^31 - 480 * q^32 + 628 * q^33 - 32 * q^34 + 182 * q^35 + 656 * q^36 + 462 * q^37 - 1022 * q^38 - 208 * q^40 + 1082 * q^41 + 70 * q^42 + 919 * q^43 + 28 * q^44 + 2118 * q^45 + 536 * q^46 + 595 * q^47 - 80 * q^48 + 735 * q^49 - 1278 * q^50 - 1009 * q^51 + 346 * q^53 + 1114 * q^54 - 823 * q^55 - 840 * q^56 - 1638 * q^57 + 566 * q^58 + 1162 * q^59 + 344 * q^60 + 81 * q^61 - 254 * q^62 + 1148 * q^63 + 960 * q^64 - 1256 * q^66 + 3649 * q^67 + 64 * q^68 - 1720 * q^69 - 364 * q^70 + 1152 * q^71 - 1312 * q^72 + 3983 * q^73 - 924 * q^74 + 1496 * q^75 + 2044 * q^76 + 49 * q^77 + 1333 * q^79 + 416 * q^80 + 1123 * q^81 - 2164 * q^82 + 93 * q^83 - 140 * q^84 + 628 * q^85 - 1838 * q^86 - 746 * q^87 - 56 * q^88 + 3061 * q^89 - 4236 * q^90 - 1072 * q^92 + 566 * q^93 - 1190 * q^94 - 3578 * q^95 + 160 * q^96 + 3853 * q^97 - 1470 * q^98 + 2790 * 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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	2.45772	0.472989	0.236495	−	0.971633i	$-0.424001\pi$
$3$	2.45772	0.472989	0.236495	+	0.971633i	$0.424001\pi$
$4$	4.00000	0.500000
$5$	21.4663	1.92001	0.960003	−	0.279988i	$-0.0903306\pi$
$5$	21.4663	1.92001	0.960003	+	0.279988i	$0.0903306\pi$
$6$	−4.91545	−0.334454
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−8.00000	−0.353553
$9$	−20.9596	−0.776281
$10$	−42.9327	−1.35765
$11$	−42.1847	−1.15629	−0.578144	−	0.815935i	$-0.696223\pi$
$11$	−42.1847	−1.15629	−0.578144	+	0.815935i	$0.696223\pi$
$12$	9.83090	0.236495
$13$	0	0
$13$	0	0
$14$	−14.0000	−0.267261
$15$	52.7583	0.908143
$16$	16.0000	0.250000
$17$	45.1263	0.643808	0.321904	−	0.946772i	$-0.395677\pi$
$17$	45.1263	0.643808	0.321904	+	0.946772i	$0.395677\pi$
$18$	41.9192	0.548914
$19$	−16.3522	−0.197444	−0.0987222	−	0.995115i	$-0.531476\pi$
$19$	−16.3522	−0.197444	−0.0987222	+	0.995115i	$0.531476\pi$
$20$	85.8653	0.960003
$21$	17.2041	0.178773
$22$	84.3694	0.817619
$23$	98.1670	0.889967	0.444983	−	0.895539i	$-0.353210\pi$
$23$	98.1670	0.889967	0.444983	+	0.895539i	$0.353210\pi$
$24$	−19.6618	−0.167227
$25$	335.803	2.68643
$26$	0	0
$27$	−117.871	−0.840162
$28$	28.0000	0.188982
$29$	−43.2528	−0.276960	−0.138480	−	0.990365i	$-0.544222\pi$
$29$	−43.2528	−0.276960	−0.138480	+	0.990365i	$0.544222\pi$
$30$	−105.517	−0.642154
$31$	92.9766	0.538680	0.269340	−	0.963045i	$-0.413194\pi$
$31$	92.9766	0.538680	0.269340	+	0.963045i	$0.413194\pi$
$32$	−32.0000	−0.176777
$33$	−103.678	−0.546912
$34$	−90.2526	−0.455241
$35$	150.264	0.725694
$36$	−83.8384	−0.388141
$37$	−77.6622	−0.345070	−0.172535	−	0.985003i	$-0.555196\pi$
$37$	−77.6622	−0.345070	−0.172535	+	0.985003i	$0.555196\pi$
$38$	32.7043	0.139614
$39$	0	0
$40$	−171.731	−0.678825
$41$	−453.446	−1.72723	−0.863614	−	0.504154i	$-0.831804\pi$
$41$	−453.446	−1.72723	−0.863614	+	0.504154i	$0.831804\pi$
$42$	−34.4081	−0.126412
$43$	−43.8052	−0.155354	−0.0776770	−	0.996979i	$-0.524750\pi$
$43$	−43.8052	−0.155354	−0.0776770	+	0.996979i	$0.524750\pi$
$44$	−168.739	−0.578144
$45$	−449.925	−1.49047
$46$	−196.334	−0.629301
$47$	487.388	1.51261	0.756307	−	0.654217i	$-0.227002\pi$
$47$	487.388	1.51261	0.756307	+	0.654217i	$0.227002\pi$
$48$	39.3236	0.118247
$49$	49.0000	0.142857
$50$	−671.606	−1.89959
$51$	110.908	0.304514
$52$	0	0
$53$	−428.977	−1.11178	−0.555892	−	0.831254i	$-0.687623\pi$
$53$	−428.977	−1.11178	−0.555892	+	0.831254i	$0.687623\pi$
$54$	235.743	0.594084
$55$	−905.551	−2.22008
$56$	−56.0000	−0.133631
$57$	−40.1891	−0.0933890
$58$	86.5057	0.195841
$59$	514.832	1.13603	0.568013	−	0.823020i	$-0.307712\pi$
$59$	514.832	1.13603	0.568013	+	0.823020i	$0.307712\pi$
$60$	211.033	0.454071
$61$	623.552	1.30881	0.654406	−	0.756143i	$-0.272919\pi$
$61$	623.552	1.30881	0.654406	+	0.756143i	$0.272919\pi$
$62$	−185.953	−0.380904
$63$	−146.717	−0.293407
$64$	64.0000	0.125000
$65$	0	0
$66$	207.357	0.386725
$67$	775.921	1.41483	0.707417	−	0.706797i	$-0.249860\pi$
$67$	775.921	1.41483	0.707417	+	0.706797i	$0.249860\pi$
$68$	180.505	0.321904
$69$	241.267	0.420945
$70$	−300.529	−0.513143
$71$	37.4567	0.0626097	0.0313048	−	0.999510i	$-0.490034\pi$
$71$	37.4567	0.0626097	0.0313048	+	0.999510i	$0.490034\pi$
$72$	167.677	0.274457
$73$	426.048	0.683085	0.341542	−	0.939866i	$-0.389051\pi$
$73$	426.048	0.683085	0.341542	+	0.939866i	$0.389051\pi$
$74$	155.324	0.244001
$75$	825.312	1.27065
$76$	−65.4086	−0.0987222
$77$	−295.293	−0.437036
$78$	0	0
$79$	656.199	0.934533	0.467267	−	0.884116i	$-0.345239\pi$
$79$	656.199	0.934533	0.467267	+	0.884116i	$0.345239\pi$
$80$	343.461	0.480002
$81$	276.213	0.378894
$82$	906.891	1.22133
$83$	1430.27	1.89148	0.945739	−	0.324928i	$-0.105340\pi$
$83$	1430.27	1.89148	0.945739	+	0.324928i	$0.105340\pi$
$84$	68.8163	0.0893866
$85$	968.696	1.23612
$86$	87.6103	0.109852
$87$	−106.304	−0.130999
$88$	337.478	0.408810
$89$	−527.502	−0.628260	−0.314130	−	0.949380i	$-0.601713\pi$
$89$	−527.502	−0.628260	−0.314130	+	0.949380i	$0.601713\pi$
$90$	899.851	1.05392
$91$	0	0
$92$	392.668	0.444983
$93$	228.511	0.254790
$94$	−974.776	−1.06958
$95$	−351.021	−0.379094
$96$	−78.6472	−0.0836135
$97$	−531.618	−0.556470	−0.278235	−	0.960513i	$-0.589749\pi$
$97$	−531.618	−0.556470	−0.278235	+	0.960513i	$0.589749\pi$
$98$	−98.0000	−0.101015
$99$	884.175	0.897605
$100$	1343.21	1.34321
$101$	−778.131	−0.766603	−0.383302	−	0.923623i	$-0.625213\pi$
$101$	−778.131	−0.766603	−0.383302	+	0.923623i	$0.625213\pi$
$102$	−221.816	−0.215324
$103$	−2050.80	−1.96186	−0.980930	−	0.194362i	$-0.937736\pi$
$103$	−2050.80	−1.96186	−0.980930	+	0.194362i	$0.937736\pi$
$104$	0	0
$105$	369.308	0.343246
$106$	857.955	0.786150
$107$	471.187	0.425713	0.212857	−	0.977083i	$-0.431723\pi$
$107$	471.187	0.425713	0.212857	+	0.977083i	$0.431723\pi$
$108$	−471.486	−0.420081
$109$	2152.86	1.89180	0.945900	−	0.324457i	$-0.105182\pi$
$109$	2152.86	1.89180	0.945900	+	0.324457i	$0.105182\pi$
$110$	1811.10	1.56983
$111$	−190.872	−0.163214
$112$	112.000	0.0944911
$113$	2054.43	1.71030	0.855152	−	0.518377i	$-0.173464\pi$
$113$	2054.43	1.71030	0.855152	+	0.518377i	$0.173464\pi$
$114$	80.3782	0.0660360
$115$	2107.28	1.70874
$116$	−173.011	−0.138480
$117$	0	0
$118$	−1029.66	−0.803291
$119$	315.884	0.243337
$120$	−422.066	−0.321077
$121$	448.551	0.337003
$122$	−1247.10	−0.925471
$123$	−1114.44	−0.816960
$124$	371.906	0.269340
$125$	4525.17	3.23795
$126$	293.434	0.207470
$127$	2191.29	1.53107	0.765533	−	0.643397i	$-0.222475\pi$
$127$	2191.29	1.53107	0.765533	+	0.643397i	$0.222475\pi$
$128$	−128.000	−0.0883883
$129$	−107.661	−0.0734808
$130$	0	0
$131$	−494.073	−0.329522	−0.164761	−	0.986334i	$-0.552685\pi$
$131$	−494.073	−0.329522	−0.164761	+	0.986334i	$0.552685\pi$
$132$	−414.714	−0.273456
$133$	−114.465	−0.0746269
$134$	−1551.84	−1.00044
$135$	−2530.27	−1.61312
$136$	−361.011	−0.227621
$137$	890.621	0.555408	0.277704	−	0.960667i	$-0.410427\pi$
$137$	890.621	0.555408	0.277704	+	0.960667i	$0.410427\pi$
$138$	−482.535	−0.297653
$139$	−2416.47	−1.47455	−0.737275	−	0.675592i	$-0.763888\pi$
$139$	−2416.47	−1.47455	−0.737275	+	0.675592i	$0.763888\pi$
$140$	601.057	0.362847
$141$	1197.87	0.715450
$142$	−74.9133	−0.0442717
$143$	0	0
$144$	−335.353	−0.194070
$145$	−928.480	−0.531766
$146$	−852.097	−0.483014
$147$	120.428	0.0675699
$148$	−310.649	−0.172535
$149$	1041.77	0.572787	0.286393	−	0.958112i	$-0.407544\pi$
$149$	1041.77	0.572787	0.286393	+	0.958112i	$0.407544\pi$
$150$	−1650.62	−0.898486
$151$	−135.121	−0.0728214	−0.0364107	−	0.999337i	$-0.511592\pi$
$151$	−135.121	−0.0728214	−0.0364107	+	0.999337i	$0.511592\pi$
$152$	130.817	0.0698071
$153$	−945.829	−0.499776
$154$	590.586	0.309031
$155$	1995.87	1.03427
$156$	0	0
$157$	2020.86	1.02728	0.513638	−	0.858007i	$-0.328297\pi$
$157$	2020.86	1.02728	0.513638	+	0.858007i	$0.328297\pi$
$158$	−1312.40	−0.660815
$159$	−1054.31	−0.525862
$160$	−686.922	−0.339412
$161$	687.169	0.336376
$162$	−552.427	−0.267918
$163$	127.655	0.0613418	0.0306709	−	0.999530i	$-0.490236\pi$
$163$	127.655	0.0613418	0.0306709	+	0.999530i	$0.490236\pi$
$164$	−1813.78	−0.863614
$165$	−2225.59	−1.05007
$166$	−2860.54	−1.33748
$167$	245.236	0.113634	0.0568171	−	0.998385i	$-0.481905\pi$
$167$	245.236	0.113634	0.0568171	+	0.998385i	$0.481905\pi$
$168$	−137.633	−0.0632058
$169$	0	0
$170$	−1937.39	−0.874066
$171$	342.735	0.153272
$172$	−175.221	−0.0776770
$173$	1019.24	0.447926	0.223963	−	0.974598i	$-0.428100\pi$
$173$	1019.24	0.447926	0.223963	+	0.974598i	$0.428100\pi$
$174$	212.607	0.0926305
$175$	2350.62	1.01537
$176$	−674.956	−0.289072
$177$	1265.32	0.537328
$178$	1055.00	0.444247
$179$	−2542.27	−1.06155	−0.530777	−	0.847512i	$-0.678100\pi$
$179$	−2542.27	−1.06155	−0.530777	+	0.847512i	$0.678100\pi$
$180$	−1799.70	−0.745233
$181$	−377.122	−0.154869	−0.0774344	−	0.996997i	$-0.524673\pi$
$181$	−377.122	−0.154869	−0.0774344	+	0.996997i	$0.524673\pi$
$182$	0	0
$183$	1532.52	0.619054
$184$	−785.336	−0.314651
$185$	−1667.12	−0.662537
$186$	−457.021	−0.180164
$187$	−1903.64	−0.744428
$188$	1949.55	0.756307
$189$	−825.100	−0.317551
$190$	702.042	0.268060
$191$	−3337.17	−1.26424	−0.632118	−	0.774872i	$-0.717814\pi$
$191$	−3337.17	−1.26424	−0.632118	+	0.774872i	$0.717814\pi$
$192$	157.294	0.0591237
$193$	4056.26	1.51283	0.756414	−	0.654093i	$-0.226949\pi$
$193$	4056.26	1.51283	0.756414	+	0.654093i	$0.226949\pi$
$194$	1063.24	0.393484
$195$	0	0
$196$	196.000	0.0714286
$197$	−292.748	−0.105875	−0.0529377	−	0.998598i	$-0.516858\pi$
$197$	−292.748	−0.105875	−0.0529377	+	0.998598i	$0.516858\pi$
$198$	−1768.35	−0.634703
$199$	−1145.83	−0.408170	−0.204085	−	0.978953i	$-0.565422\pi$
$199$	−1145.83	−0.408170	−0.204085	+	0.978953i	$0.565422\pi$
$200$	−2686.43	−0.949795
$201$	1907.00	0.669201
$202$	1556.26	0.542070
$203$	−302.770	−0.104681
$204$	443.632	0.152257
$205$	−9733.81	−3.31629
$206$	4101.60	1.38724
$207$	−2057.54	−0.690864
$208$	0	0
$209$	689.811	0.228303
$210$	−738.616	−0.242711
$211$	−1319.06	−0.430368	−0.215184	−	0.976573i	$-0.569035\pi$
$211$	−1319.06	−0.430368	−0.215184	+	0.976573i	$0.569035\pi$
$212$	−1715.91	−0.555892
$213$	92.0581	0.0296137
$214$	−942.373	−0.301025
$215$	−940.336	−0.298281
$216$	942.972	0.297042
$217$	650.836	0.203602
$218$	−4305.71	−1.33771
$219$	1047.11	0.323092
$220$	−3622.20	−1.11004
$221$	0	0
$222$	381.745	0.115410
$223$	−2989.54	−0.897732	−0.448866	−	0.893599i	$-0.648172\pi$
$223$	−2989.54	−0.897732	−0.448866	+	0.893599i	$0.648172\pi$
$224$	−224.000	−0.0668153
$225$	−7038.30	−2.08542
$226$	−4108.86	−1.20937
$227$	−501.158	−0.146533	−0.0732666	−	0.997312i	$-0.523342\pi$
$227$	−501.158	−0.146533	−0.0732666	+	0.997312i	$0.523342\pi$
$228$	−160.756	−0.0466945
$229$	2554.56	0.737161	0.368581	−	0.929596i	$-0.379844\pi$
$229$	2554.56	0.737161	0.368581	+	0.929596i	$0.379844\pi$
$230$	−4214.57	−1.20826
$231$	−725.749	−0.206713
$232$	346.023	0.0979203
$233$	5111.50	1.43719	0.718596	−	0.695428i	$-0.244785\pi$
$233$	5111.50	1.43719	0.718596	+	0.695428i	$0.244785\pi$
$234$	0	0
$235$	10462.4	2.90423
$236$	2059.33	0.568013
$237$	1612.76	0.442024
$238$	−631.769	−0.172065
$239$	2036.83	0.551261	0.275631	−	0.961264i	$-0.411113\pi$
$239$	2036.83	0.551261	0.275631	+	0.961264i	$0.411113\pi$
$240$	844.133	0.227036
$241$	7017.12	1.87557	0.937785	−	0.347218i	$-0.112874\pi$
$241$	7017.12	1.87557	0.937785	+	0.347218i	$0.112874\pi$
$242$	−897.102	−0.238297
$243$	3861.39	1.01937
$244$	2494.21	0.654406
$245$	1051.85	0.274287
$246$	2228.89	0.577678
$247$	0	0
$248$	−743.812	−0.190452
$249$	3515.21	0.894649
$250$	−9050.34	−2.28958
$251$	−3861.51	−0.971061	−0.485530	−	0.874220i	$-0.661374\pi$
$251$	−3861.51	−0.971061	−0.485530	+	0.874220i	$0.661374\pi$
$252$	−586.869	−0.146703
$253$	−4141.15	−1.02906
$254$	−4382.58	−1.08263
$255$	2380.79	0.584670
$256$	256.000	0.0625000
$257$	5753.13	1.39638	0.698191	−	0.715911i	$-0.253989\pi$
$257$	5753.13	1.39638	0.698191	+	0.715911i	$0.253989\pi$
$258$	215.322	0.0519588
$259$	−543.636	−0.130424
$260$	0	0
$261$	906.562	0.214999
$262$	988.145	0.233007
$263$	−2550.19	−0.597914	−0.298957	−	0.954267i	$-0.596639\pi$
$263$	−2550.19	−0.597914	−0.298957	+	0.954267i	$0.596639\pi$
$264$	829.427	0.193363
$265$	−9208.57	−2.13463
$266$	228.930	0.0527692
$267$	−1296.45	−0.297160
$268$	3103.68	0.707417
$269$	2427.64	0.550244	0.275122	−	0.961409i	$-0.411282\pi$
$269$	2427.64	0.550244	0.275122	+	0.961409i	$0.411282\pi$
$270$	5060.53	1.14065
$271$	4742.34	1.06301	0.531506	−	0.847054i	$-0.321626\pi$
$271$	4742.34	1.06301	0.531506	+	0.847054i	$0.321626\pi$
$272$	722.021	0.160952
$273$	0	0
$274$	−1781.24	−0.392733
$275$	−14165.8	−3.10628
$276$	965.069	0.210472
$277$	−4289.92	−0.930527	−0.465264	−	0.885172i	$-0.654040\pi$
$277$	−4289.92	−0.930527	−0.465264	+	0.885172i	$0.654040\pi$
$278$	4832.95	1.04266
$279$	−1948.75	−0.418167
$280$	−1202.11	−0.256572
$281$	−8976.53	−1.90568	−0.952838	−	0.303481i	$-0.901851\pi$
$281$	−8976.53	−1.90568	−0.952838	+	0.303481i	$0.901851\pi$
$282$	−2395.73	−0.505900
$283$	1172.18	0.246216	0.123108	−	0.992393i	$-0.460714\pi$
$283$	1172.18	0.246216	0.123108	+	0.992393i	$0.460714\pi$
$284$	149.827	0.0313048
$285$	−862.712	−0.179308
$286$	0	0
$287$	−3174.12	−0.652831
$288$	670.707	0.137228
$289$	−2876.62	−0.585511
$290$	1856.96	0.376015
$291$	−1306.57	−0.263205
$292$	1704.19	0.341542
$293$	7998.94	1.59489	0.797445	−	0.603391i	$-0.206184\pi$
$293$	7998.94	1.59489	0.797445	+	0.603391i	$0.206184\pi$
$294$	−240.857	−0.0477791
$295$	11051.6	2.18118
$296$	621.298	0.122001
$297$	4972.37	0.971469
$298$	−2083.54	−0.405021
$299$	0	0
$300$	3301.25	0.635325
$301$	−306.636	−0.0587183
$302$	270.243	0.0514925
$303$	−1912.43	−0.362595
$304$	−261.635	−0.0493611
$305$	13385.4	2.51293
$306$	1891.66	0.353395
$307$	452.273	0.0840801	0.0420401	−	0.999116i	$-0.486614\pi$
$307$	452.273	0.0840801	0.0420401	+	0.999116i	$0.486614\pi$
$308$	−1181.17	−0.218518
$309$	−5040.31	−0.927938
$310$	−3991.73	−0.731339
$311$	−5183.89	−0.945182	−0.472591	−	0.881282i	$-0.656681\pi$
$311$	−5183.89	−0.945182	−0.472591	+	0.881282i	$0.656681\pi$
$312$	0	0
$313$	647.884	0.116999	0.0584993	−	0.998287i	$-0.481368\pi$
$313$	647.884	0.116999	0.0584993	+	0.998287i	$0.481368\pi$
$314$	−4041.73	−0.726394
$315$	−3149.48	−0.563343
$316$	2624.80	0.467267
$317$	−3184.41	−0.564209	−0.282104	−	0.959384i	$-0.591032\pi$
$317$	−3184.41	−0.564209	−0.282104	+	0.959384i	$0.591032\pi$
$318$	2108.62	0.371841
$319$	1824.61	0.320246
$320$	1373.84	0.240001
$321$	1158.05	0.201358
$322$	−1374.34	−0.237854
$323$	−737.913	−0.127116
$324$	1104.85	0.189447
$325$	0	0
$326$	−255.310	−0.0433752
$327$	5291.13	0.894801
$328$	3627.57	0.610667
$329$	3411.72	0.571714
$330$	4451.19	0.742515
$331$	−5002.48	−0.830699	−0.415349	−	0.909662i	$-0.636341\pi$
$331$	−5002.48	−0.830699	−0.415349	+	0.909662i	$0.636341\pi$
$332$	5721.08	0.945739
$333$	1627.77	0.267871
$334$	−490.472	−0.0803515
$335$	16656.2	2.71649
$336$	275.265	0.0446933
$337$	10137.5	1.63865	0.819326	−	0.573328i	$-0.194348\pi$
$337$	10137.5	1.63865	0.819326	+	0.573328i	$0.194348\pi$
$338$	0	0
$339$	5049.22	0.808956
$340$	3874.79	0.618058
$341$	−3922.19	−0.622870
$342$	−685.469	−0.108380
$343$	343.000	0.0539949
$344$	350.441	0.0549259
$345$	5179.12	0.808216
$346$	−2038.48	−0.316732
$347$	4889.20	0.756386	0.378193	−	0.925727i	$-0.376546\pi$
$347$	4889.20	0.756386	0.378193	+	0.925727i	$0.376546\pi$
$348$	−425.214	−0.0654996
$349$	−8318.72	−1.27591	−0.637953	−	0.770075i	$-0.720219\pi$
$349$	−8318.72	−1.27591	−0.637953	+	0.770075i	$0.720219\pi$
$350$	−4701.25	−0.717978
$351$	0	0
$352$	1349.91	0.204405
$353$	8531.18	1.28631	0.643157	−	0.765734i	$-0.277624\pi$
$353$	8531.18	1.28631	0.643157	+	0.765734i	$0.277624\pi$
$354$	−2530.63	−0.379948
$355$	804.057	0.120211
$356$	−2110.01	−0.314130
$357$	776.356	0.115096
$358$	5084.54	0.750632
$359$	2441.03	0.358865	0.179433	−	0.983770i	$-0.442574\pi$
$359$	2441.03	0.358865	0.179433	+	0.983770i	$0.442574\pi$
$360$	3599.40	0.526959
$361$	−6591.61	−0.961016
$362$	754.244	0.109509
$363$	1102.41	0.159399
$364$	0	0
$365$	9145.69	1.31153
$366$	−3065.04	−0.437738
$367$	−6458.62	−0.918630	−0.459315	−	0.888274i	$-0.651905\pi$
$367$	−6458.62	−0.918630	−0.459315	+	0.888274i	$0.651905\pi$
$368$	1570.67	0.222492
$369$	9504.04	1.34081
$370$	3334.25	0.468484
$371$	−3002.84	−0.420215
$372$	914.043	0.127395
$373$	−2554.93	−0.354662	−0.177331	−	0.984151i	$-0.556746\pi$
$373$	−2554.93	−0.354662	−0.177331	+	0.984151i	$0.556746\pi$
$374$	3807.28	0.526390
$375$	11121.6	1.53151
$376$	−3899.11	−0.534790
$377$	0	0
$378$	1650.20	0.224543
$379$	−12454.6	−1.68799	−0.843995	−	0.536350i	$-0.819803\pi$
$379$	−12454.6	−1.68799	−0.843995	+	0.536350i	$0.819803\pi$
$380$	−1404.08	−0.189547
$381$	5385.58	0.724178
$382$	6674.34	0.893950
$383$	7870.26	1.05000	0.525002	−	0.851101i	$-0.324064\pi$
$383$	7870.26	1.05000	0.525002	+	0.851101i	$0.324064\pi$
$384$	−314.589	−0.0418067
$385$	−6338.86	−0.839112
$386$	−8112.52	−1.06973
$387$	918.138	0.120598
$388$	−2126.47	−0.278235
$389$	−2670.80	−0.348111	−0.174055	−	0.984736i	$-0.555687\pi$
$389$	−2670.80	−0.348111	−0.174055	+	0.984736i	$0.555687\pi$
$390$	0	0
$391$	4429.91	0.572968
$392$	−392.000	−0.0505076
$393$	−1214.29	−0.155860
$394$	585.497	0.0748652
$395$	14086.2	1.79431
$396$	3536.70	0.448802
$397$	−82.9973	−0.0104925	−0.00524624	−	0.999986i	$-0.501670\pi$
$397$	−82.9973	−0.0104925	−0.00524624	+	0.999986i	$0.501670\pi$
$398$	2291.66	0.288619
$399$	−281.324	−0.0352977
$400$	5372.85	0.671606
$401$	−11691.1	−1.45592	−0.727962	−	0.685617i	$-0.759532\pi$
$401$	−11691.1	−1.45592	−0.727962	+	0.685617i	$0.759532\pi$
$402$	−3814.00	−0.473196
$403$	0	0
$404$	−3112.52	−0.383302
$405$	5929.29	0.727478
$406$	605.540	0.0740208
$407$	3276.16	0.399000
$408$	−887.264	−0.107662
$409$	−5383.51	−0.650850	−0.325425	−	0.945568i	$-0.605507\pi$
$409$	−5383.51	−0.650850	−0.325425	+	0.945568i	$0.605507\pi$
$410$	19467.6	2.34497
$411$	2188.90	0.262702
$412$	−8203.21	−0.980930
$413$	3603.83	0.429377
$414$	4115.08	0.488515
$415$	30702.7	3.63165
$416$	0	0
$417$	−5939.02	−0.697447
$418$	−1379.62	−0.161434
$419$	−13286.9	−1.54918	−0.774591	−	0.632462i	$-0.782044\pi$
$419$	−13286.9	−1.54918	−0.774591	+	0.632462i	$0.782044\pi$
$420$	1477.23	0.171623
$421$	−663.627	−0.0768247	−0.0384123	−	0.999262i	$-0.512230\pi$
$421$	−663.627	−0.0768247	−0.0384123	+	0.999262i	$0.512230\pi$
$422$	2638.12	0.304316
$423$	−10215.5	−1.17421
$424$	3431.82	0.393075
$425$	15153.6	1.72954
$426$	−184.116	−0.0209401
$427$	4364.86	0.494685
$428$	1884.75	0.212857
$429$	0	0
$430$	1880.67	0.210916
$431$	−9952.09	−1.11224	−0.556120	−	0.831102i	$-0.687711\pi$
$431$	−9952.09	−1.11224	−0.556120	+	0.831102i	$0.687711\pi$
$432$	−1885.94	−0.210040
$433$	2926.23	0.324771	0.162385	−	0.986727i	$-0.448081\pi$
$433$	2926.23	0.324771	0.162385	+	0.986727i	$0.448081\pi$
$434$	−1301.67	−0.143968
$435$	−2281.95	−0.251520
$436$	8611.43	0.945900
$437$	−1605.24	−0.175719
$438$	−2094.22	−0.228460
$439$	6927.49	0.753146	0.376573	−	0.926387i	$-0.377102\pi$
$439$	6927.49	0.753146	0.376573	+	0.926387i	$0.377102\pi$
$440$	7244.41	0.784917
$441$	−1027.02	−0.110897
$442$	0	0
$443$	−15063.1	−1.61551	−0.807753	−	0.589521i	$-0.799316\pi$
$443$	−15063.1	−1.61551	−0.807753	+	0.589521i	$0.799316\pi$
$444$	−763.489	−0.0816072
$445$	−11323.5	−1.20626
$446$	5979.07	0.634792
$447$	2560.39	0.270922
$448$	448.000	0.0472456
$449$	11134.3	1.17029	0.585144	−	0.810929i	$-0.301038\pi$
$449$	11134.3	1.17029	0.585144	+	0.810929i	$0.301038\pi$
$450$	14076.6	1.47462
$451$	19128.5	1.99717
$452$	8217.72	0.855152
$453$	−332.091	−0.0344437
$454$	1002.32	0.103615
$455$	0	0
$456$	321.513	0.0330180
$457$	−18478.7	−1.89146	−0.945730	−	0.324953i	$-0.894651\pi$
$457$	−18478.7	−1.89146	−0.945730	+	0.324953i	$0.894651\pi$
$458$	−5109.11	−0.521252
$459$	−5319.10	−0.540903
$460$	8429.14	0.854371
$461$	−7470.13	−0.754704	−0.377352	−	0.926070i	$-0.623165\pi$
$461$	−7470.13	−0.754704	−0.377352	+	0.926070i	$0.623165\pi$
$462$	1451.50	0.146168
$463$	14194.1	1.42474	0.712369	−	0.701805i	$-0.247622\pi$
$463$	14194.1	1.42474	0.712369	+	0.701805i	$0.247622\pi$
$464$	−692.046	−0.0692401
$465$	4905.29	0.489198
$466$	−10223.0	−1.01625
$467$	9488.66	0.940220	0.470110	−	0.882608i	$-0.344214\pi$
$467$	9488.66	0.940220	0.470110	+	0.882608i	$0.344214\pi$
$468$	0	0
$469$	5431.45	0.534757
$470$	−20924.9	−2.05360
$471$	4966.72	0.485891
$472$	−4118.66	−0.401646
$473$	1847.91	0.179634
$474$	−3225.51	−0.312558
$475$	−5491.11	−0.530420
$476$	1263.54	0.121668
$477$	8991.19	0.863057
$478$	−4073.66	−0.389801
$479$	5262.47	0.501980	0.250990	−	0.967990i	$-0.419244\pi$
$479$	5262.47	0.501980	0.250990	+	0.967990i	$0.419244\pi$
$480$	−1688.27	−0.160538
$481$	0	0
$482$	−14034.2	−1.32623
$483$	1688.87	0.159102
$484$	1794.20	0.168501
$485$	−11411.9	−1.06843
$486$	−7722.77	−0.720807
$487$	−8449.29	−0.786188	−0.393094	−	0.919498i	$-0.628595\pi$
$487$	−8449.29	−0.786188	−0.393094	+	0.919498i	$0.628595\pi$
$488$	−4988.41	−0.462735
$489$	313.741	0.0290140
$490$	−2103.70	−0.193950
$491$	7096.25	0.652239	0.326120	−	0.945329i	$-0.394259\pi$
$491$	7096.25	0.652239	0.326120	+	0.945329i	$0.394259\pi$
$492$	−4457.78	−0.408480
$493$	−1951.84	−0.178309
$494$	0	0
$495$	18980.0	1.72341
$496$	1487.62	0.134670
$497$	262.197	0.0236642
$498$	−7030.42	−0.632612
$499$	−21196.7	−1.90159	−0.950795	−	0.309820i	$-0.899731\pi$
$499$	−21196.7	−1.90159	−0.950795	+	0.309820i	$0.899731\pi$
$500$	18100.7	1.61897
$501$	602.722	0.0537478
$502$	7723.02	0.686644
$503$	−1101.11	−0.0976061	−0.0488031	−	0.998808i	$-0.515541\pi$
$503$	−1101.11	−0.0976061	−0.0488031	+	0.998808i	$0.515541\pi$
$504$	1173.74	0.103735
$505$	−16703.6	−1.47188
$506$	8282.29	0.727654
$507$	0	0
$508$	8765.15	0.765533
$509$	16000.9	1.39338	0.696688	−	0.717374i	$-0.254656\pi$
$509$	16000.9	1.39338	0.696688	+	0.717374i	$0.254656\pi$
$510$	−4761.58	−0.413424
$511$	2982.34	0.258182
$512$	−512.000	−0.0441942
$513$	1927.45	0.165885
$514$	−11506.3	−0.987391
$515$	−44023.2	−3.76678
$516$	−430.644	−0.0367404
$517$	−20560.3	−1.74902
$518$	1087.27	0.0922238
$519$	2505.01	0.211864
$520$	0	0
$521$	−12150.4	−1.02172	−0.510861	−	0.859664i	$-0.670673\pi$
$521$	−12150.4	−1.02172	−0.510861	+	0.859664i	$0.670673\pi$
$522$	−1813.12	−0.152027
$523$	7856.36	0.656854	0.328427	−	0.944529i	$-0.393481\pi$
$523$	7856.36	0.656854	0.328427	+	0.944529i	$0.393481\pi$
$524$	−1976.29	−0.164761
$525$	5777.18	0.480261
$526$	5100.37	0.422789
$527$	4195.69	0.346807
$528$	−1658.85	−0.136728
$529$	−2530.24	−0.207960
$530$	18417.1	1.50941
$531$	−10790.7	−0.881875
$532$	−457.860	−0.0373135
$533$	0	0
$534$	2592.91	0.210124
$535$	10114.6	0.817373
$536$	−6207.37	−0.500219
$537$	−6248.19	−0.502103
$538$	−4855.28	−0.389081
$539$	−2067.05	−0.165184
$540$	−10121.1	−0.806558
$541$	13035.7	1.03595	0.517976	−	0.855395i	$-0.326686\pi$
$541$	13035.7	1.03595	0.517976	+	0.855395i	$0.326686\pi$
$542$	−9484.67	−0.751663
$543$	−926.862	−0.0732513
$544$	−1444.04	−0.113810
$545$	46213.9	3.63227
$546$	0	0
$547$	−21225.0	−1.65908	−0.829540	−	0.558448i	$-0.811397\pi$
$547$	−21225.0	−1.65908	−0.829540	+	0.558448i	$0.811397\pi$
$548$	3562.48	0.277704
$549$	−13069.4	−1.01601
$550$	28331.5	2.19647
$551$	707.277	0.0546843
$552$	−1930.14	−0.148826
$553$	4593.39	0.353220
$554$	8579.83	0.657982
$555$	−4097.33	−0.313373
$556$	−9665.89	−0.737275
$557$	19654.1	1.49510	0.747552	−	0.664203i	$-0.231229\pi$
$557$	19654.1	1.49510	0.747552	+	0.664203i	$0.231229\pi$
$558$	3897.50	0.295689
$559$	0	0
$560$	2404.23	0.181424
$561$	−4678.63	−0.352106
$562$	17953.1	1.34752
$563$	−12386.7	−0.927240	−0.463620	−	0.886034i	$-0.653450\pi$
$563$	−12386.7	−0.927240	−0.463620	+	0.886034i	$0.653450\pi$
$564$	4791.46	0.357725
$565$	44101.0	3.28380
$566$	−2344.37	−0.174101
$567$	1933.49	0.143208
$568$	−299.653	−0.0221359
$569$	−957.883	−0.0705739	−0.0352869	−	0.999377i	$-0.511235\pi$
$569$	−957.883	−0.0705739	−0.0352869	+	0.999377i	$0.511235\pi$
$570$	1725.42	0.126790
$571$	2533.56	0.185685	0.0928424	−	0.995681i	$-0.470405\pi$
$571$	2533.56	0.185685	0.0928424	+	0.995681i	$0.470405\pi$
$572$	0	0
$573$	−8201.84	−0.597970
$574$	6348.24	0.461621
$575$	32964.8	2.39083
$576$	−1341.41	−0.0970351
$577$	20857.3	1.50485	0.752426	−	0.658677i	$-0.228884\pi$
$577$	20857.3	1.50485	0.752426	+	0.658677i	$0.228884\pi$
$578$	5753.23	0.414019
$579$	9969.17	0.715552
$580$	−3713.92	−0.265883
$581$	10011.9	0.714911
$582$	2613.14	0.186114
$583$	18096.3	1.28554
$584$	−3408.39	−0.241507
$585$	0	0
$586$	−15997.9	−1.12776
$587$	−11074.0	−0.778656	−0.389328	−	0.921099i	$-0.627293\pi$
$587$	−11074.0	−0.778656	−0.389328	+	0.921099i	$0.627293\pi$
$588$	481.714	0.0337849
$589$	−1520.37	−0.106359
$590$	−22103.1	−1.54232
$591$	−719.495	−0.0500779
$592$	−1242.60	−0.0862675
$593$	−16008.7	−1.10860	−0.554298	−	0.832318i	$-0.687013\pi$
$593$	−16008.7	−1.10860	−0.554298	+	0.832318i	$0.687013\pi$
$594$	−9944.75	−0.686933
$595$	6780.87	0.467208
$596$	4167.09	0.286393
$597$	−2816.13	−0.193060
$598$	0	0
$599$	−11262.6	−0.768242	−0.384121	−	0.923283i	$-0.625495\pi$
$599$	−11262.6	−0.768242	−0.384121	+	0.923283i	$0.625495\pi$
$600$	−6602.49	−0.449243
$601$	−4326.39	−0.293639	−0.146820	−	0.989163i	$-0.546904\pi$
$601$	−4326.39	−0.293639	−0.146820	+	0.989163i	$0.546904\pi$
$602$	613.272	0.0415201
$603$	−16263.0	−1.09831
$604$	−540.486	−0.0364107
$605$	9628.74	0.647048
$606$	3824.86	0.256393
$607$	−19610.4	−1.31130	−0.655652	−	0.755063i	$-0.727606\pi$
$607$	−19610.4	−1.31130	−0.655652	+	0.755063i	$0.727606\pi$
$608$	523.269	0.0349036
$609$	−744.125	−0.0495131
$610$	−26770.7	−1.77691
$611$	0	0
$612$	−3783.32	−0.249888
$613$	24741.6	1.63019	0.815093	−	0.579331i	$-0.196686\pi$
$613$	24741.6	1.63019	0.815093	+	0.579331i	$0.196686\pi$
$614$	−904.546	−0.0594536
$615$	−23923.0	−1.56857
$616$	2362.34	0.154516
$617$	21446.0	1.39933	0.699664	−	0.714472i	$-0.253333\pi$
$617$	21446.0	1.39933	0.699664	+	0.714472i	$0.253333\pi$
$618$	10080.6	0.656152
$619$	883.388	0.0573609	0.0286804	−	0.999589i	$-0.490869\pi$
$619$	883.388	0.0573609	0.0286804	+	0.999589i	$0.490869\pi$
$620$	7983.46	0.517135
$621$	−11571.1	−0.747716
$622$	10367.8	0.668344
$623$	−3692.52	−0.237460
$624$	0	0
$625$	55163.4	3.53046
$626$	−1295.77	−0.0827305
$627$	1695.37	0.107985
$628$	8083.45	0.513638
$629$	−3504.61	−0.222159
$630$	6298.96	0.398344
$631$	17489.2	1.10339	0.551693	−	0.834048i	$-0.313982\pi$
$631$	17489.2	1.10339	0.551693	+	0.834048i	$0.313982\pi$
$632$	−5249.59	−0.330407
$633$	−3241.88	−0.203560
$634$	6368.82	0.398956
$635$	47038.9	2.93966
$636$	−4217.23	−0.262931
$637$	0	0
$638$	−3649.22	−0.226448
$639$	−785.076	−0.0486027
$640$	−2747.69	−0.169706
$641$	16602.4	1.02302	0.511511	−	0.859277i	$-0.329086\pi$
$641$	16602.4	1.02302	0.511511	+	0.859277i	$0.329086\pi$
$642$	−2316.09	−0.142382
$643$	922.835	0.0565989	0.0282994	−	0.999599i	$-0.490991\pi$
$643$	922.835	0.0565989	0.0282994	+	0.999599i	$0.490991\pi$
$644$	2748.68	0.168188
$645$	−2311.09	−0.141084
$646$	1475.83	0.0898848
$647$	15242.5	0.926191	0.463095	−	0.886308i	$-0.346739\pi$
$647$	15242.5	0.926191	0.463095	+	0.886308i	$0.346739\pi$
$648$	−2209.71	−0.133959
$649$	−21718.1	−1.31357
$650$	0	0
$651$	1599.58	0.0963015
$652$	510.620	0.0306709
$653$	23977.3	1.43691	0.718456	−	0.695572i	$-0.244849\pi$
$653$	23977.3	1.43691	0.718456	+	0.695572i	$0.244849\pi$
$654$	−10582.3	−0.632720
$655$	−10605.9	−0.632684
$656$	−7255.13	−0.431807
$657$	−8929.80	−0.530266
$658$	−6823.43	−0.404263
$659$	−7234.38	−0.427635	−0.213817	−	0.976874i	$-0.568590\pi$
$659$	−7234.38	−0.427635	−0.213817	+	0.976874i	$0.568590\pi$
$660$	−8902.38	−0.525037
$661$	21957.7	1.29207	0.646033	−	0.763310i	$-0.276427\pi$
$661$	21957.7	1.29207	0.646033	+	0.763310i	$0.276427\pi$
$662$	10005.0	0.587393
$663$	0	0
$664$	−11442.2	−0.668738
$665$	−2457.15	−0.143284
$666$	−3255.54	−0.189414
$667$	−4246.00	−0.246485
$668$	980.943	0.0568171
$669$	−7347.46	−0.424617
$670$	−33312.3	−1.92085
$671$	−26304.3	−1.51337
$672$	−550.530	−0.0316029
$673$	3433.58	0.196664	0.0983319	−	0.995154i	$-0.468649\pi$
$673$	3433.58	0.196664	0.0983319	+	0.995154i	$0.468649\pi$
$674$	−20275.0	−1.15870
$675$	−39581.6	−2.25703
$676$	0	0
$677$	19403.6	1.10154	0.550770	−	0.834657i	$-0.314334\pi$
$677$	19403.6	1.10154	0.550770	+	0.834657i	$0.314334\pi$
$678$	−10098.4	−0.572018
$679$	−3721.33	−0.210326
$680$	−7749.57	−0.437033
$681$	−1231.71	−0.0693086
$682$	7844.38	0.440435
$683$	−13088.9	−0.733286	−0.366643	−	0.930362i	$-0.619493\pi$
$683$	−13088.9	−0.733286	−0.366643	+	0.930362i	$0.619493\pi$
$684$	1370.94	0.0766362
$685$	19118.4	1.06639
$686$	−686.000	−0.0381802
$687$	6278.40	0.348669
$688$	−700.882	−0.0388385
$689$	0	0
$690$	−10358.2	−0.571495
$691$	29138.8	1.60418	0.802092	−	0.597200i	$-0.203720\pi$
$691$	29138.8	1.60418	0.802092	+	0.597200i	$0.203720\pi$
$692$	4076.95	0.223963
$693$	6189.22	0.339263
$694$	−9778.40	−0.534846
$695$	−51872.8	−2.83115
$696$	850.429	0.0463152
$697$	−20462.3	−1.11200
$698$	16637.4	0.902202
$699$	12562.7	0.679776
$700$	9402.49	0.507687
$701$	−231.696	−0.0124836	−0.00624182	−	0.999981i	$-0.501987\pi$
$701$	−231.696	−0.0124836	−0.00624182	+	0.999981i	$0.501987\pi$
$702$	0	0
$703$	1269.94	0.0681321
$704$	−2699.82	−0.144536
$705$	25713.8	1.37367
$706$	−17062.4	−0.909562
$707$	−5446.92	−0.289749
$708$	5061.26	0.268664
$709$	−5259.46	−0.278594	−0.139297	−	0.990251i	$-0.544484\pi$
$709$	−5259.46	−0.278594	−0.139297	+	0.990251i	$0.544484\pi$
$710$	−1608.11	−0.0850020
$711$	−13753.7	−0.725461
$712$	4220.02	0.222123
$713$	9127.23	0.479407
$714$	−1552.71	−0.0813849
$715$	0	0
$716$	−10169.1	−0.530777
$717$	5005.96	0.260741
$718$	−4882.06	−0.253756
$719$	−29001.7	−1.50428	−0.752141	−	0.659002i	$-0.770979\pi$
$719$	−29001.7	−1.50428	−0.752141	+	0.659002i	$0.770979\pi$
$720$	−7198.81	−0.372616
$721$	−14355.6	−0.741513
$722$	13183.2	0.679541
$723$	17246.1	0.887124
$724$	−1508.49	−0.0774344
$725$	−14524.4	−0.744034
$726$	−2204.83	−0.112712
$727$	25032.2	1.27702	0.638510	−	0.769613i	$-0.279551\pi$
$727$	25032.2	1.27702	0.638510	+	0.769613i	$0.279551\pi$
$728$	0	0
$729$	2032.46	0.103259
$730$	−18291.4	−0.927390
$731$	−1976.77	−0.100018
$732$	6130.07	0.309527
$733$	13976.2	0.704261	0.352130	−	0.935951i	$-0.385457\pi$
$733$	13976.2	0.704261	0.352130	+	0.935951i	$0.385457\pi$
$734$	12917.2	0.649569
$735$	2585.16	0.129735
$736$	−3141.34	−0.157325
$737$	−32732.0	−1.63596
$738$	−19008.1	−0.948099
$739$	14790.5	0.736234	0.368117	−	0.929779i	$-0.380003\pi$
$739$	14790.5	0.736234	0.368117	+	0.929779i	$0.380003\pi$
$740$	−6668.49	−0.331268
$741$	0	0
$742$	6005.68	0.297137
$743$	2705.94	0.133609	0.0668044	−	0.997766i	$-0.478720\pi$
$743$	2705.94	0.133609	0.0668044	+	0.997766i	$0.478720\pi$
$744$	−1828.09	−0.0900818
$745$	22363.0	1.09975
$746$	5109.85	0.250784
$747$	−29977.9	−1.46832
$748$	−7614.57	−0.372214
$749$	3298.31	0.160905
$750$	−22243.2	−1.08294
$751$	−12594.2	−0.611944	−0.305972	−	0.952041i	$-0.598981\pi$
$751$	−12594.2	−0.611944	−0.305972	+	0.952041i	$0.598981\pi$
$752$	7798.21	0.378154
$753$	−9490.52	−0.459301
$754$	0	0
$755$	−2900.56	−0.139818
$756$	−3300.40	−0.158776
$757$	−9734.02	−0.467357	−0.233678	−	0.972314i	$-0.575076\pi$
$757$	−9734.02	−0.467357	−0.233678	+	0.972314i	$0.575076\pi$
$758$	24909.1	1.19359
$759$	−10177.8	−0.486733
$760$	2808.17	0.134030
$761$	−16203.7	−0.771857	−0.385928	−	0.922529i	$-0.626119\pi$
$761$	−16203.7	−0.771857	−0.385928	+	0.922529i	$0.626119\pi$
$762$	−10771.2	−0.512071
$763$	15070.0	0.715033
$764$	−13348.7	−0.632118
$765$	−20303.5	−0.959574
$766$	−15740.5	−0.742465
$767$	0	0
$768$	629.177	0.0295618
$769$	39079.8	1.83258	0.916289	−	0.400517i	$-0.131170\pi$
$769$	39079.8	1.83258	0.916289	+	0.400517i	$0.131170\pi$
$770$	12677.7	0.593342
$771$	14139.6	0.660474
$772$	16225.0	0.756414
$773$	−10840.4	−0.504400	−0.252200	−	0.967675i	$-0.581154\pi$
$773$	−10840.4	−0.504400	−0.252200	+	0.967675i	$0.581154\pi$
$774$	−1836.28	−0.0852759
$775$	31221.8	1.44712
$776$	4252.94	0.196742
$777$	−1336.11	−0.0616892
$778$	5341.61	0.246151
$779$	7414.82	0.341031
$780$	0	0
$781$	−1580.10	−0.0723949
$782$	−8859.83	−0.405149
$783$	5098.28	0.232692
$784$	784.000	0.0357143
$785$	43380.5	1.97238
$786$	2428.59	0.110210
$787$	19016.4	0.861323	0.430662	−	0.902513i	$-0.358280\pi$
$787$	19016.4	0.861323	0.430662	+	0.902513i	$0.358280\pi$
$788$	−1170.99	−0.0529377
$789$	−6267.66	−0.282807
$790$	−28172.4	−1.26877
$791$	14381.0	0.646434
$792$	−7073.40	−0.317351
$793$	0	0
$794$	165.995	0.00741930
$795$	−22632.1	−1.00966
$796$	−4583.32	−0.204085
$797$	14170.2	0.629779	0.314889	−	0.949128i	$-0.398033\pi$
$797$	14170.2	0.629779	0.314889	+	0.949128i	$0.398033\pi$
$798$	562.647	0.0249593
$799$	21994.0	0.973834
$800$	−10745.7	−0.474897
$801$	11056.2	0.487706
$802$	23382.2	1.02949
$803$	−17972.7	−0.789843
$804$	7628.00	0.334600
$805$	14751.0	0.645844
$806$	0	0
$807$	5966.46	0.260260
$808$	6225.05	0.271035
$809$	31986.6	1.39010	0.695049	−	0.718962i	$-0.255383\pi$
$809$	31986.6	1.39010	0.695049	+	0.718962i	$0.255383\pi$
$810$	−11858.6	−0.514405
$811$	−16119.2	−0.697929	−0.348964	−	0.937136i	$-0.613467\pi$
$811$	−16119.2	−0.697929	−0.348964	+	0.937136i	$0.613467\pi$
$812$	−1211.08	−0.0523406
$813$	11655.4	0.502793
$814$	−6552.32	−0.282136
$815$	2740.29	0.117777
$816$	1774.53	0.0761286
$817$	716.309	0.0306738
$818$	10767.0	0.460220
$819$	0	0
$820$	−38935.3	−1.65814
$821$	−17781.8	−0.755894	−0.377947	−	0.925827i	$-0.623370\pi$
$821$	−17781.8	−0.755894	−0.377947	+	0.925827i	$0.623370\pi$
$822$	−4377.80	−0.185758
$823$	−45323.4	−1.91965	−0.959826	−	0.280596i	$-0.909468\pi$
$823$	−45323.4	−1.91965	−0.959826	+	0.280596i	$0.909468\pi$
$824$	16406.4	0.693622
$825$	−34815.5	−1.46924
$826$	−7207.65	−0.303615
$827$	−20084.2	−0.844495	−0.422248	−	0.906481i	$-0.638759\pi$
$827$	−20084.2	−0.844495	−0.422248	+	0.906481i	$0.638759\pi$
$828$	−8230.16	−0.345432
$829$	−29869.7	−1.25141	−0.625705	−	0.780060i	$-0.715188\pi$
$829$	−29869.7	−1.25141	−0.625705	+	0.780060i	$0.715188\pi$
$830$	−61405.3	−2.56796
$831$	−10543.4	−0.440129
$832$	0	0
$833$	2211.19	0.0919726
$834$	11878.0	0.493169
$835$	5264.31	0.218178
$836$	2759.24	0.114151
$837$	−10959.3	−0.452578
$838$	26573.8	1.09544
$839$	14416.0	0.593200	0.296600	−	0.955002i	$-0.404147\pi$
$839$	14416.0	0.593200	0.296600	+	0.955002i	$0.404147\pi$
$840$	−2954.47	−0.121356
$841$	−22518.2	−0.923293
$842$	1327.25	0.0543232
$843$	−22061.8	−0.901364
$844$	−5276.23	−0.215184
$845$	0	0
$846$	20430.9	0.830295
$847$	3139.86	0.127375
$848$	−6863.64	−0.277946
$849$	2880.90	0.116457
$850$	−30307.1	−1.22297
$851$	−7623.87	−0.307101
$852$	368.233	0.0148069
$853$	31303.7	1.25653	0.628264	−	0.778000i	$-0.283766\pi$
$853$	31303.7	1.25653	0.628264	+	0.778000i	$0.283766\pi$
$854$	−8729.72	−0.349795
$855$	7357.25	0.294284
$856$	−3769.49	−0.150512
$857$	−10963.9	−0.437014	−0.218507	−	0.975835i	$-0.570119\pi$
$857$	−10963.9	−0.437014	−0.218507	+	0.975835i	$0.570119\pi$
$858$	0	0
$859$	−14414.7	−0.572555	−0.286277	−	0.958147i	$-0.592418\pi$
$859$	−14414.7	−0.572555	−0.286277	+	0.958147i	$0.592418\pi$
$860$	−3761.34	−0.149140
$861$	−7801.11	−0.308782
$862$	19904.2	0.786472
$863$	21754.8	0.858103	0.429051	−	0.903280i	$-0.358848\pi$
$863$	21754.8	0.858103	0.429051	+	0.903280i	$0.358848\pi$
$864$	3771.89	0.148521
$865$	21879.3	0.860022
$866$	−5852.47	−0.229648
$867$	−7069.93	−0.276940
$868$	2603.34	0.101801
$869$	−27681.6	−1.08059
$870$	4563.89	0.177851
$871$	0	0
$872$	−17222.9	−0.668853
$873$	11142.5	0.431978
$874$	3210.48	0.124252
$875$	31676.2	1.22383
$876$	4188.44	0.161546
$877$	−23486.3	−0.904306	−0.452153	−	0.891940i	$-0.649344\pi$
$877$	−23486.3	−0.904306	−0.452153	+	0.891940i	$0.649344\pi$
$878$	−13855.0	−0.532555
$879$	19659.2	0.754366
$880$	−14488.8	−0.555020
$881$	41012.9	1.56840	0.784199	−	0.620509i	$-0.213074\pi$
$881$	41012.9	1.56840	0.784199	+	0.620509i	$0.213074\pi$
$882$	2054.04	0.0784162
$883$	−9316.41	−0.355065	−0.177532	−	0.984115i	$-0.556811\pi$
$883$	−9316.41	−0.355065	−0.177532	+	0.984115i	$0.556811\pi$
$884$	0	0
$885$	27161.7	1.03167
$886$	30126.2	1.14234
$887$	37008.3	1.40092	0.700460	−	0.713692i	$-0.252978\pi$
$887$	37008.3	1.40092	0.700460	+	0.713692i	$0.252978\pi$
$888$	1526.98	0.0577050
$889$	15339.0	0.578689
$890$	22647.1	0.852957
$891$	−11652.0	−0.438110
$892$	−11958.1	−0.448866
$893$	−7969.85	−0.298657
$894$	−5120.77	−0.191571
$895$	−54573.2	−2.03819
$896$	−896.000	−0.0334077
$897$	0	0
$898$	−22268.6	−0.827519
$899$	−4021.50	−0.149193
$900$	−28153.2	−1.04271
$901$	−19358.2	−0.715776
$902$	−38257.0	−1.41221
$903$	−753.627	−0.0277731
$904$	−16435.4	−0.604684
$905$	−8095.42	−0.297349
$906$	664.183	0.0243554
$907$	2918.81	0.106855	0.0534276	−	0.998572i	$-0.482985\pi$
$907$	2918.81	0.106855	0.0534276	+	0.998572i	$0.482985\pi$
$908$	−2004.63	−0.0732666
$909$	16309.3	0.595100
$910$	0	0
$911$	−39007.3	−1.41863	−0.709314	−	0.704893i	$-0.750995\pi$
$911$	−39007.3	−1.41863	−0.709314	+	0.704893i	$0.750995\pi$
$912$	−643.025	−0.0233473
$913$	−60335.6	−2.18709
$914$	36957.4	1.33746
$915$	32897.5	1.18859
$916$	10218.2	0.368581
$917$	−3458.51	−0.124547
$918$	10638.2	0.382476
$919$	43324.9	1.55512	0.777561	−	0.628807i	$-0.216457\pi$
$919$	43324.9	1.55512	0.777561	+	0.628807i	$0.216457\pi$
$920$	−16858.3	−0.604131
$921$	1111.56	0.0397690
$922$	14940.3	0.533656
$923$	0	0
$924$	−2903.00	−0.103357
$925$	−26079.2	−0.927005
$926$	−28388.1	−1.00744
$927$	42984.0	1.52295
$928$	1384.09	0.0489601
$929$	−55772.3	−1.96968	−0.984838	−	0.173478i	$-0.944499\pi$
$929$	−55772.3	−1.96968	−0.984838	+	0.173478i	$0.944499\pi$
$930$	−9810.57	−0.345915
$931$	−801.256	−0.0282063
$932$	20446.0	0.718596
$933$	−12740.6	−0.447061
$934$	−18977.3	−0.664836
$935$	−40864.2	−1.42931
$936$	0	0
$937$	12478.6	0.435067	0.217534	−	0.976053i	$-0.430199\pi$
$937$	12478.6	0.435067	0.217534	+	0.976053i	$0.430199\pi$
$938$	−10862.9	−0.378130
$939$	1592.32	0.0553391
$940$	41849.7	1.45211
$941$	−19838.5	−0.687267	−0.343633	−	0.939104i	$-0.611658\pi$
$941$	−19838.5	−0.687267	−0.343633	+	0.939104i	$0.611658\pi$
$942$	−9933.45	−0.343577
$943$	−44513.4	−1.53717
$944$	8237.32	0.284006
$945$	−17711.9	−0.609701
$946$	−3695.82	−0.127020
$947$	12942.0	0.444096	0.222048	−	0.975036i	$-0.428726\pi$
$947$	12942.0	0.444096	0.222048	+	0.975036i	$0.428726\pi$
$948$	6451.02	0.221012
$949$	0	0
$950$	10982.2	0.375063
$951$	−7826.40	−0.266865
$952$	−2527.07	−0.0860325
$953$	−52682.6	−1.79072	−0.895361	−	0.445342i	$-0.853082\pi$
$953$	−52682.6	−1.79072	−0.895361	+	0.445342i	$0.853082\pi$
$954$	−17982.4	−0.610274
$955$	−71636.8	−2.42734
$956$	8147.31	0.275631
$957$	4484.39	0.151473
$958$	−10524.9	−0.354954
$959$	6234.35	0.209924
$960$	3376.53	0.113518
$961$	−21146.4	−0.709824
$962$	0	0
$963$	−9875.88	−0.330473
$964$	28068.5	0.937785
$965$	87073.0	2.90464
$966$	−3377.74	−0.112502
$967$	26795.2	0.891082	0.445541	−	0.895262i	$-0.353011\pi$
$967$	26795.2	0.891082	0.445541	+	0.895262i	$0.353011\pi$
$968$	−3588.41	−0.119149
$969$	−1813.59	−0.0601246
$970$	22823.8	0.755492
$971$	26952.3	0.890772	0.445386	−	0.895339i	$-0.353066\pi$
$971$	26952.3	0.890772	0.445386	+	0.895339i	$0.353066\pi$
$972$	15445.5	0.509687
$973$	−16915.3	−0.557328
$974$	16898.6	0.555919
$975$	0	0
$976$	9976.82	0.327203
$977$	7690.39	0.251830	0.125915	−	0.992041i	$-0.459813\pi$
$977$	7690.39	0.251830	0.125915	+	0.992041i	$0.459813\pi$
$978$	−627.482	−0.0205160
$979$	22252.5	0.726449
$980$	4207.40	0.137143
$981$	−45123.0	−1.46857
$982$	−14192.5	−0.461203
$983$	−24910.0	−0.808245	−0.404123	−	0.914705i	$-0.632423\pi$
$983$	−24910.0	−0.808245	−0.404123	+	0.914705i	$0.632423\pi$
$984$	8915.56	0.288839
$985$	−6284.23	−0.203281
$986$	3903.68	0.126084
$987$	8385.06	0.270415
$988$	0	0
$989$	−4300.22	−0.138260
$990$	−37960.0	−1.21863
$991$	35112.3	1.12551	0.562754	−	0.826625i	$-0.309742\pi$
$991$	35112.3	1.12551	0.562754	+	0.826625i	$0.309742\pi$
$992$	−2975.25	−0.0952261
$993$	−12294.7	−0.392912
$994$	−524.393	−0.0167331
$995$	−24596.8	−0.783688
$996$	14060.8	0.447324
$997$	6660.43	0.211573	0.105786	−	0.994389i	$-0.466264\pi$
$997$	6660.43	0.211573	0.105786	+	0.994389i	$0.466264\pi$
$998$	42393.4	1.34463
$999$	9154.16	0.289915

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2366.4.a.bi.1.9	✓	15
13.12	even	2		2366.4.a.bk.1.9	yes	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2366.4.a.bi.1.9	✓	15	1.1	even	1	trivial
2366.4.a.bk.1.9	yes	15	13.12	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2366.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +2.45772 q^{3} +4.00000 q^{4} +21.4663 q^{5} -4.91545 q^{6} +7.00000 q^{7} -8.00000 q^{8} -20.9596 q^{9} -42.9327 q^{10} -42.1847 q^{11} +9.83090 q^{12} -14.0000 q^{14} +52.7583 q^{15} +16.0000 q^{16} +45.1263 q^{17} +41.9192 q^{18} -16.3522 q^{19} +85.8653 q^{20} +17.2041 q^{21} +84.3694 q^{22} +98.1670 q^{23} -19.6618 q^{24} +335.803 q^{25} -117.871 q^{27} +28.0000 q^{28} -43.2528 q^{29} -105.517 q^{30} +92.9766 q^{31} -32.0000 q^{32} -103.678 q^{33} -90.2526 q^{34} +150.264 q^{35} -83.8384 q^{36} -77.6622 q^{37} +32.7043 q^{38} -171.731 q^{40} -453.446 q^{41} -34.4081 q^{42} -43.8052 q^{43} -168.739 q^{44} -449.925 q^{45} -196.334 q^{46} +487.388 q^{47} +39.3236 q^{48} +49.0000 q^{49} -671.606 q^{50} +110.908 q^{51} -428.977 q^{53} +235.743 q^{54} -905.551 q^{55} -56.0000 q^{56} -40.1891 q^{57} +86.5057 q^{58} +514.832 q^{59} +211.033 q^{60} +623.552 q^{61} -185.953 q^{62} -146.717 q^{63} +64.0000 q^{64} +207.357 q^{66} +775.921 q^{67} +180.505 q^{68} +241.267 q^{69} -300.529 q^{70} +37.4567 q^{71} +167.677 q^{72} +426.048 q^{73} +155.324 q^{74} +825.312 q^{75} -65.4086 q^{76} -295.293 q^{77} +656.199 q^{79} +343.461 q^{80} +276.213 q^{81} +906.891 q^{82} +1430.27 q^{83} +68.8163 q^{84} +968.696 q^{85} +87.6103 q^{86} -106.304 q^{87} +337.478 q^{88} -527.502 q^{89} +899.851 q^{90} +392.668 q^{92} +228.511 q^{93} -974.776 q^{94} -351.021 q^{95} -78.6472 q^{96} -531.618 q^{97} -98.0000 q^{98} +884.175 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 30 q^{2} - 5 q^{3} + 60 q^{4} + 26 q^{5} + 10 q^{6} + 105 q^{7} - 120 q^{8} + 164 q^{9} - 52 q^{10} + 7 q^{11} - 20 q^{12} - 210 q^{14} + 86 q^{15} + 240 q^{16} + 16 q^{17} - 328 q^{18} + 511 q^{19}+ \cdots + 2790 q^{99}+O(q^{100}) \) 15 * q - 30 * q^2 - 5 * q^3 + 60 * q^4 + 26 * q^5 + 10 * q^6 + 105 * q^7 - 120 * q^8 + 164 * q^9 - 52 * q^10 + 7 * q^11 - 20 * q^12 - 210 * q^14 + 86 * q^15 + 240 * q^16 + 16 * q^17 - 328 * q^18 + 511 * q^19 + 104 * q^20 - 35 * q^21 - 14 * q^22 - 268 * q^23 + 40 * q^24 + 639 * q^25 - 557 * q^27 + 420 * q^28 - 283 * q^29 - 172 * q^30 + 127 * q^31 - 480 * q^32 + 628 * q^33 - 32 * q^34 + 182 * q^35 + 656 * q^36 + 462 * q^37 - 1022 * q^38 - 208 * q^40 + 1082 * q^41 + 70 * q^42 + 919 * q^43 + 28 * q^44 + 2118 * q^45 + 536 * q^46 + 595 * q^47 - 80 * q^48 + 735 * q^49 - 1278 * q^50 - 1009 * q^51 + 346 * q^53 + 1114 * q^54 - 823 * q^55 - 840 * q^56 - 1638 * q^57 + 566 * q^58 + 1162 * q^59 + 344 * q^60 + 81 * q^61 - 254 * q^62 + 1148 * q^63 + 960 * q^64 - 1256 * q^66 + 3649 * q^67 + 64 * q^68 - 1720 * q^69 - 364 * q^70 + 1152 * q^71 - 1312 * q^72 + 3983 * q^73 - 924 * q^74 + 1496 * q^75 + 2044 * q^76 + 49 * q^77 + 1333 * q^79 + 416 * q^80 + 1123 * q^81 - 2164 * q^82 + 93 * q^83 - 140 * q^84 + 628 * q^85 - 1838 * q^86 - 746 * q^87 - 56 * q^88 + 3061 * q^89 - 4236 * q^90 - 1072 * q^92 + 566 * q^93 - 1190 * q^94 - 3578 * q^95 + 160 * q^96 + 3853 * q^97 - 1470 * q^98 + 2790 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more