LMFDB - Embedded newform 1250.4.a.c.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1250,4,Mod(1,1250)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1250, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1250.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1250 = 2 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1250.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:	$73.7523875072$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 145x^{4} + 120x^{3} + 5125x^{2} - 2431x - 1069 $ x^6 - x^5 - 145x^4 + 120x^3 + 5125x^2 - 2431x - 1069
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 5 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.278104$ of defining polynomial
Character	$\chi$	$=$	1250.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} +0.660070 q^{3} +4.00000 q^{4} -1.32014 q^{6} -15.8349 q^{7} -8.00000 q^{8} -26.5643 q^{9} -5.50104 q^{11} +2.64028 q^{12} +63.8038 q^{13} +31.6697 q^{14} +16.0000 q^{16} -81.0213 q^{17} +53.1286 q^{18} +20.0588 q^{19} -10.4521 q^{21} +11.0021 q^{22} -84.1633 q^{23} -5.28056 q^{24} -127.608 q^{26} -35.3562 q^{27} -63.3395 q^{28} -98.1075 q^{29} +197.566 q^{31} -32.0000 q^{32} -3.63107 q^{33} +162.043 q^{34} -106.257 q^{36} -96.5325 q^{37} -40.1175 q^{38} +42.1149 q^{39} -29.9555 q^{41} +20.9042 q^{42} -247.303 q^{43} -22.0041 q^{44} +168.327 q^{46} -339.902 q^{47} +10.5611 q^{48} -92.2570 q^{49} -53.4797 q^{51} +255.215 q^{52} -106.039 q^{53} +70.7124 q^{54} +126.679 q^{56} +13.2402 q^{57} +196.215 q^{58} +417.834 q^{59} +780.727 q^{61} -395.131 q^{62} +420.642 q^{63} +64.0000 q^{64} +7.26213 q^{66} -645.674 q^{67} -324.085 q^{68} -55.5537 q^{69} -34.6744 q^{71} +212.514 q^{72} +1142.77 q^{73} +193.065 q^{74} +80.2350 q^{76} +87.1082 q^{77} -84.2299 q^{78} +1038.37 q^{79} +693.899 q^{81} +59.9110 q^{82} +457.171 q^{83} -41.8085 q^{84} +494.606 q^{86} -64.7578 q^{87} +44.0083 q^{88} +331.800 q^{89} -1010.32 q^{91} -336.653 q^{92} +130.407 q^{93} +679.803 q^{94} -21.1222 q^{96} +630.243 q^{97} +184.514 q^{98} +146.131 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 12 q^{2} + 8 q^{3} + 24 q^{4} - 16 q^{6} - 6 q^{7} - 48 q^{8} + 142 q^{9} + 107 q^{11} + 32 q^{12} - 42 q^{13} + 12 q^{14} + 96 q^{16} - 156 q^{17} - 284 q^{18} + 50 q^{19} + 122 q^{21} - 214 q^{22}+ \cdots + 2389 q^{99}+O(q^{100}) $ 6 * q - 12 * q^2 + 8 * q^3 + 24 * q^4 - 16 * q^6 - 6 * q^7 - 48 * q^8 + 142 * q^9 + 107 * q^11 + 32 * q^12 - 42 * q^13 + 12 * q^14 + 96 * q^16 - 156 * q^17 - 284 * q^18 + 50 * q^19 + 122 * q^21 - 214 * q^22 - 2 * q^23 - 64 * q^24 + 84 * q^26 + 680 * q^27 - 24 * q^28 + 100 * q^29 + 742 * q^31 - 192 * q^32 + 601 * q^33 + 312 * q^34 + 568 * q^36 - 71 * q^37 - 100 * q^38 + 259 * q^39 - 513 * q^41 - 244 * q^42 - 92 * q^43 + 428 * q^44 + 4 * q^46 - 311 * q^47 + 128 * q^48 + 348 * q^49 - 1998 * q^51 - 168 * q^52 - 1892 * q^53 - 1360 * q^54 + 48 * q^56 + 1515 * q^57 - 200 * q^58 + 665 * q^59 + 1577 * q^61 - 1484 * q^62 - 747 * q^63 + 384 * q^64 - 1202 * q^66 - 2201 * q^67 - 624 * q^68 + 4044 * q^69 + 1312 * q^71 - 1136 * q^72 + 708 * q^73 + 142 * q^74 + 200 * q^76 + 2318 * q^77 - 518 * q^78 - 1820 * q^79 + 5126 * q^81 + 1026 * q^82 + 1013 * q^83 + 488 * q^84 + 184 * q^86 - 1230 * q^87 - 856 * q^88 - 80 * q^89 + 1402 * q^91 - 8 * q^92 + 396 * q^93 + 622 * q^94 - 256 * q^96 - 996 * q^97 - 696 * q^98 + 2389 * 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
$3$	0.660070	0.127030	0.0635152	−	0.997981i	$-0.479769\pi$
$3$	0.660070	0.127030	0.0635152	+	0.997981i	$0.479769\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.32014	−0.0898241
$7$	−15.8349	−0.855002	−0.427501	−	0.904015i	$-0.640606\pi$
$7$	−15.8349	−0.855002	−0.427501	+	0.904015i	$0.640606\pi$
$8$	−8.00000	−0.353553
$9$	−26.5643	−0.983863
$10$	0	0
$11$	−5.50104	−0.150784	−0.0753920	−	0.997154i	$-0.524021\pi$
$11$	−5.50104	−0.150784	−0.0753920	+	0.997154i	$0.524021\pi$
$12$	2.64028	0.0635152
$13$	63.8038	1.36123	0.680615	−	0.732641i	$-0.261713\pi$
$13$	63.8038	1.36123	0.680615	+	0.732641i	$0.261713\pi$
$14$	31.6697	0.604578
$15$	0	0
$16$	16.0000	0.250000
$17$	−81.0213	−1.15591	−0.577957	−	0.816067i	$-0.696150\pi$
$17$	−81.0213	−1.15591	−0.577957	+	0.816067i	$0.696150\pi$
$18$	53.1286	0.695696
$19$	20.0588	0.242200	0.121100	−	0.992640i	$-0.461358\pi$
$19$	20.0588	0.242200	0.121100	+	0.992640i	$0.461358\pi$
$20$	0	0
$21$	−10.4521	−0.108611
$22$	11.0021	0.106620
$23$	−84.1633	−0.763012	−0.381506	−	0.924366i	$-0.624594\pi$
$23$	−84.1633	−0.763012	−0.381506	+	0.924366i	$0.624594\pi$
$24$	−5.28056	−0.0449121
$25$	0	0
$26$	−127.608	−0.962535
$27$	−35.3562	−0.252011
$28$	−63.3395	−0.427501
$29$	−98.1075	−0.628211	−0.314105	−	0.949388i	$-0.601704\pi$
$29$	−98.1075	−0.628211	−0.314105	+	0.949388i	$0.601704\pi$
$30$	0	0
$31$	197.566	1.14464	0.572320	−	0.820031i	$-0.306044\pi$
$31$	197.566	1.14464	0.572320	+	0.820031i	$0.306044\pi$
$32$	−32.0000	−0.176777
$33$	−3.63107	−0.0191542
$34$	162.043	0.817355
$35$	0	0
$36$	−106.257	−0.491932
$37$	−96.5325	−0.428915	−0.214457	−	0.976733i	$-0.568798\pi$
$37$	−96.5325	−0.428915	−0.214457	+	0.976733i	$0.568798\pi$
$38$	−40.1175	−0.171261
$39$	42.1149	0.172918
$40$	0	0
$41$	−29.9555	−0.114104	−0.0570520	−	0.998371i	$-0.518170\pi$
$41$	−29.9555	−0.114104	−0.0570520	+	0.998371i	$0.518170\pi$
$42$	20.9042	0.0767998
$43$	−247.303	−0.877055	−0.438527	−	0.898718i	$-0.644500\pi$
$43$	−247.303	−0.877055	−0.438527	+	0.898718i	$0.644500\pi$
$44$	−22.0041	−0.0753920
$45$	0	0
$46$	168.327	0.539531
$47$	−339.902	−1.05489	−0.527444	−	0.849590i	$-0.676850\pi$
$47$	−339.902	−1.05489	−0.527444	+	0.849590i	$0.676850\pi$
$48$	10.5611	0.0317576
$49$	−92.2570	−0.268971
$50$	0	0
$51$	−53.4797	−0.146836
$52$	255.215	0.680615
$53$	−106.039	−0.274821	−0.137411	−	0.990514i	$-0.543878\pi$
$53$	−106.039	−0.274821	−0.137411	+	0.990514i	$0.543878\pi$
$54$	70.7124	0.178199
$55$	0	0
$56$	126.679	0.302289
$57$	13.2402	0.0307667
$58$	196.215	0.444212
$59$	417.834	0.921989	0.460994	−	0.887403i	$-0.347493\pi$
$59$	417.834	0.921989	0.460994	+	0.887403i	$0.347493\pi$
$60$	0	0
$61$	780.727	1.63872	0.819359	−	0.573280i	$-0.194329\pi$
$61$	780.727	1.63872	0.819359	+	0.573280i	$0.194329\pi$
$62$	−395.131	−0.809382
$63$	420.642	0.841206
$64$	64.0000	0.125000
$65$	0	0
$66$	7.26213	0.0135440
$67$	−645.674	−1.17734	−0.588669	−	0.808374i	$-0.700348\pi$
$67$	−645.674	−1.17734	−0.588669	+	0.808374i	$0.700348\pi$
$68$	−324.085	−0.577957
$69$	−55.5537	−0.0969257
$70$	0	0
$71$	−34.6744	−0.0579590	−0.0289795	−	0.999580i	$-0.509226\pi$
$71$	−34.6744	−0.0579590	−0.0289795	+	0.999580i	$0.509226\pi$
$72$	212.514	0.347848
$73$	1142.77	1.83221	0.916107	−	0.400934i	$-0.131314\pi$
$73$	1142.77	1.83221	0.916107	+	0.400934i	$0.131314\pi$
$74$	193.065	0.303288
$75$	0	0
$76$	80.2350	0.121100
$77$	87.1082	0.128921
$78$	−84.2299	−0.122271
$79$	1038.37	1.47881	0.739403	−	0.673263i	$-0.235108\pi$
$79$	1038.37	1.47881	0.739403	+	0.673263i	$0.235108\pi$
$80$	0	0
$81$	693.899	0.951850
$82$	59.9110	0.0806837
$83$	457.171	0.604591	0.302296	−	0.953214i	$-0.402247\pi$
$83$	457.171	0.604591	0.302296	+	0.953214i	$0.402247\pi$
$84$	−41.8085	−0.0543057
$85$	0	0
$86$	494.606	0.620171
$87$	−64.7578	−0.0798019
$88$	44.0083	0.0533102
$89$	331.800	0.395177	0.197588	−	0.980285i	$-0.436689\pi$
$89$	331.800	0.395177	0.197588	+	0.980285i	$0.436689\pi$
$90$	0	0
$91$	−1010.32	−1.16385
$92$	−336.653	−0.381506
$93$	130.407	0.145404
$94$	679.803	0.745919
$95$	0	0
$96$	−21.1222	−0.0224560
$97$	630.243	0.659706	0.329853	−	0.944032i	$-0.393001\pi$
$97$	630.243	0.659706	0.329853	+	0.944032i	$0.393001\pi$
$98$	184.514	0.190191
$99$	146.131	0.148351
$100$	0	0
$101$	−1840.58	−1.81331	−0.906655	−	0.421872i	$-0.861373\pi$
$101$	−1840.58	−1.81331	−0.906655	+	0.421872i	$0.861373\pi$
$102$	106.959	0.103829
$103$	−1279.88	−1.22437	−0.612185	−	0.790714i	$-0.709709\pi$
$103$	−1279.88	−1.22437	−0.612185	+	0.790714i	$0.709709\pi$
$104$	−510.430	−0.481267
$105$	0	0
$106$	212.077	0.194328
$107$	446.971	0.403835	0.201917	−	0.979403i	$-0.435283\pi$
$107$	446.971	0.403835	0.201917	+	0.979403i	$0.435283\pi$
$108$	−141.425	−0.126006
$109$	−295.021	−0.259247	−0.129623	−	0.991563i	$-0.541377\pi$
$109$	−295.021	−0.259247	−0.129623	+	0.991563i	$0.541377\pi$
$110$	0	0
$111$	−63.7182	−0.0544852
$112$	−253.358	−0.213751
$113$	−194.457	−0.161885	−0.0809425	−	0.996719i	$-0.525793\pi$
$113$	−194.457	−0.161885	−0.0809425	+	0.996719i	$0.525793\pi$
$114$	−26.4804	−0.0217554
$115$	0	0
$116$	−392.430	−0.314105
$117$	−1694.90	−1.33926
$118$	−835.668	−0.651944
$119$	1282.96	0.988310
$120$	0	0
$121$	−1300.74	−0.977264
$122$	−1561.45	−1.15875
$123$	−19.7727	−0.0144947
$124$	790.262	0.572320
$125$	0	0
$126$	−841.285	−0.594822
$127$	2355.03	1.64547	0.822737	−	0.568422i	$-0.192446\pi$
$127$	2355.03	1.64547	0.822737	+	0.568422i	$0.192446\pi$
$128$	−128.000	−0.0883883
$129$	−163.237	−0.111413
$130$	0	0
$131$	1828.59	1.21958	0.609790	−	0.792563i	$-0.291254\pi$
$131$	1828.59	1.21958	0.609790	+	0.792563i	$0.291254\pi$
$132$	−14.5243	−0.00957709
$133$	−317.628	−0.207081
$134$	1291.35	0.832503
$135$	0	0
$136$	648.170	0.408678
$137$	−2666.23	−1.66271	−0.831355	−	0.555742i	$-0.812434\pi$
$137$	−2666.23	−1.66271	−0.831355	+	0.555742i	$0.812434\pi$
$138$	111.107	0.0685368
$139$	1661.15	1.01365	0.506823	−	0.862050i	$-0.330820\pi$
$139$	1661.15	1.01365	0.506823	+	0.862050i	$0.330820\pi$
$140$	0	0
$141$	−224.359	−0.134003
$142$	69.3487	0.0409832
$143$	−350.987	−0.205252
$144$	−425.029	−0.245966
$145$	0	0
$146$	−2285.55	−1.29557
$147$	−60.8960	−0.0341675
$148$	−386.130	−0.214457
$149$	3305.09	1.81721	0.908603	−	0.417660i	$-0.137150\pi$
$149$	3305.09	1.81721	0.908603	+	0.417660i	$0.137150\pi$
$150$	0	0
$151$	−998.295	−0.538014	−0.269007	−	0.963138i	$-0.586695\pi$
$151$	−998.295	−0.538014	−0.269007	+	0.963138i	$0.586695\pi$
$152$	−160.470	−0.0856305
$153$	2152.27	1.13726
$154$	−174.216	−0.0911607
$155$	0	0
$156$	168.460	0.0864588
$157$	62.1627	0.0315995	0.0157998	−	0.999875i	$-0.494971\pi$
$157$	62.1627	0.0315995	0.0157998	+	0.999875i	$0.494971\pi$
$158$	−2076.74	−1.04567
$159$	−69.9929	−0.0349107
$160$	0	0
$161$	1332.71	0.652377
$162$	−1387.80	−0.673060
$163$	97.1898	0.0467024	0.0233512	−	0.999727i	$-0.492566\pi$
$163$	97.1898	0.0467024	0.0233512	+	0.999727i	$0.492566\pi$
$164$	−119.822	−0.0570520
$165$	0	0
$166$	−914.343	−0.427511
$167$	−4112.06	−1.90539	−0.952696	−	0.303924i	$-0.901703\pi$
$167$	−4112.06	−1.90539	−0.952696	+	0.303924i	$0.901703\pi$
$168$	83.6169	0.0383999
$169$	1873.92	0.852946
$170$	0	0
$171$	−532.847	−0.238291
$172$	−989.212	−0.438527
$173$	794.694	0.349245	0.174623	−	0.984635i	$-0.444129\pi$
$173$	794.694	0.349245	0.174623	+	0.984635i	$0.444129\pi$
$174$	129.516	0.0564285
$175$	0	0
$176$	−88.0166	−0.0376960
$177$	275.799	0.117121
$178$	−663.600	−0.279432
$179$	4716.85	1.96957	0.984787	−	0.173763i	$-0.0555927\pi$
$179$	4716.85	1.96957	0.984787	+	0.173763i	$0.0555927\pi$
$180$	0	0
$181$	1678.68	0.689364	0.344682	−	0.938719i	$-0.387987\pi$
$181$	1678.68	0.689364	0.344682	+	0.938719i	$0.387987\pi$
$182$	2020.65	0.822970
$183$	515.334	0.208167
$184$	673.307	0.269765
$185$	0	0
$186$	−260.814	−0.102816
$187$	445.701	0.174293
$188$	−1359.61	−0.527444
$189$	559.860	0.215470
$190$	0	0
$191$	2621.45	0.993098	0.496549	−	0.868009i	$-0.334600\pi$
$191$	2621.45	0.993098	0.496549	+	0.868009i	$0.334600\pi$
$192$	42.2445	0.0158788
$193$	1271.64	0.474274	0.237137	−	0.971476i	$-0.423791\pi$
$193$	1271.64	0.474274	0.237137	+	0.971476i	$0.423791\pi$
$194$	−1260.49	−0.466482
$195$	0	0
$196$	−369.028	−0.134485
$197$	3664.05	1.32514	0.662570	−	0.749000i	$-0.269466\pi$
$197$	3664.05	1.32514	0.662570	+	0.749000i	$0.269466\pi$
$198$	−292.262	−0.104900
$199$	−2278.43	−0.811626	−0.405813	−	0.913956i	$-0.633012\pi$
$199$	−2278.43	−0.811626	−0.405813	+	0.913956i	$0.633012\pi$
$200$	0	0
$201$	−426.190	−0.149558
$202$	3681.16	1.28220
$203$	1553.52	0.537122
$204$	−213.919	−0.0734182
$205$	0	0
$206$	2559.76	0.865761
$207$	2235.74	0.750699
$208$	1020.86	0.340307
$209$	−110.344	−0.0365199
$210$	0	0
$211$	5210.61	1.70006	0.850032	−	0.526731i	$-0.176583\pi$
$211$	5210.61	1.70006	0.850032	+	0.526731i	$0.176583\pi$
$212$	−424.154	−0.137411
$213$	−22.8875	−0.00736256
$214$	−893.942	−0.285554
$215$	0	0
$216$	282.849	0.0890994
$217$	−3128.42	−0.978669
$218$	590.042	0.183315
$219$	754.311	0.232747
$220$	0	0
$221$	−5169.46	−1.57347
$222$	127.436	0.0385269
$223$	−801.962	−0.240822	−0.120411	−	0.992724i	$-0.538421\pi$
$223$	−801.962	−0.240822	−0.120411	+	0.992724i	$0.538421\pi$
$224$	506.716	0.151145
$225$	0	0
$226$	388.915	0.114470
$227$	2943.54	0.860660	0.430330	−	0.902672i	$-0.358397\pi$
$227$	2943.54	0.860660	0.430330	+	0.902672i	$0.358397\pi$
$228$	52.9607	0.0153834
$229$	5430.56	1.56708	0.783541	−	0.621340i	$-0.213412\pi$
$229$	5430.56	1.56708	0.783541	+	0.621340i	$0.213412\pi$
$230$	0	0
$231$	57.4975	0.0163769
$232$	784.860	0.222106
$233$	2356.15	0.662474	0.331237	−	0.943548i	$-0.392534\pi$
$233$	2356.15	0.662474	0.331237	+	0.943548i	$0.392534\pi$
$234$	3389.81	0.947003
$235$	0	0
$236$	1671.34	0.460994
$237$	685.396	0.187853
$238$	−2565.92	−0.698841
$239$	5153.04	1.39465	0.697327	−	0.716753i	$-0.254373\pi$
$239$	5153.04	1.39465	0.697327	+	0.716753i	$0.254373\pi$
$240$	0	0
$241$	−3074.54	−0.821778	−0.410889	−	0.911685i	$-0.634782\pi$
$241$	−3074.54	−0.821778	−0.410889	+	0.911685i	$0.634782\pi$
$242$	2601.48	0.691030
$243$	1412.64	0.372925
$244$	3122.91	0.819359
$245$	0	0
$246$	39.5454	0.0102493
$247$	1279.82	0.329689
$248$	−1580.52	−0.404691
$249$	301.765	0.0768015
$250$	0	0
$251$	3095.48	0.778426	0.389213	−	0.921148i	$-0.372747\pi$
$251$	3095.48	0.778426	0.389213	+	0.921148i	$0.372747\pi$
$252$	1682.57	0.420603
$253$	462.985	0.115050
$254$	−4710.06	−1.16353
$255$	0	0
$256$	256.000	0.0625000
$257$	4559.39	1.10664	0.553321	−	0.832968i	$-0.313360\pi$
$257$	4559.39	1.10664	0.553321	+	0.832968i	$0.313360\pi$
$258$	326.475	0.0787807
$259$	1528.58	0.366723
$260$	0	0
$261$	2606.16	0.618073
$262$	−3657.19	−0.862374
$263$	4239.53	0.993994	0.496997	−	0.867752i	$-0.334436\pi$
$263$	4239.53	0.993994	0.496997	+	0.867752i	$0.334436\pi$
$264$	29.0485	0.00677202
$265$	0	0
$266$	635.255	0.146429
$267$	219.011	0.0501995
$268$	−2582.70	−0.588669
$269$	−612.740	−0.138883	−0.0694413	−	0.997586i	$-0.522122\pi$
$269$	−612.740	−0.138883	−0.0694413	+	0.997586i	$0.522122\pi$
$270$	0	0
$271$	−7582.91	−1.69974	−0.849869	−	0.526994i	$-0.823319\pi$
$271$	−7582.91	−1.69974	−0.849869	+	0.526994i	$0.823319\pi$
$272$	−1296.34	−0.288979
$273$	−666.885	−0.147845
$274$	5332.46	1.17571
$275$	0	0
$276$	−222.215	−0.0484629
$277$	2701.65	0.586015	0.293008	−	0.956110i	$-0.405344\pi$
$277$	2701.65	0.586015	0.293008	+	0.956110i	$0.405344\pi$
$278$	−3322.30	−0.716757
$279$	−5248.19	−1.12617
$280$	0	0
$281$	−5892.76	−1.25101	−0.625503	−	0.780222i	$-0.715106\pi$
$281$	−5892.76	−1.25101	−0.625503	+	0.780222i	$0.715106\pi$
$282$	448.717	0.0947544
$283$	−2995.93	−0.629293	−0.314646	−	0.949209i	$-0.601886\pi$
$283$	−2995.93	−0.629293	−0.314646	+	0.949209i	$0.601886\pi$
$284$	−138.697	−0.0289795
$285$	0	0
$286$	701.974	0.145135
$287$	474.341	0.0975591
$288$	850.058	0.173924
$289$	1651.45	0.336139
$290$	0	0
$291$	416.004	0.0838027
$292$	4571.10	0.916107
$293$	4031.15	0.803763	0.401881	−	0.915692i	$-0.368356\pi$
$293$	4031.15	0.803763	0.401881	+	0.915692i	$0.368356\pi$
$294$	121.792	0.0241601
$295$	0	0
$296$	772.260	0.151644
$297$	194.496	0.0379993
$298$	−6610.19	−1.28496
$299$	−5369.94	−1.03863
$300$	0	0
$301$	3916.01	0.749884
$302$	1996.59	0.380433
$303$	−1214.91	−0.230346
$304$	320.940	0.0605499
$305$	0	0
$306$	−4304.55	−0.804166
$307$	−3707.11	−0.689173	−0.344587	−	0.938755i	$-0.611981\pi$
$307$	−3707.11	−0.689173	−0.344587	+	0.938755i	$0.611981\pi$
$308$	348.433	0.0644604
$309$	−844.809	−0.155532
$310$	0	0
$311$	−14.0723	−0.00256581	−0.00128291	−	0.999999i	$-0.500408\pi$
$311$	−14.0723	−0.00256581	−0.00128291	+	0.999999i	$0.500408\pi$
$312$	−336.920	−0.0611356
$313$	865.165	0.156236	0.0781182	−	0.996944i	$-0.475109\pi$
$313$	865.165	0.156236	0.0781182	+	0.996944i	$0.475109\pi$
$314$	−124.325	−0.0223442
$315$	0	0
$316$	4153.48	0.739403
$317$	−7354.40	−1.30304	−0.651521	−	0.758631i	$-0.725869\pi$
$317$	−7354.40	−1.30304	−0.651521	+	0.758631i	$0.725869\pi$
$318$	139.986	0.0246856
$319$	539.693	0.0947241
$320$	0	0
$321$	295.032	0.0512993
$322$	−2665.43	−0.461300
$323$	−1625.19	−0.279962
$324$	2775.60	0.475925
$325$	0	0
$326$	−194.380	−0.0330236
$327$	−194.734	−0.0329322
$328$	239.644	0.0403418
$329$	5382.30	0.901932
$330$	0	0
$331$	10052.2	1.66924	0.834620	−	0.550827i	$-0.185687\pi$
$331$	10052.2	1.66924	0.834620	+	0.550827i	$0.185687\pi$
$332$	1828.69	0.302296
$333$	2564.32	0.421993
$334$	8224.11	1.34732
$335$	0	0
$336$	−167.234	−0.0271528
$337$	8909.55	1.44016	0.720080	−	0.693891i	$-0.244105\pi$
$337$	8909.55	1.44016	0.720080	+	0.693891i	$0.244105\pi$
$338$	−3747.85	−0.603124
$339$	−128.355	−0.0205643
$340$	0	0
$341$	−1086.82	−0.172593
$342$	1065.69	0.168497
$343$	6892.24	1.08497
$344$	1978.42	0.310086
$345$	0	0
$346$	−1589.39	−0.246954
$347$	10035.8	1.55260	0.776298	−	0.630366i	$-0.217095\pi$
$347$	10035.8	1.55260	0.776298	+	0.630366i	$0.217095\pi$
$348$	−259.031	−0.0399009
$349$	−999.782	−0.153344	−0.0766721	−	0.997056i	$-0.524429\pi$
$349$	−999.782	−0.153344	−0.0766721	+	0.997056i	$0.524429\pi$
$350$	0	0
$351$	−2255.86	−0.343045
$352$	176.033	0.0266551
$353$	6654.40	1.00334	0.501669	−	0.865060i	$-0.332720\pi$
$353$	6654.40	1.00334	0.501669	+	0.865060i	$0.332720\pi$
$354$	−551.599	−0.0828168
$355$	0	0
$356$	1327.20	0.197588
$357$	846.844	0.125545
$358$	−9433.70	−1.39270
$359$	−10416.8	−1.53142	−0.765709	−	0.643187i	$-0.777612\pi$
$359$	−10416.8	−1.53142	−0.765709	+	0.643187i	$0.777612\pi$
$360$	0	0
$361$	−6456.65	−0.941339
$362$	−3357.35	−0.487454
$363$	−858.578	−0.124142
$364$	−4041.30	−0.581927
$365$	0	0
$366$	−1030.67	−0.147196
$367$	−1614.50	−0.229636	−0.114818	−	0.993387i	$-0.536628\pi$
$367$	−1614.50	−0.229636	−0.114818	+	0.993387i	$0.536628\pi$
$368$	−1346.61	−0.190753
$369$	795.747	0.112263
$370$	0	0
$371$	1679.11	0.234973
$372$	521.628	0.0727020
$373$	−1308.52	−0.181642	−0.0908209	−	0.995867i	$-0.528949\pi$
$373$	−1308.52	−0.181642	−0.0908209	+	0.995867i	$0.528949\pi$
$374$	−891.402	−0.123244
$375$	0	0
$376$	2719.21	0.372959
$377$	−6259.63	−0.855139
$378$	−1119.72	−0.152360
$379$	5971.07	0.809270	0.404635	−	0.914478i	$-0.367399\pi$
$379$	5971.07	0.809270	0.404635	+	0.914478i	$0.367399\pi$
$380$	0	0
$381$	1554.48	0.209025
$382$	−5242.91	−0.702226
$383$	8021.89	1.07023	0.535117	−	0.844778i	$-0.320268\pi$
$383$	8021.89	1.07023	0.535117	+	0.844778i	$0.320268\pi$
$384$	−84.4889	−0.0112280
$385$	0	0
$386$	−2543.29	−0.335362
$387$	6569.43	0.862902
$388$	2520.97	0.329853
$389$	2599.92	0.338872	0.169436	−	0.985541i	$-0.445805\pi$
$389$	2599.92	0.338872	0.169436	+	0.985541i	$0.445805\pi$
$390$	0	0
$391$	6819.02	0.881976
$392$	738.056	0.0950955
$393$	1207.00	0.154924
$394$	−7328.09	−0.937015
$395$	0	0
$396$	584.525	0.0741754
$397$	−3934.06	−0.497343	−0.248671	−	0.968588i	$-0.579994\pi$
$397$	−3934.06	−0.497343	−0.248671	+	0.968588i	$0.579994\pi$
$398$	4556.86	0.573906
$399$	−209.656	−0.0263056
$400$	0	0
$401$	−2617.21	−0.325929	−0.162964	−	0.986632i	$-0.552106\pi$
$401$	−2617.21	−0.325929	−0.162964	+	0.986632i	$0.552106\pi$
$402$	852.380	0.105753
$403$	12605.4	1.55812
$404$	−7362.31	−0.906655
$405$	0	0
$406$	−3107.04	−0.379802
$407$	531.029	0.0646735
$408$	427.838	0.0519145
$409$	−7160.34	−0.865663	−0.432832	−	0.901475i	$-0.642486\pi$
$409$	−7160.34	−0.865663	−0.432832	+	0.901475i	$0.642486\pi$
$410$	0	0
$411$	−1759.90	−0.211215
$412$	−5119.51	−0.612185
$413$	−6616.34	−0.788303
$414$	−4471.48	−0.530824
$415$	0	0
$416$	−2041.72	−0.240634
$417$	1096.48	0.128764
$418$	220.688	0.0258234
$419$	−8203.43	−0.956476	−0.478238	−	0.878230i	$-0.658724\pi$
$419$	−8203.43	−0.956476	−0.478238	+	0.878230i	$0.658724\pi$
$420$	0	0
$421$	13503.1	1.56318	0.781590	−	0.623793i	$-0.214409\pi$
$421$	13503.1	1.56318	0.781590	+	0.623793i	$0.214409\pi$
$422$	−10421.2	−1.20213
$423$	9029.25	1.03787
$424$	848.309	0.0971640
$425$	0	0
$426$	45.7750	0.00520612
$427$	−12362.7	−1.40111
$428$	1787.88	0.201917
$429$	−231.676	−0.0260732
$430$	0	0
$431$	−11298.0	−1.26265	−0.631327	−	0.775517i	$-0.717489\pi$
$431$	−11298.0	−1.26265	−0.631327	+	0.775517i	$0.717489\pi$
$432$	−565.699	−0.0630028
$433$	494.980	0.0549359	0.0274679	−	0.999623i	$-0.491256\pi$
$433$	494.980	0.0549359	0.0274679	+	0.999623i	$0.491256\pi$
$434$	6256.85	0.692024
$435$	0	0
$436$	−1180.08	−0.129623
$437$	−1688.21	−0.184801
$438$	−1508.62	−0.164577
$439$	−9269.10	−1.00772	−0.503861	−	0.863785i	$-0.668088\pi$
$439$	−9269.10	−1.00772	−0.503861	+	0.863785i	$0.668088\pi$
$440$	0	0
$441$	2450.74	0.264630
$442$	10338.9	1.11261
$443$	−14313.6	−1.53512	−0.767561	−	0.640976i	$-0.778530\pi$
$443$	−14313.6	−1.53512	−0.767561	+	0.640976i	$0.778530\pi$
$444$	−254.873	−0.0272426
$445$	0	0
$446$	1603.92	0.170287
$447$	2181.59	0.230841
$448$	−1013.43	−0.106875
$449$	−8501.11	−0.893524	−0.446762	−	0.894653i	$-0.647423\pi$
$449$	−8501.11	−0.893524	−0.446762	+	0.894653i	$0.647423\pi$
$450$	0	0
$451$	164.786	0.0172051
$452$	−777.829	−0.0809425
$453$	−658.944	−0.0683442
$454$	−5887.09	−0.608579
$455$	0	0
$456$	−105.921	−0.0108777
$457$	−16042.6	−1.64211	−0.821053	−	0.570852i	$-0.806613\pi$
$457$	−16042.6	−1.64211	−0.821053	+	0.570852i	$0.806613\pi$
$458$	−10861.1	−1.10809
$459$	2864.60	0.291303
$460$	0	0
$461$	−2936.38	−0.296662	−0.148331	−	0.988938i	$-0.547390\pi$
$461$	−2936.38	−0.296662	−0.148331	+	0.988938i	$0.547390\pi$
$462$	−114.995	−0.0115802
$463$	8714.07	0.874681	0.437340	−	0.899296i	$-0.355921\pi$
$463$	8714.07	0.874681	0.437340	+	0.899296i	$0.355921\pi$
$464$	−1569.72	−0.157053
$465$	0	0
$466$	−4712.30	−0.468440
$467$	10347.5	1.02532	0.512662	−	0.858591i	$-0.328660\pi$
$467$	10347.5	1.02532	0.512662	+	0.858591i	$0.328660\pi$
$468$	−6779.61	−0.669632
$469$	10224.2	1.00663
$470$	0	0
$471$	41.0317	0.00401410
$472$	−3342.67	−0.325972
$473$	1360.42	0.132246
$474$	−1370.79	−0.132832
$475$	0	0
$476$	5131.85	0.494155
$477$	2816.84	0.270386
$478$	−10306.1	−0.986169
$479$	−2017.43	−0.192440	−0.0962200	−	0.995360i	$-0.530675\pi$
$479$	−2017.43	−0.192440	−0.0962200	+	0.995360i	$0.530675\pi$
$480$	0	0
$481$	−6159.14	−0.583851
$482$	6149.08	0.581085
$483$	879.685	0.0828717
$484$	−5202.95	−0.488632
$485$	0	0
$486$	−2825.28	−0.263698
$487$	5149.95	0.479192	0.239596	−	0.970873i	$-0.422985\pi$
$487$	5149.95	0.479192	0.239596	+	0.970873i	$0.422985\pi$
$488$	−6245.82	−0.579375
$489$	64.1520	0.00593263
$490$	0	0
$491$	−9008.67	−0.828015	−0.414008	−	0.910273i	$-0.635871\pi$
$491$	−9008.67	−0.828015	−0.414008	+	0.910273i	$0.635871\pi$
$492$	−79.0908	−0.00724734
$493$	7948.80	0.726158
$494$	−2559.65	−0.233126
$495$	0	0
$496$	3161.05	0.286160
$497$	549.064	0.0495551
$498$	−603.530	−0.0543069
$499$	6847.91	0.614337	0.307169	−	0.951655i	$-0.400618\pi$
$499$	6847.91	0.614337	0.307169	+	0.951655i	$0.400618\pi$
$500$	0	0
$501$	−2714.24	−0.242043
$502$	−6190.96	−0.550430
$503$	−17526.3	−1.55360	−0.776798	−	0.629750i	$-0.783157\pi$
$503$	−17526.3	−1.55360	−0.776798	+	0.629750i	$0.783157\pi$
$504$	−3365.14	−0.297411
$505$	0	0
$506$	−925.971	−0.0813526
$507$	1236.92	0.108350
$508$	9420.12	0.822737
$509$	−10921.0	−0.951015	−0.475508	−	0.879712i	$-0.657736\pi$
$509$	−10921.0	−0.951015	−0.475508	+	0.879712i	$0.657736\pi$
$510$	0	0
$511$	−18095.7	−1.56655
$512$	−512.000	−0.0441942
$513$	−709.201	−0.0610370
$514$	−9118.78	−0.782514
$515$	0	0
$516$	−652.949	−0.0557064
$517$	1869.81	0.159060
$518$	−3057.16	−0.259312
$519$	524.553	0.0443648
$520$	0	0
$521$	21731.3	1.82738	0.913689	−	0.406415i	$-0.133221\pi$
$521$	21731.3	1.82738	0.913689	+	0.406415i	$0.133221\pi$
$522$	−5212.32	−0.437044
$523$	21974.2	1.83722	0.918610	−	0.395166i	$-0.129313\pi$
$523$	21974.2	1.83722	0.918610	+	0.395166i	$0.129313\pi$
$524$	7314.38	0.609790
$525$	0	0
$526$	−8479.05	−0.702860
$527$	−16007.0	−1.32311
$528$	−58.0971	−0.00478854
$529$	−5083.54	−0.417813
$530$	0	0
$531$	−11099.5	−0.907111
$532$	−1270.51	−0.103541
$533$	−1911.27	−0.155322
$534$	−438.022	−0.0354964
$535$	0	0
$536$	5165.39	0.416252
$537$	3113.45	0.250196
$538$	1225.48	0.0982048
$539$	507.509	0.0405565
$540$	0	0
$541$	−5318.63	−0.422672	−0.211336	−	0.977413i	$-0.567781\pi$
$541$	−5318.63	−0.422672	−0.211336	+	0.977413i	$0.567781\pi$
$542$	15165.8	1.20190
$543$	1108.04	0.0875703
$544$	2592.68	0.204339
$545$	0	0
$546$	1333.77	0.104542
$547$	13549.1	1.05908	0.529542	−	0.848284i	$-0.322364\pi$
$547$	13549.1	1.05908	0.529542	+	0.848284i	$0.322364\pi$
$548$	−10664.9	−0.831355
$549$	−20739.5	−1.61228
$550$	0	0
$551$	−1967.91	−0.152152
$552$	444.429	0.0342684
$553$	−16442.4	−1.26438
$554$	−5403.29	−0.414375
$555$	0	0
$556$	6644.60	0.506823
$557$	−10078.9	−0.766707	−0.383354	−	0.923602i	$-0.625231\pi$
$557$	−10078.9	−0.766707	−0.383354	+	0.923602i	$0.625231\pi$
$558$	10496.4	0.796321
$559$	−15778.9	−1.19387
$560$	0	0
$561$	294.194	0.0221406
$562$	11785.5	0.884594
$563$	12267.1	0.918287	0.459143	−	0.888362i	$-0.348156\pi$
$563$	12267.1	0.918287	0.459143	+	0.888362i	$0.348156\pi$
$564$	−897.435	−0.0670015
$565$	0	0
$566$	5991.87	0.444977
$567$	−10987.8	−0.813834
$568$	277.395	0.0204916
$569$	−20017.6	−1.47484	−0.737419	−	0.675436i	$-0.763956\pi$
$569$	−20017.6	−1.47484	−0.737419	+	0.675436i	$0.763956\pi$
$570$	0	0
$571$	19167.6	1.40479	0.702397	−	0.711785i	$-0.252113\pi$
$571$	19167.6	1.40479	0.702397	+	0.711785i	$0.252113\pi$
$572$	−1403.95	−0.102626
$573$	1730.34	0.126154
$574$	−948.682	−0.0689847
$575$	0	0
$576$	−1700.12	−0.122983
$577$	−22491.1	−1.62273	−0.811365	−	0.584540i	$-0.801275\pi$
$577$	−22491.1	−1.62273	−0.811365	+	0.584540i	$0.801275\pi$
$578$	−3302.90	−0.237686
$579$	839.373	0.0602473
$580$	0	0
$581$	−7239.25	−0.516927
$582$	−832.008	−0.0592575
$583$	583.322	0.0414387
$584$	−9142.19	−0.647785
$585$	0	0
$586$	−8062.31	−0.568346
$587$	9659.60	0.679207	0.339604	−	0.940569i	$-0.389707\pi$
$587$	9659.60	0.679207	0.339604	+	0.940569i	$0.389707\pi$
$588$	−243.584	−0.0170837
$589$	3962.92	0.277231
$590$	0	0
$591$	2418.53	0.168333
$592$	−1544.52	−0.107229
$593$	−5572.36	−0.385884	−0.192942	−	0.981210i	$-0.561803\pi$
$593$	−5572.36	−0.385884	−0.192942	+	0.981210i	$0.561803\pi$
$594$	−388.991	−0.0268695
$595$	0	0
$596$	13220.4	0.908603
$597$	−1503.92	−0.103101
$598$	10739.9	0.734425
$599$	−7911.99	−0.539692	−0.269846	−	0.962904i	$-0.586973\pi$
$599$	−7911.99	−0.539692	−0.269846	+	0.962904i	$0.586973\pi$
$600$	0	0
$601$	19247.9	1.30638	0.653192	−	0.757192i	$-0.273429\pi$
$601$	19247.9	1.30638	0.653192	+	0.757192i	$0.273429\pi$
$602$	−7832.02	−0.530248
$603$	17151.9	1.15834
$604$	−3993.18	−0.269007
$605$	0	0
$606$	2429.82	0.162879
$607$	−5839.82	−0.390496	−0.195248	−	0.980754i	$-0.562551\pi$
$607$	−5839.82	−0.390496	−0.195248	+	0.980754i	$0.562551\pi$
$608$	−641.880	−0.0428153
$609$	1025.43	0.0682308
$610$	0	0
$611$	−21687.0	−1.43595
$612$	8609.10	0.568631
$613$	12124.1	0.798835	0.399418	−	0.916769i	$-0.369212\pi$
$613$	12124.1	0.798835	0.399418	+	0.916769i	$0.369212\pi$
$614$	7414.23	0.487319
$615$	0	0
$616$	−696.865	−0.0455804
$617$	11825.0	0.771565	0.385783	−	0.922590i	$-0.373932\pi$
$617$	11825.0	0.771565	0.385783	+	0.922590i	$0.373932\pi$
$618$	1689.62	0.109978
$619$	−14009.0	−0.909645	−0.454822	−	0.890582i	$-0.650297\pi$
$619$	−14009.0	−0.909645	−0.454822	+	0.890582i	$0.650297\pi$
$620$	0	0
$621$	2975.69	0.192287
$622$	28.1446	0.00181430
$623$	−5254.01	−0.337877
$624$	673.839	0.0432294
$625$	0	0
$626$	−1730.33	−0.110476
$627$	−72.8347	−0.00463913
$628$	248.651	0.0157998
$629$	7821.19	0.495789
$630$	0	0
$631$	2071.70	0.130702	0.0653510	−	0.997862i	$-0.479183\pi$
$631$	2071.70	0.130702	0.0653510	+	0.997862i	$0.479183\pi$
$632$	−8306.96	−0.522837
$633$	3439.37	0.215960
$634$	14708.8	0.921390
$635$	0	0
$636$	−279.971	−0.0174553
$637$	−5886.34	−0.366131
$638$	−1079.39	−0.0669801
$639$	921.101	0.0570237
$640$	0	0
$641$	−875.590	−0.0539528	−0.0269764	−	0.999636i	$-0.508588\pi$
$641$	−875.590	−0.0539528	−0.0269764	+	0.999636i	$0.508588\pi$
$642$	−590.064	−0.0362741
$643$	−7757.55	−0.475782	−0.237891	−	0.971292i	$-0.576456\pi$
$643$	−7757.55	−0.475782	−0.237891	+	0.971292i	$0.576456\pi$
$644$	5330.86	0.326188
$645$	0	0
$646$	3250.37	0.197963
$647$	−22628.2	−1.37497	−0.687485	−	0.726198i	$-0.741285\pi$
$647$	−22628.2	−1.37497	−0.687485	+	0.726198i	$0.741285\pi$
$648$	−5551.19	−0.336530
$649$	−2298.52	−0.139021
$650$	0	0
$651$	−2064.98	−0.124321
$652$	388.759	0.0233512
$653$	6584.08	0.394571	0.197286	−	0.980346i	$-0.436787\pi$
$653$	6584.08	0.394571	0.197286	+	0.980346i	$0.436787\pi$
$654$	389.469	0.0232866
$655$	0	0
$656$	−479.288	−0.0285260
$657$	−30357.0	−1.80265
$658$	−10764.6	−0.637762
$659$	215.219	0.0127219	0.00636094	−	0.999980i	$-0.497975\pi$
$659$	215.219	0.0127219	0.00636094	+	0.999980i	$0.497975\pi$
$660$	0	0
$661$	26476.8	1.55799	0.778994	−	0.627032i	$-0.215730\pi$
$661$	26476.8	1.55799	0.778994	+	0.627032i	$0.215730\pi$
$662$	−20104.4	−1.18033
$663$	−3412.21	−0.199878
$664$	−3657.37	−0.213755
$665$	0	0
$666$	−5128.64	−0.298394
$667$	8257.05	0.479332
$668$	−16448.2	−0.952696
$669$	−529.351	−0.0305918
$670$	0	0
$671$	−4294.81	−0.247093
$672$	334.468	0.0192000
$673$	−13606.4	−0.779327	−0.389663	−	0.920957i	$-0.627409\pi$
$673$	−13606.4	−0.779327	−0.389663	+	0.920957i	$0.627409\pi$
$674$	−17819.1	−1.01835
$675$	0	0
$676$	7495.69	0.426473
$677$	−17729.5	−1.00650	−0.503251	−	0.864140i	$-0.667863\pi$
$677$	−17729.5	−1.00650	−0.503251	+	0.864140i	$0.667863\pi$
$678$	256.711	0.0145412
$679$	−9979.81	−0.564050
$680$	0	0
$681$	1942.94	0.109330
$682$	2173.63	0.122042
$683$	−17693.6	−0.991256	−0.495628	−	0.868535i	$-0.665062\pi$
$683$	−17693.6	−0.991256	−0.495628	+	0.868535i	$0.665062\pi$
$684$	−2131.39	−0.119146
$685$	0	0
$686$	−13784.5	−0.767192
$687$	3584.55	0.199067
$688$	−3956.85	−0.219264
$689$	−6765.66	−0.374095
$690$	0	0
$691$	21596.3	1.18895	0.594473	−	0.804116i	$-0.297361\pi$
$691$	21596.3	1.18895	0.594473	+	0.804116i	$0.297361\pi$
$692$	3178.78	0.174623
$693$	−2313.97	−0.126840
$694$	−20071.6	−1.09785
$695$	0	0
$696$	518.062	0.0282142
$697$	2427.03	0.131894
$698$	1999.56	0.108431
$699$	1555.22	0.0841544
$700$	0	0
$701$	−13380.6	−0.720939	−0.360469	−	0.932771i	$-0.617384\pi$
$701$	−13380.6	−0.720939	−0.360469	+	0.932771i	$0.617384\pi$
$702$	4511.72	0.242569
$703$	−1936.32	−0.103883
$704$	−352.066	−0.0188480
$705$	0	0
$706$	−13308.8	−0.709467
$707$	29145.3	1.55039
$708$	1103.20	0.0585603
$709$	17755.9	0.940530	0.470265	−	0.882525i	$-0.344158\pi$
$709$	17755.9	0.940530	0.470265	+	0.882525i	$0.344158\pi$
$710$	0	0
$711$	−27583.6	−1.45494
$712$	−2654.40	−0.139716
$713$	−16627.8	−0.873373
$714$	−1693.69	−0.0887741
$715$	0	0
$716$	18867.4	0.984787
$717$	3401.36	0.177164
$718$	20833.7	1.08288
$719$	5858.00	0.303848	0.151924	−	0.988392i	$-0.451453\pi$
$719$	5858.00	0.303848	0.151924	+	0.988392i	$0.451453\pi$
$720$	0	0
$721$	20266.7	1.04684
$722$	12913.3	0.665627
$723$	−2029.41	−0.104391
$724$	6714.70	0.344682
$725$	0	0
$726$	1717.16	0.0877819
$727$	2258.37	0.115211	0.0576056	−	0.998339i	$-0.481653\pi$
$727$	2258.37	0.115211	0.0576056	+	0.998339i	$0.481653\pi$
$728$	8082.60	0.411485
$729$	−17802.8	−0.904477
$730$	0	0
$731$	20036.8	1.01380
$732$	2061.34	0.104084
$733$	21005.7	1.05847	0.529237	−	0.848474i	$-0.322478\pi$
$733$	21005.7	1.05847	0.529237	+	0.848474i	$0.322478\pi$
$734$	3229.01	0.162377
$735$	0	0
$736$	2693.23	0.134883
$737$	3551.87	0.177524
$738$	−1591.49	−0.0793817
$739$	−30135.7	−1.50008	−0.750040	−	0.661393i	$-0.769966\pi$
$739$	−30135.7	−1.50008	−0.750040	+	0.661393i	$0.769966\pi$
$740$	0	0
$741$	844.773	0.0418806
$742$	−3358.21	−0.166151
$743$	−11814.8	−0.583367	−0.291683	−	0.956515i	$-0.594215\pi$
$743$	−11814.8	−0.583367	−0.291683	+	0.956515i	$0.594215\pi$
$744$	−1043.26	−0.0514081
$745$	0	0
$746$	2617.03	0.128440
$747$	−12144.4	−0.594835
$748$	1782.80	0.0871467
$749$	−7077.73	−0.345280
$750$	0	0
$751$	22130.5	1.07530	0.537651	−	0.843168i	$-0.319312\pi$
$751$	22130.5	1.07530	0.537651	+	0.843168i	$0.319312\pi$
$752$	−5438.43	−0.263722
$753$	2043.23	0.0988839
$754$	12519.3	0.604674
$755$	0	0
$756$	2239.44	0.107735
$757$	8303.35	0.398666	0.199333	−	0.979932i	$-0.436122\pi$
$757$	8303.35	0.398666	0.199333	+	0.979932i	$0.436122\pi$
$758$	−11942.1	−0.572241
$759$	305.603	0.0146149
$760$	0	0
$761$	8025.03	0.382270	0.191135	−	0.981564i	$-0.438783\pi$
$761$	8025.03	0.382270	0.191135	+	0.981564i	$0.438783\pi$
$762$	−3108.97	−0.147803
$763$	4671.62	0.221657
$764$	10485.8	0.496549
$765$	0	0
$766$	−16043.8	−0.756769
$767$	26659.4	1.25504
$768$	168.978	0.00793941
$769$	−5180.39	−0.242925	−0.121463	−	0.992596i	$-0.538758\pi$
$769$	−5180.39	−0.242925	−0.121463	+	0.992596i	$0.538758\pi$
$770$	0	0
$771$	3009.52	0.140577
$772$	5086.57	0.237137
$773$	−26023.3	−1.21086	−0.605430	−	0.795899i	$-0.706999\pi$
$773$	−26023.3	−1.21086	−0.605430	+	0.795899i	$0.706999\pi$
$774$	−13138.9	−0.610164
$775$	0	0
$776$	−5041.94	−0.233241
$777$	1008.97	0.0465850
$778$	−5199.84	−0.239619
$779$	−600.870	−0.0276359
$780$	0	0
$781$	190.745	0.00873930
$782$	−13638.0	−0.623651
$783$	3468.71	0.158316
$784$	−1476.11	−0.0672427
$785$	0	0
$786$	−2414.00	−0.109548
$787$	20158.3	0.913044	0.456522	−	0.889712i	$-0.349095\pi$
$787$	20158.3	0.913044	0.456522	+	0.889712i	$0.349095\pi$
$788$	14656.2	0.662570
$789$	2798.38	0.126268
$790$	0	0
$791$	3079.21	0.138412
$792$	−1169.05	−0.0524500
$793$	49813.3	2.23067
$794$	7868.13	0.351674
$795$	0	0
$796$	−9113.72	−0.405813
$797$	1492.92	0.0663511	0.0331755	−	0.999450i	$-0.489438\pi$
$797$	1492.92	0.0663511	0.0331755	+	0.999450i	$0.489438\pi$
$798$	419.313	0.0186009
$799$	27539.3	1.21936
$800$	0	0
$801$	−8814.04	−0.388800
$802$	5234.43	0.230466
$803$	−6286.44	−0.276269
$804$	−1704.76	−0.0747789
$805$	0	0
$806$	−25210.9	−1.10176
$807$	−404.451	−0.0176423
$808$	14724.6	0.641102
$809$	−32899.3	−1.42976	−0.714881	−	0.699246i	$-0.753519\pi$
$809$	−32899.3	−1.42976	−0.714881	+	0.699246i	$0.753519\pi$
$810$	0	0
$811$	13832.3	0.598914	0.299457	−	0.954110i	$-0.403194\pi$
$811$	13832.3	0.598914	0.299457	+	0.954110i	$0.403194\pi$
$812$	6214.08	0.268561
$813$	−5005.25	−0.215918
$814$	−1062.06	−0.0457311
$815$	0	0
$816$	−855.675	−0.0367091
$817$	−4960.59	−0.212422
$818$	14320.7	0.612116
$819$	26838.6	1.14507
$820$	0	0
$821$	−34809.0	−1.47971	−0.739855	−	0.672766i	$-0.765106\pi$
$821$	−34809.0	−1.47971	−0.739855	+	0.672766i	$0.765106\pi$
$822$	3519.79	0.149351
$823$	28468.8	1.20579	0.602893	−	0.797822i	$-0.294015\pi$
$823$	28468.8	1.20579	0.602893	+	0.797822i	$0.294015\pi$
$824$	10239.0	0.432880
$825$	0	0
$826$	13232.7	0.557414
$827$	26950.0	1.13319	0.566593	−	0.823998i	$-0.308261\pi$
$827$	26950.0	1.13319	0.566593	+	0.823998i	$0.308261\pi$
$828$	8942.96	0.375349
$829$	775.483	0.0324893	0.0162446	−	0.999868i	$-0.494829\pi$
$829$	775.483	0.0324893	0.0162446	+	0.999868i	$0.494829\pi$
$830$	0	0
$831$	1783.28	0.0744418
$832$	4083.44	0.170154
$833$	7474.78	0.310907
$834$	−2192.95	−0.0910499
$835$	0	0
$836$	−441.376	−0.0182599
$837$	−6985.16	−0.288462
$838$	16406.9	0.676331
$839$	37781.8	1.55468	0.777338	−	0.629084i	$-0.216570\pi$
$839$	37781.8	1.55468	0.777338	+	0.629084i	$0.216570\pi$
$840$	0	0
$841$	−14763.9	−0.605352
$842$	−27006.1	−1.10533
$843$	−3889.63	−0.158916
$844$	20842.5	0.850032
$845$	0	0
$846$	−18058.5	−0.733882
$847$	20597.0	0.835563
$848$	−1696.62	−0.0687053
$849$	−1977.53	−0.0799393
$850$	0	0
$851$	8124.49	0.327267
$852$	−91.5500	−0.00368128
$853$	−10089.8	−0.405005	−0.202503	−	0.979282i	$-0.564907\pi$
$853$	−10089.8	−0.405005	−0.202503	+	0.979282i	$0.564907\pi$
$854$	24725.4	0.990733
$855$	0	0
$856$	−3575.77	−0.142777
$857$	29927.3	1.19288	0.596440	−	0.802658i	$-0.296581\pi$
$857$	29927.3	1.19288	0.596440	+	0.802658i	$0.296581\pi$
$858$	463.352	0.0184366
$859$	−8506.89	−0.337894	−0.168947	−	0.985625i	$-0.554037\pi$
$859$	−8506.89	−0.337894	−0.168947	+	0.985625i	$0.554037\pi$
$860$	0	0
$861$	313.098	0.0123930
$862$	22595.9	0.892831
$863$	−16995.9	−0.670392	−0.335196	−	0.942148i	$-0.608803\pi$
$863$	−16995.9	−0.670392	−0.335196	+	0.942148i	$0.608803\pi$
$864$	1131.40	0.0445497
$865$	0	0
$866$	−989.960	−0.0388455
$867$	1090.07	0.0426998
$868$	−12513.7	−0.489335
$869$	−5712.11	−0.222980
$870$	0	0
$871$	−41196.4	−1.60263
$872$	2360.17	0.0916575
$873$	−16742.0	−0.649060
$874$	3376.42	0.130674
$875$	0	0
$876$	3017.24	0.116373
$877$	30356.9	1.16885	0.584424	−	0.811448i	$-0.301320\pi$
$877$	30356.9	1.16885	0.584424	+	0.811448i	$0.301320\pi$
$878$	18538.2	0.712567
$879$	2660.84	0.102102
$880$	0	0
$881$	−42101.5	−1.61003	−0.805015	−	0.593255i	$-0.797843\pi$
$881$	−42101.5	−1.61003	−0.805015	+	0.593255i	$0.797843\pi$
$882$	−4901.49	−0.187122
$883$	−39597.1	−1.50912	−0.754558	−	0.656234i	$-0.772148\pi$
$883$	−39597.1	−1.50912	−0.754558	+	0.656234i	$0.772148\pi$
$884$	−20677.9	−0.786733
$885$	0	0
$886$	28627.2	1.08550
$887$	−4172.22	−0.157936	−0.0789680	−	0.996877i	$-0.525163\pi$
$887$	−4172.22	−0.157936	−0.0789680	+	0.996877i	$0.525163\pi$
$888$	509.745	0.0192634
$889$	−37291.6	−1.40688
$890$	0	0
$891$	−3817.16	−0.143524
$892$	−3207.85	−0.120411
$893$	−6818.00	−0.255494
$894$	−4363.18	−0.163229
$895$	0	0
$896$	2026.86	0.0755723
$897$	−3544.53	−0.131938
$898$	17002.2	0.631817
$899$	−19382.7	−0.719074
$900$	0	0
$901$	8591.38	0.317670
$902$	−329.572	−0.0121658
$903$	2584.84	0.0952581
$904$	1555.66	0.0572350
$905$	0	0
$906$	1317.89	0.0483266
$907$	6248.25	0.228743	0.114371	−	0.993438i	$-0.463515\pi$
$907$	6248.25	0.228743	0.114371	+	0.993438i	$0.463515\pi$
$908$	11774.2	0.430330
$909$	48893.7	1.78405
$910$	0	0
$911$	5191.71	0.188813	0.0944066	−	0.995534i	$-0.469905\pi$
$911$	5191.71	0.188813	0.0944066	+	0.995534i	$0.469905\pi$
$912$	211.843	0.00769169
$913$	−2514.92	−0.0911627
$914$	32085.2	1.16114
$915$	0	0
$916$	21722.3	0.783541
$917$	−28955.6	−1.04274
$918$	−5729.21	−0.205983
$919$	2826.18	0.101444	0.0507221	−	0.998713i	$-0.483848\pi$
$919$	2826.18	0.101444	0.0507221	+	0.998713i	$0.483848\pi$
$920$	0	0
$921$	−2446.95	−0.0875460
$922$	5872.77	0.209771
$923$	−2212.36	−0.0788955
$924$	229.990	0.00818843
$925$	0	0
$926$	−17428.1	−0.618493
$927$	33999.1	1.20461
$928$	3139.44	0.111053
$929$	−12961.0	−0.457735	−0.228868	−	0.973458i	$-0.573502\pi$
$929$	−12961.0	−0.457735	−0.228868	+	0.973458i	$0.573502\pi$
$930$	0	0
$931$	−1850.56	−0.0651446
$932$	9424.60	0.331237
$933$	−9.28871	−0.000325937 0
$934$	−20695.0	−0.725013
$935$	0	0
$936$	13559.2	0.473501
$937$	−34128.2	−1.18988	−0.594941	−	0.803769i	$-0.702825\pi$
$937$	−34128.2	−1.18988	−0.594941	+	0.803769i	$0.702825\pi$
$938$	−20448.3	−0.711792
$939$	571.069	0.0198468
$940$	0	0
$941$	−41298.5	−1.43071	−0.715353	−	0.698763i	$-0.753734\pi$
$941$	−41298.5	−1.43071	−0.715353	+	0.698763i	$0.753734\pi$
$942$	−82.0635	−0.00283840
$943$	2521.15	0.0870626
$944$	6685.34	0.230497
$945$	0	0
$946$	−2720.85	−0.0935120
$947$	21393.9	0.734116	0.367058	−	0.930198i	$-0.380365\pi$
$947$	21393.9	0.734116	0.367058	+	0.930198i	$0.380365\pi$
$948$	2741.59	0.0939267
$949$	72913.3	2.49406
$950$	0	0
$951$	−4854.42	−0.165526
$952$	−10263.7	−0.349420
$953$	24381.0	0.828728	0.414364	−	0.910111i	$-0.364004\pi$
$953$	24381.0	0.828728	0.414364	+	0.910111i	$0.364004\pi$
$954$	−5633.68	−0.191192
$955$	0	0
$956$	20612.1	0.697327
$957$	356.235	0.0120329
$958$	4034.86	0.136076
$959$	42219.4	1.42162
$960$	0	0
$961$	9241.14	0.310199
$962$	12318.3	0.412845
$963$	−11873.5	−0.397318
$964$	−12298.2	−0.410889
$965$	0	0
$966$	−1759.37	−0.0585992
$967$	25361.5	0.843402	0.421701	−	0.906735i	$-0.361433\pi$
$967$	25361.5	0.843402	0.421701	+	0.906735i	$0.361433\pi$
$968$	10405.9	0.345515
$969$	−1072.74	−0.0355637
$970$	0	0
$971$	−40071.3	−1.32436	−0.662178	−	0.749347i	$-0.730368\pi$
$971$	−40071.3	−1.32436	−0.662178	+	0.749347i	$0.730368\pi$
$972$	5650.55	0.186463
$973$	−26304.1	−0.866671
$974$	−10299.9	−0.338840
$975$	0	0
$976$	12491.6	0.409680
$977$	−4847.90	−0.158749	−0.0793746	−	0.996845i	$-0.525292\pi$
$977$	−4847.90	−0.158749	−0.0793746	+	0.996845i	$0.525292\pi$
$978$	−128.304	−0.00419500
$979$	−1825.24	−0.0595864
$980$	0	0
$981$	7837.03	0.255063
$982$	18017.3	0.585495
$983$	−575.902	−0.0186861	−0.00934304	−	0.999956i	$-0.502974\pi$
$983$	−575.902	−0.0186861	−0.00934304	+	0.999956i	$0.502974\pi$
$984$	158.182	0.00512464
$985$	0	0
$986$	−15897.6	−0.513471
$987$	3552.69	0.114573
$988$	5119.30	0.164845
$989$	20813.8	0.669203
$990$	0	0
$991$	49749.5	1.59470	0.797348	−	0.603520i	$-0.206235\pi$
$991$	49749.5	1.59470	0.797348	+	0.603520i	$0.206235\pi$
$992$	−6322.10	−0.202346
$993$	6635.15	0.212044
$994$	−1098.13	−0.0350408
$995$	0	0
$996$	1207.06	0.0384008
$997$	−42066.3	−1.33626	−0.668130	−	0.744044i	$-0.732905\pi$
$997$	−42066.3	−1.33626	−0.668130	+	0.744044i	$0.732905\pi$
$998$	−13695.8	−0.434402
$999$	3413.02	0.108091

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1250.4.a.c.1.3	✓	6
5.4	even	2		1250.4.a.f.1.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1250.4.a.c.1.3	✓	6	1.1	even	1	trivial
1250.4.a.f.1.4	yes	6	5.4	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:	\( N \)	\(=\)	\( 1250 = 2 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1250.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +0.660070 q^{3} +4.00000 q^{4} -1.32014 q^{6} -15.8349 q^{7} -8.00000 q^{8} -26.5643 q^{9} -5.50104 q^{11} +2.64028 q^{12} +63.8038 q^{13} +31.6697 q^{14} +16.0000 q^{16} -81.0213 q^{17} +53.1286 q^{18} +20.0588 q^{19} -10.4521 q^{21} +11.0021 q^{22} -84.1633 q^{23} -5.28056 q^{24} -127.608 q^{26} -35.3562 q^{27} -63.3395 q^{28} -98.1075 q^{29} +197.566 q^{31} -32.0000 q^{32} -3.63107 q^{33} +162.043 q^{34} -106.257 q^{36} -96.5325 q^{37} -40.1175 q^{38} +42.1149 q^{39} -29.9555 q^{41} +20.9042 q^{42} -247.303 q^{43} -22.0041 q^{44} +168.327 q^{46} -339.902 q^{47} +10.5611 q^{48} -92.2570 q^{49} -53.4797 q^{51} +255.215 q^{52} -106.039 q^{53} +70.7124 q^{54} +126.679 q^{56} +13.2402 q^{57} +196.215 q^{58} +417.834 q^{59} +780.727 q^{61} -395.131 q^{62} +420.642 q^{63} +64.0000 q^{64} +7.26213 q^{66} -645.674 q^{67} -324.085 q^{68} -55.5537 q^{69} -34.6744 q^{71} +212.514 q^{72} +1142.77 q^{73} +193.065 q^{74} +80.2350 q^{76} +87.1082 q^{77} -84.2299 q^{78} +1038.37 q^{79} +693.899 q^{81} +59.9110 q^{82} +457.171 q^{83} -41.8085 q^{84} +494.606 q^{86} -64.7578 q^{87} +44.0083 q^{88} +331.800 q^{89} -1010.32 q^{91} -336.653 q^{92} +130.407 q^{93} +679.803 q^{94} -21.1222 q^{96} +630.243 q^{97} +184.514 q^{98} +146.131 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 12 q^{2} + 8 q^{3} + 24 q^{4} - 16 q^{6} - 6 q^{7} - 48 q^{8} + 142 q^{9} + 107 q^{11} + 32 q^{12} - 42 q^{13} + 12 q^{14} + 96 q^{16} - 156 q^{17} - 284 q^{18} + 50 q^{19} + 122 q^{21} - 214 q^{22}+ \cdots + 2389 q^{99}+O(q^{100}) \) 6 * q - 12 * q^2 + 8 * q^3 + 24 * q^4 - 16 * q^6 - 6 * q^7 - 48 * q^8 + 142 * q^9 + 107 * q^11 + 32 * q^12 - 42 * q^13 + 12 * q^14 + 96 * q^16 - 156 * q^17 - 284 * q^18 + 50 * q^19 + 122 * q^21 - 214 * q^22 - 2 * q^23 - 64 * q^24 + 84 * q^26 + 680 * q^27 - 24 * q^28 + 100 * q^29 + 742 * q^31 - 192 * q^32 + 601 * q^33 + 312 * q^34 + 568 * q^36 - 71 * q^37 - 100 * q^38 + 259 * q^39 - 513 * q^41 - 244 * q^42 - 92 * q^43 + 428 * q^44 + 4 * q^46 - 311 * q^47 + 128 * q^48 + 348 * q^49 - 1998 * q^51 - 168 * q^52 - 1892 * q^53 - 1360 * q^54 + 48 * q^56 + 1515 * q^57 - 200 * q^58 + 665 * q^59 + 1577 * q^61 - 1484 * q^62 - 747 * q^63 + 384 * q^64 - 1202 * q^66 - 2201 * q^67 - 624 * q^68 + 4044 * q^69 + 1312 * q^71 - 1136 * q^72 + 708 * q^73 + 142 * q^74 + 200 * q^76 + 2318 * q^77 - 518 * q^78 - 1820 * q^79 + 5126 * q^81 + 1026 * q^82 + 1013 * q^83 + 488 * q^84 + 184 * q^86 - 1230 * q^87 - 856 * q^88 - 80 * q^89 + 1402 * q^91 - 8 * q^92 + 396 * q^93 + 622 * q^94 - 256 * q^96 - 996 * q^97 - 696 * q^98 + 2389 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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