LMFDB - Embedded newform 1150.4.a.s.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1150,4,Mod(1,1150)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1150.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1150 = 2 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1150.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,-10,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$67.8521965066$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 38x^{3} + 38x^{2} + 202x + 101 $ x^5 - x^4 - 38x^3 + 38x^2 + 202*x + 101
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.50693$ of defining polynomial
Character	$\chi$	$=$	1150.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} +5.31348 q^{3} +4.00000 q^{4} -10.6270 q^{6} -34.1697 q^{7} -8.00000 q^{8} +1.23302 q^{9} -58.9953 q^{11} +21.2539 q^{12} -57.0300 q^{13} +68.3394 q^{14} +16.0000 q^{16} +116.881 q^{17} -2.46604 q^{18} -35.0292 q^{19} -181.560 q^{21} +117.991 q^{22} -23.0000 q^{23} -42.5078 q^{24} +114.060 q^{26} -136.912 q^{27} -136.679 q^{28} -171.371 q^{29} +131.300 q^{31} -32.0000 q^{32} -313.470 q^{33} -233.761 q^{34} +4.93208 q^{36} +325.843 q^{37} +70.0584 q^{38} -303.028 q^{39} -72.3772 q^{41} +363.120 q^{42} +59.4075 q^{43} -235.981 q^{44} +46.0000 q^{46} +479.286 q^{47} +85.0156 q^{48} +824.568 q^{49} +621.043 q^{51} -228.120 q^{52} -230.887 q^{53} +273.824 q^{54} +273.358 q^{56} -186.127 q^{57} +342.743 q^{58} +495.740 q^{59} +267.995 q^{61} -262.601 q^{62} -42.1319 q^{63} +64.0000 q^{64} +626.940 q^{66} +11.1110 q^{67} +467.523 q^{68} -122.210 q^{69} -353.044 q^{71} -9.86416 q^{72} +466.427 q^{73} -651.686 q^{74} -140.117 q^{76} +2015.85 q^{77} +606.056 q^{78} -773.967 q^{79} -760.771 q^{81} +144.754 q^{82} +22.4688 q^{83} -726.239 q^{84} -118.815 q^{86} -910.578 q^{87} +471.962 q^{88} +1172.86 q^{89} +1948.70 q^{91} -92.0000 q^{92} +697.661 q^{93} -958.572 q^{94} -170.031 q^{96} +1318.71 q^{97} -1649.14 q^{98} -72.7424 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 10 q^{2} + 12 q^{3} + 20 q^{4} - 24 q^{6} - 24 q^{7} - 40 q^{8} + 11 q^{9} - 54 q^{11} + 48 q^{12} + 36 q^{13} + 48 q^{14} + 80 q^{16} + 132 q^{17} - 22 q^{18} - 50 q^{19} - 158 q^{21} + 108 q^{22}+ \cdots - 1740 q^{99}+O(q^{100}) $ 5 * q - 10 * q^2 + 12 * q^3 + 20 * q^4 - 24 * q^6 - 24 * q^7 - 40 * q^8 + 11 * q^9 - 54 * q^11 + 48 * q^12 + 36 * q^13 + 48 * q^14 + 80 * q^16 + 132 * q^17 - 22 * q^18 - 50 * q^19 - 158 * q^21 + 108 * q^22 - 115 * q^23 - 96 * q^24 - 72 * q^26 + 489 * q^27 - 96 * q^28 - 104 * q^29 - 342 * q^31 - 160 * q^32 - 348 * q^33 - 264 * q^34 + 44 * q^36 + 380 * q^37 + 100 * q^38 + 193 * q^39 + 2 * q^41 + 316 * q^42 - 114 * q^43 - 216 * q^44 + 230 * q^46 + 966 * q^47 + 192 * q^48 - 441 * q^49 + 1200 * q^51 + 144 * q^52 + 308 * q^53 - 978 * q^54 + 192 * q^56 + 1410 * q^57 + 208 * q^58 - 277 * q^59 + 482 * q^61 + 684 * q^62 - 222 * q^63 + 320 * q^64 + 696 * q^66 + 2044 * q^67 + 528 * q^68 - 276 * q^69 + 208 * q^71 - 88 * q^72 + 646 * q^73 - 760 * q^74 - 200 * q^76 + 2124 * q^77 - 386 * q^78 - 3444 * q^79 + 2933 * q^81 - 4 * q^82 + 1218 * q^83 - 632 * q^84 + 228 * q^86 + 103 * q^87 + 432 * q^88 + 140 * q^89 + 1976 * q^91 - 460 * q^92 + 473 * q^93 - 1932 * q^94 - 384 * q^96 + 2816 * q^97 + 882 * q^98 - 1740 * 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
$3$	5.31348	1.02258	0.511289	−	0.859409i	$-0.329168\pi$
$3$	5.31348	1.02258	0.511289	+	0.859409i	$0.329168\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−10.6270	−0.723072
$7$	−34.1697	−1.84499	−0.922495	−	0.386009i	$-0.873853\pi$
$7$	−34.1697	−1.84499	−0.922495	+	0.386009i	$0.873853\pi$
$8$	−8.00000	−0.353553
$9$	1.23302	0.0456674
$10$	0	0
$11$	−58.9953	−1.61707	−0.808534	−	0.588449i	$-0.799739\pi$
$11$	−58.9953	−1.61707	−0.808534	+	0.588449i	$0.799739\pi$
$12$	21.2539	0.511289
$13$	−57.0300	−1.21671	−0.608357	−	0.793663i	$-0.708171\pi$
$13$	−57.0300	−1.21671	−0.608357	+	0.793663i	$0.708171\pi$
$14$	68.3394	1.30460
$15$	0	0
$16$	16.0000	0.250000
$17$	116.881	1.66751	0.833757	−	0.552132i	$-0.186186\pi$
$17$	116.881	1.66751	0.833757	+	0.552132i	$0.186186\pi$
$18$	−2.46604	−0.0322917
$19$	−35.0292	−0.422960	−0.211480	−	0.977382i	$-0.567828\pi$
$19$	−35.0292	−0.422960	−0.211480	+	0.977382i	$0.567828\pi$
$20$	0	0
$21$	−181.560	−1.88665
$22$	117.991	1.14344
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	−42.5078	−0.361536
$25$	0	0
$26$	114.060	0.860347
$27$	−136.912	−0.975880
$28$	−136.679	−0.922495
$29$	−171.371	−1.09734	−0.548670	−	0.836039i	$-0.684866\pi$
$29$	−171.371	−1.09734	−0.548670	+	0.836039i	$0.684866\pi$
$30$	0	0
$31$	131.300	0.760717	0.380359	−	0.924839i	$-0.375801\pi$
$31$	131.300	0.760717	0.380359	+	0.924839i	$0.375801\pi$
$32$	−32.0000	−0.176777
$33$	−313.470	−1.65358
$34$	−233.761	−1.17911
$35$	0	0
$36$	4.93208	0.0228337
$37$	325.843	1.44779	0.723896	−	0.689910i	$-0.242350\pi$
$37$	325.843	1.44779	0.723896	+	0.689910i	$0.242350\pi$
$38$	70.0584	0.299078
$39$	−303.028	−1.24419
$40$	0	0
$41$	−72.3772	−0.275693	−0.137847	−	0.990454i	$-0.544018\pi$
$41$	−72.3772	−0.275693	−0.137847	+	0.990454i	$0.544018\pi$
$42$	363.120	1.33406
$43$	59.4075	0.210688	0.105344	−	0.994436i	$-0.466406\pi$
$43$	59.4075	0.210688	0.105344	+	0.994436i	$0.466406\pi$
$44$	−235.981	−0.808534
$45$	0	0
$46$	46.0000	0.147442
$47$	479.286	1.48747	0.743735	−	0.668475i	$-0.233053\pi$
$47$	479.286	1.48747	0.743735	+	0.668475i	$0.233053\pi$
$48$	85.0156	0.255645
$49$	824.568	2.40399
$50$	0	0
$51$	621.043	1.70516
$52$	−228.120	−0.608357
$53$	−230.887	−0.598392	−0.299196	−	0.954192i	$-0.596718\pi$
$53$	−230.887	−0.598392	−0.299196	+	0.954192i	$0.596718\pi$
$54$	273.824	0.690052
$55$	0	0
$56$	273.358	0.652302
$57$	−186.127	−0.432510
$58$	342.743	0.775937
$59$	495.740	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$59$	495.740	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$60$	0	0
$61$	267.995	0.562512	0.281256	−	0.959633i	$-0.409249\pi$
$61$	267.995	0.562512	0.281256	+	0.959633i	$0.409249\pi$
$62$	−262.601	−0.537908
$63$	−42.1319	−0.0842559
$64$	64.0000	0.125000
$65$	0	0
$66$	626.940	1.16926
$67$	11.1110	0.0202601	0.0101301	−	0.999949i	$-0.496775\pi$
$67$	11.1110	0.0202601	0.0101301	+	0.999949i	$0.496775\pi$
$68$	467.523	0.833757
$69$	−122.210	−0.213222
$70$	0	0
$71$	−353.044	−0.590121	−0.295060	−	0.955479i	$-0.595340\pi$
$71$	−353.044	−0.590121	−0.295060	+	0.955479i	$0.595340\pi$
$72$	−9.86416	−0.0161459
$73$	466.427	0.747824	0.373912	−	0.927464i	$-0.378016\pi$
$73$	466.427	0.747824	0.373912	+	0.927464i	$0.378016\pi$
$74$	−651.686	−1.02374
$75$	0	0
$76$	−140.117	−0.211480
$77$	2015.85	2.98348
$78$	606.056	0.879773
$79$	−773.967	−1.10225	−0.551127	−	0.834421i	$-0.685802\pi$
$79$	−773.967	−1.10225	−0.551127	+	0.834421i	$0.685802\pi$
$80$	0	0
$81$	−760.771	−1.04358
$82$	144.754	0.194945
$83$	22.4688	0.0297141	0.0148571	−	0.999890i	$-0.495271\pi$
$83$	22.4688	0.0297141	0.0148571	+	0.999890i	$0.495271\pi$
$84$	−726.239	−0.943324
$85$	0	0
$86$	−118.815	−0.148979
$87$	−910.578	−1.12212
$88$	471.962	0.571720
$89$	1172.86	1.39688	0.698441	−	0.715667i	$-0.253877\pi$
$89$	1172.86	1.39688	0.698441	+	0.715667i	$0.253877\pi$
$90$	0	0
$91$	1948.70	2.24483
$92$	−92.0000	−0.104257
$93$	697.661	0.777893
$94$	−958.572	−1.05180
$95$	0	0
$96$	−170.031	−0.180768
$97$	1318.71	1.38035	0.690177	−	0.723641i	$-0.257533\pi$
$97$	1318.71	1.38035	0.690177	+	0.723641i	$0.257533\pi$
$98$	−1649.14	−1.69988
$99$	−72.7424	−0.0738473
$100$	0	0
$101$	−1452.56	−1.43104	−0.715522	−	0.698590i	$-0.753811\pi$
$101$	−1452.56	−1.43104	−0.715522	+	0.698590i	$0.753811\pi$
$102$	−1242.09	−1.20573
$103$	−481.080	−0.460216	−0.230108	−	0.973165i	$-0.573908\pi$
$103$	−481.080	−0.460216	−0.230108	+	0.973165i	$0.573908\pi$
$104$	456.240	0.430174
$105$	0	0
$106$	461.774	0.423127
$107$	176.518	0.159483	0.0797415	−	0.996816i	$-0.474591\pi$
$107$	176.518	0.159483	0.0797415	+	0.996816i	$0.474591\pi$
$108$	−547.649	−0.487940
$109$	−336.029	−0.295282	−0.147641	−	0.989041i	$-0.547168\pi$
$109$	−336.029	−0.295282	−0.147641	+	0.989041i	$0.547168\pi$
$110$	0	0
$111$	1731.36	1.48048
$112$	−546.715	−0.461247
$113$	1140.89	0.949790	0.474895	−	0.880043i	$-0.342486\pi$
$113$	1140.89	0.949790	0.474895	+	0.880043i	$0.342486\pi$
$114$	372.253	0.305831
$115$	0	0
$116$	−685.486	−0.548670
$117$	−70.3192	−0.0555642
$118$	−991.481	−0.773502
$119$	−3993.78	−3.07654
$120$	0	0
$121$	2149.45	1.61491
$122$	−535.990	−0.397756
$123$	−384.575	−0.281918
$124$	525.201	0.380359
$125$	0	0
$126$	84.2638	0.0595779
$127$	1919.42	1.34111	0.670556	−	0.741859i	$-0.266056\pi$
$127$	1919.42	1.34111	0.670556	+	0.741859i	$0.266056\pi$
$128$	−128.000	−0.0883883
$129$	315.660	0.215445
$130$	0	0
$131$	−2728.18	−1.81956	−0.909778	−	0.415094i	$-0.863749\pi$
$131$	−2728.18	−1.81956	−0.909778	+	0.415094i	$0.863749\pi$
$132$	−1253.88	−0.826790
$133$	1196.94	0.780358
$134$	−22.2221	−0.0143261
$135$	0	0
$136$	−935.045	−0.589555
$137$	−1498.15	−0.934271	−0.467136	−	0.884186i	$-0.654714\pi$
$137$	−1498.15	−0.934271	−0.467136	+	0.884186i	$0.654714\pi$
$138$	244.420	0.150771
$139$	−3070.10	−1.87340	−0.936700	−	0.350132i	$-0.886137\pi$
$139$	−3070.10	−1.87340	−0.936700	+	0.350132i	$0.886137\pi$
$140$	0	0
$141$	2546.68	1.52105
$142$	706.087	0.417278
$143$	3364.51	1.96751
$144$	19.7283	0.0114169
$145$	0	0
$146$	−932.854	−0.528791
$147$	4381.32	2.45827
$148$	1303.37	0.723896
$149$	−2446.56	−1.34517	−0.672585	−	0.740020i	$-0.734816\pi$
$149$	−2446.56	−1.34517	−0.672585	+	0.740020i	$0.734816\pi$
$150$	0	0
$151$	1398.67	0.753789	0.376895	−	0.926256i	$-0.376992\pi$
$151$	1398.67	0.753789	0.376895	+	0.926256i	$0.376992\pi$
$152$	280.234	0.149539
$153$	144.116	0.0761510
$154$	−4031.70	−2.10964
$155$	0	0
$156$	−1212.11	−0.622093
$157$	−1629.82	−0.828495	−0.414248	−	0.910164i	$-0.635955\pi$
$157$	−1629.82	−0.828495	−0.414248	+	0.910164i	$0.635955\pi$
$158$	1547.93	0.779412
$159$	−1226.81	−0.611903
$160$	0	0
$161$	785.903	0.384707
$162$	1521.54	0.737924
$163$	−209.019	−0.100439	−0.0502197	−	0.998738i	$-0.515992\pi$
$163$	−209.019	−0.100439	−0.0502197	+	0.998738i	$0.515992\pi$
$164$	−289.509	−0.137847
$165$	0	0
$166$	−44.9376	−0.0210111
$167$	891.381	0.413037	0.206518	−	0.978443i	$-0.433787\pi$
$167$	891.381	0.413037	0.206518	+	0.978443i	$0.433787\pi$
$168$	1452.48	0.667031
$169$	1055.43	0.480394
$170$	0	0
$171$	−43.1917	−0.0193155
$172$	237.630	0.105344
$173$	1198.99	0.526922	0.263461	−	0.964670i	$-0.415136\pi$
$173$	1198.99	0.526922	0.263461	+	0.964670i	$0.415136\pi$
$174$	1821.16	0.793457
$175$	0	0
$176$	−943.925	−0.404267
$177$	2634.10	1.11860
$178$	−2345.71	−0.987745
$179$	−1993.46	−0.832392	−0.416196	−	0.909275i	$-0.636637\pi$
$179$	−1993.46	−0.832392	−0.416196	+	0.909275i	$0.636637\pi$
$180$	0	0
$181$	−1964.42	−0.806707	−0.403354	−	0.915044i	$-0.632156\pi$
$181$	−1964.42	−0.806707	−0.403354	+	0.915044i	$0.632156\pi$
$182$	−3897.40	−1.58733
$183$	1423.98	0.575213
$184$	184.000	0.0737210
$185$	0	0
$186$	−1395.32	−0.550054
$187$	−6895.41	−2.69648
$188$	1917.14	0.743735
$189$	4678.25	1.80049
$190$	0	0
$191$	−1282.64	−0.485910	−0.242955	−	0.970038i	$-0.578117\pi$
$191$	−1282.64	−0.485910	−0.242955	+	0.970038i	$0.578117\pi$
$192$	340.062	0.127822
$193$	561.448	0.209399	0.104699	−	0.994504i	$-0.466612\pi$
$193$	561.448	0.209399	0.104699	+	0.994504i	$0.466612\pi$
$194$	−2637.41	−0.976058
$195$	0	0
$196$	3298.27	1.20199
$197$	4599.35	1.66340	0.831701	−	0.555224i	$-0.187367\pi$
$197$	4599.35	1.66340	0.831701	+	0.555224i	$0.187367\pi$
$198$	145.485	0.0522180
$199$	331.109	0.117948	0.0589740	−	0.998260i	$-0.481217\pi$
$199$	331.109	0.117948	0.0589740	+	0.998260i	$0.481217\pi$
$200$	0	0
$201$	59.0382	0.0207176
$202$	2905.13	1.01190
$203$	5855.71	2.02458
$204$	2484.17	0.852582
$205$	0	0
$206$	962.160	0.325422
$207$	−28.3595	−0.00952231
$208$	−912.481	−0.304179
$209$	2066.56	0.683956
$210$	0	0
$211$	−3949.25	−1.28852	−0.644260	−	0.764807i	$-0.722834\pi$
$211$	−3949.25	−1.28852	−0.644260	+	0.764807i	$0.722834\pi$
$212$	−923.549	−0.299196
$213$	−1875.89	−0.603445
$214$	−353.037	−0.112772
$215$	0	0
$216$	1095.30	0.345026
$217$	−4486.49	−1.40352
$218$	672.059	0.208796
$219$	2478.35	0.764709
$220$	0	0
$221$	−6665.71	−2.02889
$222$	−3462.72	−1.04686
$223$	−3592.14	−1.07869	−0.539344	−	0.842085i	$-0.681328\pi$
$223$	−3592.14	−1.07869	−0.539344	+	0.842085i	$0.681328\pi$
$224$	1093.43	0.326151
$225$	0	0
$226$	−2281.79	−0.671603
$227$	−3349.51	−0.979359	−0.489680	−	0.871902i	$-0.662886\pi$
$227$	−3349.51	−0.979359	−0.489680	+	0.871902i	$0.662886\pi$
$228$	−744.507	−0.216255
$229$	2763.74	0.797525	0.398763	−	0.917054i	$-0.369440\pi$
$229$	2763.74	0.797525	0.398763	+	0.917054i	$0.369440\pi$
$230$	0	0
$231$	10711.2	3.05084
$232$	1370.97	0.387968
$233$	2085.17	0.586285	0.293142	−	0.956069i	$-0.405299\pi$
$233$	2085.17	0.586285	0.293142	+	0.956069i	$0.405299\pi$
$234$	140.638	0.0392898
$235$	0	0
$236$	1982.96	0.546948
$237$	−4112.45	−1.12714
$238$	7987.55	2.17545
$239$	4763.85	1.28932	0.644660	−	0.764469i	$-0.276999\pi$
$239$	4763.85	1.28932	0.644660	+	0.764469i	$0.276999\pi$
$240$	0	0
$241$	6928.92	1.85200	0.925998	−	0.377528i	$-0.123226\pi$
$241$	6928.92	1.85200	0.925998	+	0.377528i	$0.123226\pi$
$242$	−4298.89	−1.14191
$243$	−345.709	−0.0912645
$244$	1071.98	0.281256
$245$	0	0
$246$	769.149	0.199346
$247$	1997.72	0.514622
$248$	−1050.40	−0.268954
$249$	119.387	0.0303850
$250$	0	0
$251$	4718.91	1.18667	0.593336	−	0.804955i	$-0.297810\pi$
$251$	4718.91	1.18667	0.593336	+	0.804955i	$0.297810\pi$
$252$	−168.528	−0.0421280
$253$	1356.89	0.337182
$254$	−3838.84	−0.948309
$255$	0	0
$256$	256.000	0.0625000
$257$	7890.24	1.91510	0.957548	−	0.288275i	$-0.0930817\pi$
$257$	7890.24	1.91510	0.957548	+	0.288275i	$0.0930817\pi$
$258$	−631.321	−0.152342
$259$	−11134.0	−2.67116
$260$	0	0
$261$	−211.304	−0.0501127
$262$	5456.35	1.28662
$263$	129.509	0.0303645	0.0151822	−	0.999885i	$-0.495167\pi$
$263$	129.509	0.0303645	0.0151822	+	0.999885i	$0.495167\pi$
$264$	2507.76	0.584629
$265$	0	0
$266$	−2393.87	−0.551796
$267$	6231.95	1.42842
$268$	44.4441	0.0101301
$269$	5625.75	1.27512	0.637561	−	0.770400i	$-0.279943\pi$
$269$	5625.75	1.27512	0.637561	+	0.770400i	$0.279943\pi$
$270$	0	0
$271$	5008.19	1.12260	0.561302	−	0.827611i	$-0.310301\pi$
$271$	5008.19	1.12260	0.561302	+	0.827611i	$0.310301\pi$
$272$	1870.09	0.416878
$273$	10354.4	2.29551
$274$	2996.29	0.660630
$275$	0	0
$276$	−488.840	−0.106611
$277$	−6191.44	−1.34299	−0.671494	−	0.741010i	$-0.734347\pi$
$277$	−6191.44	−1.34299	−0.671494	+	0.741010i	$0.734347\pi$
$278$	6140.21	1.32469
$279$	161.896	0.0347400
$280$	0	0
$281$	7739.13	1.64298	0.821491	−	0.570221i	$-0.193142\pi$
$281$	7739.13	1.64298	0.821491	+	0.570221i	$0.193142\pi$
$282$	−5093.35	−1.07555
$283$	4201.98	0.882621	0.441310	−	0.897354i	$-0.354514\pi$
$283$	4201.98	0.882621	0.441310	+	0.897354i	$0.354514\pi$
$284$	−1412.17	−0.295060
$285$	0	0
$286$	−6729.01	−1.39124
$287$	2473.11	0.508651
$288$	−39.4566	−0.00807293
$289$	8748.09	1.78060
$290$	0	0
$291$	7006.91	1.41152
$292$	1865.71	0.373912
$293$	−3293.70	−0.656724	−0.328362	−	0.944552i	$-0.606497\pi$
$293$	−3293.70	−0.656724	−0.328362	+	0.944552i	$0.606497\pi$
$294$	−8762.64	−1.73826
$295$	0	0
$296$	−2606.74	−0.511871
$297$	8077.18	1.57807
$298$	4893.12	0.951178
$299$	1311.69	0.253703
$300$	0	0
$301$	−2029.94	−0.388716
$302$	−2797.34	−0.533009
$303$	−7718.16	−1.46336
$304$	−560.467	−0.105740
$305$	0	0
$306$	−288.232	−0.0538469
$307$	1088.67	0.202390	0.101195	−	0.994867i	$-0.467733\pi$
$307$	1088.67	0.202390	0.101195	+	0.994867i	$0.467733\pi$
$308$	8063.41	1.49174
$309$	−2556.21	−0.470607
$310$	0	0
$311$	3853.58	0.702625	0.351313	−	0.936258i	$-0.385735\pi$
$311$	3853.58	0.702625	0.351313	+	0.936258i	$0.385735\pi$
$312$	2424.22	0.439886
$313$	−8534.78	−1.54126	−0.770630	−	0.637282i	$-0.780058\pi$
$313$	−8534.78	−1.54126	−0.770630	+	0.637282i	$0.780058\pi$
$314$	3259.64	0.585835
$315$	0	0
$316$	−3095.87	−0.551127
$317$	4063.86	0.720029	0.360015	−	0.932947i	$-0.382772\pi$
$317$	4063.86	0.720029	0.360015	+	0.932947i	$0.382772\pi$
$318$	2453.63	0.432681
$319$	10110.1	1.77447
$320$	0	0
$321$	937.926	0.163084
$322$	−1571.81	−0.272029
$323$	−4094.23	−0.705292
$324$	−3043.08	−0.521791
$325$	0	0
$326$	418.038	0.0710214
$327$	−1785.48	−0.301949
$328$	579.018	0.0974723
$329$	−16377.1	−2.74437
$330$	0	0
$331$	−7528.47	−1.25016	−0.625079	−	0.780562i	$-0.714933\pi$
$331$	−7528.47	−1.25016	−0.625079	+	0.780562i	$0.714933\pi$
$332$	89.8753	0.0148571
$333$	401.771	0.0661169
$334$	−1782.76	−0.292061
$335$	0	0
$336$	−2904.96	−0.471662
$337$	5076.22	0.820533	0.410266	−	0.911966i	$-0.365436\pi$
$337$	5076.22	0.820533	0.410266	+	0.911966i	$0.365436\pi$
$338$	−2110.85	−0.339690
$339$	6062.11	0.971235
$340$	0	0
$341$	−7746.10	−1.23013
$342$	86.3834	0.0136581
$343$	−16455.0	−2.59034
$344$	−475.260	−0.0744893
$345$	0	0
$346$	−2397.98	−0.372590
$347$	1637.97	0.253404	0.126702	−	0.991941i	$-0.459561\pi$
$347$	1637.97	0.253404	0.126702	+	0.991941i	$0.459561\pi$
$348$	−3642.31	−0.561059
$349$	−9786.85	−1.50108	−0.750542	−	0.660823i	$-0.770208\pi$
$349$	−9786.85	−1.50108	−0.750542	+	0.660823i	$0.770208\pi$
$350$	0	0
$351$	7808.11	1.18737
$352$	1887.85	0.285860
$353$	−11697.5	−1.76372	−0.881860	−	0.471512i	$-0.843708\pi$
$353$	−11697.5	−1.76372	−0.881860	+	0.471512i	$0.843708\pi$
$354$	−5268.21	−0.790967
$355$	0	0
$356$	4691.43	0.698441
$357$	−21220.8	−3.14601
$358$	3986.92	0.588590
$359$	1498.68	0.220327	0.110164	−	0.993913i	$-0.464863\pi$
$359$	1498.68	0.220327	0.110164	+	0.993913i	$0.464863\pi$
$360$	0	0
$361$	−5631.96	−0.821105
$362$	3928.84	0.570428
$363$	11421.0	1.65137
$364$	7794.80	1.12241
$365$	0	0
$366$	−2847.97	−0.406737
$367$	750.417	0.106734	0.0533671	−	0.998575i	$-0.483005\pi$
$367$	750.417	0.106734	0.0533671	+	0.998575i	$0.483005\pi$
$368$	−368.000	−0.0521286
$369$	−89.2426	−0.0125902
$370$	0	0
$371$	7889.34	1.10403
$372$	2790.64	0.388947
$373$	−12191.6	−1.69238	−0.846190	−	0.532882i	$-0.821109\pi$
$373$	−12191.6	−1.69238	−0.846190	+	0.532882i	$0.821109\pi$
$374$	13790.8	1.90670
$375$	0	0
$376$	−3834.29	−0.525900
$377$	9773.32	1.33515
$378$	−9356.50	−1.27314
$379$	2267.64	0.307337	0.153668	−	0.988122i	$-0.450891\pi$
$379$	2267.64	0.307337	0.153668	+	0.988122i	$0.450891\pi$
$380$	0	0
$381$	10198.8	1.37139
$382$	2565.29	0.343590
$383$	7320.76	0.976693	0.488346	−	0.872650i	$-0.337600\pi$
$383$	7320.76	0.976693	0.488346	+	0.872650i	$0.337600\pi$
$384$	−680.125	−0.0903841
$385$	0	0
$386$	−1122.90	−0.148067
$387$	73.2507	0.00962155
$388$	5274.82	0.690177
$389$	−3099.78	−0.404024	−0.202012	−	0.979383i	$-0.564748\pi$
$389$	−3099.78	−0.404024	−0.202012	+	0.979383i	$0.564748\pi$
$390$	0	0
$391$	−2688.26	−0.347701
$392$	−6596.54	−0.849938
$393$	−14496.1	−1.86064
$394$	−9198.70	−1.17620
$395$	0	0
$396$	−290.970	−0.0369237
$397$	6476.52	0.818759	0.409380	−	0.912364i	$-0.365745\pi$
$397$	6476.52	0.818759	0.409380	+	0.912364i	$0.365745\pi$
$398$	−662.217	−0.0834019
$399$	6359.89	0.797977
$400$	0	0
$401$	9038.72	1.12562	0.562808	−	0.826588i	$-0.309721\pi$
$401$	9038.72	1.12562	0.562808	+	0.826588i	$0.309721\pi$
$402$	−118.076	−0.0146495
$403$	−7488.06	−0.925576
$404$	−5810.25	−0.715522
$405$	0	0
$406$	−11711.4	−1.43160
$407$	−19223.2	−2.34118
$408$	−4968.34	−0.602866
$409$	1463.62	0.176948	0.0884738	−	0.996079i	$-0.471801\pi$
$409$	1463.62	0.176948	0.0884738	+	0.996079i	$0.471801\pi$
$410$	0	0
$411$	−7960.36	−0.955366
$412$	−1924.32	−0.230108
$413$	−16939.3	−2.01823
$414$	56.7189	0.00673329
$415$	0	0
$416$	1824.96	0.215087
$417$	−16312.9	−1.91570
$418$	−4133.12	−0.483630
$419$	−3240.74	−0.377853	−0.188926	−	0.981991i	$-0.560501\pi$
$419$	−3240.74	−0.377853	−0.188926	+	0.981991i	$0.560501\pi$
$420$	0	0
$421$	11983.1	1.38722	0.693608	−	0.720352i	$-0.256020\pi$
$421$	11983.1	1.38722	0.693608	+	0.720352i	$0.256020\pi$
$422$	7898.50	0.911121
$423$	590.969	0.0679289
$424$	1847.10	0.211564
$425$	0	0
$426$	3751.78	0.426700
$427$	−9157.30	−1.03783
$428$	706.074	0.0797415
$429$	17877.2	2.01194
$430$	0	0
$431$	−7554.30	−0.844265	−0.422132	−	0.906534i	$-0.638718\pi$
$431$	−7554.30	−0.844265	−0.422132	+	0.906534i	$0.638718\pi$
$432$	−2190.60	−0.243970
$433$	13051.5	1.44854	0.724268	−	0.689519i	$-0.242178\pi$
$433$	13051.5	1.44854	0.724268	+	0.689519i	$0.242178\pi$
$434$	8972.98	0.992435
$435$	0	0
$436$	−1344.12	−0.147641
$437$	805.671	0.0881933
$438$	−4956.70	−0.540731
$439$	−6821.34	−0.741606	−0.370803	−	0.928712i	$-0.620917\pi$
$439$	−6821.34	−0.741606	−0.370803	+	0.928712i	$0.620917\pi$
$440$	0	0
$441$	1016.71	0.109784
$442$	13331.4	1.43464
$443$	10760.1	1.15402	0.577008	−	0.816739i	$-0.304220\pi$
$443$	10760.1	1.15402	0.577008	+	0.816739i	$0.304220\pi$
$444$	6925.44	0.740240
$445$	0	0
$446$	7184.28	0.762748
$447$	−12999.7	−1.37554
$448$	−2186.86	−0.230624
$449$	7503.52	0.788671	0.394335	−	0.918967i	$-0.370975\pi$
$449$	7503.52	0.788671	0.394335	+	0.918967i	$0.370975\pi$
$450$	0	0
$451$	4269.92	0.445815
$452$	4563.57	0.474895
$453$	7431.80	0.770809
$454$	6699.01	0.692512
$455$	0	0
$456$	1489.01	0.152915
$457$	18746.2	1.91884	0.959421	−	0.281977i	$-0.0909903\pi$
$457$	18746.2	1.91884	0.959421	+	0.281977i	$0.0909903\pi$
$458$	−5527.48	−0.563935
$459$	−16002.4	−1.62729
$460$	0	0
$461$	−3461.92	−0.349756	−0.174878	−	0.984590i	$-0.555953\pi$
$461$	−3461.92	−0.349756	−0.174878	+	0.984590i	$0.555953\pi$
$462$	−21422.4	−2.15727
$463$	−12567.6	−1.26148	−0.630741	−	0.775993i	$-0.717249\pi$
$463$	−12567.6	−1.26148	−0.630741	+	0.775993i	$0.717249\pi$
$464$	−2741.94	−0.274335
$465$	0	0
$466$	−4170.35	−0.414566
$467$	15137.8	1.49999	0.749995	−	0.661443i	$-0.230056\pi$
$467$	15137.8	1.49999	0.749995	+	0.661443i	$0.230056\pi$
$468$	−281.277	−0.0277821
$469$	−379.661	−0.0373797
$470$	0	0
$471$	−8660.01	−0.847202
$472$	−3965.92	−0.386751
$473$	−3504.77	−0.340696
$474$	8224.91	0.797010
$475$	0	0
$476$	−15975.1	−1.53827
$477$	−284.689	−0.0273270
$478$	−9527.69	−0.911687
$479$	10179.6	0.971021	0.485511	−	0.874231i	$-0.338634\pi$
$479$	10179.6	0.971021	0.485511	+	0.874231i	$0.338634\pi$
$480$	0	0
$481$	−18582.8	−1.76155
$482$	−13857.8	−1.30956
$483$	4175.88	0.393393
$484$	8597.79	0.807456
$485$	0	0
$486$	691.419	0.0645337
$487$	10705.5	0.996123	0.498061	−	0.867142i	$-0.334045\pi$
$487$	10705.5	0.996123	0.498061	+	0.867142i	$0.334045\pi$
$488$	−2143.96	−0.198878
$489$	−1110.62	−0.102707
$490$	0	0
$491$	1389.39	0.127704	0.0638518	−	0.997959i	$-0.479661\pi$
$491$	1389.39	0.127704	0.0638518	+	0.997959i	$0.479661\pi$
$492$	−1538.30	−0.140959
$493$	−20030.0	−1.82983
$494$	−3995.43	−0.363893
$495$	0	0
$496$	2100.80	0.190179
$497$	12063.4	1.08877
$498$	−238.775	−0.0214855
$499$	6024.03	0.540426	0.270213	−	0.962801i	$-0.412906\pi$
$499$	6024.03	0.540426	0.270213	+	0.962801i	$0.412906\pi$
$500$	0	0
$501$	4736.33	0.422363
$502$	−9437.82	−0.839104
$503$	897.694	0.0795749	0.0397875	−	0.999208i	$-0.487332\pi$
$503$	897.694	0.0795749	0.0397875	+	0.999208i	$0.487332\pi$
$504$	337.055	0.0297890
$505$	0	0
$506$	−2713.78	−0.238424
$507$	5607.98	0.491241
$508$	7677.69	0.670556
$509$	−2501.06	−0.217795	−0.108897	−	0.994053i	$-0.534732\pi$
$509$	−2501.06	−0.217795	−0.108897	+	0.994053i	$0.534732\pi$
$510$	0	0
$511$	−15937.7	−1.37973
$512$	−512.000	−0.0441942
$513$	4795.92	0.412759
$514$	−15780.5	−1.35418
$515$	0	0
$516$	1262.64	0.107722
$517$	−28275.6	−2.40534
$518$	22267.9	1.88880
$519$	6370.80	0.538819
$520$	0	0
$521$	−2231.18	−0.187619	−0.0938097	−	0.995590i	$-0.529905\pi$
$521$	−2231.18	−0.187619	−0.0938097	+	0.995590i	$0.529905\pi$
$522$	422.609	0.0354350
$523$	19156.5	1.60163	0.800816	−	0.598910i	$-0.204399\pi$
$523$	19156.5	1.60163	0.800816	+	0.598910i	$0.204399\pi$
$524$	−10912.7	−0.909778
$525$	0	0
$526$	−259.018	−0.0214709
$527$	15346.5	1.26851
$528$	−5015.52	−0.413395
$529$	529.000	0.0434783
$530$	0	0
$531$	611.258	0.0499554
$532$	4787.75	0.390179
$533$	4127.68	0.335440
$534$	−12463.9	−1.01005
$535$	0	0
$536$	−88.8883	−0.00716304
$537$	−10592.2	−0.851187
$538$	−11251.5	−0.901648
$539$	−48645.6	−3.88741
$540$	0	0
$541$	−3507.07	−0.278708	−0.139354	−	0.990243i	$-0.544503\pi$
$541$	−3507.07	−0.278708	−0.139354	+	0.990243i	$0.544503\pi$
$542$	−10016.4	−0.793801
$543$	−10437.9	−0.824922
$544$	−3740.18	−0.294777
$545$	0	0
$546$	−20708.7	−1.62317
$547$	18849.4	1.47339	0.736694	−	0.676227i	$-0.236386\pi$
$547$	18849.4	1.47339	0.736694	+	0.676227i	$0.236386\pi$
$548$	−5992.58	−0.467136
$549$	330.443	0.0256885
$550$	0	0
$551$	6003.00	0.464132
$552$	977.679	0.0753855
$553$	26446.2	2.03365
$554$	12382.9	0.949635
$555$	0	0
$556$	−12280.4	−0.936700
$557$	532.356	0.0404966	0.0202483	−	0.999795i	$-0.493554\pi$
$557$	532.356	0.0404966	0.0202483	+	0.999795i	$0.493554\pi$
$558$	−323.792	−0.0245649
$559$	−3388.01	−0.256347
$560$	0	0
$561$	−36638.6	−2.75737
$562$	−15478.3	−1.16176
$563$	820.550	0.0614246	0.0307123	−	0.999528i	$-0.490222\pi$
$563$	820.550	0.0614246	0.0307123	+	0.999528i	$0.490222\pi$
$564$	10186.7	0.760527
$565$	0	0
$566$	−8403.96	−0.624107
$567$	25995.3	1.92540
$568$	2824.35	0.208639
$569$	24200.5	1.78302	0.891508	−	0.453006i	$-0.149648\pi$
$569$	24200.5	1.78302	0.891508	+	0.453006i	$0.149648\pi$
$570$	0	0
$571$	18192.0	1.33329	0.666646	−	0.745374i	$-0.267729\pi$
$571$	18192.0	1.33329	0.666646	+	0.745374i	$0.267729\pi$
$572$	13458.0	0.983756
$573$	−6815.29	−0.496881
$574$	−4946.21	−0.359671
$575$	0	0
$576$	78.9133	0.00570843
$577$	−21178.1	−1.52800	−0.763998	−	0.645218i	$-0.776766\pi$
$577$	−21178.1	−1.52800	−0.763998	+	0.645218i	$0.776766\pi$
$578$	−17496.2	−1.25907
$579$	2983.24	0.214127
$580$	0	0
$581$	−767.752	−0.0548223
$582$	−14013.8	−0.998096
$583$	13621.3	0.967641
$584$	−3731.42	−0.264396
$585$	0	0
$586$	6587.40	0.464374
$587$	−10093.9	−0.709745	−0.354873	−	0.934915i	$-0.615476\pi$
$587$	−10093.9	−0.709745	−0.354873	+	0.934915i	$0.615476\pi$
$588$	17525.3	1.22913
$589$	−4599.34	−0.321753
$590$	0	0
$591$	24438.5	1.70096
$592$	5213.49	0.361948
$593$	−14438.3	−0.999851	−0.499926	−	0.866068i	$-0.666639\pi$
$593$	−14438.3	−0.999851	−0.499926	+	0.866068i	$0.666639\pi$
$594$	−16154.4	−1.11586
$595$	0	0
$596$	−9786.25	−0.672585
$597$	1759.34	0.120611
$598$	−2623.38	−0.179395
$599$	3466.27	0.236440	0.118220	−	0.992987i	$-0.462281\pi$
$599$	3466.27	0.236440	0.118220	+	0.992987i	$0.462281\pi$
$600$	0	0
$601$	12108.7	0.821839	0.410919	−	0.911672i	$-0.365208\pi$
$601$	12108.7	0.821839	0.410919	+	0.911672i	$0.365208\pi$
$602$	4059.87	0.274864
$603$	13.7001	0.000925228 0
$604$	5594.68	0.376895
$605$	0	0
$606$	15436.3	1.03475
$607$	−5997.00	−0.401006	−0.200503	−	0.979693i	$-0.564258\pi$
$607$	−5997.00	−0.401006	−0.200503	+	0.979693i	$0.564258\pi$
$608$	1120.93	0.0747695
$609$	31114.2	2.07029
$610$	0	0
$611$	−27333.7	−1.80983
$612$	576.465	0.0380755
$613$	−7989.11	−0.526390	−0.263195	−	0.964743i	$-0.584776\pi$
$613$	−7989.11	−0.526390	−0.263195	+	0.964743i	$0.584776\pi$
$614$	−2177.34	−0.143111
$615$	0	0
$616$	−16126.8	−1.05482
$617$	−53.6752	−0.00350224	−0.00175112	−	0.999998i	$-0.500557\pi$
$617$	−53.6752	−0.00350224	−0.00175112	+	0.999998i	$0.500557\pi$
$618$	5112.41	0.332769
$619$	−40.1805	−0.00260903	−0.00130452	−	0.999999i	$-0.500415\pi$
$619$	−40.1805	−0.00260903	−0.00130452	+	0.999999i	$0.500415\pi$
$620$	0	0
$621$	3148.98	0.203485
$622$	−7707.16	−0.496831
$623$	−40076.2	−2.57723
$624$	−4848.44	−0.311047
$625$	0	0
$626$	17069.6	1.08984
$627$	10980.6	0.699399
$628$	−6519.28	−0.414248
$629$	38084.8	2.41421
$630$	0	0
$631$	−9418.81	−0.594227	−0.297113	−	0.954842i	$-0.596024\pi$
$631$	−9418.81	−0.594227	−0.297113	+	0.954842i	$0.596024\pi$
$632$	6191.74	0.389706
$633$	−20984.2	−1.31761
$634$	−8127.73	−0.509138
$635$	0	0
$636$	−4907.25	−0.305952
$637$	−47025.1	−2.92497
$638$	−20220.2	−1.25474
$639$	−435.310	−0.0269493
$640$	0	0
$641$	991.573	0.0610995	0.0305497	−	0.999533i	$-0.490274\pi$
$641$	991.573	0.0610995	0.0305497	+	0.999533i	$0.490274\pi$
$642$	−1875.85	−0.115318
$643$	23370.9	1.43337	0.716687	−	0.697395i	$-0.245657\pi$
$643$	23370.9	1.43337	0.716687	+	0.697395i	$0.245657\pi$
$644$	3143.61	0.192353
$645$	0	0
$646$	8188.47	0.498717
$647$	17039.2	1.03536	0.517682	−	0.855573i	$-0.326795\pi$
$647$	17039.2	1.03536	0.517682	+	0.855573i	$0.326795\pi$
$648$	6086.17	0.368962
$649$	−29246.4	−1.76891
$650$	0	0
$651$	−23838.9	−1.43520
$652$	−836.075	−0.0502197
$653$	−31674.7	−1.89820	−0.949102	−	0.314968i	$-0.898006\pi$
$653$	−31674.7	−1.89820	−0.949102	+	0.314968i	$0.898006\pi$
$654$	3570.97	0.213510
$655$	0	0
$656$	−1158.04	−0.0689233
$657$	575.114	0.0341512
$658$	32754.1	1.94056
$659$	25107.8	1.48416	0.742080	−	0.670311i	$-0.233839\pi$
$659$	25107.8	1.48416	0.742080	+	0.670311i	$0.233839\pi$
$660$	0	0
$661$	−21628.9	−1.27272	−0.636358	−	0.771394i	$-0.719560\pi$
$661$	−21628.9	−1.27272	−0.636358	+	0.771394i	$0.719560\pi$
$662$	15056.9	0.883995
$663$	−35418.1	−2.07470
$664$	−179.751	−0.0105055
$665$	0	0
$666$	−803.542	−0.0467517
$667$	3941.54	0.228811
$668$	3565.52	0.206518
$669$	−19086.8	−1.10304
$670$	0	0
$671$	−15810.4	−0.909621
$672$	5809.91	0.333515
$673$	−21063.2	−1.20643	−0.603214	−	0.797579i	$-0.706114\pi$
$673$	−21063.2	−1.20643	−0.603214	+	0.797579i	$0.706114\pi$
$674$	−10152.4	−0.580204
$675$	0	0
$676$	4221.71	0.240197
$677$	−1518.27	−0.0861921	−0.0430960	−	0.999071i	$-0.513722\pi$
$677$	−1518.27	−0.0861921	−0.0430960	+	0.999071i	$0.513722\pi$
$678$	−12124.2	−0.686767
$679$	−45059.8	−2.54674
$680$	0	0
$681$	−17797.5	−1.00147
$682$	15492.2	0.869834
$683$	−27013.3	−1.51337	−0.756687	−	0.653777i	$-0.773183\pi$
$683$	−27013.3	−1.51337	−0.756687	+	0.653777i	$0.773183\pi$
$684$	−172.767	−0.00965775
$685$	0	0
$686$	32910.0	1.83165
$687$	14685.1	0.815532
$688$	950.521	0.0526719
$689$	13167.5	0.728073
$690$	0	0
$691$	−29988.3	−1.65095	−0.825477	−	0.564435i	$-0.809094\pi$
$691$	−29988.3	−1.65095	−0.825477	+	0.564435i	$0.809094\pi$
$692$	4795.96	0.263461
$693$	2485.59	0.136248
$694$	−3275.95	−0.179183
$695$	0	0
$696$	7284.62	0.396728
$697$	−8459.50	−0.459722
$698$	19573.7	1.06143
$699$	11079.5	0.599522
$700$	0	0
$701$	7127.00	0.383999	0.191999	−	0.981395i	$-0.438503\pi$
$701$	7127.00	0.383999	0.191999	+	0.981395i	$0.438503\pi$
$702$	−15616.2	−0.839596
$703$	−11414.0	−0.612358
$704$	−3775.70	−0.202134
$705$	0	0
$706$	23394.9	1.24714
$707$	49633.6	2.64026
$708$	10536.4	0.559298
$709$	19983.2	1.05851	0.529257	−	0.848462i	$-0.322471\pi$
$709$	19983.2	1.05851	0.529257	+	0.848462i	$0.322471\pi$
$710$	0	0
$711$	−954.317	−0.0503371
$712$	−9382.85	−0.493873
$713$	−3019.91	−0.158620
$714$	42441.7	2.22456
$715$	0	0
$716$	−7973.84	−0.416196
$717$	25312.6	1.31843
$718$	−2997.36	−0.155795
$719$	−10043.0	−0.520921	−0.260460	−	0.965485i	$-0.583874\pi$
$719$	−10043.0	−0.520921	−0.260460	+	0.965485i	$0.583874\pi$
$720$	0	0
$721$	16438.4	0.849093
$722$	11263.9	0.580609
$723$	36816.7	1.89381
$724$	−7857.67	−0.403354
$725$	0	0
$726$	−22842.1	−1.16770
$727$	6665.79	0.340056	0.170028	−	0.985439i	$-0.445614\pi$
$727$	6665.79	0.340056	0.170028	+	0.985439i	$0.445614\pi$
$728$	−15589.6	−0.793666
$729$	18703.9	0.950257
$730$	0	0
$731$	6943.59	0.351324
$732$	5695.94	0.287606
$733$	18197.2	0.916955	0.458477	−	0.888706i	$-0.348395\pi$
$733$	18197.2	0.916955	0.458477	+	0.888706i	$0.348395\pi$
$734$	−1500.83	−0.0754725
$735$	0	0
$736$	736.000	0.0368605
$737$	−655.499	−0.0327620
$738$	178.485	0.00890261
$739$	−1847.66	−0.0919722	−0.0459861	−	0.998942i	$-0.514643\pi$
$739$	−1847.66	−0.0919722	−0.0459861	+	0.998942i	$0.514643\pi$
$740$	0	0
$741$	10614.8	0.526242
$742$	−15778.7	−0.780665
$743$	25291.0	1.24877	0.624385	−	0.781117i	$-0.285350\pi$
$743$	25291.0	1.24877	0.624385	+	0.781117i	$0.285350\pi$
$744$	−5581.29	−0.275027
$745$	0	0
$746$	24383.2	1.19669
$747$	27.7045	0.00135697
$748$	−27581.6	−1.34824
$749$	−6031.58	−0.294245
$750$	0	0
$751$	15059.5	0.731731	0.365865	−	0.930668i	$-0.380773\pi$
$751$	15059.5	0.731731	0.365865	+	0.930668i	$0.380773\pi$
$752$	7668.58	0.371867
$753$	25073.8	1.21347
$754$	−19546.6	−0.944094
$755$	0	0
$756$	18713.0	0.900245
$757$	−23808.2	−1.14310	−0.571548	−	0.820569i	$-0.693657\pi$
$757$	−23808.2	−1.14310	−0.571548	+	0.820569i	$0.693657\pi$
$758$	−4535.27	−0.217320
$759$	7209.81	0.344795
$760$	0	0
$761$	15673.8	0.746614	0.373307	−	0.927708i	$-0.378224\pi$
$761$	15673.8	0.746614	0.373307	+	0.927708i	$0.378224\pi$
$762$	−20397.6	−0.969721
$763$	11482.0	0.544793
$764$	−5130.57	−0.242955
$765$	0	0
$766$	−14641.5	−0.690626
$767$	−28272.1	−1.33096
$768$	1360.25	0.0639112
$769$	24213.5	1.13545	0.567724	−	0.823219i	$-0.307824\pi$
$769$	24213.5	1.13545	0.567724	+	0.823219i	$0.307824\pi$
$770$	0	0
$771$	41924.6	1.95834
$772$	2245.79	0.104699
$773$	34754.7	1.61713	0.808563	−	0.588410i	$-0.200246\pi$
$773$	34754.7	1.61713	0.808563	+	0.588410i	$0.200246\pi$
$774$	−146.501	−0.00680347
$775$	0	0
$776$	−10549.6	−0.488029
$777$	−59160.0	−2.73147
$778$	6199.56	0.285688
$779$	2535.32	0.116607
$780$	0	0
$781$	20827.9	0.954266
$782$	5376.51	0.245861
$783$	23462.8	1.07087
$784$	13193.1	0.600997
$785$	0	0
$786$	28992.2	1.31567
$787$	−7625.98	−0.345409	−0.172704	−	0.984974i	$-0.555251\pi$
$787$	−7625.98	−0.345409	−0.172704	+	0.984974i	$0.555251\pi$
$788$	18397.4	0.831701
$789$	688.142	0.0310501
$790$	0	0
$791$	−38984.0	−1.75235
$792$	581.939	0.0261090
$793$	−15283.8	−0.684417
$794$	−12953.0	−0.578950
$795$	0	0
$796$	1324.43	0.0589740
$797$	1536.17	0.0682733	0.0341367	−	0.999417i	$-0.489132\pi$
$797$	1536.17	0.0682733	0.0341367	+	0.999417i	$0.489132\pi$
$798$	−12719.8	−0.564255
$799$	56019.3	2.48038
$800$	0	0
$801$	1446.16	0.0637920
$802$	−18077.4	−0.795931
$803$	−27517.0	−1.20928
$804$	236.153	0.0103588
$805$	0	0
$806$	14976.1	0.654481
$807$	29892.3	1.30391
$808$	11620.5	0.505950
$809$	−2722.59	−0.118320	−0.0591601	−	0.998249i	$-0.518842\pi$
$809$	−2722.59	−0.118320	−0.0591601	+	0.998249i	$0.518842\pi$
$810$	0	0
$811$	828.222	0.0358604	0.0179302	−	0.999839i	$-0.494292\pi$
$811$	828.222	0.0358604	0.0179302	+	0.999839i	$0.494292\pi$
$812$	23422.8	1.01229
$813$	26610.9	1.14795
$814$	38446.4	1.65546
$815$	0	0
$816$	9936.68	0.426291
$817$	−2081.00	−0.0891125
$818$	−2927.25	−0.125121
$819$	2402.79	0.102515
$820$	0	0
$821$	32894.5	1.39832	0.699162	−	0.714963i	$-0.253556\pi$
$821$	32894.5	1.39832	0.699162	+	0.714963i	$0.253556\pi$
$822$	15920.7	0.675546
$823$	−18307.6	−0.775412	−0.387706	−	0.921783i	$-0.626733\pi$
$823$	−18307.6	−0.775412	−0.387706	+	0.921783i	$0.626733\pi$
$824$	3848.64	0.162711
$825$	0	0
$826$	33878.6	1.42710
$827$	−5237.10	−0.220208	−0.110104	−	0.993920i	$-0.535118\pi$
$827$	−5237.10	−0.220208	−0.110104	+	0.993920i	$0.535118\pi$
$828$	−113.438	−0.00476116
$829$	−2235.56	−0.0936601	−0.0468300	−	0.998903i	$-0.514912\pi$
$829$	−2235.56	−0.0936601	−0.0468300	+	0.998903i	$0.514912\pi$
$830$	0	0
$831$	−32898.1	−1.37331
$832$	−3649.92	−0.152089
$833$	96376.0	4.00868
$834$	32625.8	1.35460
$835$	0	0
$836$	8266.23	0.341978
$837$	−17976.6	−0.742369
$838$	6481.47	0.267182
$839$	−23031.3	−0.947711	−0.473856	−	0.880603i	$-0.657138\pi$
$839$	−23031.3	−0.947711	−0.473856	+	0.880603i	$0.657138\pi$
$840$	0	0
$841$	4979.16	0.204156
$842$	−23966.1	−0.980910
$843$	41121.7	1.68008
$844$	−15797.0	−0.644260
$845$	0	0
$846$	−1181.94	−0.0480330
$847$	−73445.9	−2.97949
$848$	−3694.19	−0.149598
$849$	22327.1	0.902550
$850$	0	0
$851$	−7494.39	−0.301885
$852$	−7503.55	−0.301722
$853$	−4326.89	−0.173681	−0.0868405	−	0.996222i	$-0.527677\pi$
$853$	−4326.89	−0.173681	−0.0868405	+	0.996222i	$0.527677\pi$
$854$	18314.6	0.733856
$855$	0	0
$856$	−1412.15	−0.0563858
$857$	−20912.6	−0.833562	−0.416781	−	0.909007i	$-0.636842\pi$
$857$	−20912.6	−0.833562	−0.416781	+	0.909007i	$0.636842\pi$
$858$	−35754.4	−1.42265
$859$	−18002.3	−0.715054	−0.357527	−	0.933903i	$-0.616380\pi$
$859$	−18002.3	−0.715054	−0.357527	+	0.933903i	$0.616380\pi$
$860$	0	0
$861$	13140.8	0.520136
$862$	15108.6	0.596985
$863$	−37389.4	−1.47480	−0.737399	−	0.675457i	$-0.763946\pi$
$863$	−37389.4	−1.47480	−0.737399	+	0.675457i	$0.763946\pi$
$864$	4381.19	0.172513
$865$	0	0
$866$	−26103.0	−1.02427
$867$	46482.8	1.82080
$868$	−17946.0	−0.701758
$869$	45660.4	1.78242
$870$	0	0
$871$	−633.663	−0.0246508
$872$	2688.23	0.104398
$873$	1625.99	0.0630372
$874$	−1611.34	−0.0623621
$875$	0	0
$876$	9913.39	0.382355
$877$	−3952.88	−0.152200	−0.0760999	−	0.997100i	$-0.524247\pi$
$877$	−3952.88	−0.152200	−0.0760999	+	0.997100i	$0.524247\pi$
$878$	13642.7	0.524395
$879$	−17501.0	−0.671552
$880$	0	0
$881$	51900.7	1.98477	0.992383	−	0.123188i	$-0.0393119\pi$
$881$	51900.7	1.98477	0.992383	+	0.123188i	$0.0393119\pi$
$882$	−2033.42	−0.0776289
$883$	8033.47	0.306170	0.153085	−	0.988213i	$-0.451079\pi$
$883$	8033.47	0.306170	0.153085	+	0.988213i	$0.451079\pi$
$884$	−26662.8	−1.01444
$885$	0	0
$886$	−21520.2	−0.816012
$887$	16089.7	0.609062	0.304531	−	0.952502i	$-0.401500\pi$
$887$	16089.7	0.609062	0.304531	+	0.952502i	$0.401500\pi$
$888$	−13850.9	−0.523429
$889$	−65586.0	−2.47434
$890$	0	0
$891$	44881.9	1.68754
$892$	−14368.6	−0.539344
$893$	−16789.0	−0.629141
$894$	25999.5	0.972655
$895$	0	0
$896$	4373.72	0.163076
$897$	6969.64	0.259431
$898$	−15007.0	−0.557675
$899$	−22501.1	−0.834766
$900$	0	0
$901$	−26986.2	−0.997827
$902$	−8539.83	−0.315239
$903$	−10786.0	−0.397493
$904$	−9127.15	−0.335801
$905$	0	0
$906$	−14863.6	−0.545044
$907$	10734.2	0.392969	0.196485	−	0.980507i	$-0.437047\pi$
$907$	10734.2	0.392969	0.196485	+	0.980507i	$0.437047\pi$
$908$	−13398.0	−0.489680
$909$	−1791.04	−0.0653521
$910$	0	0
$911$	7368.29	0.267972	0.133986	−	0.990983i	$-0.457222\pi$
$911$	7368.29	0.267972	0.133986	+	0.990983i	$0.457222\pi$
$912$	−2978.03	−0.108128
$913$	−1325.55	−0.0480498
$914$	−37492.4	−1.35683
$915$	0	0
$916$	11055.0	0.398763
$917$	93221.0	3.35706
$918$	32004.8	1.15067
$919$	−12106.3	−0.434549	−0.217275	−	0.976111i	$-0.569717\pi$
$919$	−12106.3	−0.434549	−0.217275	+	0.976111i	$0.569717\pi$
$920$	0	0
$921$	5784.62	0.206960
$922$	6923.83	0.247315
$923$	20134.1	0.718008
$924$	42844.7	1.52542
$925$	0	0
$926$	25135.2	0.892003
$927$	−593.181	−0.0210169
$928$	5483.89	0.193984
$929$	14622.7	0.516421	0.258210	−	0.966089i	$-0.416867\pi$
$929$	14622.7	0.516421	0.258210	+	0.966089i	$0.416867\pi$
$930$	0	0
$931$	−28883.9	−1.01679
$932$	8340.70	0.293142
$933$	20475.9	0.718490
$934$	−30275.7	−1.06065
$935$	0	0
$936$	562.554	0.0196449
$937$	−54950.3	−1.91585	−0.957924	−	0.287023i	$-0.907334\pi$
$937$	−54950.3	−1.91585	−0.957924	+	0.287023i	$0.907334\pi$
$938$	759.321	0.0264315
$939$	−45349.4	−1.57606
$940$	0	0
$941$	36477.3	1.26368	0.631841	−	0.775098i	$-0.282299\pi$
$941$	36477.3	1.26368	0.631841	+	0.775098i	$0.282299\pi$
$942$	17320.0	0.599062
$943$	1664.68	0.0574860
$944$	7931.85	0.273474
$945$	0	0
$946$	7009.53	0.240909
$947$	20311.8	0.696985	0.348492	−	0.937312i	$-0.386694\pi$
$947$	20311.8	0.696985	0.348492	+	0.937312i	$0.386694\pi$
$948$	−16449.8	−0.563571
$949$	−26600.4	−0.909888
$950$	0	0
$951$	21593.2	0.736287
$952$	31950.2	1.08772
$953$	−4320.54	−0.146858	−0.0734292	−	0.997300i	$-0.523394\pi$
$953$	−4320.54	−0.146858	−0.0734292	+	0.997300i	$0.523394\pi$
$954$	569.377	0.0193231
$955$	0	0
$956$	19055.4	0.644660
$957$	53719.8	1.81454
$958$	−20359.3	−0.686616
$959$	51191.2	1.72372
$960$	0	0
$961$	−12551.2	−0.421310
$962$	37165.7	1.24560
$963$	217.651	0.00728318
$964$	27715.7	0.925998
$965$	0	0
$966$	−8351.75	−0.278171
$967$	−19455.1	−0.646986	−0.323493	−	0.946231i	$-0.604857\pi$
$967$	−19455.1	−0.646986	−0.323493	+	0.946231i	$0.604857\pi$
$968$	−17195.6	−0.570957
$969$	−21754.6	−0.721217
$970$	0	0
$971$	−19512.5	−0.644888	−0.322444	−	0.946588i	$-0.604504\pi$
$971$	−19512.5	−0.644888	−0.322444	+	0.946588i	$0.604504\pi$
$972$	−1382.84	−0.0456322
$973$	104904.	3.45641
$974$	−21411.0	−0.704365
$975$	0	0
$976$	4287.92	0.140628
$977$	23683.1	0.775526	0.387763	−	0.921759i	$-0.373248\pi$
$977$	23683.1	0.775526	0.387763	+	0.921759i	$0.373248\pi$
$978$	2221.23	0.0726249
$979$	−69193.1	−2.25886
$980$	0	0
$981$	−414.331	−0.0134848
$982$	−2778.79	−0.0903001
$983$	−19378.5	−0.628767	−0.314384	−	0.949296i	$-0.601798\pi$
$983$	−19378.5	−0.628767	−0.314384	+	0.949296i	$0.601798\pi$
$984$	3076.60	0.0996731
$985$	0	0
$986$	40060.0	1.29388
$987$	−87019.1	−2.80633
$988$	7990.87	0.257311
$989$	−1366.37	−0.0439314
$990$	0	0
$991$	2734.03	0.0876380	0.0438190	−	0.999039i	$-0.486048\pi$
$991$	2734.03	0.0876380	0.0438190	+	0.999039i	$0.486048\pi$
$992$	−4201.61	−0.134477
$993$	−40002.3	−1.27838
$994$	−24126.8	−0.769874
$995$	0	0
$996$	477.550	0.0151925
$997$	35626.3	1.13169	0.565846	−	0.824511i	$-0.308550\pi$
$997$	35626.3	1.13169	0.565846	+	0.824511i	$0.308550\pi$
$998$	−12048.1	−0.382139
$999$	−44611.9	−1.41287

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1150.4.a.s.1.4	✓	5
5.2	odd	4		1150.4.b.p.599.2		10
5.3	odd	4		1150.4.b.p.599.9		10
5.4	even	2		1150.4.a.t.1.2	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1150.4.a.s.1.4	✓	5	1.1	even	1	trivial
1150.4.a.t.1.2	yes	5	5.4	even	2
1150.4.b.p.599.2		10	5.2	odd	4
1150.4.b.p.599.9		10	5.3	odd	4

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:	\( N \)	\(=\)	\( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1150.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +5.31348 q^{3} +4.00000 q^{4} -10.6270 q^{6} -34.1697 q^{7} -8.00000 q^{8} +1.23302 q^{9} -58.9953 q^{11} +21.2539 q^{12} -57.0300 q^{13} +68.3394 q^{14} +16.0000 q^{16} +116.881 q^{17} -2.46604 q^{18} -35.0292 q^{19} -181.560 q^{21} +117.991 q^{22} -23.0000 q^{23} -42.5078 q^{24} +114.060 q^{26} -136.912 q^{27} -136.679 q^{28} -171.371 q^{29} +131.300 q^{31} -32.0000 q^{32} -313.470 q^{33} -233.761 q^{34} +4.93208 q^{36} +325.843 q^{37} +70.0584 q^{38} -303.028 q^{39} -72.3772 q^{41} +363.120 q^{42} +59.4075 q^{43} -235.981 q^{44} +46.0000 q^{46} +479.286 q^{47} +85.0156 q^{48} +824.568 q^{49} +621.043 q^{51} -228.120 q^{52} -230.887 q^{53} +273.824 q^{54} +273.358 q^{56} -186.127 q^{57} +342.743 q^{58} +495.740 q^{59} +267.995 q^{61} -262.601 q^{62} -42.1319 q^{63} +64.0000 q^{64} +626.940 q^{66} +11.1110 q^{67} +467.523 q^{68} -122.210 q^{69} -353.044 q^{71} -9.86416 q^{72} +466.427 q^{73} -651.686 q^{74} -140.117 q^{76} +2015.85 q^{77} +606.056 q^{78} -773.967 q^{79} -760.771 q^{81} +144.754 q^{82} +22.4688 q^{83} -726.239 q^{84} -118.815 q^{86} -910.578 q^{87} +471.962 q^{88} +1172.86 q^{89} +1948.70 q^{91} -92.0000 q^{92} +697.661 q^{93} -958.572 q^{94} -170.031 q^{96} +1318.71 q^{97} -1649.14 q^{98} -72.7424 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 10 q^{2} + 12 q^{3} + 20 q^{4} - 24 q^{6} - 24 q^{7} - 40 q^{8} + 11 q^{9} - 54 q^{11} + 48 q^{12} + 36 q^{13} + 48 q^{14} + 80 q^{16} + 132 q^{17} - 22 q^{18} - 50 q^{19} - 158 q^{21} + 108 q^{22}+ \cdots - 1740 q^{99}+O(q^{100}) \) 5 * q - 10 * q^2 + 12 * q^3 + 20 * q^4 - 24 * q^6 - 24 * q^7 - 40 * q^8 + 11 * q^9 - 54 * q^11 + 48 * q^12 + 36 * q^13 + 48 * q^14 + 80 * q^16 + 132 * q^17 - 22 * q^18 - 50 * q^19 - 158 * q^21 + 108 * q^22 - 115 * q^23 - 96 * q^24 - 72 * q^26 + 489 * q^27 - 96 * q^28 - 104 * q^29 - 342 * q^31 - 160 * q^32 - 348 * q^33 - 264 * q^34 + 44 * q^36 + 380 * q^37 + 100 * q^38 + 193 * q^39 + 2 * q^41 + 316 * q^42 - 114 * q^43 - 216 * q^44 + 230 * q^46 + 966 * q^47 + 192 * q^48 - 441 * q^49 + 1200 * q^51 + 144 * q^52 + 308 * q^53 - 978 * q^54 + 192 * q^56 + 1410 * q^57 + 208 * q^58 - 277 * q^59 + 482 * q^61 + 684 * q^62 - 222 * q^63 + 320 * q^64 + 696 * q^66 + 2044 * q^67 + 528 * q^68 - 276 * q^69 + 208 * q^71 - 88 * q^72 + 646 * q^73 - 760 * q^74 - 200 * q^76 + 2124 * q^77 - 386 * q^78 - 3444 * q^79 + 2933 * q^81 - 4 * q^82 + 1218 * q^83 - 632 * q^84 + 228 * q^86 + 103 * q^87 + 432 * q^88 + 140 * q^89 + 1976 * q^91 - 460 * q^92 + 473 * q^93 - 1932 * q^94 - 384 * q^96 + 2816 * q^97 + 882 * q^98 - 1740 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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