LMFDB - Embedded newform 2254.4.a.y.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2254,4,Mod(1,2254)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2254, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2254.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2254 = 2 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2254.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:	$132.990305153$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 4 x^{10} - 212 x^{9} + 487 x^{8} + 16315 x^{7} - 9025 x^{6} - 516068 x^{5} - 504693 x^{4} + \cdots - 11394027 $ x^11 - 4x^10 - 212x^9 + 487x^8 + 16315x^7 - 9025x^6 - 516068x^5 - 504693x^4 + 5020566x^3 + 8901846x^2 - 4796415x - 11394027
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 322)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.85313$ of defining polynomial
Character	$\chi$	$=$	2254.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000 q^{2} -1.85313 q^{3} +4.00000 q^{4} +2.81000 q^{5} -3.70626 q^{6} +8.00000 q^{8} -23.5659 q^{9} +5.62000 q^{10} +18.6231 q^{11} -7.41251 q^{12} +39.9118 q^{13} -5.20729 q^{15} +16.0000 q^{16} -34.3817 q^{17} -47.1318 q^{18} -60.1884 q^{19} +11.2400 q^{20} +37.2462 q^{22} +23.0000 q^{23} -14.8250 q^{24} -117.104 q^{25} +79.8236 q^{26} +93.7051 q^{27} +145.133 q^{29} -10.4146 q^{30} +94.1204 q^{31} +32.0000 q^{32} -34.5110 q^{33} -68.7634 q^{34} -94.2637 q^{36} +43.0744 q^{37} -120.377 q^{38} -73.9617 q^{39} +22.4800 q^{40} +206.230 q^{41} +153.983 q^{43} +74.4925 q^{44} -66.2202 q^{45} +46.0000 q^{46} -274.386 q^{47} -29.6500 q^{48} -234.208 q^{50} +63.7137 q^{51} +159.647 q^{52} -609.365 q^{53} +187.410 q^{54} +52.3309 q^{55} +111.537 q^{57} +290.267 q^{58} +495.942 q^{59} -20.8291 q^{60} +818.182 q^{61} +188.241 q^{62} +64.0000 q^{64} +112.152 q^{65} -69.0221 q^{66} -616.576 q^{67} -137.527 q^{68} -42.6219 q^{69} +975.636 q^{71} -188.527 q^{72} -149.117 q^{73} +86.1487 q^{74} +217.009 q^{75} -240.754 q^{76} -147.923 q^{78} -272.456 q^{79} +44.9600 q^{80} +462.632 q^{81} +412.461 q^{82} -1.18964 q^{83} -96.6125 q^{85} +307.967 q^{86} -268.951 q^{87} +148.985 q^{88} +150.473 q^{89} -132.440 q^{90} +92.0000 q^{92} -174.417 q^{93} -548.772 q^{94} -169.129 q^{95} -59.3001 q^{96} +1666.17 q^{97} -438.871 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 22 q^{2} + 18 q^{3} + 44 q^{4} + 33 q^{5} + 36 q^{6} + 88 q^{8} + 171 q^{9} + 66 q^{10} + 8 q^{11} + 72 q^{12} + 185 q^{13} - 186 q^{15} + 176 q^{16} + 107 q^{17} + 342 q^{18} + 114 q^{19} + 132 q^{20}+ \cdots - 1729 q^{99}+O(q^{100}) $ 11 * q + 22 * q^2 + 18 * q^3 + 44 * q^4 + 33 * q^5 + 36 * q^6 + 88 * q^8 + 171 * q^9 + 66 * q^10 + 8 * q^11 + 72 * q^12 + 185 * q^13 - 186 * q^15 + 176 * q^16 + 107 * q^17 + 342 * q^18 + 114 * q^19 + 132 * q^20 + 16 * q^22 + 253 * q^23 + 144 * q^24 + 494 * q^25 + 370 * q^26 + 561 * q^27 - 79 * q^29 - 372 * q^30 + 488 * q^31 + 352 * q^32 + 304 * q^33 + 214 * q^34 + 684 * q^36 - 41 * q^37 + 228 * q^38 + 425 * q^39 + 264 * q^40 + 812 * q^41 + 219 * q^43 + 32 * q^44 + 619 * q^45 + 506 * q^46 + 1141 * q^47 + 288 * q^48 + 988 * q^50 + 83 * q^51 + 740 * q^52 + 579 * q^53 + 1122 * q^54 + 951 * q^55 + 383 * q^57 - 158 * q^58 + 1695 * q^59 - 744 * q^60 + 1105 * q^61 + 976 * q^62 + 704 * q^64 - 347 * q^65 + 608 * q^66 + 695 * q^67 + 428 * q^68 + 414 * q^69 + 65 * q^71 + 1368 * q^72 + 1935 * q^73 - 82 * q^74 + 1712 * q^75 + 456 * q^76 + 850 * q^78 + 447 * q^79 + 528 * q^80 + 2203 * q^81 + 1624 * q^82 + 1389 * q^83 + 153 * q^85 + 438 * q^86 + 3354 * q^87 + 64 * q^88 + 2698 * q^89 + 1238 * q^90 + 1012 * q^92 - 686 * q^93 + 2282 * q^94 + 827 * q^95 + 576 * q^96 + 1589 * q^97 - 1729 * 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
$3$	−1.85313	−0.356635	−0.178317	−	0.983973i	$-0.557065\pi$
$3$	−1.85313	−0.356635	−0.178317	+	0.983973i	$0.557065\pi$
$4$	4.00000	0.500000
$5$	2.81000	0.251334	0.125667	−	0.992072i	$-0.459893\pi$
$5$	2.81000	0.251334	0.125667	+	0.992072i	$0.459893\pi$
$6$	−3.70626	−0.252179
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	−23.5659	−0.872812
$10$	5.62000	0.177720
$11$	18.6231	0.510462	0.255231	−	0.966880i	$-0.417848\pi$
$11$	18.6231	0.510462	0.255231	+	0.966880i	$0.417848\pi$
$12$	−7.41251	−0.178317
$13$	39.9118	0.851504	0.425752	−	0.904840i	$-0.360010\pi$
$13$	39.9118	0.851504	0.425752	+	0.904840i	$0.360010\pi$
$14$	0	0
$15$	−5.20729	−0.0896344
$16$	16.0000	0.250000
$17$	−34.3817	−0.490517	−0.245258	−	0.969458i	$-0.578873\pi$
$17$	−34.3817	−0.490517	−0.245258	+	0.969458i	$0.578873\pi$
$18$	−47.1318	−0.617171
$19$	−60.1884	−0.726746	−0.363373	−	0.931644i	$-0.618375\pi$
$19$	−60.1884	−0.726746	−0.363373	+	0.931644i	$0.618375\pi$
$20$	11.2400	0.125667
$21$	0	0
$22$	37.2462	0.360951
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	−14.8250	−0.126089
$25$	−117.104	−0.936831
$26$	79.8236	0.602104
$27$	93.7051	0.667910
$28$	0	0
$29$	145.133	0.929331	0.464666	−	0.885486i	$-0.346175\pi$
$29$	145.133	0.929331	0.464666	+	0.885486i	$0.346175\pi$
$30$	−10.4146	−0.0633811
$31$	94.1204	0.545307	0.272654	−	0.962112i	$-0.412099\pi$
$31$	94.1204	0.545307	0.272654	+	0.962112i	$0.412099\pi$
$32$	32.0000	0.176777
$33$	−34.5110	−0.182048
$34$	−68.7634	−0.346848
$35$	0	0
$36$	−94.2637	−0.436406
$37$	43.0744	0.191389	0.0956944	−	0.995411i	$-0.469493\pi$
$37$	43.0744	0.191389	0.0956944	+	0.995411i	$0.469493\pi$
$38$	−120.377	−0.513887
$39$	−73.9617	−0.303676
$40$	22.4800	0.0888600
$41$	206.230	0.785555	0.392777	−	0.919634i	$-0.371514\pi$
$41$	206.230	0.785555	0.392777	+	0.919634i	$0.371514\pi$
$42$	0	0
$43$	153.983	0.546098	0.273049	−	0.962000i	$-0.411968\pi$
$43$	153.983	0.546098	0.273049	+	0.962000i	$0.411968\pi$
$44$	74.4925	0.255231
$45$	−66.2202	−0.219367
$46$	46.0000	0.147442
$47$	−274.386	−0.851560	−0.425780	−	0.904827i	$-0.640000\pi$
$47$	−274.386	−0.851560	−0.425780	+	0.904827i	$0.640000\pi$
$48$	−29.6500	−0.0891587
$49$	0	0
$50$	−234.208	−0.662440
$51$	63.7137	0.174935
$52$	159.647	0.425752
$53$	−609.365	−1.57930	−0.789649	−	0.613559i	$-0.789737\pi$
$53$	−609.365	−1.57930	−0.789649	+	0.613559i	$0.789737\pi$
$54$	187.410	0.472283
$55$	52.3309	0.128296
$56$	0	0
$57$	111.537	0.259183
$58$	290.267	0.657136
$59$	495.942	1.09434	0.547171	−	0.837021i	$-0.315705\pi$
$59$	495.942	1.09434	0.547171	+	0.837021i	$0.315705\pi$
$60$	−20.8291	−0.0448172
$61$	818.182	1.71733	0.858667	−	0.512533i	$-0.171293\pi$
$61$	818.182	1.71733	0.858667	+	0.512533i	$0.171293\pi$
$62$	188.241	0.385591
$63$	0	0
$64$	64.0000	0.125000
$65$	112.152	0.214012
$66$	−69.0221	−0.128728
$67$	−616.576	−1.12428	−0.562140	−	0.827042i	$-0.690022\pi$
$67$	−616.576	−1.12428	−0.562140	+	0.827042i	$0.690022\pi$
$68$	−137.527	−0.245258
$69$	−42.6219	−0.0743635
$70$	0	0
$71$	975.636	1.63080	0.815399	−	0.578900i	$-0.196518\pi$
$71$	975.636	1.63080	0.815399	+	0.578900i	$0.196518\pi$
$72$	−188.527	−0.308586
$73$	−149.117	−0.239079	−0.119540	−	0.992829i	$-0.538142\pi$
$73$	−149.117	−0.239079	−0.119540	+	0.992829i	$0.538142\pi$
$74$	86.1487	0.135332
$75$	217.009	0.334106
$76$	−240.754	−0.363373
$77$	0	0
$78$	−147.923	−0.214731
$79$	−272.456	−0.388022	−0.194011	−	0.980999i	$-0.562150\pi$
$79$	−272.456	−0.388022	−0.194011	+	0.980999i	$0.562150\pi$
$80$	44.9600	0.0628335
$81$	462.632	0.634612
$82$	412.461	0.555471
$83$	−1.18964	−0.00157326	−0.000786629	−	1.00000i	$-0.500250\pi$
$83$	−1.18964	−0.00157326	−0.000786629		1.00000i	$0.500250\pi$
$84$	0	0
$85$	−96.6125	−0.123283
$86$	307.967	0.386150
$87$	−268.951	−0.331432
$88$	148.985	0.180476
$89$	150.473	0.179214	0.0896072	−	0.995977i	$-0.471439\pi$
$89$	150.473	0.179214	0.0896072	+	0.995977i	$0.471439\pi$
$90$	−132.440	−0.155116
$91$	0	0
$92$	92.0000	0.104257
$93$	−174.417	−0.194475
$94$	−548.772	−0.602144
$95$	−169.129	−0.182656
$96$	−59.3001	−0.0630447
$97$	1666.17	1.74406	0.872032	−	0.489448i	$-0.162802\pi$
$97$	1666.17	1.74406	0.872032	+	0.489448i	$0.162802\pi$
$98$	0	0
$99$	−438.871	−0.445537
$100$	−468.416	−0.468416
$101$	1499.65	1.47743	0.738716	−	0.674017i	$-0.235433\pi$
$101$	1499.65	1.47743	0.738716	+	0.674017i	$0.235433\pi$
$102$	127.427	0.123698
$103$	−305.699	−0.292440	−0.146220	−	0.989252i	$-0.546711\pi$
$103$	−305.699	−0.292440	−0.146220	+	0.989252i	$0.546711\pi$
$104$	319.295	0.301052
$105$	0	0
$106$	−1218.73	−1.11673
$107$	−1733.16	−1.56589	−0.782946	−	0.622089i	$-0.786284\pi$
$107$	−1733.16	−1.56589	−0.782946	+	0.622089i	$0.786284\pi$
$108$	374.820	0.333955
$109$	61.3913	0.0539470	0.0269735	−	0.999636i	$-0.491413\pi$
$109$	61.3913	0.0539470	0.0269735	+	0.999636i	$0.491413\pi$
$110$	104.662	0.0907193
$111$	−79.8223	−0.0682559
$112$	0	0
$113$	−934.132	−0.777661	−0.388831	−	0.921309i	$-0.627121\pi$
$113$	−934.132	−0.777661	−0.388831	+	0.921309i	$0.627121\pi$
$114$	223.074	0.183270
$115$	64.6300	0.0524067
$116$	580.534	0.464666
$117$	−940.559	−0.743202
$118$	991.885	0.773817
$119$	0	0
$120$	−41.6583	−0.0316905
$121$	−984.179	−0.739428
$122$	1636.36	1.21434
$123$	−382.171	−0.280156
$124$	376.482	0.272654
$125$	−680.312	−0.486791
$126$	0	0
$127$	410.664	0.286933	0.143467	−	0.989655i	$-0.454175\pi$
$127$	410.664	0.286933	0.143467	+	0.989655i	$0.454175\pi$
$128$	128.000	0.0883883
$129$	−285.351	−0.194758
$130$	224.304	0.151329
$131$	1089.02	0.726323	0.363161	−	0.931726i	$-0.381697\pi$
$131$	1089.02	0.726323	0.363161	+	0.931726i	$0.381697\pi$
$132$	−138.044	−0.0910242
$133$	0	0
$134$	−1233.15	−0.794985
$135$	263.311	0.167868
$136$	−275.053	−0.173424
$137$	1995.05	1.24415	0.622077	−	0.782956i	$-0.286289\pi$
$137$	1995.05	1.24415	0.622077	+	0.782956i	$0.286289\pi$
$138$	−85.2439	−0.0525829
$139$	2746.68	1.67604	0.838022	−	0.545636i	$-0.183712\pi$
$139$	2746.68	1.67604	0.838022	+	0.545636i	$0.183712\pi$
$140$	0	0
$141$	508.472	0.303696
$142$	1951.27	1.15315
$143$	743.283	0.434660
$144$	−377.055	−0.218203
$145$	407.825	0.233572
$146$	−298.234	−0.169055
$147$	0	0
$148$	172.297	0.0956944
$149$	−644.638	−0.354435	−0.177217	−	0.984172i	$-0.556710\pi$
$149$	−644.638	−0.354435	−0.177217	+	0.984172i	$0.556710\pi$
$150$	434.017	0.236249
$151$	2491.93	1.34298	0.671492	−	0.741012i	$-0.265654\pi$
$151$	2491.93	1.34298	0.671492	+	0.741012i	$0.265654\pi$
$152$	−481.507	−0.256943
$153$	810.236	0.428129
$154$	0	0
$155$	264.478	0.137054
$156$	−295.847	−0.151838
$157$	1652.38	0.839963	0.419981	−	0.907533i	$-0.362037\pi$
$157$	1652.38	0.839963	0.419981	+	0.907533i	$0.362037\pi$
$158$	−544.912	−0.274373
$159$	1129.23	0.563232
$160$	89.9200	0.0444300
$161$	0	0
$162$	925.264	0.448738
$163$	−819.167	−0.393632	−0.196816	−	0.980440i	$-0.563060\pi$
$163$	−819.167	−0.393632	−0.196816	+	0.980440i	$0.563060\pi$
$164$	824.921	0.392777
$165$	−96.9759	−0.0457550
$166$	−2.37929	−0.00111246
$167$	−2687.31	−1.24521	−0.622606	−	0.782536i	$-0.713926\pi$
$167$	−2687.31	−1.24521	−0.622606	+	0.782536i	$0.713926\pi$
$168$	0	0
$169$	−604.046	−0.274941
$170$	−193.225	−0.0871746
$171$	1418.39	0.634312
$172$	615.933	0.273049
$173$	1701.03	0.747554	0.373777	−	0.927519i	$-0.378063\pi$
$173$	1701.03	0.747554	0.373777	+	0.927519i	$0.378063\pi$
$174$	−537.902	−0.234358
$175$	0	0
$176$	297.970	0.127616
$177$	−919.045	−0.390280
$178$	300.946	0.126724
$179$	2498.24	1.04317	0.521584	−	0.853200i	$-0.325341\pi$
$179$	2498.24	1.04317	0.521584	+	0.853200i	$0.325341\pi$
$180$	−264.881	−0.109684
$181$	2201.82	0.904199	0.452100	−	0.891968i	$-0.350675\pi$
$181$	2201.82	0.904199	0.452100	+	0.891968i	$0.350675\pi$
$182$	0	0
$183$	−1516.20	−0.612461
$184$	184.000	0.0737210
$185$	121.039	0.0481025
$186$	−348.834	−0.137515
$187$	−640.294	−0.250390
$188$	−1097.54	−0.425780
$189$	0	0
$190$	−338.259	−0.129157
$191$	2647.73	1.00305	0.501527	−	0.865142i	$-0.332772\pi$
$191$	2647.73	1.00305	0.501527	+	0.865142i	$0.332772\pi$
$192$	−118.600	−0.0445793
$193$	−1203.35	−0.448803	−0.224402	−	0.974497i	$-0.572043\pi$
$193$	−1203.35	−0.448803	−0.224402	+	0.974497i	$0.572043\pi$
$194$	3332.35	1.23324
$195$	−207.832	−0.0763240
$196$	0	0
$197$	675.018	0.244127	0.122064	−	0.992522i	$-0.461049\pi$
$197$	675.018	0.244127	0.122064	+	0.992522i	$0.461049\pi$
$198$	−877.742	−0.315042
$199$	811.923	0.289225	0.144612	−	0.989488i	$-0.453807\pi$
$199$	811.923	0.289225	0.144612	+	0.989488i	$0.453807\pi$
$200$	−936.831	−0.331220
$201$	1142.59	0.400957
$202$	2999.30	1.04470
$203$	0	0
$204$	254.855	0.0874676
$205$	579.507	0.197437
$206$	−611.397	−0.206787
$207$	−542.016	−0.181994
$208$	638.589	0.212876
$209$	−1120.90	−0.370976
$210$	0	0
$211$	5321.80	1.73634	0.868170	−	0.496267i	$-0.165296\pi$
$211$	5321.80	1.73634	0.868170	+	0.496267i	$0.165296\pi$
$212$	−2437.46	−0.789649
$213$	−1807.98	−0.581599
$214$	−3466.31	−1.10725
$215$	432.693	0.137253
$216$	749.641	0.236142
$217$	0	0
$218$	122.783	0.0381463
$219$	276.333	0.0852640
$220$	209.324	0.0641482
$221$	−1372.24	−0.417677
$222$	−159.645	−0.0482642
$223$	1449.97	0.435413	0.217707	−	0.976014i	$-0.430142\pi$
$223$	1449.97	0.435413	0.217707	+	0.976014i	$0.430142\pi$
$224$	0	0
$225$	2759.66	0.817677
$226$	−1868.26	−0.549890
$227$	−864.947	−0.252901	−0.126451	−	0.991973i	$-0.540358\pi$
$227$	−864.947	−0.252901	−0.126451	+	0.991973i	$0.540358\pi$
$228$	446.147	0.129591
$229$	4625.19	1.33468	0.667338	−	0.744755i	$-0.267433\pi$
$229$	4625.19	1.33468	0.667338	+	0.744755i	$0.267433\pi$
$230$	129.260	0.0370572
$231$	0	0
$232$	1161.07	0.328568
$233$	−117.582	−0.0330602	−0.0165301	−	0.999863i	$-0.505262\pi$
$233$	−117.582	−0.0330602	−0.0165301	+	0.999863i	$0.505262\pi$
$234$	−1881.12	−0.525523
$235$	−771.024	−0.214026
$236$	1983.77	0.547171
$237$	504.896	0.138382
$238$	0	0
$239$	23.2085	0.00628131	0.00314065	−	0.999995i	$-0.499000\pi$
$239$	23.2085	0.00628131	0.00314065	+	0.999995i	$0.499000\pi$
$240$	−83.3166	−0.0224086
$241$	−309.852	−0.0828187	−0.0414094	−	0.999142i	$-0.513185\pi$
$241$	−309.852	−0.0828187	−0.0414094	+	0.999142i	$0.513185\pi$
$242$	−1968.36	−0.522855
$243$	−3387.35	−0.894234
$244$	3272.73	0.858667
$245$	0	0
$246$	−764.342	−0.198100
$247$	−2402.23	−0.618827
$248$	752.963	0.192795
$249$	2.20456	0.000561078 0
$250$	−1360.62	−0.344213
$251$	−4709.05	−1.18419	−0.592097	−	0.805867i	$-0.701700\pi$
$251$	−4709.05	−1.18419	−0.592097	+	0.805867i	$0.701700\pi$
$252$	0	0
$253$	428.332	0.106439
$254$	821.328	0.202893
$255$	179.035	0.0439672
$256$	256.000	0.0625000
$257$	5064.23	1.22918	0.614588	−	0.788848i	$-0.289322\pi$
$257$	5064.23	1.22918	0.614588	+	0.788848i	$0.289322\pi$
$258$	−570.701	−0.137714
$259$	0	0
$260$	448.609	0.107006
$261$	−3420.20	−0.811131
$262$	2178.04	0.513588
$263$	−1624.06	−0.380775	−0.190388	−	0.981709i	$-0.560974\pi$
$263$	−1624.06	−0.380775	−0.190388	+	0.981709i	$0.560974\pi$
$264$	−276.088	−0.0643639
$265$	−1712.32	−0.396931
$266$	0	0
$267$	−278.845	−0.0639141
$268$	−2466.30	−0.562140
$269$	−5316.00	−1.20492	−0.602458	−	0.798151i	$-0.705812\pi$
$269$	−5316.00	−1.20492	−0.602458	+	0.798151i	$0.705812\pi$
$270$	526.622	0.118701
$271$	3948.73	0.885124	0.442562	−	0.896738i	$-0.354070\pi$
$271$	3948.73	0.885124	0.442562	+	0.896738i	$0.354070\pi$
$272$	−550.107	−0.122629
$273$	0	0
$274$	3990.11	0.879749
$275$	−2180.84	−0.478217
$276$	−170.488	−0.0371817
$277$	−7074.24	−1.53448	−0.767238	−	0.641362i	$-0.778370\pi$
$277$	−7074.24	−1.53448	−0.767238	+	0.641362i	$0.778370\pi$
$278$	5493.36	1.18514
$279$	−2218.03	−0.475951
$280$	0	0
$281$	1608.61	0.341501	0.170751	−	0.985314i	$-0.445381\pi$
$281$	1608.61	0.341501	0.170751	+	0.985314i	$0.445381\pi$
$282$	1016.94	0.214745
$283$	−1822.79	−0.382876	−0.191438	−	0.981505i	$-0.561315\pi$
$283$	−1822.79	−0.382876	−0.191438	+	0.981505i	$0.561315\pi$
$284$	3902.54	0.815399
$285$	313.418	0.0651414
$286$	1486.57	0.307351
$287$	0	0
$288$	−754.109	−0.154293
$289$	−3730.90	−0.759393
$290$	815.649	0.165161
$291$	−3087.63	−0.621994
$292$	−596.467	−0.119540
$293$	6920.73	1.37991	0.689954	−	0.723853i	$-0.257631\pi$
$293$	6920.73	1.37991	0.689954	+	0.723853i	$0.257631\pi$
$294$	0	0
$295$	1393.60	0.275045
$296$	344.595	0.0676661
$297$	1745.08	0.340943
$298$	−1289.28	−0.250623
$299$	917.972	0.177551
$300$	868.034	0.167053
$301$	0	0
$302$	4983.87	0.949633
$303$	−2779.04	−0.526903
$304$	−963.014	−0.181686
$305$	2299.09	0.431624
$306$	1620.47	0.302733
$307$	−4389.48	−0.816029	−0.408014	−	0.912976i	$-0.633779\pi$
$307$	−4389.48	−0.816029	−0.408014	+	0.912976i	$0.633779\pi$
$308$	0	0
$309$	566.498	0.104294
$310$	528.956	0.0969120
$311$	6619.28	1.20690	0.603448	−	0.797402i	$-0.293793\pi$
$311$	6619.28	1.20690	0.603448	+	0.797402i	$0.293793\pi$
$312$	−591.694	−0.107366
$313$	6438.82	1.16276	0.581380	−	0.813632i	$-0.302513\pi$
$313$	6438.82	1.16276	0.581380	+	0.813632i	$0.302513\pi$
$314$	3304.76	0.593943
$315$	0	0
$316$	−1089.82	−0.194011
$317$	4933.73	0.874152	0.437076	−	0.899425i	$-0.356014\pi$
$317$	4933.73	0.874152	0.437076	+	0.899425i	$0.356014\pi$
$318$	2258.46	0.398265
$319$	2702.84	0.474388
$320$	179.840	0.0314167
$321$	3211.76	0.558452
$322$	0	0
$323$	2069.38	0.356481
$324$	1850.53	0.317306
$325$	−4673.83	−0.797715
$326$	−1638.33	−0.278340
$327$	−113.766	−0.0192394
$328$	1649.84	0.277736
$329$	0	0
$330$	−193.952	−0.0323536
$331$	4067.36	0.675415	0.337708	−	0.941251i	$-0.390348\pi$
$331$	4067.36	0.675415	0.337708	+	0.941251i	$0.390348\pi$
$332$	−4.75858	−0.000786629 0
$333$	−1015.09	−0.167046
$334$	−5374.62	−0.880497
$335$	−1732.58	−0.282569
$336$	0	0
$337$	9944.83	1.60751	0.803753	−	0.594963i	$-0.202833\pi$
$337$	9944.83	1.60751	0.803753	+	0.594963i	$0.202833\pi$
$338$	−1208.09	−0.194413
$339$	1731.07	0.277341
$340$	−386.450	−0.0616417
$341$	1752.82	0.278359
$342$	2836.79	0.448526
$343$	0	0
$344$	1231.87	0.193075
$345$	−119.768	−0.0186901
$346$	3402.06	0.528600
$347$	11240.3	1.73893	0.869466	−	0.493994i	$-0.164463\pi$
$347$	11240.3	1.73893	0.869466	+	0.493994i	$0.164463\pi$
$348$	−1075.80	−0.165716
$349$	−8047.97	−1.23438	−0.617189	−	0.786815i	$-0.711729\pi$
$349$	−8047.97	−1.23438	−0.617189	+	0.786815i	$0.711729\pi$
$350$	0	0
$351$	3739.94	0.568727
$352$	595.940	0.0902378
$353$	5144.71	0.775709	0.387855	−	0.921721i	$-0.373216\pi$
$353$	5144.71	0.775709	0.387855	+	0.921721i	$0.373216\pi$
$354$	−1838.09	−0.275970
$355$	2741.53	0.409875
$356$	601.891	0.0896072
$357$	0	0
$358$	4996.48	0.737631
$359$	3894.54	0.572552	0.286276	−	0.958147i	$-0.407583\pi$
$359$	3894.54	0.572552	0.286276	+	0.958147i	$0.407583\pi$
$360$	−529.762	−0.0775580
$361$	−3236.36	−0.471841
$362$	4403.64	0.639365
$363$	1823.81	0.263706
$364$	0	0
$365$	−419.018	−0.0600888
$366$	−3032.39	−0.433075
$367$	−4053.20	−0.576500	−0.288250	−	0.957555i	$-0.593073\pi$
$367$	−4053.20	−0.576500	−0.288250	+	0.957555i	$0.593073\pi$
$368$	368.000	0.0521286
$369$	−4860.00	−0.685642
$370$	242.078	0.0340136
$371$	0	0
$372$	−697.669	−0.0972377
$373$	−8800.78	−1.22168	−0.610841	−	0.791754i	$-0.709168\pi$
$373$	−8800.78	−1.22168	−0.610841	+	0.791754i	$0.709168\pi$
$374$	−1280.59	−0.177053
$375$	1260.70	0.173607
$376$	−2195.09	−0.301072
$377$	5792.54	0.791329
$378$	0	0
$379$	−1005.08	−0.136220	−0.0681098	−	0.997678i	$-0.521697\pi$
$379$	−1005.08	−0.136220	−0.0681098	+	0.997678i	$0.521697\pi$
$380$	−676.517	−0.0913279
$381$	−761.013	−0.102330
$382$	5295.46	0.709266
$383$	2379.44	0.317451	0.158726	−	0.987323i	$-0.449261\pi$
$383$	2379.44	0.317451	0.158726	+	0.987323i	$0.449261\pi$
$384$	−237.200	−0.0315223
$385$	0	0
$386$	−2406.70	−0.317352
$387$	−3628.76	−0.476641
$388$	6664.69	0.872032
$389$	−7189.06	−0.937017	−0.468508	−	0.883459i	$-0.655208\pi$
$389$	−7189.06	−0.937017	−0.468508	+	0.883459i	$0.655208\pi$
$390$	−415.665	−0.0539692
$391$	−790.779	−0.102280
$392$	0	0
$393$	−2018.10	−0.259032
$394$	1350.04	0.172624
$395$	−765.601	−0.0975230
$396$	−1755.48	−0.222769
$397$	−1530.43	−0.193476	−0.0967379	−	0.995310i	$-0.530841\pi$
$397$	−1530.43	−0.193476	−0.0967379	+	0.995310i	$0.530841\pi$
$398$	1623.85	0.204513
$399$	0	0
$400$	−1873.66	−0.234208
$401$	−7077.08	−0.881328	−0.440664	−	0.897672i	$-0.645257\pi$
$401$	−7077.08	−0.881328	−0.440664	+	0.897672i	$0.645257\pi$
$402$	2285.19	0.283519
$403$	3756.52	0.464331
$404$	5998.59	0.738716
$405$	1300.00	0.159500
$406$	0	0
$407$	802.179	0.0976967
$408$	509.709	0.0618489
$409$	5683.55	0.687123	0.343562	−	0.939130i	$-0.388367\pi$
$409$	5683.55	0.687123	0.343562	+	0.939130i	$0.388367\pi$
$410$	1159.01	0.139609
$411$	−3697.09	−0.443708
$412$	−1222.79	−0.146220
$413$	0	0
$414$	−1084.03	−0.128689
$415$	−3.34290	−0.000395413 0
$416$	1277.18	0.150526
$417$	−5089.95	−0.597736
$418$	−2241.79	−0.262320
$419$	−9874.34	−1.15130	−0.575648	−	0.817698i	$-0.695250\pi$
$419$	−9874.34	−1.15130	−0.575648	+	0.817698i	$0.695250\pi$
$420$	0	0
$421$	1765.47	0.204379	0.102190	−	0.994765i	$-0.467415\pi$
$421$	1765.47	0.204379	0.102190	+	0.994765i	$0.467415\pi$
$422$	10643.6	1.22778
$423$	6466.16	0.743251
$424$	−4874.92	−0.558366
$425$	4026.23	0.459531
$426$	−3615.96	−0.411253
$427$	0	0
$428$	−6932.62	−0.782946
$429$	−1377.40	−0.155015
$430$	865.385	0.0970525
$431$	4207.66	0.470246	0.235123	−	0.971966i	$-0.424451\pi$
$431$	4207.66	0.470246	0.235123	+	0.971966i	$0.424451\pi$
$432$	1499.28	0.166977
$433$	−474.563	−0.0526698	−0.0263349	−	0.999653i	$-0.508384\pi$
$433$	−474.563	−0.0526698	−0.0263349	+	0.999653i	$0.508384\pi$
$434$	0	0
$435$	−755.751	−0.0833000
$436$	245.565	0.0269735
$437$	−1384.33	−0.151537
$438$	552.665	0.0602908
$439$	16270.6	1.76891	0.884457	−	0.466622i	$-0.154529\pi$
$439$	16270.6	1.76891	0.884457	+	0.466622i	$0.154529\pi$
$440$	418.648	0.0453596
$441$	0	0
$442$	−2744.47	−0.295342
$443$	4484.57	0.480967	0.240483	−	0.970653i	$-0.422694\pi$
$443$	4484.57	0.480967	0.240483	+	0.970653i	$0.422694\pi$
$444$	−319.289	−0.0341279
$445$	422.828	0.0450427
$446$	2899.94	0.307884
$447$	1194.60	0.126404
$448$	0	0
$449$	1916.53	0.201441	0.100720	−	0.994915i	$-0.467885\pi$
$449$	1916.53	0.201441	0.100720	+	0.994915i	$0.467885\pi$
$450$	5519.32	0.578185
$451$	3840.65	0.400996
$452$	−3736.53	−0.388831
$453$	−4617.87	−0.478955
$454$	−1729.89	−0.178828
$455$	0	0
$456$	892.295	0.0916349
$457$	10138.9	1.03780	0.518902	−	0.854834i	$-0.326341\pi$
$457$	10138.9	1.03780	0.518902	+	0.854834i	$0.326341\pi$
$458$	9250.38	0.943759
$459$	−3221.74	−0.327621
$460$	258.520	0.0262034
$461$	−5763.64	−0.582298	−0.291149	−	0.956678i	$-0.594038\pi$
$461$	−5763.64	−0.582298	−0.291149	+	0.956678i	$0.594038\pi$
$462$	0	0
$463$	12452.2	1.24990	0.624949	−	0.780666i	$-0.285120\pi$
$463$	12452.2	1.24990	0.624949	+	0.780666i	$0.285120\pi$
$464$	2322.13	0.232333
$465$	−490.112	−0.0488783
$466$	−235.163	−0.0233771
$467$	7305.83	0.723926	0.361963	−	0.932192i	$-0.382107\pi$
$467$	7305.83	0.723926	0.361963	+	0.932192i	$0.382107\pi$
$468$	−3762.23	−0.371601
$469$	0	0
$470$	−1542.05	−0.151339
$471$	−3062.07	−0.299560
$472$	3967.54	0.386908
$473$	2867.65	0.278762
$474$	1009.79	0.0978508
$475$	7048.30	0.680838
$476$	0	0
$477$	14360.3	1.37843
$478$	46.4170	0.00444155
$479$	6078.18	0.579790	0.289895	−	0.957059i	$-0.406380\pi$
$479$	6078.18	0.579790	0.289895	+	0.957059i	$0.406380\pi$
$480$	−166.633	−0.0158453
$481$	1719.18	0.162968
$482$	−619.704	−0.0585617
$483$	0	0
$484$	−3936.72	−0.369714
$485$	4681.95	0.438343
$486$	−6774.71	−0.632319
$487$	−15944.9	−1.48364	−0.741818	−	0.670601i	$-0.766036\pi$
$487$	−15944.9	−1.48364	−0.741818	+	0.670601i	$0.766036\pi$
$488$	6545.45	0.607169
$489$	1518.02	0.140383
$490$	0	0
$491$	−13095.3	−1.20363	−0.601817	−	0.798634i	$-0.705556\pi$
$491$	−13095.3	−1.20363	−0.601817	+	0.798634i	$0.705556\pi$
$492$	−1528.68	−0.140078
$493$	−4989.93	−0.455852
$494$	−4804.46	−0.437577
$495$	−1233.23	−0.111979
$496$	1505.93	0.136327
$497$	0	0
$498$	4.40913	0.000396742 0
$499$	−4230.74	−0.379547	−0.189774	−	0.981828i	$-0.560775\pi$
$499$	−4230.74	−0.379547	−0.189774	+	0.981828i	$0.560775\pi$
$500$	−2721.25	−0.243396
$501$	4979.93	0.444085
$502$	−9418.10	−0.837351
$503$	−22512.5	−1.99559	−0.997797	−	0.0663437i	$-0.978867\pi$
$503$	−22512.5	−1.99559	−0.997797	+	0.0663437i	$0.978867\pi$
$504$	0	0
$505$	4214.01	0.371329
$506$	856.664	0.0752635
$507$	1119.38	0.0980536
$508$	1642.66	0.143467
$509$	−7913.74	−0.689136	−0.344568	−	0.938761i	$-0.611975\pi$
$509$	−7913.74	−0.689136	−0.344568	+	0.938761i	$0.611975\pi$
$510$	358.071	0.0310895
$511$	0	0
$512$	512.000	0.0441942
$513$	−5639.96	−0.485400
$514$	10128.5	0.869159
$515$	−859.012	−0.0735002
$516$	−1141.40	−0.0973788
$517$	−5109.92	−0.434689
$518$	0	0
$519$	−3152.22	−0.266604
$520$	897.217	0.0756646
$521$	−290.126	−0.0243966	−0.0121983	−	0.999926i	$-0.503883\pi$
$521$	−290.126	−0.0243966	−0.0121983	+	0.999926i	$0.503883\pi$
$522$	−6840.40	−0.573556
$523$	−3625.32	−0.303105	−0.151553	−	0.988449i	$-0.548427\pi$
$523$	−3625.32	−0.303105	−0.151553	+	0.988449i	$0.548427\pi$
$524$	4356.09	0.363161
$525$	0	0
$526$	−3248.12	−0.269249
$527$	−3236.02	−0.267482
$528$	−552.176	−0.0455121
$529$	529.000	0.0434783
$530$	−3424.63	−0.280673
$531$	−11687.3	−0.955155
$532$	0	0
$533$	8231.03	0.668903
$534$	−557.691	−0.0451941
$535$	−4870.17	−0.393562
$536$	−4932.61	−0.397493
$537$	−4629.55	−0.372030
$538$	−10632.0	−0.852004
$539$	0	0
$540$	1053.24	0.0839342
$541$	−4970.02	−0.394968	−0.197484	−	0.980306i	$-0.563277\pi$
$541$	−4970.02	−0.394968	−0.197484	+	0.980306i	$0.563277\pi$
$542$	7897.47	0.625877
$543$	−4080.26	−0.322469
$544$	−1100.21	−0.0867119
$545$	172.509	0.0135587
$546$	0	0
$547$	4837.90	0.378161	0.189080	−	0.981962i	$-0.439449\pi$
$547$	4837.90	0.378161	0.189080	+	0.981962i	$0.439449\pi$
$548$	7980.22	0.622077
$549$	−19281.2	−1.49891
$550$	−4361.68	−0.338150
$551$	−8735.35	−0.675387
$552$	−340.976	−0.0262915
$553$	0	0
$554$	−14148.5	−1.08504
$555$	−224.301	−0.0171550
$556$	10986.7	0.838022
$557$	15648.8	1.19042	0.595209	−	0.803571i	$-0.297069\pi$
$557$	15648.8	1.19042	0.595209	+	0.803571i	$0.297069\pi$
$558$	−4436.07	−0.336548
$559$	6145.75	0.465005
$560$	0	0
$561$	1186.55	0.0892978
$562$	3217.23	0.241478
$563$	1268.45	0.0949532	0.0474766	−	0.998872i	$-0.484882\pi$
$563$	1268.45	0.0949532	0.0474766	+	0.998872i	$0.484882\pi$
$564$	2033.89	0.151848
$565$	−2624.91	−0.195453
$566$	−3645.59	−0.270734
$567$	0	0
$568$	7805.08	0.576574
$569$	10490.1	0.772878	0.386439	−	0.922315i	$-0.373705\pi$
$569$	10490.1	0.772878	0.386439	+	0.922315i	$0.373705\pi$
$570$	626.837	0.0460619
$571$	9206.76	0.674765	0.337383	−	0.941368i	$-0.390458\pi$
$571$	9206.76	0.674765	0.337383	+	0.941368i	$0.390458\pi$
$572$	2973.13	0.217330
$573$	−4906.59	−0.357724
$574$	0	0
$575$	−2693.39	−0.195343
$576$	−1508.22	−0.109101
$577$	8009.57	0.577890	0.288945	−	0.957346i	$-0.406695\pi$
$577$	8009.57	0.577890	0.288945	+	0.957346i	$0.406695\pi$
$578$	−7461.80	−0.536972
$579$	2229.96	0.160059
$580$	1631.30	0.116786
$581$	0	0
$582$	−6175.27	−0.439816
$583$	−11348.3	−0.806172
$584$	−1192.93	−0.0845274
$585$	−2642.97	−0.186792
$586$	13841.5	0.975743
$587$	5277.47	0.371081	0.185540	−	0.982637i	$-0.440596\pi$
$587$	5277.47	0.371081	0.185540	+	0.982637i	$0.440596\pi$
$588$	0	0
$589$	−5664.96	−0.396300
$590$	2787.19	0.194486
$591$	−1250.90	−0.0870642
$592$	689.190	0.0478472
$593$	−16290.5	−1.12811	−0.564057	−	0.825736i	$-0.690760\pi$
$593$	−16290.5	−1.12811	−0.564057	+	0.825736i	$0.690760\pi$
$594$	3490.16	0.241083
$595$	0	0
$596$	−2578.55	−0.177217
$597$	−1504.60	−0.103147
$598$	1835.94	0.125547
$599$	−22889.1	−1.56131	−0.780654	−	0.624964i	$-0.785114\pi$
$599$	−22889.1	−1.56131	−0.780654	+	0.624964i	$0.785114\pi$
$600$	1736.07	0.118124
$601$	22607.8	1.53443	0.767213	−	0.641392i	$-0.221643\pi$
$601$	22607.8	1.53443	0.767213	+	0.641392i	$0.221643\pi$
$602$	0	0
$603$	14530.2	0.981284
$604$	9967.73	0.671492
$605$	−2765.54	−0.185843
$606$	−5558.08	−0.372577
$607$	−2369.05	−0.158413	−0.0792067	−	0.996858i	$-0.525239\pi$
$607$	−2369.05	−0.158413	−0.0792067	+	0.996858i	$0.525239\pi$
$608$	−1926.03	−0.128472
$609$	0	0
$610$	4598.18	0.305205
$611$	−10951.2	−0.725106
$612$	3240.94	0.214064
$613$	−13700.9	−0.902730	−0.451365	−	0.892339i	$-0.649063\pi$
$613$	−13700.9	−0.902730	−0.451365	+	0.892339i	$0.649063\pi$
$614$	−8778.96	−0.577019
$615$	−1073.90	−0.0704127
$616$	0	0
$617$	8426.94	0.549847	0.274924	−	0.961466i	$-0.411347\pi$
$617$	8426.94	0.549847	0.274924	+	0.961466i	$0.411347\pi$
$618$	1133.00	0.0737473
$619$	1602.35	0.104045	0.0520225	−	0.998646i	$-0.483433\pi$
$619$	1602.35	0.104045	0.0520225	+	0.998646i	$0.483433\pi$
$620$	1057.91	0.0685271
$621$	2155.22	0.139269
$622$	13238.6	0.853405
$623$	0	0
$624$	−1183.39	−0.0759189
$625$	12726.3	0.814484
$626$	12877.6	0.822195
$627$	2077.16	0.132303
$628$	6609.51	0.419981
$629$	−1480.97	−0.0938794
$630$	0	0
$631$	20975.3	1.32332	0.661660	−	0.749804i	$-0.269852\pi$
$631$	20975.3	1.32332	0.661660	+	0.749804i	$0.269852\pi$
$632$	−2179.65	−0.137186
$633$	−9861.97	−0.619239
$634$	9867.47	0.618119
$635$	1153.97	0.0721161
$636$	4516.93	0.281616
$637$	0	0
$638$	5405.68	0.335443
$639$	−22991.7	−1.42338
$640$	359.680	0.0222150
$641$	−17597.6	−1.08434	−0.542172	−	0.840267i	$-0.682398\pi$
$641$	−17597.6	−1.08434	−0.542172	+	0.840267i	$0.682398\pi$
$642$	6423.52	0.394885
$643$	−379.731	−0.0232895	−0.0116447	−	0.999932i	$-0.503707\pi$
$643$	−379.731	−0.0232895	−0.0116447	+	0.999932i	$0.503707\pi$
$644$	0	0
$645$	−801.835	−0.0489492
$646$	4138.76	0.252070
$647$	−5644.50	−0.342980	−0.171490	−	0.985186i	$-0.554858\pi$
$647$	−5644.50	−0.342980	−0.171490	+	0.985186i	$0.554858\pi$
$648$	3701.06	0.224369
$649$	9235.99	0.558620
$650$	−9347.66	−0.564070
$651$	0	0
$652$	−3276.67	−0.196816
$653$	−23152.5	−1.38748	−0.693742	−	0.720223i	$-0.744039\pi$
$653$	−23152.5	−1.38748	−0.693742	+	0.720223i	$0.744039\pi$
$654$	−227.532	−0.0136043
$655$	3060.15	0.182550
$656$	3299.68	0.196389
$657$	3514.07	0.208671
$658$	0	0
$659$	18788.9	1.11064	0.555320	−	0.831637i	$-0.312596\pi$
$659$	18788.9	1.11064	0.555320	+	0.831637i	$0.312596\pi$
$660$	−387.904	−0.0228775
$661$	−19538.4	−1.14970	−0.574852	−	0.818257i	$-0.694940\pi$
$661$	−19538.4	−1.14970	−0.574852	+	0.818257i	$0.694940\pi$
$662$	8134.73	0.477591
$663$	2542.93	0.148958
$664$	−9.51715	−0.000556231 0
$665$	0	0
$666$	−2030.17	−0.118120
$667$	3338.07	0.193779
$668$	−10749.2	−0.622606
$669$	−2686.98	−0.155283
$670$	−3465.15	−0.199807
$671$	15237.1	0.876634
$672$	0	0
$673$	−19828.4	−1.13570	−0.567852	−	0.823130i	$-0.692226\pi$
$673$	−19828.4	−1.13570	−0.567852	+	0.823130i	$0.692226\pi$
$674$	19889.7	1.13668
$675$	−10973.2	−0.625719
$676$	−2416.18	−0.137471
$677$	−5055.39	−0.286993	−0.143497	−	0.989651i	$-0.545835\pi$
$677$	−5055.39	−0.286993	−0.143497	+	0.989651i	$0.545835\pi$
$678$	3462.13	0.196110
$679$	0	0
$680$	−772.900	−0.0435873
$681$	1602.86	0.0901933
$682$	3505.63	0.196829
$683$	14558.0	0.815585	0.407793	−	0.913075i	$-0.366299\pi$
$683$	14558.0	0.815585	0.407793	+	0.913075i	$0.366299\pi$
$684$	5673.58	0.317156
$685$	5606.10	0.312698
$686$	0	0
$687$	−8571.06	−0.475992
$688$	2463.73	0.136525
$689$	−24320.9	−1.34478
$690$	−239.535	−0.0132159
$691$	−11411.5	−0.628240	−0.314120	−	0.949383i	$-0.601709\pi$
$691$	−11411.5	−0.628240	−0.314120	+	0.949383i	$0.601709\pi$
$692$	6804.11	0.373777
$693$	0	0
$694$	22480.5	1.22961
$695$	7718.16	0.421247
$696$	−2151.61	−0.117179
$697$	−7090.54	−0.385328
$698$	−16095.9	−0.872837
$699$	217.894	0.0117904
$700$	0	0
$701$	−36915.1	−1.98896	−0.994482	−	0.104909i	$-0.966545\pi$
$701$	−36915.1	−1.98896	−0.994482	+	0.104909i	$0.966545\pi$
$702$	7479.88	0.402151
$703$	−2592.58	−0.139091
$704$	1191.88	0.0638078
$705$	1428.81	0.0763290
$706$	10289.4	0.548509
$707$	0	0
$708$	−3676.18	−0.195140
$709$	15981.0	0.846517	0.423259	−	0.906009i	$-0.360886\pi$
$709$	15981.0	0.846517	0.423259	+	0.906009i	$0.360886\pi$
$710$	5483.07	0.289825
$711$	6420.68	0.338670
$712$	1203.78	0.0633619
$713$	2164.77	0.113704
$714$	0	0
$715$	2088.62	0.109245
$716$	9992.95	0.521584
$717$	−43.0083	−0.00224013
$718$	7789.08	0.404855
$719$	6526.68	0.338531	0.169266	−	0.985570i	$-0.445860\pi$
$719$	6526.68	0.338531	0.169266	+	0.985570i	$0.445860\pi$
$720$	−1059.52	−0.0548418
$721$	0	0
$722$	−6472.71	−0.333642
$723$	574.195	0.0295360
$724$	8807.28	0.452100
$725$	−16995.7	−0.870626
$726$	3647.62	0.186468
$727$	−18670.9	−0.952500	−0.476250	−	0.879310i	$-0.658004\pi$
$727$	−18670.9	−0.952500	−0.476250	+	0.879310i	$0.658004\pi$
$728$	0	0
$729$	−6213.87	−0.315697
$730$	−838.036	−0.0424892
$731$	−5294.20	−0.267870
$732$	−6064.78	−0.306231
$733$	−38561.0	−1.94309	−0.971544	−	0.236858i	$-0.923882\pi$
$733$	−38561.0	−1.94309	−0.971544	+	0.236858i	$0.923882\pi$
$734$	−8106.40	−0.407647
$735$	0	0
$736$	736.000	0.0368605
$737$	−11482.6	−0.573902
$738$	−9720.01	−0.484822
$739$	−4948.98	−0.246348	−0.123174	−	0.992385i	$-0.539307\pi$
$739$	−4948.98	−0.246348	−0.123174	+	0.992385i	$0.539307\pi$
$740$	484.156	0.0240512
$741$	4451.64	0.220695
$742$	0	0
$743$	−11730.7	−0.579217	−0.289608	−	0.957145i	$-0.593525\pi$
$743$	−11730.7	−0.579217	−0.289608	+	0.957145i	$0.593525\pi$
$744$	−1395.34	−0.0687575
$745$	−1811.43	−0.0890815
$746$	−17601.6	−0.863859
$747$	28.0351	0.00137316
$748$	−2561.18	−0.125195
$749$	0	0
$750$	2521.41	0.122758
$751$	−30622.3	−1.48791	−0.743956	−	0.668229i	$-0.767053\pi$
$751$	−30622.3	−1.48791	−0.743956	+	0.668229i	$0.767053\pi$
$752$	−4390.18	−0.212890
$753$	8726.47	0.422324
$754$	11585.1	0.559554
$755$	7002.33	0.337537
$756$	0	0
$757$	−15308.6	−0.735010	−0.367505	−	0.930022i	$-0.619788\pi$
$757$	−15308.6	−0.735010	−0.367505	+	0.930022i	$0.619788\pi$
$758$	−2010.15	−0.0963218
$759$	−793.754	−0.0379597
$760$	−1353.03	−0.0645786
$761$	18956.7	0.902994	0.451497	−	0.892273i	$-0.350890\pi$
$761$	18956.7	0.902994	0.451497	+	0.892273i	$0.350890\pi$
$762$	−1522.03	−0.0723585
$763$	0	0
$764$	10590.9	0.501527
$765$	2276.76	0.107603
$766$	4758.89	0.224472
$767$	19794.0	0.931836
$768$	−474.401	−0.0222897
$769$	−31576.8	−1.48074	−0.740370	−	0.672199i	$-0.765350\pi$
$769$	−31576.8	−1.48074	−0.740370	+	0.672199i	$0.765350\pi$
$770$	0	0
$771$	−9384.67	−0.438367
$772$	−4813.40	−0.224402
$773$	−9127.49	−0.424700	−0.212350	−	0.977194i	$-0.568112\pi$
$773$	−9127.49	−0.424700	−0.212350	+	0.977194i	$0.568112\pi$
$774$	−7257.51	−0.337036
$775$	−11021.9	−0.510861
$776$	13329.4	0.616620
$777$	0	0
$778$	−14378.1	−0.662571
$779$	−12412.7	−0.570899
$780$	−831.329	−0.0381620
$781$	18169.4	0.832460
$782$	−1581.56	−0.0723227
$783$	13599.7	0.620709
$784$	0	0
$785$	4643.18	0.211111
$786$	−4036.19	−0.183163
$787$	−6066.87	−0.274791	−0.137395	−	0.990516i	$-0.543873\pi$
$787$	−6066.87	−0.274791	−0.137395	+	0.990516i	$0.543873\pi$
$788$	2700.07	0.122064
$789$	3009.59	0.135798
$790$	−1531.20	−0.0689592
$791$	0	0
$792$	−3510.97	−0.157521
$793$	32655.1	1.46232
$794$	−3060.85	−0.136808
$795$	3173.14	0.141559
$796$	3247.69	0.144612
$797$	39942.9	1.77522	0.887611	−	0.460594i	$-0.152364\pi$
$797$	39942.9	1.77522	0.887611	+	0.460594i	$0.152364\pi$
$798$	0	0
$799$	9433.85	0.417704
$800$	−3747.33	−0.165610
$801$	−3546.03	−0.156420
$802$	−14154.2	−0.623193
$803$	−2777.02	−0.122041
$804$	4570.37	0.200478
$805$	0	0
$806$	7513.04	0.328332
$807$	9851.23	0.429715
$808$	11997.2	0.522351
$809$	17048.9	0.740924	0.370462	−	0.928848i	$-0.379199\pi$
$809$	17048.9	0.740924	0.370462	+	0.928848i	$0.379199\pi$
$810$	2599.99	0.112783
$811$	−43025.5	−1.86292	−0.931462	−	0.363839i	$-0.881466\pi$
$811$	−43025.5	−1.86292	−0.931462	+	0.363839i	$0.881466\pi$
$812$	0	0
$813$	−7317.51	−0.315666
$814$	1604.36	0.0690820
$815$	−2301.86	−0.0989332
$816$	1019.42	0.0437338
$817$	−9268.01	−0.396875
$818$	11367.1	0.485869
$819$	0	0
$820$	2318.03	0.0987183
$821$	−30724.0	−1.30606	−0.653030	−	0.757332i	$-0.726503\pi$
$821$	−30724.0	−1.30606	−0.653030	+	0.757332i	$0.726503\pi$
$822$	−7394.18	−0.313749
$823$	−13338.2	−0.564936	−0.282468	−	0.959277i	$-0.591153\pi$
$823$	−13338.2	−0.564936	−0.282468	+	0.959277i	$0.591153\pi$
$824$	−2445.59	−0.103393
$825$	4041.38	0.170549
$826$	0	0
$827$	−29240.6	−1.22950	−0.614750	−	0.788722i	$-0.710743\pi$
$827$	−29240.6	−1.22950	−0.614750	+	0.788722i	$0.710743\pi$
$828$	−2168.06	−0.0909969
$829$	−35576.4	−1.49050	−0.745248	−	0.666788i	$-0.767669\pi$
$829$	−35576.4	−1.49050	−0.745248	+	0.666788i	$0.767669\pi$
$830$	−6.68580	−0.000279599 0
$831$	13109.5	0.547248
$832$	2554.36	0.106438
$833$	0	0
$834$	−10179.9	−0.422663
$835$	−7551.34	−0.312964
$836$	−4483.58	−0.185488
$837$	8819.57	0.364216
$838$	−19748.7	−0.814089
$839$	−24395.1	−1.00383	−0.501915	−	0.864917i	$-0.667371\pi$
$839$	−24395.1	−1.00383	−0.501915	+	0.864917i	$0.667371\pi$
$840$	0	0
$841$	−3325.29	−0.136344
$842$	3530.93	0.144518
$843$	−2980.97	−0.121791
$844$	21287.2	0.868170
$845$	−1697.37	−0.0691021
$846$	12932.3	0.525558
$847$	0	0
$848$	−9749.85	−0.394824
$849$	3377.87	0.136547
$850$	8052.46	0.324938
$851$	990.710	0.0399073
$852$	−7231.91	−0.290799
$853$	−44158.8	−1.77253	−0.886266	−	0.463177i	$-0.846709\pi$
$853$	−44158.8	−1.77253	−0.886266	+	0.463177i	$0.846709\pi$
$854$	0	0
$855$	3985.69	0.159424
$856$	−13865.2	−0.553627
$857$	31335.3	1.24900	0.624500	−	0.781025i	$-0.285303\pi$
$857$	31335.3	1.24900	0.624500	+	0.781025i	$0.285303\pi$
$858$	−2754.80	−0.109612
$859$	8363.71	0.332207	0.166104	−	0.986108i	$-0.446881\pi$
$859$	8363.71	0.332207	0.166104	+	0.986108i	$0.446881\pi$
$860$	1730.77	0.0686265
$861$	0	0
$862$	8415.32	0.332514
$863$	37339.2	1.47282	0.736408	−	0.676538i	$-0.236520\pi$
$863$	37339.2	1.47282	0.736408	+	0.676538i	$0.236520\pi$
$864$	2998.56	0.118071
$865$	4779.89	0.187886
$866$	−949.126	−0.0372432
$867$	6913.84	0.270826
$868$	0	0
$869$	−5073.98	−0.198070
$870$	−1511.50	−0.0589020
$871$	−24608.7	−0.957328
$872$	491.130	0.0190731
$873$	−39264.9	−1.52224
$874$	−2768.67	−0.107153
$875$	0	0
$876$	1105.33	0.0426320
$877$	−45581.4	−1.75504	−0.877522	−	0.479536i	$-0.840805\pi$
$877$	−45581.4	−1.75504	−0.877522	+	0.479536i	$0.840805\pi$
$878$	32541.2	1.25081
$879$	−12825.0	−0.492123
$880$	837.295	0.0320741
$881$	−39745.3	−1.51993	−0.759963	−	0.649967i	$-0.774783\pi$
$881$	−39745.3	−1.51993	−0.759963	+	0.649967i	$0.774783\pi$
$882$	0	0
$883$	15480.4	0.589984	0.294992	−	0.955500i	$-0.404683\pi$
$883$	15480.4	0.589984	0.294992	+	0.955500i	$0.404683\pi$
$884$	−5488.94	−0.208838
$885$	−2582.51	−0.0980907
$886$	8969.14	0.340095
$887$	−10055.3	−0.380637	−0.190319	−	0.981722i	$-0.560952\pi$
$887$	−10055.3	−0.380637	−0.190319	+	0.981722i	$0.560952\pi$
$888$	−638.579	−0.0241321
$889$	0	0
$890$	845.657	0.0318500
$891$	8615.66	0.323945
$892$	5799.88	0.217707
$893$	16514.9	0.618867
$894$	2389.19	0.0893810
$895$	7020.04	0.262183
$896$	0	0
$897$	−1701.12	−0.0633208
$898$	3833.07	0.142440
$899$	13660.0	0.506771
$900$	11038.6	0.408839
$901$	20951.0	0.774672
$902$	7681.30	0.283547
$903$	0	0
$904$	−7473.06	−0.274945
$905$	6187.11	0.227256
$906$	−9235.74	−0.338672
$907$	41037.7	1.50235	0.751177	−	0.660101i	$-0.229487\pi$
$907$	41037.7	1.50235	0.751177	+	0.660101i	$0.229487\pi$
$908$	−3459.79	−0.126451
$909$	−35340.6	−1.28952
$910$	0	0
$911$	−19336.8	−0.703247	−0.351624	−	0.936141i	$-0.614370\pi$
$911$	−19336.8	−0.703247	−0.351624	+	0.936141i	$0.614370\pi$
$912$	1784.59	0.0647957
$913$	−22.1549	−0.000803089 0
$914$	20277.7	0.733838
$915$	−4260.51	−0.153932
$916$	18500.8	0.667338
$917$	0	0
$918$	−6443.48	−0.231663
$919$	49626.3	1.78131	0.890654	−	0.454683i	$-0.150247\pi$
$919$	49626.3	1.78131	0.890654	+	0.454683i	$0.150247\pi$
$920$	517.040	0.0185286
$921$	8134.27	0.291024
$922$	−11527.3	−0.411747
$923$	38939.4	1.38863
$924$	0	0
$925$	−5044.18	−0.179299
$926$	24904.4	0.883811
$927$	7204.07	0.255245
$928$	4644.27	0.164284
$929$	−15209.7	−0.537152	−0.268576	−	0.963258i	$-0.586553\pi$
$929$	−15209.7	−0.537152	−0.268576	+	0.963258i	$0.586553\pi$
$930$	−980.224	−0.0345622
$931$	0	0
$932$	−470.327	−0.0165301
$933$	−12266.4	−0.430421
$934$	14611.7	0.511893
$935$	−1799.23	−0.0629315
$936$	−7524.47	−0.262762
$937$	−12658.4	−0.441337	−0.220669	−	0.975349i	$-0.570824\pi$
$937$	−12658.4	−0.441337	−0.220669	+	0.975349i	$0.570824\pi$
$938$	0	0
$939$	−11932.0	−0.414680
$940$	−3084.10	−0.107013
$941$	20893.6	0.723815	0.361908	−	0.932214i	$-0.382126\pi$
$941$	20893.6	0.723815	0.361908	+	0.932214i	$0.382126\pi$
$942$	−6124.14	−0.211821
$943$	4743.30	0.163800
$944$	7935.08	0.273586
$945$	0	0
$946$	5735.30	0.197115
$947$	−34190.9	−1.17324	−0.586618	−	0.809864i	$-0.699541\pi$
$947$	−34190.9	−1.17324	−0.586618	+	0.809864i	$0.699541\pi$
$948$	2019.58	0.0691910
$949$	−5951.52	−0.203577
$950$	14096.6	0.481425
$951$	−9142.84	−0.311753
$952$	0	0
$953$	2536.75	0.0862262	0.0431131	−	0.999070i	$-0.486272\pi$
$953$	2536.75	0.0862262	0.0431131	+	0.999070i	$0.486272\pi$
$954$	28720.5	0.974697
$955$	7440.12	0.252101
$956$	92.8340	0.00314065
$957$	−5008.70	−0.169183
$958$	12156.4	0.409973
$959$	0	0
$960$	−333.266	−0.0112043
$961$	−20932.3	−0.702640
$962$	3438.35	0.115236
$963$	40843.4	1.36673
$964$	−1239.41	−0.0414094
$965$	−3381.41	−0.112800
$966$	0	0
$967$	−21632.0	−0.719378	−0.359689	−	0.933072i	$-0.617117\pi$
$967$	−21632.0	−0.719378	−0.359689	+	0.933072i	$0.617117\pi$
$968$	−7873.43	−0.261427
$969$	−3834.82	−0.127133
$970$	9363.89	0.309955
$971$	−8754.33	−0.289331	−0.144665	−	0.989481i	$-0.546211\pi$
$971$	−8754.33	−0.289331	−0.144665	+	0.989481i	$0.546211\pi$
$972$	−13549.4	−0.447117
$973$	0	0
$974$	−31889.7	−1.04909
$975$	8661.21	0.284493
$976$	13090.9	0.429334
$977$	12419.8	0.406698	0.203349	−	0.979106i	$-0.434817\pi$
$977$	12419.8	0.406698	0.203349	+	0.979106i	$0.434817\pi$
$978$	3036.04	0.0992657
$979$	2802.27	0.0914822
$980$	0	0
$981$	−1446.74	−0.0470856
$982$	−26190.7	−0.851097
$983$	−6967.80	−0.226082	−0.113041	−	0.993590i	$-0.536059\pi$
$983$	−6967.80	−0.226082	−0.113041	+	0.993590i	$0.536059\pi$
$984$	−3057.37	−0.0990502
$985$	1896.80	0.0613574
$986$	−9979.86	−0.322336
$987$	0	0
$988$	−9608.92	−0.309413
$989$	3541.61	0.113869
$990$	−2466.45	−0.0791808
$991$	−1073.43	−0.0344083	−0.0172041	−	0.999852i	$-0.505477\pi$
$991$	−1073.43	−0.0344083	−0.0172041	+	0.999852i	$0.505477\pi$
$992$	3011.85	0.0963976
$993$	−7537.34	−0.240877
$994$	0	0
$995$	2281.50	0.0726919
$996$	8.81825	0.000280539 0
$997$	3416.02	0.108512	0.0542560	−	0.998527i	$-0.482721\pi$
$997$	3416.02	0.108512	0.0542560	+	0.998527i	$0.482721\pi$
$998$	−8461.49	−0.268380
$999$	4036.29	0.127830

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.y.1.4		11
7.2	even	3		322.4.e.a.277.8	yes	22
7.4	even	3		322.4.e.a.93.8	✓	22
7.6	odd	2		2254.4.a.v.1.8		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.4.e.a.93.8	✓	22	7.4	even	3
322.4.e.a.277.8	yes	22	7.2	even	3
2254.4.a.v.1.8		11	7.6	odd	2
2254.4.a.y.1.4		11	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2254 = 2 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2254.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -1.85313 q^{3} +4.00000 q^{4} +2.81000 q^{5} -3.70626 q^{6} +8.00000 q^{8} -23.5659 q^{9} +5.62000 q^{10} +18.6231 q^{11} -7.41251 q^{12} +39.9118 q^{13} -5.20729 q^{15} +16.0000 q^{16} -34.3817 q^{17} -47.1318 q^{18} -60.1884 q^{19} +11.2400 q^{20} +37.2462 q^{22} +23.0000 q^{23} -14.8250 q^{24} -117.104 q^{25} +79.8236 q^{26} +93.7051 q^{27} +145.133 q^{29} -10.4146 q^{30} +94.1204 q^{31} +32.0000 q^{32} -34.5110 q^{33} -68.7634 q^{34} -94.2637 q^{36} +43.0744 q^{37} -120.377 q^{38} -73.9617 q^{39} +22.4800 q^{40} +206.230 q^{41} +153.983 q^{43} +74.4925 q^{44} -66.2202 q^{45} +46.0000 q^{46} -274.386 q^{47} -29.6500 q^{48} -234.208 q^{50} +63.7137 q^{51} +159.647 q^{52} -609.365 q^{53} +187.410 q^{54} +52.3309 q^{55} +111.537 q^{57} +290.267 q^{58} +495.942 q^{59} -20.8291 q^{60} +818.182 q^{61} +188.241 q^{62} +64.0000 q^{64} +112.152 q^{65} -69.0221 q^{66} -616.576 q^{67} -137.527 q^{68} -42.6219 q^{69} +975.636 q^{71} -188.527 q^{72} -149.117 q^{73} +86.1487 q^{74} +217.009 q^{75} -240.754 q^{76} -147.923 q^{78} -272.456 q^{79} +44.9600 q^{80} +462.632 q^{81} +412.461 q^{82} -1.18964 q^{83} -96.6125 q^{85} +307.967 q^{86} -268.951 q^{87} +148.985 q^{88} +150.473 q^{89} -132.440 q^{90} +92.0000 q^{92} -174.417 q^{93} -548.772 q^{94} -169.129 q^{95} -59.3001 q^{96} +1666.17 q^{97} -438.871 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 22 q^{2} + 18 q^{3} + 44 q^{4} + 33 q^{5} + 36 q^{6} + 88 q^{8} + 171 q^{9} + 66 q^{10} + 8 q^{11} + 72 q^{12} + 185 q^{13} - 186 q^{15} + 176 q^{16} + 107 q^{17} + 342 q^{18} + 114 q^{19} + 132 q^{20}+ \cdots - 1729 q^{99}+O(q^{100}) \) 11 * q + 22 * q^2 + 18 * q^3 + 44 * q^4 + 33 * q^5 + 36 * q^6 + 88 * q^8 + 171 * q^9 + 66 * q^10 + 8 * q^11 + 72 * q^12 + 185 * q^13 - 186 * q^15 + 176 * q^16 + 107 * q^17 + 342 * q^18 + 114 * q^19 + 132 * q^20 + 16 * q^22 + 253 * q^23 + 144 * q^24 + 494 * q^25 + 370 * q^26 + 561 * q^27 - 79 * q^29 - 372 * q^30 + 488 * q^31 + 352 * q^32 + 304 * q^33 + 214 * q^34 + 684 * q^36 - 41 * q^37 + 228 * q^38 + 425 * q^39 + 264 * q^40 + 812 * q^41 + 219 * q^43 + 32 * q^44 + 619 * q^45 + 506 * q^46 + 1141 * q^47 + 288 * q^48 + 988 * q^50 + 83 * q^51 + 740 * q^52 + 579 * q^53 + 1122 * q^54 + 951 * q^55 + 383 * q^57 - 158 * q^58 + 1695 * q^59 - 744 * q^60 + 1105 * q^61 + 976 * q^62 + 704 * q^64 - 347 * q^65 + 608 * q^66 + 695 * q^67 + 428 * q^68 + 414 * q^69 + 65 * q^71 + 1368 * q^72 + 1935 * q^73 - 82 * q^74 + 1712 * q^75 + 456 * q^76 + 850 * q^78 + 447 * q^79 + 528 * q^80 + 2203 * q^81 + 1624 * q^82 + 1389 * q^83 + 153 * q^85 + 438 * q^86 + 3354 * q^87 + 64 * q^88 + 2698 * q^89 + 1238 * q^90 + 1012 * q^92 - 686 * q^93 + 2282 * q^94 + 827 * q^95 + 576 * q^96 + 1589 * q^97 - 1729 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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