LMFDB - Embedded newform 2254.4.a.s.1.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2254,4,Mod(1,2254)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2254.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [11,-22,0,44,-17] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$132.990305153$
Analytic rank:	$1$
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} - 178 x^{9} - 29 x^{8} + 11409 x^{7} + 5153 x^{6} - 327598 x^{5} - 270393 x^{4} + 4277164 x^{3} + \cdots - 33055073 $ x^11 - 178x^9 - 29x^8 + 11409x^7 + 5153x^6 - 327598x^5 - 270393x^4 + 4277164x^3 + 5208998x^2 - 20738379*x - 33055073
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 322)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$5.44340$ 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)

$f(q)$	$=$	\(q-2.00000 q^{2} +5.44340 q^{3} +4.00000 q^{4} -7.06685 q^{5} -10.8868 q^{6} -8.00000 q^{8} +2.63057 q^{9} +14.1337 q^{10} +59.2166 q^{11} +21.7736 q^{12} -20.3418 q^{13} -38.4677 q^{15} +16.0000 q^{16} +10.5614 q^{17} -5.26114 q^{18} +76.8327 q^{19} -28.2674 q^{20} -118.433 q^{22} -23.0000 q^{23} -43.5472 q^{24} -75.0596 q^{25} +40.6835 q^{26} -132.652 q^{27} +153.342 q^{29} +76.9353 q^{30} -338.533 q^{31} -32.0000 q^{32} +322.339 q^{33} -21.1228 q^{34} +10.5223 q^{36} -69.0064 q^{37} -153.665 q^{38} -110.728 q^{39} +56.5348 q^{40} -377.866 q^{41} -238.549 q^{43} +236.866 q^{44} -18.5898 q^{45} +46.0000 q^{46} +190.609 q^{47} +87.0944 q^{48} +150.119 q^{50} +57.4899 q^{51} -81.3670 q^{52} -11.8805 q^{53} +265.305 q^{54} -418.475 q^{55} +418.231 q^{57} -306.684 q^{58} +289.361 q^{59} -153.871 q^{60} -561.791 q^{61} +677.065 q^{62} +64.0000 q^{64} +143.752 q^{65} -644.679 q^{66} +416.218 q^{67} +42.2456 q^{68} -125.198 q^{69} +125.156 q^{71} -21.0446 q^{72} -535.502 q^{73} +138.013 q^{74} -408.579 q^{75} +307.331 q^{76} +221.456 q^{78} +1084.99 q^{79} -113.070 q^{80} -793.105 q^{81} +755.732 q^{82} +828.120 q^{83} -74.6358 q^{85} +477.099 q^{86} +834.702 q^{87} -473.733 q^{88} +883.731 q^{89} +37.1797 q^{90} -92.0000 q^{92} -1842.77 q^{93} -381.217 q^{94} -542.965 q^{95} -174.189 q^{96} -551.245 q^{97} +155.773 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 22 q^{2} + 44 q^{4} - 17 q^{5} - 88 q^{8} + 59 q^{9} + 34 q^{10} - 3 q^{13} - 144 q^{15} + 176 q^{16} - 47 q^{17} - 118 q^{18} + 138 q^{19} - 68 q^{20} - 253 q^{23} + 350 q^{25} + 6 q^{26} + 87 q^{27}+ \cdots + 3525 q^{99}+O(q^{100}) $ 11 * q - 22 * q^2 + 44 * q^4 - 17 * q^5 - 88 * q^8 + 59 * q^9 + 34 * q^10 - 3 * q^13 - 144 * q^15 + 176 * q^16 - 47 * q^17 - 118 * q^18 + 138 * q^19 - 68 * q^20 - 253 * q^23 + 350 * q^25 + 6 * q^26 + 87 * q^27 - 251 * q^29 + 288 * q^30 + 46 * q^31 - 352 * q^32 + 222 * q^33 + 94 * q^34 + 236 * q^36 + 475 * q^37 - 276 * q^38 - 115 * q^39 + 136 * q^40 - 740 * q^41 + 17 * q^43 - 875 * q^45 + 506 * q^46 + 33 * q^47 - 700 * q^50 + 1017 * q^51 - 12 * q^52 - 835 * q^53 - 174 * q^54 + 1323 * q^55 + 439 * q^57 + 502 * q^58 - 1723 * q^59 - 576 * q^60 + 253 * q^61 - 92 * q^62 + 704 * q^64 + 1753 * q^65 - 444 * q^66 - 2573 * q^67 - 188 * q^68 + 1319 * q^71 - 472 * q^72 - 1159 * q^73 - 950 * q^74 - 268 * q^75 + 552 * q^76 + 230 * q^78 + 1095 * q^79 - 272 * q^80 - 3085 * q^81 + 1480 * q^82 + 451 * q^83 + 2507 * q^85 - 34 * q^86 - 3540 * q^87 - 3634 * q^89 + 1750 * q^90 - 1012 * q^92 + 3132 * q^93 - 66 * q^94 - 4249 * q^95 + 1513 * q^97 + 3525 * 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
$3$	5.44340	1.04758	0.523791	−	0.851847i	$-0.324517\pi$
$3$	5.44340	1.04758	0.523791	+	0.851847i	$0.324517\pi$
$4$	4.00000	0.500000
$5$	−7.06685	−0.632078	−0.316039	−	0.948746i	$-0.602353\pi$
$5$	−7.06685	−0.632078	−0.316039	+	0.948746i	$0.602353\pi$
$6$	−10.8868	−0.740752
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	2.63057	0.0974285
$10$	14.1337	0.446947
$11$	59.2166	1.62313	0.811567	−	0.584260i	$-0.198615\pi$
$11$	59.2166	1.62313	0.811567	+	0.584260i	$0.198615\pi$
$12$	21.7736	0.523791
$13$	−20.3418	−0.433984	−0.216992	−	0.976173i	$-0.569624\pi$
$13$	−20.3418	−0.433984	−0.216992	+	0.976173i	$0.569624\pi$
$14$	0	0
$15$	−38.4677	−0.662154
$16$	16.0000	0.250000
$17$	10.5614	0.150677	0.0753387	−	0.997158i	$-0.475996\pi$
$17$	10.5614	0.150677	0.0753387	+	0.997158i	$0.475996\pi$
$18$	−5.26114	−0.0688924
$19$	76.8327	0.927717	0.463858	−	0.885909i	$-0.346465\pi$
$19$	76.8327	0.927717	0.463858	+	0.885909i	$0.346465\pi$
$20$	−28.2674	−0.316039
$21$	0	0
$22$	−118.433	−1.14773
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	−43.5472	−0.370376
$25$	−75.0596	−0.600477
$26$	40.6835	0.306873
$27$	−132.652	−0.945518
$28$	0	0
$29$	153.342	0.981893	0.490947	−	0.871190i	$-0.336651\pi$
$29$	153.342	0.981893	0.490947	+	0.871190i	$0.336651\pi$
$30$	76.9353	0.468214
$31$	−338.533	−1.96136	−0.980681	−	0.195613i	$-0.937330\pi$
$31$	−338.533	−1.96136	−0.980681	+	0.195613i	$0.937330\pi$
$32$	−32.0000	−0.176777
$33$	322.339	1.70037
$34$	−21.1228	−0.106545
$35$	0	0
$36$	10.5223	0.0487143
$37$	−69.0064	−0.306610	−0.153305	−	0.988179i	$-0.548992\pi$
$37$	−69.0064	−0.306610	−0.153305	+	0.988179i	$0.548992\pi$
$38$	−153.665	−0.655995
$39$	−110.728	−0.454634
$40$	56.5348	0.223473
$41$	−377.866	−1.43933	−0.719667	−	0.694319i	$-0.755706\pi$
$41$	−377.866	−1.43933	−0.719667	+	0.694319i	$0.755706\pi$
$42$	0	0
$43$	−238.549	−0.846010	−0.423005	−	0.906127i	$-0.639025\pi$
$43$	−238.549	−0.846010	−0.423005	+	0.906127i	$0.639025\pi$
$44$	236.866	0.811567
$45$	−18.5898	−0.0615825
$46$	46.0000	0.147442
$47$	190.609	0.591556	0.295778	−	0.955257i	$-0.404421\pi$
$47$	190.609	0.591556	0.295778	+	0.955257i	$0.404421\pi$
$48$	87.0944	0.261896
$49$	0	0
$50$	150.119	0.424601
$51$	57.4899	0.157847
$52$	−81.3670	−0.216992
$53$	−11.8805	−0.0307907	−0.0153953	−	0.999881i	$-0.504901\pi$
$53$	−11.8805	−0.0307907	−0.0153953	+	0.999881i	$0.504901\pi$
$54$	265.305	0.668582
$55$	−418.475	−1.02595
$56$	0	0
$57$	418.231	0.971860
$58$	−306.684	−0.694304
$59$	289.361	0.638501	0.319250	−	0.947670i	$-0.396569\pi$
$59$	289.361	0.638501	0.319250	+	0.947670i	$0.396569\pi$
$60$	−153.871	−0.331077
$61$	−561.791	−1.17918	−0.589590	−	0.807703i	$-0.700711\pi$
$61$	−561.791	−1.17918	−0.589590	+	0.807703i	$0.700711\pi$
$62$	677.065	1.38689
$63$	0	0
$64$	64.0000	0.125000
$65$	143.752	0.274312
$66$	−644.679	−1.20234
$67$	416.218	0.758943	0.379472	−	0.925203i	$-0.376106\pi$
$67$	416.218	0.758943	0.379472	+	0.925203i	$0.376106\pi$
$68$	42.2456	0.0753387
$69$	−125.198	−0.218436
$70$	0	0
$71$	125.156	0.209201	0.104600	−	0.994514i	$-0.466644\pi$
$71$	125.156	0.209201	0.104600	+	0.994514i	$0.466644\pi$
$72$	−21.0446	−0.0344462
$73$	−535.502	−0.858572	−0.429286	−	0.903169i	$-0.641235\pi$
$73$	−535.502	−0.858572	−0.429286	+	0.903169i	$0.641235\pi$
$74$	138.013	0.216806
$75$	−408.579	−0.629049
$76$	307.331	0.463858
$77$	0	0
$78$	221.456	0.321474
$79$	1084.99	1.54520	0.772601	−	0.634892i	$-0.218955\pi$
$79$	1084.99	1.54520	0.772601	+	0.634892i	$0.218955\pi$
$80$	−113.070	−0.158020
$81$	−793.105	−1.08794
$82$	755.732	1.01776
$83$	828.120	1.09516	0.547578	−	0.836754i	$-0.315550\pi$
$83$	828.120	1.09516	0.547578	+	0.836754i	$0.315550\pi$
$84$	0	0
$85$	−74.6358	−0.0952399
$86$	477.099	0.598219
$87$	834.702	1.02861
$88$	−473.733	−0.573864
$89$	883.731	1.05253	0.526265	−	0.850320i	$-0.323592\pi$
$89$	883.731	1.05253	0.526265	+	0.850320i	$0.323592\pi$
$90$	37.1797	0.0435454
$91$	0	0
$92$	−92.0000	−0.104257
$93$	−1842.77	−2.05469
$94$	−381.217	−0.418293
$95$	−542.965	−0.586390
$96$	−174.189	−0.185188
$97$	−551.245	−0.577015	−0.288507	−	0.957478i	$-0.593159\pi$
$97$	−551.245	−0.577015	−0.288507	+	0.957478i	$0.593159\pi$
$98$	0	0
$99$	155.773	0.158140
$100$	−300.238	−0.300238
$101$	−99.0429	−0.0975756	−0.0487878	−	0.998809i	$-0.515536\pi$
$101$	−99.0429	−0.0975756	−0.0487878	+	0.998809i	$0.515536\pi$
$102$	−114.980	−0.111615
$103$	1055.87	1.01007	0.505036	−	0.863098i	$-0.331479\pi$
$103$	1055.87	1.01007	0.505036	+	0.863098i	$0.331479\pi$
$104$	162.734	0.153436
$105$	0	0
$106$	23.7609	0.0217723
$107$	277.184	0.250434	0.125217	−	0.992129i	$-0.460037\pi$
$107$	277.184	0.250434	0.125217	+	0.992129i	$0.460037\pi$
$108$	−530.610	−0.472759
$109$	345.367	0.303488	0.151744	−	0.988420i	$-0.451511\pi$
$109$	345.367	0.303488	0.151744	+	0.988420i	$0.451511\pi$
$110$	836.949	0.725454
$111$	−375.629	−0.321200
$112$	0	0
$113$	−1326.83	−1.10458	−0.552289	−	0.833653i	$-0.686245\pi$
$113$	−1326.83	−1.10458	−0.552289	+	0.833653i	$0.686245\pi$
$114$	−836.461	−0.687209
$115$	162.538	0.131797
$116$	613.368	0.490947
$117$	−53.5104	−0.0422824
$118$	−578.721	−0.451488
$119$	0	0
$120$	307.741	0.234107
$121$	2175.60	1.63456
$122$	1123.58	0.833806
$123$	−2056.87	−1.50782
$124$	−1354.13	−0.980681
$125$	1413.79	1.01163
$126$	0	0
$127$	355.779	0.248585	0.124292	−	0.992246i	$-0.460334\pi$
$127$	355.779	0.248585	0.124292	+	0.992246i	$0.460334\pi$
$128$	−128.000	−0.0883883
$129$	−1298.52	−0.886265
$130$	−287.504	−0.193968
$131$	−1515.62	−1.01085	−0.505423	−	0.862872i	$-0.668663\pi$
$131$	−1515.62	−1.01085	−0.505423	+	0.862872i	$0.668663\pi$
$132$	1289.36	0.850183
$133$	0	0
$134$	−832.437	−0.536654
$135$	937.435	0.597641
$136$	−84.4912	−0.0532725
$137$	−2983.54	−1.86059	−0.930294	−	0.366814i	$-0.880449\pi$
$137$	−2983.54	−1.86059	−0.930294	+	0.366814i	$0.880449\pi$
$138$	250.396	0.154458
$139$	−1507.52	−0.919898	−0.459949	−	0.887945i	$-0.652132\pi$
$139$	−1507.52	−0.919898	−0.459949	+	0.887945i	$0.652132\pi$
$140$	0	0
$141$	1037.56	0.619704
$142$	−250.311	−0.147927
$143$	−1204.57	−0.704413
$144$	42.0891	0.0243571
$145$	−1083.65	−0.620634
$146$	1071.00	0.607102
$147$	0	0
$148$	−276.026	−0.153305
$149$	−460.395	−0.253135	−0.126567	−	0.991958i	$-0.540396\pi$
$149$	−460.395	−0.253135	−0.126567	+	0.991958i	$0.540396\pi$
$150$	817.159	0.444805
$151$	−1563.82	−0.842792	−0.421396	−	0.906877i	$-0.638460\pi$
$151$	−1563.82	−0.842792	−0.421396	+	0.906877i	$0.638460\pi$
$152$	−614.661	−0.327997
$153$	27.7825	0.0146803
$154$	0	0
$155$	2392.36	1.23973
$156$	−442.913	−0.227317
$157$	2815.84	1.43139	0.715696	−	0.698412i	$-0.246110\pi$
$157$	2815.84	1.43139	0.715696	+	0.698412i	$0.246110\pi$
$158$	−2169.98	−1.09262
$159$	−64.6701	−0.0322558
$160$	226.139	0.111737
$161$	0	0
$162$	1586.21	0.769287
$163$	−1281.50	−0.615796	−0.307898	−	0.951419i	$-0.599626\pi$
$163$	−1281.50	−0.615796	−0.307898	+	0.951419i	$0.599626\pi$
$164$	−1511.46	−0.719667
$165$	−2277.92	−1.07476
$166$	−1656.24	−0.774393
$167$	−3743.59	−1.73466	−0.867329	−	0.497735i	$-0.834165\pi$
$167$	−3743.59	−1.73466	−0.867329	+	0.497735i	$0.834165\pi$
$168$	0	0
$169$	−1783.21	−0.811658
$170$	149.272	0.0673448
$171$	202.114	0.0903861
$172$	−954.197	−0.423005
$173$	−3125.92	−1.37375	−0.686877	−	0.726773i	$-0.741019\pi$
$173$	−3125.92	−1.37375	−0.686877	+	0.726773i	$0.741019\pi$
$174$	−1669.40	−0.727340
$175$	0	0
$176$	947.465	0.405783
$177$	1575.11	0.668882
$178$	−1767.46	−0.744252
$179$	−4545.56	−1.89805	−0.949025	−	0.315201i	$-0.897928\pi$
$179$	−4545.56	−1.89805	−0.949025	+	0.315201i	$0.897928\pi$
$180$	−74.3594	−0.0307912
$181$	−495.143	−0.203335	−0.101668	−	0.994818i	$-0.532418\pi$
$181$	−495.143	−0.203335	−0.101668	+	0.994818i	$0.532418\pi$
$182$	0	0
$183$	−3058.05	−1.23529
$184$	184.000	0.0737210
$185$	487.658	0.193802
$186$	3685.53	1.45288
$187$	625.410	0.244569
$188$	762.435	0.295778
$189$	0	0
$190$	1085.93	0.414640
$191$	−794.876	−0.301127	−0.150563	−	0.988600i	$-0.548109\pi$
$191$	−794.876	−0.301127	−0.150563	+	0.988600i	$0.548109\pi$
$192$	348.377	0.130948
$193$	555.154	0.207051	0.103525	−	0.994627i	$-0.466988\pi$
$193$	555.154	0.207051	0.103525	+	0.994627i	$0.466988\pi$
$194$	1102.49	0.408011
$195$	782.500	0.287364
$196$	0	0
$197$	4487.43	1.62293	0.811463	−	0.584404i	$-0.198672\pi$
$197$	4487.43	1.62293	0.811463	+	0.584404i	$0.198672\pi$
$198$	−311.547	−0.111822
$199$	−1574.35	−0.560817	−0.280409	−	0.959881i	$-0.590470\pi$
$199$	−1574.35	−0.560817	−0.280409	+	0.959881i	$0.590470\pi$
$200$	600.477	0.212301
$201$	2265.64	0.795055
$202$	198.086	0.0689964
$203$	0	0
$204$	229.959	0.0789235
$205$	2670.32	0.909772
$206$	−2111.73	−0.714229
$207$	−60.5031	−0.0203153
$208$	−325.468	−0.108496
$209$	4549.77	1.50581
$210$	0	0
$211$	−4680.39	−1.52707	−0.763534	−	0.645768i	$-0.776537\pi$
$211$	−4680.39	−1.52707	−0.763534	+	0.645768i	$0.776537\pi$
$212$	−47.5218	−0.0153953
$213$	681.272	0.219155
$214$	−554.369	−0.177084
$215$	1685.79	0.534744
$216$	1061.22	0.334291
$217$	0	0
$218$	−690.734	−0.214598
$219$	−2914.95	−0.899425
$220$	−1673.90	−0.512974
$221$	−214.837	−0.0653915
$222$	751.259	0.227122
$223$	3455.88	1.03777	0.518885	−	0.854844i	$-0.326347\pi$
$223$	3455.88	1.03777	0.518885	+	0.854844i	$0.326347\pi$
$224$	0	0
$225$	−197.450	−0.0585036
$226$	2653.65	0.781054
$227$	−3684.08	−1.07718	−0.538592	−	0.842566i	$-0.681044\pi$
$227$	−3684.08	−1.07718	−0.538592	+	0.842566i	$0.681044\pi$
$228$	1672.92	0.485930
$229$	−1640.87	−0.473500	−0.236750	−	0.971571i	$-0.576082\pi$
$229$	−1640.87	−0.473500	−0.236750	+	0.971571i	$0.576082\pi$
$230$	−325.075	−0.0931949
$231$	0	0
$232$	−1226.74	−0.347152
$233$	−6717.06	−1.88862	−0.944311	−	0.329053i	$-0.893270\pi$
$233$	−6717.06	−1.88862	−0.944311	+	0.329053i	$0.893270\pi$
$234$	107.021	0.0298982
$235$	−1347.00	−0.373910
$236$	1157.44	0.319250
$237$	5906.03	1.61873
$238$	0	0
$239$	−3592.73	−0.972361	−0.486180	−	0.873859i	$-0.661610\pi$
$239$	−3592.73	−0.972361	−0.486180	+	0.873859i	$0.661610\pi$
$240$	−615.483	−0.165539
$241$	1111.08	0.296975	0.148488	−	0.988914i	$-0.452559\pi$
$241$	1111.08	0.296975	0.148488	+	0.988914i	$0.452559\pi$
$242$	−4351.21	−1.15581
$243$	−735.571	−0.194185
$244$	−2247.16	−0.589590
$245$	0	0
$246$	4113.75	1.06619
$247$	−1562.91	−0.402614
$248$	2708.26	0.693446
$249$	4507.79	1.14727
$250$	−2827.58	−0.715328
$251$	−4174.36	−1.04973	−0.524867	−	0.851184i	$-0.675885\pi$
$251$	−4174.36	−1.04973	−0.524867	+	0.851184i	$0.675885\pi$
$252$	0	0
$253$	−1361.98	−0.338447
$254$	−711.557	−0.175776
$255$	−406.272	−0.0997716
$256$	256.000	0.0625000
$257$	−6123.65	−1.48631	−0.743157	−	0.669117i	$-0.766672\pi$
$257$	−6123.65	−1.48631	−0.743157	+	0.669117i	$0.766672\pi$
$258$	2597.04	0.626684
$259$	0	0
$260$	575.009	0.137156
$261$	403.377	0.0956644
$262$	3031.25	0.714776
$263$	4152.70	0.973637	0.486818	−	0.873503i	$-0.338157\pi$
$263$	4152.70	0.973637	0.486818	+	0.873503i	$0.338157\pi$
$264$	−2578.71	−0.601170
$265$	83.9575	0.0194621
$266$	0	0
$267$	4810.50	1.10261
$268$	1664.87	0.379472
$269$	3060.74	0.693742	0.346871	−	0.937913i	$-0.387244\pi$
$269$	3060.74	0.693742	0.346871	+	0.937913i	$0.387244\pi$
$270$	−1874.87	−0.422596
$271$	−2661.63	−0.596615	−0.298307	−	0.954470i	$-0.596422\pi$
$271$	−2661.63	−0.596615	−0.298307	+	0.954470i	$0.596422\pi$
$272$	168.982	0.0376693
$273$	0	0
$274$	5967.07	1.31563
$275$	−4444.77	−0.974654
$276$	−500.793	−0.109218
$277$	4100.93	0.889534	0.444767	−	0.895646i	$-0.353286\pi$
$277$	4100.93	0.889534	0.444767	+	0.895646i	$0.353286\pi$
$278$	3015.03	0.650466
$279$	−890.534	−0.191093
$280$	0	0
$281$	8582.08	1.82194	0.910968	−	0.412477i	$-0.135336\pi$
$281$	8582.08	1.82194	0.910968	+	0.412477i	$0.135336\pi$
$282$	−2075.12	−0.438197
$283$	−3518.63	−0.739085	−0.369542	−	0.929214i	$-0.620486\pi$
$283$	−3518.63	−0.739085	−0.369542	+	0.929214i	$0.620486\pi$
$284$	500.623	0.104600
$285$	−2955.57	−0.614292
$286$	2409.14	0.498095
$287$	0	0
$288$	−84.1782	−0.0172231
$289$	−4801.46	−0.977296
$290$	2167.29	0.438854
$291$	−3000.64	−0.604470
$292$	−2142.01	−0.429286
$293$	−3157.18	−0.629503	−0.314752	−	0.949174i	$-0.601921\pi$
$293$	−3157.18	−0.629503	−0.314752	+	0.949174i	$0.601921\pi$
$294$	0	0
$295$	−2044.87	−0.403583
$296$	552.051	0.108403
$297$	−7855.23	−1.53470
$298$	920.791	0.178993
$299$	467.860	0.0904918
$300$	−1634.32	−0.314524
$301$	0	0
$302$	3127.63	0.595944
$303$	−539.130	−0.102218
$304$	1229.32	0.231929
$305$	3970.09	0.745334
$306$	−55.5650	−0.0103805
$307$	−601.675	−0.111855	−0.0559274	−	0.998435i	$-0.517812\pi$
$307$	−601.675	−0.111855	−0.0559274	+	0.998435i	$0.517812\pi$
$308$	0	0
$309$	5747.49	1.05813
$310$	−4784.72	−0.876625
$311$	4133.26	0.753620	0.376810	−	0.926291i	$-0.377021\pi$
$311$	4133.26	0.753620	0.376810	+	0.926291i	$0.377021\pi$
$312$	885.826	0.160737
$313$	5544.68	1.00129	0.500645	−	0.865653i	$-0.333096\pi$
$313$	5544.68	1.00129	0.500645	+	0.865653i	$0.333096\pi$
$314$	−5631.68	−1.01215
$315$	0	0
$316$	4339.96	0.772601
$317$	6753.57	1.19659	0.598294	−	0.801277i	$-0.295845\pi$
$317$	6753.57	1.19659	0.598294	+	0.801277i	$0.295845\pi$
$318$	129.340	0.0228083
$319$	9080.39	1.59374
$320$	−452.278	−0.0790098
$321$	1508.82	0.262350
$322$	0	0
$323$	811.460	0.139786
$324$	−3172.42	−0.543968
$325$	1526.84	0.260597
$326$	2563.00	0.435434
$327$	1879.97	0.317928
$328$	3022.93	0.508882
$329$	0	0
$330$	4555.85	0.759973
$331$	−3556.00	−0.590499	−0.295250	−	0.955420i	$-0.595403\pi$
$331$	−3556.00	−0.590499	−0.295250	+	0.955420i	$0.595403\pi$
$332$	3312.48	0.547578
$333$	−181.526	−0.0298726
$334$	7487.18	1.22659
$335$	−2941.35	−0.479711
$336$	0	0
$337$	7607.26	1.22966	0.614828	−	0.788661i	$-0.289226\pi$
$337$	7607.26	1.22966	0.614828	+	0.788661i	$0.289226\pi$
$338$	3566.43	0.573929
$339$	−7222.44	−1.15714
$340$	−298.543	−0.0476199
$341$	−20046.7	−3.18355
$342$	−404.227	−0.0639126
$343$	0	0
$344$	1908.39	0.299110
$345$	884.757	0.138069
$346$	6251.85	0.971391
$347$	−5967.32	−0.923178	−0.461589	−	0.887094i	$-0.652720\pi$
$347$	−5967.32	−0.923178	−0.461589	+	0.887094i	$0.652720\pi$
$348$	3338.81	0.514307
$349$	−11089.3	−1.70085	−0.850426	−	0.526094i	$-0.823656\pi$
$349$	−11089.3	−1.70085	−0.850426	+	0.526094i	$0.823656\pi$
$350$	0	0
$351$	2698.38	0.410339
$352$	−1894.93	−0.286932
$353$	−203.155	−0.0306312	−0.0153156	−	0.999883i	$-0.504875\pi$
$353$	−203.155	−0.0306312	−0.0153156	+	0.999883i	$0.504875\pi$
$354$	−3150.21	−0.472971
$355$	−884.457	−0.132231
$356$	3534.92	0.526265
$357$	0	0
$358$	9091.11	1.34212
$359$	3630.09	0.533674	0.266837	−	0.963742i	$-0.414021\pi$
$359$	3630.09	0.533674	0.266837	+	0.963742i	$0.414021\pi$
$360$	148.719	0.0217727
$361$	−955.742	−0.139341
$362$	990.287	0.143780
$363$	11842.7	1.71234
$364$	0	0
$365$	3784.31	0.542685
$366$	6116.10	0.873481
$367$	−5035.11	−0.716159	−0.358080	−	0.933691i	$-0.616568\pi$
$367$	−5035.11	−0.716159	−0.358080	+	0.933691i	$0.616568\pi$
$368$	−368.000	−0.0521286
$369$	−994.002	−0.140232
$370$	−975.316	−0.137039
$371$	0	0
$372$	−7371.07	−1.02734
$373$	13545.8	1.88036	0.940181	−	0.340676i	$-0.110656\pi$
$373$	13545.8	1.88036	0.940181	+	0.340676i	$0.110656\pi$
$374$	−1250.82	−0.172937
$375$	7695.83	1.05976
$376$	−1524.87	−0.209147
$377$	−3119.25	−0.426126
$378$	0	0
$379$	732.186	0.0992344	0.0496172	−	0.998768i	$-0.484200\pi$
$379$	732.186	0.0992344	0.0496172	+	0.998768i	$0.484200\pi$
$380$	−2171.86	−0.293195
$381$	1936.64	0.260413
$382$	1589.75	0.212929
$383$	1840.77	0.245584	0.122792	−	0.992432i	$-0.460815\pi$
$383$	1840.77	0.245584	0.122792	+	0.992432i	$0.460815\pi$
$384$	−696.755	−0.0925941
$385$	0	0
$386$	−1110.31	−0.146407
$387$	−627.521	−0.0824255
$388$	−2204.98	−0.288507
$389$	4831.83	0.629777	0.314889	−	0.949129i	$-0.398033\pi$
$389$	4831.83	0.629777	0.314889	+	0.949129i	$0.398033\pi$
$390$	−1565.00	−0.203197
$391$	−242.912	−0.0314184
$392$	0	0
$393$	−8250.15	−1.05894
$394$	−8974.87	−1.14758
$395$	−7667.47	−0.976689
$396$	623.093	0.0790698
$397$	−5612.33	−0.709508	−0.354754	−	0.934960i	$-0.615435\pi$
$397$	−5612.33	−0.709508	−0.354754	+	0.934960i	$0.615435\pi$
$398$	3148.70	0.396558
$399$	0	0
$400$	−1200.95	−0.150119
$401$	6542.72	0.814782	0.407391	−	0.913254i	$-0.366439\pi$
$401$	6542.72	0.814782	0.407391	+	0.913254i	$0.366439\pi$
$402$	−4531.28	−0.562189
$403$	6886.35	0.851199
$404$	−396.172	−0.0487878
$405$	5604.76	0.687661
$406$	0	0
$407$	−4086.32	−0.497670
$408$	−459.919	−0.0558073
$409$	−4260.40	−0.515069	−0.257535	−	0.966269i	$-0.582910\pi$
$409$	−4260.40	−0.515069	−0.257535	+	0.966269i	$0.582910\pi$
$410$	−5340.64	−0.643306
$411$	−16240.6	−1.94912
$412$	4223.46	0.505036
$413$	0	0
$414$	121.006	0.0143651
$415$	−5852.20	−0.692225
$416$	650.936	0.0767182
$417$	−8206.01	−0.963669
$418$	−9099.53	−1.06477
$419$	−16467.6	−1.92004	−0.960020	−	0.279933i	$-0.909688\pi$
$419$	−16467.6	−1.92004	−0.960020	+	0.279933i	$0.909688\pi$
$420$	0	0
$421$	−4252.38	−0.492276	−0.246138	−	0.969235i	$-0.579162\pi$
$421$	−4252.38	−0.492276	−0.246138	+	0.969235i	$0.579162\pi$
$422$	9360.78	1.07980
$423$	501.409	0.0576344
$424$	95.0437	0.0108862
$425$	−792.734	−0.0904783
$426$	−1362.54	−0.154966
$427$	0	0
$428$	1108.74	0.125217
$429$	−6556.95	−0.737931
$430$	−3371.58	−0.378121
$431$	−7688.72	−0.859287	−0.429643	−	0.902999i	$-0.641361\pi$
$431$	−7688.72	−0.859287	−0.429643	+	0.902999i	$0.641361\pi$
$432$	−2122.44	−0.236379
$433$	−2752.17	−0.305452	−0.152726	−	0.988269i	$-0.548805\pi$
$433$	−2752.17	−0.305452	−0.152726	+	0.988269i	$0.548805\pi$
$434$	0	0
$435$	−5898.71	−0.650165
$436$	1381.47	0.151744
$437$	−1767.15	−0.193442
$438$	5829.90	0.635989
$439$	−13737.0	−1.49347	−0.746733	−	0.665123i	$-0.768379\pi$
$439$	−13737.0	−1.49347	−0.746733	+	0.665123i	$0.768379\pi$
$440$	3347.80	0.362727
$441$	0	0
$442$	429.675	0.0462388
$443$	−15205.6	−1.63079	−0.815396	−	0.578903i	$-0.803481\pi$
$443$	−15205.6	−1.63079	−0.815396	+	0.578903i	$0.803481\pi$
$444$	−1502.52	−0.160600
$445$	−6245.19	−0.665282
$446$	−6911.75	−0.733814
$447$	−2506.12	−0.265179
$448$	0	0
$449$	1130.68	0.118842	0.0594209	−	0.998233i	$-0.481075\pi$
$449$	1130.68	0.118842	0.0594209	+	0.998233i	$0.481075\pi$
$450$	394.899	0.0413683
$451$	−22375.9	−2.33623
$452$	−5307.30	−0.552289
$453$	−8512.47	−0.882894
$454$	7368.16	0.761685
$455$	0	0
$456$	−3345.85	−0.343604
$457$	9179.63	0.939617	0.469809	−	0.882768i	$-0.344323\pi$
$457$	9179.63	0.939617	0.469809	+	0.882768i	$0.344323\pi$
$458$	3281.73	0.334815
$459$	−1401.00	−0.142468
$460$	650.150	0.0658987
$461$	−13727.6	−1.38689	−0.693445	−	0.720510i	$-0.743908\pi$
$461$	−13727.6	−1.38689	−0.693445	+	0.720510i	$0.743908\pi$
$462$	0	0
$463$	11817.0	1.18614	0.593069	−	0.805151i	$-0.297916\pi$
$463$	11817.0	1.18614	0.593069	+	0.805151i	$0.297916\pi$
$464$	2453.47	0.245473
$465$	13022.6	1.29872
$466$	13434.1	1.33546
$467$	3923.65	0.388790	0.194395	−	0.980923i	$-0.437726\pi$
$467$	3923.65	0.388790	0.194395	+	0.980923i	$0.437726\pi$
$468$	−214.042	−0.0211412
$469$	0	0
$470$	2694.01	0.264394
$471$	15327.7	1.49950
$472$	−2314.89	−0.225744
$473$	−14126.1	−1.37319
$474$	−11812.1	−1.14461
$475$	−5767.03	−0.557073
$476$	0	0
$477$	−31.2524	−0.00299989
$478$	7185.45	0.687563
$479$	−13108.9	−1.25044	−0.625220	−	0.780448i	$-0.714991\pi$
$479$	−13108.9	−1.25044	−0.625220	+	0.780448i	$0.714991\pi$
$480$	1230.97	0.117053
$481$	1403.71	0.133064
$482$	−2222.16	−0.209993
$483$	0	0
$484$	8702.41	0.817281
$485$	3895.56	0.364718
$486$	1471.14	0.137309
$487$	3623.20	0.337132	0.168566	−	0.985690i	$-0.446086\pi$
$487$	3623.20	0.337132	0.168566	+	0.985690i	$0.446086\pi$
$488$	4494.33	0.416903
$489$	−6975.71	−0.645097
$490$	0	0
$491$	6124.55	0.562927	0.281464	−	0.959572i	$-0.409180\pi$
$491$	6124.55	0.562927	0.281464	+	0.959572i	$0.409180\pi$
$492$	−8227.49	−0.753911
$493$	1619.51	0.147949
$494$	3125.82	0.284691
$495$	−1100.83	−0.0999566
$496$	−5416.52	−0.490341
$497$	0	0
$498$	−9015.58	−0.811240
$499$	11726.7	1.05202	0.526010	−	0.850478i	$-0.323687\pi$
$499$	11726.7	1.05202	0.526010	+	0.850478i	$0.323687\pi$
$500$	5655.17	0.505813
$501$	−20377.9	−1.81720
$502$	8348.72	0.742274
$503$	1144.58	0.101460	0.0507298	−	0.998712i	$-0.483845\pi$
$503$	1144.58	0.101460	0.0507298	+	0.998712i	$0.483845\pi$
$504$	0	0
$505$	699.921	0.0616754
$506$	2723.96	0.239318
$507$	−9706.74	−0.850279
$508$	1423.11	0.124292
$509$	−18457.0	−1.60725	−0.803627	−	0.595134i	$-0.797099\pi$
$509$	−18457.0	−1.60725	−0.803627	+	0.595134i	$0.797099\pi$
$510$	812.545	0.0705492
$511$	0	0
$512$	−512.000	−0.0441942
$513$	−10192.0	−0.877173
$514$	12247.3	1.05098
$515$	−7461.64	−0.638445
$516$	−5194.07	−0.443132
$517$	11287.2	0.960174
$518$	0	0
$519$	−17015.6	−1.43912
$520$	−1150.02	−0.0969838
$521$	−10760.1	−0.904812	−0.452406	−	0.891812i	$-0.649434\pi$
$521$	−10760.1	−0.904812	−0.452406	+	0.891812i	$0.649434\pi$
$522$	−806.754	−0.0676450
$523$	16378.2	1.36934	0.684672	−	0.728851i	$-0.259946\pi$
$523$	16378.2	1.36934	0.684672	+	0.728851i	$0.259946\pi$
$524$	−6062.50	−0.505423
$525$	0	0
$526$	−8305.40	−0.688465
$527$	−3575.38	−0.295533
$528$	5157.43	0.425091
$529$	529.000	0.0434783
$530$	−167.915	−0.0137618
$531$	761.184	0.0622082
$532$	0	0
$533$	7686.45	0.624648
$534$	−9620.99	−0.779665
$535$	−1958.82	−0.158294
$536$	−3329.75	−0.268327
$537$	−24743.3	−1.98836
$538$	−6121.48	−0.490549
$539$	0	0
$540$	3749.74	0.298821
$541$	−9279.77	−0.737465	−0.368732	−	0.929536i	$-0.620208\pi$
$541$	−9279.77	−0.737465	−0.368732	+	0.929536i	$0.620208\pi$
$542$	5323.27	0.421871
$543$	−2695.26	−0.213011
$544$	−337.965	−0.0266362
$545$	−2440.66	−0.191828
$546$	0	0
$547$	12380.3	0.967718	0.483859	−	0.875146i	$-0.339235\pi$
$547$	12380.3	0.967718	0.483859	+	0.875146i	$0.339235\pi$
$548$	−11934.1	−0.930294
$549$	−1477.83	−0.114886
$550$	8889.55	0.689185
$551$	11781.7	0.910919
$552$	1001.59	0.0772288
$553$	0	0
$554$	−8201.86	−0.628996
$555$	2654.52	0.203023
$556$	−6030.06	−0.459949
$557$	−5768.90	−0.438844	−0.219422	−	0.975630i	$-0.570417\pi$
$557$	−5768.90	−0.438844	−0.219422	+	0.975630i	$0.570417\pi$
$558$	1781.07	0.135123
$559$	4852.51	0.367154
$560$	0	0
$561$	3404.35	0.256207
$562$	−17164.2	−1.28830
$563$	−3119.36	−0.233509	−0.116754	−	0.993161i	$-0.537249\pi$
$563$	−3119.36	−0.233509	−0.116754	+	0.993161i	$0.537249\pi$
$564$	4150.23	0.309852
$565$	9376.48	0.698180
$566$	7037.27	0.522612
$567$	0	0
$568$	−1001.25	−0.0739636
$569$	4753.28	0.350207	0.175104	−	0.984550i	$-0.443974\pi$
$569$	4753.28	0.350207	0.175104	+	0.984550i	$0.443974\pi$
$570$	5911.15	0.434370
$571$	−23437.0	−1.71770	−0.858851	−	0.512226i	$-0.828821\pi$
$571$	−23437.0	−1.71770	−0.858851	+	0.512226i	$0.828821\pi$
$572$	−4818.28	−0.352207
$573$	−4326.82	−0.315455
$574$	0	0
$575$	1726.37	0.125208
$576$	168.356	0.0121786
$577$	−2561.82	−0.184835	−0.0924175	−	0.995720i	$-0.529459\pi$
$577$	−2561.82	−0.184835	−0.0924175	+	0.995720i	$0.529459\pi$
$578$	9602.91	0.691053
$579$	3021.92	0.216903
$580$	−4334.58	−0.310317
$581$	0	0
$582$	6001.29	0.427425
$583$	−703.520	−0.0499774
$584$	4284.02	0.303551
$585$	378.150	0.0267258
$586$	6314.36	0.445126
$587$	−6248.52	−0.439359	−0.219680	−	0.975572i	$-0.570501\pi$
$587$	−6248.52	−0.439359	−0.219680	+	0.975572i	$0.570501\pi$
$588$	0	0
$589$	−26010.4	−1.81959
$590$	4089.74	0.285376
$591$	24426.9	1.70015
$592$	−1104.10	−0.0766526
$593$	18250.0	1.26381	0.631904	−	0.775047i	$-0.282274\pi$
$593$	18250.0	1.26381	0.631904	+	0.775047i	$0.282274\pi$
$594$	15710.5	1.08520
$595$	0	0
$596$	−1841.58	−0.126567
$597$	−8569.80	−0.587502
$598$	−935.721	−0.0639874
$599$	−9696.04	−0.661385	−0.330692	−	0.943739i	$-0.607282\pi$
$599$	−9696.04	−0.661385	−0.330692	+	0.943739i	$0.607282\pi$
$600$	3268.63	0.222402
$601$	9471.96	0.642877	0.321439	−	0.946930i	$-0.395834\pi$
$601$	9471.96	0.642877	0.321439	+	0.946930i	$0.395834\pi$
$602$	0	0
$603$	1094.89	0.0739427
$604$	−6255.26	−0.421396
$605$	−15374.7	−1.03317
$606$	1078.26	0.0722794
$607$	13000.1	0.869288	0.434644	−	0.900602i	$-0.356874\pi$
$607$	13000.1	0.869288	0.434644	+	0.900602i	$0.356874\pi$
$608$	−2458.65	−0.163999
$609$	0	0
$610$	−7940.19	−0.527031
$611$	−3877.31	−0.256726
$612$	111.130	0.00734014
$613$	18348.3	1.20894	0.604471	−	0.796627i	$-0.293385\pi$
$613$	18348.3	1.20894	0.604471	+	0.796627i	$0.293385\pi$
$614$	1203.35	0.0790933
$615$	14535.6	0.953061
$616$	0	0
$617$	11707.8	0.763922	0.381961	−	0.924179i	$-0.375249\pi$
$617$	11707.8	0.763922	0.381961	+	0.924179i	$0.375249\pi$
$618$	−11495.0	−0.748214
$619$	16394.5	1.06454	0.532271	−	0.846574i	$-0.321339\pi$
$619$	16394.5	1.06454	0.532271	+	0.846574i	$0.321339\pi$
$620$	9569.44	0.619867
$621$	3051.01	0.197154
$622$	−8266.52	−0.532890
$623$	0	0
$624$	−1771.65	−0.113658
$625$	−608.601	−0.0389505
$626$	−11089.4	−0.708019
$627$	24766.2	1.57746
$628$	11263.4	0.715696
$629$	−728.804	−0.0461992
$630$	0	0
$631$	−20099.8	−1.26808	−0.634041	−	0.773299i	$-0.718605\pi$
$631$	−20099.8	−1.26808	−0.634041	+	0.773299i	$0.718605\pi$
$632$	−8679.92	−0.546311
$633$	−25477.2	−1.59973
$634$	−13507.1	−0.846116
$635$	−2514.23	−0.157125
$636$	−258.680	−0.0161279
$637$	0	0
$638$	−18160.8	−1.12695
$639$	329.231	0.0203821
$640$	904.557	0.0558684
$641$	16590.3	1.02227	0.511136	−	0.859500i	$-0.329225\pi$
$641$	16590.3	1.02227	0.511136	+	0.859500i	$0.329225\pi$
$642$	−3017.65	−0.185510
$643$	−23133.2	−1.41879	−0.709396	−	0.704810i	$-0.751032\pi$
$643$	−23133.2	−1.41879	−0.709396	+	0.704810i	$0.751032\pi$
$644$	0	0
$645$	9176.43	0.560189
$646$	−1622.92	−0.0988436
$647$	22965.2	1.39545	0.697725	−	0.716365i	$-0.254196\pi$
$647$	22965.2	1.39545	0.697725	+	0.716365i	$0.254196\pi$
$648$	6344.84	0.384644
$649$	17135.0	1.03637
$650$	−3053.69	−0.184270
$651$	0	0
$652$	−5126.00	−0.307898
$653$	24646.6	1.47702	0.738512	−	0.674240i	$-0.235529\pi$
$653$	24646.6	1.47702	0.738512	+	0.674240i	$0.235529\pi$
$654$	−3759.94	−0.224809
$655$	10710.7	0.638934
$656$	−6045.85	−0.359834
$657$	−1408.68	−0.0836494
$658$	0	0
$659$	15924.1	0.941295	0.470647	−	0.882321i	$-0.344020\pi$
$659$	15924.1	0.941295	0.470647	+	0.882321i	$0.344020\pi$
$660$	−9111.70	−0.537382
$661$	−10671.6	−0.627951	−0.313975	−	0.949431i	$-0.601661\pi$
$661$	−10671.6	−0.627951	−0.313975	+	0.949431i	$0.601661\pi$
$662$	7111.99	0.417546
$663$	−1169.44	−0.0685030
$664$	−6624.96	−0.387196
$665$	0	0
$666$	363.053	0.0211231
$667$	−3526.87	−0.204739
$668$	−14974.4	−0.867329
$669$	18811.7	1.08715
$670$	5882.71	0.339207
$671$	−33267.3	−1.91397
$672$	0	0
$673$	−23453.7	−1.34335	−0.671675	−	0.740846i	$-0.734425\pi$
$673$	−23453.7	−1.34335	−0.671675	+	0.740846i	$0.734425\pi$
$674$	−15214.5	−0.869498
$675$	9956.84	0.567762
$676$	−7132.85	−0.405829
$677$	−14408.5	−0.817965	−0.408982	−	0.912542i	$-0.634116\pi$
$677$	−14408.5	−0.817965	−0.408982	+	0.912542i	$0.634116\pi$
$678$	14444.9	0.818219
$679$	0	0
$680$	597.086	0.0336724
$681$	−20053.9	−1.12844
$682$	40093.5	2.25111
$683$	−12553.4	−0.703283	−0.351641	−	0.936135i	$-0.614376\pi$
$683$	−12553.4	−0.703283	−0.351641	+	0.936135i	$0.614376\pi$
$684$	808.455	0.0451930
$685$	21084.2	1.17604
$686$	0	0
$687$	−8931.89	−0.496030
$688$	−3816.79	−0.211502
$689$	241.669	0.0133627
$690$	−1769.51	−0.0976293
$691$	22274.2	1.22627	0.613134	−	0.789979i	$-0.289909\pi$
$691$	22274.2	1.22627	0.613134	+	0.789979i	$0.289909\pi$
$692$	−12503.7	−0.686877
$693$	0	0
$694$	11934.6	0.652785
$695$	10653.4	0.581448
$696$	−6677.61	−0.363670
$697$	−3990.79	−0.216875
$698$	22178.6	1.20268
$699$	−36563.6	−1.97849
$700$	0	0
$701$	16964.2	0.914024	0.457012	−	0.889461i	$-0.348920\pi$
$701$	16964.2	0.914024	0.457012	+	0.889461i	$0.348920\pi$
$702$	−5396.77	−0.290154
$703$	−5301.95	−0.284448
$704$	3789.86	0.202892
$705$	−7332.27	−0.391701
$706$	406.309	0.0216596
$707$	0	0
$708$	6300.42	0.334441
$709$	26987.4	1.42953	0.714763	−	0.699366i	$-0.246534\pi$
$709$	26987.4	1.42953	0.714763	+	0.699366i	$0.246534\pi$
$710$	1768.91	0.0935016
$711$	2854.14	0.150547
$712$	−7069.84	−0.372126
$713$	7786.25	0.408972
$714$	0	0
$715$	8512.51	0.445244
$716$	−18182.2	−0.949025
$717$	−19556.6	−1.01863
$718$	−7260.19	−0.377365
$719$	13507.6	0.700622	0.350311	−	0.936633i	$-0.386076\pi$
$719$	13507.6	0.700622	0.350311	+	0.936633i	$0.386076\pi$
$720$	−297.438	−0.0153956
$721$	0	0
$722$	1911.48	0.0985292
$723$	6048.05	0.311106
$724$	−1980.57	−0.101668
$725$	−11509.8	−0.589604
$726$	−23685.3	−1.21081
$727$	12991.8	0.662777	0.331388	−	0.943494i	$-0.392483\pi$
$727$	12991.8	0.662777	0.331388	+	0.943494i	$0.392483\pi$
$728$	0	0
$729$	17409.8	0.884512
$730$	−7568.62	−0.383736
$731$	−2519.41	−0.127475
$732$	−12232.2	−0.617644
$733$	−1354.79	−0.0682678	−0.0341339	−	0.999417i	$-0.510867\pi$
$733$	−1354.79	−0.0682678	−0.0341339	+	0.999417i	$0.510867\pi$
$734$	10070.2	0.506401
$735$	0	0
$736$	736.000	0.0368605
$737$	24647.0	1.23187
$738$	1988.00	0.0991592
$739$	−16588.9	−0.825754	−0.412877	−	0.910787i	$-0.635476\pi$
$739$	−16588.9	−0.825754	−0.412877	+	0.910787i	$0.635476\pi$
$740$	1950.63	0.0969009
$741$	−8507.55	−0.421771
$742$	0	0
$743$	−29256.6	−1.44458	−0.722288	−	0.691593i	$-0.756909\pi$
$743$	−29256.6	−1.44458	−0.722288	+	0.691593i	$0.756909\pi$
$744$	14742.1	0.726442
$745$	3253.55	0.160001
$746$	−27091.6	−1.32962
$747$	2178.43	0.106700
$748$	2501.64	0.122285
$749$	0	0
$750$	−15391.7	−0.749365
$751$	−14214.3	−0.690663	−0.345332	−	0.938481i	$-0.612234\pi$
$751$	−14214.3	−0.690663	−0.345332	+	0.938481i	$0.612234\pi$
$752$	3049.74	0.147889
$753$	−22722.7	−1.09968
$754$	6238.49	0.301316
$755$	11051.3	0.532710
$756$	0	0
$757$	16229.5	0.779223	0.389612	−	0.920979i	$-0.372609\pi$
$757$	16229.5	0.779223	0.389612	+	0.920979i	$0.372609\pi$
$758$	−1464.37	−0.0701693
$759$	−7413.80	−0.354551
$760$	4343.72	0.207320
$761$	6073.91	0.289328	0.144664	−	0.989481i	$-0.453790\pi$
$761$	6073.91	0.289328	0.144664	+	0.989481i	$0.453790\pi$
$762$	−3873.29	−0.184140
$763$	0	0
$764$	−3179.50	−0.150563
$765$	−196.335	−0.00927908
$766$	−3681.53	−0.173654
$767$	−5886.10	−0.277099
$768$	1393.51	0.0654739
$769$	−9079.48	−0.425766	−0.212883	−	0.977078i	$-0.568285\pi$
$769$	−9079.48	−0.425766	−0.212883	+	0.977078i	$0.568285\pi$
$770$	0	0
$771$	−33333.5	−1.55704
$772$	2220.61	0.103525
$773$	2612.80	0.121573	0.0607866	−	0.998151i	$-0.480639\pi$
$773$	2612.80	0.121573	0.0607866	+	0.998151i	$0.480639\pi$
$774$	1255.04	0.0582836
$775$	25410.1	1.17775
$776$	4409.96	0.204005
$777$	0	0
$778$	−9663.66	−0.445320
$779$	−29032.4	−1.33530
$780$	3130.00	0.143682
$781$	7411.29	0.339561
$782$	485.824	0.0222162
$783$	−20341.2	−0.928398
$784$	0	0
$785$	−19899.1	−0.904752
$786$	16500.3	0.748786
$787$	−8243.19	−0.373365	−0.186682	−	0.982420i	$-0.559774\pi$
$787$	−8243.19	−0.373365	−0.186682	+	0.982420i	$0.559774\pi$
$788$	17949.7	0.811463
$789$	22604.8	1.01996
$790$	15334.9	0.690623
$791$	0	0
$792$	−1246.19	−0.0559108
$793$	11427.8	0.511745
$794$	11224.7	0.501698
$795$	457.014	0.0203882
$796$	−6297.39	−0.280409
$797$	−9141.47	−0.406283	−0.203141	−	0.979149i	$-0.565115\pi$
$797$	−9141.47	−0.406283	−0.203141	+	0.979149i	$0.565115\pi$
$798$	0	0
$799$	2013.09	0.0891341
$800$	2401.91	0.106150
$801$	2324.72	0.102547
$802$	−13085.4	−0.576138
$803$	−31710.6	−1.39358
$804$	9062.57	0.397528
$805$	0	0
$806$	−13772.7	−0.601889
$807$	16660.8	0.726752
$808$	792.343	0.0344982
$809$	13550.8	0.588902	0.294451	−	0.955667i	$-0.404863\pi$
$809$	13550.8	0.588902	0.294451	+	0.955667i	$0.404863\pi$
$810$	−11209.5	−0.486250
$811$	−8593.49	−0.372082	−0.186041	−	0.982542i	$-0.559566\pi$
$811$	−8593.49	−0.372082	−0.186041	+	0.982542i	$0.559566\pi$
$812$	0	0
$813$	−14488.3	−0.625003
$814$	8172.65	0.351906
$815$	9056.16	0.389231
$816$	919.838	0.0394617
$817$	−18328.4	−0.784858
$818$	8520.81	0.364209
$819$	0	0
$820$	10681.3	0.454886
$821$	14588.6	0.620152	0.310076	−	0.950712i	$-0.399645\pi$
$821$	14588.6	0.620152	0.310076	+	0.950712i	$0.399645\pi$
$822$	32481.1	1.37824
$823$	−23934.6	−1.01374	−0.506869	−	0.862023i	$-0.669197\pi$
$823$	−23934.6	−1.01374	−0.506869	+	0.862023i	$0.669197\pi$
$824$	−8446.92	−0.357115
$825$	−24194.7	−1.02103
$826$	0	0
$827$	−39607.7	−1.66541	−0.832706	−	0.553715i	$-0.813210\pi$
$827$	−39607.7	−1.66541	−0.832706	+	0.553715i	$0.813210\pi$
$828$	−242.012	−0.0101576
$829$	−30922.4	−1.29551	−0.647755	−	0.761849i	$-0.724292\pi$
$829$	−30922.4	−1.29551	−0.647755	+	0.761849i	$0.724292\pi$
$830$	11704.4	0.489477
$831$	22323.0	0.931860
$832$	−1301.87	−0.0542480
$833$	0	0
$834$	16412.0	0.681417
$835$	26455.4	1.09644
$836$	18199.1	0.752904
$837$	44907.2	1.85450
$838$	32935.3	1.35767
$839$	40349.1	1.66032	0.830158	−	0.557528i	$-0.188250\pi$
$839$	40349.1	1.66032	0.830158	+	0.557528i	$0.188250\pi$
$840$	0	0
$841$	−875.206	−0.0358853
$842$	8504.76	0.348092
$843$	46715.7	1.90863
$844$	−18721.6	−0.763534
$845$	12601.7	0.513032
$846$	−1002.82	−0.0407537
$847$	0	0
$848$	−190.087	−0.00769767
$849$	−19153.3	−0.774252
$850$	1585.47	0.0639778
$851$	1587.15	0.0639327
$852$	2725.09	0.109577
$853$	37021.2	1.48603	0.743014	−	0.669275i	$-0.233395\pi$
$853$	37021.2	1.48603	0.743014	+	0.669275i	$0.233395\pi$
$854$	0	0
$855$	−1428.31	−0.0571311
$856$	−2217.48	−0.0885418
$857$	−35476.3	−1.41406	−0.707029	−	0.707184i	$-0.749965\pi$
$857$	−35476.3	−1.41406	−0.707029	+	0.707184i	$0.749965\pi$
$858$	13113.9	0.521796
$859$	32047.3	1.27292	0.636461	−	0.771308i	$-0.280397\pi$
$859$	32047.3	1.27292	0.636461	+	0.771308i	$0.280397\pi$
$860$	6743.17	0.267372
$861$	0	0
$862$	15377.4	0.607608
$863$	35127.9	1.38559	0.692796	−	0.721133i	$-0.256379\pi$
$863$	35127.9	1.38559	0.692796	+	0.721133i	$0.256379\pi$
$864$	4244.88	0.167146
$865$	22090.4	0.868320
$866$	5504.34	0.215987
$867$	−26136.2	−1.02380
$868$	0	0
$869$	64249.4	2.50807
$870$	11797.4	0.459736
$871$	−8466.61	−0.329369
$872$	−2762.94	−0.107299
$873$	−1450.09	−0.0562177
$874$	3534.30	0.136784
$875$	0	0
$876$	−11659.8	−0.449712
$877$	15660.7	0.602991	0.301495	−	0.953468i	$-0.402514\pi$
$877$	15660.7	0.602991	0.301495	+	0.953468i	$0.402514\pi$
$878$	27474.0	1.05604
$879$	−17185.8	−0.659456
$880$	−6695.60	−0.256487
$881$	−1733.30	−0.0662842	−0.0331421	−	0.999451i	$-0.510551\pi$
$881$	−1733.30	−0.0662842	−0.0331421	+	0.999451i	$0.510551\pi$
$882$	0	0
$883$	−31775.4	−1.21102	−0.605509	−	0.795839i	$-0.707030\pi$
$883$	−31775.4	−1.21102	−0.605509	+	0.795839i	$0.707030\pi$
$884$	−859.349	−0.0326958
$885$	−11131.0	−0.422786
$886$	30411.2	1.15314
$887$	−23869.5	−0.903562	−0.451781	−	0.892129i	$-0.649211\pi$
$887$	−23869.5	−0.903562	−0.451781	+	0.892129i	$0.649211\pi$
$888$	3005.04	0.113561
$889$	0	0
$890$	12490.4	0.470425
$891$	−46965.0	−1.76587
$892$	13823.5	0.518885
$893$	14645.0	0.548796
$894$	5012.23	0.187510
$895$	32122.8	1.19972
$896$	0	0
$897$	2546.75	0.0947977
$898$	−2261.36	−0.0840338
$899$	−51911.3	−1.92585
$900$	−789.798	−0.0292518
$901$	−125.474	−0.00463946
$902$	44751.8	1.65197
$903$	0	0
$904$	10614.6	0.390527
$905$	3499.10	0.128524
$906$	17024.9	0.624300
$907$	−7496.99	−0.274458	−0.137229	−	0.990539i	$-0.543820\pi$
$907$	−7496.99	−0.274458	−0.137229	+	0.990539i	$0.543820\pi$
$908$	−14736.3	−0.538592
$909$	−260.539	−0.00950665
$910$	0	0
$911$	33271.5	1.21003	0.605013	−	0.796216i	$-0.293168\pi$
$911$	33271.5	1.21003	0.605013	+	0.796216i	$0.293168\pi$
$912$	6691.69	0.242965
$913$	49038.5	1.77759
$914$	−18359.3	−0.664410
$915$	21610.8	0.780799
$916$	−6563.47	−0.236750
$917$	0	0
$918$	2801.99	0.100740
$919$	39590.0	1.42106	0.710529	−	0.703668i	$-0.248456\pi$
$919$	39590.0	1.42106	0.710529	+	0.703668i	$0.248456\pi$
$920$	−1300.30	−0.0465974
$921$	−3275.16	−0.117177
$922$	27455.1	0.980679
$923$	−2545.89	−0.0907897
$924$	0	0
$925$	5179.60	0.184113
$926$	−23634.0	−0.838727
$927$	2777.53	0.0984099
$928$	−4906.95	−0.173576
$929$	867.034	0.0306205	0.0153103	−	0.999883i	$-0.495126\pi$
$929$	867.034	0.0306205	0.0153103	+	0.999883i	$0.495126\pi$
$930$	−26045.1	−0.918337
$931$	0	0
$932$	−26868.2	−0.944311
$933$	22499.0	0.789479
$934$	−7847.31	−0.274916
$935$	−4419.68	−0.154587
$936$	428.083	0.0149491
$937$	13213.5	0.460689	0.230344	−	0.973109i	$-0.426015\pi$
$937$	13213.5	0.460689	0.230344	+	0.973109i	$0.426015\pi$
$938$	0	0
$939$	30181.9	1.04893
$940$	−5388.01	−0.186955
$941$	−20850.0	−0.722307	−0.361153	−	0.932506i	$-0.617617\pi$
$941$	−20850.0	−0.722307	−0.361153	+	0.932506i	$0.617617\pi$
$942$	−30655.5	−1.06031
$943$	8690.91	0.300122
$944$	4629.77	0.159625
$945$	0	0
$946$	28252.1	0.970990
$947$	−18237.9	−0.625821	−0.312910	−	0.949783i	$-0.601304\pi$
$947$	−18237.9	−0.625821	−0.312910	+	0.949783i	$0.601304\pi$
$948$	23624.1	0.809363
$949$	10893.0	0.372606
$950$	11534.1	0.393910
$951$	36762.4	1.25352
$952$	0	0
$953$	−27324.0	−0.928762	−0.464381	−	0.885636i	$-0.653723\pi$
$953$	−27324.0	−0.928762	−0.464381	+	0.885636i	$0.653723\pi$
$954$	62.5048	0.00212124
$955$	5617.27	0.190336
$956$	−14370.9	−0.486180
$957$	49428.2	1.66958
$958$	26217.8	0.884195
$959$	0	0
$960$	−2461.93	−0.0827693
$961$	84813.3	2.84694
$962$	−2807.42	−0.0940904
$963$	729.153	0.0243994
$964$	4444.32	0.148488
$965$	−3923.19	−0.130872
$966$	0	0
$967$	−18434.0	−0.613029	−0.306514	−	0.951866i	$-0.599163\pi$
$967$	−18434.0	−0.613029	−0.306514	+	0.951866i	$0.599163\pi$
$968$	−17404.8	−0.577905
$969$	4417.10	0.146437
$970$	−7791.13	−0.257895
$971$	−7682.52	−0.253907	−0.126953	−	0.991909i	$-0.540520\pi$
$971$	−7682.52	−0.253907	−0.126953	+	0.991909i	$0.540520\pi$
$972$	−2942.28	−0.0970924
$973$	0	0
$974$	−7246.41	−0.238388
$975$	8311.22	0.272997
$976$	−8988.66	−0.294795
$977$	30106.8	0.985877	0.492939	−	0.870064i	$-0.335923\pi$
$977$	30106.8	0.985877	0.492939	+	0.870064i	$0.335923\pi$
$978$	13951.4	0.456152
$979$	52331.5	1.70840
$980$	0	0
$981$	908.513	0.0295684
$982$	−12249.1	−0.398050
$983$	15437.6	0.500899	0.250449	−	0.968130i	$-0.419422\pi$
$983$	15437.6	0.500899	0.250449	+	0.968130i	$0.419422\pi$
$984$	16455.0	0.533095
$985$	−31712.0	−1.02582
$986$	−3239.01	−0.104616
$987$	0	0
$988$	−6251.64	−0.201307
$989$	5486.63	0.176405
$990$	2201.65	0.0706800
$991$	51193.6	1.64099	0.820494	−	0.571656i	$-0.193699\pi$
$991$	51193.6	1.64099	0.820494	+	0.571656i	$0.193699\pi$
$992$	10833.0	0.346723
$993$	−19356.7	−0.618597
$994$	0	0
$995$	11125.7	0.354480
$996$	18031.2	0.573633
$997$	16730.8	0.531465	0.265732	−	0.964047i	$-0.414386\pi$
$997$	16730.8	0.531465	0.265732	+	0.964047i	$0.414386\pi$
$998$	−23453.4	−0.743891
$999$	9153.87	0.289906

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.s.1.9		11
7.3	odd	6		322.4.e.c.93.9	✓	22
7.5	odd	6		322.4.e.c.277.9	yes	22
7.6	odd	2		2254.4.a.t.1.3		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.4.e.c.93.9	✓	22	7.3	odd	6
322.4.e.c.277.9	yes	22	7.5	odd	6
2254.4.a.s.1.9		11	1.1	even	1	trivial
2254.4.a.t.1.3		11	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +5.44340 q^{3} +4.00000 q^{4} -7.06685 q^{5} -10.8868 q^{6} -8.00000 q^{8} +2.63057 q^{9} +14.1337 q^{10} +59.2166 q^{11} +21.7736 q^{12} -20.3418 q^{13} -38.4677 q^{15} +16.0000 q^{16} +10.5614 q^{17} -5.26114 q^{18} +76.8327 q^{19} -28.2674 q^{20} -118.433 q^{22} -23.0000 q^{23} -43.5472 q^{24} -75.0596 q^{25} +40.6835 q^{26} -132.652 q^{27} +153.342 q^{29} +76.9353 q^{30} -338.533 q^{31} -32.0000 q^{32} +322.339 q^{33} -21.1228 q^{34} +10.5223 q^{36} -69.0064 q^{37} -153.665 q^{38} -110.728 q^{39} +56.5348 q^{40} -377.866 q^{41} -238.549 q^{43} +236.866 q^{44} -18.5898 q^{45} +46.0000 q^{46} +190.609 q^{47} +87.0944 q^{48} +150.119 q^{50} +57.4899 q^{51} -81.3670 q^{52} -11.8805 q^{53} +265.305 q^{54} -418.475 q^{55} +418.231 q^{57} -306.684 q^{58} +289.361 q^{59} -153.871 q^{60} -561.791 q^{61} +677.065 q^{62} +64.0000 q^{64} +143.752 q^{65} -644.679 q^{66} +416.218 q^{67} +42.2456 q^{68} -125.198 q^{69} +125.156 q^{71} -21.0446 q^{72} -535.502 q^{73} +138.013 q^{74} -408.579 q^{75} +307.331 q^{76} +221.456 q^{78} +1084.99 q^{79} -113.070 q^{80} -793.105 q^{81} +755.732 q^{82} +828.120 q^{83} -74.6358 q^{85} +477.099 q^{86} +834.702 q^{87} -473.733 q^{88} +883.731 q^{89} +37.1797 q^{90} -92.0000 q^{92} -1842.77 q^{93} -381.217 q^{94} -542.965 q^{95} -174.189 q^{96} -551.245 q^{97} +155.773 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 22 q^{2} + 44 q^{4} - 17 q^{5} - 88 q^{8} + 59 q^{9} + 34 q^{10} - 3 q^{13} - 144 q^{15} + 176 q^{16} - 47 q^{17} - 118 q^{18} + 138 q^{19} - 68 q^{20} - 253 q^{23} + 350 q^{25} + 6 q^{26} + 87 q^{27}+ \cdots + 3525 q^{99}+O(q^{100}) \) 11 * q - 22 * q^2 + 44 * q^4 - 17 * q^5 - 88 * q^8 + 59 * q^9 + 34 * q^10 - 3 * q^13 - 144 * q^15 + 176 * q^16 - 47 * q^17 - 118 * q^18 + 138 * q^19 - 68 * q^20 - 253 * q^23 + 350 * q^25 + 6 * q^26 + 87 * q^27 - 251 * q^29 + 288 * q^30 + 46 * q^31 - 352 * q^32 + 222 * q^33 + 94 * q^34 + 236 * q^36 + 475 * q^37 - 276 * q^38 - 115 * q^39 + 136 * q^40 - 740 * q^41 + 17 * q^43 - 875 * q^45 + 506 * q^46 + 33 * q^47 - 700 * q^50 + 1017 * q^51 - 12 * q^52 - 835 * q^53 - 174 * q^54 + 1323 * q^55 + 439 * q^57 + 502 * q^58 - 1723 * q^59 - 576 * q^60 + 253 * q^61 - 92 * q^62 + 704 * q^64 + 1753 * q^65 - 444 * q^66 - 2573 * q^67 - 188 * q^68 + 1319 * q^71 - 472 * q^72 - 1159 * q^73 - 950 * q^74 - 268 * q^75 + 552 * q^76 + 230 * q^78 + 1095 * q^79 - 272 * q^80 - 3085 * q^81 + 1480 * q^82 + 451 * q^83 + 2507 * q^85 - 34 * q^86 - 3540 * q^87 - 3634 * q^89 + 1750 * q^90 - 1012 * q^92 + 3132 * q^93 - 66 * q^94 - 4249 * q^95 + 1513 * q^97 + 3525 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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