LMFDB - Embedded newform 2254.4.a.u.1.11

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} - 234 x^{9} - 105 x^{8} + 18997 x^{7} + 16513 x^{6} - 621598 x^{5} - 743169 x^{4} + \cdots - 12103441 $ x^11 - 234x^9 - 105x^8 + 18997x^7 + 16513x^6 - 621598x^5 - 743169x^4 + 7229916x^3 + 7324254x^2 - 26794327*x - 12103441
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 7 $
Twist minimal:	no (minimal twist has level 322)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$9.96286$ 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)

$f(q)$	$=$	\(q-2.00000 q^{2} +9.96286 q^{3} +4.00000 q^{4} +11.8037 q^{5} -19.9257 q^{6} -8.00000 q^{8} +72.2585 q^{9} -23.6074 q^{10} +4.60635 q^{11} +39.8514 q^{12} -44.6962 q^{13} +117.598 q^{15} +16.0000 q^{16} +42.8996 q^{17} -144.517 q^{18} +80.8065 q^{19} +47.2147 q^{20} -9.21269 q^{22} +23.0000 q^{23} -79.7029 q^{24} +14.3270 q^{25} +89.3924 q^{26} +450.904 q^{27} +83.4085 q^{29} -235.197 q^{30} -29.1874 q^{31} -32.0000 q^{32} +45.8924 q^{33} -85.7992 q^{34} +289.034 q^{36} -11.7683 q^{37} -161.613 q^{38} -445.302 q^{39} -94.4295 q^{40} +385.859 q^{41} +60.7982 q^{43} +18.4254 q^{44} +852.917 q^{45} -46.0000 q^{46} -113.269 q^{47} +159.406 q^{48} -28.6541 q^{50} +427.403 q^{51} -178.785 q^{52} -652.574 q^{53} -901.808 q^{54} +54.3719 q^{55} +805.063 q^{57} -166.817 q^{58} +318.100 q^{59} +470.394 q^{60} +266.802 q^{61} +58.3747 q^{62} +64.0000 q^{64} -527.580 q^{65} -91.7847 q^{66} +957.595 q^{67} +171.598 q^{68} +229.146 q^{69} -540.145 q^{71} -578.068 q^{72} -989.786 q^{73} +23.5365 q^{74} +142.738 q^{75} +323.226 q^{76} +890.604 q^{78} +240.319 q^{79} +188.859 q^{80} +2541.31 q^{81} -771.718 q^{82} -1281.34 q^{83} +506.374 q^{85} -121.596 q^{86} +830.987 q^{87} -36.8508 q^{88} +372.062 q^{89} -1705.83 q^{90} +92.0000 q^{92} -290.789 q^{93} +226.538 q^{94} +953.815 q^{95} -318.811 q^{96} +1477.84 q^{97} +332.848 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 22 q^{2} + 44 q^{4} + 23 q^{5} - 88 q^{8} + 171 q^{9} - 46 q^{10} - 48 q^{11} + 77 q^{13} + 104 q^{15} + 176 q^{16} + 97 q^{17} - 342 q^{18} + 138 q^{19} + 92 q^{20} + 96 q^{22} + 253 q^{23} + 30 q^{25}+ \cdots - 2545 q^{99}+O(q^{100}) $ 11 * q - 22 * q^2 + 44 * q^4 + 23 * q^5 - 88 * q^8 + 171 * q^9 - 46 * q^10 - 48 * q^11 + 77 * q^13 + 104 * q^15 + 176 * q^16 + 97 * q^17 - 342 * q^18 + 138 * q^19 + 92 * q^20 + 96 * q^22 + 253 * q^23 + 30 * q^25 - 154 * q^26 + 315 * q^27 + 137 * q^29 - 208 * q^30 + 234 * q^31 - 352 * q^32 + 28 * q^33 - 194 * q^34 + 684 * q^36 + 87 * q^37 - 276 * q^38 + 129 * q^39 - 184 * q^40 + 612 * q^41 + 187 * q^43 - 192 * q^44 + 597 * q^45 - 506 * q^46 + 793 * q^47 - 60 * q^50 - 1257 * q^51 + 308 * q^52 + 575 * q^53 - 630 * q^54 - 549 * q^55 + 39 * q^57 - 274 * q^58 + 1241 * q^59 + 416 * q^60 + 187 * q^61 - 468 * q^62 + 704 * q^64 - 937 * q^65 - 56 * q^66 + 2275 * q^67 + 388 * q^68 + 115 * q^71 - 1368 * q^72 + 2385 * q^73 - 174 * q^74 + 196 * q^75 + 552 * q^76 - 258 * q^78 - 1225 * q^79 + 368 * q^80 + 3635 * q^81 - 1224 * q^82 - 597 * q^83 - 2493 * q^85 - 374 * q^86 + 1428 * q^87 + 384 * q^88 + 602 * q^89 - 1194 * q^90 + 1012 * q^92 - 1360 * q^93 - 1586 * q^94 + 4835 * q^95 - 1039 * q^97 - 2545 * 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
$3$	9.96286	1.91735	0.958676	−	0.284499i	$-0.0918273\pi$
$3$	9.96286	1.91735	0.958676	+	0.284499i	$0.0918273\pi$
$4$	4.00000	0.500000
$5$	11.8037	1.05575	0.527877	−	0.849321i	$-0.322988\pi$
$5$	11.8037	1.05575	0.527877	+	0.849321i	$0.322988\pi$
$6$	−19.9257	−1.35577
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	72.2585	2.67624
$10$	−23.6074	−0.746531
$11$	4.60635	0.126260	0.0631302	−	0.998005i	$-0.479892\pi$
$11$	4.60635	0.126260	0.0631302	+	0.998005i	$0.479892\pi$
$12$	39.8514	0.958676
$13$	−44.6962	−0.953576	−0.476788	−	0.879018i	$-0.658199\pi$
$13$	−44.6962	−0.953576	−0.476788	+	0.879018i	$0.658199\pi$
$14$	0	0
$15$	117.598	2.02425
$16$	16.0000	0.250000
$17$	42.8996	0.612040	0.306020	−	0.952025i	$-0.401002\pi$
$17$	42.8996	0.612040	0.306020	+	0.952025i	$0.401002\pi$
$18$	−144.517	−1.89239
$19$	80.8065	0.975699	0.487849	−	0.872928i	$-0.337781\pi$
$19$	80.8065	0.975699	0.487849	+	0.872928i	$0.337781\pi$
$20$	47.2147	0.527877
$21$	0	0
$22$	−9.21269	−0.0892796
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	−79.7029	−0.677887
$25$	14.3270	0.114616
$26$	89.3924	0.674280
$27$	450.904	3.21395
$28$	0	0
$29$	83.4085	0.534089	0.267044	−	0.963684i	$-0.413953\pi$
$29$	83.4085	0.534089	0.267044	+	0.963684i	$0.413953\pi$
$30$	−235.197	−1.43136
$31$	−29.1874	−0.169103	−0.0845517	−	0.996419i	$-0.526946\pi$
$31$	−29.1874	−0.169103	−0.0845517	+	0.996419i	$0.526946\pi$
$32$	−32.0000	−0.176777
$33$	45.8924	0.242086
$34$	−85.7992	−0.432778
$35$	0	0
$36$	289.034	1.33812
$37$	−11.7683	−0.0522889	−0.0261445	−	0.999658i	$-0.508323\pi$
$37$	−11.7683	−0.0522889	−0.0261445	+	0.999658i	$0.508323\pi$
$38$	−161.613	−0.689923
$39$	−445.302	−1.82834
$40$	−94.4295	−0.373265
$41$	385.859	1.46978	0.734891	−	0.678186i	$-0.237233\pi$
$41$	385.859	1.46978	0.734891	+	0.678186i	$0.237233\pi$
$42$	0	0
$43$	60.7982	0.215619	0.107810	−	0.994172i	$-0.465616\pi$
$43$	60.7982	0.215619	0.107810	+	0.994172i	$0.465616\pi$
$44$	18.4254	0.0631302
$45$	852.917	2.82545
$46$	−46.0000	−0.147442
$47$	−113.269	−0.351532	−0.175766	−	0.984432i	$-0.556240\pi$
$47$	−113.269	−0.351532	−0.175766	+	0.984432i	$0.556240\pi$
$48$	159.406	0.479338
$49$	0	0
$50$	−28.6541	−0.0810460
$51$	427.403	1.17350
$52$	−178.785	−0.476788
$53$	−652.574	−1.69128	−0.845641	−	0.533752i	$-0.820782\pi$
$53$	−652.574	−1.69128	−0.845641	+	0.533752i	$0.820782\pi$
$54$	−901.808	−2.27260
$55$	54.3719	0.133300
$56$	0	0
$57$	805.063	1.87076
$58$	−166.817	−0.377658
$59$	318.100	0.701917	0.350958	−	0.936391i	$-0.385856\pi$
$59$	318.100	0.701917	0.350958	+	0.936391i	$0.385856\pi$
$60$	470.394	1.01213
$61$	266.802	0.560008	0.280004	−	0.959999i	$-0.409664\pi$
$61$	266.802	0.560008	0.280004	+	0.959999i	$0.409664\pi$
$62$	58.3747	0.119574
$63$	0	0
$64$	64.0000	0.125000
$65$	−527.580	−1.00674
$66$	−91.7847	−0.171181
$67$	957.595	1.74610	0.873051	−	0.487629i	$-0.162138\pi$
$67$	957.595	1.74610	0.873051	+	0.487629i	$0.162138\pi$
$68$	171.598	0.306020
$69$	229.146	0.399796
$70$	0	0
$71$	−540.145	−0.902866	−0.451433	−	0.892305i	$-0.649087\pi$
$71$	−540.145	−0.902866	−0.451433	+	0.892305i	$0.649087\pi$
$72$	−578.068	−0.946194
$73$	−989.786	−1.58693	−0.793464	−	0.608618i	$-0.791724\pi$
$73$	−989.786	−1.58693	−0.793464	+	0.608618i	$0.791724\pi$
$74$	23.5365	0.0369739
$75$	142.738	0.219760
$76$	323.226	0.487849
$77$	0	0
$78$	890.604	1.29283
$79$	240.319	0.342253	0.171126	−	0.985249i	$-0.445259\pi$
$79$	240.319	0.342253	0.171126	+	0.985249i	$0.445259\pi$
$80$	188.859	0.263938
$81$	2541.31	3.48603
$82$	−771.718	−1.03929
$83$	−1281.34	−1.69452	−0.847262	−	0.531176i	$-0.821750\pi$
$83$	−1281.34	−1.69452	−0.847262	+	0.531176i	$0.821750\pi$
$84$	0	0
$85$	506.374	0.646164
$86$	−121.596	−0.152466
$87$	830.987	1.02404
$88$	−36.8508	−0.0446398
$89$	372.062	0.443129	0.221565	−	0.975146i	$-0.428884\pi$
$89$	372.062	0.443129	0.221565	+	0.975146i	$0.428884\pi$
$90$	−1705.83	−1.99790
$91$	0	0
$92$	92.0000	0.104257
$93$	−290.789	−0.324231
$94$	226.538	0.248571
$95$	953.815	1.03010
$96$	−318.811	−0.338943
$97$	1477.84	1.54692	0.773462	−	0.633843i	$-0.218524\pi$
$97$	1477.84	1.54692	0.773462	+	0.633843i	$0.218524\pi$
$98$	0	0
$99$	332.848	0.337903
$100$	57.3081	0.0573081
$101$	−1524.82	−1.50223	−0.751116	−	0.660171i	$-0.770484\pi$
$101$	−1524.82	−1.50223	−0.751116	+	0.660171i	$0.770484\pi$
$102$	−854.806	−0.829788
$103$	−635.654	−0.608086	−0.304043	−	0.952658i	$-0.598337\pi$
$103$	−635.654	−0.608086	−0.304043	+	0.952658i	$0.598337\pi$
$104$	357.570	0.337140
$105$	0	0
$106$	1305.15	1.19592
$107$	1826.01	1.64978	0.824892	−	0.565291i	$-0.191236\pi$
$107$	1826.01	1.64978	0.824892	+	0.565291i	$0.191236\pi$
$108$	1803.62	1.60697
$109$	2108.02	1.85240	0.926200	−	0.377032i	$-0.123055\pi$
$109$	2108.02	1.85240	0.926200	+	0.377032i	$0.123055\pi$
$110$	−108.744	−0.0942573
$111$	−117.246	−0.100256
$112$	0	0
$113$	342.787	0.285369	0.142684	−	0.989768i	$-0.454427\pi$
$113$	342.787	0.285369	0.142684	+	0.989768i	$0.454427\pi$
$114$	−1610.13	−1.32283
$115$	271.485	0.220140
$116$	333.634	0.267044
$117$	−3229.68	−2.55200
$118$	−636.200	−0.496330
$119$	0	0
$120$	−940.788	−0.715681
$121$	−1309.78	−0.984058
$122$	−533.603	−0.395985
$123$	3844.26	2.81809
$124$	−116.749	−0.0845517
$125$	−1306.35	−0.934747
$126$	0	0
$127$	−2405.93	−1.68104	−0.840520	−	0.541780i	$-0.817751\pi$
$127$	−2405.93	−1.68104	−0.840520	+	0.541780i	$0.817751\pi$
$128$	−128.000	−0.0883883
$129$	605.724	0.413419
$130$	1055.16	0.711874
$131$	2047.24	1.36541	0.682704	−	0.730695i	$-0.260804\pi$
$131$	2047.24	1.36541	0.682704	+	0.730695i	$0.260804\pi$
$132$	183.569	0.121043
$133$	0	0
$134$	−1915.19	−1.23468
$135$	5322.33	3.39314
$136$	−343.197	−0.216389
$137$	−174.150	−0.108603	−0.0543017	−	0.998525i	$-0.517293\pi$
$137$	−174.150	−0.108603	−0.0543017	+	0.998525i	$0.517293\pi$
$138$	−458.291	−0.282698
$139$	688.080	0.419872	0.209936	−	0.977715i	$-0.432674\pi$
$139$	688.080	0.419872	0.209936	+	0.977715i	$0.432674\pi$
$140$	0	0
$141$	−1128.48	−0.674011
$142$	1080.29	0.638422
$143$	−205.886	−0.120399
$144$	1156.14	0.669060
$145$	984.528	0.563866
$146$	1979.57	1.12213
$147$	0	0
$148$	−47.0731	−0.0261445
$149$	−2128.33	−1.17020	−0.585100	−	0.810961i	$-0.698945\pi$
$149$	−2128.33	−1.17020	−0.585100	+	0.810961i	$0.698945\pi$
$150$	−285.476	−0.155394
$151$	3145.53	1.69523	0.847613	−	0.530614i	$-0.178039\pi$
$151$	3145.53	1.69523	0.847613	+	0.530614i	$0.178039\pi$
$152$	−646.452	−0.344962
$153$	3099.86	1.63797
$154$	0	0
$155$	−344.518	−0.178532
$156$	−1781.21	−0.914171
$157$	1580.39	0.803367	0.401684	−	0.915778i	$-0.368425\pi$
$157$	1580.39	0.803367	0.401684	+	0.915778i	$0.368425\pi$
$158$	−480.637	−0.242009
$159$	−6501.50	−3.24278
$160$	−377.718	−0.186633
$161$	0	0
$162$	−5082.63	−2.46499
$163$	1334.83	0.641422	0.320711	−	0.947177i	$-0.396078\pi$
$163$	1334.83	0.641422	0.320711	+	0.947177i	$0.396078\pi$
$164$	1543.44	0.734891
$165$	541.699	0.255583
$166$	2562.68	1.19821
$167$	−650.417	−0.301382	−0.150691	−	0.988581i	$-0.548150\pi$
$167$	−650.417	−0.301382	−0.150691	+	0.988581i	$0.548150\pi$
$168$	0	0
$169$	−199.250	−0.0906920
$170$	−1012.75	−0.456907
$171$	5838.96	2.61121
$172$	243.193	0.107810
$173$	−72.0315	−0.0316558	−0.0158279	−	0.999875i	$-0.505038\pi$
$173$	−72.0315	−0.0316558	−0.0158279	+	0.999875i	$0.505038\pi$
$174$	−1661.97	−0.724103
$175$	0	0
$176$	73.7015	0.0315651
$177$	3169.18	1.34582
$178$	−744.124	−0.313340
$179$	2189.74	0.914351	0.457176	−	0.889376i	$-0.348861\pi$
$179$	2189.74	0.914351	0.457176	+	0.889376i	$0.348861\pi$
$180$	3411.67	1.41273
$181$	1197.04	0.491578	0.245789	−	0.969323i	$-0.420953\pi$
$181$	1197.04	0.491578	0.245789	+	0.969323i	$0.420953\pi$
$182$	0	0
$183$	2658.11	1.07373
$184$	−184.000	−0.0737210
$185$	−138.909	−0.0552042
$186$	581.579	0.229266
$187$	197.610	0.0772765
$188$	−453.077	−0.175766
$189$	0	0
$190$	−1907.63	−0.728389
$191$	3172.81	1.20197	0.600986	−	0.799259i	$-0.294775\pi$
$191$	3172.81	1.20197	0.600986	+	0.799259i	$0.294775\pi$
$192$	637.623	0.239669
$193$	−2797.24	−1.04326	−0.521631	−	0.853171i	$-0.674676\pi$
$193$	−2797.24	−1.04326	−0.521631	+	0.853171i	$0.674676\pi$
$194$	−2955.67	−1.09384
$195$	−5256.20	−1.93028
$196$	0	0
$197$	4600.10	1.66367	0.831836	−	0.555022i	$-0.187290\pi$
$197$	4600.10	1.66367	0.831836	+	0.555022i	$0.187290\pi$
$198$	−665.695	−0.238934
$199$	−1775.34	−0.632416	−0.316208	−	0.948690i	$-0.602410\pi$
$199$	−1775.34	−0.632416	−0.316208	+	0.948690i	$0.602410\pi$
$200$	−114.616	−0.0405230
$201$	9540.38	3.34789
$202$	3049.64	1.06224
$203$	0	0
$204$	1709.61	0.586749
$205$	4554.56	1.55173
$206$	1271.31	0.429981
$207$	1661.95	0.558035
$208$	−715.139	−0.238394
$209$	372.223	0.123192
$210$	0	0
$211$	4329.50	1.41258	0.706292	−	0.707920i	$-0.250366\pi$
$211$	4329.50	1.41258	0.706292	+	0.707920i	$0.250366\pi$
$212$	−2610.30	−0.845641
$213$	−5381.39	−1.73111
$214$	−3652.02	−1.16657
$215$	717.643	0.227641
$216$	−3607.23	−1.13630
$217$	0	0
$218$	−4216.04	−1.30985
$219$	−9861.10	−3.04270
$220$	217.487	0.0666500
$221$	−1917.45	−0.583627
$222$	234.491	0.0708919
$223$	−2457.40	−0.737935	−0.368967	−	0.929442i	$-0.620289\pi$
$223$	−2457.40	−0.737935	−0.368967	+	0.929442i	$0.620289\pi$
$224$	0	0
$225$	1035.25	0.306741
$226$	−685.573	−0.201786
$227$	−737.033	−0.215500	−0.107750	−	0.994178i	$-0.534365\pi$
$227$	−737.033	−0.215500	−0.107750	+	0.994178i	$0.534365\pi$
$228$	3220.25	0.935379
$229$	−1117.06	−0.322347	−0.161173	−	0.986926i	$-0.551528\pi$
$229$	−1117.06	−0.322347	−0.161173	+	0.986926i	$0.551528\pi$
$230$	−542.970	−0.155662
$231$	0	0
$232$	−667.268	−0.188829
$233$	−1660.79	−0.466961	−0.233481	−	0.972361i	$-0.575012\pi$
$233$	−1660.79	−0.466961	−0.233481	+	0.972361i	$0.575012\pi$
$234$	6459.36	1.80454
$235$	−1336.99	−0.371131
$236$	1272.40	0.350958
$237$	2394.26	0.656219
$238$	0	0
$239$	−6359.15	−1.72109	−0.860543	−	0.509379i	$-0.829875\pi$
$239$	−6359.15	−1.72109	−0.860543	+	0.509379i	$0.829875\pi$
$240$	1881.58	0.506063
$241$	−5325.63	−1.42346	−0.711731	−	0.702452i	$-0.752088\pi$
$241$	−5325.63	−1.42346	−0.711731	+	0.702452i	$0.752088\pi$
$242$	2619.56	0.695834
$243$	13144.3	3.46999
$244$	1067.21	0.280004
$245$	0	0
$246$	−7688.51	−1.99269
$247$	−3611.74	−0.930404
$248$	233.499	0.0597871
$249$	−12765.8	−3.24900
$250$	2612.70	0.660966
$251$	2613.61	0.657250	0.328625	−	0.944461i	$-0.393415\pi$
$251$	2613.61	0.657250	0.328625	+	0.944461i	$0.393415\pi$
$252$	0	0
$253$	105.946	0.0263271
$254$	4811.87	1.18868
$255$	5044.93	1.23892
$256$	256.000	0.0625000
$257$	−4998.11	−1.21313	−0.606564	−	0.795035i	$-0.707453\pi$
$257$	−4998.11	−1.21313	−0.606564	+	0.795035i	$0.707453\pi$
$258$	−1211.45	−0.292331
$259$	0	0
$260$	−2110.32	−0.503371
$261$	6026.97	1.42935
$262$	−4094.49	−0.965490
$263$	−5547.29	−1.30061	−0.650306	−	0.759673i	$-0.725359\pi$
$263$	−5547.29	−1.30061	−0.650306	+	0.759673i	$0.725359\pi$
$264$	−367.139	−0.0855903
$265$	−7702.78	−1.78558
$266$	0	0
$267$	3706.80	0.849635
$268$	3830.38	0.873051
$269$	5278.63	1.19644	0.598222	−	0.801330i	$-0.295874\pi$
$269$	5278.63	1.19644	0.598222	+	0.801330i	$0.295874\pi$
$270$	−10644.7	−2.39931
$271$	−4868.81	−1.09136	−0.545681	−	0.837993i	$-0.683729\pi$
$271$	−4868.81	−1.09136	−0.545681	+	0.837993i	$0.683729\pi$
$272$	686.394	0.153010
$273$	0	0
$274$	348.301	0.0767942
$275$	65.9953	0.0144715
$276$	916.583	0.199898
$277$	1826.24	0.396131	0.198066	−	0.980189i	$-0.436534\pi$
$277$	1826.24	0.396131	0.198066	+	0.980189i	$0.436534\pi$
$278$	−1376.16	−0.296894
$279$	−2109.04	−0.452561
$280$	0	0
$281$	−4630.71	−0.983078	−0.491539	−	0.870856i	$-0.663565\pi$
$281$	−4630.71	−0.983078	−0.491539	+	0.870856i	$0.663565\pi$
$282$	2256.97	0.476598
$283$	−6261.47	−1.31521	−0.657607	−	0.753361i	$-0.728431\pi$
$283$	−6261.47	−1.31521	−0.657607	+	0.753361i	$0.728431\pi$
$284$	−2160.58	−0.451433
$285$	9502.72	1.97506
$286$	411.772	0.0851350
$287$	0	0
$288$	−2312.27	−0.473097
$289$	−3072.62	−0.625407
$290$	−1969.06	−0.398714
$291$	14723.5	2.96600
$292$	−3959.14	−0.793464
$293$	−2024.03	−0.403568	−0.201784	−	0.979430i	$-0.564674\pi$
$293$	−2024.03	−0.403568	−0.201784	+	0.979430i	$0.564674\pi$
$294$	0	0
$295$	3754.75	0.741051
$296$	94.1461	0.0184869
$297$	2077.02	0.405794
$298$	4256.67	0.827457
$299$	−1028.01	−0.198834
$300$	570.953	0.109880
$301$	0	0
$302$	−6291.05	−1.19871
$303$	−15191.6	−2.88031
$304$	1292.90	0.243925
$305$	3149.24	0.591230
$306$	−6199.73	−1.15822
$307$	−7703.59	−1.43214	−0.716070	−	0.698028i	$-0.754061\pi$
$307$	−7703.59	−1.43214	−0.716070	+	0.698028i	$0.754061\pi$
$308$	0	0
$309$	−6332.93	−1.16591
$310$	689.037	0.126241
$311$	2858.35	0.521164	0.260582	−	0.965452i	$-0.416085\pi$
$311$	2858.35	0.521164	0.260582	+	0.965452i	$0.416085\pi$
$312$	3562.41	0.646417
$313$	−749.127	−0.135282	−0.0676408	−	0.997710i	$-0.521547\pi$
$313$	−749.127	−0.135282	−0.0676408	+	0.997710i	$0.521547\pi$
$314$	−3160.78	−0.568067
$315$	0	0
$316$	961.275	0.171126
$317$	7316.56	1.29634	0.648169	−	0.761497i	$-0.275535\pi$
$317$	7316.56	1.29634	0.648169	+	0.761497i	$0.275535\pi$
$318$	13003.0	2.29299
$319$	384.208	0.0674343
$320$	755.436	0.131969
$321$	18192.3	3.16322
$322$	0	0
$323$	3466.57	0.597167
$324$	10165.3	1.74301
$325$	−640.364	−0.109295
$326$	−2669.66	−0.453554
$327$	21001.9	3.55171
$328$	−3086.87	−0.519646
$329$	0	0
$330$	−1083.40	−0.180725
$331$	3045.14	0.505667	0.252834	−	0.967510i	$-0.418637\pi$
$331$	3045.14	0.505667	0.252834	+	0.967510i	$0.418637\pi$
$332$	−5125.36	−0.847262
$333$	−850.357	−0.139938
$334$	1300.83	0.213109
$335$	11303.2	1.84345
$336$	0	0
$337$	2125.20	0.343523	0.171761	−	0.985139i	$-0.445054\pi$
$337$	2125.20	0.343523	0.171761	+	0.985139i	$0.445054\pi$
$338$	398.501	0.0641289
$339$	3415.13	0.547152
$340$	2025.49	0.323082
$341$	−134.447	−0.0213511
$342$	−11677.9	−1.84640
$343$	0	0
$344$	−486.386	−0.0762330
$345$	2704.76	0.422086
$346$	144.063	0.0223840
$347$	−9606.09	−1.48611	−0.743057	−	0.669228i	$-0.766625\pi$
$347$	−9606.09	−1.48611	−0.743057	+	0.669228i	$0.766625\pi$
$348$	3323.95	0.512018
$349$	−636.545	−0.0976317	−0.0488159	−	0.998808i	$-0.515545\pi$
$349$	−636.545	−0.0976317	−0.0488159	+	0.998808i	$0.515545\pi$
$350$	0	0
$351$	−20153.7	−3.06474
$352$	−147.403	−0.0223199
$353$	5977.38	0.901258	0.450629	−	0.892711i	$-0.351200\pi$
$353$	5977.38	0.901258	0.450629	+	0.892711i	$0.351200\pi$
$354$	−6338.37	−0.951640
$355$	−6375.71	−0.953204
$356$	1488.25	0.221565
$357$	0	0
$358$	−4379.48	−0.646544
$359$	−13016.3	−1.91357	−0.956785	−	0.290797i	$-0.906080\pi$
$359$	−13016.3	−1.91357	−0.956785	+	0.290797i	$0.906080\pi$
$360$	−6823.33	−0.998948
$361$	−329.311	−0.0480116
$362$	−2394.09	−0.347598
$363$	−13049.2	−1.88679
$364$	0	0
$365$	−11683.1	−1.67540
$366$	−5316.21	−0.759243
$367$	6167.72	0.877255	0.438627	−	0.898669i	$-0.355465\pi$
$367$	6167.72	0.877255	0.438627	+	0.898669i	$0.355465\pi$
$368$	368.000	0.0521286
$369$	27881.6	3.93349
$370$	277.818	0.0390353
$371$	0	0
$372$	−1163.16	−0.162115
$373$	−1964.93	−0.272761	−0.136381	−	0.990657i	$-0.543547\pi$
$373$	−1964.93	−0.272761	−0.136381	+	0.990657i	$0.543547\pi$
$374$	−395.221	−0.0546427
$375$	−13015.0	−1.79224
$376$	906.153	0.124285
$377$	−3728.04	−0.509294
$378$	0	0
$379$	−5891.81	−0.798528	−0.399264	−	0.916836i	$-0.630734\pi$
$379$	−5891.81	−0.798528	−0.399264	+	0.916836i	$0.630734\pi$
$380$	3815.26	0.515049
$381$	−23970.0	−3.22315
$382$	−6345.63	−0.849923
$383$	−13514.7	−1.80306	−0.901529	−	0.432718i	$-0.857555\pi$
$383$	−13514.7	−1.80306	−0.901529	+	0.432718i	$0.857555\pi$
$384$	−1275.25	−0.169472
$385$	0	0
$386$	5594.48	0.737698
$387$	4393.19	0.577050
$388$	5911.35	0.773462
$389$	−41.5038	−0.00540958	−0.00270479	−	0.999996i	$-0.500861\pi$
$389$	−41.5038	−0.00540958	−0.00270479	+	0.999996i	$0.500861\pi$
$390$	10512.4	1.36491
$391$	986.691	0.127619
$392$	0	0
$393$	20396.4	2.61797
$394$	−9200.19	−1.17639
$395$	2836.65	0.361335
$396$	1331.39	0.168952
$397$	6465.08	0.817313	0.408656	−	0.912688i	$-0.365997\pi$
$397$	6465.08	0.817313	0.408656	+	0.912688i	$0.365997\pi$
$398$	3550.69	0.447185
$399$	0	0
$400$	229.233	0.0286541
$401$	−2444.04	−0.304363	−0.152181	−	0.988353i	$-0.548630\pi$
$401$	−2444.04	−0.304363	−0.152181	+	0.988353i	$0.548630\pi$
$402$	−19080.8	−2.36732
$403$	1304.56	0.161253
$404$	−6099.28	−0.751116
$405$	29996.9	3.68038
$406$	0	0
$407$	−54.2087	−0.00660203
$408$	−3419.22	−0.414894
$409$	6908.70	0.835241	0.417620	−	0.908622i	$-0.362864\pi$
$409$	6908.70	0.835241	0.417620	+	0.908622i	$0.362864\pi$
$410$	−9109.11	−1.09724
$411$	−1735.03	−0.208231
$412$	−2542.61	−0.304043
$413$	0	0
$414$	−3323.89	−0.394590
$415$	−15124.5	−1.78900
$416$	1430.28	0.168570
$417$	6855.25	0.805043
$418$	−744.445	−0.0871101
$419$	2222.30	0.259109	0.129554	−	0.991572i	$-0.458645\pi$
$419$	2222.30	0.259109	0.129554	+	0.991572i	$0.458645\pi$
$420$	0	0
$421$	−1014.89	−0.117489	−0.0587445	−	0.998273i	$-0.518710\pi$
$421$	−1014.89	−0.117489	−0.0587445	+	0.998273i	$0.518710\pi$
$422$	−8659.01	−0.998848
$423$	−8184.66	−0.940784
$424$	5220.59	0.597959
$425$	614.624	0.0701498
$426$	10762.8	1.22408
$427$	0	0
$428$	7304.03	0.824892
$429$	−2051.21	−0.230847
$430$	−1435.29	−0.160967
$431$	−3350.13	−0.374409	−0.187204	−	0.982321i	$-0.559943\pi$
$431$	−3350.13	−0.374409	−0.187204	+	0.982321i	$0.559943\pi$
$432$	7214.46	0.803486
$433$	1204.51	0.133684	0.0668419	−	0.997764i	$-0.478708\pi$
$433$	1204.51	0.133684	0.0668419	+	0.997764i	$0.478708\pi$
$434$	0	0
$435$	9808.71	1.08113
$436$	8432.08	0.926200
$437$	1858.55	0.203447
$438$	19722.2	2.15151
$439$	5226.22	0.568186	0.284093	−	0.958797i	$-0.408308\pi$
$439$	5226.22	0.568186	0.284093	+	0.958797i	$0.408308\pi$
$440$	−434.975	−0.0471287
$441$	0	0
$442$	3834.90	0.412687
$443$	4858.51	0.521072	0.260536	−	0.965464i	$-0.416101\pi$
$443$	4858.51	0.521072	0.260536	+	0.965464i	$0.416101\pi$
$444$	−468.982	−0.0501282
$445$	4391.70	0.467835
$446$	4914.79	0.521799
$447$	−21204.3	−2.24369
$448$	0	0
$449$	11797.4	1.23999	0.619994	−	0.784606i	$-0.287135\pi$
$449$	11797.4	1.23999	0.619994	+	0.784606i	$0.287135\pi$
$450$	−2070.50	−0.216899
$451$	1777.40	0.185575
$452$	1371.15	0.142684
$453$	31338.4	3.25035
$454$	1474.07	0.152382
$455$	0	0
$456$	−6440.51	−0.661413
$457$	2317.80	0.237247	0.118624	−	0.992939i	$-0.462152\pi$
$457$	2317.80	0.237247	0.118624	+	0.992939i	$0.462152\pi$
$458$	2234.12	0.227934
$459$	19343.6	1.96706
$460$	1085.94	0.110070
$461$	6605.88	0.667389	0.333695	−	0.942681i	$-0.391705\pi$
$461$	6605.88	0.667389	0.333695	+	0.942681i	$0.391705\pi$
$462$	0	0
$463$	13744.4	1.37961	0.689804	−	0.723996i	$-0.257697\pi$
$463$	13744.4	1.37961	0.689804	+	0.723996i	$0.257697\pi$
$464$	1334.54	0.133522
$465$	−3432.39	−0.342308
$466$	3321.58	0.330191
$467$	−11563.1	−1.14578	−0.572889	−	0.819633i	$-0.694177\pi$
$467$	−11563.1	−1.14578	−0.572889	+	0.819633i	$0.694177\pi$
$468$	−12918.7	−1.27600
$469$	0	0
$470$	2673.99	0.262429
$471$	15745.2	1.54034
$472$	−2544.80	−0.248165
$473$	280.057	0.0272242
$474$	−4788.52	−0.464017
$475$	1157.72	0.111831
$476$	0	0
$477$	−47154.0	−4.52628
$478$	12718.3	1.21699
$479$	17590.9	1.67797	0.838985	−	0.544155i	$-0.183150\pi$
$479$	17590.9	1.67797	0.838985	+	0.544155i	$0.183150\pi$
$480$	−3763.15	−0.357841
$481$	525.997	0.0498615
$482$	10651.3	1.00654
$483$	0	0
$484$	−5239.13	−0.492029
$485$	17443.9	1.63317
$486$	−26288.6	−2.45366
$487$	−8379.26	−0.779672	−0.389836	−	0.920884i	$-0.627468\pi$
$487$	−8379.26	−0.779672	−0.389836	+	0.920884i	$0.627468\pi$
$488$	−2134.41	−0.197993
$489$	13298.7	1.22983
$490$	0	0
$491$	−15435.2	−1.41869	−0.709347	−	0.704859i	$-0.751010\pi$
$491$	−15435.2	−1.41869	−0.709347	+	0.704859i	$0.751010\pi$
$492$	15377.0	1.40904
$493$	3578.19	0.326884
$494$	7223.48	0.657895
$495$	3928.83	0.356743
$496$	−466.998	−0.0422758
$497$	0	0
$498$	25531.6	2.29739
$499$	12303.0	1.10372	0.551859	−	0.833937i	$-0.313918\pi$
$499$	12303.0	1.10372	0.551859	+	0.833937i	$0.313918\pi$
$500$	−5225.40	−0.467374
$501$	−6480.02	−0.577856
$502$	−5227.22	−0.464746
$503$	−1978.55	−0.175386	−0.0876931	−	0.996148i	$-0.527949\pi$
$503$	−1978.55	−0.175386	−0.0876931	+	0.996148i	$0.527949\pi$
$504$	0	0
$505$	−17998.5	−1.58599
$506$	−211.892	−0.0186161
$507$	−1985.10	−0.173888
$508$	−9623.74	−0.840520
$509$	−2138.21	−0.186197	−0.0930987	−	0.995657i	$-0.529677\pi$
$509$	−2138.21	−0.186197	−0.0930987	+	0.995657i	$0.529677\pi$
$510$	−10089.9	−0.876052
$511$	0	0
$512$	−512.000	−0.0441942
$513$	36436.0	3.13584
$514$	9996.23	0.857811
$515$	−7503.06	−0.641989
$516$	2422.89	0.206709
$517$	−521.757	−0.0443846
$518$	0	0
$519$	−717.640	−0.0606953
$520$	4220.64	0.355937
$521$	14496.3	1.21899	0.609495	−	0.792790i	$-0.291372\pi$
$521$	14496.3	1.21899	0.609495	+	0.792790i	$0.291372\pi$
$522$	−12053.9	−1.01070
$523$	−1064.47	−0.0889984	−0.0444992	−	0.999009i	$-0.514169\pi$
$523$	−1064.47	−0.0889984	−0.0444992	+	0.999009i	$0.514169\pi$
$524$	8188.98	0.682704
$525$	0	0
$526$	11094.6	0.919671
$527$	−1252.13	−0.103498
$528$	734.278	0.0605215
$529$	529.000	0.0434783
$530$	15405.6	1.26259
$531$	22985.4	1.87850
$532$	0	0
$533$	−17246.4	−1.40155
$534$	−7413.60	−0.600782
$535$	21553.6	1.74177
$536$	−7660.76	−0.617340
$537$	21816.1	1.75313
$538$	−10557.3	−0.846014
$539$	0	0
$540$	21289.3	1.69657
$541$	−5778.57	−0.459224	−0.229612	−	0.973282i	$-0.573746\pi$
$541$	−5778.57	−0.459224	−0.229612	+	0.973282i	$0.573746\pi$
$542$	9737.61	0.771709
$543$	11926.0	0.942528
$544$	−1372.79	−0.108194
$545$	24882.4	1.95568
$546$	0	0
$547$	683.854	0.0534543	0.0267271	−	0.999643i	$-0.491491\pi$
$547$	683.854	0.0534543	0.0267271	+	0.999643i	$0.491491\pi$
$548$	−696.601	−0.0543017
$549$	19278.7	1.49872
$550$	−131.991	−0.0102329
$551$	6739.95	0.521110
$552$	−1833.17	−0.141349
$553$	0	0
$554$	−3652.49	−0.280107
$555$	−1383.93	−0.105846
$556$	2752.32	0.209936
$557$	11916.8	0.906516	0.453258	−	0.891379i	$-0.350262\pi$
$557$	11916.8	0.906516	0.453258	+	0.891379i	$0.350262\pi$
$558$	4218.07	0.320009
$559$	−2717.45	−0.205610
$560$	0	0
$561$	1968.76	0.148166
$562$	9261.42	0.695141
$563$	11461.3	0.857967	0.428984	−	0.903312i	$-0.358872\pi$
$563$	11461.3	0.857967	0.428984	+	0.903312i	$0.358872\pi$
$564$	−4513.94	−0.337005
$565$	4046.15	0.301279
$566$	12522.9	0.929997
$567$	0	0
$568$	4321.16	0.319211
$569$	−24135.3	−1.77822	−0.889109	−	0.457695i	$-0.848675\pi$
$569$	−24135.3	−1.77822	−0.889109	+	0.457695i	$0.848675\pi$
$570$	−19005.4	−1.39658
$571$	24832.3	1.81996	0.909982	−	0.414647i	$-0.136095\pi$
$571$	24832.3	1.81996	0.909982	+	0.414647i	$0.136095\pi$
$572$	−823.544	−0.0601995
$573$	31610.3	2.30461
$574$	0	0
$575$	329.522	0.0238991
$576$	4624.54	0.334530
$577$	−5663.64	−0.408632	−0.204316	−	0.978905i	$-0.565497\pi$
$577$	−5663.64	−0.408632	−0.204316	+	0.978905i	$0.565497\pi$
$578$	6145.24	0.442229
$579$	−27868.5	−2.00030
$580$	3938.11	0.281933
$581$	0	0
$582$	−29446.9	−2.09728
$583$	−3005.98	−0.213542
$584$	7918.29	0.561063
$585$	−38122.1	−2.69428
$586$	4048.07	0.285365
$587$	6199.90	0.435940	0.217970	−	0.975955i	$-0.430056\pi$
$587$	6199.90	0.435940	0.217970	+	0.975955i	$0.430056\pi$
$588$	0	0
$589$	−2358.53	−0.164994
$590$	−7509.51	−0.524003
$591$	45830.1	3.18985
$592$	−188.292	−0.0130722
$593$	−17208.9	−1.19171	−0.595857	−	0.803090i	$-0.703188\pi$
$593$	−17208.9	−1.19171	−0.595857	+	0.803090i	$0.703188\pi$
$594$	−4154.04	−0.286940
$595$	0	0
$596$	−8513.33	−0.585100
$597$	−17687.5	−1.21256
$598$	2056.02	0.140597
$599$	12255.1	0.835943	0.417971	−	0.908460i	$-0.362741\pi$
$599$	12255.1	0.835943	0.417971	+	0.908460i	$0.362741\pi$
$600$	−1141.91	−0.0776968
$601$	−21764.9	−1.47722	−0.738610	−	0.674133i	$-0.764517\pi$
$601$	−21764.9	−1.47722	−0.738610	+	0.674133i	$0.764517\pi$
$602$	0	0
$603$	69194.4	4.67299
$604$	12582.1	0.847613
$605$	−15460.3	−1.03892
$606$	30383.1	2.03668
$607$	−15469.7	−1.03442	−0.517211	−	0.855858i	$-0.673030\pi$
$607$	−15469.7	−1.03442	−0.517211	+	0.855858i	$0.673030\pi$
$608$	−2585.81	−0.172481
$609$	0	0
$610$	−6298.49	−0.418063
$611$	5062.70	0.335213
$612$	12399.5	0.818984
$613$	−10199.0	−0.671996	−0.335998	−	0.941863i	$-0.609074\pi$
$613$	−10199.0	−0.671996	−0.335998	+	0.941863i	$0.609074\pi$
$614$	15407.2	1.01268
$615$	45376.4	2.97521
$616$	0	0
$617$	16429.1	1.07198	0.535989	−	0.844225i	$-0.319939\pi$
$617$	16429.1	1.07198	0.535989	+	0.844225i	$0.319939\pi$
$618$	12665.9	0.824426
$619$	26369.9	1.71227	0.856135	−	0.516753i	$-0.172859\pi$
$619$	26369.9	1.71227	0.856135	+	0.516753i	$0.172859\pi$
$620$	−1378.07	−0.0892658
$621$	10370.8	0.670154
$622$	−5716.70	−0.368519
$623$	0	0
$624$	−7124.83	−0.457086
$625$	−17210.6	−1.10148
$626$	1498.25	0.0956586
$627$	3708.40	0.236203
$628$	6321.55	0.401684
$629$	−504.854	−0.0320029
$630$	0	0
$631$	22428.4	1.41499	0.707497	−	0.706717i	$-0.249824\pi$
$631$	22428.4	1.41499	0.707497	+	0.706717i	$0.249824\pi$
$632$	−1922.55	−0.121005
$633$	43134.2	2.70842
$634$	−14633.1	−0.916649
$635$	−28398.9	−1.77477
$636$	−26006.0	−1.62139
$637$	0	0
$638$	−768.417	−0.0476832
$639$	−39030.1	−2.41629
$640$	−1510.87	−0.0933163
$641$	20783.9	1.28068	0.640340	−	0.768092i	$-0.278794\pi$
$641$	20783.9	1.28068	0.640340	+	0.768092i	$0.278794\pi$
$642$	−36384.5	−2.23673
$643$	−6962.28	−0.427007	−0.213503	−	0.976942i	$-0.568487\pi$
$643$	−6962.28	−0.427007	−0.213503	+	0.976942i	$0.568487\pi$
$644$	0	0
$645$	7149.77	0.436468
$646$	−6933.14	−0.422261
$647$	−29508.1	−1.79302	−0.896508	−	0.443027i	$-0.853905\pi$
$647$	−29508.1	−1.79302	−0.896508	+	0.443027i	$0.853905\pi$
$648$	−20330.5	−1.23250
$649$	1465.28	0.0886244
$650$	1280.73	0.0772835
$651$	0	0
$652$	5339.31	0.320711
$653$	−16503.7	−0.989036	−0.494518	−	0.869167i	$-0.664655\pi$
$653$	−16503.7	−0.989036	−0.494518	+	0.869167i	$0.664655\pi$
$654$	−42003.8	−2.51144
$655$	24165.0	1.44154
$656$	6173.74	0.367445
$657$	−71520.5	−4.24700
$658$	0	0
$659$	−1536.50	−0.0908247	−0.0454123	−	0.998968i	$-0.514460\pi$
$659$	−1536.50	−0.0908247	−0.0454123	+	0.998968i	$0.514460\pi$
$660$	2166.80	0.127792
$661$	3901.31	0.229566	0.114783	−	0.993391i	$-0.463383\pi$
$661$	3901.31	0.229566	0.114783	+	0.993391i	$0.463383\pi$
$662$	−6090.27	−0.357561
$663$	−19103.3	−1.11902
$664$	10250.7	0.599104
$665$	0	0
$666$	1700.71	0.0989510
$667$	1918.40	0.111365
$668$	−2601.67	−0.150691
$669$	−24482.7	−1.41488
$670$	−22606.3	−1.30352
$671$	1228.98	0.0707068
$672$	0	0
$673$	−31727.1	−1.81722	−0.908611	−	0.417643i	$-0.862856\pi$
$673$	−31727.1	−1.81722	−0.908611	+	0.417643i	$0.862856\pi$
$674$	−4250.40	−0.242907
$675$	6460.12	0.368370
$676$	−797.001	−0.0453460
$677$	−15653.5	−0.888646	−0.444323	−	0.895867i	$-0.646556\pi$
$677$	−15653.5	−0.888646	−0.444323	+	0.895867i	$0.646556\pi$
$678$	−6830.27	−0.386895
$679$	0	0
$680$	−4050.99	−0.228453
$681$	−7342.95	−0.413190
$682$	268.894	0.0150975
$683$	11400.0	0.638669	0.319334	−	0.947642i	$-0.396541\pi$
$683$	11400.0	0.638669	0.319334	+	0.947642i	$0.396541\pi$
$684$	23355.8	1.30560
$685$	−2055.62	−0.114658
$686$	0	0
$687$	−11129.1	−0.618053
$688$	972.771	0.0539049
$689$	29167.6	1.61277
$690$	−5409.53	−0.298460
$691$	−6450.93	−0.355145	−0.177572	−	0.984108i	$-0.556824\pi$
$691$	−6450.93	−0.355145	−0.177572	+	0.984108i	$0.556824\pi$
$692$	−288.126	−0.0158279
$693$	0	0
$694$	19212.2	1.05084
$695$	8121.89	0.443282
$696$	−6647.89	−0.362051
$697$	16553.2	0.899565
$698$	1273.09	0.0690361
$699$	−16546.2	−0.895329
$700$	0	0
$701$	10471.5	0.564197	0.282098	−	0.959385i	$-0.408970\pi$
$701$	10471.5	0.564197	0.282098	+	0.959385i	$0.408970\pi$
$702$	40307.4	2.16710
$703$	−950.952	−0.0510183
$704$	294.806	0.0157826
$705$	−13320.3	−0.711589
$706$	−11954.8	−0.637286
$707$	0	0
$708$	12676.7	0.672911
$709$	−7948.32	−0.421023	−0.210512	−	0.977591i	$-0.567513\pi$
$709$	−7948.32	−0.421023	−0.210512	+	0.977591i	$0.567513\pi$
$710$	12751.4	0.674017
$711$	17365.1	0.915951
$712$	−2976.50	−0.156670
$713$	−671.309	−0.0352605
$714$	0	0
$715$	−2430.22	−0.127112
$716$	8758.96	0.457176
$717$	−63355.3	−3.29993
$718$	26032.5	1.35310
$719$	5747.90	0.298137	0.149068	−	0.988827i	$-0.452373\pi$
$719$	5747.90	0.298137	0.149068	+	0.988827i	$0.452373\pi$
$720$	13646.7	0.706363
$721$	0	0
$722$	658.623	0.0339493
$723$	−53058.5	−2.72928
$724$	4788.17	0.245789
$725$	1195.00	0.0612152
$726$	26098.3	1.33416
$727$	−7627.21	−0.389103	−0.194551	−	0.980892i	$-0.562325\pi$
$727$	−7627.21	−0.389103	−0.194551	+	0.980892i	$0.562325\pi$
$728$	0	0
$729$	62339.6	3.16718
$730$	23366.2	1.18469
$731$	2608.22	0.131968
$732$	10632.4	0.536866
$733$	−9862.40	−0.496966	−0.248483	−	0.968636i	$-0.579932\pi$
$733$	−9862.40	−0.496966	−0.248483	+	0.968636i	$0.579932\pi$
$734$	−12335.4	−0.620313
$735$	0	0
$736$	−736.000	−0.0368605
$737$	4411.01	0.220464
$738$	−55763.2	−2.78140
$739$	12312.1	0.612867	0.306434	−	0.951892i	$-0.400864\pi$
$739$	12312.1	0.612867	0.306434	+	0.951892i	$0.400864\pi$
$740$	−555.636	−0.0276021
$741$	−35983.3	−1.78391
$742$	0	0
$743$	1575.54	0.0777941	0.0388971	−	0.999243i	$-0.487616\pi$
$743$	1575.54	0.0777941	0.0388971	+	0.999243i	$0.487616\pi$
$744$	2326.32	0.114633
$745$	−25122.2	−1.23544
$746$	3929.85	0.192871
$747$	−92587.8	−4.53495
$748$	790.442	0.0386383
$749$	0	0
$750$	26029.9	1.26731
$751$	−9080.79	−0.441229	−0.220614	−	0.975361i	$-0.570806\pi$
$751$	−9080.79	−0.441229	−0.220614	+	0.975361i	$0.570806\pi$
$752$	−1812.31	−0.0878830
$753$	26039.0	1.26018
$754$	7456.08	0.360125
$755$	37128.8	1.78974
$756$	0	0
$757$	−8814.72	−0.423219	−0.211609	−	0.977354i	$-0.567870\pi$
$757$	−8814.72	−0.423219	−0.211609	+	0.977354i	$0.567870\pi$
$758$	11783.6	0.564645
$759$	1055.52	0.0504784
$760$	−7630.52	−0.364195
$761$	1939.62	0.0923932	0.0461966	−	0.998932i	$-0.485290\pi$
$761$	1939.62	0.0923932	0.0461966	+	0.998932i	$0.485290\pi$
$762$	47940.0	2.27911
$763$	0	0
$764$	12691.3	0.600986
$765$	36589.8	1.72929
$766$	27029.5	1.27495
$767$	−14217.9	−0.669331
$768$	2550.49	0.119835
$769$	−35373.4	−1.65877	−0.829387	−	0.558674i	$-0.811310\pi$
$769$	−35373.4	−1.65877	−0.829387	+	0.558674i	$0.811310\pi$
$770$	0	0
$771$	−49795.5	−2.32599
$772$	−11189.0	−0.521631
$773$	−7121.74	−0.331373	−0.165686	−	0.986178i	$-0.552984\pi$
$773$	−7121.74	−0.331373	−0.165686	+	0.986178i	$0.552984\pi$
$774$	−8786.37	−0.408036
$775$	−418.168	−0.0193820
$776$	−11822.7	−0.546920
$777$	0	0
$778$	83.0076	0.00382515
$779$	31179.9	1.43406
$780$	−21024.8	−0.965140
$781$	−2488.10	−0.113996
$782$	−1973.38	−0.0902404
$783$	37609.2	1.71653
$784$	0	0
$785$	18654.4	0.848158
$786$	−40792.8	−1.85118
$787$	12339.2	0.558888	0.279444	−	0.960162i	$-0.409850\pi$
$787$	12339.2	0.558888	0.279444	+	0.960162i	$0.409850\pi$
$788$	18400.4	0.831836
$789$	−55266.9	−2.49373
$790$	−5673.29	−0.255502
$791$	0	0
$792$	−2662.78	−0.119467
$793$	−11925.0	−0.534010
$794$	−12930.2	−0.577928
$795$	−76741.7	−3.42358
$796$	−7101.37	−0.316208
$797$	4533.69	0.201495	0.100748	−	0.994912i	$-0.467877\pi$
$797$	4533.69	0.201495	0.100748	+	0.994912i	$0.467877\pi$
$798$	0	0
$799$	−4859.20	−0.215152
$800$	−458.465	−0.0202615
$801$	26884.6	1.18592
$802$	4888.08	0.215217
$803$	−4559.30	−0.200366
$804$	38161.5	1.67395
$805$	0	0
$806$	−2609.13	−0.114023
$807$	52590.2	2.29401
$808$	12198.6	0.531119
$809$	31358.8	1.36281	0.681407	−	0.731905i	$-0.261369\pi$
$809$	31358.8	1.36281	0.681407	+	0.731905i	$0.261369\pi$
$810$	−59993.7	−2.60243
$811$	−41598.6	−1.80114	−0.900571	−	0.434709i	$-0.856851\pi$
$811$	−41598.6	−1.80114	−0.900571	+	0.434709i	$0.856851\pi$
$812$	0	0
$813$	−48507.2	−2.09252
$814$	108.417	0.00466834
$815$	15755.9	0.677184
$816$	6838.44	0.293374
$817$	4912.89	0.210380
$818$	−13817.4	−0.590604
$819$	0	0
$820$	18218.2	0.775864
$821$	−34783.4	−1.47862	−0.739311	−	0.673365i	$-0.764848\pi$
$821$	−34783.4	−1.47862	−0.739311	+	0.673365i	$0.764848\pi$
$822$	3470.07	0.147242
$823$	−37165.3	−1.57412	−0.787060	−	0.616876i	$-0.788398\pi$
$823$	−37165.3	−1.57412	−0.787060	+	0.616876i	$0.788398\pi$
$824$	5085.23	0.214991
$825$	657.501	0.0277470
$826$	0	0
$827$	−28232.1	−1.18709	−0.593546	−	0.804800i	$-0.702273\pi$
$827$	−28232.1	−1.18709	−0.593546	+	0.804800i	$0.702273\pi$
$828$	6647.78	0.279017
$829$	−30435.1	−1.27510	−0.637549	−	0.770410i	$-0.720051\pi$
$829$	−30435.1	−1.27510	−0.637549	+	0.770410i	$0.720051\pi$
$830$	30249.1	1.26501
$831$	18194.6	0.759523
$832$	−2860.56	−0.119197
$833$	0	0
$834$	−13710.5	−0.569251
$835$	−7677.32	−0.318185
$836$	1488.89	0.0615961
$837$	−13160.7	−0.543489
$838$	−4444.61	−0.183218
$839$	45535.7	1.87374	0.936870	−	0.349679i	$-0.113709\pi$
$839$	45535.7	1.87374	0.936870	+	0.349679i	$0.113709\pi$
$840$	0	0
$841$	−17432.0	−0.714749
$842$	2029.79	0.0830772
$843$	−46135.1	−1.88491
$844$	17318.0	0.706292
$845$	−2351.89	−0.0957484
$846$	16369.3	0.665235
$847$	0	0
$848$	−10441.2	−0.422821
$849$	−62382.1	−2.52173
$850$	−1229.25	−0.0496034
$851$	−270.670	−0.0109030
$852$	−21525.6	−0.865556
$853$	11248.5	0.451515	0.225757	−	0.974184i	$-0.427514\pi$
$853$	11248.5	0.451515	0.225757	+	0.974184i	$0.427514\pi$
$854$	0	0
$855$	68921.2	2.75679
$856$	−14608.1	−0.583287
$857$	24226.4	0.965647	0.482823	−	0.875718i	$-0.339611\pi$
$857$	24226.4	0.965647	0.482823	+	0.875718i	$0.339611\pi$
$858$	4102.43	0.163234
$859$	24266.8	0.963879	0.481939	−	0.876205i	$-0.339933\pi$
$859$	24266.8	0.963879	0.481939	+	0.876205i	$0.339933\pi$
$860$	2870.57	0.113821
$861$	0	0
$862$	6700.27	0.264747
$863$	−9960.81	−0.392897	−0.196448	−	0.980514i	$-0.562941\pi$
$863$	−9960.81	−0.392897	−0.196448	+	0.980514i	$0.562941\pi$
$864$	−14428.9	−0.568151
$865$	−850.237	−0.0334207
$866$	−2409.02	−0.0945287
$867$	−30612.1	−1.19912
$868$	0	0
$869$	1106.99	0.0432130
$870$	−19617.4	−0.764474
$871$	−42800.9	−1.66504
$872$	−16864.2	−0.654923
$873$	106786.	4.13994
$874$	−3717.10	−0.143859
$875$	0	0
$876$	−39444.4	−1.52135
$877$	−22575.7	−0.869243	−0.434622	−	0.900613i	$-0.643118\pi$
$877$	−22575.7	−0.869243	−0.434622	+	0.900613i	$0.643118\pi$
$878$	−10452.4	−0.401768
$879$	−20165.2	−0.773781
$880$	869.950	0.0333250
$881$	−5032.20	−0.192440	−0.0962198	−	0.995360i	$-0.530675\pi$
$881$	−5032.20	−0.192440	−0.0962198	+	0.995360i	$0.530675\pi$
$882$	0	0
$883$	44961.8	1.71357	0.856787	−	0.515671i	$-0.172457\pi$
$883$	44961.8	1.71357	0.856787	+	0.515671i	$0.172457\pi$
$884$	−7669.80	−0.291814
$885$	37408.1	1.42086
$886$	−9717.03	−0.368454
$887$	12229.9	0.462954	0.231477	−	0.972840i	$-0.425644\pi$
$887$	12229.9	0.462954	0.231477	+	0.972840i	$0.425644\pi$
$888$	937.964	0.0354460
$889$	0	0
$890$	−8783.40	−0.330809
$891$	11706.2	0.440147
$892$	−9829.58	−0.368967
$893$	−9152.88	−0.342989
$894$	42408.6	1.58653
$895$	25847.0	0.965330
$896$	0	0
$897$	−10241.9	−0.381236
$898$	−23594.9	−0.876805
$899$	−2434.47	−0.0903162
$900$	4141.00	0.153370
$901$	−27995.2	−1.03513
$902$	−3554.80	−0.131222
$903$	0	0
$904$	−2742.29	−0.100893
$905$	14129.5	0.518985
$906$	−62676.8	−2.29834
$907$	−14640.9	−0.535991	−0.267995	−	0.963420i	$-0.586361\pi$
$907$	−14640.9	−0.535991	−0.267995	+	0.963420i	$0.586361\pi$
$908$	−2948.13	−0.107750
$909$	−110181.	−4.02033
$910$	0	0
$911$	7735.65	0.281332	0.140666	−	0.990057i	$-0.455076\pi$
$911$	7735.65	0.281332	0.140666	+	0.990057i	$0.455076\pi$
$912$	12881.0	0.467690
$913$	−5902.30	−0.213951
$914$	−4635.60	−0.167759
$915$	31375.5	1.13360
$916$	−4468.24	−0.161173
$917$	0	0
$918$	−38687.2	−1.39092
$919$	20281.0	0.727973	0.363986	−	0.931404i	$-0.381415\pi$
$919$	20281.0	0.727973	0.363986	+	0.931404i	$0.381415\pi$
$920$	−2171.88	−0.0778312
$921$	−76749.8	−2.74592
$922$	−13211.8	−0.471915
$923$	24142.4	0.860951
$924$	0	0
$925$	−168.604	−0.00599316
$926$	−27488.9	−0.975530
$927$	−45931.4	−1.62738
$928$	−2669.07	−0.0944144
$929$	−10956.5	−0.386946	−0.193473	−	0.981106i	$-0.561975\pi$
$929$	−10956.5	−0.386946	−0.193473	+	0.981106i	$0.561975\pi$
$930$	6864.78	0.242048
$931$	0	0
$932$	−6643.16	−0.233481
$933$	28477.3	0.999256
$934$	23126.3	0.810187
$935$	2332.53	0.0815850
$936$	25837.4	0.902268
$937$	1853.60	0.0646258	0.0323129	−	0.999478i	$-0.489713\pi$
$937$	1853.60	0.0646258	0.0323129	+	0.999478i	$0.489713\pi$
$938$	0	0
$939$	−7463.44	−0.259383
$940$	−5347.97	−0.185566
$941$	−5012.93	−0.173663	−0.0868315	−	0.996223i	$-0.527674\pi$
$941$	−5012.93	−0.173663	−0.0868315	+	0.996223i	$0.527674\pi$
$942$	−31490.4	−1.08918
$943$	8874.75	0.306471
$944$	5089.60	0.175479
$945$	0	0
$946$	−560.115	−0.0192504
$947$	−17071.1	−0.585783	−0.292892	−	0.956146i	$-0.594618\pi$
$947$	−17071.1	−0.585783	−0.292892	+	0.956146i	$0.594618\pi$
$948$	9577.04	0.328110
$949$	44239.7	1.51326
$950$	−2315.43	−0.0790764
$951$	72893.8	2.48554
$952$	0	0
$953$	−41078.3	−1.39628	−0.698141	−	0.715960i	$-0.745989\pi$
$953$	−41078.3	−1.39628	−0.698141	+	0.715960i	$0.745989\pi$
$954$	94308.1	3.20056
$955$	37450.9	1.26899
$956$	−25436.6	−0.860543
$957$	3827.81	0.129295
$958$	−35181.7	−1.18650
$959$	0	0
$960$	7526.30	0.253032
$961$	−28939.1	−0.971404
$962$	−1051.99	−0.0352574
$963$	131945.	4.41522
$964$	−21302.5	−0.711731
$965$	−33017.7	−1.10143
$966$	0	0
$967$	−52684.5	−1.75204	−0.876018	−	0.482279i	$-0.839809\pi$
$967$	−52684.5	−1.75204	−0.876018	+	0.482279i	$0.839809\pi$
$968$	10478.3	0.347917
$969$	34536.9	1.14498
$970$	−34887.8	−1.15483
$971$	−40592.2	−1.34157	−0.670786	−	0.741651i	$-0.734043\pi$
$971$	−40592.2	−1.34157	−0.670786	+	0.741651i	$0.734043\pi$
$972$	52577.3	1.73500
$973$	0	0
$974$	16758.5	0.551312
$975$	−6379.85	−0.209558
$976$	4268.83	0.140002
$977$	14073.1	0.460837	0.230419	−	0.973092i	$-0.425990\pi$
$977$	14073.1	0.460837	0.230419	+	0.973092i	$0.425990\pi$
$978$	−26597.4	−0.869623
$979$	1713.85	0.0559497
$980$	0	0
$981$	152322.	4.95747
$982$	30870.3	1.00317
$983$	15345.0	0.497893	0.248946	−	0.968517i	$-0.419916\pi$
$983$	15345.0	0.497893	0.248946	+	0.968517i	$0.419916\pi$
$984$	−30754.1	−0.996345
$985$	54298.1	1.75643
$986$	−7156.39	−0.231142
$987$	0	0
$988$	−14447.0	−0.465202
$989$	1398.36	0.0449598
$990$	−7857.66	−0.252255
$991$	−17131.1	−0.549130	−0.274565	−	0.961568i	$-0.588534\pi$
$991$	−17131.1	−0.549130	−0.274565	+	0.961568i	$0.588534\pi$
$992$	933.996	0.0298935
$993$	30338.3	0.969542
$994$	0	0
$995$	−20955.6	−0.667675
$996$	−51063.3	−1.62450
$997$	9120.39	0.289715	0.144857	−	0.989453i	$-0.453728\pi$
$997$	9120.39	0.289715	0.144857	+	0.989453i	$0.453728\pi$
$998$	−24605.9	−0.780447
$999$	−5306.36	−0.168054

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.u.1.11		11
7.3	odd	6		322.4.e.d.93.11	✓	22
7.5	odd	6		322.4.e.d.277.11	yes	22
7.6	odd	2		2254.4.a.r.1.1		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.4.e.d.93.11	✓	22	7.3	odd	6
322.4.e.d.277.11	yes	22	7.5	odd	6
2254.4.a.r.1.1		11	7.6	odd	2
2254.4.a.u.1.11		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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +9.96286 q^{3} +4.00000 q^{4} +11.8037 q^{5} -19.9257 q^{6} -8.00000 q^{8} +72.2585 q^{9} -23.6074 q^{10} +4.60635 q^{11} +39.8514 q^{12} -44.6962 q^{13} +117.598 q^{15} +16.0000 q^{16} +42.8996 q^{17} -144.517 q^{18} +80.8065 q^{19} +47.2147 q^{20} -9.21269 q^{22} +23.0000 q^{23} -79.7029 q^{24} +14.3270 q^{25} +89.3924 q^{26} +450.904 q^{27} +83.4085 q^{29} -235.197 q^{30} -29.1874 q^{31} -32.0000 q^{32} +45.8924 q^{33} -85.7992 q^{34} +289.034 q^{36} -11.7683 q^{37} -161.613 q^{38} -445.302 q^{39} -94.4295 q^{40} +385.859 q^{41} +60.7982 q^{43} +18.4254 q^{44} +852.917 q^{45} -46.0000 q^{46} -113.269 q^{47} +159.406 q^{48} -28.6541 q^{50} +427.403 q^{51} -178.785 q^{52} -652.574 q^{53} -901.808 q^{54} +54.3719 q^{55} +805.063 q^{57} -166.817 q^{58} +318.100 q^{59} +470.394 q^{60} +266.802 q^{61} +58.3747 q^{62} +64.0000 q^{64} -527.580 q^{65} -91.7847 q^{66} +957.595 q^{67} +171.598 q^{68} +229.146 q^{69} -540.145 q^{71} -578.068 q^{72} -989.786 q^{73} +23.5365 q^{74} +142.738 q^{75} +323.226 q^{76} +890.604 q^{78} +240.319 q^{79} +188.859 q^{80} +2541.31 q^{81} -771.718 q^{82} -1281.34 q^{83} +506.374 q^{85} -121.596 q^{86} +830.987 q^{87} -36.8508 q^{88} +372.062 q^{89} -1705.83 q^{90} +92.0000 q^{92} -290.789 q^{93} +226.538 q^{94} +953.815 q^{95} -318.811 q^{96} +1477.84 q^{97} +332.848 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 22 q^{2} + 44 q^{4} + 23 q^{5} - 88 q^{8} + 171 q^{9} - 46 q^{10} - 48 q^{11} + 77 q^{13} + 104 q^{15} + 176 q^{16} + 97 q^{17} - 342 q^{18} + 138 q^{19} + 92 q^{20} + 96 q^{22} + 253 q^{23} + 30 q^{25}+ \cdots - 2545 q^{99}+O(q^{100}) \) 11 * q - 22 * q^2 + 44 * q^4 + 23 * q^5 - 88 * q^8 + 171 * q^9 - 46 * q^10 - 48 * q^11 + 77 * q^13 + 104 * q^15 + 176 * q^16 + 97 * q^17 - 342 * q^18 + 138 * q^19 + 92 * q^20 + 96 * q^22 + 253 * q^23 + 30 * q^25 - 154 * q^26 + 315 * q^27 + 137 * q^29 - 208 * q^30 + 234 * q^31 - 352 * q^32 + 28 * q^33 - 194 * q^34 + 684 * q^36 + 87 * q^37 - 276 * q^38 + 129 * q^39 - 184 * q^40 + 612 * q^41 + 187 * q^43 - 192 * q^44 + 597 * q^45 - 506 * q^46 + 793 * q^47 - 60 * q^50 - 1257 * q^51 + 308 * q^52 + 575 * q^53 - 630 * q^54 - 549 * q^55 + 39 * q^57 - 274 * q^58 + 1241 * q^59 + 416 * q^60 + 187 * q^61 - 468 * q^62 + 704 * q^64 - 937 * q^65 - 56 * q^66 + 2275 * q^67 + 388 * q^68 + 115 * q^71 - 1368 * q^72 + 2385 * q^73 - 174 * q^74 + 196 * q^75 + 552 * q^76 - 258 * q^78 - 1225 * q^79 + 368 * q^80 + 3635 * q^81 - 1224 * q^82 - 597 * q^83 - 2493 * q^85 - 374 * q^86 + 1428 * q^87 + 384 * q^88 + 602 * q^89 - 1194 * q^90 + 1012 * q^92 - 1360 * q^93 - 1586 * q^94 + 4835 * q^95 - 1039 * q^97 - 2545 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more