LMFDB - Embedded newform 2254.4.a.x.1.6

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} - 5 x^{10} - 165 x^{9} + 798 x^{8} + 8769 x^{7} - 38472 x^{6} - 184213 x^{5} + 644009 x^{4} + \cdots + 2848203 $ x^11 - 5x^10 - 165x^9 + 798x^8 + 8769x^7 - 38472x^6 - 184213x^5 + 644009x^4 + 1428336x^3 - 2764269x^2 - 1864863x + 2848203
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 322)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.05420$ 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)

$f(q)$	$=$	\(q+2.00000 q^{2} -0.0542033 q^{3} +4.00000 q^{4} -13.0870 q^{5} -0.108407 q^{6} +8.00000 q^{8} -26.9971 q^{9} -26.1741 q^{10} -31.4054 q^{11} -0.216813 q^{12} -19.0210 q^{13} +0.709361 q^{15} +16.0000 q^{16} +73.1836 q^{17} -53.9941 q^{18} -55.6496 q^{19} -52.3481 q^{20} -62.8108 q^{22} -23.0000 q^{23} -0.433627 q^{24} +46.2704 q^{25} -38.0421 q^{26} +2.92682 q^{27} -193.443 q^{29} +1.41872 q^{30} +14.1796 q^{31} +32.0000 q^{32} +1.70228 q^{33} +146.367 q^{34} -107.988 q^{36} +343.112 q^{37} -111.299 q^{38} +1.03100 q^{39} -104.696 q^{40} -289.370 q^{41} -372.570 q^{43} -125.622 q^{44} +353.311 q^{45} -46.0000 q^{46} +33.2197 q^{47} -0.867254 q^{48} +92.5408 q^{50} -3.96679 q^{51} -76.0841 q^{52} -459.874 q^{53} +5.85364 q^{54} +411.003 q^{55} +3.01640 q^{57} -386.887 q^{58} +749.606 q^{59} +2.83744 q^{60} -4.40654 q^{61} +28.3593 q^{62} +64.0000 q^{64} +248.929 q^{65} +3.40455 q^{66} -283.350 q^{67} +292.734 q^{68} +1.24668 q^{69} -487.770 q^{71} -215.976 q^{72} +385.512 q^{73} +686.225 q^{74} -2.50801 q^{75} -222.599 q^{76} +2.06201 q^{78} +706.336 q^{79} -209.392 q^{80} +728.762 q^{81} -578.740 q^{82} +1253.54 q^{83} -957.756 q^{85} -745.140 q^{86} +10.4853 q^{87} -251.243 q^{88} +13.9601 q^{89} +706.623 q^{90} -92.0000 q^{92} -0.768584 q^{93} +66.4394 q^{94} +728.289 q^{95} -1.73451 q^{96} -256.212 q^{97} +847.853 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 22 q^{2} + 6 q^{3} + 44 q^{4} + 27 q^{5} + 12 q^{6} + 88 q^{8} + 59 q^{9} + 54 q^{10} + 56 q^{11} + 24 q^{12} + 103 q^{13} + 62 q^{15} + 176 q^{16} + 157 q^{17} + 118 q^{18} + 266 q^{19} + 108 q^{20}+ \cdots + 101 q^{99}+O(q^{100}) $ 11 * q + 22 * q^2 + 6 * q^3 + 44 * q^4 + 27 * q^5 + 12 * q^6 + 88 * q^8 + 59 * q^9 + 54 * q^10 + 56 * q^11 + 24 * q^12 + 103 * q^13 + 62 * q^15 + 176 * q^16 + 157 * q^17 + 118 * q^18 + 266 * q^19 + 108 * q^20 + 112 * q^22 - 253 * q^23 + 48 * q^24 + 142 * q^25 + 206 * q^26 + 531 * q^27 - 195 * q^29 + 124 * q^30 + 316 * q^31 + 352 * q^32 + 682 * q^33 + 314 * q^34 + 236 * q^36 + 347 * q^37 + 532 * q^38 - 99 * q^39 + 216 * q^40 + 540 * q^41 + 193 * q^43 + 224 * q^44 + 609 * q^45 - 506 * q^46 + 731 * q^47 + 96 * q^48 + 284 * q^50 - 303 * q^51 + 412 * q^52 + 413 * q^53 + 1062 * q^54 + 1809 * q^55 + 343 * q^57 - 390 * q^58 + 1061 * q^59 + 248 * q^60 + 1157 * q^61 + 632 * q^62 + 704 * q^64 - 37 * q^65 + 1364 * q^66 - 841 * q^67 + 628 * q^68 - 138 * q^69 + 869 * q^71 + 472 * q^72 + 409 * q^73 + 694 * q^74 + 2024 * q^75 + 1064 * q^76 - 198 * q^78 + 247 * q^79 + 432 * q^80 + 1019 * q^81 + 1080 * q^82 + 2863 * q^83 + 785 * q^85 + 386 * q^86 + 670 * q^87 + 448 * q^88 + 1610 * q^89 + 1218 * q^90 - 1012 * q^92 - 1290 * q^93 + 1462 * q^94 - 209 * q^95 + 192 * q^96 + 1783 * q^97 + 101 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	−0.0542033	−0.0104314	−0.00521572	−	0.999986i	$-0.501660\pi$
$3$	−0.0542033	−0.0104314	−0.00521572	+	0.999986i	$0.501660\pi$
$4$	4.00000	0.500000
$5$	−13.0870	−1.17054	−0.585270	−	0.810839i	$-0.699011\pi$
$5$	−13.0870	−1.17054	−0.585270	+	0.810839i	$0.699011\pi$
$6$	−0.108407	−0.00737614
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	−26.9971	−0.999891
$10$	−26.1741	−0.827697
$11$	−31.4054	−0.860825	−0.430413	−	0.902632i	$-0.641632\pi$
$11$	−31.4054	−0.860825	−0.430413	+	0.902632i	$0.641632\pi$
$12$	−0.216813	−0.00521572
$13$	−19.0210	−0.405807	−0.202903	−	0.979199i	$-0.565038\pi$
$13$	−19.0210	−0.405807	−0.202903	+	0.979199i	$0.565038\pi$
$14$	0	0
$15$	0.709361	0.0122104
$16$	16.0000	0.250000
$17$	73.1836	1.04410	0.522048	−	0.852916i	$-0.325168\pi$
$17$	73.1836	1.04410	0.522048	+	0.852916i	$0.325168\pi$
$18$	−53.9941	−0.707030
$19$	−55.6496	−0.671942	−0.335971	−	0.941872i	$-0.609064\pi$
$19$	−55.6496	−0.671942	−0.335971	+	0.941872i	$0.609064\pi$
$20$	−52.3481	−0.585270
$21$	0	0
$22$	−62.8108	−0.608695
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	−0.433627	−0.00368807
$25$	46.2704	0.370163
$26$	−38.0421	−0.286949
$27$	2.92682	0.0208617
$28$	0	0
$29$	−193.443	−1.23867	−0.619337	−	0.785125i	$-0.712599\pi$
$29$	−193.443	−1.23867	−0.619337	+	0.785125i	$0.712599\pi$
$30$	1.41872	0.00863407
$31$	14.1796	0.0821528	0.0410764	−	0.999156i	$-0.486921\pi$
$31$	14.1796	0.0821528	0.0410764	+	0.999156i	$0.486921\pi$
$32$	32.0000	0.176777
$33$	1.70228	0.00897965
$34$	146.367	0.738287
$35$	0	0
$36$	−107.988	−0.499946
$37$	343.112	1.52452	0.762261	−	0.647269i	$-0.224089\pi$
$37$	343.112	1.52452	0.762261	+	0.647269i	$0.224089\pi$
$38$	−111.299	−0.475135
$39$	1.03100	0.00423315
$40$	−104.696	−0.413848
$41$	−289.370	−1.10224	−0.551122	−	0.834425i	$-0.685800\pi$
$41$	−289.370	−1.10224	−0.551122	+	0.834425i	$0.685800\pi$
$42$	0	0
$43$	−372.570	−1.32131	−0.660656	−	0.750689i	$-0.729722\pi$
$43$	−372.570	−1.32131	−0.660656	+	0.750689i	$0.729722\pi$
$44$	−125.622	−0.430413
$45$	353.311	1.17041
$46$	−46.0000	−0.147442
$47$	33.2197	0.103098	0.0515489	−	0.998670i	$-0.483584\pi$
$47$	33.2197	0.103098	0.0515489	+	0.998670i	$0.483584\pi$
$48$	−0.867254	−0.00260786
$49$	0	0
$50$	92.5408	0.261745
$51$	−3.96679	−0.0108914
$52$	−76.0841	−0.202903
$53$	−459.874	−1.19186	−0.595929	−	0.803037i	$-0.703216\pi$
$53$	−459.874	−1.19186	−0.595929	+	0.803037i	$0.703216\pi$
$54$	5.85364	0.0147515
$55$	411.003	1.00763
$56$	0	0
$57$	3.01640	0.00700933
$58$	−386.887	−0.875875
$59$	749.606	1.65407	0.827037	−	0.562148i	$-0.190025\pi$
$59$	749.606	1.65407	0.827037	+	0.562148i	$0.190025\pi$
$60$	2.83744	0.00610521
$61$	−4.40654	−0.00924918	−0.00462459	−	0.999989i	$-0.501472\pi$
$61$	−4.40654	−0.00924918	−0.00462459	+	0.999989i	$0.501472\pi$
$62$	28.3593	0.0580908
$63$	0	0
$64$	64.0000	0.125000
$65$	248.929	0.475013
$66$	3.40455	0.00634957
$67$	−283.350	−0.516667	−0.258334	−	0.966056i	$-0.583173\pi$
$67$	−283.350	−0.516667	−0.258334	+	0.966056i	$0.583173\pi$
$68$	292.734	0.522048
$69$	1.24668	0.00217511
$70$	0	0
$71$	−487.770	−0.815319	−0.407659	−	0.913134i	$-0.633655\pi$
$71$	−487.770	−0.815319	−0.407659	+	0.913134i	$0.633655\pi$
$72$	−215.976	−0.353515
$73$	385.512	0.618093	0.309046	−	0.951047i	$-0.399990\pi$
$73$	385.512	0.618093	0.309046	+	0.951047i	$0.399990\pi$
$74$	686.225	1.07800
$75$	−2.50801	−0.00386133
$76$	−222.599	−0.335971
$77$	0	0
$78$	2.06201	0.00299329
$79$	706.336	1.00594	0.502969	−	0.864305i	$-0.332241\pi$
$79$	706.336	1.00594	0.502969	+	0.864305i	$0.332241\pi$
$80$	−209.392	−0.292635
$81$	728.762	0.999674
$82$	−578.740	−0.779404
$83$	1253.54	1.65775	0.828877	−	0.559431i	$-0.188980\pi$
$83$	1253.54	1.65775	0.828877	+	0.559431i	$0.188980\pi$
$84$	0	0
$85$	−957.756	−1.22216
$86$	−745.140	−0.934309
$87$	10.4853	0.0129212
$88$	−251.243	−0.304348
$89$	13.9601	0.0166266	0.00831331	−	0.999965i	$-0.497354\pi$
$89$	13.9601	0.0166266	0.00831331	+	0.999965i	$0.497354\pi$
$90$	706.623	0.827606
$91$	0	0
$92$	−92.0000	−0.104257
$93$	−0.768584	−0.000856972 0
$94$	66.4394	0.0729011
$95$	728.289	0.786535
$96$	−1.73451	−0.00184404
$97$	−256.212	−0.268190	−0.134095	−	0.990969i	$-0.542813\pi$
$97$	−256.212	−0.268190	−0.134095	+	0.990969i	$0.542813\pi$
$98$	0	0
$99$	847.853	0.860732
$100$	185.082	0.185082
$101$	1627.32	1.60321	0.801606	−	0.597853i	$-0.203979\pi$
$101$	1627.32	1.60321	0.801606	+	0.597853i	$0.203979\pi$
$102$	−7.93359	−0.00770140
$103$	−57.6098	−0.0551113	−0.0275557	−	0.999620i	$-0.508772\pi$
$103$	−57.6098	−0.0551113	−0.0275557	+	0.999620i	$0.508772\pi$
$104$	−152.168	−0.143474
$105$	0	0
$106$	−919.747	−0.842771
$107$	−752.393	−0.679781	−0.339890	−	0.940465i	$-0.610390\pi$
$107$	−752.393	−0.679781	−0.339890	+	0.940465i	$0.610390\pi$
$108$	11.7073	0.0104309
$109$	−1403.77	−1.23355	−0.616776	−	0.787139i	$-0.711562\pi$
$109$	−1403.77	−1.23355	−0.616776	+	0.787139i	$0.711562\pi$
$110$	822.006	0.712502
$111$	−18.5978	−0.0159030
$112$	0	0
$113$	1791.53	1.49144	0.745721	−	0.666258i	$-0.232105\pi$
$113$	1791.53	1.49144	0.745721	+	0.666258i	$0.232105\pi$
$114$	6.03279	0.00495634
$115$	301.002	0.244074
$116$	−773.774	−0.619337
$117$	513.512	0.405762
$118$	1499.21	1.16961
$119$	0	0
$120$	5.67489	0.00431703
$121$	−344.702	−0.258980
$122$	−8.81309	−0.00654016
$123$	15.6848	0.0114980
$124$	56.7185	0.0410764
$125$	1030.34	0.737249
$126$	0	0
$127$	2170.48	1.51653	0.758264	−	0.651947i	$-0.226048\pi$
$127$	2170.48	1.51653	0.758264	+	0.651947i	$0.226048\pi$
$128$	128.000	0.0883883
$129$	20.1945	0.0137832
$130$	497.858	0.335885
$131$	2309.83	1.54054	0.770269	−	0.637720i	$-0.220122\pi$
$131$	2309.83	1.54054	0.770269	+	0.637720i	$0.220122\pi$
$132$	6.80911	0.00448982
$133$	0	0
$134$	−566.700	−0.365339
$135$	−38.3034	−0.0244195
$136$	585.469	0.369144
$137$	−1606.60	−1.00191	−0.500954	−	0.865474i	$-0.667017\pi$
$137$	−1606.60	−1.00191	−0.500954	+	0.865474i	$0.667017\pi$
$138$	2.49335	0.00153803
$139$	−840.040	−0.512599	−0.256300	−	0.966597i	$-0.582503\pi$
$139$	−840.040	−0.512599	−0.256300	+	0.966597i	$0.582503\pi$
$140$	0	0
$141$	−1.80062	−0.00107546
$142$	−975.540	−0.576517
$143$	597.363	0.349329
$144$	−431.953	−0.249973
$145$	2531.60	1.44992
$146$	771.024	0.437058
$147$	0	0
$148$	1372.45	0.762261
$149$	3553.48	1.95377	0.976886	−	0.213759i	$-0.0685706\pi$
$149$	3553.48	1.95377	0.976886	+	0.213759i	$0.0685706\pi$
$150$	−5.01602	−0.00273037
$151$	1201.08	0.647303	0.323651	−	0.946176i	$-0.395090\pi$
$151$	1201.08	0.647303	0.323651	+	0.946176i	$0.395090\pi$
$152$	−445.197	−0.237567
$153$	−1975.74	−1.04398
$154$	0	0
$155$	−185.569	−0.0961631
$156$	4.12401	0.00211657
$157$	−787.465	−0.400297	−0.200148	−	0.979766i	$-0.564142\pi$
$157$	−787.465	−0.400297	−0.200148	+	0.979766i	$0.564142\pi$
$158$	1412.67	0.711305
$159$	24.9267	0.0124328
$160$	−418.785	−0.206924
$161$	0	0
$162$	1457.52	0.706876
$163$	1779.91	0.855297	0.427649	−	0.903945i	$-0.359342\pi$
$163$	1779.91	0.855297	0.427649	+	0.903945i	$0.359342\pi$
$164$	−1157.48	−0.551122
$165$	−22.2777	−0.0105110
$166$	2507.07	1.17221
$167$	−1493.18	−0.691892	−0.345946	−	0.938254i	$-0.612442\pi$
$167$	−1493.18	−0.691892	−0.345946	+	0.938254i	$0.612442\pi$
$168$	0	0
$169$	−1835.20	−0.835321
$170$	−1915.51	−0.864194
$171$	1502.38	0.671869
$172$	−1490.28	−0.660656
$173$	3992.74	1.75470	0.877349	−	0.479854i	$-0.159310\pi$
$173$	3992.74	1.75470	0.877349	+	0.479854i	$0.159310\pi$
$174$	20.9706	0.00913664
$175$	0	0
$176$	−502.486	−0.215206
$177$	−40.6311	−0.0172544
$178$	27.9202	0.0117568
$179$	102.217	0.0426818	0.0213409	−	0.999772i	$-0.493206\pi$
$179$	102.217	0.0426818	0.0213409	+	0.999772i	$0.493206\pi$
$180$	1413.25	0.585206
$181$	2286.03	0.938781	0.469391	−	0.882991i	$-0.344474\pi$
$181$	2286.03	0.938781	0.469391	+	0.882991i	$0.344474\pi$
$182$	0	0
$183$	0.238849	9.64823e−5 0
$184$	−184.000	−0.0737210
$185$	−4490.32	−1.78451
$186$	−1.53717	−0.000605971 0
$187$	−2298.36	−0.898784
$188$	132.879	0.0515489
$189$	0	0
$190$	1456.58	0.556164
$191$	−3303.96	−1.25165	−0.625827	−	0.779962i	$-0.715239\pi$
$191$	−3303.96	−1.25165	−0.625827	+	0.779962i	$0.715239\pi$
$192$	−3.46901	−0.00130393
$193$	1552.55	0.579041	0.289520	−	0.957172i	$-0.406504\pi$
$193$	1552.55	0.579041	0.289520	+	0.957172i	$0.406504\pi$
$194$	−512.424	−0.189639
$195$	−13.4928	−0.00495507
$196$	0	0
$197$	−4545.08	−1.64378	−0.821888	−	0.569649i	$-0.807079\pi$
$197$	−4545.08	−1.64378	−0.821888	+	0.569649i	$0.807079\pi$
$198$	1695.71	0.608629
$199$	−3538.22	−1.26039	−0.630195	−	0.776437i	$-0.717025\pi$
$199$	−3538.22	−1.26039	−0.630195	+	0.776437i	$0.717025\pi$
$200$	370.163	0.130872
$201$	15.3585	0.00538958
$202$	3254.64	1.13364
$203$	0	0
$204$	−15.8672	−0.00544571
$205$	3786.99	1.29022
$206$	−115.220	−0.0389696
$207$	620.932	0.208492
$208$	−304.337	−0.101452
$209$	1747.70	0.578425
$210$	0	0
$211$	1930.46	0.629851	0.314925	−	0.949116i	$-0.398020\pi$
$211$	1930.46	0.629851	0.314925	+	0.949116i	$0.398020\pi$
$212$	−1839.49	−0.595929
$213$	26.4388	0.00850495
$214$	−1504.79	−0.480678
$215$	4875.84	1.54665
$216$	23.4146	0.00737574
$217$	0	0
$218$	−2807.55	−0.872253
$219$	−20.8960	−0.00644760
$220$	1644.01	0.503815
$221$	−1392.03	−0.423701
$222$	−37.1957	−0.0112451
$223$	−3600.95	−1.08133	−0.540667	−	0.841236i	$-0.681828\pi$
$223$	−3600.95	−1.08133	−0.540667	+	0.841236i	$0.681828\pi$
$224$	0	0
$225$	−1249.16	−0.370123
$226$	3583.06	1.05461
$227$	3762.62	1.10015	0.550075	−	0.835115i	$-0.314599\pi$
$227$	3762.62	1.10015	0.550075	+	0.835115i	$0.314599\pi$
$228$	12.0656	0.00350466
$229$	−1796.42	−0.518388	−0.259194	−	0.965825i	$-0.583457\pi$
$229$	−1796.42	−0.518388	−0.259194	+	0.965825i	$0.583457\pi$
$230$	602.003	0.172587
$231$	0	0
$232$	−1547.55	−0.437937
$233$	4300.36	1.20912	0.604562	−	0.796558i	$-0.293348\pi$
$233$	4300.36	1.20912	0.604562	+	0.796558i	$0.293348\pi$
$234$	1027.02	0.286917
$235$	−434.747	−0.120680
$236$	2998.42	0.827037
$237$	−38.2858	−0.0104934
$238$	0	0
$239$	−3512.97	−0.950775	−0.475388	−	0.879776i	$-0.657692\pi$
$239$	−3512.97	−0.950775	−0.475388	+	0.879776i	$0.657692\pi$
$240$	11.3498	0.00305260
$241$	−28.2210	−0.00754305	−0.00377152	−	0.999993i	$-0.501201\pi$
$241$	−28.2210	−0.00754305	−0.00377152	+	0.999993i	$0.501201\pi$
$242$	−689.404	−0.183126
$243$	−118.526	−0.0312898
$244$	−17.6262	−0.00462459
$245$	0	0
$246$	31.3696	0.00813030
$247$	1058.51	0.272679
$248$	113.437	0.0290454
$249$	−67.9459	−0.0172928
$250$	2060.67	0.521314
$251$	−2488.66	−0.625828	−0.312914	−	0.949781i	$-0.601305\pi$
$251$	−2488.66	−0.625828	−0.312914	+	0.949781i	$0.601305\pi$
$252$	0	0
$253$	722.324	0.179494
$254$	4340.96	1.07235
$255$	51.9136	0.0127488
$256$	256.000	0.0625000
$257$	−2331.33	−0.565854	−0.282927	−	0.959141i	$-0.591305\pi$
$257$	−2331.33	−0.565854	−0.282927	+	0.959141i	$0.591305\pi$
$258$	40.3891	0.00974618
$259$	0	0
$260$	995.715	0.237506
$261$	5222.41	1.23854
$262$	4619.65	1.08932
$263$	3559.90	0.834650	0.417325	−	0.908757i	$-0.362968\pi$
$263$	3559.90	0.834650	0.417325	+	0.908757i	$0.362968\pi$
$264$	13.6182	0.00317478
$265$	6018.38	1.39512
$266$	0	0
$267$	−0.756685	−0.000173440 0
$268$	−1133.40	−0.258334
$269$	−3397.67	−0.770110	−0.385055	−	0.922894i	$-0.625818\pi$
$269$	−3397.67	−0.770110	−0.385055	+	0.922894i	$0.625818\pi$
$270$	−76.6068	−0.0172672
$271$	4530.79	1.01559	0.507797	−	0.861477i	$-0.330460\pi$
$271$	4530.79	1.01559	0.507797	+	0.861477i	$0.330460\pi$
$272$	1170.94	0.261024
$273$	0	0
$274$	−3213.21	−0.708456
$275$	−1453.14	−0.318646
$276$	4.98671	0.00108755
$277$	4404.10	0.955295	0.477648	−	0.878552i	$-0.341490\pi$
$277$	4404.10	0.955295	0.477648	+	0.878552i	$0.341490\pi$
$278$	−1680.08	−0.362462
$279$	−382.808	−0.0821439
$280$	0	0
$281$	1754.73	0.372520	0.186260	−	0.982500i	$-0.440363\pi$
$281$	1754.73	0.372520	0.186260	+	0.982500i	$0.440363\pi$
$282$	−3.60124	−0.000760463 0
$283$	2164.05	0.454556	0.227278	−	0.973830i	$-0.427017\pi$
$283$	2164.05	0.454556	0.227278	+	0.973830i	$0.427017\pi$
$284$	−1951.08	−0.407659
$285$	−39.4757	−0.00820469
$286$	1194.73	0.247013
$287$	0	0
$288$	−863.906	−0.176757
$289$	442.836	0.0901356
$290$	5063.20	1.02525
$291$	13.8876	0.00279760
$292$	1542.05	0.309046
$293$	5199.98	1.03681	0.518407	−	0.855134i	$-0.326525\pi$
$293$	5199.98	1.03681	0.518407	+	0.855134i	$0.326525\pi$
$294$	0	0
$295$	−9810.11	−1.93616
$296$	2744.90	0.539000
$297$	−91.9179	−0.0179583
$298$	7106.95	1.38153
$299$	437.484	0.0846165
$300$	−10.0320	−0.00193067
$301$	0	0
$302$	2402.16	0.457712
$303$	−88.2062	−0.0167238
$304$	−890.394	−0.167986
$305$	57.6686	0.0108265
$306$	−3951.48	−0.738207
$307$	2568.71	0.477538	0.238769	−	0.971076i	$-0.423256\pi$
$307$	2568.71	0.477538	0.238769	+	0.971076i	$0.423256\pi$
$308$	0	0
$309$	3.12265	0.000574890 0
$310$	−371.139	−0.0679976
$311$	−2161.40	−0.394088	−0.197044	−	0.980395i	$-0.563134\pi$
$311$	−2161.40	−0.394088	−0.197044	+	0.980395i	$0.563134\pi$
$312$	8.24803	0.00149664
$313$	−610.293	−0.110210	−0.0551051	−	0.998481i	$-0.517549\pi$
$313$	−610.293	−0.110210	−0.0551051	+	0.998481i	$0.517549\pi$
$314$	−1574.93	−0.283052
$315$	0	0
$316$	2825.34	0.502969
$317$	1498.64	0.265527	0.132763	−	0.991148i	$-0.457615\pi$
$317$	1498.64	0.265527	0.132763	+	0.991148i	$0.457615\pi$
$318$	49.8534	0.00879132
$319$	6075.17	1.06628
$320$	−837.570	−0.146317
$321$	40.7822	0.00709109
$322$	0	0
$323$	−4072.64	−0.701572
$324$	2915.05	0.499837
$325$	−880.110	−0.150215
$326$	3559.82	0.604786
$327$	76.0893	0.0128677
$328$	−2314.96	−0.389702
$329$	0	0
$330$	−44.5555	−0.00743242
$331$	−1272.10	−0.211242	−0.105621	−	0.994406i	$-0.533683\pi$
$331$	−1272.10	−0.211242	−0.105621	+	0.994406i	$0.533683\pi$
$332$	5014.15	0.828877
$333$	−9263.03	−1.52436
$334$	−2986.37	−0.489242
$335$	3708.21	0.604780
$336$	0	0
$337$	3190.71	0.515754	0.257877	−	0.966178i	$-0.416977\pi$
$337$	3190.71	0.515754	0.257877	+	0.966178i	$0.416977\pi$
$338$	−3670.40	−0.590661
$339$	−97.1069	−0.0155579
$340$	−3831.02	−0.611078
$341$	−445.317	−0.0707192
$342$	3004.75	0.475083
$343$	0	0
$344$	−2980.56	−0.467154
$345$	−16.3153	−0.00254605
$346$	7985.49	1.24076
$347$	−1242.97	−0.192294	−0.0961472	−	0.995367i	$-0.530652\pi$
$347$	−1242.97	−0.192294	−0.0961472	+	0.995367i	$0.530652\pi$
$348$	41.9411	0.00646058
$349$	1839.96	0.282209	0.141105	−	0.989995i	$-0.454935\pi$
$349$	1839.96	0.282209	0.141105	+	0.989995i	$0.454935\pi$
$350$	0	0
$351$	−55.6712	−0.00846583
$352$	−1004.97	−0.152174
$353$	12687.6	1.91301	0.956505	−	0.291716i	$-0.0942262\pi$
$353$	12687.6	1.91301	0.956505	+	0.291716i	$0.0942262\pi$
$354$	−81.2623	−0.0122007
$355$	6383.46	0.954363
$356$	55.8405	0.00831331
$357$	0	0
$358$	204.433	0.0301806
$359$	−10645.9	−1.56510	−0.782548	−	0.622591i	$-0.786080\pi$
$359$	−10645.9	−1.56510	−0.782548	+	0.622591i	$0.786080\pi$
$360$	2826.49	0.413803
$361$	−3762.12	−0.548494
$362$	4572.06	0.663819
$363$	18.6840	0.00270153
$364$	0	0
$365$	−5045.21	−0.723502
$366$	0.477699	6.82233e−5 0
$367$	8787.22	1.24983	0.624917	−	0.780691i	$-0.285133\pi$
$367$	8787.22	1.24983	0.624917	+	0.780691i	$0.285133\pi$
$368$	−368.000	−0.0521286
$369$	7812.14	1.10212
$370$	−8980.65	−1.26184
$371$	0	0
$372$	−3.07433	−0.000428486 0
$373$	8283.03	1.14981	0.574905	−	0.818220i	$-0.305039\pi$
$373$	8283.03	1.14981	0.574905	+	0.818220i	$0.305039\pi$
$374$	−4596.72	−0.635536
$375$	−55.8477	−0.00769057
$376$	265.758	0.0364505
$377$	3679.50	0.502662
$378$	0	0
$379$	−5492.70	−0.744435	−0.372218	−	0.928146i	$-0.621402\pi$
$379$	−5492.70	−0.744435	−0.372218	+	0.928146i	$0.621402\pi$
$380$	2913.15	0.393268
$381$	−117.647	−0.0158196
$382$	−6607.92	−0.885053
$383$	8652.83	1.15441	0.577205	−	0.816599i	$-0.304143\pi$
$383$	8652.83	1.15441	0.577205	+	0.816599i	$0.304143\pi$
$384$	−6.93803	−0.000922018 0
$385$	0	0
$386$	3105.10	0.409444
$387$	10058.3	1.32117
$388$	−1024.85	−0.134095
$389$	949.379	0.123741	0.0618707	−	0.998084i	$-0.480293\pi$
$389$	949.379	0.123741	0.0618707	+	0.998084i	$0.480293\pi$
$390$	−26.9856	−0.00350376
$391$	−1683.22	−0.217709
$392$	0	0
$393$	−125.200	−0.0160700
$394$	−9090.17	−1.16232
$395$	−9243.84	−1.17749
$396$	3391.41	0.430366
$397$	−3126.90	−0.395302	−0.197651	−	0.980272i	$-0.563331\pi$
$397$	−3126.90	−0.395302	−0.197651	+	0.980272i	$0.563331\pi$
$398$	−7076.43	−0.891230
$399$	0	0
$400$	740.326	0.0925408
$401$	−1675.24	−0.208622	−0.104311	−	0.994545i	$-0.533264\pi$
$401$	−1675.24	−0.208622	−0.104311	+	0.994545i	$0.533264\pi$
$402$	30.7170	0.00381101
$403$	−269.711	−0.0333382
$404$	6509.28	0.801606
$405$	−9537.33	−1.17016
$406$	0	0
$407$	−10775.6	−1.31235
$408$	−31.7344	−0.00385070
$409$	−8951.41	−1.08220	−0.541099	−	0.840959i	$-0.681991\pi$
$409$	−8951.41	−1.08220	−0.541099	+	0.840959i	$0.681991\pi$
$410$	7573.98	0.912323
$411$	87.0833	0.0104513
$412$	−230.439	−0.0275557
$413$	0	0
$414$	1241.86	0.147426
$415$	−16405.1	−1.94047
$416$	−608.673	−0.0717371
$417$	45.5330	0.00534715
$418$	3495.40	0.409008
$419$	−3811.27	−0.444375	−0.222187	−	0.975004i	$-0.571320\pi$
$419$	−3811.27	−0.444375	−0.222187	+	0.975004i	$0.571320\pi$
$420$	0	0
$421$	−3830.34	−0.443419	−0.221710	−	0.975113i	$-0.571164\pi$
$421$	−3830.34	−0.443419	−0.221710	+	0.975113i	$0.571164\pi$
$422$	3860.93	0.445372
$423$	−896.835	−0.103086
$424$	−3678.99	−0.421386
$425$	3386.23	0.386486
$426$	52.8775	0.00601391
$427$	0	0
$428$	−3009.57	−0.339890
$429$	−32.3791	−0.00364400
$430$	9751.67	1.09365
$431$	8756.31	0.978600	0.489300	−	0.872116i	$-0.337252\pi$
$431$	8756.31	0.978600	0.489300	+	0.872116i	$0.337252\pi$
$432$	46.8291	0.00521544
$433$	260.904	0.0289567	0.0144783	−	0.999895i	$-0.495391\pi$
$433$	260.904	0.0289567	0.0144783	+	0.999895i	$0.495391\pi$
$434$	0	0
$435$	−137.221	−0.0151247
$436$	−5615.10	−0.616776
$437$	1279.94	0.140110
$438$	−41.7921	−0.00455914
$439$	−12907.1	−1.40324	−0.701619	−	0.712553i	$-0.747539\pi$
$439$	−12907.1	−1.40324	−0.701619	+	0.712553i	$0.747539\pi$
$440$	3288.03	0.356251
$441$	0	0
$442$	−2784.05	−0.299602
$443$	−388.326	−0.0416476	−0.0208238	−	0.999783i	$-0.506629\pi$
$443$	−388.326	−0.0416476	−0.0208238	+	0.999783i	$0.506629\pi$
$444$	−74.3914	−0.00795148
$445$	−182.697	−0.0194621
$446$	−7201.91	−0.764619
$447$	−192.610	−0.0203807
$448$	0	0
$449$	−4058.85	−0.426612	−0.213306	−	0.976985i	$-0.568423\pi$
$449$	−4058.85	−0.426612	−0.213306	+	0.976985i	$0.568423\pi$
$450$	−2498.33	−0.261716
$451$	9087.77	0.948839
$452$	7166.12	0.745721
$453$	−65.1027	−0.00675230
$454$	7525.25	0.777923
$455$	0	0
$456$	24.1312	0.00247817
$457$	564.408	0.0577722	0.0288861	−	0.999583i	$-0.490804\pi$
$457$	564.408	0.0577722	0.0288861	+	0.999583i	$0.490804\pi$
$458$	−3592.84	−0.366556
$459$	214.195	0.0217817
$460$	1204.01	0.122037
$461$	−12838.4	−1.29706	−0.648528	−	0.761191i	$-0.724615\pi$
$461$	−12838.4	−1.29706	−0.648528	+	0.761191i	$0.724615\pi$
$462$	0	0
$463$	−6743.25	−0.676858	−0.338429	−	0.940992i	$-0.609896\pi$
$463$	−6743.25	−0.676858	−0.338429	+	0.940992i	$0.609896\pi$
$464$	−3095.10	−0.309669
$465$	10.0585	0.00100312
$466$	8600.72	0.854980
$467$	4063.32	0.402630	0.201315	−	0.979527i	$-0.435479\pi$
$467$	4063.32	0.402630	0.201315	+	0.979527i	$0.435479\pi$
$468$	2054.05	0.202881
$469$	0	0
$470$	−869.495	−0.0853336
$471$	42.6832	0.00417567
$472$	5996.84	0.584803
$473$	11700.7	1.13742
$474$	−76.5716	−0.00741993
$475$	−2574.93	−0.248728
$476$	0	0
$477$	12415.2	1.19173
$478$	−7025.94	−0.672300
$479$	−8803.20	−0.839726	−0.419863	−	0.907588i	$-0.637922\pi$
$479$	−8803.20	−0.839726	−0.419863	+	0.907588i	$0.637922\pi$
$480$	22.6995	0.00215852
$481$	−6526.35	−0.618661
$482$	−56.4420	−0.00533374
$483$	0	0
$484$	−1378.81	−0.129490
$485$	3353.06	0.313927
$486$	−237.051	−0.0221252
$487$	−14124.4	−1.31425	−0.657125	−	0.753782i	$-0.728228\pi$
$487$	−14124.4	−1.31425	−0.657125	+	0.753782i	$0.728228\pi$
$488$	−35.2523	−0.00327008
$489$	−96.4772	−0.00892198
$490$	0	0
$491$	−10425.2	−0.958211	−0.479106	−	0.877757i	$-0.659039\pi$
$491$	−10425.2	−0.958211	−0.479106	+	0.877757i	$0.659039\pi$
$492$	62.7393	0.00574899
$493$	−14156.9	−1.29329
$494$	2117.03	0.192813
$495$	−11095.9	−1.00752
$496$	226.874	0.0205382
$497$	0	0
$498$	−135.892	−0.0122278
$499$	−13995.6	−1.25557	−0.627784	−	0.778388i	$-0.716038\pi$
$499$	−13995.6	−1.25557	−0.627784	+	0.778388i	$0.716038\pi$
$500$	4121.35	0.368625
$501$	80.9355	0.00721743
$502$	−4977.32	−0.442527
$503$	−18506.9	−1.64052	−0.820259	−	0.571993i	$-0.806171\pi$
$503$	−18506.9	−1.64052	−0.820259	+	0.571993i	$0.806171\pi$
$504$	0	0
$505$	−21296.8	−1.87662
$506$	1444.65	0.126922
$507$	99.4740	0.00871360
$508$	8681.93	0.758264
$509$	22823.6	1.98750	0.993750	−	0.111632i	$-0.0356078\pi$
$509$	22823.6	1.98750	0.993750	+	0.111632i	$0.0356078\pi$
$510$	103.827	0.00901479
$511$	0	0
$512$	512.000	0.0441942
$513$	−162.877	−0.0140179
$514$	−4662.66	−0.400119
$515$	753.941	0.0645100
$516$	80.7782	0.00689159
$517$	−1043.28	−0.0887491
$518$	0	0
$519$	−216.420	−0.0183040
$520$	1991.43	0.167942
$521$	−10632.1	−0.894048	−0.447024	−	0.894522i	$-0.647516\pi$
$521$	−10632.1	−0.894048	−0.447024	+	0.894522i	$0.647516\pi$
$522$	10444.8	0.875780
$523$	21476.3	1.79559	0.897793	−	0.440417i	$-0.145170\pi$
$523$	21476.3	1.79559	0.897793	+	0.440417i	$0.145170\pi$
$524$	9239.30	0.770269
$525$	0	0
$526$	7119.80	0.590187
$527$	1037.72	0.0857754
$528$	27.2364	0.00224491
$529$	529.000	0.0434783
$530$	12036.8	0.986497
$531$	−20237.1	−1.65389
$532$	0	0
$533$	5504.11	0.447298
$534$	−1.51337	−0.000122640 0
$535$	9846.58	0.795710
$536$	−2266.80	−0.182670
$537$	−5.54049	−0.000445232 0
$538$	−6795.34	−0.544550
$539$	0	0
$540$	−153.214	−0.0122097
$541$	13368.5	1.06239	0.531197	−	0.847248i	$-0.321742\pi$
$541$	13368.5	1.06239	0.531197	+	0.847248i	$0.321742\pi$
$542$	9061.58	0.718133
$543$	−123.911	−0.00979284
$544$	2341.87	0.184572
$545$	18371.2	1.44392
$546$	0	0
$547$	5937.19	0.464088	0.232044	−	0.972705i	$-0.425459\pi$
$547$	5937.19	0.464088	0.232044	+	0.972705i	$0.425459\pi$
$548$	−6426.41	−0.500954
$549$	118.964	0.00924817
$550$	−2906.28	−0.225317
$551$	10765.1	0.832318
$552$	9.97342	0.000769016 0
$553$	0	0
$554$	8808.20	0.675496
$555$	243.391	0.0186151
$556$	−3360.16	−0.256300
$557$	13625.3	1.03649	0.518245	−	0.855232i	$-0.326586\pi$
$557$	13625.3	1.03649	0.518245	+	0.855232i	$0.326586\pi$
$558$	−765.617	−0.0580845
$559$	7086.67	0.536197
$560$	0	0
$561$	124.579	0.00937561
$562$	3509.45	0.263411
$563$	24917.9	1.86530	0.932652	−	0.360777i	$-0.117488\pi$
$563$	24917.9	1.86530	0.932652	+	0.360777i	$0.117488\pi$
$564$	−7.20248	−0.000537729 0
$565$	−23445.8	−1.74579
$566$	4328.10	0.321420
$567$	0	0
$568$	−3902.16	−0.288259
$569$	8188.17	0.603279	0.301640	−	0.953422i	$-0.402466\pi$
$569$	8188.17	0.603279	0.301640	+	0.953422i	$0.402466\pi$
$570$	−78.9514	−0.00580159
$571$	6038.51	0.442563	0.221282	−	0.975210i	$-0.428976\pi$
$571$	6038.51	0.442563	0.221282	+	0.975210i	$0.428976\pi$
$572$	2389.45	0.174664
$573$	179.086	0.0130566
$574$	0	0
$575$	−1064.22	−0.0771843
$576$	−1727.81	−0.124986
$577$	17532.8	1.26499	0.632495	−	0.774564i	$-0.282031\pi$
$577$	17532.8	1.26499	0.632495	+	0.774564i	$0.282031\pi$
$578$	885.672	0.0637355
$579$	−84.1533	−0.00604023
$580$	10126.4	0.724959
$581$	0	0
$582$	27.7751	0.00197821
$583$	14442.5	1.02598
$584$	3084.10	0.218529
$585$	−6720.35	−0.474961
$586$	10400.0	0.733138
$587$	−9011.11	−0.633608	−0.316804	−	0.948491i	$-0.602610\pi$
$587$	−9011.11	−0.633608	−0.316804	+	0.948491i	$0.602610\pi$
$588$	0	0
$589$	−789.092	−0.0552020
$590$	−19620.2	−1.36907
$591$	246.359	0.0171469
$592$	5489.80	0.381131
$593$	−9716.20	−0.672844	−0.336422	−	0.941711i	$-0.609217\pi$
$593$	−9716.20	−0.672844	−0.336422	+	0.941711i	$0.609217\pi$
$594$	−183.836	−0.0126984
$595$	0	0
$596$	14213.9	0.976886
$597$	191.783	0.0131477
$598$	874.968	0.0598329
$599$	7809.19	0.532679	0.266340	−	0.963879i	$-0.414186\pi$
$599$	7809.19	0.532679	0.266340	+	0.963879i	$0.414186\pi$
$600$	−20.0641	−0.00136519
$601$	−13546.6	−0.919428	−0.459714	−	0.888067i	$-0.652048\pi$
$601$	−13546.6	−0.919428	−0.459714	+	0.888067i	$0.652048\pi$
$602$	0	0
$603$	7649.62	0.516611
$604$	4804.33	0.323651
$605$	4511.13	0.303146
$606$	−176.412	−0.0118255
$607$	−9931.68	−0.664110	−0.332055	−	0.943260i	$-0.607742\pi$
$607$	−9931.68	−0.664110	−0.332055	+	0.943260i	$0.607742\pi$
$608$	−1780.79	−0.118784
$609$	0	0
$610$	115.337	0.00765551
$611$	−631.873	−0.0418377
$612$	−7902.97	−0.521991
$613$	10142.9	0.668299	0.334149	−	0.942520i	$-0.391551\pi$
$613$	10142.9	0.668299	0.334149	+	0.942520i	$0.391551\pi$
$614$	5137.42	0.337670
$615$	−205.268	−0.0134588
$616$	0	0
$617$	30295.3	1.97673	0.988366	−	0.152091i	$-0.0486008\pi$
$617$	30295.3	1.97673	0.988366	+	0.152091i	$0.0486008\pi$
$618$	6.24529	0.000406509 0
$619$	−4060.58	−0.263665	−0.131832	−	0.991272i	$-0.542086\pi$
$619$	−4060.58	−0.263665	−0.131832	+	0.991272i	$0.542086\pi$
$620$	−742.277	−0.0480816
$621$	−67.3169	−0.00434997
$622$	−4322.79	−0.278663
$623$	0	0
$624$	16.4961	0.00105829
$625$	−19267.8	−1.23314
$626$	−1220.59	−0.0779303
$627$	−94.7311	−0.00603380
$628$	−3149.86	−0.200148
$629$	25110.2	1.59175
$630$	0	0
$631$	−2461.05	−0.155266	−0.0776331	−	0.996982i	$-0.524736\pi$
$631$	−2461.05	−0.155266	−0.0776331	+	0.996982i	$0.524736\pi$
$632$	5650.69	0.355652
$633$	−104.638	−0.00657025
$634$	2997.28	0.187756
$635$	−28405.2	−1.77516
$636$	99.7068	0.00621640
$637$	0	0
$638$	12150.3	0.753975
$639$	13168.4	0.815230
$640$	−1675.14	−0.103462
$641$	2333.86	0.143810	0.0719048	−	0.997412i	$-0.477092\pi$
$641$	2333.86	0.143810	0.0719048	+	0.997412i	$0.477092\pi$
$642$	81.5644	0.00501416
$643$	−15498.1	−0.950524	−0.475262	−	0.879844i	$-0.657647\pi$
$643$	−15498.1	−0.950524	−0.475262	+	0.879844i	$0.657647\pi$
$644$	0	0
$645$	−264.287	−0.0161338
$646$	−8145.28	−0.496086
$647$	−23277.1	−1.41440	−0.707199	−	0.707014i	$-0.750042\pi$
$647$	−23277.1	−1.41440	−0.707199	+	0.707014i	$0.750042\pi$
$648$	5830.10	0.353438
$649$	−23541.6	−1.42387
$650$	−1760.22	−0.106218
$651$	0	0
$652$	7119.65	0.427649
$653$	10483.8	0.628276	0.314138	−	0.949377i	$-0.398285\pi$
$653$	10483.8	0.628276	0.314138	+	0.949377i	$0.398285\pi$
$654$	152.179	0.00909886
$655$	−30228.8	−1.80326
$656$	−4629.92	−0.275561
$657$	−10407.7	−0.618026
$658$	0	0
$659$	−11847.5	−0.700323	−0.350162	−	0.936689i	$-0.613873\pi$
$659$	−11847.5	−0.700323	−0.350162	+	0.936689i	$0.613873\pi$
$660$	−89.1110	−0.00525552
$661$	−19718.1	−1.16028	−0.580139	−	0.814518i	$-0.697002\pi$
$661$	−19718.1	−1.16028	−0.580139	+	0.814518i	$0.697002\pi$
$662$	−2544.20	−0.149370
$663$	75.4525	0.00441981
$664$	10028.3	0.586105
$665$	0	0
$666$	−18526.1	−1.07788
$667$	4449.20	0.258281
$668$	−5972.73	−0.345946
$669$	195.184	0.0112799
$670$	7416.42	0.427644
$671$	138.389	0.00796193
$672$	0	0
$673$	−19678.2	−1.12710	−0.563550	−	0.826082i	$-0.690565\pi$
$673$	−19678.2	−1.12710	−0.563550	+	0.826082i	$0.690565\pi$
$674$	6381.42	0.364693
$675$	135.425	0.00772225
$676$	−7340.80	−0.417661
$677$	−13907.3	−0.789512	−0.394756	−	0.918786i	$-0.629171\pi$
$677$	−13907.3	−0.789512	−0.394756	+	0.918786i	$0.629171\pi$
$678$	−194.214	−0.0110011
$679$	0	0
$680$	−7662.05	−0.432097
$681$	−203.947	−0.0114761
$682$	−890.634	−0.0500061
$683$	−124.832	−0.00699348	−0.00349674	−	0.999994i	$-0.501113\pi$
$683$	−124.832	−0.00699348	−0.00349674	+	0.999994i	$0.501113\pi$
$684$	6009.51	0.335935
$685$	21025.7	1.17277
$686$	0	0
$687$	97.3721	0.00540753
$688$	−5961.12	−0.330328
$689$	8747.27	0.483664
$690$	−32.6306	−0.00180033
$691$	27514.2	1.51475	0.757373	−	0.652983i	$-0.226483\pi$
$691$	27514.2	1.51475	0.757373	+	0.652983i	$0.226483\pi$
$692$	15971.0	0.877349
$693$	0	0
$694$	−2485.94	−0.135973
$695$	10993.6	0.600018
$696$	83.8823	0.00456832
$697$	−21177.1	−1.15085
$698$	3679.93	0.199552
$699$	−233.094	−0.0126129
$700$	0	0
$701$	21289.4	1.14706	0.573532	−	0.819183i	$-0.305573\pi$
$701$	21289.4	1.14706	0.573532	+	0.819183i	$0.305573\pi$
$702$	−111.342	−0.00598625
$703$	−19094.1	−1.02439
$704$	−2009.94	−0.107603
$705$	23.5648	0.00125887
$706$	25375.2	1.35270
$707$	0	0
$708$	−162.525	−0.00862718
$709$	14503.4	0.768244	0.384122	−	0.923282i	$-0.374504\pi$
$709$	14503.4	0.768244	0.384122	+	0.923282i	$0.374504\pi$
$710$	12766.9	0.674837
$711$	−19069.0	−1.00583
$712$	111.681	0.00587840
$713$	−326.132	−0.0171300
$714$	0	0
$715$	−7817.71	−0.408903
$716$	408.867	0.0213409
$717$	190.415	0.00991795
$718$	−21291.8	−1.10669
$719$	2462.39	0.127722	0.0638608	−	0.997959i	$-0.479659\pi$
$719$	2462.39	0.127722	0.0638608	+	0.997959i	$0.479659\pi$
$720$	5652.98	0.292603
$721$	0	0
$722$	−7524.23	−0.387844
$723$	1.52967	7.86848e−5 0
$724$	9144.13	0.469391
$725$	−8950.70	−0.458511
$726$	37.3680	0.00191027
$727$	13804.6	0.704240	0.352120	−	0.935955i	$-0.385461\pi$
$727$	13804.6	0.704240	0.352120	+	0.935955i	$0.385461\pi$
$728$	0	0
$729$	−19670.2	−0.999347
$730$	−10090.4	−0.511593
$731$	−27266.0	−1.37958
$732$	0.955397	4.82411e−5 0
$733$	28457.7	1.43398	0.716992	−	0.697082i	$-0.245519\pi$
$733$	28457.7	1.43398	0.716992	+	0.697082i	$0.245519\pi$
$734$	17574.4	0.883766
$735$	0	0
$736$	−736.000	−0.0368605
$737$	8898.72	0.444760
$738$	15624.3	0.779319
$739$	25496.4	1.26915	0.634573	−	0.772863i	$-0.281176\pi$
$739$	25496.4	1.26915	0.634573	+	0.772863i	$0.281176\pi$
$740$	−17961.3	−0.892257
$741$	−57.3750	−0.00284443
$742$	0	0
$743$	24682.0	1.21870	0.609351	−	0.792901i	$-0.291430\pi$
$743$	24682.0	1.21870	0.609351	+	0.792901i	$0.291430\pi$
$744$	−6.14867	−0.000302985 0
$745$	−46504.5	−2.28697
$746$	16566.1	0.813038
$747$	−33841.8	−1.65757
$748$	−9193.43	−0.449392
$749$	0	0
$750$	−111.695	−0.00543805
$751$	10796.2	0.524579	0.262290	−	0.964989i	$-0.415522\pi$
$751$	10796.2	0.524579	0.262290	+	0.964989i	$0.415522\pi$
$752$	531.515	0.0257744
$753$	134.894	0.00652829
$754$	7358.99	0.355436
$755$	−15718.6	−0.757693
$756$	0	0
$757$	26337.9	1.26455	0.632276	−	0.774743i	$-0.282121\pi$
$757$	26337.9	1.26455	0.632276	+	0.774743i	$0.282121\pi$
$758$	−10985.4	−0.526395
$759$	−39.1524	−0.00187239
$760$	5826.31	0.278082
$761$	−28788.9	−1.37135	−0.685674	−	0.727909i	$-0.740492\pi$
$761$	−28788.9	−1.37135	−0.685674	+	0.727909i	$0.740492\pi$
$762$	−235.295	−0.0111861
$763$	0	0
$764$	−13215.8	−0.625827
$765$	25856.6	1.22202
$766$	17305.7	0.816291
$767$	−14258.3	−0.671234
$768$	−13.8761	−0.000651965 0
$769$	−10675.6	−0.500612	−0.250306	−	0.968167i	$-0.580531\pi$
$769$	−10675.6	−0.500612	−0.250306	+	0.968167i	$0.580531\pi$
$770$	0	0
$771$	126.366	0.00590267
$772$	6210.19	0.289520
$773$	8432.58	0.392366	0.196183	−	0.980567i	$-0.437145\pi$
$773$	8432.58	0.392366	0.196183	+	0.980567i	$0.437145\pi$
$774$	20116.6	0.934207
$775$	656.097	0.0304099
$776$	−2049.70	−0.0948194
$777$	0	0
$778$	1898.76	0.0874984
$779$	16103.3	0.740644
$780$	−53.9711	−0.00247753
$781$	15318.6	0.701847
$782$	−3366.44	−0.153943
$783$	−566.175	−0.0258409
$784$	0	0
$785$	10305.6	0.468563
$786$	−250.401	−0.0113632
$787$	10992.3	0.497883	0.248942	−	0.968518i	$-0.419917\pi$
$787$	10992.3	0.497883	0.248942	+	0.968518i	$0.419917\pi$
$788$	−18180.3	−0.821888
$789$	−192.959	−0.00870660
$790$	−18487.7	−0.832611
$791$	0	0
$792$	6782.82	0.304315
$793$	83.8170	0.00375338
$794$	−6253.81	−0.279521
$795$	−326.216	−0.0145531
$796$	−14152.9	−0.630195
$797$	−30559.6	−1.35819	−0.679094	−	0.734051i	$-0.737627\pi$
$797$	−30559.6	−1.35819	−0.679094	+	0.734051i	$0.737627\pi$
$798$	0	0
$799$	2431.14	0.107644
$800$	1480.65	0.0654362
$801$	−376.882	−0.0166248
$802$	−3350.48	−0.147518
$803$	−12107.2	−0.532070
$804$	61.4341	0.00269479
$805$	0	0
$806$	−539.423	−0.0235736
$807$	184.165	0.00803336
$808$	13018.6	0.566821
$809$	−779.749	−0.0338869	−0.0169435	−	0.999856i	$-0.505394\pi$
$809$	−779.749	−0.0338869	−0.0169435	+	0.999856i	$0.505394\pi$
$810$	−19074.7	−0.827426
$811$	14098.7	0.610445	0.305222	−	0.952281i	$-0.401269\pi$
$811$	14098.7	0.610445	0.305222	+	0.952281i	$0.401269\pi$
$812$	0	0
$813$	−245.584	−0.0105941
$814$	−21551.2	−0.927970
$815$	−23293.8	−1.00116
$816$	−63.4687	−0.00272285
$817$	20733.4	0.887845
$818$	−17902.8	−0.765229
$819$	0	0
$820$	15148.0	0.645110
$821$	4473.53	0.190167	0.0950837	−	0.995469i	$-0.469688\pi$
$821$	4473.53	0.190167	0.0950837	+	0.995469i	$0.469688\pi$
$822$	174.167	0.00739022
$823$	11771.4	0.498574	0.249287	−	0.968430i	$-0.419804\pi$
$823$	11771.4	0.498574	0.249287	+	0.968430i	$0.419804\pi$
$824$	−460.879	−0.0194848
$825$	78.7650	0.00332393
$826$	0	0
$827$	9400.65	0.395275	0.197638	−	0.980275i	$-0.436673\pi$
$827$	9400.65	0.395275	0.197638	+	0.980275i	$0.436673\pi$
$828$	2483.73	0.104246
$829$	−42378.6	−1.77548	−0.887738	−	0.460349i	$-0.847724\pi$
$829$	−42378.6	−1.77548	−0.887738	+	0.460349i	$0.847724\pi$
$830$	−32810.2	−1.37212
$831$	−238.717	−0.00996510
$832$	−1217.35	−0.0507258
$833$	0	0
$834$	91.0660	0.00378100
$835$	19541.3	0.809887
$836$	6990.79	0.289212
$837$	41.5013	0.00171385
$838$	−7622.55	−0.314220
$839$	7200.79	0.296304	0.148152	−	0.988965i	$-0.452668\pi$
$839$	7200.79	0.296304	0.148152	+	0.988965i	$0.452668\pi$
$840$	0	0
$841$	13031.4	0.534314
$842$	−7660.69	−0.313545
$843$	−95.1120	−0.00388592
$844$	7721.85	0.314925
$845$	24017.3	0.977776
$846$	−1793.67	−0.0728932
$847$	0	0
$848$	−7357.98	−0.297965
$849$	−117.299	−0.00474168
$850$	6772.46	0.273287
$851$	−7891.59	−0.317885
$852$	105.755	0.00425247
$853$	−18633.5	−0.747946	−0.373973	−	0.927440i	$-0.622005\pi$
$853$	−18633.5	−0.747946	−0.373973	+	0.927440i	$0.622005\pi$
$854$	0	0
$855$	−19661.7	−0.786449
$856$	−6019.14	−0.240339
$857$	10637.9	0.424017	0.212009	−	0.977268i	$-0.431999\pi$
$857$	10637.9	0.424017	0.212009	+	0.977268i	$0.431999\pi$
$858$	−64.7581	−0.00257670
$859$	7232.74	0.287285	0.143643	−	0.989630i	$-0.454118\pi$
$859$	7232.74	0.287285	0.143643	+	0.989630i	$0.454118\pi$
$860$	19503.3	0.773324
$861$	0	0
$862$	17512.6	0.691975
$863$	42247.1	1.66641	0.833203	−	0.552967i	$-0.186505\pi$
$863$	42247.1	1.66641	0.833203	+	0.552967i	$0.186505\pi$
$864$	93.6583	0.00368787
$865$	−52253.1	−2.05394
$866$	521.808	0.0204755
$867$	−24.0032	−0.000940244 0
$868$	0	0
$869$	−22182.8	−0.865936
$870$	−274.442	−0.0106948
$871$	5389.61	0.209667
$872$	−11230.2	−0.436127
$873$	6916.98	0.268161
$874$	2559.88	0.0990725
$875$	0	0
$876$	−83.5842	−0.00322380
$877$	−47245.6	−1.81912	−0.909560	−	0.415572i	$-0.863582\pi$
$877$	−47245.6	−1.81912	−0.909560	+	0.415572i	$0.863582\pi$
$878$	−25814.2	−0.992239
$879$	−281.857	−0.0108155
$880$	6576.05	0.251908
$881$	8004.88	0.306120	0.153060	−	0.988217i	$-0.451087\pi$
$881$	8004.88	0.306120	0.153060	+	0.988217i	$0.451087\pi$
$882$	0	0
$883$	−43800.6	−1.66932	−0.834659	−	0.550767i	$-0.814335\pi$
$883$	−43800.6	−1.66932	−0.834659	+	0.550767i	$0.814335\pi$
$884$	−5568.11	−0.211850
$885$	531.741	0.0201969
$886$	−776.651	−0.0294493
$887$	−1929.00	−0.0730208	−0.0365104	−	0.999333i	$-0.511624\pi$
$887$	−1929.00	−0.0730208	−0.0365104	+	0.999333i	$0.511624\pi$
$888$	−148.783	−0.00562255
$889$	0	0
$890$	−365.393	−0.0137618
$891$	−22887.0	−0.860544
$892$	−14403.8	−0.540667
$893$	−1848.66	−0.0692757
$894$	−385.221	−0.0144113
$895$	−1337.71	−0.0499607
$896$	0	0
$897$	−23.7131	−0.000882672 0
$898$	−8117.70	−0.301660
$899$	−2742.96	−0.101761
$900$	−4996.66	−0.185061
$901$	−33655.2	−1.24441
$902$	18175.5	0.670930
$903$	0	0
$904$	14332.2	0.527305
$905$	−29917.4	−1.09888
$906$	−130.205	−0.00477459
$907$	3445.37	0.126132	0.0630660	−	0.998009i	$-0.479912\pi$
$907$	3445.37	0.126132	0.0630660	+	0.998009i	$0.479912\pi$
$908$	15050.5	0.550075
$909$	−43932.9	−1.60304
$910$	0	0
$911$	30571.3	1.11183	0.555913	−	0.831241i	$-0.312369\pi$
$911$	30571.3	1.11183	0.555913	+	0.831241i	$0.312369\pi$
$912$	48.2624	0.00175233
$913$	−39367.8	−1.42704
$914$	1128.82	0.0408511
$915$	−3.12583	−0.000112936 0
$916$	−7185.69	−0.259194
$917$	0	0
$918$	428.391	0.0154020
$919$	−29854.7	−1.07162	−0.535809	−	0.844339i	$-0.679993\pi$
$919$	−29854.7	−1.07162	−0.535809	+	0.844339i	$0.679993\pi$
$920$	2408.01	0.0862933
$921$	−139.233	−0.00498140
$922$	−25676.7	−0.917156
$923$	9277.89	0.330862
$924$	0	0
$925$	15875.9	0.564322
$926$	−13486.5	−0.478611
$927$	1555.30	0.0551053
$928$	−6190.19	−0.218969
$929$	−37797.8	−1.33488	−0.667441	−	0.744663i	$-0.732610\pi$
$929$	−37797.8	−1.33488	−0.667441	+	0.744663i	$0.732610\pi$
$930$	20.1170	0.000709313 0
$931$	0	0
$932$	17201.4	0.604562
$933$	117.155	0.00411091
$934$	8126.64	0.284702
$935$	30078.7	1.05206
$936$	4108.10	0.143459
$937$	2885.94	0.100618	0.0503092	−	0.998734i	$-0.483979\pi$
$937$	2885.94	0.100618	0.0503092	+	0.998734i	$0.483979\pi$
$938$	0	0
$939$	33.0799	0.00114965
$940$	−1738.99	−0.0603400
$941$	−13767.0	−0.476929	−0.238465	−	0.971151i	$-0.576644\pi$
$941$	−13767.0	−0.476929	−0.238465	+	0.971151i	$0.576644\pi$
$942$	85.3665	0.00295264
$943$	6655.51	0.229834
$944$	11993.7	0.413518
$945$	0	0
$946$	23401.4	0.804277
$947$	−54210.7	−1.86020	−0.930102	−	0.367302i	$-0.880281\pi$
$947$	−54210.7	−1.86020	−0.930102	+	0.367302i	$0.880281\pi$
$948$	−153.143	−0.00524669
$949$	−7332.84	−0.250826
$950$	−5149.86	−0.175877
$951$	−81.2313	−0.00276983
$952$	0	0
$953$	33288.7	1.13151	0.565753	−	0.824574i	$-0.308585\pi$
$953$	33288.7	1.13151	0.565753	+	0.824574i	$0.308585\pi$
$954$	24830.5	0.842680
$955$	43239.0	1.46511
$956$	−14051.9	−0.475388
$957$	−329.294	−0.0111229
$958$	−17606.4	−0.593776
$959$	0	0
$960$	45.3991	0.00152630
$961$	−29589.9	−0.993251
$962$	−13052.7	−0.437460
$963$	20312.4	0.679707
$964$	−112.884	−0.00377152
$965$	−20318.3	−0.677790
$966$	0	0
$967$	−29821.4	−0.991719	−0.495860	−	0.868403i	$-0.665147\pi$
$967$	−29821.4	−0.991719	−0.495860	+	0.868403i	$0.665147\pi$
$968$	−2757.62	−0.0915632
$969$	220.751	0.00731841
$970$	6706.11	0.221980
$971$	59734.7	1.97423	0.987115	−	0.160011i	$-0.0511528\pi$
$971$	59734.7	1.97423	0.987115	+	0.160011i	$0.0511528\pi$
$972$	−474.102	−0.0156449
$973$	0	0
$974$	−28248.9	−0.929315
$975$	47.7049	0.00156695
$976$	−70.5047	−0.00231230
$977$	−13895.1	−0.455010	−0.227505	−	0.973777i	$-0.573057\pi$
$977$	−13895.1	−0.455010	−0.227505	+	0.973777i	$0.573057\pi$
$978$	−192.954	−0.00630879
$979$	−438.423	−0.0143126
$980$	0	0
$981$	37897.8	1.23342
$982$	−20850.4	−0.677558
$983$	46977.7	1.52427	0.762134	−	0.647419i	$-0.224152\pi$
$983$	46977.7	1.52427	0.762134	+	0.647419i	$0.224152\pi$
$984$	125.479	0.00406515
$985$	59481.6	1.92410
$986$	−28313.8	−0.914497
$987$	0	0
$988$	4234.06	0.136339
$989$	8569.11	0.275513
$990$	−22191.8	−0.712425
$991$	−17180.5	−0.550712	−0.275356	−	0.961342i	$-0.588796\pi$
$991$	−17180.5	−0.550712	−0.275356	+	0.961342i	$0.588796\pi$
$992$	453.748	0.0145227
$993$	68.9521	0.00220355
$994$	0	0
$995$	46304.7	1.47534
$996$	−271.784	−0.00864638
$997$	19426.4	0.617091	0.308545	−	0.951210i	$-0.400158\pi$
$997$	19426.4	0.617091	0.308545	+	0.951210i	$0.400158\pi$
$998$	−27991.2	−0.887820
$999$	1004.23	0.0318042

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.x.1.6		11
7.3	odd	6		322.4.e.b.93.6	✓	22
7.5	odd	6		322.4.e.b.277.6	yes	22
7.6	odd	2		2254.4.a.w.1.6		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.4.e.b.93.6	✓	22	7.3	odd	6
322.4.e.b.277.6	yes	22	7.5	odd	6
2254.4.a.w.1.6		11	7.6	odd	2
2254.4.a.x.1.6		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)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -0.0542033 q^{3} +4.00000 q^{4} -13.0870 q^{5} -0.108407 q^{6} +8.00000 q^{8} -26.9971 q^{9} -26.1741 q^{10} -31.4054 q^{11} -0.216813 q^{12} -19.0210 q^{13} +0.709361 q^{15} +16.0000 q^{16} +73.1836 q^{17} -53.9941 q^{18} -55.6496 q^{19} -52.3481 q^{20} -62.8108 q^{22} -23.0000 q^{23} -0.433627 q^{24} +46.2704 q^{25} -38.0421 q^{26} +2.92682 q^{27} -193.443 q^{29} +1.41872 q^{30} +14.1796 q^{31} +32.0000 q^{32} +1.70228 q^{33} +146.367 q^{34} -107.988 q^{36} +343.112 q^{37} -111.299 q^{38} +1.03100 q^{39} -104.696 q^{40} -289.370 q^{41} -372.570 q^{43} -125.622 q^{44} +353.311 q^{45} -46.0000 q^{46} +33.2197 q^{47} -0.867254 q^{48} +92.5408 q^{50} -3.96679 q^{51} -76.0841 q^{52} -459.874 q^{53} +5.85364 q^{54} +411.003 q^{55} +3.01640 q^{57} -386.887 q^{58} +749.606 q^{59} +2.83744 q^{60} -4.40654 q^{61} +28.3593 q^{62} +64.0000 q^{64} +248.929 q^{65} +3.40455 q^{66} -283.350 q^{67} +292.734 q^{68} +1.24668 q^{69} -487.770 q^{71} -215.976 q^{72} +385.512 q^{73} +686.225 q^{74} -2.50801 q^{75} -222.599 q^{76} +2.06201 q^{78} +706.336 q^{79} -209.392 q^{80} +728.762 q^{81} -578.740 q^{82} +1253.54 q^{83} -957.756 q^{85} -745.140 q^{86} +10.4853 q^{87} -251.243 q^{88} +13.9601 q^{89} +706.623 q^{90} -92.0000 q^{92} -0.768584 q^{93} +66.4394 q^{94} +728.289 q^{95} -1.73451 q^{96} -256.212 q^{97} +847.853 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 22 q^{2} + 6 q^{3} + 44 q^{4} + 27 q^{5} + 12 q^{6} + 88 q^{8} + 59 q^{9} + 54 q^{10} + 56 q^{11} + 24 q^{12} + 103 q^{13} + 62 q^{15} + 176 q^{16} + 157 q^{17} + 118 q^{18} + 266 q^{19} + 108 q^{20}+ \cdots + 101 q^{99}+O(q^{100}) \) 11 * q + 22 * q^2 + 6 * q^3 + 44 * q^4 + 27 * q^5 + 12 * q^6 + 88 * q^8 + 59 * q^9 + 54 * q^10 + 56 * q^11 + 24 * q^12 + 103 * q^13 + 62 * q^15 + 176 * q^16 + 157 * q^17 + 118 * q^18 + 266 * q^19 + 108 * q^20 + 112 * q^22 - 253 * q^23 + 48 * q^24 + 142 * q^25 + 206 * q^26 + 531 * q^27 - 195 * q^29 + 124 * q^30 + 316 * q^31 + 352 * q^32 + 682 * q^33 + 314 * q^34 + 236 * q^36 + 347 * q^37 + 532 * q^38 - 99 * q^39 + 216 * q^40 + 540 * q^41 + 193 * q^43 + 224 * q^44 + 609 * q^45 - 506 * q^46 + 731 * q^47 + 96 * q^48 + 284 * q^50 - 303 * q^51 + 412 * q^52 + 413 * q^53 + 1062 * q^54 + 1809 * q^55 + 343 * q^57 - 390 * q^58 + 1061 * q^59 + 248 * q^60 + 1157 * q^61 + 632 * q^62 + 704 * q^64 - 37 * q^65 + 1364 * q^66 - 841 * q^67 + 628 * q^68 - 138 * q^69 + 869 * q^71 + 472 * q^72 + 409 * q^73 + 694 * q^74 + 2024 * q^75 + 1064 * q^76 - 198 * q^78 + 247 * q^79 + 432 * q^80 + 1019 * q^81 + 1080 * q^82 + 2863 * q^83 + 785 * q^85 + 386 * q^86 + 670 * q^87 + 448 * q^88 + 1610 * q^89 + 1218 * q^90 - 1012 * q^92 - 1290 * q^93 + 1462 * q^94 - 209 * q^95 + 192 * q^96 + 1783 * q^97 + 101 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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