LMFDB - Embedded newform 2254.4.a.g.1.1

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{57}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 14 $ x^2 - x - 14
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 322)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.27492$ 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)

$f(q)$	$=$	\(q+2.00000 q^{2} -4.27492 q^{3} +4.00000 q^{4} +16.5498 q^{5} -8.54983 q^{6} +8.00000 q^{8} -8.72508 q^{9} +33.0997 q^{10} -69.6495 q^{11} -17.0997 q^{12} +47.3746 q^{13} -70.7492 q^{15} +16.0000 q^{16} +49.8488 q^{17} -17.4502 q^{18} -90.7974 q^{19} +66.1993 q^{20} -139.299 q^{22} +23.0000 q^{23} -34.1993 q^{24} +148.897 q^{25} +94.7492 q^{26} +152.722 q^{27} -15.8762 q^{29} -141.498 q^{30} -55.9244 q^{31} +32.0000 q^{32} +297.746 q^{33} +99.6977 q^{34} -34.9003 q^{36} -9.90364 q^{37} -181.595 q^{38} -202.522 q^{39} +132.399 q^{40} -146.571 q^{41} -15.2026 q^{43} -278.598 q^{44} -144.399 q^{45} +46.0000 q^{46} +511.169 q^{47} -68.3987 q^{48} +297.794 q^{50} -213.100 q^{51} +189.498 q^{52} -291.897 q^{53} +305.444 q^{54} -1152.69 q^{55} +388.151 q^{57} -31.7525 q^{58} -536.640 q^{59} -282.997 q^{60} -479.945 q^{61} -111.849 q^{62} +64.0000 q^{64} +784.042 q^{65} +595.492 q^{66} -924.234 q^{67} +199.395 q^{68} -98.3231 q^{69} +235.815 q^{71} -69.8007 q^{72} -813.127 q^{73} -19.8073 q^{74} -636.522 q^{75} -363.189 q^{76} -405.045 q^{78} -1036.14 q^{79} +264.797 q^{80} -417.296 q^{81} -293.141 q^{82} +328.846 q^{83} +824.990 q^{85} -30.4053 q^{86} +67.8696 q^{87} -557.196 q^{88} +1580.83 q^{89} -288.797 q^{90} +92.0000 q^{92} +239.072 q^{93} +1022.34 q^{94} -1502.68 q^{95} -136.797 q^{96} -344.460 q^{97} +607.698 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} - q^{3} + 8 q^{4} + 18 q^{5} - 2 q^{6} + 16 q^{8} - 25 q^{9} + 36 q^{10} - 94 q^{11} - 4 q^{12} + 57 q^{13} - 66 q^{15} + 32 q^{16} - 6 q^{17} - 50 q^{18} + 60 q^{19} + 72 q^{20} - 188 q^{22}+ \cdots + 1004 q^{99}+O(q^{100}) $ 2 * q + 4 * q^2 - q^3 + 8 * q^4 + 18 * q^5 - 2 * q^6 + 16 * q^8 - 25 * q^9 + 36 * q^10 - 94 * q^11 - 4 * q^12 + 57 * q^13 - 66 * q^15 + 32 * q^16 - 6 * q^17 - 50 * q^18 + 60 * q^19 + 72 * q^20 - 188 * q^22 + 46 * q^23 - 8 * q^24 + 26 * q^25 + 114 * q^26 + 11 * q^27 - 145 * q^29 - 132 * q^30 - 59 * q^31 + 64 * q^32 + 218 * q^33 - 12 * q^34 - 100 * q^36 - 352 * q^37 + 120 * q^38 - 171 * q^39 + 144 * q^40 + 107 * q^41 - 272 * q^43 - 376 * q^44 - 168 * q^45 + 92 * q^46 + 441 * q^47 - 16 * q^48 + 52 * q^50 - 396 * q^51 + 228 * q^52 - 312 * q^53 + 22 * q^54 - 1188 * q^55 + 882 * q^57 - 290 * q^58 - 122 * q^59 - 264 * q^60 - 522 * q^61 - 118 * q^62 + 128 * q^64 + 798 * q^65 + 436 * q^66 - 414 * q^67 - 24 * q^68 - 23 * q^69 - 457 * q^71 - 200 * q^72 - 1815 * q^73 - 704 * q^74 - 1039 * q^75 + 240 * q^76 - 342 * q^78 - 970 * q^79 + 288 * q^80 - 442 * q^81 + 214 * q^82 + 250 * q^83 + 744 * q^85 - 544 * q^86 - 355 * q^87 - 752 * q^88 + 1546 * q^89 - 336 * q^90 + 184 * q^92 + 229 * q^93 + 882 * q^94 - 1284 * q^95 - 32 * q^96 - 1610 * q^97 + 1004 * 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
$3$	−4.27492	−0.822708	−0.411354	−	0.911476i	$-0.634944\pi$
$3$	−4.27492	−0.822708	−0.411354	+	0.911476i	$0.634944\pi$
$4$	4.00000	0.500000
$5$	16.5498	1.48026	0.740131	−	0.672463i	$-0.234763\pi$
$5$	16.5498	1.48026	0.740131	+	0.672463i	$0.234763\pi$
$6$	−8.54983	−0.581743
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	−8.72508	−0.323151
$10$	33.0997	1.04670
$11$	−69.6495	−1.90910	−0.954551	−	0.298049i	$-0.903664\pi$
$11$	−69.6495	−1.90910	−0.954551	+	0.298049i	$0.903664\pi$
$12$	−17.0997	−0.411354
$13$	47.3746	1.01072	0.505359	−	0.862909i	$-0.331360\pi$
$13$	47.3746	1.01072	0.505359	+	0.862909i	$0.331360\pi$
$14$	0	0
$15$	−70.7492	−1.21782
$16$	16.0000	0.250000
$17$	49.8488	0.711184	0.355592	−	0.934641i	$-0.384279\pi$
$17$	49.8488	0.711184	0.355592	+	0.934641i	$0.384279\pi$
$18$	−17.4502	−0.228502
$19$	−90.7974	−1.09633	−0.548167	−	0.836369i	$-0.684674\pi$
$19$	−90.7974	−1.09633	−0.548167	+	0.836369i	$0.684674\pi$
$20$	66.1993	0.740131
$21$	0	0
$22$	−139.299	−1.34994
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	−34.1993	−0.290871
$25$	148.897	1.19118
$26$	94.7492	0.714686
$27$	152.722	1.08857
$28$	0	0
$29$	−15.8762	−0.101660	−0.0508301	−	0.998707i	$-0.516187\pi$
$29$	−15.8762	−0.101660	−0.0508301	+	0.998707i	$0.516187\pi$
$30$	−141.498	−0.861132
$31$	−55.9244	−0.324010	−0.162005	−	0.986790i	$-0.551796\pi$
$31$	−55.9244	−0.324010	−0.162005	+	0.986790i	$0.551796\pi$
$32$	32.0000	0.176777
$33$	297.746	1.57063
$34$	99.6977	0.502883
$35$	0	0
$36$	−34.9003	−0.161576
$37$	−9.90364	−0.0440040	−0.0220020	−	0.999758i	$-0.507004\pi$
$37$	−9.90364	−0.0440040	−0.0220020	+	0.999758i	$0.507004\pi$
$38$	−181.595	−0.775225
$39$	−202.522	−0.831527
$40$	132.399	0.523352
$41$	−146.571	−0.558304	−0.279152	−	0.960247i	$-0.590053\pi$
$41$	−146.571	−0.558304	−0.279152	+	0.960247i	$0.590053\pi$
$42$	0	0
$43$	−15.2026	−0.0539159	−0.0269579	−	0.999637i	$-0.508582\pi$
$43$	−15.2026	−0.0539159	−0.0269579	+	0.999637i	$0.508582\pi$
$44$	−278.598	−0.954551
$45$	−144.399	−0.478349
$46$	46.0000	0.147442
$47$	511.169	1.58642	0.793209	−	0.608950i	$-0.208409\pi$
$47$	511.169	1.58642	0.793209	+	0.608950i	$0.208409\pi$
$48$	−68.3987	−0.205677
$49$	0	0
$50$	297.794	0.842289
$51$	−213.100	−0.585097
$52$	189.498	0.505359
$53$	−291.897	−0.756512	−0.378256	−	0.925701i	$-0.623476\pi$
$53$	−291.897	−0.756512	−0.378256	+	0.925701i	$0.623476\pi$
$54$	305.444	0.769733
$55$	−1152.69	−2.82597
$56$	0	0
$57$	388.151	0.901963
$58$	−31.7525	−0.0718846
$59$	−536.640	−1.18414	−0.592072	−	0.805885i	$-0.701690\pi$
$59$	−536.640	−1.18414	−0.592072	+	0.805885i	$0.701690\pi$
$60$	−282.997	−0.608912
$61$	−479.945	−1.00739	−0.503694	−	0.863882i	$-0.668026\pi$
$61$	−479.945	−1.00739	−0.503694	+	0.863882i	$0.668026\pi$
$62$	−111.849	−0.229110
$63$	0	0
$64$	64.0000	0.125000
$65$	784.042	1.49613
$66$	595.492	1.11061
$67$	−924.234	−1.68527	−0.842636	−	0.538484i	$-0.818997\pi$
$67$	−924.234	−1.68527	−0.842636	+	0.538484i	$0.818997\pi$
$68$	199.395	0.355592
$69$	−98.3231	−0.171547
$70$	0	0
$71$	235.815	0.394170	0.197085	−	0.980386i	$-0.436853\pi$
$71$	235.815	0.394170	0.197085	+	0.980386i	$0.436853\pi$
$72$	−69.8007	−0.114251
$73$	−813.127	−1.30369	−0.651845	−	0.758352i	$-0.726005\pi$
$73$	−813.127	−1.30369	−0.651845	+	0.758352i	$0.726005\pi$
$74$	−19.8073	−0.0311155
$75$	−636.522	−0.979990
$76$	−363.189	−0.548167
$77$	0	0
$78$	−405.045	−0.587978
$79$	−1036.14	−1.47563	−0.737814	−	0.675004i	$-0.764142\pi$
$79$	−1036.14	−1.47563	−0.737814	+	0.675004i	$0.764142\pi$
$80$	264.797	0.370066
$81$	−417.296	−0.572422
$82$	−293.141	−0.394781
$83$	328.846	0.434885	0.217443	−	0.976073i	$-0.430228\pi$
$83$	328.846	0.434885	0.217443	+	0.976073i	$0.430228\pi$
$84$	0	0
$85$	824.990	1.05274
$86$	−30.4053	−0.0381243
$87$	67.8696	0.0836366
$88$	−557.196	−0.674969
$89$	1580.83	1.88279	0.941393	−	0.337313i	$-0.109518\pi$
$89$	1580.83	1.88279	0.941393	+	0.337313i	$0.109518\pi$
$90$	−288.797	−0.338243
$91$	0	0
$92$	92.0000	0.104257
$93$	239.072	0.266566
$94$	1022.34	1.12177
$95$	−1502.68	−1.62286
$96$	−136.797	−0.145436
$97$	−344.460	−0.360563	−0.180282	−	0.983615i	$-0.557701\pi$
$97$	−344.460	−0.360563	−0.180282	+	0.983615i	$0.557701\pi$
$98$	0	0
$99$	607.698	0.616928
$100$	595.588	0.595588
$101$	−1264.25	−1.24552	−0.622759	−	0.782414i	$-0.713988\pi$
$101$	−1264.25	−1.24552	−0.622759	+	0.782414i	$0.713988\pi$
$102$	−426.199	−0.413726
$103$	33.5815	0.0321250	0.0160625	−	0.999871i	$-0.494887\pi$
$103$	33.5815	0.0321250	0.0160625	+	0.999871i	$0.494887\pi$
$104$	378.997	0.357343
$105$	0	0
$106$	−583.794	−0.534935
$107$	−1283.82	−1.15992	−0.579962	−	0.814644i	$-0.696932\pi$
$107$	−1283.82	−1.15992	−0.579962	+	0.814644i	$0.696932\pi$
$108$	610.887	0.544284
$109$	92.4336	0.0812251	0.0406125	−	0.999175i	$-0.487069\pi$
$109$	92.4336	0.0812251	0.0406125	+	0.999175i	$0.487069\pi$
$110$	−2305.38	−1.99826
$111$	42.3373	0.0362025
$112$	0	0
$113$	−408.061	−0.339710	−0.169855	−	0.985469i	$-0.554330\pi$
$113$	−408.061	−0.339710	−0.169855	+	0.985469i	$0.554330\pi$
$114$	776.302	0.637784
$115$	380.646	0.308656
$116$	−63.5050	−0.0508301
$117$	−413.347	−0.326615
$118$	−1073.28	−0.837317
$119$	0	0
$120$	−565.993	−0.430566
$121$	3520.05	2.64467
$122$	−959.890	−0.712331
$123$	626.577	0.459322
$124$	−223.698	−0.162005
$125$	395.492	0.282991
$126$	0	0
$127$	2065.05	1.44286	0.721432	−	0.692485i	$-0.243484\pi$
$127$	2065.05	1.44286	0.721432	+	0.692485i	$0.243484\pi$
$128$	128.000	0.0883883
$129$	64.9901	0.0443570
$130$	1568.08	1.05792
$131$	1428.10	0.952469	0.476234	−	0.879318i	$-0.342001\pi$
$131$	1428.10	0.952469	0.476234	+	0.879318i	$0.342001\pi$
$132$	1190.98	0.785317
$133$	0	0
$134$	−1848.47	−1.19167
$135$	2527.52	1.61137
$136$	398.791	0.251441
$137$	880.145	0.548875	0.274437	−	0.961605i	$-0.411508\pi$
$137$	880.145	0.548875	0.274437	+	0.961605i	$0.411508\pi$
$138$	−196.646	−0.121302
$139$	−1232.30	−0.751961	−0.375981	−	0.926628i	$-0.622694\pi$
$139$	−1232.30	−0.751961	−0.375981	+	0.926628i	$0.622694\pi$
$140$	0	0
$141$	−2185.20	−1.30516
$142$	471.630	0.278720
$143$	−3299.62	−1.92956
$144$	−139.601	−0.0807878
$145$	−262.749	−0.150484
$146$	−1626.25	−0.921848
$147$	0	0
$148$	−39.6146	−0.0220020
$149$	−447.827	−0.246224	−0.123112	−	0.992393i	$-0.539287\pi$
$149$	−447.827	−0.246224	−0.123112	+	0.992393i	$0.539287\pi$
$150$	−1273.04	−0.692958
$151$	−1461.73	−0.787773	−0.393886	−	0.919159i	$-0.628870\pi$
$151$	−1461.73	−0.787773	−0.393886	+	0.919159i	$0.628870\pi$
$152$	−726.379	−0.387613
$153$	−434.935	−0.229820
$154$	0	0
$155$	−925.540	−0.479620
$156$	−810.090	−0.415763
$157$	1034.17	0.525707	0.262853	−	0.964836i	$-0.415336\pi$
$157$	1034.17	0.525707	0.262853	+	0.964836i	$0.415336\pi$
$158$	−2072.28	−1.04343
$159$	1247.84	0.622389
$160$	529.595	0.261676
$161$	0	0
$162$	−834.591	−0.404764
$163$	−3843.27	−1.84680	−0.923398	−	0.383843i	$-0.874600\pi$
$163$	−3843.27	−1.84680	−0.923398	+	0.383843i	$0.874600\pi$
$164$	−586.282	−0.279152
$165$	4927.64	2.32495
$166$	657.691	0.307510
$167$	−3832.24	−1.77573	−0.887866	−	0.460102i	$-0.847813\pi$
$167$	−3832.24	−1.77573	−0.887866	+	0.460102i	$0.847813\pi$
$168$	0	0
$169$	47.3514	0.0215528
$170$	1649.98	0.744398
$171$	792.214	0.354282
$172$	−60.8106	−0.0269579
$173$	1042.76	0.458265	0.229132	−	0.973395i	$-0.426411\pi$
$173$	1042.76	0.458265	0.229132	+	0.973395i	$0.426411\pi$
$174$	135.739	0.0591400
$175$	0	0
$176$	−1114.39	−0.477275
$177$	2294.09	0.974205
$178$	3161.66	1.33133
$179$	1505.07	0.628457	0.314229	−	0.949347i	$-0.398254\pi$
$179$	1505.07	0.628457	0.314229	+	0.949347i	$0.398254\pi$
$180$	−577.595	−0.239174
$181$	1145.17	0.470274	0.235137	−	0.971962i	$-0.424446\pi$
$181$	1145.17	0.470274	0.235137	+	0.971962i	$0.424446\pi$
$182$	0	0
$183$	2051.73	0.828787
$184$	184.000	0.0737210
$185$	−163.904	−0.0651375
$186$	478.145	0.188491
$187$	−3471.95	−1.35772
$188$	2044.67	0.793209
$189$	0	0
$190$	−3005.36	−1.14754
$191$	−259.038	−0.0981327	−0.0490664	−	0.998796i	$-0.515625\pi$
$191$	−259.038	−0.0981327	−0.0490664	+	0.998796i	$0.515625\pi$
$192$	−273.595	−0.102839
$193$	1886.95	0.703761	0.351881	−	0.936045i	$-0.385542\pi$
$193$	1886.95	0.703761	0.351881	+	0.936045i	$0.385542\pi$
$194$	−688.920	−0.254957
$195$	−3351.71	−1.23088
$196$	0	0
$197$	−3761.16	−1.36026	−0.680130	−	0.733092i	$-0.738077\pi$
$197$	−3761.16	−1.36026	−0.680130	+	0.733092i	$0.738077\pi$
$198$	1215.40	0.436234
$199$	−1799.92	−0.641170	−0.320585	−	0.947220i	$-0.603880\pi$
$199$	−1799.92	−0.641170	−0.320585	+	0.947220i	$0.603880\pi$
$200$	1191.18	0.421144
$201$	3951.02	1.38649
$202$	−2528.50	−0.880714
$203$	0	0
$204$	−852.399	−0.292548
$205$	−2425.72	−0.826437
$206$	67.1629	0.0227158
$207$	−200.677	−0.0673817
$208$	757.993	0.252680
$209$	6323.99	2.09301
$210$	0	0
$211$	−5736.90	−1.87177	−0.935887	−	0.352300i	$-0.885400\pi$
$211$	−5736.90	−1.87177	−0.935887	+	0.352300i	$0.885400\pi$
$212$	−1167.59	−0.378256
$213$	−1008.09	−0.324287
$214$	−2567.64	−0.820190
$215$	−251.601	−0.0798096
$216$	1221.77	0.384867
$217$	0	0
$218$	184.867	0.0574348
$219$	3476.05	1.07256
$220$	−4610.75	−1.41299
$221$	2361.57	0.718807
$222$	84.6745	0.0255990
$223$	−4840.19	−1.45347	−0.726734	−	0.686919i	$-0.758963\pi$
$223$	−4840.19	−1.45347	−0.726734	+	0.686919i	$0.758963\pi$
$224$	0	0
$225$	−1299.14	−0.384930
$226$	−816.123	−0.240211
$227$	−2402.95	−0.702596	−0.351298	−	0.936264i	$-0.614260\pi$
$227$	−2402.95	−0.702596	−0.351298	+	0.936264i	$0.614260\pi$
$228$	1552.60	0.450981
$229$	3738.48	1.07880	0.539400	−	0.842049i	$-0.318651\pi$
$229$	3738.48	1.07880	0.539400	+	0.842049i	$0.318651\pi$
$230$	761.292	0.218253
$231$	0	0
$232$	−127.010	−0.0359423
$233$	−1892.45	−0.532097	−0.266049	−	0.963960i	$-0.585718\pi$
$233$	−1892.45	−0.532097	−0.266049	+	0.963960i	$0.585718\pi$
$234$	−826.694	−0.230952
$235$	8459.76	2.34831
$236$	−2146.56	−0.592072
$237$	4429.40	1.21401
$238$	0	0
$239$	2127.34	0.575757	0.287879	−	0.957667i	$-0.407050\pi$
$239$	2127.34	0.575757	0.287879	+	0.957667i	$0.407050\pi$
$240$	−1131.99	−0.304456
$241$	6777.26	1.81146	0.905730	−	0.423856i	$-0.139324\pi$
$241$	6777.26	1.81146	0.905730	+	0.423856i	$0.139324\pi$
$242$	7040.11	1.87006
$243$	−2339.58	−0.617631
$244$	−1919.78	−0.503694
$245$	0	0
$246$	1253.15	0.324789
$247$	−4301.49	−1.10809
$248$	−447.395	−0.114555
$249$	−1405.79	−0.357784
$250$	790.983	0.200105
$251$	−94.3137	−0.0237173	−0.0118586	−	0.999930i	$-0.503775\pi$
$251$	−94.3137	−0.0237173	−0.0118586	+	0.999930i	$0.503775\pi$
$252$	0	0
$253$	−1601.94	−0.398075
$254$	4130.10	1.02026
$255$	−3526.76	−0.866096
$256$	256.000	0.0625000
$257$	2564.12	0.622357	0.311178	−	0.950352i	$-0.399276\pi$
$257$	2564.12	0.622357	0.311178	+	0.950352i	$0.399276\pi$
$258$	129.980	0.0313652
$259$	0	0
$260$	3136.17	0.748065
$261$	138.522	0.0328516
$262$	2856.19	0.673497
$263$	−5576.13	−1.30737	−0.653686	−	0.756766i	$-0.726778\pi$
$263$	−5576.13	−1.30737	−0.653686	+	0.756766i	$0.726778\pi$
$264$	2381.97	0.555303
$265$	−4830.85	−1.11984
$266$	0	0
$267$	−6757.93	−1.54898
$268$	−3696.94	−0.842636
$269$	−5025.71	−1.13912	−0.569560	−	0.821950i	$-0.692886\pi$
$269$	−5025.71	−1.13912	−0.569560	+	0.821950i	$0.692886\pi$
$270$	5055.04	1.13941
$271$	−3210.70	−0.719690	−0.359845	−	0.933012i	$-0.617170\pi$
$271$	−3210.70	−0.719690	−0.359845	+	0.933012i	$0.617170\pi$
$272$	797.581	0.177796
$273$	0	0
$274$	1760.29	0.388113
$275$	−10370.6	−2.27408
$276$	−393.292	−0.0857733
$277$	−5020.87	−1.08908	−0.544539	−	0.838736i	$-0.683295\pi$
$277$	−5020.87	−1.08908	−0.544539	+	0.838736i	$0.683295\pi$
$278$	−2464.61	−0.531717
$279$	487.945	0.104704
$280$	0	0
$281$	5733.10	1.21711	0.608555	−	0.793512i	$-0.291749\pi$
$281$	5733.10	1.21711	0.608555	+	0.793512i	$0.291749\pi$
$282$	−4370.41	−0.922886
$283$	3103.76	0.651941	0.325970	−	0.945380i	$-0.394309\pi$
$283$	3103.76	0.651941	0.325970	+	0.945380i	$0.394309\pi$
$284$	943.259	0.197085
$285$	6423.84	1.33514
$286$	−6599.23	−1.36441
$287$	0	0
$288$	−279.203	−0.0571256
$289$	−2428.09	−0.494218
$290$	−525.498	−0.106408
$291$	1472.54	0.296638
$292$	−3252.51	−0.651845
$293$	−5521.17	−1.10085	−0.550427	−	0.834883i	$-0.685535\pi$
$293$	−5521.17	−1.10085	−0.550427	+	0.834883i	$0.685535\pi$
$294$	0	0
$295$	−8881.30	−1.75284
$296$	−79.2291	−0.0155578
$297$	−10637.0	−2.07819
$298$	−895.654	−0.174107
$299$	1089.62	0.210749
$300$	−2546.09	−0.489995
$301$	0	0
$302$	−2923.45	−0.557039
$303$	5404.55	1.02470
$304$	−1452.76	−0.274083
$305$	−7943.01	−1.49120
$306$	−869.871	−0.162507
$307$	8965.97	1.66682	0.833412	−	0.552653i	$-0.186384\pi$
$307$	8965.97	1.66682	0.833412	+	0.552653i	$0.186384\pi$
$308$	0	0
$309$	−143.558	−0.0264295
$310$	−1851.08	−0.339143
$311$	4581.45	0.835339	0.417670	−	0.908599i	$-0.362847\pi$
$311$	4581.45	0.835339	0.417670	+	0.908599i	$0.362847\pi$
$312$	−1620.18	−0.293989
$313$	5708.68	1.03091	0.515453	−	0.856918i	$-0.327624\pi$
$313$	5708.68	1.03091	0.515453	+	0.856918i	$0.327624\pi$
$314$	2068.35	0.371731
$315$	0	0
$316$	−4144.55	−0.737814
$317$	−9254.54	−1.63971	−0.819853	−	0.572574i	$-0.805945\pi$
$317$	−9254.54	−1.63971	−0.819853	+	0.572574i	$0.805945\pi$
$318$	2495.67	0.440095
$319$	1105.77	0.194080
$320$	1059.19	0.185033
$321$	5488.23	0.954278
$322$	0	0
$323$	−4526.14	−0.779694
$324$	−1669.18	−0.286211
$325$	7053.93	1.20394
$326$	−7686.53	−1.30588
$327$	−395.146	−0.0668245
$328$	−1172.56	−0.197390
$329$	0	0
$330$	9855.29	1.64399
$331$	−5003.49	−0.830865	−0.415433	−	0.909624i	$-0.636370\pi$
$331$	−5003.49	−0.830865	−0.415433	+	0.909624i	$0.636370\pi$
$332$	1315.38	0.217443
$333$	86.4101	0.0142200
$334$	−7664.47	−1.25563
$335$	−15295.9	−2.49464
$336$	0	0
$337$	3428.92	0.554259	0.277129	−	0.960833i	$-0.410617\pi$
$337$	3428.92	0.554259	0.277129	+	0.960833i	$0.410617\pi$
$338$	94.7028	0.0152401
$339$	1744.43	0.279482
$340$	3299.96	0.526369
$341$	3895.11	0.618569
$342$	1584.43	0.250515
$343$	0	0
$344$	−121.621	−0.0190621
$345$	−1627.23	−0.253934
$346$	2085.52	0.324042
$347$	−6784.50	−1.04960	−0.524799	−	0.851226i	$-0.675860\pi$
$347$	−6784.50	−1.04960	−0.524799	+	0.851226i	$0.675860\pi$
$348$	271.478	0.0418183
$349$	1140.34	0.174903	0.0874513	−	0.996169i	$-0.472128\pi$
$349$	1140.34	0.174903	0.0874513	+	0.996169i	$0.472128\pi$
$350$	0	0
$351$	7235.13	1.10024
$352$	−2228.78	−0.337485
$353$	11.2017	0.00168898	0.000844488	−	1.00000i	$-0.499731\pi$
$353$	11.2017	0.00168898	0.000844488		1.00000i	$0.499731\pi$
$354$	4588.18	0.688867
$355$	3902.70	0.583475
$356$	6323.33	0.941393
$357$	0	0
$358$	3010.13	0.444387
$359$	8360.60	1.22912	0.614562	−	0.788869i	$-0.289333\pi$
$359$	8360.60	1.22912	0.614562	+	0.788869i	$0.289333\pi$
$360$	−1155.19	−0.169122
$361$	1385.16	0.201948
$362$	2290.34	0.332534
$363$	−15047.9	−2.17579
$364$	0	0
$365$	−13457.1	−1.92980
$366$	4103.45	0.586041
$367$	−8589.92	−1.22177	−0.610886	−	0.791719i	$-0.709186\pi$
$367$	−8589.92	−1.22177	−0.610886	+	0.791719i	$0.709186\pi$
$368$	368.000	0.0521286
$369$	1278.84	0.180417
$370$	−327.807	−0.0460592
$371$	0	0
$372$	956.289	0.133283
$373$	6524.79	0.905740	0.452870	−	0.891577i	$-0.350400\pi$
$373$	6524.79	0.905740	0.452870	+	0.891577i	$0.350400\pi$
$374$	−6943.89	−0.960054
$375$	−1690.69	−0.232819
$376$	4089.35	0.560883
$377$	−752.130	−0.102750
$378$	0	0
$379$	5457.54	0.739670	0.369835	−	0.929097i	$-0.379414\pi$
$379$	5457.54	0.739670	0.369835	+	0.929097i	$0.379414\pi$
$380$	−6010.72	−0.811431
$381$	−8827.93	−1.18706
$382$	−518.076	−0.0693903
$383$	13629.3	1.81834	0.909172	−	0.416421i	$-0.136716\pi$
$383$	13629.3	1.81834	0.909172	+	0.416421i	$0.136716\pi$
$384$	−547.189	−0.0727178
$385$	0	0
$386$	3773.91	0.497634
$387$	132.644	0.0174230
$388$	−1377.84	−0.180282
$389$	−282.051	−0.0367624	−0.0183812	−	0.999831i	$-0.505851\pi$
$389$	−282.051	−0.0367624	−0.0183812	+	0.999831i	$0.505851\pi$
$390$	−6703.43	−0.870362
$391$	1146.52	0.148292
$392$	0	0
$393$	−6105.00	−0.783604
$394$	−7522.31	−0.961849
$395$	−17147.9	−2.18432
$396$	2430.79	0.308464
$397$	−9681.76	−1.22396	−0.611982	−	0.790872i	$-0.709627\pi$
$397$	−9681.76	−1.22396	−0.611982	+	0.790872i	$0.709627\pi$
$398$	−3599.84	−0.453376
$399$	0	0
$400$	2382.35	0.297794
$401$	−8285.22	−1.03178	−0.515891	−	0.856654i	$-0.672539\pi$
$401$	−8285.22	−1.03178	−0.515891	+	0.856654i	$0.672539\pi$
$402$	7902.05	0.980394
$403$	−2649.40	−0.327483
$404$	−5056.99	−0.622759
$405$	−6906.17	−0.847335
$406$	0	0
$407$	689.784	0.0840081
$408$	−1704.80	−0.206863
$409$	10448.7	1.26322	0.631609	−	0.775287i	$-0.282395\pi$
$409$	10448.7	1.26322	0.631609	+	0.775287i	$0.282395\pi$
$410$	−4851.44	−0.584379
$411$	−3762.55	−0.451564
$412$	134.326	0.0160625
$413$	0	0
$414$	−401.354	−0.0476460
$415$	5442.34	0.643744
$416$	1515.99	0.178672
$417$	5267.99	0.618645
$418$	12648.0	1.47998
$419$	−4158.98	−0.484915	−0.242457	−	0.970162i	$-0.577953\pi$
$419$	−4158.98	−0.484915	−0.242457	+	0.970162i	$0.577953\pi$
$420$	0	0
$421$	−8981.73	−1.03977	−0.519884	−	0.854237i	$-0.674025\pi$
$421$	−8981.73	−1.03977	−0.519884	+	0.854237i	$0.674025\pi$
$422$	−11473.8	−1.32354
$423$	−4459.99	−0.512653
$424$	−2335.18	−0.267467
$425$	7422.34	0.847145
$426$	−2016.18	−0.229305
$427$	0	0
$428$	−5135.29	−0.579962
$429$	14105.6	1.58747
$430$	−503.203	−0.0564339
$431$	4982.65	0.556858	0.278429	−	0.960457i	$-0.410186\pi$
$431$	4982.65	0.556858	0.278429	+	0.960457i	$0.410186\pi$
$432$	2443.55	0.272142
$433$	−8314.43	−0.922785	−0.461392	−	0.887196i	$-0.652650\pi$
$433$	−8314.43	−0.922785	−0.461392	+	0.887196i	$0.652650\pi$
$434$	0	0
$435$	1123.23	0.123804
$436$	369.734	0.0406125
$437$	−2088.34	−0.228601
$438$	6952.10	0.758412
$439$	−10101.7	−1.09824	−0.549122	−	0.835742i	$-0.685038\pi$
$439$	−10101.7	−1.09824	−0.549122	+	0.835742i	$0.685038\pi$
$440$	−9221.50	−0.999131
$441$	0	0
$442$	4723.14	0.508273
$443$	−13940.3	−1.49509	−0.747544	−	0.664212i	$-0.768767\pi$
$443$	−13940.3	−1.49509	−0.747544	+	0.664212i	$0.768767\pi$
$444$	169.349	0.0181012
$445$	26162.5	2.78702
$446$	−9680.39	−1.02776
$447$	1914.42	0.202571
$448$	0	0
$449$	−16667.7	−1.75188	−0.875942	−	0.482416i	$-0.839759\pi$
$449$	−16667.7	−1.75188	−0.875942	+	0.482416i	$0.839759\pi$
$450$	−2598.28	−0.272187
$451$	10208.6	1.06586
$452$	−1632.25	−0.169855
$453$	6248.76	0.648107
$454$	−4805.90	−0.496810
$455$	0	0
$456$	3105.21	0.318892
$457$	4095.33	0.419194	0.209597	−	0.977788i	$-0.432785\pi$
$457$	4095.33	0.419194	0.209597	+	0.977788i	$0.432785\pi$
$458$	7476.95	0.762827
$459$	7613.00	0.774171
$460$	1522.58	0.154328
$461$	3128.82	0.316103	0.158052	−	0.987431i	$-0.449479\pi$
$461$	3128.82	0.316103	0.158052	+	0.987431i	$0.449479\pi$
$462$	0	0
$463$	−3612.65	−0.362622	−0.181311	−	0.983426i	$-0.558034\pi$
$463$	−3612.65	−0.362622	−0.181311	+	0.983426i	$0.558034\pi$
$464$	−254.020	−0.0254150
$465$	3956.61	0.394588
$466$	−3784.91	−0.376250
$467$	11625.8	1.15199	0.575996	−	0.817453i	$-0.304614\pi$
$467$	11625.8	1.15199	0.575996	+	0.817453i	$0.304614\pi$
$468$	−1653.39	−0.163308
$469$	0	0
$470$	16919.5	1.66051
$471$	−4421.00	−0.432503
$472$	−4293.12	−0.418658
$473$	1058.86	0.102931
$474$	8858.81	0.858436
$475$	−13519.5	−1.30593
$476$	0	0
$477$	2546.83	0.244468
$478$	4254.67	0.407122
$479$	−11893.7	−1.13453	−0.567264	−	0.823536i	$-0.691998\pi$
$479$	−11893.7	−1.13453	−0.567264	+	0.823536i	$0.691998\pi$
$480$	−2263.97	−0.215283
$481$	−469.181	−0.0444757
$482$	13554.5	1.28090
$483$	0	0
$484$	14080.2	1.32233
$485$	−5700.76	−0.533728
$486$	−4679.17	−0.436731
$487$	12344.8	1.14866	0.574329	−	0.818624i	$-0.305263\pi$
$487$	12344.8	1.14866	0.574329	+	0.818624i	$0.305263\pi$
$488$	−3839.56	−0.356166
$489$	16429.6	1.51937
$490$	0	0
$491$	16721.7	1.53694	0.768471	−	0.639884i	$-0.221018\pi$
$491$	16721.7	1.53694	0.768471	+	0.639884i	$0.221018\pi$
$492$	2506.31	0.229661
$493$	−791.412	−0.0722990
$494$	−8602.97	−0.783535
$495$	10057.3	0.913216
$496$	−894.791	−0.0810026
$497$	0	0
$498$	−2811.57	−0.252991
$499$	−16623.2	−1.49130	−0.745650	−	0.666338i	$-0.767861\pi$
$499$	−16623.2	−1.49130	−0.745650	+	0.666338i	$0.767861\pi$
$500$	1581.97	0.141495
$501$	16382.5	1.46091
$502$	−188.627	−0.0167706
$503$	−7775.55	−0.689254	−0.344627	−	0.938740i	$-0.611995\pi$
$503$	−7775.55	−0.689254	−0.344627	+	0.938740i	$0.611995\pi$
$504$	0	0
$505$	−20923.1	−1.84369
$506$	−3203.88	−0.281482
$507$	−202.423	−0.0177316
$508$	8260.21	0.721432
$509$	−20295.0	−1.76731	−0.883654	−	0.468140i	$-0.844924\pi$
$509$	−20295.0	−1.76731	−0.883654	+	0.468140i	$0.844924\pi$
$510$	−7053.53	−0.612423
$511$	0	0
$512$	512.000	0.0441942
$513$	−13866.7	−1.19343
$514$	5128.25	0.440073
$515$	555.768	0.0475535
$516$	259.960	0.0221785
$517$	−35602.6	−3.02863
$518$	0	0
$519$	−4457.72	−0.377018
$520$	6272.33	0.528961
$521$	3576.53	0.300750	0.150375	−	0.988629i	$-0.451952\pi$
$521$	3576.53	0.300750	0.150375	+	0.988629i	$0.451952\pi$
$522$	277.043	0.0232296
$523$	−270.391	−0.0226068	−0.0113034	−	0.999936i	$-0.503598\pi$
$523$	−270.391	−0.0226068	−0.0113034	+	0.999936i	$0.503598\pi$
$524$	5712.39	0.476234
$525$	0	0
$526$	−11152.3	−0.924452
$527$	−2787.77	−0.230431
$528$	4763.93	0.392658
$529$	529.000	0.0434783
$530$	−9661.69	−0.791844
$531$	4682.22	0.382658
$532$	0	0
$533$	−6943.72	−0.564289
$534$	−13515.9	−1.09530
$535$	−21247.0	−1.71699
$536$	−7393.87	−0.595833
$537$	−6434.03	−0.517037
$538$	−10051.4	−0.805479
$539$	0	0
$540$	10110.1	0.805683
$541$	14438.1	1.14740	0.573701	−	0.819065i	$-0.305507\pi$
$541$	14438.1	1.14740	0.573701	+	0.819065i	$0.305507\pi$
$542$	−6421.40	−0.508898
$543$	−4895.50	−0.386899
$544$	1595.16	0.125721
$545$	1529.76	0.120234
$546$	0	0
$547$	21656.6	1.69281	0.846407	−	0.532537i	$-0.178761\pi$
$547$	21656.6	1.69281	0.846407	+	0.532537i	$0.178761\pi$
$548$	3520.58	0.274437
$549$	4187.56	0.325539
$550$	−20741.2	−1.60801
$551$	1441.52	0.111453
$552$	−786.585	−0.0606509
$553$	0	0
$554$	−10041.7	−0.770094
$555$	700.675	0.0535891
$556$	−4929.21	−0.375981
$557$	5635.65	0.428708	0.214354	−	0.976756i	$-0.431235\pi$
$557$	5635.65	0.428708	0.214354	+	0.976756i	$0.431235\pi$
$558$	975.890	0.0740372
$559$	−720.219	−0.0544938
$560$	0	0
$561$	14842.3	1.11701
$562$	11466.2	0.860627
$563$	18250.2	1.36617	0.683084	−	0.730340i	$-0.260638\pi$
$563$	18250.2	1.36617	0.683084	+	0.730340i	$0.260638\pi$
$564$	−8740.81	−0.652579
$565$	−6753.35	−0.502859
$566$	6207.51	0.460992
$567$	0	0
$568$	1886.52	0.139360
$569$	116.360	0.00857302	0.00428651	−	0.999991i	$-0.498636\pi$
$569$	116.360	0.00857302	0.00428651	+	0.999991i	$0.498636\pi$
$570$	12847.7	0.944088
$571$	2142.29	0.157009	0.0785045	−	0.996914i	$-0.474986\pi$
$571$	2142.29	0.157009	0.0785045	+	0.996914i	$0.474986\pi$
$572$	−13198.5	−0.964782
$573$	1107.37	0.0807346
$574$	0	0
$575$	3424.63	0.248377
$576$	−558.405	−0.0403939
$577$	5115.43	0.369078	0.184539	−	0.982825i	$-0.440921\pi$
$577$	5115.43	0.369078	0.184539	+	0.982825i	$0.440921\pi$
$578$	−4856.19	−0.349465
$579$	−8066.57	−0.578990
$580$	−1051.00	−0.0752418
$581$	0	0
$582$	2945.08	0.209755
$583$	20330.5	1.44426
$584$	−6505.02	−0.460924
$585$	−6840.83	−0.483476
$586$	−11042.3	−0.778422
$587$	10411.5	0.732075	0.366037	−	0.930600i	$-0.380714\pi$
$587$	10411.5	0.732075	0.366037	+	0.930600i	$0.380714\pi$
$588$	0	0
$589$	5077.79	0.355224
$590$	−17762.6	−1.23945
$591$	16078.6	1.11910
$592$	−158.458	−0.0110010
$593$	1707.70	0.118258	0.0591288	−	0.998250i	$-0.481168\pi$
$593$	1707.70	0.118258	0.0591288	+	0.998250i	$0.481168\pi$
$594$	−21274.0	−1.46950
$595$	0	0
$596$	−1791.31	−0.123112
$597$	7694.50	0.527496
$598$	2179.23	0.149022
$599$	−5529.49	−0.377177	−0.188588	−	0.982056i	$-0.560391\pi$
$599$	−5529.49	−0.377177	−0.188588	+	0.982056i	$0.560391\pi$
$600$	−5092.18	−0.346479
$601$	3976.19	0.269870	0.134935	−	0.990854i	$-0.456917\pi$
$601$	3976.19	0.269870	0.134935	+	0.990854i	$0.456917\pi$
$602$	0	0
$603$	8064.02	0.544598
$604$	−5846.91	−0.393886
$605$	58256.3	3.91480
$606$	10809.1	0.724571
$607$	−25927.8	−1.73374	−0.866868	−	0.498538i	$-0.833870\pi$
$607$	−25927.8	−1.73374	−0.866868	+	0.498538i	$0.833870\pi$
$608$	−2905.52	−0.193806
$609$	0	0
$610$	−15886.0	−1.05444
$611$	24216.4	1.60342
$612$	−1739.74	−0.114910
$613$	6153.62	0.405453	0.202726	−	0.979235i	$-0.435020\pi$
$613$	6153.62	0.405453	0.202726	+	0.979235i	$0.435020\pi$
$614$	17931.9	1.17862
$615$	10369.7	0.679917
$616$	0	0
$617$	25144.4	1.64064	0.820320	−	0.571904i	$-0.193795\pi$
$617$	25144.4	1.64064	0.820320	+	0.571904i	$0.193795\pi$
$618$	−287.116	−0.0186885
$619$	−25006.2	−1.62372	−0.811860	−	0.583852i	$-0.801545\pi$
$619$	−25006.2	−1.62372	−0.811860	+	0.583852i	$0.801545\pi$
$620$	−3702.16	−0.239810
$621$	3512.60	0.226982
$622$	9162.91	0.590674
$623$	0	0
$624$	−3240.36	−0.207882
$625$	−12066.8	−0.772276
$626$	11417.4	0.728961
$627$	−27034.5	−1.72194
$628$	4136.69	0.262853
$629$	−493.685	−0.0312949
$630$	0	0
$631$	5044.71	0.318267	0.159134	−	0.987257i	$-0.449130\pi$
$631$	5044.71	0.318267	0.159134	+	0.987257i	$0.449130\pi$
$632$	−8289.10	−0.521713
$633$	24524.8	1.53992
$634$	−18509.1	−1.15945
$635$	34176.3	2.13582
$636$	4991.34	0.311194
$637$	0	0
$638$	2211.54	0.137235
$639$	−2057.50	−0.127377
$640$	2118.38	0.130838
$641$	−1661.99	−0.102410	−0.0512049	−	0.998688i	$-0.516306\pi$
$641$	−1661.99	−0.102410	−0.0512049	+	0.998688i	$0.516306\pi$
$642$	10976.5	0.674777
$643$	−24928.8	−1.52892	−0.764462	−	0.644669i	$-0.776995\pi$
$643$	−24928.8	−1.52892	−0.764462	+	0.644669i	$0.776995\pi$
$644$	0	0
$645$	1075.57	0.0656600
$646$	−9052.29	−0.551327
$647$	18562.3	1.12791	0.563955	−	0.825806i	$-0.309279\pi$
$647$	18562.3	1.12791	0.563955	+	0.825806i	$0.309279\pi$
$648$	−3338.37	−0.202382
$649$	37376.7	2.26065
$650$	14107.9	0.851317
$651$	0	0
$652$	−15373.1	−0.923398
$653$	14415.0	0.863861	0.431931	−	0.901907i	$-0.357833\pi$
$653$	14415.0	0.863861	0.431931	+	0.901907i	$0.357833\pi$
$654$	−790.292	−0.0472521
$655$	23634.8	1.40990
$656$	−2345.13	−0.139576
$657$	7094.60	0.421289
$658$	0	0
$659$	−20501.3	−1.21186	−0.605931	−	0.795518i	$-0.707199\pi$
$659$	−20501.3	−1.21186	−0.605931	+	0.795518i	$0.707199\pi$
$660$	19710.6	1.16247
$661$	24331.7	1.43176	0.715881	−	0.698223i	$-0.246026\pi$
$661$	24331.7	1.43176	0.715881	+	0.698223i	$0.246026\pi$
$662$	−10007.0	−0.587511
$663$	−10095.5	−0.591368
$664$	2630.76	0.153755
$665$	0	0
$666$	172.820	0.0100550
$667$	−365.154	−0.0211976
$668$	−15328.9	−0.887866
$669$	20691.4	1.19578
$670$	−30591.8	−1.76398
$671$	33427.9	1.92321
$672$	0	0
$673$	−11878.1	−0.680337	−0.340169	−	0.940364i	$-0.610484\pi$
$673$	−11878.1	−0.680337	−0.340169	+	0.940364i	$0.610484\pi$
$674$	6857.84	0.391920
$675$	22739.8	1.29668
$676$	189.406	0.0107764
$677$	12375.0	0.702526	0.351263	−	0.936277i	$-0.385752\pi$
$677$	12375.0	0.702526	0.351263	+	0.936277i	$0.385752\pi$
$678$	3488.86	0.197624
$679$	0	0
$680$	6599.92	0.372199
$681$	10272.4	0.578032
$682$	7790.22	0.437394
$683$	31113.7	1.74309	0.871545	−	0.490315i	$-0.163118\pi$
$683$	31113.7	1.74309	0.871545	+	0.490315i	$0.163118\pi$
$684$	3168.86	0.177141
$685$	14566.2	0.812478
$686$	0	0
$687$	−15981.7	−0.887538
$688$	−243.242	−0.0134790
$689$	−13828.5	−0.764621
$690$	−3254.46	−0.179558
$691$	29283.7	1.61216	0.806081	−	0.591805i	$-0.201585\pi$
$691$	29283.7	1.61216	0.806081	+	0.591805i	$0.201585\pi$
$692$	4171.05	0.229132
$693$	0	0
$694$	−13569.0	−0.742178
$695$	−20394.4	−1.11310
$696$	542.957	0.0295700
$697$	−7306.38	−0.397057
$698$	2280.68	0.123675
$699$	8090.08	0.437761
$700$	0	0
$701$	−35863.5	−1.93231	−0.966154	−	0.257968i	$-0.916947\pi$
$701$	−35863.5	−1.93231	−0.966154	+	0.257968i	$0.916947\pi$
$702$	14470.3	0.777984
$703$	899.224	0.0482431
$704$	−4457.57	−0.238638
$705$	−36164.8	−1.93198
$706$	22.4035	0.00119429
$707$	0	0
$708$	9176.36	0.487103
$709$	19457.1	1.03064	0.515322	−	0.856997i	$-0.327672\pi$
$709$	19457.1	1.03064	0.515322	+	0.856997i	$0.327672\pi$
$710$	7805.39	0.412579
$711$	9040.39	0.476851
$712$	12646.7	0.665665
$713$	−1286.26	−0.0675608
$714$	0	0
$715$	−54608.1	−2.85626
$716$	6020.26	0.314229
$717$	−9094.19	−0.473680
$718$	16721.2	0.869122
$719$	31205.7	1.61860	0.809302	−	0.587393i	$-0.199846\pi$
$719$	31205.7	1.61860	0.809302	+	0.587393i	$0.199846\pi$
$720$	−2310.38	−0.119587
$721$	0	0
$722$	2770.32	0.142799
$723$	−28972.2	−1.49030
$724$	4580.67	0.235137
$725$	−2363.93	−0.121095
$726$	−30095.9	−1.53852
$727$	12061.2	0.615303	0.307651	−	0.951499i	$-0.400457\pi$
$727$	12061.2	0.615303	0.307651	+	0.951499i	$0.400457\pi$
$728$	0	0
$729$	21268.5	1.08055
$730$	−26914.2	−1.36458
$731$	−757.834	−0.0383441
$732$	8206.90	0.414393
$733$	7873.46	0.396744	0.198372	−	0.980127i	$-0.436435\pi$
$733$	7873.46	0.396744	0.198372	+	0.980127i	$0.436435\pi$
$734$	−17179.8	−0.863923
$735$	0	0
$736$	736.000	0.0368605
$737$	64372.5	3.21735
$738$	2557.68	0.127574
$739$	37979.9	1.89055	0.945273	−	0.326281i	$-0.105796\pi$
$739$	37979.9	1.89055	0.945273	+	0.326281i	$0.105796\pi$
$740$	−655.615	−0.0325687
$741$	18388.5	0.911631
$742$	0	0
$743$	27163.4	1.34122	0.670611	−	0.741809i	$-0.266032\pi$
$743$	27163.4	1.34122	0.670611	+	0.741809i	$0.266032\pi$
$744$	1912.58	0.0942453
$745$	−7411.47	−0.364477
$746$	13049.6	0.640455
$747$	−2869.20	−0.140534
$748$	−13887.8	−0.678861
$749$	0	0
$750$	−3381.39	−0.164628
$751$	37879.2	1.84052	0.920261	−	0.391305i	$-0.127976\pi$
$751$	37879.2	1.84052	0.920261	+	0.391305i	$0.127976\pi$
$752$	8178.70	0.396604
$753$	403.183	0.0195124
$754$	−1504.26	−0.0726551
$755$	−24191.3	−1.16611
$756$	0	0
$757$	20608.1	0.989450	0.494725	−	0.869049i	$-0.335269\pi$
$757$	20608.1	0.989450	0.494725	+	0.869049i	$0.335269\pi$
$758$	10915.1	0.523026
$759$	6848.15	0.327500
$760$	−12021.4	−0.573768
$761$	−16681.7	−0.794629	−0.397315	−	0.917682i	$-0.630058\pi$
$761$	−16681.7	−0.794629	−0.397315	+	0.917682i	$0.630058\pi$
$762$	−17655.9	−0.839376
$763$	0	0
$764$	−1036.15	−0.0490664
$765$	−7198.11	−0.340194
$766$	27258.6	1.28576
$767$	−25423.1	−1.19684
$768$	−1094.38	−0.0514193
$769$	−1227.39	−0.0575561	−0.0287780	−	0.999586i	$-0.509162\pi$
$769$	−1227.39	−0.0575561	−0.0287780	+	0.999586i	$0.509162\pi$
$770$	0	0
$771$	−10961.4	−0.512018
$772$	7547.82	0.351881
$773$	−28842.7	−1.34204	−0.671021	−	0.741438i	$-0.734144\pi$
$773$	−28842.7	−1.34204	−0.671021	+	0.741438i	$0.734144\pi$
$774$	265.289	0.0123199
$775$	−8326.98	−0.385953
$776$	−2755.68	−0.127478
$777$	0	0
$778$	−564.102	−0.0259949
$779$	13308.2	0.612088
$780$	−13406.9	−0.615439
$781$	−16424.4	−0.752510
$782$	2293.05	0.104858
$783$	−2424.65	−0.110664
$784$	0	0
$785$	17115.4	0.778184
$786$	−12210.0	−0.554092
$787$	−35236.0	−1.59597	−0.797984	−	0.602678i	$-0.794100\pi$
$787$	−35236.0	−1.59597	−0.797984	+	0.602678i	$0.794100\pi$
$788$	−15044.6	−0.680130
$789$	23837.5	1.07559
$790$	−34295.8	−1.54455
$791$	0	0
$792$	4861.58	0.218117
$793$	−22737.2	−1.01819
$794$	−19363.5	−0.865473
$795$	20651.5	0.921298
$796$	−7199.67	−0.320585
$797$	7775.38	0.345569	0.172784	−	0.984960i	$-0.444724\pi$
$797$	7775.38	0.345569	0.172784	+	0.984960i	$0.444724\pi$
$798$	0	0
$799$	25481.2	1.12823
$800$	4764.70	0.210572
$801$	−13792.9	−0.608424
$802$	−16570.4	−0.729580
$803$	56633.9	2.48888
$804$	15804.1	0.693243
$805$	0	0
$806$	−5298.79	−0.231566
$807$	21484.5	0.937163
$808$	−10114.0	−0.440357
$809$	22756.3	0.988962	0.494481	−	0.869188i	$-0.335358\pi$
$809$	22756.3	0.988962	0.494481	+	0.869188i	$0.335358\pi$
$810$	−13812.3	−0.599156
$811$	29622.7	1.28260	0.641302	−	0.767288i	$-0.278394\pi$
$811$	29622.7	1.28260	0.641302	+	0.767288i	$0.278394\pi$
$812$	0	0
$813$	13725.5	0.592095
$814$	1379.57	0.0594027
$815$	−63605.4	−2.73374
$816$	−3409.59	−0.146274
$817$	1380.36	0.0591098
$818$	20897.4	0.893230
$819$	0	0
$820$	−9702.88	−0.413219
$821$	18489.2	0.785967	0.392983	−	0.919546i	$-0.371443\pi$
$821$	18489.2	0.785967	0.392983	+	0.919546i	$0.371443\pi$
$822$	−7525.09	−0.319304
$823$	−29486.2	−1.24887	−0.624437	−	0.781075i	$-0.714672\pi$
$823$	−29486.2	−1.24887	−0.624437	+	0.781075i	$0.714672\pi$
$824$	268.652	0.0113579
$825$	44333.5	1.87090
$826$	0	0
$827$	33231.6	1.39731	0.698656	−	0.715458i	$-0.253782\pi$
$827$	33231.6	1.39731	0.698656	+	0.715458i	$0.253782\pi$
$828$	−802.708	−0.0336908
$829$	−12663.3	−0.530535	−0.265267	−	0.964175i	$-0.585460\pi$
$829$	−12663.3	−0.530535	−0.265267	+	0.964175i	$0.585460\pi$
$830$	10884.7	0.455196
$831$	21463.8	0.895993
$832$	3031.97	0.126340
$833$	0	0
$834$	10536.0	0.437448
$835$	−63422.9	−2.62855
$836$	25296.0	1.04651
$837$	−8540.88	−0.352707
$838$	−8317.95	−0.342886
$839$	42098.8	1.73231	0.866156	−	0.499773i	$-0.166583\pi$
$839$	42098.8	1.73231	0.866156	+	0.499773i	$0.166583\pi$
$840$	0	0
$841$	−24136.9	−0.989665
$842$	−17963.5	−0.735228
$843$	−24508.5	−1.00133
$844$	−22947.6	−0.935887
$845$	783.658	0.0319037
$846$	−8919.98	−0.362500
$847$	0	0
$848$	−4670.35	−0.189128
$849$	−13268.3	−0.536357
$850$	14844.7	0.599022
$851$	−227.784	−0.00917547
$852$	−4032.36	−0.162143
$853$	37986.2	1.52476	0.762382	−	0.647127i	$-0.224030\pi$
$853$	37986.2	1.52476	0.762382	+	0.647127i	$0.224030\pi$
$854$	0	0
$855$	13111.0	0.524430
$856$	−10270.6	−0.410095
$857$	−9153.59	−0.364855	−0.182427	−	0.983219i	$-0.558395\pi$
$857$	−9153.59	−0.364855	−0.182427	+	0.983219i	$0.558395\pi$
$858$	28211.2	1.12251
$859$	−6963.50	−0.276591	−0.138296	−	0.990391i	$-0.544162\pi$
$859$	−6963.50	−0.276591	−0.138296	+	0.990391i	$0.544162\pi$
$860$	−1006.41	−0.0399048
$861$	0	0
$862$	9965.30	0.393758
$863$	11060.7	0.436280	0.218140	−	0.975917i	$-0.430001\pi$
$863$	11060.7	0.436280	0.218140	+	0.975917i	$0.430001\pi$
$864$	4887.10	0.192433
$865$	17257.5	0.678352
$866$	−16628.9	−0.652507
$867$	10379.9	0.406597
$868$	0	0
$869$	72166.5	2.81712
$870$	2246.46	0.0875427
$871$	−43785.2	−1.70334
$872$	739.469	0.0287174
$873$	3005.44	0.116516
$874$	−4176.68	−0.161646
$875$	0	0
$876$	13904.2	0.536278
$877$	39829.6	1.53358	0.766791	−	0.641897i	$-0.221852\pi$
$877$	39829.6	1.53358	0.766791	+	0.641897i	$0.221852\pi$
$878$	−20203.5	−0.776576
$879$	23602.6	0.905682
$880$	−18443.0	−0.706493
$881$	−27464.3	−1.05028	−0.525139	−	0.851016i	$-0.675987\pi$
$881$	−27464.3	−1.05028	−0.525139	+	0.851016i	$0.675987\pi$
$882$	0	0
$883$	−42891.2	−1.63466	−0.817329	−	0.576171i	$-0.804546\pi$
$883$	−42891.2	−1.63466	−0.817329	+	0.576171i	$0.804546\pi$
$884$	9446.27	0.359403
$885$	37966.8	1.44208
$886$	−27880.6	−1.05719
$887$	10250.3	0.388018	0.194009	−	0.981000i	$-0.437851\pi$
$887$	10250.3	0.388018	0.194009	+	0.981000i	$0.437851\pi$
$888$	338.698	0.0127995
$889$	0	0
$890$	52325.0	1.97072
$891$	29064.4	1.09281
$892$	−19360.8	−0.726734
$893$	−46412.8	−1.73924
$894$	3828.85	0.143239
$895$	24908.6	0.930282
$896$	0	0
$897$	−4658.02	−0.173385
$898$	−33335.3	−1.23877
$899$	887.870	0.0329389
$900$	−5196.56	−0.192465
$901$	−14550.7	−0.538019
$902$	20417.1	0.753677
$903$	0	0
$904$	−3264.49	−0.120105
$905$	18952.3	0.696129
$906$	12497.5	0.458281
$907$	−19908.4	−0.728828	−0.364414	−	0.931237i	$-0.618731\pi$
$907$	−19908.4	−0.728828	−0.364414	+	0.931237i	$0.618731\pi$
$908$	−9611.79	−0.351298
$909$	11030.7	0.402491
$910$	0	0
$911$	1504.75	0.0547250	0.0273625	−	0.999626i	$-0.491289\pi$
$911$	1504.75	0.0547250	0.0273625	+	0.999626i	$0.491289\pi$
$912$	6210.42	0.225491
$913$	−22903.9	−0.830240
$914$	8190.66	0.296415
$915$	33955.7	1.22682
$916$	14953.9	0.539400
$917$	0	0
$918$	15226.0	0.547422
$919$	55073.1	1.97682	0.988408	−	0.151819i	$-0.0485132\pi$
$919$	55073.1	1.97682	0.988408	+	0.151819i	$0.0485132\pi$
$920$	3045.17	0.109126
$921$	−38328.8	−1.37131
$922$	6257.64	0.223519
$923$	11171.6	0.398395
$924$	0	0
$925$	−1474.62	−0.0524165
$926$	−7225.30	−0.256413
$927$	−293.001	−0.0103812
$928$	−508.040	−0.0179711
$929$	−22894.5	−0.808551	−0.404276	−	0.914637i	$-0.632476\pi$
$929$	−22894.5	−0.808551	−0.404276	+	0.914637i	$0.632476\pi$
$930$	7913.21	0.279016
$931$	0	0
$932$	−7569.81	−0.266049
$933$	−19585.3	−0.687240
$934$	23251.7	0.814581
$935$	−57460.1	−2.00978
$936$	−3306.78	−0.115476
$937$	−14976.1	−0.522144	−0.261072	−	0.965319i	$-0.584076\pi$
$937$	−14976.1	−0.522144	−0.261072	+	0.965319i	$0.584076\pi$
$938$	0	0
$939$	−24404.1	−0.848135
$940$	33839.0	1.17416
$941$	−127.874	−0.00442992	−0.00221496	−	0.999998i	$-0.500705\pi$
$941$	−127.874	−0.00442992	−0.00221496	+	0.999998i	$0.500705\pi$
$942$	−8842.01	−0.305826
$943$	−3371.12	−0.116415
$944$	−8586.23	−0.296036
$945$	0	0
$946$	2117.71	0.0727831
$947$	−25726.5	−0.882788	−0.441394	−	0.897314i	$-0.645516\pi$
$947$	−25726.5	−0.882788	−0.441394	+	0.897314i	$0.645516\pi$
$948$	17717.6	0.607006
$949$	−38521.6	−1.31766
$950$	−27038.9	−0.923430
$951$	39562.4	1.34900
$952$	0	0
$953$	−35350.8	−1.20160	−0.600799	−	0.799400i	$-0.705151\pi$
$953$	−35350.8	−1.20160	−0.600799	+	0.799400i	$0.705151\pi$
$954$	5093.65	0.172865
$955$	−4287.04	−0.145262
$956$	8509.35	0.287879
$957$	−4727.09	−0.159671
$958$	−23787.5	−0.802232
$959$	0	0
$960$	−4527.95	−0.152228
$961$	−26663.5	−0.895017
$962$	−938.362	−0.0314491
$963$	11201.5	0.374831
$964$	27109.0	0.905730
$965$	31228.8	1.04175
$966$	0	0
$967$	−12332.0	−0.410105	−0.205052	−	0.978751i	$-0.565736\pi$
$967$	−12332.0	−0.410105	−0.205052	+	0.978751i	$0.565736\pi$
$968$	28160.4	0.935031
$969$	19348.9	0.641461
$970$	−11401.5	−0.377403
$971$	−25110.0	−0.829884	−0.414942	−	0.909848i	$-0.636198\pi$
$971$	−25110.0	−0.829884	−0.414942	+	0.909848i	$0.636198\pi$
$972$	−9358.33	−0.308816
$973$	0	0
$974$	24689.6	0.812224
$975$	−30155.0	−0.990495
$976$	−7679.12	−0.251847
$977$	16058.7	0.525860	0.262930	−	0.964815i	$-0.415311\pi$
$977$	16058.7	0.525860	0.262930	+	0.964815i	$0.415311\pi$
$978$	32859.3	1.07436
$979$	−110104.	−3.59443
$980$	0	0
$981$	−806.491	−0.0262480
$982$	33443.4	1.08678
$983$	19727.7	0.640098	0.320049	−	0.947401i	$-0.396301\pi$
$983$	19727.7	0.640098	0.320049	+	0.947401i	$0.396301\pi$
$984$	5012.62	0.162395
$985$	−62246.5	−2.01354
$986$	−1582.82	−0.0511231
$987$	0	0
$988$	−17205.9	−0.554043
$989$	−349.661	−0.0112422
$990$	20114.6	0.645741
$991$	−27848.5	−0.892670	−0.446335	−	0.894866i	$-0.647271\pi$
$991$	−27848.5	−0.892670	−0.446335	+	0.894866i	$0.647271\pi$
$992$	−1789.58	−0.0572775
$993$	21389.5	0.683560
$994$	0	0
$995$	−29788.4	−0.949100
$996$	−5623.15	−0.178892
$997$	17190.5	0.546066	0.273033	−	0.962005i	$-0.411973\pi$
$997$	17190.5	0.546066	0.273033	+	0.962005i	$0.411973\pi$
$998$	−33246.5	−1.05451
$999$	−1512.50	−0.0479013

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.g.1.1		2
7.6	odd	2		322.4.a.c.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.4.a.c.1.2	✓	2	7.6	odd	2
2254.4.a.g.1.1		2	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)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -4.27492 q^{3} +4.00000 q^{4} +16.5498 q^{5} -8.54983 q^{6} +8.00000 q^{8} -8.72508 q^{9} +33.0997 q^{10} -69.6495 q^{11} -17.0997 q^{12} +47.3746 q^{13} -70.7492 q^{15} +16.0000 q^{16} +49.8488 q^{17} -17.4502 q^{18} -90.7974 q^{19} +66.1993 q^{20} -139.299 q^{22} +23.0000 q^{23} -34.1993 q^{24} +148.897 q^{25} +94.7492 q^{26} +152.722 q^{27} -15.8762 q^{29} -141.498 q^{30} -55.9244 q^{31} +32.0000 q^{32} +297.746 q^{33} +99.6977 q^{34} -34.9003 q^{36} -9.90364 q^{37} -181.595 q^{38} -202.522 q^{39} +132.399 q^{40} -146.571 q^{41} -15.2026 q^{43} -278.598 q^{44} -144.399 q^{45} +46.0000 q^{46} +511.169 q^{47} -68.3987 q^{48} +297.794 q^{50} -213.100 q^{51} +189.498 q^{52} -291.897 q^{53} +305.444 q^{54} -1152.69 q^{55} +388.151 q^{57} -31.7525 q^{58} -536.640 q^{59} -282.997 q^{60} -479.945 q^{61} -111.849 q^{62} +64.0000 q^{64} +784.042 q^{65} +595.492 q^{66} -924.234 q^{67} +199.395 q^{68} -98.3231 q^{69} +235.815 q^{71} -69.8007 q^{72} -813.127 q^{73} -19.8073 q^{74} -636.522 q^{75} -363.189 q^{76} -405.045 q^{78} -1036.14 q^{79} +264.797 q^{80} -417.296 q^{81} -293.141 q^{82} +328.846 q^{83} +824.990 q^{85} -30.4053 q^{86} +67.8696 q^{87} -557.196 q^{88} +1580.83 q^{89} -288.797 q^{90} +92.0000 q^{92} +239.072 q^{93} +1022.34 q^{94} -1502.68 q^{95} -136.797 q^{96} -344.460 q^{97} +607.698 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} - q^{3} + 8 q^{4} + 18 q^{5} - 2 q^{6} + 16 q^{8} - 25 q^{9} + 36 q^{10} - 94 q^{11} - 4 q^{12} + 57 q^{13} - 66 q^{15} + 32 q^{16} - 6 q^{17} - 50 q^{18} + 60 q^{19} + 72 q^{20} - 188 q^{22}+ \cdots + 1004 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 - q^3 + 8 * q^4 + 18 * q^5 - 2 * q^6 + 16 * q^8 - 25 * q^9 + 36 * q^10 - 94 * q^11 - 4 * q^12 + 57 * q^13 - 66 * q^15 + 32 * q^16 - 6 * q^17 - 50 * q^18 + 60 * q^19 + 72 * q^20 - 188 * q^22 + 46 * q^23 - 8 * q^24 + 26 * q^25 + 114 * q^26 + 11 * q^27 - 145 * q^29 - 132 * q^30 - 59 * q^31 + 64 * q^32 + 218 * q^33 - 12 * q^34 - 100 * q^36 - 352 * q^37 + 120 * q^38 - 171 * q^39 + 144 * q^40 + 107 * q^41 - 272 * q^43 - 376 * q^44 - 168 * q^45 + 92 * q^46 + 441 * q^47 - 16 * q^48 + 52 * q^50 - 396 * q^51 + 228 * q^52 - 312 * q^53 + 22 * q^54 - 1188 * q^55 + 882 * q^57 - 290 * q^58 - 122 * q^59 - 264 * q^60 - 522 * q^61 - 118 * q^62 + 128 * q^64 + 798 * q^65 + 436 * q^66 - 414 * q^67 - 24 * q^68 - 23 * q^69 - 457 * q^71 - 200 * q^72 - 1815 * q^73 - 704 * q^74 - 1039 * q^75 + 240 * q^76 - 342 * q^78 - 970 * q^79 + 288 * q^80 - 442 * q^81 + 214 * q^82 + 250 * q^83 + 744 * q^85 - 544 * q^86 - 355 * q^87 - 752 * q^88 + 1546 * q^89 - 336 * q^90 + 184 * q^92 + 229 * q^93 + 882 * q^94 - 1284 * q^95 - 32 * q^96 - 1610 * q^97 + 1004 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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