LMFDB - Embedded newform 2254.4.a.x.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("2254.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$132.990305153$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 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			1.4
Root			$5.45902$ 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)

$f(q)$	$=$	\(q+2.00000 q^{2} -4.45902 q^{3} +4.00000 q^{4} +11.8862 q^{5} -8.91803 q^{6} +8.00000 q^{8} -7.11717 q^{9} +23.7724 q^{10} -10.2805 q^{11} -17.8361 q^{12} -43.4321 q^{13} -53.0007 q^{15} +16.0000 q^{16} +54.9334 q^{17} -14.2343 q^{18} -47.0673 q^{19} +47.5447 q^{20} -20.5610 q^{22} -23.0000 q^{23} -35.6721 q^{24} +16.2813 q^{25} -86.8641 q^{26} +152.129 q^{27} +245.857 q^{29} -106.001 q^{30} -287.824 q^{31} +32.0000 q^{32} +45.8410 q^{33} +109.867 q^{34} -28.4687 q^{36} -254.867 q^{37} -94.1345 q^{38} +193.664 q^{39} +95.0895 q^{40} +35.0262 q^{41} +257.685 q^{43} -41.1220 q^{44} -84.5960 q^{45} -46.0000 q^{46} +618.984 q^{47} -71.3443 q^{48} +32.5626 q^{50} -244.949 q^{51} -173.728 q^{52} -349.480 q^{53} +304.258 q^{54} -122.196 q^{55} +209.874 q^{57} +491.715 q^{58} +626.493 q^{59} -212.003 q^{60} +626.512 q^{61} -575.648 q^{62} +64.0000 q^{64} -516.241 q^{65} +91.6819 q^{66} +435.715 q^{67} +219.734 q^{68} +102.557 q^{69} -727.972 q^{71} -56.9374 q^{72} +460.719 q^{73} -509.734 q^{74} -72.5987 q^{75} -188.269 q^{76} +387.329 q^{78} +322.728 q^{79} +190.179 q^{80} -486.182 q^{81} +70.0525 q^{82} +995.707 q^{83} +652.948 q^{85} +515.371 q^{86} -1096.28 q^{87} -82.2441 q^{88} +1167.23 q^{89} -169.192 q^{90} -92.0000 q^{92} +1283.41 q^{93} +1237.97 q^{94} -559.450 q^{95} -142.689 q^{96} +550.000 q^{97} +73.1681 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−4.45902	−0.858138	−0.429069	−	0.903272i	$-0.641158\pi$
$3$	−4.45902	−0.858138	−0.429069	+	0.903272i	$0.641158\pi$
$4$	4.00000	0.500000
$5$	11.8862	1.06313	0.531566	−	0.847017i	$-0.321604\pi$
$5$	11.8862	1.06313	0.531566	+	0.847017i	$0.321604\pi$
$6$	−8.91803	−0.606795
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	−7.11717	−0.263599
$10$	23.7724	0.751748
$11$	−10.2805	−0.281790	−0.140895	−	0.990025i	$-0.544998\pi$
$11$	−10.2805	−0.281790	−0.140895	+	0.990025i	$0.544998\pi$
$12$	−17.8361	−0.429069
$13$	−43.4321	−0.926607	−0.463303	−	0.886200i	$-0.653336\pi$
$13$	−43.4321	−0.926607	−0.463303	+	0.886200i	$0.653336\pi$
$14$	0	0
$15$	−53.0007	−0.912315
$16$	16.0000	0.250000
$17$	54.9334	0.783724	0.391862	−	0.920024i	$-0.371831\pi$
$17$	54.9334	0.783724	0.391862	+	0.920024i	$0.371831\pi$
$18$	−14.2343	−0.186393
$19$	−47.0673	−0.568314	−0.284157	−	0.958778i	$-0.591714\pi$
$19$	−47.0673	−0.568314	−0.284157	+	0.958778i	$0.591714\pi$
$20$	47.5447	0.531566
$21$	0	0
$22$	−20.5610	−0.199256
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	−35.6721	−0.303398
$25$	16.2813	0.130251
$26$	−86.8641	−0.655210
$27$	152.129	1.08434
$28$	0	0
$29$	245.857	1.57430	0.787148	−	0.616764i	$-0.211557\pi$
$29$	245.857	1.57430	0.787148	+	0.616764i	$0.211557\pi$
$30$	−106.001	−0.645104
$31$	−287.824	−1.66757	−0.833786	−	0.552087i	$-0.813832\pi$
$31$	−287.824	−1.66757	−0.833786	+	0.552087i	$0.813832\pi$
$32$	32.0000	0.176777
$33$	45.8410	0.241815
$34$	109.867	0.554176
$35$	0	0
$36$	−28.4687	−0.131799
$37$	−254.867	−1.13243	−0.566215	−	0.824258i	$-0.691593\pi$
$37$	−254.867	−1.13243	−0.566215	+	0.824258i	$0.691593\pi$
$38$	−94.1345	−0.401859
$39$	193.664	0.795157
$40$	95.0895	0.375874
$41$	35.0262	0.133419	0.0667095	−	0.997772i	$-0.478750\pi$
$41$	35.0262	0.133419	0.0667095	+	0.997772i	$0.478750\pi$
$42$	0	0
$43$	257.685	0.913875	0.456938	−	0.889499i	$-0.348946\pi$
$43$	257.685	0.913875	0.456938	+	0.889499i	$0.348946\pi$
$44$	−41.1220	−0.140895
$45$	−84.5960	−0.280240
$46$	−46.0000	−0.147442
$47$	618.984	1.92102	0.960511	−	0.278241i	$-0.0897513\pi$
$47$	618.984	1.92102	0.960511	+	0.278241i	$0.0897513\pi$
$48$	−71.3443	−0.214535
$49$	0	0
$50$	32.5626	0.0921011
$51$	−244.949	−0.672543
$52$	−173.728	−0.463303
$53$	−349.480	−0.905751	−0.452875	−	0.891574i	$-0.649602\pi$
$53$	−349.480	−0.905751	−0.452875	+	0.891574i	$0.649602\pi$
$54$	304.258	0.766746
$55$	−122.196	−0.299580
$56$	0	0
$57$	209.874	0.487692
$58$	491.715	1.11320
$59$	626.493	1.38241	0.691207	−	0.722657i	$-0.257079\pi$
$59$	626.493	1.38241	0.691207	+	0.722657i	$0.257079\pi$
$60$	−212.003	−0.456157
$61$	626.512	1.31503	0.657513	−	0.753443i	$-0.271608\pi$
$61$	626.512	1.31503	0.657513	+	0.753443i	$0.271608\pi$
$62$	−575.648	−1.17915
$63$	0	0
$64$	64.0000	0.125000
$65$	−516.241	−0.985106
$66$	91.6819	0.170989
$67$	435.715	0.794494	0.397247	−	0.917712i	$-0.369966\pi$
$67$	435.715	0.794494	0.397247	+	0.917712i	$0.369966\pi$
$68$	219.734	0.391862
$69$	102.557	0.178934
$70$	0	0
$71$	−727.972	−1.21682	−0.608411	−	0.793622i	$-0.708193\pi$
$71$	−727.972	−1.21682	−0.608411	+	0.793622i	$0.708193\pi$
$72$	−56.9374	−0.0931963
$73$	460.719	0.738672	0.369336	−	0.929296i	$-0.379585\pi$
$73$	460.719	0.738672	0.369336	+	0.929296i	$0.379585\pi$
$74$	−509.734	−0.800748
$75$	−72.5987	−0.111773
$76$	−188.269	−0.284157
$77$	0	0
$78$	387.329	0.562261
$79$	322.728	0.459617	0.229808	−	0.973236i	$-0.426190\pi$
$79$	322.728	0.459617	0.229808	+	0.973236i	$0.426190\pi$
$80$	190.179	0.265783
$81$	−486.182	−0.666917
$82$	70.0525	0.0943415
$83$	995.707	1.31678	0.658391	−	0.752676i	$-0.271237\pi$
$83$	995.707	1.31678	0.658391	+	0.752676i	$0.271237\pi$
$84$	0	0
$85$	652.948	0.833202
$86$	515.371	0.646207
$87$	−1096.28	−1.35096
$88$	−82.2441	−0.0996278
$89$	1167.23	1.39018	0.695089	−	0.718923i	$-0.255365\pi$
$89$	1167.23	1.39018	0.695089	+	0.718923i	$0.255365\pi$
$90$	−169.192	−0.198160
$91$	0	0
$92$	−92.0000	−0.104257
$93$	1283.41	1.43101
$94$	1237.97	1.35837
$95$	−559.450	−0.604193
$96$	−142.689	−0.151699
$97$	550.000	0.575711	0.287856	−	0.957674i	$-0.407058\pi$
$97$	550.000	0.575711	0.287856	+	0.957674i	$0.407058\pi$
$98$	0	0
$99$	73.1681	0.0742795
$100$	65.1253	0.0651253
$101$	−1545.51	−1.52261	−0.761306	−	0.648393i	$-0.775441\pi$
$101$	−1545.51	−1.52261	−0.761306	+	0.648393i	$0.775441\pi$
$102$	−489.898	−0.475560
$103$	1146.99	1.09724	0.548621	−	0.836071i	$-0.315153\pi$
$103$	1146.99	1.09724	0.548621	+	0.836071i	$0.315153\pi$
$104$	−347.457	−0.327605
$105$	0	0
$106$	−698.960	−0.640463
$107$	−1024.30	−0.925447	−0.462724	−	0.886503i	$-0.653128\pi$
$107$	−1024.30	−0.925447	−0.462724	+	0.886503i	$0.653128\pi$
$108$	608.516	0.542171
$109$	114.016	0.100191	0.0500953	−	0.998744i	$-0.484048\pi$
$109$	114.016	0.100191	0.0500953	+	0.998744i	$0.484048\pi$
$110$	−244.392	−0.211835
$111$	1136.46	0.971781
$112$	0	0
$113$	−416.934	−0.347096	−0.173548	−	0.984825i	$-0.555523\pi$
$113$	−416.934	−0.347096	−0.173548	+	0.984825i	$0.555523\pi$
$114$	419.747	0.344850
$115$	−273.382	−0.221678
$116$	983.430	0.787148
$117$	309.113	0.244252
$118$	1252.99	0.977515
$119$	0	0
$120$	−424.005	−0.322552
$121$	−1225.31	−0.920594
$122$	1253.02	0.929864
$123$	−156.183	−0.114492
$124$	−1151.30	−0.833786
$125$	−1292.25	−0.924659
$126$	0	0
$127$	1801.69	1.25885	0.629425	−	0.777061i	$-0.283291\pi$
$127$	1801.69	1.25885	0.629425	+	0.777061i	$0.283291\pi$
$128$	128.000	0.0883883
$129$	−1149.02	−0.784231
$130$	−1032.48	−0.696575
$131$	−1796.16	−1.19795	−0.598975	−	0.800767i	$-0.704425\pi$
$131$	−1796.16	−1.19795	−0.598975	+	0.800767i	$0.704425\pi$
$132$	183.364	0.120907
$133$	0	0
$134$	871.431	0.561792
$135$	1808.23	1.15280
$136$	439.467	0.277088
$137$	−2014.64	−1.25637	−0.628183	−	0.778065i	$-0.716201\pi$
$137$	−2014.64	−1.25637	−0.628183	+	0.778065i	$0.716201\pi$
$138$	205.115	0.126526
$139$	2195.66	1.33981	0.669903	−	0.742448i	$-0.266336\pi$
$139$	2195.66	1.33981	0.669903	+	0.742448i	$0.266336\pi$
$140$	0	0
$141$	−2760.06	−1.64850
$142$	−1455.94	−0.860424
$143$	446.504	0.261108
$144$	−113.875	−0.0658997
$145$	2922.31	1.67369
$146$	921.437	0.522320
$147$	0	0
$148$	−1019.47	−0.566215
$149$	1126.72	0.619492	0.309746	−	0.950819i	$-0.399756\pi$
$149$	1126.72	0.619492	0.309746	+	0.950819i	$0.399756\pi$
$150$	−145.197	−0.0790354
$151$	−511.205	−0.275505	−0.137753	−	0.990467i	$-0.543988\pi$
$151$	−511.205	−0.275505	−0.137753	+	0.990467i	$0.543988\pi$
$152$	−376.538	−0.200929
$153$	−390.970	−0.206589
$154$	0	0
$155$	−3421.13	−1.77285
$156$	774.657	0.397578
$157$	1.99715	0.00101522	0.000507611	−	1.00000i	$-0.499838\pi$
$157$	1.99715	0.00101522	0.000507611		1.00000i	$0.499838\pi$
$158$	645.456	0.324998
$159$	1558.34	0.777259
$160$	380.358	0.187937
$161$	0	0
$162$	−972.365	−0.471581
$163$	1497.10	0.719398	0.359699	−	0.933068i	$-0.382879\pi$
$163$	1497.10	0.719398	0.359699	+	0.933068i	$0.382879\pi$
$164$	140.105	0.0667095
$165$	544.874	0.257081
$166$	1991.41	0.931106
$167$	2450.82	1.13563	0.567815	−	0.823156i	$-0.307789\pi$
$167$	2450.82	1.13563	0.567815	+	0.823156i	$0.307789\pi$
$168$	0	0
$169$	−310.656	−0.141400
$170$	1305.90	0.589163
$171$	334.986	0.149807
$172$	1030.74	0.456938
$173$	553.775	0.243368	0.121684	−	0.992569i	$-0.461171\pi$
$173$	553.775	0.243368	0.121684	+	0.992569i	$0.461171\pi$
$174$	−2192.57	−0.955276
$175$	0	0
$176$	−164.488	−0.0704475
$177$	−2793.54	−1.18630
$178$	2334.46	0.983005
$179$	−2992.35	−1.24949	−0.624746	−	0.780828i	$-0.714797\pi$
$179$	−2992.35	−1.24949	−0.624746	+	0.780828i	$0.714797\pi$
$180$	−338.384	−0.140120
$181$	−2213.30	−0.908912	−0.454456	−	0.890769i	$-0.650166\pi$
$181$	−2213.30	−0.908912	−0.454456	+	0.890769i	$0.650166\pi$
$182$	0	0
$183$	−2793.63	−1.12847
$184$	−184.000	−0.0737210
$185$	−3029.40	−1.20392
$186$	2566.83	1.01188
$187$	−564.743	−0.220846
$188$	2475.94	0.960511
$189$	0	0
$190$	−1118.90	−0.427229
$191$	4390.68	1.66334	0.831672	−	0.555267i	$-0.187384\pi$
$191$	4390.68	1.66334	0.831672	+	0.555267i	$0.187384\pi$
$192$	−285.377	−0.107267
$193$	1824.21	0.680358	0.340179	−	0.940361i	$-0.389512\pi$
$193$	1824.21	0.680358	0.340179	+	0.940361i	$0.389512\pi$
$194$	1100.00	0.407089
$195$	2301.93	0.845357
$196$	0	0
$197$	−2099.84	−0.759429	−0.379715	−	0.925104i	$-0.623978\pi$
$197$	−2099.84	−0.759429	−0.379715	+	0.925104i	$0.623978\pi$
$198$	146.336	0.0525235
$199$	3572.24	1.27251	0.636254	−	0.771479i	$-0.280483\pi$
$199$	3572.24	1.27251	0.636254	+	0.771479i	$0.280483\pi$
$200$	130.251	0.0460505
$201$	−1942.86	−0.681786
$202$	−3091.02	−1.07665
$203$	0	0
$204$	−979.796	−0.336272
$205$	416.328	0.141842
$206$	2293.97	0.775868
$207$	163.695	0.0549642
$208$	−694.913	−0.231652
$209$	483.875	0.160145
$210$	0	0
$211$	3332.11	1.08716	0.543582	−	0.839356i	$-0.317068\pi$
$211$	3332.11	1.08716	0.543582	+	0.839356i	$0.317068\pi$
$212$	−1397.92	−0.452875
$213$	3246.04	1.04420
$214$	−2048.60	−0.654390
$215$	3062.89	0.971570
$216$	1217.03	0.383373
$217$	0	0
$218$	228.032	0.0708454
$219$	−2054.35	−0.633882
$220$	−488.784	−0.149790
$221$	−2385.87	−0.726204
$222$	2272.91	0.687153
$223$	−2977.46	−0.894105	−0.447052	−	0.894508i	$-0.647526\pi$
$223$	−2977.46	−0.894105	−0.447052	+	0.894508i	$0.647526\pi$
$224$	0	0
$225$	−115.877	−0.0343339
$226$	−833.869	−0.245434
$227$	5408.03	1.58125	0.790625	−	0.612301i	$-0.209756\pi$
$227$	5408.03	1.58125	0.790625	+	0.612301i	$0.209756\pi$
$228$	839.495	0.243846
$229$	894.318	0.258071	0.129035	−	0.991640i	$-0.458812\pi$
$229$	894.318	0.258071	0.129035	+	0.991640i	$0.458812\pi$
$230$	−546.764	−0.156750
$231$	0	0
$232$	1966.86	0.556598
$233$	5151.85	1.44854	0.724268	−	0.689518i	$-0.242178\pi$
$233$	5151.85	1.44854	0.724268	+	0.689518i	$0.242178\pi$
$234$	618.227	0.172713
$235$	7357.35	2.04230
$236$	2505.97	0.691207
$237$	−1439.05	−0.394415
$238$	0	0
$239$	1365.86	0.369665	0.184833	−	0.982770i	$-0.440826\pi$
$239$	1365.86	0.369665	0.184833	+	0.982770i	$0.440826\pi$
$240$	−848.011	−0.228079
$241$	4639.50	1.24007	0.620034	−	0.784575i	$-0.287119\pi$
$241$	4639.50	1.24007	0.620034	+	0.784575i	$0.287119\pi$
$242$	−2450.62	−0.650959
$243$	−1939.59	−0.512036
$244$	2506.05	0.657513
$245$	0	0
$246$	−312.365	−0.0809580
$247$	2044.23	0.526604
$248$	−2302.59	−0.589576
$249$	−4439.87	−1.12998
$250$	−2584.50	−0.653833
$251$	−2698.14	−0.678507	−0.339253	−	0.940695i	$-0.610174\pi$
$251$	−2698.14	−0.678507	−0.339253	+	0.940695i	$0.610174\pi$
$252$	0	0
$253$	236.452	0.0587573
$254$	3603.37	0.890141
$255$	−2911.51	−0.715003
$256$	256.000	0.0625000
$257$	237.755	0.0577072	0.0288536	−	0.999584i	$-0.490814\pi$
$257$	237.755	0.0577072	0.0288536	+	0.999584i	$0.490814\pi$
$258$	−2298.05	−0.554535
$259$	0	0
$260$	−2064.97	−0.492553
$261$	−1749.81	−0.414983
$262$	−3592.33	−0.847079
$263$	−790.865	−0.185425	−0.0927126	−	0.995693i	$-0.529554\pi$
$263$	−790.865	−0.185425	−0.0927126	+	0.995693i	$0.529554\pi$
$264$	366.728	0.0854944
$265$	−4153.99	−0.962933
$266$	0	0
$267$	−5204.69	−1.19297
$268$	1742.86	0.397247
$269$	5894.42	1.33602	0.668009	−	0.744153i	$-0.267147\pi$
$269$	5894.42	1.33602	0.668009	+	0.744153i	$0.267147\pi$
$270$	3616.47	0.815152
$271$	6474.66	1.45132	0.725660	−	0.688053i	$-0.241535\pi$
$271$	6474.66	1.45132	0.725660	+	0.688053i	$0.241535\pi$
$272$	878.934	0.195931
$273$	0	0
$274$	−4029.28	−0.888386
$275$	−167.380	−0.0367033
$276$	410.230	0.0894671
$277$	6638.11	1.43987	0.719937	−	0.694039i	$-0.244170\pi$
$277$	6638.11	1.43987	0.719937	+	0.694039i	$0.244170\pi$
$278$	4391.31	0.947386
$279$	2048.49	0.439570
$280$	0	0
$281$	5701.81	1.21047	0.605234	−	0.796048i	$-0.293080\pi$
$281$	5701.81	1.21047	0.605234	+	0.796048i	$0.293080\pi$
$282$	−5520.12	−1.16567
$283$	−6975.15	−1.46512	−0.732561	−	0.680701i	$-0.761675\pi$
$283$	−6975.15	−1.46512	−0.732561	+	0.680701i	$0.761675\pi$
$284$	−2911.89	−0.608411
$285$	2494.60	0.518481
$286$	893.007	0.184632
$287$	0	0
$288$	−227.749	−0.0465981
$289$	−1895.32	−0.385777
$290$	5844.61	1.18347
$291$	−2452.46	−0.494040
$292$	1842.87	0.369336
$293$	6802.09	1.35625	0.678127	−	0.734945i	$-0.262792\pi$
$293$	6802.09	1.35625	0.678127	+	0.734945i	$0.262792\pi$
$294$	0	0
$295$	7446.61	1.46969
$296$	−2038.94	−0.400374
$297$	−1563.96	−0.305557
$298$	2253.44	0.438047
$299$	998.937	0.193211
$300$	−290.395	−0.0558865
$301$	0	0
$302$	−1022.41	−0.194812
$303$	6891.44	1.30661
$304$	−753.076	−0.142079
$305$	7446.83	1.39805
$306$	−781.941	−0.146080
$307$	−2688.94	−0.499889	−0.249945	−	0.968260i	$-0.580412\pi$
$307$	−2688.94	−0.499889	−0.249945	+	0.968260i	$0.580412\pi$
$308$	0	0
$309$	−5114.43	−0.941586
$310$	−6842.26	−1.25359
$311$	−5172.53	−0.943111	−0.471555	−	0.881836i	$-0.656307\pi$
$311$	−5172.53	−0.943111	−0.471555	+	0.881836i	$0.656307\pi$
$312$	1549.31	0.281130
$313$	6588.73	1.18983	0.594915	−	0.803789i	$-0.297186\pi$
$313$	6588.73	1.18983	0.594915	+	0.803789i	$0.297186\pi$
$314$	3.99430	0.000717870 0
$315$	0	0
$316$	1290.91	0.229808
$317$	2858.53	0.506470	0.253235	−	0.967405i	$-0.418505\pi$
$317$	2858.53	0.506470	0.253235	+	0.967405i	$0.418505\pi$
$318$	3116.68	0.549605
$319$	−2527.54	−0.443621
$320$	760.716	0.132892
$321$	4567.37	0.794161
$322$	0	0
$323$	−2585.57	−0.445402
$324$	−1944.73	−0.333458
$325$	−707.131	−0.120691
$326$	2994.20	0.508691
$327$	−508.400	−0.0859773
$328$	280.210	0.0471707
$329$	0	0
$330$	1089.75	0.181784
$331$	6503.65	1.07998	0.539989	−	0.841672i	$-0.318428\pi$
$331$	6503.65	1.07998	0.539989	+	0.841672i	$0.318428\pi$
$332$	3982.83	0.658391
$333$	1813.93	0.298507
$334$	4901.65	0.803012
$335$	5178.99	0.844653
$336$	0	0
$337$	−952.242	−0.153923	−0.0769614	−	0.997034i	$-0.524522\pi$
$337$	−952.242	−0.153923	−0.0769614	+	0.997034i	$0.524522\pi$
$338$	−621.312	−0.0999849
$339$	1859.12	0.297857
$340$	2611.79	0.416601
$341$	2958.98	0.469905
$342$	669.971	0.105930
$343$	0	0
$344$	2061.48	0.323104
$345$	1219.02	0.190231
$346$	1107.55	0.172087
$347$	−292.017	−0.0451766	−0.0225883	−	0.999745i	$-0.507191\pi$
$347$	−292.017	−0.0451766	−0.0225883	+	0.999745i	$0.507191\pi$
$348$	−4385.13	−0.675482
$349$	4360.15	0.668750	0.334375	−	0.942440i	$-0.391475\pi$
$349$	4360.15	0.668750	0.334375	+	0.942440i	$0.391475\pi$
$350$	0	0
$351$	−6607.28	−1.00476
$352$	−328.976	−0.0498139
$353$	−3666.20	−0.552783	−0.276391	−	0.961045i	$-0.589139\pi$
$353$	−3666.20	−0.552783	−0.276391	+	0.961045i	$0.589139\pi$
$354$	−5587.09	−0.838843
$355$	−8652.81	−1.29364
$356$	4668.91	0.695089
$357$	0	0
$358$	−5984.71	−0.883524
$359$	−8159.92	−1.19962	−0.599811	−	0.800142i	$-0.704757\pi$
$359$	−8159.92	−1.19962	−0.599811	+	0.800142i	$0.704757\pi$
$360$	−676.768	−0.0990800
$361$	−4643.67	−0.677019
$362$	−4426.59	−0.642698
$363$	5463.68	0.789997
$364$	0	0
$365$	5476.18	0.785306
$366$	−5587.25	−0.797952
$367$	−1030.25	−0.146535	−0.0732677	−	0.997312i	$-0.523343\pi$
$367$	−1030.25	−0.146535	−0.0732677	+	0.997312i	$0.523343\pi$
$368$	−368.000	−0.0521286
$369$	−249.288	−0.0351691
$370$	−6058.79	−0.851302
$371$	0	0
$372$	5133.65	0.715504
$373$	7572.88	1.05123	0.525615	−	0.850723i	$-0.323835\pi$
$373$	7572.88	1.05123	0.525615	+	0.850723i	$0.323835\pi$
$374$	−1129.49	−0.156161
$375$	5762.16	0.793485
$376$	4951.87	0.679184
$377$	−10678.1	−1.45875
$378$	0	0
$379$	−3957.77	−0.536403	−0.268202	−	0.963363i	$-0.586429\pi$
$379$	−3957.77	−0.536403	−0.268202	+	0.963363i	$0.586429\pi$
$380$	−2237.80	−0.302097
$381$	−8033.75	−1.08027
$382$	8781.37	1.17616
$383$	−6797.77	−0.906918	−0.453459	−	0.891277i	$-0.649810\pi$
$383$	−6797.77	−0.906918	−0.453459	+	0.891277i	$0.649810\pi$
$384$	−570.754	−0.0758494
$385$	0	0
$386$	3648.41	0.481086
$387$	−1833.99	−0.240896
$388$	2200.00	0.287856
$389$	−5733.86	−0.747347	−0.373674	−	0.927560i	$-0.621902\pi$
$389$	−5733.86	−0.747347	−0.373674	+	0.927560i	$0.621902\pi$
$390$	4603.86	0.597757
$391$	−1263.47	−0.163418
$392$	0	0
$393$	8009.12	1.02801
$394$	−4199.69	−0.536998
$395$	3836.00	0.488633
$396$	292.672	0.0371398
$397$	−2412.97	−0.305047	−0.152524	−	0.988300i	$-0.548740\pi$
$397$	−2412.97	−0.305047	−0.152524	+	0.988300i	$0.548740\pi$
$398$	7144.47	0.899799
$399$	0	0
$400$	260.501	0.0325626
$401$	3889.19	0.484331	0.242165	−	0.970235i	$-0.422142\pi$
$401$	3889.19	0.484331	0.242165	+	0.970235i	$0.422142\pi$
$402$	−3885.72	−0.482095
$403$	12500.8	1.54518
$404$	−6182.03	−0.761306
$405$	−5778.85	−0.709021
$406$	0	0
$407$	2620.16	0.319107
$408$	−1959.59	−0.237780
$409$	6627.60	0.801256	0.400628	−	0.916241i	$-0.368792\pi$
$409$	6627.60	0.801256	0.400628	+	0.916241i	$0.368792\pi$
$410$	832.657	0.100297
$411$	8983.31	1.07814
$412$	4587.95	0.548621
$413$	0	0
$414$	327.390	0.0388655
$415$	11835.1	1.39991
$416$	−1389.83	−0.163802
$417$	−9790.47	−1.14974
$418$	967.751	0.113240
$419$	6921.05	0.806958	0.403479	−	0.914989i	$-0.367801\pi$
$419$	6921.05	0.806958	0.403479	+	0.914989i	$0.367801\pi$
$420$	0	0
$421$	−11130.6	−1.28853	−0.644264	−	0.764803i	$-0.722836\pi$
$421$	−11130.6	−1.28853	−0.644264	+	0.764803i	$0.722836\pi$
$422$	6664.21	0.768741
$423$	−4405.41	−0.506379
$424$	−2795.84	−0.320231
$425$	894.388	0.102080
$426$	6492.08	0.738362
$427$	0	0
$428$	−4097.20	−0.462724
$429$	−1990.97	−0.224067
$430$	6125.79	0.687004
$431$	371.467	0.0415150	0.0207575	−	0.999785i	$-0.493392\pi$
$431$	371.467	0.0415150	0.0207575	+	0.999785i	$0.493392\pi$
$432$	2434.06	0.271086
$433$	−9497.51	−1.05409	−0.527045	−	0.849837i	$-0.676700\pi$
$433$	−9497.51	−1.05409	−0.527045	+	0.849837i	$0.676700\pi$
$434$	0	0
$435$	−13030.6	−1.43625
$436$	456.065	0.0500953
$437$	1082.55	0.118502
$438$	−4108.70	−0.448222
$439$	2286.99	0.248638	0.124319	−	0.992242i	$-0.460325\pi$
$439$	2286.99	0.248638	0.124319	+	0.992242i	$0.460325\pi$
$440$	−977.568	−0.105918
$441$	0	0
$442$	−4771.74	−0.513504
$443$	18453.0	1.97907	0.989537	−	0.144281i	$-0.0460869\pi$
$443$	18453.0	1.97907	0.989537	+	0.144281i	$0.0460869\pi$
$444$	4545.83	0.485890
$445$	13873.9	1.47794
$446$	−5954.92	−0.632228
$447$	−5024.05	−0.531610
$448$	0	0
$449$	−8638.93	−0.908010	−0.454005	−	0.890999i	$-0.650005\pi$
$449$	−8638.93	−0.908010	−0.454005	+	0.890999i	$0.650005\pi$
$450$	−231.754	−0.0242777
$451$	−360.087	−0.0375961
$452$	−1667.74	−0.173548
$453$	2279.47	0.236422
$454$	10816.1	1.11811
$455$	0	0
$456$	1678.99	0.172425
$457$	−4287.79	−0.438893	−0.219447	−	0.975625i	$-0.570425\pi$
$457$	−4287.79	−0.438893	−0.219447	+	0.975625i	$0.570425\pi$
$458$	1788.64	0.182483
$459$	8356.96	0.849825
$460$	−1093.53	−0.110839
$461$	−1737.71	−0.175560	−0.0877800	−	0.996140i	$-0.527977\pi$
$461$	−1737.71	−0.175560	−0.0877800	+	0.996140i	$0.527977\pi$
$462$	0	0
$463$	5751.48	0.577309	0.288655	−	0.957433i	$-0.406792\pi$
$463$	5751.48	0.577309	0.288655	+	0.957433i	$0.406792\pi$
$464$	3933.72	0.393574
$465$	15254.9	1.52135
$466$	10303.7	1.02427
$467$	−3939.03	−0.390313	−0.195157	−	0.980772i	$-0.562522\pi$
$467$	−3939.03	−0.390313	−0.195157	+	0.980772i	$0.562522\pi$
$468$	1236.45	0.122126
$469$	0	0
$470$	14714.7	1.44413
$471$	−8.90532	−0.000871200 0
$472$	5011.94	0.488757
$473$	−2649.14	−0.257521
$474$	−2878.10	−0.278893
$475$	−766.317	−0.0740233
$476$	0	0
$477$	2487.31	0.238755
$478$	2731.71	0.261393
$479$	5976.41	0.570081	0.285041	−	0.958515i	$-0.407993\pi$
$479$	5976.41	0.570081	0.285041	+	0.958515i	$0.407993\pi$
$480$	−1696.02	−0.161276
$481$	11069.4	1.04932
$482$	9279.00	0.876861
$483$	0	0
$484$	−4901.24	−0.460297
$485$	6537.40	0.612058
$486$	−3879.18	−0.362064
$487$	17410.3	1.61999	0.809996	−	0.586436i	$-0.199469\pi$
$487$	17410.3	1.61999	0.809996	+	0.586436i	$0.199469\pi$
$488$	5012.10	0.464932
$489$	−6675.59	−0.617343
$490$	0	0
$491$	7602.40	0.698761	0.349380	−	0.936981i	$-0.386392\pi$
$491$	7602.40	0.698761	0.349380	+	0.936981i	$0.386392\pi$
$492$	−624.730	−0.0572460
$493$	13505.8	1.23381
$494$	4088.46	0.372365
$495$	869.689	0.0789690
$496$	−4605.19	−0.416893
$497$	0	0
$498$	−8879.74	−0.799018
$499$	16620.4	1.49104	0.745520	−	0.666483i	$-0.232201\pi$
$499$	16620.4	1.49104	0.745520	+	0.666483i	$0.232201\pi$
$500$	−5169.00	−0.462329
$501$	−10928.3	−0.974528
$502$	−5396.28	−0.479777
$503$	−2923.40	−0.259141	−0.129571	−	0.991570i	$-0.541360\pi$
$503$	−2923.40	−0.259141	−0.129571	+	0.991570i	$0.541360\pi$
$504$	0	0
$505$	−18370.2	−1.61874
$506$	472.903	0.0415477
$507$	1385.22	0.121341
$508$	7206.75	0.629425
$509$	52.6787	0.00458731	0.00229365	−	0.999997i	$-0.499270\pi$
$509$	52.6787	0.00458731	0.00229365	+	0.999997i	$0.499270\pi$
$510$	−5823.02	−0.505583
$511$	0	0
$512$	512.000	0.0441942
$513$	−7160.30	−0.616247
$514$	475.510	0.0408051
$515$	13633.3	1.16651
$516$	−4596.09	−0.392116
$517$	−6363.47	−0.541325
$518$	0	0
$519$	−2469.29	−0.208844
$520$	−4129.93	−0.348287
$521$	−1466.38	−0.123308	−0.0616538	−	0.998098i	$-0.519637\pi$
$521$	−1466.38	−0.123308	−0.0616538	+	0.998098i	$0.519637\pi$
$522$	−3499.62	−0.293437
$523$	−4876.05	−0.407676	−0.203838	−	0.979005i	$-0.565342\pi$
$523$	−4876.05	−0.407676	−0.203838	+	0.979005i	$0.565342\pi$
$524$	−7184.65	−0.598975
$525$	0	0
$526$	−1581.73	−0.131115
$527$	−15811.2	−1.30692
$528$	733.455	0.0604537
$529$	529.000	0.0434783
$530$	−8307.97	−0.680897
$531$	−4458.86	−0.364403
$532$	0	0
$533$	−1521.26	−0.123627
$534$	−10409.4	−0.843554
$535$	−12175.0	−0.983873
$536$	3485.72	0.280896
$537$	13343.0	1.07224
$538$	11788.8	0.944708
$539$	0	0
$540$	7232.93	0.576400
$541$	−5178.12	−0.411506	−0.205753	−	0.978604i	$-0.565964\pi$
$541$	−5178.12	−0.411506	−0.205753	+	0.978604i	$0.565964\pi$
$542$	12949.3	1.02624
$543$	9869.13	0.779972
$544$	1757.87	0.138544
$545$	1355.22	0.106516
$546$	0	0
$547$	−6431.74	−0.502745	−0.251372	−	0.967890i	$-0.580882\pi$
$547$	−6431.74	−0.502745	−0.251372	+	0.967890i	$0.580882\pi$
$548$	−8058.56	−0.628183
$549$	−4458.99	−0.346640
$550$	−334.760	−0.0259532
$551$	−11571.8	−0.894695
$552$	820.459	0.0632628
$553$	0	0
$554$	13276.2	1.01815
$555$	13508.1	1.03313
$556$	8782.63	0.669903
$557$	10927.2	0.831237	0.415618	−	0.909539i	$-0.363565\pi$
$557$	10927.2	0.831237	0.415618	+	0.909539i	$0.363565\pi$
$558$	4096.99	0.310823
$559$	−11191.8	−0.846803
$560$	0	0
$561$	2518.20	0.189516
$562$	11403.6	0.855930
$563$	−16255.3	−1.21684	−0.608418	−	0.793617i	$-0.708195\pi$
$563$	−16255.3	−1.21684	−0.608418	+	0.793617i	$0.708195\pi$
$564$	−11040.2	−0.824252
$565$	−4955.76	−0.369009
$566$	−13950.3	−1.03600
$567$	0	0
$568$	−5823.78	−0.430212
$569$	−6111.27	−0.450259	−0.225130	−	0.974329i	$-0.572281\pi$
$569$	−6111.27	−0.450259	−0.225130	+	0.974329i	$0.572281\pi$
$570$	4989.19	0.366622
$571$	−4997.15	−0.366242	−0.183121	−	0.983090i	$-0.558620\pi$
$571$	−4997.15	−0.366242	−0.183121	+	0.983090i	$0.558620\pi$
$572$	1786.01	0.130554
$573$	−19578.1	−1.42738
$574$	0	0
$575$	−374.470	−0.0271591
$576$	−455.499	−0.0329499
$577$	−18172.2	−1.31112	−0.655561	−	0.755142i	$-0.727568\pi$
$577$	−18172.2	−1.31112	−0.655561	+	0.755142i	$0.727568\pi$
$578$	−3790.64	−0.272785
$579$	−8134.16	−0.583842
$580$	11689.2	0.836843
$581$	0	0
$582$	−4904.92	−0.349339
$583$	3592.83	0.255231
$584$	3685.75	0.261160
$585$	3674.18	0.259673
$586$	13604.2	0.959016
$587$	25446.4	1.78924	0.894622	−	0.446823i	$-0.147445\pi$
$587$	25446.4	1.78924	0.894622	+	0.446823i	$0.147445\pi$
$588$	0	0
$589$	13547.1	0.947705
$590$	14893.2	1.03923
$591$	9363.23	0.651695
$592$	−4077.87	−0.283107
$593$	−1692.73	−0.117221	−0.0586106	−	0.998281i	$-0.518667\pi$
$593$	−1692.73	−0.117221	−0.0586106	+	0.998281i	$0.518667\pi$
$594$	−3127.93	−0.216061
$595$	0	0
$596$	4506.87	0.309746
$597$	−15928.7	−1.09199
$598$	1997.87	0.136621
$599$	−10867.9	−0.741323	−0.370661	−	0.928768i	$-0.620869\pi$
$599$	−10867.9	−0.741323	−0.370661	+	0.928768i	$0.620869\pi$
$600$	−580.789	−0.0395177
$601$	13484.0	0.915184	0.457592	−	0.889162i	$-0.348712\pi$
$601$	13484.0	0.915184	0.457592	+	0.889162i	$0.348712\pi$
$602$	0	0
$603$	−3101.06	−0.209428
$604$	−2044.82	−0.137753
$605$	−14564.3	−0.978714
$606$	13782.9	0.923914
$607$	19759.4	1.32127	0.660634	−	0.750708i	$-0.270287\pi$
$607$	19759.4	1.32127	0.660634	+	0.750708i	$0.270287\pi$
$608$	−1506.15	−0.100465
$609$	0	0
$610$	14893.7	0.988569
$611$	−26883.7	−1.78003
$612$	−1563.88	−0.103294
$613$	−24881.3	−1.63939	−0.819696	−	0.572799i	$-0.805857\pi$
$613$	−24881.3	−1.63939	−0.819696	+	0.572799i	$0.805857\pi$
$614$	−5377.88	−0.353475
$615$	−1856.41	−0.121720
$616$	0	0
$617$	−27257.9	−1.77854	−0.889272	−	0.457379i	$-0.848788\pi$
$617$	−27257.9	−1.77854	−0.889272	+	0.457379i	$0.848788\pi$
$618$	−10228.9	−0.665802
$619$	−1319.47	−0.0856766	−0.0428383	−	0.999082i	$-0.513640\pi$
$619$	−1319.47	−0.0856766	−0.0428383	+	0.999082i	$0.513640\pi$
$620$	−13684.5	−0.886425
$621$	−3498.97	−0.226101
$622$	−10345.1	−0.666880
$623$	0	0
$624$	3098.63	0.198789
$625$	−17395.1	−1.11329
$626$	13177.5	0.841337
$627$	−2157.61	−0.137427
$628$	7.98859	0.000507611 0
$629$	−14000.7	−0.887512
$630$	0	0
$631$	−13073.1	−0.824774	−0.412387	−	0.911009i	$-0.635305\pi$
$631$	−13073.1	−0.824774	−0.412387	+	0.911009i	$0.635305\pi$
$632$	2581.82	0.162499
$633$	−14857.9	−0.932937
$634$	5717.06	0.358128
$635$	21415.2	1.33832
$636$	6233.35	0.388630
$637$	0	0
$638$	−5055.08	−0.313687
$639$	5181.10	0.320753
$640$	1521.43	0.0939685
$641$	−20467.5	−1.26118	−0.630591	−	0.776116i	$-0.717187\pi$
$641$	−20467.5	−1.26118	−0.630591	+	0.776116i	$0.717187\pi$
$642$	9134.74	0.561557
$643$	−22069.8	−1.35357	−0.676787	−	0.736179i	$-0.736628\pi$
$643$	−22069.8	−1.35357	−0.676787	+	0.736179i	$0.736628\pi$
$644$	0	0
$645$	−13657.5	−0.833742
$646$	−5171.13	−0.314946
$647$	−2121.94	−0.128937	−0.0644684	−	0.997920i	$-0.520535\pi$
$647$	−2121.94	−0.128937	−0.0644684	+	0.997920i	$0.520535\pi$
$648$	−3889.46	−0.235791
$649$	−6440.67	−0.389550
$650$	−1414.26	−0.0853415
$651$	0	0
$652$	5988.39	0.359699
$653$	9739.64	0.583678	0.291839	−	0.956467i	$-0.405733\pi$
$653$	9739.64	0.583678	0.291839	+	0.956467i	$0.405733\pi$
$654$	−1016.80	−0.0607952
$655$	−21349.5	−1.27358
$656$	560.420	0.0333548
$657$	−3279.01	−0.194713
$658$	0	0
$659$	−14344.0	−0.847896	−0.423948	−	0.905687i	$-0.639356\pi$
$659$	−14344.0	−0.847896	−0.423948	+	0.905687i	$0.639356\pi$
$660$	2179.50	0.128541
$661$	−3748.54	−0.220577	−0.110289	−	0.993900i	$-0.535178\pi$
$661$	−3748.54	−0.220577	−0.110289	+	0.993900i	$0.535178\pi$
$662$	13007.3	0.763660
$663$	10638.6	0.623183
$664$	7965.65	0.465553
$665$	0	0
$666$	3627.86	0.211076
$667$	−5654.72	−0.328263
$668$	9803.29	0.567815
$669$	13276.5	0.767265
$670$	10358.0	0.597260
$671$	−6440.86	−0.370561
$672$	0	0
$673$	−31694.6	−1.81536	−0.907680	−	0.419663i	$-0.862148\pi$
$673$	−31694.6	−1.81536	−0.907680	+	0.419663i	$0.862148\pi$
$674$	−1904.48	−0.108840
$675$	2476.86	0.141236
$676$	−1242.62	−0.0707000
$677$	907.289	0.0515066	0.0257533	−	0.999668i	$-0.491802\pi$
$677$	907.289	0.0515066	0.0257533	+	0.999668i	$0.491802\pi$
$678$	3718.23	0.210616
$679$	0	0
$680$	5223.59	0.294582
$681$	−24114.5	−1.35693
$682$	5917.96	0.332273
$683$	−10055.9	−0.563367	−0.281683	−	0.959507i	$-0.590893\pi$
$683$	−10055.9	−0.563367	−0.281683	+	0.959507i	$0.590893\pi$
$684$	1339.94	0.0749035
$685$	−23946.4	−1.33568
$686$	0	0
$687$	−3987.78	−0.221460
$688$	4122.96	0.228469
$689$	15178.6	0.839275
$690$	2438.03	0.134513
$691$	−6789.68	−0.373794	−0.186897	−	0.982380i	$-0.559843\pi$
$691$	−6789.68	−0.373794	−0.186897	+	0.982380i	$0.559843\pi$
$692$	2215.10	0.121684
$693$	0	0
$694$	−584.033	−0.0319447
$695$	26098.0	1.42439
$696$	−8770.26	−0.477638
$697$	1924.11	0.104564
$698$	8720.31	0.472878
$699$	−22972.2	−1.24304
$700$	0	0
$701$	−20750.0	−1.11800	−0.559000	−	0.829168i	$-0.688815\pi$
$701$	−20750.0	−1.11800	−0.559000	+	0.829168i	$0.688815\pi$
$702$	−13214.6	−0.710472
$703$	11995.9	0.643576
$704$	−657.952	−0.0352237
$705$	−32806.6	−1.75258
$706$	−7332.41	−0.390877
$707$	0	0
$708$	−11174.2	−0.593151
$709$	−25057.6	−1.32730	−0.663650	−	0.748043i	$-0.730994\pi$
$709$	−25057.6	−1.32730	−0.663650	+	0.748043i	$0.730994\pi$
$710$	−17305.6	−0.914744
$711$	−2296.91	−0.121154
$712$	9337.82	0.491502
$713$	6619.96	0.347713
$714$	0	0
$715$	5307.22	0.277593
$716$	−11969.4	−0.624746
$717$	−6090.38	−0.317224
$718$	−16319.8	−0.848260
$719$	−2692.89	−0.139677	−0.0698386	−	0.997558i	$-0.522248\pi$
$719$	−2692.89	−0.139677	−0.0698386	+	0.997558i	$0.522248\pi$
$720$	−1353.54	−0.0700601
$721$	0	0
$722$	−9287.34	−0.478725
$723$	−20687.6	−1.06415
$724$	−8853.19	−0.454456
$725$	4002.88	0.205053
$726$	10927.4	0.558612
$727$	31861.3	1.62541	0.812704	−	0.582676i	$-0.197994\pi$
$727$	31861.3	1.62541	0.812704	+	0.582676i	$0.197994\pi$
$728$	0	0
$729$	21775.6	1.10631
$730$	10952.4	0.555295
$731$	14155.5	0.716226
$732$	−11174.5	−0.564237
$733$	6284.48	0.316675	0.158337	−	0.987385i	$-0.449387\pi$
$733$	6284.48	0.316675	0.158337	+	0.987385i	$0.449387\pi$
$734$	−2060.49	−0.103616
$735$	0	0
$736$	−736.000	−0.0368605
$737$	−4479.38	−0.223880
$738$	−498.575	−0.0248683
$739$	−13472.9	−0.670646	−0.335323	−	0.942103i	$-0.608845\pi$
$739$	−13472.9	−0.670646	−0.335323	+	0.942103i	$0.608845\pi$
$740$	−12117.6	−0.601961
$741$	−9115.25	−0.451899
$742$	0	0
$743$	28593.5	1.41184	0.705918	−	0.708293i	$-0.250535\pi$
$743$	28593.5	1.41184	0.705918	+	0.708293i	$0.250535\pi$
$744$	10267.3	0.505938
$745$	13392.4	0.658602
$746$	15145.8	0.743332
$747$	−7086.61	−0.347102
$748$	−2258.97	−0.110423
$749$	0	0
$750$	11524.3	0.561079
$751$	31544.4	1.53272	0.766360	−	0.642412i	$-0.222066\pi$
$751$	31544.4	1.53272	0.766360	+	0.642412i	$0.222066\pi$
$752$	9903.74	0.480256
$753$	12031.1	0.582253
$754$	−21356.2	−1.03149
$755$	−6076.28	−0.292899
$756$	0	0
$757$	21644.2	1.03919	0.519597	−	0.854411i	$-0.326082\pi$
$757$	21644.2	1.03919	0.519597	+	0.854411i	$0.326082\pi$
$758$	−7915.53	−0.379294
$759$	−1054.34	−0.0504219
$760$	−4475.60	−0.213615
$761$	7309.90	0.348204	0.174102	−	0.984728i	$-0.444298\pi$
$761$	7309.90	0.348204	0.174102	+	0.984728i	$0.444298\pi$
$762$	−16067.5	−0.763864
$763$	0	0
$764$	17562.7	0.831672
$765$	−4647.14	−0.219631
$766$	−13595.5	−0.641288
$767$	−27209.9	−1.28095
$768$	−1141.51	−0.0536336
$769$	−13959.6	−0.654611	−0.327306	−	0.944919i	$-0.606141\pi$
$769$	−13959.6	−0.654611	−0.327306	+	0.944919i	$0.606141\pi$
$770$	0	0
$771$	−1060.15	−0.0495207
$772$	7296.82	0.340179
$773$	−32291.7	−1.50253	−0.751263	−	0.660004i	$-0.770555\pi$
$773$	−32291.7	−1.50253	−0.751263	+	0.660004i	$0.770555\pi$
$774$	−3667.98	−0.170340
$775$	−4686.16	−0.217202
$776$	4400.00	0.203545
$777$	0	0
$778$	−11467.7	−0.528454
$779$	−1648.59	−0.0758239
$780$	9207.72	0.422678
$781$	7483.92	0.342888
$782$	−2526.94	−0.115554
$783$	37402.1	1.70708
$784$	0	0
$785$	23.7385	0.00107931
$786$	16018.2	0.726911
$787$	−27872.1	−1.26243	−0.631216	−	0.775607i	$-0.717444\pi$
$787$	−27872.1	−1.26243	−0.631216	+	0.775607i	$0.717444\pi$
$788$	−8399.37	−0.379715
$789$	3526.48	0.159120
$790$	7672.00	0.345516
$791$	0	0
$792$	585.345	0.0262618
$793$	−27210.7	−1.21851
$794$	−4825.95	−0.215701
$795$	18522.7	0.826330
$796$	14288.9	0.636254
$797$	−3535.94	−0.157151	−0.0785756	−	0.996908i	$-0.525037\pi$
$797$	−3535.94	−0.157151	−0.0785756	+	0.996908i	$0.525037\pi$
$798$	0	0
$799$	34002.9	1.50555
$800$	521.002	0.0230253
$801$	−8307.36	−0.366449
$802$	7778.37	0.342474
$803$	−4736.42	−0.208150
$804$	−7771.45	−0.340893
$805$	0	0
$806$	25001.6	1.09261
$807$	−26283.3	−1.14649
$808$	−12364.1	−0.538324
$809$	−40884.9	−1.77681	−0.888404	−	0.459063i	$-0.848185\pi$
$809$	−40884.9	−1.77681	−0.888404	+	0.459063i	$0.848185\pi$
$810$	−11557.7	−0.501353
$811$	−29290.2	−1.26821	−0.634106	−	0.773246i	$-0.718632\pi$
$811$	−29290.2	−1.26821	−0.634106	+	0.773246i	$0.718632\pi$
$812$	0	0
$813$	−28870.6	−1.24543
$814$	5240.32	0.225643
$815$	17794.8	0.764815
$816$	−3919.18	−0.168136
$817$	−12128.5	−0.519368
$818$	13255.2	0.566573
$819$	0	0
$820$	1665.31	0.0709210
$821$	42923.8	1.82467	0.912334	−	0.409448i	$-0.134279\pi$
$821$	42923.8	1.82467	0.912334	+	0.409448i	$0.134279\pi$
$822$	17966.6	0.762358
$823$	−4030.32	−0.170703	−0.0853513	−	0.996351i	$-0.527201\pi$
$823$	−4030.32	−0.170703	−0.0853513	+	0.996351i	$0.527201\pi$
$824$	9175.90	0.387934
$825$	746.351	0.0314965
$826$	0	0
$827$	6589.68	0.277081	0.138540	−	0.990357i	$-0.455759\pi$
$827$	6589.68	0.277081	0.138540	+	0.990357i	$0.455759\pi$
$828$	654.780	0.0274821
$829$	1413.71	0.0592280	0.0296140	−	0.999561i	$-0.490572\pi$
$829$	1413.71	0.0592280	0.0296140	+	0.999561i	$0.490572\pi$
$830$	23670.3	0.989889
$831$	−29599.4	−1.23561
$832$	−2779.65	−0.115826
$833$	0	0
$834$	−19580.9	−0.812989
$835$	29130.9	1.20733
$836$	1935.50	0.0800726
$837$	−43786.4	−1.80822
$838$	13842.1	0.570606
$839$	2035.63	0.0837636	0.0418818	−	0.999123i	$-0.486665\pi$
$839$	2035.63	0.0837636	0.0418818	+	0.999123i	$0.486665\pi$
$840$	0	0
$841$	36056.9	1.47841
$842$	−22261.1	−0.911127
$843$	−25424.5	−1.03875
$844$	13328.4	0.543582
$845$	−3692.51	−0.150327
$846$	−8810.82	−0.358064
$847$	0	0
$848$	−5591.68	−0.226438
$849$	31102.3	1.25728
$850$	1788.78	0.0721818
$851$	5861.94	0.236128
$852$	12984.2	0.522101
$853$	33999.8	1.36475	0.682374	−	0.731003i	$-0.260947\pi$
$853$	33999.8	1.36475	0.682374	+	0.731003i	$0.260947\pi$
$854$	0	0
$855$	3981.70	0.159265
$856$	−8194.40	−0.327195
$857$	7043.33	0.280742	0.140371	−	0.990099i	$-0.455171\pi$
$857$	7043.33	0.280742	0.140371	+	0.990099i	$0.455171\pi$
$858$	−3981.93	−0.158439
$859$	4240.00	0.168413	0.0842065	−	0.996448i	$-0.473164\pi$
$859$	4240.00	0.168413	0.0842065	+	0.996448i	$0.473164\pi$
$860$	12251.6	0.485785
$861$	0	0
$862$	742.934	0.0293555
$863$	−22072.8	−0.870647	−0.435324	−	0.900274i	$-0.643366\pi$
$863$	−22072.8	−0.870647	−0.435324	+	0.900274i	$0.643366\pi$
$864$	4868.13	0.191686
$865$	6582.26	0.258733
$866$	−18995.0	−0.745355
$867$	8451.27	0.331050
$868$	0	0
$869$	−3317.81	−0.129515
$870$	−26061.2	−1.01558
$871$	−18924.0	−0.736184
$872$	912.130	0.0354227
$873$	−3914.44	−0.151757
$874$	2165.09	0.0837934
$875$	0	0
$876$	−8217.41	−0.316941
$877$	29563.1	1.13828	0.569142	−	0.822239i	$-0.307275\pi$
$877$	29563.1	1.13828	0.569142	+	0.822239i	$0.307275\pi$
$878$	4573.99	0.175814
$879$	−30330.6	−1.16385
$880$	−1955.14	−0.0748950
$881$	23588.9	0.902077	0.451039	−	0.892504i	$-0.351054\pi$
$881$	23588.9	0.902077	0.451039	+	0.892504i	$0.351054\pi$
$882$	0	0
$883$	27447.6	1.04608	0.523039	−	0.852309i	$-0.324798\pi$
$883$	27447.6	1.04608	0.523039	+	0.852309i	$0.324798\pi$
$884$	−9543.48	−0.363102
$885$	−33204.6	−1.26120
$886$	36906.0	1.39942
$887$	−39987.4	−1.51369	−0.756847	−	0.653592i	$-0.773261\pi$
$887$	−39987.4	−1.51369	−0.756847	+	0.653592i	$0.773261\pi$
$888$	9091.65	0.343576
$889$	0	0
$890$	27747.8	1.04506
$891$	4998.20	0.187930
$892$	−11909.8	−0.447052
$893$	−29133.9	−1.09174
$894$	−10048.1	−0.375905
$895$	−35567.7	−1.32838
$896$	0	0
$897$	−4454.28	−0.165802
$898$	−17277.9	−0.642060
$899$	−70763.7	−2.62525
$900$	−463.508	−0.0171669
$901$	−19198.1	−0.709859
$902$	−720.175	−0.0265845
$903$	0	0
$904$	−3335.47	−0.122717
$905$	−26307.6	−0.966293
$906$	4558.95	0.167175
$907$	19489.0	0.713474	0.356737	−	0.934205i	$-0.383889\pi$
$907$	19489.0	0.713474	0.356737	+	0.934205i	$0.383889\pi$
$908$	21632.1	0.790625
$909$	10999.6	0.401359
$910$	0	0
$911$	20571.8	0.748161	0.374081	−	0.927396i	$-0.377958\pi$
$911$	20571.8	0.748161	0.374081	+	0.927396i	$0.377958\pi$
$912$	3357.98	0.121923
$913$	−10236.4	−0.371056
$914$	−8575.57	−0.310344
$915$	−33205.6	−1.19972
$916$	3577.27	0.129035
$917$	0	0
$918$	16713.9	0.600917
$919$	45895.8	1.64740	0.823700	−	0.567025i	$-0.191906\pi$
$919$	45895.8	1.64740	0.823700	+	0.567025i	$0.191906\pi$
$920$	−2187.06	−0.0783752
$921$	11990.0	0.428974
$922$	−3475.42	−0.124140
$923$	31617.3	1.12752
$924$	0	0
$925$	−4149.57	−0.147500
$926$	11503.0	0.408219
$927$	−8163.30	−0.289232
$928$	7867.44	0.278299
$929$	−34121.8	−1.20506	−0.602529	−	0.798097i	$-0.705840\pi$
$929$	−34121.8	−1.20506	−0.602529	+	0.798097i	$0.705840\pi$
$930$	30509.8	1.07576
$931$	0	0
$932$	20607.4	0.724268
$933$	23064.4	0.809319
$934$	−7878.05	−0.275993
$935$	−6712.64	−0.234788
$936$	2472.91	0.0863563
$937$	26695.0	0.930723	0.465362	−	0.885121i	$-0.345924\pi$
$937$	26695.0	0.930723	0.465362	+	0.885121i	$0.345924\pi$
$938$	0	0
$939$	−29379.2	−1.02104
$940$	29429.4	1.02115
$941$	−31859.6	−1.10371	−0.551857	−	0.833939i	$-0.686081\pi$
$941$	−31859.6	−1.10371	−0.551857	+	0.833939i	$0.686081\pi$
$942$	−17.8106	−0.000616032 0
$943$	−805.604	−0.0278198
$944$	10023.9	0.345604
$945$	0	0
$946$	−5298.27	−0.182095
$947$	8444.38	0.289763	0.144882	−	0.989449i	$-0.453720\pi$
$947$	8444.38	0.289763	0.144882	+	0.989449i	$0.453720\pi$
$948$	−5756.19	−0.197207
$949$	−20010.0	−0.684458
$950$	−1532.63	−0.0523423
$951$	−12746.2	−0.434621
$952$	0	0
$953$	−3711.39	−0.126153	−0.0630765	−	0.998009i	$-0.520091\pi$
$953$	−3711.39	−0.126153	−0.0630765	+	0.998009i	$0.520091\pi$
$954$	4974.62	0.168825
$955$	52188.5	1.76835
$956$	5463.43	0.184833
$957$	11270.3	0.380688
$958$	11952.8	0.403108
$959$	0	0
$960$	−3392.04	−0.114039
$961$	53051.8	1.78080
$962$	22138.8	0.741979
$963$	7290.12	0.243947
$964$	18558.0	0.620034
$965$	21682.8	0.723311
$966$	0	0
$967$	13826.4	0.459801	0.229901	−	0.973214i	$-0.426160\pi$
$967$	13826.4	0.459801	0.229901	+	0.973214i	$0.426160\pi$
$968$	−9802.49	−0.325479
$969$	11529.1	0.382216
$970$	13074.8	0.432790
$971$	32815.5	1.08455	0.542276	−	0.840201i	$-0.317563\pi$
$971$	32815.5	1.08455	0.542276	+	0.840201i	$0.317563\pi$
$972$	−7758.35	−0.256018
$973$	0	0
$974$	34820.6	1.14551
$975$	3153.11	0.103570
$976$	10024.2	0.328757
$977$	−23155.5	−0.758250	−0.379125	−	0.925345i	$-0.623775\pi$
$977$	−23155.5	−0.758250	−0.379125	+	0.925345i	$0.623775\pi$
$978$	−13351.2	−0.436527
$979$	−11999.7	−0.391738
$980$	0	0
$981$	−811.472	−0.0264101
$982$	15204.8	0.494099
$983$	42979.1	1.39453	0.697263	−	0.716815i	$-0.254401\pi$
$983$	42979.1	1.39453	0.697263	+	0.716815i	$0.254401\pi$
$984$	−1249.46	−0.0404790
$985$	−24959.1	−0.807374
$986$	27011.6	0.872438
$987$	0	0
$988$	8176.91	0.263302
$989$	−5926.76	−0.190556
$990$	1739.38	0.0558395
$991$	11590.5	0.371528	0.185764	−	0.982594i	$-0.440524\pi$
$991$	11590.5	0.371528	0.185764	+	0.982594i	$0.440524\pi$
$992$	−9210.37	−0.294788
$993$	−28999.9	−0.926771
$994$	0	0
$995$	42460.3	1.35285
$996$	−17759.5	−0.564991
$997$	−28069.5	−0.891644	−0.445822	−	0.895122i	$-0.647089\pi$
$997$	−28069.5	−0.891644	−0.445822	+	0.895122i	$0.647089\pi$
$998$	33240.7	1.05432
$999$	−38772.7	−1.22794

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.x.1.4		11
7.3	odd	6		322.4.e.b.93.4	✓	22
7.5	odd	6		322.4.e.b.277.4	yes	22
7.6	odd	2		2254.4.a.w.1.8		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.4.e.b.93.4	✓	22	7.3	odd	6
322.4.e.b.277.4	yes	22	7.5	odd	6
2254.4.a.w.1.8		11	7.6	odd	2
2254.4.a.x.1.4		11	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -4.45902 q^{3} +4.00000 q^{4} +11.8862 q^{5} -8.91803 q^{6} +8.00000 q^{8} -7.11717 q^{9} +23.7724 q^{10} -10.2805 q^{11} -17.8361 q^{12} -43.4321 q^{13} -53.0007 q^{15} +16.0000 q^{16} +54.9334 q^{17} -14.2343 q^{18} -47.0673 q^{19} +47.5447 q^{20} -20.5610 q^{22} -23.0000 q^{23} -35.6721 q^{24} +16.2813 q^{25} -86.8641 q^{26} +152.129 q^{27} +245.857 q^{29} -106.001 q^{30} -287.824 q^{31} +32.0000 q^{32} +45.8410 q^{33} +109.867 q^{34} -28.4687 q^{36} -254.867 q^{37} -94.1345 q^{38} +193.664 q^{39} +95.0895 q^{40} +35.0262 q^{41} +257.685 q^{43} -41.1220 q^{44} -84.5960 q^{45} -46.0000 q^{46} +618.984 q^{47} -71.3443 q^{48} +32.5626 q^{50} -244.949 q^{51} -173.728 q^{52} -349.480 q^{53} +304.258 q^{54} -122.196 q^{55} +209.874 q^{57} +491.715 q^{58} +626.493 q^{59} -212.003 q^{60} +626.512 q^{61} -575.648 q^{62} +64.0000 q^{64} -516.241 q^{65} +91.6819 q^{66} +435.715 q^{67} +219.734 q^{68} +102.557 q^{69} -727.972 q^{71} -56.9374 q^{72} +460.719 q^{73} -509.734 q^{74} -72.5987 q^{75} -188.269 q^{76} +387.329 q^{78} +322.728 q^{79} +190.179 q^{80} -486.182 q^{81} +70.0525 q^{82} +995.707 q^{83} +652.948 q^{85} +515.371 q^{86} -1096.28 q^{87} -82.2441 q^{88} +1167.23 q^{89} -169.192 q^{90} -92.0000 q^{92} +1283.41 q^{93} +1237.97 q^{94} -559.450 q^{95} -142.689 q^{96} +550.000 q^{97} +73.1681 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 22 q^{2} + 6 q^{3} + 44 q^{4} + 27 q^{5} + 12 q^{6} + 88 q^{8} + 59 q^{9} + 54 q^{10} + 56 q^{11} + 24 q^{12} + 103 q^{13} + 62 q^{15} + 176 q^{16} + 157 q^{17} + 118 q^{18} + 266 q^{19} + 108 q^{20}+ \cdots + 101 q^{99}+O(q^{100}) \) 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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