LMFDB - Embedded newform 2254.4.a.d.1.2

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{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 46)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.70156$ 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)

$f(q)$	$=$	\(q-2.00000 q^{2} +10.1047 q^{3} +4.00000 q^{4} -11.4031 q^{5} -20.2094 q^{6} -8.00000 q^{8} +75.1047 q^{9} +22.8062 q^{10} +10.3875 q^{11} +40.4187 q^{12} +33.0891 q^{13} -115.225 q^{15} +16.0000 q^{16} -138.837 q^{17} -150.209 q^{18} -68.6281 q^{19} -45.6125 q^{20} -20.7750 q^{22} -23.0000 q^{23} -80.8375 q^{24} +5.03124 q^{25} -66.1781 q^{26} +486.083 q^{27} +215.602 q^{29} +230.450 q^{30} -87.9266 q^{31} -32.0000 q^{32} +104.962 q^{33} +277.675 q^{34} +300.419 q^{36} -215.109 q^{37} +137.256 q^{38} +334.355 q^{39} +91.2250 q^{40} -175.267 q^{41} -40.7125 q^{43} +41.5500 q^{44} -856.428 q^{45} +46.0000 q^{46} -405.245 q^{47} +161.675 q^{48} -10.0625 q^{50} -1402.91 q^{51} +132.356 q^{52} -276.994 q^{53} -972.166 q^{54} -118.450 q^{55} -693.466 q^{57} -431.203 q^{58} -293.550 q^{59} -460.900 q^{60} +450.731 q^{61} +175.853 q^{62} +64.0000 q^{64} -377.319 q^{65} -209.925 q^{66} +273.675 q^{67} -555.350 q^{68} -232.408 q^{69} -643.842 q^{71} -600.837 q^{72} -106.345 q^{73} +430.219 q^{74} +50.8391 q^{75} -274.512 q^{76} -668.709 q^{78} +60.0844 q^{79} -182.450 q^{80} +2883.89 q^{81} +350.534 q^{82} +372.878 q^{83} +1583.18 q^{85} +81.4250 q^{86} +2178.59 q^{87} -83.1000 q^{88} +543.947 q^{89} +1712.86 q^{90} -92.0000 q^{92} -888.470 q^{93} +810.491 q^{94} +782.575 q^{95} -323.350 q^{96} -550.944 q^{97} +780.150 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + q^{3} + 8 q^{4} - 10 q^{5} - 2 q^{6} - 16 q^{8} + 131 q^{9} + 20 q^{10} + 72 q^{11} + 4 q^{12} + 111 q^{13} - 128 q^{15} + 32 q^{16} - 124 q^{17} - 262 q^{18} - 22 q^{19} - 40 q^{20} - 144 q^{22}+ \cdots + 4224 q^{99}+O(q^{100}) $ 2 * q - 4 * q^2 + q^3 + 8 * q^4 - 10 * q^5 - 2 * q^6 - 16 * q^8 + 131 * q^9 + 20 * q^10 + 72 * q^11 + 4 * q^12 + 111 * q^13 - 128 * q^15 + 32 * q^16 - 124 * q^17 - 262 * q^18 - 22 * q^19 - 40 * q^20 - 144 * q^22 - 46 * q^23 - 8 * q^24 - 118 * q^25 - 222 * q^26 + 223 * q^27 + 15 * q^29 + 256 * q^30 - 67 * q^31 - 64 * q^32 - 456 * q^33 + 248 * q^34 + 524 * q^36 + 18 * q^37 + 44 * q^38 - 375 * q^39 + 80 * q^40 - 485 * q^41 - 440 * q^43 + 288 * q^44 - 778 * q^45 + 92 * q^46 - 215 * q^47 + 16 * q^48 + 236 * q^50 - 1538 * q^51 + 444 * q^52 + 240 * q^53 - 446 * q^54 - 32 * q^55 - 1118 * q^57 - 30 * q^58 - 792 * q^59 - 512 * q^60 - 456 * q^61 + 134 * q^62 + 128 * q^64 - 268 * q^65 + 912 * q^66 + 240 * q^67 - 496 * q^68 - 23 * q^69 - 705 * q^71 - 1048 * q^72 - 27 * q^73 - 36 * q^74 + 1171 * q^75 - 88 * q^76 + 750 * q^78 + 594 * q^79 - 160 * q^80 + 3770 * q^81 + 970 * q^82 - 394 * q^83 + 1604 * q^85 + 880 * q^86 + 4005 * q^87 - 576 * q^88 + 486 * q^89 + 1556 * q^90 - 184 * q^92 - 1079 * q^93 + 430 * q^94 + 848 * q^95 - 32 * q^96 - 2152 * q^97 + 4224 * 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
$3$	10.1047	1.94465	0.972324	−	0.233637i	$-0.0750627\pi$
$3$	10.1047	1.94465	0.972324	+	0.233637i	$0.0750627\pi$
$4$	4.00000	0.500000
$5$	−11.4031	−1.01993	−0.509963	−	0.860196i	$-0.670341\pi$
$5$	−11.4031	−1.01993	−0.509963	+	0.860196i	$0.670341\pi$
$6$	−20.2094	−1.37507
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	75.1047	2.78166
$10$	22.8062	0.721197
$11$	10.3875	0.284723	0.142361	−	0.989815i	$-0.454530\pi$
$11$	10.3875	0.284723	0.142361	+	0.989815i	$0.454530\pi$
$12$	40.4187	0.972324
$13$	33.0891	0.705943	0.352971	−	0.935634i	$-0.385171\pi$
$13$	33.0891	0.705943	0.352971	+	0.935634i	$0.385171\pi$
$14$	0	0
$15$	−115.225	−1.98340
$16$	16.0000	0.250000
$17$	−138.837	−1.98077	−0.990383	−	0.138350i	$-0.955820\pi$
$17$	−138.837	−1.98077	−0.990383	+	0.138350i	$0.955820\pi$
$18$	−150.209	−1.96693
$19$	−68.6281	−0.828651	−0.414326	−	0.910129i	$-0.635982\pi$
$19$	−68.6281	−0.828651	−0.414326	+	0.910129i	$0.635982\pi$
$20$	−45.6125	−0.509963
$21$	0	0
$22$	−20.7750	−0.201329
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	−80.8375	−0.687537
$25$	5.03124	0.0402499
$26$	−66.1781	−0.499177
$27$	486.083	3.46469
$28$	0	0
$29$	215.602	1.38056	0.690279	−	0.723543i	$-0.257488\pi$
$29$	215.602	1.38056	0.690279	+	0.723543i	$0.257488\pi$
$30$	230.450	1.40247
$31$	−87.9266	−0.509422	−0.254711	−	0.967017i	$-0.581980\pi$
$31$	−87.9266	−0.509422	−0.254711	+	0.967017i	$0.581980\pi$
$32$	−32.0000	−0.176777
$33$	104.962	0.553685
$34$	277.675	1.40061
$35$	0	0
$36$	300.419	1.39083
$37$	−215.109	−0.955777	−0.477889	−	0.878420i	$-0.658598\pi$
$37$	−215.109	−0.955777	−0.477889	+	0.878420i	$0.658598\pi$
$38$	137.256	0.585945
$39$	334.355	1.37281
$40$	91.2250	0.360598
$41$	−175.267	−0.667613	−0.333807	−	0.942642i	$-0.608333\pi$
$41$	−175.267	−0.667613	−0.333807	+	0.942642i	$0.608333\pi$
$42$	0	0
$43$	−40.7125	−0.144386	−0.0721930	−	0.997391i	$-0.523000\pi$
$43$	−40.7125	−0.144386	−0.0721930	+	0.997391i	$0.523000\pi$
$44$	41.5500	0.142361
$45$	−856.428	−2.83708
$46$	46.0000	0.147442
$47$	−405.245	−1.25768	−0.628841	−	0.777534i	$-0.716471\pi$
$47$	−405.245	−1.25768	−0.628841	+	0.777534i	$0.716471\pi$
$48$	161.675	0.486162
$49$	0	0
$50$	−10.0625	−0.0284610
$51$	−1402.91	−3.85189
$52$	132.356	0.352971
$53$	−276.994	−0.717887	−0.358944	−	0.933359i	$-0.616863\pi$
$53$	−276.994	−0.717887	−0.358944	+	0.933359i	$0.616863\pi$
$54$	−972.166	−2.44991
$55$	−118.450	−0.290396
$56$	0	0
$57$	−693.466	−1.61143
$58$	−431.203	−0.976202
$59$	−293.550	−0.647745	−0.323873	−	0.946101i	$-0.604985\pi$
$59$	−293.550	−0.647745	−0.323873	+	0.946101i	$0.604985\pi$
$60$	−460.900	−0.991699
$61$	450.731	0.946069	0.473035	−	0.881044i	$-0.343159\pi$
$61$	450.731	0.946069	0.473035	+	0.881044i	$0.343159\pi$
$62$	175.853	0.360216
$63$	0	0
$64$	64.0000	0.125000
$65$	−377.319	−0.720010
$66$	−209.925	−0.391515
$67$	273.675	0.499026	0.249513	−	0.968371i	$-0.419730\pi$
$67$	273.675	0.499026	0.249513	+	0.968371i	$0.419730\pi$
$68$	−555.350	−0.990383
$69$	−232.408	−0.405487
$70$	0	0
$71$	−643.842	−1.07620	−0.538099	−	0.842882i	$-0.680857\pi$
$71$	−643.842	−1.07620	−0.538099	+	0.842882i	$0.680857\pi$
$72$	−600.837	−0.983464
$73$	−106.345	−0.170504	−0.0852519	−	0.996359i	$-0.527169\pi$
$73$	−106.345	−0.170504	−0.0852519	+	0.996359i	$0.527169\pi$
$74$	430.219	0.675837
$75$	50.8391	0.0782720
$76$	−274.512	−0.414326
$77$	0	0
$78$	−668.709	−0.970723
$79$	60.0844	0.0855699	0.0427850	−	0.999084i	$-0.486377\pi$
$79$	60.0844	0.0855699	0.0427850	+	0.999084i	$0.486377\pi$
$80$	−182.450	−0.254982
$81$	2883.89	3.95595
$82$	350.534	0.472074
$83$	372.878	0.493117	0.246558	−	0.969128i	$-0.420700\pi$
$83$	372.878	0.493117	0.246558	+	0.969128i	$0.420700\pi$
$84$	0	0
$85$	1583.18	2.02024
$86$	81.4250	0.102096
$87$	2178.59	2.68470
$88$	−83.1000	−0.100665
$89$	543.947	0.647846	0.323923	−	0.946084i	$-0.394998\pi$
$89$	543.947	0.647846	0.323923	+	0.946084i	$0.394998\pi$
$90$	1712.86	2.00612
$91$	0	0
$92$	−92.0000	−0.104257
$93$	−888.470	−0.990646
$94$	810.491	0.889316
$95$	782.575	0.845163
$96$	−323.350	−0.343768
$97$	−550.944	−0.576700	−0.288350	−	0.957525i	$-0.593107\pi$
$97$	−550.944	−0.576700	−0.288350	+	0.957525i	$0.593107\pi$
$98$	0	0
$99$	780.150	0.792000
$100$	20.1250	0.0201250
$101$	−1084.57	−1.06851	−0.534254	−	0.845324i	$-0.679407\pi$
$101$	−1084.57	−1.06851	−0.534254	+	0.845324i	$0.679407\pi$
$102$	2805.82	2.72370
$103$	−1369.19	−1.30981	−0.654905	−	0.755712i	$-0.727291\pi$
$103$	−1369.19	−1.30981	−0.654905	+	0.755712i	$0.727291\pi$
$104$	−264.713	−0.249588
$105$	0	0
$106$	553.987	0.507623
$107$	−1193.82	−1.07861	−0.539304	−	0.842111i	$-0.681313\pi$
$107$	−1193.82	−1.07861	−0.539304	+	0.842111i	$0.681313\pi$
$108$	1944.33	1.73235
$109$	−893.309	−0.784986	−0.392493	−	0.919755i	$-0.628387\pi$
$109$	−893.309	−0.784986	−0.392493	+	0.919755i	$0.628387\pi$
$110$	236.900	0.205341
$111$	−2173.61	−1.85865
$112$	0	0
$113$	1842.16	1.53359	0.766797	−	0.641890i	$-0.221849\pi$
$113$	1842.16	1.53359	0.766797	+	0.641890i	$0.221849\pi$
$114$	1386.93	1.13946
$115$	262.272	0.212669
$116$	862.406	0.690279
$117$	2485.14	1.96369
$118$	587.100	0.458025
$119$	0	0
$120$	921.800	0.701237
$121$	−1223.10	−0.918933
$122$	−901.462	−0.668972
$123$	−1771.02	−1.29827
$124$	−351.706	−0.254711
$125$	1368.02	0.978874
$126$	0	0
$127$	599.361	0.418777	0.209389	−	0.977833i	$-0.432853\pi$
$127$	599.361	0.418777	0.209389	+	0.977833i	$0.432853\pi$
$128$	−128.000	−0.0883883
$129$	−411.387	−0.280780
$130$	754.637	0.509124
$131$	560.136	0.373582	0.186791	−	0.982400i	$-0.440191\pi$
$131$	560.136	0.373582	0.186791	+	0.982400i	$0.440191\pi$
$132$	419.850	0.276843
$133$	0	0
$134$	−547.350	−0.352864
$135$	−5542.86	−3.53373
$136$	1110.70	0.700307
$137$	−2111.69	−1.31689	−0.658445	−	0.752629i	$-0.728785\pi$
$137$	−2111.69	−1.31689	−0.658445	+	0.752629i	$0.728785\pi$
$138$	464.816	0.286723
$139$	944.952	0.576617	0.288308	−	0.957538i	$-0.406907\pi$
$139$	944.952	0.576617	0.288308	+	0.957538i	$0.406907\pi$
$140$	0	0
$141$	−4094.88	−2.44575
$142$	1287.68	0.760986
$143$	343.713	0.200998
$144$	1201.67	0.695414
$145$	−2458.53	−1.40807
$146$	212.691	0.120564
$147$	0	0
$148$	−860.437	−0.477889
$149$	−2480.47	−1.36381	−0.681906	−	0.731440i	$-0.738849\pi$
$149$	−2480.47	−1.36381	−0.681906	+	0.731440i	$0.738849\pi$
$150$	−101.678	−0.0553466
$151$	1164.53	0.627605	0.313802	−	0.949488i	$-0.398397\pi$
$151$	1164.53	0.627605	0.313802	+	0.949488i	$0.398397\pi$
$152$	549.025	0.292972
$153$	−10427.3	−5.50981
$154$	0	0
$155$	1002.64	0.519573
$156$	1337.42	0.686405
$157$	1525.27	0.775347	0.387673	−	0.921797i	$-0.373279\pi$
$157$	1525.27	0.775347	0.387673	+	0.921797i	$0.373279\pi$
$158$	−120.169	−0.0605071
$159$	−2798.93	−1.39604
$160$	364.900	0.180299
$161$	0	0
$162$	−5767.77	−2.79728
$163$	1944.47	0.934371	0.467185	−	0.884159i	$-0.345268\pi$
$163$	1944.47	0.934371	0.467185	+	0.884159i	$0.345268\pi$
$164$	−701.069	−0.333807
$165$	−1196.90	−0.564718
$166$	−745.756	−0.348686
$167$	2632.58	1.21985	0.609926	−	0.792458i	$-0.291199\pi$
$167$	2632.58	1.21985	0.609926	+	0.792458i	$0.291199\pi$
$168$	0	0
$169$	−1102.11	−0.501645
$170$	−3166.36	−1.42852
$171$	−5154.29	−2.30502
$172$	−162.850	−0.0721930
$173$	−1189.63	−0.522806	−0.261403	−	0.965230i	$-0.584185\pi$
$173$	−1189.63	−0.522806	−0.261403	+	0.965230i	$0.584185\pi$
$174$	−4357.17	−1.89837
$175$	0	0
$176$	166.200	0.0711807
$177$	−2966.23	−1.25964
$178$	−1087.89	−0.458096
$179$	−1548.97	−0.646790	−0.323395	−	0.946264i	$-0.604824\pi$
$179$	−1548.97	−0.646790	−0.323395	+	0.946264i	$0.604824\pi$
$180$	−3425.71	−1.41854
$181$	1908.38	0.783697	0.391848	−	0.920030i	$-0.371836\pi$
$181$	1908.38	0.783697	0.391848	+	0.920030i	$0.371836\pi$
$182$	0	0
$183$	4554.50	1.83977
$184$	184.000	0.0737210
$185$	2452.92	0.974823
$186$	1776.94	0.700492
$187$	−1442.17	−0.563969
$188$	−1620.98	−0.628841
$189$	0	0
$190$	−1565.15	−0.597621
$191$	352.625	0.133587	0.0667933	−	0.997767i	$-0.478723\pi$
$191$	352.625	0.133587	0.0667933	+	0.997767i	$0.478723\pi$
$192$	646.700	0.243081
$193$	−1414.28	−0.527472	−0.263736	−	0.964595i	$-0.584955\pi$
$193$	−1414.28	−0.527472	−0.263736	+	0.964595i	$0.584955\pi$
$194$	1101.89	0.407788
$195$	−3812.69	−1.40017
$196$	0	0
$197$	−432.530	−0.156429	−0.0782144	−	0.996937i	$-0.524922\pi$
$197$	−432.530	−0.156429	−0.0782144	+	0.996937i	$0.524922\pi$
$198$	−1560.30	−0.560029
$199$	108.519	0.0386567	0.0193283	−	0.999813i	$-0.493847\pi$
$199$	108.519	0.0386567	0.0193283	+	0.999813i	$0.493847\pi$
$200$	−40.2499	−0.0142305
$201$	2765.40	0.970429
$202$	2169.15	0.755549
$203$	0	0
$204$	−5611.64	−1.92595
$205$	1998.59	0.680916
$206$	2738.38	0.926175
$207$	−1727.41	−0.580015
$208$	529.425	0.176486
$209$	−712.875	−0.235936
$210$	0	0
$211$	−1082.37	−0.353144	−0.176572	−	0.984288i	$-0.556501\pi$
$211$	−1082.37	−0.353144	−0.176572	+	0.984288i	$0.556501\pi$
$212$	−1107.97	−0.358944
$213$	−6505.82	−2.09282
$214$	2387.64	0.762691
$215$	464.250	0.147263
$216$	−3888.66	−1.22495
$217$	0	0
$218$	1786.62	0.555069
$219$	−1074.59	−0.331570
$220$	−473.800	−0.145198
$221$	−4594.00	−1.39831
$222$	4347.22	1.31426
$223$	1619.44	0.486303	0.243151	−	0.969988i	$-0.421819\pi$
$223$	1619.44	0.486303	0.243151	+	0.969988i	$0.421819\pi$
$224$	0	0
$225$	377.870	0.111961
$226$	−3684.32	−1.08441
$227$	−6746.10	−1.97249	−0.986244	−	0.165299i	$-0.947141\pi$
$227$	−6746.10	−1.97249	−0.986244	+	0.165299i	$0.947141\pi$
$228$	−2773.86	−0.805717
$229$	−1192.63	−0.344155	−0.172077	−	0.985083i	$-0.555048\pi$
$229$	−1192.63	−0.344155	−0.172077	+	0.985083i	$0.555048\pi$
$230$	−524.544	−0.150380
$231$	0	0
$232$	−1724.81	−0.488101
$233$	−2860.16	−0.804187	−0.402093	−	0.915599i	$-0.631717\pi$
$233$	−2860.16	−0.804187	−0.402093	+	0.915599i	$0.631717\pi$
$234$	−4970.29	−1.38854
$235$	4621.06	1.28274
$236$	−1174.20	−0.323873
$237$	607.134	0.166403
$238$	0	0
$239$	−594.373	−0.160865	−0.0804327	−	0.996760i	$-0.525630\pi$
$239$	−594.373	−0.160865	−0.0804327	+	0.996760i	$0.525630\pi$
$240$	−1843.60	−0.495849
$241$	−3930.15	−1.05047	−0.525234	−	0.850958i	$-0.676022\pi$
$241$	−3930.15	−1.05047	−0.525234	+	0.850958i	$0.676022\pi$
$242$	2446.20	0.649784
$243$	16016.5	4.22824
$244$	1802.92	0.473035
$245$	0	0
$246$	3542.04	0.918017
$247$	−2270.84	−0.584980
$248$	703.412	0.180108
$249$	3767.82	0.958938
$250$	−2736.04	−0.692169
$251$	−2122.07	−0.533641	−0.266821	−	0.963746i	$-0.585973\pi$
$251$	−2122.07	−0.533641	−0.266821	+	0.963746i	$0.585973\pi$
$252$	0	0
$253$	−238.913	−0.0593688
$254$	−1198.72	−0.296120
$255$	15997.5	3.92865
$256$	256.000	0.0625000
$257$	−1143.68	−0.277591	−0.138795	−	0.990321i	$-0.544323\pi$
$257$	−1143.68	−0.277591	−0.138795	+	0.990321i	$0.544323\pi$
$258$	822.775	0.198541
$259$	0	0
$260$	−1509.27	−0.360005
$261$	16192.7	3.84024
$262$	−1120.27	−0.264163
$263$	−947.534	−0.222158	−0.111079	−	0.993812i	$-0.535431\pi$
$263$	−947.534	−0.222158	−0.111079	+	0.993812i	$0.535431\pi$
$264$	−839.700	−0.195757
$265$	3158.59	0.732192
$266$	0	0
$267$	5496.41	1.25983
$268$	1094.70	0.249513
$269$	−3217.70	−0.729317	−0.364659	−	0.931141i	$-0.618814\pi$
$269$	−3217.70	−0.729317	−0.364659	+	0.931141i	$0.618814\pi$
$270$	11085.7	2.49872
$271$	7284.05	1.63275	0.816374	−	0.577524i	$-0.195981\pi$
$271$	7284.05	1.63275	0.816374	+	0.577524i	$0.195981\pi$
$272$	−2221.40	−0.495192
$273$	0	0
$274$	4223.38	0.931182
$275$	52.2620	0.0114601
$276$	−929.631	−0.202744
$277$	2524.08	0.547498	0.273749	−	0.961801i	$-0.411736\pi$
$277$	2524.08	0.547498	0.273749	+	0.961801i	$0.411736\pi$
$278$	−1889.90	−0.407730
$279$	−6603.70	−1.41704
$280$	0	0
$281$	−6277.44	−1.33267	−0.666336	−	0.745652i	$-0.732138\pi$
$281$	−6277.44	−1.33267	−0.666336	+	0.745652i	$0.732138\pi$
$282$	8189.75	1.72941
$283$	8.34415	0.00175268	0.000876340	−	1.00000i	$-0.499721\pi$
$283$	8.34415	0.00175268	0.000876340		1.00000i	$0.499721\pi$
$284$	−2575.37	−0.538099
$285$	7907.67	1.64354
$286$	−687.426	−0.142127
$287$	0	0
$288$	−2403.35	−0.491732
$289$	14362.8	2.92344
$290$	4917.06	0.995655
$291$	−5567.11	−1.12148
$292$	−425.381	−0.0852519
$293$	−3075.15	−0.613147	−0.306573	−	0.951847i	$-0.599182\pi$
$293$	−3075.15	−0.613147	−0.306573	+	0.951847i	$0.599182\pi$
$294$	0	0
$295$	3347.39	0.660652
$296$	1720.87	0.337918
$297$	5049.19	0.986476
$298$	4960.94	0.964360
$299$	−761.048	−0.147199
$300$	203.357	0.0391360
$301$	0	0
$302$	−2329.07	−0.443784
$303$	−10959.3	−2.07787
$304$	−1098.05	−0.207163
$305$	−5139.74	−0.964921
$306$	20854.7	3.89602
$307$	1039.90	0.193323	0.0966616	−	0.995317i	$-0.469184\pi$
$307$	1039.90	0.193323	0.0966616	+	0.995317i	$0.469184\pi$
$308$	0	0
$309$	−13835.2	−2.54712
$310$	−2005.27	−0.367393
$311$	−5917.12	−1.07887	−0.539436	−	0.842027i	$-0.681362\pi$
$311$	−5917.12	−1.07887	−0.539436	+	0.842027i	$0.681362\pi$
$312$	−2674.84	−0.485362
$313$	−2581.67	−0.466213	−0.233107	−	0.972451i	$-0.574889\pi$
$313$	−2581.67	−0.466213	−0.233107	+	0.972451i	$0.574889\pi$
$314$	−3050.53	−0.548253
$315$	0	0
$316$	240.338	0.0427850
$317$	7203.26	1.27626	0.638132	−	0.769927i	$-0.279708\pi$
$317$	7203.26	1.27626	0.638132	+	0.769927i	$0.279708\pi$
$318$	5597.87	0.987147
$319$	2239.56	0.393076
$320$	−729.800	−0.127491
$321$	−12063.2	−2.09751
$322$	0	0
$323$	9528.16	1.64136
$324$	11535.5	1.97797
$325$	166.479	0.0284142
$326$	−3888.93	−0.660700
$327$	−9026.61	−1.52652
$328$	1402.14	0.236037
$329$	0	0
$330$	2393.80	0.399316
$331$	−11007.6	−1.82789	−0.913943	−	0.405843i	$-0.866978\pi$
$331$	−11007.6	−1.82789	−0.913943	+	0.405843i	$0.866978\pi$
$332$	1491.51	0.246558
$333$	−16155.7	−2.65864
$334$	−5265.16	−0.862565
$335$	−3120.75	−0.508969
$336$	0	0
$337$	1436.15	0.232143	0.116072	−	0.993241i	$-0.462970\pi$
$337$	1436.15	0.232143	0.116072	+	0.993241i	$0.462970\pi$
$338$	2204.23	0.354716
$339$	18614.5	2.98230
$340$	6332.72	1.01012
$341$	−913.337	−0.145044
$342$	10308.6	1.62990
$343$	0	0
$344$	325.700	0.0510482
$345$	2650.17	0.413567
$346$	2379.25	0.369680
$347$	4296.61	0.664710	0.332355	−	0.943154i	$-0.392157\pi$
$347$	4296.61	0.664710	0.332355	+	0.943154i	$0.392157\pi$
$348$	8714.34	1.34235
$349$	−3496.28	−0.536250	−0.268125	−	0.963384i	$-0.586404\pi$
$349$	−3496.28	−0.536250	−0.268125	+	0.963384i	$0.586404\pi$
$350$	0	0
$351$	16084.0	2.44587
$352$	−332.400	−0.0503323
$353$	−2884.90	−0.434980	−0.217490	−	0.976063i	$-0.569787\pi$
$353$	−2884.90	−0.434980	−0.217490	+	0.976063i	$0.569787\pi$
$354$	5932.46	0.890697
$355$	7341.81	1.09764
$356$	2175.79	0.323923
$357$	0	0
$358$	3097.94	0.457350
$359$	−9774.21	−1.43694	−0.718472	−	0.695556i	$-0.755158\pi$
$359$	−9774.21	−1.43694	−0.718472	+	0.695556i	$0.755158\pi$
$360$	6851.42	1.00306
$361$	−2149.18	−0.313337
$362$	−3816.77	−0.554157
$363$	−12359.0	−1.78700
$364$	0	0
$365$	1212.67	0.173901
$366$	−9108.99	−1.30091
$367$	5245.42	0.746073	0.373036	−	0.927817i	$-0.378317\pi$
$367$	5245.42	0.746073	0.373036	+	0.927817i	$0.378317\pi$
$368$	−368.000	−0.0521286
$369$	−13163.4	−1.85707
$370$	−4905.84	−0.689304
$371$	0	0
$372$	−3553.88	−0.495323
$373$	−12320.9	−1.71032	−0.855160	−	0.518364i	$-0.826541\pi$
$373$	−12320.9	−1.71032	−0.855160	+	0.518364i	$0.826541\pi$
$374$	2884.35	0.398787
$375$	13823.4	1.90357
$376$	3241.96	0.444658
$377$	7134.05	0.974595
$378$	0	0
$379$	−1216.68	−0.164899	−0.0824495	−	0.996595i	$-0.526274\pi$
$379$	−1216.68	−0.164899	−0.0824495	+	0.996595i	$0.526274\pi$
$380$	3130.30	0.422582
$381$	6056.35	0.814374
$382$	−705.250	−0.0944600
$383$	3989.13	0.532206	0.266103	−	0.963945i	$-0.414264\pi$
$383$	3989.13	0.532206	0.266103	+	0.963945i	$0.414264\pi$
$384$	−1293.40	−0.171884
$385$	0	0
$386$	2828.56	0.372979
$387$	−3057.70	−0.401632
$388$	−2203.78	−0.288350
$389$	15077.7	1.96522	0.982608	−	0.185693i	$-0.0594530\pi$
$389$	15077.7	1.96522	0.982608	+	0.185693i	$0.0594530\pi$
$390$	7625.37	0.990066
$391$	3193.26	0.413018
$392$	0	0
$393$	5660.00	0.726486
$394$	865.060	0.110612
$395$	−685.150	−0.0872750
$396$	3120.60	0.396000
$397$	12767.4	1.61405	0.807026	−	0.590515i	$-0.201075\pi$
$397$	12767.4	1.61405	0.807026	+	0.590515i	$0.201075\pi$
$398$	−217.037	−0.0273344
$399$	0	0
$400$	80.4999	0.0100625
$401$	−8846.67	−1.10170	−0.550850	−	0.834604i	$-0.685696\pi$
$401$	−8846.67	−1.10170	−0.550850	+	0.834604i	$0.685696\pi$
$402$	−5530.80	−0.686197
$403$	−2909.41	−0.359623
$404$	−4338.30	−0.534254
$405$	−32885.3	−4.03478
$406$	0	0
$407$	−2234.45	−0.272132
$408$	11223.3	1.36185
$409$	3945.27	0.476970	0.238485	−	0.971146i	$-0.423349\pi$
$409$	3945.27	0.476970	0.238485	+	0.971146i	$0.423349\pi$
$410$	−3997.19	−0.481481
$411$	−21338.0	−2.56089
$412$	−5476.76	−0.654905
$413$	0	0
$414$	3454.82	0.410133
$415$	−4251.97	−0.502943
$416$	−1058.85	−0.124794
$417$	9548.44	1.12132
$418$	1425.75	0.166832
$419$	−13937.2	−1.62500	−0.812502	−	0.582958i	$-0.801895\pi$
$419$	−13937.2	−1.62500	−0.812502	+	0.582958i	$0.801895\pi$
$420$	0	0
$421$	−2471.55	−0.286119	−0.143059	−	0.989714i	$-0.545694\pi$
$421$	−2471.55	−0.286119	−0.143059	+	0.989714i	$0.545694\pi$
$422$	2164.74	0.249710
$423$	−30435.8	−3.49844
$424$	2215.95	0.253811
$425$	−698.525	−0.0797257
$426$	13011.6	1.47985
$427$	0	0
$428$	−4775.29	−0.539304
$429$	3473.11	0.390870
$430$	−928.500	−0.104131
$431$	7340.81	0.820404	0.410202	−	0.911995i	$-0.365458\pi$
$431$	7340.81	0.820404	0.410202	+	0.911995i	$0.365458\pi$
$432$	7777.32	0.866173
$433$	8838.13	0.980909	0.490455	−	0.871467i	$-0.336831\pi$
$433$	8838.13	0.980909	0.490455	+	0.871467i	$0.336831\pi$
$434$	0	0
$435$	−24842.7	−2.73820
$436$	−3573.24	−0.392493
$437$	1578.45	0.172786
$438$	2149.17	0.234455
$439$	10403.7	1.13107	0.565534	−	0.824725i	$-0.308670\pi$
$439$	10403.7	1.13107	0.565534	+	0.824725i	$0.308670\pi$
$440$	947.600	0.102671
$441$	0	0
$442$	9188.01	0.988753
$443$	−4415.27	−0.473535	−0.236767	−	0.971566i	$-0.576088\pi$
$443$	−4415.27	−0.473535	−0.236767	+	0.971566i	$0.576088\pi$
$444$	−8694.45	−0.929325
$445$	−6202.69	−0.660755
$446$	−3238.88	−0.343868
$447$	−25064.4	−2.65213
$448$	0	0
$449$	3711.34	0.390087	0.195044	−	0.980795i	$-0.437515\pi$
$449$	3711.34	0.390087	0.195044	+	0.980795i	$0.437515\pi$
$450$	−755.740	−0.0791687
$451$	−1820.59	−0.190085
$452$	7368.65	0.766797
$453$	11767.2	1.22047
$454$	13492.2	1.39476
$455$	0	0
$456$	5547.72	0.569728
$457$	15711.9	1.60825	0.804124	−	0.594461i	$-0.202635\pi$
$457$	15711.9	1.60825	0.804124	+	0.594461i	$0.202635\pi$
$458$	2385.27	0.243354
$459$	−67486.5	−6.86275
$460$	1049.09	0.106335
$461$	−3177.04	−0.320975	−0.160488	−	0.987038i	$-0.551307\pi$
$461$	−3177.04	−0.320975	−0.160488	+	0.987038i	$0.551307\pi$
$462$	0	0
$463$	−14774.8	−1.48303	−0.741516	−	0.670935i	$-0.765893\pi$
$463$	−14774.8	−1.48303	−0.741516	+	0.670935i	$0.765893\pi$
$464$	3449.62	0.345140
$465$	10131.3	1.01039
$466$	5720.33	0.568646
$467$	2408.12	0.238618	0.119309	−	0.992857i	$-0.461932\pi$
$467$	2408.12	0.238618	0.119309	+	0.992857i	$0.461932\pi$
$468$	9940.58	0.981845
$469$	0	0
$470$	−9242.12	−0.907037
$471$	15412.3	1.50778
$472$	2348.40	0.229012
$473$	−422.901	−0.0411100
$474$	−1214.27	−0.117665
$475$	−345.285	−0.0333532
$476$	0	0
$477$	−20803.5	−1.99691
$478$	1188.75	0.113749
$479$	12528.3	1.19506	0.597528	−	0.801848i	$-0.296150\pi$
$479$	12528.3	1.19506	0.597528	+	0.801848i	$0.296150\pi$
$480$	3687.20	0.350618
$481$	−7117.77	−0.674724
$482$	7860.29	0.742794
$483$	0	0
$484$	−4892.40	−0.459466
$485$	6282.48	0.588191
$486$	−32033.1	−2.98982
$487$	16798.6	1.56307	0.781537	−	0.623858i	$-0.214436\pi$
$487$	16798.6	1.56307	0.781537	+	0.623858i	$0.214436\pi$
$488$	−3605.85	−0.334486
$489$	19648.2	1.81702
$490$	0	0
$491$	−18007.7	−1.65514	−0.827571	−	0.561361i	$-0.810278\pi$
$491$	−18007.7	−1.65514	−0.827571	+	0.561361i	$0.810278\pi$
$492$	−7084.08	−0.649136
$493$	−29933.6	−2.73456
$494$	4541.68	0.413643
$495$	−8896.15	−0.807782
$496$	−1406.82	−0.127355
$497$	0	0
$498$	−7535.63	−0.678072
$499$	696.627	0.0624956	0.0312478	−	0.999512i	$-0.490052\pi$
$499$	696.627	0.0624956	0.0312478	+	0.999512i	$0.490052\pi$
$500$	5472.07	0.489437
$501$	26601.4	2.37218
$502$	4244.14	0.377341
$503$	20469.0	1.81444	0.907222	−	0.420652i	$-0.138199\pi$
$503$	20469.0	1.81444	0.907222	+	0.420652i	$0.138199\pi$
$504$	0	0
$505$	12367.5	1.08980
$506$	477.825	0.0419801
$507$	−11136.5	−0.975523
$508$	2397.44	0.209389
$509$	−13437.5	−1.17015	−0.585073	−	0.810980i	$-0.698934\pi$
$509$	−13437.5	−1.17015	−0.585073	+	0.810980i	$0.698934\pi$
$510$	−31995.1	−2.77797
$511$	0	0
$512$	−512.000	−0.0441942
$513$	−33358.9	−2.87102
$514$	2287.36	0.196286
$515$	15613.0	1.33591
$516$	−1645.55	−0.140390
$517$	−4209.49	−0.358091
$518$	0	0
$519$	−12020.8	−1.01667
$520$	3018.55	0.254562
$521$	4917.10	0.413478	0.206739	−	0.978396i	$-0.433715\pi$
$521$	4917.10	0.413478	0.206739	+	0.978396i	$0.433715\pi$
$522$	−32385.4	−2.71546
$523$	−15959.4	−1.33434	−0.667168	−	0.744908i	$-0.732493\pi$
$523$	−15959.4	−1.33434	−0.667168	+	0.744908i	$0.732493\pi$
$524$	2240.54	0.186791
$525$	0	0
$526$	1895.07	0.157089
$527$	12207.5	1.00905
$528$	1679.40	0.138421
$529$	529.000	0.0434783
$530$	−6317.19	−0.517738
$531$	−22047.0	−1.80180
$532$	0	0
$533$	−5799.43	−0.471297
$534$	−10992.8	−0.890835
$535$	13613.3	1.10010
$536$	−2189.40	−0.176432
$537$	−15651.9	−1.25778
$538$	6435.39	0.515705
$539$	0	0
$540$	−22171.4	−1.76687
$541$	−8018.32	−0.637217	−0.318608	−	0.947886i	$-0.603216\pi$
$541$	−8018.32	−0.637217	−0.318608	+	0.947886i	$0.603216\pi$
$542$	−14568.1	−1.15453
$543$	19283.6	1.52401
$544$	4442.80	0.350153
$545$	10186.5	0.800628
$546$	0	0
$547$	13255.0	1.03609	0.518047	−	0.855352i	$-0.326659\pi$
$547$	13255.0	1.03609	0.518047	+	0.855352i	$0.326659\pi$
$548$	−8446.76	−0.658445
$549$	33852.0	2.63164
$550$	−104.524	−0.00810349
$551$	−14796.3	−1.14400
$552$	1859.26	0.143361
$553$	0	0
$554$	−5048.15	−0.387140
$555$	24786.0	1.89569
$556$	3779.81	0.288308
$557$	−10410.3	−0.791918	−0.395959	−	0.918268i	$-0.629588\pi$
$557$	−10410.3	−0.791918	−0.395959	+	0.918268i	$0.629588\pi$
$558$	13207.4	1.00200
$559$	−1347.14	−0.101928
$560$	0	0
$561$	−14572.7	−1.09672
$562$	12554.9	0.942341
$563$	−19009.9	−1.42304	−0.711520	−	0.702666i	$-0.751993\pi$
$563$	−19009.9	−1.42304	−0.711520	+	0.702666i	$0.751993\pi$
$564$	−16379.5	−1.22288
$565$	−21006.4	−1.56415
$566$	−16.6883	−0.00123933
$567$	0	0
$568$	5150.74	0.380493
$569$	11671.7	0.859934	0.429967	−	0.902845i	$-0.358525\pi$
$569$	11671.7	0.859934	0.429967	+	0.902845i	$0.358525\pi$
$570$	−15815.3	−1.16216
$571$	−21359.8	−1.56546	−0.782732	−	0.622359i	$-0.786174\pi$
$571$	−21359.8	−1.56546	−0.782732	+	0.622359i	$0.786174\pi$
$572$	1374.85	0.100499
$573$	3563.16	0.259779
$574$	0	0
$575$	−115.719	−0.00839269
$576$	4806.70	0.347707
$577$	−13416.4	−0.967996	−0.483998	−	0.875069i	$-0.660816\pi$
$577$	−13416.4	−0.967996	−0.483998	+	0.875069i	$0.660816\pi$
$578$	−28725.7	−2.06718
$579$	−14290.9	−1.02575
$580$	−9834.12	−0.704034
$581$	0	0
$582$	11134.2	0.793005
$583$	−2877.27	−0.204399
$584$	850.762	0.0602822
$585$	−28338.4	−2.00282
$586$	6150.29	0.433560
$587$	11542.3	0.811586	0.405793	−	0.913965i	$-0.366995\pi$
$587$	11542.3	0.811586	0.405793	+	0.913965i	$0.366995\pi$
$588$	0	0
$589$	6034.23	0.422133
$590$	−6694.77	−0.467152
$591$	−4370.58	−0.304199
$592$	−3441.75	−0.238944
$593$	−3151.02	−0.218207	−0.109104	−	0.994030i	$-0.534798\pi$
$593$	−3151.02	−0.218207	−0.109104	+	0.994030i	$0.534798\pi$
$594$	−10098.4	−0.697544
$595$	0	0
$596$	−9921.87	−0.681906
$597$	1096.55	0.0751736
$598$	1522.10	0.104086
$599$	−293.676	−0.0200322	−0.0100161	−	0.999950i	$-0.503188\pi$
$599$	−293.676	−0.0200322	−0.0100161	+	0.999950i	$0.503188\pi$
$600$	−406.713	−0.0276733
$601$	19297.6	1.30976	0.654881	−	0.755732i	$-0.272719\pi$
$601$	19297.6	1.30976	0.654881	+	0.755732i	$0.272719\pi$
$602$	0	0
$603$	20554.3	1.38812
$604$	4658.13	0.313802
$605$	13947.2	0.937244
$606$	21918.6	1.46928
$607$	24573.1	1.64315	0.821573	−	0.570103i	$-0.193097\pi$
$607$	24573.1	1.64315	0.821573	+	0.570103i	$0.193097\pi$
$608$	2196.10	0.146486
$609$	0	0
$610$	10279.5	0.682302
$611$	−13409.2	−0.887852
$612$	−41709.4	−2.75491
$613$	14171.5	0.933738	0.466869	−	0.884326i	$-0.345382\pi$
$613$	14171.5	0.933738	0.466869	+	0.884326i	$0.345382\pi$
$614$	−2079.80	−0.136700
$615$	20195.2	1.32414
$616$	0	0
$617$	−3693.48	−0.240995	−0.120498	−	0.992714i	$-0.538449\pi$
$617$	−3693.48	−0.240995	−0.120498	+	0.992714i	$0.538449\pi$
$618$	27670.5	1.80108
$619$	−29215.7	−1.89705	−0.948527	−	0.316697i	$-0.897426\pi$
$619$	−29215.7	−1.89705	−0.948527	+	0.316697i	$0.897426\pi$
$620$	4010.55	0.259786
$621$	−11179.9	−0.722438
$622$	11834.2	0.762878
$623$	0	0
$624$	5349.67	0.343202
$625$	−16228.6	−1.03863
$626$	5163.34	0.329663
$627$	−7203.38	−0.458812
$628$	6101.06	0.387673
$629$	29865.2	1.89317
$630$	0	0
$631$	−10216.8	−0.644571	−0.322286	−	0.946643i	$-0.604451\pi$
$631$	−10216.8	−0.644571	−0.322286	+	0.946643i	$0.604451\pi$
$632$	−480.675	−0.0302535
$633$	−10937.0	−0.686740
$634$	−14406.5	−0.902454
$635$	−6834.59	−0.427122
$636$	−11195.7	−0.698019
$637$	0	0
$638$	−4479.12	−0.277947
$639$	−48355.6	−2.99361
$640$	1459.60	0.0901496
$641$	11330.1	0.698148	0.349074	−	0.937095i	$-0.386496\pi$
$641$	11330.1	0.698148	0.349074	+	0.937095i	$0.386496\pi$
$642$	24126.4	1.48317
$643$	18572.6	1.13908	0.569542	−	0.821962i	$-0.307121\pi$
$643$	18572.6	1.13908	0.569542	+	0.821962i	$0.307121\pi$
$644$	0	0
$645$	4691.10	0.286375
$646$	−19056.3	−1.16062
$647$	13995.8	0.850436	0.425218	−	0.905091i	$-0.360198\pi$
$647$	13995.8	0.850436	0.425218	+	0.905091i	$0.360198\pi$
$648$	−23071.1	−1.39864
$649$	−3049.25	−0.184428
$650$	−332.958	−0.0200918
$651$	0	0
$652$	7777.87	0.467185
$653$	20473.1	1.22691	0.613456	−	0.789729i	$-0.289779\pi$
$653$	20473.1	1.22691	0.613456	+	0.789729i	$0.289779\pi$
$654$	18053.2	1.07941
$655$	−6387.30	−0.381027
$656$	−2804.28	−0.166903
$657$	−7987.03	−0.474283
$658$	0	0
$659$	−27951.8	−1.65227	−0.826135	−	0.563473i	$-0.809465\pi$
$659$	−27951.8	−1.65227	−0.826135	+	0.563473i	$0.809465\pi$
$660$	−4787.60	−0.282359
$661$	−20698.3	−1.21796	−0.608980	−	0.793185i	$-0.708421\pi$
$661$	−20698.3	−1.21796	−0.608980	+	0.793185i	$0.708421\pi$
$662$	22015.1	1.29251
$663$	−46421.0	−2.71922
$664$	−2983.02	−0.174343
$665$	0	0
$666$	32311.4	1.87994
$667$	−4958.84	−0.287866
$668$	10530.3	0.609926
$669$	16363.9	0.945688
$670$	6241.50	0.359896
$671$	4681.97	0.269367
$672$	0	0
$673$	28428.7	1.62830	0.814150	−	0.580654i	$-0.197203\pi$
$673$	28428.7	1.62830	0.814150	+	0.580654i	$0.197203\pi$
$674$	−2872.31	−0.164150
$675$	2445.60	0.139454
$676$	−4408.46	−0.250822
$677$	7179.44	0.407575	0.203787	−	0.979015i	$-0.434675\pi$
$677$	7179.44	0.407575	0.203787	+	0.979015i	$0.434675\pi$
$678$	−37228.9	−2.10880
$679$	0	0
$680$	−12665.4	−0.714261
$681$	−68167.3	−3.83579
$682$	1826.67	0.102562
$683$	5146.87	0.288345	0.144172	−	0.989553i	$-0.453948\pi$
$683$	5146.87	0.288345	0.144172	+	0.989553i	$0.453948\pi$
$684$	−20617.2	−1.15251
$685$	24079.9	1.34313
$686$	0	0
$687$	−12051.2	−0.669260
$688$	−651.400	−0.0360965
$689$	−9165.46	−0.506787
$690$	−5300.35	−0.292436
$691$	−29679.6	−1.63396	−0.816978	−	0.576669i	$-0.804352\pi$
$691$	−29679.6	−1.63396	−0.816978	+	0.576669i	$0.804352\pi$
$692$	−4758.50	−0.261403
$693$	0	0
$694$	−8593.23	−0.470021
$695$	−10775.4	−0.588107
$696$	−17428.7	−0.949185
$697$	24333.7	1.32239
$698$	6992.55	0.379186
$699$	−28901.1	−1.56386
$700$	0	0
$701$	9627.78	0.518739	0.259370	−	0.965778i	$-0.416485\pi$
$701$	9627.78	0.518739	0.259370	+	0.965778i	$0.416485\pi$
$702$	−32168.0	−1.72949
$703$	14762.5	0.792006
$704$	664.800	0.0355903
$705$	46694.4	2.49449
$706$	5769.80	0.307577
$707$	0	0
$708$	−11864.9	−0.629818
$709$	11527.4	0.610606	0.305303	−	0.952255i	$-0.401242\pi$
$709$	11527.4	0.610606	0.305303	+	0.952255i	$0.401242\pi$
$710$	−14683.6	−0.776150
$711$	4512.62	0.238026
$712$	−4351.57	−0.229048
$713$	2022.31	0.106222
$714$	0	0
$715$	−3919.40	−0.205003
$716$	−6195.88	−0.323395
$717$	−6005.96	−0.312826
$718$	19548.4	1.01607
$719$	7837.87	0.406542	0.203271	−	0.979123i	$-0.434843\pi$
$719$	7837.87	0.406542	0.203271	+	0.979123i	$0.434843\pi$
$720$	−13702.8	−0.709271
$721$	0	0
$722$	4298.36	0.221563
$723$	−39712.9	−2.04279
$724$	7633.54	0.391848
$725$	1084.74	0.0555674
$726$	24718.1	1.26360
$727$	−726.845	−0.0370800	−0.0185400	−	0.999828i	$-0.505902\pi$
$727$	−726.845	−0.0370800	−0.0185400	+	0.999828i	$0.505902\pi$
$728$	0	0
$729$	83977.2	4.26648
$730$	−2425.34	−0.122967
$731$	5652.42	0.285995
$732$	18218.0	0.919886
$733$	378.674	0.0190814	0.00954069	−	0.999954i	$-0.496963\pi$
$733$	378.674	0.0190814	0.00954069	+	0.999954i	$0.496963\pi$
$734$	−10490.8	−0.527553
$735$	0	0
$736$	736.000	0.0368605
$737$	2842.80	0.142084
$738$	26326.8	1.31315
$739$	18952.9	0.943428	0.471714	−	0.881752i	$-0.343636\pi$
$739$	18952.9	0.943428	0.471714	+	0.881752i	$0.343636\pi$
$740$	9811.67	0.487411
$741$	−22946.1	−1.13758
$742$	0	0
$743$	−19317.8	−0.953838	−0.476919	−	0.878947i	$-0.658246\pi$
$743$	−19317.8	−0.953838	−0.476919	+	0.878947i	$0.658246\pi$
$744$	7107.76	0.350246
$745$	28285.1	1.39099
$746$	24641.7	1.20938
$747$	28004.9	1.37168
$748$	−5768.70	−0.281985
$749$	0	0
$750$	−27646.8	−1.34602
$751$	−25592.6	−1.24353	−0.621763	−	0.783206i	$-0.713583\pi$
$751$	−25592.6	−1.24353	−0.621763	+	0.783206i	$0.713583\pi$
$752$	−6483.92	−0.314421
$753$	−21442.9	−1.03774
$754$	−14268.1	−0.689143
$755$	−13279.3	−0.640111
$756$	0	0
$757$	−36790.6	−1.76642	−0.883208	−	0.468981i	$-0.844621\pi$
$757$	−36790.6	−1.76642	−0.883208	+	0.468981i	$0.844621\pi$
$758$	2433.36	0.116601
$759$	−2414.14	−0.115451
$760$	−6260.60	−0.298810
$761$	−2257.71	−0.107545	−0.0537726	−	0.998553i	$-0.517125\pi$
$761$	−2257.71	−0.107545	−0.0537726	+	0.998553i	$0.517125\pi$
$762$	−12112.7	−0.575849
$763$	0	0
$764$	1410.50	0.0667933
$765$	118904.	5.61960
$766$	−7978.26	−0.376327
$767$	−9713.30	−0.457271
$768$	2586.80	0.121540
$769$	−6903.74	−0.323739	−0.161869	−	0.986812i	$-0.551752\pi$
$769$	−6903.74	−0.323739	−0.161869	+	0.986812i	$0.551752\pi$
$770$	0	0
$771$	−11556.5	−0.539816
$772$	−5657.12	−0.263736
$773$	30096.1	1.40036	0.700182	−	0.713965i	$-0.253102\pi$
$773$	30096.1	1.40036	0.700182	+	0.713965i	$0.253102\pi$
$774$	6115.40	0.283997
$775$	−442.380	−0.0205042
$776$	4407.55	0.203894
$777$	0	0
$778$	−30155.4	−1.38962
$779$	12028.3	0.553218
$780$	−15250.7	−0.700083
$781$	−6687.91	−0.306418
$782$	−6386.52	−0.292048
$783$	104800.	4.78321
$784$	0	0
$785$	−17392.8	−0.790797
$786$	−11320.0	−0.513703
$787$	6753.37	0.305885	0.152943	−	0.988235i	$-0.451125\pi$
$787$	6753.37	0.305885	0.152943	+	0.988235i	$0.451125\pi$
$788$	−1730.12	−0.0782144
$789$	−9574.54	−0.432018
$790$	1370.30	0.0617128
$791$	0	0
$792$	−6241.20	−0.280014
$793$	14914.3	0.667871
$794$	−25534.9	−1.14131
$795$	31916.6	1.42386
$796$	434.074	0.0193283
$797$	−26666.0	−1.18514	−0.592570	−	0.805519i	$-0.701887\pi$
$797$	−26666.0	−1.18514	−0.592570	+	0.805519i	$0.701887\pi$
$798$	0	0
$799$	56263.2	2.49118
$800$	−161.000	−0.00711525
$801$	40853.0	1.80208
$802$	17693.3	0.779019
$803$	−1104.66	−0.0485463
$804$	11061.6	0.485215
$805$	0	0
$806$	5818.82	0.254292
$807$	−32513.8	−1.41827
$808$	8676.60	0.377774
$809$	−16430.2	−0.714035	−0.357018	−	0.934098i	$-0.616206\pi$
$809$	−16430.2	−0.714035	−0.357018	+	0.934098i	$0.616206\pi$
$810$	65770.7	2.85302
$811$	23394.6	1.01294	0.506470	−	0.862257i	$-0.330950\pi$
$811$	23394.6	1.01294	0.506470	+	0.862257i	$0.330950\pi$
$812$	0	0
$813$	73603.0	3.17512
$814$	4468.90	0.192426
$815$	−22173.0	−0.952989
$816$	−22446.5	−0.962974
$817$	2794.02	0.119646
$818$	−7890.54	−0.337269
$819$	0	0
$820$	7994.37	0.340458
$821$	30615.7	1.30145	0.650727	−	0.759312i	$-0.274464\pi$
$821$	30615.7	1.30145	0.650727	+	0.759312i	$0.274464\pi$
$822$	42675.9	1.81082
$823$	−959.744	−0.0406496	−0.0203248	−	0.999793i	$-0.506470\pi$
$823$	−959.744	−0.0406496	−0.0203248	+	0.999793i	$0.506470\pi$
$824$	10953.5	0.463087
$825$	528.092	0.0222858
$826$	0	0
$827$	37507.5	1.57710	0.788551	−	0.614970i	$-0.210832\pi$
$827$	37507.5	1.57710	0.788551	+	0.614970i	$0.210832\pi$
$828$	−6909.63	−0.290008
$829$	27029.9	1.13243	0.566217	−	0.824256i	$-0.308406\pi$
$829$	27029.9	1.13243	0.566217	+	0.824256i	$0.308406\pi$
$830$	8503.95	0.355634
$831$	25505.0	1.06469
$832$	2117.70	0.0882428
$833$	0	0
$834$	−19096.9	−0.792891
$835$	−30019.6	−1.24416
$836$	−2851.50	−0.117968
$837$	−42739.6	−1.76499
$838$	27874.4	1.14905
$839$	−38982.8	−1.60409	−0.802047	−	0.597261i	$-0.796256\pi$
$839$	−38982.8	−1.60409	−0.802047	+	0.597261i	$0.796256\pi$
$840$	0	0
$841$	22095.0	0.905942
$842$	4943.10	0.202316
$843$	−63431.6	−2.59158
$844$	−4329.48	−0.176572
$845$	12567.5	0.511641
$846$	60871.6	2.47377
$847$	0	0
$848$	−4431.90	−0.179472
$849$	84.3151	0.00340835
$850$	1397.05	0.0563746
$851$	4947.52	0.199293
$852$	−26023.3	−1.04641
$853$	16634.7	0.667715	0.333858	−	0.942624i	$-0.391650\pi$
$853$	16634.7	0.667715	0.333858	+	0.942624i	$0.391650\pi$
$854$	0	0
$855$	58775.0	2.35095
$856$	9550.57	0.381346
$857$	15234.4	0.607230	0.303615	−	0.952795i	$-0.401806\pi$
$857$	15234.4	0.607230	0.303615	+	0.952795i	$0.401806\pi$
$858$	−6946.22	−0.276387
$859$	−31925.2	−1.26807	−0.634036	−	0.773303i	$-0.718603\pi$
$859$	−31925.2	−1.26807	−0.634036	+	0.773303i	$0.718603\pi$
$860$	1857.00	0.0736316
$861$	0	0
$862$	−14681.6	−0.580113
$863$	−19518.9	−0.769910	−0.384955	−	0.922935i	$-0.625783\pi$
$863$	−19518.9	−0.769910	−0.384955	+	0.922935i	$0.625783\pi$
$864$	−15554.6	−0.612477
$865$	13565.4	0.533224
$866$	−17676.3	−0.693607
$867$	145132.	5.68506
$868$	0	0
$869$	624.127	0.0243637
$870$	49685.4	1.93620
$871$	9055.65	0.352284
$872$	7146.48	0.277535
$873$	−41378.5	−1.60418
$874$	−3156.89	−0.122178
$875$	0	0
$876$	−4298.34	−0.165785
$877$	16853.3	0.648913	0.324456	−	0.945901i	$-0.394819\pi$
$877$	16853.3	0.648913	0.324456	+	0.945901i	$0.394819\pi$
$878$	−20807.3	−0.799786
$879$	−31073.4	−1.19235
$880$	−1895.20	−0.0725991
$881$	−10834.3	−0.414320	−0.207160	−	0.978307i	$-0.566422\pi$
$881$	−10834.3	−0.414320	−0.207160	+	0.978307i	$0.566422\pi$
$882$	0	0
$883$	−23506.2	−0.895863	−0.447931	−	0.894068i	$-0.647839\pi$
$883$	−23506.2	−0.895863	−0.447931	+	0.894068i	$0.647839\pi$
$884$	−18376.0	−0.699154
$885$	33824.3	1.28474
$886$	8830.54	0.334840
$887$	−34287.2	−1.29792	−0.648958	−	0.760824i	$-0.724795\pi$
$887$	−34287.2	−1.29792	−0.648958	+	0.760824i	$0.724795\pi$
$888$	17388.9	0.657132
$889$	0	0
$890$	12405.4	0.467224
$891$	29956.4	1.12635
$892$	6477.75	0.243151
$893$	27811.2	1.04218
$894$	50128.7	1.87534
$895$	17663.1	0.659679
$896$	0	0
$897$	−7690.16	−0.286251
$898$	−7422.69	−0.275833
$899$	−18957.1	−0.703287
$900$	1511.48	0.0559807
$901$	38457.1	1.42197
$902$	3641.18	0.134410
$903$	0	0
$904$	−14737.3	−0.542207
$905$	−21761.5	−0.799313
$906$	−23534.5	−0.863003
$907$	47930.9	1.75471	0.877354	−	0.479844i	$-0.159307\pi$
$907$	47930.9	1.75471	0.877354	+	0.479844i	$0.159307\pi$
$908$	−26984.4	−0.986244
$909$	−81456.7	−2.97222
$910$	0	0
$911$	9787.98	0.355972	0.177986	−	0.984033i	$-0.443042\pi$
$911$	9787.98	0.355972	0.177986	+	0.984033i	$0.443042\pi$
$912$	−11095.4	−0.402859
$913$	3873.27	0.140402
$914$	−31423.7	−1.13720
$915$	−51935.5	−1.87643
$916$	−4770.54	−0.172077
$917$	0	0
$918$	134973.	4.85269
$919$	25226.6	0.905495	0.452748	−	0.891639i	$-0.350444\pi$
$919$	25226.6	0.905495	0.452748	+	0.891639i	$0.350444\pi$
$920$	−2098.17	−0.0751900
$921$	10507.9	0.375946
$922$	6354.08	0.226964
$923$	−21304.1	−0.759734
$924$	0	0
$925$	−1082.27	−0.0384700
$926$	29549.6	1.04866
$927$	−102833.	−3.64344
$928$	−6899.25	−0.244051
$929$	−38214.5	−1.34960	−0.674800	−	0.738001i	$-0.735770\pi$
$929$	−38214.5	−1.34960	−0.674800	+	0.738001i	$0.735770\pi$
$930$	−20262.7	−0.714451
$931$	0	0
$932$	−11440.7	−0.402093
$933$	−59790.6	−2.09803
$934$	−4816.25	−0.168728
$935$	16445.3	0.575207
$936$	−19881.2	−0.694269
$937$	19742.9	0.688338	0.344169	−	0.938908i	$-0.388161\pi$
$937$	19742.9	0.688338	0.344169	+	0.938908i	$0.388161\pi$
$938$	0	0
$939$	−26087.0	−0.906621
$940$	18484.2	0.641372
$941$	39767.5	1.37767	0.688833	−	0.724920i	$-0.258123\pi$
$941$	39767.5	1.37767	0.688833	+	0.724920i	$0.258123\pi$
$942$	−30824.7	−1.06616
$943$	4031.15	0.139207
$944$	−4696.80	−0.161936
$945$	0	0
$946$	845.803	0.0290692
$947$	32884.3	1.12840	0.564200	−	0.825638i	$-0.309185\pi$
$947$	32884.3	1.12840	0.564200	+	0.825638i	$0.309185\pi$
$948$	2428.54	0.0832017
$949$	−3518.87	−0.120366
$950$	690.569	0.0235842
$951$	72786.7	2.48188
$952$	0	0
$953$	4674.79	0.158899	0.0794497	−	0.996839i	$-0.474684\pi$
$953$	4674.79	0.158899	0.0794497	+	0.996839i	$0.474684\pi$
$954$	41607.1	1.41203
$955$	−4021.02	−0.136248
$956$	−2377.49	−0.0804327
$957$	22630.1	0.764395
$958$	−25056.6	−0.845032
$959$	0	0
$960$	−7374.40	−0.247925
$961$	−22059.9	−0.740489
$962$	14235.5	0.477102
$963$	−89661.6	−3.00032
$964$	−15720.6	−0.525234
$965$	16127.2	0.537982
$966$	0	0
$967$	−27165.3	−0.903390	−0.451695	−	0.892173i	$-0.649180\pi$
$967$	−27165.3	−0.903390	−0.451695	+	0.892173i	$0.649180\pi$
$968$	9784.80	0.324892
$969$	96279.0	3.19188
$970$	−12565.0	−0.415914
$971$	−14597.4	−0.482443	−0.241221	−	0.970470i	$-0.577548\pi$
$971$	−14597.4	−0.482443	−0.241221	+	0.970470i	$0.577548\pi$
$972$	64066.2	2.11412
$973$	0	0
$974$	−33597.2	−1.10526
$975$	1682.22	0.0552555
$976$	7211.70	0.236517
$977$	33327.5	1.09134	0.545670	−	0.838000i	$-0.316275\pi$
$977$	33327.5	1.09134	0.545670	+	0.838000i	$0.316275\pi$
$978$	−39296.5	−1.28483
$979$	5650.25	0.184456
$980$	0	0
$981$	−67091.7	−2.18356
$982$	36015.4	1.17036
$983$	21834.8	0.708465	0.354232	−	0.935157i	$-0.384742\pi$
$983$	21834.8	0.708465	0.354232	+	0.935157i	$0.384742\pi$
$984$	14168.2	0.459009
$985$	4932.19	0.159546
$986$	59867.2	1.93363
$987$	0	0
$988$	−9083.36	−0.292490
$989$	936.388	0.0301066
$990$	17792.3	0.571188
$991$	45191.9	1.44861	0.724303	−	0.689482i	$-0.242162\pi$
$991$	45191.9	1.44861	0.724303	+	0.689482i	$0.242162\pi$
$992$	2813.65	0.0900539
$993$	−111228.	−3.55459
$994$	0	0
$995$	−1237.45	−0.0394269
$996$	15071.3	0.479469
$997$	−18760.0	−0.595924	−0.297962	−	0.954578i	$-0.596307\pi$
$997$	−18760.0	−0.595924	−0.297962	+	0.954578i	$0.596307\pi$
$998$	−1393.25	−0.0441911
$999$	−104561.	−3.31147

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.d.1.2		2
7.6	odd	2		46.4.a.c.1.1	✓	2
21.20	even	2		414.4.a.j.1.1		2
28.27	even	2		368.4.a.g.1.2		2
35.13	even	4		1150.4.b.i.599.3		4
35.27	even	4		1150.4.b.i.599.2		4
35.34	odd	2		1150.4.a.k.1.2		2
56.13	odd	2		1472.4.a.m.1.2		2
56.27	even	2		1472.4.a.l.1.1		2
161.160	even	2		1058.4.a.f.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
46.4.a.c.1.1	✓	2	7.6	odd	2
368.4.a.g.1.2		2	28.27	even	2
414.4.a.j.1.1		2	21.20	even	2
1058.4.a.f.1.1		2	161.160	even	2
1150.4.a.k.1.2		2	35.34	odd	2
1150.4.b.i.599.2		4	35.27	even	4
1150.4.b.i.599.3		4	35.13	even	4
1472.4.a.l.1.1		2	56.27	even	2
1472.4.a.m.1.2		2	56.13	odd	2
2254.4.a.d.1.2		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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +10.1047 q^{3} +4.00000 q^{4} -11.4031 q^{5} -20.2094 q^{6} -8.00000 q^{8} +75.1047 q^{9} +22.8062 q^{10} +10.3875 q^{11} +40.4187 q^{12} +33.0891 q^{13} -115.225 q^{15} +16.0000 q^{16} -138.837 q^{17} -150.209 q^{18} -68.6281 q^{19} -45.6125 q^{20} -20.7750 q^{22} -23.0000 q^{23} -80.8375 q^{24} +5.03124 q^{25} -66.1781 q^{26} +486.083 q^{27} +215.602 q^{29} +230.450 q^{30} -87.9266 q^{31} -32.0000 q^{32} +104.962 q^{33} +277.675 q^{34} +300.419 q^{36} -215.109 q^{37} +137.256 q^{38} +334.355 q^{39} +91.2250 q^{40} -175.267 q^{41} -40.7125 q^{43} +41.5500 q^{44} -856.428 q^{45} +46.0000 q^{46} -405.245 q^{47} +161.675 q^{48} -10.0625 q^{50} -1402.91 q^{51} +132.356 q^{52} -276.994 q^{53} -972.166 q^{54} -118.450 q^{55} -693.466 q^{57} -431.203 q^{58} -293.550 q^{59} -460.900 q^{60} +450.731 q^{61} +175.853 q^{62} +64.0000 q^{64} -377.319 q^{65} -209.925 q^{66} +273.675 q^{67} -555.350 q^{68} -232.408 q^{69} -643.842 q^{71} -600.837 q^{72} -106.345 q^{73} +430.219 q^{74} +50.8391 q^{75} -274.512 q^{76} -668.709 q^{78} +60.0844 q^{79} -182.450 q^{80} +2883.89 q^{81} +350.534 q^{82} +372.878 q^{83} +1583.18 q^{85} +81.4250 q^{86} +2178.59 q^{87} -83.1000 q^{88} +543.947 q^{89} +1712.86 q^{90} -92.0000 q^{92} -888.470 q^{93} +810.491 q^{94} +782.575 q^{95} -323.350 q^{96} -550.944 q^{97} +780.150 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + q^{3} + 8 q^{4} - 10 q^{5} - 2 q^{6} - 16 q^{8} + 131 q^{9} + 20 q^{10} + 72 q^{11} + 4 q^{12} + 111 q^{13} - 128 q^{15} + 32 q^{16} - 124 q^{17} - 262 q^{18} - 22 q^{19} - 40 q^{20} - 144 q^{22}+ \cdots + 4224 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 + q^3 + 8 * q^4 - 10 * q^5 - 2 * q^6 - 16 * q^8 + 131 * q^9 + 20 * q^10 + 72 * q^11 + 4 * q^12 + 111 * q^13 - 128 * q^15 + 32 * q^16 - 124 * q^17 - 262 * q^18 - 22 * q^19 - 40 * q^20 - 144 * q^22 - 46 * q^23 - 8 * q^24 - 118 * q^25 - 222 * q^26 + 223 * q^27 + 15 * q^29 + 256 * q^30 - 67 * q^31 - 64 * q^32 - 456 * q^33 + 248 * q^34 + 524 * q^36 + 18 * q^37 + 44 * q^38 - 375 * q^39 + 80 * q^40 - 485 * q^41 - 440 * q^43 + 288 * q^44 - 778 * q^45 + 92 * q^46 - 215 * q^47 + 16 * q^48 + 236 * q^50 - 1538 * q^51 + 444 * q^52 + 240 * q^53 - 446 * q^54 - 32 * q^55 - 1118 * q^57 - 30 * q^58 - 792 * q^59 - 512 * q^60 - 456 * q^61 + 134 * q^62 + 128 * q^64 - 268 * q^65 + 912 * q^66 + 240 * q^67 - 496 * q^68 - 23 * q^69 - 705 * q^71 - 1048 * q^72 - 27 * q^73 - 36 * q^74 + 1171 * q^75 - 88 * q^76 + 750 * q^78 + 594 * q^79 - 160 * q^80 + 3770 * q^81 + 970 * q^82 - 394 * q^83 + 1604 * q^85 + 880 * q^86 + 4005 * q^87 - 576 * q^88 + 486 * q^89 + 1556 * q^90 - 184 * q^92 - 1079 * q^93 + 430 * q^94 + 848 * q^95 - 32 * q^96 - 2152 * q^97 + 4224 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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