LMFDB - Embedded newform 2254.4.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("2254.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$132.990305153$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{73}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 18 $ x^2 - x - 18
Coefficient ring:	$\Z[a_1, a_2, a_3]$
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			1.1
Root			$4.77200$ 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)

$f(q)$	$=$	\(q+2.00000 q^{2} -5.77200 q^{3} +4.00000 q^{4} +3.54400 q^{5} -11.5440 q^{6} +8.00000 q^{8} +6.31601 q^{9} +7.08801 q^{10} -6.00000 q^{11} -23.0880 q^{12} -21.4920 q^{13} -20.4560 q^{15} +16.0000 q^{16} +7.36799 q^{17} +12.6320 q^{18} +93.4400 q^{19} +14.1760 q^{20} -12.0000 q^{22} +23.0000 q^{23} -46.1760 q^{24} -112.440 q^{25} -42.9840 q^{26} +119.388 q^{27} +112.476 q^{29} -40.9120 q^{30} -286.268 q^{31} +32.0000 q^{32} +34.6320 q^{33} +14.7360 q^{34} +25.2640 q^{36} -59.3361 q^{37} +186.880 q^{38} +124.052 q^{39} +28.3520 q^{40} -62.6119 q^{41} +507.304 q^{43} -24.0000 q^{44} +22.3839 q^{45} +46.0000 q^{46} -536.300 q^{47} -92.3520 q^{48} -224.880 q^{50} -42.5280 q^{51} -85.9681 q^{52} +187.336 q^{53} +238.776 q^{54} -21.2640 q^{55} -539.336 q^{57} +224.952 q^{58} -49.6799 q^{59} -81.8240 q^{60} +778.776 q^{61} -572.536 q^{62} +64.0000 q^{64} -76.1678 q^{65} +69.2640 q^{66} -661.232 q^{67} +29.4720 q^{68} -132.756 q^{69} +289.212 q^{71} +50.5280 q^{72} +651.316 q^{73} -118.672 q^{74} +649.004 q^{75} +373.760 q^{76} +248.104 q^{78} -50.0720 q^{79} +56.7041 q^{80} -859.640 q^{81} -125.224 q^{82} -807.016 q^{83} +26.1122 q^{85} +1014.61 q^{86} -649.212 q^{87} -48.0000 q^{88} -946.104 q^{89} +44.7679 q^{90} +92.0000 q^{92} +1652.34 q^{93} -1072.60 q^{94} +331.152 q^{95} -184.704 q^{96} -715.529 q^{97} -37.8960 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} - 3 q^{3} + 8 q^{4} - 10 q^{5} - 6 q^{6} + 16 q^{8} - 13 q^{9} - 20 q^{10} - 12 q^{11} - 12 q^{12} + 51 q^{13} - 58 q^{15} + 32 q^{16} + 66 q^{17} - 26 q^{18} + 16 q^{19} - 40 q^{20} - 24 q^{22}+ \cdots + 78 q^{99}+O(q^{100}) $ 2 * q + 4 * q^2 - 3 * q^3 + 8 * q^4 - 10 * q^5 - 6 * q^6 + 16 * q^8 - 13 * q^9 - 20 * q^10 - 12 * q^11 - 12 * q^12 + 51 * q^13 - 58 * q^15 + 32 * q^16 + 66 * q^17 - 26 * q^18 + 16 * q^19 - 40 * q^20 - 24 * q^22 + 46 * q^23 - 24 * q^24 - 54 * q^25 + 102 * q^26 - 9 * q^27 - 57 * q^29 - 116 * q^30 + 17 * q^31 + 64 * q^32 + 18 * q^33 + 132 * q^34 - 52 * q^36 + 206 * q^37 + 32 * q^38 + 325 * q^39 - 80 * q^40 - 373 * q^41 + 314 * q^43 - 48 * q^44 + 284 * q^45 + 92 * q^46 - 859 * q^47 - 48 * q^48 - 108 * q^50 + 120 * q^51 + 204 * q^52 + 50 * q^53 - 18 * q^54 + 60 * q^55 - 754 * q^57 - 114 * q^58 - 612 * q^59 - 232 * q^60 + 1062 * q^61 + 34 * q^62 + 128 * q^64 - 1058 * q^65 + 36 * q^66 - 844 * q^67 + 264 * q^68 - 69 * q^69 + 399 * q^71 - 104 * q^72 + 1277 * q^73 + 412 * q^74 + 811 * q^75 + 64 * q^76 + 650 * q^78 + 122 * q^79 - 160 * q^80 - 694 * q^81 - 746 * q^82 - 1802 * q^83 - 768 * q^85 + 628 * q^86 - 1119 * q^87 - 96 * q^88 - 2046 * q^89 + 568 * q^90 + 184 * q^92 + 2493 * q^93 - 1718 * q^94 + 1380 * q^95 - 96 * q^96 + 910 * q^97 + 78 * 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
$3$	−5.77200	−1.11082	−0.555411	−	0.831576i	$-0.687439\pi$
$3$	−5.77200	−1.11082	−0.555411	+	0.831576i	$0.687439\pi$
$4$	4.00000	0.500000
$5$	3.54400	0.316985	0.158493	−	0.987360i	$-0.449337\pi$
$5$	3.54400	0.316985	0.158493	+	0.987360i	$0.449337\pi$
$6$	−11.5440	−0.785470
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	6.31601	0.233926
$10$	7.08801	0.224142
$11$	−6.00000	−0.164461	−0.0822304	−	0.996613i	$-0.526204\pi$
$11$	−6.00000	−0.164461	−0.0822304	+	0.996613i	$0.526204\pi$
$12$	−23.0880	−0.555411
$13$	−21.4920	−0.458524	−0.229262	−	0.973365i	$-0.573631\pi$
$13$	−21.4920	−0.458524	−0.229262	+	0.973365i	$0.573631\pi$
$14$	0	0
$15$	−20.4560	−0.352114
$16$	16.0000	0.250000
$17$	7.36799	0.105118	0.0525588	−	0.998618i	$-0.483262\pi$
$17$	7.36799	0.105118	0.0525588	+	0.998618i	$0.483262\pi$
$18$	12.6320	0.165411
$19$	93.4400	1.12824	0.564121	−	0.825692i	$-0.309215\pi$
$19$	93.4400	1.12824	0.564121	+	0.825692i	$0.309215\pi$
$20$	14.1760	0.158493
$21$	0	0
$22$	−12.0000	−0.116291
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	−46.1760	−0.392735
$25$	−112.440	−0.899520
$26$	−42.9840	−0.324226
$27$	119.388	0.850972
$28$	0	0
$29$	112.476	0.720217	0.360108	−	0.932911i	$-0.382740\pi$
$29$	112.476	0.720217	0.360108	+	0.932911i	$0.382740\pi$
$30$	−40.9120	−0.248982
$31$	−286.268	−1.65856	−0.829279	−	0.558835i	$-0.811248\pi$
$31$	−286.268	−1.65856	−0.829279	+	0.558835i	$0.811248\pi$
$32$	32.0000	0.176777
$33$	34.6320	0.182687
$34$	14.7360	0.0743294
$35$	0	0
$36$	25.2640	0.116963
$37$	−59.3361	−0.263643	−0.131821	−	0.991273i	$-0.542083\pi$
$37$	−59.3361	−0.263643	−0.131821	+	0.991273i	$0.542083\pi$
$38$	186.880	0.797788
$39$	124.052	0.509339
$40$	28.3520	0.112071
$41$	−62.6119	−0.238496	−0.119248	−	0.992864i	$-0.538048\pi$
$41$	−62.6119	−0.238496	−0.119248	+	0.992864i	$0.538048\pi$
$42$	0	0
$43$	507.304	1.79914	0.899572	−	0.436773i	$-0.143879\pi$
$43$	507.304	1.79914	0.899572	+	0.436773i	$0.143879\pi$
$44$	−24.0000	−0.0822304
$45$	22.3839	0.0741512
$46$	46.0000	0.147442
$47$	−536.300	−1.66441	−0.832206	−	0.554466i	$-0.812923\pi$
$47$	−536.300	−1.66441	−0.832206	+	0.554466i	$0.812923\pi$
$48$	−92.3520	−0.277706
$49$	0	0
$50$	−224.880	−0.636057
$51$	−42.5280	−0.116767
$52$	−85.9681	−0.229262
$53$	187.336	0.485521	0.242760	−	0.970086i	$-0.421947\pi$
$53$	187.336	0.485521	0.242760	+	0.970086i	$0.421947\pi$
$54$	238.776	0.601728
$55$	−21.2640	−0.0521316
$56$	0	0
$57$	−539.336	−1.25328
$58$	224.952	0.509270
$59$	−49.6799	−0.109623	−0.0548116	−	0.998497i	$-0.517456\pi$
$59$	−49.6799	−0.109623	−0.0548116	+	0.998497i	$0.517456\pi$
$60$	−81.8240	−0.176057
$61$	778.776	1.63462	0.817312	−	0.576195i	$-0.195463\pi$
$61$	778.776	1.63462	0.817312	+	0.576195i	$0.195463\pi$
$62$	−572.536	−1.17278
$63$	0	0
$64$	64.0000	0.125000
$65$	−76.1678	−0.145345
$66$	69.2640	0.129179
$67$	−661.232	−1.20571	−0.602853	−	0.797852i	$-0.705970\pi$
$67$	−661.232	−1.20571	−0.602853	+	0.797852i	$0.705970\pi$
$68$	29.4720	0.0525588
$69$	−132.756	−0.231622
$70$	0	0
$71$	289.212	0.483425	0.241712	−	0.970348i	$-0.422291\pi$
$71$	289.212	0.483425	0.241712	+	0.970348i	$0.422291\pi$
$72$	50.5280	0.0827054
$73$	651.316	1.04426	0.522129	−	0.852867i	$-0.325138\pi$
$73$	651.316	1.04426	0.522129	+	0.852867i	$0.325138\pi$
$74$	−118.672	−0.186424
$75$	649.004	0.999207
$76$	373.760	0.564121
$77$	0	0
$78$	248.104	0.360157
$79$	−50.0720	−0.0713107	−0.0356554	−	0.999364i	$-0.511352\pi$
$79$	−50.0720	−0.0713107	−0.0356554	+	0.999364i	$0.511352\pi$
$80$	56.7041	0.0792463
$81$	−859.640	−1.17920
$82$	−125.224	−0.168642
$83$	−807.016	−1.06725	−0.533624	−	0.845722i	$-0.679170\pi$
$83$	−807.016	−1.06725	−0.533624	+	0.845722i	$0.679170\pi$
$84$	0	0
$85$	26.1122	0.0333207
$86$	1014.61	1.27219
$87$	−649.212	−0.800033
$88$	−48.0000	−0.0581456
$89$	−946.104	−1.12682	−0.563409	−	0.826178i	$-0.690510\pi$
$89$	−946.104	−1.12682	−0.563409	+	0.826178i	$0.690510\pi$
$90$	44.7679	0.0524328
$91$	0	0
$92$	92.0000	0.104257
$93$	1652.34	1.84236
$94$	−1072.60	−1.17692
$95$	331.152	0.357636
$96$	−184.704	−0.196367
$97$	−715.529	−0.748978	−0.374489	−	0.927231i	$-0.622182\pi$
$97$	−715.529	−0.748978	−0.374489	+	0.927231i	$0.622182\pi$
$98$	0	0
$99$	−37.8960	−0.0384717
$100$	−449.760	−0.449760
$101$	272.544	0.268507	0.134253	−	0.990947i	$-0.457136\pi$
$101$	272.544	0.268507	0.134253	+	0.990947i	$0.457136\pi$
$102$	−85.0561	−0.0825667
$103$	609.904	0.583453	0.291726	−	0.956502i	$-0.405770\pi$
$103$	609.904	0.583453	0.291726	+	0.956502i	$0.405770\pi$
$104$	−171.936	−0.162113
$105$	0	0
$106$	374.672	0.343315
$107$	−212.240	−0.191757	−0.0958785	−	0.995393i	$-0.530566\pi$
$107$	−212.240	−0.191757	−0.0958785	+	0.995393i	$0.530566\pi$
$108$	477.552	0.425486
$109$	−1287.18	−1.13110	−0.565550	−	0.824714i	$-0.691336\pi$
$109$	−1287.18	−1.13110	−0.565550	+	0.824714i	$0.691336\pi$
$110$	−42.5280	−0.0368626
$111$	342.488	0.292860
$112$	0	0
$113$	−313.336	−0.260851	−0.130426	−	0.991458i	$-0.541634\pi$
$113$	−313.336	−0.260851	−0.130426	+	0.991458i	$0.541634\pi$
$114$	−1078.67	−0.886201
$115$	81.5121	0.0660960
$116$	449.904	0.360108
$117$	−135.744	−0.107261
$118$	−99.3598	−0.0775153
$119$	0	0
$120$	−163.648	−0.124491
$121$	−1295.00	−0.972953
$122$	1557.55	1.15585
$123$	361.396	0.264927
$124$	−1145.07	−0.829279
$125$	−841.488	−0.602120
$126$	0	0
$127$	−1152.43	−0.805208	−0.402604	−	0.915374i	$-0.631895\pi$
$127$	−1152.43	−0.805208	−0.402604	+	0.915374i	$0.631895\pi$
$128$	128.000	0.0883883
$129$	−2928.16	−1.99853
$130$	−152.336	−0.102775
$131$	2368.37	1.57959	0.789793	−	0.613374i	$-0.210188\pi$
$131$	2368.37	1.57959	0.789793	+	0.613374i	$0.210188\pi$
$132$	138.528	0.0913433
$133$	0	0
$134$	−1322.46	−0.852563
$135$	423.112	0.269746
$136$	58.9439	0.0371647
$137$	−2872.44	−1.79131	−0.895654	−	0.444752i	$-0.853292\pi$
$137$	−2872.44	−1.79131	−0.895654	+	0.444752i	$0.853292\pi$
$138$	−265.512	−0.163782
$139$	−1637.24	−0.999054	−0.499527	−	0.866298i	$-0.666493\pi$
$139$	−1637.24	−0.999054	−0.499527	+	0.866298i	$0.666493\pi$
$140$	0	0
$141$	3095.52	1.84887
$142$	578.424	0.341833
$143$	128.952	0.0754092
$144$	101.056	0.0584815
$145$	398.616	0.228298
$146$	1302.63	0.738401
$147$	0	0
$148$	−237.344	−0.131821
$149$	3125.40	1.71841	0.859204	−	0.511633i	$-0.170959\pi$
$149$	3125.40	1.71841	0.859204	+	0.511633i	$0.170959\pi$
$150$	1298.01	0.706546
$151$	−370.436	−0.199640	−0.0998200	−	0.995006i	$-0.531827\pi$
$151$	−370.436	−0.199640	−0.0998200	+	0.995006i	$0.531827\pi$
$152$	747.520	0.398894
$153$	46.5363	0.0245898
$154$	0	0
$155$	−1014.54	−0.525738
$156$	496.208	0.254669
$157$	1993.44	1.01334	0.506669	−	0.862141i	$-0.330877\pi$
$157$	1993.44	1.01334	0.506669	+	0.862141i	$0.330877\pi$
$158$	−100.144	−0.0504243
$159$	−1081.30	−0.539327
$160$	113.408	0.0560356
$161$	0	0
$162$	−1719.28	−0.833824
$163$	1722.10	0.827517	0.413759	−	0.910387i	$-0.364216\pi$
$163$	1722.10	0.827517	0.413759	+	0.910387i	$0.364216\pi$
$164$	−250.448	−0.119248
$165$	122.736	0.0579090
$166$	−1614.03	−0.754658
$167$	1870.88	0.866904	0.433452	−	0.901177i	$-0.357295\pi$
$167$	1870.88	0.866904	0.433452	+	0.901177i	$0.357295\pi$
$168$	0	0
$169$	−1735.09	−0.789756
$170$	52.2244	0.0235613
$171$	590.168	0.263925
$172$	2029.22	0.899572
$173$	700.512	0.307855	0.153927	−	0.988082i	$-0.450808\pi$
$173$	700.512	0.307855	0.153927	+	0.988082i	$0.450808\pi$
$174$	−1298.42	−0.565708
$175$	0	0
$176$	−96.0000	−0.0411152
$177$	286.752	0.121772
$178$	−1892.21	−0.796781
$179$	3804.58	1.58865	0.794323	−	0.607495i	$-0.207826\pi$
$179$	3804.58	1.58865	0.794323	+	0.607495i	$0.207826\pi$
$180$	89.5358	0.0370756
$181$	−3660.10	−1.50306	−0.751529	−	0.659700i	$-0.770683\pi$
$181$	−3660.10	−1.50306	−0.751529	+	0.659700i	$0.770683\pi$
$182$	0	0
$183$	−4495.10	−1.81578
$184$	184.000	0.0737210
$185$	−210.287	−0.0835710
$186$	3304.68	1.30275
$187$	−44.2079	−0.0172877
$188$	−2145.20	−0.832206
$189$	0	0
$190$	662.304	0.252887
$191$	−1999.46	−0.757467	−0.378733	−	0.925506i	$-0.623640\pi$
$191$	−1999.46	−0.757467	−0.378733	+	0.925506i	$0.623640\pi$
$192$	−369.408	−0.138853
$193$	879.885	0.328163	0.164082	−	0.986447i	$-0.447534\pi$
$193$	879.885	0.328163	0.164082	+	0.986447i	$0.447534\pi$
$194$	−1431.06	−0.529608
$195$	439.641	0.161453
$196$	0	0
$197$	−2760.80	−0.998473	−0.499236	−	0.866466i	$-0.666386\pi$
$197$	−2760.80	−0.998473	−0.499236	+	0.866466i	$0.666386\pi$
$198$	−75.7921	−0.0272036
$199$	−3377.19	−1.20303	−0.601515	−	0.798862i	$-0.705436\pi$
$199$	−3377.19	−1.20303	−0.601515	+	0.798862i	$0.705436\pi$
$200$	−899.520	−0.318028
$201$	3816.63	1.33933
$202$	545.089	0.189863
$203$	0	0
$204$	−170.112	−0.0583835
$205$	−221.897	−0.0755998
$206$	1219.81	0.412564
$207$	145.268	0.0487770
$208$	−343.872	−0.114631
$209$	−560.640	−0.185552
$210$	0	0
$211$	−1072.19	−0.349823	−0.174912	−	0.984584i	$-0.555964\pi$
$211$	−1072.19	−0.349823	−0.174912	+	0.984584i	$0.555964\pi$
$212$	749.344	0.242760
$213$	−1669.33	−0.536999
$214$	−424.480	−0.135593
$215$	1797.89	0.570302
$216$	955.104	0.300864
$217$	0	0
$218$	−2574.37	−0.799809
$219$	−3759.40	−1.15998
$220$	−85.0561	−0.0260658
$221$	−158.353	−0.0481990
$222$	684.976	0.207084
$223$	−1494.88	−0.448899	−0.224450	−	0.974486i	$-0.572058\pi$
$223$	−1494.88	−0.448899	−0.224450	+	0.974486i	$0.572058\pi$
$224$	0	0
$225$	−710.172	−0.210421
$226$	−626.672	−0.184450
$227$	−2714.01	−0.793547	−0.396773	−	0.917917i	$-0.629870\pi$
$227$	−2714.01	−0.793547	−0.396773	+	0.917917i	$0.629870\pi$
$228$	−2157.34	−0.626639
$229$	−2512.00	−0.724881	−0.362441	−	0.932007i	$-0.618056\pi$
$229$	−2512.00	−0.724881	−0.362441	+	0.932007i	$0.618056\pi$
$230$	163.024	0.0467369
$231$	0	0
$232$	899.808	0.254635
$233$	−1728.36	−0.485961	−0.242981	−	0.970031i	$-0.578125\pi$
$233$	−1728.36	−0.485961	−0.242981	+	0.970031i	$0.578125\pi$
$234$	−271.487	−0.0758448
$235$	−1900.65	−0.527594
$236$	−198.720	−0.0548116
$237$	289.016	0.0792135
$238$	0	0
$239$	−1689.48	−0.457252	−0.228626	−	0.973514i	$-0.573423\pi$
$239$	−1689.48	−0.457252	−0.228626	+	0.973514i	$0.573423\pi$
$240$	−327.296	−0.0880286
$241$	−3155.01	−0.843286	−0.421643	−	0.906762i	$-0.638546\pi$
$241$	−3155.01	−0.843286	−0.421643	+	0.906762i	$0.638546\pi$
$242$	−2590.00	−0.687981
$243$	1738.37	0.458915
$244$	3115.10	0.817312
$245$	0	0
$246$	722.793	0.187332
$247$	−2008.22	−0.517327
$248$	−2290.15	−0.586389
$249$	4658.10	1.18552
$250$	−1682.98	−0.425763
$251$	−6341.93	−1.59482	−0.797408	−	0.603440i	$-0.793796\pi$
$251$	−6341.93	−1.59482	−0.797408	+	0.603440i	$0.793796\pi$
$252$	0	0
$253$	−138.000	−0.0342924
$254$	−2304.86	−0.569368
$255$	−150.720	−0.0370134
$256$	256.000	0.0625000
$257$	5956.53	1.44575	0.722875	−	0.690979i	$-0.242820\pi$
$257$	5956.53	1.44575	0.722875	+	0.690979i	$0.242820\pi$
$258$	−5856.32	−1.41317
$259$	0	0
$260$	−304.671	−0.0726727
$261$	710.399	0.168477
$262$	4736.75	1.11694
$263$	−7147.68	−1.67583	−0.837917	−	0.545797i	$-0.816227\pi$
$263$	−7147.68	−1.67583	−0.837917	+	0.545797i	$0.816227\pi$
$264$	277.056	0.0645895
$265$	663.920	0.153903
$266$	0	0
$267$	5460.91	1.25169
$268$	−2644.93	−0.602853
$269$	−7432.72	−1.68469	−0.842343	−	0.538941i	$-0.818824\pi$
$269$	−7432.72	−1.68469	−0.842343	+	0.538941i	$0.818824\pi$
$270$	846.223	0.190739
$271$	−3236.06	−0.725376	−0.362688	−	0.931911i	$-0.618141\pi$
$271$	−3236.06	−0.725376	−0.362688	+	0.931911i	$0.618141\pi$
$272$	117.888	0.0262794
$273$	0	0
$274$	−5744.88	−1.26665
$275$	674.640	0.147936
$276$	−531.024	−0.115811
$277$	5437.77	1.17951	0.589755	−	0.807582i	$-0.299224\pi$
$277$	5437.77	1.17951	0.589755	+	0.807582i	$0.299224\pi$
$278$	−3274.47	−0.706438
$279$	−1808.07	−0.387980
$280$	0	0
$281$	244.617	0.0519312	0.0259656	−	0.999663i	$-0.491734\pi$
$281$	244.617	0.0519312	0.0259656	+	0.999663i	$0.491734\pi$
$282$	6191.05	1.30735
$283$	1050.75	0.220710	0.110355	−	0.993892i	$-0.464801\pi$
$283$	1050.75	0.220710	0.110355	+	0.993892i	$0.464801\pi$
$284$	1156.85	0.241712
$285$	−1911.41	−0.397271
$286$	257.904	0.0533224
$287$	0	0
$288$	202.112	0.0413527
$289$	−4858.71	−0.988950
$290$	797.231	0.161431
$291$	4130.03	0.831982
$292$	2605.26	0.522129
$293$	−5643.46	−1.12524	−0.562618	−	0.826717i	$-0.690206\pi$
$293$	−5643.46	−1.12524	−0.562618	+	0.826717i	$0.690206\pi$
$294$	0	0
$295$	−176.066	−0.0347490
$296$	−474.689	−0.0932119
$297$	−716.328	−0.139951
$298$	6250.80	1.21510
$299$	−494.316	−0.0956089
$300$	2596.02	0.499604
$301$	0	0
$302$	−740.872	−0.141167
$303$	−1573.13	−0.298263
$304$	1495.04	0.282061
$305$	2759.99	0.518152
$306$	93.0725	0.0173876
$307$	−5876.55	−1.09248	−0.546241	−	0.837628i	$-0.683942\pi$
$307$	−5876.55	−1.09248	−0.546241	+	0.837628i	$0.683942\pi$
$308$	0	0
$309$	−3520.37	−0.648113
$310$	−2029.07	−0.371753
$311$	−8748.00	−1.59503	−0.797514	−	0.603301i	$-0.793852\pi$
$311$	−8748.00	−1.59503	−0.797514	+	0.603301i	$0.793852\pi$
$312$	992.416	0.180078
$313$	5029.42	0.908241	0.454120	−	0.890940i	$-0.349954\pi$
$313$	5029.42	0.908241	0.454120	+	0.890940i	$0.349954\pi$
$314$	3986.88	0.716537
$315$	0	0
$316$	−200.288	−0.0356554
$317$	−7199.63	−1.27562	−0.637810	−	0.770193i	$-0.720160\pi$
$317$	−7199.63	−1.27562	−0.637810	+	0.770193i	$0.720160\pi$
$318$	−2162.61	−0.381362
$319$	−674.856	−0.118447
$320$	226.816	0.0396232
$321$	1225.05	0.213008
$322$	0	0
$323$	688.465	0.118598
$324$	−3438.56	−0.589602
$325$	2416.56	0.412452
$326$	3444.20	0.585143
$327$	7429.63	1.25645
$328$	−500.896	−0.0843211
$329$	0	0
$330$	245.472	0.0409478
$331$	−772.092	−0.128212	−0.0641058	−	0.997943i	$-0.520420\pi$
$331$	−772.092	−0.128212	−0.0641058	+	0.997943i	$0.520420\pi$
$332$	−3228.06	−0.533624
$333$	−374.767	−0.0616730
$334$	3741.76	0.612994
$335$	−2343.41	−0.382191
$336$	0	0
$337$	−11069.6	−1.78932	−0.894659	−	0.446750i	$-0.852581\pi$
$337$	−11069.6	−1.78932	−0.894659	+	0.446750i	$0.852581\pi$
$338$	−3470.19	−0.558442
$339$	1808.58	0.289759
$340$	104.449	0.0166604
$341$	1717.61	0.272768
$342$	1180.34	0.186624
$343$	0	0
$344$	4058.43	0.636093
$345$	−470.488	−0.0734209
$346$	1401.02	0.217686
$347$	−1202.99	−0.186109	−0.0930547	−	0.995661i	$-0.529663\pi$
$347$	−1202.99	−0.186109	−0.0930547	+	0.995661i	$0.529663\pi$
$348$	−2596.85	−0.400016
$349$	4573.69	0.701501	0.350750	−	0.936469i	$-0.385927\pi$
$349$	4573.69	0.701501	0.350750	+	0.936469i	$0.385927\pi$
$350$	0	0
$351$	−2565.89	−0.390191
$352$	−192.000	−0.0290728
$353$	974.740	0.146969	0.0734846	−	0.997296i	$-0.476588\pi$
$353$	974.740	0.146969	0.0734846	+	0.997296i	$0.476588\pi$
$354$	573.505	0.0861058
$355$	1024.97	0.153239
$356$	−3784.42	−0.563409
$357$	0	0
$358$	7609.16	1.12334
$359$	−4872.46	−0.716319	−0.358160	−	0.933660i	$-0.616596\pi$
$359$	−4872.46	−0.716319	−0.358160	+	0.933660i	$0.616596\pi$
$360$	179.072	0.0262164
$361$	1872.04	0.272932
$362$	−7320.21	−1.06282
$363$	7474.74	1.08078
$364$	0	0
$365$	2308.27	0.331014
$366$	−8990.19	−1.28395
$367$	−11508.8	−1.63693	−0.818464	−	0.574557i	$-0.805174\pi$
$367$	−11508.8	−1.63693	−0.818464	+	0.574557i	$0.805174\pi$
$368$	368.000	0.0521286
$369$	−395.457	−0.0557905
$370$	−420.575	−0.0590936
$371$	0	0
$372$	6609.36	0.921181
$373$	−3989.01	−0.553735	−0.276868	−	0.960908i	$-0.589296\pi$
$373$	−3989.01	−0.553735	−0.276868	+	0.960908i	$0.589296\pi$
$374$	−88.4159	−0.0122243
$375$	4857.07	0.668848
$376$	−4290.40	−0.588459
$377$	−2417.34	−0.330237
$378$	0	0
$379$	2191.96	0.297080	0.148540	−	0.988906i	$-0.452543\pi$
$379$	2191.96	0.297080	0.148540	+	0.988906i	$0.452543\pi$
$380$	1324.61	0.178818
$381$	6651.81	0.894443
$382$	−3998.93	−0.535610
$383$	−10659.4	−1.42212	−0.711060	−	0.703131i	$-0.751785\pi$
$383$	−10659.4	−1.42212	−0.711060	+	0.703131i	$0.751785\pi$
$384$	−738.816	−0.0981837
$385$	0	0
$386$	1759.77	0.232046
$387$	3204.14	0.420867
$388$	−2862.11	−0.374489
$389$	8828.21	1.15066	0.575332	−	0.817920i	$-0.304873\pi$
$389$	8828.21	1.15066	0.575332	+	0.817920i	$0.304873\pi$
$390$	879.281	0.114164
$391$	169.464	0.0219185
$392$	0	0
$393$	−13670.3	−1.75464
$394$	−5521.61	−0.706027
$395$	−177.456	−0.0226044
$396$	−151.584	−0.0192358
$397$	4309.68	0.544828	0.272414	−	0.962180i	$-0.412178\pi$
$397$	4309.68	0.544828	0.272414	+	0.962180i	$0.412178\pi$
$398$	−6754.39	−0.850670
$399$	0	0
$400$	−1799.04	−0.224880
$401$	10650.5	1.32634	0.663169	−	0.748470i	$-0.269211\pi$
$401$	10650.5	1.32634	0.663169	+	0.748470i	$0.269211\pi$
$402$	7633.27	0.947047
$403$	6152.48	0.760489
$404$	1090.18	0.134253
$405$	−3046.57	−0.373791
$406$	0	0
$407$	356.016	0.0433589
$408$	−340.224	−0.0412834
$409$	299.234	0.0361764	0.0180882	−	0.999836i	$-0.494242\pi$
$409$	299.234	0.0361764	0.0180882	+	0.999836i	$0.494242\pi$
$410$	−443.794	−0.0534571
$411$	16579.7	1.98982
$412$	2439.62	0.291726
$413$	0	0
$414$	290.536	0.0344905
$415$	−2860.07	−0.338302
$416$	−687.745	−0.0810564
$417$	9450.13	1.10977
$418$	−1121.28	−0.131205
$419$	9596.39	1.11889	0.559445	−	0.828868i	$-0.311014\pi$
$419$	9596.39	1.11889	0.559445	+	0.828868i	$0.311014\pi$
$420$	0	0
$421$	14049.2	1.62641	0.813203	−	0.581980i	$-0.197722\pi$
$421$	14049.2	1.62641	0.813203	+	0.581980i	$0.197722\pi$
$422$	−2144.38	−0.247362
$423$	−3387.27	−0.389350
$424$	1498.69	0.171657
$425$	−828.457	−0.0945554
$426$	−3338.66	−0.379716
$427$	0	0
$428$	−848.959	−0.0958785
$429$	−744.312	−0.0837662
$430$	3595.78	0.403264
$431$	−5348.79	−0.597777	−0.298889	−	0.954288i	$-0.596616\pi$
$431$	−5348.79	−0.597777	−0.298889	+	0.954288i	$0.596616\pi$
$432$	1910.21	0.212743
$433$	−10727.2	−1.19057	−0.595284	−	0.803516i	$-0.702960\pi$
$433$	−10727.2	−1.19057	−0.595284	+	0.803516i	$0.702960\pi$
$434$	0	0
$435$	−2300.81	−0.253599
$436$	−5148.74	−0.565550
$437$	2149.12	0.235255
$438$	−7518.79	−0.820233
$439$	1425.53	0.154981	0.0774907	−	0.996993i	$-0.475309\pi$
$439$	1425.53	0.154981	0.0774907	+	0.996993i	$0.475309\pi$
$440$	−170.112	−0.0184313
$441$	0	0
$442$	−316.706	−0.0340818
$443$	−13585.6	−1.45705	−0.728524	−	0.685020i	$-0.759793\pi$
$443$	−13585.6	−1.45705	−0.728524	+	0.685020i	$0.759793\pi$
$444$	1369.95	0.146430
$445$	−3353.00	−0.357185
$446$	−2989.76	−0.317420
$447$	−18039.8	−1.90885
$448$	0	0
$449$	16471.6	1.73128	0.865638	−	0.500670i	$-0.166913\pi$
$449$	16471.6	1.73128	0.865638	+	0.500670i	$0.166913\pi$
$450$	−1420.34	−0.148790
$451$	375.672	0.0392233
$452$	−1253.34	−0.130426
$453$	2138.16	0.221765
$454$	−5428.02	−0.561122
$455$	0	0
$456$	−4314.69	−0.443100
$457$	1032.87	0.105724	0.0528619	−	0.998602i	$-0.483166\pi$
$457$	1032.87	0.105724	0.0528619	+	0.998602i	$0.483166\pi$
$458$	−5024.00	−0.512568
$459$	879.650	0.0894522
$460$	326.048	0.0330480
$461$	−239.436	−0.0241902	−0.0120951	−	0.999927i	$-0.503850\pi$
$461$	−239.436	−0.0241902	−0.0120951	+	0.999927i	$0.503850\pi$
$462$	0	0
$463$	−16596.4	−1.66587	−0.832935	−	0.553370i	$-0.813341\pi$
$463$	−16596.4	−1.66587	−0.832935	+	0.553370i	$0.813341\pi$
$464$	1799.62	0.180054
$465$	5855.90	0.584002
$466$	−3456.73	−0.343626
$467$	−10251.7	−1.01583	−0.507917	−	0.861406i	$-0.669584\pi$
$467$	−10251.7	−1.01583	−0.507917	+	0.861406i	$0.669584\pi$
$468$	−542.975	−0.0536304
$469$	0	0
$470$	−3801.30	−0.373066
$471$	−11506.1	−1.12564
$472$	−397.439	−0.0387577
$473$	−3043.82	−0.295888
$474$	578.032	0.0560124
$475$	−10506.4	−1.01488
$476$	0	0
$477$	1183.22	0.113576
$478$	−3378.95	−0.323326
$479$	4030.99	0.384511	0.192256	−	0.981345i	$-0.438420\pi$
$479$	4030.99	0.384511	0.192256	+	0.981345i	$0.438420\pi$
$480$	−654.592	−0.0622456
$481$	1275.25	0.120887
$482$	−6310.02	−0.596293
$483$	0	0
$484$	−5180.00	−0.486476
$485$	−2535.84	−0.237415
$486$	3476.74	0.324502
$487$	6054.93	0.563399	0.281699	−	0.959503i	$-0.409102\pi$
$487$	6054.93	0.563399	0.281699	+	0.959503i	$0.409102\pi$
$488$	6230.21	0.577927
$489$	−9939.97	−0.919225
$490$	0	0
$491$	−19864.4	−1.82580	−0.912902	−	0.408178i	$-0.866164\pi$
$491$	−19864.4	−1.82580	−0.912902	+	0.408178i	$0.866164\pi$
$492$	1445.59	0.132463
$493$	828.722	0.0757075
$494$	−4016.43	−0.365805
$495$	−134.304	−0.0121950
$496$	−4580.29	−0.414639
$497$	0	0
$498$	9316.20	0.838291
$499$	−12128.9	−1.08811	−0.544053	−	0.839051i	$-0.683111\pi$
$499$	−12128.9	−1.08811	−0.544053	+	0.839051i	$0.683111\pi$
$500$	−3365.95	−0.301060
$501$	−10798.7	−0.962977
$502$	−12683.9	−1.12771
$503$	18351.3	1.62673	0.813363	−	0.581756i	$-0.197634\pi$
$503$	18351.3	1.62673	0.813363	+	0.581756i	$0.197634\pi$
$504$	0	0
$505$	965.899	0.0851127
$506$	−276.000	−0.0242484
$507$	10015.0	0.877278
$508$	−4609.71	−0.402604
$509$	10876.1	0.947106	0.473553	−	0.880765i	$-0.342971\pi$
$509$	10876.1	0.947106	0.473553	+	0.880765i	$0.342971\pi$
$510$	−301.439	−0.0261724
$511$	0	0
$512$	512.000	0.0441942
$513$	11155.6	0.960103
$514$	11913.1	1.02230
$515$	2161.50	0.184946
$516$	−11712.6	−0.999264
$517$	3217.80	0.273731
$518$	0	0
$519$	−4043.35	−0.341972
$520$	−609.342	−0.0513874
$521$	−4317.61	−0.363067	−0.181533	−	0.983385i	$-0.558106\pi$
$521$	−4317.61	−0.363067	−0.181533	+	0.983385i	$0.558106\pi$
$522$	1420.80	0.119132
$523$	11789.9	0.985733	0.492866	−	0.870105i	$-0.335949\pi$
$523$	11789.9	0.985733	0.492866	+	0.870105i	$0.335949\pi$
$524$	9473.49	0.789793
$525$	0	0
$526$	−14295.4	−1.18499
$527$	−2109.22	−0.174344
$528$	554.112	0.0456717
$529$	529.000	0.0434783
$530$	1327.84	0.108826
$531$	−313.778	−0.0256437
$532$	0	0
$533$	1345.66	0.109356
$534$	10921.8	0.885082
$535$	−752.179	−0.0607842
$536$	−5289.86	−0.426282
$537$	−21960.0	−1.76470
$538$	−14865.4	−1.19125
$539$	0	0
$540$	1692.45	0.134873
$541$	8944.99	0.710860	0.355430	−	0.934703i	$-0.384334\pi$
$541$	8944.99	0.710860	0.355430	+	0.934703i	$0.384334\pi$
$542$	−6472.13	−0.512918
$543$	21126.1	1.66963
$544$	235.776	0.0185823
$545$	−4561.79	−0.358542
$546$	0	0
$547$	16376.3	1.28007	0.640036	−	0.768345i	$-0.278919\pi$
$547$	16376.3	1.28007	0.640036	+	0.768345i	$0.278919\pi$
$548$	−11489.8	−0.895654
$549$	4918.75	0.382381
$550$	1349.28	0.104606
$551$	10509.8	0.812579
$552$	−1062.05	−0.0818909
$553$	0	0
$554$	10875.5	0.834039
$555$	1213.78	0.0928325
$556$	−6548.94	−0.499527
$557$	18239.2	1.38747	0.693736	−	0.720230i	$-0.255964\pi$
$557$	18239.2	1.38747	0.693736	+	0.720230i	$0.255964\pi$
$558$	−3616.14	−0.274343
$559$	−10903.0	−0.824951
$560$	0	0
$561$	255.168	0.0192036
$562$	489.235	0.0367209
$563$	286.757	0.0214660	0.0107330	−	0.999942i	$-0.496584\pi$
$563$	286.757	0.0214660	0.0107330	+	0.999942i	$0.496584\pi$
$564$	12382.1	0.924433
$565$	−1110.46	−0.0826860
$566$	2101.51	0.156065
$567$	0	0
$568$	2313.70	0.170916
$569$	13951.6	1.02791	0.513955	−	0.857817i	$-0.328180\pi$
$569$	13951.6	1.02791	0.513955	+	0.857817i	$0.328180\pi$
$570$	−3822.82	−0.280913
$571$	−2020.68	−0.148096	−0.0740481	−	0.997255i	$-0.523592\pi$
$571$	−2020.68	−0.148096	−0.0740481	+	0.997255i	$0.523592\pi$
$572$	515.808	0.0377046
$573$	11540.9	0.841411
$574$	0	0
$575$	−2586.12	−0.187563
$576$	404.224	0.0292408
$577$	21630.4	1.56064	0.780318	−	0.625383i	$-0.215057\pi$
$577$	21630.4	1.56064	0.780318	+	0.625383i	$0.215057\pi$
$578$	−9717.43	−0.699293
$579$	−5078.70	−0.364531
$580$	1594.46	0.114149
$581$	0	0
$582$	8260.06	0.588300
$583$	−1124.02	−0.0798491
$584$	5210.53	0.369201
$585$	−481.076	−0.0340001
$586$	−11286.9	−0.795663
$587$	−1510.94	−0.106240	−0.0531202	−	0.998588i	$-0.516917\pi$
$587$	−1510.94	−0.106240	−0.0531202	+	0.998588i	$0.516917\pi$
$588$	0	0
$589$	−26748.9	−1.87126
$590$	−352.131	−0.0245712
$591$	15935.4	1.10913
$592$	−949.377	−0.0659107
$593$	19464.5	1.34791	0.673957	−	0.738771i	$-0.264593\pi$
$593$	19464.5	1.34791	0.673957	+	0.738771i	$0.264593\pi$
$594$	−1432.66	−0.0989606
$595$	0	0
$596$	12501.6	0.859204
$597$	19493.2	1.33635
$598$	−988.633	−0.0676057
$599$	−11515.5	−0.785497	−0.392748	−	0.919646i	$-0.628476\pi$
$599$	−11515.5	−0.785497	−0.392748	+	0.919646i	$0.628476\pi$
$600$	5192.03	0.353273
$601$	−372.589	−0.0252882	−0.0126441	−	0.999920i	$-0.504025\pi$
$601$	−372.589	−0.0252882	−0.0126441	+	0.999920i	$0.504025\pi$
$602$	0	0
$603$	−4176.35	−0.282046
$604$	−1481.74	−0.0998200
$605$	−4589.48	−0.308412
$606$	−3146.25	−0.210904
$607$	19564.0	1.30820	0.654099	−	0.756409i	$-0.273048\pi$
$607$	19564.0	1.30820	0.654099	+	0.756409i	$0.273048\pi$
$608$	2990.08	0.199447
$609$	0	0
$610$	5519.97	0.366389
$611$	11526.2	0.763173
$612$	186.145	0.0122949
$613$	−14060.7	−0.926435	−0.463217	−	0.886245i	$-0.653305\pi$
$613$	−14060.7	−0.926435	−0.463217	+	0.886245i	$0.653305\pi$
$614$	−11753.1	−0.772502
$615$	1280.79	0.0839779
$616$	0	0
$617$	−282.447	−0.0184293	−0.00921465	−	0.999958i	$-0.502933\pi$
$617$	−282.447	−0.0184293	−0.00921465	+	0.999958i	$0.502933\pi$
$618$	−7040.74	−0.458285
$619$	−6071.66	−0.394250	−0.197125	−	0.980378i	$-0.563160\pi$
$619$	−6071.66	−0.394250	−0.197125	+	0.980378i	$0.563160\pi$
$620$	−4058.14	−0.262869
$621$	2745.93	0.177440
$622$	−17496.0	−1.12785
$623$	0	0
$624$	1984.83	0.127335
$625$	11072.8	0.708657
$626$	10058.8	0.642223
$627$	3236.02	0.206115
$628$	7973.76	0.506669
$629$	−437.188	−0.0277135
$630$	0	0
$631$	3977.43	0.250933	0.125467	−	0.992098i	$-0.459957\pi$
$631$	3977.43	0.250933	0.125467	+	0.992098i	$0.459957\pi$
$632$	−400.576	−0.0252121
$633$	6188.69	0.388591
$634$	−14399.3	−0.902000
$635$	−4084.21	−0.255239
$636$	−4325.22	−0.269664
$637$	0	0
$638$	−1349.71	−0.0837549
$639$	1826.66	0.113086
$640$	453.632	0.0280178
$641$	18024.3	1.11064	0.555318	−	0.831638i	$-0.312597\pi$
$641$	18024.3	1.11064	0.555318	+	0.831638i	$0.312597\pi$
$642$	2450.10	0.150619
$643$	−9926.77	−0.608824	−0.304412	−	0.952541i	$-0.598460\pi$
$643$	−9926.77	−0.608824	−0.304412	+	0.952541i	$0.598460\pi$
$644$	0	0
$645$	−10377.4	−0.633504
$646$	1376.93	0.0838616
$647$	−2274.51	−0.138207	−0.0691036	−	0.997609i	$-0.522014\pi$
$647$	−2274.51	−0.138207	−0.0691036	+	0.997609i	$0.522014\pi$
$648$	−6877.12	−0.416912
$649$	298.079	0.0180287
$650$	4833.13	0.291647
$651$	0	0
$652$	6888.40	0.413759
$653$	1794.68	0.107552	0.0537758	−	0.998553i	$-0.482874\pi$
$653$	1794.68	0.107552	0.0537758	+	0.998553i	$0.482874\pi$
$654$	14859.3	0.888445
$655$	8393.52	0.500705
$656$	−1001.79	−0.0596240
$657$	4113.72	0.244279
$658$	0	0
$659$	4081.99	0.241292	0.120646	−	0.992696i	$-0.461503\pi$
$659$	4081.99	0.241292	0.120646	+	0.992696i	$0.461503\pi$
$660$	490.944	0.0289545
$661$	15350.1	0.903253	0.451626	−	0.892207i	$-0.350844\pi$
$661$	15350.1	0.903253	0.451626	+	0.892207i	$0.350844\pi$
$662$	−1544.18	−0.0906593
$663$	914.014	0.0535405
$664$	−6456.13	−0.377329
$665$	0	0
$666$	−749.534	−0.0436094
$667$	2586.95	0.150176
$668$	7483.52	0.433452
$669$	8628.45	0.498647
$670$	−4686.82	−0.270250
$671$	−4672.66	−0.268831
$672$	0	0
$673$	26310.5	1.50698	0.753488	−	0.657461i	$-0.228370\pi$
$673$	26310.5	1.50698	0.753488	+	0.657461i	$0.228370\pi$
$674$	−22139.2	−1.26524
$675$	−13424.0	−0.765467
$676$	−6940.37	−0.394878
$677$	22299.1	1.26591	0.632957	−	0.774187i	$-0.281841\pi$
$677$	22299.1	1.26591	0.632957	+	0.774187i	$0.281841\pi$
$678$	3617.15	0.204891
$679$	0	0
$680$	208.897	0.0117807
$681$	15665.3	0.881489
$682$	3435.22	0.192876
$683$	24001.7	1.34466	0.672328	−	0.740253i	$-0.265294\pi$
$683$	24001.7	1.34466	0.672328	+	0.740253i	$0.265294\pi$
$684$	2360.67	0.131963
$685$	−10179.9	−0.567818
$686$	0	0
$687$	14499.3	0.805214
$688$	8116.87	0.449786
$689$	−4026.23	−0.222623
$690$	−940.976	−0.0519164
$691$	−9627.58	−0.530030	−0.265015	−	0.964244i	$-0.585377\pi$
$691$	−9627.58	−0.530030	−0.265015	+	0.964244i	$0.585377\pi$
$692$	2802.05	0.153927
$693$	0	0
$694$	−2405.98	−0.131599
$695$	−5802.37	−0.316686
$696$	−5193.70	−0.282854
$697$	−461.324	−0.0250702
$698$	9147.37	0.496036
$699$	9976.12	0.539816
$700$	0	0
$701$	11750.7	0.633121	0.316561	−	0.948572i	$-0.397472\pi$
$701$	11750.7	0.633121	0.316561	+	0.948572i	$0.397472\pi$
$702$	−5131.78	−0.275907
$703$	−5544.36	−0.297453
$704$	−384.000	−0.0205576
$705$	10970.6	0.586064
$706$	1949.48	0.103923
$707$	0	0
$708$	1147.01	0.0608860
$709$	1045.77	0.0553946	0.0276973	−	0.999616i	$-0.491183\pi$
$709$	1045.77	0.0553946	0.0276973	+	0.999616i	$0.491183\pi$
$710$	2049.94	0.108356
$711$	−316.255	−0.0166814
$712$	−7568.83	−0.398390
$713$	−6584.17	−0.345833
$714$	0	0
$715$	457.007	0.0239036
$716$	15218.3	0.794323
$717$	9751.66	0.507925
$718$	−9744.92	−0.506514
$719$	−9367.97	−0.485906	−0.242953	−	0.970038i	$-0.578116\pi$
$719$	−9367.97	−0.485906	−0.242953	+	0.970038i	$0.578116\pi$
$720$	358.143	0.0185378
$721$	0	0
$722$	3744.08	0.192992
$723$	18210.7	0.936741
$724$	−14640.4	−0.751529
$725$	−12646.8	−0.647849
$726$	14949.5	0.764225
$727$	24447.9	1.24721	0.623605	−	0.781740i	$-0.285668\pi$
$727$	24447.9	1.24721	0.623605	+	0.781740i	$0.285668\pi$
$728$	0	0
$729$	13176.4	0.669432
$730$	4616.53	0.234062
$731$	3737.81	0.189122
$732$	−17980.4	−0.907888
$733$	−24144.2	−1.21662	−0.608312	−	0.793698i	$-0.708153\pi$
$733$	−24144.2	−1.21662	−0.608312	+	0.793698i	$0.708153\pi$
$734$	−23017.5	−1.15748
$735$	0	0
$736$	736.000	0.0368605
$737$	3967.39	0.198291
$738$	−790.915	−0.0394498
$739$	10006.7	0.498108	0.249054	−	0.968490i	$-0.419880\pi$
$739$	10006.7	0.498108	0.249054	+	0.968490i	$0.419880\pi$
$740$	−841.149	−0.0417855
$741$	11591.4	0.574658
$742$	0	0
$743$	−7313.96	−0.361135	−0.180567	−	0.983563i	$-0.557793\pi$
$743$	−7313.96	−0.361135	−0.180567	+	0.983563i	$0.557793\pi$
$744$	13218.7	0.651373
$745$	11076.4	0.544710
$746$	−7978.03	−0.391550
$747$	−5097.12	−0.249657
$748$	−176.832	−0.00864386
$749$	0	0
$750$	9714.15	0.472947
$751$	−3297.61	−0.160228	−0.0801141	−	0.996786i	$-0.525528\pi$
$751$	−3297.61	−0.160228	−0.0801141	+	0.996786i	$0.525528\pi$
$752$	−8580.80	−0.416103
$753$	36605.6	1.77156
$754$	−4834.68	−0.233513
$755$	−1312.83	−0.0632830
$756$	0	0
$757$	−23745.7	−1.14009	−0.570047	−	0.821612i	$-0.693075\pi$
$757$	−23745.7	−1.14009	−0.570047	+	0.821612i	$0.693075\pi$
$758$	4383.92	0.210068
$759$	796.536	0.0380928
$760$	2649.21	0.126444
$761$	−25723.6	−1.22533	−0.612666	−	0.790342i	$-0.709903\pi$
$761$	−25723.6	−1.22533	−0.612666	+	0.790342i	$0.709903\pi$
$762$	13303.6	0.632467
$763$	0	0
$764$	−7997.85	−0.378733
$765$	164.925	0.00779459
$766$	−21318.9	−1.00559
$767$	1067.72	0.0502649
$768$	−1477.63	−0.0694264
$769$	36423.8	1.70803	0.854016	−	0.520246i	$-0.174160\pi$
$769$	36423.8	1.70803	0.854016	+	0.520246i	$0.174160\pi$
$770$	0	0
$771$	−34381.1	−1.60597
$772$	3519.54	0.164082
$773$	20401.3	0.949265	0.474633	−	0.880184i	$-0.342581\pi$
$773$	20401.3	0.949265	0.474633	+	0.880184i	$0.342581\pi$
$774$	6408.27	0.297598
$775$	32188.0	1.49191
$776$	−5724.23	−0.264804
$777$	0	0
$778$	17656.4	0.813642
$779$	−5850.46	−0.269082
$780$	1758.56	0.0807265
$781$	−1735.27	−0.0795044
$782$	338.927	0.0154987
$783$	13428.3	0.612884
$784$	0	0
$785$	7064.76	0.321213
$786$	−27340.5	−1.24072
$787$	−23639.8	−1.07073	−0.535367	−	0.844620i	$-0.679827\pi$
$787$	−23639.8	−1.07073	−0.535367	+	0.844620i	$0.679827\pi$
$788$	−11043.2	−0.499236
$789$	41256.4	1.86155
$790$	−354.911	−0.0159838
$791$	0	0
$792$	−303.168	−0.0136018
$793$	−16737.5	−0.749515
$794$	8619.36	0.385251
$795$	−3832.15	−0.170959
$796$	−13508.8	−0.601515
$797$	41015.8	1.82291	0.911453	−	0.411405i	$-0.134962\pi$
$797$	41015.8	1.82291	0.911453	+	0.411405i	$0.134962\pi$
$798$	0	0
$799$	−3951.45	−0.174959
$800$	−3598.08	−0.159014
$801$	−5975.60	−0.263592
$802$	21301.0	0.937862
$803$	−3907.90	−0.171739
$804$	15266.5	0.669663
$805$	0	0
$806$	12305.0	0.537747
$807$	42901.7	1.87139
$808$	2180.36	0.0949315
$809$	−30619.8	−1.33070	−0.665349	−	0.746532i	$-0.731717\pi$
$809$	−30619.8	−1.33070	−0.665349	+	0.746532i	$0.731717\pi$
$810$	−6093.14	−0.264310
$811$	7000.16	0.303093	0.151547	−	0.988450i	$-0.451575\pi$
$811$	7000.16	0.303093	0.151547	+	0.988450i	$0.451575\pi$
$812$	0	0
$813$	18678.6	0.805764
$814$	712.033	0.0306594
$815$	6103.13	0.262311
$816$	−680.449	−0.0291918
$817$	47402.5	2.02987
$818$	598.468	0.0255806
$819$	0	0
$820$	−887.588	−0.0377999
$821$	31972.1	1.35912	0.679558	−	0.733621i	$-0.262171\pi$
$821$	31972.1	1.35912	0.679558	+	0.733621i	$0.262171\pi$
$822$	33159.5	1.40702
$823$	6616.25	0.280228	0.140114	−	0.990135i	$-0.455253\pi$
$823$	6616.25	0.280228	0.140114	+	0.990135i	$0.455253\pi$
$824$	4879.23	0.206282
$825$	−3894.02	−0.164330
$826$	0	0
$827$	−18261.5	−0.767852	−0.383926	−	0.923364i	$-0.625428\pi$
$827$	−18261.5	−0.767852	−0.383926	+	0.923364i	$0.625428\pi$
$828$	581.073	0.0243885
$829$	−16935.3	−0.709513	−0.354757	−	0.934959i	$-0.615436\pi$
$829$	−16935.3	−0.709513	−0.354757	+	0.934959i	$0.615436\pi$
$830$	−5720.14	−0.239215
$831$	−31386.8	−1.31023
$832$	−1375.49	−0.0573155
$833$	0	0
$834$	18900.3	0.784727
$835$	6630.41	0.274796
$836$	−2242.56	−0.0927758
$837$	−34177.0	−1.41139
$838$	19192.8	0.791174
$839$	−11971.2	−0.492598	−0.246299	−	0.969194i	$-0.579215\pi$
$839$	−11971.2	−0.492598	−0.246299	+	0.969194i	$0.579215\pi$
$840$	0	0
$841$	−11738.1	−0.481288
$842$	28098.4	1.15004
$843$	−1411.93	−0.0576863
$844$	−4288.77	−0.174912
$845$	−6149.18	−0.250341
$846$	−6774.55	−0.275312
$847$	0	0
$848$	2997.38	0.121380
$849$	−6064.95	−0.245169
$850$	−1656.91	−0.0668608
$851$	−1364.73	−0.0549734
$852$	−6677.33	−0.268499
$853$	−14844.9	−0.595872	−0.297936	−	0.954586i	$-0.596298\pi$
$853$	−14844.9	−0.595872	−0.297936	+	0.954586i	$0.596298\pi$
$854$	0	0
$855$	2091.56	0.0836605
$856$	−1697.92	−0.0677963
$857$	−45234.9	−1.80303	−0.901514	−	0.432751i	$-0.857543\pi$
$857$	−45234.9	−1.80303	−0.901514	+	0.432751i	$0.857543\pi$
$858$	−1488.62	−0.0592317
$859$	43687.6	1.73528	0.867639	−	0.497195i	$-0.165637\pi$
$859$	43687.6	1.73528	0.867639	+	0.497195i	$0.165637\pi$
$860$	7191.55	0.285151
$861$	0	0
$862$	−10697.6	−0.422692
$863$	22267.6	0.878331	0.439165	−	0.898406i	$-0.355274\pi$
$863$	22267.6	0.878331	0.439165	+	0.898406i	$0.355274\pi$
$864$	3820.42	0.150432
$865$	2482.62	0.0975855
$866$	−21454.4	−0.841858
$867$	28044.5	1.09855
$868$	0	0
$869$	300.432	0.0117278
$870$	−4601.62	−0.179321
$871$	14211.2	0.552846
$872$	−10297.5	−0.399904
$873$	−4519.28	−0.175206
$874$	4298.24	0.166350
$875$	0	0
$876$	−15037.6	−0.579992
$877$	−17550.1	−0.675743	−0.337871	−	0.941192i	$-0.609707\pi$
$877$	−17550.1	−0.675743	−0.337871	+	0.941192i	$0.609707\pi$
$878$	2851.06	0.109588
$879$	32574.1	1.24994
$880$	−340.224	−0.0130329
$881$	28114.7	1.07515	0.537577	−	0.843215i	$-0.319340\pi$
$881$	28114.7	1.07515	0.537577	+	0.843215i	$0.319340\pi$
$882$	0	0
$883$	46811.6	1.78407	0.892036	−	0.451964i	$-0.149277\pi$
$883$	46811.6	1.78407	0.892036	+	0.451964i	$0.149277\pi$
$884$	−633.412	−0.0240995
$885$	1016.25	0.0385999
$886$	−27171.2	−1.03029
$887$	−1604.14	−0.0607234	−0.0303617	−	0.999539i	$-0.509666\pi$
$887$	−1604.14	−0.0607234	−0.0303617	+	0.999539i	$0.509666\pi$
$888$	2739.90	0.103542
$889$	0	0
$890$	−6705.99	−0.252568
$891$	5157.84	0.193933
$892$	−5979.52	−0.224450
$893$	−50111.9	−1.87786
$894$	−36079.6	−1.34976
$895$	13483.4	0.503578
$896$	0	0
$897$	2853.20	0.106204
$898$	32943.2	1.22420
$899$	−32198.3	−1.19452
$900$	−2840.69	−0.105211
$901$	1380.29	0.0510368
$902$	751.343	0.0277350
$903$	0	0
$904$	−2506.69	−0.0922248
$905$	−12971.4	−0.476447
$906$	4276.31	0.156811
$907$	−1200.10	−0.0439345	−0.0219672	−	0.999759i	$-0.506993\pi$
$907$	−1200.10	−0.0439345	−0.0219672	+	0.999759i	$0.506993\pi$
$908$	−10856.0	−0.396773
$909$	1721.39	0.0628108
$910$	0	0
$911$	34847.5	1.26734	0.633672	−	0.773602i	$-0.281547\pi$
$911$	34847.5	1.26734	0.633672	+	0.773602i	$0.281547\pi$
$912$	−8629.38	−0.313319
$913$	4842.10	0.175520
$914$	2065.75	0.0747580
$915$	−15930.6	−0.575575
$916$	−10048.0	−0.362441
$917$	0	0
$918$	1759.30	0.0632522
$919$	−44895.1	−1.61148	−0.805741	−	0.592268i	$-0.798233\pi$
$919$	−44895.1	−1.61148	−0.805741	+	0.592268i	$0.798233\pi$
$920$	652.097	0.0233685
$921$	33919.4	1.21355
$922$	−478.873	−0.0171050
$923$	−6215.75	−0.221662
$924$	0	0
$925$	6671.75	0.237152
$926$	−33192.7	−1.17795
$927$	3852.16	0.136485
$928$	3599.23	0.127318
$929$	−17783.9	−0.628063	−0.314032	−	0.949413i	$-0.601680\pi$
$929$	−17783.9	−0.628063	−0.314032	+	0.949413i	$0.601680\pi$
$930$	11711.8	0.412952
$931$	0	0
$932$	−6913.46	−0.242981
$933$	50493.5	1.77179
$934$	−20503.5	−0.718303
$935$	−156.673	−0.00547995
$936$	−1085.95	−0.0379224
$937$	25058.9	0.873682	0.436841	−	0.899539i	$-0.356097\pi$
$937$	25058.9	0.873682	0.436841	+	0.899539i	$0.356097\pi$
$938$	0	0
$939$	−29029.8	−1.00889
$940$	−7602.60	−0.263797
$941$	−29903.4	−1.03594	−0.517972	−	0.855398i	$-0.673313\pi$
$941$	−29903.4	−1.03594	−0.517972	+	0.855398i	$0.673313\pi$
$942$	−23012.3	−0.795946
$943$	−1440.07	−0.0497299
$944$	−794.878	−0.0274058
$945$	0	0
$946$	−6087.65	−0.209225
$947$	17921.2	0.614952	0.307476	−	0.951556i	$-0.400516\pi$
$947$	17921.2	0.614952	0.307476	+	0.951556i	$0.400516\pi$
$948$	1156.06	0.0396068
$949$	−13998.1	−0.478817
$950$	−21012.8	−0.717627
$951$	41556.3	1.41699
$952$	0	0
$953$	31122.3	1.05787	0.528935	−	0.848662i	$-0.322592\pi$
$953$	31122.3	1.05787	0.528935	+	0.848662i	$0.322592\pi$
$954$	2366.43	0.0803103
$955$	−7086.11	−0.240106
$956$	−6757.91	−0.228626
$957$	3895.27	0.131574
$958$	8061.99	0.271890
$959$	0	0
$960$	−1309.18	−0.0440143
$961$	52158.4	1.75081
$962$	2550.50	0.0854798
$963$	−1340.51	−0.0448570
$964$	−12620.0	−0.421643
$965$	3118.32	0.104023
$966$	0	0
$967$	6627.78	0.220409	0.110204	−	0.993909i	$-0.464849\pi$
$967$	6627.78	0.220409	0.110204	+	0.993909i	$0.464849\pi$
$968$	−10360.0	−0.343991
$969$	−3973.82	−0.131742
$970$	−5071.67	−0.167878
$971$	−38645.1	−1.27722	−0.638609	−	0.769531i	$-0.720490\pi$
$971$	−38645.1	−1.27722	−0.638609	+	0.769531i	$0.720490\pi$
$972$	6953.47	0.229457
$973$	0	0
$974$	12109.9	0.398383
$975$	−13948.4	−0.458161
$976$	12460.4	0.408656
$977$	−14137.0	−0.462930	−0.231465	−	0.972843i	$-0.574352\pi$
$977$	−14137.0	−0.462930	−0.231465	+	0.972843i	$0.574352\pi$
$978$	−19879.9	−0.649990
$979$	5676.62	0.185317
$980$	0	0
$981$	−8129.87	−0.264594
$982$	−39728.9	−1.29104
$983$	−32966.5	−1.06965	−0.534825	−	0.844963i	$-0.679623\pi$
$983$	−32966.5	−1.06965	−0.534825	+	0.844963i	$0.679623\pi$
$984$	2891.17	0.0936658
$985$	−9784.30	−0.316501
$986$	1657.44	0.0535333
$987$	0	0
$988$	−8032.86	−0.258663
$989$	11668.0	0.375147
$990$	−268.607	−0.00862313
$991$	47592.2	1.52555	0.762774	−	0.646666i	$-0.223837\pi$
$991$	47592.2	1.52555	0.762774	+	0.646666i	$0.223837\pi$
$992$	−9160.58	−0.293194
$993$	4456.52	0.142420
$994$	0	0
$995$	−11968.8	−0.381343
$996$	18632.4	0.592761
$997$	−11153.1	−0.354285	−0.177142	−	0.984185i	$-0.556685\pi$
$997$	−11153.1	−0.354285	−0.177142	+	0.984185i	$0.556685\pi$
$998$	−24257.8	−0.769407
$999$	−7084.02	−0.224353

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -5.77200 q^{3} +4.00000 q^{4} +3.54400 q^{5} -11.5440 q^{6} +8.00000 q^{8} +6.31601 q^{9} +7.08801 q^{10} -6.00000 q^{11} -23.0880 q^{12} -21.4920 q^{13} -20.4560 q^{15} +16.0000 q^{16} +7.36799 q^{17} +12.6320 q^{18} +93.4400 q^{19} +14.1760 q^{20} -12.0000 q^{22} +23.0000 q^{23} -46.1760 q^{24} -112.440 q^{25} -42.9840 q^{26} +119.388 q^{27} +112.476 q^{29} -40.9120 q^{30} -286.268 q^{31} +32.0000 q^{32} +34.6320 q^{33} +14.7360 q^{34} +25.2640 q^{36} -59.3361 q^{37} +186.880 q^{38} +124.052 q^{39} +28.3520 q^{40} -62.6119 q^{41} +507.304 q^{43} -24.0000 q^{44} +22.3839 q^{45} +46.0000 q^{46} -536.300 q^{47} -92.3520 q^{48} -224.880 q^{50} -42.5280 q^{51} -85.9681 q^{52} +187.336 q^{53} +238.776 q^{54} -21.2640 q^{55} -539.336 q^{57} +224.952 q^{58} -49.6799 q^{59} -81.8240 q^{60} +778.776 q^{61} -572.536 q^{62} +64.0000 q^{64} -76.1678 q^{65} +69.2640 q^{66} -661.232 q^{67} +29.4720 q^{68} -132.756 q^{69} +289.212 q^{71} +50.5280 q^{72} +651.316 q^{73} -118.672 q^{74} +649.004 q^{75} +373.760 q^{76} +248.104 q^{78} -50.0720 q^{79} +56.7041 q^{80} -859.640 q^{81} -125.224 q^{82} -807.016 q^{83} +26.1122 q^{85} +1014.61 q^{86} -649.212 q^{87} -48.0000 q^{88} -946.104 q^{89} +44.7679 q^{90} +92.0000 q^{92} +1652.34 q^{93} -1072.60 q^{94} +331.152 q^{95} -184.704 q^{96} -715.529 q^{97} -37.8960 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} - 3 q^{3} + 8 q^{4} - 10 q^{5} - 6 q^{6} + 16 q^{8} - 13 q^{9} - 20 q^{10} - 12 q^{11} - 12 q^{12} + 51 q^{13} - 58 q^{15} + 32 q^{16} + 66 q^{17} - 26 q^{18} + 16 q^{19} - 40 q^{20} - 24 q^{22}+ \cdots + 78 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 - 3 * q^3 + 8 * q^4 - 10 * q^5 - 6 * q^6 + 16 * q^8 - 13 * q^9 - 20 * q^10 - 12 * q^11 - 12 * q^12 + 51 * q^13 - 58 * q^15 + 32 * q^16 + 66 * q^17 - 26 * q^18 + 16 * q^19 - 40 * q^20 - 24 * q^22 + 46 * q^23 - 24 * q^24 - 54 * q^25 + 102 * q^26 - 9 * q^27 - 57 * q^29 - 116 * q^30 + 17 * q^31 + 64 * q^32 + 18 * q^33 + 132 * q^34 - 52 * q^36 + 206 * q^37 + 32 * q^38 + 325 * q^39 - 80 * q^40 - 373 * q^41 + 314 * q^43 - 48 * q^44 + 284 * q^45 + 92 * q^46 - 859 * q^47 - 48 * q^48 - 108 * q^50 + 120 * q^51 + 204 * q^52 + 50 * q^53 - 18 * q^54 + 60 * q^55 - 754 * q^57 - 114 * q^58 - 612 * q^59 - 232 * q^60 + 1062 * q^61 + 34 * q^62 + 128 * q^64 - 1058 * q^65 + 36 * q^66 - 844 * q^67 + 264 * q^68 - 69 * q^69 + 399 * q^71 - 104 * q^72 + 1277 * q^73 + 412 * q^74 + 811 * q^75 + 64 * q^76 + 650 * q^78 + 122 * q^79 - 160 * q^80 - 694 * q^81 - 746 * q^82 - 1802 * q^83 - 768 * q^85 + 628 * q^86 - 1119 * q^87 - 96 * q^88 - 2046 * q^89 + 568 * q^90 + 184 * q^92 + 2493 * q^93 - 1718 * q^94 + 1380 * q^95 - 96 * q^96 + 910 * q^97 + 78 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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