LMFDB - Embedded newform 2254.2.a.r.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2254,2,Mod(1,2254)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2254.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2254, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

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

Newform invariants

comment:select newform

sage:traces = [4,-4,-1,4,5,1,0,-4,-1,-5,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$17.9982806156$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.8957.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + x + 1 $ x^4 - x^3 - 5*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 322)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.84483$ 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)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.84483 q^{3} +1.00000 q^{4} +3.70618 q^{5} -1.84483 q^{6} -1.00000 q^{8} +0.403401 q^{9} -3.70618 q^{10} +0.357320 q^{11} +1.84483 q^{12} +2.14761 q^{13} +6.83727 q^{15} +1.00000 q^{16} -0.706176 q^{17} -0.403401 q^{18} -2.79029 q^{19} +3.70618 q^{20} -0.357320 q^{22} -1.00000 q^{23} -1.84483 q^{24} +8.73574 q^{25} -2.14761 q^{26} -4.79029 q^{27} +5.79924 q^{29} -6.83727 q^{30} +3.16412 q^{31} -1.00000 q^{32} +0.659195 q^{33} +0.706176 q^{34} +0.403401 q^{36} +8.15656 q^{37} +2.79029 q^{38} +3.96197 q^{39} -3.70618 q^{40} +6.51708 q^{41} -6.33234 q^{43} +0.357320 q^{44} +1.49507 q^{45} +1.00000 q^{46} -5.64407 q^{47} +1.84483 q^{48} -8.73574 q^{50} -1.30278 q^{51} +2.14761 q^{52} +11.5345 q^{53} +4.79029 q^{54} +1.32429 q^{55} -5.14761 q^{57} -5.79924 q^{58} +9.44143 q^{59} +6.83727 q^{60} -0.983485 q^{61} -3.16412 q^{62} +1.00000 q^{64} +7.95941 q^{65} -0.659195 q^{66} +15.0310 q^{67} -0.706176 q^{68} -1.84483 q^{69} -14.1655 q^{71} -0.403401 q^{72} -2.21520 q^{73} -8.15656 q^{74} +16.1160 q^{75} -2.79029 q^{76} -3.96197 q^{78} +12.8923 q^{79} +3.70618 q^{80} -10.0475 q^{81} -6.51708 q^{82} -4.38689 q^{83} -2.61721 q^{85} +6.33234 q^{86} +10.6986 q^{87} -0.357320 q^{88} +5.20971 q^{89} -1.49507 q^{90} -1.00000 q^{92} +5.83727 q^{93} +5.64407 q^{94} -10.3413 q^{95} -1.84483 q^{96} -15.7000 q^{97} +0.144143 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - q^{3} + 4 q^{4} + 5 q^{5} + q^{6} - 4 q^{8} - q^{9} - 5 q^{10} - 2 q^{11} - q^{12} - 7 q^{13} - 5 q^{15} + 4 q^{16} + 7 q^{17} + q^{18} + q^{19} + 5 q^{20} + 2 q^{22} - 4 q^{23} + q^{24}+ \cdots - 35 q^{99}+O(q^{100}) $ 4 * q - 4 * q^2 - q^3 + 4 * q^4 + 5 * q^5 + q^6 - 4 * q^8 - q^9 - 5 * q^10 - 2 * q^11 - q^12 - 7 * q^13 - 5 * q^15 + 4 * q^16 + 7 * q^17 + q^18 + q^19 + 5 * q^20 + 2 * q^22 - 4 * q^23 + q^24 + 19 * q^25 + 7 * q^26 - 7 * q^27 - 6 * q^29 + 5 * q^30 + 4 * q^31 - 4 * q^32 + 13 * q^33 - 7 * q^34 - q^36 - q^38 + 19 * q^39 - 5 * q^40 + 15 * q^41 - 12 * q^43 - 2 * q^44 + 25 * q^45 + 4 * q^46 + 15 * q^47 - q^48 - 19 * q^50 + 2 * q^51 - 7 * q^52 + 21 * q^53 + 7 * q^54 - 12 * q^55 - 5 * q^57 + 6 * q^58 + 32 * q^59 - 5 * q^60 + 3 * q^61 - 4 * q^62 + 4 * q^64 - 6 * q^65 - 13 * q^66 + 13 * q^67 + 7 * q^68 + q^69 - 7 * q^71 + q^72 + 16 * q^73 - 6 * q^75 + q^76 - 19 * q^78 + 3 * q^79 + 5 * q^80 - 15 * q^82 - 8 * q^83 - 24 * q^85 + 12 * q^86 + 9 * q^87 + 2 * q^88 + 33 * q^89 - 25 * q^90 - 4 * q^92 - 9 * q^93 - 15 * q^94 - 11 * q^95 + q^96 + 12 * q^97 - 35 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.84483	1.06511	0.532557	−	0.846394i	$-0.321231\pi$
$3$	1.84483	1.06511	0.532557	+	0.846394i	$0.321231\pi$
$4$	1.00000	0.500000
$5$	3.70618	1.65745	0.828726	−	0.559654i	$-0.189066\pi$
$5$	3.70618	1.65745	0.828726	+	0.559654i	$0.189066\pi$
$6$	−1.84483	−0.753149
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	0.403401	0.134467
$10$	−3.70618	−1.17200
$11$	0.357320	0.107736	0.0538680	−	0.998548i	$-0.482845\pi$
$11$	0.357320	0.107736	0.0538680	+	0.998548i	$0.482845\pi$
$12$	1.84483	0.532557
$13$	2.14761	0.595639	0.297819	−	0.954622i	$-0.403741\pi$
$13$	2.14761	0.595639	0.297819	+	0.954622i	$0.403741\pi$
$14$	0	0
$15$	6.83727	1.76537
$16$	1.00000	0.250000
$17$	−0.706176	−0.171273	−0.0856364	−	0.996326i	$-0.527292\pi$
$17$	−0.706176	−0.171273	−0.0856364	+	0.996326i	$0.527292\pi$
$18$	−0.403401	−0.0950824
$19$	−2.79029	−0.640136	−0.320068	−	0.947395i	$-0.603706\pi$
$19$	−2.79029	−0.640136	−0.320068	+	0.947395i	$0.603706\pi$
$20$	3.70618	0.828726
$21$	0	0
$22$	−0.357320	−0.0761809
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.84483	−0.376575
$25$	8.73574	1.74715
$26$	−2.14761	−0.421180
$27$	−4.79029	−0.921891
$28$	0	0
$29$	5.79924	1.07689	0.538446	−	0.842660i	$-0.319012\pi$
$29$	5.79924	1.07689	0.538446	+	0.842660i	$0.319012\pi$
$30$	−6.83727	−1.24831
$31$	3.16412	0.568293	0.284146	−	0.958781i	$-0.408290\pi$
$31$	3.16412	0.568293	0.284146	+	0.958781i	$0.408290\pi$
$32$	−1.00000	−0.176777
$33$	0.659195	0.114751
$34$	0.706176	0.121108
$35$	0	0
$36$	0.403401	0.0672334
$37$	8.15656	1.34093	0.670465	−	0.741941i	$-0.266095\pi$
$37$	8.15656	1.34093	0.670465	+	0.741941i	$0.266095\pi$
$38$	2.79029	0.452644
$39$	3.96197	0.634423
$40$	−3.70618	−0.585998
$41$	6.51708	1.01780	0.508898	−	0.860827i	$-0.330053\pi$
$41$	6.51708	1.01780	0.508898	+	0.860827i	$0.330053\pi$
$42$	0	0
$43$	−6.33234	−0.965673	−0.482837	−	0.875711i	$-0.660393\pi$
$43$	−6.33234	−0.965673	−0.482837	+	0.875711i	$0.660393\pi$
$44$	0.357320	0.0538680
$45$	1.49507	0.222872
$46$	1.00000	0.147442
$47$	−5.64407	−0.823272	−0.411636	−	0.911348i	$-0.635043\pi$
$47$	−5.64407	−0.823272	−0.411636	+	0.911348i	$0.635043\pi$
$48$	1.84483	0.266278
$49$	0	0
$50$	−8.73574	−1.23542
$51$	−1.30278	−0.182425
$52$	2.14761	0.297819
$53$	11.5345	1.58438	0.792192	−	0.610272i	$-0.208940\pi$
$53$	11.5345	1.58438	0.792192	+	0.610272i	$0.208940\pi$
$54$	4.79029	0.651875
$55$	1.32429	0.178567
$56$	0	0
$57$	−5.14761	−0.681817
$58$	−5.79924	−0.761477
$59$	9.44143	1.22917	0.614585	−	0.788851i	$-0.289324\pi$
$59$	9.44143	1.22917	0.614585	+	0.788851i	$0.289324\pi$
$60$	6.83727	0.882687
$61$	−0.983485	−0.125922	−0.0629612	−	0.998016i	$-0.520054\pi$
$61$	−0.983485	−0.125922	−0.0629612	+	0.998016i	$0.520054\pi$
$62$	−3.16412	−0.401844
$63$	0	0
$64$	1.00000	0.125000
$65$	7.95941	0.987243
$66$	−0.659195	−0.0811413
$67$	15.0310	1.83632	0.918162	−	0.396206i	$-0.129673\pi$
$67$	15.0310	1.83632	0.918162	+	0.396206i	$0.129673\pi$
$68$	−0.706176	−0.0856364
$69$	−1.84483	−0.222092
$70$	0	0
$71$	−14.1655	−1.68114	−0.840568	−	0.541705i	$-0.817779\pi$
$71$	−14.1655	−1.68114	−0.840568	+	0.541705i	$0.817779\pi$
$72$	−0.403401	−0.0475412
$73$	−2.21520	−0.259270	−0.129635	−	0.991562i	$-0.541381\pi$
$73$	−2.21520	−0.259270	−0.129635	+	0.991562i	$0.541381\pi$
$74$	−8.15656	−0.948181
$75$	16.1160	1.86091
$76$	−2.79029	−0.320068
$77$	0	0
$78$	−3.96197	−0.448605
$79$	12.8923	1.45050	0.725249	−	0.688487i	$-0.241725\pi$
$79$	12.8923	1.45050	0.725249	+	0.688487i	$0.241725\pi$
$80$	3.70618	0.414363
$81$	−10.0475	−1.11639
$82$	−6.51708	−0.719691
$83$	−4.38689	−0.481523	−0.240762	−	0.970584i	$-0.577397\pi$
$83$	−4.38689	−0.481523	−0.240762	+	0.970584i	$0.577397\pi$
$84$	0	0
$85$	−2.61721	−0.283877
$86$	6.33234	0.682834
$87$	10.6986	1.14701
$88$	−0.357320	−0.0380904
$89$	5.20971	0.552229	0.276114	−	0.961125i	$-0.410953\pi$
$89$	5.20971	0.552229	0.276114	+	0.961125i	$0.410953\pi$
$90$	−1.49507	−0.157595
$91$	0	0
$92$	−1.00000	−0.104257
$93$	5.83727	0.605296
$94$	5.64407	0.582141
$95$	−10.3413	−1.06099
$96$	−1.84483	−0.188287
$97$	−15.7000	−1.59409	−0.797047	−	0.603917i	$-0.793606\pi$
$97$	−15.7000	−1.59409	−0.797047	+	0.603917i	$0.793606\pi$
$98$	0	0
$99$	0.144143	0.0144869
$100$	8.73574	0.873574
$101$	13.9718	1.39025	0.695124	−	0.718890i	$-0.255349\pi$
$101$	13.9718	1.39025	0.695124	+	0.718890i	$0.255349\pi$
$102$	1.30278	0.128994
$103$	10.9970	1.08357	0.541785	−	0.840517i	$-0.317749\pi$
$103$	10.9970	1.08357	0.541785	+	0.840517i	$0.317749\pi$
$104$	−2.14761	−0.210590
$105$	0	0
$106$	−11.5345	−1.12033
$107$	−16.8421	−1.62819	−0.814095	−	0.580732i	$-0.802766\pi$
$107$	−16.8421	−1.62819	−0.814095	+	0.580732i	$0.802766\pi$
$108$	−4.79029	−0.460946
$109$	−10.4715	−1.00299	−0.501493	−	0.865162i	$-0.667216\pi$
$109$	−10.4715	−1.00299	−0.501493	+	0.865162i	$0.667216\pi$
$110$	−1.32429	−0.126266
$111$	15.0475	1.42824
$112$	0	0
$113$	−2.13865	−0.201188	−0.100594	−	0.994928i	$-0.532074\pi$
$113$	−2.13865	−0.201188	−0.100594	+	0.994928i	$0.532074\pi$
$114$	5.14761	0.482118
$115$	−3.70618	−0.345603
$116$	5.79924	0.538446
$117$	0.866346	0.0800937
$118$	−9.44143	−0.869154
$119$	0	0
$120$	−6.83727	−0.624154
$121$	−10.8723	−0.988393
$122$	0.983485	0.0890406
$123$	12.0229	1.08407
$124$	3.16412	0.284146
$125$	13.8453	1.23836
$126$	0	0
$127$	1.66420	0.147673	0.0738367	−	0.997270i	$-0.476476\pi$
$127$	1.66420	0.147673	0.0738367	+	0.997270i	$0.476476\pi$
$128$	−1.00000	−0.0883883
$129$	−11.6821	−1.02855
$130$	−7.95941	−0.698086
$131$	−12.8047	−1.11875	−0.559377	−	0.828913i	$-0.688960\pi$
$131$	−12.8047	−1.11875	−0.559377	+	0.828913i	$0.688960\pi$
$132$	0.659195	0.0573755
$133$	0	0
$134$	−15.0310	−1.29848
$135$	−17.7536	−1.52799
$136$	0.706176	0.0605541
$137$	−10.3883	−0.887530	−0.443765	−	0.896143i	$-0.646358\pi$
$137$	−10.3883	−0.887530	−0.443765	+	0.896143i	$0.646358\pi$
$138$	1.84483	0.157042
$139$	−18.1536	−1.53977	−0.769883	−	0.638185i	$-0.779686\pi$
$139$	−18.1536	−1.53977	−0.769883	+	0.638185i	$0.779686\pi$
$140$	0	0
$141$	−10.4124	−0.876878
$142$	14.1655	1.18874
$143$	0.767383	0.0641718
$144$	0.403401	0.0336167
$145$	21.4930	1.78490
$146$	2.21520	0.183331
$147$	0	0
$148$	8.15656	0.670465
$149$	−19.6745	−1.61180	−0.805900	−	0.592051i	$-0.798318\pi$
$149$	−19.6745	−1.61180	−0.805900	+	0.592051i	$0.798318\pi$
$150$	−16.1160	−1.31586
$151$	2.56267	0.208547	0.104274	−	0.994549i	$-0.466748\pi$
$151$	2.56267	0.208547	0.104274	+	0.994549i	$0.466748\pi$
$152$	2.79029	0.226322
$153$	−0.284872	−0.0230305
$154$	0	0
$155$	11.7268	0.941918
$156$	3.96197	0.317212
$157$	−16.4433	−1.31232	−0.656159	−	0.754622i	$-0.727820\pi$
$157$	−16.4433	−1.31232	−0.656159	+	0.754622i	$0.727820\pi$
$158$	−12.8923	−1.02566
$159$	21.2792	1.68755
$160$	−3.70618	−0.292999
$161$	0	0
$162$	10.0475	0.789404
$163$	−19.0530	−1.49234	−0.746171	−	0.665754i	$-0.768110\pi$
$163$	−19.0530	−1.49234	−0.746171	+	0.665754i	$0.768110\pi$
$164$	6.51708	0.508898
$165$	2.44309	0.190194
$166$	4.38689	0.340488
$167$	11.6154	0.898827	0.449413	−	0.893324i	$-0.351633\pi$
$167$	11.6154	0.898827	0.449413	+	0.893324i	$0.351633\pi$
$168$	0	0
$169$	−8.38779	−0.645214
$170$	2.61721	0.200731
$171$	−1.12560	−0.0860770
$172$	−6.33234	−0.482837
$173$	−12.1951	−0.927174	−0.463587	−	0.886051i	$-0.653438\pi$
$173$	−12.1951	−0.927174	−0.463587	+	0.886051i	$0.653438\pi$
$174$	−10.6986	−0.811060
$175$	0	0
$176$	0.357320	0.0269340
$177$	17.4178	1.30921
$178$	−5.20971	−0.390485
$179$	2.15107	0.160779	0.0803893	−	0.996764i	$-0.474384\pi$
$179$	2.15107	0.160779	0.0803893	+	0.996764i	$0.474384\pi$
$180$	1.49507	0.111436
$181$	15.7236	1.16873	0.584363	−	0.811493i	$-0.301345\pi$
$181$	15.7236	1.16873	0.584363	+	0.811493i	$0.301345\pi$
$182$	0	0
$183$	−1.81436	−0.134122
$184$	1.00000	0.0737210
$185$	30.2296	2.22253
$186$	−5.83727	−0.428009
$187$	−0.252331	−0.0184523
$188$	−5.64407	−0.411636
$189$	0	0
$190$	10.3413	0.750236
$191$	−6.69376	−0.484344	−0.242172	−	0.970233i	$-0.577860\pi$
$191$	−6.69376	−0.484344	−0.242172	+	0.970233i	$0.577860\pi$
$192$	1.84483	0.133139
$193$	5.09855	0.367002	0.183501	−	0.983020i	$-0.441257\pi$
$193$	5.09855	0.367002	0.183501	+	0.983020i	$0.441257\pi$
$194$	15.7000	1.12719
$195$	14.6838	1.05153
$196$	0	0
$197$	4.36217	0.310792	0.155396	−	0.987852i	$-0.450335\pi$
$197$	4.36217	0.310792	0.155396	+	0.987852i	$0.450335\pi$
$198$	−0.144143	−0.0102438
$199$	−1.71374	−0.121484	−0.0607419	−	0.998154i	$-0.519347\pi$
$199$	−1.71374	−0.121484	−0.0607419	+	0.998154i	$0.519347\pi$
$200$	−8.73574	−0.617710
$201$	27.7296	1.95589
$202$	−13.9718	−0.983054
$203$	0	0
$204$	−1.30278	−0.0912125
$205$	24.1534	1.68695
$206$	−10.9970	−0.766199
$207$	−0.403401	−0.0280383
$208$	2.14761	0.148910
$209$	−0.997025	−0.0689657
$210$	0	0
$211$	1.64129	0.112991	0.0564956	−	0.998403i	$-0.482007\pi$
$211$	1.64129	0.112991	0.0564956	+	0.998403i	$0.482007\pi$
$212$	11.5345	0.792192
$213$	−26.1330	−1.79060
$214$	16.8421	1.15130
$215$	−23.4688	−1.60056
$216$	4.79029	0.325938
$217$	0	0
$218$	10.4715	0.709218
$219$	−4.08667	−0.276152
$220$	1.32429	0.0892837
$221$	−1.51659	−0.102017
$222$	−15.0475	−1.00992
$223$	3.51361	0.235289	0.117644	−	0.993056i	$-0.462466\pi$
$223$	3.51361	0.235289	0.117644	+	0.993056i	$0.462466\pi$
$224$	0	0
$225$	3.52400	0.234934
$226$	2.13865	0.142261
$227$	20.7126	1.37474	0.687371	−	0.726307i	$-0.258765\pi$
$227$	20.7126	1.37474	0.687371	+	0.726307i	$0.258765\pi$
$228$	−5.14761	−0.340909
$229$	14.4353	0.953909	0.476954	−	0.878928i	$-0.341741\pi$
$229$	14.4353	0.953909	0.476954	+	0.878928i	$0.341741\pi$
$230$	3.70618	0.244378
$231$	0	0
$232$	−5.79924	−0.380739
$233$	−5.58543	−0.365913	−0.182957	−	0.983121i	$-0.558567\pi$
$233$	−5.58543	−0.365913	−0.182957	+	0.983121i	$0.558567\pi$
$234$	−0.866346	−0.0566348
$235$	−20.9179	−1.36453
$236$	9.44143	0.614585
$237$	23.7841	1.54494
$238$	0	0
$239$	10.5760	0.684104	0.342052	−	0.939681i	$-0.388878\pi$
$239$	10.5760	0.684104	0.342052	+	0.939681i	$0.388878\pi$
$240$	6.83727	0.441344
$241$	1.43158	0.0922160	0.0461080	−	0.998936i	$-0.485318\pi$
$241$	1.43158	0.0922160	0.0461080	+	0.998936i	$0.485318\pi$
$242$	10.8723	0.698899
$243$	−4.16502	−0.267186
$244$	−0.983485	−0.0629612
$245$	0	0
$246$	−12.0229	−0.766552
$247$	−5.99244	−0.381290
$248$	−3.16412	−0.200922
$249$	−8.09306	−0.512877
$250$	−13.8453	−0.875655
$251$	−17.0163	−1.07406	−0.537030	−	0.843563i	$-0.680454\pi$
$251$	−17.0163	−1.07406	−0.537030	+	0.843563i	$0.680454\pi$
$252$	0	0
$253$	−0.357320	−0.0224645
$254$	−1.66420	−0.104421
$255$	−4.82832	−0.302361
$256$	1.00000	0.0625000
$257$	27.7670	1.73206	0.866028	−	0.499996i	$-0.166665\pi$
$257$	27.7670	1.73206	0.866028	+	0.499996i	$0.166665\pi$
$258$	11.6821	0.727296
$259$	0	0
$260$	7.95941	0.493622
$261$	2.33942	0.144806
$262$	12.8047	0.791079
$263$	11.2072	0.691067	0.345534	−	0.938406i	$-0.387698\pi$
$263$	11.2072	0.691067	0.345534	+	0.938406i	$0.387698\pi$
$264$	−0.659195	−0.0405706
$265$	42.7489	2.62604
$266$	0	0
$267$	9.61104	0.588186
$268$	15.0310	0.918162
$269$	29.5104	1.79928	0.899641	−	0.436631i	$-0.143828\pi$
$269$	29.5104	1.79928	0.899641	+	0.436631i	$0.143828\pi$
$270$	17.7536	1.08045
$271$	−19.0435	−1.15681	−0.578406	−	0.815749i	$-0.696325\pi$
$271$	−19.0435	−1.15681	−0.578406	+	0.815749i	$0.696325\pi$
$272$	−0.706176	−0.0428182
$273$	0	0
$274$	10.3883	0.627579
$275$	3.12145	0.188231
$276$	−1.84483	−0.111046
$277$	3.79119	0.227790	0.113895	−	0.993493i	$-0.463667\pi$
$277$	3.79119	0.227790	0.113895	+	0.993493i	$0.463667\pi$
$278$	18.1536	1.08878
$279$	1.27641	0.0764166
$280$	0	0
$281$	−14.5489	−0.867917	−0.433958	−	0.900933i	$-0.642883\pi$
$281$	−14.5489	−0.867917	−0.433958	+	0.900933i	$0.642883\pi$
$282$	10.4124	0.620047
$283$	0.452945	0.0269248	0.0134624	−	0.999909i	$-0.495715\pi$
$283$	0.452945	0.0269248	0.0134624	+	0.999909i	$0.495715\pi$
$284$	−14.1655	−0.840568
$285$	−19.0779	−1.13008
$286$	−0.767383	−0.0453763
$287$	0	0
$288$	−0.403401	−0.0237706
$289$	−16.5013	−0.970666
$290$	−21.4930	−1.26211
$291$	−28.9638	−1.69789
$292$	−2.21520	−0.129635
$293$	−8.36766	−0.488844	−0.244422	−	0.969669i	$-0.578598\pi$
$293$	−8.36766	−0.488844	−0.244422	+	0.969669i	$0.578598\pi$
$294$	0	0
$295$	34.9916	2.03729
$296$	−8.15656	−0.474090
$297$	−1.71166	−0.0993209
$298$	19.6745	1.13972
$299$	−2.14761	−0.124199
$300$	16.1160	0.930456
$301$	0	0
$302$	−2.56267	−0.147465
$303$	25.7756	1.48077
$304$	−2.79029	−0.160034
$305$	−3.64497	−0.208710
$306$	0.284872	0.0162850
$307$	−24.8636	−1.41904	−0.709521	−	0.704684i	$-0.751089\pi$
$307$	−24.8636	−1.41904	−0.709521	+	0.704684i	$0.751089\pi$
$308$	0	0
$309$	20.2877	1.15412
$310$	−11.7268	−0.666037
$311$	−16.2246	−0.920015	−0.460008	−	0.887915i	$-0.652153\pi$
$311$	−16.2246	−0.920015	−0.460008	+	0.887915i	$0.652153\pi$
$312$	−3.96197	−0.224302
$313$	0.957606	0.0541271	0.0270636	−	0.999634i	$-0.491384\pi$
$313$	0.957606	0.0541271	0.0270636	+	0.999634i	$0.491384\pi$
$314$	16.4433	0.927950
$315$	0	0
$316$	12.8923	0.725249
$317$	17.3051	0.971954	0.485977	−	0.873972i	$-0.338464\pi$
$317$	17.3051	0.971954	0.485977	+	0.873972i	$0.338464\pi$
$318$	−21.2792	−1.19328
$319$	2.07218	0.116020
$320$	3.70618	0.207182
$321$	−31.0709	−1.73421
$322$	0	0
$323$	1.97043	0.109638
$324$	−10.0475	−0.558193
$325$	18.7609	1.04067
$326$	19.0530	1.05525
$327$	−19.3181	−1.06829
$328$	−6.51708	−0.359845
$329$	0	0
$330$	−2.44309	−0.134488
$331$	4.70208	0.258450	0.129225	−	0.991615i	$-0.458751\pi$
$331$	4.70208	0.258450	0.129225	+	0.991615i	$0.458751\pi$
$332$	−4.38689	−0.240762
$333$	3.29036	0.180311
$334$	−11.6154	−0.635567
$335$	55.7074	3.04362
$336$	0	0
$337$	−34.5382	−1.88142	−0.940709	−	0.339214i	$-0.889839\pi$
$337$	−34.5382	−1.88142	−0.940709	+	0.339214i	$0.889839\pi$
$338$	8.38779	0.456235
$339$	−3.94546	−0.214288
$340$	−2.61721	−0.141938
$341$	1.13060	0.0612256
$342$	1.12560	0.0608657
$343$	0	0
$344$	6.33234	0.341417
$345$	−6.83727	−0.368106
$346$	12.1951	0.655611
$347$	−3.01700	−0.161961	−0.0809806	−	0.996716i	$-0.525805\pi$
$347$	−3.01700	−0.161961	−0.0809806	+	0.996716i	$0.525805\pi$
$348$	10.6986	0.573506
$349$	−19.1069	−1.02277	−0.511384	−	0.859353i	$-0.670867\pi$
$349$	−19.1069	−1.02277	−0.511384	+	0.859353i	$0.670867\pi$
$350$	0	0
$351$	−10.2877	−0.549114
$352$	−0.357320	−0.0190452
$353$	−4.99515	−0.265865	−0.132932	−	0.991125i	$-0.542439\pi$
$353$	−4.99515	−0.265865	−0.132932	+	0.991125i	$0.542439\pi$
$354$	−17.4178	−0.925748
$355$	−52.4999	−2.78640
$356$	5.20971	0.276114
$357$	0	0
$358$	−2.15107	−0.113688
$359$	−17.8137	−0.940170	−0.470085	−	0.882621i	$-0.655777\pi$
$359$	−17.8137	−0.940170	−0.470085	+	0.882621i	$0.655777\pi$
$360$	−1.49507	−0.0787973
$361$	−11.2143	−0.590226
$362$	−15.7236	−0.826414
$363$	−20.0576	−1.05275
$364$	0	0
$365$	−8.20993	−0.429727
$366$	1.81436	0.0948383
$367$	−8.99244	−0.469401	−0.234701	−	0.972068i	$-0.575411\pi$
$367$	−8.99244	−0.469401	−0.234701	+	0.972068i	$0.575411\pi$
$368$	−1.00000	−0.0521286
$369$	2.62899	0.136860
$370$	−30.2296	−1.57156
$371$	0	0
$372$	5.83727	0.302648
$373$	9.61909	0.498058	0.249029	−	0.968496i	$-0.419889\pi$
$373$	9.61909	0.498058	0.249029	+	0.968496i	$0.419889\pi$
$374$	0.252331	0.0130477
$375$	25.5423	1.31900
$376$	5.64407	0.291071
$377$	12.4545	0.641438
$378$	0	0
$379$	−4.72457	−0.242685	−0.121342	−	0.992611i	$-0.538720\pi$
$379$	−4.72457	−0.242685	−0.121342	+	0.992611i	$0.538720\pi$
$380$	−10.3413	−0.530497
$381$	3.07016	0.157289
$382$	6.69376	0.342483
$383$	−5.38501	−0.275161	−0.137581	−	0.990491i	$-0.543933\pi$
$383$	−5.38501	−0.275161	−0.137581	+	0.990491i	$0.543933\pi$
$384$	−1.84483	−0.0941436
$385$	0	0
$386$	−5.09855	−0.259509
$387$	−2.55447	−0.129851
$388$	−15.7000	−0.797047
$389$	33.8055	1.71401	0.857003	−	0.515312i	$-0.172324\pi$
$389$	33.8055	1.71401	0.857003	+	0.515312i	$0.172324\pi$
$390$	−14.6838	−0.743541
$391$	0.706176	0.0357129
$392$	0	0
$393$	−23.6226	−1.19160
$394$	−4.36217	−0.219763
$395$	47.7811	2.40413
$396$	0.144143	0.00724346
$397$	−9.32685	−0.468101	−0.234051	−	0.972224i	$-0.575198\pi$
$397$	−9.32685	−0.468101	−0.234051	+	0.972224i	$0.575198\pi$
$398$	1.71374	0.0859020
$399$	0	0
$400$	8.73574	0.436787
$401$	−13.9152	−0.694892	−0.347446	−	0.937700i	$-0.612951\pi$
$401$	−13.9152	−0.694892	−0.347446	+	0.937700i	$0.612951\pi$
$402$	−27.7296	−1.38303
$403$	6.79529	0.338497
$404$	13.9718	0.695124
$405$	−37.2377	−1.85036
$406$	0	0
$407$	2.91450	0.144466
$408$	1.30278	0.0644970
$409$	25.1840	1.24527	0.622636	−	0.782512i	$-0.286062\pi$
$409$	25.1840	1.24527	0.622636	+	0.782512i	$0.286062\pi$
$410$	−24.1534	−1.19285
$411$	−19.1646	−0.945320
$412$	10.9970	0.541785
$413$	0	0
$414$	0.403401	0.0198261
$415$	−16.2586	−0.798102
$416$	−2.14761	−0.105295
$417$	−33.4903	−1.64003
$418$	0.997025	0.0487661
$419$	26.6155	1.30025	0.650126	−	0.759827i	$-0.274716\pi$
$419$	26.6155	1.30025	0.650126	+	0.759827i	$0.274716\pi$
$420$	0	0
$421$	13.2888	0.647655	0.323828	−	0.946116i	$-0.395030\pi$
$421$	13.2888	0.647655	0.323828	+	0.946116i	$0.395030\pi$
$422$	−1.64129	−0.0798968
$423$	−2.27682	−0.110703
$424$	−11.5345	−0.560164
$425$	−6.16897	−0.299239
$426$	26.1330	1.26615
$427$	0	0
$428$	−16.8421	−0.814095
$429$	1.41569	0.0683502
$430$	23.4688	1.13176
$431$	−7.64704	−0.368345	−0.184173	−	0.982894i	$-0.558961\pi$
$431$	−7.64704	−0.368345	−0.184173	+	0.982894i	$0.558961\pi$
$432$	−4.79029	−0.230473
$433$	15.5384	0.746730	0.373365	−	0.927685i	$-0.378204\pi$
$433$	15.5384	0.746730	0.373365	+	0.927685i	$0.378204\pi$
$434$	0	0
$435$	39.6509	1.90112
$436$	−10.4715	−0.501493
$437$	2.79029	0.133478
$438$	4.08667	0.195269
$439$	−36.9647	−1.76423	−0.882116	−	0.471033i	$-0.843881\pi$
$439$	−36.9647	−1.76423	−0.882116	+	0.471033i	$0.843881\pi$
$440$	−1.32429	−0.0631331
$441$	0	0
$442$	1.51659	0.0721368
$443$	35.8529	1.70343	0.851713	−	0.524009i	$-0.175564\pi$
$443$	35.8529	1.70343	0.851713	+	0.524009i	$0.175564\pi$
$444$	15.0475	0.714121
$445$	19.3081	0.915292
$446$	−3.51361	−0.166374
$447$	−36.2962	−1.71675
$448$	0	0
$449$	26.6931	1.25972	0.629862	−	0.776707i	$-0.283111\pi$
$449$	26.6931	1.25972	0.629862	+	0.776707i	$0.283111\pi$
$450$	−3.52400	−0.166123
$451$	2.32868	0.109653
$452$	−2.13865	−0.100594
$453$	4.72769	0.222126
$454$	−20.7126	−0.972089
$455$	0	0
$456$	5.14761	0.241059
$457$	−37.1541	−1.73799	−0.868997	−	0.494817i	$-0.835235\pi$
$457$	−37.1541	−1.73799	−0.868997	+	0.494817i	$0.835235\pi$
$458$	−14.4353	−0.674515
$459$	3.38279	0.157895
$460$	−3.70618	−0.172801
$461$	8.56316	0.398826	0.199413	−	0.979916i	$-0.436096\pi$
$461$	8.56316	0.398826	0.199413	+	0.979916i	$0.436096\pi$
$462$	0	0
$463$	−32.2691	−1.49967	−0.749835	−	0.661625i	$-0.769867\pi$
$463$	−32.2691	−1.49967	−0.749835	+	0.661625i	$0.769867\pi$
$464$	5.79924	0.269223
$465$	21.6339	1.00325
$466$	5.58543	0.258740
$467$	31.5664	1.46072	0.730360	−	0.683062i	$-0.239352\pi$
$467$	31.5664	1.46072	0.730360	+	0.683062i	$0.239352\pi$
$468$	0.866346	0.0400468
$469$	0	0
$470$	20.9179	0.964871
$471$	−30.3351	−1.39777
$472$	−9.44143	−0.434577
$473$	−2.26267	−0.104038
$474$	−23.7841	−1.09244
$475$	−24.3752	−1.11841
$476$	0	0
$477$	4.65302	0.213047
$478$	−10.5760	−0.483734
$479$	4.55511	0.208128	0.104064	−	0.994571i	$-0.466815\pi$
$479$	4.55511	0.208128	0.104064	+	0.994571i	$0.466815\pi$
$480$	−6.83727	−0.312077
$481$	17.5171	0.798710
$482$	−1.43158	−0.0652066
$483$	0	0
$484$	−10.8723	−0.494196
$485$	−58.1870	−2.64213
$486$	4.16502	0.188929
$487$	−28.2113	−1.27838	−0.639189	−	0.769050i	$-0.720730\pi$
$487$	−28.2113	−1.27838	−0.639189	+	0.769050i	$0.720730\pi$
$488$	0.983485	0.0445203
$489$	−35.1495	−1.58951
$490$	0	0
$491$	21.1781	0.955756	0.477878	−	0.878426i	$-0.341406\pi$
$491$	21.1781	0.955756	0.477878	+	0.878426i	$0.341406\pi$
$492$	12.0229	0.542034
$493$	−4.09528	−0.184442
$494$	5.99244	0.269612
$495$	0.534220	0.0240114
$496$	3.16412	0.142073
$497$	0	0
$498$	8.09306	0.362659
$499$	41.3353	1.85042	0.925212	−	0.379450i	$-0.123887\pi$
$499$	41.3353	1.85042	0.925212	+	0.379450i	$0.123887\pi$
$500$	13.8453	0.619181
$501$	21.4285	0.957353
$502$	17.0163	0.759475
$503$	10.1646	0.453218	0.226609	−	0.973986i	$-0.427236\pi$
$503$	10.1646	0.453218	0.226609	+	0.973986i	$0.427236\pi$
$504$	0	0
$505$	51.7820	2.30427
$506$	0.357320	0.0158848
$507$	−15.4740	−0.687227
$508$	1.66420	0.0738367
$509$	2.37654	0.105339	0.0526693	−	0.998612i	$-0.483227\pi$
$509$	2.37654	0.105339	0.0526693	+	0.998612i	$0.483227\pi$
$510$	4.82832	0.213801
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	13.3663	0.590135
$514$	−27.7670	−1.22475
$515$	40.7569	1.79596
$516$	−11.6821	−0.514276
$517$	−2.01674	−0.0886960
$518$	0	0
$519$	−22.4979	−0.987546
$520$	−7.95941	−0.349043
$521$	−44.8249	−1.96382	−0.981908	−	0.189360i	$-0.939359\pi$
$521$	−44.8249	−1.96382	−0.981908	+	0.189360i	$0.939359\pi$
$522$	−2.33942	−0.102393
$523$	−30.1568	−1.31866	−0.659332	−	0.751852i	$-0.729161\pi$
$523$	−30.1568	−1.31866	−0.659332	+	0.751852i	$0.729161\pi$
$524$	−12.8047	−0.559377
$525$	0	0
$526$	−11.2072	−0.488658
$527$	−2.23443	−0.0973332
$528$	0.659195	0.0286878
$529$	1.00000	0.0434783
$530$	−42.7489	−1.85689
$531$	3.80868	0.165283
$532$	0	0
$533$	13.9961	0.606239
$534$	−9.61104	−0.415910
$535$	−62.4199	−2.69865
$536$	−15.0310	−0.649238
$537$	3.96836	0.171247
$538$	−29.5104	−1.27228
$539$	0	0
$540$	−17.7536	−0.763995
$541$	−26.0317	−1.11919	−0.559595	−	0.828766i	$-0.689043\pi$
$541$	−26.0317	−1.11919	−0.559595	+	0.828766i	$0.689043\pi$
$542$	19.0435	0.817989
$543$	29.0074	1.24483
$544$	0.706176	0.0302771
$545$	−38.8092	−1.66240
$546$	0	0
$547$	1.81734	0.0777038	0.0388519	−	0.999245i	$-0.487630\pi$
$547$	1.81734	0.0777038	0.0388519	+	0.999245i	$0.487630\pi$
$548$	−10.3883	−0.443765
$549$	−0.396739	−0.0169324
$550$	−3.12145	−0.133099
$551$	−16.1815	−0.689357
$552$	1.84483	0.0785212
$553$	0	0
$554$	−3.79119	−0.161072
$555$	55.7686	2.36724
$556$	−18.1536	−0.769883
$557$	22.3523	0.947098	0.473549	−	0.880767i	$-0.342973\pi$
$557$	22.3523	0.947098	0.473549	+	0.880767i	$0.342973\pi$
$558$	−1.27641	−0.0540347
$559$	−13.5994	−0.575192
$560$	0	0
$561$	−0.465508	−0.0196537
$562$	14.5489	0.613710
$563$	−30.0445	−1.26622	−0.633112	−	0.774060i	$-0.718223\pi$
$563$	−30.0445	−1.26622	−0.633112	+	0.774060i	$0.718223\pi$
$564$	−10.4124	−0.438439
$565$	−7.92623	−0.333459
$566$	−0.452945	−0.0190387
$567$	0	0
$568$	14.1655	0.594372
$569$	4.31722	0.180987	0.0904936	−	0.995897i	$-0.471156\pi$
$569$	4.31722	0.180987	0.0904936	+	0.995897i	$0.471156\pi$
$570$	19.0779	0.799087
$571$	−24.5801	−1.02864	−0.514322	−	0.857597i	$-0.671956\pi$
$571$	−24.5801	−1.02864	−0.514322	+	0.857597i	$0.671956\pi$
$572$	0.767383	0.0320859
$573$	−12.3489	−0.515881
$574$	0	0
$575$	−8.73574	−0.364306
$576$	0.403401	0.0168084
$577$	−10.8024	−0.449711	−0.224856	−	0.974392i	$-0.572191\pi$
$577$	−10.8024	−0.449711	−0.224856	+	0.974392i	$0.572191\pi$
$578$	16.5013	0.686364
$579$	9.40596	0.390898
$580$	21.4930	0.892448
$581$	0	0
$582$	28.9638	1.20059
$583$	4.12150	0.170695
$584$	2.21520	0.0916657
$585$	3.21083	0.132751
$586$	8.36766	0.345665
$587$	−22.8859	−0.944601	−0.472300	−	0.881438i	$-0.656576\pi$
$587$	−22.8859	−0.944601	−0.472300	+	0.881438i	$0.656576\pi$
$588$	0	0
$589$	−8.82880	−0.363785
$590$	−34.9916	−1.44058
$591$	8.04747	0.331029
$592$	8.15656	0.335232
$593$	−20.1132	−0.825950	−0.412975	−	0.910742i	$-0.635510\pi$
$593$	−20.1132	−0.825950	−0.412975	+	0.910742i	$0.635510\pi$
$594$	1.71166	0.0702305
$595$	0	0
$596$	−19.6745	−0.805900
$597$	−3.16156	−0.129394
$598$	2.14761	0.0878222
$599$	−4.86620	−0.198827	−0.0994137	−	0.995046i	$-0.531697\pi$
$599$	−4.86620	−0.198827	−0.0994137	+	0.995046i	$0.531697\pi$
$600$	−16.1160	−0.657932
$601$	27.3566	1.11590	0.557950	−	0.829874i	$-0.311588\pi$
$601$	27.3566	1.11590	0.557950	+	0.829874i	$0.311588\pi$
$602$	0	0
$603$	6.06350	0.246925
$604$	2.56267	0.104274
$605$	−40.2947	−1.63821
$606$	−25.7756	−1.04706
$607$	−2.61609	−0.106184	−0.0530919	−	0.998590i	$-0.516908\pi$
$607$	−2.61609	−0.106184	−0.0530919	+	0.998590i	$0.516908\pi$
$608$	2.79029	0.113161
$609$	0	0
$610$	3.64497	0.147581
$611$	−12.1212	−0.490373
$612$	−0.284872	−0.0115153
$613$	−28.3739	−1.14601	−0.573006	−	0.819551i	$-0.694223\pi$
$613$	−28.3739	−1.14601	−0.573006	+	0.819551i	$0.694223\pi$
$614$	24.8636	1.00341
$615$	44.5590	1.79679
$616$	0	0
$617$	−0.343632	−0.0138341	−0.00691706	−	0.999976i	$-0.502202\pi$
$617$	−0.343632	−0.0138341	−0.00691706	+	0.999976i	$0.502202\pi$
$618$	−20.2877	−0.816089
$619$	35.8082	1.43925	0.719626	−	0.694362i	$-0.244313\pi$
$619$	35.8082	1.43925	0.719626	+	0.694362i	$0.244313\pi$
$620$	11.7268	0.470959
$621$	4.79029	0.192228
$622$	16.2246	0.650549
$623$	0	0
$624$	3.96197	0.158606
$625$	7.63448	0.305379
$626$	−0.957606	−0.0382736
$627$	−1.83934	−0.0734563
$628$	−16.4433	−0.656159
$629$	−5.75997	−0.229665
$630$	0	0
$631$	11.2216	0.446726	0.223363	−	0.974735i	$-0.428296\pi$
$631$	11.2216	0.446726	0.223363	+	0.974735i	$0.428296\pi$
$632$	−12.8923	−0.512828
$633$	3.02790	0.120348
$634$	−17.3051	−0.687275
$635$	6.16780	0.244762
$636$	21.2792	0.843775
$637$	0	0
$638$	−2.07218	−0.0820385
$639$	−5.71438	−0.226057
$640$	−3.70618	−0.146499
$641$	−44.4507	−1.75570	−0.877848	−	0.478939i	$-0.841021\pi$
$641$	−44.4507	−1.75570	−0.877848	+	0.478939i	$0.841021\pi$
$642$	31.0709	1.22627
$643$	33.8157	1.33356	0.666781	−	0.745254i	$-0.267672\pi$
$643$	33.8157	1.33356	0.666781	+	0.745254i	$0.267672\pi$
$644$	0	0
$645$	−43.2959	−1.70478
$646$	−1.97043	−0.0775257
$647$	−34.8325	−1.36941	−0.684703	−	0.728822i	$-0.740068\pi$
$647$	−34.8325	−1.36941	−0.684703	+	0.728822i	$0.740068\pi$
$648$	10.0475	0.394702
$649$	3.37361	0.132426
$650$	−18.7609	−0.735864
$651$	0	0
$652$	−19.0530	−0.746171
$653$	42.5042	1.66332	0.831660	−	0.555286i	$-0.187391\pi$
$653$	42.5042	1.66332	0.831660	+	0.555286i	$0.187391\pi$
$654$	19.3181	0.755398
$655$	−47.4566	−1.85428
$656$	6.51708	0.254449
$657$	−0.893614	−0.0348632
$658$	0	0
$659$	5.96821	0.232489	0.116244	−	0.993221i	$-0.462914\pi$
$659$	5.96821	0.232489	0.116244	+	0.993221i	$0.462914\pi$
$660$	2.44309	0.0950972
$661$	−44.2990	−1.72303	−0.861515	−	0.507732i	$-0.830484\pi$
$661$	−44.2990	−1.72303	−0.861515	+	0.507732i	$0.830484\pi$
$662$	−4.70208	−0.182751
$663$	−2.79785	−0.108659
$664$	4.38689	0.170244
$665$	0	0
$666$	−3.29036	−0.127499
$667$	−5.79924	−0.224547
$668$	11.6154	0.449413
$669$	6.48202	0.250609
$670$	−55.7074	−2.15216
$671$	−0.351419	−0.0135664
$672$	0	0
$673$	−29.5675	−1.13974	−0.569872	−	0.821733i	$-0.693007\pi$
$673$	−29.5675	−1.13974	−0.569872	+	0.821733i	$0.693007\pi$
$674$	34.5382	1.33036
$675$	−41.8467	−1.61068
$676$	−8.38779	−0.322607
$677$	36.8707	1.41706	0.708529	−	0.705682i	$-0.249359\pi$
$677$	36.8707	1.41706	0.708529	+	0.705682i	$0.249359\pi$
$678$	3.94546	0.151524
$679$	0	0
$680$	2.61721	0.100366
$681$	38.2112	1.46426
$682$	−1.13060	−0.0432930
$683$	−19.5932	−0.749713	−0.374857	−	0.927083i	$-0.622308\pi$
$683$	−19.5932	−0.749713	−0.374857	+	0.927083i	$0.622308\pi$
$684$	−1.12560	−0.0430385
$685$	−38.5008	−1.47104
$686$	0	0
$687$	26.6306	1.01602
$688$	−6.33234	−0.241418
$689$	24.7715	0.943721
$690$	6.83727	0.260290
$691$	17.5149	0.666296	0.333148	−	0.942874i	$-0.391889\pi$
$691$	17.5149	0.666296	0.333148	+	0.942874i	$0.391889\pi$
$692$	−12.1951	−0.463587
$693$	0	0
$694$	3.01700	0.114524
$695$	−67.2804	−2.55209
$696$	−10.6986	−0.405530
$697$	−4.60220	−0.174321
$698$	19.1069	0.723206
$699$	−10.3042	−0.389739
$700$	0	0
$701$	3.82144	0.144334	0.0721669	−	0.997393i	$-0.477009\pi$
$701$	3.82144	0.144334	0.0721669	+	0.997393i	$0.477009\pi$
$702$	10.2877	0.388282
$703$	−22.7591	−0.858377
$704$	0.357320	0.0134670
$705$	−38.5900	−1.45338
$706$	4.99515	0.187995
$707$	0	0
$708$	17.4178	0.654603
$709$	17.0586	0.640651	0.320325	−	0.947308i	$-0.396208\pi$
$709$	17.0586	0.640651	0.320325	+	0.947308i	$0.396208\pi$
$710$	52.4999	1.97029
$711$	5.20076	0.195044
$712$	−5.20971	−0.195242
$713$	−3.16412	−0.118497
$714$	0	0
$715$	2.84405	0.106362
$716$	2.15107	0.0803893
$717$	19.5109	0.728648
$718$	17.8137	0.664801
$719$	9.51320	0.354783	0.177391	−	0.984140i	$-0.443234\pi$
$719$	9.51320	0.354783	0.177391	+	0.984140i	$0.443234\pi$
$720$	1.49507	0.0557181
$721$	0	0
$722$	11.2143	0.417353
$723$	2.64102	0.0982205
$724$	15.7236	0.584363
$725$	50.6607	1.88149
$726$	20.0576	0.744407
$727$	10.6189	0.393835	0.196917	−	0.980420i	$-0.436907\pi$
$727$	10.6189	0.393835	0.196917	+	0.980420i	$0.436907\pi$
$728$	0	0
$729$	22.4586	0.831802
$730$	8.20993	0.303863
$731$	4.47175	0.165394
$732$	−1.81436	−0.0670608
$733$	−21.0303	−0.776773	−0.388386	−	0.921497i	$-0.626967\pi$
$733$	−21.0303	−0.776773	−0.388386	+	0.921497i	$0.626967\pi$
$734$	8.99244	0.331917
$735$	0	0
$736$	1.00000	0.0368605
$737$	5.37086	0.197838
$738$	−2.62899	−0.0967746
$739$	−35.2670	−1.29732	−0.648659	−	0.761079i	$-0.724670\pi$
$739$	−35.2670	−1.29732	−0.648659	+	0.761079i	$0.724670\pi$
$740$	30.2296	1.11126
$741$	−11.0550	−0.406117
$742$	0	0
$743$	−7.12534	−0.261403	−0.130702	−	0.991422i	$-0.541723\pi$
$743$	−7.12534	−0.261403	−0.130702	+	0.991422i	$0.541723\pi$
$744$	−5.83727	−0.214005
$745$	−72.9173	−2.67148
$746$	−9.61909	−0.352180
$747$	−1.76967	−0.0647489
$748$	−0.252331	−0.00922613
$749$	0	0
$750$	−25.5423	−0.932672
$751$	11.8613	0.432827	0.216413	−	0.976302i	$-0.430564\pi$
$751$	11.8613	0.432827	0.216413	+	0.976302i	$0.430564\pi$
$752$	−5.64407	−0.205818
$753$	−31.3922	−1.14400
$754$	−12.4545	−0.453565
$755$	9.49770	0.345657
$756$	0	0
$757$	−42.2269	−1.53476	−0.767382	−	0.641190i	$-0.778441\pi$
$757$	−42.2269	−1.53476	−0.767382	+	0.641190i	$0.778441\pi$
$758$	4.72457	0.171604
$759$	−0.659195	−0.0239273
$760$	10.3413	0.375118
$761$	39.2591	1.42314	0.711570	−	0.702615i	$-0.247984\pi$
$761$	39.2591	1.42314	0.711570	+	0.702615i	$0.247984\pi$
$762$	−3.07016	−0.111220
$763$	0	0
$764$	−6.69376	−0.242172
$765$	−1.05579	−0.0381720
$766$	5.38501	0.194568
$767$	20.2765	0.732141
$768$	1.84483	0.0665696
$769$	29.8728	1.07724	0.538621	−	0.842548i	$-0.318946\pi$
$769$	29.8728	1.07724	0.538621	+	0.842548i	$0.318946\pi$
$770$	0	0
$771$	51.2253	1.84484
$772$	5.09855	0.183501
$773$	39.9927	1.43844	0.719219	−	0.694783i	$-0.244500\pi$
$773$	39.9927	1.43844	0.719219	+	0.694783i	$0.244500\pi$
$774$	2.55447	0.0918185
$775$	27.6409	0.992892
$776$	15.7000	0.563597
$777$	0	0
$778$	−33.8055	−1.21199
$779$	−18.1845	−0.651528
$780$	14.6838	0.525763
$781$	−5.06162	−0.181119
$782$	−0.706176	−0.0252528
$783$	−27.7800	−0.992777
$784$	0	0
$785$	−60.9418	−2.17511
$786$	23.6226	0.842589
$787$	14.4288	0.514332	0.257166	−	0.966367i	$-0.417211\pi$
$787$	14.4288	0.514332	0.257166	+	0.966367i	$0.417211\pi$
$788$	4.36217	0.155396
$789$	20.6754	0.736065
$790$	−47.7811	−1.69998
$791$	0	0
$792$	−0.144143	−0.00512190
$793$	−2.11214	−0.0750043
$794$	9.32685	0.330998
$795$	78.8644	2.79703
$796$	−1.71374	−0.0607419
$797$	−20.9087	−0.740626	−0.370313	−	0.928907i	$-0.620749\pi$
$797$	−20.9087	−0.740626	−0.370313	+	0.928907i	$0.620749\pi$
$798$	0	0
$799$	3.98571	0.141004
$800$	−8.73574	−0.308855
$801$	2.10160	0.0742564
$802$	13.9152	0.491363
$803$	−0.791536	−0.0279327
$804$	27.7296	0.977946
$805$	0	0
$806$	−6.79529	−0.239354
$807$	54.4417	1.91644
$808$	−13.9718	−0.491527
$809$	26.3090	0.924976	0.462488	−	0.886626i	$-0.346957\pi$
$809$	26.3090	0.924976	0.462488	+	0.886626i	$0.346957\pi$
$810$	37.2377	1.30840
$811$	−2.50790	−0.0880643	−0.0440322	−	0.999030i	$-0.514020\pi$
$811$	−2.50790	−0.0880643	−0.0440322	+	0.999030i	$0.514020\pi$
$812$	0	0
$813$	−35.1321	−1.23214
$814$	−2.91450	−0.102153
$815$	−70.6136	−2.47349
$816$	−1.30278	−0.0456063
$817$	17.6690	0.618162
$818$	−25.1840	−0.880540
$819$	0	0
$820$	24.1534	0.843475
$821$	−22.8349	−0.796945	−0.398472	−	0.917180i	$-0.630459\pi$
$821$	−22.8349	−0.796945	−0.398472	+	0.917180i	$0.630459\pi$
$822$	19.1646	0.668443
$823$	−9.72818	−0.339103	−0.169552	−	0.985521i	$-0.554232\pi$
$823$	−9.72818	−0.339103	−0.169552	+	0.985521i	$0.554232\pi$
$824$	−10.9970	−0.383100
$825$	5.75856	0.200487
$826$	0	0
$827$	53.9493	1.87600	0.938000	−	0.346635i	$-0.112676\pi$
$827$	53.9493	1.87600	0.938000	+	0.346635i	$0.112676\pi$
$828$	−0.403401	−0.0140191
$829$	38.6876	1.34368	0.671838	−	0.740698i	$-0.265505\pi$
$829$	38.6876	1.34368	0.671838	+	0.740698i	$0.265505\pi$
$830$	16.2586	0.564343
$831$	6.99410	0.242623
$832$	2.14761	0.0744549
$833$	0	0
$834$	33.4903	1.15967
$835$	43.0487	1.48976
$836$	−0.997025	−0.0344828
$837$	−15.1570	−0.523904
$838$	−26.6155	−0.919416
$839$	33.0600	1.14136	0.570680	−	0.821173i	$-0.306680\pi$
$839$	33.0600	1.14136	0.570680	+	0.821173i	$0.306680\pi$
$840$	0	0
$841$	4.63117	0.159695
$842$	−13.2888	−0.457961
$843$	−26.8403	−0.924430
$844$	1.64129	0.0564956
$845$	−31.0866	−1.06941
$846$	2.27682	0.0782787
$847$	0	0
$848$	11.5345	0.396096
$849$	0.835606	0.0286779
$850$	6.16897	0.211594
$851$	−8.15656	−0.279603
$852$	−26.1330	−0.895301
$853$	22.0257	0.754145	0.377072	−	0.926184i	$-0.376931\pi$
$853$	22.0257	0.754145	0.377072	+	0.926184i	$0.376931\pi$
$854$	0	0
$855$	−4.17168	−0.142669
$856$	16.8421	0.575652
$857$	19.6258	0.670405	0.335203	−	0.942146i	$-0.391195\pi$
$857$	19.6258	0.670405	0.335203	+	0.942146i	$0.391195\pi$
$858$	−1.41569	−0.0483309
$859$	17.5735	0.599600	0.299800	−	0.954002i	$-0.403080\pi$
$859$	17.5735	0.599600	0.299800	+	0.954002i	$0.403080\pi$
$860$	−23.4688	−0.800279
$861$	0	0
$862$	7.64704	0.260459
$863$	−40.9092	−1.39257	−0.696283	−	0.717767i	$-0.745164\pi$
$863$	−40.9092	−1.39257	−0.696283	+	0.717767i	$0.745164\pi$
$864$	4.79029	0.162969
$865$	−45.1971	−1.53675
$866$	−15.5384	−0.528018
$867$	−30.4421	−1.03387
$868$	0	0
$869$	4.60668	0.156271
$870$	−39.6509	−1.34429
$871$	32.2806	1.09379
$872$	10.4715	0.354609
$873$	−6.33339	−0.214353
$874$	−2.79029	−0.0943828
$875$	0	0
$876$	−4.08667	−0.138076
$877$	−19.8401	−0.669953	−0.334977	−	0.942226i	$-0.608728\pi$
$877$	−19.8401	−0.669953	−0.334977	+	0.942226i	$0.608728\pi$
$878$	36.9647	1.24750
$879$	−15.4369	−0.520674
$880$	1.32429	0.0446418
$881$	9.43289	0.317802	0.158901	−	0.987295i	$-0.449205\pi$
$881$	9.43289	0.317802	0.158901	+	0.987295i	$0.449205\pi$
$882$	0	0
$883$	−19.2276	−0.647061	−0.323530	−	0.946218i	$-0.604870\pi$
$883$	−19.2276	−0.647061	−0.323530	+	0.946218i	$0.604870\pi$
$884$	−1.51659	−0.0510084
$885$	64.5536	2.16995
$886$	−35.8529	−1.20450
$887$	−29.9117	−1.00434	−0.502168	−	0.864770i	$-0.667464\pi$
$887$	−29.9117	−1.00434	−0.502168	+	0.864770i	$0.667464\pi$
$888$	−15.0475	−0.504960
$889$	0	0
$890$	−19.3081	−0.647210
$891$	−3.59016	−0.120275
$892$	3.51361	0.117644
$893$	15.7486	0.527006
$894$	36.2962	1.21393
$895$	7.97224	0.266483
$896$	0	0
$897$	−3.96197	−0.132286
$898$	−26.6931	−0.890759
$899$	18.3495	0.611990
$900$	3.52400	0.117467
$901$	−8.14538	−0.271362
$902$	−2.32868	−0.0775366
$903$	0	0
$904$	2.13865	0.0711306
$905$	58.2744	1.93711
$906$	−4.72769	−0.157067
$907$	−21.4948	−0.713723	−0.356862	−	0.934157i	$-0.616153\pi$
$907$	−21.4948	−0.713723	−0.356862	+	0.934157i	$0.616153\pi$
$908$	20.7126	0.687371
$909$	5.63624	0.186942
$910$	0	0
$911$	−7.80285	−0.258520	−0.129260	−	0.991611i	$-0.541260\pi$
$911$	−7.80285	−0.258520	−0.129260	+	0.991611i	$0.541260\pi$
$912$	−5.14761	−0.170454
$913$	−1.56752	−0.0518774
$914$	37.1541	1.22895
$915$	−6.72435	−0.222300
$916$	14.4353	0.476954
$917$	0	0
$918$	−3.38279	−0.111649
$919$	13.4341	0.443150	0.221575	−	0.975143i	$-0.428880\pi$
$919$	13.4341	0.443150	0.221575	+	0.975143i	$0.428880\pi$
$920$	3.70618	0.122189
$921$	−45.8692	−1.51144
$922$	−8.56316	−0.282013
$923$	−30.4219	−1.00135
$924$	0	0
$925$	71.2536	2.34280
$926$	32.2691	1.06043
$927$	4.43621	0.145704
$928$	−5.79924	−0.190369
$929$	13.3685	0.438606	0.219303	−	0.975657i	$-0.429622\pi$
$929$	13.3685	0.438606	0.219303	+	0.975657i	$0.429622\pi$
$930$	−21.6339	−0.709405
$931$	0	0
$932$	−5.58543	−0.182957
$933$	−29.9317	−0.979920
$934$	−31.5664	−1.03288
$935$	−0.935183	−0.0305837
$936$	−0.866346	−0.0283174
$937$	28.2477	0.922814	0.461407	−	0.887189i	$-0.347345\pi$
$937$	28.2477	0.922814	0.461407	+	0.887189i	$0.347345\pi$
$938$	0	0
$939$	1.76662	0.0576515
$940$	−20.9179	−0.682267
$941$	42.8686	1.39748	0.698739	−	0.715377i	$-0.253745\pi$
$941$	42.8686	1.39748	0.698739	+	0.715377i	$0.253745\pi$
$942$	30.3351	0.988372
$943$	−6.51708	−0.212225
$944$	9.44143	0.307292
$945$	0	0
$946$	2.26267	0.0735658
$947$	−43.1245	−1.40136	−0.700679	−	0.713477i	$-0.747119\pi$
$947$	−43.1245	−1.40136	−0.700679	+	0.713477i	$0.747119\pi$
$948$	23.7841	0.772472
$949$	−4.75738	−0.154431
$950$	24.3752	0.790837
$951$	31.9251	1.03524
$952$	0	0
$953$	6.64034	0.215102	0.107551	−	0.994200i	$-0.465699\pi$
$953$	6.64034	0.215102	0.107551	+	0.994200i	$0.465699\pi$
$954$	−4.65302	−0.150647
$955$	−24.8083	−0.802776
$956$	10.5760	0.342052
$957$	3.82283	0.123574
$958$	−4.55511	−0.147169
$959$	0	0
$960$	6.83727	0.220672
$961$	−20.9883	−0.677043
$962$	−17.5171	−0.564773
$963$	−6.79412	−0.218937
$964$	1.43158	0.0461080
$965$	18.8961	0.608288
$966$	0	0
$967$	−18.5854	−0.597667	−0.298833	−	0.954305i	$-0.596598\pi$
$967$	−18.5854	−0.597667	−0.298833	+	0.954305i	$0.596598\pi$
$968$	10.8723	0.349450
$969$	3.63512	0.116777
$970$	58.1870	1.86827
$971$	9.92201	0.318413	0.159206	−	0.987245i	$-0.449107\pi$
$971$	9.92201	0.318413	0.159206	+	0.987245i	$0.449107\pi$
$972$	−4.16502	−0.133593
$973$	0	0
$974$	28.2113	0.903949
$975$	34.6108	1.10843
$976$	−0.983485	−0.0314806
$977$	0.716741	0.0229306	0.0114653	−	0.999934i	$-0.496350\pi$
$977$	0.716741	0.0229306	0.0114653	+	0.999934i	$0.496350\pi$
$978$	35.1495	1.12396
$979$	1.86153	0.0594949
$980$	0	0
$981$	−4.22420	−0.134868
$982$	−21.1781	−0.675822
$983$	23.1139	0.737220	0.368610	−	0.929584i	$-0.379834\pi$
$983$	23.1139	0.737220	0.368610	+	0.929584i	$0.379834\pi$
$984$	−12.0229	−0.383276
$985$	16.1670	0.515123
$986$	4.09528	0.130420
$987$	0	0
$988$	−5.99244	−0.190645
$989$	6.33234	0.201357
$990$	−0.534220	−0.0169786
$991$	−25.6970	−0.816291	−0.408145	−	0.912917i	$-0.633824\pi$
$991$	−25.6970	−0.816291	−0.408145	+	0.912917i	$0.633824\pi$
$992$	−3.16412	−0.100461
$993$	8.67454	0.275278
$994$	0	0
$995$	−6.35142	−0.201354
$996$	−8.09306	−0.256438
$997$	−0.923377	−0.0292436	−0.0146218	−	0.999893i	$-0.504654\pi$
$997$	−0.923377	−0.0292436	−0.0146218	+	0.999893i	$0.504654\pi$
$998$	−41.3353	−1.30845
$999$	−39.0723	−1.23619

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.2.a.r.1.4		4
7.3	odd	6		322.2.e.d.93.4	✓	8
7.5	odd	6		322.2.e.d.277.4	yes	8
7.6	odd	2		2254.2.a.u.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.e.d.93.4	✓	8	7.3	odd	6
322.2.e.d.277.4	yes	8	7.5	odd	6
2254.2.a.r.1.4		4	1.1	even	1	trivial
2254.2.a.u.1.1		4	7.6	odd	2

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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.84483 q^{3} +1.00000 q^{4} +3.70618 q^{5} -1.84483 q^{6} -1.00000 q^{8} +0.403401 q^{9} -3.70618 q^{10} +0.357320 q^{11} +1.84483 q^{12} +2.14761 q^{13} +6.83727 q^{15} +1.00000 q^{16} -0.706176 q^{17} -0.403401 q^{18} -2.79029 q^{19} +3.70618 q^{20} -0.357320 q^{22} -1.00000 q^{23} -1.84483 q^{24} +8.73574 q^{25} -2.14761 q^{26} -4.79029 q^{27} +5.79924 q^{29} -6.83727 q^{30} +3.16412 q^{31} -1.00000 q^{32} +0.659195 q^{33} +0.706176 q^{34} +0.403401 q^{36} +8.15656 q^{37} +2.79029 q^{38} +3.96197 q^{39} -3.70618 q^{40} +6.51708 q^{41} -6.33234 q^{43} +0.357320 q^{44} +1.49507 q^{45} +1.00000 q^{46} -5.64407 q^{47} +1.84483 q^{48} -8.73574 q^{50} -1.30278 q^{51} +2.14761 q^{52} +11.5345 q^{53} +4.79029 q^{54} +1.32429 q^{55} -5.14761 q^{57} -5.79924 q^{58} +9.44143 q^{59} +6.83727 q^{60} -0.983485 q^{61} -3.16412 q^{62} +1.00000 q^{64} +7.95941 q^{65} -0.659195 q^{66} +15.0310 q^{67} -0.706176 q^{68} -1.84483 q^{69} -14.1655 q^{71} -0.403401 q^{72} -2.21520 q^{73} -8.15656 q^{74} +16.1160 q^{75} -2.79029 q^{76} -3.96197 q^{78} +12.8923 q^{79} +3.70618 q^{80} -10.0475 q^{81} -6.51708 q^{82} -4.38689 q^{83} -2.61721 q^{85} +6.33234 q^{86} +10.6986 q^{87} -0.357320 q^{88} +5.20971 q^{89} -1.49507 q^{90} -1.00000 q^{92} +5.83727 q^{93} +5.64407 q^{94} -10.3413 q^{95} -1.84483 q^{96} -15.7000 q^{97} +0.144143 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - q^{3} + 4 q^{4} + 5 q^{5} + q^{6} - 4 q^{8} - q^{9} - 5 q^{10} - 2 q^{11} - q^{12} - 7 q^{13} - 5 q^{15} + 4 q^{16} + 7 q^{17} + q^{18} + q^{19} + 5 q^{20} + 2 q^{22} - 4 q^{23} + q^{24}+ \cdots - 35 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 - q^3 + 4 * q^4 + 5 * q^5 + q^6 - 4 * q^8 - q^9 - 5 * q^10 - 2 * q^11 - q^12 - 7 * q^13 - 5 * q^15 + 4 * q^16 + 7 * q^17 + q^18 + q^19 + 5 * q^20 + 2 * q^22 - 4 * q^23 + q^24 + 19 * q^25 + 7 * q^26 - 7 * q^27 - 6 * q^29 + 5 * q^30 + 4 * q^31 - 4 * q^32 + 13 * q^33 - 7 * q^34 - q^36 - q^38 + 19 * q^39 - 5 * q^40 + 15 * q^41 - 12 * q^43 - 2 * q^44 + 25 * q^45 + 4 * q^46 + 15 * q^47 - q^48 - 19 * q^50 + 2 * q^51 - 7 * q^52 + 21 * q^53 + 7 * q^54 - 12 * q^55 - 5 * q^57 + 6 * q^58 + 32 * q^59 - 5 * q^60 + 3 * q^61 - 4 * q^62 + 4 * q^64 - 6 * q^65 - 13 * q^66 + 13 * q^67 + 7 * q^68 + q^69 - 7 * q^71 + q^72 + 16 * q^73 - 6 * q^75 + q^76 - 19 * q^78 + 3 * q^79 + 5 * q^80 - 15 * q^82 - 8 * q^83 - 24 * q^85 + 12 * q^86 + 9 * q^87 + 2 * q^88 + 33 * q^89 - 25 * q^90 - 4 * q^92 - 9 * q^93 - 15 * q^94 - 11 * q^95 + q^96 + 12 * q^97 - 35 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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