LMFDB - Embedded newform 2275.2.a.y.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2275,2,Mod(1,2275)] 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("2275.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2275 = 5^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2275.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.1659664598$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 8x^{5} - 2x^{4} + 16x^{3} + 5x^{2} - 6x - 2 $ x^7 - 8x^5 - 2x^4 + 16x^3 + 5x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 455)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.51372$ of defining polynomial
Character	$\chi$	$=$	2275.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.51372 q^{2} +1.55652 q^{3} +0.291349 q^{4} +2.35614 q^{6} -1.00000 q^{7} -2.58642 q^{8} -0.577244 q^{9} -1.70619 q^{11} +0.453490 q^{12} +1.00000 q^{13} -1.51372 q^{14} -4.49781 q^{16} -4.39648 q^{17} -0.873786 q^{18} -5.08214 q^{19} -1.55652 q^{21} -2.58270 q^{22} -5.25967 q^{23} -4.02582 q^{24} +1.51372 q^{26} -5.56805 q^{27} -0.291349 q^{28} +0.226354 q^{29} -1.70807 q^{31} -1.63559 q^{32} -2.65572 q^{33} -6.65504 q^{34} -0.168179 q^{36} +5.41532 q^{37} -7.69293 q^{38} +1.55652 q^{39} -0.800562 q^{41} -2.35614 q^{42} +1.43855 q^{43} -0.497097 q^{44} -7.96167 q^{46} +9.52006 q^{47} -7.00094 q^{48} +1.00000 q^{49} -6.84321 q^{51} +0.291349 q^{52} -0.943553 q^{53} -8.42847 q^{54} +2.58642 q^{56} -7.91045 q^{57} +0.342636 q^{58} +13.3456 q^{59} +5.89886 q^{61} -2.58553 q^{62} +0.577244 q^{63} +6.51980 q^{64} -4.02002 q^{66} -9.53119 q^{67} -1.28091 q^{68} -8.18678 q^{69} -1.82999 q^{71} +1.49300 q^{72} -6.40325 q^{73} +8.19727 q^{74} -1.48067 q^{76} +1.70619 q^{77} +2.35614 q^{78} +17.3261 q^{79} -6.93506 q^{81} -1.21183 q^{82} -12.7108 q^{83} -0.453490 q^{84} +2.17756 q^{86} +0.352324 q^{87} +4.41293 q^{88} -17.9355 q^{89} -1.00000 q^{91} -1.53240 q^{92} -2.65864 q^{93} +14.4107 q^{94} -2.54583 q^{96} -6.64792 q^{97} +1.51372 q^{98} +0.984890 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{3} + 2 q^{4} - 8 q^{6} - 7 q^{7} + 6 q^{8} + q^{9} - 6 q^{11} - 3 q^{12} + 7 q^{13} - 4 q^{16} - 2 q^{17} + q^{18} - 10 q^{19} - 2 q^{21} - 18 q^{22} - 10 q^{23} - 5 q^{24} + 8 q^{27} - 2 q^{28}+ \cdots - 18 q^{99}+O(q^{100}) $ 7 * q + 2 * q^3 + 2 * q^4 - 8 * q^6 - 7 * q^7 + 6 * q^8 + q^9 - 6 * q^11 - 3 * q^12 + 7 * q^13 - 4 * q^16 - 2 * q^17 + q^18 - 10 * q^19 - 2 * q^21 - 18 * q^22 - 10 * q^23 - 5 * q^24 + 8 * q^27 - 2 * q^28 - 26 * q^29 - 6 * q^31 + 7 * q^32 - 4 * q^33 - q^34 - 6 * q^36 + 4 * q^37 + q^38 + 2 * q^39 - 28 * q^41 + 8 * q^42 - 2 * q^43 - 18 * q^46 + 20 * q^47 - 20 * q^48 + 7 * q^49 - 32 * q^51 + 2 * q^52 - 8 * q^53 + 11 * q^54 - 6 * q^56 - 14 * q^57 + 12 * q^58 - 28 * q^59 - 26 * q^61 + 22 * q^62 - q^63 - 8 * q^64 + 2 * q^66 - 2 * q^67 + 2 * q^68 - 22 * q^69 - 14 * q^71 - 4 * q^72 + 2 * q^73 - 12 * q^74 - 36 * q^76 + 6 * q^77 - 8 * q^78 + 14 * q^79 - 33 * q^81 - 19 * q^82 + 28 * q^83 + 3 * q^84 - 4 * q^86 - 22 * q^87 - 24 * q^88 - 30 * q^89 - 7 * q^91 - 42 * q^92 - 2 * q^93 + 14 * q^94 + 18 * q^97 - 18 * 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 $

$2$	1.51372	1.07036	0.535181	−	0.844737i	$-0.320243\pi$
$2$	1.51372	1.07036	0.535181	+	0.844737i	$0.320243\pi$
$3$	1.55652	0.898657	0.449329	−	0.893366i	$-0.351663\pi$
$3$	1.55652	0.898657	0.449329	+	0.893366i	$0.351663\pi$
$4$	0.291349	0.145674
$5$	0	0
$5$	0	0
$6$	2.35614	0.961889
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.58642	−0.914438
$9$	−0.577244	−0.192415
$10$	0	0
$11$	−1.70619	−0.514436	−0.257218	−	0.966353i	$-0.582806\pi$
$11$	−1.70619	−0.514436	−0.257218	+	0.966353i	$0.582806\pi$
$12$	0.453490	0.130911
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−1.51372	−0.404559
$15$	0	0
$16$	−4.49781	−1.12445
$17$	−4.39648	−1.06630	−0.533151	−	0.846020i	$-0.678992\pi$
$17$	−4.39648	−1.06630	−0.533151	+	0.846020i	$0.678992\pi$
$18$	−0.873786	−0.205953
$19$	−5.08214	−1.16592	−0.582961	−	0.812500i	$-0.698106\pi$
$19$	−5.08214	−1.16592	−0.582961	+	0.812500i	$0.698106\pi$
$20$	0	0
$21$	−1.55652	−0.339661
$22$	−2.58270	−0.550633
$23$	−5.25967	−1.09672	−0.548358	−	0.836243i	$-0.684747\pi$
$23$	−5.25967	−1.09672	−0.548358	+	0.836243i	$0.684747\pi$
$24$	−4.02582	−0.821766
$25$	0	0
$26$	1.51372	0.296865
$27$	−5.56805	−1.07157
$28$	−0.291349	−0.0550597
$29$	0.226354	0.0420329	0.0210164	−	0.999779i	$-0.493310\pi$
$29$	0.226354	0.0420329	0.0210164	+	0.999779i	$0.493310\pi$
$30$	0	0
$31$	−1.70807	−0.306778	−0.153389	−	0.988166i	$-0.549019\pi$
$31$	−1.70807	−0.306778	−0.153389	+	0.988166i	$0.549019\pi$
$32$	−1.63559	−0.289134
$33$	−2.65572	−0.462302
$34$	−6.65504	−1.14133
$35$	0	0
$36$	−0.168179	−0.0280299
$37$	5.41532	0.890273	0.445136	−	0.895463i	$-0.353155\pi$
$37$	5.41532	0.890273	0.445136	+	0.895463i	$0.353155\pi$
$38$	−7.69293	−1.24796
$39$	1.55652	0.249243
$40$	0	0
$41$	−0.800562	−0.125027	−0.0625134	−	0.998044i	$-0.519912\pi$
$41$	−0.800562	−0.125027	−0.0625134	+	0.998044i	$0.519912\pi$
$42$	−2.35614	−0.363560
$43$	1.43855	0.219377	0.109688	−	0.993966i	$-0.465015\pi$
$43$	1.43855	0.219377	0.109688	+	0.993966i	$0.465015\pi$
$44$	−0.497097	−0.0749401
$45$	0	0
$46$	−7.96167	−1.17388
$47$	9.52006	1.38864	0.694322	−	0.719665i	$-0.255704\pi$
$47$	9.52006	1.38864	0.694322	+	0.719665i	$0.255704\pi$
$48$	−7.00094	−1.01050
$49$	1.00000	0.142857
$50$	0	0
$51$	−6.84321	−0.958241
$52$	0.291349	0.0404028
$53$	−0.943553	−0.129607	−0.0648035	−	0.997898i	$-0.520642\pi$
$53$	−0.943553	−0.129607	−0.0648035	+	0.997898i	$0.520642\pi$
$54$	−8.42847	−1.14697
$55$	0	0
$56$	2.58642	0.345625
$57$	−7.91045	−1.04776
$58$	0.342636	0.0449904
$59$	13.3456	1.73745	0.868723	−	0.495298i	$-0.164941\pi$
$59$	13.3456	1.73745	0.868723	+	0.495298i	$0.164941\pi$
$60$	0	0
$61$	5.89886	0.755271	0.377636	−	0.925954i	$-0.376737\pi$
$61$	5.89886	0.755271	0.377636	+	0.925954i	$0.376737\pi$
$62$	−2.58553	−0.328363
$63$	0.577244	0.0727259
$64$	6.51980	0.814975
$65$	0	0
$66$	−4.02002	−0.494830
$67$	−9.53119	−1.16442	−0.582210	−	0.813039i	$-0.697812\pi$
$67$	−9.53119	−1.16442	−0.582210	+	0.813039i	$0.697812\pi$
$68$	−1.28091	−0.155333
$69$	−8.18678	−0.985573
$70$	0	0
$71$	−1.82999	−0.217180	−0.108590	−	0.994087i	$-0.534634\pi$
$71$	−1.82999	−0.217180	−0.108590	+	0.994087i	$0.534634\pi$
$72$	1.49300	0.175951
$73$	−6.40325	−0.749444	−0.374722	−	0.927137i	$-0.622262\pi$
$73$	−6.40325	−0.749444	−0.374722	+	0.927137i	$0.622262\pi$
$74$	8.19727	0.952914
$75$	0	0
$76$	−1.48067	−0.169845
$77$	1.70619	0.194439
$78$	2.35614	0.266780
$79$	17.3261	1.94934	0.974668	−	0.223656i	$-0.0717991\pi$
$79$	17.3261	1.94934	0.974668	+	0.223656i	$0.0717991\pi$
$80$	0	0
$81$	−6.93506	−0.770562
$82$	−1.21183	−0.133824
$83$	−12.7108	−1.39520	−0.697598	−	0.716489i	$-0.745748\pi$
$83$	−12.7108	−1.39520	−0.697598	+	0.716489i	$0.745748\pi$
$84$	−0.453490	−0.0494798
$85$	0	0
$86$	2.17756	0.234813
$87$	0.352324	0.0377731
$88$	4.41293	0.470420
$89$	−17.9355	−1.90116	−0.950581	−	0.310477i	$-0.899511\pi$
$89$	−17.9355	−1.90116	−0.950581	+	0.310477i	$0.899511\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	−1.53240	−0.159763
$93$	−2.65864	−0.275688
$94$	14.4107	1.48635
$95$	0	0
$96$	−2.54583	−0.259833
$97$	−6.64792	−0.674994	−0.337497	−	0.941327i	$-0.609580\pi$
$97$	−6.64792	−0.674994	−0.337497	+	0.941327i	$0.609580\pi$
$98$	1.51372	0.152909
$99$	0.984890	0.0989851
$100$	0	0
$101$	−16.7051	−1.66222	−0.831108	−	0.556111i	$-0.812293\pi$
$101$	−16.7051	−1.66222	−0.831108	+	0.556111i	$0.812293\pi$
$102$	−10.3587	−1.02566
$103$	3.98923	0.393070	0.196535	−	0.980497i	$-0.437031\pi$
$103$	3.98923	0.393070	0.196535	+	0.980497i	$0.437031\pi$
$104$	−2.58642	−0.253619
$105$	0	0
$106$	−1.42828	−0.138726
$107$	7.64615	0.739181	0.369590	−	0.929195i	$-0.379498\pi$
$107$	7.64615	0.739181	0.369590	+	0.929195i	$0.379498\pi$
$108$	−1.62224	−0.156101
$109$	−4.48989	−0.430054	−0.215027	−	0.976608i	$-0.568984\pi$
$109$	−4.48989	−0.430054	−0.215027	+	0.976608i	$0.568984\pi$
$110$	0	0
$111$	8.42905	0.800050
$112$	4.49781	0.425003
$113$	1.06988	0.100646	0.0503229	−	0.998733i	$-0.483975\pi$
$113$	1.06988	0.100646	0.0503229	+	0.998733i	$0.483975\pi$
$114$	−11.9742	−1.12149
$115$	0	0
$116$	0.0659479	0.00612311
$117$	−0.577244	−0.0533663
$118$	20.2015	1.85970
$119$	4.39648	0.403024
$120$	0	0
$121$	−8.08891	−0.735355
$122$	8.92922	0.808414
$123$	−1.24609	−0.112356
$124$	−0.497643	−0.0446896
$125$	0	0
$126$	0.873786	0.0778431
$127$	−1.85900	−0.164960	−0.0824798	−	0.996593i	$-0.526284\pi$
$127$	−1.85900	−0.164960	−0.0824798	+	0.996593i	$0.526284\pi$
$128$	13.1403	1.16145
$129$	2.23913	0.197145
$130$	0	0
$131$	−18.5414	−1.61997	−0.809983	−	0.586454i	$-0.800524\pi$
$131$	−18.5414	−1.61997	−0.809983	+	0.586454i	$0.800524\pi$
$132$	−0.773741	−0.0673455
$133$	5.08214	0.440677
$134$	−14.4275	−1.24635
$135$	0	0
$136$	11.3711	0.975067
$137$	1.06152	0.0906920	0.0453460	−	0.998971i	$-0.485561\pi$
$137$	1.06152	0.0906920	0.0453460	+	0.998971i	$0.485561\pi$
$138$	−12.3925	−1.05492
$139$	6.66480	0.565301	0.282651	−	0.959223i	$-0.408786\pi$
$139$	6.66480	0.565301	0.282651	+	0.959223i	$0.408786\pi$
$140$	0	0
$141$	14.8182	1.24791
$142$	−2.77009	−0.232461
$143$	−1.70619	−0.142679
$144$	2.59634	0.216361
$145$	0	0
$146$	−9.69273	−0.802176
$147$	1.55652	0.128380
$148$	1.57774	0.129690
$149$	−4.77917	−0.391525	−0.195762	−	0.980651i	$-0.562718\pi$
$149$	−4.77917	−0.391525	−0.195762	+	0.980651i	$0.562718\pi$
$150$	0	0
$151$	−12.5301	−1.01968	−0.509842	−	0.860268i	$-0.670296\pi$
$151$	−12.5301	−1.01968	−0.509842	+	0.860268i	$0.670296\pi$
$152$	13.1445	1.06616
$153$	2.53784	0.205172
$154$	2.58270	0.208120
$155$	0	0
$156$	0.453490	0.0363083
$157$	6.77795	0.540939	0.270469	−	0.962729i	$-0.412821\pi$
$157$	6.77795	0.540939	0.270469	+	0.962729i	$0.412821\pi$
$158$	26.2268	2.08650
$159$	−1.46866	−0.116472
$160$	0	0
$161$	5.25967	0.414520
$162$	−10.4977	−0.824780
$163$	9.13986	0.715889	0.357945	−	0.933743i	$-0.383478\pi$
$163$	9.13986	0.715889	0.357945	+	0.933743i	$0.383478\pi$
$164$	−0.233243	−0.0182132
$165$	0	0
$166$	−19.2407	−1.49336
$167$	14.9071	1.15355	0.576774	−	0.816904i	$-0.304311\pi$
$167$	14.9071	1.15355	0.576774	+	0.816904i	$0.304311\pi$
$168$	4.02582	0.310598
$169$	1.00000	0.0769231
$170$	0	0
$171$	2.93363	0.224341
$172$	0.419120	0.0319576
$173$	11.8631	0.901935	0.450968	−	0.892540i	$-0.351079\pi$
$173$	11.8631	0.901935	0.450968	+	0.892540i	$0.351079\pi$
$174$	0.533320	0.0404309
$175$	0	0
$176$	7.67413	0.578460
$177$	20.7727	1.56137
$178$	−27.1494	−2.03493
$179$	−16.9588	−1.26756	−0.633780	−	0.773513i	$-0.718498\pi$
$179$	−16.9588	−1.26756	−0.633780	+	0.773513i	$0.718498\pi$
$180$	0	0
$181$	−10.5220	−0.782095	−0.391047	−	0.920371i	$-0.627887\pi$
$181$	−10.5220	−0.782095	−0.391047	+	0.920371i	$0.627887\pi$
$182$	−1.51372	−0.112204
$183$	9.18169	0.678730
$184$	13.6037	1.00288
$185$	0	0
$186$	−4.02444	−0.295086
$187$	7.50123	0.548545
$188$	2.77366	0.202290
$189$	5.56805	0.405016
$190$	0	0
$191$	1.03609	0.0749686	0.0374843	−	0.999297i	$-0.488066\pi$
$191$	1.03609	0.0749686	0.0374843	+	0.999297i	$0.488066\pi$
$192$	10.1482	0.732383
$193$	4.35059	0.313162	0.156581	−	0.987665i	$-0.449953\pi$
$193$	4.35059	0.313162	0.156581	+	0.987665i	$0.449953\pi$
$194$	−10.0631	−0.722488
$195$	0	0
$196$	0.291349	0.0208106
$197$	15.8940	1.13240	0.566199	−	0.824269i	$-0.308413\pi$
$197$	15.8940	1.13240	0.566199	+	0.824269i	$0.308413\pi$
$198$	1.49085	0.105950
$199$	−2.53134	−0.179442	−0.0897210	−	0.995967i	$-0.528598\pi$
$199$	−2.53134	−0.179442	−0.0897210	+	0.995967i	$0.528598\pi$
$200$	0	0
$201$	−14.8355	−1.04641
$202$	−25.2868	−1.77917
$203$	−0.226354	−0.0158869
$204$	−1.99376	−0.139591
$205$	0	0
$206$	6.03857	0.420727
$207$	3.03611	0.211024
$208$	−4.49781	−0.311867
$209$	8.67110	0.599792
$210$	0	0
$211$	−9.68225	−0.666554	−0.333277	−	0.942829i	$-0.608154\pi$
$211$	−9.68225	−0.666554	−0.333277	+	0.942829i	$0.608154\pi$
$212$	−0.274903	−0.0188804
$213$	−2.84842	−0.195170
$214$	11.5741	0.791191
$215$	0	0
$216$	14.4013	0.979886
$217$	1.70807	0.115951
$218$	−6.79644	−0.460313
$219$	−9.96679	−0.673493
$220$	0	0
$221$	−4.39648	−0.295739
$222$	12.7592	0.856343
$223$	5.88496	0.394086	0.197043	−	0.980395i	$-0.436866\pi$
$223$	5.88496	0.394086	0.197043	+	0.980395i	$0.436866\pi$
$224$	1.63559	0.109282
$225$	0	0
$226$	1.61950	0.107727
$227$	15.0986	1.00213	0.501064	−	0.865410i	$-0.332942\pi$
$227$	15.0986	1.00213	0.501064	+	0.865410i	$0.332942\pi$
$228$	−2.30470	−0.152632
$229$	−2.98638	−0.197345	−0.0986727	−	0.995120i	$-0.531460\pi$
$229$	−2.98638	−0.197345	−0.0986727	+	0.995120i	$0.531460\pi$
$230$	0	0
$231$	2.65572	0.174734
$232$	−0.585446	−0.0384364
$233$	−25.0444	−1.64071	−0.820355	−	0.571854i	$-0.806224\pi$
$233$	−25.0444	−1.64071	−0.820355	+	0.571854i	$0.806224\pi$
$234$	−0.873786	−0.0571212
$235$	0	0
$236$	3.88822	0.253101
$237$	26.9684	1.75179
$238$	6.65504	0.431382
$239$	−23.7820	−1.53833	−0.769164	−	0.639051i	$-0.779327\pi$
$239$	−23.7820	−1.53833	−0.769164	+	0.639051i	$0.779327\pi$
$240$	0	0
$241$	29.4866	1.89940	0.949699	−	0.313164i	$-0.101389\pi$
$241$	29.4866	1.89940	0.949699	+	0.313164i	$0.101389\pi$
$242$	−12.2443	−0.787096
$243$	5.90960	0.379101
$244$	1.71862	0.110024
$245$	0	0
$246$	−1.88623	−0.120262
$247$	−5.08214	−0.323369
$248$	4.41778	0.280529
$249$	−19.7847	−1.25380
$250$	0	0
$251$	20.1321	1.27072	0.635362	−	0.772214i	$-0.280851\pi$
$251$	20.1321	1.27072	0.635362	+	0.772214i	$0.280851\pi$
$252$	0.168179	0.0105943
$253$	8.97400	0.564191
$254$	−2.81401	−0.176566
$255$	0	0
$256$	6.85119	0.428199
$257$	−1.26460	−0.0788834	−0.0394417	−	0.999222i	$-0.512558\pi$
$257$	−1.26460	−0.0788834	−0.0394417	+	0.999222i	$0.512558\pi$
$258$	3.38942	0.211016
$259$	−5.41532	−0.336491
$260$	0	0
$261$	−0.130661	−0.00808774
$262$	−28.0664	−1.73395
$263$	−30.6283	−1.88862	−0.944310	−	0.329057i	$-0.893269\pi$
$263$	−30.6283	−1.88862	−0.944310	+	0.329057i	$0.893269\pi$
$264$	6.86881	0.422746
$265$	0	0
$266$	7.69293	0.471684
$267$	−27.9170	−1.70849
$268$	−2.77690	−0.169626
$269$	−17.6007	−1.07313	−0.536567	−	0.843858i	$-0.680279\pi$
$269$	−17.6007	−1.07313	−0.536567	+	0.843858i	$0.680279\pi$
$270$	0	0
$271$	−19.9768	−1.21350	−0.606752	−	0.794891i	$-0.707528\pi$
$271$	−19.9768	−1.21350	−0.606752	+	0.794891i	$0.707528\pi$
$272$	19.7745	1.19901
$273$	−1.55652	−0.0942049
$274$	1.60685	0.0970733
$275$	0	0
$276$	−2.38521	−0.143573
$277$	−4.60917	−0.276938	−0.138469	−	0.990367i	$-0.544218\pi$
$277$	−4.60917	−0.276938	−0.138469	+	0.990367i	$0.544218\pi$
$278$	10.0886	0.605077
$279$	0.985972	0.0590286
$280$	0	0
$281$	−5.29016	−0.315584	−0.157792	−	0.987472i	$-0.550438\pi$
$281$	−5.29016	−0.315584	−0.157792	+	0.987472i	$0.550438\pi$
$282$	22.4306	1.33572
$283$	18.9403	1.12589	0.562943	−	0.826496i	$-0.309669\pi$
$283$	18.9403	1.12589	0.562943	+	0.826496i	$0.309669\pi$
$284$	−0.533165	−0.0316375
$285$	0	0
$286$	−2.58270	−0.152718
$287$	0.800562	0.0472557
$288$	0.944135	0.0556337
$289$	2.32901	0.137001
$290$	0	0
$291$	−10.3476	−0.606588
$292$	−1.86558	−0.109175
$293$	26.5845	1.55308	0.776542	−	0.630065i	$-0.216972\pi$
$293$	26.5845	1.55308	0.776542	+	0.630065i	$0.216972\pi$
$294$	2.35614	0.137413
$295$	0	0
$296$	−14.0063	−0.814099
$297$	9.50017	0.551256
$298$	−7.23433	−0.419073
$299$	−5.25967	−0.304174
$300$	0	0
$301$	−1.43855	−0.0829167
$302$	−18.9671	−1.09143
$303$	−26.0018	−1.49376
$304$	22.8585	1.31102
$305$	0	0
$306$	3.84158	0.219609
$307$	−11.7278	−0.669339	−0.334669	−	0.942336i	$-0.608625\pi$
$307$	−11.7278	−0.669339	−0.334669	+	0.942336i	$0.608625\pi$
$308$	0.497097	0.0283247
$309$	6.20931	0.353235
$310$	0	0
$311$	17.0918	0.969186	0.484593	−	0.874740i	$-0.338968\pi$
$311$	17.0918	0.969186	0.484593	+	0.874740i	$0.338968\pi$
$312$	−4.02582	−0.227917
$313$	11.3677	0.642539	0.321270	−	0.946988i	$-0.395890\pi$
$313$	11.3677	0.642539	0.321270	+	0.946988i	$0.395890\pi$
$314$	10.2599	0.579000
$315$	0	0
$316$	5.04793	0.283968
$317$	−32.1249	−1.80432	−0.902158	−	0.431405i	$-0.858018\pi$
$317$	−32.1249	−1.80432	−0.902158	+	0.431405i	$0.858018\pi$
$318$	−2.22314	−0.124668
$319$	−0.386203	−0.0216232
$320$	0	0
$321$	11.9014	0.664271
$322$	7.96167	0.443686
$323$	22.3435	1.24323
$324$	−2.02052	−0.112251
$325$	0	0
$326$	13.8352	0.766260
$327$	−6.98861	−0.386471
$328$	2.07059	0.114329
$329$	−9.52006	−0.524858
$330$	0	0
$331$	34.3780	1.88959	0.944793	−	0.327668i	$-0.106263\pi$
$331$	34.3780	1.88959	0.944793	+	0.327668i	$0.106263\pi$
$332$	−3.70329	−0.203244
$333$	−3.12596	−0.171302
$334$	22.5652	1.23471
$335$	0	0
$336$	7.00094	0.381932
$337$	−17.2161	−0.937821	−0.468911	−	0.883246i	$-0.655353\pi$
$337$	−17.2161	−0.937821	−0.468911	+	0.883246i	$0.655353\pi$
$338$	1.51372	0.0823355
$339$	1.66529	0.0904461
$340$	0	0
$341$	2.91429	0.157818
$342$	4.44070	0.240126
$343$	−1.00000	−0.0539949
$344$	−3.72070	−0.200607
$345$	0	0
$346$	17.9574	0.965397
$347$	6.25201	0.335626	0.167813	−	0.985819i	$-0.446330\pi$
$347$	6.25201	0.335626	0.167813	+	0.985819i	$0.446330\pi$
$348$	0.102649	0.00550257
$349$	−20.0836	−1.07505	−0.537525	−	0.843248i	$-0.680641\pi$
$349$	−20.0836	−1.07505	−0.537525	+	0.843248i	$0.680641\pi$
$350$	0	0
$351$	−5.56805	−0.297201
$352$	2.79063	0.148741
$353$	18.2290	0.970231	0.485116	−	0.874450i	$-0.338778\pi$
$353$	18.2290	0.970231	0.485116	+	0.874450i	$0.338778\pi$
$354$	31.4440	1.67123
$355$	0	0
$356$	−5.22549	−0.276950
$357$	6.84321	0.362181
$358$	−25.6709	−1.35675
$359$	26.0113	1.37282	0.686411	−	0.727214i	$-0.259185\pi$
$359$	26.0113	1.37282	0.686411	+	0.727214i	$0.259185\pi$
$360$	0	0
$361$	6.82810	0.359374
$362$	−15.9274	−0.837124
$363$	−12.5906	−0.660833
$364$	−0.291349	−0.0152708
$365$	0	0
$366$	13.8985	0.726487
$367$	−22.7780	−1.18900	−0.594501	−	0.804095i	$-0.702650\pi$
$367$	−22.7780	−1.18900	−0.594501	+	0.804095i	$0.702650\pi$
$368$	23.6570	1.23321
$369$	0.462120	0.0240570
$370$	0	0
$371$	0.943553	0.0489869
$372$	−0.774591	−0.0401607
$373$	10.2829	0.532429	0.266215	−	0.963914i	$-0.414227\pi$
$373$	10.2829	0.532429	0.266215	+	0.963914i	$0.414227\pi$
$374$	11.3548	0.587141
$375$	0	0
$376$	−24.6229	−1.26983
$377$	0.226354	0.0116578
$378$	8.42847	0.433514
$379$	−16.0420	−0.824024	−0.412012	−	0.911178i	$-0.635174\pi$
$379$	−16.0420	−0.824024	−0.412012	+	0.911178i	$0.635174\pi$
$380$	0	0
$381$	−2.89357	−0.148242
$382$	1.56834	0.0802435
$383$	26.7546	1.36709	0.683547	−	0.729906i	$-0.260436\pi$
$383$	26.7546	1.36709	0.683547	+	0.729906i	$0.260436\pi$
$384$	20.4532	1.04375
$385$	0	0
$386$	6.58557	0.335197
$387$	−0.830395	−0.0422114
$388$	−1.93686	−0.0983293
$389$	−34.4266	−1.74550	−0.872749	−	0.488169i	$-0.837665\pi$
$389$	−34.4266	−1.74550	−0.872749	+	0.488169i	$0.837665\pi$
$390$	0	0
$391$	23.1240	1.16943
$392$	−2.58642	−0.130634
$393$	−28.8600	−1.45579
$394$	24.0590	1.21208
$395$	0	0
$396$	0.286946	0.0144196
$397$	24.9782	1.25362	0.626811	−	0.779171i	$-0.284360\pi$
$397$	24.9782	1.25362	0.626811	+	0.779171i	$0.284360\pi$
$398$	−3.83174	−0.192068
$399$	7.91045	0.396018
$400$	0	0
$401$	−31.1933	−1.55772	−0.778859	−	0.627199i	$-0.784201\pi$
$401$	−31.1933	−1.55772	−0.778859	+	0.627199i	$0.784201\pi$
$402$	−22.4568	−1.12004
$403$	−1.70807	−0.0850848
$404$	−4.86700	−0.242142
$405$	0	0
$406$	−0.342636	−0.0170048
$407$	−9.23957	−0.457988
$408$	17.6994	0.876251
$409$	−21.5441	−1.06529	−0.532644	−	0.846339i	$-0.678802\pi$
$409$	−21.5441	−1.06529	−0.532644	+	0.846339i	$0.678802\pi$
$410$	0	0
$411$	1.65228	0.0815011
$412$	1.16226	0.0572602
$413$	−13.3456	−0.656693
$414$	4.59583	0.225873
$415$	0	0
$416$	−1.63559	−0.0801914
$417$	10.3739	0.508012
$418$	13.1256	0.641995
$419$	−3.66042	−0.178823	−0.0894116	−	0.995995i	$-0.528499\pi$
$419$	−3.66042	−0.178823	−0.0894116	+	0.995995i	$0.528499\pi$
$420$	0	0
$421$	19.3543	0.943270	0.471635	−	0.881794i	$-0.343664\pi$
$421$	19.3543	0.943270	0.471635	+	0.881794i	$0.343664\pi$
$422$	−14.6562	−0.713454
$423$	−5.49540	−0.267196
$424$	2.44043	0.118518
$425$	0	0
$426$	−4.31171	−0.208903
$427$	−5.89886	−0.285466
$428$	2.22769	0.107680
$429$	−2.65572	−0.128219
$430$	0	0
$431$	−2.27901	−0.109776	−0.0548880	−	0.998493i	$-0.517480\pi$
$431$	−2.27901	−0.109776	−0.0548880	+	0.998493i	$0.517480\pi$
$432$	25.0441	1.20493
$433$	−4.93000	−0.236921	−0.118460	−	0.992959i	$-0.537796\pi$
$433$	−4.93000	−0.236921	−0.118460	+	0.992959i	$0.537796\pi$
$434$	2.58553	0.124110
$435$	0	0
$436$	−1.30812	−0.0626478
$437$	26.7303	1.27869
$438$	−15.0869	−0.720881
$439$	−10.5141	−0.501810	−0.250905	−	0.968012i	$-0.580728\pi$
$439$	−10.5141	−0.501810	−0.250905	+	0.968012i	$0.580728\pi$
$440$	0	0
$441$	−0.577244	−0.0274878
$442$	−6.65504	−0.316548
$443$	35.9299	1.70708	0.853541	−	0.521026i	$-0.174451\pi$
$443$	35.9299	1.70708	0.853541	+	0.521026i	$0.174451\pi$
$444$	2.45579	0.116547
$445$	0	0
$446$	8.90819	0.421815
$447$	−7.43888	−0.351847
$448$	−6.51980	−0.308032
$449$	33.9052	1.60008	0.800042	−	0.599944i	$-0.204810\pi$
$449$	33.9052	1.60008	0.800042	+	0.599944i	$0.204810\pi$
$450$	0	0
$451$	1.36591	0.0643183
$452$	0.311708	0.0146615
$453$	−19.5034	−0.916347
$454$	22.8550	1.07264
$455$	0	0
$456$	20.4597	0.958115
$457$	38.3708	1.79491	0.897456	−	0.441104i	$-0.145413\pi$
$457$	38.3708	1.79491	0.897456	+	0.441104i	$0.145413\pi$
$458$	−4.52054	−0.211231
$459$	24.4798	1.14262
$460$	0	0
$461$	−9.97988	−0.464809	−0.232405	−	0.972619i	$-0.574659\pi$
$461$	−9.97988	−0.464809	−0.232405	+	0.972619i	$0.574659\pi$
$462$	4.02002	0.187028
$463$	−28.9445	−1.34516	−0.672582	−	0.740023i	$-0.734815\pi$
$463$	−28.9445	−1.34516	−0.672582	+	0.740023i	$0.734815\pi$
$464$	−1.01810	−0.0472640
$465$	0	0
$466$	−37.9102	−1.75615
$467$	−18.0277	−0.834221	−0.417111	−	0.908856i	$-0.636957\pi$
$467$	−18.0277	−0.834221	−0.417111	+	0.908856i	$0.636957\pi$
$468$	−0.168179	−0.00777409
$469$	9.53119	0.440109
$470$	0	0
$471$	10.5500	0.486119
$472$	−34.5173	−1.58879
$473$	−2.45444	−0.112855
$474$	40.8226	1.87504
$475$	0	0
$476$	1.28091	0.0587103
$477$	0.544661	0.0249383
$478$	−35.9993	−1.64657
$479$	−3.57336	−0.163271	−0.0816355	−	0.996662i	$-0.526014\pi$
$479$	−3.57336	−0.163271	−0.0816355	+	0.996662i	$0.526014\pi$
$480$	0	0
$481$	5.41532	0.246917
$482$	44.6344	2.03304
$483$	8.18678	0.372511
$484$	−2.35669	−0.107122
$485$	0	0
$486$	8.94549	0.405776
$487$	3.28440	0.148830	0.0744152	−	0.997227i	$-0.476291\pi$
$487$	3.28440	0.148830	0.0744152	+	0.997227i	$0.476291\pi$
$488$	−15.2569	−0.690648
$489$	14.2264	0.643339
$490$	0	0
$491$	−11.2735	−0.508766	−0.254383	−	0.967104i	$-0.581872\pi$
$491$	−11.2735	−0.508766	−0.254383	+	0.967104i	$0.581872\pi$
$492$	−0.363047	−0.0163674
$493$	−0.995159	−0.0448197
$494$	−7.69293	−0.346121
$495$	0	0
$496$	7.68256	0.344957
$497$	1.82999	0.0820863
$498$	−29.9485	−1.34202
$499$	−18.6317	−0.834071	−0.417036	−	0.908890i	$-0.636931\pi$
$499$	−18.6317	−0.834071	−0.417036	+	0.908890i	$0.636931\pi$
$500$	0	0
$501$	23.2033	1.03664
$502$	30.4743	1.36014
$503$	−21.1876	−0.944708	−0.472354	−	0.881409i	$-0.656596\pi$
$503$	−21.1876	−0.944708	−0.472354	+	0.881409i	$0.656596\pi$
$504$	−1.49300	−0.0665033
$505$	0	0
$506$	13.5841	0.603888
$507$	1.55652	0.0691275
$508$	−0.541617	−0.0240304
$509$	−22.5662	−1.00023	−0.500115	−	0.865959i	$-0.666709\pi$
$509$	−22.5662	−1.00023	−0.500115	+	0.865959i	$0.666709\pi$
$510$	0	0
$511$	6.40325	0.283263
$512$	−15.9099	−0.703124
$513$	28.2976	1.24937
$514$	−1.91425	−0.0844337
$515$	0	0
$516$	0.652369	0.0287189
$517$	−16.2431	−0.714369
$518$	−8.19727	−0.360168
$519$	18.4652	0.810531
$520$	0	0
$521$	−24.3788	−1.06806	−0.534028	−	0.845467i	$-0.679322\pi$
$521$	−24.3788	−1.06806	−0.534028	+	0.845467i	$0.679322\pi$
$522$	−0.197785	−0.00865681
$523$	−2.45291	−0.107258	−0.0536292	−	0.998561i	$-0.517079\pi$
$523$	−2.45291	−0.107258	−0.0536292	+	0.998561i	$0.517079\pi$
$524$	−5.40200	−0.235987
$525$	0	0
$526$	−46.3626	−2.02151
$527$	7.50947	0.327118
$528$	11.9449	0.519837
$529$	4.66411	0.202787
$530$	0	0
$531$	−7.70366	−0.334310
$532$	1.48067	0.0641953
$533$	−0.800562	−0.0346762
$534$	−42.2585	−1.82871
$535$	0	0
$536$	24.6517	1.06479
$537$	−26.3967	−1.13910
$538$	−26.6425	−1.14864
$539$	−1.70619	−0.0734909
$540$	0	0
$541$	−12.4593	−0.535667	−0.267833	−	0.963465i	$-0.586308\pi$
$541$	−12.4593	−0.535667	−0.267833	+	0.963465i	$0.586308\pi$
$542$	−30.2393	−1.29889
$543$	−16.3777	−0.702835
$544$	7.19083	0.308305
$545$	0	0
$546$	−2.35614	−0.100833
$547$	−41.5251	−1.77548	−0.887742	−	0.460341i	$-0.847727\pi$
$547$	−41.5251	−1.77548	−0.887742	+	0.460341i	$0.847727\pi$
$548$	0.309273	0.0132115
$549$	−3.40508	−0.145325
$550$	0	0
$551$	−1.15036	−0.0490070
$552$	21.1745	0.901245
$553$	−17.3261	−0.736780
$554$	−6.97700	−0.296424
$555$	0	0
$556$	1.94178	0.0823499
$557$	1.69359	0.0717599	0.0358799	−	0.999356i	$-0.488577\pi$
$557$	1.69359	0.0717599	0.0358799	+	0.999356i	$0.488577\pi$
$558$	1.49249	0.0631819
$559$	1.43855	0.0608442
$560$	0	0
$561$	11.6758	0.492954
$562$	−8.00782	−0.337789
$563$	23.6757	0.997813	0.498907	−	0.866656i	$-0.333735\pi$
$563$	23.6757	0.997813	0.498907	+	0.866656i	$0.333735\pi$
$564$	4.31725	0.181789
$565$	0	0
$566$	28.6703	1.20510
$567$	6.93506	0.291245
$568$	4.73313	0.198598
$569$	15.3206	0.642273	0.321137	−	0.947033i	$-0.395935\pi$
$569$	15.3206	0.642273	0.321137	+	0.947033i	$0.395935\pi$
$570$	0	0
$571$	−42.0192	−1.75845	−0.879225	−	0.476407i	$-0.841939\pi$
$571$	−42.0192	−1.75845	−0.879225	+	0.476407i	$0.841939\pi$
$572$	−0.497097	−0.0207847
$573$	1.61269	0.0673711
$574$	1.21183	0.0505807
$575$	0	0
$576$	−3.76352	−0.156813
$577$	16.0835	0.669565	0.334782	−	0.942296i	$-0.391337\pi$
$577$	16.0835	0.669565	0.334782	+	0.942296i	$0.391337\pi$
$578$	3.52547	0.146640
$579$	6.77178	0.281426
$580$	0	0
$581$	12.7108	0.527335
$582$	−15.6634	−0.649269
$583$	1.60988	0.0666746
$584$	16.5615	0.685320
$585$	0	0
$586$	40.2415	1.66236
$587$	21.9499	0.905971	0.452985	−	0.891518i	$-0.350359\pi$
$587$	21.9499	0.905971	0.452985	+	0.891518i	$0.350359\pi$
$588$	0.453490	0.0187016
$589$	8.68062	0.357679
$590$	0	0
$591$	24.7393	1.01764
$592$	−24.3571	−1.00107
$593$	8.55916	0.351483	0.175741	−	0.984436i	$-0.443768\pi$
$593$	8.55916	0.351483	0.175741	+	0.984436i	$0.443768\pi$
$594$	14.3806	0.590043
$595$	0	0
$596$	−1.39240	−0.0570351
$597$	−3.94008	−0.161257
$598$	−7.96167	−0.325577
$599$	−15.9205	−0.650495	−0.325247	−	0.945629i	$-0.605448\pi$
$599$	−15.9205	−0.650495	−0.325247	+	0.945629i	$0.605448\pi$
$600$	0	0
$601$	32.7877	1.33744	0.668719	−	0.743515i	$-0.266843\pi$
$601$	32.7877	1.33744	0.668719	+	0.743515i	$0.266843\pi$
$602$	−2.17756	−0.0887509
$603$	5.50182	0.224052
$604$	−3.65063	−0.148542
$605$	0	0
$606$	−39.3594	−1.59887
$607$	−28.5734	−1.15976	−0.579879	−	0.814702i	$-0.696900\pi$
$607$	−28.5734	−1.15976	−0.579879	+	0.814702i	$0.696900\pi$
$608$	8.31229	0.337108
$609$	−0.352324	−0.0142769
$610$	0	0
$611$	9.52006	0.385140
$612$	0.739396	0.0298883
$613$	−28.8195	−1.16401	−0.582004	−	0.813186i	$-0.697731\pi$
$613$	−28.8195	−1.16401	−0.582004	+	0.813186i	$0.697731\pi$
$614$	−17.7525	−0.716434
$615$	0	0
$616$	−4.41293	−0.177802
$617$	−19.0391	−0.766487	−0.383244	−	0.923647i	$-0.625193\pi$
$617$	−19.0391	−0.766487	−0.383244	+	0.923647i	$0.625193\pi$
$618$	9.39916	0.378090
$619$	−3.44554	−0.138488	−0.0692440	−	0.997600i	$-0.522059\pi$
$619$	−3.44554	−0.138488	−0.0692440	+	0.997600i	$0.522059\pi$
$620$	0	0
$621$	29.2861	1.17521
$622$	25.8722	1.03738
$623$	17.9355	0.718572
$624$	−7.00094	−0.280262
$625$	0	0
$626$	17.2075	0.687749
$627$	13.4967	0.539008
$628$	1.97474	0.0788009
$629$	−23.8083	−0.949300
$630$	0	0
$631$	16.4992	0.656823	0.328412	−	0.944535i	$-0.393487\pi$
$631$	16.4992	0.656823	0.328412	+	0.944535i	$0.393487\pi$
$632$	−44.8125	−1.78255
$633$	−15.0706	−0.599004
$634$	−48.6282	−1.93127
$635$	0	0
$636$	−0.427892	−0.0169670
$637$	1.00000	0.0396214
$638$	−0.584603	−0.0231447
$639$	1.05635	0.0417886
$640$	0	0
$641$	−43.7044	−1.72622	−0.863111	−	0.505014i	$-0.831487\pi$
$641$	−43.7044	−1.72622	−0.863111	+	0.505014i	$0.831487\pi$
$642$	18.0154	0.711010
$643$	47.8876	1.88850	0.944251	−	0.329227i	$-0.106788\pi$
$643$	47.8876	1.88850	0.944251	+	0.329227i	$0.106788\pi$
$644$	1.53240	0.0603849
$645$	0	0
$646$	33.8218	1.33070
$647$	−29.5957	−1.16353	−0.581763	−	0.813359i	$-0.697637\pi$
$647$	−29.5957	−1.16353	−0.581763	+	0.813359i	$0.697637\pi$
$648$	17.9370	0.704631
$649$	−22.7701	−0.893805
$650$	0	0
$651$	2.65864	0.104200
$652$	2.66289	0.104287
$653$	−22.9358	−0.897548	−0.448774	−	0.893645i	$-0.648139\pi$
$653$	−22.9358	−0.897548	−0.448774	+	0.893645i	$0.648139\pi$
$654$	−10.5788	−0.413664
$655$	0	0
$656$	3.60078	0.140587
$657$	3.69624	0.144204
$658$	−14.4107	−0.561788
$659$	6.72588	0.262003	0.131002	−	0.991382i	$-0.458181\pi$
$659$	6.72588	0.262003	0.131002	+	0.991382i	$0.458181\pi$
$660$	0	0
$661$	−20.9389	−0.814427	−0.407214	−	0.913333i	$-0.633500\pi$
$661$	−20.9389	−0.814427	−0.407214	+	0.913333i	$0.633500\pi$
$662$	52.0387	2.02254
$663$	−6.84321	−0.265768
$664$	32.8756	1.27582
$665$	0	0
$666$	−4.73183	−0.183355
$667$	−1.19055	−0.0460981
$668$	4.34317	0.168042
$669$	9.16007	0.354149
$670$	0	0
$671$	−10.0646	−0.388539
$672$	2.54583	0.0982075
$673$	−30.9682	−1.19374	−0.596868	−	0.802339i	$-0.703589\pi$
$673$	−30.9682	−1.19374	−0.596868	+	0.802339i	$0.703589\pi$
$674$	−26.0604	−1.00381
$675$	0	0
$676$	0.291349	0.0112057
$677$	5.48268	0.210716	0.105358	−	0.994434i	$-0.466401\pi$
$677$	5.48268	0.210716	0.105358	+	0.994434i	$0.466401\pi$
$678$	2.52078	0.0968100
$679$	6.64792	0.255124
$680$	0	0
$681$	23.5013	0.900571
$682$	4.41142	0.168922
$683$	−46.2394	−1.76930	−0.884651	−	0.466254i	$-0.845603\pi$
$683$	−46.2394	−1.76930	−0.884651	+	0.466254i	$0.845603\pi$
$684$	0.854710	0.0326807
$685$	0	0
$686$	−1.51372	−0.0577941
$687$	−4.64836	−0.177346
$688$	−6.47033	−0.246679
$689$	−0.943553	−0.0359465
$690$	0	0
$691$	46.6224	1.77360	0.886799	−	0.462155i	$-0.152924\pi$
$691$	46.6224	1.77360	0.886799	+	0.462155i	$0.152924\pi$
$692$	3.45630	0.131389
$693$	−0.984890	−0.0374129
$694$	9.46380	0.359241
$695$	0	0
$696$	−0.911259	−0.0345412
$697$	3.51965	0.133316
$698$	−30.4009	−1.15069
$699$	−38.9821	−1.47444
$700$	0	0
$701$	35.9176	1.35659	0.678295	−	0.734790i	$-0.262719\pi$
$701$	35.9176	1.35659	0.678295	+	0.734790i	$0.262719\pi$
$702$	−8.42847	−0.318112
$703$	−27.5214	−1.03799
$704$	−11.1240	−0.419253
$705$	0	0
$706$	27.5936	1.03850
$707$	16.7051	0.628259
$708$	6.05209	0.227451
$709$	19.0532	0.715558	0.357779	−	0.933806i	$-0.383534\pi$
$709$	19.0532	0.715558	0.357779	+	0.933806i	$0.383534\pi$
$710$	0	0
$711$	−10.0014	−0.375081
$712$	46.3888	1.73849
$713$	8.98386	0.336448
$714$	10.3587	0.387665
$715$	0	0
$716$	−4.94092	−0.184651
$717$	−37.0171	−1.38243
$718$	39.3738	1.46942
$719$	29.5537	1.10217	0.551084	−	0.834450i	$-0.314214\pi$
$719$	29.5537	1.10217	0.551084	+	0.834450i	$0.314214\pi$
$720$	0	0
$721$	−3.98923	−0.148567
$722$	10.3358	0.384660
$723$	45.8965	1.70691
$724$	−3.06557	−0.113931
$725$	0	0
$726$	−19.0586	−0.707330
$727$	39.3834	1.46065	0.730325	−	0.683099i	$-0.239368\pi$
$727$	39.3834	1.46065	0.730325	+	0.683099i	$0.239368\pi$
$728$	2.58642	0.0958591
$729$	30.0036	1.11124
$730$	0	0
$731$	−6.32456	−0.233922
$732$	2.67507	0.0988735
$733$	−19.2063	−0.709401	−0.354701	−	0.934980i	$-0.615417\pi$
$733$	−19.2063	−0.709401	−0.354701	+	0.934980i	$0.615417\pi$
$734$	−34.4795	−1.27266
$735$	0	0
$736$	8.60266	0.317098
$737$	16.2620	0.599020
$738$	0.699520	0.0257497
$739$	24.0650	0.885246	0.442623	−	0.896708i	$-0.354048\pi$
$739$	24.0650	0.885246	0.442623	+	0.896708i	$0.354048\pi$
$740$	0	0
$741$	−7.91045	−0.290598
$742$	1.42828	0.0524337
$743$	13.1055	0.480795	0.240397	−	0.970675i	$-0.422722\pi$
$743$	13.1055	0.480795	0.240397	+	0.970675i	$0.422722\pi$
$744$	6.87636	0.252100
$745$	0	0
$746$	15.5655	0.569892
$747$	7.33726	0.268456
$748$	2.18547	0.0799088
$749$	−7.64615	−0.279384
$750$	0	0
$751$	31.8582	1.16252	0.581261	−	0.813717i	$-0.302560\pi$
$751$	31.8582	1.16252	0.581261	+	0.813717i	$0.302560\pi$
$752$	−42.8195	−1.56146
$753$	31.3360	1.14195
$754$	0.342636	0.0124781
$755$	0	0
$756$	1.62224	0.0590005
$757$	10.5369	0.382972	0.191486	−	0.981495i	$-0.438669\pi$
$757$	10.5369	0.382972	0.191486	+	0.981495i	$0.438669\pi$
$758$	−24.2832	−0.882004
$759$	13.9682	0.507014
$760$	0	0
$761$	−35.0667	−1.27117	−0.635584	−	0.772032i	$-0.719241\pi$
$761$	−35.0667	−1.27117	−0.635584	+	0.772032i	$0.719241\pi$
$762$	−4.38006	−0.158673
$763$	4.48989	0.162545
$764$	0.301862	0.0109210
$765$	0	0
$766$	40.4989	1.46329
$767$	13.3456	0.481881
$768$	10.6640	0.384804
$769$	−1.81955	−0.0656145	−0.0328073	−	0.999462i	$-0.510445\pi$
$769$	−1.81955	−0.0656145	−0.0328073	+	0.999462i	$0.510445\pi$
$770$	0	0
$771$	−1.96837	−0.0708891
$772$	1.26754	0.0456197
$773$	15.8577	0.570364	0.285182	−	0.958473i	$-0.407946\pi$
$773$	15.8577	0.570364	0.285182	+	0.958473i	$0.407946\pi$
$774$	−1.25699	−0.0451814
$775$	0	0
$776$	17.1943	0.617240
$777$	−8.42905	−0.302391
$778$	−52.1122	−1.86831
$779$	4.06856	0.145771
$780$	0	0
$781$	3.12232	0.111725
$782$	35.0033	1.25171
$783$	−1.26035	−0.0450412
$784$	−4.49781	−0.160636
$785$	0	0
$786$	−43.6860	−1.55823
$787$	−2.33975	−0.0834032	−0.0417016	−	0.999130i	$-0.513278\pi$
$787$	−2.33975	−0.0834032	−0.0417016	+	0.999130i	$0.513278\pi$
$788$	4.63068	0.164961
$789$	−47.6735	−1.69722
$790$	0	0
$791$	−1.06988	−0.0380405
$792$	−2.54734	−0.0905157
$793$	5.89886	0.209475
$794$	37.8101	1.34183
$795$	0	0
$796$	−0.737502	−0.0261401
$797$	48.4541	1.71633	0.858165	−	0.513374i	$-0.171604\pi$
$797$	48.4541	1.71633	0.858165	+	0.513374i	$0.171604\pi$
$798$	11.9742	0.423882
$799$	−41.8547	−1.48071
$800$	0	0
$801$	10.3532	0.365812
$802$	−47.2179	−1.66732
$803$	10.9252	0.385541
$804$	−4.32230	−0.152436
$805$	0	0
$806$	−2.58553	−0.0910716
$807$	−27.3958	−0.964379
$808$	43.2063	1.51999
$809$	−5.13889	−0.180674	−0.0903369	−	0.995911i	$-0.528794\pi$
$809$	−5.13889	−0.180674	−0.0903369	+	0.995911i	$0.528794\pi$
$810$	0	0
$811$	1.93873	0.0680781	0.0340391	−	0.999421i	$-0.489163\pi$
$811$	1.93873	0.0680781	0.0340391	+	0.999421i	$0.489163\pi$
$812$	−0.0659479	−0.00231432
$813$	−31.0943	−1.09052
$814$	−13.9861	−0.490213
$815$	0	0
$816$	30.7795	1.07750
$817$	−7.31091	−0.255776
$818$	−32.6118	−1.14024
$819$	0.577244	0.0201705
$820$	0	0
$821$	−38.3122	−1.33710	−0.668552	−	0.743665i	$-0.733086\pi$
$821$	−38.3122	−1.33710	−0.668552	+	0.743665i	$0.733086\pi$
$822$	2.50109	0.0872356
$823$	−10.7296	−0.374011	−0.187006	−	0.982359i	$-0.559878\pi$
$823$	−10.7296	−0.374011	−0.187006	+	0.982359i	$0.559878\pi$
$824$	−10.3178	−0.359438
$825$	0	0
$826$	−20.2015	−0.702899
$827$	6.48204	0.225402	0.112701	−	0.993629i	$-0.464050\pi$
$827$	6.48204	0.225402	0.112701	+	0.993629i	$0.464050\pi$
$828$	0.884567	0.0307408
$829$	−45.2952	−1.57317	−0.786584	−	0.617483i	$-0.788152\pi$
$829$	−45.2952	−1.57317	−0.786584	+	0.617483i	$0.788152\pi$
$830$	0	0
$831$	−7.17427	−0.248873
$832$	6.51980	0.226033
$833$	−4.39648	−0.152329
$834$	15.7032	0.543757
$835$	0	0
$836$	2.52631	0.0873743
$837$	9.51061	0.328735
$838$	−5.54085	−0.191406
$839$	21.3026	0.735447	0.367723	−	0.929935i	$-0.380137\pi$
$839$	21.3026	0.735447	0.367723	+	0.929935i	$0.380137\pi$
$840$	0	0
$841$	−28.9488	−0.998233
$842$	29.2970	1.00964
$843$	−8.23424	−0.283602
$844$	−2.82091	−0.0970998
$845$	0	0
$846$	−8.31850	−0.285996
$847$	8.08891	0.277938
$848$	4.24393	0.145737
$849$	29.4810	1.01179
$850$	0	0
$851$	−28.4828	−0.976377
$852$	−0.829883	−0.0284313
$853$	36.0094	1.23294	0.616469	−	0.787379i	$-0.288563\pi$
$853$	36.0094	1.23294	0.616469	+	0.787379i	$0.288563\pi$
$854$	−8.92922	−0.305552
$855$	0	0
$856$	−19.7761	−0.675935
$857$	−10.3074	−0.352095	−0.176047	−	0.984382i	$-0.556331\pi$
$857$	−10.3074	−0.352095	−0.176047	+	0.984382i	$0.556331\pi$
$858$	−4.02002	−0.137241
$859$	20.0469	0.683991	0.341995	−	0.939702i	$-0.388897\pi$
$859$	20.0469	0.683991	0.341995	+	0.939702i	$0.388897\pi$
$860$	0	0
$861$	1.24609	0.0424667
$862$	−3.44978	−0.117500
$863$	−16.6861	−0.568003	−0.284001	−	0.958824i	$-0.591662\pi$
$863$	−16.6861	−0.568003	−0.284001	+	0.958824i	$0.591662\pi$
$864$	9.10705	0.309828
$865$	0	0
$866$	−7.46264	−0.253591
$867$	3.62515	0.123117
$868$	0.497643	0.0168911
$869$	−29.5616	−1.00281
$870$	0	0
$871$	−9.53119	−0.322952
$872$	11.6127	0.393257
$873$	3.83747	0.129879
$874$	40.4623	1.36866
$875$	0	0
$876$	−2.90381	−0.0981107
$877$	40.8612	1.37978	0.689892	−	0.723913i	$-0.257658\pi$
$877$	40.8612	1.37978	0.689892	+	0.723913i	$0.257658\pi$
$878$	−15.9154	−0.537118
$879$	41.3793	1.39569
$880$	0	0
$881$	−32.7472	−1.10328	−0.551641	−	0.834082i	$-0.685998\pi$
$881$	−32.7472	−1.10328	−0.551641	+	0.834082i	$0.685998\pi$
$882$	−0.873786	−0.0294219
$883$	8.25936	0.277949	0.138975	−	0.990296i	$-0.455619\pi$
$883$	8.25936	0.277949	0.138975	+	0.990296i	$0.455619\pi$
$884$	−1.28091	−0.0430816
$885$	0	0
$886$	54.3878	1.82719
$887$	−30.6994	−1.03078	−0.515392	−	0.856954i	$-0.672354\pi$
$887$	−30.6994	−1.03078	−0.515392	+	0.856954i	$0.672354\pi$
$888$	−21.8011	−0.731596
$889$	1.85900	0.0623489
$890$	0	0
$891$	11.8325	0.396405
$892$	1.71458	0.0574082
$893$	−48.3822	−1.61905
$894$	−11.2604	−0.376603
$895$	0	0
$896$	−13.1403	−0.438988
$897$	−8.18678	−0.273349
$898$	51.3230	1.71267
$899$	−0.386627	−0.0128947
$900$	0	0
$901$	4.14831	0.138200
$902$	2.06761	0.0688439
$903$	−2.23913	−0.0745137
$904$	−2.76716	−0.0920343
$905$	0	0
$906$	−29.5226	−0.980823
$907$	−13.6056	−0.451765	−0.225883	−	0.974155i	$-0.572527\pi$
$907$	−13.6056	−0.451765	−0.225883	+	0.974155i	$0.572527\pi$
$908$	4.39895	0.145984
$909$	9.64291	0.319835
$910$	0	0
$911$	19.6092	0.649682	0.324841	−	0.945769i	$-0.394689\pi$
$911$	19.6092	0.649682	0.324841	+	0.945769i	$0.394689\pi$
$912$	35.5797	1.17816
$913$	21.6871	0.717739
$914$	58.0827	1.92121
$915$	0	0
$916$	−0.870077	−0.0287482
$917$	18.5414	0.612289
$918$	37.0556	1.22302
$919$	11.9566	0.394411	0.197205	−	0.980362i	$-0.436813\pi$
$919$	11.9566	0.394411	0.197205	+	0.980362i	$0.436813\pi$
$920$	0	0
$921$	−18.2545	−0.601506
$922$	−15.1067	−0.497514
$923$	−1.82999	−0.0602349
$924$	0.773741	0.0254542
$925$	0	0
$926$	−43.8138	−1.43981
$927$	−2.30276	−0.0756325
$928$	−0.370222	−0.0121531
$929$	11.6935	0.383651	0.191826	−	0.981429i	$-0.438559\pi$
$929$	11.6935	0.383651	0.191826	+	0.981429i	$0.438559\pi$
$930$	0	0
$931$	−5.08214	−0.166560
$932$	−7.29664	−0.239009
$933$	26.6037	0.870966
$934$	−27.2888	−0.892918
$935$	0	0
$936$	1.49300	0.0488001
$937$	−12.8300	−0.419138	−0.209569	−	0.977794i	$-0.567206\pi$
$937$	−12.8300	−0.419138	−0.209569	+	0.977794i	$0.567206\pi$
$938$	14.4275	0.471076
$939$	17.6940	0.577423
$940$	0	0
$941$	30.8795	1.00664	0.503322	−	0.864099i	$-0.332111\pi$
$941$	30.8795	1.00664	0.503322	+	0.864099i	$0.332111\pi$
$942$	15.9698	0.520323
$943$	4.21069	0.137119
$944$	−60.0259	−1.95368
$945$	0	0
$946$	−3.71534	−0.120796
$947$	8.78770	0.285562	0.142781	−	0.989754i	$-0.454396\pi$
$947$	8.78770	0.285562	0.142781	+	0.989754i	$0.454396\pi$
$948$	7.85720	0.255190
$949$	−6.40325	−0.207858
$950$	0	0
$951$	−50.0031	−1.62146
$952$	−11.3711	−0.368541
$953$	−14.3205	−0.463886	−0.231943	−	0.972729i	$-0.574508\pi$
$953$	−14.3205	−0.463886	−0.231943	+	0.972729i	$0.574508\pi$
$954$	0.824464	0.0266930
$955$	0	0
$956$	−6.92884	−0.224095
$957$	−0.601133	−0.0194319
$958$	−5.40907	−0.174759
$959$	−1.06152	−0.0342784
$960$	0	0
$961$	−28.0825	−0.905887
$962$	8.19727	0.264291
$963$	−4.41370	−0.142229
$964$	8.59087	0.276693
$965$	0	0
$966$	12.3925	0.398722
$967$	−0.228291	−0.00734134	−0.00367067	−	0.999993i	$-0.501168\pi$
$967$	−0.228291	−0.00734134	−0.00367067	+	0.999993i	$0.501168\pi$
$968$	20.9213	0.672437
$969$	34.7781	1.11723
$970$	0	0
$971$	6.81054	0.218561	0.109280	−	0.994011i	$-0.465145\pi$
$971$	6.81054	0.218561	0.109280	+	0.994011i	$0.465145\pi$
$972$	1.72175	0.0552253
$973$	−6.66480	−0.213664
$974$	4.97167	0.159302
$975$	0	0
$976$	−26.5320	−0.849267
$977$	−26.2214	−0.838896	−0.419448	−	0.907779i	$-0.637776\pi$
$977$	−26.2214	−0.838896	−0.419448	+	0.907779i	$0.637776\pi$
$978$	21.5348	0.688606
$979$	30.6014	0.978027
$980$	0	0
$981$	2.59176	0.0827487
$982$	−17.0649	−0.544564
$983$	47.7344	1.52249	0.761245	−	0.648464i	$-0.224589\pi$
$983$	47.7344	1.52249	0.761245	+	0.648464i	$0.224589\pi$
$984$	3.22292	0.102743
$985$	0	0
$986$	−1.50639	−0.0479733
$987$	−14.8182	−0.471667
$988$	−1.48067	−0.0471065
$989$	−7.56630	−0.240594
$990$	0	0
$991$	21.9811	0.698254	0.349127	−	0.937075i	$-0.386478\pi$
$991$	21.9811	0.698254	0.349127	+	0.937075i	$0.386478\pi$
$992$	2.79370	0.0887000
$993$	53.5101	1.69809
$994$	2.77009	0.0878620
$995$	0	0
$996$	−5.76424	−0.182647
$997$	−41.5448	−1.31574	−0.657869	−	0.753132i	$-0.728542\pi$
$997$	−41.5448	−1.31574	−0.657869	+	0.753132i	$0.728542\pi$
$998$	−28.2032	−0.892758
$999$	−30.1528	−0.953992

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2275.2.a.y.1.6		7
5.2	odd	4		455.2.c.b.274.11	yes	14
5.3	odd	4		455.2.c.b.274.4	✓	14
5.4	even	2		2275.2.a.w.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
455.2.c.b.274.4	✓	14	5.3	odd	4
455.2.c.b.274.11	yes	14	5.2	odd	4
2275.2.a.w.1.2		7	5.4	even	2
2275.2.a.y.1.6		7	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2275 = 5^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2275.a (trivial)

\(f(q)\)	\(=\)	\(q+1.51372 q^{2} +1.55652 q^{3} +0.291349 q^{4} +2.35614 q^{6} -1.00000 q^{7} -2.58642 q^{8} -0.577244 q^{9} -1.70619 q^{11} +0.453490 q^{12} +1.00000 q^{13} -1.51372 q^{14} -4.49781 q^{16} -4.39648 q^{17} -0.873786 q^{18} -5.08214 q^{19} -1.55652 q^{21} -2.58270 q^{22} -5.25967 q^{23} -4.02582 q^{24} +1.51372 q^{26} -5.56805 q^{27} -0.291349 q^{28} +0.226354 q^{29} -1.70807 q^{31} -1.63559 q^{32} -2.65572 q^{33} -6.65504 q^{34} -0.168179 q^{36} +5.41532 q^{37} -7.69293 q^{38} +1.55652 q^{39} -0.800562 q^{41} -2.35614 q^{42} +1.43855 q^{43} -0.497097 q^{44} -7.96167 q^{46} +9.52006 q^{47} -7.00094 q^{48} +1.00000 q^{49} -6.84321 q^{51} +0.291349 q^{52} -0.943553 q^{53} -8.42847 q^{54} +2.58642 q^{56} -7.91045 q^{57} +0.342636 q^{58} +13.3456 q^{59} +5.89886 q^{61} -2.58553 q^{62} +0.577244 q^{63} +6.51980 q^{64} -4.02002 q^{66} -9.53119 q^{67} -1.28091 q^{68} -8.18678 q^{69} -1.82999 q^{71} +1.49300 q^{72} -6.40325 q^{73} +8.19727 q^{74} -1.48067 q^{76} +1.70619 q^{77} +2.35614 q^{78} +17.3261 q^{79} -6.93506 q^{81} -1.21183 q^{82} -12.7108 q^{83} -0.453490 q^{84} +2.17756 q^{86} +0.352324 q^{87} +4.41293 q^{88} -17.9355 q^{89} -1.00000 q^{91} -1.53240 q^{92} -2.65864 q^{93} +14.4107 q^{94} -2.54583 q^{96} -6.64792 q^{97} +1.51372 q^{98} +0.984890 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{3} + 2 q^{4} - 8 q^{6} - 7 q^{7} + 6 q^{8} + q^{9} - 6 q^{11} - 3 q^{12} + 7 q^{13} - 4 q^{16} - 2 q^{17} + q^{18} - 10 q^{19} - 2 q^{21} - 18 q^{22} - 10 q^{23} - 5 q^{24} + 8 q^{27} - 2 q^{28}+ \cdots - 18 q^{99}+O(q^{100}) \) 7 * q + 2 * q^3 + 2 * q^4 - 8 * q^6 - 7 * q^7 + 6 * q^8 + q^9 - 6 * q^11 - 3 * q^12 + 7 * q^13 - 4 * q^16 - 2 * q^17 + q^18 - 10 * q^19 - 2 * q^21 - 18 * q^22 - 10 * q^23 - 5 * q^24 + 8 * q^27 - 2 * q^28 - 26 * q^29 - 6 * q^31 + 7 * q^32 - 4 * q^33 - q^34 - 6 * q^36 + 4 * q^37 + q^38 + 2 * q^39 - 28 * q^41 + 8 * q^42 - 2 * q^43 - 18 * q^46 + 20 * q^47 - 20 * q^48 + 7 * q^49 - 32 * q^51 + 2 * q^52 - 8 * q^53 + 11 * q^54 - 6 * q^56 - 14 * q^57 + 12 * q^58 - 28 * q^59 - 26 * q^61 + 22 * q^62 - q^63 - 8 * q^64 + 2 * q^66 - 2 * q^67 + 2 * q^68 - 22 * q^69 - 14 * q^71 - 4 * q^72 + 2 * q^73 - 12 * q^74 - 36 * q^76 + 6 * q^77 - 8 * q^78 + 14 * q^79 - 33 * q^81 - 19 * q^82 + 28 * q^83 + 3 * q^84 - 4 * q^86 - 22 * q^87 - 24 * q^88 - 30 * q^89 - 7 * q^91 - 42 * q^92 - 2 * q^93 + 14 * q^94 + 18 * q^97 - 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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