LMFDB - Embedded newform 1274.2.a.u.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1274,2,Mod(1,1274)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1274, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1274.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1274 = 2 \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1274.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-4,2,4,0,-2,0,-4,6,0,-6,2,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$10.1729412175$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.16448.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 8x + 14 $ x^4 - 2x^3 - 7x^2 + 8*x + 14
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.18398$ of defining polynomial
Character	$\chi$	$=$	1274.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +2.18398 q^{3} +1.00000 q^{4} +3.08861 q^{5} -2.18398 q^{6} -1.00000 q^{8} +1.76977 q^{9} -3.08861 q^{10} +1.85838 q^{11} +2.18398 q^{12} +1.00000 q^{13} +6.74547 q^{15} +1.00000 q^{16} -6.50283 q^{17} -1.76977 q^{18} +4.58579 q^{19} +3.08861 q^{20} -1.85838 q^{22} +1.02995 q^{23} -2.18398 q^{24} +4.53953 q^{25} -1.00000 q^{26} -2.68681 q^{27} +8.74547 q^{29} -6.74547 q^{30} +0.141621 q^{31} -1.00000 q^{32} +4.05866 q^{33} +6.50283 q^{34} +1.76977 q^{36} -3.23023 q^{37} -4.58579 q^{38} +2.18398 q^{39} -3.08861 q^{40} +9.64055 q^{41} +4.10777 q^{43} +1.85838 q^{44} +5.46612 q^{45} -1.02995 q^{46} -11.0548 q^{47} +2.18398 q^{48} -4.53953 q^{50} -14.2020 q^{51} +1.00000 q^{52} -2.43741 q^{53} +2.68681 q^{54} +5.73981 q^{55} +10.0153 q^{57} -8.74547 q^{58} +9.84208 q^{59} +6.74547 q^{60} +8.11857 q^{61} -0.141621 q^{62} +1.00000 q^{64} +3.08861 q^{65} -4.05866 q^{66} -12.2263 q^{67} -6.50283 q^{68} +2.24939 q^{69} -8.65296 q^{71} -1.76977 q^{72} -11.2974 q^{73} +3.23023 q^{74} +9.91424 q^{75} +4.58579 q^{76} -2.18398 q^{78} -0.877538 q^{79} +3.08861 q^{80} -11.1772 q^{81} -9.64055 q^{82} -8.78783 q^{83} -20.0847 q^{85} -4.10777 q^{86} +19.0999 q^{87} -1.85838 q^{88} -12.4103 q^{89} -5.46612 q^{90} +1.02995 q^{92} +0.309297 q^{93} +11.0548 q^{94} +14.1637 q^{95} -2.18398 q^{96} +5.79296 q^{97} +3.28890 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{6} - 4 q^{8} + 6 q^{9} - 6 q^{11} + 2 q^{12} + 4 q^{13} - 8 q^{15} + 4 q^{16} - 8 q^{17} - 6 q^{18} + 24 q^{19} + 6 q^{22} + 2 q^{23} - 2 q^{24} + 16 q^{25} - 4 q^{26}+ \cdots + 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 2 * q^6 - 4 * q^8 + 6 * q^9 - 6 * q^11 + 2 * q^12 + 4 * q^13 - 8 * q^15 + 4 * q^16 - 8 * q^17 - 6 * q^18 + 24 * q^19 + 6 * q^22 + 2 * q^23 - 2 * q^24 + 16 * q^25 - 4 * q^26 + 14 * q^27 + 8 * q^30 + 14 * q^31 - 4 * q^32 + 6 * q^33 + 8 * q^34 + 6 * q^36 - 14 * q^37 - 24 * q^38 + 2 * q^39 + 6 * q^41 + 4 * q^43 - 6 * q^44 - 12 * q^45 - 2 * q^46 - 6 * q^47 + 2 * q^48 - 16 * q^50 + 4 * q^51 + 4 * q^52 + 16 * q^53 - 14 * q^54 + 24 * q^55 + 12 * q^57 + 16 * q^59 - 8 * q^60 + 18 * q^61 - 14 * q^62 + 4 * q^64 - 6 * q^66 - 22 * q^67 - 8 * q^68 + 10 * q^69 + 16 * q^71 - 6 * q^72 + 10 * q^73 + 14 * q^74 + 42 * q^75 + 24 * q^76 - 2 * q^78 + 10 * q^79 - 20 * q^81 - 6 * q^82 + 20 * q^83 - 40 * q^85 - 4 * q^86 - 8 * q^87 + 6 * q^88 - 16 * q^89 + 12 * q^90 + 2 * q^92 - 2 * q^93 + 6 * q^94 - 4 * q^95 - 2 * q^96 + 2 * q^97 + 4 * 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
$3$	2.18398	1.26092	0.630461	−	0.776221i	$-0.282866\pi$
$3$	2.18398	1.26092	0.630461	+	0.776221i	$0.282866\pi$
$4$	1.00000	0.500000
$5$	3.08861	1.38127	0.690635	−	0.723204i	$-0.257331\pi$
$5$	3.08861	1.38127	0.690635	+	0.723204i	$0.257331\pi$
$6$	−2.18398	−0.891606
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.76977	0.589922
$10$	−3.08861	−0.976705
$11$	1.85838	0.560322	0.280161	−	0.959953i	$-0.409612\pi$
$11$	1.85838	0.560322	0.280161	+	0.959953i	$0.409612\pi$
$12$	2.18398	0.630461
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	6.74547	1.74167
$16$	1.00000	0.250000
$17$	−6.50283	−1.57717	−0.788584	−	0.614928i	$-0.789185\pi$
$17$	−6.50283	−1.57717	−0.788584	+	0.614928i	$0.789185\pi$
$18$	−1.76977	−0.417138
$19$	4.58579	1.05205	0.526026	−	0.850469i	$-0.323682\pi$
$19$	4.58579	1.05205	0.526026	+	0.850469i	$0.323682\pi$
$20$	3.08861	0.690635
$21$	0	0
$22$	−1.85838	−0.396208
$23$	1.02995	0.214760	0.107380	−	0.994218i	$-0.465754\pi$
$23$	1.02995	0.214760	0.107380	+	0.994218i	$0.465754\pi$
$24$	−2.18398	−0.445803
$25$	4.53953	0.907906
$26$	−1.00000	−0.196116
$27$	−2.68681	−0.517076
$28$	0	0
$29$	8.74547	1.62399	0.811996	−	0.583663i	$-0.198381\pi$
$29$	8.74547	1.62399	0.811996	+	0.583663i	$0.198381\pi$
$30$	−6.74547	−1.23155
$31$	0.141621	0.0254359	0.0127179	−	0.999919i	$-0.495952\pi$
$31$	0.141621	0.0254359	0.0127179	+	0.999919i	$0.495952\pi$
$32$	−1.00000	−0.176777
$33$	4.05866	0.706522
$34$	6.50283	1.11523
$35$	0	0
$36$	1.76977	0.294961
$37$	−3.23023	−0.531047	−0.265524	−	0.964104i	$-0.585545\pi$
$37$	−3.23023	−0.531047	−0.265524	+	0.964104i	$0.585545\pi$
$38$	−4.58579	−0.743913
$39$	2.18398	0.349717
$40$	−3.08861	−0.488353
$41$	9.64055	1.50560	0.752801	−	0.658249i	$-0.228702\pi$
$41$	9.64055	1.50560	0.752801	+	0.658249i	$0.228702\pi$
$42$	0	0
$43$	4.10777	0.626429	0.313215	−	0.949682i	$-0.398594\pi$
$43$	4.10777	0.626429	0.313215	+	0.949682i	$0.398594\pi$
$44$	1.85838	0.280161
$45$	5.46612	0.814841
$46$	−1.02995	−0.151858
$47$	−11.0548	−1.61250	−0.806252	−	0.591573i	$-0.798507\pi$
$47$	−11.0548	−1.61250	−0.806252	+	0.591573i	$0.798507\pi$
$48$	2.18398	0.315230
$49$	0	0
$50$	−4.53953	−0.641987
$51$	−14.2020	−1.98868
$52$	1.00000	0.138675
$53$	−2.43741	−0.334804	−0.167402	−	0.985889i	$-0.553538\pi$
$53$	−2.43741	−0.334804	−0.167402	+	0.985889i	$0.553538\pi$
$54$	2.68681	0.365628
$55$	5.73981	0.773956
$56$	0	0
$57$	10.0153	1.32655
$58$	−8.74547	−1.14834
$59$	9.84208	1.28133	0.640665	−	0.767821i	$-0.278659\pi$
$59$	9.84208	1.28133	0.640665	+	0.767821i	$0.278659\pi$
$60$	6.74547	0.870836
$61$	8.11857	1.03948	0.519738	−	0.854326i	$-0.326030\pi$
$61$	8.11857	1.03948	0.519738	+	0.854326i	$0.326030\pi$
$62$	−0.141621	−0.0179859
$63$	0	0
$64$	1.00000	0.125000
$65$	3.08861	0.383095
$66$	−4.05866	−0.499587
$67$	−12.2263	−1.49368	−0.746842	−	0.665001i	$-0.768431\pi$
$67$	−12.2263	−1.49368	−0.746842	+	0.665001i	$0.768431\pi$
$68$	−6.50283	−0.788584
$69$	2.24939	0.270795
$70$	0	0
$71$	−8.65296	−1.02692	−0.513459	−	0.858114i	$-0.671636\pi$
$71$	−8.65296	−1.02692	−0.513459	+	0.858114i	$0.671636\pi$
$72$	−1.76977	−0.208569
$73$	−11.2974	−1.32226	−0.661131	−	0.750271i	$-0.729923\pi$
$73$	−11.2974	−1.32226	−0.661131	+	0.750271i	$0.729923\pi$
$74$	3.23023	0.375507
$75$	9.91424	1.14480
$76$	4.58579	0.526026
$77$	0	0
$78$	−2.18398	−0.247287
$79$	−0.877538	−0.0987308	−0.0493654	−	0.998781i	$-0.515720\pi$
$79$	−0.877538	−0.0987308	−0.0493654	+	0.998781i	$0.515720\pi$
$80$	3.08861	0.345317
$81$	−11.1772	−1.24191
$82$	−9.64055	−1.06462
$83$	−8.78783	−0.964589	−0.482295	−	0.876009i	$-0.660197\pi$
$83$	−8.78783	−0.964589	−0.482295	+	0.876009i	$0.660197\pi$
$84$	0	0
$85$	−20.0847	−2.17849
$86$	−4.10777	−0.442953
$87$	19.0999	2.04773
$88$	−1.85838	−0.198104
$89$	−12.4103	−1.31549	−0.657745	−	0.753240i	$-0.728490\pi$
$89$	−12.4103	−1.31549	−0.657745	+	0.753240i	$0.728490\pi$
$90$	−5.46612	−0.576180
$91$	0	0
$92$	1.02995	0.107380
$93$	0.309297	0.0320726
$94$	11.0548	1.14021
$95$	14.1637	1.45317
$96$	−2.18398	−0.222901
$97$	5.79296	0.588186	0.294093	−	0.955777i	$-0.404982\pi$
$97$	5.79296	0.588186	0.294093	+	0.955777i	$0.404982\pi$
$98$	0	0
$99$	3.28890	0.330546
$100$	4.53953	0.453953
$101$	0.900594	0.0896125	0.0448062	−	0.998996i	$-0.485733\pi$
$101$	0.900594	0.0896125	0.0448062	+	0.998996i	$0.485733\pi$
$102$	14.2020	1.40621
$103$	19.6817	1.93929	0.969646	−	0.244512i	$-0.0786279\pi$
$103$	19.6817	1.93929	0.969646	+	0.244512i	$0.0786279\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	2.43741	0.236742
$107$	−10.9961	−1.06303	−0.531517	−	0.847048i	$-0.678378\pi$
$107$	−10.9961	−1.06303	−0.531517	+	0.847048i	$0.678378\pi$
$108$	−2.68681	−0.258538
$109$	−14.8532	−1.42268	−0.711341	−	0.702847i	$-0.751912\pi$
$109$	−14.8532	−1.42268	−0.711341	+	0.702847i	$0.751912\pi$
$110$	−5.73981	−0.547270
$111$	−7.05476	−0.669609
$112$	0	0
$113$	5.75061	0.540972	0.270486	−	0.962724i	$-0.412816\pi$
$113$	5.75061	0.540972	0.270486	+	0.962724i	$0.412816\pi$
$114$	−10.0153	−0.938015
$115$	3.18112	0.296641
$116$	8.74547	0.811996
$117$	1.76977	0.163615
$118$	−9.84208	−0.906037
$119$	0	0
$120$	−6.74547	−0.615774
$121$	−7.54643	−0.686039
$122$	−8.11857	−0.735020
$123$	21.0548	1.89844
$124$	0.141621	0.0127179
$125$	−1.42221	−0.127206
$126$	0	0
$127$	10.6868	0.948301	0.474150	−	0.880444i	$-0.342755\pi$
$127$	10.6868	0.948301	0.474150	+	0.880444i	$0.342755\pi$
$128$	−1.00000	−0.0883883
$129$	8.97129	0.789878
$130$	−3.08861	−0.270889
$131$	14.1597	1.23714	0.618569	−	0.785731i	$-0.287713\pi$
$131$	14.1597	1.23714	0.618569	+	0.785731i	$0.287713\pi$
$132$	4.05866	0.353261
$133$	0	0
$134$	12.2263	1.05619
$135$	−8.29850	−0.714222
$136$	6.50283	0.557613
$137$	−11.1964	−0.956572	−0.478286	−	0.878204i	$-0.658742\pi$
$137$	−11.1964	−0.956572	−0.478286	+	0.878204i	$0.658742\pi$
$138$	−2.24939	−0.191481
$139$	14.3121	1.21394	0.606968	−	0.794726i	$-0.292386\pi$
$139$	14.3121	1.21394	0.606968	+	0.794726i	$0.292386\pi$
$140$	0	0
$141$	−24.1434	−2.03324
$142$	8.65296	0.726140
$143$	1.85838	0.155405
$144$	1.76977	0.147480
$145$	27.0114	2.24317
$146$	11.2974	0.934980
$147$	0	0
$148$	−3.23023	−0.265524
$149$	21.9910	1.80157	0.900785	−	0.434265i	$-0.142992\pi$
$149$	21.9910	1.80157	0.900785	+	0.434265i	$0.142992\pi$
$150$	−9.91424	−0.809495
$151$	−5.38712	−0.438397	−0.219199	−	0.975680i	$-0.570344\pi$
$151$	−5.38712	−0.438397	−0.219199	+	0.975680i	$0.570344\pi$
$152$	−4.58579	−0.371956
$153$	−11.5085	−0.930405
$154$	0	0
$155$	0.437413	0.0351338
$156$	2.18398	0.174858
$157$	−1.40746	−0.112328	−0.0561638	−	0.998422i	$-0.517887\pi$
$157$	−1.40746	−0.112328	−0.0561638	+	0.998422i	$0.517887\pi$
$158$	0.877538	0.0698132
$159$	−5.32326	−0.422162
$160$	−3.08861	−0.244176
$161$	0	0
$162$	11.1772	0.878166
$163$	6.08472	0.476592	0.238296	−	0.971193i	$-0.423411\pi$
$163$	6.08472	0.476592	0.238296	+	0.971193i	$0.423411\pi$
$164$	9.64055	0.752801
$165$	12.5356	0.975898
$166$	8.78783	0.682068
$167$	1.52603	0.118087	0.0590437	−	0.998255i	$-0.481195\pi$
$167$	1.52603	0.118087	0.0590437	+	0.998255i	$0.481195\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	20.0847	1.54043
$171$	8.11577	0.620628
$172$	4.10777	0.313215
$173$	−19.5892	−1.48934	−0.744668	−	0.667435i	$-0.767392\pi$
$173$	−19.5892	−1.48934	−0.744668	+	0.667435i	$0.767392\pi$
$174$	−19.0999	−1.44796
$175$	0	0
$176$	1.85838	0.140081
$177$	21.4949	1.61565
$178$	12.4103	0.930193
$179$	−24.8071	−1.85417	−0.927086	−	0.374849i	$-0.877695\pi$
$179$	−24.8071	−1.85417	−0.927086	+	0.374849i	$0.877695\pi$
$180$	5.46612	0.407421
$181$	1.17282	0.0871747	0.0435873	−	0.999050i	$-0.486121\pi$
$181$	1.17282	0.0871747	0.0435873	+	0.999050i	$0.486121\pi$
$182$	0	0
$183$	17.7308	1.31070
$184$	−1.02995	−0.0759291
$185$	−9.97694	−0.733520
$186$	−0.309297	−0.0226788
$187$	−12.0847	−0.883722
$188$	−11.0548	−0.806252
$189$	0	0
$190$	−14.1637	−1.02754
$191$	8.48528	0.613973	0.306987	−	0.951714i	$-0.400679\pi$
$191$	8.48528	0.613973	0.306987	+	0.951714i	$0.400679\pi$
$192$	2.18398	0.157615
$193$	7.43352	0.535076	0.267538	−	0.963547i	$-0.413790\pi$
$193$	7.43352	0.535076	0.267538	+	0.963547i	$0.413790\pi$
$194$	−5.79296	−0.415911
$195$	6.74547	0.483053
$196$	0	0
$197$	−20.5688	−1.46546	−0.732732	−	0.680518i	$-0.761755\pi$
$197$	−20.5688	−1.46546	−0.732732	+	0.680518i	$0.761755\pi$
$198$	−3.28890	−0.233732
$199$	−10.3449	−0.733331	−0.366665	−	0.930353i	$-0.619501\pi$
$199$	−10.3449	−0.733331	−0.366665	+	0.930353i	$0.619501\pi$
$200$	−4.53953	−0.320993
$201$	−26.7021	−1.88342
$202$	−0.900594	−0.0633656
$203$	0	0
$204$	−14.2020	−0.994342
$205$	29.7759	2.07964
$206$	−19.6817	−1.37129
$207$	1.82277	0.126692
$208$	1.00000	0.0693375
$209$	8.52213	0.589488
$210$	0	0
$211$	−4.41583	−0.303998	−0.151999	−	0.988381i	$-0.548571\pi$
$211$	−4.41583	−0.303998	−0.151999	+	0.988381i	$0.548571\pi$
$212$	−2.43741	−0.167402
$213$	−18.8979	−1.29486
$214$	10.9961	0.751678
$215$	12.6873	0.865268
$216$	2.68681	0.182814
$217$	0	0
$218$	14.8532	1.00599
$219$	−24.6733	−1.66727
$220$	5.73981	0.386978
$221$	−6.50283	−0.437427
$222$	7.05476	0.473485
$223$	20.5592	1.37675	0.688373	−	0.725357i	$-0.258325\pi$
$223$	20.5592	1.37675	0.688373	+	0.725357i	$0.258325\pi$
$224$	0	0
$225$	8.03391	0.535594
$226$	−5.75061	−0.382525
$227$	22.0073	1.46067	0.730337	−	0.683087i	$-0.239363\pi$
$227$	22.0073	1.46067	0.730337	+	0.683087i	$0.239363\pi$
$228$	10.0153	0.663277
$229$	−0.996103	−0.0658244	−0.0329122	−	0.999458i	$-0.510478\pi$
$229$	−0.996103	−0.0658244	−0.0329122	+	0.999458i	$0.510478\pi$
$230$	−3.18112	−0.209757
$231$	0	0
$232$	−8.74547	−0.574168
$233$	15.7403	1.03118	0.515592	−	0.856834i	$-0.327572\pi$
$233$	15.7403	1.03118	0.515592	+	0.856834i	$0.327572\pi$
$234$	−1.76977	−0.115693
$235$	−34.1439	−2.22730
$236$	9.84208	0.640665
$237$	−1.91653	−0.124492
$238$	0	0
$239$	1.10388	0.0714038	0.0357019	−	0.999362i	$-0.488633\pi$
$239$	1.10388	0.0714038	0.0357019	+	0.999362i	$0.488633\pi$
$240$	6.74547	0.435418
$241$	−4.68005	−0.301469	−0.150734	−	0.988574i	$-0.548164\pi$
$241$	−4.68005	−0.301469	−0.150734	+	0.988574i	$0.548164\pi$
$242$	7.54643	0.485103
$243$	−16.3504	−1.04888
$244$	8.11857	0.519738
$245$	0	0
$246$	−21.0548	−1.34240
$247$	4.58579	0.291787
$248$	−0.141621	−0.00899294
$249$	−19.1924	−1.21627
$250$	1.42221	0.0899484
$251$	7.84083	0.494909	0.247455	−	0.968900i	$-0.420406\pi$
$251$	7.84083	0.494909	0.247455	+	0.968900i	$0.420406\pi$
$252$	0	0
$253$	1.91404	0.120335
$254$	−10.6868	−0.670550
$255$	−43.8646	−2.74691
$256$	1.00000	0.0625000
$257$	−25.8516	−1.61258	−0.806290	−	0.591520i	$-0.798528\pi$
$257$	−25.8516	−1.61258	−0.806290	+	0.591520i	$0.798528\pi$
$258$	−8.97129	−0.558528
$259$	0	0
$260$	3.08861	0.191548
$261$	15.4774	0.958029
$262$	−14.1597	−0.874788
$263$	−15.3488	−0.946448	−0.473224	−	0.880942i	$-0.656910\pi$
$263$	−15.3488	−0.946448	−0.473224	+	0.880942i	$0.656910\pi$
$264$	−4.05866	−0.249793
$265$	−7.52822	−0.462455
$266$	0	0
$267$	−27.1039	−1.65873
$268$	−12.2263	−0.746842
$269$	−23.3997	−1.42670	−0.713351	−	0.700806i	$-0.752824\pi$
$269$	−23.3997	−1.42670	−0.713351	+	0.700806i	$0.752824\pi$
$270$	8.29850	0.505031
$271$	6.75450	0.410307	0.205153	−	0.978730i	$-0.434231\pi$
$271$	6.75450	0.410307	0.205153	+	0.978730i	$0.434231\pi$
$272$	−6.50283	−0.394292
$273$	0	0
$274$	11.1964	0.676398
$275$	8.43617	0.508720
$276$	2.24939	0.135398
$277$	15.5955	0.937042	0.468521	−	0.883452i	$-0.344787\pi$
$277$	15.5955	0.937042	0.468521	+	0.883452i	$0.344787\pi$
$278$	−14.3121	−0.858382
$279$	0.250636	0.0150052
$280$	0	0
$281$	−20.2499	−1.20801	−0.604004	−	0.796981i	$-0.706429\pi$
$281$	−20.2499	−1.20801	−0.604004	+	0.796981i	$0.706429\pi$
$282$	24.1434	1.43772
$283$	24.0103	1.42726	0.713631	−	0.700522i	$-0.247049\pi$
$283$	24.0103	1.42726	0.713631	+	0.700522i	$0.247049\pi$
$284$	−8.65296	−0.513459
$285$	30.9333	1.83233
$286$	−1.85838	−0.109888
$287$	0	0
$288$	−1.76977	−0.104284
$289$	25.2868	1.48746
$290$	−27.0114	−1.58616
$291$	12.6517	0.741657
$292$	−11.2974	−0.661131
$293$	−19.9266	−1.16412	−0.582062	−	0.813144i	$-0.697754\pi$
$293$	−19.9266	−1.16412	−0.582062	+	0.813144i	$0.697754\pi$
$294$	0	0
$295$	30.3984	1.76986
$296$	3.23023	0.187754
$297$	−4.99310	−0.289729
$298$	−21.9910	−1.27390
$299$	1.02995	0.0595637
$300$	9.91424	0.572399
$301$	0	0
$302$	5.38712	0.309994
$303$	1.96688	0.112994
$304$	4.58579	0.263013
$305$	25.0751	1.43580
$306$	11.5085	0.657896
$307$	−16.6570	−0.950665	−0.475333	−	0.879806i	$-0.657672\pi$
$307$	−16.6570	−0.950665	−0.475333	+	0.879806i	$0.657672\pi$
$308$	0	0
$309$	42.9844	2.44529
$310$	−0.437413	−0.0248434
$311$	−26.5931	−1.50795	−0.753977	−	0.656901i	$-0.771867\pi$
$311$	−26.5931	−1.50795	−0.753977	+	0.656901i	$0.771867\pi$
$312$	−2.18398	−0.123643
$313$	−6.53543	−0.369405	−0.184702	−	0.982795i	$-0.559132\pi$
$313$	−6.53543	−0.369405	−0.184702	+	0.982795i	$0.559132\pi$
$314$	1.40746	0.0794276
$315$	0	0
$316$	−0.877538	−0.0493654
$317$	−27.7676	−1.55959	−0.779793	−	0.626038i	$-0.784676\pi$
$317$	−27.7676	−1.55959	−0.779793	+	0.626038i	$0.784676\pi$
$318$	5.32326	0.298514
$319$	16.2524	0.909959
$320$	3.08861	0.172659
$321$	−24.0153	−1.34040
$322$	0	0
$323$	−29.8206	−1.65926
$324$	−11.1772	−0.620957
$325$	4.53953	0.251808
$326$	−6.08472	−0.337001
$327$	−32.4392	−1.79389
$328$	−9.64055	−0.532310
$329$	0	0
$330$	−12.5356	−0.690064
$331$	1.92608	0.105867	0.0529334	−	0.998598i	$-0.483143\pi$
$331$	1.92608	0.105867	0.0529334	+	0.998598i	$0.483143\pi$
$332$	−8.78783	−0.482295
$333$	−5.71676	−0.313276
$334$	−1.52603	−0.0835004
$335$	−37.7624	−2.06318
$336$	0	0
$337$	16.5361	0.900781	0.450391	−	0.892832i	$-0.351285\pi$
$337$	16.5361	0.900781	0.450391	+	0.892832i	$0.351285\pi$
$338$	−1.00000	−0.0543928
$339$	12.5592	0.682122
$340$	−20.0847	−1.08925
$341$	0.263186	0.0142523
$342$	−8.11577	−0.438850
$343$	0	0
$344$	−4.10777	−0.221476
$345$	6.94751	0.374041
$346$	19.5892	1.05312
$347$	−6.50687	−0.349307	−0.174653	−	0.984630i	$-0.555881\pi$
$347$	−6.50687	−0.349307	−0.174653	+	0.984630i	$0.555881\pi$
$348$	19.0999	1.02386
$349$	−20.0266	−1.07200	−0.535999	−	0.844219i	$-0.680065\pi$
$349$	−20.0266	−1.07200	−0.535999	+	0.844219i	$0.680065\pi$
$350$	0	0
$351$	−2.68681	−0.143411
$352$	−1.85838	−0.0990519
$353$	19.2162	1.02278	0.511388	−	0.859350i	$-0.329132\pi$
$353$	19.2162	1.02278	0.511388	+	0.859350i	$0.329132\pi$
$354$	−21.4949	−1.14244
$355$	−26.7256	−1.41845
$356$	−12.4103	−0.657745
$357$	0	0
$358$	24.8071	1.31110
$359$	−18.5795	−0.980591	−0.490296	−	0.871556i	$-0.663111\pi$
$359$	−18.5795	−0.980591	−0.490296	+	0.871556i	$0.663111\pi$
$360$	−5.46612	−0.288090
$361$	2.02944	0.106812
$362$	−1.17282	−0.0616418
$363$	−16.4812	−0.865041
$364$	0	0
$365$	−34.8933	−1.82640
$366$	−17.7308	−0.926803
$367$	11.5356	0.602152	0.301076	−	0.953600i	$-0.402654\pi$
$367$	11.5356	0.602152	0.301076	+	0.953600i	$0.402654\pi$
$368$	1.02995	0.0536900
$369$	17.0615	0.888187
$370$	9.97694	0.518677
$371$	0	0
$372$	0.309297	0.0160363
$373$	−31.7433	−1.64361	−0.821804	−	0.569771i	$-0.807032\pi$
$373$	−31.7433	−1.64361	−0.821804	+	0.569771i	$0.807032\pi$
$374$	12.0847	0.624886
$375$	−3.10607	−0.160397
$376$	11.0548	0.570106
$377$	8.74547	0.450414
$378$	0	0
$379$	−9.75185	−0.500919	−0.250459	−	0.968127i	$-0.580582\pi$
$379$	−9.75185	−0.500919	−0.250459	+	0.968127i	$0.580582\pi$
$380$	14.1637	0.726584
$381$	23.3398	1.19573
$382$	−8.48528	−0.434145
$383$	−11.3571	−0.580321	−0.290161	−	0.956978i	$-0.593709\pi$
$383$	−11.3571	−0.580321	−0.290161	+	0.956978i	$0.593709\pi$
$384$	−2.18398	−0.111451
$385$	0	0
$386$	−7.43352	−0.378356
$387$	7.26980	0.369544
$388$	5.79296	0.294093
$389$	3.80927	0.193138	0.0965688	−	0.995326i	$-0.469213\pi$
$389$	3.80927	0.193138	0.0965688	+	0.995326i	$0.469213\pi$
$390$	−6.74547	−0.341570
$391$	−6.69760	−0.338712
$392$	0	0
$393$	30.9245	1.55993
$394$	20.5688	1.03624
$395$	−2.71038	−0.136374
$396$	3.28890	0.165273
$397$	20.8916	1.04852	0.524259	−	0.851559i	$-0.324343\pi$
$397$	20.8916	1.04852	0.524259	+	0.851559i	$0.324343\pi$
$398$	10.3449	0.518543
$399$	0	0
$400$	4.53953	0.226977
$401$	−19.8596	−0.991742	−0.495871	−	0.868396i	$-0.665151\pi$
$401$	−19.8596	−0.991742	−0.495871	+	0.868396i	$0.665151\pi$
$402$	26.7021	1.33178
$403$	0.141621	0.00705464
$404$	0.900594	0.0448062
$405$	−34.5221	−1.71542
$406$	0	0
$407$	−6.00300	−0.297558
$408$	14.2020	0.703106
$409$	26.7896	1.32466	0.662330	−	0.749212i	$-0.269568\pi$
$409$	26.7896	1.32466	0.662330	+	0.749212i	$0.269568\pi$
$410$	−29.7759	−1.47053
$411$	−24.4527	−1.20616
$412$	19.6817	0.969646
$413$	0	0
$414$	−1.82277	−0.0895844
$415$	−27.1422	−1.33236
$416$	−1.00000	−0.0490290
$417$	31.2573	1.53068
$418$	−8.52213	−0.416831
$419$	18.6884	0.912986	0.456493	−	0.889727i	$-0.349105\pi$
$419$	18.6884	0.912986	0.456493	+	0.889727i	$0.349105\pi$
$420$	0	0
$421$	−22.7269	−1.10764	−0.553820	−	0.832636i	$-0.686831\pi$
$421$	−22.7269	−1.10764	−0.553820	+	0.832636i	$0.686831\pi$
$422$	4.41583	0.214959
$423$	−19.5643	−0.951251
$424$	2.43741	0.118371
$425$	−29.5198	−1.43192
$426$	18.8979	0.915605
$427$	0	0
$428$	−10.9961	−0.531517
$429$	4.05866	0.195954
$430$	−12.6873	−0.611837
$431$	11.6817	0.562686	0.281343	−	0.959607i	$-0.409220\pi$
$431$	11.6817	0.562686	0.281343	+	0.959607i	$0.409220\pi$
$432$	−2.68681	−0.129269
$433$	−0.728650	−0.0350167	−0.0175083	−	0.999847i	$-0.505573\pi$
$433$	−0.728650	−0.0350167	−0.0175083	+	0.999847i	$0.505573\pi$
$434$	0	0
$435$	58.9923	2.82846
$436$	−14.8532	−0.711341
$437$	4.72314	0.225938
$438$	24.6733	1.17894
$439$	38.4271	1.83403	0.917014	−	0.398856i	$-0.130593\pi$
$439$	38.4271	1.83403	0.917014	+	0.398856i	$0.130593\pi$
$440$	−5.73981	−0.273635
$441$	0	0
$442$	6.50283	0.309308
$443$	−23.7511	−1.12845	−0.564225	−	0.825621i	$-0.690825\pi$
$443$	−23.7511	−1.12845	−0.564225	+	0.825621i	$0.690825\pi$
$444$	−7.05476	−0.334804
$445$	−38.3307	−1.81705
$446$	−20.5592	−0.973507
$447$	48.0278	2.27164
$448$	0	0
$449$	−14.7398	−0.695612	−0.347806	−	0.937567i	$-0.613073\pi$
$449$	−14.7398	−0.695612	−0.347806	+	0.937567i	$0.613073\pi$
$450$	−8.03391	−0.378722
$451$	17.9158	0.843622
$452$	5.75061	0.270486
$453$	−11.7654	−0.552785
$454$	−22.0073	−1.03285
$455$	0	0
$456$	−10.0153	−0.469008
$457$	37.3619	1.74771	0.873857	−	0.486183i	$-0.161611\pi$
$457$	37.3619	1.74771	0.873857	+	0.486183i	$0.161611\pi$
$458$	0.996103	0.0465449
$459$	17.4718	0.815515
$460$	3.18112	0.148321
$461$	−25.8709	−1.20493	−0.602465	−	0.798146i	$-0.705815\pi$
$461$	−25.8709	−1.20493	−0.602465	+	0.798146i	$0.705815\pi$
$462$	0	0
$463$	24.4041	1.13415	0.567077	−	0.823665i	$-0.308074\pi$
$463$	24.4041	1.13415	0.567077	+	0.823665i	$0.308074\pi$
$464$	8.74547	0.405998
$465$	0.955300	0.0443010
$466$	−15.7403	−0.729157
$467$	36.5660	1.69207	0.846035	−	0.533127i	$-0.178983\pi$
$467$	36.5660	1.69207	0.846035	+	0.533127i	$0.178983\pi$
$468$	1.76977	0.0818075
$469$	0	0
$470$	34.1439	1.57494
$471$	−3.07386	−0.141636
$472$	−9.84208	−0.453018
$473$	7.63380	0.351002
$474$	1.91653	0.0880289
$475$	20.8173	0.955164
$476$	0	0
$477$	−4.31365	−0.197508
$478$	−1.10388	−0.0504901
$479$	−5.50693	−0.251618	−0.125809	−	0.992054i	$-0.540153\pi$
$479$	−5.50693	−0.251618	−0.125809	+	0.992054i	$0.540153\pi$
$480$	−6.74547	−0.307887
$481$	−3.23023	−0.147286
$482$	4.68005	0.213171
$483$	0	0
$484$	−7.54643	−0.343019
$485$	17.8922	0.812444
$486$	16.3504	0.741670
$487$	−15.1981	−0.688694	−0.344347	−	0.938843i	$-0.611900\pi$
$487$	−15.1981	−0.688694	−0.344347	+	0.938843i	$0.611900\pi$
$488$	−8.11857	−0.367510
$489$	13.2889	0.600945
$490$	0	0
$491$	−41.7930	−1.88609	−0.943045	−	0.332666i	$-0.892052\pi$
$491$	−41.7930	−1.88609	−0.943045	+	0.332666i	$0.892052\pi$
$492$	21.0548	0.949222
$493$	−56.8703	−2.56131
$494$	−4.58579	−0.206324
$495$	10.1581	0.456574
$496$	0.141621	0.00635897
$497$	0	0
$498$	19.1924	0.860033
$499$	6.54790	0.293124	0.146562	−	0.989201i	$-0.453179\pi$
$499$	6.54790	0.293124	0.146562	+	0.989201i	$0.453179\pi$
$500$	−1.42221	−0.0636031
$501$	3.33281	0.148899
$502$	−7.84083	−0.349954
$503$	−0.920937	−0.0410625	−0.0205313	−	0.999789i	$-0.506536\pi$
$503$	−0.920937	−0.0410625	−0.0205313	+	0.999789i	$0.506536\pi$
$504$	0	0
$505$	2.78159	0.123779
$506$	−1.91404	−0.0850895
$507$	2.18398	0.0969939
$508$	10.6868	0.474150
$509$	5.23323	0.231959	0.115980	−	0.993252i	$-0.462999\pi$
$509$	5.23323	0.231959	0.115980	+	0.993252i	$0.462999\pi$
$510$	43.8646	1.94236
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−12.3211	−0.543991
$514$	25.8516	1.14027
$515$	60.7891	2.67869
$516$	8.97129	0.394939
$517$	−20.5439	−0.903522
$518$	0	0
$519$	−42.7823	−1.87794
$520$	−3.08861	−0.135445
$521$	36.3465	1.59237	0.796184	−	0.605055i	$-0.206849\pi$
$521$	36.3465	1.59237	0.796184	+	0.605055i	$0.206849\pi$
$522$	−15.4774	−0.677429
$523$	−7.01812	−0.306881	−0.153440	−	0.988158i	$-0.549035\pi$
$523$	−7.01812	−0.306881	−0.153440	+	0.988158i	$0.549035\pi$
$524$	14.1597	0.618569
$525$	0	0
$526$	15.3488	0.669239
$527$	−0.920937	−0.0401166
$528$	4.05866	0.176631
$529$	−21.9392	−0.953878
$530$	7.52822	0.327005
$531$	17.4182	0.755884
$532$	0	0
$533$	9.64055	0.417579
$534$	27.1039	1.17290
$535$	−33.9627	−1.46834
$536$	12.2263	0.528097
$537$	−54.1783	−2.33796
$538$	23.3997	1.00883
$539$	0	0
$540$	−8.29850	−0.357111
$541$	−0.329640	−0.0141723	−0.00708616	−	0.999975i	$-0.502256\pi$
$541$	−0.329640	−0.0141723	−0.00708616	+	0.999975i	$0.502256\pi$
$542$	−6.75450	−0.290131
$543$	2.56140	0.109920
$544$	6.50283	0.278806
$545$	−45.8759	−1.96511
$546$	0	0
$547$	37.5335	1.60482	0.802408	−	0.596776i	$-0.203552\pi$
$547$	37.5335	1.60482	0.802408	+	0.596776i	$0.203552\pi$
$548$	−11.1964	−0.478286
$549$	14.3680	0.613210
$550$	−8.43617	−0.359720
$551$	40.1048	1.70852
$552$	−2.24939	−0.0957406
$553$	0	0
$554$	−15.5955	−0.662588
$555$	−21.7894	−0.924910
$556$	14.3121	0.606968
$557$	−16.7587	−0.710091	−0.355045	−	0.934849i	$-0.615535\pi$
$557$	−16.7587	−0.710091	−0.355045	+	0.934849i	$0.615535\pi$
$558$	−0.250636	−0.0106103
$559$	4.10777	0.173740
$560$	0	0
$561$	−26.3928	−1.11430
$562$	20.2499	0.854191
$563$	20.4785	0.863067	0.431534	−	0.902097i	$-0.357973\pi$
$563$	20.4785	0.863067	0.431534	+	0.902097i	$0.357973\pi$
$564$	−24.1434	−1.01662
$565$	17.7614	0.747228
$566$	−24.0103	−1.00923
$567$	0	0
$568$	8.65296	0.363070
$569$	−10.7290	−0.449781	−0.224891	−	0.974384i	$-0.572203\pi$
$569$	−10.7290	−0.449781	−0.224891	+	0.974384i	$0.572203\pi$
$570$	−30.9333	−1.29565
$571$	5.00712	0.209542	0.104771	−	0.994496i	$-0.466589\pi$
$571$	5.00712	0.209542	0.104771	+	0.994496i	$0.466589\pi$
$572$	1.85838	0.0777027
$573$	18.5317	0.774172
$574$	0	0
$575$	4.67550	0.194982
$576$	1.76977	0.0737402
$577$	11.8133	0.491794	0.245897	−	0.969296i	$-0.420917\pi$
$577$	11.8133	0.491794	0.245897	+	0.969296i	$0.420917\pi$
$578$	−25.2868	−1.05179
$579$	16.2346	0.674689
$580$	27.0114	1.12159
$581$	0	0
$582$	−12.6517	−0.524430
$583$	−4.52964	−0.187598
$584$	11.2974	0.467490
$585$	5.46612	0.225996
$586$	19.9266	0.823160
$587$	31.6010	1.30431	0.652156	−	0.758085i	$-0.273865\pi$
$587$	31.6010	1.30431	0.652156	+	0.758085i	$0.273865\pi$
$588$	0	0
$589$	0.649444	0.0267599
$590$	−30.3984	−1.25148
$591$	−44.9217	−1.84783
$592$	−3.23023	−0.132762
$593$	7.07282	0.290446	0.145223	−	0.989399i	$-0.453610\pi$
$593$	7.07282	0.290446	0.145223	+	0.989399i	$0.453610\pi$
$594$	4.99310	0.204870
$595$	0	0
$596$	21.9910	0.900785
$597$	−22.5931	−0.924672
$598$	−1.02995	−0.0421179
$599$	18.5344	0.757295	0.378647	−	0.925541i	$-0.376389\pi$
$599$	18.5344	0.757295	0.378647	+	0.925541i	$0.376389\pi$
$600$	−9.91424	−0.404747
$601$	28.8126	1.17529	0.587646	−	0.809118i	$-0.300055\pi$
$601$	28.8126	1.17529	0.587646	+	0.809118i	$0.300055\pi$
$602$	0	0
$603$	−21.6378	−0.881157
$604$	−5.38712	−0.219199
$605$	−23.3080	−0.947605
$606$	−1.96688	−0.0798990
$607$	0.840462	0.0341133	0.0170567	−	0.999855i	$-0.494570\pi$
$607$	0.840462	0.0341133	0.0170567	+	0.999855i	$0.494570\pi$
$608$	−4.58579	−0.185978
$609$	0	0
$610$	−25.0751	−1.01526
$611$	−11.0548	−0.447228
$612$	−11.5085	−0.465203
$613$	−0.434413	−0.0175458	−0.00877289	−	0.999962i	$-0.502793\pi$
$613$	−0.434413	−0.0175458	−0.00877289	+	0.999962i	$0.502793\pi$
$614$	16.6570	0.672222
$615$	65.0300	2.62226
$616$	0	0
$617$	−8.42713	−0.339264	−0.169632	−	0.985508i	$-0.554258\pi$
$617$	−8.42713	−0.339264	−0.169632	+	0.985508i	$0.554258\pi$
$618$	−42.9844	−1.72908
$619$	−14.0306	−0.563938	−0.281969	−	0.959424i	$-0.590988\pi$
$619$	−14.0306	−0.563938	−0.281969	+	0.959424i	$0.590988\pi$
$620$	0.437413	0.0175669
$621$	−2.76728	−0.111047
$622$	26.5931	1.06628
$623$	0	0
$624$	2.18398	0.0874291
$625$	−27.0903	−1.08361
$626$	6.53543	0.261208
$627$	18.6122	0.743298
$628$	−1.40746	−0.0561638
$629$	21.0057	0.837550
$630$	0	0
$631$	−18.4144	−0.733064	−0.366532	−	0.930405i	$-0.619455\pi$
$631$	−18.4144	−0.733064	−0.366532	+	0.930405i	$0.619455\pi$
$632$	0.877538	0.0349066
$633$	−9.64408	−0.383318
$634$	27.7676	1.10279
$635$	33.0074	1.30986
$636$	−5.32326	−0.211081
$637$	0	0
$638$	−16.2524	−0.643438
$639$	−15.3137	−0.605801
$640$	−3.08861	−0.122088
$641$	28.6996	1.13357	0.566783	−	0.823867i	$-0.308188\pi$
$641$	28.6996	1.13357	0.566783	+	0.823867i	$0.308188\pi$
$642$	24.0153	0.947807
$643$	21.2363	0.837476	0.418738	−	0.908107i	$-0.362473\pi$
$643$	21.2363	0.837476	0.418738	+	0.908107i	$0.362473\pi$
$644$	0	0
$645$	27.7088	1.09103
$646$	29.8206	1.17327
$647$	18.0592	0.709979	0.354990	−	0.934870i	$-0.384484\pi$
$647$	18.0592	0.709979	0.354990	+	0.934870i	$0.384484\pi$
$648$	11.1772	0.439083
$649$	18.2903	0.717957
$650$	−4.53953	−0.178055
$651$	0	0
$652$	6.08472	0.238296
$653$	2.53772	0.0993085	0.0496542	−	0.998766i	$-0.484188\pi$
$653$	2.53772	0.0993085	0.0496542	+	0.998766i	$0.484188\pi$
$654$	32.4392	1.26847
$655$	43.7338	1.70882
$656$	9.64055	0.376400
$657$	−19.9938	−0.780031
$658$	0	0
$659$	46.3562	1.80578	0.902891	−	0.429870i	$-0.141441\pi$
$659$	46.3562	1.80578	0.902891	+	0.429870i	$0.141441\pi$
$660$	12.5356	0.487949
$661$	6.02877	0.234492	0.117246	−	0.993103i	$-0.462593\pi$
$661$	6.02877	0.234492	0.117246	+	0.993103i	$0.462593\pi$
$662$	−1.92608	−0.0748591
$663$	−14.2020	−0.551562
$664$	8.78783	0.341034
$665$	0	0
$666$	5.71676	0.221520
$667$	9.00741	0.348768
$668$	1.52603	0.0590437
$669$	44.9009	1.73597
$670$	37.7624	1.45889
$671$	15.0874	0.582441
$672$	0	0
$673$	−26.9122	−1.03739	−0.518694	−	0.854960i	$-0.673582\pi$
$673$	−26.9122	−1.03739	−0.518694	+	0.854960i	$0.673582\pi$
$674$	−16.5361	−0.636949
$675$	−12.1968	−0.469457
$676$	1.00000	0.0384615
$677$	−20.7021	−0.795645	−0.397823	−	0.917462i	$-0.630234\pi$
$677$	−20.7021	−0.795645	−0.397823	+	0.917462i	$0.630234\pi$
$678$	−12.5592	−0.482333
$679$	0	0
$680$	20.0847	0.770214
$681$	48.0634	1.84179
$682$	−0.263186	−0.0100779
$683$	33.0987	1.26649	0.633243	−	0.773953i	$-0.281723\pi$
$683$	33.0987	1.26649	0.633243	+	0.773953i	$0.281723\pi$
$684$	8.11577	0.310314
$685$	−34.5813	−1.32128
$686$	0	0
$687$	−2.17547	−0.0829993
$688$	4.10777	0.156607
$689$	−2.43741	−0.0928580
$690$	−6.94751	−0.264487
$691$	−5.50289	−0.209340	−0.104670	−	0.994507i	$-0.533379\pi$
$691$	−5.50289	−0.209340	−0.104670	+	0.994507i	$0.533379\pi$
$692$	−19.5892	−0.744668
$693$	0	0
$694$	6.50687	0.246997
$695$	44.2045	1.67677
$696$	−19.0999	−0.723981
$697$	−62.6908	−2.37458
$698$	20.0266	0.758017
$699$	34.3766	1.30024
$700$	0	0
$701$	−35.0184	−1.32263	−0.661314	−	0.750109i	$-0.730001\pi$
$701$	−35.0184	−1.32263	−0.661314	+	0.750109i	$0.730001\pi$
$702$	2.68681	0.101407
$703$	−14.8132	−0.558689
$704$	1.85838	0.0700403
$705$	−74.5696	−2.80845
$706$	−19.2162	−0.723211
$707$	0	0
$708$	21.4949	0.807827
$709$	−36.2529	−1.36151	−0.680753	−	0.732513i	$-0.738347\pi$
$709$	−36.2529	−1.36151	−0.680753	+	0.732513i	$0.738347\pi$
$710$	26.7256	1.00300
$711$	−1.55304	−0.0582435
$712$	12.4103	0.465096
$713$	0.145863	0.00546261
$714$	0	0
$715$	5.73981	0.214657
$716$	−24.8071	−0.927086
$717$	2.41084	0.0900346
$718$	18.5795	0.693383
$719$	12.4878	0.465715	0.232858	−	0.972511i	$-0.425192\pi$
$719$	12.4878	0.465715	0.232858	+	0.972511i	$0.425192\pi$
$720$	5.46612	0.203710
$721$	0	0
$722$	−2.02944	−0.0755278
$723$	−10.2211	−0.380128
$724$	1.17282	0.0435873
$725$	39.7003	1.47443
$726$	16.4812	0.611676
$727$	4.67279	0.173304	0.0866520	−	0.996239i	$-0.472383\pi$
$727$	4.67279	0.173304	0.0866520	+	0.996239i	$0.472383\pi$
$728$	0	0
$729$	−2.17729	−0.0806402
$730$	34.8933	1.29146
$731$	−26.7121	−0.987984
$732$	17.7308	0.655348
$733$	3.62425	0.133865	0.0669323	−	0.997758i	$-0.478679\pi$
$733$	3.62425	0.133865	0.0669323	+	0.997758i	$0.478679\pi$
$734$	−11.5356	−0.425786
$735$	0	0
$736$	−1.02995	−0.0379645
$737$	−22.7212	−0.836945
$738$	−17.0615	−0.628043
$739$	−13.8171	−0.508269	−0.254134	−	0.967169i	$-0.581791\pi$
$739$	−13.8171	−0.508269	−0.254134	+	0.967169i	$0.581791\pi$
$740$	−9.97694	−0.366760
$741$	10.0153	0.367920
$742$	0	0
$743$	22.8590	0.838614	0.419307	−	0.907845i	$-0.362273\pi$
$743$	22.8590	0.838614	0.419307	+	0.907845i	$0.362273\pi$
$744$	−0.309297	−0.0113394
$745$	67.9216	2.48845
$746$	31.7433	1.16221
$747$	−15.5524	−0.569032
$748$	−12.0847	−0.441861
$749$	0	0
$750$	3.10607	0.113418
$751$	−13.9814	−0.510189	−0.255095	−	0.966916i	$-0.582107\pi$
$751$	−13.9814	−0.510189	−0.255095	+	0.966916i	$0.582107\pi$
$752$	−11.0548	−0.403126
$753$	17.1242	0.624041
$754$	−8.74547	−0.318491
$755$	−16.6387	−0.605545
$756$	0	0
$757$	16.8475	0.612334	0.306167	−	0.951978i	$-0.400953\pi$
$757$	16.8475	0.612334	0.306167	+	0.951978i	$0.400953\pi$
$758$	9.75185	0.354203
$759$	4.18023	0.151733
$760$	−14.1637	−0.513772
$761$	28.6820	1.03972	0.519862	−	0.854251i	$-0.325984\pi$
$761$	28.6820	1.03972	0.519862	+	0.854251i	$0.325984\pi$
$762$	−23.3398	−0.845510
$763$	0	0
$764$	8.48528	0.306987
$765$	−35.5452	−1.28514
$766$	11.3571	0.410349
$767$	9.84208	0.355377
$768$	2.18398	0.0788076
$769$	17.1879	0.619811	0.309905	−	0.950767i	$-0.399703\pi$
$769$	17.1879	0.619811	0.309905	+	0.950767i	$0.399703\pi$
$770$	0	0
$771$	−56.4594	−2.03334
$772$	7.43352	0.267538
$773$	−18.3194	−0.658902	−0.329451	−	0.944173i	$-0.606864\pi$
$773$	−18.3194	−0.658902	−0.329451	+	0.944173i	$0.606864\pi$
$774$	−7.26980	−0.261307
$775$	0.642893	0.0230934
$776$	−5.79296	−0.207955
$777$	0	0
$778$	−3.80927	−0.136569
$779$	44.2095	1.58397
$780$	6.74547	0.241526
$781$	−16.0805	−0.575405
$782$	6.69760	0.239506
$783$	−23.4974	−0.839728
$784$	0	0
$785$	−4.34710	−0.155155
$786$	−30.9245	−1.10304
$787$	44.6046	1.58998	0.794990	−	0.606622i	$-0.207476\pi$
$787$	44.6046	1.58998	0.794990	+	0.606622i	$0.207476\pi$
$788$	−20.5688	−0.732732
$789$	−33.5215	−1.19340
$790$	2.71038	0.0964309
$791$	0	0
$792$	−3.28890	−0.116866
$793$	8.11857	0.288299
$794$	−20.8916	−0.741414
$795$	−16.4415	−0.583119
$796$	−10.3449	−0.366665
$797$	12.3667	0.438052	0.219026	−	0.975719i	$-0.429712\pi$
$797$	12.3667	0.438052	0.219026	+	0.975719i	$0.429712\pi$
$798$	0	0
$799$	71.8872	2.54319
$800$	−4.53953	−0.160497
$801$	−21.9634	−0.776037
$802$	19.8596	0.701268
$803$	−20.9949	−0.740893
$804$	−26.7021	−0.941709
$805$	0	0
$806$	−0.141621	−0.00498839
$807$	−51.1044	−1.79896
$808$	−0.900594	−0.0316828
$809$	41.9220	1.47390	0.736950	−	0.675947i	$-0.236265\pi$
$809$	41.9220	1.47390	0.736950	+	0.675947i	$0.236265\pi$
$810$	34.5221	1.21298
$811$	−26.3617	−0.925685	−0.462843	−	0.886440i	$-0.653170\pi$
$811$	−26.3617	−0.925685	−0.462843	+	0.886440i	$0.653170\pi$
$812$	0	0
$813$	14.7517	0.517365
$814$	6.00300	0.210405
$815$	18.7933	0.658302
$816$	−14.2020	−0.497171
$817$	18.8374	0.659036
$818$	−26.7896	−0.936676
$819$	0	0
$820$	29.7759	1.03982
$821$	4.68409	0.163476	0.0817380	−	0.996654i	$-0.473953\pi$
$821$	4.68409	0.163476	0.0817380	+	0.996654i	$0.473953\pi$
$822$	24.4527	0.852885
$823$	−33.2320	−1.15839	−0.579197	−	0.815187i	$-0.696634\pi$
$823$	−33.2320	−1.15839	−0.579197	+	0.815187i	$0.696634\pi$
$824$	−19.6817	−0.685643
$825$	18.4244	0.641456
$826$	0	0
$827$	4.33535	0.150755	0.0753775	−	0.997155i	$-0.475984\pi$
$827$	4.33535	0.150755	0.0753775	+	0.997155i	$0.475984\pi$
$828$	1.82277	0.0633458
$829$	−12.1942	−0.423524	−0.211762	−	0.977321i	$-0.567920\pi$
$829$	−12.1942	−0.423524	−0.211762	+	0.977321i	$0.567920\pi$
$830$	27.1422	0.942119
$831$	34.0602	1.18154
$832$	1.00000	0.0346688
$833$	0	0
$834$	−31.2573	−1.08235
$835$	4.71330	0.163111
$836$	8.52213	0.294744
$837$	−0.380508	−0.0131523
$838$	−18.6884	−0.645579
$839$	30.4397	1.05089	0.525447	−	0.850826i	$-0.323898\pi$
$839$	30.4397	1.05089	0.525447	+	0.850826i	$0.323898\pi$
$840$	0	0
$841$	47.4832	1.63735
$842$	22.7269	0.783220
$843$	−44.2254	−1.52320
$844$	−4.41583	−0.151999
$845$	3.08861	0.106252
$846$	19.5643	0.672636
$847$	0	0
$848$	−2.43741	−0.0837011
$849$	52.4379	1.79967
$850$	29.5198	1.01252
$851$	−3.32699	−0.114048
$852$	−18.8979	−0.647431
$853$	−17.3470	−0.593951	−0.296976	−	0.954885i	$-0.595978\pi$
$853$	−17.3470	−0.593951	−0.296976	+	0.954885i	$0.595978\pi$
$854$	0	0
$855$	25.0665	0.857255
$856$	10.9961	0.375839
$857$	13.0678	0.446389	0.223194	−	0.974774i	$-0.428352\pi$
$857$	13.0678	0.446389	0.223194	+	0.974774i	$0.428352\pi$
$858$	−4.05866	−0.138560
$859$	−25.3127	−0.863657	−0.431829	−	0.901956i	$-0.642131\pi$
$859$	−25.3127	−0.863657	−0.431829	+	0.901956i	$0.642131\pi$
$860$	12.6873	0.432634
$861$	0	0
$862$	−11.6817	−0.397879
$863$	−40.5891	−1.38167	−0.690834	−	0.723013i	$-0.742757\pi$
$863$	−40.5891	−1.38167	−0.690834	+	0.723013i	$0.742757\pi$
$864$	2.68681	0.0914070
$865$	−60.5033	−2.05717
$866$	0.728650	0.0247605
$867$	55.2258	1.87556
$868$	0	0
$869$	−1.63080	−0.0553211
$870$	−58.9923	−2.00003
$871$	−12.2263	−0.414274
$872$	14.8532	0.502994
$873$	10.2522	0.346984
$874$	−4.72314	−0.159763
$875$	0	0
$876$	−24.6733	−0.833634
$877$	3.22775	0.108993	0.0544967	−	0.998514i	$-0.482645\pi$
$877$	3.22775	0.108993	0.0544967	+	0.998514i	$0.482645\pi$
$878$	−38.4271	−1.29685
$879$	−43.5193	−1.46787
$880$	5.73981	0.193489
$881$	41.5142	1.39865	0.699324	−	0.714804i	$-0.253484\pi$
$881$	41.5142	1.39865	0.699324	+	0.714804i	$0.253484\pi$
$882$	0	0
$883$	13.7281	0.461986	0.230993	−	0.972955i	$-0.425803\pi$
$883$	13.7281	0.461986	0.230993	+	0.972955i	$0.425803\pi$
$884$	−6.50283	−0.218714
$885$	66.3894	2.23166
$886$	23.7511	0.797935
$887$	−42.9159	−1.44098	−0.720488	−	0.693467i	$-0.756082\pi$
$887$	−42.9159	−1.44098	−0.720488	+	0.693467i	$0.756082\pi$
$888$	7.05476	0.236742
$889$	0	0
$890$	38.3307	1.28485
$891$	−20.7715	−0.695872
$892$	20.5592	0.688373
$893$	−50.6948	−1.69644
$894$	−48.0278	−1.60629
$895$	−76.6196	−2.56111
$896$	0	0
$897$	2.24939	0.0751051
$898$	14.7398	0.491872
$899$	1.23854	0.0413077
$900$	8.03391	0.267797
$901$	15.8501	0.528042
$902$	−17.9158	−0.596531
$903$	0	0
$904$	−5.75061	−0.191262
$905$	3.62237	0.120412
$906$	11.7654	0.390878
$907$	32.8071	1.08934	0.544671	−	0.838650i	$-0.316654\pi$
$907$	32.8071	1.08934	0.544671	+	0.838650i	$0.316654\pi$
$908$	22.0073	0.730337
$909$	1.59384	0.0528644
$910$	0	0
$911$	1.87240	0.0620354	0.0310177	−	0.999519i	$-0.490125\pi$
$911$	1.87240	0.0620354	0.0310177	+	0.999519i	$0.490125\pi$
$912$	10.0153	0.331638
$913$	−16.3311	−0.540481
$914$	−37.3619	−1.23582
$915$	54.7635	1.81043
$916$	−0.996103	−0.0329122
$917$	0	0
$918$	−17.4718	−0.576656
$919$	46.1835	1.52345	0.761726	−	0.647900i	$-0.224352\pi$
$919$	46.1835	1.52345	0.761726	+	0.647900i	$0.224352\pi$
$920$	−3.18112	−0.104879
$921$	−36.3785	−1.19871
$922$	25.8709	0.852014
$923$	−8.65296	−0.284816
$924$	0	0
$925$	−14.6638	−0.482141
$926$	−24.4041	−0.801968
$927$	34.8319	1.14403
$928$	−8.74547	−0.287084
$929$	−38.6352	−1.26758	−0.633790	−	0.773505i	$-0.718502\pi$
$929$	−38.6352	−1.26758	−0.633790	+	0.773505i	$0.718502\pi$
$930$	−0.955300	−0.0313255
$931$	0	0
$932$	15.7403	0.515592
$933$	−58.0787	−1.90141
$934$	−36.5660	−1.19647
$935$	−37.3250	−1.22066
$936$	−1.76977	−0.0578466
$937$	2.65276	0.0866617	0.0433309	−	0.999061i	$-0.486203\pi$
$937$	2.65276	0.0866617	0.0433309	+	0.999061i	$0.486203\pi$
$938$	0	0
$939$	−14.2733	−0.465790
$940$	−34.1439	−1.11365
$941$	−24.1574	−0.787509	−0.393754	−	0.919216i	$-0.628824\pi$
$941$	−24.1574	−0.787509	−0.393754	+	0.919216i	$0.628824\pi$
$942$	3.07386	0.100152
$943$	9.92930	0.323343
$944$	9.84208	0.320332
$945$	0	0
$946$	−7.63380	−0.248196
$947$	−30.7990	−1.00083	−0.500417	−	0.865784i	$-0.666820\pi$
$947$	−30.7990	−1.00083	−0.500417	+	0.865784i	$0.666820\pi$
$948$	−1.91653	−0.0622459
$949$	−11.2974	−0.366729
$950$	−20.8173	−0.675403
$951$	−60.6439	−1.96651
$952$	0	0
$953$	36.9808	1.19793	0.598963	−	0.800776i	$-0.295579\pi$
$953$	36.9808	1.19793	0.598963	+	0.800776i	$0.295579\pi$
$954$	4.31365	0.139660
$955$	26.2078	0.848063
$956$	1.10388	0.0357019
$957$	35.4949	1.14739
$958$	5.50693	0.177921
$959$	0	0
$960$	6.74547	0.217709
$961$	−30.9799	−0.999353
$962$	3.23023	0.104147
$963$	−19.4605	−0.627107
$964$	−4.68005	−0.150734
$965$	22.9593	0.739085
$966$	0	0
$967$	−10.8805	−0.349895	−0.174947	−	0.984578i	$-0.555975\pi$
$967$	−10.8805	−0.349895	−0.174947	+	0.984578i	$0.555975\pi$
$968$	7.54643	0.242551
$969$	−65.1275	−2.09220
$970$	−17.8922	−0.574485
$971$	−19.1195	−0.613572	−0.306786	−	0.951778i	$-0.599254\pi$
$971$	−19.1195	−0.613572	−0.306786	+	0.951778i	$0.599254\pi$
$972$	−16.3504	−0.524440
$973$	0	0
$974$	15.1981	0.486980
$975$	9.91424	0.317510
$976$	8.11857	0.259869
$977$	14.2754	0.456712	0.228356	−	0.973578i	$-0.426665\pi$
$977$	14.2754	0.456712	0.228356	+	0.973578i	$0.426665\pi$
$978$	−13.2889	−0.424932
$979$	−23.0631	−0.737099
$980$	0	0
$981$	−26.2868	−0.839272
$982$	41.7930	1.33367
$983$	−16.7685	−0.534833	−0.267416	−	0.963581i	$-0.586170\pi$
$983$	−16.7685	−0.534833	−0.267416	+	0.963581i	$0.586170\pi$
$984$	−21.0548	−0.671201
$985$	−63.5289	−2.02420
$986$	56.8703	1.81112
$987$	0	0
$988$	4.58579	0.145893
$989$	4.23081	0.134532
$990$	−10.1581	−0.322846
$991$	49.7063	1.57897	0.789486	−	0.613769i	$-0.210347\pi$
$991$	49.7063	1.57897	0.789486	+	0.613769i	$0.210347\pi$
$992$	−0.141621	−0.00449647
$993$	4.20651	0.133490
$994$	0	0
$995$	−31.9514	−1.01293
$996$	−19.1924	−0.608135
$997$	−10.1594	−0.321750	−0.160875	−	0.986975i	$-0.551432\pi$
$997$	−10.1594	−0.321750	−0.160875	+	0.986975i	$0.551432\pi$
$998$	−6.54790	−0.207270
$999$	8.67901	0.274592

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1274.2.a.u.1.3	yes	4
7.2	even	3		1274.2.f.z.1145.2		8
7.3	odd	6		1274.2.f.ba.79.3		8
7.4	even	3		1274.2.f.z.79.2		8
7.5	odd	6		1274.2.f.ba.1145.3		8
7.6	odd	2		1274.2.a.t.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1274.2.a.t.1.2	✓	4	7.6	odd	2
1274.2.a.u.1.3	yes	4	1.1	even	1	trivial
1274.2.f.z.79.2		8	7.4	even	3
1274.2.f.z.1145.2		8	7.2	even	3
1274.2.f.ba.79.3		8	7.3	odd	6
1274.2.f.ba.1145.3		8	7.5	odd	6

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:	\( N \)	\(=\)	\( 1274 = 2 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1274.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +2.18398 q^{3} +1.00000 q^{4} +3.08861 q^{5} -2.18398 q^{6} -1.00000 q^{8} +1.76977 q^{9} -3.08861 q^{10} +1.85838 q^{11} +2.18398 q^{12} +1.00000 q^{13} +6.74547 q^{15} +1.00000 q^{16} -6.50283 q^{17} -1.76977 q^{18} +4.58579 q^{19} +3.08861 q^{20} -1.85838 q^{22} +1.02995 q^{23} -2.18398 q^{24} +4.53953 q^{25} -1.00000 q^{26} -2.68681 q^{27} +8.74547 q^{29} -6.74547 q^{30} +0.141621 q^{31} -1.00000 q^{32} +4.05866 q^{33} +6.50283 q^{34} +1.76977 q^{36} -3.23023 q^{37} -4.58579 q^{38} +2.18398 q^{39} -3.08861 q^{40} +9.64055 q^{41} +4.10777 q^{43} +1.85838 q^{44} +5.46612 q^{45} -1.02995 q^{46} -11.0548 q^{47} +2.18398 q^{48} -4.53953 q^{50} -14.2020 q^{51} +1.00000 q^{52} -2.43741 q^{53} +2.68681 q^{54} +5.73981 q^{55} +10.0153 q^{57} -8.74547 q^{58} +9.84208 q^{59} +6.74547 q^{60} +8.11857 q^{61} -0.141621 q^{62} +1.00000 q^{64} +3.08861 q^{65} -4.05866 q^{66} -12.2263 q^{67} -6.50283 q^{68} +2.24939 q^{69} -8.65296 q^{71} -1.76977 q^{72} -11.2974 q^{73} +3.23023 q^{74} +9.91424 q^{75} +4.58579 q^{76} -2.18398 q^{78} -0.877538 q^{79} +3.08861 q^{80} -11.1772 q^{81} -9.64055 q^{82} -8.78783 q^{83} -20.0847 q^{85} -4.10777 q^{86} +19.0999 q^{87} -1.85838 q^{88} -12.4103 q^{89} -5.46612 q^{90} +1.02995 q^{92} +0.309297 q^{93} +11.0548 q^{94} +14.1637 q^{95} -2.18398 q^{96} +5.79296 q^{97} +3.28890 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{6} - 4 q^{8} + 6 q^{9} - 6 q^{11} + 2 q^{12} + 4 q^{13} - 8 q^{15} + 4 q^{16} - 8 q^{17} - 6 q^{18} + 24 q^{19} + 6 q^{22} + 2 q^{23} - 2 q^{24} + 16 q^{25} - 4 q^{26}+ \cdots + 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 + 2 * q^3 + 4 * q^4 - 2 * q^6 - 4 * q^8 + 6 * q^9 - 6 * q^11 + 2 * q^12 + 4 * q^13 - 8 * q^15 + 4 * q^16 - 8 * q^17 - 6 * q^18 + 24 * q^19 + 6 * q^22 + 2 * q^23 - 2 * q^24 + 16 * q^25 - 4 * q^26 + 14 * q^27 + 8 * q^30 + 14 * q^31 - 4 * q^32 + 6 * q^33 + 8 * q^34 + 6 * q^36 - 14 * q^37 - 24 * q^38 + 2 * q^39 + 6 * q^41 + 4 * q^43 - 6 * q^44 - 12 * q^45 - 2 * q^46 - 6 * q^47 + 2 * q^48 - 16 * q^50 + 4 * q^51 + 4 * q^52 + 16 * q^53 - 14 * q^54 + 24 * q^55 + 12 * q^57 + 16 * q^59 - 8 * q^60 + 18 * q^61 - 14 * q^62 + 4 * q^64 - 6 * q^66 - 22 * q^67 - 8 * q^68 + 10 * q^69 + 16 * q^71 - 6 * q^72 + 10 * q^73 + 14 * q^74 + 42 * q^75 + 24 * q^76 - 2 * q^78 + 10 * q^79 - 20 * q^81 - 6 * q^82 + 20 * q^83 - 40 * q^85 - 4 * q^86 - 8 * q^87 + 6 * q^88 - 16 * q^89 + 12 * q^90 + 2 * q^92 - 2 * q^93 + 6 * q^94 - 4 * q^95 - 2 * q^96 + 2 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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