LMFDB - Embedded newform 1911.2.a.s.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1911,2,Mod(1,1911)] 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("1911.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1911, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1911 = 3 \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1911.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,1,-4,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$15.2594118263$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.17428.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 4x + 6 $ x^4 - x^3 - 6x^2 + 4x + 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 273)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.36865$ of defining polynomial
Character	$\chi$	$=$	1911.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.61050 q^{2} -1.00000 q^{3} +4.81471 q^{4} +3.81471 q^{5} -2.61050 q^{6} +7.34780 q^{8} +1.00000 q^{9} +9.95830 q^{10} -4.73730 q^{11} -4.81471 q^{12} -1.00000 q^{13} -3.81471 q^{15} +9.55201 q^{16} +5.22100 q^{17} +2.61050 q^{18} -2.92259 q^{19} +18.3667 q^{20} -12.3667 q^{22} +3.33101 q^{23} -7.34780 q^{24} +9.55201 q^{25} -2.61050 q^{26} -1.00000 q^{27} -0.922589 q^{29} -9.95830 q^{30} +7.51941 q^{31} +10.2399 q^{32} +4.73730 q^{33} +13.6294 q^{34} +4.81471 q^{36} +0.154821 q^{37} -7.62942 q^{38} +1.00000 q^{39} +28.0297 q^{40} -6.36672 q^{41} -6.55201 q^{43} -22.8087 q^{44} +3.81471 q^{45} +8.69560 q^{46} -9.03571 q^{47} -9.55201 q^{48} +24.9355 q^{50} -5.22100 q^{51} -4.81471 q^{52} +8.55201 q^{53} -2.61050 q^{54} -18.0714 q^{55} +2.92259 q^{57} -2.40842 q^{58} -3.95830 q^{59} -18.3667 q^{60} -12.4420 q^{61} +19.6294 q^{62} +7.62729 q^{64} -3.81471 q^{65} +12.3667 q^{66} -10.6620 q^{67} +25.1376 q^{68} -3.33101 q^{69} -6.58248 q^{71} +7.34780 q^{72} +7.73517 q^{73} +0.404161 q^{74} -9.55201 q^{75} -14.0714 q^{76} +2.61050 q^{78} +13.3646 q^{79} +36.4381 q^{80} +1.00000 q^{81} -16.6203 q^{82} +1.40629 q^{83} +19.9166 q^{85} -17.1040 q^{86} +0.922589 q^{87} -34.8087 q^{88} +1.96953 q^{89} +9.95830 q^{90} +16.0378 q^{92} -7.51941 q^{93} -23.5877 q^{94} -11.1488 q^{95} -10.2399 q^{96} +2.11001 q^{97} -4.73730 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} - 4 q^{3} + 7 q^{4} + 3 q^{5} - q^{6} + 3 q^{8} + 4 q^{9} + 4 q^{10} - 2 q^{11} - 7 q^{12} - 4 q^{13} - 3 q^{15} + 9 q^{16} + 2 q^{17} + q^{18} - 7 q^{19} + 32 q^{20} - 8 q^{22} + 3 q^{23}+ \cdots - 2 q^{99}+O(q^{100}) $ 4 * q + q^2 - 4 * q^3 + 7 * q^4 + 3 * q^5 - q^6 + 3 * q^8 + 4 * q^9 + 4 * q^10 - 2 * q^11 - 7 * q^12 - 4 * q^13 - 3 * q^15 + 9 * q^16 + 2 * q^17 + q^18 - 7 * q^19 + 32 * q^20 - 8 * q^22 + 3 * q^23 - 3 * q^24 + 9 * q^25 - q^26 - 4 * q^27 + q^29 - 4 * q^30 - 3 * q^31 + 7 * q^32 + 2 * q^33 + 30 * q^34 + 7 * q^36 + 10 * q^37 - 6 * q^38 + 4 * q^39 + 14 * q^40 + 16 * q^41 + 3 * q^43 - 12 * q^44 + 3 * q^45 - 18 * q^46 - 5 * q^47 - 9 * q^48 + 13 * q^50 - 2 * q^51 - 7 * q^52 + 5 * q^53 - q^54 - 10 * q^55 + 7 * q^57 - 4 * q^58 + 20 * q^59 - 32 * q^60 - 12 * q^61 + 54 * q^62 + 5 * q^64 - 3 * q^65 + 8 * q^66 - 22 * q^67 + 10 * q^68 - 3 * q^69 + 3 * q^72 + 13 * q^73 - 6 * q^74 - 9 * q^75 + 6 * q^76 + q^78 + 11 * q^79 + 42 * q^80 + 4 * q^81 - 10 * q^82 - q^83 + 8 * q^85 - 10 * q^86 - q^87 - 60 * q^88 + 5 * q^89 + 4 * q^90 + 34 * q^92 + 3 * q^93 - 34 * q^94 + 13 * q^95 - 7 * q^96 + 17 * q^97 - 2 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.61050	1.84590	0.922951	−	0.384917i	$-0.125770\pi$
$2$	2.61050	1.84590	0.922951	+	0.384917i	$0.125770\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	4.81471	2.40735
$5$	3.81471	1.70599	0.852995	−	0.521919i	$-0.174784\pi$
$5$	3.81471	1.70599	0.852995	+	0.521919i	$0.174784\pi$
$6$	−2.61050	−1.06573
$7$	0	0
$7$	0	0
$8$	7.34780	2.59784
$9$	1.00000	0.333333
$10$	9.95830	3.14909
$11$	−4.73730	−1.42835	−0.714175	−	0.699968i	$-0.753198\pi$
$11$	−4.73730	−1.42835	−0.714175	+	0.699968i	$0.753198\pi$
$12$	−4.81471	−1.38989
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−3.81471	−0.984954
$16$	9.55201	2.38800
$17$	5.22100	1.26628	0.633139	−	0.774038i	$-0.281766\pi$
$17$	5.22100	1.26628	0.633139	+	0.774038i	$0.281766\pi$
$18$	2.61050	0.615301
$19$	−2.92259	−0.670488	−0.335244	−	0.942131i	$-0.608819\pi$
$19$	−2.92259	−0.670488	−0.335244	+	0.942131i	$0.608819\pi$
$20$	18.3667	4.10692
$21$	0	0
$22$	−12.3667	−2.63659
$23$	3.33101	0.694563	0.347282	−	0.937761i	$-0.387105\pi$
$23$	3.33101	0.694563	0.347282	+	0.937761i	$0.387105\pi$
$24$	−7.34780	−1.49986
$25$	9.55201	1.91040
$26$	−2.61050	−0.511961
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−0.922589	−0.171321	−0.0856603	−	0.996324i	$-0.527300\pi$
$29$	−0.922589	−0.171321	−0.0856603	+	0.996324i	$0.527300\pi$
$30$	−9.95830	−1.81813
$31$	7.51941	1.35053	0.675263	−	0.737577i	$-0.264030\pi$
$31$	7.51941	1.35053	0.675263	+	0.737577i	$0.264030\pi$
$32$	10.2399	1.81018
$33$	4.73730	0.824658
$34$	13.6294	2.33743
$35$	0	0
$36$	4.81471	0.802452
$37$	0.154821	0.0254525	0.0127262	−	0.999919i	$-0.495949\pi$
$37$	0.154821	0.0254525	0.0127262	+	0.999919i	$0.495949\pi$
$38$	−7.62942	−1.23766
$39$	1.00000	0.160128
$40$	28.0297	4.43189
$41$	−6.36672	−0.994314	−0.497157	−	0.867661i	$-0.665623\pi$
$41$	−6.36672	−0.994314	−0.497157	+	0.867661i	$0.665623\pi$
$42$	0	0
$43$	−6.55201	−0.999172	−0.499586	−	0.866264i	$-0.666514\pi$
$43$	−6.55201	−0.999172	−0.499586	+	0.866264i	$0.666514\pi$
$44$	−22.8087	−3.43854
$45$	3.81471	0.568663
$46$	8.69560	1.28210
$47$	−9.03571	−1.31799	−0.658997	−	0.752146i	$-0.729019\pi$
$47$	−9.03571	−1.31799	−0.658997	+	0.752146i	$0.729019\pi$
$48$	−9.55201	−1.37871
$49$	0	0
$50$	24.9355	3.52641
$51$	−5.22100	−0.731086
$52$	−4.81471	−0.667680
$53$	8.55201	1.17471	0.587354	−	0.809330i	$-0.300170\pi$
$53$	8.55201	1.17471	0.587354	+	0.809330i	$0.300170\pi$
$54$	−2.61050	−0.355244
$55$	−18.0714	−2.43675
$56$	0	0
$57$	2.92259	0.387106
$58$	−2.40842	−0.316241
$59$	−3.95830	−0.515327	−0.257663	−	0.966235i	$-0.582953\pi$
$59$	−3.95830	−0.515327	−0.257663	+	0.966235i	$0.582953\pi$
$60$	−18.3667	−2.37113
$61$	−12.4420	−1.59303	−0.796517	−	0.604616i	$-0.793327\pi$
$61$	−12.4420	−1.59303	−0.796517	+	0.604616i	$0.793327\pi$
$62$	19.6294	2.49294
$63$	0	0
$64$	7.62729	0.953411
$65$	−3.81471	−0.473156
$66$	12.3667	1.52224
$67$	−10.6620	−1.30257	−0.651286	−	0.758832i	$-0.725770\pi$
$67$	−10.6620	−1.30257	−0.651286	+	0.758832i	$0.725770\pi$
$68$	25.1376	3.04838
$69$	−3.33101	−0.401006
$70$	0	0
$71$	−6.58248	−0.781196	−0.390598	−	0.920561i	$-0.627732\pi$
$71$	−6.58248	−0.781196	−0.390598	+	0.920561i	$0.627732\pi$
$72$	7.34780	0.865946
$73$	7.73517	0.905333	0.452667	−	0.891680i	$-0.350473\pi$
$73$	7.73517	0.905333	0.452667	+	0.891680i	$0.350473\pi$
$74$	0.404161	0.0469828
$75$	−9.55201	−1.10297
$76$	−14.0714	−1.61410
$77$	0	0
$78$	2.61050	0.295581
$79$	13.3646	1.50363	0.751817	−	0.659372i	$-0.229178\pi$
$79$	13.3646	1.50363	0.751817	+	0.659372i	$0.229178\pi$
$80$	36.4381	4.07391
$81$	1.00000	0.111111
$82$	−16.6203	−1.83541
$83$	1.40629	0.154360	0.0771802	−	0.997017i	$-0.475408\pi$
$83$	1.40629	0.154360	0.0771802	+	0.997017i	$0.475408\pi$
$84$	0	0
$85$	19.9166	2.16026
$86$	−17.1040	−1.84437
$87$	0.922589	0.0989120
$88$	−34.8087	−3.71062
$89$	1.96953	0.208770	0.104385	−	0.994537i	$-0.466713\pi$
$89$	1.96953	0.208770	0.104385	+	0.994537i	$0.466713\pi$
$90$	9.95830	1.04970
$91$	0	0
$92$	16.0378	1.67206
$93$	−7.51941	−0.779727
$94$	−23.5877	−2.43289
$95$	−11.1488	−1.14385
$96$	−10.2399	−1.04511
$97$	2.11001	0.214239	0.107119	−	0.994246i	$-0.465837\pi$
$97$	2.11001	0.214239	0.107119	+	0.994246i	$0.465837\pi$
$98$	0	0
$99$	−4.73730	−0.476116
$100$	45.9901	4.59901
$101$	−0.850419	−0.0846198	−0.0423099	−	0.999105i	$-0.513472\pi$
$101$	−0.850419	−0.0846198	−0.0423099	+	0.999105i	$0.513472\pi$
$102$	−13.6294	−1.34951
$103$	1.47460	0.145296	0.0726482	−	0.997358i	$-0.476855\pi$
$103$	1.47460	0.145296	0.0726482	+	0.997358i	$0.476855\pi$
$104$	−7.34780	−0.720511
$105$	0	0
$106$	22.3250	2.16840
$107$	2.62418	0.253689	0.126844	−	0.991923i	$-0.459515\pi$
$107$	2.62418	0.253689	0.126844	+	0.991923i	$0.459515\pi$
$108$	−4.81471	−0.463296
$109$	−17.9166	−1.71610	−0.858049	−	0.513567i	$-0.828324\pi$
$109$	−17.9166	−1.71610	−0.858049	+	0.513567i	$0.828324\pi$
$110$	−47.1754	−4.49800
$111$	−0.154821	−0.0146950
$112$	0	0
$113$	0.922589	0.0867899	0.0433950	−	0.999058i	$-0.486183\pi$
$113$	0.922589	0.0867899	0.0433950	+	0.999058i	$0.486183\pi$
$114$	7.62942	0.714561
$115$	12.7068	1.18492
$116$	−4.44200	−0.412429
$117$	−1.00000	−0.0924500
$118$	−10.3331	−0.951242
$119$	0	0
$120$	−28.0297	−2.55875
$121$	11.4420	1.04018
$122$	−32.4798	−2.94059
$123$	6.36672	0.574068
$124$	36.2038	3.25119
$125$	17.3646	1.55314
$126$	0	0
$127$	17.4746	1.55062	0.775310	−	0.631581i	$-0.217594\pi$
$127$	17.4746	1.55062	0.775310	+	0.631581i	$0.217594\pi$
$128$	−0.568798	−0.0502751
$129$	6.55201	0.576872
$130$	−9.95830	−0.873401
$131$	−0.967402	−0.0845223	−0.0422611	−	0.999107i	$-0.513456\pi$
$131$	−0.967402	−0.0845223	−0.0422611	+	0.999107i	$0.513456\pi$
$132$	22.8087	1.98524
$133$	0	0
$134$	−27.8332	−2.40442
$135$	−3.81471	−0.328318
$136$	38.3629	3.28959
$137$	3.29628	0.281620	0.140810	−	0.990037i	$-0.455029\pi$
$137$	3.29628	0.281620	0.140810	+	0.990037i	$0.455029\pi$
$138$	−8.69560	−0.740218
$139$	−0.370581	−0.0314323	−0.0157161	−	0.999876i	$-0.505003\pi$
$139$	−0.370581	−0.0314323	−0.0157161	+	0.999876i	$0.505003\pi$
$140$	0	0
$141$	9.03571	0.760944
$142$	−17.1836	−1.44201
$143$	4.73730	0.396153
$144$	9.55201	0.796001
$145$	−3.51941	−0.292271
$146$	20.1927	1.67116
$147$	0	0
$148$	0.745420	0.0612732
$149$	−15.7425	−1.28968	−0.644840	−	0.764318i	$-0.723076\pi$
$149$	−15.7425	−1.28968	−0.644840	+	0.764318i	$0.723076\pi$
$150$	−24.9355	−2.03598
$151$	10.2914	0.837505	0.418753	−	0.908100i	$-0.362467\pi$
$151$	10.2914	0.837505	0.418753	+	0.908100i	$0.362467\pi$
$152$	−21.4746	−1.74182
$153$	5.22100	0.422093
$154$	0	0
$155$	28.6844	2.30398
$156$	4.81471	0.385485
$157$	11.4137	0.910909	0.455455	−	0.890259i	$-0.349477\pi$
$157$	11.4137	0.910909	0.455455	+	0.890259i	$0.349477\pi$
$158$	34.8883	2.77556
$159$	−8.55201	−0.678218
$160$	39.0623	3.08815
$161$	0	0
$162$	2.61050	0.205100
$163$	−13.4746	−1.05541	−0.527706	−	0.849427i	$-0.676948\pi$
$163$	−13.4746	−1.05541	−0.527706	+	0.849427i	$0.676948\pi$
$164$	−30.6539	−2.39367
$165$	18.0714	1.40686
$166$	3.67112	0.284934
$167$	−19.1905	−1.48501	−0.742504	−	0.669842i	$-0.766362\pi$
$167$	−19.1905	−1.48501	−0.742504	+	0.669842i	$0.766362\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	51.9923	3.98763
$171$	−2.92259	−0.223496
$172$	−31.5460	−2.40536
$173$	19.5124	1.48350	0.741752	−	0.670675i	$-0.233995\pi$
$173$	19.5124	1.48350	0.741752	+	0.670675i	$0.233995\pi$
$174$	2.40842	0.182582
$175$	0	0
$176$	−45.2507	−3.41090
$177$	3.95830	0.297524
$178$	5.14146	0.385369
$179$	−16.5856	−1.23967	−0.619833	−	0.784734i	$-0.712799\pi$
$179$	−16.5856	−1.23967	−0.619833	+	0.784734i	$0.712799\pi$
$180$	18.3667	1.36897
$181$	2.81684	0.209374	0.104687	−	0.994505i	$-0.466616\pi$
$181$	2.81684	0.209374	0.104687	+	0.994505i	$0.466616\pi$
$182$	0	0
$183$	12.4420	0.919739
$184$	24.4756	1.80436
$185$	0.590599	0.0434217
$186$	−19.6294	−1.43930
$187$	−24.7334	−1.80869
$188$	−43.5043	−3.17288
$189$	0	0
$190$	−29.1040	−2.11143
$191$	15.1601	1.09694	0.548472	−	0.836169i	$-0.315210\pi$
$191$	15.1601	1.09694	0.548472	+	0.836169i	$0.315210\pi$
$192$	−7.62729	−0.550452
$193$	0.0651962	0.00469293	0.00234646	−	0.999997i	$-0.499253\pi$
$193$	0.0651962	0.00469293	0.00234646	+	0.999997i	$0.499253\pi$
$194$	5.50818	0.395464
$195$	3.81471	0.273177
$196$	0	0
$197$	−17.1415	−1.22128	−0.610639	−	0.791909i	$-0.709087\pi$
$197$	−17.1415	−1.22128	−0.610639	+	0.791909i	$0.709087\pi$
$198$	−12.3667	−0.878864
$199$	−6.44200	−0.456661	−0.228331	−	0.973584i	$-0.573327\pi$
$199$	−6.44200	−0.456661	−0.228331	+	0.973584i	$0.573327\pi$
$200$	70.1862	4.96292
$201$	10.6620	0.752041
$202$	−2.22002	−0.156200
$203$	0	0
$204$	−25.1376	−1.75998
$205$	−24.2872	−1.69629
$206$	3.84944	0.268203
$207$	3.33101	0.231521
$208$	−9.55201	−0.662313
$209$	13.8452	0.957691
$210$	0	0
$211$	−16.0266	−1.10332	−0.551659	−	0.834070i	$-0.686005\pi$
$211$	−16.0266	−1.10332	−0.551659	+	0.834070i	$0.686005\pi$
$212$	41.1754	2.82794
$213$	6.58248	0.451024
$214$	6.85042	0.468285
$215$	−24.9940	−1.70458
$216$	−7.34780	−0.499954
$217$	0	0
$218$	−46.7713	−3.16775
$219$	−7.73517	−0.522694
$220$	−87.0086	−5.86612
$221$	−5.22100	−0.351202
$222$	−0.404161	−0.0271255
$223$	−20.0266	−1.34108	−0.670540	−	0.741873i	$-0.733938\pi$
$223$	−20.0266	−1.34108	−0.670540	+	0.741873i	$0.733938\pi$
$224$	0	0
$225$	9.55201	0.636801
$226$	2.40842	0.160206
$227$	−19.2171	−1.27549	−0.637743	−	0.770249i	$-0.720132\pi$
$227$	−19.2171	−1.27549	−0.637743	+	0.770249i	$0.720132\pi$
$228$	14.0714	0.931902
$229$	−5.25884	−0.347514	−0.173757	−	0.984789i	$-0.555591\pi$
$229$	−5.25884	−0.347514	−0.173757	+	0.984789i	$0.555591\pi$
$230$	33.1712	2.18724
$231$	0	0
$232$	−6.77900	−0.445063
$233$	−26.6234	−1.74416	−0.872079	−	0.489365i	$-0.837229\pi$
$233$	−26.6234	−1.74416	−0.872079	+	0.489365i	$0.837229\pi$
$234$	−2.61050	−0.170654
$235$	−34.4686	−2.24848
$236$	−19.0581	−1.24057
$237$	−13.3646	−0.868123
$238$	0	0
$239$	4.29104	0.277564	0.138782	−	0.990323i	$-0.455681\pi$
$239$	4.29104	0.277564	0.138782	+	0.990323i	$0.455681\pi$
$240$	−36.4381	−2.35207
$241$	−7.52367	−0.484642	−0.242321	−	0.970196i	$-0.577909\pi$
$241$	−7.52367	−0.484642	−0.242321	+	0.970196i	$0.577909\pi$
$242$	29.8693	1.92007
$243$	−1.00000	−0.0641500
$244$	−59.9046	−3.83500
$245$	0	0
$246$	16.6203	1.05967
$247$	2.92259	0.185960
$248$	55.2511	3.50845
$249$	−1.40629	−0.0891200
$250$	45.3303	2.86694
$251$	−11.3198	−0.714498	−0.357249	−	0.934009i	$-0.616285\pi$
$251$	−11.3198	−0.714498	−0.357249	+	0.934009i	$0.616285\pi$
$252$	0	0
$253$	−15.7800	−0.992079
$254$	45.6174	2.86229
$255$	−19.9166	−1.24723
$256$	−16.7394	−1.04621
$257$	24.8504	1.55013	0.775063	−	0.631884i	$-0.217718\pi$
$257$	24.8504	1.55013	0.775063	+	0.631884i	$0.217718\pi$
$258$	17.1040	1.06485
$259$	0	0
$260$	−18.3667	−1.13906
$261$	−0.922589	−0.0571068
$262$	−2.52540	−0.156020
$263$	17.1762	1.05913	0.529565	−	0.848270i	$-0.322355\pi$
$263$	17.1762	1.05913	0.529565	+	0.848270i	$0.322355\pi$
$264$	34.8087	2.14233
$265$	32.6234	2.00404
$266$	0	0
$267$	−1.96953	−0.120533
$268$	−51.3345	−3.13575
$269$	7.74640	0.472306	0.236153	−	0.971716i	$-0.424113\pi$
$269$	7.74640	0.472306	0.236153	+	0.971716i	$0.424113\pi$
$270$	−9.95830	−0.606043
$271$	29.7008	1.80420	0.902099	−	0.431530i	$-0.142026\pi$
$271$	29.7008	1.80420	0.902099	+	0.431530i	$0.142026\pi$
$272$	49.8710	3.02388
$273$	0	0
$274$	8.60494	0.519844
$275$	−45.2507	−2.72872
$276$	−16.0378	−0.965364
$277$	25.1488	1.51105	0.755523	−	0.655122i	$-0.227383\pi$
$277$	25.1488	1.51105	0.755523	+	0.655122i	$0.227383\pi$
$278$	−0.967402	−0.0580209
$279$	7.51941	0.450175
$280$	0	0
$281$	8.40030	0.501120	0.250560	−	0.968101i	$-0.419385\pi$
$281$	8.40030	0.501120	0.250560	+	0.968101i	$0.419385\pi$
$282$	23.5877	1.40463
$283$	21.3912	1.27157	0.635787	−	0.771864i	$-0.280676\pi$
$283$	21.3912	1.27157	0.635787	+	0.771864i	$0.280676\pi$
$284$	−31.6927	−1.88062
$285$	11.1488	0.660400
$286$	12.3667	0.731259
$287$	0	0
$288$	10.2399	0.603393
$289$	10.2588	0.603461
$290$	−9.18742	−0.539504
$291$	−2.11001	−0.123691
$292$	37.2426	2.17946
$293$	9.59895	0.560777	0.280388	−	0.959887i	$-0.409537\pi$
$293$	9.59895	0.560777	0.280388	+	0.959887i	$0.409537\pi$
$294$	0	0
$295$	−15.0998	−0.879142
$296$	1.13760	0.0661215
$297$	4.73730	0.274886
$298$	−41.0959	−2.38062
$299$	−3.33101	−0.192637
$300$	−45.9901	−2.65524
$301$	0	0
$302$	26.8658	1.54595
$303$	0.850419	0.0488553
$304$	−27.9166	−1.60113
$305$	−47.4626	−2.71770
$306$	13.6294	0.779142
$307$	15.1488	0.864589	0.432295	−	0.901732i	$-0.357704\pi$
$307$	15.1488	0.864589	0.432295	+	0.901732i	$0.357704\pi$
$308$	0	0
$309$	−1.47460	−0.0838869
$310$	74.8805	4.25293
$311$	4.37058	0.247833	0.123916	−	0.992293i	$-0.460455\pi$
$311$	4.37058	0.247833	0.123916	+	0.992293i	$0.460455\pi$
$312$	7.34780	0.415987
$313$	1.49280	0.0843783	0.0421891	−	0.999110i	$-0.486567\pi$
$313$	1.49280	0.0843783	0.0421891	+	0.999110i	$0.486567\pi$
$314$	29.7954	1.68145
$315$	0	0
$316$	64.3466	3.61978
$317$	23.2129	1.30377	0.651883	−	0.758320i	$-0.273979\pi$
$317$	23.2129	1.30377	0.651883	+	0.758320i	$0.273979\pi$
$318$	−22.3250	−1.25192
$319$	4.37058	0.244706
$320$	29.0959	1.62651
$321$	−2.62418	−0.146467
$322$	0	0
$323$	−15.2588	−0.849024
$324$	4.81471	0.267484
$325$	−9.55201	−0.529850
$326$	−35.1754	−1.94819
$327$	17.9166	0.990790
$328$	−46.7814	−2.58307
$329$	0	0
$330$	47.1754	2.59692
$331$	1.10402	0.0606822	0.0303411	−	0.999540i	$-0.490341\pi$
$331$	1.10402	0.0606822	0.0303411	+	0.999540i	$0.490341\pi$
$332$	6.77088	0.371600
$333$	0.154821	0.00848416
$334$	−50.0969	−2.74118
$335$	−40.6725	−2.22218
$336$	0	0
$337$	−4.24237	−0.231096	−0.115548	−	0.993302i	$-0.536862\pi$
$337$	−4.24237	−0.231096	−0.115548	+	0.993302i	$0.536862\pi$
$338$	2.61050	0.141992
$339$	−0.922589	−0.0501082
$340$	95.8926	5.20051
$341$	−35.6217	−1.92902
$342$	−7.62942	−0.412552
$343$	0	0
$344$	−48.1428	−2.59569
$345$	−12.7068	−0.684113
$346$	50.9372	2.73840
$347$	14.0336	0.753362	0.376681	−	0.926343i	$-0.377065\pi$
$347$	14.0336	0.753362	0.376681	+	0.926343i	$0.377065\pi$
$348$	4.44200	0.238116
$349$	−4.10575	−0.219776	−0.109888	−	0.993944i	$-0.535049\pi$
$349$	−4.10575	−0.219776	−0.109888	+	0.993944i	$0.535049\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	−48.5096	−2.58557
$353$	16.0753	0.855601	0.427800	−	0.903873i	$-0.359289\pi$
$353$	16.0753	0.855601	0.427800	+	0.903873i	$0.359289\pi$
$354$	10.3331	0.549200
$355$	−25.1102	−1.33271
$356$	9.48272	0.502583
$357$	0	0
$358$	−43.2967	−2.28830
$359$	−18.1510	−0.957971	−0.478985	−	0.877823i	$-0.658995\pi$
$359$	−18.1510	−0.957971	−0.478985	+	0.877823i	$0.658995\pi$
$360$	28.0297	1.47730
$361$	−10.4585	−0.550446
$362$	7.35336	0.386484
$363$	−11.4420	−0.600549
$364$	0	0
$365$	29.5074	1.54449
$366$	32.4798	1.69775
$367$	23.3955	1.22123	0.610616	−	0.791927i	$-0.290922\pi$
$367$	23.3955	1.22123	0.610616	+	0.791927i	$0.290922\pi$
$368$	31.8178	1.65862
$369$	−6.36672	−0.331438
$370$	1.54176	0.0801522
$371$	0	0
$372$	−36.2038	−1.87708
$373$	−4.75164	−0.246031	−0.123015	−	0.992405i	$-0.539256\pi$
$373$	−4.75164	−0.246031	−0.123015	+	0.992405i	$0.539256\pi$
$374$	−64.5666	−3.33866
$375$	−17.3646	−0.896704
$376$	−66.3926	−3.42394
$377$	0.922589	0.0475158
$378$	0	0
$379$	−6.81258	−0.349939	−0.174969	−	0.984574i	$-0.555983\pi$
$379$	−6.81258	−0.349939	−0.174969	+	0.984574i	$0.555983\pi$
$380$	−53.6784	−2.75364
$381$	−17.4746	−0.895251
$382$	39.5753	2.02485
$383$	−2.25746	−0.115351	−0.0576754	−	0.998335i	$-0.518369\pi$
$383$	−2.25746	−0.115351	−0.0576754	+	0.998335i	$0.518369\pi$
$384$	0.568798	0.0290264
$385$	0	0
$386$	0.170195	0.00866268
$387$	−6.55201	−0.333057
$388$	10.1591	0.515749
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	9.95830	0.504258
$391$	17.3912	0.879511
$392$	0	0
$393$	0.967402	0.0487990
$394$	−44.7478	−2.25436
$395$	50.9820	2.56518
$396$	−22.8087	−1.14618
$397$	−24.9897	−1.25420	−0.627100	−	0.778939i	$-0.715758\pi$
$397$	−24.9897	−1.25420	−0.627100	+	0.778939i	$0.715758\pi$
$398$	−16.8168	−0.842952
$399$	0	0
$400$	91.2409	4.56204
$401$	27.6529	1.38092	0.690460	−	0.723370i	$-0.257408\pi$
$401$	27.6529	1.38092	0.690460	+	0.723370i	$0.257408\pi$
$402$	27.8332	1.38819
$403$	−7.51941	−0.374568
$404$	−4.09452	−0.203710
$405$	3.81471	0.189554
$406$	0	0
$407$	−0.733435	−0.0363550
$408$	−38.3629	−1.89924
$409$	−13.1488	−0.650168	−0.325084	−	0.945685i	$-0.605393\pi$
$409$	−13.1488	−0.650168	−0.325084	+	0.945685i	$0.605393\pi$
$410$	−63.4017	−3.13119
$411$	−3.29628	−0.162594
$412$	7.09976	0.349780
$413$	0	0
$414$	8.69560	0.427365
$415$	5.36459	0.263337
$416$	−10.2399	−0.502053
$417$	0.370581	0.0181474
$418$	36.1428	1.76780
$419$	5.69462	0.278200	0.139100	−	0.990278i	$-0.455579\pi$
$419$	5.69462	0.278200	0.139100	+	0.990278i	$0.455579\pi$
$420$	0	0
$421$	−23.0206	−1.12196	−0.560978	−	0.827831i	$-0.689575\pi$
$421$	−23.0206	−1.12196	−0.560978	+	0.827831i	$0.689575\pi$
$422$	−41.8375	−2.03662
$423$	−9.03571	−0.439331
$424$	62.8384	3.05170
$425$	49.8710	2.41910
$426$	17.1836	0.832546
$427$	0	0
$428$	12.6347	0.610719
$429$	−4.73730	−0.228719
$430$	−65.2469	−3.14648
$431$	20.8045	1.00212	0.501058	−	0.865414i	$-0.332944\pi$
$431$	20.8045	1.00212	0.501058	+	0.865414i	$0.332944\pi$
$432$	−9.55201	−0.459571
$433$	−31.4808	−1.51287	−0.756436	−	0.654068i	$-0.773061\pi$
$433$	−31.4808	−1.51287	−0.756436	+	0.654068i	$0.773061\pi$
$434$	0	0
$435$	3.51941	0.168743
$436$	−86.2632	−4.13126
$437$	−9.73517	−0.465696
$438$	−20.1927	−0.964843
$439$	−22.3811	−1.06819	−0.534095	−	0.845425i	$-0.679347\pi$
$439$	−22.3811	−1.06819	−0.534095	+	0.845425i	$0.679347\pi$
$440$	−132.785	−6.33028
$441$	0	0
$442$	−13.6294	−0.648285
$443$	8.52465	0.405018	0.202509	−	0.979280i	$-0.435090\pi$
$443$	8.52465	0.405018	0.202509	+	0.979280i	$0.435090\pi$
$444$	−0.745420	−0.0353761
$445$	7.51319	0.356159
$446$	−52.2795	−2.47550
$447$	15.7425	0.744597
$448$	0	0
$449$	18.7142	0.883178	0.441589	−	0.897218i	$-0.354415\pi$
$449$	18.7142	0.883178	0.441589	+	0.897218i	$0.354415\pi$
$450$	24.9355	1.17547
$451$	30.1610	1.42023
$452$	4.44200	0.208934
$453$	−10.2914	−0.483534
$454$	−50.1663	−2.35442
$455$	0	0
$456$	21.4746	1.00564
$457$	−14.1366	−0.661283	−0.330641	−	0.943756i	$-0.607265\pi$
$457$	−14.1366	−0.661283	−0.330641	+	0.943756i	$0.607265\pi$
$458$	−13.7282	−0.641476
$459$	−5.22100	−0.243695
$460$	61.1797	2.85252
$461$	2.67636	0.124651	0.0623253	−	0.998056i	$-0.480148\pi$
$461$	2.67636	0.124651	0.0623253	+	0.998056i	$0.480148\pi$
$462$	0	0
$463$	−2.53162	−0.117655	−0.0588273	−	0.998268i	$-0.518736\pi$
$463$	−2.53162	−0.117655	−0.0588273	+	0.998268i	$0.518736\pi$
$464$	−8.81258	−0.409114
$465$	−28.6844	−1.33021
$466$	−69.5005	−3.21955
$467$	−2.00426	−0.0927460	−0.0463730	−	0.998924i	$-0.514766\pi$
$467$	−2.00426	−0.0927460	−0.0463730	+	0.998924i	$0.514766\pi$
$468$	−4.81471	−0.222560
$469$	0	0
$470$	−89.9803	−4.15048
$471$	−11.4137	−0.525914
$472$	−29.0848	−1.33874
$473$	31.0388	1.42717
$474$	−34.8883	−1.60247
$475$	−27.9166	−1.28090
$476$	0	0
$477$	8.55201	0.391570
$478$	11.2018	0.512357
$479$	14.6609	0.669872	0.334936	−	0.942241i	$-0.391285\pi$
$479$	14.6609	0.669872	0.334936	+	0.942241i	$0.391285\pi$
$480$	−39.0623	−1.78294
$481$	−0.154821	−0.00705925
$482$	−19.6405	−0.894602
$483$	0	0
$484$	55.0899	2.50409
$485$	8.04907	0.365489
$486$	−2.61050	−0.118415
$487$	38.2143	1.73165	0.865827	−	0.500344i	$-0.166793\pi$
$487$	38.2143	1.73165	0.865827	+	0.500344i	$0.166793\pi$
$488$	−91.4213	−4.13845
$489$	13.4746	0.609342
$490$	0	0
$491$	21.3758	0.964677	0.482339	−	0.875985i	$-0.339787\pi$
$491$	21.3758	0.964677	0.482339	+	0.875985i	$0.339787\pi$
$492$	30.6539	1.38198
$493$	−4.81684	−0.216939
$494$	7.62942	0.343264
$495$	−18.0714	−0.812250
$496$	71.8255	3.22506
$497$	0	0
$498$	−3.67112	−0.164507
$499$	27.0920	1.21281	0.606403	−	0.795158i	$-0.292612\pi$
$499$	27.0920	1.21281	0.606403	+	0.795158i	$0.292612\pi$
$500$	83.6054	3.73895
$501$	19.1905	0.857370
$502$	−29.5503	−1.31889
$503$	−24.8778	−1.10925	−0.554623	−	0.832102i	$-0.687137\pi$
$503$	−24.8778	−1.10925	−0.554623	+	0.832102i	$0.687137\pi$
$504$	0	0
$505$	−3.24410	−0.144361
$506$	−41.1936	−1.83128
$507$	−1.00000	−0.0444116
$508$	84.1351	3.73289
$509$	−2.70297	−0.119807	−0.0599034	−	0.998204i	$-0.519079\pi$
$509$	−2.70297	−0.119807	−0.0599034	+	0.998204i	$0.519079\pi$
$510$	−51.9923	−2.30226
$511$	0	0
$512$	−42.5607	−1.88093
$513$	2.92259	0.129035
$514$	64.8720	2.86138
$515$	5.62516	0.247874
$516$	31.5460	1.38874
$517$	42.8049	1.88256
$518$	0	0
$519$	−19.5124	−0.856501
$520$	−28.0297	−1.22918
$521$	33.4472	1.46535	0.732675	−	0.680579i	$-0.238272\pi$
$521$	33.4472	1.46535	0.732675	+	0.680579i	$0.238272\pi$
$522$	−2.40842	−0.105414
$523$	−19.3198	−0.844795	−0.422397	−	0.906411i	$-0.638811\pi$
$523$	−19.3198	−0.844795	−0.422397	+	0.906411i	$0.638811\pi$
$524$	−4.65776	−0.203475
$525$	0	0
$526$	44.8384	1.95505
$527$	39.2588	1.71014
$528$	45.2507	1.96928
$529$	−11.9044	−0.517582
$530$	85.1634	3.69926
$531$	−3.95830	−0.171776
$532$	0	0
$533$	6.36672	0.275773
$534$	−5.14146	−0.222493
$535$	10.0105	0.432791
$536$	−78.3424	−3.38387
$537$	16.5856	0.715721
$538$	20.2220	0.871832
$539$	0	0
$540$	−18.3667	−0.790378
$541$	−24.9554	−1.07292	−0.536459	−	0.843927i	$-0.680238\pi$
$541$	−24.9554	−1.07292	−0.536459	+	0.843927i	$0.680238\pi$
$542$	77.5340	3.33037
$543$	−2.81684	−0.120882
$544$	53.4626	2.29219
$545$	−68.3466	−2.92765
$546$	0	0
$547$	3.80037	0.162492	0.0812460	−	0.996694i	$-0.474110\pi$
$547$	3.80037	0.162492	0.0812460	+	0.996694i	$0.474110\pi$
$548$	15.8706	0.677960
$549$	−12.4420	−0.531012
$550$	−118.127	−5.03695
$551$	2.69635	0.114868
$552$	−24.4756	−1.04175
$553$	0	0
$554$	65.6510	2.78924
$555$	−0.590599	−0.0250695
$556$	−1.78424	−0.0756686
$557$	43.8792	1.85922	0.929610	−	0.368546i	$-0.120144\pi$
$557$	43.8792	1.85922	0.929610	+	0.368546i	$0.120144\pi$
$558$	19.6294	0.830980
$559$	6.55201	0.277120
$560$	0	0
$561$	24.7334	1.04425
$562$	21.9290	0.925018
$563$	−12.3768	−0.521620	−0.260810	−	0.965390i	$-0.583990\pi$
$563$	−12.3768	−0.521620	−0.260810	+	0.965390i	$0.583990\pi$
$564$	43.5043	1.83186
$565$	3.51941	0.148063
$566$	55.8417	2.34720
$567$	0	0
$568$	−48.3667	−2.02942
$569$	−26.6234	−1.11611	−0.558056	−	0.829803i	$-0.688453\pi$
$569$	−26.6234	−1.11611	−0.558056	+	0.829803i	$0.688453\pi$
$570$	29.1040	1.21903
$571$	10.9983	0.460263	0.230132	−	0.973160i	$-0.426084\pi$
$571$	10.9983	0.460263	0.230132	+	0.973160i	$0.426084\pi$
$572$	22.8087	0.953680
$573$	−15.1601	−0.633321
$574$	0	0
$575$	31.8178	1.32689
$576$	7.62729	0.317804
$577$	−30.4238	−1.26656	−0.633280	−	0.773923i	$-0.718292\pi$
$577$	−30.4238	−1.26656	−0.633280	+	0.773923i	$0.718292\pi$
$578$	26.7807	1.11393
$579$	−0.0651962	−0.00270946
$580$	−16.9449	−0.703600
$581$	0	0
$582$	−5.50818	−0.228321
$583$	−40.5134	−1.67789
$584$	56.8365	2.35191
$585$	−3.81471	−0.157719
$586$	25.0581	1.03514
$587$	−3.84207	−0.158579	−0.0792895	−	0.996852i	$-0.525265\pi$
$587$	−3.84207	−0.158579	−0.0792895	+	0.996852i	$0.525265\pi$
$588$	0	0
$589$	−21.9761	−0.905511
$590$	−39.4179	−1.62281
$591$	17.1415	0.705105
$592$	1.47886	0.0607806
$593$	23.8904	0.981061	0.490530	−	0.871424i	$-0.336803\pi$
$593$	23.8904	0.981061	0.490530	+	0.871424i	$0.336803\pi$
$594$	12.3667	0.507413
$595$	0	0
$596$	−75.7958	−3.10471
$597$	6.44200	0.263653
$598$	−8.69560	−0.355589
$599$	−15.4206	−0.630070	−0.315035	−	0.949080i	$-0.602016\pi$
$599$	−15.4206	−0.630070	−0.315035	+	0.949080i	$0.602016\pi$
$600$	−70.1862	−2.86534
$601$	1.49280	0.0608928	0.0304464	−	0.999536i	$-0.490307\pi$
$601$	1.49280	0.0608928	0.0304464	+	0.999536i	$0.490307\pi$
$602$	0	0
$603$	−10.6620	−0.434191
$604$	49.5503	2.01617
$605$	43.6479	1.77454
$606$	2.22002	0.0901821
$607$	4.44626	0.180468	0.0902340	−	0.995921i	$-0.471239\pi$
$607$	4.44626	0.180468	0.0902340	+	0.995921i	$0.471239\pi$
$608$	−29.9271	−1.21370
$609$	0	0
$610$	−123.901	−5.01661
$611$	9.03571	0.365546
$612$	25.1376	1.01613
$613$	26.6640	1.07695	0.538474	−	0.842642i	$-0.319001\pi$
$613$	26.6640	1.07695	0.538474	+	0.842642i	$0.319001\pi$
$614$	39.5460	1.59595
$615$	24.2872	0.979354
$616$	0	0
$617$	27.3495	1.10105	0.550525	−	0.834819i	$-0.314428\pi$
$617$	27.3495	1.10105	0.550525	+	0.834819i	$0.314428\pi$
$618$	−3.84944	−0.154847
$619$	9.24836	0.371723	0.185861	−	0.982576i	$-0.440492\pi$
$619$	9.24836	0.371723	0.185861	+	0.982576i	$0.440492\pi$
$620$	138.107	5.54651
$621$	−3.33101	−0.133669
$622$	11.4094	0.457475
$623$	0	0
$624$	9.55201	0.382386
$625$	18.4808	0.739233
$626$	3.89697	0.155754
$627$	−13.8452	−0.552923
$628$	54.9535	2.19288
$629$	0.808323	0.0322299
$630$	0	0
$631$	24.5134	0.975864	0.487932	−	0.872882i	$-0.337751\pi$
$631$	24.5134	0.975864	0.487932	+	0.872882i	$0.337751\pi$
$632$	98.2003	3.90620
$633$	16.0266	0.637000
$634$	60.5972	2.40662
$635$	66.6605	2.64534
$636$	−41.1754	−1.63271
$637$	0	0
$638$	11.4094	0.451703
$639$	−6.58248	−0.260399
$640$	−2.16980	−0.0857689
$641$	34.3972	1.35861	0.679304	−	0.733857i	$-0.262282\pi$
$641$	34.3972	1.35861	0.679304	+	0.733857i	$0.262282\pi$
$642$	−6.85042	−0.270364
$643$	7.69036	0.303278	0.151639	−	0.988436i	$-0.451545\pi$
$643$	7.69036	0.303278	0.151639	+	0.988436i	$0.451545\pi$
$644$	0	0
$645$	24.9940	0.984138
$646$	−39.8332	−1.56722
$647$	−41.9123	−1.64774	−0.823872	−	0.566776i	$-0.808191\pi$
$647$	−41.9123	−1.64774	−0.823872	+	0.566776i	$0.808191\pi$
$648$	7.34780	0.288649
$649$	18.7516	0.736066
$650$	−24.9355	−0.978051
$651$	0	0
$652$	−64.8763	−2.54075
$653$	34.2262	1.33938	0.669688	−	0.742642i	$-0.266428\pi$
$653$	34.2262	1.33938	0.669688	+	0.742642i	$0.266428\pi$
$654$	46.7713	1.82890
$655$	−3.69036	−0.144194
$656$	−60.8149	−2.37442
$657$	7.73517	0.301778
$658$	0	0
$659$	45.1685	1.75951	0.879757	−	0.475424i	$-0.157705\pi$
$659$	45.1685	1.75951	0.879757	+	0.475424i	$0.157705\pi$
$660$	87.0086	3.38681
$661$	−11.9839	−0.466119	−0.233059	−	0.972463i	$-0.574874\pi$
$661$	−11.9839	−0.466119	−0.233059	+	0.972463i	$0.574874\pi$
$662$	2.88203	0.112013
$663$	5.22100	0.202767
$664$	10.3331	0.401004
$665$	0	0
$666$	0.404161	0.0156609
$667$	−3.07315	−0.118993
$668$	−92.3968	−3.57494
$669$	20.0266	0.774273
$670$	−106.176	−4.10192
$671$	58.9415	2.27541
$672$	0	0
$673$	−29.5117	−1.13759	−0.568796	−	0.822479i	$-0.692591\pi$
$673$	−29.5117	−1.13759	−0.568796	+	0.822479i	$0.692591\pi$
$674$	−11.0747	−0.426581
$675$	−9.55201	−0.367657
$676$	4.81471	0.185181
$677$	31.0662	1.19397	0.596985	−	0.802252i	$-0.296365\pi$
$677$	31.0662	1.19397	0.596985	+	0.802252i	$0.296365\pi$
$678$	−2.40842	−0.0924948
$679$	0	0
$680$	146.343	5.61200
$681$	19.2171	0.736402
$682$	−92.9904	−3.56079
$683$	5.79433	0.221714	0.110857	−	0.993836i	$-0.464640\pi$
$683$	5.79433	0.221714	0.110857	+	0.993836i	$0.464640\pi$
$684$	−14.0714	−0.538034
$685$	12.5744	0.480441
$686$	0	0
$687$	5.25884	0.200637
$688$	−62.5848	−2.38602
$689$	−8.55201	−0.325806
$690$	−33.1712	−1.26281
$691$	−18.8617	−0.717531	−0.358766	−	0.933428i	$-0.616802\pi$
$691$	−18.8617	−0.717531	−0.358766	+	0.933428i	$0.616802\pi$
$692$	93.9467	3.57132
$693$	0	0
$694$	36.6347	1.39063
$695$	−1.41366	−0.0536232
$696$	6.77900	0.256957
$697$	−33.2406	−1.25908
$698$	−10.7181	−0.405685
$699$	26.6234	1.00699
$700$	0	0
$701$	−32.3180	−1.22064	−0.610318	−	0.792157i	$-0.708958\pi$
$701$	−32.3180	−1.22064	−0.610318	+	0.792157i	$0.708958\pi$
$702$	2.61050	0.0985270
$703$	−0.452479	−0.0170656
$704$	−36.1328	−1.36180
$705$	34.4686	1.29816
$706$	41.9645	1.57936
$707$	0	0
$708$	19.0581	0.716246
$709$	4.72296	0.177374	0.0886872	−	0.996060i	$-0.471733\pi$
$709$	4.72296	0.177374	0.0886872	+	0.996060i	$0.471733\pi$
$710$	−65.5503	−2.46006
$711$	13.3646	0.501211
$712$	14.4717	0.542350
$713$	25.0472	0.938026
$714$	0	0
$715$	18.0714	0.675833
$716$	−79.8548	−2.98431
$717$	−4.29104	−0.160252
$718$	−47.3831	−1.76832
$719$	40.8902	1.52495	0.762474	−	0.647019i	$-0.223985\pi$
$719$	40.8902	1.52495	0.762474	+	0.647019i	$0.223985\pi$
$720$	36.4381	1.35797
$721$	0	0
$722$	−27.3018	−1.01607
$723$	7.52367	0.279808
$724$	13.5623	0.504037
$725$	−8.81258	−0.327291
$726$	−29.8693	−1.10856
$727$	−18.7292	−0.694627	−0.347313	−	0.937749i	$-0.612906\pi$
$727$	−18.7292	−0.694627	−0.347313	+	0.937749i	$0.612906\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	77.0291	2.85098
$731$	−34.2080	−1.26523
$732$	59.9046	2.21414
$733$	14.1100	0.521165	0.260583	−	0.965452i	$-0.416085\pi$
$733$	14.1100	0.521165	0.260583	+	0.965452i	$0.416085\pi$
$734$	61.0738	2.25428
$735$	0	0
$736$	34.1093	1.25728
$737$	50.5092	1.86053
$738$	−16.6203	−0.611802
$739$	−49.9209	−1.83637	−0.918184	−	0.396154i	$-0.870345\pi$
$739$	−49.9209	−1.83637	−0.918184	+	0.396154i	$0.870345\pi$
$740$	2.84356	0.104531
$741$	−2.92259	−0.107364
$742$	0	0
$743$	2.15868	0.0791945	0.0395972	−	0.999216i	$-0.487393\pi$
$743$	2.15868	0.0791945	0.0395972	+	0.999216i	$0.487393\pi$
$744$	−55.2511	−2.02560
$745$	−60.0532	−2.20018
$746$	−12.4042	−0.454149
$747$	1.40629	0.0514535
$748$	−119.084	−4.35415
$749$	0	0
$750$	−45.3303	−1.65523
$751$	16.8372	0.614399	0.307199	−	0.951645i	$-0.400608\pi$
$751$	16.8372	0.614399	0.307199	+	0.951645i	$0.400608\pi$
$752$	−86.3092	−3.14737
$753$	11.3198	0.412516
$754$	2.40842	0.0877095
$755$	39.2588	1.42878
$756$	0	0
$757$	−17.6028	−0.639785	−0.319893	−	0.947454i	$-0.603647\pi$
$757$	−17.6028	−0.639785	−0.319893	+	0.947454i	$0.603647\pi$
$758$	−17.7842	−0.645953
$759$	15.7800	0.572777
$760$	−81.9193	−2.97153
$761$	−19.2179	−0.696648	−0.348324	−	0.937374i	$-0.613249\pi$
$761$	−19.2179	−0.696648	−0.348324	+	0.937374i	$0.613249\pi$
$762$	−45.6174	−1.65255
$763$	0	0
$764$	72.9913	2.64073
$765$	19.9166	0.720086
$766$	−5.89310	−0.212926
$767$	3.95830	0.142926
$768$	16.7394	0.604032
$769$	14.0406	0.506315	0.253158	−	0.967425i	$-0.418531\pi$
$769$	14.0406	0.506315	0.253158	+	0.967425i	$0.418531\pi$
$770$	0	0
$771$	−24.8504	−0.894966
$772$	0.313901	0.0112975
$773$	−22.4339	−0.806891	−0.403445	−	0.915004i	$-0.632187\pi$
$773$	−22.4339	−0.806891	−0.403445	+	0.915004i	$0.632187\pi$
$774$	−17.1040	−0.614791
$775$	71.8255	2.58005
$776$	15.5039	0.556558
$777$	0	0
$778$	15.6630	0.561546
$779$	18.6073	0.666676
$780$	18.3667	0.657634
$781$	31.1832	1.11582
$782$	45.3997	1.62349
$783$	0.922589	0.0329707
$784$	0	0
$785$	43.5398	1.55400
$786$	2.52540	0.0900781
$787$	0.926847	0.0330385	0.0165193	−	0.999864i	$-0.494742\pi$
$787$	0.926847	0.0330385	0.0165193	+	0.999864i	$0.494742\pi$
$788$	−82.5311	−2.94005
$789$	−17.1762	−0.611488
$790$	133.089	4.73508
$791$	0	0
$792$	−34.8087	−1.23687
$793$	12.4420	0.441828
$794$	−65.2357	−2.31513
$795$	−32.6234	−1.15703
$796$	−31.0164	−1.09935
$797$	26.0154	0.921512	0.460756	−	0.887527i	$-0.347578\pi$
$797$	26.0154	0.921512	0.460756	+	0.887527i	$0.347578\pi$
$798$	0	0
$799$	−47.1754	−1.66895
$800$	97.8118	3.45817
$801$	1.96953	0.0695900
$802$	72.1879	2.54904
$803$	−36.6438	−1.29313
$804$	51.3345	1.81043
$805$	0	0
$806$	−19.6294	−0.691417
$807$	−7.74640	−0.272686
$808$	−6.24870	−0.219829
$809$	−44.8539	−1.57698	−0.788490	−	0.615048i	$-0.789137\pi$
$809$	−44.8539	−1.57698	−0.788490	+	0.615048i	$0.789137\pi$
$810$	9.95830	0.349899
$811$	−27.2511	−0.956916	−0.478458	−	0.878110i	$-0.658804\pi$
$811$	−27.2511	−0.956916	−0.478458	+	0.878110i	$0.658804\pi$
$812$	0	0
$813$	−29.7008	−1.04165
$814$	−1.91463	−0.0671078
$815$	−51.4017	−1.80052
$816$	−49.8710	−1.74584
$817$	19.1488	0.669933
$818$	−34.3250	−1.20015
$819$	0	0
$820$	−116.936	−4.08357
$821$	20.4003	0.711975	0.355988	−	0.934491i	$-0.384145\pi$
$821$	20.4003	0.711975	0.355988	+	0.934491i	$0.384145\pi$
$822$	−8.60494	−0.300132
$823$	54.3509	1.89455	0.947276	−	0.320418i	$-0.103824\pi$
$823$	54.3509	1.89455	0.947276	+	0.320418i	$0.103824\pi$
$824$	10.8350	0.377457
$825$	45.2507	1.57543
$826$	0	0
$827$	3.48272	0.121106	0.0605530	−	0.998165i	$-0.480714\pi$
$827$	3.48272	0.121106	0.0605530	+	0.998165i	$0.480714\pi$
$828$	16.0378	0.557353
$829$	29.8417	1.03645	0.518223	−	0.855246i	$-0.326594\pi$
$829$	29.8417	1.03645	0.518223	+	0.855246i	$0.326594\pi$
$830$	14.0043	0.486095
$831$	−25.1488	−0.872403
$832$	−7.62729	−0.264429
$833$	0	0
$834$	0.967402	0.0334984
$835$	−73.2063	−2.53341
$836$	66.6605	2.30550
$837$	−7.51941	−0.259909
$838$	14.8658	0.513530
$839$	43.0033	1.48464	0.742320	−	0.670045i	$-0.233725\pi$
$839$	43.0033	1.48464	0.742320	+	0.670045i	$0.233725\pi$
$840$	0	0
$841$	−28.1488	−0.970649
$842$	−60.0953	−2.07102
$843$	−8.40030	−0.289322
$844$	−77.1634	−2.65608
$845$	3.81471	0.131230
$846$	−23.5877	−0.810962
$847$	0	0
$848$	81.6889	2.80521
$849$	−21.3912	−0.734144
$850$	130.188	4.46542
$851$	0.515711	0.0176784
$852$	31.6927	1.08577
$853$	−34.1023	−1.16764	−0.583820	−	0.811883i	$-0.698443\pi$
$853$	−34.1023	−1.16764	−0.583820	+	0.811883i	$0.698443\pi$
$854$	0	0
$855$	−11.1488	−0.381282
$856$	19.2819	0.659043
$857$	45.7344	1.56226	0.781129	−	0.624370i	$-0.214644\pi$
$857$	45.7344	1.56226	0.781129	+	0.624370i	$0.214644\pi$
$858$	−12.3667	−0.422193
$859$	−15.9861	−0.545437	−0.272719	−	0.962094i	$-0.587923\pi$
$859$	−15.9861	−0.545437	−0.272719	+	0.962094i	$0.587923\pi$
$860$	−120.339	−4.10352
$861$	0	0
$862$	54.3100	1.84981
$863$	−24.5096	−0.834315	−0.417157	−	0.908834i	$-0.636974\pi$
$863$	−24.5096	−0.834315	−0.417157	+	0.908834i	$0.636974\pi$
$864$	−10.2399	−0.348369
$865$	74.4343	2.53084
$866$	−82.1807	−2.79261
$867$	−10.2588	−0.348408
$868$	0	0
$869$	−63.3120	−2.14771
$870$	9.18742	0.311483
$871$	10.6620	0.361269
$872$	−131.648	−4.45815
$873$	2.11001	0.0714130
$874$	−25.4137	−0.859630
$875$	0	0
$876$	−37.2426	−1.25831
$877$	53.4913	1.80627	0.903136	−	0.429354i	$-0.141259\pi$
$877$	53.4913	1.80627	0.903136	+	0.429354i	$0.141259\pi$
$878$	−58.4258	−1.97177
$879$	−9.59895	−0.323765
$880$	−172.618	−5.81896
$881$	−26.1745	−0.881840	−0.440920	−	0.897546i	$-0.645348\pi$
$881$	−26.1745	−0.881840	−0.440920	+	0.897546i	$0.645348\pi$
$882$	0	0
$883$	23.1840	0.780202	0.390101	−	0.920772i	$-0.372440\pi$
$883$	23.1840	0.780202	0.390101	+	0.920772i	$0.372440\pi$
$884$	−25.1376	−0.845469
$885$	15.0998	0.507573
$886$	22.2536	0.747624
$887$	13.7008	0.460029	0.230015	−	0.973187i	$-0.426123\pi$
$887$	13.7008	0.460029	0.230015	+	0.973187i	$0.426123\pi$
$888$	−1.13760	−0.0381752
$889$	0	0
$890$	19.6132	0.657435
$891$	−4.73730	−0.158705
$892$	−96.4223	−3.22846
$893$	26.4077	0.883699
$894$	41.0959	1.37445
$895$	−63.2692	−2.11486
$896$	0	0
$897$	3.33101	0.111219
$898$	48.8534	1.63026
$899$	−6.93733	−0.231373
$900$	45.9901	1.53300
$901$	44.6500	1.48751
$902$	78.7354	2.62160
$903$	0	0
$904$	6.77900	0.225466
$905$	10.7454	0.357190
$906$	−26.8658	−0.892556
$907$	25.5621	0.848777	0.424388	−	0.905480i	$-0.360489\pi$
$907$	25.5621	0.848777	0.424388	+	0.905480i	$0.360489\pi$
$908$	−92.5249	−3.07055
$909$	−0.850419	−0.0282066
$910$	0	0
$911$	2.07643	0.0687951	0.0343976	−	0.999408i	$-0.489049\pi$
$911$	2.07643	0.0687951	0.0343976	+	0.999408i	$0.489049\pi$
$912$	27.9166	0.924411
$913$	−6.66202	−0.220481
$914$	−36.9036	−1.22066
$915$	47.4626	1.56907
$916$	−25.3198	−0.836589
$917$	0	0
$918$	−13.6294	−0.449838
$919$	17.8514	0.588863	0.294432	−	0.955673i	$-0.404870\pi$
$919$	17.8514	0.588863	0.294432	+	0.955673i	$0.404870\pi$
$920$	93.3672	3.07823
$921$	−15.1488	−0.499171
$922$	6.98664	0.230093
$923$	6.58248	0.216665
$924$	0	0
$925$	1.47886	0.0486245
$926$	−6.60881	−0.217179
$927$	1.47460	0.0484321
$928$	−9.44724	−0.310121
$929$	37.0553	1.21575	0.607873	−	0.794034i	$-0.292023\pi$
$929$	37.0553	1.21575	0.607873	+	0.794034i	$0.292023\pi$
$930$	−74.8805	−2.45543
$931$	0	0
$932$	−128.184	−4.19881
$933$	−4.37058	−0.143086
$934$	−5.23212	−0.171200
$935$	−94.3509	−3.08560
$936$	−7.34780	−0.240170
$937$	−39.7540	−1.29871	−0.649354	−	0.760486i	$-0.724961\pi$
$937$	−39.7540	−1.29871	−0.649354	+	0.760486i	$0.724961\pi$
$938$	0	0
$939$	−1.49280	−0.0487158
$940$	−165.956	−5.41290
$941$	47.3530	1.54366	0.771832	−	0.635827i	$-0.219341\pi$
$941$	47.3530	1.54366	0.771832	+	0.635827i	$0.219341\pi$
$942$	−29.7954	−0.970785
$943$	−21.2076	−0.690614
$944$	−37.8097	−1.23060
$945$	0	0
$946$	81.0268	2.63441
$947$	8.80446	0.286106	0.143053	−	0.989715i	$-0.454308\pi$
$947$	8.80446	0.286106	0.143053	+	0.989715i	$0.454308\pi$
$948$	−64.3466	−2.08988
$949$	−7.73517	−0.251094
$950$	−72.8763	−2.36442
$951$	−23.2129	−0.752729
$952$	0	0
$953$	7.72895	0.250365	0.125183	−	0.992134i	$-0.460048\pi$
$953$	7.72895	0.250365	0.125183	+	0.992134i	$0.460048\pi$
$954$	22.3250	0.722799
$955$	57.8312	1.87137
$956$	20.6601	0.668196
$957$	−4.37058	−0.141281
$958$	38.2722	1.23652
$959$	0	0
$960$	−29.0959	−0.939066
$961$	25.5415	0.823920
$962$	−0.404161	−0.0130307
$963$	2.62418	0.0845630
$964$	−36.2243	−1.16671
$965$	0.248705	0.00800608
$966$	0	0
$967$	14.3054	0.460030	0.230015	−	0.973187i	$-0.426122\pi$
$967$	14.3054	0.460030	0.230015	+	0.973187i	$0.426122\pi$
$968$	84.0735	2.70222
$969$	15.2588	0.490184
$970$	21.0121	0.674658
$971$	15.1692	0.486803	0.243402	−	0.969926i	$-0.421737\pi$
$971$	15.1692	0.486803	0.243402	+	0.969926i	$0.421737\pi$
$972$	−4.81471	−0.154432
$973$	0	0
$974$	99.7583	3.19646
$975$	9.55201	0.305909
$976$	−118.846	−3.80417
$977$	−8.25746	−0.264180	−0.132090	−	0.991238i	$-0.542169\pi$
$977$	−8.25746	−0.264180	−0.132090	+	0.991238i	$0.542169\pi$
$978$	35.1754	1.12479
$979$	−9.33026	−0.298196
$980$	0	0
$981$	−17.9166	−0.572033
$982$	55.8016	1.78070
$983$	25.8525	0.824568	0.412284	−	0.911055i	$-0.364731\pi$
$983$	25.8525	0.824568	0.412284	+	0.911055i	$0.364731\pi$
$984$	46.7814	1.49134
$985$	−65.3897	−2.08349
$986$	−12.5744	−0.400449
$987$	0	0
$988$	14.0714	0.447671
$989$	−21.8248	−0.693988
$990$	−47.1754	−1.49933
$991$	−56.2100	−1.78557	−0.892785	−	0.450484i	$-0.851252\pi$
$991$	−56.2100	−1.78557	−0.892785	+	0.450484i	$0.851252\pi$
$992$	76.9981	2.44469
$993$	−1.10402	−0.0350349
$994$	0	0
$995$	−24.5744	−0.779059
$996$	−6.77088	−0.214544
$997$	−37.4789	−1.18697	−0.593484	−	0.804846i	$-0.702248\pi$
$997$	−37.4789	−1.18697	−0.593484	+	0.804846i	$0.702248\pi$
$998$	70.7237	2.23872
$999$	−0.154821	−0.00489833

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1911.2.a.s.1.4		4
3.2	odd	2		5733.2.a.bf.1.1		4
7.6	odd	2		273.2.a.e.1.4	✓	4
21.20	even	2		819.2.a.k.1.1		4
28.27	even	2		4368.2.a.br.1.1		4
35.34	odd	2		6825.2.a.bg.1.1		4
91.90	odd	2		3549.2.a.w.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.a.e.1.4	✓	4	7.6	odd	2
819.2.a.k.1.1		4	21.20	even	2
1911.2.a.s.1.4		4	1.1	even	1	trivial
3549.2.a.w.1.1		4	91.90	odd	2
4368.2.a.br.1.1		4	28.27	even	2
5733.2.a.bf.1.1		4	3.2	odd	2
6825.2.a.bg.1.1		4	35.34	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:	\( N \)	\(=\)	\( 1911 = 3 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1911.a (trivial)

\(f(q)\)	\(=\)	\(q+2.61050 q^{2} -1.00000 q^{3} +4.81471 q^{4} +3.81471 q^{5} -2.61050 q^{6} +7.34780 q^{8} +1.00000 q^{9} +9.95830 q^{10} -4.73730 q^{11} -4.81471 q^{12} -1.00000 q^{13} -3.81471 q^{15} +9.55201 q^{16} +5.22100 q^{17} +2.61050 q^{18} -2.92259 q^{19} +18.3667 q^{20} -12.3667 q^{22} +3.33101 q^{23} -7.34780 q^{24} +9.55201 q^{25} -2.61050 q^{26} -1.00000 q^{27} -0.922589 q^{29} -9.95830 q^{30} +7.51941 q^{31} +10.2399 q^{32} +4.73730 q^{33} +13.6294 q^{34} +4.81471 q^{36} +0.154821 q^{37} -7.62942 q^{38} +1.00000 q^{39} +28.0297 q^{40} -6.36672 q^{41} -6.55201 q^{43} -22.8087 q^{44} +3.81471 q^{45} +8.69560 q^{46} -9.03571 q^{47} -9.55201 q^{48} +24.9355 q^{50} -5.22100 q^{51} -4.81471 q^{52} +8.55201 q^{53} -2.61050 q^{54} -18.0714 q^{55} +2.92259 q^{57} -2.40842 q^{58} -3.95830 q^{59} -18.3667 q^{60} -12.4420 q^{61} +19.6294 q^{62} +7.62729 q^{64} -3.81471 q^{65} +12.3667 q^{66} -10.6620 q^{67} +25.1376 q^{68} -3.33101 q^{69} -6.58248 q^{71} +7.34780 q^{72} +7.73517 q^{73} +0.404161 q^{74} -9.55201 q^{75} -14.0714 q^{76} +2.61050 q^{78} +13.3646 q^{79} +36.4381 q^{80} +1.00000 q^{81} -16.6203 q^{82} +1.40629 q^{83} +19.9166 q^{85} -17.1040 q^{86} +0.922589 q^{87} -34.8087 q^{88} +1.96953 q^{89} +9.95830 q^{90} +16.0378 q^{92} -7.51941 q^{93} -23.5877 q^{94} -11.1488 q^{95} -10.2399 q^{96} +2.11001 q^{97} -4.73730 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} - 4 q^{3} + 7 q^{4} + 3 q^{5} - q^{6} + 3 q^{8} + 4 q^{9} + 4 q^{10} - 2 q^{11} - 7 q^{12} - 4 q^{13} - 3 q^{15} + 9 q^{16} + 2 q^{17} + q^{18} - 7 q^{19} + 32 q^{20} - 8 q^{22} + 3 q^{23}+ \cdots - 2 q^{99}+O(q^{100}) \) 4 * q + q^2 - 4 * q^3 + 7 * q^4 + 3 * q^5 - q^6 + 3 * q^8 + 4 * q^9 + 4 * q^10 - 2 * q^11 - 7 * q^12 - 4 * q^13 - 3 * q^15 + 9 * q^16 + 2 * q^17 + q^18 - 7 * q^19 + 32 * q^20 - 8 * q^22 + 3 * q^23 - 3 * q^24 + 9 * q^25 - q^26 - 4 * q^27 + q^29 - 4 * q^30 - 3 * q^31 + 7 * q^32 + 2 * q^33 + 30 * q^34 + 7 * q^36 + 10 * q^37 - 6 * q^38 + 4 * q^39 + 14 * q^40 + 16 * q^41 + 3 * q^43 - 12 * q^44 + 3 * q^45 - 18 * q^46 - 5 * q^47 - 9 * q^48 + 13 * q^50 - 2 * q^51 - 7 * q^52 + 5 * q^53 - q^54 - 10 * q^55 + 7 * q^57 - 4 * q^58 + 20 * q^59 - 32 * q^60 - 12 * q^61 + 54 * q^62 + 5 * q^64 - 3 * q^65 + 8 * q^66 - 22 * q^67 + 10 * q^68 - 3 * q^69 + 3 * q^72 + 13 * q^73 - 6 * q^74 - 9 * q^75 + 6 * q^76 + q^78 + 11 * q^79 + 42 * q^80 + 4 * q^81 - 10 * q^82 - q^83 + 8 * q^85 - 10 * q^86 - q^87 - 60 * q^88 + 5 * q^89 + 4 * q^90 + 34 * q^92 + 3 * q^93 - 34 * q^94 + 13 * q^95 - 7 * q^96 + 17 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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