LMFDB - Embedded newform 8037.2.a.o.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8037,2,Mod(1,8037)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8037, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8037.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8037 = 3^{2} \cdot 19 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8037.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.1757681045$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - x^{17} - 28 x^{16} + 25 x^{15} + 322 x^{14} - 247 x^{13} - 1971 x^{12} + 1231 x^{11} + 6953 x^{10} + \cdots + 9 $ x^18 - x^17 - 28x^16 + 25x^15 + 322x^14 - 247x^13 - 1971x^12 + 1231x^11 + 6953x^10 - 3283x^9 - 14235x^8 + 4562x^7 + 15962x^6 - 2882x^5 - 8159x^4 + 606x^3 + 890x^2 - 179x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 893)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.33085$ of defining polynomial
Character	$\chi$	$=$	8037.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.33085 q^{2} +3.43286 q^{4} -1.49796 q^{5} +0.334615 q^{7} -3.33977 q^{8} +O(q^{10})$	\(q-2.33085 q^{2} +3.43286 q^{4} -1.49796 q^{5} +0.334615 q^{7} -3.33977 q^{8} +3.49151 q^{10} +3.28150 q^{11} -2.34142 q^{13} -0.779937 q^{14} +0.918788 q^{16} -4.10178 q^{17} -1.00000 q^{19} -5.14227 q^{20} -7.64869 q^{22} -4.14851 q^{23} -2.75612 q^{25} +5.45750 q^{26} +1.14869 q^{28} -8.24616 q^{29} -1.56835 q^{31} +4.53799 q^{32} +9.56062 q^{34} -0.501239 q^{35} -11.4954 q^{37} +2.33085 q^{38} +5.00283 q^{40} -1.31768 q^{41} +2.79334 q^{43} +11.2649 q^{44} +9.66954 q^{46} -1.00000 q^{47} -6.88803 q^{49} +6.42411 q^{50} -8.03776 q^{52} -3.35606 q^{53} -4.91555 q^{55} -1.11754 q^{56} +19.2206 q^{58} +4.15235 q^{59} -8.65980 q^{61} +3.65559 q^{62} -12.4149 q^{64} +3.50735 q^{65} +9.64565 q^{67} -14.0808 q^{68} +1.16831 q^{70} -12.2828 q^{71} +14.9246 q^{73} +26.7941 q^{74} -3.43286 q^{76} +1.09804 q^{77} +8.02708 q^{79} -1.37630 q^{80} +3.07131 q^{82} +1.34448 q^{83} +6.14428 q^{85} -6.51086 q^{86} -10.9595 q^{88} +11.3611 q^{89} -0.783475 q^{91} -14.2412 q^{92} +2.33085 q^{94} +1.49796 q^{95} +15.0547 q^{97} +16.0550 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8}+O(q^{10}) $ 18 * q - q^2 + 21 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8	$ 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8} + 7 q^{10} - 6 q^{11} + 21 q^{13} + 9 q^{14} + 23 q^{16} + 4 q^{17} - 18 q^{19} - 9 q^{20} + 28 q^{22} - 9 q^{23} + 11 q^{25} + q^{26} + 7 q^{28} - 12 q^{29} + 26 q^{31} + 7 q^{32} + 26 q^{34} - 9 q^{35} + 8 q^{37} + q^{38} + 16 q^{40} - 12 q^{41} + 28 q^{43} - 2 q^{44} - 33 q^{46} - 18 q^{47} + 17 q^{49} + 29 q^{50} + 30 q^{52} - 5 q^{53} + 28 q^{55} + 77 q^{56} - 6 q^{58} - 30 q^{59} - 16 q^{61} - 16 q^{62} + 28 q^{64} + 22 q^{65} + 45 q^{67} + 96 q^{68} - 2 q^{70} + q^{71} - 24 q^{73} + 19 q^{74} - 21 q^{76} - 2 q^{77} + 33 q^{79} + 25 q^{80} + 18 q^{82} + 13 q^{83} - 7 q^{85} - 3 q^{86} + 27 q^{88} + 6 q^{89} + 42 q^{91} + 11 q^{92} + q^{94} + 5 q^{95} + 44 q^{97} - 58 q^{98}+O(q^{100}) $ 18 * q - q^2 + 21 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8 + 7 * q^10 - 6 * q^11 + 21 * q^13 + 9 * q^14 + 23 * q^16 + 4 * q^17 - 18 * q^19 - 9 * q^20 + 28 * q^22 - 9 * q^23 + 11 * q^25 + q^26 + 7 * q^28 - 12 * q^29 + 26 * q^31 + 7 * q^32 + 26 * q^34 - 9 * q^35 + 8 * q^37 + q^38 + 16 * q^40 - 12 * q^41 + 28 * q^43 - 2 * q^44 - 33 * q^46 - 18 * q^47 + 17 * q^49 + 29 * q^50 + 30 * q^52 - 5 * q^53 + 28 * q^55 + 77 * q^56 - 6 * q^58 - 30 * q^59 - 16 * q^61 - 16 * q^62 + 28 * q^64 + 22 * q^65 + 45 * q^67 + 96 * q^68 - 2 * q^70 + q^71 - 24 * q^73 + 19 * q^74 - 21 * q^76 - 2 * q^77 + 33 * q^79 + 25 * q^80 + 18 * q^82 + 13 * q^83 - 7 * q^85 - 3 * q^86 + 27 * q^88 + 6 * q^89 + 42 * q^91 + 11 * q^92 + q^94 + 5 * q^95 + 44 * q^97 - 58 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−2.33085	−1.64816	−0.824079	−	0.566474i	$-0.808307\pi$
$2$	−2.33085	−1.64816	−0.824079	+	0.566474i	$0.808307\pi$
$3$	0	0
$3$	0	0
$4$	3.43286	1.71643
$5$	−1.49796	−0.669907	−0.334953	−	0.942235i	$-0.608721\pi$
$5$	−1.49796	−0.669907	−0.334953	+	0.942235i	$0.608721\pi$
$6$	0	0
$7$	0.334615	0.126473	0.0632363	−	0.997999i	$-0.479858\pi$
$7$	0.334615	0.126473	0.0632363	+	0.997999i	$0.479858\pi$
$8$	−3.33977	−1.18079
$9$	0	0
$10$	3.49151	1.10411
$11$	3.28150	0.989410	0.494705	−	0.869061i	$-0.335276\pi$
$11$	3.28150	0.989410	0.494705	+	0.869061i	$0.335276\pi$
$12$	0	0
$13$	−2.34142	−0.649393	−0.324697	−	0.945818i	$-0.605262\pi$
$13$	−2.34142	−0.649393	−0.324697	+	0.945818i	$0.605262\pi$
$14$	−0.779937	−0.208447
$15$	0	0
$16$	0.918788	0.229697
$17$	−4.10178	−0.994827	−0.497413	−	0.867514i	$-0.665717\pi$
$17$	−4.10178	−0.994827	−0.497413	+	0.867514i	$0.665717\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−5.14227	−1.14985
$21$	0	0
$22$	−7.64869	−1.63071
$23$	−4.14851	−0.865023	−0.432512	−	0.901628i	$-0.642373\pi$
$23$	−4.14851	−0.865023	−0.432512	+	0.901628i	$0.642373\pi$
$24$	0	0
$25$	−2.75612	−0.551225
$26$	5.45750	1.07030
$27$	0	0
$28$	1.14869	0.217081
$29$	−8.24616	−1.53127	−0.765637	−	0.643273i	$-0.777576\pi$
$29$	−8.24616	−1.53127	−0.765637	+	0.643273i	$0.777576\pi$
$30$	0	0
$31$	−1.56835	−0.281684	−0.140842	−	0.990032i	$-0.544981\pi$
$31$	−1.56835	−0.281684	−0.140842	+	0.990032i	$0.544981\pi$
$32$	4.53799	0.802210
$33$	0	0
$34$	9.56062	1.63963
$35$	−0.501239	−0.0847249
$36$	0	0
$37$	−11.4954	−1.88984	−0.944918	−	0.327307i	$-0.893859\pi$
$37$	−11.4954	−1.88984	−0.944918	+	0.327307i	$0.893859\pi$
$38$	2.33085	0.378114
$39$	0	0
$40$	5.00283	0.791017
$41$	−1.31768	−0.205787	−0.102894	−	0.994692i	$-0.532810\pi$
$41$	−1.31768	−0.205787	−0.102894	+	0.994692i	$0.532810\pi$
$42$	0	0
$43$	2.79334	0.425981	0.212990	−	0.977054i	$-0.431680\pi$
$43$	2.79334	0.425981	0.212990	+	0.977054i	$0.431680\pi$
$44$	11.2649	1.69825
$45$	0	0
$46$	9.66954	1.42570
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	−6.88803	−0.984005
$50$	6.42411	0.908506
$51$	0	0
$52$	−8.03776	−1.11464
$53$	−3.35606	−0.460990	−0.230495	−	0.973073i	$-0.574035\pi$
$53$	−3.35606	−0.460990	−0.230495	+	0.973073i	$0.574035\pi$
$54$	0	0
$55$	−4.91555	−0.662813
$56$	−1.11754	−0.149337
$57$	0	0
$58$	19.2206	2.52378
$59$	4.15235	0.540591	0.270295	−	0.962777i	$-0.412879\pi$
$59$	4.15235	0.540591	0.270295	+	0.962777i	$0.412879\pi$
$60$	0	0
$61$	−8.65980	−1.10877	−0.554387	−	0.832259i	$-0.687047\pi$
$61$	−8.65980	−1.10877	−0.554387	+	0.832259i	$0.687047\pi$
$62$	3.65559	0.464261
$63$	0	0
$64$	−12.4149	−1.55187
$65$	3.50735	0.435033
$66$	0	0
$67$	9.64565	1.17840	0.589202	−	0.807986i	$-0.299442\pi$
$67$	9.64565	1.17840	0.589202	+	0.807986i	$0.299442\pi$
$68$	−14.0808	−1.70755
$69$	0	0
$70$	1.16831	0.139640
$71$	−12.2828	−1.45770	−0.728852	−	0.684671i	$-0.759946\pi$
$71$	−12.2828	−1.45770	−0.728852	+	0.684671i	$0.759946\pi$
$72$	0	0
$73$	14.9246	1.74679	0.873394	−	0.487014i	$-0.161914\pi$
$73$	14.9246	1.74679	0.873394	+	0.487014i	$0.161914\pi$
$74$	26.7941	3.11475
$75$	0	0
$76$	−3.43286	−0.393776
$77$	1.09804	0.125133
$78$	0	0
$79$	8.02708	0.903117	0.451559	−	0.892241i	$-0.350868\pi$
$79$	8.02708	0.903117	0.451559	+	0.892241i	$0.350868\pi$
$80$	−1.37630	−0.153876
$81$	0	0
$82$	3.07131	0.339170
$83$	1.34448	0.147576	0.0737881	−	0.997274i	$-0.476491\pi$
$83$	1.34448	0.147576	0.0737881	+	0.997274i	$0.476491\pi$
$84$	0	0
$85$	6.14428	0.666441
$86$	−6.51086	−0.702084
$87$	0	0
$88$	−10.9595	−1.16828
$89$	11.3611	1.20428	0.602139	−	0.798391i	$-0.294315\pi$
$89$	11.3611	1.20428	0.602139	+	0.798391i	$0.294315\pi$
$90$	0	0
$91$	−0.783475	−0.0821305
$92$	−14.2412	−1.48475
$93$	0	0
$94$	2.33085	0.240409
$95$	1.49796	0.153687
$96$	0	0
$97$	15.0547	1.52857	0.764285	−	0.644878i	$-0.223092\pi$
$97$	15.0547	1.52857	0.764285	+	0.644878i	$0.223092\pi$
$98$	16.0550	1.62180
$99$	0	0
$100$	−9.46138	−0.946138
$101$	−12.7241	−1.26609	−0.633047	−	0.774113i	$-0.718196\pi$
$101$	−12.7241	−1.26609	−0.633047	+	0.774113i	$0.718196\pi$
$102$	0	0
$103$	−17.6419	−1.73831	−0.869155	−	0.494539i	$-0.835337\pi$
$103$	−17.6419	−1.73831	−0.869155	+	0.494539i	$0.835337\pi$
$104$	7.81981	0.766795
$105$	0	0
$106$	7.82246	0.759785
$107$	−5.20222	−0.502918	−0.251459	−	0.967868i	$-0.580910\pi$
$107$	−5.20222	−0.502918	−0.251459	+	0.967868i	$0.580910\pi$
$108$	0	0
$109$	17.2706	1.65423	0.827113	−	0.562036i	$-0.189982\pi$
$109$	17.2706	1.65423	0.827113	+	0.562036i	$0.189982\pi$
$110$	11.4574	1.09242
$111$	0	0
$112$	0.307440	0.0290504
$113$	15.7039	1.47730	0.738651	−	0.674088i	$-0.235463\pi$
$113$	15.7039	1.47730	0.738651	+	0.674088i	$0.235463\pi$
$114$	0	0
$115$	6.21428	0.579485
$116$	−28.3079	−2.62832
$117$	0	0
$118$	−9.67851	−0.890979
$119$	−1.37252	−0.125818
$120$	0	0
$121$	−0.231737	−0.0210670
$122$	20.1847	1.82744
$123$	0	0
$124$	−5.38393	−0.483491
$125$	11.6183	1.03918
$126$	0	0
$127$	4.64980	0.412603	0.206302	−	0.978488i	$-0.433857\pi$
$127$	4.64980	0.412603	0.206302	+	0.978488i	$0.433857\pi$
$128$	19.8614	1.75551
$129$	0	0
$130$	−8.17510	−0.717003
$131$	−10.6567	−0.931081	−0.465541	−	0.885027i	$-0.654140\pi$
$131$	−10.6567	−0.931081	−0.465541	+	0.885027i	$0.654140\pi$
$132$	0	0
$133$	−0.334615	−0.0290148
$134$	−22.4825	−1.94220
$135$	0	0
$136$	13.6990	1.17468
$137$	8.21156	0.701561	0.350781	−	0.936458i	$-0.385916\pi$
$137$	8.21156	0.701561	0.350781	+	0.936458i	$0.385916\pi$
$138$	0	0
$139$	−8.38231	−0.710978	−0.355489	−	0.934680i	$-0.615686\pi$
$139$	−8.38231	−0.710978	−0.355489	+	0.934680i	$0.615686\pi$
$140$	−1.72068	−0.145424
$141$	0	0
$142$	28.6294	2.40253
$143$	−7.68338	−0.642516
$144$	0	0
$145$	12.3524	1.02581
$146$	−34.7869	−2.87899
$147$	0	0
$148$	−39.4621	−3.24377
$149$	1.37829	0.112914	0.0564570	−	0.998405i	$-0.482020\pi$
$149$	1.37829	0.112914	0.0564570	+	0.998405i	$0.482020\pi$
$150$	0	0
$151$	−8.77967	−0.714480	−0.357240	−	0.934013i	$-0.616282\pi$
$151$	−8.77967	−0.714480	−0.357240	+	0.934013i	$0.616282\pi$
$152$	3.33977	0.270891
$153$	0	0
$154$	−2.55937	−0.206240
$155$	2.34933	0.188702
$156$	0	0
$157$	5.90542	0.471304	0.235652	−	0.971838i	$-0.424277\pi$
$157$	5.90542	0.471304	0.235652	+	0.971838i	$0.424277\pi$
$158$	−18.7099	−1.48848
$159$	0	0
$160$	−6.79771	−0.537406
$161$	−1.38815	−0.109402
$162$	0	0
$163$	11.3049	0.885464	0.442732	−	0.896654i	$-0.354009\pi$
$163$	11.3049	0.885464	0.442732	+	0.896654i	$0.354009\pi$
$164$	−4.52341	−0.353219
$165$	0	0
$166$	−3.13379	−0.243229
$167$	6.59927	0.510667	0.255333	−	0.966853i	$-0.417815\pi$
$167$	6.59927	0.510667	0.255333	+	0.966853i	$0.417815\pi$
$168$	0	0
$169$	−7.51775	−0.578289
$170$	−14.3214	−1.09840
$171$	0	0
$172$	9.58914	0.731165
$173$	5.31719	0.404259	0.202129	−	0.979359i	$-0.435214\pi$
$173$	5.31719	0.404259	0.202129	+	0.979359i	$0.435214\pi$
$174$	0	0
$175$	−0.922241	−0.0697149
$176$	3.01500	0.227265
$177$	0	0
$178$	−26.4811	−1.98484
$179$	2.32992	0.174146	0.0870731	−	0.996202i	$-0.472249\pi$
$179$	2.32992	0.174146	0.0870731	+	0.996202i	$0.472249\pi$
$180$	0	0
$181$	11.5762	0.860450	0.430225	−	0.902722i	$-0.358434\pi$
$181$	11.5762	0.860450	0.430225	+	0.902722i	$0.358434\pi$
$182$	1.82616	0.135364
$183$	0	0
$184$	13.8551	1.02141
$185$	17.2197	1.26601
$186$	0	0
$187$	−13.4600	−0.984292
$188$	−3.43286	−0.250367
$189$	0	0
$190$	−3.49151	−0.253301
$191$	−23.2217	−1.68026	−0.840132	−	0.542381i	$-0.817523\pi$
$191$	−23.2217	−1.68026	−0.840132	+	0.542381i	$0.817523\pi$
$192$	0	0
$193$	−4.30508	−0.309886	−0.154943	−	0.987923i	$-0.549519\pi$
$193$	−4.30508	−0.309886	−0.154943	+	0.987923i	$0.549519\pi$
$194$	−35.0902	−2.51933
$195$	0	0
$196$	−23.6456	−1.68897
$197$	17.3546	1.23646	0.618232	−	0.785996i	$-0.287849\pi$
$197$	17.3546	1.23646	0.618232	+	0.785996i	$0.287849\pi$
$198$	0	0
$199$	−14.5856	−1.03394	−0.516972	−	0.856002i	$-0.672941\pi$
$199$	−14.5856	−1.03394	−0.516972	+	0.856002i	$0.672941\pi$
$200$	9.20482	0.650879
$201$	0	0
$202$	29.6579	2.08672
$203$	−2.75929	−0.193664
$204$	0	0
$205$	1.97383	0.137858
$206$	41.1207	2.86501
$207$	0	0
$208$	−2.15127	−0.149164
$209$	−3.28150	−0.226986
$210$	0	0
$211$	−0.585363	−0.0402980	−0.0201490	−	0.999797i	$-0.506414\pi$
$211$	−0.585363	−0.0402980	−0.0201490	+	0.999797i	$0.506414\pi$
$212$	−11.5209	−0.791256
$213$	0	0
$214$	12.1256	0.828889
$215$	−4.18431	−0.285367
$216$	0	0
$217$	−0.524795	−0.0356254
$218$	−40.2552	−2.72643
$219$	0	0
$220$	−16.8744	−1.13767
$221$	9.60398	0.646034
$222$	0	0
$223$	15.6544	1.04830	0.524149	−	0.851626i	$-0.324383\pi$
$223$	15.6544	1.04830	0.524149	+	0.851626i	$0.324383\pi$
$224$	1.51848	0.101458
$225$	0	0
$226$	−36.6035	−2.43483
$227$	−9.17772	−0.609147	−0.304573	−	0.952489i	$-0.598514\pi$
$227$	−9.17772	−0.609147	−0.304573	+	0.952489i	$0.598514\pi$
$228$	0	0
$229$	8.42223	0.556556	0.278278	−	0.960501i	$-0.410236\pi$
$229$	8.42223	0.556556	0.278278	+	0.960501i	$0.410236\pi$
$230$	−14.4846	−0.955083
$231$	0	0
$232$	27.5403	1.80811
$233$	17.4806	1.14519	0.572596	−	0.819838i	$-0.305936\pi$
$233$	17.4806	1.14519	0.572596	+	0.819838i	$0.305936\pi$
$234$	0	0
$235$	1.49796	0.0977159
$236$	14.2544	0.927885
$237$	0	0
$238$	3.19913	0.207369
$239$	−17.9567	−1.16153	−0.580763	−	0.814073i	$-0.697246\pi$
$239$	−17.9567	−1.16153	−0.580763	+	0.814073i	$0.697246\pi$
$240$	0	0
$241$	10.3380	0.665932	0.332966	−	0.942939i	$-0.391951\pi$
$241$	10.3380	0.665932	0.332966	+	0.942939i	$0.391951\pi$
$242$	0.540143	0.0347217
$243$	0	0
$244$	−29.7279	−1.90313
$245$	10.3180	0.659191
$246$	0	0
$247$	2.34142	0.148981
$248$	5.23794	0.332609
$249$	0	0
$250$	−27.0806	−1.71273
$251$	−0.548338	−0.0346108	−0.0173054	−	0.999850i	$-0.505509\pi$
$251$	−0.548338	−0.0346108	−0.0173054	+	0.999850i	$0.505509\pi$
$252$	0	0
$253$	−13.6133	−0.855863
$254$	−10.8380	−0.680036
$255$	0	0
$256$	−21.4640	−1.34150
$257$	−15.8574	−0.989155	−0.494577	−	0.869134i	$-0.664677\pi$
$257$	−15.8574	−0.989155	−0.494577	+	0.869134i	$0.664677\pi$
$258$	0	0
$259$	−3.84654	−0.239013
$260$	12.0402	0.746703
$261$	0	0
$262$	24.8392	1.53457
$263$	17.9910	1.10937	0.554687	−	0.832059i	$-0.312838\pi$
$263$	17.9910	1.10937	0.554687	+	0.832059i	$0.312838\pi$
$264$	0	0
$265$	5.02723	0.308820
$266$	0.779937	0.0478210
$267$	0	0
$268$	33.1121	2.02264
$269$	−10.6325	−0.648273	−0.324136	−	0.946010i	$-0.605074\pi$
$269$	−10.6325	−0.648273	−0.324136	+	0.946010i	$0.605074\pi$
$270$	0	0
$271$	−17.5754	−1.06763	−0.533815	−	0.845602i	$-0.679242\pi$
$271$	−17.5754	−1.06763	−0.533815	+	0.845602i	$0.679242\pi$
$272$	−3.76866	−0.228509
$273$	0	0
$274$	−19.1399	−1.15628
$275$	−9.04423	−0.545388
$276$	0	0
$277$	13.3895	0.804495	0.402247	−	0.915531i	$-0.368229\pi$
$277$	13.3895	0.804495	0.402247	+	0.915531i	$0.368229\pi$
$278$	19.5379	1.17180
$279$	0	0
$280$	1.67402	0.100042
$281$	17.8889	1.06716	0.533580	−	0.845750i	$-0.320846\pi$
$281$	17.8889	1.06716	0.533580	+	0.845750i	$0.320846\pi$
$282$	0	0
$283$	27.1083	1.61142	0.805712	−	0.592308i	$-0.201783\pi$
$283$	27.1083	1.61142	0.805712	+	0.592308i	$0.201783\pi$
$284$	−42.1652	−2.50204
$285$	0	0
$286$	17.9088	1.05897
$287$	−0.440916	−0.0260264
$288$	0	0
$289$	−0.175440	−0.0103200
$290$	−28.7916	−1.69070
$291$	0	0
$292$	51.2339	2.99824
$293$	−4.36150	−0.254801	−0.127401	−	0.991851i	$-0.540663\pi$
$293$	−4.36150	−0.254801	−0.127401	+	0.991851i	$0.540663\pi$
$294$	0	0
$295$	−6.22005	−0.362145
$296$	38.3921	2.23149
$297$	0	0
$298$	−3.21259	−0.186100
$299$	9.71339	0.561740
$300$	0	0
$301$	0.934694	0.0538749
$302$	20.4641	1.17758
$303$	0	0
$304$	−0.918788	−0.0526961
$305$	12.9720	0.742775
$306$	0	0
$307$	−20.2628	−1.15646	−0.578230	−	0.815874i	$-0.696256\pi$
$307$	−20.2628	−1.15646	−0.578230	+	0.815874i	$0.696256\pi$
$308$	3.76941	0.214782
$309$	0	0
$310$	−5.47592	−0.311011
$311$	22.8790	1.29735	0.648675	−	0.761066i	$-0.275324\pi$
$311$	22.8790	1.29735	0.648675	+	0.761066i	$0.275324\pi$
$312$	0	0
$313$	2.75508	0.155726	0.0778632	−	0.996964i	$-0.475190\pi$
$313$	2.75508	0.155726	0.0778632	+	0.996964i	$0.475190\pi$
$314$	−13.7646	−0.776784
$315$	0	0
$316$	27.5558	1.55014
$317$	6.50108	0.365137	0.182569	−	0.983193i	$-0.441559\pi$
$317$	6.50108	0.365137	0.182569	+	0.983193i	$0.441559\pi$
$318$	0	0
$319$	−27.0598	−1.51506
$320$	18.5970	1.03961
$321$	0	0
$322$	3.23557	0.180311
$323$	4.10178	0.228229
$324$	0	0
$325$	6.45325	0.357962
$326$	−26.3499	−1.45939
$327$	0	0
$328$	4.40075	0.242991
$329$	−0.334615	−0.0184479
$330$	0	0
$331$	24.1890	1.32955	0.664774	−	0.747044i	$-0.268528\pi$
$331$	24.1890	1.32955	0.664774	+	0.747044i	$0.268528\pi$
$332$	4.61542	0.253304
$333$	0	0
$334$	−15.3819	−0.841660
$335$	−14.4488	−0.789420
$336$	0	0
$337$	17.3199	0.943476	0.471738	−	0.881739i	$-0.343627\pi$
$337$	17.3199	0.943476	0.471738	+	0.881739i	$0.343627\pi$
$338$	17.5227	0.953111
$339$	0	0
$340$	21.0924	1.14390
$341$	−5.14655	−0.278702
$342$	0	0
$343$	−4.64715	−0.250922
$344$	−9.32912	−0.502992
$345$	0	0
$346$	−12.3936	−0.666283
$347$	−6.11356	−0.328193	−0.164097	−	0.986444i	$-0.552471\pi$
$347$	−6.11356	−0.328193	−0.164097	+	0.986444i	$0.552471\pi$
$348$	0	0
$349$	9.39594	0.502953	0.251477	−	0.967863i	$-0.419084\pi$
$349$	9.39594	0.502953	0.251477	+	0.967863i	$0.419084\pi$
$350$	2.14960	0.114901
$351$	0	0
$352$	14.8914	0.793715
$353$	14.3497	0.763757	0.381878	−	0.924213i	$-0.375277\pi$
$353$	14.3497	0.763757	0.381878	+	0.924213i	$0.375277\pi$
$354$	0	0
$355$	18.3992	0.976526
$356$	39.0012	2.06706
$357$	0	0
$358$	−5.43068	−0.287020
$359$	−4.99757	−0.263761	−0.131881	−	0.991266i	$-0.542102\pi$
$359$	−4.99757	−0.263761	−0.131881	+	0.991266i	$0.542102\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−26.9823	−1.41816
$363$	0	0
$364$	−2.68956	−0.140971
$365$	−22.3564	−1.17019
$366$	0	0
$367$	5.48934	0.286541	0.143271	−	0.989684i	$-0.454238\pi$
$367$	5.48934	0.286541	0.143271	+	0.989684i	$0.454238\pi$
$368$	−3.81160	−0.198693
$369$	0	0
$370$	−40.1364	−2.08659
$371$	−1.12299	−0.0583026
$372$	0	0
$373$	−32.3430	−1.67466	−0.837328	−	0.546700i	$-0.815884\pi$
$373$	−32.3430	−1.67466	−0.837328	+	0.546700i	$0.815884\pi$
$374$	31.3732	1.62227
$375$	0	0
$376$	3.33977	0.172235
$377$	19.3077	0.994399
$378$	0	0
$379$	7.57119	0.388906	0.194453	−	0.980912i	$-0.437707\pi$
$379$	7.57119	0.388906	0.194453	+	0.980912i	$0.437707\pi$
$380$	5.14227	0.263793
$381$	0	0
$382$	54.1263	2.76934
$383$	−12.4927	−0.638349	−0.319174	−	0.947696i	$-0.603406\pi$
$383$	−12.4927	−0.638349	−0.319174	+	0.947696i	$0.603406\pi$
$384$	0	0
$385$	−1.64482	−0.0838277
$386$	10.0345	0.510742
$387$	0	0
$388$	51.6805	2.62368
$389$	−32.7368	−1.65982	−0.829912	−	0.557895i	$-0.811609\pi$
$389$	−32.7368	−1.65982	−0.829912	+	0.557895i	$0.811609\pi$
$390$	0	0
$391$	17.0162	0.860548
$392$	23.0044	1.16190
$393$	0	0
$394$	−40.4509	−2.03789
$395$	−12.0242	−0.605004
$396$	0	0
$397$	−11.2986	−0.567061	−0.283531	−	0.958963i	$-0.591506\pi$
$397$	−11.2986	−0.567061	−0.283531	+	0.958963i	$0.591506\pi$
$398$	33.9968	1.70410
$399$	0	0
$400$	−2.53229	−0.126615
$401$	−2.48280	−0.123985	−0.0619925	−	0.998077i	$-0.519745\pi$
$401$	−2.48280	−0.123985	−0.0619925	+	0.998077i	$0.519745\pi$
$402$	0	0
$403$	3.67217	0.182924
$404$	−43.6800	−2.17316
$405$	0	0
$406$	6.43149	0.319189
$407$	−37.7223	−1.86982
$408$	0	0
$409$	0.956473	0.0472945	0.0236473	−	0.999720i	$-0.492472\pi$
$409$	0.956473	0.0472945	0.0236473	+	0.999720i	$0.492472\pi$
$410$	−4.60070	−0.227212
$411$	0	0
$412$	−60.5622	−2.98368
$413$	1.38944	0.0683699
$414$	0	0
$415$	−2.01398	−0.0988623
$416$	−10.6253	−0.520950
$417$	0	0
$418$	7.64869	0.374110
$419$	−15.7619	−0.770020	−0.385010	−	0.922912i	$-0.625802\pi$
$419$	−15.7619	−0.770020	−0.385010	+	0.922912i	$0.625802\pi$
$420$	0	0
$421$	2.76756	0.134883	0.0674414	−	0.997723i	$-0.478516\pi$
$421$	2.76756	0.134883	0.0674414	+	0.997723i	$0.478516\pi$
$422$	1.36439	0.0664176
$423$	0	0
$424$	11.2085	0.544331
$425$	11.3050	0.548373
$426$	0	0
$427$	−2.89770	−0.140230
$428$	−17.8585	−0.863222
$429$	0	0
$430$	9.75298	0.470331
$431$	25.7762	1.24160	0.620799	−	0.783970i	$-0.286808\pi$
$431$	25.7762	1.24160	0.620799	+	0.783970i	$0.286808\pi$
$432$	0	0
$433$	−29.3849	−1.41215	−0.706074	−	0.708138i	$-0.749535\pi$
$433$	−29.3849	−1.41215	−0.706074	+	0.708138i	$0.749535\pi$
$434$	1.22322	0.0587163
$435$	0	0
$436$	59.2875	2.83936
$437$	4.14851	0.198450
$438$	0	0
$439$	−9.85760	−0.470478	−0.235239	−	0.971938i	$-0.575587\pi$
$439$	−9.85760	−0.470478	−0.235239	+	0.971938i	$0.575587\pi$
$440$	16.4168	0.782641
$441$	0	0
$442$	−22.3854	−1.06477
$443$	39.5518	1.87916	0.939582	−	0.342325i	$-0.111214\pi$
$443$	39.5518	1.87916	0.939582	+	0.342325i	$0.111214\pi$
$444$	0	0
$445$	−17.0185	−0.806755
$446$	−36.4881	−1.72776
$447$	0	0
$448$	−4.15422	−0.196269
$449$	−10.4162	−0.491569	−0.245784	−	0.969325i	$-0.579046\pi$
$449$	−10.4162	−0.491569	−0.245784	+	0.969325i	$0.579046\pi$
$450$	0	0
$451$	−4.32397	−0.203608
$452$	53.9093	2.53568
$453$	0	0
$454$	21.3919	1.00397
$455$	1.17361	0.0550197
$456$	0	0
$457$	−24.1696	−1.13061	−0.565304	−	0.824883i	$-0.691241\pi$
$457$	−24.1696	−1.13061	−0.565304	+	0.824883i	$0.691241\pi$
$458$	−19.6309	−0.917293
$459$	0	0
$460$	21.3327	0.994644
$461$	31.4793	1.46614	0.733068	−	0.680155i	$-0.238087\pi$
$461$	31.4793	1.46614	0.733068	+	0.680155i	$0.238087\pi$
$462$	0	0
$463$	−29.5842	−1.37489	−0.687447	−	0.726235i	$-0.741269\pi$
$463$	−29.5842	−1.37489	−0.687447	+	0.726235i	$0.741269\pi$
$464$	−7.57647	−0.351729
$465$	0	0
$466$	−40.7446	−1.88746
$467$	−16.8378	−0.779158	−0.389579	−	0.920993i	$-0.627380\pi$
$467$	−16.8378	−0.779158	−0.389579	+	0.920993i	$0.627380\pi$
$468$	0	0
$469$	3.22758	0.149036
$470$	−3.49151	−0.161051
$471$	0	0
$472$	−13.8679	−0.638322
$473$	9.16636	0.421470
$474$	0	0
$475$	2.75612	0.126460
$476$	−4.71165	−0.215958
$477$	0	0
$478$	41.8544	1.91438
$479$	35.4299	1.61883	0.809417	−	0.587234i	$-0.199783\pi$
$479$	35.4299	1.61883	0.809417	+	0.587234i	$0.199783\pi$
$480$	0	0
$481$	26.9156	1.22725
$482$	−24.0964	−1.09756
$483$	0	0
$484$	−0.795519	−0.0361600
$485$	−22.5513	−1.02400
$486$	0	0
$487$	−20.1441	−0.912815	−0.456407	−	0.889771i	$-0.650864\pi$
$487$	−20.1441	−0.912815	−0.456407	+	0.889771i	$0.650864\pi$
$488$	28.9218	1.30923
$489$	0	0
$490$	−24.0496	−1.08645
$491$	−18.0984	−0.816771	−0.408386	−	0.912809i	$-0.633908\pi$
$491$	−18.0984	−0.816771	−0.408386	+	0.912809i	$0.633908\pi$
$492$	0	0
$493$	33.8239	1.52335
$494$	−5.45750	−0.245544
$495$	0	0
$496$	−1.44098	−0.0647021
$497$	−4.11002	−0.184360
$498$	0	0
$499$	35.7080	1.59851	0.799255	−	0.600993i	$-0.205228\pi$
$499$	35.7080	1.59851	0.799255	+	0.600993i	$0.205228\pi$
$500$	39.8841	1.78367
$501$	0	0
$502$	1.27809	0.0570441
$503$	−31.0735	−1.38550	−0.692748	−	0.721179i	$-0.743600\pi$
$503$	−31.0735	−1.38550	−0.692748	+	0.721179i	$0.743600\pi$
$504$	0	0
$505$	19.0601	0.848165
$506$	31.7306	1.41060
$507$	0	0
$508$	15.9621	0.708204
$509$	−18.8860	−0.837109	−0.418554	−	0.908192i	$-0.637463\pi$
$509$	−18.8860	−0.837109	−0.418554	+	0.908192i	$0.637463\pi$
$510$	0	0
$511$	4.99399	0.220921
$512$	10.3065	0.455488
$513$	0	0
$514$	36.9611	1.63028
$515$	26.4268	1.16451
$516$	0	0
$517$	−3.28150	−0.144320
$518$	8.96571	0.393931
$519$	0	0
$520$	−11.7137	−0.513681
$521$	−11.0431	−0.483808	−0.241904	−	0.970300i	$-0.577772\pi$
$521$	−11.0431	−0.483808	−0.241904	+	0.970300i	$0.577772\pi$
$522$	0	0
$523$	12.8964	0.563921	0.281961	−	0.959426i	$-0.409015\pi$
$523$	12.8964	0.563921	0.281961	+	0.959426i	$0.409015\pi$
$524$	−36.5829	−1.59813
$525$	0	0
$526$	−41.9343	−1.82842
$527$	6.43303	0.280227
$528$	0	0
$529$	−5.78990	−0.251735
$530$	−11.7177	−0.508985
$531$	0	0
$532$	−1.14869	−0.0498018
$533$	3.08524	0.133637
$534$	0	0
$535$	7.79271	0.336908
$536$	−32.2142	−1.39144
$537$	0	0
$538$	24.7827	1.06846
$539$	−22.6031	−0.973585
$540$	0	0
$541$	−35.6559	−1.53297	−0.766483	−	0.642265i	$-0.777995\pi$
$541$	−35.6559	−1.53297	−0.766483	+	0.642265i	$0.777995\pi$
$542$	40.9656	1.75962
$543$	0	0
$544$	−18.6138	−0.798060
$545$	−25.8706	−1.10818
$546$	0	0
$547$	27.1927	1.16267	0.581337	−	0.813663i	$-0.302530\pi$
$547$	27.1927	1.16267	0.581337	+	0.813663i	$0.302530\pi$
$548$	28.1891	1.20418
$549$	0	0
$550$	21.0807	0.898886
$551$	8.24616	0.351298
$552$	0	0
$553$	2.68598	0.114220
$554$	−31.2088	−1.32594
$555$	0	0
$556$	−28.7752	−1.22034
$557$	24.9853	1.05866	0.529331	−	0.848416i	$-0.322443\pi$
$557$	24.9853	1.05866	0.529331	+	0.848416i	$0.322443\pi$
$558$	0	0
$559$	−6.54039	−0.276629
$560$	−0.460532	−0.0194610
$561$	0	0
$562$	−41.6962	−1.75885
$563$	−12.2659	−0.516948	−0.258474	−	0.966018i	$-0.583220\pi$
$563$	−12.2659	−0.516948	−0.258474	+	0.966018i	$0.583220\pi$
$564$	0	0
$565$	−23.5238	−0.989654
$566$	−63.1854	−2.65588
$567$	0	0
$568$	41.0218	1.72124
$569$	12.4611	0.522397	0.261198	−	0.965285i	$-0.415882\pi$
$569$	12.4611	0.522397	0.261198	+	0.965285i	$0.415882\pi$
$570$	0	0
$571$	1.85036	0.0774353	0.0387176	−	0.999250i	$-0.487673\pi$
$571$	1.85036	0.0774353	0.0387176	+	0.999250i	$0.487673\pi$
$572$	−26.3759	−1.10283
$573$	0	0
$574$	1.02771	0.0428957
$575$	11.4338	0.476822
$576$	0	0
$577$	37.8908	1.57742	0.788708	−	0.614768i	$-0.210750\pi$
$577$	37.8908	1.57742	0.788708	+	0.614768i	$0.210750\pi$
$578$	0.408924	0.0170090
$579$	0	0
$580$	42.4040	1.76073
$581$	0.449884	0.0186643
$582$	0	0
$583$	−11.0129	−0.456108
$584$	−49.8446	−2.06259
$585$	0	0
$586$	10.1660	0.419953
$587$	14.4353	0.595809	0.297904	−	0.954596i	$-0.403712\pi$
$587$	14.4353	0.595809	0.297904	+	0.954596i	$0.403712\pi$
$588$	0	0
$589$	1.56835	0.0646229
$590$	14.4980	0.596873
$591$	0	0
$592$	−10.5619	−0.434089
$593$	−9.68469	−0.397703	−0.198851	−	0.980030i	$-0.563721\pi$
$593$	−9.68469	−0.397703	−0.198851	+	0.980030i	$0.563721\pi$
$594$	0	0
$595$	2.05597	0.0842865
$596$	4.73148	0.193809
$597$	0	0
$598$	−22.6405	−0.925837
$599$	1.02060	0.0417005	0.0208503	−	0.999783i	$-0.493363\pi$
$599$	1.02060	0.0417005	0.0208503	+	0.999783i	$0.493363\pi$
$600$	0	0
$601$	45.7516	1.86624	0.933122	−	0.359559i	$-0.117073\pi$
$601$	45.7516	1.86624	0.933122	+	0.359559i	$0.117073\pi$
$602$	−2.17863	−0.0887944
$603$	0	0
$604$	−30.1394	−1.22635
$605$	0.347132	0.0141129
$606$	0	0
$607$	30.3822	1.23318	0.616588	−	0.787286i	$-0.288515\pi$
$607$	30.3822	1.23318	0.616588	+	0.787286i	$0.288515\pi$
$608$	−4.53799	−0.184040
$609$	0	0
$610$	−30.2358	−1.22421
$611$	2.34142	0.0947237
$612$	0	0
$613$	−41.9653	−1.69496	−0.847481	−	0.530825i	$-0.821882\pi$
$613$	−41.9653	−1.69496	−0.847481	+	0.530825i	$0.821882\pi$
$614$	47.2296	1.90603
$615$	0	0
$616$	−3.66720	−0.147756
$617$	−30.3725	−1.22275	−0.611375	−	0.791341i	$-0.709383\pi$
$617$	−30.3725	−1.22275	−0.611375	+	0.791341i	$0.709383\pi$
$618$	0	0
$619$	25.9986	1.04497	0.522485	−	0.852648i	$-0.325005\pi$
$619$	25.9986	1.04497	0.522485	+	0.852648i	$0.325005\pi$
$620$	8.06490	0.323894
$621$	0	0
$622$	−53.3275	−2.13824
$623$	3.80161	0.152308
$624$	0	0
$625$	−3.62316	−0.144926
$626$	−6.42168	−0.256662
$627$	0	0
$628$	20.2725	0.808959
$629$	47.1516	1.88006
$630$	0	0
$631$	−3.12901	−0.124564	−0.0622821	−	0.998059i	$-0.519838\pi$
$631$	−3.12901	−0.124564	−0.0622821	+	0.998059i	$0.519838\pi$
$632$	−26.8086	−1.06639
$633$	0	0
$634$	−15.1530	−0.601804
$635$	−6.96521	−0.276406
$636$	0	0
$637$	16.1278	0.639006
$638$	63.0723	2.49706
$639$	0	0
$640$	−29.7515	−1.17603
$641$	−5.39170	−0.212960	−0.106480	−	0.994315i	$-0.533958\pi$
$641$	−5.39170	−0.212960	−0.106480	+	0.994315i	$0.533958\pi$
$642$	0	0
$643$	−2.28839	−0.0902453	−0.0451227	−	0.998981i	$-0.514368\pi$
$643$	−2.28839	−0.0902453	−0.0451227	+	0.998981i	$0.514368\pi$
$644$	−4.76533	−0.187780
$645$	0	0
$646$	−9.56062	−0.376157
$647$	−47.6353	−1.87274	−0.936368	−	0.351020i	$-0.885835\pi$
$647$	−47.6353	−1.87274	−0.936368	+	0.351020i	$0.885835\pi$
$648$	0	0
$649$	13.6260	0.534866
$650$	−15.0415	−0.589978
$651$	0	0
$652$	38.8079	1.51984
$653$	−9.27165	−0.362828	−0.181414	−	0.983407i	$-0.558067\pi$
$653$	−9.27165	−0.362828	−0.181414	+	0.983407i	$0.558067\pi$
$654$	0	0
$655$	15.9633	0.623738
$656$	−1.21067	−0.0472687
$657$	0	0
$658$	0.779937	0.0304051
$659$	3.79035	0.147651	0.0738256	−	0.997271i	$-0.476479\pi$
$659$	3.79035	0.147651	0.0738256	+	0.997271i	$0.476479\pi$
$660$	0	0
$661$	−30.4674	−1.18504	−0.592522	−	0.805554i	$-0.701868\pi$
$661$	−30.4674	−1.18504	−0.592522	+	0.805554i	$0.701868\pi$
$662$	−56.3809	−2.19131
$663$	0	0
$664$	−4.49026	−0.174256
$665$	0.501239	0.0194372
$666$	0	0
$667$	34.2092	1.32459
$668$	22.6543	0.876523
$669$	0	0
$670$	33.6779	1.30109
$671$	−28.4172	−1.09703
$672$	0	0
$673$	−15.7058	−0.605415	−0.302708	−	0.953083i	$-0.597891\pi$
$673$	−15.7058	−0.605415	−0.302708	+	0.953083i	$0.597891\pi$
$674$	−40.3701	−1.55500
$675$	0	0
$676$	−25.8074	−0.992591
$677$	44.0264	1.69207	0.846037	−	0.533125i	$-0.178982\pi$
$677$	44.0264	1.69207	0.846037	+	0.533125i	$0.178982\pi$
$678$	0	0
$679$	5.03752	0.193322
$680$	−20.5205	−0.786925
$681$	0	0
$682$	11.9958	0.459344
$683$	32.8882	1.25843	0.629216	−	0.777231i	$-0.283376\pi$
$683$	32.8882	1.25843	0.629216	+	0.777231i	$0.283376\pi$
$684$	0	0
$685$	−12.3006	−0.469981
$686$	10.8318	0.413560
$687$	0	0
$688$	2.56649	0.0978464
$689$	7.85794	0.299364
$690$	0	0
$691$	38.0424	1.44720	0.723600	−	0.690219i	$-0.242486\pi$
$691$	38.0424	1.44720	0.723600	+	0.690219i	$0.242486\pi$
$692$	18.2532	0.693881
$693$	0	0
$694$	14.2498	0.540914
$695$	12.5563	0.476289
$696$	0	0
$697$	5.40483	0.204723
$698$	−21.9005	−0.828947
$699$	0	0
$700$	−3.16592	−0.119661
$701$	15.1975	0.574000	0.287000	−	0.957931i	$-0.407342\pi$
$701$	15.1975	0.574000	0.287000	+	0.957931i	$0.407342\pi$
$702$	0	0
$703$	11.4954	0.433558
$704$	−40.7396	−1.53543
$705$	0	0
$706$	−33.4470	−1.25879
$707$	−4.25767	−0.160126
$708$	0	0
$709$	−13.3288	−0.500573	−0.250287	−	0.968172i	$-0.580525\pi$
$709$	−13.3288	−0.500573	−0.250287	+	0.968172i	$0.580525\pi$
$710$	−42.8856	−1.60947
$711$	0	0
$712$	−37.9436	−1.42200
$713$	6.50632	0.243664
$714$	0	0
$715$	11.5094	0.430426
$716$	7.99827	0.298909
$717$	0	0
$718$	11.6486	0.434721
$719$	−25.8085	−0.962496	−0.481248	−	0.876585i	$-0.659816\pi$
$719$	−25.8085	−0.962496	−0.481248	+	0.876585i	$0.659816\pi$
$720$	0	0
$721$	−5.90325	−0.219849
$722$	−2.33085	−0.0867452
$723$	0	0
$724$	39.7393	1.47690
$725$	22.7274	0.844076
$726$	0	0
$727$	52.2550	1.93803	0.969015	−	0.247002i	$-0.0794454\pi$
$727$	52.2550	1.93803	0.969015	+	0.247002i	$0.0794454\pi$
$728$	2.61663	0.0969786
$729$	0	0
$730$	52.1093	1.92865
$731$	−11.4577	−0.423777
$732$	0	0
$733$	−49.2614	−1.81951	−0.909756	−	0.415144i	$-0.863731\pi$
$733$	−49.2614	−1.81951	−0.909756	+	0.415144i	$0.863731\pi$
$734$	−12.7948	−0.472265
$735$	0	0
$736$	−18.8259	−0.693930
$737$	31.6522	1.16592
$738$	0	0
$739$	−4.26421	−0.156862	−0.0784308	−	0.996920i	$-0.524991\pi$
$739$	−4.26421	−0.156862	−0.0784308	+	0.996920i	$0.524991\pi$
$740$	59.1126	2.17302
$741$	0	0
$742$	2.61751	0.0960920
$743$	24.1636	0.886475	0.443238	−	0.896404i	$-0.353830\pi$
$743$	24.1636	0.886475	0.443238	+	0.896404i	$0.353830\pi$
$744$	0	0
$745$	−2.06462	−0.0756419
$746$	75.3866	2.76010
$747$	0	0
$748$	−46.2062	−1.68947
$749$	−1.74074	−0.0636053
$750$	0	0
$751$	25.5791	0.933393	0.466697	−	0.884417i	$-0.345444\pi$
$751$	25.5791	0.933393	0.466697	+	0.884417i	$0.345444\pi$
$752$	−0.918788	−0.0335047
$753$	0	0
$754$	−45.0034	−1.63893
$755$	13.1516	0.478635
$756$	0	0
$757$	12.4689	0.453189	0.226594	−	0.973989i	$-0.427241\pi$
$757$	12.4689	0.453189	0.226594	+	0.973989i	$0.427241\pi$
$758$	−17.6473	−0.640979
$759$	0	0
$760$	−5.00283	−0.181472
$761$	−21.5350	−0.780643	−0.390322	−	0.920679i	$-0.627636\pi$
$761$	−21.5350	−0.780643	−0.390322	+	0.920679i	$0.627636\pi$
$762$	0	0
$763$	5.77901	0.209214
$764$	−79.7168	−2.88405
$765$	0	0
$766$	29.1187	1.05210
$767$	−9.72241	−0.351056
$768$	0	0
$769$	−28.3724	−1.02314	−0.511568	−	0.859243i	$-0.670935\pi$
$769$	−28.3724	−1.02314	−0.511568	+	0.859243i	$0.670935\pi$
$770$	3.83382	0.138161
$771$	0	0
$772$	−14.7787	−0.531897
$773$	28.9668	1.04186	0.520932	−	0.853598i	$-0.325584\pi$
$773$	28.9668	1.04186	0.520932	+	0.853598i	$0.325584\pi$
$774$	0	0
$775$	4.32258	0.155272
$776$	−50.2791	−1.80492
$777$	0	0
$778$	76.3046	2.73565
$779$	1.31768	0.0472108
$780$	0	0
$781$	−40.3061	−1.44227
$782$	−39.6623	−1.41832
$783$	0	0
$784$	−6.32864	−0.226023
$785$	−8.84607	−0.315730
$786$	0	0
$787$	34.9927	1.24736	0.623678	−	0.781681i	$-0.285638\pi$
$787$	34.9927	1.24736	0.623678	+	0.781681i	$0.285638\pi$
$788$	59.5758	2.12230
$789$	0	0
$790$	28.0266	0.997143
$791$	5.25477	0.186838
$792$	0	0
$793$	20.2762	0.720031
$794$	26.3354	0.934607
$795$	0	0
$796$	−50.0702	−1.77469
$797$	32.4142	1.14817	0.574084	−	0.818796i	$-0.305358\pi$
$797$	32.4142	1.14817	0.574084	+	0.818796i	$0.305358\pi$
$798$	0	0
$799$	4.10178	0.145110
$800$	−12.5073	−0.442198
$801$	0	0
$802$	5.78702	0.204347
$803$	48.9750	1.72829
$804$	0	0
$805$	2.07939	0.0732890
$806$	−8.55928	−0.301488
$807$	0	0
$808$	42.4955	1.49499
$809$	5.35007	0.188098	0.0940492	−	0.995568i	$-0.470019\pi$
$809$	5.35007	0.188098	0.0940492	+	0.995568i	$0.470019\pi$
$810$	0	0
$811$	−30.1450	−1.05853	−0.529267	−	0.848455i	$-0.677533\pi$
$811$	−30.1450	−1.05853	−0.529267	+	0.848455i	$0.677533\pi$
$812$	−9.47224	−0.332411
$813$	0	0
$814$	87.9249	3.08177
$815$	−16.9342	−0.593179
$816$	0	0
$817$	−2.79334	−0.0977267
$818$	−2.22939	−0.0779489
$819$	0	0
$820$	6.77587	0.236624
$821$	−20.6278	−0.719914	−0.359957	−	0.932969i	$-0.617209\pi$
$821$	−20.6278	−0.719914	−0.359957	+	0.932969i	$0.617209\pi$
$822$	0	0
$823$	−24.5216	−0.854768	−0.427384	−	0.904070i	$-0.640565\pi$
$823$	−24.5216	−0.854768	−0.427384	+	0.904070i	$0.640565\pi$
$824$	58.9200	2.05257
$825$	0	0
$826$	−3.23858	−0.112684
$827$	37.9589	1.31996	0.659981	−	0.751282i	$-0.270564\pi$
$827$	37.9589	1.31996	0.659981	+	0.751282i	$0.270564\pi$
$828$	0	0
$829$	−6.48391	−0.225196	−0.112598	−	0.993641i	$-0.535917\pi$
$829$	−6.48391	−0.225196	−0.112598	+	0.993641i	$0.535917\pi$
$830$	4.69428	0.162941
$831$	0	0
$832$	29.0686	1.00777
$833$	28.2532	0.978914
$834$	0	0
$835$	−9.88542	−0.342099
$836$	−11.2649	−0.389606
$837$	0	0
$838$	36.7386	1.26911
$839$	12.1603	0.419821	0.209910	−	0.977721i	$-0.432683\pi$
$839$	12.1603	0.419821	0.209910	+	0.977721i	$0.432683\pi$
$840$	0	0
$841$	38.9992	1.34480
$842$	−6.45077	−0.222308
$843$	0	0
$844$	−2.00947	−0.0691687
$845$	11.2613	0.387399
$846$	0	0
$847$	−0.0775426	−0.00266440
$848$	−3.08350	−0.105888
$849$	0	0
$850$	−26.3503	−0.903806
$851$	47.6888	1.63475
$852$	0	0
$853$	36.5824	1.25256	0.626279	−	0.779599i	$-0.284577\pi$
$853$	36.5824	1.25256	0.626279	+	0.779599i	$0.284577\pi$
$854$	6.75410	0.231121
$855$	0	0
$856$	17.3742	0.593839
$857$	21.6715	0.740283	0.370141	−	0.928975i	$-0.379309\pi$
$857$	21.6715	0.740283	0.370141	+	0.928975i	$0.379309\pi$
$858$	0	0
$859$	−32.1130	−1.09568	−0.547841	−	0.836583i	$-0.684550\pi$
$859$	−32.1130	−1.09568	−0.547841	+	0.836583i	$0.684550\pi$
$860$	−14.3641	−0.489812
$861$	0	0
$862$	−60.0805	−2.04635
$863$	7.96777	0.271226	0.135613	−	0.990762i	$-0.456700\pi$
$863$	7.96777	0.271226	0.135613	+	0.990762i	$0.456700\pi$
$864$	0	0
$865$	−7.96493	−0.270816
$866$	68.4918	2.32744
$867$	0	0
$868$	−1.80154	−0.0611484
$869$	26.3409	0.893554
$870$	0	0
$871$	−22.5845	−0.765247
$872$	−57.6799	−1.95329
$873$	0	0
$874$	−9.66954	−0.327077
$875$	3.88767	0.131427
$876$	0	0
$877$	−51.5888	−1.74203	−0.871015	−	0.491256i	$-0.836538\pi$
$877$	−51.5888	−1.74203	−0.871015	+	0.491256i	$0.836538\pi$
$878$	22.9766	0.775422
$879$	0	0
$880$	−4.51635	−0.152246
$881$	33.7064	1.13560	0.567798	−	0.823168i	$-0.307795\pi$
$881$	33.7064	1.13560	0.567798	+	0.823168i	$0.307795\pi$
$882$	0	0
$883$	−51.4647	−1.73192	−0.865962	−	0.500110i	$-0.833293\pi$
$883$	−51.4647	−1.73192	−0.865962	+	0.500110i	$0.833293\pi$
$884$	32.9691	1.10887
$885$	0	0
$886$	−92.1893	−3.09716
$887$	−24.0819	−0.808591	−0.404295	−	0.914628i	$-0.632483\pi$
$887$	−24.0819	−0.808591	−0.404295	+	0.914628i	$0.632483\pi$
$888$	0	0
$889$	1.55589	0.0521830
$890$	39.6676	1.32966
$891$	0	0
$892$	53.7394	1.79933
$893$	1.00000	0.0334637
$894$	0	0
$895$	−3.49011	−0.116662
$896$	6.64591	0.222024
$897$	0	0
$898$	24.2785	0.810184
$899$	12.9329	0.431336
$900$	0	0
$901$	13.7658	0.458605
$902$	10.0785	0.335578
$903$	0	0
$904$	−52.4475	−1.74438
$905$	−17.3406	−0.576421
$906$	0	0
$907$	28.5272	0.947231	0.473616	−	0.880732i	$-0.342949\pi$
$907$	28.5272	0.947231	0.473616	+	0.880732i	$0.342949\pi$
$908$	−31.5058	−1.04556
$909$	0	0
$910$	−2.73551	−0.0906813
$911$	53.6852	1.77867	0.889335	−	0.457256i	$-0.151168\pi$
$911$	53.6852	1.77867	0.889335	+	0.457256i	$0.151168\pi$
$912$	0	0
$913$	4.41193	0.146013
$914$	56.3357	1.86342
$915$	0	0
$916$	28.9123	0.955289
$917$	−3.56590	−0.117756
$918$	0	0
$919$	−36.7539	−1.21240	−0.606199	−	0.795313i	$-0.707306\pi$
$919$	−36.7539	−1.21240	−0.606199	+	0.795313i	$0.707306\pi$
$920$	−20.7543	−0.684248
$921$	0	0
$922$	−73.3735	−2.41643
$923$	28.7593	0.946623
$924$	0	0
$925$	31.6828	1.04172
$926$	68.9563	2.26604
$927$	0	0
$928$	−37.4210	−1.22840
$929$	41.4855	1.36110	0.680548	−	0.732703i	$-0.261742\pi$
$929$	41.4855	1.36110	0.680548	+	0.732703i	$0.261742\pi$
$930$	0	0
$931$	6.88803	0.225746
$932$	60.0083	1.96564
$933$	0	0
$934$	39.2463	1.28418
$935$	20.1625	0.659384
$936$	0	0
$937$	−29.1044	−0.950799	−0.475400	−	0.879770i	$-0.657697\pi$
$937$	−29.1044	−0.950799	−0.475400	+	0.879770i	$0.657697\pi$
$938$	−7.52300	−0.245635
$939$	0	0
$940$	5.14227	0.167722
$941$	24.4186	0.796022	0.398011	−	0.917381i	$-0.369701\pi$
$941$	24.4186	0.796022	0.398011	+	0.917381i	$0.369701\pi$
$942$	0	0
$943$	5.46641	0.178011
$944$	3.81513	0.124172
$945$	0	0
$946$	−21.3654	−0.694649
$947$	−31.4950	−1.02345	−0.511725	−	0.859149i	$-0.670993\pi$
$947$	−31.4950	−1.02345	−0.511725	+	0.859149i	$0.670993\pi$
$948$	0	0
$949$	−34.9447	−1.13435
$950$	−6.42411	−0.208426
$951$	0	0
$952$	4.58389	0.148565
$953$	27.3498	0.885947	0.442973	−	0.896535i	$-0.353924\pi$
$953$	27.3498	0.885947	0.442973	+	0.896535i	$0.353924\pi$
$954$	0	0
$955$	34.7851	1.12562
$956$	−61.6429	−1.99367
$957$	0	0
$958$	−82.5818	−2.66810
$959$	2.74771	0.0887283
$960$	0	0
$961$	−28.5403	−0.920654
$962$	−62.7362	−2.02270
$963$	0	0
$964$	35.4890	1.14302
$965$	6.44882	0.207595
$966$	0	0
$967$	50.7123	1.63080	0.815399	−	0.578900i	$-0.196518\pi$
$967$	50.7123	1.63080	0.815399	+	0.578900i	$0.196518\pi$
$968$	0.773948	0.0248756
$969$	0	0
$970$	52.5636	1.68771
$971$	−19.2200	−0.616800	−0.308400	−	0.951257i	$-0.599793\pi$
$971$	−19.2200	−0.616800	−0.308400	+	0.951257i	$0.599793\pi$
$972$	0	0
$973$	−2.80485	−0.0899192
$974$	46.9528	1.50446
$975$	0	0
$976$	−7.95652	−0.254682
$977$	26.5102	0.848138	0.424069	−	0.905630i	$-0.360601\pi$
$977$	26.5102	0.848138	0.424069	+	0.905630i	$0.360601\pi$
$978$	0	0
$979$	37.2816	1.19153
$980$	35.4201	1.13145
$981$	0	0
$982$	42.1847	1.34617
$983$	31.3591	1.00020	0.500101	−	0.865967i	$-0.333296\pi$
$983$	31.3591	1.00020	0.500101	+	0.865967i	$0.333296\pi$
$984$	0	0
$985$	−25.9964	−0.828315
$986$	−78.8384	−2.51073
$987$	0	0
$988$	8.03776	0.255715
$989$	−11.5882	−0.368483
$990$	0	0
$991$	20.8337	0.661803	0.330902	−	0.943665i	$-0.392647\pi$
$991$	20.8337	0.661803	0.330902	+	0.943665i	$0.392647\pi$
$992$	−7.11716	−0.225970
$993$	0	0
$994$	9.57984	0.303854
$995$	21.8486	0.692646
$996$	0	0
$997$	0.972329	0.0307940	0.0153970	−	0.999881i	$-0.495099\pi$
$997$	0.972329	0.0307940	0.0153970	+	0.999881i	$0.495099\pi$
$998$	−83.2299	−2.63460
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.o.1.4		18
3.2	odd	2		893.2.a.c.1.15	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
893.2.a.c.1.15	✓	18	3.2	odd	2
8037.2.a.o.1.4		18	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q-2.33085 q^{2} +3.43286 q^{4} -1.49796 q^{5} +0.334615 q^{7} -3.33977 q^{8} +O(q^{10})\)	\(q-2.33085 q^{2} +3.43286 q^{4} -1.49796 q^{5} +0.334615 q^{7} -3.33977 q^{8} +3.49151 q^{10} +3.28150 q^{11} -2.34142 q^{13} -0.779937 q^{14} +0.918788 q^{16} -4.10178 q^{17} -1.00000 q^{19} -5.14227 q^{20} -7.64869 q^{22} -4.14851 q^{23} -2.75612 q^{25} +5.45750 q^{26} +1.14869 q^{28} -8.24616 q^{29} -1.56835 q^{31} +4.53799 q^{32} +9.56062 q^{34} -0.501239 q^{35} -11.4954 q^{37} +2.33085 q^{38} +5.00283 q^{40} -1.31768 q^{41} +2.79334 q^{43} +11.2649 q^{44} +9.66954 q^{46} -1.00000 q^{47} -6.88803 q^{49} +6.42411 q^{50} -8.03776 q^{52} -3.35606 q^{53} -4.91555 q^{55} -1.11754 q^{56} +19.2206 q^{58} +4.15235 q^{59} -8.65980 q^{61} +3.65559 q^{62} -12.4149 q^{64} +3.50735 q^{65} +9.64565 q^{67} -14.0808 q^{68} +1.16831 q^{70} -12.2828 q^{71} +14.9246 q^{73} +26.7941 q^{74} -3.43286 q^{76} +1.09804 q^{77} +8.02708 q^{79} -1.37630 q^{80} +3.07131 q^{82} +1.34448 q^{83} +6.14428 q^{85} -6.51086 q^{86} -10.9595 q^{88} +11.3611 q^{89} -0.783475 q^{91} -14.2412 q^{92} +2.33085 q^{94} +1.49796 q^{95} +15.0547 q^{97} +16.0550 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8}+O(q^{10}) \) 18 * q - q^2 + 21 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8	\( 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8} + 7 q^{10} - 6 q^{11} + 21 q^{13} + 9 q^{14} + 23 q^{16} + 4 q^{17} - 18 q^{19} - 9 q^{20} + 28 q^{22} - 9 q^{23} + 11 q^{25} + q^{26} + 7 q^{28} - 12 q^{29} + 26 q^{31} + 7 q^{32} + 26 q^{34} - 9 q^{35} + 8 q^{37} + q^{38} + 16 q^{40} - 12 q^{41} + 28 q^{43} - 2 q^{44} - 33 q^{46} - 18 q^{47} + 17 q^{49} + 29 q^{50} + 30 q^{52} - 5 q^{53} + 28 q^{55} + 77 q^{56} - 6 q^{58} - 30 q^{59} - 16 q^{61} - 16 q^{62} + 28 q^{64} + 22 q^{65} + 45 q^{67} + 96 q^{68} - 2 q^{70} + q^{71} - 24 q^{73} + 19 q^{74} - 21 q^{76} - 2 q^{77} + 33 q^{79} + 25 q^{80} + 18 q^{82} + 13 q^{83} - 7 q^{85} - 3 q^{86} + 27 q^{88} + 6 q^{89} + 42 q^{91} + 11 q^{92} + q^{94} + 5 q^{95} + 44 q^{97} - 58 q^{98}+O(q^{100}) \) 18 * q - q^2 + 21 * q^4 - 5 * q^5 + 3 * q^7 - 6 * q^8 + 7 * q^10 - 6 * q^11 + 21 * q^13 + 9 * q^14 + 23 * q^16 + 4 * q^17 - 18 * q^19 - 9 * q^20 + 28 * q^22 - 9 * q^23 + 11 * q^25 + q^26 + 7 * q^28 - 12 * q^29 + 26 * q^31 + 7 * q^32 + 26 * q^34 - 9 * q^35 + 8 * q^37 + q^38 + 16 * q^40 - 12 * q^41 + 28 * q^43 - 2 * q^44 - 33 * q^46 - 18 * q^47 + 17 * q^49 + 29 * q^50 + 30 * q^52 - 5 * q^53 + 28 * q^55 + 77 * q^56 - 6 * q^58 - 30 * q^59 - 16 * q^61 - 16 * q^62 + 28 * q^64 + 22 * q^65 + 45 * q^67 + 96 * q^68 - 2 * q^70 + q^71 - 24 * q^73 + 19 * q^74 - 21 * q^76 - 2 * q^77 + 33 * q^79 + 25 * q^80 + 18 * q^82 + 13 * q^83 - 7 * q^85 - 3 * q^86 + 27 * q^88 + 6 * q^89 + 42 * q^91 + 11 * q^92 + q^94 + 5 * q^95 + 44 * q^97 - 58 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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