LMFDB - Embedded newform 8037.2.a.o.1.10

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			1.10
Root			$0.0898969$ 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)

$f(q)$	$=$	$q-0.0898969 q^{2} -1.99192 q^{4} -2.73841 q^{5} +4.94566 q^{7} +0.358861 q^{8} +O(q^{10})$	\(q-0.0898969 q^{2} -1.99192 q^{4} -2.73841 q^{5} +4.94566 q^{7} +0.358861 q^{8} +0.246175 q^{10} +0.583086 q^{11} +1.98136 q^{13} -0.444600 q^{14} +3.95158 q^{16} +0.412514 q^{17} -1.00000 q^{19} +5.45469 q^{20} -0.0524177 q^{22} +5.19146 q^{23} +2.49889 q^{25} -0.178118 q^{26} -9.85135 q^{28} +1.76148 q^{29} +7.75468 q^{31} -1.07296 q^{32} -0.0370837 q^{34} -13.5432 q^{35} +7.04040 q^{37} +0.0898969 q^{38} -0.982709 q^{40} -3.64462 q^{41} +2.82019 q^{43} -1.16146 q^{44} -0.466696 q^{46} -1.00000 q^{47} +17.4596 q^{49} -0.224642 q^{50} -3.94671 q^{52} -1.17270 q^{53} -1.59673 q^{55} +1.77481 q^{56} -0.158352 q^{58} -0.783404 q^{59} -4.82063 q^{61} -0.697122 q^{62} -7.80670 q^{64} -5.42577 q^{65} +8.57132 q^{67} -0.821693 q^{68} +1.21750 q^{70} -8.22491 q^{71} -0.408728 q^{73} -0.632910 q^{74} +1.99192 q^{76} +2.88375 q^{77} +9.20351 q^{79} -10.8210 q^{80} +0.327640 q^{82} +6.84384 q^{83} -1.12963 q^{85} -0.253527 q^{86} +0.209247 q^{88} -7.22108 q^{89} +9.79913 q^{91} -10.3410 q^{92} +0.0898969 q^{94} +2.73841 q^{95} -16.9644 q^{97} -1.56956 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.0898969	−0.0635667	−0.0317834	−	0.999495i	$-0.510119\pi$
$2$	−0.0898969	−0.0635667	−0.0317834	+	0.999495i	$0.510119\pi$
$3$	0	0
$3$	0	0
$4$	−1.99192	−0.995959
$5$	−2.73841	−1.22465	−0.612327	−	0.790605i	$-0.709766\pi$
$5$	−2.73841	−1.22465	−0.612327	+	0.790605i	$0.709766\pi$
$6$	0	0
$7$	4.94566	1.86928	0.934642	−	0.355590i	$-0.115720\pi$
$7$	4.94566	1.86928	0.934642	+	0.355590i	$0.115720\pi$
$8$	0.358861	0.126877
$9$	0	0
$10$	0.246175	0.0778473
$11$	0.583086	0.175807	0.0879036	−	0.996129i	$-0.471983\pi$
$11$	0.583086	0.175807	0.0879036	+	0.996129i	$0.471983\pi$
$12$	0	0
$13$	1.98136	0.549530	0.274765	−	0.961511i	$-0.411400\pi$
$13$	1.98136	0.549530	0.274765	+	0.961511i	$0.411400\pi$
$14$	−0.444600	−0.118824
$15$	0	0
$16$	3.95158	0.987894
$17$	0.412514	0.100049	0.0500246	−	0.998748i	$-0.484070\pi$
$17$	0.412514	0.100049	0.0500246	+	0.998748i	$0.484070\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	5.45469	1.21971
$21$	0	0
$22$	−0.0524177	−0.0111755
$23$	5.19146	1.08249	0.541247	−	0.840864i	$-0.317952\pi$
$23$	5.19146	1.08249	0.541247	+	0.840864i	$0.317952\pi$
$24$	0	0
$25$	2.49889	0.499778
$26$	−0.178118	−0.0349318
$27$	0	0
$28$	−9.85135	−1.86173
$29$	1.76148	0.327099	0.163550	−	0.986535i	$-0.447706\pi$
$29$	1.76148	0.327099	0.163550	+	0.986535i	$0.447706\pi$
$30$	0	0
$31$	7.75468	1.39278	0.696391	−	0.717663i	$-0.254788\pi$
$31$	7.75468	1.39278	0.696391	+	0.717663i	$0.254788\pi$
$32$	−1.07296	−0.189674
$33$	0	0
$34$	−0.0370837	−0.00635980
$35$	−13.5432	−2.28923
$36$	0	0
$37$	7.04040	1.15743	0.578717	−	0.815528i	$-0.303554\pi$
$37$	7.04040	1.15743	0.578717	+	0.815528i	$0.303554\pi$
$38$	0.0898969	0.0145832
$39$	0	0
$40$	−0.982709	−0.155380
$41$	−3.64462	−0.569193	−0.284597	−	0.958647i	$-0.591860\pi$
$41$	−3.64462	−0.569193	−0.284597	+	0.958647i	$0.591860\pi$
$42$	0	0
$43$	2.82019	0.430075	0.215038	−	0.976606i	$-0.431013\pi$
$43$	2.82019	0.430075	0.215038	+	0.976606i	$0.431013\pi$
$44$	−1.16146	−0.175097
$45$	0	0
$46$	−0.466696	−0.0688106
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	17.4596	2.49422
$50$	−0.224642	−0.0317692
$51$	0	0
$52$	−3.94671	−0.547310
$53$	−1.17270	−0.161083	−0.0805417	−	0.996751i	$-0.525665\pi$
$53$	−1.17270	−0.161083	−0.0805417	+	0.996751i	$0.525665\pi$
$54$	0	0
$55$	−1.59673	−0.215303
$56$	1.77481	0.237168
$57$	0	0
$58$	−0.158352	−0.0207926
$59$	−0.783404	−0.101991	−0.0509953	−	0.998699i	$-0.516239\pi$
$59$	−0.783404	−0.101991	−0.0509953	+	0.998699i	$0.516239\pi$
$60$	0	0
$61$	−4.82063	−0.617218	−0.308609	−	0.951189i	$-0.599863\pi$
$61$	−4.82063	−0.617218	−0.308609	+	0.951189i	$0.599863\pi$
$62$	−0.697122	−0.0885345
$63$	0	0
$64$	−7.80670	−0.975837
$65$	−5.42577	−0.672985
$66$	0	0
$67$	8.57132	1.04715	0.523577	−	0.851978i	$-0.324597\pi$
$67$	8.57132	1.04715	0.523577	+	0.851978i	$0.324597\pi$
$68$	−0.821693	−0.0996450
$69$	0	0
$70$	1.21750	0.145519
$71$	−8.22491	−0.976117	−0.488059	−	0.872811i	$-0.662295\pi$
$71$	−8.22491	−0.976117	−0.488059	+	0.872811i	$0.662295\pi$
$72$	0	0
$73$	−0.408728	−0.0478379	−0.0239190	−	0.999714i	$-0.507614\pi$
$73$	−0.408728	−0.0478379	−0.0239190	+	0.999714i	$0.507614\pi$
$74$	−0.632910	−0.0735743
$75$	0	0
$76$	1.99192	0.228489
$77$	2.88375	0.328633
$78$	0	0
$79$	9.20351	1.03548	0.517738	−	0.855539i	$-0.326774\pi$
$79$	9.20351	1.03548	0.517738	+	0.855539i	$0.326774\pi$
$80$	−10.8210	−1.20983
$81$	0	0
$82$	0.327640	0.0361818
$83$	6.84384	0.751210	0.375605	−	0.926780i	$-0.377435\pi$
$83$	6.84384	0.751210	0.375605	+	0.926780i	$0.377435\pi$
$84$	0	0
$85$	−1.12963	−0.122526
$86$	−0.253527	−0.0273385
$87$	0	0
$88$	0.209247	0.0223058
$89$	−7.22108	−0.765433	−0.382716	−	0.923866i	$-0.625011\pi$
$89$	−7.22108	−0.765433	−0.382716	+	0.923866i	$0.625011\pi$
$90$	0	0
$91$	9.79913	1.02723
$92$	−10.3410	−1.07812
$93$	0	0
$94$	0.0898969	0.00927216
$95$	2.73841	0.280955
$96$	0	0
$97$	−16.9644	−1.72247	−0.861237	−	0.508203i	$-0.830310\pi$
$97$	−16.9644	−1.72247	−0.861237	+	0.508203i	$0.830310\pi$
$98$	−1.56956	−0.158550
$99$	0	0
$100$	−4.97758	−0.497758
$101$	−2.03825	−0.202814	−0.101407	−	0.994845i	$-0.532334\pi$
$101$	−2.03825	−0.202814	−0.101407	+	0.994845i	$0.532334\pi$
$102$	0	0
$103$	1.25562	0.123720	0.0618601	−	0.998085i	$-0.480297\pi$
$103$	1.25562	0.123720	0.0618601	+	0.998085i	$0.480297\pi$
$104$	0.711033	0.0697225
$105$	0	0
$106$	0.105423	0.0102395
$107$	−7.73716	−0.747980	−0.373990	−	0.927433i	$-0.622011\pi$
$107$	−7.73716	−0.747980	−0.373990	+	0.927433i	$0.622011\pi$
$108$	0	0
$109$	5.68010	0.544055	0.272027	−	0.962290i	$-0.412306\pi$
$109$	5.68010	0.544055	0.272027	+	0.962290i	$0.412306\pi$
$110$	0.143541	0.0136861
$111$	0	0
$112$	19.5432	1.84666
$113$	7.24790	0.681825	0.340913	−	0.940095i	$-0.389264\pi$
$113$	7.24790	0.681825	0.340913	+	0.940095i	$0.389264\pi$
$114$	0	0
$115$	−14.2163	−1.32568
$116$	−3.50873	−0.325778
$117$	0	0
$118$	0.0704256	0.00648320
$119$	2.04015	0.187020
$120$	0	0
$121$	−10.6600	−0.969092
$122$	0.433359	0.0392345
$123$	0	0
$124$	−15.4467	−1.38715
$125$	6.84907	0.612599
$126$	0	0
$127$	−14.3404	−1.27251	−0.636254	−	0.771480i	$-0.719517\pi$
$127$	−14.3404	−1.27251	−0.636254	+	0.771480i	$0.719517\pi$
$128$	2.84771	0.251705
$129$	0	0
$130$	0.487760	0.0427794
$131$	−6.35386	−0.555139	−0.277570	−	0.960705i	$-0.589529\pi$
$131$	−6.35386	−0.555139	−0.277570	+	0.960705i	$0.589529\pi$
$132$	0	0
$133$	−4.94566	−0.428843
$134$	−0.770536	−0.0665641
$135$	0	0
$136$	0.148035	0.0126939
$137$	−15.6792	−1.33956	−0.669781	−	0.742558i	$-0.733612\pi$
$137$	−15.6792	−1.33956	−0.669781	+	0.742558i	$0.733612\pi$
$138$	0	0
$139$	−4.42523	−0.375343	−0.187671	−	0.982232i	$-0.560094\pi$
$139$	−4.42523	−0.375343	−0.187671	+	0.982232i	$0.560094\pi$
$140$	26.9770	2.27998
$141$	0	0
$142$	0.739394	0.0620486
$143$	1.15530	0.0966113
$144$	0	0
$145$	−4.82367	−0.400584
$146$	0.0367433	0.00304090
$147$	0	0
$148$	−14.0239	−1.15276
$149$	23.8345	1.95260	0.976299	−	0.216428i	$-0.0694406\pi$
$149$	23.8345	1.95260	0.976299	+	0.216428i	$0.0694406\pi$
$150$	0	0
$151$	6.63821	0.540210	0.270105	−	0.962831i	$-0.412942\pi$
$151$	6.63821	0.540210	0.270105	+	0.962831i	$0.412942\pi$
$152$	−0.358861	−0.0291075
$153$	0	0
$154$	−0.259240	−0.0208902
$155$	−21.2355	−1.70568
$156$	0	0
$157$	−3.81863	−0.304760	−0.152380	−	0.988322i	$-0.548694\pi$
$157$	−3.81863	−0.304760	−0.152380	+	0.988322i	$0.548694\pi$
$158$	−0.827367	−0.0658218
$159$	0	0
$160$	2.93820	0.232285
$161$	25.6752	2.02349
$162$	0	0
$163$	7.53496	0.590183	0.295092	−	0.955469i	$-0.404650\pi$
$163$	7.53496	0.590183	0.295092	+	0.955469i	$0.404650\pi$
$164$	7.25978	0.566893
$165$	0	0
$166$	−0.615241	−0.0477519
$167$	20.2115	1.56402	0.782008	−	0.623268i	$-0.214196\pi$
$167$	20.2115	1.56402	0.782008	+	0.623268i	$0.214196\pi$
$168$	0	0
$169$	−9.07421	−0.698016
$170$	0.101550	0.00778856
$171$	0	0
$172$	−5.61759	−0.428337
$173$	1.32347	0.100622	0.0503109	−	0.998734i	$-0.483979\pi$
$173$	1.32347	0.100622	0.0503109	+	0.998734i	$0.483979\pi$
$174$	0	0
$175$	12.3587	0.934227
$176$	2.30411	0.173679
$177$	0	0
$178$	0.649153	0.0486560
$179$	−20.4729	−1.53021	−0.765106	−	0.643904i	$-0.777314\pi$
$179$	−20.4729	−1.53021	−0.765106	+	0.643904i	$0.777314\pi$
$180$	0	0
$181$	17.8750	1.32864	0.664319	−	0.747450i	$-0.268722\pi$
$181$	17.8750	1.32864	0.664319	+	0.747450i	$0.268722\pi$
$182$	−0.880912	−0.0652975
$183$	0	0
$184$	1.86301	0.137343
$185$	−19.2795	−1.41746
$186$	0	0
$187$	0.240531	0.0175894
$188$	1.99192	0.145276
$189$	0	0
$190$	−0.246175	−0.0178594
$191$	−18.0646	−1.30711	−0.653553	−	0.756881i	$-0.726722\pi$
$191$	−18.0646	−1.30711	−0.653553	+	0.756881i	$0.726722\pi$
$192$	0	0
$193$	−11.0507	−0.795449	−0.397724	−	0.917505i	$-0.630200\pi$
$193$	−11.0507	−0.795449	−0.397724	+	0.917505i	$0.630200\pi$
$194$	1.52505	0.109492
$195$	0	0
$196$	−34.7780	−2.48415
$197$	18.9606	1.35088	0.675442	−	0.737413i	$-0.263953\pi$
$197$	18.9606	1.35088	0.675442	+	0.737413i	$0.263953\pi$
$198$	0	0
$199$	−10.3270	−0.732065	−0.366032	−	0.930602i	$-0.619284\pi$
$199$	−10.3270	−0.732065	−0.366032	+	0.930602i	$0.619284\pi$
$200$	0.896754	0.0634101
$201$	0	0
$202$	0.183232	0.0128922
$203$	8.71171	0.611442
$204$	0	0
$205$	9.98045	0.697065
$206$	−0.112877	−0.00786449
$207$	0	0
$208$	7.82949	0.542878
$209$	−0.583086	−0.0403329
$210$	0	0
$211$	−15.5877	−1.07310	−0.536550	−	0.843868i	$-0.680273\pi$
$211$	−15.5877	−1.07310	−0.536550	+	0.843868i	$0.680273\pi$
$212$	2.33593	0.160433
$213$	0	0
$214$	0.695547	0.0475466
$215$	−7.72284	−0.526693
$216$	0	0
$217$	38.3520	2.60350
$218$	−0.510623	−0.0345838
$219$	0	0
$220$	3.18055	0.214433
$221$	0.817338	0.0549801
$222$	0	0
$223$	23.0326	1.54238	0.771189	−	0.636606i	$-0.219662\pi$
$223$	23.0326	1.54238	0.771189	+	0.636606i	$0.219662\pi$
$224$	−5.30648	−0.354554
$225$	0	0
$226$	−0.651564	−0.0433414
$227$	22.6755	1.50503	0.752514	−	0.658576i	$-0.228841\pi$
$227$	22.6755	1.50503	0.752514	+	0.658576i	$0.228841\pi$
$228$	0	0
$229$	2.03502	0.134478	0.0672391	−	0.997737i	$-0.478581\pi$
$229$	2.03502	0.134478	0.0672391	+	0.997737i	$0.478581\pi$
$230$	1.27801	0.0842692
$231$	0	0
$232$	0.632128	0.0415013
$233$	28.0001	1.83435	0.917173	−	0.398490i	$-0.130466\pi$
$233$	28.0001	1.83435	0.917173	+	0.398490i	$0.130466\pi$
$234$	0	0
$235$	2.73841	0.178634
$236$	1.56048	0.101578
$237$	0	0
$238$	−0.183403	−0.0118883
$239$	−1.14836	−0.0742814	−0.0371407	−	0.999310i	$-0.511825\pi$
$239$	−1.14836	−0.0742814	−0.0371407	+	0.999310i	$0.511825\pi$
$240$	0	0
$241$	3.17652	0.204618	0.102309	−	0.994753i	$-0.467377\pi$
$241$	3.17652	0.204618	0.102309	+	0.994753i	$0.467377\pi$
$242$	0.958302	0.0616020
$243$	0	0
$244$	9.60230	0.614724
$245$	−47.8115	−3.05456
$246$	0	0
$247$	−1.98136	−0.126071
$248$	2.78285	0.176711
$249$	0	0
$250$	−0.615710	−0.0389409
$251$	27.0314	1.70621	0.853104	−	0.521742i	$-0.174717\pi$
$251$	27.0314	1.70621	0.853104	+	0.521742i	$0.174717\pi$
$252$	0	0
$253$	3.02707	0.190310
$254$	1.28916	0.0808891
$255$	0	0
$256$	15.3574	0.959837
$257$	−20.1857	−1.25915	−0.629574	−	0.776941i	$-0.716771\pi$
$257$	−20.1857	−1.25915	−0.629574	+	0.776941i	$0.716771\pi$
$258$	0	0
$259$	34.8194	2.16357
$260$	10.8077	0.670265
$261$	0	0
$262$	0.571193	0.0352884
$263$	−15.6528	−0.965193	−0.482596	−	0.875843i	$-0.660306\pi$
$263$	−15.6528	−0.965193	−0.482596	+	0.875843i	$0.660306\pi$
$264$	0	0
$265$	3.21135	0.197271
$266$	0.444600	0.0272602
$267$	0	0
$268$	−17.0734	−1.04292
$269$	8.47292	0.516603	0.258302	−	0.966064i	$-0.416837\pi$
$269$	8.47292	0.516603	0.258302	+	0.966064i	$0.416837\pi$
$270$	0	0
$271$	8.04208	0.488521	0.244261	−	0.969710i	$-0.421455\pi$
$271$	8.04208	0.488521	0.244261	+	0.969710i	$0.421455\pi$
$272$	1.63008	0.0988381
$273$	0	0
$274$	1.40951	0.0851516
$275$	1.45707	0.0878645
$276$	0	0
$277$	21.9947	1.32153	0.660766	−	0.750592i	$-0.270232\pi$
$277$	21.9947	1.32153	0.660766	+	0.750592i	$0.270232\pi$
$278$	0.397814	0.0238593
$279$	0	0
$280$	−4.86015	−0.290449
$281$	11.1920	0.667661	0.333831	−	0.942633i	$-0.391659\pi$
$281$	11.1920	0.667661	0.333831	+	0.942633i	$0.391659\pi$
$282$	0	0
$283$	11.0699	0.658038	0.329019	−	0.944323i	$-0.393282\pi$
$283$	11.0699	0.658038	0.329019	+	0.944323i	$0.393282\pi$
$284$	16.3834	0.972173
$285$	0	0
$286$	−0.103858	−0.00614127
$287$	−18.0250	−1.06398
$288$	0	0
$289$	−16.8298	−0.989990
$290$	0.433633	0.0254638
$291$	0	0
$292$	0.814152	0.0476446
$293$	16.3889	0.957450	0.478725	−	0.877965i	$-0.341099\pi$
$293$	16.3889	0.957450	0.478725	+	0.877965i	$0.341099\pi$
$294$	0	0
$295$	2.14528	0.124903
$296$	2.52653	0.146851
$297$	0	0
$298$	−2.14265	−0.124120
$299$	10.2861	0.594863
$300$	0	0
$301$	13.9477	0.803933
$302$	−0.596754	−0.0343393
$303$	0	0
$304$	−3.95158	−0.226638
$305$	13.2009	0.755879
$306$	0	0
$307$	−17.4404	−0.995376	−0.497688	−	0.867356i	$-0.665818\pi$
$307$	−17.4404	−0.995376	−0.497688	+	0.867356i	$0.665818\pi$
$308$	−5.74419	−0.327306
$309$	0	0
$310$	1.90900	0.108424
$311$	13.5936	0.770823	0.385411	−	0.922745i	$-0.374060\pi$
$311$	13.5936	0.770823	0.385411	+	0.922745i	$0.374060\pi$
$312$	0	0
$313$	−32.1760	−1.81869	−0.909346	−	0.416040i	$-0.863418\pi$
$313$	−32.1760	−1.81869	−0.909346	+	0.416040i	$0.863418\pi$
$314$	0.343283	0.0193726
$315$	0	0
$316$	−18.3326	−1.03129
$317$	−23.1646	−1.30105	−0.650526	−	0.759484i	$-0.725451\pi$
$317$	−23.1646	−1.30105	−0.650526	+	0.759484i	$0.725451\pi$
$318$	0	0
$319$	1.02710	0.0575064
$320$	21.3779	1.19506
$321$	0	0
$322$	−2.30812	−0.128627
$323$	−0.412514	−0.0229529
$324$	0	0
$325$	4.95120	0.274643
$326$	−0.677370	−0.0375160
$327$	0	0
$328$	−1.30791	−0.0722173
$329$	−4.94566	−0.272663
$330$	0	0
$331$	7.13898	0.392394	0.196197	−	0.980565i	$-0.437141\pi$
$331$	7.13898	0.392394	0.196197	+	0.980565i	$0.437141\pi$
$332$	−13.6324	−0.748174
$333$	0	0
$334$	−1.81696	−0.0994194
$335$	−23.4718	−1.28240
$336$	0	0
$337$	−10.4807	−0.570919	−0.285459	−	0.958391i	$-0.592146\pi$
$337$	−10.4807	−0.570919	−0.285459	+	0.958391i	$0.592146\pi$
$338$	0.815744	0.0443706
$339$	0	0
$340$	2.25013	0.122031
$341$	4.52164	0.244861
$342$	0	0
$343$	51.7295	2.79313
$344$	1.01206	0.0545665
$345$	0	0
$346$	−0.118976	−0.00639620
$347$	−21.9259	−1.17704	−0.588522	−	0.808481i	$-0.700290\pi$
$347$	−21.9259	−1.17704	−0.588522	+	0.808481i	$0.700290\pi$
$348$	0	0
$349$	−1.34402	−0.0719435	−0.0359718	−	0.999353i	$-0.511453\pi$
$349$	−1.34402	−0.0719435	−0.0359718	+	0.999353i	$0.511453\pi$
$350$	−1.11101	−0.0593857
$351$	0	0
$352$	−0.625626	−0.0333460
$353$	21.1144	1.12381	0.561903	−	0.827203i	$-0.310070\pi$
$353$	21.1144	1.12381	0.561903	+	0.827203i	$0.310070\pi$
$354$	0	0
$355$	22.5232	1.19541
$356$	14.3838	0.762340
$357$	0	0
$358$	1.84045	0.0972706
$359$	−7.04404	−0.371770	−0.185885	−	0.982571i	$-0.559515\pi$
$359$	−7.04404	−0.371770	−0.185885	+	0.982571i	$0.559515\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−1.60691	−0.0844571
$363$	0	0
$364$	−19.5191	−1.02308
$365$	1.11926	0.0585849
$366$	0	0
$367$	32.7657	1.71036	0.855179	−	0.518333i	$-0.173447\pi$
$367$	32.7657	1.71036	0.855179	+	0.518333i	$0.173447\pi$
$368$	20.5144	1.06939
$369$	0	0
$370$	1.73317	0.0901031
$371$	−5.79980	−0.301111
$372$	0	0
$373$	−1.23048	−0.0637116	−0.0318558	−	0.999492i	$-0.510142\pi$
$373$	−1.23048	−0.0637116	−0.0318558	+	0.999492i	$0.510142\pi$
$374$	−0.0216230	−0.00111810
$375$	0	0
$376$	−0.358861	−0.0185069
$377$	3.49013	0.179751
$378$	0	0
$379$	−16.2949	−0.837012	−0.418506	−	0.908214i	$-0.637446\pi$
$379$	−16.2949	−0.837012	−0.418506	+	0.908214i	$0.637446\pi$
$380$	−5.45469	−0.279820
$381$	0	0
$382$	1.62395	0.0830884
$383$	4.56830	0.233429	0.116715	−	0.993165i	$-0.462764\pi$
$383$	4.56830	0.233429	0.116715	+	0.993165i	$0.462764\pi$
$384$	0	0
$385$	−7.89688	−0.402462
$386$	0.993426	0.0505641
$387$	0	0
$388$	33.7917	1.71551
$389$	−5.43211	−0.275419	−0.137709	−	0.990473i	$-0.543974\pi$
$389$	−5.43211	−0.275419	−0.137709	+	0.990473i	$0.543974\pi$
$390$	0	0
$391$	2.14155	0.108303
$392$	6.26556	0.316459
$393$	0	0
$394$	−1.70450	−0.0858713
$395$	−25.2030	−1.26810
$396$	0	0
$397$	26.1687	1.31337	0.656686	−	0.754164i	$-0.271958\pi$
$397$	26.1687	1.31337	0.656686	+	0.754164i	$0.271958\pi$
$398$	0.928370	0.0465350
$399$	0	0
$400$	9.87455	0.493727
$401$	32.8619	1.64105	0.820523	−	0.571614i	$-0.193683\pi$
$401$	32.8619	1.64105	0.820523	+	0.571614i	$0.193683\pi$
$402$	0	0
$403$	15.3648	0.765375
$404$	4.06003	0.201994
$405$	0	0
$406$	−0.783156	−0.0388674
$407$	4.10516	0.203485
$408$	0	0
$409$	−12.9780	−0.641723	−0.320862	−	0.947126i	$-0.603972\pi$
$409$	−12.9780	−0.641723	−0.320862	+	0.947126i	$0.603972\pi$
$410$	−0.897212	−0.0443101
$411$	0	0
$412$	−2.50110	−0.123220
$413$	−3.87445	−0.190649
$414$	0	0
$415$	−18.7413	−0.919972
$416$	−2.12591	−0.104231
$417$	0	0
$418$	0.0524177	0.00256383
$419$	−11.0751	−0.541056	−0.270528	−	0.962712i	$-0.587198\pi$
$419$	−11.0751	−0.541056	−0.270528	+	0.962712i	$0.587198\pi$
$420$	0	0
$421$	−16.3368	−0.796207	−0.398103	−	0.917341i	$-0.630331\pi$
$421$	−16.3368	−0.796207	−0.398103	+	0.917341i	$0.630331\pi$
$422$	1.40128	0.0682135
$423$	0	0
$424$	−0.420838	−0.0204377
$425$	1.03083	0.0500024
$426$	0	0
$427$	−23.8412	−1.15376
$428$	15.4118	0.744957
$429$	0	0
$430$	0.694260	0.0334802
$431$	24.3746	1.17408	0.587042	−	0.809557i	$-0.300292\pi$
$431$	24.3746	1.17408	0.587042	+	0.809557i	$0.300292\pi$
$432$	0	0
$433$	−8.36745	−0.402114	−0.201057	−	0.979580i	$-0.564438\pi$
$433$	−8.36745	−0.402114	−0.201057	+	0.979580i	$0.564438\pi$
$434$	−3.44773	−0.165496
$435$	0	0
$436$	−11.3143	−0.541856
$437$	−5.19146	−0.248341
$438$	0	0
$439$	9.40051	0.448662	0.224331	−	0.974513i	$-0.427980\pi$
$439$	9.40051	0.448662	0.224331	+	0.974513i	$0.427980\pi$
$440$	−0.573004	−0.0273169
$441$	0	0
$442$	−0.0734761	−0.00349490
$443$	−4.89629	−0.232630	−0.116315	−	0.993212i	$-0.537108\pi$
$443$	−4.89629	−0.232630	−0.116315	+	0.993212i	$0.537108\pi$
$444$	0	0
$445$	19.7743	0.937390
$446$	−2.07056	−0.0980439
$447$	0	0
$448$	−38.6093	−1.82412
$449$	32.8492	1.55025	0.775125	−	0.631808i	$-0.217687\pi$
$449$	32.8492	1.55025	0.775125	+	0.631808i	$0.217687\pi$
$450$	0	0
$451$	−2.12512	−0.100068
$452$	−14.4372	−0.679070
$453$	0	0
$454$	−2.03846	−0.0956697
$455$	−26.8340	−1.25800
$456$	0	0
$457$	−25.3710	−1.18681	−0.593403	−	0.804905i	$-0.702216\pi$
$457$	−25.3710	−1.18681	−0.593403	+	0.804905i	$0.702216\pi$
$458$	−0.182942	−0.00854834
$459$	0	0
$460$	28.3178	1.32032
$461$	23.7774	1.10742	0.553712	−	0.832708i	$-0.313211\pi$
$461$	23.7774	1.10742	0.553712	+	0.832708i	$0.313211\pi$
$462$	0	0
$463$	−7.63592	−0.354871	−0.177436	−	0.984132i	$-0.556780\pi$
$463$	−7.63592	−0.354871	−0.177436	+	0.984132i	$0.556780\pi$
$464$	6.96064	0.323140
$465$	0	0
$466$	−2.51712	−0.116603
$467$	32.3674	1.49778	0.748892	−	0.662692i	$-0.230586\pi$
$467$	32.3674	1.49778	0.748892	+	0.662692i	$0.230586\pi$
$468$	0	0
$469$	42.3909	1.95743
$470$	−0.246175	−0.0113552
$471$	0	0
$472$	−0.281133	−0.0129402
$473$	1.64442	0.0756103
$474$	0	0
$475$	−2.49889	−0.114657
$476$	−4.06382	−0.186265
$477$	0	0
$478$	0.103234	0.00472183
$479$	26.7427	1.22191	0.610953	−	0.791667i	$-0.290786\pi$
$479$	26.7427	1.22191	0.610953	+	0.791667i	$0.290786\pi$
$480$	0	0
$481$	13.9496	0.636045
$482$	−0.285559	−0.0130069
$483$	0	0
$484$	21.2339	0.965176
$485$	46.4555	2.10944
$486$	0	0
$487$	13.4920	0.611380	0.305690	−	0.952131i	$-0.401113\pi$
$487$	13.4920	0.611380	0.305690	+	0.952131i	$0.401113\pi$
$488$	−1.72994	−0.0783105
$489$	0	0
$490$	4.29810	0.194168
$491$	2.00538	0.0905015	0.0452508	−	0.998976i	$-0.485591\pi$
$491$	2.00538	0.0905015	0.0452508	+	0.998976i	$0.485591\pi$
$492$	0	0
$493$	0.726636	0.0327260
$494$	0.178118	0.00801391
$495$	0	0
$496$	30.6432	1.37592
$497$	−40.6776	−1.82464
$498$	0	0
$499$	30.0284	1.34426	0.672128	−	0.740435i	$-0.265380\pi$
$499$	30.0284	1.34426	0.672128	+	0.740435i	$0.265380\pi$
$500$	−13.6428	−0.610124
$501$	0	0
$502$	−2.43004	−0.108458
$503$	15.9105	0.709414	0.354707	−	0.934978i	$-0.384581\pi$
$503$	15.9105	0.709414	0.354707	+	0.934978i	$0.384581\pi$
$504$	0	0
$505$	5.58157	0.248376
$506$	−0.272124	−0.0120974
$507$	0	0
$508$	28.5650	1.26737
$509$	27.2206	1.20653	0.603267	−	0.797539i	$-0.293865\pi$
$509$	27.2206	1.20653	0.603267	+	0.797539i	$0.293865\pi$
$510$	0	0
$511$	−2.02143	−0.0894227
$512$	−7.07601	−0.312718
$513$	0	0
$514$	1.81463	0.0800399
$515$	−3.43841	−0.151515
$516$	0	0
$517$	−0.583086	−0.0256441
$518$	−3.13016	−0.137531
$519$	0	0
$520$	−1.94710	−0.0853860
$521$	23.2240	1.01746	0.508732	−	0.860925i	$-0.330114\pi$
$521$	23.2240	1.01746	0.508732	+	0.860925i	$0.330114\pi$
$522$	0	0
$523$	44.6506	1.95243	0.976217	−	0.216794i	$-0.0695600\pi$
$523$	44.6506	1.95243	0.976217	+	0.216794i	$0.0695600\pi$
$524$	12.6564	0.552896
$525$	0	0
$526$	1.40714	0.0613542
$527$	3.19891	0.139347
$528$	0	0
$529$	3.95123	0.171793
$530$	−0.288690	−0.0125399
$531$	0	0
$532$	9.85135	0.427110
$533$	−7.22129	−0.312789
$534$	0	0
$535$	21.1875	0.916017
$536$	3.07592	0.132859
$537$	0	0
$538$	−0.761689	−0.0328388
$539$	10.1804	0.438502
$540$	0	0
$541$	15.0888	0.648718	0.324359	−	0.945934i	$-0.394851\pi$
$541$	15.0888	0.648718	0.324359	+	0.945934i	$0.394851\pi$
$542$	−0.722958	−0.0310537
$543$	0	0
$544$	−0.442609	−0.0189767
$545$	−15.5544	−0.666279
$546$	0	0
$547$	9.26722	0.396238	0.198119	−	0.980178i	$-0.436517\pi$
$547$	9.26722	0.396238	0.198119	+	0.980178i	$0.436517\pi$
$548$	31.2316	1.33415
$549$	0	0
$550$	−0.130986	−0.00558526
$551$	−1.76148	−0.0750418
$552$	0	0
$553$	45.5174	1.93560
$554$	−1.97725	−0.0840054
$555$	0	0
$556$	8.81469	0.373826
$557$	−44.0293	−1.86558	−0.932791	−	0.360418i	$-0.882634\pi$
$557$	−44.0293	−1.86558	−0.932791	+	0.360418i	$0.882634\pi$
$558$	0	0
$559$	5.58782	0.236339
$560$	−53.5172	−2.26151
$561$	0	0
$562$	−1.00613	−0.0424410
$563$	7.44308	0.313688	0.156844	−	0.987623i	$-0.449868\pi$
$563$	7.44308	0.313688	0.156844	+	0.987623i	$0.449868\pi$
$564$	0	0
$565$	−19.8477	−0.835000
$566$	−0.995151	−0.0418293
$567$	0	0
$568$	−2.95160	−0.123846
$569$	−4.34529	−0.182164	−0.0910821	−	0.995843i	$-0.529033\pi$
$569$	−4.34529	−0.182164	−0.0910821	+	0.995843i	$0.529033\pi$
$570$	0	0
$571$	2.63781	0.110389	0.0551945	−	0.998476i	$-0.482422\pi$
$571$	2.63781	0.110389	0.0551945	+	0.998476i	$0.482422\pi$
$572$	−2.30127	−0.0962209
$573$	0	0
$574$	1.62040	0.0676340
$575$	12.9729	0.541006
$576$	0	0
$577$	−23.0972	−0.961551	−0.480776	−	0.876844i	$-0.659645\pi$
$577$	−23.0972	−0.961551	−0.480776	+	0.876844i	$0.659645\pi$
$578$	1.51295	0.0629304
$579$	0	0
$580$	9.60835	0.398965
$581$	33.8473	1.40422
$582$	0	0
$583$	−0.683788	−0.0283196
$584$	−0.146676	−0.00606952
$585$	0	0
$586$	−1.47331	−0.0608620
$587$	−32.7537	−1.35189	−0.675945	−	0.736952i	$-0.736264\pi$
$587$	−32.7537	−1.35189	−0.675945	+	0.736952i	$0.736264\pi$
$588$	0	0
$589$	−7.75468	−0.319526
$590$	−0.192854	−0.00793968
$591$	0	0
$592$	27.8207	1.14342
$593$	−15.0725	−0.618953	−0.309477	−	0.950907i	$-0.600154\pi$
$593$	−15.0725	−0.618953	−0.309477	+	0.950907i	$0.600154\pi$
$594$	0	0
$595$	−5.58677	−0.229035
$596$	−47.4764	−1.94471
$597$	0	0
$598$	−0.924693	−0.0378135
$599$	−32.6731	−1.33499	−0.667493	−	0.744616i	$-0.732633\pi$
$599$	−32.6731	−1.33499	−0.667493	+	0.744616i	$0.732633\pi$
$600$	0	0
$601$	8.71454	0.355474	0.177737	−	0.984078i	$-0.443122\pi$
$601$	8.71454	0.355474	0.177737	+	0.984078i	$0.443122\pi$
$602$	−1.25386	−0.0511034
$603$	0	0
$604$	−13.2228	−0.538027
$605$	29.1915	1.18680
$606$	0	0
$607$	−32.3909	−1.31470	−0.657352	−	0.753583i	$-0.728324\pi$
$607$	−32.3909	−1.31470	−0.657352	+	0.753583i	$0.728324\pi$
$608$	1.07296	0.0435142
$609$	0	0
$610$	−1.18672	−0.0480487
$611$	−1.98136	−0.0801572
$612$	0	0
$613$	28.6470	1.15704	0.578520	−	0.815668i	$-0.303630\pi$
$613$	28.6470	1.15704	0.578520	+	0.815668i	$0.303630\pi$
$614$	1.56784	0.0632728
$615$	0	0
$616$	1.03486	0.0416959
$617$	45.3331	1.82504	0.912521	−	0.409030i	$-0.134133\pi$
$617$	45.3331	1.82504	0.912521	+	0.409030i	$0.134133\pi$
$618$	0	0
$619$	7.55200	0.303541	0.151770	−	0.988416i	$-0.451503\pi$
$619$	7.55200	0.303541	0.151770	+	0.988416i	$0.451503\pi$
$620$	42.2994	1.69878
$621$	0	0
$622$	−1.22202	−0.0489987
$623$	−35.7130	−1.43081
$624$	0	0
$625$	−31.2500	−1.25000
$626$	2.89252	0.115608
$627$	0	0
$628$	7.60640	0.303528
$629$	2.90426	0.115800
$630$	0	0
$631$	−41.2222	−1.64103	−0.820515	−	0.571626i	$-0.806313\pi$
$631$	−41.2222	−1.64103	−0.820515	+	0.571626i	$0.806313\pi$
$632$	3.30278	0.131378
$633$	0	0
$634$	2.08242	0.0827036
$635$	39.2700	1.55838
$636$	0	0
$637$	34.5937	1.37065
$638$	−0.0923329	−0.00365549
$639$	0	0
$640$	−7.79820	−0.308251
$641$	28.2312	1.11507	0.557533	−	0.830155i	$-0.311748\pi$
$641$	28.2312	1.11507	0.557533	+	0.830155i	$0.311748\pi$
$642$	0	0
$643$	−6.77992	−0.267374	−0.133687	−	0.991024i	$-0.542682\pi$
$643$	−6.77992	−0.267374	−0.133687	+	0.991024i	$0.542682\pi$
$644$	−51.1429	−2.01531
$645$	0	0
$646$	0.0370837	0.00145904
$647$	−20.6839	−0.813169	−0.406585	−	0.913613i	$-0.633280\pi$
$647$	−20.6839	−0.813169	−0.406585	+	0.913613i	$0.633280\pi$
$648$	0	0
$649$	−0.456792	−0.0179307
$650$	−0.445097	−0.0174582
$651$	0	0
$652$	−15.0090	−0.587799
$653$	−43.4916	−1.70196	−0.850980	−	0.525199i	$-0.823991\pi$
$653$	−43.4916	−1.70196	−0.850980	+	0.525199i	$0.823991\pi$
$654$	0	0
$655$	17.3995	0.679854
$656$	−14.4020	−0.562303
$657$	0	0
$658$	0.444600	0.0173323
$659$	46.4789	1.81056	0.905280	−	0.424816i	$-0.139661\pi$
$659$	46.4789	1.81056	0.905280	+	0.424816i	$0.139661\pi$
$660$	0	0
$661$	34.4489	1.33991	0.669954	−	0.742403i	$-0.266314\pi$
$661$	34.4489	1.33991	0.669954	+	0.742403i	$0.266314\pi$
$662$	−0.641772	−0.0249432
$663$	0	0
$664$	2.45599	0.0953109
$665$	13.5432	0.525185
$666$	0	0
$667$	9.14467	0.354083
$668$	−40.2597	−1.55770
$669$	0	0
$670$	2.11004	0.0815181
$671$	−2.81084	−0.108511
$672$	0	0
$673$	20.4266	0.787387	0.393694	−	0.919242i	$-0.371197\pi$
$673$	20.4266	0.787387	0.393694	+	0.919242i	$0.371197\pi$
$674$	0.942180	0.0362914
$675$	0	0
$676$	18.0751	0.695196
$677$	38.3966	1.47570	0.737850	−	0.674965i	$-0.235841\pi$
$677$	38.3966	1.47570	0.737850	+	0.674965i	$0.235841\pi$
$678$	0	0
$679$	−83.9002	−3.21979
$680$	−0.405381	−0.0155456
$681$	0	0
$682$	−0.406482	−0.0155650
$683$	−26.0524	−0.996868	−0.498434	−	0.866928i	$-0.666091\pi$
$683$	−26.0524	−0.996868	−0.498434	+	0.866928i	$0.666091\pi$
$684$	0	0
$685$	42.9360	1.64050
$686$	−4.65032	−0.177550
$687$	0	0
$688$	11.1442	0.424869
$689$	−2.32355	−0.0885202
$690$	0	0
$691$	−41.9263	−1.59495	−0.797477	−	0.603350i	$-0.793832\pi$
$691$	−41.9263	−1.59495	−0.797477	+	0.603350i	$0.793832\pi$
$692$	−2.63625	−0.100215
$693$	0	0
$694$	1.97107	0.0748208
$695$	12.1181	0.459665
$696$	0	0
$697$	−1.50345	−0.0569474
$698$	0.120823	0.00457322
$699$	0	0
$700$	−24.6174	−0.930452
$701$	14.7117	0.555652	0.277826	−	0.960631i	$-0.410386\pi$
$701$	14.7117	0.555652	0.277826	+	0.960631i	$0.410386\pi$
$702$	0	0
$703$	−7.04040	−0.265534
$704$	−4.55198	−0.171559
$705$	0	0
$706$	−1.89812	−0.0714366
$707$	−10.0805	−0.379116
$708$	0	0
$709$	6.84752	0.257164	0.128582	−	0.991699i	$-0.458957\pi$
$709$	6.84752	0.257164	0.128582	+	0.991699i	$0.458957\pi$
$710$	−2.02476	−0.0759881
$711$	0	0
$712$	−2.59136	−0.0971155
$713$	40.2581	1.50768
$714$	0	0
$715$	−3.16369	−0.118315
$716$	40.7803	1.52403
$717$	0	0
$718$	0.633237	0.0236322
$719$	−10.6832	−0.398417	−0.199209	−	0.979957i	$-0.563837\pi$
$719$	−10.6832	−0.398417	−0.199209	+	0.979957i	$0.563837\pi$
$720$	0	0
$721$	6.20989	0.231268
$722$	−0.0898969	−0.00334562
$723$	0	0
$724$	−35.6055	−1.32327
$725$	4.40175	0.163477
$726$	0	0
$727$	−35.0620	−1.30038	−0.650188	−	0.759773i	$-0.725310\pi$
$727$	−35.0620	−1.30038	−0.650188	+	0.759773i	$0.725310\pi$
$728$	3.51653	0.130331
$729$	0	0
$730$	−0.100618	−0.00372405
$731$	1.16337	0.0430287
$732$	0	0
$733$	33.0356	1.22020	0.610099	−	0.792325i	$-0.291129\pi$
$733$	33.0356	1.22020	0.610099	+	0.792325i	$0.291129\pi$
$734$	−2.94554	−0.108722
$735$	0	0
$736$	−5.57021	−0.205321
$737$	4.99782	0.184097
$738$	0	0
$739$	−48.1286	−1.77044	−0.885219	−	0.465174i	$-0.845992\pi$
$739$	−48.1286	−1.77044	−0.885219	+	0.465174i	$0.845992\pi$
$740$	38.4032	1.41173
$741$	0	0
$742$	0.521384	0.0191406
$743$	−39.5488	−1.45090	−0.725452	−	0.688273i	$-0.758369\pi$
$743$	−39.5488	−1.45090	−0.725452	+	0.688273i	$0.758369\pi$
$744$	0	0
$745$	−65.2686	−2.39126
$746$	0.110616	0.00404994
$747$	0	0
$748$	−0.479118	−0.0175183
$749$	−38.2654	−1.39819
$750$	0	0
$751$	2.67320	0.0975464	0.0487732	−	0.998810i	$-0.484469\pi$
$751$	2.67320	0.0975464	0.0487732	+	0.998810i	$0.484469\pi$
$752$	−3.95158	−0.144099
$753$	0	0
$754$	−0.313752	−0.0114262
$755$	−18.1781	−0.661570
$756$	0	0
$757$	19.5051	0.708925	0.354462	−	0.935070i	$-0.384664\pi$
$757$	19.5051	0.708925	0.354462	+	0.935070i	$0.384664\pi$
$758$	1.46486	0.0532061
$759$	0	0
$760$	0.982709	0.0356466
$761$	7.11381	0.257875	0.128938	−	0.991653i	$-0.458843\pi$
$761$	7.11381	0.257875	0.128938	+	0.991653i	$0.458843\pi$
$762$	0	0
$763$	28.0918	1.01699
$764$	35.9831	1.30182
$765$	0	0
$766$	−0.410676	−0.0148383
$767$	−1.55220	−0.0560469
$768$	0	0
$769$	43.6123	1.57270	0.786350	−	0.617782i	$-0.211968\pi$
$769$	43.6123	1.57270	0.786350	+	0.617782i	$0.211968\pi$
$770$	0.709905	0.0255832
$771$	0	0
$772$	22.0121	0.792234
$773$	30.4279	1.09442	0.547208	−	0.836997i	$-0.315691\pi$
$773$	30.4279	1.09442	0.547208	+	0.836997i	$0.315691\pi$
$774$	0	0
$775$	19.3781	0.696081
$776$	−6.08787	−0.218542
$777$	0	0
$778$	0.488330	0.0175075
$779$	3.64462	0.130582
$780$	0	0
$781$	−4.79583	−0.171608
$782$	−0.192518	−0.00688445
$783$	0	0
$784$	68.9928	2.46403
$785$	10.4570	0.373225
$786$	0	0
$787$	−33.8557	−1.20683	−0.603413	−	0.797429i	$-0.706193\pi$
$787$	−33.8557	−1.20683	−0.603413	+	0.797429i	$0.706193\pi$
$788$	−37.7679	−1.34543
$789$	0	0
$790$	2.26567	0.0806089
$791$	35.8457	1.27453
$792$	0	0
$793$	−9.55139	−0.339180
$794$	−2.35249	−0.0834867
$795$	0	0
$796$	20.5706	0.729107
$797$	15.9275	0.564180	0.282090	−	0.959388i	$-0.408972\pi$
$797$	15.9275	0.564180	0.282090	+	0.959388i	$0.408972\pi$
$798$	0	0
$799$	−0.412514	−0.0145937
$800$	−2.68120	−0.0947947
$801$	0	0
$802$	−2.95418	−0.104316
$803$	−0.238323	−0.00841025
$804$	0	0
$805$	−70.3092	−2.47807
$806$	−1.38125	−0.0486524
$807$	0	0
$808$	−0.731449	−0.0257323
$809$	−10.0596	−0.353678	−0.176839	−	0.984240i	$-0.556587\pi$
$809$	−10.0596	−0.353678	−0.176839	+	0.984240i	$0.556587\pi$
$810$	0	0
$811$	39.2219	1.37727	0.688634	−	0.725109i	$-0.258211\pi$
$811$	39.2219	1.37727	0.688634	+	0.725109i	$0.258211\pi$
$812$	−17.3530	−0.608971
$813$	0	0
$814$	−0.369041	−0.0129349
$815$	−20.6338	−0.722771
$816$	0	0
$817$	−2.82019	−0.0986660
$818$	1.16669	0.0407922
$819$	0	0
$820$	−19.8802	−0.694248
$821$	−1.66291	−0.0580359	−0.0290180	−	0.999579i	$-0.509238\pi$
$821$	−1.66291	−0.0580359	−0.0290180	+	0.999579i	$0.509238\pi$
$822$	0	0
$823$	3.84053	0.133872	0.0669362	−	0.997757i	$-0.478678\pi$
$823$	3.84053	0.133872	0.0669362	+	0.997757i	$0.478678\pi$
$824$	0.450595	0.0156972
$825$	0	0
$826$	0.348301	0.0121189
$827$	−42.5494	−1.47959	−0.739793	−	0.672834i	$-0.765077\pi$
$827$	−42.5494	−1.47959	−0.739793	+	0.672834i	$0.765077\pi$
$828$	0	0
$829$	−18.8293	−0.653970	−0.326985	−	0.945030i	$-0.606033\pi$
$829$	−18.8293	−0.653970	−0.326985	+	0.945030i	$0.606033\pi$
$830$	1.68478	0.0584796
$831$	0	0
$832$	−15.4679	−0.536252
$833$	7.20231	0.249545
$834$	0	0
$835$	−55.3475	−1.91538
$836$	1.16146	0.0401699
$837$	0	0
$838$	0.995622	0.0343932
$839$	−49.1335	−1.69628	−0.848138	−	0.529775i	$-0.822276\pi$
$839$	−49.1335	−1.69628	−0.848138	+	0.529775i	$0.822276\pi$
$840$	0	0
$841$	−25.8972	−0.893006
$842$	1.46863	0.0506123
$843$	0	0
$844$	31.0494	1.06876
$845$	24.8489	0.854829
$846$	0	0
$847$	−52.7208	−1.81151
$848$	−4.63403	−0.159133
$849$	0	0
$850$	−0.0926680	−0.00317849
$851$	36.5499	1.25291
$852$	0	0
$853$	29.6650	1.01571	0.507854	−	0.861443i	$-0.330439\pi$
$853$	29.6650	1.01571	0.507854	+	0.861443i	$0.330439\pi$
$854$	2.14325	0.0733405
$855$	0	0
$856$	−2.77657	−0.0949011
$857$	−30.2463	−1.03319	−0.516597	−	0.856229i	$-0.672802\pi$
$857$	−30.2463	−1.03319	−0.516597	+	0.856229i	$0.672802\pi$
$858$	0	0
$859$	−13.3024	−0.453873	−0.226936	−	0.973910i	$-0.572871\pi$
$859$	−13.3024	−0.453873	−0.226936	+	0.973910i	$0.572871\pi$
$860$	15.3833	0.524565
$861$	0	0
$862$	−2.19120	−0.0746326
$863$	9.60500	0.326958	0.163479	−	0.986547i	$-0.447728\pi$
$863$	9.60500	0.326958	0.163479	+	0.986547i	$0.447728\pi$
$864$	0	0
$865$	−3.62421	−0.123227
$866$	0.752208	0.0255611
$867$	0	0
$868$	−76.3941	−2.59298
$869$	5.36644	0.182044
$870$	0	0
$871$	16.9829	0.575443
$872$	2.03837	0.0690278
$873$	0	0
$874$	0.466696	0.0157862
$875$	33.8732	1.14512
$876$	0	0
$877$	45.8745	1.54907	0.774535	−	0.632531i	$-0.217984\pi$
$877$	45.8745	1.54907	0.774535	+	0.632531i	$0.217984\pi$
$878$	−0.845077	−0.0285200
$879$	0	0
$880$	−6.30960	−0.212696
$881$	7.65602	0.257938	0.128969	−	0.991649i	$-0.458833\pi$
$881$	7.65602	0.257938	0.128969	+	0.991649i	$0.458833\pi$
$882$	0	0
$883$	5.91366	0.199010	0.0995052	−	0.995037i	$-0.468274\pi$
$883$	5.91366	0.199010	0.0995052	+	0.995037i	$0.468274\pi$
$884$	−1.62807	−0.0547579
$885$	0	0
$886$	0.440161	0.0147875
$887$	30.9118	1.03792	0.518958	−	0.854799i	$-0.326320\pi$
$887$	30.9118	1.03792	0.518958	+	0.854799i	$0.326320\pi$
$888$	0	0
$889$	−70.9229	−2.37868
$890$	−1.77765	−0.0595868
$891$	0	0
$892$	−45.8791	−1.53615
$893$	1.00000	0.0334637
$894$	0	0
$895$	56.0631	1.87398
$896$	14.0838	0.470507
$897$	0	0
$898$	−2.95304	−0.0985443
$899$	13.6597	0.455578
$900$	0	0
$901$	−0.483757	−0.0161163
$902$	0.191042	0.00636101
$903$	0	0
$904$	2.60099	0.0865077
$905$	−48.9490	−1.62712
$906$	0	0
$907$	−52.9861	−1.75938	−0.879688	−	0.475552i	$-0.842249\pi$
$907$	−52.9861	−1.75938	−0.879688	+	0.475552i	$0.842249\pi$
$908$	−45.1678	−1.49895
$909$	0	0
$910$	2.41230	0.0799669
$911$	−20.3856	−0.675406	−0.337703	−	0.941253i	$-0.609650\pi$
$911$	−20.3856	−0.675406	−0.337703	+	0.941253i	$0.609650\pi$
$912$	0	0
$913$	3.99055	0.132068
$914$	2.28078	0.0754414
$915$	0	0
$916$	−4.05360	−0.133935
$917$	−31.4240	−1.03771
$918$	0	0
$919$	−40.8662	−1.34805	−0.674025	−	0.738708i	$-0.735436\pi$
$919$	−40.8662	−1.34805	−0.674025	+	0.738708i	$0.735436\pi$
$920$	−5.10169	−0.168198
$921$	0	0
$922$	−2.13751	−0.0703953
$923$	−16.2965	−0.536406
$924$	0	0
$925$	17.5932	0.578460
$926$	0.686446	0.0225580
$927$	0	0
$928$	−1.89000	−0.0620422
$929$	−42.8296	−1.40519	−0.702597	−	0.711588i	$-0.747976\pi$
$929$	−42.8296	−1.40519	−0.702597	+	0.711588i	$0.747976\pi$
$930$	0	0
$931$	−17.4596	−0.572214
$932$	−55.7738	−1.82693
$933$	0	0
$934$	−2.90973	−0.0952093
$935$	−0.658672	−0.0215409
$936$	0	0
$937$	8.34114	0.272493	0.136247	−	0.990675i	$-0.456496\pi$
$937$	8.34114	0.272493	0.136247	+	0.990675i	$0.456496\pi$
$938$	−3.81081	−0.124427
$939$	0	0
$940$	−5.45469	−0.177912
$941$	−22.8057	−0.743446	−0.371723	−	0.928344i	$-0.621233\pi$
$941$	−22.8057	−0.743446	−0.371723	+	0.928344i	$0.621233\pi$
$942$	0	0
$943$	−18.9209	−0.616148
$944$	−3.09568	−0.100756
$945$	0	0
$946$	−0.147828	−0.00480630
$947$	28.2122	0.916774	0.458387	−	0.888753i	$-0.348427\pi$
$947$	28.2122	0.916774	0.458387	+	0.888753i	$0.348427\pi$
$948$	0	0
$949$	−0.809836	−0.0262884
$950$	0.224642	0.00728836
$951$	0	0
$952$	0.732132	0.0237285
$953$	9.06005	0.293484	0.146742	−	0.989175i	$-0.453121\pi$
$953$	9.06005	0.293484	0.146742	+	0.989175i	$0.453121\pi$
$954$	0	0
$955$	49.4682	1.60075
$956$	2.28745	0.0739813
$957$	0	0
$958$	−2.40409	−0.0776726
$959$	−77.5439	−2.50402
$960$	0	0
$961$	29.1350	0.939839
$962$	−1.25402	−0.0404313
$963$	0	0
$964$	−6.32737	−0.203791
$965$	30.2614	0.974149
$966$	0	0
$967$	−3.78010	−0.121560	−0.0607799	−	0.998151i	$-0.519359\pi$
$967$	−3.78010	−0.121560	−0.0607799	+	0.998151i	$0.519359\pi$
$968$	−3.82546	−0.122955
$969$	0	0
$970$	−4.17621	−0.134090
$971$	−8.97298	−0.287957	−0.143978	−	0.989581i	$-0.545990\pi$
$971$	−8.97298	−0.287957	−0.143978	+	0.989581i	$0.545990\pi$
$972$	0	0
$973$	−21.8857	−0.701623
$974$	−1.21289	−0.0388634
$975$	0	0
$976$	−19.0491	−0.609746
$977$	38.8785	1.24383	0.621917	−	0.783083i	$-0.286354\pi$
$977$	38.8785	1.24383	0.621917	+	0.783083i	$0.286354\pi$
$978$	0	0
$979$	−4.21051	−0.134568
$980$	95.2365	3.04222
$981$	0	0
$982$	−0.180277	−0.00575289
$983$	−42.8459	−1.36657	−0.683286	−	0.730151i	$-0.739450\pi$
$983$	−42.8459	−1.36657	−0.683286	+	0.730151i	$0.739450\pi$
$984$	0	0
$985$	−51.9218	−1.65437
$986$	−0.0653224	−0.00208029
$987$	0	0
$988$	3.94671	0.125561
$989$	14.6409	0.465554
$990$	0	0
$991$	18.0149	0.572261	0.286130	−	0.958191i	$-0.407631\pi$
$991$	18.0149	0.572261	0.286130	+	0.958191i	$0.407631\pi$
$992$	−8.32043	−0.264174
$993$	0	0
$994$	3.65679	0.115986
$995$	28.2797	0.896526
$996$	0	0
$997$	6.91277	0.218930	0.109465	−	0.993991i	$-0.465086\pi$
$997$	6.91277	0.218930	0.109465	+	0.993991i	$0.465086\pi$
$998$	−2.69946	−0.0854500
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.o.1.10		18
3.2	odd	2		893.2.a.c.1.9	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
893.2.a.c.1.9	✓	18	3.2	odd	2
8037.2.a.o.1.10		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)

\(f(q)\)	\(=\)	\(q-0.0898969 q^{2} -1.99192 q^{4} -2.73841 q^{5} +4.94566 q^{7} +0.358861 q^{8} +O(q^{10})\)	\(q-0.0898969 q^{2} -1.99192 q^{4} -2.73841 q^{5} +4.94566 q^{7} +0.358861 q^{8} +0.246175 q^{10} +0.583086 q^{11} +1.98136 q^{13} -0.444600 q^{14} +3.95158 q^{16} +0.412514 q^{17} -1.00000 q^{19} +5.45469 q^{20} -0.0524177 q^{22} +5.19146 q^{23} +2.49889 q^{25} -0.178118 q^{26} -9.85135 q^{28} +1.76148 q^{29} +7.75468 q^{31} -1.07296 q^{32} -0.0370837 q^{34} -13.5432 q^{35} +7.04040 q^{37} +0.0898969 q^{38} -0.982709 q^{40} -3.64462 q^{41} +2.82019 q^{43} -1.16146 q^{44} -0.466696 q^{46} -1.00000 q^{47} +17.4596 q^{49} -0.224642 q^{50} -3.94671 q^{52} -1.17270 q^{53} -1.59673 q^{55} +1.77481 q^{56} -0.158352 q^{58} -0.783404 q^{59} -4.82063 q^{61} -0.697122 q^{62} -7.80670 q^{64} -5.42577 q^{65} +8.57132 q^{67} -0.821693 q^{68} +1.21750 q^{70} -8.22491 q^{71} -0.408728 q^{73} -0.632910 q^{74} +1.99192 q^{76} +2.88375 q^{77} +9.20351 q^{79} -10.8210 q^{80} +0.327640 q^{82} +6.84384 q^{83} -1.12963 q^{85} -0.253527 q^{86} +0.209247 q^{88} -7.22108 q^{89} +9.79913 q^{91} -10.3410 q^{92} +0.0898969 q^{94} +2.73841 q^{95} -16.9644 q^{97} -1.56956 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8}+O(q^{10}) \) 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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Groups

Database

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