LMFDB - Embedded newform 2835.2.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2835,2,Mod(1,2835)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2835, 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("2835.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2835 = 3^{4} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2835.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:	$22.6375889730$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{15})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 4x + 1 $ x^4 - x^3 - 4x^2 + 4x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 315)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.95630$ of defining polynomial
Character	$\chi$	$=$	2835.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.209057 q^{2} -1.95630 q^{4} +1.00000 q^{5} -1.00000 q^{7} +0.827091 q^{8} +O(q^{10})$	\(q-0.209057 q^{2} -1.95630 q^{4} +1.00000 q^{5} -1.00000 q^{7} +0.827091 q^{8} -0.209057 q^{10} -2.20906 q^{11} -2.48883 q^{13} +0.209057 q^{14} +3.73968 q^{16} +0.209057 q^{17} +7.22851 q^{19} -1.95630 q^{20} +0.461819 q^{22} +2.85857 q^{23} +1.00000 q^{25} +0.520307 q^{26} +1.95630 q^{28} -4.83274 q^{29} -3.98331 q^{31} -2.43599 q^{32} -0.0437048 q^{34} -1.00000 q^{35} +0.739681 q^{37} -1.51117 q^{38} +0.827091 q^{40} -2.19427 q^{41} -5.30369 q^{43} +4.32157 q^{44} -0.597604 q^{46} +8.61515 q^{47} +1.00000 q^{49} -0.209057 q^{50} +4.86889 q^{52} +7.93863 q^{53} -2.20906 q^{55} -0.827091 q^{56} +1.01032 q^{58} -14.0342 q^{59} +0.413648 q^{61} +0.832738 q^{62} -6.97010 q^{64} -2.48883 q^{65} -4.38197 q^{67} -0.408977 q^{68} +0.209057 q^{70} +8.07039 q^{71} -16.0446 q^{73} -0.154636 q^{74} -14.1411 q^{76} +2.20906 q^{77} -15.7220 q^{79} +3.73968 q^{80} +0.458728 q^{82} -3.12729 q^{83} +0.209057 q^{85} +1.10877 q^{86} -1.82709 q^{88} -10.1477 q^{89} +2.48883 q^{91} -5.59220 q^{92} -1.80106 q^{94} +7.22851 q^{95} -11.9060 q^{97} -0.209057 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + q^{4} + 4 q^{5} - 4 q^{7} - 3 q^{8}+O(q^{10}) $ 4 * q + q^2 + q^4 + 4 * q^5 - 4 * q^7 - 3 * q^8	$ 4 q + q^{2} + q^{4} + 4 q^{5} - 4 q^{7} - 3 q^{8} + q^{10} - 7 q^{11} - 8 q^{13} - q^{14} - 9 q^{16} - q^{17} + 3 q^{19} + q^{20} + 7 q^{22} + 8 q^{23} + 4 q^{25} + 3 q^{26} - q^{28} - q^{29} - 9 q^{34} - 4 q^{35} - 21 q^{37} - 8 q^{38} - 3 q^{40} - 20 q^{41} - 7 q^{43} - 3 q^{44} - 23 q^{46} + 2 q^{47} + 4 q^{49} + q^{50} - 7 q^{52} + 8 q^{53} - 7 q^{55} + 3 q^{56} - 19 q^{58} - 19 q^{59} - 12 q^{61} - 15 q^{62} - 7 q^{64} - 8 q^{65} - 22 q^{67} + q^{68} - q^{70} - 13 q^{71} - 4 q^{73} - 9 q^{74} - 13 q^{76} + 7 q^{77} - 24 q^{79} - 9 q^{80} + 19 q^{83} - q^{85} + 27 q^{86} - q^{88} - 15 q^{89} + 8 q^{91} - 3 q^{92} - 7 q^{94} + 3 q^{95} - 12 q^{97} + q^{98}+O(q^{100}) $ 4 * q + q^2 + q^4 + 4 * q^5 - 4 * q^7 - 3 * q^8 + q^10 - 7 * q^11 - 8 * q^13 - q^14 - 9 * q^16 - q^17 + 3 * q^19 + q^20 + 7 * q^22 + 8 * q^23 + 4 * q^25 + 3 * q^26 - q^28 - q^29 - 9 * q^34 - 4 * q^35 - 21 * q^37 - 8 * q^38 - 3 * q^40 - 20 * q^41 - 7 * q^43 - 3 * q^44 - 23 * q^46 + 2 * q^47 + 4 * q^49 + q^50 - 7 * q^52 + 8 * q^53 - 7 * q^55 + 3 * q^56 - 19 * q^58 - 19 * q^59 - 12 * q^61 - 15 * q^62 - 7 * q^64 - 8 * q^65 - 22 * q^67 + q^68 - q^70 - 13 * q^71 - 4 * q^73 - 9 * q^74 - 13 * q^76 + 7 * q^77 - 24 * q^79 - 9 * q^80 + 19 * q^83 - q^85 + 27 * q^86 - q^88 - 15 * q^89 + 8 * q^91 - 3 * q^92 - 7 * q^94 + 3 * q^95 - 12 * q^97 + 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 $

$2$	−0.209057	−0.147826	−0.0739128	−	0.997265i	$-0.523549\pi$
$2$	−0.209057	−0.147826	−0.0739128	+	0.997265i	$0.523549\pi$
$3$	0	0
$3$	0	0
$4$	−1.95630	−0.978148
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0.827091	0.292421
$9$	0	0
$10$	−0.209057	−0.0661096
$11$	−2.20906	−0.666056	−0.333028	−	0.942917i	$-0.608070\pi$
$11$	−2.20906	−0.666056	−0.333028	+	0.942917i	$0.608070\pi$
$12$	0	0
$13$	−2.48883	−0.690277	−0.345139	−	0.938552i	$-0.612168\pi$
$13$	−2.48883	−0.690277	−0.345139	+	0.938552i	$0.612168\pi$
$14$	0.209057	0.0558728
$15$	0	0
$16$	3.73968	0.934920
$17$	0.209057	0.0507038	0.0253519	−	0.999679i	$-0.491929\pi$
$17$	0.209057	0.0507038	0.0253519	+	0.999679i	$0.491929\pi$
$18$	0	0
$19$	7.22851	1.65833	0.829167	−	0.559001i	$-0.188815\pi$
$19$	7.22851	1.65833	0.829167	+	0.559001i	$0.188815\pi$
$20$	−1.95630	−0.437441
$21$	0	0
$22$	0.461819	0.0984601
$23$	2.85857	0.596053	0.298026	−	0.954558i	$-0.403672\pi$
$23$	2.85857	0.596053	0.298026	+	0.954558i	$0.403672\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0.520307	0.102041
$27$	0	0
$28$	1.95630	0.369705
$29$	−4.83274	−0.897417	−0.448708	−	0.893678i	$-0.648116\pi$
$29$	−4.83274	−0.897417	−0.448708	+	0.893678i	$0.648116\pi$
$30$	0	0
$31$	−3.98331	−0.715423	−0.357711	−	0.933832i	$-0.616443\pi$
$31$	−3.98331	−0.715423	−0.357711	+	0.933832i	$0.616443\pi$
$32$	−2.43599	−0.430626
$33$	0	0
$34$	−0.0437048	−0.00749531
$35$	−1.00000	−0.169031
$36$	0	0
$37$	0.739681	0.121603	0.0608014	−	0.998150i	$-0.480634\pi$
$37$	0.739681	0.121603	0.0608014	+	0.998150i	$0.480634\pi$
$38$	−1.51117	−0.245144
$39$	0	0
$40$	0.827091	0.130775
$41$	−2.19427	−0.342688	−0.171344	−	0.985211i	$-0.554811\pi$
$41$	−2.19427	−0.342688	−0.171344	+	0.985211i	$0.554811\pi$
$42$	0	0
$43$	−5.30369	−0.808806	−0.404403	−	0.914581i	$-0.632521\pi$
$43$	−5.30369	−0.808806	−0.404403	+	0.914581i	$0.632521\pi$
$44$	4.32157	0.651501
$45$	0	0
$46$	−0.597604	−0.0881118
$47$	8.61515	1.25665	0.628324	−	0.777952i	$-0.283741\pi$
$47$	8.61515	1.25665	0.628324	+	0.777952i	$0.283741\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−0.209057	−0.0295651
$51$	0	0
$52$	4.86889	0.675193
$53$	7.93863	1.09045	0.545227	−	0.838288i	$-0.316443\pi$
$53$	7.93863	1.09045	0.545227	+	0.838288i	$0.316443\pi$
$54$	0	0
$55$	−2.20906	−0.297869
$56$	−0.827091	−0.110525
$57$	0	0
$58$	1.01032	0.132661
$59$	−14.0342	−1.82710	−0.913551	−	0.406724i	$-0.866671\pi$
$59$	−14.0342	−1.82710	−0.913551	+	0.406724i	$0.866671\pi$
$60$	0	0
$61$	0.413648	0.0529621	0.0264811	−	0.999649i	$-0.491570\pi$
$61$	0.413648	0.0529621	0.0264811	+	0.999649i	$0.491570\pi$
$62$	0.832738	0.105758
$63$	0	0
$64$	−6.97010	−0.871263
$65$	−2.48883	−0.308701
$66$	0	0
$67$	−4.38197	−0.535342	−0.267671	−	0.963510i	$-0.586254\pi$
$67$	−4.38197	−0.535342	−0.267671	+	0.963510i	$0.586254\pi$
$68$	−0.408977	−0.0495958
$69$	0	0
$70$	0.209057	0.0249871
$71$	8.07039	0.957779	0.478889	−	0.877875i	$-0.341040\pi$
$71$	8.07039	0.957779	0.478889	+	0.877875i	$0.341040\pi$
$72$	0	0
$73$	−16.0446	−1.87787	−0.938937	−	0.344090i	$-0.888187\pi$
$73$	−16.0446	−1.87787	−0.938937	+	0.344090i	$0.888187\pi$
$74$	−0.154636	−0.0179760
$75$	0	0
$76$	−14.1411	−1.62210
$77$	2.20906	0.251745
$78$	0	0
$79$	−15.7220	−1.76886	−0.884432	−	0.466669i	$-0.845454\pi$
$79$	−15.7220	−1.76886	−0.884432	+	0.466669i	$0.845454\pi$
$80$	3.73968	0.418109
$81$	0	0
$82$	0.458728	0.0506580
$83$	−3.12729	−0.343265	−0.171633	−	0.985161i	$-0.554904\pi$
$83$	−3.12729	−0.343265	−0.171633	+	0.985161i	$0.554904\pi$
$84$	0	0
$85$	0.209057	0.0226754
$86$	1.10877	0.119562
$87$	0	0
$88$	−1.82709	−0.194769
$89$	−10.1477	−1.07565	−0.537826	−	0.843056i	$-0.680754\pi$
$89$	−10.1477	−1.07565	−0.537826	+	0.843056i	$0.680754\pi$
$90$	0	0
$91$	2.48883	0.260900
$92$	−5.59220	−0.583028
$93$	0	0
$94$	−1.80106	−0.185765
$95$	7.22851	0.741630
$96$	0	0
$97$	−11.9060	−1.20887	−0.604436	−	0.796654i	$-0.706602\pi$
$97$	−11.9060	−1.20887	−0.604436	+	0.796654i	$0.706602\pi$
$98$	−0.209057	−0.0211179
$99$	0	0
$100$	−1.95630	−0.195630
$101$	11.4671	1.14102	0.570511	−	0.821290i	$-0.306745\pi$
$101$	11.4671	1.14102	0.570511	+	0.821290i	$0.306745\pi$
$102$	0	0
$103$	5.39326	0.531414	0.265707	−	0.964054i	$-0.414395\pi$
$103$	5.39326	0.531414	0.265707	+	0.964054i	$0.414395\pi$
$104$	−2.05849	−0.201851
$105$	0	0
$106$	−1.65962	−0.161197
$107$	−17.6475	−1.70605	−0.853023	−	0.521873i	$-0.825233\pi$
$107$	−17.6475	−1.70605	−0.853023	+	0.521873i	$0.825233\pi$
$108$	0	0
$109$	7.37559	0.706453	0.353227	−	0.935538i	$-0.385084\pi$
$109$	7.37559	0.706453	0.353227	+	0.935538i	$0.385084\pi$
$110$	0.461819	0.0440327
$111$	0	0
$112$	−3.73968	−0.353367
$113$	13.5511	1.27478	0.637388	−	0.770543i	$-0.280015\pi$
$113$	13.5511	1.27478	0.637388	+	0.770543i	$0.280015\pi$
$114$	0	0
$115$	2.85857	0.266563
$116$	9.45426	0.877806
$117$	0	0
$118$	2.93395	0.270092
$119$	−0.209057	−0.0191642
$120$	0	0
$121$	−6.12007	−0.556370
$122$	−0.0864759	−0.00782916
$123$	0	0
$124$	7.79252	0.699789
$125$	1.00000	0.0894427
$126$	0	0
$127$	−7.93802	−0.704386	−0.352193	−	0.935927i	$-0.614564\pi$
$127$	−7.93802	−0.704386	−0.352193	+	0.935927i	$0.614564\pi$
$128$	6.32912	0.559421
$129$	0	0
$130$	0.520307	0.0456340
$131$	13.9276	1.21686	0.608429	−	0.793608i	$-0.291800\pi$
$131$	13.9276	1.21686	0.608429	+	0.793608i	$0.291800\pi$
$132$	0	0
$133$	−7.22851	−0.626791
$134$	0.916080	0.0791373
$135$	0	0
$136$	0.172909	0.0148268
$137$	−2.71734	−0.232158	−0.116079	−	0.993240i	$-0.537033\pi$
$137$	−2.71734	−0.232158	−0.116079	+	0.993240i	$0.537033\pi$
$138$	0	0
$139$	−2.04528	−0.173479	−0.0867394	−	0.996231i	$-0.527645\pi$
$139$	−2.04528	−0.173479	−0.0867394	+	0.996231i	$0.527645\pi$
$140$	1.95630	0.165337
$141$	0	0
$142$	−1.68717	−0.141584
$143$	5.49797	0.459763
$144$	0	0
$145$	−4.83274	−0.401337
$146$	3.35423	0.277598
$147$	0	0
$148$	−1.44704	−0.118946
$149$	−3.28424	−0.269055	−0.134528	−	0.990910i	$-0.542952\pi$
$149$	−3.28424	−0.269055	−0.134528	+	0.990910i	$0.542952\pi$
$150$	0	0
$151$	−2.23449	−0.181840	−0.0909200	−	0.995858i	$-0.528981\pi$
$151$	−2.23449	−0.181840	−0.0909200	+	0.995858i	$0.528981\pi$
$152$	5.97864	0.484931
$153$	0	0
$154$	−0.461819	−0.0372144
$155$	−3.98331	−0.319947
$156$	0	0
$157$	−0.750125	−0.0598665	−0.0299332	−	0.999552i	$-0.509529\pi$
$157$	−0.750125	−0.0598665	−0.0299332	+	0.999552i	$0.509529\pi$
$158$	3.28680	0.261483
$159$	0	0
$160$	−2.43599	−0.192582
$161$	−2.85857	−0.225287
$162$	0	0
$163$	−21.0896	−1.65187	−0.825934	−	0.563767i	$-0.809352\pi$
$163$	−21.0896	−1.65187	−0.825934	+	0.563767i	$0.809352\pi$
$164$	4.29265	0.335199
$165$	0	0
$166$	0.653783	0.0507434
$167$	−4.94874	−0.382945	−0.191472	−	0.981498i	$-0.561326\pi$
$167$	−4.94874	−0.382945	−0.191472	+	0.981498i	$0.561326\pi$
$168$	0	0
$169$	−6.80573	−0.523517
$170$	−0.0437048	−0.00335201
$171$	0	0
$172$	10.3756	0.791131
$173$	−17.1351	−1.30276	−0.651380	−	0.758752i	$-0.725809\pi$
$173$	−17.1351	−1.30276	−0.651380	+	0.758752i	$0.725809\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	−8.26117	−0.622709
$177$	0	0
$178$	2.12144	0.159009
$179$	−22.4030	−1.67448	−0.837239	−	0.546837i	$-0.815832\pi$
$179$	−22.4030	−1.67448	−0.837239	+	0.546837i	$0.815832\pi$
$180$	0	0
$181$	−5.69164	−0.423056	−0.211528	−	0.977372i	$-0.567844\pi$
$181$	−5.69164	−0.423056	−0.211528	+	0.977372i	$0.567844\pi$
$182$	−0.520307	−0.0385677
$183$	0	0
$184$	2.36430	0.174298
$185$	0.739681	0.0543825
$186$	0	0
$187$	−0.461819	−0.0337715
$188$	−16.8538	−1.22919
$189$	0	0
$190$	−1.51117	−0.109632
$191$	24.1961	1.75077	0.875384	−	0.483428i	$-0.160608\pi$
$191$	24.1961	1.75077	0.875384	+	0.483428i	$0.160608\pi$
$192$	0	0
$193$	20.1166	1.44803	0.724014	−	0.689785i	$-0.242295\pi$
$193$	20.1166	1.44803	0.724014	+	0.689785i	$0.242295\pi$
$194$	2.48903	0.178702
$195$	0	0
$196$	−1.95630	−0.139735
$197$	−2.57340	−0.183347	−0.0916734	−	0.995789i	$-0.529222\pi$
$197$	−2.57340	−0.183347	−0.0916734	+	0.995789i	$0.529222\pi$
$198$	0	0
$199$	−24.5589	−1.74093	−0.870467	−	0.492227i	$-0.836183\pi$
$199$	−24.5589	−1.74093	−0.870467	+	0.492227i	$0.836183\pi$
$200$	0.827091	0.0584842
$201$	0	0
$202$	−2.39728	−0.168672
$203$	4.83274	0.339192
$204$	0	0
$205$	−2.19427	−0.153255
$206$	−1.12750	−0.0785565
$207$	0	0
$208$	−9.30743	−0.645354
$209$	−15.9682	−1.10454
$210$	0	0
$211$	−11.9565	−0.823119	−0.411560	−	0.911383i	$-0.635016\pi$
$211$	−11.9565	−0.823119	−0.411560	+	0.911383i	$0.635016\pi$
$212$	−15.5303	−1.06662
$213$	0	0
$214$	3.68933	0.252197
$215$	−5.30369	−0.361709
$216$	0	0
$217$	3.98331	0.270404
$218$	−1.54192	−0.104432
$219$	0	0
$220$	4.32157	0.291360
$221$	−0.520307	−0.0349996
$222$	0	0
$223$	−6.80849	−0.455930	−0.227965	−	0.973669i	$-0.573207\pi$
$223$	−6.80849	−0.455930	−0.227965	+	0.973669i	$0.573207\pi$
$224$	2.43599	0.162761
$225$	0	0
$226$	−2.83294	−0.188444
$227$	20.0471	1.33057	0.665287	−	0.746588i	$-0.268309\pi$
$227$	20.0471	1.33057	0.665287	+	0.746588i	$0.268309\pi$
$228$	0	0
$229$	−21.2649	−1.40522	−0.702611	−	0.711574i	$-0.747982\pi$
$229$	−21.2649	−1.40522	−0.702611	+	0.711574i	$0.747982\pi$
$230$	−0.597604	−0.0394048
$231$	0	0
$232$	−3.99711	−0.262423
$233$	5.44631	0.356799	0.178400	−	0.983958i	$-0.442908\pi$
$233$	5.44631	0.356799	0.178400	+	0.983958i	$0.442908\pi$
$234$	0	0
$235$	8.61515	0.561990
$236$	27.4551	1.78718
$237$	0	0
$238$	0.0437048	0.00283296
$239$	18.6686	1.20757	0.603787	−	0.797146i	$-0.293658\pi$
$239$	18.6686	1.20757	0.603787	+	0.797146i	$0.293658\pi$
$240$	0	0
$241$	−4.63342	−0.298465	−0.149232	−	0.988802i	$-0.547680\pi$
$241$	−4.63342	−0.298465	−0.149232	+	0.988802i	$0.547680\pi$
$242$	1.27944	0.0822457
$243$	0	0
$244$	−0.809217	−0.0518048
$245$	1.00000	0.0638877
$246$	0	0
$247$	−17.9905	−1.14471
$248$	−3.29456	−0.209205
$249$	0	0
$250$	−0.209057	−0.0132219
$251$	−9.53378	−0.601767	−0.300884	−	0.953661i	$-0.597282\pi$
$251$	−9.53378	−0.601767	−0.300884	+	0.953661i	$0.597282\pi$
$252$	0	0
$253$	−6.31474	−0.397004
$254$	1.65950	0.104126
$255$	0	0
$256$	12.6171	0.788566
$257$	−30.3941	−1.89593	−0.947967	−	0.318368i	$-0.896865\pi$
$257$	−30.3941	−1.89593	−0.947967	+	0.318368i	$0.896865\pi$
$258$	0	0
$259$	−0.739681	−0.0459616
$260$	4.86889	0.301955
$261$	0	0
$262$	−2.91166	−0.179883
$263$	−9.78866	−0.603595	−0.301797	−	0.953372i	$-0.597587\pi$
$263$	−9.78866	−0.603595	−0.301797	+	0.953372i	$0.597587\pi$
$264$	0	0
$265$	7.93863	0.487666
$266$	1.51117	0.0926558
$267$	0	0
$268$	8.57242	0.523644
$269$	−23.2210	−1.41581	−0.707903	−	0.706309i	$-0.750359\pi$
$269$	−23.2210	−1.41581	−0.707903	+	0.706309i	$0.750359\pi$
$270$	0	0
$271$	−3.68277	−0.223713	−0.111856	−	0.993724i	$-0.535680\pi$
$271$	−3.68277	−0.223713	−0.111856	+	0.993724i	$0.535680\pi$
$272$	0.781806	0.0474040
$273$	0	0
$274$	0.568079	0.0343189
$275$	−2.20906	−0.133211
$276$	0	0
$277$	3.22602	0.193833	0.0969165	−	0.995293i	$-0.469102\pi$
$277$	3.22602	0.193833	0.0969165	+	0.995293i	$0.469102\pi$
$278$	0.427581	0.0256446
$279$	0	0
$280$	−0.827091	−0.0494281
$281$	6.21340	0.370660	0.185330	−	0.982676i	$-0.440665\pi$
$281$	6.21340	0.370660	0.185330	+	0.982676i	$0.440665\pi$
$282$	0	0
$283$	26.6420	1.58370	0.791852	−	0.610713i	$-0.209117\pi$
$283$	26.6420	1.58370	0.791852	+	0.610713i	$0.209117\pi$
$284$	−15.7881	−0.936849
$285$	0	0
$286$	−1.14939	−0.0679647
$287$	2.19427	0.129524
$288$	0	0
$289$	−16.9563	−0.997429
$290$	1.01032	0.0593279
$291$	0	0
$292$	31.3879	1.83684
$293$	27.3014	1.59496	0.797482	−	0.603343i	$-0.206165\pi$
$293$	27.3014	1.59496	0.797482	+	0.603343i	$0.206165\pi$
$294$	0	0
$295$	−14.0342	−0.817105
$296$	0.611784	0.0355592
$297$	0	0
$298$	0.686593	0.0397733
$299$	−7.11449	−0.411442
$300$	0	0
$301$	5.30369	0.305700
$302$	0.467135	0.0268806
$303$	0	0
$304$	27.0323	1.55041
$305$	0.413648	0.0236854
$306$	0	0
$307$	−0.831362	−0.0474483	−0.0237242	−	0.999719i	$-0.507552\pi$
$307$	−0.831362	−0.0474483	−0.0237242	+	0.999719i	$0.507552\pi$
$308$	−4.32157	−0.246244
$309$	0	0
$310$	0.832738	0.0472963
$311$	−26.6541	−1.51142	−0.755708	−	0.654909i	$-0.772707\pi$
$311$	−26.6541	−1.51142	−0.755708	+	0.654909i	$0.772707\pi$
$312$	0	0
$313$	3.49854	0.197749	0.0988747	−	0.995100i	$-0.468476\pi$
$313$	3.49854	0.197749	0.0988747	+	0.995100i	$0.468476\pi$
$314$	0.156819	0.00884980
$315$	0	0
$316$	30.7569	1.73021
$317$	−14.4407	−0.811072	−0.405536	−	0.914079i	$-0.632915\pi$
$317$	−14.4407	−0.811072	−0.405536	+	0.914079i	$0.632915\pi$
$318$	0	0
$319$	10.6758	0.597730
$320$	−6.97010	−0.389641
$321$	0	0
$322$	0.597604	0.0333031
$323$	1.51117	0.0840838
$324$	0	0
$325$	−2.48883	−0.138055
$326$	4.40893	0.244188
$327$	0	0
$328$	−1.81486	−0.100209
$329$	−8.61515	−0.474968
$330$	0	0
$331$	5.39366	0.296462	0.148231	−	0.988953i	$-0.452642\pi$
$331$	5.39366	0.296462	0.148231	+	0.988953i	$0.452642\pi$
$332$	6.11791	0.335764
$333$	0	0
$334$	1.03457	0.0566090
$335$	−4.38197	−0.239412
$336$	0	0
$337$	−11.6730	−0.635871	−0.317935	−	0.948112i	$-0.602989\pi$
$337$	−11.6730	−0.635871	−0.317935	+	0.948112i	$0.602989\pi$
$338$	1.42278	0.0773893
$339$	0	0
$340$	−0.408977	−0.0221799
$341$	8.79935	0.476512
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−4.38664	−0.236512
$345$	0	0
$346$	3.58222	0.192581
$347$	35.2129	1.89033	0.945163	−	0.326599i	$-0.105903\pi$
$347$	35.2129	1.89033	0.945163	+	0.326599i	$0.105903\pi$
$348$	0	0
$349$	−30.1584	−1.61434	−0.807172	−	0.590316i	$-0.799003\pi$
$349$	−30.1584	−1.61434	−0.807172	+	0.590316i	$0.799003\pi$
$350$	0.209057	0.0111746
$351$	0	0
$352$	5.38124	0.286821
$353$	2.89065	0.153854	0.0769269	−	0.997037i	$-0.475489\pi$
$353$	2.89065	0.153854	0.0769269	+	0.997037i	$0.475489\pi$
$354$	0	0
$355$	8.07039	0.428332
$356$	19.8519	1.05215
$357$	0	0
$358$	4.68350	0.247531
$359$	29.6466	1.56469	0.782345	−	0.622846i	$-0.214024\pi$
$359$	29.6466	1.56469	0.782345	+	0.622846i	$0.214024\pi$
$360$	0	0
$361$	33.2514	1.75007
$362$	1.18988	0.0625385
$363$	0	0
$364$	−4.86889	−0.255199
$365$	−16.0446	−0.839810
$366$	0	0
$367$	27.3376	1.42701	0.713506	−	0.700650i	$-0.247106\pi$
$367$	27.3376	1.42701	0.713506	+	0.700650i	$0.247106\pi$
$368$	10.6901	0.557262
$369$	0	0
$370$	−0.154636	−0.00803912
$371$	−7.93863	−0.412153
$372$	0	0
$373$	−19.9324	−1.03206	−0.516030	−	0.856571i	$-0.672591\pi$
$373$	−19.9324	−1.03206	−0.516030	+	0.856571i	$0.672591\pi$
$374$	0.0965464	0.00499229
$375$	0	0
$376$	7.12551	0.367470
$377$	12.0279	0.619466
$378$	0	0
$379$	21.2461	1.09134	0.545668	−	0.838001i	$-0.316276\pi$
$379$	21.2461	1.09134	0.545668	+	0.838001i	$0.316276\pi$
$380$	−14.1411	−0.725423
$381$	0	0
$382$	−5.05836	−0.258808
$383$	0.915807	0.0467956	0.0233978	−	0.999726i	$-0.492552\pi$
$383$	0.915807	0.0467956	0.0233978	+	0.999726i	$0.492552\pi$
$384$	0	0
$385$	2.20906	0.112584
$386$	−4.20552	−0.214056
$387$	0	0
$388$	23.2917	1.18246
$389$	−24.4596	−1.24015	−0.620075	−	0.784543i	$-0.712898\pi$
$389$	−24.4596	−1.24015	−0.620075	+	0.784543i	$0.712898\pi$
$390$	0	0
$391$	0.597604	0.0302221
$392$	0.827091	0.0417744
$393$	0	0
$394$	0.537986	0.0271034
$395$	−15.7220	−0.791060
$396$	0	0
$397$	10.0648	0.505137	0.252568	−	0.967579i	$-0.418725\pi$
$397$	10.0648	0.505137	0.252568	+	0.967579i	$0.418725\pi$
$398$	5.13421	0.257354
$399$	0	0
$400$	3.73968	0.186984
$401$	−8.54134	−0.426534	−0.213267	−	0.976994i	$-0.568410\pi$
$401$	−8.54134	−0.426534	−0.213267	+	0.976994i	$0.568410\pi$
$402$	0	0
$403$	9.91377	0.493840
$404$	−22.4331	−1.11609
$405$	0	0
$406$	−1.01032	−0.0501412
$407$	−1.63400	−0.0809943
$408$	0	0
$409$	6.03672	0.298497	0.149248	−	0.988800i	$-0.452315\pi$
$409$	6.03672	0.298497	0.149248	+	0.988800i	$0.452315\pi$
$410$	0.458728	0.0226550
$411$	0	0
$412$	−10.5508	−0.519801
$413$	14.0342	0.690580
$414$	0	0
$415$	−3.12729	−0.153513
$416$	6.06276	0.297251
$417$	0	0
$418$	3.33826	0.163280
$419$	−19.8742	−0.970918	−0.485459	−	0.874259i	$-0.661348\pi$
$419$	−19.8742	−0.970918	−0.485459	+	0.874259i	$0.661348\pi$
$420$	0	0
$421$	−32.2097	−1.56980	−0.784902	−	0.619619i	$-0.787287\pi$
$421$	−32.2097	−1.56980	−0.784902	+	0.619619i	$0.787287\pi$
$422$	2.49959	0.121678
$423$	0	0
$424$	6.56596	0.318871
$425$	0.209057	0.0101408
$426$	0	0
$427$	−0.413648	−0.0200178
$428$	34.5237	1.66876
$429$	0	0
$430$	1.10877	0.0534698
$431$	−8.89203	−0.428314	−0.214157	−	0.976799i	$-0.568700\pi$
$431$	−8.89203	−0.428314	−0.214157	+	0.976799i	$0.568700\pi$
$432$	0	0
$433$	−6.06725	−0.291574	−0.145787	−	0.989316i	$-0.546571\pi$
$433$	−6.06725	−0.291574	−0.145787	+	0.989316i	$0.546571\pi$
$434$	−0.832738	−0.0399727
$435$	0	0
$436$	−14.4288	−0.691016
$437$	20.6632	0.988455
$438$	0	0
$439$	27.7705	1.32541	0.662706	−	0.748880i	$-0.269408\pi$
$439$	27.7705	1.32541	0.662706	+	0.748880i	$0.269408\pi$
$440$	−1.82709	−0.0871031
$441$	0	0
$442$	0.108774	0.00517384
$443$	34.0505	1.61779	0.808893	−	0.587955i	$-0.200067\pi$
$443$	34.0505	1.61779	0.808893	+	0.587955i	$0.200067\pi$
$444$	0	0
$445$	−10.1477	−0.481046
$446$	1.42336	0.0673981
$447$	0	0
$448$	6.97010	0.329306
$449$	0.508887	0.0240159	0.0120079	−	0.999928i	$-0.496178\pi$
$449$	0.508887	0.0240159	0.0120079	+	0.999928i	$0.496178\pi$
$450$	0	0
$451$	4.84727	0.228249
$452$	−26.5099	−1.24692
$453$	0	0
$454$	−4.19099	−0.196693
$455$	2.48883	0.116678
$456$	0	0
$457$	19.0096	0.889232	0.444616	−	0.895721i	$-0.353340\pi$
$457$	19.0096	0.889232	0.444616	+	0.895721i	$0.353340\pi$
$458$	4.44557	0.207728
$459$	0	0
$460$	−5.59220	−0.260738
$461$	−12.9470	−0.603003	−0.301502	−	0.953466i	$-0.597488\pi$
$461$	−12.9470	−0.603003	−0.301502	+	0.953466i	$0.597488\pi$
$462$	0	0
$463$	−17.0252	−0.791227	−0.395613	−	0.918417i	$-0.629468\pi$
$463$	−17.0252	−0.791227	−0.395613	+	0.918417i	$0.629468\pi$
$464$	−18.0729	−0.839013
$465$	0	0
$466$	−1.13859	−0.0527441
$467$	31.1794	1.44281	0.721405	−	0.692513i	$-0.243496\pi$
$467$	31.1794	1.44281	0.721405	+	0.692513i	$0.243496\pi$
$468$	0	0
$469$	4.38197	0.202340
$470$	−1.80106	−0.0830765
$471$	0	0
$472$	−11.6076	−0.534283
$473$	11.7162	0.538710
$474$	0	0
$475$	7.22851	0.331667
$476$	0.408977	0.0187454
$477$	0	0
$478$	−3.90281	−0.178510
$479$	13.5557	0.619375	0.309687	−	0.950838i	$-0.399776\pi$
$479$	13.5557	0.619375	0.309687	+	0.950838i	$0.399776\pi$
$480$	0	0
$481$	−1.84094	−0.0839397
$482$	0.968649	0.0441207
$483$	0	0
$484$	11.9727	0.544212
$485$	−11.9060	−0.540624
$486$	0	0
$487$	5.50906	0.249639	0.124820	−	0.992179i	$-0.460165\pi$
$487$	5.50906	0.249639	0.124820	+	0.992179i	$0.460165\pi$
$488$	0.342124	0.0154872
$489$	0	0
$490$	−0.209057	−0.00944423
$491$	−10.2637	−0.463192	−0.231596	−	0.972812i	$-0.574395\pi$
$491$	−10.2637	−0.463192	−0.231596	+	0.972812i	$0.574395\pi$
$492$	0	0
$493$	−1.01032	−0.0455024
$494$	3.76105	0.169217
$495$	0	0
$496$	−14.8963	−0.668863
$497$	−8.07039	−0.362006
$498$	0	0
$499$	24.1331	1.08035	0.540174	−	0.841553i	$-0.318358\pi$
$499$	24.1331	1.08035	0.540174	+	0.841553i	$0.318358\pi$
$500$	−1.95630	−0.0874882
$501$	0	0
$502$	1.99310	0.0889566
$503$	33.9679	1.51455	0.757277	−	0.653094i	$-0.226529\pi$
$503$	33.9679	1.51455	0.757277	+	0.653094i	$0.226529\pi$
$504$	0	0
$505$	11.4671	0.510281
$506$	1.32014	0.0586874
$507$	0	0
$508$	15.5291	0.688993
$509$	1.03850	0.0460307	0.0230154	−	0.999735i	$-0.492673\pi$
$509$	1.03850	0.0460307	0.0230154	+	0.999735i	$0.492673\pi$
$510$	0	0
$511$	16.0446	0.709769
$512$	−15.2959	−0.675991
$513$	0	0
$514$	6.35410	0.280268
$515$	5.39326	0.237655
$516$	0	0
$517$	−19.0314	−0.836998
$518$	0.154636	0.00679429
$519$	0	0
$520$	−2.05849	−0.0902707
$521$	32.6154	1.42891	0.714453	−	0.699683i	$-0.246676\pi$
$521$	32.6154	1.42891	0.714453	+	0.699683i	$0.246676\pi$
$522$	0	0
$523$	27.0228	1.18162	0.590812	−	0.806809i	$-0.298807\pi$
$523$	27.0228	1.18162	0.590812	+	0.806809i	$0.298807\pi$
$524$	−27.2465	−1.19027
$525$	0	0
$526$	2.04639	0.0892267
$527$	−0.832738	−0.0362746
$528$	0	0
$529$	−14.8286	−0.644721
$530$	−1.65962	−0.0720895
$531$	0	0
$532$	14.1411	0.613095
$533$	5.46117	0.236550
$534$	0	0
$535$	−17.6475	−0.762967
$536$	−3.62428	−0.156545
$537$	0	0
$538$	4.85450	0.209292
$539$	−2.20906	−0.0951508
$540$	0	0
$541$	−35.7007	−1.53489	−0.767447	−	0.641113i	$-0.778473\pi$
$541$	−35.7007	−1.53489	−0.767447	+	0.641113i	$0.778473\pi$
$542$	0.769909	0.0330704
$543$	0	0
$544$	−0.509260	−0.0218344
$545$	7.37559	0.315936
$546$	0	0
$547$	18.4077	0.787057	0.393528	−	0.919312i	$-0.371254\pi$
$547$	18.4077	0.787057	0.393528	+	0.919312i	$0.371254\pi$
$548$	5.31592	0.227085
$549$	0	0
$550$	0.461819	0.0196920
$551$	−34.9335	−1.48822
$552$	0	0
$553$	15.7220	0.668568
$554$	−0.674423	−0.0286535
$555$	0	0
$556$	4.00118	0.169688
$557$	−35.9397	−1.52281	−0.761407	−	0.648275i	$-0.775491\pi$
$557$	−35.9397	−1.52281	−0.761407	+	0.648275i	$0.775491\pi$
$558$	0	0
$559$	13.2000	0.558300
$560$	−3.73968	−0.158030
$561$	0	0
$562$	−1.29895	−0.0547931
$563$	−11.0737	−0.466700	−0.233350	−	0.972393i	$-0.574969\pi$
$563$	−11.0737	−0.466700	−0.233350	+	0.972393i	$0.574969\pi$
$564$	0	0
$565$	13.5511	0.570097
$566$	−5.56970	−0.234112
$567$	0	0
$568$	6.67494	0.280074
$569$	−7.16824	−0.300508	−0.150254	−	0.988647i	$-0.548009\pi$
$569$	−7.16824	−0.300508	−0.150254	+	0.988647i	$0.548009\pi$
$570$	0	0
$571$	−9.35909	−0.391666	−0.195833	−	0.980637i	$-0.562741\pi$
$571$	−9.35909	−0.391666	−0.195833	+	0.980637i	$0.562741\pi$
$572$	−10.7556	−0.449716
$573$	0	0
$574$	−0.458728	−0.0191469
$575$	2.85857	0.119211
$576$	0	0
$577$	−8.63400	−0.359438	−0.179719	−	0.983718i	$-0.557519\pi$
$577$	−8.63400	−0.359438	−0.179719	+	0.983718i	$0.557519\pi$
$578$	3.54483	0.147446
$579$	0	0
$580$	9.45426	0.392567
$581$	3.12729	0.129742
$582$	0	0
$583$	−17.5369	−0.726303
$584$	−13.2703	−0.549129
$585$	0	0
$586$	−5.70754	−0.235776
$587$	42.8141	1.76713	0.883563	−	0.468312i	$-0.155138\pi$
$587$	42.8141	1.76713	0.883563	+	0.468312i	$0.155138\pi$
$588$	0	0
$589$	−28.7934	−1.18641
$590$	2.93395	0.120789
$591$	0	0
$592$	2.76617	0.113689
$593$	21.9382	0.900895	0.450447	−	0.892803i	$-0.351264\pi$
$593$	21.9382	0.900895	0.450447	+	0.892803i	$0.351264\pi$
$594$	0	0
$595$	−0.209057	−0.00857050
$596$	6.42494	0.263176
$597$	0	0
$598$	1.48733	0.0608216
$599$	−18.4664	−0.754515	−0.377258	−	0.926108i	$-0.623133\pi$
$599$	−18.4664	−0.754515	−0.377258	+	0.926108i	$0.623133\pi$
$600$	0	0
$601$	−1.72839	−0.0705024	−0.0352512	−	0.999378i	$-0.511223\pi$
$601$	−1.72839	−0.0705024	−0.0352512	+	0.999378i	$0.511223\pi$
$602$	−1.10877	−0.0451902
$603$	0	0
$604$	4.37132	0.177866
$605$	−6.12007	−0.248816
$606$	0	0
$607$	22.6798	0.920546	0.460273	−	0.887777i	$-0.347751\pi$
$607$	22.6798	0.920546	0.460273	+	0.887777i	$0.347751\pi$
$608$	−17.6086	−0.714122
$609$	0	0
$610$	−0.0864759	−0.00350131
$611$	−21.4416	−0.867436
$612$	0	0
$613$	18.5227	0.748125	0.374062	−	0.927404i	$-0.377965\pi$
$613$	18.5227	0.748125	0.374062	+	0.927404i	$0.377965\pi$
$614$	0.173802	0.00701408
$615$	0	0
$616$	1.82709	0.0736156
$617$	22.1878	0.893246	0.446623	−	0.894722i	$-0.352627\pi$
$617$	22.1878	0.893246	0.446623	+	0.894722i	$0.352627\pi$
$618$	0	0
$619$	9.28791	0.373312	0.186656	−	0.982425i	$-0.440235\pi$
$619$	9.28791	0.373312	0.186656	+	0.982425i	$0.440235\pi$
$620$	7.79252	0.312955
$621$	0	0
$622$	5.57222	0.223426
$623$	10.1477	0.406558
$624$	0	0
$625$	1.00000	0.0400000
$626$	−0.731395	−0.0292324
$627$	0	0
$628$	1.46747	0.0585582
$629$	0.154636	0.00616572
$630$	0	0
$631$	26.6838	1.06227	0.531133	−	0.847288i	$-0.321766\pi$
$631$	26.6838	1.06227	0.531133	+	0.847288i	$0.321766\pi$
$632$	−13.0035	−0.517253
$633$	0	0
$634$	3.01893	0.119897
$635$	−7.93802	−0.315011
$636$	0	0
$637$	−2.48883	−0.0986110
$638$	−2.23185	−0.0883597
$639$	0	0
$640$	6.32912	0.250181
$641$	6.57745	0.259794	0.129897	−	0.991528i	$-0.458535\pi$
$641$	6.57745	0.259794	0.129897	+	0.991528i	$0.458535\pi$
$642$	0	0
$643$	7.36430	0.290419	0.145210	−	0.989401i	$-0.453614\pi$
$643$	7.36430	0.290419	0.145210	+	0.989401i	$0.453614\pi$
$644$	5.59220	0.220364
$645$	0	0
$646$	−0.315921	−0.0124297
$647$	−11.8899	−0.467438	−0.233719	−	0.972304i	$-0.575090\pi$
$647$	−11.8899	−0.467438	−0.233719	+	0.972304i	$0.575090\pi$
$648$	0	0
$649$	31.0024	1.21695
$650$	0.520307	0.0204081
$651$	0	0
$652$	41.2576	1.61577
$653$	5.15622	0.201778	0.100889	−	0.994898i	$-0.467831\pi$
$653$	5.15622	0.201778	0.100889	+	0.994898i	$0.467831\pi$
$654$	0	0
$655$	13.9276	0.544196
$656$	−8.20588	−0.320386
$657$	0	0
$658$	1.80106	0.0702125
$659$	−5.82933	−0.227079	−0.113539	−	0.993534i	$-0.536219\pi$
$659$	−5.82933	−0.227079	−0.113539	+	0.993534i	$0.536219\pi$
$660$	0	0
$661$	13.1491	0.511441	0.255720	−	0.966751i	$-0.417687\pi$
$661$	13.1491	0.511441	0.255720	+	0.966751i	$0.417687\pi$
$662$	−1.12758	−0.0438247
$663$	0	0
$664$	−2.58656	−0.100378
$665$	−7.22851	−0.280310
$666$	0	0
$667$	−13.8147	−0.534908
$668$	9.68119	0.374577
$669$	0	0
$670$	0.916080	0.0353913
$671$	−0.913771	−0.0352757
$672$	0	0
$673$	−22.0175	−0.848712	−0.424356	−	0.905495i	$-0.639500\pi$
$673$	−22.0175	−0.848712	−0.424356	+	0.905495i	$0.639500\pi$
$674$	2.44033	0.0939979
$675$	0	0
$676$	13.3140	0.512077
$677$	3.16495	0.121639	0.0608195	−	0.998149i	$-0.480629\pi$
$677$	3.16495	0.121639	0.0608195	+	0.998149i	$0.480629\pi$
$678$	0	0
$679$	11.9060	0.456911
$680$	0.172909	0.00663076
$681$	0	0
$682$	−1.83957	−0.0704406
$683$	23.4933	0.898946	0.449473	−	0.893294i	$-0.351612\pi$
$683$	23.4933	0.898946	0.449473	+	0.893294i	$0.351612\pi$
$684$	0	0
$685$	−2.71734	−0.103824
$686$	0.209057	0.00798183
$687$	0	0
$688$	−19.8341	−0.756169
$689$	−19.7579	−0.752716
$690$	0	0
$691$	50.6077	1.92521	0.962604	−	0.270913i	$-0.0873255\pi$
$691$	50.6077	1.92521	0.962604	+	0.270913i	$0.0873255\pi$
$692$	33.5214	1.27429
$693$	0	0
$694$	−7.36149	−0.279439
$695$	−2.04528	−0.0775821
$696$	0	0
$697$	−0.458728	−0.0173756
$698$	6.30483	0.238641
$699$	0	0
$700$	1.95630	0.0739410
$701$	47.3661	1.78899	0.894496	−	0.447076i	$-0.147535\pi$
$701$	47.3661	1.78899	0.894496	+	0.447076i	$0.147535\pi$
$702$	0	0
$703$	5.34679	0.201658
$704$	15.3974	0.580310
$705$	0	0
$706$	−0.604310	−0.0227435
$707$	−11.4671	−0.431266
$708$	0	0
$709$	−27.1166	−1.01839	−0.509193	−	0.860653i	$-0.670056\pi$
$709$	−27.1166	−1.01839	−0.509193	+	0.860653i	$0.670056\pi$
$710$	−1.68717	−0.0633184
$711$	0	0
$712$	−8.39306	−0.314543
$713$	−11.3866	−0.426430
$714$	0	0
$715$	5.49797	0.205612
$716$	43.8269	1.63789
$717$	0	0
$718$	−6.19783	−0.231301
$719$	13.6974	0.510828	0.255414	−	0.966832i	$-0.417788\pi$
$719$	13.6974	0.510828	0.255414	+	0.966832i	$0.417788\pi$
$720$	0	0
$721$	−5.39326	−0.200855
$722$	−6.95143	−0.258705
$723$	0	0
$724$	11.1345	0.413811
$725$	−4.83274	−0.179483
$726$	0	0
$727$	0.912191	0.0338313	0.0169156	−	0.999857i	$-0.494615\pi$
$727$	0.912191	0.0338313	0.0169156	+	0.999857i	$0.494615\pi$
$728$	2.05849	0.0762927
$729$	0	0
$730$	3.35423	0.124145
$731$	−1.10877	−0.0410095
$732$	0	0
$733$	−25.5827	−0.944918	−0.472459	−	0.881353i	$-0.656633\pi$
$733$	−25.5827	−0.944918	−0.472459	+	0.881353i	$0.656633\pi$
$734$	−5.71511	−0.210949
$735$	0	0
$736$	−6.96344	−0.256676
$737$	9.68001	0.356568
$738$	0	0
$739$	−46.2781	−1.70237	−0.851183	−	0.524869i	$-0.824114\pi$
$739$	−46.2781	−1.70237	−0.851183	+	0.524869i	$0.824114\pi$
$740$	−1.44704	−0.0531941
$741$	0	0
$742$	1.65962	0.0609267
$743$	−22.2441	−0.816056	−0.408028	−	0.912969i	$-0.633783\pi$
$743$	−22.2441	−0.816056	−0.408028	+	0.912969i	$0.633783\pi$
$744$	0	0
$745$	−3.28424	−0.120325
$746$	4.16700	0.152565
$747$	0	0
$748$	0.903454	0.0330335
$749$	17.6475	0.644825
$750$	0	0
$751$	20.8059	0.759217	0.379609	−	0.925147i	$-0.376059\pi$
$751$	20.8059	0.759217	0.379609	+	0.925147i	$0.376059\pi$
$752$	32.2179	1.17487
$753$	0	0
$754$	−2.51451	−0.0915730
$755$	−2.23449	−0.0813213
$756$	0	0
$757$	−41.4984	−1.50828	−0.754142	−	0.656712i	$-0.771947\pi$
$757$	−41.4984	−1.50828	−0.754142	+	0.656712i	$0.771947\pi$
$758$	−4.44163	−0.161327
$759$	0	0
$760$	5.97864	0.216868
$761$	19.9353	0.722655	0.361328	−	0.932439i	$-0.382324\pi$
$761$	19.9353	0.722655	0.361328	+	0.932439i	$0.382324\pi$
$762$	0	0
$763$	−7.37559	−0.267014
$764$	−47.3347	−1.71251
$765$	0	0
$766$	−0.191456	−0.00691758
$767$	34.9288	1.26121
$768$	0	0
$769$	−11.1490	−0.402043	−0.201021	−	0.979587i	$-0.564426\pi$
$769$	−11.1490	−0.402043	−0.201021	+	0.979587i	$0.564426\pi$
$770$	−0.461819	−0.0166428
$771$	0	0
$772$	−39.3541	−1.41639
$773$	−34.9747	−1.25795	−0.628976	−	0.777425i	$-0.716526\pi$
$773$	−34.9747	−1.25795	−0.628976	+	0.777425i	$0.716526\pi$
$774$	0	0
$775$	−3.98331	−0.143085
$776$	−9.84735	−0.353499
$777$	0	0
$778$	5.11344	0.183326
$779$	−15.8613	−0.568291
$780$	0	0
$781$	−17.8279	−0.637934
$782$	−0.124933	−0.00446760
$783$	0	0
$784$	3.73968	0.133560
$785$	−0.750125	−0.0267731
$786$	0	0
$787$	42.6584	1.52061	0.760304	−	0.649568i	$-0.225050\pi$
$787$	42.6584	1.52061	0.760304	+	0.649568i	$0.225050\pi$
$788$	5.03432	0.179340
$789$	0	0
$790$	3.28680	0.116939
$791$	−13.5511	−0.481820
$792$	0	0
$793$	−1.02950	−0.0365586
$794$	−2.10411	−0.0746722
$795$	0	0
$796$	48.0444	1.70289
$797$	8.26807	0.292870	0.146435	−	0.989220i	$-0.453220\pi$
$797$	8.26807	0.292870	0.146435	+	0.989220i	$0.453220\pi$
$798$	0	0
$799$	1.80106	0.0637168
$800$	−2.43599	−0.0861252
$801$	0	0
$802$	1.78563	0.0630527
$803$	35.4433	1.25077
$804$	0	0
$805$	−2.85857	−0.100751
$806$	−2.07254	−0.0730022
$807$	0	0
$808$	9.48436	0.333659
$809$	7.79155	0.273936	0.136968	−	0.990575i	$-0.456264\pi$
$809$	7.79155	0.273936	0.136968	+	0.990575i	$0.456264\pi$
$810$	0	0
$811$	−31.4280	−1.10359	−0.551793	−	0.833981i	$-0.686056\pi$
$811$	−31.4280	−1.10359	−0.551793	+	0.833981i	$0.686056\pi$
$812$	−9.45426	−0.331780
$813$	0	0
$814$	0.341599	0.0119730
$815$	−21.0896	−0.738738
$816$	0	0
$817$	−38.3378	−1.34127
$818$	−1.26202	−0.0441255
$819$	0	0
$820$	4.29265	0.149906
$821$	−41.3256	−1.44227	−0.721136	−	0.692793i	$-0.756380\pi$
$821$	−41.3256	−1.44227	−0.721136	+	0.692793i	$0.756380\pi$
$822$	0	0
$823$	−45.2630	−1.57777	−0.788885	−	0.614541i	$-0.789341\pi$
$823$	−45.2630	−1.57777	−0.788885	+	0.614541i	$0.789341\pi$
$824$	4.46072	0.155396
$825$	0	0
$826$	−2.93395	−0.102085
$827$	−39.9978	−1.39086	−0.695430	−	0.718594i	$-0.744786\pi$
$827$	−39.9978	−1.39086	−0.695430	+	0.718594i	$0.744786\pi$
$828$	0	0
$829$	41.0978	1.42739	0.713693	−	0.700459i	$-0.247021\pi$
$829$	41.0978	1.42739	0.713693	+	0.700459i	$0.247021\pi$
$830$	0.653783	0.0226931
$831$	0	0
$832$	17.3474	0.601413
$833$	0.209057	0.00724339
$834$	0	0
$835$	−4.94874	−0.171258
$836$	31.2385	1.08041
$837$	0	0
$838$	4.15484	0.143527
$839$	1.20113	0.0414675	0.0207338	−	0.999785i	$-0.493400\pi$
$839$	1.20113	0.0414675	0.0207338	+	0.999785i	$0.493400\pi$
$840$	0	0
$841$	−5.64465	−0.194643
$842$	6.73366	0.232057
$843$	0	0
$844$	23.3904	0.805132
$845$	−6.80573	−0.234124
$846$	0	0
$847$	6.12007	0.210288
$848$	29.6879	1.01949
$849$	0	0
$850$	−0.0437048	−0.00149906
$851$	2.11443	0.0724817
$852$	0	0
$853$	34.8214	1.19226	0.596131	−	0.802887i	$-0.296704\pi$
$853$	34.8214	1.19226	0.596131	+	0.802887i	$0.296704\pi$
$854$	0.0864759	0.00295914
$855$	0	0
$856$	−14.5961	−0.498883
$857$	13.7445	0.469505	0.234752	−	0.972055i	$-0.424572\pi$
$857$	13.7445	0.469505	0.234752	+	0.972055i	$0.424572\pi$
$858$	0	0
$859$	20.6520	0.704638	0.352319	−	0.935880i	$-0.385393\pi$
$859$	20.6520	0.704638	0.352319	+	0.935880i	$0.385393\pi$
$860$	10.3756	0.353805
$861$	0	0
$862$	1.85894	0.0633158
$863$	−35.9288	−1.22303	−0.611515	−	0.791233i	$-0.709440\pi$
$863$	−35.9288	−1.22303	−0.611515	+	0.791233i	$0.709440\pi$
$864$	0	0
$865$	−17.1351	−0.582612
$866$	1.26840	0.0431020
$867$	0	0
$868$	−7.79252	−0.264495
$869$	34.7308	1.17816
$870$	0	0
$871$	10.9060	0.369535
$872$	6.10028	0.206582
$873$	0	0
$874$	−4.31978	−0.146119
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−33.8341	−1.14250	−0.571249	−	0.820777i	$-0.693541\pi$
$877$	−33.8341	−1.14250	−0.571249	+	0.820777i	$0.693541\pi$
$878$	−5.80561	−0.195930
$879$	0	0
$880$	−8.26117	−0.278484
$881$	−39.0617	−1.31602	−0.658011	−	0.753008i	$-0.728602\pi$
$881$	−39.0617	−1.31602	−0.658011	+	0.753008i	$0.728602\pi$
$882$	0	0
$883$	−20.1056	−0.676606	−0.338303	−	0.941037i	$-0.609853\pi$
$883$	−20.1056	−0.676606	−0.338303	+	0.941037i	$0.609853\pi$
$884$	1.01787	0.0342348
$885$	0	0
$886$	−7.11849	−0.239150
$887$	−44.6880	−1.50047	−0.750237	−	0.661168i	$-0.770061\pi$
$887$	−44.6880	−1.50047	−0.750237	+	0.661168i	$0.770061\pi$
$888$	0	0
$889$	7.93802	0.266233
$890$	2.12144	0.0711109
$891$	0	0
$892$	13.3194	0.445967
$893$	62.2747	2.08394
$894$	0	0
$895$	−22.4030	−0.748850
$896$	−6.32912	−0.211441
$897$	0	0
$898$	−0.106386	−0.00355016
$899$	19.2503	0.642033
$900$	0	0
$901$	1.65962	0.0552901
$902$	−1.01336	−0.0337411
$903$	0	0
$904$	11.2080	0.372771
$905$	−5.69164	−0.189196
$906$	0	0
$907$	31.0586	1.03128	0.515642	−	0.856804i	$-0.327553\pi$
$907$	31.0586	1.03128	0.515642	+	0.856804i	$0.327553\pi$
$908$	−39.2181	−1.30150
$909$	0	0
$910$	−0.520307	−0.0172480
$911$	−24.8454	−0.823165	−0.411583	−	0.911372i	$-0.635024\pi$
$911$	−24.8454	−0.823165	−0.411583	+	0.911372i	$0.635024\pi$
$912$	0	0
$913$	6.90837	0.228634
$914$	−3.97409	−0.131451
$915$	0	0
$916$	41.6003	1.37451
$917$	−13.9276	−0.459929
$918$	0	0
$919$	−15.5187	−0.511914	−0.255957	−	0.966688i	$-0.582391\pi$
$919$	−15.5187	−0.511914	−0.255957	+	0.966688i	$0.582391\pi$
$920$	2.36430	0.0779485
$921$	0	0
$922$	2.70667	0.0891393
$923$	−20.0858	−0.661133
$924$	0	0
$925$	0.739681	0.0243206
$926$	3.55923	0.116964
$927$	0	0
$928$	11.7725	0.386451
$929$	−13.5478	−0.444488	−0.222244	−	0.974991i	$-0.571338\pi$
$929$	−13.5478	−0.444488	−0.222244	+	0.974991i	$0.571338\pi$
$930$	0	0
$931$	7.22851	0.236905
$932$	−10.6546	−0.349002
$933$	0	0
$934$	−6.51827	−0.213284
$935$	−0.461819	−0.0151031
$936$	0	0
$937$	12.8207	0.418833	0.209416	−	0.977827i	$-0.432844\pi$
$937$	12.8207	0.418833	0.209416	+	0.977827i	$0.432844\pi$
$938$	−0.916080	−0.0299111
$939$	0	0
$940$	−16.8538	−0.549709
$941$	25.4370	0.829224	0.414612	−	0.909998i	$-0.363917\pi$
$941$	25.4370	0.829224	0.414612	+	0.909998i	$0.363917\pi$
$942$	0	0
$943$	−6.27248	−0.204260
$944$	−52.4836	−1.70819
$945$	0	0
$946$	−2.44934	−0.0796351
$947$	37.6020	1.22190	0.610950	−	0.791669i	$-0.290788\pi$
$947$	37.6020	1.22190	0.610950	+	0.791669i	$0.290788\pi$
$948$	0	0
$949$	39.9322	1.29625
$950$	−1.51117	−0.0490288
$951$	0	0
$952$	−0.172909	−0.00560402
$953$	39.7536	1.28775	0.643873	−	0.765133i	$-0.277327\pi$
$953$	39.7536	1.28775	0.643873	+	0.765133i	$0.277327\pi$
$954$	0	0
$955$	24.1961	0.782967
$956$	−36.5214	−1.18119
$957$	0	0
$958$	−2.83391	−0.0915595
$959$	2.71734	0.0877475
$960$	0	0
$961$	−15.1333	−0.488170
$962$	0.384861	0.0124084
$963$	0	0
$964$	9.06434	0.291943
$965$	20.1166	0.647578
$966$	0	0
$967$	11.7788	0.378780	0.189390	−	0.981902i	$-0.439349\pi$
$967$	11.7788	0.378780	0.189390	+	0.981902i	$0.439349\pi$
$968$	−5.06185	−0.162694
$969$	0	0
$970$	2.48903	0.0799181
$971$	−53.8520	−1.72819	−0.864096	−	0.503327i	$-0.832109\pi$
$971$	−53.8520	−1.72819	−0.864096	+	0.503327i	$0.832109\pi$
$972$	0	0
$973$	2.04528	0.0655688
$974$	−1.15171	−0.0369031
$975$	0	0
$976$	1.54691	0.0495154
$977$	30.9580	0.990434	0.495217	−	0.868769i	$-0.335089\pi$
$977$	30.9580	0.990434	0.495217	+	0.868769i	$0.335089\pi$
$978$	0	0
$979$	22.4168	0.716444
$980$	−1.95630	−0.0624916
$981$	0	0
$982$	2.14569	0.0684717
$983$	−40.6307	−1.29592	−0.647959	−	0.761675i	$-0.724377\pi$
$983$	−40.6307	−1.29592	−0.647959	+	0.761675i	$0.724377\pi$
$984$	0	0
$985$	−2.57340	−0.0819952
$986$	0.211214	0.00672642
$987$	0	0
$988$	35.1948	1.11970
$989$	−15.1610	−0.482091
$990$	0	0
$991$	−54.5615	−1.73320	−0.866602	−	0.499000i	$-0.833701\pi$
$991$	−54.5615	−1.73320	−0.866602	+	0.499000i	$0.833701\pi$
$992$	9.70329	0.308080
$993$	0	0
$994$	1.68717	0.0535138
$995$	−24.5589	−0.778569
$996$	0	0
$997$	30.1750	0.955652	0.477826	−	0.878454i	$-0.341425\pi$
$997$	30.1750	0.955652	0.477826	+	0.878454i	$0.341425\pi$
$998$	−5.04520	−0.159703
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2835.2.a.q.1.2		4
3.2	odd	2		2835.2.a.l.1.3		4
9.2	odd	6		315.2.i.d.211.2	yes	8
9.4	even	3		945.2.i.c.316.3		8
9.5	odd	6		315.2.i.d.106.2	✓	8
9.7	even	3		945.2.i.c.631.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.i.d.106.2	✓	8	9.5	odd	6
315.2.i.d.211.2	yes	8	9.2	odd	6
945.2.i.c.316.3		8	9.4	even	3
945.2.i.c.631.3		8	9.7	even	3
2835.2.a.l.1.3		4	3.2	odd	2
2835.2.a.q.1.2		4	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:	\( N \)	\(=\)	\( 2835 = 3^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2835.a (trivial)

\(f(q)\)	\(=\)	\(q-0.209057 q^{2} -1.95630 q^{4} +1.00000 q^{5} -1.00000 q^{7} +0.827091 q^{8} +O(q^{10})\)	\(q-0.209057 q^{2} -1.95630 q^{4} +1.00000 q^{5} -1.00000 q^{7} +0.827091 q^{8} -0.209057 q^{10} -2.20906 q^{11} -2.48883 q^{13} +0.209057 q^{14} +3.73968 q^{16} +0.209057 q^{17} +7.22851 q^{19} -1.95630 q^{20} +0.461819 q^{22} +2.85857 q^{23} +1.00000 q^{25} +0.520307 q^{26} +1.95630 q^{28} -4.83274 q^{29} -3.98331 q^{31} -2.43599 q^{32} -0.0437048 q^{34} -1.00000 q^{35} +0.739681 q^{37} -1.51117 q^{38} +0.827091 q^{40} -2.19427 q^{41} -5.30369 q^{43} +4.32157 q^{44} -0.597604 q^{46} +8.61515 q^{47} +1.00000 q^{49} -0.209057 q^{50} +4.86889 q^{52} +7.93863 q^{53} -2.20906 q^{55} -0.827091 q^{56} +1.01032 q^{58} -14.0342 q^{59} +0.413648 q^{61} +0.832738 q^{62} -6.97010 q^{64} -2.48883 q^{65} -4.38197 q^{67} -0.408977 q^{68} +0.209057 q^{70} +8.07039 q^{71} -16.0446 q^{73} -0.154636 q^{74} -14.1411 q^{76} +2.20906 q^{77} -15.7220 q^{79} +3.73968 q^{80} +0.458728 q^{82} -3.12729 q^{83} +0.209057 q^{85} +1.10877 q^{86} -1.82709 q^{88} -10.1477 q^{89} +2.48883 q^{91} -5.59220 q^{92} -1.80106 q^{94} +7.22851 q^{95} -11.9060 q^{97} -0.209057 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + q^{4} + 4 q^{5} - 4 q^{7} - 3 q^{8}+O(q^{10}) \) 4 * q + q^2 + q^4 + 4 * q^5 - 4 * q^7 - 3 * q^8	\( 4 q + q^{2} + q^{4} + 4 q^{5} - 4 q^{7} - 3 q^{8} + q^{10} - 7 q^{11} - 8 q^{13} - q^{14} - 9 q^{16} - q^{17} + 3 q^{19} + q^{20} + 7 q^{22} + 8 q^{23} + 4 q^{25} + 3 q^{26} - q^{28} - q^{29} - 9 q^{34} - 4 q^{35} - 21 q^{37} - 8 q^{38} - 3 q^{40} - 20 q^{41} - 7 q^{43} - 3 q^{44} - 23 q^{46} + 2 q^{47} + 4 q^{49} + q^{50} - 7 q^{52} + 8 q^{53} - 7 q^{55} + 3 q^{56} - 19 q^{58} - 19 q^{59} - 12 q^{61} - 15 q^{62} - 7 q^{64} - 8 q^{65} - 22 q^{67} + q^{68} - q^{70} - 13 q^{71} - 4 q^{73} - 9 q^{74} - 13 q^{76} + 7 q^{77} - 24 q^{79} - 9 q^{80} + 19 q^{83} - q^{85} + 27 q^{86} - q^{88} - 15 q^{89} + 8 q^{91} - 3 q^{92} - 7 q^{94} + 3 q^{95} - 12 q^{97} + q^{98}+O(q^{100}) \) 4 * q + q^2 + q^4 + 4 * q^5 - 4 * q^7 - 3 * q^8 + q^10 - 7 * q^11 - 8 * q^13 - q^14 - 9 * q^16 - q^17 + 3 * q^19 + q^20 + 7 * q^22 + 8 * q^23 + 4 * q^25 + 3 * q^26 - q^28 - q^29 - 9 * q^34 - 4 * q^35 - 21 * q^37 - 8 * q^38 - 3 * q^40 - 20 * q^41 - 7 * q^43 - 3 * q^44 - 23 * q^46 + 2 * q^47 + 4 * q^49 + q^50 - 7 * q^52 + 8 * q^53 - 7 * q^55 + 3 * q^56 - 19 * q^58 - 19 * q^59 - 12 * q^61 - 15 * q^62 - 7 * q^64 - 8 * q^65 - 22 * q^67 + q^68 - q^70 - 13 * q^71 - 4 * q^73 - 9 * q^74 - 13 * q^76 + 7 * q^77 - 24 * q^79 - 9 * q^80 + 19 * q^83 - q^85 + 27 * q^86 - q^88 - 15 * q^89 + 8 * q^91 - 3 * q^92 - 7 * q^94 + 3 * q^95 - 12 * q^97 + q^98

Introduction

L-functions

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