LMFDB - Embedded newform 4527.2.a.k.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4527,2,Mod(1,4527)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4527, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4527.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4527 = 3^{2} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4527.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:	$36.1482769950$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 9x^{8} + 14x^{7} + 27x^{6} - 27x^{5} - 34x^{4} + 14x^{3} + 17x^{2} + x - 1 $ x^10 - 2x^9 - 9x^8 + 14x^7 + 27x^6 - 27x^5 - 34x^4 + 14x^3 + 17x^2 + x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 503)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-2.07227$ of defining polynomial
Character	$\chi$	$=$	4527.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.756417 q^{2} -1.42783 q^{4} -0.386144 q^{5} +0.194914 q^{7} -2.59287 q^{8} +O(q^{10})$	\(q+0.756417 q^{2} -1.42783 q^{4} -0.386144 q^{5} +0.194914 q^{7} -2.59287 q^{8} -0.292086 q^{10} -2.36326 q^{11} -1.22636 q^{13} +0.147436 q^{14} +0.894372 q^{16} +5.04830 q^{17} +4.24460 q^{19} +0.551349 q^{20} -1.78761 q^{22} -1.53457 q^{23} -4.85089 q^{25} -0.927639 q^{26} -0.278304 q^{28} +7.31602 q^{29} +3.33893 q^{31} +5.86226 q^{32} +3.81863 q^{34} -0.0752647 q^{35} -2.17138 q^{37} +3.21069 q^{38} +1.00122 q^{40} +1.04840 q^{41} -1.53067 q^{43} +3.37434 q^{44} -1.16077 q^{46} +1.08422 q^{47} -6.96201 q^{49} -3.66930 q^{50} +1.75104 q^{52} -1.55308 q^{53} +0.912558 q^{55} -0.505386 q^{56} +5.53396 q^{58} -14.8451 q^{59} -5.45056 q^{61} +2.52563 q^{62} +2.64557 q^{64} +0.473551 q^{65} -11.3202 q^{67} -7.20814 q^{68} -0.0569315 q^{70} +11.4236 q^{71} -0.902229 q^{73} -1.64247 q^{74} -6.06059 q^{76} -0.460631 q^{77} -12.7431 q^{79} -0.345356 q^{80} +0.793029 q^{82} -6.10241 q^{83} -1.94937 q^{85} -1.15782 q^{86} +6.12763 q^{88} -6.44432 q^{89} -0.239034 q^{91} +2.19111 q^{92} +0.820122 q^{94} -1.63903 q^{95} -12.9886 q^{97} -5.26618 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 4 q^{2} + 4 q^{4} + q^{5} - 5 q^{7} + 3 q^{8}+O(q^{10}) $ 10 * q + 4 * q^2 + 4 * q^4 + q^5 - 5 * q^7 + 3 * q^8	$ 10 q + 4 q^{2} + 4 q^{4} + q^{5} - 5 q^{7} + 3 q^{8} - 4 q^{10} + 3 q^{11} - 18 q^{13} - q^{14} - 4 q^{16} + 11 q^{17} + 3 q^{20} - 18 q^{22} + 2 q^{23} - 27 q^{25} - 11 q^{26} - 22 q^{28} + 9 q^{29} - 22 q^{31} + 10 q^{32} - 10 q^{34} + 6 q^{35} - 35 q^{37} - 2 q^{38} - 19 q^{40} + 4 q^{41} - 20 q^{43} - 9 q^{44} - q^{46} - 7 q^{47} - 27 q^{49} - 16 q^{50} - 7 q^{52} + 24 q^{53} - 11 q^{55} - 12 q^{56} + 2 q^{58} - 17 q^{59} - 4 q^{61} - 8 q^{62} + 3 q^{64} + 16 q^{65} - 6 q^{67} - 28 q^{68} + 26 q^{70} + q^{71} - 31 q^{73} - 11 q^{74} + 20 q^{76} - 3 q^{77} - 10 q^{79} - 24 q^{80} - 9 q^{82} - 22 q^{83} - 6 q^{85} - 38 q^{86} - 3 q^{88} - q^{89} + 10 q^{91} - 27 q^{92} + 33 q^{94} - 39 q^{95} - 57 q^{97} - 40 q^{98}+O(q^{100}) $ 10 * q + 4 * q^2 + 4 * q^4 + q^5 - 5 * q^7 + 3 * q^8 - 4 * q^10 + 3 * q^11 - 18 * q^13 - q^14 - 4 * q^16 + 11 * q^17 + 3 * q^20 - 18 * q^22 + 2 * q^23 - 27 * q^25 - 11 * q^26 - 22 * q^28 + 9 * q^29 - 22 * q^31 + 10 * q^32 - 10 * q^34 + 6 * q^35 - 35 * q^37 - 2 * q^38 - 19 * q^40 + 4 * q^41 - 20 * q^43 - 9 * q^44 - q^46 - 7 * q^47 - 27 * q^49 - 16 * q^50 - 7 * q^52 + 24 * q^53 - 11 * q^55 - 12 * q^56 + 2 * q^58 - 17 * q^59 - 4 * q^61 - 8 * q^62 + 3 * q^64 + 16 * q^65 - 6 * q^67 - 28 * q^68 + 26 * q^70 + q^71 - 31 * q^73 - 11 * q^74 + 20 * q^76 - 3 * q^77 - 10 * q^79 - 24 * q^80 - 9 * q^82 - 22 * q^83 - 6 * q^85 - 38 * q^86 - 3 * q^88 - q^89 + 10 * q^91 - 27 * q^92 + 33 * q^94 - 39 * q^95 - 57 * q^97 - 40 * 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 $

$2$	0.756417	0.534868	0.267434	−	0.963576i	$-0.413824\pi$
$2$	0.756417	0.534868	0.267434	+	0.963576i	$0.413824\pi$
$3$	0	0
$3$	0	0
$4$	−1.42783	−0.713916
$5$	−0.386144	−0.172689	−0.0863445	−	0.996265i	$-0.527519\pi$
$5$	−0.386144	−0.172689	−0.0863445	+	0.996265i	$0.527519\pi$
$6$	0	0
$7$	0.194914	0.0736704	0.0368352	−	0.999321i	$-0.488272\pi$
$7$	0.194914	0.0736704	0.0368352	+	0.999321i	$0.488272\pi$
$8$	−2.59287	−0.916719
$9$	0	0
$10$	−0.292086	−0.0923657
$11$	−2.36326	−0.712549	−0.356275	−	0.934381i	$-0.615953\pi$
$11$	−2.36326	−0.712549	−0.356275	+	0.934381i	$0.615953\pi$
$12$	0	0
$13$	−1.22636	−0.340131	−0.170065	−	0.985433i	$-0.554398\pi$
$13$	−1.22636	−0.340131	−0.170065	+	0.985433i	$0.554398\pi$
$14$	0.147436	0.0394039
$15$	0	0
$16$	0.894372	0.223593
$17$	5.04830	1.22439	0.612197	−	0.790705i	$-0.290286\pi$
$17$	5.04830	1.22439	0.612197	+	0.790705i	$0.290286\pi$
$18$	0	0
$19$	4.24460	0.973779	0.486890	−	0.873464i	$-0.338131\pi$
$19$	4.24460	0.973779	0.486890	+	0.873464i	$0.338131\pi$
$20$	0.551349	0.123285
$21$	0	0
$22$	−1.78761	−0.381120
$23$	−1.53457	−0.319980	−0.159990	−	0.987119i	$-0.551146\pi$
$23$	−1.53457	−0.319980	−0.159990	+	0.987119i	$0.551146\pi$
$24$	0	0
$25$	−4.85089	−0.970179
$26$	−0.927639	−0.181925
$27$	0	0
$28$	−0.278304	−0.0525945
$29$	7.31602	1.35855	0.679275	−	0.733883i	$-0.262294\pi$
$29$	7.31602	1.35855	0.679275	+	0.733883i	$0.262294\pi$
$30$	0	0
$31$	3.33893	0.599690	0.299845	−	0.953988i	$-0.403065\pi$
$31$	3.33893	0.599690	0.299845	+	0.953988i	$0.403065\pi$
$32$	5.86226	1.03631
$33$	0	0
$34$	3.81863	0.654889
$35$	−0.0752647	−0.0127221
$36$	0	0
$37$	−2.17138	−0.356973	−0.178487	−	0.983942i	$-0.557120\pi$
$37$	−2.17138	−0.356973	−0.178487	+	0.983942i	$0.557120\pi$
$38$	3.21069	0.520843
$39$	0	0
$40$	1.00122	0.158307
$41$	1.04840	0.163733	0.0818663	−	0.996643i	$-0.473912\pi$
$41$	1.04840	0.163733	0.0818663	+	0.996643i	$0.473912\pi$
$42$	0	0
$43$	−1.53067	−0.233425	−0.116712	−	0.993166i	$-0.537236\pi$
$43$	−1.53067	−0.233425	−0.116712	+	0.993166i	$0.537236\pi$
$44$	3.37434	0.508701
$45$	0	0
$46$	−1.16077	−0.171147
$47$	1.08422	0.158150	0.0790748	−	0.996869i	$-0.474803\pi$
$47$	1.08422	0.158150	0.0790748	+	0.996869i	$0.474803\pi$
$48$	0	0
$49$	−6.96201	−0.994573
$50$	−3.66930	−0.518917
$51$	0	0
$52$	1.75104	0.242825
$53$	−1.55308	−0.213332	−0.106666	−	0.994295i	$-0.534017\pi$
$53$	−1.55308	−0.213332	−0.106666	+	0.994295i	$0.534017\pi$
$54$	0	0
$55$	0.912558	0.123049
$56$	−0.505386	−0.0675350
$57$	0	0
$58$	5.53396	0.726645
$59$	−14.8451	−1.93267	−0.966334	−	0.257293i	$-0.917170\pi$
$59$	−14.8451	−1.93267	−0.966334	+	0.257293i	$0.917170\pi$
$60$	0	0
$61$	−5.45056	−0.697873	−0.348936	−	0.937146i	$-0.613457\pi$
$61$	−5.45056	−0.697873	−0.348936	+	0.937146i	$0.613457\pi$
$62$	2.52563	0.320755
$63$	0	0
$64$	2.64557	0.330697
$65$	0.473551	0.0587368
$66$	0	0
$67$	−11.3202	−1.38298	−0.691490	−	0.722386i	$-0.743045\pi$
$67$	−11.3202	−1.38298	−0.691490	+	0.722386i	$0.743045\pi$
$68$	−7.20814	−0.874115
$69$	0	0
$70$	−0.0569315	−0.00680462
$71$	11.4236	1.35573	0.677864	−	0.735187i	$-0.262906\pi$
$71$	11.4236	1.35573	0.677864	+	0.735187i	$0.262906\pi$
$72$	0	0
$73$	−0.902229	−0.105598	−0.0527990	−	0.998605i	$-0.516814\pi$
$73$	−0.902229	−0.105598	−0.0527990	+	0.998605i	$0.516814\pi$
$74$	−1.64247	−0.190933
$75$	0	0
$76$	−6.06059	−0.695197
$77$	−0.460631	−0.0524938
$78$	0	0
$79$	−12.7431	−1.43371	−0.716853	−	0.697225i	$-0.754418\pi$
$79$	−12.7431	−1.43371	−0.716853	+	0.697225i	$0.754418\pi$
$80$	−0.345356	−0.0386120
$81$	0	0
$82$	0.793029	0.0875753
$83$	−6.10241	−0.669826	−0.334913	−	0.942249i	$-0.608707\pi$
$83$	−6.10241	−0.669826	−0.334913	+	0.942249i	$0.608707\pi$
$84$	0	0
$85$	−1.94937	−0.211439
$86$	−1.15782	−0.124851
$87$	0	0
$88$	6.12763	0.653207
$89$	−6.44432	−0.683097	−0.341548	−	0.939864i	$-0.610951\pi$
$89$	−6.44432	−0.683097	−0.341548	+	0.939864i	$0.610951\pi$
$90$	0	0
$91$	−0.239034	−0.0250576
$92$	2.19111	0.228439
$93$	0	0
$94$	0.820122	0.0845892
$95$	−1.63903	−0.168161
$96$	0	0
$97$	−12.9886	−1.31879	−0.659394	−	0.751797i	$-0.729187\pi$
$97$	−12.9886	−1.31879	−0.659394	+	0.751797i	$0.729187\pi$
$98$	−5.26618	−0.531965
$99$	0	0
$100$	6.92626	0.692626
$101$	−6.18139	−0.615072	−0.307536	−	0.951537i	$-0.599504\pi$
$101$	−6.18139	−0.615072	−0.307536	+	0.951537i	$0.599504\pi$
$102$	0	0
$103$	7.78250	0.766833	0.383416	−	0.923576i	$-0.374747\pi$
$103$	7.78250	0.766833	0.383416	+	0.923576i	$0.374747\pi$
$104$	3.17979	0.311804
$105$	0	0
$106$	−1.17477	−0.114104
$107$	8.18217	0.791000	0.395500	−	0.918466i	$-0.370571\pi$
$107$	8.18217	0.791000	0.395500	+	0.918466i	$0.370571\pi$
$108$	0	0
$109$	−5.32486	−0.510029	−0.255015	−	0.966937i	$-0.582080\pi$
$109$	−5.32486	−0.510029	−0.255015	+	0.966937i	$0.582080\pi$
$110$	0.690275	0.0658151
$111$	0	0
$112$	0.174325	0.0164722
$113$	3.60573	0.339199	0.169599	−	0.985513i	$-0.445753\pi$
$113$	3.60573	0.339199	0.169599	+	0.985513i	$0.445753\pi$
$114$	0	0
$115$	0.592565	0.0552570
$116$	−10.4461	−0.969892
$117$	0	0
$118$	−11.2291	−1.03372
$119$	0.983983	0.0902016
$120$	0	0
$121$	−5.41501	−0.492274
$122$	−4.12290	−0.373270
$123$	0	0
$124$	−4.76744	−0.428129
$125$	3.80386	0.340228
$126$	0	0
$127$	−17.8352	−1.58262	−0.791308	−	0.611418i	$-0.790599\pi$
$127$	−17.8352	−1.58262	−0.791308	+	0.611418i	$0.790599\pi$
$128$	−9.72337	−0.859432
$129$	0	0
$130$	0.358202	0.0314164
$131$	12.1841	1.06453	0.532267	−	0.846577i	$-0.321340\pi$
$131$	12.1841	1.06453	0.532267	+	0.846577i	$0.321340\pi$
$132$	0	0
$133$	0.827331	0.0717387
$134$	−8.56278	−0.739711
$135$	0	0
$136$	−13.0896	−1.12242
$137$	13.9009	1.18764	0.593819	−	0.804599i	$-0.297620\pi$
$137$	13.9009	1.18764	0.593819	+	0.804599i	$0.297620\pi$
$138$	0	0
$139$	4.99920	0.424027	0.212013	−	0.977267i	$-0.431998\pi$
$139$	4.99920	0.424027	0.212013	+	0.977267i	$0.431998\pi$
$140$	0.107465	0.00908249
$141$	0	0
$142$	8.64099	0.725135
$143$	2.89820	0.242360
$144$	0	0
$145$	−2.82504	−0.234607
$146$	−0.682462	−0.0564809
$147$	0	0
$148$	3.10037	0.254849
$149$	−17.4937	−1.43314	−0.716570	−	0.697515i	$-0.754289\pi$
$149$	−17.4937	−1.43314	−0.716570	+	0.697515i	$0.754289\pi$
$150$	0	0
$151$	0.480306	0.0390867	0.0195434	−	0.999809i	$-0.493779\pi$
$151$	0.480306	0.0390867	0.0195434	+	0.999809i	$0.493779\pi$
$152$	−11.0057	−0.892682
$153$	0	0
$154$	−0.348429	−0.0280772
$155$	−1.28931	−0.103560
$156$	0	0
$157$	0.117389	0.00936869	0.00468434	−	0.999989i	$-0.498509\pi$
$157$	0.117389	0.00936869	0.00468434	+	0.999989i	$0.498509\pi$
$158$	−9.63907	−0.766843
$159$	0	0
$160$	−2.26368	−0.178959
$161$	−0.299108	−0.0235730
$162$	0	0
$163$	−12.5273	−0.981210	−0.490605	−	0.871382i	$-0.663224\pi$
$163$	−12.5273	−0.981210	−0.490605	+	0.871382i	$0.663224\pi$
$164$	−1.49694	−0.116891
$165$	0	0
$166$	−4.61597	−0.358268
$167$	0.744061	0.0575772	0.0287886	−	0.999586i	$-0.490835\pi$
$167$	0.744061	0.0575772	0.0287886	+	0.999586i	$0.490835\pi$
$168$	0	0
$169$	−11.4960	−0.884311
$170$	−1.47454	−0.113092
$171$	0	0
$172$	2.18554	0.166646
$173$	−11.6073	−0.882484	−0.441242	−	0.897388i	$-0.645462\pi$
$173$	−11.6073	−0.882484	−0.441242	+	0.897388i	$0.645462\pi$
$174$	0	0
$175$	−0.945505	−0.0714734
$176$	−2.11363	−0.159321
$177$	0	0
$178$	−4.87460	−0.365367
$179$	21.1974	1.58437	0.792183	−	0.610284i	$-0.208945\pi$
$179$	21.1974	1.58437	0.792183	+	0.610284i	$0.208945\pi$
$180$	0	0
$181$	11.9508	0.888299	0.444150	−	0.895953i	$-0.353506\pi$
$181$	11.9508	0.888299	0.444150	+	0.895953i	$0.353506\pi$
$182$	−0.180809	−0.0134025
$183$	0	0
$184$	3.97894	0.293332
$185$	0.838467	0.0616453
$186$	0	0
$187$	−11.9304	−0.872441
$188$	−1.54808	−0.112906
$189$	0	0
$190$	−1.23979	−0.0899438
$191$	−24.2243	−1.75281	−0.876406	−	0.481574i	$-0.840065\pi$
$191$	−24.2243	−1.75281	−0.876406	+	0.481574i	$0.840065\pi$
$192$	0	0
$193$	−5.78545	−0.416446	−0.208223	−	0.978081i	$-0.566768\pi$
$193$	−5.78545	−0.416446	−0.208223	+	0.978081i	$0.566768\pi$
$194$	−9.82478	−0.705378
$195$	0	0
$196$	9.94058	0.710042
$197$	−4.06793	−0.289828	−0.144914	−	0.989444i	$-0.546291\pi$
$197$	−4.06793	−0.289828	−0.144914	+	0.989444i	$0.546291\pi$
$198$	0	0
$199$	−11.4204	−0.809571	−0.404785	−	0.914412i	$-0.632654\pi$
$199$	−11.4204	−0.809571	−0.404785	+	0.914412i	$0.632654\pi$
$200$	12.5777	0.889381
$201$	0	0
$202$	−4.67571	−0.328982
$203$	1.42599	0.100085
$204$	0	0
$205$	−0.404834	−0.0282748
$206$	5.88682	0.410154
$207$	0	0
$208$	−1.09682	−0.0760509
$209$	−10.0311	−0.693866
$210$	0	0
$211$	18.0941	1.24565	0.622824	−	0.782362i	$-0.285985\pi$
$211$	18.0941	1.24565	0.622824	+	0.782362i	$0.285985\pi$
$212$	2.21753	0.152301
$213$	0	0
$214$	6.18913	0.423080
$215$	0.591058	0.0403098
$216$	0	0
$217$	0.650804	0.0441794
$218$	−4.02782	−0.272798
$219$	0	0
$220$	−1.30298	−0.0878469
$221$	−6.19103	−0.416454
$222$	0	0
$223$	−21.1665	−1.41741	−0.708706	−	0.705504i	$-0.750721\pi$
$223$	−21.1665	−1.41741	−0.708706	+	0.705504i	$0.750721\pi$
$224$	1.14263	0.0763455
$225$	0	0
$226$	2.72744	0.181426
$227$	13.3828	0.888248	0.444124	−	0.895965i	$-0.353515\pi$
$227$	13.3828	0.888248	0.444124	+	0.895965i	$0.353515\pi$
$228$	0	0
$229$	−12.5484	−0.829225	−0.414612	−	0.909998i	$-0.636083\pi$
$229$	−12.5484	−0.829225	−0.414612	+	0.909998i	$0.636083\pi$
$230$	0.448226	0.0295552
$231$	0	0
$232$	−18.9695	−1.24541
$233$	−15.6538	−1.02551	−0.512757	−	0.858534i	$-0.671376\pi$
$233$	−15.6538	−1.02551	−0.512757	+	0.858534i	$0.671376\pi$
$234$	0	0
$235$	−0.418665	−0.0273107
$236$	21.1963	1.37976
$237$	0	0
$238$	0.744302	0.0482459
$239$	−3.45659	−0.223588	−0.111794	−	0.993731i	$-0.535660\pi$
$239$	−3.45659	−0.223588	−0.111794	+	0.993731i	$0.535660\pi$
$240$	0	0
$241$	28.7843	1.85416	0.927080	−	0.374864i	$-0.122311\pi$
$241$	28.7843	1.85416	0.927080	+	0.374864i	$0.122311\pi$
$242$	−4.09601	−0.263301
$243$	0	0
$244$	7.78249	0.498223
$245$	2.68834	0.171752
$246$	0	0
$247$	−5.20541	−0.331212
$248$	−8.65743	−0.549747
$249$	0	0
$250$	2.87731	0.181977
$251$	1.82824	0.115397	0.0576986	−	0.998334i	$-0.481624\pi$
$251$	1.82824	0.115397	0.0576986	+	0.998334i	$0.481624\pi$
$252$	0	0
$253$	3.62658	0.228001
$254$	−13.4908	−0.846490
$255$	0	0
$256$	−12.6461	−0.790380
$257$	−3.65882	−0.228231	−0.114116	−	0.993467i	$-0.536403\pi$
$257$	−3.65882	−0.228231	−0.114116	+	0.993467i	$0.536403\pi$
$258$	0	0
$259$	−0.423232	−0.0262983
$260$	−0.676152	−0.0419332
$261$	0	0
$262$	9.21630	0.569385
$263$	13.1581	0.811362	0.405681	−	0.914015i	$-0.367034\pi$
$263$	13.1581	0.811362	0.405681	+	0.914015i	$0.367034\pi$
$264$	0	0
$265$	0.599712	0.0368400
$266$	0.625807	0.0383707
$267$	0	0
$268$	16.1633	0.987332
$269$	30.0225	1.83050	0.915251	−	0.402885i	$-0.131992\pi$
$269$	30.0225	1.83050	0.915251	+	0.402885i	$0.131992\pi$
$270$	0	0
$271$	−9.04876	−0.549673	−0.274837	−	0.961491i	$-0.588624\pi$
$271$	−9.04876	−0.549673	−0.274837	+	0.961491i	$0.588624\pi$
$272$	4.51506	0.273766
$273$	0	0
$274$	10.5149	0.635229
$275$	11.4639	0.691300
$276$	0	0
$277$	10.3482	0.621762	0.310881	−	0.950449i	$-0.399376\pi$
$277$	10.3482	0.621762	0.310881	+	0.950449i	$0.399376\pi$
$278$	3.78148	0.226798
$279$	0	0
$280$	0.195152	0.0116625
$281$	19.3686	1.15543	0.577716	−	0.816238i	$-0.303944\pi$
$281$	19.3686	1.15543	0.577716	+	0.816238i	$0.303944\pi$
$282$	0	0
$283$	9.72949	0.578358	0.289179	−	0.957275i	$-0.406618\pi$
$283$	9.72949	0.578358	0.289179	+	0.957275i	$0.406618\pi$
$284$	−16.3109	−0.967876
$285$	0	0
$286$	2.19225	0.129631
$287$	0.204347	0.0120622
$288$	0	0
$289$	8.48538	0.499140
$290$	−2.13691	−0.125484
$291$	0	0
$292$	1.28823	0.0753881
$293$	−26.2321	−1.53249	−0.766247	−	0.642546i	$-0.777878\pi$
$293$	−26.2321	−1.53249	−0.766247	+	0.642546i	$0.777878\pi$
$294$	0	0
$295$	5.73235	0.333750
$296$	5.63012	0.327244
$297$	0	0
$298$	−13.2325	−0.766540
$299$	1.88193	0.108835
$300$	0	0
$301$	−0.298348	−0.0171965
$302$	0.363312	0.0209062
$303$	0	0
$304$	3.79626	0.217730
$305$	2.10470	0.120515
$306$	0	0
$307$	−13.7097	−0.782455	−0.391228	−	0.920294i	$-0.627950\pi$
$307$	−13.7097	−0.782455	−0.391228	+	0.920294i	$0.627950\pi$
$308$	0.657704	0.0374762
$309$	0	0
$310$	−0.975257	−0.0553909
$311$	−30.9043	−1.75242	−0.876212	−	0.481926i	$-0.839937\pi$
$311$	−30.9043	−1.75242	−0.876212	+	0.481926i	$0.839937\pi$
$312$	0	0
$313$	−28.2287	−1.59558	−0.797790	−	0.602935i	$-0.793998\pi$
$313$	−28.2287	−1.59558	−0.797790	+	0.602935i	$0.793998\pi$
$314$	0.0887953	0.00501101
$315$	0	0
$316$	18.1949	1.02355
$317$	7.55793	0.424496	0.212248	−	0.977216i	$-0.431922\pi$
$317$	7.55793	0.424496	0.212248	+	0.977216i	$0.431922\pi$
$318$	0	0
$319$	−17.2896	−0.968034
$320$	−1.02157	−0.0571077
$321$	0	0
$322$	−0.226251	−0.0126085
$323$	21.4281	1.19229
$324$	0	0
$325$	5.94894	0.329988
$326$	−9.47583	−0.524818
$327$	0	0
$328$	−2.71837	−0.150097
$329$	0.211329	0.0116509
$330$	0	0
$331$	20.4225	1.12252	0.561260	−	0.827640i	$-0.310317\pi$
$331$	20.4225	1.12252	0.561260	+	0.827640i	$0.310317\pi$
$332$	8.71322	0.478200
$333$	0	0
$334$	0.562821	0.0307962
$335$	4.37122	0.238825
$336$	0	0
$337$	16.7603	0.912994	0.456497	−	0.889725i	$-0.349104\pi$
$337$	16.7603	0.912994	0.456497	+	0.889725i	$0.349104\pi$
$338$	−8.69581	−0.472990
$339$	0	0
$340$	2.78338	0.150950
$341$	−7.89077	−0.427309
$342$	0	0
$343$	−2.72138	−0.146941
$344$	3.96882	0.213985
$345$	0	0
$346$	−8.77994	−0.472012
$347$	−14.1418	−0.759171	−0.379586	−	0.925157i	$-0.623933\pi$
$347$	−14.1418	−0.759171	−0.379586	+	0.925157i	$0.623933\pi$
$348$	0	0
$349$	−22.1523	−1.18579	−0.592893	−	0.805282i	$-0.702014\pi$
$349$	−22.1523	−1.18579	−0.592893	+	0.805282i	$0.702014\pi$
$350$	−0.715196	−0.0382288
$351$	0	0
$352$	−13.8540	−0.738423
$353$	−17.2209	−0.916574	−0.458287	−	0.888804i	$-0.651537\pi$
$353$	−17.2209	−0.916574	−0.458287	+	0.888804i	$0.651537\pi$
$354$	0	0
$355$	−4.41114	−0.234119
$356$	9.20141	0.487674
$357$	0	0
$358$	16.0341	0.847426
$359$	−21.9672	−1.15938	−0.579692	−	0.814836i	$-0.696827\pi$
$359$	−21.9672	−1.15938	−0.579692	+	0.814836i	$0.696827\pi$
$360$	0	0
$361$	−0.983329	−0.0517541
$362$	9.03983	0.475123
$363$	0	0
$364$	0.341300	0.0178890
$365$	0.348390	0.0182356
$366$	0	0
$367$	−13.1841	−0.688202	−0.344101	−	0.938933i	$-0.611816\pi$
$367$	−13.1841	−0.688202	−0.344101	+	0.938933i	$0.611816\pi$
$368$	−1.37248	−0.0715452
$369$	0	0
$370$	0.634231	0.0329721
$371$	−0.302716	−0.0157162
$372$	0	0
$373$	−35.2516	−1.82526	−0.912630	−	0.408787i	$-0.865952\pi$
$373$	−35.2516	−1.82526	−0.912630	+	0.408787i	$0.865952\pi$
$374$	−9.02440	−0.466641
$375$	0	0
$376$	−2.81124	−0.144979
$377$	−8.97207	−0.462085
$378$	0	0
$379$	23.6003	1.21227	0.606134	−	0.795363i	$-0.292720\pi$
$379$	23.6003	1.21227	0.606134	+	0.795363i	$0.292720\pi$
$380$	2.34026	0.120053
$381$	0	0
$382$	−18.3237	−0.937522
$383$	13.1311	0.670970	0.335485	−	0.942046i	$-0.391100\pi$
$383$	13.1311	0.670970	0.335485	+	0.942046i	$0.391100\pi$
$384$	0	0
$385$	0.177870	0.00906509
$386$	−4.37622	−0.222744
$387$	0	0
$388$	18.5455	0.941505
$389$	−29.2216	−1.48159	−0.740797	−	0.671729i	$-0.765552\pi$
$389$	−29.2216	−1.48159	−0.740797	+	0.671729i	$0.765552\pi$
$390$	0	0
$391$	−7.74697	−0.391781
$392$	18.0516	0.911743
$393$	0	0
$394$	−3.07706	−0.155020
$395$	4.92065	0.247585
$396$	0	0
$397$	26.2129	1.31559	0.657795	−	0.753197i	$-0.271489\pi$
$397$	26.2129	1.31559	0.657795	+	0.753197i	$0.271489\pi$
$398$	−8.63859	−0.433013
$399$	0	0
$400$	−4.33850	−0.216925
$401$	24.3910	1.21803	0.609014	−	0.793160i	$-0.291565\pi$
$401$	24.3910	1.21803	0.609014	+	0.793160i	$0.291565\pi$
$402$	0	0
$403$	−4.09473	−0.203973
$404$	8.82599	0.439110
$405$	0	0
$406$	1.07864	0.0535322
$407$	5.13154	0.254361
$408$	0	0
$409$	13.2071	0.653051	0.326526	−	0.945188i	$-0.394122\pi$
$409$	13.2071	0.653051	0.326526	+	0.945188i	$0.394122\pi$
$410$	−0.306223	−0.0151233
$411$	0	0
$412$	−11.1121	−0.547454
$413$	−2.89351	−0.142380
$414$	0	0
$415$	2.35641	0.115672
$416$	−7.18924	−0.352481
$417$	0	0
$418$	−7.58770	−0.371126
$419$	−15.8597	−0.774799	−0.387400	−	0.921912i	$-0.626627\pi$
$419$	−15.8597	−0.774799	−0.387400	+	0.921912i	$0.626627\pi$
$420$	0	0
$421$	−4.53957	−0.221245	−0.110622	−	0.993863i	$-0.535284\pi$
$421$	−4.53957	−0.221245	−0.110622	+	0.993863i	$0.535284\pi$
$422$	13.6867	0.666257
$423$	0	0
$424$	4.02693	0.195565
$425$	−24.4888	−1.18788
$426$	0	0
$427$	−1.06239	−0.0514126
$428$	−11.6828	−0.564708
$429$	0	0
$430$	0.447087	0.0215604
$431$	24.0460	1.15825	0.579126	−	0.815238i	$-0.303394\pi$
$431$	24.0460	1.15825	0.579126	+	0.815238i	$0.303394\pi$
$432$	0	0
$433$	−20.8069	−0.999914	−0.499957	−	0.866050i	$-0.666651\pi$
$433$	−20.8069	−0.999914	−0.499957	+	0.866050i	$0.666651\pi$
$434$	0.492279	0.0236302
$435$	0	0
$436$	7.60301	0.364118
$437$	−6.51364	−0.311590
$438$	0	0
$439$	−27.0822	−1.29256	−0.646282	−	0.763098i	$-0.723677\pi$
$439$	−27.0822	−1.29256	−0.646282	+	0.763098i	$0.723677\pi$
$440$	−2.36615	−0.112802
$441$	0	0
$442$	−4.68301	−0.222748
$443$	−26.1108	−1.24056	−0.620282	−	0.784379i	$-0.712982\pi$
$443$	−26.1108	−1.24056	−0.620282	+	0.784379i	$0.712982\pi$
$444$	0	0
$445$	2.48844	0.117963
$446$	−16.0107	−0.758129
$447$	0	0
$448$	0.515658	0.0243626
$449$	−18.4224	−0.869407	−0.434703	−	0.900574i	$-0.643147\pi$
$449$	−18.4224	−0.869407	−0.434703	+	0.900574i	$0.643147\pi$
$450$	0	0
$451$	−2.47764	−0.116668
$452$	−5.14838	−0.242159
$453$	0	0
$454$	10.1230	0.475095
$455$	0.0923016	0.00432716
$456$	0	0
$457$	−3.55147	−0.166131	−0.0830654	−	0.996544i	$-0.526471\pi$
$457$	−3.55147	−0.166131	−0.0830654	+	0.996544i	$0.526471\pi$
$458$	−9.49186	−0.443526
$459$	0	0
$460$	−0.846084	−0.0394489
$461$	2.86665	0.133513	0.0667565	−	0.997769i	$-0.478735\pi$
$461$	2.86665	0.133513	0.0667565	+	0.997769i	$0.478735\pi$
$462$	0	0
$463$	30.5287	1.41879	0.709395	−	0.704811i	$-0.248968\pi$
$463$	30.5287	1.41879	0.709395	+	0.704811i	$0.248968\pi$
$464$	6.54324	0.303762
$465$	0	0
$466$	−11.8408	−0.548515
$467$	−18.7000	−0.865332	−0.432666	−	0.901554i	$-0.642427\pi$
$467$	−18.7000	−0.865332	−0.432666	+	0.901554i	$0.642427\pi$
$468$	0	0
$469$	−2.20646	−0.101885
$470$	−0.316685	−0.0146076
$471$	0	0
$472$	38.4914	1.77171
$473$	3.61736	0.166326
$474$	0	0
$475$	−20.5901	−0.944740
$476$	−1.40496	−0.0643964
$477$	0	0
$478$	−2.61463	−0.119590
$479$	−7.50976	−0.343130	−0.171565	−	0.985173i	$-0.554882\pi$
$479$	−7.50976	−0.343130	−0.171565	+	0.985173i	$0.554882\pi$
$480$	0	0
$481$	2.66289	0.121418
$482$	21.7729	0.991730
$483$	0	0
$484$	7.73173	0.351442
$485$	5.01546	0.227740
$486$	0	0
$487$	32.6786	1.48081	0.740404	−	0.672162i	$-0.234634\pi$
$487$	32.6786	1.48081	0.740404	+	0.672162i	$0.234634\pi$
$488$	14.1326	0.639753
$489$	0	0
$490$	2.03351	0.0918644
$491$	−2.81716	−0.127137	−0.0635683	−	0.997977i	$-0.520248\pi$
$491$	−2.81716	−0.127137	−0.0635683	+	0.997977i	$0.520248\pi$
$492$	0	0
$493$	36.9335	1.66340
$494$	−3.93746	−0.177155
$495$	0	0
$496$	2.98625	0.134087
$497$	2.22661	0.0998770
$498$	0	0
$499$	29.0591	1.30086	0.650432	−	0.759564i	$-0.274588\pi$
$499$	29.0591	1.30086	0.650432	+	0.759564i	$0.274588\pi$
$500$	−5.43128	−0.242894
$501$	0	0
$502$	1.38291	0.0617223
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	2.38691	0.106216
$506$	2.74321	0.121951
$507$	0	0
$508$	25.4656	1.12986
$509$	−7.58042	−0.335996	−0.167998	−	0.985787i	$-0.553730\pi$
$509$	−7.58042	−0.335996	−0.167998	+	0.985787i	$0.553730\pi$
$510$	0	0
$511$	−0.175857	−0.00777944
$512$	9.88103	0.436684
$513$	0	0
$514$	−2.76760	−0.122074
$515$	−3.00517	−0.132423
$516$	0	0
$517$	−2.56229	−0.112689
$518$	−0.320140	−0.0140661
$519$	0	0
$520$	−1.22786	−0.0538451
$521$	8.90148	0.389981	0.194990	−	0.980805i	$-0.437532\pi$
$521$	8.90148	0.389981	0.194990	+	0.980805i	$0.437532\pi$
$522$	0	0
$523$	37.7311	1.64986	0.824932	−	0.565232i	$-0.191213\pi$
$523$	37.7311	1.64986	0.824932	+	0.565232i	$0.191213\pi$
$524$	−17.3969	−0.759988
$525$	0	0
$526$	9.95301	0.433972
$527$	16.8560	0.734257
$528$	0	0
$529$	−20.6451	−0.897613
$530$	0.453632	0.0197045
$531$	0	0
$532$	−1.18129	−0.0512154
$533$	−1.28572	−0.0556905
$534$	0	0
$535$	−3.15950	−0.136597
$536$	29.3518	1.26780
$537$	0	0
$538$	22.7095	0.979076
$539$	16.4530	0.708682
$540$	0	0
$541$	11.6293	0.499981	0.249990	−	0.968248i	$-0.419573\pi$
$541$	11.6293	0.499981	0.249990	+	0.968248i	$0.419573\pi$
$542$	−6.84464	−0.294003
$543$	0	0
$544$	29.5945	1.26885
$545$	2.05616	0.0880764
$546$	0	0
$547$	−13.4958	−0.577040	−0.288520	−	0.957474i	$-0.593163\pi$
$547$	−13.4958	−0.577040	−0.288520	+	0.957474i	$0.593163\pi$
$548$	−19.8482	−0.847874
$549$	0	0
$550$	8.67150	0.369754
$551$	31.0536	1.32293
$552$	0	0
$553$	−2.48379	−0.105622
$554$	7.82754	0.332560
$555$	0	0
$556$	−7.13803	−0.302720
$557$	34.2002	1.44911	0.724554	−	0.689218i	$-0.242046\pi$
$557$	34.2002	1.44911	0.724554	+	0.689218i	$0.242046\pi$
$558$	0	0
$559$	1.87715	0.0793949
$560$	−0.0673146	−0.00284456
$561$	0	0
$562$	14.6507	0.618003
$563$	−25.6005	−1.07893	−0.539467	−	0.842007i	$-0.681374\pi$
$563$	−25.6005	−1.07893	−0.539467	+	0.842007i	$0.681374\pi$
$564$	0	0
$565$	−1.39233	−0.0585758
$566$	7.35956	0.309345
$567$	0	0
$568$	−29.6199	−1.24282
$569$	17.2311	0.722363	0.361182	−	0.932495i	$-0.382373\pi$
$569$	17.2311	0.722363	0.361182	+	0.932495i	$0.382373\pi$
$570$	0	0
$571$	2.79325	0.116894	0.0584470	−	0.998291i	$-0.481385\pi$
$571$	2.79325	0.116894	0.0584470	+	0.998291i	$0.481385\pi$
$572$	−4.13815	−0.173025
$573$	0	0
$574$	0.154572	0.00645171
$575$	7.44403	0.310438
$576$	0	0
$577$	−40.1143	−1.66998	−0.834991	−	0.550264i	$-0.814527\pi$
$577$	−40.1143	−1.66998	−0.834991	+	0.550264i	$0.814527\pi$
$578$	6.41849	0.266974
$579$	0	0
$580$	4.03368	0.167490
$581$	−1.18944	−0.0493464
$582$	0	0
$583$	3.67032	0.152009
$584$	2.33936	0.0968036
$585$	0	0
$586$	−19.8424	−0.819682
$587$	38.5445	1.59090	0.795451	−	0.606018i	$-0.207234\pi$
$587$	38.5445	1.59090	0.795451	+	0.606018i	$0.207234\pi$
$588$	0	0
$589$	14.1725	0.583966
$590$	4.33605	0.178512
$591$	0	0
$592$	−1.94202	−0.0798167
$593$	25.0372	1.02816	0.514078	−	0.857744i	$-0.328134\pi$
$593$	25.0372	1.02816	0.514078	+	0.857744i	$0.328134\pi$
$594$	0	0
$595$	−0.379959	−0.0155768
$596$	24.9781	1.02314
$597$	0	0
$598$	1.42353	0.0582123
$599$	23.0220	0.940655	0.470328	−	0.882492i	$-0.344136\pi$
$599$	23.0220	0.940655	0.470328	+	0.882492i	$0.344136\pi$
$600$	0	0
$601$	−33.9451	−1.38465	−0.692326	−	0.721585i	$-0.743414\pi$
$601$	−33.9451	−1.38465	−0.692326	+	0.721585i	$0.743414\pi$
$602$	−0.225675	−0.00919784
$603$	0	0
$604$	−0.685796	−0.0279046
$605$	2.09097	0.0850102
$606$	0	0
$607$	24.3039	0.986467	0.493233	−	0.869897i	$-0.335815\pi$
$607$	24.3039	0.986467	0.493233	+	0.869897i	$0.335815\pi$
$608$	24.8830	1.00914
$609$	0	0
$610$	1.59203	0.0644595
$611$	−1.32964	−0.0537916
$612$	0	0
$613$	34.5604	1.39588	0.697940	−	0.716156i	$-0.254100\pi$
$613$	34.5604	1.39588	0.697940	+	0.716156i	$0.254100\pi$
$614$	−10.3703	−0.418510
$615$	0	0
$616$	1.19436	0.0481220
$617$	−34.3607	−1.38331	−0.691655	−	0.722228i	$-0.743118\pi$
$617$	−34.3607	−1.38331	−0.691655	+	0.722228i	$0.743118\pi$
$618$	0	0
$619$	1.08451	0.0435902	0.0217951	−	0.999762i	$-0.493062\pi$
$619$	1.08451	0.0435902	0.0217951	+	0.999762i	$0.493062\pi$
$620$	1.84092	0.0739331
$621$	0	0
$622$	−23.3766	−0.937315
$623$	−1.25609	−0.0503240
$624$	0	0
$625$	22.7856	0.911425
$626$	−21.3527	−0.853425
$627$	0	0
$628$	−0.167612	−0.00668846
$629$	−10.9618	−0.437076
$630$	0	0
$631$	−6.03426	−0.240220	−0.120110	−	0.992761i	$-0.538325\pi$
$631$	−6.03426	−0.240220	−0.120110	+	0.992761i	$0.538325\pi$
$632$	33.0411	1.31430
$633$	0	0
$634$	5.71695	0.227049
$635$	6.88695	0.273300
$636$	0	0
$637$	8.53792	0.338285
$638$	−13.0782	−0.517770
$639$	0	0
$640$	3.75462	0.148414
$641$	22.0716	0.871778	0.435889	−	0.900001i	$-0.356434\pi$
$641$	22.0716	0.871778	0.435889	+	0.900001i	$0.356434\pi$
$642$	0	0
$643$	−40.6373	−1.60258	−0.801290	−	0.598277i	$-0.795852\pi$
$643$	−40.6373	−1.60258	−0.801290	+	0.598277i	$0.795852\pi$
$644$	0.427077	0.0168292
$645$	0	0
$646$	16.2086	0.637717
$647$	−22.9719	−0.903118	−0.451559	−	0.892241i	$-0.649132\pi$
$647$	−22.9719	−0.903118	−0.451559	+	0.892241i	$0.649132\pi$
$648$	0	0
$649$	35.0828	1.37712
$650$	4.49988	0.176500
$651$	0	0
$652$	17.8868	0.700502
$653$	39.0459	1.52798	0.763992	−	0.645226i	$-0.223237\pi$
$653$	39.0459	1.52798	0.763992	+	0.645226i	$0.223237\pi$
$654$	0	0
$655$	−4.70483	−0.183833
$656$	0.937660	0.0366095
$657$	0	0
$658$	0.159853	0.00623172
$659$	26.6054	1.03640	0.518199	−	0.855260i	$-0.326603\pi$
$659$	26.6054	1.03640	0.518199	+	0.855260i	$0.326603\pi$
$660$	0	0
$661$	−39.3661	−1.53116	−0.765582	−	0.643338i	$-0.777549\pi$
$661$	−39.3661	−1.53116	−0.765582	+	0.643338i	$0.777549\pi$
$662$	15.4479	0.600400
$663$	0	0
$664$	15.8228	0.614042
$665$	−0.319469	−0.0123885
$666$	0	0
$667$	−11.2269	−0.434709
$668$	−1.06239	−0.0411053
$669$	0	0
$670$	3.30647	0.127740
$671$	12.8811	0.497269
$672$	0	0
$673$	−13.4145	−0.517090	−0.258545	−	0.965999i	$-0.583243\pi$
$673$	−13.4145	−0.517090	−0.258545	+	0.965999i	$0.583243\pi$
$674$	12.6778	0.488331
$675$	0	0
$676$	16.4144	0.631324
$677$	2.13435	0.0820299	0.0410149	−	0.999159i	$-0.486941\pi$
$677$	2.13435	0.0820299	0.0410149	+	0.999159i	$0.486941\pi$
$678$	0	0
$679$	−2.53165	−0.0971557
$680$	5.05448	0.193830
$681$	0	0
$682$	−5.96871	−0.228554
$683$	−21.3544	−0.817105	−0.408552	−	0.912735i	$-0.633966\pi$
$683$	−21.3544	−0.817105	−0.408552	+	0.912735i	$0.633966\pi$
$684$	0	0
$685$	−5.36777	−0.205092
$686$	−2.05850	−0.0785940
$687$	0	0
$688$	−1.36899	−0.0521921
$689$	1.90463	0.0725606
$690$	0	0
$691$	13.8167	0.525614	0.262807	−	0.964848i	$-0.415352\pi$
$691$	13.8167	0.525614	0.262807	+	0.964848i	$0.415352\pi$
$692$	16.5732	0.630020
$693$	0	0
$694$	−10.6971	−0.406056
$695$	−1.93041	−0.0732247
$696$	0	0
$697$	5.29265	0.200473
$698$	−16.7564	−0.634238
$699$	0	0
$700$	1.35002	0.0510260
$701$	−45.1209	−1.70419	−0.852097	−	0.523385i	$-0.824669\pi$
$701$	−45.1209	−1.70419	−0.852097	+	0.523385i	$0.824669\pi$
$702$	0	0
$703$	−9.21666	−0.347613
$704$	−6.25217	−0.235638
$705$	0	0
$706$	−13.0262	−0.490246
$707$	−1.20484	−0.0453126
$708$	0	0
$709$	−37.6350	−1.41341	−0.706706	−	0.707507i	$-0.749820\pi$
$709$	−37.6350	−1.41341	−0.706706	+	0.707507i	$0.749820\pi$
$710$	−3.33667	−0.125223
$711$	0	0
$712$	16.7093	0.626208
$713$	−5.12383	−0.191889
$714$	0	0
$715$	−1.11912	−0.0418529
$716$	−30.2663	−1.13110
$717$	0	0
$718$	−16.6164	−0.620117
$719$	34.9860	1.30476	0.652380	−	0.757892i	$-0.273771\pi$
$719$	34.9860	1.30476	0.652380	+	0.757892i	$0.273771\pi$
$720$	0	0
$721$	1.51691	0.0564929
$722$	−0.743807	−0.0276816
$723$	0	0
$724$	−17.0638	−0.634171
$725$	−35.4892	−1.31804
$726$	0	0
$727$	0.0594046	0.00220319	0.00110160	−	0.999999i	$-0.499649\pi$
$727$	0.0594046	0.00220319	0.00110160	+	0.999999i	$0.499649\pi$
$728$	0.619784	0.0229707
$729$	0	0
$730$	0.263529	0.00975363
$731$	−7.72727	−0.285804
$732$	0	0
$733$	−23.6914	−0.875060	−0.437530	−	0.899204i	$-0.644147\pi$
$733$	−23.6914	−0.875060	−0.437530	+	0.899204i	$0.644147\pi$
$734$	−9.97265	−0.368097
$735$	0	0
$736$	−8.99605	−0.331599
$737$	26.7525	0.985441
$738$	0	0
$739$	−13.2539	−0.487551	−0.243776	−	0.969832i	$-0.578386\pi$
$739$	−13.2539	−0.487551	−0.243776	+	0.969832i	$0.578386\pi$
$740$	−1.19719	−0.0440096
$741$	0	0
$742$	−0.228979	−0.00840610
$743$	13.7983	0.506210	0.253105	−	0.967439i	$-0.418548\pi$
$743$	13.7983	0.506210	0.253105	+	0.967439i	$0.418548\pi$
$744$	0	0
$745$	6.75509	0.247487
$746$	−26.6649	−0.976273
$747$	0	0
$748$	17.0347	0.622850
$749$	1.59481	0.0582733
$750$	0	0
$751$	3.82043	0.139410	0.0697048	−	0.997568i	$-0.477794\pi$
$751$	3.82043	0.139410	0.0697048	+	0.997568i	$0.477794\pi$
$752$	0.969695	0.0353611
$753$	0	0
$754$	−6.78663	−0.247154
$755$	−0.185467	−0.00674984
$756$	0	0
$757$	19.6215	0.713157	0.356578	−	0.934265i	$-0.383943\pi$
$757$	19.6215	0.713157	0.356578	+	0.934265i	$0.383943\pi$
$758$	17.8517	0.648403
$759$	0	0
$760$	4.24979	0.154156
$761$	−47.5141	−1.72238	−0.861192	−	0.508280i	$-0.830281\pi$
$761$	−47.5141	−1.72238	−0.861192	+	0.508280i	$0.830281\pi$
$762$	0	0
$763$	−1.03789	−0.0375741
$764$	34.5883	1.25136
$765$	0	0
$766$	9.93262	0.358880
$767$	18.2054	0.657360
$768$	0	0
$769$	−10.0139	−0.361111	−0.180556	−	0.983565i	$-0.557790\pi$
$769$	−10.0139	−0.361111	−0.180556	+	0.983565i	$0.557790\pi$
$770$	0.134544	0.00484863
$771$	0	0
$772$	8.26066	0.297308
$773$	35.8956	1.29108	0.645538	−	0.763728i	$-0.276633\pi$
$773$	35.8956	1.29108	0.645538	+	0.763728i	$0.276633\pi$
$774$	0	0
$775$	−16.1968	−0.581807
$776$	33.6777	1.20896
$777$	0	0
$778$	−22.1037	−0.792457
$779$	4.45005	0.159439
$780$	0	0
$781$	−26.9968	−0.966023
$782$	−5.85995	−0.209551
$783$	0	0
$784$	−6.22662	−0.222379
$785$	−0.0453292	−0.00161787
$786$	0	0
$787$	1.89211	0.0674464	0.0337232	−	0.999431i	$-0.489264\pi$
$787$	1.89211	0.0674464	0.0337232	+	0.999431i	$0.489264\pi$
$788$	5.80833	0.206913
$789$	0	0
$790$	3.72207	0.132425
$791$	0.702806	0.0249889
$792$	0	0
$793$	6.68434	0.237368
$794$	19.8279	0.703667
$795$	0	0
$796$	16.3064	0.577966
$797$	−24.4028	−0.864393	−0.432197	−	0.901779i	$-0.642261\pi$
$797$	−24.4028	−0.864393	−0.432197	+	0.901779i	$0.642261\pi$
$798$	0	0
$799$	5.47347	0.193637
$800$	−28.4372	−1.00541
$801$	0	0
$802$	18.4498	0.651484
$803$	2.13220	0.0752437
$804$	0	0
$805$	0.115499	0.00407080
$806$	−3.09733	−0.109099
$807$	0	0
$808$	16.0276	0.563848
$809$	32.9341	1.15790	0.578951	−	0.815363i	$-0.303462\pi$
$809$	32.9341	1.15790	0.578951	+	0.815363i	$0.303462\pi$
$810$	0	0
$811$	−42.8145	−1.50342	−0.751711	−	0.659493i	$-0.770771\pi$
$811$	−42.8145	−1.50342	−0.751711	+	0.659493i	$0.770771\pi$
$812$	−2.03608	−0.0714523
$813$	0	0
$814$	3.88158	0.136049
$815$	4.83732	0.169444
$816$	0	0
$817$	−6.49708	−0.227304
$818$	9.99011	0.349296
$819$	0	0
$820$	0.578035	0.0201859
$821$	28.2632	0.986393	0.493196	−	0.869918i	$-0.335828\pi$
$821$	28.2632	0.986393	0.493196	+	0.869918i	$0.335828\pi$
$822$	0	0
$823$	10.4127	0.362962	0.181481	−	0.983394i	$-0.441911\pi$
$823$	10.4127	0.362962	0.181481	+	0.983394i	$0.441911\pi$
$824$	−20.1790	−0.702970
$825$	0	0
$826$	−2.18870	−0.0761547
$827$	0.0841446	0.00292600	0.00146300	−	0.999999i	$-0.499534\pi$
$827$	0.0841446	0.00292600	0.00146300	+	0.999999i	$0.499534\pi$
$828$	0	0
$829$	46.3470	1.60970	0.804850	−	0.593479i	$-0.202246\pi$
$829$	46.3470	1.60970	0.804850	+	0.593479i	$0.202246\pi$
$830$	1.78243	0.0618690
$831$	0	0
$832$	−3.24442	−0.112480
$833$	−35.1463	−1.21775
$834$	0	0
$835$	−0.287315	−0.00994294
$836$	14.3227	0.495362
$837$	0	0
$838$	−11.9966	−0.414415
$839$	13.9611	0.481990	0.240995	−	0.970526i	$-0.422526\pi$
$839$	13.9611	0.481990	0.240995	+	0.970526i	$0.422526\pi$
$840$	0	0
$841$	24.5241	0.845660
$842$	−3.43381	−0.118337
$843$	0	0
$844$	−25.8353	−0.889289
$845$	4.43913	0.152711
$846$	0	0
$847$	−1.05546	−0.0362660
$848$	−1.38903	−0.0476994
$849$	0	0
$850$	−18.5237	−0.635359
$851$	3.33214	0.114224
$852$	0	0
$853$	−54.5527	−1.86785	−0.933924	−	0.357472i	$-0.883639\pi$
$853$	−54.5527	−1.86785	−0.933924	+	0.357472i	$0.883639\pi$
$854$	−0.803609	−0.0274989
$855$	0	0
$856$	−21.2153	−0.725124
$857$	−27.6798	−0.945525	−0.472762	−	0.881190i	$-0.656743\pi$
$857$	−27.6798	−0.945525	−0.472762	+	0.881190i	$0.656743\pi$
$858$	0	0
$859$	27.1497	0.926336	0.463168	−	0.886270i	$-0.346713\pi$
$859$	27.1497	0.926336	0.463168	+	0.886270i	$0.346713\pi$
$860$	−0.843932	−0.0287778
$861$	0	0
$862$	18.1888	0.619512
$863$	−25.9641	−0.883827	−0.441913	−	0.897058i	$-0.645700\pi$
$863$	−25.9641	−0.883827	−0.441913	+	0.897058i	$0.645700\pi$
$864$	0	0
$865$	4.48208	0.152395
$866$	−15.7387	−0.534822
$867$	0	0
$868$	−0.929239	−0.0315404
$869$	30.1151	1.02159
$870$	0	0
$871$	13.8826	0.470394
$872$	13.8067	0.467554
$873$	0	0
$874$	−4.92703	−0.166659
$875$	0.741425	0.0250647
$876$	0	0
$877$	37.2176	1.25675	0.628374	−	0.777911i	$-0.283721\pi$
$877$	37.2176	1.25675	0.628374	+	0.777911i	$0.283721\pi$
$878$	−20.4855	−0.691351
$879$	0	0
$880$	0.816166	0.0275130
$881$	6.44185	0.217031	0.108516	−	0.994095i	$-0.465390\pi$
$881$	6.44185	0.217031	0.108516	+	0.994095i	$0.465390\pi$
$882$	0	0
$883$	45.9727	1.54711	0.773553	−	0.633732i	$-0.218478\pi$
$883$	45.9727	1.54711	0.773553	+	0.633732i	$0.218478\pi$
$884$	8.83976	0.297313
$885$	0	0
$886$	−19.7507	−0.663538
$887$	28.3187	0.950849	0.475425	−	0.879756i	$-0.342294\pi$
$887$	28.3187	0.950849	0.475425	+	0.879756i	$0.342294\pi$
$888$	0	0
$889$	−3.47632	−0.116592
$890$	1.88230	0.0630947
$891$	0	0
$892$	30.2222	1.01191
$893$	4.60208	0.154003
$894$	0	0
$895$	−8.18524	−0.273602
$896$	−1.89522	−0.0633147
$897$	0	0
$898$	−13.9350	−0.465018
$899$	24.4277	0.814710
$900$	0	0
$901$	−7.84041	−0.261202
$902$	−1.87413	−0.0624017
$903$	0	0
$904$	−9.34920	−0.310950
$905$	−4.61475	−0.153399
$906$	0	0
$907$	−12.5099	−0.415384	−0.207692	−	0.978194i	$-0.566595\pi$
$907$	−12.5099	−0.415384	−0.207692	+	0.978194i	$0.566595\pi$
$908$	−19.1084	−0.634135
$909$	0	0
$910$	0.0698185	0.00231446
$911$	−2.41940	−0.0801584	−0.0400792	−	0.999197i	$-0.512761\pi$
$911$	−2.41940	−0.0801584	−0.0400792	+	0.999197i	$0.512761\pi$
$912$	0	0
$913$	14.4216	0.477284
$914$	−2.68639	−0.0888580
$915$	0	0
$916$	17.9171	0.591997
$917$	2.37485	0.0784246
$918$	0	0
$919$	50.3462	1.66077	0.830384	−	0.557192i	$-0.188121\pi$
$919$	50.3462	1.66077	0.830384	+	0.557192i	$0.188121\pi$
$920$	−1.53645	−0.0506551
$921$	0	0
$922$	2.16838	0.0714118
$923$	−14.0094	−0.461125
$924$	0	0
$925$	10.5331	0.346328
$926$	23.0924	0.758865
$927$	0	0
$928$	42.8884	1.40788
$929$	32.9858	1.08223	0.541115	−	0.840948i	$-0.318002\pi$
$929$	32.9858	1.08223	0.541115	+	0.840948i	$0.318002\pi$
$930$	0	0
$931$	−29.5510	−0.968494
$932$	22.3510	0.732131
$933$	0	0
$934$	−14.1450	−0.462838
$935$	4.60687	0.150661
$936$	0	0
$937$	47.9023	1.56490	0.782450	−	0.622714i	$-0.213970\pi$
$937$	47.9023	1.56490	0.782450	+	0.622714i	$0.213970\pi$
$938$	−1.66900	−0.0544948
$939$	0	0
$940$	0.597783	0.0194975
$941$	4.42362	0.144206	0.0721030	−	0.997397i	$-0.477029\pi$
$941$	4.42362	0.144206	0.0721030	+	0.997397i	$0.477029\pi$
$942$	0	0
$943$	−1.60884	−0.0523912
$944$	−13.2770	−0.432131
$945$	0	0
$946$	2.73624	0.0889627
$947$	−19.3253	−0.627987	−0.313994	−	0.949425i	$-0.601667\pi$
$947$	−19.3253	−0.627987	−0.313994	+	0.949425i	$0.601667\pi$
$948$	0	0
$949$	1.10646	0.0359171
$950$	−15.5747	−0.505311
$951$	0	0
$952$	−2.55134	−0.0826895
$953$	−30.0968	−0.974931	−0.487465	−	0.873142i	$-0.662078\pi$
$953$	−30.0968	−0.974931	−0.487465	+	0.873142i	$0.662078\pi$
$954$	0	0
$955$	9.35408	0.302691
$956$	4.93544	0.159623
$957$	0	0
$958$	−5.68051	−0.183529
$959$	2.70948	0.0874937
$960$	0	0
$961$	−19.8515	−0.640371
$962$	2.01426	0.0649423
$963$	0	0
$964$	−41.0992	−1.32372
$965$	2.23402	0.0719156
$966$	0	0
$967$	−42.6607	−1.37187	−0.685937	−	0.727661i	$-0.740608\pi$
$967$	−42.6607	−1.37187	−0.685937	+	0.727661i	$0.740608\pi$
$968$	14.0404	0.451277
$969$	0	0
$970$	3.79378	0.121811
$971$	−33.0050	−1.05918	−0.529591	−	0.848253i	$-0.677655\pi$
$971$	−33.0050	−1.05918	−0.529591	+	0.848253i	$0.677655\pi$
$972$	0	0
$973$	0.974412	0.0312382
$974$	24.7187	0.792037
$975$	0	0
$976$	−4.87483	−0.156039
$977$	5.79949	0.185542	0.0927710	−	0.995687i	$-0.470428\pi$
$977$	5.79949	0.185542	0.0927710	+	0.995687i	$0.470428\pi$
$978$	0	0
$979$	15.2296	0.486740
$980$	−3.83850	−0.122616
$981$	0	0
$982$	−2.13095	−0.0680013
$983$	35.2304	1.12367	0.561837	−	0.827248i	$-0.310095\pi$
$983$	35.2304	1.12367	0.561837	+	0.827248i	$0.310095\pi$
$984$	0	0
$985$	1.57081	0.0500501
$986$	27.9371	0.889700
$987$	0	0
$988$	7.43245	0.236458
$989$	2.34891	0.0746911
$990$	0	0
$991$	9.08023	0.288443	0.144221	−	0.989545i	$-0.453932\pi$
$991$	9.08023	0.288443	0.144221	+	0.989545i	$0.453932\pi$
$992$	19.5737	0.621466
$993$	0	0
$994$	1.68424	0.0534210
$995$	4.40992	0.139804
$996$	0	0
$997$	59.6440	1.88894	0.944472	−	0.328591i	$-0.106574\pi$
$997$	59.6440	1.88894	0.944472	+	0.328591i	$0.106574\pi$
$998$	21.9808	0.695791
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4527.2.a.k.1.6		10
3.2	odd	2		503.2.a.e.1.5	✓	10
12.11	even	2		8048.2.a.p.1.10		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.a.e.1.5	✓	10	3.2	odd	2
4527.2.a.k.1.6		10	1.1	even	1	trivial
8048.2.a.p.1.10		10	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4527 = 3^{2} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4527.a (trivial)

\(f(q)\)	\(=\)	\(q+0.756417 q^{2} -1.42783 q^{4} -0.386144 q^{5} +0.194914 q^{7} -2.59287 q^{8} +O(q^{10})\)	\(q+0.756417 q^{2} -1.42783 q^{4} -0.386144 q^{5} +0.194914 q^{7} -2.59287 q^{8} -0.292086 q^{10} -2.36326 q^{11} -1.22636 q^{13} +0.147436 q^{14} +0.894372 q^{16} +5.04830 q^{17} +4.24460 q^{19} +0.551349 q^{20} -1.78761 q^{22} -1.53457 q^{23} -4.85089 q^{25} -0.927639 q^{26} -0.278304 q^{28} +7.31602 q^{29} +3.33893 q^{31} +5.86226 q^{32} +3.81863 q^{34} -0.0752647 q^{35} -2.17138 q^{37} +3.21069 q^{38} +1.00122 q^{40} +1.04840 q^{41} -1.53067 q^{43} +3.37434 q^{44} -1.16077 q^{46} +1.08422 q^{47} -6.96201 q^{49} -3.66930 q^{50} +1.75104 q^{52} -1.55308 q^{53} +0.912558 q^{55} -0.505386 q^{56} +5.53396 q^{58} -14.8451 q^{59} -5.45056 q^{61} +2.52563 q^{62} +2.64557 q^{64} +0.473551 q^{65} -11.3202 q^{67} -7.20814 q^{68} -0.0569315 q^{70} +11.4236 q^{71} -0.902229 q^{73} -1.64247 q^{74} -6.06059 q^{76} -0.460631 q^{77} -12.7431 q^{79} -0.345356 q^{80} +0.793029 q^{82} -6.10241 q^{83} -1.94937 q^{85} -1.15782 q^{86} +6.12763 q^{88} -6.44432 q^{89} -0.239034 q^{91} +2.19111 q^{92} +0.820122 q^{94} -1.63903 q^{95} -12.9886 q^{97} -5.26618 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 4 q^{2} + 4 q^{4} + q^{5} - 5 q^{7} + 3 q^{8}+O(q^{10}) \) 10 * q + 4 * q^2 + 4 * q^4 + q^5 - 5 * q^7 + 3 * q^8	\( 10 q + 4 q^{2} + 4 q^{4} + q^{5} - 5 q^{7} + 3 q^{8} - 4 q^{10} + 3 q^{11} - 18 q^{13} - q^{14} - 4 q^{16} + 11 q^{17} + 3 q^{20} - 18 q^{22} + 2 q^{23} - 27 q^{25} - 11 q^{26} - 22 q^{28} + 9 q^{29} - 22 q^{31} + 10 q^{32} - 10 q^{34} + 6 q^{35} - 35 q^{37} - 2 q^{38} - 19 q^{40} + 4 q^{41} - 20 q^{43} - 9 q^{44} - q^{46} - 7 q^{47} - 27 q^{49} - 16 q^{50} - 7 q^{52} + 24 q^{53} - 11 q^{55} - 12 q^{56} + 2 q^{58} - 17 q^{59} - 4 q^{61} - 8 q^{62} + 3 q^{64} + 16 q^{65} - 6 q^{67} - 28 q^{68} + 26 q^{70} + q^{71} - 31 q^{73} - 11 q^{74} + 20 q^{76} - 3 q^{77} - 10 q^{79} - 24 q^{80} - 9 q^{82} - 22 q^{83} - 6 q^{85} - 38 q^{86} - 3 q^{88} - q^{89} + 10 q^{91} - 27 q^{92} + 33 q^{94} - 39 q^{95} - 57 q^{97} - 40 q^{98}+O(q^{100}) \) 10 * q + 4 * q^2 + 4 * q^4 + q^5 - 5 * q^7 + 3 * q^8 - 4 * q^10 + 3 * q^11 - 18 * q^13 - q^14 - 4 * q^16 + 11 * q^17 + 3 * q^20 - 18 * q^22 + 2 * q^23 - 27 * q^25 - 11 * q^26 - 22 * q^28 + 9 * q^29 - 22 * q^31 + 10 * q^32 - 10 * q^34 + 6 * q^35 - 35 * q^37 - 2 * q^38 - 19 * q^40 + 4 * q^41 - 20 * q^43 - 9 * q^44 - q^46 - 7 * q^47 - 27 * q^49 - 16 * q^50 - 7 * q^52 + 24 * q^53 - 11 * q^55 - 12 * q^56 + 2 * q^58 - 17 * q^59 - 4 * q^61 - 8 * q^62 + 3 * q^64 + 16 * q^65 - 6 * q^67 - 28 * q^68 + 26 * q^70 + q^71 - 31 * q^73 - 11 * q^74 + 20 * q^76 - 3 * q^77 - 10 * q^79 - 24 * q^80 - 9 * q^82 - 22 * q^83 - 6 * q^85 - 38 * q^86 - 3 * q^88 - q^89 + 10 * q^91 - 27 * q^92 + 33 * q^94 - 39 * q^95 - 57 * q^97 - 40 * q^98

Introduction

L-functions

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