LMFDB - Embedded newform 5265.2.a.bl.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5265,2,Mod(1,5265)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5265, 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("5265.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5265 = 3^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5265.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:	$42.0412366642$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 25 x^{13} + 24 x^{12} + 244 x^{11} - 226 x^{10} - 1170 x^{9} + 1051 x^{8} + 2842 x^{7} + \cdots + 144 $ x^15 - x^14 - 25x^13 + 24x^12 + 244x^11 - 226x^10 - 1170x^9 + 1051x^8 + 2842x^7 - 2478x^6 - 3252x^5 + 2697x^4 + 1497x^3 - 1107x^2 - 225*x + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 585)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$1.25728$ of defining polynomial
Character	$\chi$	$=$	5265.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.25728 q^{2} -0.419240 q^{4} +1.00000 q^{5} +4.56890 q^{7} -3.04167 q^{8} +O(q^{10})$	\(q+1.25728 q^{2} -0.419240 q^{4} +1.00000 q^{5} +4.56890 q^{7} -3.04167 q^{8} +1.25728 q^{10} +5.18598 q^{11} +1.00000 q^{13} +5.74440 q^{14} -2.98576 q^{16} -4.28682 q^{17} +4.84071 q^{19} -0.419240 q^{20} +6.52024 q^{22} +7.56437 q^{23} +1.00000 q^{25} +1.25728 q^{26} -1.91547 q^{28} +0.218538 q^{29} -1.23812 q^{31} +2.32940 q^{32} -5.38974 q^{34} +4.56890 q^{35} +2.15479 q^{37} +6.08614 q^{38} -3.04167 q^{40} -11.9785 q^{41} -1.65781 q^{43} -2.17417 q^{44} +9.51055 q^{46} -11.0630 q^{47} +13.8748 q^{49} +1.25728 q^{50} -0.419240 q^{52} +9.54923 q^{53} +5.18598 q^{55} -13.8971 q^{56} +0.274765 q^{58} +4.00950 q^{59} +7.95550 q^{61} -1.55667 q^{62} +8.90022 q^{64} +1.00000 q^{65} -3.56109 q^{67} +1.79721 q^{68} +5.74440 q^{70} -10.0496 q^{71} -9.80752 q^{73} +2.70918 q^{74} -2.02942 q^{76} +23.6942 q^{77} -3.36870 q^{79} -2.98576 q^{80} -15.0603 q^{82} -4.25736 q^{83} -4.28682 q^{85} -2.08434 q^{86} -15.7740 q^{88} +0.708626 q^{89} +4.56890 q^{91} -3.17129 q^{92} -13.9093 q^{94} +4.84071 q^{95} -3.56098 q^{97} +17.4446 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + q^{2} + 21 q^{4} + 15 q^{5} + 10 q^{7}+O(q^{10}) $ 15 * q + q^2 + 21 * q^4 + 15 * q^5 + 10 * q^7	$ 15 q + q^{2} + 21 q^{4} + 15 q^{5} + 10 q^{7} + q^{10} + 9 q^{11} + 15 q^{13} + 3 q^{14} + 33 q^{16} - 3 q^{17} + 15 q^{19} + 21 q^{20} + 10 q^{22} - 6 q^{23} + 15 q^{25} + q^{26} + 35 q^{28} + 8 q^{29} + 22 q^{31} + 21 q^{32} + 9 q^{34} + 10 q^{35} + 4 q^{37} - 14 q^{38} + 13 q^{41} + 24 q^{43} - 5 q^{44} - 3 q^{46} - q^{47} + 37 q^{49} + q^{50} + 21 q^{52} - 7 q^{53} + 9 q^{55} + 17 q^{56} + 22 q^{58} + 19 q^{59} + 16 q^{61} - 13 q^{62} + 36 q^{64} + 15 q^{65} + 11 q^{67} - 28 q^{68} + 3 q^{70} + 28 q^{71} + 26 q^{73} + 8 q^{74} + 18 q^{76} - 24 q^{77} + 44 q^{79} + 33 q^{80} + 35 q^{82} - 3 q^{83} - 3 q^{85} + 40 q^{86} + 37 q^{88} + 4 q^{89} + 10 q^{91} - 74 q^{92} + 2 q^{94} + 15 q^{95} + 33 q^{97} - 3 q^{98}+O(q^{100}) $ 15 * q + q^2 + 21 * q^4 + 15 * q^5 + 10 * q^7 + q^10 + 9 * q^11 + 15 * q^13 + 3 * q^14 + 33 * q^16 - 3 * q^17 + 15 * q^19 + 21 * q^20 + 10 * q^22 - 6 * q^23 + 15 * q^25 + q^26 + 35 * q^28 + 8 * q^29 + 22 * q^31 + 21 * q^32 + 9 * q^34 + 10 * q^35 + 4 * q^37 - 14 * q^38 + 13 * q^41 + 24 * q^43 - 5 * q^44 - 3 * q^46 - q^47 + 37 * q^49 + q^50 + 21 * q^52 - 7 * q^53 + 9 * q^55 + 17 * q^56 + 22 * q^58 + 19 * q^59 + 16 * q^61 - 13 * q^62 + 36 * q^64 + 15 * q^65 + 11 * q^67 - 28 * q^68 + 3 * q^70 + 28 * q^71 + 26 * q^73 + 8 * q^74 + 18 * q^76 - 24 * q^77 + 44 * q^79 + 33 * q^80 + 35 * q^82 - 3 * q^83 - 3 * q^85 + 40 * q^86 + 37 * q^88 + 4 * q^89 + 10 * q^91 - 74 * q^92 + 2 * q^94 + 15 * q^95 + 33 * q^97 - 3 * 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 $

$2$	1.25728	0.889033	0.444517	−	0.895771i	$-0.353375\pi$
$2$	1.25728	0.889033	0.444517	+	0.895771i	$0.353375\pi$
$3$	0	0
$3$	0	0
$4$	−0.419240	−0.209620
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.56890	1.72688	0.863441	−	0.504450i	$-0.168305\pi$
$7$	4.56890	1.72688	0.863441	+	0.504450i	$0.168305\pi$
$8$	−3.04167	−1.07539
$9$	0	0
$10$	1.25728	0.397588
$11$	5.18598	1.56363	0.781816	−	0.623509i	$-0.214294\pi$
$11$	5.18598	1.56363	0.781816	+	0.623509i	$0.214294\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	5.74440	1.53525
$15$	0	0
$16$	−2.98576	−0.746439
$17$	−4.28682	−1.03971	−0.519853	−	0.854256i	$-0.674013\pi$
$17$	−4.28682	−1.03971	−0.519853	+	0.854256i	$0.674013\pi$
$18$	0	0
$19$	4.84071	1.11054	0.555268	−	0.831672i	$-0.312616\pi$
$19$	4.84071	1.11054	0.555268	+	0.831672i	$0.312616\pi$
$20$	−0.419240	−0.0937450
$21$	0	0
$22$	6.52024	1.39012
$23$	7.56437	1.57728	0.788640	−	0.614855i	$-0.210785\pi$
$23$	7.56437	1.57728	0.788640	+	0.614855i	$0.210785\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.25728	0.246573
$27$	0	0
$28$	−1.91547	−0.361989
$29$	0.218538	0.0405816	0.0202908	−	0.999794i	$-0.493541\pi$
$29$	0.218538	0.0405816	0.0202908	+	0.999794i	$0.493541\pi$
$30$	0	0
$31$	−1.23812	−0.222373	−0.111187	−	0.993800i	$-0.535465\pi$
$31$	−1.23812	−0.222373	−0.111187	+	0.993800i	$0.535465\pi$
$32$	2.32940	0.411783
$33$	0	0
$34$	−5.38974	−0.924333
$35$	4.56890	0.772285
$36$	0	0
$37$	2.15479	0.354246	0.177123	−	0.984189i	$-0.443321\pi$
$37$	2.15479	0.354246	0.177123	+	0.984189i	$0.443321\pi$
$38$	6.08614	0.987302
$39$	0	0
$40$	−3.04167	−0.480930
$41$	−11.9785	−1.87072	−0.935362	−	0.353691i	$-0.884927\pi$
$41$	−11.9785	−1.87072	−0.935362	+	0.353691i	$0.884927\pi$
$42$	0	0
$43$	−1.65781	−0.252814	−0.126407	−	0.991978i	$-0.540345\pi$
$43$	−1.65781	−0.252814	−0.126407	+	0.991978i	$0.540345\pi$
$44$	−2.17417	−0.327769
$45$	0	0
$46$	9.51055	1.40225
$47$	−11.0630	−1.61371	−0.806853	−	0.590752i	$-0.798831\pi$
$47$	−11.0630	−1.61371	−0.806853	+	0.590752i	$0.798831\pi$
$48$	0	0
$49$	13.8748	1.98212
$50$	1.25728	0.177807
$51$	0	0
$52$	−0.419240	−0.0581382
$53$	9.54923	1.31169	0.655844	−	0.754896i	$-0.272313\pi$
$53$	9.54923	1.31169	0.655844	+	0.754896i	$0.272313\pi$
$54$	0	0
$55$	5.18598	0.699278
$56$	−13.8971	−1.85708
$57$	0	0
$58$	0.274765	0.0360784
$59$	4.00950	0.521992	0.260996	−	0.965340i	$-0.415949\pi$
$59$	4.00950	0.521992	0.260996	+	0.965340i	$0.415949\pi$
$60$	0	0
$61$	7.95550	1.01860	0.509299	−	0.860590i	$-0.329905\pi$
$61$	7.95550	1.01860	0.509299	+	0.860590i	$0.329905\pi$
$62$	−1.55667	−0.197697
$63$	0	0
$64$	8.90022	1.11253
$65$	1.00000	0.124035
$66$	0	0
$67$	−3.56109	−0.435057	−0.217528	−	0.976054i	$-0.569800\pi$
$67$	−3.56109	−0.435057	−0.217528	+	0.976054i	$0.569800\pi$
$68$	1.79721	0.217943
$69$	0	0
$70$	5.74440	0.686587
$71$	−10.0496	−1.19267	−0.596333	−	0.802737i	$-0.703376\pi$
$71$	−10.0496	−1.19267	−0.596333	+	0.802737i	$0.703376\pi$
$72$	0	0
$73$	−9.80752	−1.14788	−0.573942	−	0.818896i	$-0.694586\pi$
$73$	−9.80752	−1.14788	−0.573942	+	0.818896i	$0.694586\pi$
$74$	2.70918	0.314936
$75$	0	0
$76$	−2.02942	−0.232791
$77$	23.6942	2.70021
$78$	0	0
$79$	−3.36870	−0.379008	−0.189504	−	0.981880i	$-0.560688\pi$
$79$	−3.36870	−0.379008	−0.189504	+	0.981880i	$0.560688\pi$
$80$	−2.98576	−0.333818
$81$	0	0
$82$	−15.0603	−1.66314
$83$	−4.25736	−0.467306	−0.233653	−	0.972320i	$-0.575068\pi$
$83$	−4.25736	−0.467306	−0.233653	+	0.972320i	$0.575068\pi$
$84$	0	0
$85$	−4.28682	−0.464970
$86$	−2.08434	−0.224760
$87$	0	0
$88$	−15.7740	−1.68152
$89$	0.708626	0.0751142	0.0375571	−	0.999294i	$-0.488042\pi$
$89$	0.708626	0.0751142	0.0375571	+	0.999294i	$0.488042\pi$
$90$	0	0
$91$	4.56890	0.478951
$92$	−3.17129	−0.330630
$93$	0	0
$94$	−13.9093	−1.43464
$95$	4.84071	0.496646
$96$	0	0
$97$	−3.56098	−0.361563	−0.180782	−	0.983523i	$-0.557863\pi$
$97$	−3.56098	−0.361563	−0.180782	+	0.983523i	$0.557863\pi$
$98$	17.4446	1.76217
$99$	0	0
$100$	−0.419240	−0.0419240
$101$	15.0455	1.49708	0.748540	−	0.663090i	$-0.230755\pi$
$101$	15.0455	1.49708	0.748540	+	0.663090i	$0.230755\pi$
$102$	0	0
$103$	19.1186	1.88381	0.941907	−	0.335873i	$-0.109031\pi$
$103$	19.1186	1.88381	0.941907	+	0.335873i	$0.109031\pi$
$104$	−3.04167	−0.298260
$105$	0	0
$106$	12.0061	1.16613
$107$	−1.48195	−0.143266	−0.0716328	−	0.997431i	$-0.522821\pi$
$107$	−1.48195	−0.143266	−0.0716328	+	0.997431i	$0.522821\pi$
$108$	0	0
$109$	−1.15271	−0.110409	−0.0552047	−	0.998475i	$-0.517581\pi$
$109$	−1.15271	−0.110409	−0.0552047	+	0.998475i	$0.517581\pi$
$110$	6.52024	0.621681
$111$	0	0
$112$	−13.6416	−1.28901
$113$	−5.56294	−0.523317	−0.261659	−	0.965160i	$-0.584269\pi$
$113$	−5.56294	−0.523317	−0.261659	+	0.965160i	$0.584269\pi$
$114$	0	0
$115$	7.56437	0.705381
$116$	−0.0916201	−0.00850671
$117$	0	0
$118$	5.04107	0.464069
$119$	−19.5860	−1.79545
$120$	0	0
$121$	15.8944	1.44495
$122$	10.0023	0.905567
$123$	0	0
$124$	0.519070	0.0466139
$125$	1.00000	0.0894427
$126$	0	0
$127$	−13.3818	−1.18744	−0.593720	−	0.804672i	$-0.702341\pi$
$127$	−13.3818	−1.18744	−0.593720	+	0.804672i	$0.702341\pi$
$128$	6.53130	0.577291
$129$	0	0
$130$	1.25728	0.110271
$131$	7.21542	0.630414	0.315207	−	0.949023i	$-0.397926\pi$
$131$	7.21542	0.630414	0.315207	+	0.949023i	$0.397926\pi$
$132$	0	0
$133$	22.1167	1.91776
$134$	−4.47730	−0.386780
$135$	0	0
$136$	13.0391	1.11809
$137$	0.675467	0.0577091	0.0288545	−	0.999584i	$-0.490814\pi$
$137$	0.675467	0.0577091	0.0288545	+	0.999584i	$0.490814\pi$
$138$	0	0
$139$	16.4793	1.39775	0.698877	−	0.715242i	$-0.253683\pi$
$139$	16.4793	1.39775	0.698877	+	0.715242i	$0.253683\pi$
$140$	−1.91547	−0.161886
$141$	0	0
$142$	−12.6352	−1.06032
$143$	5.18598	0.433674
$144$	0	0
$145$	0.218538	0.0181486
$146$	−12.3308	−1.02051
$147$	0	0
$148$	−0.903376	−0.0742571
$149$	4.64270	0.380345	0.190173	−	0.981751i	$-0.439095\pi$
$149$	4.64270	0.380345	0.190173	+	0.981751i	$0.439095\pi$
$150$	0	0
$151$	−9.69775	−0.789192	−0.394596	−	0.918855i	$-0.629115\pi$
$151$	−9.69775	−0.789192	−0.394596	+	0.918855i	$0.629115\pi$
$152$	−14.7238	−1.19426
$153$	0	0
$154$	29.7903	2.40057
$155$	−1.23812	−0.0994483
$156$	0	0
$157$	−4.25608	−0.339672	−0.169836	−	0.985472i	$-0.554324\pi$
$157$	−4.25608	−0.339672	−0.169836	+	0.985472i	$0.554324\pi$
$158$	−4.23540	−0.336951
$159$	0	0
$160$	2.32940	0.184155
$161$	34.5609	2.72378
$162$	0	0
$163$	14.1023	1.10458	0.552289	−	0.833653i	$-0.313754\pi$
$163$	14.1023	1.10458	0.552289	+	0.833653i	$0.313754\pi$
$164$	5.02186	0.392142
$165$	0	0
$166$	−5.35271	−0.415451
$167$	−8.92876	−0.690928	−0.345464	−	0.938432i	$-0.612279\pi$
$167$	−8.92876	−0.690928	−0.345464	+	0.938432i	$0.612279\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−5.38974	−0.413374
$171$	0	0
$172$	0.695022	0.0529949
$173$	20.9797	1.59506	0.797530	−	0.603280i	$-0.206140\pi$
$173$	20.9797	1.59506	0.797530	+	0.603280i	$0.206140\pi$
$174$	0	0
$175$	4.56890	0.345376
$176$	−15.4841	−1.16716
$177$	0	0
$178$	0.890943	0.0667790
$179$	3.67145	0.274417	0.137209	−	0.990542i	$-0.456187\pi$
$179$	3.67145	0.274417	0.137209	+	0.990542i	$0.456187\pi$
$180$	0	0
$181$	−25.7170	−1.91153	−0.955764	−	0.294134i	$-0.904969\pi$
$181$	−25.7170	−1.91153	−0.955764	+	0.294134i	$0.904969\pi$
$182$	5.74440	0.425803
$183$	0	0
$184$	−23.0083	−1.69620
$185$	2.15479	0.158424
$186$	0	0
$187$	−22.2313	−1.62572
$188$	4.63806	0.338265
$189$	0	0
$190$	6.08614	0.441535
$191$	−19.0513	−1.37851	−0.689253	−	0.724521i	$-0.742061\pi$
$191$	−19.0513	−1.37851	−0.689253	+	0.724521i	$0.742061\pi$
$192$	0	0
$193$	1.50135	0.108069	0.0540346	−	0.998539i	$-0.482792\pi$
$193$	1.50135	0.108069	0.0540346	+	0.998539i	$0.482792\pi$
$194$	−4.47716	−0.321442
$195$	0	0
$196$	−5.81689	−0.415492
$197$	−19.3725	−1.38024	−0.690118	−	0.723697i	$-0.742441\pi$
$197$	−19.3725	−1.38024	−0.690118	+	0.723697i	$0.742441\pi$
$198$	0	0
$199$	12.3685	0.876782	0.438391	−	0.898784i	$-0.355548\pi$
$199$	12.3685	0.876782	0.438391	+	0.898784i	$0.355548\pi$
$200$	−3.04167	−0.215078
$201$	0	0
$202$	18.9164	1.33095
$203$	0.998480	0.0700796
$204$	0	0
$205$	−11.9785	−0.836614
$206$	24.0375	1.67477
$207$	0	0
$208$	−2.98576	−0.207025
$209$	25.1038	1.73647
$210$	0	0
$211$	−7.73728	−0.532656	−0.266328	−	0.963882i	$-0.585810\pi$
$211$	−7.73728	−0.532656	−0.266328	+	0.963882i	$0.585810\pi$
$212$	−4.00342	−0.274956
$213$	0	0
$214$	−1.86323	−0.127368
$215$	−1.65781	−0.113062
$216$	0	0
$217$	−5.65685	−0.384012
$218$	−1.44928	−0.0981576
$219$	0	0
$220$	−2.17417	−0.146583
$221$	−4.28682	−0.288362
$222$	0	0
$223$	−1.85214	−0.124029	−0.0620143	−	0.998075i	$-0.519752\pi$
$223$	−1.85214	−0.124029	−0.0620143	+	0.998075i	$0.519752\pi$
$224$	10.6428	0.711101
$225$	0	0
$226$	−6.99419	−0.465246
$227$	6.85446	0.454946	0.227473	−	0.973784i	$-0.426954\pi$
$227$	6.85446	0.454946	0.227473	+	0.973784i	$0.426954\pi$
$228$	0	0
$229$	25.5871	1.69085	0.845423	−	0.534097i	$-0.179348\pi$
$229$	25.5871	1.69085	0.845423	+	0.534097i	$0.179348\pi$
$230$	9.51055	0.627107
$231$	0	0
$232$	−0.664722	−0.0436411
$233$	−1.99925	−0.130975	−0.0654876	−	0.997853i	$-0.520860\pi$
$233$	−1.99925	−0.130975	−0.0654876	+	0.997853i	$0.520860\pi$
$234$	0	0
$235$	−11.0630	−0.721672
$236$	−1.68094	−0.109420
$237$	0	0
$238$	−24.6252	−1.59621
$239$	−8.22190	−0.531830	−0.265915	−	0.963996i	$-0.585674\pi$
$239$	−8.22190	−0.531830	−0.265915	+	0.963996i	$0.585674\pi$
$240$	0	0
$241$	15.4013	0.992085	0.496043	−	0.868298i	$-0.334786\pi$
$241$	15.4013	0.992085	0.496043	+	0.868298i	$0.334786\pi$
$242$	19.9838	1.28460
$243$	0	0
$244$	−3.33527	−0.213519
$245$	13.8748	0.886431
$246$	0	0
$247$	4.84071	0.308007
$248$	3.76595	0.239138
$249$	0	0
$250$	1.25728	0.0795175
$251$	10.4035	0.656666	0.328333	−	0.944562i	$-0.393513\pi$
$251$	10.4035	0.656666	0.328333	+	0.944562i	$0.393513\pi$
$252$	0	0
$253$	39.2287	2.46629
$254$	−16.8247	−1.05567
$255$	0	0
$256$	−9.58875	−0.599297
$257$	0.964165	0.0601430	0.0300715	−	0.999548i	$-0.490427\pi$
$257$	0.964165	0.0601430	0.0300715	+	0.999548i	$0.490427\pi$
$258$	0	0
$259$	9.84504	0.611741
$260$	−0.419240	−0.0260002
$261$	0	0
$262$	9.07182	0.560459
$263$	−23.3246	−1.43826	−0.719129	−	0.694876i	$-0.755459\pi$
$263$	−23.3246	−1.43826	−0.719129	+	0.694876i	$0.755459\pi$
$264$	0	0
$265$	9.54923	0.586605
$266$	27.8070	1.70495
$267$	0	0
$268$	1.49295	0.0911967
$269$	3.58613	0.218650	0.109325	−	0.994006i	$-0.465131\pi$
$269$	3.58613	0.218650	0.109325	+	0.994006i	$0.465131\pi$
$270$	0	0
$271$	2.22891	0.135397	0.0676984	−	0.997706i	$-0.478434\pi$
$271$	2.22891	0.135397	0.0676984	+	0.997706i	$0.478434\pi$
$272$	12.7994	0.776077
$273$	0	0
$274$	0.849253	0.0513053
$275$	5.18598	0.312726
$276$	0	0
$277$	20.0207	1.20293	0.601463	−	0.798901i	$-0.294585\pi$
$277$	20.0207	1.20293	0.601463	+	0.798901i	$0.294585\pi$
$278$	20.7191	1.24265
$279$	0	0
$280$	−13.8971	−0.830509
$281$	1.79403	0.107023	0.0535114	−	0.998567i	$-0.482959\pi$
$281$	1.79403	0.107023	0.0535114	+	0.998567i	$0.482959\pi$
$282$	0	0
$283$	19.7115	1.17172	0.585862	−	0.810411i	$-0.300756\pi$
$283$	19.7115	1.17172	0.585862	+	0.810411i	$0.300756\pi$
$284$	4.21319	0.250007
$285$	0	0
$286$	6.52024	0.385550
$287$	−54.7285	−3.23052
$288$	0	0
$289$	1.37679	0.0809875
$290$	0.274765	0.0161347
$291$	0	0
$292$	4.11171	0.240620
$293$	−7.97787	−0.466072	−0.233036	−	0.972468i	$-0.574866\pi$
$293$	−7.97787	−0.466072	−0.233036	+	0.972468i	$0.574866\pi$
$294$	0	0
$295$	4.00950	0.233442
$296$	−6.55417	−0.380953
$297$	0	0
$298$	5.83719	0.338139
$299$	7.56437	0.437459
$300$	0	0
$301$	−7.57438	−0.436580
$302$	−12.1928	−0.701618
$303$	0	0
$304$	−14.4532	−0.828947
$305$	7.95550	0.455531
$306$	0	0
$307$	−24.8134	−1.41618	−0.708088	−	0.706125i	$-0.750442\pi$
$307$	−24.8134	−1.41618	−0.708088	+	0.706125i	$0.750442\pi$
$308$	−9.93358	−0.566018
$309$	0	0
$310$	−1.55667	−0.0884128
$311$	3.09335	0.175408	0.0877040	−	0.996147i	$-0.472047\pi$
$311$	3.09335	0.175408	0.0877040	+	0.996147i	$0.472047\pi$
$312$	0	0
$313$	12.1670	0.687720	0.343860	−	0.939021i	$-0.388265\pi$
$313$	12.1670	0.687720	0.343860	+	0.939021i	$0.388265\pi$
$314$	−5.35110	−0.301980
$315$	0	0
$316$	1.41229	0.0794477
$317$	14.0982	0.791836	0.395918	−	0.918286i	$-0.370426\pi$
$317$	14.0982	0.791836	0.395918	+	0.918286i	$0.370426\pi$
$318$	0	0
$319$	1.13334	0.0634546
$320$	8.90022	0.497538
$321$	0	0
$322$	43.4528	2.42153
$323$	−20.7512	−1.15463
$324$	0	0
$325$	1.00000	0.0554700
$326$	17.7306	0.982007
$327$	0	0
$328$	36.4346	2.01176
$329$	−50.5458	−2.78668
$330$	0	0
$331$	−9.16487	−0.503747	−0.251873	−	0.967760i	$-0.581047\pi$
$331$	−9.16487	−0.503747	−0.251873	+	0.967760i	$0.581047\pi$
$332$	1.78486	0.0979568
$333$	0	0
$334$	−11.2260	−0.614258
$335$	−3.56109	−0.194563
$336$	0	0
$337$	8.22730	0.448170	0.224085	−	0.974570i	$-0.428061\pi$
$337$	8.22730	0.448170	0.224085	+	0.974570i	$0.428061\pi$
$338$	1.25728	0.0683872
$339$	0	0
$340$	1.79721	0.0974672
$341$	−6.42087	−0.347710
$342$	0	0
$343$	31.4105	1.69601
$344$	5.04252	0.271874
$345$	0	0
$346$	26.3775	1.41806
$347$	−16.4710	−0.884208	−0.442104	−	0.896964i	$-0.645768\pi$
$347$	−16.4710	−0.884208	−0.442104	+	0.896964i	$0.645768\pi$
$348$	0	0
$349$	12.7601	0.683033	0.341516	−	0.939876i	$-0.389059\pi$
$349$	12.7601	0.683033	0.341516	+	0.939876i	$0.389059\pi$
$350$	5.74440	0.307051
$351$	0	0
$352$	12.0802	0.643877
$353$	21.6224	1.15084	0.575422	−	0.817857i	$-0.304838\pi$
$353$	21.6224	1.15084	0.575422	+	0.817857i	$0.304838\pi$
$354$	0	0
$355$	−10.0496	−0.533377
$356$	−0.297085	−0.0157455
$357$	0	0
$358$	4.61605	0.243966
$359$	−32.3390	−1.70679	−0.853393	−	0.521268i	$-0.825459\pi$
$359$	−32.3390	−1.70679	−0.853393	+	0.521268i	$0.825459\pi$
$360$	0	0
$361$	4.43247	0.233288
$362$	−32.3335	−1.69941
$363$	0	0
$364$	−1.91547	−0.100398
$365$	−9.80752	−0.513349
$366$	0	0
$367$	18.5866	0.970214	0.485107	−	0.874455i	$-0.338781\pi$
$367$	18.5866	0.970214	0.485107	+	0.874455i	$0.338781\pi$
$368$	−22.5854	−1.17734
$369$	0	0
$370$	2.70918	0.140844
$371$	43.6295	2.26513
$372$	0	0
$373$	14.9605	0.774626	0.387313	−	0.921948i	$-0.373403\pi$
$373$	14.9605	0.774626	0.387313	+	0.921948i	$0.373403\pi$
$374$	−27.9511	−1.44532
$375$	0	0
$376$	33.6500	1.73537
$377$	0.218538	0.0112553
$378$	0	0
$379$	24.4519	1.25601	0.628005	−	0.778209i	$-0.283872\pi$
$379$	24.4519	1.25601	0.628005	+	0.778209i	$0.283872\pi$
$380$	−2.02942	−0.104107
$381$	0	0
$382$	−23.9529	−1.22554
$383$	7.52683	0.384603	0.192301	−	0.981336i	$-0.438405\pi$
$383$	7.52683	0.384603	0.192301	+	0.981336i	$0.438405\pi$
$384$	0	0
$385$	23.6942	1.20757
$386$	1.88762	0.0960772
$387$	0	0
$388$	1.49291	0.0757909
$389$	−22.2781	−1.12955	−0.564773	−	0.825247i	$-0.691036\pi$
$389$	−22.2781	−1.12955	−0.564773	+	0.825247i	$0.691036\pi$
$390$	0	0
$391$	−32.4271	−1.63991
$392$	−42.2027	−2.13156
$393$	0	0
$394$	−24.3568	−1.22708
$395$	−3.36870	−0.169497
$396$	0	0
$397$	9.09552	0.456491	0.228246	−	0.973604i	$-0.426701\pi$
$397$	9.09552	0.456491	0.228246	+	0.973604i	$0.426701\pi$
$398$	15.5507	0.779488
$399$	0	0
$400$	−2.98576	−0.149288
$401$	−13.0101	−0.649692	−0.324846	−	0.945767i	$-0.605313\pi$
$401$	−13.0101	−0.649692	−0.324846	+	0.945767i	$0.605313\pi$
$402$	0	0
$403$	−1.23812	−0.0616752
$404$	−6.30767	−0.313818
$405$	0	0
$406$	1.25537	0.0623031
$407$	11.1747	0.553910
$408$	0	0
$409$	9.15078	0.452477	0.226238	−	0.974072i	$-0.427357\pi$
$409$	9.15078	0.452477	0.226238	+	0.974072i	$0.427357\pi$
$410$	−15.0603	−0.743777
$411$	0	0
$412$	−8.01530	−0.394885
$413$	18.3190	0.901419
$414$	0	0
$415$	−4.25736	−0.208986
$416$	2.32940	0.114208
$417$	0	0
$418$	31.5626	1.54378
$419$	7.94029	0.387909	0.193954	−	0.981011i	$-0.437869\pi$
$419$	7.94029	0.387909	0.193954	+	0.981011i	$0.437869\pi$
$420$	0	0
$421$	−1.32068	−0.0643659	−0.0321830	−	0.999482i	$-0.510246\pi$
$421$	−1.32068	−0.0643659	−0.0321830	+	0.999482i	$0.510246\pi$
$422$	−9.72794	−0.473549
$423$	0	0
$424$	−29.0456	−1.41058
$425$	−4.28682	−0.207941
$426$	0	0
$427$	36.3479	1.75900
$428$	0.621294	0.0300314
$429$	0	0
$430$	−2.08434	−0.100516
$431$	−25.7492	−1.24029	−0.620147	−	0.784486i	$-0.712927\pi$
$431$	−25.7492	−1.24029	−0.620147	+	0.784486i	$0.712927\pi$
$432$	0	0
$433$	−15.5552	−0.747535	−0.373767	−	0.927523i	$-0.621934\pi$
$433$	−15.5552	−0.747535	−0.373767	+	0.927523i	$0.621934\pi$
$434$	−7.11226	−0.341399
$435$	0	0
$436$	0.483262	0.0231440
$437$	36.6169	1.75163
$438$	0	0
$439$	−0.633602	−0.0302402	−0.0151201	−	0.999886i	$-0.504813\pi$
$439$	−0.633602	−0.0302402	−0.0151201	+	0.999886i	$0.504813\pi$
$440$	−15.7740	−0.751998
$441$	0	0
$442$	−5.38974	−0.256364
$443$	1.14732	0.0545106	0.0272553	−	0.999629i	$-0.491323\pi$
$443$	1.14732	0.0545106	0.0272553	+	0.999629i	$0.491323\pi$
$444$	0	0
$445$	0.708626	0.0335921
$446$	−2.32867	−0.110265
$447$	0	0
$448$	40.6642	1.92120
$449$	−10.9248	−0.515572	−0.257786	−	0.966202i	$-0.582993\pi$
$449$	−10.9248	−0.515572	−0.257786	+	0.966202i	$0.582993\pi$
$450$	0	0
$451$	−62.1202	−2.92513
$452$	2.33221	0.109698
$453$	0	0
$454$	8.61799	0.404462
$455$	4.56890	0.214193
$456$	0	0
$457$	18.2007	0.851393	0.425697	−	0.904866i	$-0.360029\pi$
$457$	18.2007	0.851393	0.425697	+	0.904866i	$0.360029\pi$
$458$	32.1703	1.50322
$459$	0	0
$460$	−3.17129	−0.147862
$461$	−30.3849	−1.41517	−0.707584	−	0.706630i	$-0.750215\pi$
$461$	−30.3849	−1.41517	−0.707584	+	0.706630i	$0.750215\pi$
$462$	0	0
$463$	−18.0376	−0.838279	−0.419140	−	0.907922i	$-0.637668\pi$
$463$	−18.0376	−0.838279	−0.419140	+	0.907922i	$0.637668\pi$
$464$	−0.652503	−0.0302917
$465$	0	0
$466$	−2.51362	−0.116441
$467$	−21.6738	−1.00294	−0.501471	−	0.865174i	$-0.667208\pi$
$467$	−21.6738	−1.00294	−0.501471	+	0.865174i	$0.667208\pi$
$468$	0	0
$469$	−16.2703	−0.751292
$470$	−13.9093	−0.641590
$471$	0	0
$472$	−12.1956	−0.561347
$473$	−8.59739	−0.395308
$474$	0	0
$475$	4.84071	0.222107
$476$	8.21125	0.376362
$477$	0	0
$478$	−10.3372	−0.472815
$479$	−26.7843	−1.22381	−0.611904	−	0.790932i	$-0.709596\pi$
$479$	−26.7843	−1.22381	−0.611904	+	0.790932i	$0.709596\pi$
$480$	0	0
$481$	2.15479	0.0982501
$482$	19.3638	0.881996
$483$	0	0
$484$	−6.66358	−0.302890
$485$	−3.56098	−0.161696
$486$	0	0
$487$	−15.3605	−0.696051	−0.348025	−	0.937485i	$-0.613148\pi$
$487$	−15.3605	−0.696051	−0.348025	+	0.937485i	$0.613148\pi$
$488$	−24.1980	−1.09539
$489$	0	0
$490$	17.4446	0.788067
$491$	14.8892	0.671941	0.335970	−	0.941873i	$-0.390936\pi$
$491$	14.8892	0.671941	0.335970	+	0.941873i	$0.390936\pi$
$492$	0	0
$493$	−0.936834	−0.0421929
$494$	6.08614	0.273828
$495$	0	0
$496$	3.69673	0.165988
$497$	−45.9156	−2.05959
$498$	0	0
$499$	−22.7942	−1.02041	−0.510205	−	0.860053i	$-0.670431\pi$
$499$	−22.7942	−1.02041	−0.510205	+	0.860053i	$0.670431\pi$
$500$	−0.419240	−0.0187490
$501$	0	0
$502$	13.0802	0.583798
$503$	−14.2644	−0.636018	−0.318009	−	0.948088i	$-0.603014\pi$
$503$	−14.2644	−0.636018	−0.318009	+	0.948088i	$0.603014\pi$
$504$	0	0
$505$	15.0455	0.669515
$506$	49.3216	2.19261
$507$	0	0
$508$	5.61018	0.248911
$509$	2.23042	0.0988615	0.0494308	−	0.998778i	$-0.484259\pi$
$509$	2.23042	0.0988615	0.0494308	+	0.998778i	$0.484259\pi$
$510$	0	0
$511$	−44.8096	−1.98226
$512$	−25.1184	−1.11009
$513$	0	0
$514$	1.21223	0.0534691
$515$	19.1186	0.842467
$516$	0	0
$517$	−57.3726	−2.52324
$518$	12.3780	0.543858
$519$	0	0
$520$	−3.04167	−0.133386
$521$	−14.7841	−0.647703	−0.323851	−	0.946108i	$-0.604978\pi$
$521$	−14.7841	−0.647703	−0.323851	+	0.946108i	$0.604978\pi$
$522$	0	0
$523$	18.8788	0.825513	0.412756	−	0.910842i	$-0.364566\pi$
$523$	18.8788	0.825513	0.412756	+	0.910842i	$0.364566\pi$
$524$	−3.02499	−0.132147
$525$	0	0
$526$	−29.3257	−1.27866
$527$	5.30760	0.231203
$528$	0	0
$529$	34.2197	1.48781
$530$	12.0061	0.521511
$531$	0	0
$532$	−9.27222	−0.402002
$533$	−11.9785	−0.518846
$534$	0	0
$535$	−1.48195	−0.0640703
$536$	10.8317	0.467857
$537$	0	0
$538$	4.50878	0.194387
$539$	71.9547	3.09931
$540$	0	0
$541$	34.4090	1.47936	0.739679	−	0.672960i	$-0.234977\pi$
$541$	34.4090	1.47936	0.739679	+	0.672960i	$0.234977\pi$
$542$	2.80237	0.120372
$543$	0	0
$544$	−9.98570	−0.428133
$545$	−1.15271	−0.0493766
$546$	0	0
$547$	34.3421	1.46836	0.734180	−	0.678955i	$-0.237567\pi$
$547$	34.3421	1.46836	0.734180	+	0.678955i	$0.237567\pi$
$548$	−0.283183	−0.0120970
$549$	0	0
$550$	6.52024	0.278024
$551$	1.05788	0.0450673
$552$	0	0
$553$	−15.3912	−0.654502
$554$	25.1716	1.06944
$555$	0	0
$556$	−6.90878	−0.292997
$557$	−0.532967	−0.0225825	−0.0112913	−	0.999936i	$-0.503594\pi$
$557$	−0.532967	−0.0225825	−0.0112913	+	0.999936i	$0.503594\pi$
$558$	0	0
$559$	−1.65781	−0.0701180
$560$	−13.6416	−0.576464
$561$	0	0
$562$	2.25560	0.0951469
$563$	−38.7746	−1.63415	−0.817076	−	0.576530i	$-0.804407\pi$
$563$	−38.7746	−1.63415	−0.817076	+	0.576530i	$0.804407\pi$
$564$	0	0
$565$	−5.56294	−0.234035
$566$	24.7829	1.04170
$567$	0	0
$568$	30.5675	1.28258
$569$	14.9024	0.624741	0.312371	−	0.949960i	$-0.398877\pi$
$569$	14.9024	0.624741	0.312371	+	0.949960i	$0.398877\pi$
$570$	0	0
$571$	20.2247	0.846377	0.423189	−	0.906042i	$-0.360911\pi$
$571$	20.2247	0.846377	0.423189	+	0.906042i	$0.360911\pi$
$572$	−2.17417	−0.0909067
$573$	0	0
$574$	−68.8092	−2.87204
$575$	7.56437	0.315456
$576$	0	0
$577$	−11.0995	−0.462078	−0.231039	−	0.972945i	$-0.574212\pi$
$577$	−11.0995	−0.462078	−0.231039	+	0.972945i	$0.574212\pi$
$578$	1.73101	0.0720005
$579$	0	0
$580$	−0.0916201	−0.00380432
$581$	−19.4515	−0.806983
$582$	0	0
$583$	49.5222	2.05100
$584$	29.8312	1.23443
$585$	0	0
$586$	−10.0304	−0.414354
$587$	6.44346	0.265950	0.132975	−	0.991119i	$-0.457547\pi$
$587$	6.44346	0.265950	0.132975	+	0.991119i	$0.457547\pi$
$588$	0	0
$589$	−5.99339	−0.246953
$590$	5.04107	0.207538
$591$	0	0
$592$	−6.43369	−0.264423
$593$	−21.5980	−0.886923	−0.443462	−	0.896293i	$-0.646250\pi$
$593$	−21.5980	−0.886923	−0.443462	+	0.896293i	$0.646250\pi$
$594$	0	0
$595$	−19.5860	−0.802949
$596$	−1.94641	−0.0797280
$597$	0	0
$598$	9.51055	0.388915
$599$	−24.7733	−1.01221	−0.506104	−	0.862472i	$-0.668915\pi$
$599$	−24.7733	−1.01221	−0.506104	+	0.862472i	$0.668915\pi$
$600$	0	0
$601$	−37.2755	−1.52050	−0.760250	−	0.649631i	$-0.774924\pi$
$601$	−37.2755	−1.52050	−0.760250	+	0.649631i	$0.774924\pi$
$602$	−9.52313	−0.388134
$603$	0	0
$604$	4.06569	0.165430
$605$	15.8944	0.646200
$606$	0	0
$607$	29.6207	1.20227	0.601133	−	0.799149i	$-0.294716\pi$
$607$	29.6207	1.20227	0.601133	+	0.799149i	$0.294716\pi$
$608$	11.2759	0.457300
$609$	0	0
$610$	10.0023	0.404982
$611$	−11.0630	−0.447562
$612$	0	0
$613$	−20.7223	−0.836966	−0.418483	−	0.908225i	$-0.637438\pi$
$613$	−20.7223	−0.836966	−0.418483	+	0.908225i	$0.637438\pi$
$614$	−31.1975	−1.25903
$615$	0	0
$616$	−72.0700	−2.90378
$617$	−24.3517	−0.980361	−0.490181	−	0.871621i	$-0.663069\pi$
$617$	−24.3517	−0.980361	−0.490181	+	0.871621i	$0.663069\pi$
$618$	0	0
$619$	−2.45515	−0.0986809	−0.0493404	−	0.998782i	$-0.515712\pi$
$619$	−2.45515	−0.0986809	−0.0493404	+	0.998782i	$0.515712\pi$
$620$	0.519070	0.0208464
$621$	0	0
$622$	3.88922	0.155943
$623$	3.23764	0.129713
$624$	0	0
$625$	1.00000	0.0400000
$626$	15.2974	0.611406
$627$	0	0
$628$	1.78432	0.0712022
$629$	−9.23720	−0.368311
$630$	0	0
$631$	−34.2854	−1.36488	−0.682441	−	0.730941i	$-0.739082\pi$
$631$	−34.2854	−1.36488	−0.682441	+	0.730941i	$0.739082\pi$
$632$	10.2465	0.407582
$633$	0	0
$634$	17.7255	0.703969
$635$	−13.3818	−0.531039
$636$	0	0
$637$	13.8748	0.549741
$638$	1.42492	0.0564133
$639$	0	0
$640$	6.53130	0.258172
$641$	−3.18572	−0.125828	−0.0629142	−	0.998019i	$-0.520039\pi$
$641$	−3.18572	−0.125828	−0.0629142	+	0.998019i	$0.520039\pi$
$642$	0	0
$643$	−36.8165	−1.45190	−0.725951	−	0.687747i	$-0.758600\pi$
$643$	−36.8165	−1.45190	−0.725951	+	0.687747i	$0.758600\pi$
$644$	−14.4893	−0.570959
$645$	0	0
$646$	−26.0902	−1.02650
$647$	47.4992	1.86739	0.933693	−	0.358075i	$-0.116567\pi$
$647$	47.4992	1.86739	0.933693	+	0.358075i	$0.116567\pi$
$648$	0	0
$649$	20.7932	0.816204
$650$	1.25728	0.0493147
$651$	0	0
$652$	−5.91226	−0.231542
$653$	−14.9394	−0.584624	−0.292312	−	0.956323i	$-0.594425\pi$
$653$	−14.9394	−0.584624	−0.292312	+	0.956323i	$0.594425\pi$
$654$	0	0
$655$	7.21542	0.281930
$656$	35.7648	1.39638
$657$	0	0
$658$	−63.5504	−2.47745
$659$	−12.4998	−0.486921	−0.243461	−	0.969911i	$-0.578283\pi$
$659$	−12.4998	−0.486921	−0.243461	+	0.969911i	$0.578283\pi$
$660$	0	0
$661$	30.4001	1.18243	0.591213	−	0.806515i	$-0.298649\pi$
$661$	30.4001	1.18243	0.591213	+	0.806515i	$0.298649\pi$
$662$	−11.5228	−0.447848
$663$	0	0
$664$	12.9495	0.502538
$665$	22.1167	0.857650
$666$	0	0
$667$	1.65311	0.0640085
$668$	3.74330	0.144833
$669$	0	0
$670$	−4.47730	−0.172973
$671$	41.2571	1.59271
$672$	0	0
$673$	−31.7416	−1.22355	−0.611774	−	0.791032i	$-0.709544\pi$
$673$	−31.7416	−1.22355	−0.611774	+	0.791032i	$0.709544\pi$
$674$	10.3440	0.398438
$675$	0	0
$676$	−0.419240	−0.0161246
$677$	−23.6232	−0.907914	−0.453957	−	0.891024i	$-0.649988\pi$
$677$	−23.6232	−0.907914	−0.453957	+	0.891024i	$0.649988\pi$
$678$	0	0
$679$	−16.2698	−0.624377
$680$	13.0391	0.500026
$681$	0	0
$682$	−8.07285	−0.309126
$683$	−24.0792	−0.921366	−0.460683	−	0.887565i	$-0.652395\pi$
$683$	−24.0792	−0.921366	−0.460683	+	0.887565i	$0.652395\pi$
$684$	0	0
$685$	0.675467	0.0258083
$686$	39.4918	1.50781
$687$	0	0
$688$	4.94983	0.188710
$689$	9.54923	0.363797
$690$	0	0
$691$	0.539717	0.0205318	0.0102659	−	0.999947i	$-0.496732\pi$
$691$	0.539717	0.0205318	0.0102659	+	0.999947i	$0.496732\pi$
$692$	−8.79555	−0.334357
$693$	0	0
$694$	−20.7087	−0.786090
$695$	16.4793	0.625095
$696$	0	0
$697$	51.3495	1.94500
$698$	16.0431	0.607239
$699$	0	0
$700$	−1.91547	−0.0723978
$701$	7.61552	0.287634	0.143817	−	0.989604i	$-0.454062\pi$
$701$	7.61552	0.287634	0.143817	+	0.989604i	$0.454062\pi$
$702$	0	0
$703$	10.4307	0.393402
$704$	46.1564	1.73958
$705$	0	0
$706$	27.1855	1.02314
$707$	68.7412	2.58528
$708$	0	0
$709$	−32.8373	−1.23323	−0.616616	−	0.787264i	$-0.711497\pi$
$709$	−32.8373	−1.23323	−0.616616	+	0.787264i	$0.711497\pi$
$710$	−12.6352	−0.474190
$711$	0	0
$712$	−2.15541	−0.0807773
$713$	−9.36561	−0.350745
$714$	0	0
$715$	5.18598	0.193945
$716$	−1.53922	−0.0575233
$717$	0	0
$718$	−40.6592	−1.51739
$719$	27.6099	1.02967	0.514837	−	0.857288i	$-0.327852\pi$
$719$	27.6099	1.02967	0.514837	+	0.857288i	$0.327852\pi$
$720$	0	0
$721$	87.3511	3.25312
$722$	5.57287	0.207401
$723$	0	0
$724$	10.7816	0.400695
$725$	0.218538	0.00811631
$726$	0	0
$727$	−10.8981	−0.404189	−0.202094	−	0.979366i	$-0.564775\pi$
$727$	−10.8981	−0.404189	−0.202094	+	0.979366i	$0.564775\pi$
$728$	−13.8971	−0.515060
$729$	0	0
$730$	−12.3308	−0.456385
$731$	7.10674	0.262852
$732$	0	0
$733$	1.50270	0.0555035	0.0277517	−	0.999615i	$-0.491165\pi$
$733$	1.50270	0.0555035	0.0277517	+	0.999615i	$0.491165\pi$
$734$	23.3686	0.862552
$735$	0	0
$736$	17.6204	0.649498
$737$	−18.4678	−0.680269
$738$	0	0
$739$	−32.8385	−1.20798	−0.603991	−	0.796991i	$-0.706424\pi$
$739$	−32.8385	−1.20798	−0.603991	+	0.796991i	$0.706424\pi$
$740$	−0.903376	−0.0332088
$741$	0	0
$742$	54.8546	2.01378
$743$	−0.463514	−0.0170047	−0.00850233	−	0.999964i	$-0.502706\pi$
$743$	−0.463514	−0.0170047	−0.00850233	+	0.999964i	$0.502706\pi$
$744$	0	0
$745$	4.64270	0.170096
$746$	18.8096	0.688668
$747$	0	0
$748$	9.32028	0.340783
$749$	−6.77089	−0.247403
$750$	0	0
$751$	−27.6596	−1.00931	−0.504657	−	0.863320i	$-0.668381\pi$
$751$	−27.6596	−1.00931	−0.504657	+	0.863320i	$0.668381\pi$
$752$	33.0315	1.20453
$753$	0	0
$754$	0.274765	0.0100063
$755$	−9.69775	−0.352937
$756$	0	0
$757$	0.643945	0.0234046	0.0117023	−	0.999932i	$-0.496275\pi$
$757$	0.643945	0.0234046	0.0117023	+	0.999932i	$0.496275\pi$
$758$	30.7429	1.11663
$759$	0	0
$760$	−14.7238	−0.534090
$761$	1.72092	0.0623833	0.0311916	−	0.999513i	$-0.490070\pi$
$761$	1.72092	0.0623833	0.0311916	+	0.999513i	$0.490070\pi$
$762$	0	0
$763$	−5.26661	−0.190664
$764$	7.98709	0.288963
$765$	0	0
$766$	9.46335	0.341925
$767$	4.00950	0.144775
$768$	0	0
$769$	−35.0152	−1.26268	−0.631341	−	0.775506i	$-0.717495\pi$
$769$	−35.0152	−1.26268	−0.631341	+	0.775506i	$0.717495\pi$
$770$	29.7903	1.07357
$771$	0	0
$772$	−0.629425	−0.0226535
$773$	−44.9701	−1.61746	−0.808731	−	0.588179i	$-0.799845\pi$
$773$	−44.9701	−1.61746	−0.808731	+	0.588179i	$0.799845\pi$
$774$	0	0
$775$	−1.23812	−0.0444746
$776$	10.8313	0.388822
$777$	0	0
$778$	−28.0099	−1.00420
$779$	−57.9844	−2.07751
$780$	0	0
$781$	−52.1170	−1.86489
$782$	−40.7700	−1.45793
$783$	0	0
$784$	−41.4269	−1.47953
$785$	−4.25608	−0.151906
$786$	0	0
$787$	−26.2495	−0.935694	−0.467847	−	0.883809i	$-0.654970\pi$
$787$	−26.2495	−0.935694	−0.467847	+	0.883809i	$0.654970\pi$
$788$	8.12175	0.289325
$789$	0	0
$790$	−4.23540	−0.150689
$791$	−25.4165	−0.903707
$792$	0	0
$793$	7.95550	0.282508
$794$	11.4356	0.405836
$795$	0	0
$796$	−5.18539	−0.183791
$797$	−7.58246	−0.268585	−0.134292	−	0.990942i	$-0.542876\pi$
$797$	−7.58246	−0.268585	−0.134292	+	0.990942i	$0.542876\pi$
$798$	0	0
$799$	47.4251	1.67778
$800$	2.32940	0.0823566
$801$	0	0
$802$	−16.3573	−0.577598
$803$	−50.8616	−1.79487
$804$	0	0
$805$	34.5609	1.21811
$806$	−1.55667	−0.0548313
$807$	0	0
$808$	−45.7633	−1.60995
$809$	−24.8033	−0.872036	−0.436018	−	0.899938i	$-0.643612\pi$
$809$	−24.8033	−0.872036	−0.436018	+	0.899938i	$0.643612\pi$
$810$	0	0
$811$	4.73351	0.166216	0.0831080	−	0.996541i	$-0.473515\pi$
$811$	4.73351	0.166216	0.0831080	+	0.996541i	$0.473515\pi$
$812$	−0.418603	−0.0146901
$813$	0	0
$814$	14.0498	0.492445
$815$	14.1023	0.493983
$816$	0	0
$817$	−8.02499	−0.280759
$818$	11.5051	0.402267
$819$	0	0
$820$	5.02186	0.175371
$821$	−19.9424	−0.695993	−0.347997	−	0.937496i	$-0.613138\pi$
$821$	−19.9424	−0.695993	−0.347997	+	0.937496i	$0.613138\pi$
$822$	0	0
$823$	5.10773	0.178044	0.0890222	−	0.996030i	$-0.471626\pi$
$823$	5.10773	0.178044	0.0890222	+	0.996030i	$0.471626\pi$
$824$	−58.1525	−2.02584
$825$	0	0
$826$	23.0322	0.801391
$827$	−32.7290	−1.13810	−0.569050	−	0.822303i	$-0.692689\pi$
$827$	−32.7290	−1.13810	−0.569050	+	0.822303i	$0.692689\pi$
$828$	0	0
$829$	−14.7155	−0.511091	−0.255546	−	0.966797i	$-0.582255\pi$
$829$	−14.7155	−0.511091	−0.255546	+	0.966797i	$0.582255\pi$
$830$	−5.35271	−0.185795
$831$	0	0
$832$	8.90022	0.308560
$833$	−59.4789	−2.06082
$834$	0	0
$835$	−8.92876	−0.308993
$836$	−10.5245	−0.363999
$837$	0	0
$838$	9.98319	0.344864
$839$	5.28342	0.182404	0.0912019	−	0.995832i	$-0.470929\pi$
$839$	5.28342	0.182404	0.0912019	+	0.995832i	$0.470929\pi$
$840$	0	0
$841$	−28.9522	−0.998353
$842$	−1.66047	−0.0572235
$843$	0	0
$844$	3.24378	0.111655
$845$	1.00000	0.0344010
$846$	0	0
$847$	72.6199	2.49525
$848$	−28.5117	−0.979096
$849$	0	0
$850$	−5.38974	−0.184867
$851$	16.2997	0.558745
$852$	0	0
$853$	−50.0056	−1.71216	−0.856080	−	0.516843i	$-0.827107\pi$
$853$	−50.0056	−1.71216	−0.856080	+	0.516843i	$0.827107\pi$
$854$	45.6996	1.56381
$855$	0	0
$856$	4.50760	0.154067
$857$	−40.6303	−1.38790	−0.693952	−	0.720022i	$-0.744132\pi$
$857$	−40.6303	−1.38790	−0.693952	+	0.720022i	$0.744132\pi$
$858$	0	0
$859$	55.3353	1.88802	0.944008	−	0.329921i	$-0.107022\pi$
$859$	55.3353	1.88802	0.944008	+	0.329921i	$0.107022\pi$
$860$	0.695022	0.0237000
$861$	0	0
$862$	−32.3740	−1.10266
$863$	−7.98763	−0.271902	−0.135951	−	0.990716i	$-0.543409\pi$
$863$	−7.98763	−0.271902	−0.135951	+	0.990716i	$0.543409\pi$
$864$	0	0
$865$	20.9797	0.713332
$866$	−19.5573	−0.664583
$867$	0	0
$868$	2.37158	0.0804967
$869$	−17.4700	−0.592629
$870$	0	0
$871$	−3.56109	−0.120663
$872$	3.50616	0.118733
$873$	0	0
$874$	46.0378	1.55725
$875$	4.56890	0.154457
$876$	0	0
$877$	−52.5013	−1.77284	−0.886422	−	0.462878i	$-0.846817\pi$
$877$	−52.5013	−1.77284	−0.886422	+	0.462878i	$0.846817\pi$
$878$	−0.796617	−0.0268845
$879$	0	0
$880$	−15.4841	−0.521968
$881$	0.0697999	0.00235162	0.00117581	−	0.999999i	$-0.499626\pi$
$881$	0.0697999	0.00235162	0.00117581	+	0.999999i	$0.499626\pi$
$882$	0	0
$883$	19.2936	0.649280	0.324640	−	0.945838i	$-0.394757\pi$
$883$	19.2936	0.649280	0.324640	+	0.945838i	$0.394757\pi$
$884$	1.79721	0.0604466
$885$	0	0
$886$	1.44250	0.0484618
$887$	−26.7853	−0.899362	−0.449681	−	0.893189i	$-0.648462\pi$
$887$	−26.7853	−0.899362	−0.449681	+	0.893189i	$0.648462\pi$
$888$	0	0
$889$	−61.1400	−2.05057
$890$	0.890943	0.0298645
$891$	0	0
$892$	0.776492	0.0259989
$893$	−53.5528	−1.79208
$894$	0	0
$895$	3.67145	0.122723
$896$	29.8409	0.996914
$897$	0	0
$898$	−13.7355	−0.458360
$899$	−0.270577	−0.00902425
$900$	0	0
$901$	−40.9358	−1.36377
$902$	−78.1026	−2.60053
$903$	0	0
$904$	16.9206	0.562771
$905$	−25.7170	−0.854862
$906$	0	0
$907$	24.5052	0.813682	0.406841	−	0.913499i	$-0.366630\pi$
$907$	24.5052	0.813682	0.406841	+	0.913499i	$0.366630\pi$
$908$	−2.87366	−0.0953659
$909$	0	0
$910$	5.74440	0.190425
$911$	6.81348	0.225741	0.112870	−	0.993610i	$-0.463996\pi$
$911$	6.81348	0.225741	0.112870	+	0.993610i	$0.463996\pi$
$912$	0	0
$913$	−22.0786	−0.730695
$914$	22.8834	0.756917
$915$	0	0
$916$	−10.7272	−0.354435
$917$	32.9665	1.08865
$918$	0	0
$919$	42.9532	1.41690	0.708448	−	0.705763i	$-0.249396\pi$
$919$	42.9532	1.41690	0.708448	+	0.705763i	$0.249396\pi$
$920$	−23.0083	−0.758562
$921$	0	0
$922$	−38.2025	−1.25813
$923$	−10.0496	−0.330786
$924$	0	0
$925$	2.15479	0.0708492
$926$	−22.6784	−0.745258
$927$	0	0
$928$	0.509063	0.0167108
$929$	22.9418	0.752695	0.376348	−	0.926479i	$-0.377180\pi$
$929$	22.9418	0.752695	0.376348	+	0.926479i	$0.377180\pi$
$930$	0	0
$931$	67.1641	2.20121
$932$	0.838167	0.0274551
$933$	0	0
$934$	−27.2501	−0.891649
$935$	−22.2313	−0.727043
$936$	0	0
$937$	−47.1996	−1.54194	−0.770971	−	0.636870i	$-0.780229\pi$
$937$	−47.1996	−1.54194	−0.770971	+	0.636870i	$0.780229\pi$
$938$	−20.4563	−0.667923
$939$	0	0
$940$	4.63806	0.151277
$941$	22.1145	0.720912	0.360456	−	0.932776i	$-0.382621\pi$
$941$	22.1145	0.720912	0.360456	+	0.932776i	$0.382621\pi$
$942$	0	0
$943$	−90.6097	−2.95066
$944$	−11.9714	−0.389636
$945$	0	0
$946$	−10.8093	−0.351442
$947$	−5.01286	−0.162896	−0.0814480	−	0.996678i	$-0.525954\pi$
$947$	−5.01286	−0.162896	−0.0814480	+	0.996678i	$0.525954\pi$
$948$	0	0
$949$	−9.80752	−0.318366
$950$	6.08614	0.197460
$951$	0	0
$952$	59.5742	1.93081
$953$	−13.2349	−0.428720	−0.214360	−	0.976755i	$-0.568767\pi$
$953$	−13.2349	−0.428720	−0.214360	+	0.976755i	$0.568767\pi$
$954$	0	0
$955$	−19.0513	−0.616487
$956$	3.44695	0.111482
$957$	0	0
$958$	−33.6755	−1.08801
$959$	3.08614	0.0996567
$960$	0	0
$961$	−29.4671	−0.950550
$962$	2.70918	0.0873476
$963$	0	0
$964$	−6.45685	−0.207961
$965$	1.50135	0.0483301
$966$	0	0
$967$	−44.7614	−1.43943	−0.719715	−	0.694269i	$-0.755728\pi$
$967$	−44.7614	−1.43943	−0.719715	+	0.694269i	$0.755728\pi$
$968$	−48.3455	−1.55388
$969$	0	0
$970$	−4.47716	−0.143753
$971$	10.6241	0.340942	0.170471	−	0.985363i	$-0.445471\pi$
$971$	10.6241	0.340942	0.170471	+	0.985363i	$0.445471\pi$
$972$	0	0
$973$	75.2922	2.41376
$974$	−19.3125	−0.618812
$975$	0	0
$976$	−23.7532	−0.760321
$977$	15.7768	0.504744	0.252372	−	0.967630i	$-0.418789\pi$
$977$	15.7768	0.504744	0.252372	+	0.967630i	$0.418789\pi$
$978$	0	0
$979$	3.67492	0.117451
$980$	−5.81689	−0.185814
$981$	0	0
$982$	18.7199	0.597377
$983$	29.1464	0.929625	0.464812	−	0.885409i	$-0.346122\pi$
$983$	29.1464	0.929625	0.464812	+	0.885409i	$0.346122\pi$
$984$	0	0
$985$	−19.3725	−0.617260
$986$	−1.17787	−0.0375109
$987$	0	0
$988$	−2.02942	−0.0645645
$989$	−12.5403	−0.398759
$990$	0	0
$991$	−0.603228	−0.0191622	−0.00958109	−	0.999954i	$-0.503050\pi$
$991$	−0.603228	−0.0191622	−0.00958109	+	0.999954i	$0.503050\pi$
$992$	−2.88408	−0.0915695
$993$	0	0
$994$	−57.7288	−1.83105
$995$	12.3685	0.392109
$996$	0	0
$997$	−43.3472	−1.37282	−0.686410	−	0.727215i	$-0.740814\pi$
$997$	−43.3472	−1.37282	−0.686410	+	0.727215i	$0.740814\pi$
$998$	−28.6588	−0.907179
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5265.2.a.bl.1.11		15
3.2	odd	2		5265.2.a.bk.1.5		15
9.2	odd	6		585.2.i.h.391.11	yes	30
9.4	even	3		1755.2.i.h.586.5		30
9.5	odd	6		585.2.i.h.196.11	✓	30
9.7	even	3		1755.2.i.h.1171.5		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.i.h.196.11	✓	30	9.5	odd	6
585.2.i.h.391.11	yes	30	9.2	odd	6
1755.2.i.h.586.5		30	9.4	even	3
1755.2.i.h.1171.5		30	9.7	even	3
5265.2.a.bk.1.5		15	3.2	odd	2
5265.2.a.bl.1.11		15	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:	\( N \)	\(=\)	\( 5265 = 3^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5265.a (trivial)

\(f(q)\)	\(=\)	\(q+1.25728 q^{2} -0.419240 q^{4} +1.00000 q^{5} +4.56890 q^{7} -3.04167 q^{8} +O(q^{10})\)	\(q+1.25728 q^{2} -0.419240 q^{4} +1.00000 q^{5} +4.56890 q^{7} -3.04167 q^{8} +1.25728 q^{10} +5.18598 q^{11} +1.00000 q^{13} +5.74440 q^{14} -2.98576 q^{16} -4.28682 q^{17} +4.84071 q^{19} -0.419240 q^{20} +6.52024 q^{22} +7.56437 q^{23} +1.00000 q^{25} +1.25728 q^{26} -1.91547 q^{28} +0.218538 q^{29} -1.23812 q^{31} +2.32940 q^{32} -5.38974 q^{34} +4.56890 q^{35} +2.15479 q^{37} +6.08614 q^{38} -3.04167 q^{40} -11.9785 q^{41} -1.65781 q^{43} -2.17417 q^{44} +9.51055 q^{46} -11.0630 q^{47} +13.8748 q^{49} +1.25728 q^{50} -0.419240 q^{52} +9.54923 q^{53} +5.18598 q^{55} -13.8971 q^{56} +0.274765 q^{58} +4.00950 q^{59} +7.95550 q^{61} -1.55667 q^{62} +8.90022 q^{64} +1.00000 q^{65} -3.56109 q^{67} +1.79721 q^{68} +5.74440 q^{70} -10.0496 q^{71} -9.80752 q^{73} +2.70918 q^{74} -2.02942 q^{76} +23.6942 q^{77} -3.36870 q^{79} -2.98576 q^{80} -15.0603 q^{82} -4.25736 q^{83} -4.28682 q^{85} -2.08434 q^{86} -15.7740 q^{88} +0.708626 q^{89} +4.56890 q^{91} -3.17129 q^{92} -13.9093 q^{94} +4.84071 q^{95} -3.56098 q^{97} +17.4446 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + q^{2} + 21 q^{4} + 15 q^{5} + 10 q^{7}+O(q^{10}) \) 15 * q + q^2 + 21 * q^4 + 15 * q^5 + 10 * q^7	\( 15 q + q^{2} + 21 q^{4} + 15 q^{5} + 10 q^{7} + q^{10} + 9 q^{11} + 15 q^{13} + 3 q^{14} + 33 q^{16} - 3 q^{17} + 15 q^{19} + 21 q^{20} + 10 q^{22} - 6 q^{23} + 15 q^{25} + q^{26} + 35 q^{28} + 8 q^{29} + 22 q^{31} + 21 q^{32} + 9 q^{34} + 10 q^{35} + 4 q^{37} - 14 q^{38} + 13 q^{41} + 24 q^{43} - 5 q^{44} - 3 q^{46} - q^{47} + 37 q^{49} + q^{50} + 21 q^{52} - 7 q^{53} + 9 q^{55} + 17 q^{56} + 22 q^{58} + 19 q^{59} + 16 q^{61} - 13 q^{62} + 36 q^{64} + 15 q^{65} + 11 q^{67} - 28 q^{68} + 3 q^{70} + 28 q^{71} + 26 q^{73} + 8 q^{74} + 18 q^{76} - 24 q^{77} + 44 q^{79} + 33 q^{80} + 35 q^{82} - 3 q^{83} - 3 q^{85} + 40 q^{86} + 37 q^{88} + 4 q^{89} + 10 q^{91} - 74 q^{92} + 2 q^{94} + 15 q^{95} + 33 q^{97} - 3 q^{98}+O(q^{100}) \) 15 * q + q^2 + 21 * q^4 + 15 * q^5 + 10 * q^7 + q^10 + 9 * q^11 + 15 * q^13 + 3 * q^14 + 33 * q^16 - 3 * q^17 + 15 * q^19 + 21 * q^20 + 10 * q^22 - 6 * q^23 + 15 * q^25 + q^26 + 35 * q^28 + 8 * q^29 + 22 * q^31 + 21 * q^32 + 9 * q^34 + 10 * q^35 + 4 * q^37 - 14 * q^38 + 13 * q^41 + 24 * q^43 - 5 * q^44 - 3 * q^46 - q^47 + 37 * q^49 + q^50 + 21 * q^52 - 7 * q^53 + 9 * q^55 + 17 * q^56 + 22 * q^58 + 19 * q^59 + 16 * q^61 - 13 * q^62 + 36 * q^64 + 15 * q^65 + 11 * q^67 - 28 * q^68 + 3 * q^70 + 28 * q^71 + 26 * q^73 + 8 * q^74 + 18 * q^76 - 24 * q^77 + 44 * q^79 + 33 * q^80 + 35 * q^82 - 3 * q^83 - 3 * q^85 + 40 * q^86 + 37 * q^88 + 4 * q^89 + 10 * q^91 - 74 * q^92 + 2 * q^94 + 15 * q^95 + 33 * q^97 - 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more