LMFDB - Embedded newform 5265.2.a.bl.1.13

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			1.13
Root			$2.35360$ 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)

$f(q)$	$=$	$q+2.35360 q^{2} +3.53946 q^{4} +1.00000 q^{5} -0.0779966 q^{7} +3.62327 q^{8} +O(q^{10})$	\(q+2.35360 q^{2} +3.53946 q^{4} +1.00000 q^{5} -0.0779966 q^{7} +3.62327 q^{8} +2.35360 q^{10} +1.03749 q^{11} +1.00000 q^{13} -0.183573 q^{14} +1.44884 q^{16} +3.01674 q^{17} -1.33883 q^{19} +3.53946 q^{20} +2.44184 q^{22} +7.32023 q^{23} +1.00000 q^{25} +2.35360 q^{26} -0.276066 q^{28} +4.09342 q^{29} -6.98260 q^{31} -3.83655 q^{32} +7.10022 q^{34} -0.0779966 q^{35} +5.18283 q^{37} -3.15108 q^{38} +3.62327 q^{40} +3.31340 q^{41} +5.35402 q^{43} +3.67215 q^{44} +17.2289 q^{46} +9.61821 q^{47} -6.99392 q^{49} +2.35360 q^{50} +3.53946 q^{52} -10.9978 q^{53} +1.03749 q^{55} -0.282603 q^{56} +9.63430 q^{58} +0.0478566 q^{59} +4.57173 q^{61} -16.4343 q^{62} -11.9274 q^{64} +1.00000 q^{65} +8.38035 q^{67} +10.6776 q^{68} -0.183573 q^{70} -2.31094 q^{71} -2.28031 q^{73} +12.1983 q^{74} -4.73873 q^{76} -0.0809206 q^{77} +2.44912 q^{79} +1.44884 q^{80} +7.79844 q^{82} +1.81706 q^{83} +3.01674 q^{85} +12.6012 q^{86} +3.75910 q^{88} +9.90169 q^{89} -0.0779966 q^{91} +25.9096 q^{92} +22.6375 q^{94} -1.33883 q^{95} +9.26477 q^{97} -16.4609 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.35360	1.66425	0.832125	−	0.554588i	$-0.187124\pi$
$2$	2.35360	1.66425	0.832125	+	0.554588i	$0.187124\pi$
$3$	0	0
$3$	0	0
$4$	3.53946	1.76973
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.0779966	−0.0294799	−0.0147400	−	0.999891i	$-0.504692\pi$
$7$	−0.0779966	−0.0294799	−0.0147400	+	0.999891i	$0.504692\pi$
$8$	3.62327	1.28102
$9$	0	0
$10$	2.35360	0.744275
$11$	1.03749	0.312815	0.156407	−	0.987693i	$-0.450009\pi$
$11$	1.03749	0.312815	0.156407	+	0.987693i	$0.450009\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−0.183573	−0.0490620
$15$	0	0
$16$	1.44884	0.362210
$17$	3.01674	0.731667	0.365834	−	0.930680i	$-0.380784\pi$
$17$	3.01674	0.731667	0.365834	+	0.930680i	$0.380784\pi$
$18$	0	0
$19$	−1.33883	−0.307149	−0.153574	−	0.988137i	$-0.549078\pi$
$19$	−1.33883	−0.307149	−0.153574	+	0.988137i	$0.549078\pi$
$20$	3.53946	0.791447
$21$	0	0
$22$	2.44184	0.520602
$23$	7.32023	1.52637	0.763186	−	0.646178i	$-0.223634\pi$
$23$	7.32023	1.52637	0.763186	+	0.646178i	$0.223634\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.35360	0.461580
$27$	0	0
$28$	−0.276066	−0.0521715
$29$	4.09342	0.760129	0.380065	−	0.924960i	$-0.375902\pi$
$29$	4.09342	0.760129	0.380065	+	0.924960i	$0.375902\pi$
$30$	0	0
$31$	−6.98260	−1.25411	−0.627056	−	0.778974i	$-0.715740\pi$
$31$	−6.98260	−1.25411	−0.627056	+	0.778974i	$0.715740\pi$
$32$	−3.83655	−0.678213
$33$	0	0
$34$	7.10022	1.21768
$35$	−0.0779966	−0.0131838
$36$	0	0
$37$	5.18283	0.852052	0.426026	−	0.904711i	$-0.359913\pi$
$37$	5.18283	0.852052	0.426026	+	0.904711i	$0.359913\pi$
$38$	−3.15108	−0.511172
$39$	0	0
$40$	3.62327	0.572890
$41$	3.31340	0.517467	0.258733	−	0.965949i	$-0.416695\pi$
$41$	3.31340	0.517467	0.258733	+	0.965949i	$0.416695\pi$
$42$	0	0
$43$	5.35402	0.816480	0.408240	−	0.912875i	$-0.366143\pi$
$43$	5.35402	0.816480	0.408240	+	0.912875i	$0.366143\pi$
$44$	3.67215	0.553597
$45$	0	0
$46$	17.2289	2.54027
$47$	9.61821	1.40296	0.701480	−	0.712689i	$-0.252523\pi$
$47$	9.61821	1.40296	0.701480	+	0.712689i	$0.252523\pi$
$48$	0	0
$49$	−6.99392	−0.999131
$50$	2.35360	0.332850
$51$	0	0
$52$	3.53946	0.490834
$53$	−10.9978	−1.51067	−0.755333	−	0.655342i	$-0.772525\pi$
$53$	−10.9978	−1.51067	−0.755333	+	0.655342i	$0.772525\pi$
$54$	0	0
$55$	1.03749	0.139895
$56$	−0.282603	−0.0377644
$57$	0	0
$58$	9.63430	1.26505
$59$	0.0478566	0.00623039	0.00311520	−	0.999995i	$-0.499008\pi$
$59$	0.0478566	0.00623039	0.00311520	+	0.999995i	$0.499008\pi$
$60$	0	0
$61$	4.57173	0.585351	0.292675	−	0.956212i	$-0.405454\pi$
$61$	4.57173	0.585351	0.292675	+	0.956212i	$0.405454\pi$
$62$	−16.4343	−2.08716
$63$	0	0
$64$	−11.9274	−1.49093
$65$	1.00000	0.124035
$66$	0	0
$67$	8.38035	1.02382	0.511912	−	0.859038i	$-0.328938\pi$
$67$	8.38035	1.02382	0.511912	+	0.859038i	$0.328938\pi$
$68$	10.6776	1.29485
$69$	0	0
$70$	−0.183573	−0.0219412
$71$	−2.31094	−0.274259	−0.137129	−	0.990553i	$-0.543788\pi$
$71$	−2.31094	−0.274259	−0.137129	+	0.990553i	$0.543788\pi$
$72$	0	0
$73$	−2.28031	−0.266890	−0.133445	−	0.991056i	$-0.542604\pi$
$73$	−2.28031	−0.266890	−0.133445	+	0.991056i	$0.542604\pi$
$74$	12.1983	1.41803
$75$	0	0
$76$	−4.73873	−0.543570
$77$	−0.0809206	−0.00922176
$78$	0	0
$79$	2.44912	0.275547	0.137773	−	0.990464i	$-0.456005\pi$
$79$	2.44912	0.275547	0.137773	+	0.990464i	$0.456005\pi$
$80$	1.44884	0.161985
$81$	0	0
$82$	7.79844	0.861194
$83$	1.81706	0.199448	0.0997239	−	0.995015i	$-0.468204\pi$
$83$	1.81706	0.199448	0.0997239	+	0.995015i	$0.468204\pi$
$84$	0	0
$85$	3.01674	0.327212
$86$	12.6012	1.35883
$87$	0	0
$88$	3.75910	0.400722
$89$	9.90169	1.04958	0.524789	−	0.851233i	$-0.324144\pi$
$89$	9.90169	1.04958	0.524789	+	0.851233i	$0.324144\pi$
$90$	0	0
$91$	−0.0779966	−0.00817626
$92$	25.9096	2.70126
$93$	0	0
$94$	22.6375	2.33488
$95$	−1.33883	−0.137361
$96$	0	0
$97$	9.26477	0.940695	0.470347	−	0.882481i	$-0.344129\pi$
$97$	9.26477	0.940695	0.470347	+	0.882481i	$0.344129\pi$
$98$	−16.4609	−1.66280
$99$	0	0
$100$	3.53946	0.353946
$101$	−5.43031	−0.540336	−0.270168	−	0.962813i	$-0.587079\pi$
$101$	−5.43031	−0.540336	−0.270168	+	0.962813i	$0.587079\pi$
$102$	0	0
$103$	13.0711	1.28794	0.643969	−	0.765052i	$-0.277287\pi$
$103$	13.0711	1.28794	0.643969	+	0.765052i	$0.277287\pi$
$104$	3.62327	0.355291
$105$	0	0
$106$	−25.8845	−2.51412
$107$	−16.5067	−1.59576	−0.797882	−	0.602814i	$-0.794046\pi$
$107$	−16.5067	−1.59576	−0.797882	+	0.602814i	$0.794046\pi$
$108$	0	0
$109$	−15.2455	−1.46025	−0.730126	−	0.683312i	$-0.760539\pi$
$109$	−15.2455	−1.46025	−0.730126	+	0.683312i	$0.760539\pi$
$110$	2.44184	0.232820
$111$	0	0
$112$	−0.113004	−0.0106779
$113$	10.4248	0.980684	0.490342	−	0.871530i	$-0.336872\pi$
$113$	10.4248	0.980684	0.490342	+	0.871530i	$0.336872\pi$
$114$	0	0
$115$	7.32023	0.682615
$116$	14.4885	1.34522
$117$	0	0
$118$	0.112635	0.0103689
$119$	−0.235296	−0.0215695
$120$	0	0
$121$	−9.92362	−0.902147
$122$	10.7601	0.974170
$123$	0	0
$124$	−24.7146	−2.21944
$125$	1.00000	0.0894427
$126$	0	0
$127$	−14.9255	−1.32442	−0.662211	−	0.749317i	$-0.730382\pi$
$127$	−14.9255	−1.32442	−0.662211	+	0.749317i	$0.730382\pi$
$128$	−20.3993	−1.80306
$129$	0	0
$130$	2.35360	0.206425
$131$	−12.8136	−1.11953	−0.559763	−	0.828653i	$-0.689108\pi$
$131$	−12.8136	−1.11953	−0.559763	+	0.828653i	$0.689108\pi$
$132$	0	0
$133$	0.104424	0.00905473
$134$	19.7240	1.70390
$135$	0	0
$136$	10.9305	0.937281
$137$	13.3005	1.13634	0.568170	−	0.822911i	$-0.307652\pi$
$137$	13.3005	1.13634	0.568170	+	0.822911i	$0.307652\pi$
$138$	0	0
$139$	4.49804	0.381519	0.190760	−	0.981637i	$-0.438905\pi$
$139$	4.49804	0.381519	0.190760	+	0.981637i	$0.438905\pi$
$140$	−0.276066	−0.0233318
$141$	0	0
$142$	−5.43905	−0.456435
$143$	1.03749	0.0867592
$144$	0	0
$145$	4.09342	0.339940
$146$	−5.36694	−0.444171
$147$	0	0
$148$	18.3444	1.50790
$149$	−1.66167	−0.136129	−0.0680647	−	0.997681i	$-0.521682\pi$
$149$	−1.66167	−0.136129	−0.0680647	+	0.997681i	$0.521682\pi$
$150$	0	0
$151$	18.5440	1.50909	0.754546	−	0.656247i	$-0.227857\pi$
$151$	18.5440	1.50909	0.754546	+	0.656247i	$0.227857\pi$
$152$	−4.85095	−0.393464
$153$	0	0
$154$	−0.190455	−0.0153473
$155$	−6.98260	−0.560856
$156$	0	0
$157$	10.6046	0.846341	0.423170	−	0.906050i	$-0.360917\pi$
$157$	10.6046	0.846341	0.423170	+	0.906050i	$0.360917\pi$
$158$	5.76425	0.458579
$159$	0	0
$160$	−3.83655	−0.303306
$161$	−0.570953	−0.0449974
$162$	0	0
$163$	2.10755	0.165076	0.0825379	−	0.996588i	$-0.473697\pi$
$163$	2.10755	0.165076	0.0825379	+	0.996588i	$0.473697\pi$
$164$	11.7276	0.915775
$165$	0	0
$166$	4.27663	0.331931
$167$	−6.40821	−0.495882	−0.247941	−	0.968775i	$-0.579754\pi$
$167$	−6.40821	−0.495882	−0.247941	+	0.968775i	$0.579754\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	7.10022	0.544562
$171$	0	0
$172$	18.9503	1.44495
$173$	−9.84018	−0.748135	−0.374067	−	0.927402i	$-0.622037\pi$
$173$	−9.84018	−0.748135	−0.374067	+	0.927402i	$0.622037\pi$
$174$	0	0
$175$	−0.0779966	−0.00589599
$176$	1.50315	0.113305
$177$	0	0
$178$	23.3047	1.74676
$179$	−13.5026	−1.00923	−0.504614	−	0.863345i	$-0.668365\pi$
$179$	−13.5026	−1.00923	−0.504614	+	0.863345i	$0.668365\pi$
$180$	0	0
$181$	−2.07776	−0.154439	−0.0772194	−	0.997014i	$-0.524604\pi$
$181$	−2.07776	−0.154439	−0.0772194	+	0.997014i	$0.524604\pi$
$182$	−0.183573	−0.0136073
$183$	0	0
$184$	26.5232	1.95531
$185$	5.18283	0.381049
$186$	0	0
$187$	3.12984	0.228876
$188$	34.0432	2.48286
$189$	0	0
$190$	−3.15108	−0.228603
$191$	−10.5100	−0.760480	−0.380240	−	0.924888i	$-0.624159\pi$
$191$	−10.5100	−0.760480	−0.380240	+	0.924888i	$0.624159\pi$
$192$	0	0
$193$	−18.3262	−1.31915	−0.659575	−	0.751639i	$-0.729264\pi$
$193$	−18.3262	−1.31915	−0.659575	+	0.751639i	$0.729264\pi$
$194$	21.8056	1.56555
$195$	0	0
$196$	−24.7547	−1.76819
$197$	7.66870	0.546372	0.273186	−	0.961961i	$-0.411923\pi$
$197$	7.66870	0.546372	0.273186	+	0.961961i	$0.411923\pi$
$198$	0	0
$199$	16.8297	1.19302	0.596512	−	0.802604i	$-0.296553\pi$
$199$	16.8297	1.19302	0.596512	+	0.802604i	$0.296553\pi$
$200$	3.62327	0.256204
$201$	0	0
$202$	−12.7808	−0.899255
$203$	−0.319273	−0.0224086
$204$	0	0
$205$	3.31340	0.231418
$206$	30.7643	2.14345
$207$	0	0
$208$	1.44884	0.100459
$209$	−1.38902	−0.0960806
$210$	0	0
$211$	−1.63317	−0.112432	−0.0562159	−	0.998419i	$-0.517904\pi$
$211$	−1.63317	−0.112432	−0.0562159	+	0.998419i	$0.517904\pi$
$212$	−38.9263	−2.67347
$213$	0	0
$214$	−38.8503	−2.65575
$215$	5.35402	0.365141
$216$	0	0
$217$	0.544619	0.0369712
$218$	−35.8819	−2.43023
$219$	0	0
$220$	3.67215	0.247576
$221$	3.01674	0.202928
$222$	0	0
$223$	−12.2022	−0.817123	−0.408561	−	0.912731i	$-0.633969\pi$
$223$	−12.2022	−0.817123	−0.408561	+	0.912731i	$0.633969\pi$
$224$	0.299238	0.0199937
$225$	0	0
$226$	24.5359	1.63210
$227$	−9.00385	−0.597607	−0.298803	−	0.954315i	$-0.596587\pi$
$227$	−9.00385	−0.597607	−0.298803	+	0.954315i	$0.596587\pi$
$228$	0	0
$229$	−23.6934	−1.56570	−0.782852	−	0.622208i	$-0.786236\pi$
$229$	−23.6934	−1.56570	−0.782852	+	0.622208i	$0.786236\pi$
$230$	17.2289	1.13604
$231$	0	0
$232$	14.8316	0.973741
$233$	−25.4172	−1.66514	−0.832569	−	0.553921i	$-0.813131\pi$
$233$	−25.4172	−1.66514	−0.832569	+	0.553921i	$0.813131\pi$
$234$	0	0
$235$	9.61821	0.627423
$236$	0.169386	0.0110261
$237$	0	0
$238$	−0.553793	−0.0358971
$239$	0.285218	0.0184492	0.00922462	−	0.999957i	$-0.497064\pi$
$239$	0.285218	0.0184492	0.00922462	+	0.999957i	$0.497064\pi$
$240$	0	0
$241$	−11.6001	−0.747228	−0.373614	−	0.927584i	$-0.621881\pi$
$241$	−11.6001	−0.747228	−0.373614	+	0.927584i	$0.621881\pi$
$242$	−23.3563	−1.50140
$243$	0	0
$244$	16.1815	1.03591
$245$	−6.99392	−0.446825
$246$	0	0
$247$	−1.33883	−0.0851877
$248$	−25.2999	−1.60654
$249$	0	0
$250$	2.35360	0.148855
$251$	22.4343	1.41604	0.708019	−	0.706193i	$-0.249589\pi$
$251$	22.4343	1.41604	0.708019	+	0.706193i	$0.249589\pi$
$252$	0	0
$253$	7.59465	0.477472
$254$	−35.1287	−2.20417
$255$	0	0
$256$	−24.1571	−1.50982
$257$	−1.91961	−0.119742	−0.0598709	−	0.998206i	$-0.519069\pi$
$257$	−1.91961	−0.119742	−0.0598709	+	0.998206i	$0.519069\pi$
$258$	0	0
$259$	−0.404243	−0.0251184
$260$	3.53946	0.219508
$261$	0	0
$262$	−30.1580	−1.86317
$263$	24.8220	1.53059	0.765295	−	0.643679i	$-0.222593\pi$
$263$	24.8220	1.53059	0.765295	+	0.643679i	$0.222593\pi$
$264$	0	0
$265$	−10.9978	−0.675590
$266$	0.245773	0.0150693
$267$	0	0
$268$	29.6619	1.81189
$269$	−10.3259	−0.629584	−0.314792	−	0.949161i	$-0.601935\pi$
$269$	−10.3259	−0.629584	−0.314792	+	0.949161i	$0.601935\pi$
$270$	0	0
$271$	−28.5144	−1.73213	−0.866065	−	0.499932i	$-0.833358\pi$
$271$	−28.5144	−1.73213	−0.866065	+	0.499932i	$0.833358\pi$
$272$	4.37077	0.265017
$273$	0	0
$274$	31.3042	1.89115
$275$	1.03749	0.0625629
$276$	0	0
$277$	19.1472	1.15044	0.575222	−	0.817997i	$-0.304916\pi$
$277$	19.1472	1.15044	0.575222	+	0.817997i	$0.304916\pi$
$278$	10.5866	0.634943
$279$	0	0
$280$	−0.282603	−0.0168888
$281$	−12.4009	−0.739776	−0.369888	−	0.929076i	$-0.620604\pi$
$281$	−12.4009	−0.739776	−0.369888	+	0.929076i	$0.620604\pi$
$282$	0	0
$283$	23.1153	1.37406	0.687032	−	0.726627i	$-0.258913\pi$
$283$	23.1153	1.37406	0.687032	+	0.726627i	$0.258913\pi$
$284$	−8.17948	−0.485363
$285$	0	0
$286$	2.44184	0.144389
$287$	−0.258434	−0.0152549
$288$	0	0
$289$	−7.89927	−0.464663
$290$	9.63430	0.565745
$291$	0	0
$292$	−8.07105	−0.472322
$293$	−19.6947	−1.15058	−0.575290	−	0.817950i	$-0.695111\pi$
$293$	−19.6947	−1.15058	−0.575290	+	0.817950i	$0.695111\pi$
$294$	0	0
$295$	0.0478566	0.00278632
$296$	18.7788	1.09150
$297$	0	0
$298$	−3.91092	−0.226553
$299$	7.32023	0.423340
$300$	0	0
$301$	−0.417595	−0.0240698
$302$	43.6453	2.51151
$303$	0	0
$304$	−1.93975	−0.111252
$305$	4.57173	0.261777
$306$	0	0
$307$	−26.9891	−1.54035	−0.770175	−	0.637833i	$-0.779831\pi$
$307$	−26.9891	−1.54035	−0.770175	+	0.637833i	$0.779831\pi$
$308$	−0.286415	−0.0163200
$309$	0	0
$310$	−16.4343	−0.933405
$311$	20.0020	1.13421	0.567106	−	0.823645i	$-0.308063\pi$
$311$	20.0020	1.13421	0.567106	+	0.823645i	$0.308063\pi$
$312$	0	0
$313$	21.5456	1.21783	0.608915	−	0.793236i	$-0.291605\pi$
$313$	21.5456	1.21783	0.608915	+	0.793236i	$0.291605\pi$
$314$	24.9591	1.40852
$315$	0	0
$316$	8.66854	0.487643
$317$	−16.2187	−0.910936	−0.455468	−	0.890252i	$-0.650528\pi$
$317$	−16.2187	−0.910936	−0.455468	+	0.890252i	$0.650528\pi$
$318$	0	0
$319$	4.24688	0.237780
$320$	−11.9274	−0.666762
$321$	0	0
$322$	−1.34380	−0.0748869
$323$	−4.03891	−0.224731
$324$	0	0
$325$	1.00000	0.0554700
$326$	4.96033	0.274727
$327$	0	0
$328$	12.0054	0.662885
$329$	−0.750187	−0.0413592
$330$	0	0
$331$	4.97071	0.273215	0.136607	−	0.990625i	$-0.456380\pi$
$331$	4.97071	0.273215	0.136607	+	0.990625i	$0.456380\pi$
$332$	6.43139	0.352968
$333$	0	0
$334$	−15.0824	−0.825272
$335$	8.38035	0.457868
$336$	0	0
$337$	−25.5116	−1.38971	−0.694853	−	0.719152i	$-0.744530\pi$
$337$	−25.5116	−1.38971	−0.694853	+	0.719152i	$0.744530\pi$
$338$	2.35360	0.128019
$339$	0	0
$340$	10.6776	0.579076
$341$	−7.24437	−0.392305
$342$	0	0
$343$	1.09148	0.0589343
$344$	19.3991	1.04593
$345$	0	0
$346$	−23.1599	−1.24508
$347$	−16.1966	−0.869477	−0.434738	−	0.900557i	$-0.643159\pi$
$347$	−16.1966	−0.869477	−0.434738	+	0.900557i	$0.643159\pi$
$348$	0	0
$349$	15.6506	0.837756	0.418878	−	0.908043i	$-0.362423\pi$
$349$	15.6506	0.837756	0.418878	+	0.908043i	$0.362423\pi$
$350$	−0.183573	−0.00981240
$351$	0	0
$352$	−3.98038	−0.212155
$353$	−11.4906	−0.611584	−0.305792	−	0.952098i	$-0.598921\pi$
$353$	−11.4906	−0.611584	−0.305792	+	0.952098i	$0.598921\pi$
$354$	0	0
$355$	−2.31094	−0.122652
$356$	35.0466	1.85747
$357$	0	0
$358$	−31.7797	−1.67961
$359$	29.3245	1.54769	0.773844	−	0.633376i	$-0.218331\pi$
$359$	29.3245	1.54769	0.773844	+	0.633376i	$0.218331\pi$
$360$	0	0
$361$	−17.2075	−0.905660
$362$	−4.89023	−0.257025
$363$	0	0
$364$	−0.276066	−0.0144698
$365$	−2.28031	−0.119357
$366$	0	0
$367$	3.36332	0.175564	0.0877820	−	0.996140i	$-0.472022\pi$
$367$	3.36332	0.175564	0.0877820	+	0.996140i	$0.472022\pi$
$368$	10.6058	0.552867
$369$	0	0
$370$	12.1983	0.634161
$371$	0.857791	0.0445343
$372$	0	0
$373$	7.87780	0.407897	0.203949	−	0.978982i	$-0.434622\pi$
$373$	7.87780	0.407897	0.203949	+	0.978982i	$0.434622\pi$
$374$	7.36640	0.380907
$375$	0	0
$376$	34.8494	1.79722
$377$	4.09342	0.210822
$378$	0	0
$379$	15.4148	0.791805	0.395903	−	0.918293i	$-0.370432\pi$
$379$	15.4148	0.791805	0.395903	+	0.918293i	$0.370432\pi$
$380$	−4.73873	−0.243092
$381$	0	0
$382$	−24.7365	−1.26563
$383$	−34.2315	−1.74915	−0.874575	−	0.484890i	$-0.838860\pi$
$383$	−34.2315	−1.74915	−0.874575	+	0.484890i	$0.838860\pi$
$384$	0	0
$385$	−0.0809206	−0.00412410
$386$	−43.1327	−2.19539
$387$	0	0
$388$	32.7922	1.66477
$389$	28.3996	1.43992	0.719959	−	0.694017i	$-0.244161\pi$
$389$	28.3996	1.43992	0.719959	+	0.694017i	$0.244161\pi$
$390$	0	0
$391$	22.0832	1.11680
$392$	−25.3409	−1.27991
$393$	0	0
$394$	18.0491	0.909300
$395$	2.44912	0.123228
$396$	0	0
$397$	−29.4099	−1.47604	−0.738021	−	0.674778i	$-0.764239\pi$
$397$	−29.4099	−1.47604	−0.738021	+	0.674778i	$0.764239\pi$
$398$	39.6104	1.98549
$399$	0	0
$400$	1.44884	0.0724419
$401$	−32.8090	−1.63840	−0.819201	−	0.573506i	$-0.805583\pi$
$401$	−32.8090	−1.63840	−0.819201	+	0.573506i	$0.805583\pi$
$402$	0	0
$403$	−6.98260	−0.347828
$404$	−19.2204	−0.956248
$405$	0	0
$406$	−0.751442	−0.0372935
$407$	5.37713	0.266534
$408$	0	0
$409$	14.4931	0.716639	0.358320	−	0.933599i	$-0.383350\pi$
$409$	14.4931	0.716639	0.358320	+	0.933599i	$0.383350\pi$
$410$	7.79844	0.385138
$411$	0	0
$412$	46.2647	2.27930
$413$	−0.00373265	−0.000183672 0
$414$	0	0
$415$	1.81706	0.0891957
$416$	−3.83655	−0.188102
$417$	0	0
$418$	−3.26921	−0.159902
$419$	12.7191	0.621370	0.310685	−	0.950513i	$-0.399442\pi$
$419$	12.7191	0.621370	0.310685	+	0.950513i	$0.399442\pi$
$420$	0	0
$421$	−10.7558	−0.524205	−0.262103	−	0.965040i	$-0.584416\pi$
$421$	−10.7558	−0.524205	−0.262103	+	0.965040i	$0.584416\pi$
$422$	−3.84383	−0.187115
$423$	0	0
$424$	−39.8481	−1.93519
$425$	3.01674	0.146333
$426$	0	0
$427$	−0.356580	−0.0172561
$428$	−58.4247	−2.82407
$429$	0	0
$430$	12.6012	0.607686
$431$	−33.6881	−1.62270	−0.811351	−	0.584560i	$-0.801267\pi$
$431$	−33.6881	−1.62270	−0.811351	+	0.584560i	$0.801267\pi$
$432$	0	0
$433$	40.2183	1.93277	0.966384	−	0.257104i	$-0.0827683\pi$
$433$	40.2183	1.93277	0.966384	+	0.257104i	$0.0827683\pi$
$434$	1.28182	0.0615293
$435$	0	0
$436$	−53.9607	−2.58425
$437$	−9.80054	−0.468823
$438$	0	0
$439$	18.1536	0.866423	0.433212	−	0.901292i	$-0.357380\pi$
$439$	18.1536	0.866423	0.433212	+	0.901292i	$0.357380\pi$
$440$	3.75910	0.179208
$441$	0	0
$442$	7.10022	0.337723
$443$	−8.60991	−0.409069	−0.204534	−	0.978859i	$-0.565568\pi$
$443$	−8.60991	−0.409069	−0.204534	+	0.978859i	$0.565568\pi$
$444$	0	0
$445$	9.90169	0.469385
$446$	−28.7193	−1.35990
$447$	0	0
$448$	0.930297	0.0439524
$449$	38.0939	1.79776	0.898881	−	0.438192i	$-0.144381\pi$
$449$	38.0939	1.79776	0.898881	+	0.438192i	$0.144381\pi$
$450$	0	0
$451$	3.43762	0.161871
$452$	36.8982	1.73554
$453$	0	0
$454$	−21.1915	−0.994567
$455$	−0.0779966	−0.00365654
$456$	0	0
$457$	−29.5088	−1.38036	−0.690182	−	0.723636i	$-0.742470\pi$
$457$	−29.5088	−1.38036	−0.690182	+	0.723636i	$0.742470\pi$
$458$	−55.7649	−2.60572
$459$	0	0
$460$	25.9096	1.20804
$461$	−33.4991	−1.56021	−0.780104	−	0.625650i	$-0.784834\pi$
$461$	−33.4991	−1.56021	−0.780104	+	0.625650i	$0.784834\pi$
$462$	0	0
$463$	11.0244	0.512346	0.256173	−	0.966631i	$-0.417538\pi$
$463$	11.0244	0.512346	0.256173	+	0.966631i	$0.417538\pi$
$464$	5.93071	0.275326
$465$	0	0
$466$	−59.8221	−2.77121
$467$	−23.9869	−1.10998	−0.554990	−	0.831857i	$-0.687278\pi$
$467$	−23.9869	−1.10998	−0.554990	+	0.831857i	$0.687278\pi$
$468$	0	0
$469$	−0.653639	−0.0301822
$470$	22.6375	1.04419
$471$	0	0
$472$	0.173397	0.00798126
$473$	5.55473	0.255407
$474$	0	0
$475$	−1.33883	−0.0614298
$476$	−0.832818	−0.0381722
$477$	0	0
$478$	0.671291	0.0307041
$479$	29.5751	1.35132	0.675660	−	0.737213i	$-0.263859\pi$
$479$	29.5751	1.35132	0.675660	+	0.737213i	$0.263859\pi$
$480$	0	0
$481$	5.18283	0.236317
$482$	−27.3020	−1.24357
$483$	0	0
$484$	−35.1242	−1.59655
$485$	9.26477	0.420692
$486$	0	0
$487$	8.53551	0.386781	0.193390	−	0.981122i	$-0.438052\pi$
$487$	8.53551	0.386781	0.193390	+	0.981122i	$0.438052\pi$
$488$	16.5646	0.749846
$489$	0	0
$490$	−16.4609	−0.743628
$491$	−6.31022	−0.284776	−0.142388	−	0.989811i	$-0.545478\pi$
$491$	−6.31022	−0.284776	−0.142388	+	0.989811i	$0.545478\pi$
$492$	0	0
$493$	12.3488	0.556162
$494$	−3.15108	−0.141774
$495$	0	0
$496$	−10.1167	−0.454252
$497$	0.180246	0.00808513
$498$	0	0
$499$	1.85092	0.0828585	0.0414293	−	0.999141i	$-0.486809\pi$
$499$	1.85092	0.0828585	0.0414293	+	0.999141i	$0.486809\pi$
$500$	3.53946	0.158289
$501$	0	0
$502$	52.8014	2.35664
$503$	−43.5312	−1.94096	−0.970480	−	0.241181i	$-0.922465\pi$
$503$	−43.5312	−1.94096	−0.970480	+	0.241181i	$0.922465\pi$
$504$	0	0
$505$	−5.43031	−0.241646
$506$	17.8748	0.794632
$507$	0	0
$508$	−52.8281	−2.34387
$509$	−27.4875	−1.21836	−0.609181	−	0.793031i	$-0.708502\pi$
$509$	−27.4875	−1.21836	−0.609181	+	0.793031i	$0.708502\pi$
$510$	0	0
$511$	0.177856	0.00786789
$512$	−16.0576	−0.709653
$513$	0	0
$514$	−4.51800	−0.199280
$515$	13.0711	0.575983
$516$	0	0
$517$	9.97878	0.438866
$518$	−0.951428	−0.0418033
$519$	0	0
$520$	3.62327	0.158891
$521$	−24.5536	−1.07571	−0.537857	−	0.843036i	$-0.680766\pi$
$521$	−24.5536	−1.07571	−0.537857	+	0.843036i	$0.680766\pi$
$522$	0	0
$523$	41.0287	1.79406	0.897030	−	0.441970i	$-0.145720\pi$
$523$	41.0287	1.79406	0.897030	+	0.441970i	$0.145720\pi$
$524$	−45.3530	−1.98126
$525$	0	0
$526$	58.4212	2.54729
$527$	−21.0647	−0.917593
$528$	0	0
$529$	30.5857	1.32981
$530$	−25.8845	−1.12435
$531$	0	0
$532$	0.369605	0.0160244
$533$	3.31340	0.143519
$534$	0	0
$535$	−16.5067	−0.713647
$536$	30.3643	1.31154
$537$	0	0
$538$	−24.3032	−1.04778
$539$	−7.25611	−0.312543
$540$	0	0
$541$	−14.2877	−0.614278	−0.307139	−	0.951665i	$-0.599372\pi$
$541$	−14.2877	−0.614278	−0.307139	+	0.951665i	$0.599372\pi$
$542$	−67.1117	−2.88270
$543$	0	0
$544$	−11.5739	−0.496226
$545$	−15.2455	−0.653045
$546$	0	0
$547$	−26.9702	−1.15316	−0.576580	−	0.817040i	$-0.695613\pi$
$547$	−26.9702	−1.15316	−0.576580	+	0.817040i	$0.695613\pi$
$548$	47.0766	2.01101
$549$	0	0
$550$	2.44184	0.104120
$551$	−5.48040	−0.233473
$552$	0	0
$553$	−0.191023	−0.00812311
$554$	45.0650	1.91463
$555$	0	0
$556$	15.9206	0.675185
$557$	−27.5795	−1.16858	−0.584290	−	0.811545i	$-0.698627\pi$
$557$	−27.5795	−1.16858	−0.584290	+	0.811545i	$0.698627\pi$
$558$	0	0
$559$	5.35402	0.226451
$560$	−0.113004	−0.00477531
$561$	0	0
$562$	−29.1868	−1.23117
$563$	−5.84732	−0.246435	−0.123218	−	0.992380i	$-0.539321\pi$
$563$	−5.84732	−0.246435	−0.123218	+	0.992380i	$0.539321\pi$
$564$	0	0
$565$	10.4248	0.438575
$566$	54.4043	2.28679
$567$	0	0
$568$	−8.37318	−0.351331
$569$	−27.1564	−1.13846	−0.569228	−	0.822180i	$-0.692758\pi$
$569$	−27.1564	−1.13846	−0.569228	+	0.822180i	$0.692758\pi$
$570$	0	0
$571$	−15.1755	−0.635073	−0.317537	−	0.948246i	$-0.602856\pi$
$571$	−15.1755	−0.635073	−0.317537	+	0.948246i	$0.602856\pi$
$572$	3.67215	0.153540
$573$	0	0
$574$	−0.608252	−0.0253879
$575$	7.32023	0.305275
$576$	0	0
$577$	9.39043	0.390929	0.195464	−	0.980711i	$-0.437379\pi$
$577$	9.39043	0.390929	0.195464	+	0.980711i	$0.437379\pi$
$578$	−18.5918	−0.773315
$579$	0	0
$580$	14.4885	0.601602
$581$	−0.141724	−0.00587971
$582$	0	0
$583$	−11.4101	−0.472558
$584$	−8.26217	−0.341891
$585$	0	0
$586$	−46.3537	−1.91485
$587$	34.2126	1.41211	0.706054	−	0.708158i	$-0.250474\pi$
$587$	34.2126	1.41211	0.706054	+	0.708158i	$0.250474\pi$
$588$	0	0
$589$	9.34852	0.385199
$590$	0.112635	0.00463713
$591$	0	0
$592$	7.50908	0.308621
$593$	34.8084	1.42941	0.714704	−	0.699427i	$-0.246561\pi$
$593$	34.8084	1.42941	0.714704	+	0.699427i	$0.246561\pi$
$594$	0	0
$595$	−0.235296	−0.00964618
$596$	−5.88141	−0.240912
$597$	0	0
$598$	17.2289	0.704543
$599$	−26.4645	−1.08131	−0.540655	−	0.841244i	$-0.681824\pi$
$599$	−26.4645	−1.08131	−0.540655	+	0.841244i	$0.681824\pi$
$600$	0	0
$601$	28.4305	1.15970	0.579852	−	0.814722i	$-0.303110\pi$
$601$	28.4305	1.15970	0.579852	+	0.814722i	$0.303110\pi$
$602$	−0.982854	−0.0400581
$603$	0	0
$604$	65.6358	2.67068
$605$	−9.92362	−0.403452
$606$	0	0
$607$	−29.5420	−1.19907	−0.599537	−	0.800347i	$-0.704649\pi$
$607$	−29.5420	−1.19907	−0.599537	+	0.800347i	$0.704649\pi$
$608$	5.13649	0.208312
$609$	0	0
$610$	10.7601	0.435662
$611$	9.61821	0.389111
$612$	0	0
$613$	34.7033	1.40165	0.700826	−	0.713332i	$-0.252815\pi$
$613$	34.7033	1.40165	0.700826	+	0.713332i	$0.252815\pi$
$614$	−63.5217	−2.56353
$615$	0	0
$616$	−0.293197	−0.0118133
$617$	18.7462	0.754692	0.377346	−	0.926072i	$-0.376837\pi$
$617$	18.7462	0.754692	0.377346	+	0.926072i	$0.376837\pi$
$618$	0	0
$619$	31.1834	1.25337	0.626684	−	0.779274i	$-0.284412\pi$
$619$	31.1834	1.25337	0.626684	+	0.779274i	$0.284412\pi$
$620$	−24.7146	−0.992563
$621$	0	0
$622$	47.0769	1.88761
$623$	−0.772298	−0.0309415
$624$	0	0
$625$	1.00000	0.0400000
$626$	50.7098	2.02677
$627$	0	0
$628$	37.5346	1.49779
$629$	15.6353	0.623418
$630$	0	0
$631$	−8.29360	−0.330163	−0.165082	−	0.986280i	$-0.552789\pi$
$631$	−8.29360	−0.330163	−0.165082	+	0.986280i	$0.552789\pi$
$632$	8.87381	0.352981
$633$	0	0
$634$	−38.1725	−1.51602
$635$	−14.9255	−0.592300
$636$	0	0
$637$	−6.99392	−0.277109
$638$	9.99548	0.395725
$639$	0	0
$640$	−20.3993	−0.806353
$641$	−10.0897	−0.398521	−0.199260	−	0.979947i	$-0.563854\pi$
$641$	−10.0897	−0.398521	−0.199260	+	0.979947i	$0.563854\pi$
$642$	0	0
$643$	37.6129	1.48331	0.741654	−	0.670783i	$-0.234042\pi$
$643$	37.6129	1.48331	0.741654	+	0.670783i	$0.234042\pi$
$644$	−2.02086	−0.0796331
$645$	0	0
$646$	−9.50599	−0.374008
$647$	−24.9035	−0.979058	−0.489529	−	0.871987i	$-0.662831\pi$
$647$	−24.9035	−0.979058	−0.489529	+	0.871987i	$0.662831\pi$
$648$	0	0
$649$	0.0496506	0.00194896
$650$	2.35360	0.0923160
$651$	0	0
$652$	7.45957	0.292139
$653$	−2.62240	−0.102623	−0.0513113	−	0.998683i	$-0.516340\pi$
$653$	−2.62240	−0.102623	−0.0513113	+	0.998683i	$0.516340\pi$
$654$	0	0
$655$	−12.8136	−0.500667
$656$	4.80059	0.187431
$657$	0	0
$658$	−1.76564	−0.0688320
$659$	−9.58078	−0.373214	−0.186607	−	0.982435i	$-0.559749\pi$
$659$	−9.58078	−0.373214	−0.186607	+	0.982435i	$0.559749\pi$
$660$	0	0
$661$	28.6492	1.11432	0.557162	−	0.830404i	$-0.311890\pi$
$661$	28.6492	1.11432	0.557162	+	0.830404i	$0.311890\pi$
$662$	11.6991	0.454698
$663$	0	0
$664$	6.58369	0.255497
$665$	0.104424	0.00404940
$666$	0	0
$667$	29.9648	1.16024
$668$	−22.6816	−0.877576
$669$	0	0
$670$	19.7240	0.762006
$671$	4.74312	0.183106
$672$	0	0
$673$	−41.9727	−1.61793	−0.808965	−	0.587856i	$-0.799972\pi$
$673$	−41.9727	−1.61793	−0.808965	+	0.587856i	$0.799972\pi$
$674$	−60.0442	−2.31282
$675$	0	0
$676$	3.53946	0.136133
$677$	−31.6441	−1.21618	−0.608091	−	0.793867i	$-0.708064\pi$
$677$	−31.6441	−1.21618	−0.608091	+	0.793867i	$0.708064\pi$
$678$	0	0
$679$	−0.722620	−0.0277316
$680$	10.9305	0.419165
$681$	0	0
$682$	−17.0504	−0.652893
$683$	−11.7455	−0.449429	−0.224714	−	0.974425i	$-0.572145\pi$
$683$	−11.7455	−0.449429	−0.224714	+	0.974425i	$0.572145\pi$
$684$	0	0
$685$	13.3005	0.508187
$686$	2.56891	0.0980813
$687$	0	0
$688$	7.75711	0.295737
$689$	−10.9978	−0.418983
$690$	0	0
$691$	7.16916	0.272728	0.136364	−	0.990659i	$-0.456458\pi$
$691$	7.16916	0.272728	0.136364	+	0.990659i	$0.456458\pi$
$692$	−34.8289	−1.32400
$693$	0	0
$694$	−38.1203	−1.44703
$695$	4.49804	0.170620
$696$	0	0
$697$	9.99568	0.378613
$698$	36.8353	1.39424
$699$	0	0
$700$	−0.276066	−0.0104343
$701$	−15.0370	−0.567939	−0.283970	−	0.958833i	$-0.591652\pi$
$701$	−15.0370	−0.567939	−0.283970	+	0.958833i	$0.591652\pi$
$702$	0	0
$703$	−6.93893	−0.261707
$704$	−12.3745	−0.466383
$705$	0	0
$706$	−27.0444	−1.01783
$707$	0.423546	0.0159291
$708$	0	0
$709$	−38.6285	−1.45072	−0.725362	−	0.688367i	$-0.758328\pi$
$709$	−38.6285	−1.45072	−0.725362	+	0.688367i	$0.758328\pi$
$710$	−5.43905	−0.204124
$711$	0	0
$712$	35.8765	1.34453
$713$	−51.1142	−1.91424
$714$	0	0
$715$	1.03749	0.0387999
$716$	−47.7917	−1.78606
$717$	0	0
$718$	69.0183	2.57574
$719$	0.942162	0.0351367	0.0175683	−	0.999846i	$-0.494408\pi$
$719$	0.942162	0.0351367	0.0175683	+	0.999846i	$0.494408\pi$
$720$	0	0
$721$	−1.01950	−0.0379683
$722$	−40.4997	−1.50724
$723$	0	0
$724$	−7.35415	−0.273315
$725$	4.09342	0.152026
$726$	0	0
$727$	−32.0837	−1.18992	−0.594959	−	0.803756i	$-0.702832\pi$
$727$	−32.0837	−1.18992	−0.594959	+	0.803756i	$0.702832\pi$
$728$	−0.282603	−0.0104740
$729$	0	0
$730$	−5.36694	−0.198639
$731$	16.1517	0.597392
$732$	0	0
$733$	37.4939	1.38487	0.692434	−	0.721481i	$-0.256539\pi$
$733$	37.4939	1.38487	0.692434	+	0.721481i	$0.256539\pi$
$734$	7.91593	0.292182
$735$	0	0
$736$	−28.0844	−1.03521
$737$	8.69453	0.320267
$738$	0	0
$739$	34.3049	1.26193	0.630963	−	0.775813i	$-0.282660\pi$
$739$	34.3049	1.26193	0.630963	+	0.775813i	$0.282660\pi$
$740$	18.3444	0.674353
$741$	0	0
$742$	2.01890	0.0741162
$743$	−44.6352	−1.63751	−0.818754	−	0.574145i	$-0.805335\pi$
$743$	−44.6352	−1.63751	−0.818754	+	0.574145i	$0.805335\pi$
$744$	0	0
$745$	−1.66167	−0.0608789
$746$	18.5412	0.678843
$747$	0	0
$748$	11.0779	0.405049
$749$	1.28747	0.0470430
$750$	0	0
$751$	32.3406	1.18013	0.590063	−	0.807358i	$-0.299103\pi$
$751$	32.3406	1.18013	0.590063	+	0.807358i	$0.299103\pi$
$752$	13.9352	0.508166
$753$	0	0
$754$	9.63430	0.350860
$755$	18.5440	0.674887
$756$	0	0
$757$	30.7792	1.11869	0.559344	−	0.828936i	$-0.311053\pi$
$757$	30.7792	1.11869	0.559344	+	0.828936i	$0.311053\pi$
$758$	36.2803	1.31776
$759$	0	0
$760$	−4.85095	−0.175962
$761$	1.57326	0.0570306	0.0285153	−	0.999593i	$-0.490922\pi$
$761$	1.57326	0.0570306	0.0285153	+	0.999593i	$0.490922\pi$
$762$	0	0
$763$	1.18910	0.0430482
$764$	−37.1999	−1.34584
$765$	0	0
$766$	−80.5675	−2.91102
$767$	0.0478566	0.00172800
$768$	0	0
$769$	−30.6892	−1.10668	−0.553341	−	0.832955i	$-0.686647\pi$
$769$	−30.6892	−1.10668	−0.553341	+	0.832955i	$0.686647\pi$
$770$	−0.190455	−0.00686353
$771$	0	0
$772$	−64.8648	−2.33454
$773$	0.437296	0.0157284	0.00786422	−	0.999969i	$-0.497497\pi$
$773$	0.437296	0.0157284	0.00786422	+	0.999969i	$0.497497\pi$
$774$	0	0
$775$	−6.98260	−0.250823
$776$	33.5688	1.20505
$777$	0	0
$778$	66.8415	2.39638
$779$	−4.43608	−0.158939
$780$	0	0
$781$	−2.39758	−0.0857921
$782$	51.9752	1.85863
$783$	0	0
$784$	−10.1331	−0.361895
$785$	10.6046	0.378495
$786$	0	0
$787$	−23.5844	−0.840692	−0.420346	−	0.907364i	$-0.638091\pi$
$787$	−23.5844	−0.840692	−0.420346	+	0.907364i	$0.638091\pi$
$788$	27.1430	0.966930
$789$	0	0
$790$	5.76425	0.205083
$791$	−0.813100	−0.0289105
$792$	0	0
$793$	4.57173	0.162347
$794$	−69.2193	−2.45650
$795$	0	0
$796$	59.5679	2.11133
$797$	5.25251	0.186053	0.0930267	−	0.995664i	$-0.470346\pi$
$797$	5.25251	0.186053	0.0930267	+	0.995664i	$0.470346\pi$
$798$	0	0
$799$	29.0157	1.02650
$800$	−3.83655	−0.135643
$801$	0	0
$802$	−77.2194	−2.72671
$803$	−2.36579	−0.0834870
$804$	0	0
$805$	−0.570953	−0.0201234
$806$	−16.4343	−0.578873
$807$	0	0
$808$	−19.6755	−0.692182
$809$	40.3038	1.41701	0.708503	−	0.705708i	$-0.249371\pi$
$809$	40.3038	1.41701	0.708503	+	0.705708i	$0.249371\pi$
$810$	0	0
$811$	−13.7105	−0.481441	−0.240721	−	0.970594i	$-0.577384\pi$
$811$	−13.7105	−0.481441	−0.240721	+	0.970594i	$0.577384\pi$
$812$	−1.13005	−0.0396571
$813$	0	0
$814$	12.6556	0.443580
$815$	2.10755	0.0738241
$816$	0	0
$817$	−7.16812	−0.250781
$818$	34.1111	1.19267
$819$	0	0
$820$	11.7276	0.409547
$821$	43.6806	1.52446	0.762232	−	0.647304i	$-0.224104\pi$
$821$	43.6806	1.52446	0.762232	+	0.647304i	$0.224104\pi$
$822$	0	0
$823$	52.1533	1.81795	0.908974	−	0.416853i	$-0.136867\pi$
$823$	52.1533	1.81795	0.908974	+	0.416853i	$0.136867\pi$
$824$	47.3603	1.64987
$825$	0	0
$826$	−0.00878518	−0.000305675 0
$827$	−35.2190	−1.22468	−0.612342	−	0.790593i	$-0.709772\pi$
$827$	−35.2190	−1.22468	−0.612342	+	0.790593i	$0.709772\pi$
$828$	0	0
$829$	30.9645	1.07544	0.537721	−	0.843123i	$-0.319285\pi$
$829$	30.9645	1.07544	0.537721	+	0.843123i	$0.319285\pi$
$830$	4.27663	0.148444
$831$	0	0
$832$	−11.9274	−0.413508
$833$	−21.0988	−0.731032
$834$	0	0
$835$	−6.40821	−0.221765
$836$	−4.91638	−0.170037
$837$	0	0
$838$	29.9358	1.03411
$839$	−56.2912	−1.94339	−0.971694	−	0.236243i	$-0.924084\pi$
$839$	−56.2912	−1.94339	−0.971694	+	0.236243i	$0.924084\pi$
$840$	0	0
$841$	−12.2439	−0.422204
$842$	−25.3149	−0.872409
$843$	0	0
$844$	−5.78052	−0.198974
$845$	1.00000	0.0344010
$846$	0	0
$847$	0.774008	0.0265952
$848$	−15.9340	−0.547178
$849$	0	0
$850$	7.10022	0.243536
$851$	37.9395	1.30055
$852$	0	0
$853$	−15.4026	−0.527375	−0.263688	−	0.964608i	$-0.584939\pi$
$853$	−15.4026	−0.527375	−0.263688	+	0.964608i	$0.584939\pi$
$854$	−0.839248	−0.0287185
$855$	0	0
$856$	−59.8083	−2.04420
$857$	−15.0100	−0.512732	−0.256366	−	0.966580i	$-0.582525\pi$
$857$	−15.0100	−0.512732	−0.256366	+	0.966580i	$0.582525\pi$
$858$	0	0
$859$	49.2981	1.68203	0.841015	−	0.541012i	$-0.181959\pi$
$859$	49.2981	1.68203	0.841015	+	0.541012i	$0.181959\pi$
$860$	18.9503	0.646200
$861$	0	0
$862$	−79.2886	−2.70058
$863$	38.8540	1.32261	0.661303	−	0.750119i	$-0.270004\pi$
$863$	38.8540	1.32261	0.661303	+	0.750119i	$0.270004\pi$
$864$	0	0
$865$	−9.84018	−0.334576
$866$	94.6580	3.21661
$867$	0	0
$868$	1.92766	0.0654289
$869$	2.54093	0.0861951
$870$	0	0
$871$	8.38035	0.283957
$872$	−55.2385	−1.87061
$873$	0	0
$874$	−23.0666	−0.780239
$875$	−0.0779966	−0.00263677
$876$	0	0
$877$	24.0713	0.812830	0.406415	−	0.913688i	$-0.366779\pi$
$877$	24.0713	0.812830	0.406415	+	0.913688i	$0.366779\pi$
$878$	42.7264	1.44194
$879$	0	0
$880$	1.50315	0.0506713
$881$	51.5561	1.73697	0.868484	−	0.495716i	$-0.165094\pi$
$881$	51.5561	1.73697	0.868484	+	0.495716i	$0.165094\pi$
$882$	0	0
$883$	−53.3239	−1.79449	−0.897246	−	0.441531i	$-0.854436\pi$
$883$	−53.3239	−1.79449	−0.897246	+	0.441531i	$0.854436\pi$
$884$	10.6776	0.359127
$885$	0	0
$886$	−20.2643	−0.680793
$887$	−28.1756	−0.946044	−0.473022	−	0.881051i	$-0.656837\pi$
$887$	−28.1756	−0.946044	−0.473022	+	0.881051i	$0.656837\pi$
$888$	0	0
$889$	1.16414	0.0390439
$890$	23.3047	0.781174
$891$	0	0
$892$	−43.1893	−1.44609
$893$	−12.8771	−0.430917
$894$	0	0
$895$	−13.5026	−0.451341
$896$	1.59107	0.0531541
$897$	0	0
$898$	89.6580	2.99193
$899$	−28.5827	−0.953288
$900$	0	0
$901$	−33.1776	−1.10530
$902$	8.09080	0.269394
$903$	0	0
$904$	37.7719	1.25628
$905$	−2.07776	−0.0690671
$906$	0	0
$907$	−14.9538	−0.496532	−0.248266	−	0.968692i	$-0.579861\pi$
$907$	−14.9538	−0.496532	−0.248266	+	0.968692i	$0.579861\pi$
$908$	−31.8687	−1.05760
$909$	0	0
$910$	−0.183573	−0.00608539
$911$	29.3496	0.972395	0.486197	−	0.873849i	$-0.338384\pi$
$911$	29.3496	0.972395	0.486197	+	0.873849i	$0.338384\pi$
$912$	0	0
$913$	1.88517	0.0623902
$914$	−69.4521	−2.29727
$915$	0	0
$916$	−83.8617	−2.77087
$917$	0.999413	0.0330035
$918$	0	0
$919$	−4.84694	−0.159886	−0.0799430	−	0.996799i	$-0.525474\pi$
$919$	−4.84694	−0.159886	−0.0799430	+	0.996799i	$0.525474\pi$
$920$	26.5232	0.874443
$921$	0	0
$922$	−78.8436	−2.59658
$923$	−2.31094	−0.0760656
$924$	0	0
$925$	5.18283	0.170410
$926$	25.9470	0.852671
$927$	0	0
$928$	−15.7046	−0.515529
$929$	−44.8699	−1.47213	−0.736067	−	0.676909i	$-0.763319\pi$
$929$	−44.8699	−1.47213	−0.736067	+	0.676909i	$0.763319\pi$
$930$	0	0
$931$	9.36367	0.306882
$932$	−89.9632	−2.94684
$933$	0	0
$934$	−56.4557	−1.84729
$935$	3.12984	0.102357
$936$	0	0
$937$	1.39738	0.0456504	0.0228252	−	0.999739i	$-0.492734\pi$
$937$	1.39738	0.0456504	0.0228252	+	0.999739i	$0.492734\pi$
$938$	−1.53841	−0.0502308
$939$	0	0
$940$	34.0432	1.11037
$941$	−3.33563	−0.108738	−0.0543692	−	0.998521i	$-0.517315\pi$
$941$	−3.33563	−0.108738	−0.0543692	+	0.998521i	$0.517315\pi$
$942$	0	0
$943$	24.2549	0.789847
$944$	0.0693364	0.00225671
$945$	0	0
$946$	13.0737	0.425061
$947$	−11.6341	−0.378058	−0.189029	−	0.981972i	$-0.560534\pi$
$947$	−11.6341	−0.378058	−0.189029	+	0.981972i	$0.560534\pi$
$948$	0	0
$949$	−2.28031	−0.0740219
$950$	−3.15108	−0.102234
$951$	0	0
$952$	−0.852540	−0.0276310
$953$	−26.6894	−0.864554	−0.432277	−	0.901741i	$-0.642290\pi$
$953$	−26.6894	−0.864554	−0.432277	+	0.901741i	$0.642290\pi$
$954$	0	0
$955$	−10.5100	−0.340097
$956$	1.00952	0.0326501
$957$	0	0
$958$	69.6081	2.24894
$959$	−1.03740	−0.0334992
$960$	0	0
$961$	17.7567	0.572798
$962$	12.1983	0.393290
$963$	0	0
$964$	−41.0580	−1.32239
$965$	−18.3262	−0.589942
$966$	0	0
$967$	32.6909	1.05127	0.525634	−	0.850711i	$-0.323828\pi$
$967$	32.6909	1.05127	0.525634	+	0.850711i	$0.323828\pi$
$968$	−35.9560	−1.15567
$969$	0	0
$970$	21.8056	0.700136
$971$	−23.4851	−0.753673	−0.376836	−	0.926280i	$-0.622988\pi$
$971$	−23.4851	−0.753673	−0.376836	+	0.926280i	$0.622988\pi$
$972$	0	0
$973$	−0.350832	−0.0112472
$974$	20.0892	0.643700
$975$	0	0
$976$	6.62371	0.212020
$977$	24.0417	0.769162	0.384581	−	0.923091i	$-0.374346\pi$
$977$	24.0417	0.769162	0.384581	+	0.923091i	$0.374346\pi$
$978$	0	0
$979$	10.2729	0.328323
$980$	−24.7547	−0.790759
$981$	0	0
$982$	−14.8518	−0.473939
$983$	−3.42485	−0.109236	−0.0546178	−	0.998507i	$-0.517394\pi$
$983$	−3.42485	−0.109236	−0.0546178	+	0.998507i	$0.517394\pi$
$984$	0	0
$985$	7.66870	0.244345
$986$	29.0642	0.925592
$987$	0	0
$988$	−4.73873	−0.150759
$989$	39.1926	1.24625
$990$	0	0
$991$	19.2845	0.612593	0.306297	−	0.951936i	$-0.400910\pi$
$991$	19.2845	0.612593	0.306297	+	0.951936i	$0.400910\pi$
$992$	26.7891	0.850555
$993$	0	0
$994$	0.424227	0.0134557
$995$	16.8297	0.533537
$996$	0	0
$997$	14.8669	0.470841	0.235420	−	0.971894i	$-0.424353\pi$
$997$	14.8669	0.470841	0.235420	+	0.971894i	$0.424353\pi$
$998$	4.35633	0.137897
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5265.2.a.bl.1.13		15
3.2	odd	2		5265.2.a.bk.1.3		15
9.2	odd	6		585.2.i.h.391.13	yes	30
9.4	even	3		1755.2.i.h.586.3		30
9.5	odd	6		585.2.i.h.196.13	✓	30
9.7	even	3		1755.2.i.h.1171.3		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.i.h.196.13	✓	30	9.5	odd	6
585.2.i.h.391.13	yes	30	9.2	odd	6
1755.2.i.h.586.3		30	9.4	even	3
1755.2.i.h.1171.3		30	9.7	even	3
5265.2.a.bk.1.3		15	3.2	odd	2
5265.2.a.bl.1.13		15	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q+2.35360 q^{2} +3.53946 q^{4} +1.00000 q^{5} -0.0779966 q^{7} +3.62327 q^{8} +O(q^{10})\)	\(q+2.35360 q^{2} +3.53946 q^{4} +1.00000 q^{5} -0.0779966 q^{7} +3.62327 q^{8} +2.35360 q^{10} +1.03749 q^{11} +1.00000 q^{13} -0.183573 q^{14} +1.44884 q^{16} +3.01674 q^{17} -1.33883 q^{19} +3.53946 q^{20} +2.44184 q^{22} +7.32023 q^{23} +1.00000 q^{25} +2.35360 q^{26} -0.276066 q^{28} +4.09342 q^{29} -6.98260 q^{31} -3.83655 q^{32} +7.10022 q^{34} -0.0779966 q^{35} +5.18283 q^{37} -3.15108 q^{38} +3.62327 q^{40} +3.31340 q^{41} +5.35402 q^{43} +3.67215 q^{44} +17.2289 q^{46} +9.61821 q^{47} -6.99392 q^{49} +2.35360 q^{50} +3.53946 q^{52} -10.9978 q^{53} +1.03749 q^{55} -0.282603 q^{56} +9.63430 q^{58} +0.0478566 q^{59} +4.57173 q^{61} -16.4343 q^{62} -11.9274 q^{64} +1.00000 q^{65} +8.38035 q^{67} +10.6776 q^{68} -0.183573 q^{70} -2.31094 q^{71} -2.28031 q^{73} +12.1983 q^{74} -4.73873 q^{76} -0.0809206 q^{77} +2.44912 q^{79} +1.44884 q^{80} +7.79844 q^{82} +1.81706 q^{83} +3.01674 q^{85} +12.6012 q^{86} +3.75910 q^{88} +9.90169 q^{89} -0.0779966 q^{91} +25.9096 q^{92} +22.6375 q^{94} -1.33883 q^{95} +9.26477 q^{97} -16.4609 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + q^{2} + 21 q^{4} + 15 q^{5} + 10 q^{7}+O(q^{10}) \) 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

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Database

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