LMFDB - Embedded newform 5265.2.a.bk.1.14

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			1.14
Root			$-2.52000$ 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)

$f(q)$	$=$	$q+2.52000 q^{2} +4.35041 q^{4} -1.00000 q^{5} +3.01221 q^{7} +5.92303 q^{8} +O(q^{10})$	\(q+2.52000 q^{2} +4.35041 q^{4} -1.00000 q^{5} +3.01221 q^{7} +5.92303 q^{8} -2.52000 q^{10} -5.74814 q^{11} +1.00000 q^{13} +7.59077 q^{14} +6.22524 q^{16} +5.46275 q^{17} +5.89643 q^{19} -4.35041 q^{20} -14.4853 q^{22} +8.39117 q^{23} +1.00000 q^{25} +2.52000 q^{26} +13.1043 q^{28} +3.37479 q^{29} -4.29543 q^{31} +3.84154 q^{32} +13.7661 q^{34} -3.01221 q^{35} -4.29747 q^{37} +14.8590 q^{38} -5.92303 q^{40} -6.56253 q^{41} -0.654059 q^{43} -25.0068 q^{44} +21.1458 q^{46} -10.4100 q^{47} +2.07340 q^{49} +2.52000 q^{50} +4.35041 q^{52} +6.03463 q^{53} +5.74814 q^{55} +17.8414 q^{56} +8.50448 q^{58} +7.55872 q^{59} +6.02155 q^{61} -10.8245 q^{62} -2.76979 q^{64} -1.00000 q^{65} +5.69643 q^{67} +23.7652 q^{68} -7.59077 q^{70} +2.00130 q^{71} +12.4518 q^{73} -10.8296 q^{74} +25.6519 q^{76} -17.3146 q^{77} +13.6473 q^{79} -6.22524 q^{80} -16.5376 q^{82} -6.16250 q^{83} -5.46275 q^{85} -1.64823 q^{86} -34.0464 q^{88} +2.25804 q^{89} +3.01221 q^{91} +36.5050 q^{92} -26.2333 q^{94} -5.89643 q^{95} +6.24936 q^{97} +5.22496 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.52000	1.78191	0.890955	−	0.454091i	$-0.150036\pi$
$2$	2.52000	1.78191	0.890955	+	0.454091i	$0.150036\pi$
$3$	0	0
$3$	0	0
$4$	4.35041	2.17520
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.01221	1.13851	0.569254	−	0.822162i	$-0.307232\pi$
$7$	3.01221	1.13851	0.569254	+	0.822162i	$0.307232\pi$
$8$	5.92303	2.09411
$9$	0	0
$10$	−2.52000	−0.796894
$11$	−5.74814	−1.73313	−0.866565	−	0.499064i	$-0.833677\pi$
$11$	−5.74814	−1.73313	−0.866565	+	0.499064i	$0.833677\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	7.59077	2.02872
$15$	0	0
$16$	6.22524	1.55631
$17$	5.46275	1.32491	0.662456	−	0.749101i	$-0.269514\pi$
$17$	5.46275	1.32491	0.662456	+	0.749101i	$0.269514\pi$
$18$	0	0
$19$	5.89643	1.35273	0.676367	−	0.736565i	$-0.263553\pi$
$19$	5.89643	1.35273	0.676367	+	0.736565i	$0.263553\pi$
$20$	−4.35041	−0.972781
$21$	0	0
$22$	−14.4853	−3.08828
$23$	8.39117	1.74968	0.874840	−	0.484412i	$-0.160967\pi$
$23$	8.39117	1.74968	0.874840	+	0.484412i	$0.160967\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.52000	0.494213
$27$	0	0
$28$	13.1043	2.47649
$29$	3.37479	0.626683	0.313342	−	0.949640i	$-0.398552\pi$
$29$	3.37479	0.626683	0.313342	+	0.949640i	$0.398552\pi$
$30$	0	0
$31$	−4.29543	−0.771482	−0.385741	−	0.922607i	$-0.626054\pi$
$31$	−4.29543	−0.771482	−0.385741	+	0.922607i	$0.626054\pi$
$32$	3.84154	0.679094
$33$	0	0
$34$	13.7661	2.36087
$35$	−3.01221	−0.509156
$36$	0	0
$37$	−4.29747	−0.706500	−0.353250	−	0.935529i	$-0.614923\pi$
$37$	−4.29747	−0.706500	−0.353250	+	0.935529i	$0.614923\pi$
$38$	14.8590	2.41045
$39$	0	0
$40$	−5.92303	−0.936514
$41$	−6.56253	−1.02489	−0.512447	−	0.858719i	$-0.671261\pi$
$41$	−6.56253	−1.02489	−0.512447	+	0.858719i	$0.671261\pi$
$42$	0	0
$43$	−0.654059	−0.0997431	−0.0498716	−	0.998756i	$-0.515881\pi$
$43$	−0.654059	−0.0997431	−0.0498716	+	0.998756i	$0.515881\pi$
$44$	−25.0068	−3.76991
$45$	0	0
$46$	21.1458	3.11777
$47$	−10.4100	−1.51846	−0.759231	−	0.650821i	$-0.774425\pi$
$47$	−10.4100	−1.51846	−0.759231	+	0.650821i	$0.774425\pi$
$48$	0	0
$49$	2.07340	0.296199
$50$	2.52000	0.356382
$51$	0	0
$52$	4.35041	0.603293
$53$	6.03463	0.828920	0.414460	−	0.910068i	$-0.363970\pi$
$53$	6.03463	0.828920	0.414460	+	0.910068i	$0.363970\pi$
$54$	0	0
$55$	5.74814	0.775080
$56$	17.8414	2.38416
$57$	0	0
$58$	8.50448	1.11669
$59$	7.55872	0.984062	0.492031	−	0.870578i	$-0.336255\pi$
$59$	7.55872	0.984062	0.492031	+	0.870578i	$0.336255\pi$
$60$	0	0
$61$	6.02155	0.770981	0.385490	−	0.922712i	$-0.374032\pi$
$61$	6.02155	0.770981	0.385490	+	0.922712i	$0.374032\pi$
$62$	−10.8245	−1.37471
$63$	0	0
$64$	−2.76979	−0.346224
$65$	−1.00000	−0.124035
$66$	0	0
$67$	5.69643	0.695930	0.347965	−	0.937508i	$-0.386873\pi$
$67$	5.69643	0.695930	0.347965	+	0.937508i	$0.386873\pi$
$68$	23.7652	2.88195
$69$	0	0
$70$	−7.59077	−0.907270
$71$	2.00130	0.237511	0.118755	−	0.992924i	$-0.462110\pi$
$71$	2.00130	0.237511	0.118755	+	0.992924i	$0.462110\pi$
$72$	0	0
$73$	12.4518	1.45738	0.728689	−	0.684845i	$-0.240130\pi$
$73$	12.4518	1.45738	0.728689	+	0.684845i	$0.240130\pi$
$74$	−10.8296	−1.25892
$75$	0	0
$76$	25.6519	2.94247
$77$	−17.3146	−1.97318
$78$	0	0
$79$	13.6473	1.53544	0.767721	−	0.640784i	$-0.221390\pi$
$79$	13.6473	1.53544	0.767721	+	0.640784i	$0.221390\pi$
$80$	−6.22524	−0.696002
$81$	0	0
$82$	−16.5376	−1.82627
$83$	−6.16250	−0.676422	−0.338211	−	0.941070i	$-0.609822\pi$
$83$	−6.16250	−0.676422	−0.338211	+	0.941070i	$0.609822\pi$
$84$	0	0
$85$	−5.46275	−0.592518
$86$	−1.64823	−0.177733
$87$	0	0
$88$	−34.0464	−3.62936
$89$	2.25804	0.239351	0.119676	−	0.992813i	$-0.461815\pi$
$89$	2.25804	0.239351	0.119676	+	0.992813i	$0.461815\pi$
$90$	0	0
$91$	3.01221	0.315765
$92$	36.5050	3.80591
$93$	0	0
$94$	−26.2333	−2.70576
$95$	−5.89643	−0.604961
$96$	0	0
$97$	6.24936	0.634527	0.317263	−	0.948337i	$-0.397236\pi$
$97$	6.24936	0.634527	0.317263	+	0.948337i	$0.397236\pi$
$98$	5.22496	0.527801
$99$	0	0
$100$	4.35041	0.435041
$101$	−5.95578	−0.592622	−0.296311	−	0.955091i	$-0.595757\pi$
$101$	−5.95578	−0.592622	−0.296311	+	0.955091i	$0.595757\pi$
$102$	0	0
$103$	−6.36814	−0.627471	−0.313736	−	0.949510i	$-0.601581\pi$
$103$	−6.36814	−0.627471	−0.313736	+	0.949510i	$0.601581\pi$
$104$	5.92303	0.580801
$105$	0	0
$106$	15.2073	1.47706
$107$	12.7993	1.23735	0.618675	−	0.785647i	$-0.287670\pi$
$107$	12.7993	1.23735	0.618675	+	0.785647i	$0.287670\pi$
$108$	0	0
$109$	9.44687	0.904846	0.452423	−	0.891804i	$-0.350560\pi$
$109$	9.44687	0.904846	0.452423	+	0.891804i	$0.350560\pi$
$110$	14.4853	1.38112
$111$	0	0
$112$	18.7517	1.77187
$113$	−13.6674	−1.28572	−0.642860	−	0.765983i	$-0.722253\pi$
$113$	−13.6674	−1.28572	−0.642860	+	0.765983i	$0.722253\pi$
$114$	0	0
$115$	−8.39117	−0.782480
$116$	14.6817	1.36316
$117$	0	0
$118$	19.0480	1.75351
$119$	16.4549	1.50842
$120$	0	0
$121$	22.0412	2.00374
$122$	15.1743	1.37382
$123$	0	0
$124$	−18.6869	−1.67813
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	4.37449	0.388173	0.194086	−	0.980984i	$-0.437826\pi$
$127$	4.37449	0.388173	0.194086	+	0.980984i	$0.437826\pi$
$128$	−14.6629	−1.29603
$129$	0	0
$130$	−2.52000	−0.221019
$131$	−13.2663	−1.15908	−0.579542	−	0.814942i	$-0.696769\pi$
$131$	−13.2663	−1.15908	−0.579542	+	0.814942i	$0.696769\pi$
$132$	0	0
$133$	17.7613	1.54010
$134$	14.3550	1.24008
$135$	0	0
$136$	32.3560	2.77451
$137$	−10.6214	−0.907451	−0.453725	−	0.891142i	$-0.649905\pi$
$137$	−10.6214	−0.907451	−0.453725	+	0.891142i	$0.649905\pi$
$138$	0	0
$139$	4.45256	0.377661	0.188831	−	0.982010i	$-0.439530\pi$
$139$	4.45256	0.377661	0.188831	+	0.982010i	$0.439530\pi$
$140$	−13.1043	−1.10752
$141$	0	0
$142$	5.04329	0.423223
$143$	−5.74814	−0.480684
$144$	0	0
$145$	−3.37479	−0.280261
$146$	31.3786	2.59692
$147$	0	0
$148$	−18.6958	−1.53678
$149$	−16.3616	−1.34040	−0.670198	−	0.742182i	$-0.733791\pi$
$149$	−16.3616	−1.34040	−0.670198	+	0.742182i	$0.733791\pi$
$150$	0	0
$151$	−5.98419	−0.486987	−0.243493	−	0.969903i	$-0.578293\pi$
$151$	−5.98419	−0.486987	−0.243493	+	0.969903i	$0.578293\pi$
$152$	34.9247	2.83277
$153$	0	0
$154$	−43.6328	−3.51603
$155$	4.29543	0.345017
$156$	0	0
$157$	6.22902	0.497130	0.248565	−	0.968615i	$-0.420041\pi$
$157$	6.22902	0.497130	0.248565	+	0.968615i	$0.420041\pi$
$158$	34.3913	2.73602
$159$	0	0
$160$	−3.84154	−0.303700
$161$	25.2759	1.99202
$162$	0	0
$163$	−2.42340	−0.189815	−0.0949076	−	0.995486i	$-0.530256\pi$
$163$	−2.42340	−0.189815	−0.0949076	+	0.995486i	$0.530256\pi$
$164$	−28.5497	−2.22935
$165$	0	0
$166$	−15.5295	−1.20532
$167$	4.96480	0.384188	0.192094	−	0.981377i	$-0.438472\pi$
$167$	4.96480	0.384188	0.192094	+	0.981377i	$0.438472\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−13.7661	−1.05581
$171$	0	0
$172$	−2.84542	−0.216962
$173$	−2.66799	−0.202843	−0.101422	−	0.994844i	$-0.532339\pi$
$173$	−2.66799	−0.202843	−0.101422	+	0.994844i	$0.532339\pi$
$174$	0	0
$175$	3.01221	0.227702
$176$	−35.7835	−2.69729
$177$	0	0
$178$	5.69026	0.426503
$179$	−13.9929	−1.04588	−0.522938	−	0.852371i	$-0.675164\pi$
$179$	−13.9929	−1.04588	−0.522938	+	0.852371i	$0.675164\pi$
$180$	0	0
$181$	8.28746	0.616002	0.308001	−	0.951386i	$-0.400340\pi$
$181$	8.28746	0.616002	0.308001	+	0.951386i	$0.400340\pi$
$182$	7.59077	0.562665
$183$	0	0
$184$	49.7012	3.66402
$185$	4.29747	0.315956
$186$	0	0
$187$	−31.4007	−2.29624
$188$	−45.2880	−3.30296
$189$	0	0
$190$	−14.8590	−1.07799
$191$	0.712091	0.0515251	0.0257625	−	0.999668i	$-0.491799\pi$
$191$	0.712091	0.0515251	0.0257625	+	0.999668i	$0.491799\pi$
$192$	0	0
$193$	−20.6987	−1.48992	−0.744961	−	0.667108i	$-0.767532\pi$
$193$	−20.6987	−1.48992	−0.744961	+	0.667108i	$0.767532\pi$
$194$	15.7484	1.13067
$195$	0	0
$196$	9.02012	0.644294
$197$	−25.1924	−1.79489	−0.897444	−	0.441129i	$-0.854578\pi$
$197$	−25.1924	−1.79489	−0.897444	+	0.441129i	$0.854578\pi$
$198$	0	0
$199$	−0.692437	−0.0490855	−0.0245428	−	0.999699i	$-0.507813\pi$
$199$	−0.692437	−0.0490855	−0.0245428	+	0.999699i	$0.507813\pi$
$200$	5.92303	0.418822
$201$	0	0
$202$	−15.0086	−1.05600
$203$	10.1656	0.713484
$204$	0	0
$205$	6.56253	0.458347
$206$	−16.0477	−1.11810
$207$	0	0
$208$	6.22524	0.431642
$209$	−33.8935	−2.34446
$210$	0	0
$211$	3.54860	0.244296	0.122148	−	0.992512i	$-0.461022\pi$
$211$	3.54860	0.244296	0.122148	+	0.992512i	$0.461022\pi$
$212$	26.2531	1.80307
$213$	0	0
$214$	32.2541	2.20485
$215$	0.654059	0.0446065
$216$	0	0
$217$	−12.9387	−0.878338
$218$	23.8061	1.61235
$219$	0	0
$220$	25.0068	1.68596
$221$	5.46275	0.367464
$222$	0	0
$223$	−0.722762	−0.0483997	−0.0241999	−	0.999707i	$-0.507704\pi$
$223$	−0.722762	−0.0483997	−0.0241999	+	0.999707i	$0.507704\pi$
$224$	11.5715	0.773154
$225$	0	0
$226$	−34.4419	−2.29104
$227$	−19.6746	−1.30585	−0.652925	−	0.757423i	$-0.726458\pi$
$227$	−19.6746	−1.30585	−0.652925	+	0.757423i	$0.726458\pi$
$228$	0	0
$229$	4.51675	0.298475	0.149238	−	0.988801i	$-0.452318\pi$
$229$	4.51675	0.298475	0.149238	+	0.988801i	$0.452318\pi$
$230$	−21.1458	−1.39431
$231$	0	0
$232$	19.9890	1.31234
$233$	1.13516	0.0743669	0.0371835	−	0.999308i	$-0.488161\pi$
$233$	1.13516	0.0743669	0.0371835	+	0.999308i	$0.488161\pi$
$234$	0	0
$235$	10.4100	0.679077
$236$	32.8835	2.14054
$237$	0	0
$238$	41.4665	2.68787
$239$	10.8080	0.699113	0.349556	−	0.936915i	$-0.386332\pi$
$239$	10.8080	0.699113	0.349556	+	0.936915i	$0.386332\pi$
$240$	0	0
$241$	−7.63513	−0.491822	−0.245911	−	0.969292i	$-0.579087\pi$
$241$	−7.63513	−0.491822	−0.245911	+	0.969292i	$0.579087\pi$
$242$	55.5438	3.57049
$243$	0	0
$244$	26.1962	1.67704
$245$	−2.07340	−0.132464
$246$	0	0
$247$	5.89643	0.375181
$248$	−25.4420	−1.61557
$249$	0	0
$250$	−2.52000	−0.159379
$251$	6.71696	0.423971	0.211985	−	0.977273i	$-0.432007\pi$
$251$	6.71696	0.423971	0.211985	+	0.977273i	$0.432007\pi$
$252$	0	0
$253$	−48.2336	−3.03242
$254$	11.0237	0.691689
$255$	0	0
$256$	−31.4111	−1.96319
$257$	22.5325	1.40554	0.702770	−	0.711417i	$-0.251946\pi$
$257$	22.5325	1.40554	0.702770	+	0.711417i	$0.251946\pi$
$258$	0	0
$259$	−12.9449	−0.804355
$260$	−4.35041	−0.269801
$261$	0	0
$262$	−33.4312	−2.06539
$263$	13.8641	0.854896	0.427448	−	0.904040i	$-0.359413\pi$
$263$	13.8641	0.854896	0.427448	+	0.904040i	$0.359413\pi$
$264$	0	0
$265$	−6.03463	−0.370704
$266$	44.7584	2.74432
$267$	0	0
$268$	24.7818	1.51379
$269$	3.78932	0.231039	0.115520	−	0.993305i	$-0.463147\pi$
$269$	3.78932	0.231039	0.115520	+	0.993305i	$0.463147\pi$
$270$	0	0
$271$	−8.66850	−0.526574	−0.263287	−	0.964718i	$-0.584807\pi$
$271$	−8.66850	−0.526574	−0.263287	+	0.964718i	$0.584807\pi$
$272$	34.0069	2.06197
$273$	0	0
$274$	−26.7660	−1.61700
$275$	−5.74814	−0.346626
$276$	0	0
$277$	−12.3427	−0.741603	−0.370801	−	0.928712i	$-0.620917\pi$
$277$	−12.3427	−0.741603	−0.370801	+	0.928712i	$0.620917\pi$
$278$	11.2205	0.672958
$279$	0	0
$280$	−17.8414	−1.06623
$281$	−4.72605	−0.281933	−0.140966	−	0.990014i	$-0.545021\pi$
$281$	−4.72605	−0.281933	−0.140966	+	0.990014i	$0.545021\pi$
$282$	0	0
$283$	−12.8718	−0.765151	−0.382576	−	0.923924i	$-0.624963\pi$
$283$	−12.8718	−0.765151	−0.382576	+	0.923924i	$0.624963\pi$
$284$	8.70648	0.516635
$285$	0	0
$286$	−14.4853	−0.856536
$287$	−19.7677	−1.16685
$288$	0	0
$289$	12.8416	0.755391
$290$	−8.50448	−0.499400
$291$	0	0
$292$	54.1706	3.17009
$293$	−25.4833	−1.48875	−0.744375	−	0.667762i	$-0.767252\pi$
$293$	−25.4833	−1.48875	−0.744375	+	0.667762i	$0.767252\pi$
$294$	0	0
$295$	−7.55872	−0.440086
$296$	−25.4541	−1.47949
$297$	0	0
$298$	−41.2313	−2.38847
$299$	8.39117	0.485274
$300$	0	0
$301$	−1.97016	−0.113558
$302$	−15.0802	−0.867766
$303$	0	0
$304$	36.7067	2.10527
$305$	−6.02155	−0.344793
$306$	0	0
$307$	−31.4528	−1.79511	−0.897553	−	0.440906i	$-0.854657\pi$
$307$	−31.4528	−1.79511	−0.897553	+	0.440906i	$0.854657\pi$
$308$	−75.3256	−4.29207
$309$	0	0
$310$	10.8245	0.614790
$311$	−6.08011	−0.344771	−0.172386	−	0.985030i	$-0.555148\pi$
$311$	−6.08011	−0.344771	−0.172386	+	0.985030i	$0.555148\pi$
$312$	0	0
$313$	−7.87478	−0.445109	−0.222554	−	0.974920i	$-0.571439\pi$
$313$	−7.87478	−0.445109	−0.222554	+	0.974920i	$0.571439\pi$
$314$	15.6971	0.885841
$315$	0	0
$316$	59.3714	3.33990
$317$	−19.6354	−1.10284	−0.551418	−	0.834229i	$-0.685913\pi$
$317$	−19.6354	−1.10284	−0.551418	+	0.834229i	$0.685913\pi$
$318$	0	0
$319$	−19.3988	−1.08612
$320$	2.76979	0.154836
$321$	0	0
$322$	63.6954	3.54961
$323$	32.2107	1.79225
$324$	0	0
$325$	1.00000	0.0554700
$326$	−6.10697	−0.338234
$327$	0	0
$328$	−38.8701	−2.14624
$329$	−31.3572	−1.72878
$330$	0	0
$331$	−3.32613	−0.182820	−0.0914102	−	0.995813i	$-0.529137\pi$
$331$	−3.32613	−0.182820	−0.0914102	+	0.995813i	$0.529137\pi$
$332$	−26.8094	−1.47136
$333$	0	0
$334$	12.5113	0.684588
$335$	−5.69643	−0.311229
$336$	0	0
$337$	−27.3511	−1.48991	−0.744954	−	0.667116i	$-0.767528\pi$
$337$	−27.3511	−1.48991	−0.744954	+	0.667116i	$0.767528\pi$
$338$	2.52000	0.137070
$339$	0	0
$340$	−23.7652	−1.28885
$341$	24.6908	1.33708
$342$	0	0
$343$	−14.8400	−0.801282
$344$	−3.87401	−0.208873
$345$	0	0
$346$	−6.72334	−0.361449
$347$	−24.8433	−1.33366	−0.666828	−	0.745211i	$-0.732349\pi$
$347$	−24.8433	−1.33366	−0.666828	+	0.745211i	$0.732349\pi$
$348$	0	0
$349$	−1.83113	−0.0980183	−0.0490092	−	0.998798i	$-0.515606\pi$
$349$	−1.83113	−0.0980183	−0.0490092	+	0.998798i	$0.515606\pi$
$350$	7.59077	0.405744
$351$	0	0
$352$	−22.0817	−1.17696
$353$	21.2604	1.13158	0.565788	−	0.824551i	$-0.308572\pi$
$353$	21.2604	1.13158	0.565788	+	0.824551i	$0.308572\pi$
$354$	0	0
$355$	−2.00130	−0.106218
$356$	9.82338	0.520638
$357$	0	0
$358$	−35.2620	−1.86366
$359$	−4.46395	−0.235598	−0.117799	−	0.993037i	$-0.537584\pi$
$359$	−4.46395	−0.235598	−0.117799	+	0.993037i	$0.537584\pi$
$360$	0	0
$361$	15.7679	0.829889
$362$	20.8844	1.09766
$363$	0	0
$364$	13.1043	0.686854
$365$	−12.4518	−0.651759
$366$	0	0
$367$	22.8656	1.19358	0.596788	−	0.802399i	$-0.296443\pi$
$367$	22.8656	1.19358	0.596788	+	0.802399i	$0.296443\pi$
$368$	52.2370	2.72304
$369$	0	0
$370$	10.8296	0.563006
$371$	18.1776	0.943732
$372$	0	0
$373$	17.8391	0.923673	0.461836	−	0.886965i	$-0.347191\pi$
$373$	17.8391	0.923673	0.461836	+	0.886965i	$0.347191\pi$
$374$	−79.1298	−4.09170
$375$	0	0
$376$	−61.6591	−3.17982
$377$	3.37479	0.173811
$378$	0	0
$379$	24.4173	1.25423	0.627117	−	0.778925i	$-0.284235\pi$
$379$	24.4173	1.25423	0.627117	+	0.778925i	$0.284235\pi$
$380$	−25.6519	−1.31591
$381$	0	0
$382$	1.79447	0.0918130
$383$	−32.2621	−1.64852	−0.824258	−	0.566214i	$-0.808408\pi$
$383$	−32.2621	−1.64852	−0.824258	+	0.566214i	$0.808408\pi$
$384$	0	0
$385$	17.3146	0.882434
$386$	−52.1607	−2.65491
$387$	0	0
$388$	27.1873	1.38023
$389$	−19.8608	−1.00698	−0.503491	−	0.864001i	$-0.667951\pi$
$389$	−19.8608	−1.00698	−0.503491	+	0.864001i	$0.667951\pi$
$390$	0	0
$391$	45.8389	2.31817
$392$	12.2808	0.620274
$393$	0	0
$394$	−63.4850	−3.19833
$395$	−13.6473	−0.686671
$396$	0	0
$397$	0.215714	0.0108264	0.00541319	−	0.999985i	$-0.498277\pi$
$397$	0.215714	0.0108264	0.00541319	+	0.999985i	$0.498277\pi$
$398$	−1.74494	−0.0874660
$399$	0	0
$400$	6.22524	0.311262
$401$	18.8074	0.939197	0.469598	−	0.882880i	$-0.344399\pi$
$401$	18.8074	0.939197	0.469598	+	0.882880i	$0.344399\pi$
$402$	0	0
$403$	−4.29543	−0.213971
$404$	−25.9101	−1.28907
$405$	0	0
$406$	25.6173	1.27136
$407$	24.7025	1.22446
$408$	0	0
$409$	1.24419	0.0615212	0.0307606	−	0.999527i	$-0.490207\pi$
$409$	1.24419	0.0615212	0.0307606	+	0.999527i	$0.490207\pi$
$410$	16.5376	0.816733
$411$	0	0
$412$	−27.7040	−1.36488
$413$	22.7684	1.12036
$414$	0	0
$415$	6.16250	0.302505
$416$	3.84154	0.188347
$417$	0	0
$418$	−85.4118	−4.17763
$419$	33.9802	1.66004	0.830020	−	0.557733i	$-0.188329\pi$
$419$	33.9802	1.66004	0.830020	+	0.557733i	$0.188329\pi$
$420$	0	0
$421$	−11.0622	−0.539141	−0.269570	−	0.962981i	$-0.586882\pi$
$421$	−11.0622	−0.539141	−0.269570	+	0.962981i	$0.586882\pi$
$422$	8.94248	0.435313
$423$	0	0
$424$	35.7433	1.73585
$425$	5.46275	0.264982
$426$	0	0
$427$	18.1382	0.877767
$428$	55.6820	2.69149
$429$	0	0
$430$	1.64823	0.0794847
$431$	−29.3780	−1.41509	−0.707545	−	0.706669i	$-0.750197\pi$
$431$	−29.3780	−1.41509	−0.707545	+	0.706669i	$0.750197\pi$
$432$	0	0
$433$	−32.7711	−1.57488	−0.787439	−	0.616393i	$-0.788593\pi$
$433$	−32.7711	−1.57488	−0.787439	+	0.616393i	$0.788593\pi$
$434$	−32.6056	−1.56512
$435$	0	0
$436$	41.0977	1.96822
$437$	49.4779	2.36685
$438$	0	0
$439$	4.05016	0.193303	0.0966517	−	0.995318i	$-0.469187\pi$
$439$	4.05016	0.193303	0.0966517	+	0.995318i	$0.469187\pi$
$440$	34.0464	1.62310
$441$	0	0
$442$	13.7661	0.654789
$443$	−4.67219	−0.221983	−0.110991	−	0.993821i	$-0.535403\pi$
$443$	−4.67219	−0.221983	−0.110991	+	0.993821i	$0.535403\pi$
$444$	0	0
$445$	−2.25804	−0.107041
$446$	−1.82136	−0.0862439
$447$	0	0
$448$	−8.34318	−0.394178
$449$	−25.3333	−1.19555	−0.597775	−	0.801664i	$-0.703948\pi$
$449$	−25.3333	−1.19555	−0.597775	+	0.801664i	$0.703948\pi$
$450$	0	0
$451$	37.7223	1.77628
$452$	−59.4588	−2.79671
$453$	0	0
$454$	−49.5800	−2.32691
$455$	−3.01221	−0.141214
$456$	0	0
$457$	1.32675	0.0620626	0.0310313	−	0.999518i	$-0.490121\pi$
$457$	1.32675	0.0620626	0.0310313	+	0.999518i	$0.490121\pi$
$458$	11.3822	0.531856
$459$	0	0
$460$	−36.5050	−1.70205
$461$	7.72464	0.359772	0.179886	−	0.983687i	$-0.442427\pi$
$461$	7.72464	0.359772	0.179886	+	0.983687i	$0.442427\pi$
$462$	0	0
$463$	−10.6826	−0.496461	−0.248231	−	0.968701i	$-0.579849\pi$
$463$	−10.6826	−0.496461	−0.248231	+	0.968701i	$0.579849\pi$
$464$	21.0089	0.975313
$465$	0	0
$466$	2.86061	0.132515
$467$	2.08913	0.0966736	0.0483368	−	0.998831i	$-0.484608\pi$
$467$	2.08913	0.0966736	0.0483368	+	0.998831i	$0.484608\pi$
$468$	0	0
$469$	17.1588	0.792321
$470$	26.2333	1.21005
$471$	0	0
$472$	44.7706	2.06073
$473$	3.75963	0.172868
$474$	0	0
$475$	5.89643	0.270547
$476$	71.5857	3.28113
$477$	0	0
$478$	27.2362	1.24576
$479$	1.26601	0.0578453	0.0289226	−	0.999582i	$-0.490792\pi$
$479$	1.26601	0.0578453	0.0289226	+	0.999582i	$0.490792\pi$
$480$	0	0
$481$	−4.29747	−0.195948
$482$	−19.2405	−0.876382
$483$	0	0
$484$	95.8881	4.35855
$485$	−6.24936	−0.283769
$486$	0	0
$487$	20.6599	0.936191	0.468096	−	0.883678i	$-0.344940\pi$
$487$	20.6599	0.936191	0.468096	+	0.883678i	$0.344940\pi$
$488$	35.6658	1.61452
$489$	0	0
$490$	−5.22496	−0.236040
$491$	−12.1258	−0.547231	−0.273616	−	0.961839i	$-0.588220\pi$
$491$	−12.1258	−0.547231	−0.273616	+	0.961839i	$0.588220\pi$
$492$	0	0
$493$	18.4357	0.830300
$494$	14.8590	0.668539
$495$	0	0
$496$	−26.7401	−1.20066
$497$	6.02834	0.270408
$498$	0	0
$499$	8.37801	0.375051	0.187526	−	0.982260i	$-0.439953\pi$
$499$	8.37801	0.375051	0.187526	+	0.982260i	$0.439953\pi$
$500$	−4.35041	−0.194556
$501$	0	0
$502$	16.9267	0.755477
$503$	33.8591	1.50970	0.754851	−	0.655897i	$-0.227709\pi$
$503$	33.8591	1.50970	0.754851	+	0.655897i	$0.227709\pi$
$504$	0	0
$505$	5.95578	0.265029
$506$	−121.549	−5.40351
$507$	0	0
$508$	19.0308	0.844355
$509$	11.1315	0.493393	0.246697	−	0.969093i	$-0.420655\pi$
$509$	11.1315	0.493393	0.246697	+	0.969093i	$0.420655\pi$
$510$	0	0
$511$	37.5075	1.65923
$512$	−49.8301	−2.20220
$513$	0	0
$514$	56.7820	2.50455
$515$	6.36814	0.280614
$516$	0	0
$517$	59.8385	2.63169
$518$	−32.6211	−1.43329
$519$	0	0
$520$	−5.92303	−0.259742
$521$	−7.38081	−0.323359	−0.161679	−	0.986843i	$-0.551691\pi$
$521$	−7.38081	−0.323359	−0.161679	+	0.986843i	$0.551691\pi$
$522$	0	0
$523$	−27.1586	−1.18756	−0.593781	−	0.804626i	$-0.702366\pi$
$523$	−27.1586	−1.18756	−0.593781	+	0.804626i	$0.702366\pi$
$524$	−57.7140	−2.52125
$525$	0	0
$526$	34.9375	1.52335
$527$	−23.4649	−1.02215
$528$	0	0
$529$	47.4117	2.06138
$530$	−15.2073	−0.660562
$531$	0	0
$532$	77.2688	3.35003
$533$	−6.56253	−0.284255
$534$	0	0
$535$	−12.7993	−0.553360
$536$	33.7401	1.45735
$537$	0	0
$538$	9.54910	0.411691
$539$	−11.9182	−0.513352
$540$	0	0
$541$	17.0809	0.734365	0.367182	−	0.930149i	$-0.380322\pi$
$541$	17.0809	0.734365	0.367182	+	0.930149i	$0.380322\pi$
$542$	−21.8446	−0.938307
$543$	0	0
$544$	20.9854	0.899740
$545$	−9.44687	−0.404659
$546$	0	0
$547$	−20.8070	−0.889645	−0.444822	−	0.895619i	$-0.646733\pi$
$547$	−20.8070	−0.889645	−0.444822	+	0.895619i	$0.646733\pi$
$548$	−46.2076	−1.97389
$549$	0	0
$550$	−14.4853	−0.617657
$551$	19.8992	0.847736
$552$	0	0
$553$	41.1085	1.74811
$554$	−31.1037	−1.32147
$555$	0	0
$556$	19.3704	0.821490
$557$	−29.9187	−1.26770	−0.633848	−	0.773458i	$-0.718525\pi$
$557$	−29.9187	−1.26770	−0.633848	+	0.773458i	$0.718525\pi$
$558$	0	0
$559$	−0.654059	−0.0276638
$560$	−18.7517	−0.792404
$561$	0	0
$562$	−11.9097	−0.502379
$563$	33.6159	1.41674	0.708372	−	0.705840i	$-0.249430\pi$
$563$	33.6159	1.41674	0.708372	+	0.705840i	$0.249430\pi$
$564$	0	0
$565$	13.6674	0.574992
$566$	−32.4370	−1.36343
$567$	0	0
$568$	11.8538	0.497374
$569$	39.2056	1.64358	0.821792	−	0.569788i	$-0.192975\pi$
$569$	39.2056	1.64358	0.821792	+	0.569788i	$0.192975\pi$
$570$	0	0
$571$	−13.7782	−0.576599	−0.288299	−	0.957540i	$-0.593090\pi$
$571$	−13.7782	−0.576599	−0.288299	+	0.957540i	$0.593090\pi$
$572$	−25.0068	−1.04559
$573$	0	0
$574$	−49.8146	−2.07922
$575$	8.39117	0.349936
$576$	0	0
$577$	25.4974	1.06147	0.530736	−	0.847537i	$-0.321916\pi$
$577$	25.4974	1.06147	0.530736	+	0.847537i	$0.321916\pi$
$578$	32.3610	1.34604
$579$	0	0
$580$	−14.6817	−0.609626
$581$	−18.5627	−0.770112
$582$	0	0
$583$	−34.6879	−1.43663
$584$	73.7526	3.05191
$585$	0	0
$586$	−64.2179	−2.65282
$587$	−30.6283	−1.26417	−0.632083	−	0.774901i	$-0.717800\pi$
$587$	−30.6283	−1.26417	−0.632083	+	0.774901i	$0.717800\pi$
$588$	0	0
$589$	−25.3277	−1.04361
$590$	−19.0480	−0.784194
$591$	0	0
$592$	−26.7528	−1.09953
$593$	29.5710	1.21433	0.607167	−	0.794574i	$-0.292306\pi$
$593$	29.5710	1.21433	0.607167	+	0.794574i	$0.292306\pi$
$594$	0	0
$595$	−16.4549	−0.674587
$596$	−71.1797	−2.91564
$597$	0	0
$598$	21.1458	0.864714
$599$	1.25104	0.0511159	0.0255580	−	0.999673i	$-0.491864\pi$
$599$	1.25104	0.0511159	0.0255580	+	0.999673i	$0.491864\pi$
$600$	0	0
$601$	−10.7362	−0.437937	−0.218969	−	0.975732i	$-0.570269\pi$
$601$	−10.7362	−0.437937	−0.218969	+	0.975732i	$0.570269\pi$
$602$	−4.96481	−0.202351
$603$	0	0
$604$	−26.0337	−1.05930
$605$	−22.0412	−0.896101
$606$	0	0
$607$	35.6778	1.44812	0.724059	−	0.689738i	$-0.242274\pi$
$607$	35.6778	1.44812	0.724059	+	0.689738i	$0.242274\pi$
$608$	22.6514	0.918634
$609$	0	0
$610$	−15.1743	−0.614390
$611$	−10.4100	−0.421146
$612$	0	0
$613$	40.1588	1.62200	0.810999	−	0.585048i	$-0.198924\pi$
$613$	40.1588	1.62200	0.810999	+	0.585048i	$0.198924\pi$
$614$	−79.2611	−3.19872
$615$	0	0
$616$	−102.555	−4.13206
$617$	−9.14730	−0.368256	−0.184128	−	0.982902i	$-0.558946\pi$
$617$	−9.14730	−0.368256	−0.184128	+	0.982902i	$0.558946\pi$
$618$	0	0
$619$	−38.5377	−1.54896	−0.774481	−	0.632597i	$-0.781989\pi$
$619$	−38.5377	−1.54896	−0.774481	+	0.632597i	$0.781989\pi$
$620$	18.6869	0.750483
$621$	0	0
$622$	−15.3219	−0.614351
$623$	6.80168	0.272503
$624$	0	0
$625$	1.00000	0.0400000
$626$	−19.8445	−0.793144
$627$	0	0
$628$	27.0988	1.08136
$629$	−23.4760	−0.936050
$630$	0	0
$631$	−20.8431	−0.829749	−0.414875	−	0.909879i	$-0.636175\pi$
$631$	−20.8431	−0.829749	−0.414875	+	0.909879i	$0.636175\pi$
$632$	80.8335	3.21538
$633$	0	0
$634$	−49.4813	−1.96515
$635$	−4.37449	−0.173596
$636$	0	0
$637$	2.07340	0.0821509
$638$	−48.8850	−1.93538
$639$	0	0
$640$	14.6629	0.579604
$641$	−5.05834	−0.199792	−0.0998962	−	0.994998i	$-0.531851\pi$
$641$	−5.05834	−0.199792	−0.0998962	+	0.994998i	$0.531851\pi$
$642$	0	0
$643$	1.11156	0.0438355	0.0219177	−	0.999760i	$-0.493023\pi$
$643$	1.11156	0.0438355	0.0219177	+	0.999760i	$0.493023\pi$
$644$	109.961	4.33306
$645$	0	0
$646$	81.1711	3.19363
$647$	−4.82790	−0.189804	−0.0949022	−	0.995487i	$-0.530254\pi$
$647$	−4.82790	−0.189804	−0.0949022	+	0.995487i	$0.530254\pi$
$648$	0	0
$649$	−43.4486	−1.70551
$650$	2.52000	0.0988426
$651$	0	0
$652$	−10.5428	−0.412887
$653$	−18.0563	−0.706596	−0.353298	−	0.935511i	$-0.614940\pi$
$653$	−18.0563	−0.706596	−0.353298	+	0.935511i	$0.614940\pi$
$654$	0	0
$655$	13.2663	0.518359
$656$	−40.8533	−1.59505
$657$	0	0
$658$	−79.0203	−3.08053
$659$	−31.8348	−1.24011	−0.620054	−	0.784559i	$-0.712889\pi$
$659$	−31.8348	−1.24011	−0.620054	+	0.784559i	$0.712889\pi$
$660$	0	0
$661$	13.8225	0.537634	0.268817	−	0.963191i	$-0.413367\pi$
$661$	13.8225	0.537634	0.268817	+	0.963191i	$0.413367\pi$
$662$	−8.38184	−0.325770
$663$	0	0
$664$	−36.5007	−1.41650
$665$	−17.7613	−0.688753
$666$	0	0
$667$	28.3185	1.09649
$668$	21.5989	0.835687
$669$	0	0
$670$	−14.3550	−0.554582
$671$	−34.6127	−1.33621
$672$	0	0
$673$	−4.58531	−0.176751	−0.0883754	−	0.996087i	$-0.528168\pi$
$673$	−4.58531	−0.176751	−0.0883754	+	0.996087i	$0.528168\pi$
$674$	−68.9247	−2.65488
$675$	0	0
$676$	4.35041	0.167323
$677$	−4.17449	−0.160439	−0.0802193	−	0.996777i	$-0.525562\pi$
$677$	−4.17449	−0.160439	−0.0802193	+	0.996777i	$0.525562\pi$
$678$	0	0
$679$	18.8244	0.722413
$680$	−32.3560	−1.24080
$681$	0	0
$682$	62.2208	2.38256
$683$	18.6465	0.713489	0.356744	−	0.934202i	$-0.383887\pi$
$683$	18.6465	0.713489	0.356744	+	0.934202i	$0.383887\pi$
$684$	0	0
$685$	10.6214	0.405824
$686$	−37.3967	−1.42781
$687$	0	0
$688$	−4.07167	−0.155231
$689$	6.03463	0.229901
$690$	0	0
$691$	−1.83720	−0.0698905	−0.0349453	−	0.999389i	$-0.511126\pi$
$691$	−1.83720	−0.0698905	−0.0349453	+	0.999389i	$0.511126\pi$
$692$	−11.6068	−0.441226
$693$	0	0
$694$	−62.6051	−2.37646
$695$	−4.45256	−0.168895
$696$	0	0
$697$	−35.8494	−1.35789
$698$	−4.61446	−0.174660
$699$	0	0
$700$	13.1043	0.495297
$701$	−48.5393	−1.83330	−0.916652	−	0.399687i	$-0.869119\pi$
$701$	−48.5393	−1.83330	−0.916652	+	0.399687i	$0.869119\pi$
$702$	0	0
$703$	−25.3397	−0.955706
$704$	15.9211	0.600051
$705$	0	0
$706$	53.5762	2.01637
$707$	−17.9401	−0.674705
$708$	0	0
$709$	−40.0208	−1.50301	−0.751507	−	0.659725i	$-0.770673\pi$
$709$	−40.0208	−1.50301	−0.751507	+	0.659725i	$0.770673\pi$
$710$	−5.04329	−0.189271
$711$	0	0
$712$	13.3744	0.501228
$713$	−36.0437	−1.34985
$714$	0	0
$715$	5.74814	0.214968
$716$	−60.8746	−2.27499
$717$	0	0
$718$	−11.2492	−0.419815
$719$	−28.6704	−1.06923	−0.534613	−	0.845097i	$-0.679543\pi$
$719$	−28.6704	−1.06923	−0.534613	+	0.845097i	$0.679543\pi$
$720$	0	0
$721$	−19.1822	−0.714381
$722$	39.7351	1.47879
$723$	0	0
$724$	36.0538	1.33993
$725$	3.37479	0.125337
$726$	0	0
$727$	37.3060	1.38360	0.691802	−	0.722087i	$-0.256817\pi$
$727$	37.3060	1.38360	0.691802	+	0.722087i	$0.256817\pi$
$728$	17.8414	0.661246
$729$	0	0
$730$	−31.3786	−1.16138
$731$	−3.57296	−0.132151
$732$	0	0
$733$	−12.7628	−0.471404	−0.235702	−	0.971825i	$-0.575739\pi$
$733$	−12.7628	−0.471404	−0.235702	+	0.971825i	$0.575739\pi$
$734$	57.6214	2.12685
$735$	0	0
$736$	32.2350	1.18820
$737$	−32.7439	−1.20614
$738$	0	0
$739$	20.5383	0.755512	0.377756	−	0.925905i	$-0.376696\pi$
$739$	20.5383	0.755512	0.377756	+	0.925905i	$0.376696\pi$
$740$	18.6958	0.687269
$741$	0	0
$742$	45.8075	1.68165
$743$	21.0520	0.772322	0.386161	−	0.922431i	$-0.373801\pi$
$743$	21.0520	0.772322	0.386161	+	0.922431i	$0.373801\pi$
$744$	0	0
$745$	16.3616	0.599443
$746$	44.9545	1.64590
$747$	0	0
$748$	−136.606	−4.99480
$749$	38.5540	1.40873
$750$	0	0
$751$	8.09437	0.295368	0.147684	−	0.989035i	$-0.452818\pi$
$751$	8.09437	0.295368	0.147684	+	0.989035i	$0.452818\pi$
$752$	−64.8050	−2.36320
$753$	0	0
$754$	8.50448	0.309715
$755$	5.98419	0.217787
$756$	0	0
$757$	16.9554	0.616254	0.308127	−	0.951345i	$-0.400298\pi$
$757$	16.9554	0.616254	0.308127	+	0.951345i	$0.400298\pi$
$758$	61.5317	2.23493
$759$	0	0
$760$	−34.9247	−1.26685
$761$	38.6271	1.40023	0.700115	−	0.714030i	$-0.253132\pi$
$761$	38.6271	1.40023	0.700115	+	0.714030i	$0.253132\pi$
$762$	0	0
$763$	28.4559	1.03017
$764$	3.09788	0.112078
$765$	0	0
$766$	−81.3006	−2.93751
$767$	7.55872	0.272930
$768$	0	0
$769$	21.5095	0.775651	0.387825	−	0.921733i	$-0.373226\pi$
$769$	21.5095	0.775651	0.387825	+	0.921733i	$0.373226\pi$
$770$	43.6328	1.57242
$771$	0	0
$772$	−90.0477	−3.24089
$773$	−38.8254	−1.39645	−0.698226	−	0.715878i	$-0.746027\pi$
$773$	−38.8254	−1.39645	−0.698226	+	0.715878i	$0.746027\pi$
$774$	0	0
$775$	−4.29543	−0.154296
$776$	37.0152	1.32877
$777$	0	0
$778$	−50.0492	−1.79435
$779$	−38.6955	−1.38641
$780$	0	0
$781$	−11.5038	−0.411637
$782$	115.514	4.13077
$783$	0	0
$784$	12.9074	0.460978
$785$	−6.22902	−0.222323
$786$	0	0
$787$	37.7221	1.34465	0.672324	−	0.740257i	$-0.265296\pi$
$787$	37.7221	1.34465	0.672324	+	0.740257i	$0.265296\pi$
$788$	−109.597	−3.90425
$789$	0	0
$790$	−34.3913	−1.22359
$791$	−41.1691	−1.46380
$792$	0	0
$793$	6.02155	0.213832
$794$	0.543600	0.0192916
$795$	0	0
$796$	−3.01238	−0.106771
$797$	−33.6805	−1.19303	−0.596513	−	0.802603i	$-0.703448\pi$
$797$	−33.6805	−1.19303	−0.596513	+	0.802603i	$0.703448\pi$
$798$	0	0
$799$	−56.8675	−2.01183
$800$	3.84154	0.135819
$801$	0	0
$802$	47.3947	1.67356
$803$	−71.5749	−2.52582
$804$	0	0
$805$	−25.2759	−0.890860
$806$	−10.8245	−0.381277
$807$	0	0
$808$	−35.2763	−1.24102
$809$	−27.7726	−0.976432	−0.488216	−	0.872723i	$-0.662352\pi$
$809$	−27.7726	−0.976432	−0.488216	+	0.872723i	$0.662352\pi$
$810$	0	0
$811$	19.6682	0.690645	0.345323	−	0.938484i	$-0.387770\pi$
$811$	19.6682	0.690645	0.345323	+	0.938484i	$0.387770\pi$
$812$	44.2244	1.55197
$813$	0	0
$814$	62.2503	2.18187
$815$	2.42340	0.0848879
$816$	0	0
$817$	−3.85662	−0.134926
$818$	3.13536	0.109625
$819$	0	0
$820$	28.5497	0.996998
$821$	−15.0928	−0.526744	−0.263372	−	0.964694i	$-0.584835\pi$
$821$	−15.0928	−0.526744	−0.263372	+	0.964694i	$0.584835\pi$
$822$	0	0
$823$	−16.5382	−0.576485	−0.288242	−	0.957557i	$-0.593071\pi$
$823$	−16.5382	−0.576485	−0.288242	+	0.957557i	$0.593071\pi$
$824$	−37.7187	−1.31399
$825$	0	0
$826$	57.3765	1.99638
$827$	50.4211	1.75331	0.876656	−	0.481117i	$-0.159769\pi$
$827$	50.4211	1.75331	0.876656	+	0.481117i	$0.159769\pi$
$828$	0	0
$829$	−27.0835	−0.940650	−0.470325	−	0.882493i	$-0.655863\pi$
$829$	−27.0835	−0.940650	−0.470325	+	0.882493i	$0.655863\pi$
$830$	15.5295	0.539037
$831$	0	0
$832$	−2.76979	−0.0960251
$833$	11.3264	0.392438
$834$	0	0
$835$	−4.96480	−0.171814
$836$	−147.451	−5.09969
$837$	0	0
$838$	85.6301	2.95804
$839$	35.1426	1.21326	0.606628	−	0.794986i	$-0.292522\pi$
$839$	35.1426	1.21326	0.606628	+	0.794986i	$0.292522\pi$
$840$	0	0
$841$	−17.6108	−0.607268
$842$	−27.8769	−0.960701
$843$	0	0
$844$	15.4379	0.531393
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	66.3926	2.28128
$848$	37.5670	1.29006
$849$	0	0
$850$	13.7661	0.472175
$851$	−36.0608	−1.23615
$852$	0	0
$853$	−47.1648	−1.61489	−0.807446	−	0.589941i	$-0.799151\pi$
$853$	−47.1648	−1.61489	−0.807446	+	0.589941i	$0.799151\pi$
$854$	45.7082	1.56410
$855$	0	0
$856$	75.8104	2.59115
$857$	54.4819	1.86106	0.930532	−	0.366209i	$-0.119345\pi$
$857$	54.4819	1.86106	0.930532	+	0.366209i	$0.119345\pi$
$858$	0	0
$859$	−26.5007	−0.904192	−0.452096	−	0.891969i	$-0.649324\pi$
$859$	−26.5007	−0.904192	−0.452096	+	0.891969i	$0.649324\pi$
$860$	2.84542	0.0970282
$861$	0	0
$862$	−74.0326	−2.52156
$863$	−34.4594	−1.17301	−0.586506	−	0.809945i	$-0.699497\pi$
$863$	−34.4594	−1.17301	−0.586506	+	0.809945i	$0.699497\pi$
$864$	0	0
$865$	2.66799	0.0907144
$866$	−82.5832	−2.80629
$867$	0	0
$868$	−56.2888	−1.91057
$869$	−78.4467	−2.66112
$870$	0	0
$871$	5.69643	0.193016
$872$	55.9541	1.89485
$873$	0	0
$874$	124.684	4.21752
$875$	−3.01221	−0.101831
$876$	0	0
$877$	−21.5922	−0.729117	−0.364558	−	0.931181i	$-0.618780\pi$
$877$	−21.5922	−0.729117	−0.364558	+	0.931181i	$0.618780\pi$
$878$	10.2064	0.344449
$879$	0	0
$880$	35.7835	1.20626
$881$	47.3112	1.59396	0.796978	−	0.604009i	$-0.206431\pi$
$881$	47.3112	1.59396	0.796978	+	0.604009i	$0.206431\pi$
$882$	0	0
$883$	−5.56289	−0.187206	−0.0936031	−	0.995610i	$-0.529838\pi$
$883$	−5.56289	−0.187206	−0.0936031	+	0.995610i	$0.529838\pi$
$884$	23.7652	0.799310
$885$	0	0
$886$	−11.7739	−0.395553
$887$	36.4132	1.22264	0.611318	−	0.791385i	$-0.290640\pi$
$887$	36.4132	1.22264	0.611318	+	0.791385i	$0.290640\pi$
$888$	0	0
$889$	13.1769	0.441938
$890$	−5.69026	−0.190738
$891$	0	0
$892$	−3.14431	−0.105279
$893$	−61.3821	−2.05407
$894$	0	0
$895$	13.9929	0.467730
$896$	−44.1678	−1.47554
$897$	0	0
$898$	−63.8399	−2.13036
$899$	−14.4962	−0.483475
$900$	0	0
$901$	32.9657	1.09825
$902$	95.0604	3.16516
$903$	0	0
$904$	−80.9525	−2.69244
$905$	−8.28746	−0.275484
$906$	0	0
$907$	44.0920	1.46405	0.732025	−	0.681277i	$-0.238575\pi$
$907$	44.0920	1.46405	0.732025	+	0.681277i	$0.238575\pi$
$908$	−85.5925	−2.84049
$909$	0	0
$910$	−7.59077	−0.251632
$911$	−22.6860	−0.751620	−0.375810	−	0.926697i	$-0.622635\pi$
$911$	−22.6860	−0.751620	−0.375810	+	0.926697i	$0.622635\pi$
$912$	0	0
$913$	35.4229	1.17233
$914$	3.34340	0.110590
$915$	0	0
$916$	19.6497	0.649245
$917$	−39.9610	−1.31963
$918$	0	0
$919$	−45.1040	−1.48784	−0.743922	−	0.668267i	$-0.767036\pi$
$919$	−45.1040	−1.48784	−0.743922	+	0.668267i	$0.767036\pi$
$920$	−49.7012	−1.63860
$921$	0	0
$922$	19.4661	0.641082
$923$	2.00130	0.0658737
$924$	0	0
$925$	−4.29747	−0.141300
$926$	−26.9201	−0.884650
$927$	0	0
$928$	12.9644	0.425577
$929$	35.0613	1.15032	0.575162	−	0.818039i	$-0.304939\pi$
$929$	35.0613	1.15032	0.575162	+	0.818039i	$0.304939\pi$
$930$	0	0
$931$	12.2256	0.400679
$932$	4.93842	0.161763
$933$	0	0
$934$	5.26462	0.172264
$935$	31.4007	1.02691
$936$	0	0
$937$	−41.2551	−1.34775	−0.673873	−	0.738847i	$-0.735370\pi$
$937$	−41.2551	−1.34775	−0.673873	+	0.738847i	$0.735370\pi$
$938$	43.2403	1.41184
$939$	0	0
$940$	45.2880	1.47713
$941$	24.3564	0.793995	0.396998	−	0.917820i	$-0.370052\pi$
$941$	24.3564	0.793995	0.396998	+	0.917820i	$0.370052\pi$
$942$	0	0
$943$	−55.0673	−1.79324
$944$	47.0548	1.53150
$945$	0	0
$946$	9.47427	0.308035
$947$	53.6142	1.74223	0.871114	−	0.491081i	$-0.163398\pi$
$947$	53.6142	1.74223	0.871114	+	0.491081i	$0.163398\pi$
$948$	0	0
$949$	12.4518	0.404204
$950$	14.8590	0.482090
$951$	0	0
$952$	97.4631	3.15880
$953$	49.0812	1.58990	0.794948	−	0.606677i	$-0.207498\pi$
$953$	49.0812	1.58990	0.794948	+	0.606677i	$0.207498\pi$
$954$	0	0
$955$	−0.712091	−0.0230427
$956$	47.0193	1.52071
$957$	0	0
$958$	3.19034	0.103075
$959$	−31.9940	−1.03314
$960$	0	0
$961$	−12.5493	−0.404815
$962$	−10.8296	−0.349161
$963$	0	0
$964$	−33.2159	−1.06981
$965$	20.6987	0.666314
$966$	0	0
$967$	−5.70799	−0.183556	−0.0917782	−	0.995779i	$-0.529255\pi$
$967$	−5.70799	−0.183556	−0.0917782	+	0.995779i	$0.529255\pi$
$968$	130.551	4.19605
$969$	0	0
$970$	−15.7484	−0.505651
$971$	−14.0392	−0.450538	−0.225269	−	0.974297i	$-0.572326\pi$
$971$	−14.0392	−0.450538	−0.225269	+	0.974297i	$0.572326\pi$
$972$	0	0
$973$	13.4120	0.429970
$974$	52.0631	1.66821
$975$	0	0
$976$	37.4856	1.19988
$977$	9.34180	0.298871	0.149435	−	0.988772i	$-0.452254\pi$
$977$	9.34180	0.298871	0.149435	+	0.988772i	$0.452254\pi$
$978$	0	0
$979$	−12.9795	−0.414827
$980$	−9.02012	−0.288137
$981$	0	0
$982$	−30.5571	−0.975117
$983$	32.1912	1.02674	0.513370	−	0.858167i	$-0.328397\pi$
$983$	32.1912	1.02674	0.513370	+	0.858167i	$0.328397\pi$
$984$	0	0
$985$	25.1924	0.802698
$986$	46.4579	1.47952
$987$	0	0
$988$	25.6519	0.816095
$989$	−5.48832	−0.174518
$990$	0	0
$991$	13.6054	0.432189	0.216095	−	0.976372i	$-0.430668\pi$
$991$	13.6054	0.432189	0.216095	+	0.976372i	$0.430668\pi$
$992$	−16.5011	−0.523909
$993$	0	0
$994$	15.1914	0.481843
$995$	0.692437	0.0219517
$996$	0	0
$997$	9.89944	0.313518	0.156759	−	0.987637i	$-0.449895\pi$
$997$	9.89944	0.313518	0.156759	+	0.987637i	$0.449895\pi$
$998$	21.1126	0.668308
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5265.2.a.bk.1.14		15
3.2	odd	2		5265.2.a.bl.1.2		15
9.2	odd	6		1755.2.i.h.1171.14		30
9.4	even	3		585.2.i.h.196.2	✓	30
9.5	odd	6		1755.2.i.h.586.14		30
9.7	even	3		585.2.i.h.391.2	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.i.h.196.2	✓	30	9.4	even	3
585.2.i.h.391.2	yes	30	9.7	even	3
1755.2.i.h.586.14		30	9.5	odd	6
1755.2.i.h.1171.14		30	9.2	odd	6
5265.2.a.bk.1.14		15	1.1	even	1	trivial
5265.2.a.bl.1.2		15	3.2	odd	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:	\( N \)	\(=\)	\( 5265 = 3^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5265.a (trivial)

\(f(q)\)	\(=\)	\(q+2.52000 q^{2} +4.35041 q^{4} -1.00000 q^{5} +3.01221 q^{7} +5.92303 q^{8} +O(q^{10})\)	\(q+2.52000 q^{2} +4.35041 q^{4} -1.00000 q^{5} +3.01221 q^{7} +5.92303 q^{8} -2.52000 q^{10} -5.74814 q^{11} +1.00000 q^{13} +7.59077 q^{14} +6.22524 q^{16} +5.46275 q^{17} +5.89643 q^{19} -4.35041 q^{20} -14.4853 q^{22} +8.39117 q^{23} +1.00000 q^{25} +2.52000 q^{26} +13.1043 q^{28} +3.37479 q^{29} -4.29543 q^{31} +3.84154 q^{32} +13.7661 q^{34} -3.01221 q^{35} -4.29747 q^{37} +14.8590 q^{38} -5.92303 q^{40} -6.56253 q^{41} -0.654059 q^{43} -25.0068 q^{44} +21.1458 q^{46} -10.4100 q^{47} +2.07340 q^{49} +2.52000 q^{50} +4.35041 q^{52} +6.03463 q^{53} +5.74814 q^{55} +17.8414 q^{56} +8.50448 q^{58} +7.55872 q^{59} +6.02155 q^{61} -10.8245 q^{62} -2.76979 q^{64} -1.00000 q^{65} +5.69643 q^{67} +23.7652 q^{68} -7.59077 q^{70} +2.00130 q^{71} +12.4518 q^{73} -10.8296 q^{74} +25.6519 q^{76} -17.3146 q^{77} +13.6473 q^{79} -6.22524 q^{80} -16.5376 q^{82} -6.16250 q^{83} -5.46275 q^{85} -1.64823 q^{86} -34.0464 q^{88} +2.25804 q^{89} +3.01221 q^{91} +36.5050 q^{92} -26.2333 q^{94} -5.89643 q^{95} +6.24936 q^{97} +5.22496 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{2} + 21 q^{4} - 15 q^{5} + 10 q^{7}+O(q^{10}) \) 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

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