LMFDB - Embedded newform 5265.2.a.bi.1.8

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:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} - 20 x^{12} + 36 x^{11} + 156 x^{10} - 242 x^{9} - 601 x^{8} + 750 x^{7} + 1188 x^{6} + \cdots + 6 $ x^14 - 2x^13 - 20x^12 + 36x^11 + 156x^10 - 242x^9 - 601x^8 + 750x^7 + 1188x^6 - 1038x^5 - 1133x^4 + 438x^3 + 423x^2 + 90*x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.199325$ 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)

$f(q)$	$=$	$q+0.199325 q^{2} -1.96027 q^{4} -1.00000 q^{5} -0.890678 q^{7} -0.789381 q^{8} +O(q^{10})$	\(q+0.199325 q^{2} -1.96027 q^{4} -1.00000 q^{5} -0.890678 q^{7} -0.789381 q^{8} -0.199325 q^{10} -4.56330 q^{11} +1.00000 q^{13} -0.177534 q^{14} +3.76320 q^{16} -2.07667 q^{17} +5.91283 q^{19} +1.96027 q^{20} -0.909579 q^{22} -1.14532 q^{23} +1.00000 q^{25} +0.199325 q^{26} +1.74597 q^{28} +6.75122 q^{29} +4.82517 q^{31} +2.32886 q^{32} -0.413933 q^{34} +0.890678 q^{35} +7.64768 q^{37} +1.17858 q^{38} +0.789381 q^{40} +1.48515 q^{41} -3.83975 q^{43} +8.94529 q^{44} -0.228291 q^{46} +0.669298 q^{47} -6.20669 q^{49} +0.199325 q^{50} -1.96027 q^{52} -0.947701 q^{53} +4.56330 q^{55} +0.703084 q^{56} +1.34569 q^{58} -5.07462 q^{59} +8.20304 q^{61} +0.961778 q^{62} -7.06219 q^{64} -1.00000 q^{65} -14.0388 q^{67} +4.07084 q^{68} +0.177534 q^{70} +5.17522 q^{71} +1.48251 q^{73} +1.52437 q^{74} -11.5907 q^{76} +4.06442 q^{77} -1.94268 q^{79} -3.76320 q^{80} +0.296028 q^{82} -14.3419 q^{83} +2.07667 q^{85} -0.765358 q^{86} +3.60218 q^{88} -2.08793 q^{89} -0.890678 q^{91} +2.24513 q^{92} +0.133408 q^{94} -5.91283 q^{95} -10.6233 q^{97} -1.23715 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 2 q^{2} + 16 q^{4} - 14 q^{5} + 4 q^{7} - 12 q^{8}+O(q^{10}) $ 14 * q - 2 * q^2 + 16 * q^4 - 14 * q^5 + 4 * q^7 - 12 * q^8	$ 14 q - 2 q^{2} + 16 q^{4} - 14 q^{5} + 4 q^{7} - 12 q^{8} + 2 q^{10} - 8 q^{11} + 14 q^{13} - 12 q^{14} + 16 q^{16} - 10 q^{17} + 10 q^{19} - 16 q^{20} - 10 q^{22} - 34 q^{23} + 14 q^{25} - 2 q^{26} + 2 q^{28} - 16 q^{29} + 6 q^{31} - 26 q^{32} + 8 q^{34} - 4 q^{35} + 4 q^{37} - 12 q^{38} + 12 q^{40} - 10 q^{41} - 8 q^{43} - 8 q^{44} - 16 q^{46} - 46 q^{47} + 2 q^{49} - 2 q^{50} + 16 q^{52} - 14 q^{53} + 8 q^{55} - 20 q^{56} - 28 q^{58} - 16 q^{59} - 30 q^{62} + 14 q^{64} - 14 q^{65} + 6 q^{67} - 4 q^{68} + 12 q^{70} - 34 q^{71} - 4 q^{73} - 8 q^{74} + 50 q^{76} - 24 q^{77} - 14 q^{79} - 16 q^{80} - 16 q^{82} - 4 q^{83} + 10 q^{85} - 6 q^{86} - 68 q^{88} - 18 q^{89} + 4 q^{91} - 90 q^{92} - 10 q^{95} + 12 q^{97} + 16 q^{98}+O(q^{100}) $ 14 * q - 2 * q^2 + 16 * q^4 - 14 * q^5 + 4 * q^7 - 12 * q^8 + 2 * q^10 - 8 * q^11 + 14 * q^13 - 12 * q^14 + 16 * q^16 - 10 * q^17 + 10 * q^19 - 16 * q^20 - 10 * q^22 - 34 * q^23 + 14 * q^25 - 2 * q^26 + 2 * q^28 - 16 * q^29 + 6 * q^31 - 26 * q^32 + 8 * q^34 - 4 * q^35 + 4 * q^37 - 12 * q^38 + 12 * q^40 - 10 * q^41 - 8 * q^43 - 8 * q^44 - 16 * q^46 - 46 * q^47 + 2 * q^49 - 2 * q^50 + 16 * q^52 - 14 * q^53 + 8 * q^55 - 20 * q^56 - 28 * q^58 - 16 * q^59 - 30 * q^62 + 14 * q^64 - 14 * q^65 + 6 * q^67 - 4 * q^68 + 12 * q^70 - 34 * q^71 - 4 * q^73 - 8 * q^74 + 50 * q^76 - 24 * q^77 - 14 * q^79 - 16 * q^80 - 16 * q^82 - 4 * q^83 + 10 * q^85 - 6 * q^86 - 68 * q^88 - 18 * q^89 + 4 * q^91 - 90 * q^92 - 10 * q^95 + 12 * q^97 + 16 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.199325	0.140944	0.0704721	−	0.997514i	$-0.477549\pi$
$2$	0.199325	0.140944	0.0704721	+	0.997514i	$0.477549\pi$
$3$	0	0
$3$	0	0
$4$	−1.96027	−0.980135
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.890678	−0.336644	−0.168322	−	0.985732i	$-0.553835\pi$
$7$	−0.890678	−0.336644	−0.168322	+	0.985732i	$0.553835\pi$
$8$	−0.789381	−0.279088
$9$	0	0
$10$	−0.199325	−0.0630321
$11$	−4.56330	−1.37589	−0.687943	−	0.725765i	$-0.741486\pi$
$11$	−4.56330	−1.37589	−0.687943	+	0.725765i	$0.741486\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−0.177534	−0.0474481
$15$	0	0
$16$	3.76320	0.940799
$17$	−2.07667	−0.503668	−0.251834	−	0.967770i	$-0.581034\pi$
$17$	−2.07667	−0.503668	−0.251834	+	0.967770i	$0.581034\pi$
$18$	0	0
$19$	5.91283	1.35650	0.678248	−	0.734833i	$-0.262739\pi$
$19$	5.91283	1.35650	0.678248	+	0.734833i	$0.262739\pi$
$20$	1.96027	0.438330
$21$	0	0
$22$	−0.909579	−0.193923
$23$	−1.14532	−0.238815	−0.119408	−	0.992845i	$-0.538100\pi$
$23$	−1.14532	−0.238815	−0.119408	+	0.992845i	$0.538100\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0.199325	0.0390909
$27$	0	0
$28$	1.74597	0.329957
$29$	6.75122	1.25367	0.626835	−	0.779152i	$-0.284350\pi$
$29$	6.75122	1.25367	0.626835	+	0.779152i	$0.284350\pi$
$30$	0	0
$31$	4.82517	0.866627	0.433313	−	0.901243i	$-0.357344\pi$
$31$	4.82517	0.866627	0.433313	+	0.901243i	$0.357344\pi$
$32$	2.32886	0.411688
$33$	0	0
$34$	−0.413933	−0.0709890
$35$	0.890678	0.150552
$36$	0	0
$37$	7.64768	1.25727	0.628636	−	0.777700i	$-0.283614\pi$
$37$	7.64768	1.25727	0.628636	+	0.777700i	$0.283614\pi$
$38$	1.17858	0.191190
$39$	0	0
$40$	0.789381	0.124812
$41$	1.48515	0.231942	0.115971	−	0.993253i	$-0.463002\pi$
$41$	1.48515	0.231942	0.115971	+	0.993253i	$0.463002\pi$
$42$	0	0
$43$	−3.83975	−0.585556	−0.292778	−	0.956180i	$-0.594580\pi$
$43$	−3.83975	−0.585556	−0.292778	+	0.956180i	$0.594580\pi$
$44$	8.94529	1.34855
$45$	0	0
$46$	−0.228291	−0.0336596
$47$	0.669298	0.0976271	0.0488136	−	0.998808i	$-0.484456\pi$
$47$	0.669298	0.0976271	0.0488136	+	0.998808i	$0.484456\pi$
$48$	0	0
$49$	−6.20669	−0.886671
$50$	0.199325	0.0281888
$51$	0	0
$52$	−1.96027	−0.271840
$53$	−0.947701	−0.130177	−0.0650884	−	0.997880i	$-0.520733\pi$
$53$	−0.947701	−0.130177	−0.0650884	+	0.997880i	$0.520733\pi$
$54$	0	0
$55$	4.56330	0.615315
$56$	0.703084	0.0939535
$57$	0	0
$58$	1.34569	0.176697
$59$	−5.07462	−0.660659	−0.330330	−	0.943866i	$-0.607160\pi$
$59$	−5.07462	−0.660659	−0.330330	+	0.943866i	$0.607160\pi$
$60$	0	0
$61$	8.20304	1.05029	0.525146	−	0.851012i	$-0.324011\pi$
$61$	8.20304	1.05029	0.525146	+	0.851012i	$0.324011\pi$
$62$	0.961778	0.122146
$63$	0	0
$64$	−7.06219	−0.882774
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−14.0388	−1.71511	−0.857554	−	0.514394i	$-0.828017\pi$
$67$	−14.0388	−1.71511	−0.857554	+	0.514394i	$0.828017\pi$
$68$	4.07084	0.493662
$69$	0	0
$70$	0.177534	0.0212194
$71$	5.17522	0.614185	0.307093	−	0.951680i	$-0.400644\pi$
$71$	5.17522	0.614185	0.307093	+	0.951680i	$0.400644\pi$
$72$	0	0
$73$	1.48251	0.173515	0.0867576	−	0.996229i	$-0.472349\pi$
$73$	1.48251	0.173515	0.0867576	+	0.996229i	$0.472349\pi$
$74$	1.52437	0.177205
$75$	0	0
$76$	−11.5907	−1.32955
$77$	4.06442	0.463184
$78$	0	0
$79$	−1.94268	−0.218568	−0.109284	−	0.994011i	$-0.534856\pi$
$79$	−1.94268	−0.218568	−0.109284	+	0.994011i	$0.534856\pi$
$80$	−3.76320	−0.420738
$81$	0	0
$82$	0.296028	0.0326908
$83$	−14.3419	−1.57423	−0.787115	−	0.616806i	$-0.788426\pi$
$83$	−14.3419	−1.57423	−0.787115	+	0.616806i	$0.788426\pi$
$84$	0	0
$85$	2.07667	0.225247
$86$	−0.765358	−0.0825307
$87$	0	0
$88$	3.60218	0.383994
$89$	−2.08793	−0.221320	−0.110660	−	0.993858i	$-0.535296\pi$
$89$	−2.08793	−0.221320	−0.110660	+	0.993858i	$0.535296\pi$
$90$	0	0
$91$	−0.890678	−0.0933684
$92$	2.24513	0.234071
$93$	0	0
$94$	0.133408	0.0137600
$95$	−5.91283	−0.606644
$96$	0	0
$97$	−10.6233	−1.07863	−0.539316	−	0.842104i	$-0.681317\pi$
$97$	−10.6233	−1.07863	−0.539316	+	0.842104i	$0.681317\pi$
$98$	−1.23715	−0.124971
$99$	0	0
$100$	−1.96027	−0.196027
$101$	−14.7680	−1.46947	−0.734735	−	0.678354i	$-0.762694\pi$
$101$	−14.7680	−1.46947	−0.734735	+	0.678354i	$0.762694\pi$
$102$	0	0
$103$	−15.5829	−1.53542	−0.767712	−	0.640794i	$-0.778605\pi$
$103$	−15.5829	−1.53542	−0.767712	+	0.640794i	$0.778605\pi$
$104$	−0.789381	−0.0774052
$105$	0	0
$106$	−0.188901	−0.0183476
$107$	11.9265	1.15298	0.576491	−	0.817103i	$-0.304422\pi$
$107$	11.9265	1.15298	0.576491	+	0.817103i	$0.304422\pi$
$108$	0	0
$109$	16.7323	1.60266	0.801332	−	0.598220i	$-0.204125\pi$
$109$	16.7323	1.60266	0.801332	+	0.598220i	$0.204125\pi$
$110$	0.909579	0.0867250
$111$	0	0
$112$	−3.35179	−0.316715
$113$	15.6850	1.47552	0.737762	−	0.675061i	$-0.235883\pi$
$113$	15.6850	1.47552	0.737762	+	0.675061i	$0.235883\pi$
$114$	0	0
$115$	1.14532	0.106801
$116$	−13.2342	−1.22877
$117$	0	0
$118$	−1.01150	−0.0931160
$119$	1.84965	0.169557
$120$	0	0
$121$	9.82366	0.893060
$122$	1.63507	0.148032
$123$	0	0
$124$	−9.45864	−0.849411
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	8.24919	0.731997	0.365999	−	0.930615i	$-0.380728\pi$
$127$	8.24919	0.731997	0.365999	+	0.930615i	$0.380728\pi$
$128$	−6.06539	−0.536110
$129$	0	0
$130$	−0.199325	−0.0174820
$131$	−12.7774	−1.11637	−0.558183	−	0.829718i	$-0.688501\pi$
$131$	−12.7774	−1.11637	−0.558183	+	0.829718i	$0.688501\pi$
$132$	0	0
$133$	−5.26643	−0.456657
$134$	−2.79828	−0.241734
$135$	0	0
$136$	1.63929	0.140568
$137$	−20.2060	−1.72631	−0.863157	−	0.504936i	$-0.831516\pi$
$137$	−20.2060	−1.72631	−0.863157	+	0.504936i	$0.831516\pi$
$138$	0	0
$139$	−14.5867	−1.23723	−0.618613	−	0.785696i	$-0.712305\pi$
$139$	−14.5867	−1.23723	−0.618613	+	0.785696i	$0.712305\pi$
$140$	−1.74597	−0.147561
$141$	0	0
$142$	1.03155	0.0865658
$143$	−4.56330	−0.381602
$144$	0	0
$145$	−6.75122	−0.560658
$146$	0.295502	0.0244559
$147$	0	0
$148$	−14.9915	−1.23230
$149$	−8.80169	−0.721062	−0.360531	−	0.932747i	$-0.617405\pi$
$149$	−8.80169	−0.721062	−0.360531	+	0.932747i	$0.617405\pi$
$150$	0	0
$151$	10.6909	0.870015	0.435008	−	0.900427i	$-0.356746\pi$
$151$	10.6909	0.870015	0.435008	+	0.900427i	$0.356746\pi$
$152$	−4.66748	−0.378582
$153$	0	0
$154$	0.810142	0.0652831
$155$	−4.82517	−0.387567
$156$	0	0
$157$	4.88523	0.389884	0.194942	−	0.980815i	$-0.437548\pi$
$157$	4.88523	0.389884	0.194942	+	0.980815i	$0.437548\pi$
$158$	−0.387224	−0.0308059
$159$	0	0
$160$	−2.32886	−0.184113
$161$	1.02011	0.0803959
$162$	0	0
$163$	15.2181	1.19198	0.595988	−	0.802994i	$-0.296761\pi$
$163$	15.2181	1.19198	0.595988	+	0.802994i	$0.296761\pi$
$164$	−2.91130	−0.227334
$165$	0	0
$166$	−2.85870	−0.221878
$167$	2.21533	0.171427	0.0857136	−	0.996320i	$-0.472683\pi$
$167$	2.21533	0.171427	0.0857136	+	0.996320i	$0.472683\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0.413933	0.0317472
$171$	0	0
$172$	7.52694	0.573924
$173$	12.3117	0.936041	0.468020	−	0.883718i	$-0.344967\pi$
$173$	12.3117	0.936041	0.468020	+	0.883718i	$0.344967\pi$
$174$	0	0
$175$	−0.890678	−0.0673289
$176$	−17.1726	−1.29443
$177$	0	0
$178$	−0.416176	−0.0311937
$179$	−6.85395	−0.512288	−0.256144	−	0.966639i	$-0.582452\pi$
$179$	−6.85395	−0.512288	−0.256144	+	0.966639i	$0.582452\pi$
$180$	0	0
$181$	3.95222	0.293766	0.146883	−	0.989154i	$-0.453076\pi$
$181$	3.95222	0.293766	0.146883	+	0.989154i	$0.453076\pi$
$182$	−0.177534	−0.0131597
$183$	0	0
$184$	0.904093	0.0666506
$185$	−7.64768	−0.562269
$186$	0	0
$187$	9.47648	0.692989
$188$	−1.31200	−0.0956877
$189$	0	0
$190$	−1.17858	−0.0855028
$191$	19.8926	1.43938	0.719690	−	0.694296i	$-0.244284\pi$
$191$	19.8926	1.43938	0.719690	+	0.694296i	$0.244284\pi$
$192$	0	0
$193$	4.74333	0.341433	0.170716	−	0.985320i	$-0.445392\pi$
$193$	4.74333	0.341433	0.170716	+	0.985320i	$0.445392\pi$
$194$	−2.11749	−0.152027
$195$	0	0
$196$	12.1668	0.869057
$197$	−27.8805	−1.98641	−0.993203	−	0.116397i	$-0.962866\pi$
$197$	−27.8805	−1.98641	−0.993203	+	0.116397i	$0.962866\pi$
$198$	0	0
$199$	−18.6826	−1.32437	−0.662187	−	0.749339i	$-0.730371\pi$
$199$	−18.6826	−1.32437	−0.662187	+	0.749339i	$0.730371\pi$
$200$	−0.789381	−0.0558177
$201$	0	0
$202$	−2.94363	−0.207113
$203$	−6.01316	−0.422041
$204$	0	0
$205$	−1.48515	−0.103727
$206$	−3.10605	−0.216409
$207$	0	0
$208$	3.76320	0.260931
$209$	−26.9820	−1.86638
$210$	0	0
$211$	−2.85361	−0.196451	−0.0982253	−	0.995164i	$-0.531317\pi$
$211$	−2.85361	−0.196451	−0.0982253	+	0.995164i	$0.531317\pi$
$212$	1.85775	0.127591
$213$	0	0
$214$	2.37726	0.162506
$215$	3.83975	0.261869
$216$	0	0
$217$	−4.29767	−0.291745
$218$	3.33517	0.225886
$219$	0	0
$220$	−8.94529	−0.603091
$221$	−2.07667	−0.139692
$222$	0	0
$223$	−2.71907	−0.182082	−0.0910411	−	0.995847i	$-0.529019\pi$
$223$	−2.71907	−0.182082	−0.0910411	+	0.995847i	$0.529019\pi$
$224$	−2.07426	−0.138593
$225$	0	0
$226$	3.12642	0.207966
$227$	−5.57436	−0.369984	−0.184992	−	0.982740i	$-0.559226\pi$
$227$	−5.57436	−0.369984	−0.184992	+	0.982740i	$0.559226\pi$
$228$	0	0
$229$	−17.3583	−1.14707	−0.573534	−	0.819182i	$-0.694428\pi$
$229$	−17.3583	−1.14707	−0.573534	+	0.819182i	$0.694428\pi$
$230$	0.228291	0.0150530
$231$	0	0
$232$	−5.32929	−0.349885
$233$	−23.8695	−1.56375	−0.781873	−	0.623438i	$-0.785735\pi$
$233$	−23.8695	−1.56375	−0.781873	+	0.623438i	$0.785735\pi$
$234$	0	0
$235$	−0.669298	−0.0436602
$236$	9.94762	0.647535
$237$	0	0
$238$	0.368681	0.0238980
$239$	−14.7192	−0.952109	−0.476054	−	0.879416i	$-0.657933\pi$
$239$	−14.7192	−0.952109	−0.476054	+	0.879416i	$0.657933\pi$
$240$	0	0
$241$	6.99010	0.450272	0.225136	−	0.974327i	$-0.427717\pi$
$241$	6.99010	0.450272	0.225136	+	0.974327i	$0.427717\pi$
$242$	1.95810	0.125872
$243$	0	0
$244$	−16.0802	−1.02943
$245$	6.20669	0.396531
$246$	0	0
$247$	5.91283	0.376224
$248$	−3.80890	−0.241865
$249$	0	0
$250$	−0.199325	−0.0126064
$251$	−17.8541	−1.12694	−0.563472	−	0.826135i	$-0.690535\pi$
$251$	−17.8541	−1.12694	−0.563472	+	0.826135i	$0.690535\pi$
$252$	0	0
$253$	5.22643	0.328583
$254$	1.64427	0.103171
$255$	0	0
$256$	12.9154	0.807212
$257$	2.12255	0.132401	0.0662006	−	0.997806i	$-0.478912\pi$
$257$	2.12255	0.132401	0.0662006	+	0.997806i	$0.478912\pi$
$258$	0	0
$259$	−6.81162	−0.423253
$260$	1.96027	0.121571
$261$	0	0
$262$	−2.54685	−0.157345
$263$	0.801921	0.0494486	0.0247243	−	0.999694i	$-0.492129\pi$
$263$	0.801921	0.0494486	0.0247243	+	0.999694i	$0.492129\pi$
$264$	0	0
$265$	0.947701	0.0582168
$266$	−1.04973	−0.0643631
$267$	0	0
$268$	27.5198	1.68104
$269$	22.2769	1.35825	0.679123	−	0.734024i	$-0.262360\pi$
$269$	22.2769	1.35825	0.679123	+	0.734024i	$0.262360\pi$
$270$	0	0
$271$	11.1924	0.679890	0.339945	−	0.940445i	$-0.389591\pi$
$271$	11.1924	0.679890	0.339945	+	0.940445i	$0.389591\pi$
$272$	−7.81493	−0.473850
$273$	0	0
$274$	−4.02756	−0.243314
$275$	−4.56330	−0.275177
$276$	0	0
$277$	1.84568	0.110896	0.0554481	−	0.998462i	$-0.482341\pi$
$277$	1.84568	0.110896	0.0554481	+	0.998462i	$0.482341\pi$
$278$	−2.90749	−0.174380
$279$	0	0
$280$	−0.703084	−0.0420173
$281$	−4.88977	−0.291699	−0.145850	−	0.989307i	$-0.546592\pi$
$281$	−4.88977	−0.291699	−0.145850	+	0.989307i	$0.546592\pi$
$282$	0	0
$283$	25.8914	1.53908	0.769541	−	0.638597i	$-0.220485\pi$
$283$	25.8914	1.53908	0.769541	+	0.638597i	$0.220485\pi$
$284$	−10.1448	−0.601984
$285$	0	0
$286$	−0.909579	−0.0537845
$287$	−1.32279	−0.0780818
$288$	0	0
$289$	−12.6874	−0.746319
$290$	−1.34569	−0.0790215
$291$	0	0
$292$	−2.90613	−0.170068
$293$	7.36429	0.430226	0.215113	−	0.976589i	$-0.430988\pi$
$293$	7.36429	0.430226	0.215113	+	0.976589i	$0.430988\pi$
$294$	0	0
$295$	5.07462	0.295456
$296$	−6.03693	−0.350890
$297$	0	0
$298$	−1.75440	−0.101629
$299$	−1.14532	−0.0662355
$300$	0	0
$301$	3.41998	0.197124
$302$	2.13097	0.122623
$303$	0	0
$304$	22.2511	1.27619
$305$	−8.20304	−0.469705
$306$	0	0
$307$	−12.3190	−0.703081	−0.351540	−	0.936173i	$-0.614342\pi$
$307$	−12.3190	−0.703081	−0.351540	+	0.936173i	$0.614342\pi$
$308$	−7.96737	−0.453983
$309$	0	0
$310$	−0.961778	−0.0546253
$311$	0.842169	0.0477550	0.0238775	−	0.999715i	$-0.492399\pi$
$311$	0.842169	0.0477550	0.0238775	+	0.999715i	$0.492399\pi$
$312$	0	0
$313$	20.2862	1.14664	0.573321	−	0.819331i	$-0.305655\pi$
$313$	20.2862	1.14664	0.573321	+	0.819331i	$0.305655\pi$
$314$	0.973750	0.0549519
$315$	0	0
$316$	3.80817	0.214226
$317$	−8.00301	−0.449494	−0.224747	−	0.974417i	$-0.572156\pi$
$317$	−8.00301	−0.449494	−0.224747	+	0.974417i	$0.572156\pi$
$318$	0	0
$319$	−30.8078	−1.72491
$320$	7.06219	0.394788
$321$	0	0
$322$	0.203333	0.0113313
$323$	−12.2790	−0.683223
$324$	0	0
$325$	1.00000	0.0554700
$326$	3.03335	0.168002
$327$	0	0
$328$	−1.17235	−0.0647322
$329$	−0.596129	−0.0328656
$330$	0	0
$331$	−8.67840	−0.477008	−0.238504	−	0.971142i	$-0.576657\pi$
$331$	−8.67840	−0.477008	−0.238504	+	0.971142i	$0.576657\pi$
$332$	28.1140	1.54296
$333$	0	0
$334$	0.441570	0.0241617
$335$	14.0388	0.767019
$336$	0	0
$337$	−32.8395	−1.78888	−0.894441	−	0.447187i	$-0.852426\pi$
$337$	−32.8395	−1.78888	−0.894441	+	0.447187i	$0.852426\pi$
$338$	0.199325	0.0108419
$339$	0	0
$340$	−4.07084	−0.220772
$341$	−22.0187	−1.19238
$342$	0	0
$343$	11.7629	0.635137
$344$	3.03102	0.163422
$345$	0	0
$346$	2.45403	0.131929
$347$	−23.7285	−1.27382	−0.636908	−	0.770940i	$-0.719787\pi$
$347$	−23.7285	−1.27382	−0.636908	+	0.770940i	$0.719787\pi$
$348$	0	0
$349$	0.998659	0.0534570	0.0267285	−	0.999643i	$-0.491491\pi$
$349$	0.998659	0.0534570	0.0267285	+	0.999643i	$0.491491\pi$
$350$	−0.177534	−0.00948961
$351$	0	0
$352$	−10.6273	−0.566436
$353$	8.17804	0.435273	0.217637	−	0.976030i	$-0.430165\pi$
$353$	8.17804	0.435273	0.217637	+	0.976030i	$0.430165\pi$
$354$	0	0
$355$	−5.17522	−0.274672
$356$	4.09290	0.216923
$357$	0	0
$358$	−1.36616	−0.0722040
$359$	6.43727	0.339746	0.169873	−	0.985466i	$-0.445664\pi$
$359$	6.43727	0.339746	0.169873	+	0.985466i	$0.445664\pi$
$360$	0	0
$361$	15.9616	0.840082
$362$	0.787777	0.0414046
$363$	0	0
$364$	1.74597	0.0915136
$365$	−1.48251	−0.0775983
$366$	0	0
$367$	−18.9178	−0.987503	−0.493751	−	0.869603i	$-0.664375\pi$
$367$	−18.9178	−0.987503	−0.493751	+	0.869603i	$0.664375\pi$
$368$	−4.31006	−0.224677
$369$	0	0
$370$	−1.52437	−0.0792485
$371$	0.844096	0.0438233
$372$	0	0
$373$	−27.3757	−1.41746	−0.708731	−	0.705479i	$-0.750732\pi$
$373$	−27.3757	−1.41746	−0.708731	+	0.705479i	$0.750732\pi$
$374$	1.88890	0.0976727
$375$	0	0
$376$	−0.528331	−0.0272466
$377$	6.75122	0.347706
$378$	0	0
$379$	2.98070	0.153108	0.0765541	−	0.997065i	$-0.475608\pi$
$379$	2.98070	0.153108	0.0765541	+	0.997065i	$0.475608\pi$
$380$	11.5907	0.594592
$381$	0	0
$382$	3.96510	0.202872
$383$	22.7895	1.16449	0.582244	−	0.813014i	$-0.302175\pi$
$383$	22.7895	1.16449	0.582244	+	0.813014i	$0.302175\pi$
$384$	0	0
$385$	−4.06442	−0.207142
$386$	0.945465	0.0481229
$387$	0	0
$388$	20.8245	1.05720
$389$	−16.5527	−0.839258	−0.419629	−	0.907696i	$-0.637840\pi$
$389$	−16.5527	−0.839258	−0.419629	+	0.907696i	$0.637840\pi$
$390$	0	0
$391$	2.37845	0.120284
$392$	4.89945	0.247459
$393$	0	0
$394$	−5.55729	−0.279972
$395$	1.94268	0.0977466
$396$	0	0
$397$	1.09815	0.0551145	0.0275573	−	0.999620i	$-0.491227\pi$
$397$	1.09815	0.0551145	0.0275573	+	0.999620i	$0.491227\pi$
$398$	−3.72391	−0.186663
$399$	0	0
$400$	3.76320	0.188160
$401$	−5.90739	−0.295001	−0.147501	−	0.989062i	$-0.547123\pi$
$401$	−5.90739	−0.295001	−0.147501	+	0.989062i	$0.547123\pi$
$402$	0	0
$403$	4.82517	0.240359
$404$	28.9493	1.44028
$405$	0	0
$406$	−1.19857	−0.0594842
$407$	−34.8986	−1.72986
$408$	0	0
$409$	13.5553	0.670268	0.335134	−	0.942171i	$-0.391218\pi$
$409$	13.5553	0.670268	0.335134	+	0.942171i	$0.391218\pi$
$410$	−0.296028	−0.0146198
$411$	0	0
$412$	30.5466	1.50492
$413$	4.51985	0.222407
$414$	0	0
$415$	14.3419	0.704017
$416$	2.32886	0.114182
$417$	0	0
$418$	−5.37819	−0.263056
$419$	−12.5051	−0.610915	−0.305458	−	0.952206i	$-0.598809\pi$
$419$	−12.5051	−0.610915	−0.305458	+	0.952206i	$0.598809\pi$
$420$	0	0
$421$	14.0403	0.684281	0.342141	−	0.939649i	$-0.388848\pi$
$421$	14.0403	0.684281	0.342141	+	0.939649i	$0.388848\pi$
$422$	−0.568796	−0.0276885
$423$	0	0
$424$	0.748097	0.0363308
$425$	−2.07667	−0.100734
$426$	0	0
$427$	−7.30626	−0.353575
$428$	−23.3792	−1.13008
$429$	0	0
$430$	0.765358	0.0369088
$431$	−23.6912	−1.14117	−0.570583	−	0.821240i	$-0.693283\pi$
$431$	−23.6912	−1.14117	−0.570583	+	0.821240i	$0.693283\pi$
$432$	0	0
$433$	−14.3918	−0.691624	−0.345812	−	0.938304i	$-0.612396\pi$
$433$	−14.3918	−0.691624	−0.345812	+	0.938304i	$0.612396\pi$
$434$	−0.856634	−0.0411198
$435$	0	0
$436$	−32.7998	−1.57083
$437$	−6.77207	−0.323952
$438$	0	0
$439$	−10.8404	−0.517383	−0.258692	−	0.965960i	$-0.583291\pi$
$439$	−10.8404	−0.517383	−0.258692	+	0.965960i	$0.583291\pi$
$440$	−3.60218	−0.171727
$441$	0	0
$442$	−0.413933	−0.0196888
$443$	−19.0936	−0.907162	−0.453581	−	0.891215i	$-0.649854\pi$
$443$	−19.0936	−0.907162	−0.453581	+	0.891215i	$0.649854\pi$
$444$	0	0
$445$	2.08793	0.0989773
$446$	−0.541978	−0.0256634
$447$	0	0
$448$	6.29013	0.297181
$449$	−37.3176	−1.76113	−0.880564	−	0.473927i	$-0.842836\pi$
$449$	−37.3176	−1.76113	−0.880564	+	0.473927i	$0.842836\pi$
$450$	0	0
$451$	−6.77718	−0.319125
$452$	−30.7469	−1.44621
$453$	0	0
$454$	−1.11111	−0.0521470
$455$	0.890678	0.0417556
$456$	0	0
$457$	13.5847	0.635464	0.317732	−	0.948181i	$-0.397079\pi$
$457$	13.5847	0.635464	0.317732	+	0.948181i	$0.397079\pi$
$458$	−3.45994	−0.161672
$459$	0	0
$460$	−2.24513	−0.104680
$461$	15.9632	0.743479	0.371740	−	0.928337i	$-0.378761\pi$
$461$	15.9632	0.743479	0.371740	+	0.928337i	$0.378761\pi$
$462$	0	0
$463$	35.4223	1.64622	0.823108	−	0.567885i	$-0.192238\pi$
$463$	35.4223	1.64622	0.823108	+	0.567885i	$0.192238\pi$
$464$	25.4062	1.17945
$465$	0	0
$466$	−4.75780	−0.220401
$467$	−16.2421	−0.751595	−0.375797	−	0.926702i	$-0.622631\pi$
$467$	−16.2421	−0.751595	−0.375797	+	0.926702i	$0.622631\pi$
$468$	0	0
$469$	12.5040	0.577381
$470$	−0.133408	−0.00615364
$471$	0	0
$472$	4.00581	0.184382
$473$	17.5219	0.805658
$474$	0	0
$475$	5.91283	0.271299
$476$	−3.62581	−0.166189
$477$	0	0
$478$	−2.93391	−0.134194
$479$	−40.6390	−1.85685	−0.928423	−	0.371526i	$-0.878835\pi$
$479$	−40.6390	−1.85685	−0.928423	+	0.371526i	$0.878835\pi$
$480$	0	0
$481$	7.64768	0.348704
$482$	1.39330	0.0634632
$483$	0	0
$484$	−19.2570	−0.875319
$485$	10.6233	0.482379
$486$	0	0
$487$	−17.9850	−0.814977	−0.407489	−	0.913210i	$-0.633595\pi$
$487$	−17.9850	−0.814977	−0.407489	+	0.913210i	$0.633595\pi$
$488$	−6.47532	−0.293124
$489$	0	0
$490$	1.23715	0.0558887
$491$	−9.38891	−0.423715	−0.211858	−	0.977301i	$-0.567951\pi$
$491$	−9.38891	−0.423715	−0.211858	+	0.977301i	$0.567951\pi$
$492$	0	0
$493$	−14.0201	−0.631433
$494$	1.17858	0.0530266
$495$	0	0
$496$	18.1581	0.815322
$497$	−4.60945	−0.206762
$498$	0	0
$499$	−12.1852	−0.545483	−0.272742	−	0.962087i	$-0.587930\pi$
$499$	−12.1852	−0.545483	−0.272742	+	0.962087i	$0.587930\pi$
$500$	1.96027	0.0876659
$501$	0	0
$502$	−3.55878	−0.158836
$503$	−17.6551	−0.787204	−0.393602	−	0.919281i	$-0.628771\pi$
$503$	−17.6551	−0.787204	−0.393602	+	0.919281i	$0.628771\pi$
$504$	0	0
$505$	14.7680	0.657167
$506$	1.04176	0.0463118
$507$	0	0
$508$	−16.1706	−0.717456
$509$	17.6590	0.782723	0.391362	−	0.920237i	$-0.372004\pi$
$509$	17.6590	0.782723	0.391362	+	0.920237i	$0.372004\pi$
$510$	0	0
$511$	−1.32044	−0.0584129
$512$	14.7052	0.649882
$513$	0	0
$514$	0.423078	0.0186612
$515$	15.5829	0.686663
$516$	0	0
$517$	−3.05420	−0.134324
$518$	−1.35773	−0.0596551
$519$	0	0
$520$	0.789381	0.0346166
$521$	24.2797	1.06371	0.531857	−	0.846834i	$-0.321495\pi$
$521$	24.2797	1.06371	0.531857	+	0.846834i	$0.321495\pi$
$522$	0	0
$523$	−11.8523	−0.518267	−0.259133	−	0.965842i	$-0.583437\pi$
$523$	−11.8523	−0.518267	−0.259133	+	0.965842i	$0.583437\pi$
$524$	25.0471	1.09419
$525$	0	0
$526$	0.159843	0.00696948
$527$	−10.0203	−0.436492
$528$	0	0
$529$	−21.6882	−0.942967
$530$	0.188901	0.00820532
$531$	0	0
$532$	10.3236	0.447585
$533$	1.48515	0.0643290
$534$	0	0
$535$	−11.9265	−0.515629
$536$	11.0819	0.478666
$537$	0	0
$538$	4.44034	0.191437
$539$	28.3230	1.21996
$540$	0	0
$541$	30.9497	1.33063	0.665315	−	0.746563i	$-0.268297\pi$
$541$	30.9497	1.33063	0.665315	+	0.746563i	$0.268297\pi$
$542$	2.23093	0.0958265
$543$	0	0
$544$	−4.83629	−0.207354
$545$	−16.7323	−0.716733
$546$	0	0
$547$	−32.1964	−1.37662	−0.688309	−	0.725418i	$-0.741646\pi$
$547$	−32.1964	−1.37662	−0.688309	+	0.725418i	$0.741646\pi$
$548$	39.6092	1.69202
$549$	0	0
$550$	−0.909579	−0.0387846
$551$	39.9188	1.70060
$552$	0	0
$553$	1.73030	0.0735797
$554$	0.367891	0.0156302
$555$	0	0
$556$	28.5938	1.21265
$557$	−15.2444	−0.645928	−0.322964	−	0.946411i	$-0.604679\pi$
$557$	−15.2444	−0.645928	−0.322964	+	0.946411i	$0.604679\pi$
$558$	0	0
$559$	−3.83975	−0.162404
$560$	3.35179	0.141639
$561$	0	0
$562$	−0.974654	−0.0411133
$563$	−2.65370	−0.111840	−0.0559200	−	0.998435i	$-0.517809\pi$
$563$	−2.65370	−0.111840	−0.0559200	+	0.998435i	$0.517809\pi$
$564$	0	0
$565$	−15.6850	−0.659874
$566$	5.16080	0.216925
$567$	0	0
$568$	−4.08522	−0.171412
$569$	−0.486553	−0.0203974	−0.0101987	−	0.999948i	$-0.503246\pi$
$569$	−0.486553	−0.0203974	−0.0101987	+	0.999948i	$0.503246\pi$
$570$	0	0
$571$	21.0564	0.881182	0.440591	−	0.897708i	$-0.354769\pi$
$571$	21.0564	0.881182	0.440591	+	0.897708i	$0.354769\pi$
$572$	8.94529	0.374021
$573$	0	0
$574$	−0.263665	−0.0110052
$575$	−1.14532	−0.0477631
$576$	0	0
$577$	−40.9541	−1.70494	−0.852470	−	0.522776i	$-0.824896\pi$
$577$	−40.9541	−1.70494	−0.852470	+	0.522776i	$0.824896\pi$
$578$	−2.52892	−0.105189
$579$	0	0
$580$	13.2342	0.549521
$581$	12.7740	0.529956
$582$	0	0
$583$	4.32464	0.179108
$584$	−1.17027	−0.0484261
$585$	0	0
$586$	1.46789	0.0606379
$587$	−12.0529	−0.497476	−0.248738	−	0.968571i	$-0.580016\pi$
$587$	−12.0529	−0.497476	−0.248738	+	0.968571i	$0.580016\pi$
$588$	0	0
$589$	28.5304	1.17558
$590$	1.01150	0.0416427
$591$	0	0
$592$	28.7797	1.18284
$593$	30.3292	1.24547	0.622735	−	0.782433i	$-0.286022\pi$
$593$	30.3292	1.24547	0.622735	+	0.782433i	$0.286022\pi$
$594$	0	0
$595$	−1.84965	−0.0758282
$596$	17.2537	0.706738
$597$	0	0
$598$	−0.228291	−0.00933550
$599$	−40.8245	−1.66804	−0.834021	−	0.551732i	$-0.813967\pi$
$599$	−40.8245	−1.66804	−0.834021	+	0.551732i	$0.813967\pi$
$600$	0	0
$601$	−27.3834	−1.11699	−0.558495	−	0.829508i	$-0.688621\pi$
$601$	−27.3834	−1.11699	−0.558495	+	0.829508i	$0.688621\pi$
$602$	0.681687	0.0277835
$603$	0	0
$604$	−20.9571	−0.852732
$605$	−9.82366	−0.399389
$606$	0	0
$607$	34.4872	1.39979	0.699895	−	0.714246i	$-0.253230\pi$
$607$	34.4872	1.39979	0.699895	+	0.714246i	$0.253230\pi$
$608$	13.7702	0.558454
$609$	0	0
$610$	−1.63507	−0.0662021
$611$	0.669298	0.0270769
$612$	0	0
$613$	−16.4445	−0.664186	−0.332093	−	0.943247i	$-0.607755\pi$
$613$	−16.4445	−0.664186	−0.332093	+	0.943247i	$0.607755\pi$
$614$	−2.45548	−0.0990951
$615$	0	0
$616$	−3.20838	−0.129269
$617$	40.0599	1.61275	0.806376	−	0.591404i	$-0.201426\pi$
$617$	40.0599	1.61275	0.806376	+	0.591404i	$0.201426\pi$
$618$	0	0
$619$	−23.9068	−0.960895	−0.480447	−	0.877024i	$-0.659526\pi$
$619$	−23.9068	−0.960895	−0.480447	+	0.877024i	$0.659526\pi$
$620$	9.45864	0.379868
$621$	0	0
$622$	0.167865	0.00673079
$623$	1.85967	0.0745061
$624$	0	0
$625$	1.00000	0.0400000
$626$	4.04354	0.161612
$627$	0	0
$628$	−9.57637	−0.382139
$629$	−15.8817	−0.633247
$630$	0	0
$631$	46.6892	1.85867	0.929334	−	0.369241i	$-0.120382\pi$
$631$	46.6892	1.85867	0.929334	+	0.369241i	$0.120382\pi$
$632$	1.53351	0.0609998
$633$	0	0
$634$	−1.59520	−0.0633535
$635$	−8.24919	−0.327359
$636$	0	0
$637$	−6.20669	−0.245918
$638$	−6.14077	−0.243115
$639$	0	0
$640$	6.06539	0.239756
$641$	34.0302	1.34411	0.672057	−	0.740500i	$-0.265411\pi$
$641$	34.0302	1.34411	0.672057	+	0.740500i	$0.265411\pi$
$642$	0	0
$643$	18.5198	0.730349	0.365175	−	0.930939i	$-0.381009\pi$
$643$	18.5198	0.730349	0.365175	+	0.930939i	$0.381009\pi$
$644$	−1.99969	−0.0787988
$645$	0	0
$646$	−2.44752	−0.0962963
$647$	13.0318	0.512333	0.256167	−	0.966633i	$-0.417540\pi$
$647$	13.0318	0.512333	0.256167	+	0.966633i	$0.417540\pi$
$648$	0	0
$649$	23.1570	0.908991
$650$	0.199325	0.00781817
$651$	0	0
$652$	−29.8316	−1.16830
$653$	−37.3371	−1.46112	−0.730558	−	0.682851i	$-0.760740\pi$
$653$	−37.3371	−1.46112	−0.730558	+	0.682851i	$0.760740\pi$
$654$	0	0
$655$	12.7774	0.499254
$656$	5.58891	0.218210
$657$	0	0
$658$	−0.118823	−0.00463222
$659$	13.2886	0.517649	0.258825	−	0.965924i	$-0.416665\pi$
$659$	13.2886	0.517649	0.258825	+	0.965924i	$0.416665\pi$
$660$	0	0
$661$	33.8164	1.31531	0.657654	−	0.753320i	$-0.271549\pi$
$661$	33.8164	1.31531	0.657654	+	0.753320i	$0.271549\pi$
$662$	−1.72982	−0.0672315
$663$	0	0
$664$	11.3212	0.439349
$665$	5.26643	0.204223
$666$	0	0
$667$	−7.73230	−0.299396
$668$	−4.34264	−0.168022
$669$	0	0
$670$	2.79828	0.108107
$671$	−37.4329	−1.44508
$672$	0	0
$673$	−36.6875	−1.41420	−0.707100	−	0.707114i	$-0.749997\pi$
$673$	−36.6875	−1.41420	−0.707100	+	0.707114i	$0.749997\pi$
$674$	−6.54573	−0.252132
$675$	0	0
$676$	−1.96027	−0.0753950
$677$	26.5275	1.01953	0.509767	−	0.860313i	$-0.329732\pi$
$677$	26.5275	1.01953	0.509767	+	0.860313i	$0.329732\pi$
$678$	0	0
$679$	9.46193	0.363115
$680$	−1.63929	−0.0628638
$681$	0	0
$682$	−4.38888	−0.168059
$683$	12.2513	0.468782	0.234391	−	0.972142i	$-0.424690\pi$
$683$	12.2513	0.468782	0.234391	+	0.972142i	$0.424690\pi$
$684$	0	0
$685$	20.2060	0.772031
$686$	2.34464	0.0895188
$687$	0	0
$688$	−14.4497	−0.550890
$689$	−0.947701	−0.0361045
$690$	0	0
$691$	48.5977	1.84874	0.924372	−	0.381492i	$-0.124590\pi$
$691$	48.5977	1.84874	0.924372	+	0.381492i	$0.124590\pi$
$692$	−24.1342	−0.917446
$693$	0	0
$694$	−4.72969	−0.179537
$695$	14.5867	0.553305
$696$	0	0
$697$	−3.08417	−0.116821
$698$	0.199058	0.00753445
$699$	0	0
$700$	1.74597	0.0659914
$701$	1.64559	0.0621530	0.0310765	−	0.999517i	$-0.490106\pi$
$701$	1.64559	0.0621530	0.0310765	+	0.999517i	$0.490106\pi$
$702$	0	0
$703$	45.2194	1.70548
$704$	32.2269	1.21460
$705$	0	0
$706$	1.63009	0.0613492
$707$	13.1535	0.494689
$708$	0	0
$709$	41.6125	1.56279	0.781394	−	0.624038i	$-0.214509\pi$
$709$	41.6125	1.56279	0.781394	+	0.624038i	$0.214509\pi$
$710$	−1.03155	−0.0387134
$711$	0	0
$712$	1.64817	0.0617678
$713$	−5.52636	−0.206964
$714$	0	0
$715$	4.56330	0.170658
$716$	13.4356	0.502111
$717$	0	0
$718$	1.28311	0.0478852
$719$	4.37856	0.163293	0.0816464	−	0.996661i	$-0.473982\pi$
$719$	4.37856	0.163293	0.0816464	+	0.996661i	$0.473982\pi$
$720$	0	0
$721$	13.8793	0.516892
$722$	3.18154	0.118405
$723$	0	0
$724$	−7.74742	−0.287931
$725$	6.75122	0.250734
$726$	0	0
$727$	−13.0083	−0.482451	−0.241226	−	0.970469i	$-0.577549\pi$
$727$	−13.0083	−0.482451	−0.241226	+	0.970469i	$0.577549\pi$
$728$	0.703084	0.0260580
$729$	0	0
$730$	−0.295502	−0.0109370
$731$	7.97391	0.294926
$732$	0	0
$733$	−28.0176	−1.03485	−0.517427	−	0.855727i	$-0.673110\pi$
$733$	−28.0176	−1.03485	−0.517427	+	0.855727i	$0.673110\pi$
$734$	−3.77080	−0.139183
$735$	0	0
$736$	−2.66729	−0.0983175
$737$	64.0630	2.35979
$738$	0	0
$739$	8.54608	0.314373	0.157186	−	0.987569i	$-0.449758\pi$
$739$	8.54608	0.314373	0.157186	+	0.987569i	$0.449758\pi$
$740$	14.9915	0.551099
$741$	0	0
$742$	0.168250	0.00617663
$743$	26.4393	0.969962	0.484981	−	0.874525i	$-0.338826\pi$
$743$	26.4393	0.969962	0.484981	+	0.874525i	$0.338826\pi$
$744$	0	0
$745$	8.80169	0.322469
$746$	−5.45667	−0.199783
$747$	0	0
$748$	−18.5765	−0.679223
$749$	−10.6227	−0.388145
$750$	0	0
$751$	6.51799	0.237845	0.118922	−	0.992904i	$-0.462056\pi$
$751$	6.51799	0.237845	0.118922	+	0.992904i	$0.462056\pi$
$752$	2.51870	0.0918475
$753$	0	0
$754$	1.34569	0.0490071
$755$	−10.6909	−0.389083
$756$	0	0
$757$	0.166340	0.00604573	0.00302286	−	0.999995i	$-0.499038\pi$
$757$	0.166340	0.00604573	0.00302286	+	0.999995i	$0.499038\pi$
$758$	0.594128	0.0215797
$759$	0	0
$760$	4.66748	0.169307
$761$	−39.9549	−1.44836	−0.724181	−	0.689610i	$-0.757782\pi$
$761$	−39.9549	−1.44836	−0.724181	+	0.689610i	$0.757782\pi$
$762$	0	0
$763$	−14.9031	−0.539528
$764$	−38.9949	−1.41079
$765$	0	0
$766$	4.54252	0.164128
$767$	−5.07462	−0.183234
$768$	0	0
$769$	14.8073	0.533967	0.266983	−	0.963701i	$-0.413973\pi$
$769$	14.8073	0.533967	0.266983	+	0.963701i	$0.413973\pi$
$770$	−0.810142	−0.0291955
$771$	0	0
$772$	−9.29821	−0.334650
$773$	9.00742	0.323974	0.161987	−	0.986793i	$-0.448210\pi$
$773$	9.00742	0.323974	0.161987	+	0.986793i	$0.448210\pi$
$774$	0	0
$775$	4.82517	0.173325
$776$	8.38582	0.301034
$777$	0	0
$778$	−3.29938	−0.118288
$779$	8.78144	0.314628
$780$	0	0
$781$	−23.6160	−0.845049
$782$	0.474085	0.0169533
$783$	0	0
$784$	−23.3570	−0.834179
$785$	−4.88523	−0.174361
$786$	0	0
$787$	−21.9184	−0.781308	−0.390654	−	0.920538i	$-0.627751\pi$
$787$	−21.9184	−0.781308	−0.390654	+	0.920538i	$0.627751\pi$
$788$	54.6534	1.94695
$789$	0	0
$790$	0.387224	0.0137768
$791$	−13.9703	−0.496727
$792$	0	0
$793$	8.20304	0.291298
$794$	0.218889	0.00776807
$795$	0	0
$796$	36.6229	1.29806
$797$	28.1392	0.996743	0.498371	−	0.866964i	$-0.333932\pi$
$797$	28.1392	0.996743	0.498371	+	0.866964i	$0.333932\pi$
$798$	0	0
$799$	−1.38991	−0.0491716
$800$	2.32886	0.0823377
$801$	0	0
$802$	−1.17749	−0.0415787
$803$	−6.76515	−0.238737
$804$	0	0
$805$	−1.02011	−0.0359541
$806$	0.961778	0.0338772
$807$	0	0
$808$	11.6576	0.410112
$809$	25.6202	0.900757	0.450379	−	0.892838i	$-0.351289\pi$
$809$	25.6202	0.900757	0.450379	+	0.892838i	$0.351289\pi$
$810$	0	0
$811$	−35.1864	−1.23556	−0.617781	−	0.786350i	$-0.711968\pi$
$811$	−35.1864	−1.23556	−0.617781	+	0.786350i	$0.711968\pi$
$812$	11.7874	0.413657
$813$	0	0
$814$	−6.95617	−0.243814
$815$	−15.2181	−0.533068
$816$	0	0
$817$	−22.7038	−0.794305
$818$	2.70192	0.0944703
$819$	0	0
$820$	2.91130	0.101667
$821$	23.3430	0.814675	0.407337	−	0.913278i	$-0.366457\pi$
$821$	23.3430	0.814675	0.407337	+	0.913278i	$0.366457\pi$
$822$	0	0
$823$	−3.55282	−0.123844	−0.0619218	−	0.998081i	$-0.519723\pi$
$823$	−3.55282	−0.123844	−0.0619218	+	0.998081i	$0.519723\pi$
$824$	12.3008	0.428519
$825$	0	0
$826$	0.900919	0.0313470
$827$	−31.9838	−1.11219	−0.556093	−	0.831120i	$-0.687700\pi$
$827$	−31.9838	−1.11219	−0.556093	+	0.831120i	$0.687700\pi$
$828$	0	0
$829$	15.9083	0.552519	0.276260	−	0.961083i	$-0.410905\pi$
$829$	15.9083	0.552519	0.276260	+	0.961083i	$0.410905\pi$
$830$	2.85870	0.0992271
$831$	0	0
$832$	−7.06219	−0.244837
$833$	12.8893	0.446587
$834$	0	0
$835$	−2.21533	−0.0766646
$836$	52.8920	1.82931
$837$	0	0
$838$	−2.49258	−0.0861049
$839$	26.1754	0.903677	0.451838	−	0.892100i	$-0.350768\pi$
$839$	26.1754	0.903677	0.451838	+	0.892100i	$0.350768\pi$
$840$	0	0
$841$	16.5790	0.571690
$842$	2.79858	0.0964454
$843$	0	0
$844$	5.59384	0.192548
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−8.74972	−0.300644
$848$	−3.56638	−0.122470
$849$	0	0
$850$	−0.413933	−0.0141978
$851$	−8.75903	−0.300256
$852$	0	0
$853$	−49.9178	−1.70915	−0.854577	−	0.519325i	$-0.826183\pi$
$853$	−49.9178	−1.70915	−0.854577	+	0.519325i	$0.826183\pi$
$854$	−1.45632	−0.0498343
$855$	0	0
$856$	−9.41459	−0.321784
$857$	−25.8073	−0.881562	−0.440781	−	0.897615i	$-0.645298\pi$
$857$	−25.8073	−0.881562	−0.440781	+	0.897615i	$0.645298\pi$
$858$	0	0
$859$	36.7796	1.25491	0.627453	−	0.778655i	$-0.284098\pi$
$859$	36.7796	1.25491	0.627453	+	0.778655i	$0.284098\pi$
$860$	−7.52694	−0.256667
$861$	0	0
$862$	−4.72225	−0.160841
$863$	−45.3741	−1.54455	−0.772275	−	0.635288i	$-0.780881\pi$
$863$	−45.3741	−1.54455	−0.772275	+	0.635288i	$0.780881\pi$
$864$	0	0
$865$	−12.3117	−0.418610
$866$	−2.86864	−0.0974803
$867$	0	0
$868$	8.42460	0.285950
$869$	8.86500	0.300725
$870$	0	0
$871$	−14.0388	−0.475685
$872$	−13.2082	−0.447285
$873$	0	0
$874$	−1.34984	−0.0456592
$875$	0.890678	0.0301104
$876$	0	0
$877$	−14.5730	−0.492096	−0.246048	−	0.969258i	$-0.579132\pi$
$877$	−14.5730	−0.492096	−0.246048	+	0.969258i	$0.579132\pi$
$878$	−2.16076	−0.0729222
$879$	0	0
$880$	17.1726	0.578887
$881$	−16.1754	−0.544962	−0.272481	−	0.962161i	$-0.587844\pi$
$881$	−16.1754	−0.544962	−0.272481	+	0.962161i	$0.587844\pi$
$882$	0	0
$883$	3.88668	0.130797	0.0653987	−	0.997859i	$-0.479168\pi$
$883$	3.88668	0.130797	0.0653987	+	0.997859i	$0.479168\pi$
$884$	4.07084	0.136917
$885$	0	0
$886$	−3.80583	−0.127859
$887$	−22.3280	−0.749700	−0.374850	−	0.927086i	$-0.622306\pi$
$887$	−22.3280	−0.749700	−0.374850	+	0.927086i	$0.622306\pi$
$888$	0	0
$889$	−7.34737	−0.246423
$890$	0.416176	0.0139503
$891$	0	0
$892$	5.33010	0.178465
$893$	3.95744	0.132431
$894$	0	0
$895$	6.85395	0.229102
$896$	5.40231	0.180479
$897$	0	0
$898$	−7.43834	−0.248221
$899$	32.5758	1.08646
$900$	0	0
$901$	1.96807	0.0655658
$902$	−1.35086	−0.0449788
$903$	0	0
$904$	−12.3815	−0.411801
$905$	−3.95222	−0.131376
$906$	0	0
$907$	−22.0236	−0.731282	−0.365641	−	0.930756i	$-0.619150\pi$
$907$	−22.0236	−0.731282	−0.365641	+	0.930756i	$0.619150\pi$
$908$	10.9273	0.362634
$909$	0	0
$910$	0.177534	0.00588521
$911$	26.1323	0.865800	0.432900	−	0.901442i	$-0.357490\pi$
$911$	26.1323	0.865800	0.432900	+	0.901442i	$0.357490\pi$
$912$	0	0
$913$	65.4464	2.16596
$914$	2.70777	0.0895649
$915$	0	0
$916$	34.0269	1.12428
$917$	11.3805	0.375818
$918$	0	0
$919$	−40.4801	−1.33532	−0.667658	−	0.744468i	$-0.732703\pi$
$919$	−40.4801	−1.33532	−0.667658	+	0.744468i	$0.732703\pi$
$920$	−0.904093	−0.0298070
$921$	0	0
$922$	3.18186	0.104789
$923$	5.17522	0.170344
$924$	0	0
$925$	7.64768	0.251454
$926$	7.06056	0.232024
$927$	0	0
$928$	15.7227	0.516122
$929$	−36.5168	−1.19808	−0.599039	−	0.800720i	$-0.704450\pi$
$929$	−36.5168	−1.19808	−0.599039	+	0.800720i	$0.704450\pi$
$930$	0	0
$931$	−36.6991	−1.20277
$932$	46.7907	1.53268
$933$	0	0
$934$	−3.23746	−0.105933
$935$	−9.47648	−0.309914
$936$	0	0
$937$	−48.5570	−1.58629	−0.793143	−	0.609035i	$-0.791557\pi$
$937$	−48.5570	−1.58629	−0.793143	+	0.609035i	$0.791557\pi$
$938$	2.49236	0.0813785
$939$	0	0
$940$	1.31200	0.0427929
$941$	−33.5882	−1.09494	−0.547472	−	0.836824i	$-0.684410\pi$
$941$	−33.5882	−1.09494	−0.547472	+	0.836824i	$0.684410\pi$
$942$	0	0
$943$	−1.70097	−0.0553912
$944$	−19.0968	−0.621547
$945$	0	0
$946$	3.49255	0.113553
$947$	−56.7510	−1.84416	−0.922080	−	0.386998i	$-0.873512\pi$
$947$	−56.7510	−1.84416	−0.922080	+	0.386998i	$0.873512\pi$
$948$	0	0
$949$	1.48251	0.0481244
$950$	1.17858	0.0382380
$951$	0	0
$952$	−1.46008	−0.0473214
$953$	26.5202	0.859072	0.429536	−	0.903050i	$-0.358677\pi$
$953$	26.5202	0.859072	0.429536	+	0.903050i	$0.358677\pi$
$954$	0	0
$955$	−19.8926	−0.643710
$956$	28.8537	0.933195
$957$	0	0
$958$	−8.10038	−0.261711
$959$	17.9970	0.581154
$960$	0	0
$961$	−7.71770	−0.248958
$962$	1.52437	0.0491478
$963$	0	0
$964$	−13.7025	−0.441327
$965$	−4.74333	−0.152693
$966$	0	0
$967$	16.5301	0.531573	0.265786	−	0.964032i	$-0.414368\pi$
$967$	16.5301	0.531573	0.265786	+	0.964032i	$0.414368\pi$
$968$	−7.75461	−0.249243
$969$	0	0
$970$	2.11749	0.0679884
$971$	9.67727	0.310558	0.155279	−	0.987871i	$-0.450372\pi$
$971$	9.67727	0.310558	0.155279	+	0.987871i	$0.450372\pi$
$972$	0	0
$973$	12.9920	0.416505
$974$	−3.58486	−0.114866
$975$	0	0
$976$	30.8696	0.988113
$977$	57.5844	1.84229	0.921144	−	0.389222i	$-0.127256\pi$
$977$	57.5844	1.84229	0.921144	+	0.389222i	$0.127256\pi$
$978$	0	0
$979$	9.52783	0.304511
$980$	−12.1668	−0.388654
$981$	0	0
$982$	−1.87144	−0.0597202
$983$	−21.0130	−0.670209	−0.335105	−	0.942181i	$-0.608772\pi$
$983$	−21.0130	−0.670209	−0.335105	+	0.942181i	$0.608772\pi$
$984$	0	0
$985$	27.8805	0.888348
$986$	−2.79456	−0.0889968
$987$	0	0
$988$	−11.5907	−0.368751
$989$	4.39773	0.139840
$990$	0	0
$991$	−36.1892	−1.14959	−0.574794	−	0.818298i	$-0.694918\pi$
$991$	−36.1892	−1.14959	−0.574794	+	0.818298i	$0.694918\pi$
$992$	11.2372	0.356780
$993$	0	0
$994$	−0.918779	−0.0291419
$995$	18.6826	0.592278
$996$	0	0
$997$	3.01779	0.0955743	0.0477871	−	0.998858i	$-0.484783\pi$
$997$	3.01779	0.0955743	0.0477871	+	0.998858i	$0.484783\pi$
$998$	−2.42881	−0.0768826
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5265.2.a.bi.1.8	✓	14
3.2	odd	2		5265.2.a.bj.1.7	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5265.2.a.bi.1.8	✓	14	1.1	even	1	trivial
5265.2.a.bj.1.7	yes	14	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)

\(f(q)\)	\(=\)	\(q+0.199325 q^{2} -1.96027 q^{4} -1.00000 q^{5} -0.890678 q^{7} -0.789381 q^{8} +O(q^{10})\)	\(q+0.199325 q^{2} -1.96027 q^{4} -1.00000 q^{5} -0.890678 q^{7} -0.789381 q^{8} -0.199325 q^{10} -4.56330 q^{11} +1.00000 q^{13} -0.177534 q^{14} +3.76320 q^{16} -2.07667 q^{17} +5.91283 q^{19} +1.96027 q^{20} -0.909579 q^{22} -1.14532 q^{23} +1.00000 q^{25} +0.199325 q^{26} +1.74597 q^{28} +6.75122 q^{29} +4.82517 q^{31} +2.32886 q^{32} -0.413933 q^{34} +0.890678 q^{35} +7.64768 q^{37} +1.17858 q^{38} +0.789381 q^{40} +1.48515 q^{41} -3.83975 q^{43} +8.94529 q^{44} -0.228291 q^{46} +0.669298 q^{47} -6.20669 q^{49} +0.199325 q^{50} -1.96027 q^{52} -0.947701 q^{53} +4.56330 q^{55} +0.703084 q^{56} +1.34569 q^{58} -5.07462 q^{59} +8.20304 q^{61} +0.961778 q^{62} -7.06219 q^{64} -1.00000 q^{65} -14.0388 q^{67} +4.07084 q^{68} +0.177534 q^{70} +5.17522 q^{71} +1.48251 q^{73} +1.52437 q^{74} -11.5907 q^{76} +4.06442 q^{77} -1.94268 q^{79} -3.76320 q^{80} +0.296028 q^{82} -14.3419 q^{83} +2.07667 q^{85} -0.765358 q^{86} +3.60218 q^{88} -2.08793 q^{89} -0.890678 q^{91} +2.24513 q^{92} +0.133408 q^{94} -5.91283 q^{95} -10.6233 q^{97} -1.23715 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 2 q^{2} + 16 q^{4} - 14 q^{5} + 4 q^{7} - 12 q^{8}+O(q^{10}) \) 14 * q - 2 * q^2 + 16 * q^4 - 14 * q^5 + 4 * q^7 - 12 * q^8	\( 14 q - 2 q^{2} + 16 q^{4} - 14 q^{5} + 4 q^{7} - 12 q^{8} + 2 q^{10} - 8 q^{11} + 14 q^{13} - 12 q^{14} + 16 q^{16} - 10 q^{17} + 10 q^{19} - 16 q^{20} - 10 q^{22} - 34 q^{23} + 14 q^{25} - 2 q^{26} + 2 q^{28} - 16 q^{29} + 6 q^{31} - 26 q^{32} + 8 q^{34} - 4 q^{35} + 4 q^{37} - 12 q^{38} + 12 q^{40} - 10 q^{41} - 8 q^{43} - 8 q^{44} - 16 q^{46} - 46 q^{47} + 2 q^{49} - 2 q^{50} + 16 q^{52} - 14 q^{53} + 8 q^{55} - 20 q^{56} - 28 q^{58} - 16 q^{59} - 30 q^{62} + 14 q^{64} - 14 q^{65} + 6 q^{67} - 4 q^{68} + 12 q^{70} - 34 q^{71} - 4 q^{73} - 8 q^{74} + 50 q^{76} - 24 q^{77} - 14 q^{79} - 16 q^{80} - 16 q^{82} - 4 q^{83} + 10 q^{85} - 6 q^{86} - 68 q^{88} - 18 q^{89} + 4 q^{91} - 90 q^{92} - 10 q^{95} + 12 q^{97} + 16 q^{98}+O(q^{100}) \) 14 * q - 2 * q^2 + 16 * q^4 - 14 * q^5 + 4 * q^7 - 12 * q^8 + 2 * q^10 - 8 * q^11 + 14 * q^13 - 12 * q^14 + 16 * q^16 - 10 * q^17 + 10 * q^19 - 16 * q^20 - 10 * q^22 - 34 * q^23 + 14 * q^25 - 2 * q^26 + 2 * q^28 - 16 * q^29 + 6 * q^31 - 26 * q^32 + 8 * q^34 - 4 * q^35 + 4 * q^37 - 12 * q^38 + 12 * q^40 - 10 * q^41 - 8 * q^43 - 8 * q^44 - 16 * q^46 - 46 * q^47 + 2 * q^49 - 2 * q^50 + 16 * q^52 - 14 * q^53 + 8 * q^55 - 20 * q^56 - 28 * q^58 - 16 * q^59 - 30 * q^62 + 14 * q^64 - 14 * q^65 + 6 * q^67 - 4 * q^68 + 12 * q^70 - 34 * q^71 - 4 * q^73 - 8 * q^74 + 50 * q^76 - 24 * q^77 - 14 * q^79 - 16 * q^80 - 16 * q^82 - 4 * q^83 + 10 * q^85 - 6 * q^86 - 68 * q^88 - 18 * q^89 + 4 * q^91 - 90 * q^92 - 10 * q^95 + 12 * q^97 + 16 * q^98

Introduction

L-functions

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