LMFDB - Embedded newform 5265.2.a.s.1.2

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:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.347296$ of defining polynomial
Character	$\chi$	$=$	5265.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.34730 q^{2} -0.184793 q^{4} +1.00000 q^{5} -2.18479 q^{7} +2.94356 q^{8} +O(q^{10})$	\(q-1.34730 q^{2} -0.184793 q^{4} +1.00000 q^{5} -2.18479 q^{7} +2.94356 q^{8} -1.34730 q^{10} +3.87939 q^{11} +1.00000 q^{13} +2.94356 q^{14} -3.59627 q^{16} +3.71688 q^{17} +2.41147 q^{19} -0.184793 q^{20} -5.22668 q^{22} -4.75877 q^{23} +1.00000 q^{25} -1.34730 q^{26} +0.403733 q^{28} -3.57398 q^{29} -10.8229 q^{31} -1.04189 q^{32} -5.00774 q^{34} -2.18479 q^{35} -7.04963 q^{37} -3.24897 q^{38} +2.94356 q^{40} -1.97771 q^{41} -1.14290 q^{43} -0.716881 q^{44} +6.41147 q^{46} -1.04189 q^{47} -2.22668 q^{49} -1.34730 q^{50} -0.184793 q^{52} -1.95811 q^{53} +3.87939 q^{55} -6.43107 q^{56} +4.81521 q^{58} -12.1061 q^{59} +12.5030 q^{61} +14.5817 q^{62} +8.59627 q^{64} +1.00000 q^{65} +14.8726 q^{67} -0.686852 q^{68} +2.94356 q^{70} -3.46791 q^{71} +4.63816 q^{73} +9.49794 q^{74} -0.445622 q^{76} -8.47565 q^{77} -17.2344 q^{79} -3.59627 q^{80} +2.66456 q^{82} +10.5030 q^{83} +3.71688 q^{85} +1.53983 q^{86} +11.4192 q^{88} +17.3327 q^{89} -2.18479 q^{91} +0.879385 q^{92} +1.40373 q^{94} +2.41147 q^{95} -10.7733 q^{97} +3.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - 3 q^{7} - 6 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - 3 * q^7 - 6 * q^8	$ 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - 3 q^{7} - 6 q^{8} - 3 q^{10} + 6 q^{11} + 3 q^{13} - 6 q^{14} + 3 q^{16} + 3 q^{17} - 3 q^{19} + 3 q^{20} - 9 q^{22} - 3 q^{23} + 3 q^{25} - 3 q^{26} + 15 q^{28} - 3 q^{29} - 12 q^{31} + 9 q^{34} - 3 q^{35} + 6 q^{37} + 3 q^{38} - 6 q^{40} - 12 q^{41} - 3 q^{43} + 6 q^{44} + 9 q^{46} - 3 q^{50} + 3 q^{52} - 9 q^{53} + 6 q^{55} - 12 q^{56} + 18 q^{58} - 24 q^{59} - 3 q^{61} + 12 q^{62} + 12 q^{64} + 3 q^{65} - 3 q^{67} - 24 q^{68} - 6 q^{70} - 15 q^{71} - 3 q^{73} + 3 q^{74} - 12 q^{76} - 6 q^{77} - 21 q^{79} + 3 q^{80} + 36 q^{82} - 9 q^{83} + 3 q^{85} - 24 q^{86} + 33 q^{89} - 3 q^{91} - 3 q^{92} + 18 q^{94} - 3 q^{95} - 39 q^{97} + 9 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - 3 * q^7 - 6 * q^8 - 3 * q^10 + 6 * q^11 + 3 * q^13 - 6 * q^14 + 3 * q^16 + 3 * q^17 - 3 * q^19 + 3 * q^20 - 9 * q^22 - 3 * q^23 + 3 * q^25 - 3 * q^26 + 15 * q^28 - 3 * q^29 - 12 * q^31 + 9 * q^34 - 3 * q^35 + 6 * q^37 + 3 * q^38 - 6 * q^40 - 12 * q^41 - 3 * q^43 + 6 * q^44 + 9 * q^46 - 3 * q^50 + 3 * q^52 - 9 * q^53 + 6 * q^55 - 12 * q^56 + 18 * q^58 - 24 * q^59 - 3 * q^61 + 12 * q^62 + 12 * q^64 + 3 * q^65 - 3 * q^67 - 24 * q^68 - 6 * q^70 - 15 * q^71 - 3 * q^73 + 3 * q^74 - 12 * q^76 - 6 * q^77 - 21 * q^79 + 3 * q^80 + 36 * q^82 - 9 * q^83 + 3 * q^85 - 24 * q^86 + 33 * q^89 - 3 * q^91 - 3 * q^92 + 18 * q^94 - 3 * q^95 - 39 * q^97 + 9 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.34730	−0.952682	−0.476341	−	0.879261i	$-0.658037\pi$
$2$	−1.34730	−0.952682	−0.476341	+	0.879261i	$0.658037\pi$
$3$	0	0
$3$	0	0
$4$	−0.184793	−0.0923963
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.18479	−0.825774	−0.412887	−	0.910782i	$-0.635480\pi$
$7$	−2.18479	−0.825774	−0.412887	+	0.910782i	$0.635480\pi$
$8$	2.94356	1.04071
$9$	0	0
$10$	−1.34730	−0.426053
$11$	3.87939	1.16968	0.584839	−	0.811149i	$-0.301158\pi$
$11$	3.87939	1.16968	0.584839	+	0.811149i	$0.301158\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	2.94356	0.786700
$15$	0	0
$16$	−3.59627	−0.899067
$17$	3.71688	0.901476	0.450738	−	0.892656i	$-0.351161\pi$
$17$	3.71688	0.901476	0.450738	+	0.892656i	$0.351161\pi$
$18$	0	0
$19$	2.41147	0.553230	0.276615	−	0.960981i	$-0.410787\pi$
$19$	2.41147	0.553230	0.276615	+	0.960981i	$0.410787\pi$
$20$	−0.184793	−0.0413209
$21$	0	0
$22$	−5.22668	−1.11433
$23$	−4.75877	−0.992272	−0.496136	−	0.868245i	$-0.665248\pi$
$23$	−4.75877	−0.992272	−0.496136	+	0.868245i	$0.665248\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.34730	−0.264227
$27$	0	0
$28$	0.403733	0.0762984
$29$	−3.57398	−0.663671	−0.331836	−	0.943337i	$-0.607668\pi$
$29$	−3.57398	−0.663671	−0.331836	+	0.943337i	$0.607668\pi$
$30$	0	0
$31$	−10.8229	−1.94386	−0.971929	−	0.235273i	$-0.924402\pi$
$31$	−10.8229	−1.94386	−0.971929	+	0.235273i	$0.924402\pi$
$32$	−1.04189	−0.184182
$33$	0	0
$34$	−5.00774	−0.858820
$35$	−2.18479	−0.369297
$36$	0	0
$37$	−7.04963	−1.15895	−0.579476	−	0.814989i	$-0.696743\pi$
$37$	−7.04963	−1.15895	−0.579476	+	0.814989i	$0.696743\pi$
$38$	−3.24897	−0.527053
$39$	0	0
$40$	2.94356	0.465418
$41$	−1.97771	−0.308867	−0.154433	−	0.988003i	$-0.549355\pi$
$41$	−1.97771	−0.308867	−0.154433	+	0.988003i	$0.549355\pi$
$42$	0	0
$43$	−1.14290	−0.174291	−0.0871456	−	0.996196i	$-0.527775\pi$
$43$	−1.14290	−0.174291	−0.0871456	+	0.996196i	$0.527775\pi$
$44$	−0.716881	−0.108074
$45$	0	0
$46$	6.41147	0.945320
$47$	−1.04189	−0.151975	−0.0759876	−	0.997109i	$-0.524211\pi$
$47$	−1.04189	−0.151975	−0.0759876	+	0.997109i	$0.524211\pi$
$48$	0	0
$49$	−2.22668	−0.318097
$50$	−1.34730	−0.190536
$51$	0	0
$52$	−0.184793	−0.0256261
$53$	−1.95811	−0.268967	−0.134484	−	0.990916i	$-0.542938\pi$
$53$	−1.95811	−0.268967	−0.134484	+	0.990916i	$0.542938\pi$
$54$	0	0
$55$	3.87939	0.523096
$56$	−6.43107	−0.859388
$57$	0	0
$58$	4.81521	0.632268
$59$	−12.1061	−1.57608	−0.788038	−	0.615627i	$-0.788903\pi$
$59$	−12.1061	−1.57608	−0.788038	+	0.615627i	$0.788903\pi$
$60$	0	0
$61$	12.5030	1.60084	0.800422	−	0.599437i	$-0.204609\pi$
$61$	12.5030	1.60084	0.800422	+	0.599437i	$0.204609\pi$
$62$	14.5817	1.85188
$63$	0	0
$64$	8.59627	1.07453
$65$	1.00000	0.124035
$66$	0	0
$67$	14.8726	1.81697	0.908487	−	0.417912i	$-0.137238\pi$
$67$	14.8726	1.81697	0.908487	+	0.417912i	$0.137238\pi$
$68$	−0.686852	−0.0832930
$69$	0	0
$70$	2.94356	0.351823
$71$	−3.46791	−0.411565	−0.205783	−	0.978598i	$-0.565974\pi$
$71$	−3.46791	−0.411565	−0.205783	+	0.978598i	$0.565974\pi$
$72$	0	0
$73$	4.63816	0.542855	0.271428	−	0.962459i	$-0.412504\pi$
$73$	4.63816	0.542855	0.271428	+	0.962459i	$0.412504\pi$
$74$	9.49794	1.10411
$75$	0	0
$76$	−0.445622	−0.0511164
$77$	−8.47565	−0.965890
$78$	0	0
$79$	−17.2344	−1.93902	−0.969512	−	0.245044i	$-0.921198\pi$
$79$	−17.2344	−1.93902	−0.969512	+	0.245044i	$0.921198\pi$
$80$	−3.59627	−0.402075
$81$	0	0
$82$	2.66456	0.294252
$83$	10.5030	1.15285	0.576427	−	0.817149i	$-0.304447\pi$
$83$	10.5030	1.15285	0.576427	+	0.817149i	$0.304447\pi$
$84$	0	0
$85$	3.71688	0.403152
$86$	1.53983	0.166044
$87$	0	0
$88$	11.4192	1.21729
$89$	17.3327	1.83727	0.918634	−	0.395110i	$-0.129294\pi$
$89$	17.3327	1.83727	0.918634	+	0.395110i	$0.129294\pi$
$90$	0	0
$91$	−2.18479	−0.229028
$92$	0.879385	0.0916822
$93$	0	0
$94$	1.40373	0.144784
$95$	2.41147	0.247412
$96$	0	0
$97$	−10.7733	−1.09386	−0.546932	−	0.837177i	$-0.684205\pi$
$97$	−10.7733	−1.09386	−0.546932	+	0.837177i	$0.684205\pi$
$98$	3.00000	0.303046
$99$	0	0
$100$	−0.184793	−0.0184793
$101$	−13.3969	−1.33304	−0.666522	−	0.745485i	$-0.732218\pi$
$101$	−13.3969	−1.33304	−0.666522	+	0.745485i	$0.732218\pi$
$102$	0	0
$103$	18.2344	1.79669	0.898346	−	0.439290i	$-0.144770\pi$
$103$	18.2344	1.79669	0.898346	+	0.439290i	$0.144770\pi$
$104$	2.94356	0.288640
$105$	0	0
$106$	2.63816	0.256240
$107$	−12.9094	−1.24800	−0.624000	−	0.781424i	$-0.714494\pi$
$107$	−12.9094	−1.24800	−0.624000	+	0.781424i	$0.714494\pi$
$108$	0	0
$109$	12.6459	1.21126	0.605629	−	0.795747i	$-0.292922\pi$
$109$	12.6459	1.21126	0.605629	+	0.795747i	$0.292922\pi$
$110$	−5.22668	−0.498345
$111$	0	0
$112$	7.85710	0.742426
$113$	1.63310	0.153629	0.0768147	−	0.997045i	$-0.475525\pi$
$113$	1.63310	0.153629	0.0768147	+	0.997045i	$0.475525\pi$
$114$	0	0
$115$	−4.75877	−0.443758
$116$	0.660444	0.0613207
$117$	0	0
$118$	16.3105	1.50150
$119$	−8.12061	−0.744416
$120$	0	0
$121$	4.04963	0.368148
$122$	−16.8452	−1.52510
$123$	0	0
$124$	2.00000	0.179605
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	−9.49794	−0.839507
$129$	0	0
$130$	−1.34730	−0.118166
$131$	−19.0915	−1.66803	−0.834017	−	0.551739i	$-0.813964\pi$
$131$	−19.0915	−1.66803	−0.834017	+	0.551739i	$0.813964\pi$
$132$	0	0
$133$	−5.26857	−0.456843
$134$	−20.0378	−1.73100
$135$	0	0
$136$	10.9409	0.938172
$137$	−1.65270	−0.141200	−0.0706000	−	0.997505i	$-0.522491\pi$
$137$	−1.65270	−0.141200	−0.0706000	+	0.997505i	$0.522491\pi$
$138$	0	0
$139$	21.1584	1.79463	0.897315	−	0.441390i	$-0.145514\pi$
$139$	21.1584	1.79463	0.897315	+	0.441390i	$0.145514\pi$
$140$	0.403733	0.0341217
$141$	0	0
$142$	4.67230	0.392091
$143$	3.87939	0.324410
$144$	0	0
$145$	−3.57398	−0.296803
$146$	−6.24897	−0.517168
$147$	0	0
$148$	1.30272	0.107083
$149$	−21.4243	−1.75514	−0.877572	−	0.479445i	$-0.840838\pi$
$149$	−21.4243	−1.75514	−0.877572	+	0.479445i	$0.840838\pi$
$150$	0	0
$151$	−11.1088	−0.904018	−0.452009	−	0.892013i	$-0.649292\pi$
$151$	−11.1088	−0.904018	−0.452009	+	0.892013i	$0.649292\pi$
$152$	7.09833	0.575750
$153$	0	0
$154$	11.4192	0.920187
$155$	−10.8229	−0.869320
$156$	0	0
$157$	−5.04189	−0.402387	−0.201193	−	0.979552i	$-0.564482\pi$
$157$	−5.04189	−0.402387	−0.201193	+	0.979552i	$0.564482\pi$
$158$	23.2199	1.84727
$159$	0	0
$160$	−1.04189	−0.0823686
$161$	10.3969	0.819393
$162$	0	0
$163$	−0.714193	−0.0559399	−0.0279700	−	0.999609i	$-0.508904\pi$
$163$	−0.714193	−0.0559399	−0.0279700	+	0.999609i	$0.508904\pi$
$164$	0.365466	0.0285381
$165$	0	0
$166$	−14.1506	−1.09830
$167$	−14.9436	−1.15637	−0.578184	−	0.815907i	$-0.696238\pi$
$167$	−14.9436	−1.15637	−0.578184	+	0.815907i	$0.696238\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−5.00774	−0.384076
$171$	0	0
$172$	0.211200	0.0161039
$173$	3.06923	0.233349	0.116675	−	0.993170i	$-0.462777\pi$
$173$	3.06923	0.233349	0.116675	+	0.993170i	$0.462777\pi$
$174$	0	0
$175$	−2.18479	−0.165155
$176$	−13.9513	−1.05162
$177$	0	0
$178$	−23.3523	−1.75033
$179$	−0.255777	−0.0191177	−0.00955885	−	0.999954i	$-0.503043\pi$
$179$	−0.255777	−0.0191177	−0.00955885	+	0.999954i	$0.503043\pi$
$180$	0	0
$181$	6.01548	0.447127	0.223564	−	0.974689i	$-0.428231\pi$
$181$	6.01548	0.447127	0.223564	+	0.974689i	$0.428231\pi$
$182$	2.94356	0.218191
$183$	0	0
$184$	−14.0077	−1.03266
$185$	−7.04963	−0.518299
$186$	0	0
$187$	14.4192	1.05444
$188$	0.192533	0.0140419
$189$	0	0
$190$	−3.24897	−0.235705
$191$	14.0770	1.01857	0.509287	−	0.860597i	$-0.329909\pi$
$191$	14.0770	1.01857	0.509287	+	0.860597i	$0.329909\pi$
$192$	0	0
$193$	−1.46110	−0.105173	−0.0525863	−	0.998616i	$-0.516746\pi$
$193$	−1.46110	−0.105173	−0.0525863	+	0.998616i	$0.516746\pi$
$194$	14.5149	1.04211
$195$	0	0
$196$	0.411474	0.0293910
$197$	−15.4543	−1.10107	−0.550537	−	0.834811i	$-0.685577\pi$
$197$	−15.4543	−1.10107	−0.550537	+	0.834811i	$0.685577\pi$
$198$	0	0
$199$	−1.00000	−0.0708881	−0.0354441	−	0.999372i	$-0.511285\pi$
$199$	−1.00000	−0.0708881	−0.0354441	+	0.999372i	$0.511285\pi$
$200$	2.94356	0.208141
$201$	0	0
$202$	18.0496	1.26997
$203$	7.80840	0.548042
$204$	0	0
$205$	−1.97771	−0.138129
$206$	−24.5672	−1.71168
$207$	0	0
$208$	−3.59627	−0.249356
$209$	9.35504	0.647101
$210$	0	0
$211$	−19.2918	−1.32810	−0.664051	−	0.747687i	$-0.731164\pi$
$211$	−19.2918	−1.32810	−0.664051	+	0.747687i	$0.731164\pi$
$212$	0.361844	0.0248516
$213$	0	0
$214$	17.3928	1.18895
$215$	−1.14290	−0.0779454
$216$	0	0
$217$	23.6459	1.60519
$218$	−17.0378	−1.15394
$219$	0	0
$220$	−0.716881	−0.0483321
$221$	3.71688	0.250025
$222$	0	0
$223$	−12.9736	−0.868776	−0.434388	−	0.900726i	$-0.643035\pi$
$223$	−12.9736	−0.868776	−0.434388	+	0.900726i	$0.643035\pi$
$224$	2.27631	0.152092
$225$	0	0
$226$	−2.20027	−0.146360
$227$	−9.29860	−0.617170	−0.308585	−	0.951197i	$-0.599855\pi$
$227$	−9.29860	−0.617170	−0.308585	+	0.951197i	$0.599855\pi$
$228$	0	0
$229$	−8.91447	−0.589085	−0.294542	−	0.955638i	$-0.595167\pi$
$229$	−8.91447	−0.589085	−0.294542	+	0.955638i	$0.595167\pi$
$230$	6.41147	0.422760
$231$	0	0
$232$	−10.5202	−0.690687
$233$	21.9222	1.43617	0.718086	−	0.695955i	$-0.245019\pi$
$233$	21.9222	1.43617	0.718086	+	0.695955i	$0.245019\pi$
$234$	0	0
$235$	−1.04189	−0.0679653
$236$	2.23711	0.145624
$237$	0	0
$238$	10.9409	0.709192
$239$	−4.32770	−0.279935	−0.139968	−	0.990156i	$-0.544700\pi$
$239$	−4.32770	−0.279935	−0.139968	+	0.990156i	$0.544700\pi$
$240$	0	0
$241$	−11.1088	−0.715578	−0.357789	−	0.933802i	$-0.616469\pi$
$241$	−11.1088	−0.715578	−0.357789	+	0.933802i	$0.616469\pi$
$242$	−5.45605	−0.350728
$243$	0	0
$244$	−2.31046	−0.147912
$245$	−2.22668	−0.142257
$246$	0	0
$247$	2.41147	0.153438
$248$	−31.8580	−2.02299
$249$	0	0
$250$	−1.34730	−0.0852105
$251$	−8.64496	−0.545665	−0.272833	−	0.962062i	$-0.587961\pi$
$251$	−8.64496	−0.545665	−0.272833	+	0.962062i	$0.587961\pi$
$252$	0	0
$253$	−18.4611	−1.16064
$254$	5.38919	0.338148
$255$	0	0
$256$	−4.39599	−0.274750
$257$	−13.3473	−0.832581	−0.416291	−	0.909232i	$-0.636670\pi$
$257$	−13.3473	−0.832581	−0.416291	+	0.909232i	$0.636670\pi$
$258$	0	0
$259$	15.4020	0.957032
$260$	−0.184793	−0.0114603
$261$	0	0
$262$	25.7219	1.58911
$263$	−7.68273	−0.473738	−0.236869	−	0.971542i	$-0.576121\pi$
$263$	−7.68273	−0.473738	−0.236869	+	0.971542i	$0.576121\pi$
$264$	0	0
$265$	−1.95811	−0.120286
$266$	7.09833	0.435226
$267$	0	0
$268$	−2.74834	−0.167882
$269$	19.8084	1.20774	0.603870	−	0.797083i	$-0.293625\pi$
$269$	19.8084	1.20774	0.603870	+	0.797083i	$0.293625\pi$
$270$	0	0
$271$	−28.6536	−1.74058	−0.870292	−	0.492536i	$-0.836070\pi$
$271$	−28.6536	−1.74058	−0.870292	+	0.492536i	$0.836070\pi$
$272$	−13.3669	−0.810487
$273$	0	0
$274$	2.22668	0.134519
$275$	3.87939	0.233936
$276$	0	0
$277$	6.88207	0.413504	0.206752	−	0.978393i	$-0.433711\pi$
$277$	6.88207	0.413504	0.206752	+	0.978393i	$0.433711\pi$
$278$	−28.5066	−1.70971
$279$	0	0
$280$	−6.43107	−0.384330
$281$	−11.5621	−0.689738	−0.344869	−	0.938651i	$-0.612077\pi$
$281$	−11.5621	−0.689738	−0.344869	+	0.938651i	$0.612077\pi$
$282$	0	0
$283$	26.2249	1.55891	0.779455	−	0.626458i	$-0.215496\pi$
$283$	26.2249	1.55891	0.779455	+	0.626458i	$0.215496\pi$
$284$	0.640844	0.0380271
$285$	0	0
$286$	−5.22668	−0.309060
$287$	4.32089	0.255054
$288$	0	0
$289$	−3.18479	−0.187341
$290$	4.81521	0.282759
$291$	0	0
$292$	−0.857097	−0.0501578
$293$	32.4884	1.89800	0.948998	−	0.315283i	$-0.102100\pi$
$293$	32.4884	1.89800	0.948998	+	0.315283i	$0.102100\pi$
$294$	0	0
$295$	−12.1061	−0.704842
$296$	−20.7510	−1.20613
$297$	0	0
$298$	28.8648	1.67210
$299$	−4.75877	−0.275207
$300$	0	0
$301$	2.49701	0.143925
$302$	14.9668	0.861242
$303$	0	0
$304$	−8.67230	−0.497391
$305$	12.5030	0.715919
$306$	0	0
$307$	−7.09926	−0.405176	−0.202588	−	0.979264i	$-0.564935\pi$
$307$	−7.09926	−0.405176	−0.202588	+	0.979264i	$0.564935\pi$
$308$	1.56624	0.0892446
$309$	0	0
$310$	14.5817	0.828186
$311$	4.52940	0.256839	0.128419	−	0.991720i	$-0.459010\pi$
$311$	4.52940	0.256839	0.128419	+	0.991720i	$0.459010\pi$
$312$	0	0
$313$	−11.5790	−0.654485	−0.327243	−	0.944940i	$-0.606119\pi$
$313$	−11.5790	−0.654485	−0.327243	+	0.944940i	$0.606119\pi$
$314$	6.79292	0.383347
$315$	0	0
$316$	3.18479	0.179159
$317$	−10.7888	−0.605959	−0.302980	−	0.952997i	$-0.597981\pi$
$317$	−10.7888	−0.605959	−0.302980	+	0.952997i	$0.597981\pi$
$318$	0	0
$319$	−13.8648	−0.776282
$320$	8.59627	0.480546
$321$	0	0
$322$	−14.0077	−0.780621
$323$	8.96316	0.498724
$324$	0	0
$325$	1.00000	0.0554700
$326$	0.962230	0.0532930
$327$	0	0
$328$	−5.82152	−0.321440
$329$	2.27631	0.125497
$330$	0	0
$331$	−9.22668	−0.507144	−0.253572	−	0.967316i	$-0.581606\pi$
$331$	−9.22668	−0.507144	−0.253572	+	0.967316i	$0.581606\pi$
$332$	−1.94087	−0.106519
$333$	0	0
$334$	20.1334	1.10165
$335$	14.8726	0.812576
$336$	0	0
$337$	22.9736	1.25145	0.625726	−	0.780043i	$-0.284803\pi$
$337$	22.9736	1.25145	0.625726	+	0.780043i	$0.284803\pi$
$338$	−1.34730	−0.0732833
$339$	0	0
$340$	−0.686852	−0.0372498
$341$	−41.9864	−2.27369
$342$	0	0
$343$	20.1584	1.08845
$344$	−3.36421	−0.181386
$345$	0	0
$346$	−4.13516	−0.222308
$347$	−14.6013	−0.783840	−0.391920	−	0.919999i	$-0.628189\pi$
$347$	−14.6013	−0.783840	−0.391920	+	0.919999i	$0.628189\pi$
$348$	0	0
$349$	−16.2686	−0.870837	−0.435418	−	0.900228i	$-0.643400\pi$
$349$	−16.2686	−0.870837	−0.435418	+	0.900228i	$0.643400\pi$
$350$	2.94356	0.157340
$351$	0	0
$352$	−4.04189	−0.215433
$353$	−7.05232	−0.375357	−0.187679	−	0.982231i	$-0.560096\pi$
$353$	−7.05232	−0.375357	−0.187679	+	0.982231i	$0.560096\pi$
$354$	0	0
$355$	−3.46791	−0.184058
$356$	−3.20296	−0.169757
$357$	0	0
$358$	0.344608	0.0182131
$359$	−17.1830	−0.906886	−0.453443	−	0.891285i	$-0.649804\pi$
$359$	−17.1830	−0.906886	−0.453443	+	0.891285i	$0.649804\pi$
$360$	0	0
$361$	−13.1848	−0.693936
$362$	−8.10464	−0.425970
$363$	0	0
$364$	0.403733	0.0211614
$365$	4.63816	0.242772
$366$	0	0
$367$	17.7469	0.926381	0.463191	−	0.886259i	$-0.346705\pi$
$367$	17.7469	0.926381	0.463191	+	0.886259i	$0.346705\pi$
$368$	17.1138	0.892119
$369$	0	0
$370$	9.49794	0.493774
$371$	4.27807	0.222106
$372$	0	0
$373$	−29.0155	−1.50236	−0.751182	−	0.660095i	$-0.770516\pi$
$373$	−29.0155	−1.50236	−0.751182	+	0.660095i	$0.770516\pi$
$374$	−19.4270	−1.00454
$375$	0	0
$376$	−3.06687	−0.158162
$377$	−3.57398	−0.184069
$378$	0	0
$379$	8.42871	0.432954	0.216477	−	0.976288i	$-0.430543\pi$
$379$	8.42871	0.432954	0.216477	+	0.976288i	$0.430543\pi$
$380$	−0.445622	−0.0228599
$381$	0	0
$382$	−18.9659	−0.970377
$383$	19.6159	1.00232	0.501162	−	0.865353i	$-0.332906\pi$
$383$	19.6159	1.00232	0.501162	+	0.865353i	$0.332906\pi$
$384$	0	0
$385$	−8.47565	−0.431959
$386$	1.96854	0.100196
$387$	0	0
$388$	1.99083	0.101069
$389$	−19.4270	−0.984986	−0.492493	−	0.870316i	$-0.663914\pi$
$389$	−19.4270	−0.984986	−0.492493	+	0.870316i	$0.663914\pi$
$390$	0	0
$391$	−17.6878	−0.894510
$392$	−6.55438	−0.331046
$393$	0	0
$394$	20.8215	1.04897
$395$	−17.2344	−0.867158
$396$	0	0
$397$	−6.35235	−0.318815	−0.159408	−	0.987213i	$-0.550958\pi$
$397$	−6.35235	−0.318815	−0.159408	+	0.987213i	$0.550958\pi$
$398$	1.34730	0.0675339
$399$	0	0
$400$	−3.59627	−0.179813
$401$	5.76146	0.287714	0.143857	−	0.989599i	$-0.454050\pi$
$401$	5.76146	0.287714	0.143857	+	0.989599i	$0.454050\pi$
$402$	0	0
$403$	−10.8229	−0.539129
$404$	2.47565	0.123168
$405$	0	0
$406$	−10.5202	−0.522110
$407$	−27.3482	−1.35560
$408$	0	0
$409$	35.5604	1.75835	0.879173	−	0.476502i	$-0.158095\pi$
$409$	35.5604	1.75835	0.879173	+	0.476502i	$0.158095\pi$
$410$	2.66456	0.131593
$411$	0	0
$412$	−3.36959	−0.166008
$413$	26.4492	1.30148
$414$	0	0
$415$	10.5030	0.515572
$416$	−1.04189	−0.0510828
$417$	0	0
$418$	−12.6040	−0.616482
$419$	−0.206148	−0.0100710	−0.00503548	−	0.999987i	$-0.501603\pi$
$419$	−0.206148	−0.0100710	−0.00503548	+	0.999987i	$0.501603\pi$
$420$	0	0
$421$	7.44562	0.362877	0.181439	−	0.983402i	$-0.441925\pi$
$421$	7.44562	0.362877	0.181439	+	0.983402i	$0.441925\pi$
$422$	25.9918	1.26526
$423$	0	0
$424$	−5.76382	−0.279916
$425$	3.71688	0.180295
$426$	0	0
$427$	−27.3164	−1.32194
$428$	2.38556	0.115311
$429$	0	0
$430$	1.53983	0.0742572
$431$	−14.0009	−0.674401	−0.337201	−	0.941433i	$-0.609480\pi$
$431$	−14.0009	−0.674401	−0.337201	+	0.941433i	$0.609480\pi$
$432$	0	0
$433$	6.11793	0.294009	0.147004	−	0.989136i	$-0.453037\pi$
$433$	6.11793	0.294009	0.147004	+	0.989136i	$0.453037\pi$
$434$	−31.8580	−1.52923
$435$	0	0
$436$	−2.33687	−0.111916
$437$	−11.4757	−0.548955
$438$	0	0
$439$	−13.4611	−0.642463	−0.321232	−	0.947001i	$-0.604097\pi$
$439$	−13.4611	−0.642463	−0.321232	+	0.947001i	$0.604097\pi$
$440$	11.4192	0.544390
$441$	0	0
$442$	−5.00774	−0.238194
$443$	28.8972	1.37295	0.686474	−	0.727154i	$-0.259157\pi$
$443$	28.8972	1.37295	0.686474	+	0.727154i	$0.259157\pi$
$444$	0	0
$445$	17.3327	0.821651
$446$	17.4793	0.827668
$447$	0	0
$448$	−18.7811	−0.887322
$449$	16.8348	0.794484	0.397242	−	0.917714i	$-0.369967\pi$
$449$	16.8348	0.794484	0.397242	+	0.917714i	$0.369967\pi$
$450$	0	0
$451$	−7.67230	−0.361275
$452$	−0.301785	−0.0141948
$453$	0	0
$454$	12.5280	0.587967
$455$	−2.18479	−0.102425
$456$	0	0
$457$	−8.16756	−0.382062	−0.191031	−	0.981584i	$-0.561183\pi$
$457$	−8.16756	−0.382062	−0.191031	+	0.981584i	$0.561183\pi$
$458$	12.0104	0.561210
$459$	0	0
$460$	0.879385	0.0410015
$461$	−0.762889	−0.0355313	−0.0177656	−	0.999842i	$-0.505655\pi$
$461$	−0.762889	−0.0355313	−0.0177656	+	0.999842i	$0.505655\pi$
$462$	0	0
$463$	16.4020	0.762265	0.381132	−	0.924521i	$-0.375534\pi$
$463$	16.4020	0.762265	0.381132	+	0.924521i	$0.375534\pi$
$464$	12.8530	0.596685
$465$	0	0
$466$	−29.5357	−1.36822
$467$	−5.17705	−0.239565	−0.119783	−	0.992800i	$-0.538220\pi$
$467$	−5.17705	−0.239565	−0.119783	+	0.992800i	$0.538220\pi$
$468$	0	0
$469$	−32.4935	−1.50041
$470$	1.40373	0.0647494
$471$	0	0
$472$	−35.6350	−1.64023
$473$	−4.43376	−0.203865
$474$	0	0
$475$	2.41147	0.110646
$476$	1.50063	0.0687812
$477$	0	0
$478$	5.83069	0.266690
$479$	4.05468	0.185263	0.0926316	−	0.995700i	$-0.470472\pi$
$479$	4.05468	0.185263	0.0926316	+	0.995700i	$0.470472\pi$
$480$	0	0
$481$	−7.04963	−0.321435
$482$	14.9668	0.681718
$483$	0	0
$484$	−0.748341	−0.0340155
$485$	−10.7733	−0.489191
$486$	0	0
$487$	41.1052	1.86266	0.931328	−	0.364181i	$-0.118651\pi$
$487$	41.1052	1.86266	0.931328	+	0.364181i	$0.118651\pi$
$488$	36.8033	1.66601
$489$	0	0
$490$	3.00000	0.135526
$491$	−32.7570	−1.47830	−0.739152	−	0.673539i	$-0.764773\pi$
$491$	−32.7570	−1.47830	−0.739152	+	0.673539i	$0.764773\pi$
$492$	0	0
$493$	−13.2841	−0.598284
$494$	−3.24897	−0.146178
$495$	0	0
$496$	38.9222	1.74766
$497$	7.57667	0.339860
$498$	0	0
$499$	−29.9581	−1.34111	−0.670555	−	0.741860i	$-0.733944\pi$
$499$	−29.9581	−1.34111	−0.670555	+	0.741860i	$0.733944\pi$
$500$	−0.184793	−0.00826417
$501$	0	0
$502$	11.6473	0.519846
$503$	−35.0966	−1.56488	−0.782439	−	0.622727i	$-0.786025\pi$
$503$	−35.0966	−1.56488	−0.782439	+	0.622727i	$0.786025\pi$
$504$	0	0
$505$	−13.3969	−0.596155
$506$	24.8726	1.10572
$507$	0	0
$508$	0.739170	0.0327954
$509$	18.4911	0.819605	0.409803	−	0.912174i	$-0.365598\pi$
$509$	18.4911	0.819605	0.409803	+	0.912174i	$0.365598\pi$
$510$	0	0
$511$	−10.1334	−0.448276
$512$	24.9186	1.10126
$513$	0	0
$514$	17.9828	0.793186
$515$	18.2344	0.803505
$516$	0	0
$517$	−4.04189	−0.177762
$518$	−20.7510	−0.911748
$519$	0	0
$520$	2.94356	0.129084
$521$	−10.4037	−0.455796	−0.227898	−	0.973685i	$-0.573185\pi$
$521$	−10.4037	−0.455796	−0.227898	+	0.973685i	$0.573185\pi$
$522$	0	0
$523$	−22.4611	−0.982156	−0.491078	−	0.871116i	$-0.663397\pi$
$523$	−22.4611	−0.982156	−0.491078	+	0.871116i	$0.663397\pi$
$524$	3.52797	0.154120
$525$	0	0
$526$	10.3509	0.451321
$527$	−40.2276	−1.75234
$528$	0	0
$529$	−0.354103	−0.0153958
$530$	2.63816	0.114594
$531$	0	0
$532$	0.973593	0.0422106
$533$	−1.97771	−0.0856642
$534$	0	0
$535$	−12.9094	−0.558123
$536$	43.7784	1.89094
$537$	0	0
$538$	−26.6878	−1.15059
$539$	−8.63816	−0.372072
$540$	0	0
$541$	22.8135	0.980827	0.490413	−	0.871490i	$-0.336846\pi$
$541$	22.8135	0.980827	0.490413	+	0.871490i	$0.336846\pi$
$542$	38.6049	1.65822
$543$	0	0
$544$	−3.87258	−0.166035
$545$	12.6459	0.541691
$546$	0	0
$547$	17.1925	0.735100	0.367550	−	0.930004i	$-0.380197\pi$
$547$	17.1925	0.735100	0.367550	+	0.930004i	$0.380197\pi$
$548$	0.305407	0.0130464
$549$	0	0
$550$	−5.22668	−0.222866
$551$	−8.61856	−0.367163
$552$	0	0
$553$	37.6536	1.60120
$554$	−9.27219	−0.393938
$555$	0	0
$556$	−3.90991	−0.165817
$557$	−15.0428	−0.637385	−0.318692	−	0.947858i	$-0.603244\pi$
$557$	−15.0428	−0.637385	−0.318692	+	0.947858i	$0.603244\pi$
$558$	0	0
$559$	−1.14290	−0.0483397
$560$	7.85710	0.332023
$561$	0	0
$562$	15.5776	0.657101
$563$	−17.7442	−0.747830	−0.373915	−	0.927463i	$-0.621985\pi$
$563$	−17.7442	−0.747830	−0.373915	+	0.927463i	$0.621985\pi$
$564$	0	0
$565$	1.63310	0.0687052
$566$	−35.3327	−1.48515
$567$	0	0
$568$	−10.2080	−0.428319
$569$	−14.5990	−0.612020	−0.306010	−	0.952028i	$-0.598994\pi$
$569$	−14.5990	−0.612020	−0.306010	+	0.952028i	$0.598994\pi$
$570$	0	0
$571$	−25.1257	−1.05148	−0.525738	−	0.850646i	$-0.676211\pi$
$571$	−25.1257	−1.05148	−0.525738	+	0.850646i	$0.676211\pi$
$572$	−0.716881	−0.0299743
$573$	0	0
$574$	−5.82152	−0.242985
$575$	−4.75877	−0.198454
$576$	0	0
$577$	−11.3601	−0.472927	−0.236463	−	0.971640i	$-0.575988\pi$
$577$	−11.3601	−0.472927	−0.236463	+	0.971640i	$0.575988\pi$
$578$	4.29086	0.178476
$579$	0	0
$580$	0.660444	0.0274235
$581$	−22.9469	−0.951996
$582$	0	0
$583$	−7.59627	−0.314605
$584$	13.6527	0.564953
$585$	0	0
$586$	−43.7716	−1.80819
$587$	−1.37370	−0.0566988	−0.0283494	−	0.999598i	$-0.509025\pi$
$587$	−1.37370	−0.0566988	−0.0283494	+	0.999598i	$0.509025\pi$
$588$	0	0
$589$	−26.0993	−1.07540
$590$	16.3105	0.671491
$591$	0	0
$592$	25.3523	1.04198
$593$	−31.3073	−1.28564	−0.642818	−	0.766019i	$-0.722235\pi$
$593$	−31.3073	−1.28564	−0.642818	+	0.766019i	$0.722235\pi$
$594$	0	0
$595$	−8.12061	−0.332913
$596$	3.95904	0.162169
$597$	0	0
$598$	6.41147	0.262185
$599$	−26.3259	−1.07565	−0.537824	−	0.843057i	$-0.680754\pi$
$599$	−26.3259	−1.07565	−0.537824	+	0.843057i	$0.680754\pi$
$600$	0	0
$601$	15.0247	0.612868	0.306434	−	0.951892i	$-0.400864\pi$
$601$	15.0247	0.612868	0.306434	+	0.951892i	$0.400864\pi$
$602$	−3.36421	−0.137115
$603$	0	0
$604$	2.05281	0.0835279
$605$	4.04963	0.164641
$606$	0	0
$607$	−5.77156	−0.234261	−0.117130	−	0.993117i	$-0.537370\pi$
$607$	−5.77156	−0.234261	−0.117130	+	0.993117i	$0.537370\pi$
$608$	−2.51249	−0.101895
$609$	0	0
$610$	−16.8452	−0.682044
$611$	−1.04189	−0.0421503
$612$	0	0
$613$	−46.8462	−1.89210	−0.946050	−	0.324022i	$-0.894965\pi$
$613$	−46.8462	−1.89210	−0.946050	+	0.324022i	$0.894965\pi$
$614$	9.56481	0.386004
$615$	0	0
$616$	−24.9486	−1.00521
$617$	22.2104	0.894156	0.447078	−	0.894495i	$-0.352465\pi$
$617$	22.2104	0.894156	0.447078	+	0.894495i	$0.352465\pi$
$618$	0	0
$619$	8.19253	0.329286	0.164643	−	0.986353i	$-0.447353\pi$
$619$	8.19253	0.329286	0.164643	+	0.986353i	$0.447353\pi$
$620$	2.00000	0.0803219
$621$	0	0
$622$	−6.10244	−0.244686
$623$	−37.8685	−1.51717
$624$	0	0
$625$	1.00000	0.0400000
$626$	15.6004	0.623517
$627$	0	0
$628$	0.931703	0.0371790
$629$	−26.2026	−1.04477
$630$	0	0
$631$	9.66313	0.384683	0.192342	−	0.981328i	$-0.438392\pi$
$631$	9.66313	0.384683	0.192342	+	0.981328i	$0.438392\pi$
$632$	−50.7306	−2.01796
$633$	0	0
$634$	14.5357	0.577287
$635$	−4.00000	−0.158735
$636$	0	0
$637$	−2.22668	−0.0882243
$638$	18.6800	0.739550
$639$	0	0
$640$	−9.49794	−0.375439
$641$	12.1857	0.481307	0.240654	−	0.970611i	$-0.422638\pi$
$641$	12.1857	0.481307	0.240654	+	0.970611i	$0.422638\pi$
$642$	0	0
$643$	27.6195	1.08921	0.544603	−	0.838694i	$-0.316680\pi$
$643$	27.6195	1.08921	0.544603	+	0.838694i	$0.316680\pi$
$644$	−1.92127	−0.0757088
$645$	0	0
$646$	−12.0760	−0.475125
$647$	−27.4311	−1.07843	−0.539213	−	0.842169i	$-0.681278\pi$
$647$	−27.4311	−1.07843	−0.539213	+	0.842169i	$0.681278\pi$
$648$	0	0
$649$	−46.9641	−1.84350
$650$	−1.34730	−0.0528453
$651$	0	0
$652$	0.131978	0.00516864
$653$	−29.1875	−1.14219	−0.571097	−	0.820882i	$-0.693482\pi$
$653$	−29.1875	−1.14219	−0.571097	+	0.820882i	$0.693482\pi$
$654$	0	0
$655$	−19.0915	−0.745967
$656$	7.11238	0.277692
$657$	0	0
$658$	−3.06687	−0.119559
$659$	8.21389	0.319968	0.159984	−	0.987120i	$-0.448856\pi$
$659$	8.21389	0.319968	0.159984	+	0.987120i	$0.448856\pi$
$660$	0	0
$661$	1.07461	0.0417974	0.0208987	−	0.999782i	$-0.493347\pi$
$661$	1.07461	0.0417974	0.0208987	+	0.999782i	$0.493347\pi$
$662$	12.4311	0.483147
$663$	0	0
$664$	30.9162	1.19978
$665$	−5.26857	−0.204306
$666$	0	0
$667$	17.0077	0.658542
$668$	2.76146	0.106844
$669$	0	0
$670$	−20.0378	−0.774127
$671$	48.5039	1.87247
$672$	0	0
$673$	28.7547	1.10841	0.554205	−	0.832380i	$-0.313022\pi$
$673$	28.7547	1.10841	0.554205	+	0.832380i	$0.313022\pi$
$674$	−30.9522	−1.19224
$675$	0	0
$676$	−0.184793	−0.00710741
$677$	34.2404	1.31597	0.657983	−	0.753033i	$-0.271410\pi$
$677$	34.2404	1.31597	0.657983	+	0.753033i	$0.271410\pi$
$678$	0	0
$679$	23.5375	0.903285
$680$	10.9409	0.419563
$681$	0	0
$682$	56.5681	2.16610
$683$	−35.3732	−1.35352	−0.676759	−	0.736205i	$-0.736616\pi$
$683$	−35.3732	−1.35352	−0.676759	+	0.736205i	$0.736616\pi$
$684$	0	0
$685$	−1.65270	−0.0631466
$686$	−27.1593	−1.03695
$687$	0	0
$688$	4.11019	0.156699
$689$	−1.95811	−0.0745981
$690$	0	0
$691$	24.2576	0.922804	0.461402	−	0.887191i	$-0.347347\pi$
$691$	24.2576	0.922804	0.461402	+	0.887191i	$0.347347\pi$
$692$	−0.567171	−0.0215606
$693$	0	0
$694$	19.6723	0.746750
$695$	21.1584	0.802583
$696$	0	0
$697$	−7.35092	−0.278436
$698$	21.9186	0.829631
$699$	0	0
$700$	0.403733	0.0152597
$701$	−50.4597	−1.90584	−0.952918	−	0.303229i	$-0.901935\pi$
$701$	−50.4597	−1.90584	−0.952918	+	0.303229i	$0.901935\pi$
$702$	0	0
$703$	−17.0000	−0.641167
$704$	33.3482	1.25686
$705$	0	0
$706$	9.50156	0.357596
$707$	29.2695	1.10079
$708$	0	0
$709$	−12.4861	−0.468925	−0.234462	−	0.972125i	$-0.575333\pi$
$709$	−12.4861	−0.468925	−0.234462	+	0.972125i	$0.575333\pi$
$710$	4.67230	0.175348
$711$	0	0
$712$	51.0200	1.91206
$713$	51.5039	1.92884
$714$	0	0
$715$	3.87939	0.145081
$716$	0.0472658	0.00176640
$717$	0	0
$718$	23.1506	0.863974
$719$	−21.0797	−0.786139	−0.393069	−	0.919509i	$-0.628587\pi$
$719$	−21.0797	−0.786139	−0.393069	+	0.919509i	$0.628587\pi$
$720$	0	0
$721$	−39.8384	−1.48366
$722$	17.7638	0.661101
$723$	0	0
$724$	−1.11162	−0.0413129
$725$	−3.57398	−0.132734
$726$	0	0
$727$	11.2950	0.418908	0.209454	−	0.977819i	$-0.432831\pi$
$727$	11.2950	0.418908	0.209454	+	0.977819i	$0.432831\pi$
$728$	−6.43107	−0.238351
$729$	0	0
$730$	−6.24897	−0.231285
$731$	−4.24804	−0.157119
$732$	0	0
$733$	13.7374	0.507403	0.253702	−	0.967283i	$-0.418352\pi$
$733$	13.7374	0.507403	0.253702	+	0.967283i	$0.418352\pi$
$734$	−23.9103	−0.882547
$735$	0	0
$736$	4.95811	0.182758
$737$	57.6965	2.12528
$738$	0	0
$739$	−26.3541	−0.969451	−0.484726	−	0.874666i	$-0.661081\pi$
$739$	−26.3541	−0.969451	−0.484726	+	0.874666i	$0.661081\pi$
$740$	1.30272	0.0478889
$741$	0	0
$742$	−5.76382	−0.211597
$743$	39.5904	1.45243	0.726215	−	0.687467i	$-0.241278\pi$
$743$	39.5904	1.45243	0.726215	+	0.687467i	$0.241278\pi$
$744$	0	0
$745$	−21.4243	−0.784924
$746$	39.0925	1.43128
$747$	0	0
$748$	−2.66456	−0.0974261
$749$	28.2044	1.03057
$750$	0	0
$751$	−3.49525	−0.127544	−0.0637718	−	0.997965i	$-0.520313\pi$
$751$	−3.49525	−0.127544	−0.0637718	+	0.997965i	$0.520313\pi$
$752$	3.74691	0.136636
$753$	0	0
$754$	4.81521	0.175360
$755$	−11.1088	−0.404289
$756$	0	0
$757$	−17.2517	−0.627022	−0.313511	−	0.949585i	$-0.601505\pi$
$757$	−17.2517	−0.627022	−0.313511	+	0.949585i	$0.601505\pi$
$758$	−11.3560	−0.412467
$759$	0	0
$760$	7.09833	0.257483
$761$	51.3783	1.86246	0.931230	−	0.364431i	$-0.118737\pi$
$761$	51.3783	1.86246	0.931230	+	0.364431i	$0.118737\pi$
$762$	0	0
$763$	−27.6287	−1.00022
$764$	−2.60132	−0.0941124
$765$	0	0
$766$	−26.4284	−0.954896
$767$	−12.1061	−0.437125
$768$	0	0
$769$	2.29498	0.0827590	0.0413795	−	0.999144i	$-0.486825\pi$
$769$	2.29498	0.0827590	0.0413795	+	0.999144i	$0.486825\pi$
$770$	11.4192	0.411520
$771$	0	0
$772$	0.270001	0.00971755
$773$	39.0601	1.40489	0.702446	−	0.711737i	$-0.252091\pi$
$773$	39.0601	1.40489	0.702446	+	0.711737i	$0.252091\pi$
$774$	0	0
$775$	−10.8229	−0.388772
$776$	−31.7119	−1.13839
$777$	0	0
$778$	26.1739	0.938379
$779$	−4.76920	−0.170874
$780$	0	0
$781$	−13.4534	−0.481399
$782$	23.8307	0.852184
$783$	0	0
$784$	8.00774	0.285991
$785$	−5.04189	−0.179953
$786$	0	0
$787$	−31.4935	−1.12262	−0.561311	−	0.827605i	$-0.689703\pi$
$787$	−31.4935	−1.12262	−0.561311	+	0.827605i	$0.689703\pi$
$788$	2.85584	0.101735
$789$	0	0
$790$	23.2199	0.826126
$791$	−3.56799	−0.126863
$792$	0	0
$793$	12.5030	0.443994
$794$	8.55850	0.303730
$795$	0	0
$796$	0.184793	0.00654980
$797$	−49.5827	−1.75631	−0.878154	−	0.478378i	$-0.841225\pi$
$797$	−49.5827	−1.75631	−0.878154	+	0.478378i	$0.841225\pi$
$798$	0	0
$799$	−3.87258	−0.137002
$800$	−1.04189	−0.0368363
$801$	0	0
$802$	−7.76239	−0.274100
$803$	17.9932	0.634966
$804$	0	0
$805$	10.3969	0.366443
$806$	14.5817	0.513619
$807$	0	0
$808$	−39.4347	−1.38731
$809$	−30.8749	−1.08551	−0.542753	−	0.839893i	$-0.682618\pi$
$809$	−30.8749	−1.08551	−0.542753	+	0.839893i	$0.682618\pi$
$810$	0	0
$811$	17.9982	0.632004	0.316002	−	0.948759i	$-0.397659\pi$
$811$	17.9982	0.632004	0.316002	+	0.948759i	$0.397659\pi$
$812$	−1.44293	−0.0506371
$813$	0	0
$814$	36.8462	1.29146
$815$	−0.714193	−0.0250171
$816$	0	0
$817$	−2.75608	−0.0964231
$818$	−47.9103	−1.67515
$819$	0	0
$820$	0.365466	0.0127626
$821$	−13.2772	−0.463379	−0.231690	−	0.972790i	$-0.574425\pi$
$821$	−13.2772	−0.463379	−0.231690	+	0.972790i	$0.574425\pi$
$822$	0	0
$823$	−2.10876	−0.0735066	−0.0367533	−	0.999324i	$-0.511702\pi$
$823$	−2.10876	−0.0735066	−0.0367533	+	0.999324i	$0.511702\pi$
$824$	53.6742	1.86983
$825$	0	0
$826$	−35.6350	−1.23990
$827$	−8.10700	−0.281908	−0.140954	−	0.990016i	$-0.545017\pi$
$827$	−8.10700	−0.281908	−0.140954	+	0.990016i	$0.545017\pi$
$828$	0	0
$829$	−46.3587	−1.61010	−0.805051	−	0.593205i	$-0.797862\pi$
$829$	−46.3587	−1.61010	−0.805051	+	0.593205i	$0.797862\pi$
$830$	−14.1506	−0.491176
$831$	0	0
$832$	8.59627	0.298022
$833$	−8.27631	−0.286757
$834$	0	0
$835$	−14.9436	−0.517143
$836$	−1.72874	−0.0597898
$837$	0	0
$838$	0.277742	0.00959443
$839$	−38.3236	−1.32308	−0.661538	−	0.749911i	$-0.730096\pi$
$839$	−38.3236	−1.32308	−0.661538	+	0.749911i	$0.730096\pi$
$840$	0	0
$841$	−16.2267	−0.559541
$842$	−10.0315	−0.345707
$843$	0	0
$844$	3.56498	0.122712
$845$	1.00000	0.0344010
$846$	0	0
$847$	−8.84760	−0.304007
$848$	7.04189	0.241819
$849$	0	0
$850$	−5.00774	−0.171764
$851$	33.5476	1.15000
$852$	0	0
$853$	49.1070	1.68139	0.840696	−	0.541508i	$-0.182146\pi$
$853$	49.1070	1.68139	0.840696	+	0.541508i	$0.182146\pi$
$854$	36.8033	1.25938
$855$	0	0
$856$	−37.9997	−1.29880
$857$	10.8557	0.370822	0.185411	−	0.982661i	$-0.440638\pi$
$857$	10.8557	0.370822	0.185411	+	0.982661i	$0.440638\pi$
$858$	0	0
$859$	−36.1070	−1.23195	−0.615977	−	0.787764i	$-0.711239\pi$
$859$	−36.1070	−1.23195	−0.615977	+	0.787764i	$0.711239\pi$
$860$	0.211200	0.00720186
$861$	0	0
$862$	18.8634	0.642490
$863$	40.3397	1.37318	0.686589	−	0.727046i	$-0.259107\pi$
$863$	40.3397	1.37318	0.686589	+	0.727046i	$0.259107\pi$
$864$	0	0
$865$	3.06923	0.104357
$866$	−8.24266	−0.280097
$867$	0	0
$868$	−4.36959	−0.148313
$869$	−66.8590	−2.26804
$870$	0	0
$871$	14.8726	0.503938
$872$	37.2240	1.26056
$873$	0	0
$874$	15.4611	0.522980
$875$	−2.18479	−0.0738595
$876$	0	0
$877$	6.30447	0.212887	0.106443	−	0.994319i	$-0.466054\pi$
$877$	6.30447	0.212887	0.106443	+	0.994319i	$0.466054\pi$
$878$	18.1361	0.612064
$879$	0	0
$880$	−13.9513	−0.470298
$881$	−26.9992	−0.909625	−0.454813	−	0.890587i	$-0.650294\pi$
$881$	−26.9992	−0.909625	−0.454813	+	0.890587i	$0.650294\pi$
$882$	0	0
$883$	−15.5449	−0.523127	−0.261563	−	0.965186i	$-0.584238\pi$
$883$	−15.5449	−0.523127	−0.261563	+	0.965186i	$0.584238\pi$
$884$	−0.686852	−0.0231013
$885$	0	0
$886$	−38.9331	−1.30798
$887$	−43.6483	−1.46557	−0.732783	−	0.680463i	$-0.761779\pi$
$887$	−43.6483	−1.46557	−0.732783	+	0.680463i	$0.761779\pi$
$888$	0	0
$889$	8.73917	0.293102
$890$	−23.3523	−0.782773
$891$	0	0
$892$	2.39742	0.0802717
$893$	−2.51249	−0.0840772
$894$	0	0
$895$	−0.255777	−0.00854970
$896$	20.7510	0.693243
$897$	0	0
$898$	−22.6815	−0.756891
$899$	38.6810	1.29008
$900$	0	0
$901$	−7.27807	−0.242468
$902$	10.3369	0.344180
$903$	0	0
$904$	4.80714	0.159883
$905$	6.01548	0.199961
$906$	0	0
$907$	−3.15663	−0.104814	−0.0524071	−	0.998626i	$-0.516689\pi$
$907$	−3.15663	−0.104814	−0.0524071	+	0.998626i	$0.516689\pi$
$908$	1.71831	0.0570242
$909$	0	0
$910$	2.94356	0.0975782
$911$	−40.5931	−1.34491	−0.672454	−	0.740139i	$-0.734760\pi$
$911$	−40.5931	−1.34491	−0.672454	+	0.740139i	$0.734760\pi$
$912$	0	0
$913$	40.7452	1.34847
$914$	11.0041	0.363984
$915$	0	0
$916$	1.64733	0.0544292
$917$	41.7110	1.37742
$918$	0	0
$919$	17.1601	0.566061	0.283030	−	0.959111i	$-0.408660\pi$
$919$	17.1601	0.566061	0.283030	+	0.959111i	$0.408660\pi$
$920$	−14.0077	−0.461822
$921$	0	0
$922$	1.02784	0.0338500
$923$	−3.46791	−0.114148
$924$	0	0
$925$	−7.04963	−0.231790
$926$	−22.0983	−0.726196
$927$	0	0
$928$	3.72369	0.122236
$929$	−23.9691	−0.786402	−0.393201	−	0.919452i	$-0.628632\pi$
$929$	−23.9691	−0.786402	−0.393201	+	0.919452i	$0.628632\pi$
$930$	0	0
$931$	−5.36959	−0.175981
$932$	−4.05106	−0.132697
$933$	0	0
$934$	6.97502	0.228230
$935$	14.4192	0.471559
$936$	0	0
$937$	40.6614	1.32835	0.664175	−	0.747577i	$-0.268783\pi$
$937$	40.6614	1.32835	0.664175	+	0.747577i	$0.268783\pi$
$938$	43.7784	1.42941
$939$	0	0
$940$	0.192533	0.00627974
$941$	50.6786	1.65208	0.826038	−	0.563615i	$-0.190590\pi$
$941$	50.6786	1.65208	0.826038	+	0.563615i	$0.190590\pi$
$942$	0	0
$943$	9.41147	0.306480
$944$	43.5366	1.41700
$945$	0	0
$946$	5.97359	0.194218
$947$	−2.60813	−0.0847527	−0.0423764	−	0.999102i	$-0.513493\pi$
$947$	−2.60813	−0.0847527	−0.0423764	+	0.999102i	$0.513493\pi$
$948$	0	0
$949$	4.63816	0.150561
$950$	−3.24897	−0.105411
$951$	0	0
$952$	−23.9035	−0.774718
$953$	39.6408	1.28409	0.642046	−	0.766666i	$-0.278086\pi$
$953$	39.6408	1.28409	0.642046	+	0.766666i	$0.278086\pi$
$954$	0	0
$955$	14.0770	0.455520
$956$	0.799726	0.0258650
$957$	0	0
$958$	−5.46286	−0.176497
$959$	3.61081	0.116599
$960$	0	0
$961$	86.1362	2.77859
$962$	9.49794	0.306226
$963$	0	0
$964$	2.05281	0.0661167
$965$	−1.46110	−0.0470346
$966$	0	0
$967$	18.4534	0.593420	0.296710	−	0.954968i	$-0.404111\pi$
$967$	18.4534	0.593420	0.296710	+	0.954968i	$0.404111\pi$
$968$	11.9203	0.383134
$969$	0	0
$970$	14.5149	0.466044
$971$	21.7665	0.698521	0.349260	−	0.937026i	$-0.386433\pi$
$971$	21.7665	0.698521	0.349260	+	0.937026i	$0.386433\pi$
$972$	0	0
$973$	−46.2267	−1.48196
$974$	−55.3809	−1.77452
$975$	0	0
$976$	−44.9641	−1.43927
$977$	10.4338	0.333806	0.166903	−	0.985973i	$-0.446623\pi$
$977$	10.4338	0.333806	0.166903	+	0.985973i	$0.446623\pi$
$978$	0	0
$979$	67.2404	2.14901
$980$	0.411474	0.0131441
$981$	0	0
$982$	44.1334	1.40835
$983$	9.04046	0.288346	0.144173	−	0.989553i	$-0.453948\pi$
$983$	9.04046	0.288346	0.144173	+	0.989553i	$0.453948\pi$
$984$	0	0
$985$	−15.4543	−0.492415
$986$	17.8976	0.569974
$987$	0	0
$988$	−0.445622	−0.0141771
$989$	5.43882	0.172944
$990$	0	0
$991$	−8.86484	−0.281601	−0.140800	−	0.990038i	$-0.544968\pi$
$991$	−8.86484	−0.281601	−0.140800	+	0.990038i	$0.544968\pi$
$992$	11.2763	0.358023
$993$	0	0
$994$	−10.2080	−0.323779
$995$	−1.00000	−0.0317021
$996$	0	0
$997$	5.30304	0.167949	0.0839745	−	0.996468i	$-0.473239\pi$
$997$	5.30304	0.167949	0.0839745	+	0.996468i	$0.473239\pi$
$998$	40.3625	1.27765
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5265.2.a.s.1.2	✓	3
3.2	odd	2		5265.2.a.t.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5265.2.a.s.1.2	✓	3	1.1	even	1	trivial
5265.2.a.t.1.2	yes	3	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)

\(f(q)\)	\(=\)	\(q-1.34730 q^{2} -0.184793 q^{4} +1.00000 q^{5} -2.18479 q^{7} +2.94356 q^{8} +O(q^{10})\)	\(q-1.34730 q^{2} -0.184793 q^{4} +1.00000 q^{5} -2.18479 q^{7} +2.94356 q^{8} -1.34730 q^{10} +3.87939 q^{11} +1.00000 q^{13} +2.94356 q^{14} -3.59627 q^{16} +3.71688 q^{17} +2.41147 q^{19} -0.184793 q^{20} -5.22668 q^{22} -4.75877 q^{23} +1.00000 q^{25} -1.34730 q^{26} +0.403733 q^{28} -3.57398 q^{29} -10.8229 q^{31} -1.04189 q^{32} -5.00774 q^{34} -2.18479 q^{35} -7.04963 q^{37} -3.24897 q^{38} +2.94356 q^{40} -1.97771 q^{41} -1.14290 q^{43} -0.716881 q^{44} +6.41147 q^{46} -1.04189 q^{47} -2.22668 q^{49} -1.34730 q^{50} -0.184793 q^{52} -1.95811 q^{53} +3.87939 q^{55} -6.43107 q^{56} +4.81521 q^{58} -12.1061 q^{59} +12.5030 q^{61} +14.5817 q^{62} +8.59627 q^{64} +1.00000 q^{65} +14.8726 q^{67} -0.686852 q^{68} +2.94356 q^{70} -3.46791 q^{71} +4.63816 q^{73} +9.49794 q^{74} -0.445622 q^{76} -8.47565 q^{77} -17.2344 q^{79} -3.59627 q^{80} +2.66456 q^{82} +10.5030 q^{83} +3.71688 q^{85} +1.53983 q^{86} +11.4192 q^{88} +17.3327 q^{89} -2.18479 q^{91} +0.879385 q^{92} +1.40373 q^{94} +2.41147 q^{95} -10.7733 q^{97} +3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - 3 q^{7} - 6 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - 3 * q^7 - 6 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - 3 q^{7} - 6 q^{8} - 3 q^{10} + 6 q^{11} + 3 q^{13} - 6 q^{14} + 3 q^{16} + 3 q^{17} - 3 q^{19} + 3 q^{20} - 9 q^{22} - 3 q^{23} + 3 q^{25} - 3 q^{26} + 15 q^{28} - 3 q^{29} - 12 q^{31} + 9 q^{34} - 3 q^{35} + 6 q^{37} + 3 q^{38} - 6 q^{40} - 12 q^{41} - 3 q^{43} + 6 q^{44} + 9 q^{46} - 3 q^{50} + 3 q^{52} - 9 q^{53} + 6 q^{55} - 12 q^{56} + 18 q^{58} - 24 q^{59} - 3 q^{61} + 12 q^{62} + 12 q^{64} + 3 q^{65} - 3 q^{67} - 24 q^{68} - 6 q^{70} - 15 q^{71} - 3 q^{73} + 3 q^{74} - 12 q^{76} - 6 q^{77} - 21 q^{79} + 3 q^{80} + 36 q^{82} - 9 q^{83} + 3 q^{85} - 24 q^{86} + 33 q^{89} - 3 q^{91} - 3 q^{92} + 18 q^{94} - 3 q^{95} - 39 q^{97} + 9 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - 3 * q^7 - 6 * q^8 - 3 * q^10 + 6 * q^11 + 3 * q^13 - 6 * q^14 + 3 * q^16 + 3 * q^17 - 3 * q^19 + 3 * q^20 - 9 * q^22 - 3 * q^23 + 3 * q^25 - 3 * q^26 + 15 * q^28 - 3 * q^29 - 12 * q^31 + 9 * q^34 - 3 * q^35 + 6 * q^37 + 3 * q^38 - 6 * q^40 - 12 * q^41 - 3 * q^43 + 6 * q^44 + 9 * q^46 - 3 * q^50 + 3 * q^52 - 9 * q^53 + 6 * q^55 - 12 * q^56 + 18 * q^58 - 24 * q^59 - 3 * q^61 + 12 * q^62 + 12 * q^64 + 3 * q^65 - 3 * q^67 - 24 * q^68 - 6 * q^70 - 15 * q^71 - 3 * q^73 + 3 * q^74 - 12 * q^76 - 6 * q^77 - 21 * q^79 + 3 * q^80 + 36 * q^82 - 9 * q^83 + 3 * q^85 - 24 * q^86 + 33 * q^89 - 3 * q^91 - 3 * q^92 + 18 * q^94 - 3 * q^95 - 39 * q^97 + 9 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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