LMFDB - Embedded newform 4225.2.a.bv.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4225,2,Mod(1,4225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4225 = 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4225.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:	$33.7367948540$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 16x^{8} + 84x^{6} - 163x^{4} + 118x^{2} - 27 $ x^10 - 16x^8 + 84x^6 - 163x^4 + 118x^2 - 27
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 325)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.36551$ of defining polynomial
Character	$\chi$	$=$	4225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.36551 q^{2} -0.399918 q^{3} -0.135381 q^{4} +0.546092 q^{6} -3.64365 q^{7} +2.91589 q^{8} -2.84007 q^{9} +O(q^{10})$	\(q-1.36551 q^{2} -0.399918 q^{3} -0.135381 q^{4} +0.546092 q^{6} -3.64365 q^{7} +2.91589 q^{8} -2.84007 q^{9} -3.27711 q^{11} +0.0541412 q^{12} +4.97545 q^{14} -3.71091 q^{16} -2.17561 q^{17} +3.87814 q^{18} +0.857697 q^{19} +1.45716 q^{21} +4.47493 q^{22} -0.765051 q^{23} -1.16611 q^{24} +2.33555 q^{27} +0.493281 q^{28} -3.06437 q^{29} -8.41833 q^{31} -0.764484 q^{32} +1.31058 q^{33} +2.97082 q^{34} +0.384491 q^{36} -10.8576 q^{37} -1.17119 q^{38} -7.61344 q^{41} -1.98977 q^{42} +3.83606 q^{43} +0.443659 q^{44} +1.04469 q^{46} +4.68303 q^{47} +1.48406 q^{48} +6.27621 q^{49} +0.870065 q^{51} -3.04646 q^{53} -3.18921 q^{54} -10.6245 q^{56} -0.343008 q^{57} +4.18443 q^{58} -13.3227 q^{59} -13.0306 q^{61} +11.4953 q^{62} +10.3482 q^{63} +8.46573 q^{64} -1.78960 q^{66} +4.64794 q^{67} +0.294536 q^{68} +0.305958 q^{69} -10.8356 q^{71} -8.28131 q^{72} -7.68313 q^{73} +14.8262 q^{74} -0.116116 q^{76} +11.9407 q^{77} -10.2106 q^{79} +7.58617 q^{81} +10.3962 q^{82} +1.18065 q^{83} -0.197272 q^{84} -5.23818 q^{86} +1.22550 q^{87} -9.55569 q^{88} -3.26258 q^{89} +0.103573 q^{92} +3.36664 q^{93} -6.39472 q^{94} +0.305731 q^{96} +5.25271 q^{97} -8.57023 q^{98} +9.30722 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 6 q^{3} + 12 q^{4} + 16 q^{9}+O(q^{10}) $ 10 * q + 6 * q^3 + 12 * q^4 + 16 * q^9	$ 10 q + 6 q^{3} + 12 q^{4} + 16 q^{9} + 28 q^{12} - 8 q^{14} + 24 q^{16} + 16 q^{17} + 24 q^{22} + 26 q^{23} + 30 q^{27} - 14 q^{29} - 6 q^{36} + 62 q^{38} - 64 q^{42} + 8 q^{43} + 52 q^{48} - 2 q^{49} - 16 q^{51} + 24 q^{53} - 42 q^{56} - 26 q^{61} + 34 q^{62} + 34 q^{64} - 42 q^{66} + 26 q^{68} + 40 q^{69} + 52 q^{74} + 48 q^{77} + 4 q^{79} + 34 q^{81} - 2 q^{82} + 98 q^{87} - 12 q^{88} + 34 q^{92} - 10 q^{94}+O(q^{100}) $ 10 * q + 6 * q^3 + 12 * q^4 + 16 * q^9 + 28 * q^12 - 8 * q^14 + 24 * q^16 + 16 * q^17 + 24 * q^22 + 26 * q^23 + 30 * q^27 - 14 * q^29 - 6 * q^36 + 62 * q^38 - 64 * q^42 + 8 * q^43 + 52 * q^48 - 2 * q^49 - 16 * q^51 + 24 * q^53 - 42 * q^56 - 26 * q^61 + 34 * q^62 + 34 * q^64 - 42 * q^66 + 26 * q^68 + 40 * q^69 + 52 * q^74 + 48 * q^77 + 4 * q^79 + 34 * q^81 - 2 * q^82 + 98 * q^87 - 12 * q^88 + 34 * q^92 - 10 * q^94

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 $

$2$	−1.36551	−0.965562	−0.482781	−	0.875741i	$-0.660373\pi$
$2$	−1.36551	−0.965562	−0.482781	+	0.875741i	$0.660373\pi$
$3$	−0.399918	−0.230893	−0.115446	−	0.993314i	$-0.536830\pi$
$3$	−0.399918	−0.230893	−0.115446	+	0.993314i	$0.536830\pi$
$4$	−0.135381	−0.0676905
$5$	0	0
$5$	0	0
$6$	0.546092	0.222941
$7$	−3.64365	−1.37717	−0.688586	−	0.725155i	$-0.741768\pi$
$7$	−3.64365	−1.37717	−0.688586	+	0.725155i	$0.741768\pi$
$8$	2.91589	1.03092
$9$	−2.84007	−0.946689
$10$	0	0
$11$	−3.27711	−0.988087	−0.494043	−	0.869437i	$-0.664482\pi$
$11$	−3.27711	−0.988087	−0.494043	+	0.869437i	$0.664482\pi$
$12$	0.0541412	0.0156292
$13$	0	0
$13$	0	0
$14$	4.97545	1.32974
$15$	0	0
$16$	−3.71091	−0.927728
$17$	−2.17561	−0.527663	−0.263832	−	0.964569i	$-0.584986\pi$
$17$	−2.17561	−0.527663	−0.263832	+	0.964569i	$0.584986\pi$
$18$	3.87814	0.914086
$19$	0.857697	0.196769	0.0983846	−	0.995148i	$-0.468632\pi$
$19$	0.857697	0.196769	0.0983846	+	0.995148i	$0.468632\pi$
$20$	0	0
$21$	1.45716	0.317979
$22$	4.47493	0.954059
$23$	−0.765051	−0.159524	−0.0797621	−	0.996814i	$-0.525416\pi$
$23$	−0.765051	−0.159524	−0.0797621	+	0.996814i	$0.525416\pi$
$24$	−1.16611	−0.238032
$25$	0	0
$26$	0	0
$27$	2.33555	0.449476
$28$	0.493281	0.0932214
$29$	−3.06437	−0.569040	−0.284520	−	0.958670i	$-0.591834\pi$
$29$	−3.06437	−0.569040	−0.284520	+	0.958670i	$0.591834\pi$
$30$	0	0
$31$	−8.41833	−1.51198	−0.755988	−	0.654585i	$-0.772843\pi$
$31$	−8.41833	−1.51198	−0.755988	+	0.654585i	$0.772843\pi$
$32$	−0.764484	−0.135143
$33$	1.31058	0.228142
$34$	2.97082	0.509491
$35$	0	0
$36$	0.384491	0.0640818
$37$	−10.8576	−1.78498	−0.892488	−	0.451070i	$-0.851042\pi$
$37$	−10.8576	−1.78498	−0.892488	+	0.451070i	$0.851042\pi$
$38$	−1.17119	−0.189993
$39$	0	0
$40$	0	0
$41$	−7.61344	−1.18902	−0.594510	−	0.804088i	$-0.702654\pi$
$41$	−7.61344	−1.18902	−0.594510	+	0.804088i	$0.702654\pi$
$42$	−1.98977	−0.307028
$43$	3.83606	0.584994	0.292497	−	0.956266i	$-0.405514\pi$
$43$	3.83606	0.584994	0.292497	+	0.956266i	$0.405514\pi$
$44$	0.443659	0.0668841
$45$	0	0
$46$	1.04469	0.154030
$47$	4.68303	0.683090	0.341545	−	0.939865i	$-0.389050\pi$
$47$	4.68303	0.683090	0.341545	+	0.939865i	$0.389050\pi$
$48$	1.48406	0.214205
$49$	6.27621	0.896601
$50$	0	0
$51$	0.870065	0.121834
$52$	0	0
$53$	−3.04646	−0.418463	−0.209231	−	0.977866i	$-0.567096\pi$
$53$	−3.04646	−0.418463	−0.209231	+	0.977866i	$0.567096\pi$
$54$	−3.18921	−0.433997
$55$	0	0
$56$	−10.6245	−1.41976
$57$	−0.343008	−0.0454325
$58$	4.18443	0.549443
$59$	−13.3227	−1.73447	−0.867235	−	0.497898i	$-0.834105\pi$
$59$	−13.3227	−1.73447	−0.867235	+	0.497898i	$0.834105\pi$
$60$	0	0
$61$	−13.0306	−1.66840	−0.834200	−	0.551462i	$-0.814070\pi$
$61$	−13.0306	−1.66840	−0.834200	+	0.551462i	$0.814070\pi$
$62$	11.4953	1.45991
$63$	10.3482	1.30375
$64$	8.46573	1.05822
$65$	0	0
$66$	−1.78960	−0.220285
$67$	4.64794	0.567836	0.283918	−	0.958849i	$-0.408366\pi$
$67$	4.64794	0.567836	0.283918	+	0.958849i	$0.408366\pi$
$68$	0.294536	0.0357178
$69$	0.305958	0.0368330
$70$	0	0
$71$	−10.8356	−1.28595	−0.642976	−	0.765886i	$-0.722300\pi$
$71$	−10.8356	−1.28595	−0.642976	+	0.765886i	$0.722300\pi$
$72$	−8.28131	−0.975961
$73$	−7.68313	−0.899242	−0.449621	−	0.893219i	$-0.648441\pi$
$73$	−7.68313	−0.899242	−0.449621	+	0.893219i	$0.648441\pi$
$74$	14.8262	1.72351
$75$	0	0
$76$	−0.116116	−0.0133194
$77$	11.9407	1.36076
$78$	0	0
$79$	−10.2106	−1.14879	−0.574394	−	0.818579i	$-0.694762\pi$
$79$	−10.2106	−1.14879	−0.574394	+	0.818579i	$0.694762\pi$
$80$	0	0
$81$	7.58617	0.842908
$82$	10.3962	1.14807
$83$	1.18065	0.129593	0.0647964	−	0.997899i	$-0.479360\pi$
$83$	1.18065	0.129593	0.0647964	+	0.997899i	$0.479360\pi$
$84$	−0.197272	−0.0215241
$85$	0	0
$86$	−5.23818	−0.564848
$87$	1.22550	0.131387
$88$	−9.55569	−1.01864
$89$	−3.26258	−0.345833	−0.172916	−	0.984937i	$-0.555319\pi$
$89$	−3.26258	−0.345833	−0.172916	+	0.984937i	$0.555319\pi$
$90$	0	0
$91$	0	0
$92$	0.103573	0.0107983
$93$	3.36664	0.349104
$94$	−6.39472	−0.659565
$95$	0	0
$96$	0.305731	0.0312035
$97$	5.25271	0.533332	0.266666	−	0.963789i	$-0.414078\pi$
$97$	5.25271	0.533332	0.266666	+	0.963789i	$0.414078\pi$
$98$	−8.57023	−0.865724
$99$	9.30722	0.935410
$100$	0	0
$101$	17.0763	1.69915	0.849577	−	0.527464i	$-0.176857\pi$
$101$	17.0763	1.69915	0.849577	+	0.527464i	$0.176857\pi$
$102$	−1.18808	−0.117638
$103$	−0.810478	−0.0798588	−0.0399294	−	0.999203i	$-0.512713\pi$
$103$	−0.810478	−0.0798588	−0.0399294	+	0.999203i	$0.512713\pi$
$104$	0	0
$105$	0	0
$106$	4.15997	0.404052
$107$	−13.3870	−1.29417	−0.647087	−	0.762416i	$-0.724013\pi$
$107$	−13.3870	−1.29417	−0.647087	+	0.762416i	$0.724013\pi$
$108$	−0.316188	−0.0304252
$109$	5.61232	0.537562	0.268781	−	0.963201i	$-0.413379\pi$
$109$	5.61232	0.537562	0.268781	+	0.963201i	$0.413379\pi$
$110$	0	0
$111$	4.34214	0.412138
$112$	13.5213	1.27764
$113$	18.2323	1.71515	0.857575	−	0.514359i	$-0.171970\pi$
$113$	18.2323	1.71515	0.857575	+	0.514359i	$0.171970\pi$
$114$	0.468381	0.0438679
$115$	0	0
$116$	0.414858	0.0385186
$117$	0	0
$118$	18.1923	1.67474
$119$	7.92717	0.726683
$120$	0	0
$121$	−0.260531	−0.0236846
$122$	17.7934	1.61094
$123$	3.04475	0.274536
$124$	1.13968	0.102346
$125$	0	0
$126$	−14.1306	−1.25885
$127$	16.9703	1.50587	0.752933	−	0.658097i	$-0.228638\pi$
$127$	16.9703	1.50587	0.752933	+	0.658097i	$0.228638\pi$
$128$	−10.0311	−0.886630
$129$	−1.53411	−0.135071
$130$	0	0
$131$	−17.7226	−1.54843	−0.774215	−	0.632923i	$-0.781855\pi$
$131$	−17.7226	−1.54843	−0.774215	+	0.632923i	$0.781855\pi$
$132$	−0.177427	−0.0154430
$133$	−3.12515	−0.270985
$134$	−6.34681	−0.548280
$135$	0	0
$136$	−6.34383	−0.543979
$137$	−16.1087	−1.37626	−0.688129	−	0.725589i	$-0.741568\pi$
$137$	−16.1087	−1.37626	−0.688129	+	0.725589i	$0.741568\pi$
$138$	−0.417788	−0.0355645
$139$	−3.43095	−0.291009	−0.145505	−	0.989358i	$-0.546481\pi$
$139$	−3.43095	−0.291009	−0.145505	+	0.989358i	$0.546481\pi$
$140$	0	0
$141$	−1.87282	−0.157720
$142$	14.7962	1.24167
$143$	0	0
$144$	10.5392	0.878269
$145$	0	0
$146$	10.4914	0.868273
$147$	−2.50997	−0.207019
$148$	1.46991	0.120826
$149$	−18.6604	−1.52872	−0.764358	−	0.644792i	$-0.776944\pi$
$149$	−18.6604	−1.52872	−0.764358	+	0.644792i	$0.776944\pi$
$150$	0	0
$151$	6.63134	0.539651	0.269826	−	0.962909i	$-0.413034\pi$
$151$	6.63134	0.539651	0.269826	+	0.962909i	$0.413034\pi$
$152$	2.50095	0.202853
$153$	6.17888	0.499533
$154$	−16.3051	−1.31390
$155$	0	0
$156$	0	0
$157$	4.61502	0.368319	0.184159	−	0.982896i	$-0.441044\pi$
$157$	4.61502	0.368319	0.184159	+	0.982896i	$0.441044\pi$
$158$	13.9427	1.10923
$159$	1.21833	0.0966200
$160$	0	0
$161$	2.78758	0.219692
$162$	−10.3590	−0.813880
$163$	−9.75031	−0.763703	−0.381852	−	0.924224i	$-0.624714\pi$
$163$	−9.75031	−0.763703	−0.381852	+	0.924224i	$0.624714\pi$
$164$	1.03072	0.0804853
$165$	0	0
$166$	−1.61219	−0.125130
$167$	−10.3849	−0.803605	−0.401803	−	0.915726i	$-0.631616\pi$
$167$	−10.3849	−0.803605	−0.401803	+	0.915726i	$0.631616\pi$
$168$	4.24892	0.327811
$169$	0	0
$170$	0	0
$171$	−2.43592	−0.186279
$172$	−0.519330	−0.0395985
$173$	15.2909	1.16254	0.581271	−	0.813710i	$-0.302556\pi$
$173$	15.2909	1.16254	0.581271	+	0.813710i	$0.302556\pi$
$174$	−1.67343	−0.126862
$175$	0	0
$176$	12.1611	0.916675
$177$	5.32799	0.400476
$178$	4.45509	0.333923
$179$	11.6061	0.867484	0.433742	−	0.901037i	$-0.357193\pi$
$179$	11.6061	0.867484	0.433742	+	0.901037i	$0.357193\pi$
$180$	0	0
$181$	−4.01543	−0.298464	−0.149232	−	0.988802i	$-0.547680\pi$
$181$	−4.01543	−0.298464	−0.149232	+	0.988802i	$0.547680\pi$
$182$	0	0
$183$	5.21117	0.385221
$184$	−2.23080	−0.164457
$185$	0	0
$186$	−4.59718	−0.337082
$187$	7.12972	0.521377
$188$	−0.633993	−0.0462387
$189$	−8.50992	−0.619005
$190$	0	0
$191$	−1.92899	−0.139577	−0.0697884	−	0.997562i	$-0.522232\pi$
$191$	−1.92899	−0.139577	−0.0697884	+	0.997562i	$0.522232\pi$
$192$	−3.38560	−0.244334
$193$	−5.32239	−0.383114	−0.191557	−	0.981481i	$-0.561354\pi$
$193$	−5.32239	−0.383114	−0.191557	+	0.981481i	$0.561354\pi$
$194$	−7.17263	−0.514965
$195$	0	0
$196$	−0.849679	−0.0606914
$197$	−15.3428	−1.09313	−0.546566	−	0.837416i	$-0.684065\pi$
$197$	−15.3428	−1.09313	−0.546566	+	0.837416i	$0.684065\pi$
$198$	−12.7091	−0.903197
$199$	21.2871	1.50900	0.754500	−	0.656300i	$-0.227880\pi$
$199$	21.2871	1.50900	0.754500	+	0.656300i	$0.227880\pi$
$200$	0	0
$201$	−1.85879	−0.131109
$202$	−23.3179	−1.64064
$203$	11.1655	0.783665
$204$	−0.117790	−0.00824697
$205$	0	0
$206$	1.10672	0.0771086
$207$	2.17280	0.151020
$208$	0	0
$209$	−2.81077	−0.194425
$210$	0	0
$211$	−13.6417	−0.939131	−0.469566	−	0.882898i	$-0.655589\pi$
$211$	−13.6417	−0.939131	−0.469566	+	0.882898i	$0.655589\pi$
$212$	0.412432	0.0283260
$213$	4.33336	0.296917
$214$	18.2801	1.24960
$215$	0	0
$216$	6.81018	0.463374
$217$	30.6735	2.08225
$218$	−7.66368	−0.519050
$219$	3.07262	0.207628
$220$	0	0
$221$	0	0
$222$	−5.92924	−0.397945
$223$	−24.7121	−1.65484	−0.827422	−	0.561581i	$-0.810193\pi$
$223$	−24.7121	−1.65484	−0.827422	+	0.561581i	$0.810193\pi$
$224$	2.78551	0.186115
$225$	0	0
$226$	−24.8964	−1.65608
$227$	11.8182	0.784403	0.392201	−	0.919879i	$-0.371714\pi$
$227$	11.8182	0.784403	0.392201	+	0.919879i	$0.371714\pi$
$228$	0.0464368	0.00307535
$229$	−2.98379	−0.197174	−0.0985872	−	0.995128i	$-0.531432\pi$
$229$	−2.98379	−0.197174	−0.0985872	+	0.995128i	$0.531432\pi$
$230$	0	0
$231$	−4.77528	−0.314190
$232$	−8.93536	−0.586635
$233$	−6.88071	−0.450770	−0.225385	−	0.974270i	$-0.572364\pi$
$233$	−6.88071	−0.450770	−0.225385	+	0.974270i	$0.572364\pi$
$234$	0	0
$235$	0	0
$236$	1.80364	0.117407
$237$	4.08342	0.265246
$238$	−10.8246	−0.701657
$239$	18.3427	1.18649	0.593246	−	0.805021i	$-0.297846\pi$
$239$	18.3427	1.18649	0.593246	+	0.805021i	$0.297846\pi$
$240$	0	0
$241$	16.6366	1.07165	0.535827	−	0.844328i	$-0.320000\pi$
$241$	16.6366	1.07165	0.535827	+	0.844328i	$0.320000\pi$
$242$	0.355758	0.0228690
$243$	−10.0405	−0.644097
$244$	1.76410	0.112935
$245$	0	0
$246$	−4.15764	−0.265081
$247$	0	0
$248$	−24.5469	−1.55873
$249$	−0.472161	−0.0299220
$250$	0	0
$251$	−2.80970	−0.177347	−0.0886733	−	0.996061i	$-0.528263\pi$
$251$	−2.80970	−0.177347	−0.0886733	+	0.996061i	$0.528263\pi$
$252$	−1.40095	−0.0882516
$253$	2.50716	0.157624
$254$	−23.1731	−1.45401
$255$	0	0
$256$	−3.23392	−0.202120
$257$	−3.31893	−0.207029	−0.103515	−	0.994628i	$-0.533009\pi$
$257$	−3.31893	−0.207029	−0.103515	+	0.994628i	$0.533009\pi$
$258$	2.09484	0.130419
$259$	39.5613	2.45822
$260$	0	0
$261$	8.70302	0.538703
$262$	24.2004	1.49510
$263$	24.9618	1.53921	0.769606	−	0.638519i	$-0.220453\pi$
$263$	24.9618	1.53921	0.769606	+	0.638519i	$0.220453\pi$
$264$	3.82149	0.235196
$265$	0	0
$266$	4.26743	0.261653
$267$	1.30476	0.0798502
$268$	−0.629242	−0.0384371
$269$	11.5367	0.703407	0.351704	−	0.936111i	$-0.385602\pi$
$269$	11.5367	0.703407	0.351704	+	0.936111i	$0.385602\pi$
$270$	0	0
$271$	19.9817	1.21380	0.606902	−	0.794777i	$-0.292412\pi$
$271$	19.9817	1.21380	0.606902	+	0.794777i	$0.292412\pi$
$272$	8.07350	0.489528
$273$	0	0
$274$	21.9966	1.32886
$275$	0	0
$276$	−0.0414208	−0.00249324
$277$	6.12106	0.367779	0.183889	−	0.982947i	$-0.441131\pi$
$277$	6.12106	0.367779	0.183889	+	0.982947i	$0.441131\pi$
$278$	4.68499	0.280987
$279$	23.9086	1.43137
$280$	0	0
$281$	−25.6283	−1.52885	−0.764427	−	0.644711i	$-0.776978\pi$
$281$	−25.6283	−1.52885	−0.764427	+	0.644711i	$0.776978\pi$
$282$	2.55736	0.152289
$283$	17.6824	1.05111	0.525553	−	0.850761i	$-0.323858\pi$
$283$	17.6824	1.05111	0.525553	+	0.850761i	$0.323858\pi$
$284$	1.46694	0.0870467
$285$	0	0
$286$	0	0
$287$	27.7407	1.63748
$288$	2.17118	0.127938
$289$	−12.2667	−0.721571
$290$	0	0
$291$	−2.10065	−0.123142
$292$	1.04015	0.0608701
$293$	22.0044	1.28551	0.642757	−	0.766070i	$-0.277791\pi$
$293$	22.0044	1.28551	0.642757	+	0.766070i	$0.277791\pi$
$294$	3.42739	0.199889
$295$	0	0
$296$	−31.6595	−1.84017
$297$	−7.65385	−0.444121
$298$	25.4809	1.47607
$299$	0	0
$300$	0	0
$301$	−13.9773	−0.805637
$302$	−9.05517	−0.521067
$303$	−6.82911	−0.392322
$304$	−3.18284	−0.182548
$305$	0	0
$306$	−8.43733	−0.482330
$307$	1.63084	0.0930769	0.0465384	−	0.998917i	$-0.485181\pi$
$307$	1.63084	0.0930769	0.0465384	+	0.998917i	$0.485181\pi$
$308$	−1.61654	−0.0921108
$309$	0.324124	0.0184388
$310$	0	0
$311$	−19.0053	−1.07769	−0.538845	−	0.842405i	$-0.681139\pi$
$311$	−19.0053	−1.07769	−0.538845	+	0.842405i	$0.681139\pi$
$312$	0	0
$313$	−11.1004	−0.627434	−0.313717	−	0.949517i	$-0.601574\pi$
$313$	−11.1004	−0.627434	−0.313717	+	0.949517i	$0.601574\pi$
$314$	−6.30186	−0.355635
$315$	0	0
$316$	1.38233	0.0777620
$317$	−5.22062	−0.293219	−0.146610	−	0.989194i	$-0.546836\pi$
$317$	−5.22062	−0.293219	−0.146610	+	0.989194i	$0.546836\pi$
$318$	−1.66364	−0.0932926
$319$	10.0423	0.562260
$320$	0	0
$321$	5.35371	0.298815
$322$	−3.80647	−0.212126
$323$	−1.86602	−0.103828
$324$	−1.02702	−0.0570568
$325$	0	0
$326$	13.3142	0.737403
$327$	−2.24446	−0.124119
$328$	−22.1999	−1.22579
$329$	−17.0633	−0.940731
$330$	0	0
$331$	25.6375	1.40916	0.704582	−	0.709623i	$-0.251135\pi$
$331$	25.6375	1.40916	0.704582	+	0.709623i	$0.251135\pi$
$332$	−0.159837	−0.00877220
$333$	30.8363	1.68982
$334$	14.1806	0.775930
$335$	0	0
$336$	−5.40739	−0.294998
$337$	−36.1447	−1.96892	−0.984462	−	0.175596i	$-0.943815\pi$
$337$	−36.1447	−1.96892	−0.984462	+	0.175596i	$0.943815\pi$
$338$	0	0
$339$	−7.29142	−0.396015
$340$	0	0
$341$	27.5878	1.49396
$342$	3.32627	0.179864
$343$	2.63725	0.142398
$344$	11.1855	0.603082
$345$	0	0
$346$	−20.8798	−1.12251
$347$	14.4108	0.773613	0.386806	−	0.922161i	$-0.373578\pi$
$347$	14.4108	0.773613	0.386806	+	0.922161i	$0.373578\pi$
$348$	−0.165909	−0.00889365
$349$	3.33936	0.178752	0.0893758	−	0.995998i	$-0.471513\pi$
$349$	3.33936	0.178752	0.0893758	+	0.995998i	$0.471513\pi$
$350$	0	0
$351$	0	0
$352$	2.50530	0.133533
$353$	11.6806	0.621694	0.310847	−	0.950460i	$-0.399387\pi$
$353$	11.6806	0.621694	0.310847	+	0.950460i	$0.399387\pi$
$354$	−7.27543	−0.386685
$355$	0	0
$356$	0.441691	0.0234096
$357$	−3.17022	−0.167786
$358$	−15.8483	−0.837609
$359$	6.54679	0.345526	0.172763	−	0.984963i	$-0.444730\pi$
$359$	6.54679	0.345526	0.172763	+	0.984963i	$0.444730\pi$
$360$	0	0
$361$	−18.2644	−0.961282
$362$	5.48311	0.288186
$363$	0.104191	0.00546861
$364$	0	0
$365$	0	0
$366$	−7.11591	−0.371955
$367$	−22.6218	−1.18085	−0.590425	−	0.807093i	$-0.701040\pi$
$367$	−22.6218	−1.18085	−0.590425	+	0.807093i	$0.701040\pi$
$368$	2.83904	0.147995
$369$	21.6227	1.12563
$370$	0	0
$371$	11.1002	0.576295
$372$	−0.455779	−0.0236310
$373$	−4.25166	−0.220142	−0.110071	−	0.993924i	$-0.535108\pi$
$373$	−4.25166	−0.220142	−0.110071	+	0.993924i	$0.535108\pi$
$374$	−9.73571	−0.503422
$375$	0	0
$376$	13.6552	0.704211
$377$	0	0
$378$	11.6204	0.597688
$379$	11.1415	0.572299	0.286150	−	0.958185i	$-0.407625\pi$
$379$	11.1415	0.572299	0.286150	+	0.958185i	$0.407625\pi$
$380$	0	0
$381$	−6.78670	−0.347693
$382$	2.63406	0.134770
$383$	5.48926	0.280488	0.140244	−	0.990117i	$-0.455211\pi$
$383$	5.48926	0.280488	0.140244	+	0.990117i	$0.455211\pi$
$384$	4.01161	0.204716
$385$	0	0
$386$	7.26779	0.369921
$387$	−10.8947	−0.553807
$388$	−0.711117	−0.0361015
$389$	−25.9475	−1.31559	−0.657794	−	0.753198i	$-0.728510\pi$
$389$	−25.9475	−1.31559	−0.657794	+	0.753198i	$0.728510\pi$
$390$	0	0
$391$	1.66445	0.0841751
$392$	18.3007	0.924325
$393$	7.08757	0.357521
$394$	20.9508	1.05549
$395$	0	0
$396$	−1.26002	−0.0633184
$397$	−13.8706	−0.696145	−0.348073	−	0.937468i	$-0.613164\pi$
$397$	−13.8706	−0.696145	−0.348073	+	0.937468i	$0.613164\pi$
$398$	−29.0677	−1.45703
$399$	1.24980	0.0625684
$400$	0	0
$401$	1.26330	0.0630861	0.0315430	−	0.999502i	$-0.489958\pi$
$401$	1.26330	0.0630861	0.0315430	+	0.999502i	$0.489958\pi$
$402$	2.53820	0.126594
$403$	0	0
$404$	−2.31181	−0.115017
$405$	0	0
$406$	−15.2466	−0.756677
$407$	35.5816	1.76371
$408$	2.53701	0.125601
$409$	−22.0486	−1.09023	−0.545117	−	0.838360i	$-0.683515\pi$
$409$	−22.0486	−1.09023	−0.545117	+	0.838360i	$0.683515\pi$
$410$	0	0
$411$	6.44215	0.317768
$412$	0.109723	0.00540568
$413$	48.5434	2.38866
$414$	−2.96698	−0.145819
$415$	0	0
$416$	0	0
$417$	1.37210	0.0671918
$418$	3.83814	0.187729
$419$	11.6804	0.570624	0.285312	−	0.958435i	$-0.407903\pi$
$419$	11.6804	0.570624	0.285312	+	0.958435i	$0.407903\pi$
$420$	0	0
$421$	21.9773	1.07111	0.535555	−	0.844501i	$-0.320103\pi$
$421$	21.9773	1.07111	0.535555	+	0.844501i	$0.320103\pi$
$422$	18.6278	0.906789
$423$	−13.3001	−0.646673
$424$	−8.88312	−0.431402
$425$	0	0
$426$	−5.91724	−0.286691
$427$	47.4790	2.29767
$428$	1.81235	0.0876032
$429$	0	0
$430$	0	0
$431$	24.1166	1.16166	0.580828	−	0.814027i	$-0.302729\pi$
$431$	24.1166	1.16166	0.580828	+	0.814027i	$0.302729\pi$
$432$	−8.66700	−0.416991
$433$	6.69180	0.321588	0.160794	−	0.986988i	$-0.448595\pi$
$433$	6.69180	0.321588	0.160794	+	0.986988i	$0.448595\pi$
$434$	−41.8850	−2.01054
$435$	0	0
$436$	−0.759801	−0.0363878
$437$	−0.656182	−0.0313894
$438$	−4.19569	−0.200478
$439$	−27.9775	−1.33529	−0.667646	−	0.744479i	$-0.732698\pi$
$439$	−27.9775	−1.33529	−0.667646	+	0.744479i	$0.732698\pi$
$440$	0	0
$441$	−17.8248	−0.848802
$442$	0	0
$443$	19.4755	0.925309	0.462655	−	0.886539i	$-0.346897\pi$
$443$	19.4755	0.925309	0.462655	+	0.886539i	$0.346897\pi$
$444$	−0.587843	−0.0278978
$445$	0	0
$446$	33.7446	1.59785
$447$	7.46261	0.352969
$448$	−30.8462	−1.45735
$449$	13.1870	0.622332	0.311166	−	0.950356i	$-0.399280\pi$
$449$	13.1870	0.622332	0.311166	+	0.950356i	$0.399280\pi$
$450$	0	0
$451$	24.9501	1.17486
$452$	−2.46831	−0.116099
$453$	−2.65199	−0.124601
$454$	−16.1379	−0.757389
$455$	0	0
$456$	−1.00017	−0.0468374
$457$	−13.8580	−0.648248	−0.324124	−	0.946015i	$-0.605070\pi$
$457$	−13.8580	−0.648248	−0.324124	+	0.946015i	$0.605070\pi$
$458$	4.07440	0.190384
$459$	−5.08124	−0.237172
$460$	0	0
$461$	14.0669	0.655159	0.327580	−	0.944824i	$-0.393767\pi$
$461$	14.0669	0.655159	0.327580	+	0.944824i	$0.393767\pi$
$462$	6.52070	0.303370
$463$	4.86555	0.226121	0.113061	−	0.993588i	$-0.463935\pi$
$463$	4.86555	0.226121	0.113061	+	0.993588i	$0.463935\pi$
$464$	11.3716	0.527914
$465$	0	0
$466$	9.39568	0.435246
$467$	26.6409	1.23279	0.616397	−	0.787436i	$-0.288592\pi$
$467$	26.6409	1.23279	0.616397	+	0.787436i	$0.288592\pi$
$468$	0	0
$469$	−16.9355	−0.782007
$470$	0	0
$471$	−1.84563	−0.0850421
$472$	−38.8475	−1.78810
$473$	−12.5712	−0.578024
$474$	−5.57595	−0.256112
$475$	0	0
$476$	−1.07319	−0.0491895
$477$	8.65213	0.396154
$478$	−25.0472	−1.14563
$479$	3.02307	0.138127	0.0690637	−	0.997612i	$-0.477999\pi$
$479$	3.02307	0.138127	0.0690637	+	0.997612i	$0.477999\pi$
$480$	0	0
$481$	0	0
$482$	−22.7174	−1.03475
$483$	−1.11480	−0.0507253
$484$	0.0352709	0.00160322
$485$	0	0
$486$	13.7104	0.621916
$487$	11.8201	0.535618	0.267809	−	0.963472i	$-0.413700\pi$
$487$	11.8201	0.535618	0.267809	+	0.963472i	$0.413700\pi$
$488$	−37.9958	−1.71999
$489$	3.89932	0.176333
$490$	0	0
$491$	9.64629	0.435331	0.217665	−	0.976023i	$-0.430156\pi$
$491$	9.64629	0.435331	0.217665	+	0.976023i	$0.430156\pi$
$492$	−0.412201	−0.0185835
$493$	6.66688	0.300261
$494$	0	0
$495$	0	0
$496$	31.2397	1.40270
$497$	39.4812	1.77098
$498$	0.644741	0.0288916
$499$	10.0807	0.451274	0.225637	−	0.974211i	$-0.427554\pi$
$499$	10.0807	0.451274	0.225637	+	0.974211i	$0.427554\pi$
$500$	0	0
$501$	4.15309	0.185546
$502$	3.83667	0.171239
$503$	16.7467	0.746700	0.373350	−	0.927690i	$-0.378209\pi$
$503$	16.7467	0.746700	0.373350	+	0.927690i	$0.378209\pi$
$504$	30.1742	1.34407
$505$	0	0
$506$	−3.42355	−0.152195
$507$	0	0
$508$	−2.29745	−0.101933
$509$	31.4625	1.39455	0.697276	−	0.716803i	$-0.254395\pi$
$509$	31.4625	1.39455	0.697276	+	0.716803i	$0.254395\pi$
$510$	0	0
$511$	27.9946	1.23841
$512$	24.4781	1.08179
$513$	2.00319	0.0884430
$514$	4.53203	0.199899
$515$	0	0
$516$	0.207689	0.00914300
$517$	−15.3468	−0.674952
$518$	−54.0214	−2.37356
$519$	−6.11508	−0.268422
$520$	0	0
$521$	6.44923	0.282546	0.141273	−	0.989971i	$-0.454880\pi$
$521$	6.44923	0.282546	0.141273	+	0.989971i	$0.454880\pi$
$522$	−11.8841	−0.520151
$523$	−11.6815	−0.510798	−0.255399	−	0.966836i	$-0.582207\pi$
$523$	−11.6815	−0.510798	−0.255399	+	0.966836i	$0.582207\pi$
$524$	2.39930	0.104814
$525$	0	0
$526$	−34.0856	−1.48620
$527$	18.3150	0.797815
$528$	−4.86343	−0.211654
$529$	−22.4147	−0.974552
$530$	0	0
$531$	37.8374	1.64200
$532$	0.423086	0.0183431
$533$	0	0
$534$	−1.78167	−0.0771003
$535$	0	0
$536$	13.5528	0.585394
$537$	−4.64150	−0.200296
$538$	−15.7535	−0.679183
$539$	−20.5678	−0.885920
$540$	0	0
$541$	−10.5675	−0.454330	−0.227165	−	0.973856i	$-0.572946\pi$
$541$	−10.5675	−0.454330	−0.227165	+	0.973856i	$0.572946\pi$
$542$	−27.2852	−1.17200
$543$	1.60584	0.0689132
$544$	1.66322	0.0713099
$545$	0	0
$546$	0	0
$547$	11.1484	0.476669	0.238335	−	0.971183i	$-0.423399\pi$
$547$	11.1484	0.476669	0.238335	+	0.971183i	$0.423399\pi$
$548$	2.18081	0.0931595
$549$	37.0078	1.57945
$550$	0	0
$551$	−2.62830	−0.111969
$552$	0.892137	0.0379719
$553$	37.2041	1.58208
$554$	−8.35837	−0.355113
$555$	0	0
$556$	0.464485	0.0196985
$557$	−12.4354	−0.526905	−0.263453	−	0.964672i	$-0.584861\pi$
$557$	−12.4354	−0.526905	−0.263453	+	0.964672i	$0.584861\pi$
$558$	−32.6475	−1.38208
$559$	0	0
$560$	0	0
$561$	−2.85130	−0.120382
$562$	34.9957	1.47620
$563$	−17.5547	−0.739844	−0.369922	−	0.929063i	$-0.620616\pi$
$563$	−17.5547	−0.739844	−0.369922	+	0.929063i	$0.620616\pi$
$564$	0.253545	0.0106762
$565$	0	0
$566$	−24.1454	−1.01491
$567$	−27.6414	−1.16083
$568$	−31.5954	−1.32571
$569$	−30.6169	−1.28353	−0.641763	−	0.766903i	$-0.721797\pi$
$569$	−30.6169	−1.28353	−0.641763	+	0.766903i	$0.721797\pi$
$570$	0	0
$571$	5.33906	0.223433	0.111716	−	0.993740i	$-0.464365\pi$
$571$	5.33906	0.223433	0.111716	+	0.993740i	$0.464365\pi$
$572$	0	0
$573$	0.771438	0.0322273
$574$	−37.8803	−1.58109
$575$	0	0
$576$	−24.0432	−1.00180
$577$	2.82977	0.117805	0.0589024	−	0.998264i	$-0.481240\pi$
$577$	2.82977	0.117805	0.0589024	+	0.998264i	$0.481240\pi$
$578$	16.7503	0.696722
$579$	2.12852	0.0884583
$580$	0	0
$581$	−4.30187	−0.178471
$582$	2.86846	0.118902
$583$	9.98358	0.413478
$584$	−22.4031	−0.927047
$585$	0	0
$586$	−30.0473	−1.24124
$587$	8.59517	0.354761	0.177380	−	0.984142i	$-0.443238\pi$
$587$	8.59517	0.354761	0.177380	+	0.984142i	$0.443238\pi$
$588$	0.339802	0.0140132
$589$	−7.22038	−0.297510
$590$	0	0
$591$	6.13588	0.252396
$592$	40.2915	1.65597
$593$	−15.6269	−0.641720	−0.320860	−	0.947127i	$-0.603972\pi$
$593$	−15.6269	−0.641720	−0.320860	+	0.947127i	$0.603972\pi$
$594$	10.4514	0.428826
$595$	0	0
$596$	2.52626	0.103480
$597$	−8.51307	−0.348417
$598$	0	0
$599$	5.84438	0.238795	0.119398	−	0.992847i	$-0.461904\pi$
$599$	5.84438	0.238795	0.119398	+	0.992847i	$0.461904\pi$
$600$	0	0
$601$	23.7515	0.968843	0.484421	−	0.874835i	$-0.339030\pi$
$601$	23.7515	0.968843	0.484421	+	0.874835i	$0.339030\pi$
$602$	19.0861	0.777892
$603$	−13.2004	−0.537564
$604$	−0.897758	−0.0365292
$605$	0	0
$606$	9.32522	0.378811
$607$	−37.3079	−1.51428	−0.757141	−	0.653251i	$-0.773404\pi$
$607$	−37.3079	−1.51428	−0.757141	+	0.653251i	$0.773404\pi$
$608$	−0.655695	−0.0265920
$609$	−4.46528	−0.180942
$610$	0	0
$611$	0	0
$612$	−0.836503	−0.0338136
$613$	−41.4546	−1.67434	−0.837168	−	0.546946i	$-0.815790\pi$
$613$	−41.4546	−1.67434	−0.837168	+	0.546946i	$0.815790\pi$
$614$	−2.22693	−0.0898715
$615$	0	0
$616$	34.8176	1.40284
$617$	−9.11172	−0.366824	−0.183412	−	0.983036i	$-0.558714\pi$
$617$	−9.11172	−0.366824	−0.183412	+	0.983036i	$0.558714\pi$
$618$	−0.442595	−0.0178038
$619$	−33.6828	−1.35382	−0.676912	−	0.736064i	$-0.736682\pi$
$619$	−33.6828	−1.35382	−0.676912	+	0.736064i	$0.736682\pi$
$620$	0	0
$621$	−1.78681	−0.0717023
$622$	25.9519	1.04058
$623$	11.8877	0.476271
$624$	0	0
$625$	0	0
$626$	15.1578	0.605826
$627$	1.12408	0.0448913
$628$	−0.624786	−0.0249317
$629$	23.6219	0.941867
$630$	0	0
$631$	25.8213	1.02793	0.513965	−	0.857811i	$-0.328176\pi$
$631$	25.8213	1.02793	0.513965	+	0.857811i	$0.328176\pi$
$632$	−29.7731	−1.18431
$633$	5.45554	0.216838
$634$	7.12881	0.283121
$635$	0	0
$636$	−0.164939	−0.00654025
$637$	0	0
$638$	−13.7129	−0.542897
$639$	30.7739	1.21740
$640$	0	0
$641$	13.2356	0.522773	0.261387	−	0.965234i	$-0.415820\pi$
$641$	13.2356	0.522773	0.261387	+	0.965234i	$0.415820\pi$
$642$	−7.31055	−0.288524
$643$	−22.8723	−0.901995	−0.450998	−	0.892525i	$-0.648932\pi$
$643$	−22.8723	−0.901995	−0.450998	+	0.892525i	$0.648932\pi$
$644$	−0.377385	−0.0148711
$645$	0	0
$646$	2.54806	0.100252
$647$	−4.26033	−0.167491	−0.0837454	−	0.996487i	$-0.526688\pi$
$647$	−4.26033	−0.167491	−0.0837454	+	0.996487i	$0.526688\pi$
$648$	22.1204	0.868972
$649$	43.6601	1.71381
$650$	0	0
$651$	−12.2669	−0.480776
$652$	1.32001	0.0516955
$653$	12.6642	0.495589	0.247794	−	0.968813i	$-0.420294\pi$
$653$	12.6642	0.495589	0.247794	+	0.968813i	$0.420294\pi$
$654$	3.06484	0.119845
$655$	0	0
$656$	28.2528	1.10309
$657$	21.8206	0.851302
$658$	23.3001	0.908334
$659$	−27.2660	−1.06213	−0.531066	−	0.847331i	$-0.678208\pi$
$659$	−27.2660	−1.06213	−0.531066	+	0.847331i	$0.678208\pi$
$660$	0	0
$661$	4.55273	0.177081	0.0885403	−	0.996073i	$-0.471780\pi$
$661$	4.55273	0.177081	0.0885403	+	0.996073i	$0.471780\pi$
$662$	−35.0082	−1.36063
$663$	0	0
$664$	3.44263	0.133600
$665$	0	0
$666$	−42.1073	−1.63162
$667$	2.34440	0.0907756
$668$	1.40591	0.0543964
$669$	9.88280	0.382091
$670$	0	0
$671$	42.7028	1.64852
$672$	−1.11398	−0.0429726
$673$	−20.6613	−0.796434	−0.398217	−	0.917291i	$-0.630371\pi$
$673$	−20.6613	−0.796434	−0.398217	+	0.917291i	$0.630371\pi$
$674$	49.3559	1.90112
$675$	0	0
$676$	0	0
$677$	−36.1528	−1.38947	−0.694733	−	0.719268i	$-0.744477\pi$
$677$	−36.1528	−1.38947	−0.694733	+	0.719268i	$0.744477\pi$
$678$	9.95651	0.382377
$679$	−19.1391	−0.734490
$680$	0	0
$681$	−4.72632	−0.181113
$682$	−37.6715	−1.44251
$683$	36.6280	1.40153	0.700765	−	0.713392i	$-0.252842\pi$
$683$	36.6280	1.40153	0.700765	+	0.713392i	$0.252842\pi$
$684$	0.329777	0.0126093
$685$	0	0
$686$	−3.60119	−0.137494
$687$	1.19327	0.0455261
$688$	−14.2353	−0.542715
$689$	0	0
$690$	0	0
$691$	−29.3931	−1.11816	−0.559082	−	0.829112i	$-0.688846\pi$
$691$	−29.3931	−1.11816	−0.559082	+	0.829112i	$0.688846\pi$
$692$	−2.07009	−0.0786930
$693$	−33.9123	−1.28822
$694$	−19.6781	−0.746971
$695$	0	0
$696$	3.57341	0.135450
$697$	16.5639	0.627402
$698$	−4.55993	−0.172596
$699$	2.75172	0.104079
$700$	0	0
$701$	19.8074	0.748116	0.374058	−	0.927405i	$-0.377966\pi$
$701$	19.8074	0.748116	0.374058	+	0.927405i	$0.377966\pi$
$702$	0	0
$703$	−9.31252	−0.351228
$704$	−27.7432	−1.04561
$705$	0	0
$706$	−15.9499	−0.600284
$707$	−62.2201	−2.34003
$708$	−0.721309	−0.0271084
$709$	13.4434	0.504877	0.252438	−	0.967613i	$-0.418768\pi$
$709$	13.4434	0.504877	0.252438	+	0.967613i	$0.418768\pi$
$710$	0	0
$711$	28.9989	1.08754
$712$	−9.51331	−0.356526
$713$	6.44045	0.241197
$714$	4.32896	0.162007
$715$	0	0
$716$	−1.57125	−0.0587204
$717$	−7.33558	−0.273952
$718$	−8.93971	−0.333627
$719$	−14.2224	−0.530404	−0.265202	−	0.964193i	$-0.585439\pi$
$719$	−14.2224	−0.530404	−0.265202	+	0.964193i	$0.585439\pi$
$720$	0	0
$721$	2.95310	0.109979
$722$	24.9402	0.928177
$723$	−6.65325	−0.247437
$724$	0.543612	0.0202032
$725$	0	0
$726$	−0.142274	−0.00528028
$727$	−47.6220	−1.76620	−0.883101	−	0.469183i	$-0.844548\pi$
$727$	−47.6220	−1.76620	−0.883101	+	0.469183i	$0.844548\pi$
$728$	0	0
$729$	−18.7431	−0.694191
$730$	0	0
$731$	−8.34578	−0.308680
$732$	−0.705494	−0.0260758
$733$	−16.0484	−0.592762	−0.296381	−	0.955070i	$-0.595780\pi$
$733$	−16.0484	−0.592762	−0.296381	+	0.955070i	$0.595780\pi$
$734$	30.8903	1.14018
$735$	0	0
$736$	0.584869	0.0215586
$737$	−15.2318	−0.561071
$738$	−29.5260	−1.08687
$739$	9.02452	0.331972	0.165986	−	0.986128i	$-0.446919\pi$
$739$	9.02452	0.331972	0.165986	+	0.986128i	$0.446919\pi$
$740$	0	0
$741$	0	0
$742$	−15.1575	−0.556449
$743$	−36.9372	−1.35509	−0.677547	−	0.735480i	$-0.736957\pi$
$743$	−36.9372	−1.35509	−0.677547	+	0.735480i	$0.736957\pi$
$744$	9.81673	0.359899
$745$	0	0
$746$	5.80568	0.212561
$747$	−3.35311	−0.122684
$748$	−0.965229	−0.0352923
$749$	48.7777	1.78230
$750$	0	0
$751$	4.59877	0.167812	0.0839058	−	0.996474i	$-0.473261\pi$
$751$	4.59877	0.167812	0.0839058	+	0.996474i	$0.473261\pi$
$752$	−17.3783	−0.633721
$753$	1.12365	0.0409480
$754$	0	0
$755$	0	0
$756$	1.15208	0.0419008
$757$	−15.6853	−0.570092	−0.285046	−	0.958514i	$-0.592009\pi$
$757$	−15.6853	−0.570092	−0.285046	+	0.958514i	$0.592009\pi$
$758$	−15.2138	−0.552590
$759$	−1.00266	−0.0363942
$760$	0	0
$761$	−23.4099	−0.848608	−0.424304	−	0.905520i	$-0.639481\pi$
$761$	−23.4099	−0.848608	−0.424304	+	0.905520i	$0.639481\pi$
$762$	9.26731	0.335719
$763$	−20.4493	−0.740315
$764$	0.261149	0.00944803
$765$	0	0
$766$	−7.49564	−0.270829
$767$	0	0
$768$	1.29330	0.0466680
$769$	−28.6748	−1.03404	−0.517020	−	0.855974i	$-0.672959\pi$
$769$	−28.6748	−1.03404	−0.517020	+	0.855974i	$0.672959\pi$
$770$	0	0
$771$	1.32730	0.0478015
$772$	0.720551	0.0259332
$773$	28.5519	1.02694	0.513470	−	0.858108i	$-0.328360\pi$
$773$	28.5519	1.02694	0.513470	+	0.858108i	$0.328360\pi$
$774$	14.8768	0.534735
$775$	0	0
$776$	15.3163	0.549823
$777$	−15.8213	−0.567585
$778$	35.4315	1.27028
$779$	−6.53003	−0.233963
$780$	0	0
$781$	35.5096	1.27063
$782$	−2.27283	−0.0812762
$783$	−7.15698	−0.255770
$784$	−23.2904	−0.831802
$785$	0	0
$786$	−9.67816	−0.345208
$787$	−7.74259	−0.275993	−0.137997	−	0.990433i	$-0.544066\pi$
$787$	−7.74259	−0.275993	−0.137997	+	0.990433i	$0.544066\pi$
$788$	2.07713	0.0739947
$789$	−9.98268	−0.355393
$790$	0	0
$791$	−66.4322	−2.36206
$792$	27.1388	0.964334
$793$	0	0
$794$	18.9404	0.672171
$795$	0	0
$796$	−2.88186	−0.102145
$797$	−32.8491	−1.16357	−0.581787	−	0.813341i	$-0.697646\pi$
$797$	−32.8491	−1.16357	−0.581787	+	0.813341i	$0.697646\pi$
$798$	−1.70662	−0.0604137
$799$	−10.1884	−0.360441
$800$	0	0
$801$	9.26594	0.327396
$802$	−1.72505	−0.0609135
$803$	25.1785	0.888529
$804$	0.251645	0.00887484
$805$	0	0
$806$	0	0
$807$	−4.61375	−0.162412
$808$	49.7925	1.75169
$809$	−25.5043	−0.896682	−0.448341	−	0.893863i	$-0.647985\pi$
$809$	−25.5043	−0.896682	−0.448341	+	0.893863i	$0.647985\pi$
$810$	0	0
$811$	−2.12653	−0.0746726	−0.0373363	−	0.999303i	$-0.511887\pi$
$811$	−2.12653	−0.0746726	−0.0373363	+	0.999303i	$0.511887\pi$
$812$	−1.51160	−0.0530467
$813$	−7.99104	−0.280258
$814$	−48.5870	−1.70297
$815$	0	0
$816$	−3.22873	−0.113028
$817$	3.29018	0.115109
$818$	30.1076	1.05269
$819$	0	0
$820$	0	0
$821$	−15.6715	−0.546938	−0.273469	−	0.961881i	$-0.588171\pi$
$821$	−15.6715	−0.546938	−0.273469	+	0.961881i	$0.588171\pi$
$822$	−8.79682	−0.306824
$823$	34.4278	1.20008	0.600040	−	0.799970i	$-0.295151\pi$
$823$	34.4278	1.20008	0.600040	+	0.799970i	$0.295151\pi$
$824$	−2.36326	−0.0823281
$825$	0	0
$826$	−66.2865	−2.30640
$827$	−37.6478	−1.30914	−0.654572	−	0.756000i	$-0.727151\pi$
$827$	−37.6478	−1.30914	−0.654572	+	0.756000i	$0.727151\pi$
$828$	−0.294155	−0.0102226
$829$	−33.3338	−1.15773	−0.578865	−	0.815423i	$-0.696504\pi$
$829$	−33.3338	−1.15773	−0.578865	+	0.815423i	$0.696504\pi$
$830$	0	0
$831$	−2.44792	−0.0849174
$832$	0	0
$833$	−13.6546	−0.473104
$834$	−1.87361	−0.0648779
$835$	0	0
$836$	0.380525	0.0131607
$837$	−19.6614	−0.679597
$838$	−15.9497	−0.550973
$839$	−4.35384	−0.150311	−0.0751556	−	0.997172i	$-0.523945\pi$
$839$	−4.35384	−0.150311	−0.0751556	+	0.997172i	$0.523945\pi$
$840$	0	0
$841$	−19.6096	−0.676194
$842$	−30.0103	−1.03422
$843$	10.2492	0.353001
$844$	1.84682	0.0635702
$845$	0	0
$846$	18.1614	0.624403
$847$	0.949285	0.0326178
$848$	11.3051	0.388220
$849$	−7.07148	−0.242693
$850$	0	0
$851$	8.30661	0.284747
$852$	−0.586654	−0.0200984
$853$	11.2955	0.386751	0.193375	−	0.981125i	$-0.438056\pi$
$853$	11.2955	0.386751	0.193375	+	0.981125i	$0.438056\pi$
$854$	−64.8331	−2.21854
$855$	0	0
$856$	−39.0351	−1.33419
$857$	9.88043	0.337509	0.168754	−	0.985658i	$-0.446026\pi$
$857$	9.88043	0.337509	0.168754	+	0.985658i	$0.446026\pi$
$858$	0	0
$859$	0.0855154	0.00291775	0.00145887	−	0.999999i	$-0.499536\pi$
$859$	0.0855154	0.00291775	0.00145887	+	0.999999i	$0.499536\pi$
$860$	0	0
$861$	−11.0940	−0.378083
$862$	−32.9315	−1.12165
$863$	43.5513	1.48250	0.741252	−	0.671227i	$-0.234233\pi$
$863$	43.5513	1.48250	0.741252	+	0.671227i	$0.234233\pi$
$864$	−1.78549	−0.0607435
$865$	0	0
$866$	−9.13773	−0.310513
$867$	4.90568	0.166605
$868$	−4.15260	−0.140949
$869$	33.4614	1.13510
$870$	0	0
$871$	0	0
$872$	16.3649	0.554184
$873$	−14.9181	−0.504900
$874$	0.896024	0.0303085
$875$	0	0
$876$	−0.415974	−0.0140545
$877$	12.3438	0.416821	0.208411	−	0.978041i	$-0.433171\pi$
$877$	12.3438	0.416821	0.208411	+	0.978041i	$0.433171\pi$
$878$	38.2035	1.28931
$879$	−8.79997	−0.296816
$880$	0	0
$881$	50.5310	1.70243	0.851217	−	0.524814i	$-0.175865\pi$
$881$	50.5310	1.70243	0.851217	+	0.524814i	$0.175865\pi$
$882$	24.3400	0.819571
$883$	−23.8742	−0.803432	−0.401716	−	0.915764i	$-0.631586\pi$
$883$	−23.8742	−0.803432	−0.401716	+	0.915764i	$0.631586\pi$
$884$	0	0
$885$	0	0
$886$	−26.5940	−0.893443
$887$	36.3197	1.21950	0.609748	−	0.792595i	$-0.291271\pi$
$887$	36.3197	1.21950	0.609748	+	0.792595i	$0.291271\pi$
$888$	12.6612	0.424882
$889$	−61.8337	−2.07384
$890$	0	0
$891$	−24.8607	−0.832866
$892$	3.34555	0.112017
$893$	4.01662	0.134411
$894$	−10.1903	−0.340814
$895$	0	0
$896$	36.5498	1.22104
$897$	0	0
$898$	−18.0070	−0.600900
$899$	25.7969	0.860375
$900$	0	0
$901$	6.62790	0.220807
$902$	−34.0696	−1.13440
$903$	5.58976	0.186016
$904$	53.1633	1.76818
$905$	0	0
$906$	3.62132	0.120310
$907$	30.4522	1.01115	0.505575	−	0.862783i	$-0.331280\pi$
$907$	30.4522	1.01115	0.505575	+	0.862783i	$0.331280\pi$
$908$	−1.59996	−0.0530966
$909$	−48.4978	−1.60857
$910$	0	0
$911$	29.8054	0.987498	0.493749	−	0.869604i	$-0.335626\pi$
$911$	29.8054	0.987498	0.493749	+	0.869604i	$0.335626\pi$
$912$	1.27287	0.0421490
$913$	−3.86911	−0.128049
$914$	18.9232	0.625924
$915$	0	0
$916$	0.403948	0.0133468
$917$	64.5749	2.13245
$918$	6.93849	0.229004
$919$	−9.72223	−0.320707	−0.160353	−	0.987060i	$-0.551263\pi$
$919$	−9.72223	−0.320707	−0.160353	+	0.987060i	$0.551263\pi$
$920$	0	0
$921$	−0.652201	−0.0214908
$922$	−19.2084	−0.632597
$923$	0	0
$924$	0.646482	0.0212677
$925$	0	0
$926$	−6.64396	−0.218334
$927$	2.30181	0.0756014
$928$	2.34266	0.0769017
$929$	−4.29074	−0.140775	−0.0703874	−	0.997520i	$-0.522424\pi$
$929$	−4.29074	−0.140775	−0.0703874	+	0.997520i	$0.522424\pi$
$930$	0	0
$931$	5.38308	0.176423
$932$	0.931517	0.0305128
$933$	7.60055	0.248831
$934$	−36.3784	−1.19034
$935$	0	0
$936$	0	0
$937$	8.43750	0.275641	0.137821	−	0.990457i	$-0.455990\pi$
$937$	8.43750	0.275641	0.137821	+	0.990457i	$0.455990\pi$
$938$	23.1256	0.755076
$939$	4.43926	0.144870
$940$	0	0
$941$	−22.1898	−0.723368	−0.361684	−	0.932301i	$-0.617798\pi$
$941$	−22.1898	−0.723368	−0.361684	+	0.932301i	$0.617798\pi$
$942$	2.52023	0.0821134
$943$	5.82468	0.189678
$944$	49.4394	1.60912
$945$	0	0
$946$	17.1661	0.558118
$947$	−12.1616	−0.395199	−0.197599	−	0.980283i	$-0.563315\pi$
$947$	−12.1616	−0.395199	−0.197599	+	0.980283i	$0.563315\pi$
$948$	−0.552817	−0.0179547
$949$	0	0
$950$	0	0
$951$	2.08782	0.0677021
$952$	23.1147	0.749153
$953$	55.7841	1.80702	0.903512	−	0.428564i	$-0.140980\pi$
$953$	55.7841	1.80702	0.903512	+	0.428564i	$0.140980\pi$
$954$	−11.8146	−0.382511
$955$	0	0
$956$	−2.48326	−0.0803143
$957$	−4.01609	−0.129822
$958$	−4.12803	−0.133371
$959$	58.6944	1.89534
$960$	0	0
$961$	39.8683	1.28607
$962$	0	0
$963$	38.0201	1.22518
$964$	−2.25227	−0.0725408
$965$	0	0
$966$	1.52228	0.0489784
$967$	−12.5378	−0.403190	−0.201595	−	0.979469i	$-0.564612\pi$
$967$	−12.5378	−0.403190	−0.201595	+	0.979469i	$0.564612\pi$
$968$	−0.759679	−0.0244170
$969$	0.746252	0.0239731
$970$	0	0
$971$	20.6534	0.662799	0.331400	−	0.943490i	$-0.392479\pi$
$971$	20.6534	0.662799	0.331400	+	0.943490i	$0.392479\pi$
$972$	1.35929	0.0435992
$973$	12.5012	0.400769
$974$	−16.1404	−0.517172
$975$	0	0
$976$	48.3554	1.54782
$977$	−19.2410	−0.615575	−0.307787	−	0.951455i	$-0.599589\pi$
$977$	−19.2410	−0.615575	−0.307787	+	0.951455i	$0.599589\pi$
$978$	−5.32457	−0.170261
$979$	10.6918	0.341713
$980$	0	0
$981$	−15.9393	−0.508904
$982$	−13.1721	−0.420339
$983$	−28.9070	−0.921990	−0.460995	−	0.887403i	$-0.652507\pi$
$983$	−28.9070	−0.921990	−0.460995	+	0.887403i	$0.652507\pi$
$984$	8.87814	0.283025
$985$	0	0
$986$	−9.10370	−0.289921
$987$	6.82392	0.217208
$988$	0	0
$989$	−2.93478	−0.0933207
$990$	0	0
$991$	−38.4009	−1.21984	−0.609922	−	0.792462i	$-0.708799\pi$
$991$	−38.4009	−1.21984	−0.609922	+	0.792462i	$0.708799\pi$
$992$	6.43568	0.204333
$993$	−10.2529	−0.325365
$994$	−53.9121	−1.70999
$995$	0	0
$996$	0.0639217	0.00202544
$997$	−52.9800	−1.67789	−0.838947	−	0.544213i	$-0.816828\pi$
$997$	−52.9800	−1.67789	−0.838947	+	0.544213i	$0.816828\pi$
$998$	−13.7653	−0.435733
$999$	−25.3584	−0.802304

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4225.2.a.bv.1.3		10
5.4	even	2		4225.2.a.bu.1.8		10
13.2	odd	12		325.2.n.e.251.2	yes	10
13.7	odd	12		325.2.n.e.101.2	✓	10
13.12	even	2	inner	4225.2.a.bv.1.8		10
65.2	even	12		325.2.m.d.199.8		20
65.7	even	12		325.2.m.d.49.3		20
65.28	even	12		325.2.m.d.199.3		20
65.33	even	12		325.2.m.d.49.8		20
65.54	odd	12		325.2.n.f.251.4	yes	10
65.59	odd	12		325.2.n.f.101.4	yes	10
65.64	even	2		4225.2.a.bu.1.3		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.2.m.d.49.3		20	65.7	even	12
325.2.m.d.49.8		20	65.33	even	12
325.2.m.d.199.3		20	65.28	even	12
325.2.m.d.199.8		20	65.2	even	12
325.2.n.e.101.2	✓	10	13.7	odd	12
325.2.n.e.251.2	yes	10	13.2	odd	12
325.2.n.f.101.4	yes	10	65.59	odd	12
325.2.n.f.251.4	yes	10	65.54	odd	12
4225.2.a.bu.1.3		10	65.64	even	2
4225.2.a.bu.1.8		10	5.4	even	2
4225.2.a.bv.1.3		10	1.1	even	1	trivial
4225.2.a.bv.1.8		10	13.12	even	2	inner

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:	\( N \)	\(=\)	\( 4225 = 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4225.a (trivial)

\(f(q)\)	\(=\)	\(q-1.36551 q^{2} -0.399918 q^{3} -0.135381 q^{4} +0.546092 q^{6} -3.64365 q^{7} +2.91589 q^{8} -2.84007 q^{9} +O(q^{10})\)	\(q-1.36551 q^{2} -0.399918 q^{3} -0.135381 q^{4} +0.546092 q^{6} -3.64365 q^{7} +2.91589 q^{8} -2.84007 q^{9} -3.27711 q^{11} +0.0541412 q^{12} +4.97545 q^{14} -3.71091 q^{16} -2.17561 q^{17} +3.87814 q^{18} +0.857697 q^{19} +1.45716 q^{21} +4.47493 q^{22} -0.765051 q^{23} -1.16611 q^{24} +2.33555 q^{27} +0.493281 q^{28} -3.06437 q^{29} -8.41833 q^{31} -0.764484 q^{32} +1.31058 q^{33} +2.97082 q^{34} +0.384491 q^{36} -10.8576 q^{37} -1.17119 q^{38} -7.61344 q^{41} -1.98977 q^{42} +3.83606 q^{43} +0.443659 q^{44} +1.04469 q^{46} +4.68303 q^{47} +1.48406 q^{48} +6.27621 q^{49} +0.870065 q^{51} -3.04646 q^{53} -3.18921 q^{54} -10.6245 q^{56} -0.343008 q^{57} +4.18443 q^{58} -13.3227 q^{59} -13.0306 q^{61} +11.4953 q^{62} +10.3482 q^{63} +8.46573 q^{64} -1.78960 q^{66} +4.64794 q^{67} +0.294536 q^{68} +0.305958 q^{69} -10.8356 q^{71} -8.28131 q^{72} -7.68313 q^{73} +14.8262 q^{74} -0.116116 q^{76} +11.9407 q^{77} -10.2106 q^{79} +7.58617 q^{81} +10.3962 q^{82} +1.18065 q^{83} -0.197272 q^{84} -5.23818 q^{86} +1.22550 q^{87} -9.55569 q^{88} -3.26258 q^{89} +0.103573 q^{92} +3.36664 q^{93} -6.39472 q^{94} +0.305731 q^{96} +5.25271 q^{97} -8.57023 q^{98} +9.30722 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 6 q^{3} + 12 q^{4} + 16 q^{9}+O(q^{10}) \) 10 * q + 6 * q^3 + 12 * q^4 + 16 * q^9	\( 10 q + 6 q^{3} + 12 q^{4} + 16 q^{9} + 28 q^{12} - 8 q^{14} + 24 q^{16} + 16 q^{17} + 24 q^{22} + 26 q^{23} + 30 q^{27} - 14 q^{29} - 6 q^{36} + 62 q^{38} - 64 q^{42} + 8 q^{43} + 52 q^{48} - 2 q^{49} - 16 q^{51} + 24 q^{53} - 42 q^{56} - 26 q^{61} + 34 q^{62} + 34 q^{64} - 42 q^{66} + 26 q^{68} + 40 q^{69} + 52 q^{74} + 48 q^{77} + 4 q^{79} + 34 q^{81} - 2 q^{82} + 98 q^{87} - 12 q^{88} + 34 q^{92} - 10 q^{94}+O(q^{100}) \) 10 * q + 6 * q^3 + 12 * q^4 + 16 * q^9 + 28 * q^12 - 8 * q^14 + 24 * q^16 + 16 * q^17 + 24 * q^22 + 26 * q^23 + 30 * q^27 - 14 * q^29 - 6 * q^36 + 62 * q^38 - 64 * q^42 + 8 * q^43 + 52 * q^48 - 2 * q^49 - 16 * q^51 + 24 * q^53 - 42 * q^56 - 26 * q^61 + 34 * q^62 + 34 * q^64 - 42 * q^66 + 26 * q^68 + 40 * q^69 + 52 * q^74 + 48 * q^77 + 4 * q^79 + 34 * q^81 - 2 * q^82 + 98 * q^87 - 12 * q^88 + 34 * q^92 - 10 * q^94

Introduction

L-functions

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