LMFDB - Embedded newform 9025.2.a.co.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9025,2,Mod(1,9025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9025, 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("9025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9025 = 5^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9025.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:	$72.0649878242$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 32 x^{18} + 426 x^{16} - 3061 x^{14} + 12909 x^{12} - 32678 x^{10} + 49159 x^{8} - 42549 x^{6} + \cdots + 405 $ x^20 - 32x^18 + 426x^16 - 3061x^14 + 12909x^12 - 32678x^10 + 49159x^8 - 42549x^6 + 20056x^4 - 4635*x^2 + 405
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-0.566010$ of defining polynomial
Character	$\chi$	$=$	9025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.566010 q^{2} -3.10364 q^{3} -1.67963 q^{4} +1.75669 q^{6} +3.12047 q^{7} +2.08271 q^{8} +6.63258 q^{9} +O(q^{10})$	\(q-0.566010 q^{2} -3.10364 q^{3} -1.67963 q^{4} +1.75669 q^{6} +3.12047 q^{7} +2.08271 q^{8} +6.63258 q^{9} -1.90725 q^{11} +5.21297 q^{12} -5.79625 q^{13} -1.76621 q^{14} +2.18043 q^{16} +7.75726 q^{17} -3.75410 q^{18} -9.68480 q^{21} +1.07952 q^{22} -1.43350 q^{23} -6.46398 q^{24} +3.28074 q^{26} -11.2742 q^{27} -5.24124 q^{28} -1.18185 q^{29} +0.520495 q^{31} -5.39956 q^{32} +5.91940 q^{33} -4.39069 q^{34} -11.1403 q^{36} +10.1250 q^{37} +17.9895 q^{39} +6.25872 q^{41} +5.48169 q^{42} +5.86116 q^{43} +3.20347 q^{44} +0.811375 q^{46} +0.458650 q^{47} -6.76727 q^{48} +2.73731 q^{49} -24.0757 q^{51} +9.73558 q^{52} -6.70216 q^{53} +6.38131 q^{54} +6.49902 q^{56} +0.668936 q^{58} +4.95071 q^{59} -1.87838 q^{61} -0.294605 q^{62} +20.6967 q^{63} -1.30466 q^{64} -3.35044 q^{66} +13.3783 q^{67} -13.0293 q^{68} +4.44907 q^{69} +6.31236 q^{71} +13.8137 q^{72} +5.77080 q^{73} -5.73085 q^{74} -5.95150 q^{77} -10.1822 q^{78} -6.02369 q^{79} +15.0933 q^{81} -3.54250 q^{82} +15.0284 q^{83} +16.2669 q^{84} -3.31748 q^{86} +3.66802 q^{87} -3.97224 q^{88} -5.42789 q^{89} -18.0870 q^{91} +2.40775 q^{92} -1.61543 q^{93} -0.259600 q^{94} +16.7583 q^{96} +1.63225 q^{97} -1.54934 q^{98} -12.6500 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 24 q^{4} - 20 q^{6} + 8 q^{7} + 16 q^{9}+O(q^{10}) $ 20 * q + 24 * q^4 - 20 * q^6 + 8 * q^7 + 16 * q^9	$ 20 q + 24 q^{4} - 20 q^{6} + 8 q^{7} + 16 q^{9} + 10 q^{11} + 40 q^{16} + 38 q^{17} + 56 q^{23} - 54 q^{24} - 22 q^{26} - 16 q^{28} - 32 q^{36} + 80 q^{39} + 44 q^{43} - 16 q^{44} + 98 q^{47} - 4 q^{49} - 54 q^{54} + 50 q^{58} + 40 q^{61} + 90 q^{62} + 6 q^{63} + 46 q^{64} + 8 q^{66} + 4 q^{68} + 34 q^{73} + 20 q^{74} + 80 q^{77} + 60 q^{81} - 10 q^{82} + 84 q^{83} + 90 q^{87} + 78 q^{92} + 20 q^{93} - 120 q^{96} - 76 q^{99}+O(q^{100}) $ 20 * q + 24 * q^4 - 20 * q^6 + 8 * q^7 + 16 * q^9 + 10 * q^11 + 40 * q^16 + 38 * q^17 + 56 * q^23 - 54 * q^24 - 22 * q^26 - 16 * q^28 - 32 * q^36 + 80 * q^39 + 44 * q^43 - 16 * q^44 + 98 * q^47 - 4 * q^49 - 54 * q^54 + 50 * q^58 + 40 * q^61 + 90 * q^62 + 6 * q^63 + 46 * q^64 + 8 * q^66 + 4 * q^68 + 34 * q^73 + 20 * q^74 + 80 * q^77 + 60 * q^81 - 10 * q^82 + 84 * q^83 + 90 * q^87 + 78 * q^92 + 20 * q^93 - 120 * q^96 - 76 * q^99

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 $

$2$	−0.566010	−0.400229	−0.200115	−	0.979772i	$-0.564131\pi$
$2$	−0.566010	−0.400229	−0.200115	+	0.979772i	$0.564131\pi$
$3$	−3.10364	−1.79189	−0.895943	−	0.444168i	$-0.853499\pi$
$3$	−3.10364	−1.79189	−0.895943	+	0.444168i	$0.853499\pi$
$4$	−1.67963	−0.839816
$5$	0	0
$5$	0	0
$6$	1.75669	0.717166
$7$	3.12047	1.17943	0.589713	−	0.807613i	$-0.299241\pi$
$7$	3.12047	1.17943	0.589713	+	0.807613i	$0.299241\pi$
$8$	2.08271	0.736349
$9$	6.63258	2.21086
$10$	0	0
$11$	−1.90725	−0.575056	−0.287528	−	0.957772i	$-0.592833\pi$
$11$	−1.90725	−0.575056	−0.287528	+	0.957772i	$0.592833\pi$
$12$	5.21297	1.50486
$13$	−5.79625	−1.60759	−0.803796	−	0.594905i	$-0.797190\pi$
$13$	−5.79625	−1.60759	−0.803796	+	0.594905i	$0.797190\pi$
$14$	−1.76621	−0.472041
$15$	0	0
$16$	2.18043	0.545108
$17$	7.75726	1.88141	0.940706	−	0.339223i	$-0.110164\pi$
$17$	7.75726	1.88141	0.940706	+	0.339223i	$0.110164\pi$
$18$	−3.75410	−0.884851
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−9.68480	−2.11340
$22$	1.07952	0.230154
$23$	−1.43350	−0.298905	−0.149453	−	0.988769i	$-0.547751\pi$
$23$	−1.43350	−0.298905	−0.149453	+	0.988769i	$0.547751\pi$
$24$	−6.46398	−1.31945
$25$	0	0
$26$	3.28074	0.643406
$27$	−11.2742	−2.16972
$28$	−5.24124	−0.990501
$29$	−1.18185	−0.219463	−0.109732	−	0.993961i	$-0.534999\pi$
$29$	−1.18185	−0.219463	−0.109732	+	0.993961i	$0.534999\pi$
$30$	0	0
$31$	0.520495	0.0934836	0.0467418	−	0.998907i	$-0.485116\pi$
$31$	0.520495	0.0934836	0.0467418	+	0.998907i	$0.485116\pi$
$32$	−5.39956	−0.954517
$33$	5.91940	1.03044
$34$	−4.39069	−0.752996
$35$	0	0
$36$	−11.1403	−1.85672
$37$	10.1250	1.66454	0.832271	−	0.554370i	$-0.187041\pi$
$37$	10.1250	1.66454	0.832271	+	0.554370i	$0.187041\pi$
$38$	0	0
$39$	17.9895	2.88062
$40$	0	0
$41$	6.25872	0.977448	0.488724	−	0.872439i	$-0.337463\pi$
$41$	6.25872	0.977448	0.488724	+	0.872439i	$0.337463\pi$
$42$	5.48169	0.845844
$43$	5.86116	0.893819	0.446910	−	0.894579i	$-0.352525\pi$
$43$	5.86116	0.893819	0.446910	+	0.894579i	$0.352525\pi$
$44$	3.20347	0.482942
$45$	0	0
$46$	0.811375	0.119631
$47$	0.458650	0.0669009	0.0334505	−	0.999440i	$-0.489350\pi$
$47$	0.458650	0.0669009	0.0334505	+	0.999440i	$0.489350\pi$
$48$	−6.76727	−0.976772
$49$	2.73731	0.391044
$50$	0	0
$51$	−24.0757	−3.37128
$52$	9.73558	1.35008
$53$	−6.70216	−0.920613	−0.460307	−	0.887760i	$-0.652260\pi$
$53$	−6.70216	−0.920613	−0.460307	+	0.887760i	$0.652260\pi$
$54$	6.38131	0.868387
$55$	0	0
$56$	6.49902	0.868468
$57$	0	0
$58$	0.668936	0.0878356
$59$	4.95071	0.644528	0.322264	−	0.946650i	$-0.395556\pi$
$59$	4.95071	0.644528	0.322264	+	0.946650i	$0.395556\pi$
$60$	0	0
$61$	−1.87838	−0.240501	−0.120251	−	0.992744i	$-0.538370\pi$
$61$	−1.87838	−0.240501	−0.120251	+	0.992744i	$0.538370\pi$
$62$	−0.294605	−0.0374149
$63$	20.6967	2.60754
$64$	−1.30466	−0.163082
$65$	0	0
$66$	−3.35044	−0.412411
$67$	13.3783	1.63442	0.817211	−	0.576339i	$-0.195519\pi$
$67$	13.3783	1.63442	0.817211	+	0.576339i	$0.195519\pi$
$68$	−13.0293	−1.58004
$69$	4.44907	0.535605
$70$	0	0
$71$	6.31236	0.749139	0.374569	−	0.927199i	$-0.377791\pi$
$71$	6.31236	0.749139	0.374569	+	0.927199i	$0.377791\pi$
$72$	13.8137	1.62796
$73$	5.77080	0.675421	0.337710	−	0.941250i	$-0.390348\pi$
$73$	5.77080	0.675421	0.337710	+	0.941250i	$0.390348\pi$
$74$	−5.73085	−0.666198
$75$	0	0
$76$	0	0
$77$	−5.95150	−0.678236
$78$	−10.1822	−1.15291
$79$	−6.02369	−0.677718	−0.338859	−	0.940837i	$-0.610041\pi$
$79$	−6.02369	−0.677718	−0.338859	+	0.940837i	$0.610041\pi$
$80$	0	0
$81$	15.0933	1.67704
$82$	−3.54250	−0.391203
$83$	15.0284	1.64958	0.824791	−	0.565438i	$-0.191293\pi$
$83$	15.0284	1.64958	0.824791	+	0.565438i	$0.191293\pi$
$84$	16.2669	1.77487
$85$	0	0
$86$	−3.31748	−0.357733
$87$	3.66802	0.393253
$88$	−3.97224	−0.423442
$89$	−5.42789	−0.575355	−0.287678	−	0.957727i	$-0.592883\pi$
$89$	−5.42789	−0.575355	−0.287678	+	0.957727i	$0.592883\pi$
$90$	0	0
$91$	−18.0870	−1.89603
$92$	2.40775	0.251026
$93$	−1.61543	−0.167512
$94$	−0.259600	−0.0267757
$95$	0	0
$96$	16.7583	1.71039
$97$	1.63225	0.165730	0.0828648	−	0.996561i	$-0.473593\pi$
$97$	1.63225	0.165730	0.0828648	+	0.996561i	$0.473593\pi$
$98$	−1.54934	−0.156507
$99$	−12.6500	−1.27137
$100$	0	0
$101$	15.3315	1.52554	0.762769	−	0.646671i	$-0.223839\pi$
$101$	15.3315	1.52554	0.762769	+	0.646671i	$0.223839\pi$
$102$	13.6271	1.34928
$103$	8.63663	0.850993	0.425496	−	0.904960i	$-0.360100\pi$
$103$	8.63663	0.850993	0.425496	+	0.904960i	$0.360100\pi$
$104$	−12.0719	−1.18375
$105$	0	0
$106$	3.79349	0.368456
$107$	2.81932	0.272554	0.136277	−	0.990671i	$-0.456486\pi$
$107$	2.81932	0.272554	0.136277	+	0.990671i	$0.456486\pi$
$108$	18.9365	1.82217
$109$	−11.3376	−1.08594	−0.542970	−	0.839752i	$-0.682700\pi$
$109$	−11.3376	−1.08594	−0.542970	+	0.839752i	$0.682700\pi$
$110$	0	0
$111$	−31.4244	−2.98267
$112$	6.80396	0.642914
$113$	4.42536	0.416303	0.208152	−	0.978097i	$-0.433255\pi$
$113$	4.42536	0.416303	0.208152	+	0.978097i	$0.433255\pi$
$114$	0	0
$115$	0	0
$116$	1.98507	0.184309
$117$	−38.4441	−3.55416
$118$	−2.80215	−0.257959
$119$	24.2063	2.21898
$120$	0	0
$121$	−7.36241	−0.669310
$122$	1.06318	0.0962557
$123$	−19.4248	−1.75148
$124$	−0.874240	−0.0785091
$125$	0	0
$126$	−11.7146	−1.04362
$127$	11.4226	1.01359	0.506796	−	0.862066i	$-0.330830\pi$
$127$	11.4226	1.01359	0.506796	+	0.862066i	$0.330830\pi$
$128$	11.5376	1.01979
$129$	−18.1909	−1.60162
$130$	0	0
$131$	−17.9137	−1.56513	−0.782563	−	0.622571i	$-0.786088\pi$
$131$	−17.9137	−1.56513	−0.782563	+	0.622571i	$0.786088\pi$
$132$	−9.94243	−0.865377
$133$	0	0
$134$	−7.57226	−0.654144
$135$	0	0
$136$	16.1561	1.38537
$137$	−8.56010	−0.731339	−0.365669	−	0.930745i	$-0.619160\pi$
$137$	−8.56010	−0.731339	−0.365669	+	0.930745i	$0.619160\pi$
$138$	−2.51822	−0.214365
$139$	−3.17002	−0.268878	−0.134439	−	0.990922i	$-0.542923\pi$
$139$	−3.17002	−0.268878	−0.134439	+	0.990922i	$0.542923\pi$
$140$	0	0
$141$	−1.42348	−0.119879
$142$	−3.57286	−0.299827
$143$	11.0549	0.924456
$144$	14.4619	1.20516
$145$	0	0
$146$	−3.26633	−0.270323
$147$	−8.49561	−0.700706
$148$	−17.0063	−1.39791
$149$	−9.79749	−0.802641	−0.401321	−	0.915938i	$-0.631449\pi$
$149$	−9.79749	−0.802641	−0.401321	+	0.915938i	$0.631449\pi$
$150$	0	0
$151$	−10.6381	−0.865717	−0.432858	−	0.901462i	$-0.642495\pi$
$151$	−10.6381	−0.865717	−0.432858	+	0.901462i	$0.642495\pi$
$152$	0	0
$153$	51.4506	4.15954
$154$	3.36861	0.271450
$155$	0	0
$156$	−30.2157	−2.41919
$157$	−20.2450	−1.61572	−0.807862	−	0.589371i	$-0.799375\pi$
$157$	−20.2450	−1.61572	−0.807862	+	0.589371i	$0.799375\pi$
$158$	3.40947	0.271243
$159$	20.8011	1.64963
$160$	0	0
$161$	−4.47319	−0.352536
$162$	−8.54298	−0.671200
$163$	12.4283	0.973458	0.486729	−	0.873553i	$-0.338190\pi$
$163$	12.4283	0.973458	0.486729	+	0.873553i	$0.338190\pi$
$164$	−10.5123	−0.820876
$165$	0	0
$166$	−8.50622	−0.660211
$167$	16.0396	1.24119	0.620593	−	0.784133i	$-0.286892\pi$
$167$	16.0396	1.24119	0.620593	+	0.784133i	$0.286892\pi$
$168$	−20.1706	−1.55620
$169$	20.5966	1.58435
$170$	0	0
$171$	0	0
$172$	−9.84460	−0.750644
$173$	6.32141	0.480608	0.240304	−	0.970698i	$-0.422753\pi$
$173$	6.32141	0.480608	0.240304	+	0.970698i	$0.422753\pi$
$174$	−2.07614	−0.157392
$175$	0	0
$176$	−4.15862	−0.313468
$177$	−15.3652	−1.15492
$178$	3.07224	0.230274
$179$	−24.3287	−1.81841	−0.909205	−	0.416348i	$-0.863310\pi$
$179$	−24.3287	−1.81841	−0.909205	+	0.416348i	$0.863310\pi$
$180$	0	0
$181$	−19.5206	−1.45096	−0.725478	−	0.688245i	$-0.758381\pi$
$181$	−19.5206	−1.45096	−0.725478	+	0.688245i	$0.758381\pi$
$182$	10.2374	0.758849
$183$	5.82980	0.430951
$184$	−2.98556	−0.220099
$185$	0	0
$186$	0.914348	0.0670432
$187$	−14.7950	−1.08192
$188$	−0.770363	−0.0561845
$189$	−35.1808	−2.55903
$190$	0	0
$191$	7.34808	0.531689	0.265844	−	0.964016i	$-0.414349\pi$
$191$	7.34808	0.531689	0.265844	+	0.964016i	$0.414349\pi$
$192$	4.04919	0.292225
$193$	2.80671	0.202032	0.101016	−	0.994885i	$-0.467791\pi$
$193$	2.80671	0.202032	0.101016	+	0.994885i	$0.467791\pi$
$194$	−0.923869	−0.0663299
$195$	0	0
$196$	−4.59767	−0.328405
$197$	−18.8587	−1.34362	−0.671812	−	0.740722i	$-0.734484\pi$
$197$	−18.8587	−1.34362	−0.671812	+	0.740722i	$0.734484\pi$
$198$	7.16000	0.508839
$199$	−5.60015	−0.396984	−0.198492	−	0.980103i	$-0.563604\pi$
$199$	−5.60015	−0.396984	−0.198492	+	0.980103i	$0.563604\pi$
$200$	0	0
$201$	−41.5215	−2.92870
$202$	−8.67776	−0.610565
$203$	−3.68791	−0.258840
$204$	40.4384	2.83125
$205$	0	0
$206$	−4.88842	−0.340592
$207$	−9.50780	−0.660838
$208$	−12.6383	−0.876311
$209$	0	0
$210$	0	0
$211$	14.8314	1.02104	0.510519	−	0.859866i	$-0.329453\pi$
$211$	14.8314	1.02104	0.510519	+	0.859866i	$0.329453\pi$
$212$	11.2572	0.773146
$213$	−19.5913	−1.34237
$214$	−1.59576	−0.109084
$215$	0	0
$216$	−23.4809	−1.59767
$217$	1.62419	0.110257
$218$	6.41717	0.434625
$219$	−17.9105	−1.21028
$220$	0	0
$221$	−44.9631	−3.02454
$222$	17.7865	1.19375
$223$	−14.5374	−0.973495	−0.486748	−	0.873543i	$-0.661817\pi$
$223$	−14.5374	−0.973495	−0.486748	+	0.873543i	$0.661817\pi$
$224$	−16.8492	−1.12578
$225$	0	0
$226$	−2.50480	−0.166617
$227$	−22.9609	−1.52397	−0.761984	−	0.647595i	$-0.775775\pi$
$227$	−22.9609	−1.52397	−0.761984	+	0.647595i	$0.775775\pi$
$228$	0	0
$229$	16.9391	1.11937	0.559685	−	0.828705i	$-0.310922\pi$
$229$	16.9391	1.11937	0.559685	+	0.828705i	$0.310922\pi$
$230$	0	0
$231$	18.4713	1.21532
$232$	−2.46144	−0.161601
$233$	−20.9704	−1.37381	−0.686907	−	0.726745i	$-0.741032\pi$
$233$	−20.9704	−1.37381	−0.686907	+	0.726745i	$0.741032\pi$
$234$	21.7597	1.42248
$235$	0	0
$236$	−8.31538	−0.541285
$237$	18.6954	1.21439
$238$	−13.7010	−0.888103
$239$	7.63744	0.494025	0.247012	−	0.969012i	$-0.420551\pi$
$239$	7.63744	0.494025	0.247012	+	0.969012i	$0.420551\pi$
$240$	0	0
$241$	3.94117	0.253873	0.126937	−	0.991911i	$-0.459486\pi$
$241$	3.94117	0.253873	0.126937	+	0.991911i	$0.459486\pi$
$242$	4.16720	0.267878
$243$	−13.0217	−0.835341
$244$	3.15498	0.201977
$245$	0	0
$246$	10.9946	0.700992
$247$	0	0
$248$	1.08404	0.0688365
$249$	−46.6427	−2.95586
$250$	0	0
$251$	10.9761	0.692804	0.346402	−	0.938086i	$-0.387403\pi$
$251$	10.9761	0.692804	0.346402	+	0.938086i	$0.387403\pi$
$252$	−34.7629	−2.18986
$253$	2.73404	0.171887
$254$	−6.46530	−0.405669
$255$	0	0
$256$	−3.92107	−0.245067
$257$	−19.1968	−1.19747	−0.598733	−	0.800949i	$-0.704329\pi$
$257$	−19.1968	−1.19747	−0.598733	+	0.800949i	$0.704329\pi$
$258$	10.2963	0.641017
$259$	31.5947	1.96320
$260$	0	0
$261$	−7.83868	−0.485202
$262$	10.1393	0.626410
$263$	11.5735	0.713652	0.356826	−	0.934171i	$-0.383859\pi$
$263$	11.5735	0.713652	0.356826	+	0.934171i	$0.383859\pi$
$264$	12.3284	0.758760
$265$	0	0
$266$	0	0
$267$	16.8462	1.03097
$268$	−22.4707	−1.37261
$269$	−11.2213	−0.684176	−0.342088	−	0.939668i	$-0.611134\pi$
$269$	−11.2213	−0.684176	−0.342088	+	0.939668i	$0.611134\pi$
$270$	0	0
$271$	0.696303	0.0422974	0.0211487	−	0.999776i	$-0.493268\pi$
$271$	0.696303	0.0422974	0.0211487	+	0.999776i	$0.493268\pi$
$272$	16.9142	1.02557
$273$	56.1356	3.39748
$274$	4.84510	0.292703
$275$	0	0
$276$	−7.47280	−0.449810
$277$	14.4855	0.870350	0.435175	−	0.900346i	$-0.356687\pi$
$277$	14.4855	0.870350	0.435175	+	0.900346i	$0.356687\pi$
$278$	1.79426	0.107613
$279$	3.45222	0.206679
$280$	0	0
$281$	−6.33223	−0.377749	−0.188875	−	0.982001i	$-0.560484\pi$
$281$	−6.33223	−0.377749	−0.188875	+	0.982001i	$0.560484\pi$
$282$	0.805706	0.0479791
$283$	−6.46180	−0.384114	−0.192057	−	0.981384i	$-0.561516\pi$
$283$	−6.46180	−0.384114	−0.192057	+	0.981384i	$0.561516\pi$
$284$	−10.6024	−0.629139
$285$	0	0
$286$	−6.25717	−0.369994
$287$	19.5301	1.15283
$288$	−35.8130	−2.11030
$289$	43.1751	2.53971
$290$	0	0
$291$	−5.06591	−0.296969
$292$	−9.69282	−0.567229
$293$	−4.57752	−0.267422	−0.133711	−	0.991020i	$-0.542689\pi$
$293$	−4.57752	−0.267422	−0.133711	+	0.991020i	$0.542689\pi$
$294$	4.80860	0.280443
$295$	0	0
$296$	21.0874	1.22568
$297$	21.5027	1.24771
$298$	5.54547	0.321241
$299$	8.30893	0.480518
$300$	0	0
$301$	18.2896	1.05419
$302$	6.02127	0.346485
$303$	−47.5833	−2.73359
$304$	0	0
$305$	0	0
$306$	−29.1216	−1.66477
$307$	−15.3116	−0.873877	−0.436938	−	0.899491i	$-0.643937\pi$
$307$	−15.3116	−0.873877	−0.436938	+	0.899491i	$0.643937\pi$
$308$	9.99633	0.569594
$309$	−26.8050	−1.52488
$310$	0	0
$311$	2.39113	0.135589	0.0677943	−	0.997699i	$-0.478404\pi$
$311$	2.39113	0.135589	0.0677943	+	0.997699i	$0.478404\pi$
$312$	37.4669	2.12114
$313$	−4.63299	−0.261872	−0.130936	−	0.991391i	$-0.541798\pi$
$313$	−4.63299	−0.261872	−0.130936	+	0.991391i	$0.541798\pi$
$314$	11.4589	0.646660
$315$	0	0
$316$	10.1176	0.569159
$317$	25.6000	1.43784	0.718920	−	0.695093i	$-0.244637\pi$
$317$	25.6000	1.43784	0.718920	+	0.695093i	$0.244637\pi$
$318$	−11.7736	−0.660232
$319$	2.25407	0.126204
$320$	0	0
$321$	−8.75015	−0.488386
$322$	2.53187	0.141095
$323$	0	0
$324$	−25.3513	−1.40840
$325$	0	0
$326$	−7.03453	−0.389606
$327$	35.1877	1.94588
$328$	13.0351	0.719742
$329$	1.43120	0.0789047
$330$	0	0
$331$	12.7712	0.701966	0.350983	−	0.936382i	$-0.385847\pi$
$331$	12.7712	0.701966	0.350983	+	0.936382i	$0.385847\pi$
$332$	−25.2422	−1.38535
$333$	67.1549	3.68007
$334$	−9.07860	−0.496759
$335$	0	0
$336$	−21.1170	−1.15203
$337$	27.0888	1.47562	0.737809	−	0.675009i	$-0.235860\pi$
$337$	27.0888	1.47562	0.737809	+	0.675009i	$0.235860\pi$
$338$	−11.6579	−0.634104
$339$	−13.7347	−0.745968
$340$	0	0
$341$	−0.992711	−0.0537583
$342$	0	0
$343$	−13.3016	−0.718218
$344$	12.2071	0.658162
$345$	0	0
$346$	−3.57798	−0.192353
$347$	16.7438	0.898856	0.449428	−	0.893317i	$-0.351628\pi$
$347$	16.7438	0.898856	0.449428	+	0.893317i	$0.351628\pi$
$348$	−6.16093	−0.330261
$349$	2.73752	0.146536	0.0732681	−	0.997312i	$-0.476657\pi$
$349$	2.73752	0.146536	0.0732681	+	0.997312i	$0.476657\pi$
$350$	0	0
$351$	65.3482	3.48803
$352$	10.2983	0.548901
$353$	4.35759	0.231931	0.115966	−	0.993253i	$-0.463004\pi$
$353$	4.35759	0.231931	0.115966	+	0.993253i	$0.463004\pi$
$354$	8.69687	0.462234
$355$	0	0
$356$	9.11686	0.483193
$357$	−75.1275	−3.97617
$358$	13.7703	0.727782
$359$	−15.3729	−0.811352	−0.405676	−	0.914017i	$-0.632964\pi$
$359$	−15.3729	−0.811352	−0.405676	+	0.914017i	$0.632964\pi$
$360$	0	0
$361$	0	0
$362$	11.0489	0.580716
$363$	22.8503	1.19933
$364$	30.3795	1.59232
$365$	0	0
$366$	−3.29972	−0.172479
$367$	−0.237481	−0.0123964	−0.00619821	−	0.999981i	$-0.501973\pi$
$367$	−0.237481	−0.0123964	−0.00619821	+	0.999981i	$0.501973\pi$
$368$	−3.12565	−0.162936
$369$	41.5114	2.16100
$370$	0	0
$371$	−20.9139	−1.08579
$372$	2.71333	0.140679
$373$	6.14667	0.318262	0.159131	−	0.987257i	$-0.449131\pi$
$373$	6.14667	0.318262	0.159131	+	0.987257i	$0.449131\pi$
$374$	8.37412	0.433015
$375$	0	0
$376$	0.955234	0.0492624
$377$	6.85028	0.352807
$378$	19.9127	1.02420
$379$	16.7408	0.859915	0.429957	−	0.902849i	$-0.358529\pi$
$379$	16.7408	0.859915	0.429957	+	0.902849i	$0.358529\pi$
$380$	0	0
$381$	−35.4516	−1.81624
$382$	−4.15909	−0.212797
$383$	−11.3181	−0.578329	−0.289165	−	0.957279i	$-0.593377\pi$
$383$	−11.3181	−0.578329	−0.289165	+	0.957279i	$0.593377\pi$
$384$	−35.8085	−1.82734
$385$	0	0
$386$	−1.58863	−0.0808590
$387$	38.8746	1.97611
$388$	−2.74158	−0.139183
$389$	−15.5137	−0.786574	−0.393287	−	0.919416i	$-0.628662\pi$
$389$	−15.5137	−0.786574	−0.393287	+	0.919416i	$0.628662\pi$
$390$	0	0
$391$	−11.1200	−0.562364
$392$	5.70101	0.287945
$393$	55.5976	2.80453
$394$	10.6742	0.537758
$395$	0	0
$396$	21.2473	1.06772
$397$	2.28451	0.114656	0.0573282	−	0.998355i	$-0.481742\pi$
$397$	2.28451	0.114656	0.0573282	+	0.998355i	$0.481742\pi$
$398$	3.16974	0.158885
$399$	0	0
$400$	0	0
$401$	11.3323	0.565910	0.282955	−	0.959133i	$-0.408685\pi$
$401$	11.3323	0.565910	0.282955	+	0.959133i	$0.408685\pi$
$402$	23.5016	1.17215
$403$	−3.01692	−0.150283
$404$	−25.7512	−1.28117
$405$	0	0
$406$	2.08739	0.103596
$407$	−19.3109	−0.957205
$408$	−50.1427	−2.48244
$409$	10.1823	0.503484	0.251742	−	0.967794i	$-0.418997\pi$
$409$	10.1823	0.503484	0.251742	+	0.967794i	$0.418997\pi$
$410$	0	0
$411$	26.5675	1.31048
$412$	−14.5064	−0.714678
$413$	15.4485	0.760173
$414$	5.38151	0.264487
$415$	0	0
$416$	31.2972	1.53447
$417$	9.83860	0.481798
$418$	0	0
$419$	22.2437	1.08668	0.543338	−	0.839514i	$-0.317160\pi$
$419$	22.2437	1.08668	0.543338	+	0.839514i	$0.317160\pi$
$420$	0	0
$421$	−11.5477	−0.562800	−0.281400	−	0.959591i	$-0.590799\pi$
$421$	−11.5477	−0.562800	−0.281400	+	0.959591i	$0.590799\pi$
$422$	−8.39474	−0.408649
$423$	3.04203	0.147909
$424$	−13.9587	−0.677892
$425$	0	0
$426$	11.0889	0.537257
$427$	−5.86141	−0.283653
$428$	−4.73542	−0.228895
$429$	−34.3104	−1.65652
$430$	0	0
$431$	−0.204743	−0.00986211	−0.00493106	−	0.999988i	$-0.501570\pi$
$431$	−0.204743	−0.00986211	−0.00493106	+	0.999988i	$0.501570\pi$
$432$	−24.5826	−1.18273
$433$	11.4707	0.551248	0.275624	−	0.961266i	$-0.411116\pi$
$433$	11.4707	0.551248	0.275624	+	0.961266i	$0.411116\pi$
$434$	−0.919305	−0.0441281
$435$	0	0
$436$	19.0429	0.911991
$437$	0	0
$438$	10.1375	0.484389
$439$	12.0275	0.574040	0.287020	−	0.957925i	$-0.407335\pi$
$439$	12.0275	0.574040	0.287020	+	0.957925i	$0.407335\pi$
$440$	0	0
$441$	18.1554	0.864543
$442$	25.4495	1.21051
$443$	38.4412	1.82640	0.913198	−	0.407517i	$-0.133605\pi$
$443$	38.4412	1.82640	0.913198	+	0.407517i	$0.133605\pi$
$444$	52.7814	2.50489
$445$	0	0
$446$	8.22830	0.389621
$447$	30.4079	1.43824
$448$	−4.07114	−0.192343
$449$	4.75952	0.224616	0.112308	−	0.993673i	$-0.464176\pi$
$449$	4.75952	0.224616	0.112308	+	0.993673i	$0.464176\pi$
$450$	0	0
$451$	−11.9369	−0.562087
$452$	−7.43298	−0.349618
$453$	33.0168	1.55127
$454$	12.9961	0.609937
$455$	0	0
$456$	0	0
$457$	−9.67436	−0.452547	−0.226274	−	0.974064i	$-0.572654\pi$
$457$	−9.67436	−0.452547	−0.226274	+	0.974064i	$0.572654\pi$
$458$	−9.58773	−0.448005
$459$	−87.4570	−4.08214
$460$	0	0
$461$	20.6812	0.963219	0.481609	−	0.876386i	$-0.340052\pi$
$461$	20.6812	0.963219	0.481609	+	0.876386i	$0.340052\pi$
$462$	−10.4549	−0.486408
$463$	11.0423	0.513178	0.256589	−	0.966521i	$-0.417401\pi$
$463$	11.0423	0.513178	0.256589	+	0.966521i	$0.417401\pi$
$464$	−2.57693	−0.119631
$465$	0	0
$466$	11.8694	0.549841
$467$	−34.2415	−1.58451	−0.792254	−	0.610192i	$-0.791092\pi$
$467$	−34.2415	−1.58451	−0.792254	+	0.610192i	$0.791092\pi$
$468$	64.5720	2.98484
$469$	41.7466	1.92768
$470$	0	0
$471$	62.8331	2.89520
$472$	10.3109	0.474597
$473$	−11.1787	−0.513996
$474$	−10.5818	−0.486036
$475$	0	0
$476$	−40.6576	−1.86354
$477$	−44.4526	−2.03535
$478$	−4.32286	−0.197723
$479$	−11.1641	−0.510099	−0.255050	−	0.966928i	$-0.582092\pi$
$479$	−11.1641	−0.510099	−0.255050	+	0.966928i	$0.582092\pi$
$480$	0	0
$481$	−58.6871	−2.67590
$482$	−2.23074	−0.101607
$483$	13.8832	0.631706
$484$	12.3661	0.562098
$485$	0	0
$486$	7.37040	0.334328
$487$	2.07704	0.0941197	0.0470599	−	0.998892i	$-0.485015\pi$
$487$	2.07704	0.0941197	0.0470599	+	0.998892i	$0.485015\pi$
$488$	−3.91211	−0.177093
$489$	−38.5729	−1.74433
$490$	0	0
$491$	−14.6442	−0.660883	−0.330441	−	0.943827i	$-0.607198\pi$
$491$	−14.6442	−0.660883	−0.330441	+	0.943827i	$0.607198\pi$
$492$	32.6265	1.47092
$493$	−9.16788	−0.412901
$494$	0	0
$495$	0	0
$496$	1.13490	0.0509587
$497$	19.6975	0.883553
$498$	26.4003	1.18302
$499$	−22.0256	−0.986003	−0.493001	−	0.870029i	$-0.664100\pi$
$499$	−22.0256	−0.986003	−0.493001	+	0.870029i	$0.664100\pi$
$500$	0	0
$501$	−49.7813	−2.22406
$502$	−6.21257	−0.277281
$503$	41.1927	1.83669	0.918346	−	0.395778i	$-0.129525\pi$
$503$	41.1927	1.83669	0.918346	+	0.395778i	$0.129525\pi$
$504$	43.1053	1.92006
$505$	0	0
$506$	−1.54749	−0.0687944
$507$	−63.9243	−2.83898
$508$	−19.1858	−0.851231
$509$	−6.77080	−0.300110	−0.150055	−	0.988678i	$-0.547945\pi$
$509$	−6.77080	−0.300110	−0.150055	+	0.988678i	$0.547945\pi$
$510$	0	0
$511$	18.0076	0.796608
$512$	−20.8558	−0.921704
$513$	0	0
$514$	10.8656	0.479261
$515$	0	0
$516$	30.5541	1.34507
$517$	−0.874758	−0.0384718
$518$	−17.8829	−0.785731
$519$	−19.6194	−0.861195
$520$	0	0
$521$	−18.7550	−0.821672	−0.410836	−	0.911709i	$-0.634763\pi$
$521$	−18.7550	−0.821672	−0.410836	+	0.911709i	$0.634763\pi$
$522$	4.43677	0.194192
$523$	−30.4431	−1.33118	−0.665592	−	0.746316i	$-0.731821\pi$
$523$	−30.4431	−1.33118	−0.665592	+	0.746316i	$0.731821\pi$
$524$	30.0884	1.31442
$525$	0	0
$526$	−6.55071	−0.285624
$527$	4.03761	0.175881
$528$	12.9069	0.561699
$529$	−20.9451	−0.910656
$530$	0	0
$531$	32.8360	1.42496
$532$	0	0
$533$	−36.2771	−1.57134
$534$	−9.53512	−0.412625
$535$	0	0
$536$	27.8631	1.20350
$537$	75.5074	3.25839
$538$	6.35137	0.273827
$539$	−5.22072	−0.224872
$540$	0	0
$541$	−28.2790	−1.21581	−0.607905	−	0.794010i	$-0.707990\pi$
$541$	−28.2790	−1.21581	−0.607905	+	0.794010i	$0.707990\pi$
$542$	−0.394114	−0.0169287
$543$	60.5850	2.59995
$544$	−41.8858	−1.79584
$545$	0	0
$546$	−31.7733	−1.35977
$547$	−27.4761	−1.17479	−0.587397	−	0.809299i	$-0.699847\pi$
$547$	−27.4761	−1.17479	−0.587397	+	0.809299i	$0.699847\pi$
$548$	14.3778	0.614190
$549$	−12.4585	−0.531714
$550$	0	0
$551$	0	0
$552$	9.26611	0.394392
$553$	−18.7967	−0.799318
$554$	−8.19894	−0.348340
$555$	0	0
$556$	5.32447	0.225808
$557$	29.3429	1.24330	0.621650	−	0.783295i	$-0.286463\pi$
$557$	29.3429	1.24330	0.621650	+	0.783295i	$0.286463\pi$
$558$	−1.95399	−0.0827190
$559$	−33.9728	−1.43690
$560$	0	0
$561$	45.9184	1.93867
$562$	3.58410	0.151186
$563$	28.1756	1.18746	0.593731	−	0.804664i	$-0.297654\pi$
$563$	28.1756	1.18746	0.593731	+	0.804664i	$0.297654\pi$
$564$	2.39093	0.100676
$565$	0	0
$566$	3.65744	0.153734
$567$	47.0983	1.97794
$568$	13.1468	0.551627
$569$	−4.80258	−0.201335	−0.100667	−	0.994920i	$-0.532098\pi$
$569$	−4.80258	−0.201335	−0.100667	+	0.994920i	$0.532098\pi$
$570$	0	0
$571$	−13.0419	−0.545788	−0.272894	−	0.962044i	$-0.587981\pi$
$571$	−13.0419	−0.545788	−0.272894	+	0.962044i	$0.587981\pi$
$572$	−18.5681	−0.776373
$573$	−22.8058	−0.952726
$574$	−11.0542	−0.461395
$575$	0	0
$576$	−8.65324	−0.360552
$577$	−1.30814	−0.0544587	−0.0272294	−	0.999629i	$-0.508668\pi$
$577$	−1.30814	−0.0544587	−0.0272294	+	0.999629i	$0.508668\pi$
$578$	−24.4375	−1.01647
$579$	−8.71103	−0.362018
$580$	0	0
$581$	46.8956	1.94556
$582$	2.86735	0.118856
$583$	12.7827	0.529404
$584$	12.0189	0.497345
$585$	0	0
$586$	2.59092	0.107030
$587$	6.16988	0.254658	0.127329	−	0.991861i	$-0.459360\pi$
$587$	6.16988	0.254658	0.127329	+	0.991861i	$0.459360\pi$
$588$	14.2695	0.588465
$589$	0	0
$590$	0	0
$591$	58.5305	2.40762
$592$	22.0769	0.907355
$593$	13.2023	0.542154	0.271077	−	0.962558i	$-0.412620\pi$
$593$	13.2023	0.542154	0.271077	+	0.962558i	$0.412620\pi$
$594$	−12.1707	−0.499371
$595$	0	0
$596$	16.4562	0.674071
$597$	17.3808	0.711350
$598$	−4.70294	−0.192317
$599$	−11.3153	−0.462332	−0.231166	−	0.972914i	$-0.574254\pi$
$599$	−11.3153	−0.462332	−0.231166	+	0.972914i	$0.574254\pi$
$600$	0	0
$601$	38.7884	1.58221	0.791107	−	0.611678i	$-0.209505\pi$
$601$	38.7884	1.58221	0.791107	+	0.611678i	$0.209505\pi$
$602$	−10.3521	−0.421919
$603$	88.7327	3.61348
$604$	17.8681	0.727043
$605$	0	0
$606$	26.9326	1.09406
$607$	−6.60507	−0.268091	−0.134046	−	0.990975i	$-0.542797\pi$
$607$	−6.60507	−0.268091	−0.134046	+	0.990975i	$0.542797\pi$
$608$	0	0
$609$	11.4459	0.463813
$610$	0	0
$611$	−2.65845	−0.107549
$612$	−86.4181	−3.49325
$613$	−40.9493	−1.65393	−0.826963	−	0.562257i	$-0.809933\pi$
$613$	−40.9493	−1.65393	−0.826963	+	0.562257i	$0.809933\pi$
$614$	8.66649	0.349751
$615$	0	0
$616$	−12.3952	−0.499418
$617$	−8.30584	−0.334380	−0.167190	−	0.985925i	$-0.553469\pi$
$617$	−8.30584	−0.334380	−0.167190	+	0.985925i	$0.553469\pi$
$618$	15.1719	0.610303
$619$	46.6706	1.87585	0.937926	−	0.346836i	$-0.112744\pi$
$619$	46.6706	1.87585	0.937926	+	0.346836i	$0.112744\pi$
$620$	0	0
$621$	16.1616	0.648542
$622$	−1.35340	−0.0542665
$623$	−16.9375	−0.678588
$624$	39.2248	1.57025
$625$	0	0
$626$	2.62232	0.104809
$627$	0	0
$628$	34.0041	1.35691
$629$	78.5423	3.13169
$630$	0	0
$631$	48.6998	1.93871	0.969354	−	0.245669i	$-0.0790077\pi$
$631$	48.6998	1.93871	0.969354	+	0.245669i	$0.0790077\pi$
$632$	−12.5456	−0.499037
$633$	−46.0314	−1.82958
$634$	−14.4899	−0.575466
$635$	0	0
$636$	−34.9382	−1.38539
$637$	−15.8661	−0.628639
$638$	−1.27583	−0.0505104
$639$	41.8672	1.65624
$640$	0	0
$641$	32.7873	1.29502	0.647510	−	0.762057i	$-0.275810\pi$
$641$	32.7873	1.29502	0.647510	+	0.762057i	$0.275810\pi$
$642$	4.95267	0.195466
$643$	−22.4539	−0.885497	−0.442749	−	0.896646i	$-0.645997\pi$
$643$	−22.4539	−0.885497	−0.442749	+	0.896646i	$0.645997\pi$
$644$	7.51331	0.296066
$645$	0	0
$646$	0	0
$647$	45.9065	1.80477	0.902385	−	0.430931i	$-0.141815\pi$
$647$	45.9065	1.80477	0.902385	+	0.430931i	$0.141815\pi$
$648$	31.4350	1.23489
$649$	−9.44223	−0.370640
$650$	0	0
$651$	−5.04089	−0.197568
$652$	−20.8749	−0.817526
$653$	−16.7906	−0.657066	−0.328533	−	0.944493i	$-0.606554\pi$
$653$	−16.7906	−0.657066	−0.328533	+	0.944493i	$0.606554\pi$
$654$	−19.9166	−0.778800
$655$	0	0
$656$	13.6467	0.532814
$657$	38.2753	1.49326
$658$	−0.810074	−0.0315800
$659$	−1.77434	−0.0691183	−0.0345591	−	0.999403i	$-0.511003\pi$
$659$	−1.77434	−0.0691183	−0.0345591	+	0.999403i	$0.511003\pi$
$660$	0	0
$661$	27.5585	1.07190	0.535950	−	0.844249i	$-0.319953\pi$
$661$	27.5585	1.07190	0.535950	+	0.844249i	$0.319953\pi$
$662$	−7.22860	−0.280948
$663$	139.549	5.41964
$664$	31.2998	1.21467
$665$	0	0
$666$	−38.0103	−1.47287
$667$	1.69418	0.0655987
$668$	−26.9407	−1.04237
$669$	45.1188	1.74439
$670$	0	0
$671$	3.58252	0.138302
$672$	52.2937	2.01727
$673$	30.4962	1.17554	0.587771	−	0.809027i	$-0.300005\pi$
$673$	30.4962	1.17554	0.587771	+	0.809027i	$0.300005\pi$
$674$	−15.3325	−0.590586
$675$	0	0
$676$	−34.5947	−1.33056
$677$	27.6675	1.06335	0.531673	−	0.846949i	$-0.321563\pi$
$677$	27.6675	1.06335	0.531673	+	0.846949i	$0.321563\pi$
$678$	7.77399	0.298558
$679$	5.09337	0.195466
$680$	0	0
$681$	71.2624	2.73078
$682$	0.561884	0.0215157
$683$	−17.3695	−0.664627	−0.332313	−	0.943169i	$-0.607829\pi$
$683$	−17.3695	−0.664627	−0.332313	+	0.943169i	$0.607829\pi$
$684$	0	0
$685$	0	0
$686$	7.52883	0.287452
$687$	−52.5730	−2.00579
$688$	12.7799	0.487228
$689$	38.8475	1.47997
$690$	0	0
$691$	−22.6009	−0.859779	−0.429890	−	0.902881i	$-0.641448\pi$
$691$	−22.6009	−0.859779	−0.429890	+	0.902881i	$0.641448\pi$
$692$	−10.6176	−0.403622
$693$	−39.4738	−1.49948
$694$	−9.47717	−0.359748
$695$	0	0
$696$	7.63942	0.289571
$697$	48.5505	1.83898
$698$	−1.54946	−0.0586481
$699$	65.0844	2.46172
$700$	0	0
$701$	−24.7089	−0.933243	−0.466621	−	0.884457i	$-0.654529\pi$
$701$	−24.7089	−0.933243	−0.466621	+	0.884457i	$0.654529\pi$
$702$	−36.9877	−1.39601
$703$	0	0
$704$	2.48830	0.0937815
$705$	0	0
$706$	−2.46644	−0.0928257
$707$	47.8413	1.79926
$708$	25.8079	0.969922
$709$	−2.71216	−0.101857	−0.0509286	−	0.998702i	$-0.516218\pi$
$709$	−2.71216	−0.101857	−0.0509286	+	0.998702i	$0.516218\pi$
$710$	0	0
$711$	−39.9526	−1.49834
$712$	−11.3047	−0.423662
$713$	−0.746129	−0.0279427
$714$	42.5229	1.59138
$715$	0	0
$716$	40.8632	1.52713
$717$	−23.7038	−0.885236
$718$	8.70122	0.324727
$719$	2.81367	0.104932	0.0524660	−	0.998623i	$-0.483292\pi$
$719$	2.81367	0.104932	0.0524660	+	0.998623i	$0.483292\pi$
$720$	0	0
$721$	26.9503	1.00368
$722$	0	0
$723$	−12.2320	−0.454912
$724$	32.7875	1.21854
$725$	0	0
$726$	−12.9335	−0.480006
$727$	16.2817	0.603856	0.301928	−	0.953331i	$-0.402370\pi$
$727$	16.2817	0.603856	0.301928	+	0.953331i	$0.402370\pi$
$728$	−37.6700	−1.39614
$729$	−4.86545	−0.180202
$730$	0	0
$731$	45.4666	1.68164
$732$	−9.79192	−0.361920
$733$	−25.4087	−0.938493	−0.469247	−	0.883067i	$-0.655474\pi$
$733$	−25.4087	−0.938493	−0.469247	+	0.883067i	$0.655474\pi$
$734$	0.134417	0.00496141
$735$	0	0
$736$	7.74027	0.285310
$737$	−25.5157	−0.939885
$738$	−23.4959	−0.864895
$739$	7.41718	0.272845	0.136423	−	0.990651i	$-0.456439\pi$
$739$	7.41718	0.272845	0.136423	+	0.990651i	$0.456439\pi$
$740$	0	0
$741$	0	0
$742$	11.8375	0.434567
$743$	27.6001	1.01255	0.506275	−	0.862372i	$-0.331022\pi$
$743$	27.6001	1.01255	0.506275	+	0.862372i	$0.331022\pi$
$744$	−3.36446	−0.123347
$745$	0	0
$746$	−3.47907	−0.127378
$747$	99.6770	3.64699
$748$	24.8502	0.908612
$749$	8.79759	0.321457
$750$	0	0
$751$	20.4914	0.747743	0.373872	−	0.927480i	$-0.378030\pi$
$751$	20.4914	0.747743	0.373872	+	0.927480i	$0.378030\pi$
$752$	1.00005	0.0364682
$753$	−34.0658	−1.24143
$754$	−3.87732	−0.141204
$755$	0	0
$756$	59.0908	2.14911
$757$	−35.2204	−1.28011	−0.640054	−	0.768330i	$-0.721088\pi$
$757$	−35.2204	−1.28011	−0.640054	+	0.768330i	$0.721088\pi$
$758$	−9.47543	−0.344163
$759$	−8.48546	−0.308003
$760$	0	0
$761$	29.2908	1.06179	0.530895	−	0.847438i	$-0.321856\pi$
$761$	29.2908	1.06179	0.530895	+	0.847438i	$0.321856\pi$
$762$	20.0660	0.726913
$763$	−35.3785	−1.28079
$764$	−12.3421	−0.446521
$765$	0	0
$766$	6.40617	0.231464
$767$	−28.6956	−1.03614
$768$	12.1696	0.439132
$769$	−37.4753	−1.35140	−0.675698	−	0.737179i	$-0.736158\pi$
$769$	−37.4753	−1.35140	−0.675698	+	0.737179i	$0.736158\pi$
$770$	0	0
$771$	59.5800	2.14572
$772$	−4.71425	−0.169669
$773$	18.1431	0.652563	0.326281	−	0.945273i	$-0.394204\pi$
$773$	18.1431	0.652563	0.326281	+	0.945273i	$0.394204\pi$
$774$	−22.0034	−0.790897
$775$	0	0
$776$	3.39950	0.122035
$777$	−98.0587	−3.51784
$778$	8.78089	0.314810
$779$	0	0
$780$	0	0
$781$	−12.0392	−0.430797
$782$	6.29405	0.225075
$783$	13.3244	0.476174
$784$	5.96851	0.213161
$785$	0	0
$786$	−31.4688	−1.12246
$787$	32.8976	1.17267	0.586337	−	0.810067i	$-0.300570\pi$
$787$	32.8976	1.17267	0.586337	+	0.810067i	$0.300570\pi$
$788$	31.6756	1.12840
$789$	−35.9199	−1.27878
$790$	0	0
$791$	13.8092	0.490998
$792$	−26.3462	−0.936171
$793$	10.8875	0.386628
$794$	−1.29306	−0.0458889
$795$	0	0
$796$	9.40619	0.333394
$797$	26.9073	0.953107	0.476553	−	0.879146i	$-0.341886\pi$
$797$	26.9073	0.953107	0.476553	+	0.879146i	$0.341886\pi$
$798$	0	0
$799$	3.55787	0.125868
$800$	0	0
$801$	−36.0009	−1.27203
$802$	−6.41421	−0.226494
$803$	−11.0063	−0.388405
$804$	69.7408	2.45957
$805$	0	0
$806$	1.70761	0.0601479
$807$	34.8269	1.22597
$808$	31.9310	1.12333
$809$	−44.2608	−1.55613	−0.778064	−	0.628186i	$-0.783798\pi$
$809$	−44.2608	−1.55613	−0.778064	+	0.628186i	$0.783798\pi$
$810$	0	0
$811$	28.9424	1.01631	0.508153	−	0.861267i	$-0.330328\pi$
$811$	28.9424	1.01631	0.508153	+	0.861267i	$0.330328\pi$
$812$	6.19433	0.217378
$813$	−2.16107	−0.0757922
$814$	10.9302	0.383102
$815$	0	0
$816$	−52.4955	−1.83771
$817$	0	0
$818$	−5.76331	−0.201509
$819$	−119.964	−4.19186
$820$	0	0
$821$	35.0982	1.22494	0.612468	−	0.790495i	$-0.290177\pi$
$821$	35.0982	1.22494	0.612468	+	0.790495i	$0.290177\pi$
$822$	−15.0374	−0.524491
$823$	−28.2713	−0.985475	−0.492737	−	0.870178i	$-0.664004\pi$
$823$	−28.2713	−0.985475	−0.492737	+	0.870178i	$0.664004\pi$
$824$	17.9876	0.626627
$825$	0	0
$826$	−8.74402	−0.304243
$827$	−26.6922	−0.928178	−0.464089	−	0.885788i	$-0.653618\pi$
$827$	−26.6922	−0.928178	−0.464089	+	0.885788i	$0.653618\pi$
$828$	15.9696	0.554982
$829$	35.2964	1.22590	0.612948	−	0.790123i	$-0.289983\pi$
$829$	35.2964	1.22590	0.612948	+	0.790123i	$0.289983\pi$
$830$	0	0
$831$	−44.9578	−1.55957
$832$	7.56213	0.262170
$833$	21.2340	0.735714
$834$	−5.56874	−0.192830
$835$	0	0
$836$	0	0
$837$	−5.86816	−0.202833
$838$	−12.5901	−0.434919
$839$	9.80307	0.338440	0.169220	−	0.985578i	$-0.445875\pi$
$839$	9.80307	0.338440	0.169220	+	0.985578i	$0.445875\pi$
$840$	0	0
$841$	−27.6032	−0.951836
$842$	6.53610	0.225249
$843$	19.6530	0.676884
$844$	−24.9114	−0.857484
$845$	0	0
$846$	−1.72182	−0.0591974
$847$	−22.9742	−0.789401
$848$	−14.6136	−0.501834
$849$	20.0551	0.688289
$850$	0	0
$851$	−14.5142	−0.497540
$852$	32.9062	1.12735
$853$	−22.4833	−0.769812	−0.384906	−	0.922956i	$-0.625766\pi$
$853$	−22.4833	−0.769812	−0.384906	+	0.922956i	$0.625766\pi$
$854$	3.31761	0.113526
$855$	0	0
$856$	5.87182	0.200695
$857$	−9.97119	−0.340609	−0.170305	−	0.985391i	$-0.554475\pi$
$857$	−9.97119	−0.340609	−0.170305	+	0.985391i	$0.554475\pi$
$858$	19.4200	0.662988
$859$	−8.06015	−0.275009	−0.137504	−	0.990501i	$-0.543908\pi$
$859$	−8.06015	−0.275009	−0.137504	+	0.990501i	$0.543908\pi$
$860$	0	0
$861$	−60.6144	−2.06573
$862$	0.115886	0.00394711
$863$	35.6330	1.21296	0.606482	−	0.795098i	$-0.292580\pi$
$863$	35.6330	1.21296	0.606482	+	0.795098i	$0.292580\pi$
$864$	60.8758	2.07104
$865$	0	0
$866$	−6.49255	−0.220626
$867$	−134.000	−4.55087
$868$	−2.72804	−0.0925956
$869$	11.4887	0.389726
$870$	0	0
$871$	−77.5441	−2.62748
$872$	−23.6128	−0.799631
$873$	10.8260	0.366405
$874$	0	0
$875$	0	0
$876$	30.0830	1.01641
$877$	14.2717	0.481921	0.240960	−	0.970535i	$-0.422538\pi$
$877$	14.2717	0.481921	0.240960	+	0.970535i	$0.422538\pi$
$878$	−6.80766	−0.229748
$879$	14.2070	0.479190
$880$	0	0
$881$	−15.8766	−0.534896	−0.267448	−	0.963572i	$-0.586180\pi$
$881$	−15.8766	−0.534896	−0.267448	+	0.963572i	$0.586180\pi$
$882$	−10.2761	−0.346015
$883$	13.1143	0.441330	0.220665	−	0.975350i	$-0.429177\pi$
$883$	13.1143	0.441330	0.220665	+	0.975350i	$0.429177\pi$
$884$	75.5214	2.54006
$885$	0	0
$886$	−21.7581	−0.730977
$887$	36.9062	1.23919	0.619594	−	0.784922i	$-0.287297\pi$
$887$	36.9062	1.23919	0.619594	+	0.784922i	$0.287297\pi$
$888$	−65.4478	−2.19628
$889$	35.6438	1.19546
$890$	0	0
$891$	−28.7867	−0.964392
$892$	24.4175	0.817557
$893$	0	0
$894$	−17.2112	−0.575627
$895$	0	0
$896$	36.0026	1.20276
$897$	−25.7879	−0.861034
$898$	−2.69393	−0.0898977
$899$	−0.615144	−0.0205162
$900$	0	0
$901$	−51.9904	−1.73205
$902$	6.75641	0.224964
$903$	−56.7642	−1.88899
$904$	9.21674	0.306544
$905$	0	0
$906$	−18.6879	−0.620863
$907$	−31.9934	−1.06232	−0.531162	−	0.847270i	$-0.678245\pi$
$907$	−31.9934	−1.06232	−0.531162	+	0.847270i	$0.678245\pi$
$908$	38.5659	1.27985
$909$	101.687	3.37275
$910$	0	0
$911$	20.9529	0.694202	0.347101	−	0.937828i	$-0.387166\pi$
$911$	20.9529	0.694202	0.347101	+	0.937828i	$0.387166\pi$
$912$	0	0
$913$	−28.6629	−0.948602
$914$	5.47578	0.181123
$915$	0	0
$916$	−28.4515	−0.940066
$917$	−55.8991	−1.84595
$918$	49.5015	1.63379
$919$	19.9519	0.658154	0.329077	−	0.944303i	$-0.393262\pi$
$919$	19.9519	0.658154	0.329077	+	0.944303i	$0.393262\pi$
$920$	0	0
$921$	47.5215	1.56589
$922$	−11.7058	−0.385508
$923$	−36.5880	−1.20431
$924$	−31.0250	−1.02065
$925$	0	0
$926$	−6.25003	−0.205389
$927$	57.2831	1.88142
$928$	6.38145	0.209481
$929$	48.0073	1.57507	0.787534	−	0.616271i	$-0.211357\pi$
$929$	48.0073	1.57507	0.787534	+	0.616271i	$0.211357\pi$
$930$	0	0
$931$	0	0
$932$	35.2225	1.15375
$933$	−7.42121	−0.242959
$934$	19.3810	0.634167
$935$	0	0
$936$	−80.0679	−2.61710
$937$	−32.6729	−1.06738	−0.533689	−	0.845681i	$-0.679195\pi$
$937$	−32.6729	−1.06738	−0.533689	+	0.845681i	$0.679195\pi$
$938$	−23.6290	−0.771514
$939$	14.3791	0.469245
$940$	0	0
$941$	−0.931053	−0.0303515	−0.0151757	−	0.999885i	$-0.504831\pi$
$941$	−0.931053	−0.0303515	−0.0151757	+	0.999885i	$0.504831\pi$
$942$	−35.5641	−1.15874
$943$	−8.97187	−0.292164
$944$	10.7947	0.351337
$945$	0	0
$946$	6.32724	0.205716
$947$	−53.8855	−1.75104	−0.875522	−	0.483179i	$-0.839482\pi$
$947$	−53.8855	−1.75104	−0.875522	+	0.483179i	$0.839482\pi$
$948$	−31.4013	−1.01987
$949$	−33.4490	−1.08580
$950$	0	0
$951$	−79.4532	−2.57645
$952$	50.4146	1.63395
$953$	−5.05788	−0.163841	−0.0819204	−	0.996639i	$-0.526105\pi$
$953$	−5.05788	−0.163841	−0.0819204	+	0.996639i	$0.526105\pi$
$954$	25.1606	0.814605
$955$	0	0
$956$	−12.8281	−0.414890
$957$	−6.99582	−0.226143
$958$	6.31897	0.204157
$959$	−26.7115	−0.862559
$960$	0	0
$961$	−30.7291	−0.991261
$962$	33.2175	1.07098
$963$	18.6993	0.602578
$964$	−6.61972	−0.213207
$965$	0	0
$966$	−7.85800	−0.252827
$967$	33.7202	1.08437	0.542184	−	0.840260i	$-0.317598\pi$
$967$	33.7202	1.08437	0.542184	+	0.840260i	$0.317598\pi$
$968$	−15.3338	−0.492846
$969$	0	0
$970$	0	0
$971$	−30.3543	−0.974117	−0.487059	−	0.873369i	$-0.661930\pi$
$971$	−30.3543	−0.974117	−0.487059	+	0.873369i	$0.661930\pi$
$972$	21.8716	0.701533
$973$	−9.89194	−0.317121
$974$	−1.17563	−0.0376695
$975$	0	0
$976$	−4.09567	−0.131099
$977$	−28.9126	−0.924996	−0.462498	−	0.886620i	$-0.653047\pi$
$977$	−28.9126	−0.924996	−0.462498	+	0.886620i	$0.653047\pi$
$978$	21.8326	0.698131
$979$	10.3523	0.330862
$980$	0	0
$981$	−75.1972	−2.40086
$982$	8.28875	0.264505
$983$	−13.3298	−0.425156	−0.212578	−	0.977144i	$-0.568186\pi$
$983$	−13.3298	−0.425156	−0.212578	+	0.977144i	$0.568186\pi$
$984$	−40.4562	−1.28970
$985$	0	0
$986$	5.18911	0.165255
$987$	−4.44193	−0.141388
$988$	0	0
$989$	−8.40198	−0.267167
$990$	0	0
$991$	−56.7803	−1.80369	−0.901844	−	0.432063i	$-0.857786\pi$
$991$	−56.7803	−1.80369	−0.901844	+	0.432063i	$0.857786\pi$
$992$	−2.81044	−0.0892317
$993$	−39.6371	−1.25784
$994$	−11.1490	−0.353624
$995$	0	0
$996$	78.3427	2.48238
$997$	−26.2095	−0.830064	−0.415032	−	0.909807i	$-0.636230\pi$
$997$	−26.2095	−0.830064	−0.415032	+	0.909807i	$0.636230\pi$
$998$	12.4667	0.394627
$999$	−114.151	−3.61159

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.co.1.9	yes	20
5.4	even	2		9025.2.a.cn.1.12	yes	20
19.18	odd	2	inner	9025.2.a.co.1.12	yes	20
95.94	odd	2		9025.2.a.cn.1.9	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9025.2.a.cn.1.9	✓	20	95.94	odd	2
9025.2.a.cn.1.12	yes	20	5.4	even	2
9025.2.a.co.1.9	yes	20	1.1	even	1	trivial
9025.2.a.co.1.12	yes	20	19.18	odd	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:	\( N \)	\(=\)	\( 9025 = 5^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9025.a (trivial)

\(f(q)\)	\(=\)	\(q-0.566010 q^{2} -3.10364 q^{3} -1.67963 q^{4} +1.75669 q^{6} +3.12047 q^{7} +2.08271 q^{8} +6.63258 q^{9} +O(q^{10})\)	\(q-0.566010 q^{2} -3.10364 q^{3} -1.67963 q^{4} +1.75669 q^{6} +3.12047 q^{7} +2.08271 q^{8} +6.63258 q^{9} -1.90725 q^{11} +5.21297 q^{12} -5.79625 q^{13} -1.76621 q^{14} +2.18043 q^{16} +7.75726 q^{17} -3.75410 q^{18} -9.68480 q^{21} +1.07952 q^{22} -1.43350 q^{23} -6.46398 q^{24} +3.28074 q^{26} -11.2742 q^{27} -5.24124 q^{28} -1.18185 q^{29} +0.520495 q^{31} -5.39956 q^{32} +5.91940 q^{33} -4.39069 q^{34} -11.1403 q^{36} +10.1250 q^{37} +17.9895 q^{39} +6.25872 q^{41} +5.48169 q^{42} +5.86116 q^{43} +3.20347 q^{44} +0.811375 q^{46} +0.458650 q^{47} -6.76727 q^{48} +2.73731 q^{49} -24.0757 q^{51} +9.73558 q^{52} -6.70216 q^{53} +6.38131 q^{54} +6.49902 q^{56} +0.668936 q^{58} +4.95071 q^{59} -1.87838 q^{61} -0.294605 q^{62} +20.6967 q^{63} -1.30466 q^{64} -3.35044 q^{66} +13.3783 q^{67} -13.0293 q^{68} +4.44907 q^{69} +6.31236 q^{71} +13.8137 q^{72} +5.77080 q^{73} -5.73085 q^{74} -5.95150 q^{77} -10.1822 q^{78} -6.02369 q^{79} +15.0933 q^{81} -3.54250 q^{82} +15.0284 q^{83} +16.2669 q^{84} -3.31748 q^{86} +3.66802 q^{87} -3.97224 q^{88} -5.42789 q^{89} -18.0870 q^{91} +2.40775 q^{92} -1.61543 q^{93} -0.259600 q^{94} +16.7583 q^{96} +1.63225 q^{97} -1.54934 q^{98} -12.6500 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 24 q^{4} - 20 q^{6} + 8 q^{7} + 16 q^{9}+O(q^{10}) \) 20 * q + 24 * q^4 - 20 * q^6 + 8 * q^7 + 16 * q^9	\( 20 q + 24 q^{4} - 20 q^{6} + 8 q^{7} + 16 q^{9} + 10 q^{11} + 40 q^{16} + 38 q^{17} + 56 q^{23} - 54 q^{24} - 22 q^{26} - 16 q^{28} - 32 q^{36} + 80 q^{39} + 44 q^{43} - 16 q^{44} + 98 q^{47} - 4 q^{49} - 54 q^{54} + 50 q^{58} + 40 q^{61} + 90 q^{62} + 6 q^{63} + 46 q^{64} + 8 q^{66} + 4 q^{68} + 34 q^{73} + 20 q^{74} + 80 q^{77} + 60 q^{81} - 10 q^{82} + 84 q^{83} + 90 q^{87} + 78 q^{92} + 20 q^{93} - 120 q^{96} - 76 q^{99}+O(q^{100}) \) 20 * q + 24 * q^4 - 20 * q^6 + 8 * q^7 + 16 * q^9 + 10 * q^11 + 40 * q^16 + 38 * q^17 + 56 * q^23 - 54 * q^24 - 22 * q^26 - 16 * q^28 - 32 * q^36 + 80 * q^39 + 44 * q^43 - 16 * q^44 + 98 * q^47 - 4 * q^49 - 54 * q^54 + 50 * q^58 + 40 * q^61 + 90 * q^62 + 6 * q^63 + 46 * q^64 + 8 * q^66 + 4 * q^68 + 34 * q^73 + 20 * q^74 + 80 * q^77 + 60 * q^81 - 10 * q^82 + 84 * q^83 + 90 * q^87 + 78 * q^92 + 20 * q^93 - 120 * q^96 - 76 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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