LMFDB - Embedded newform 9025.2.a.bc.1.1

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:	$1$
Dimension:	$3$
Coefficient field:	3.3.169.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x - 1 $ x^3 - x^2 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 475)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.65109$ 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)

$f(q)$	$=$	$q-1.65109 q^{2} +2.37720 q^{3} +0.726109 q^{4} -3.92498 q^{6} +0.377203 q^{7} +2.10331 q^{8} +2.65109 q^{9} +O(q^{10})$	\(q-1.65109 q^{2} +2.37720 q^{3} +0.726109 q^{4} -3.92498 q^{6} +0.377203 q^{7} +2.10331 q^{8} +2.65109 q^{9} -1.37720 q^{11} +1.72611 q^{12} +2.82167 q^{13} -0.622797 q^{14} -4.92498 q^{16} -6.37720 q^{17} -4.37720 q^{18} +0.896688 q^{21} +2.27389 q^{22} -6.19887 q^{23} +5.00000 q^{24} -4.65884 q^{26} -0.829422 q^{27} +0.273891 q^{28} +3.37720 q^{29} -2.48052 q^{31} +3.92498 q^{32} -3.27389 q^{33} +10.5294 q^{34} +1.92498 q^{36} +5.58383 q^{37} +6.70769 q^{39} -8.50106 q^{41} -1.48052 q^{42} +12.1522 q^{43} -1.00000 q^{44} +10.2349 q^{46} +6.87826 q^{47} -11.7077 q^{48} -6.85772 q^{49} -15.1599 q^{51} +2.04884 q^{52} +11.5478 q^{53} +1.36945 q^{54} +0.793375 q^{56} -5.57608 q^{58} -6.05659 q^{59} +5.02830 q^{61} +4.09556 q^{62} +1.00000 q^{63} +3.36945 q^{64} +5.40550 q^{66} +3.22717 q^{67} -4.63055 q^{68} -14.7360 q^{69} +2.30219 q^{71} +5.57608 q^{72} +3.19887 q^{73} -9.21942 q^{74} -0.519485 q^{77} -11.0750 q^{78} +6.71836 q^{79} -9.92498 q^{81} +14.0360 q^{82} -18.2165 q^{83} +0.651093 q^{84} -20.0643 q^{86} +8.02830 q^{87} -2.89669 q^{88} -1.50106 q^{89} +1.06434 q^{91} -4.50106 q^{92} -5.89669 q^{93} -11.3567 q^{94} +9.33048 q^{96} -11.7827 q^{97} +11.3227 q^{98} -3.65109 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} + 2 q^{3} + 4 q^{4} - 3 q^{6} - 4 q^{7} + 3 q^{8} + q^{9}+O(q^{10}) $ 3 * q + 2 * q^2 + 2 * q^3 + 4 * q^4 - 3 * q^6 - 4 * q^7 + 3 * q^8 + q^9	$ 3 q + 2 q^{2} + 2 q^{3} + 4 q^{4} - 3 q^{6} - 4 q^{7} + 3 q^{8} + q^{9} + q^{11} + 7 q^{12} + 3 q^{13} - 7 q^{14} - 6 q^{16} - 14 q^{17} - 8 q^{18} + 6 q^{21} + 5 q^{22} - 8 q^{23} + 15 q^{24} - 11 q^{26} - q^{27} - q^{28} + 5 q^{29} + q^{31} + 3 q^{32} - 8 q^{33} - 5 q^{34} - 3 q^{36} + 5 q^{37} - 11 q^{39} - q^{41} + 4 q^{42} + 5 q^{43} - 3 q^{44} + 12 q^{46} - 9 q^{47} - 4 q^{48} - 7 q^{49} - 18 q^{51} - 22 q^{52} + 31 q^{53} - 5 q^{54} + 9 q^{56} - q^{58} + 6 q^{59} + 3 q^{61} + 5 q^{62} + 3 q^{63} + q^{64} - q^{66} - 13 q^{67} - 23 q^{68} - q^{69} - 7 q^{71} + q^{72} - q^{73} - q^{74} - 10 q^{77} - 42 q^{78} + 18 q^{79} - 21 q^{81} + 34 q^{82} - 3 q^{83} - 5 q^{84} - 40 q^{86} + 12 q^{87} - 12 q^{88} + 20 q^{89} - 17 q^{91} + 11 q^{92} - 21 q^{93} - 45 q^{94} + 2 q^{96} - 13 q^{97} + 4 q^{98} - 4 q^{99}+O(q^{100}) $ 3 * q + 2 * q^2 + 2 * q^3 + 4 * q^4 - 3 * q^6 - 4 * q^7 + 3 * q^8 + q^9 + q^11 + 7 * q^12 + 3 * q^13 - 7 * q^14 - 6 * q^16 - 14 * q^17 - 8 * q^18 + 6 * q^21 + 5 * q^22 - 8 * q^23 + 15 * q^24 - 11 * q^26 - q^27 - q^28 + 5 * q^29 + q^31 + 3 * q^32 - 8 * q^33 - 5 * q^34 - 3 * q^36 + 5 * q^37 - 11 * q^39 - q^41 + 4 * q^42 + 5 * q^43 - 3 * q^44 + 12 * q^46 - 9 * q^47 - 4 * q^48 - 7 * q^49 - 18 * q^51 - 22 * q^52 + 31 * q^53 - 5 * q^54 + 9 * q^56 - q^58 + 6 * q^59 + 3 * q^61 + 5 * q^62 + 3 * q^63 + q^64 - q^66 - 13 * q^67 - 23 * q^68 - q^69 - 7 * q^71 + q^72 - q^73 - q^74 - 10 * q^77 - 42 * q^78 + 18 * q^79 - 21 * q^81 + 34 * q^82 - 3 * q^83 - 5 * q^84 - 40 * q^86 + 12 * q^87 - 12 * q^88 + 20 * q^89 - 17 * q^91 + 11 * q^92 - 21 * q^93 - 45 * q^94 + 2 * q^96 - 13 * q^97 + 4 * q^98 - 4 * 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 $

$2$	−1.65109	−1.16750	−0.583750	−	0.811934i	$-0.698415\pi$
$2$	−1.65109	−1.16750	−0.583750	+	0.811934i	$0.698415\pi$
$3$	2.37720	1.37248	0.686239	−	0.727376i	$-0.259260\pi$
$3$	2.37720	1.37248	0.686239	+	0.727376i	$0.259260\pi$
$4$	0.726109	0.363055
$5$	0	0
$5$	0	0
$6$	−3.92498	−1.60237
$7$	0.377203	0.142569	0.0712846	−	0.997456i	$-0.477290\pi$
$7$	0.377203	0.142569	0.0712846	+	0.997456i	$0.477290\pi$
$8$	2.10331	0.743633
$9$	2.65109	0.883698
$10$	0	0
$11$	−1.37720	−0.415242	−0.207621	−	0.978209i	$-0.566572\pi$
$11$	−1.37720	−0.415242	−0.207621	+	0.978209i	$0.566572\pi$
$12$	1.72611	0.498285
$13$	2.82167	0.782591	0.391295	−	0.920265i	$-0.372027\pi$
$13$	2.82167	0.782591	0.391295	+	0.920265i	$0.372027\pi$
$14$	−0.622797	−0.166450
$15$	0	0
$16$	−4.92498	−1.23125
$17$	−6.37720	−1.54670	−0.773349	−	0.633980i	$-0.781420\pi$
$17$	−6.37720	−1.54670	−0.773349	+	0.633980i	$0.781420\pi$
$18$	−4.37720	−1.03172
$19$	0	0
$19$	0	0
$20$	0	0
$21$	0.896688	0.195673
$22$	2.27389	0.484795
$23$	−6.19887	−1.29255	−0.646277	−	0.763103i	$-0.723675\pi$
$23$	−6.19887	−1.29255	−0.646277	+	0.763103i	$0.723675\pi$
$24$	5.00000	1.02062
$25$	0	0
$26$	−4.65884	−0.913674
$27$	−0.829422	−0.159622
$28$	0.273891	0.0517604
$29$	3.37720	0.627131	0.313565	−	0.949567i	$-0.398476\pi$
$29$	3.37720	0.627131	0.313565	+	0.949567i	$0.398476\pi$
$30$	0	0
$31$	−2.48052	−0.445514	−0.222757	−	0.974874i	$-0.571506\pi$
$31$	−2.48052	−0.445514	−0.222757	+	0.974874i	$0.571506\pi$
$32$	3.92498	0.693846
$33$	−3.27389	−0.569911
$34$	10.5294	1.80577
$35$	0	0
$36$	1.92498	0.320831
$37$	5.58383	0.917976	0.458988	−	0.888443i	$-0.348212\pi$
$37$	5.58383	0.917976	0.458988	+	0.888443i	$0.348212\pi$
$38$	0	0
$39$	6.70769	1.07409
$40$	0	0
$41$	−8.50106	−1.32764	−0.663821	−	0.747891i	$-0.731066\pi$
$41$	−8.50106	−1.32764	−0.663821	+	0.747891i	$0.731066\pi$
$42$	−1.48052	−0.228448
$43$	12.1522	1.85319	0.926593	−	0.376065i	$-0.122723\pi$
$43$	12.1522	1.85319	0.926593	+	0.376065i	$0.122723\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	10.2349	1.50906
$47$	6.87826	1.00330	0.501649	−	0.865071i	$-0.332727\pi$
$47$	6.87826	1.00330	0.501649	+	0.865071i	$0.332727\pi$
$48$	−11.7077	−1.68986
$49$	−6.85772	−0.979674
$50$	0	0
$51$	−15.1599	−2.12281
$52$	2.04884	0.284123
$53$	11.5478	1.58621	0.793105	−	0.609085i	$-0.208463\pi$
$53$	11.5478	1.58621	0.793105	+	0.609085i	$0.208463\pi$
$54$	1.36945	0.186359
$55$	0	0
$56$	0.793375	0.106019
$57$	0	0
$58$	−5.57608	−0.732175
$59$	−6.05659	−0.788501	−0.394251	−	0.919003i	$-0.628996\pi$
$59$	−6.05659	−0.788501	−0.394251	+	0.919003i	$0.628996\pi$
$60$	0	0
$61$	5.02830	0.643807	0.321904	−	0.946772i	$-0.395677\pi$
$61$	5.02830	0.643807	0.321904	+	0.946772i	$0.395677\pi$
$62$	4.09556	0.520137
$63$	1.00000	0.125988
$64$	3.36945	0.421182
$65$	0	0
$66$	5.40550	0.665371
$67$	3.22717	0.394262	0.197131	−	0.980377i	$-0.436838\pi$
$67$	3.22717	0.394262	0.197131	+	0.980377i	$0.436838\pi$
$68$	−4.63055	−0.561536
$69$	−14.7360	−1.77400
$70$	0	0
$71$	2.30219	0.273219	0.136610	−	0.990625i	$-0.456379\pi$
$71$	2.30219	0.273219	0.136610	+	0.990625i	$0.456379\pi$
$72$	5.57608	0.657147
$73$	3.19887	0.374400	0.187200	−	0.982322i	$-0.440059\pi$
$73$	3.19887	0.374400	0.187200	+	0.982322i	$0.440059\pi$
$74$	−9.21942	−1.07174
$75$	0	0
$76$	0	0
$77$	−0.519485	−0.0592008
$78$	−11.0750	−1.25400
$79$	6.71836	0.755874	0.377937	−	0.925831i	$-0.376634\pi$
$79$	6.71836	0.755874	0.377937	+	0.925831i	$0.376634\pi$
$80$	0	0
$81$	−9.92498	−1.10278
$82$	14.0360	1.55002
$83$	−18.2165	−1.99952	−0.999760	−	0.0218996i	$-0.993029\pi$
$83$	−18.2165	−1.99952	−0.999760	+	0.0218996i	$0.993029\pi$
$84$	0.651093	0.0710401
$85$	0	0
$86$	−20.0643	−2.16359
$87$	8.02830	0.860724
$88$	−2.89669	−0.308788
$89$	−1.50106	−0.159112	−0.0795561	−	0.996830i	$-0.525350\pi$
$89$	−1.50106	−0.159112	−0.0795561	+	0.996830i	$0.525350\pi$
$90$	0	0
$91$	1.06434	0.111573
$92$	−4.50106	−0.469268
$93$	−5.89669	−0.611458
$94$	−11.3567	−1.17135
$95$	0	0
$96$	9.33048	0.952288
$97$	−11.7827	−1.19635	−0.598176	−	0.801365i	$-0.704108\pi$
$97$	−11.7827	−1.19635	−0.598176	+	0.801365i	$0.704108\pi$
$98$	11.3227	1.14377
$99$	−3.65109	−0.366949
$100$	0	0
$101$	−9.18820	−0.914260	−0.457130	−	0.889400i	$-0.651123\pi$
$101$	−9.18820	−0.914260	−0.457130	+	0.889400i	$0.651123\pi$
$102$	25.0304	2.47838
$103$	−9.47277	−0.933379	−0.466690	−	0.884421i	$-0.654553\pi$
$103$	−9.47277	−0.933379	−0.466690	+	0.884421i	$0.654553\pi$
$104$	5.93486	0.581961
$105$	0	0
$106$	−19.0665	−1.85190
$107$	−15.3510	−1.48404	−0.742020	−	0.670378i	$-0.766132\pi$
$107$	−15.3510	−1.48404	−0.742020	+	0.670378i	$0.766132\pi$
$108$	−0.602251	−0.0579516
$109$	16.7643	1.60573	0.802863	−	0.596163i	$-0.203309\pi$
$109$	16.7643	1.60573	0.802863	+	0.596163i	$0.203309\pi$
$110$	0	0
$111$	13.2739	1.25990
$112$	−1.85772	−0.175538
$113$	−13.3305	−1.25403	−0.627013	−	0.779009i	$-0.715723\pi$
$113$	−13.3305	−1.25403	−0.627013	+	0.779009i	$0.715723\pi$
$114$	0	0
$115$	0	0
$116$	2.45222	0.227683
$117$	7.48052	0.691574
$118$	10.0000	0.920575
$119$	−2.40550	−0.220512
$120$	0	0
$121$	−9.10331	−0.827574
$122$	−8.30219	−0.751645
$123$	−20.2087	−1.82216
$124$	−1.80113	−0.161746
$125$	0	0
$126$	−1.65109	−0.147091
$127$	−3.04672	−0.270353	−0.135176	−	0.990822i	$-0.543160\pi$
$127$	−3.04672	−0.270353	−0.135176	+	0.990822i	$0.543160\pi$
$128$	−13.4132	−1.18557
$129$	28.8881	2.54346
$130$	0	0
$131$	−8.70769	−0.760794	−0.380397	−	0.924823i	$-0.624213\pi$
$131$	−8.70769	−0.760794	−0.380397	+	0.924823i	$0.624213\pi$
$132$	−2.37720	−0.206909
$133$	0	0
$134$	−5.32836	−0.460300
$135$	0	0
$136$	−13.4132	−1.15018
$137$	−6.59450	−0.563406	−0.281703	−	0.959502i	$-0.590899\pi$
$137$	−6.59450	−0.563406	−0.281703	+	0.959502i	$0.590899\pi$
$138$	24.3305	2.07115
$139$	−10.1677	−0.862409	−0.431205	−	0.902254i	$-0.641911\pi$
$139$	−10.1677	−0.862409	−0.431205	+	0.902254i	$0.641911\pi$
$140$	0	0
$141$	16.3510	1.37701
$142$	−3.80113	−0.318983
$143$	−3.88601	−0.324965
$144$	−13.0566	−1.08805
$145$	0	0
$146$	−5.28164	−0.437112
$147$	−16.3022	−1.34458
$148$	4.05447	0.333275
$149$	−5.54003	−0.453857	−0.226929	−	0.973911i	$-0.572868\pi$
$149$	−5.54003	−0.453857	−0.226929	+	0.973911i	$0.572868\pi$
$150$	0	0
$151$	−5.12386	−0.416974	−0.208487	−	0.978025i	$-0.566854\pi$
$151$	−5.12386	−0.416974	−0.208487	+	0.978025i	$0.566854\pi$
$152$	0	0
$153$	−16.9066	−1.36681
$154$	0.857718	0.0691169
$155$	0	0
$156$	4.87051	0.389953
$157$	−19.7643	−1.57736	−0.788681	−	0.614803i	$-0.789235\pi$
$157$	−19.7643	−1.57736	−0.788681	+	0.614803i	$0.789235\pi$
$158$	−11.0926	−0.882483
$159$	27.4514	2.17704
$160$	0	0
$161$	−2.33823	−0.184279
$162$	16.3871	1.28749
$163$	−13.5195	−1.05893	−0.529464	−	0.848332i	$-0.677607\pi$
$163$	−13.5195	−1.05893	−0.529464	+	0.848332i	$0.677607\pi$
$164$	−6.17270	−0.482007
$165$	0	0
$166$	30.0771	2.33444
$167$	13.1054	1.01413	0.507065	−	0.861908i	$-0.330731\pi$
$167$	13.1054	1.01413	0.507065	+	0.861908i	$0.330731\pi$
$168$	1.88601	0.145509
$169$	−5.03817	−0.387551
$170$	0	0
$171$	0	0
$172$	8.82379	0.672808
$173$	1.48827	0.113151	0.0565754	−	0.998398i	$-0.481982\pi$
$173$	1.48827	0.113151	0.0565754	+	0.998398i	$0.481982\pi$
$174$	−13.2555	−1.00489
$175$	0	0
$176$	6.78270	0.511265
$177$	−14.3977	−1.08220
$178$	2.47839	0.185763
$179$	17.1132	1.27910	0.639550	−	0.768750i	$-0.279121\pi$
$179$	17.1132	1.27910	0.639550	+	0.768750i	$0.279121\pi$
$180$	0	0
$181$	5.73891	0.426569	0.213285	−	0.976990i	$-0.431584\pi$
$181$	5.73891	0.426569	0.213285	+	0.976990i	$0.431584\pi$
$182$	−1.75733	−0.130262
$183$	11.9533	0.883612
$184$	−13.0382	−0.961187
$185$	0	0
$186$	9.73598	0.713877
$187$	8.78270	0.642255
$188$	4.99437	0.364252
$189$	−0.312860	−0.0227572
$190$	0	0
$191$	22.3948	1.62043	0.810216	−	0.586131i	$-0.199350\pi$
$191$	22.3948	1.62043	0.810216	+	0.586131i	$0.199350\pi$
$192$	8.00987	0.578063
$193$	−21.3687	−1.53815	−0.769075	−	0.639159i	$-0.779283\pi$
$193$	−21.3687	−1.53815	−0.769075	+	0.639159i	$0.779283\pi$
$194$	19.4543	1.39674
$195$	0	0
$196$	−4.97945	−0.355675
$197$	−1.11399	−0.0793682	−0.0396841	−	0.999212i	$-0.512635\pi$
$197$	−1.11399	−0.0793682	−0.0396841	+	0.999212i	$0.512635\pi$
$198$	6.02830	0.428412
$199$	−2.22505	−0.157729	−0.0788647	−	0.996885i	$-0.525130\pi$
$199$	−2.22505	−0.157729	−0.0788647	+	0.996885i	$0.525130\pi$
$200$	0	0
$201$	7.67164	0.541116
$202$	15.1706	1.06740
$203$	1.27389	0.0894096
$204$	−11.0078	−0.770697
$205$	0	0
$206$	15.6404	1.08972
$207$	−16.4338	−1.14223
$208$	−13.8967	−0.963562
$209$	0	0
$210$	0	0
$211$	9.75441	0.671521	0.335760	−	0.941947i	$-0.391007\pi$
$211$	9.75441	0.671521	0.335760	+	0.941947i	$0.391007\pi$
$212$	8.38495	0.575881
$213$	5.47277	0.374988
$214$	25.3460	1.73262
$215$	0	0
$216$	−1.74453	−0.118700
$217$	−0.935657	−0.0635166
$218$	−27.6794	−1.87468
$219$	7.60437	0.513856
$220$	0	0
$221$	−17.9944	−1.21043
$222$	−21.9164	−1.47093
$223$	−18.6433	−1.24845	−0.624225	−	0.781244i	$-0.714585\pi$
$223$	−18.6433	−1.24845	−0.624225	+	0.781244i	$0.714585\pi$
$224$	1.48052	0.0989211
$225$	0	0
$226$	22.0099	1.46407
$227$	−8.04672	−0.534080	−0.267040	−	0.963686i	$-0.586046\pi$
$227$	−8.04672	−0.534080	−0.267040	+	0.963686i	$0.586046\pi$
$228$	0	0
$229$	20.1415	1.33099	0.665493	−	0.746404i	$-0.268221\pi$
$229$	20.1415	1.33099	0.665493	+	0.746404i	$0.268221\pi$
$230$	0	0
$231$	−1.23492	−0.0812518
$232$	7.10331	0.466355
$233$	1.73386	0.113589	0.0567945	−	0.998386i	$-0.481912\pi$
$233$	1.73386	0.113589	0.0567945	+	0.998386i	$0.481912\pi$
$234$	−12.3510	−0.807412
$235$	0	0
$236$	−4.39775	−0.286269
$237$	15.9709	1.03742
$238$	3.97170	0.257447
$239$	−18.3150	−1.18470	−0.592349	−	0.805682i	$-0.701799\pi$
$239$	−18.3150	−1.18470	−0.592349	+	0.805682i	$0.701799\pi$
$240$	0	0
$241$	2.19675	0.141505	0.0707526	−	0.997494i	$-0.477460\pi$
$241$	2.19675	0.141505	0.0707526	+	0.997494i	$0.477460\pi$
$242$	15.0304	0.966192
$243$	−21.1054	−1.35391
$244$	3.65109	0.233737
$245$	0	0
$246$	33.3665	2.12737
$247$	0	0
$248$	−5.21730	−0.331299
$249$	−43.3043	−2.74430
$250$	0	0
$251$	−13.6716	−0.862946	−0.431473	−	0.902126i	$-0.642006\pi$
$251$	−13.6716	−0.862946	−0.431473	+	0.902126i	$0.642006\pi$
$252$	0.726109	0.0457406
$253$	8.53711	0.536723
$254$	5.03042	0.315637
$255$	0	0
$256$	15.4076	0.962976
$257$	28.4904	1.77718	0.888591	−	0.458701i	$-0.151685\pi$
$257$	28.4904	1.77718	0.888591	+	0.458701i	$0.151685\pi$
$258$	−47.6970	−2.96949
$259$	2.10624	0.130875
$260$	0	0
$261$	8.95328	0.554194
$262$	14.3772	0.888227
$263$	−5.36945	−0.331095	−0.165547	−	0.986202i	$-0.552939\pi$
$263$	−5.36945	−0.331095	−0.165547	+	0.986202i	$0.552939\pi$
$264$	−6.88601	−0.423805
$265$	0	0
$266$	0	0
$267$	−3.56833	−0.218378
$268$	2.34328	0.143139
$269$	−7.06727	−0.430899	−0.215449	−	0.976515i	$-0.569122\pi$
$269$	−7.06727	−0.430899	−0.215449	+	0.976515i	$0.569122\pi$
$270$	0	0
$271$	21.6970	1.31800	0.659000	−	0.752143i	$-0.270980\pi$
$271$	21.6970	1.31800	0.659000	+	0.752143i	$0.270980\pi$
$272$	31.4076	1.90437
$273$	2.53016	0.153132
$274$	10.8881	0.657776
$275$	0	0
$276$	−10.6999	−0.644060
$277$	−25.1132	−1.50891	−0.754453	−	0.656355i	$-0.772098\pi$
$277$	−25.1132	−1.50891	−0.754453	+	0.656355i	$0.772098\pi$
$278$	16.7877	1.00686
$279$	−6.57608	−0.393699
$280$	0	0
$281$	10.0411	0.599001	0.299501	−	0.954096i	$-0.403180\pi$
$281$	10.0411	0.599001	0.299501	+	0.954096i	$0.403180\pi$
$282$	−26.9971	−1.60765
$283$	20.9143	1.24323	0.621613	−	0.783324i	$-0.286478\pi$
$283$	20.9143	1.24323	0.621613	+	0.783324i	$0.286478\pi$
$284$	1.67164	0.0991936
$285$	0	0
$286$	6.41617	0.379396
$287$	−3.20662	−0.189281
$288$	10.4055	0.613150
$289$	23.6687	1.39228
$290$	0	0
$291$	−28.0099	−1.64197
$292$	2.32273	0.135928
$293$	−0.818748	−0.0478318	−0.0239159	−	0.999714i	$-0.507613\pi$
$293$	−0.818748	−0.0478318	−0.0239159	+	0.999714i	$0.507613\pi$
$294$	26.9164	1.56980
$295$	0	0
$296$	11.7445	0.682637
$297$	1.14228	0.0662819
$298$	9.14711	0.529878
$299$	−17.4912	−1.01154
$300$	0	0
$301$	4.58383	0.264207
$302$	8.45997	0.486817
$303$	−21.8422	−1.25480
$304$	0	0
$305$	0	0
$306$	27.9143	1.59575
$307$	−7.44447	−0.424878	−0.212439	−	0.977174i	$-0.568141\pi$
$307$	−7.44447	−0.424878	−0.212439	+	0.977174i	$0.568141\pi$
$308$	−0.377203	−0.0214931
$309$	−22.5187	−1.28104
$310$	0	0
$311$	−0.956204	−0.0542213	−0.0271107	−	0.999632i	$-0.508631\pi$
$311$	−0.956204	−0.0542213	−0.0271107	+	0.999632i	$0.508631\pi$
$312$	14.1084	0.798729
$313$	−5.98158	−0.338099	−0.169049	−	0.985608i	$-0.554070\pi$
$313$	−5.98158	−0.338099	−0.169049	+	0.985608i	$0.554070\pi$
$314$	32.6327	1.84157
$315$	0	0
$316$	4.87826	0.274424
$317$	22.6511	1.27221	0.636106	−	0.771602i	$-0.280544\pi$
$317$	22.6511	1.27221	0.636106	+	0.771602i	$0.280544\pi$
$318$	−45.3249	−2.54169
$319$	−4.65109	−0.260411
$320$	0	0
$321$	−36.4925	−2.03681
$322$	3.86064	0.215145
$323$	0	0
$324$	−7.20662	−0.400368
$325$	0	0
$326$	22.3219	1.23630
$327$	39.8521	2.20383
$328$	−17.8804	−0.987279
$329$	2.59450	0.143039
$330$	0	0
$331$	−4.16283	−0.228810	−0.114405	−	0.993434i	$-0.536496\pi$
$331$	−4.16283	−0.228810	−0.114405	+	0.993434i	$0.536496\pi$
$332$	−13.2272	−0.725935
$333$	14.8032	0.811213
$334$	−21.6383	−1.18399
$335$	0	0
$336$	−4.41617	−0.240922
$337$	−18.0304	−0.982180	−0.491090	−	0.871109i	$-0.663401\pi$
$337$	−18.0304	−0.982180	−0.491090	+	0.871109i	$0.663401\pi$
$338$	8.31849	0.452466
$339$	−31.6893	−1.72112
$340$	0	0
$341$	3.41617	0.184996
$342$	0	0
$343$	−5.22717	−0.282241
$344$	25.5598	1.37809
$345$	0	0
$346$	−2.45726	−0.132103
$347$	−7.43380	−0.399067	−0.199534	−	0.979891i	$-0.563943\pi$
$347$	−7.43380	−0.399067	−0.199534	+	0.979891i	$0.563943\pi$
$348$	5.82942	0.312490
$349$	21.9194	1.17332	0.586658	−	0.809835i	$-0.300443\pi$
$349$	21.9194	1.17332	0.586658	+	0.809835i	$0.300443\pi$
$350$	0	0
$351$	−2.34036	−0.124919
$352$	−5.40550	−0.288114
$353$	−10.3695	−0.551910	−0.275955	−	0.961171i	$-0.588994\pi$
$353$	−10.3695	−0.551910	−0.275955	+	0.961171i	$0.588994\pi$
$354$	23.7720	1.26347
$355$	0	0
$356$	−1.08993	−0.0577664
$357$	−5.71836	−0.302648
$358$	−28.2555	−1.49335
$359$	−23.9893	−1.26611	−0.633054	−	0.774108i	$-0.718199\pi$
$359$	−23.9893	−1.26611	−0.633054	+	0.774108i	$0.718199\pi$
$360$	0	0
$361$	0	0
$362$	−9.47547	−0.498020
$363$	−21.6404	−1.13583
$364$	0.772829	0.0405073
$365$	0	0
$366$	−19.7360	−1.03162
$367$	−35.0275	−1.82842	−0.914210	−	0.405240i	$-0.867188\pi$
$367$	−35.0275	−1.82842	−0.914210	+	0.405240i	$0.867188\pi$
$368$	30.5294	1.59145
$369$	−22.5371	−1.17323
$370$	0	0
$371$	4.35586	0.226145
$372$	−4.28164	−0.221993
$373$	30.8003	1.59478	0.797390	−	0.603464i	$-0.206213\pi$
$373$	30.8003	1.59478	0.797390	+	0.603464i	$0.206213\pi$
$374$	−14.5011	−0.749832
$375$	0	0
$376$	14.4671	0.746086
$377$	9.52936	0.490787
$378$	0.516561	0.0265691
$379$	−0.671640	−0.0344998	−0.0172499	−	0.999851i	$-0.505491\pi$
$379$	−0.671640	−0.0344998	−0.0172499	+	0.999851i	$0.505491\pi$
$380$	0	0
$381$	−7.24267	−0.371053
$382$	−36.9759	−1.89185
$383$	−30.1805	−1.54215	−0.771075	−	0.636745i	$-0.780280\pi$
$383$	−30.1805	−1.54215	−0.771075	+	0.636745i	$0.780280\pi$
$384$	−31.8860	−1.62718
$385$	0	0
$386$	35.2816	1.79579
$387$	32.2165	1.63766
$388$	−8.55553	−0.434341
$389$	−31.5109	−1.59767	−0.798834	−	0.601552i	$-0.794549\pi$
$389$	−31.5109	−1.59767	−0.798834	+	0.601552i	$0.794549\pi$
$390$	0	0
$391$	39.5315	1.99919
$392$	−14.4239	−0.728518
$393$	−20.6999	−1.04417
$394$	1.83929	0.0926623
$395$	0	0
$396$	−2.65109	−0.133222
$397$	12.5761	0.631175	0.315588	−	0.948896i	$-0.397798\pi$
$397$	12.5761	0.631175	0.315588	+	0.948896i	$0.397798\pi$
$398$	3.67376	0.184149
$399$	0	0
$400$	0	0
$401$	−24.6036	−1.22864	−0.614322	−	0.789056i	$-0.710570\pi$
$401$	−24.6036	−1.22864	−0.614322	+	0.789056i	$0.710570\pi$
$402$	−12.6666	−0.631752
$403$	−6.99920	−0.348655
$404$	−6.67164	−0.331926
$405$	0	0
$406$	−2.10331	−0.104386
$407$	−7.69006	−0.381182
$408$	−31.8860	−1.57859
$409$	−13.6666	−0.675770	−0.337885	−	0.941187i	$-0.609711\pi$
$409$	−13.6666	−0.675770	−0.337885	+	0.941187i	$0.609711\pi$
$410$	0	0
$411$	−15.6765	−0.773263
$412$	−6.87826	−0.338868
$413$	−2.28456	−0.112416
$414$	27.1337	1.33355
$415$	0	0
$416$	11.0750	0.542997
$417$	−24.1706	−1.18364
$418$	0	0
$419$	12.6893	0.619911	0.309956	−	0.950751i	$-0.399686\pi$
$419$	12.6893	0.619911	0.309956	+	0.950751i	$0.399686\pi$
$420$	0	0
$421$	−29.1826	−1.42227	−0.711136	−	0.703055i	$-0.751819\pi$
$421$	−29.1826	−1.42227	−0.711136	+	0.703055i	$0.751819\pi$
$422$	−16.1054	−0.784000
$423$	18.2349	0.886612
$424$	24.2886	1.17956
$425$	0	0
$426$	−9.03605	−0.437798
$427$	1.89669	0.0917872
$428$	−11.1465	−0.538788
$429$	−9.23784	−0.446007
$430$	0	0
$431$	29.9816	1.44416	0.722081	−	0.691809i	$-0.243186\pi$
$431$	29.9816	1.44416	0.722081	+	0.691809i	$0.243186\pi$
$432$	4.08489	0.196534
$433$	−4.76216	−0.228855	−0.114427	−	0.993432i	$-0.536503\pi$
$433$	−4.76216	−0.228855	−0.114427	+	0.993432i	$0.536503\pi$
$434$	1.54486	0.0741555
$435$	0	0
$436$	12.1727	0.582967
$437$	0	0
$438$	−12.5555	−0.599926
$439$	−6.88601	−0.328652	−0.164326	−	0.986406i	$-0.552545\pi$
$439$	−6.88601	−0.328652	−0.164326	+	0.986406i	$0.552545\pi$
$440$	0	0
$441$	−18.1805	−0.865736
$442$	29.7104	1.41318
$443$	−8.65109	−0.411026	−0.205513	−	0.978654i	$-0.565886\pi$
$443$	−8.65109	−0.411026	−0.205513	+	0.978654i	$0.565886\pi$
$444$	9.63830	0.457413
$445$	0	0
$446$	30.7819	1.45757
$447$	−13.1698	−0.622909
$448$	1.27097	0.0600476
$449$	−8.63055	−0.407301	−0.203650	−	0.979044i	$-0.565281\pi$
$449$	−8.63055	−0.407301	−0.203650	+	0.979044i	$0.565281\pi$
$450$	0	0
$451$	11.7077	0.551293
$452$	−9.67939	−0.455280
$453$	−12.1805	−0.572288
$454$	13.2859	0.623538
$455$	0	0
$456$	0	0
$457$	28.6092	1.33828	0.669141	−	0.743135i	$-0.266662\pi$
$457$	28.6092	1.33828	0.669141	+	0.743135i	$0.266662\pi$
$458$	−33.2555	−1.55393
$459$	5.28939	0.246888
$460$	0	0
$461$	−39.1025	−1.82119	−0.910593	−	0.413305i	$-0.864374\pi$
$461$	−39.1025	−1.82119	−0.910593	+	0.413305i	$0.864374\pi$
$462$	2.03897	0.0948615
$463$	8.04380	0.373827	0.186913	−	0.982376i	$-0.440152\pi$
$463$	8.04380	0.373827	0.186913	+	0.982376i	$0.440152\pi$
$464$	−16.6327	−0.772152
$465$	0	0
$466$	−2.86276	−0.132615
$467$	−17.0488	−0.788926	−0.394463	−	0.918912i	$-0.629069\pi$
$467$	−17.0488	−0.788926	−0.394463	+	0.918912i	$0.629069\pi$
$468$	5.43167	0.251079
$469$	1.21730	0.0562096
$470$	0	0
$471$	−46.9837	−2.16489
$472$	−12.7389	−0.586356
$473$	−16.7360	−0.769521
$474$	−26.3695	−1.21119
$475$	0	0
$476$	−1.74666	−0.0800578
$477$	30.6142	1.40173
$478$	30.2397	1.38313
$479$	22.5478	1.03023	0.515117	−	0.857120i	$-0.327748\pi$
$479$	22.5478	1.03023	0.515117	+	0.857120i	$0.327748\pi$
$480$	0	0
$481$	15.7557	0.718399
$482$	−3.62704	−0.165207
$483$	−5.55845	−0.252918
$484$	−6.61000	−0.300455
$485$	0	0
$486$	34.8470	1.58069
$487$	−24.5577	−1.11281	−0.556407	−	0.830910i	$-0.687820\pi$
$487$	−24.5577	−1.11281	−0.556407	+	0.830910i	$0.687820\pi$
$488$	10.5761	0.478757
$489$	−32.1386	−1.45336
$490$	0	0
$491$	−16.9298	−0.764032	−0.382016	−	0.924156i	$-0.624770\pi$
$491$	−16.9298	−0.764032	−0.382016	+	0.924156i	$0.624770\pi$
$492$	−14.6738	−0.661544
$493$	−21.5371	−0.969983
$494$	0	0
$495$	0	0
$496$	12.2165	0.548537
$497$	0.868391	0.0389527
$498$	71.4995	3.20397
$499$	0.418295	0.0187255	0.00936273	−	0.999956i	$-0.497020\pi$
$499$	0.418295	0.0187255	0.00936273	+	0.999956i	$0.497020\pi$
$500$	0	0
$501$	31.1543	1.39187
$502$	22.5732	1.00749
$503$	12.5993	0.561776	0.280888	−	0.959741i	$-0.409371\pi$
$503$	12.5993	0.561776	0.280888	+	0.959741i	$0.409371\pi$
$504$	2.10331	0.0936890
$505$	0	0
$506$	−14.0956	−0.626624
$507$	−11.9767	−0.531906
$508$	−2.21225	−0.0981528
$509$	−12.8209	−0.568275	−0.284138	−	0.958784i	$-0.591707\pi$
$509$	−12.8209	−0.568275	−0.284138	+	0.958784i	$0.591707\pi$
$510$	0	0
$511$	1.20662	0.0533779
$512$	1.38708	0.0613007
$513$	0	0
$514$	−47.0403	−2.07486
$515$	0	0
$516$	20.9759	0.923415
$517$	−9.47277	−0.416612
$518$	−3.47759	−0.152797
$519$	3.53791	0.155297
$520$	0	0
$521$	−11.3743	−0.498316	−0.249158	−	0.968463i	$-0.580154\pi$
$521$	−11.3743	−0.498316	−0.249158	+	0.968463i	$0.580154\pi$
$522$	−14.7827	−0.647021
$523$	33.0948	1.44713	0.723566	−	0.690255i	$-0.242502\pi$
$523$	33.0948	1.44713	0.723566	+	0.690255i	$0.242502\pi$
$524$	−6.32273	−0.276210
$525$	0	0
$526$	8.86547	0.386553
$527$	15.8187	0.689075
$528$	16.1239	0.701701
$529$	15.4260	0.670698
$530$	0	0
$531$	−16.0566	−0.696797
$532$	0	0
$533$	−23.9872	−1.03900
$534$	5.89164	0.254956
$535$	0	0
$536$	6.78775	0.293186
$537$	40.6815	1.75554
$538$	11.6687	0.503074
$539$	9.44447	0.406802
$540$	0	0
$541$	31.4338	1.35144	0.675722	−	0.737156i	$-0.263832\pi$
$541$	31.4338	1.35144	0.675722	+	0.737156i	$0.263832\pi$
$542$	−35.8238	−1.53876
$543$	13.6425	0.585458
$544$	−25.0304	−1.07317
$545$	0	0
$546$	−4.17753	−0.178782
$547$	−23.3460	−0.998202	−0.499101	−	0.866544i	$-0.666336\pi$
$547$	−23.3460	−0.998202	−0.499101	+	0.866544i	$0.666336\pi$
$548$	−4.78833	−0.204547
$549$	13.3305	0.568931
$550$	0	0
$551$	0	0
$552$	−30.9944	−1.31921
$553$	2.53418	0.107764
$554$	41.4642	1.76165
$555$	0	0
$556$	−7.38283	−0.313102
$557$	−15.9229	−0.674673	−0.337337	−	0.941384i	$-0.609526\pi$
$557$	−15.9229	−0.674673	−0.337337	+	0.941384i	$0.609526\pi$
$558$	10.8577	0.459644
$559$	34.2894	1.45029
$560$	0	0
$561$	20.8783	0.881481
$562$	−16.5788	−0.699334
$563$	−29.6404	−1.24919	−0.624597	−	0.780947i	$-0.714737\pi$
$563$	−29.6404	−1.24919	−0.624597	+	0.780947i	$0.714737\pi$
$564$	11.8726	0.499928
$565$	0	0
$566$	−34.5315	−1.45147
$567$	−3.74373	−0.157222
$568$	4.84222	0.203175
$569$	7.64334	0.320426	0.160213	−	0.987082i	$-0.448782\pi$
$569$	7.64334	0.320426	0.160213	+	0.987082i	$0.448782\pi$
$570$	0	0
$571$	−10.5577	−0.441824	−0.220912	−	0.975294i	$-0.570903\pi$
$571$	−10.5577	−0.441824	−0.220912	+	0.975294i	$0.570903\pi$
$572$	−2.82167	−0.117980
$573$	53.2370	2.22401
$574$	5.29444	0.220986
$575$	0	0
$576$	8.93273	0.372197
$577$	−38.9447	−1.62129	−0.810645	−	0.585538i	$-0.800883\pi$
$577$	−38.9447	−1.62129	−0.810645	+	0.585538i	$0.800883\pi$
$578$	−39.0793	−1.62548
$579$	−50.7976	−2.11108
$580$	0	0
$581$	−6.87131	−0.285070
$582$	46.2469	1.91700
$583$	−15.9036	−0.658661
$584$	6.72823	0.278416
$585$	0	0
$586$	1.35183	0.0558436
$587$	10.9992	0.453986	0.226993	−	0.973896i	$-0.427111\pi$
$587$	10.9992	0.453986	0.226993	+	0.973896i	$0.427111\pi$
$588$	−11.8372	−0.488157
$589$	0	0
$590$	0	0
$591$	−2.64817	−0.108931
$592$	−27.5003	−1.13025
$593$	−26.7848	−1.09992	−0.549960	−	0.835191i	$-0.685357\pi$
$593$	−26.7848	−1.09992	−0.549960	+	0.835191i	$0.685357\pi$
$594$	−1.88601	−0.0773841
$595$	0	0
$596$	−4.02267	−0.164775
$597$	−5.28939	−0.216480
$598$	28.8796	1.18097
$599$	3.44930	0.140934	0.0704672	−	0.997514i	$-0.477551\pi$
$599$	3.44930	0.140934	0.0704672	+	0.997514i	$0.477551\pi$
$600$	0	0
$601$	37.0686	1.51206	0.756030	−	0.654537i	$-0.227137\pi$
$601$	37.0686	1.51206	0.756030	+	0.654537i	$0.227137\pi$
$602$	−7.56833	−0.308462
$603$	8.55553	0.348408
$604$	−3.72048	−0.151384
$605$	0	0
$606$	36.0635	1.46498
$607$	−14.8732	−0.603685	−0.301843	−	0.953358i	$-0.597602\pi$
$607$	−14.8732	−0.603685	−0.301843	+	0.953358i	$0.597602\pi$
$608$	0	0
$609$	3.02830	0.122713
$610$	0	0
$611$	19.4082	0.785172
$612$	−12.2760	−0.496228
$613$	40.4671	1.63445	0.817226	−	0.576317i	$-0.195511\pi$
$613$	40.4671	1.63445	0.817226	+	0.576317i	$0.195511\pi$
$614$	12.2915	0.496045
$615$	0	0
$616$	−1.09264	−0.0440237
$617$	−8.76991	−0.353063	−0.176532	−	0.984295i	$-0.556488\pi$
$617$	−8.76991	−0.353063	−0.176532	+	0.984295i	$0.556488\pi$
$618$	37.1805	1.49562
$619$	32.0510	1.28824	0.644119	−	0.764926i	$-0.277224\pi$
$619$	32.0510	1.28824	0.644119	+	0.764926i	$0.277224\pi$
$620$	0	0
$621$	5.14148	0.206321
$622$	1.57878	0.0633034
$623$	−0.566205	−0.0226845
$624$	−33.0352	−1.32247
$625$	0	0
$626$	9.87614	0.394730
$627$	0	0
$628$	−14.3510	−0.572668
$629$	−35.6092	−1.41983
$630$	0	0
$631$	−18.9709	−0.755220	−0.377610	−	0.925965i	$-0.623254\pi$
$631$	−18.9709	−0.755220	−0.377610	+	0.925965i	$0.623254\pi$
$632$	14.1308	0.562093
$633$	23.1882	0.921648
$634$	−37.3991	−1.48531
$635$	0	0
$636$	19.9327	0.790384
$637$	−19.3502	−0.766684
$638$	7.67939	0.304030
$639$	6.10331	0.241443
$640$	0	0
$641$	44.3433	1.75145	0.875727	−	0.482806i	$-0.160383\pi$
$641$	44.3433	1.75145	0.875727	+	0.482806i	$0.160383\pi$
$642$	60.2525	2.37798
$643$	−42.9397	−1.69338	−0.846688	−	0.532090i	$-0.821407\pi$
$643$	−42.9397	−1.69338	−0.846688	+	0.532090i	$0.821407\pi$
$644$	−1.69781	−0.0669032
$645$	0	0
$646$	0	0
$647$	−17.9864	−0.707118	−0.353559	−	0.935412i	$-0.615029\pi$
$647$	−17.9864	−0.707118	−0.353559	+	0.935412i	$0.615029\pi$
$648$	−20.8753	−0.820061
$649$	8.34116	0.327419
$650$	0	0
$651$	−2.22425	−0.0871751
$652$	−9.81663	−0.384449
$653$	21.7274	0.850260	0.425130	−	0.905132i	$-0.360228\pi$
$653$	21.7274	0.850260	0.425130	+	0.905132i	$0.360228\pi$
$654$	−65.7995	−2.57297
$655$	0	0
$656$	41.8676	1.63465
$657$	8.48052	0.330856
$658$	−4.28376	−0.166998
$659$	40.8590	1.59164	0.795821	−	0.605532i	$-0.207040\pi$
$659$	40.8590	1.59164	0.795821	+	0.605532i	$0.207040\pi$
$660$	0	0
$661$	−33.7282	−1.31188	−0.655938	−	0.754815i	$-0.727727\pi$
$661$	−33.7282	−1.31188	−0.655938	+	0.754815i	$0.727727\pi$
$662$	6.87322	0.267135
$663$	−42.7763	−1.66129
$664$	−38.3150	−1.48691
$665$	0	0
$666$	−24.4415	−0.947091
$667$	−20.9349	−0.810601
$668$	9.51598	0.368184
$669$	−44.3190	−1.71347
$670$	0	0
$671$	−6.92498	−0.267336
$672$	3.51948	0.135767
$673$	−11.0622	−0.426417	−0.213209	−	0.977007i	$-0.568391\pi$
$673$	−11.0622	−0.426417	−0.213209	+	0.977007i	$0.568391\pi$
$674$	29.7699	1.14669
$675$	0	0
$676$	−3.65826	−0.140702
$677$	−6.14711	−0.236253	−0.118126	−	0.992999i	$-0.537689\pi$
$677$	−6.14711	−0.236253	−0.118126	+	0.992999i	$0.537689\pi$
$678$	52.3219	2.00941
$679$	−4.44447	−0.170563
$680$	0	0
$681$	−19.1287	−0.733013
$682$	−5.64042	−0.215983
$683$	−13.6073	−0.520669	−0.260334	−	0.965519i	$-0.583833\pi$
$683$	−13.6073	−0.520669	−0.260334	+	0.965519i	$0.583833\pi$
$684$	0	0
$685$	0	0
$686$	8.63055	0.329516
$687$	47.8804	1.82675
$688$	−59.8492	−2.28173
$689$	32.5840	1.24135
$690$	0	0
$691$	−40.3043	−1.53325	−0.766624	−	0.642096i	$-0.778065\pi$
$691$	−40.3043	−1.53325	−0.766624	+	0.642096i	$0.778065\pi$
$692$	1.08064	0.0410799
$693$	−1.37720	−0.0523156
$694$	12.2739	0.465911
$695$	0	0
$696$	16.8860	0.640063
$697$	54.2130	2.05346
$698$	−36.1909	−1.36985
$699$	4.12174	0.155898
$700$	0	0
$701$	−3.23704	−0.122261	−0.0611307	−	0.998130i	$-0.519471\pi$
$701$	−3.23704	−0.122261	−0.0611307	+	0.998130i	$0.519471\pi$
$702$	3.86415	0.145843
$703$	0	0
$704$	−4.64042	−0.174892
$705$	0	0
$706$	17.1209	0.644355
$707$	−3.46582	−0.130345
$708$	−10.4543	−0.392898
$709$	20.2370	0.760018	0.380009	−	0.924983i	$-0.375921\pi$
$709$	20.2370	0.760018	0.380009	+	0.924983i	$0.375921\pi$
$710$	0	0
$711$	17.8110	0.667965
$712$	−3.15720	−0.118321
$713$	15.3764	0.575851
$714$	9.44155	0.353341
$715$	0	0
$716$	12.4260	0.464383
$717$	−43.5384	−1.62597
$718$	39.6086	1.47818
$719$	47.8804	1.78564	0.892819	−	0.450416i	$-0.148724\pi$
$719$	47.8804	1.78564	0.892819	+	0.450416i	$0.148724\pi$
$720$	0	0
$721$	−3.57315	−0.133071
$722$	0	0
$723$	5.22212	0.194213
$724$	4.16707	0.154868
$725$	0	0
$726$	35.7304	1.32608
$727$	−17.7926	−0.659890	−0.329945	−	0.944000i	$-0.607030\pi$
$727$	−17.7926	−0.659890	−0.329945	+	0.944000i	$0.607030\pi$
$728$	2.23864	0.0829697
$729$	−20.3969	−0.755443
$730$	0	0
$731$	−77.4968	−2.86632
$732$	8.67939	0.320799
$733$	5.66097	0.209093	0.104546	−	0.994520i	$-0.466661\pi$
$733$	5.66097	0.209093	0.104546	+	0.994520i	$0.466661\pi$
$734$	57.8337	2.13468
$735$	0	0
$736$	−24.3305	−0.896834
$737$	−4.44447	−0.163714
$738$	37.2109	1.36975
$739$	−5.59955	−0.205983	−0.102991	−	0.994682i	$-0.532841\pi$
$739$	−5.59955	−0.205983	−0.102991	+	0.994682i	$0.532841\pi$
$740$	0	0
$741$	0	0
$742$	−7.19193	−0.264024
$743$	5.81663	0.213391	0.106696	−	0.994292i	$-0.465973\pi$
$743$	5.81663	0.213391	0.106696	+	0.994292i	$0.465973\pi$
$744$	−12.4026	−0.454700
$745$	0	0
$746$	−50.8542	−1.86191
$747$	−48.2936	−1.76697
$748$	6.37720	0.233174
$749$	−5.79045	−0.211579
$750$	0	0
$751$	41.1797	1.50267	0.751333	−	0.659923i	$-0.229411\pi$
$751$	41.1797	1.50267	0.751333	+	0.659923i	$0.229411\pi$
$752$	−33.8753	−1.23531
$753$	−32.5003	−1.18438
$754$	−15.7339	−0.572993
$755$	0	0
$756$	−0.227171	−0.00826212
$757$	33.5208	1.21833	0.609167	−	0.793042i	$-0.291504\pi$
$757$	33.5208	1.21833	0.609167	+	0.793042i	$0.291504\pi$
$758$	1.10894	0.0402785
$759$	20.2944	0.736641
$760$	0	0
$761$	31.5032	1.14199	0.570995	−	0.820954i	$-0.306558\pi$
$761$	31.5032	1.14199	0.570995	+	0.820954i	$0.306558\pi$
$762$	11.9583	0.433204
$763$	6.32353	0.228927
$764$	16.2611	0.588306
$765$	0	0
$766$	49.8307	1.80046
$767$	−17.0897	−0.617074
$768$	36.6270	1.32166
$769$	−1.22425	−0.0441475	−0.0220737	−	0.999756i	$-0.507027\pi$
$769$	−1.22425	−0.0441475	−0.0220737	+	0.999756i	$0.507027\pi$
$770$	0	0
$771$	67.7274	2.43914
$772$	−15.5160	−0.558432
$773$	40.6687	1.46275	0.731376	−	0.681974i	$-0.238878\pi$
$773$	40.6687	1.46275	0.731376	+	0.681974i	$0.238878\pi$
$774$	−53.1924	−1.91196
$775$	0	0
$776$	−24.7827	−0.889647
$777$	5.00695	0.179623
$778$	52.0275	1.86528
$779$	0	0
$780$	0	0
$781$	−3.17058	−0.113452
$782$	−65.2702	−2.33406
$783$	−2.80113	−0.100104
$784$	33.7742	1.20622
$785$	0	0
$786$	34.1775	1.21907
$787$	−39.9525	−1.42415	−0.712076	−	0.702102i	$-0.752245\pi$
$787$	−39.9525	−1.42415	−0.712076	+	0.702102i	$0.752245\pi$
$788$	−0.808876	−0.0288150
$789$	−12.7643	−0.454420
$790$	0	0
$791$	−5.02830	−0.178786
$792$	−7.67939	−0.272875
$793$	14.1882	0.503838
$794$	−20.7643	−0.736897
$795$	0	0
$796$	−1.61563	−0.0572644
$797$	−4.53791	−0.160741	−0.0803705	−	0.996765i	$-0.525610\pi$
$797$	−4.53791	−0.160741	−0.0803705	+	0.996765i	$0.525610\pi$
$798$	0	0
$799$	−43.8641	−1.55180
$800$	0	0
$801$	−3.97945	−0.140607
$802$	40.6228	1.43444
$803$	−4.40550	−0.155467
$804$	5.57045	0.196455
$805$	0	0
$806$	11.5563	0.407054
$807$	−16.8003	−0.591399
$808$	−19.3257	−0.679874
$809$	55.5958	1.95465	0.977323	−	0.211756i	$-0.0679183\pi$
$809$	55.5958	1.95465	0.977323	+	0.211756i	$0.0679183\pi$
$810$	0	0
$811$	−42.3326	−1.48650	−0.743249	−	0.669014i	$-0.766716\pi$
$811$	−42.3326	−1.48650	−0.743249	+	0.669014i	$0.766716\pi$
$812$	0.924984	0.0324606
$813$	51.5782	1.80893
$814$	12.6970	0.445030
$815$	0	0
$816$	74.6623	2.61370
$817$	0	0
$818$	22.5648	0.788961
$819$	2.82167	0.0985972
$820$	0	0
$821$	−3.10543	−0.108380	−0.0541902	−	0.998531i	$-0.517258\pi$
$821$	−3.10543	−0.108380	−0.0541902	+	0.998531i	$0.517258\pi$
$822$	25.8833	0.902784
$823$	45.2002	1.57558	0.787790	−	0.615944i	$-0.211225\pi$
$823$	45.2002	1.57558	0.787790	+	0.615944i	$0.211225\pi$
$824$	−19.9242	−0.694092
$825$	0	0
$826$	3.77203	0.131246
$827$	−35.6503	−1.23968	−0.619841	−	0.784727i	$-0.712803\pi$
$827$	−35.6503	−1.23968	−0.619841	+	0.784727i	$0.712803\pi$
$828$	−11.9327	−0.414691
$829$	18.4330	0.640204	0.320102	−	0.947383i	$-0.396283\pi$
$829$	18.4330	0.640204	0.320102	+	0.947383i	$0.396283\pi$
$830$	0	0
$831$	−59.6991	−2.07094
$832$	9.50749	0.329613
$833$	43.7331	1.51526
$834$	39.9079	1.38190
$835$	0	0
$836$	0	0
$837$	2.05739	0.0711139
$838$	−20.9512	−0.723746
$839$	14.1805	0.489564	0.244782	−	0.969578i	$-0.421284\pi$
$839$	14.1805	0.489564	0.244782	+	0.969578i	$0.421284\pi$
$840$	0	0
$841$	−17.5945	−0.606707
$842$	48.1832	1.66050
$843$	23.8697	0.822117
$844$	7.08277	0.243799
$845$	0	0
$846$	−30.1076	−1.03512
$847$	−3.43380	−0.117987
$848$	−56.8726	−1.95301
$849$	49.7176	1.70630
$850$	0	0
$851$	−34.6134	−1.18653
$852$	3.97383	0.136141
$853$	−42.6639	−1.46078	−0.730392	−	0.683028i	$-0.760663\pi$
$853$	−42.6639	−1.46078	−0.730392	+	0.683028i	$0.760663\pi$
$854$	−3.13161	−0.107161
$855$	0	0
$856$	−32.2880	−1.10358
$857$	−1.42392	−0.0486403	−0.0243201	−	0.999704i	$-0.507742\pi$
$857$	−1.42392	−0.0486403	−0.0243201	+	0.999704i	$0.507742\pi$
$858$	15.2525	0.520713
$859$	−8.27894	−0.282474	−0.141237	−	0.989976i	$-0.545108\pi$
$859$	−8.27894	−0.282474	−0.141237	+	0.989976i	$0.545108\pi$
$860$	0	0
$861$	−7.62280	−0.259784
$862$	−49.5024	−1.68606
$863$	30.4154	1.03535	0.517676	−	0.855577i	$-0.326797\pi$
$863$	30.4154	1.03535	0.517676	+	0.855577i	$0.326797\pi$
$864$	−3.25547	−0.110753
$865$	0	0
$866$	7.86276	0.267188
$867$	56.2653	1.91087
$868$	−0.679390	−0.0230600
$869$	−9.25254	−0.313871
$870$	0	0
$871$	9.10602	0.308546
$872$	35.2605	1.19407
$873$	−31.2370	−1.05721
$874$	0	0
$875$	0	0
$876$	5.52161	0.186558
$877$	57.8484	1.95340	0.976700	−	0.214608i	$-0.0688474\pi$
$877$	57.8484	1.95340	0.976700	+	0.214608i	$0.0688474\pi$
$878$	11.3695	0.383700
$879$	−1.94633	−0.0656481
$880$	0	0
$881$	18.4055	0.620097	0.310049	−	0.950721i	$-0.399655\pi$
$881$	18.4055	0.620097	0.310049	+	0.950721i	$0.399655\pi$
$882$	30.0176	1.01075
$883$	18.7947	0.632492	0.316246	−	0.948677i	$-0.397578\pi$
$883$	18.7947	0.632492	0.316246	+	0.948677i	$0.397578\pi$
$884$	−13.0659	−0.439453
$885$	0	0
$886$	14.2838	0.479872
$887$	−11.1471	−0.374283	−0.187142	−	0.982333i	$-0.559922\pi$
$887$	−11.1471	−0.374283	−0.187142	+	0.982333i	$0.559922\pi$
$888$	27.9191	0.936905
$889$	−1.14923	−0.0385440
$890$	0	0
$891$	13.6687	0.457919
$892$	−13.5371	−0.453256
$893$	0	0
$894$	21.7445	0.727246
$895$	0	0
$896$	−5.05952	−0.169027
$897$	−41.5801	−1.38832
$898$	14.2498	0.475523
$899$	−8.37720	−0.279395
$900$	0	0
$901$	−73.6425	−2.45339
$902$	−19.3305	−0.643635
$903$	10.8967	0.362619
$904$	−28.0382	−0.932536
$905$	0	0
$906$	20.1111	0.668145
$907$	38.5598	1.28036	0.640178	−	0.768226i	$-0.278861\pi$
$907$	38.5598	1.28036	0.640178	+	0.768226i	$0.278861\pi$
$908$	−5.84280	−0.193900
$909$	−24.3588	−0.807930
$910$	0	0
$911$	1.79820	0.0595771	0.0297885	−	0.999556i	$-0.490517\pi$
$911$	1.79820	0.0595771	0.0297885	+	0.999556i	$0.490517\pi$
$912$	0	0
$913$	25.0878	0.830285
$914$	−47.2365	−1.56244
$915$	0	0
$916$	14.6249	0.483221
$917$	−3.28456	−0.108466
$918$	−8.73328	−0.288241
$919$	−32.4458	−1.07029	−0.535144	−	0.844761i	$-0.679743\pi$
$919$	−32.4458	−1.07029	−0.535144	+	0.844761i	$0.679743\pi$
$920$	0	0
$921$	−17.6970	−0.583136
$922$	64.5619	2.12623
$923$	6.49602	0.213819
$924$	−0.896688	−0.0294989
$925$	0	0
$926$	−13.2811	−0.436443
$927$	−25.1132	−0.824825
$928$	13.2555	0.435132
$929$	−18.4231	−0.604443	−0.302222	−	0.953238i	$-0.597728\pi$
$929$	−18.4231	−0.604443	−0.302222	+	0.953238i	$0.597728\pi$
$930$	0	0
$931$	0	0
$932$	1.25897	0.0412390
$933$	−2.27309	−0.0744176
$934$	28.1492	0.921071
$935$	0	0
$936$	15.7339	0.514277
$937$	−8.29926	−0.271125	−0.135563	−	0.990769i	$-0.543284\pi$
$937$	−8.29926	−0.271125	−0.135563	+	0.990769i	$0.543284\pi$
$938$	−2.00987	−0.0656247
$939$	−14.2194	−0.464033
$940$	0	0
$941$	−33.6687	−1.09757	−0.548784	−	0.835964i	$-0.684909\pi$
$941$	−33.6687	−1.09757	−0.548784	+	0.835964i	$0.684909\pi$
$942$	77.5745	2.52751
$943$	52.6970	1.71605
$944$	29.8286	0.970839
$945$	0	0
$946$	27.6327	0.898416
$947$	−1.35103	−0.0439026	−0.0219513	−	0.999759i	$-0.506988\pi$
$947$	−1.35103	−0.0439026	−0.0219513	+	0.999759i	$0.506988\pi$
$948$	11.5966	0.376641
$949$	9.02617	0.293002
$950$	0	0
$951$	53.8462	1.74608
$952$	−5.05952	−0.163980
$953$	−6.70769	−0.217283	−0.108642	−	0.994081i	$-0.534650\pi$
$953$	−6.70769	−0.217283	−0.108642	+	0.994081i	$0.534650\pi$
$954$	−50.5470	−1.63652
$955$	0	0
$956$	−13.2987	−0.430110
$957$	−11.0566	−0.357409
$958$	−37.2285	−1.20280
$959$	−2.48746	−0.0803244
$960$	0	0
$961$	−24.8470	−0.801518
$962$	−26.0142	−0.838731
$963$	−40.6970	−1.31144
$964$	1.59508	0.0513741
$965$	0	0
$966$	9.17753	0.295282
$967$	−19.5174	−0.627636	−0.313818	−	0.949483i	$-0.601608\pi$
$967$	−19.5174	−0.627636	−0.313818	+	0.949483i	$0.601608\pi$
$968$	−19.1471	−0.615411
$969$	0	0
$970$	0	0
$971$	8.50669	0.272993	0.136496	−	0.990641i	$-0.456416\pi$
$971$	8.50669	0.272993	0.136496	+	0.990641i	$0.456416\pi$
$972$	−15.3249	−0.491545
$973$	−3.83527	−0.122953
$974$	40.5470	1.29921
$975$	0	0
$976$	−24.7643	−0.792685
$977$	19.7048	0.630411	0.315206	−	0.949023i	$-0.397927\pi$
$977$	19.7048	0.630411	0.315206	+	0.949023i	$0.397927\pi$
$978$	53.0638	1.69679
$979$	2.06727	0.0660701
$980$	0	0
$981$	44.4437	1.41898
$982$	27.9527	0.892006
$983$	−12.4677	−0.397658	−0.198829	−	0.980034i	$-0.563714\pi$
$983$	−12.4677	−0.397658	−0.198829	+	0.980034i	$0.563714\pi$
$984$	−42.5053	−1.35502
$985$	0	0
$986$	35.5598	1.13245
$987$	6.16765	0.196319
$988$	0	0
$989$	−75.3297	−2.39534
$990$	0	0
$991$	−25.0841	−0.796822	−0.398411	−	0.917207i	$-0.630438\pi$
$991$	−25.0841	−0.796822	−0.398411	+	0.917207i	$0.630438\pi$
$992$	−9.73598	−0.309118
$993$	−9.89589	−0.314036
$994$	−1.43380	−0.0454772
$995$	0	0
$996$	−31.4437	−0.996331
$997$	−11.8775	−0.376163	−0.188082	−	0.982153i	$-0.560227\pi$
$997$	−11.8775	−0.376163	−0.188082	+	0.982153i	$0.560227\pi$
$998$	−0.690644	−0.0218620
$999$	−4.63135	−0.146529

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.bc.1.1		3
5.4	even	2		9025.2.a.y.1.3		3
19.18	odd	2		475.2.a.e.1.3	✓	3
57.56	even	2		4275.2.a.bm.1.1		3
76.75	even	2		7600.2.a.cc.1.3		3
95.18	even	4		475.2.b.b.324.2		6
95.37	even	4		475.2.b.b.324.5		6
95.94	odd	2		475.2.a.g.1.1	yes	3
285.284	even	2		4275.2.a.ba.1.3		3
380.379	even	2		7600.2.a.bh.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.a.e.1.3	✓	3	19.18	odd	2
475.2.a.g.1.1	yes	3	95.94	odd	2
475.2.b.b.324.2		6	95.18	even	4
475.2.b.b.324.5		6	95.37	even	4
4275.2.a.ba.1.3		3	285.284	even	2
4275.2.a.bm.1.1		3	57.56	even	2
7600.2.a.bh.1.1		3	380.379	even	2
7600.2.a.cc.1.3		3	76.75	even	2
9025.2.a.y.1.3		3	5.4	even	2
9025.2.a.bc.1.1		3	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-1.65109 q^{2} +2.37720 q^{3} +0.726109 q^{4} -3.92498 q^{6} +0.377203 q^{7} +2.10331 q^{8} +2.65109 q^{9} +O(q^{10})\)	\(q-1.65109 q^{2} +2.37720 q^{3} +0.726109 q^{4} -3.92498 q^{6} +0.377203 q^{7} +2.10331 q^{8} +2.65109 q^{9} -1.37720 q^{11} +1.72611 q^{12} +2.82167 q^{13} -0.622797 q^{14} -4.92498 q^{16} -6.37720 q^{17} -4.37720 q^{18} +0.896688 q^{21} +2.27389 q^{22} -6.19887 q^{23} +5.00000 q^{24} -4.65884 q^{26} -0.829422 q^{27} +0.273891 q^{28} +3.37720 q^{29} -2.48052 q^{31} +3.92498 q^{32} -3.27389 q^{33} +10.5294 q^{34} +1.92498 q^{36} +5.58383 q^{37} +6.70769 q^{39} -8.50106 q^{41} -1.48052 q^{42} +12.1522 q^{43} -1.00000 q^{44} +10.2349 q^{46} +6.87826 q^{47} -11.7077 q^{48} -6.85772 q^{49} -15.1599 q^{51} +2.04884 q^{52} +11.5478 q^{53} +1.36945 q^{54} +0.793375 q^{56} -5.57608 q^{58} -6.05659 q^{59} +5.02830 q^{61} +4.09556 q^{62} +1.00000 q^{63} +3.36945 q^{64} +5.40550 q^{66} +3.22717 q^{67} -4.63055 q^{68} -14.7360 q^{69} +2.30219 q^{71} +5.57608 q^{72} +3.19887 q^{73} -9.21942 q^{74} -0.519485 q^{77} -11.0750 q^{78} +6.71836 q^{79} -9.92498 q^{81} +14.0360 q^{82} -18.2165 q^{83} +0.651093 q^{84} -20.0643 q^{86} +8.02830 q^{87} -2.89669 q^{88} -1.50106 q^{89} +1.06434 q^{91} -4.50106 q^{92} -5.89669 q^{93} -11.3567 q^{94} +9.33048 q^{96} -11.7827 q^{97} +11.3227 q^{98} -3.65109 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 2 q^{3} + 4 q^{4} - 3 q^{6} - 4 q^{7} + 3 q^{8} + q^{9}+O(q^{10}) \) 3 * q + 2 * q^2 + 2 * q^3 + 4 * q^4 - 3 * q^6 - 4 * q^7 + 3 * q^8 + q^9	\( 3 q + 2 q^{2} + 2 q^{3} + 4 q^{4} - 3 q^{6} - 4 q^{7} + 3 q^{8} + q^{9} + q^{11} + 7 q^{12} + 3 q^{13} - 7 q^{14} - 6 q^{16} - 14 q^{17} - 8 q^{18} + 6 q^{21} + 5 q^{22} - 8 q^{23} + 15 q^{24} - 11 q^{26} - q^{27} - q^{28} + 5 q^{29} + q^{31} + 3 q^{32} - 8 q^{33} - 5 q^{34} - 3 q^{36} + 5 q^{37} - 11 q^{39} - q^{41} + 4 q^{42} + 5 q^{43} - 3 q^{44} + 12 q^{46} - 9 q^{47} - 4 q^{48} - 7 q^{49} - 18 q^{51} - 22 q^{52} + 31 q^{53} - 5 q^{54} + 9 q^{56} - q^{58} + 6 q^{59} + 3 q^{61} + 5 q^{62} + 3 q^{63} + q^{64} - q^{66} - 13 q^{67} - 23 q^{68} - q^{69} - 7 q^{71} + q^{72} - q^{73} - q^{74} - 10 q^{77} - 42 q^{78} + 18 q^{79} - 21 q^{81} + 34 q^{82} - 3 q^{83} - 5 q^{84} - 40 q^{86} + 12 q^{87} - 12 q^{88} + 20 q^{89} - 17 q^{91} + 11 q^{92} - 21 q^{93} - 45 q^{94} + 2 q^{96} - 13 q^{97} + 4 q^{98} - 4 q^{99}+O(q^{100}) \) 3 * q + 2 * q^2 + 2 * q^3 + 4 * q^4 - 3 * q^6 - 4 * q^7 + 3 * q^8 + q^9 + q^11 + 7 * q^12 + 3 * q^13 - 7 * q^14 - 6 * q^16 - 14 * q^17 - 8 * q^18 + 6 * q^21 + 5 * q^22 - 8 * q^23 + 15 * q^24 - 11 * q^26 - q^27 - q^28 + 5 * q^29 + q^31 + 3 * q^32 - 8 * q^33 - 5 * q^34 - 3 * q^36 + 5 * q^37 - 11 * q^39 - q^41 + 4 * q^42 + 5 * q^43 - 3 * q^44 + 12 * q^46 - 9 * q^47 - 4 * q^48 - 7 * q^49 - 18 * q^51 - 22 * q^52 + 31 * q^53 - 5 * q^54 + 9 * q^56 - q^58 + 6 * q^59 + 3 * q^61 + 5 * q^62 + 3 * q^63 + q^64 - q^66 - 13 * q^67 - 23 * q^68 - q^69 - 7 * q^71 + q^72 - q^73 - q^74 - 10 * q^77 - 42 * q^78 + 18 * q^79 - 21 * q^81 + 34 * q^82 - 3 * q^83 - 5 * q^84 - 40 * q^86 + 12 * q^87 - 12 * q^88 + 20 * q^89 - 17 * q^91 + 11 * q^92 - 21 * q^93 - 45 * q^94 + 2 * q^96 - 13 * q^97 + 4 * q^98 - 4 * q^99

Introduction

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