LMFDB - Embedded newform 7225.2.a.r.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7225,2,Mod(1,7225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7225, 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("7225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7225 = 5^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7225.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:	$57.6919154604$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	7225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.48119 q^{2} +1.67513 q^{3} +0.193937 q^{4} +2.48119 q^{6} +1.28726 q^{7} -2.67513 q^{8} -0.193937 q^{9} -0.481194 q^{11} +0.324869 q^{12} +2.15633 q^{13} +1.90668 q^{14} -4.35026 q^{16} -0.287258 q^{18} -3.35026 q^{19} +2.15633 q^{21} -0.712742 q^{22} -8.24965 q^{23} -4.48119 q^{24} +3.19394 q^{26} -5.35026 q^{27} +0.249646 q^{28} -0.649738 q^{29} -1.83146 q^{31} -1.09332 q^{32} -0.806063 q^{33} -0.0376114 q^{36} +4.31265 q^{37} -4.96239 q^{38} +3.61213 q^{39} -11.2750 q^{41} +3.19394 q^{42} +8.15633 q^{43} -0.0933212 q^{44} -12.2193 q^{46} +6.54420 q^{47} -7.28726 q^{48} -5.34297 q^{49} +0.418190 q^{52} -8.57452 q^{53} -7.92478 q^{54} -3.44358 q^{56} -5.61213 q^{57} -0.962389 q^{58} -4.96239 q^{59} +2.83638 q^{61} -2.71274 q^{62} -0.249646 q^{63} +7.08110 q^{64} -1.19394 q^{66} -4.93207 q^{67} -13.8192 q^{69} +14.5320 q^{71} +0.518806 q^{72} -13.3503 q^{73} +6.38787 q^{74} -0.649738 q^{76} -0.619421 q^{77} +5.35026 q^{78} +9.05571 q^{79} -8.38058 q^{81} -16.7005 q^{82} -13.4314 q^{83} +0.418190 q^{84} +12.0811 q^{86} -1.08840 q^{87} +1.28726 q^{88} -16.7816 q^{89} +2.77575 q^{91} -1.59991 q^{92} -3.06793 q^{93} +9.69323 q^{94} -1.83146 q^{96} -3.66291 q^{97} -7.91397 q^{98} +0.0933212 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + q^{4} + 2 q^{6} - 2 q^{7} - 3 q^{8} - q^{9} + 4 q^{11} + 6 q^{12} - 4 q^{13} + 12 q^{14} - 3 q^{16} + 5 q^{18} - 4 q^{21} - 8 q^{22} - 8 q^{23} - 8 q^{24} + 10 q^{26} - 6 q^{27} - 16 q^{28}+ \cdots - 6 q^{99}+O(q^{100}) $ 3 * q - q^2 + q^4 + 2 * q^6 - 2 * q^7 - 3 * q^8 - q^9 + 4 * q^11 + 6 * q^12 - 4 * q^13 + 12 * q^14 - 3 * q^16 + 5 * q^18 - 4 * q^21 - 8 * q^22 - 8 * q^23 - 8 * q^24 + 10 * q^26 - 6 * q^27 - 16 * q^28 - 12 * q^29 + 10 * q^31 + 3 * q^32 - 2 * q^33 - 11 * q^36 - 8 * q^37 - 4 * q^38 + 10 * q^39 - 2 * q^41 + 10 * q^42 + 14 * q^43 + 6 * q^44 - 22 * q^46 + 10 * q^47 - 16 * q^48 + 7 * q^49 - 14 * q^53 - 2 * q^54 + 6 * q^56 - 16 * q^57 + 8 * q^58 - 4 * q^59 + 6 * q^61 - 14 * q^62 + 16 * q^63 - 11 * q^64 - 4 * q^66 - 6 * q^67 + 8 * q^71 + 7 * q^72 - 30 * q^73 + 20 * q^74 - 12 * q^76 - 14 * q^77 + 6 * q^78 + 10 * q^79 - 13 * q^81 - 30 * q^82 + 2 * q^83 + 4 * q^86 + 16 * q^87 - 2 * q^88 + 2 * q^89 + 10 * q^91 + 22 * q^92 - 18 * q^93 - 4 * q^94 + 10 * q^96 + 20 * q^97 - 43 * q^98 - 6 * q^99

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 $

$2$	1.48119	1.04736	0.523681	−	0.851914i	$-0.324558\pi$
$2$	1.48119	1.04736	0.523681	+	0.851914i	$0.324558\pi$
$3$	1.67513	0.967137	0.483569	−	0.875306i	$-0.339340\pi$
$3$	1.67513	0.967137	0.483569	+	0.875306i	$0.339340\pi$
$4$	0.193937	0.0969683
$5$	0	0
$5$	0	0
$6$	2.48119	1.01294
$7$	1.28726	0.486538	0.243269	−	0.969959i	$-0.421780\pi$
$7$	1.28726	0.486538	0.243269	+	0.969959i	$0.421780\pi$
$8$	−2.67513	−0.945802
$9$	−0.193937	−0.0646455
$10$	0	0
$11$	−0.481194	−0.145086	−0.0725428	−	0.997365i	$-0.523111\pi$
$11$	−0.481194	−0.145086	−0.0725428	+	0.997365i	$0.523111\pi$
$12$	0.324869	0.0937816
$13$	2.15633	0.598057	0.299028	−	0.954244i	$-0.403337\pi$
$13$	2.15633	0.598057	0.299028	+	0.954244i	$0.403337\pi$
$14$	1.90668	0.509581
$15$	0	0
$16$	−4.35026	−1.08757
$17$	0	0
$17$	0	0
$18$	−0.287258	−0.0677073
$19$	−3.35026	−0.768603	−0.384301	−	0.923208i	$-0.625558\pi$
$19$	−3.35026	−0.768603	−0.384301	+	0.923208i	$0.625558\pi$
$20$	0	0
$21$	2.15633	0.470549
$22$	−0.712742	−0.151957
$23$	−8.24965	−1.72017	−0.860085	−	0.510151i	$-0.829590\pi$
$23$	−8.24965	−1.72017	−0.860085	+	0.510151i	$0.829590\pi$
$24$	−4.48119	−0.914720
$25$	0	0
$26$	3.19394	0.626382
$27$	−5.35026	−1.02966
$28$	0.249646	0.0471787
$29$	−0.649738	−0.120653	−0.0603267	−	0.998179i	$-0.519214\pi$
$29$	−0.649738	−0.120653	−0.0603267	+	0.998179i	$0.519214\pi$
$30$	0	0
$31$	−1.83146	−0.328939	−0.164470	−	0.986382i	$-0.552591\pi$
$31$	−1.83146	−0.328939	−0.164470	+	0.986382i	$0.552591\pi$
$32$	−1.09332	−0.193274
$33$	−0.806063	−0.140318
$34$	0	0
$35$	0	0
$36$	−0.0376114	−0.00626857
$37$	4.31265	0.708995	0.354498	−	0.935057i	$-0.384652\pi$
$37$	4.31265	0.708995	0.354498	+	0.935057i	$0.384652\pi$
$38$	−4.96239	−0.805006
$39$	3.61213	0.578403
$40$	0	0
$41$	−11.2750	−1.76087	−0.880433	−	0.474171i	$-0.842748\pi$
$41$	−11.2750	−1.76087	−0.880433	+	0.474171i	$0.842748\pi$
$42$	3.19394	0.492835
$43$	8.15633	1.24383	0.621914	−	0.783086i	$-0.286355\pi$
$43$	8.15633	1.24383	0.621914	+	0.783086i	$0.286355\pi$
$44$	−0.0933212	−0.0140687
$45$	0	0
$46$	−12.2193	−1.80164
$47$	6.54420	0.954569	0.477285	−	0.878749i	$-0.341621\pi$
$47$	6.54420	0.954569	0.477285	+	0.878749i	$0.341621\pi$
$48$	−7.28726	−1.05183
$49$	−5.34297	−0.763281
$50$	0	0
$51$	0	0
$52$	0.418190	0.0579926
$53$	−8.57452	−1.17780	−0.588900	−	0.808206i	$-0.700439\pi$
$53$	−8.57452	−1.17780	−0.588900	+	0.808206i	$0.700439\pi$
$54$	−7.92478	−1.07843
$55$	0	0
$56$	−3.44358	−0.460168
$57$	−5.61213	−0.743344
$58$	−0.962389	−0.126368
$59$	−4.96239	−0.646048	−0.323024	−	0.946391i	$-0.604699\pi$
$59$	−4.96239	−0.646048	−0.323024	+	0.946391i	$0.604699\pi$
$60$	0	0
$61$	2.83638	0.363161	0.181581	−	0.983376i	$-0.441879\pi$
$61$	2.83638	0.363161	0.181581	+	0.983376i	$0.441879\pi$
$62$	−2.71274	−0.344519
$63$	−0.249646	−0.0314525
$64$	7.08110	0.885138
$65$	0	0
$66$	−1.19394	−0.146963
$67$	−4.93207	−0.602548	−0.301274	−	0.953538i	$-0.597412\pi$
$67$	−4.93207	−0.602548	−0.301274	+	0.953538i	$0.597412\pi$
$68$	0	0
$69$	−13.8192	−1.66364
$70$	0	0
$71$	14.5320	1.72463	0.862314	−	0.506373i	$-0.169014\pi$
$71$	14.5320	1.72463	0.862314	+	0.506373i	$0.169014\pi$
$72$	0.518806	0.0611418
$73$	−13.3503	−1.56253	−0.781265	−	0.624200i	$-0.785425\pi$
$73$	−13.3503	−1.56253	−0.781265	+	0.624200i	$0.785425\pi$
$74$	6.38787	0.742575
$75$	0	0
$76$	−0.649738	−0.0745301
$77$	−0.619421	−0.0705896
$78$	5.35026	0.605798
$79$	9.05571	1.01885	0.509423	−	0.860516i	$-0.329859\pi$
$79$	9.05571	1.01885	0.509423	+	0.860516i	$0.329859\pi$
$80$	0	0
$81$	−8.38058	−0.931175
$82$	−16.7005	−1.84426
$83$	−13.4314	−1.47428	−0.737142	−	0.675738i	$-0.763825\pi$
$83$	−13.4314	−1.47428	−0.737142	+	0.675738i	$0.763825\pi$
$84$	0.418190	0.0456283
$85$	0	0
$86$	12.0811	1.30274
$87$	−1.08840	−0.116688
$88$	1.28726	0.137222
$89$	−16.7816	−1.77885	−0.889424	−	0.457082i	$-0.848894\pi$
$89$	−16.7816	−1.77885	−0.889424	+	0.457082i	$0.848894\pi$
$90$	0	0
$91$	2.77575	0.290977
$92$	−1.59991	−0.166802
$93$	−3.06793	−0.318129
$94$	9.69323	0.999780
$95$	0	0
$96$	−1.83146	−0.186922
$97$	−3.66291	−0.371912	−0.185956	−	0.982558i	$-0.559538\pi$
$97$	−3.66291	−0.371912	−0.185956	+	0.982558i	$0.559538\pi$
$98$	−7.91397	−0.799432
$99$	0.0933212	0.00937913
$100$	0	0
$101$	6.85685	0.682282	0.341141	−	0.940012i	$-0.389187\pi$
$101$	6.85685	0.682282	0.341141	+	0.940012i	$0.389187\pi$
$102$	0	0
$103$	−7.04349	−0.694016	−0.347008	−	0.937862i	$-0.612802\pi$
$103$	−7.04349	−0.694016	−0.347008	+	0.937862i	$0.612802\pi$
$104$	−5.76845	−0.565643
$105$	0	0
$106$	−12.7005	−1.23358
$107$	−5.86177	−0.566679	−0.283340	−	0.959020i	$-0.591442\pi$
$107$	−5.86177	−0.566679	−0.283340	+	0.959020i	$0.591442\pi$
$108$	−1.03761	−0.0998442
$109$	12.7005	1.21649	0.608245	−	0.793750i	$-0.291874\pi$
$109$	12.7005	1.21649	0.608245	+	0.793750i	$0.291874\pi$
$110$	0	0
$111$	7.22425	0.685696
$112$	−5.59991	−0.529142
$113$	−4.88717	−0.459746	−0.229873	−	0.973221i	$-0.573831\pi$
$113$	−4.88717	−0.459746	−0.229873	+	0.973221i	$0.573831\pi$
$114$	−8.31265	−0.778551
$115$	0	0
$116$	−0.126008	−0.0116995
$117$	−0.418190	−0.0386617
$118$	−7.35026	−0.676646
$119$	0	0
$120$	0	0
$121$	−10.7685	−0.978950
$122$	4.20123	0.380362
$123$	−18.8872	−1.70300
$124$	−0.355186	−0.0318967
$125$	0	0
$126$	−0.369775	−0.0329422
$127$	1.76845	0.156925	0.0784624	−	0.996917i	$-0.474999\pi$
$127$	1.76845	0.156925	0.0784624	+	0.996917i	$0.474999\pi$
$128$	12.6751	1.12033
$129$	13.6629	1.20295
$130$	0	0
$131$	−11.1065	−0.970379	−0.485189	−	0.874409i	$-0.661249\pi$
$131$	−11.1065	−0.970379	−0.485189	+	0.874409i	$0.661249\pi$
$132$	−0.156325	−0.0136064
$133$	−4.31265	−0.373954
$134$	−7.30536	−0.631087
$135$	0	0
$136$	0	0
$137$	−11.7685	−1.00545	−0.502723	−	0.864447i	$-0.667669\pi$
$137$	−11.7685	−1.00545	−0.502723	+	0.864447i	$0.667669\pi$
$138$	−20.4690	−1.74243
$139$	11.7054	0.992843	0.496422	−	0.868082i	$-0.334647\pi$
$139$	11.7054	0.992843	0.496422	+	0.868082i	$0.334647\pi$
$140$	0	0
$141$	10.9624	0.923200
$142$	21.5247	1.80631
$143$	−1.03761	−0.0867694
$144$	0.843675	0.0703062
$145$	0	0
$146$	−19.7743	−1.63654
$147$	−8.95017	−0.738198
$148$	0.836381	0.0687501
$149$	17.1998	1.40906	0.704532	−	0.709672i	$-0.251157\pi$
$149$	17.1998	1.40906	0.704532	+	0.709672i	$0.251157\pi$
$150$	0	0
$151$	13.5877	1.10575	0.552875	−	0.833264i	$-0.313531\pi$
$151$	13.5877	1.10575	0.552875	+	0.833264i	$0.313531\pi$
$152$	8.96239	0.726946
$153$	0	0
$154$	−0.917483	−0.0739329
$155$	0	0
$156$	0.700523	0.0560868
$157$	8.57452	0.684321	0.342160	−	0.939642i	$-0.388841\pi$
$157$	8.57452	0.684321	0.342160	+	0.939642i	$0.388841\pi$
$158$	13.4133	1.06710
$159$	−14.3634	−1.13909
$160$	0	0
$161$	−10.6194	−0.836928
$162$	−12.4133	−0.975278
$163$	−14.3757	−1.12599	−0.562994	−	0.826461i	$-0.690351\pi$
$163$	−14.3757	−1.12599	−0.562994	+	0.826461i	$0.690351\pi$
$164$	−2.18664	−0.170748
$165$	0	0
$166$	−19.8945	−1.54411
$167$	7.54912	0.584169	0.292084	−	0.956393i	$-0.405651\pi$
$167$	7.54912	0.584169	0.292084	+	0.956393i	$0.405651\pi$
$168$	−5.76845	−0.445046
$169$	−8.35026	−0.642328
$170$	0	0
$171$	0.649738	0.0496867
$172$	1.58181	0.120612
$173$	−11.8496	−0.900905	−0.450452	−	0.892800i	$-0.648737\pi$
$173$	−11.8496	−0.900905	−0.450452	+	0.892800i	$0.648737\pi$
$174$	−1.61213	−0.122215
$175$	0	0
$176$	2.09332	0.157790
$177$	−8.31265	−0.624817
$178$	−24.8568	−1.86310
$179$	3.22425	0.240992	0.120496	−	0.992714i	$-0.461551\pi$
$179$	3.22425	0.240992	0.120496	+	0.992714i	$0.461551\pi$
$180$	0	0
$181$	6.88717	0.511919	0.255960	−	0.966688i	$-0.417609\pi$
$181$	6.88717	0.511919	0.255960	+	0.966688i	$0.417609\pi$
$182$	4.11142	0.304759
$183$	4.75131	0.351227
$184$	22.0689	1.62694
$185$	0	0
$186$	−4.54420	−0.333197
$187$	0	0
$188$	1.26916	0.0925630
$189$	−6.88717	−0.500968
$190$	0	0
$191$	7.19982	0.520960	0.260480	−	0.965479i	$-0.416119\pi$
$191$	7.19982	0.520960	0.260480	+	0.965479i	$0.416119\pi$
$192$	11.8618	0.856050
$193$	20.1114	1.44765	0.723826	−	0.689983i	$-0.242382\pi$
$193$	20.1114	1.44765	0.723826	+	0.689983i	$0.242382\pi$
$194$	−5.42548	−0.389527
$195$	0	0
$196$	−1.03620	−0.0740141
$197$	−3.16362	−0.225399	−0.112699	−	0.993629i	$-0.535950\pi$
$197$	−3.16362	−0.225399	−0.112699	+	0.993629i	$0.535950\pi$
$198$	0.138227	0.00982335
$199$	21.4944	1.52370	0.761848	−	0.647756i	$-0.224292\pi$
$199$	21.4944	1.52370	0.761848	+	0.647756i	$0.224292\pi$
$200$	0	0
$201$	−8.26187	−0.582747
$202$	10.1563	0.714597
$203$	−0.836381	−0.0587024
$204$	0	0
$205$	0	0
$206$	−10.4328	−0.726886
$207$	1.59991	0.111201
$208$	−9.38058	−0.650426
$209$	1.61213	0.111513
$210$	0	0
$211$	20.8691	1.43669	0.718343	−	0.695689i	$-0.244901\pi$
$211$	20.8691	1.43669	0.718343	+	0.695689i	$0.244901\pi$
$212$	−1.66291	−0.114209
$213$	24.3430	1.66795
$214$	−8.68243	−0.593518
$215$	0	0
$216$	14.3127	0.973853
$217$	−2.35756	−0.160041
$218$	18.8119	1.27411
$219$	−22.3634	−1.51118
$220$	0	0
$221$	0	0
$222$	10.7005	0.718172
$223$	−24.3430	−1.63013	−0.815063	−	0.579373i	$-0.803298\pi$
$223$	−24.3430	−1.63013	−0.815063	+	0.579373i	$0.803298\pi$
$224$	−1.40739	−0.0940349
$225$	0	0
$226$	−7.23884	−0.481521
$227$	−11.2120	−0.744169	−0.372084	−	0.928199i	$-0.621357\pi$
$227$	−11.2120	−0.744169	−0.372084	+	0.928199i	$0.621357\pi$
$228$	−1.08840	−0.0720808
$229$	−12.1563	−0.803313	−0.401656	−	0.915790i	$-0.631565\pi$
$229$	−12.1563	−0.803313	−0.401656	+	0.915790i	$0.631565\pi$
$230$	0	0
$231$	−1.03761	−0.0682698
$232$	1.73813	0.114114
$233$	11.9756	0.784545	0.392273	−	0.919849i	$-0.371689\pi$
$233$	11.9756	0.784545	0.392273	+	0.919849i	$0.371689\pi$
$234$	−0.619421	−0.0404928
$235$	0	0
$236$	−0.962389	−0.0626462
$237$	15.1695	0.985365
$238$	0	0
$239$	−8.83638	−0.571578	−0.285789	−	0.958293i	$-0.592256\pi$
$239$	−8.83638	−0.571578	−0.285789	+	0.958293i	$0.592256\pi$
$240$	0	0
$241$	0.261865	0.0168682	0.00843411	−	0.999964i	$-0.497315\pi$
$241$	0.261865	0.0168682	0.00843411	+	0.999964i	$0.497315\pi$
$242$	−15.9502	−1.02532
$243$	2.01222	0.129084
$244$	0.550078	0.0352151
$245$	0	0
$246$	−27.9756	−1.78366
$247$	−7.22425	−0.459668
$248$	4.89938	0.311111
$249$	−22.4993	−1.42583
$250$	0	0
$251$	−23.6629	−1.49359	−0.746795	−	0.665054i	$-0.768408\pi$
$251$	−23.6629	−1.49359	−0.746795	+	0.665054i	$0.768408\pi$
$252$	−0.0484156	−0.00304989
$253$	3.96968	0.249572
$254$	2.61942	0.164357
$255$	0	0
$256$	4.61213	0.288258
$257$	19.2447	1.20045	0.600226	−	0.799830i	$-0.295077\pi$
$257$	19.2447	1.20045	0.600226	+	0.799830i	$0.295077\pi$
$258$	20.2374	1.25993
$259$	5.55149	0.344953
$260$	0	0
$261$	0.126008	0.00779970
$262$	−16.4509	−1.01634
$263$	−17.2447	−1.06336	−0.531678	−	0.846947i	$-0.678438\pi$
$263$	−17.2447	−1.06336	−0.531678	+	0.846947i	$0.678438\pi$
$264$	2.15633	0.132713
$265$	0	0
$266$	−6.38787	−0.391666
$267$	−28.1114	−1.72039
$268$	−0.956509	−0.0584281
$269$	−26.6253	−1.62337	−0.811687	−	0.584093i	$-0.801450\pi$
$269$	−26.6253	−1.62337	−0.811687	+	0.584093i	$0.801450\pi$
$270$	0	0
$271$	−6.70052	−0.407028	−0.203514	−	0.979072i	$-0.565236\pi$
$271$	−6.70052	−0.407028	−0.203514	+	0.979072i	$0.565236\pi$
$272$	0	0
$273$	4.64974	0.281415
$274$	−17.4314	−1.05307
$275$	0	0
$276$	−2.68006	−0.161320
$277$	−16.7005	−1.00344	−0.501719	−	0.865031i	$-0.667299\pi$
$277$	−16.7005	−1.00344	−0.501719	+	0.865031i	$0.667299\pi$
$278$	17.3380	1.03987
$279$	0.355186	0.0212644
$280$	0	0
$281$	28.4241	1.69564	0.847819	−	0.530286i	$-0.177915\pi$
$281$	28.4241	1.69564	0.847819	+	0.530286i	$0.177915\pi$
$282$	16.2374	0.966925
$283$	−10.0630	−0.598183	−0.299092	−	0.954224i	$-0.596684\pi$
$283$	−10.0630	−0.598183	−0.299092	+	0.954224i	$0.596684\pi$
$284$	2.81828	0.167234
$285$	0	0
$286$	−1.53690	−0.0908790
$287$	−14.5139	−0.856727
$288$	0.212035	0.0124943
$289$	0	0
$290$	0	0
$291$	−6.13586	−0.359690
$292$	−2.58910	−0.151516
$293$	−14.3127	−0.836154	−0.418077	−	0.908412i	$-0.637296\pi$
$293$	−14.3127	−0.836154	−0.418077	+	0.908412i	$0.637296\pi$
$294$	−13.2569	−0.773160
$295$	0	0
$296$	−11.5369	−0.670569
$297$	2.57452	0.149389
$298$	25.4763	1.47580
$299$	−17.7889	−1.02876
$300$	0	0
$301$	10.4993	0.605169
$302$	20.1260	1.15812
$303$	11.4861	0.659860
$304$	14.5745	0.835906
$305$	0	0
$306$	0	0
$307$	−29.2809	−1.67115	−0.835575	−	0.549376i	$-0.814865\pi$
$307$	−29.2809	−1.67115	−0.835575	+	0.549376i	$0.814865\pi$
$308$	−0.120128	−0.00684495
$309$	−11.7988	−0.671209
$310$	0	0
$311$	−4.54183	−0.257543	−0.128772	−	0.991674i	$-0.541103\pi$
$311$	−4.54183	−0.257543	−0.128772	+	0.991674i	$0.541103\pi$
$312$	−9.66291	−0.547055
$313$	0.826531	0.0467183	0.0233592	−	0.999727i	$-0.492564\pi$
$313$	0.826531	0.0467183	0.0233592	+	0.999727i	$0.492564\pi$
$314$	12.7005	0.716732
$315$	0	0
$316$	1.75623	0.0987958
$317$	32.2374	1.81063	0.905317	−	0.424736i	$-0.139633\pi$
$317$	32.2374	1.81063	0.905317	+	0.424736i	$0.139633\pi$
$318$	−21.2750	−1.19304
$319$	0.312650	0.0175051
$320$	0	0
$321$	−9.81924	−0.548056
$322$	−15.7294	−0.876567
$323$	0	0
$324$	−1.62530	−0.0902945
$325$	0	0
$326$	−21.2931	−1.17932
$327$	21.2750	1.17651
$328$	30.1622	1.66543
$329$	8.42407	0.464434
$330$	0	0
$331$	6.82653	0.375220	0.187610	−	0.982244i	$-0.439926\pi$
$331$	6.82653	0.375220	0.187610	+	0.982244i	$0.439926\pi$
$332$	−2.60483	−0.142959
$333$	−0.836381	−0.0458334
$334$	11.1817	0.611836
$335$	0	0
$336$	−9.38058	−0.511753
$337$	18.0508	0.983289	0.491644	−	0.870796i	$-0.336396\pi$
$337$	18.0508	0.983289	0.491644	+	0.870796i	$0.336396\pi$
$338$	−12.3684	−0.672750
$339$	−8.18664	−0.444637
$340$	0	0
$341$	0.881286	0.0477243
$342$	0.962389	0.0520400
$343$	−15.8886	−0.857903
$344$	−21.8192	−1.17641
$345$	0	0
$346$	−17.5515	−0.943574
$347$	18.3512	0.985145	0.492572	−	0.870271i	$-0.336057\pi$
$347$	18.3512	0.985145	0.492572	+	0.870271i	$0.336057\pi$
$348$	−0.211080	−0.0113151
$349$	13.6023	0.728113	0.364057	−	0.931377i	$-0.381391\pi$
$349$	13.6023	0.728113	0.364057	+	0.931377i	$0.381391\pi$
$350$	0	0
$351$	−11.5369	−0.615794
$352$	0.526100	0.0280412
$353$	−27.4010	−1.45841	−0.729205	−	0.684295i	$-0.760110\pi$
$353$	−27.4010	−1.45841	−0.729205	+	0.684295i	$0.760110\pi$
$354$	−12.3127	−0.654410
$355$	0	0
$356$	−3.25457	−0.172492
$357$	0	0
$358$	4.77575	0.252406
$359$	9.08840	0.479667	0.239834	−	0.970814i	$-0.422907\pi$
$359$	9.08840	0.479667	0.239834	+	0.970814i	$0.422907\pi$
$360$	0	0
$361$	−7.77575	−0.409250
$362$	10.2012	0.536165
$363$	−18.0386	−0.946779
$364$	0.538319	0.0282156
$365$	0	0
$366$	7.03761	0.367862
$367$	12.7635	0.666251	0.333125	−	0.942883i	$-0.391897\pi$
$367$	12.7635	0.666251	0.333125	+	0.942883i	$0.391897\pi$
$368$	35.8881	1.87080
$369$	2.18664	0.113832
$370$	0	0
$371$	−11.0376	−0.573044
$372$	−0.594984	−0.0308485
$373$	−10.0957	−0.522735	−0.261368	−	0.965239i	$-0.584173\pi$
$373$	−10.0957	−0.522735	−0.261368	+	0.965239i	$0.584173\pi$
$374$	0	0
$375$	0	0
$376$	−17.5066	−0.902833
$377$	−1.40105	−0.0721576
$378$	−10.2012	−0.524695
$379$	−9.18172	−0.471633	−0.235817	−	0.971798i	$-0.575776\pi$
$379$	−9.18172	−0.471633	−0.235817	+	0.971798i	$0.575776\pi$
$380$	0	0
$381$	2.96239	0.151768
$382$	10.6643	0.545634
$383$	−1.83383	−0.0937041	−0.0468521	−	0.998902i	$-0.514919\pi$
$383$	−1.83383	−0.0937041	−0.0468521	+	0.998902i	$0.514919\pi$
$384$	21.2325	1.08352
$385$	0	0
$386$	29.7889	1.51622
$387$	−1.58181	−0.0804079
$388$	−0.710373	−0.0360637
$389$	−11.6570	−0.591035	−0.295518	−	0.955337i	$-0.595492\pi$
$389$	−11.6570	−0.591035	−0.295518	+	0.955337i	$0.595492\pi$
$390$	0	0
$391$	0	0
$392$	14.2931	0.721912
$393$	−18.6048	−0.938490
$394$	−4.68594	−0.236074
$395$	0	0
$396$	0.0180984	0.000909478 0
$397$	9.37470	0.470503	0.235251	−	0.971935i	$-0.424409\pi$
$397$	9.37470	0.470503	0.235251	+	0.971935i	$0.424409\pi$
$398$	31.8373	1.59586
$399$	−7.22425	−0.361665
$400$	0	0
$401$	7.93937	0.396473	0.198237	−	0.980154i	$-0.436479\pi$
$401$	7.93937	0.396473	0.198237	+	0.980154i	$0.436479\pi$
$402$	−12.2374	−0.610347
$403$	−3.94921	−0.196724
$404$	1.32979	0.0661597
$405$	0	0
$406$	−1.23884	−0.0614827
$407$	−2.07522	−0.102865
$408$	0	0
$409$	9.07381	0.448671	0.224335	−	0.974512i	$-0.427979\pi$
$409$	9.07381	0.448671	0.224335	+	0.974512i	$0.427979\pi$
$410$	0	0
$411$	−19.7137	−0.972405
$412$	−1.36599	−0.0672975
$413$	−6.38787	−0.314327
$414$	2.36977	0.116468
$415$	0	0
$416$	−2.35756	−0.115589
$417$	19.6082	0.960216
$418$	2.38787	0.116795
$419$	−8.99508	−0.439438	−0.219719	−	0.975563i	$-0.570514\pi$
$419$	−8.99508	−0.439438	−0.219719	+	0.975563i	$0.570514\pi$
$420$	0	0
$421$	16.9076	0.824028	0.412014	−	0.911178i	$-0.364826\pi$
$421$	16.9076	0.824028	0.412014	+	0.911178i	$0.364826\pi$
$422$	30.9111	1.50473
$423$	−1.26916	−0.0617086
$424$	22.9380	1.11397
$425$	0	0
$426$	36.0567	1.74695
$427$	3.65115	0.176692
$428$	−1.13681	−0.0549499
$429$	−1.73813	−0.0839179
$430$	0	0
$431$	14.6678	0.706525	0.353262	−	0.935524i	$-0.385072\pi$
$431$	14.6678	0.706525	0.353262	+	0.935524i	$0.385072\pi$
$432$	23.2750	1.11982
$433$	−6.68006	−0.321023	−0.160511	−	0.987034i	$-0.551314\pi$
$433$	−6.68006	−0.321023	−0.160511	+	0.987034i	$0.551314\pi$
$434$	−3.49200	−0.167621
$435$	0	0
$436$	2.46310	0.117961
$437$	27.6385	1.32213
$438$	−33.1246	−1.58275
$439$	−9.56959	−0.456732	−0.228366	−	0.973575i	$-0.573338\pi$
$439$	−9.56959	−0.456732	−0.228366	+	0.973575i	$0.573338\pi$
$440$	0	0
$441$	1.03620	0.0493427
$442$	0	0
$443$	14.7915	0.702764	0.351382	−	0.936232i	$-0.385712\pi$
$443$	14.7915	0.702764	0.351382	+	0.936232i	$0.385712\pi$
$444$	1.40105	0.0664907
$445$	0	0
$446$	−36.0567	−1.70733
$447$	28.8119	1.36276
$448$	9.11520	0.430653
$449$	10.9018	0.514486	0.257243	−	0.966347i	$-0.417186\pi$
$449$	10.9018	0.514486	0.257243	+	0.966347i	$0.417186\pi$
$450$	0	0
$451$	5.42548	0.255476
$452$	−0.947800	−0.0445808
$453$	22.7612	1.06941
$454$	−16.6072	−0.779415
$455$	0	0
$456$	15.0132	0.703056
$457$	7.61801	0.356355	0.178178	−	0.983998i	$-0.442980\pi$
$457$	7.61801	0.356355	0.178178	+	0.983998i	$0.442980\pi$
$458$	−18.0059	−0.841360
$459$	0	0
$460$	0	0
$461$	−4.57452	−0.213056	−0.106528	−	0.994310i	$-0.533973\pi$
$461$	−4.57452	−0.213056	−0.106528	+	0.994310i	$0.533973\pi$
$462$	−1.53690	−0.0715032
$463$	−2.56864	−0.119375	−0.0596873	−	0.998217i	$-0.519010\pi$
$463$	−2.56864	−0.119375	−0.0596873	+	0.998217i	$0.519010\pi$
$464$	2.82653	0.131218
$465$	0	0
$466$	17.7381	0.821703
$467$	1.55737	0.0720666	0.0360333	−	0.999351i	$-0.488528\pi$
$467$	1.55737	0.0720666	0.0360333	+	0.999351i	$0.488528\pi$
$468$	−0.0811024	−0.00374896
$469$	−6.34885	−0.293163
$470$	0	0
$471$	14.3634	0.661832
$472$	13.2750	0.611033
$473$	−3.92478	−0.180461
$474$	22.4690	1.03203
$475$	0	0
$476$	0	0
$477$	1.66291	0.0761395
$478$	−13.0884	−0.598649
$479$	20.4060	0.932373	0.466186	−	0.884687i	$-0.345628\pi$
$479$	20.4060	0.932373	0.466186	+	0.884687i	$0.345628\pi$
$480$	0	0
$481$	9.29948	0.424020
$482$	0.387873	0.0176671
$483$	−17.7889	−0.809424
$484$	−2.08840	−0.0949271
$485$	0	0
$486$	2.98049	0.135198
$487$	−0.162664	−0.00737103	−0.00368551	−	0.999993i	$-0.501173\pi$
$487$	−0.162664	−0.00737103	−0.00368551	+	0.999993i	$0.501173\pi$
$488$	−7.58769	−0.343479
$489$	−24.0811	−1.08899
$490$	0	0
$491$	−12.8364	−0.579298	−0.289649	−	0.957133i	$-0.593539\pi$
$491$	−12.8364	−0.579298	−0.289649	+	0.957133i	$0.593539\pi$
$492$	−3.66291	−0.165137
$493$	0	0
$494$	−10.7005	−0.481439
$495$	0	0
$496$	7.96731	0.357743
$497$	18.7064	0.839097
$498$	−33.3258	−1.49337
$499$	−17.7416	−0.794225	−0.397113	−	0.917770i	$-0.629988\pi$
$499$	−17.7416	−0.794225	−0.397113	+	0.917770i	$0.629988\pi$
$500$	0	0
$501$	12.6458	0.564971
$502$	−35.0494	−1.56433
$503$	11.2120	0.499920	0.249960	−	0.968256i	$-0.419583\pi$
$503$	11.2120	0.499920	0.249960	+	0.968256i	$0.419583\pi$
$504$	0.667837	0.0297478
$505$	0	0
$506$	5.87987	0.261392
$507$	−13.9878	−0.621219
$508$	0.342968	0.0152167
$509$	−6.37328	−0.282491	−0.141245	−	0.989975i	$-0.545111\pi$
$509$	−6.37328	−0.282491	−0.141245	+	0.989975i	$0.545111\pi$
$510$	0	0
$511$	−17.1852	−0.760230
$512$	−18.5188	−0.818423
$513$	17.9248	0.791398
$514$	28.5052	1.25731
$515$	0	0
$516$	2.64974	0.116648
$517$	−3.14903	−0.138494
$518$	8.22284	0.361291
$519$	−19.8496	−0.871299
$520$	0	0
$521$	−0.986826	−0.0432336	−0.0216168	−	0.999766i	$-0.506881\pi$
$521$	−0.986826	−0.0432336	−0.0216168	+	0.999766i	$0.506881\pi$
$522$	0.186642	0.00816911
$523$	−8.99271	−0.393224	−0.196612	−	0.980481i	$-0.562994\pi$
$523$	−8.99271	−0.393224	−0.196612	+	0.980481i	$0.562994\pi$
$524$	−2.15396	−0.0940960
$525$	0	0
$526$	−25.5428	−1.11372
$527$	0	0
$528$	3.50659	0.152605
$529$	45.0567	1.95899
$530$	0	0
$531$	0.962389	0.0417641
$532$	−0.836381	−0.0362617
$533$	−24.3127	−1.05310
$534$	−41.6385	−1.80187
$535$	0	0
$536$	13.1939	0.569891
$537$	5.40105	0.233072
$538$	−39.4372	−1.70026
$539$	2.57101	0.110741
$540$	0	0
$541$	37.3766	1.60695	0.803473	−	0.595341i	$-0.202983\pi$
$541$	37.3766	1.60695	0.803473	+	0.595341i	$0.202983\pi$
$542$	−9.92478	−0.426306
$543$	11.5369	0.495096
$544$	0	0
$545$	0	0
$546$	6.88717	0.294743
$547$	−22.1744	−0.948110	−0.474055	−	0.880495i	$-0.657210\pi$
$547$	−22.1744	−0.948110	−0.474055	+	0.880495i	$0.657210\pi$
$548$	−2.28233	−0.0974964
$549$	−0.550078	−0.0234768
$550$	0	0
$551$	2.17679	0.0927345
$552$	36.9683	1.57347
$553$	11.6570	0.495707
$554$	−24.7367	−1.05096
$555$	0	0
$556$	2.27011	0.0962743
$557$	−1.16950	−0.0495533	−0.0247766	−	0.999693i	$-0.507887\pi$
$557$	−1.16950	−0.0495533	−0.0247766	+	0.999693i	$0.507887\pi$
$558$	0.526100	0.0222716
$559$	17.5877	0.743880
$560$	0	0
$561$	0	0
$562$	42.1016	1.77595
$563$	−11.1333	−0.469213	−0.234606	−	0.972090i	$-0.575380\pi$
$563$	−11.1333	−0.469213	−0.234606	+	0.972090i	$0.575380\pi$
$564$	2.12601	0.0895211
$565$	0	0
$566$	−14.9053	−0.626515
$567$	−10.7880	−0.453052
$568$	−38.8749	−1.63116
$569$	−16.7005	−0.700122	−0.350061	−	0.936727i	$-0.613839\pi$
$569$	−16.7005	−0.700122	−0.350061	+	0.936727i	$0.613839\pi$
$570$	0	0
$571$	37.2833	1.56026	0.780129	−	0.625619i	$-0.215154\pi$
$571$	37.2833	1.56026	0.780129	+	0.625619i	$0.215154\pi$
$572$	−0.201231	−0.00841388
$573$	12.0606	0.503840
$574$	−21.4979	−0.897304
$575$	0	0
$576$	−1.37328	−0.0572202
$577$	1.63259	0.0679658	0.0339829	−	0.999422i	$-0.489181\pi$
$577$	1.63259	0.0679658	0.0339829	+	0.999422i	$0.489181\pi$
$578$	0	0
$579$	33.6893	1.40008
$580$	0	0
$581$	−17.2896	−0.717295
$582$	−9.08840	−0.376726
$583$	4.12601	0.170882
$584$	35.7137	1.47784
$585$	0	0
$586$	−21.1998	−0.875756
$587$	23.0435	0.951107	0.475553	−	0.879687i	$-0.342248\pi$
$587$	23.0435	0.951107	0.475553	+	0.879687i	$0.342248\pi$
$588$	−1.73577	−0.0715818
$589$	6.13586	0.252824
$590$	0	0
$591$	−5.29948	−0.217991
$592$	−18.7612	−0.771079
$593$	44.2130	1.81561	0.907805	−	0.419393i	$-0.137757\pi$
$593$	44.2130	1.81561	0.907805	+	0.419393i	$0.137757\pi$
$594$	3.81336	0.156464
$595$	0	0
$596$	3.33567	0.136635
$597$	36.0059	1.47362
$598$	−26.3488	−1.07748
$599$	15.4518	0.631345	0.315672	−	0.948868i	$-0.397770\pi$
$599$	15.4518	0.631345	0.315672	+	0.948868i	$0.397770\pi$
$600$	0	0
$601$	−43.0640	−1.75662	−0.878308	−	0.478096i	$-0.841327\pi$
$601$	−43.0640	−1.75662	−0.878308	+	0.478096i	$0.841327\pi$
$602$	15.5515	0.633832
$603$	0.956509	0.0389521
$604$	2.63515	0.107223
$605$	0	0
$606$	17.0132	0.691113
$607$	41.5002	1.68444	0.842222	−	0.539132i	$-0.181247\pi$
$607$	41.5002	1.68444	0.842222	+	0.539132i	$0.181247\pi$
$608$	3.66291	0.148551
$609$	−1.40105	−0.0567733
$610$	0	0
$611$	14.1114	0.570887
$612$	0	0
$613$	27.7137	1.11935	0.559673	−	0.828714i	$-0.310927\pi$
$613$	27.7137	1.11935	0.559673	+	0.828714i	$0.310927\pi$
$614$	−43.3707	−1.75030
$615$	0	0
$616$	1.65703	0.0667637
$617$	−29.6629	−1.19418	−0.597092	−	0.802173i	$-0.703677\pi$
$617$	−29.6629	−1.19418	−0.597092	+	0.802173i	$0.703677\pi$
$618$	−17.4763	−0.702999
$619$	26.4422	1.06280	0.531400	−	0.847121i	$-0.321666\pi$
$619$	26.4422	1.06280	0.531400	+	0.847121i	$0.321666\pi$
$620$	0	0
$621$	44.1378	1.77119
$622$	−6.72733	−0.269741
$623$	−21.6023	−0.865477
$624$	−15.7137	−0.629051
$625$	0	0
$626$	1.22425	0.0489310
$627$	2.70052	0.107849
$628$	1.66291	0.0663574
$629$	0	0
$630$	0	0
$631$	−37.7499	−1.50280	−0.751400	−	0.659847i	$-0.770621\pi$
$631$	−37.7499	−1.50280	−0.751400	+	0.659847i	$0.770621\pi$
$632$	−24.2252	−0.963627
$633$	34.9584	1.38947
$634$	47.7499	1.89639
$635$	0	0
$636$	−2.78560	−0.110456
$637$	−11.5212	−0.456486
$638$	0.463096	0.0183341
$639$	−2.81828	−0.111490
$640$	0	0
$641$	−46.3488	−1.83067	−0.915335	−	0.402694i	$-0.868074\pi$
$641$	−46.3488	−1.83067	−0.915335	+	0.402694i	$0.868074\pi$
$642$	−14.5442	−0.574014
$643$	−34.9502	−1.37830	−0.689150	−	0.724619i	$-0.742016\pi$
$643$	−34.9502	−1.37830	−0.689150	+	0.724619i	$0.742016\pi$
$644$	−2.05949	−0.0811554
$645$	0	0
$646$	0	0
$647$	−36.5296	−1.43613	−0.718064	−	0.695978i	$-0.754971\pi$
$647$	−36.5296	−1.43613	−0.718064	+	0.695978i	$0.754971\pi$
$648$	22.4191	0.880707
$649$	2.38787	0.0937322
$650$	0	0
$651$	−3.94921	−0.154782
$652$	−2.78797	−0.109185
$653$	0.926192	0.0362447	0.0181223	−	0.999836i	$-0.494231\pi$
$653$	0.926192	0.0362447	0.0181223	+	0.999836i	$0.494231\pi$
$654$	31.5125	1.23223
$655$	0	0
$656$	49.0494	1.91506
$657$	2.58910	0.101011
$658$	12.4777	0.486431
$659$	−24.0118	−0.935365	−0.467683	−	0.883896i	$-0.654911\pi$
$659$	−24.0118	−0.935365	−0.467683	+	0.883896i	$0.654911\pi$
$660$	0	0
$661$	20.9478	0.814775	0.407387	−	0.913255i	$-0.366440\pi$
$661$	20.9478	0.814775	0.407387	+	0.913255i	$0.366440\pi$
$662$	10.1114	0.392991
$663$	0	0
$664$	35.9307	1.39438
$665$	0	0
$666$	−1.23884	−0.0480042
$667$	5.36011	0.207544
$668$	1.46405	0.0566458
$669$	−40.7777	−1.57656
$670$	0	0
$671$	−1.36485	−0.0526895
$672$	−2.35756	−0.0909447
$673$	−0.513881	−0.0198087	−0.00990433	−	0.999951i	$-0.503153\pi$
$673$	−0.513881	−0.0198087	−0.00990433	+	0.999951i	$0.503153\pi$
$674$	26.7367	1.02986
$675$	0	0
$676$	−1.61942	−0.0622854
$677$	−3.67276	−0.141156	−0.0705778	−	0.997506i	$-0.522484\pi$
$677$	−3.67276	−0.141156	−0.0705778	+	0.997506i	$0.522484\pi$
$678$	−12.1260	−0.465697
$679$	−4.71511	−0.180949
$680$	0	0
$681$	−18.7816	−0.719713
$682$	1.30536	0.0499847
$683$	10.5115	0.402212	0.201106	−	0.979570i	$-0.435546\pi$
$683$	10.5115	0.402212	0.201106	+	0.979570i	$0.435546\pi$
$684$	0.126008	0.00481804
$685$	0	0
$686$	−23.5341	−0.898535
$687$	−20.3634	−0.776914
$688$	−35.4821	−1.35274
$689$	−18.4894	−0.704392
$690$	0	0
$691$	25.5940	0.973643	0.486821	−	0.873502i	$-0.338156\pi$
$691$	25.5940	0.973643	0.486821	+	0.873502i	$0.338156\pi$
$692$	−2.29806	−0.0873592
$693$	0.120128	0.00456330
$694$	27.1817	1.03180
$695$	0	0
$696$	2.91160	0.110364
$697$	0	0
$698$	20.1476	0.762599
$699$	20.0606	0.758763
$700$	0	0
$701$	30.8324	1.16452	0.582262	−	0.813001i	$-0.302168\pi$
$701$	30.8324	1.16452	0.582262	+	0.813001i	$0.302168\pi$
$702$	−17.0884	−0.644960
$703$	−14.4485	−0.544936
$704$	−3.40739	−0.128421
$705$	0	0
$706$	−40.5863	−1.52748
$707$	8.82653	0.331956
$708$	−1.61213	−0.0605874
$709$	−28.1866	−1.05857	−0.529286	−	0.848444i	$-0.677540\pi$
$709$	−28.1866	−1.05857	−0.529286	+	0.848444i	$0.677540\pi$
$710$	0	0
$711$	−1.75623	−0.0658639
$712$	44.8930	1.68244
$713$	15.1089	0.565831
$714$	0	0
$715$	0	0
$716$	0.625301	0.0233686
$717$	−14.8021	−0.552794
$718$	13.4617	0.502385
$719$	−17.7200	−0.660846	−0.330423	−	0.943833i	$-0.607191\pi$
$719$	−17.7200	−0.660846	−0.330423	+	0.943833i	$0.607191\pi$
$720$	0	0
$721$	−9.06679	−0.337665
$722$	−11.5174	−0.428633
$723$	0.438658	0.0163139
$724$	1.33567	0.0496399
$725$	0	0
$726$	−26.7186	−0.991621
$727$	9.68338	0.359137	0.179568	−	0.983746i	$-0.442530\pi$
$727$	9.68338	0.359137	0.179568	+	0.983746i	$0.442530\pi$
$728$	−7.42548	−0.275207
$729$	28.5125	1.05602
$730$	0	0
$731$	0	0
$732$	0.921452	0.0340579
$733$	32.7123	1.20826	0.604128	−	0.796887i	$-0.293522\pi$
$733$	32.7123	1.20826	0.604128	+	0.796887i	$0.293522\pi$
$734$	18.9053	0.697806
$735$	0	0
$736$	9.01951	0.332464
$737$	2.37328	0.0874211
$738$	3.23884	0.119223
$739$	−1.42548	−0.0524373	−0.0262186	−	0.999656i	$-0.508347\pi$
$739$	−1.42548	−0.0524373	−0.0262186	+	0.999656i	$0.508347\pi$
$740$	0	0
$741$	−12.1016	−0.444562
$742$	−16.3488	−0.600185
$743$	16.8143	0.616857	0.308429	−	0.951247i	$-0.400197\pi$
$743$	16.8143	0.616857	0.308429	+	0.951247i	$0.400197\pi$
$744$	8.20711	0.300887
$745$	0	0
$746$	−14.9537	−0.547493
$747$	2.60483	0.0953058
$748$	0	0
$749$	−7.54561	−0.275711
$750$	0	0
$751$	44.0835	1.60863	0.804314	−	0.594204i	$-0.202533\pi$
$751$	44.0835	1.60863	0.804314	+	0.594204i	$0.202533\pi$
$752$	−28.4690	−1.03816
$753$	−39.6385	−1.44451
$754$	−2.07522	−0.0755752
$755$	0	0
$756$	−1.33567	−0.0485780
$757$	−39.8094	−1.44690	−0.723448	−	0.690378i	$-0.757444\pi$
$757$	−39.8094	−1.44690	−0.723448	+	0.690378i	$0.757444\pi$
$758$	−13.5999	−0.493971
$759$	6.64974	0.241370
$760$	0	0
$761$	2.81591	0.102077	0.0510384	−	0.998697i	$-0.483747\pi$
$761$	2.81591	0.102077	0.0510384	+	0.998697i	$0.483747\pi$
$762$	4.38787	0.158956
$763$	16.3488	0.591868
$764$	1.39631	0.0505166
$765$	0	0
$766$	−2.71625	−0.0981422
$767$	−10.7005	−0.386374
$768$	7.72592	0.278785
$769$	36.0665	1.30059	0.650296	−	0.759681i	$-0.274645\pi$
$769$	36.0665	1.30059	0.650296	+	0.759681i	$0.274645\pi$
$770$	0	0
$771$	32.2374	1.16100
$772$	3.90034	0.140376
$773$	11.6180	0.417871	0.208935	−	0.977929i	$-0.433000\pi$
$773$	11.6180	0.417871	0.208935	+	0.977929i	$0.433000\pi$
$774$	−2.34297	−0.0842162
$775$	0	0
$776$	9.79877	0.351755
$777$	9.29948	0.333617
$778$	−17.2663	−0.619028
$779$	37.7743	1.35341
$780$	0	0
$781$	−6.99271	−0.250219
$782$	0	0
$783$	3.47627	0.124232
$784$	23.2433	0.830118
$785$	0	0
$786$	−27.5574	−0.982939
$787$	28.4626	1.01458	0.507292	−	0.861774i	$-0.330647\pi$
$787$	28.4626	1.01458	0.507292	+	0.861774i	$0.330647\pi$
$788$	−0.613541	−0.0218565
$789$	−28.8872	−1.02841
$790$	0	0
$791$	−6.29104	−0.223684
$792$	−0.249646	−0.00887080
$793$	6.11616	0.217191
$794$	13.8858	0.492787
$795$	0	0
$796$	4.16854	0.147750
$797$	−50.5355	−1.79006	−0.895029	−	0.446007i	$-0.852846\pi$
$797$	−50.5355	−1.79006	−0.895029	+	0.446007i	$0.852846\pi$
$798$	−10.7005	−0.378794
$799$	0	0
$800$	0	0
$801$	3.25457	0.114995
$802$	11.7597	0.415251
$803$	6.42407	0.226701
$804$	−1.60228	−0.0565080
$805$	0	0
$806$	−5.84955	−0.206042
$807$	−44.6009	−1.57002
$808$	−18.3430	−0.645303
$809$	2.50914	0.0882167	0.0441084	−	0.999027i	$-0.485955\pi$
$809$	2.50914	0.0882167	0.0441084	+	0.999027i	$0.485955\pi$
$810$	0	0
$811$	18.0933	0.635342	0.317671	−	0.948201i	$-0.397099\pi$
$811$	18.0933	0.635342	0.317671	+	0.948201i	$0.397099\pi$
$812$	−0.162205	−0.00569227
$813$	−11.2243	−0.393652
$814$	−3.07381	−0.107737
$815$	0	0
$816$	0	0
$817$	−27.3258	−0.956010
$818$	13.4401	0.469921
$819$	−0.538319	−0.0188104
$820$	0	0
$821$	−7.02776	−0.245271	−0.122635	−	0.992452i	$-0.539135\pi$
$821$	−7.02776	−0.245271	−0.122635	+	0.992452i	$0.539135\pi$
$822$	−29.1998	−1.01846
$823$	28.5379	0.994767	0.497384	−	0.867531i	$-0.334294\pi$
$823$	28.5379	0.994767	0.497384	+	0.867531i	$0.334294\pi$
$824$	18.8423	0.656401
$825$	0	0
$826$	−9.46168	−0.329214
$827$	−26.3004	−0.914556	−0.457278	−	0.889324i	$-0.651175\pi$
$827$	−26.3004	−0.914556	−0.457278	+	0.889324i	$0.651175\pi$
$828$	0.310281	0.0107830
$829$	−27.2506	−0.946453	−0.473226	−	0.880941i	$-0.656911\pi$
$829$	−27.2506	−0.946453	−0.473226	+	0.880941i	$0.656911\pi$
$830$	0	0
$831$	−27.9756	−0.970462
$832$	15.2692	0.529363
$833$	0	0
$834$	29.0435	1.00569
$835$	0	0
$836$	0.312650	0.0108132
$837$	9.79877	0.338695
$838$	−13.3235	−0.460251
$839$	38.6580	1.33462	0.667311	−	0.744779i	$-0.267445\pi$
$839$	38.6580	1.33462	0.667311	+	0.744779i	$0.267445\pi$
$840$	0	0
$841$	−28.5778	−0.985443
$842$	25.0435	0.863056
$843$	47.6140	1.63991
$844$	4.04728	0.139313
$845$	0	0
$846$	−1.87987	−0.0646313
$847$	−13.8618	−0.476296
$848$	37.3014	1.28093
$849$	−16.8568	−0.578526
$850$	0	0
$851$	−35.5778	−1.21959
$852$	4.72099	0.161739
$853$	−5.46168	−0.187004	−0.0935022	−	0.995619i	$-0.529806\pi$
$853$	−5.46168	−0.187004	−0.0935022	+	0.995619i	$0.529806\pi$
$854$	5.40807	0.185060
$855$	0	0
$856$	15.6810	0.535966
$857$	−8.30280	−0.283618	−0.141809	−	0.989894i	$-0.545292\pi$
$857$	−8.30280	−0.283618	−0.141809	+	0.989894i	$0.545292\pi$
$858$	−2.57452	−0.0878925
$859$	−10.7612	−0.367166	−0.183583	−	0.983004i	$-0.558770\pi$
$859$	−10.7612	−0.367166	−0.183583	+	0.983004i	$0.558770\pi$
$860$	0	0
$861$	−24.3127	−0.828573
$862$	21.7259	0.739988
$863$	−43.7342	−1.48873	−0.744364	−	0.667774i	$-0.767247\pi$
$863$	−43.7342	−1.48873	−0.744364	+	0.667774i	$0.767247\pi$
$864$	5.84955	0.199006
$865$	0	0
$866$	−9.89446	−0.336227
$867$	0	0
$868$	−0.457216	−0.0155189
$869$	−4.35756	−0.147820
$870$	0	0
$871$	−10.6351	−0.360358
$872$	−33.9756	−1.15056
$873$	0.710373	0.0240425
$874$	40.9380	1.38475
$875$	0	0
$876$	−4.33709	−0.146537
$877$	−43.5633	−1.47103	−0.735513	−	0.677510i	$-0.763059\pi$
$877$	−43.5633	−1.47103	−0.735513	+	0.677510i	$0.763059\pi$
$878$	−14.1744	−0.478364
$879$	−23.9756	−0.808676
$880$	0	0
$881$	43.4372	1.46344	0.731719	−	0.681606i	$-0.238718\pi$
$881$	43.4372	1.46344	0.731719	+	0.681606i	$0.238718\pi$
$882$	1.53481	0.0516797
$883$	6.56864	0.221052	0.110526	−	0.993873i	$-0.464746\pi$
$883$	6.56864	0.221052	0.110526	+	0.993873i	$0.464746\pi$
$884$	0	0
$885$	0	0
$886$	21.9090	0.736049
$887$	21.4396	0.719872	0.359936	−	0.932977i	$-0.382799\pi$
$887$	21.4396	0.719872	0.359936	+	0.932977i	$0.382799\pi$
$888$	−19.3258	−0.648532
$889$	2.27645	0.0763498
$890$	0	0
$891$	4.03269	0.135100
$892$	−4.72099	−0.158070
$893$	−21.9248	−0.733685
$894$	42.6761	1.42730
$895$	0	0
$896$	16.3162	0.545085
$897$	−29.7988	−0.994952
$898$	16.1476	0.538853
$899$	1.18997	0.0396876
$900$	0	0
$901$	0	0
$902$	8.03620	0.267576
$903$	17.5877	0.585282
$904$	13.0738	0.434828
$905$	0	0
$906$	33.7137	1.12006
$907$	21.2628	0.706020	0.353010	−	0.935619i	$-0.385158\pi$
$907$	21.2628	0.706020	0.353010	+	0.935619i	$0.385158\pi$
$908$	−2.17442	−0.0721608
$909$	−1.32979	−0.0441065
$910$	0	0
$911$	−48.6678	−1.61244	−0.806219	−	0.591618i	$-0.798490\pi$
$911$	−48.6678	−1.61244	−0.806219	+	0.591618i	$0.798490\pi$
$912$	24.4142	0.808436
$913$	6.46310	0.213897
$914$	11.2837	0.373233
$915$	0	0
$916$	−2.35756	−0.0778958
$917$	−14.2969	−0.472126
$918$	0	0
$919$	4.55993	0.150418	0.0752091	−	0.997168i	$-0.476038\pi$
$919$	4.55993	0.150418	0.0752091	+	0.997168i	$0.476038\pi$
$920$	0	0
$921$	−49.0494	−1.61623
$922$	−6.77575	−0.223147
$923$	31.3357	1.03143
$924$	−0.201231	−0.00662001
$925$	0	0
$926$	−3.80465	−0.125029
$927$	1.36599	0.0448650
$928$	0.710373	0.0233191
$929$	−23.3963	−0.767608	−0.383804	−	0.923415i	$-0.625386\pi$
$929$	−23.3963	−0.767608	−0.383804	+	0.923415i	$0.625386\pi$
$930$	0	0
$931$	17.9003	0.586660
$932$	2.32250	0.0760760
$933$	−7.60816	−0.249080
$934$	2.30677	0.0754798
$935$	0	0
$936$	1.11871	0.0365663
$937$	−30.3898	−0.992791	−0.496395	−	0.868097i	$-0.665343\pi$
$937$	−30.3898	−0.992791	−0.496395	+	0.868097i	$0.665343\pi$
$938$	−9.40388	−0.307047
$939$	1.38455	0.0451830
$940$	0	0
$941$	25.9610	0.846304	0.423152	−	0.906059i	$-0.360924\pi$
$941$	25.9610	0.846304	0.423152	+	0.906059i	$0.360924\pi$
$942$	21.2750	0.693178
$943$	93.0151	3.02899
$944$	21.5877	0.702619
$945$	0	0
$946$	−5.81336	−0.189009
$947$	−16.8289	−0.546866	−0.273433	−	0.961891i	$-0.588159\pi$
$947$	−16.8289	−0.546866	−0.273433	+	0.961891i	$0.588159\pi$
$948$	2.94192	0.0955491
$949$	−28.7875	−0.934482
$950$	0	0
$951$	54.0019	1.75113
$952$	0	0
$953$	12.7553	0.413184	0.206592	−	0.978427i	$-0.433763\pi$
$953$	12.7553	0.413184	0.206592	+	0.978427i	$0.433763\pi$
$954$	2.46310	0.0797457
$955$	0	0
$956$	−1.71370	−0.0554249
$957$	0.523730	0.0169298
$958$	30.2252	0.976532
$959$	−15.1490	−0.489188
$960$	0	0
$961$	−27.6458	−0.891799
$962$	13.7743	0.444102
$963$	1.13681	0.0366333
$964$	0.0507852	0.00163568
$965$	0	0
$966$	−26.3488	−0.847760
$967$	42.5560	1.36851	0.684254	−	0.729244i	$-0.260128\pi$
$967$	42.5560	1.36851	0.684254	+	0.729244i	$0.260128\pi$
$968$	28.8070	0.925893
$969$	0	0
$970$	0	0
$971$	18.4485	0.592041	0.296020	−	0.955182i	$-0.404340\pi$
$971$	18.4485	0.592041	0.296020	+	0.955182i	$0.404340\pi$
$972$	0.390243	0.0125170
$973$	15.0679	0.483056
$974$	−0.240938	−0.00772014
$975$	0	0
$976$	−12.3390	−0.394962
$977$	24.8383	0.794647	0.397324	−	0.917679i	$-0.369939\pi$
$977$	24.8383	0.794647	0.397324	+	0.917679i	$0.369939\pi$
$978$	−35.6688	−1.14056
$979$	8.07522	0.258085
$980$	0	0
$981$	−2.46310	−0.0786406
$982$	−19.0132	−0.606735
$983$	4.32487	0.137942	0.0689710	−	0.997619i	$-0.478028\pi$
$983$	4.32487	0.137942	0.0689710	+	0.997619i	$0.478028\pi$
$984$	50.5256	1.61070
$985$	0	0
$986$	0	0
$987$	14.1114	0.449171
$988$	−1.40105	−0.0445732
$989$	−67.2868	−2.13960
$990$	0	0
$991$	31.8432	1.01153	0.505767	−	0.862670i	$-0.331210\pi$
$991$	31.8432	1.01153	0.505767	+	0.862670i	$0.331210\pi$
$992$	2.00237	0.0635753
$993$	11.4353	0.362889
$994$	27.7078	0.878839
$995$	0	0
$996$	−4.36344	−0.138261
$997$	−29.0278	−0.919318	−0.459659	−	0.888095i	$-0.652028\pi$
$997$	−29.0278	−0.919318	−0.459659	+	0.888095i	$0.652028\pi$
$998$	−26.2788	−0.831842
$999$	−23.0738	−0.730023

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.r.1.3		3
5.4	even	2		1445.2.a.k.1.1		3
17.4	even	4		425.2.d.c.101.1		6
17.13	even	4		425.2.d.c.101.2		6
17.16	even	2		7225.2.a.q.1.3		3
85.4	even	4		85.2.d.a.16.6	yes	6
85.13	odd	4		425.2.c.b.424.5		6
85.38	odd	4		425.2.c.a.424.5		6
85.47	odd	4		425.2.c.a.424.2		6
85.64	even	4		85.2.d.a.16.5	✓	6
85.72	odd	4		425.2.c.b.424.2		6
85.84	even	2		1445.2.a.j.1.1		3
255.89	odd	4		765.2.g.b.271.2		6
255.149	odd	4		765.2.g.b.271.1		6
340.259	odd	4		1360.2.c.f.1121.2		6
340.319	odd	4		1360.2.c.f.1121.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.d.a.16.5	✓	6	85.64	even	4
85.2.d.a.16.6	yes	6	85.4	even	4
425.2.c.a.424.2		6	85.47	odd	4
425.2.c.a.424.5		6	85.38	odd	4
425.2.c.b.424.2		6	85.72	odd	4
425.2.c.b.424.5		6	85.13	odd	4
425.2.d.c.101.1		6	17.4	even	4
425.2.d.c.101.2		6	17.13	even	4
765.2.g.b.271.1		6	255.149	odd	4
765.2.g.b.271.2		6	255.89	odd	4
1360.2.c.f.1121.2		6	340.259	odd	4
1360.2.c.f.1121.5		6	340.319	odd	4
1445.2.a.j.1.1		3	85.84	even	2
1445.2.a.k.1.1		3	5.4	even	2
7225.2.a.q.1.3		3	17.16	even	2
7225.2.a.r.1.3		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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7225 = 5^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7225.a (trivial)

\(f(q)\)	\(=\)	\(q+1.48119 q^{2} +1.67513 q^{3} +0.193937 q^{4} +2.48119 q^{6} +1.28726 q^{7} -2.67513 q^{8} -0.193937 q^{9} -0.481194 q^{11} +0.324869 q^{12} +2.15633 q^{13} +1.90668 q^{14} -4.35026 q^{16} -0.287258 q^{18} -3.35026 q^{19} +2.15633 q^{21} -0.712742 q^{22} -8.24965 q^{23} -4.48119 q^{24} +3.19394 q^{26} -5.35026 q^{27} +0.249646 q^{28} -0.649738 q^{29} -1.83146 q^{31} -1.09332 q^{32} -0.806063 q^{33} -0.0376114 q^{36} +4.31265 q^{37} -4.96239 q^{38} +3.61213 q^{39} -11.2750 q^{41} +3.19394 q^{42} +8.15633 q^{43} -0.0933212 q^{44} -12.2193 q^{46} +6.54420 q^{47} -7.28726 q^{48} -5.34297 q^{49} +0.418190 q^{52} -8.57452 q^{53} -7.92478 q^{54} -3.44358 q^{56} -5.61213 q^{57} -0.962389 q^{58} -4.96239 q^{59} +2.83638 q^{61} -2.71274 q^{62} -0.249646 q^{63} +7.08110 q^{64} -1.19394 q^{66} -4.93207 q^{67} -13.8192 q^{69} +14.5320 q^{71} +0.518806 q^{72} -13.3503 q^{73} +6.38787 q^{74} -0.649738 q^{76} -0.619421 q^{77} +5.35026 q^{78} +9.05571 q^{79} -8.38058 q^{81} -16.7005 q^{82} -13.4314 q^{83} +0.418190 q^{84} +12.0811 q^{86} -1.08840 q^{87} +1.28726 q^{88} -16.7816 q^{89} +2.77575 q^{91} -1.59991 q^{92} -3.06793 q^{93} +9.69323 q^{94} -1.83146 q^{96} -3.66291 q^{97} -7.91397 q^{98} +0.0933212 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + q^{4} + 2 q^{6} - 2 q^{7} - 3 q^{8} - q^{9} + 4 q^{11} + 6 q^{12} - 4 q^{13} + 12 q^{14} - 3 q^{16} + 5 q^{18} - 4 q^{21} - 8 q^{22} - 8 q^{23} - 8 q^{24} + 10 q^{26} - 6 q^{27} - 16 q^{28}+ \cdots - 6 q^{99}+O(q^{100}) \) 3 * q - q^2 + q^4 + 2 * q^6 - 2 * q^7 - 3 * q^8 - q^9 + 4 * q^11 + 6 * q^12 - 4 * q^13 + 12 * q^14 - 3 * q^16 + 5 * q^18 - 4 * q^21 - 8 * q^22 - 8 * q^23 - 8 * q^24 + 10 * q^26 - 6 * q^27 - 16 * q^28 - 12 * q^29 + 10 * q^31 + 3 * q^32 - 2 * q^33 - 11 * q^36 - 8 * q^37 - 4 * q^38 + 10 * q^39 - 2 * q^41 + 10 * q^42 + 14 * q^43 + 6 * q^44 - 22 * q^46 + 10 * q^47 - 16 * q^48 + 7 * q^49 - 14 * q^53 - 2 * q^54 + 6 * q^56 - 16 * q^57 + 8 * q^58 - 4 * q^59 + 6 * q^61 - 14 * q^62 + 16 * q^63 - 11 * q^64 - 4 * q^66 - 6 * q^67 + 8 * q^71 + 7 * q^72 - 30 * q^73 + 20 * q^74 - 12 * q^76 - 14 * q^77 + 6 * q^78 + 10 * q^79 - 13 * q^81 - 30 * q^82 + 2 * q^83 + 4 * q^86 + 16 * q^87 - 2 * q^88 + 2 * q^89 + 10 * q^91 + 22 * q^92 - 18 * q^93 - 4 * q^94 + 10 * q^96 + 20 * q^97 - 43 * q^98 - 6 * q^99

Introduction

L-functions

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