LMFDB - Embedded newform 7245.2.a.bv.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7245,2,Mod(1,7245)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7245, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7245.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7245 = 3^{2} \cdot 5 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7245.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:	$57.8516162644$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 16x^{8} + 30x^{7} + 87x^{6} - 143x^{5} - 196x^{4} + 244x^{3} + 160x^{2} - 89x - 12 $ x^10 - 2x^9 - 16x^8 + 30x^7 + 87x^6 - 143x^5 - 196x^4 + 244x^3 + 160x^2 - 89*x - 12
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2415)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-1.90103$ of defining polynomial
Character	$\chi$	$=$	7245.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.90103 q^{2} +1.61393 q^{4} -1.00000 q^{5} +1.00000 q^{7} -0.733935 q^{8} +O(q^{10})$	\(q+1.90103 q^{2} +1.61393 q^{4} -1.00000 q^{5} +1.00000 q^{7} -0.733935 q^{8} -1.90103 q^{10} -3.08250 q^{11} -4.36980 q^{13} +1.90103 q^{14} -4.62309 q^{16} +3.91348 q^{17} +0.209598 q^{19} -1.61393 q^{20} -5.85994 q^{22} +1.00000 q^{23} +1.00000 q^{25} -8.30714 q^{26} +1.61393 q^{28} -2.84119 q^{29} +9.13845 q^{31} -7.32078 q^{32} +7.43965 q^{34} -1.00000 q^{35} +6.74648 q^{37} +0.398452 q^{38} +0.733935 q^{40} +3.97914 q^{41} +9.51553 q^{43} -4.97494 q^{44} +1.90103 q^{46} -7.70980 q^{47} +1.00000 q^{49} +1.90103 q^{50} -7.05255 q^{52} +12.7231 q^{53} +3.08250 q^{55} -0.733935 q^{56} -5.40119 q^{58} -3.22294 q^{59} -1.71549 q^{61} +17.3725 q^{62} -4.67087 q^{64} +4.36980 q^{65} +4.68562 q^{67} +6.31607 q^{68} -1.90103 q^{70} -4.54168 q^{71} -0.767751 q^{73} +12.8253 q^{74} +0.338276 q^{76} -3.08250 q^{77} +6.64506 q^{79} +4.62309 q^{80} +7.56448 q^{82} +15.8348 q^{83} -3.91348 q^{85} +18.0893 q^{86} +2.26236 q^{88} -7.44650 q^{89} -4.36980 q^{91} +1.61393 q^{92} -14.6566 q^{94} -0.209598 q^{95} +4.06148 q^{97} +1.90103 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{2} + 16 q^{4} - 10 q^{5} + 10 q^{7} - 6 q^{8}+O(q^{10}) $ 10 * q - 2 * q^2 + 16 * q^4 - 10 * q^5 + 10 * q^7 - 6 * q^8	$ 10 q - 2 q^{2} + 16 q^{4} - 10 q^{5} + 10 q^{7} - 6 q^{8} + 2 q^{10} - 9 q^{11} + 14 q^{13} - 2 q^{14} + 20 q^{16} - 8 q^{17} + 13 q^{19} - 16 q^{20} + 10 q^{23} + 10 q^{25} + 11 q^{26} + 16 q^{28} - 10 q^{29} + 8 q^{31} + 11 q^{32} - 5 q^{34} - 10 q^{35} + 8 q^{37} + 10 q^{38} + 6 q^{40} + 5 q^{41} + 4 q^{43} - 3 q^{44} - 2 q^{46} - q^{47} + 10 q^{49} - 2 q^{50} + 14 q^{52} - 9 q^{53} + 9 q^{55} - 6 q^{56} - 28 q^{58} + 17 q^{59} + 19 q^{61} + 28 q^{62} + 24 q^{64} - 14 q^{65} - 8 q^{68} + 2 q^{70} + 6 q^{73} - 3 q^{74} + 15 q^{76} - 9 q^{77} + 32 q^{79} - 20 q^{80} + 14 q^{82} + 2 q^{83} + 8 q^{85} - 2 q^{86} - 3 q^{88} - 10 q^{89} + 14 q^{91} + 16 q^{92} - 10 q^{94} - 13 q^{95} + 18 q^{97} - 2 q^{98}+O(q^{100}) $ 10 * q - 2 * q^2 + 16 * q^4 - 10 * q^5 + 10 * q^7 - 6 * q^8 + 2 * q^10 - 9 * q^11 + 14 * q^13 - 2 * q^14 + 20 * q^16 - 8 * q^17 + 13 * q^19 - 16 * q^20 + 10 * q^23 + 10 * q^25 + 11 * q^26 + 16 * q^28 - 10 * q^29 + 8 * q^31 + 11 * q^32 - 5 * q^34 - 10 * q^35 + 8 * q^37 + 10 * q^38 + 6 * q^40 + 5 * q^41 + 4 * q^43 - 3 * q^44 - 2 * q^46 - q^47 + 10 * q^49 - 2 * q^50 + 14 * q^52 - 9 * q^53 + 9 * q^55 - 6 * q^56 - 28 * q^58 + 17 * q^59 + 19 * q^61 + 28 * q^62 + 24 * q^64 - 14 * q^65 - 8 * q^68 + 2 * q^70 + 6 * q^73 - 3 * q^74 + 15 * q^76 - 9 * q^77 + 32 * q^79 - 20 * q^80 + 14 * q^82 + 2 * q^83 + 8 * q^85 - 2 * q^86 - 3 * q^88 - 10 * q^89 + 14 * q^91 + 16 * q^92 - 10 * q^94 - 13 * q^95 + 18 * q^97 - 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.90103	1.34423	0.672117	−	0.740445i	$-0.265385\pi$
$2$	1.90103	1.34423	0.672117	+	0.740445i	$0.265385\pi$
$3$	0	0
$3$	0	0
$4$	1.61393	0.806964
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−0.733935	−0.259485
$9$	0	0
$10$	−1.90103	−0.601160
$11$	−3.08250	−0.929410	−0.464705	−	0.885466i	$-0.653840\pi$
$11$	−3.08250	−0.929410	−0.464705	+	0.885466i	$0.653840\pi$
$12$	0	0
$13$	−4.36980	−1.21197	−0.605983	−	0.795478i	$-0.707220\pi$
$13$	−4.36980	−1.21197	−0.605983	+	0.795478i	$0.707220\pi$
$14$	1.90103	0.508073
$15$	0	0
$16$	−4.62309	−1.15577
$17$	3.91348	0.949157	0.474579	−	0.880213i	$-0.342600\pi$
$17$	3.91348	0.949157	0.474579	+	0.880213i	$0.342600\pi$
$18$	0	0
$19$	0.209598	0.0480850	0.0240425	−	0.999711i	$-0.492346\pi$
$19$	0.209598	0.0480850	0.0240425	+	0.999711i	$0.492346\pi$
$20$	−1.61393	−0.360885
$21$	0	0
$22$	−5.85994	−1.24934
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	−8.30714	−1.62916
$27$	0	0
$28$	1.61393	0.305004
$29$	−2.84119	−0.527595	−0.263798	−	0.964578i	$-0.584975\pi$
$29$	−2.84119	−0.527595	−0.263798	+	0.964578i	$0.584975\pi$
$30$	0	0
$31$	9.13845	1.64131	0.820657	−	0.571421i	$-0.193608\pi$
$31$	9.13845	1.64131	0.820657	+	0.571421i	$0.193608\pi$
$32$	−7.32078	−1.29414
$33$	0	0
$34$	7.43965	1.27589
$35$	−1.00000	−0.169031
$36$	0	0
$37$	6.74648	1.10911	0.554557	−	0.832145i	$-0.312888\pi$
$37$	6.74648	1.10911	0.554557	+	0.832145i	$0.312888\pi$
$38$	0.398452	0.0646375
$39$	0	0
$40$	0.733935	0.116045
$41$	3.97914	0.621437	0.310719	−	0.950502i	$-0.399430\pi$
$41$	3.97914	0.621437	0.310719	+	0.950502i	$0.399430\pi$
$42$	0	0
$43$	9.51553	1.45110	0.725552	−	0.688167i	$-0.241584\pi$
$43$	9.51553	1.45110	0.725552	+	0.688167i	$0.241584\pi$
$44$	−4.97494	−0.750000
$45$	0	0
$46$	1.90103	0.280292
$47$	−7.70980	−1.12459	−0.562295	−	0.826937i	$-0.690081\pi$
$47$	−7.70980	−1.12459	−0.562295	+	0.826937i	$0.690081\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.90103	0.268847
$51$	0	0
$52$	−7.05255	−0.978013
$53$	12.7231	1.74766	0.873829	−	0.486234i	$-0.161630\pi$
$53$	12.7231	1.74766	0.873829	+	0.486234i	$0.161630\pi$
$54$	0	0
$55$	3.08250	0.415645
$56$	−0.733935	−0.0980762
$57$	0	0
$58$	−5.40119	−0.709211
$59$	−3.22294	−0.419591	−0.209796	−	0.977745i	$-0.567280\pi$
$59$	−3.22294	−0.419591	−0.209796	+	0.977745i	$0.567280\pi$
$60$	0	0
$61$	−1.71549	−0.219646	−0.109823	−	0.993951i	$-0.535028\pi$
$61$	−1.71549	−0.219646	−0.109823	+	0.993951i	$0.535028\pi$
$62$	17.3725	2.20631
$63$	0	0
$64$	−4.67087	−0.583859
$65$	4.36980	0.542007
$66$	0	0
$67$	4.68562	0.572440	0.286220	−	0.958164i	$-0.407601\pi$
$67$	4.68562	0.572440	0.286220	+	0.958164i	$0.407601\pi$
$68$	6.31607	0.765936
$69$	0	0
$70$	−1.90103	−0.227217
$71$	−4.54168	−0.538999	−0.269499	−	0.963001i	$-0.586858\pi$
$71$	−4.54168	−0.538999	−0.269499	+	0.963001i	$0.586858\pi$
$72$	0	0
$73$	−0.767751	−0.0898585	−0.0449292	−	0.998990i	$-0.514306\pi$
$73$	−0.767751	−0.0898585	−0.0449292	+	0.998990i	$0.514306\pi$
$74$	12.8253	1.49091
$75$	0	0
$76$	0.338276	0.0388029
$77$	−3.08250	−0.351284
$78$	0	0
$79$	6.64506	0.747628	0.373814	−	0.927504i	$-0.378050\pi$
$79$	6.64506	0.747628	0.373814	+	0.927504i	$0.378050\pi$
$80$	4.62309	0.516877
$81$	0	0
$82$	7.56448	0.835357
$83$	15.8348	1.73809	0.869047	−	0.494730i	$-0.164733\pi$
$83$	15.8348	1.73809	0.869047	+	0.494730i	$0.164733\pi$
$84$	0	0
$85$	−3.91348	−0.424476
$86$	18.0893	1.95062
$87$	0	0
$88$	2.26236	0.241168
$89$	−7.44650	−0.789328	−0.394664	−	0.918826i	$-0.629139\pi$
$89$	−7.44650	−0.789328	−0.394664	+	0.918826i	$0.629139\pi$
$90$	0	0
$91$	−4.36980	−0.458080
$92$	1.61393	0.168264
$93$	0	0
$94$	−14.6566	−1.51171
$95$	−0.209598	−0.0215043
$96$	0	0
$97$	4.06148	0.412380	0.206190	−	0.978512i	$-0.433893\pi$
$97$	4.06148	0.412380	0.206190	+	0.978512i	$0.433893\pi$
$98$	1.90103	0.192033
$99$	0	0
$100$	1.61393	0.161393
$101$	10.3687	1.03173	0.515864	−	0.856670i	$-0.327471\pi$
$101$	10.3687	1.03173	0.515864	+	0.856670i	$0.327471\pi$
$102$	0	0
$103$	9.76343	0.962020	0.481010	−	0.876715i	$-0.340270\pi$
$103$	9.76343	0.962020	0.481010	+	0.876715i	$0.340270\pi$
$104$	3.20715	0.314487
$105$	0	0
$106$	24.1871	2.34926
$107$	−11.9192	−1.15227	−0.576134	−	0.817355i	$-0.695440\pi$
$107$	−11.9192	−1.15227	−0.576134	+	0.817355i	$0.695440\pi$
$108$	0	0
$109$	16.2721	1.55859	0.779293	−	0.626660i	$-0.215579\pi$
$109$	16.2721	1.55859	0.779293	+	0.626660i	$0.215579\pi$
$110$	5.85994	0.558723
$111$	0	0
$112$	−4.62309	−0.436841
$113$	7.03810	0.662089	0.331045	−	0.943615i	$-0.392599\pi$
$113$	7.03810	0.662089	0.331045	+	0.943615i	$0.392599\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	−4.58547	−0.425750
$117$	0	0
$118$	−6.12692	−0.564028
$119$	3.91348	0.358748
$120$	0	0
$121$	−1.49818	−0.136198
$122$	−3.26121	−0.295256
$123$	0	0
$124$	14.7488	1.32448
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−5.61942	−0.498642	−0.249321	−	0.968421i	$-0.580208\pi$
$127$	−5.61942	−0.498642	−0.249321	+	0.968421i	$0.580208\pi$
$128$	5.76208	0.509301
$129$	0	0
$130$	8.30714	0.728585
$131$	6.97507	0.609415	0.304707	−	0.952446i	$-0.401441\pi$
$131$	6.97507	0.609415	0.304707	+	0.952446i	$0.401441\pi$
$132$	0	0
$133$	0.209598	0.0181744
$134$	8.90752	0.769492
$135$	0	0
$136$	−2.87224	−0.246292
$137$	−9.50150	−0.811768	−0.405884	−	0.913925i	$-0.633036\pi$
$137$	−9.50150	−0.811768	−0.405884	+	0.913925i	$0.633036\pi$
$138$	0	0
$139$	12.7424	1.08079	0.540397	−	0.841410i	$-0.318274\pi$
$139$	12.7424	1.08079	0.540397	+	0.841410i	$0.318274\pi$
$140$	−1.61393	−0.136402
$141$	0	0
$142$	−8.63390	−0.724540
$143$	13.4699	1.12641
$144$	0	0
$145$	2.84119	0.235948
$146$	−1.45952	−0.120791
$147$	0	0
$148$	10.8883	0.895016
$149$	−5.60930	−0.459532	−0.229766	−	0.973246i	$-0.573796\pi$
$149$	−5.60930	−0.459532	−0.229766	+	0.973246i	$0.573796\pi$
$150$	0	0
$151$	8.79336	0.715594	0.357797	−	0.933799i	$-0.383528\pi$
$151$	8.79336	0.715594	0.357797	+	0.933799i	$0.383528\pi$
$152$	−0.153831	−0.0124774
$153$	0	0
$154$	−5.85994	−0.472208
$155$	−9.13845	−0.734018
$156$	0	0
$157$	−4.41110	−0.352044	−0.176022	−	0.984386i	$-0.556323\pi$
$157$	−4.41110	−0.352044	−0.176022	+	0.984386i	$0.556323\pi$
$158$	12.6325	1.00499
$159$	0	0
$160$	7.32078	0.578759
$161$	1.00000	0.0788110
$162$	0	0
$163$	6.73572	0.527582	0.263791	−	0.964580i	$-0.415027\pi$
$163$	6.73572	0.527582	0.263791	+	0.964580i	$0.415027\pi$
$164$	6.42205	0.501478
$165$	0	0
$166$	30.1025	2.33640
$167$	−2.01945	−0.156269	−0.0781347	−	0.996943i	$-0.524896\pi$
$167$	−2.01945	−0.156269	−0.0781347	+	0.996943i	$0.524896\pi$
$168$	0	0
$169$	6.09518	0.468860
$170$	−7.43965	−0.570595
$171$	0	0
$172$	15.3574	1.17099
$173$	5.66981	0.431068	0.215534	−	0.976496i	$-0.430851\pi$
$173$	5.66981	0.431068	0.215534	+	0.976496i	$0.430851\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	14.2507	1.07419
$177$	0	0
$178$	−14.1561	−1.06104
$179$	−4.87447	−0.364335	−0.182168	−	0.983267i	$-0.558311\pi$
$179$	−4.87447	−0.364335	−0.182168	+	0.983267i	$0.558311\pi$
$180$	0	0
$181$	−11.3864	−0.846348	−0.423174	−	0.906048i	$-0.639084\pi$
$181$	−11.3864	−0.846348	−0.423174	+	0.906048i	$0.639084\pi$
$182$	−8.30714	−0.615766
$183$	0	0
$184$	−0.733935	−0.0541064
$185$	−6.74648	−0.496011
$186$	0	0
$187$	−12.0633	−0.882156
$188$	−12.4431	−0.907503
$189$	0	0
$190$	−0.398452	−0.0289068
$191$	−22.0335	−1.59429	−0.797144	−	0.603790i	$-0.793657\pi$
$191$	−22.0335	−1.59429	−0.797144	+	0.603790i	$0.793657\pi$
$192$	0	0
$193$	−15.2006	−1.09416	−0.547081	−	0.837080i	$-0.684261\pi$
$193$	−15.2006	−1.09416	−0.547081	+	0.837080i	$0.684261\pi$
$194$	7.72100	0.554336
$195$	0	0
$196$	1.61393	0.115281
$197$	−8.03612	−0.572550	−0.286275	−	0.958148i	$-0.592417\pi$
$197$	−8.03612	−0.572550	−0.286275	+	0.958148i	$0.592417\pi$
$198$	0	0
$199$	−25.5617	−1.81202	−0.906011	−	0.423254i	$-0.860888\pi$
$199$	−25.5617	−1.81202	−0.906011	+	0.423254i	$0.860888\pi$
$200$	−0.733935	−0.0518970
$201$	0	0
$202$	19.7113	1.38688
$203$	−2.84119	−0.199412
$204$	0	0
$205$	−3.97914	−0.277915
$206$	18.5606	1.29318
$207$	0	0
$208$	20.2020	1.40076
$209$	−0.646086	−0.0446907
$210$	0	0
$211$	11.9115	0.820023	0.410011	−	0.912080i	$-0.365525\pi$
$211$	11.9115	0.820023	0.410011	+	0.912080i	$0.365525\pi$
$212$	20.5342	1.41030
$213$	0	0
$214$	−22.6587	−1.54892
$215$	−9.51553	−0.648954
$216$	0	0
$217$	9.13845	0.620358
$218$	30.9338	2.09510
$219$	0	0
$220$	4.97494	0.335410
$221$	−17.1011	−1.15035
$222$	0	0
$223$	9.10602	0.609784	0.304892	−	0.952387i	$-0.401380\pi$
$223$	9.10602	0.609784	0.304892	+	0.952387i	$0.401380\pi$
$224$	−7.32078	−0.489140
$225$	0	0
$226$	13.3797	0.890003
$227$	6.99649	0.464373	0.232187	−	0.972671i	$-0.425412\pi$
$227$	6.99649	0.464373	0.232187	+	0.972671i	$0.425412\pi$
$228$	0	0
$229$	16.7572	1.10735	0.553675	−	0.832733i	$-0.313225\pi$
$229$	16.7572	1.10735	0.553675	+	0.832733i	$0.313225\pi$
$230$	−1.90103	−0.125350
$231$	0	0
$232$	2.08525	0.136903
$233$	20.2492	1.32657	0.663283	−	0.748368i	$-0.269162\pi$
$233$	20.2492	1.32657	0.663283	+	0.748368i	$0.269162\pi$
$234$	0	0
$235$	7.70980	0.502932
$236$	−5.20160	−0.338595
$237$	0	0
$238$	7.43965	0.482241
$239$	−22.1015	−1.42963	−0.714814	−	0.699314i	$-0.753489\pi$
$239$	−22.1015	−1.42963	−0.714814	+	0.699314i	$0.753489\pi$
$240$	0	0
$241$	21.9020	1.41083	0.705416	−	0.708794i	$-0.250760\pi$
$241$	21.9020	1.41083	0.705416	+	0.708794i	$0.250760\pi$
$242$	−2.84808	−0.183082
$243$	0	0
$244$	−2.76868	−0.177247
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−0.915901	−0.0582774
$248$	−6.70703	−0.425897
$249$	0	0
$250$	−1.90103	−0.120232
$251$	4.01352	0.253331	0.126666	−	0.991945i	$-0.459573\pi$
$251$	4.01352	0.253331	0.126666	+	0.991945i	$0.459573\pi$
$252$	0	0
$253$	−3.08250	−0.193795
$254$	−10.6827	−0.670292
$255$	0	0
$256$	20.2957	1.26848
$257$	−3.79537	−0.236749	−0.118374	−	0.992969i	$-0.537768\pi$
$257$	−3.79537	−0.236749	−0.118374	+	0.992969i	$0.537768\pi$
$258$	0	0
$259$	6.74648	0.419206
$260$	7.05255	0.437381
$261$	0	0
$262$	13.2598	0.819196
$263$	5.72874	0.353249	0.176625	−	0.984278i	$-0.443482\pi$
$263$	5.72874	0.353249	0.176625	+	0.984278i	$0.443482\pi$
$264$	0	0
$265$	−12.7231	−0.781576
$266$	0.398452	0.0244307
$267$	0	0
$268$	7.56225	0.461938
$269$	3.25219	0.198289	0.0991446	−	0.995073i	$-0.468389\pi$
$269$	3.25219	0.198289	0.0991446	+	0.995073i	$0.468389\pi$
$270$	0	0
$271$	12.1949	0.740789	0.370394	−	0.928875i	$-0.379223\pi$
$271$	12.1949	0.740789	0.370394	+	0.928875i	$0.379223\pi$
$272$	−18.0924	−1.09701
$273$	0	0
$274$	−18.0627	−1.09121
$275$	−3.08250	−0.185882
$276$	0	0
$277$	−17.8849	−1.07460	−0.537301	−	0.843391i	$-0.680556\pi$
$277$	−17.8849	−1.07460	−0.537301	+	0.843391i	$0.680556\pi$
$278$	24.2237	1.45284
$279$	0	0
$280$	0.733935	0.0438610
$281$	−27.6266	−1.64807	−0.824034	−	0.566541i	$-0.808281\pi$
$281$	−27.6266	−1.64807	−0.824034	+	0.566541i	$0.808281\pi$
$282$	0	0
$283$	−27.9862	−1.66361	−0.831803	−	0.555071i	$-0.812691\pi$
$283$	−27.9862	−1.66361	−0.831803	+	0.555071i	$0.812691\pi$
$284$	−7.32995	−0.434953
$285$	0	0
$286$	25.6068	1.51416
$287$	3.97914	0.234881
$288$	0	0
$289$	−1.68470	−0.0991002
$290$	5.40119	0.317169
$291$	0	0
$292$	−1.23910	−0.0725126
$293$	23.8737	1.39471	0.697357	−	0.716724i	$-0.254359\pi$
$293$	23.8737	1.39471	0.697357	+	0.716724i	$0.254359\pi$
$294$	0	0
$295$	3.22294	0.187647
$296$	−4.95148	−0.287799
$297$	0	0
$298$	−10.6635	−0.617718
$299$	−4.36980	−0.252712
$300$	0	0
$301$	9.51553	0.548466
$302$	16.7165	0.961925
$303$	0	0
$304$	−0.968990	−0.0555754
$305$	1.71549	0.0982289
$306$	0	0
$307$	25.9704	1.48221	0.741105	−	0.671390i	$-0.234302\pi$
$307$	25.9704	1.48221	0.741105	+	0.671390i	$0.234302\pi$
$308$	−4.97494	−0.283473
$309$	0	0
$310$	−17.3725	−0.986692
$311$	2.17066	0.123087	0.0615434	−	0.998104i	$-0.480398\pi$
$311$	2.17066	0.123087	0.0615434	+	0.998104i	$0.480398\pi$
$312$	0	0
$313$	30.0595	1.69906	0.849532	−	0.527537i	$-0.176884\pi$
$313$	30.0595	1.69906	0.849532	+	0.527537i	$0.176884\pi$
$314$	−8.38566	−0.473230
$315$	0	0
$316$	10.7247	0.603309
$317$	1.65752	0.0930954	0.0465477	−	0.998916i	$-0.485178\pi$
$317$	1.65752	0.0930954	0.0465477	+	0.998916i	$0.485178\pi$
$318$	0	0
$319$	8.75797	0.490352
$320$	4.67087	0.261110
$321$	0	0
$322$	1.90103	0.105940
$323$	0.820256	0.0456403
$324$	0	0
$325$	−4.36980	−0.242393
$326$	12.8048	0.709194
$327$	0	0
$328$	−2.92043	−0.161254
$329$	−7.70980	−0.425055
$330$	0	0
$331$	20.7489	1.14046	0.570232	−	0.821483i	$-0.306853\pi$
$331$	20.7489	1.14046	0.570232	+	0.821483i	$0.306853\pi$
$332$	25.5562	1.40258
$333$	0	0
$334$	−3.83903	−0.210063
$335$	−4.68562	−0.256003
$336$	0	0
$337$	−24.4181	−1.33014	−0.665068	−	0.746782i	$-0.731598\pi$
$337$	−24.4181	−1.33014	−0.665068	+	0.746782i	$0.731598\pi$
$338$	11.5871	0.630257
$339$	0	0
$340$	−6.31607	−0.342537
$341$	−28.1693	−1.52545
$342$	0	0
$343$	1.00000	0.0539949
$344$	−6.98378	−0.376540
$345$	0	0
$346$	10.7785	0.579456
$347$	−9.13516	−0.490401	−0.245201	−	0.969472i	$-0.578854\pi$
$347$	−9.13516	−0.490401	−0.245201	+	0.969472i	$0.578854\pi$
$348$	0	0
$349$	21.6578	1.15932	0.579658	−	0.814860i	$-0.303186\pi$
$349$	21.6578	1.15932	0.579658	+	0.814860i	$0.303186\pi$
$350$	1.90103	0.101615
$351$	0	0
$352$	22.5663	1.20279
$353$	19.4080	1.03298	0.516491	−	0.856293i	$-0.327238\pi$
$353$	19.4080	1.03298	0.516491	+	0.856293i	$0.327238\pi$
$354$	0	0
$355$	4.54168	0.241048
$356$	−12.0181	−0.636959
$357$	0	0
$358$	−9.26654	−0.489752
$359$	20.2121	1.06675	0.533377	−	0.845878i	$-0.320923\pi$
$359$	20.2121	1.06675	0.533377	+	0.845878i	$0.320923\pi$
$360$	0	0
$361$	−18.9561	−0.997688
$362$	−21.6460	−1.13769
$363$	0	0
$364$	−7.05255	−0.369654
$365$	0.767751	0.0401859
$366$	0	0
$367$	−6.58805	−0.343893	−0.171947	−	0.985106i	$-0.555006\pi$
$367$	−6.58805	−0.343893	−0.171947	+	0.985106i	$0.555006\pi$
$368$	−4.62309	−0.240995
$369$	0	0
$370$	−12.8253	−0.666755
$371$	12.7231	0.660552
$372$	0	0
$373$	−17.0919	−0.884986	−0.442493	−	0.896772i	$-0.645906\pi$
$373$	−17.0919	−0.884986	−0.442493	+	0.896772i	$0.645906\pi$
$374$	−22.9327	−1.18582
$375$	0	0
$376$	5.65849	0.291814
$377$	12.4154	0.639427
$378$	0	0
$379$	13.0912	0.672449	0.336224	−	0.941782i	$-0.390850\pi$
$379$	13.0912	0.672449	0.336224	+	0.941782i	$0.390850\pi$
$380$	−0.338276	−0.0173532
$381$	0	0
$382$	−41.8864	−2.14309
$383$	−11.9650	−0.611382	−0.305691	−	0.952131i	$-0.598887\pi$
$383$	−11.9650	−0.611382	−0.305691	+	0.952131i	$0.598887\pi$
$384$	0	0
$385$	3.08250	0.157099
$386$	−28.8968	−1.47081
$387$	0	0
$388$	6.55493	0.332776
$389$	14.7054	0.745591	0.372796	−	0.927914i	$-0.378399\pi$
$389$	14.7054	0.745591	0.372796	+	0.927914i	$0.378399\pi$
$390$	0	0
$391$	3.91348	0.197913
$392$	−0.733935	−0.0370693
$393$	0	0
$394$	−15.2769	−0.769640
$395$	−6.64506	−0.334349
$396$	0	0
$397$	36.3058	1.82213	0.911067	−	0.412258i	$-0.135260\pi$
$397$	36.3058	1.82213	0.911067	+	0.412258i	$0.135260\pi$
$398$	−48.5937	−2.43578
$399$	0	0
$400$	−4.62309	−0.231155
$401$	−28.9819	−1.44729	−0.723644	−	0.690173i	$-0.757534\pi$
$401$	−28.9819	−1.44729	−0.723644	+	0.690173i	$0.757534\pi$
$402$	0	0
$403$	−39.9332	−1.98922
$404$	16.7344	0.832568
$405$	0	0
$406$	−5.40119	−0.268057
$407$	−20.7961	−1.03082
$408$	0	0
$409$	31.0744	1.53653	0.768264	−	0.640133i	$-0.221121\pi$
$409$	31.0744	1.53653	0.768264	+	0.640133i	$0.221121\pi$
$410$	−7.56448	−0.373583
$411$	0	0
$412$	15.7575	0.776316
$413$	−3.22294	−0.158591
$414$	0	0
$415$	−15.8348	−0.777299
$416$	31.9904	1.56846
$417$	0	0
$418$	−1.22823	−0.0600747
$419$	−18.8076	−0.918811	−0.459405	−	0.888227i	$-0.651937\pi$
$419$	−18.8076	−0.918811	−0.459405	+	0.888227i	$0.651937\pi$
$420$	0	0
$421$	−21.3350	−1.03980	−0.519902	−	0.854226i	$-0.674032\pi$
$421$	−21.3350	−1.03980	−0.519902	+	0.854226i	$0.674032\pi$
$422$	22.6442	1.10230
$423$	0	0
$424$	−9.33795	−0.453491
$425$	3.91348	0.189831
$426$	0	0
$427$	−1.71549	−0.0830186
$428$	−19.2367	−0.929840
$429$	0	0
$430$	−18.0893	−0.872345
$431$	23.3085	1.12273	0.561365	−	0.827569i	$-0.310277\pi$
$431$	23.3085	1.12273	0.561365	+	0.827569i	$0.310277\pi$
$432$	0	0
$433$	−3.79975	−0.182604	−0.0913022	−	0.995823i	$-0.529103\pi$
$433$	−3.79975	−0.182604	−0.0913022	+	0.995823i	$0.529103\pi$
$434$	17.3725	0.833907
$435$	0	0
$436$	26.2620	1.25772
$437$	0.209598	0.0100264
$438$	0	0
$439$	−33.5648	−1.60196	−0.800979	−	0.598692i	$-0.795687\pi$
$439$	−33.5648	−1.60196	−0.800979	+	0.598692i	$0.795687\pi$
$440$	−2.26236	−0.107854
$441$	0	0
$442$	−32.5098	−1.54633
$443$	−2.73658	−0.130019	−0.0650094	−	0.997885i	$-0.520708\pi$
$443$	−2.73658	−0.130019	−0.0650094	+	0.997885i	$0.520708\pi$
$444$	0	0
$445$	7.44650	0.352998
$446$	17.3108	0.819692
$447$	0	0
$448$	−4.67087	−0.220678
$449$	5.40737	0.255189	0.127595	−	0.991826i	$-0.459274\pi$
$449$	5.40737	0.255189	0.127595	+	0.991826i	$0.459274\pi$
$450$	0	0
$451$	−12.2657	−0.577570
$452$	11.3590	0.534282
$453$	0	0
$454$	13.3006	0.624226
$455$	4.36980	0.204860
$456$	0	0
$457$	−12.5890	−0.588888	−0.294444	−	0.955669i	$-0.595134\pi$
$457$	−12.5890	−0.588888	−0.294444	+	0.955669i	$0.595134\pi$
$458$	31.8561	1.48854
$459$	0	0
$460$	−1.61393	−0.0752498
$461$	14.5901	0.679529	0.339764	−	0.940511i	$-0.389653\pi$
$461$	14.5901	0.679529	0.339764	+	0.940511i	$0.389653\pi$
$462$	0	0
$463$	32.9600	1.53178	0.765890	−	0.642972i	$-0.222299\pi$
$463$	32.9600	1.53178	0.765890	+	0.642972i	$0.222299\pi$
$464$	13.1351	0.609780
$465$	0	0
$466$	38.4943	1.78322
$467$	−4.16884	−0.192911	−0.0964554	−	0.995337i	$-0.530751\pi$
$467$	−4.16884	−0.192911	−0.0964554	+	0.995337i	$0.530751\pi$
$468$	0	0
$469$	4.68562	0.216362
$470$	14.6566	0.676058
$471$	0	0
$472$	2.36543	0.108878
$473$	−29.3316	−1.34867
$474$	0	0
$475$	0.209598	0.00961701
$476$	6.31607	0.289497
$477$	0	0
$478$	−42.0157	−1.92176
$479$	36.1536	1.65190	0.825949	−	0.563745i	$-0.190640\pi$
$479$	36.1536	1.65190	0.825949	+	0.563745i	$0.190640\pi$
$480$	0	0
$481$	−29.4808	−1.34421
$482$	41.6364	1.89649
$483$	0	0
$484$	−2.41795	−0.109907
$485$	−4.06148	−0.184422
$486$	0	0
$487$	−39.3649	−1.78379	−0.891896	−	0.452240i	$-0.850625\pi$
$487$	−39.3649	−1.78379	−0.891896	+	0.452240i	$0.850625\pi$
$488$	1.25906	0.0569950
$489$	0	0
$490$	−1.90103	−0.0858799
$491$	−16.1377	−0.728284	−0.364142	−	0.931344i	$-0.618638\pi$
$491$	−16.1377	−0.728284	−0.364142	+	0.931344i	$0.618638\pi$
$492$	0	0
$493$	−11.1189	−0.500771
$494$	−1.74116	−0.0783384
$495$	0	0
$496$	−42.2479	−1.89699
$497$	−4.54168	−0.203722
$498$	0	0
$499$	−12.0077	−0.537539	−0.268770	−	0.963205i	$-0.586617\pi$
$499$	−12.0077	−0.537539	−0.268770	+	0.963205i	$0.586617\pi$
$500$	−1.61393	−0.0721771
$501$	0	0
$502$	7.62984	0.340536
$503$	14.0978	0.628588	0.314294	−	0.949326i	$-0.398232\pi$
$503$	14.0978	0.628588	0.314294	+	0.949326i	$0.398232\pi$
$504$	0	0
$505$	−10.3687	−0.461403
$506$	−5.85994	−0.260506
$507$	0	0
$508$	−9.06933	−0.402387
$509$	2.17861	0.0965653	0.0482826	−	0.998834i	$-0.484625\pi$
$509$	2.17861	0.0965653	0.0482826	+	0.998834i	$0.484625\pi$
$510$	0	0
$511$	−0.767751	−0.0339633
$512$	27.0586	1.19583
$513$	0	0
$514$	−7.21512	−0.318245
$515$	−9.76343	−0.430228
$516$	0	0
$517$	23.7655	1.04520
$518$	12.8253	0.563511
$519$	0	0
$520$	−3.20715	−0.140643
$521$	29.3160	1.28436	0.642178	−	0.766555i	$-0.278031\pi$
$521$	29.3160	1.28436	0.642178	+	0.766555i	$0.278031\pi$
$522$	0	0
$523$	−5.47392	−0.239358	−0.119679	−	0.992813i	$-0.538187\pi$
$523$	−5.47392	−0.239358	−0.119679	+	0.992813i	$0.538187\pi$
$524$	11.2573	0.491776
$525$	0	0
$526$	10.8905	0.474849
$527$	35.7631	1.55787
$528$	0	0
$529$	1.00000	0.0434783
$530$	−24.1871	−1.05062
$531$	0	0
$532$	0.338276	0.0146661
$533$	−17.3881	−0.753160
$534$	0	0
$535$	11.9192	0.515310
$536$	−3.43894	−0.148540
$537$	0	0
$538$	6.18251	0.266547
$539$	−3.08250	−0.132773
$540$	0	0
$541$	24.1677	1.03905	0.519525	−	0.854455i	$-0.326109\pi$
$541$	24.1677	1.03905	0.519525	+	0.854455i	$0.326109\pi$
$542$	23.1830	0.995794
$543$	0	0
$544$	−28.6497	−1.22835
$545$	−16.2721	−0.697021
$546$	0	0
$547$	8.49967	0.363419	0.181710	−	0.983352i	$-0.441837\pi$
$547$	8.49967	0.363419	0.181710	+	0.983352i	$0.441837\pi$
$548$	−15.3347	−0.655068
$549$	0	0
$550$	−5.85994	−0.249869
$551$	−0.595507	−0.0253694
$552$	0	0
$553$	6.64506	0.282577
$554$	−33.9999	−1.44452
$555$	0	0
$556$	20.5653	0.872162
$557$	17.4128	0.737803	0.368902	−	0.929468i	$-0.379734\pi$
$557$	17.4128	0.737803	0.368902	+	0.929468i	$0.379734\pi$
$558$	0	0
$559$	−41.5810	−1.75869
$560$	4.62309	0.195361
$561$	0	0
$562$	−52.5192	−2.21539
$563$	7.29010	0.307241	0.153620	−	0.988130i	$-0.450907\pi$
$563$	7.29010	0.307241	0.153620	+	0.988130i	$0.450907\pi$
$564$	0	0
$565$	−7.03810	−0.296095
$566$	−53.2027	−2.23627
$567$	0	0
$568$	3.33330	0.139862
$569$	28.1750	1.18116	0.590578	−	0.806980i	$-0.298900\pi$
$569$	28.1750	1.18116	0.590578	+	0.806980i	$0.298900\pi$
$570$	0	0
$571$	32.1338	1.34476	0.672380	−	0.740206i	$-0.265272\pi$
$571$	32.1338	1.34476	0.672380	+	0.740206i	$0.265272\pi$
$572$	21.7395	0.908974
$573$	0	0
$574$	7.56448	0.315735
$575$	1.00000	0.0417029
$576$	0	0
$577$	12.3009	0.512095	0.256047	−	0.966664i	$-0.417580\pi$
$577$	12.3009	0.512095	0.256047	+	0.966664i	$0.417580\pi$
$578$	−3.20268	−0.133214
$579$	0	0
$580$	4.58547	0.190401
$581$	15.8348	0.656938
$582$	0	0
$583$	−39.2191	−1.62429
$584$	0.563479	0.0233169
$585$	0	0
$586$	45.3846	1.87482
$587$	19.9075	0.821671	0.410836	−	0.911709i	$-0.365237\pi$
$587$	19.9075	0.821671	0.410836	+	0.911709i	$0.365237\pi$
$588$	0	0
$589$	1.91540	0.0789226
$590$	6.12692	0.252241
$591$	0	0
$592$	−31.1896	−1.28188
$593$	1.27208	0.0522382	0.0261191	−	0.999659i	$-0.491685\pi$
$593$	1.27208	0.0522382	0.0261191	+	0.999659i	$0.491685\pi$
$594$	0	0
$595$	−3.91348	−0.160437
$596$	−9.05301	−0.370826
$597$	0	0
$598$	−8.30714	−0.339704
$599$	12.3041	0.502732	0.251366	−	0.967892i	$-0.419120\pi$
$599$	12.3041	0.502732	0.251366	+	0.967892i	$0.419120\pi$
$600$	0	0
$601$	29.9449	1.22148	0.610739	−	0.791832i	$-0.290872\pi$
$601$	29.9449	1.22148	0.610739	+	0.791832i	$0.290872\pi$
$602$	18.0893	0.737266
$603$	0	0
$604$	14.1919	0.577459
$605$	1.49818	0.0609095
$606$	0	0
$607$	3.66335	0.148691	0.0743454	−	0.997233i	$-0.476313\pi$
$607$	3.66335	0.148691	0.0743454	+	0.997233i	$0.476313\pi$
$608$	−1.53442	−0.0622290
$609$	0	0
$610$	3.26121	0.132043
$611$	33.6903	1.36296
$612$	0	0
$613$	13.7083	0.553672	0.276836	−	0.960917i	$-0.410714\pi$
$613$	13.7083	0.553672	0.276836	+	0.960917i	$0.410714\pi$
$614$	49.3706	1.99244
$615$	0	0
$616$	2.26236	0.0911529
$617$	−27.5979	−1.11105	−0.555525	−	0.831500i	$-0.687483\pi$
$617$	−27.5979	−1.11105	−0.555525	+	0.831500i	$0.687483\pi$
$618$	0	0
$619$	−32.6525	−1.31241	−0.656207	−	0.754581i	$-0.727840\pi$
$619$	−32.6525	−1.31241	−0.656207	+	0.754581i	$0.727840\pi$
$620$	−14.7488	−0.592326
$621$	0	0
$622$	4.12649	0.165457
$623$	−7.44650	−0.298338
$624$	0	0
$625$	1.00000	0.0400000
$626$	57.1441	2.28394
$627$	0	0
$628$	−7.11921	−0.284087
$629$	26.4022	1.05272
$630$	0	0
$631$	−26.2164	−1.04366	−0.521830	−	0.853050i	$-0.674750\pi$
$631$	−26.2164	−1.04366	−0.521830	+	0.853050i	$0.674750\pi$
$632$	−4.87704	−0.193998
$633$	0	0
$634$	3.15099	0.125142
$635$	5.61942	0.223000
$636$	0	0
$637$	−4.36980	−0.173138
$638$	16.6492	0.659148
$639$	0	0
$640$	−5.76208	−0.227766
$641$	−1.61002	−0.0635919	−0.0317959	−	0.999494i	$-0.510123\pi$
$641$	−1.61002	−0.0635919	−0.0317959	+	0.999494i	$0.510123\pi$
$642$	0	0
$643$	16.6094	0.655011	0.327505	−	0.944849i	$-0.393792\pi$
$643$	16.6094	0.655011	0.327505	+	0.944849i	$0.393792\pi$
$644$	1.61393	0.0635977
$645$	0	0
$646$	1.55933	0.0613512
$647$	2.13466	0.0839220	0.0419610	−	0.999119i	$-0.486639\pi$
$647$	2.13466	0.0839220	0.0419610	+	0.999119i	$0.486639\pi$
$648$	0	0
$649$	9.93472	0.389972
$650$	−8.30714	−0.325833
$651$	0	0
$652$	10.8710	0.425740
$653$	−33.3242	−1.30408	−0.652039	−	0.758185i	$-0.726086\pi$
$653$	−33.3242	−1.30408	−0.652039	+	0.758185i	$0.726086\pi$
$654$	0	0
$655$	−6.97507	−0.272539
$656$	−18.3959	−0.718240
$657$	0	0
$658$	−14.6566	−0.571373
$659$	21.3629	0.832181	0.416091	−	0.909323i	$-0.363400\pi$
$659$	21.3629	0.832181	0.416091	+	0.909323i	$0.363400\pi$
$660$	0	0
$661$	−36.3297	−1.41306	−0.706530	−	0.707683i	$-0.749741\pi$
$661$	−36.3297	−1.41306	−0.706530	+	0.707683i	$0.749741\pi$
$662$	39.4444	1.53305
$663$	0	0
$664$	−11.6217	−0.451009
$665$	−0.209598	−0.00812786
$666$	0	0
$667$	−2.84119	−0.110011
$668$	−3.25924	−0.126104
$669$	0	0
$670$	−8.90752	−0.344127
$671$	5.28801	0.204142
$672$	0	0
$673$	−46.2707	−1.78360	−0.891801	−	0.452427i	$-0.850558\pi$
$673$	−46.2707	−1.78360	−0.891801	+	0.452427i	$0.850558\pi$
$674$	−46.4196	−1.78801
$675$	0	0
$676$	9.83718	0.378353
$677$	49.3486	1.89662	0.948310	−	0.317345i	$-0.102791\pi$
$677$	49.3486	1.89662	0.948310	+	0.317345i	$0.102791\pi$
$678$	0	0
$679$	4.06148	0.155865
$680$	2.87224	0.110145
$681$	0	0
$682$	−53.5508	−2.05057
$683$	31.0717	1.18893	0.594463	−	0.804123i	$-0.297365\pi$
$683$	31.0717	1.18893	0.594463	+	0.804123i	$0.297365\pi$
$684$	0	0
$685$	9.50150	0.363034
$686$	1.90103	0.0725818
$687$	0	0
$688$	−43.9912	−1.67715
$689$	−55.5976	−2.11810
$690$	0	0
$691$	−18.1085	−0.688879	−0.344440	−	0.938808i	$-0.611931\pi$
$691$	−18.1085	−0.688879	−0.344440	+	0.938808i	$0.611931\pi$
$692$	9.15067	0.347856
$693$	0	0
$694$	−17.3663	−0.659214
$695$	−12.7424	−0.483346
$696$	0	0
$697$	15.5723	0.589842
$698$	41.1722	1.55839
$699$	0	0
$700$	1.61393	0.0610008
$701$	−12.5359	−0.473476	−0.236738	−	0.971573i	$-0.576078\pi$
$701$	−12.5359	−0.473476	−0.236738	+	0.971573i	$0.576078\pi$
$702$	0	0
$703$	1.41405	0.0533318
$704$	14.3980	0.542644
$705$	0	0
$706$	36.8952	1.38857
$707$	10.3687	0.389957
$708$	0	0
$709$	24.6520	0.925826	0.462913	−	0.886404i	$-0.346804\pi$
$709$	24.6520	0.925826	0.462913	+	0.886404i	$0.346804\pi$
$710$	8.63390	0.324024
$711$	0	0
$712$	5.46525	0.204819
$713$	9.13845	0.342238
$714$	0	0
$715$	−13.4699	−0.503747
$716$	−7.86705	−0.294005
$717$	0	0
$718$	38.4239	1.43397
$719$	−3.39866	−0.126749	−0.0633743	−	0.997990i	$-0.520186\pi$
$719$	−3.39866	−0.126749	−0.0633743	+	0.997990i	$0.520186\pi$
$720$	0	0
$721$	9.76343	0.363609
$722$	−36.0361	−1.34113
$723$	0	0
$724$	−18.3769	−0.682972
$725$	−2.84119	−0.105519
$726$	0	0
$727$	−24.0284	−0.891165	−0.445583	−	0.895241i	$-0.647003\pi$
$727$	−24.0284	−0.891165	−0.445583	+	0.895241i	$0.647003\pi$
$728$	3.20715	0.118865
$729$	0	0
$730$	1.45952	0.0540193
$731$	37.2388	1.37733
$732$	0	0
$733$	19.0389	0.703218	0.351609	−	0.936147i	$-0.385635\pi$
$733$	19.0389	0.703218	0.351609	+	0.936147i	$0.385635\pi$
$734$	−12.5241	−0.462273
$735$	0	0
$736$	−7.32078	−0.269848
$737$	−14.4434	−0.532031
$738$	0	0
$739$	10.8761	0.400083	0.200041	−	0.979787i	$-0.435892\pi$
$739$	10.8761	0.400083	0.200041	+	0.979787i	$0.435892\pi$
$740$	−10.8883	−0.400263
$741$	0	0
$742$	24.1871	0.887937
$743$	−31.0934	−1.14071	−0.570353	−	0.821400i	$-0.693193\pi$
$743$	−31.0934	−1.14071	−0.570353	+	0.821400i	$0.693193\pi$
$744$	0	0
$745$	5.60930	0.205509
$746$	−32.4923	−1.18963
$747$	0	0
$748$	−19.4693	−0.711868
$749$	−11.9192	−0.435517
$750$	0	0
$751$	0.377023	0.0137578	0.00687889	−	0.999976i	$-0.497810\pi$
$751$	0.377023	0.0137578	0.00687889	+	0.999976i	$0.497810\pi$
$752$	35.6431	1.29977
$753$	0	0
$754$	23.6021	0.859539
$755$	−8.79336	−0.320023
$756$	0	0
$757$	−49.5370	−1.80045	−0.900227	−	0.435421i	$-0.856599\pi$
$757$	−49.5370	−1.80045	−0.900227	+	0.435421i	$0.856599\pi$
$758$	24.8868	0.903928
$759$	0	0
$760$	0.153831	0.00558004
$761$	−23.1228	−0.838201	−0.419100	−	0.907940i	$-0.637654\pi$
$761$	−23.1228	−0.838201	−0.419100	+	0.907940i	$0.637654\pi$
$762$	0	0
$763$	16.2721	0.589090
$764$	−35.5605	−1.28653
$765$	0	0
$766$	−22.7458	−0.821841
$767$	14.0836	0.508530
$768$	0	0
$769$	3.97204	0.143235	0.0716177	−	0.997432i	$-0.477184\pi$
$769$	3.97204	0.143235	0.0716177	+	0.997432i	$0.477184\pi$
$770$	5.85994	0.211178
$771$	0	0
$772$	−24.5326	−0.882949
$773$	15.7764	0.567438	0.283719	−	0.958907i	$-0.408432\pi$
$773$	15.7764	0.567438	0.283719	+	0.958907i	$0.408432\pi$
$774$	0	0
$775$	9.13845	0.328263
$776$	−2.98086	−0.107007
$777$	0	0
$778$	27.9554	1.00225
$779$	0.834019	0.0298818
$780$	0	0
$781$	13.9998	0.500951
$782$	7.43965	0.266041
$783$	0	0
$784$	−4.62309	−0.165110
$785$	4.41110	0.157439
$786$	0	0
$787$	23.8046	0.848541	0.424271	−	0.905535i	$-0.360531\pi$
$787$	23.8046	0.848541	0.424271	+	0.905535i	$0.360531\pi$
$788$	−12.9697	−0.462027
$789$	0	0
$790$	−12.6325	−0.449444
$791$	7.03810	0.250246
$792$	0	0
$793$	7.49637	0.266204
$794$	69.0185	2.44937
$795$	0	0
$796$	−41.2548	−1.46224
$797$	−29.2994	−1.03784	−0.518918	−	0.854824i	$-0.673665\pi$
$797$	−29.2994	−1.03784	−0.518918	+	0.854824i	$0.673665\pi$
$798$	0	0
$799$	−30.1721	−1.06741
$800$	−7.32078	−0.258829
$801$	0	0
$802$	−55.0956	−1.94549
$803$	2.36660	0.0835153
$804$	0	0
$805$	−1.00000	−0.0352454
$806$	−75.9144	−2.67397
$807$	0	0
$808$	−7.60998	−0.267718
$809$	30.0197	1.05544	0.527718	−	0.849419i	$-0.323048\pi$
$809$	30.0197	1.05544	0.527718	+	0.849419i	$0.323048\pi$
$810$	0	0
$811$	−8.04869	−0.282628	−0.141314	−	0.989965i	$-0.545133\pi$
$811$	−8.04869	−0.282628	−0.141314	+	0.989965i	$0.545133\pi$
$812$	−4.58547	−0.160919
$813$	0	0
$814$	−39.5340	−1.38567
$815$	−6.73572	−0.235942
$816$	0	0
$817$	1.99443	0.0697764
$818$	59.0734	2.06545
$819$	0	0
$820$	−6.42205	−0.224268
$821$	10.3262	0.360388	0.180194	−	0.983631i	$-0.442327\pi$
$821$	10.3262	0.360388	0.180194	+	0.983631i	$0.442327\pi$
$822$	0	0
$823$	−22.7733	−0.793827	−0.396914	−	0.917856i	$-0.629919\pi$
$823$	−22.7733	−0.793827	−0.396914	+	0.917856i	$0.629919\pi$
$824$	−7.16573	−0.249630
$825$	0	0
$826$	−6.12692	−0.213183
$827$	17.0953	0.594461	0.297230	−	0.954806i	$-0.403937\pi$
$827$	17.0953	0.594461	0.297230	+	0.954806i	$0.403937\pi$
$828$	0	0
$829$	−23.9580	−0.832097	−0.416049	−	0.909342i	$-0.636585\pi$
$829$	−23.9580	−0.832097	−0.416049	+	0.909342i	$0.636585\pi$
$830$	−30.1025	−1.04487
$831$	0	0
$832$	20.4108	0.707617
$833$	3.91348	0.135594
$834$	0	0
$835$	2.01945	0.0698858
$836$	−1.04274	−0.0360638
$837$	0	0
$838$	−35.7539	−1.23510
$839$	−9.81942	−0.339004	−0.169502	−	0.985530i	$-0.554216\pi$
$839$	−9.81942	−0.339004	−0.169502	+	0.985530i	$0.554216\pi$
$840$	0	0
$841$	−20.9277	−0.721643
$842$	−40.5585	−1.39774
$843$	0	0
$844$	19.2243	0.661729
$845$	−6.09518	−0.209681
$846$	0	0
$847$	−1.49818	−0.0514779
$848$	−58.8202	−2.01990
$849$	0	0
$850$	7.43965	0.255178
$851$	6.74648	0.231266
$852$	0	0
$853$	−28.4065	−0.972621	−0.486311	−	0.873786i	$-0.661658\pi$
$853$	−28.4065	−0.972621	−0.486311	+	0.873786i	$0.661658\pi$
$854$	−3.26121	−0.111596
$855$	0	0
$856$	8.74789	0.298997
$857$	−52.0850	−1.77919	−0.889594	−	0.456752i	$-0.849013\pi$
$857$	−52.0850	−1.77919	−0.889594	+	0.456752i	$0.849013\pi$
$858$	0	0
$859$	−42.1751	−1.43900	−0.719498	−	0.694494i	$-0.755628\pi$
$859$	−42.1751	−1.43900	−0.719498	+	0.694494i	$0.755628\pi$
$860$	−15.3574	−0.523682
$861$	0	0
$862$	44.3102	1.50921
$863$	35.5510	1.21017	0.605085	−	0.796161i	$-0.293139\pi$
$863$	35.5510	1.21017	0.605085	+	0.796161i	$0.293139\pi$
$864$	0	0
$865$	−5.66981	−0.192779
$866$	−7.22346	−0.245463
$867$	0	0
$868$	14.7488	0.500607
$869$	−20.4834	−0.694853
$870$	0	0
$871$	−20.4752	−0.693777
$872$	−11.9427	−0.404430
$873$	0	0
$874$	0.398452	0.0134779
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−12.2578	−0.413917	−0.206959	−	0.978350i	$-0.566357\pi$
$877$	−12.2578	−0.413917	−0.206959	+	0.978350i	$0.566357\pi$
$878$	−63.8077	−2.15341
$879$	0	0
$880$	−14.2507	−0.480391
$881$	5.90662	0.198999	0.0994996	−	0.995038i	$-0.468276\pi$
$881$	5.90662	0.198999	0.0994996	+	0.995038i	$0.468276\pi$
$882$	0	0
$883$	37.4513	1.26033	0.630167	−	0.776459i	$-0.282986\pi$
$883$	37.4513	1.26033	0.630167	+	0.776459i	$0.282986\pi$
$884$	−27.6000	−0.928288
$885$	0	0
$886$	−5.20233	−0.174776
$887$	−6.18188	−0.207567	−0.103784	−	0.994600i	$-0.533095\pi$
$887$	−6.18188	−0.207567	−0.103784	+	0.994600i	$0.533095\pi$
$888$	0	0
$889$	−5.61942	−0.188469
$890$	14.1561	0.474512
$891$	0	0
$892$	14.6965	0.492074
$893$	−1.61596	−0.0540759
$894$	0	0
$895$	4.87447	0.162936
$896$	5.76208	0.192498
$897$	0	0
$898$	10.2796	0.343034
$899$	−25.9640	−0.865949
$900$	0	0
$901$	49.7917	1.65880
$902$	−23.3175	−0.776389
$903$	0	0
$904$	−5.16551	−0.171802
$905$	11.3864	0.378498
$906$	0	0
$907$	−12.5773	−0.417622	−0.208811	−	0.977956i	$-0.566959\pi$
$907$	−12.5773	−0.417622	−0.208811	+	0.977956i	$0.566959\pi$
$908$	11.2918	0.374733
$909$	0	0
$910$	8.30714	0.275379
$911$	30.2652	1.00273	0.501365	−	0.865236i	$-0.332831\pi$
$911$	30.2652	1.00273	0.501365	+	0.865236i	$0.332831\pi$
$912$	0	0
$913$	−48.8108	−1.61540
$914$	−23.9321	−0.791603
$915$	0	0
$916$	27.0450	0.893592
$917$	6.97507	0.230337
$918$	0	0
$919$	39.2049	1.29325	0.646625	−	0.762808i	$-0.276180\pi$
$919$	39.2049	1.29325	0.646625	+	0.762808i	$0.276180\pi$
$920$	0.733935	0.0241971
$921$	0	0
$922$	27.7363	0.913445
$923$	19.8463	0.653248
$924$	0	0
$925$	6.74648	0.221823
$926$	62.6580	2.05907
$927$	0	0
$928$	20.7997	0.682784
$929$	−51.0030	−1.67335	−0.836677	−	0.547697i	$-0.815505\pi$
$929$	−51.0030	−1.67335	−0.836677	+	0.547697i	$0.815505\pi$
$930$	0	0
$931$	0.209598	0.00686929
$932$	32.6807	1.07049
$933$	0	0
$934$	−7.92510	−0.259317
$935$	12.0633	0.394512
$936$	0	0
$937$	45.8018	1.49628	0.748139	−	0.663542i	$-0.230947\pi$
$937$	45.8018	1.49628	0.748139	+	0.663542i	$0.230947\pi$
$938$	8.90752	0.290841
$939$	0	0
$940$	12.4431	0.405848
$941$	−2.75338	−0.0897576	−0.0448788	−	0.998992i	$-0.514290\pi$
$941$	−2.75338	−0.0897576	−0.0448788	+	0.998992i	$0.514290\pi$
$942$	0	0
$943$	3.97914	0.129579
$944$	14.8999	0.484952
$945$	0	0
$946$	−55.7604	−1.81293
$947$	−16.0802	−0.522538	−0.261269	−	0.965266i	$-0.584141\pi$
$947$	−16.0802	−0.522538	−0.261269	+	0.965266i	$0.584141\pi$
$948$	0	0
$949$	3.35492	0.108905
$950$	0.398452	0.0129275
$951$	0	0
$952$	−2.87224	−0.0930897
$953$	30.6082	0.991497	0.495748	−	0.868466i	$-0.334894\pi$
$953$	30.6082	0.991497	0.495748	+	0.868466i	$0.334894\pi$
$954$	0	0
$955$	22.0335	0.712987
$956$	−35.6703	−1.15366
$957$	0	0
$958$	68.7291	2.22054
$959$	−9.50150	−0.306819
$960$	0	0
$961$	52.5113	1.69391
$962$	−56.0440	−1.80693
$963$	0	0
$964$	35.3482	1.13849
$965$	15.2006	0.489324
$966$	0	0
$967$	18.7317	0.602370	0.301185	−	0.953566i	$-0.402618\pi$
$967$	18.7317	0.602370	0.301185	+	0.953566i	$0.402618\pi$
$968$	1.09956	0.0353413
$969$	0	0
$970$	−7.72100	−0.247906
$971$	23.1564	0.743124	0.371562	−	0.928408i	$-0.378822\pi$
$971$	23.1564	0.743124	0.371562	+	0.928408i	$0.378822\pi$
$972$	0	0
$973$	12.7424	0.408502
$974$	−74.8340	−2.39783
$975$	0	0
$976$	7.93088	0.253861
$977$	−19.1370	−0.612246	−0.306123	−	0.951992i	$-0.599032\pi$
$977$	−19.1370	−0.612246	−0.306123	+	0.951992i	$0.599032\pi$
$978$	0	0
$979$	22.9539	0.733609
$980$	−1.61393	−0.0515551
$981$	0	0
$982$	−30.6783	−0.978983
$983$	−42.4715	−1.35463	−0.677315	−	0.735693i	$-0.736857\pi$
$983$	−42.4715	−1.35463	−0.677315	+	0.735693i	$0.736857\pi$
$984$	0	0
$985$	8.03612	0.256052
$986$	−21.1374	−0.673153
$987$	0	0
$988$	−1.47820	−0.0470278
$989$	9.51553	0.302576
$990$	0	0
$991$	−5.46923	−0.173736	−0.0868679	−	0.996220i	$-0.527686\pi$
$991$	−5.46923	−0.173736	−0.0868679	+	0.996220i	$0.527686\pi$
$992$	−66.9006	−2.12410
$993$	0	0
$994$	−8.63390	−0.273851
$995$	25.5617	0.810361
$996$	0	0
$997$	−6.14896	−0.194740	−0.0973698	−	0.995248i	$-0.531043\pi$
$997$	−6.14896	−0.194740	−0.0973698	+	0.995248i	$0.531043\pi$
$998$	−22.8271	−0.722578
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7245.2.a.bv.1.9		10
3.2	odd	2		2415.2.a.w.1.2	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2415.2.a.w.1.2	✓	10	3.2	odd	2
7245.2.a.bv.1.9		10	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:	\( N \)	\(=\)	\( 7245 = 3^{2} \cdot 5 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7245.a (trivial)

\(f(q)\)	\(=\)	\(q+1.90103 q^{2} +1.61393 q^{4} -1.00000 q^{5} +1.00000 q^{7} -0.733935 q^{8} +O(q^{10})\)	\(q+1.90103 q^{2} +1.61393 q^{4} -1.00000 q^{5} +1.00000 q^{7} -0.733935 q^{8} -1.90103 q^{10} -3.08250 q^{11} -4.36980 q^{13} +1.90103 q^{14} -4.62309 q^{16} +3.91348 q^{17} +0.209598 q^{19} -1.61393 q^{20} -5.85994 q^{22} +1.00000 q^{23} +1.00000 q^{25} -8.30714 q^{26} +1.61393 q^{28} -2.84119 q^{29} +9.13845 q^{31} -7.32078 q^{32} +7.43965 q^{34} -1.00000 q^{35} +6.74648 q^{37} +0.398452 q^{38} +0.733935 q^{40} +3.97914 q^{41} +9.51553 q^{43} -4.97494 q^{44} +1.90103 q^{46} -7.70980 q^{47} +1.00000 q^{49} +1.90103 q^{50} -7.05255 q^{52} +12.7231 q^{53} +3.08250 q^{55} -0.733935 q^{56} -5.40119 q^{58} -3.22294 q^{59} -1.71549 q^{61} +17.3725 q^{62} -4.67087 q^{64} +4.36980 q^{65} +4.68562 q^{67} +6.31607 q^{68} -1.90103 q^{70} -4.54168 q^{71} -0.767751 q^{73} +12.8253 q^{74} +0.338276 q^{76} -3.08250 q^{77} +6.64506 q^{79} +4.62309 q^{80} +7.56448 q^{82} +15.8348 q^{83} -3.91348 q^{85} +18.0893 q^{86} +2.26236 q^{88} -7.44650 q^{89} -4.36980 q^{91} +1.61393 q^{92} -14.6566 q^{94} -0.209598 q^{95} +4.06148 q^{97} +1.90103 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{2} + 16 q^{4} - 10 q^{5} + 10 q^{7} - 6 q^{8}+O(q^{10}) \) 10 * q - 2 * q^2 + 16 * q^4 - 10 * q^5 + 10 * q^7 - 6 * q^8	\( 10 q - 2 q^{2} + 16 q^{4} - 10 q^{5} + 10 q^{7} - 6 q^{8} + 2 q^{10} - 9 q^{11} + 14 q^{13} - 2 q^{14} + 20 q^{16} - 8 q^{17} + 13 q^{19} - 16 q^{20} + 10 q^{23} + 10 q^{25} + 11 q^{26} + 16 q^{28} - 10 q^{29} + 8 q^{31} + 11 q^{32} - 5 q^{34} - 10 q^{35} + 8 q^{37} + 10 q^{38} + 6 q^{40} + 5 q^{41} + 4 q^{43} - 3 q^{44} - 2 q^{46} - q^{47} + 10 q^{49} - 2 q^{50} + 14 q^{52} - 9 q^{53} + 9 q^{55} - 6 q^{56} - 28 q^{58} + 17 q^{59} + 19 q^{61} + 28 q^{62} + 24 q^{64} - 14 q^{65} - 8 q^{68} + 2 q^{70} + 6 q^{73} - 3 q^{74} + 15 q^{76} - 9 q^{77} + 32 q^{79} - 20 q^{80} + 14 q^{82} + 2 q^{83} + 8 q^{85} - 2 q^{86} - 3 q^{88} - 10 q^{89} + 14 q^{91} + 16 q^{92} - 10 q^{94} - 13 q^{95} + 18 q^{97} - 2 q^{98}+O(q^{100}) \) 10 * q - 2 * q^2 + 16 * q^4 - 10 * q^5 + 10 * q^7 - 6 * q^8 + 2 * q^10 - 9 * q^11 + 14 * q^13 - 2 * q^14 + 20 * q^16 - 8 * q^17 + 13 * q^19 - 16 * q^20 + 10 * q^23 + 10 * q^25 + 11 * q^26 + 16 * q^28 - 10 * q^29 + 8 * q^31 + 11 * q^32 - 5 * q^34 - 10 * q^35 + 8 * q^37 + 10 * q^38 + 6 * q^40 + 5 * q^41 + 4 * q^43 - 3 * q^44 - 2 * q^46 - q^47 + 10 * q^49 - 2 * q^50 + 14 * q^52 - 9 * q^53 + 9 * q^55 - 6 * q^56 - 28 * q^58 + 17 * q^59 + 19 * q^61 + 28 * q^62 + 24 * q^64 - 14 * q^65 - 8 * q^68 + 2 * q^70 + 6 * q^73 - 3 * q^74 + 15 * q^76 - 9 * q^77 + 32 * q^79 - 20 * q^80 + 14 * q^82 + 2 * q^83 + 8 * q^85 - 2 * q^86 - 3 * q^88 - 10 * q^89 + 14 * q^91 + 16 * q^92 - 10 * q^94 - 13 * q^95 + 18 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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