LMFDB - Embedded newform 9025.2.a.bn.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:	$4$
Coefficient field:	4.4.7168.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 7 $ x^4 - 6*x^2 + 7
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1805)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.10100$ 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)

$f(q)$	$=$	\(q-2.10100 q^{2} +2.97127 q^{3} +2.41421 q^{4} -6.24264 q^{6} -4.82843 q^{7} -0.870264 q^{8} +5.82843 q^{9} +2.00000 q^{11} +7.17327 q^{12} -1.23074 q^{13} +10.1445 q^{14} -3.00000 q^{16} +3.65685 q^{17} -12.2455 q^{18} -14.3465 q^{21} -4.20201 q^{22} -4.82843 q^{23} -2.58579 q^{24} +2.58579 q^{26} +8.40401 q^{27} -11.6569 q^{28} +2.46148 q^{29} -5.94253 q^{31} +8.04354 q^{32} +5.94253 q^{33} -7.68306 q^{34} +14.0711 q^{36} -7.17327 q^{37} -3.65685 q^{39} +30.1421 q^{42} -0.828427 q^{43} +4.82843 q^{44} +10.1445 q^{46} +6.48528 q^{47} -8.91380 q^{48} +16.3137 q^{49} +10.8655 q^{51} -2.97127 q^{52} -4.71179 q^{53} -17.6569 q^{54} +4.20201 q^{56} -5.17157 q^{58} +11.8851 q^{59} +2.82843 q^{61} +12.4853 q^{62} -28.1421 q^{63} -10.8995 q^{64} -12.4853 q^{66} -5.43275 q^{67} +8.82843 q^{68} -14.3465 q^{69} +10.8655 q^{71} -5.07227 q^{72} +0.343146 q^{73} +15.0711 q^{74} -9.65685 q^{77} +7.68306 q^{78} -11.8851 q^{79} +7.48528 q^{81} -3.17157 q^{83} -34.6356 q^{84} +1.74053 q^{86} +7.31371 q^{87} -1.74053 q^{88} +9.42359 q^{89} +5.94253 q^{91} -11.6569 q^{92} -17.6569 q^{93} -13.6256 q^{94} +23.8995 q^{96} -1.23074 q^{97} -34.2752 q^{98} +11.6569 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} - 8 q^{6} - 8 q^{7} + 12 q^{9} + 8 q^{11} - 12 q^{16} - 8 q^{17} - 8 q^{23} - 16 q^{24} + 16 q^{26} - 24 q^{28} + 28 q^{36} + 8 q^{39} + 64 q^{42} + 8 q^{43} + 8 q^{44} - 8 q^{47} + 20 q^{49}+ \cdots + 24 q^{99}+O(q^{100}) $ 4 * q + 4 * q^4 - 8 * q^6 - 8 * q^7 + 12 * q^9 + 8 * q^11 - 12 * q^16 - 8 * q^17 - 8 * q^23 - 16 * q^24 + 16 * q^26 - 24 * q^28 + 28 * q^36 + 8 * q^39 + 64 * q^42 + 8 * q^43 + 8 * q^44 - 8 * q^47 + 20 * q^49 - 48 * q^54 - 32 * q^58 + 16 * q^62 - 56 * q^63 - 4 * q^64 - 16 * q^66 + 24 * q^68 + 24 * q^73 + 32 * q^74 - 16 * q^77 - 4 * q^81 - 24 * q^83 - 16 * q^87 - 24 * q^92 - 48 * q^93 + 56 * q^96 + 24 * 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 $

$2$	−2.10100	−1.48563	−0.742817	−	0.669495i	$-0.766511\pi$
$2$	−2.10100	−1.48563	−0.742817	+	0.669495i	$0.766511\pi$
$3$	2.97127	1.71546	0.857731	−	0.514099i	$-0.171874\pi$
$3$	2.97127	1.71546	0.857731	+	0.514099i	$0.171874\pi$
$4$	2.41421	1.20711
$5$	0	0
$5$	0	0
$6$	−6.24264	−2.54855
$7$	−4.82843	−1.82497	−0.912487	−	0.409106i	$-0.865841\pi$
$7$	−4.82843	−1.82497	−0.912487	+	0.409106i	$0.865841\pi$
$8$	−0.870264	−0.307685
$9$	5.82843	1.94281
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	7.17327	2.07075
$13$	−1.23074	−0.341346	−0.170673	−	0.985328i	$-0.554594\pi$
$13$	−1.23074	−0.341346	−0.170673	+	0.985328i	$0.554594\pi$
$14$	10.1445	2.71124
$15$	0	0
$16$	−3.00000	−0.750000
$17$	3.65685	0.886917	0.443459	−	0.896295i	$-0.353751\pi$
$17$	3.65685	0.886917	0.443459	+	0.896295i	$0.353751\pi$
$18$	−12.2455	−2.88630
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−14.3465	−3.13067
$22$	−4.20201	−0.895871
$23$	−4.82843	−1.00680	−0.503398	−	0.864054i	$-0.667917\pi$
$23$	−4.82843	−1.00680	−0.503398	+	0.864054i	$0.667917\pi$
$24$	−2.58579	−0.527821
$25$	0	0
$26$	2.58579	0.507114
$27$	8.40401	1.61735
$28$	−11.6569	−2.20294
$29$	2.46148	0.457085	0.228543	−	0.973534i	$-0.426604\pi$
$29$	2.46148	0.457085	0.228543	+	0.973534i	$0.426604\pi$
$30$	0	0
$31$	−5.94253	−1.06731	−0.533655	−	0.845702i	$-0.679182\pi$
$31$	−5.94253	−1.06731	−0.533655	+	0.845702i	$0.679182\pi$
$32$	8.04354	1.42191
$33$	5.94253	1.03446
$34$	−7.68306	−1.31763
$35$	0	0
$36$	14.0711	2.34518
$37$	−7.17327	−1.17928	−0.589639	−	0.807667i	$-0.700730\pi$
$37$	−7.17327	−1.17928	−0.589639	+	0.807667i	$0.700730\pi$
$38$	0	0
$39$	−3.65685	−0.585565
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	30.1421	4.65103
$43$	−0.828427	−0.126334	−0.0631670	−	0.998003i	$-0.520120\pi$
$43$	−0.828427	−0.126334	−0.0631670	+	0.998003i	$0.520120\pi$
$44$	4.82843	0.727913
$45$	0	0
$46$	10.1445	1.49573
$47$	6.48528	0.945976	0.472988	−	0.881069i	$-0.343175\pi$
$47$	6.48528	0.945976	0.472988	+	0.881069i	$0.343175\pi$
$48$	−8.91380	−1.28660
$49$	16.3137	2.33053
$50$	0	0
$51$	10.8655	1.52147
$52$	−2.97127	−0.412041
$53$	−4.71179	−0.647215	−0.323607	−	0.946191i	$-0.604896\pi$
$53$	−4.71179	−0.647215	−0.323607	+	0.946191i	$0.604896\pi$
$54$	−17.6569	−2.40279
$55$	0	0
$56$	4.20201	0.561517
$57$	0	0
$58$	−5.17157	−0.679061
$59$	11.8851	1.54730	0.773652	−	0.633611i	$-0.218428\pi$
$59$	11.8851	1.54730	0.773652	+	0.633611i	$0.218428\pi$
$60$	0	0
$61$	2.82843	0.362143	0.181071	−	0.983470i	$-0.442043\pi$
$61$	2.82843	0.362143	0.181071	+	0.983470i	$0.442043\pi$
$62$	12.4853	1.58563
$63$	−28.1421	−3.54558
$64$	−10.8995	−1.36244
$65$	0	0
$66$	−12.4853	−1.53683
$67$	−5.43275	−0.663715	−0.331858	−	0.943329i	$-0.607675\pi$
$67$	−5.43275	−0.663715	−0.331858	+	0.943329i	$0.607675\pi$
$68$	8.82843	1.07060
$69$	−14.3465	−1.72712
$70$	0	0
$71$	10.8655	1.28950	0.644748	−	0.764395i	$-0.276962\pi$
$71$	10.8655	1.28950	0.644748	+	0.764395i	$0.276962\pi$
$72$	−5.07227	−0.597773
$73$	0.343146	0.0401622	0.0200811	−	0.999798i	$-0.493608\pi$
$73$	0.343146	0.0401622	0.0200811	+	0.999798i	$0.493608\pi$
$74$	15.0711	1.75198
$75$	0	0
$76$	0	0
$77$	−9.65685	−1.10050
$78$	7.68306	0.869935
$79$	−11.8851	−1.33717	−0.668587	−	0.743634i	$-0.733101\pi$
$79$	−11.8851	−1.33717	−0.668587	+	0.743634i	$0.733101\pi$
$80$	0	0
$81$	7.48528	0.831698
$82$	0	0
$83$	−3.17157	−0.348125	−0.174063	−	0.984735i	$-0.555690\pi$
$83$	−3.17157	−0.348125	−0.174063	+	0.984735i	$0.555690\pi$
$84$	−34.6356	−3.77906
$85$	0	0
$86$	1.74053	0.187686
$87$	7.31371	0.784112
$88$	−1.74053	−0.185541
$89$	9.42359	0.998898	0.499449	−	0.866343i	$-0.333536\pi$
$89$	9.42359	0.998898	0.499449	+	0.866343i	$0.333536\pi$
$90$	0	0
$91$	5.94253	0.622947
$92$	−11.6569	−1.21531
$93$	−17.6569	−1.83093
$94$	−13.6256	−1.40537
$95$	0	0
$96$	23.8995	2.43923
$97$	−1.23074	−0.124963	−0.0624813	−	0.998046i	$-0.519901\pi$
$97$	−1.23074	−0.124963	−0.0624813	+	0.998046i	$0.519901\pi$
$98$	−34.2752	−3.46231
$99$	11.6569	1.17156
$100$	0	0
$101$	−16.4853	−1.64035	−0.820173	−	0.572115i	$-0.806123\pi$
$101$	−16.4853	−1.64035	−0.820173	+	0.572115i	$0.806123\pi$
$102$	−22.8284	−2.26035
$103$	−8.91380	−0.878303	−0.439151	−	0.898413i	$-0.644721\pi$
$103$	−8.91380	−0.878303	−0.439151	+	0.898413i	$0.644721\pi$
$104$	1.07107	0.105027
$105$	0	0
$106$	9.89949	0.961524
$107$	−6.45232	−0.623770	−0.311885	−	0.950120i	$-0.600960\pi$
$107$	−6.45232	−0.623770	−0.311885	+	0.950120i	$0.600960\pi$
$108$	20.2891	1.95232
$109$	−19.2695	−1.84568	−0.922842	−	0.385179i	$-0.874140\pi$
$109$	−19.2695	−1.84568	−0.922842	+	0.385179i	$0.874140\pi$
$110$	0	0
$111$	−21.3137	−2.02301
$112$	14.4853	1.36873
$113$	9.63475	0.906361	0.453181	−	0.891419i	$-0.350289\pi$
$113$	9.63475	0.906361	0.453181	+	0.891419i	$0.350289\pi$
$114$	0	0
$115$	0	0
$116$	5.94253	0.551750
$117$	−7.17327	−0.663169
$118$	−24.9706	−2.29873
$119$	−17.6569	−1.61860
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−5.94253	−0.538012
$123$	0	0
$124$	−14.3465	−1.28836
$125$	0	0
$126$	59.1267	5.26743
$127$	−12.3949	−1.09987	−0.549933	−	0.835209i	$-0.685347\pi$
$127$	−12.3949	−1.09987	−0.549933	+	0.835209i	$0.685347\pi$
$128$	6.81280	0.602172
$129$	−2.46148	−0.216721
$130$	0	0
$131$	−7.31371	−0.639002	−0.319501	−	0.947586i	$-0.603515\pi$
$131$	−7.31371	−0.639002	−0.319501	+	0.947586i	$0.603515\pi$
$132$	14.3465	1.24871
$133$	0	0
$134$	11.4142	0.986038
$135$	0	0
$136$	−3.18243	−0.272891
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	30.1421	2.56587
$139$	8.34315	0.707656	0.353828	−	0.935310i	$-0.384880\pi$
$139$	8.34315	0.707656	0.353828	+	0.935310i	$0.384880\pi$
$140$	0	0
$141$	19.2695	1.62278
$142$	−22.8284	−1.91572
$143$	−2.46148	−0.205839
$144$	−17.4853	−1.45711
$145$	0	0
$146$	−0.720950	−0.0596663
$147$	48.4724	3.99793
$148$	−17.3178	−1.42352
$149$	−9.17157	−0.751365	−0.375682	−	0.926749i	$-0.622592\pi$
$149$	−9.17157	−0.751365	−0.375682	+	0.926749i	$0.622592\pi$
$150$	0	0
$151$	17.8276	1.45079	0.725395	−	0.688333i	$-0.241657\pi$
$151$	17.8276	1.45079	0.725395	+	0.688333i	$0.241657\pi$
$152$	0	0
$153$	21.3137	1.72311
$154$	20.2891	1.63494
$155$	0	0
$156$	−8.82843	−0.706840
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	24.9706	1.98655
$159$	−14.0000	−1.11027
$160$	0	0
$161$	23.3137	1.83738
$162$	−15.7266	−1.23560
$163$	−12.8284	−1.00480	−0.502400	−	0.864635i	$-0.667549\pi$
$163$	−12.8284	−1.00480	−0.502400	+	0.864635i	$0.667549\pi$
$164$	0	0
$165$	0	0
$166$	6.66348	0.517187
$167$	22.2408	1.72104	0.860521	−	0.509415i	$-0.170138\pi$
$167$	22.2408	1.72104	0.860521	+	0.509415i	$0.170138\pi$
$168$	12.4853	0.963260
$169$	−11.4853	−0.883483
$170$	0	0
$171$	0	0
$172$	−2.00000	−0.152499
$173$	13.1158	0.997176	0.498588	−	0.866839i	$-0.333852\pi$
$173$	13.1158	0.997176	0.498588	+	0.866839i	$0.333852\pi$
$174$	−15.3661	−1.16490
$175$	0	0
$176$	−6.00000	−0.452267
$177$	35.3137	2.65434
$178$	−19.7990	−1.48400
$179$	4.92296	0.367959	0.183980	−	0.982930i	$-0.441102\pi$
$179$	4.92296	0.367959	0.183980	+	0.982930i	$0.441102\pi$
$180$	0	0
$181$	−16.8080	−1.24933	−0.624665	−	0.780893i	$-0.714765\pi$
$181$	−16.8080	−1.24933	−0.624665	+	0.780893i	$0.714765\pi$
$182$	−12.4853	−0.925471
$183$	8.40401	0.621242
$184$	4.20201	0.309776
$185$	0	0
$186$	37.0971	2.72009
$187$	7.31371	0.534831
$188$	15.6569	1.14189
$189$	−40.5782	−2.95163
$190$	0	0
$191$	−13.6569	−0.988175	−0.494088	−	0.869412i	$-0.664498\pi$
$191$	−13.6569	−0.988175	−0.494088	+	0.869412i	$0.664498\pi$
$192$	−32.3853	−2.33721
$193$	−8.19285	−0.589734	−0.294867	−	0.955538i	$-0.595275\pi$
$193$	−8.19285	−0.589734	−0.294867	+	0.955538i	$0.595275\pi$
$194$	2.58579	0.185649
$195$	0	0
$196$	39.3848	2.81320
$197$	13.3137	0.948562	0.474281	−	0.880373i	$-0.342708\pi$
$197$	13.3137	0.948562	0.474281	+	0.880373i	$0.342708\pi$
$198$	−24.4911	−1.74051
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	−16.1421	−1.13858
$202$	34.6356	2.43695
$203$	−11.8851	−0.834168
$204$	26.2316	1.83658
$205$	0	0
$206$	18.7279	1.30484
$207$	−28.1421	−1.95601
$208$	3.69222	0.256009
$209$	0	0
$210$	0	0
$211$	−10.8655	−0.748011	−0.374006	−	0.927426i	$-0.622016\pi$
$211$	−10.8655	−0.748011	−0.374006	+	0.927426i	$0.622016\pi$
$212$	−11.3753	−0.781257
$213$	32.2843	2.21208
$214$	13.5563	0.926693
$215$	0	0
$216$	−7.31371	−0.497635
$217$	28.6931	1.94781
$218$	40.4853	2.74201
$219$	1.01958	0.0688967
$220$	0	0
$221$	−4.50063	−0.302745
$222$	44.7802	3.00545
$223$	17.3178	1.15969	0.579843	−	0.814728i	$-0.303114\pi$
$223$	17.3178	1.15969	0.579843	+	0.814728i	$0.303114\pi$
$224$	−38.8376	−2.59495
$225$	0	0
$226$	−20.2426	−1.34652
$227$	22.2408	1.47617	0.738086	−	0.674707i	$-0.235730\pi$
$227$	22.2408	1.47617	0.738086	+	0.674707i	$0.235730\pi$
$228$	0	0
$229$	12.4853	0.825051	0.412525	−	0.910946i	$-0.364647\pi$
$229$	12.4853	0.825051	0.412525	+	0.910946i	$0.364647\pi$
$230$	0	0
$231$	−28.6931	−1.88787
$232$	−2.14214	−0.140638
$233$	−13.3137	−0.872210	−0.436105	−	0.899896i	$-0.643642\pi$
$233$	−13.3137	−0.872210	−0.436105	+	0.899896i	$0.643642\pi$
$234$	15.0711	0.985227
$235$	0	0
$236$	28.6931	1.86776
$237$	−35.3137	−2.29387
$238$	37.0971	2.40465
$239$	−1.65685	−0.107173	−0.0535865	−	0.998563i	$-0.517065\pi$
$239$	−1.65685	−0.107173	−0.0535865	+	0.998563i	$0.517065\pi$
$240$	0	0
$241$	−6.96211	−0.448469	−0.224235	−	0.974535i	$-0.571988\pi$
$241$	−6.96211	−0.448469	−0.224235	+	0.974535i	$0.571988\pi$
$242$	14.7070	0.945403
$243$	−2.97127	−0.190607
$244$	6.82843	0.437145
$245$	0	0
$246$	0	0
$247$	0	0
$248$	5.17157	0.328395
$249$	−9.42359	−0.597196
$250$	0	0
$251$	−12.9706	−0.818695	−0.409347	−	0.912379i	$-0.634244\pi$
$251$	−12.9706	−0.818695	−0.409347	+	0.912379i	$0.634244\pi$
$252$	−67.9411	−4.27989
$253$	−9.65685	−0.607121
$254$	26.0416	1.63400
$255$	0	0
$256$	7.48528	0.467830
$257$	8.19285	0.511056	0.255528	−	0.966802i	$-0.417751\pi$
$257$	8.19285	0.511056	0.255528	+	0.966802i	$0.417751\pi$
$258$	5.17157	0.321968
$259$	34.6356	2.15215
$260$	0	0
$261$	14.3465	0.888029
$262$	15.3661	0.949322
$263$	−12.1421	−0.748716	−0.374358	−	0.927284i	$-0.622137\pi$
$263$	−12.1421	−0.748716	−0.374358	+	0.927284i	$0.622137\pi$
$264$	−5.17157	−0.318288
$265$	0	0
$266$	0	0
$267$	28.0000	1.71357
$268$	−13.1158	−0.801175
$269$	2.46148	0.150079	0.0750395	−	0.997181i	$-0.476092\pi$
$269$	2.46148	0.150079	0.0750395	+	0.997181i	$0.476092\pi$
$270$	0	0
$271$	−2.97056	−0.180449	−0.0902244	−	0.995921i	$-0.528758\pi$
$271$	−2.97056	−0.180449	−0.0902244	+	0.995921i	$0.528758\pi$
$272$	−10.9706	−0.665188
$273$	17.6569	1.06864
$274$	−4.20201	−0.253852
$275$	0	0
$276$	−34.6356	−2.08482
$277$	−5.31371	−0.319270	−0.159635	−	0.987176i	$-0.551032\pi$
$277$	−5.31371	−0.319270	−0.159635	+	0.987176i	$0.551032\pi$
$278$	−17.5290	−1.05132
$279$	−34.6356	−2.07358
$280$	0	0
$281$	−23.7701	−1.41801	−0.709004	−	0.705205i	$-0.750855\pi$
$281$	−23.7701	−1.41801	−0.709004	+	0.705205i	$0.750855\pi$
$282$	−40.4853	−2.41086
$283$	−29.7990	−1.77137	−0.885683	−	0.464290i	$-0.846309\pi$
$283$	−29.7990	−1.77137	−0.885683	+	0.464290i	$0.846309\pi$
$284$	26.2316	1.55656
$285$	0	0
$286$	5.17157	0.305802
$287$	0	0
$288$	46.8812	2.76250
$289$	−3.62742	−0.213377
$290$	0	0
$291$	−3.65685	−0.214369
$292$	0.828427	0.0484800
$293$	12.0962	0.706669	0.353335	−	0.935497i	$-0.385048\pi$
$293$	12.0962	0.706669	0.353335	+	0.935497i	$0.385048\pi$
$294$	−101.841	−5.93947
$295$	0	0
$296$	6.24264	0.362846
$297$	16.8080	0.975300
$298$	19.2695	1.11625
$299$	5.94253	0.343666
$300$	0	0
$301$	4.00000	0.230556
$302$	−37.4558	−2.15534
$303$	−48.9822	−2.81395
$304$	0	0
$305$	0	0
$306$	−44.7802	−2.55991
$307$	−11.3753	−0.649221	−0.324611	−	0.945848i	$-0.605233\pi$
$307$	−11.3753	−0.649221	−0.324611	+	0.945848i	$0.605233\pi$
$308$	−23.3137	−1.32842
$309$	−26.4853	−1.50670
$310$	0	0
$311$	−31.6569	−1.79510	−0.897548	−	0.440917i	$-0.854653\pi$
$311$	−31.6569	−1.79510	−0.897548	+	0.440917i	$0.854653\pi$
$312$	3.18243	0.180170
$313$	20.6274	1.16593	0.582965	−	0.812497i	$-0.301892\pi$
$313$	20.6274	1.16593	0.582965	+	0.812497i	$0.301892\pi$
$314$	37.8181	2.13420
$315$	0	0
$316$	−28.6931	−1.61411
$317$	15.5773	0.874907	0.437454	−	0.899241i	$-0.355880\pi$
$317$	15.5773	0.874907	0.437454	+	0.899241i	$0.355880\pi$
$318$	29.4140	1.64946
$319$	4.92296	0.275633
$320$	0	0
$321$	−19.1716	−1.07005
$322$	−48.9822	−2.72967
$323$	0	0
$324$	18.0711	1.00395
$325$	0	0
$326$	26.9526	1.49276
$327$	−57.2548	−3.16620
$328$	0	0
$329$	−31.3137	−1.72638
$330$	0	0
$331$	29.7127	1.63316	0.816578	−	0.577235i	$-0.195868\pi$
$331$	29.7127	1.63316	0.816578	+	0.577235i	$0.195868\pi$
$332$	−7.65685	−0.420224
$333$	−41.8089	−2.29111
$334$	−46.7279	−2.55684
$335$	0	0
$336$	43.0396	2.34800
$337$	−30.9434	−1.68559	−0.842797	−	0.538231i	$-0.819093\pi$
$337$	−30.9434	−1.68559	−0.842797	+	0.538231i	$0.819093\pi$
$338$	24.1306	1.31253
$339$	28.6274	1.55483
$340$	0	0
$341$	−11.8851	−0.643612
$342$	0	0
$343$	−44.9706	−2.42818
$344$	0.720950	0.0388710
$345$	0	0
$346$	−27.5563	−1.48144
$347$	−24.8284	−1.33286	−0.666430	−	0.745568i	$-0.732178\pi$
$347$	−24.8284	−1.33286	−0.666430	+	0.745568i	$0.732178\pi$
$348$	17.6569	0.946507
$349$	−6.68629	−0.357909	−0.178954	−	0.983857i	$-0.557271\pi$
$349$	−6.68629	−0.357909	−0.178954	+	0.983857i	$0.557271\pi$
$350$	0	0
$351$	−10.3431	−0.552076
$352$	16.0871	0.857444
$353$	3.65685	0.194635	0.0973174	−	0.995253i	$-0.468974\pi$
$353$	3.65685	0.194635	0.0973174	+	0.995253i	$0.468974\pi$
$354$	−74.1942	−3.94338
$355$	0	0
$356$	22.7506	1.20578
$357$	−52.4632	−2.77665
$358$	−10.3431	−0.546652
$359$	−25.3137	−1.33601	−0.668003	−	0.744158i	$-0.732851\pi$
$359$	−25.3137	−1.33601	−0.668003	+	0.744158i	$0.732851\pi$
$360$	0	0
$361$	0	0
$362$	35.3137	1.85605
$363$	−20.7989	−1.09166
$364$	14.3465	0.751963
$365$	0	0
$366$	−17.6569	−0.922939
$367$	−8.82843	−0.460840	−0.230420	−	0.973091i	$-0.574010\pi$
$367$	−8.82843	−0.460840	−0.230420	+	0.973091i	$0.574010\pi$
$368$	14.4853	0.755097
$369$	0	0
$370$	0	0
$371$	22.7506	1.18115
$372$	−42.6274	−2.21013
$373$	23.9813	1.24170	0.620852	−	0.783928i	$-0.286787\pi$
$373$	23.9813	1.24170	0.620852	+	0.783928i	$0.286787\pi$
$374$	−15.3661	−0.794563
$375$	0	0
$376$	−5.64391	−0.291062
$377$	−3.02944	−0.156024
$378$	85.2548	4.38504
$379$	−4.92296	−0.252875	−0.126438	−	0.991975i	$-0.540354\pi$
$379$	−4.92296	−0.252875	−0.126438	+	0.991975i	$0.540354\pi$
$380$	0	0
$381$	−36.8284	−1.88678
$382$	28.6931	1.46807
$383$	−12.8172	−0.654927	−0.327464	−	0.944864i	$-0.606194\pi$
$383$	−12.8172	−0.654927	−0.327464	+	0.944864i	$0.606194\pi$
$384$	20.2426	1.03300
$385$	0	0
$386$	17.2132	0.876129
$387$	−4.82843	−0.245443
$388$	−2.97127	−0.150843
$389$	24.6274	1.24866	0.624330	−	0.781161i	$-0.285372\pi$
$389$	24.6274	1.24866	0.624330	+	0.781161i	$0.285372\pi$
$390$	0	0
$391$	−17.6569	−0.892946
$392$	−14.1972	−0.717069
$393$	−21.7310	−1.09618
$394$	−27.9721	−1.40922
$395$	0	0
$396$	28.1421	1.41420
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	8.40401	0.421255
$399$	0	0
$400$	0	0
$401$	−16.8080	−0.839353	−0.419676	−	0.907674i	$-0.637856\pi$
$401$	−16.8080	−0.839353	−0.419676	+	0.907674i	$0.637856\pi$
$402$	33.9147	1.69151
$403$	7.31371	0.364322
$404$	−39.7990	−1.98007
$405$	0	0
$406$	24.9706	1.23927
$407$	−14.3465	−0.711132
$408$	−9.45584	−0.468134
$409$	−38.1167	−1.88475	−0.942374	−	0.334560i	$-0.891412\pi$
$409$	−38.1167	−1.88475	−0.942374	+	0.334560i	$0.891412\pi$
$410$	0	0
$411$	5.94253	0.293124
$412$	−21.5198	−1.06021
$413$	−57.3862	−2.82379
$414$	59.1267	2.90592
$415$	0	0
$416$	−9.89949	−0.485363
$417$	24.7897	1.21396
$418$	0	0
$419$	8.97056	0.438241	0.219120	−	0.975698i	$-0.429681\pi$
$419$	8.97056	0.438241	0.219120	+	0.975698i	$0.429681\pi$
$420$	0	0
$421$	−16.8080	−0.819173	−0.409586	−	0.912271i	$-0.634327\pi$
$421$	−16.8080	−0.819173	−0.409586	+	0.912271i	$0.634327\pi$
$422$	22.8284	1.11127
$423$	37.7990	1.83785
$424$	4.10051	0.199138
$425$	0	0
$426$	−67.8294	−3.28634
$427$	−13.6569	−0.660901
$428$	−15.5773	−0.752956
$429$	−7.31371	−0.353109
$430$	0	0
$431$	−5.94253	−0.286242	−0.143121	−	0.989705i	$-0.545714\pi$
$431$	−5.94253	−0.286242	−0.143121	+	0.989705i	$0.545714\pi$
$432$	−25.2120	−1.21301
$433$	33.4049	1.60534	0.802668	−	0.596426i	$-0.203413\pi$
$433$	33.4049	1.60534	0.802668	+	0.596426i	$0.203413\pi$
$434$	−60.2843	−2.89374
$435$	0	0
$436$	−46.5207	−2.22794
$437$	0	0
$438$	−2.14214	−0.102355
$439$	28.6931	1.36945	0.684723	−	0.728803i	$-0.259923\pi$
$439$	28.6931	1.36945	0.684723	+	0.728803i	$0.259923\pi$
$440$	0	0
$441$	95.0833	4.52777
$442$	9.45584	0.449769
$443$	8.14214	0.386845	0.193422	−	0.981116i	$-0.438041\pi$
$443$	8.14214	0.386845	0.193422	+	0.981116i	$0.438041\pi$
$444$	−51.4558	−2.44199
$445$	0	0
$446$	−36.3848	−1.72287
$447$	−27.2512	−1.28894
$448$	52.6274	2.48641
$449$	−2.46148	−0.116164	−0.0580822	−	0.998312i	$-0.518499\pi$
$449$	−2.46148	−0.116164	−0.0580822	+	0.998312i	$0.518499\pi$
$450$	0	0
$451$	0	0
$452$	23.2603	1.09407
$453$	52.9706	2.48877
$454$	−46.7279	−2.19305
$455$	0	0
$456$	0	0
$457$	17.3137	0.809901	0.404951	−	0.914339i	$-0.367289\pi$
$457$	17.3137	0.809901	0.404951	+	0.914339i	$0.367289\pi$
$458$	−26.2316	−1.22572
$459$	30.7322	1.43446
$460$	0	0
$461$	−33.3137	−1.55157	−0.775787	−	0.630995i	$-0.782647\pi$
$461$	−33.3137	−1.55157	−0.775787	+	0.630995i	$0.782647\pi$
$462$	60.2843	2.80468
$463$	27.1716	1.26277	0.631385	−	0.775469i	$-0.282487\pi$
$463$	27.1716	1.26277	0.631385	+	0.775469i	$0.282487\pi$
$464$	−7.38443	−0.342814
$465$	0	0
$466$	27.9721	1.29578
$467$	−21.5147	−0.995582	−0.497791	−	0.867297i	$-0.665855\pi$
$467$	−21.5147	−0.995582	−0.497791	+	0.867297i	$0.665855\pi$
$468$	−17.3178	−0.800516
$469$	26.2316	1.21126
$470$	0	0
$471$	−53.4828	−2.46436
$472$	−10.3431	−0.476082
$473$	−1.65685	−0.0761822
$474$	74.1942	3.40785
$475$	0	0
$476$	−42.6274	−1.95382
$477$	−27.4624	−1.25741
$478$	3.48106	0.159220
$479$	10.0000	0.456912	0.228456	−	0.973554i	$-0.426632\pi$
$479$	10.0000	0.456912	0.228456	+	0.973554i	$0.426632\pi$
$480$	0	0
$481$	8.82843	0.402542
$482$	14.6274	0.666261
$483$	69.2713	3.15195
$484$	−16.8995	−0.768159
$485$	0	0
$486$	6.24264	0.283172
$487$	−15.8759	−0.719406	−0.359703	−	0.933067i	$-0.617122\pi$
$487$	−15.8759	−0.719406	−0.359703	+	0.933067i	$0.617122\pi$
$488$	−2.46148	−0.111426
$489$	−38.1167	−1.72370
$490$	0	0
$491$	−24.2843	−1.09593	−0.547967	−	0.836500i	$-0.684598\pi$
$491$	−24.2843	−1.09593	−0.547967	+	0.836500i	$0.684598\pi$
$492$	0	0
$493$	9.00127	0.405397
$494$	0	0
$495$	0	0
$496$	17.8276	0.800483
$497$	−52.4632	−2.35330
$498$	19.7990	0.887214
$499$	34.9706	1.56550	0.782749	−	0.622338i	$-0.213817\pi$
$499$	34.9706	1.56550	0.782749	+	0.622338i	$0.213817\pi$
$500$	0	0
$501$	66.0833	2.95238
$502$	27.2512	1.21628
$503$	14.4853	0.645867	0.322933	−	0.946422i	$-0.395331\pi$
$503$	14.4853	0.645867	0.322933	+	0.946422i	$0.395331\pi$
$504$	24.4911	1.09092
$505$	0	0
$506$	20.2891	0.901960
$507$	−34.1258	−1.51558
$508$	−29.9238	−1.32766
$509$	4.50063	0.199487	0.0997435	−	0.995013i	$-0.468198\pi$
$509$	4.50063	0.199487	0.0997435	+	0.995013i	$0.468198\pi$
$510$	0	0
$511$	−1.65685	−0.0732949
$512$	−29.3522	−1.29720
$513$	0	0
$514$	−17.2132	−0.759242
$515$	0	0
$516$	−5.94253	−0.261605
$517$	12.9706	0.570445
$518$	−72.7696	−3.19731
$519$	38.9706	1.71062
$520$	0	0
$521$	−4.92296	−0.215679	−0.107839	−	0.994168i	$-0.534393\pi$
$521$	−4.92296	−0.215679	−0.107839	+	0.994168i	$0.534393\pi$
$522$	−30.1421	−1.31929
$523$	−7.89422	−0.345190	−0.172595	−	0.984993i	$-0.555215\pi$
$523$	−7.89422	−0.345190	−0.172595	+	0.984993i	$0.555215\pi$
$524$	−17.6569	−0.771343
$525$	0	0
$526$	25.5107	1.11232
$527$	−21.7310	−0.946616
$528$	−17.8276	−0.775847
$529$	0.313708	0.0136395
$530$	0	0
$531$	69.2713	3.00612
$532$	0	0
$533$	0	0
$534$	−58.8281	−2.54574
$535$	0	0
$536$	4.72792	0.204215
$537$	14.6274	0.631220
$538$	−5.17157	−0.222962
$539$	32.6274	1.40536
$540$	0	0
$541$	−6.14214	−0.264071	−0.132036	−	0.991245i	$-0.542151\pi$
$541$	−6.14214	−0.264071	−0.132036	+	0.991245i	$0.542151\pi$
$542$	6.24116	0.268081
$543$	−49.9411	−2.14318
$544$	29.4140	1.26112
$545$	0	0
$546$	−37.0971	−1.58761
$547$	−13.8368	−0.591617	−0.295809	−	0.955247i	$-0.595589\pi$
$547$	−13.8368	−0.591617	−0.295809	+	0.955247i	$0.595589\pi$
$548$	4.82843	0.206260
$549$	16.4853	0.703575
$550$	0	0
$551$	0	0
$552$	12.4853	0.531409
$553$	57.3862	2.44031
$554$	11.1641	0.474318
$555$	0	0
$556$	20.1421	0.854217
$557$	−16.6274	−0.704526	−0.352263	−	0.935901i	$-0.614588\pi$
$557$	−16.6274	−0.704526	−0.352263	+	0.935901i	$0.614588\pi$
$558$	72.7696	3.08058
$559$	1.01958	0.0431235
$560$	0	0
$561$	21.7310	0.917483
$562$	49.9411	2.10664
$563$	−9.93338	−0.418642	−0.209321	−	0.977847i	$-0.567125\pi$
$563$	−9.93338	−0.418642	−0.209321	+	0.977847i	$0.567125\pi$
$564$	46.5207	1.95887
$565$	0	0
$566$	62.6078	2.63160
$567$	−36.1421	−1.51783
$568$	−9.45584	−0.396758
$569$	9.42359	0.395057	0.197529	−	0.980297i	$-0.436708\pi$
$569$	9.42359	0.395057	0.197529	+	0.980297i	$0.436708\pi$
$570$	0	0
$571$	−9.31371	−0.389767	−0.194883	−	0.980826i	$-0.562433\pi$
$571$	−9.31371	−0.389767	−0.194883	+	0.980826i	$0.562433\pi$
$572$	−5.94253	−0.248470
$573$	−40.5782	−1.69518
$574$	0	0
$575$	0	0
$576$	−63.5269	−2.64695
$577$	−10.0000	−0.416305	−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000	−0.416305	−0.208153	+	0.978096i	$0.566745\pi$
$578$	7.62121	0.317001
$579$	−24.3431	−1.01167
$580$	0	0
$581$	15.3137	0.635320
$582$	7.68306	0.318473
$583$	−9.42359	−0.390285
$584$	−0.298627	−0.0123573
$585$	0	0
$586$	−25.4142	−1.04985
$587$	−1.51472	−0.0625191	−0.0312596	−	0.999511i	$-0.509952\pi$
$587$	−1.51472	−0.0625191	−0.0312596	+	0.999511i	$0.509952\pi$
$588$	117.023	4.82593
$589$	0	0
$590$	0	0
$591$	39.5586	1.62722
$592$	21.5198	0.884459
$593$	−13.3137	−0.546728	−0.273364	−	0.961911i	$-0.588136\pi$
$593$	−13.3137	−0.546728	−0.273364	+	0.961911i	$0.588136\pi$
$594$	−35.3137	−1.44894
$595$	0	0
$596$	−22.1421	−0.906977
$597$	−11.8851	−0.486423
$598$	−12.4853	−0.510561
$599$	−9.84591	−0.402293	−0.201147	−	0.979561i	$-0.564467\pi$
$599$	−9.84591	−0.402293	−0.201147	+	0.979561i	$0.564467\pi$
$600$	0	0
$601$	−4.92296	−0.200812	−0.100406	−	0.994947i	$-0.532014\pi$
$601$	−4.92296	−0.200812	−0.100406	+	0.994947i	$0.532014\pi$
$602$	−8.40401	−0.342522
$603$	−31.6644	−1.28947
$604$	43.0396	1.75126
$605$	0	0
$606$	102.912	4.18050
$607$	21.8184	0.885583	0.442792	−	0.896625i	$-0.353988\pi$
$607$	21.8184	0.885583	0.442792	+	0.896625i	$0.353988\pi$
$608$	0	0
$609$	−35.3137	−1.43098
$610$	0	0
$611$	−7.98169	−0.322905
$612$	51.4558	2.07998
$613$	−42.2843	−1.70785	−0.853923	−	0.520400i	$-0.825783\pi$
$613$	−42.2843	−1.70785	−0.853923	+	0.520400i	$0.825783\pi$
$614$	23.8995	0.964505
$615$	0	0
$616$	8.40401	0.338607
$617$	26.2843	1.05816	0.529082	−	0.848570i	$-0.322536\pi$
$617$	26.2843	1.05816	0.529082	+	0.848570i	$0.322536\pi$
$618$	55.6457	2.23840
$619$	14.9706	0.601718	0.300859	−	0.953669i	$-0.402727\pi$
$619$	14.9706	0.601718	0.300859	+	0.953669i	$0.402727\pi$
$620$	0	0
$621$	−40.5782	−1.62835
$622$	66.5111	2.66685
$623$	−45.5011	−1.82296
$624$	10.9706	0.439174
$625$	0	0
$626$	−43.3383	−1.73215
$627$	0	0
$628$	−43.4558	−1.73408
$629$	−26.2316	−1.04592
$630$	0	0
$631$	−38.2843	−1.52407	−0.762036	−	0.647534i	$-0.775800\pi$
$631$	−38.2843	−1.52407	−0.762036	+	0.647534i	$0.775800\pi$
$632$	10.3431	0.411428
$633$	−32.2843	−1.28318
$634$	−32.7279	−1.29979
$635$	0	0
$636$	−33.7990	−1.34022
$637$	−20.0779	−0.795516
$638$	−10.3431	−0.409489
$639$	63.3287	2.50525
$640$	0	0
$641$	35.6552	1.40830	0.704148	−	0.710053i	$-0.251329\pi$
$641$	35.6552	1.40830	0.704148	+	0.710053i	$0.251329\pi$
$642$	40.2795	1.58971
$643$	−26.4853	−1.04448	−0.522239	−	0.852799i	$-0.674903\pi$
$643$	−26.4853	−1.04448	−0.522239	+	0.852799i	$0.674903\pi$
$644$	56.2843	2.21791
$645$	0	0
$646$	0	0
$647$	32.8284	1.29062	0.645309	−	0.763921i	$-0.276728\pi$
$647$	32.8284	1.29062	0.645309	+	0.763921i	$0.276728\pi$
$648$	−6.51417	−0.255901
$649$	23.7701	0.933059
$650$	0	0
$651$	85.2548	3.34140
$652$	−30.9706	−1.21290
$653$	−16.3431	−0.639557	−0.319778	−	0.947492i	$-0.603608\pi$
$653$	−16.3431	−0.639557	−0.319778	+	0.947492i	$0.603608\pi$
$654$	120.293	4.70381
$655$	0	0
$656$	0	0
$657$	2.00000	0.0780274
$658$	65.7902	2.56477
$659$	−33.6160	−1.30950	−0.654748	−	0.755848i	$-0.727225\pi$
$659$	−33.6160	−1.30950	−0.654748	+	0.755848i	$0.727225\pi$
$660$	0	0
$661$	28.6931	1.11603	0.558016	−	0.829830i	$-0.311563\pi$
$661$	28.6931	1.11603	0.558016	+	0.829830i	$0.311563\pi$
$662$	−62.4264	−2.42627
$663$	−13.3726	−0.519348
$664$	2.76011	0.107113
$665$	0	0
$666$	87.8406	3.40375
$667$	−11.8851	−0.460192
$668$	53.6940	2.07748
$669$	51.4558	1.98940
$670$	0	0
$671$	5.65685	0.218380
$672$	−115.397	−4.45153
$673$	−7.59560	−0.292789	−0.146394	−	0.989226i	$-0.546767\pi$
$673$	−7.59560	−0.292789	−0.146394	+	0.989226i	$0.546767\pi$
$674$	65.0122	2.50418
$675$	0	0
$676$	−27.7279	−1.06646
$677$	−21.5198	−0.827074	−0.413537	−	0.910487i	$-0.635707\pi$
$677$	−21.5198	−0.827074	−0.413537	+	0.910487i	$0.635707\pi$
$678$	−60.1463	−2.30990
$679$	5.94253	0.228054
$680$	0	0
$681$	66.0833	2.53232
$682$	24.9706	0.956172
$683$	13.8368	0.529449	0.264724	−	0.964324i	$-0.414719\pi$
$683$	13.8368	0.529449	0.264724	+	0.964324i	$0.414719\pi$
$684$	0	0
$685$	0	0
$686$	94.4833	3.60739
$687$	37.0971	1.41534
$688$	2.48528	0.0947505
$689$	5.79899	0.220924
$690$	0	0
$691$	−28.6274	−1.08904	−0.544519	−	0.838748i	$-0.683288\pi$
$691$	−28.6274	−1.08904	−0.544519	+	0.838748i	$0.683288\pi$
$692$	31.6644	1.20370
$693$	−56.2843	−2.13806
$694$	52.1646	1.98014
$695$	0	0
$696$	−6.36486	−0.241259
$697$	0	0
$698$	14.0479	0.531722
$699$	−39.5586	−1.49624
$700$	0	0
$701$	−7.51472	−0.283827	−0.141914	−	0.989879i	$-0.545325\pi$
$701$	−7.51472	−0.283827	−0.141914	+	0.989879i	$0.545325\pi$
$702$	21.7310	0.820183
$703$	0	0
$704$	−21.7990	−0.821580
$705$	0	0
$706$	−7.68306	−0.289156
$707$	79.5980	2.99359
$708$	85.2548	3.20407
$709$	17.3137	0.650230	0.325115	−	0.945674i	$-0.394597\pi$
$709$	17.3137	0.650230	0.325115	+	0.945674i	$0.394597\pi$
$710$	0	0
$711$	−69.2713	−2.59787
$712$	−8.20101	−0.307346
$713$	28.6931	1.07456
$714$	110.225	4.12508
$715$	0	0
$716$	11.8851	0.444166
$717$	−4.92296	−0.183851
$718$	53.1842	1.98482
$719$	−35.9411	−1.34038	−0.670189	−	0.742191i	$-0.733787\pi$
$719$	−35.9411	−1.34038	−0.670189	+	0.742191i	$0.733787\pi$
$720$	0	0
$721$	43.0396	1.60288
$722$	0	0
$723$	−20.6863	−0.769331
$724$	−40.5782	−1.50808
$725$	0	0
$726$	43.6985	1.62180
$727$	10.4853	0.388878	0.194439	−	0.980915i	$-0.437711\pi$
$727$	10.4853	0.388878	0.194439	+	0.980915i	$0.437711\pi$
$728$	−5.17157	−0.191671
$729$	−31.2843	−1.15868
$730$	0	0
$731$	−3.02944	−0.112048
$732$	20.2891	0.749906
$733$	24.3431	0.899135	0.449567	−	0.893246i	$-0.351578\pi$
$733$	24.3431	0.899135	0.449567	+	0.893246i	$0.351578\pi$
$734$	18.5486	0.684640
$735$	0	0
$736$	−38.8376	−1.43157
$737$	−10.8655	−0.400235
$738$	0	0
$739$	46.6274	1.71522	0.857609	−	0.514303i	$-0.171949\pi$
$739$	46.6274	1.71522	0.857609	+	0.514303i	$0.171949\pi$
$740$	0	0
$741$	0	0
$742$	−47.7990	−1.75476
$743$	−5.43275	−0.199308	−0.0996540	−	0.995022i	$-0.531774\pi$
$743$	−5.43275	−0.199308	−0.0996540	+	0.995022i	$0.531774\pi$
$744$	15.3661	0.563349
$745$	0	0
$746$	−50.3848	−1.84472
$747$	−18.4853	−0.676341
$748$	17.6569	0.645599
$749$	31.1546	1.13836
$750$	0	0
$751$	−22.7506	−0.830180	−0.415090	−	0.909780i	$-0.636250\pi$
$751$	−22.7506	−0.830180	−0.415090	+	0.909780i	$0.636250\pi$
$752$	−19.4558	−0.709482
$753$	−38.5390	−1.40444
$754$	6.36486	0.231794
$755$	0	0
$756$	−97.9643	−3.56293
$757$	29.3137	1.06542	0.532712	−	0.846296i	$-0.321173\pi$
$757$	29.3137	1.06542	0.532712	+	0.846296i	$0.321173\pi$
$758$	10.3431	0.375680
$759$	−28.6931	−1.04149
$760$	0	0
$761$	−14.8284	−0.537530	−0.268765	−	0.963206i	$-0.586616\pi$
$761$	−14.8284	−0.537530	−0.268765	+	0.963206i	$0.586616\pi$
$762$	77.3766	2.80306
$763$	93.0414	3.36832
$764$	−32.9706	−1.19283
$765$	0	0
$766$	26.9289	0.972982
$767$	−14.6274	−0.528165
$768$	22.2408	0.802545
$769$	−14.1421	−0.509978	−0.254989	−	0.966944i	$-0.582072\pi$
$769$	−14.1421	−0.509978	−0.254989	+	0.966944i	$0.582072\pi$
$770$	0	0
$771$	24.3431	0.876697
$772$	−19.7793	−0.711872
$773$	22.5394	0.810686	0.405343	−	0.914165i	$-0.367152\pi$
$773$	22.5394	0.810686	0.405343	+	0.914165i	$0.367152\pi$
$774$	10.1445	0.364638
$775$	0	0
$776$	1.07107	0.0384491
$777$	102.912	3.69194
$778$	−51.7423	−1.85505
$779$	0	0
$780$	0	0
$781$	21.7310	0.777596
$782$	37.0971	1.32659
$783$	20.6863	0.739268
$784$	−48.9411	−1.74790
$785$	0	0
$786$	45.6569	1.62853
$787$	−23.6827	−0.844196	−0.422098	−	0.906550i	$-0.638706\pi$
$787$	−23.6827	−0.844196	−0.422098	+	0.906550i	$0.638706\pi$
$788$	32.1421	1.14502
$789$	−36.0775	−1.28439
$790$	0	0
$791$	−46.5207	−1.65409
$792$	−10.1445	−0.360471
$793$	−3.48106	−0.123616
$794$	−29.4140	−1.04387
$795$	0	0
$796$	−9.65685	−0.342278
$797$	7.17327	0.254090	0.127045	−	0.991897i	$-0.459451\pi$
$797$	7.17327	0.254090	0.127045	+	0.991897i	$0.459451\pi$
$798$	0	0
$799$	23.7157	0.839002
$800$	0	0
$801$	54.9247	1.94067
$802$	35.3137	1.24697
$803$	0.686292	0.0242187
$804$	−38.9706	−1.37439
$805$	0	0
$806$	−15.3661	−0.541249
$807$	7.31371	0.257455
$808$	14.3465	0.504710
$809$	9.31371	0.327453	0.163726	−	0.986506i	$-0.447649\pi$
$809$	9.31371	0.327453	0.163726	+	0.986506i	$0.447649\pi$
$810$	0	0
$811$	−44.4815	−1.56196	−0.780979	−	0.624557i	$-0.785279\pi$
$811$	−44.4815	−1.56196	−0.780979	+	0.624557i	$0.785279\pi$
$812$	−28.6931	−1.00693
$813$	−8.82633	−0.309553
$814$	30.1421	1.05648
$815$	0	0
$816$	−32.5965	−1.14110
$817$	0	0
$818$	80.0833	2.80005
$819$	34.6356	1.21027
$820$	0	0
$821$	25.3137	0.883455	0.441727	−	0.897149i	$-0.354366\pi$
$821$	25.3137	0.883455	0.441727	+	0.897149i	$0.354366\pi$
$822$	−12.4853	−0.435474
$823$	54.0833	1.88522	0.942612	−	0.333890i	$-0.108361\pi$
$823$	54.0833	1.88522	0.942612	+	0.333890i	$0.108361\pi$
$824$	7.75736	0.270240
$825$	0	0
$826$	120.569	4.19512
$827$	1.95169	0.0678669	0.0339334	−	0.999424i	$-0.489197\pi$
$827$	1.95169	0.0678669	0.0339334	+	0.999424i	$0.489197\pi$
$828$	−67.9411	−2.36112
$829$	12.3074	0.427453	0.213727	−	0.976893i	$-0.431440\pi$
$829$	12.3074	0.427453	0.213727	+	0.976893i	$0.431440\pi$
$830$	0	0
$831$	−15.7884	−0.547695
$832$	13.4144	0.465062
$833$	59.6569	2.06699
$834$	−52.0833	−1.80350
$835$	0	0
$836$	0	0
$837$	−49.9411	−1.72622
$838$	−18.8472	−0.651065
$839$	18.8472	0.650677	0.325338	−	0.945598i	$-0.394522\pi$
$839$	18.8472	0.650677	0.325338	+	0.945598i	$0.394522\pi$
$840$	0	0
$841$	−22.9411	−0.791073
$842$	35.3137	1.21699
$843$	−70.6274	−2.43254
$844$	−26.2316	−0.902929
$845$	0	0
$846$	−79.4158	−2.73037
$847$	33.7990	1.16135
$848$	14.1354	0.485411
$849$	−88.5408	−3.03871
$850$	0	0
$851$	34.6356	1.18729
$852$	77.9411	2.67022
$853$	−47.6569	−1.63174	−0.815870	−	0.578236i	$-0.803741\pi$
$853$	−47.6569	−1.63174	−0.815870	+	0.578236i	$0.803741\pi$
$854$	28.6931	0.981857
$855$	0	0
$856$	5.61522	0.191924
$857$	36.8859	1.26000	0.630000	−	0.776595i	$-0.283055\pi$
$857$	36.8859	1.26000	0.630000	+	0.776595i	$0.283055\pi$
$858$	15.3661	0.524591
$859$	12.2843	0.419134	0.209567	−	0.977794i	$-0.432795\pi$
$859$	12.2843	0.419134	0.209567	+	0.977794i	$0.432795\pi$
$860$	0	0
$861$	0	0
$862$	12.4853	0.425250
$863$	−24.2799	−0.826498	−0.413249	−	0.910618i	$-0.635606\pi$
$863$	−24.2799	−0.826498	−0.413249	+	0.910618i	$0.635606\pi$
$864$	67.5980	2.29973
$865$	0	0
$866$	−70.1838	−2.38494
$867$	−10.7780	−0.366041
$868$	69.2713	2.35122
$869$	−23.7701	−0.806347
$870$	0	0
$871$	6.68629	0.226556
$872$	16.7696	0.567889
$873$	−7.17327	−0.242779
$874$	0	0
$875$	0	0
$876$	2.46148	0.0831656
$877$	23.9813	0.809791	0.404895	−	0.914363i	$-0.367308\pi$
$877$	23.9813	0.809791	0.404895	+	0.914363i	$0.367308\pi$
$878$	−60.2843	−2.03450
$879$	35.9411	1.21226
$880$	0	0
$881$	−12.4853	−0.420640	−0.210320	−	0.977633i	$-0.567451\pi$
$881$	−12.4853	−0.420640	−0.210320	+	0.977633i	$0.567451\pi$
$882$	−199.770	−6.72661
$883$	−32.1421	−1.08167	−0.540834	−	0.841129i	$-0.681891\pi$
$883$	−32.1421	−1.08167	−0.540834	+	0.841129i	$0.681891\pi$
$884$	−10.8655	−0.365446
$885$	0	0
$886$	−17.1067	−0.574709
$887$	−31.6644	−1.06319	−0.531593	−	0.847000i	$-0.678406\pi$
$887$	−31.6644	−1.06319	−0.531593	+	0.847000i	$0.678406\pi$
$888$	18.5486	0.622449
$889$	59.8477	2.00723
$890$	0	0
$891$	14.9706	0.501533
$892$	41.8089	1.39987
$893$	0	0
$894$	57.2548	1.91489
$895$	0	0
$896$	−32.8951	−1.09895
$897$	17.6569	0.589545
$898$	5.17157	0.172578
$899$	−14.6274	−0.487852
$900$	0	0
$901$	−17.2303	−0.574026
$902$	0	0
$903$	11.8851	0.395510
$904$	−8.38478	−0.278874
$905$	0	0
$906$	−111.291	−3.69741
$907$	−33.1063	−1.09928	−0.549638	−	0.835403i	$-0.685234\pi$
$907$	−33.1063	−1.09928	−0.549638	+	0.835403i	$0.685234\pi$
$908$	53.6940	1.78190
$909$	−96.0833	−3.18688
$910$	0	0
$911$	−56.3666	−1.86751	−0.933754	−	0.357914i	$-0.883488\pi$
$911$	−56.3666	−1.86751	−0.933754	+	0.357914i	$0.883488\pi$
$912$	0	0
$913$	−6.34315	−0.209927
$914$	−36.3762	−1.20322
$915$	0	0
$916$	30.1421	0.995924
$917$	35.3137	1.16616
$918$	−64.5685	−2.13108
$919$	−56.2843	−1.85665	−0.928323	−	0.371774i	$-0.878750\pi$
$919$	−56.2843	−1.85665	−0.928323	+	0.371774i	$0.878750\pi$
$920$	0	0
$921$	−33.7990	−1.11371
$922$	69.9922	2.30507
$923$	−13.3726	−0.440164
$924$	−69.2713	−2.27886
$925$	0	0
$926$	−57.0876	−1.87601
$927$	−51.9534	−1.70637
$928$	19.7990	0.649934
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	0	0
$932$	−32.1421	−1.05285
$933$	−94.0610	−3.07942
$934$	45.2025	1.47907
$935$	0	0
$936$	6.24264	0.204047
$937$	22.9706	0.750416	0.375208	−	0.926941i	$-0.377571\pi$
$937$	22.9706	0.750416	0.375208	+	0.926941i	$0.377571\pi$
$938$	−55.1127	−1.79949
$939$	61.2896	2.00011
$940$	0	0
$941$	28.6931	0.935368	0.467684	−	0.883896i	$-0.345089\pi$
$941$	28.6931	0.935368	0.467684	+	0.883896i	$0.345089\pi$
$942$	112.368	3.66113
$943$	0	0
$944$	−35.6552	−1.16048
$945$	0	0
$946$	3.48106	0.113179
$947$	−24.1421	−0.784514	−0.392257	−	0.919856i	$-0.628306\pi$
$947$	−24.1421	−0.784514	−0.392257	+	0.919856i	$0.628306\pi$
$948$	−85.2548	−2.76895
$949$	−0.422323	−0.0137092
$950$	0	0
$951$	46.2843	1.50087
$952$	15.3661	0.498019
$953$	−27.4624	−0.889593	−0.444796	−	0.895632i	$-0.646724\pi$
$953$	−27.4624	−0.889593	−0.444796	+	0.895632i	$0.646724\pi$
$954$	57.6985	1.86806
$955$	0	0
$956$	−4.00000	−0.129369
$957$	14.6274	0.472837
$958$	−21.0100	−0.678803
$959$	−9.65685	−0.311836
$960$	0	0
$961$	4.31371	0.139152
$962$	−18.5486	−0.598029
$963$	−37.6069	−1.21187
$964$	−16.8080	−0.541350
$965$	0	0
$966$	−145.539	−4.68264
$967$	−38.4853	−1.23760	−0.618802	−	0.785547i	$-0.712382\pi$
$967$	−38.4853	−1.23760	−0.618802	+	0.785547i	$0.712382\pi$
$968$	6.09185	0.195799
$969$	0	0
$970$	0	0
$971$	−46.5207	−1.49292	−0.746460	−	0.665430i	$-0.768248\pi$
$971$	−46.5207	−1.49292	−0.746460	+	0.665430i	$0.768248\pi$
$972$	−7.17327	−0.230083
$973$	−40.2843	−1.29145
$974$	33.3553	1.06877
$975$	0	0
$976$	−8.48528	−0.271607
$977$	21.0975	0.674969	0.337484	−	0.941331i	$-0.390424\pi$
$977$	21.0975	0.674969	0.337484	+	0.941331i	$0.390424\pi$
$978$	80.0833	2.56078
$979$	18.8472	0.602358
$980$	0	0
$981$	−112.311	−3.58581
$982$	51.0213	1.62816
$983$	56.4541	1.80061	0.900303	−	0.435265i	$-0.143345\pi$
$983$	56.4541	1.80061	0.900303	+	0.435265i	$0.143345\pi$
$984$	0	0
$985$	0	0
$986$	−18.9117	−0.602271
$987$	−93.0414	−2.96154
$988$	0	0
$989$	4.00000	0.127193
$990$	0	0
$991$	−12.9046	−0.409930	−0.204965	−	0.978769i	$-0.565708\pi$
$991$	−12.9046	−0.409930	−0.204965	+	0.978769i	$0.565708\pi$
$992$	−47.7990	−1.51762
$993$	88.2843	2.80162
$994$	110.225	3.49614
$995$	0	0
$996$	−22.7506	−0.720879
$997$	−3.65685	−0.115814	−0.0579069	−	0.998322i	$-0.518443\pi$
$997$	−3.65685	−0.115814	−0.0579069	+	0.998322i	$0.518443\pi$
$998$	−73.4733	−2.32576
$999$	−60.2843	−1.90731

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.bn.1.1		4
5.4	even	2		1805.2.a.m.1.4	yes	4
19.18	odd	2	inner	9025.2.a.bn.1.4		4
95.94	odd	2		1805.2.a.m.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1805.2.a.m.1.1	✓	4	95.94	odd	2
1805.2.a.m.1.4	yes	4	5.4	even	2
9025.2.a.bn.1.1		4	1.1	even	1	trivial
9025.2.a.bn.1.4		4	19.18	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 9025 = 5^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9025.a (trivial)

\(f(q)\)	\(=\)	\(q-2.10100 q^{2} +2.97127 q^{3} +2.41421 q^{4} -6.24264 q^{6} -4.82843 q^{7} -0.870264 q^{8} +5.82843 q^{9} +2.00000 q^{11} +7.17327 q^{12} -1.23074 q^{13} +10.1445 q^{14} -3.00000 q^{16} +3.65685 q^{17} -12.2455 q^{18} -14.3465 q^{21} -4.20201 q^{22} -4.82843 q^{23} -2.58579 q^{24} +2.58579 q^{26} +8.40401 q^{27} -11.6569 q^{28} +2.46148 q^{29} -5.94253 q^{31} +8.04354 q^{32} +5.94253 q^{33} -7.68306 q^{34} +14.0711 q^{36} -7.17327 q^{37} -3.65685 q^{39} +30.1421 q^{42} -0.828427 q^{43} +4.82843 q^{44} +10.1445 q^{46} +6.48528 q^{47} -8.91380 q^{48} +16.3137 q^{49} +10.8655 q^{51} -2.97127 q^{52} -4.71179 q^{53} -17.6569 q^{54} +4.20201 q^{56} -5.17157 q^{58} +11.8851 q^{59} +2.82843 q^{61} +12.4853 q^{62} -28.1421 q^{63} -10.8995 q^{64} -12.4853 q^{66} -5.43275 q^{67} +8.82843 q^{68} -14.3465 q^{69} +10.8655 q^{71} -5.07227 q^{72} +0.343146 q^{73} +15.0711 q^{74} -9.65685 q^{77} +7.68306 q^{78} -11.8851 q^{79} +7.48528 q^{81} -3.17157 q^{83} -34.6356 q^{84} +1.74053 q^{86} +7.31371 q^{87} -1.74053 q^{88} +9.42359 q^{89} +5.94253 q^{91} -11.6569 q^{92} -17.6569 q^{93} -13.6256 q^{94} +23.8995 q^{96} -1.23074 q^{97} -34.2752 q^{98} +11.6569 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} - 8 q^{6} - 8 q^{7} + 12 q^{9} + 8 q^{11} - 12 q^{16} - 8 q^{17} - 8 q^{23} - 16 q^{24} + 16 q^{26} - 24 q^{28} + 28 q^{36} + 8 q^{39} + 64 q^{42} + 8 q^{43} + 8 q^{44} - 8 q^{47} + 20 q^{49}+ \cdots + 24 q^{99}+O(q^{100}) \) 4 * q + 4 * q^4 - 8 * q^6 - 8 * q^7 + 12 * q^9 + 8 * q^11 - 12 * q^16 - 8 * q^17 - 8 * q^23 - 16 * q^24 + 16 * q^26 - 24 * q^28 + 28 * q^36 + 8 * q^39 + 64 * q^42 + 8 * q^43 + 8 * q^44 - 8 * q^47 + 20 * q^49 - 48 * q^54 - 32 * q^58 + 16 * q^62 - 56 * q^63 - 4 * q^64 - 16 * q^66 + 24 * q^68 + 24 * q^73 + 32 * q^74 - 16 * q^77 - 4 * q^81 - 24 * q^83 - 16 * q^87 - 24 * q^92 - 48 * q^93 + 56 * q^96 + 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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