LMFDB - Embedded newform 9025.2.a.bn.1.4

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			1.4
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)

$f(q)$	$=$	$q+2.10100 q^{2} -2.97127 q^{3} +2.41421 q^{4} -6.24264 q^{6} -4.82843 q^{7} +0.870264 q^{8} +5.82843 q^{9} +O(q^{10})$	\(q+2.10100 q^{2} -2.97127 q^{3} +2.41421 q^{4} -6.24264 q^{6} -4.82843 q^{7} +0.870264 q^{8} +5.82843 q^{9} +2.00000 q^{11} -7.17327 q^{12} +1.23074 q^{13} -10.1445 q^{14} -3.00000 q^{16} +3.65685 q^{17} +12.2455 q^{18} +14.3465 q^{21} +4.20201 q^{22} -4.82843 q^{23} -2.58579 q^{24} +2.58579 q^{26} -8.40401 q^{27} -11.6569 q^{28} -2.46148 q^{29} +5.94253 q^{31} -8.04354 q^{32} -5.94253 q^{33} +7.68306 q^{34} +14.0711 q^{36} +7.17327 q^{37} -3.65685 q^{39} +30.1421 q^{42} -0.828427 q^{43} +4.82843 q^{44} -10.1445 q^{46} +6.48528 q^{47} +8.91380 q^{48} +16.3137 q^{49} -10.8655 q^{51} +2.97127 q^{52} +4.71179 q^{53} -17.6569 q^{54} -4.20201 q^{56} -5.17157 q^{58} -11.8851 q^{59} +2.82843 q^{61} +12.4853 q^{62} -28.1421 q^{63} -10.8995 q^{64} -12.4853 q^{66} +5.43275 q^{67} +8.82843 q^{68} +14.3465 q^{69} -10.8655 q^{71} +5.07227 q^{72} +0.343146 q^{73} +15.0711 q^{74} -9.65685 q^{77} -7.68306 q^{78} +11.8851 q^{79} +7.48528 q^{81} -3.17157 q^{83} +34.6356 q^{84} -1.74053 q^{86} +7.31371 q^{87} +1.74053 q^{88} -9.42359 q^{89} -5.94253 q^{91} -11.6569 q^{92} -17.6569 q^{93} +13.6256 q^{94} +23.8995 q^{96} +1.23074 q^{97} +34.2752 q^{98} +11.6569 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} - 8 q^{6} - 8 q^{7} + 12 q^{9}+O(q^{10}) $ 4 * q + 4 * q^4 - 8 * q^6 - 8 * q^7 + 12 * q^9	$ 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}+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

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 $

$2$	2.10100	1.48563	0.742817	−	0.669495i	$-0.233489\pi$
$2$	2.10100	1.48563	0.742817	+	0.669495i	$0.233489\pi$
$3$	−2.97127	−1.71546	−0.857731	−	0.514099i	$-0.828126\pi$
$3$	−2.97127	−1.71546	−0.857731	+	0.514099i	$0.828126\pi$
$4$	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	$-0.445406\pi$
$13$	1.23074	0.341346	0.170673	+	0.985328i	$0.445406\pi$
$14$	−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	$-0.573396\pi$
$29$	−2.46148	−0.457085	−0.228543	+	0.973534i	$0.573396\pi$
$30$	0	0
$31$	5.94253	1.06731	0.533655	−	0.845702i	$-0.320818\pi$
$31$	5.94253	1.06731	0.533655	+	0.845702i	$0.320818\pi$
$32$	−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	$-0.299270\pi$
$37$	7.17327	1.17928	0.589639	+	0.807667i	$0.299270\pi$
$38$	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	$-0.395104\pi$
$53$	4.71179	0.647215	0.323607	+	0.946191i	$0.395104\pi$
$54$	−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	$-0.781572\pi$
$59$	−11.8851	−1.54730	−0.773652	+	0.633611i	$0.781572\pi$
$60$	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	$-0.392325\pi$
$67$	5.43275	0.663715	0.331858	+	0.943329i	$0.392325\pi$
$68$	8.82843	1.07060
$69$	14.3465	1.72712
$70$	0	0
$71$	−10.8655	−1.28950	−0.644748	−	0.764395i	$-0.723038\pi$
$71$	−10.8655	−1.28950	−0.644748	+	0.764395i	$0.723038\pi$
$72$	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	$-0.266899\pi$
$79$	11.8851	1.33717	0.668587	+	0.743634i	$0.266899\pi$
$80$	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	$-0.666464\pi$
$89$	−9.42359	−0.998898	−0.499449	+	0.866343i	$0.666464\pi$
$90$	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	$-0.480099\pi$
$97$	1.23074	0.124963	0.0624813	+	0.998046i	$0.480099\pi$
$98$	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	$-0.355279\pi$
$103$	8.91380	0.878303	0.439151	+	0.898413i	$0.355279\pi$
$104$	1.07107	0.105027
$105$	0	0
$106$	9.89949	0.961524
$107$	6.45232	0.623770	0.311885	−	0.950120i	$-0.399040\pi$
$107$	6.45232	0.623770	0.311885	+	0.950120i	$0.399040\pi$
$108$	−20.2891	−1.95232
$109$	19.2695	1.84568	0.922842	−	0.385179i	$-0.125860\pi$
$109$	19.2695	1.84568	0.922842	+	0.385179i	$0.125860\pi$
$110$	0	0
$111$	−21.3137	−2.02301
$112$	14.4853	1.36873
$113$	−9.63475	−0.906361	−0.453181	−	0.891419i	$-0.649711\pi$
$113$	−9.63475	−0.906361	−0.453181	+	0.891419i	$0.649711\pi$
$114$	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	$-0.314653\pi$
$127$	12.3949	1.09987	0.549933	+	0.835209i	$0.314653\pi$
$128$	−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	$-0.758343\pi$
$151$	−17.8276	−1.45079	−0.725395	+	0.688333i	$0.758343\pi$
$152$	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	$-0.829862\pi$
$167$	−22.2408	−1.72104	−0.860521	+	0.509415i	$0.829862\pi$
$168$	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	$-0.666148\pi$
$173$	−13.1158	−0.997176	−0.498588	+	0.866839i	$0.666148\pi$
$174$	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	$-0.558898\pi$
$179$	−4.92296	−0.367959	−0.183980	+	0.982930i	$0.558898\pi$
$180$	0	0
$181$	16.8080	1.24933	0.624665	−	0.780893i	$-0.285235\pi$
$181$	16.8080	1.24933	0.624665	+	0.780893i	$0.285235\pi$
$182$	−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	$-0.404725\pi$
$193$	8.19285	0.589734	0.294867	+	0.955538i	$0.404725\pi$
$194$	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	$-0.377984\pi$
$211$	10.8655	0.748011	0.374006	+	0.927426i	$0.377984\pi$
$212$	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	$-0.696886\pi$
$223$	−17.3178	−1.15969	−0.579843	+	0.814728i	$0.696886\pi$
$224$	38.8376	2.59495
$225$	0	0
$226$	−20.2426	−1.34652
$227$	−22.2408	−1.47617	−0.738086	−	0.674707i	$-0.764270\pi$
$227$	−22.2408	−1.47617	−0.738086	+	0.674707i	$0.764270\pi$
$228$	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	$-0.428012\pi$
$241$	6.96211	0.448469	0.224235	+	0.974535i	$0.428012\pi$
$242$	−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	$-0.582249\pi$
$257$	−8.19285	−0.511056	−0.255528	+	0.966802i	$0.582249\pi$
$258$	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	$-0.523908\pi$
$269$	−2.46148	−0.150079	−0.0750395	+	0.997181i	$0.523908\pi$
$270$	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	$-0.249145\pi$
$281$	23.7701	1.41801	0.709004	+	0.705205i	$0.249145\pi$
$282$	−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	$-0.614952\pi$
$293$	−12.0962	−0.706669	−0.353335	+	0.935497i	$0.614952\pi$
$294$	−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	$-0.394767\pi$
$307$	11.3753	0.649221	0.324611	+	0.945848i	$0.394767\pi$
$308$	−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	$-0.644120\pi$
$317$	−15.5773	−0.874907	−0.437454	+	0.899241i	$0.644120\pi$
$318$	−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	$-0.804132\pi$
$331$	−29.7127	−1.63316	−0.816578	+	0.577235i	$0.804132\pi$
$332$	−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	$-0.180907\pi$
$337$	30.9434	1.68559	0.842797	+	0.538231i	$0.180907\pi$
$338$	−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	$-0.713213\pi$
$373$	−23.9813	−1.24170	−0.620852	+	0.783928i	$0.713213\pi$
$374$	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	$-0.459646\pi$
$379$	4.92296	0.252875	0.126438	+	0.991975i	$0.459646\pi$
$380$	0	0
$381$	−36.8284	−1.88678
$382$	−28.6931	−1.46807
$383$	12.8172	0.654927	0.327464	−	0.944864i	$-0.393806\pi$
$383$	12.8172	0.654927	0.327464	+	0.944864i	$0.393806\pi$
$384$	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	$-0.362144\pi$
$401$	16.8080	0.839353	0.419676	+	0.907674i	$0.362144\pi$
$402$	−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	$-0.108588\pi$
$409$	38.1167	1.88475	0.942374	+	0.334560i	$0.108588\pi$
$410$	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	$-0.365673\pi$
$421$	16.8080	0.819173	0.409586	+	0.912271i	$0.365673\pi$
$422$	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	$-0.454286\pi$
$431$	5.94253	0.286242	0.143121	+	0.989705i	$0.454286\pi$
$432$	25.2120	1.21301
$433$	−33.4049	−1.60534	−0.802668	−	0.596426i	$-0.796587\pi$
$433$	−33.4049	−1.60534	−0.802668	+	0.596426i	$0.796587\pi$
$434$	−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	$-0.740077\pi$
$439$	−28.6931	−1.36945	−0.684723	+	0.728803i	$0.740077\pi$
$440$	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	$-0.481501\pi$
$449$	2.46148	0.116164	0.0580822	+	0.998312i	$0.481501\pi$
$450$	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	$-0.382878\pi$
$487$	15.8759	0.719406	0.359703	+	0.933067i	$0.382878\pi$
$488$	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	$-0.531802\pi$
$509$	−4.50063	−0.199487	−0.0997435	+	0.995013i	$0.531802\pi$
$510$	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	$-0.465607\pi$
$521$	4.92296	0.215679	0.107839	+	0.994168i	$0.465607\pi$
$522$	−30.1421	−1.31929
$523$	7.89422	0.345190	0.172595	−	0.984993i	$-0.444785\pi$
$523$	7.89422	0.345190	0.172595	+	0.984993i	$0.444785\pi$
$524$	−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	$-0.404411\pi$
$547$	13.8368	0.591617	0.295809	+	0.955247i	$0.404411\pi$
$548$	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	$-0.432875\pi$
$563$	9.93338	0.418642	0.209321	+	0.977847i	$0.432875\pi$
$564$	−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	$-0.563292\pi$
$569$	−9.42359	−0.395057	−0.197529	+	0.980297i	$0.563292\pi$
$570$	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	$-0.435533\pi$
$599$	9.84591	0.402293	0.201147	+	0.979561i	$0.435533\pi$
$600$	0	0
$601$	4.92296	0.200812	0.100406	−	0.994947i	$-0.467986\pi$
$601$	4.92296	0.200812	0.100406	+	0.994947i	$0.467986\pi$
$602$	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	$-0.646012\pi$
$607$	−21.8184	−0.885583	−0.442792	+	0.896625i	$0.646012\pi$
$608$	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	$-0.748671\pi$
$641$	−35.6552	−1.40830	−0.704148	+	0.710053i	$0.748671\pi$
$642$	−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	$-0.272775\pi$
$659$	33.6160	1.30950	0.654748	+	0.755848i	$0.272775\pi$
$660$	0	0
$661$	−28.6931	−1.11603	−0.558016	−	0.829830i	$-0.688437\pi$
$661$	−28.6931	−1.11603	−0.558016	+	0.829830i	$0.688437\pi$
$662$	−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	$-0.453233\pi$
$673$	7.59560	0.292789	0.146394	+	0.989226i	$0.453233\pi$
$674$	65.0122	2.50418
$675$	0	0
$676$	−27.7279	−1.06646
$677$	21.5198	0.827074	0.413537	−	0.910487i	$-0.364293\pi$
$677$	21.5198	0.827074	0.413537	+	0.910487i	$0.364293\pi$
$678$	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	$-0.585281\pi$
$683$	−13.8368	−0.529449	−0.264724	+	0.964324i	$0.585281\pi$
$684$	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	$-0.468226\pi$
$743$	5.43275	0.199308	0.0996540	+	0.995022i	$0.468226\pi$
$744$	−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	$-0.363750\pi$
$751$	22.7506	0.830180	0.415090	+	0.909780i	$0.363750\pi$
$752$	−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	$-0.632848\pi$
$773$	−22.5394	−0.810686	−0.405343	+	0.914165i	$0.632848\pi$
$774$	−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	$-0.361294\pi$
$787$	23.6827	0.844196	0.422098	+	0.906550i	$0.361294\pi$
$788$	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	$-0.540549\pi$
$797$	−7.17327	−0.254090	−0.127045	+	0.991897i	$0.540549\pi$
$798$	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	$-0.214721\pi$
$811$	44.4815	1.56196	0.780979	+	0.624557i	$0.214721\pi$
$812$	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	$-0.510803\pi$
$827$	−1.95169	−0.0678669	−0.0339334	+	0.999424i	$0.510803\pi$
$828$	−67.9411	−2.36112
$829$	−12.3074	−0.427453	−0.213727	−	0.976893i	$-0.568560\pi$
$829$	−12.3074	−0.427453	−0.213727	+	0.976893i	$0.568560\pi$
$830$	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	$-0.605478\pi$
$839$	−18.8472	−0.650677	−0.325338	+	0.945598i	$0.605478\pi$
$840$	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	$-0.716945\pi$
$857$	−36.8859	−1.26000	−0.630000	+	0.776595i	$0.716945\pi$
$858$	−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	$-0.364394\pi$
$863$	24.2799	0.826498	0.413249	+	0.910618i	$0.364394\pi$
$864$	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	$-0.632692\pi$
$877$	−23.9813	−0.809791	−0.404895	+	0.914363i	$0.632692\pi$
$878$	−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	$-0.321594\pi$
$887$	31.6644	1.06319	0.531593	+	0.847000i	$0.321594\pi$
$888$	−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	$-0.314766\pi$
$907$	33.1063	1.09928	0.549638	+	0.835403i	$0.314766\pi$
$908$	−53.6940	−1.78190
$909$	−96.0833	−3.18688
$910$	0	0
$911$	56.3666	1.86751	0.933754	−	0.357914i	$-0.116512\pi$
$911$	56.3666	1.86751	0.933754	+	0.357914i	$0.116512\pi$
$912$	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	$-0.654911\pi$
$941$	−28.6931	−0.935368	−0.467684	+	0.883896i	$0.654911\pi$
$942$	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	$-0.353276\pi$
$953$	27.4624	0.889593	0.444796	+	0.895632i	$0.353276\pi$
$954$	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	$-0.231752\pi$
$971$	46.5207	1.49292	0.746460	+	0.665430i	$0.231752\pi$
$972$	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	$-0.609576\pi$
$977$	−21.0975	−0.674969	−0.337484	+	0.941331i	$0.609576\pi$
$978$	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	$-0.856655\pi$
$983$	−56.4541	−1.80061	−0.900303	+	0.435265i	$0.856655\pi$
$984$	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	$-0.434292\pi$
$991$	12.9046	0.409930	0.204965	+	0.978769i	$0.434292\pi$
$992$	−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.1.4		4
5.4	even	2		1805.2.a.m.1.1	✓	4
19.18	odd	2	inner	9025.2.a.bn.1.1		4
95.94	odd	2		1805.2.a.m.1.4	yes	4

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.10100 q^{2} -2.97127 q^{3} +2.41421 q^{4} -6.24264 q^{6} -4.82843 q^{7} +0.870264 q^{8} +5.82843 q^{9} +O(q^{10})\)	\(q+2.10100 q^{2} -2.97127 q^{3} +2.41421 q^{4} -6.24264 q^{6} -4.82843 q^{7} +0.870264 q^{8} +5.82843 q^{9} +2.00000 q^{11} -7.17327 q^{12} +1.23074 q^{13} -10.1445 q^{14} -3.00000 q^{16} +3.65685 q^{17} +12.2455 q^{18} +14.3465 q^{21} +4.20201 q^{22} -4.82843 q^{23} -2.58579 q^{24} +2.58579 q^{26} -8.40401 q^{27} -11.6569 q^{28} -2.46148 q^{29} +5.94253 q^{31} -8.04354 q^{32} -5.94253 q^{33} +7.68306 q^{34} +14.0711 q^{36} +7.17327 q^{37} -3.65685 q^{39} +30.1421 q^{42} -0.828427 q^{43} +4.82843 q^{44} -10.1445 q^{46} +6.48528 q^{47} +8.91380 q^{48} +16.3137 q^{49} -10.8655 q^{51} +2.97127 q^{52} +4.71179 q^{53} -17.6569 q^{54} -4.20201 q^{56} -5.17157 q^{58} -11.8851 q^{59} +2.82843 q^{61} +12.4853 q^{62} -28.1421 q^{63} -10.8995 q^{64} -12.4853 q^{66} +5.43275 q^{67} +8.82843 q^{68} +14.3465 q^{69} -10.8655 q^{71} +5.07227 q^{72} +0.343146 q^{73} +15.0711 q^{74} -9.65685 q^{77} -7.68306 q^{78} +11.8851 q^{79} +7.48528 q^{81} -3.17157 q^{83} +34.6356 q^{84} -1.74053 q^{86} +7.31371 q^{87} +1.74053 q^{88} -9.42359 q^{89} -5.94253 q^{91} -11.6569 q^{92} -17.6569 q^{93} +13.6256 q^{94} +23.8995 q^{96} +1.23074 q^{97} +34.2752 q^{98} +11.6569 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} - 8 q^{6} - 8 q^{7} + 12 q^{9}+O(q^{10}) \) 4 * q + 4 * q^4 - 8 * q^6 - 8 * q^7 + 12 * q^9	\( 4 q + 4 q^{4} - 8 q^{6} - 8 q^{7} + 12 q^{9} + 8 q^{11} - 12 q^{16} - 8 q^{17} - 8 q^{23} - 16 q^{24} + 16 q^{26} - 24 q^{28} + 28 q^{36} + 8 q^{39} + 64 q^{42} + 8 q^{43} + 8 q^{44} - 8 q^{47} + 20 q^{49} - 48 q^{54} - 32 q^{58} + 16 q^{62} - 56 q^{63} - 4 q^{64} - 16 q^{66} + 24 q^{68} + 24 q^{73} + 32 q^{74} - 16 q^{77} - 4 q^{81} - 24 q^{83} - 16 q^{87} - 24 q^{92} - 48 q^{93} + 56 q^{96} + 24 q^{99}+O(q^{100}) \) 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

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Database

Properties

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