LMFDB - Embedded newform 495.2.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,2,Mod(1,495)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("495.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	495.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:	$3.95259490005$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	495.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.70928 q^{2} +5.34017 q^{4} -1.00000 q^{5} +1.07838 q^{7} +9.04945 q^{8} +O(q^{10})$	\(q+2.70928 q^{2} +5.34017 q^{4} -1.00000 q^{5} +1.07838 q^{7} +9.04945 q^{8} -2.70928 q^{10} -1.00000 q^{11} -4.34017 q^{13} +2.92162 q^{14} +13.8371 q^{16} -7.75872 q^{17} +5.26180 q^{19} -5.34017 q^{20} -2.70928 q^{22} +2.15676 q^{23} +1.00000 q^{25} -11.7587 q^{26} +5.75872 q^{28} -1.41855 q^{29} -4.68035 q^{31} +19.3896 q^{32} -21.0205 q^{34} -1.07838 q^{35} -2.00000 q^{37} +14.2557 q^{38} -9.04945 q^{40} +9.41855 q^{41} +7.60197 q^{43} -5.34017 q^{44} +5.84324 q^{46} -4.68035 q^{47} -5.83710 q^{49} +2.70928 q^{50} -23.1773 q^{52} -0.156755 q^{53} +1.00000 q^{55} +9.75872 q^{56} -3.84324 q^{58} -6.15676 q^{59} -4.15676 q^{61} -12.6803 q^{62} +24.8576 q^{64} +4.34017 q^{65} -8.68035 q^{67} -41.4329 q^{68} -2.92162 q^{70} +4.68035 q^{71} -10.4969 q^{73} -5.41855 q^{74} +28.0989 q^{76} -1.07838 q^{77} -8.09890 q^{79} -13.8371 q^{80} +25.5174 q^{82} +11.0205 q^{83} +7.75872 q^{85} +20.5958 q^{86} -9.04945 q^{88} +12.8371 q^{89} -4.68035 q^{91} +11.5174 q^{92} -12.6803 q^{94} -5.26180 q^{95} +14.6803 q^{97} -15.8143 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 5 q^{4} - 3 q^{5} + 9 q^{8}+O(q^{10}) $ 3 * q + q^2 + 5 * q^4 - 3 * q^5 + 9 * q^8	$ 3 q + q^{2} + 5 q^{4} - 3 q^{5} + 9 q^{8} - q^{10} - 3 q^{11} - 2 q^{13} + 12 q^{14} + 13 q^{16} + 2 q^{17} + 8 q^{19} - 5 q^{20} - q^{22} + 3 q^{25} - 10 q^{26} - 8 q^{28} + 10 q^{29} + 8 q^{31} + 29 q^{32} - 30 q^{34} - 6 q^{37} - 9 q^{40} + 14 q^{41} + 4 q^{43} - 5 q^{44} + 24 q^{46} + 8 q^{47} + 11 q^{49} + q^{50} - 30 q^{52} + 6 q^{53} + 3 q^{55} + 4 q^{56} - 18 q^{58} - 12 q^{59} - 6 q^{61} - 16 q^{62} + 13 q^{64} + 2 q^{65} - 4 q^{67} - 42 q^{68} - 12 q^{70} - 8 q^{71} - 14 q^{73} - 2 q^{74} + 48 q^{76} + 12 q^{79} - 13 q^{80} + 26 q^{82} - 2 q^{85} + 8 q^{86} - 9 q^{88} + 10 q^{89} + 8 q^{91} - 16 q^{92} - 16 q^{94} - 8 q^{95} + 22 q^{97} - 39 q^{98}+O(q^{100}) $ 3 * q + q^2 + 5 * q^4 - 3 * q^5 + 9 * q^8 - q^10 - 3 * q^11 - 2 * q^13 + 12 * q^14 + 13 * q^16 + 2 * q^17 + 8 * q^19 - 5 * q^20 - q^22 + 3 * q^25 - 10 * q^26 - 8 * q^28 + 10 * q^29 + 8 * q^31 + 29 * q^32 - 30 * q^34 - 6 * q^37 - 9 * q^40 + 14 * q^41 + 4 * q^43 - 5 * q^44 + 24 * q^46 + 8 * q^47 + 11 * q^49 + q^50 - 30 * q^52 + 6 * q^53 + 3 * q^55 + 4 * q^56 - 18 * q^58 - 12 * q^59 - 6 * q^61 - 16 * q^62 + 13 * q^64 + 2 * q^65 - 4 * q^67 - 42 * q^68 - 12 * q^70 - 8 * q^71 - 14 * q^73 - 2 * q^74 + 48 * q^76 + 12 * q^79 - 13 * q^80 + 26 * q^82 - 2 * q^85 + 8 * q^86 - 9 * q^88 + 10 * q^89 + 8 * q^91 - 16 * q^92 - 16 * q^94 - 8 * q^95 + 22 * q^97 - 39 * 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 $

$2$	2.70928	1.91575	0.957873	−	0.287190i	$-0.0927213\pi$
$2$	2.70928	1.91575	0.957873	+	0.287190i	$0.0927213\pi$
$3$	0	0
$3$	0	0
$4$	5.34017	2.67009
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.07838	0.407588	0.203794	−	0.979014i	$-0.434673\pi$
$7$	1.07838	0.407588	0.203794	+	0.979014i	$0.434673\pi$
$8$	9.04945	3.19946
$9$	0	0
$10$	−2.70928	−0.856748
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−4.34017	−1.20375	−0.601874	−	0.798591i	$-0.705579\pi$
$13$	−4.34017	−1.20375	−0.601874	+	0.798591i	$0.705579\pi$
$14$	2.92162	0.780836
$15$	0	0
$16$	13.8371	3.45928
$17$	−7.75872	−1.88177	−0.940883	−	0.338730i	$-0.890003\pi$
$17$	−7.75872	−1.88177	−0.940883	+	0.338730i	$0.890003\pi$
$18$	0	0
$19$	5.26180	1.20714	0.603569	−	0.797311i	$-0.293745\pi$
$19$	5.26180	1.20714	0.603569	+	0.797311i	$0.293745\pi$
$20$	−5.34017	−1.19410
$21$	0	0
$22$	−2.70928	−0.577619
$23$	2.15676	0.449715	0.224857	−	0.974392i	$-0.427808\pi$
$23$	2.15676	0.449715	0.224857	+	0.974392i	$0.427808\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−11.7587	−2.30608
$27$	0	0
$28$	5.75872	1.08830
$29$	−1.41855	−0.263418	−0.131709	−	0.991288i	$-0.542046\pi$
$29$	−1.41855	−0.263418	−0.131709	+	0.991288i	$0.542046\pi$
$30$	0	0
$31$	−4.68035	−0.840615	−0.420307	−	0.907382i	$-0.638078\pi$
$31$	−4.68035	−0.840615	−0.420307	+	0.907382i	$0.638078\pi$
$32$	19.3896	3.42763
$33$	0	0
$34$	−21.0205	−3.60499
$35$	−1.07838	−0.182279
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	14.2557	2.31257
$39$	0	0
$40$	−9.04945	−1.43084
$41$	9.41855	1.47093	0.735465	−	0.677562i	$-0.236964\pi$
$41$	9.41855	1.47093	0.735465	+	0.677562i	$0.236964\pi$
$42$	0	0
$43$	7.60197	1.15929	0.579645	−	0.814869i	$-0.303191\pi$
$43$	7.60197	1.15929	0.579645	+	0.814869i	$0.303191\pi$
$44$	−5.34017	−0.805061
$45$	0	0
$46$	5.84324	0.861539
$47$	−4.68035	−0.682699	−0.341349	−	0.939937i	$-0.610884\pi$
$47$	−4.68035	−0.682699	−0.341349	+	0.939937i	$0.610884\pi$
$48$	0	0
$49$	−5.83710	−0.833872
$50$	2.70928	0.383149
$51$	0	0
$52$	−23.1773	−3.21411
$53$	−0.156755	−0.0215320	−0.0107660	−	0.999942i	$-0.503427\pi$
$53$	−0.156755	−0.0215320	−0.0107660	+	0.999942i	$0.503427\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	9.75872	1.30406
$57$	0	0
$58$	−3.84324	−0.504643
$59$	−6.15676	−0.801541	−0.400771	−	0.916178i	$-0.631258\pi$
$59$	−6.15676	−0.801541	−0.400771	+	0.916178i	$0.631258\pi$
$60$	0	0
$61$	−4.15676	−0.532218	−0.266109	−	0.963943i	$-0.585738\pi$
$61$	−4.15676	−0.532218	−0.266109	+	0.963943i	$0.585738\pi$
$62$	−12.6803	−1.61041
$63$	0	0
$64$	24.8576	3.10720
$65$	4.34017	0.538332
$66$	0	0
$67$	−8.68035	−1.06047	−0.530237	−	0.847850i	$-0.677897\pi$
$67$	−8.68035	−1.06047	−0.530237	+	0.847850i	$0.677897\pi$
$68$	−41.4329	−5.02448
$69$	0	0
$70$	−2.92162	−0.349201
$71$	4.68035	0.555455	0.277727	−	0.960660i	$-0.410419\pi$
$71$	4.68035	0.555455	0.277727	+	0.960660i	$0.410419\pi$
$72$	0	0
$73$	−10.4969	−1.22857	−0.614286	−	0.789083i	$-0.710556\pi$
$73$	−10.4969	−1.22857	−0.614286	+	0.789083i	$0.710556\pi$
$74$	−5.41855	−0.629894
$75$	0	0
$76$	28.0989	3.22316
$77$	−1.07838	−0.122893
$78$	0	0
$79$	−8.09890	−0.911197	−0.455599	−	0.890185i	$-0.650575\pi$
$79$	−8.09890	−0.911197	−0.455599	+	0.890185i	$0.650575\pi$
$80$	−13.8371	−1.54703
$81$	0	0
$82$	25.5174	2.81793
$83$	11.0205	1.20966	0.604830	−	0.796355i	$-0.293241\pi$
$83$	11.0205	1.20966	0.604830	+	0.796355i	$0.293241\pi$
$84$	0	0
$85$	7.75872	0.841552
$86$	20.5958	2.22090
$87$	0	0
$88$	−9.04945	−0.964674
$89$	12.8371	1.36073	0.680365	−	0.732873i	$-0.261821\pi$
$89$	12.8371	1.36073	0.680365	+	0.732873i	$0.261821\pi$
$90$	0	0
$91$	−4.68035	−0.490634
$92$	11.5174	1.20078
$93$	0	0
$94$	−12.6803	−1.30788
$95$	−5.26180	−0.539849
$96$	0	0
$97$	14.6803	1.49056	0.745282	−	0.666750i	$-0.232315\pi$
$97$	14.6803	1.49056	0.745282	+	0.666750i	$0.232315\pi$
$98$	−15.8143	−1.59749
$99$	0	0
$100$	5.34017	0.534017
$101$	15.5753	1.54980	0.774900	−	0.632083i	$-0.217800\pi$
$101$	15.5753	1.54980	0.774900	+	0.632083i	$0.217800\pi$
$102$	0	0
$103$	6.83710	0.673680	0.336840	−	0.941562i	$-0.390642\pi$
$103$	6.83710	0.673680	0.336840	+	0.941562i	$0.390642\pi$
$104$	−39.2762	−3.85135
$105$	0	0
$106$	−0.424694	−0.0412499
$107$	−6.34017	−0.612928	−0.306464	−	0.951882i	$-0.599146\pi$
$107$	−6.34017	−0.612928	−0.306464	+	0.951882i	$0.599146\pi$
$108$	0	0
$109$	2.31351	0.221594	0.110797	−	0.993843i	$-0.464660\pi$
$109$	2.31351	0.221594	0.110797	+	0.993843i	$0.464660\pi$
$110$	2.70928	0.258319
$111$	0	0
$112$	14.9216	1.40996
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	−2.15676	−0.201118
$116$	−7.57531	−0.703350
$117$	0	0
$118$	−16.6803	−1.53555
$119$	−8.36683	−0.766987
$120$	0	0
$121$	1.00000	0.0909091
$122$	−11.2618	−1.01960
$123$	0	0
$124$	−24.9939	−2.24451
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−2.24128	−0.198881	−0.0994406	−	0.995044i	$-0.531705\pi$
$127$	−2.24128	−0.198881	−0.0994406	+	0.995044i	$0.531705\pi$
$128$	28.5669	2.52498
$129$	0	0
$130$	11.7587	1.03131
$131$	−8.68035	−0.758405	−0.379203	−	0.925314i	$-0.623802\pi$
$131$	−8.68035	−0.758405	−0.379203	+	0.925314i	$0.623802\pi$
$132$	0	0
$133$	5.67420	0.492016
$134$	−23.5174	−2.03160
$135$	0	0
$136$	−70.2122	−6.02064
$137$	15.3607	1.31235	0.656176	−	0.754608i	$-0.272173\pi$
$137$	15.3607	1.31235	0.656176	+	0.754608i	$0.272173\pi$
$138$	0	0
$139$	8.58145	0.727869	0.363935	−	0.931425i	$-0.381433\pi$
$139$	8.58145	0.727869	0.363935	+	0.931425i	$0.381433\pi$
$140$	−5.75872	−0.486701
$141$	0	0
$142$	12.6803	1.06411
$143$	4.34017	0.362943
$144$	0	0
$145$	1.41855	0.117804
$146$	−28.4391	−2.35363
$147$	0	0
$148$	−10.6803	−0.877919
$149$	18.0989	1.48272	0.741360	−	0.671108i	$-0.234181\pi$
$149$	18.0989	1.48272	0.741360	+	0.671108i	$0.234181\pi$
$150$	0	0
$151$	22.9360	1.86651	0.933253	−	0.359221i	$-0.116958\pi$
$151$	22.9360	1.86651	0.933253	+	0.359221i	$0.116958\pi$
$152$	47.6163	3.86220
$153$	0	0
$154$	−2.92162	−0.235431
$155$	4.68035	0.375934
$156$	0	0
$157$	−10.9939	−0.877405	−0.438703	−	0.898632i	$-0.644562\pi$
$157$	−10.9939	−0.877405	−0.438703	+	0.898632i	$0.644562\pi$
$158$	−21.9421	−1.74562
$159$	0	0
$160$	−19.3896	−1.53288
$161$	2.32580	0.183298
$162$	0	0
$163$	−6.52359	−0.510967	−0.255484	−	0.966813i	$-0.582235\pi$
$163$	−6.52359	−0.510967	−0.255484	+	0.966813i	$0.582235\pi$
$164$	50.2967	3.92751
$165$	0	0
$166$	29.8576	2.31740
$167$	−1.97334	−0.152701	−0.0763507	−	0.997081i	$-0.524327\pi$
$167$	−1.97334	−0.152701	−0.0763507	+	0.997081i	$0.524327\pi$
$168$	0	0
$169$	5.83710	0.449008
$170$	21.0205	1.61220
$171$	0	0
$172$	40.5958	3.09540
$173$	−3.75872	−0.285770	−0.142885	−	0.989739i	$-0.545638\pi$
$173$	−3.75872	−0.285770	−0.142885	+	0.989739i	$0.545638\pi$
$174$	0	0
$175$	1.07838	0.0815177
$176$	−13.8371	−1.04301
$177$	0	0
$178$	34.7792	2.60681
$179$	−15.1506	−1.13241	−0.566205	−	0.824264i	$-0.691589\pi$
$179$	−15.1506	−1.13241	−0.566205	+	0.824264i	$0.691589\pi$
$180$	0	0
$181$	4.83710	0.359539	0.179769	−	0.983709i	$-0.442465\pi$
$181$	4.83710	0.359539	0.179769	+	0.983709i	$0.442465\pi$
$182$	−12.6803	−0.939930
$183$	0	0
$184$	19.5174	1.43885
$185$	2.00000	0.147043
$186$	0	0
$187$	7.75872	0.567374
$188$	−24.9939	−1.82286
$189$	0	0
$190$	−14.2557	−1.03421
$191$	−2.52359	−0.182601	−0.0913003	−	0.995823i	$-0.529102\pi$
$191$	−2.52359	−0.182601	−0.0913003	+	0.995823i	$0.529102\pi$
$192$	0	0
$193$	0.0266620	0.00191917	0.000959586	−	1.00000i	$-0.499695\pi$
$193$	0.0266620	0.00191917	0.000959586		1.00000i	$0.499695\pi$
$194$	39.7731	2.85554
$195$	0	0
$196$	−31.1711	−2.22651
$197$	−21.1194	−1.50470	−0.752348	−	0.658766i	$-0.771079\pi$
$197$	−21.1194	−1.50470	−0.752348	+	0.658766i	$0.771079\pi$
$198$	0	0
$199$	10.5236	0.745998	0.372999	−	0.927832i	$-0.378330\pi$
$199$	10.5236	0.745998	0.372999	+	0.927832i	$0.378330\pi$
$200$	9.04945	0.639893
$201$	0	0
$202$	42.1978	2.96903
$203$	−1.52973	−0.107366
$204$	0	0
$205$	−9.41855	−0.657820
$206$	18.5236	1.29060
$207$	0	0
$208$	−60.0554	−4.16409
$209$	−5.26180	−0.363966
$210$	0	0
$211$	9.57531	0.659191	0.329596	−	0.944122i	$-0.393088\pi$
$211$	9.57531	0.659191	0.329596	+	0.944122i	$0.393088\pi$
$212$	−0.837101	−0.0574924
$213$	0	0
$214$	−17.1773	−1.17421
$215$	−7.60197	−0.518450
$216$	0	0
$217$	−5.04718	−0.342625
$218$	6.26794	0.424518
$219$	0	0
$220$	5.34017	0.360034
$221$	33.6742	2.26517
$222$	0	0
$223$	−2.15676	−0.144427	−0.0722135	−	0.997389i	$-0.523006\pi$
$223$	−2.15676	−0.144427	−0.0722135	+	0.997389i	$0.523006\pi$
$224$	20.9093	1.39706
$225$	0	0
$226$	16.2557	1.08131
$227$	−9.65983	−0.641145	−0.320573	−	0.947224i	$-0.603875\pi$
$227$	−9.65983	−0.641145	−0.320573	+	0.947224i	$0.603875\pi$
$228$	0	0
$229$	−3.36069	−0.222081	−0.111040	−	0.993816i	$-0.535418\pi$
$229$	−3.36069	−0.222081	−0.111040	+	0.993816i	$0.535418\pi$
$230$	−5.84324	−0.385292
$231$	0	0
$232$	−12.8371	−0.842797
$233$	2.39803	0.157100	0.0785501	−	0.996910i	$-0.474971\pi$
$233$	2.39803	0.157100	0.0785501	+	0.996910i	$0.474971\pi$
$234$	0	0
$235$	4.68035	0.305312
$236$	−32.8781	−2.14018
$237$	0	0
$238$	−22.6681	−1.46935
$239$	7.20394	0.465984	0.232992	−	0.972479i	$-0.425148\pi$
$239$	7.20394	0.465984	0.232992	+	0.972479i	$0.425148\pi$
$240$	0	0
$241$	−5.20394	−0.335215	−0.167608	−	0.985854i	$-0.553604\pi$
$241$	−5.20394	−0.335215	−0.167608	+	0.985854i	$0.553604\pi$
$242$	2.70928	0.174159
$243$	0	0
$244$	−22.1978	−1.42107
$245$	5.83710	0.372919
$246$	0	0
$247$	−22.8371	−1.45309
$248$	−42.3545	−2.68952
$249$	0	0
$250$	−2.70928	−0.171350
$251$	−15.3197	−0.966968	−0.483484	−	0.875353i	$-0.660629\pi$
$251$	−15.3197	−0.966968	−0.483484	+	0.875353i	$0.660629\pi$
$252$	0	0
$253$	−2.15676	−0.135594
$254$	−6.07223	−0.381006
$255$	0	0
$256$	27.6803	1.73002
$257$	−4.15676	−0.259291	−0.129646	−	0.991560i	$-0.541384\pi$
$257$	−4.15676	−0.259291	−0.129646	+	0.991560i	$0.541384\pi$
$258$	0	0
$259$	−2.15676	−0.134014
$260$	23.1773	1.43739
$261$	0	0
$262$	−23.5174	−1.45291
$263$	18.7070	1.15352	0.576762	−	0.816912i	$-0.304316\pi$
$263$	18.7070	1.15352	0.576762	+	0.816912i	$0.304316\pi$
$264$	0	0
$265$	0.156755	0.00962941
$266$	15.3730	0.942578
$267$	0	0
$268$	−46.3545	−2.83155
$269$	−23.3607	−1.42433	−0.712163	−	0.702014i	$-0.752284\pi$
$269$	−23.3607	−1.42433	−0.712163	+	0.702014i	$0.752284\pi$
$270$	0	0
$271$	−5.57531	−0.338676	−0.169338	−	0.985558i	$-0.554163\pi$
$271$	−5.57531	−0.338676	−0.169338	+	0.985558i	$0.554163\pi$
$272$	−107.358	−6.50955
$273$	0	0
$274$	41.6163	2.51414
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−26.0144	−1.56305	−0.781526	−	0.623872i	$-0.785558\pi$
$277$	−26.0144	−1.56305	−0.781526	+	0.623872i	$0.785558\pi$
$278$	23.2495	1.39441
$279$	0	0
$280$	−9.75872	−0.583195
$281$	9.41855	0.561864	0.280932	−	0.959728i	$-0.409357\pi$
$281$	9.41855	0.561864	0.280932	+	0.959728i	$0.409357\pi$
$282$	0	0
$283$	14.2413	0.846556	0.423278	−	0.906000i	$-0.360879\pi$
$283$	14.2413	0.846556	0.423278	+	0.906000i	$0.360879\pi$
$284$	24.9939	1.48311
$285$	0	0
$286$	11.7587	0.695308
$287$	10.1568	0.599534
$288$	0	0
$289$	43.1978	2.54105
$290$	3.84324	0.225683
$291$	0	0
$292$	−56.0554	−3.28039
$293$	15.7587	0.920634	0.460317	−	0.887754i	$-0.347736\pi$
$293$	15.7587	0.920634	0.460317	+	0.887754i	$0.347736\pi$
$294$	0	0
$295$	6.15676	0.358460
$296$	−18.0989	−1.05198
$297$	0	0
$298$	49.0349	2.84052
$299$	−9.36069	−0.541343
$300$	0	0
$301$	8.19779	0.472513
$302$	62.1399	3.57575
$303$	0	0
$304$	72.8080	4.17582
$305$	4.15676	0.238015
$306$	0	0
$307$	−18.9216	−1.07991	−0.539957	−	0.841693i	$-0.681560\pi$
$307$	−18.9216	−1.07991	−0.539957	+	0.841693i	$0.681560\pi$
$308$	−5.75872	−0.328134
$309$	0	0
$310$	12.6803	0.720195
$311$	20.8781	1.18389	0.591945	−	0.805978i	$-0.298360\pi$
$311$	20.8781	1.18389	0.591945	+	0.805978i	$0.298360\pi$
$312$	0	0
$313$	6.31351	0.356861	0.178430	−	0.983953i	$-0.442898\pi$
$313$	6.31351	0.356861	0.178430	+	0.983953i	$0.442898\pi$
$314$	−29.7854	−1.68089
$315$	0	0
$316$	−43.2495	−2.43297
$317$	−31.3607	−1.76139	−0.880696	−	0.473682i	$-0.842925\pi$
$317$	−31.3607	−1.76139	−0.880696	+	0.473682i	$0.842925\pi$
$318$	0	0
$319$	1.41855	0.0794236
$320$	−24.8576	−1.38958
$321$	0	0
$322$	6.30122	0.351154
$323$	−40.8248	−2.27155
$324$	0	0
$325$	−4.34017	−0.240749
$326$	−17.6742	−0.978884
$327$	0	0
$328$	85.2327	4.70619
$329$	−5.04718	−0.278260
$330$	0	0
$331$	19.2039	1.05554	0.527772	−	0.849386i	$-0.323028\pi$
$331$	19.2039	1.05554	0.527772	+	0.849386i	$0.323028\pi$
$332$	58.8515	3.22989
$333$	0	0
$334$	−5.34632	−0.292537
$335$	8.68035	0.474258
$336$	0	0
$337$	13.5031	0.735559	0.367780	−	0.929913i	$-0.380118\pi$
$337$	13.5031	0.735559	0.367780	+	0.929913i	$0.380118\pi$
$338$	15.8143	0.860185
$339$	0	0
$340$	41.4329	2.24702
$341$	4.68035	0.253455
$342$	0	0
$343$	−13.8432	−0.747465
$344$	68.7936	3.70910
$345$	0	0
$346$	−10.1834	−0.547464
$347$	−6.34017	−0.340358	−0.170179	−	0.985413i	$-0.554435\pi$
$347$	−6.34017	−0.340358	−0.170179	+	0.985413i	$0.554435\pi$
$348$	0	0
$349$	16.1568	0.864851	0.432426	−	0.901670i	$-0.357658\pi$
$349$	16.1568	0.864851	0.432426	+	0.901670i	$0.357658\pi$
$350$	2.92162	0.156167
$351$	0	0
$352$	−19.3896	−1.03347
$353$	13.2039	0.702775	0.351387	−	0.936230i	$-0.385710\pi$
$353$	13.2039	0.702775	0.351387	+	0.936230i	$0.385710\pi$
$354$	0	0
$355$	−4.68035	−0.248407
$356$	68.5523	3.63327
$357$	0	0
$358$	−41.0472	−2.16941
$359$	−3.31965	−0.175205	−0.0876023	−	0.996156i	$-0.527920\pi$
$359$	−3.31965	−0.175205	−0.0876023	+	0.996156i	$0.527920\pi$
$360$	0	0
$361$	8.68649	0.457184
$362$	13.1050	0.688786
$363$	0	0
$364$	−24.9939	−1.31003
$365$	10.4969	0.549434
$366$	0	0
$367$	−36.1445	−1.88673	−0.943363	−	0.331762i	$-0.892357\pi$
$367$	−36.1445	−1.88673	−0.943363	+	0.331762i	$0.892357\pi$
$368$	29.8432	1.55569
$369$	0	0
$370$	5.41855	0.281697
$371$	−0.169042	−0.00877620
$372$	0	0
$373$	−2.81044	−0.145519	−0.0727595	−	0.997350i	$-0.523181\pi$
$373$	−2.81044	−0.145519	−0.0727595	+	0.997350i	$0.523181\pi$
$374$	21.0205	1.08695
$375$	0	0
$376$	−42.3545	−2.18427
$377$	6.15676	0.317089
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	−28.0989	−1.44144
$381$	0	0
$382$	−6.83710	−0.349817
$383$	−33.5585	−1.71476	−0.857379	−	0.514685i	$-0.827909\pi$
$383$	−33.5585	−1.71476	−0.857379	+	0.514685i	$0.827909\pi$
$384$	0	0
$385$	1.07838	0.0549592
$386$	0.0722347	0.00367665
$387$	0	0
$388$	78.3956	3.97993
$389$	−12.8371	−0.650867	−0.325433	−	0.945565i	$-0.605510\pi$
$389$	−12.8371	−0.650867	−0.325433	+	0.945565i	$0.605510\pi$
$390$	0	0
$391$	−16.7337	−0.846258
$392$	−52.8225	−2.66794
$393$	0	0
$394$	−57.2183	−2.88262
$395$	8.09890	0.407500
$396$	0	0
$397$	−5.31965	−0.266986	−0.133493	−	0.991050i	$-0.542619\pi$
$397$	−5.31965	−0.266986	−0.133493	+	0.991050i	$0.542619\pi$
$398$	28.5113	1.42914
$399$	0	0
$400$	13.8371	0.691855
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	0	0
$403$	20.3135	1.01189
$404$	83.1748	4.13810
$405$	0	0
$406$	−4.14447	−0.205687
$407$	2.00000	0.0991363
$408$	0	0
$409$	26.1978	1.29540	0.647699	−	0.761897i	$-0.275732\pi$
$409$	26.1978	1.29540	0.647699	+	0.761897i	$0.275732\pi$
$410$	−25.5174	−1.26022
$411$	0	0
$412$	36.5113	1.79878
$413$	−6.63931	−0.326699
$414$	0	0
$415$	−11.0205	−0.540976
$416$	−84.1543	−4.12600
$417$	0	0
$418$	−14.2557	−0.697267
$419$	2.83710	0.138601	0.0693007	−	0.997596i	$-0.477923\pi$
$419$	2.83710	0.138601	0.0693007	+	0.997596i	$0.477923\pi$
$420$	0	0
$421$	11.4764	0.559326	0.279663	−	0.960098i	$-0.409777\pi$
$421$	11.4764	0.559326	0.279663	+	0.960098i	$0.409777\pi$
$422$	25.9421	1.26284
$423$	0	0
$424$	−1.41855	−0.0688909
$425$	−7.75872	−0.376353
$426$	0	0
$427$	−4.48255	−0.216926
$428$	−33.8576	−1.63657
$429$	0	0
$430$	−20.5958	−0.993219
$431$	−23.5708	−1.13536	−0.567682	−	0.823248i	$-0.692160\pi$
$431$	−23.5708	−1.13536	−0.567682	+	0.823248i	$0.692160\pi$
$432$	0	0
$433$	−14.9939	−0.720559	−0.360279	−	0.932844i	$-0.617319\pi$
$433$	−14.9939	−0.720559	−0.360279	+	0.932844i	$0.617319\pi$
$434$	−13.6742	−0.656383
$435$	0	0
$436$	12.3545	0.591676
$437$	11.3484	0.542868
$438$	0	0
$439$	4.77924	0.228101	0.114050	−	0.993475i	$-0.463617\pi$
$439$	4.77924	0.228101	0.114050	+	0.993475i	$0.463617\pi$
$440$	9.04945	0.431416
$441$	0	0
$442$	91.2327	4.33950
$443$	−20.1978	−0.959626	−0.479813	−	0.877371i	$-0.659296\pi$
$443$	−20.1978	−0.959626	−0.479813	+	0.877371i	$0.659296\pi$
$444$	0	0
$445$	−12.8371	−0.608537
$446$	−5.84324	−0.276686
$447$	0	0
$448$	26.8059	1.26646
$449$	21.5708	1.01799	0.508994	−	0.860770i	$-0.330018\pi$
$449$	21.5708	1.01799	0.508994	+	0.860770i	$0.330018\pi$
$450$	0	0
$451$	−9.41855	−0.443502
$452$	32.0410	1.50708
$453$	0	0
$454$	−26.1711	−1.22827
$455$	4.68035	0.219418
$456$	0	0
$457$	28.1711	1.31779	0.658895	−	0.752235i	$-0.271024\pi$
$457$	28.1711	1.31779	0.658895	+	0.752235i	$0.271024\pi$
$458$	−9.10504	−0.425451
$459$	0	0
$460$	−11.5174	−0.537004
$461$	1.47187	0.0685520	0.0342760	−	0.999412i	$-0.489087\pi$
$461$	1.47187	0.0685520	0.0342760	+	0.999412i	$0.489087\pi$
$462$	0	0
$463$	−23.2039	−1.07838	−0.539189	−	0.842185i	$-0.681269\pi$
$463$	−23.2039	−1.07838	−0.539189	+	0.842185i	$0.681269\pi$
$464$	−19.6286	−0.911236
$465$	0	0
$466$	6.49693	0.300964
$467$	−14.1568	−0.655097	−0.327548	−	0.944834i	$-0.606222\pi$
$467$	−14.1568	−0.655097	−0.327548	+	0.944834i	$0.606222\pi$
$468$	0	0
$469$	−9.36069	−0.432237
$470$	12.6803	0.584901
$471$	0	0
$472$	−55.7152	−2.56450
$473$	−7.60197	−0.349539
$474$	0	0
$475$	5.26180	0.241428
$476$	−44.6803	−2.04792
$477$	0	0
$478$	19.5174	0.892707
$479$	13.8432	0.632514	0.316257	−	0.948674i	$-0.397574\pi$
$479$	13.8432	0.632514	0.316257	+	0.948674i	$0.397574\pi$
$480$	0	0
$481$	8.68035	0.395790
$482$	−14.0989	−0.642187
$483$	0	0
$484$	5.34017	0.242735
$485$	−14.6803	−0.666600
$486$	0	0
$487$	−40.9939	−1.85761	−0.928804	−	0.370570i	$-0.879162\pi$
$487$	−40.9939	−1.85761	−0.928804	+	0.370570i	$0.879162\pi$
$488$	−37.6163	−1.70281
$489$	0	0
$490$	15.8143	0.714418
$491$	−34.8371	−1.57218	−0.786088	−	0.618114i	$-0.787897\pi$
$491$	−34.8371	−1.57218	−0.786088	+	0.618114i	$0.787897\pi$
$492$	0	0
$493$	11.0061	0.495692
$494$	−61.8720	−2.78375
$495$	0	0
$496$	−64.7624	−2.90792
$497$	5.04718	0.226397
$498$	0	0
$499$	15.1506	0.678235	0.339117	−	0.940744i	$-0.389872\pi$
$499$	15.1506	0.678235	0.339117	+	0.940744i	$0.389872\pi$
$500$	−5.34017	−0.238820
$501$	0	0
$502$	−41.5052	−1.85247
$503$	6.65368	0.296673	0.148337	−	0.988937i	$-0.452608\pi$
$503$	6.65368	0.296673	0.148337	+	0.988937i	$0.452608\pi$
$504$	0	0
$505$	−15.5753	−0.693092
$506$	−5.84324	−0.259764
$507$	0	0
$508$	−11.9688	−0.531030
$509$	−41.3484	−1.83274	−0.916368	−	0.400337i	$-0.868893\pi$
$509$	−41.3484	−1.83274	−0.916368	+	0.400337i	$0.868893\pi$
$510$	0	0
$511$	−11.3197	−0.500752
$512$	17.8599	0.789303
$513$	0	0
$514$	−11.2618	−0.496736
$515$	−6.83710	−0.301279
$516$	0	0
$517$	4.68035	0.205841
$518$	−5.84324	−0.256737
$519$	0	0
$520$	39.2762	1.72237
$521$	−7.67420	−0.336213	−0.168106	−	0.985769i	$-0.553765\pi$
$521$	−7.67420	−0.336213	−0.168106	+	0.985769i	$0.553765\pi$
$522$	0	0
$523$	23.2351	1.01600	0.508001	−	0.861357i	$-0.330385\pi$
$523$	23.2351	1.01600	0.508001	+	0.861357i	$0.330385\pi$
$524$	−46.3545	−2.02501
$525$	0	0
$526$	50.6824	2.20986
$527$	36.3135	1.58184
$528$	0	0
$529$	−18.3484	−0.797757
$530$	0.424694	0.0184475
$531$	0	0
$532$	30.3012	1.31372
$533$	−40.8781	−1.77063
$534$	0	0
$535$	6.34017	0.274110
$536$	−78.5523	−3.39294
$537$	0	0
$538$	−63.2905	−2.72865
$539$	5.83710	0.251422
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	−15.1050	−0.648817
$543$	0	0
$544$	−150.439	−6.45001
$545$	−2.31351	−0.0990999
$546$	0	0
$547$	−23.0661	−0.986235	−0.493117	−	0.869963i	$-0.664143\pi$
$547$	−23.0661	−0.986235	−0.493117	+	0.869963i	$0.664143\pi$
$548$	82.0288	3.50409
$549$	0	0
$550$	−2.70928	−0.115524
$551$	−7.46412	−0.317982
$552$	0	0
$553$	−8.73367	−0.371393
$554$	−70.4801	−2.99441
$555$	0	0
$556$	45.8264	1.94347
$557$	−10.5958	−0.448960	−0.224480	−	0.974479i	$-0.572068\pi$
$557$	−10.5958	−0.448960	−0.224480	+	0.974479i	$0.572068\pi$
$558$	0	0
$559$	−32.9939	−1.39549
$560$	−14.9216	−0.630554
$561$	0	0
$562$	25.5174	1.07639
$563$	36.2122	1.52616	0.763080	−	0.646303i	$-0.223686\pi$
$563$	36.2122	1.52616	0.763080	+	0.646303i	$0.223686\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	38.5835	1.62179
$567$	0	0
$568$	42.3545	1.77716
$569$	27.5753	1.15602	0.578008	−	0.816031i	$-0.303830\pi$
$569$	27.5753	1.15602	0.578008	+	0.816031i	$0.303830\pi$
$570$	0	0
$571$	−27.9299	−1.16883	−0.584414	−	0.811456i	$-0.698676\pi$
$571$	−27.9299	−1.16883	−0.584414	+	0.811456i	$0.698676\pi$
$572$	23.1773	0.969091
$573$	0	0
$574$	27.5174	1.14856
$575$	2.15676	0.0899429
$576$	0	0
$577$	41.4017	1.72358	0.861788	−	0.507268i	$-0.169345\pi$
$577$	41.4017	1.72358	0.861788	+	0.507268i	$0.169345\pi$
$578$	117.035	4.86800
$579$	0	0
$580$	7.57531	0.314547
$581$	11.8843	0.493043
$582$	0	0
$583$	0.156755	0.00649215
$584$	−94.9914	−3.93077
$585$	0	0
$586$	42.6947	1.76370
$587$	−8.48255	−0.350112	−0.175056	−	0.984558i	$-0.556011\pi$
$587$	−8.48255	−0.350112	−0.175056	+	0.984558i	$0.556011\pi$
$588$	0	0
$589$	−24.6270	−1.01474
$590$	16.6803	0.686719
$591$	0	0
$592$	−27.6742	−1.13740
$593$	−7.56093	−0.310490	−0.155245	−	0.987876i	$-0.549617\pi$
$593$	−7.56093	−0.310490	−0.155245	+	0.987876i	$0.549617\pi$
$594$	0	0
$595$	8.36683	0.343007
$596$	96.6512	3.95899
$597$	0	0
$598$	−25.3607	−1.03708
$599$	−5.67420	−0.231842	−0.115921	−	0.993258i	$-0.536982\pi$
$599$	−5.67420	−0.231842	−0.115921	+	0.993258i	$0.536982\pi$
$600$	0	0
$601$	−1.31965	−0.0538298	−0.0269149	−	0.999638i	$-0.508568\pi$
$601$	−1.31965	−0.0538298	−0.0269149	+	0.999638i	$0.508568\pi$
$602$	22.2101	0.905215
$603$	0	0
$604$	122.482	4.98373
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−2.24128	−0.0909706	−0.0454853	−	0.998965i	$-0.514483\pi$
$607$	−2.24128	−0.0909706	−0.0454853	+	0.998965i	$0.514483\pi$
$608$	102.024	4.13763
$609$	0	0
$610$	11.2618	0.455977
$611$	20.3135	0.821797
$612$	0	0
$613$	42.8638	1.73125	0.865626	−	0.500692i	$-0.166921\pi$
$613$	42.8638	1.73125	0.865626	+	0.500692i	$0.166921\pi$
$614$	−51.2639	−2.06884
$615$	0	0
$616$	−9.75872	−0.393190
$617$	−11.3607	−0.457364	−0.228682	−	0.973501i	$-0.573442\pi$
$617$	−11.3607	−0.457364	−0.228682	+	0.973501i	$0.573442\pi$
$618$	0	0
$619$	−45.1917	−1.81641	−0.908203	−	0.418530i	$-0.862545\pi$
$619$	−45.1917	−1.81641	−0.908203	+	0.418530i	$0.862545\pi$
$620$	24.9939	1.00378
$621$	0	0
$622$	56.5646	2.26803
$623$	13.8432	0.554618
$624$	0	0
$625$	1.00000	0.0400000
$626$	17.1050	0.683655
$627$	0	0
$628$	−58.7091	−2.34275
$629$	15.5174	0.618721
$630$	0	0
$631$	−9.78992	−0.389731	−0.194865	−	0.980830i	$-0.562427\pi$
$631$	−9.78992	−0.389731	−0.194865	+	0.980830i	$0.562427\pi$
$632$	−73.2905	−2.91534
$633$	0	0
$634$	−84.9647	−3.37438
$635$	2.24128	0.0889423
$636$	0	0
$637$	25.3340	1.00377
$638$	3.84324	0.152156
$639$	0	0
$640$	−28.5669	−1.12921
$641$	−0.210079	−0.00829764	−0.00414882	−	0.999991i	$-0.501321\pi$
$641$	−0.210079	−0.00829764	−0.00414882	+	0.999991i	$0.501321\pi$
$642$	0	0
$643$	14.5236	0.572754	0.286377	−	0.958117i	$-0.407549\pi$
$643$	14.5236	0.572754	0.286377	+	0.958117i	$0.407549\pi$
$644$	12.4202	0.489423
$645$	0	0
$646$	−110.606	−4.35172
$647$	−15.4641	−0.607957	−0.303979	−	0.952679i	$-0.598315\pi$
$647$	−15.4641	−0.607957	−0.303979	+	0.952679i	$0.598315\pi$
$648$	0	0
$649$	6.15676	0.241674
$650$	−11.7587	−0.461215
$651$	0	0
$652$	−34.8371	−1.36433
$653$	17.8310	0.697779	0.348890	−	0.937164i	$-0.386559\pi$
$653$	17.8310	0.697779	0.348890	+	0.937164i	$0.386559\pi$
$654$	0	0
$655$	8.68035	0.339169
$656$	130.325	5.08835
$657$	0	0
$658$	−13.6742	−0.533076
$659$	32.3135	1.25876	0.629378	−	0.777099i	$-0.283310\pi$
$659$	32.3135	1.25876	0.629378	+	0.777099i	$0.283310\pi$
$660$	0	0
$661$	−5.68649	−0.221179	−0.110589	−	0.993866i	$-0.535274\pi$
$661$	−5.68649	−0.221179	−0.110589	+	0.993866i	$0.535274\pi$
$662$	52.0288	2.02215
$663$	0	0
$664$	99.7296	3.87026
$665$	−5.67420	−0.220036
$666$	0	0
$667$	−3.05947	−0.118463
$668$	−10.5380	−0.407726
$669$	0	0
$670$	23.5174	0.908558
$671$	4.15676	0.160470
$672$	0	0
$673$	−21.0205	−0.810281	−0.405141	−	0.914254i	$-0.632777\pi$
$673$	−21.0205	−0.810281	−0.405141	+	0.914254i	$0.632777\pi$
$674$	36.5835	1.40915
$675$	0	0
$676$	31.1711	1.19889
$677$	−36.7526	−1.41252	−0.706258	−	0.707954i	$-0.749618\pi$
$677$	−36.7526	−1.41252	−0.706258	+	0.707954i	$0.749618\pi$
$678$	0	0
$679$	15.8310	0.607536
$680$	70.2122	2.69251
$681$	0	0
$682$	12.6803	0.485556
$683$	17.3074	0.662248	0.331124	−	0.943587i	$-0.392572\pi$
$683$	17.3074	0.662248	0.331124	+	0.943587i	$0.392572\pi$
$684$	0	0
$685$	−15.3607	−0.586902
$686$	−37.5052	−1.43195
$687$	0	0
$688$	105.189	4.01030
$689$	0.680346	0.0259191
$690$	0	0
$691$	17.6742	0.672358	0.336179	−	0.941798i	$-0.390865\pi$
$691$	17.6742	0.672358	0.336179	+	0.941798i	$0.390865\pi$
$692$	−20.0722	−0.763032
$693$	0	0
$694$	−17.1773	−0.652040
$695$	−8.58145	−0.325513
$696$	0	0
$697$	−73.0759	−2.76795
$698$	43.7731	1.65684
$699$	0	0
$700$	5.75872	0.217659
$701$	17.1050	0.646048	0.323024	−	0.946391i	$-0.395300\pi$
$701$	17.1050	0.646048	0.323024	+	0.946391i	$0.395300\pi$
$702$	0	0
$703$	−10.5236	−0.396905
$704$	−24.8576	−0.936857
$705$	0	0
$706$	35.7731	1.34634
$707$	16.7961	0.631681
$708$	0	0
$709$	25.1506	0.944551	0.472276	−	0.881451i	$-0.343433\pi$
$709$	25.1506	0.944551	0.472276	+	0.881451i	$0.343433\pi$
$710$	−12.6803	−0.475885
$711$	0	0
$712$	116.169	4.35361
$713$	−10.0944	−0.378037
$714$	0	0
$715$	−4.34017	−0.162313
$716$	−80.9069	−3.02363
$717$	0	0
$718$	−8.99386	−0.335648
$719$	1.78992	0.0667528	0.0333764	−	0.999443i	$-0.489374\pi$
$719$	1.78992	0.0667528	0.0333764	+	0.999443i	$0.489374\pi$
$720$	0	0
$721$	7.37298	0.274584
$722$	23.5341	0.875848
$723$	0	0
$724$	25.8310	0.960000
$725$	−1.41855	−0.0526837
$726$	0	0
$727$	25.9877	0.963831	0.481915	−	0.876218i	$-0.339941\pi$
$727$	25.9877	0.963831	0.481915	+	0.876218i	$0.339941\pi$
$728$	−42.3545	−1.56976
$729$	0	0
$730$	28.4391	1.05258
$731$	−58.9816	−2.18151
$732$	0	0
$733$	−41.0205	−1.51513	−0.757564	−	0.652761i	$-0.773610\pi$
$733$	−41.0205	−1.51513	−0.757564	+	0.652761i	$0.773610\pi$
$734$	−97.9253	−3.61449
$735$	0	0
$736$	41.8187	1.54146
$737$	8.68035	0.319745
$738$	0	0
$739$	−47.6163	−1.75160	−0.875798	−	0.482678i	$-0.839664\pi$
$739$	−47.6163	−1.75160	−0.875798	+	0.482678i	$0.839664\pi$
$740$	10.6803	0.392617
$741$	0	0
$742$	−0.457980	−0.0168130
$743$	−0.550252	−0.0201868	−0.0100934	−	0.999949i	$-0.503213\pi$
$743$	−0.550252	−0.0201868	−0.0100934	+	0.999949i	$0.503213\pi$
$744$	0	0
$745$	−18.0989	−0.663092
$746$	−7.61425	−0.278778
$747$	0	0
$748$	41.4329	1.51494
$749$	−6.83710	−0.249822
$750$	0	0
$751$	41.5585	1.51649	0.758245	−	0.651969i	$-0.226057\pi$
$751$	41.5585	1.51649	0.758245	+	0.651969i	$0.226057\pi$
$752$	−64.7624	−2.36164
$753$	0	0
$754$	16.6803	0.607462
$755$	−22.9360	−0.834726
$756$	0	0
$757$	1.31965	0.0479636	0.0239818	−	0.999712i	$-0.492366\pi$
$757$	1.31965	0.0479636	0.0239818	+	0.999712i	$0.492366\pi$
$758$	−54.1855	−1.96811
$759$	0	0
$760$	−47.6163	−1.72723
$761$	2.21461	0.0802797	0.0401399	−	0.999194i	$-0.487220\pi$
$761$	2.21461	0.0802797	0.0401399	+	0.999194i	$0.487220\pi$
$762$	0	0
$763$	2.49484	0.0903192
$764$	−13.4764	−0.487559
$765$	0	0
$766$	−90.9192	−3.28504
$767$	26.7214	0.964853
$768$	0	0
$769$	−14.3668	−0.518081	−0.259041	−	0.965866i	$-0.583406\pi$
$769$	−14.3668	−0.518081	−0.259041	+	0.965866i	$0.583406\pi$
$770$	2.92162	0.105288
$771$	0	0
$772$	0.142380	0.00512436
$773$	−40.1568	−1.44434	−0.722169	−	0.691717i	$-0.756855\pi$
$773$	−40.1568	−1.44434	−0.722169	+	0.691717i	$0.756855\pi$
$774$	0	0
$775$	−4.68035	−0.168123
$776$	132.849	4.76900
$777$	0	0
$778$	−34.7792	−1.24690
$779$	49.5585	1.77562
$780$	0	0
$781$	−4.68035	−0.167476
$782$	−45.3361	−1.62122
$783$	0	0
$784$	−80.7686	−2.88459
$785$	10.9939	0.392388
$786$	0	0
$787$	49.5897	1.76768	0.883841	−	0.467788i	$-0.154949\pi$
$787$	49.5897	1.76768	0.883841	+	0.467788i	$0.154949\pi$
$788$	−112.781	−4.01767
$789$	0	0
$790$	21.9421	0.780666
$791$	6.47027	0.230056
$792$	0	0
$793$	18.0410	0.640656
$794$	−14.4124	−0.511477
$795$	0	0
$796$	56.1978	1.99188
$797$	46.7091	1.65452	0.827261	−	0.561818i	$-0.189898\pi$
$797$	46.7091	1.65452	0.827261	+	0.561818i	$0.189898\pi$
$798$	0	0
$799$	36.3135	1.28468
$800$	19.3896	0.685527
$801$	0	0
$802$	−5.41855	−0.191336
$803$	10.4969	0.370429
$804$	0	0
$805$	−2.32580	−0.0819736
$806$	55.0349	1.93852
$807$	0	0
$808$	140.948	4.95853
$809$	18.5814	0.653289	0.326644	−	0.945147i	$-0.394082\pi$
$809$	18.5814	0.653289	0.326644	+	0.945147i	$0.394082\pi$
$810$	0	0
$811$	27.3028	0.958732	0.479366	−	0.877615i	$-0.340867\pi$
$811$	27.3028	0.958732	0.479366	+	0.877615i	$0.340867\pi$
$812$	−8.16904	−0.286677
$813$	0	0
$814$	5.41855	0.189920
$815$	6.52359	0.228511
$816$	0	0
$817$	40.0000	1.39942
$818$	70.9770	2.48165
$819$	0	0
$820$	−50.2967	−1.75644
$821$	31.2085	1.08918	0.544592	−	0.838701i	$-0.316685\pi$
$821$	31.2085	1.08918	0.544592	+	0.838701i	$0.316685\pi$
$822$	0	0
$823$	−50.1855	−1.74936	−0.874678	−	0.484704i	$-0.838927\pi$
$823$	−50.1855	−1.74936	−0.874678	+	0.484704i	$0.838927\pi$
$824$	61.8720	2.15541
$825$	0	0
$826$	−17.9877	−0.625873
$827$	−27.3874	−0.952352	−0.476176	−	0.879350i	$-0.657977\pi$
$827$	−27.3874	−0.952352	−0.476176	+	0.879350i	$0.657977\pi$
$828$	0	0
$829$	−26.1978	−0.909887	−0.454943	−	0.890520i	$-0.650341\pi$
$829$	−26.1978	−0.909887	−0.454943	+	0.890520i	$0.650341\pi$
$830$	−29.8576	−1.03637
$831$	0	0
$832$	−107.886	−3.74029
$833$	45.2885	1.56915
$834$	0	0
$835$	1.97334	0.0682902
$836$	−28.0989	−0.971821
$837$	0	0
$838$	7.68649	0.265525
$839$	−7.20394	−0.248708	−0.124354	−	0.992238i	$-0.539686\pi$
$839$	−7.20394	−0.248708	−0.124354	+	0.992238i	$0.539686\pi$
$840$	0	0
$841$	−26.9877	−0.930611
$842$	31.0928	1.07153
$843$	0	0
$844$	51.1338	1.76010
$845$	−5.83710	−0.200802
$846$	0	0
$847$	1.07838	0.0370535
$848$	−2.16904	−0.0744852
$849$	0	0
$850$	−21.0205	−0.720998
$851$	−4.31351	−0.147865
$852$	0	0
$853$	39.8043	1.36287	0.681437	−	0.731877i	$-0.261356\pi$
$853$	39.8043	1.36287	0.681437	+	0.731877i	$0.261356\pi$
$854$	−12.1445	−0.415575
$855$	0	0
$856$	−57.3751	−1.96104
$857$	36.9504	1.26220	0.631100	−	0.775701i	$-0.282604\pi$
$857$	36.9504	1.26220	0.631100	+	0.775701i	$0.282604\pi$
$858$	0	0
$859$	57.5052	1.96205	0.981025	−	0.193879i	$-0.0621070\pi$
$859$	57.5052	1.96205	0.981025	+	0.193879i	$0.0621070\pi$
$860$	−40.5958	−1.38431
$861$	0	0
$862$	−63.8597	−2.17507
$863$	1.89657	0.0645599	0.0322800	−	0.999479i	$-0.489723\pi$
$863$	1.89657	0.0645599	0.0322800	+	0.999479i	$0.489723\pi$
$864$	0	0
$865$	3.75872	0.127800
$866$	−40.6225	−1.38041
$867$	0	0
$868$	−26.9528	−0.914838
$869$	8.09890	0.274736
$870$	0	0
$871$	37.6742	1.27654
$872$	20.9360	0.708982
$873$	0	0
$874$	30.7460	1.04000
$875$	−1.07838	−0.0364558
$876$	0	0
$877$	32.5380	1.09873	0.549365	−	0.835583i	$-0.314870\pi$
$877$	32.5380	1.09873	0.549365	+	0.835583i	$0.314870\pi$
$878$	12.9483	0.436983
$879$	0	0
$880$	13.8371	0.466449
$881$	−18.1978	−0.613099	−0.306550	−	0.951855i	$-0.599175\pi$
$881$	−18.1978	−0.613099	−0.306550	+	0.951855i	$0.599175\pi$
$882$	0	0
$883$	36.3956	1.22481	0.612405	−	0.790545i	$-0.290202\pi$
$883$	36.3956	1.22481	0.612405	+	0.790545i	$0.290202\pi$
$884$	179.826	6.04821
$885$	0	0
$886$	−54.7214	−1.83840
$887$	−27.8699	−0.935780	−0.467890	−	0.883787i	$-0.654986\pi$
$887$	−27.8699	−0.935780	−0.467890	+	0.883787i	$0.654986\pi$
$888$	0	0
$889$	−2.41694	−0.0810616
$890$	−34.7792	−1.16580
$891$	0	0
$892$	−11.5174	−0.385633
$893$	−24.6270	−0.824112
$894$	0	0
$895$	15.1506	0.506429
$896$	30.8059	1.02915
$897$	0	0
$898$	58.4412	1.95021
$899$	6.63931	0.221433
$900$	0	0
$901$	1.21622	0.0405182
$902$	−25.5174	−0.849638
$903$	0	0
$904$	54.2967	1.80588
$905$	−4.83710	−0.160791
$906$	0	0
$907$	−27.9376	−0.927653	−0.463826	−	0.885926i	$-0.653524\pi$
$907$	−27.9376	−0.927653	−0.463826	+	0.885926i	$0.653524\pi$
$908$	−51.5851	−1.71191
$909$	0	0
$910$	12.6803	0.420349
$911$	11.8843	0.393744	0.196872	−	0.980429i	$-0.436922\pi$
$911$	11.8843	0.393744	0.196872	+	0.980429i	$0.436922\pi$
$912$	0	0
$913$	−11.0205	−0.364726
$914$	76.3234	2.52455
$915$	0	0
$916$	−17.9467	−0.592975
$917$	−9.36069	−0.309117
$918$	0	0
$919$	−45.6041	−1.50434	−0.752170	−	0.658970i	$-0.770993\pi$
$919$	−45.6041	−1.50434	−0.752170	+	0.658970i	$0.770993\pi$
$920$	−19.5174	−0.643471
$921$	0	0
$922$	3.98771	0.131328
$923$	−20.3135	−0.668627
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	−62.8659	−2.06590
$927$	0	0
$928$	−27.5052	−0.902901
$929$	25.1506	0.825165	0.412582	−	0.910920i	$-0.364627\pi$
$929$	25.1506	0.825165	0.412582	+	0.910920i	$0.364627\pi$
$930$	0	0
$931$	−30.7136	−1.00660
$932$	12.8059	0.419471
$933$	0	0
$934$	−38.3545	−1.25500
$935$	−7.75872	−0.253737
$936$	0	0
$937$	5.33403	0.174255	0.0871276	−	0.996197i	$-0.472231\pi$
$937$	5.33403	0.174255	0.0871276	+	0.996197i	$0.472231\pi$
$938$	−25.3607	−0.828056
$939$	0	0
$940$	24.9939	0.815210
$941$	−56.8203	−1.85229	−0.926144	−	0.377170i	$-0.876897\pi$
$941$	−56.8203	−1.85229	−0.926144	+	0.377170i	$0.876897\pi$
$942$	0	0
$943$	20.3135	0.661499
$944$	−85.1917	−2.77275
$945$	0	0
$946$	−20.5958	−0.669628
$947$	20.9939	0.682209	0.341104	−	0.940025i	$-0.389199\pi$
$947$	20.9939	0.682209	0.341104	+	0.940025i	$0.389199\pi$
$948$	0	0
$949$	45.5585	1.47889
$950$	14.2557	0.462514
$951$	0	0
$952$	−75.7152	−2.45395
$953$	25.2351	0.817446	0.408723	−	0.912658i	$-0.365974\pi$
$953$	25.2351	0.817446	0.408723	+	0.912658i	$0.365974\pi$
$954$	0	0
$955$	2.52359	0.0816615
$956$	38.4703	1.24422
$957$	0	0
$958$	37.5052	1.21174
$959$	16.5646	0.534900
$960$	0	0
$961$	−9.09436	−0.293367
$962$	23.5174	0.758233
$963$	0	0
$964$	−27.7899	−0.895053
$965$	−0.0266620	−0.000858280 0
$966$	0	0
$967$	13.1317	0.422287	0.211144	−	0.977455i	$-0.432281\pi$
$967$	13.1317	0.422287	0.211144	+	0.977455i	$0.432281\pi$
$968$	9.04945	0.290860
$969$	0	0
$970$	−39.7731	−1.27704
$971$	−8.94053	−0.286915	−0.143458	−	0.989656i	$-0.545822\pi$
$971$	−8.94053	−0.286915	−0.143458	+	0.989656i	$0.545822\pi$
$972$	0	0
$973$	9.25404	0.296671
$974$	−111.064	−3.55871
$975$	0	0
$976$	−57.5174	−1.84109
$977$	−50.3956	−1.61230	−0.806149	−	0.591713i	$-0.798452\pi$
$977$	−50.3956	−1.61230	−0.806149	+	0.591713i	$0.798452\pi$
$978$	0	0
$979$	−12.8371	−0.410276
$980$	31.1711	0.995725
$981$	0	0
$982$	−94.3833	−3.01189
$983$	32.1978	1.02695	0.513475	−	0.858105i	$-0.328358\pi$
$983$	32.1978	1.02695	0.513475	+	0.858105i	$0.328358\pi$
$984$	0	0
$985$	21.1194	0.672921
$986$	29.8187	0.949620
$987$	0	0
$988$	−121.954	−3.87988
$989$	16.3956	0.521349
$990$	0	0
$991$	−46.7747	−1.48585	−0.742924	−	0.669376i	$-0.766562\pi$
$991$	−46.7747	−1.48585	−0.742924	+	0.669376i	$0.766562\pi$
$992$	−90.7501	−2.88132
$993$	0	0
$994$	13.6742	0.433719
$995$	−10.5236	−0.333620
$996$	0	0
$997$	38.2122	1.21019	0.605096	−	0.796153i	$-0.293135\pi$
$997$	38.2122	1.21019	0.605096	+	0.796153i	$0.293135\pi$
$998$	41.0472	1.29933
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.2.a.e.1.3		3
3.2	odd	2		165.2.a.c.1.1	✓	3
4.3	odd	2		7920.2.a.cj.1.2		3
5.2	odd	4		2475.2.c.r.199.6		6
5.3	odd	4		2475.2.c.r.199.1		6
5.4	even	2		2475.2.a.bb.1.1		3
11.10	odd	2		5445.2.a.z.1.1		3
12.11	even	2		2640.2.a.be.1.2		3
15.2	even	4		825.2.c.g.199.1		6
15.8	even	4		825.2.c.g.199.6		6
15.14	odd	2		825.2.a.k.1.3		3
21.20	even	2		8085.2.a.bk.1.1		3
33.32	even	2		1815.2.a.m.1.3		3
165.164	even	2		9075.2.a.cf.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.a.c.1.1	✓	3	3.2	odd	2
495.2.a.e.1.3		3	1.1	even	1	trivial
825.2.a.k.1.3		3	15.14	odd	2
825.2.c.g.199.1		6	15.2	even	4
825.2.c.g.199.6		6	15.8	even	4
1815.2.a.m.1.3		3	33.32	even	2
2475.2.a.bb.1.1		3	5.4	even	2
2475.2.c.r.199.1		6	5.3	odd	4
2475.2.c.r.199.6		6	5.2	odd	4
2640.2.a.be.1.2		3	12.11	even	2
5445.2.a.z.1.1		3	11.10	odd	2
7920.2.a.cj.1.2		3	4.3	odd	2
8085.2.a.bk.1.1		3	21.20	even	2
9075.2.a.cf.1.1		3	165.164	even	2

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:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	495.a (trivial)

\(f(q)\)	\(=\)	\(q+2.70928 q^{2} +5.34017 q^{4} -1.00000 q^{5} +1.07838 q^{7} +9.04945 q^{8} +O(q^{10})\)	\(q+2.70928 q^{2} +5.34017 q^{4} -1.00000 q^{5} +1.07838 q^{7} +9.04945 q^{8} -2.70928 q^{10} -1.00000 q^{11} -4.34017 q^{13} +2.92162 q^{14} +13.8371 q^{16} -7.75872 q^{17} +5.26180 q^{19} -5.34017 q^{20} -2.70928 q^{22} +2.15676 q^{23} +1.00000 q^{25} -11.7587 q^{26} +5.75872 q^{28} -1.41855 q^{29} -4.68035 q^{31} +19.3896 q^{32} -21.0205 q^{34} -1.07838 q^{35} -2.00000 q^{37} +14.2557 q^{38} -9.04945 q^{40} +9.41855 q^{41} +7.60197 q^{43} -5.34017 q^{44} +5.84324 q^{46} -4.68035 q^{47} -5.83710 q^{49} +2.70928 q^{50} -23.1773 q^{52} -0.156755 q^{53} +1.00000 q^{55} +9.75872 q^{56} -3.84324 q^{58} -6.15676 q^{59} -4.15676 q^{61} -12.6803 q^{62} +24.8576 q^{64} +4.34017 q^{65} -8.68035 q^{67} -41.4329 q^{68} -2.92162 q^{70} +4.68035 q^{71} -10.4969 q^{73} -5.41855 q^{74} +28.0989 q^{76} -1.07838 q^{77} -8.09890 q^{79} -13.8371 q^{80} +25.5174 q^{82} +11.0205 q^{83} +7.75872 q^{85} +20.5958 q^{86} -9.04945 q^{88} +12.8371 q^{89} -4.68035 q^{91} +11.5174 q^{92} -12.6803 q^{94} -5.26180 q^{95} +14.6803 q^{97} -15.8143 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 5 q^{4} - 3 q^{5} + 9 q^{8}+O(q^{10}) \) 3 * q + q^2 + 5 * q^4 - 3 * q^5 + 9 * q^8	\( 3 q + q^{2} + 5 q^{4} - 3 q^{5} + 9 q^{8} - q^{10} - 3 q^{11} - 2 q^{13} + 12 q^{14} + 13 q^{16} + 2 q^{17} + 8 q^{19} - 5 q^{20} - q^{22} + 3 q^{25} - 10 q^{26} - 8 q^{28} + 10 q^{29} + 8 q^{31} + 29 q^{32} - 30 q^{34} - 6 q^{37} - 9 q^{40} + 14 q^{41} + 4 q^{43} - 5 q^{44} + 24 q^{46} + 8 q^{47} + 11 q^{49} + q^{50} - 30 q^{52} + 6 q^{53} + 3 q^{55} + 4 q^{56} - 18 q^{58} - 12 q^{59} - 6 q^{61} - 16 q^{62} + 13 q^{64} + 2 q^{65} - 4 q^{67} - 42 q^{68} - 12 q^{70} - 8 q^{71} - 14 q^{73} - 2 q^{74} + 48 q^{76} + 12 q^{79} - 13 q^{80} + 26 q^{82} - 2 q^{85} + 8 q^{86} - 9 q^{88} + 10 q^{89} + 8 q^{91} - 16 q^{92} - 16 q^{94} - 8 q^{95} + 22 q^{97} - 39 q^{98}+O(q^{100}) \) 3 * q + q^2 + 5 * q^4 - 3 * q^5 + 9 * q^8 - q^10 - 3 * q^11 - 2 * q^13 + 12 * q^14 + 13 * q^16 + 2 * q^17 + 8 * q^19 - 5 * q^20 - q^22 + 3 * q^25 - 10 * q^26 - 8 * q^28 + 10 * q^29 + 8 * q^31 + 29 * q^32 - 30 * q^34 - 6 * q^37 - 9 * q^40 + 14 * q^41 + 4 * q^43 - 5 * q^44 + 24 * q^46 + 8 * q^47 + 11 * q^49 + q^50 - 30 * q^52 + 6 * q^53 + 3 * q^55 + 4 * q^56 - 18 * q^58 - 12 * q^59 - 6 * q^61 - 16 * q^62 + 13 * q^64 + 2 * q^65 - 4 * q^67 - 42 * q^68 - 12 * q^70 - 8 * q^71 - 14 * q^73 - 2 * q^74 + 48 * q^76 + 12 * q^79 - 13 * q^80 + 26 * q^82 - 2 * q^85 + 8 * q^86 - 9 * q^88 + 10 * q^89 + 8 * q^91 - 16 * q^92 - 16 * q^94 - 8 * q^95 + 22 * q^97 - 39 * q^98

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