LMFDB - Embedded newform 5445.2.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5445,2,Mod(1,5445)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5445, 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("5445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5445 = 3^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5445.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:	$43.4785439006$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	5445.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.70928 q^{2} +5.34017 q^{4} -1.00000 q^{5} -1.07838 q^{7} -9.04945 q^{8} +O(q^{10})$	$q-2.70928 q^{2} +5.34017 q^{4} -1.00000 q^{5} -1.07838 q^{7} -9.04945 q^{8} +2.70928 q^{10} +4.34017 q^{13} +2.92162 q^{14} +13.8371 q^{16} +7.75872 q^{17} -5.26180 q^{19} -5.34017 q^{20} +2.15676 q^{23} +1.00000 q^{25} -11.7587 q^{26} -5.75872 q^{28} +1.41855 q^{29} -4.68035 q^{31} -19.3896 q^{32} -21.0205 q^{34} +1.07838 q^{35} -2.00000 q^{37} +14.2557 q^{38} +9.04945 q^{40} -9.41855 q^{41} -7.60197 q^{43} -5.84324 q^{46} -4.68035 q^{47} -5.83710 q^{49} -2.70928 q^{50} +23.1773 q^{52} -0.156755 q^{53} +9.75872 q^{56} -3.84324 q^{58} -6.15676 q^{59} +4.15676 q^{61} +12.6803 q^{62} +24.8576 q^{64} -4.34017 q^{65} -8.68035 q^{67} +41.4329 q^{68} -2.92162 q^{70} +4.68035 q^{71} +10.4969 q^{73} +5.41855 q^{74} -28.0989 q^{76} +8.09890 q^{79} -13.8371 q^{80} +25.5174 q^{82} -11.0205 q^{83} -7.75872 q^{85} +20.5958 q^{86} +12.8371 q^{89} -4.68035 q^{91} +11.5174 q^{92} +12.6803 q^{94} +5.26180 q^{95} +14.6803 q^{97} +15.8143 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 2 q^{13} + 12 q^{14} + 13 q^{16} - 2 q^{17} - 8 q^{19} - 5 q^{20} + 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} - 24 q^{46} + 8 q^{47} + 11 q^{49} - q^{50} + 30 q^{52} + 6 q^{53} + 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} + 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 + 2 * q^13 + 12 * q^14 + 13 * q^16 - 2 * q^17 - 8 * q^19 - 5 * q^20 + 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 - 24 * q^46 + 8 * q^47 + 11 * q^49 - q^50 + 30 * q^52 + 6 * q^53 + 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 + 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	$-0.907279\pi$
$2$	−2.70928	−1.91575	−0.957873	+	0.287190i	$0.907279\pi$
$3$	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	$-0.565327\pi$
$7$	−1.07838	−0.407588	−0.203794	+	0.979014i	$0.565327\pi$
$8$	−9.04945	−3.19946
$9$	0	0
$10$	2.70928	0.856748
$11$	0	0
$11$	0	0
$12$	0	0
$13$	4.34017	1.20375	0.601874	−	0.798591i	$-0.294421\pi$
$13$	4.34017	1.20375	0.601874	+	0.798591i	$0.294421\pi$
$14$	2.92162	0.780836
$15$	0	0
$16$	13.8371	3.45928
$17$	7.75872	1.88177	0.940883	−	0.338730i	$-0.109997\pi$
$17$	7.75872	1.88177	0.940883	+	0.338730i	$0.109997\pi$
$18$	0	0
$19$	−5.26180	−1.20714	−0.603569	−	0.797311i	$-0.706255\pi$
$19$	−5.26180	−1.20714	−0.603569	+	0.797311i	$0.706255\pi$
$20$	−5.34017	−1.19410
$21$	0	0
$22$	0	0
$23$	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	$-0.457954\pi$
$29$	1.41855	0.263418	0.131709	+	0.991288i	$0.457954\pi$
$30$	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	$-0.763036\pi$
$41$	−9.41855	−1.47093	−0.735465	+	0.677562i	$0.763036\pi$
$42$	0	0
$43$	−7.60197	−1.15929	−0.579645	−	0.814869i	$-0.696809\pi$
$43$	−7.60197	−1.15929	−0.579645	+	0.814869i	$0.696809\pi$
$44$	0	0
$45$	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
$55$	0	0
$56$	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	$-0.414262\pi$
$61$	4.15676	0.532218	0.266109	+	0.963943i	$0.414262\pi$
$62$	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	$-0.289444\pi$
$73$	10.4969	1.22857	0.614286	+	0.789083i	$0.289444\pi$
$74$	5.41855	0.629894
$75$	0	0
$76$	−28.0989	−3.22316
$77$	0	0
$78$	0	0
$79$	8.09890	0.911197	0.455599	−	0.890185i	$-0.349425\pi$
$79$	8.09890	0.911197	0.455599	+	0.890185i	$0.349425\pi$
$80$	−13.8371	−1.54703
$81$	0	0
$82$	25.5174	2.81793
$83$	−11.0205	−1.20966	−0.604830	−	0.796355i	$-0.706759\pi$
$83$	−11.0205	−1.20966	−0.604830	+	0.796355i	$0.706759\pi$
$84$	0	0
$85$	−7.75872	−0.841552
$86$	20.5958	2.22090
$87$	0	0
$88$	0	0
$89$	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	$-0.782200\pi$
$101$	−15.5753	−1.54980	−0.774900	+	0.632083i	$0.782200\pi$
$102$	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	$-0.400854\pi$
$107$	6.34017	0.612928	0.306464	+	0.951882i	$0.400854\pi$
$108$	0	0
$109$	−2.31351	−0.221594	−0.110797	−	0.993843i	$-0.535340\pi$
$109$	−2.31351	−0.221594	−0.110797	+	0.993843i	$0.535340\pi$
$110$	0	0
$111$	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
$121$	0	0
$122$	−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	$-0.468295\pi$
$127$	2.24128	0.198881	0.0994406	+	0.995044i	$0.468295\pi$
$128$	−28.5669	−2.52498
$129$	0	0
$130$	11.7587	1.03131
$131$	8.68035	0.758405	0.379203	−	0.925314i	$-0.376198\pi$
$131$	8.68035	0.758405	0.379203	+	0.925314i	$0.376198\pi$
$132$	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	$-0.618567\pi$
$139$	−8.58145	−0.727869	−0.363935	+	0.931425i	$0.618567\pi$
$140$	5.75872	0.486701
$141$	0	0
$142$	−12.6803	−1.06411
$143$	0	0
$144$	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	$-0.765819\pi$
$149$	−18.0989	−1.48272	−0.741360	+	0.671108i	$0.765819\pi$
$150$	0	0
$151$	−22.9360	−1.86651	−0.933253	−	0.359221i	$-0.883042\pi$
$151$	−22.9360	−1.86651	−0.933253	+	0.359221i	$0.883042\pi$
$152$	47.6163	3.86220
$153$	0	0
$154$	0	0
$155$	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	$-0.475673\pi$
$167$	1.97334	0.152701	0.0763507	+	0.997081i	$0.475673\pi$
$168$	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	$-0.454362\pi$
$173$	3.75872	0.285770	0.142885	+	0.989739i	$0.454362\pi$
$174$	0	0
$175$	−1.07838	−0.0815177
$176$	0	0
$177$	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
$187$	0	0
$188$	−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	$-0.500305\pi$
$193$	−0.0266620	−0.00191917	−0.000959586		1.00000i	$0.500305\pi$
$194$	−39.7731	−2.85554
$195$	0	0
$196$	−31.1711	−2.22651
$197$	21.1194	1.50470	0.752348	−	0.658766i	$-0.228921\pi$
$197$	21.1194	1.50470	0.752348	+	0.658766i	$0.228921\pi$
$198$	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
$209$	0	0
$210$	0	0
$211$	−9.57531	−0.659191	−0.329596	−	0.944122i	$-0.606912\pi$
$211$	−9.57531	−0.659191	−0.329596	+	0.944122i	$0.606912\pi$
$212$	−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
$220$	0	0
$221$	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	$-0.396125\pi$
$227$	9.65983	0.641145	0.320573	+	0.947224i	$0.396125\pi$
$228$	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	$-0.525029\pi$
$233$	−2.39803	−0.157100	−0.0785501	+	0.996910i	$0.525029\pi$
$234$	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	$-0.574852\pi$
$239$	−7.20394	−0.465984	−0.232992	+	0.972479i	$0.574852\pi$
$240$	0	0
$241$	5.20394	0.335215	0.167608	−	0.985854i	$-0.446396\pi$
$241$	5.20394	0.335215	0.167608	+	0.985854i	$0.446396\pi$
$242$	0	0
$243$	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
$253$	0	0
$254$	−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	$-0.695684\pi$
$263$	−18.7070	−1.15352	−0.576762	+	0.816912i	$0.695684\pi$
$264$	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	$-0.445837\pi$
$271$	5.57531	0.338676	0.169338	+	0.985558i	$0.445837\pi$
$272$	107.358	6.50955
$273$	0	0
$274$	−41.6163	−2.51414
$275$	0	0
$276$	0	0
$277$	26.0144	1.56305	0.781526	−	0.623872i	$-0.214442\pi$
$277$	26.0144	1.56305	0.781526	+	0.623872i	$0.214442\pi$
$278$	23.2495	1.39441
$279$	0	0
$280$	−9.75872	−0.583195
$281$	−9.41855	−0.561864	−0.280932	−	0.959728i	$-0.590643\pi$
$281$	−9.41855	−0.561864	−0.280932	+	0.959728i	$0.590643\pi$
$282$	0	0
$283$	−14.2413	−0.846556	−0.423278	−	0.906000i	$-0.639121\pi$
$283$	−14.2413	−0.846556	−0.423278	+	0.906000i	$0.639121\pi$
$284$	24.9939	1.48311
$285$	0	0
$286$	0	0
$287$	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	$-0.652264\pi$
$293$	−15.7587	−0.920634	−0.460317	+	0.887754i	$0.652264\pi$
$294$	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	$-0.318440\pi$
$307$	18.9216	1.07991	0.539957	+	0.841693i	$0.318440\pi$
$308$	0	0
$309$	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
$319$	0	0
$320$	−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	$-0.619882\pi$
$337$	−13.5031	−0.735559	−0.367780	+	0.929913i	$0.619882\pi$
$338$	−15.8143	−0.860185
$339$	0	0
$340$	−41.4329	−2.24702
$341$	0	0
$342$	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	$-0.445565\pi$
$347$	6.34017	0.340358	0.170179	+	0.985413i	$0.445565\pi$
$348$	0	0
$349$	−16.1568	−0.864851	−0.432426	−	0.901670i	$-0.642342\pi$
$349$	−16.1568	−0.864851	−0.432426	+	0.901670i	$0.642342\pi$
$350$	2.92162	0.156167
$351$	0	0
$352$	0	0
$353$	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	$-0.472080\pi$
$359$	3.31965	0.175205	0.0876023	+	0.996156i	$0.472080\pi$
$360$	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	$-0.476819\pi$
$373$	2.81044	0.145519	0.0727595	+	0.997350i	$0.476819\pi$
$374$	0	0
$375$	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
$385$	0	0
$386$	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
$407$	0	0
$408$	0	0
$409$	−26.1978	−1.29540	−0.647699	−	0.761897i	$-0.724268\pi$
$409$	−26.1978	−1.29540	−0.647699	+	0.761897i	$0.724268\pi$
$410$	−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
$418$	0	0
$419$	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	$-0.307840\pi$
$431$	23.5708	1.13536	0.567682	+	0.823248i	$0.307840\pi$
$432$	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	$-0.536383\pi$
$439$	−4.77924	−0.228101	−0.114050	+	0.993475i	$0.536383\pi$
$440$	0	0
$441$	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
$451$	0	0
$452$	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	$-0.728976\pi$
$457$	−28.1711	−1.31779	−0.658895	+	0.752235i	$0.728976\pi$
$458$	9.10504	0.425451
$459$	0	0
$460$	−11.5174	−0.537004
$461$	−1.47187	−0.0685520	−0.0342760	−	0.999412i	$-0.510913\pi$
$461$	−1.47187	−0.0685520	−0.0342760	+	0.999412i	$0.510913\pi$
$462$	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
$473$	0	0
$474$	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	$-0.602426\pi$
$479$	−13.8432	−0.632514	−0.316257	+	0.948674i	$0.602426\pi$
$480$	0	0
$481$	−8.68035	−0.395790
$482$	−14.0989	−0.642187
$483$	0	0
$484$	0	0
$485$	−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	$-0.212103\pi$
$491$	34.8371	1.57218	0.786088	+	0.618114i	$0.212103\pi$
$492$	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	$-0.547392\pi$
$503$	−6.65368	−0.296673	−0.148337	+	0.988937i	$0.547392\pi$
$504$	0	0
$505$	15.5753	0.693092
$506$	0	0
$507$	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
$517$	0	0
$518$	−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	$-0.669615\pi$
$523$	−23.2351	−1.01600	−0.508001	+	0.861357i	$0.669615\pi$
$524$	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
$539$	0	0
$540$	0	0
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	−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	$-0.335857\pi$
$547$	23.0661	0.986235	0.493117	+	0.869963i	$0.335857\pi$
$548$	82.0288	3.50409
$549$	0	0
$550$	0	0
$551$	−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	$-0.427932\pi$
$557$	10.5958	0.448960	0.224480	+	0.974479i	$0.427932\pi$
$558$	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	$-0.776314\pi$
$563$	−36.2122	−1.52616	−0.763080	+	0.646303i	$0.776314\pi$
$564$	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	$-0.696170\pi$
$569$	−27.5753	−1.15602	−0.578008	+	0.816031i	$0.696170\pi$
$570$	0	0
$571$	27.9299	1.16883	0.584414	−	0.811456i	$-0.301324\pi$
$571$	27.9299	1.16883	0.584414	+	0.811456i	$0.301324\pi$
$572$	0	0
$573$	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	0
$584$	−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	$-0.450383\pi$
$593$	7.56093	0.310490	0.155245	+	0.987876i	$0.450383\pi$
$594$	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	$-0.491432\pi$
$601$	1.31965	0.0538298	0.0269149	+	0.999638i	$0.491432\pi$
$602$	−22.2101	−0.905215
$603$	0	0
$604$	−122.482	−4.98373
$605$	0	0
$606$	0	0
$607$	2.24128	0.0909706	0.0454853	−	0.998965i	$-0.485517\pi$
$607$	2.24128	0.0909706	0.0454853	+	0.998965i	$0.485517\pi$
$608$	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	$-0.833079\pi$
$613$	−42.8638	−1.73125	−0.865626	+	0.500692i	$0.833079\pi$
$614$	−51.2639	−2.06884
$615$	0	0
$616$	0	0
$617$	−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
$638$	0	0
$639$	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
$649$	0	0
$650$	−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	$-0.716690\pi$
$659$	−32.3135	−1.25876	−0.629378	+	0.777099i	$0.716690\pi$
$660$	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
$671$	0	0
$672$	0	0
$673$	21.0205	0.810281	0.405141	−	0.914254i	$-0.367223\pi$
$673$	21.0205	0.810281	0.405141	+	0.914254i	$0.367223\pi$
$674$	36.5835	1.40915
$675$	0	0
$676$	31.1711	1.19889
$677$	36.7526	1.41252	0.706258	−	0.707954i	$-0.250382\pi$
$677$	36.7526	1.41252	0.706258	+	0.707954i	$0.250382\pi$
$678$	0	0
$679$	−15.8310	−0.607536
$680$	70.2122	2.69251
$681$	0	0
$682$	0	0
$683$	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	$-0.604700\pi$
$701$	−17.1050	−0.646048	−0.323024	+	0.946391i	$0.604700\pi$
$702$	0	0
$703$	10.5236	0.396905
$704$	0	0
$705$	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
$715$	0	0
$716$	−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	$-0.226390\pi$
$733$	41.0205	1.51513	0.757564	+	0.652761i	$0.226390\pi$
$734$	97.9253	3.61449
$735$	0	0
$736$	−41.8187	−1.54146
$737$	0	0
$738$	0	0
$739$	47.6163	1.75160	0.875798	−	0.482678i	$-0.160336\pi$
$739$	47.6163	1.75160	0.875798	+	0.482678i	$0.160336\pi$
$740$	10.6803	0.392617
$741$	0	0
$742$	−0.457980	−0.0168130
$743$	0.550252	0.0201868	0.0100934	−	0.999949i	$-0.496787\pi$
$743$	0.550252	0.0201868	0.0100934	+	0.999949i	$0.496787\pi$
$744$	0	0
$745$	18.0989	0.663092
$746$	−7.61425	−0.278778
$747$	0	0
$748$	0	0
$749$	−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	$-0.512780\pi$
$761$	−2.21461	−0.0802797	−0.0401399	+	0.999194i	$0.512780\pi$
$762$	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	$-0.416594\pi$
$769$	14.3668	0.518081	0.259041	+	0.965866i	$0.416594\pi$
$770$	0	0
$771$	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
$781$	0	0
$782$	−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	$-0.845051\pi$
$787$	−49.5897	−1.76768	−0.883841	+	0.467788i	$0.845051\pi$
$788$	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
$803$	0	0
$804$	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	$-0.605918\pi$
$809$	−18.5814	−0.653289	−0.326644	+	0.945147i	$0.605918\pi$
$810$	0	0
$811$	−27.3028	−0.958732	−0.479366	−	0.877615i	$-0.659133\pi$
$811$	−27.3028	−0.958732	−0.479366	+	0.877615i	$0.659133\pi$
$812$	−8.16904	−0.286677
$813$	0	0
$814$	0	0
$815$	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	$-0.683315\pi$
$821$	−31.2085	−1.08918	−0.544592	+	0.838701i	$0.683315\pi$
$822$	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	$-0.342023\pi$
$827$	27.3874	0.952352	0.476176	+	0.879350i	$0.342023\pi$
$828$	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
$836$	0	0
$837$	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
$847$	0	0
$848$	−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	$-0.738644\pi$
$853$	−39.8043	−1.36287	−0.681437	+	0.731877i	$0.738644\pi$
$854$	12.1445	0.415575
$855$	0	0
$856$	−57.3751	−1.96104
$857$	−36.9504	−1.26220	−0.631100	−	0.775701i	$-0.717396\pi$
$857$	−36.9504	−1.26220	−0.631100	+	0.775701i	$0.717396\pi$
$858$	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
$869$	0	0
$870$	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	$-0.685130\pi$
$877$	−32.5380	−1.09873	−0.549365	+	0.835583i	$0.685130\pi$
$878$	12.9483	0.436983
$879$	0	0
$880$	0	0
$881$	−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	$-0.345014\pi$
$887$	27.8699	0.935780	0.467890	+	0.883787i	$0.345014\pi$
$888$	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
$902$	0	0
$903$	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
$913$	0	0
$914$	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	$-0.229007\pi$
$919$	45.6041	1.50434	0.752170	+	0.658970i	$0.229007\pi$
$920$	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
$935$	0	0
$936$	0	0
$937$	−5.33403	−0.174255	−0.0871276	−	0.996197i	$-0.527769\pi$
$937$	−5.33403	−0.174255	−0.0871276	+	0.996197i	$0.527769\pi$
$938$	−25.3607	−0.828056
$939$	0	0
$940$	24.9939	0.815210
$941$	56.8203	1.85229	0.926144	−	0.377170i	$-0.123103\pi$
$941$	56.8203	1.85229	0.926144	+	0.377170i	$0.123103\pi$
$942$	0	0
$943$	−20.3135	−0.661499
$944$	−85.1917	−2.77275
$945$	0	0
$946$	0	0
$947$	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	$-0.634026\pi$
$953$	−25.2351	−0.817446	−0.408723	+	0.912658i	$0.634026\pi$
$954$	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	$-0.567719\pi$
$967$	−13.1317	−0.422287	−0.211144	+	0.977455i	$0.567719\pi$
$968$	0	0
$969$	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
$979$	0	0
$980$	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	$-0.706865\pi$
$997$	−38.2122	−1.21019	−0.605096	+	0.796153i	$0.706865\pi$
$998$	−41.0472	−1.29933
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.z.1.1		3
3.2	odd	2		1815.2.a.m.1.3		3
11.10	odd	2		495.2.a.e.1.3		3
15.14	odd	2		9075.2.a.cf.1.1		3
33.32	even	2		165.2.a.c.1.1	✓	3
44.43	even	2		7920.2.a.cj.1.2		3
55.32	even	4		2475.2.c.r.199.6		6
55.43	even	4		2475.2.c.r.199.1		6
55.54	odd	2		2475.2.a.bb.1.1		3
132.131	odd	2		2640.2.a.be.1.2		3
165.32	odd	4		825.2.c.g.199.1		6
165.98	odd	4		825.2.c.g.199.6		6
165.164	even	2		825.2.a.k.1.3		3
231.230	odd	2		8085.2.a.bk.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.a.c.1.1	✓	3	33.32	even	2
495.2.a.e.1.3		3	11.10	odd	2
825.2.a.k.1.3		3	165.164	even	2
825.2.c.g.199.1		6	165.32	odd	4
825.2.c.g.199.6		6	165.98	odd	4
1815.2.a.m.1.3		3	3.2	odd	2
2475.2.a.bb.1.1		3	55.54	odd	2
2475.2.c.r.199.1		6	55.43	even	4
2475.2.c.r.199.6		6	55.32	even	4
2640.2.a.be.1.2		3	132.131	odd	2
5445.2.a.z.1.1		3	1.1	even	1	trivial
7920.2.a.cj.1.2		3	44.43	even	2
8085.2.a.bk.1.1		3	231.230	odd	2
9075.2.a.cf.1.1		3	15.14	odd	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:	\( N \)	\(=\)	\( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5445.a (trivial)

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