LMFDB - Embedded newform 162.5.b.b.161.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,5,Mod(161,162)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("162.161");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	162.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$16.7459340196$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.1
Root			$-0.517638i$ of defining polynomial
Character	$\chi$	$=$	162.161
Dual form			162.5.b.b.161.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{2} -8.00000 q^{4} -0.416102i q^{5} +2.58846 q^{7} +22.6274i q^{8} +O(q^{10})$	\(q-2.82843i q^{2} -8.00000 q^{4} -0.416102i q^{5} +2.58846 q^{7} +22.6274i q^{8} -1.17691 q^{10} -54.3221i q^{11} -228.885 q^{13} -7.32126i q^{14} +64.0000 q^{16} +380.263i q^{17} -298.242 q^{19} +3.32882i q^{20} -153.646 q^{22} +631.827i q^{23} +624.827 q^{25} +647.383i q^{26} -20.7077 q^{28} +656.703i q^{29} -1544.59 q^{31} -181.019i q^{32} +1075.55 q^{34} -1.07706i q^{35} -187.842 q^{37} +843.556i q^{38} +9.41532 q^{40} +2134.91i q^{41} +2304.97 q^{43} +434.577i q^{44} +1787.08 q^{46} +2005.51i q^{47} -2394.30 q^{49} -1767.28i q^{50} +1831.08 q^{52} -3293.71i q^{53} -22.6036 q^{55} +58.5701i q^{56} +1857.44 q^{58} +3211.14i q^{59} -1822.22 q^{61} +4368.77i q^{62} -512.000 q^{64} +95.2394i q^{65} +2098.09 q^{67} -3042.10i q^{68} -3.04639 q^{70} -2181.30i q^{71} -5527.36 q^{73} +531.297i q^{74} +2385.94 q^{76} -140.610i q^{77} -11026.3 q^{79} -26.6305i q^{80} +6038.44 q^{82} -12058.7i q^{83} +158.228 q^{85} -6519.45i q^{86} +1229.17 q^{88} -2905.35i q^{89} -592.458 q^{91} -5054.62i q^{92} +5672.45 q^{94} +124.099i q^{95} -12892.8 q^{97} +6772.10i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{4} - 52 q^{7}+O(q^{10}) $ 4 * q - 32 * q^4 - 52 * q^7	$ 4 q - 32 q^{4} - 52 q^{7} + 120 q^{10} - 292 q^{13} + 256 q^{16} + 740 q^{19} - 864 q^{22} + 1564 q^{25} + 416 q^{28} - 5056 q^{31} - 312 q^{34} + 4424 q^{37} - 960 q^{40} + 428 q^{43} + 2160 q^{46} - 7956 q^{49} + 2336 q^{52} + 4212 q^{55} + 72 q^{58} + 4496 q^{61} - 2048 q^{64} + 16436 q^{67} - 3504 q^{70} - 20800 q^{73} - 5920 q^{76} - 16732 q^{79} - 1536 q^{82} + 19152 q^{85} + 6912 q^{88} - 5924 q^{91} + 29424 q^{94} - 3808 q^{97}+O(q^{100}) $ 4 * q - 32 * q^4 - 52 * q^7 + 120 * q^10 - 292 * q^13 + 256 * q^16 + 740 * q^19 - 864 * q^22 + 1564 * q^25 + 416 * q^28 - 5056 * q^31 - 312 * q^34 + 4424 * q^37 - 960 * q^40 + 428 * q^43 + 2160 * q^46 - 7956 * q^49 + 2336 * q^52 + 4212 * q^55 + 72 * q^58 + 4496 * q^61 - 2048 * q^64 + 16436 * q^67 - 3504 * q^70 - 20800 * q^73 - 5920 * q^76 - 16732 * q^79 - 1536 * q^82 + 19152 * q^85 + 6912 * q^88 - 5924 * q^91 + 29424 * q^94 - 3808 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$.

$n$	$83$
$\chi(n)$	$-1$

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 $

$2$	− 2.82843i	− 0.707107i
$2$	− 2.82843i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−8.00000	−0.500000
$5$	− 0.416102i	− 0.0166441i	−0.999965	−	0.00832204i	$-0.997351\pi$
$5$	− 0.416102i	− 0.0166441i	0.999965	−	0.00832204i	$-0.00264902\pi$
$6$	0	0
$7$	2.58846	0.0528257	0.0264128	−	0.999651i	$-0.491592\pi$
$7$	2.58846	0.0528257	0.0264128	+	0.999651i	$0.491592\pi$
$8$	22.6274i	0.353553i
$9$	0	0
$10$	−1.17691	−0.0117691
$11$	− 54.3221i	− 0.448943i	−0.974481	−	0.224472i	$-0.927934\pi$
$11$	− 54.3221i	− 0.448943i	0.974481	−	0.224472i	$-0.0720656\pi$
$12$	0	0
$13$	−228.885	−1.35435	−0.677173	−	0.735824i	$-0.736795\pi$
$13$	−228.885	−1.35435	−0.677173	+	0.735824i	$0.736795\pi$
$14$	− 7.32126i	− 0.0373534i
$15$	0	0
$16$	64.0000	0.250000
$17$	380.263i	1.31579i	0.753110	+	0.657894i	$0.228553\pi$
$17$	380.263i	1.31579i	−0.753110	+	0.657894i	$0.771447\pi$
$18$	0	0
$19$	−298.242	−0.826156	−0.413078	−	0.910696i	$-0.635546\pi$
$19$	−298.242	−0.826156	−0.413078	+	0.910696i	$0.635546\pi$
$20$	3.32882i	0.00832204i
$21$	0	0
$22$	−153.646	−0.317451
$23$	631.827i	1.19438i	0.802100	+	0.597190i	$0.203716\pi$
$23$	631.827i	1.19438i	−0.802100	+	0.597190i	$0.796284\pi$
$24$	0	0
$25$	624.827	0.999723
$26$	647.383i	0.957668i
$27$	0	0
$28$	−20.7077	−0.0264128
$29$	656.703i	0.780860i	0.920633	+	0.390430i	$0.127674\pi$
$29$	656.703i	0.780860i	−0.920633	+	0.390430i	$0.872326\pi$
$30$	0	0
$31$	−1544.59	−1.60728	−0.803638	−	0.595118i	$-0.797105\pi$
$31$	−1544.59	−1.60728	−0.803638	+	0.595118i	$0.797105\pi$
$32$	− 181.019i	− 0.176777i
$33$	0	0
$34$	1075.55	0.930403
$35$	− 1.07706i	0 0.000879235i
$36$	0	0
$37$	−187.842	−0.137211	−0.0686055	−	0.997644i	$-0.521855\pi$
$37$	−187.842	−0.137211	−0.0686055	+	0.997644i	$0.521855\pi$
$38$	843.556i	0.584180i
$39$	0	0
$40$	9.41532	0.00588457
$41$	2134.91i	1.27003i	0.772502	+	0.635013i	$0.219005\pi$
$41$	2134.91i	1.27003i	−0.772502	+	0.635013i	$0.780995\pi$
$42$	0	0
$43$	2304.97	1.24660	0.623302	−	0.781981i	$-0.285791\pi$
$43$	2304.97	1.24660	0.623302	+	0.781981i	$0.285791\pi$
$44$	434.577i	0.224472i
$45$	0	0
$46$	1787.08	0.844554
$47$	2005.51i	0.907883i	0.891032	+	0.453941i	$0.149982\pi$
$47$	2005.51i	0.907883i	−0.891032	+	0.453941i	$0.850018\pi$
$48$	0	0
$49$	−2394.30	−0.997209
$50$	− 1767.28i	− 0.706911i
$51$	0	0
$52$	1831.08	0.677173
$53$	− 3293.71i	− 1.17256i	−0.810110	−	0.586278i	$-0.800593\pi$
$53$	− 3293.71i	− 1.17256i	0.810110	−	0.586278i	$-0.199407\pi$
$54$	0	0
$55$	−22.6036	−0.00747225
$56$	58.5701i	0.0186767i
$57$	0	0
$58$	1857.44	0.552152
$59$	3211.14i	0.922476i	0.887276	+	0.461238i	$0.152595\pi$
$59$	3211.14i	0.922476i	−0.887276	+	0.461238i	$0.847405\pi$
$60$	0	0
$61$	−1822.22	−0.489712	−0.244856	−	0.969559i	$-0.578741\pi$
$61$	−1822.22	−0.489712	−0.244856	+	0.969559i	$0.578741\pi$
$62$	4368.77i	1.13652i
$63$	0	0
$64$	−512.000	−0.125000
$65$	95.2394i	0.0225419i
$66$	0	0
$67$	2098.09	0.467384	0.233692	−	0.972311i	$-0.424919\pi$
$67$	2098.09	0.467384	0.233692	+	0.972311i	$0.424919\pi$
$68$	− 3042.10i	− 0.657894i
$69$	0	0
$70$	−3.04639	−0.000621713 0
$71$	− 2181.30i	− 0.432713i	−0.976314	−	0.216356i	$-0.930583\pi$
$71$	− 2181.30i	− 0.432713i	0.976314	−	0.216356i	$-0.0694173\pi$
$72$	0	0
$73$	−5527.36	−1.03722	−0.518611	−	0.855010i	$-0.673551\pi$
$73$	−5527.36	−1.03722	−0.518611	+	0.855010i	$0.673551\pi$
$74$	531.297i	0.0970229i
$75$	0	0
$76$	2385.94	0.413078
$77$	− 140.610i	− 0.0237157i
$78$	0	0
$79$	−11026.3	−1.76676	−0.883379	−	0.468660i	$-0.844737\pi$
$79$	−11026.3	−1.76676	−0.883379	+	0.468660i	$0.844737\pi$
$80$	− 26.6305i	− 0.00416102i
$81$	0	0
$82$	6038.44	0.898043
$83$	− 12058.7i	− 1.75043i	−0.483737	−	0.875214i	$-0.660721\pi$
$83$	− 12058.7i	− 1.75043i	0.483737	−	0.875214i	$-0.339279\pi$
$84$	0	0
$85$	158.228	0.0219001
$86$	− 6519.45i	− 0.881483i
$87$	0	0
$88$	1229.17	0.158725
$89$	− 2905.35i	− 0.366791i	−0.983039	−	0.183395i	$-0.941291\pi$
$89$	− 2905.35i	− 0.366791i	0.983039	−	0.183395i	$-0.0587089\pi$
$90$	0	0
$91$	−592.458	−0.0715443
$92$	− 5054.62i	− 0.597190i
$93$	0	0
$94$	5672.45	0.641970
$95$	124.099i	0.0137506i
$96$	0	0
$97$	−12892.8	−1.37026	−0.685129	−	0.728422i	$-0.740254\pi$
$97$	−12892.8	−1.37026	−0.685129	+	0.728422i	$0.740254\pi$
$98$	6772.10i	0.705134i
$99$	0	0
$100$	−4998.61	−0.499861
$101$	− 3603.96i	− 0.353295i	−0.984274	−	0.176647i	$-0.943475\pi$
$101$	− 3603.96i	− 0.353295i	0.984274	−	0.176647i	$-0.0565252\pi$
$102$	0	0
$103$	3214.12	0.302962	0.151481	−	0.988460i	$-0.451596\pi$
$103$	3214.12	0.302962	0.151481	+	0.988460i	$0.451596\pi$
$104$	− 5179.07i	− 0.478834i
$105$	0	0
$106$	−9316.02	−0.829122
$107$	− 13043.6i	− 1.13928i	−0.821895	−	0.569639i	$-0.807083\pi$
$107$	− 13043.6i	− 1.13928i	0.821895	−	0.569639i	$-0.192917\pi$
$108$	0	0
$109$	3214.31	0.270542	0.135271	−	0.990809i	$-0.456810\pi$
$109$	3214.31	0.270542	0.135271	+	0.990809i	$0.456810\pi$
$110$	63.9325i	0.00528368i
$111$	0	0
$112$	165.661	0.0132064
$113$	− 11248.3i	− 0.880904i	−0.897776	−	0.440452i	$-0.854818\pi$
$113$	− 11248.3i	− 0.880904i	0.897776	−	0.440452i	$-0.145182\pi$
$114$	0	0
$115$	262.905	0.0198794
$116$	− 5253.63i	− 0.390430i
$117$	0	0
$118$	9082.47	0.652289
$119$	984.294i	0.0695074i
$120$	0	0
$121$	11690.1	0.798450
$122$	5154.01i	0.346279i
$123$	0	0
$124$	12356.7	0.803638
$125$	− 520.056i	− 0.0332836i
$126$	0	0
$127$	−1395.11	−0.0864973	−0.0432487	−	0.999064i	$-0.513771\pi$
$127$	−1395.11	−0.0864973	−0.0432487	+	0.999064i	$0.513771\pi$
$128$	1448.15i	0.0883883i
$129$	0	0
$130$	269.378	0.0159395
$131$	16755.1i	0.976349i	0.872746	+	0.488175i	$0.162337\pi$
$131$	16755.1i	0.976349i	−0.872746	+	0.488175i	$0.837663\pi$
$132$	0	0
$133$	−771.987	−0.0436422
$134$	− 5934.29i	− 0.330491i
$135$	0	0
$136$	−8604.37	−0.465201
$137$	18322.4i	0.976204i	0.872787	+	0.488102i	$0.162311\pi$
$137$	18322.4i	0.976204i	−0.872787	+	0.488102i	$0.837689\pi$
$138$	0	0
$139$	8280.03	0.428551	0.214275	−	0.976773i	$-0.431261\pi$
$139$	8280.03	0.428551	0.214275	+	0.976773i	$0.431261\pi$
$140$	8.61650i	0 0.000439617i
$141$	0	0
$142$	−6169.66	−0.305974
$143$	12433.5i	0.608025i
$144$	0	0
$145$	273.256	0.0129967
$146$	15633.7i	0.733427i
$147$	0	0
$148$	1502.74	0.0686055
$149$	− 11610.6i	− 0.522976i	−0.965207	−	0.261488i	$-0.915787\pi$
$149$	− 11610.6i	− 0.522976i	0.965207	−	0.261488i	$-0.0842133\pi$
$150$	0	0
$151$	43909.3	1.92576	0.962881	−	0.269926i	$-0.0869993\pi$
$151$	43909.3	1.92576	0.962881	+	0.269926i	$0.0869993\pi$
$152$	− 6748.45i	− 0.292090i
$153$	0	0
$154$	−397.707	−0.0167695
$155$	642.708i	0.0267516i
$156$	0	0
$157$	−36481.6	−1.48004	−0.740022	−	0.672583i	$-0.765185\pi$
$157$	−36481.6	−1.48004	−0.740022	+	0.672583i	$0.765185\pi$
$158$	31187.2i	1.24929i
$159$	0	0
$160$	−75.3225	−0.00294229
$161$	1635.46i	0.0630939i
$162$	0	0
$163$	34529.1	1.29960	0.649800	−	0.760105i	$-0.274853\pi$
$163$	34529.1	1.29960	0.649800	+	0.760105i	$0.274853\pi$
$164$	− 17079.3i	− 0.635013i
$165$	0	0
$166$	−34107.1	−1.23774
$167$	− 3364.81i	− 0.120650i	−0.998179	−	0.0603250i	$-0.980786\pi$
$167$	− 3364.81i	− 0.120650i	0.998179	−	0.0603250i	$-0.0192137\pi$
$168$	0	0
$169$	23827.1	0.834255
$170$	− 447.537i	− 0.0154857i
$171$	0	0
$172$	−18439.8	−0.623302
$173$	44486.7i	1.48641i	0.669064	+	0.743204i	$0.266695\pi$
$173$	44486.7i	1.48641i	−0.669064	+	0.743204i	$0.733305\pi$
$174$	0	0
$175$	1617.34	0.0528110
$176$	− 3476.62i	− 0.112236i
$177$	0	0
$178$	−8217.57	−0.259360
$179$	57987.0i	1.80977i	0.425652	+	0.904887i	$0.360045\pi$
$179$	57987.0i	1.80977i	−0.425652	+	0.904887i	$0.639955\pi$
$180$	0	0
$181$	−33114.4	−1.01079	−0.505393	−	0.862889i	$-0.668653\pi$
$181$	−33114.4	−1.01079	−0.505393	+	0.862889i	$0.668653\pi$
$182$	1675.72i	0.0505894i
$183$	0	0
$184$	−14296.6	−0.422277
$185$	78.1614i	0.00228375i
$186$	0	0
$187$	20656.7	0.590714
$188$	− 16044.1i	− 0.453941i
$189$	0	0
$190$	351.006	0.00972315
$191$	− 61638.8i	− 1.68961i	−0.535072	−	0.844807i	$-0.679715\pi$
$191$	− 61638.8i	− 1.68961i	0.535072	−	0.844807i	$-0.320285\pi$
$192$	0	0
$193$	−37678.8	−1.01154	−0.505769	−	0.862669i	$-0.668791\pi$
$193$	−37678.8	−1.01154	−0.505769	+	0.862669i	$0.668791\pi$
$194$	36466.2i	0.968919i
$195$	0	0
$196$	19154.4	0.498605
$197$	53748.5i	1.38495i	0.721442	+	0.692475i	$0.243480\pi$
$197$	53748.5i	1.38495i	−0.721442	+	0.692475i	$0.756520\pi$
$198$	0	0
$199$	16462.2	0.415702	0.207851	−	0.978161i	$-0.433353\pi$
$199$	16462.2	0.415702	0.207851	+	0.978161i	$0.433353\pi$
$200$	14138.2i	0.353455i
$201$	0	0
$202$	−10193.5	−0.249817
$203$	1699.85i	0.0412495i
$204$	0	0
$205$	888.342	0.0211384
$206$	− 9090.91i	− 0.214226i
$207$	0	0
$208$	−14648.6	−0.338587
$209$	16201.1i	0.370897i
$210$	0	0
$211$	8567.60	0.192439	0.0962197	−	0.995360i	$-0.469325\pi$
$211$	8567.60	0.192439	0.0962197	+	0.995360i	$0.469325\pi$
$212$	26349.7i	0.586278i
$213$	0	0
$214$	−36892.8	−0.805591
$215$	− 959.104i	− 0.0207486i
$216$	0	0
$217$	−3998.11	−0.0849054
$218$	− 9091.43i	− 0.191302i
$219$	0	0
$220$	180.828	0.00373612
$221$	− 87036.3i	− 1.78203i
$222$	0	0
$223$	8325.28	0.167413	0.0837065	−	0.996490i	$-0.473324\pi$
$223$	8325.28	0.167413	0.0837065	+	0.996490i	$0.473324\pi$
$224$	− 468.561i	− 0.00933835i
$225$	0	0
$226$	−31814.9	−0.622893
$227$	30679.4i	0.595381i	0.954662	+	0.297690i	$0.0962163\pi$
$227$	30679.4i	0.595381i	−0.954662	+	0.297690i	$0.903784\pi$
$228$	0	0
$229$	50446.1	0.961960	0.480980	−	0.876732i	$-0.340281\pi$
$229$	50446.1	0.961960	0.480980	+	0.876732i	$0.340281\pi$
$230$	− 743.606i	− 0.0140568i
$231$	0	0
$232$	−14859.5	−0.276076
$233$	7603.00i	0.140047i	0.997545	+	0.0700234i	$0.0223074\pi$
$233$	7603.00i	0.140047i	−0.997545	+	0.0700234i	$0.977693\pi$
$234$	0	0
$235$	834.498	0.0151109
$236$	− 25689.1i	− 0.461238i
$237$	0	0
$238$	2784.00	0.0491491
$239$	100313.i	1.75615i	0.478525	+	0.878074i	$0.341172\pi$
$239$	100313.i	1.75615i	−0.478525	+	0.878074i	$0.658828\pi$
$240$	0	0
$241$	−37939.3	−0.653214	−0.326607	−	0.945160i	$-0.605905\pi$
$241$	−37939.3	−0.653214	−0.326607	+	0.945160i	$0.605905\pi$
$242$	− 33064.6i	− 0.564589i
$243$	0	0
$244$	14577.7	0.244856
$245$	996.273i	0.0165976i
$246$	0	0
$247$	68263.0	1.11890
$248$	− 34950.1i	− 0.568258i
$249$	0	0
$250$	−1470.94	−0.0235350
$251$	− 58448.0i	− 0.927732i	−0.885905	−	0.463866i	$-0.846462\pi$
$251$	− 58448.0i	− 0.927732i	0.885905	−	0.463866i	$-0.153538\pi$
$252$	0	0
$253$	34322.2	0.536209
$254$	3945.98i	0.0611628i
$255$	0	0
$256$	4096.00	0.0625000
$257$	− 118183.i	− 1.78932i	−0.446752	−	0.894658i	$-0.647419\pi$
$257$	− 118183.i	− 1.78932i	0.446752	−	0.894658i	$-0.352581\pi$
$258$	0	0
$259$	−486.221	−0.00724827
$260$	− 761.915i	− 0.0112709i
$261$	0	0
$262$	47390.7	0.690383
$263$	− 106350.i	− 1.53754i	−0.639527	−	0.768769i	$-0.720870\pi$
$263$	− 106350.i	− 1.53754i	0.639527	−	0.768769i	$-0.279130\pi$
$264$	0	0
$265$	−1370.52	−0.0195161
$266$	2183.51i	0.0308597i
$267$	0	0
$268$	−16784.7	−0.233692
$269$	− 137219.i	− 1.89631i	−0.317813	−	0.948154i	$-0.602948\pi$
$269$	− 137219.i	− 1.89631i	0.317813	−	0.948154i	$-0.397052\pi$
$270$	0	0
$271$	60463.3	0.823291	0.411646	−	0.911344i	$-0.364954\pi$
$271$	60463.3	0.823291	0.411646	+	0.911344i	$0.364954\pi$
$272$	24336.8i	0.328947i
$273$	0	0
$274$	51823.5	0.690280
$275$	− 33941.9i	− 0.448819i
$276$	0	0
$277$	−119243.	−1.55408	−0.777039	−	0.629453i	$-0.783279\pi$
$277$	−119243.	−1.55408	−0.777039	+	0.629453i	$0.783279\pi$
$278$	− 23419.5i	− 0.303031i
$279$	0	0
$280$	24.3711	0.000310856 0
$281$	103690.i	1.31318i	0.754250	+	0.656588i	$0.228001\pi$
$281$	103690.i	1.31318i	−0.754250	+	0.656588i	$0.771999\pi$
$282$	0	0
$283$	67641.7	0.844582	0.422291	−	0.906460i	$-0.361226\pi$
$283$	67641.7	0.844582	0.422291	+	0.906460i	$0.361226\pi$
$284$	17450.4i	0.216356i
$285$	0	0
$286$	35167.2	0.429938
$287$	5526.13i	0.0670899i
$288$	0	0
$289$	−61078.9	−0.731299
$290$	− 772.884i	− 0.00919006i
$291$	0	0
$292$	44218.9	0.518611
$293$	49294.0i	0.574194i	0.957902	+	0.287097i	$0.0926902\pi$
$293$	49294.0i	0.574194i	−0.957902	+	0.287097i	$0.907310\pi$
$294$	0	0
$295$	1336.16	0.0153538
$296$	− 4250.38i	− 0.0485114i
$297$	0	0
$298$	−32839.7	−0.369800
$299$	− 144615.i	− 1.61760i
$300$	0	0
$301$	5966.32	0.0658527
$302$	− 124194.i	− 1.36172i
$303$	0	0
$304$	−19087.5	−0.206539
$305$	758.229i	0.00815081i
$306$	0	0
$307$	19565.2	0.207591	0.103795	−	0.994599i	$-0.466901\pi$
$307$	19565.2	0.207591	0.103795	+	0.994599i	$0.466901\pi$
$308$	1124.88i	0.0118579i
$309$	0	0
$310$	1817.85	0.0189163
$311$	49857.4i	0.515476i	0.966215	+	0.257738i	$0.0829772\pi$
$311$	49857.4i	0.515476i	−0.966215	+	0.257738i	$0.917023\pi$
$312$	0	0
$313$	−106184.	−1.08385	−0.541925	−	0.840427i	$-0.682304\pi$
$313$	−106184.	−1.08385	−0.541925	+	0.840427i	$0.682304\pi$
$314$	103186.i	1.04655i
$315$	0	0
$316$	88210.7	0.883379
$317$	87455.5i	0.870299i	0.900358	+	0.435149i	$0.143304\pi$
$317$	87455.5i	0.870299i	−0.900358	+	0.435149i	$0.856696\pi$
$318$	0	0
$319$	35673.5	0.350562
$320$	213.044i	0.00208051i
$321$	0	0
$322$	4625.77	0.0446141
$323$	− 113410.i	− 1.08705i
$324$	0	0
$325$	−143013.	−1.35397
$326$	− 97663.0i	− 0.918956i
$327$	0	0
$328$	−48307.6	−0.449022
$329$	5191.18i	0.0479595i
$330$	0	0
$331$	21875.4	0.199664	0.0998319	−	0.995004i	$-0.468169\pi$
$331$	21875.4	0.199664	0.0998319	+	0.995004i	$0.468169\pi$
$332$	96469.5i	0.875214i
$333$	0	0
$334$	−9517.11	−0.0853124
$335$	− 873.019i	− 0.00777919i
$336$	0	0
$337$	−188043.	−1.65576	−0.827880	−	0.560906i	$-0.810453\pi$
$337$	−188043.	−1.65576	−0.827880	+	0.560906i	$0.810453\pi$
$338$	− 67393.4i	− 0.589907i
$339$	0	0
$340$	−1265.83	−0.0109500
$341$	83905.5i	0.721576i
$342$	0	0
$343$	−12412.4	−0.105504
$344$	52155.6i	0.440741i
$345$	0	0
$346$	125827.	1.05105
$347$	188187.i	1.56290i	0.623968	+	0.781450i	$0.285519\pi$
$347$	188187.i	1.56290i	−0.623968	+	0.781450i	$0.714481\pi$
$348$	0	0
$349$	78656.6	0.645780	0.322890	−	0.946437i	$-0.395346\pi$
$349$	78656.6	0.645780	0.322890	+	0.946437i	$0.395346\pi$
$350$	− 4574.52i	− 0.0373430i
$351$	0	0
$352$	−9833.35	−0.0793627
$353$	− 115870.i	− 0.929866i	−0.885346	−	0.464933i	$-0.846078\pi$
$353$	− 115870.i	− 0.929866i	0.885346	−	0.464933i	$-0.153922\pi$
$354$	0	0
$355$	−907.646	−0.00720211
$356$	23242.8i	0.183395i
$357$	0	0
$358$	164012.	1.27970
$359$	187042.i	1.45128i	0.688075	+	0.725640i	$0.258456\pi$
$359$	187042.i	1.45128i	−0.688075	+	0.725640i	$0.741544\pi$
$360$	0	0
$361$	−41372.6	−0.317467
$362$	93661.6i	0.714734i
$363$	0	0
$364$	4739.66	0.0357721
$365$	2299.95i	0.0172636i
$366$	0	0
$367$	−60140.0	−0.446510	−0.223255	−	0.974760i	$-0.571668\pi$
$367$	−60140.0	−0.446510	−0.223255	+	0.974760i	$0.571668\pi$
$368$	40436.9i	0.298595i
$369$	0	0
$370$	221.074	0.00161486
$371$	− 8525.62i	− 0.0619410i
$372$	0	0
$373$	−242647.	−1.74404	−0.872022	−	0.489467i	$-0.837191\pi$
$373$	−242647.	−1.74404	−0.872022	+	0.489467i	$0.837191\pi$
$374$	− 58425.9i	− 0.417698i
$375$	0	0
$376$	−45379.6	−0.320985
$377$	− 150309.i	− 1.05756i
$378$	0	0
$379$	339.729	0.00236513	0.00118256	−	0.999999i	$-0.499624\pi$
$379$	339.729	0.00236513	0.00118256	+	0.999999i	$0.499624\pi$
$380$	− 992.794i	− 0.00687530i
$381$	0	0
$382$	−174341.	−1.19474
$383$	41726.4i	0.284455i	0.989834	+	0.142227i	$0.0454264\pi$
$383$	41726.4i	0.284455i	−0.989834	+	0.142227i	$0.954574\pi$
$384$	0	0
$385$	−58.5083	−0.000394726 0
$386$	106572.i	0.715265i
$387$	0	0
$388$	103142.	0.685129
$389$	175724.i	1.16127i	0.814164	+	0.580635i	$0.197196\pi$
$389$	175724.i	1.16127i	−0.814164	+	0.580635i	$0.802804\pi$
$390$	0	0
$391$	−240260.	−1.57155
$392$	− 54176.8i	− 0.352567i
$393$	0	0
$394$	152024.	0.979308
$395$	4588.08i	0.0294061i
$396$	0	0
$397$	−13497.3	−0.0856379	−0.0428190	−	0.999083i	$-0.513634\pi$
$397$	−13497.3	−0.0856379	−0.0428190	+	0.999083i	$0.513634\pi$
$398$	− 46562.2i	− 0.293946i
$399$	0	0
$400$	39988.9	0.249931
$401$	68479.6i	0.425865i	0.977067	+	0.212933i	$0.0683015\pi$
$401$	68479.6i	0.425865i	−0.977067	+	0.212933i	$0.931699\pi$
$402$	0	0
$403$	353533.	2.17681
$404$	28831.7i	0.176647i
$405$	0	0
$406$	4807.90	0.0291678
$407$	10204.0i	0.0616000i
$408$	0	0
$409$	179755.	1.07457	0.537283	−	0.843402i	$-0.319451\pi$
$409$	179755.	1.07457	0.537283	+	0.843402i	$0.319451\pi$
$410$	− 2512.61i	− 0.0149471i
$411$	0	0
$412$	−25713.0	−0.151481
$413$	8311.90i	0.0487304i
$414$	0	0
$415$	−5017.65	−0.0291343
$416$	41432.5i	0.239417i
$417$	0	0
$418$	45823.8	0.262264
$419$	290453.i	1.65443i	0.561887	+	0.827214i	$0.310076\pi$
$419$	290453.i	1.65443i	−0.561887	+	0.827214i	$0.689924\pi$
$420$	0	0
$421$	39307.9	0.221776	0.110888	−	0.993833i	$-0.464630\pi$
$421$	39307.9	0.221776	0.110888	+	0.993833i	$0.464630\pi$
$422$	− 24232.8i	− 0.136075i
$423$	0	0
$424$	74528.1	0.414561
$425$	237598.i	1.31542i
$426$	0	0
$427$	−4716.73	−0.0258694
$428$	104349.i	0.569639i
$429$	0	0
$430$	−2712.76	−0.0146715
$431$	− 144937.i	− 0.780235i	−0.920765	−	0.390117i	$-0.872434\pi$
$431$	− 144937.i	− 0.780235i	0.920765	−	0.390117i	$-0.127566\pi$
$432$	0	0
$433$	34162.1	0.182208	0.0911042	−	0.995841i	$-0.470960\pi$
$433$	34162.1	0.182208	0.0911042	+	0.995841i	$0.470960\pi$
$434$	11308.4i	0.0600372i
$435$	0	0
$436$	−25714.5	−0.135271
$437$	− 188437.i	− 0.986744i
$438$	0	0
$439$	−21972.4	−0.114012	−0.0570058	−	0.998374i	$-0.518155\pi$
$439$	−21972.4	−0.114012	−0.0570058	+	0.998374i	$0.518155\pi$
$440$	− 511.460i	− 0.00264184i
$441$	0	0
$442$	−246176.	−1.26009
$443$	64062.5i	0.326435i	0.986590	+	0.163217i	$0.0521872\pi$
$443$	64062.5i	0.326435i	−0.986590	+	0.163217i	$0.947813\pi$
$444$	0	0
$445$	−1208.92	−0.00610490
$446$	− 23547.4i	− 0.118379i
$447$	0	0
$448$	−1325.29	−0.00660321
$449$	162975.i	0.808403i	0.914670	+	0.404202i	$0.132451\pi$
$449$	162975.i	0.808403i	−0.914670	+	0.404202i	$0.867549\pi$
$450$	0	0
$451$	115973.	0.570169
$452$	89986.1i	0.440452i
$453$	0	0
$454$	86774.4	0.420998
$455$	246.523i	0.00119079i
$456$	0	0
$457$	148356.	0.710352	0.355176	−	0.934799i	$-0.384421\pi$
$457$	148356.	0.710352	0.355176	+	0.934799i	$0.384421\pi$
$458$	− 142683.i	− 0.680208i
$459$	0	0
$460$	−2103.24	−0.00993968
$461$	− 59560.8i	− 0.280258i	−0.990133	−	0.140129i	$-0.955248\pi$
$461$	− 59560.8i	− 0.280258i	0.990133	−	0.140129i	$-0.0447518\pi$
$462$	0	0
$463$	67777.3	0.316171	0.158086	−	0.987425i	$-0.449468\pi$
$463$	67777.3	0.316171	0.158086	+	0.987425i	$0.449468\pi$
$464$	42029.0i	0.195215i
$465$	0	0
$466$	21504.5	0.0990280
$467$	− 57234.8i	− 0.262438i	−0.991353	−	0.131219i	$-0.958111\pi$
$467$	− 57234.8i	− 0.262438i	0.991353	−	0.131219i	$-0.0418891\pi$
$468$	0	0
$469$	5430.81	0.0246899
$470$	− 2360.32i	− 0.0106850i
$471$	0	0
$472$	−72659.8	−0.326145
$473$	− 125211.i	− 0.559655i
$474$	0	0
$475$	−186350.	−0.825927
$476$	− 7874.35i	− 0.0347537i
$477$	0	0
$478$	283728.	1.24178
$479$	− 190463.i	− 0.830118i	−0.909794	−	0.415059i	$-0.863761\pi$
$479$	− 190463.i	− 0.830118i	0.909794	−	0.415059i	$-0.136239\pi$
$480$	0	0
$481$	42994.1	0.185831
$482$	107309.i	0.461892i
$483$	0	0
$484$	−93520.9	−0.399225
$485$	5364.70i	0.0228067i
$486$	0	0
$487$	−167195.	−0.704962	−0.352481	−	0.935819i	$-0.614662\pi$
$487$	−167195.	−0.704962	−0.352481	+	0.935819i	$0.614662\pi$
$488$	− 41232.1i	− 0.173139i
$489$	0	0
$490$	2817.89	0.0117363
$491$	− 247763.i	− 1.02772i	−0.857875	−	0.513858i	$-0.828216\pi$
$491$	− 247763.i	− 1.02772i	0.857875	−	0.513858i	$-0.171784\pi$
$492$	0	0
$493$	−249720.	−1.02745
$494$	− 193077.i	− 0.791183i
$495$	0	0
$496$	−98853.9	−0.401819
$497$	− 5646.21i	− 0.0228583i
$498$	0	0
$499$	411030.	1.65072	0.825359	−	0.564609i	$-0.190973\pi$
$499$	411030.	1.65072	0.825359	+	0.564609i	$0.190973\pi$
$500$	4160.44i	0.0166418i
$501$	0	0
$502$	−165316.	−0.656006
$503$	108804.i	0.430039i	0.976610	+	0.215020i	$0.0689816\pi$
$503$	108804.i	0.430039i	−0.976610	+	0.215020i	$0.931018\pi$
$504$	0	0
$505$	−1499.61	−0.00588026
$506$	− 97077.8i	− 0.379157i
$507$	0	0
$508$	11160.9	0.0432487
$509$	79302.0i	0.306090i	0.988219	+	0.153045i	$0.0489079\pi$
$509$	79302.0i	0.306090i	−0.988219	+	0.153045i	$0.951092\pi$
$510$	0	0
$511$	−14307.3	−0.0547920
$512$	− 11585.2i	− 0.0441942i
$513$	0	0
$514$	−334271.	−1.26524
$515$	− 1337.40i	− 0.00504252i
$516$	0	0
$517$	108944.	0.407588
$518$	1375.24i	0.00512530i
$519$	0	0
$520$	−2155.02	−0.00796975
$521$	− 55028.4i	− 0.202727i	−0.994849	−	0.101364i	$-0.967679\pi$
$521$	− 55028.4i	− 0.202727i	0.994849	−	0.101364i	$-0.0323205\pi$
$522$	0	0
$523$	473394.	1.73069	0.865346	−	0.501176i	$-0.167099\pi$
$523$	473394.	1.73069	0.865346	+	0.501176i	$0.167099\pi$
$524$	− 134041.i	− 0.488175i
$525$	0	0
$526$	−300803.	−1.08720
$527$	− 587351.i	− 2.11484i
$528$	0	0
$529$	−119364.	−0.426543
$530$	3876.41i	0.0138000i
$531$	0	0
$532$	6175.90	0.0218211
$533$	− 488649.i	− 1.72005i
$534$	0	0
$535$	−5427.47	−0.0189622
$536$	47474.3i	0.165245i
$537$	0	0
$538$	−388113.	−1.34089
$539$	130063.i	0.447690i
$540$	0	0
$541$	239116.	0.816984	0.408492	−	0.912762i	$-0.366055\pi$
$541$	239116.	0.816984	0.408492	+	0.912762i	$0.366055\pi$
$542$	− 171016.i	− 0.582155i
$543$	0	0
$544$	68834.9	0.232601
$545$	− 1337.48i	− 0.00450292i
$546$	0	0
$547$	−182974.	−0.611527	−0.305763	−	0.952108i	$-0.598912\pi$
$547$	−182974.	−0.611527	−0.305763	+	0.952108i	$0.598912\pi$
$548$	− 146579.i	− 0.488102i
$549$	0	0
$550$	−96002.3	−0.317363
$551$	− 195857.i	− 0.645112i
$552$	0	0
$553$	−28541.2	−0.0933301
$554$	337270.i	1.09890i
$555$	0	0
$556$	−66240.3	−0.214275
$557$	− 63739.1i	− 0.205445i	−0.994710	−	0.102722i	$-0.967245\pi$
$557$	− 63739.1i	− 0.205445i	0.994710	−	0.102722i	$-0.0327553\pi$
$558$	0	0
$559$	−527573.	−1.68834
$560$	− 68.9320i	0 0.000219809i
$561$	0	0
$562$	293279.	0.928555
$563$	− 353735.i	− 1.11599i	−0.829844	−	0.557995i	$-0.811571\pi$
$563$	− 353735.i	− 1.11599i	0.829844	−	0.557995i	$-0.188429\pi$
$564$	0	0
$565$	−4680.42	−0.0146618
$566$	− 191320.i	− 0.597210i
$567$	0	0
$568$	49357.3	0.152987
$569$	− 87882.2i	− 0.271442i	−0.990747	−	0.135721i	$-0.956665\pi$
$569$	− 87882.2i	− 0.271442i	0.990747	−	0.135721i	$-0.0433350\pi$
$570$	0	0
$571$	200332.	0.614439	0.307220	−	0.951639i	$-0.400601\pi$
$571$	200332.	0.614439	0.307220	+	0.951639i	$0.400601\pi$
$572$	− 99468.0i	− 0.304012i
$573$	0	0
$574$	15630.3	0.0474397
$575$	394782.i	1.19405i
$576$	0	0
$577$	−3625.36	−0.0108893	−0.00544464	−	0.999985i	$-0.501733\pi$
$577$	−3625.36	−0.0108893	−0.00544464	+	0.999985i	$0.501733\pi$
$578$	172757.i	0.517107i
$579$	0	0
$580$	−2186.05	−0.00649835
$581$	− 31213.4i	− 0.0924675i
$582$	0	0
$583$	−178921.	−0.526411
$584$	− 125070.i	− 0.366713i
$585$	0	0
$586$	139424.	0.406016
$587$	507123.i	1.47176i	0.677112	+	0.735880i	$0.263231\pi$
$587$	507123.i	1.47176i	−0.677112	+	0.735880i	$0.736769\pi$
$588$	0	0
$589$	460663.	1.32786
$590$	− 3779.24i	− 0.0108568i
$591$	0	0
$592$	−12021.9	−0.0343028
$593$	255264.i	0.725906i	0.931807	+	0.362953i	$0.118231\pi$
$593$	255264.i	0.725906i	−0.931807	+	0.362953i	$0.881769\pi$
$594$	0	0
$595$	409.567	0.00115689
$596$	92884.8i	0.261488i
$597$	0	0
$598$	−409034.	−1.14382
$599$	287137.i	0.800268i	0.916457	+	0.400134i	$0.131036\pi$
$599$	287137.i	0.800268i	−0.916457	+	0.400134i	$0.868964\pi$
$600$	0	0
$601$	6704.79	0.0185625	0.00928125	−	0.999957i	$-0.497046\pi$
$601$	6704.79	0.0185625	0.00928125	+	0.999957i	$0.497046\pi$
$602$	− 16875.3i	− 0.0465649i
$603$	0	0
$604$	−351274.	−0.962881
$605$	− 4864.28i	− 0.0132895i
$606$	0	0
$607$	−168976.	−0.458615	−0.229307	−	0.973354i	$-0.573646\pi$
$607$	−168976.	−0.458615	−0.229307	+	0.973354i	$0.573646\pi$
$608$	53987.6i	0.146045i
$609$	0	0
$610$	2144.60	0.00576349
$611$	− 459031.i	− 1.22959i
$612$	0	0
$613$	110387.	0.293763	0.146882	−	0.989154i	$-0.453076\pi$
$613$	110387.	0.293763	0.146882	+	0.989154i	$0.453076\pi$
$614$	− 55338.8i	− 0.146789i
$615$	0	0
$616$	3181.65	0.00838477
$617$	− 134090.i	− 0.352230i	−0.984370	−	0.176115i	$-0.943647\pi$
$617$	− 134090.i	− 0.352230i	0.984370	−	0.176115i	$-0.0563530\pi$
$618$	0	0
$619$	359201.	0.937468	0.468734	−	0.883339i	$-0.344710\pi$
$619$	359201.	0.937468	0.468734	+	0.883339i	$0.344710\pi$
$620$	− 5141.66i	− 0.0133758i
$621$	0	0
$622$	141018.	0.364497
$623$	− 7520.38i	− 0.0193760i
$624$	0	0
$625$	390300.	0.999169
$626$	300333.i	0.766397i
$627$	0	0
$628$	291853.	0.740022
$629$	− 71429.3i	− 0.180541i
$630$	0	0
$631$	431012.	1.08251	0.541253	−	0.840860i	$-0.317950\pi$
$631$	431012.	1.08251	0.541253	+	0.840860i	$0.317950\pi$
$632$	− 249497.i	− 0.624643i
$633$	0	0
$634$	247361.	0.615394
$635$	580.510i	0.00143967i
$636$	0	0
$637$	548018.	1.35057
$638$	− 100900.i	− 0.247885i
$639$	0	0
$640$	602.580	0.00147114
$641$	− 63621.0i	− 0.154841i	−0.996999	−	0.0774203i	$-0.975332\pi$
$641$	− 63621.0i	− 0.154841i	0.996999	−	0.0774203i	$-0.0246683\pi$
$642$	0	0
$643$	−797158.	−1.92807	−0.964034	−	0.265779i	$-0.914371\pi$
$643$	−797158.	−1.92807	−0.964034	+	0.265779i	$0.914371\pi$
$644$	− 13083.7i	− 0.0315470i
$645$	0	0
$646$	−320773.	−0.768658
$647$	185467.i	0.443054i	0.975154	+	0.221527i	$0.0711042\pi$
$647$	185467.i	0.443054i	−0.975154	+	0.221527i	$0.928896\pi$
$648$	0	0
$649$	174436.	0.414139
$650$	404502.i	0.957402i
$651$	0	0
$652$	−276233.	−0.649800
$653$	106978.i	0.250881i	0.992101	+	0.125441i	$0.0400345\pi$
$653$	106978.i	0.250881i	−0.992101	+	0.125441i	$0.959966\pi$
$654$	0	0
$655$	6971.85	0.0162504
$656$	136634.i	0.317506i
$657$	0	0
$658$	14682.9	0.0339125
$659$	761592.i	1.75369i	0.480777	+	0.876843i	$0.340355\pi$
$659$	761592.i	1.75369i	−0.480777	+	0.876843i	$0.659645\pi$
$660$	0	0
$661$	−108759.	−0.248921	−0.124461	−	0.992225i	$-0.539720\pi$
$661$	−108759.	−0.248921	−0.124461	+	0.992225i	$0.539720\pi$
$662$	− 61872.9i	− 0.141184i
$663$	0	0
$664$	272857.	0.618869
$665$	321.225i	0 0.000726385i
$666$	0	0
$667$	−414923.	−0.932644
$668$	26918.5i	0.0603250i
$669$	0	0
$670$	−2469.27	−0.00550072
$671$	98986.8i	0.219853i
$672$	0	0
$673$	554349.	1.22392	0.611961	−	0.790888i	$-0.290381\pi$
$673$	554349.	1.22392	0.611961	+	0.790888i	$0.290381\pi$
$674$	531866.i	1.17080i
$675$	0	0
$676$	−190617.	−0.417127
$677$	385406.i	0.840893i	0.907317	+	0.420446i	$0.138127\pi$
$677$	385406.i	0.840893i	−0.907317	+	0.420446i	$0.861873\pi$
$678$	0	0
$679$	−33372.4	−0.0723848
$680$	3580.30i	0.00774285i
$681$	0	0
$682$	237321.	0.510231
$683$	− 731269.i	− 1.56760i	−0.621012	−	0.783801i	$-0.713278\pi$
$683$	− 731269.i	− 1.56760i	0.621012	−	0.783801i	$-0.286722\pi$
$684$	0	0
$685$	7623.98	0.0162480
$686$	35107.7i	0.0746025i
$687$	0	0
$688$	147518.	0.311651
$689$	753879.i	1.58805i
$690$	0	0
$691$	51383.9	0.107614	0.0538072	−	0.998551i	$-0.482864\pi$
$691$	51383.9	0.107614	0.0538072	+	0.998551i	$0.482864\pi$
$692$	− 355894.i	− 0.743204i
$693$	0	0
$694$	532274.	1.10514
$695$	− 3445.34i	− 0.00713284i
$696$	0	0
$697$	−811828.	−1.67108
$698$	− 222475.i	− 0.456635i
$699$	0	0
$700$	−12938.7	−0.0264055
$701$	538501.i	1.09585i	0.836528	+	0.547924i	$0.184582\pi$
$701$	538501.i	1.09585i	−0.836528	+	0.547924i	$0.815418\pi$
$702$	0	0
$703$	56022.4	0.113358
$704$	27812.9i	0.0561179i
$705$	0	0
$706$	−327729.	−0.657514
$707$	− 9328.69i	− 0.0186630i
$708$	0	0
$709$	323504.	0.643557	0.321779	−	0.946815i	$-0.395719\pi$
$709$	323504.	0.643557	0.321779	+	0.946815i	$0.395719\pi$
$710$	2567.21i	0.00509266i
$711$	0	0
$712$	65740.6	0.129680
$713$	− 975915.i	− 1.91970i
$714$	0	0
$715$	5173.60	0.0101200
$716$	− 463896.i	− 0.904887i
$717$	0	0
$718$	529036.	1.02621
$719$	759369.i	1.46891i	0.678658	+	0.734455i	$0.262562\pi$
$719$	759369.i	1.46891i	−0.678658	+	0.734455i	$0.737438\pi$
$720$	0	0
$721$	8319.62	0.0160042
$722$	117019.i	0.224483i
$723$	0	0
$724$	264915.	0.505393
$725$	410326.i	0.780644i
$726$	0	0
$727$	−487394.	−0.922172	−0.461086	−	0.887356i	$-0.652540\pi$
$727$	−487394.	−0.922172	−0.461086	+	0.887356i	$0.652540\pi$
$728$	− 13405.8i	− 0.0252947i
$729$	0	0
$730$	6505.23	0.0122072
$731$	876495.i	1.64027i
$732$	0	0
$733$	−460066.	−0.856274	−0.428137	−	0.903714i	$-0.640830\pi$
$733$	−460066.	−0.856274	−0.428137	+	0.903714i	$0.640830\pi$
$734$	170102.i	0.315731i
$735$	0	0
$736$	114373.	0.211139
$737$	− 113973.i	− 0.209829i
$738$	0	0
$739$	−870177.	−1.59338	−0.796689	−	0.604390i	$-0.793417\pi$
$739$	−870177.	−1.59338	−0.796689	+	0.604390i	$0.793417\pi$
$740$	− 625.291i	− 0.00114188i
$741$	0	0
$742$	−24114.1	−0.0437989
$743$	− 223645.i	− 0.405117i	−0.979270	−	0.202559i	$-0.935074\pi$
$743$	− 223645.i	− 0.405117i	0.979270	−	0.202559i	$-0.0649257\pi$
$744$	0	0
$745$	−4831.20	−0.00870446
$746$	686309.i	1.23323i
$747$	0	0
$748$	−165253.	−0.295357
$749$	− 33762.8i	− 0.0601831i
$750$	0	0
$751$	−717599.	−1.27234	−0.636169	−	0.771550i	$-0.719482\pi$
$751$	−717599.	−1.27234	−0.636169	+	0.771550i	$0.719482\pi$
$752$	128353.i	0.226971i
$753$	0	0
$754$	−425139.	−0.747805
$755$	− 18270.8i	− 0.0320525i
$756$	0	0
$757$	248013.	0.432795	0.216398	−	0.976305i	$-0.430569\pi$
$757$	248013.	0.432795	0.216398	+	0.976305i	$0.430569\pi$
$758$	− 960.899i	− 0.00167240i
$759$	0	0
$760$	−2808.04	−0.00486157
$761$	869443.i	1.50132i	0.660691	+	0.750658i	$0.270263\pi$
$761$	869443.i	1.50132i	−0.660691	+	0.750658i	$0.729737\pi$
$762$	0	0
$763$	8320.10	0.0142916
$764$	493110.i	0.844807i
$765$	0	0
$766$	118020.	0.201140
$767$	− 734980.i	− 1.24935i
$768$	0	0
$769$	949705.	1.60597	0.802983	−	0.596002i	$-0.203245\pi$
$769$	949705.	1.60597	0.802983	+	0.596002i	$0.203245\pi$
$770$	165.487i	0 0.000279114i
$771$	0	0
$772$	301430.	0.505769
$773$	− 608953.i	− 1.01912i	−0.860435	−	0.509560i	$-0.829808\pi$
$773$	− 608953.i	− 1.01912i	0.860435	−	0.509560i	$-0.170192\pi$
$774$	0	0
$775$	−965103.	−1.60683
$776$	− 291730.i	− 0.484459i
$777$	0	0
$778$	497024.	0.821142
$779$	− 636721.i	− 1.04924i
$780$	0	0
$781$	−118493.	−0.194263
$782$	679559.i	1.11125i
$783$	0	0
$784$	−153235.	−0.249302
$785$	15180.1i	0.0246340i
$786$	0	0
$787$	151254.	0.244207	0.122104	−	0.992517i	$-0.461036\pi$
$787$	151254.	0.244207	0.122104	+	0.992517i	$0.461036\pi$
$788$	− 429988.i	− 0.692475i
$789$	0	0
$790$	12977.1	0.0207932
$791$	− 29115.6i	− 0.0465343i
$792$	0	0
$793$	417078.	0.663240
$794$	38176.2i	0.0605552i
$795$	0	0
$796$	−131698.	−0.207851
$797$	894130.i	1.40761i	0.710391	+	0.703807i	$0.248518\pi$
$797$	894130.i	1.40761i	−0.710391	+	0.703807i	$0.751482\pi$
$798$	0	0
$799$	−762622.	−1.19458
$800$	− 113106.i	− 0.176728i
$801$	0	0
$802$	193689.	0.301132
$803$	300258.i	0.465654i
$804$	0	0
$805$	680.517	0.00105014
$806$	− 999943.i	− 1.53924i
$807$	0	0
$808$	81548.3	0.124908
$809$	61045.4i	0.0932730i	0.998912	+	0.0466365i	$0.0148502\pi$
$809$	61045.4i	0.0932730i	−0.998912	+	0.0466365i	$0.985150\pi$
$810$	0	0
$811$	716958.	1.09006	0.545032	−	0.838415i	$-0.316517\pi$
$811$	716958.	1.09006	0.545032	+	0.838415i	$0.316517\pi$
$812$	− 13598.8i	− 0.0206247i
$813$	0	0
$814$	28861.2	0.0435578
$815$	− 14367.6i	− 0.0216307i
$816$	0	0
$817$	−687440.	−1.02989
$818$	− 508423.i	− 0.759834i
$819$	0	0
$820$	−7106.73	−0.0105692
$821$	− 991122.i	− 1.47042i	−0.677841	−	0.735209i	$-0.737084\pi$
$821$	− 991122.i	− 1.47042i	0.677841	−	0.735209i	$-0.262916\pi$
$822$	0	0
$823$	−976629.	−1.44188	−0.720941	−	0.692996i	$-0.756290\pi$
$823$	−976629.	−1.44188	−0.720941	+	0.692996i	$0.756290\pi$
$824$	72727.3i	0.107113i
$825$	0	0
$826$	23509.6	0.0344576
$827$	− 997499.i	− 1.45848i	−0.684256	−	0.729242i	$-0.739873\pi$
$827$	− 997499.i	− 1.45848i	0.684256	−	0.729242i	$-0.260127\pi$
$828$	0	0
$829$	−1.02815e6	−1.49606	−0.748029	−	0.663666i	$-0.768999\pi$
$829$	−1.02815e6	−1.49606	−0.748029	+	0.663666i	$0.768999\pi$
$830$	14192.1i	0.0206010i
$831$	0	0
$832$	117189.	0.169293
$833$	− 910463.i	− 1.31212i
$834$	0	0
$835$	−1400.10	−0.00200811
$836$	− 129609.i	− 0.185448i
$837$	0	0
$838$	821525.	1.16986
$839$	426268.i	0.605562i	0.953060	+	0.302781i	$0.0979151\pi$
$839$	426268.i	0.605562i	−0.953060	+	0.302781i	$0.902085\pi$
$840$	0	0
$841$	276022.	0.390257
$842$	− 111179.i	− 0.156820i
$843$	0	0
$844$	−68540.8	−0.0962197
$845$	− 9914.53i	− 0.0138854i
$846$	0	0
$847$	30259.3	0.0421786
$848$	− 210797.i	− 0.293139i
$849$	0	0
$850$	672030.	0.930145
$851$	− 118684.i	− 0.163882i
$852$	0	0
$853$	−508676.	−0.699107	−0.349553	−	0.936916i	$-0.613667\pi$
$853$	−508676.	−0.699107	−0.349553	+	0.936916i	$0.613667\pi$
$854$	13340.9i	0.0182924i
$855$	0	0
$856$	295143.	0.402795
$857$	− 464207.i	− 0.632047i	−0.948751	−	0.316024i	$-0.897652\pi$
$857$	− 464207.i	− 0.632047i	0.948751	−	0.316024i	$-0.102348\pi$
$858$	0	0
$859$	1.07546e6	1.45750	0.728749	−	0.684781i	$-0.240102\pi$
$859$	1.07546e6	1.45750	0.728749	+	0.684781i	$0.240102\pi$
$860$	7672.83i	0.0103743i
$861$	0	0
$862$	−409944.	−0.551709
$863$	353636.i	0.474827i	0.971409	+	0.237413i	$0.0762996\pi$
$863$	353636.i	0.474827i	−0.971409	+	0.237413i	$0.923700\pi$
$864$	0	0
$865$	18511.0	0.0247399
$866$	− 96624.9i	− 0.128841i
$867$	0	0
$868$	31984.9	0.0424527
$869$	598974.i	0.793174i
$870$	0	0
$871$	−480220.	−0.633001
$872$	72731.5i	0.0956510i
$873$	0	0
$874$	−532982.	−0.697733
$875$	− 1346.14i	− 0.00175823i
$876$	0	0
$877$	1.18720e6	1.54357	0.771785	−	0.635883i	$-0.219364\pi$
$877$	1.18720e6	1.54357	0.771785	+	0.635883i	$0.219364\pi$
$878$	62147.4i	0.0806183i
$879$	0	0
$880$	−1446.63	−0.00186806
$881$	− 1.51991e6i	− 1.95824i	−0.203294	−	0.979118i	$-0.565165\pi$
$881$	− 1.51991e6i	− 1.95824i	0.203294	−	0.979118i	$-0.434835\pi$
$882$	0	0
$883$	794738.	1.01930	0.509650	−	0.860382i	$-0.329775\pi$
$883$	794738.	1.01930	0.509650	+	0.860382i	$0.329775\pi$
$884$	696290.i	0.891017i
$885$	0	0
$886$	181196.	0.230824
$887$	− 819736.i	− 1.04190i	−0.853587	−	0.520951i	$-0.825577\pi$
$887$	− 819736.i	− 1.04190i	0.853587	−	0.520951i	$-0.174423\pi$
$888$	0	0
$889$	−3611.20	−0.00456928
$890$	3419.35i	0.00431682i
$891$	0	0
$892$	−66602.2	−0.0837065
$893$	− 598128.i	− 0.750052i
$894$	0	0
$895$	24128.5	0.0301220
$896$	3748.49i	0.00466917i
$897$	0	0
$898$	460963.	0.571627
$899$	− 1.01434e6i	− 1.25506i
$900$	0	0
$901$	1.25248e6	1.54284
$902$	− 328021.i	− 0.403170i
$903$	0	0
$904$	254519.	0.311447
$905$	13779.0i	0.0168236i
$906$	0	0
$907$	299229.	0.363738	0.181869	−	0.983323i	$-0.441785\pi$
$907$	299229.	0.363738	0.181869	+	0.983323i	$0.441785\pi$
$908$	− 245435.i	− 0.297690i
$909$	0	0
$910$	697.272	0.000842015 0
$911$	288132.i	0.347181i	0.984818	+	0.173590i	$0.0555369\pi$
$911$	288132.i	0.347181i	−0.984818	+	0.173590i	$0.944463\pi$
$912$	0	0
$913$	−655054.	−0.785842
$914$	− 419615.i	− 0.502295i
$915$	0	0
$916$	−403569.	−0.480980
$917$	43369.9i	0.0515763i
$918$	0	0
$919$	−1.26206e6	−1.49434	−0.747172	−	0.664631i	$-0.768589\pi$
$919$	−1.26206e6	−1.49434	−0.747172	+	0.664631i	$0.768589\pi$
$920$	5948.85i	0.00702842i
$921$	0	0
$922$	−168463.	−0.198173
$923$	499267.i	0.586043i
$924$	0	0
$925$	−117369.	−0.137173
$926$	− 191703.i	− 0.223567i
$927$	0	0
$928$	118876.	0.138038
$929$	742695.i	0.860556i	0.902696	+	0.430278i	$0.141584\pi$
$929$	742695.i	0.860556i	−0.902696	+	0.430278i	$0.858416\pi$
$930$	0	0
$931$	714081.	0.823850
$932$	− 60824.0i	− 0.0700234i
$933$	0	0
$934$	−161885.	−0.185572
$935$	− 8595.29i	− 0.00983190i
$936$	0	0
$937$	359577.	0.409555	0.204778	−	0.978809i	$-0.434353\pi$
$937$	359577.	0.409555	0.204778	+	0.978809i	$0.434353\pi$
$938$	− 15360.7i	− 0.0174584i
$939$	0	0
$940$	−6675.98	−0.00755544
$941$	− 1.23219e6i	− 1.39155i	−0.718261	−	0.695774i	$-0.755062\pi$
$941$	− 1.23219e6i	− 1.39155i	0.718261	−	0.695774i	$-0.244938\pi$
$942$	0	0
$943$	−1.34890e6	−1.51689
$944$	205513.i	0.230619i
$945$	0	0
$946$	−354150.	−0.395736
$947$	892643.i	0.995354i	0.867362	+	0.497677i	$0.165814\pi$
$947$	892643.i	0.995354i	−0.867362	+	0.497677i	$0.834186\pi$
$948$	0	0
$949$	1.26513e6	1.40476
$950$	527077.i	0.584018i
$951$	0	0
$952$	−22272.0	−0.0245746
$953$	737750.i	0.812313i	0.913803	+	0.406157i	$0.133131\pi$
$953$	737750.i	0.812313i	−0.913803	+	0.406157i	$0.866869\pi$
$954$	0	0
$955$	−25648.0	−0.0281221
$956$	− 802503.i	− 0.878074i
$957$	0	0
$958$	−538711.	−0.586982
$959$	47426.7i	0.0515686i
$960$	0	0
$961$	1.46224e6	1.58334
$962$	− 121606.i	− 0.131403i
$963$	0	0
$964$	303515.	0.326607
$965$	15678.2i	0.0168361i
$966$	0	0
$967$	48161.3	0.0515045	0.0257523	−	0.999668i	$-0.491802\pi$
$967$	48161.3	0.0515045	0.0257523	+	0.999668i	$0.491802\pi$
$968$	264517.i	0.282295i
$969$	0	0
$970$	15173.7	0.0161268
$971$	80929.3i	0.0858356i	0.999079	+	0.0429178i	$0.0136654\pi$
$971$	80929.3i	0.0858356i	−0.999079	+	0.0429178i	$0.986335\pi$
$972$	0	0
$973$	21432.5	0.0226385
$974$	472899.i	0.498484i
$975$	0	0
$976$	−116622.	−0.122428
$977$	607005.i	0.635920i	0.948104	+	0.317960i	$0.102998\pi$
$977$	607005.i	0.635920i	−0.948104	+	0.317960i	$0.897002\pi$
$978$	0	0
$979$	−157825.	−0.164668
$980$	− 7970.19i	− 0.00829882i
$981$	0	0
$982$	−700779.	−0.726705
$983$	− 1.51719e6i	− 1.57012i	−0.619420	−	0.785060i	$-0.712632\pi$
$983$	− 1.51719e6i	− 1.57012i	0.619420	−	0.785060i	$-0.287368\pi$
$984$	0	0
$985$	22364.9	0.0230512
$986$	706315.i	0.726515i
$987$	0	0
$988$	−546104.	−0.559451
$989$	1.45634e6i	1.48892i
$990$	0	0
$991$	−1.27300e6	−1.29623	−0.648116	−	0.761542i	$-0.724443\pi$
$991$	−1.27300e6	−1.29623	−0.648116	+	0.761542i	$0.724443\pi$
$992$	279601.i	0.284129i
$993$	0	0
$994$	−15969.9	−0.0161633
$995$	− 6849.96i	− 0.00691898i
$996$	0	0
$997$	1.19262e6	1.19981	0.599903	−	0.800073i	$-0.295206\pi$
$997$	1.19262e6	1.19981	0.599903	+	0.800073i	$0.295206\pi$
$998$	− 1.16257e6i	− 1.16723i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.5.b.b.161.1	✓	4
3.2	odd	2	inner	162.5.b.b.161.4	yes	4
4.3	odd	2		1296.5.e.a.161.2		4
9.2	odd	6		162.5.d.e.53.3		8
9.4	even	3		162.5.d.e.107.3		8
9.5	odd	6		162.5.d.e.107.2		8
9.7	even	3		162.5.d.e.53.2		8
12.11	even	2		1296.5.e.a.161.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.5.b.b.161.1	✓	4	1.1	even	1	trivial
162.5.b.b.161.4	yes	4	3.2	odd	2	inner
162.5.d.e.53.2		8	9.7	even	3
162.5.d.e.53.3		8	9.2	odd	6
162.5.d.e.107.2		8	9.5	odd	6
162.5.d.e.107.3		8	9.4	even	3
1296.5.e.a.161.2		4	4.3	odd	2
1296.5.e.a.161.3		4	12.11	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:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	162.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.82843i q^{2} -8.00000 q^{4} -0.416102i q^{5} +2.58846 q^{7} +22.6274i q^{8} +O(q^{10})\)	\(q-2.82843i q^{2} -8.00000 q^{4} -0.416102i q^{5} +2.58846 q^{7} +22.6274i q^{8} -1.17691 q^{10} -54.3221i q^{11} -228.885 q^{13} -7.32126i q^{14} +64.0000 q^{16} +380.263i q^{17} -298.242 q^{19} +3.32882i q^{20} -153.646 q^{22} +631.827i q^{23} +624.827 q^{25} +647.383i q^{26} -20.7077 q^{28} +656.703i q^{29} -1544.59 q^{31} -181.019i q^{32} +1075.55 q^{34} -1.07706i q^{35} -187.842 q^{37} +843.556i q^{38} +9.41532 q^{40} +2134.91i q^{41} +2304.97 q^{43} +434.577i q^{44} +1787.08 q^{46} +2005.51i q^{47} -2394.30 q^{49} -1767.28i q^{50} +1831.08 q^{52} -3293.71i q^{53} -22.6036 q^{55} +58.5701i q^{56} +1857.44 q^{58} +3211.14i q^{59} -1822.22 q^{61} +4368.77i q^{62} -512.000 q^{64} +95.2394i q^{65} +2098.09 q^{67} -3042.10i q^{68} -3.04639 q^{70} -2181.30i q^{71} -5527.36 q^{73} +531.297i q^{74} +2385.94 q^{76} -140.610i q^{77} -11026.3 q^{79} -26.6305i q^{80} +6038.44 q^{82} -12058.7i q^{83} +158.228 q^{85} -6519.45i q^{86} +1229.17 q^{88} -2905.35i q^{89} -592.458 q^{91} -5054.62i q^{92} +5672.45 q^{94} +124.099i q^{95} -12892.8 q^{97} +6772.10i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{4} - 52 q^{7}+O(q^{10}) \) 4 * q - 32 * q^4 - 52 * q^7	\( 4 q - 32 q^{4} - 52 q^{7} + 120 q^{10} - 292 q^{13} + 256 q^{16} + 740 q^{19} - 864 q^{22} + 1564 q^{25} + 416 q^{28} - 5056 q^{31} - 312 q^{34} + 4424 q^{37} - 960 q^{40} + 428 q^{43} + 2160 q^{46} - 7956 q^{49} + 2336 q^{52} + 4212 q^{55} + 72 q^{58} + 4496 q^{61} - 2048 q^{64} + 16436 q^{67} - 3504 q^{70} - 20800 q^{73} - 5920 q^{76} - 16732 q^{79} - 1536 q^{82} + 19152 q^{85} + 6912 q^{88} - 5924 q^{91} + 29424 q^{94} - 3808 q^{97}+O(q^{100}) \) 4 * q - 32 * q^4 - 52 * q^7 + 120 * q^10 - 292 * q^13 + 256 * q^16 + 740 * q^19 - 864 * q^22 + 1564 * q^25 + 416 * q^28 - 5056 * q^31 - 312 * q^34 + 4424 * q^37 - 960 * q^40 + 428 * q^43 + 2160 * q^46 - 7956 * q^49 + 2336 * q^52 + 4212 * q^55 + 72 * q^58 + 4496 * q^61 - 2048 * q^64 + 16436 * q^67 - 3504 * q^70 - 20800 * q^73 - 5920 * q^76 - 16732 * q^79 - 1536 * q^82 + 19152 * q^85 + 6912 * q^88 - 5924 * q^91 + 29424 * q^94 - 3808 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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