LMFDB - Embedded newform 141.4.c.a.140.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [141,4,Mod(140,141)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(141, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("141.140");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 141 = 3 \cdot 47 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	141.c (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:	$8.31926931081$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	10.0.2239697333984375.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 9x^{5} + 32 $ x^10 - 9*x^5 + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			140.4
Root			$1.40224 + 0.183603i$ of defining polynomial
Character	$\chi$	$=$	141.140
Dual form			141.4.c.a.140.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.15372i q^{2} +(-4.87197 - 1.80662i) q^{3} +3.36148 q^{4} +(-3.89095 + 10.4929i) q^{6} +27.7516 q^{7} -24.4695i q^{8} +(20.4723 + 17.6036i) q^{9} +O(q^{10})$	\(q-2.15372i q^{2} +(-4.87197 - 1.80662i) q^{3} +3.36148 q^{4} +(-3.89095 + 10.4929i) q^{6} +27.7516 q^{7} -24.4695i q^{8} +(20.4723 + 17.6036i) q^{9} +(-16.3771 - 6.07291i) q^{12} -59.7692i q^{14} -25.8086 q^{16} -123.262i q^{17} +(37.9132 - 44.0916i) q^{18} +(-135.205 - 50.1365i) q^{21} +(-44.2070 + 119.215i) q^{24} -125.000 q^{25} +(-67.9375 - 122.750i) q^{27} +93.2865 q^{28} -140.171i q^{32} -265.471 q^{34} +(68.8172 + 59.1741i) q^{36} +355.336 q^{37} +(-107.980 + 291.194i) q^{42} -322.216i q^{47} +(125.739 + 46.6262i) q^{48} +427.150 q^{49} +269.215i q^{50} +(-222.687 + 600.528i) q^{51} +748.678i q^{53} +(-264.369 + 146.318i) q^{54} -679.067i q^{56} -176.561i q^{59} +809.290 q^{61} +(568.138 + 488.527i) q^{63} -508.359 q^{64} -414.342i q^{68} +198.496i q^{71} +(430.750 - 500.946i) q^{72} -765.295i q^{74} +(608.997 + 225.827i) q^{75} -1225.77 q^{79} +(109.228 + 720.771i) q^{81} +836.390i q^{83} +(-454.489 - 168.533i) q^{84} +185.710i q^{89} -693.963 q^{94} +(-253.236 + 682.911i) q^{96} -803.860 q^{97} -919.963i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 80 q^{4}+O(q^{10}) $ 10 * q - 80 * q^4	$ 10 q - 80 q^{4} - 175 q^{12} + 640 q^{16} + 725 q^{18} - 805 q^{24} - 1250 q^{25} + 275 q^{42} + 1400 q^{48} + 3430 q^{49} - 2740 q^{51} + 455 q^{54} + 5000 q^{63} - 5120 q^{64} - 5800 q^{72} - 3145 q^{84} + 6440 q^{96}+O(q^{100}) $ 10 * q - 80 * q^4 - 175 * q^12 + 640 * q^16 + 725 * q^18 - 805 * q^24 - 1250 * q^25 + 275 * q^42 + 1400 * q^48 + 3430 * q^49 - 2740 * q^51 + 455 * q^54 + 5000 * q^63 - 5120 * q^64 - 5800 * q^72 - 3145 * q^84 + 6440 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/141\mathbb{Z}\right)^\times$.

$n$	$52$	$95$
$\chi(n)$	$-1$	$-1$

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 $

$2$		−	2.15372i		−	0.761456i	−0.924687	−	0.380728i	$-0.875673\pi$
$2$		−	2.15372i		−	0.761456i	0.924687	−	0.380728i	$-0.124327\pi$
$3$	−4.87197	−	1.80662i	−0.937612	−	0.347684i
$3$	−4.87197	−	1.80662i	−0.937612	−	0.347684i
$4$	3.36148			0.420185
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$	−3.89095	+	10.4929i	−0.264746	+	0.713950i
$7$	27.7516			1.49844			0.749222	−	0.662319i	$-0.230427\pi$
$7$	27.7516			1.49844			0.749222	+	0.662319i	$0.230427\pi$
$8$		−	24.4695i		−	1.08141i
$9$	20.4723	+	17.6036i	0.758232	+	0.651984i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	−16.3771	−	6.07291i	−0.393971	−	0.146092i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		−	59.7692i		−	1.14100i
$15$		0			0
$16$	−25.8086			−0.403259
$17$		−	123.262i		−	1.75855i	−0.476314	−	0.879275i	$-0.658028\pi$
$17$		−	123.262i		−	1.75855i	0.476314	−	0.879275i	$-0.341972\pi$
$18$	37.9132	−	44.0916i	0.496457	−	0.577360i
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$	−135.205	−	50.1365i	−1.40496	−	0.520985i
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$	−44.2070	+	119.215i	−0.375988	+	1.01394i
$25$	−125.000			−1.00000
$26$		0			0
$27$	−67.9375	−	122.750i	−0.484244	−	0.874933i
$28$	93.2865			0.629625
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		−	140.171i		−	0.774344i
$33$		0			0
$34$	−265.471			−1.33906
$35$		0			0
$36$	68.8172	+	59.1741i	0.318598	+	0.273954i
$37$	355.336			1.57884			0.789418	−	0.613856i	$-0.210383\pi$
$37$	355.336			1.57884			0.789418	+	0.613856i	$0.210383\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$	−107.980	+	291.194i	−0.396707	+	1.06981i
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	322.216i		−	1.00000i
$47$		−	322.216i		−	1.00000i
$48$	125.739	+	46.6262i	0.378100	+	0.140206i
$49$	427.150			1.24534
$50$			269.215i			0.761456i
$51$	−222.687	+	600.528i	−0.611419	+	1.64884i
$52$		0			0
$53$			748.678i			1.94036i	0.242394	+	0.970178i	$0.422067\pi$
$53$			748.678i			1.94036i	−0.242394	+	0.970178i	$0.577933\pi$
$54$	−264.369	+	146.318i	−0.666223	+	0.368730i
$55$		0			0
$56$		−	679.067i		−	1.62043i
$57$		0			0
$58$		0			0
$59$		−	176.561i		−	0.389597i	−0.980843	−	0.194799i	$-0.937595\pi$
$59$		−	176.561i		−	0.389597i	0.980843	−	0.194799i	$-0.0624053\pi$
$60$		0			0
$61$	809.290			1.69867			0.849335	−	0.527854i	$-0.177003\pi$
$61$	809.290			1.69867			0.849335	+	0.527854i	$0.177003\pi$
$62$		0			0
$63$	568.138	+	488.527i	1.13617	+	0.976963i
$64$	−508.359			−0.992888
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		−	414.342i		−	0.738917i
$69$		0			0
$70$		0			0
$71$			198.496i			0.331791i	0.986143	+	0.165896i	$0.0530515\pi$
$71$			198.496i			0.331791i	−0.986143	+	0.165896i	$0.946949\pi$
$72$	430.750	−	500.946i	0.705061	−	0.819959i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		−	765.295i		−	1.20221i
$75$	608.997	+	225.827i	0.937612	+	0.347684i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−1225.77			−1.74569			−0.872845	−	0.487997i	$-0.837728\pi$
$79$	−1225.77			−1.74569			−0.872845	+	0.487997i	$0.837728\pi$
$80$		0			0
$81$	109.228	+	720.771i	0.149833	+	0.988711i
$82$		0			0
$83$			836.390i			1.10609i	0.833150	+	0.553046i	$0.186535\pi$
$83$			836.390i			1.10609i	−0.833150	+	0.553046i	$0.813465\pi$
$84$	−454.489	−	168.533i	−0.590343	−	0.218910i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$			185.710i			0.221182i	0.993866	+	0.110591i	$0.0352744\pi$
$89$			185.710i			0.221182i	−0.993866	+	0.110591i	$0.964726\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$	−693.963			−0.761456
$95$		0			0
$96$	−253.236	+	682.911i	−0.269227	+	0.726035i
$97$	−803.860			−0.841439			−0.420720	−	0.907191i	$-0.638222\pi$
$97$	−803.860			−0.841439			−0.420720	+	0.907191i	$0.638222\pi$
$98$		−	919.963i		−	0.948269i
$99$		0			0
$100$	−420.185			−0.420185
$101$			1537.60i			1.51482i	0.652937	+	0.757412i	$0.273537\pi$
$101$			1537.60i			1.51482i	−0.652937	+	0.757412i	$0.726463\pi$
$102$	1293.37	+	479.605i	1.25552	+	0.465568i
$103$	1437.50			1.37515			0.687576	−	0.726112i	$-0.258675\pi$
$103$	1437.50			1.37515			0.687576	+	0.726112i	$0.258675\pi$
$104$		0			0
$105$		0			0
$106$	1612.44			1.47749
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	−228.371	−	412.621i	−0.203472	−	0.367634i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$	−1731.19	−	641.956i	−1.48034	−	0.548935i
$112$	−716.229			−0.604261
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−380.262			−0.296661
$119$		−	3420.71i		−	2.63509i
$120$		0			0
$121$	−1331.00			−1.00000
$122$		−	1742.98i		−	1.29346i
$123$		0			0
$124$		0			0
$125$		0			0
$126$	1052.15	−	1223.61i	0.743914	−	0.865143i
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		−	26.5076i		−	0.0183044i
$129$		0			0
$130$		0			0
$131$		−	1898.97i		−	1.26652i	−0.773939	−	0.633260i	$-0.781716\pi$
$131$		−	1898.97i		−	1.26652i	0.773939	−	0.633260i	$-0.218284\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−3016.15			−1.90171
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$	−582.120	+	1569.83i	−0.347684	+	0.937612i
$142$	427.506			0.252644
$143$		0			0
$144$	−528.360	−	454.323i	−0.305764	−	0.262919i
$145$		0			0
$146$		0			0
$147$	−2081.07	−	771.697i	−1.16764	−	0.432983i
$148$	1194.46			0.663404
$149$			3545.86i			1.94959i	0.223111	+	0.974793i	$0.428379\pi$
$149$			3545.86i			1.94959i	−0.223111	+	0.974793i	$0.571621\pi$
$150$	486.369	−	1311.61i	0.264746	−	0.713950i
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	2169.85	−	2523.45i	1.14655	−	1.33339i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	3608.44			1.83430			0.917149	−	0.398543i	$-0.130484\pi$
$157$	3608.44			1.83430			0.917149	+	0.398543i	$0.130484\pi$
$158$			2639.96i			1.32927i
$159$	1352.57	−	3647.54i	0.674630	−	1.81930i
$160$		0			0
$161$		0			0
$162$	1552.34	−	235.247i	0.752860	−	0.114091i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$	1801.35			0.842241
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$	−1226.81	+	3308.40i	−0.563397	+	1.51933i
$169$	2197.00			1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$			4456.50i			1.95850i	0.202648	+	0.979252i	$0.435045\pi$
$173$			4456.50i			1.95850i	−0.202648	+	0.979252i	$0.564955\pi$
$174$		0			0
$175$	−3468.95			−1.49844
$176$		0			0
$177$	−318.977	+	860.199i	−0.135456	+	0.365291i
$178$	399.968			0.168421
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$	−3942.84	−	1462.08i	−1.59269	−	0.590600i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		−	1083.12i		−	0.420185i
$189$	−1885.37	−	3406.50i	−0.725612	−	1.31104i
$190$		0			0
$191$			5278.85i			1.99981i	0.0136381	+	0.999907i	$0.495659\pi$
$191$			5278.85i			1.99981i	−0.0136381	+	0.999907i	$0.504341\pi$
$192$	2476.71	+	918.409i	0.930943	+	0.345211i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$			1731.29i			0.640719i
$195$		0			0
$196$	1435.86			0.523272
$197$			4415.04i			1.59674i	0.602164	+	0.798372i	$0.294305\pi$
$197$			4415.04i			1.59674i	−0.602164	+	0.798372i	$0.705695\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$			3058.68i			1.08141i
$201$		0			0
$202$	3311.57			1.15347
$203$		0			0
$204$	−748.557	+	2018.66i	−0.256909	+	0.692817i
$205$		0			0
$206$		−	3095.97i		−	1.04712i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$			2516.67i			0.815309i
$213$	358.606	−	967.068i	0.115358	−	0.311091i
$214$		0			0
$215$		0			0
$216$	−3003.62	+	1662.39i	−0.946160	+	0.523665i
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$	−1382.60	+	3728.50i	−0.417990	+	1.12721i
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		−	3889.98i		−	1.16031i
$225$	−2559.03	−	2200.45i	−0.758232	−	0.651984i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$		0			0
$236$		−	593.505i		−	0.163703i
$237$	5971.91	+	2214.49i	1.63678	+	0.606948i
$238$	−7367.25			−2.00650
$239$		−	7030.31i		−	1.90273i	−0.308064	−	0.951366i	$-0.599681\pi$
$239$		−	7030.31i		−	1.90273i	0.308064	−	0.951366i	$-0.400319\pi$
$240$		0			0
$241$	−3202.48			−0.855973			−0.427987	−	0.903785i	$-0.640777\pi$
$241$	−3202.48			−0.855973			−0.427987	+	0.903785i	$0.640777\pi$
$242$			2866.60i			0.761456i
$243$	770.000	−	3708.91i	0.203274	−	0.979122i
$244$	2720.41			0.713756
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	1511.04	−	4074.87i	0.384570	−	1.03709i
$250$		0			0
$251$		−	7110.68i		−	1.78814i	−0.447931	−	0.894068i	$-0.647839\pi$
$251$		−	7110.68i		−	1.78814i	0.447931	−	0.894068i	$-0.352161\pi$
$252$	1909.79	+	1642.18i	0.477402	+	0.410505i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−4123.96			−1.00683
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$	9861.14			2.36580
$260$		0			0
$261$		0			0
$262$	−4089.86			−0.964398
$263$			8334.53i			1.95410i	0.213003	+	0.977052i	$0.431676\pi$
$263$			8334.53i			1.95410i	−0.213003	+	0.977052i	$0.568324\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	335.507	−	904.775i	0.0769015	−	0.207383i
$268$		0			0
$269$		−	1508.24i		−	0.341856i	−0.985284	−	0.170928i	$-0.945323\pi$
$269$		−	1508.24i		−	0.341856i	0.985284	−	0.170928i	$-0.0546765\pi$
$270$		0			0
$271$	−2545.11			−0.570497			−0.285249	−	0.958454i	$-0.592076\pi$
$271$	−2545.11			−0.570497			−0.285249	+	0.958454i	$0.592076\pi$
$272$			3181.21i			0.709151i
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	8612.13			1.86806			0.934030	−	0.357195i	$-0.116267\pi$
$277$	8612.13			1.86806			0.934030	+	0.357195i	$0.116267\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$	3380.97	+	1253.73i	0.713950	+	0.264746i
$283$	4157.05			0.873185			0.436592	−	0.899659i	$-0.356185\pi$
$283$	4157.05			0.873185			0.436592	+	0.899659i	$0.356185\pi$
$284$			667.242i			0.139414i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	2467.52	−	2869.63i	0.504860	−	0.587133i
$289$	−10280.4			−2.09250
$290$		0			0
$291$	3916.39	+	1452.27i	0.788944	+	0.292555i
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	−1662.02	+	4482.04i	−0.329697	+	0.889108i
$295$		0			0
$296$		−	8694.89i		−	1.70737i
$297$		0			0
$298$	7636.80			1.48452
$299$		0			0
$300$	2047.13	+	759.114i	0.393971	+	0.146092i
$301$		0			0
$302$		0			0
$303$	2777.86	−	7491.16i	0.526679	−	1.42032i
$304$		0			0
$305$		0			0
$306$	−5434.80	−	4673.25i	−1.01532	−	0.873045i
$307$	−10690.3			−1.98739			−0.993695	−	0.112113i	$-0.964238\pi$
$307$	−10690.3			−1.98739			−0.993695	+	0.112113i	$0.964238\pi$
$308$		0			0
$309$	−7003.44	−	2597.00i	−1.28936	−	0.478118i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		−	7771.58i		−	1.39674i
$315$		0			0
$316$	−4120.39			−0.733514
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$	−7855.79	−	2913.07i	−1.38532	−	0.513701i
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	367.168	+	2422.86i	0.0629575	+	0.415442i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	8942.00i		−	1.49844i
$330$		0			0
$331$	−8464.69			−1.40562			−0.702812	−	0.711376i	$-0.748072\pi$
$331$	−8464.69			−1.40562			−0.702812	+	0.711376i	$0.748072\pi$
$332$			2811.51i			0.464764i
$333$	7274.54	+	6255.19i	1.19712	+	1.02938i
$334$		0			0
$335$		0			0
$336$	3489.45	+	1293.95i	0.566563	+	0.210092i
$337$	−10296.0			−1.66426			−0.832132	−	0.554577i	$-0.812880\pi$
$337$	−10296.0			−1.66426			−0.832132	+	0.554577i	$0.812880\pi$
$338$		−	4731.73i		−	0.761456i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	2335.31			0.367624
$344$		0			0
$345$		0			0
$346$	9598.05			1.49131
$347$		−	3336.83i		−	0.516226i	−0.966115	−	0.258113i	$-0.916899\pi$
$347$		−	3336.83i		−	0.516226i	0.966115	−	0.258113i	$-0.0831006\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$			7471.15i			1.14100i
$351$		0			0
$352$		0			0
$353$		−	13263.6i		−	1.99986i	−0.0119921	−	0.999928i	$-0.503817\pi$
$353$		−	13263.6i		−	1.99986i	0.0119921	−	0.999928i	$-0.496183\pi$
$354$	1852.63	+	686.988i	0.278153	+	0.103144i
$355$		0			0
$356$			624.261i			0.0929376i
$357$	−6179.91	+	16665.6i	−0.916177	+	2.47069i
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	6859.00			1.00000
$362$		0			0
$363$	6484.60	+	2404.61i	0.937612	+	0.347684i
$364$		0			0
$365$		0			0
$366$	−3148.90	+	8491.78i	−0.449715	+	1.21277i
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			20777.0i			2.90752i
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$	−7884.45			−1.08141
$377$		0			0
$378$	−7336.65	+	4060.57i	−0.998298	+	0.552521i
$379$	−13825.1			−1.87374			−0.936868	−	0.349684i	$-0.886289\pi$
$379$	−13825.1			−1.87374			−0.936868	+	0.349684i	$0.886289\pi$
$380$		0			0
$381$		0			0
$382$	11369.2			1.52277
$383$		−	14982.5i		−	1.99887i	−0.0335940	−	0.999436i	$-0.510695\pi$
$383$		−	14982.5i		−	1.99887i	0.0335940	−	0.999436i	$-0.489305\pi$
$384$	−47.8890	+	129.144i	−0.00636413	+	0.0171624i
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−2702.16			−0.353560
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$		−	10452.1i		−	1.34672i
$393$	−3430.72	+	9251.75i	−0.440348	+	1.18750i
$394$	9508.77			1.21585
$395$		0			0
$396$		0			0
$397$	104.405			0.0131989			0.00659944	−	0.999978i	$-0.497899\pi$
$397$	104.405			0.0131989			0.00659944	+	0.999978i	$0.497899\pi$
$398$		0			0
$399$		0			0
$400$	3226.07			0.403259
$401$			3845.71i			0.478916i	0.970907	+	0.239458i	$0.0769698\pi$
$401$			3845.71i			0.478916i	−0.970907	+	0.239458i	$0.923030\pi$
$402$		0			0
$403$		0			0
$404$			5168.63i			0.636507i
$405$		0			0
$406$		0			0
$407$		0			0
$408$	14694.6	+	5449.02i	1.78307	+	0.661193i
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$	4832.12			0.577819
$413$		−	4899.84i		−	0.583790i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$	5672.15	−	6596.49i	0.651984	−	0.758232i
$424$	18319.8			2.09832
$425$			15407.7i			1.75855i
$426$	−2082.80	−	772.339i	−0.236882	−	0.0878402i
$427$	22459.1			2.54536
$428$		0			0
$429$		0			0
$430$		0			0
$431$			11293.7i			1.26217i	0.775713	+	0.631086i	$0.217391\pi$
$431$			11293.7i			1.26217i	−0.775713	+	0.631086i	$0.782609\pi$
$432$	1753.37	+	3168.00i	0.195276	+	0.352825i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−10024.0			−1.08979			−0.544897	−	0.838503i	$-0.683431\pi$
$439$	−10024.0			−1.08979			−0.544897	+	0.838503i	$0.683431\pi$
$440$		0			0
$441$	8744.74	+	7519.38i	0.944255	+	0.811940i
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$	−5819.36	−	2157.93i	−0.622015	−	0.230654i
$445$		0			0
$446$		0			0
$447$	6406.01	−	17275.3i	0.677839	−	1.82796i
$448$	−14107.8			−1.48779
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$	−4739.15	+	5511.45i	−0.496457	+	0.577360i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	19512.2			1.99725			0.998624	−	0.0524363i	$-0.0166986\pi$
$457$	19512.2			1.99725			0.998624	+	0.0524363i	$0.0166986\pi$
$458$		0			0
$459$	−15130.3	+	8374.09i	−1.53861	+	0.851567i
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−17580.2	−	6519.07i	−1.71986	−	0.637755i
$472$	−4320.34			−0.421313
$473$		0			0
$474$	4769.40	−	12861.8i	0.462164	−	1.24634i
$475$		0			0
$476$		−	11498.7i		−	1.10723i
$477$	−13179.4	+	15327.1i	−1.26508	+	1.47124i
$478$	−15141.3			−1.44885
$479$			16902.7i			1.61232i	0.591696	+	0.806161i	$0.298458\pi$
$479$			16902.7i			1.61232i	−0.591696	+	0.806161i	$0.701542\pi$
$480$		0			0
$481$		0			0
$482$			6897.24i			0.651786i
$483$		0			0
$484$	−4474.13			−0.420185
$485$		0			0
$486$	−7987.96	−	1658.37i	−0.745558	−	0.154784i
$487$	19280.0			1.79396			0.896982	−	0.442068i	$-0.145755\pi$
$487$	19280.0			1.79396			0.896982	+	0.442068i	$0.145755\pi$
$488$		−	19802.9i		−	1.83696i
$489$		0			0
$490$		0			0
$491$		−	21550.7i		−	1.98079i	−0.138250	−	0.990397i	$-0.544148\pi$
$491$		−	21550.7i		−	1.98079i	0.138250	−	0.990397i	$-0.455852\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$			5508.58i			0.497171i
$498$	−8776.14	−	3254.35i	−0.789695	−	0.292833i
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$		0			0
$501$		0			0
$502$	−15314.4			−1.36159
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$	11954.0	−	13902.0i	1.05650	−	1.22866i
$505$		0			0
$506$		0			0
$507$	−10703.7	−	3969.14i	−0.937612	−	0.347684i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$			8669.80i			0.748349i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		−	21238.2i		−	1.80145i
$519$	8051.18	−	21711.9i	0.680939	−	1.83632i
$520$		0			0
$521$			20808.5i			1.74978i	0.484319	+	0.874891i	$0.339067\pi$
$521$			20808.5i			1.74978i	−0.484319	+	0.874891i	$0.660933\pi$
$522$		0			0
$523$	−23380.0			−1.95475			−0.977377	−	0.211506i	$-0.932163\pi$
$523$	−23380.0			−1.95475			−0.977377	+	0.211506i	$0.932163\pi$
$524$		−	6383.37i		−	0.532173i
$525$	16900.6	+	6267.06i	1.40496	+	0.520985i
$526$	17950.3			1.48796
$527$		0			0
$528$		0			0
$529$	−12167.0			−1.00000
$530$		0			0
$531$	3108.10	−	3614.60i	0.254011	−	0.295405i
$532$		0			0
$533$		0			0
$534$	−1948.63	−	722.589i	−0.157913	−	0.0585571i
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−3248.34			−0.260308
$539$		0			0
$540$		0			0
$541$	24054.5			1.91162			0.955808	−	0.293991i	$-0.0949834\pi$
$541$	24054.5			1.91162			0.955808	+	0.293991i	$0.0949834\pi$
$542$			5481.47i			0.434408i
$543$		0			0
$544$	−17277.8			−1.36172
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$	16568.0	+	14246.4i	1.28799	+	1.10751i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−34017.0			−2.61582
$554$		−	18548.1i		−	1.42244i
$555$		0			0
$556$		0			0
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$	−1956.79	+	5276.95i	−0.146092	+	0.393971i
$565$		0			0
$566$		−	8953.14i		−	0.664891i
$567$	3031.25	+	20002.5i	0.224516	+	1.48153i
$568$	4857.10			0.358802
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$	−6667.49			−0.488661			−0.244331	−	0.969692i	$-0.578568\pi$
$571$	−6667.49			−0.488661			−0.244331	+	0.969692i	$0.578568\pi$
$572$		0			0
$573$	9536.86	−	25718.4i	0.695302	−	1.87505i
$574$		0			0
$575$		0			0
$576$	−10407.3	−	8948.93i	−0.752840	−	0.647347i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$			22141.2i			1.59335i
$579$		0			0
$580$		0			0
$581$			23211.1i			1.65742i
$582$	3127.78	−	8434.80i	0.222767	−	0.600745i
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	−6995.47	−	2594.05i	−0.490626	−	0.181933i
$589$		0			0
$590$		0			0
$591$	7976.29	−	21510.0i	0.555162	−	1.49713i
$592$	−9170.72			−0.636680
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$			11919.4i			0.819188i
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$	5525.87	−	14901.8i	0.375988	−	1.01394i
$601$	29338.7			1.99127			0.995633	−	0.0933503i	$-0.0297576\pi$
$601$	29338.7			1.99127			0.995633	+	0.0933503i	$0.0297576\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$	−16133.9	−	5982.74i	−1.08151	−	0.401043i
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	7293.90	−	8482.53i	0.481762	−	0.560271i
$613$	−9074.91			−0.597931			−0.298966	−	0.954264i	$-0.596642\pi$
$613$	−9074.91			−0.597931			−0.298966	+	0.954264i	$0.596642\pi$
$614$			23024.0i			1.51331i
$615$		0			0
$616$		0			0
$617$		−	20464.9i		−	1.33531i	−0.744470	−	0.667656i	$-0.767298\pi$
$617$		−	20464.9i		−	1.33531i	0.744470	−	0.667656i	$-0.232702\pi$
$618$	−5593.22	+	15083.5i	−0.364065	+	0.981790i
$619$	30044.0			1.95084			0.975420	−	0.220353i	$-0.0707208\pi$
$619$	30044.0			1.95084			0.975420	+	0.220353i	$0.0707208\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			5153.75i			0.331430i
$624$		0			0
$625$	15625.0			1.00000
$626$		0			0
$627$		0			0
$628$	12129.7			0.770746
$629$		−	43799.4i		−	2.77646i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$			29993.9i			1.88780i
$633$		0			0
$634$		0			0
$635$		0			0
$636$	4546.65	−	12261.1i	0.283469	−	0.764443i
$637$		0			0
$638$		0			0
$639$	−3494.24	+	4063.67i	−0.216323	+	0.251575i
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$	−10093.5			−0.619048			−0.309524	−	0.950892i	$-0.600170\pi$
$643$	−10093.5			−0.619048			−0.309524	+	0.950892i	$0.600170\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			14028.8i			0.852442i	0.904619	+	0.426221i	$0.140155\pi$
$647$			14028.8i			0.852442i	−0.904619	+	0.426221i	$0.859845\pi$
$648$	17636.9	−	2672.75i	1.06920	−	0.162030i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	4289.50i		−	0.257061i	−0.991706	−	0.128531i	$-0.958974\pi$
$653$		−	4289.50i		−	0.257061i	0.991706	−	0.128531i	$-0.0410260\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	−19258.6			−1.14100
$659$			24098.4i			1.42449i	0.701931	+	0.712245i	$0.252321\pi$
$659$			24098.4i			1.42449i	−0.701931	+	0.712245i	$0.747679\pi$
$660$		0			0
$661$	−23217.4			−1.36619			−0.683097	−	0.730328i	$-0.739367\pi$
$661$	−23217.4			−1.36619			−0.683097	+	0.730328i	$0.739367\pi$
$662$			18230.6i			1.07032i
$663$		0			0
$664$	20466.0			1.19614
$665$		0			0
$666$	13471.9	−	15667.3i	0.783824	−	0.911557i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$	−7027.69	+	18951.9i	−0.403421	+	1.08792i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$			22174.7i			1.26726i
$675$	8492.19	+	15343.7i	0.484244	+	0.874933i
$676$	7385.18			0.420185
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$	−22308.4			−1.26085
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	15697.9i		−	0.879448i	−0.898133	−	0.439724i	$-0.855076\pi$
$683$		−	15697.9i		−	0.879448i	0.898133	−	0.439724i	$-0.144924\pi$
$684$		0			0
$685$		0			0
$686$		−	5029.61i		−	0.279929i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$			14980.4i			0.822934i
$693$		0			0
$694$	−7186.60			−0.393083
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−11660.8			−0.629625
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−28566.1			−1.52280
$707$			42670.9i			2.26988i
$708$	−1072.24	+	2891.54i	−0.0569168	+	0.153490i
$709$	−29733.9			−1.57501			−0.787503	−	0.616311i	$-0.788626\pi$
$709$	−29733.9			−1.57501			−0.787503	+	0.616311i	$0.788626\pi$
$710$		0			0
$711$	−25094.2	−	21577.9i	−1.32364	−	1.13816i
$712$	4544.23			0.239188
$713$		0			0
$714$	35893.1	+	13309.8i	1.88132	+	0.697628i
$715$		0			0
$716$		0			0
$717$	−12701.1	+	34251.5i	−0.661548	+	1.78402i
$718$		0			0
$719$		−	36136.4i		−	1.87435i	−0.348857	−	0.937176i	$-0.613430\pi$
$719$		−	36136.4i		−	1.87435i	0.348857	−	0.937176i	$-0.386570\pi$
$720$		0			0
$721$	39892.8			2.06059
$722$		−	14772.4i		−	0.761456i
$723$	15602.4	+	5785.64i	0.802571	+	0.297608i
$724$		0			0
$725$		0			0
$726$	5178.85	−	13966.0i	0.264746	−	0.713950i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	−10452.0	+	16678.6i	−0.531016	+	0.847362i
$730$		0			0
$731$		0			0
$732$	−13253.8	−	4914.74i	−0.669227	−	0.248161i
$733$	20180.5			1.01690			0.508448	−	0.861093i	$-0.330219\pi$
$733$	20180.5			1.01690			0.508448	+	0.861093i	$0.330219\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	33359.8			1.66057			0.830283	−	0.557342i	$-0.188179\pi$
$739$	33359.8			1.66057			0.830283	+	0.557342i	$0.188179\pi$
$740$		0			0
$741$		0			0
$742$	44747.9			2.21394
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−14723.5	+	17122.8i	−0.721155	+	0.838675i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$			8315.93i			0.403259i
$753$	−12846.3	+	34643.1i	−0.621706	+	1.67658i
$754$		0			0
$755$		0			0
$756$	−6337.65	−	11450.9i	−0.304892	−	0.550879i
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$			29775.3i			1.42677i
$759$		0			0
$760$		0			0
$761$		−	39762.8i		−	1.89409i	−0.321105	−	0.947044i	$-0.604054\pi$
$761$		−	39762.8i		−	1.89409i	0.321105	−	0.947044i	$-0.395946\pi$
$762$		0			0
$763$		0			0
$764$			17744.8i			0.840293i
$765$		0			0
$766$	−32268.0			−1.52205
$767$		0			0
$768$	20091.8	+	7450.41i	0.944012	+	0.350057i
$769$	−9729.25			−0.456236			−0.228118	−	0.973633i	$-0.573257\pi$
$769$	−9729.25			−0.456236			−0.228118	+	0.973633i	$0.573257\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	35364.2i		−	1.64549i	−0.568414	−	0.822743i	$-0.692443\pi$
$773$		−	35364.2i		−	1.64549i	0.568414	−	0.822743i	$-0.307557\pi$
$774$		0			0
$775$		0			0
$776$			19670.0i			0.909939i
$777$	−48043.2	−	17815.3i	−2.21820	−	0.822549i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−11024.1			−0.502193
$785$		0			0
$786$	19925.7	+	7388.81i	0.904231	+	0.335305i
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$			14841.1i			0.670929i
$789$	15057.3	−	40605.6i	0.679409	−	1.83219i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		−	224.860i		−	0.0100504i
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$	−39716.9			−1.75855
$800$			17521.4i			0.774344i
$801$	−3269.16	+	3801.91i	−0.144207	+	0.167708i
$802$	8282.58			0.364673
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−2724.82	+	7348.13i	−0.118858	+	0.320528i
$808$	37624.3			1.63814
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$	24255.0			1.05019			0.525097	−	0.851042i	$-0.324029\pi$
$811$	24255.0			1.05019			0.525097	+	0.851042i	$0.324029\pi$
$812$		0			0
$813$	12399.7	+	4598.05i	0.534905	+	0.198352i
$814$		0			0
$815$		0			0
$816$	5747.22	−	15498.8i	0.246560	−	0.664909i
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$		0			0
$823$	34760.0			1.47224			0.736122	−	0.676848i	$-0.236655\pi$
$823$	34760.0			1.47224			0.736122	+	0.676848i	$0.236655\pi$
$824$		−	35174.8i		−	1.48710i
$825$		0			0
$826$	−10552.9			−0.444530
$827$		−	38658.2i		−	1.62549i	−0.582622	−	0.812743i	$-0.697973\pi$
$827$		−	38658.2i		−	1.62549i	0.582622	−	0.812743i	$-0.302027\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$	−41958.1	−	15558.8i	−1.75151	−	0.649493i
$832$		0			0
$833$		−	52651.3i		−	2.18999i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−24389.0			−1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$	−14207.0	−	12216.2i	−0.577360	−	0.496457i
$847$	−36937.4			−1.49844
$848$		−	19322.3i		−	0.782466i
$849$	−20253.1	−	7510.20i	−0.818708	−	0.303592i
$850$	33183.9			1.33906
$851$		0			0
$852$	1205.45	−	3250.78i	0.0484719	−	0.130716i
$853$	−48273.7			−1.93770			−0.968851	−	0.247646i	$-0.920343\pi$
$853$	−48273.7			−1.93770			−0.968851	+	0.247646i	$0.920343\pi$
$854$		−	48370.6i		−	1.93818i
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$	24323.4			0.961088
$863$		−	46757.7i		−	1.84432i	−0.386805	−	0.922162i	$-0.626421\pi$
$863$		−	46757.7i		−	1.84432i	0.386805	−	0.922162i	$-0.373579\pi$
$864$	−17206.0	+	9522.89i	−0.677500	+	0.374971i
$865$		0			0
$866$		0			0
$867$	50086.1	+	18572.8i	1.96195	+	0.727527i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−16456.8	−	14150.8i	−0.638007	−	0.548605i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$			21588.9i			0.829829i
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$	16194.6	−	18833.7i	0.618256	−	0.719008i
$883$	31460.6			1.19902			0.599509	−	0.800368i	$-0.295363\pi$
$883$	31460.6			1.19902			0.599509	+	0.800368i	$0.295363\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$	−15708.3	+	42361.3i	−0.593623	+	1.60085i
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$	−37206.3	−	13796.8i	−1.39191	−	0.516144i
$895$		0			0
$896$		−	735.627i		−	0.0274281i
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−8602.15	−	7396.77i	−0.318598	−	0.273954i
$901$	92283.3			3.41221
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−34540.0			−1.26448			−0.632239	−	0.774773i	$-0.717864\pi$
$907$	−34540.0			−1.26448			−0.632239	+	0.774773i	$0.717864\pi$
$908$		0			0
$909$	−27067.3	+	31478.2i	−0.987642	+	1.14859i
$910$		0			0
$911$			17700.2i			0.643725i	0.946786	+	0.321862i	$0.104309\pi$
$911$			17700.2i			0.643725i	−0.946786	+	0.321862i	$0.895691\pi$
$912$		0			0
$913$		0			0
$914$		−	42023.9i		−	1.52082i
$915$		0			0
$916$		0			0
$917$		−	52699.5i		−	1.89781i
$918$	18035.5	+	32586.5i	0.648430	+	1.17159i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	52083.0	+	19313.3i	1.86340	+	0.690983i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−44417.0			−1.57884
$926$		0			0
$927$	29428.8	+	25305.1i	1.04268	+	0.896578i
$928$		0			0
$929$			4177.63i			0.147539i	0.997275	+	0.0737694i	$0.0235029\pi$
$929$			4177.63i			0.147539i	−0.997275	+	0.0737694i	$0.976497\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			52901.9i			1.83268i	0.400398	+	0.916341i	$0.368872\pi$
$941$			52901.9i			1.83268i	−0.400398	+	0.916341i	$0.631128\pi$
$942$	−14040.3	+	37862.9i	−0.485622	+	1.30960i
$943$		0			0
$944$			4556.78i			0.157108i
$945$		0			0
$946$		0			0
$947$			36376.1i			1.24822i	0.781336	+	0.624110i	$0.214538\pi$
$947$			36376.1i			1.24822i	−0.781336	+	0.624110i	$0.785462\pi$
$948$	20074.5	+	7443.97i	0.687751	+	0.255031i
$949$		0			0
$950$		0			0
$951$		0			0
$952$	−83702.9			−2.84961
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$	33010.4	+	28384.8i	1.12028	+	0.963304i
$955$		0			0
$956$		−	23632.3i		−	0.799500i
$957$		0			0
$958$	36403.6			1.22771
$959$		0			0
$960$		0			0
$961$	29791.0			1.00000
$962$		0			0
$963$		0			0
$964$	−10765.1			−0.359667
$965$		0			0
$966$		0			0
$967$	−41448.6			−1.37838			−0.689192	−	0.724579i	$-0.742034\pi$
$967$	−41448.6			−1.37838			−0.689192	+	0.724579i	$0.742034\pi$
$968$			32568.9i			1.08141i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	2588.34	−	12467.4i	0.0854127	−	0.411413i
$973$		0			0
$974$		−	41523.8i		−	1.36602i
$975$		0			0
$976$	−20886.6			−0.685004
$977$		−	4223.08i		−	0.138289i	−0.997607	−	0.0691445i	$-0.977973\pi$
$977$		−	4223.08i		−	0.138289i	0.997607	−	0.0691445i	$-0.0220270\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−46414.2			−1.50829
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−16154.8	+	43565.2i	−0.520985	+	1.40496i
$988$		0			0
$989$		0			0
$990$		0			0
$991$	15662.2			0.502043			0.251022	−	0.967981i	$-0.419233\pi$
$991$	15662.2			0.502043			0.251022	+	0.967981i	$0.419233\pi$
$992$		0			0
$993$	41239.7	+	15292.4i	1.31793	+	0.488712i
$994$	11864.0			0.378573
$995$		0			0
$996$	5079.32	−	13697.6i	0.161591	−	0.435768i
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$		0			0
$999$	−24140.7	−	43617.4i	−0.764541	−	1.38138i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	141.4.c.a.140.4	✓	10
3.2	odd	2	inner	141.4.c.a.140.7	yes	10
47.46	odd	2	CM	141.4.c.a.140.4	✓	10
141.140	even	2	inner	141.4.c.a.140.7	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
141.4.c.a.140.4	✓	10	1.1	even	1	trivial
141.4.c.a.140.4	✓	10	47.46	odd	2	CM
141.4.c.a.140.7	yes	10	3.2	odd	2	inner
141.4.c.a.140.7	yes	10	141.140	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 141 = 3 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	141.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.15372i q^{2} +(-4.87197 - 1.80662i) q^{3} +3.36148 q^{4} +(-3.89095 + 10.4929i) q^{6} +27.7516 q^{7} -24.4695i q^{8} +(20.4723 + 17.6036i) q^{9} +O(q^{10})\)	\(q-2.15372i q^{2} +(-4.87197 - 1.80662i) q^{3} +3.36148 q^{4} +(-3.89095 + 10.4929i) q^{6} +27.7516 q^{7} -24.4695i q^{8} +(20.4723 + 17.6036i) q^{9} +(-16.3771 - 6.07291i) q^{12} -59.7692i q^{14} -25.8086 q^{16} -123.262i q^{17} +(37.9132 - 44.0916i) q^{18} +(-135.205 - 50.1365i) q^{21} +(-44.2070 + 119.215i) q^{24} -125.000 q^{25} +(-67.9375 - 122.750i) q^{27} +93.2865 q^{28} -140.171i q^{32} -265.471 q^{34} +(68.8172 + 59.1741i) q^{36} +355.336 q^{37} +(-107.980 + 291.194i) q^{42} -322.216i q^{47} +(125.739 + 46.6262i) q^{48} +427.150 q^{49} +269.215i q^{50} +(-222.687 + 600.528i) q^{51} +748.678i q^{53} +(-264.369 + 146.318i) q^{54} -679.067i q^{56} -176.561i q^{59} +809.290 q^{61} +(568.138 + 488.527i) q^{63} -508.359 q^{64} -414.342i q^{68} +198.496i q^{71} +(430.750 - 500.946i) q^{72} -765.295i q^{74} +(608.997 + 225.827i) q^{75} -1225.77 q^{79} +(109.228 + 720.771i) q^{81} +836.390i q^{83} +(-454.489 - 168.533i) q^{84} +185.710i q^{89} -693.963 q^{94} +(-253.236 + 682.911i) q^{96} -803.860 q^{97} -919.963i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 80 q^{4}+O(q^{10}) \) 10 * q - 80 * q^4	\( 10 q - 80 q^{4} - 175 q^{12} + 640 q^{16} + 725 q^{18} - 805 q^{24} - 1250 q^{25} + 275 q^{42} + 1400 q^{48} + 3430 q^{49} - 2740 q^{51} + 455 q^{54} + 5000 q^{63} - 5120 q^{64} - 5800 q^{72} - 3145 q^{84} + 6440 q^{96}+O(q^{100}) \) 10 * q - 80 * q^4 - 175 * q^12 + 640 * q^16 + 725 * q^18 - 805 * q^24 - 1250 * q^25 + 275 * q^42 + 1400 * q^48 + 3430 * q^49 - 2740 * q^51 + 455 * q^54 + 5000 * q^63 - 5120 * q^64 - 5800 * q^72 - 3145 * q^84 + 6440 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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