LMFDB - Embedded newform 8001.2.a.v.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 4 x^{18} - 22 x^{17} + 101 x^{16} + 178 x^{15} - 1035 x^{14} - 583 x^{13} + 5572 x^{12} + \cdots + 210 $ x^19 - 4x^18 - 22x^17 + 101x^16 + 178x^15 - 1035x^14 - 583x^13 + 5572x^12 + 21x^11 - 17032x^10 + 4985x^9 + 29792x^8 - 13249x^7 - 28600x^6 + 14000x^5 + 13725x^4 - 5723x^3 - 2913x^2 + 608x + 210
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$0.395540$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.395540 q^{2} -1.84355 q^{4} -1.95228 q^{5} +1.00000 q^{7} +1.52028 q^{8} +O(q^{10})$	\(q-0.395540 q^{2} -1.84355 q^{4} -1.95228 q^{5} +1.00000 q^{7} +1.52028 q^{8} +0.772204 q^{10} +5.34735 q^{11} +6.49250 q^{13} -0.395540 q^{14} +3.08577 q^{16} -7.92778 q^{17} -5.50935 q^{19} +3.59912 q^{20} -2.11509 q^{22} +3.09814 q^{23} -1.18861 q^{25} -2.56804 q^{26} -1.84355 q^{28} +3.39563 q^{29} +10.2114 q^{31} -4.26110 q^{32} +3.13575 q^{34} -1.95228 q^{35} +7.25011 q^{37} +2.17917 q^{38} -2.96800 q^{40} +10.1950 q^{41} +12.7584 q^{43} -9.85809 q^{44} -1.22544 q^{46} +1.66284 q^{47} +1.00000 q^{49} +0.470144 q^{50} -11.9692 q^{52} -9.51844 q^{53} -10.4395 q^{55} +1.52028 q^{56} -1.34311 q^{58} +5.51303 q^{59} +0.0570727 q^{61} -4.03903 q^{62} -4.48610 q^{64} -12.6752 q^{65} -10.0453 q^{67} +14.6152 q^{68} +0.772204 q^{70} -7.73577 q^{71} +1.79288 q^{73} -2.86771 q^{74} +10.1568 q^{76} +5.34735 q^{77} +1.81852 q^{79} -6.02427 q^{80} -4.03252 q^{82} -6.90635 q^{83} +15.4772 q^{85} -5.04644 q^{86} +8.12944 q^{88} -17.5417 q^{89} +6.49250 q^{91} -5.71157 q^{92} -0.657719 q^{94} +10.7558 q^{95} +4.44649 q^{97} -0.395540 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8}+O(q^{10}) $ 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8	$ 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8} + 9 q^{11} + 24 q^{13} - 4 q^{14} + 20 q^{16} - 17 q^{17} + 23 q^{19} - 5 q^{20} - 3 q^{22} + 17 q^{23} + 38 q^{25} - 28 q^{26} + 22 q^{28} - 2 q^{29} + 16 q^{31} - 17 q^{32} + 29 q^{34} - 5 q^{35} + 56 q^{37} - 2 q^{38} - 13 q^{40} + 7 q^{41} + 19 q^{43} + 29 q^{44} + 10 q^{46} - 25 q^{47} + 19 q^{49} + 9 q^{50} + 16 q^{52} - 18 q^{53} + 10 q^{55} - 9 q^{56} + 31 q^{58} - 11 q^{59} + 26 q^{61} - 26 q^{62} + 45 q^{64} - 27 q^{65} + 24 q^{67} - 14 q^{68} + 32 q^{71} + 51 q^{73} + 12 q^{76} + 9 q^{77} + 30 q^{79} + 30 q^{80} - 52 q^{82} - q^{83} + 44 q^{85} + 24 q^{86} - 30 q^{88} - 5 q^{89} + 24 q^{91} + 88 q^{92} + 7 q^{94} + 24 q^{95} + 5 q^{97} - 4 q^{98}+O(q^{100}) $ 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8 + 9 * q^11 + 24 * q^13 - 4 * q^14 + 20 * q^16 - 17 * q^17 + 23 * q^19 - 5 * q^20 - 3 * q^22 + 17 * q^23 + 38 * q^25 - 28 * q^26 + 22 * q^28 - 2 * q^29 + 16 * q^31 - 17 * q^32 + 29 * q^34 - 5 * q^35 + 56 * q^37 - 2 * q^38 - 13 * q^40 + 7 * q^41 + 19 * q^43 + 29 * q^44 + 10 * q^46 - 25 * q^47 + 19 * q^49 + 9 * q^50 + 16 * q^52 - 18 * q^53 + 10 * q^55 - 9 * q^56 + 31 * q^58 - 11 * q^59 + 26 * q^61 - 26 * q^62 + 45 * q^64 - 27 * q^65 + 24 * q^67 - 14 * q^68 + 32 * q^71 + 51 * q^73 + 12 * q^76 + 9 * q^77 + 30 * q^79 + 30 * q^80 - 52 * q^82 - q^83 + 44 * q^85 + 24 * q^86 - 30 * q^88 - 5 * q^89 + 24 * q^91 + 88 * q^92 + 7 * q^94 + 24 * q^95 + 5 * q^97 - 4 * 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 $

$2$	−0.395540	−0.279689	−0.139844	−	0.990173i	$-0.544660\pi$
$2$	−0.395540	−0.279689	−0.139844	+	0.990173i	$0.544660\pi$
$3$	0	0
$3$	0	0
$4$	−1.84355	−0.921774
$5$	−1.95228	−0.873085	−0.436543	−	0.899684i	$-0.643797\pi$
$5$	−1.95228	−0.873085	−0.436543	+	0.899684i	$0.643797\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.52028	0.537499
$9$	0	0
$10$	0.772204	0.244192
$11$	5.34735	1.61229	0.806143	−	0.591721i	$-0.201551\pi$
$11$	5.34735	1.61229	0.806143	+	0.591721i	$0.201551\pi$
$12$	0	0
$13$	6.49250	1.80070	0.900348	−	0.435170i	$-0.143312\pi$
$13$	6.49250	1.80070	0.900348	+	0.435170i	$0.143312\pi$
$14$	−0.395540	−0.105712
$15$	0	0
$16$	3.08577	0.771442
$17$	−7.92778	−1.92277	−0.961385	−	0.275208i	$-0.911253\pi$
$17$	−7.92778	−1.92277	−0.961385	+	0.275208i	$0.911253\pi$
$18$	0	0
$19$	−5.50935	−1.26393	−0.631966	−	0.774996i	$-0.717752\pi$
$19$	−5.50935	−1.26393	−0.631966	+	0.774996i	$0.717752\pi$
$20$	3.59912	0.804787
$21$	0	0
$22$	−2.11509	−0.450938
$23$	3.09814	0.646006	0.323003	−	0.946398i	$-0.395308\pi$
$23$	3.09814	0.646006	0.323003	+	0.946398i	$0.395308\pi$
$24$	0	0
$25$	−1.18861	−0.237723
$26$	−2.56804	−0.503635
$27$	0	0
$28$	−1.84355	−0.348398
$29$	3.39563	0.630552	0.315276	−	0.949000i	$-0.397903\pi$
$29$	3.39563	0.630552	0.315276	+	0.949000i	$0.397903\pi$
$30$	0	0
$31$	10.2114	1.83403	0.917014	−	0.398855i	$-0.130592\pi$
$31$	10.2114	1.83403	0.917014	+	0.398855i	$0.130592\pi$
$32$	−4.26110	−0.753263
$33$	0	0
$34$	3.13575	0.537777
$35$	−1.95228	−0.329995
$36$	0	0
$37$	7.25011	1.19191	0.595955	−	0.803018i	$-0.296774\pi$
$37$	7.25011	1.19191	0.595955	+	0.803018i	$0.296774\pi$
$38$	2.17917	0.353508
$39$	0	0
$40$	−2.96800	−0.469282
$41$	10.1950	1.59219	0.796093	−	0.605174i	$-0.206896\pi$
$41$	10.1950	1.59219	0.796093	+	0.605174i	$0.206896\pi$
$42$	0	0
$43$	12.7584	1.94563	0.972816	−	0.231581i	$-0.0743899\pi$
$43$	12.7584	1.94563	0.972816	+	0.231581i	$0.0743899\pi$
$44$	−9.85809	−1.48616
$45$	0	0
$46$	−1.22544	−0.180681
$47$	1.66284	0.242550	0.121275	−	0.992619i	$-0.461302\pi$
$47$	1.66284	0.242550	0.121275	+	0.992619i	$0.461302\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0.470144	0.0664884
$51$	0	0
$52$	−11.9692	−1.65984
$53$	−9.51844	−1.30746	−0.653729	−	0.756729i	$-0.726796\pi$
$53$	−9.51844	−1.30746	−0.653729	+	0.756729i	$0.726796\pi$
$54$	0	0
$55$	−10.4395	−1.40766
$56$	1.52028	0.203156
$57$	0	0
$58$	−1.34311	−0.176358
$59$	5.51303	0.717735	0.358868	−	0.933388i	$-0.383163\pi$
$59$	5.51303	0.717735	0.358868	+	0.933388i	$0.383163\pi$
$60$	0	0
$61$	0.0570727	0.00730741	0.00365370	−	0.999993i	$-0.498837\pi$
$61$	0.0570727	0.00730741	0.00365370	+	0.999993i	$0.498837\pi$
$62$	−4.03903	−0.512957
$63$	0	0
$64$	−4.48610	−0.560762
$65$	−12.6752	−1.57216
$66$	0	0
$67$	−10.0453	−1.22723	−0.613617	−	0.789604i	$-0.710286\pi$
$67$	−10.0453	−1.22723	−0.613617	+	0.789604i	$0.710286\pi$
$68$	14.6152	1.77236
$69$	0	0
$70$	0.772204	0.0922960
$71$	−7.73577	−0.918067	−0.459033	−	0.888419i	$-0.651804\pi$
$71$	−7.73577	−0.918067	−0.459033	+	0.888419i	$0.651804\pi$
$72$	0	0
$73$	1.79288	0.209841	0.104921	−	0.994481i	$-0.466541\pi$
$73$	1.79288	0.209841	0.104921	+	0.994481i	$0.466541\pi$
$74$	−2.86771	−0.333364
$75$	0	0
$76$	10.1568	1.16506
$77$	5.34735	0.609387
$78$	0	0
$79$	1.81852	0.204600	0.102300	−	0.994754i	$-0.467380\pi$
$79$	1.81852	0.204600	0.102300	+	0.994754i	$0.467380\pi$
$80$	−6.02427	−0.673534
$81$	0	0
$82$	−4.03252	−0.445317
$83$	−6.90635	−0.758070	−0.379035	−	0.925382i	$-0.623744\pi$
$83$	−6.90635	−0.758070	−0.379035	+	0.925382i	$0.623744\pi$
$84$	0	0
$85$	15.4772	1.67874
$86$	−5.04644	−0.544171
$87$	0	0
$88$	8.12944	0.866602
$89$	−17.5417	−1.85941	−0.929706	−	0.368303i	$-0.879939\pi$
$89$	−17.5417	−1.85941	−0.929706	+	0.368303i	$0.879939\pi$
$90$	0	0
$91$	6.49250	0.680599
$92$	−5.71157	−0.595472
$93$	0	0
$94$	−0.657719	−0.0678385
$95$	10.7558	1.10352
$96$	0	0
$97$	4.44649	0.451473	0.225736	−	0.974188i	$-0.427521\pi$
$97$	4.44649	0.451473	0.225736	+	0.974188i	$0.427521\pi$
$98$	−0.395540	−0.0399556
$99$	0	0
$100$	2.19127	0.219127
$101$	4.31554	0.429412	0.214706	−	0.976679i	$-0.431121\pi$
$101$	4.31554	0.429412	0.214706	+	0.976679i	$0.431121\pi$
$102$	0	0
$103$	−3.54414	−0.349214	−0.174607	−	0.984638i	$-0.555866\pi$
$103$	−3.54414	−0.349214	−0.174607	+	0.984638i	$0.555866\pi$
$104$	9.87040	0.967872
$105$	0	0
$106$	3.76492	0.365682
$107$	−12.4575	−1.20431	−0.602155	−	0.798379i	$-0.705691\pi$
$107$	−12.4575	−1.20431	−0.602155	+	0.798379i	$0.705691\pi$
$108$	0	0
$109$	4.55460	0.436251	0.218126	−	0.975921i	$-0.430006\pi$
$109$	4.55460	0.436251	0.218126	+	0.975921i	$0.430006\pi$
$110$	4.12924	0.393707
$111$	0	0
$112$	3.08577	0.291578
$113$	0.943880	0.0887928	0.0443964	−	0.999014i	$-0.485864\pi$
$113$	0.943880	0.0887928	0.0443964	+	0.999014i	$0.485864\pi$
$114$	0	0
$115$	−6.04843	−0.564019
$116$	−6.26000	−0.581227
$117$	0	0
$118$	−2.18062	−0.200743
$119$	−7.92778	−0.726739
$120$	0	0
$121$	17.5941	1.59946
$122$	−0.0225745	−0.00204380
$123$	0	0
$124$	−18.8253	−1.69056
$125$	12.0819	1.08064
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	10.2966	0.910102
$129$	0	0
$130$	5.01353	0.439716
$131$	−4.18883	−0.365980	−0.182990	−	0.983115i	$-0.558578\pi$
$131$	−4.18883	−0.365980	−0.182990	+	0.983115i	$0.558578\pi$
$132$	0	0
$133$	−5.50935	−0.477721
$134$	3.97333	0.343244
$135$	0	0
$136$	−12.0524	−1.03349
$137$	−4.87863	−0.416809	−0.208405	−	0.978043i	$-0.566827\pi$
$137$	−4.87863	−0.416809	−0.208405	+	0.978043i	$0.566827\pi$
$138$	0	0
$139$	3.61685	0.306777	0.153389	−	0.988166i	$-0.450981\pi$
$139$	3.61685	0.306777	0.153389	+	0.988166i	$0.450981\pi$
$140$	3.59912	0.304181
$141$	0	0
$142$	3.05980	0.256773
$143$	34.7177	2.90324
$144$	0	0
$145$	−6.62921	−0.550526
$146$	−0.709157	−0.0586903
$147$	0	0
$148$	−13.3659	−1.09867
$149$	5.32520	0.436257	0.218129	−	0.975920i	$-0.430005\pi$
$149$	5.32520	0.436257	0.218129	+	0.975920i	$0.430005\pi$
$150$	0	0
$151$	−11.1323	−0.905935	−0.452967	−	0.891527i	$-0.649635\pi$
$151$	−11.1323	−0.905935	−0.452967	+	0.891527i	$0.649635\pi$
$152$	−8.37574	−0.679362
$153$	0	0
$154$	−2.11509	−0.170439
$155$	−19.9356	−1.60126
$156$	0	0
$157$	−21.8326	−1.74243	−0.871214	−	0.490903i	$-0.836667\pi$
$157$	−21.8326	−1.74243	−0.871214	+	0.490903i	$0.836667\pi$
$158$	−0.719298	−0.0572243
$159$	0	0
$160$	8.31884	0.657662
$161$	3.09814	0.244168
$162$	0	0
$163$	21.6255	1.69384	0.846921	−	0.531719i	$-0.178454\pi$
$163$	21.6255	1.69384	0.846921	+	0.531719i	$0.178454\pi$
$164$	−18.7949	−1.46764
$165$	0	0
$166$	2.73174	0.212024
$167$	−6.90780	−0.534542	−0.267271	−	0.963621i	$-0.586122\pi$
$167$	−6.90780	−0.534542	−0.267271	+	0.963621i	$0.586122\pi$
$168$	0	0
$169$	29.1526	2.24251
$170$	−6.12186	−0.469525
$171$	0	0
$172$	−23.5206	−1.79343
$173$	−19.7315	−1.50016	−0.750080	−	0.661347i	$-0.769985\pi$
$173$	−19.7315	−1.50016	−0.750080	+	0.661347i	$0.769985\pi$
$174$	0	0
$175$	−1.18861	−0.0898507
$176$	16.5007	1.24378
$177$	0	0
$178$	6.93843	0.520057
$179$	23.0568	1.72334	0.861672	−	0.507465i	$-0.169417\pi$
$179$	23.0568	1.72334	0.861672	+	0.507465i	$0.169417\pi$
$180$	0	0
$181$	1.92103	0.142789	0.0713946	−	0.997448i	$-0.477255\pi$
$181$	1.92103	0.142789	0.0713946	+	0.997448i	$0.477255\pi$
$182$	−2.56804	−0.190356
$183$	0	0
$184$	4.71003	0.347228
$185$	−14.1542	−1.04064
$186$	0	0
$187$	−42.3926	−3.10005
$188$	−3.06552	−0.223576
$189$	0	0
$190$	−4.25434	−0.308642
$191$	5.70825	0.413034	0.206517	−	0.978443i	$-0.433787\pi$
$191$	5.70825	0.413034	0.206517	+	0.978443i	$0.433787\pi$
$192$	0	0
$193$	−15.2985	−1.10121	−0.550606	−	0.834765i	$-0.685603\pi$
$193$	−15.2985	−1.10121	−0.550606	+	0.834765i	$0.685603\pi$
$194$	−1.75877	−0.126272
$195$	0	0
$196$	−1.84355	−0.131682
$197$	−2.27770	−0.162279	−0.0811396	−	0.996703i	$-0.525856\pi$
$197$	−2.27770	−0.162279	−0.0811396	+	0.996703i	$0.525856\pi$
$198$	0	0
$199$	−24.2047	−1.71583	−0.857913	−	0.513794i	$-0.828240\pi$
$199$	−24.2047	−1.71583	−0.857913	+	0.513794i	$0.828240\pi$
$200$	−1.80702	−0.127776
$201$	0	0
$202$	−1.70697	−0.120102
$203$	3.39563	0.238326
$204$	0	0
$205$	−19.9034	−1.39011
$206$	1.40185	0.0976714
$207$	0	0
$208$	20.0343	1.38913
$209$	−29.4604	−2.03782
$210$	0	0
$211$	3.91191	0.269307	0.134654	−	0.990893i	$-0.457008\pi$
$211$	3.91191	0.269307	0.134654	+	0.990893i	$0.457008\pi$
$212$	17.5477	1.20518
$213$	0	0
$214$	4.92743	0.336832
$215$	−24.9079	−1.69870
$216$	0	0
$217$	10.2114	0.693197
$218$	−1.80152	−0.122015
$219$	0	0
$220$	19.2457	1.29755
$221$	−51.4712	−3.46232
$222$	0	0
$223$	−2.42028	−0.162074	−0.0810371	−	0.996711i	$-0.525823\pi$
$223$	−2.42028	−0.162074	−0.0810371	+	0.996711i	$0.525823\pi$
$224$	−4.26110	−0.284707
$225$	0	0
$226$	−0.373342	−0.0248344
$227$	10.9159	0.724512	0.362256	−	0.932079i	$-0.382007\pi$
$227$	10.9159	0.724512	0.362256	+	0.932079i	$0.382007\pi$
$228$	0	0
$229$	8.16418	0.539504	0.269752	−	0.962930i	$-0.413058\pi$
$229$	8.16418	0.539504	0.269752	+	0.962930i	$0.413058\pi$
$230$	2.39239	0.157750
$231$	0	0
$232$	5.16229	0.338921
$233$	20.3575	1.33366	0.666831	−	0.745209i	$-0.267650\pi$
$233$	20.3575	1.33366	0.666831	+	0.745209i	$0.267650\pi$
$234$	0	0
$235$	−3.24632	−0.211767
$236$	−10.1635	−0.661590
$237$	0	0
$238$	3.13575	0.203261
$239$	9.63093	0.622973	0.311486	−	0.950251i	$-0.399173\pi$
$239$	9.63093	0.622973	0.311486	+	0.950251i	$0.399173\pi$
$240$	0	0
$241$	28.2537	1.81998	0.909992	−	0.414626i	$-0.136088\pi$
$241$	28.2537	1.81998	0.909992	+	0.414626i	$0.136088\pi$
$242$	−6.95917	−0.447352
$243$	0	0
$244$	−0.105216	−0.00673578
$245$	−1.95228	−0.124726
$246$	0	0
$247$	−35.7695	−2.27596
$248$	15.5242	0.985788
$249$	0	0
$250$	−4.77887	−0.302242
$251$	5.81771	0.367210	0.183605	−	0.983000i	$-0.441223\pi$
$251$	5.81771	0.367210	0.183605	+	0.983000i	$0.441223\pi$
$252$	0	0
$253$	16.5668	1.04155
$254$	−0.395540	−0.0248184
$255$	0	0
$256$	4.89947	0.306217
$257$	9.99962	0.623759	0.311880	−	0.950122i	$-0.399041\pi$
$257$	9.99962	0.623759	0.311880	+	0.950122i	$0.399041\pi$
$258$	0	0
$259$	7.25011	0.450500
$260$	23.3673	1.44918
$261$	0	0
$262$	1.65685	0.102360
$263$	6.44764	0.397579	0.198789	−	0.980042i	$-0.436299\pi$
$263$	6.44764	0.397579	0.198789	+	0.980042i	$0.436299\pi$
$264$	0	0
$265$	18.5826	1.14152
$266$	2.17917	0.133613
$267$	0	0
$268$	18.5191	1.13123
$269$	0.309677	0.0188814	0.00944068	−	0.999955i	$-0.496995\pi$
$269$	0.309677	0.0188814	0.00944068	+	0.999955i	$0.496995\pi$
$270$	0	0
$271$	−8.35899	−0.507772	−0.253886	−	0.967234i	$-0.581709\pi$
$271$	−8.35899	−0.507772	−0.253886	+	0.967234i	$0.581709\pi$
$272$	−24.4633	−1.48330
$273$	0	0
$274$	1.92969	0.116577
$275$	−6.35592	−0.383277
$276$	0	0
$277$	19.4246	1.16711	0.583554	−	0.812074i	$-0.301662\pi$
$277$	19.4246	1.16711	0.583554	+	0.812074i	$0.301662\pi$
$278$	−1.43061	−0.0858022
$279$	0	0
$280$	−2.96800	−0.177372
$281$	−6.46823	−0.385862	−0.192931	−	0.981212i	$-0.561799\pi$
$281$	−6.46823	−0.385862	−0.192931	+	0.981212i	$0.561799\pi$
$282$	0	0
$283$	−14.6422	−0.870390	−0.435195	−	0.900336i	$-0.643321\pi$
$283$	−14.6422	−0.870390	−0.435195	+	0.900336i	$0.643321\pi$
$284$	14.2613	0.846250
$285$	0	0
$286$	−13.7322	−0.812003
$287$	10.1950	0.601790
$288$	0	0
$289$	45.8497	2.69704
$290$	2.62212	0.153976
$291$	0	0
$292$	−3.30527	−0.193426
$293$	4.71917	0.275697	0.137848	−	0.990453i	$-0.455981\pi$
$293$	4.71917	0.275697	0.137848	+	0.990453i	$0.455981\pi$
$294$	0	0
$295$	−10.7630	−0.626644
$296$	11.0222	0.640651
$297$	0	0
$298$	−2.10633	−0.122016
$299$	20.1147	1.16326
$300$	0	0
$301$	12.7584	0.735379
$302$	4.40327	0.253380
$303$	0	0
$304$	−17.0006	−0.975049
$305$	−0.111422	−0.00637999
$306$	0	0
$307$	−6.36282	−0.363145	−0.181573	−	0.983378i	$-0.558119\pi$
$307$	−6.36282	−0.363145	−0.181573	+	0.983378i	$0.558119\pi$
$308$	−9.85809	−0.561717
$309$	0	0
$310$	7.88531	0.447855
$311$	−0.442606	−0.0250979	−0.0125489	−	0.999921i	$-0.503995\pi$
$311$	−0.442606	−0.0250979	−0.0125489	+	0.999921i	$0.503995\pi$
$312$	0	0
$313$	27.8781	1.57576	0.787881	−	0.615827i	$-0.211178\pi$
$313$	27.8781	1.57576	0.787881	+	0.615827i	$0.211178\pi$
$314$	8.63565	0.487338
$315$	0	0
$316$	−3.35253	−0.188595
$317$	27.0519	1.51938	0.759692	−	0.650283i	$-0.225350\pi$
$317$	27.0519	1.51938	0.759692	+	0.650283i	$0.225350\pi$
$318$	0	0
$319$	18.1576	1.01663
$320$	8.75811	0.489593
$321$	0	0
$322$	−1.22544	−0.0682909
$323$	43.6769	2.43025
$324$	0	0
$325$	−7.71707	−0.428066
$326$	−8.55376	−0.473749
$327$	0	0
$328$	15.4992	0.855799
$329$	1.66284	0.0916753
$330$	0	0
$331$	25.1442	1.38205	0.691026	−	0.722830i	$-0.257159\pi$
$331$	25.1442	1.38205	0.691026	+	0.722830i	$0.257159\pi$
$332$	12.7322	0.698770
$333$	0	0
$334$	2.73231	0.149505
$335$	19.6113	1.07148
$336$	0	0
$337$	−0.0529758	−0.00288578	−0.00144289	−	0.999999i	$-0.500459\pi$
$337$	−0.0529758	−0.00288578	−0.00144289	+	0.999999i	$0.500459\pi$
$338$	−11.5310	−0.627204
$339$	0	0
$340$	−28.5330	−1.54742
$341$	54.6041	2.95698
$342$	0	0
$343$	1.00000	0.0539949
$344$	19.3962	1.04577
$345$	0	0
$346$	7.80461	0.419578
$347$	19.4800	1.04574	0.522870	−	0.852412i	$-0.324861\pi$
$347$	19.4800	1.04574	0.522870	+	0.852412i	$0.324861\pi$
$348$	0	0
$349$	0.0195389	0.00104589	0.000522947	−	1.00000i	$-0.499834\pi$
$349$	0.0195389	0.00104589	0.000522947		1.00000i	$0.499834\pi$
$350$	0.470144	0.0251302
$351$	0	0
$352$	−22.7856	−1.21447
$353$	22.1141	1.17701	0.588507	−	0.808492i	$-0.299716\pi$
$353$	22.1141	1.17701	0.588507	+	0.808492i	$0.299716\pi$
$354$	0	0
$355$	15.1024	0.801550
$356$	32.3389	1.71396
$357$	0	0
$358$	−9.11988	−0.482000
$359$	−2.17615	−0.114853	−0.0574264	−	0.998350i	$-0.518289\pi$
$359$	−2.17615	−0.114853	−0.0574264	+	0.998350i	$0.518289\pi$
$360$	0	0
$361$	11.3529	0.597523
$362$	−0.759844	−0.0399365
$363$	0	0
$364$	−11.9692	−0.627359
$365$	−3.50021	−0.183209
$366$	0	0
$367$	−12.6682	−0.661276	−0.330638	−	0.943758i	$-0.607264\pi$
$367$	−12.6682	−0.661276	−0.330638	+	0.943758i	$0.607264\pi$
$368$	9.56013	0.498356
$369$	0	0
$370$	5.59856	0.291055
$371$	−9.51844	−0.494173
$372$	0	0
$373$	−31.9234	−1.65293	−0.826466	−	0.562987i	$-0.809652\pi$
$373$	−31.9234	−1.65293	−0.826466	+	0.562987i	$0.809652\pi$
$374$	16.7680	0.867051
$375$	0	0
$376$	2.52797	0.130370
$377$	22.0461	1.13543
$378$	0	0
$379$	6.66072	0.342138	0.171069	−	0.985259i	$-0.445278\pi$
$379$	6.66072	0.342138	0.171069	+	0.985259i	$0.445278\pi$
$380$	−19.8288	−1.01720
$381$	0	0
$382$	−2.25784	−0.115521
$383$	16.8149	0.859203	0.429602	−	0.903019i	$-0.358654\pi$
$383$	16.8149	0.859203	0.429602	+	0.903019i	$0.358654\pi$
$384$	0	0
$385$	−10.4395	−0.532046
$386$	6.05118	0.307997
$387$	0	0
$388$	−8.19732	−0.416156
$389$	1.76018	0.0892446	0.0446223	−	0.999004i	$-0.485792\pi$
$389$	1.76018	0.0892446	0.0446223	+	0.999004i	$0.485792\pi$
$390$	0	0
$391$	−24.5614	−1.24212
$392$	1.52028	0.0767856
$393$	0	0
$394$	0.900920	0.0453877
$395$	−3.55026	−0.178633
$396$	0	0
$397$	12.5865	0.631698	0.315849	−	0.948809i	$-0.397711\pi$
$397$	12.5865	0.631698	0.315849	+	0.948809i	$0.397711\pi$
$398$	9.57393	0.479898
$399$	0	0
$400$	−3.66778	−0.183389
$401$	8.90988	0.444938	0.222469	−	0.974940i	$-0.428588\pi$
$401$	8.90988	0.444938	0.222469	+	0.974940i	$0.428588\pi$
$402$	0	0
$403$	66.2978	3.30253
$404$	−7.95590	−0.395821
$405$	0	0
$406$	−1.34311	−0.0666572
$407$	38.7688	1.92170
$408$	0	0
$409$	−30.6626	−1.51617	−0.758084	−	0.652157i	$-0.773864\pi$
$409$	−30.6626	−1.51617	−0.758084	+	0.652157i	$0.773864\pi$
$410$	7.87259	0.388800
$411$	0	0
$412$	6.53379	0.321897
$413$	5.51303	0.271278
$414$	0	0
$415$	13.4831	0.661860
$416$	−27.6652	−1.35640
$417$	0	0
$418$	11.6528	0.569955
$419$	−3.03425	−0.148233	−0.0741163	−	0.997250i	$-0.523614\pi$
$419$	−3.03425	−0.148233	−0.0741163	+	0.997250i	$0.523614\pi$
$420$	0	0
$421$	5.88620	0.286876	0.143438	−	0.989659i	$-0.454184\pi$
$421$	5.88620	0.286876	0.143438	+	0.989659i	$0.454184\pi$
$422$	−1.54732	−0.0753222
$423$	0	0
$424$	−14.4707	−0.702757
$425$	9.42306	0.457086
$426$	0	0
$427$	0.0570727	0.00276194
$428$	22.9660	1.11010
$429$	0	0
$430$	9.85205	0.475108
$431$	−20.9852	−1.01082	−0.505412	−	0.862878i	$-0.668659\pi$
$431$	−20.9852	−1.01082	−0.505412	+	0.862878i	$0.668659\pi$
$432$	0	0
$433$	−9.07998	−0.436356	−0.218178	−	0.975909i	$-0.570011\pi$
$433$	−9.07998	−0.436356	−0.218178	+	0.975909i	$0.570011\pi$
$434$	−4.03903	−0.193880
$435$	0	0
$436$	−8.39662	−0.402125
$437$	−17.0687	−0.816508
$438$	0	0
$439$	−16.3936	−0.782425	−0.391212	−	0.920300i	$-0.627944\pi$
$439$	−16.3936	−0.782425	−0.391212	+	0.920300i	$0.627944\pi$
$440$	−15.8709	−0.756617
$441$	0	0
$442$	20.3589	0.968374
$443$	−24.2993	−1.15449	−0.577247	−	0.816570i	$-0.695873\pi$
$443$	−24.2993	−1.15449	−0.577247	+	0.816570i	$0.695873\pi$
$444$	0	0
$445$	34.2462	1.62342
$446$	0.957318	0.0453303
$447$	0	0
$448$	−4.48610	−0.211948
$449$	−22.0774	−1.04190	−0.520948	−	0.853588i	$-0.674421\pi$
$449$	−22.0774	−1.04190	−0.520948	+	0.853588i	$0.674421\pi$
$450$	0	0
$451$	54.5160	2.56706
$452$	−1.74009	−0.0818469
$453$	0	0
$454$	−4.31766	−0.202638
$455$	−12.6752	−0.594221
$456$	0	0
$457$	14.4817	0.677423	0.338712	−	0.940890i	$-0.390009\pi$
$457$	14.4817	0.677423	0.338712	+	0.940890i	$0.390009\pi$
$458$	−3.22926	−0.150893
$459$	0	0
$460$	11.1506	0.519898
$461$	32.1184	1.49590	0.747951	−	0.663754i	$-0.231038\pi$
$461$	32.1184	1.49590	0.747951	+	0.663754i	$0.231038\pi$
$462$	0	0
$463$	−15.8911	−0.738523	−0.369261	−	0.929326i	$-0.620389\pi$
$463$	−15.8911	−0.738523	−0.369261	+	0.929326i	$0.620389\pi$
$464$	10.4781	0.486434
$465$	0	0
$466$	−8.05219	−0.373010
$467$	1.85900	0.0860242	0.0430121	−	0.999075i	$-0.486305\pi$
$467$	1.85900	0.0860242	0.0430121	+	0.999075i	$0.486305\pi$
$468$	0	0
$469$	−10.0453	−0.463851
$470$	1.28405	0.0592288
$471$	0	0
$472$	8.38133	0.385782
$473$	68.2233	3.13691
$474$	0	0
$475$	6.54848	0.300465
$476$	14.6152	0.669889
$477$	0	0
$478$	−3.80942	−0.174239
$479$	21.7411	0.993376	0.496688	−	0.867929i	$-0.334549\pi$
$479$	21.7411	0.993376	0.496688	+	0.867929i	$0.334549\pi$
$480$	0	0
$481$	47.0714	2.14627
$482$	−11.1755	−0.509029
$483$	0	0
$484$	−32.4356	−1.47434
$485$	−8.68079	−0.394174
$486$	0	0
$487$	−33.5681	−1.52112	−0.760558	−	0.649270i	$-0.775074\pi$
$487$	−33.5681	−1.52112	−0.760558	+	0.649270i	$0.775074\pi$
$488$	0.0867663	0.00392772
$489$	0	0
$490$	0.772204	0.0348846
$491$	18.1365	0.818490	0.409245	−	0.912425i	$-0.365792\pi$
$491$	18.1365	0.818490	0.409245	+	0.912425i	$0.365792\pi$
$492$	0	0
$493$	−26.9198	−1.21241
$494$	14.1483	0.636560
$495$	0	0
$496$	31.5101	1.41485
$497$	−7.73577	−0.346997
$498$	0	0
$499$	−22.7742	−1.01951	−0.509757	−	0.860318i	$-0.670265\pi$
$499$	−22.7742	−1.01951	−0.509757	+	0.860318i	$0.670265\pi$
$500$	−22.2735	−0.996103
$501$	0	0
$502$	−2.30113	−0.102705
$503$	−30.7610	−1.37157	−0.685784	−	0.727805i	$-0.740540\pi$
$503$	−30.7610	−1.37157	−0.685784	+	0.727805i	$0.740540\pi$
$504$	0	0
$505$	−8.42513	−0.374913
$506$	−6.55284	−0.291309
$507$	0	0
$508$	−1.84355	−0.0817942
$509$	7.29123	0.323178	0.161589	−	0.986858i	$-0.448338\pi$
$509$	7.29123	0.323178	0.161589	+	0.986858i	$0.448338\pi$
$510$	0	0
$511$	1.79288	0.0793125
$512$	−22.5312	−0.995747
$513$	0	0
$514$	−3.95525	−0.174459
$515$	6.91914	0.304894
$516$	0	0
$517$	8.89177	0.391060
$518$	−2.86771	−0.126000
$519$	0	0
$520$	−19.2698	−0.845035
$521$	14.8186	0.649213	0.324606	−	0.945849i	$-0.394768\pi$
$521$	14.8186	0.649213	0.324606	+	0.945849i	$0.394768\pi$
$522$	0	0
$523$	−29.2907	−1.28079	−0.640396	−	0.768045i	$-0.721230\pi$
$523$	−29.2907	−1.28079	−0.640396	+	0.768045i	$0.721230\pi$
$524$	7.72231	0.337351
$525$	0	0
$526$	−2.55030	−0.111198
$527$	−80.9540	−3.52641
$528$	0	0
$529$	−13.4015	−0.582676
$530$	−7.35017	−0.319271
$531$	0	0
$532$	10.1568	0.440351
$533$	66.1909	2.86705
$534$	0	0
$535$	24.3205	1.05147
$536$	−15.2717	−0.659637
$537$	0	0
$538$	−0.122490	−0.00528091
$539$	5.34735	0.230326
$540$	0	0
$541$	43.0311	1.85005	0.925026	−	0.379903i	$-0.124043\pi$
$541$	43.0311	1.85005	0.925026	+	0.379903i	$0.124043\pi$
$542$	3.30631	0.142018
$543$	0	0
$544$	33.7810	1.44835
$545$	−8.89184	−0.380884
$546$	0	0
$547$	28.9462	1.23765	0.618826	−	0.785528i	$-0.287609\pi$
$547$	28.9462	1.23765	0.618826	+	0.785528i	$0.287609\pi$
$548$	8.99398	0.384204
$549$	0	0
$550$	2.51402	0.107198
$551$	−18.7077	−0.796975
$552$	0	0
$553$	1.81852	0.0773314
$554$	−7.68319	−0.326427
$555$	0	0
$556$	−6.66784	−0.282779
$557$	30.0717	1.27418	0.637089	−	0.770791i	$-0.280139\pi$
$557$	30.0717	1.27418	0.637089	+	0.770791i	$0.280139\pi$
$558$	0	0
$559$	82.8337	3.50349
$560$	−6.02427	−0.254572
$561$	0	0
$562$	2.55844	0.107921
$563$	15.2506	0.642738	0.321369	−	0.946954i	$-0.395857\pi$
$563$	15.2506	0.642738	0.321369	+	0.946954i	$0.395857\pi$
$564$	0	0
$565$	−1.84272	−0.0775236
$566$	5.79158	0.243438
$567$	0	0
$568$	−11.7605	−0.493460
$569$	24.0976	1.01022	0.505112	−	0.863054i	$-0.331451\pi$
$569$	24.0976	1.01022	0.505112	+	0.863054i	$0.331451\pi$
$570$	0	0
$571$	3.54378	0.148302	0.0741512	−	0.997247i	$-0.476375\pi$
$571$	3.54378	0.148302	0.0741512	+	0.997247i	$0.476375\pi$
$572$	−64.0037	−2.67613
$573$	0	0
$574$	−4.03252	−0.168314
$575$	−3.68249	−0.153570
$576$	0	0
$577$	−34.4496	−1.43415	−0.717077	−	0.696994i	$-0.754520\pi$
$577$	−34.4496	−1.43415	−0.717077	+	0.696994i	$0.754520\pi$
$578$	−18.1354	−0.754333
$579$	0	0
$580$	12.2213	0.507460
$581$	−6.90635	−0.286524
$582$	0	0
$583$	−50.8984	−2.10800
$584$	2.72568	0.112789
$585$	0	0
$586$	−1.86662	−0.0771093
$587$	−5.82459	−0.240407	−0.120203	−	0.992749i	$-0.538355\pi$
$587$	−5.82459	−0.240407	−0.120203	+	0.992749i	$0.538355\pi$
$588$	0	0
$589$	−56.2584	−2.31809
$590$	4.25718	0.175265
$591$	0	0
$592$	22.3721	0.919489
$593$	17.5879	0.722250	0.361125	−	0.932517i	$-0.382393\pi$
$593$	17.5879	0.722250	0.361125	+	0.932517i	$0.382393\pi$
$594$	0	0
$595$	15.4772	0.634505
$596$	−9.81726	−0.402131
$597$	0	0
$598$	−7.95615	−0.325351
$599$	33.7610	1.37944	0.689718	−	0.724078i	$-0.257734\pi$
$599$	33.7610	1.37944	0.689718	+	0.724078i	$0.257734\pi$
$600$	0	0
$601$	−34.6238	−1.41233	−0.706167	−	0.708045i	$-0.749577\pi$
$601$	−34.6238	−1.41233	−0.706167	+	0.708045i	$0.749577\pi$
$602$	−5.04644	−0.205677
$603$	0	0
$604$	20.5230	0.835067
$605$	−34.3486	−1.39647
$606$	0	0
$607$	2.20615	0.0895450	0.0447725	−	0.998997i	$-0.485744\pi$
$607$	2.20615	0.0895450	0.0447725	+	0.998997i	$0.485744\pi$
$608$	23.4759	0.952072
$609$	0	0
$610$	0.0440717	0.00178441
$611$	10.7960	0.436759
$612$	0	0
$613$	18.7651	0.757914	0.378957	−	0.925414i	$-0.376283\pi$
$613$	18.7651	0.757914	0.378957	+	0.925414i	$0.376283\pi$
$614$	2.51675	0.101568
$615$	0	0
$616$	8.12944	0.327545
$617$	−19.8323	−0.798419	−0.399209	−	0.916860i	$-0.630715\pi$
$617$	−19.8323	−0.798419	−0.399209	+	0.916860i	$0.630715\pi$
$618$	0	0
$619$	9.77604	0.392932	0.196466	−	0.980511i	$-0.437053\pi$
$619$	9.77604	0.392932	0.196466	+	0.980511i	$0.437053\pi$
$620$	36.7522	1.47600
$621$	0	0
$622$	0.175068	0.00701960
$623$	−17.5417	−0.702792
$624$	0	0
$625$	−17.6441	−0.705765
$626$	−11.0269	−0.440723
$627$	0	0
$628$	40.2494	1.60613
$629$	−57.4773	−2.29177
$630$	0	0
$631$	−7.95483	−0.316677	−0.158338	−	0.987385i	$-0.550614\pi$
$631$	−7.95483	−0.316677	−0.158338	+	0.987385i	$0.550614\pi$
$632$	2.76466	0.109972
$633$	0	0
$634$	−10.7001	−0.424955
$635$	−1.95228	−0.0774738
$636$	0	0
$637$	6.49250	0.257242
$638$	−7.18205	−0.284340
$639$	0	0
$640$	−20.1019	−0.794596
$641$	34.9817	1.38169	0.690847	−	0.723001i	$-0.257238\pi$
$641$	34.9817	1.38169	0.690847	+	0.723001i	$0.257238\pi$
$642$	0	0
$643$	15.8737	0.625999	0.312999	−	0.949753i	$-0.398666\pi$
$643$	15.8737	0.625999	0.312999	+	0.949753i	$0.398666\pi$
$644$	−5.71157	−0.225067
$645$	0	0
$646$	−17.2760	−0.679714
$647$	21.9320	0.862234	0.431117	−	0.902296i	$-0.358120\pi$
$647$	21.9320	0.862234	0.431117	+	0.902296i	$0.358120\pi$
$648$	0	0
$649$	29.4801	1.15719
$650$	3.05241	0.119725
$651$	0	0
$652$	−39.8677	−1.56134
$653$	1.16252	0.0454928	0.0227464	−	0.999741i	$-0.492759\pi$
$653$	1.16252	0.0454928	0.0227464	+	0.999741i	$0.492759\pi$
$654$	0	0
$655$	8.17776	0.319531
$656$	31.4593	1.22828
$657$	0	0
$658$	−0.657719	−0.0256406
$659$	45.3968	1.76841	0.884205	−	0.467099i	$-0.154701\pi$
$659$	45.3968	1.76841	0.884205	+	0.467099i	$0.154701\pi$
$660$	0	0
$661$	7.35732	0.286167	0.143083	−	0.989711i	$-0.454298\pi$
$661$	7.35732	0.286167	0.143083	+	0.989711i	$0.454298\pi$
$662$	−9.94555	−0.386545
$663$	0	0
$664$	−10.4996	−0.407462
$665$	10.7558	0.417091
$666$	0	0
$667$	10.5201	0.407341
$668$	12.7349	0.492727
$669$	0	0
$670$	−7.75705	−0.299681
$671$	0.305187	0.0117816
$672$	0	0
$673$	−38.8530	−1.49767	−0.748837	−	0.662755i	$-0.769387\pi$
$673$	−38.8530	−1.49767	−0.748837	+	0.662755i	$0.769387\pi$
$674$	0.0209540	0.000807120 0
$675$	0	0
$676$	−53.7442	−2.06709
$677$	−39.7794	−1.52885	−0.764424	−	0.644714i	$-0.776977\pi$
$677$	−39.7794	−1.52885	−0.764424	+	0.644714i	$0.776977\pi$
$678$	0	0
$679$	4.44649	0.170641
$680$	23.5297	0.902322
$681$	0	0
$682$	−21.5981	−0.827034
$683$	22.0096	0.842176	0.421088	−	0.907020i	$-0.361648\pi$
$683$	22.0096	0.842176	0.421088	+	0.907020i	$0.361648\pi$
$684$	0	0
$685$	9.52443	0.363910
$686$	−0.395540	−0.0151018
$687$	0	0
$688$	39.3693	1.50094
$689$	−61.7985	−2.35434
$690$	0	0
$691$	−9.52843	−0.362479	−0.181239	−	0.983439i	$-0.558011\pi$
$691$	−9.52843	−0.362479	−0.181239	+	0.983439i	$0.558011\pi$
$692$	36.3760	1.38281
$693$	0	0
$694$	−7.70511	−0.292482
$695$	−7.06110	−0.267843
$696$	0	0
$697$	−80.8235	−3.06141
$698$	−0.00772841	−0.000292525 0
$699$	0	0
$700$	2.19127	0.0828220
$701$	21.9732	0.829915	0.414957	−	0.909841i	$-0.363796\pi$
$701$	21.9732	0.829915	0.414957	+	0.909841i	$0.363796\pi$
$702$	0	0
$703$	−39.9434	−1.50649
$704$	−23.9887	−0.904109
$705$	0	0
$706$	−8.74700	−0.329198
$707$	4.31554	0.162303
$708$	0	0
$709$	−28.0969	−1.05520	−0.527600	−	0.849493i	$-0.676908\pi$
$709$	−28.0969	−1.05520	−0.527600	+	0.849493i	$0.676908\pi$
$710$	−5.97359	−0.224185
$711$	0	0
$712$	−26.6682	−0.999432
$713$	31.6364	1.18479
$714$	0	0
$715$	−67.7785	−2.53477
$716$	−42.5063	−1.58853
$717$	0	0
$718$	0.860755	0.0321231
$719$	25.8815	0.965218	0.482609	−	0.875836i	$-0.339689\pi$
$719$	25.8815	0.965218	0.482609	+	0.875836i	$0.339689\pi$
$720$	0	0
$721$	−3.54414	−0.131991
$722$	−4.49054	−0.167121
$723$	0	0
$724$	−3.54151	−0.131619
$725$	−4.03609	−0.149896
$726$	0	0
$727$	4.62598	0.171568	0.0857841	−	0.996314i	$-0.472660\pi$
$727$	4.62598	0.171568	0.0857841	+	0.996314i	$0.472660\pi$
$728$	9.87040	0.365821
$729$	0	0
$730$	1.38447	0.0512416
$731$	−101.145	−3.74100
$732$	0	0
$733$	−3.75379	−0.138649	−0.0693247	−	0.997594i	$-0.522084\pi$
$733$	−3.75379	−0.138649	−0.0693247	+	0.997594i	$0.522084\pi$
$734$	5.01079	0.184952
$735$	0	0
$736$	−13.2015	−0.486613
$737$	−53.7159	−1.97865
$738$	0	0
$739$	−4.26601	−0.156928	−0.0784638	−	0.996917i	$-0.525002\pi$
$739$	−4.26601	−0.156928	−0.0784638	+	0.996917i	$0.525002\pi$
$740$	26.0940	0.959234
$741$	0	0
$742$	3.76492	0.138215
$743$	5.16502	0.189486	0.0947431	−	0.995502i	$-0.469797\pi$
$743$	5.16502	0.189486	0.0947431	+	0.995502i	$0.469797\pi$
$744$	0	0
$745$	−10.3963	−0.380890
$746$	12.6270	0.462307
$747$	0	0
$748$	78.1528	2.85755
$749$	−12.4575	−0.455187
$750$	0	0
$751$	22.8108	0.832377	0.416189	−	0.909278i	$-0.363366\pi$
$751$	22.8108	0.832377	0.416189	+	0.909278i	$0.363366\pi$
$752$	5.13113	0.187113
$753$	0	0
$754$	−8.72012	−0.317568
$755$	21.7334	0.790958
$756$	0	0
$757$	7.90395	0.287274	0.143637	−	0.989630i	$-0.454120\pi$
$757$	7.90395	0.287274	0.143637	+	0.989630i	$0.454120\pi$
$758$	−2.63458	−0.0956922
$759$	0	0
$760$	16.3518	0.593141
$761$	−1.19183	−0.0432039	−0.0216020	−	0.999767i	$-0.506877\pi$
$761$	−1.19183	−0.0432039	−0.0216020	+	0.999767i	$0.506877\pi$
$762$	0	0
$763$	4.55460	0.164887
$764$	−10.5234	−0.380724
$765$	0	0
$766$	−6.65098	−0.240310
$767$	35.7934	1.29242
$768$	0	0
$769$	−1.86185	−0.0671400	−0.0335700	−	0.999436i	$-0.510688\pi$
$769$	−1.86185	−0.0671400	−0.0335700	+	0.999436i	$0.510688\pi$
$770$	4.12924	0.148807
$771$	0	0
$772$	28.2036	1.01507
$773$	−33.4527	−1.20321	−0.601604	−	0.798794i	$-0.705472\pi$
$773$	−33.4527	−1.20321	−0.601604	+	0.798794i	$0.705472\pi$
$774$	0	0
$775$	−12.1374	−0.435990
$776$	6.75990	0.242666
$777$	0	0
$778$	−0.696221	−0.0249607
$779$	−56.1677	−2.01242
$780$	0	0
$781$	−41.3658	−1.48019
$782$	9.71500	0.347408
$783$	0	0
$784$	3.08577	0.110206
$785$	42.6232	1.52129
$786$	0	0
$787$	20.8078	0.741718	0.370859	−	0.928689i	$-0.379063\pi$
$787$	20.8078	0.741718	0.370859	+	0.928689i	$0.379063\pi$
$788$	4.19904	0.149585
$789$	0	0
$790$	1.40427	0.0499617
$791$	0.943880	0.0335605
$792$	0	0
$793$	0.370545	0.0131584
$794$	−4.97846	−0.176679
$795$	0	0
$796$	44.6226	1.58160
$797$	34.8605	1.23482	0.617410	−	0.786641i	$-0.288182\pi$
$797$	34.8605	1.23482	0.617410	+	0.786641i	$0.288182\pi$
$798$	0	0
$799$	−13.1826	−0.466368
$800$	5.06479	0.179068
$801$	0	0
$802$	−3.52421	−0.124444
$803$	9.58717	0.338324
$804$	0	0
$805$	−6.04843	−0.213179
$806$	−26.2234	−0.923680
$807$	0	0
$808$	6.56081	0.230809
$809$	−2.21699	−0.0779452	−0.0389726	−	0.999240i	$-0.512409\pi$
$809$	−2.21699	−0.0779452	−0.0389726	+	0.999240i	$0.512409\pi$
$810$	0	0
$811$	10.5287	0.369712	0.184856	−	0.982766i	$-0.440818\pi$
$811$	10.5287	0.369712	0.184856	+	0.982766i	$0.440818\pi$
$812$	−6.26000	−0.219683
$813$	0	0
$814$	−15.3346	−0.537478
$815$	−42.2190	−1.47887
$816$	0	0
$817$	−70.2903	−2.45914
$818$	12.1283	0.424056
$819$	0	0
$820$	36.6929	1.28137
$821$	3.61333	0.126106	0.0630530	−	0.998010i	$-0.479916\pi$
$821$	3.61333	0.126106	0.0630530	+	0.998010i	$0.479916\pi$
$822$	0	0
$823$	10.2088	0.355857	0.177928	−	0.984043i	$-0.443060\pi$
$823$	10.2088	0.355857	0.177928	+	0.984043i	$0.443060\pi$
$824$	−5.38807	−0.187702
$825$	0	0
$826$	−2.18062	−0.0758736
$827$	15.0646	0.523849	0.261924	−	0.965088i	$-0.415643\pi$
$827$	15.0646	0.523849	0.261924	+	0.965088i	$0.415643\pi$
$828$	0	0
$829$	26.0848	0.905962	0.452981	−	0.891520i	$-0.350361\pi$
$829$	26.0848	0.905962	0.452981	+	0.891520i	$0.350361\pi$
$830$	−5.33311	−0.185115
$831$	0	0
$832$	−29.1260	−1.00976
$833$	−7.92778	−0.274681
$834$	0	0
$835$	13.4859	0.466700
$836$	54.3117	1.87841
$837$	0	0
$838$	1.20017	0.0414590
$839$	−15.8729	−0.547993	−0.273996	−	0.961731i	$-0.588346\pi$
$839$	−15.8729	−0.547993	−0.273996	+	0.961731i	$0.588346\pi$
$840$	0	0
$841$	−17.4697	−0.602404
$842$	−2.32823	−0.0802360
$843$	0	0
$844$	−7.21180	−0.248240
$845$	−56.9140	−1.95790
$846$	0	0
$847$	17.5941	0.604540
$848$	−29.3717	−1.00863
$849$	0	0
$850$	−3.72720	−0.127842
$851$	22.4618	0.769982
$852$	0	0
$853$	36.2530	1.24128	0.620639	−	0.784096i	$-0.286873\pi$
$853$	36.2530	1.24128	0.620639	+	0.784096i	$0.286873\pi$
$854$	−0.0225745	−0.000772484 0
$855$	0	0
$856$	−18.9388	−0.647316
$857$	−0.282539	−0.00965134	−0.00482567	−	0.999988i	$-0.501536\pi$
$857$	−0.282539	−0.00965134	−0.00482567	+	0.999988i	$0.501536\pi$
$858$	0	0
$859$	21.8107	0.744172	0.372086	−	0.928198i	$-0.378643\pi$
$859$	21.8107	0.744172	0.372086	+	0.928198i	$0.378643\pi$
$860$	45.9188	1.56582
$861$	0	0
$862$	8.30050	0.282716
$863$	−49.3581	−1.68017	−0.840084	−	0.542456i	$-0.817494\pi$
$863$	−49.3581	−1.68017	−0.840084	+	0.542456i	$0.817494\pi$
$864$	0	0
$865$	38.5214	1.30977
$866$	3.59150	0.122044
$867$	0	0
$868$	−18.8253	−0.638971
$869$	9.72427	0.329873
$870$	0	0
$871$	−65.2194	−2.20987
$872$	6.92425	0.234485
$873$	0	0
$874$	6.75136	0.228368
$875$	12.0819	0.408442
$876$	0	0
$877$	3.95664	0.133606	0.0668032	−	0.997766i	$-0.478720\pi$
$877$	3.95664	0.133606	0.0668032	+	0.997766i	$0.478720\pi$
$878$	6.48433	0.218836
$879$	0	0
$880$	−32.2139	−1.08593
$881$	−3.54601	−0.119468	−0.0597341	−	0.998214i	$-0.519025\pi$
$881$	−3.54601	−0.119468	−0.0597341	+	0.998214i	$0.519025\pi$
$882$	0	0
$883$	49.6627	1.67128	0.835642	−	0.549274i	$-0.185096\pi$
$883$	49.6627	1.67128	0.835642	+	0.549274i	$0.185096\pi$
$884$	94.8895	3.19148
$885$	0	0
$886$	9.61133	0.322899
$887$	55.3611	1.85884	0.929421	−	0.369021i	$-0.120307\pi$
$887$	55.3611	1.85884	0.929421	+	0.369021i	$0.120307\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−13.5457	−0.454054
$891$	0	0
$892$	4.46191	0.149396
$893$	−9.16116	−0.306567
$894$	0	0
$895$	−45.0132	−1.50463
$896$	10.2966	0.343986
$897$	0	0
$898$	8.73248	0.291407
$899$	34.6742	1.15645
$900$	0	0
$901$	75.4601	2.51394
$902$	−21.5633	−0.717978
$903$	0	0
$904$	1.43496	0.0477260
$905$	−3.75039	−0.124667
$906$	0	0
$907$	−14.0958	−0.468042	−0.234021	−	0.972231i	$-0.575189\pi$
$907$	−14.0958	−0.468042	−0.234021	+	0.972231i	$0.575189\pi$
$908$	−20.1239	−0.667836
$909$	0	0
$910$	5.01353	0.166197
$911$	−53.7470	−1.78072	−0.890359	−	0.455259i	$-0.849547\pi$
$911$	−53.7470	−1.78072	−0.890359	+	0.455259i	$0.849547\pi$
$912$	0	0
$913$	−36.9306	−1.22223
$914$	−5.72807	−0.189468
$915$	0	0
$916$	−15.0511	−0.497301
$917$	−4.18883	−0.138327
$918$	0	0
$919$	−16.6028	−0.547675	−0.273837	−	0.961776i	$-0.588293\pi$
$919$	−16.6028	−0.547675	−0.273837	+	0.961776i	$0.588293\pi$
$920$	−9.19528	−0.303159
$921$	0	0
$922$	−12.7041	−0.418387
$923$	−50.2245	−1.65316
$924$	0	0
$925$	−8.61757	−0.283344
$926$	6.28557	0.206557
$927$	0	0
$928$	−14.4691	−0.474971
$929$	−47.3772	−1.55440	−0.777198	−	0.629256i	$-0.783360\pi$
$929$	−47.3772	−1.55440	−0.777198	+	0.629256i	$0.783360\pi$
$930$	0	0
$931$	−5.50935	−0.180562
$932$	−37.5300	−1.22933
$933$	0	0
$934$	−0.735308	−0.0240600
$935$	82.7621	2.70661
$936$	0	0
$937$	20.7488	0.677832	0.338916	−	0.940817i	$-0.389940\pi$
$937$	20.7488	0.677832	0.338916	+	0.940817i	$0.389940\pi$
$938$	3.97333	0.129734
$939$	0	0
$940$	5.98475	0.195201
$941$	−43.3850	−1.41431	−0.707155	−	0.707059i	$-0.750022\pi$
$941$	−43.3850	−1.41431	−0.707155	+	0.707059i	$0.750022\pi$
$942$	0	0
$943$	31.5854	1.02856
$944$	17.0119	0.553691
$945$	0	0
$946$	−26.9850	−0.877360
$947$	−17.1077	−0.555924	−0.277962	−	0.960592i	$-0.589659\pi$
$947$	−17.1077	−0.555924	−0.277962	+	0.960592i	$0.589659\pi$
$948$	0	0
$949$	11.6403	0.377860
$950$	−2.59019	−0.0840368
$951$	0	0
$952$	−12.0524	−0.390621
$953$	−10.4394	−0.338165	−0.169082	−	0.985602i	$-0.554080\pi$
$953$	−10.4394	−0.338165	−0.169082	+	0.985602i	$0.554080\pi$
$954$	0	0
$955$	−11.1441	−0.360614
$956$	−17.7551	−0.574240
$957$	0	0
$958$	−8.59947	−0.277836
$959$	−4.87863	−0.157539
$960$	0	0
$961$	73.2734	2.36366
$962$	−18.6186	−0.600288
$963$	0	0
$964$	−52.0871	−1.67761
$965$	29.8670	0.961452
$966$	0	0
$967$	13.6829	0.440013	0.220007	−	0.975498i	$-0.429392\pi$
$967$	13.6829	0.440013	0.220007	+	0.975498i	$0.429392\pi$
$968$	26.7479	0.859710
$969$	0	0
$970$	3.43360	0.110246
$971$	−47.4849	−1.52386	−0.761931	−	0.647658i	$-0.775749\pi$
$971$	−47.4849	−1.52386	−0.761931	+	0.647658i	$0.775749\pi$
$972$	0	0
$973$	3.61685	0.115951
$974$	13.2775	0.425439
$975$	0	0
$976$	0.176113	0.00563724
$977$	35.7720	1.14445	0.572223	−	0.820098i	$-0.306081\pi$
$977$	35.7720	1.14445	0.572223	+	0.820098i	$0.306081\pi$
$978$	0	0
$979$	−93.8013	−2.99790
$980$	3.59912	0.114970
$981$	0	0
$982$	−7.17372	−0.228922
$983$	4.08379	0.130253	0.0651263	−	0.997877i	$-0.479255\pi$
$983$	4.08379	0.130253	0.0651263	+	0.997877i	$0.479255\pi$
$984$	0	0
$985$	4.44670	0.141683
$986$	10.6479	0.339097
$987$	0	0
$988$	65.9427	2.09792
$989$	39.5272	1.25689
$990$	0	0
$991$	7.24609	0.230180	0.115090	−	0.993355i	$-0.463284\pi$
$991$	7.24609	0.230180	0.115090	+	0.993355i	$0.463284\pi$
$992$	−43.5119	−1.38150
$993$	0	0
$994$	3.05980	0.0970511
$995$	47.2543	1.49806
$996$	0	0
$997$	4.41907	0.139953	0.0699767	−	0.997549i	$-0.477708\pi$
$997$	4.41907	0.139953	0.0699767	+	0.997549i	$0.477708\pi$
$998$	9.00812	0.285147
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.v.1.10		19
3.2	odd	2		2667.2.a.q.1.10	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.q.1.10	✓	19	3.2	odd	2
8001.2.a.v.1.10		19	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q-0.395540 q^{2} -1.84355 q^{4} -1.95228 q^{5} +1.00000 q^{7} +1.52028 q^{8} +O(q^{10})\)	\(q-0.395540 q^{2} -1.84355 q^{4} -1.95228 q^{5} +1.00000 q^{7} +1.52028 q^{8} +0.772204 q^{10} +5.34735 q^{11} +6.49250 q^{13} -0.395540 q^{14} +3.08577 q^{16} -7.92778 q^{17} -5.50935 q^{19} +3.59912 q^{20} -2.11509 q^{22} +3.09814 q^{23} -1.18861 q^{25} -2.56804 q^{26} -1.84355 q^{28} +3.39563 q^{29} +10.2114 q^{31} -4.26110 q^{32} +3.13575 q^{34} -1.95228 q^{35} +7.25011 q^{37} +2.17917 q^{38} -2.96800 q^{40} +10.1950 q^{41} +12.7584 q^{43} -9.85809 q^{44} -1.22544 q^{46} +1.66284 q^{47} +1.00000 q^{49} +0.470144 q^{50} -11.9692 q^{52} -9.51844 q^{53} -10.4395 q^{55} +1.52028 q^{56} -1.34311 q^{58} +5.51303 q^{59} +0.0570727 q^{61} -4.03903 q^{62} -4.48610 q^{64} -12.6752 q^{65} -10.0453 q^{67} +14.6152 q^{68} +0.772204 q^{70} -7.73577 q^{71} +1.79288 q^{73} -2.86771 q^{74} +10.1568 q^{76} +5.34735 q^{77} +1.81852 q^{79} -6.02427 q^{80} -4.03252 q^{82} -6.90635 q^{83} +15.4772 q^{85} -5.04644 q^{86} +8.12944 q^{88} -17.5417 q^{89} +6.49250 q^{91} -5.71157 q^{92} -0.657719 q^{94} +10.7558 q^{95} +4.44649 q^{97} -0.395540 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8}+O(q^{10}) \) 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8	\( 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8} + 9 q^{11} + 24 q^{13} - 4 q^{14} + 20 q^{16} - 17 q^{17} + 23 q^{19} - 5 q^{20} - 3 q^{22} + 17 q^{23} + 38 q^{25} - 28 q^{26} + 22 q^{28} - 2 q^{29} + 16 q^{31} - 17 q^{32} + 29 q^{34} - 5 q^{35} + 56 q^{37} - 2 q^{38} - 13 q^{40} + 7 q^{41} + 19 q^{43} + 29 q^{44} + 10 q^{46} - 25 q^{47} + 19 q^{49} + 9 q^{50} + 16 q^{52} - 18 q^{53} + 10 q^{55} - 9 q^{56} + 31 q^{58} - 11 q^{59} + 26 q^{61} - 26 q^{62} + 45 q^{64} - 27 q^{65} + 24 q^{67} - 14 q^{68} + 32 q^{71} + 51 q^{73} + 12 q^{76} + 9 q^{77} + 30 q^{79} + 30 q^{80} - 52 q^{82} - q^{83} + 44 q^{85} + 24 q^{86} - 30 q^{88} - 5 q^{89} + 24 q^{91} + 88 q^{92} + 7 q^{94} + 24 q^{95} + 5 q^{97} - 4 q^{98}+O(q^{100}) \) 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8 + 9 * q^11 + 24 * q^13 - 4 * q^14 + 20 * q^16 - 17 * q^17 + 23 * q^19 - 5 * q^20 - 3 * q^22 + 17 * q^23 + 38 * q^25 - 28 * q^26 + 22 * q^28 - 2 * q^29 + 16 * q^31 - 17 * q^32 + 29 * q^34 - 5 * q^35 + 56 * q^37 - 2 * q^38 - 13 * q^40 + 7 * q^41 + 19 * q^43 + 29 * q^44 + 10 * q^46 - 25 * q^47 + 19 * q^49 + 9 * q^50 + 16 * q^52 - 18 * q^53 + 10 * q^55 - 9 * q^56 + 31 * q^58 - 11 * q^59 + 26 * q^61 - 26 * q^62 + 45 * q^64 - 27 * q^65 + 24 * q^67 - 14 * q^68 + 32 * q^71 + 51 * q^73 + 12 * q^76 + 9 * q^77 + 30 * q^79 + 30 * q^80 - 52 * q^82 - q^83 + 44 * q^85 + 24 * q^86 - 30 * q^88 - 5 * q^89 + 24 * q^91 + 88 * q^92 + 7 * q^94 + 24 * q^95 + 5 * q^97 - 4 * q^98

Introduction

L-functions

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