LMFDB - Embedded newform 8001.2.a.q.1.13

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:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 22 x^{13} + 186 x^{11} - 763 x^{9} - 7 x^{8} + 1588 x^{7} + 64 x^{6} - 1625 x^{5} - 185 x^{4} + \cdots - 13 $ x^15 - 22x^13 + 186x^11 - 763x^9 - 7x^8 + 1588x^7 + 64x^6 - 1625x^5 - 185x^4 + 726x^3 + 145x^2 - 83*x - 13
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$2.16969$ 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)

$f(q)$	$=$	$q+2.16969 q^{2} +2.70756 q^{4} +2.09371 q^{5} -1.00000 q^{7} +1.53520 q^{8} +O(q^{10})$	\(q+2.16969 q^{2} +2.70756 q^{4} +2.09371 q^{5} -1.00000 q^{7} +1.53520 q^{8} +4.54272 q^{10} -6.57883 q^{11} +2.39437 q^{13} -2.16969 q^{14} -2.08422 q^{16} +1.17908 q^{17} -3.96173 q^{19} +5.66887 q^{20} -14.2740 q^{22} -0.848810 q^{23} -0.616358 q^{25} +5.19506 q^{26} -2.70756 q^{28} -8.50037 q^{29} +8.20411 q^{31} -7.59252 q^{32} +2.55825 q^{34} -2.09371 q^{35} -4.04576 q^{37} -8.59575 q^{38} +3.21427 q^{40} +6.47720 q^{41} -0.335632 q^{43} -17.8126 q^{44} -1.84166 q^{46} -10.7732 q^{47} +1.00000 q^{49} -1.33731 q^{50} +6.48292 q^{52} -3.76105 q^{53} -13.7742 q^{55} -1.53520 q^{56} -18.4432 q^{58} -4.27318 q^{59} +15.6127 q^{61} +17.8004 q^{62} -12.3050 q^{64} +5.01314 q^{65} -9.44445 q^{67} +3.19244 q^{68} -4.54272 q^{70} -12.4140 q^{71} +16.1518 q^{73} -8.77806 q^{74} -10.7267 q^{76} +6.57883 q^{77} -10.8216 q^{79} -4.36377 q^{80} +14.0535 q^{82} -12.3415 q^{83} +2.46866 q^{85} -0.728217 q^{86} -10.0998 q^{88} -5.20838 q^{89} -2.39437 q^{91} -2.29821 q^{92} -23.3744 q^{94} -8.29474 q^{95} -3.10665 q^{97} +2.16969 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7}+O(q^{10}) $ 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7	$ 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7} + 10 q^{10} - 14 q^{11} + 6 q^{13} + 20 q^{16} - 10 q^{17} + 13 q^{19} - 8 q^{20} - 11 q^{22} - 15 q^{23} - 22 q^{26} - 14 q^{28} - 16 q^{29} + 22 q^{31} - 15 q^{34} + 7 q^{35} - 14 q^{37} + 6 q^{38} + 22 q^{40} - 19 q^{41} - q^{43} - 25 q^{44} - 28 q^{46} - 49 q^{47} + 15 q^{49} - 24 q^{50} - 17 q^{52} + 28 q^{53} + 39 q^{55} - 10 q^{58} - 43 q^{59} + 27 q^{61} - 14 q^{62} + 18 q^{64} + 8 q^{65} + 3 q^{67} - 13 q^{68} - 10 q^{70} - 55 q^{71} - 3 q^{73} + 12 q^{74} - 20 q^{76} + 14 q^{77} + 18 q^{79} - 29 q^{80} + 14 q^{82} - 17 q^{83} + 7 q^{85} - 4 q^{86} - 114 q^{88} - 36 q^{89} - 6 q^{91} - 45 q^{92} - 15 q^{94} - 59 q^{95} - 2 q^{97}+O(q^{100}) $ 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7 + 10 * q^10 - 14 * q^11 + 6 * q^13 + 20 * q^16 - 10 * q^17 + 13 * q^19 - 8 * q^20 - 11 * q^22 - 15 * q^23 - 22 * q^26 - 14 * q^28 - 16 * q^29 + 22 * q^31 - 15 * q^34 + 7 * q^35 - 14 * q^37 + 6 * q^38 + 22 * q^40 - 19 * q^41 - q^43 - 25 * q^44 - 28 * q^46 - 49 * q^47 + 15 * q^49 - 24 * q^50 - 17 * q^52 + 28 * q^53 + 39 * q^55 - 10 * q^58 - 43 * q^59 + 27 * q^61 - 14 * q^62 + 18 * q^64 + 8 * q^65 + 3 * q^67 - 13 * q^68 - 10 * q^70 - 55 * q^71 - 3 * q^73 + 12 * q^74 - 20 * q^76 + 14 * q^77 + 18 * q^79 - 29 * q^80 + 14 * q^82 - 17 * q^83 + 7 * q^85 - 4 * q^86 - 114 * q^88 - 36 * q^89 - 6 * q^91 - 45 * q^92 - 15 * q^94 - 59 * q^95 - 2 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.16969	1.53420	0.767102	−	0.641525i	$-0.221698\pi$
$2$	2.16969	1.53420	0.767102	+	0.641525i	$0.221698\pi$
$3$	0	0
$3$	0	0
$4$	2.70756	1.35378
$5$	2.09371	0.936338	0.468169	−	0.883639i	$-0.344914\pi$
$5$	2.09371	0.936338	0.468169	+	0.883639i	$0.344914\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.53520	0.542775
$9$	0	0
$10$	4.54272	1.43653
$11$	−6.57883	−1.98359	−0.991796	−	0.127832i	$-0.959198\pi$
$11$	−6.57883	−1.98359	−0.991796	+	0.127832i	$0.959198\pi$
$12$	0	0
$13$	2.39437	0.664080	0.332040	−	0.943265i	$-0.392263\pi$
$13$	2.39437	0.664080	0.332040	+	0.943265i	$0.392263\pi$
$14$	−2.16969	−0.579875
$15$	0	0
$16$	−2.08422	−0.521056
$17$	1.17908	0.285969	0.142985	−	0.989725i	$-0.454330\pi$
$17$	1.17908	0.285969	0.142985	+	0.989725i	$0.454330\pi$
$18$	0	0
$19$	−3.96173	−0.908884	−0.454442	−	0.890776i	$-0.650161\pi$
$19$	−3.96173	−0.908884	−0.454442	+	0.890776i	$0.650161\pi$
$20$	5.66887	1.26760
$21$	0	0
$22$	−14.2740	−3.04323
$23$	−0.848810	−0.176989	−0.0884946	−	0.996077i	$-0.528206\pi$
$23$	−0.848810	−0.176989	−0.0884946	+	0.996077i	$0.528206\pi$
$24$	0	0
$25$	−0.616358	−0.123272
$26$	5.19506	1.01883
$27$	0	0
$28$	−2.70756	−0.511682
$29$	−8.50037	−1.57848	−0.789239	−	0.614086i	$-0.789525\pi$
$29$	−8.50037	−1.57848	−0.789239	+	0.614086i	$0.789525\pi$
$30$	0	0
$31$	8.20411	1.47350	0.736751	−	0.676164i	$-0.236359\pi$
$31$	8.20411	1.47350	0.736751	+	0.676164i	$0.236359\pi$
$32$	−7.59252	−1.34218
$33$	0	0
$34$	2.55825	0.438735
$35$	−2.09371	−0.353902
$36$	0	0
$37$	−4.04576	−0.665119	−0.332560	−	0.943082i	$-0.607912\pi$
$37$	−4.04576	−0.665119	−0.332560	+	0.943082i	$0.607912\pi$
$38$	−8.59575	−1.39441
$39$	0	0
$40$	3.21427	0.508220
$41$	6.47720	1.01157	0.505785	−	0.862660i	$-0.331203\pi$
$41$	6.47720	1.01157	0.505785	+	0.862660i	$0.331203\pi$
$42$	0	0
$43$	−0.335632	−0.0511833	−0.0255917	−	0.999672i	$-0.508147\pi$
$43$	−0.335632	−0.0511833	−0.0255917	+	0.999672i	$0.508147\pi$
$44$	−17.8126	−2.68535
$45$	0	0
$46$	−1.84166	−0.271538
$47$	−10.7732	−1.57143	−0.785713	−	0.618591i	$-0.787704\pi$
$47$	−10.7732	−1.57143	−0.785713	+	0.618591i	$0.787704\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.33731	−0.189124
$51$	0	0
$52$	6.48292	0.899020
$53$	−3.76105	−0.516619	−0.258310	−	0.966062i	$-0.583165\pi$
$53$	−3.76105	−0.516619	−0.258310	+	0.966062i	$0.583165\pi$
$54$	0	0
$55$	−13.7742	−1.85731
$56$	−1.53520	−0.205149
$57$	0	0
$58$	−18.4432	−2.42171
$59$	−4.27318	−0.556321	−0.278160	−	0.960535i	$-0.589725\pi$
$59$	−4.27318	−0.556321	−0.278160	+	0.960535i	$0.589725\pi$
$60$	0	0
$61$	15.6127	1.99901	0.999503	−	0.0315280i	$-0.0100373\pi$
$61$	15.6127	1.99901	0.999503	+	0.0315280i	$0.0100373\pi$
$62$	17.8004	2.26065
$63$	0	0
$64$	−12.3050	−1.53812
$65$	5.01314	0.621803
$66$	0	0
$67$	−9.44445	−1.15382	−0.576912	−	0.816807i	$-0.695742\pi$
$67$	−9.44445	−1.15382	−0.576912	+	0.816807i	$0.695742\pi$
$68$	3.19244	0.387140
$69$	0	0
$70$	−4.54272	−0.542959
$71$	−12.4140	−1.47328	−0.736638	−	0.676287i	$-0.763588\pi$
$71$	−12.4140	−1.47328	−0.736638	+	0.676287i	$0.763588\pi$
$72$	0	0
$73$	16.1518	1.89043	0.945215	−	0.326447i	$-0.105852\pi$
$73$	16.1518	1.89043	0.945215	+	0.326447i	$0.105852\pi$
$74$	−8.77806	−1.02043
$75$	0	0
$76$	−10.7267	−1.23043
$77$	6.57883	0.749727
$78$	0	0
$79$	−10.8216	−1.21753	−0.608764	−	0.793351i	$-0.708334\pi$
$79$	−10.8216	−1.21753	−0.608764	+	0.793351i	$0.708334\pi$
$80$	−4.36377	−0.487884
$81$	0	0
$82$	14.0535	1.55195
$83$	−12.3415	−1.35466	−0.677329	−	0.735680i	$-0.736863\pi$
$83$	−12.3415	−1.35466	−0.677329	+	0.735680i	$0.736863\pi$
$84$	0	0
$85$	2.46866	0.267764
$86$	−0.728217	−0.0785257
$87$	0	0
$88$	−10.0998	−1.07664
$89$	−5.20838	−0.552087	−0.276044	−	0.961145i	$-0.589023\pi$
$89$	−5.20838	−0.552087	−0.276044	+	0.961145i	$0.589023\pi$
$90$	0	0
$91$	−2.39437	−0.250999
$92$	−2.29821	−0.239605
$93$	0	0
$94$	−23.3744	−2.41089
$95$	−8.29474	−0.851023
$96$	0	0
$97$	−3.10665	−0.315432	−0.157716	−	0.987484i	$-0.550413\pi$
$97$	−3.10665	−0.315432	−0.157716	+	0.987484i	$0.550413\pi$
$98$	2.16969	0.219172
$99$	0	0
$100$	−1.66883	−0.166883
$101$	−3.46958	−0.345236	−0.172618	−	0.984989i	$-0.555223\pi$
$101$	−3.46958	−0.345236	−0.172618	+	0.984989i	$0.555223\pi$
$102$	0	0
$103$	−5.27117	−0.519384	−0.259692	−	0.965692i	$-0.583621\pi$
$103$	−5.27117	−0.519384	−0.259692	+	0.965692i	$0.583621\pi$
$104$	3.67584	0.360446
$105$	0	0
$106$	−8.16031	−0.792599
$107$	14.7861	1.42942	0.714712	−	0.699419i	$-0.246558\pi$
$107$	14.7861	1.42942	0.714712	+	0.699419i	$0.246558\pi$
$108$	0	0
$109$	−12.0655	−1.15566	−0.577831	−	0.816157i	$-0.696101\pi$
$109$	−12.0655	−1.15566	−0.577831	+	0.816157i	$0.696101\pi$
$110$	−29.8858	−2.84950
$111$	0	0
$112$	2.08422	0.196940
$113$	7.86163	0.739560	0.369780	−	0.929119i	$-0.379433\pi$
$113$	7.86163	0.739560	0.369780	+	0.929119i	$0.379433\pi$
$114$	0	0
$115$	−1.77717	−0.165722
$116$	−23.0153	−2.13692
$117$	0	0
$118$	−9.27149	−0.853510
$119$	−1.17908	−0.108086
$120$	0	0
$121$	32.2810	2.93464
$122$	33.8748	3.06688
$123$	0	0
$124$	22.2132	1.99480
$125$	−11.7591	−1.05176
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−11.5130	−1.01761
$129$	0	0
$130$	10.8770	0.953973
$131$	0.944582	0.0825285	0.0412643	−	0.999148i	$-0.486861\pi$
$131$	0.944582	0.0825285	0.0412643	+	0.999148i	$0.486861\pi$
$132$	0	0
$133$	3.96173	0.343526
$134$	−20.4915	−1.77020
$135$	0	0
$136$	1.81012	0.155217
$137$	7.99653	0.683190	0.341595	−	0.939847i	$-0.389033\pi$
$137$	7.99653	0.683190	0.341595	+	0.939847i	$0.389033\pi$
$138$	0	0
$139$	10.5965	0.898787	0.449393	−	0.893334i	$-0.351640\pi$
$139$	10.5965	0.898787	0.449393	+	0.893334i	$0.351640\pi$
$140$	−5.66887	−0.479107
$141$	0	0
$142$	−26.9347	−2.26031
$143$	−15.7522	−1.31726
$144$	0	0
$145$	−17.7973	−1.47799
$146$	35.0445	2.90031
$147$	0	0
$148$	−10.9542	−0.900427
$149$	−9.22129	−0.755437	−0.377719	−	0.925920i	$-0.623291\pi$
$149$	−9.22129	−0.755437	−0.377719	+	0.925920i	$0.623291\pi$
$150$	0	0
$151$	−13.7018	−1.11504	−0.557518	−	0.830165i	$-0.688246\pi$
$151$	−13.7018	−1.11504	−0.557518	+	0.830165i	$0.688246\pi$
$152$	−6.08205	−0.493319
$153$	0	0
$154$	14.2740	1.15023
$155$	17.1771	1.37970
$156$	0	0
$157$	−0.664318	−0.0530184	−0.0265092	−	0.999649i	$-0.508439\pi$
$157$	−0.664318	−0.0530184	−0.0265092	+	0.999649i	$0.508439\pi$
$158$	−23.4796	−1.86794
$159$	0	0
$160$	−15.8966	−1.25673
$161$	0.848810	0.0668956
$162$	0	0
$163$	−6.58565	−0.515828	−0.257914	−	0.966168i	$-0.583035\pi$
$163$	−6.58565	−0.515828	−0.257914	+	0.966168i	$0.583035\pi$
$164$	17.5375	1.36944
$165$	0	0
$166$	−26.7773	−2.07832
$167$	−7.11915	−0.550896	−0.275448	−	0.961316i	$-0.588826\pi$
$167$	−7.11915	−0.550896	−0.275448	+	0.961316i	$0.588826\pi$
$168$	0	0
$169$	−7.26697	−0.558998
$170$	5.35624	0.410805
$171$	0	0
$172$	−0.908744	−0.0692911
$173$	−2.27352	−0.172853	−0.0864263	−	0.996258i	$-0.527545\pi$
$173$	−2.27352	−0.172853	−0.0864263	+	0.996258i	$0.527545\pi$
$174$	0	0
$175$	0.616358	0.0465923
$176$	13.7117	1.03356
$177$	0	0
$178$	−11.3006	−0.847015
$179$	−5.60890	−0.419229	−0.209614	−	0.977784i	$-0.567221\pi$
$179$	−5.60890	−0.419229	−0.209614	+	0.977784i	$0.567221\pi$
$180$	0	0
$181$	9.95237	0.739754	0.369877	−	0.929081i	$-0.379400\pi$
$181$	9.95237	0.739754	0.369877	+	0.929081i	$0.379400\pi$
$182$	−5.19506	−0.385083
$183$	0	0
$184$	−1.30309	−0.0960652
$185$	−8.47067	−0.622776
$186$	0	0
$187$	−7.75698	−0.567247
$188$	−29.1690	−2.12737
$189$	0	0
$190$	−17.9970	−1.30564
$191$	24.1579	1.74801	0.874003	−	0.485920i	$-0.161515\pi$
$191$	24.1579	1.74801	0.874003	+	0.485920i	$0.161515\pi$
$192$	0	0
$193$	12.5406	0.902692	0.451346	−	0.892349i	$-0.350944\pi$
$193$	12.5406	0.902692	0.451346	+	0.892349i	$0.350944\pi$
$194$	−6.74047	−0.483938
$195$	0	0
$196$	2.70756	0.193397
$197$	0.250024	0.0178135	0.00890674	−	0.999960i	$-0.497165\pi$
$197$	0.250024	0.0178135	0.00890674	+	0.999960i	$0.497165\pi$
$198$	0	0
$199$	−22.6063	−1.60252	−0.801259	−	0.598317i	$-0.795836\pi$
$199$	−22.6063	−1.60252	−0.801259	+	0.598317i	$0.795836\pi$
$200$	−0.946232	−0.0669087
$201$	0	0
$202$	−7.52791	−0.529662
$203$	8.50037	0.596609
$204$	0	0
$205$	13.5614	0.947171
$206$	−11.4368	−0.796840
$207$	0	0
$208$	−4.99041	−0.346023
$209$	26.0636	1.80286
$210$	0	0
$211$	6.21178	0.427637	0.213818	−	0.976873i	$-0.431410\pi$
$211$	6.21178	0.427637	0.213818	+	0.976873i	$0.431410\pi$
$212$	−10.1833	−0.699390
$213$	0	0
$214$	32.0812	2.19303
$215$	−0.702717	−0.0479249
$216$	0	0
$217$	−8.20411	−0.556932
$218$	−26.1783	−1.77302
$219$	0	0
$220$	−37.2945	−2.51440
$221$	2.82316	0.189907
$222$	0	0
$223$	−14.6561	−0.981448	−0.490724	−	0.871315i	$-0.663268\pi$
$223$	−14.6561	−0.981448	−0.490724	+	0.871315i	$0.663268\pi$
$224$	7.59252	0.507296
$225$	0	0
$226$	17.0573	1.13464
$227$	27.6162	1.83295	0.916474	−	0.400093i	$-0.131022\pi$
$227$	27.6162	1.83295	0.916474	+	0.400093i	$0.131022\pi$
$228$	0	0
$229$	4.27058	0.282208	0.141104	−	0.989995i	$-0.454935\pi$
$229$	4.27058	0.282208	0.141104	+	0.989995i	$0.454935\pi$
$230$	−3.85590	−0.254251
$231$	0	0
$232$	−13.0497	−0.856758
$233$	18.1131	1.18663	0.593315	−	0.804970i	$-0.297819\pi$
$233$	18.1131	1.18663	0.593315	+	0.804970i	$0.297819\pi$
$234$	0	0
$235$	−22.5559	−1.47139
$236$	−11.5699	−0.753137
$237$	0	0
$238$	−2.55825	−0.165826
$239$	−10.6946	−0.691776	−0.345888	−	0.938276i	$-0.612422\pi$
$239$	−10.6946	−0.691776	−0.345888	+	0.938276i	$0.612422\pi$
$240$	0	0
$241$	−4.90676	−0.316072	−0.158036	−	0.987433i	$-0.550516\pi$
$241$	−4.90676	−0.316072	−0.158036	+	0.987433i	$0.550516\pi$
$242$	70.0398	4.50233
$243$	0	0
$244$	42.2725	2.70622
$245$	2.09371	0.133763
$246$	0	0
$247$	−9.48588	−0.603572
$248$	12.5949	0.799780
$249$	0	0
$250$	−25.5135	−1.61362
$251$	17.7355	1.11946	0.559729	−	0.828676i	$-0.310905\pi$
$251$	17.7355	1.11946	0.559729	+	0.828676i	$0.310905\pi$
$252$	0	0
$253$	5.58418	0.351074
$254$	2.16969	0.136139
$255$	0	0
$256$	−0.369686	−0.0231054
$257$	−7.66946	−0.478408	−0.239204	−	0.970969i	$-0.576886\pi$
$257$	−7.66946	−0.478408	−0.239204	+	0.970969i	$0.576886\pi$
$258$	0	0
$259$	4.04576	0.251391
$260$	13.5734	0.841786
$261$	0	0
$262$	2.04945	0.126616
$263$	2.50055	0.154190	0.0770952	−	0.997024i	$-0.475435\pi$
$263$	2.50055	0.154190	0.0770952	+	0.997024i	$0.475435\pi$
$264$	0	0
$265$	−7.87456	−0.483730
$266$	8.59575	0.527039
$267$	0	0
$268$	−25.5715	−1.56203
$269$	−1.77278	−0.108088	−0.0540442	−	0.998539i	$-0.517211\pi$
$269$	−1.77278	−0.108088	−0.0540442	+	0.998539i	$0.517211\pi$
$270$	0	0
$271$	5.69192	0.345759	0.172880	−	0.984943i	$-0.444693\pi$
$271$	5.69192	0.345759	0.172880	+	0.984943i	$0.444693\pi$
$272$	−2.45747	−0.149006
$273$	0	0
$274$	17.3500	1.04815
$275$	4.05492	0.244521
$276$	0	0
$277$	−13.6249	−0.818639	−0.409320	−	0.912391i	$-0.634234\pi$
$277$	−13.6249	−0.818639	−0.409320	+	0.912391i	$0.634234\pi$
$278$	22.9912	1.37892
$279$	0	0
$280$	−3.21427	−0.192089
$281$	−6.15332	−0.367077	−0.183538	−	0.983013i	$-0.558755\pi$
$281$	−6.15332	−0.367077	−0.183538	+	0.983013i	$0.558755\pi$
$282$	0	0
$283$	23.3811	1.38986	0.694931	−	0.719076i	$-0.255435\pi$
$283$	23.3811	1.38986	0.694931	+	0.719076i	$0.255435\pi$
$284$	−33.6118	−1.99450
$285$	0	0
$286$	−34.1774	−2.02095
$287$	−6.47720	−0.382337
$288$	0	0
$289$	−15.6098	−0.918221
$290$	−38.6148	−2.26754
$291$	0	0
$292$	43.7322	2.55923
$293$	8.34496	0.487518	0.243759	−	0.969836i	$-0.421619\pi$
$293$	8.34496	0.487518	0.243759	+	0.969836i	$0.421619\pi$
$294$	0	0
$295$	−8.94682	−0.520904
$296$	−6.21105	−0.361010
$297$	0	0
$298$	−20.0074	−1.15900
$299$	−2.03237	−0.117535
$300$	0	0
$301$	0.335632	0.0193455
$302$	−29.7287	−1.71069
$303$	0	0
$304$	8.25714	0.473579
$305$	32.6886	1.87174
$306$	0	0
$307$	2.38253	0.135978	0.0679890	−	0.997686i	$-0.478342\pi$
$307$	2.38253	0.135978	0.0679890	+	0.997686i	$0.478342\pi$
$308$	17.8126	1.01497
$309$	0	0
$310$	37.2690	2.11674
$311$	30.6677	1.73901	0.869503	−	0.493929i	$-0.164440\pi$
$311$	30.6677	1.73901	0.869503	+	0.493929i	$0.164440\pi$
$312$	0	0
$313$	−20.5686	−1.16261	−0.581303	−	0.813687i	$-0.697457\pi$
$313$	−20.5686	−1.16261	−0.581303	+	0.813687i	$0.697457\pi$
$314$	−1.44137	−0.0813410
$315$	0	0
$316$	−29.3003	−1.64827
$317$	−19.4946	−1.09493	−0.547463	−	0.836830i	$-0.684406\pi$
$317$	−19.4946	−1.09493	−0.547463	+	0.836830i	$0.684406\pi$
$318$	0	0
$319$	55.9225	3.13106
$320$	−25.7631	−1.44020
$321$	0	0
$322$	1.84166	0.102632
$323$	−4.67121	−0.259913
$324$	0	0
$325$	−1.47579	−0.0818623
$326$	−14.2888	−0.791385
$327$	0	0
$328$	9.94379	0.549054
$329$	10.7732	0.593943
$330$	0	0
$331$	−11.6348	−0.639509	−0.319755	−	0.947500i	$-0.603600\pi$
$331$	−11.6348	−0.639509	−0.319755	+	0.947500i	$0.603600\pi$
$332$	−33.4155	−1.83391
$333$	0	0
$334$	−15.4464	−0.845187
$335$	−19.7740	−1.08037
$336$	0	0
$337$	−20.4963	−1.11650	−0.558251	−	0.829672i	$-0.688527\pi$
$337$	−20.4963	−1.11650	−0.558251	+	0.829672i	$0.688527\pi$
$338$	−15.7671	−0.857617
$339$	0	0
$340$	6.68406	0.362494
$341$	−53.9735	−2.92283
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−0.515261	−0.0277810
$345$	0	0
$346$	−4.93284	−0.265191
$347$	8.74194	0.469292	0.234646	−	0.972081i	$-0.424607\pi$
$347$	8.74194	0.469292	0.234646	+	0.972081i	$0.424607\pi$
$348$	0	0
$349$	15.7151	0.841210	0.420605	−	0.907244i	$-0.361818\pi$
$349$	15.7151	0.841210	0.420605	+	0.907244i	$0.361818\pi$
$350$	1.33731	0.0714821
$351$	0	0
$352$	49.9499	2.66234
$353$	19.0770	1.01537	0.507684	−	0.861543i	$-0.330502\pi$
$353$	19.0770	1.01537	0.507684	+	0.861543i	$0.330502\pi$
$354$	0	0
$355$	−25.9915	−1.37948
$356$	−14.1020	−0.747406
$357$	0	0
$358$	−12.1696	−0.643182
$359$	−26.2829	−1.38716	−0.693580	−	0.720380i	$-0.743968\pi$
$359$	−26.2829	−1.38716	−0.693580	+	0.720380i	$0.743968\pi$
$360$	0	0
$361$	−3.30466	−0.173929
$362$	21.5936	1.13493
$363$	0	0
$364$	−6.48292	−0.339798
$365$	33.8174	1.77008
$366$	0	0
$367$	3.06881	0.160191	0.0800954	−	0.996787i	$-0.474478\pi$
$367$	3.06881	0.160191	0.0800954	+	0.996787i	$0.474478\pi$
$368$	1.76911	0.0922212
$369$	0	0
$370$	−18.3787	−0.955466
$371$	3.76105	0.195264
$372$	0	0
$373$	−17.9361	−0.928695	−0.464347	−	0.885653i	$-0.653711\pi$
$373$	−17.9361	−0.928695	−0.464347	+	0.885653i	$0.653711\pi$
$374$	−16.8303	−0.870272
$375$	0	0
$376$	−16.5389	−0.852930
$377$	−20.3531	−1.04824
$378$	0	0
$379$	22.7237	1.16724	0.583619	−	0.812028i	$-0.301636\pi$
$379$	22.7237	1.16724	0.583619	+	0.812028i	$0.301636\pi$
$380$	−22.4586	−1.15210
$381$	0	0
$382$	52.4153	2.68180
$383$	28.1027	1.43598	0.717990	−	0.696054i	$-0.245062\pi$
$383$	28.1027	1.43598	0.717990	+	0.696054i	$0.245062\pi$
$384$	0	0
$385$	13.7742	0.701998
$386$	27.2093	1.38491
$387$	0	0
$388$	−8.41145	−0.427027
$389$	7.17477	0.363775	0.181888	−	0.983319i	$-0.441779\pi$
$389$	7.17477	0.363775	0.181888	+	0.983319i	$0.441779\pi$
$390$	0	0
$391$	−1.00082	−0.0506135
$392$	1.53520	0.0775392
$393$	0	0
$394$	0.542476	0.0273295
$395$	−22.6574	−1.14002
$396$	0	0
$397$	18.2622	0.916553	0.458276	−	0.888810i	$-0.348467\pi$
$397$	18.2622	0.916553	0.458276	+	0.888810i	$0.348467\pi$
$398$	−49.0487	−2.45859
$399$	0	0
$400$	1.28463	0.0642314
$401$	35.6986	1.78270	0.891351	−	0.453314i	$-0.149758\pi$
$401$	35.6986	1.78270	0.891351	+	0.453314i	$0.149758\pi$
$402$	0	0
$403$	19.6437	0.978524
$404$	−9.39410	−0.467374
$405$	0	0
$406$	18.4432	0.915320
$407$	26.6164	1.31932
$408$	0	0
$409$	25.5188	1.26183	0.630913	−	0.775854i	$-0.282681\pi$
$409$	25.5188	1.26183	0.630913	+	0.775854i	$0.282681\pi$
$410$	29.4241	1.45315
$411$	0	0
$412$	−14.2720	−0.703132
$413$	4.27318	0.210269
$414$	0	0
$415$	−25.8396	−1.26842
$416$	−18.1793	−0.891315
$417$	0	0
$418$	56.5499	2.76595
$419$	2.28552	0.111655	0.0558274	−	0.998440i	$-0.482220\pi$
$419$	2.28552	0.111655	0.0558274	+	0.998440i	$0.482220\pi$
$420$	0	0
$421$	−12.3449	−0.601655	−0.300828	−	0.953679i	$-0.597263\pi$
$421$	−12.3449	−0.601655	−0.300828	+	0.953679i	$0.597263\pi$
$422$	13.4777	0.656082
$423$	0	0
$424$	−5.77395	−0.280408
$425$	−0.726737	−0.0352519
$426$	0	0
$427$	−15.6127	−0.755553
$428$	40.0343	1.93513
$429$	0	0
$430$	−1.52468	−0.0735266
$431$	−28.1635	−1.35659	−0.678293	−	0.734791i	$-0.737280\pi$
$431$	−28.1635	−1.35659	−0.678293	+	0.734791i	$0.737280\pi$
$432$	0	0
$433$	−6.66806	−0.320447	−0.160223	−	0.987081i	$-0.551221\pi$
$433$	−6.66806	−0.320447	−0.160223	+	0.987081i	$0.551221\pi$
$434$	−17.8004	−0.854447
$435$	0	0
$436$	−32.6680	−1.56451
$437$	3.36276	0.160863
$438$	0	0
$439$	−10.7259	−0.511918	−0.255959	−	0.966688i	$-0.582391\pi$
$439$	−10.7259	−0.511918	−0.255959	+	0.966688i	$0.582391\pi$
$440$	−21.1461	−1.00810
$441$	0	0
$442$	6.12540	0.291355
$443$	9.23838	0.438929	0.219464	−	0.975621i	$-0.429569\pi$
$443$	9.23838	0.438929	0.219464	+	0.975621i	$0.429569\pi$
$444$	0	0
$445$	−10.9049	−0.516940
$446$	−31.7993	−1.50574
$447$	0	0
$448$	12.3050	0.581356
$449$	−10.6923	−0.504602	−0.252301	−	0.967649i	$-0.581187\pi$
$449$	−10.6923	−0.504602	−0.252301	+	0.967649i	$0.581187\pi$
$450$	0	0
$451$	−42.6124	−2.00654
$452$	21.2859	1.00120
$453$	0	0
$454$	59.9186	2.81212
$455$	−5.01314	−0.235020
$456$	0	0
$457$	−3.78695	−0.177146	−0.0885730	−	0.996070i	$-0.528231\pi$
$457$	−3.78695	−0.177146	−0.0885730	+	0.996070i	$0.528231\pi$
$458$	9.26584	0.432964
$459$	0	0
$460$	−4.81179	−0.224351
$461$	−19.7102	−0.917993	−0.458997	−	0.888438i	$-0.651791\pi$
$461$	−19.7102	−0.917993	−0.458997	+	0.888438i	$0.651791\pi$
$462$	0	0
$463$	−16.0653	−0.746617	−0.373309	−	0.927707i	$-0.621777\pi$
$463$	−16.0653	−0.746617	−0.373309	+	0.927707i	$0.621777\pi$
$464$	17.7167	0.822475
$465$	0	0
$466$	39.2999	1.82053
$467$	16.9347	0.783645	0.391823	−	0.920041i	$-0.371845\pi$
$467$	16.9347	0.783645	0.391823	+	0.920041i	$0.371845\pi$
$468$	0	0
$469$	9.44445	0.436104
$470$	−48.9394	−2.25741
$471$	0	0
$472$	−6.56018	−0.301957
$473$	2.20806	0.101527
$474$	0	0
$475$	2.44185	0.112040
$476$	−3.19244	−0.146325
$477$	0	0
$478$	−23.2040	−1.06133
$479$	−10.9235	−0.499109	−0.249554	−	0.968361i	$-0.580284\pi$
$479$	−10.9235	−0.499109	−0.249554	+	0.968361i	$0.580284\pi$
$480$	0	0
$481$	−9.68707	−0.441692
$482$	−10.6462	−0.484919
$483$	0	0
$484$	87.4029	3.97286
$485$	−6.50444	−0.295351
$486$	0	0
$487$	1.41526	0.0641315	0.0320658	−	0.999486i	$-0.489791\pi$
$487$	1.41526	0.0641315	0.0320658	+	0.999486i	$0.489791\pi$
$488$	23.9686	1.08501
$489$	0	0
$490$	4.54272	0.205219
$491$	−12.8773	−0.581143	−0.290571	−	0.956853i	$-0.593845\pi$
$491$	−12.8773	−0.581143	−0.290571	+	0.956853i	$0.593845\pi$
$492$	0	0
$493$	−10.0226	−0.451397
$494$	−20.5814	−0.926003
$495$	0	0
$496$	−17.0992	−0.767777
$497$	12.4140	0.556846
$498$	0	0
$499$	4.39397	0.196701	0.0983505	−	0.995152i	$-0.468643\pi$
$499$	4.39397	0.196701	0.0983505	+	0.995152i	$0.468643\pi$
$500$	−31.8384	−1.42386
$501$	0	0
$502$	38.4807	1.71748
$503$	7.50901	0.334810	0.167405	−	0.985888i	$-0.446461\pi$
$503$	7.50901	0.334810	0.167405	+	0.985888i	$0.446461\pi$
$504$	0	0
$505$	−7.26430	−0.323257
$506$	12.1159	0.538620
$507$	0	0
$508$	2.70756	0.120129
$509$	2.49156	0.110436	0.0552182	−	0.998474i	$-0.482415\pi$
$509$	2.49156	0.110436	0.0552182	+	0.998474i	$0.482415\pi$
$510$	0	0
$511$	−16.1518	−0.714516
$512$	22.2239	0.982166
$513$	0	0
$514$	−16.6404	−0.733975
$515$	−11.0363	−0.486318
$516$	0	0
$517$	70.8747	3.11707
$518$	8.77806	0.385686
$519$	0	0
$520$	7.69616	0.337499
$521$	−20.6140	−0.903114	−0.451557	−	0.892242i	$-0.649131\pi$
$521$	−20.6140	−0.903114	−0.451557	+	0.892242i	$0.649131\pi$
$522$	0	0
$523$	4.13489	0.180806	0.0904031	−	0.995905i	$-0.471184\pi$
$523$	4.13489	0.180806	0.0904031	+	0.995905i	$0.471184\pi$
$524$	2.55752	0.111726
$525$	0	0
$526$	5.42542	0.236560
$527$	9.67332	0.421377
$528$	0	0
$529$	−22.2795	−0.968675
$530$	−17.0854	−0.742141
$531$	0	0
$532$	10.7267	0.465059
$533$	15.5089	0.671763
$534$	0	0
$535$	30.9578	1.33842
$536$	−14.4991	−0.626266
$537$	0	0
$538$	−3.84639	−0.165830
$539$	−6.57883	−0.283370
$540$	0	0
$541$	5.81468	0.249993	0.124996	−	0.992157i	$-0.460108\pi$
$541$	5.81468	0.249993	0.124996	+	0.992157i	$0.460108\pi$
$542$	12.3497	0.530466
$543$	0	0
$544$	−8.95220	−0.383822
$545$	−25.2616	−1.08209
$546$	0	0
$547$	−0.778611	−0.0332910	−0.0166455	−	0.999861i	$-0.505299\pi$
$547$	−0.778611	−0.0332910	−0.0166455	+	0.999861i	$0.505299\pi$
$548$	21.6511	0.924890
$549$	0	0
$550$	8.79792	0.375145
$551$	33.6762	1.43465
$552$	0	0
$553$	10.8216	0.460182
$554$	−29.5618	−1.25596
$555$	0	0
$556$	28.6908	1.21676
$557$	16.9696	0.719027	0.359514	−	0.933140i	$-0.382943\pi$
$557$	16.9696	0.719027	0.359514	+	0.933140i	$0.382943\pi$
$558$	0	0
$559$	−0.803628	−0.0339898
$560$	4.36377	0.184403
$561$	0	0
$562$	−13.3508	−0.563170
$563$	−22.8665	−0.963707	−0.481853	−	0.876252i	$-0.660036\pi$
$563$	−22.8665	−0.963707	−0.481853	+	0.876252i	$0.660036\pi$
$564$	0	0
$565$	16.4600	0.692478
$566$	50.7298	2.13233
$567$	0	0
$568$	−19.0580	−0.799657
$569$	−6.17146	−0.258721	−0.129360	−	0.991598i	$-0.541292\pi$
$569$	−6.17146	−0.258721	−0.129360	+	0.991598i	$0.541292\pi$
$570$	0	0
$571$	32.0018	1.33923	0.669616	−	0.742707i	$-0.266459\pi$
$571$	32.0018	1.33923	0.669616	+	0.742707i	$0.266459\pi$
$572$	−42.6501	−1.78329
$573$	0	0
$574$	−14.0535	−0.586583
$575$	0.523171	0.0218178
$576$	0	0
$577$	−36.3765	−1.51437	−0.757186	−	0.653199i	$-0.773426\pi$
$577$	−36.3765	−1.51437	−0.757186	+	0.653199i	$0.773426\pi$
$578$	−33.8684	−1.40874
$579$	0	0
$580$	−48.1875	−2.00088
$581$	12.3415	0.512013
$582$	0	0
$583$	24.7433	1.02476
$584$	24.7963	1.02608
$585$	0	0
$586$	18.1060	0.747952
$587$	−13.9096	−0.574112	−0.287056	−	0.957914i	$-0.592677\pi$
$587$	−13.9096	−0.574112	−0.287056	+	0.957914i	$0.592677\pi$
$588$	0	0
$589$	−32.5025	−1.33924
$590$	−19.4119	−0.799173
$591$	0	0
$592$	8.43226	0.346564
$593$	−4.17747	−0.171548	−0.0857740	−	0.996315i	$-0.527336\pi$
$593$	−4.17747	−0.171548	−0.0857740	+	0.996315i	$0.527336\pi$
$594$	0	0
$595$	−2.46866	−0.101205
$596$	−24.9672	−1.02270
$597$	0	0
$598$	−4.40962	−0.180323
$599$	36.6664	1.49815	0.749074	−	0.662486i	$-0.230499\pi$
$599$	36.6664	1.49815	0.749074	+	0.662486i	$0.230499\pi$
$600$	0	0
$601$	−26.8146	−1.09379	−0.546896	−	0.837200i	$-0.684191\pi$
$601$	−26.8146	−1.09379	−0.546896	+	0.837200i	$0.684191\pi$
$602$	0.728217	0.0296799
$603$	0	0
$604$	−37.0985	−1.50952
$605$	67.5872	2.74781
$606$	0	0
$607$	2.70166	0.109657	0.0548285	−	0.998496i	$-0.482539\pi$
$607$	2.70166	0.109657	0.0548285	+	0.998496i	$0.482539\pi$
$608$	30.0795	1.21989
$609$	0	0
$610$	70.9242	2.87164
$611$	−25.7950	−1.04355
$612$	0	0
$613$	−13.8194	−0.558159	−0.279079	−	0.960268i	$-0.590029\pi$
$613$	−13.8194	−0.558159	−0.279079	+	0.960268i	$0.590029\pi$
$614$	5.16935	0.208618
$615$	0	0
$616$	10.0998	0.406933
$617$	−12.8565	−0.517585	−0.258793	−	0.965933i	$-0.583325\pi$
$617$	−12.8565	−0.517585	−0.258793	+	0.965933i	$0.583325\pi$
$618$	0	0
$619$	5.64489	0.226887	0.113444	−	0.993544i	$-0.463812\pi$
$619$	5.64489	0.226887	0.113444	+	0.993544i	$0.463812\pi$
$620$	46.5080	1.86781
$621$	0	0
$622$	66.5394	2.66799
$623$	5.20838	0.208669
$624$	0	0
$625$	−21.5383	−0.861532
$626$	−44.6276	−1.78368
$627$	0	0
$628$	−1.79868	−0.0717753
$629$	−4.77028	−0.190204
$630$	0	0
$631$	−0.184789	−0.00735635	−0.00367818	−	0.999993i	$-0.501171\pi$
$631$	−0.184789	−0.00735635	−0.00367818	+	0.999993i	$0.501171\pi$
$632$	−16.6133	−0.660843
$633$	0	0
$634$	−42.2973	−1.67984
$635$	2.09371	0.0830865
$636$	0	0
$637$	2.39437	0.0948686
$638$	121.335	4.80368
$639$	0	0
$640$	−24.1049	−0.952831
$641$	−40.3321	−1.59302	−0.796510	−	0.604625i	$-0.793323\pi$
$641$	−40.3321	−1.59302	−0.796510	+	0.604625i	$0.793323\pi$
$642$	0	0
$643$	10.3860	0.409586	0.204793	−	0.978805i	$-0.434348\pi$
$643$	10.3860	0.409586	0.204793	+	0.978805i	$0.434348\pi$
$644$	2.29821	0.0905621
$645$	0	0
$646$	−10.1351	−0.398760
$647$	−47.4642	−1.86601	−0.933005	−	0.359865i	$-0.882823\pi$
$647$	−47.4642	−1.86601	−0.933005	+	0.359865i	$0.882823\pi$
$648$	0	0
$649$	28.1125	1.10351
$650$	−3.20202	−0.125593
$651$	0	0
$652$	−17.8311	−0.698319
$653$	−8.72606	−0.341477	−0.170739	−	0.985316i	$-0.554615\pi$
$653$	−8.72606	−0.341477	−0.170739	+	0.985316i	$0.554615\pi$
$654$	0	0
$655$	1.97769	0.0772745
$656$	−13.4999	−0.527084
$657$	0	0
$658$	23.3744	0.911230
$659$	39.4366	1.53623	0.768116	−	0.640311i	$-0.221195\pi$
$659$	39.4366	1.53623	0.768116	+	0.640311i	$0.221195\pi$
$660$	0	0
$661$	42.6447	1.65869	0.829343	−	0.558740i	$-0.188715\pi$
$661$	42.6447	1.65869	0.829343	+	0.558740i	$0.188715\pi$
$662$	−25.2440	−0.981137
$663$	0	0
$664$	−18.9467	−0.735274
$665$	8.29474	0.321656
$666$	0	0
$667$	7.21520	0.279374
$668$	−19.2756	−0.745793
$669$	0	0
$670$	−42.9035	−1.65751
$671$	−102.714	−3.96521
$672$	0	0
$673$	−12.6588	−0.487959	−0.243980	−	0.969780i	$-0.578453\pi$
$673$	−12.6588	−0.487959	−0.243980	+	0.969780i	$0.578453\pi$
$674$	−44.4706	−1.71294
$675$	0	0
$676$	−19.6758	−0.756761
$677$	42.3219	1.62656	0.813281	−	0.581872i	$-0.197679\pi$
$677$	42.3219	1.62656	0.813281	+	0.581872i	$0.197679\pi$
$678$	0	0
$679$	3.10665	0.119222
$680$	3.78988	0.145335
$681$	0	0
$682$	−117.106	−4.48421
$683$	48.6648	1.86211	0.931054	−	0.364881i	$-0.118890\pi$
$683$	48.6648	1.86211	0.931054	+	0.364881i	$0.118890\pi$
$684$	0	0
$685$	16.7425	0.639696
$686$	−2.16969	−0.0828392
$687$	0	0
$688$	0.699531	0.0266694
$689$	−9.00535	−0.343077
$690$	0	0
$691$	−8.28985	−0.315361	−0.157680	−	0.987490i	$-0.550402\pi$
$691$	−8.28985	−0.315361	−0.157680	+	0.987490i	$0.550402\pi$
$692$	−6.15570	−0.234005
$693$	0	0
$694$	18.9673	0.719990
$695$	22.1861	0.841568
$696$	0	0
$697$	7.63716	0.289278
$698$	34.0969	1.29059
$699$	0	0
$700$	1.66883	0.0630759
$701$	2.85419	0.107801	0.0539006	−	0.998546i	$-0.482835\pi$
$701$	2.85419	0.107801	0.0539006	+	0.998546i	$0.482835\pi$
$702$	0	0
$703$	16.0282	0.604516
$704$	80.9524	3.05101
$705$	0	0
$706$	41.3913	1.55778
$707$	3.46958	0.130487
$708$	0	0
$709$	−4.03335	−0.151476	−0.0757378	−	0.997128i	$-0.524131\pi$
$709$	−4.03335	−0.151476	−0.0757378	+	0.997128i	$0.524131\pi$
$710$	−56.3935	−2.11641
$711$	0	0
$712$	−7.99590	−0.299659
$713$	−6.96374	−0.260794
$714$	0	0
$715$	−32.9806	−1.23340
$716$	−15.1865	−0.567545
$717$	0	0
$718$	−57.0259	−2.12819
$719$	−33.4924	−1.24906	−0.624528	−	0.781002i	$-0.714709\pi$
$719$	−33.4924	−1.24906	−0.624528	+	0.781002i	$0.714709\pi$
$720$	0	0
$721$	5.27117	0.196309
$722$	−7.17009	−0.266843
$723$	0	0
$724$	26.9467	1.00147
$725$	5.23927	0.194582
$726$	0	0
$727$	−45.1915	−1.67606	−0.838030	−	0.545625i	$-0.816292\pi$
$727$	−45.1915	−1.67606	−0.838030	+	0.545625i	$0.816292\pi$
$728$	−3.67584	−0.136236
$729$	0	0
$730$	73.3733	2.71567
$731$	−0.395737	−0.0146369
$732$	0	0
$733$	29.1000	1.07483	0.537417	−	0.843316i	$-0.319400\pi$
$733$	29.1000	1.07483	0.537417	+	0.843316i	$0.319400\pi$
$734$	6.65838	0.245765
$735$	0	0
$736$	6.44461	0.237551
$737$	62.1334	2.28871
$738$	0	0
$739$	41.3614	1.52150	0.760752	−	0.649042i	$-0.224830\pi$
$739$	41.3614	1.52150	0.760752	+	0.649042i	$0.224830\pi$
$740$	−22.9349	−0.843103
$741$	0	0
$742$	8.16031	0.299574
$743$	−48.7497	−1.78845	−0.894226	−	0.447615i	$-0.852274\pi$
$743$	−48.7497	−1.78845	−0.894226	+	0.447615i	$0.852274\pi$
$744$	0	0
$745$	−19.3067	−0.707344
$746$	−38.9158	−1.42481
$747$	0	0
$748$	−21.0025	−0.767928
$749$	−14.7861	−0.540271
$750$	0	0
$751$	47.5238	1.73417	0.867084	−	0.498163i	$-0.165992\pi$
$751$	47.5238	1.73417	0.867084	+	0.498163i	$0.165992\pi$
$752$	22.4536	0.818800
$753$	0	0
$754$	−44.1599	−1.60821
$755$	−28.6877	−1.04405
$756$	0	0
$757$	−41.1484	−1.49557	−0.747783	−	0.663943i	$-0.768882\pi$
$757$	−41.1484	−1.49557	−0.747783	+	0.663943i	$0.768882\pi$
$758$	49.3034	1.79078
$759$	0	0
$760$	−12.7341	−0.461913
$761$	−36.1118	−1.30905	−0.654526	−	0.756040i	$-0.727132\pi$
$761$	−36.1118	−1.30905	−0.654526	+	0.756040i	$0.727132\pi$
$762$	0	0
$763$	12.0655	0.436799
$764$	65.4092	2.36642
$765$	0	0
$766$	60.9741	2.20309
$767$	−10.2316	−0.369442
$768$	0	0
$769$	−26.6364	−0.960532	−0.480266	−	0.877123i	$-0.659460\pi$
$769$	−26.6364	−0.960532	−0.480266	+	0.877123i	$0.659460\pi$
$770$	29.8858	1.07701
$771$	0	0
$772$	33.9545	1.22205
$773$	20.0947	0.722756	0.361378	−	0.932419i	$-0.382306\pi$
$773$	20.0947	0.722756	0.361378	+	0.932419i	$0.382306\pi$
$774$	0	0
$775$	−5.05668	−0.181641
$776$	−4.76932	−0.171209
$777$	0	0
$778$	15.5671	0.558106
$779$	−25.6610	−0.919400
$780$	0	0
$781$	81.6699	2.92238
$782$	−2.17146	−0.0776514
$783$	0	0
$784$	−2.08422	−0.0744365
$785$	−1.39089	−0.0496431
$786$	0	0
$787$	−0.378082	−0.0134772	−0.00673858	−	0.999977i	$-0.502145\pi$
$787$	−0.378082	−0.0134772	−0.00673858	+	0.999977i	$0.502145\pi$
$788$	0.676957	0.0241156
$789$	0	0
$790$	−49.1596	−1.74902
$791$	−7.86163	−0.279527
$792$	0	0
$793$	37.3827	1.32750
$794$	39.6233	1.40618
$795$	0	0
$796$	−61.2080	−2.16946
$797$	21.5740	0.764190	0.382095	−	0.924123i	$-0.375203\pi$
$797$	21.5740	0.764190	0.382095	+	0.924123i	$0.375203\pi$
$798$	0	0
$799$	−12.7024	−0.449380
$800$	4.67971	0.165453
$801$	0	0
$802$	77.4549	2.73503
$803$	−106.260	−3.74984
$804$	0	0
$805$	1.77717	0.0626369
$806$	42.6208	1.50125
$807$	0	0
$808$	−5.32649	−0.187385
$809$	40.1769	1.41254	0.706271	−	0.707941i	$-0.250376\pi$
$809$	40.1769	1.41254	0.706271	+	0.707941i	$0.250376\pi$
$810$	0	0
$811$	−42.4590	−1.49094	−0.745469	−	0.666540i	$-0.767774\pi$
$811$	−42.4590	−1.49094	−0.745469	+	0.666540i	$0.767774\pi$
$812$	23.0153	0.807679
$813$	0	0
$814$	57.7493	2.02411
$815$	−13.7885	−0.482989
$816$	0	0
$817$	1.32968	0.0465197
$818$	55.3680	1.93590
$819$	0	0
$820$	36.7184	1.28226
$821$	−26.5731	−0.927407	−0.463703	−	0.885991i	$-0.653480\pi$
$821$	−26.5731	−0.927407	−0.463703	+	0.885991i	$0.653480\pi$
$822$	0	0
$823$	−18.3867	−0.640920	−0.320460	−	0.947262i	$-0.603838\pi$
$823$	−18.3867	−0.640920	−0.320460	+	0.947262i	$0.603838\pi$
$824$	−8.09229	−0.281908
$825$	0	0
$826$	9.27149	0.322596
$827$	16.4634	0.572487	0.286244	−	0.958157i	$-0.407593\pi$
$827$	16.4634	0.572487	0.286244	+	0.958157i	$0.407593\pi$
$828$	0	0
$829$	11.2305	0.390052	0.195026	−	0.980798i	$-0.437521\pi$
$829$	11.2305	0.390052	0.195026	+	0.980798i	$0.437521\pi$
$830$	−56.0640	−1.94601
$831$	0	0
$832$	−29.4627	−1.02144
$833$	1.17908	0.0408528
$834$	0	0
$835$	−14.9055	−0.515825
$836$	70.5688	2.44067
$837$	0	0
$838$	4.95887	0.171301
$839$	−44.9241	−1.55095	−0.775476	−	0.631377i	$-0.782490\pi$
$839$	−44.9241	−1.55095	−0.775476	+	0.631377i	$0.782490\pi$
$840$	0	0
$841$	43.2562	1.49159
$842$	−26.7847	−0.923062
$843$	0	0
$844$	16.8188	0.578927
$845$	−15.2150	−0.523411
$846$	0	0
$847$	−32.2810	−1.10919
$848$	7.83885	0.269187
$849$	0	0
$850$	−1.57680	−0.0540837
$851$	3.43408	0.117719
$852$	0	0
$853$	−11.9190	−0.408097	−0.204049	−	0.978961i	$-0.565410\pi$
$853$	−11.9190	−0.408097	−0.204049	+	0.978961i	$0.565410\pi$
$854$	−33.8748	−1.15917
$855$	0	0
$856$	22.6996	0.775855
$857$	−22.0855	−0.754427	−0.377214	−	0.926126i	$-0.623118\pi$
$857$	−22.0855	−0.754427	−0.377214	+	0.926126i	$0.623118\pi$
$858$	0	0
$859$	11.6217	0.396527	0.198264	−	0.980149i	$-0.436470\pi$
$859$	11.6217	0.396527	0.198264	+	0.980149i	$0.436470\pi$
$860$	−1.90265	−0.0648799
$861$	0	0
$862$	−61.1061	−2.08128
$863$	−43.7004	−1.48758	−0.743789	−	0.668414i	$-0.766973\pi$
$863$	−43.7004	−1.48758	−0.743789	+	0.668414i	$0.766973\pi$
$864$	0	0
$865$	−4.76010	−0.161848
$866$	−14.4676	−0.491631
$867$	0	0
$868$	−22.2132	−0.753964
$869$	71.1936	2.41508
$870$	0	0
$871$	−22.6135	−0.766231
$872$	−18.5229	−0.627264
$873$	0	0
$874$	7.29616	0.246796
$875$	11.7591	0.397529
$876$	0	0
$877$	−28.2832	−0.955057	−0.477528	−	0.878616i	$-0.658467\pi$
$877$	−28.2832	−0.955057	−0.477528	+	0.878616i	$0.658467\pi$
$878$	−23.2718	−0.785386
$879$	0	0
$880$	28.7085	0.967762
$881$	31.3271	1.05544	0.527719	−	0.849419i	$-0.323048\pi$
$881$	31.3271	1.05544	0.527719	+	0.849419i	$0.323048\pi$
$882$	0	0
$883$	−55.5102	−1.86807	−0.934033	−	0.357186i	$-0.883736\pi$
$883$	−55.5102	−1.86807	−0.934033	+	0.357186i	$0.883736\pi$
$884$	7.64390	0.257092
$885$	0	0
$886$	20.0444	0.673406
$887$	−26.3164	−0.883618	−0.441809	−	0.897109i	$-0.645663\pi$
$887$	−26.3164	−0.883618	−0.441809	+	0.897109i	$0.645663\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−23.6602	−0.793092
$891$	0	0
$892$	−39.6825	−1.32867
$893$	42.6804	1.42824
$894$	0	0
$895$	−11.7434	−0.392540
$896$	11.5130	0.384622
$897$	0	0
$898$	−23.1990	−0.774162
$899$	−69.7380	−2.32589
$900$	0	0
$901$	−4.43458	−0.147737
$902$	−92.4558	−3.07844
$903$	0	0
$904$	12.0692	0.401414
$905$	20.8374	0.692659
$906$	0	0
$907$	55.8611	1.85484	0.927418	−	0.374027i	$-0.122023\pi$
$907$	55.8611	1.85484	0.927418	+	0.374027i	$0.122023\pi$
$908$	74.7725	2.48141
$909$	0	0
$910$	−10.8770	−0.360568
$911$	−1.29829	−0.0430142	−0.0215071	−	0.999769i	$-0.506846\pi$
$911$	−1.29829	−0.0430142	−0.0215071	+	0.999769i	$0.506846\pi$
$912$	0	0
$913$	81.1928	2.68709
$914$	−8.21652	−0.271778
$915$	0	0
$916$	11.5629	0.382048
$917$	−0.944582	−0.0311928
$918$	0	0
$919$	57.6721	1.90243	0.951213	−	0.308534i	$-0.0998385\pi$
$919$	57.6721	1.90243	0.951213	+	0.308534i	$0.0998385\pi$
$920$	−2.72830	−0.0899495
$921$	0	0
$922$	−42.7650	−1.40839
$923$	−29.7239	−0.978373
$924$	0	0
$925$	2.49364	0.0819904
$926$	−34.8567	−1.14546
$927$	0	0
$928$	64.5392	2.11860
$929$	−8.78823	−0.288333	−0.144166	−	0.989553i	$-0.546050\pi$
$929$	−8.78823	−0.288333	−0.144166	+	0.989553i	$0.546050\pi$
$930$	0	0
$931$	−3.96173	−0.129841
$932$	49.0425	1.60644
$933$	0	0
$934$	36.7431	1.20227
$935$	−16.2409	−0.531134
$936$	0	0
$937$	44.3141	1.44768	0.723840	−	0.689968i	$-0.242375\pi$
$937$	44.3141	1.44768	0.723840	+	0.689968i	$0.242375\pi$
$938$	20.4915	0.669073
$939$	0	0
$940$	−61.0716	−1.99194
$941$	38.1958	1.24515	0.622574	−	0.782561i	$-0.286087\pi$
$941$	38.1958	1.24515	0.622574	+	0.782561i	$0.286087\pi$
$942$	0	0
$943$	−5.49792	−0.179037
$944$	8.90626	0.289874
$945$	0	0
$946$	4.79082	0.155763
$947$	−23.9647	−0.778748	−0.389374	−	0.921080i	$-0.627309\pi$
$947$	−23.9647	−0.778748	−0.389374	+	0.921080i	$0.627309\pi$
$948$	0	0
$949$	38.6736	1.25540
$950$	5.29806	0.171892
$951$	0	0
$952$	−1.81012	−0.0586665
$953$	1.74783	0.0566176	0.0283088	−	0.999599i	$-0.490988\pi$
$953$	1.74783	0.0566176	0.0283088	+	0.999599i	$0.490988\pi$
$954$	0	0
$955$	50.5798	1.63672
$956$	−28.9563	−0.936514
$957$	0	0
$958$	−23.7007	−0.765735
$959$	−7.99653	−0.258221
$960$	0	0
$961$	36.3075	1.17121
$962$	−21.0180	−0.677646
$963$	0	0
$964$	−13.2854	−0.427893
$965$	26.2564	0.845225
$966$	0	0
$967$	5.95338	0.191448	0.0957239	−	0.995408i	$-0.469483\pi$
$967$	5.95338	0.191448	0.0957239	+	0.995408i	$0.469483\pi$
$968$	49.5577	1.59285
$969$	0	0
$970$	−14.1126	−0.453129
$971$	3.67743	0.118014	0.0590071	−	0.998258i	$-0.481207\pi$
$971$	3.67743	0.118014	0.0590071	+	0.998258i	$0.481207\pi$
$972$	0	0
$973$	−10.5965	−0.339710
$974$	3.07068	0.0983909
$975$	0	0
$976$	−32.5404	−1.04159
$977$	−42.1390	−1.34815	−0.674074	−	0.738664i	$-0.735457\pi$
$977$	−42.1390	−1.34815	−0.674074	+	0.738664i	$0.735457\pi$
$978$	0	0
$979$	34.2650	1.09512
$980$	5.66887	0.181085
$981$	0	0
$982$	−27.9397	−0.891592
$983$	33.0426	1.05390	0.526948	−	0.849898i	$-0.323336\pi$
$983$	33.0426	1.05390	0.526948	+	0.849898i	$0.323336\pi$
$984$	0	0
$985$	0.523479	0.0166794
$986$	−21.7460	−0.692535
$987$	0	0
$988$	−25.6836	−0.817105
$989$	0.284888	0.00905890
$990$	0	0
$991$	35.5541	1.12941	0.564707	−	0.825291i	$-0.308989\pi$
$991$	35.5541	1.12941	0.564707	+	0.825291i	$0.308989\pi$
$992$	−62.2899	−1.97771
$993$	0	0
$994$	26.9347	0.854315
$995$	−47.3312	−1.50050
$996$	0	0
$997$	−46.3710	−1.46858	−0.734292	−	0.678834i	$-0.762486\pi$
$997$	−46.3710	−1.46858	−0.734292	+	0.678834i	$0.762486\pi$
$998$	9.53356	0.301779
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.q.1.13		15
3.2	odd	2		889.2.a.b.1.3	✓	15
21.20	even	2		6223.2.a.j.1.3		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.b.1.3	✓	15	3.2	odd	2
6223.2.a.j.1.3		15	21.20	even	2
8001.2.a.q.1.13		15	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)

\(f(q)\)	\(=\)	\(q+2.16969 q^{2} +2.70756 q^{4} +2.09371 q^{5} -1.00000 q^{7} +1.53520 q^{8} +O(q^{10})\)	\(q+2.16969 q^{2} +2.70756 q^{4} +2.09371 q^{5} -1.00000 q^{7} +1.53520 q^{8} +4.54272 q^{10} -6.57883 q^{11} +2.39437 q^{13} -2.16969 q^{14} -2.08422 q^{16} +1.17908 q^{17} -3.96173 q^{19} +5.66887 q^{20} -14.2740 q^{22} -0.848810 q^{23} -0.616358 q^{25} +5.19506 q^{26} -2.70756 q^{28} -8.50037 q^{29} +8.20411 q^{31} -7.59252 q^{32} +2.55825 q^{34} -2.09371 q^{35} -4.04576 q^{37} -8.59575 q^{38} +3.21427 q^{40} +6.47720 q^{41} -0.335632 q^{43} -17.8126 q^{44} -1.84166 q^{46} -10.7732 q^{47} +1.00000 q^{49} -1.33731 q^{50} +6.48292 q^{52} -3.76105 q^{53} -13.7742 q^{55} -1.53520 q^{56} -18.4432 q^{58} -4.27318 q^{59} +15.6127 q^{61} +17.8004 q^{62} -12.3050 q^{64} +5.01314 q^{65} -9.44445 q^{67} +3.19244 q^{68} -4.54272 q^{70} -12.4140 q^{71} +16.1518 q^{73} -8.77806 q^{74} -10.7267 q^{76} +6.57883 q^{77} -10.8216 q^{79} -4.36377 q^{80} +14.0535 q^{82} -12.3415 q^{83} +2.46866 q^{85} -0.728217 q^{86} -10.0998 q^{88} -5.20838 q^{89} -2.39437 q^{91} -2.29821 q^{92} -23.3744 q^{94} -8.29474 q^{95} -3.10665 q^{97} +2.16969 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7}+O(q^{10}) \) 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7	\( 15 q + 14 q^{4} - 7 q^{5} - 15 q^{7} + 10 q^{10} - 14 q^{11} + 6 q^{13} + 20 q^{16} - 10 q^{17} + 13 q^{19} - 8 q^{20} - 11 q^{22} - 15 q^{23} - 22 q^{26} - 14 q^{28} - 16 q^{29} + 22 q^{31} - 15 q^{34} + 7 q^{35} - 14 q^{37} + 6 q^{38} + 22 q^{40} - 19 q^{41} - q^{43} - 25 q^{44} - 28 q^{46} - 49 q^{47} + 15 q^{49} - 24 q^{50} - 17 q^{52} + 28 q^{53} + 39 q^{55} - 10 q^{58} - 43 q^{59} + 27 q^{61} - 14 q^{62} + 18 q^{64} + 8 q^{65} + 3 q^{67} - 13 q^{68} - 10 q^{70} - 55 q^{71} - 3 q^{73} + 12 q^{74} - 20 q^{76} + 14 q^{77} + 18 q^{79} - 29 q^{80} + 14 q^{82} - 17 q^{83} + 7 q^{85} - 4 q^{86} - 114 q^{88} - 36 q^{89} - 6 q^{91} - 45 q^{92} - 15 q^{94} - 59 q^{95} - 2 q^{97}+O(q^{100}) \) 15 * q + 14 * q^4 - 7 * q^5 - 15 * q^7 + 10 * q^10 - 14 * q^11 + 6 * q^13 + 20 * q^16 - 10 * q^17 + 13 * q^19 - 8 * q^20 - 11 * q^22 - 15 * q^23 - 22 * q^26 - 14 * q^28 - 16 * q^29 + 22 * q^31 - 15 * q^34 + 7 * q^35 - 14 * q^37 + 6 * q^38 + 22 * q^40 - 19 * q^41 - q^43 - 25 * q^44 - 28 * q^46 - 49 * q^47 + 15 * q^49 - 24 * q^50 - 17 * q^52 + 28 * q^53 + 39 * q^55 - 10 * q^58 - 43 * q^59 + 27 * q^61 - 14 * q^62 + 18 * q^64 + 8 * q^65 + 3 * q^67 - 13 * q^68 - 10 * q^70 - 55 * q^71 - 3 * q^73 + 12 * q^74 - 20 * q^76 + 14 * q^77 + 18 * q^79 - 29 * q^80 + 14 * q^82 - 17 * q^83 + 7 * q^85 - 4 * q^86 - 114 * q^88 - 36 * q^89 - 6 * q^91 - 45 * q^92 - 15 * q^94 - 59 * q^95 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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