LMFDB - Embedded newform 8001.2.a.m.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:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 2 x^{10} - 15 x^{9} + 25 x^{8} + 88 x^{7} - 112 x^{6} - 247 x^{5} + 215 x^{4} + 313 x^{3} + \cdots + 57 $ x^11 - 2x^10 - 15x^9 + 25x^8 + 88x^7 - 112x^6 - 247x^5 + 215x^4 + 313x^3 - 170x^2 - 121x + 57
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$2.22659$ 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)

$f(q)$	$=$	$q+2.22659 q^{2} +2.95768 q^{4} +0.700608 q^{5} -1.00000 q^{7} +2.13237 q^{8} +O(q^{10})$	\(q+2.22659 q^{2} +2.95768 q^{4} +0.700608 q^{5} -1.00000 q^{7} +2.13237 q^{8} +1.55996 q^{10} -2.84067 q^{11} -1.64764 q^{13} -2.22659 q^{14} -1.16747 q^{16} -1.42000 q^{17} +3.55007 q^{19} +2.07218 q^{20} -6.32501 q^{22} +9.09099 q^{23} -4.50915 q^{25} -3.66860 q^{26} -2.95768 q^{28} -5.14303 q^{29} -5.38528 q^{31} -6.86421 q^{32} -3.16175 q^{34} -0.700608 q^{35} -8.77130 q^{37} +7.90453 q^{38} +1.49395 q^{40} -7.79318 q^{41} -2.96091 q^{43} -8.40182 q^{44} +20.2419 q^{46} +10.2547 q^{47} +1.00000 q^{49} -10.0400 q^{50} -4.87319 q^{52} -1.24092 q^{53} -1.99020 q^{55} -2.13237 q^{56} -11.4514 q^{58} -12.0755 q^{59} -3.35472 q^{61} -11.9908 q^{62} -12.9488 q^{64} -1.15435 q^{65} +7.32764 q^{67} -4.19991 q^{68} -1.55996 q^{70} +0.417897 q^{71} -7.52196 q^{73} -19.5300 q^{74} +10.5000 q^{76} +2.84067 q^{77} +9.40271 q^{79} -0.817939 q^{80} -17.3522 q^{82} +0.994782 q^{83} -0.994863 q^{85} -6.59273 q^{86} -6.05736 q^{88} +11.2314 q^{89} +1.64764 q^{91} +26.8883 q^{92} +22.8329 q^{94} +2.48720 q^{95} -12.2388 q^{97} +2.22659 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 2 q^{2} + 12 q^{4} - q^{5} - 11 q^{7} + 15 q^{8}+O(q^{10}) $ 11 * q + 2 * q^2 + 12 * q^4 - q^5 - 11 * q^7 + 15 * q^8	$ 11 q + 2 q^{2} + 12 q^{4} - q^{5} - 11 q^{7} + 15 q^{8} - 12 q^{10} + 7 q^{11} - 24 q^{13} - 2 q^{14} - 6 q^{16} + 15 q^{17} - 19 q^{19} - 3 q^{20} - 3 q^{22} + 11 q^{23} + 10 q^{25} - 10 q^{26} - 12 q^{28} + 10 q^{29} - 20 q^{31} + 27 q^{32} - 9 q^{34} + q^{35} - 22 q^{37} - 8 q^{38} - 29 q^{40} - 9 q^{41} - 17 q^{43} + 9 q^{44} - 18 q^{46} + 7 q^{47} + 11 q^{49} + 47 q^{50} - 66 q^{52} + 28 q^{53} - 24 q^{55} - 15 q^{56} - 39 q^{58} - 35 q^{59} - 6 q^{61} - 18 q^{62} + 11 q^{64} + 43 q^{65} - 22 q^{67} + 12 q^{68} + 12 q^{70} + 22 q^{71} - 29 q^{73} - 14 q^{74} + 10 q^{76} - 7 q^{77} - 20 q^{79} - 66 q^{80} - 24 q^{82} - 17 q^{83} - 50 q^{85} + 12 q^{86} + 2 q^{88} + q^{89} + 24 q^{91} + 22 q^{92} + q^{94} - 10 q^{95} - 45 q^{97} + 2 q^{98}+O(q^{100}) $ 11 * q + 2 * q^2 + 12 * q^4 - q^5 - 11 * q^7 + 15 * q^8 - 12 * q^10 + 7 * q^11 - 24 * q^13 - 2 * q^14 - 6 * q^16 + 15 * q^17 - 19 * q^19 - 3 * q^20 - 3 * q^22 + 11 * q^23 + 10 * q^25 - 10 * q^26 - 12 * q^28 + 10 * q^29 - 20 * q^31 + 27 * q^32 - 9 * q^34 + q^35 - 22 * q^37 - 8 * q^38 - 29 * q^40 - 9 * q^41 - 17 * q^43 + 9 * q^44 - 18 * q^46 + 7 * q^47 + 11 * q^49 + 47 * q^50 - 66 * q^52 + 28 * q^53 - 24 * q^55 - 15 * q^56 - 39 * q^58 - 35 * q^59 - 6 * q^61 - 18 * q^62 + 11 * q^64 + 43 * q^65 - 22 * q^67 + 12 * q^68 + 12 * q^70 + 22 * q^71 - 29 * q^73 - 14 * q^74 + 10 * q^76 - 7 * q^77 - 20 * q^79 - 66 * q^80 - 24 * q^82 - 17 * q^83 - 50 * q^85 + 12 * q^86 + 2 * q^88 + q^89 + 24 * q^91 + 22 * q^92 + q^94 - 10 * q^95 - 45 * q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.22659	1.57443	0.787217	−	0.616676i	$-0.211521\pi$
$2$	2.22659	1.57443	0.787217	+	0.616676i	$0.211521\pi$
$3$	0	0
$3$	0	0
$4$	2.95768	1.47884
$5$	0.700608	0.313321	0.156661	−	0.987652i	$-0.449927\pi$
$5$	0.700608	0.313321	0.156661	+	0.987652i	$0.449927\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.13237	0.753906
$9$	0	0
$10$	1.55996	0.493304
$11$	−2.84067	−0.856496	−0.428248	−	0.903661i	$-0.640869\pi$
$11$	−2.84067	−0.856496	−0.428248	+	0.903661i	$0.640869\pi$
$12$	0	0
$13$	−1.64764	−0.456972	−0.228486	−	0.973547i	$-0.573377\pi$
$13$	−1.64764	−0.456972	−0.228486	+	0.973547i	$0.573377\pi$
$14$	−2.22659	−0.595080
$15$	0	0
$16$	−1.16747	−0.291868
$17$	−1.42000	−0.344401	−0.172200	−	0.985062i	$-0.555088\pi$
$17$	−1.42000	−0.344401	−0.172200	+	0.985062i	$0.555088\pi$
$18$	0	0
$19$	3.55007	0.814442	0.407221	−	0.913330i	$-0.366498\pi$
$19$	3.55007	0.814442	0.407221	+	0.913330i	$0.366498\pi$
$20$	2.07218	0.463353
$21$	0	0
$22$	−6.32501	−1.34850
$23$	9.09099	1.89560	0.947802	−	0.318860i	$-0.103300\pi$
$23$	9.09099	1.89560	0.947802	+	0.318860i	$0.103300\pi$
$24$	0	0
$25$	−4.50915	−0.901830
$26$	−3.66860	−0.719472
$27$	0	0
$28$	−2.95768	−0.558950
$29$	−5.14303	−0.955037	−0.477519	−	0.878622i	$-0.658464\pi$
$29$	−5.14303	−0.955037	−0.477519	+	0.878622i	$0.658464\pi$
$30$	0	0
$31$	−5.38528	−0.967224	−0.483612	−	0.875282i	$-0.660675\pi$
$31$	−5.38528	−0.967224	−0.483612	+	0.875282i	$0.660675\pi$
$32$	−6.86421	−1.21343
$33$	0	0
$34$	−3.16175	−0.542236
$35$	−0.700608	−0.118424
$36$	0	0
$37$	−8.77130	−1.44199	−0.720996	−	0.692939i	$-0.756315\pi$
$37$	−8.77130	−1.44199	−0.720996	+	0.692939i	$0.756315\pi$
$38$	7.90453	1.28228
$39$	0	0
$40$	1.49395	0.236215
$41$	−7.79318	−1.21709	−0.608545	−	0.793519i	$-0.708247\pi$
$41$	−7.79318	−1.21709	−0.608545	+	0.793519i	$0.708247\pi$
$42$	0	0
$43$	−2.96091	−0.451535	−0.225767	−	0.974181i	$-0.572489\pi$
$43$	−2.96091	−0.451535	−0.225767	+	0.974181i	$0.572489\pi$
$44$	−8.40182	−1.26662
$45$	0	0
$46$	20.2419	2.98450
$47$	10.2547	1.49580	0.747899	−	0.663812i	$-0.231063\pi$
$47$	10.2547	1.49580	0.747899	+	0.663812i	$0.231063\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−10.0400	−1.41987
$51$	0	0
$52$	−4.87319	−0.675789
$53$	−1.24092	−0.170453	−0.0852266	−	0.996362i	$-0.527161\pi$
$53$	−1.24092	−0.170453	−0.0852266	+	0.996362i	$0.527161\pi$
$54$	0	0
$55$	−1.99020	−0.268358
$56$	−2.13237	−0.284950
$57$	0	0
$58$	−11.4514	−1.50364
$59$	−12.0755	−1.57210	−0.786049	−	0.618165i	$-0.787877\pi$
$59$	−12.0755	−1.57210	−0.786049	+	0.618165i	$0.787877\pi$
$60$	0	0
$61$	−3.35472	−0.429528	−0.214764	−	0.976666i	$-0.568898\pi$
$61$	−3.35472	−0.429528	−0.214764	+	0.976666i	$0.568898\pi$
$62$	−11.9908	−1.52283
$63$	0	0
$64$	−12.9488	−1.61860
$65$	−1.15435	−0.143179
$66$	0	0
$67$	7.32764	0.895213	0.447607	−	0.894231i	$-0.352276\pi$
$67$	7.32764	0.895213	0.447607	+	0.894231i	$0.352276\pi$
$68$	−4.19991	−0.509314
$69$	0	0
$70$	−1.55996	−0.186451
$71$	0.417897	0.0495953	0.0247976	−	0.999692i	$-0.492106\pi$
$71$	0.417897	0.0495953	0.0247976	+	0.999692i	$0.492106\pi$
$72$	0	0
$73$	−7.52196	−0.880379	−0.440189	−	0.897905i	$-0.645089\pi$
$73$	−7.52196	−0.880379	−0.440189	+	0.897905i	$0.645089\pi$
$74$	−19.5300	−2.27032
$75$	0	0
$76$	10.5000	1.20443
$77$	2.84067	0.323725
$78$	0	0
$79$	9.40271	1.05789	0.528944	−	0.848657i	$-0.322588\pi$
$79$	9.40271	1.05789	0.528944	+	0.848657i	$0.322588\pi$
$80$	−0.817939	−0.0914484
$81$	0	0
$82$	−17.3522	−1.91623
$83$	0.994782	0.109192	0.0545958	−	0.998509i	$-0.482613\pi$
$83$	0.994782	0.109192	0.0545958	+	0.998509i	$0.482613\pi$
$84$	0	0
$85$	−0.994863	−0.107908
$86$	−6.59273	−0.710912
$87$	0	0
$88$	−6.05736	−0.645717
$89$	11.2314	1.19052	0.595262	−	0.803532i	$-0.297048\pi$
$89$	11.2314	1.19052	0.595262	+	0.803532i	$0.297048\pi$
$90$	0	0
$91$	1.64764	0.172719
$92$	26.8883	2.80330
$93$	0	0
$94$	22.8329	2.35504
$95$	2.48720	0.255182
$96$	0	0
$97$	−12.2388	−1.24266	−0.621331	−	0.783548i	$-0.713408\pi$
$97$	−12.2388	−1.24266	−0.621331	+	0.783548i	$0.713408\pi$
$98$	2.22659	0.224919
$99$	0	0
$100$	−13.3366	−1.33366
$101$	14.1553	1.40850	0.704250	−	0.709952i	$-0.251283\pi$
$101$	14.1553	1.40850	0.704250	+	0.709952i	$0.251283\pi$
$102$	0	0
$103$	−8.52311	−0.839807	−0.419903	−	0.907569i	$-0.637936\pi$
$103$	−8.52311	−0.839807	−0.419903	+	0.907569i	$0.637936\pi$
$104$	−3.51336	−0.344514
$105$	0	0
$106$	−2.76301	−0.268367
$107$	6.22395	0.601692	0.300846	−	0.953673i	$-0.402731\pi$
$107$	6.22395	0.601692	0.300846	+	0.953673i	$0.402731\pi$
$108$	0	0
$109$	−12.7062	−1.21703	−0.608516	−	0.793541i	$-0.708235\pi$
$109$	−12.7062	−1.21703	−0.608516	+	0.793541i	$0.708235\pi$
$110$	−4.43135	−0.422512
$111$	0	0
$112$	1.16747	0.110316
$113$	−15.3591	−1.44486	−0.722429	−	0.691445i	$-0.756975\pi$
$113$	−15.3591	−1.44486	−0.722429	+	0.691445i	$0.756975\pi$
$114$	0	0
$115$	6.36922	0.593933
$116$	−15.2115	−1.41235
$117$	0	0
$118$	−26.8872	−2.47516
$119$	1.42000	0.130171
$120$	0	0
$121$	−2.93057	−0.266415
$122$	−7.46958	−0.676264
$123$	0	0
$124$	−15.9279	−1.43037
$125$	−6.66218	−0.595884
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−15.1032	−1.33495
$129$	0	0
$130$	−2.57025	−0.225426
$131$	−17.0346	−1.48832	−0.744159	−	0.668003i	$-0.767150\pi$
$131$	−17.0346	−1.48832	−0.744159	+	0.668003i	$0.767150\pi$
$132$	0	0
$133$	−3.55007	−0.307830
$134$	16.3156	1.40945
$135$	0	0
$136$	−3.02796	−0.259645
$137$	−11.4427	−0.977613	−0.488807	−	0.872392i	$-0.662568\pi$
$137$	−11.4427	−0.977613	−0.488807	+	0.872392i	$0.662568\pi$
$138$	0	0
$139$	−13.3083	−1.12879	−0.564396	−	0.825504i	$-0.690891\pi$
$139$	−13.3083	−1.12879	−0.564396	+	0.825504i	$0.690891\pi$
$140$	−2.07218	−0.175131
$141$	0	0
$142$	0.930484	0.0780845
$143$	4.68040	0.391394
$144$	0	0
$145$	−3.60325	−0.299233
$146$	−16.7483	−1.38610
$147$	0	0
$148$	−25.9427	−2.13248
$149$	12.2019	0.999615	0.499808	−	0.866136i	$-0.333404\pi$
$149$	12.2019	0.999615	0.499808	+	0.866136i	$0.333404\pi$
$150$	0	0
$151$	15.5718	1.26721	0.633606	−	0.773656i	$-0.281574\pi$
$151$	15.5718	1.26721	0.633606	+	0.773656i	$0.281574\pi$
$152$	7.57005	0.614012
$153$	0	0
$154$	6.32501	0.509684
$155$	−3.77297	−0.303052
$156$	0	0
$157$	9.94117	0.793392	0.396696	−	0.917950i	$-0.370157\pi$
$157$	9.94117	0.793392	0.396696	+	0.917950i	$0.370157\pi$
$158$	20.9359	1.66557
$159$	0	0
$160$	−4.80912	−0.380194
$161$	−9.09099	−0.716471
$162$	0	0
$163$	−3.04587	−0.238571	−0.119286	−	0.992860i	$-0.538060\pi$
$163$	−3.04587	−0.238571	−0.119286	+	0.992860i	$0.538060\pi$
$164$	−23.0498	−1.79989
$165$	0	0
$166$	2.21497	0.171915
$167$	11.2197	0.868209	0.434105	−	0.900862i	$-0.357065\pi$
$167$	11.2197	0.868209	0.434105	+	0.900862i	$0.357065\pi$
$168$	0	0
$169$	−10.2853	−0.791177
$170$	−2.21515	−0.169894
$171$	0	0
$172$	−8.75745	−0.667749
$173$	0.308017	0.0234181	0.0117090	−	0.999931i	$-0.496273\pi$
$173$	0.308017	0.0234181	0.0117090	+	0.999931i	$0.496273\pi$
$174$	0	0
$175$	4.50915	0.340860
$176$	3.31641	0.249984
$177$	0	0
$178$	25.0076	1.87440
$179$	5.03397	0.376257	0.188128	−	0.982144i	$-0.439758\pi$
$179$	5.03397	0.376257	0.188128	+	0.982144i	$0.439758\pi$
$180$	0	0
$181$	14.2745	1.06102	0.530508	−	0.847680i	$-0.322001\pi$
$181$	14.2745	1.06102	0.530508	+	0.847680i	$0.322001\pi$
$182$	3.66860	0.271935
$183$	0	0
$184$	19.3853	1.42911
$185$	−6.14524	−0.451807
$186$	0	0
$187$	4.03376	0.294978
$188$	30.3301	2.21205
$189$	0	0
$190$	5.53798	0.401767
$191$	10.0711	0.728720	0.364360	−	0.931258i	$-0.381288\pi$
$191$	10.0711	0.728720	0.364360	+	0.931258i	$0.381288\pi$
$192$	0	0
$193$	10.0929	0.726501	0.363251	−	0.931691i	$-0.381667\pi$
$193$	10.0929	0.726501	0.363251	+	0.931691i	$0.381667\pi$
$194$	−27.2507	−1.95649
$195$	0	0
$196$	2.95768	0.211263
$197$	−12.5215	−0.892120	−0.446060	−	0.895003i	$-0.647173\pi$
$197$	−12.5215	−0.892120	−0.446060	+	0.895003i	$0.647173\pi$
$198$	0	0
$199$	−17.6906	−1.25406	−0.627028	−	0.778997i	$-0.715729\pi$
$199$	−17.6906	−1.25406	−0.627028	+	0.778997i	$0.715729\pi$
$200$	−9.61516	−0.679894
$201$	0	0
$202$	31.5179	2.21759
$203$	5.14303	0.360970
$204$	0	0
$205$	−5.45996	−0.381340
$206$	−18.9774	−1.32222
$207$	0	0
$208$	1.92357	0.133375
$209$	−10.0846	−0.697566
$210$	0	0
$211$	−21.6795	−1.49248	−0.746238	−	0.665680i	$-0.768142\pi$
$211$	−21.6795	−1.49248	−0.746238	+	0.665680i	$0.768142\pi$
$212$	−3.67024	−0.252073
$213$	0	0
$214$	13.8582	0.947324
$215$	−2.07444	−0.141475
$216$	0	0
$217$	5.38528	0.365576
$218$	−28.2914	−1.91614
$219$	0	0
$220$	−5.88638	−0.396860
$221$	2.33964	0.157381
$222$	0	0
$223$	−28.6440	−1.91815	−0.959073	−	0.283157i	$-0.908618\pi$
$223$	−28.6440	−1.91815	−0.959073	+	0.283157i	$0.908618\pi$
$224$	6.86421	0.458634
$225$	0	0
$226$	−34.1983	−2.27483
$227$	−1.86131	−0.123539	−0.0617697	−	0.998090i	$-0.519674\pi$
$227$	−1.86131	−0.123539	−0.0617697	+	0.998090i	$0.519674\pi$
$228$	0	0
$229$	17.2643	1.14086	0.570429	−	0.821347i	$-0.306777\pi$
$229$	17.2643	1.14086	0.570429	+	0.821347i	$0.306777\pi$
$230$	14.1816	0.935108
$231$	0	0
$232$	−10.9668	−0.720008
$233$	−7.93576	−0.519889	−0.259945	−	0.965624i	$-0.583704\pi$
$233$	−7.93576	−0.519889	−0.259945	+	0.965624i	$0.583704\pi$
$234$	0	0
$235$	7.18450	0.468665
$236$	−35.7155	−2.32488
$237$	0	0
$238$	3.16175	0.204946
$239$	26.2092	1.69533	0.847665	−	0.530532i	$-0.178008\pi$
$239$	26.2092	1.69533	0.847665	+	0.530532i	$0.178008\pi$
$240$	0	0
$241$	−22.5982	−1.45568	−0.727838	−	0.685749i	$-0.759475\pi$
$241$	−22.5982	−1.45568	−0.727838	+	0.685749i	$0.759475\pi$
$242$	−6.52516	−0.419453
$243$	0	0
$244$	−9.92222	−0.635205
$245$	0.700608	0.0447602
$246$	0	0
$247$	−5.84922	−0.372177
$248$	−11.4834	−0.729196
$249$	0	0
$250$	−14.8339	−0.938179
$251$	−19.2653	−1.21601	−0.608006	−	0.793933i	$-0.708030\pi$
$251$	−19.2653	−1.21601	−0.608006	+	0.793933i	$0.708030\pi$
$252$	0	0
$253$	−25.8246	−1.62358
$254$	2.22659	0.139708
$255$	0	0
$256$	−7.73099	−0.483187
$257$	9.47986	0.591338	0.295669	−	0.955291i	$-0.404458\pi$
$257$	9.47986	0.591338	0.295669	+	0.955291i	$0.404458\pi$
$258$	0	0
$259$	8.77130	0.545022
$260$	−3.41419	−0.211739
$261$	0	0
$262$	−37.9290	−2.34326
$263$	25.0231	1.54299	0.771494	−	0.636236i	$-0.219510\pi$
$263$	25.0231	1.54299	0.771494	+	0.636236i	$0.219510\pi$
$264$	0	0
$265$	−0.869396	−0.0534066
$266$	−7.90453	−0.484658
$267$	0	0
$268$	21.6728	1.32388
$269$	0.342395	0.0208762	0.0104381	−	0.999946i	$-0.496677\pi$
$269$	0.342395	0.0208762	0.0104381	+	0.999946i	$0.496677\pi$
$270$	0	0
$271$	−0.905345	−0.0549958	−0.0274979	−	0.999622i	$-0.508754\pi$
$271$	−0.905345	−0.0549958	−0.0274979	+	0.999622i	$0.508754\pi$
$272$	1.65781	0.100519
$273$	0	0
$274$	−25.4781	−1.53919
$275$	12.8090	0.772413
$276$	0	0
$277$	−2.97806	−0.178934	−0.0894672	−	0.995990i	$-0.528516\pi$
$277$	−2.97806	−0.178934	−0.0894672	+	0.995990i	$0.528516\pi$
$278$	−29.6320	−1.77721
$279$	0	0
$280$	−1.49395	−0.0892807
$281$	22.3174	1.33135	0.665674	−	0.746243i	$-0.268144\pi$
$281$	22.3174	1.33135	0.665674	+	0.746243i	$0.268144\pi$
$282$	0	0
$283$	−14.2390	−0.846420	−0.423210	−	0.906032i	$-0.639097\pi$
$283$	−14.2390	−0.846420	−0.423210	+	0.906032i	$0.639097\pi$
$284$	1.23601	0.0733436
$285$	0	0
$286$	10.4213	0.616225
$287$	7.79318	0.460017
$288$	0	0
$289$	−14.9836	−0.881388
$290$	−8.02294	−0.471123
$291$	0	0
$292$	−22.2476	−1.30194
$293$	7.81886	0.456783	0.228391	−	0.973569i	$-0.426653\pi$
$293$	7.81886	0.456783	0.228391	+	0.973569i	$0.426653\pi$
$294$	0	0
$295$	−8.46019	−0.492571
$296$	−18.7036	−1.08713
$297$	0	0
$298$	27.1685	1.57383
$299$	−14.9786	−0.866237
$300$	0	0
$301$	2.96091	0.170664
$302$	34.6719	1.99514
$303$	0	0
$304$	−4.14460	−0.237709
$305$	−2.35035	−0.134580
$306$	0	0
$307$	19.3432	1.10398	0.551988	−	0.833852i	$-0.313869\pi$
$307$	19.3432	1.10398	0.551988	+	0.833852i	$0.313869\pi$
$308$	8.40182	0.478738
$309$	0	0
$310$	−8.40083	−0.477135
$311$	−3.25829	−0.184761	−0.0923804	−	0.995724i	$-0.529448\pi$
$311$	−3.25829	−0.184761	−0.0923804	+	0.995724i	$0.529448\pi$
$312$	0	0
$313$	−20.3151	−1.14828	−0.574140	−	0.818757i	$-0.694663\pi$
$313$	−20.3151	−1.14828	−0.574140	+	0.818757i	$0.694663\pi$
$314$	22.1349	1.24914
$315$	0	0
$316$	27.8103	1.56445
$317$	22.0628	1.23917	0.619585	−	0.784929i	$-0.287301\pi$
$317$	22.0628	1.23917	0.619585	+	0.784929i	$0.287301\pi$
$318$	0	0
$319$	14.6097	0.817985
$320$	−9.07203	−0.507142
$321$	0	0
$322$	−20.2419	−1.12804
$323$	−5.04110	−0.280494
$324$	0	0
$325$	7.42943	0.412111
$326$	−6.78190	−0.375615
$327$	0	0
$328$	−16.6179	−0.917572
$329$	−10.2547	−0.565359
$330$	0	0
$331$	19.0855	1.04904	0.524518	−	0.851399i	$-0.324246\pi$
$331$	19.0855	1.04904	0.524518	+	0.851399i	$0.324246\pi$
$332$	2.94225	0.161477
$333$	0	0
$334$	24.9817	1.36694
$335$	5.13380	0.280489
$336$	0	0
$337$	10.4496	0.569225	0.284612	−	0.958643i	$-0.408135\pi$
$337$	10.4496	0.569225	0.284612	+	0.958643i	$0.408135\pi$
$338$	−22.9011	−1.24566
$339$	0	0
$340$	−2.94249	−0.159579
$341$	15.2978	0.828423
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−6.31375	−0.340415
$345$	0	0
$346$	0.685825	0.0368702
$347$	12.9364	0.694460	0.347230	−	0.937780i	$-0.387122\pi$
$347$	12.9364	0.694460	0.347230	+	0.937780i	$0.387122\pi$
$348$	0	0
$349$	17.7137	0.948195	0.474097	−	0.880472i	$-0.342774\pi$
$349$	17.7137	0.948195	0.474097	+	0.880472i	$0.342774\pi$
$350$	10.0400	0.536661
$351$	0	0
$352$	19.4990	1.03930
$353$	14.3725	0.764970	0.382485	−	0.923962i	$-0.375068\pi$
$353$	14.3725	0.764970	0.382485	+	0.923962i	$0.375068\pi$
$354$	0	0
$355$	0.292782	0.0155392
$356$	33.2189	1.76060
$357$	0	0
$358$	11.2086	0.592392
$359$	29.5307	1.55857	0.779286	−	0.626668i	$-0.215582\pi$
$359$	29.5307	1.55857	0.779286	+	0.626668i	$0.215582\pi$
$360$	0	0
$361$	−6.39701	−0.336685
$362$	31.7834	1.67050
$363$	0	0
$364$	4.87319	0.255424
$365$	−5.26994	−0.275841
$366$	0	0
$367$	−27.7621	−1.44917	−0.724584	−	0.689187i	$-0.757968\pi$
$367$	−27.7621	−1.44917	−0.724584	+	0.689187i	$0.757968\pi$
$368$	−10.6135	−0.553266
$369$	0	0
$370$	−13.6829	−0.711340
$371$	1.24092	0.0644252
$372$	0	0
$373$	33.4355	1.73123	0.865614	−	0.500713i	$-0.166929\pi$
$373$	33.4355	1.73123	0.865614	+	0.500713i	$0.166929\pi$
$374$	8.98151	0.464423
$375$	0	0
$376$	21.8667	1.12769
$377$	8.47384	0.436425
$378$	0	0
$379$	11.5924	0.595460	0.297730	−	0.954650i	$-0.403771\pi$
$379$	11.5924	0.595460	0.297730	+	0.954650i	$0.403771\pi$
$380$	7.35637	0.377374
$381$	0	0
$382$	22.4242	1.14732
$383$	−29.5265	−1.50873	−0.754367	−	0.656453i	$-0.772056\pi$
$383$	−29.5265	−1.50873	−0.754367	+	0.656453i	$0.772056\pi$
$384$	0	0
$385$	1.99020	0.101430
$386$	22.4727	1.14383
$387$	0	0
$388$	−36.1985	−1.83770
$389$	9.06944	0.459839	0.229919	−	0.973210i	$-0.426154\pi$
$389$	9.06944	0.459839	0.229919	+	0.973210i	$0.426154\pi$
$390$	0	0
$391$	−12.9092	−0.652847
$392$	2.13237	0.107701
$393$	0	0
$394$	−27.8802	−1.40458
$395$	6.58761	0.331459
$396$	0	0
$397$	−1.50250	−0.0754083	−0.0377041	−	0.999289i	$-0.512004\pi$
$397$	−1.50250	−0.0754083	−0.0377041	+	0.999289i	$0.512004\pi$
$398$	−39.3897	−1.97443
$399$	0	0
$400$	5.26430	0.263215
$401$	4.23177	0.211324	0.105662	−	0.994402i	$-0.466304\pi$
$401$	4.23177	0.211324	0.105662	+	0.994402i	$0.466304\pi$
$402$	0	0
$403$	8.87297	0.441994
$404$	41.8668	2.08295
$405$	0	0
$406$	11.4514	0.568324
$407$	24.9164	1.23506
$408$	0	0
$409$	18.2286	0.901348	0.450674	−	0.892689i	$-0.351184\pi$
$409$	18.2286	0.901348	0.450674	+	0.892689i	$0.351184\pi$
$410$	−12.1571	−0.600395
$411$	0	0
$412$	−25.2087	−1.24194
$413$	12.0755	0.594197
$414$	0	0
$415$	0.696952	0.0342120
$416$	11.3097	0.554504
$417$	0	0
$418$	−22.4542	−1.09827
$419$	26.6637	1.30261	0.651304	−	0.758817i	$-0.274222\pi$
$419$	26.6637	1.30261	0.651304	+	0.758817i	$0.274222\pi$
$420$	0	0
$421$	−25.3227	−1.23415	−0.617077	−	0.786902i	$-0.711684\pi$
$421$	−25.3227	−1.23415	−0.617077	+	0.786902i	$0.711684\pi$
$422$	−48.2712	−2.34980
$423$	0	0
$424$	−2.64609	−0.128506
$425$	6.40299	0.310591
$426$	0	0
$427$	3.35472	0.162347
$428$	18.4085	0.889807
$429$	0	0
$430$	−4.61891	−0.222744
$431$	−29.0684	−1.40018	−0.700088	−	0.714056i	$-0.746856\pi$
$431$	−29.0684	−1.40018	−0.700088	+	0.714056i	$0.746856\pi$
$432$	0	0
$433$	−3.57066	−0.171595	−0.0857975	−	0.996313i	$-0.527344\pi$
$433$	−3.57066	−0.171595	−0.0857975	+	0.996313i	$0.527344\pi$
$434$	11.9908	0.575576
$435$	0	0
$436$	−37.5809	−1.79980
$437$	32.2737	1.54386
$438$	0	0
$439$	9.77463	0.466518	0.233259	−	0.972415i	$-0.425061\pi$
$439$	9.77463	0.466518	0.233259	+	0.972415i	$0.425061\pi$
$440$	−4.24383	−0.202317
$441$	0	0
$442$	5.20941	0.247787
$443$	30.6561	1.45652	0.728258	−	0.685303i	$-0.240330\pi$
$443$	30.6561	1.45652	0.728258	+	0.685303i	$0.240330\pi$
$444$	0	0
$445$	7.86879	0.373017
$446$	−63.7784	−3.02000
$447$	0	0
$448$	12.9488	0.611774
$449$	19.0103	0.897149	0.448575	−	0.893745i	$-0.351932\pi$
$449$	19.0103	0.897149	0.448575	+	0.893745i	$0.351932\pi$
$450$	0	0
$451$	22.1379	1.04243
$452$	−45.4273	−2.13672
$453$	0	0
$454$	−4.14436	−0.194505
$455$	1.15435	0.0541166
$456$	0	0
$457$	−41.2978	−1.93183	−0.965915	−	0.258859i	$-0.916654\pi$
$457$	−41.2978	−1.93183	−0.965915	+	0.258859i	$0.916654\pi$
$458$	38.4404	1.79620
$459$	0	0
$460$	18.8381	0.878333
$461$	5.04250	0.234853	0.117426	−	0.993082i	$-0.462536\pi$
$461$	5.04250	0.234853	0.117426	+	0.993082i	$0.462536\pi$
$462$	0	0
$463$	−24.9181	−1.15804	−0.579021	−	0.815312i	$-0.696565\pi$
$463$	−24.9181	−1.15804	−0.579021	+	0.815312i	$0.696565\pi$
$464$	6.00434	0.278745
$465$	0	0
$466$	−17.6697	−0.818531
$467$	−11.9933	−0.554983	−0.277492	−	0.960728i	$-0.589503\pi$
$467$	−11.9933	−0.554983	−0.277492	+	0.960728i	$0.589503\pi$
$468$	0	0
$469$	−7.32764	−0.338359
$470$	15.9969	0.737883
$471$	0	0
$472$	−25.7494	−1.18521
$473$	8.41099	0.386738
$474$	0	0
$475$	−16.0078	−0.734488
$476$	4.19991	0.192503
$477$	0	0
$478$	58.3570	2.66918
$479$	13.5604	0.619592	0.309796	−	0.950803i	$-0.399739\pi$
$479$	13.5604	0.619592	0.309796	+	0.950803i	$0.399739\pi$
$480$	0	0
$481$	14.4519	0.658950
$482$	−50.3168	−2.29187
$483$	0	0
$484$	−8.66769	−0.393986
$485$	−8.57459	−0.389352
$486$	0	0
$487$	1.88027	0.0852030	0.0426015	−	0.999092i	$-0.486435\pi$
$487$	1.88027	0.0852030	0.0426015	+	0.999092i	$0.486435\pi$
$488$	−7.15350	−0.323824
$489$	0	0
$490$	1.55996	0.0704719
$491$	16.1027	0.726705	0.363353	−	0.931652i	$-0.381632\pi$
$491$	16.1027	0.726705	0.363353	+	0.931652i	$0.381632\pi$
$492$	0	0
$493$	7.30311	0.328915
$494$	−13.0238	−0.585968
$495$	0	0
$496$	6.28715	0.282302
$497$	−0.417897	−0.0187452
$498$	0	0
$499$	35.1949	1.57554	0.787770	−	0.615969i	$-0.211235\pi$
$499$	35.1949	1.57554	0.787770	+	0.615969i	$0.211235\pi$
$500$	−19.7046	−0.881218
$501$	0	0
$502$	−42.8957	−1.91453
$503$	−21.5688	−0.961705	−0.480853	−	0.876801i	$-0.659673\pi$
$503$	−21.5688	−0.961705	−0.480853	+	0.876801i	$0.659673\pi$
$504$	0	0
$505$	9.91728	0.441313
$506$	−57.5006	−2.55621
$507$	0	0
$508$	2.95768	0.131226
$509$	−3.91252	−0.173419	−0.0867096	−	0.996234i	$-0.527635\pi$
$509$	−3.91252	−0.173419	−0.0867096	+	0.996234i	$0.527635\pi$
$510$	0	0
$511$	7.52196	0.332752
$512$	12.9927	0.574203
$513$	0	0
$514$	21.1077	0.931022
$515$	−5.97135	−0.263129
$516$	0	0
$517$	−29.1302	−1.28114
$518$	19.5300	0.858101
$519$	0	0
$520$	−2.46149	−0.107943
$521$	−41.8612	−1.83397	−0.916986	−	0.398918i	$-0.869386\pi$
$521$	−41.8612	−1.83397	−0.916986	+	0.398918i	$0.869386\pi$
$522$	0	0
$523$	12.6055	0.551202	0.275601	−	0.961272i	$-0.411123\pi$
$523$	12.6055	0.551202	0.275601	+	0.961272i	$0.411123\pi$
$524$	−50.3829	−2.20099
$525$	0	0
$526$	55.7160	2.42933
$527$	7.64709	0.333113
$528$	0	0
$529$	59.6462	2.59331
$530$	−1.93578	−0.0840851
$531$	0	0
$532$	−10.5000	−0.455232
$533$	12.8403	0.556176
$534$	0	0
$535$	4.36054	0.188523
$536$	15.6252	0.674906
$537$	0	0
$538$	0.762372	0.0328682
$539$	−2.84067	−0.122357
$540$	0	0
$541$	28.3946	1.22078	0.610389	−	0.792102i	$-0.291013\pi$
$541$	28.3946	1.22078	0.610389	+	0.792102i	$0.291013\pi$
$542$	−2.01583	−0.0865873
$543$	0	0
$544$	9.74717	0.417907
$545$	−8.90206	−0.381322
$546$	0	0
$547$	−5.78416	−0.247313	−0.123656	−	0.992325i	$-0.539462\pi$
$547$	−5.78416	−0.247313	−0.123656	+	0.992325i	$0.539462\pi$
$548$	−33.8438	−1.44574
$549$	0	0
$550$	28.5204	1.21611
$551$	−18.2581	−0.777822
$552$	0	0
$553$	−9.40271	−0.399844
$554$	−6.63091	−0.281721
$555$	0	0
$556$	−39.3617	−1.66931
$557$	−35.1847	−1.49082	−0.745412	−	0.666604i	$-0.767747\pi$
$557$	−35.1847	−1.49082	−0.745412	+	0.666604i	$0.767747\pi$
$558$	0	0
$559$	4.87850	0.206339
$560$	0.817939	0.0345642
$561$	0	0
$562$	49.6917	2.09612
$563$	−44.6800	−1.88304	−0.941519	−	0.336959i	$-0.890601\pi$
$563$	−44.6800	−1.88304	−0.941519	+	0.336959i	$0.890601\pi$
$564$	0	0
$565$	−10.7607	−0.452705
$566$	−31.7043	−1.33263
$567$	0	0
$568$	0.891110	0.0373901
$569$	25.2289	1.05765	0.528826	−	0.848730i	$-0.322632\pi$
$569$	25.2289	1.05765	0.528826	+	0.848730i	$0.322632\pi$
$570$	0	0
$571$	−7.75014	−0.324333	−0.162167	−	0.986763i	$-0.551848\pi$
$571$	−7.75014	−0.324333	−0.162167	+	0.986763i	$0.551848\pi$
$572$	13.8431	0.578811
$573$	0	0
$574$	17.3522	0.724267
$575$	−40.9926	−1.70951
$576$	0	0
$577$	−5.72703	−0.238419	−0.119210	−	0.992869i	$-0.538036\pi$
$577$	−5.72703	−0.238419	−0.119210	+	0.992869i	$0.538036\pi$
$578$	−33.3623	−1.38769
$579$	0	0
$580$	−10.6573	−0.442519
$581$	−0.994782	−0.0412705
$582$	0	0
$583$	3.52504	0.145992
$584$	−16.0396	−0.663723
$585$	0	0
$586$	17.4094	0.719174
$587$	−11.6079	−0.479111	−0.239555	−	0.970883i	$-0.577002\pi$
$587$	−11.6079	−0.479111	−0.239555	+	0.970883i	$0.577002\pi$
$588$	0	0
$589$	−19.1181	−0.787748
$590$	−18.8373	−0.775521
$591$	0	0
$592$	10.2402	0.420871
$593$	5.90952	0.242675	0.121337	−	0.992611i	$-0.461282\pi$
$593$	5.90952	0.242675	0.121337	+	0.992611i	$0.461282\pi$
$594$	0	0
$595$	0.994863	0.0407854
$596$	36.0892	1.47827
$597$	0	0
$598$	−33.3512	−1.36383
$599$	−36.4127	−1.48778	−0.743892	−	0.668300i	$-0.767022\pi$
$599$	−36.4127	−1.48778	−0.743892	+	0.668300i	$0.767022\pi$
$600$	0	0
$601$	−44.1472	−1.80080	−0.900400	−	0.435063i	$-0.856726\pi$
$601$	−44.1472	−1.80080	−0.900400	+	0.435063i	$0.856726\pi$
$602$	6.59273	0.268699
$603$	0	0
$604$	46.0564	1.87401
$605$	−2.05318	−0.0834735
$606$	0	0
$607$	21.0796	0.855594	0.427797	−	0.903875i	$-0.359290\pi$
$607$	21.0796	0.855594	0.427797	+	0.903875i	$0.359290\pi$
$608$	−24.3684	−0.988270
$609$	0	0
$610$	−5.23325	−0.211888
$611$	−16.8960	−0.683538
$612$	0	0
$613$	−32.9086	−1.32917	−0.664583	−	0.747214i	$-0.731391\pi$
$613$	−32.9086	−1.32917	−0.664583	+	0.747214i	$0.731391\pi$
$614$	43.0694	1.73814
$615$	0	0
$616$	6.05736	0.244058
$617$	20.1159	0.809836	0.404918	−	0.914353i	$-0.367300\pi$
$617$	20.1159	0.809836	0.404918	+	0.914353i	$0.367300\pi$
$618$	0	0
$619$	−7.98022	−0.320752	−0.160376	−	0.987056i	$-0.551271\pi$
$619$	−7.98022	−0.320752	−0.160376	+	0.987056i	$0.551271\pi$
$620$	−11.1592	−0.448166
$621$	0	0
$622$	−7.25487	−0.290894
$623$	−11.2314	−0.449976
$624$	0	0
$625$	17.8782	0.715127
$626$	−45.2334	−1.80789
$627$	0	0
$628$	29.4029	1.17330
$629$	12.4552	0.496623
$630$	0	0
$631$	−26.4102	−1.05137	−0.525687	−	0.850678i	$-0.676192\pi$
$631$	−26.4102	−1.05137	−0.525687	+	0.850678i	$0.676192\pi$
$632$	20.0500	0.797547
$633$	0	0
$634$	49.1247	1.95099
$635$	0.700608	0.0278028
$636$	0	0
$637$	−1.64764	−0.0652817
$638$	32.5297	1.28786
$639$	0	0
$640$	−10.5814	−0.418268
$641$	24.9710	0.986295	0.493147	−	0.869946i	$-0.335846\pi$
$641$	24.9710	0.986295	0.493147	+	0.869946i	$0.335846\pi$
$642$	0	0
$643$	15.3502	0.605353	0.302677	−	0.953093i	$-0.402120\pi$
$643$	15.3502	0.605353	0.302677	+	0.953093i	$0.402120\pi$
$644$	−26.8883	−1.05955
$645$	0	0
$646$	−11.2244	−0.441619
$647$	−4.43876	−0.174506	−0.0872529	−	0.996186i	$-0.527809\pi$
$647$	−4.43876	−0.174506	−0.0872529	+	0.996186i	$0.527809\pi$
$648$	0	0
$649$	34.3026	1.34649
$650$	16.5423	0.648841
$651$	0	0
$652$	−9.00873	−0.352809
$653$	−12.5403	−0.490739	−0.245370	−	0.969430i	$-0.578909\pi$
$653$	−12.5403	−0.490739	−0.245370	+	0.969430i	$0.578909\pi$
$654$	0	0
$655$	−11.9346	−0.466322
$656$	9.09832	0.355230
$657$	0	0
$658$	−22.8329	−0.890120
$659$	−34.5712	−1.34670	−0.673351	−	0.739323i	$-0.735145\pi$
$659$	−34.5712	−1.34670	−0.673351	+	0.739323i	$0.735145\pi$
$660$	0	0
$661$	40.7366	1.58447	0.792236	−	0.610215i	$-0.208917\pi$
$661$	40.7366	1.58447	0.792236	+	0.610215i	$0.208917\pi$
$662$	42.4956	1.65164
$663$	0	0
$664$	2.12124	0.0823201
$665$	−2.48720	−0.0964497
$666$	0	0
$667$	−46.7553	−1.81037
$668$	33.1844	1.28394
$669$	0	0
$670$	11.4308	0.441612
$671$	9.52968	0.367889
$672$	0	0
$673$	45.2386	1.74382	0.871909	−	0.489667i	$-0.162882\pi$
$673$	45.2386	1.74382	0.871909	+	0.489667i	$0.162882\pi$
$674$	23.2669	0.896207
$675$	0	0
$676$	−30.4207	−1.17003
$677$	−34.4342	−1.32341	−0.661707	−	0.749763i	$-0.730168\pi$
$677$	−34.4342	−1.32341	−0.661707	+	0.749763i	$0.730168\pi$
$678$	0	0
$679$	12.2388	0.469682
$680$	−2.12141	−0.0813524
$681$	0	0
$682$	34.0619	1.30430
$683$	6.51358	0.249235	0.124618	−	0.992205i	$-0.460230\pi$
$683$	6.51358	0.249235	0.124618	+	0.992205i	$0.460230\pi$
$684$	0	0
$685$	−8.01682	−0.306307
$686$	−2.22659	−0.0850114
$687$	0	0
$688$	3.45678	0.131789
$689$	2.04458	0.0778923
$690$	0	0
$691$	39.0856	1.48688	0.743442	−	0.668800i	$-0.233192\pi$
$691$	39.0856	1.48688	0.743442	+	0.668800i	$0.233192\pi$
$692$	0.911016	0.0346316
$693$	0	0
$694$	28.8039	1.09338
$695$	−9.32387	−0.353675
$696$	0	0
$697$	11.0663	0.419167
$698$	39.4412	1.49287
$699$	0	0
$700$	13.3366	0.504078
$701$	−21.4535	−0.810286	−0.405143	−	0.914253i	$-0.632778\pi$
$701$	−21.4535	−0.810286	−0.405143	+	0.914253i	$0.632778\pi$
$702$	0	0
$703$	−31.1387	−1.17442
$704$	36.7834	1.38632
$705$	0	0
$706$	32.0016	1.20440
$707$	−14.1553	−0.532363
$708$	0	0
$709$	19.5341	0.733619	0.366810	−	0.930296i	$-0.380450\pi$
$709$	19.5341	0.733619	0.366810	+	0.930296i	$0.380450\pi$
$710$	0.651904	0.0244655
$711$	0	0
$712$	23.9494	0.897543
$713$	−48.9575	−1.83347
$714$	0	0
$715$	3.27912	0.122632
$716$	14.8889	0.556425
$717$	0	0
$718$	65.7527	2.45387
$719$	−36.6120	−1.36540	−0.682699	−	0.730699i	$-0.739194\pi$
$719$	−36.6120	−1.36540	−0.682699	+	0.730699i	$0.739194\pi$
$720$	0	0
$721$	8.52311	0.317417
$722$	−14.2435	−0.530088
$723$	0	0
$724$	42.2195	1.56907
$725$	23.1907	0.861281
$726$	0	0
$727$	19.5176	0.723867	0.361934	−	0.932204i	$-0.382117\pi$
$727$	19.5176	0.723867	0.361934	+	0.932204i	$0.382117\pi$
$728$	3.51336	0.130214
$729$	0	0
$730$	−11.7340	−0.434294
$731$	4.20449	0.155509
$732$	0	0
$733$	9.77763	0.361145	0.180572	−	0.983562i	$-0.442205\pi$
$733$	9.77763	0.361145	0.180572	+	0.983562i	$0.442205\pi$
$734$	−61.8146	−2.28162
$735$	0	0
$736$	−62.4025	−2.30019
$737$	−20.8154	−0.766746
$738$	0	0
$739$	−15.6746	−0.576600	−0.288300	−	0.957540i	$-0.593090\pi$
$739$	−15.6746	−0.576600	−0.288300	+	0.957540i	$0.593090\pi$
$740$	−18.1757	−0.668151
$741$	0	0
$742$	2.76301	0.101433
$743$	40.9943	1.50394	0.751968	−	0.659200i	$-0.229105\pi$
$743$	40.9943	1.50394	0.751968	+	0.659200i	$0.229105\pi$
$744$	0	0
$745$	8.54871	0.313201
$746$	74.4471	2.72570
$747$	0	0
$748$	11.9306	0.436225
$749$	−6.22395	−0.227418
$750$	0	0
$751$	−2.52718	−0.0922180	−0.0461090	−	0.998936i	$-0.514682\pi$
$751$	−2.52718	−0.0922180	−0.0461090	+	0.998936i	$0.514682\pi$
$752$	−11.9720	−0.436575
$753$	0	0
$754$	18.8677	0.687123
$755$	10.9097	0.397044
$756$	0	0
$757$	11.1400	0.404891	0.202445	−	0.979294i	$-0.435111\pi$
$757$	11.1400	0.404891	0.202445	+	0.979294i	$0.435111\pi$
$758$	25.8114	0.937512
$759$	0	0
$760$	5.30363	0.192383
$761$	2.99589	0.108601	0.0543005	−	0.998525i	$-0.482707\pi$
$761$	2.99589	0.108601	0.0543005	+	0.998525i	$0.482707\pi$
$762$	0	0
$763$	12.7062	0.459995
$764$	29.7872	1.07766
$765$	0	0
$766$	−65.7433	−2.37540
$767$	19.8960	0.718404
$768$	0	0
$769$	−32.1976	−1.16108	−0.580538	−	0.814233i	$-0.697158\pi$
$769$	−32.1976	−1.16108	−0.580538	+	0.814233i	$0.697158\pi$
$770$	4.43135	0.159695
$771$	0	0
$772$	29.8516	1.07438
$773$	−44.5912	−1.60383	−0.801917	−	0.597435i	$-0.796187\pi$
$773$	−44.5912	−1.60383	−0.801917	+	0.597435i	$0.796187\pi$
$774$	0	0
$775$	24.2830	0.872272
$776$	−26.0976	−0.936849
$777$	0	0
$778$	20.1939	0.723986
$779$	−27.6663	−0.991249
$780$	0	0
$781$	−1.18711	−0.0424781
$782$	−28.7435	−1.02786
$783$	0	0
$784$	−1.16747	−0.0416954
$785$	6.96486	0.248587
$786$	0	0
$787$	−26.2513	−0.935758	−0.467879	−	0.883792i	$-0.654982\pi$
$787$	−26.2513	−0.935758	−0.467879	+	0.883792i	$0.654982\pi$
$788$	−37.0346	−1.31930
$789$	0	0
$790$	14.6679	0.521860
$791$	15.3591	0.546105
$792$	0	0
$793$	5.52736	0.196282
$794$	−3.34544	−0.118725
$795$	0	0
$796$	−52.3233	−1.85455
$797$	−18.3266	−0.649162	−0.324581	−	0.945858i	$-0.605223\pi$
$797$	−18.3266	−0.649162	−0.324581	+	0.945858i	$0.605223\pi$
$798$	0	0
$799$	−14.5616	−0.515154
$800$	30.9517	1.09431
$801$	0	0
$802$	9.42240	0.332716
$803$	21.3674	0.754041
$804$	0	0
$805$	−6.36922	−0.224485
$806$	19.7564	0.695891
$807$	0	0
$808$	30.1842	1.06188
$809$	−24.7287	−0.869415	−0.434707	−	0.900572i	$-0.643148\pi$
$809$	−24.7287	−0.869415	−0.434707	+	0.900572i	$0.643148\pi$
$810$	0	0
$811$	17.1100	0.600815	0.300407	−	0.953811i	$-0.402877\pi$
$811$	17.1100	0.600815	0.300407	+	0.953811i	$0.402877\pi$
$812$	15.2115	0.533818
$813$	0	0
$814$	55.4785	1.94452
$815$	−2.13396	−0.0747494
$816$	0	0
$817$	−10.5114	−0.367749
$818$	40.5876	1.41911
$819$	0	0
$820$	−16.1488	−0.563942
$821$	43.9674	1.53447	0.767236	−	0.641364i	$-0.221631\pi$
$821$	43.9674	1.53447	0.767236	+	0.641364i	$0.221631\pi$
$822$	0	0
$823$	17.4501	0.608272	0.304136	−	0.952629i	$-0.401632\pi$
$823$	17.4501	0.608272	0.304136	+	0.952629i	$0.401632\pi$
$824$	−18.1744	−0.633135
$825$	0	0
$826$	26.8872	0.935524
$827$	−9.02653	−0.313883	−0.156942	−	0.987608i	$-0.550163\pi$
$827$	−9.02653	−0.313883	−0.156942	+	0.987608i	$0.550163\pi$
$828$	0	0
$829$	21.4721	0.745758	0.372879	−	0.927880i	$-0.378371\pi$
$829$	21.4721	0.745758	0.372879	+	0.927880i	$0.378371\pi$
$830$	1.55182	0.0538646
$831$	0	0
$832$	21.3349	0.739655
$833$	−1.42000	−0.0492001
$834$	0	0
$835$	7.86063	0.272028
$836$	−29.8270	−1.03159
$837$	0	0
$838$	59.3691	2.05087
$839$	16.8359	0.581241	0.290621	−	0.956838i	$-0.406138\pi$
$839$	16.8359	0.581241	0.290621	+	0.956838i	$0.406138\pi$
$840$	0	0
$841$	−2.54921	−0.0879038
$842$	−56.3833	−1.94310
$843$	0	0
$844$	−64.1210	−2.20714
$845$	−7.20596	−0.247892
$846$	0	0
$847$	2.93057	0.100695
$848$	1.44874	0.0497498
$849$	0	0
$850$	14.2568	0.489004
$851$	−79.7398	−2.73345
$852$	0	0
$853$	−4.72596	−0.161814	−0.0809070	−	0.996722i	$-0.525782\pi$
$853$	−4.72596	−0.161814	−0.0809070	+	0.996722i	$0.525782\pi$
$854$	7.46958	0.255604
$855$	0	0
$856$	13.2717	0.453619
$857$	38.2455	1.30644	0.653221	−	0.757167i	$-0.273417\pi$
$857$	38.2455	1.30644	0.653221	+	0.757167i	$0.273417\pi$
$858$	0	0
$859$	−32.0219	−1.09257	−0.546287	−	0.837598i	$-0.683959\pi$
$859$	−32.0219	−1.09257	−0.546287	+	0.837598i	$0.683959\pi$
$860$	−6.13553	−0.209220
$861$	0	0
$862$	−64.7233	−2.20449
$863$	−49.5269	−1.68592	−0.842958	−	0.537979i	$-0.819188\pi$
$863$	−49.5269	−1.68592	−0.842958	+	0.537979i	$0.819188\pi$
$864$	0	0
$865$	0.215799	0.00733738
$866$	−7.95039	−0.270165
$867$	0	0
$868$	15.9279	0.540630
$869$	−26.7100	−0.906076
$870$	0	0
$871$	−12.0733	−0.409087
$872$	−27.0943	−0.917528
$873$	0	0
$874$	71.8601	2.43070
$875$	6.66218	0.225223
$876$	0	0
$877$	−1.61087	−0.0543952	−0.0271976	−	0.999630i	$-0.508658\pi$
$877$	−1.61087	−0.0543952	−0.0271976	+	0.999630i	$0.508658\pi$
$878$	21.7641	0.734502
$879$	0	0
$880$	2.32350	0.0783251
$881$	32.1731	1.08394	0.541970	−	0.840398i	$-0.317679\pi$
$881$	32.1731	1.08394	0.541970	+	0.840398i	$0.317679\pi$
$882$	0	0
$883$	30.6226	1.03053	0.515267	−	0.857030i	$-0.327693\pi$
$883$	30.6226	1.03053	0.515267	+	0.857030i	$0.327693\pi$
$884$	6.91992	0.232742
$885$	0	0
$886$	68.2585	2.29319
$887$	3.02012	0.101406	0.0507028	−	0.998714i	$-0.483854\pi$
$887$	3.02012	0.101406	0.0507028	+	0.998714i	$0.483854\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	17.5205	0.587290
$891$	0	0
$892$	−84.7200	−2.83664
$893$	36.4048	1.21824
$894$	0	0
$895$	3.52684	0.117889
$896$	15.1032	0.504563
$897$	0	0
$898$	42.3280	1.41250
$899$	27.6967	0.923735
$900$	0	0
$901$	1.76210	0.0587041
$902$	49.2919	1.64124
$903$	0	0
$904$	−32.7511	−1.08929
$905$	10.0008	0.332439
$906$	0	0
$907$	−46.8581	−1.55590	−0.777949	−	0.628328i	$-0.783740\pi$
$907$	−46.8581	−1.55590	−0.777949	+	0.628328i	$0.783740\pi$
$908$	−5.50516	−0.182695
$909$	0	0
$910$	2.57025	0.0852030
$911$	−26.4540	−0.876460	−0.438230	−	0.898863i	$-0.644394\pi$
$911$	−26.4540	−0.876460	−0.438230	+	0.898863i	$0.644394\pi$
$912$	0	0
$913$	−2.82585	−0.0935221
$914$	−91.9531	−3.04154
$915$	0	0
$916$	51.0624	1.68715
$917$	17.0346	0.562531
$918$	0	0
$919$	−54.5679	−1.80003	−0.900015	−	0.435859i	$-0.856445\pi$
$919$	−54.5679	−1.80003	−0.900015	+	0.435859i	$0.856445\pi$
$920$	13.5815	0.447769
$921$	0	0
$922$	11.2276	0.369760
$923$	−0.688542	−0.0226636
$924$	0	0
$925$	39.5511	1.30043
$926$	−55.4823	−1.82326
$927$	0	0
$928$	35.3029	1.15887
$929$	−21.5164	−0.705931	−0.352966	−	0.935636i	$-0.614827\pi$
$929$	−21.5164	−0.705931	−0.352966	+	0.935636i	$0.614827\pi$
$930$	0	0
$931$	3.55007	0.116349
$932$	−23.4715	−0.768834
$933$	0	0
$934$	−26.7041	−0.873784
$935$	2.82608	0.0924227
$936$	0	0
$937$	2.73408	0.0893184	0.0446592	−	0.999002i	$-0.485780\pi$
$937$	2.73408	0.0893184	0.0446592	+	0.999002i	$0.485780\pi$
$938$	−16.3156	−0.532724
$939$	0	0
$940$	21.2495	0.693082
$941$	59.7777	1.94870	0.974349	−	0.225041i	$-0.0722517\pi$
$941$	59.7777	1.94870	0.974349	+	0.225041i	$0.0722517\pi$
$942$	0	0
$943$	−70.8478	−2.30712
$944$	14.0978	0.458845
$945$	0	0
$946$	18.7278	0.608893
$947$	44.4391	1.44408	0.722038	−	0.691853i	$-0.243205\pi$
$947$	44.4391	1.44408	0.722038	+	0.691853i	$0.243205\pi$
$948$	0	0
$949$	12.3934	0.402308
$950$	−35.6427	−1.15640
$951$	0	0
$952$	3.02796	0.0981368
$953$	−8.49316	−0.275120	−0.137560	−	0.990493i	$-0.543926\pi$
$953$	−8.49316	−0.275120	−0.137560	+	0.990493i	$0.543926\pi$
$954$	0	0
$955$	7.05590	0.228323
$956$	77.5184	2.50713
$957$	0	0
$958$	30.1935	0.975506
$959$	11.4427	0.369503
$960$	0	0
$961$	−1.99880	−0.0644774
$962$	32.1784	1.03747
$963$	0	0
$964$	−66.8383	−2.15272
$965$	7.07115	0.227628
$966$	0	0
$967$	−0.101854	−0.00327542	−0.00163771	−	0.999999i	$-0.500521\pi$
$967$	−0.101854	−0.00327542	−0.00163771	+	0.999999i	$0.500521\pi$
$968$	−6.24904	−0.200852
$969$	0	0
$970$	−19.0921	−0.613009
$971$	−28.8741	−0.926615	−0.463307	−	0.886198i	$-0.653337\pi$
$971$	−28.8741	−0.926615	−0.463307	+	0.886198i	$0.653337\pi$
$972$	0	0
$973$	13.3083	0.426643
$974$	4.18657	0.134147
$975$	0	0
$976$	3.91654	0.125366
$977$	34.6379	1.10817	0.554083	−	0.832461i	$-0.313069\pi$
$977$	34.6379	1.10817	0.554083	+	0.832461i	$0.313069\pi$
$978$	0	0
$979$	−31.9047	−1.01968
$980$	2.07218	0.0661932
$981$	0	0
$982$	35.8541	1.14415
$983$	−34.9097	−1.11345	−0.556723	−	0.830698i	$-0.687941\pi$
$983$	−34.9097	−1.11345	−0.556723	+	0.830698i	$0.687941\pi$
$984$	0	0
$985$	−8.77266	−0.279520
$986$	16.2610	0.517855
$987$	0	0
$988$	−17.3001	−0.550391
$989$	−26.9176	−0.855931
$990$	0	0
$991$	56.8414	1.80563	0.902813	−	0.430034i	$-0.141498\pi$
$991$	56.8414	1.80563	0.902813	+	0.430034i	$0.141498\pi$
$992$	36.9657	1.17366
$993$	0	0
$994$	−0.930484	−0.0295132
$995$	−12.3942	−0.392922
$996$	0	0
$997$	9.91403	0.313980	0.156990	−	0.987600i	$-0.449821\pi$
$997$	9.91403	0.313980	0.156990	+	0.987600i	$0.449821\pi$
$998$	78.3645	2.48059
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.m.1.10		11
3.2	odd	2		2667.2.a.k.1.2	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.k.1.2	✓	11	3.2	odd	2
8001.2.a.m.1.10		11	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)

\(f(q)\)	\(=\)	\(q+2.22659 q^{2} +2.95768 q^{4} +0.700608 q^{5} -1.00000 q^{7} +2.13237 q^{8} +O(q^{10})\)	\(q+2.22659 q^{2} +2.95768 q^{4} +0.700608 q^{5} -1.00000 q^{7} +2.13237 q^{8} +1.55996 q^{10} -2.84067 q^{11} -1.64764 q^{13} -2.22659 q^{14} -1.16747 q^{16} -1.42000 q^{17} +3.55007 q^{19} +2.07218 q^{20} -6.32501 q^{22} +9.09099 q^{23} -4.50915 q^{25} -3.66860 q^{26} -2.95768 q^{28} -5.14303 q^{29} -5.38528 q^{31} -6.86421 q^{32} -3.16175 q^{34} -0.700608 q^{35} -8.77130 q^{37} +7.90453 q^{38} +1.49395 q^{40} -7.79318 q^{41} -2.96091 q^{43} -8.40182 q^{44} +20.2419 q^{46} +10.2547 q^{47} +1.00000 q^{49} -10.0400 q^{50} -4.87319 q^{52} -1.24092 q^{53} -1.99020 q^{55} -2.13237 q^{56} -11.4514 q^{58} -12.0755 q^{59} -3.35472 q^{61} -11.9908 q^{62} -12.9488 q^{64} -1.15435 q^{65} +7.32764 q^{67} -4.19991 q^{68} -1.55996 q^{70} +0.417897 q^{71} -7.52196 q^{73} -19.5300 q^{74} +10.5000 q^{76} +2.84067 q^{77} +9.40271 q^{79} -0.817939 q^{80} -17.3522 q^{82} +0.994782 q^{83} -0.994863 q^{85} -6.59273 q^{86} -6.05736 q^{88} +11.2314 q^{89} +1.64764 q^{91} +26.8883 q^{92} +22.8329 q^{94} +2.48720 q^{95} -12.2388 q^{97} +2.22659 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 2 q^{2} + 12 q^{4} - q^{5} - 11 q^{7} + 15 q^{8}+O(q^{10}) \) 11 * q + 2 * q^2 + 12 * q^4 - q^5 - 11 * q^7 + 15 * q^8	\( 11 q + 2 q^{2} + 12 q^{4} - q^{5} - 11 q^{7} + 15 q^{8} - 12 q^{10} + 7 q^{11} - 24 q^{13} - 2 q^{14} - 6 q^{16} + 15 q^{17} - 19 q^{19} - 3 q^{20} - 3 q^{22} + 11 q^{23} + 10 q^{25} - 10 q^{26} - 12 q^{28} + 10 q^{29} - 20 q^{31} + 27 q^{32} - 9 q^{34} + q^{35} - 22 q^{37} - 8 q^{38} - 29 q^{40} - 9 q^{41} - 17 q^{43} + 9 q^{44} - 18 q^{46} + 7 q^{47} + 11 q^{49} + 47 q^{50} - 66 q^{52} + 28 q^{53} - 24 q^{55} - 15 q^{56} - 39 q^{58} - 35 q^{59} - 6 q^{61} - 18 q^{62} + 11 q^{64} + 43 q^{65} - 22 q^{67} + 12 q^{68} + 12 q^{70} + 22 q^{71} - 29 q^{73} - 14 q^{74} + 10 q^{76} - 7 q^{77} - 20 q^{79} - 66 q^{80} - 24 q^{82} - 17 q^{83} - 50 q^{85} + 12 q^{86} + 2 q^{88} + q^{89} + 24 q^{91} + 22 q^{92} + q^{94} - 10 q^{95} - 45 q^{97} + 2 q^{98}+O(q^{100}) \) 11 * q + 2 * q^2 + 12 * q^4 - q^5 - 11 * q^7 + 15 * q^8 - 12 * q^10 + 7 * q^11 - 24 * q^13 - 2 * q^14 - 6 * q^16 + 15 * q^17 - 19 * q^19 - 3 * q^20 - 3 * q^22 + 11 * q^23 + 10 * q^25 - 10 * q^26 - 12 * q^28 + 10 * q^29 - 20 * q^31 + 27 * q^32 - 9 * q^34 + q^35 - 22 * q^37 - 8 * q^38 - 29 * q^40 - 9 * q^41 - 17 * q^43 + 9 * q^44 - 18 * q^46 + 7 * q^47 + 11 * q^49 + 47 * q^50 - 66 * q^52 + 28 * q^53 - 24 * q^55 - 15 * q^56 - 39 * q^58 - 35 * q^59 - 6 * q^61 - 18 * q^62 + 11 * q^64 + 43 * q^65 - 22 * q^67 + 12 * q^68 + 12 * q^70 + 22 * q^71 - 29 * q^73 - 14 * q^74 + 10 * q^76 - 7 * q^77 - 20 * q^79 - 66 * q^80 - 24 * q^82 - 17 * q^83 - 50 * q^85 + 12 * q^86 + 2 * q^88 + q^89 + 24 * q^91 + 22 * q^92 + q^94 - 10 * q^95 - 45 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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