LMFDB - Embedded newform 9009.2.a.w.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9009,2,Mod(1,9009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9009, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9009 = 3^{2} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9009.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:	$71.9372271810$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.23301.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 2x + 1 $ x^4 - x^3 - 6x^2 + 2x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1001)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.13939$ of defining polynomial
Character	$\chi$	$=$	9009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.532576 q^{2} -1.71636 q^{4} +3.67196 q^{5} +1.00000 q^{7} -1.97925 q^{8} +O(q^{10})$	\(q+0.532576 q^{2} -1.71636 q^{4} +3.67196 q^{5} +1.00000 q^{7} -1.97925 q^{8} +1.95560 q^{10} +1.00000 q^{11} -1.00000 q^{13} +0.532576 q^{14} +2.37862 q^{16} -4.60681 q^{17} +0.0443994 q^{19} -6.30242 q^{20} +0.532576 q^{22} -8.32317 q^{23} +8.48331 q^{25} -0.532576 q^{26} -1.71636 q^{28} +3.51182 q^{29} +8.34393 q^{31} +5.22529 q^{32} -2.45348 q^{34} +3.67196 q^{35} -0.348790 q^{37} +0.0236460 q^{38} -7.26772 q^{40} -0.467424 q^{41} +4.92090 q^{43} -1.71636 q^{44} -4.43272 q^{46} -10.3232 q^{47} +1.00000 q^{49} +4.51801 q^{50} +1.71636 q^{52} -5.46256 q^{53} +3.67196 q^{55} -1.97925 q^{56} +1.87031 q^{58} +3.76695 q^{59} +11.8467 q^{61} +4.44378 q^{62} -1.97438 q^{64} -3.67196 q^{65} +2.15395 q^{67} +7.90696 q^{68} +1.95560 q^{70} +10.5575 q^{71} +5.62756 q^{73} -0.185757 q^{74} -0.0762054 q^{76} +1.00000 q^{77} +5.60681 q^{79} +8.73422 q^{80} -0.248939 q^{82} +9.99514 q^{83} -16.9160 q^{85} +2.62076 q^{86} -1.97925 q^{88} +11.8156 q^{89} -1.00000 q^{91} +14.2856 q^{92} -5.49788 q^{94} +0.163033 q^{95} +5.74620 q^{97} +0.532576 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 8 q^{4} + 5 q^{5} + 4 q^{7} - 3 q^{8}+O(q^{10}) $ 4 * q + 2 * q^2 + 8 * q^4 + 5 * q^5 + 4 * q^7 - 3 * q^8	$ 4 q + 2 q^{2} + 8 q^{4} + 5 q^{5} + 4 q^{7} - 3 q^{8} + 13 q^{10} + 4 q^{11} - 4 q^{13} + 2 q^{14} + 20 q^{16} - 9 q^{17} - 5 q^{19} + 4 q^{20} + 2 q^{22} - 9 q^{23} + 5 q^{25} - 2 q^{26} + 8 q^{28} + 9 q^{29} + 14 q^{31} + 16 q^{32} + 15 q^{34} + 5 q^{35} - 16 q^{37} - 10 q^{38} + 30 q^{40} - 2 q^{41} - 5 q^{43} + 8 q^{44} + 12 q^{46} - 17 q^{47} + 4 q^{49} + 19 q^{50} - 8 q^{52} + 12 q^{53} + 5 q^{55} - 3 q^{56} - 18 q^{58} + q^{59} + 32 q^{61} + 28 q^{62} + 31 q^{64} - 5 q^{65} - 2 q^{67} - 18 q^{68} + 13 q^{70} + 4 q^{71} + 18 q^{73} - 35 q^{74} - 40 q^{76} + 4 q^{77} + 13 q^{79} + 40 q^{80} + 14 q^{82} + 6 q^{83} - 9 q^{85} + 26 q^{86} - 3 q^{88} - 15 q^{89} - 4 q^{91} + 18 q^{92} + 8 q^{94} - 19 q^{95} + 4 q^{97} + 2 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 8 * q^4 + 5 * q^5 + 4 * q^7 - 3 * q^8 + 13 * q^10 + 4 * q^11 - 4 * q^13 + 2 * q^14 + 20 * q^16 - 9 * q^17 - 5 * q^19 + 4 * q^20 + 2 * q^22 - 9 * q^23 + 5 * q^25 - 2 * q^26 + 8 * q^28 + 9 * q^29 + 14 * q^31 + 16 * q^32 + 15 * q^34 + 5 * q^35 - 16 * q^37 - 10 * q^38 + 30 * q^40 - 2 * q^41 - 5 * q^43 + 8 * q^44 + 12 * q^46 - 17 * q^47 + 4 * q^49 + 19 * q^50 - 8 * q^52 + 12 * q^53 + 5 * q^55 - 3 * q^56 - 18 * q^58 + q^59 + 32 * q^61 + 28 * q^62 + 31 * q^64 - 5 * q^65 - 2 * q^67 - 18 * q^68 + 13 * q^70 + 4 * q^71 + 18 * q^73 - 35 * q^74 - 40 * q^76 + 4 * q^77 + 13 * q^79 + 40 * q^80 + 14 * q^82 + 6 * q^83 - 9 * q^85 + 26 * q^86 - 3 * q^88 - 15 * q^89 - 4 * q^91 + 18 * q^92 + 8 * q^94 - 19 * q^95 + 4 * 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 $

$2$	0.532576	0.376588	0.188294	−	0.982113i	$-0.439704\pi$
$2$	0.532576	0.376588	0.188294	+	0.982113i	$0.439704\pi$
$3$	0	0
$3$	0	0
$4$	−1.71636	−0.858181
$5$	3.67196	1.64215	0.821076	−	0.570819i	$-0.193374\pi$
$5$	3.67196	1.64215	0.821076	+	0.570819i	$0.193374\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.97925	−0.699769
$9$	0	0
$10$	1.95560	0.618415
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0.532576	0.142337
$15$	0	0
$16$	2.37862	0.594656
$17$	−4.60681	−1.11732	−0.558658	−	0.829398i	$-0.688683\pi$
$17$	−4.60681	−1.11732	−0.558658	+	0.829398i	$0.688683\pi$
$18$	0	0
$19$	0.0443994	0.0101859	0.00509296	−	0.999987i	$-0.498379\pi$
$19$	0.0443994	0.0101859	0.00509296	+	0.999987i	$0.498379\pi$
$20$	−6.30242	−1.40926
$21$	0	0
$22$	0.532576	0.113546
$23$	−8.32317	−1.73550	−0.867751	−	0.497000i	$-0.834435\pi$
$23$	−8.32317	−1.73550	−0.867751	+	0.497000i	$0.834435\pi$
$24$	0	0
$25$	8.48331	1.69666
$26$	−0.532576	−0.104447
$27$	0	0
$28$	−1.71636	−0.324362
$29$	3.51182	0.652129	0.326065	−	0.945347i	$-0.394277\pi$
$29$	3.51182	0.652129	0.326065	+	0.945347i	$0.394277\pi$
$30$	0	0
$31$	8.34393	1.49861	0.749307	−	0.662223i	$-0.230387\pi$
$31$	8.34393	1.49861	0.749307	+	0.662223i	$0.230387\pi$
$32$	5.22529	0.923710
$33$	0	0
$34$	−2.45348	−0.420768
$35$	3.67196	0.620675
$36$	0	0
$37$	−0.348790	−0.0573408	−0.0286704	−	0.999589i	$-0.509127\pi$
$37$	−0.348790	−0.0573408	−0.0286704	+	0.999589i	$0.509127\pi$
$38$	0.0236460	0.00383590
$39$	0	0
$40$	−7.26772	−1.14913
$41$	−0.467424	−0.0729993	−0.0364997	−	0.999334i	$-0.511621\pi$
$41$	−0.467424	−0.0729993	−0.0364997	+	0.999334i	$0.511621\pi$
$42$	0	0
$43$	4.92090	0.750431	0.375215	−	0.926938i	$-0.377569\pi$
$43$	4.92090	0.750431	0.375215	+	0.926938i	$0.377569\pi$
$44$	−1.71636	−0.258751
$45$	0	0
$46$	−4.43272	−0.653570
$47$	−10.3232	−1.50579	−0.752895	−	0.658141i	$-0.771343\pi$
$47$	−10.3232	−1.50579	−0.752895	+	0.658141i	$0.771343\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	4.51801	0.638943
$51$	0	0
$52$	1.71636	0.238017
$53$	−5.46256	−0.750340	−0.375170	−	0.926956i	$-0.622416\pi$
$53$	−5.46256	−0.750340	−0.375170	+	0.926956i	$0.622416\pi$
$54$	0	0
$55$	3.67196	0.495127
$56$	−1.97925	−0.264488
$57$	0	0
$58$	1.87031	0.245584
$59$	3.76695	0.490415	0.245208	−	0.969471i	$-0.421144\pi$
$59$	3.76695	0.490415	0.245208	+	0.969471i	$0.421144\pi$
$60$	0	0
$61$	11.8467	1.51681	0.758405	−	0.651783i	$-0.225979\pi$
$61$	11.8467	1.51681	0.758405	+	0.651783i	$0.225979\pi$
$62$	4.44378	0.564360
$63$	0	0
$64$	−1.97438	−0.246798
$65$	−3.67196	−0.455451
$66$	0	0
$67$	2.15395	0.263147	0.131574	−	0.991306i	$-0.457997\pi$
$67$	2.15395	0.263147	0.131574	+	0.991306i	$0.457997\pi$
$68$	7.90696	0.958859
$69$	0	0
$70$	1.95560	0.233739
$71$	10.5575	1.25295	0.626475	−	0.779441i	$-0.284497\pi$
$71$	10.5575	1.25295	0.626475	+	0.779441i	$0.284497\pi$
$72$	0	0
$73$	5.62756	0.658657	0.329328	−	0.944215i	$-0.393178\pi$
$73$	5.62756	0.658657	0.329328	+	0.944215i	$0.393178\pi$
$74$	−0.185757	−0.0215939
$75$	0	0
$76$	−0.0762054	−0.00874136
$77$	1.00000	0.113961
$78$	0	0
$79$	5.60681	0.630815	0.315408	−	0.948956i	$-0.397859\pi$
$79$	5.60681	0.630815	0.315408	+	0.948956i	$0.397859\pi$
$80$	8.73422	0.976516
$81$	0	0
$82$	−0.248939	−0.0274907
$83$	9.99514	1.09711	0.548554	−	0.836115i	$-0.315178\pi$
$83$	9.99514	1.09711	0.548554	+	0.836115i	$0.315178\pi$
$84$	0	0
$85$	−16.9160	−1.83480
$86$	2.62076	0.282603
$87$	0	0
$88$	−1.97925	−0.210988
$89$	11.8156	1.25245	0.626224	−	0.779643i	$-0.284600\pi$
$89$	11.8156	1.25245	0.626224	+	0.779643i	$0.284600\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	14.2856	1.48937
$93$	0	0
$94$	−5.49788	−0.567063
$95$	0.163033	0.0167268
$96$	0	0
$97$	5.74620	0.583438	0.291719	−	0.956504i	$-0.405773\pi$
$97$	5.74620	0.583438	0.291719	+	0.956504i	$0.405773\pi$
$98$	0.532576	0.0537983
$99$	0	0
$100$	−14.5604	−1.45604
$101$	−11.2088	−1.11531	−0.557656	−	0.830072i	$-0.688299\pi$
$101$	−11.2088	−1.11531	−0.557656	+	0.830072i	$0.688299\pi$
$102$	0	0
$103$	9.32804	0.919119	0.459559	−	0.888147i	$-0.348007\pi$
$103$	9.32804	0.919119	0.459559	+	0.888147i	$0.348007\pi$
$104$	1.97925	0.194081
$105$	0	0
$106$	−2.90923	−0.282569
$107$	−0.568600	−0.0549686	−0.0274843	−	0.999622i	$-0.508750\pi$
$107$	−0.568600	−0.0549686	−0.0274843	+	0.999622i	$0.508750\pi$
$108$	0	0
$109$	−5.14847	−0.493134	−0.246567	−	0.969126i	$-0.579303\pi$
$109$	−5.14847	−0.493134	−0.246567	+	0.969126i	$0.579303\pi$
$110$	1.95560	0.186459
$111$	0	0
$112$	2.37862	0.224759
$113$	14.1283	1.32908	0.664541	−	0.747252i	$-0.268627\pi$
$113$	14.1283	1.32908	0.664541	+	0.747252i	$0.268627\pi$
$114$	0	0
$115$	−30.5624	−2.84996
$116$	−6.02756	−0.559645
$117$	0	0
$118$	2.00619	0.184685
$119$	−4.60681	−0.422306
$120$	0	0
$121$	1.00000	0.0909091
$122$	6.30925	0.571213
$123$	0	0
$124$	−14.3212	−1.28608
$125$	12.7906	1.14403
$126$	0	0
$127$	12.5958	1.11769	0.558846	−	0.829271i	$-0.311244\pi$
$127$	12.5958	1.11769	0.558846	+	0.829271i	$0.311244\pi$
$128$	−11.5021	−1.01665
$129$	0	0
$130$	−1.95560	−0.171518
$131$	7.76984	0.678854	0.339427	−	0.940632i	$-0.389767\pi$
$131$	7.76984	0.678854	0.339427	+	0.940632i	$0.389767\pi$
$132$	0	0
$133$	0.0443994	0.00384991
$134$	1.14714	0.0990981
$135$	0	0
$136$	9.11801	0.781863
$137$	−13.9792	−1.19433	−0.597164	−	0.802119i	$-0.703706\pi$
$137$	−13.9792	−1.19433	−0.597164	+	0.802119i	$0.703706\pi$
$138$	0	0
$139$	−8.61784	−0.730955	−0.365478	−	0.930820i	$-0.619094\pi$
$139$	−8.61784	−0.730955	−0.365478	+	0.930820i	$0.619094\pi$
$140$	−6.30242	−0.532652
$141$	0	0
$142$	5.62270	0.471846
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	12.8953	1.07090
$146$	2.99711	0.248042
$147$	0	0
$148$	0.598650	0.0492088
$149$	20.8807	1.71062	0.855308	−	0.518121i	$-0.173368\pi$
$149$	20.8807	1.71062	0.855308	+	0.518121i	$0.173368\pi$
$150$	0	0
$151$	10.5485	0.858422	0.429211	−	0.903204i	$-0.358792\pi$
$151$	10.5485	0.858422	0.429211	+	0.903204i	$0.358792\pi$
$152$	−0.0878773	−0.00712779
$153$	0	0
$154$	0.532576	0.0429162
$155$	30.6386	2.46095
$156$	0	0
$157$	13.7857	1.10022	0.550111	−	0.835092i	$-0.314586\pi$
$157$	13.7857	1.10022	0.550111	+	0.835092i	$0.314586\pi$
$158$	2.98605	0.237558
$159$	0	0
$160$	19.1871	1.51687
$161$	−8.32317	−0.655958
$162$	0	0
$163$	−18.8924	−1.47977	−0.739883	−	0.672735i	$-0.765119\pi$
$163$	−18.8924	−1.47977	−0.739883	+	0.672735i	$0.765119\pi$
$164$	0.802268	0.0626466
$165$	0	0
$166$	5.32317	0.413158
$167$	6.85089	0.530137	0.265069	−	0.964230i	$-0.414605\pi$
$167$	6.85089	0.530137	0.265069	+	0.964230i	$0.414605\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−9.00908	−0.690965
$171$	0	0
$172$	−8.44605	−0.644005
$173$	−7.50407	−0.570524	−0.285262	−	0.958450i	$-0.592081\pi$
$173$	−7.50407	−0.570524	−0.285262	+	0.958450i	$0.592081\pi$
$174$	0	0
$175$	8.48331	0.641278
$176$	2.37862	0.179296
$177$	0	0
$178$	6.29269	0.471657
$179$	9.75106	0.728829	0.364414	−	0.931237i	$-0.381269\pi$
$179$	9.75106	0.728829	0.364414	+	0.931237i	$0.381269\pi$
$180$	0	0
$181$	−8.88620	−0.660506	−0.330253	−	0.943892i	$-0.607134\pi$
$181$	−8.88620	−0.660506	−0.330253	+	0.943892i	$0.607134\pi$
$182$	−0.532576	−0.0394772
$183$	0	0
$184$	16.4736	1.21445
$185$	−1.28074	−0.0941622
$186$	0	0
$187$	−4.60681	−0.336883
$188$	17.7183	1.29224
$189$	0	0
$190$	0.0868274	0.00629912
$191$	7.40846	0.536057	0.268029	−	0.963411i	$-0.413628\pi$
$191$	7.40846	0.536057	0.268029	+	0.963411i	$0.413628\pi$
$192$	0	0
$193$	22.5076	1.62013	0.810065	−	0.586339i	$-0.199432\pi$
$193$	22.5076	1.62013	0.810065	+	0.586339i	$0.199432\pi$
$194$	3.06029	0.219716
$195$	0	0
$196$	−1.71636	−0.122597
$197$	−22.0756	−1.57282	−0.786409	−	0.617706i	$-0.788062\pi$
$197$	−22.0756	−1.57282	−0.786409	+	0.617706i	$0.788062\pi$
$198$	0	0
$199$	−9.85027	−0.698267	−0.349133	−	0.937073i	$-0.613524\pi$
$199$	−9.85027	−0.698267	−0.349133	+	0.937073i	$0.613524\pi$
$200$	−16.7906	−1.18727
$201$	0	0
$202$	−5.96952	−0.420014
$203$	3.51182	0.246482
$204$	0	0
$205$	−1.71636	−0.119876
$206$	4.96789	0.346129
$207$	0	0
$208$	−2.37862	−0.164928
$209$	0.0443994	0.00307117
$210$	0	0
$211$	−1.94587	−0.133959	−0.0669797	−	0.997754i	$-0.521336\pi$
$211$	−1.94587	−0.133959	−0.0669797	+	0.997754i	$0.521336\pi$
$212$	9.37573	0.643928
$213$	0	0
$214$	−0.302823	−0.0207005
$215$	18.0694	1.23232
$216$	0	0
$217$	8.34393	0.566423
$218$	−2.74195	−0.185708
$219$	0	0
$220$	−6.30242	−0.424909
$221$	4.60681	0.309888
$222$	0	0
$223$	−11.9783	−0.802126	−0.401063	−	0.916050i	$-0.631359\pi$
$223$	−11.9783	−0.802126	−0.401063	+	0.916050i	$0.631359\pi$
$224$	5.22529	0.349130
$225$	0	0
$226$	7.52442	0.500517
$227$	18.8120	1.24859	0.624297	−	0.781187i	$-0.285385\pi$
$227$	18.8120	1.24859	0.624297	+	0.781187i	$0.285385\pi$
$228$	0	0
$229$	13.5076	0.892606	0.446303	−	0.894882i	$-0.352740\pi$
$229$	13.5076	0.892606	0.446303	+	0.894882i	$0.352740\pi$
$230$	−16.2768	−1.07326
$231$	0	0
$232$	−6.95076	−0.456340
$233$	−22.9841	−1.50574	−0.752869	−	0.658171i	$-0.771330\pi$
$233$	−22.9841	−1.50574	−0.752869	+	0.658171i	$0.771330\pi$
$234$	0	0
$235$	−37.9063	−2.47273
$236$	−6.46545	−0.420865
$237$	0	0
$238$	−2.45348	−0.159035
$239$	−4.33290	−0.280272	−0.140136	−	0.990132i	$-0.544754\pi$
$239$	−4.33290	−0.280272	−0.140136	+	0.990132i	$0.544754\pi$
$240$	0	0
$241$	17.4979	1.12714	0.563569	−	0.826069i	$-0.309428\pi$
$241$	17.4979	1.12714	0.563569	+	0.826069i	$0.309428\pi$
$242$	0.532576	0.0342353
$243$	0	0
$244$	−20.3332	−1.30170
$245$	3.67196	0.234593
$246$	0	0
$247$	−0.0443994	−0.00282506
$248$	−16.5147	−1.04868
$249$	0	0
$250$	6.81197	0.430827
$251$	−3.75301	−0.236888	−0.118444	−	0.992961i	$-0.537791\pi$
$251$	−3.75301	−0.236888	−0.118444	+	0.992961i	$0.537791\pi$
$252$	0	0
$253$	−8.32317	−0.523273
$254$	6.70820	0.420910
$255$	0	0
$256$	−2.17698	−0.136061
$257$	−0.104688	−0.00653026	−0.00326513	−	0.999995i	$-0.501039\pi$
$257$	−0.104688	−0.00653026	−0.00326513	+	0.999995i	$0.501039\pi$
$258$	0	0
$259$	−0.348790	−0.0216728
$260$	6.30242	0.390859
$261$	0	0
$262$	4.13803	0.255649
$263$	−20.6463	−1.27311	−0.636554	−	0.771232i	$-0.719641\pi$
$263$	−20.6463	−1.27311	−0.636554	+	0.771232i	$0.719641\pi$
$264$	0	0
$265$	−20.0583	−1.23217
$266$	0.0236460	0.00144983
$267$	0	0
$268$	−3.69696	−0.225828
$269$	−5.06577	−0.308866	−0.154433	−	0.988003i	$-0.549355\pi$
$269$	−5.06577	−0.308866	−0.154433	+	0.988003i	$0.549355\pi$
$270$	0	0
$271$	−6.68104	−0.405845	−0.202922	−	0.979195i	$-0.565044\pi$
$271$	−6.68104	−0.405845	−0.202922	+	0.979195i	$0.565044\pi$
$272$	−10.9579	−0.664419
$273$	0	0
$274$	−7.44502	−0.449770
$275$	8.48331	0.511563
$276$	0	0
$277$	−7.01589	−0.421544	−0.210772	−	0.977535i	$-0.567598\pi$
$277$	−7.01589	−0.421544	−0.210772	+	0.977535i	$0.567598\pi$
$278$	−4.58966	−0.275269
$279$	0	0
$280$	−7.26772	−0.434329
$281$	11.1851	0.667248	0.333624	−	0.942706i	$-0.391728\pi$
$281$	11.1851	0.667248	0.333624	+	0.942706i	$0.391728\pi$
$282$	0	0
$283$	−1.87650	−0.111546	−0.0557732	−	0.998443i	$-0.517762\pi$
$283$	−1.87650	−0.111546	−0.0557732	+	0.998443i	$0.517762\pi$
$284$	−18.1206	−1.07526
$285$	0	0
$286$	−0.532576	−0.0314919
$287$	−0.467424	−0.0275911
$288$	0	0
$289$	4.22270	0.248394
$290$	6.86772	0.403287
$291$	0	0
$292$	−9.65894	−0.565247
$293$	−30.7018	−1.79362	−0.896808	−	0.442419i	$-0.854120\pi$
$293$	−30.7018	−1.79362	−0.896808	+	0.442419i	$0.854120\pi$
$294$	0	0
$295$	13.8321	0.805336
$296$	0.690342	0.0401253
$297$	0	0
$298$	11.1206	0.644198
$299$	8.32317	0.481342
$300$	0	0
$301$	4.92090	0.283636
$302$	5.61786	0.323272
$303$	0	0
$304$	0.105609	0.00605712
$305$	43.5005	2.49083
$306$	0	0
$307$	7.62076	0.434940	0.217470	−	0.976067i	$-0.430220\pi$
$307$	7.62076	0.434940	0.217470	+	0.976067i	$0.430220\pi$
$308$	−1.71636	−0.0977988
$309$	0	0
$310$	16.3174	0.926765
$311$	14.6130	0.828626	0.414313	−	0.910135i	$-0.364022\pi$
$311$	14.6130	0.828626	0.414313	+	0.910135i	$0.364022\pi$
$312$	0	0
$313$	7.64018	0.431849	0.215924	−	0.976410i	$-0.430724\pi$
$313$	7.64018	0.431849	0.215924	+	0.976410i	$0.430724\pi$
$314$	7.34195	0.414330
$315$	0	0
$316$	−9.62332	−0.541354
$317$	30.8453	1.73245	0.866223	−	0.499658i	$-0.166541\pi$
$317$	30.8453	1.73245	0.866223	+	0.499658i	$0.166541\pi$
$318$	0	0
$319$	3.51182	0.196624
$320$	−7.24986	−0.405280
$321$	0	0
$322$	−4.43272	−0.247026
$323$	−0.204539	−0.0113809
$324$	0	0
$325$	−8.48331	−0.470570
$326$	−10.0616	−0.557263
$327$	0	0
$328$	0.925147	0.0510827
$329$	−10.3232	−0.569135
$330$	0	0
$331$	1.64770	0.0905657	0.0452828	−	0.998974i	$-0.485581\pi$
$331$	1.64770	0.0905657	0.0452828	+	0.998974i	$0.485581\pi$
$332$	−17.1553	−0.941518
$333$	0	0
$334$	3.64862	0.199644
$335$	7.90923	0.432127
$336$	0	0
$337$	−28.1780	−1.53495	−0.767476	−	0.641078i	$-0.778488\pi$
$337$	−28.1780	−1.53495	−0.767476	+	0.641078i	$0.778488\pi$
$338$	0.532576	0.0289683
$339$	0	0
$340$	29.0341	1.57459
$341$	8.34393	0.451849
$342$	0	0
$343$	1.00000	0.0539949
$344$	−9.73968	−0.525128
$345$	0	0
$346$	−3.99649	−0.214853
$347$	−25.1242	−1.34874	−0.674368	−	0.738395i	$-0.735584\pi$
$347$	−25.1242	−1.34874	−0.674368	+	0.738395i	$0.735584\pi$
$348$	0	0
$349$	−29.0490	−1.55496	−0.777478	−	0.628910i	$-0.783502\pi$
$349$	−29.0490	−1.55496	−0.777478	+	0.628910i	$0.783502\pi$
$350$	4.51801	0.241498
$351$	0	0
$352$	5.22529	0.278509
$353$	−36.4881	−1.94207	−0.971034	−	0.238941i	$-0.923200\pi$
$353$	−36.4881	−1.94207	−0.971034	+	0.238941i	$0.923200\pi$
$354$	0	0
$355$	38.7669	2.05753
$356$	−20.2798	−1.07483
$357$	0	0
$358$	5.19318	0.274468
$359$	28.7468	1.51720	0.758599	−	0.651558i	$-0.225884\pi$
$359$	28.7468	1.51720	0.758599	+	0.651558i	$0.225884\pi$
$360$	0	0
$361$	−18.9980	−0.999896
$362$	−4.73258	−0.248739
$363$	0	0
$364$	1.71636	0.0899618
$365$	20.6642	1.08161
$366$	0	0
$367$	21.1498	1.10401	0.552005	−	0.833841i	$-0.313863\pi$
$367$	21.1498	1.10401	0.552005	+	0.833841i	$0.313863\pi$
$368$	−19.7977	−1.03203
$369$	0	0
$370$	−0.682094	−0.0354604
$371$	−5.46256	−0.283602
$372$	0	0
$373$	11.0894	0.574188	0.287094	−	0.957902i	$-0.407311\pi$
$373$	11.0894	0.574188	0.287094	+	0.957902i	$0.407311\pi$
$374$	−2.45348	−0.126866
$375$	0	0
$376$	20.4321	1.05371
$377$	−3.51182	−0.180868
$378$	0	0
$379$	−13.6395	−0.700616	−0.350308	−	0.936635i	$-0.613923\pi$
$379$	−13.6395	−0.700616	−0.350308	+	0.936635i	$0.613923\pi$
$380$	−0.279823	−0.0143546
$381$	0	0
$382$	3.94557	0.201873
$383$	15.6846	0.801443	0.400722	−	0.916200i	$-0.368759\pi$
$383$	15.6846	0.801443	0.400722	+	0.916200i	$0.368759\pi$
$384$	0	0
$385$	3.67196	0.187141
$386$	11.9870	0.610122
$387$	0	0
$388$	−9.86256	−0.500695
$389$	38.3119	1.94249	0.971245	−	0.238084i	$-0.0765193\pi$
$389$	38.3119	1.94249	0.971245	+	0.238084i	$0.0765193\pi$
$390$	0	0
$391$	38.3433	1.93910
$392$	−1.97925	−0.0999671
$393$	0	0
$394$	−11.7569	−0.592305
$395$	20.5880	1.03589
$396$	0	0
$397$	2.36600	0.118746	0.0593732	−	0.998236i	$-0.481090\pi$
$397$	2.36600	0.118746	0.0593732	+	0.998236i	$0.481090\pi$
$398$	−5.24602	−0.262959
$399$	0	0
$400$	20.1786	1.00893
$401$	31.2648	1.56129	0.780646	−	0.624974i	$-0.214890\pi$
$401$	31.2648	1.56129	0.780646	+	0.624974i	$0.214890\pi$
$402$	0	0
$403$	−8.34393	−0.415641
$404$	19.2383	0.957141
$405$	0	0
$406$	1.87031	0.0928221
$407$	−0.348790	−0.0172889
$408$	0	0
$409$	−27.4438	−1.35701	−0.678503	−	0.734598i	$-0.737371\pi$
$409$	−27.4438	−1.35701	−0.678503	+	0.734598i	$0.737371\pi$
$410$	−0.914094	−0.0451439
$411$	0	0
$412$	−16.0103	−0.788770
$413$	3.76695	0.185360
$414$	0	0
$415$	36.7018	1.80162
$416$	−5.22529	−0.256191
$417$	0	0
$418$	0.0236460	0.00115657
$419$	−35.3433	−1.72663	−0.863316	−	0.504664i	$-0.831617\pi$
$419$	−35.3433	−1.72663	−0.863316	+	0.504664i	$0.831617\pi$
$420$	0	0
$421$	34.8198	1.69701	0.848506	−	0.529185i	$-0.177502\pi$
$421$	34.8198	1.69701	0.848506	+	0.529185i	$0.177502\pi$
$422$	−1.03633	−0.0504475
$423$	0	0
$424$	10.8118	0.525065
$425$	−39.0810	−1.89571
$426$	0	0
$427$	11.8467	0.573300
$428$	0.975923	0.0471730
$429$	0	0
$430$	9.62332	0.464078
$431$	32.1776	1.54994	0.774970	−	0.631999i	$-0.217765\pi$
$431$	32.1776	1.54994	0.774970	+	0.631999i	$0.217765\pi$
$432$	0	0
$433$	18.1218	0.870880	0.435440	−	0.900218i	$-0.356593\pi$
$433$	18.1218	0.870880	0.435440	+	0.900218i	$0.356593\pi$
$434$	4.44378	0.213308
$435$	0	0
$436$	8.83664	0.423198
$437$	−0.369544	−0.0176777
$438$	0	0
$439$	32.5679	1.55438	0.777190	−	0.629266i	$-0.216645\pi$
$439$	32.5679	1.55438	0.777190	+	0.629266i	$0.216645\pi$
$440$	−7.26772	−0.346475
$441$	0	0
$442$	2.45348	0.116700
$443$	29.3037	1.39226	0.696131	−	0.717915i	$-0.254903\pi$
$443$	29.3037	1.39226	0.696131	+	0.717915i	$0.254903\pi$
$444$	0	0
$445$	43.3863	2.05671
$446$	−6.37936	−0.302071
$447$	0	0
$448$	−1.97438	−0.0932808
$449$	−3.82724	−0.180619	−0.0903093	−	0.995914i	$-0.528786\pi$
$449$	−3.82724	−0.180619	−0.0903093	+	0.995914i	$0.528786\pi$
$450$	0	0
$451$	−0.467424	−0.0220101
$452$	−24.2493	−1.14059
$453$	0	0
$454$	10.0188	0.470206
$455$	−3.67196	−0.172144
$456$	0	0
$457$	−6.68496	−0.312709	−0.156355	−	0.987701i	$-0.549974\pi$
$457$	−6.68496	−0.312709	−0.156355	+	0.987701i	$0.549974\pi$
$458$	7.19382	0.336145
$459$	0	0
$460$	52.4561	2.44578
$461$	−22.0921	−1.02893	−0.514465	−	0.857511i	$-0.672009\pi$
$461$	−22.0921	−1.02893	−0.514465	+	0.857511i	$0.672009\pi$
$462$	0	0
$463$	18.8736	0.877130	0.438565	−	0.898700i	$-0.355487\pi$
$463$	18.8736	0.877130	0.438565	+	0.898700i	$0.355487\pi$
$464$	8.35331	0.387793
$465$	0	0
$466$	−12.2408	−0.567043
$467$	−13.4023	−0.620183	−0.310092	−	0.950707i	$-0.600360\pi$
$467$	−13.4023	−0.620183	−0.310092	+	0.950707i	$0.600360\pi$
$468$	0	0
$469$	2.15395	0.0994602
$470$	−20.1880	−0.931203
$471$	0	0
$472$	−7.45572	−0.343178
$473$	4.92090	0.226263
$474$	0	0
$475$	0.376654	0.0172821
$476$	7.90696	0.362415
$477$	0	0
$478$	−2.30760	−0.105547
$479$	−31.0810	−1.42013	−0.710064	−	0.704137i	$-0.751334\pi$
$479$	−31.0810	−1.42013	−0.710064	+	0.704137i	$0.751334\pi$
$480$	0	0
$481$	0.348790	0.0159035
$482$	9.31896	0.424467
$483$	0	0
$484$	−1.71636	−0.0780165
$485$	21.0998	0.958094
$486$	0	0
$487$	−14.9030	−0.675321	−0.337661	−	0.941268i	$-0.609636\pi$
$487$	−14.9030	−0.675321	−0.337661	+	0.941268i	$0.609636\pi$
$488$	−23.4475	−1.06142
$489$	0	0
$490$	1.95560	0.0883450
$491$	33.3307	1.50419	0.752096	−	0.659054i	$-0.229043\pi$
$491$	33.3307	1.50419	0.752096	+	0.659054i	$0.229043\pi$
$492$	0	0
$493$	−16.1783	−0.728634
$494$	−0.0236460	−0.00106389
$495$	0	0
$496$	19.8471	0.891160
$497$	10.5575	0.473571
$498$	0	0
$499$	38.7789	1.73598	0.867991	−	0.496580i	$-0.165411\pi$
$499$	38.7789	1.73598	0.867991	+	0.496580i	$0.165411\pi$
$500$	−21.9533	−0.981781
$501$	0	0
$502$	−1.99876	−0.0892091
$503$	−10.0820	−0.449534	−0.224767	−	0.974413i	$-0.572162\pi$
$503$	−10.0820	−0.449534	−0.224767	+	0.974413i	$0.572162\pi$
$504$	0	0
$505$	−41.1581	−1.83151
$506$	−4.43272	−0.197059
$507$	0	0
$508$	−21.6189	−0.959183
$509$	−16.9037	−0.749245	−0.374623	−	0.927177i	$-0.622228\pi$
$509$	−16.9037	−0.749245	−0.374623	+	0.927177i	$0.622228\pi$
$510$	0	0
$511$	5.62756	0.248949
$512$	21.8448	0.965412
$513$	0	0
$514$	−0.0557543	−0.00245922
$515$	34.2522	1.50933
$516$	0	0
$517$	−10.3232	−0.454013
$518$	−0.185757	−0.00816171
$519$	0	0
$520$	7.26772	0.318711
$521$	32.0742	1.40520	0.702599	−	0.711586i	$-0.252023\pi$
$521$	32.0742	1.40520	0.702599	+	0.711586i	$0.252023\pi$
$522$	0	0
$523$	33.8619	1.48068	0.740339	−	0.672234i	$-0.234665\pi$
$523$	33.8619	1.48068	0.740339	+	0.672234i	$0.234665\pi$
$524$	−13.3359	−0.582580
$525$	0	0
$526$	−10.9958	−0.479438
$527$	−38.4389	−1.67442
$528$	0	0
$529$	46.2752	2.01197
$530$	−10.6826	−0.464022
$531$	0	0
$532$	−0.0762054	−0.00330392
$533$	0.467424	0.0202464
$534$	0	0
$535$	−2.08788	−0.0902668
$536$	−4.26320	−0.184142
$537$	0	0
$538$	−2.69791	−0.116315
$539$	1.00000	0.0430730
$540$	0	0
$541$	14.2878	0.614279	0.307140	−	0.951664i	$-0.400628\pi$
$541$	14.2878	0.614279	0.307140	+	0.951664i	$0.400628\pi$
$542$	−3.55817	−0.152836
$543$	0	0
$544$	−24.0719	−1.03208
$545$	−18.9050	−0.809801
$546$	0	0
$547$	−15.6269	−0.668156	−0.334078	−	0.942545i	$-0.608425\pi$
$547$	−15.6269	−0.668156	−0.334078	+	0.942545i	$0.608425\pi$
$548$	23.9935	1.02495
$549$	0	0
$550$	4.51801	0.192649
$551$	0.155923	0.00664253
$552$	0	0
$553$	5.60681	0.238426
$554$	−3.73650	−0.158749
$555$	0	0
$556$	14.7913	0.627292
$557$	13.5404	0.573725	0.286862	−	0.957972i	$-0.407388\pi$
$557$	13.5404	0.573725	0.286862	+	0.957972i	$0.407388\pi$
$558$	0	0
$559$	−4.92090	−0.208132
$560$	8.73422	0.369088
$561$	0	0
$562$	5.95693	0.251278
$563$	−45.0110	−1.89699	−0.948494	−	0.316796i	$-0.897393\pi$
$563$	−45.0110	−1.89699	−0.948494	+	0.316796i	$0.897393\pi$
$564$	0	0
$565$	51.8787	2.18255
$566$	−0.999381	−0.0420071
$567$	0	0
$568$	−20.8960	−0.876776
$569$	−20.1319	−0.843975	−0.421987	−	0.906602i	$-0.638667\pi$
$569$	−20.1319	−0.843975	−0.421987	+	0.906602i	$0.638667\pi$
$570$	0	0
$571$	0.816833	0.0341834	0.0170917	−	0.999854i	$-0.494559\pi$
$571$	0.816833	0.0341834	0.0170917	+	0.999854i	$0.494559\pi$
$572$	1.71636	0.0717647
$573$	0	0
$574$	−0.248939	−0.0103905
$575$	−70.6081	−2.94456
$576$	0	0
$577$	−2.86837	−0.119412	−0.0597059	−	0.998216i	$-0.519016\pi$
$577$	−2.86837	−0.119412	−0.0597059	+	0.998216i	$0.519016\pi$
$578$	2.24891	0.0935424
$579$	0	0
$580$	−22.1330	−0.919022
$581$	9.99514	0.414668
$582$	0	0
$583$	−5.46256	−0.226236
$584$	−11.1383	−0.460908
$585$	0	0
$586$	−16.3510	−0.675455
$587$	16.7786	0.692528	0.346264	−	0.938137i	$-0.387450\pi$
$587$	16.7786	0.692528	0.346264	+	0.938137i	$0.387450\pi$
$588$	0	0
$589$	0.370465	0.0152647
$590$	7.36665	0.303280
$591$	0	0
$592$	−0.829641	−0.0340980
$593$	−21.9381	−0.900890	−0.450445	−	0.892804i	$-0.648735\pi$
$593$	−21.9381	−0.900890	−0.450445	+	0.892804i	$0.648735\pi$
$594$	0	0
$595$	−16.9160	−0.693490
$596$	−35.8389	−1.46802
$597$	0	0
$598$	4.43272	0.181268
$599$	13.4182	0.548251	0.274126	−	0.961694i	$-0.411612\pi$
$599$	13.4182	0.548251	0.274126	+	0.961694i	$0.411612\pi$
$600$	0	0
$601$	18.9189	0.771719	0.385860	−	0.922557i	$-0.373905\pi$
$601$	18.9189	0.771719	0.385860	+	0.922557i	$0.373905\pi$
$602$	2.62076	0.106814
$603$	0	0
$604$	−18.1050	−0.736682
$605$	3.67196	0.149287
$606$	0	0
$607$	−39.6262	−1.60838	−0.804189	−	0.594374i	$-0.797400\pi$
$607$	−39.6262	−1.60838	−0.804189	+	0.594374i	$0.797400\pi$
$608$	0.232000	0.00940883
$609$	0	0
$610$	23.1674	0.938019
$611$	10.3232	0.417631
$612$	0	0
$613$	6.36043	0.256896	0.128448	−	0.991716i	$-0.459001\pi$
$613$	6.36043	0.256896	0.128448	+	0.991716i	$0.459001\pi$
$614$	4.05863	0.163793
$615$	0	0
$616$	−1.97925	−0.0797461
$617$	15.5196	0.624797	0.312399	−	0.949951i	$-0.398868\pi$
$617$	15.5196	0.624797	0.312399	+	0.949951i	$0.398868\pi$
$618$	0	0
$619$	−5.39316	−0.216769	−0.108385	−	0.994109i	$-0.534568\pi$
$619$	−5.39316	−0.216769	−0.108385	+	0.994109i	$0.534568\pi$
$620$	−52.5869	−2.11194
$621$	0	0
$622$	7.78252	0.312051
$623$	11.8156	0.473381
$624$	0	0
$625$	4.55003	0.182001
$626$	4.06898	0.162629
$627$	0	0
$628$	−23.6613	−0.944189
$629$	1.60681	0.0640677
$630$	0	0
$631$	21.9916	0.875473	0.437736	−	0.899103i	$-0.355780\pi$
$631$	21.9916	0.875473	0.437736	+	0.899103i	$0.355780\pi$
$632$	−11.0973	−0.441425
$633$	0	0
$634$	16.4275	0.652419
$635$	46.2512	1.83542
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	1.87031	0.0740464
$639$	0	0
$640$	−42.2353	−1.66950
$641$	25.4321	1.00451	0.502254	−	0.864720i	$-0.332504\pi$
$641$	25.4321	1.00451	0.502254	+	0.864720i	$0.332504\pi$
$642$	0	0
$643$	22.2794	0.878613	0.439307	−	0.898337i	$-0.355224\pi$
$643$	22.2794	0.878613	0.439307	+	0.898337i	$0.355224\pi$
$644$	14.2856	0.562931
$645$	0	0
$646$	−0.108933	−0.00428591
$647$	−3.76430	−0.147990	−0.0739950	−	0.997259i	$-0.523575\pi$
$647$	−3.76430	−0.147990	−0.0739950	+	0.997259i	$0.523575\pi$
$648$	0	0
$649$	3.76695	0.147866
$650$	−4.51801	−0.177211
$651$	0	0
$652$	32.4262	1.26991
$653$	−33.9916	−1.33019	−0.665096	−	0.746757i	$-0.731610\pi$
$653$	−33.9916	−1.33019	−0.665096	+	0.746757i	$0.731610\pi$
$654$	0	0
$655$	28.5306	1.11478
$656$	−1.11183	−0.0434095
$657$	0	0
$658$	−5.49788	−0.214330
$659$	−14.0091	−0.545716	−0.272858	−	0.962054i	$-0.587969\pi$
$659$	−14.0091	−0.545716	−0.272858	+	0.962054i	$0.587969\pi$
$660$	0	0
$661$	−5.50139	−0.213979	−0.106990	−	0.994260i	$-0.534121\pi$
$661$	−5.50139	−0.213979	−0.106990	+	0.994260i	$0.534121\pi$
$662$	0.877525	0.0341060
$663$	0	0
$664$	−19.7828	−0.767723
$665$	0.163033	0.00632214
$666$	0	0
$667$	−29.2295	−1.13177
$668$	−11.7586	−0.454954
$669$	0	0
$670$	4.21227	0.162734
$671$	11.8467	0.457336
$672$	0	0
$673$	−27.8036	−1.07175	−0.535874	−	0.844298i	$-0.680018\pi$
$673$	−27.8036	−1.07175	−0.535874	+	0.844298i	$0.680018\pi$
$674$	−15.0069	−0.578045
$675$	0	0
$676$	−1.71636	−0.0660139
$677$	25.5864	0.983366	0.491683	−	0.870774i	$-0.336382\pi$
$677$	25.5864	0.983366	0.491683	+	0.870774i	$0.336382\pi$
$678$	0	0
$679$	5.74620	0.220519
$680$	33.4810	1.28394
$681$	0	0
$682$	4.44378	0.170161
$683$	44.6690	1.70921	0.854607	−	0.519276i	$-0.173798\pi$
$683$	44.6690	1.70921	0.854607	+	0.519276i	$0.173798\pi$
$684$	0	0
$685$	−51.3313	−1.96127
$686$	0.532576	0.0203339
$687$	0	0
$688$	11.7050	0.446248
$689$	5.46256	0.208107
$690$	0	0
$691$	37.2681	1.41775	0.708873	−	0.705337i	$-0.249204\pi$
$691$	37.2681	1.41775	0.708873	+	0.705337i	$0.249204\pi$
$692$	12.8797	0.489613
$693$	0	0
$694$	−13.3805	−0.507918
$695$	−31.6444	−1.20034
$696$	0	0
$697$	2.15333	0.0815633
$698$	−15.4708	−0.585579
$699$	0	0
$700$	−14.5604	−0.550333
$701$	−19.4911	−0.736168	−0.368084	−	0.929793i	$-0.619986\pi$
$701$	−19.4911	−0.736168	−0.368084	+	0.929793i	$0.619986\pi$
$702$	0	0
$703$	−0.0154861	−0.000584068 0
$704$	−1.97438	−0.0744123
$705$	0	0
$706$	−19.4327	−0.731360
$707$	−11.2088	−0.421549
$708$	0	0
$709$	−42.1568	−1.58323	−0.791616	−	0.611019i	$-0.790760\pi$
$709$	−42.1568	−1.58323	−0.791616	+	0.611019i	$0.790760\pi$
$710$	20.6463	0.774844
$711$	0	0
$712$	−23.3859	−0.876424
$713$	−69.4479	−2.60085
$714$	0	0
$715$	−3.67196	−0.137324
$716$	−16.7364	−0.625467
$717$	0	0
$718$	15.3099	0.571359
$719$	32.0185	1.19409	0.597045	−	0.802208i	$-0.296341\pi$
$719$	32.0185	1.19409	0.597045	+	0.802208i	$0.296341\pi$
$720$	0	0
$721$	9.32804	0.347394
$722$	−10.1179	−0.376549
$723$	0	0
$724$	15.2519	0.566834
$725$	29.7919	1.10644
$726$	0	0
$727$	1.57673	0.0584778	0.0292389	−	0.999572i	$-0.490692\pi$
$727$	1.57673	0.0584778	0.0292389	+	0.999572i	$0.490692\pi$
$728$	1.97925	0.0733558
$729$	0	0
$730$	11.0053	0.407323
$731$	−22.6697	−0.838468
$732$	0	0
$733$	−13.1465	−0.485577	−0.242788	−	0.970079i	$-0.578062\pi$
$733$	−13.1465	−0.485577	−0.242788	+	0.970079i	$0.578062\pi$
$734$	11.2639	0.415757
$735$	0	0
$736$	−43.4910	−1.60310
$737$	2.15395	0.0793418
$738$	0	0
$739$	32.2318	1.18566	0.592832	−	0.805326i	$-0.298010\pi$
$739$	32.2318	1.18566	0.592832	+	0.805326i	$0.298010\pi$
$740$	2.19822	0.0808083
$741$	0	0
$742$	−2.90923	−0.106801
$743$	−31.3730	−1.15096	−0.575482	−	0.817814i	$-0.695186\pi$
$743$	−31.3730	−1.15096	−0.575482	+	0.817814i	$0.695186\pi$
$744$	0	0
$745$	76.6732	2.80909
$746$	5.90596	0.216233
$747$	0	0
$748$	7.90696	0.289107
$749$	−0.568600	−0.0207762
$750$	0	0
$751$	12.0950	0.441352	0.220676	−	0.975347i	$-0.429174\pi$
$751$	12.0950	0.441352	0.220676	+	0.975347i	$0.429174\pi$
$752$	−24.5550	−0.895427
$753$	0	0
$754$	−1.87031	−0.0681128
$755$	38.7336	1.40966
$756$	0	0
$757$	−44.3383	−1.61150	−0.805752	−	0.592253i	$-0.798239\pi$
$757$	−44.3383	−1.61150	−0.805752	+	0.592253i	$0.798239\pi$
$758$	−7.26410	−0.263844
$759$	0	0
$760$	−0.322682	−0.0117049
$761$	11.0194	0.399454	0.199727	−	0.979852i	$-0.435994\pi$
$761$	11.0194	0.399454	0.199727	+	0.979852i	$0.435994\pi$
$762$	0	0
$763$	−5.14847	−0.186387
$764$	−12.7156	−0.460034
$765$	0	0
$766$	8.35322	0.301814
$767$	−3.76695	−0.136017
$768$	0	0
$769$	−45.8000	−1.65159	−0.825796	−	0.563970i	$-0.809274\pi$
$769$	−45.8000	−1.65159	−0.825796	+	0.563970i	$0.809274\pi$
$770$	1.95560	0.0704750
$771$	0	0
$772$	−38.6312	−1.39037
$773$	3.43429	0.123523	0.0617615	−	0.998091i	$-0.480328\pi$
$773$	3.43429	0.123523	0.0617615	+	0.998091i	$0.480328\pi$
$774$	0	0
$775$	70.7841	2.54264
$776$	−11.3731	−0.408272
$777$	0	0
$778$	20.4040	0.731519
$779$	−0.0207533	−0.000743565 0
$780$	0	0
$781$	10.5575	0.377779
$782$	20.4207	0.730244
$783$	0	0
$784$	2.37862	0.0849509
$785$	50.6207	1.80673
$786$	0	0
$787$	39.5358	1.40930	0.704650	−	0.709555i	$-0.251104\pi$
$787$	39.5358	1.40930	0.704650	+	0.709555i	$0.251104\pi$
$788$	37.8897	1.34976
$789$	0	0
$790$	10.9647	0.390106
$791$	14.1283	0.502346
$792$	0	0
$793$	−11.8467	−0.420688
$794$	1.26008	0.0447185
$795$	0	0
$796$	16.9066	0.599239
$797$	39.0972	1.38490	0.692448	−	0.721468i	$-0.256532\pi$
$797$	39.0972	1.38490	0.692448	+	0.721468i	$0.256532\pi$
$798$	0	0
$799$	47.5569	1.68244
$800$	44.3278	1.56722
$801$	0	0
$802$	16.6509	0.587964
$803$	5.62756	0.198592
$804$	0	0
$805$	−30.5624	−1.07718
$806$	−4.44378	−0.156525
$807$	0	0
$808$	22.1849	0.780462
$809$	−9.90353	−0.348190	−0.174095	−	0.984729i	$-0.555700\pi$
$809$	−9.90353	−0.348190	−0.174095	+	0.984729i	$0.555700\pi$
$810$	0	0
$811$	−40.9845	−1.43916	−0.719581	−	0.694408i	$-0.755666\pi$
$811$	−40.9845	−1.43916	−0.719581	+	0.694408i	$0.755666\pi$
$812$	−6.02756	−0.211526
$813$	0	0
$814$	−0.185757	−0.00651079
$815$	−69.3722	−2.43000
$816$	0	0
$817$	0.218485	0.00764382
$818$	−14.6159	−0.511033
$819$	0	0
$820$	2.94590	0.102875
$821$	10.3018	0.359535	0.179768	−	0.983709i	$-0.442465\pi$
$821$	10.3018	0.359535	0.179768	+	0.983709i	$0.442465\pi$
$822$	0	0
$823$	−38.5549	−1.34394	−0.671970	−	0.740578i	$-0.734552\pi$
$823$	−38.5549	−1.34394	−0.671970	+	0.740578i	$0.734552\pi$
$824$	−18.4625	−0.643171
$825$	0	0
$826$	2.00619	0.0698042
$827$	6.07008	0.211077	0.105539	−	0.994415i	$-0.466343\pi$
$827$	6.07008	0.211077	0.105539	+	0.994415i	$0.466343\pi$
$828$	0	0
$829$	−5.52741	−0.191975	−0.0959874	−	0.995383i	$-0.530601\pi$
$829$	−5.52741	−0.191975	−0.0959874	+	0.995383i	$0.530601\pi$
$830$	19.5465	0.678469
$831$	0	0
$832$	1.97438	0.0684494
$833$	−4.60681	−0.159617
$834$	0	0
$835$	25.1562	0.870566
$836$	−0.0762054	−0.00263562
$837$	0	0
$838$	−18.8230	−0.650230
$839$	7.21489	0.249086	0.124543	−	0.992214i	$-0.460254\pi$
$839$	7.21489	0.249086	0.124543	+	0.992214i	$0.460254\pi$
$840$	0	0
$841$	−16.6671	−0.574728
$842$	18.5442	0.639075
$843$	0	0
$844$	3.33982	0.114961
$845$	3.67196	0.126319
$846$	0	0
$847$	1.00000	0.0343604
$848$	−12.9934	−0.446195
$849$	0	0
$850$	−20.8136	−0.713901
$851$	2.90304	0.0995150
$852$	0	0
$853$	57.3320	1.96301	0.981505	−	0.191437i	$-0.0613148\pi$
$853$	57.3320	1.96301	0.981505	+	0.191437i	$0.0613148\pi$
$854$	6.30925	0.215898
$855$	0	0
$856$	1.12540	0.0384654
$857$	−36.0408	−1.23113	−0.615565	−	0.788086i	$-0.711072\pi$
$857$	−36.0408	−1.23113	−0.615565	+	0.788086i	$0.711072\pi$
$858$	0	0
$859$	−12.5118	−0.426897	−0.213448	−	0.976954i	$-0.568470\pi$
$859$	−12.5118	−0.426897	−0.213448	+	0.976954i	$0.568470\pi$
$860$	−31.0136	−1.05755
$861$	0	0
$862$	17.1370	0.583689
$863$	45.8363	1.56029	0.780143	−	0.625602i	$-0.215146\pi$
$863$	45.8363	1.56029	0.780143	+	0.625602i	$0.215146\pi$
$864$	0	0
$865$	−27.5547	−0.936886
$866$	9.65126	0.327963
$867$	0	0
$868$	−14.3212	−0.486093
$869$	5.60681	0.190198
$870$	0	0
$871$	−2.15395	−0.0729839
$872$	10.1901	0.345080
$873$	0	0
$874$	−0.196810	−0.00665720
$875$	12.7906	0.432401
$876$	0	0
$877$	31.3355	1.05812	0.529062	−	0.848583i	$-0.322544\pi$
$877$	31.3355	1.05812	0.529062	+	0.848583i	$0.322544\pi$
$878$	17.3449	0.585361
$879$	0	0
$880$	8.73422	0.294431
$881$	−19.9929	−0.673577	−0.336788	−	0.941580i	$-0.609341\pi$
$881$	−19.9929	−0.673577	−0.336788	+	0.941580i	$0.609341\pi$
$882$	0	0
$883$	−11.3536	−0.382080	−0.191040	−	0.981582i	$-0.561186\pi$
$883$	−11.3536	−0.382080	−0.191040	+	0.981582i	$0.561186\pi$
$884$	−7.90696	−0.265940
$885$	0	0
$886$	15.6065	0.524309
$887$	−9.18705	−0.308471	−0.154236	−	0.988034i	$-0.549291\pi$
$887$	−9.18705	−0.308471	−0.154236	+	0.988034i	$0.549291\pi$
$888$	0	0
$889$	12.5958	0.422448
$890$	23.1065	0.774533
$891$	0	0
$892$	20.5591	0.688370
$893$	−0.458342	−0.0153378
$894$	0	0
$895$	35.8055	1.19685
$896$	−11.5021	−0.384258
$897$	0	0
$898$	−2.03830	−0.0680189
$899$	29.3024	0.977289
$900$	0	0
$901$	25.1650	0.838367
$902$	−0.248939	−0.00828875
$903$	0	0
$904$	−27.9635	−0.930051
$905$	−32.6298	−1.08465
$906$	0	0
$907$	−26.0963	−0.866513	−0.433256	−	0.901271i	$-0.642636\pi$
$907$	−26.0963	−0.866513	−0.433256	+	0.901271i	$0.642636\pi$
$908$	−32.2882	−1.07152
$909$	0	0
$910$	−1.95560	−0.0648275
$911$	1.44378	0.0478345	0.0239172	−	0.999714i	$-0.492386\pi$
$911$	1.44378	0.0478345	0.0239172	+	0.999714i	$0.492386\pi$
$912$	0	0
$913$	9.99514	0.330791
$914$	−3.56025	−0.117763
$915$	0	0
$916$	−23.1839	−0.766018
$917$	7.76984	0.256583
$918$	0	0
$919$	−44.5014	−1.46796	−0.733982	−	0.679169i	$-0.762340\pi$
$919$	−44.5014	−1.46796	−0.733982	+	0.679169i	$0.762340\pi$
$920$	60.4905	1.99431
$921$	0	0
$922$	−11.7657	−0.387483
$923$	−10.5575	−0.347506
$924$	0	0
$925$	−2.95890	−0.0972879
$926$	10.0516	0.330317
$927$	0	0
$928$	18.3503	0.602378
$929$	10.4037	0.341334	0.170667	−	0.985329i	$-0.445408\pi$
$929$	10.4037	0.341334	0.170667	+	0.985329i	$0.445408\pi$
$930$	0	0
$931$	0.0443994	0.00145513
$932$	39.4490	1.29220
$933$	0	0
$934$	−7.13773	−0.233554
$935$	−16.9160	−0.553214
$936$	0	0
$937$	−49.1484	−1.60561	−0.802804	−	0.596243i	$-0.796660\pi$
$937$	−49.1484	−1.60561	−0.802804	+	0.596243i	$0.796660\pi$
$938$	1.14714	0.0374556
$939$	0	0
$940$	65.0610	2.12205
$941$	−19.5803	−0.638301	−0.319150	−	0.947704i	$-0.603397\pi$
$941$	−19.5803	−0.638301	−0.319150	+	0.947704i	$0.603397\pi$
$942$	0	0
$943$	3.89045	0.126690
$944$	8.96016	0.291628
$945$	0	0
$946$	2.62076	0.0852081
$947$	−58.0226	−1.88548	−0.942741	−	0.333525i	$-0.891762\pi$
$947$	−58.0226	−1.88548	−0.942741	+	0.333525i	$0.891762\pi$
$948$	0	0
$949$	−5.62756	−0.182678
$950$	0.200597	0.00650822
$951$	0	0
$952$	9.11801	0.295517
$953$	33.7475	1.09319	0.546594	−	0.837398i	$-0.315924\pi$
$953$	33.7475	1.09319	0.546594	+	0.837398i	$0.315924\pi$
$954$	0	0
$955$	27.2036	0.880288
$956$	7.43683	0.240524
$957$	0	0
$958$	−16.5530	−0.534804
$959$	−13.9792	−0.451413
$960$	0	0
$961$	38.6211	1.24584
$962$	0.185757	0.00598906
$963$	0	0
$964$	−30.0327	−0.967288
$965$	82.6470	2.66050
$966$	0	0
$967$	6.71606	0.215974	0.107987	−	0.994152i	$-0.465560\pi$
$967$	6.71606	0.215974	0.107987	+	0.994152i	$0.465560\pi$
$968$	−1.97925	−0.0636154
$969$	0	0
$970$	11.2373	0.360807
$971$	46.7608	1.50062	0.750312	−	0.661084i	$-0.229903\pi$
$971$	46.7608	1.50062	0.750312	+	0.661084i	$0.229903\pi$
$972$	0	0
$973$	−8.61784	−0.276275
$974$	−7.93701	−0.254318
$975$	0	0
$976$	28.1788	0.901981
$977$	−14.9998	−0.479885	−0.239942	−	0.970787i	$-0.577129\pi$
$977$	−14.9998	−0.479885	−0.239942	+	0.970787i	$0.577129\pi$
$978$	0	0
$979$	11.8156	0.377627
$980$	−6.30242	−0.201323
$981$	0	0
$982$	17.7511	0.566461
$983$	2.74715	0.0876204	0.0438102	−	0.999040i	$-0.486050\pi$
$983$	2.74715	0.0876204	0.0438102	+	0.999040i	$0.486050\pi$
$984$	0	0
$985$	−81.0606	−2.58281
$986$	−8.61618	−0.274395
$987$	0	0
$988$	0.0762054	0.00242442
$989$	−40.9575	−1.30237
$990$	0	0
$991$	−49.9682	−1.58729	−0.793646	−	0.608380i	$-0.791820\pi$
$991$	−49.9682	−1.58729	−0.793646	+	0.608380i	$0.791820\pi$
$992$	43.5995	1.38428
$993$	0	0
$994$	5.62270	0.178341
$995$	−36.1698	−1.14666
$996$	0	0
$997$	6.81011	0.215678	0.107839	−	0.994168i	$-0.465607\pi$
$997$	6.81011	0.215678	0.107839	+	0.994168i	$0.465607\pi$
$998$	20.6527	0.653751
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9009.2.a.w.1.2		4
3.2	odd	2		1001.2.a.g.1.3	✓	4
21.20	even	2		7007.2.a.j.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1001.2.a.g.1.3	✓	4	3.2	odd	2
7007.2.a.j.1.3		4	21.20	even	2
9009.2.a.w.1.2		4	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:	\( N \)	\(=\)	\( 9009 = 3^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9009.a (trivial)

\(f(q)\)	\(=\)	\(q+0.532576 q^{2} -1.71636 q^{4} +3.67196 q^{5} +1.00000 q^{7} -1.97925 q^{8} +O(q^{10})\)	\(q+0.532576 q^{2} -1.71636 q^{4} +3.67196 q^{5} +1.00000 q^{7} -1.97925 q^{8} +1.95560 q^{10} +1.00000 q^{11} -1.00000 q^{13} +0.532576 q^{14} +2.37862 q^{16} -4.60681 q^{17} +0.0443994 q^{19} -6.30242 q^{20} +0.532576 q^{22} -8.32317 q^{23} +8.48331 q^{25} -0.532576 q^{26} -1.71636 q^{28} +3.51182 q^{29} +8.34393 q^{31} +5.22529 q^{32} -2.45348 q^{34} +3.67196 q^{35} -0.348790 q^{37} +0.0236460 q^{38} -7.26772 q^{40} -0.467424 q^{41} +4.92090 q^{43} -1.71636 q^{44} -4.43272 q^{46} -10.3232 q^{47} +1.00000 q^{49} +4.51801 q^{50} +1.71636 q^{52} -5.46256 q^{53} +3.67196 q^{55} -1.97925 q^{56} +1.87031 q^{58} +3.76695 q^{59} +11.8467 q^{61} +4.44378 q^{62} -1.97438 q^{64} -3.67196 q^{65} +2.15395 q^{67} +7.90696 q^{68} +1.95560 q^{70} +10.5575 q^{71} +5.62756 q^{73} -0.185757 q^{74} -0.0762054 q^{76} +1.00000 q^{77} +5.60681 q^{79} +8.73422 q^{80} -0.248939 q^{82} +9.99514 q^{83} -16.9160 q^{85} +2.62076 q^{86} -1.97925 q^{88} +11.8156 q^{89} -1.00000 q^{91} +14.2856 q^{92} -5.49788 q^{94} +0.163033 q^{95} +5.74620 q^{97} +0.532576 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 8 q^{4} + 5 q^{5} + 4 q^{7} - 3 q^{8}+O(q^{10}) \) 4 * q + 2 * q^2 + 8 * q^4 + 5 * q^5 + 4 * q^7 - 3 * q^8	\( 4 q + 2 q^{2} + 8 q^{4} + 5 q^{5} + 4 q^{7} - 3 q^{8} + 13 q^{10} + 4 q^{11} - 4 q^{13} + 2 q^{14} + 20 q^{16} - 9 q^{17} - 5 q^{19} + 4 q^{20} + 2 q^{22} - 9 q^{23} + 5 q^{25} - 2 q^{26} + 8 q^{28} + 9 q^{29} + 14 q^{31} + 16 q^{32} + 15 q^{34} + 5 q^{35} - 16 q^{37} - 10 q^{38} + 30 q^{40} - 2 q^{41} - 5 q^{43} + 8 q^{44} + 12 q^{46} - 17 q^{47} + 4 q^{49} + 19 q^{50} - 8 q^{52} + 12 q^{53} + 5 q^{55} - 3 q^{56} - 18 q^{58} + q^{59} + 32 q^{61} + 28 q^{62} + 31 q^{64} - 5 q^{65} - 2 q^{67} - 18 q^{68} + 13 q^{70} + 4 q^{71} + 18 q^{73} - 35 q^{74} - 40 q^{76} + 4 q^{77} + 13 q^{79} + 40 q^{80} + 14 q^{82} + 6 q^{83} - 9 q^{85} + 26 q^{86} - 3 q^{88} - 15 q^{89} - 4 q^{91} + 18 q^{92} + 8 q^{94} - 19 q^{95} + 4 q^{97} + 2 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 8 * q^4 + 5 * q^5 + 4 * q^7 - 3 * q^8 + 13 * q^10 + 4 * q^11 - 4 * q^13 + 2 * q^14 + 20 * q^16 - 9 * q^17 - 5 * q^19 + 4 * q^20 + 2 * q^22 - 9 * q^23 + 5 * q^25 - 2 * q^26 + 8 * q^28 + 9 * q^29 + 14 * q^31 + 16 * q^32 + 15 * q^34 + 5 * q^35 - 16 * q^37 - 10 * q^38 + 30 * q^40 - 2 * q^41 - 5 * q^43 + 8 * q^44 + 12 * q^46 - 17 * q^47 + 4 * q^49 + 19 * q^50 - 8 * q^52 + 12 * q^53 + 5 * q^55 - 3 * q^56 - 18 * q^58 + q^59 + 32 * q^61 + 28 * q^62 + 31 * q^64 - 5 * q^65 - 2 * q^67 - 18 * q^68 + 13 * q^70 + 4 * q^71 + 18 * q^73 - 35 * q^74 - 40 * q^76 + 4 * q^77 + 13 * q^79 + 40 * q^80 + 14 * q^82 + 6 * q^83 - 9 * q^85 + 26 * q^86 - 3 * q^88 - 15 * q^89 - 4 * q^91 + 18 * q^92 + 8 * q^94 - 19 * q^95 + 4 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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