LMFDB - Embedded newform 8008.2.a.p.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8008,2,Mod(1,8008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.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:	$63.9442019386$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 15x^{7} + 15x^{6} + 66x^{5} - 59x^{4} - 77x^{3} + 34x^{2} + 11x - 2 $ x^9 - x^8 - 15x^7 + 15x^6 + 66x^5 - 59x^4 - 77x^3 + 34x^2 + 11*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.536705$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.536705 q^{3} -3.30011 q^{5} -1.00000 q^{7} -2.71195 q^{9} +O(q^{10})$	$q+0.536705 q^{3} -3.30011 q^{5} -1.00000 q^{7} -2.71195 q^{9} +1.00000 q^{11} +1.00000 q^{13} -1.77118 q^{15} +3.13193 q^{17} -3.06031 q^{19} -0.536705 q^{21} +0.420528 q^{23} +5.89071 q^{25} -3.06563 q^{27} +0.161852 q^{29} +10.3651 q^{31} +0.536705 q^{33} +3.30011 q^{35} +4.54945 q^{37} +0.536705 q^{39} -9.30747 q^{41} +2.24787 q^{43} +8.94972 q^{45} +2.05740 q^{47} +1.00000 q^{49} +1.68092 q^{51} +8.66441 q^{53} -3.30011 q^{55} -1.64248 q^{57} +4.24396 q^{59} -6.87493 q^{61} +2.71195 q^{63} -3.30011 q^{65} +8.01827 q^{67} +0.225699 q^{69} -16.2909 q^{71} +4.62059 q^{73} +3.16157 q^{75} -1.00000 q^{77} -6.36899 q^{79} +6.49051 q^{81} +3.20265 q^{83} -10.3357 q^{85} +0.0868669 q^{87} +5.49762 q^{89} -1.00000 q^{91} +5.56299 q^{93} +10.0993 q^{95} +10.8560 q^{97} -2.71195 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + q^{3} - 4 q^{5} - 9 q^{7} + 4 q^{9}+O(q^{10}) $ 9 * q + q^3 - 4 * q^5 - 9 * q^7 + 4 * q^9	$ 9 q + q^{3} - 4 q^{5} - 9 q^{7} + 4 q^{9} + 9 q^{11} + 9 q^{13} - 9 q^{15} - 11 q^{17} + 10 q^{19} - q^{21} - 14 q^{23} - q^{25} - 5 q^{27} - 10 q^{29} + 5 q^{31} + q^{33} + 4 q^{35} - 16 q^{37} + q^{39} + 2 q^{41} + 4 q^{43} - 30 q^{45} + 9 q^{49} + 3 q^{51} - 23 q^{53} - 4 q^{55} + 14 q^{57} + 9 q^{59} - 14 q^{61} - 4 q^{63} - 4 q^{65} + 8 q^{67} - 26 q^{69} - 20 q^{71} - 23 q^{73} + 32 q^{75} - 9 q^{77} + 2 q^{79} - 11 q^{81} - 9 q^{83} - 3 q^{85} - 7 q^{87} - 6 q^{89} - 9 q^{91} - 19 q^{93} - 4 q^{95} - 3 q^{97} + 4 q^{99}+O(q^{100}) $ 9 * q + q^3 - 4 * q^5 - 9 * q^7 + 4 * q^9 + 9 * q^11 + 9 * q^13 - 9 * q^15 - 11 * q^17 + 10 * q^19 - q^21 - 14 * q^23 - q^25 - 5 * q^27 - 10 * q^29 + 5 * q^31 + q^33 + 4 * q^35 - 16 * q^37 + q^39 + 2 * q^41 + 4 * q^43 - 30 * q^45 + 9 * q^49 + 3 * q^51 - 23 * q^53 - 4 * q^55 + 14 * q^57 + 9 * q^59 - 14 * q^61 - 4 * q^63 - 4 * q^65 + 8 * q^67 - 26 * q^69 - 20 * q^71 - 23 * q^73 + 32 * q^75 - 9 * q^77 + 2 * q^79 - 11 * q^81 - 9 * q^83 - 3 * q^85 - 7 * q^87 - 6 * q^89 - 9 * q^91 - 19 * q^93 - 4 * q^95 - 3 * q^97 + 4 * q^99

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	0
$2$	0	0
$3$	0.536705	0.309867	0.154933	−	0.987925i	$-0.450484\pi$
$3$	0.536705	0.309867	0.154933	+	0.987925i	$0.450484\pi$
$4$	0	0
$5$	−3.30011	−1.47585	−0.737926	−	0.674881i	$-0.764195\pi$
$5$	−3.30011	−1.47585	−0.737926	+	0.674881i	$0.764195\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.71195	−0.903983
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−1.77118	−0.457317
$16$	0	0
$17$	3.13193	0.759605	0.379803	−	0.925068i	$-0.375992\pi$
$17$	3.13193	0.759605	0.379803	+	0.925068i	$0.375992\pi$
$18$	0	0
$19$	−3.06031	−0.702082	−0.351041	−	0.936360i	$-0.614172\pi$
$19$	−3.06031	−0.702082	−0.351041	+	0.936360i	$0.614172\pi$
$20$	0	0
$21$	−0.536705	−0.117119
$22$	0	0
$23$	0.420528	0.0876862	0.0438431	−	0.999038i	$-0.486040\pi$
$23$	0.420528	0.0876862	0.0438431	+	0.999038i	$0.486040\pi$
$24$	0	0
$25$	5.89071	1.17814
$26$	0	0
$27$	−3.06563	−0.589981
$28$	0	0
$29$	0.161852	0.0300552	0.0150276	−	0.999887i	$-0.495216\pi$
$29$	0.161852	0.0300552	0.0150276	+	0.999887i	$0.495216\pi$
$30$	0	0
$31$	10.3651	1.86162	0.930812	−	0.365498i	$-0.119101\pi$
$31$	10.3651	1.86162	0.930812	+	0.365498i	$0.119101\pi$
$32$	0	0
$33$	0.536705	0.0934283
$34$	0	0
$35$	3.30011	0.557820
$36$	0	0
$37$	4.54945	0.747924	0.373962	−	0.927444i	$-0.377999\pi$
$37$	4.54945	0.747924	0.373962	+	0.927444i	$0.377999\pi$
$38$	0	0
$39$	0.536705	0.0859415
$40$	0	0
$41$	−9.30747	−1.45358	−0.726792	−	0.686858i	$-0.758989\pi$
$41$	−9.30747	−1.45358	−0.726792	+	0.686858i	$0.758989\pi$
$42$	0	0
$43$	2.24787	0.342797	0.171398	−	0.985202i	$-0.445171\pi$
$43$	2.24787	0.342797	0.171398	+	0.985202i	$0.445171\pi$
$44$	0	0
$45$	8.94972	1.33415
$46$	0	0
$47$	2.05740	0.300102	0.150051	−	0.988678i	$-0.452056\pi$
$47$	2.05740	0.300102	0.150051	+	0.988678i	$0.452056\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.68092	0.235376
$52$	0	0
$53$	8.66441	1.19015	0.595074	−	0.803671i	$-0.297123\pi$
$53$	8.66441	1.19015	0.595074	+	0.803671i	$0.297123\pi$
$54$	0	0
$55$	−3.30011	−0.444986
$56$	0	0
$57$	−1.64248	−0.217552
$58$	0	0
$59$	4.24396	0.552517	0.276258	−	0.961083i	$-0.410905\pi$
$59$	4.24396	0.552517	0.276258	+	0.961083i	$0.410905\pi$
$60$	0	0
$61$	−6.87493	−0.880245	−0.440122	−	0.897938i	$-0.645065\pi$
$61$	−6.87493	−0.880245	−0.440122	+	0.897938i	$0.645065\pi$
$62$	0	0
$63$	2.71195	0.341673
$64$	0	0
$65$	−3.30011	−0.409328
$66$	0	0
$67$	8.01827	0.979588	0.489794	−	0.871838i	$-0.337072\pi$
$67$	8.01827	0.979588	0.489794	+	0.871838i	$0.337072\pi$
$68$	0	0
$69$	0.225699	0.0271710
$70$	0	0
$71$	−16.2909	−1.93337	−0.966686	−	0.255967i	$-0.917606\pi$
$71$	−16.2909	−1.93337	−0.966686	+	0.255967i	$0.917606\pi$
$72$	0	0
$73$	4.62059	0.540799	0.270399	−	0.962748i	$-0.412844\pi$
$73$	4.62059	0.540799	0.270399	+	0.962748i	$0.412844\pi$
$74$	0	0
$75$	3.16157	0.365067
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−6.36899	−0.716567	−0.358284	−	0.933613i	$-0.616638\pi$
$79$	−6.36899	−0.716567	−0.358284	+	0.933613i	$0.616638\pi$
$80$	0	0
$81$	6.49051	0.721167
$82$	0	0
$83$	3.20265	0.351536	0.175768	−	0.984432i	$-0.443759\pi$
$83$	3.20265	0.351536	0.175768	+	0.984432i	$0.443759\pi$
$84$	0	0
$85$	−10.3357	−1.12107
$86$	0	0
$87$	0.0868669	0.00931311
$88$	0	0
$89$	5.49762	0.582747	0.291373	−	0.956609i	$-0.405888\pi$
$89$	5.49762	0.582747	0.291373	+	0.956609i	$0.405888\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	5.56299	0.576855
$94$	0	0
$95$	10.0993	1.03617
$96$	0	0
$97$	10.8560	1.10226	0.551128	−	0.834421i	$-0.314197\pi$
$97$	10.8560	1.10226	0.551128	+	0.834421i	$0.314197\pi$
$98$	0	0
$99$	−2.71195	−0.272561
$100$	0	0
$101$	1.93119	0.192161	0.0960805	−	0.995374i	$-0.469369\pi$
$101$	1.93119	0.192161	0.0960805	+	0.995374i	$0.469369\pi$
$102$	0	0
$103$	−10.6122	−1.04565	−0.522826	−	0.852439i	$-0.675122\pi$
$103$	−10.6122	−1.04565	−0.522826	+	0.852439i	$0.675122\pi$
$104$	0	0
$105$	1.77118	0.172850
$106$	0	0
$107$	−13.0451	−1.26112	−0.630560	−	0.776141i	$-0.717175\pi$
$107$	−13.0451	−1.26112	−0.630560	+	0.776141i	$0.717175\pi$
$108$	0	0
$109$	−19.3617	−1.85452	−0.927258	−	0.374423i	$-0.877841\pi$
$109$	−19.3617	−1.85452	−0.927258	+	0.374423i	$0.877841\pi$
$110$	0	0
$111$	2.44171	0.231757
$112$	0	0
$113$	−12.5919	−1.18454	−0.592271	−	0.805739i	$-0.701769\pi$
$113$	−12.5919	−1.18454	−0.592271	+	0.805739i	$0.701769\pi$
$114$	0	0
$115$	−1.38779	−0.129412
$116$	0	0
$117$	−2.71195	−0.250720
$118$	0	0
$119$	−3.13193	−0.287104
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−4.99536	−0.450417
$124$	0	0
$125$	−2.93943	−0.262911
$126$	0	0
$127$	8.74259	0.775779	0.387890	−	0.921706i	$-0.373204\pi$
$127$	8.74259	0.775779	0.387890	+	0.921706i	$0.373204\pi$
$128$	0	0
$129$	1.20644	0.106221
$130$	0	0
$131$	−14.9292	−1.30437	−0.652184	−	0.758061i	$-0.726147\pi$
$131$	−14.9292	−1.30437	−0.652184	+	0.758061i	$0.726147\pi$
$132$	0	0
$133$	3.06031	0.265362
$134$	0	0
$135$	10.1169	0.870725
$136$	0	0
$137$	5.27810	0.450939	0.225469	−	0.974250i	$-0.427608\pi$
$137$	5.27810	0.450939	0.225469	+	0.974250i	$0.427608\pi$
$138$	0	0
$139$	−2.44425	−0.207318	−0.103659	−	0.994613i	$-0.533055\pi$
$139$	−2.44425	−0.207318	−0.103659	+	0.994613i	$0.533055\pi$
$140$	0	0
$141$	1.10421	0.0929917
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−0.534130	−0.0443571
$146$	0	0
$147$	0.536705	0.0442667
$148$	0	0
$149$	2.12900	0.174415	0.0872073	−	0.996190i	$-0.472206\pi$
$149$	2.12900	0.174415	0.0872073	+	0.996190i	$0.472206\pi$
$150$	0	0
$151$	−11.4271	−0.929921	−0.464961	−	0.885331i	$-0.653931\pi$
$151$	−11.4271	−0.929921	−0.464961	+	0.885331i	$0.653931\pi$
$152$	0	0
$153$	−8.49364	−0.686670
$154$	0	0
$155$	−34.2059	−2.74748
$156$	0	0
$157$	−3.35410	−0.267686	−0.133843	−	0.991003i	$-0.542732\pi$
$157$	−3.35410	−0.267686	−0.133843	+	0.991003i	$0.542732\pi$
$158$	0	0
$159$	4.65023	0.368787
$160$	0	0
$161$	−0.420528	−0.0331423
$162$	0	0
$163$	9.25050	0.724555	0.362278	−	0.932070i	$-0.381999\pi$
$163$	9.25050	0.724555	0.362278	+	0.932070i	$0.381999\pi$
$164$	0	0
$165$	−1.77118	−0.137886
$166$	0	0
$167$	1.03829	0.0803450	0.0401725	−	0.999193i	$-0.487209\pi$
$167$	1.03829	0.0803450	0.0401725	+	0.999193i	$0.487209\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	8.29939	0.634670
$172$	0	0
$173$	16.5134	1.25549	0.627744	−	0.778420i	$-0.283979\pi$
$173$	16.5134	1.25549	0.627744	+	0.778420i	$0.283979\pi$
$174$	0	0
$175$	−5.89071	−0.445296
$176$	0	0
$177$	2.27775	0.171206
$178$	0	0
$179$	−15.9904	−1.19518	−0.597589	−	0.801802i	$-0.703875\pi$
$179$	−15.9904	−1.19518	−0.597589	+	0.801802i	$0.703875\pi$
$180$	0	0
$181$	−1.71442	−0.127432	−0.0637159	−	0.997968i	$-0.520295\pi$
$181$	−1.71442	−0.127432	−0.0637159	+	0.997968i	$0.520295\pi$
$182$	0	0
$183$	−3.68981	−0.272758
$184$	0	0
$185$	−15.0137	−1.10383
$186$	0	0
$187$	3.13193	0.229030
$188$	0	0
$189$	3.06563	0.222992
$190$	0	0
$191$	−21.4362	−1.55107	−0.775536	−	0.631304i	$-0.782520\pi$
$191$	−21.4362	−1.55107	−0.775536	+	0.631304i	$0.782520\pi$
$192$	0	0
$193$	−5.23599	−0.376895	−0.188447	−	0.982083i	$-0.560346\pi$
$193$	−5.23599	−0.376895	−0.188447	+	0.982083i	$0.560346\pi$
$194$	0	0
$195$	−1.77118	−0.126837
$196$	0	0
$197$	3.58219	0.255221	0.127610	−	0.991824i	$-0.459269\pi$
$197$	3.58219	0.255221	0.127610	+	0.991824i	$0.459269\pi$
$198$	0	0
$199$	−11.2294	−0.796029	−0.398015	−	0.917379i	$-0.630301\pi$
$199$	−11.2294	−0.796029	−0.398015	+	0.917379i	$0.630301\pi$
$200$	0	0
$201$	4.30345	0.303542
$202$	0	0
$203$	−0.161852	−0.0113598
$204$	0	0
$205$	30.7157	2.14528
$206$	0	0
$207$	−1.14045	−0.0792668
$208$	0	0
$209$	−3.06031	−0.211686
$210$	0	0
$211$	16.8551	1.16035	0.580175	−	0.814492i	$-0.302984\pi$
$211$	16.8551	1.16035	0.580175	+	0.814492i	$0.302984\pi$
$212$	0	0
$213$	−8.74339	−0.599087
$214$	0	0
$215$	−7.41821	−0.505918
$216$	0	0
$217$	−10.3651	−0.703628
$218$	0	0
$219$	2.47989	0.167575
$220$	0	0
$221$	3.13193	0.210677
$222$	0	0
$223$	−11.7717	−0.788291	−0.394146	−	0.919048i	$-0.628959\pi$
$223$	−11.7717	−0.788291	−0.394146	+	0.919048i	$0.628959\pi$
$224$	0	0
$225$	−15.9753	−1.06502
$226$	0	0
$227$	7.66662	0.508851	0.254426	−	0.967092i	$-0.418114\pi$
$227$	7.66662	0.508851	0.254426	+	0.967092i	$0.418114\pi$
$228$	0	0
$229$	3.20926	0.212074	0.106037	−	0.994362i	$-0.466184\pi$
$229$	3.20926	0.212074	0.106037	+	0.994362i	$0.466184\pi$
$230$	0	0
$231$	−0.536705	−0.0353126
$232$	0	0
$233$	4.51393	0.295717	0.147859	−	0.989008i	$-0.452762\pi$
$233$	4.51393	0.295717	0.147859	+	0.989008i	$0.452762\pi$
$234$	0	0
$235$	−6.78963	−0.442907
$236$	0	0
$237$	−3.41827	−0.222040
$238$	0	0
$239$	−2.95772	−0.191319	−0.0956595	−	0.995414i	$-0.530496\pi$
$239$	−2.95772	−0.191319	−0.0956595	+	0.995414i	$0.530496\pi$
$240$	0	0
$241$	0.904795	0.0582830	0.0291415	−	0.999575i	$-0.490723\pi$
$241$	0.904795	0.0582830	0.0291415	+	0.999575i	$0.490723\pi$
$242$	0	0
$243$	12.6804	0.813446
$244$	0	0
$245$	−3.30011	−0.210836
$246$	0	0
$247$	−3.06031	−0.194723
$248$	0	0
$249$	1.71887	0.108929
$250$	0	0
$251$	−8.70496	−0.549452	−0.274726	−	0.961522i	$-0.588587\pi$
$251$	−8.70496	−0.549452	−0.274726	+	0.961522i	$0.588587\pi$
$252$	0	0
$253$	0.420528	0.0264384
$254$	0	0
$255$	−5.54723	−0.347381
$256$	0	0
$257$	−15.4913	−0.966324	−0.483162	−	0.875531i	$-0.660512\pi$
$257$	−15.4913	−0.966324	−0.483162	+	0.875531i	$0.660512\pi$
$258$	0	0
$259$	−4.54945	−0.282689
$260$	0	0
$261$	−0.438935	−0.0271694
$262$	0	0
$263$	−14.0389	−0.865674	−0.432837	−	0.901472i	$-0.642487\pi$
$263$	−14.0389	−0.865674	−0.432837	+	0.901472i	$0.642487\pi$
$264$	0	0
$265$	−28.5935	−1.75648
$266$	0	0
$267$	2.95060	0.180574
$268$	0	0
$269$	−10.8948	−0.664265	−0.332133	−	0.943233i	$-0.607768\pi$
$269$	−10.8948	−0.664265	−0.332133	+	0.943233i	$0.607768\pi$
$270$	0	0
$271$	−13.7704	−0.836491	−0.418246	−	0.908334i	$-0.637355\pi$
$271$	−13.7704	−0.836491	−0.418246	+	0.908334i	$0.637355\pi$
$272$	0	0
$273$	−0.536705	−0.0324828
$274$	0	0
$275$	5.89071	0.355223
$276$	0	0
$277$	−26.8078	−1.61072	−0.805361	−	0.592785i	$-0.798029\pi$
$277$	−26.8078	−1.61072	−0.805361	+	0.592785i	$0.798029\pi$
$278$	0	0
$279$	−28.1096	−1.68288
$280$	0	0
$281$	−4.68324	−0.279379	−0.139689	−	0.990195i	$-0.544610\pi$
$281$	−4.68324	−0.279379	−0.139689	+	0.990195i	$0.544610\pi$
$282$	0	0
$283$	17.5920	1.04573	0.522867	−	0.852415i	$-0.324863\pi$
$283$	17.5920	1.04573	0.522867	+	0.852415i	$0.324863\pi$
$284$	0	0
$285$	5.42036	0.321075
$286$	0	0
$287$	9.30747	0.549403
$288$	0	0
$289$	−7.19099	−0.423000
$290$	0	0
$291$	5.82645	0.341552
$292$	0	0
$293$	−12.2243	−0.714151	−0.357076	−	0.934075i	$-0.616226\pi$
$293$	−12.2243	−0.714151	−0.357076	+	0.934075i	$0.616226\pi$
$294$	0	0
$295$	−14.0055	−0.815433
$296$	0	0
$297$	−3.06563	−0.177886
$298$	0	0
$299$	0.420528	0.0243198
$300$	0	0
$301$	−2.24787	−0.129565
$302$	0	0
$303$	1.03648	0.0595443
$304$	0	0
$305$	22.6880	1.29911
$306$	0	0
$307$	11.8725	0.677602	0.338801	−	0.940858i	$-0.389979\pi$
$307$	11.8725	0.677602	0.338801	+	0.940858i	$0.389979\pi$
$308$	0	0
$309$	−5.69562	−0.324013
$310$	0	0
$311$	15.2113	0.862552	0.431276	−	0.902220i	$-0.358064\pi$
$311$	15.2113	0.862552	0.431276	+	0.902220i	$0.358064\pi$
$312$	0	0
$313$	5.34839	0.302309	0.151155	−	0.988510i	$-0.451701\pi$
$313$	5.34839	0.302309	0.151155	+	0.988510i	$0.451701\pi$
$314$	0	0
$315$	−8.94972	−0.504260
$316$	0	0
$317$	−1.40846	−0.0791071	−0.0395535	−	0.999217i	$-0.512594\pi$
$317$	−1.40846	−0.0791071	−0.0395535	+	0.999217i	$0.512594\pi$
$318$	0	0
$319$	0.161852	0.00906199
$320$	0	0
$321$	−7.00138	−0.390779
$322$	0	0
$323$	−9.58467	−0.533306
$324$	0	0
$325$	5.89071	0.326758
$326$	0	0
$327$	−10.3915	−0.574653
$328$	0	0
$329$	−2.05740	−0.113428
$330$	0	0
$331$	19.2286	1.05690	0.528451	−	0.848964i	$-0.322773\pi$
$331$	19.2286	1.05690	0.528451	+	0.848964i	$0.322773\pi$
$332$	0	0
$333$	−12.3379	−0.676111
$334$	0	0
$335$	−26.4612	−1.44573
$336$	0	0
$337$	−1.70454	−0.0928523	−0.0464262	−	0.998922i	$-0.514783\pi$
$337$	−1.70454	−0.0928523	−0.0464262	+	0.998922i	$0.514783\pi$
$338$	0	0
$339$	−6.75811	−0.367050
$340$	0	0
$341$	10.3651	0.561301
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−0.744832	−0.0401004
$346$	0	0
$347$	1.17222	0.0629282	0.0314641	−	0.999505i	$-0.489983\pi$
$347$	1.17222	0.0629282	0.0314641	+	0.999505i	$0.489983\pi$
$348$	0	0
$349$	−4.55082	−0.243600	−0.121800	−	0.992555i	$-0.538867\pi$
$349$	−4.55082	−0.243600	−0.121800	+	0.992555i	$0.538867\pi$
$350$	0	0
$351$	−3.06563	−0.163631
$352$	0	0
$353$	−21.2153	−1.12918	−0.564589	−	0.825372i	$-0.690965\pi$
$353$	−21.2153	−1.12918	−0.564589	+	0.825372i	$0.690965\pi$
$354$	0	0
$355$	53.7616	2.85337
$356$	0	0
$357$	−1.68092	−0.0889639
$358$	0	0
$359$	−0.225905	−0.0119228	−0.00596140	−	0.999982i	$-0.501898\pi$
$359$	−0.225905	−0.0119228	−0.00596140	+	0.999982i	$0.501898\pi$
$360$	0	0
$361$	−9.63453	−0.507080
$362$	0	0
$363$	0.536705	0.0281697
$364$	0	0
$365$	−15.2484	−0.798140
$366$	0	0
$367$	−10.6102	−0.553845	−0.276923	−	0.960892i	$-0.589315\pi$
$367$	−10.6102	−0.553845	−0.276923	+	0.960892i	$0.589315\pi$
$368$	0	0
$369$	25.2414	1.31401
$370$	0	0
$371$	−8.66441	−0.449834
$372$	0	0
$373$	−30.4566	−1.57698	−0.788492	−	0.615046i	$-0.789138\pi$
$373$	−30.4566	−1.57698	−0.788492	+	0.615046i	$0.789138\pi$
$374$	0	0
$375$	−1.57761	−0.0814673
$376$	0	0
$377$	0.161852	0.00833582
$378$	0	0
$379$	0.665050	0.0341613	0.0170807	−	0.999854i	$-0.494563\pi$
$379$	0.665050	0.0341613	0.0170807	+	0.999854i	$0.494563\pi$
$380$	0	0
$381$	4.69219	0.240388
$382$	0	0
$383$	4.68444	0.239364	0.119682	−	0.992812i	$-0.461813\pi$
$383$	4.68444	0.239364	0.119682	+	0.992812i	$0.461813\pi$
$384$	0	0
$385$	3.30011	0.168189
$386$	0	0
$387$	−6.09611	−0.309882
$388$	0	0
$389$	−18.8620	−0.956342	−0.478171	−	0.878267i	$-0.658700\pi$
$389$	−18.8620	−0.956342	−0.478171	+	0.878267i	$0.658700\pi$
$390$	0	0
$391$	1.31707	0.0666069
$392$	0	0
$393$	−8.01256	−0.404180
$394$	0	0
$395$	21.0183	1.05755
$396$	0	0
$397$	−38.5673	−1.93564	−0.967818	−	0.251653i	$-0.919026\pi$
$397$	−38.5673	−1.93564	−0.967818	+	0.251653i	$0.919026\pi$
$398$	0	0
$399$	1.64248	0.0822269
$400$	0	0
$401$	29.2031	1.45833	0.729165	−	0.684337i	$-0.239908\pi$
$401$	29.2031	1.45833	0.729165	+	0.684337i	$0.239908\pi$
$402$	0	0
$403$	10.3651	0.516322
$404$	0	0
$405$	−21.4194	−1.06434
$406$	0	0
$407$	4.54945	0.225508
$408$	0	0
$409$	−29.5897	−1.46312	−0.731559	−	0.681778i	$-0.761207\pi$
$409$	−29.5897	−1.46312	−0.731559	+	0.681778i	$0.761207\pi$
$410$	0	0
$411$	2.83278	0.139731
$412$	0	0
$413$	−4.24396	−0.208832
$414$	0	0
$415$	−10.5691	−0.518815
$416$	0	0
$417$	−1.31184	−0.0642410
$418$	0	0
$419$	6.03316	0.294739	0.147370	−	0.989081i	$-0.452919\pi$
$419$	6.03316	0.294739	0.147370	+	0.989081i	$0.452919\pi$
$420$	0	0
$421$	−14.1980	−0.691969	−0.345985	−	0.938240i	$-0.612455\pi$
$421$	−14.1980	−0.691969	−0.345985	+	0.938240i	$0.612455\pi$
$422$	0	0
$423$	−5.57956	−0.271287
$424$	0	0
$425$	18.4493	0.894923
$426$	0	0
$427$	6.87493	0.332701
$428$	0	0
$429$	0.536705	0.0259123
$430$	0	0
$431$	−8.47227	−0.408095	−0.204048	−	0.978961i	$-0.565410\pi$
$431$	−8.47227	−0.408095	−0.204048	+	0.978961i	$0.565410\pi$
$432$	0	0
$433$	2.72017	0.130723	0.0653616	−	0.997862i	$-0.479180\pi$
$433$	2.72017	0.130723	0.0653616	+	0.997862i	$0.479180\pi$
$434$	0	0
$435$	−0.286670	−0.0137448
$436$	0	0
$437$	−1.28694	−0.0615629
$438$	0	0
$439$	−24.7423	−1.18089	−0.590444	−	0.807079i	$-0.701047\pi$
$439$	−24.7423	−1.18089	−0.590444	+	0.807079i	$0.701047\pi$
$440$	0	0
$441$	−2.71195	−0.129140
$442$	0	0
$443$	−19.0356	−0.904411	−0.452205	−	0.891914i	$-0.649363\pi$
$443$	−19.0356	−0.904411	−0.452205	+	0.891914i	$0.649363\pi$
$444$	0	0
$445$	−18.1427	−0.860049
$446$	0	0
$447$	1.14264	0.0540452
$448$	0	0
$449$	18.6411	0.879728	0.439864	−	0.898064i	$-0.355027\pi$
$449$	18.6411	0.879728	0.439864	+	0.898064i	$0.355027\pi$
$450$	0	0
$451$	−9.30747	−0.438272
$452$	0	0
$453$	−6.13296	−0.288151
$454$	0	0
$455$	3.30011	0.154711
$456$	0	0
$457$	−1.21145	−0.0566693	−0.0283346	−	0.999598i	$-0.509020\pi$
$457$	−1.21145	−0.0566693	−0.0283346	+	0.999598i	$0.509020\pi$
$458$	0	0
$459$	−9.60135	−0.448152
$460$	0	0
$461$	−20.9911	−0.977651	−0.488826	−	0.872381i	$-0.662575\pi$
$461$	−20.9911	−0.977651	−0.488826	+	0.872381i	$0.662575\pi$
$462$	0	0
$463$	14.4073	0.669563	0.334781	−	0.942296i	$-0.391338\pi$
$463$	14.4073	0.669563	0.334781	+	0.942296i	$0.391338\pi$
$464$	0	0
$465$	−18.3585	−0.851353
$466$	0	0
$467$	6.23478	0.288511	0.144256	−	0.989540i	$-0.453921\pi$
$467$	6.23478	0.288511	0.144256	+	0.989540i	$0.453921\pi$
$468$	0	0
$469$	−8.01827	−0.370250
$470$	0	0
$471$	−1.80016	−0.0829470
$472$	0	0
$473$	2.24787	0.103357
$474$	0	0
$475$	−18.0274	−0.827152
$476$	0	0
$477$	−23.4974	−1.07587
$478$	0	0
$479$	3.12161	0.142630	0.0713151	−	0.997454i	$-0.477280\pi$
$479$	3.12161	0.142630	0.0713151	+	0.997454i	$0.477280\pi$
$480$	0	0
$481$	4.54945	0.207437
$482$	0	0
$483$	−0.225699	−0.0102697
$484$	0	0
$485$	−35.8259	−1.62677
$486$	0	0
$487$	39.5739	1.79327	0.896633	−	0.442775i	$-0.146006\pi$
$487$	39.5739	1.79327	0.896633	+	0.442775i	$0.146006\pi$
$488$	0	0
$489$	4.96479	0.224515
$490$	0	0
$491$	27.0328	1.21997	0.609986	−	0.792412i	$-0.291175\pi$
$491$	27.0328	1.21997	0.609986	+	0.792412i	$0.291175\pi$
$492$	0	0
$493$	0.506910	0.0228301
$494$	0	0
$495$	8.94972	0.402260
$496$	0	0
$497$	16.2909	0.730746
$498$	0	0
$499$	6.83496	0.305975	0.152987	−	0.988228i	$-0.451111\pi$
$499$	6.83496	0.305975	0.152987	+	0.988228i	$0.451111\pi$
$500$	0	0
$501$	0.557253	0.0248962
$502$	0	0
$503$	12.2970	0.548298	0.274149	−	0.961687i	$-0.411604\pi$
$503$	12.2970	0.548298	0.274149	+	0.961687i	$0.411604\pi$
$504$	0	0
$505$	−6.37315	−0.283601
$506$	0	0
$507$	0.536705	0.0238359
$508$	0	0
$509$	−44.1664	−1.95764	−0.978822	−	0.204715i	$-0.934373\pi$
$509$	−44.1664	−1.95764	−0.978822	+	0.204715i	$0.934373\pi$
$510$	0	0
$511$	−4.62059	−0.204403
$512$	0	0
$513$	9.38176	0.414215
$514$	0	0
$515$	35.0214	1.54323
$516$	0	0
$517$	2.05740	0.0904842
$518$	0	0
$519$	8.86280	0.389034
$520$	0	0
$521$	0.613678	0.0268857	0.0134429	−	0.999910i	$-0.495721\pi$
$521$	0.613678	0.0268857	0.0134429	+	0.999910i	$0.495721\pi$
$522$	0	0
$523$	32.2826	1.41162	0.705809	−	0.708402i	$-0.250584\pi$
$523$	32.2826	1.41162	0.705809	+	0.708402i	$0.250584\pi$
$524$	0	0
$525$	−3.16157	−0.137982
$526$	0	0
$527$	32.4628	1.41410
$528$	0	0
$529$	−22.8232	−0.992311
$530$	0	0
$531$	−11.5094	−0.499465
$532$	0	0
$533$	−9.30747	−0.403151
$534$	0	0
$535$	43.0503	1.86123
$536$	0	0
$537$	−8.58212	−0.370346
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	43.1328	1.85443	0.927213	−	0.374536i	$-0.122198\pi$
$541$	43.1328	1.85443	0.927213	+	0.374536i	$0.122198\pi$
$542$	0	0
$543$	−0.920136	−0.0394868
$544$	0	0
$545$	63.8958	2.73699
$546$	0	0
$547$	−32.5532	−1.39188	−0.695938	−	0.718102i	$-0.745011\pi$
$547$	−32.5532	−1.39188	−0.695938	+	0.718102i	$0.745011\pi$
$548$	0	0
$549$	18.6445	0.795726
$550$	0	0
$551$	−0.495317	−0.0211012
$552$	0	0
$553$	6.36899	0.270837
$554$	0	0
$555$	−8.05790	−0.342039
$556$	0	0
$557$	−32.2761	−1.36758	−0.683791	−	0.729678i	$-0.739670\pi$
$557$	−32.2761	−1.36758	−0.683791	+	0.729678i	$0.739670\pi$
$558$	0	0
$559$	2.24787	0.0950747
$560$	0	0
$561$	1.68092	0.0709686
$562$	0	0
$563$	−3.15428	−0.132937	−0.0664684	−	0.997789i	$-0.521173\pi$
$563$	−3.15428	−0.132937	−0.0664684	+	0.997789i	$0.521173\pi$
$564$	0	0
$565$	41.5545	1.74821
$566$	0	0
$567$	−6.49051	−0.272576
$568$	0	0
$569$	−25.3859	−1.06423	−0.532115	−	0.846672i	$-0.678603\pi$
$569$	−25.3859	−1.06423	−0.532115	+	0.846672i	$0.678603\pi$
$570$	0	0
$571$	37.4876	1.56881	0.784404	−	0.620251i	$-0.212969\pi$
$571$	37.4876	1.56881	0.784404	+	0.620251i	$0.212969\pi$
$572$	0	0
$573$	−11.5049	−0.480625
$574$	0	0
$575$	2.47721	0.103307
$576$	0	0
$577$	15.3663	0.639706	0.319853	−	0.947467i	$-0.396366\pi$
$577$	15.3663	0.639706	0.319853	+	0.947467i	$0.396366\pi$
$578$	0	0
$579$	−2.81018	−0.116787
$580$	0	0
$581$	−3.20265	−0.132868
$582$	0	0
$583$	8.66441	0.358843
$584$	0	0
$585$	8.94972	0.370025
$586$	0	0
$587$	37.9838	1.56776	0.783880	−	0.620912i	$-0.213238\pi$
$587$	37.9838	1.56776	0.783880	+	0.620912i	$0.213238\pi$
$588$	0	0
$589$	−31.7203	−1.30701
$590$	0	0
$591$	1.92258	0.0790843
$592$	0	0
$593$	16.2380	0.666813	0.333407	−	0.942783i	$-0.391802\pi$
$593$	16.2380	0.666813	0.333407	+	0.942783i	$0.391802\pi$
$594$	0	0
$595$	10.3357	0.423723
$596$	0	0
$597$	−6.02686	−0.246663
$598$	0	0
$599$	−20.3847	−0.832898	−0.416449	−	0.909159i	$-0.636725\pi$
$599$	−20.3847	−0.832898	−0.416449	+	0.909159i	$0.636725\pi$
$600$	0	0
$601$	26.9836	1.10068	0.550342	−	0.834939i	$-0.314497\pi$
$601$	26.9836	1.10068	0.550342	+	0.834939i	$0.314497\pi$
$602$	0	0
$603$	−21.7451	−0.885531
$604$	0	0
$605$	−3.30011	−0.134168
$606$	0	0
$607$	−30.2528	−1.22792	−0.613961	−	0.789337i	$-0.710425\pi$
$607$	−30.2528	−1.22792	−0.613961	+	0.789337i	$0.710425\pi$
$608$	0	0
$609$	−0.0868669	−0.00352002
$610$	0	0
$611$	2.05740	0.0832334
$612$	0	0
$613$	−23.3824	−0.944407	−0.472203	−	0.881490i	$-0.656541\pi$
$613$	−23.3824	−0.944407	−0.472203	+	0.881490i	$0.656541\pi$
$614$	0	0
$615$	16.4852	0.664749
$616$	0	0
$617$	−45.5404	−1.83339	−0.916694	−	0.399590i	$-0.869152\pi$
$617$	−45.5404	−1.83339	−0.916694	+	0.399590i	$0.869152\pi$
$618$	0	0
$619$	11.6923	0.469953	0.234977	−	0.972001i	$-0.424499\pi$
$619$	11.6923	0.469953	0.234977	+	0.972001i	$0.424499\pi$
$620$	0	0
$621$	−1.28918	−0.0517331
$622$	0	0
$623$	−5.49762	−0.220258
$624$	0	0
$625$	−19.7531	−0.790124
$626$	0	0
$627$	−1.64248	−0.0655943
$628$	0	0
$629$	14.2486	0.568127
$630$	0	0
$631$	−21.3113	−0.848391	−0.424195	−	0.905571i	$-0.639443\pi$
$631$	−21.3113	−0.848391	−0.424195	+	0.905571i	$0.639443\pi$
$632$	0	0
$633$	9.04619	0.359554
$634$	0	0
$635$	−28.8515	−1.14494
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	44.1800	1.74773
$640$	0	0
$641$	−19.1727	−0.757277	−0.378638	−	0.925545i	$-0.623608\pi$
$641$	−19.1727	−0.757277	−0.378638	+	0.925545i	$0.623608\pi$
$642$	0	0
$643$	22.4479	0.885261	0.442630	−	0.896704i	$-0.354045\pi$
$643$	22.4479	0.885261	0.442630	+	0.896704i	$0.354045\pi$
$644$	0	0
$645$	−3.98139	−0.156767
$646$	0	0
$647$	16.0810	0.632210	0.316105	−	0.948724i	$-0.397625\pi$
$647$	16.0810	0.632210	0.316105	+	0.948724i	$0.397625\pi$
$648$	0	0
$649$	4.24396	0.166590
$650$	0	0
$651$	−5.56299	−0.218031
$652$	0	0
$653$	40.8056	1.59685	0.798423	−	0.602097i	$-0.205668\pi$
$653$	40.8056	1.59685	0.798423	+	0.602097i	$0.205668\pi$
$654$	0	0
$655$	49.2679	1.92506
$656$	0	0
$657$	−12.5308	−0.488873
$658$	0	0
$659$	−39.0702	−1.52196	−0.760980	−	0.648776i	$-0.775281\pi$
$659$	−39.0702	−1.52196	−0.760980	+	0.648776i	$0.775281\pi$
$660$	0	0
$661$	7.68059	0.298740	0.149370	−	0.988781i	$-0.452275\pi$
$661$	7.68059	0.298740	0.149370	+	0.988781i	$0.452275\pi$
$662$	0	0
$663$	1.68092	0.0652816
$664$	0	0
$665$	−10.0993	−0.391636
$666$	0	0
$667$	0.0680634	0.00263543
$668$	0	0
$669$	−6.31792	−0.244265
$670$	0	0
$671$	−6.87493	−0.265404
$672$	0	0
$673$	18.1821	0.700870	0.350435	−	0.936587i	$-0.386034\pi$
$673$	18.1821	0.700870	0.350435	+	0.936587i	$0.386034\pi$
$674$	0	0
$675$	−18.0587	−0.695081
$676$	0	0
$677$	−12.4832	−0.479769	−0.239885	−	0.970801i	$-0.577110\pi$
$677$	−12.4832	−0.479769	−0.239885	+	0.970801i	$0.577110\pi$
$678$	0	0
$679$	−10.8560	−0.416614
$680$	0	0
$681$	4.11471	0.157676
$682$	0	0
$683$	−33.3127	−1.27468	−0.637338	−	0.770584i	$-0.719965\pi$
$683$	−33.3127	−1.27468	−0.637338	+	0.770584i	$0.719965\pi$
$684$	0	0
$685$	−17.4183	−0.665519
$686$	0	0
$687$	1.72242	0.0657146
$688$	0	0
$689$	8.66441	0.330088
$690$	0	0
$691$	43.4935	1.65457	0.827285	−	0.561782i	$-0.189884\pi$
$691$	43.4935	1.65457	0.827285	+	0.561782i	$0.189884\pi$
$692$	0	0
$693$	2.71195	0.103018
$694$	0	0
$695$	8.06628	0.305971
$696$	0	0
$697$	−29.1504	−1.10415
$698$	0	0
$699$	2.42265	0.0916330
$700$	0	0
$701$	26.4772	1.00003	0.500016	−	0.866016i	$-0.333328\pi$
$701$	26.4772	1.00003	0.500016	+	0.866016i	$0.333328\pi$
$702$	0	0
$703$	−13.9227	−0.525104
$704$	0	0
$705$	−3.64403	−0.137242
$706$	0	0
$707$	−1.93119	−0.0726300
$708$	0	0
$709$	45.7514	1.71823	0.859115	−	0.511783i	$-0.171015\pi$
$709$	45.7514	1.71823	0.859115	+	0.511783i	$0.171015\pi$
$710$	0	0
$711$	17.2724	0.647764
$712$	0	0
$713$	4.35881	0.163239
$714$	0	0
$715$	−3.30011	−0.123417
$716$	0	0
$717$	−1.58742	−0.0592833
$718$	0	0
$719$	−25.5391	−0.952449	−0.476225	−	0.879324i	$-0.657995\pi$
$719$	−25.5391	−0.952449	−0.476225	+	0.879324i	$0.657995\pi$
$720$	0	0
$721$	10.6122	0.395219
$722$	0	0
$723$	0.485608	0.0180599
$724$	0	0
$725$	0.953425	0.0354093
$726$	0	0
$727$	31.0729	1.15243	0.576215	−	0.817298i	$-0.304529\pi$
$727$	31.0729	1.15243	0.576215	+	0.817298i	$0.304529\pi$
$728$	0	0
$729$	−12.6659	−0.469108
$730$	0	0
$731$	7.04018	0.260390
$732$	0	0
$733$	43.4455	1.60469	0.802347	−	0.596857i	$-0.203584\pi$
$733$	43.4455	1.60469	0.802347	+	0.596857i	$0.203584\pi$
$734$	0	0
$735$	−1.77118	−0.0653311
$736$	0	0
$737$	8.01827	0.295357
$738$	0	0
$739$	−24.9020	−0.916035	−0.458018	−	0.888943i	$-0.651440\pi$
$739$	−24.9020	−0.916035	−0.458018	+	0.888943i	$0.651440\pi$
$740$	0	0
$741$	−1.64248	−0.0603380
$742$	0	0
$743$	−38.8381	−1.42483	−0.712416	−	0.701758i	$-0.752399\pi$
$743$	−38.8381	−1.42483	−0.712416	+	0.701758i	$0.752399\pi$
$744$	0	0
$745$	−7.02593	−0.257410
$746$	0	0
$747$	−8.68541	−0.317782
$748$	0	0
$749$	13.0451	0.476658
$750$	0	0
$751$	−47.6813	−1.73992	−0.869958	−	0.493125i	$-0.835854\pi$
$751$	−47.6813	−1.73992	−0.869958	+	0.493125i	$0.835854\pi$
$752$	0	0
$753$	−4.67199	−0.170257
$754$	0	0
$755$	37.7105	1.37243
$756$	0	0
$757$	8.19558	0.297873	0.148937	−	0.988847i	$-0.452415\pi$
$757$	8.19558	0.297873	0.148937	+	0.988847i	$0.452415\pi$
$758$	0	0
$759$	0.225699	0.00819237
$760$	0	0
$761$	−28.5644	−1.03546	−0.517730	−	0.855544i	$-0.673223\pi$
$761$	−28.5644	−1.03546	−0.517730	+	0.855544i	$0.673223\pi$
$762$	0	0
$763$	19.3617	0.700941
$764$	0	0
$765$	28.0299	1.01342
$766$	0	0
$767$	4.24396	0.153241
$768$	0	0
$769$	6.21584	0.224149	0.112075	−	0.993700i	$-0.464250\pi$
$769$	6.21584	0.224149	0.112075	+	0.993700i	$0.464250\pi$
$770$	0	0
$771$	−8.31428	−0.299431
$772$	0	0
$773$	5.97637	0.214955	0.107478	−	0.994208i	$-0.465723\pi$
$773$	5.97637	0.214955	0.107478	+	0.994208i	$0.465723\pi$
$774$	0	0
$775$	61.0577	2.19326
$776$	0	0
$777$	−2.44171	−0.0875958
$778$	0	0
$779$	28.4837	1.02054
$780$	0	0
$781$	−16.2909	−0.582933
$782$	0	0
$783$	−0.496179	−0.0177320
$784$	0	0
$785$	11.0689	0.395065
$786$	0	0
$787$	−2.13999	−0.0762824	−0.0381412	−	0.999272i	$-0.512144\pi$
$787$	−2.13999	−0.0762824	−0.0381412	+	0.999272i	$0.512144\pi$
$788$	0	0
$789$	−7.53473	−0.268243
$790$	0	0
$791$	12.5919	0.447715
$792$	0	0
$793$	−6.87493	−0.244136
$794$	0	0
$795$	−15.3462	−0.544275
$796$	0	0
$797$	−16.3602	−0.579509	−0.289754	−	0.957101i	$-0.593574\pi$
$797$	−16.3602	−0.579509	−0.289754	+	0.957101i	$0.593574\pi$
$798$	0	0
$799$	6.44363	0.227959
$800$	0	0
$801$	−14.9093	−0.526793
$802$	0	0
$803$	4.62059	0.163057
$804$	0	0
$805$	1.38779	0.0489131
$806$	0	0
$807$	−5.84727	−0.205834
$808$	0	0
$809$	54.7563	1.92513	0.962564	−	0.271055i	$-0.0873724\pi$
$809$	54.7563	1.92513	0.962564	+	0.271055i	$0.0873724\pi$
$810$	0	0
$811$	37.8235	1.32816	0.664082	−	0.747660i	$-0.268823\pi$
$811$	37.8235	1.32816	0.664082	+	0.747660i	$0.268823\pi$
$812$	0	0
$813$	−7.39063	−0.259201
$814$	0	0
$815$	−30.5277	−1.06934
$816$	0	0
$817$	−6.87917	−0.240672
$818$	0	0
$819$	2.71195	0.0947631
$820$	0	0
$821$	−21.7410	−0.758767	−0.379384	−	0.925239i	$-0.623864\pi$
$821$	−21.7410	−0.758767	−0.379384	+	0.925239i	$0.623864\pi$
$822$	0	0
$823$	27.2535	0.949997	0.474998	−	0.879987i	$-0.342449\pi$
$823$	27.2535	0.949997	0.474998	+	0.879987i	$0.342449\pi$
$824$	0	0
$825$	3.16157	0.110072
$826$	0	0
$827$	−18.6586	−0.648821	−0.324411	−	0.945916i	$-0.605166\pi$
$827$	−18.6586	−0.648821	−0.324411	+	0.945916i	$0.605166\pi$
$828$	0	0
$829$	−30.1758	−1.04805	−0.524024	−	0.851704i	$-0.675570\pi$
$829$	−30.1758	−1.04805	−0.524024	+	0.851704i	$0.675570\pi$
$830$	0	0
$831$	−14.3878	−0.499109
$832$	0	0
$833$	3.13193	0.108515
$834$	0	0
$835$	−3.42646	−0.118577
$836$	0	0
$837$	−31.7755	−1.09832
$838$	0	0
$839$	−24.8451	−0.857749	−0.428875	−	0.903364i	$-0.641090\pi$
$839$	−24.8451	−0.857749	−0.428875	+	0.903364i	$0.641090\pi$
$840$	0	0
$841$	−28.9738	−0.999097
$842$	0	0
$843$	−2.51352	−0.0865702
$844$	0	0
$845$	−3.30011	−0.113527
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	9.44168	0.324038
$850$	0	0
$851$	1.91317	0.0655826
$852$	0	0
$853$	7.25064	0.248257	0.124129	−	0.992266i	$-0.460386\pi$
$853$	7.25064	0.248257	0.124129	+	0.992266i	$0.460386\pi$
$854$	0	0
$855$	−27.3889	−0.936680
$856$	0	0
$857$	−8.30714	−0.283767	−0.141883	−	0.989883i	$-0.545316\pi$
$857$	−8.30714	−0.283767	−0.141883	+	0.989883i	$0.545316\pi$
$858$	0	0
$859$	−26.2253	−0.894797	−0.447399	−	0.894335i	$-0.647649\pi$
$859$	−26.2253	−0.894797	−0.447399	+	0.894335i	$0.647649\pi$
$860$	0	0
$861$	4.99536	0.170242
$862$	0	0
$863$	−7.44686	−0.253494	−0.126747	−	0.991935i	$-0.540454\pi$
$863$	−7.44686	−0.253494	−0.126747	+	0.991935i	$0.540454\pi$
$864$	0	0
$865$	−54.4959	−1.85291
$866$	0	0
$867$	−3.85944	−0.131073
$868$	0	0
$869$	−6.36899	−0.216053
$870$	0	0
$871$	8.01827	0.271689
$872$	0	0
$873$	−29.4408	−0.996421
$874$	0	0
$875$	2.93943	0.0993710
$876$	0	0
$877$	−0.850256	−0.0287111	−0.0143556	−	0.999897i	$-0.504570\pi$
$877$	−0.850256	−0.0287111	−0.0143556	+	0.999897i	$0.504570\pi$
$878$	0	0
$879$	−6.56084	−0.221292
$880$	0	0
$881$	10.5341	0.354903	0.177451	−	0.984130i	$-0.443215\pi$
$881$	10.5341	0.354903	0.177451	+	0.984130i	$0.443215\pi$
$882$	0	0
$883$	−5.16673	−0.173874	−0.0869372	−	0.996214i	$-0.527708\pi$
$883$	−5.16673	−0.173874	−0.0869372	+	0.996214i	$0.527708\pi$
$884$	0	0
$885$	−7.51683	−0.252675
$886$	0	0
$887$	9.84275	0.330487	0.165244	−	0.986253i	$-0.447159\pi$
$887$	9.84275	0.330487	0.165244	+	0.986253i	$0.447159\pi$
$888$	0	0
$889$	−8.74259	−0.293217
$890$	0	0
$891$	6.49051	0.217440
$892$	0	0
$893$	−6.29627	−0.210696
$894$	0	0
$895$	52.7700	1.76391
$896$	0	0
$897$	0.225699	0.00753588
$898$	0	0
$899$	1.67761	0.0559515
$900$	0	0
$901$	27.1363	0.904043
$902$	0	0
$903$	−1.20644	−0.0401479
$904$	0	0
$905$	5.65777	0.188070
$906$	0	0
$907$	15.6231	0.518755	0.259377	−	0.965776i	$-0.416483\pi$
$907$	15.6231	0.518755	0.259377	+	0.965776i	$0.416483\pi$
$908$	0	0
$909$	−5.23730	−0.173710
$910$	0	0
$911$	20.4772	0.678439	0.339220	−	0.940707i	$-0.389837\pi$
$911$	20.4772	0.678439	0.339220	+	0.940707i	$0.389837\pi$
$912$	0	0
$913$	3.20265	0.105992
$914$	0	0
$915$	12.1768	0.402551
$916$	0	0
$917$	14.9292	0.493005
$918$	0	0
$919$	−3.74457	−0.123522	−0.0617609	−	0.998091i	$-0.519672\pi$
$919$	−3.74457	−0.123522	−0.0617609	+	0.998091i	$0.519672\pi$
$920$	0	0
$921$	6.37205	0.209966
$922$	0	0
$923$	−16.2909	−0.536221
$924$	0	0
$925$	26.7995	0.881161
$926$	0	0
$927$	28.7798	0.945252
$928$	0	0
$929$	28.7883	0.944514	0.472257	−	0.881461i	$-0.343439\pi$
$929$	28.7883	0.944514	0.472257	+	0.881461i	$0.343439\pi$
$930$	0	0
$931$	−3.06031	−0.100297
$932$	0	0
$933$	8.16396	0.267276
$934$	0	0
$935$	−10.3357	−0.338014
$936$	0	0
$937$	−54.2426	−1.77203	−0.886014	−	0.463659i	$-0.846536\pi$
$937$	−54.2426	−1.77203	−0.886014	+	0.463659i	$0.846536\pi$
$938$	0	0
$939$	2.87051	0.0936755
$940$	0	0
$941$	−13.2047	−0.430462	−0.215231	−	0.976563i	$-0.569050\pi$
$941$	−13.2047	−0.430462	−0.215231	+	0.976563i	$0.569050\pi$
$942$	0	0
$943$	−3.91405	−0.127459
$944$	0	0
$945$	−10.1169	−0.329103
$946$	0	0
$947$	−43.9688	−1.42879	−0.714397	−	0.699740i	$-0.753299\pi$
$947$	−43.9688	−1.42879	−0.714397	+	0.699740i	$0.753299\pi$
$948$	0	0
$949$	4.62059	0.149991
$950$	0	0
$951$	−0.755928	−0.0245126
$952$	0	0
$953$	−36.2086	−1.17291	−0.586456	−	0.809981i	$-0.699477\pi$
$953$	−36.2086	−1.17291	−0.586456	+	0.809981i	$0.699477\pi$
$954$	0	0
$955$	70.7419	2.28915
$956$	0	0
$957$	0.0868669	0.00280801
$958$	0	0
$959$	−5.27810	−0.170439
$960$	0	0
$961$	76.4350	2.46565
$962$	0	0
$963$	35.3777	1.14003
$964$	0	0
$965$	17.2793	0.556241
$966$	0	0
$967$	8.31230	0.267306	0.133653	−	0.991028i	$-0.457329\pi$
$967$	8.31230	0.267306	0.133653	+	0.991028i	$0.457329\pi$
$968$	0	0
$969$	−5.14414	−0.165254
$970$	0	0
$971$	50.7323	1.62808	0.814039	−	0.580810i	$-0.197264\pi$
$971$	50.7323	1.62808	0.814039	+	0.580810i	$0.197264\pi$
$972$	0	0
$973$	2.44425	0.0783590
$974$	0	0
$975$	3.16157	0.101251
$976$	0	0
$977$	11.5080	0.368175	0.184087	−	0.982910i	$-0.441067\pi$
$977$	11.5080	0.368175	0.184087	+	0.982910i	$0.441067\pi$
$978$	0	0
$979$	5.49762	0.175705
$980$	0	0
$981$	52.5080	1.67645
$982$	0	0
$983$	11.6057	0.370164	0.185082	−	0.982723i	$-0.440745\pi$
$983$	11.6057	0.370164	0.185082	+	0.982723i	$0.440745\pi$
$984$	0	0
$985$	−11.8216	−0.376668
$986$	0	0
$987$	−1.10421	−0.0351475
$988$	0	0
$989$	0.945292	0.0300585
$990$	0	0
$991$	47.9015	1.52164	0.760821	−	0.648962i	$-0.224796\pi$
$991$	47.9015	1.52164	0.760821	+	0.648962i	$0.224796\pi$
$992$	0	0
$993$	10.3201	0.327498
$994$	0	0
$995$	37.0581	1.17482
$996$	0	0
$997$	1.76268	0.0558246	0.0279123	−	0.999610i	$-0.491114\pi$
$997$	1.76268	0.0558246	0.0279123	+	0.999610i	$0.491114\pi$
$998$	0	0
$999$	−13.9469	−0.441261

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.p.1.6	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.p.1.6	✓	9	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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q+0.536705 q^{3} -3.30011 q^{5} -1.00000 q^{7} -2.71195 q^{9} +O(q^{10})\)	\(q+0.536705 q^{3} -3.30011 q^{5} -1.00000 q^{7} -2.71195 q^{9} +1.00000 q^{11} +1.00000 q^{13} -1.77118 q^{15} +3.13193 q^{17} -3.06031 q^{19} -0.536705 q^{21} +0.420528 q^{23} +5.89071 q^{25} -3.06563 q^{27} +0.161852 q^{29} +10.3651 q^{31} +0.536705 q^{33} +3.30011 q^{35} +4.54945 q^{37} +0.536705 q^{39} -9.30747 q^{41} +2.24787 q^{43} +8.94972 q^{45} +2.05740 q^{47} +1.00000 q^{49} +1.68092 q^{51} +8.66441 q^{53} -3.30011 q^{55} -1.64248 q^{57} +4.24396 q^{59} -6.87493 q^{61} +2.71195 q^{63} -3.30011 q^{65} +8.01827 q^{67} +0.225699 q^{69} -16.2909 q^{71} +4.62059 q^{73} +3.16157 q^{75} -1.00000 q^{77} -6.36899 q^{79} +6.49051 q^{81} +3.20265 q^{83} -10.3357 q^{85} +0.0868669 q^{87} +5.49762 q^{89} -1.00000 q^{91} +5.56299 q^{93} +10.0993 q^{95} +10.8560 q^{97} -2.71195 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{3} - 4 q^{5} - 9 q^{7} + 4 q^{9}+O(q^{10}) \) 9 * q + q^3 - 4 * q^5 - 9 * q^7 + 4 * q^9	\( 9 q + q^{3} - 4 q^{5} - 9 q^{7} + 4 q^{9} + 9 q^{11} + 9 q^{13} - 9 q^{15} - 11 q^{17} + 10 q^{19} - q^{21} - 14 q^{23} - q^{25} - 5 q^{27} - 10 q^{29} + 5 q^{31} + q^{33} + 4 q^{35} - 16 q^{37} + q^{39} + 2 q^{41} + 4 q^{43} - 30 q^{45} + 9 q^{49} + 3 q^{51} - 23 q^{53} - 4 q^{55} + 14 q^{57} + 9 q^{59} - 14 q^{61} - 4 q^{63} - 4 q^{65} + 8 q^{67} - 26 q^{69} - 20 q^{71} - 23 q^{73} + 32 q^{75} - 9 q^{77} + 2 q^{79} - 11 q^{81} - 9 q^{83} - 3 q^{85} - 7 q^{87} - 6 q^{89} - 9 q^{91} - 19 q^{93} - 4 q^{95} - 3 q^{97} + 4 q^{99}+O(q^{100}) \) 9 * q + q^3 - 4 * q^5 - 9 * q^7 + 4 * q^9 + 9 * q^11 + 9 * q^13 - 9 * q^15 - 11 * q^17 + 10 * q^19 - q^21 - 14 * q^23 - q^25 - 5 * q^27 - 10 * q^29 + 5 * q^31 + q^33 + 4 * q^35 - 16 * q^37 + q^39 + 2 * q^41 + 4 * q^43 - 30 * q^45 + 9 * q^49 + 3 * q^51 - 23 * q^53 - 4 * q^55 + 14 * q^57 + 9 * q^59 - 14 * q^61 - 4 * q^63 - 4 * q^65 + 8 * q^67 - 26 * q^69 - 20 * q^71 - 23 * q^73 + 32 * q^75 - 9 * q^77 + 2 * q^79 - 11 * q^81 - 9 * q^83 - 3 * q^85 - 7 * q^87 - 6 * q^89 - 9 * q^91 - 19 * q^93 - 4 * q^95 - 3 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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