LMFDB - Embedded newform 8008.2.a.t.1.9

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:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} - 17x^{8} + 13x^{7} + 96x^{6} - 49x^{5} - 207x^{4} + 58x^{3} + 169x^{2} - 12x - 32 $ x^10 - x^9 - 17x^8 + 13x^7 + 96x^6 - 49x^5 - 207x^4 + 58x^3 + 169x^2 - 12x - 32
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$2.73324$ 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)

$f(q)$	$=$	$q+2.73324 q^{3} +4.21870 q^{5} +1.00000 q^{7} +4.47060 q^{9} +O(q^{10})$	$q+2.73324 q^{3} +4.21870 q^{5} +1.00000 q^{7} +4.47060 q^{9} -1.00000 q^{11} +1.00000 q^{13} +11.5307 q^{15} -3.68620 q^{17} +3.56911 q^{19} +2.73324 q^{21} +8.18576 q^{23} +12.7974 q^{25} +4.01952 q^{27} -3.03863 q^{29} -0.665094 q^{31} -2.73324 q^{33} +4.21870 q^{35} -1.15007 q^{37} +2.73324 q^{39} -0.731821 q^{41} -4.00963 q^{43} +18.8601 q^{45} -3.56384 q^{47} +1.00000 q^{49} -10.0753 q^{51} +5.77735 q^{53} -4.21870 q^{55} +9.75524 q^{57} -5.07544 q^{59} +1.37127 q^{61} +4.47060 q^{63} +4.21870 q^{65} -2.54238 q^{67} +22.3737 q^{69} +6.00418 q^{71} -0.971562 q^{73} +34.9785 q^{75} -1.00000 q^{77} -9.45503 q^{79} -2.42551 q^{81} -6.58379 q^{83} -15.5510 q^{85} -8.30532 q^{87} -11.3914 q^{89} +1.00000 q^{91} -1.81786 q^{93} +15.0570 q^{95} +4.39343 q^{97} -4.47060 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + q^{3} + 3 q^{5} + 10 q^{7} + 5 q^{9}+O(q^{10}) $ 10 * q + q^3 + 3 * q^5 + 10 * q^7 + 5 * q^9	$ 10 q + q^{3} + 3 q^{5} + 10 q^{7} + 5 q^{9} - 10 q^{11} + 10 q^{13} + 9 q^{15} + 7 q^{17} + 17 q^{19} + q^{21} + 15 q^{23} + 13 q^{25} + 7 q^{27} - q^{29} - 6 q^{31} - q^{33} + 3 q^{35} - 12 q^{37} + q^{39} + 4 q^{41} + 17 q^{43} - 15 q^{45} + 21 q^{47} + 10 q^{49} + 3 q^{51} + 20 q^{53} - 3 q^{55} + 12 q^{57} + 9 q^{59} + 6 q^{61} + 5 q^{63} + 3 q^{65} + 26 q^{67} + 30 q^{69} + 18 q^{71} - 4 q^{73} + 48 q^{75} - 10 q^{77} + 19 q^{79} - 14 q^{81} + 40 q^{83} - 25 q^{85} - 25 q^{87} - 19 q^{89} + 10 q^{91} + q^{93} + 11 q^{95} - 22 q^{97} - 5 q^{99}+O(q^{100}) $ 10 * q + q^3 + 3 * q^5 + 10 * q^7 + 5 * q^9 - 10 * q^11 + 10 * q^13 + 9 * q^15 + 7 * q^17 + 17 * q^19 + q^21 + 15 * q^23 + 13 * q^25 + 7 * q^27 - q^29 - 6 * q^31 - q^33 + 3 * q^35 - 12 * q^37 + q^39 + 4 * q^41 + 17 * q^43 - 15 * q^45 + 21 * q^47 + 10 * q^49 + 3 * q^51 + 20 * q^53 - 3 * q^55 + 12 * q^57 + 9 * q^59 + 6 * q^61 + 5 * q^63 + 3 * q^65 + 26 * q^67 + 30 * q^69 + 18 * q^71 - 4 * q^73 + 48 * q^75 - 10 * q^77 + 19 * q^79 - 14 * q^81 + 40 * q^83 - 25 * q^85 - 25 * q^87 - 19 * q^89 + 10 * q^91 + q^93 + 11 * q^95 - 22 * q^97 - 5 * 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
$3$	2.73324	1.57804	0.789019	−	0.614369i	$-0.210589\pi$
$3$	2.73324	1.57804	0.789019	+	0.614369i	$0.210589\pi$
$4$	0	0
$5$	4.21870	1.88666	0.943330	−	0.331855i	$-0.107675\pi$
$5$	4.21870	1.88666	0.943330	+	0.331855i	$0.107675\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	4.47060	1.49020
$10$	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
$15$	11.5307	2.97722
$16$	0	0
$17$	−3.68620	−0.894034	−0.447017	−	0.894525i	$-0.647514\pi$
$17$	−3.68620	−0.894034	−0.447017	+	0.894525i	$0.647514\pi$
$18$	0	0
$19$	3.56911	0.818810	0.409405	−	0.912353i	$-0.365736\pi$
$19$	3.56911	0.818810	0.409405	+	0.912353i	$0.365736\pi$
$20$	0	0
$21$	2.73324	0.596442
$22$	0	0
$23$	8.18576	1.70685	0.853424	−	0.521217i	$-0.174522\pi$
$23$	8.18576	1.70685	0.853424	+	0.521217i	$0.174522\pi$
$24$	0	0
$25$	12.7974	2.55949
$26$	0	0
$27$	4.01952	0.773556
$28$	0	0
$29$	−3.03863	−0.564260	−0.282130	−	0.959376i	$-0.591041\pi$
$29$	−3.03863	−0.564260	−0.282130	+	0.959376i	$0.591041\pi$
$30$	0	0
$31$	−0.665094	−0.119454	−0.0597272	−	0.998215i	$-0.519023\pi$
$31$	−0.665094	−0.119454	−0.0597272	+	0.998215i	$0.519023\pi$
$32$	0	0
$33$	−2.73324	−0.475796
$34$	0	0
$35$	4.21870	0.713091
$36$	0	0
$37$	−1.15007	−0.189070	−0.0945349	−	0.995522i	$-0.530136\pi$
$37$	−1.15007	−0.189070	−0.0945349	+	0.995522i	$0.530136\pi$
$38$	0	0
$39$	2.73324	0.437669
$40$	0	0
$41$	−0.731821	−0.114291	−0.0571456	−	0.998366i	$-0.518200\pi$
$41$	−0.731821	−0.114291	−0.0571456	+	0.998366i	$0.518200\pi$
$42$	0	0
$43$	−4.00963	−0.611462	−0.305731	−	0.952118i	$-0.598901\pi$
$43$	−4.00963	−0.611462	−0.305731	+	0.952118i	$0.598901\pi$
$44$	0	0
$45$	18.8601	2.81150
$46$	0	0
$47$	−3.56384	−0.519839	−0.259919	−	0.965630i	$-0.583696\pi$
$47$	−3.56384	−0.519839	−0.259919	+	0.965630i	$0.583696\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−10.0753	−1.41082
$52$	0	0
$53$	5.77735	0.793579	0.396790	−	0.917910i	$-0.370124\pi$
$53$	5.77735	0.793579	0.396790	+	0.917910i	$0.370124\pi$
$54$	0	0
$55$	−4.21870	−0.568850
$56$	0	0
$57$	9.75524	1.29211
$58$	0	0
$59$	−5.07544	−0.660765	−0.330383	−	0.943847i	$-0.607178\pi$
$59$	−5.07544	−0.660765	−0.330383	+	0.943847i	$0.607178\pi$
$60$	0	0
$61$	1.37127	0.175574	0.0877868	−	0.996139i	$-0.472021\pi$
$61$	1.37127	0.175574	0.0877868	+	0.996139i	$0.472021\pi$
$62$	0	0
$63$	4.47060	0.563243
$64$	0	0
$65$	4.21870	0.523265
$66$	0	0
$67$	−2.54238	−0.310601	−0.155300	−	0.987867i	$-0.549635\pi$
$67$	−2.54238	−0.310601	−0.155300	+	0.987867i	$0.549635\pi$
$68$	0	0
$69$	22.3737	2.69347
$70$	0	0
$71$	6.00418	0.712565	0.356283	−	0.934378i	$-0.384044\pi$
$71$	6.00418	0.712565	0.356283	+	0.934378i	$0.384044\pi$
$72$	0	0
$73$	−0.971562	−0.113713	−0.0568563	−	0.998382i	$-0.518108\pi$
$73$	−0.971562	−0.113713	−0.0568563	+	0.998382i	$0.518108\pi$
$74$	0	0
$75$	34.9785	4.03897
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−9.45503	−1.06377	−0.531887	−	0.846815i	$-0.678517\pi$
$79$	−9.45503	−1.06377	−0.531887	+	0.846815i	$0.678517\pi$
$80$	0	0
$81$	−2.42551	−0.269501
$82$	0	0
$83$	−6.58379	−0.722665	−0.361333	−	0.932437i	$-0.617678\pi$
$83$	−6.58379	−0.722665	−0.361333	+	0.932437i	$0.617678\pi$
$84$	0	0
$85$	−15.5510	−1.68674
$86$	0	0
$87$	−8.30532	−0.890423
$88$	0	0
$89$	−11.3914	−1.20749	−0.603745	−	0.797178i	$-0.706325\pi$
$89$	−11.3914	−1.20749	−0.603745	+	0.797178i	$0.706325\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−1.81786	−0.188503
$94$	0	0
$95$	15.0570	1.54482
$96$	0	0
$97$	4.39343	0.446085	0.223043	−	0.974809i	$-0.428401\pi$
$97$	4.39343	0.446085	0.223043	+	0.974809i	$0.428401\pi$
$98$	0	0
$99$	−4.47060	−0.449313
$100$	0	0
$101$	−6.01351	−0.598367	−0.299184	−	0.954196i	$-0.596714\pi$
$101$	−6.01351	−0.598367	−0.299184	+	0.954196i	$0.596714\pi$
$102$	0	0
$103$	−0.948188	−0.0934277	−0.0467139	−	0.998908i	$-0.514875\pi$
$103$	−0.948188	−0.0934277	−0.0467139	+	0.998908i	$0.514875\pi$
$104$	0	0
$105$	11.5307	1.12528
$106$	0	0
$107$	−11.3018	−1.09259	−0.546294	−	0.837593i	$-0.683962\pi$
$107$	−11.3018	−1.09259	−0.546294	+	0.837593i	$0.683962\pi$
$108$	0	0
$109$	−19.6941	−1.88635	−0.943177	−	0.332291i	$-0.892178\pi$
$109$	−19.6941	−1.88635	−0.943177	+	0.332291i	$0.892178\pi$
$110$	0	0
$111$	−3.14341	−0.298359
$112$	0	0
$113$	−6.72038	−0.632200	−0.316100	−	0.948726i	$-0.602373\pi$
$113$	−6.72038	−0.632200	−0.316100	+	0.948726i	$0.602373\pi$
$114$	0	0
$115$	34.5333	3.22024
$116$	0	0
$117$	4.47060	0.413308
$118$	0	0
$119$	−3.68620	−0.337913
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−2.00024	−0.180356
$124$	0	0
$125$	32.8951	2.94222
$126$	0	0
$127$	−8.71019	−0.772904	−0.386452	−	0.922309i	$-0.626300\pi$
$127$	−8.71019	−0.772904	−0.386452	+	0.922309i	$0.626300\pi$
$128$	0	0
$129$	−10.9593	−0.964910
$130$	0	0
$131$	3.08617	0.269640	0.134820	−	0.990870i	$-0.456954\pi$
$131$	3.08617	0.269640	0.134820	+	0.990870i	$0.456954\pi$
$132$	0	0
$133$	3.56911	0.309481
$134$	0	0
$135$	16.9571	1.45944
$136$	0	0
$137$	−10.5500	−0.901347	−0.450674	−	0.892689i	$-0.648816\pi$
$137$	−10.5500	−0.901347	−0.450674	+	0.892689i	$0.648816\pi$
$138$	0	0
$139$	6.17404	0.523675	0.261837	−	0.965112i	$-0.415672\pi$
$139$	6.17404	0.523675	0.261837	+	0.965112i	$0.415672\pi$
$140$	0	0
$141$	−9.74082	−0.820325
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−12.8191	−1.06457
$146$	0	0
$147$	2.73324	0.225434
$148$	0	0
$149$	15.3098	1.25423	0.627113	−	0.778928i	$-0.284236\pi$
$149$	15.3098	1.25423	0.627113	+	0.778928i	$0.284236\pi$
$150$	0	0
$151$	2.40869	0.196016	0.0980082	−	0.995186i	$-0.468753\pi$
$151$	2.40869	0.196016	0.0980082	+	0.995186i	$0.468753\pi$
$152$	0	0
$153$	−16.4795	−1.33229
$154$	0	0
$155$	−2.80583	−0.225370
$156$	0	0
$157$	−11.8059	−0.942214	−0.471107	−	0.882076i	$-0.656145\pi$
$157$	−11.8059	−0.942214	−0.471107	+	0.882076i	$0.656145\pi$
$158$	0	0
$159$	15.7909	1.25230
$160$	0	0
$161$	8.18576	0.645128
$162$	0	0
$163$	21.8963	1.71505	0.857526	−	0.514441i	$-0.172001\pi$
$163$	21.8963	1.71505	0.857526	+	0.514441i	$0.172001\pi$
$164$	0	0
$165$	−11.5307	−0.897666
$166$	0	0
$167$	24.8745	1.92485	0.962424	−	0.271551i	$-0.0875364\pi$
$167$	24.8745	1.92485	0.962424	+	0.271551i	$0.0875364\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	15.9561	1.22019
$172$	0	0
$173$	16.1810	1.23022	0.615109	−	0.788442i	$-0.289112\pi$
$173$	16.1810	1.23022	0.615109	+	0.788442i	$0.289112\pi$
$174$	0	0
$175$	12.7974	0.967395
$176$	0	0
$177$	−13.8724	−1.04271
$178$	0	0
$179$	2.15486	0.161062	0.0805310	−	0.996752i	$-0.474338\pi$
$179$	2.15486	0.161062	0.0805310	+	0.996752i	$0.474338\pi$
$180$	0	0
$181$	19.3700	1.43976	0.719880	−	0.694098i	$-0.244197\pi$
$181$	19.3700	1.43976	0.719880	+	0.694098i	$0.244197\pi$
$182$	0	0
$183$	3.74802	0.277062
$184$	0	0
$185$	−4.85179	−0.356710
$186$	0	0
$187$	3.68620	0.269561
$188$	0	0
$189$	4.01952	0.292377
$190$	0	0
$191$	−2.84666	−0.205977	−0.102988	−	0.994683i	$-0.532840\pi$
$191$	−2.84666	−0.205977	−0.102988	+	0.994683i	$0.532840\pi$
$192$	0	0
$193$	−23.4825	−1.69031	−0.845153	−	0.534524i	$-0.820491\pi$
$193$	−23.4825	−1.69031	−0.845153	+	0.534524i	$0.820491\pi$
$194$	0	0
$195$	11.5307	0.825732
$196$	0	0
$197$	15.3734	1.09531	0.547656	−	0.836703i	$-0.315520\pi$
$197$	15.3734	1.09531	0.547656	+	0.836703i	$0.315520\pi$
$198$	0	0
$199$	11.9795	0.849207	0.424603	−	0.905379i	$-0.360414\pi$
$199$	11.9795	0.849207	0.424603	+	0.905379i	$0.360414\pi$
$200$	0	0
$201$	−6.94893	−0.490140
$202$	0	0
$203$	−3.03863	−0.213270
$204$	0	0
$205$	−3.08734	−0.215629
$206$	0	0
$207$	36.5953	2.54355
$208$	0	0
$209$	−3.56911	−0.246881
$210$	0	0
$211$	6.91272	0.475891	0.237946	−	0.971278i	$-0.423526\pi$
$211$	6.91272	0.475891	0.237946	+	0.971278i	$0.423526\pi$
$212$	0	0
$213$	16.4109	1.12445
$214$	0	0
$215$	−16.9154	−1.15362
$216$	0	0
$217$	−0.665094	−0.0451495
$218$	0	0
$219$	−2.65551	−0.179443
$220$	0	0
$221$	−3.68620	−0.247961
$222$	0	0
$223$	7.14758	0.478638	0.239319	−	0.970941i	$-0.423076\pi$
$223$	7.14758	0.478638	0.239319	+	0.970941i	$0.423076\pi$
$224$	0	0
$225$	57.2123	3.81415
$226$	0	0
$227$	−14.5051	−0.962741	−0.481370	−	0.876517i	$-0.659861\pi$
$227$	−14.5051	−0.962741	−0.481370	+	0.876517i	$0.659861\pi$
$228$	0	0
$229$	−27.4535	−1.81418	−0.907088	−	0.420941i	$-0.861700\pi$
$229$	−27.4535	−1.81418	−0.907088	+	0.420941i	$0.861700\pi$
$230$	0	0
$231$	−2.73324	−0.179834
$232$	0	0
$233$	7.75908	0.508314	0.254157	−	0.967163i	$-0.418202\pi$
$233$	7.75908	0.508314	0.254157	+	0.967163i	$0.418202\pi$
$234$	0	0
$235$	−15.0348	−0.980760
$236$	0	0
$237$	−25.8429	−1.67868
$238$	0	0
$239$	12.6160	0.816062	0.408031	−	0.912968i	$-0.366215\pi$
$239$	12.6160	0.816062	0.408031	+	0.912968i	$0.366215\pi$
$240$	0	0
$241$	−4.80414	−0.309462	−0.154731	−	0.987957i	$-0.549451\pi$
$241$	−4.80414	−0.309462	−0.154731	+	0.987957i	$0.549451\pi$
$242$	0	0
$243$	−18.6880	−1.19884
$244$	0	0
$245$	4.21870	0.269523
$246$	0	0
$247$	3.56911	0.227097
$248$	0	0
$249$	−17.9951	−1.14039
$250$	0	0
$251$	19.9337	1.25820	0.629101	−	0.777324i	$-0.283423\pi$
$251$	19.9337	1.25820	0.629101	+	0.777324i	$0.283423\pi$
$252$	0	0
$253$	−8.18576	−0.514634
$254$	0	0
$255$	−42.5045	−2.66174
$256$	0	0
$257$	−15.6910	−0.978775	−0.489387	−	0.872066i	$-0.662780\pi$
$257$	−15.6910	−0.978775	−0.489387	+	0.872066i	$0.662780\pi$
$258$	0	0
$259$	−1.15007	−0.0714617
$260$	0	0
$261$	−13.5845	−0.840861
$262$	0	0
$263$	−1.25908	−0.0776384	−0.0388192	−	0.999246i	$-0.512360\pi$
$263$	−1.25908	−0.0776384	−0.0388192	+	0.999246i	$0.512360\pi$
$264$	0	0
$265$	24.3729	1.49721
$266$	0	0
$267$	−31.1355	−1.90546
$268$	0	0
$269$	−17.2826	−1.05374	−0.526870	−	0.849946i	$-0.676635\pi$
$269$	−17.2826	−1.05374	−0.526870	+	0.849946i	$0.676635\pi$
$270$	0	0
$271$	−8.99153	−0.546196	−0.273098	−	0.961986i	$-0.588048\pi$
$271$	−8.99153	−0.546196	−0.273098	+	0.961986i	$0.588048\pi$
$272$	0	0
$273$	2.73324	0.165423
$274$	0	0
$275$	−12.7974	−0.771715
$276$	0	0
$277$	−4.85998	−0.292008	−0.146004	−	0.989284i	$-0.546641\pi$
$277$	−4.85998	−0.292008	−0.146004	+	0.989284i	$0.546641\pi$
$278$	0	0
$279$	−2.97337	−0.178011
$280$	0	0
$281$	29.4877	1.75909	0.879543	−	0.475819i	$-0.157848\pi$
$281$	29.4877	1.75909	0.879543	+	0.475819i	$0.157848\pi$
$282$	0	0
$283$	−9.57560	−0.569210	−0.284605	−	0.958645i	$-0.591862\pi$
$283$	−9.57560	−0.569210	−0.284605	+	0.958645i	$0.591862\pi$
$284$	0	0
$285$	41.1544	2.43778
$286$	0	0
$287$	−0.731821	−0.0431980
$288$	0	0
$289$	−3.41195	−0.200703
$290$	0	0
$291$	12.0083	0.703939
$292$	0	0
$293$	33.4536	1.95438	0.977191	−	0.212362i	$-0.0681156\pi$
$293$	33.4536	1.95438	0.977191	+	0.212362i	$0.0681156\pi$
$294$	0	0
$295$	−21.4117	−1.24664
$296$	0	0
$297$	−4.01952	−0.233236
$298$	0	0
$299$	8.18576	0.473395
$300$	0	0
$301$	−4.00963	−0.231111
$302$	0	0
$303$	−16.4364	−0.944246
$304$	0	0
$305$	5.78500	0.331248
$306$	0	0
$307$	11.4832	0.655381	0.327691	−	0.944785i	$-0.393730\pi$
$307$	11.4832	0.655381	0.327691	+	0.944785i	$0.393730\pi$
$308$	0	0
$309$	−2.59163	−0.147432
$310$	0	0
$311$	7.85828	0.445602	0.222801	−	0.974864i	$-0.428480\pi$
$311$	7.85828	0.445602	0.222801	+	0.974864i	$0.428480\pi$
$312$	0	0
$313$	22.3927	1.26571	0.632855	−	0.774270i	$-0.281883\pi$
$313$	22.3927	1.26571	0.632855	+	0.774270i	$0.281883\pi$
$314$	0	0
$315$	18.8601	1.06265
$316$	0	0
$317$	−20.4517	−1.14868	−0.574341	−	0.818616i	$-0.694741\pi$
$317$	−20.4517	−1.14868	−0.574341	+	0.818616i	$0.694741\pi$
$318$	0	0
$319$	3.03863	0.170131
$320$	0	0
$321$	−30.8906	−1.72415
$322$	0	0
$323$	−13.1565	−0.732045
$324$	0	0
$325$	12.7974	0.709874
$326$	0	0
$327$	−53.8287	−2.97674
$328$	0	0
$329$	−3.56384	−0.196481
$330$	0	0
$331$	15.3643	0.844500	0.422250	−	0.906479i	$-0.361240\pi$
$331$	15.3643	0.844500	0.422250	+	0.906479i	$0.361240\pi$
$332$	0	0
$333$	−5.14149	−0.281752
$334$	0	0
$335$	−10.7255	−0.585998
$336$	0	0
$337$	−34.0710	−1.85597	−0.927983	−	0.372623i	$-0.878459\pi$
$337$	−34.0710	−1.85597	−0.927983	+	0.372623i	$0.878459\pi$
$338$	0	0
$339$	−18.3684	−0.997635
$340$	0	0
$341$	0.665094	0.0360168
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	94.3877	5.08167
$346$	0	0
$347$	−3.92407	−0.210655	−0.105327	−	0.994438i	$-0.533589\pi$
$347$	−3.92407	−0.210655	−0.105327	+	0.994438i	$0.533589\pi$
$348$	0	0
$349$	1.27611	0.0683084	0.0341542	−	0.999417i	$-0.489126\pi$
$349$	1.27611	0.0683084	0.0341542	+	0.999417i	$0.489126\pi$
$350$	0	0
$351$	4.01952	0.214546
$352$	0	0
$353$	−8.11749	−0.432050	−0.216025	−	0.976388i	$-0.569309\pi$
$353$	−8.11749	−0.432050	−0.216025	+	0.976388i	$0.569309\pi$
$354$	0	0
$355$	25.3298	1.34437
$356$	0	0
$357$	−10.0753	−0.533240
$358$	0	0
$359$	−0.622157	−0.0328362	−0.0164181	−	0.999865i	$-0.505226\pi$
$359$	−0.622157	−0.0328362	−0.0164181	+	0.999865i	$0.505226\pi$
$360$	0	0
$361$	−6.26144	−0.329550
$362$	0	0
$363$	2.73324	0.143458
$364$	0	0
$365$	−4.09873	−0.214537
$366$	0	0
$367$	−38.0741	−1.98745	−0.993727	−	0.111834i	$-0.964328\pi$
$367$	−38.0741	−1.98745	−0.993727	+	0.111834i	$0.964328\pi$
$368$	0	0
$369$	−3.27168	−0.170317
$370$	0	0
$371$	5.77735	0.299945
$372$	0	0
$373$	13.9569	0.722659	0.361329	−	0.932438i	$-0.382323\pi$
$373$	13.9569	0.722659	0.361329	+	0.932438i	$0.382323\pi$
$374$	0	0
$375$	89.9101	4.64294
$376$	0	0
$377$	−3.03863	−0.156498
$378$	0	0
$379$	28.4103	1.45934	0.729671	−	0.683799i	$-0.239673\pi$
$379$	28.4103	1.45934	0.729671	+	0.683799i	$0.239673\pi$
$380$	0	0
$381$	−23.8070	−1.21967
$382$	0	0
$383$	16.6044	0.848445	0.424223	−	0.905558i	$-0.360547\pi$
$383$	16.6044	0.848445	0.424223	+	0.905558i	$0.360547\pi$
$384$	0	0
$385$	−4.21870	−0.215005
$386$	0	0
$387$	−17.9255	−0.911202
$388$	0	0
$389$	−23.7356	−1.20344	−0.601722	−	0.798705i	$-0.705519\pi$
$389$	−23.7356	−1.20344	−0.601722	+	0.798705i	$0.705519\pi$
$390$	0	0
$391$	−30.1743	−1.52598
$392$	0	0
$393$	8.43524	0.425501
$394$	0	0
$395$	−39.8879	−2.00698
$396$	0	0
$397$	17.3241	0.869469	0.434735	−	0.900559i	$-0.356842\pi$
$397$	17.3241	0.869469	0.434735	+	0.900559i	$0.356842\pi$
$398$	0	0
$399$	9.75524	0.488373
$400$	0	0
$401$	1.89902	0.0948327	0.0474164	−	0.998875i	$-0.484901\pi$
$401$	1.89902	0.0948327	0.0474164	+	0.998875i	$0.484901\pi$
$402$	0	0
$403$	−0.665094	−0.0331307
$404$	0	0
$405$	−10.2325	−0.508457
$406$	0	0
$407$	1.15007	0.0570067
$408$	0	0
$409$	−33.6546	−1.66411	−0.832057	−	0.554690i	$-0.812837\pi$
$409$	−33.6546	−1.66411	−0.832057	+	0.554690i	$0.812837\pi$
$410$	0	0
$411$	−28.8357	−1.42236
$412$	0	0
$413$	−5.07544	−0.249746
$414$	0	0
$415$	−27.7750	−1.36342
$416$	0	0
$417$	16.8751	0.826379
$418$	0	0
$419$	40.0795	1.95801	0.979006	−	0.203830i	$-0.0653391\pi$
$419$	40.0795	1.95801	0.979006	+	0.203830i	$0.0653391\pi$
$420$	0	0
$421$	−23.1198	−1.12679	−0.563396	−	0.826187i	$-0.690505\pi$
$421$	−23.1198	−1.12679	−0.563396	+	0.826187i	$0.690505\pi$
$422$	0	0
$423$	−15.9325	−0.774665
$424$	0	0
$425$	−47.1739	−2.28827
$426$	0	0
$427$	1.37127	0.0663606
$428$	0	0
$429$	−2.73324	−0.131962
$430$	0	0
$431$	−29.0043	−1.39709	−0.698543	−	0.715568i	$-0.746168\pi$
$431$	−29.0043	−1.39709	−0.698543	+	0.715568i	$0.746168\pi$
$432$	0	0
$433$	22.8560	1.09839	0.549195	−	0.835694i	$-0.314934\pi$
$433$	22.8560	1.09839	0.549195	+	0.835694i	$0.314934\pi$
$434$	0	0
$435$	−35.0376	−1.67993
$436$	0	0
$437$	29.2159	1.39759
$438$	0	0
$439$	−8.74881	−0.417558	−0.208779	−	0.977963i	$-0.566949\pi$
$439$	−8.74881	−0.417558	−0.208779	+	0.977963i	$0.566949\pi$
$440$	0	0
$441$	4.47060	0.212886
$442$	0	0
$443$	−1.51435	−0.0719488	−0.0359744	−	0.999353i	$-0.511453\pi$
$443$	−1.51435	−0.0719488	−0.0359744	+	0.999353i	$0.511453\pi$
$444$	0	0
$445$	−48.0570	−2.27812
$446$	0	0
$447$	41.8453	1.97922
$448$	0	0
$449$	−36.7406	−1.73390	−0.866948	−	0.498399i	$-0.833921\pi$
$449$	−36.7406	−1.73390	−0.866948	+	0.498399i	$0.833921\pi$
$450$	0	0
$451$	0.731821	0.0344601
$452$	0	0
$453$	6.58353	0.309321
$454$	0	0
$455$	4.21870	0.197776
$456$	0	0
$457$	19.6682	0.920039	0.460020	−	0.887909i	$-0.347842\pi$
$457$	19.6682	0.920039	0.460020	+	0.887909i	$0.347842\pi$
$458$	0	0
$459$	−14.8167	−0.691586
$460$	0	0
$461$	−32.2791	−1.50339	−0.751694	−	0.659512i	$-0.770763\pi$
$461$	−32.2791	−1.50339	−0.751694	+	0.659512i	$0.770763\pi$
$462$	0	0
$463$	0.844993	0.0392701	0.0196351	−	0.999807i	$-0.493750\pi$
$463$	0.844993	0.0392701	0.0196351	+	0.999807i	$0.493750\pi$
$464$	0	0
$465$	−7.66901	−0.355642
$466$	0	0
$467$	6.73165	0.311504	0.155752	−	0.987796i	$-0.450220\pi$
$467$	6.73165	0.311504	0.155752	+	0.987796i	$0.450220\pi$
$468$	0	0
$469$	−2.54238	−0.117396
$470$	0	0
$471$	−32.2684	−1.48685
$472$	0	0
$473$	4.00963	0.184363
$474$	0	0
$475$	45.6755	2.09574
$476$	0	0
$477$	25.8282	1.18259
$478$	0	0
$479$	−0.546818	−0.0249848	−0.0124924	−	0.999922i	$-0.503977\pi$
$479$	−0.546818	−0.0249848	−0.0124924	+	0.999922i	$0.503977\pi$
$480$	0	0
$481$	−1.15007	−0.0524385
$482$	0	0
$483$	22.3737	1.01804
$484$	0	0
$485$	18.5346	0.841611
$486$	0	0
$487$	33.0356	1.49699	0.748493	−	0.663143i	$-0.230778\pi$
$487$	33.0356	1.49699	0.748493	+	0.663143i	$0.230778\pi$
$488$	0	0
$489$	59.8479	2.70641
$490$	0	0
$491$	−9.90152	−0.446850	−0.223425	−	0.974721i	$-0.571724\pi$
$491$	−9.90152	−0.446850	−0.223425	+	0.974721i	$0.571724\pi$
$492$	0	0
$493$	11.2010	0.504468
$494$	0	0
$495$	−18.8601	−0.847700
$496$	0	0
$497$	6.00418	0.269324
$498$	0	0
$499$	−19.6274	−0.878643	−0.439321	−	0.898330i	$-0.644781\pi$
$499$	−19.6274	−0.878643	−0.439321	+	0.898330i	$0.644781\pi$
$500$	0	0
$501$	67.9881	3.03748
$502$	0	0
$503$	−3.63642	−0.162140	−0.0810700	−	0.996708i	$-0.525834\pi$
$503$	−3.63642	−0.162140	−0.0810700	+	0.996708i	$0.525834\pi$
$504$	0	0
$505$	−25.3692	−1.12892
$506$	0	0
$507$	2.73324	0.121387
$508$	0	0
$509$	−27.3183	−1.21086	−0.605432	−	0.795897i	$-0.707000\pi$
$509$	−27.3183	−1.21086	−0.605432	+	0.795897i	$0.707000\pi$
$510$	0	0
$511$	−0.971562	−0.0429794
$512$	0	0
$513$	14.3461	0.633396
$514$	0	0
$515$	−4.00012	−0.176266
$516$	0	0
$517$	3.56384	0.156737
$518$	0	0
$519$	44.2265	1.94133
$520$	0	0
$521$	17.7418	0.777283	0.388642	−	0.921389i	$-0.372944\pi$
$521$	17.7418	0.777283	0.388642	+	0.921389i	$0.372944\pi$
$522$	0	0
$523$	−9.54135	−0.417214	−0.208607	−	0.978000i	$-0.566893\pi$
$523$	−9.54135	−0.417214	−0.208607	+	0.978000i	$0.566893\pi$
$524$	0	0
$525$	34.9785	1.52659
$526$	0	0
$527$	2.45167	0.106796
$528$	0	0
$529$	44.0067	1.91333
$530$	0	0
$531$	−22.6903	−0.984674
$532$	0	0
$533$	−0.731821	−0.0316987
$534$	0	0
$535$	−47.6790	−2.06134
$536$	0	0
$537$	5.88976	0.254162
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	29.5455	1.27026	0.635130	−	0.772405i	$-0.280946\pi$
$541$	29.5455	1.27026	0.635130	+	0.772405i	$0.280946\pi$
$542$	0	0
$543$	52.9429	2.27200
$544$	0	0
$545$	−83.0836	−3.55891
$546$	0	0
$547$	−15.2860	−0.653584	−0.326792	−	0.945096i	$-0.605968\pi$
$547$	−15.2860	−0.653584	−0.326792	+	0.945096i	$0.605968\pi$
$548$	0	0
$549$	6.13042	0.261640
$550$	0	0
$551$	−10.8452	−0.462022
$552$	0	0
$553$	−9.45503	−0.402069
$554$	0	0
$555$	−13.2611	−0.562902
$556$	0	0
$557$	−42.1064	−1.78411	−0.892054	−	0.451930i	$-0.850736\pi$
$557$	−42.1064	−1.78411	−0.892054	+	0.451930i	$0.850736\pi$
$558$	0	0
$559$	−4.00963	−0.169589
$560$	0	0
$561$	10.0753	0.425378
$562$	0	0
$563$	4.12604	0.173892	0.0869458	−	0.996213i	$-0.472289\pi$
$563$	4.12604	0.173892	0.0869458	+	0.996213i	$0.472289\pi$
$564$	0	0
$565$	−28.3513	−1.19275
$566$	0	0
$567$	−2.42551	−0.101862
$568$	0	0
$569$	43.8191	1.83699	0.918496	−	0.395431i	$-0.129405\pi$
$569$	43.8191	1.83699	0.918496	+	0.395431i	$0.129405\pi$
$570$	0	0
$571$	34.1508	1.42917	0.714583	−	0.699551i	$-0.246617\pi$
$571$	34.1508	1.42917	0.714583	+	0.699551i	$0.246617\pi$
$572$	0	0
$573$	−7.78060	−0.325039
$574$	0	0
$575$	104.757	4.36866
$576$	0	0
$577$	1.04101	0.0433380	0.0216690	−	0.999765i	$-0.493102\pi$
$577$	1.04101	0.0433380	0.0216690	+	0.999765i	$0.493102\pi$
$578$	0	0
$579$	−64.1833	−2.66737
$580$	0	0
$581$	−6.58379	−0.273142
$582$	0	0
$583$	−5.77735	−0.239273
$584$	0	0
$585$	18.8601	0.779771
$586$	0	0
$587$	−16.2166	−0.669333	−0.334666	−	0.942337i	$-0.608624\pi$
$587$	−16.2166	−0.669333	−0.334666	+	0.942337i	$0.608624\pi$
$588$	0	0
$589$	−2.37379	−0.0978105
$590$	0	0
$591$	42.0193	1.72844
$592$	0	0
$593$	−29.8966	−1.22771	−0.613854	−	0.789420i	$-0.710382\pi$
$593$	−29.8966	−1.22771	−0.613854	+	0.789420i	$0.710382\pi$
$594$	0	0
$595$	−15.5510	−0.637527
$596$	0	0
$597$	32.7430	1.34008
$598$	0	0
$599$	−22.1319	−0.904283	−0.452142	−	0.891946i	$-0.649340\pi$
$599$	−22.1319	−0.904283	−0.452142	+	0.891946i	$0.649340\pi$
$600$	0	0
$601$	29.4084	1.19959	0.599797	−	0.800152i	$-0.295248\pi$
$601$	29.4084	1.19959	0.599797	+	0.800152i	$0.295248\pi$
$602$	0	0
$603$	−11.3660	−0.462858
$604$	0	0
$605$	4.21870	0.171515
$606$	0	0
$607$	9.05345	0.367468	0.183734	−	0.982976i	$-0.441181\pi$
$607$	9.05345	0.367468	0.183734	+	0.982976i	$0.441181\pi$
$608$	0	0
$609$	−8.30532	−0.336548
$610$	0	0
$611$	−3.56384	−0.144177
$612$	0	0
$613$	−29.7272	−1.20067	−0.600336	−	0.799748i	$-0.704967\pi$
$613$	−29.7272	−1.20067	−0.600336	+	0.799748i	$0.704967\pi$
$614$	0	0
$615$	−8.43843	−0.340270
$616$	0	0
$617$	4.05540	0.163264	0.0816322	−	0.996663i	$-0.473987\pi$
$617$	4.05540	0.163264	0.0816322	+	0.996663i	$0.473987\pi$
$618$	0	0
$619$	−27.0791	−1.08840	−0.544200	−	0.838955i	$-0.683167\pi$
$619$	−27.0791	−1.08840	−0.544200	+	0.838955i	$0.683167\pi$
$620$	0	0
$621$	32.9028	1.32034
$622$	0	0
$623$	−11.3914	−0.456388
$624$	0	0
$625$	74.7872	2.99149
$626$	0	0
$627$	−9.75524	−0.389587
$628$	0	0
$629$	4.23937	0.169035
$630$	0	0
$631$	40.0291	1.59354	0.796768	−	0.604286i	$-0.206542\pi$
$631$	40.0291	1.59354	0.796768	+	0.604286i	$0.206542\pi$
$632$	0	0
$633$	18.8941	0.750974
$634$	0	0
$635$	−36.7457	−1.45821
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	26.8423	1.06187
$640$	0	0
$641$	40.5257	1.60067	0.800335	−	0.599553i	$-0.204655\pi$
$641$	40.5257	1.60067	0.800335	+	0.599553i	$0.204655\pi$
$642$	0	0
$643$	−21.2916	−0.839660	−0.419830	−	0.907603i	$-0.637910\pi$
$643$	−21.2916	−0.839660	−0.419830	+	0.907603i	$0.637910\pi$
$644$	0	0
$645$	−46.2339	−1.82046
$646$	0	0
$647$	−26.8765	−1.05662	−0.528312	−	0.849050i	$-0.677175\pi$
$647$	−26.8765	−1.05662	−0.528312	+	0.849050i	$0.677175\pi$
$648$	0	0
$649$	5.07544	0.199228
$650$	0	0
$651$	−1.81786	−0.0712476
$652$	0	0
$653$	−8.43910	−0.330248	−0.165124	−	0.986273i	$-0.552802\pi$
$653$	−8.43910	−0.330248	−0.165124	+	0.986273i	$0.552802\pi$
$654$	0	0
$655$	13.0196	0.508718
$656$	0	0
$657$	−4.34347	−0.169455
$658$	0	0
$659$	−33.6908	−1.31241	−0.656203	−	0.754585i	$-0.727838\pi$
$659$	−33.6908	−1.31241	−0.656203	+	0.754585i	$0.727838\pi$
$660$	0	0
$661$	−26.4727	−1.02967	−0.514835	−	0.857290i	$-0.672147\pi$
$661$	−26.4727	−1.02967	−0.514835	+	0.857290i	$0.672147\pi$
$662$	0	0
$663$	−10.0753	−0.391291
$664$	0	0
$665$	15.0570	0.583886
$666$	0	0
$667$	−24.8735	−0.963107
$668$	0	0
$669$	19.5361	0.755308
$670$	0	0
$671$	−1.37127	−0.0529375
$672$	0	0
$673$	−2.69773	−0.103990	−0.0519949	−	0.998647i	$-0.516558\pi$
$673$	−2.69773	−0.103990	−0.0519949	+	0.998647i	$0.516558\pi$
$674$	0	0
$675$	51.4395	1.97991
$676$	0	0
$677$	−35.7183	−1.37276	−0.686382	−	0.727241i	$-0.740802\pi$
$677$	−35.7183	−1.37276	−0.686382	+	0.727241i	$0.740802\pi$
$678$	0	0
$679$	4.39343	0.168604
$680$	0	0
$681$	−39.6461	−1.51924
$682$	0	0
$683$	10.4776	0.400914	0.200457	−	0.979703i	$-0.435757\pi$
$683$	10.4776	0.400914	0.200457	+	0.979703i	$0.435757\pi$
$684$	0	0
$685$	−44.5073	−1.70054
$686$	0	0
$687$	−75.0369	−2.86284
$688$	0	0
$689$	5.77735	0.220099
$690$	0	0
$691$	30.5097	1.16064	0.580322	−	0.814387i	$-0.302927\pi$
$691$	30.5097	1.16064	0.580322	+	0.814387i	$0.302927\pi$
$692$	0	0
$693$	−4.47060	−0.169824
$694$	0	0
$695$	26.0464	0.987997
$696$	0	0
$697$	2.69764	0.102180
$698$	0	0
$699$	21.2074	0.802139
$700$	0	0
$701$	−28.2302	−1.06624	−0.533121	−	0.846039i	$-0.678981\pi$
$701$	−28.2302	−1.06624	−0.533121	+	0.846039i	$0.678981\pi$
$702$	0	0
$703$	−4.10472	−0.154812
$704$	0	0
$705$	−41.0936	−1.54768
$706$	0	0
$707$	−6.01351	−0.226162
$708$	0	0
$709$	3.67798	0.138129	0.0690647	−	0.997612i	$-0.477999\pi$
$709$	3.67798	0.138129	0.0690647	+	0.997612i	$0.477999\pi$
$710$	0	0
$711$	−42.2697	−1.58524
$712$	0	0
$713$	−5.44430	−0.203891
$714$	0	0
$715$	−4.21870	−0.157770
$716$	0	0
$717$	34.4826	1.28778
$718$	0	0
$719$	−24.6723	−0.920120	−0.460060	−	0.887888i	$-0.652172\pi$
$719$	−24.6723	−0.920120	−0.460060	+	0.887888i	$0.652172\pi$
$720$	0	0
$721$	−0.948188	−0.0353124
$722$	0	0
$723$	−13.1309	−0.488342
$724$	0	0
$725$	−38.8867	−1.44422
$726$	0	0
$727$	−14.1930	−0.526390	−0.263195	−	0.964743i	$-0.584776\pi$
$727$	−14.1930	−0.526390	−0.263195	+	0.964743i	$0.584776\pi$
$728$	0	0
$729$	−43.8024	−1.62231
$730$	0	0
$731$	14.7803	0.546668
$732$	0	0
$733$	20.0588	0.740887	0.370444	−	0.928855i	$-0.379206\pi$
$733$	20.0588	0.740887	0.370444	+	0.928855i	$0.379206\pi$
$734$	0	0
$735$	11.5307	0.425317
$736$	0	0
$737$	2.54238	0.0936496
$738$	0	0
$739$	−2.99227	−0.110072	−0.0550362	−	0.998484i	$-0.517527\pi$
$739$	−2.99227	−0.110072	−0.0550362	+	0.998484i	$0.517527\pi$
$740$	0	0
$741$	9.75524	0.358368
$742$	0	0
$743$	12.2033	0.447695	0.223847	−	0.974624i	$-0.428138\pi$
$743$	12.2033	0.447695	0.223847	+	0.974624i	$0.428138\pi$
$744$	0	0
$745$	64.5874	2.36630
$746$	0	0
$747$	−29.4335	−1.07692
$748$	0	0
$749$	−11.3018	−0.412960
$750$	0	0
$751$	49.9893	1.82414	0.912068	−	0.410039i	$-0.134485\pi$
$751$	49.9893	1.82414	0.912068	+	0.410039i	$0.134485\pi$
$752$	0	0
$753$	54.4835	1.98549
$754$	0	0
$755$	10.1615	0.369816
$756$	0	0
$757$	−6.69756	−0.243427	−0.121714	−	0.992565i	$-0.538839\pi$
$757$	−6.69756	−0.243427	−0.121714	+	0.992565i	$0.538839\pi$
$758$	0	0
$759$	−22.3737	−0.812112
$760$	0	0
$761$	−7.76662	−0.281540	−0.140770	−	0.990042i	$-0.544958\pi$
$761$	−7.76662	−0.281540	−0.140770	+	0.990042i	$0.544958\pi$
$762$	0	0
$763$	−19.6941	−0.712975
$764$	0	0
$765$	−69.5222	−2.51358
$766$	0	0
$767$	−5.07544	−0.183263
$768$	0	0
$769$	−22.6957	−0.818428	−0.409214	−	0.912438i	$-0.634197\pi$
$769$	−22.6957	−0.818428	−0.409214	+	0.912438i	$0.634197\pi$
$770$	0	0
$771$	−42.8872	−1.54454
$772$	0	0
$773$	8.58505	0.308783	0.154391	−	0.988010i	$-0.450658\pi$
$773$	8.58505	0.308783	0.154391	+	0.988010i	$0.450658\pi$
$774$	0	0
$775$	−8.51150	−0.305742
$776$	0	0
$777$	−3.14341	−0.112769
$778$	0	0
$779$	−2.61195	−0.0935829
$780$	0	0
$781$	−6.00418	−0.214847
$782$	0	0
$783$	−12.2138	−0.436487
$784$	0	0
$785$	−49.8056	−1.77764
$786$	0	0
$787$	14.8977	0.531046	0.265523	−	0.964105i	$-0.414455\pi$
$787$	14.8977	0.531046	0.265523	+	0.964105i	$0.414455\pi$
$788$	0	0
$789$	−3.44138	−0.122516
$790$	0	0
$791$	−6.72038	−0.238949
$792$	0	0
$793$	1.37127	0.0486954
$794$	0	0
$795$	66.6170	2.36266
$796$	0	0
$797$	24.3771	0.863481	0.431741	−	0.901998i	$-0.357900\pi$
$797$	24.3771	0.863481	0.431741	+	0.901998i	$0.357900\pi$
$798$	0	0
$799$	13.1370	0.464754
$800$	0	0
$801$	−50.9266	−1.79940
$802$	0	0
$803$	0.971562	0.0342857
$804$	0	0
$805$	34.5333	1.21714
$806$	0	0
$807$	−47.2376	−1.66284
$808$	0	0
$809$	55.0108	1.93408	0.967039	−	0.254629i	$-0.0819532\pi$
$809$	55.0108	1.93408	0.967039	+	0.254629i	$0.0819532\pi$
$810$	0	0
$811$	10.2486	0.359875	0.179938	−	0.983678i	$-0.442410\pi$
$811$	10.2486	0.359875	0.179938	+	0.983678i	$0.442410\pi$
$812$	0	0
$813$	−24.5760	−0.861918
$814$	0	0
$815$	92.3740	3.23572
$816$	0	0
$817$	−14.3108	−0.500672
$818$	0	0
$819$	4.47060	0.156216
$820$	0	0
$821$	−4.99218	−0.174228	−0.0871142	−	0.996198i	$-0.527765\pi$
$821$	−4.99218	−0.174228	−0.0871142	+	0.996198i	$0.527765\pi$
$822$	0	0
$823$	25.3432	0.883408	0.441704	−	0.897161i	$-0.354374\pi$
$823$	25.3432	0.883408	0.441704	+	0.897161i	$0.354374\pi$
$824$	0	0
$825$	−34.9785	−1.21779
$826$	0	0
$827$	46.1499	1.60479	0.802395	−	0.596793i	$-0.203559\pi$
$827$	46.1499	1.60479	0.802395	+	0.596793i	$0.203559\pi$
$828$	0	0
$829$	39.2819	1.36432	0.682158	−	0.731205i	$-0.261042\pi$
$829$	39.2819	1.36432	0.682158	+	0.731205i	$0.261042\pi$
$830$	0	0
$831$	−13.2835	−0.460799
$832$	0	0
$833$	−3.68620	−0.127719
$834$	0	0
$835$	104.938	3.63154
$836$	0	0
$837$	−2.67336	−0.0924047
$838$	0	0
$839$	10.4014	0.359097	0.179548	−	0.983749i	$-0.442536\pi$
$839$	10.4014	0.359097	0.179548	+	0.983749i	$0.442536\pi$
$840$	0	0
$841$	−19.7667	−0.681611
$842$	0	0
$843$	80.5969	2.77590
$844$	0	0
$845$	4.21870	0.145128
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−26.1724	−0.898235
$850$	0	0
$851$	−9.41417	−0.322714
$852$	0	0
$853$	−3.09021	−0.105807	−0.0529033	−	0.998600i	$-0.516848\pi$
$853$	−3.09021	−0.105807	−0.0529033	+	0.998600i	$0.516848\pi$
$854$	0	0
$855$	67.3140	2.30209
$856$	0	0
$857$	−14.8772	−0.508197	−0.254098	−	0.967178i	$-0.581779\pi$
$857$	−14.8772	−0.508197	−0.254098	+	0.967178i	$0.581779\pi$
$858$	0	0
$859$	−52.0269	−1.77514	−0.887568	−	0.460678i	$-0.847606\pi$
$859$	−52.0269	−1.77514	−0.887568	+	0.460678i	$0.847606\pi$
$860$	0	0
$861$	−2.00024	−0.0681681
$862$	0	0
$863$	−24.1159	−0.820916	−0.410458	−	0.911880i	$-0.634631\pi$
$863$	−24.1159	−0.820916	−0.410458	+	0.911880i	$0.634631\pi$
$864$	0	0
$865$	68.2628	2.32100
$866$	0	0
$867$	−9.32567	−0.316716
$868$	0	0
$869$	9.45503	0.320740
$870$	0	0
$871$	−2.54238	−0.0861451
$872$	0	0
$873$	19.6413	0.664757
$874$	0	0
$875$	32.8951	1.11206
$876$	0	0
$877$	4.68755	0.158287	0.0791437	−	0.996863i	$-0.474781\pi$
$877$	4.68755	0.158287	0.0791437	+	0.996863i	$0.474781\pi$
$878$	0	0
$879$	91.4368	3.08409
$880$	0	0
$881$	−14.3670	−0.484036	−0.242018	−	0.970272i	$-0.577809\pi$
$881$	−14.3670	−0.484036	−0.242018	+	0.970272i	$0.577809\pi$
$882$	0	0
$883$	32.8209	1.10451	0.552255	−	0.833675i	$-0.313767\pi$
$883$	32.8209	1.10451	0.552255	+	0.833675i	$0.313767\pi$
$884$	0	0
$885$	−58.5235	−1.96724
$886$	0	0
$887$	−7.11090	−0.238761	−0.119380	−	0.992849i	$-0.538091\pi$
$887$	−7.11090	−0.238761	−0.119380	+	0.992849i	$0.538091\pi$
$888$	0	0
$889$	−8.71019	−0.292130
$890$	0	0
$891$	2.42551	0.0812576
$892$	0	0
$893$	−12.7197	−0.425649
$894$	0	0
$895$	9.09073	0.303869
$896$	0	0
$897$	22.3737	0.747034
$898$	0	0
$899$	2.02098	0.0674033
$900$	0	0
$901$	−21.2964	−0.709487
$902$	0	0
$903$	−10.9593	−0.364702
$904$	0	0
$905$	81.7162	2.71634
$906$	0	0
$907$	41.0560	1.36324	0.681621	−	0.731706i	$-0.261275\pi$
$907$	41.0560	1.36324	0.681621	+	0.731706i	$0.261275\pi$
$908$	0	0
$909$	−26.8840	−0.891688
$910$	0	0
$911$	0.440806	0.0146046	0.00730228	−	0.999973i	$-0.497676\pi$
$911$	0.440806	0.0146046	0.00730228	+	0.999973i	$0.497676\pi$
$912$	0	0
$913$	6.58379	0.217892
$914$	0	0
$915$	15.8118	0.522722
$916$	0	0
$917$	3.08617	0.101914
$918$	0	0
$919$	−32.4505	−1.07044	−0.535221	−	0.844712i	$-0.679772\pi$
$919$	−32.4505	−1.07044	−0.535221	+	0.844712i	$0.679772\pi$
$920$	0	0
$921$	31.3864	1.03422
$922$	0	0
$923$	6.00418	0.197630
$924$	0	0
$925$	−14.7179	−0.483922
$926$	0	0
$927$	−4.23897	−0.139226
$928$	0	0
$929$	−46.5210	−1.52630	−0.763152	−	0.646220i	$-0.776349\pi$
$929$	−46.5210	−1.52630	−0.763152	+	0.646220i	$0.776349\pi$
$930$	0	0
$931$	3.56911	0.116973
$932$	0	0
$933$	21.4786	0.703177
$934$	0	0
$935$	15.5510	0.508571
$936$	0	0
$937$	−42.4806	−1.38778	−0.693891	−	0.720080i	$-0.744105\pi$
$937$	−42.4806	−1.38778	−0.693891	+	0.720080i	$0.744105\pi$
$938$	0	0
$939$	61.2046	1.99734
$940$	0	0
$941$	−0.820708	−0.0267543	−0.0133772	−	0.999911i	$-0.504258\pi$
$941$	−0.820708	−0.0267543	−0.0133772	+	0.999911i	$0.504258\pi$
$942$	0	0
$943$	−5.99051	−0.195078
$944$	0	0
$945$	16.9571	0.551616
$946$	0	0
$947$	−35.7552	−1.16189	−0.580943	−	0.813944i	$-0.697316\pi$
$947$	−35.7552	−1.16189	−0.580943	+	0.813944i	$0.697316\pi$
$948$	0	0
$949$	−0.971562	−0.0315382
$950$	0	0
$951$	−55.8994	−1.81266
$952$	0	0
$953$	43.9941	1.42511	0.712555	−	0.701617i	$-0.247538\pi$
$953$	43.9941	1.42511	0.712555	+	0.701617i	$0.247538\pi$
$954$	0	0
$955$	−12.0092	−0.388608
$956$	0	0
$957$	8.30532	0.268473
$958$	0	0
$959$	−10.5500	−0.340677
$960$	0	0
$961$	−30.5577	−0.985731
$962$	0	0
$963$	−50.5260	−1.62818
$964$	0	0
$965$	−99.0656	−3.18903
$966$	0	0
$967$	38.5064	1.23828	0.619141	−	0.785280i	$-0.287481\pi$
$967$	38.5064	1.23828	0.619141	+	0.785280i	$0.287481\pi$
$968$	0	0
$969$	−35.9597	−1.15519
$970$	0	0
$971$	60.7789	1.95049	0.975244	−	0.221131i	$-0.0709750\pi$
$971$	60.7789	1.95049	0.975244	+	0.221131i	$0.0709750\pi$
$972$	0	0
$973$	6.17404	0.197931
$974$	0	0
$975$	34.9785	1.12021
$976$	0	0
$977$	−44.9155	−1.43697	−0.718487	−	0.695541i	$-0.755165\pi$
$977$	−44.9155	−1.43697	−0.718487	+	0.695541i	$0.755165\pi$
$978$	0	0
$979$	11.3914	0.364072
$980$	0	0
$981$	−88.0446	−2.81105
$982$	0	0
$983$	33.7448	1.07629	0.538147	−	0.842851i	$-0.319125\pi$
$983$	33.7448	1.07629	0.538147	+	0.842851i	$0.319125\pi$
$984$	0	0
$985$	64.8560	2.06648
$986$	0	0
$987$	−9.74082	−0.310054
$988$	0	0
$989$	−32.8218	−1.04367
$990$	0	0
$991$	12.9973	0.412873	0.206436	−	0.978460i	$-0.433813\pi$
$991$	12.9973	0.412873	0.206436	+	0.978460i	$0.433813\pi$
$992$	0	0
$993$	41.9944	1.33265
$994$	0	0
$995$	50.5381	1.60216
$996$	0	0
$997$	−21.7672	−0.689373	−0.344687	−	0.938718i	$-0.612015\pi$
$997$	−21.7672	−0.689373	−0.344687	+	0.938718i	$0.612015\pi$
$998$	0	0
$999$	−4.62271	−0.146256

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.t.1.9	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.t.1.9	✓	10	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)

\(f(q)\)	\(=\)	\(q+2.73324 q^{3} +4.21870 q^{5} +1.00000 q^{7} +4.47060 q^{9} +O(q^{10})\)	\(q+2.73324 q^{3} +4.21870 q^{5} +1.00000 q^{7} +4.47060 q^{9} -1.00000 q^{11} +1.00000 q^{13} +11.5307 q^{15} -3.68620 q^{17} +3.56911 q^{19} +2.73324 q^{21} +8.18576 q^{23} +12.7974 q^{25} +4.01952 q^{27} -3.03863 q^{29} -0.665094 q^{31} -2.73324 q^{33} +4.21870 q^{35} -1.15007 q^{37} +2.73324 q^{39} -0.731821 q^{41} -4.00963 q^{43} +18.8601 q^{45} -3.56384 q^{47} +1.00000 q^{49} -10.0753 q^{51} +5.77735 q^{53} -4.21870 q^{55} +9.75524 q^{57} -5.07544 q^{59} +1.37127 q^{61} +4.47060 q^{63} +4.21870 q^{65} -2.54238 q^{67} +22.3737 q^{69} +6.00418 q^{71} -0.971562 q^{73} +34.9785 q^{75} -1.00000 q^{77} -9.45503 q^{79} -2.42551 q^{81} -6.58379 q^{83} -15.5510 q^{85} -8.30532 q^{87} -11.3914 q^{89} +1.00000 q^{91} -1.81786 q^{93} +15.0570 q^{95} +4.39343 q^{97} -4.47060 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + q^{3} + 3 q^{5} + 10 q^{7} + 5 q^{9}+O(q^{10}) \) 10 * q + q^3 + 3 * q^5 + 10 * q^7 + 5 * q^9	\( 10 q + q^{3} + 3 q^{5} + 10 q^{7} + 5 q^{9} - 10 q^{11} + 10 q^{13} + 9 q^{15} + 7 q^{17} + 17 q^{19} + q^{21} + 15 q^{23} + 13 q^{25} + 7 q^{27} - q^{29} - 6 q^{31} - q^{33} + 3 q^{35} - 12 q^{37} + q^{39} + 4 q^{41} + 17 q^{43} - 15 q^{45} + 21 q^{47} + 10 q^{49} + 3 q^{51} + 20 q^{53} - 3 q^{55} + 12 q^{57} + 9 q^{59} + 6 q^{61} + 5 q^{63} + 3 q^{65} + 26 q^{67} + 30 q^{69} + 18 q^{71} - 4 q^{73} + 48 q^{75} - 10 q^{77} + 19 q^{79} - 14 q^{81} + 40 q^{83} - 25 q^{85} - 25 q^{87} - 19 q^{89} + 10 q^{91} + q^{93} + 11 q^{95} - 22 q^{97} - 5 q^{99}+O(q^{100}) \) 10 * q + q^3 + 3 * q^5 + 10 * q^7 + 5 * q^9 - 10 * q^11 + 10 * q^13 + 9 * q^15 + 7 * q^17 + 17 * q^19 + q^21 + 15 * q^23 + 13 * q^25 + 7 * q^27 - q^29 - 6 * q^31 - q^33 + 3 * q^35 - 12 * q^37 + q^39 + 4 * q^41 + 17 * q^43 - 15 * q^45 + 21 * q^47 + 10 * q^49 + 3 * q^51 + 20 * q^53 - 3 * q^55 + 12 * q^57 + 9 * q^59 + 6 * q^61 + 5 * q^63 + 3 * q^65 + 26 * q^67 + 30 * q^69 + 18 * q^71 - 4 * q^73 + 48 * q^75 - 10 * q^77 + 19 * q^79 - 14 * q^81 + 40 * q^83 - 25 * q^85 - 25 * q^87 - 19 * q^89 + 10 * q^91 + q^93 + 11 * q^95 - 22 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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