LMFDB - Embedded newform 8008.2.a.y.1.11

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:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 3 x^{13} - 27 x^{12} + 78 x^{11} + 273 x^{10} - 750 x^{9} - 1306 x^{8} + 3378 x^{7} + \cdots - 128 $ x^14 - 3x^13 - 27x^12 + 78x^11 + 273x^10 - 750x^9 - 1306x^8 + 3378x^7 + 2996x^6 - 7275x^5 - 2804x^4 + 6417x^3 + 538x^2 - 1032*x - 128
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-1.85932$ 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)

$f(q)$	$=$	$q+1.85932 q^{3} +1.13919 q^{5} +1.00000 q^{7} +0.457073 q^{9} +O(q^{10})$	$q+1.85932 q^{3} +1.13919 q^{5} +1.00000 q^{7} +0.457073 q^{9} -1.00000 q^{11} -1.00000 q^{13} +2.11812 q^{15} -2.43965 q^{17} -4.36447 q^{19} +1.85932 q^{21} +4.06162 q^{23} -3.70225 q^{25} -4.72812 q^{27} -0.412473 q^{29} -8.12679 q^{31} -1.85932 q^{33} +1.13919 q^{35} -7.51813 q^{37} -1.85932 q^{39} -0.210128 q^{41} +8.57641 q^{43} +0.520692 q^{45} -8.98496 q^{47} +1.00000 q^{49} -4.53608 q^{51} +8.77822 q^{53} -1.13919 q^{55} -8.11495 q^{57} -2.23049 q^{59} -6.23803 q^{61} +0.457073 q^{63} -1.13919 q^{65} +11.2185 q^{67} +7.55186 q^{69} -7.84719 q^{71} -15.8894 q^{73} -6.88367 q^{75} -1.00000 q^{77} -5.38936 q^{79} -10.1623 q^{81} -5.74479 q^{83} -2.77921 q^{85} -0.766920 q^{87} +0.577855 q^{89} -1.00000 q^{91} -15.1103 q^{93} -4.97195 q^{95} -2.41830 q^{97} -0.457073 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9}+O(q^{10}) $ 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9	$ 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9} - 14 q^{11} - 14 q^{13} - 6 q^{15} - 6 q^{17} - 13 q^{19} - 3 q^{21} - 9 q^{23} + 22 q^{25} - 18 q^{27} + 2 q^{29} - 2 q^{31} + 3 q^{33} - 6 q^{35} - q^{37} + 3 q^{39} - 16 q^{41} - 15 q^{43} - 44 q^{45} - 8 q^{47} + 14 q^{49} - 14 q^{51} - 6 q^{53} + 6 q^{55} - 10 q^{57} - 36 q^{59} - 19 q^{61} + 21 q^{63} + 6 q^{65} - 34 q^{67} - q^{69} - 10 q^{71} + 9 q^{73} - 44 q^{75} - 14 q^{77} - q^{79} + 42 q^{81} - 56 q^{83} + 21 q^{85} - 5 q^{87} - 14 q^{89} - 14 q^{91} - 20 q^{93} + q^{95} - 14 q^{97} - 21 q^{99}+O(q^{100}) $ 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9 - 14 * q^11 - 14 * q^13 - 6 * q^15 - 6 * q^17 - 13 * q^19 - 3 * q^21 - 9 * q^23 + 22 * q^25 - 18 * q^27 + 2 * q^29 - 2 * q^31 + 3 * q^33 - 6 * q^35 - q^37 + 3 * q^39 - 16 * q^41 - 15 * q^43 - 44 * q^45 - 8 * q^47 + 14 * q^49 - 14 * q^51 - 6 * q^53 + 6 * q^55 - 10 * q^57 - 36 * q^59 - 19 * q^61 + 21 * q^63 + 6 * q^65 - 34 * q^67 - q^69 - 10 * q^71 + 9 * q^73 - 44 * q^75 - 14 * q^77 - q^79 + 42 * q^81 - 56 * q^83 + 21 * q^85 - 5 * q^87 - 14 * q^89 - 14 * q^91 - 20 * q^93 + q^95 - 14 * q^97 - 21 * 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
$3$	1.85932	1.07348	0.536740	−	0.843748i	$-0.319656\pi$
$3$	1.85932	1.07348	0.536740	+	0.843748i	$0.319656\pi$
$4$	0	0
$5$	1.13919	0.509460	0.254730	−	0.967012i	$-0.418013\pi$
$5$	1.13919	0.509460	0.254730	+	0.967012i	$0.418013\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0.457073	0.152358
$10$	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
$15$	2.11812	0.546895
$16$	0	0
$17$	−2.43965	−0.591701	−0.295850	−	0.955234i	$-0.595603\pi$
$17$	−2.43965	−0.591701	−0.295850	+	0.955234i	$0.595603\pi$
$18$	0	0
$19$	−4.36447	−1.00128	−0.500639	−	0.865656i	$-0.666902\pi$
$19$	−4.36447	−1.00128	−0.500639	+	0.865656i	$0.666902\pi$
$20$	0	0
$21$	1.85932	0.405737
$22$	0	0
$23$	4.06162	0.846907	0.423454	−	0.905918i	$-0.360818\pi$
$23$	4.06162	0.846907	0.423454	+	0.905918i	$0.360818\pi$
$24$	0	0
$25$	−3.70225	−0.740450
$26$	0	0
$27$	−4.72812	−0.909927
$28$	0	0
$29$	−0.412473	−0.0765943	−0.0382972	−	0.999266i	$-0.512193\pi$
$29$	−0.412473	−0.0765943	−0.0382972	+	0.999266i	$0.512193\pi$
$30$	0	0
$31$	−8.12679	−1.45961	−0.729807	−	0.683653i	$-0.760390\pi$
$31$	−8.12679	−1.45961	−0.729807	+	0.683653i	$0.760390\pi$
$32$	0	0
$33$	−1.85932	−0.323666
$34$	0	0
$35$	1.13919	0.192558
$36$	0	0
$37$	−7.51813	−1.23597	−0.617986	−	0.786189i	$-0.712051\pi$
$37$	−7.51813	−1.23597	−0.617986	+	0.786189i	$0.712051\pi$
$38$	0	0
$39$	−1.85932	−0.297730
$40$	0	0
$41$	−0.210128	−0.0328165	−0.0164083	−	0.999865i	$-0.505223\pi$
$41$	−0.210128	−0.0328165	−0.0164083	+	0.999865i	$0.505223\pi$
$42$	0	0
$43$	8.57641	1.30789	0.653945	−	0.756542i	$-0.273113\pi$
$43$	8.57641	1.30789	0.653945	+	0.756542i	$0.273113\pi$
$44$	0	0
$45$	0.520692	0.0776201
$46$	0	0
$47$	−8.98496	−1.31059	−0.655296	−	0.755372i	$-0.727456\pi$
$47$	−8.98496	−1.31059	−0.655296	+	0.755372i	$0.727456\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.53608	−0.635179
$52$	0	0
$53$	8.77822	1.20578	0.602891	−	0.797824i	$-0.294016\pi$
$53$	8.77822	1.20578	0.602891	+	0.797824i	$0.294016\pi$
$54$	0	0
$55$	−1.13919	−0.153608
$56$	0	0
$57$	−8.11495	−1.07485
$58$	0	0
$59$	−2.23049	−0.290385	−0.145192	−	0.989403i	$-0.546380\pi$
$59$	−2.23049	−0.290385	−0.145192	+	0.989403i	$0.546380\pi$
$60$	0	0
$61$	−6.23803	−0.798698	−0.399349	−	0.916799i	$-0.630764\pi$
$61$	−6.23803	−0.798698	−0.399349	+	0.916799i	$0.630764\pi$
$62$	0	0
$63$	0.457073	0.0575857
$64$	0	0
$65$	−1.13919	−0.141299
$66$	0	0
$67$	11.2185	1.37056	0.685281	−	0.728279i	$-0.259679\pi$
$67$	11.2185	1.37056	0.685281	+	0.728279i	$0.259679\pi$
$68$	0	0
$69$	7.55186	0.909137
$70$	0	0
$71$	−7.84719	−0.931290	−0.465645	−	0.884972i	$-0.654178\pi$
$71$	−7.84719	−0.931290	−0.465645	+	0.884972i	$0.654178\pi$
$72$	0	0
$73$	−15.8894	−1.85971	−0.929857	−	0.367921i	$-0.880070\pi$
$73$	−15.8894	−1.85971	−0.929857	+	0.367921i	$0.880070\pi$
$74$	0	0
$75$	−6.88367	−0.794858
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−5.38936	−0.606351	−0.303175	−	0.952935i	$-0.598047\pi$
$79$	−5.38936	−0.606351	−0.303175	+	0.952935i	$0.598047\pi$
$80$	0	0
$81$	−10.1623	−1.12914
$82$	0	0
$83$	−5.74479	−0.630572	−0.315286	−	0.948997i	$-0.602100\pi$
$83$	−5.74479	−0.630572	−0.315286	+	0.948997i	$0.602100\pi$
$84$	0	0
$85$	−2.77921	−0.301448
$86$	0	0
$87$	−0.766920	−0.0822224
$88$	0	0
$89$	0.577855	0.0612525	0.0306262	−	0.999531i	$-0.490250\pi$
$89$	0.577855	0.0612525	0.0306262	+	0.999531i	$0.490250\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−15.1103	−1.56687
$94$	0	0
$95$	−4.97195	−0.510111
$96$	0	0
$97$	−2.41830	−0.245541	−0.122770	−	0.992435i	$-0.539178\pi$
$97$	−2.41830	−0.245541	−0.122770	+	0.992435i	$0.539178\pi$
$98$	0	0
$99$	−0.457073	−0.0459375
$100$	0	0
$101$	3.57116	0.355344	0.177672	−	0.984090i	$-0.443143\pi$
$101$	3.57116	0.355344	0.177672	+	0.984090i	$0.443143\pi$
$102$	0	0
$103$	−10.8044	−1.06459	−0.532295	−	0.846559i	$-0.678670\pi$
$103$	−10.8044	−1.06459	−0.532295	+	0.846559i	$0.678670\pi$
$104$	0	0
$105$	2.11812	0.206707
$106$	0	0
$107$	13.8099	1.33505	0.667526	−	0.744587i	$-0.267353\pi$
$107$	13.8099	1.33505	0.667526	+	0.744587i	$0.267353\pi$
$108$	0	0
$109$	−5.49058	−0.525903	−0.262951	−	0.964809i	$-0.584696\pi$
$109$	−5.49058	−0.525903	−0.262951	+	0.964809i	$0.584696\pi$
$110$	0	0
$111$	−13.9786	−1.32679
$112$	0	0
$113$	8.64277	0.813043	0.406522	−	0.913641i	$-0.366742\pi$
$113$	8.64277	0.813043	0.406522	+	0.913641i	$0.366742\pi$
$114$	0	0
$115$	4.62695	0.431466
$116$	0	0
$117$	−0.457073	−0.0422564
$118$	0	0
$119$	−2.43965	−0.223642
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−0.390696	−0.0352279
$124$	0	0
$125$	−9.91350	−0.886690
$126$	0	0
$127$	18.6357	1.65365	0.826824	−	0.562460i	$-0.190145\pi$
$127$	18.6357	1.65365	0.826824	+	0.562460i	$0.190145\pi$
$128$	0	0
$129$	15.9463	1.40399
$130$	0	0
$131$	−0.760163	−0.0664157	−0.0332079	−	0.999448i	$-0.510572\pi$
$131$	−0.760163	−0.0664157	−0.0332079	+	0.999448i	$0.510572\pi$
$132$	0	0
$133$	−4.36447	−0.378447
$134$	0	0
$135$	−5.38621	−0.463572
$136$	0	0
$137$	15.1918	1.29792	0.648962	−	0.760821i	$-0.275203\pi$
$137$	15.1918	1.29792	0.648962	+	0.760821i	$0.275203\pi$
$138$	0	0
$139$	−14.1907	−1.20364	−0.601820	−	0.798632i	$-0.705558\pi$
$139$	−14.1907	−1.20364	−0.601820	+	0.798632i	$0.705558\pi$
$140$	0	0
$141$	−16.7059	−1.40689
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−0.469884	−0.0390218
$146$	0	0
$147$	1.85932	0.153354
$148$	0	0
$149$	−6.84598	−0.560845	−0.280422	−	0.959877i	$-0.590475\pi$
$149$	−6.84598	−0.560845	−0.280422	+	0.959877i	$0.590475\pi$
$150$	0	0
$151$	0.713259	0.0580442	0.0290221	−	0.999579i	$-0.490761\pi$
$151$	0.713259	0.0580442	0.0290221	+	0.999579i	$0.490761\pi$
$152$	0	0
$153$	−1.11509	−0.0901501
$154$	0	0
$155$	−9.25794	−0.743616
$156$	0	0
$157$	10.6338	0.848673	0.424337	−	0.905505i	$-0.360507\pi$
$157$	10.6338	0.848673	0.424337	+	0.905505i	$0.360507\pi$
$158$	0	0
$159$	16.3215	1.29438
$160$	0	0
$161$	4.06162	0.320101
$162$	0	0
$163$	11.2140	0.878350	0.439175	−	0.898401i	$-0.355271\pi$
$163$	11.2140	0.878350	0.439175	+	0.898401i	$0.355271\pi$
$164$	0	0
$165$	−2.11812	−0.164895
$166$	0	0
$167$	3.40870	0.263773	0.131887	−	0.991265i	$-0.457896\pi$
$167$	3.40870	0.263773	0.131887	+	0.991265i	$0.457896\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−1.99488	−0.152552
$172$	0	0
$173$	−7.50127	−0.570311	−0.285155	−	0.958481i	$-0.592045\pi$
$173$	−7.50127	−0.570311	−0.285155	+	0.958481i	$0.592045\pi$
$174$	0	0
$175$	−3.70225	−0.279864
$176$	0	0
$177$	−4.14719	−0.311722
$178$	0	0
$179$	−16.0679	−1.20097	−0.600485	−	0.799636i	$-0.705026\pi$
$179$	−16.0679	−1.20097	−0.600485	+	0.799636i	$0.705026\pi$
$180$	0	0
$181$	−16.1109	−1.19751	−0.598756	−	0.800931i	$-0.704338\pi$
$181$	−16.1109	−1.19751	−0.598756	+	0.800931i	$0.704338\pi$
$182$	0	0
$183$	−11.5985	−0.857386
$184$	0	0
$185$	−8.56456	−0.629679
$186$	0	0
$187$	2.43965	0.178405
$188$	0	0
$189$	−4.72812	−0.343920
$190$	0	0
$191$	−3.59700	−0.260270	−0.130135	−	0.991496i	$-0.541541\pi$
$191$	−3.59700	−0.260270	−0.130135	+	0.991496i	$0.541541\pi$
$192$	0	0
$193$	−4.52564	−0.325762	−0.162881	−	0.986646i	$-0.552079\pi$
$193$	−4.52564	−0.325762	−0.162881	+	0.986646i	$0.552079\pi$
$194$	0	0
$195$	−2.11812	−0.151681
$196$	0	0
$197$	15.1491	1.07933	0.539666	−	0.841879i	$-0.318551\pi$
$197$	15.1491	1.07933	0.539666	+	0.841879i	$0.318551\pi$
$198$	0	0
$199$	−4.59654	−0.325840	−0.162920	−	0.986639i	$-0.552091\pi$
$199$	−4.59654	−0.325840	−0.162920	+	0.986639i	$0.552091\pi$
$200$	0	0
$201$	20.8588	1.47127
$202$	0	0
$203$	−0.412473	−0.0289499
$204$	0	0
$205$	−0.239376	−0.0167187
$206$	0	0
$207$	1.85646	0.129033
$208$	0	0
$209$	4.36447	0.301897
$210$	0	0
$211$	−7.54087	−0.519135	−0.259568	−	0.965725i	$-0.583580\pi$
$211$	−7.54087	−0.519135	−0.259568	+	0.965725i	$0.583580\pi$
$212$	0	0
$213$	−14.5904	−0.999720
$214$	0	0
$215$	9.77014	0.666318
$216$	0	0
$217$	−8.12679	−0.551682
$218$	0	0
$219$	−29.5435	−1.99636
$220$	0	0
$221$	2.43965	0.164108
$222$	0	0
$223$	−26.8067	−1.79511	−0.897555	−	0.440902i	$-0.854659\pi$
$223$	−26.8067	−1.79511	−0.897555	+	0.440902i	$0.854659\pi$
$224$	0	0
$225$	−1.69220	−0.112813
$226$	0	0
$227$	−1.96930	−0.130707	−0.0653535	−	0.997862i	$-0.520818\pi$
$227$	−1.96930	−0.130707	−0.0653535	+	0.997862i	$0.520818\pi$
$228$	0	0
$229$	20.9526	1.38459	0.692294	−	0.721616i	$-0.256600\pi$
$229$	20.9526	1.38459	0.692294	+	0.721616i	$0.256600\pi$
$230$	0	0
$231$	−1.85932	−0.122334
$232$	0	0
$233$	−15.7952	−1.03478	−0.517389	−	0.855750i	$-0.673096\pi$
$233$	−15.7952	−1.03478	−0.517389	+	0.855750i	$0.673096\pi$
$234$	0	0
$235$	−10.2356	−0.667694
$236$	0	0
$237$	−10.0206	−0.650905
$238$	0	0
$239$	24.4833	1.58369	0.791845	−	0.610722i	$-0.209121\pi$
$239$	24.4833	1.58369	0.791845	+	0.610722i	$0.209121\pi$
$240$	0	0
$241$	11.0133	0.709426	0.354713	−	0.934975i	$-0.384579\pi$
$241$	11.0133	0.709426	0.354713	+	0.934975i	$0.384579\pi$
$242$	0	0
$243$	−4.71063	−0.302187
$244$	0	0
$245$	1.13919	0.0727801
$246$	0	0
$247$	4.36447	0.277704
$248$	0	0
$249$	−10.6814	−0.676906
$250$	0	0
$251$	4.01223	0.253250	0.126625	−	0.991951i	$-0.459586\pi$
$251$	4.01223	0.253250	0.126625	+	0.991951i	$0.459586\pi$
$252$	0	0
$253$	−4.06162	−0.255352
$254$	0	0
$255$	−5.16745	−0.323598
$256$	0	0
$257$	−3.02186	−0.188498	−0.0942492	−	0.995549i	$-0.530045\pi$
$257$	−3.02186	−0.188498	−0.0942492	+	0.995549i	$0.530045\pi$
$258$	0	0
$259$	−7.51813	−0.467154
$260$	0	0
$261$	−0.188530	−0.0116697
$262$	0	0
$263$	−0.513650	−0.0316730	−0.0158365	−	0.999875i	$-0.505041\pi$
$263$	−0.513650	−0.0316730	−0.0158365	+	0.999875i	$0.505041\pi$
$264$	0	0
$265$	10.0000	0.614298
$266$	0	0
$267$	1.07442	0.0657532
$268$	0	0
$269$	−26.5185	−1.61686	−0.808429	−	0.588593i	$-0.799682\pi$
$269$	−26.5185	−1.61686	−0.808429	+	0.588593i	$0.799682\pi$
$270$	0	0
$271$	31.5252	1.91502	0.957509	−	0.288404i	$-0.0931247\pi$
$271$	31.5252	1.91502	0.957509	+	0.288404i	$0.0931247\pi$
$272$	0	0
$273$	−1.85932	−0.112531
$274$	0	0
$275$	3.70225	0.223254
$276$	0	0
$277$	27.9907	1.68180	0.840899	−	0.541192i	$-0.182027\pi$
$277$	27.9907	1.68180	0.840899	+	0.541192i	$0.182027\pi$
$278$	0	0
$279$	−3.71453	−0.222383
$280$	0	0
$281$	27.3440	1.63120	0.815602	−	0.578613i	$-0.196406\pi$
$281$	27.3440	1.63120	0.815602	+	0.578613i	$0.196406\pi$
$282$	0	0
$283$	5.18505	0.308219	0.154110	−	0.988054i	$-0.450749\pi$
$283$	5.18505	0.308219	0.154110	+	0.988054i	$0.450749\pi$
$284$	0	0
$285$	−9.24445	−0.547594
$286$	0	0
$287$	−0.210128	−0.0124035
$288$	0	0
$289$	−11.0481	−0.649890
$290$	0	0
$291$	−4.49639	−0.263583
$292$	0	0
$293$	−28.6760	−1.67527	−0.837636	−	0.546229i	$-0.816063\pi$
$293$	−28.6760	−1.67527	−0.837636	+	0.546229i	$0.816063\pi$
$294$	0	0
$295$	−2.54094	−0.147939
$296$	0	0
$297$	4.72812	0.274353
$298$	0	0
$299$	−4.06162	−0.234890
$300$	0	0
$301$	8.57641	0.494336
$302$	0	0
$303$	6.63994	0.381454
$304$	0	0
$305$	−7.10629	−0.406905
$306$	0	0
$307$	−1.21096	−0.0691130	−0.0345565	−	0.999403i	$-0.511002\pi$
$307$	−1.21096	−0.0691130	−0.0345565	+	0.999403i	$0.511002\pi$
$308$	0	0
$309$	−20.0889	−1.14282
$310$	0	0
$311$	−3.12676	−0.177302	−0.0886512	−	0.996063i	$-0.528256\pi$
$311$	−3.12676	−0.177302	−0.0886512	+	0.996063i	$0.528256\pi$
$312$	0	0
$313$	−17.1075	−0.966972	−0.483486	−	0.875352i	$-0.660630\pi$
$313$	−17.1075	−0.966972	−0.483486	+	0.875352i	$0.660630\pi$
$314$	0	0
$315$	0.520692	0.0293377
$316$	0	0
$317$	28.7586	1.61524	0.807621	−	0.589702i	$-0.200755\pi$
$317$	28.7586	1.61524	0.807621	+	0.589702i	$0.200755\pi$
$318$	0	0
$319$	0.412473	0.0230941
$320$	0	0
$321$	25.6770	1.43315
$322$	0	0
$323$	10.6478	0.592457
$324$	0	0
$325$	3.70225	0.205364
$326$	0	0
$327$	−10.2088	−0.564545
$328$	0	0
$329$	−8.98496	−0.495357
$330$	0	0
$331$	−31.1599	−1.71270	−0.856352	−	0.516392i	$-0.827275\pi$
$331$	−31.1599	−1.71270	−0.856352	+	0.516392i	$0.827275\pi$
$332$	0	0
$333$	−3.43633	−0.188310
$334$	0	0
$335$	12.7800	0.698247
$336$	0	0
$337$	−3.21637	−0.175207	−0.0876034	−	0.996155i	$-0.527921\pi$
$337$	−3.21637	−0.175207	−0.0876034	+	0.996155i	$0.527921\pi$
$338$	0	0
$339$	16.0697	0.872785
$340$	0	0
$341$	8.12679	0.440090
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	8.60299	0.463169
$346$	0	0
$347$	3.25621	0.174802	0.0874012	−	0.996173i	$-0.472144\pi$
$347$	3.25621	0.174802	0.0874012	+	0.996173i	$0.472144\pi$
$348$	0	0
$349$	4.70551	0.251880	0.125940	−	0.992038i	$-0.459805\pi$
$349$	4.70551	0.251880	0.125940	+	0.992038i	$0.459805\pi$
$350$	0	0
$351$	4.72812	0.252368
$352$	0	0
$353$	−5.76977	−0.307094	−0.153547	−	0.988141i	$-0.549070\pi$
$353$	−5.76977	−0.307094	−0.153547	+	0.988141i	$0.549070\pi$
$354$	0	0
$355$	−8.93942	−0.474455
$356$	0	0
$357$	−4.53608	−0.240075
$358$	0	0
$359$	−9.48183	−0.500432	−0.250216	−	0.968190i	$-0.580502\pi$
$359$	−9.48183	−0.500432	−0.250216	+	0.968190i	$0.580502\pi$
$360$	0	0
$361$	0.0485854	0.00255712
$362$	0	0
$363$	1.85932	0.0975890
$364$	0	0
$365$	−18.1010	−0.947451
$366$	0	0
$367$	28.0919	1.46638	0.733192	−	0.680022i	$-0.238030\pi$
$367$	28.0919	1.46638	0.733192	+	0.680022i	$0.238030\pi$
$368$	0	0
$369$	−0.0960439	−0.00499985
$370$	0	0
$371$	8.77822	0.455743
$372$	0	0
$373$	6.47251	0.335134	0.167567	−	0.985861i	$-0.446409\pi$
$373$	6.47251	0.335134	0.167567	+	0.985861i	$0.446409\pi$
$374$	0	0
$375$	−18.4324	−0.951844
$376$	0	0
$377$	0.412473	0.0212434
$378$	0	0
$379$	−22.6820	−1.16510	−0.582549	−	0.812796i	$-0.697945\pi$
$379$	−22.6820	−1.16510	−0.582549	+	0.812796i	$0.697945\pi$
$380$	0	0
$381$	34.6497	1.77516
$382$	0	0
$383$	−24.3230	−1.24285	−0.621424	−	0.783475i	$-0.713446\pi$
$383$	−24.3230	−1.24285	−0.621424	+	0.783475i	$0.713446\pi$
$384$	0	0
$385$	−1.13919	−0.0580584
$386$	0	0
$387$	3.92004	0.199267
$388$	0	0
$389$	9.23244	0.468103	0.234052	−	0.972224i	$-0.424802\pi$
$389$	9.23244	0.468103	0.234052	+	0.972224i	$0.424802\pi$
$390$	0	0
$391$	−9.90892	−0.501116
$392$	0	0
$393$	−1.41339	−0.0712959
$394$	0	0
$395$	−6.13950	−0.308912
$396$	0	0
$397$	10.7358	0.538814	0.269407	−	0.963026i	$-0.413172\pi$
$397$	10.7358	0.538814	0.269407	+	0.963026i	$0.413172\pi$
$398$	0	0
$399$	−8.11495	−0.406255
$400$	0	0
$401$	−5.44508	−0.271914	−0.135957	−	0.990715i	$-0.543411\pi$
$401$	−5.44508	−0.271914	−0.135957	+	0.990715i	$0.543411\pi$
$402$	0	0
$403$	8.12679	0.404824
$404$	0	0
$405$	−11.5768	−0.575255
$406$	0	0
$407$	7.51813	0.372660
$408$	0	0
$409$	24.7840	1.22549	0.612744	−	0.790281i	$-0.290066\pi$
$409$	24.7840	1.22549	0.612744	+	0.790281i	$0.290066\pi$
$410$	0	0
$411$	28.2465	1.39329
$412$	0	0
$413$	−2.23049	−0.109755
$414$	0	0
$415$	−6.54439	−0.321252
$416$	0	0
$417$	−26.3851	−1.29208
$418$	0	0
$419$	8.58148	0.419233	0.209616	−	0.977784i	$-0.432778\pi$
$419$	8.58148	0.419233	0.209616	+	0.977784i	$0.432778\pi$
$420$	0	0
$421$	6.89799	0.336187	0.168094	−	0.985771i	$-0.446239\pi$
$421$	6.89799	0.336187	0.168094	+	0.985771i	$0.446239\pi$
$422$	0	0
$423$	−4.10678	−0.199678
$424$	0	0
$425$	9.03218	0.438125
$426$	0	0
$427$	−6.23803	−0.301880
$428$	0	0
$429$	1.85932	0.0897688
$430$	0	0
$431$	−22.8518	−1.10073	−0.550367	−	0.834923i	$-0.685512\pi$
$431$	−22.8518	−1.10073	−0.550367	+	0.834923i	$0.685512\pi$
$432$	0	0
$433$	23.9326	1.15013	0.575063	−	0.818109i	$-0.304977\pi$
$433$	23.9326	1.15013	0.575063	+	0.818109i	$0.304977\pi$
$434$	0	0
$435$	−0.873666	−0.0418891
$436$	0	0
$437$	−17.7268	−0.847989
$438$	0	0
$439$	−4.28669	−0.204593	−0.102296	−	0.994754i	$-0.532619\pi$
$439$	−4.28669	−0.204593	−0.102296	+	0.994754i	$0.532619\pi$
$440$	0	0
$441$	0.457073	0.0217654
$442$	0	0
$443$	15.9434	0.757494	0.378747	−	0.925500i	$-0.376355\pi$
$443$	15.9434	0.757494	0.378747	+	0.925500i	$0.376355\pi$
$444$	0	0
$445$	0.658285	0.0312057
$446$	0	0
$447$	−12.7289	−0.602055
$448$	0	0
$449$	−3.69766	−0.174504	−0.0872518	−	0.996186i	$-0.527808\pi$
$449$	−3.69766	−0.174504	−0.0872518	+	0.996186i	$0.527808\pi$
$450$	0	0
$451$	0.210128	0.00989456
$452$	0	0
$453$	1.32618	0.0623093
$454$	0	0
$455$	−1.13919	−0.0534060
$456$	0	0
$457$	−25.2958	−1.18329	−0.591644	−	0.806200i	$-0.701521\pi$
$457$	−25.2958	−1.18329	−0.591644	+	0.806200i	$0.701521\pi$
$458$	0	0
$459$	11.5349	0.538404
$460$	0	0
$461$	35.3046	1.64430	0.822149	−	0.569273i	$-0.192775\pi$
$461$	35.3046	1.64430	0.822149	+	0.569273i	$0.192775\pi$
$462$	0	0
$463$	19.9855	0.928806	0.464403	−	0.885624i	$-0.346269\pi$
$463$	19.9855	0.928806	0.464403	+	0.885624i	$0.346269\pi$
$464$	0	0
$465$	−17.2135	−0.798256
$466$	0	0
$467$	31.8835	1.47539	0.737695	−	0.675134i	$-0.235914\pi$
$467$	31.8835	1.47539	0.737695	+	0.675134i	$0.235914\pi$
$468$	0	0
$469$	11.2185	0.518024
$470$	0	0
$471$	19.7717	0.911033
$472$	0	0
$473$	−8.57641	−0.394344
$474$	0	0
$475$	16.1584	0.741396
$476$	0	0
$477$	4.01228	0.183710
$478$	0	0
$479$	−35.9578	−1.64295	−0.821477	−	0.570241i	$-0.806850\pi$
$479$	−35.9578	−1.64295	−0.821477	+	0.570241i	$0.806850\pi$
$480$	0	0
$481$	7.51813	0.342797
$482$	0	0
$483$	7.55186	0.343622
$484$	0	0
$485$	−2.75489	−0.125093
$486$	0	0
$487$	−23.1605	−1.04950	−0.524752	−	0.851255i	$-0.675842\pi$
$487$	−23.1605	−1.04950	−0.524752	+	0.851255i	$0.675842\pi$
$488$	0	0
$489$	20.8505	0.942891
$490$	0	0
$491$	−29.3914	−1.32642	−0.663208	−	0.748435i	$-0.730806\pi$
$491$	−29.3914	−1.32642	−0.663208	+	0.748435i	$0.730806\pi$
$492$	0	0
$493$	1.00629	0.0453209
$494$	0	0
$495$	−0.520692	−0.0234034
$496$	0	0
$497$	−7.84719	−0.351994
$498$	0	0
$499$	−24.1855	−1.08269	−0.541346	−	0.840800i	$-0.682085\pi$
$499$	−24.1855	−1.08269	−0.541346	+	0.840800i	$0.682085\pi$
$500$	0	0
$501$	6.33787	0.283155
$502$	0	0
$503$	26.0053	1.15952	0.579760	−	0.814787i	$-0.303146\pi$
$503$	26.0053	1.15952	0.579760	+	0.814787i	$0.303146\pi$
$504$	0	0
$505$	4.06823	0.181034
$506$	0	0
$507$	1.85932	0.0825753
$508$	0	0
$509$	−20.3224	−0.900773	−0.450387	−	0.892834i	$-0.648714\pi$
$509$	−20.3224	−0.900773	−0.450387	+	0.892834i	$0.648714\pi$
$510$	0	0
$511$	−15.8894	−0.702906
$512$	0	0
$513$	20.6357	0.911089
$514$	0	0
$515$	−12.3083	−0.542367
$516$	0	0
$517$	8.98496	0.395158
$518$	0	0
$519$	−13.9473	−0.612217
$520$	0	0
$521$	−33.3536	−1.46125	−0.730625	−	0.682779i	$-0.760771\pi$
$521$	−33.3536	−1.46125	−0.730625	+	0.682779i	$0.760771\pi$
$522$	0	0
$523$	−18.3051	−0.800428	−0.400214	−	0.916422i	$-0.631064\pi$
$523$	−18.3051	−0.800428	−0.400214	+	0.916422i	$0.631064\pi$
$524$	0	0
$525$	−6.88367	−0.300428
$526$	0	0
$527$	19.8265	0.863655
$528$	0	0
$529$	−6.50320	−0.282748
$530$	0	0
$531$	−1.01949	−0.0442423
$532$	0	0
$533$	0.210128	0.00910167
$534$	0	0
$535$	15.7321	0.680156
$536$	0	0
$537$	−29.8753	−1.28922
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−7.53976	−0.324160	−0.162080	−	0.986778i	$-0.551820\pi$
$541$	−7.53976	−0.324160	−0.162080	+	0.986778i	$0.551820\pi$
$542$	0	0
$543$	−29.9553	−1.28550
$544$	0	0
$545$	−6.25481	−0.267927
$546$	0	0
$547$	7.12726	0.304740	0.152370	−	0.988324i	$-0.451310\pi$
$547$	7.12726	0.304740	0.152370	+	0.988324i	$0.451310\pi$
$548$	0	0
$549$	−2.85123	−0.121688
$550$	0	0
$551$	1.80023	0.0766922
$552$	0	0
$553$	−5.38936	−0.229179
$554$	0	0
$555$	−15.9243	−0.675947
$556$	0	0
$557$	18.7729	0.795433	0.397717	−	0.917508i	$-0.369803\pi$
$557$	18.7729	0.795433	0.397717	+	0.917508i	$0.369803\pi$
$558$	0	0
$559$	−8.57641	−0.362743
$560$	0	0
$561$	4.53608	0.191514
$562$	0	0
$563$	−18.8497	−0.794420	−0.397210	−	0.917728i	$-0.630022\pi$
$563$	−18.8497	−0.794420	−0.397210	+	0.917728i	$0.630022\pi$
$564$	0	0
$565$	9.84574	0.414213
$566$	0	0
$567$	−10.1623	−0.426777
$568$	0	0
$569$	12.8921	0.540463	0.270232	−	0.962795i	$-0.412900\pi$
$569$	12.8921	0.540463	0.270232	+	0.962795i	$0.412900\pi$
$570$	0	0
$571$	39.4783	1.65212	0.826058	−	0.563585i	$-0.190578\pi$
$571$	39.4783	1.65212	0.826058	+	0.563585i	$0.190578\pi$
$572$	0	0
$573$	−6.68798	−0.279394
$574$	0	0
$575$	−15.0372	−0.627093
$576$	0	0
$577$	−11.9135	−0.495966	−0.247983	−	0.968764i	$-0.579768\pi$
$577$	−11.9135	−0.495966	−0.247983	+	0.968764i	$0.579768\pi$
$578$	0	0
$579$	−8.41461	−0.349699
$580$	0	0
$581$	−5.74479	−0.238334
$582$	0	0
$583$	−8.77822	−0.363557
$584$	0	0
$585$	−0.520692	−0.0215280
$586$	0	0
$587$	−16.9659	−0.700259	−0.350130	−	0.936701i	$-0.613862\pi$
$587$	−16.9659	−0.700259	−0.350130	+	0.936701i	$0.613862\pi$
$588$	0	0
$589$	35.4691	1.46148
$590$	0	0
$591$	28.1671	1.15864
$592$	0	0
$593$	6.95059	0.285426	0.142713	−	0.989764i	$-0.454417\pi$
$593$	6.95059	0.285426	0.142713	+	0.989764i	$0.454417\pi$
$594$	0	0
$595$	−2.77921	−0.113937
$596$	0	0
$597$	−8.54645	−0.349783
$598$	0	0
$599$	−42.4976	−1.73640	−0.868202	−	0.496210i	$-0.834724\pi$
$599$	−42.4976	−1.73640	−0.868202	+	0.496210i	$0.834724\pi$
$600$	0	0
$601$	6.23112	0.254173	0.127086	−	0.991892i	$-0.459437\pi$
$601$	6.23112	0.254173	0.127086	+	0.991892i	$0.459437\pi$
$602$	0	0
$603$	5.12768	0.208815
$604$	0	0
$605$	1.13919	0.0463146
$606$	0	0
$607$	24.6593	1.00089	0.500445	−	0.865768i	$-0.333170\pi$
$607$	24.6593	1.00089	0.500445	+	0.865768i	$0.333170\pi$
$608$	0	0
$609$	−0.766920	−0.0310772
$610$	0	0
$611$	8.98496	0.363493
$612$	0	0
$613$	15.7158	0.634756	0.317378	−	0.948299i	$-0.397198\pi$
$613$	15.7158	0.634756	0.317378	+	0.948299i	$0.397198\pi$
$614$	0	0
$615$	−0.445076	−0.0179472
$616$	0	0
$617$	11.5791	0.466158	0.233079	−	0.972458i	$-0.425120\pi$
$617$	11.5791	0.466158	0.233079	+	0.972458i	$0.425120\pi$
$618$	0	0
$619$	7.22023	0.290205	0.145103	−	0.989417i	$-0.453649\pi$
$619$	7.22023	0.290205	0.145103	+	0.989417i	$0.453649\pi$
$620$	0	0
$621$	−19.2038	−0.770623
$622$	0	0
$623$	0.577855	0.0231513
$624$	0	0
$625$	7.21791	0.288716
$626$	0	0
$627$	8.11495	0.324080
$628$	0	0
$629$	18.3416	0.731326
$630$	0	0
$631$	22.2384	0.885298	0.442649	−	0.896695i	$-0.354039\pi$
$631$	22.2384	0.885298	0.442649	+	0.896695i	$0.354039\pi$
$632$	0	0
$633$	−14.0209	−0.557281
$634$	0	0
$635$	21.2295	0.842468
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−3.58673	−0.141889
$640$	0	0
$641$	−31.6804	−1.25130	−0.625651	−	0.780103i	$-0.715167\pi$
$641$	−31.6804	−1.25130	−0.625651	+	0.780103i	$0.715167\pi$
$642$	0	0
$643$	−37.3281	−1.47208	−0.736038	−	0.676941i	$-0.763305\pi$
$643$	−37.3281	−1.47208	−0.736038	+	0.676941i	$0.763305\pi$
$644$	0	0
$645$	18.1658	0.715279
$646$	0	0
$647$	32.0364	1.25948	0.629739	−	0.776806i	$-0.283162\pi$
$647$	32.0364	1.25948	0.629739	+	0.776806i	$0.283162\pi$
$648$	0	0
$649$	2.23049	0.0875542
$650$	0	0
$651$	−15.1103	−0.592220
$652$	0	0
$653$	−40.4097	−1.58135	−0.790676	−	0.612235i	$-0.790271\pi$
$653$	−40.4097	−1.58135	−0.790676	+	0.612235i	$0.790271\pi$
$654$	0	0
$655$	−0.865968	−0.0338362
$656$	0	0
$657$	−7.26261	−0.283341
$658$	0	0
$659$	−8.45813	−0.329482	−0.164741	−	0.986337i	$-0.552679\pi$
$659$	−8.45813	−0.329482	−0.164741	+	0.986337i	$0.552679\pi$
$660$	0	0
$661$	−17.1707	−0.667862	−0.333931	−	0.942597i	$-0.608375\pi$
$661$	−17.1707	−0.667862	−0.333931	+	0.942597i	$0.608375\pi$
$662$	0	0
$663$	4.53608	0.176167
$664$	0	0
$665$	−4.97195	−0.192804
$666$	0	0
$667$	−1.67531	−0.0648683
$668$	0	0
$669$	−49.8423	−1.92701
$670$	0	0
$671$	6.23803	0.240817
$672$	0	0
$673$	30.3578	1.17021	0.585104	−	0.810958i	$-0.301054\pi$
$673$	30.3578	1.17021	0.585104	+	0.810958i	$0.301054\pi$
$674$	0	0
$675$	17.5047	0.673755
$676$	0	0
$677$	13.6466	0.524482	0.262241	−	0.965002i	$-0.415539\pi$
$677$	13.6466	0.524482	0.262241	+	0.965002i	$0.415539\pi$
$678$	0	0
$679$	−2.41830	−0.0928057
$680$	0	0
$681$	−3.66156	−0.140311
$682$	0	0
$683$	−43.0332	−1.64662	−0.823310	−	0.567592i	$-0.807875\pi$
$683$	−43.0332	−1.64662	−0.823310	+	0.567592i	$0.807875\pi$
$684$	0	0
$685$	17.3063	0.661241
$686$	0	0
$687$	38.9576	1.48633
$688$	0	0
$689$	−8.77822	−0.334424
$690$	0	0
$691$	−11.3082	−0.430183	−0.215092	−	0.976594i	$-0.569005\pi$
$691$	−11.3082	−0.430183	−0.215092	+	0.976594i	$0.569005\pi$
$692$	0	0
$693$	−0.457073	−0.0173628
$694$	0	0
$695$	−16.1659	−0.613207
$696$	0	0
$697$	0.512639	0.0194176
$698$	0	0
$699$	−29.3684	−1.11081
$700$	0	0
$701$	27.0712	1.02246	0.511232	−	0.859442i	$-0.329189\pi$
$701$	27.0712	1.02246	0.511232	+	0.859442i	$0.329189\pi$
$702$	0	0
$703$	32.8126	1.23755
$704$	0	0
$705$	−19.0312	−0.716756
$706$	0	0
$707$	3.57116	0.134307
$708$	0	0
$709$	28.6143	1.07463	0.537316	−	0.843381i	$-0.319438\pi$
$709$	28.6143	1.07463	0.537316	+	0.843381i	$0.319438\pi$
$710$	0	0
$711$	−2.46333	−0.0923821
$712$	0	0
$713$	−33.0080	−1.23616
$714$	0	0
$715$	1.13919	0.0426032
$716$	0	0
$717$	45.5222	1.70006
$718$	0	0
$719$	7.89921	0.294591	0.147295	−	0.989093i	$-0.452943\pi$
$719$	7.89921	0.294591	0.147295	+	0.989093i	$0.452943\pi$
$720$	0	0
$721$	−10.8044	−0.402377
$722$	0	0
$723$	20.4772	0.761554
$724$	0	0
$725$	1.52708	0.0567143
$726$	0	0
$727$	−1.67249	−0.0620292	−0.0310146	−	0.999519i	$-0.509874\pi$
$727$	−1.67249	−0.0620292	−0.0310146	+	0.999519i	$0.509874\pi$
$728$	0	0
$729$	21.7283	0.804753
$730$	0	0
$731$	−20.9234	−0.773880
$732$	0	0
$733$	35.7021	1.31869	0.659344	−	0.751841i	$-0.270834\pi$
$733$	35.7021	1.31869	0.659344	+	0.751841i	$0.270834\pi$
$734$	0	0
$735$	2.11812	0.0781279
$736$	0	0
$737$	−11.2185	−0.413240
$738$	0	0
$739$	−24.9259	−0.916915	−0.458457	−	0.888716i	$-0.651598\pi$
$739$	−24.9259	−0.916915	−0.458457	+	0.888716i	$0.651598\pi$
$740$	0	0
$741$	8.11495	0.298110
$742$	0	0
$743$	10.9776	0.402729	0.201365	−	0.979516i	$-0.435462\pi$
$743$	10.9776	0.402729	0.201365	+	0.979516i	$0.435462\pi$
$744$	0	0
$745$	−7.79886	−0.285728
$746$	0	0
$747$	−2.62578	−0.0960724
$748$	0	0
$749$	13.8099	0.504602
$750$	0	0
$751$	1.54131	0.0562431	0.0281215	−	0.999605i	$-0.491047\pi$
$751$	1.54131	0.0562431	0.0281215	+	0.999605i	$0.491047\pi$
$752$	0	0
$753$	7.46002	0.271858
$754$	0	0
$755$	0.812537	0.0295712
$756$	0	0
$757$	20.8138	0.756490	0.378245	−	0.925705i	$-0.376528\pi$
$757$	20.8138	0.756490	0.378245	+	0.925705i	$0.376528\pi$
$758$	0	0
$759$	−7.55186	−0.274115
$760$	0	0
$761$	6.52690	0.236600	0.118300	−	0.992978i	$-0.462256\pi$
$761$	6.52690	0.236600	0.118300	+	0.992978i	$0.462256\pi$
$762$	0	0
$763$	−5.49058	−0.198772
$764$	0	0
$765$	−1.27030	−0.0459279
$766$	0	0
$767$	2.23049	0.0805382
$768$	0	0
$769$	28.9745	1.04485	0.522424	−	0.852686i	$-0.325028\pi$
$769$	28.9745	1.04485	0.522424	+	0.852686i	$0.325028\pi$
$770$	0	0
$771$	−5.61861	−0.202349
$772$	0	0
$773$	−10.4044	−0.374220	−0.187110	−	0.982339i	$-0.559912\pi$
$773$	−10.4044	−0.374220	−0.187110	+	0.982339i	$0.559912\pi$
$774$	0	0
$775$	30.0874	1.08077
$776$	0	0
$777$	−13.9786	−0.501480
$778$	0	0
$779$	0.917099	0.0328585
$780$	0	0
$781$	7.84719	0.280794
$782$	0	0
$783$	1.95022	0.0696952
$784$	0	0
$785$	12.1139	0.432365
$786$	0	0
$787$	11.2258	0.400155	0.200078	−	0.979780i	$-0.435881\pi$
$787$	11.2258	0.400155	0.200078	+	0.979780i	$0.435881\pi$
$788$	0	0
$789$	−0.955040	−0.0340003
$790$	0	0
$791$	8.64277	0.307301
$792$	0	0
$793$	6.23803	0.221519
$794$	0	0
$795$	18.5933	0.659436
$796$	0	0
$797$	18.2894	0.647845	0.323922	−	0.946084i	$-0.394998\pi$
$797$	18.2894	0.647845	0.323922	+	0.946084i	$0.394998\pi$
$798$	0	0
$799$	21.9201	0.775478
$800$	0	0
$801$	0.264122	0.00933228
$802$	0	0
$803$	15.8894	0.560725
$804$	0	0
$805$	4.62695	0.163079
$806$	0	0
$807$	−49.3063	−1.73566
$808$	0	0
$809$	−4.09275	−0.143893	−0.0719467	−	0.997408i	$-0.522921\pi$
$809$	−4.09275	−0.143893	−0.0719467	+	0.997408i	$0.522921\pi$
$810$	0	0
$811$	−53.3329	−1.87277	−0.936385	−	0.350973i	$-0.885851\pi$
$811$	−53.3329	−1.87277	−0.936385	+	0.350973i	$0.885851\pi$
$812$	0	0
$813$	58.6154	2.05573
$814$	0	0
$815$	12.7749	0.447485
$816$	0	0
$817$	−37.4315	−1.30956
$818$	0	0
$819$	−0.457073	−0.0159714
$820$	0	0
$821$	−31.4574	−1.09787	−0.548935	−	0.835865i	$-0.684966\pi$
$821$	−31.4574	−1.09787	−0.548935	+	0.835865i	$0.684966\pi$
$822$	0	0
$823$	55.3033	1.92775	0.963876	−	0.266351i	$-0.0858179\pi$
$823$	55.3033	1.92775	0.963876	+	0.266351i	$0.0858179\pi$
$824$	0	0
$825$	6.88367	0.239659
$826$	0	0
$827$	−52.1998	−1.81516	−0.907582	−	0.419875i	$-0.862074\pi$
$827$	−52.1998	−1.81516	−0.907582	+	0.419875i	$0.862074\pi$
$828$	0	0
$829$	19.2170	0.667435	0.333717	−	0.942673i	$-0.391697\pi$
$829$	19.2170	0.667435	0.333717	+	0.942673i	$0.391697\pi$
$830$	0	0
$831$	52.0437	1.80537
$832$	0	0
$833$	−2.43965	−0.0845287
$834$	0	0
$835$	3.88316	0.134382
$836$	0	0
$837$	38.4244	1.32814
$838$	0	0
$839$	−33.5682	−1.15890	−0.579451	−	0.815007i	$-0.696733\pi$
$839$	−33.5682	−1.15890	−0.579451	+	0.815007i	$0.696733\pi$
$840$	0	0
$841$	−28.8299	−0.994133
$842$	0	0
$843$	50.8412	1.75106
$844$	0	0
$845$	1.13919	0.0391893
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	9.64067	0.330867
$850$	0	0
$851$	−30.5358	−1.04675
$852$	0	0
$853$	28.2442	0.967064	0.483532	−	0.875327i	$-0.339354\pi$
$853$	28.2442	0.967064	0.483532	+	0.875327i	$0.339354\pi$
$854$	0	0
$855$	−2.27254	−0.0777193
$856$	0	0
$857$	10.6777	0.364742	0.182371	−	0.983230i	$-0.441623\pi$
$857$	10.6777	0.364742	0.182371	+	0.983230i	$0.441623\pi$
$858$	0	0
$859$	−30.0048	−1.02375	−0.511875	−	0.859060i	$-0.671049\pi$
$859$	−30.0048	−1.02375	−0.511875	+	0.859060i	$0.671049\pi$
$860$	0	0
$861$	−0.390696	−0.0133149
$862$	0	0
$863$	−58.1989	−1.98112	−0.990558	−	0.137097i	$-0.956223\pi$
$863$	−58.1989	−1.98112	−0.990558	+	0.137097i	$0.956223\pi$
$864$	0	0
$865$	−8.54535	−0.290551
$866$	0	0
$867$	−20.5420	−0.697643
$868$	0	0
$869$	5.38936	0.182822
$870$	0	0
$871$	−11.2185	−0.380125
$872$	0	0
$873$	−1.10534	−0.0374100
$874$	0	0
$875$	−9.91350	−0.335137
$876$	0	0
$877$	−39.8675	−1.34623	−0.673115	−	0.739538i	$-0.735044\pi$
$877$	−39.8675	−1.34623	−0.673115	+	0.739538i	$0.735044\pi$
$878$	0	0
$879$	−53.3179	−1.79837
$880$	0	0
$881$	−24.3519	−0.820437	−0.410218	−	0.911987i	$-0.634547\pi$
$881$	−24.3519	−0.820437	−0.410218	+	0.911987i	$0.634547\pi$
$882$	0	0
$883$	−15.3434	−0.516345	−0.258173	−	0.966099i	$-0.583120\pi$
$883$	−15.3434	−0.516345	−0.258173	+	0.966099i	$0.583120\pi$
$884$	0	0
$885$	−4.72443	−0.158810
$886$	0	0
$887$	−12.7397	−0.427757	−0.213878	−	0.976860i	$-0.568610\pi$
$887$	−12.7397	−0.427757	−0.213878	+	0.976860i	$0.568610\pi$
$888$	0	0
$889$	18.6357	0.625020
$890$	0	0
$891$	10.1623	0.340450
$892$	0	0
$893$	39.2146	1.31227
$894$	0	0
$895$	−18.3043	−0.611846
$896$	0	0
$897$	−7.55186	−0.252149
$898$	0	0
$899$	3.35208	0.111798
$900$	0	0
$901$	−21.4157	−0.713462
$902$	0	0
$903$	15.9463	0.530659
$904$	0	0
$905$	−18.3533	−0.610085
$906$	0	0
$907$	−29.2984	−0.972839	−0.486419	−	0.873725i	$-0.661697\pi$
$907$	−29.2984	−0.972839	−0.486419	+	0.873725i	$0.661697\pi$
$908$	0	0
$909$	1.63228	0.0541393
$910$	0	0
$911$	−26.0771	−0.863973	−0.431987	−	0.901880i	$-0.642187\pi$
$911$	−26.0771	−0.863973	−0.431987	+	0.901880i	$0.642187\pi$
$912$	0	0
$913$	5.74479	0.190125
$914$	0	0
$915$	−13.2129	−0.436804
$916$	0	0
$917$	−0.760163	−0.0251028
$918$	0	0
$919$	−7.38215	−0.243515	−0.121757	−	0.992560i	$-0.538853\pi$
$919$	−7.38215	−0.243515	−0.121757	+	0.992560i	$0.538853\pi$
$920$	0	0
$921$	−2.25156	−0.0741913
$922$	0	0
$923$	7.84719	0.258293
$924$	0	0
$925$	27.8340	0.915176
$926$	0	0
$927$	−4.93840	−0.162198
$928$	0	0
$929$	−41.2805	−1.35437	−0.677185	−	0.735813i	$-0.736800\pi$
$929$	−41.2805	−1.35437	−0.677185	+	0.735813i	$0.736800\pi$
$930$	0	0
$931$	−4.36447	−0.143040
$932$	0	0
$933$	−5.81365	−0.190330
$934$	0	0
$935$	2.77921	0.0908900
$936$	0	0
$937$	−27.6346	−0.902782	−0.451391	−	0.892326i	$-0.649072\pi$
$937$	−27.6346	−0.902782	−0.451391	+	0.892326i	$0.649072\pi$
$938$	0	0
$939$	−31.8083	−1.03802
$940$	0	0
$941$	18.2613	0.595303	0.297651	−	0.954675i	$-0.403797\pi$
$941$	18.2613	0.595303	0.297651	+	0.954675i	$0.403797\pi$
$942$	0	0
$943$	−0.853463	−0.0277926
$944$	0	0
$945$	−5.38621	−0.175214
$946$	0	0
$947$	−5.38628	−0.175031	−0.0875153	−	0.996163i	$-0.527893\pi$
$947$	−5.38628	−0.175031	−0.0875153	+	0.996163i	$0.527893\pi$
$948$	0	0
$949$	15.8894	0.515792
$950$	0	0
$951$	53.4714	1.73393
$952$	0	0
$953$	26.4840	0.857900	0.428950	−	0.903328i	$-0.358884\pi$
$953$	26.4840	0.857900	0.428950	+	0.903328i	$0.358884\pi$
$954$	0	0
$955$	−4.09766	−0.132597
$956$	0	0
$957$	0.766920	0.0247910
$958$	0	0
$959$	15.1918	0.490569
$960$	0	0
$961$	35.0447	1.13047
$962$	0	0
$963$	6.31212	0.203405
$964$	0	0
$965$	−5.15555	−0.165963
$966$	0	0
$967$	−26.4189	−0.849574	−0.424787	−	0.905293i	$-0.639651\pi$
$967$	−26.4189	−0.849574	−0.424787	+	0.905293i	$0.639651\pi$
$968$	0	0
$969$	19.7976	0.635990
$970$	0	0
$971$	7.54863	0.242247	0.121123	−	0.992637i	$-0.461350\pi$
$971$	7.54863	0.242247	0.121123	+	0.992637i	$0.461350\pi$
$972$	0	0
$973$	−14.1907	−0.454933
$974$	0	0
$975$	6.88367	0.220454
$976$	0	0
$977$	22.3574	0.715277	0.357639	−	0.933860i	$-0.383582\pi$
$977$	22.3574	0.715277	0.357639	+	0.933860i	$0.383582\pi$
$978$	0	0
$979$	−0.577855	−0.0184683
$980$	0	0
$981$	−2.50960	−0.0801252
$982$	0	0
$983$	29.6401	0.945373	0.472687	−	0.881231i	$-0.343284\pi$
$983$	29.6401	0.945373	0.472687	+	0.881231i	$0.343284\pi$
$984$	0	0
$985$	17.2577	0.549877
$986$	0	0
$987$	−16.7059	−0.531755
$988$	0	0
$989$	34.8342	1.10766
$990$	0	0
$991$	−40.4495	−1.28492	−0.642461	−	0.766318i	$-0.722087\pi$
$991$	−40.4495	−1.28492	−0.642461	+	0.766318i	$0.722087\pi$
$992$	0	0
$993$	−57.9363	−1.83855
$994$	0	0
$995$	−5.23633	−0.166003
$996$	0	0
$997$	−22.0197	−0.697372	−0.348686	−	0.937240i	$-0.613372\pi$
$997$	−22.0197	−0.697372	−0.348686	+	0.937240i	$0.613372\pi$
$998$	0	0
$999$	35.5466	1.12464

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.y.1.11	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.y.1.11	✓	14	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)

\(f(q)\)	\(=\)	\(q+1.85932 q^{3} +1.13919 q^{5} +1.00000 q^{7} +0.457073 q^{9} +O(q^{10})\)	\(q+1.85932 q^{3} +1.13919 q^{5} +1.00000 q^{7} +0.457073 q^{9} -1.00000 q^{11} -1.00000 q^{13} +2.11812 q^{15} -2.43965 q^{17} -4.36447 q^{19} +1.85932 q^{21} +4.06162 q^{23} -3.70225 q^{25} -4.72812 q^{27} -0.412473 q^{29} -8.12679 q^{31} -1.85932 q^{33} +1.13919 q^{35} -7.51813 q^{37} -1.85932 q^{39} -0.210128 q^{41} +8.57641 q^{43} +0.520692 q^{45} -8.98496 q^{47} +1.00000 q^{49} -4.53608 q^{51} +8.77822 q^{53} -1.13919 q^{55} -8.11495 q^{57} -2.23049 q^{59} -6.23803 q^{61} +0.457073 q^{63} -1.13919 q^{65} +11.2185 q^{67} +7.55186 q^{69} -7.84719 q^{71} -15.8894 q^{73} -6.88367 q^{75} -1.00000 q^{77} -5.38936 q^{79} -10.1623 q^{81} -5.74479 q^{83} -2.77921 q^{85} -0.766920 q^{87} +0.577855 q^{89} -1.00000 q^{91} -15.1103 q^{93} -4.97195 q^{95} -2.41830 q^{97} -0.457073 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9}+O(q^{10}) \) 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9	\( 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9} - 14 q^{11} - 14 q^{13} - 6 q^{15} - 6 q^{17} - 13 q^{19} - 3 q^{21} - 9 q^{23} + 22 q^{25} - 18 q^{27} + 2 q^{29} - 2 q^{31} + 3 q^{33} - 6 q^{35} - q^{37} + 3 q^{39} - 16 q^{41} - 15 q^{43} - 44 q^{45} - 8 q^{47} + 14 q^{49} - 14 q^{51} - 6 q^{53} + 6 q^{55} - 10 q^{57} - 36 q^{59} - 19 q^{61} + 21 q^{63} + 6 q^{65} - 34 q^{67} - q^{69} - 10 q^{71} + 9 q^{73} - 44 q^{75} - 14 q^{77} - q^{79} + 42 q^{81} - 56 q^{83} + 21 q^{85} - 5 q^{87} - 14 q^{89} - 14 q^{91} - 20 q^{93} + q^{95} - 14 q^{97} - 21 q^{99}+O(q^{100}) \) 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9 - 14 * q^11 - 14 * q^13 - 6 * q^15 - 6 * q^17 - 13 * q^19 - 3 * q^21 - 9 * q^23 + 22 * q^25 - 18 * q^27 + 2 * q^29 - 2 * q^31 + 3 * q^33 - 6 * q^35 - q^37 + 3 * q^39 - 16 * q^41 - 15 * q^43 - 44 * q^45 - 8 * q^47 + 14 * q^49 - 14 * q^51 - 6 * q^53 + 6 * q^55 - 10 * q^57 - 36 * q^59 - 19 * q^61 + 21 * q^63 + 6 * q^65 - 34 * q^67 - q^69 - 10 * q^71 + 9 * q^73 - 44 * q^75 - 14 * q^77 - q^79 + 42 * q^81 - 56 * q^83 + 21 * q^85 - 5 * q^87 - 14 * q^89 - 14 * q^91 - 20 * q^93 + q^95 - 14 * q^97 - 21 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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