LMFDB - Embedded newform 8008.2.a.v.1.7

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:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 2 x^{10} - 19 x^{9} + 33 x^{8} + 120 x^{7} - 178 x^{6} - 296 x^{5} + 380 x^{4} + 280 x^{3} + \cdots + 64 $ x^11 - 2x^10 - 19x^9 + 33x^8 + 120x^7 - 178x^6 - 296x^5 + 380x^4 + 280x^3 - 295x^2 - 87x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.674923$ 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)

$f(q)$	$=$	$q+0.674923 q^{3} -4.42380 q^{5} -1.00000 q^{7} -2.54448 q^{9} +O(q^{10})$	$q+0.674923 q^{3} -4.42380 q^{5} -1.00000 q^{7} -2.54448 q^{9} +1.00000 q^{11} -1.00000 q^{13} -2.98573 q^{15} +4.81611 q^{17} -5.29198 q^{19} -0.674923 q^{21} +1.44917 q^{23} +14.5700 q^{25} -3.74210 q^{27} +7.93937 q^{29} +8.10497 q^{31} +0.674923 q^{33} +4.42380 q^{35} -6.99820 q^{37} -0.674923 q^{39} +7.05801 q^{41} -7.92840 q^{43} +11.2563 q^{45} +1.44917 q^{47} +1.00000 q^{49} +3.25050 q^{51} -8.55932 q^{53} -4.42380 q^{55} -3.57168 q^{57} -0.0353024 q^{59} +9.90155 q^{61} +2.54448 q^{63} +4.42380 q^{65} -1.89639 q^{67} +0.978076 q^{69} +9.06086 q^{71} -12.8184 q^{73} +9.83364 q^{75} -1.00000 q^{77} +11.6663 q^{79} +5.10781 q^{81} +9.53027 q^{83} -21.3055 q^{85} +5.35846 q^{87} -3.76222 q^{89} +1.00000 q^{91} +5.47023 q^{93} +23.4107 q^{95} -12.6735 q^{97} -2.54448 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 2 q^{3} + 2 q^{5} - 11 q^{7} + 9 q^{9}+O(q^{10}) $ 11 * q - 2 * q^3 + 2 * q^5 - 11 * q^7 + 9 * q^9	$ 11 q - 2 q^{3} + 2 q^{5} - 11 q^{7} + 9 q^{9} + 11 q^{11} - 11 q^{13} - 7 q^{15} - 4 q^{17} - 16 q^{19} + 2 q^{21} - 3 q^{23} + 11 q^{25} - 11 q^{27} + q^{29} + 14 q^{31} - 2 q^{33} - 2 q^{35} - 8 q^{37} + 2 q^{39} + 4 q^{41} - 30 q^{43} + 13 q^{45} - 3 q^{47} + 11 q^{49} - 14 q^{51} - 5 q^{53} + 2 q^{55} - 22 q^{57} + 11 q^{59} + 15 q^{61} - 9 q^{63} - 2 q^{65} - 41 q^{67} + 12 q^{69} + q^{71} - 8 q^{73} - 24 q^{75} - 11 q^{77} - 26 q^{79} + 19 q^{81} - 31 q^{83} - 27 q^{85} - 25 q^{87} + 2 q^{89} + 11 q^{91} - 37 q^{93} - 6 q^{97} + 9 q^{99}+O(q^{100}) $ 11 * q - 2 * q^3 + 2 * q^5 - 11 * q^7 + 9 * q^9 + 11 * q^11 - 11 * q^13 - 7 * q^15 - 4 * q^17 - 16 * q^19 + 2 * q^21 - 3 * q^23 + 11 * q^25 - 11 * q^27 + q^29 + 14 * q^31 - 2 * q^33 - 2 * q^35 - 8 * q^37 + 2 * q^39 + 4 * q^41 - 30 * q^43 + 13 * q^45 - 3 * q^47 + 11 * q^49 - 14 * q^51 - 5 * q^53 + 2 * q^55 - 22 * q^57 + 11 * q^59 + 15 * q^61 - 9 * q^63 - 2 * q^65 - 41 * q^67 + 12 * q^69 + q^71 - 8 * q^73 - 24 * q^75 - 11 * q^77 - 26 * q^79 + 19 * q^81 - 31 * q^83 - 27 * q^85 - 25 * q^87 + 2 * q^89 + 11 * q^91 - 37 * q^93 - 6 * q^97 + 9 * 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
$3$	0.674923	0.389667	0.194833	−	0.980836i	$-0.437583\pi$
$3$	0.674923	0.389667	0.194833	+	0.980836i	$0.437583\pi$
$4$	0	0
$5$	−4.42380	−1.97838	−0.989192	−	0.146625i	$-0.953159\pi$
$5$	−4.42380	−1.97838	−0.989192	+	0.146625i	$0.953159\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.54448	−0.848160
$10$	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
$15$	−2.98573	−0.770911
$16$	0	0
$17$	4.81611	1.16808	0.584039	−	0.811726i	$-0.301472\pi$
$17$	4.81611	1.16808	0.584039	+	0.811726i	$0.301472\pi$
$18$	0	0
$19$	−5.29198	−1.21406	−0.607032	−	0.794677i	$-0.707640\pi$
$19$	−5.29198	−1.21406	−0.607032	+	0.794677i	$0.707640\pi$
$20$	0	0
$21$	−0.674923	−0.147280
$22$	0	0
$23$	1.44917	0.302172	0.151086	−	0.988521i	$-0.451723\pi$
$23$	1.44917	0.302172	0.151086	+	0.988521i	$0.451723\pi$
$24$	0	0
$25$	14.5700	2.91400
$26$	0	0
$27$	−3.74210	−0.720167
$28$	0	0
$29$	7.93937	1.47430	0.737152	−	0.675727i	$-0.236170\pi$
$29$	7.93937	1.47430	0.737152	+	0.675727i	$0.236170\pi$
$30$	0	0
$31$	8.10497	1.45569	0.727847	−	0.685739i	$-0.240521\pi$
$31$	8.10497	1.45569	0.727847	+	0.685739i	$0.240521\pi$
$32$	0	0
$33$	0.674923	0.117489
$34$	0	0
$35$	4.42380	0.747759
$36$	0	0
$37$	−6.99820	−1.15050	−0.575248	−	0.817979i	$-0.695095\pi$
$37$	−6.99820	−1.15050	−0.575248	+	0.817979i	$0.695095\pi$
$38$	0	0
$39$	−0.674923	−0.108074
$40$	0	0
$41$	7.05801	1.10228	0.551138	−	0.834414i	$-0.314194\pi$
$41$	7.05801	1.10228	0.551138	+	0.834414i	$0.314194\pi$
$42$	0	0
$43$	−7.92840	−1.20907	−0.604535	−	0.796579i	$-0.706641\pi$
$43$	−7.92840	−1.20907	−0.604535	+	0.796579i	$0.706641\pi$
$44$	0	0
$45$	11.2563	1.67799
$46$	0	0
$47$	1.44917	0.211383	0.105691	−	0.994399i	$-0.466294\pi$
$47$	1.44917	0.211383	0.105691	+	0.994399i	$0.466294\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.25050	0.455161
$52$	0	0
$53$	−8.55932	−1.17571	−0.587857	−	0.808965i	$-0.700028\pi$
$53$	−8.55932	−1.17571	−0.587857	+	0.808965i	$0.700028\pi$
$54$	0	0
$55$	−4.42380	−0.596505
$56$	0	0
$57$	−3.57168	−0.473081
$58$	0	0
$59$	−0.0353024	−0.00459598	−0.00229799	−	0.999997i	$-0.500731\pi$
$59$	−0.0353024	−0.00459598	−0.00229799	+	0.999997i	$0.500731\pi$
$60$	0	0
$61$	9.90155	1.26776	0.633882	−	0.773430i	$-0.281461\pi$
$61$	9.90155	1.26776	0.633882	+	0.773430i	$0.281461\pi$
$62$	0	0
$63$	2.54448	0.320574
$64$	0	0
$65$	4.42380	0.548705
$66$	0	0
$67$	−1.89639	−0.231681	−0.115840	−	0.993268i	$-0.536956\pi$
$67$	−1.89639	−0.231681	−0.115840	+	0.993268i	$0.536956\pi$
$68$	0	0
$69$	0.978076	0.117747
$70$	0	0
$71$	9.06086	1.07533	0.537663	−	0.843160i	$-0.319307\pi$
$71$	9.06086	1.07533	0.537663	+	0.843160i	$0.319307\pi$
$72$	0	0
$73$	−12.8184	−1.50028	−0.750139	−	0.661280i	$-0.770014\pi$
$73$	−12.8184	−1.50028	−0.750139	+	0.661280i	$0.770014\pi$
$74$	0	0
$75$	9.83364	1.13549
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	11.6663	1.31256	0.656280	−	0.754518i	$-0.272129\pi$
$79$	11.6663	1.31256	0.656280	+	0.754518i	$0.272129\pi$
$80$	0	0
$81$	5.10781	0.567535
$82$	0	0
$83$	9.53027	1.04608	0.523042	−	0.852307i	$-0.324797\pi$
$83$	9.53027	1.04608	0.523042	+	0.852307i	$0.324797\pi$
$84$	0	0
$85$	−21.3055	−2.31091
$86$	0	0
$87$	5.35846	0.574487
$88$	0	0
$89$	−3.76222	−0.398795	−0.199397	−	0.979919i	$-0.563898\pi$
$89$	−3.76222	−0.398795	−0.199397	+	0.979919i	$0.563898\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	5.47023	0.567236
$94$	0	0
$95$	23.4107	2.40189
$96$	0	0
$97$	−12.6735	−1.28679	−0.643397	−	0.765532i	$-0.722476\pi$
$97$	−12.6735	−1.28679	−0.643397	+	0.765532i	$0.722476\pi$
$98$	0	0
$99$	−2.54448	−0.255730
$100$	0	0
$101$	10.1777	1.01272	0.506360	−	0.862322i	$-0.330991\pi$
$101$	10.1777	1.01272	0.506360	+	0.862322i	$0.330991\pi$
$102$	0	0
$103$	9.31106	0.917446	0.458723	−	0.888579i	$-0.348307\pi$
$103$	9.31106	0.917446	0.458723	+	0.888579i	$0.348307\pi$
$104$	0	0
$105$	2.98573	0.291377
$106$	0	0
$107$	−11.9527	−1.15551	−0.577756	−	0.816210i	$-0.696071\pi$
$107$	−11.9527	−1.15551	−0.577756	+	0.816210i	$0.696071\pi$
$108$	0	0
$109$	−4.51461	−0.432421	−0.216211	−	0.976347i	$-0.569370\pi$
$109$	−4.51461	−0.432421	−0.216211	+	0.976347i	$0.569370\pi$
$110$	0	0
$111$	−4.72324	−0.448311
$112$	0	0
$113$	−9.44130	−0.888163	−0.444082	−	0.895986i	$-0.646470\pi$
$113$	−9.44130	−0.888163	−0.444082	+	0.895986i	$0.646470\pi$
$114$	0	0
$115$	−6.41083	−0.597813
$116$	0	0
$117$	2.54448	0.235237
$118$	0	0
$119$	−4.81611	−0.441492
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	4.76361	0.429521
$124$	0	0
$125$	−42.3359	−3.78664
$126$	0	0
$127$	−10.1471	−0.900412	−0.450206	−	0.892925i	$-0.648650\pi$
$127$	−10.1471	−0.900412	−0.450206	+	0.892925i	$0.648650\pi$
$128$	0	0
$129$	−5.35106	−0.471134
$130$	0	0
$131$	−12.4940	−1.09161	−0.545803	−	0.837914i	$-0.683775\pi$
$131$	−12.4940	−1.09161	−0.545803	+	0.837914i	$0.683775\pi$
$132$	0	0
$133$	5.29198	0.458873
$134$	0	0
$135$	16.5543	1.42477
$136$	0	0
$137$	−11.9240	−1.01873	−0.509367	−	0.860550i	$-0.670120\pi$
$137$	−11.9240	−1.01873	−0.509367	+	0.860550i	$0.670120\pi$
$138$	0	0
$139$	−0.714210	−0.0605785	−0.0302893	−	0.999541i	$-0.509643\pi$
$139$	−0.714210	−0.0605785	−0.0302893	+	0.999541i	$0.509643\pi$
$140$	0	0
$141$	0.978076	0.0823689
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−35.1222	−2.91674
$146$	0	0
$147$	0.674923	0.0556667
$148$	0	0
$149$	−3.41829	−0.280038	−0.140019	−	0.990149i	$-0.544716\pi$
$149$	−3.41829	−0.280038	−0.140019	+	0.990149i	$0.544716\pi$
$150$	0	0
$151$	4.15978	0.338518	0.169259	−	0.985572i	$-0.445863\pi$
$151$	4.15978	0.338518	0.169259	+	0.985572i	$0.445863\pi$
$152$	0	0
$153$	−12.2545	−0.990717
$154$	0	0
$155$	−35.8548	−2.87992
$156$	0	0
$157$	23.1118	1.84452	0.922261	−	0.386568i	$-0.126340\pi$
$157$	23.1118	1.84452	0.922261	+	0.386568i	$0.126340\pi$
$158$	0	0
$159$	−5.77688	−0.458137
$160$	0	0
$161$	−1.44917	−0.114210
$162$	0	0
$163$	−3.84516	−0.301176	−0.150588	−	0.988597i	$-0.548117\pi$
$163$	−3.84516	−0.301176	−0.150588	+	0.988597i	$0.548117\pi$
$164$	0	0
$165$	−2.98573	−0.232438
$166$	0	0
$167$	5.64659	0.436946	0.218473	−	0.975843i	$-0.429892\pi$
$167$	5.64659	0.436946	0.218473	+	0.975843i	$0.429892\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	13.4653	1.02972
$172$	0	0
$173$	−9.14183	−0.695040	−0.347520	−	0.937673i	$-0.612976\pi$
$173$	−9.14183	−0.695040	−0.347520	+	0.937673i	$0.612976\pi$
$174$	0	0
$175$	−14.5700	−1.10139
$176$	0	0
$177$	−0.0238264	−0.00179090
$178$	0	0
$179$	−19.3709	−1.44785	−0.723924	−	0.689880i	$-0.757663\pi$
$179$	−19.3709	−1.44785	−0.723924	+	0.689880i	$0.757663\pi$
$180$	0	0
$181$	1.44636	0.107507	0.0537535	−	0.998554i	$-0.482881\pi$
$181$	1.44636	0.107507	0.0537535	+	0.998554i	$0.482881\pi$
$182$	0	0
$183$	6.68278	0.494006
$184$	0	0
$185$	30.9586	2.27612
$186$	0	0
$187$	4.81611	0.352189
$188$	0	0
$189$	3.74210	0.272197
$190$	0	0
$191$	17.3629	1.25633	0.628166	−	0.778079i	$-0.283806\pi$
$191$	17.3629	1.25633	0.628166	+	0.778079i	$0.283806\pi$
$192$	0	0
$193$	15.1736	1.09222	0.546112	−	0.837712i	$-0.316107\pi$
$193$	15.1736	1.09222	0.546112	+	0.837712i	$0.316107\pi$
$194$	0	0
$195$	2.98573	0.213812
$196$	0	0
$197$	−24.5498	−1.74910	−0.874552	−	0.484932i	$-0.838844\pi$
$197$	−24.5498	−1.74910	−0.874552	+	0.484932i	$0.838844\pi$
$198$	0	0
$199$	3.34595	0.237188	0.118594	−	0.992943i	$-0.462161\pi$
$199$	3.34595	0.237188	0.118594	+	0.992943i	$0.462161\pi$
$200$	0	0
$201$	−1.27992	−0.0902783
$202$	0	0
$203$	−7.93937	−0.557234
$204$	0	0
$205$	−31.2233	−2.18073
$206$	0	0
$207$	−3.68737	−0.256290
$208$	0	0
$209$	−5.29198	−0.366054
$210$	0	0
$211$	−14.8500	−1.02232	−0.511158	−	0.859487i	$-0.670783\pi$
$211$	−14.8500	−1.02232	−0.511158	+	0.859487i	$0.670783\pi$
$212$	0	0
$213$	6.11538	0.419019
$214$	0	0
$215$	35.0737	2.39200
$216$	0	0
$217$	−8.10497	−0.550201
$218$	0	0
$219$	−8.65142	−0.584609
$220$	0	0
$221$	−4.81611	−0.323967
$222$	0	0
$223$	−20.8739	−1.39782	−0.698911	−	0.715209i	$-0.746331\pi$
$223$	−20.8739	−1.39782	−0.698911	+	0.715209i	$0.746331\pi$
$224$	0	0
$225$	−37.0731	−2.47154
$226$	0	0
$227$	−11.7135	−0.777455	−0.388728	−	0.921353i	$-0.627085\pi$
$227$	−11.7135	−0.777455	−0.388728	+	0.921353i	$0.627085\pi$
$228$	0	0
$229$	11.1255	0.735196	0.367598	−	0.929985i	$-0.380180\pi$
$229$	11.1255	0.735196	0.367598	+	0.929985i	$0.380180\pi$
$230$	0	0
$231$	−0.674923	−0.0444067
$232$	0	0
$233$	−2.91286	−0.190828	−0.0954140	−	0.995438i	$-0.530417\pi$
$233$	−2.91286	−0.190828	−0.0954140	+	0.995438i	$0.530417\pi$
$234$	0	0
$235$	−6.41083	−0.418196
$236$	0	0
$237$	7.87384	0.511461
$238$	0	0
$239$	−14.3670	−0.929321	−0.464661	−	0.885489i	$-0.653824\pi$
$239$	−14.3670	−0.929321	−0.464661	+	0.885489i	$0.653824\pi$
$240$	0	0
$241$	14.7627	0.950951	0.475476	−	0.879729i	$-0.342276\pi$
$241$	14.7627	0.950951	0.475476	+	0.879729i	$0.342276\pi$
$242$	0	0
$243$	14.6737	0.941316
$244$	0	0
$245$	−4.42380	−0.282626
$246$	0	0
$247$	5.29198	0.336721
$248$	0	0
$249$	6.43220	0.407624
$250$	0	0
$251$	8.98630	0.567210	0.283605	−	0.958941i	$-0.408469\pi$
$251$	8.98630	0.567210	0.283605	+	0.958941i	$0.408469\pi$
$252$	0	0
$253$	1.44917	0.0911083
$254$	0	0
$255$	−14.3796	−0.900484
$256$	0	0
$257$	−29.1092	−1.81578	−0.907890	−	0.419208i	$-0.862308\pi$
$257$	−29.1092	−1.81578	−0.907890	+	0.419208i	$0.862308\pi$
$258$	0	0
$259$	6.99820	0.434847
$260$	0	0
$261$	−20.2015	−1.25044
$262$	0	0
$263$	−15.1114	−0.931806	−0.465903	−	0.884836i	$-0.654270\pi$
$263$	−15.1114	−0.931806	−0.465903	+	0.884836i	$0.654270\pi$
$264$	0	0
$265$	37.8647	2.32601
$266$	0	0
$267$	−2.53921	−0.155397
$268$	0	0
$269$	22.8050	1.39044	0.695222	−	0.718795i	$-0.255306\pi$
$269$	22.8050	1.39044	0.695222	+	0.718795i	$0.255306\pi$
$270$	0	0
$271$	18.7533	1.13918	0.569591	−	0.821928i	$-0.307102\pi$
$271$	18.7533	1.13918	0.569591	+	0.821928i	$0.307102\pi$
$272$	0	0
$273$	0.674923	0.0408482
$274$	0	0
$275$	14.5700	0.878605
$276$	0	0
$277$	−3.21177	−0.192977	−0.0964884	−	0.995334i	$-0.530761\pi$
$277$	−3.21177	−0.192977	−0.0964884	+	0.995334i	$0.530761\pi$
$278$	0	0
$279$	−20.6229	−1.23466
$280$	0	0
$281$	7.26096	0.433153	0.216576	−	0.976266i	$-0.430511\pi$
$281$	7.26096	0.433153	0.216576	+	0.976266i	$0.430511\pi$
$282$	0	0
$283$	13.4704	0.800733	0.400367	−	0.916355i	$-0.368883\pi$
$283$	13.4704	0.800733	0.400367	+	0.916355i	$0.368883\pi$
$284$	0	0
$285$	15.8004	0.935935
$286$	0	0
$287$	−7.05801	−0.416621
$288$	0	0
$289$	6.19490	0.364406
$290$	0	0
$291$	−8.55361	−0.501421
$292$	0	0
$293$	−18.5449	−1.08341	−0.541703	−	0.840570i	$-0.682220\pi$
$293$	−18.5449	−1.08341	−0.541703	+	0.840570i	$0.682220\pi$
$294$	0	0
$295$	0.156171	0.00909261
$296$	0	0
$297$	−3.74210	−0.217138
$298$	0	0
$299$	−1.44917	−0.0838075
$300$	0	0
$301$	7.92840	0.456985
$302$	0	0
$303$	6.86917	0.394624
$304$	0	0
$305$	−43.8025	−2.50812
$306$	0	0
$307$	−21.1261	−1.20573	−0.602864	−	0.797844i	$-0.705974\pi$
$307$	−21.1261	−1.20573	−0.602864	+	0.797844i	$0.705974\pi$
$308$	0	0
$309$	6.28425	0.357498
$310$	0	0
$311$	−2.29335	−0.130044	−0.0650221	−	0.997884i	$-0.520712\pi$
$311$	−2.29335	−0.130044	−0.0650221	+	0.997884i	$0.520712\pi$
$312$	0	0
$313$	−6.67680	−0.377395	−0.188697	−	0.982035i	$-0.560427\pi$
$313$	−6.67680	−0.377395	−0.188697	+	0.982035i	$0.560427\pi$
$314$	0	0
$315$	−11.2563	−0.634219
$316$	0	0
$317$	−28.1307	−1.57998	−0.789989	−	0.613121i	$-0.789914\pi$
$317$	−28.1307	−1.57998	−0.789989	+	0.613121i	$0.789914\pi$
$318$	0	0
$319$	7.93937	0.444519
$320$	0	0
$321$	−8.06716	−0.450265
$322$	0	0
$323$	−25.4868	−1.41812
$324$	0	0
$325$	−14.5700	−0.808199
$326$	0	0
$327$	−3.04701	−0.168500
$328$	0	0
$329$	−1.44917	−0.0798952
$330$	0	0
$331$	−16.0175	−0.880399	−0.440200	−	0.897900i	$-0.645092\pi$
$331$	−16.0175	−0.880399	−0.440200	+	0.897900i	$0.645092\pi$
$332$	0	0
$333$	17.8068	0.975805
$334$	0	0
$335$	8.38924	0.458353
$336$	0	0
$337$	17.2084	0.937403	0.468702	−	0.883356i	$-0.344722\pi$
$337$	17.2084	0.937403	0.468702	+	0.883356i	$0.344722\pi$
$338$	0	0
$339$	−6.37215	−0.346088
$340$	0	0
$341$	8.10497	0.438909
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−4.32681	−0.232948
$346$	0	0
$347$	9.29313	0.498881	0.249441	−	0.968390i	$-0.419753\pi$
$347$	9.29313	0.498881	0.249441	+	0.968390i	$0.419753\pi$
$348$	0	0
$349$	23.1239	1.23780	0.618898	−	0.785472i	$-0.287580\pi$
$349$	23.1239	1.23780	0.618898	+	0.785472i	$0.287580\pi$
$350$	0	0
$351$	3.74210	0.199738
$352$	0	0
$353$	−13.0249	−0.693245	−0.346623	−	0.938005i	$-0.612672\pi$
$353$	−13.0249	−0.693245	−0.346623	+	0.938005i	$0.612672\pi$
$354$	0	0
$355$	−40.0835	−2.12741
$356$	0	0
$357$	−3.25050	−0.172035
$358$	0	0
$359$	12.2056	0.644189	0.322094	−	0.946708i	$-0.395613\pi$
$359$	12.2056	0.644189	0.322094	+	0.946708i	$0.395613\pi$
$360$	0	0
$361$	9.00509	0.473952
$362$	0	0
$363$	0.674923	0.0354243
$364$	0	0
$365$	56.7060	2.96813
$366$	0	0
$367$	−16.9347	−0.883983	−0.441991	−	0.897019i	$-0.645728\pi$
$367$	−16.9347	−0.883983	−0.441991	+	0.897019i	$0.645728\pi$
$368$	0	0
$369$	−17.9590	−0.934906
$370$	0	0
$371$	8.55932	0.444378
$372$	0	0
$373$	−3.19837	−0.165605	−0.0828026	−	0.996566i	$-0.526387\pi$
$373$	−3.19837	−0.165605	−0.0828026	+	0.996566i	$0.526387\pi$
$374$	0	0
$375$	−28.5735	−1.47553
$376$	0	0
$377$	−7.93937	−0.408898
$378$	0	0
$379$	16.2298	0.833669	0.416834	−	0.908982i	$-0.363140\pi$
$379$	16.2298	0.833669	0.416834	+	0.908982i	$0.363140\pi$
$380$	0	0
$381$	−6.84853	−0.350861
$382$	0	0
$383$	−24.4171	−1.24766	−0.623828	−	0.781562i	$-0.714423\pi$
$383$	−24.4171	−1.24766	−0.623828	+	0.781562i	$0.714423\pi$
$384$	0	0
$385$	4.42380	0.225458
$386$	0	0
$387$	20.1736	1.02548
$388$	0	0
$389$	−20.3401	−1.03129	−0.515643	−	0.856804i	$-0.672447\pi$
$389$	−20.3401	−1.03129	−0.515643	+	0.856804i	$0.672447\pi$
$390$	0	0
$391$	6.97935	0.352961
$392$	0	0
$393$	−8.43248	−0.425362
$394$	0	0
$395$	−51.6093	−2.59675
$396$	0	0
$397$	29.7422	1.49272	0.746358	−	0.665544i	$-0.231801\pi$
$397$	29.7422	1.49272	0.746358	+	0.665544i	$0.231801\pi$
$398$	0	0
$399$	3.57168	0.178808
$400$	0	0
$401$	−27.3109	−1.36384	−0.681921	−	0.731426i	$-0.738855\pi$
$401$	−27.3109	−1.36384	−0.681921	+	0.731426i	$0.738855\pi$
$402$	0	0
$403$	−8.10497	−0.403737
$404$	0	0
$405$	−22.5959	−1.12280
$406$	0	0
$407$	−6.99820	−0.346888
$408$	0	0
$409$	14.1376	0.699058	0.349529	−	0.936925i	$-0.386342\pi$
$409$	14.1376	0.699058	0.349529	+	0.936925i	$0.386342\pi$
$410$	0	0
$411$	−8.04776	−0.396967
$412$	0	0
$413$	0.0353024	0.00173712
$414$	0	0
$415$	−42.1600	−2.06955
$416$	0	0
$417$	−0.482037	−0.0236055
$418$	0	0
$419$	−26.0120	−1.27077	−0.635385	−	0.772195i	$-0.719159\pi$
$419$	−26.0120	−1.27077	−0.635385	+	0.772195i	$0.719159\pi$
$420$	0	0
$421$	−27.0709	−1.31935	−0.659677	−	0.751549i	$-0.729307\pi$
$421$	−27.0709	−1.31935	−0.659677	+	0.751549i	$0.729307\pi$
$422$	0	0
$423$	−3.68737	−0.179286
$424$	0	0
$425$	70.1708	3.40378
$426$	0	0
$427$	−9.90155	−0.479170
$428$	0	0
$429$	−0.674923	−0.0325856
$430$	0	0
$431$	−17.5232	−0.844061	−0.422030	−	0.906582i	$-0.638682\pi$
$431$	−17.5232	−0.844061	−0.422030	+	0.906582i	$0.638682\pi$
$432$	0	0
$433$	−5.55212	−0.266818	−0.133409	−	0.991061i	$-0.542592\pi$
$433$	−5.55212	−0.266818	−0.133409	+	0.991061i	$0.542592\pi$
$434$	0	0
$435$	−23.7048	−1.13656
$436$	0	0
$437$	−7.66897	−0.366856
$438$	0	0
$439$	−6.98133	−0.333201	−0.166600	−	0.986025i	$-0.553279\pi$
$439$	−6.98133	−0.333201	−0.166600	+	0.986025i	$0.553279\pi$
$440$	0	0
$441$	−2.54448	−0.121166
$442$	0	0
$443$	37.3655	1.77529	0.887645	−	0.460528i	$-0.152340\pi$
$443$	37.3655	1.77529	0.887645	+	0.460528i	$0.152340\pi$
$444$	0	0
$445$	16.6433	0.788969
$446$	0	0
$447$	−2.30709	−0.109121
$448$	0	0
$449$	7.56395	0.356965	0.178483	−	0.983943i	$-0.442881\pi$
$449$	7.56395	0.356965	0.178483	+	0.983943i	$0.442881\pi$
$450$	0	0
$451$	7.05801	0.332349
$452$	0	0
$453$	2.80753	0.131909
$454$	0	0
$455$	−4.42380	−0.207391
$456$	0	0
$457$	29.5994	1.38460	0.692301	−	0.721609i	$-0.256597\pi$
$457$	29.5994	1.38460	0.692301	+	0.721609i	$0.256597\pi$
$458$	0	0
$459$	−18.0223	−0.841211
$460$	0	0
$461$	38.6887	1.80191	0.900956	−	0.433911i	$-0.142867\pi$
$461$	38.6887	1.80191	0.900956	+	0.433911i	$0.142867\pi$
$462$	0	0
$463$	−10.4374	−0.485068	−0.242534	−	0.970143i	$-0.577979\pi$
$463$	−10.4374	−0.485068	−0.242534	+	0.970143i	$0.577979\pi$
$464$	0	0
$465$	−24.1992	−1.12221
$466$	0	0
$467$	−42.1488	−1.95041	−0.975207	−	0.221296i	$-0.928971\pi$
$467$	−42.1488	−1.95041	−0.975207	+	0.221296i	$0.928971\pi$
$468$	0	0
$469$	1.89639	0.0875671
$470$	0	0
$471$	15.5987	0.718749
$472$	0	0
$473$	−7.92840	−0.364548
$474$	0	0
$475$	−77.1043	−3.53779
$476$	0	0
$477$	21.7790	0.997192
$478$	0	0
$479$	−38.5075	−1.75945	−0.879726	−	0.475481i	$-0.842274\pi$
$479$	−38.5075	−1.75945	−0.879726	+	0.475481i	$0.842274\pi$
$480$	0	0
$481$	6.99820	0.319090
$482$	0	0
$483$	−0.978076	−0.0445040
$484$	0	0
$485$	56.0649	2.54577
$486$	0	0
$487$	−25.6697	−1.16320	−0.581602	−	0.813474i	$-0.697574\pi$
$487$	−25.6697	−1.16320	−0.581602	+	0.813474i	$0.697574\pi$
$488$	0	0
$489$	−2.59519	−0.117358
$490$	0	0
$491$	9.04609	0.408244	0.204122	−	0.978945i	$-0.434566\pi$
$491$	9.04609	0.408244	0.204122	+	0.978945i	$0.434566\pi$
$492$	0	0
$493$	38.2368	1.72210
$494$	0	0
$495$	11.2563	0.505932
$496$	0	0
$497$	−9.06086	−0.406435
$498$	0	0
$499$	27.9614	1.25172	0.625862	−	0.779933i	$-0.284747\pi$
$499$	27.9614	1.25172	0.625862	+	0.779933i	$0.284747\pi$
$500$	0	0
$501$	3.81101	0.170264
$502$	0	0
$503$	43.6319	1.94545	0.972724	−	0.231964i	$-0.0745150\pi$
$503$	43.6319	1.94545	0.972724	+	0.231964i	$0.0745150\pi$
$504$	0	0
$505$	−45.0242	−2.00355
$506$	0	0
$507$	0.674923	0.0299744
$508$	0	0
$509$	−5.18174	−0.229677	−0.114838	−	0.993384i	$-0.536635\pi$
$509$	−5.18174	−0.229677	−0.114838	+	0.993384i	$0.536635\pi$
$510$	0	0
$511$	12.8184	0.567052
$512$	0	0
$513$	19.8031	0.874329
$514$	0	0
$515$	−41.1903	−1.81506
$516$	0	0
$517$	1.44917	0.0637343
$518$	0	0
$519$	−6.17003	−0.270834
$520$	0	0
$521$	8.34229	0.365482	0.182741	−	0.983161i	$-0.441503\pi$
$521$	8.34229	0.365482	0.182741	+	0.983161i	$0.441503\pi$
$522$	0	0
$523$	−17.9235	−0.783740	−0.391870	−	0.920021i	$-0.628172\pi$
$523$	−17.9235	−0.783740	−0.391870	+	0.920021i	$0.628172\pi$
$524$	0	0
$525$	−9.83364	−0.429175
$526$	0	0
$527$	39.0344	1.70037
$528$	0	0
$529$	−20.8999	−0.908692
$530$	0	0
$531$	0.0898262	0.00389812
$532$	0	0
$533$	−7.05801	−0.305716
$534$	0	0
$535$	52.8764	2.28605
$536$	0	0
$537$	−13.0739	−0.564179
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−34.3819	−1.47819	−0.739097	−	0.673599i	$-0.764747\pi$
$541$	−34.3819	−1.47819	−0.739097	+	0.673599i	$0.764747\pi$
$542$	0	0
$543$	0.976180	0.0418919
$544$	0	0
$545$	19.9717	0.855495
$546$	0	0
$547$	−5.39334	−0.230602	−0.115301	−	0.993331i	$-0.536783\pi$
$547$	−5.39334	−0.230602	−0.115301	+	0.993331i	$0.536783\pi$
$548$	0	0
$549$	−25.1943	−1.07527
$550$	0	0
$551$	−42.0150	−1.78990
$552$	0	0
$553$	−11.6663	−0.496101
$554$	0	0
$555$	20.8947	0.886931
$556$	0	0
$557$	10.8664	0.460423	0.230211	−	0.973141i	$-0.426058\pi$
$557$	10.8664	0.460423	0.230211	+	0.973141i	$0.426058\pi$
$558$	0	0
$559$	7.92840	0.335336
$560$	0	0
$561$	3.25050	0.137236
$562$	0	0
$563$	39.8294	1.67861	0.839305	−	0.543661i	$-0.182962\pi$
$563$	39.8294	1.67861	0.839305	+	0.543661i	$0.182962\pi$
$564$	0	0
$565$	41.7664	1.75713
$566$	0	0
$567$	−5.10781	−0.214508
$568$	0	0
$569$	0.623097	0.0261216	0.0130608	−	0.999915i	$-0.495843\pi$
$569$	0.623097	0.0261216	0.0130608	+	0.999915i	$0.495843\pi$
$570$	0	0
$571$	31.2854	1.30925	0.654627	−	0.755952i	$-0.272826\pi$
$571$	31.2854	1.30925	0.654627	+	0.755952i	$0.272826\pi$
$572$	0	0
$573$	11.7186	0.489551
$574$	0	0
$575$	21.1144	0.880531
$576$	0	0
$577$	−11.9512	−0.497535	−0.248768	−	0.968563i	$-0.580026\pi$
$577$	−11.9512	−0.497535	−0.248768	+	0.968563i	$0.580026\pi$
$578$	0	0
$579$	10.2410	0.425603
$580$	0	0
$581$	−9.53027	−0.395382
$582$	0	0
$583$	−8.55932	−0.354491
$584$	0	0
$585$	−11.2563	−0.465390
$586$	0	0
$587$	18.5213	0.764454	0.382227	−	0.924068i	$-0.375157\pi$
$587$	18.5213	0.764454	0.382227	+	0.924068i	$0.375157\pi$
$588$	0	0
$589$	−42.8913	−1.76731
$590$	0	0
$591$	−16.5693	−0.681568
$592$	0	0
$593$	−22.2135	−0.912198	−0.456099	−	0.889929i	$-0.650754\pi$
$593$	−22.2135	−0.912198	−0.456099	+	0.889929i	$0.650754\pi$
$594$	0	0
$595$	21.3055	0.873441
$596$	0	0
$597$	2.25826	0.0924243
$598$	0	0
$599$	−43.5039	−1.77752	−0.888760	−	0.458373i	$-0.848432\pi$
$599$	−43.5039	−1.77752	−0.888760	+	0.458373i	$0.848432\pi$
$600$	0	0
$601$	28.7556	1.17296	0.586482	−	0.809962i	$-0.300512\pi$
$601$	28.7556	1.17296	0.586482	+	0.809962i	$0.300512\pi$
$602$	0	0
$603$	4.82532	0.196502
$604$	0	0
$605$	−4.42380	−0.179853
$606$	0	0
$607$	−23.5911	−0.957533	−0.478766	−	0.877942i	$-0.658916\pi$
$607$	−23.5911	−0.957533	−0.478766	+	0.877942i	$0.658916\pi$
$608$	0	0
$609$	−5.35846	−0.217136
$610$	0	0
$611$	−1.44917	−0.0586270
$612$	0	0
$613$	32.6718	1.31960	0.659801	−	0.751440i	$-0.270641\pi$
$613$	32.6718	1.31960	0.659801	+	0.751440i	$0.270641\pi$
$614$	0	0
$615$	−21.0733	−0.849757
$616$	0	0
$617$	−13.6524	−0.549626	−0.274813	−	0.961498i	$-0.588616\pi$
$617$	−13.6524	−0.549626	−0.274813	+	0.961498i	$0.588616\pi$
$618$	0	0
$619$	−40.7238	−1.63683	−0.818414	−	0.574628i	$-0.805147\pi$
$619$	−40.7238	−1.63683	−0.818414	+	0.574628i	$0.805147\pi$
$620$	0	0
$621$	−5.42292	−0.217614
$622$	0	0
$623$	3.76222	0.150730
$624$	0	0
$625$	114.435	4.57742
$626$	0	0
$627$	−3.57168	−0.142639
$628$	0	0
$629$	−33.7041	−1.34387
$630$	0	0
$631$	−26.5313	−1.05620	−0.528098	−	0.849183i	$-0.677095\pi$
$631$	−26.5313	−1.05620	−0.528098	+	0.849183i	$0.677095\pi$
$632$	0	0
$633$	−10.0226	−0.398363
$634$	0	0
$635$	44.8889	1.78136
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−23.0552	−0.912049
$640$	0	0
$641$	−38.4309	−1.51793	−0.758965	−	0.651131i	$-0.774295\pi$
$641$	−38.4309	−1.51793	−0.758965	+	0.651131i	$0.774295\pi$
$642$	0	0
$643$	18.0349	0.711227	0.355614	−	0.934633i	$-0.384272\pi$
$643$	18.0349	0.711227	0.355614	+	0.934633i	$0.384272\pi$
$644$	0	0
$645$	23.6720	0.932085
$646$	0	0
$647$	48.5846	1.91006	0.955030	−	0.296510i	$-0.0958229\pi$
$647$	48.5846	1.91006	0.955030	+	0.296510i	$0.0958229\pi$
$648$	0	0
$649$	−0.0353024	−0.00138574
$650$	0	0
$651$	−5.47023	−0.214395
$652$	0	0
$653$	−5.90005	−0.230887	−0.115443	−	0.993314i	$-0.536829\pi$
$653$	−5.90005	−0.230887	−0.115443	+	0.993314i	$0.536829\pi$
$654$	0	0
$655$	55.2710	2.15961
$656$	0	0
$657$	32.6161	1.27248
$658$	0	0
$659$	37.9221	1.47723	0.738617	−	0.674125i	$-0.235479\pi$
$659$	37.9221	1.47723	0.738617	+	0.674125i	$0.235479\pi$
$660$	0	0
$661$	34.3083	1.33444	0.667219	−	0.744862i	$-0.267485\pi$
$661$	34.3083	1.33444	0.667219	+	0.744862i	$0.267485\pi$
$662$	0	0
$663$	−3.25050	−0.126239
$664$	0	0
$665$	−23.4107	−0.907828
$666$	0	0
$667$	11.5055	0.445493
$668$	0	0
$669$	−14.0883	−0.544685
$670$	0	0
$671$	9.90155	0.382245
$672$	0	0
$673$	2.76517	0.106590	0.0532948	−	0.998579i	$-0.483028\pi$
$673$	2.76517	0.106590	0.0532948	+	0.998579i	$0.483028\pi$
$674$	0	0
$675$	−54.5224	−2.09857
$676$	0	0
$677$	17.7283	0.681353	0.340677	−	0.940181i	$-0.389344\pi$
$677$	17.7283	0.681353	0.340677	+	0.940181i	$0.389344\pi$
$678$	0	0
$679$	12.6735	0.486363
$680$	0	0
$681$	−7.90574	−0.302949
$682$	0	0
$683$	−1.98355	−0.0758985	−0.0379492	−	0.999280i	$-0.512083\pi$
$683$	−1.98355	−0.0758985	−0.0379492	+	0.999280i	$0.512083\pi$
$684$	0	0
$685$	52.7493	2.01545
$686$	0	0
$687$	7.50888	0.286482
$688$	0	0
$689$	8.55932	0.326084
$690$	0	0
$691$	39.0568	1.48579	0.742895	−	0.669408i	$-0.233452\pi$
$691$	39.0568	1.48579	0.742895	+	0.669408i	$0.233452\pi$
$692$	0	0
$693$	2.54448	0.0966568
$694$	0	0
$695$	3.15953	0.119848
$696$	0	0
$697$	33.9922	1.28754
$698$	0	0
$699$	−1.96596	−0.0743593
$700$	0	0
$701$	8.68915	0.328185	0.164092	−	0.986445i	$-0.447530\pi$
$701$	8.68915	0.328185	0.164092	+	0.986445i	$0.447530\pi$
$702$	0	0
$703$	37.0344	1.39678
$704$	0	0
$705$	−4.32681	−0.162957
$706$	0	0
$707$	−10.1777	−0.382772
$708$	0	0
$709$	9.18636	0.345001	0.172500	−	0.985009i	$-0.444815\pi$
$709$	9.18636	0.345001	0.172500	+	0.985009i	$0.444815\pi$
$710$	0	0
$711$	−29.6846	−1.11326
$712$	0	0
$713$	11.7454	0.439871
$714$	0	0
$715$	4.42380	0.165441
$716$	0	0
$717$	−9.69659	−0.362126
$718$	0	0
$719$	29.1925	1.08870	0.544349	−	0.838859i	$-0.316777\pi$
$719$	29.1925	1.08870	0.544349	+	0.838859i	$0.316777\pi$
$720$	0	0
$721$	−9.31106	−0.346762
$722$	0	0
$723$	9.96371	0.370554
$724$	0	0
$725$	115.677	4.29613
$726$	0	0
$727$	−45.8196	−1.69935	−0.849677	−	0.527303i	$-0.823203\pi$
$727$	−45.8196	−1.69935	−0.849677	+	0.527303i	$0.823203\pi$
$728$	0	0
$729$	−5.41984	−0.200735
$730$	0	0
$731$	−38.1840	−1.41229
$732$	0	0
$733$	−27.7660	−1.02556	−0.512781	−	0.858519i	$-0.671385\pi$
$733$	−27.7660	−1.02556	−0.512781	+	0.858519i	$0.671385\pi$
$734$	0	0
$735$	−2.98573	−0.110130
$736$	0	0
$737$	−1.89639	−0.0698543
$738$	0	0
$739$	5.20611	0.191510	0.0957549	−	0.995405i	$-0.469473\pi$
$739$	5.20611	0.191510	0.0957549	+	0.995405i	$0.469473\pi$
$740$	0	0
$741$	3.57168	0.131209
$742$	0	0
$743$	−27.2039	−0.998016	−0.499008	−	0.866597i	$-0.666302\pi$
$743$	−27.2039	−0.998016	−0.499008	+	0.866597i	$0.666302\pi$
$744$	0	0
$745$	15.1219	0.554022
$746$	0	0
$747$	−24.2496	−0.887245
$748$	0	0
$749$	11.9527	0.436742
$750$	0	0
$751$	−25.6555	−0.936183	−0.468091	−	0.883680i	$-0.655058\pi$
$751$	−25.6555	−0.936183	−0.468091	+	0.883680i	$0.655058\pi$
$752$	0	0
$753$	6.06506	0.221023
$754$	0	0
$755$	−18.4020	−0.669718
$756$	0	0
$757$	8.79352	0.319606	0.159803	−	0.987149i	$-0.448914\pi$
$757$	8.79352	0.319606	0.159803	+	0.987149i	$0.448914\pi$
$758$	0	0
$759$	0.978076	0.0355019
$760$	0	0
$761$	−22.4152	−0.812550	−0.406275	−	0.913751i	$-0.633173\pi$
$761$	−22.4152	−0.812550	−0.406275	+	0.913751i	$0.633173\pi$
$762$	0	0
$763$	4.51461	0.163440
$764$	0	0
$765$	54.2114	1.96002
$766$	0	0
$767$	0.0353024	0.00127470
$768$	0	0
$769$	−9.31270	−0.335824	−0.167912	−	0.985802i	$-0.553703\pi$
$769$	−9.31270	−0.335824	−0.167912	+	0.985802i	$0.553703\pi$
$770$	0	0
$771$	−19.6464	−0.707550
$772$	0	0
$773$	−35.5241	−1.27771	−0.638857	−	0.769325i	$-0.720593\pi$
$773$	−35.5241	−1.27771	−0.638857	+	0.769325i	$0.720593\pi$
$774$	0	0
$775$	118.090	4.24190
$776$	0	0
$777$	4.72324	0.169445
$778$	0	0
$779$	−37.3509	−1.33823
$780$	0	0
$781$	9.06086	0.324223
$782$	0	0
$783$	−29.7099	−1.06174
$784$	0	0
$785$	−102.242	−3.64917
$786$	0	0
$787$	31.7044	1.13014	0.565070	−	0.825043i	$-0.308849\pi$
$787$	31.7044	1.13014	0.565070	+	0.825043i	$0.308849\pi$
$788$	0	0
$789$	−10.1990	−0.363094
$790$	0	0
$791$	9.44130	0.335694
$792$	0	0
$793$	−9.90155	−0.351614
$794$	0	0
$795$	25.5558	0.906370
$796$	0	0
$797$	17.6647	0.625716	0.312858	−	0.949800i	$-0.398714\pi$
$797$	17.6647	0.625716	0.312858	+	0.949800i	$0.398714\pi$
$798$	0	0
$799$	6.97935	0.246912
$800$	0	0
$801$	9.57290	0.338242
$802$	0	0
$803$	−12.8184	−0.452351
$804$	0	0
$805$	6.41083	0.225952
$806$	0	0
$807$	15.3916	0.541810
$808$	0	0
$809$	−46.9705	−1.65140	−0.825698	−	0.564113i	$-0.809218\pi$
$809$	−46.9705	−1.65140	−0.825698	+	0.564113i	$0.809218\pi$
$810$	0	0
$811$	−36.3116	−1.27507	−0.637536	−	0.770421i	$-0.720046\pi$
$811$	−36.3116	−1.27507	−0.637536	+	0.770421i	$0.720046\pi$
$812$	0	0
$813$	12.6570	0.443902
$814$	0	0
$815$	17.0102	0.595842
$816$	0	0
$817$	41.9570	1.46789
$818$	0	0
$819$	−2.54448	−0.0889113
$820$	0	0
$821$	−42.5827	−1.48615	−0.743074	−	0.669209i	$-0.766633\pi$
$821$	−42.5827	−1.48615	−0.743074	+	0.669209i	$0.766633\pi$
$822$	0	0
$823$	21.0146	0.732522	0.366261	−	0.930512i	$-0.380638\pi$
$823$	21.0146	0.732522	0.366261	+	0.930512i	$0.380638\pi$
$824$	0	0
$825$	9.83364	0.342363
$826$	0	0
$827$	−16.2230	−0.564127	−0.282064	−	0.959396i	$-0.591019\pi$
$827$	−16.2230	−0.564127	−0.282064	+	0.959396i	$0.591019\pi$
$828$	0	0
$829$	15.1367	0.525718	0.262859	−	0.964834i	$-0.415335\pi$
$829$	15.1367	0.525718	0.262859	+	0.964834i	$0.415335\pi$
$830$	0	0
$831$	−2.16770	−0.0751967
$832$	0	0
$833$	4.81611	0.166868
$834$	0	0
$835$	−24.9794	−0.864448
$836$	0	0
$837$	−30.3296	−1.04834
$838$	0	0
$839$	−50.1889	−1.73271	−0.866357	−	0.499425i	$-0.833545\pi$
$839$	−50.1889	−1.73271	−0.866357	+	0.499425i	$0.833545\pi$
$840$	0	0
$841$	34.0335	1.17357
$842$	0	0
$843$	4.90059	0.168785
$844$	0	0
$845$	−4.42380	−0.152183
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	9.09149	0.312019
$850$	0	0
$851$	−10.1416	−0.347648
$852$	0	0
$853$	6.13538	0.210071	0.105036	−	0.994468i	$-0.466504\pi$
$853$	6.13538	0.210071	0.105036	+	0.994468i	$0.466504\pi$
$854$	0	0
$855$	−59.5680	−2.03718
$856$	0	0
$857$	−52.2350	−1.78431	−0.892157	−	0.451725i	$-0.850809\pi$
$857$	−52.2350	−1.78431	−0.892157	+	0.451725i	$0.850809\pi$
$858$	0	0
$859$	−28.2892	−0.965214	−0.482607	−	0.875837i	$-0.660310\pi$
$859$	−28.2892	−0.965214	−0.482607	+	0.875837i	$0.660310\pi$
$860$	0	0
$861$	−4.76361	−0.162344
$862$	0	0
$863$	31.0049	1.05542	0.527710	−	0.849425i	$-0.323051\pi$
$863$	31.0049	1.05542	0.527710	+	0.849425i	$0.323051\pi$
$864$	0	0
$865$	40.4416	1.37506
$866$	0	0
$867$	4.18108	0.141997
$868$	0	0
$869$	11.6663	0.395752
$870$	0	0
$871$	1.89639	0.0642567
$872$	0	0
$873$	32.2474	1.09141
$874$	0	0
$875$	42.3359	1.43121
$876$	0	0
$877$	−17.6466	−0.595884	−0.297942	−	0.954584i	$-0.596300\pi$
$877$	−17.6466	−0.595884	−0.297942	+	0.954584i	$0.596300\pi$
$878$	0	0
$879$	−12.5164	−0.422168
$880$	0	0
$881$	18.2302	0.614192	0.307096	−	0.951679i	$-0.400643\pi$
$881$	18.2302	0.614192	0.307096	+	0.951679i	$0.400643\pi$
$882$	0	0
$883$	0.897068	0.0301887	0.0150944	−	0.999886i	$-0.495195\pi$
$883$	0.897068	0.0301887	0.0150944	+	0.999886i	$0.495195\pi$
$884$	0	0
$885$	0.105403	0.00354309
$886$	0	0
$887$	−40.8563	−1.37182	−0.685910	−	0.727687i	$-0.740595\pi$
$887$	−40.8563	−1.37182	−0.685910	+	0.727687i	$0.740595\pi$
$888$	0	0
$889$	10.1471	0.340324
$890$	0	0
$891$	5.10781	0.171118
$892$	0	0
$893$	−7.66897	−0.256632
$894$	0	0
$895$	85.6930	2.86440
$896$	0	0
$897$	−0.978076	−0.0326570
$898$	0	0
$899$	64.3483	2.14614
$900$	0	0
$901$	−41.2226	−1.37332
$902$	0	0
$903$	5.35106	0.178072
$904$	0	0
$905$	−6.39840	−0.212690
$906$	0	0
$907$	−17.3319	−0.575497	−0.287749	−	0.957706i	$-0.592907\pi$
$907$	−17.3319	−0.575497	−0.287749	+	0.957706i	$0.592907\pi$
$908$	0	0
$909$	−25.8970	−0.858948
$910$	0	0
$911$	−3.89220	−0.128954	−0.0644772	−	0.997919i	$-0.520538\pi$
$911$	−3.89220	−0.128954	−0.0644772	+	0.997919i	$0.520538\pi$
$912$	0	0
$913$	9.53027	0.315406
$914$	0	0
$915$	−29.5633	−0.977333
$916$	0	0
$917$	12.4940	0.412588
$918$	0	0
$919$	33.7531	1.11341	0.556706	−	0.830710i	$-0.312065\pi$
$919$	33.7531	1.11341	0.556706	+	0.830710i	$0.312065\pi$
$920$	0	0
$921$	−14.2585	−0.469832
$922$	0	0
$923$	−9.06086	−0.298242
$924$	0	0
$925$	−101.964	−3.35255
$926$	0	0
$927$	−23.6918	−0.778141
$928$	0	0
$929$	9.32763	0.306030	0.153015	−	0.988224i	$-0.451102\pi$
$929$	9.32763	0.306030	0.153015	+	0.988224i	$0.451102\pi$
$930$	0	0
$931$	−5.29198	−0.173438
$932$	0	0
$933$	−1.54784	−0.0506739
$934$	0	0
$935$	−21.3055	−0.696765
$936$	0	0
$937$	14.0109	0.457716	0.228858	−	0.973460i	$-0.426501\pi$
$937$	14.0109	0.457716	0.228858	+	0.973460i	$0.426501\pi$
$938$	0	0
$939$	−4.50632	−0.147058
$940$	0	0
$941$	40.6157	1.32404	0.662018	−	0.749488i	$-0.269700\pi$
$941$	40.6157	1.32404	0.662018	+	0.749488i	$0.269700\pi$
$942$	0	0
$943$	10.2282	0.333077
$944$	0	0
$945$	−16.5543	−0.538511
$946$	0	0
$947$	−20.1611	−0.655148	−0.327574	−	0.944825i	$-0.606231\pi$
$947$	−20.1611	−0.655148	−0.327574	+	0.944825i	$0.606231\pi$
$948$	0	0
$949$	12.8184	0.416102
$950$	0	0
$951$	−18.9861	−0.615665
$952$	0	0
$953$	25.8170	0.836293	0.418147	−	0.908380i	$-0.362680\pi$
$953$	25.8170	0.836293	0.418147	+	0.908380i	$0.362680\pi$
$954$	0	0
$955$	−76.8098	−2.48551
$956$	0	0
$957$	5.35846	0.173214
$958$	0	0
$959$	11.9240	0.385045
$960$	0	0
$961$	34.6905	1.11905
$962$	0	0
$963$	30.4134	0.980059
$964$	0	0
$965$	−67.1252	−2.16084
$966$	0	0
$967$	19.0968	0.614113	0.307056	−	0.951691i	$-0.400656\pi$
$967$	19.0968	0.614113	0.307056	+	0.951691i	$0.400656\pi$
$968$	0	0
$969$	−17.2016	−0.552595
$970$	0	0
$971$	−7.13725	−0.229045	−0.114523	−	0.993421i	$-0.536534\pi$
$971$	−7.13725	−0.229045	−0.114523	+	0.993421i	$0.536534\pi$
$972$	0	0
$973$	0.714210	0.0228965
$974$	0	0
$975$	−9.83364	−0.314929
$976$	0	0
$977$	52.8348	1.69033	0.845167	−	0.534502i	$-0.179501\pi$
$977$	52.8348	1.69033	0.845167	+	0.534502i	$0.179501\pi$
$978$	0	0
$979$	−3.76222	−0.120241
$980$	0	0
$981$	11.4873	0.366762
$982$	0	0
$983$	−12.9107	−0.411788	−0.205894	−	0.978574i	$-0.566010\pi$
$983$	−12.9107	−0.411788	−0.205894	+	0.978574i	$0.566010\pi$
$984$	0	0
$985$	108.604	3.46040
$986$	0	0
$987$	−0.978076	−0.0311325
$988$	0	0
$989$	−11.4896	−0.365347
$990$	0	0
$991$	3.28006	0.104194	0.0520972	−	0.998642i	$-0.483409\pi$
$991$	3.28006	0.104194	0.0520972	+	0.998642i	$0.483409\pi$
$992$	0	0
$993$	−10.8106	−0.343063
$994$	0	0
$995$	−14.8018	−0.469249
$996$	0	0
$997$	50.7966	1.60875	0.804373	−	0.594125i	$-0.202501\pi$
$997$	50.7966	1.60875	0.804373	+	0.594125i	$0.202501\pi$
$998$	0	0
$999$	26.1879	0.828550

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.v.1.7	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.v.1.7	✓	11	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q+0.674923 q^{3} -4.42380 q^{5} -1.00000 q^{7} -2.54448 q^{9} +O(q^{10})\)	\(q+0.674923 q^{3} -4.42380 q^{5} -1.00000 q^{7} -2.54448 q^{9} +1.00000 q^{11} -1.00000 q^{13} -2.98573 q^{15} +4.81611 q^{17} -5.29198 q^{19} -0.674923 q^{21} +1.44917 q^{23} +14.5700 q^{25} -3.74210 q^{27} +7.93937 q^{29} +8.10497 q^{31} +0.674923 q^{33} +4.42380 q^{35} -6.99820 q^{37} -0.674923 q^{39} +7.05801 q^{41} -7.92840 q^{43} +11.2563 q^{45} +1.44917 q^{47} +1.00000 q^{49} +3.25050 q^{51} -8.55932 q^{53} -4.42380 q^{55} -3.57168 q^{57} -0.0353024 q^{59} +9.90155 q^{61} +2.54448 q^{63} +4.42380 q^{65} -1.89639 q^{67} +0.978076 q^{69} +9.06086 q^{71} -12.8184 q^{73} +9.83364 q^{75} -1.00000 q^{77} +11.6663 q^{79} +5.10781 q^{81} +9.53027 q^{83} -21.3055 q^{85} +5.35846 q^{87} -3.76222 q^{89} +1.00000 q^{91} +5.47023 q^{93} +23.4107 q^{95} -12.6735 q^{97} -2.54448 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 2 q^{3} + 2 q^{5} - 11 q^{7} + 9 q^{9}+O(q^{10}) \) 11 * q - 2 * q^3 + 2 * q^5 - 11 * q^7 + 9 * q^9	\( 11 q - 2 q^{3} + 2 q^{5} - 11 q^{7} + 9 q^{9} + 11 q^{11} - 11 q^{13} - 7 q^{15} - 4 q^{17} - 16 q^{19} + 2 q^{21} - 3 q^{23} + 11 q^{25} - 11 q^{27} + q^{29} + 14 q^{31} - 2 q^{33} - 2 q^{35} - 8 q^{37} + 2 q^{39} + 4 q^{41} - 30 q^{43} + 13 q^{45} - 3 q^{47} + 11 q^{49} - 14 q^{51} - 5 q^{53} + 2 q^{55} - 22 q^{57} + 11 q^{59} + 15 q^{61} - 9 q^{63} - 2 q^{65} - 41 q^{67} + 12 q^{69} + q^{71} - 8 q^{73} - 24 q^{75} - 11 q^{77} - 26 q^{79} + 19 q^{81} - 31 q^{83} - 27 q^{85} - 25 q^{87} + 2 q^{89} + 11 q^{91} - 37 q^{93} - 6 q^{97} + 9 q^{99}+O(q^{100}) \) 11 * q - 2 * q^3 + 2 * q^5 - 11 * q^7 + 9 * q^9 + 11 * q^11 - 11 * q^13 - 7 * q^15 - 4 * q^17 - 16 * q^19 + 2 * q^21 - 3 * q^23 + 11 * q^25 - 11 * q^27 + q^29 + 14 * q^31 - 2 * q^33 - 2 * q^35 - 8 * q^37 + 2 * q^39 + 4 * q^41 - 30 * q^43 + 13 * q^45 - 3 * q^47 + 11 * q^49 - 14 * q^51 - 5 * q^53 + 2 * q^55 - 22 * q^57 + 11 * q^59 + 15 * q^61 - 9 * q^63 - 2 * q^65 - 41 * q^67 + 12 * q^69 + q^71 - 8 * q^73 - 24 * q^75 - 11 * q^77 - 26 * q^79 + 19 * q^81 - 31 * q^83 - 27 * q^85 - 25 * q^87 + 2 * q^89 + 11 * q^91 - 37 * q^93 - 6 * q^97 + 9 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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