LMFDB - Embedded newform 8008.2.a.u.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:	$10$
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} - 2x^{9} - 16x^{8} + 26x^{7} + 93x^{6} - 113x^{5} - 230x^{4} + 197x^{3} + 201x^{2} - 115x + 8 $ x^10 - 2x^9 - 16x^8 + 26x^7 + 93x^6 - 113x^5 - 230x^4 + 197x^3 + 201x^2 - 115*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.54918$ 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)

$f(q)$	$=$	$q+1.54918 q^{3} -0.356949 q^{5} -1.00000 q^{7} -0.600028 q^{9} +O(q^{10})$	$q+1.54918 q^{3} -0.356949 q^{5} -1.00000 q^{7} -0.600028 q^{9} -1.00000 q^{11} -1.00000 q^{13} -0.552981 q^{15} -6.04878 q^{17} +8.48928 q^{19} -1.54918 q^{21} +7.98257 q^{23} -4.87259 q^{25} -5.57711 q^{27} +5.31591 q^{29} -9.77065 q^{31} -1.54918 q^{33} +0.356949 q^{35} +10.5368 q^{37} -1.54918 q^{39} +8.72233 q^{41} -7.46133 q^{43} +0.214180 q^{45} +1.45382 q^{47} +1.00000 q^{49} -9.37067 q^{51} -1.71369 q^{53} +0.356949 q^{55} +13.1515 q^{57} -0.781383 q^{59} -5.31139 q^{61} +0.600028 q^{63} +0.356949 q^{65} -6.19727 q^{67} +12.3665 q^{69} -7.65289 q^{71} -2.44764 q^{73} -7.54854 q^{75} +1.00000 q^{77} -11.8586 q^{79} -6.83988 q^{81} -5.43355 q^{83} +2.15911 q^{85} +8.23533 q^{87} +9.25740 q^{89} +1.00000 q^{91} -15.1365 q^{93} -3.03024 q^{95} +3.82384 q^{97} +0.600028 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{3} - 4 q^{5} - 10 q^{7} + 6 q^{9}+O(q^{10}) $ 10 * q + 2 * q^3 - 4 * q^5 - 10 * q^7 + 6 * q^9	$ 10 q + 2 q^{3} - 4 q^{5} - 10 q^{7} + 6 q^{9} - 10 q^{11} - 10 q^{13} - 4 q^{15} - 3 q^{17} + 13 q^{19} - 2 q^{21} + 6 q^{25} + 14 q^{27} - 3 q^{29} - q^{31} - 2 q^{33} + 4 q^{35} - 5 q^{37} - 2 q^{39} + 4 q^{41} + 19 q^{43} - 11 q^{45} + q^{47} + 10 q^{49} + 15 q^{51} - 6 q^{53} + 4 q^{55} - 6 q^{57} + 4 q^{59} - 18 q^{61} - 6 q^{63} + 4 q^{65} + q^{67} - 11 q^{69} - 21 q^{71} - 10 q^{73} + 10 q^{77} + 3 q^{79} - 30 q^{81} + 6 q^{83} - 33 q^{85} - 9 q^{87} - 22 q^{89} + 10 q^{91} - 34 q^{93} - 3 q^{95} - 9 q^{97} - 6 q^{99}+O(q^{100}) $ 10 * q + 2 * q^3 - 4 * q^5 - 10 * q^7 + 6 * q^9 - 10 * q^11 - 10 * q^13 - 4 * q^15 - 3 * q^17 + 13 * q^19 - 2 * q^21 + 6 * q^25 + 14 * q^27 - 3 * q^29 - q^31 - 2 * q^33 + 4 * q^35 - 5 * q^37 - 2 * q^39 + 4 * q^41 + 19 * q^43 - 11 * q^45 + q^47 + 10 * q^49 + 15 * q^51 - 6 * q^53 + 4 * q^55 - 6 * q^57 + 4 * q^59 - 18 * q^61 - 6 * q^63 + 4 * q^65 + q^67 - 11 * q^69 - 21 * q^71 - 10 * q^73 + 10 * q^77 + 3 * q^79 - 30 * q^81 + 6 * q^83 - 33 * q^85 - 9 * q^87 - 22 * q^89 + 10 * q^91 - 34 * q^93 - 3 * q^95 - 9 * q^97 - 6 * 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
$3$	1.54918	0.894422	0.447211	−	0.894429i	$-0.352417\pi$
$3$	1.54918	0.894422	0.447211	+	0.894429i	$0.352417\pi$
$4$	0	0
$5$	−0.356949	−0.159633	−0.0798163	−	0.996810i	$-0.525433\pi$
$5$	−0.356949	−0.159633	−0.0798163	+	0.996810i	$0.525433\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−0.600028	−0.200009
$10$	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
$15$	−0.552981	−0.142779
$16$	0	0
$17$	−6.04878	−1.46704	−0.733522	−	0.679665i	$-0.762125\pi$
$17$	−6.04878	−1.46704	−0.733522	+	0.679665i	$0.762125\pi$
$18$	0	0
$19$	8.48928	1.94757	0.973787	−	0.227462i	$-0.0730427\pi$
$19$	8.48928	1.94757	0.973787	+	0.227462i	$0.0730427\pi$
$20$	0	0
$21$	−1.54918	−0.338060
$22$	0	0
$23$	7.98257	1.66448	0.832241	−	0.554414i	$-0.187058\pi$
$23$	7.98257	1.66448	0.832241	+	0.554414i	$0.187058\pi$
$24$	0	0
$25$	−4.87259	−0.974517
$26$	0	0
$27$	−5.57711	−1.07331
$28$	0	0
$29$	5.31591	0.987140	0.493570	−	0.869706i	$-0.335692\pi$
$29$	5.31591	0.987140	0.493570	+	0.869706i	$0.335692\pi$
$30$	0	0
$31$	−9.77065	−1.75486	−0.877430	−	0.479705i	$-0.840744\pi$
$31$	−9.77065	−1.75486	−0.877430	+	0.479705i	$0.840744\pi$
$32$	0	0
$33$	−1.54918	−0.269678
$34$	0	0
$35$	0.356949	0.0603355
$36$	0	0
$37$	10.5368	1.73225	0.866123	−	0.499832i	$-0.166605\pi$
$37$	10.5368	1.73225	0.866123	+	0.499832i	$0.166605\pi$
$38$	0	0
$39$	−1.54918	−0.248068
$40$	0	0
$41$	8.72233	1.36220	0.681099	−	0.732191i	$-0.261502\pi$
$41$	8.72233	1.36220	0.681099	+	0.732191i	$0.261502\pi$
$42$	0	0
$43$	−7.46133	−1.13784	−0.568921	−	0.822392i	$-0.692639\pi$
$43$	−7.46133	−1.13784	−0.568921	+	0.822392i	$0.692639\pi$
$44$	0	0
$45$	0.214180	0.0319280
$46$	0	0
$47$	1.45382	0.212062	0.106031	−	0.994363i	$-0.466186\pi$
$47$	1.45382	0.212062	0.106031	+	0.994363i	$0.466186\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−9.37067	−1.31216
$52$	0	0
$53$	−1.71369	−0.235393	−0.117696	−	0.993050i	$-0.537551\pi$
$53$	−1.71369	−0.235393	−0.117696	+	0.993050i	$0.537551\pi$
$54$	0	0
$55$	0.356949	0.0481311
$56$	0	0
$57$	13.1515	1.74195
$58$	0	0
$59$	−0.781383	−0.101727	−0.0508637	−	0.998706i	$-0.516197\pi$
$59$	−0.781383	−0.101727	−0.0508637	+	0.998706i	$0.516197\pi$
$60$	0	0
$61$	−5.31139	−0.680053	−0.340027	−	0.940416i	$-0.610436\pi$
$61$	−5.31139	−0.680053	−0.340027	+	0.940416i	$0.610436\pi$
$62$	0	0
$63$	0.600028	0.0755964
$64$	0	0
$65$	0.356949	0.0442741
$66$	0	0
$67$	−6.19727	−0.757117	−0.378559	−	0.925577i	$-0.623580\pi$
$67$	−6.19727	−0.757117	−0.378559	+	0.925577i	$0.623580\pi$
$68$	0	0
$69$	12.3665	1.48875
$70$	0	0
$71$	−7.65289	−0.908231	−0.454115	−	0.890943i	$-0.650045\pi$
$71$	−7.65289	−0.908231	−0.454115	+	0.890943i	$0.650045\pi$
$72$	0	0
$73$	−2.44764	−0.286475	−0.143237	−	0.989688i	$-0.545751\pi$
$73$	−2.44764	−0.286475	−0.143237	+	0.989688i	$0.545751\pi$
$74$	0	0
$75$	−7.54854	−0.871630
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−11.8586	−1.33420	−0.667100	−	0.744968i	$-0.732465\pi$
$79$	−11.8586	−1.33420	−0.667100	+	0.744968i	$0.732465\pi$
$80$	0	0
$81$	−6.83988	−0.759987
$82$	0	0
$83$	−5.43355	−0.596409	−0.298205	−	0.954502i	$-0.596388\pi$
$83$	−5.43355	−0.596409	−0.298205	+	0.954502i	$0.596388\pi$
$84$	0	0
$85$	2.15911	0.234188
$86$	0	0
$87$	8.23533	0.882920
$88$	0	0
$89$	9.25740	0.981282	0.490641	−	0.871362i	$-0.336763\pi$
$89$	9.25740	0.981282	0.490641	+	0.871362i	$0.336763\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−15.1365	−1.56958
$94$	0	0
$95$	−3.03024	−0.310896
$96$	0	0
$97$	3.82384	0.388252	0.194126	−	0.980977i	$-0.437813\pi$
$97$	3.82384	0.388252	0.194126	+	0.980977i	$0.437813\pi$
$98$	0	0
$99$	0.600028	0.0603051
$100$	0	0
$101$	−6.64381	−0.661084	−0.330542	−	0.943791i	$-0.607232\pi$
$101$	−6.64381	−0.661084	−0.330542	+	0.943791i	$0.607232\pi$
$102$	0	0
$103$	−9.22802	−0.909263	−0.454632	−	0.890680i	$-0.650229\pi$
$103$	−9.22802	−0.909263	−0.454632	+	0.890680i	$0.650229\pi$
$104$	0	0
$105$	0.552981	0.0539654
$106$	0	0
$107$	−11.0799	−1.07114	−0.535569	−	0.844492i	$-0.679903\pi$
$107$	−11.0799	−1.07114	−0.535569	+	0.844492i	$0.679903\pi$
$108$	0	0
$109$	−17.6595	−1.69147	−0.845736	−	0.533602i	$-0.820838\pi$
$109$	−17.6595	−1.69147	−0.845736	+	0.533602i	$0.820838\pi$
$110$	0	0
$111$	16.3235	1.54936
$112$	0	0
$113$	−2.20701	−0.207618	−0.103809	−	0.994597i	$-0.533103\pi$
$113$	−2.20701	−0.207618	−0.103809	+	0.994597i	$0.533103\pi$
$114$	0	0
$115$	−2.84938	−0.265706
$116$	0	0
$117$	0.600028	0.0554726
$118$	0	0
$119$	6.04878	0.554491
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	13.5125	1.21838
$124$	0	0
$125$	3.52401	0.315197
$126$	0	0
$127$	6.04203	0.536144	0.268072	−	0.963399i	$-0.413614\pi$
$127$	6.04203	0.536144	0.268072	+	0.963399i	$0.413614\pi$
$128$	0	0
$129$	−11.5590	−1.01771
$130$	0	0
$131$	−21.7877	−1.90360	−0.951799	−	0.306724i	$-0.900767\pi$
$131$	−21.7877	−1.90360	−0.951799	+	0.306724i	$0.900767\pi$
$132$	0	0
$133$	−8.48928	−0.736114
$134$	0	0
$135$	1.99075	0.171336
$136$	0	0
$137$	−18.9167	−1.61616	−0.808081	−	0.589071i	$-0.799494\pi$
$137$	−18.9167	−1.61616	−0.808081	+	0.589071i	$0.799494\pi$
$138$	0	0
$139$	22.9664	1.94798	0.973991	−	0.226588i	$-0.0727572\pi$
$139$	22.9664	1.94798	0.973991	+	0.226588i	$0.0727572\pi$
$140$	0	0
$141$	2.25224	0.189673
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−1.89751	−0.157580
$146$	0	0
$147$	1.54918	0.127775
$148$	0	0
$149$	−9.37265	−0.767837	−0.383919	−	0.923367i	$-0.625426\pi$
$149$	−9.37265	−0.767837	−0.383919	+	0.923367i	$0.625426\pi$
$150$	0	0
$151$	1.64898	0.134192	0.0670961	−	0.997747i	$-0.478627\pi$
$151$	1.64898	0.134192	0.0670961	+	0.997747i	$0.478627\pi$
$152$	0	0
$153$	3.62944	0.293423
$154$	0	0
$155$	3.48763	0.280133
$156$	0	0
$157$	−14.7041	−1.17352	−0.586759	−	0.809761i	$-0.699597\pi$
$157$	−14.7041	−1.17352	−0.586759	+	0.809761i	$0.699597\pi$
$158$	0	0
$159$	−2.65482	−0.210541
$160$	0	0
$161$	−7.98257	−0.629115
$162$	0	0
$163$	−14.5583	−1.14029	−0.570147	−	0.821543i	$-0.693114\pi$
$163$	−14.5583	−1.14029	−0.570147	+	0.821543i	$0.693114\pi$
$164$	0	0
$165$	0.552981	0.0430495
$166$	0	0
$167$	−1.80194	−0.139438	−0.0697192	−	0.997567i	$-0.522210\pi$
$167$	−1.80194	−0.139438	−0.0697192	+	0.997567i	$0.522210\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−5.09381	−0.389533
$172$	0	0
$173$	13.4645	1.02369	0.511845	−	0.859078i	$-0.328962\pi$
$173$	13.4645	1.02369	0.511845	+	0.859078i	$0.328962\pi$
$174$	0	0
$175$	4.87259	0.368333
$176$	0	0
$177$	−1.21051	−0.0909872
$178$	0	0
$179$	8.43106	0.630167	0.315083	−	0.949064i	$-0.397968\pi$
$179$	8.43106	0.630167	0.315083	+	0.949064i	$0.397968\pi$
$180$	0	0
$181$	−22.4436	−1.66822	−0.834109	−	0.551599i	$-0.814018\pi$
$181$	−22.4436	−1.66822	−0.834109	+	0.551599i	$0.814018\pi$
$182$	0	0
$183$	−8.22832	−0.608255
$184$	0	0
$185$	−3.76112	−0.276523
$186$	0	0
$187$	6.04878	0.442331
$188$	0	0
$189$	5.57711	0.405675
$190$	0	0
$191$	−8.38303	−0.606574	−0.303287	−	0.952899i	$-0.598084\pi$
$191$	−8.38303	−0.606574	−0.303287	+	0.952899i	$0.598084\pi$
$192$	0	0
$193$	−8.19008	−0.589535	−0.294767	−	0.955569i	$-0.595242\pi$
$193$	−8.19008	−0.589535	−0.294767	+	0.955569i	$0.595242\pi$
$194$	0	0
$195$	0.552981	0.0395998
$196$	0	0
$197$	−12.1023	−0.862252	−0.431126	−	0.902292i	$-0.641884\pi$
$197$	−12.1023	−0.862252	−0.431126	+	0.902292i	$0.641884\pi$
$198$	0	0
$199$	24.8646	1.76261	0.881303	−	0.472551i	$-0.156667\pi$
$199$	24.8646	1.76261	0.881303	+	0.472551i	$0.156667\pi$
$200$	0	0
$201$	−9.60072	−0.677182
$202$	0	0
$203$	−5.31591	−0.373104
$204$	0	0
$205$	−3.11343	−0.217451
$206$	0	0
$207$	−4.78977	−0.332912
$208$	0	0
$209$	−8.48928	−0.587216
$210$	0	0
$211$	−25.7697	−1.77406	−0.887030	−	0.461713i	$-0.847235\pi$
$211$	−25.7697	−1.77406	−0.887030	+	0.461713i	$0.847235\pi$
$212$	0	0
$213$	−11.8557	−0.812341
$214$	0	0
$215$	2.66332	0.181637
$216$	0	0
$217$	9.77065	0.663275
$218$	0	0
$219$	−3.79185	−0.256229
$220$	0	0
$221$	6.04878	0.406885
$222$	0	0
$223$	7.74588	0.518703	0.259351	−	0.965783i	$-0.416491\pi$
$223$	7.74588	0.518703	0.259351	+	0.965783i	$0.416491\pi$
$224$	0	0
$225$	2.92369	0.194913
$226$	0	0
$227$	−7.51314	−0.498665	−0.249332	−	0.968418i	$-0.580211\pi$
$227$	−7.51314	−0.498665	−0.249332	+	0.968418i	$0.580211\pi$
$228$	0	0
$229$	−21.0839	−1.39326	−0.696630	−	0.717430i	$-0.745318\pi$
$229$	−21.0839	−1.39326	−0.696630	+	0.717430i	$0.745318\pi$
$230$	0	0
$231$	1.54918	0.101929
$232$	0	0
$233$	23.3809	1.53173	0.765866	−	0.643001i	$-0.222311\pi$
$233$	23.3809	1.53173	0.765866	+	0.643001i	$0.222311\pi$
$234$	0	0
$235$	−0.518941	−0.0338520
$236$	0	0
$237$	−18.3712	−1.19334
$238$	0	0
$239$	5.42839	0.351133	0.175567	−	0.984468i	$-0.443824\pi$
$239$	5.42839	0.351133	0.175567	+	0.984468i	$0.443824\pi$
$240$	0	0
$241$	−17.9194	−1.15429	−0.577145	−	0.816641i	$-0.695833\pi$
$241$	−17.9194	−1.15429	−0.577145	+	0.816641i	$0.695833\pi$
$242$	0	0
$243$	6.13508	0.393566
$244$	0	0
$245$	−0.356949	−0.0228047
$246$	0	0
$247$	−8.48928	−0.540160
$248$	0	0
$249$	−8.41757	−0.533442
$250$	0	0
$251$	4.32837	0.273204	0.136602	−	0.990626i	$-0.456382\pi$
$251$	4.32837	0.273204	0.136602	+	0.990626i	$0.456382\pi$
$252$	0	0
$253$	−7.98257	−0.501860
$254$	0	0
$255$	3.34486	0.209463
$256$	0	0
$257$	−11.9892	−0.747866	−0.373933	−	0.927456i	$-0.621991\pi$
$257$	−11.9892	−0.747866	−0.373933	+	0.927456i	$0.621991\pi$
$258$	0	0
$259$	−10.5368	−0.654727
$260$	0	0
$261$	−3.18970	−0.197437
$262$	0	0
$263$	−7.00402	−0.431886	−0.215943	−	0.976406i	$-0.569283\pi$
$263$	−7.00402	−0.431886	−0.215943	+	0.976406i	$0.569283\pi$
$264$	0	0
$265$	0.611699	0.0375764
$266$	0	0
$267$	14.3414	0.877680
$268$	0	0
$269$	−13.2468	−0.807672	−0.403836	−	0.914831i	$-0.632323\pi$
$269$	−13.2468	−0.807672	−0.403836	+	0.914831i	$0.632323\pi$
$270$	0	0
$271$	20.5661	1.24930	0.624650	−	0.780905i	$-0.285242\pi$
$271$	20.5661	1.24930	0.624650	+	0.780905i	$0.285242\pi$
$272$	0	0
$273$	1.54918	0.0937609
$274$	0	0
$275$	4.87259	0.293828
$276$	0	0
$277$	9.45404	0.568038	0.284019	−	0.958819i	$-0.408332\pi$
$277$	9.45404	0.568038	0.284019	+	0.958819i	$0.408332\pi$
$278$	0	0
$279$	5.86266	0.350988
$280$	0	0
$281$	0.798874	0.0476568	0.0238284	−	0.999716i	$-0.492414\pi$
$281$	0.798874	0.0476568	0.0238284	+	0.999716i	$0.492414\pi$
$282$	0	0
$283$	15.4553	0.918719	0.459360	−	0.888250i	$-0.348079\pi$
$283$	15.4553	0.918719	0.459360	+	0.888250i	$0.348079\pi$
$284$	0	0
$285$	−4.69441	−0.278073
$286$	0	0
$287$	−8.72233	−0.514863
$288$	0	0
$289$	19.5877	1.15222
$290$	0	0
$291$	5.92384	0.347261
$292$	0	0
$293$	−24.5721	−1.43552	−0.717759	−	0.696292i	$-0.754832\pi$
$293$	−24.5721	−1.43552	−0.717759	+	0.696292i	$0.754832\pi$
$294$	0	0
$295$	0.278914	0.0162390
$296$	0	0
$297$	5.57711	0.323617
$298$	0	0
$299$	−7.98257	−0.461644
$300$	0	0
$301$	7.46133	0.430064
$302$	0	0
$303$	−10.2925	−0.591288
$304$	0	0
$305$	1.89590	0.108559
$306$	0	0
$307$	−2.47883	−0.141474	−0.0707371	−	0.997495i	$-0.522535\pi$
$307$	−2.47883	−0.141474	−0.0707371	+	0.997495i	$0.522535\pi$
$308$	0	0
$309$	−14.2959	−0.813265
$310$	0	0
$311$	−30.8260	−1.74798	−0.873992	−	0.485941i	$-0.838477\pi$
$311$	−30.8260	−1.74798	−0.873992	+	0.485941i	$0.838477\pi$
$312$	0	0
$313$	22.2593	1.25817	0.629086	−	0.777336i	$-0.283429\pi$
$313$	22.2593	1.25817	0.629086	+	0.777336i	$0.283429\pi$
$314$	0	0
$315$	−0.214180	−0.0120677
$316$	0	0
$317$	2.10203	0.118062	0.0590310	−	0.998256i	$-0.481199\pi$
$317$	2.10203	0.118062	0.0590310	+	0.998256i	$0.481199\pi$
$318$	0	0
$319$	−5.31591	−0.297634
$320$	0	0
$321$	−17.1649	−0.958049
$322$	0	0
$323$	−51.3498	−2.85718
$324$	0	0
$325$	4.87259	0.270283
$326$	0	0
$327$	−27.3578	−1.51289
$328$	0	0
$329$	−1.45382	−0.0801519
$330$	0	0
$331$	30.4137	1.67169	0.835845	−	0.548966i	$-0.184978\pi$
$331$	30.4137	1.67169	0.835845	+	0.548966i	$0.184978\pi$
$332$	0	0
$333$	−6.32240	−0.346465
$334$	0	0
$335$	2.21211	0.120861
$336$	0	0
$337$	5.11263	0.278503	0.139251	−	0.990257i	$-0.455530\pi$
$337$	5.11263	0.278503	0.139251	+	0.990257i	$0.455530\pi$
$338$	0	0
$339$	−3.41907	−0.185698
$340$	0	0
$341$	9.77065	0.529110
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−4.41421	−0.237653
$346$	0	0
$347$	34.2330	1.83773	0.918863	−	0.394576i	$-0.129109\pi$
$347$	34.2330	1.83773	0.918863	+	0.394576i	$0.129109\pi$
$348$	0	0
$349$	33.2630	1.78053	0.890263	−	0.455446i	$-0.150520\pi$
$349$	33.2630	1.78053	0.890263	+	0.455446i	$0.150520\pi$
$350$	0	0
$351$	5.57711	0.297684
$352$	0	0
$353$	−18.7880	−0.999987	−0.499993	−	0.866029i	$-0.666664\pi$
$353$	−18.7880	−0.999987	−0.499993	+	0.866029i	$0.666664\pi$
$354$	0	0
$355$	2.73169	0.144983
$356$	0	0
$357$	9.37067	0.495949
$358$	0	0
$359$	2.76999	0.146195	0.0730973	−	0.997325i	$-0.476712\pi$
$359$	2.76999	0.146195	0.0730973	+	0.997325i	$0.476712\pi$
$360$	0	0
$361$	53.0678	2.79304
$362$	0	0
$363$	1.54918	0.0813111
$364$	0	0
$365$	0.873685	0.0457308
$366$	0	0
$367$	21.6481	1.13002	0.565012	−	0.825083i	$-0.308872\pi$
$367$	21.6481	1.13002	0.565012	+	0.825083i	$0.308872\pi$
$368$	0	0
$369$	−5.23364	−0.272453
$370$	0	0
$371$	1.71369	0.0889702
$372$	0	0
$373$	8.47631	0.438887	0.219443	−	0.975625i	$-0.429576\pi$
$373$	8.47631	0.438887	0.219443	+	0.975625i	$0.429576\pi$
$374$	0	0
$375$	5.45935	0.281920
$376$	0	0
$377$	−5.31591	−0.273783
$378$	0	0
$379$	12.1315	0.623152	0.311576	−	0.950221i	$-0.399143\pi$
$379$	12.1315	0.623152	0.311576	+	0.950221i	$0.399143\pi$
$380$	0	0
$381$	9.36022	0.479539
$382$	0	0
$383$	−29.1500	−1.48950	−0.744748	−	0.667346i	$-0.767430\pi$
$383$	−29.1500	−1.48950	−0.744748	+	0.667346i	$0.767430\pi$
$384$	0	0
$385$	−0.356949	−0.0181918
$386$	0	0
$387$	4.47701	0.227579
$388$	0	0
$389$	−8.15489	−0.413469	−0.206735	−	0.978397i	$-0.566284\pi$
$389$	−8.15489	−0.413469	−0.206735	+	0.978397i	$0.566284\pi$
$390$	0	0
$391$	−48.2848	−2.44187
$392$	0	0
$393$	−33.7531	−1.70262
$394$	0	0
$395$	4.23293	0.212982
$396$	0	0
$397$	−29.6320	−1.48719	−0.743595	−	0.668631i	$-0.766881\pi$
$397$	−29.6320	−1.48719	−0.743595	+	0.668631i	$0.766881\pi$
$398$	0	0
$399$	−13.1515	−0.658396
$400$	0	0
$401$	21.0032	1.04885	0.524424	−	0.851457i	$-0.324281\pi$
$401$	21.0032	1.04885	0.524424	+	0.851457i	$0.324281\pi$
$402$	0	0
$403$	9.77065	0.486711
$404$	0	0
$405$	2.44149	0.121319
$406$	0	0
$407$	−10.5368	−0.522292
$408$	0	0
$409$	−9.66645	−0.477975	−0.238988	−	0.971023i	$-0.576816\pi$
$409$	−9.66645	−0.477975	−0.238988	+	0.971023i	$0.576816\pi$
$410$	0	0
$411$	−29.3054	−1.44553
$412$	0	0
$413$	0.781383	0.0384493
$414$	0	0
$415$	1.93950	0.0952064
$416$	0	0
$417$	35.5791	1.74232
$418$	0	0
$419$	−14.0140	−0.684628	−0.342314	−	0.939586i	$-0.611211\pi$
$419$	−14.0140	−0.684628	−0.342314	+	0.939586i	$0.611211\pi$
$420$	0	0
$421$	−9.94209	−0.484548	−0.242274	−	0.970208i	$-0.577893\pi$
$421$	−9.94209	−0.484548	−0.242274	+	0.970208i	$0.577893\pi$
$422$	0	0
$423$	−0.872335	−0.0424144
$424$	0	0
$425$	29.4732	1.42966
$426$	0	0
$427$	5.31139	0.257036
$428$	0	0
$429$	1.54918	0.0747953
$430$	0	0
$431$	−6.12683	−0.295119	−0.147559	−	0.989053i	$-0.547142\pi$
$431$	−6.12683	−0.295119	−0.147559	+	0.989053i	$0.547142\pi$
$432$	0	0
$433$	22.4768	1.08017	0.540083	−	0.841612i	$-0.318393\pi$
$433$	22.4768	1.08017	0.540083	+	0.841612i	$0.318393\pi$
$434$	0	0
$435$	−2.93960	−0.140943
$436$	0	0
$437$	67.7663	3.24170
$438$	0	0
$439$	25.1531	1.20049	0.600247	−	0.799815i	$-0.295069\pi$
$439$	25.1531	1.20049	0.600247	+	0.799815i	$0.295069\pi$
$440$	0	0
$441$	−0.600028	−0.0285728
$442$	0	0
$443$	14.5891	0.693147	0.346574	−	0.938023i	$-0.387345\pi$
$443$	14.5891	0.693147	0.346574	+	0.938023i	$0.387345\pi$
$444$	0	0
$445$	−3.30442	−0.156645
$446$	0	0
$447$	−14.5200	−0.686771
$448$	0	0
$449$	−16.3003	−0.769257	−0.384629	−	0.923071i	$-0.625670\pi$
$449$	−16.3003	−0.769257	−0.384629	+	0.923071i	$0.625670\pi$
$450$	0	0
$451$	−8.72233	−0.410718
$452$	0	0
$453$	2.55458	0.120024
$454$	0	0
$455$	−0.356949	−0.0167341
$456$	0	0
$457$	3.10112	0.145064	0.0725321	−	0.997366i	$-0.476892\pi$
$457$	3.10112	0.145064	0.0725321	+	0.997366i	$0.476892\pi$
$458$	0	0
$459$	33.7347	1.57460
$460$	0	0
$461$	26.5437	1.23626	0.618131	−	0.786075i	$-0.287890\pi$
$461$	26.5437	1.23626	0.618131	+	0.786075i	$0.287890\pi$
$462$	0	0
$463$	−15.2201	−0.707340	−0.353670	−	0.935370i	$-0.615066\pi$
$463$	−15.2201	−0.707340	−0.353670	+	0.935370i	$0.615066\pi$
$464$	0	0
$465$	5.40298	0.250557
$466$	0	0
$467$	14.6005	0.675631	0.337815	−	0.941212i	$-0.390312\pi$
$467$	14.6005	0.675631	0.337815	+	0.941212i	$0.390312\pi$
$468$	0	0
$469$	6.19727	0.286163
$470$	0	0
$471$	−22.7794	−1.04962
$472$	0	0
$473$	7.46133	0.343072
$474$	0	0
$475$	−41.3647	−1.89794
$476$	0	0
$477$	1.02826	0.0470808
$478$	0	0
$479$	22.2842	1.01819	0.509095	−	0.860711i	$-0.329980\pi$
$479$	22.2842	1.01819	0.509095	+	0.860711i	$0.329980\pi$
$480$	0	0
$481$	−10.5368	−0.480438
$482$	0	0
$483$	−12.3665	−0.562694
$484$	0	0
$485$	−1.36492	−0.0619778
$486$	0	0
$487$	−36.2685	−1.64348	−0.821742	−	0.569860i	$-0.806997\pi$
$487$	−36.2685	−1.64348	−0.821742	+	0.569860i	$0.806997\pi$
$488$	0	0
$489$	−22.5535	−1.01990
$490$	0	0
$491$	5.21894	0.235528	0.117764	−	0.993042i	$-0.462427\pi$
$491$	5.21894	0.235528	0.117764	+	0.993042i	$0.462427\pi$
$492$	0	0
$493$	−32.1548	−1.44818
$494$	0	0
$495$	−0.214180	−0.00962666
$496$	0	0
$497$	7.65289	0.343279
$498$	0	0
$499$	16.5188	0.739482	0.369741	−	0.929135i	$-0.379446\pi$
$499$	16.5188	0.739482	0.369741	+	0.929135i	$0.379446\pi$
$500$	0	0
$501$	−2.79154	−0.124717
$502$	0	0
$503$	−35.4935	−1.58258	−0.791288	−	0.611444i	$-0.790589\pi$
$503$	−35.4935	−1.58258	−0.791288	+	0.611444i	$0.790589\pi$
$504$	0	0
$505$	2.37151	0.105531
$506$	0	0
$507$	1.54918	0.0688017
$508$	0	0
$509$	−43.8652	−1.94429	−0.972146	−	0.234376i	$-0.924695\pi$
$509$	−43.8652	−1.94429	−0.972146	+	0.234376i	$0.924695\pi$
$510$	0	0
$511$	2.44764	0.108277
$512$	0	0
$513$	−47.3456	−2.09036
$514$	0	0
$515$	3.29394	0.145148
$516$	0	0
$517$	−1.45382	−0.0639391
$518$	0	0
$519$	20.8591	0.915611
$520$	0	0
$521$	−11.9666	−0.524264	−0.262132	−	0.965032i	$-0.584426\pi$
$521$	−11.9666	−0.524264	−0.262132	+	0.965032i	$0.584426\pi$
$522$	0	0
$523$	15.6990	0.686470	0.343235	−	0.939249i	$-0.388477\pi$
$523$	15.6990	0.686470	0.343235	+	0.939249i	$0.388477\pi$
$524$	0	0
$525$	7.54854	0.329445
$526$	0	0
$527$	59.1005	2.57446
$528$	0	0
$529$	40.7215	1.77050
$530$	0	0
$531$	0.468852	0.0203464
$532$	0	0
$533$	−8.72233	−0.377806
$534$	0	0
$535$	3.95498	0.170988
$536$	0	0
$537$	13.0613	0.563635
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	26.7880	1.15171	0.575854	−	0.817553i	$-0.304670\pi$
$541$	26.7880	1.15171	0.575854	+	0.817553i	$0.304670\pi$
$542$	0	0
$543$	−34.7692	−1.49209
$544$	0	0
$545$	6.30354	0.270014
$546$	0	0
$547$	−9.09563	−0.388901	−0.194450	−	0.980912i	$-0.562292\pi$
$547$	−9.09563	−0.388901	−0.194450	+	0.980912i	$0.562292\pi$
$548$	0	0
$549$	3.18698	0.136017
$550$	0	0
$551$	45.1283	1.92253
$552$	0	0
$553$	11.8586	0.504280
$554$	0	0
$555$	−5.82667	−0.247328
$556$	0	0
$557$	−13.5695	−0.574958	−0.287479	−	0.957787i	$-0.592817\pi$
$557$	−13.5695	−0.574958	−0.287479	+	0.957787i	$0.592817\pi$
$558$	0	0
$559$	7.46133	0.315581
$560$	0	0
$561$	9.37067	0.395630
$562$	0	0
$563$	−6.20690	−0.261590	−0.130795	−	0.991409i	$-0.541753\pi$
$563$	−6.20690	−0.261590	−0.130795	+	0.991409i	$0.541753\pi$
$564$	0	0
$565$	0.787791	0.0331426
$566$	0	0
$567$	6.83988	0.287248
$568$	0	0
$569$	−29.4433	−1.23433	−0.617164	−	0.786834i	$-0.711719\pi$
$569$	−29.4433	−1.23433	−0.617164	+	0.786834i	$0.711719\pi$
$570$	0	0
$571$	29.8202	1.24794	0.623968	−	0.781450i	$-0.285519\pi$
$571$	29.8202	1.24794	0.623968	+	0.781450i	$0.285519\pi$
$572$	0	0
$573$	−12.9869	−0.542533
$574$	0	0
$575$	−38.8958	−1.62207
$576$	0	0
$577$	12.5995	0.524524	0.262262	−	0.964997i	$-0.415531\pi$
$577$	12.5995	0.524524	0.262262	+	0.964997i	$0.415531\pi$
$578$	0	0
$579$	−12.6879	−0.527293
$580$	0	0
$581$	5.43355	0.225422
$582$	0	0
$583$	1.71369	0.0709736
$584$	0	0
$585$	−0.214180	−0.00885524
$586$	0	0
$587$	31.3278	1.29303	0.646517	−	0.762899i	$-0.276225\pi$
$587$	31.3278	1.29303	0.646517	+	0.762899i	$0.276225\pi$
$588$	0	0
$589$	−82.9457	−3.41772
$590$	0	0
$591$	−18.7487	−0.771217
$592$	0	0
$593$	14.1972	0.583010	0.291505	−	0.956569i	$-0.405844\pi$
$593$	14.1972	0.583010	0.291505	+	0.956569i	$0.405844\pi$
$594$	0	0
$595$	−2.15911	−0.0885148
$596$	0	0
$597$	38.5199	1.57651
$598$	0	0
$599$	−35.9027	−1.46694	−0.733472	−	0.679720i	$-0.762101\pi$
$599$	−35.9027	−1.46694	−0.733472	+	0.679720i	$0.762101\pi$
$600$	0	0
$601$	6.89109	0.281094	0.140547	−	0.990074i	$-0.455114\pi$
$601$	6.89109	0.281094	0.140547	+	0.990074i	$0.455114\pi$
$602$	0	0
$603$	3.71854	0.151431
$604$	0	0
$605$	−0.356949	−0.0145121
$606$	0	0
$607$	−19.0947	−0.775032	−0.387516	−	0.921863i	$-0.626667\pi$
$607$	−19.0947	−0.775032	−0.387516	+	0.921863i	$0.626667\pi$
$608$	0	0
$609$	−8.23533	−0.333712
$610$	0	0
$611$	−1.45382	−0.0588154
$612$	0	0
$613$	11.4928	0.464189	0.232095	−	0.972693i	$-0.425442\pi$
$613$	11.4928	0.464189	0.232095	+	0.972693i	$0.425442\pi$
$614$	0	0
$615$	−4.82328	−0.194493
$616$	0	0
$617$	−32.1261	−1.29335	−0.646674	−	0.762767i	$-0.723841\pi$
$617$	−32.1261	−1.29335	−0.646674	+	0.762767i	$0.723841\pi$
$618$	0	0
$619$	24.8651	0.999411	0.499706	−	0.866195i	$-0.333442\pi$
$619$	24.8651	0.999411	0.499706	+	0.866195i	$0.333442\pi$
$620$	0	0
$621$	−44.5197	−1.78651
$622$	0	0
$623$	−9.25740	−0.370890
$624$	0	0
$625$	23.1050	0.924202
$626$	0	0
$627$	−13.1515	−0.525219
$628$	0	0
$629$	−63.7350	−2.54128
$630$	0	0
$631$	14.4588	0.575595	0.287797	−	0.957691i	$-0.407077\pi$
$631$	14.4588	0.575595	0.287797	+	0.957691i	$0.407077\pi$
$632$	0	0
$633$	−39.9220	−1.58676
$634$	0	0
$635$	−2.15670	−0.0855860
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	4.59195	0.181655
$640$	0	0
$641$	−46.4826	−1.83595	−0.917976	−	0.396637i	$-0.870177\pi$
$641$	−46.4826	−1.83595	−0.917976	+	0.396637i	$0.870177\pi$
$642$	0	0
$643$	−22.9147	−0.903666	−0.451833	−	0.892102i	$-0.649230\pi$
$643$	−22.9147	−0.903666	−0.451833	+	0.892102i	$0.649230\pi$
$644$	0	0
$645$	4.12597	0.162460
$646$	0	0
$647$	9.93310	0.390511	0.195255	−	0.980752i	$-0.437446\pi$
$647$	9.93310	0.390511	0.195255	+	0.980752i	$0.437446\pi$
$648$	0	0
$649$	0.781383	0.0306720
$650$	0	0
$651$	15.1365	0.593247
$652$	0	0
$653$	23.8734	0.934238	0.467119	−	0.884195i	$-0.345292\pi$
$653$	23.8734	0.934238	0.467119	+	0.884195i	$0.345292\pi$
$654$	0	0
$655$	7.77710	0.303876
$656$	0	0
$657$	1.46865	0.0572977
$658$	0	0
$659$	26.9548	1.05001	0.525004	−	0.851100i	$-0.324064\pi$
$659$	26.9548	1.05001	0.525004	+	0.851100i	$0.324064\pi$
$660$	0	0
$661$	−22.9740	−0.893586	−0.446793	−	0.894637i	$-0.647434\pi$
$661$	−22.9740	−0.893586	−0.446793	+	0.894637i	$0.647434\pi$
$662$	0	0
$663$	9.37067	0.363927
$664$	0	0
$665$	3.03024	0.117508
$666$	0	0
$667$	42.4347	1.64308
$668$	0	0
$669$	11.9998	0.463939
$670$	0	0
$671$	5.31139	0.205044
$672$	0	0
$673$	−6.05070	−0.233237	−0.116619	−	0.993177i	$-0.537206\pi$
$673$	−6.05070	−0.233237	−0.116619	+	0.993177i	$0.537206\pi$
$674$	0	0
$675$	27.1749	1.04596
$676$	0	0
$677$	29.1842	1.12164	0.560820	−	0.827938i	$-0.310486\pi$
$677$	29.1842	1.12164	0.560820	+	0.827938i	$0.310486\pi$
$678$	0	0
$679$	−3.82384	−0.146746
$680$	0	0
$681$	−11.6392	−0.446017
$682$	0	0
$683$	−28.6520	−1.09634	−0.548169	−	0.836367i	$-0.684675\pi$
$683$	−28.6520	−1.09634	−0.548169	+	0.836367i	$0.684675\pi$
$684$	0	0
$685$	6.75230	0.257992
$686$	0	0
$687$	−32.6628	−1.24616
$688$	0	0
$689$	1.71369	0.0652862
$690$	0	0
$691$	−13.0755	−0.497416	−0.248708	−	0.968579i	$-0.580006\pi$
$691$	−13.0755	−0.497416	−0.248708	+	0.968579i	$0.580006\pi$
$692$	0	0
$693$	−0.600028	−0.0227932
$694$	0	0
$695$	−8.19783	−0.310961
$696$	0	0
$697$	−52.7594	−1.99841
$698$	0	0
$699$	36.2213	1.37001
$700$	0	0
$701$	9.91478	0.374476	0.187238	−	0.982315i	$-0.440046\pi$
$701$	9.91478	0.374476	0.187238	+	0.982315i	$0.440046\pi$
$702$	0	0
$703$	89.4501	3.37368
$704$	0	0
$705$	−0.803936	−0.0302780
$706$	0	0
$707$	6.64381	0.249866
$708$	0	0
$709$	−37.4122	−1.40504	−0.702522	−	0.711662i	$-0.747943\pi$
$709$	−37.4122	−1.40504	−0.702522	+	0.711662i	$0.747943\pi$
$710$	0	0
$711$	7.11551	0.266853
$712$	0	0
$713$	−77.9949	−2.92093
$714$	0	0
$715$	−0.356949	−0.0133492
$716$	0	0
$717$	8.40957	0.314061
$718$	0	0
$719$	26.1404	0.974872	0.487436	−	0.873159i	$-0.337932\pi$
$719$	26.1404	0.974872	0.487436	+	0.873159i	$0.337932\pi$
$720$	0	0
$721$	9.22802	0.343669
$722$	0	0
$723$	−27.7605	−1.03242
$724$	0	0
$725$	−25.9023	−0.961985
$726$	0	0
$727$	19.2363	0.713435	0.356717	−	0.934212i	$-0.383896\pi$
$727$	19.2363	0.713435	0.356717	+	0.934212i	$0.383896\pi$
$728$	0	0
$729$	30.0240	1.11200
$730$	0	0
$731$	45.1319	1.66926
$732$	0	0
$733$	−17.8790	−0.660378	−0.330189	−	0.943915i	$-0.607112\pi$
$733$	−17.8790	−0.660378	−0.330189	+	0.943915i	$0.607112\pi$
$734$	0	0
$735$	−0.552981	−0.0203970
$736$	0	0
$737$	6.19727	0.228279
$738$	0	0
$739$	20.4069	0.750681	0.375341	−	0.926887i	$-0.377526\pi$
$739$	20.4069	0.750681	0.375341	+	0.926887i	$0.377526\pi$
$740$	0	0
$741$	−13.1515	−0.483131
$742$	0	0
$743$	−20.3952	−0.748226	−0.374113	−	0.927383i	$-0.622053\pi$
$743$	−20.3952	−0.748226	−0.374113	+	0.927383i	$0.622053\pi$
$744$	0	0
$745$	3.34556	0.122572
$746$	0	0
$747$	3.26028	0.119287
$748$	0	0
$749$	11.0799	0.404852
$750$	0	0
$751$	35.5747	1.29814	0.649070	−	0.760728i	$-0.275158\pi$
$751$	35.5747	1.29814	0.649070	+	0.760728i	$0.275158\pi$
$752$	0	0
$753$	6.70544	0.244360
$754$	0	0
$755$	−0.588603	−0.0214215
$756$	0	0
$757$	−7.55291	−0.274515	−0.137258	−	0.990535i	$-0.543829\pi$
$757$	−7.55291	−0.274515	−0.137258	+	0.990535i	$0.543829\pi$
$758$	0	0
$759$	−12.3665	−0.448875
$760$	0	0
$761$	29.6862	1.07612	0.538062	−	0.842906i	$-0.319157\pi$
$761$	29.6862	1.07612	0.538062	+	0.842906i	$0.319157\pi$
$762$	0	0
$763$	17.6595	0.639316
$764$	0	0
$765$	−1.29553	−0.0468398
$766$	0	0
$767$	0.781383	0.0282141
$768$	0	0
$769$	49.3335	1.77901	0.889506	−	0.456924i	$-0.151049\pi$
$769$	49.3335	1.77901	0.889506	+	0.456924i	$0.151049\pi$
$770$	0	0
$771$	−18.5735	−0.668907
$772$	0	0
$773$	−50.8325	−1.82832	−0.914160	−	0.405354i	$-0.867148\pi$
$773$	−50.8325	−1.82832	−0.914160	+	0.405354i	$0.867148\pi$
$774$	0	0
$775$	47.6083	1.71014
$776$	0	0
$777$	−16.3235	−0.585602
$778$	0	0
$779$	74.0463	2.65298
$780$	0	0
$781$	7.65289	0.273842
$782$	0	0
$783$	−29.6474	−1.05951
$784$	0	0
$785$	5.24864	0.187332
$786$	0	0
$787$	48.7664	1.73833	0.869167	−	0.494518i	$-0.164655\pi$
$787$	48.7664	1.73833	0.869167	+	0.494518i	$0.164655\pi$
$788$	0	0
$789$	−10.8505	−0.386289
$790$	0	0
$791$	2.20701	0.0784723
$792$	0	0
$793$	5.31139	0.188613
$794$	0	0
$795$	0.947635	0.0336092
$796$	0	0
$797$	24.3353	0.861999	0.431000	−	0.902352i	$-0.358161\pi$
$797$	24.3353	0.861999	0.431000	+	0.902352i	$0.358161\pi$
$798$	0	0
$799$	−8.79385	−0.311104
$800$	0	0
$801$	−5.55470	−0.196266
$802$	0	0
$803$	2.44764	0.0863754
$804$	0	0
$805$	2.84938	0.100427
$806$	0	0
$807$	−20.5218	−0.722400
$808$	0	0
$809$	41.2803	1.45134	0.725668	−	0.688045i	$-0.241531\pi$
$809$	41.2803	1.45134	0.725668	+	0.688045i	$0.241531\pi$
$810$	0	0
$811$	19.5033	0.684855	0.342428	−	0.939544i	$-0.388751\pi$
$811$	19.5033	0.684855	0.342428	+	0.939544i	$0.388751\pi$
$812$	0	0
$813$	31.8606	1.11740
$814$	0	0
$815$	5.19658	0.182028
$816$	0	0
$817$	−63.3413	−2.21603
$818$	0	0
$819$	−0.600028	−0.0209667
$820$	0	0
$821$	17.3464	0.605392	0.302696	−	0.953087i	$-0.402113\pi$
$821$	17.3464	0.605392	0.302696	+	0.953087i	$0.402113\pi$
$822$	0	0
$823$	−29.1458	−1.01596	−0.507980	−	0.861369i	$-0.669608\pi$
$823$	−29.1458	−1.01596	−0.507980	+	0.861369i	$0.669608\pi$
$824$	0	0
$825$	7.54854	0.262806
$826$	0	0
$827$	22.7663	0.791661	0.395830	−	0.918324i	$-0.370457\pi$
$827$	22.7663	0.791661	0.395830	+	0.918324i	$0.370457\pi$
$828$	0	0
$829$	17.2238	0.598206	0.299103	−	0.954221i	$-0.403313\pi$
$829$	17.2238	0.598206	0.299103	+	0.954221i	$0.403313\pi$
$830$	0	0
$831$	14.6460	0.508066
$832$	0	0
$833$	−6.04878	−0.209578
$834$	0	0
$835$	0.643202	0.0222589
$836$	0	0
$837$	54.4919	1.88352
$838$	0	0
$839$	30.2951	1.04590	0.522952	−	0.852362i	$-0.324831\pi$
$839$	30.2951	1.04590	0.522952	+	0.852362i	$0.324831\pi$
$840$	0	0
$841$	−0.741062	−0.0255539
$842$	0	0
$843$	1.23760	0.0426253
$844$	0	0
$845$	−0.356949	−0.0122794
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	23.9430	0.821723
$850$	0	0
$851$	84.1111	2.88329
$852$	0	0
$853$	33.4519	1.14537	0.572686	−	0.819775i	$-0.305901\pi$
$853$	33.4519	1.14537	0.572686	+	0.819775i	$0.305901\pi$
$854$	0	0
$855$	1.81823	0.0621822
$856$	0	0
$857$	4.70378	0.160678	0.0803391	−	0.996768i	$-0.474400\pi$
$857$	4.70378	0.160678	0.0803391	+	0.996768i	$0.474400\pi$
$858$	0	0
$859$	−31.7030	−1.08169	−0.540846	−	0.841122i	$-0.681896\pi$
$859$	−31.7030	−1.08169	−0.540846	+	0.841122i	$0.681896\pi$
$860$	0	0
$861$	−13.5125	−0.460505
$862$	0	0
$863$	7.32286	0.249273	0.124636	−	0.992202i	$-0.460224\pi$
$863$	7.32286	0.249273	0.124636	+	0.992202i	$0.460224\pi$
$864$	0	0
$865$	−4.80616	−0.163414
$866$	0	0
$867$	30.3450	1.03057
$868$	0	0
$869$	11.8586	0.402277
$870$	0	0
$871$	6.19727	0.209987
$872$	0	0
$873$	−2.29441	−0.0776541
$874$	0	0
$875$	−3.52401	−0.119133
$876$	0	0
$877$	−42.9628	−1.45075	−0.725375	−	0.688354i	$-0.758334\pi$
$877$	−42.9628	−1.45075	−0.725375	+	0.688354i	$0.758334\pi$
$878$	0	0
$879$	−38.0667	−1.28396
$880$	0	0
$881$	3.65959	0.123295	0.0616474	−	0.998098i	$-0.480365\pi$
$881$	3.65959	0.123295	0.0616474	+	0.998098i	$0.480365\pi$
$882$	0	0
$883$	−22.2488	−0.748731	−0.374366	−	0.927281i	$-0.622140\pi$
$883$	−22.2488	−0.748731	−0.374366	+	0.927281i	$0.622140\pi$
$884$	0	0
$885$	0.432089	0.0145245
$886$	0	0
$887$	−17.2501	−0.579202	−0.289601	−	0.957147i	$-0.593523\pi$
$887$	−17.2501	−0.579202	−0.289601	+	0.957147i	$0.593523\pi$
$888$	0	0
$889$	−6.04203	−0.202643
$890$	0	0
$891$	6.83988	0.229145
$892$	0	0
$893$	12.3419	0.413006
$894$	0	0
$895$	−3.00946	−0.100595
$896$	0	0
$897$	−12.3665	−0.412905
$898$	0	0
$899$	−51.9399	−1.73229
$900$	0	0
$901$	10.3657	0.345332
$902$	0	0
$903$	11.5590	0.384659
$904$	0	0
$905$	8.01122	0.266302
$906$	0	0
$907$	−10.7320	−0.356349	−0.178175	−	0.983999i	$-0.557019\pi$
$907$	−10.7320	−0.356349	−0.178175	+	0.983999i	$0.557019\pi$
$908$	0	0
$909$	3.98648	0.132223
$910$	0	0
$911$	17.3482	0.574771	0.287386	−	0.957815i	$-0.407214\pi$
$911$	17.3482	0.574771	0.287386	+	0.957815i	$0.407214\pi$
$912$	0	0
$913$	5.43355	0.179824
$914$	0	0
$915$	2.93709	0.0970973
$916$	0	0
$917$	21.7877	0.719492
$918$	0	0
$919$	−42.9821	−1.41785	−0.708924	−	0.705285i	$-0.750819\pi$
$919$	−42.9821	−1.41785	−0.708924	+	0.705285i	$0.750819\pi$
$920$	0	0
$921$	−3.84016	−0.126538
$922$	0	0
$923$	7.65289	0.251898
$924$	0	0
$925$	−51.3417	−1.68810
$926$	0	0
$927$	5.53707	0.181861
$928$	0	0
$929$	−48.4507	−1.58962	−0.794808	−	0.606861i	$-0.792428\pi$
$929$	−48.4507	−1.58962	−0.794808	+	0.606861i	$0.792428\pi$
$930$	0	0
$931$	8.48928	0.278225
$932$	0	0
$933$	−47.7552	−1.56343
$934$	0	0
$935$	−2.15911	−0.0706104
$936$	0	0
$937$	−33.0939	−1.08113	−0.540565	−	0.841302i	$-0.681789\pi$
$937$	−33.0939	−1.08113	−0.540565	+	0.841302i	$0.681789\pi$
$938$	0	0
$939$	34.4838	1.12534
$940$	0	0
$941$	13.8501	0.451499	0.225750	−	0.974185i	$-0.427517\pi$
$941$	13.8501	0.451499	0.225750	+	0.974185i	$0.427517\pi$
$942$	0	0
$943$	69.6266	2.26735
$944$	0	0
$945$	−1.99075	−0.0647590
$946$	0	0
$947$	−0.457427	−0.0148644	−0.00743219	−	0.999972i	$-0.502366\pi$
$947$	−0.457427	−0.0148644	−0.00743219	+	0.999972i	$0.502366\pi$
$948$	0	0
$949$	2.44764	0.0794539
$950$	0	0
$951$	3.25644	0.105597
$952$	0	0
$953$	−33.5821	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$953$	−33.5821	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$954$	0	0
$955$	2.99232	0.0968291
$956$	0	0
$957$	−8.23533	−0.266210
$958$	0	0
$959$	18.9167	0.610852
$960$	0	0
$961$	64.4655	2.07953
$962$	0	0
$963$	6.64827	0.214238
$964$	0	0
$965$	2.92345	0.0941090
$966$	0	0
$967$	−51.2960	−1.64957	−0.824784	−	0.565448i	$-0.808703\pi$
$967$	−51.2960	−1.64957	−0.824784	+	0.565448i	$0.808703\pi$
$968$	0	0
$969$	−79.5503	−2.55552
$970$	0	0
$971$	−57.5395	−1.84653	−0.923265	−	0.384163i	$-0.874490\pi$
$971$	−57.5395	−1.84653	−0.923265	+	0.384163i	$0.874490\pi$
$972$	0	0
$973$	−22.9664	−0.736268
$974$	0	0
$975$	7.54854	0.241747
$976$	0	0
$977$	3.89398	0.124579	0.0622897	−	0.998058i	$-0.480160\pi$
$977$	3.89398	0.124579	0.0622897	+	0.998058i	$0.480160\pi$
$978$	0	0
$979$	−9.25740	−0.295868
$980$	0	0
$981$	10.5962	0.338310
$982$	0	0
$983$	22.9979	0.733519	0.366760	−	0.930316i	$-0.380467\pi$
$983$	22.9979	0.733519	0.366760	+	0.930316i	$0.380467\pi$
$984$	0	0
$985$	4.31991	0.137644
$986$	0	0
$987$	−2.25224	−0.0716896
$988$	0	0
$989$	−59.5606	−1.89392
$990$	0	0
$991$	−30.3704	−0.964749	−0.482375	−	0.875965i	$-0.660226\pi$
$991$	−30.3704	−0.964749	−0.482375	+	0.875965i	$0.660226\pi$
$992$	0	0
$993$	47.1165	1.49520
$994$	0	0
$995$	−8.87541	−0.281370
$996$	0	0
$997$	3.31661	0.105038	0.0525190	−	0.998620i	$-0.483275\pi$
$997$	3.31661	0.105038	0.0525190	+	0.998620i	$0.483275\pi$
$998$	0	0
$999$	−58.7651	−1.85924

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.u.1.7	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.u.1.7	✓	10	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q+1.54918 q^{3} -0.356949 q^{5} -1.00000 q^{7} -0.600028 q^{9} +O(q^{10})\)	\(q+1.54918 q^{3} -0.356949 q^{5} -1.00000 q^{7} -0.600028 q^{9} -1.00000 q^{11} -1.00000 q^{13} -0.552981 q^{15} -6.04878 q^{17} +8.48928 q^{19} -1.54918 q^{21} +7.98257 q^{23} -4.87259 q^{25} -5.57711 q^{27} +5.31591 q^{29} -9.77065 q^{31} -1.54918 q^{33} +0.356949 q^{35} +10.5368 q^{37} -1.54918 q^{39} +8.72233 q^{41} -7.46133 q^{43} +0.214180 q^{45} +1.45382 q^{47} +1.00000 q^{49} -9.37067 q^{51} -1.71369 q^{53} +0.356949 q^{55} +13.1515 q^{57} -0.781383 q^{59} -5.31139 q^{61} +0.600028 q^{63} +0.356949 q^{65} -6.19727 q^{67} +12.3665 q^{69} -7.65289 q^{71} -2.44764 q^{73} -7.54854 q^{75} +1.00000 q^{77} -11.8586 q^{79} -6.83988 q^{81} -5.43355 q^{83} +2.15911 q^{85} +8.23533 q^{87} +9.25740 q^{89} +1.00000 q^{91} -15.1365 q^{93} -3.03024 q^{95} +3.82384 q^{97} +0.600028 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{3} - 4 q^{5} - 10 q^{7} + 6 q^{9}+O(q^{10}) \) 10 * q + 2 * q^3 - 4 * q^5 - 10 * q^7 + 6 * q^9	\( 10 q + 2 q^{3} - 4 q^{5} - 10 q^{7} + 6 q^{9} - 10 q^{11} - 10 q^{13} - 4 q^{15} - 3 q^{17} + 13 q^{19} - 2 q^{21} + 6 q^{25} + 14 q^{27} - 3 q^{29} - q^{31} - 2 q^{33} + 4 q^{35} - 5 q^{37} - 2 q^{39} + 4 q^{41} + 19 q^{43} - 11 q^{45} + q^{47} + 10 q^{49} + 15 q^{51} - 6 q^{53} + 4 q^{55} - 6 q^{57} + 4 q^{59} - 18 q^{61} - 6 q^{63} + 4 q^{65} + q^{67} - 11 q^{69} - 21 q^{71} - 10 q^{73} + 10 q^{77} + 3 q^{79} - 30 q^{81} + 6 q^{83} - 33 q^{85} - 9 q^{87} - 22 q^{89} + 10 q^{91} - 34 q^{93} - 3 q^{95} - 9 q^{97} - 6 q^{99}+O(q^{100}) \) 10 * q + 2 * q^3 - 4 * q^5 - 10 * q^7 + 6 * q^9 - 10 * q^11 - 10 * q^13 - 4 * q^15 - 3 * q^17 + 13 * q^19 - 2 * q^21 + 6 * q^25 + 14 * q^27 - 3 * q^29 - q^31 - 2 * q^33 + 4 * q^35 - 5 * q^37 - 2 * q^39 + 4 * q^41 + 19 * q^43 - 11 * q^45 + q^47 + 10 * q^49 + 15 * q^51 - 6 * q^53 + 4 * q^55 - 6 * q^57 + 4 * q^59 - 18 * q^61 - 6 * q^63 + 4 * q^65 + q^67 - 11 * q^69 - 21 * q^71 - 10 * q^73 + 10 * q^77 + 3 * q^79 - 30 * q^81 + 6 * q^83 - 33 * q^85 - 9 * q^87 - 22 * q^89 + 10 * q^91 - 34 * q^93 - 3 * q^95 - 9 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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