LMFDB - Embedded newform 8008.2.a.j.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.9442019386$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.668973.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 9x^{3} - x^{2} + 7x - 2 $ x^5 - x^4 - 9x^3 - x^2 + 7x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.376114$ 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)

$f(q)$	$=$	$q+0.293231 q^{3} -3.94142 q^{5} +1.00000 q^{7} -2.91402 q^{9} +O(q^{10})$	$q+0.293231 q^{3} -3.94142 q^{5} +1.00000 q^{7} -2.91402 q^{9} -1.00000 q^{11} -1.00000 q^{13} -1.15575 q^{15} +0.0585790 q^{17} -4.49244 q^{19} +0.293231 q^{21} -2.80394 q^{23} +10.5348 q^{25} -1.73417 q^{27} -7.01606 q^{29} -3.34473 q^{31} -0.293231 q^{33} -3.94142 q^{35} -9.50363 q^{37} -0.293231 q^{39} -9.75119 q^{41} -12.4521 q^{43} +11.4854 q^{45} +12.3205 q^{47} +1.00000 q^{49} +0.0171772 q^{51} +4.02741 q^{53} +3.94142 q^{55} -1.31732 q^{57} +11.9000 q^{59} +5.13748 q^{61} -2.91402 q^{63} +3.94142 q^{65} +9.12457 q^{67} -0.822202 q^{69} +13.5165 q^{71} -1.17401 q^{73} +3.08913 q^{75} -1.00000 q^{77} -10.3334 q^{79} +8.23353 q^{81} -8.10408 q^{83} -0.230884 q^{85} -2.05733 q^{87} -11.0789 q^{89} -1.00000 q^{91} -0.980779 q^{93} +17.7066 q^{95} -4.13149 q^{97} +2.91402 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{3} - q^{5} + 5 q^{7} + 6 q^{9}+O(q^{10}) $ 5 * q + q^3 - q^5 + 5 * q^7 + 6 * q^9	$ 5 q + q^{3} - q^{5} + 5 q^{7} + 6 q^{9} - 5 q^{11} - 5 q^{13} - 5 q^{15} + 19 q^{17} - 5 q^{19} + q^{21} + 5 q^{23} + 12 q^{25} - 11 q^{27} - 17 q^{29} + 4 q^{31} - q^{33} - q^{35} + 10 q^{37} - q^{39} + 10 q^{41} - 25 q^{43} + q^{45} + 3 q^{47} + 5 q^{49} - q^{51} + 22 q^{53} + q^{55} + 16 q^{57} + 21 q^{59} + 26 q^{61} + 6 q^{63} + q^{65} + 28 q^{67} + 16 q^{69} + 28 q^{71} - 4 q^{73} - 5 q^{77} - 11 q^{79} + 5 q^{81} + 33 q^{85} + 31 q^{87} - 37 q^{89} - 5 q^{91} - 49 q^{93} + 29 q^{95} + 18 q^{97} - 6 q^{99}+O(q^{100}) $ 5 * q + q^3 - q^5 + 5 * q^7 + 6 * q^9 - 5 * q^11 - 5 * q^13 - 5 * q^15 + 19 * q^17 - 5 * q^19 + q^21 + 5 * q^23 + 12 * q^25 - 11 * q^27 - 17 * q^29 + 4 * q^31 - q^33 - q^35 + 10 * q^37 - q^39 + 10 * q^41 - 25 * q^43 + q^45 + 3 * q^47 + 5 * q^49 - q^51 + 22 * q^53 + q^55 + 16 * q^57 + 21 * q^59 + 26 * q^61 + 6 * q^63 + q^65 + 28 * q^67 + 16 * q^69 + 28 * q^71 - 4 * q^73 - 5 * q^77 - 11 * q^79 + 5 * q^81 + 33 * q^85 + 31 * q^87 - 37 * q^89 - 5 * q^91 - 49 * q^93 + 29 * q^95 + 18 * 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
$3$	0.293231	0.169297	0.0846486	−	0.996411i	$-0.473023\pi$
$3$	0.293231	0.169297	0.0846486	+	0.996411i	$0.473023\pi$
$4$	0	0
$5$	−3.94142	−1.76266	−0.881329	−	0.472504i	$-0.843350\pi$
$5$	−3.94142	−1.76266	−0.881329	+	0.472504i	$0.843350\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.91402	−0.971338
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−1.15575	−0.298413
$16$	0	0
$17$	0.0585790	0.0142075	0.00710375	−	0.999975i	$-0.497739\pi$
$17$	0.0585790	0.0142075	0.00710375	+	0.999975i	$0.497739\pi$
$18$	0	0
$19$	−4.49244	−1.03064	−0.515318	−	0.856999i	$-0.672326\pi$
$19$	−4.49244	−1.03064	−0.515318	+	0.856999i	$0.672326\pi$
$20$	0	0
$21$	0.293231	0.0639883
$22$	0	0
$23$	−2.80394	−0.584661	−0.292331	−	0.956317i	$-0.594431\pi$
$23$	−2.80394	−0.584661	−0.292331	+	0.956317i	$0.594431\pi$
$24$	0	0
$25$	10.5348	2.10696
$26$	0	0
$27$	−1.73417	−0.333742
$28$	0	0
$29$	−7.01606	−1.30285	−0.651425	−	0.758713i	$-0.725828\pi$
$29$	−7.01606	−1.30285	−0.651425	+	0.758713i	$0.725828\pi$
$30$	0	0
$31$	−3.34473	−0.600731	−0.300366	−	0.953824i	$-0.597109\pi$
$31$	−3.34473	−0.600731	−0.300366	+	0.953824i	$0.597109\pi$
$32$	0	0
$33$	−0.293231	−0.0510450
$34$	0	0
$35$	−3.94142	−0.666222
$36$	0	0
$37$	−9.50363	−1.56239	−0.781193	−	0.624289i	$-0.785389\pi$
$37$	−9.50363	−1.56239	−0.781193	+	0.624289i	$0.785389\pi$
$38$	0	0
$39$	−0.293231	−0.0469546
$40$	0	0
$41$	−9.75119	−1.52288	−0.761440	−	0.648236i	$-0.775507\pi$
$41$	−9.75119	−1.52288	−0.761440	+	0.648236i	$0.775507\pi$
$42$	0	0
$43$	−12.4521	−1.89893	−0.949466	−	0.313870i	$-0.898374\pi$
$43$	−12.4521	−1.89893	−0.949466	+	0.313870i	$0.898374\pi$
$44$	0	0
$45$	11.4854	1.71214
$46$	0	0
$47$	12.3205	1.79713	0.898563	−	0.438845i	$-0.144612\pi$
$47$	12.3205	1.79713	0.898563	+	0.438845i	$0.144612\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.0171772	0.00240529
$52$	0	0
$53$	4.02741	0.553207	0.276603	−	0.960984i	$-0.410791\pi$
$53$	4.02741	0.553207	0.276603	+	0.960984i	$0.410791\pi$
$54$	0	0
$55$	3.94142	0.531461
$56$	0	0
$57$	−1.31732	−0.174484
$58$	0	0
$59$	11.9000	1.54925	0.774625	−	0.632421i	$-0.217938\pi$
$59$	11.9000	1.54925	0.774625	+	0.632421i	$0.217938\pi$
$60$	0	0
$61$	5.13748	0.657787	0.328894	−	0.944367i	$-0.393324\pi$
$61$	5.13748	0.657787	0.328894	+	0.944367i	$0.393324\pi$
$62$	0	0
$63$	−2.91402	−0.367131
$64$	0	0
$65$	3.94142	0.488873
$66$	0	0
$67$	9.12457	1.11474	0.557372	−	0.830263i	$-0.311810\pi$
$67$	9.12457	1.11474	0.557372	+	0.830263i	$0.311810\pi$
$68$	0	0
$69$	−0.822202	−0.0989815
$70$	0	0
$71$	13.5165	1.60412	0.802059	−	0.597245i	$-0.203738\pi$
$71$	13.5165	1.60412	0.802059	+	0.597245i	$0.203738\pi$
$72$	0	0
$73$	−1.17401	−0.137408	−0.0687039	−	0.997637i	$-0.521886\pi$
$73$	−1.17401	−0.137408	−0.0687039	+	0.997637i	$0.521886\pi$
$74$	0	0
$75$	3.08913	0.356702
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−10.3334	−1.16260	−0.581298	−	0.813691i	$-0.697455\pi$
$79$	−10.3334	−1.16260	−0.581298	+	0.813691i	$0.697455\pi$
$80$	0	0
$81$	8.23353	0.914837
$82$	0	0
$83$	−8.10408	−0.889539	−0.444769	−	0.895645i	$-0.646714\pi$
$83$	−8.10408	−0.889539	−0.444769	+	0.895645i	$0.646714\pi$
$84$	0	0
$85$	−0.230884	−0.0250429
$86$	0	0
$87$	−2.05733	−0.220569
$88$	0	0
$89$	−11.0789	−1.17436	−0.587181	−	0.809456i	$-0.699762\pi$
$89$	−11.0789	−1.17436	−0.587181	+	0.809456i	$0.699762\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−0.980779	−0.101702
$94$	0	0
$95$	17.7066	1.81666
$96$	0	0
$97$	−4.13149	−0.419489	−0.209745	−	0.977756i	$-0.567263\pi$
$97$	−4.13149	−0.419489	−0.209745	+	0.977756i	$0.567263\pi$
$98$	0	0
$99$	2.91402	0.292870
$100$	0	0
$101$	−9.53590	−0.948858	−0.474429	−	0.880294i	$-0.657345\pi$
$101$	−9.53590	−0.948858	−0.474429	+	0.880294i	$0.657345\pi$
$102$	0	0
$103$	−7.64392	−0.753178	−0.376589	−	0.926380i	$-0.622903\pi$
$103$	−7.64392	−0.753178	−0.376589	+	0.926380i	$0.622903\pi$
$104$	0	0
$105$	−1.15575	−0.112789
$106$	0	0
$107$	7.16472	0.692640	0.346320	−	0.938116i	$-0.387431\pi$
$107$	7.16472	0.692640	0.346320	+	0.938116i	$0.387431\pi$
$108$	0	0
$109$	−18.7646	−1.79732	−0.898660	−	0.438647i	$-0.855458\pi$
$109$	−18.7646	−1.79732	−0.898660	+	0.438647i	$0.855458\pi$
$110$	0	0
$111$	−2.78676	−0.264508
$112$	0	0
$113$	−13.9273	−1.31017	−0.655083	−	0.755557i	$-0.727366\pi$
$113$	−13.9273	−1.31017	−0.655083	+	0.755557i	$0.727366\pi$
$114$	0	0
$115$	11.0515	1.03056
$116$	0	0
$117$	2.91402	0.269401
$118$	0	0
$119$	0.0585790	0.00536993
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−2.85935	−0.257819
$124$	0	0
$125$	−21.8150	−1.95119
$126$	0	0
$127$	12.7443	1.13087	0.565435	−	0.824793i	$-0.308708\pi$
$127$	12.7443	1.13087	0.565435	+	0.824793i	$0.308708\pi$
$128$	0	0
$129$	−3.65135	−0.321484
$130$	0	0
$131$	−17.4397	−1.52371	−0.761856	−	0.647746i	$-0.775712\pi$
$131$	−17.4397	−1.52371	−0.761856	+	0.647746i	$0.775712\pi$
$132$	0	0
$133$	−4.49244	−0.389544
$134$	0	0
$135$	6.83511	0.588273
$136$	0	0
$137$	12.4375	1.06260	0.531302	−	0.847182i	$-0.321703\pi$
$137$	12.4375	1.06260	0.531302	+	0.847182i	$0.321703\pi$
$138$	0	0
$139$	2.41133	0.204526	0.102263	−	0.994757i	$-0.467392\pi$
$139$	2.41133	0.204526	0.102263	+	0.994757i	$0.467392\pi$
$140$	0	0
$141$	3.61275	0.304248
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	27.6532	2.29648
$146$	0	0
$147$	0.293231	0.0241853
$148$	0	0
$149$	18.4118	1.50835	0.754177	−	0.656671i	$-0.228036\pi$
$149$	18.4118	1.50835	0.754177	+	0.656671i	$0.228036\pi$
$150$	0	0
$151$	−17.6217	−1.43404	−0.717018	−	0.697055i	$-0.754493\pi$
$151$	−17.6217	−1.43404	−0.717018	+	0.697055i	$0.754493\pi$
$152$	0	0
$153$	−0.170700	−0.0138003
$154$	0	0
$155$	13.1830	1.05888
$156$	0	0
$157$	−13.4660	−1.07470	−0.537352	−	0.843358i	$-0.680575\pi$
$157$	−13.4660	−1.07470	−0.537352	+	0.843358i	$0.680575\pi$
$158$	0	0
$159$	1.18096	0.0936563
$160$	0	0
$161$	−2.80394	−0.220981
$162$	0	0
$163$	−14.8795	−1.16545	−0.582727	−	0.812668i	$-0.698014\pi$
$163$	−14.8795	−1.16545	−0.582727	+	0.812668i	$0.698014\pi$
$164$	0	0
$165$	1.15575	0.0899749
$166$	0	0
$167$	12.7765	0.988678	0.494339	−	0.869269i	$-0.335410\pi$
$167$	12.7765	0.988678	0.494339	+	0.869269i	$0.335410\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	13.0910	1.00110
$172$	0	0
$173$	−9.54394	−0.725612	−0.362806	−	0.931865i	$-0.618181\pi$
$173$	−9.54394	−0.725612	−0.362806	+	0.931865i	$0.618181\pi$
$174$	0	0
$175$	10.5348	0.796356
$176$	0	0
$177$	3.48946	0.262284
$178$	0	0
$179$	22.6135	1.69022	0.845108	−	0.534596i	$-0.179536\pi$
$179$	22.6135	1.69022	0.845108	+	0.534596i	$0.179536\pi$
$180$	0	0
$181$	5.42268	0.403065	0.201532	−	0.979482i	$-0.435408\pi$
$181$	5.42268	0.403065	0.201532	+	0.979482i	$0.435408\pi$
$182$	0	0
$183$	1.50647	0.111362
$184$	0	0
$185$	37.4578	2.75395
$186$	0	0
$187$	−0.0585790	−0.00428372
$188$	0	0
$189$	−1.73417	−0.126143
$190$	0	0
$191$	10.3973	0.752325	0.376162	−	0.926554i	$-0.377243\pi$
$191$	10.3973	0.752325	0.376162	+	0.926554i	$0.377243\pi$
$192$	0	0
$193$	−16.9858	−1.22267	−0.611333	−	0.791373i	$-0.709366\pi$
$193$	−16.9858	−1.22267	−0.611333	+	0.791373i	$0.709366\pi$
$194$	0	0
$195$	1.15575	0.0827648
$196$	0	0
$197$	−17.9413	−1.27826	−0.639131	−	0.769098i	$-0.720706\pi$
$197$	−17.9413	−1.27826	−0.639131	+	0.769098i	$0.720706\pi$
$198$	0	0
$199$	24.7269	1.75285	0.876423	−	0.481542i	$-0.159923\pi$
$199$	24.7269	1.75285	0.876423	+	0.481542i	$0.159923\pi$
$200$	0	0
$201$	2.67561	0.188723
$202$	0	0
$203$	−7.01606	−0.492431
$204$	0	0
$205$	38.4335	2.68431
$206$	0	0
$207$	8.17072	0.567904
$208$	0	0
$209$	4.49244	0.310749
$210$	0	0
$211$	−14.5148	−0.999241	−0.499620	−	0.866244i	$-0.666527\pi$
$211$	−14.5148	−0.999241	−0.499620	+	0.866244i	$0.666527\pi$
$212$	0	0
$213$	3.96347	0.271573
$214$	0	0
$215$	49.0791	3.34717
$216$	0	0
$217$	−3.34473	−0.227055
$218$	0	0
$219$	−0.344257	−0.0232627
$220$	0	0
$221$	−0.0585790	−0.00394045
$222$	0	0
$223$	26.3731	1.76607	0.883036	−	0.469304i	$-0.155495\pi$
$223$	26.3731	1.76607	0.883036	+	0.469304i	$0.155495\pi$
$224$	0	0
$225$	−30.6986	−2.04657
$226$	0	0
$227$	11.0509	0.733474	0.366737	−	0.930325i	$-0.380475\pi$
$227$	11.0509	0.733474	0.366737	+	0.930325i	$0.380475\pi$
$228$	0	0
$229$	16.9794	1.12203	0.561014	−	0.827806i	$-0.310411\pi$
$229$	16.9794	1.12203	0.561014	+	0.827806i	$0.310411\pi$
$230$	0	0
$231$	−0.293231	−0.0192932
$232$	0	0
$233$	10.8630	0.711660	0.355830	−	0.934551i	$-0.384198\pi$
$233$	10.8630	0.711660	0.355830	+	0.934551i	$0.384198\pi$
$234$	0	0
$235$	−48.5602	−3.16772
$236$	0	0
$237$	−3.03007	−0.196824
$238$	0	0
$239$	−2.97591	−0.192495	−0.0962477	−	0.995357i	$-0.530684\pi$
$239$	−2.97591	−0.192495	−0.0962477	+	0.995357i	$0.530684\pi$
$240$	0	0
$241$	−8.60458	−0.554270	−0.277135	−	0.960831i	$-0.589385\pi$
$241$	−8.60458	−0.554270	−0.277135	+	0.960831i	$0.589385\pi$
$242$	0	0
$243$	7.61685	0.488621
$244$	0	0
$245$	−3.94142	−0.251808
$246$	0	0
$247$	4.49244	0.285847
$248$	0	0
$249$	−2.37637	−0.150596
$250$	0	0
$251$	6.40284	0.404143	0.202072	−	0.979371i	$-0.435233\pi$
$251$	6.40284	0.404143	0.202072	+	0.979371i	$0.435233\pi$
$252$	0	0
$253$	2.80394	0.176282
$254$	0	0
$255$	−0.0677025	−0.00423970
$256$	0	0
$257$	13.0209	0.812223	0.406112	−	0.913823i	$-0.366884\pi$
$257$	13.0209	0.812223	0.406112	+	0.913823i	$0.366884\pi$
$258$	0	0
$259$	−9.50363	−0.590527
$260$	0	0
$261$	20.4449	1.26551
$262$	0	0
$263$	−3.01512	−0.185920	−0.0929600	−	0.995670i	$-0.529633\pi$
$263$	−3.01512	−0.185920	−0.0929600	+	0.995670i	$0.529633\pi$
$264$	0	0
$265$	−15.8737	−0.975114
$266$	0	0
$267$	−3.24868	−0.198816
$268$	0	0
$269$	5.67560	0.346047	0.173024	−	0.984918i	$-0.444646\pi$
$269$	5.67560	0.346047	0.173024	+	0.984918i	$0.444646\pi$
$270$	0	0
$271$	13.9693	0.848573	0.424286	−	0.905528i	$-0.360525\pi$
$271$	13.9693	0.848573	0.424286	+	0.905528i	$0.360525\pi$
$272$	0	0
$273$	−0.293231	−0.0177472
$274$	0	0
$275$	−10.5348	−0.635272
$276$	0	0
$277$	9.61164	0.577508	0.288754	−	0.957403i	$-0.406759\pi$
$277$	9.61164	0.577508	0.288754	+	0.957403i	$0.406759\pi$
$278$	0	0
$279$	9.74659	0.583513
$280$	0	0
$281$	30.1629	1.79936	0.899682	−	0.436545i	$-0.143798\pi$
$281$	30.1629	1.79936	0.899682	+	0.436545i	$0.143798\pi$
$282$	0	0
$283$	1.01338	0.0602390	0.0301195	−	0.999546i	$-0.490411\pi$
$283$	1.01338	0.0602390	0.0301195	+	0.999546i	$0.490411\pi$
$284$	0	0
$285$	5.19213	0.307555
$286$	0	0
$287$	−9.75119	−0.575594
$288$	0	0
$289$	−16.9966	−0.999798
$290$	0	0
$291$	−1.21148	−0.0710183
$292$	0	0
$293$	14.2404	0.831936	0.415968	−	0.909379i	$-0.363443\pi$
$293$	14.2404	0.831936	0.415968	+	0.909379i	$0.363443\pi$
$294$	0	0
$295$	−46.9030	−2.73080
$296$	0	0
$297$	1.73417	0.100627
$298$	0	0
$299$	2.80394	0.162156
$300$	0	0
$301$	−12.4521	−0.717729
$302$	0	0
$303$	−2.79623	−0.160639
$304$	0	0
$305$	−20.2490	−1.15945
$306$	0	0
$307$	3.73416	0.213120	0.106560	−	0.994306i	$-0.466016\pi$
$307$	3.73416	0.213120	0.106560	+	0.994306i	$0.466016\pi$
$308$	0	0
$309$	−2.24144	−0.127511
$310$	0	0
$311$	−0.962351	−0.0545699	−0.0272849	−	0.999628i	$-0.508686\pi$
$311$	−0.962351	−0.0545699	−0.0272849	+	0.999628i	$0.508686\pi$
$312$	0	0
$313$	25.1345	1.42068	0.710342	−	0.703856i	$-0.248540\pi$
$313$	25.1345	1.42068	0.710342	+	0.703856i	$0.248540\pi$
$314$	0	0
$315$	11.4854	0.647127
$316$	0	0
$317$	27.7071	1.55619	0.778093	−	0.628149i	$-0.216187\pi$
$317$	27.7071	1.55619	0.778093	+	0.628149i	$0.216187\pi$
$318$	0	0
$319$	7.01606	0.392824
$320$	0	0
$321$	2.10092	0.117262
$322$	0	0
$323$	−0.263163	−0.0146428
$324$	0	0
$325$	−10.5348	−0.584366
$326$	0	0
$327$	−5.50236	−0.304281
$328$	0	0
$329$	12.3205	0.679250
$330$	0	0
$331$	−25.1882	−1.38447	−0.692234	−	0.721673i	$-0.743374\pi$
$331$	−25.1882	−1.38447	−0.692234	+	0.721673i	$0.743374\pi$
$332$	0	0
$333$	27.6937	1.51761
$334$	0	0
$335$	−35.9638	−1.96491
$336$	0	0
$337$	−24.8490	−1.35361	−0.676805	−	0.736163i	$-0.736636\pi$
$337$	−24.8490	−1.35361	−0.676805	+	0.736163i	$0.736636\pi$
$338$	0	0
$339$	−4.08391	−0.221807
$340$	0	0
$341$	3.34473	0.181127
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	3.24065	0.174470
$346$	0	0
$347$	−25.4035	−1.36373	−0.681865	−	0.731478i	$-0.738831\pi$
$347$	−25.4035	−1.36373	−0.681865	+	0.731478i	$0.738831\pi$
$348$	0	0
$349$	19.9449	1.06762	0.533812	−	0.845603i	$-0.320759\pi$
$349$	19.9449	1.06762	0.533812	+	0.845603i	$0.320759\pi$
$350$	0	0
$351$	1.73417	0.0925634
$352$	0	0
$353$	9.29638	0.494796	0.247398	−	0.968914i	$-0.420424\pi$
$353$	9.29638	0.494796	0.247398	+	0.968914i	$0.420424\pi$
$354$	0	0
$355$	−53.2744	−2.82751
$356$	0	0
$357$	0.0171772	0.000909113 0
$358$	0	0
$359$	22.9714	1.21238	0.606191	−	0.795319i	$-0.292697\pi$
$359$	22.9714	1.21238	0.606191	+	0.795319i	$0.292697\pi$
$360$	0	0
$361$	1.18203	0.0622123
$362$	0	0
$363$	0.293231	0.0153907
$364$	0	0
$365$	4.62728	0.242203
$366$	0	0
$367$	−17.2817	−0.902098	−0.451049	−	0.892499i	$-0.648950\pi$
$367$	−17.2817	−0.902098	−0.451049	+	0.892499i	$0.648950\pi$
$368$	0	0
$369$	28.4151	1.47923
$370$	0	0
$371$	4.02741	0.209092
$372$	0	0
$373$	12.7227	0.658754	0.329377	−	0.944198i	$-0.393161\pi$
$373$	12.7227	0.658754	0.329377	+	0.944198i	$0.393161\pi$
$374$	0	0
$375$	−6.39683	−0.330331
$376$	0	0
$377$	7.01606	0.361345
$378$	0	0
$379$	12.9811	0.666794	0.333397	−	0.942786i	$-0.391805\pi$
$379$	12.9811	0.666794	0.333397	+	0.942786i	$0.391805\pi$
$380$	0	0
$381$	3.73701	0.191453
$382$	0	0
$383$	2.13971	0.109334	0.0546670	−	0.998505i	$-0.482590\pi$
$383$	2.13971	0.109334	0.0546670	+	0.998505i	$0.482590\pi$
$384$	0	0
$385$	3.94142	0.200873
$386$	0	0
$387$	36.2857	1.84451
$388$	0	0
$389$	−14.2044	−0.720193	−0.360096	−	0.932915i	$-0.617256\pi$
$389$	−14.2044	−0.720193	−0.360096	+	0.932915i	$0.617256\pi$
$390$	0	0
$391$	−0.164252	−0.00830657
$392$	0	0
$393$	−5.11386	−0.257960
$394$	0	0
$395$	40.7282	2.04926
$396$	0	0
$397$	−5.75387	−0.288778	−0.144389	−	0.989521i	$-0.546122\pi$
$397$	−5.75387	−0.288778	−0.144389	+	0.989521i	$0.546122\pi$
$398$	0	0
$399$	−1.31732	−0.0659487
$400$	0	0
$401$	11.1493	0.556769	0.278384	−	0.960470i	$-0.410201\pi$
$401$	11.1493	0.556769	0.278384	+	0.960470i	$0.410201\pi$
$402$	0	0
$403$	3.34473	0.166613
$404$	0	0
$405$	−32.4518	−1.61254
$406$	0	0
$407$	9.50363	0.471077
$408$	0	0
$409$	25.6577	1.26869	0.634345	−	0.773050i	$-0.281270\pi$
$409$	25.6577	1.26869	0.634345	+	0.773050i	$0.281270\pi$
$410$	0	0
$411$	3.64705	0.179896
$412$	0	0
$413$	11.9000	0.585562
$414$	0	0
$415$	31.9416	1.56795
$416$	0	0
$417$	0.707077	0.0346257
$418$	0	0
$419$	19.2479	0.940321	0.470160	−	0.882581i	$-0.344196\pi$
$419$	19.2479	0.940321	0.470160	+	0.882581i	$0.344196\pi$
$420$	0	0
$421$	−7.70725	−0.375628	−0.187814	−	0.982205i	$-0.560140\pi$
$421$	−7.70725	−0.375628	−0.187814	+	0.982205i	$0.560140\pi$
$422$	0	0
$423$	−35.9020	−1.74562
$424$	0	0
$425$	0.617118	0.0299346
$426$	0	0
$427$	5.13748	0.248620
$428$	0	0
$429$	0.293231	0.0141573
$430$	0	0
$431$	−15.0825	−0.726500	−0.363250	−	0.931692i	$-0.618333\pi$
$431$	−15.0825	−0.726500	−0.363250	+	0.931692i	$0.618333\pi$
$432$	0	0
$433$	30.4403	1.46287	0.731434	−	0.681912i	$-0.238851\pi$
$433$	30.4403	1.46287	0.731434	+	0.681912i	$0.238851\pi$
$434$	0	0
$435$	8.10879	0.388787
$436$	0	0
$437$	12.5965	0.602574
$438$	0	0
$439$	18.4164	0.878965	0.439482	−	0.898251i	$-0.355162\pi$
$439$	18.4164	0.878965	0.439482	+	0.898251i	$0.355162\pi$
$440$	0	0
$441$	−2.91402	−0.138763
$442$	0	0
$443$	−10.0778	−0.478810	−0.239405	−	0.970920i	$-0.576952\pi$
$443$	−10.0778	−0.478810	−0.239405	+	0.970920i	$0.576952\pi$
$444$	0	0
$445$	43.6666	2.07000
$446$	0	0
$447$	5.39892	0.255360
$448$	0	0
$449$	−22.2080	−1.04806	−0.524031	−	0.851699i	$-0.675572\pi$
$449$	−22.2080	−1.04806	−0.524031	+	0.851699i	$0.675572\pi$
$450$	0	0
$451$	9.75119	0.459165
$452$	0	0
$453$	−5.16724	−0.242778
$454$	0	0
$455$	3.94142	0.184777
$456$	0	0
$457$	6.03167	0.282150	0.141075	−	0.989999i	$-0.454944\pi$
$457$	6.03167	0.282150	0.141075	+	0.989999i	$0.454944\pi$
$458$	0	0
$459$	−0.101586	−0.00474164
$460$	0	0
$461$	36.3672	1.69379	0.846895	−	0.531760i	$-0.178469\pi$
$461$	36.3672	1.69379	0.846895	+	0.531760i	$0.178469\pi$
$462$	0	0
$463$	−7.93119	−0.368594	−0.184297	−	0.982871i	$-0.559001\pi$
$463$	−7.93119	−0.368594	−0.184297	+	0.982871i	$0.559001\pi$
$464$	0	0
$465$	3.86566	0.179266
$466$	0	0
$467$	−6.49546	−0.300574	−0.150287	−	0.988642i	$-0.548020\pi$
$467$	−6.49546	−0.300574	−0.150287	+	0.988642i	$0.548020\pi$
$468$	0	0
$469$	9.12457	0.421334
$470$	0	0
$471$	−3.94865	−0.181944
$472$	0	0
$473$	12.4521	0.572549
$474$	0	0
$475$	−47.3270	−2.17151
$476$	0	0
$477$	−11.7359	−0.537351
$478$	0	0
$479$	13.3721	0.610988	0.305494	−	0.952194i	$-0.401178\pi$
$479$	13.3721	0.610988	0.305494	+	0.952194i	$0.401178\pi$
$480$	0	0
$481$	9.50363	0.433328
$482$	0	0
$483$	−0.822202	−0.0374115
$484$	0	0
$485$	16.2839	0.739416
$486$	0	0
$487$	−11.7625	−0.533010	−0.266505	−	0.963834i	$-0.585869\pi$
$487$	−11.7625	−0.533010	−0.266505	+	0.963834i	$0.585869\pi$
$488$	0	0
$489$	−4.36314	−0.197308
$490$	0	0
$491$	−17.3624	−0.783554	−0.391777	−	0.920060i	$-0.628139\pi$
$491$	−17.3624	−0.783554	−0.391777	+	0.920060i	$0.628139\pi$
$492$	0	0
$493$	−0.410994	−0.0185102
$494$	0	0
$495$	−11.4854	−0.516229
$496$	0	0
$497$	13.5165	0.606299
$498$	0	0
$499$	26.8540	1.20215	0.601075	−	0.799193i	$-0.294739\pi$
$499$	26.8540	1.20215	0.601075	+	0.799193i	$0.294739\pi$
$500$	0	0
$501$	3.74648	0.167380
$502$	0	0
$503$	−11.4212	−0.509248	−0.254624	−	0.967040i	$-0.581952\pi$
$503$	−11.4212	−0.509248	−0.254624	+	0.967040i	$0.581952\pi$
$504$	0	0
$505$	37.5850	1.67251
$506$	0	0
$507$	0.293231	0.0130229
$508$	0	0
$509$	−23.6340	−1.04756	−0.523780	−	0.851854i	$-0.675478\pi$
$509$	−23.6340	−1.04756	−0.523780	+	0.851854i	$0.675478\pi$
$510$	0	0
$511$	−1.17401	−0.0519353
$512$	0	0
$513$	7.79068	0.343967
$514$	0	0
$515$	30.1279	1.32759
$516$	0	0
$517$	−12.3205	−0.541854
$518$	0	0
$519$	−2.79858	−0.122844
$520$	0	0
$521$	−10.1277	−0.443704	−0.221852	−	0.975080i	$-0.571210\pi$
$521$	−10.1277	−0.443704	−0.221852	+	0.975080i	$0.571210\pi$
$522$	0	0
$523$	−10.2101	−0.446456	−0.223228	−	0.974766i	$-0.571659\pi$
$523$	−10.2101	−0.446456	−0.223228	+	0.974766i	$0.571659\pi$
$524$	0	0
$525$	3.08913	0.134821
$526$	0	0
$527$	−0.195931	−0.00853488
$528$	0	0
$529$	−15.1379	−0.658171
$530$	0	0
$531$	−34.6768	−1.50485
$532$	0	0
$533$	9.75119	0.422371
$534$	0	0
$535$	−28.2392	−1.22089
$536$	0	0
$537$	6.63100	0.286149
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	13.5386	0.582069	0.291035	−	0.956713i	$-0.406001\pi$
$541$	13.5386	0.582069	0.291035	+	0.956713i	$0.406001\pi$
$542$	0	0
$543$	1.59010	0.0682377
$544$	0	0
$545$	73.9590	3.16806
$546$	0	0
$547$	−42.2493	−1.80645	−0.903225	−	0.429167i	$-0.858807\pi$
$547$	−42.2493	−1.80645	−0.903225	+	0.429167i	$0.858807\pi$
$548$	0	0
$549$	−14.9707	−0.638934
$550$	0	0
$551$	31.5192	1.34276
$552$	0	0
$553$	−10.3334	−0.439420
$554$	0	0
$555$	10.9838	0.466236
$556$	0	0
$557$	22.7713	0.964853	0.482426	−	0.875936i	$-0.339756\pi$
$557$	22.7713	0.964853	0.482426	+	0.875936i	$0.339756\pi$
$558$	0	0
$559$	12.4521	0.526669
$560$	0	0
$561$	−0.0171772	−0.000725222 0
$562$	0	0
$563$	16.9220	0.713179	0.356589	−	0.934261i	$-0.383940\pi$
$563$	16.9220	0.713179	0.356589	+	0.934261i	$0.383940\pi$
$564$	0	0
$565$	54.8932	2.30937
$566$	0	0
$567$	8.23353	0.345776
$568$	0	0
$569$	−18.0905	−0.758395	−0.379198	−	0.925316i	$-0.623800\pi$
$569$	−18.0905	−0.758395	−0.379198	+	0.925316i	$0.623800\pi$
$570$	0	0
$571$	7.68784	0.321726	0.160863	−	0.986977i	$-0.448572\pi$
$571$	7.68784	0.321726	0.160863	+	0.986977i	$0.448572\pi$
$572$	0	0
$573$	3.04882	0.127366
$574$	0	0
$575$	−29.5389	−1.23186
$576$	0	0
$577$	−22.5674	−0.939493	−0.469746	−	0.882801i	$-0.655655\pi$
$577$	−22.5674	−0.939493	−0.469746	+	0.882801i	$0.655655\pi$
$578$	0	0
$579$	−4.98077	−0.206994
$580$	0	0
$581$	−8.10408	−0.336214
$582$	0	0
$583$	−4.02741	−0.166798
$584$	0	0
$585$	−11.4854	−0.474861
$586$	0	0
$587$	16.1376	0.666072	0.333036	−	0.942914i	$-0.391927\pi$
$587$	16.1376	0.666072	0.333036	+	0.942914i	$0.391927\pi$
$588$	0	0
$589$	15.0260	0.619136
$590$	0	0
$591$	−5.26094	−0.216406
$592$	0	0
$593$	−17.9139	−0.735634	−0.367817	−	0.929898i	$-0.619895\pi$
$593$	−17.9139	−0.735634	−0.367817	+	0.929898i	$0.619895\pi$
$594$	0	0
$595$	−0.230884	−0.00946534
$596$	0	0
$597$	7.25071	0.296752
$598$	0	0
$599$	−33.1029	−1.35255	−0.676273	−	0.736651i	$-0.736406\pi$
$599$	−33.1029	−1.35255	−0.676273	+	0.736651i	$0.736406\pi$
$600$	0	0
$601$	−5.25939	−0.214535	−0.107268	−	0.994230i	$-0.534210\pi$
$601$	−5.25939	−0.214535	−0.107268	+	0.994230i	$0.534210\pi$
$602$	0	0
$603$	−26.5892	−1.08279
$604$	0	0
$605$	−3.94142	−0.160242
$606$	0	0
$607$	−14.0776	−0.571394	−0.285697	−	0.958320i	$-0.592225\pi$
$607$	−14.0776	−0.571394	−0.285697	+	0.958320i	$0.592225\pi$
$608$	0	0
$609$	−2.05733	−0.0833671
$610$	0	0
$611$	−12.3205	−0.498433
$612$	0	0
$613$	−10.2154	−0.412597	−0.206299	−	0.978489i	$-0.566142\pi$
$613$	−10.2154	−0.412597	−0.206299	+	0.978489i	$0.566142\pi$
$614$	0	0
$615$	11.2699	0.454447
$616$	0	0
$617$	18.2384	0.734251	0.367126	−	0.930171i	$-0.380342\pi$
$617$	18.2384	0.734251	0.367126	+	0.930171i	$0.380342\pi$
$618$	0	0
$619$	13.9994	0.562684	0.281342	−	0.959608i	$-0.409220\pi$
$619$	13.9994	0.562684	0.281342	+	0.959608i	$0.409220\pi$
$620$	0	0
$621$	4.86252	0.195126
$622$	0	0
$623$	−11.0789	−0.443867
$624$	0	0
$625$	33.3080	1.33232
$626$	0	0
$627$	1.31732	0.0526089
$628$	0	0
$629$	−0.556713	−0.0221976
$630$	0	0
$631$	14.6116	0.581679	0.290840	−	0.956772i	$-0.406065\pi$
$631$	14.6116	0.581679	0.290840	+	0.956772i	$0.406065\pi$
$632$	0	0
$633$	−4.25620	−0.169169
$634$	0	0
$635$	−50.2305	−1.99334
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−39.3874	−1.55814
$640$	0	0
$641$	−27.9244	−1.10295	−0.551474	−	0.834192i	$-0.685934\pi$
$641$	−27.9244	−1.10295	−0.551474	+	0.834192i	$0.685934\pi$
$642$	0	0
$643$	−2.57891	−0.101702	−0.0508512	−	0.998706i	$-0.516193\pi$
$643$	−2.57891	−0.101702	−0.0508512	+	0.998706i	$0.516193\pi$
$644$	0	0
$645$	14.3915	0.566666
$646$	0	0
$647$	−20.4984	−0.805875	−0.402938	−	0.915227i	$-0.632011\pi$
$647$	−20.4984	−0.805875	−0.402938	+	0.915227i	$0.632011\pi$
$648$	0	0
$649$	−11.9000	−0.467117
$650$	0	0
$651$	−0.980779	−0.0384398
$652$	0	0
$653$	19.2756	0.754312	0.377156	−	0.926150i	$-0.376902\pi$
$653$	19.2756	0.754312	0.377156	+	0.926150i	$0.376902\pi$
$654$	0	0
$655$	68.7372	2.68578
$656$	0	0
$657$	3.42109	0.133469
$658$	0	0
$659$	−9.95374	−0.387743	−0.193871	−	0.981027i	$-0.562104\pi$
$659$	−9.95374	−0.387743	−0.193871	+	0.981027i	$0.562104\pi$
$660$	0	0
$661$	−20.3055	−0.789792	−0.394896	−	0.918726i	$-0.629219\pi$
$661$	−20.3055	−0.789792	−0.394896	+	0.918726i	$0.629219\pi$
$662$	0	0
$663$	−0.0171772	−0.000667107 0
$664$	0	0
$665$	17.7066	0.686633
$666$	0	0
$667$	19.6726	0.761726
$668$	0	0
$669$	7.73342	0.298991
$670$	0	0
$671$	−5.13748	−0.198330
$672$	0	0
$673$	−34.9556	−1.34744	−0.673720	−	0.738987i	$-0.735305\pi$
$673$	−34.9556	−1.34744	−0.673720	+	0.738987i	$0.735305\pi$
$674$	0	0
$675$	−18.2692	−0.703181
$676$	0	0
$677$	−51.1105	−1.96434	−0.982169	−	0.188003i	$-0.939799\pi$
$677$	−51.1105	−1.96434	−0.982169	+	0.188003i	$0.939799\pi$
$678$	0	0
$679$	−4.13149	−0.158552
$680$	0	0
$681$	3.24047	0.124175
$682$	0	0
$683$	−19.9751	−0.764326	−0.382163	−	0.924095i	$-0.624821\pi$
$683$	−19.9751	−0.764326	−0.382163	+	0.924095i	$0.624821\pi$
$684$	0	0
$685$	−49.0213	−1.87301
$686$	0	0
$687$	4.97888	0.189956
$688$	0	0
$689$	−4.02741	−0.153432
$690$	0	0
$691$	19.7798	0.752460	0.376230	−	0.926526i	$-0.377220\pi$
$691$	19.7798	0.752460	0.376230	+	0.926526i	$0.377220\pi$
$692$	0	0
$693$	2.91402	0.110694
$694$	0	0
$695$	−9.50407	−0.360510
$696$	0	0
$697$	−0.571215	−0.0216363
$698$	0	0
$699$	3.18538	0.120482
$700$	0	0
$701$	32.5801	1.23054	0.615268	−	0.788318i	$-0.289048\pi$
$701$	32.5801	1.23054	0.615268	+	0.788318i	$0.289048\pi$
$702$	0	0
$703$	42.6945	1.61025
$704$	0	0
$705$	−14.2394	−0.536285
$706$	0	0
$707$	−9.53590	−0.358635
$708$	0	0
$709$	−1.03903	−0.0390216	−0.0195108	−	0.999810i	$-0.506211\pi$
$709$	−1.03903	−0.0390216	−0.0195108	+	0.999810i	$0.506211\pi$
$710$	0	0
$711$	30.1116	1.12927
$712$	0	0
$713$	9.37841	0.351224
$714$	0	0
$715$	−3.94142	−0.147401
$716$	0	0
$717$	−0.872629	−0.0325889
$718$	0	0
$719$	−47.5876	−1.77472	−0.887359	−	0.461080i	$-0.847462\pi$
$719$	−47.5876	−1.77472	−0.887359	+	0.461080i	$0.847462\pi$
$720$	0	0
$721$	−7.64392	−0.284675
$722$	0	0
$723$	−2.52313	−0.0938363
$724$	0	0
$725$	−73.9128	−2.74505
$726$	0	0
$727$	29.3405	1.08818	0.544089	−	0.839027i	$-0.316875\pi$
$727$	29.3405	1.08818	0.544089	+	0.839027i	$0.316875\pi$
$728$	0	0
$729$	−22.4671	−0.832115
$730$	0	0
$731$	−0.729433	−0.0269791
$732$	0	0
$733$	−33.1850	−1.22572	−0.612859	−	0.790193i	$-0.709980\pi$
$733$	−33.1850	−1.22572	−0.612859	+	0.790193i	$0.709980\pi$
$734$	0	0
$735$	−1.15575	−0.0426304
$736$	0	0
$737$	−9.12457	−0.336108
$738$	0	0
$739$	−36.9375	−1.35877	−0.679384	−	0.733783i	$-0.737753\pi$
$739$	−36.9375	−1.35877	−0.679384	+	0.733783i	$0.737753\pi$
$740$	0	0
$741$	1.31732	0.0483931
$742$	0	0
$743$	−11.6592	−0.427736	−0.213868	−	0.976863i	$-0.568606\pi$
$743$	−11.6592	−0.427736	−0.213868	+	0.976863i	$0.568606\pi$
$744$	0	0
$745$	−72.5687	−2.65871
$746$	0	0
$747$	23.6154	0.864043
$748$	0	0
$749$	7.16472	0.261793
$750$	0	0
$751$	−31.6295	−1.15418	−0.577088	−	0.816682i	$-0.695811\pi$
$751$	−31.6295	−1.15418	−0.577088	+	0.816682i	$0.695811\pi$
$752$	0	0
$753$	1.87751	0.0684203
$754$	0	0
$755$	69.4546	2.52771
$756$	0	0
$757$	32.3590	1.17611	0.588055	−	0.808821i	$-0.299894\pi$
$757$	32.3590	1.17611	0.588055	+	0.808821i	$0.299894\pi$
$758$	0	0
$759$	0.822202	0.0298441
$760$	0	0
$761$	23.5517	0.853747	0.426874	−	0.904311i	$-0.359615\pi$
$761$	23.5517	0.853747	0.426874	+	0.904311i	$0.359615\pi$
$762$	0	0
$763$	−18.7646	−0.679323
$764$	0	0
$765$	0.672801	0.0243252
$766$	0	0
$767$	−11.9000	−0.429685
$768$	0	0
$769$	21.2055	0.764691	0.382346	−	0.924019i	$-0.375116\pi$
$769$	21.2055	0.764691	0.382346	+	0.924019i	$0.375116\pi$
$770$	0	0
$771$	3.81814	0.137507
$772$	0	0
$773$	43.9246	1.57986	0.789929	−	0.613198i	$-0.210117\pi$
$773$	43.9246	1.57986	0.789929	+	0.613198i	$0.210117\pi$
$774$	0	0
$775$	−35.2361	−1.26572
$776$	0	0
$777$	−2.78676	−0.0999745
$778$	0	0
$779$	43.8066	1.56954
$780$	0	0
$781$	−13.5165	−0.483660
$782$	0	0
$783$	12.1671	0.434815
$784$	0	0
$785$	53.0751	1.89433
$786$	0	0
$787$	−38.2548	−1.36364	−0.681819	−	0.731521i	$-0.738811\pi$
$787$	−38.2548	−1.36364	−0.681819	+	0.731521i	$0.738811\pi$
$788$	0	0
$789$	−0.884126	−0.0314757
$790$	0	0
$791$	−13.9273	−0.495196
$792$	0	0
$793$	−5.13748	−0.182437
$794$	0	0
$795$	−4.65467	−0.165084
$796$	0	0
$797$	−54.6008	−1.93406	−0.967030	−	0.254663i	$-0.918035\pi$
$797$	−54.6008	−1.93406	−0.967030	+	0.254663i	$0.918035\pi$
$798$	0	0
$799$	0.721721	0.0255326
$800$	0	0
$801$	32.2841	1.14070
$802$	0	0
$803$	1.17401	0.0414300
$804$	0	0
$805$	11.0515	0.389514
$806$	0	0
$807$	1.66426	0.0585848
$808$	0	0
$809$	20.3682	0.716109	0.358055	−	0.933701i	$-0.383440\pi$
$809$	20.3682	0.716109	0.358055	+	0.933701i	$0.383440\pi$
$810$	0	0
$811$	−45.3558	−1.59266	−0.796328	−	0.604864i	$-0.793227\pi$
$811$	−45.3558	−1.59266	−0.796328	+	0.604864i	$0.793227\pi$
$812$	0	0
$813$	4.09623	0.143661
$814$	0	0
$815$	58.6465	2.05430
$816$	0	0
$817$	55.9405	1.95711
$818$	0	0
$819$	2.91402	0.101824
$820$	0	0
$821$	26.1381	0.912226	0.456113	−	0.889922i	$-0.349241\pi$
$821$	26.1381	0.912226	0.456113	+	0.889922i	$0.349241\pi$
$822$	0	0
$823$	34.2998	1.19562	0.597809	−	0.801639i	$-0.296038\pi$
$823$	34.2998	1.19562	0.597809	+	0.801639i	$0.296038\pi$
$824$	0	0
$825$	−3.08913	−0.107550
$826$	0	0
$827$	−26.7159	−0.929004	−0.464502	−	0.885572i	$-0.653767\pi$
$827$	−26.7159	−0.929004	−0.464502	+	0.885572i	$0.653767\pi$
$828$	0	0
$829$	53.7968	1.86844	0.934221	−	0.356695i	$-0.116096\pi$
$829$	53.7968	1.86844	0.934221	+	0.356695i	$0.116096\pi$
$830$	0	0
$831$	2.81843	0.0977704
$832$	0	0
$833$	0.0585790	0.00202964
$834$	0	0
$835$	−50.3577	−1.74270
$836$	0	0
$837$	5.80034	0.200489
$838$	0	0
$839$	49.8835	1.72217	0.861084	−	0.508463i	$-0.169786\pi$
$839$	49.8835	1.72217	0.861084	+	0.508463i	$0.169786\pi$
$840$	0	0
$841$	20.2251	0.697416
$842$	0	0
$843$	8.84469	0.304627
$844$	0	0
$845$	−3.94142	−0.135589
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	0.297154	0.0101983
$850$	0	0
$851$	26.6476	0.913467
$852$	0	0
$853$	−50.5409	−1.73049	−0.865244	−	0.501351i	$-0.832837\pi$
$853$	−50.5409	−1.73049	−0.865244	+	0.501351i	$0.832837\pi$
$854$	0	0
$855$	−51.5973	−1.76459
$856$	0	0
$857$	37.2721	1.27319	0.636595	−	0.771198i	$-0.280342\pi$
$857$	37.2721	1.27319	0.636595	+	0.771198i	$0.280342\pi$
$858$	0	0
$859$	−5.93782	−0.202596	−0.101298	−	0.994856i	$-0.532300\pi$
$859$	−5.93782	−0.202596	−0.101298	+	0.994856i	$0.532300\pi$
$860$	0	0
$861$	−2.85935	−0.0974465
$862$	0	0
$863$	−1.88803	−0.0642694	−0.0321347	−	0.999484i	$-0.510231\pi$
$863$	−1.88803	−0.0642694	−0.0321347	+	0.999484i	$0.510231\pi$
$864$	0	0
$865$	37.6167	1.27901
$866$	0	0
$867$	−4.98393	−0.169263
$868$	0	0
$869$	10.3334	0.350536
$870$	0	0
$871$	−9.12457	−0.309174
$872$	0	0
$873$	12.0392	0.407466
$874$	0	0
$875$	−21.8150	−0.737481
$876$	0	0
$877$	−1.24217	−0.0419452	−0.0209726	−	0.999780i	$-0.506676\pi$
$877$	−1.24217	−0.0419452	−0.0209726	+	0.999780i	$0.506676\pi$
$878$	0	0
$879$	4.17574	0.140844
$880$	0	0
$881$	−39.2205	−1.32137	−0.660686	−	0.750663i	$-0.729734\pi$
$881$	−39.2205	−1.32137	−0.660686	+	0.750663i	$0.729734\pi$
$882$	0	0
$883$	−33.6025	−1.13081	−0.565407	−	0.824812i	$-0.691281\pi$
$883$	−33.6025	−1.13081	−0.565407	+	0.824812i	$0.691281\pi$
$884$	0	0
$885$	−13.7534	−0.462316
$886$	0	0
$887$	10.7323	0.360355	0.180177	−	0.983634i	$-0.442333\pi$
$887$	10.7323	0.360355	0.180177	+	0.983634i	$0.442333\pi$
$888$	0	0
$889$	12.7443	0.427429
$890$	0	0
$891$	−8.23353	−0.275834
$892$	0	0
$893$	−55.3490	−1.85218
$894$	0	0
$895$	−89.1295	−2.97927
$896$	0	0
$897$	0.822202	0.0274525
$898$	0	0
$899$	23.4668	0.782662
$900$	0	0
$901$	0.235921	0.00785968
$902$	0	0
$903$	−3.65135	−0.121509
$904$	0	0
$905$	−21.3731	−0.710465
$906$	0	0
$907$	29.9491	0.994444	0.497222	−	0.867623i	$-0.334353\pi$
$907$	29.9491	0.994444	0.497222	+	0.867623i	$0.334353\pi$
$908$	0	0
$909$	27.7878	0.921662
$910$	0	0
$911$	−3.20882	−0.106313	−0.0531565	−	0.998586i	$-0.516928\pi$
$911$	−3.20882	−0.106313	−0.0531565	+	0.998586i	$0.516928\pi$
$912$	0	0
$913$	8.10408	0.268206
$914$	0	0
$915$	−5.93764	−0.196292
$916$	0	0
$917$	−17.4397	−0.575909
$918$	0	0
$919$	0.777809	0.0256576	0.0128288	−	0.999918i	$-0.495916\pi$
$919$	0.777809	0.0256576	0.0128288	+	0.999918i	$0.495916\pi$
$920$	0	0
$921$	1.09497	0.0360805
$922$	0	0
$923$	−13.5165	−0.444902
$924$	0	0
$925$	−100.119	−3.29189
$926$	0	0
$927$	22.2745	0.731591
$928$	0	0
$929$	8.12078	0.266434	0.133217	−	0.991087i	$-0.457469\pi$
$929$	8.12078	0.266434	0.133217	+	0.991087i	$0.457469\pi$
$930$	0	0
$931$	−4.49244	−0.147234
$932$	0	0
$933$	−0.282191	−0.00923853
$934$	0	0
$935$	0.230884	0.00755073
$936$	0	0
$937$	13.5167	0.441571	0.220786	−	0.975322i	$-0.429138\pi$
$937$	13.5167	0.441571	0.220786	+	0.975322i	$0.429138\pi$
$938$	0	0
$939$	7.37021	0.240518
$940$	0	0
$941$	53.5372	1.74526	0.872631	−	0.488380i	$-0.162412\pi$
$941$	53.5372	1.74526	0.872631	+	0.488380i	$0.162412\pi$
$942$	0	0
$943$	27.3417	0.890369
$944$	0	0
$945$	6.83511	0.222346
$946$	0	0
$947$	−46.0746	−1.49722	−0.748612	−	0.663008i	$-0.769279\pi$
$947$	−46.0746	−1.49722	−0.748612	+	0.663008i	$0.769279\pi$
$948$	0	0
$949$	1.17401	0.0381101
$950$	0	0
$951$	8.12459	0.263458
$952$	0	0
$953$	−16.9129	−0.547863	−0.273932	−	0.961749i	$-0.588324\pi$
$953$	−16.9129	−0.547863	−0.273932	+	0.961749i	$0.588324\pi$
$954$	0	0
$955$	−40.9803	−1.32609
$956$	0	0
$957$	2.05733	0.0665039
$958$	0	0
$959$	12.4375	0.401627
$960$	0	0
$961$	−19.8128	−0.639122
$962$	0	0
$963$	−20.8781	−0.672788
$964$	0	0
$965$	66.9483	2.15514
$966$	0	0
$967$	−3.07293	−0.0988187	−0.0494093	−	0.998779i	$-0.515734\pi$
$967$	−3.07293	−0.0988187	−0.0494093	+	0.998779i	$0.515734\pi$
$968$	0	0
$969$	−0.0771675	−0.00247898
$970$	0	0
$971$	48.1226	1.54433	0.772165	−	0.635423i	$-0.219174\pi$
$971$	48.1226	1.54433	0.772165	+	0.635423i	$0.219174\pi$
$972$	0	0
$973$	2.41133	0.0773037
$974$	0	0
$975$	−3.08913	−0.0989314
$976$	0	0
$977$	−5.07874	−0.162483	−0.0812416	−	0.996694i	$-0.525889\pi$
$977$	−5.07874	−0.162483	−0.0812416	+	0.996694i	$0.525889\pi$
$978$	0	0
$979$	11.0789	0.354083
$980$	0	0
$981$	54.6802	1.74581
$982$	0	0
$983$	−7.44613	−0.237495	−0.118747	−	0.992924i	$-0.537888\pi$
$983$	−7.44613	−0.237495	−0.118747	+	0.992924i	$0.537888\pi$
$984$	0	0
$985$	70.7140	2.25314
$986$	0	0
$987$	3.61275	0.114995
$988$	0	0
$989$	34.9150	1.11023
$990$	0	0
$991$	11.1893	0.355439	0.177719	−	0.984081i	$-0.443128\pi$
$991$	11.1893	0.355439	0.177719	+	0.984081i	$0.443128\pi$
$992$	0	0
$993$	−7.38597	−0.234387
$994$	0	0
$995$	−97.4592	−3.08967
$996$	0	0
$997$	−27.4077	−0.868012	−0.434006	−	0.900910i	$-0.642900\pi$
$997$	−27.4077	−0.868012	−0.434006	+	0.900910i	$0.642900\pi$
$998$	0	0
$999$	16.4809	0.521434

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.j.1.3	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.j.1.3	✓	5	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q+0.293231 q^{3} -3.94142 q^{5} +1.00000 q^{7} -2.91402 q^{9} +O(q^{10})\)	\(q+0.293231 q^{3} -3.94142 q^{5} +1.00000 q^{7} -2.91402 q^{9} -1.00000 q^{11} -1.00000 q^{13} -1.15575 q^{15} +0.0585790 q^{17} -4.49244 q^{19} +0.293231 q^{21} -2.80394 q^{23} +10.5348 q^{25} -1.73417 q^{27} -7.01606 q^{29} -3.34473 q^{31} -0.293231 q^{33} -3.94142 q^{35} -9.50363 q^{37} -0.293231 q^{39} -9.75119 q^{41} -12.4521 q^{43} +11.4854 q^{45} +12.3205 q^{47} +1.00000 q^{49} +0.0171772 q^{51} +4.02741 q^{53} +3.94142 q^{55} -1.31732 q^{57} +11.9000 q^{59} +5.13748 q^{61} -2.91402 q^{63} +3.94142 q^{65} +9.12457 q^{67} -0.822202 q^{69} +13.5165 q^{71} -1.17401 q^{73} +3.08913 q^{75} -1.00000 q^{77} -10.3334 q^{79} +8.23353 q^{81} -8.10408 q^{83} -0.230884 q^{85} -2.05733 q^{87} -11.0789 q^{89} -1.00000 q^{91} -0.980779 q^{93} +17.7066 q^{95} -4.13149 q^{97} +2.91402 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{3} - q^{5} + 5 q^{7} + 6 q^{9}+O(q^{10}) \) 5 * q + q^3 - q^5 + 5 * q^7 + 6 * q^9	\( 5 q + q^{3} - q^{5} + 5 q^{7} + 6 q^{9} - 5 q^{11} - 5 q^{13} - 5 q^{15} + 19 q^{17} - 5 q^{19} + q^{21} + 5 q^{23} + 12 q^{25} - 11 q^{27} - 17 q^{29} + 4 q^{31} - q^{33} - q^{35} + 10 q^{37} - q^{39} + 10 q^{41} - 25 q^{43} + q^{45} + 3 q^{47} + 5 q^{49} - q^{51} + 22 q^{53} + q^{55} + 16 q^{57} + 21 q^{59} + 26 q^{61} + 6 q^{63} + q^{65} + 28 q^{67} + 16 q^{69} + 28 q^{71} - 4 q^{73} - 5 q^{77} - 11 q^{79} + 5 q^{81} + 33 q^{85} + 31 q^{87} - 37 q^{89} - 5 q^{91} - 49 q^{93} + 29 q^{95} + 18 q^{97} - 6 q^{99}+O(q^{100}) \) 5 * q + q^3 - q^5 + 5 * q^7 + 6 * q^9 - 5 * q^11 - 5 * q^13 - 5 * q^15 + 19 * q^17 - 5 * q^19 + q^21 + 5 * q^23 + 12 * q^25 - 11 * q^27 - 17 * q^29 + 4 * q^31 - q^33 - q^35 + 10 * q^37 - q^39 + 10 * q^41 - 25 * q^43 + q^45 + 3 * q^47 + 5 * q^49 - q^51 + 22 * q^53 + q^55 + 16 * q^57 + 21 * q^59 + 26 * q^61 + 6 * q^63 + q^65 + 28 * q^67 + 16 * q^69 + 28 * q^71 - 4 * q^73 - 5 * q^77 - 11 * q^79 + 5 * q^81 + 33 * q^85 + 31 * q^87 - 37 * q^89 - 5 * q^91 - 49 * q^93 + 29 * q^95 + 18 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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