LMFDB - Embedded newform 8008.2.a.l.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:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 9x^{6} + 12x^{5} + 24x^{4} - 10x^{3} - 18x^{2} - 2x + 1 $ x^8 - 2x^7 - 9x^6 + 12x^5 + 24x^4 - 10x^3 - 18x^2 - 2*x + 1
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			1.3
Root			$-0.723237$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.02769 q^{3} +2.88924 q^{5} +1.00000 q^{7} +1.11151 q^{9} +O(q^{10})$	$q-2.02769 q^{3} +2.88924 q^{5} +1.00000 q^{7} +1.11151 q^{9} +1.00000 q^{11} +1.00000 q^{13} -5.85846 q^{15} +4.54289 q^{17} +4.10090 q^{19} -2.02769 q^{21} -7.64232 q^{23} +3.34769 q^{25} +3.82927 q^{27} -6.29319 q^{29} -9.32652 q^{31} -2.02769 q^{33} +2.88924 q^{35} -3.59460 q^{37} -2.02769 q^{39} +2.53997 q^{41} -9.96767 q^{43} +3.21141 q^{45} -0.779554 q^{47} +1.00000 q^{49} -9.21156 q^{51} -13.7527 q^{53} +2.88924 q^{55} -8.31533 q^{57} +0.314057 q^{59} +5.24601 q^{61} +1.11151 q^{63} +2.88924 q^{65} -14.2025 q^{67} +15.4962 q^{69} -2.27460 q^{71} +7.19802 q^{73} -6.78806 q^{75} +1.00000 q^{77} -2.06191 q^{79} -11.0991 q^{81} -0.240244 q^{83} +13.1255 q^{85} +12.7606 q^{87} -10.2922 q^{89} +1.00000 q^{91} +18.9113 q^{93} +11.8485 q^{95} -5.32147 q^{97} +1.11151 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 5 q^{3} - 7 q^{5} + 8 q^{7} + 7 q^{9}+O(q^{10}) $ 8 * q - 5 * q^3 - 7 * q^5 + 8 * q^7 + 7 * q^9	$ 8 q - 5 q^{3} - 7 q^{5} + 8 q^{7} + 7 q^{9} + 8 q^{11} + 8 q^{13} - 5 q^{15} + 3 q^{17} - 19 q^{19} - 5 q^{21} + q^{23} + 15 q^{25} - 11 q^{27} - 21 q^{29} + 2 q^{31} - 5 q^{33} - 7 q^{35} - 12 q^{37} - 5 q^{39} - 6 q^{41} - 19 q^{43} - 17 q^{45} - 7 q^{47} + 8 q^{49} - 19 q^{51} - 26 q^{53} - 7 q^{55} + 2 q^{57} - 17 q^{59} + 7 q^{63} - 7 q^{65} - 24 q^{67} + 20 q^{69} + 2 q^{71} + 2 q^{73} + 18 q^{75} + 8 q^{77} + 7 q^{79} - 4 q^{81} - 6 q^{83} + 19 q^{85} - 13 q^{87} + 13 q^{89} + 8 q^{91} + 13 q^{93} + 21 q^{95} + 18 q^{97} + 7 q^{99}+O(q^{100}) $ 8 * q - 5 * q^3 - 7 * q^5 + 8 * q^7 + 7 * q^9 + 8 * q^11 + 8 * q^13 - 5 * q^15 + 3 * q^17 - 19 * q^19 - 5 * q^21 + q^23 + 15 * q^25 - 11 * q^27 - 21 * q^29 + 2 * q^31 - 5 * q^33 - 7 * q^35 - 12 * q^37 - 5 * q^39 - 6 * q^41 - 19 * q^43 - 17 * q^45 - 7 * q^47 + 8 * q^49 - 19 * q^51 - 26 * q^53 - 7 * q^55 + 2 * q^57 - 17 * q^59 + 7 * q^63 - 7 * q^65 - 24 * q^67 + 20 * q^69 + 2 * q^71 + 2 * q^73 + 18 * q^75 + 8 * q^77 + 7 * q^79 - 4 * q^81 - 6 * q^83 + 19 * q^85 - 13 * q^87 + 13 * q^89 + 8 * q^91 + 13 * q^93 + 21 * q^95 + 18 * q^97 + 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.02769	−1.17068	−0.585342	−	0.810786i	$-0.699040\pi$
$3$	−2.02769	−1.17068	−0.585342	+	0.810786i	$0.699040\pi$
$4$	0	0
$5$	2.88924	1.29211	0.646053	−	0.763293i	$-0.276419\pi$
$5$	2.88924	1.29211	0.646053	+	0.763293i	$0.276419\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.11151	0.370503
$10$	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
$15$	−5.85846	−1.51265
$16$	0	0
$17$	4.54289	1.10181	0.550907	−	0.834567i	$-0.314282\pi$
$17$	4.54289	1.10181	0.550907	+	0.834567i	$0.314282\pi$
$18$	0	0
$19$	4.10090	0.940811	0.470405	−	0.882450i	$-0.344108\pi$
$19$	4.10090	0.940811	0.470405	+	0.882450i	$0.344108\pi$
$20$	0	0
$21$	−2.02769	−0.442477
$22$	0	0
$23$	−7.64232	−1.59353	−0.796767	−	0.604286i	$-0.793458\pi$
$23$	−7.64232	−1.59353	−0.796767	+	0.604286i	$0.793458\pi$
$24$	0	0
$25$	3.34769	0.669538
$26$	0	0
$27$	3.82927	0.736943
$28$	0	0
$29$	−6.29319	−1.16862	−0.584308	−	0.811532i	$-0.698634\pi$
$29$	−6.29319	−1.16862	−0.584308	+	0.811532i	$0.698634\pi$
$30$	0	0
$31$	−9.32652	−1.67509	−0.837547	−	0.546366i	$-0.816011\pi$
$31$	−9.32652	−1.67509	−0.837547	+	0.546366i	$0.816011\pi$
$32$	0	0
$33$	−2.02769	−0.352975
$34$	0	0
$35$	2.88924	0.488370
$36$	0	0
$37$	−3.59460	−0.590948	−0.295474	−	0.955351i	$-0.595478\pi$
$37$	−3.59460	−0.590948	−0.295474	+	0.955351i	$0.595478\pi$
$38$	0	0
$39$	−2.02769	−0.324690
$40$	0	0
$41$	2.53997	0.396677	0.198338	−	0.980134i	$-0.436446\pi$
$41$	2.53997	0.396677	0.198338	+	0.980134i	$0.436446\pi$
$42$	0	0
$43$	−9.96767	−1.52006	−0.760028	−	0.649890i	$-0.774815\pi$
$43$	−9.96767	−1.52006	−0.760028	+	0.649890i	$0.774815\pi$
$44$	0	0
$45$	3.21141	0.478729
$46$	0	0
$47$	−0.779554	−0.113710	−0.0568548	−	0.998382i	$-0.518107\pi$
$47$	−0.779554	−0.113710	−0.0568548	+	0.998382i	$0.518107\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−9.21156	−1.28988
$52$	0	0
$53$	−13.7527	−1.88908	−0.944541	−	0.328392i	$-0.893493\pi$
$53$	−13.7527	−1.88908	−0.944541	+	0.328392i	$0.893493\pi$
$54$	0	0
$55$	2.88924	0.389585
$56$	0	0
$57$	−8.31533	−1.10139
$58$	0	0
$59$	0.314057	0.0408867	0.0204433	−	0.999791i	$-0.493492\pi$
$59$	0.314057	0.0408867	0.0204433	+	0.999791i	$0.493492\pi$
$60$	0	0
$61$	5.24601	0.671683	0.335842	−	0.941918i	$-0.390979\pi$
$61$	5.24601	0.671683	0.335842	+	0.941918i	$0.390979\pi$
$62$	0	0
$63$	1.11151	0.140037
$64$	0	0
$65$	2.88924	0.358366
$66$	0	0
$67$	−14.2025	−1.73511	−0.867553	−	0.497344i	$-0.834309\pi$
$67$	−14.2025	−1.73511	−0.867553	+	0.497344i	$0.834309\pi$
$68$	0	0
$69$	15.4962	1.86553
$70$	0	0
$71$	−2.27460	−0.269945	−0.134973	−	0.990849i	$-0.543095\pi$
$71$	−2.27460	−0.269945	−0.134973	+	0.990849i	$0.543095\pi$
$72$	0	0
$73$	7.19802	0.842464	0.421232	−	0.906953i	$-0.361598\pi$
$73$	7.19802	0.842464	0.421232	+	0.906953i	$0.361598\pi$
$74$	0	0
$75$	−6.78806	−0.783818
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−2.06191	−0.231983	−0.115992	−	0.993250i	$-0.537005\pi$
$79$	−2.06191	−0.231983	−0.115992	+	0.993250i	$0.537005\pi$
$80$	0	0
$81$	−11.0991	−1.23323
$82$	0	0
$83$	−0.240244	−0.0263702	−0.0131851	−	0.999913i	$-0.504197\pi$
$83$	−0.240244	−0.0263702	−0.0131851	+	0.999913i	$0.504197\pi$
$84$	0	0
$85$	13.1255	1.42366
$86$	0	0
$87$	12.7606	1.36808
$88$	0	0
$89$	−10.2922	−1.09097	−0.545484	−	0.838122i	$-0.683654\pi$
$89$	−10.2922	−1.09097	−0.545484	+	0.838122i	$0.683654\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	18.9113	1.96101
$94$	0	0
$95$	11.8485	1.21563
$96$	0	0
$97$	−5.32147	−0.540313	−0.270157	−	0.962816i	$-0.587075\pi$
$97$	−5.32147	−0.540313	−0.270157	+	0.962816i	$0.587075\pi$
$98$	0	0
$99$	1.11151	0.111711
$100$	0	0
$101$	10.3417	1.02903	0.514517	−	0.857480i	$-0.327971\pi$
$101$	10.3417	1.02903	0.514517	+	0.857480i	$0.327971\pi$
$102$	0	0
$103$	−6.82941	−0.672922	−0.336461	−	0.941697i	$-0.609230\pi$
$103$	−6.82941	−0.672922	−0.336461	+	0.941697i	$0.609230\pi$
$104$	0	0
$105$	−5.85846	−0.571728
$106$	0	0
$107$	−2.01195	−0.194503	−0.0972514	−	0.995260i	$-0.531005\pi$
$107$	−2.01195	−0.194503	−0.0972514	+	0.995260i	$0.531005\pi$
$108$	0	0
$109$	14.2438	1.36430	0.682152	−	0.731210i	$-0.261044\pi$
$109$	14.2438	1.36430	0.682152	+	0.731210i	$0.261044\pi$
$110$	0	0
$111$	7.28871	0.691814
$112$	0	0
$113$	−11.0445	−1.03898	−0.519491	−	0.854476i	$-0.673878\pi$
$113$	−11.0445	−1.03898	−0.519491	+	0.854476i	$0.673878\pi$
$114$	0	0
$115$	−22.0805	−2.05902
$116$	0	0
$117$	1.11151	0.102759
$118$	0	0
$119$	4.54289	0.416446
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−5.15026	−0.464384
$124$	0	0
$125$	−4.77392	−0.426992
$126$	0	0
$127$	−13.3162	−1.18162	−0.590810	−	0.806811i	$-0.701192\pi$
$127$	−13.3162	−1.18162	−0.590810	+	0.806811i	$0.701192\pi$
$128$	0	0
$129$	20.2113	1.77951
$130$	0	0
$131$	−13.6557	−1.19311	−0.596553	−	0.802573i	$-0.703464\pi$
$131$	−13.6557	−1.19311	−0.596553	+	0.802573i	$0.703464\pi$
$132$	0	0
$133$	4.10090	0.355593
$134$	0	0
$135$	11.0637	0.952208
$136$	0	0
$137$	−9.38856	−0.802119	−0.401059	−	0.916052i	$-0.631358\pi$
$137$	−9.38856	−0.802119	−0.401059	+	0.916052i	$0.631358\pi$
$138$	0	0
$139$	−6.27509	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$139$	−6.27509	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$140$	0	0
$141$	1.58069	0.133118
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−18.1825	−1.50997
$146$	0	0
$147$	−2.02769	−0.167241
$148$	0	0
$149$	12.0448	0.986745	0.493373	−	0.869818i	$-0.335764\pi$
$149$	12.0448	0.986745	0.493373	+	0.869818i	$0.335764\pi$
$150$	0	0
$151$	−15.7938	−1.28529	−0.642643	−	0.766166i	$-0.722162\pi$
$151$	−15.7938	−1.28529	−0.642643	+	0.766166i	$0.722162\pi$
$152$	0	0
$153$	5.04947	0.408225
$154$	0	0
$155$	−26.9465	−2.16440
$156$	0	0
$157$	11.3698	0.907410	0.453705	−	0.891152i	$-0.350102\pi$
$157$	11.3698	0.907410	0.453705	+	0.891152i	$0.350102\pi$
$158$	0	0
$159$	27.8862	2.21152
$160$	0	0
$161$	−7.64232	−0.602300
$162$	0	0
$163$	20.1588	1.57896	0.789478	−	0.613779i	$-0.210352\pi$
$163$	20.1588	1.57896	0.789478	+	0.613779i	$0.210352\pi$
$164$	0	0
$165$	−5.85846	−0.456081
$166$	0	0
$167$	1.80750	0.139868	0.0699342	−	0.997552i	$-0.477721\pi$
$167$	1.80750	0.139868	0.0699342	+	0.997552i	$0.477721\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	4.55819	0.348573
$172$	0	0
$173$	11.5091	0.875019	0.437509	−	0.899214i	$-0.355861\pi$
$173$	11.5091	0.875019	0.437509	+	0.899214i	$0.355861\pi$
$174$	0	0
$175$	3.34769	0.253061
$176$	0	0
$177$	−0.636808	−0.0478654
$178$	0	0
$179$	4.51536	0.337493	0.168747	−	0.985659i	$-0.446028\pi$
$179$	4.51536	0.337493	0.168747	+	0.985659i	$0.446028\pi$
$180$	0	0
$181$	−11.1624	−0.829695	−0.414848	−	0.909891i	$-0.636165\pi$
$181$	−11.1624	−0.829695	−0.414848	+	0.909891i	$0.636165\pi$
$182$	0	0
$183$	−10.6373	−0.786329
$184$	0	0
$185$	−10.3856	−0.763567
$186$	0	0
$187$	4.54289	0.332209
$188$	0	0
$189$	3.82927	0.278538
$190$	0	0
$191$	14.2409	1.03043	0.515217	−	0.857060i	$-0.327711\pi$
$191$	14.2409	1.03043	0.515217	+	0.857060i	$0.327711\pi$
$192$	0	0
$193$	−0.589908	−0.0424625	−0.0212312	−	0.999775i	$-0.506759\pi$
$193$	−0.589908	−0.0424625	−0.0212312	+	0.999775i	$0.506759\pi$
$194$	0	0
$195$	−5.85846	−0.419533
$196$	0	0
$197$	1.56238	0.111315	0.0556575	−	0.998450i	$-0.482274\pi$
$197$	1.56238	0.111315	0.0556575	+	0.998450i	$0.482274\pi$
$198$	0	0
$199$	8.62986	0.611754	0.305877	−	0.952071i	$-0.401050\pi$
$199$	8.62986	0.611754	0.305877	+	0.952071i	$0.401050\pi$
$200$	0	0
$201$	28.7981	2.03126
$202$	0	0
$203$	−6.29319	−0.441695
$204$	0	0
$205$	7.33858	0.512548
$206$	0	0
$207$	−8.49451	−0.590410
$208$	0	0
$209$	4.10090	0.283665
$210$	0	0
$211$	−0.929816	−0.0640112	−0.0320056	−	0.999488i	$-0.510189\pi$
$211$	−0.929816	−0.0640112	−0.0320056	+	0.999488i	$0.510189\pi$
$212$	0	0
$213$	4.61217	0.316021
$214$	0	0
$215$	−28.7990	−1.96407
$216$	0	0
$217$	−9.32652	−0.633126
$218$	0	0
$219$	−14.5953	−0.986260
$220$	0	0
$221$	4.54289	0.305588
$222$	0	0
$223$	−0.913624	−0.0611808	−0.0305904	−	0.999532i	$-0.509739\pi$
$223$	−0.913624	−0.0611808	−0.0305904	+	0.999532i	$0.509739\pi$
$224$	0	0
$225$	3.72099	0.248066
$226$	0	0
$227$	−24.4605	−1.62350	−0.811749	−	0.584006i	$-0.801485\pi$
$227$	−24.4605	−1.62350	−0.811749	+	0.584006i	$0.801485\pi$
$228$	0	0
$229$	−0.306812	−0.0202747	−0.0101374	−	0.999949i	$-0.503227\pi$
$229$	−0.306812	−0.0202747	−0.0101374	+	0.999949i	$0.503227\pi$
$230$	0	0
$231$	−2.02769	−0.133412
$232$	0	0
$233$	18.0808	1.18451	0.592256	−	0.805750i	$-0.298237\pi$
$233$	18.0808	1.18451	0.592256	+	0.805750i	$0.298237\pi$
$234$	0	0
$235$	−2.25232	−0.146925
$236$	0	0
$237$	4.18091	0.271579
$238$	0	0
$239$	24.2776	1.57038	0.785192	−	0.619252i	$-0.212564\pi$
$239$	24.2776	1.57038	0.785192	+	0.619252i	$0.212564\pi$
$240$	0	0
$241$	8.93537	0.575578	0.287789	−	0.957694i	$-0.407080\pi$
$241$	8.93537	0.575578	0.287789	+	0.957694i	$0.407080\pi$
$242$	0	0
$243$	11.0176	0.706782
$244$	0	0
$245$	2.88924	0.184587
$246$	0	0
$247$	4.10090	0.260934
$248$	0	0
$249$	0.487138	0.0308711
$250$	0	0
$251$	−2.16485	−0.136644	−0.0683220	−	0.997663i	$-0.521765\pi$
$251$	−2.16485	−0.136644	−0.0683220	+	0.997663i	$0.521765\pi$
$252$	0	0
$253$	−7.64232	−0.480469
$254$	0	0
$255$	−26.6144	−1.66666
$256$	0	0
$257$	−12.4256	−0.775085	−0.387543	−	0.921852i	$-0.626676\pi$
$257$	−12.4256	−0.775085	−0.387543	+	0.921852i	$0.626676\pi$
$258$	0	0
$259$	−3.59460	−0.223357
$260$	0	0
$261$	−6.99493	−0.432976
$262$	0	0
$263$	10.8871	0.671325	0.335662	−	0.941982i	$-0.391040\pi$
$263$	10.8871	0.671325	0.335662	+	0.941982i	$0.391040\pi$
$264$	0	0
$265$	−39.7349	−2.44089
$266$	0	0
$267$	20.8693	1.27718
$268$	0	0
$269$	−30.5018	−1.85973	−0.929865	−	0.367901i	$-0.880076\pi$
$269$	−30.5018	−1.85973	−0.929865	+	0.367901i	$0.880076\pi$
$270$	0	0
$271$	1.11512	0.0677387	0.0338694	−	0.999426i	$-0.489217\pi$
$271$	1.11512	0.0677387	0.0338694	+	0.999426i	$0.489217\pi$
$272$	0	0
$273$	−2.02769	−0.122721
$274$	0	0
$275$	3.34769	0.201873
$276$	0	0
$277$	12.1567	0.730425	0.365213	−	0.930924i	$-0.380996\pi$
$277$	12.1567	0.730425	0.365213	+	0.930924i	$0.380996\pi$
$278$	0	0
$279$	−10.3665	−0.620627
$280$	0	0
$281$	4.64402	0.277039	0.138520	−	0.990360i	$-0.455766\pi$
$281$	4.64402	0.277039	0.138520	+	0.990360i	$0.455766\pi$
$282$	0	0
$283$	22.6546	1.34668	0.673339	−	0.739334i	$-0.264859\pi$
$283$	22.6546	1.34668	0.673339	+	0.739334i	$0.264859\pi$
$284$	0	0
$285$	−24.0250	−1.42312
$286$	0	0
$287$	2.53997	0.149930
$288$	0	0
$289$	3.63789	0.213994
$290$	0	0
$291$	10.7903	0.632536
$292$	0	0
$293$	27.7163	1.61920	0.809602	−	0.586979i	$-0.199683\pi$
$293$	27.7163	1.61920	0.809602	+	0.586979i	$0.199683\pi$
$294$	0	0
$295$	0.907384	0.0528299
$296$	0	0
$297$	3.82927	0.222197
$298$	0	0
$299$	−7.64232	−0.441967
$300$	0	0
$301$	−9.96767	−0.574527
$302$	0	0
$303$	−20.9696	−1.20467
$304$	0	0
$305$	15.1570	0.867886
$306$	0	0
$307$	17.6948	1.00990	0.504949	−	0.863149i	$-0.331511\pi$
$307$	17.6948	1.00990	0.504949	+	0.863149i	$0.331511\pi$
$308$	0	0
$309$	13.8479	0.787780
$310$	0	0
$311$	−25.6201	−1.45278	−0.726392	−	0.687280i	$-0.758804\pi$
$311$	−25.6201	−1.45278	−0.726392	+	0.687280i	$0.758804\pi$
$312$	0	0
$313$	5.53064	0.312610	0.156305	−	0.987709i	$-0.450042\pi$
$313$	5.53064	0.312610	0.156305	+	0.987709i	$0.450042\pi$
$314$	0	0
$315$	3.21141	0.180943
$316$	0	0
$317$	18.8327	1.05775	0.528874	−	0.848700i	$-0.322614\pi$
$317$	18.8327	1.05775	0.528874	+	0.848700i	$0.322614\pi$
$318$	0	0
$319$	−6.29319	−0.352351
$320$	0	0
$321$	4.07961	0.227702
$322$	0	0
$323$	18.6300	1.03660
$324$	0	0
$325$	3.34769	0.185696
$326$	0	0
$327$	−28.8819	−1.59717
$328$	0	0
$329$	−0.779554	−0.0429782
$330$	0	0
$331$	−17.7985	−0.978293	−0.489146	−	0.872202i	$-0.662692\pi$
$331$	−17.7985	−0.978293	−0.489146	+	0.872202i	$0.662692\pi$
$332$	0	0
$333$	−3.99543	−0.218948
$334$	0	0
$335$	−41.0343	−2.24194
$336$	0	0
$337$	−33.3386	−1.81607	−0.908034	−	0.418897i	$-0.862417\pi$
$337$	−33.3386	−1.81607	−0.908034	+	0.418897i	$0.862417\pi$
$338$	0	0
$339$	22.3948	1.21632
$340$	0	0
$341$	−9.32652	−0.505060
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	44.7723	2.41046
$346$	0	0
$347$	−0.245891	−0.0132001	−0.00660005	−	0.999978i	$-0.502101\pi$
$347$	−0.245891	−0.0132001	−0.00660005	+	0.999978i	$0.502101\pi$
$348$	0	0
$349$	−1.86845	−0.100016	−0.0500079	−	0.998749i	$-0.515925\pi$
$349$	−1.86845	−0.100016	−0.0500079	+	0.998749i	$0.515925\pi$
$350$	0	0
$351$	3.82927	0.204391
$352$	0	0
$353$	19.9414	1.06138	0.530688	−	0.847567i	$-0.321934\pi$
$353$	19.9414	1.06138	0.530688	+	0.847567i	$0.321934\pi$
$354$	0	0
$355$	−6.57185	−0.348798
$356$	0	0
$357$	−9.21156	−0.487528
$358$	0	0
$359$	−32.8503	−1.73377	−0.866885	−	0.498508i	$-0.833881\pi$
$359$	−32.8503	−1.73377	−0.866885	+	0.498508i	$0.833881\pi$
$360$	0	0
$361$	−2.18263	−0.114875
$362$	0	0
$363$	−2.02769	−0.106426
$364$	0	0
$365$	20.7968	1.08855
$366$	0	0
$367$	25.3042	1.32087	0.660435	−	0.750883i	$-0.270372\pi$
$367$	25.3042	1.32087	0.660435	+	0.750883i	$0.270372\pi$
$368$	0	0
$369$	2.82320	0.146970
$370$	0	0
$371$	−13.7527	−0.714006
$372$	0	0
$373$	0.578876	0.0299730	0.0149865	−	0.999888i	$-0.495229\pi$
$373$	0.578876	0.0299730	0.0149865	+	0.999888i	$0.495229\pi$
$374$	0	0
$375$	9.68001	0.499874
$376$	0	0
$377$	−6.29319	−0.324116
$378$	0	0
$379$	−26.9957	−1.38668	−0.693338	−	0.720612i	$-0.743861\pi$
$379$	−26.9957	−1.38668	−0.693338	+	0.720612i	$0.743861\pi$
$380$	0	0
$381$	27.0010	1.38330
$382$	0	0
$383$	−6.09804	−0.311595	−0.155798	−	0.987789i	$-0.549795\pi$
$383$	−6.09804	−0.311595	−0.155798	+	0.987789i	$0.549795\pi$
$384$	0	0
$385$	2.88924	0.147249
$386$	0	0
$387$	−11.0792	−0.563185
$388$	0	0
$389$	0.888481	0.0450478	0.0225239	−	0.999746i	$-0.492830\pi$
$389$	0.888481	0.0450478	0.0225239	+	0.999746i	$0.492830\pi$
$390$	0	0
$391$	−34.7183	−1.75578
$392$	0	0
$393$	27.6895	1.39675
$394$	0	0
$395$	−5.95736	−0.299747
$396$	0	0
$397$	−21.1890	−1.06345	−0.531723	−	0.846918i	$-0.678455\pi$
$397$	−21.1890	−1.06345	−0.531723	+	0.846918i	$0.678455\pi$
$398$	0	0
$399$	−8.31533	−0.416287
$400$	0	0
$401$	−1.10590	−0.0552261	−0.0276131	−	0.999619i	$-0.508791\pi$
$401$	−1.10590	−0.0552261	−0.0276131	+	0.999619i	$0.508791\pi$
$402$	0	0
$403$	−9.32652	−0.464587
$404$	0	0
$405$	−32.0679	−1.59346
$406$	0	0
$407$	−3.59460	−0.178178
$408$	0	0
$409$	25.2156	1.24683	0.623416	−	0.781890i	$-0.285744\pi$
$409$	25.2156	1.24683	0.623416	+	0.781890i	$0.285744\pi$
$410$	0	0
$411$	19.0370	0.939028
$412$	0	0
$413$	0.314057	0.0154537
$414$	0	0
$415$	−0.694121	−0.0340730
$416$	0	0
$417$	12.7239	0.623092
$418$	0	0
$419$	−12.8459	−0.627564	−0.313782	−	0.949495i	$-0.601596\pi$
$419$	−12.8459	−0.627564	−0.313782	+	0.949495i	$0.601596\pi$
$420$	0	0
$421$	−26.3123	−1.28238	−0.641191	−	0.767381i	$-0.721559\pi$
$421$	−26.3123	−1.28238	−0.641191	+	0.767381i	$0.721559\pi$
$422$	0	0
$423$	−0.866482	−0.0421298
$424$	0	0
$425$	15.2082	0.737706
$426$	0	0
$427$	5.24601	0.253872
$428$	0	0
$429$	−2.02769	−0.0978976
$430$	0	0
$431$	22.4072	1.07932	0.539659	−	0.841884i	$-0.318553\pi$
$431$	22.4072	1.07932	0.539659	+	0.841884i	$0.318553\pi$
$432$	0	0
$433$	19.5979	0.941815	0.470908	−	0.882183i	$-0.343927\pi$
$433$	19.5979	0.941815	0.470908	+	0.882183i	$0.343927\pi$
$434$	0	0
$435$	36.8684	1.76770
$436$	0	0
$437$	−31.3404	−1.49921
$438$	0	0
$439$	33.2492	1.58690	0.793448	−	0.608638i	$-0.208284\pi$
$439$	33.2492	1.58690	0.793448	+	0.608638i	$0.208284\pi$
$440$	0	0
$441$	1.11151	0.0529290
$442$	0	0
$443$	−15.9591	−0.758239	−0.379119	−	0.925348i	$-0.623773\pi$
$443$	−15.9591	−0.758239	−0.379119	+	0.925348i	$0.623773\pi$
$444$	0	0
$445$	−29.7365	−1.40965
$446$	0	0
$447$	−24.4230	−1.15517
$448$	0	0
$449$	−22.8815	−1.07985	−0.539923	−	0.841715i	$-0.681547\pi$
$449$	−22.8815	−1.07985	−0.539923	+	0.841715i	$0.681547\pi$
$450$	0	0
$451$	2.53997	0.119603
$452$	0	0
$453$	32.0250	1.50466
$454$	0	0
$455$	2.88924	0.135449
$456$	0	0
$457$	−29.3640	−1.37359	−0.686794	−	0.726852i	$-0.740983\pi$
$457$	−29.3640	−1.37359	−0.686794	+	0.726852i	$0.740983\pi$
$458$	0	0
$459$	17.3960	0.811973
$460$	0	0
$461$	−15.7642	−0.734211	−0.367105	−	0.930179i	$-0.619651\pi$
$461$	−15.7642	−0.734211	−0.367105	+	0.930179i	$0.619651\pi$
$462$	0	0
$463$	−28.3609	−1.31804	−0.659022	−	0.752123i	$-0.729030\pi$
$463$	−28.3609	−1.31804	−0.659022	+	0.752123i	$0.729030\pi$
$464$	0	0
$465$	54.6391	2.53383
$466$	0	0
$467$	−9.21265	−0.426311	−0.213155	−	0.977018i	$-0.568374\pi$
$467$	−9.21265	−0.426311	−0.213155	+	0.977018i	$0.568374\pi$
$468$	0	0
$469$	−14.2025	−0.655809
$470$	0	0
$471$	−23.0544	−1.06229
$472$	0	0
$473$	−9.96767	−0.458314
$474$	0	0
$475$	13.7285	0.629908
$476$	0	0
$477$	−15.2863	−0.699911
$478$	0	0
$479$	11.9665	0.546763	0.273381	−	0.961906i	$-0.411858\pi$
$479$	11.9665	0.546763	0.273381	+	0.961906i	$0.411858\pi$
$480$	0	0
$481$	−3.59460	−0.163899
$482$	0	0
$483$	15.4962	0.705103
$484$	0	0
$485$	−15.3750	−0.698142
$486$	0	0
$487$	−6.94505	−0.314710	−0.157355	−	0.987542i	$-0.550297\pi$
$487$	−6.94505	−0.314710	−0.157355	+	0.987542i	$0.550297\pi$
$488$	0	0
$489$	−40.8756	−1.84846
$490$	0	0
$491$	−17.6667	−0.797288	−0.398644	−	0.917106i	$-0.630519\pi$
$491$	−17.6667	−0.797288	−0.398644	+	0.917106i	$0.630519\pi$
$492$	0	0
$493$	−28.5893	−1.28760
$494$	0	0
$495$	3.21141	0.144342
$496$	0	0
$497$	−2.27460	−0.102030
$498$	0	0
$499$	15.5325	0.695328	0.347664	−	0.937619i	$-0.386975\pi$
$499$	15.5325	0.695328	0.347664	+	0.937619i	$0.386975\pi$
$500$	0	0
$501$	−3.66504	−0.163742
$502$	0	0
$503$	15.2776	0.681195	0.340598	−	0.940209i	$-0.389371\pi$
$503$	15.2776	0.681195	0.340598	+	0.940209i	$0.389371\pi$
$504$	0	0
$505$	29.8795	1.32962
$506$	0	0
$507$	−2.02769	−0.0900527
$508$	0	0
$509$	8.08052	0.358163	0.179081	−	0.983834i	$-0.442687\pi$
$509$	8.08052	0.358163	0.179081	+	0.983834i	$0.442687\pi$
$510$	0	0
$511$	7.19802	0.318422
$512$	0	0
$513$	15.7034	0.693323
$514$	0	0
$515$	−19.7318	−0.869486
$516$	0	0
$517$	−0.779554	−0.0342847
$518$	0	0
$519$	−23.3368	−1.02437
$520$	0	0
$521$	14.5176	0.636027	0.318013	−	0.948086i	$-0.396984\pi$
$521$	14.5176	0.636027	0.318013	+	0.948086i	$0.396984\pi$
$522$	0	0
$523$	−13.9002	−0.607814	−0.303907	−	0.952702i	$-0.598291\pi$
$523$	−13.9002	−0.607814	−0.303907	+	0.952702i	$0.598291\pi$
$524$	0	0
$525$	−6.78806	−0.296255
$526$	0	0
$527$	−42.3694	−1.84564
$528$	0	0
$529$	35.4051	1.53935
$530$	0	0
$531$	0.349077	0.0151486
$532$	0	0
$533$	2.53997	0.110018
$534$	0	0
$535$	−5.81301	−0.251318
$536$	0	0
$537$	−9.15572	−0.395099
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	0.0536542	0.00230677	0.00115339	−	0.999999i	$-0.499633\pi$
$541$	0.0536542	0.00230677	0.00115339	+	0.999999i	$0.499633\pi$
$542$	0	0
$543$	22.6339	0.971312
$544$	0	0
$545$	41.1536	1.76283
$546$	0	0
$547$	−28.8891	−1.23521	−0.617605	−	0.786489i	$-0.711897\pi$
$547$	−28.8891	−1.23521	−0.617605	+	0.786489i	$0.711897\pi$
$548$	0	0
$549$	5.83099	0.248861
$550$	0	0
$551$	−25.8077	−1.09945
$552$	0	0
$553$	−2.06191	−0.0876815
$554$	0	0
$555$	21.0588	0.893897
$556$	0	0
$557$	19.3899	0.821577	0.410788	−	0.911731i	$-0.365254\pi$
$557$	19.3899	0.821577	0.410788	+	0.911731i	$0.365254\pi$
$558$	0	0
$559$	−9.96767	−0.421588
$560$	0	0
$561$	−9.21156	−0.388912
$562$	0	0
$563$	−13.7051	−0.577600	−0.288800	−	0.957389i	$-0.593256\pi$
$563$	−13.7051	−0.577600	−0.288800	+	0.957389i	$0.593256\pi$
$564$	0	0
$565$	−31.9103	−1.34247
$566$	0	0
$567$	−11.0991	−0.466117
$568$	0	0
$569$	2.83174	0.118713	0.0593564	−	0.998237i	$-0.481095\pi$
$569$	2.83174	0.118713	0.0593564	+	0.998237i	$0.481095\pi$
$570$	0	0
$571$	−24.8958	−1.04186	−0.520929	−	0.853600i	$-0.674415\pi$
$571$	−24.8958	−1.04186	−0.520929	+	0.853600i	$0.674415\pi$
$572$	0	0
$573$	−28.8760	−1.20631
$574$	0	0
$575$	−25.5841	−1.06693
$576$	0	0
$577$	36.3932	1.51507	0.757535	−	0.652794i	$-0.226403\pi$
$577$	36.3932	1.51507	0.757535	+	0.652794i	$0.226403\pi$
$578$	0	0
$579$	1.19615	0.0497102
$580$	0	0
$581$	−0.240244	−0.00996698
$582$	0	0
$583$	−13.7527	−0.569580
$584$	0	0
$585$	3.21141	0.132776
$586$	0	0
$587$	−36.1438	−1.49181	−0.745907	−	0.666050i	$-0.767984\pi$
$587$	−36.1438	−1.49181	−0.745907	+	0.666050i	$0.767984\pi$
$588$	0	0
$589$	−38.2471	−1.57595
$590$	0	0
$591$	−3.16802	−0.130315
$592$	0	0
$593$	−23.9847	−0.984933	−0.492467	−	0.870331i	$-0.663905\pi$
$593$	−23.9847	−0.984933	−0.492467	+	0.870331i	$0.663905\pi$
$594$	0	0
$595$	13.1255	0.538093
$596$	0	0
$597$	−17.4986	−0.716172
$598$	0	0
$599$	8.89198	0.363316	0.181658	−	0.983362i	$-0.441854\pi$
$599$	8.89198	0.363316	0.181658	+	0.983362i	$0.441854\pi$
$600$	0	0
$601$	−20.0646	−0.818454	−0.409227	−	0.912433i	$-0.634202\pi$
$601$	−20.0646	−0.818454	−0.409227	+	0.912433i	$0.634202\pi$
$602$	0	0
$603$	−15.7862	−0.642862
$604$	0	0
$605$	2.88924	0.117464
$606$	0	0
$607$	10.2010	0.414045	0.207023	−	0.978336i	$-0.433623\pi$
$607$	10.2010	0.414045	0.207023	+	0.978336i	$0.433623\pi$
$608$	0	0
$609$	12.7606	0.517086
$610$	0	0
$611$	−0.779554	−0.0315374
$612$	0	0
$613$	−2.10767	−0.0851281	−0.0425640	−	0.999094i	$-0.513553\pi$
$613$	−2.10767	−0.0851281	−0.0425640	+	0.999094i	$0.513553\pi$
$614$	0	0
$615$	−14.8803	−0.600033
$616$	0	0
$617$	−42.4506	−1.70900	−0.854498	−	0.519454i	$-0.826135\pi$
$617$	−42.4506	−1.70900	−0.854498	+	0.519454i	$0.826135\pi$
$618$	0	0
$619$	−15.1771	−0.610021	−0.305010	−	0.952349i	$-0.598660\pi$
$619$	−15.1771	−0.610021	−0.305010	+	0.952349i	$0.598660\pi$
$620$	0	0
$621$	−29.2645	−1.17434
$622$	0	0
$623$	−10.2922	−0.412347
$624$	0	0
$625$	−30.5314	−1.22126
$626$	0	0
$627$	−8.31533	−0.332082
$628$	0	0
$629$	−16.3299	−0.651115
$630$	0	0
$631$	15.2822	0.608374	0.304187	−	0.952612i	$-0.401615\pi$
$631$	15.2822	0.608374	0.304187	+	0.952612i	$0.401615\pi$
$632$	0	0
$633$	1.88538	0.0749369
$634$	0	0
$635$	−38.4736	−1.52678
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−2.52824	−0.100015
$640$	0	0
$641$	−5.50443	−0.217412	−0.108706	−	0.994074i	$-0.534671\pi$
$641$	−5.50443	−0.217412	−0.108706	+	0.994074i	$0.534671\pi$
$642$	0	0
$643$	−38.2940	−1.51017	−0.755085	−	0.655627i	$-0.772404\pi$
$643$	−38.2940	−1.51017	−0.755085	+	0.655627i	$0.772404\pi$
$644$	0	0
$645$	58.3952	2.29931
$646$	0	0
$647$	25.3703	0.997411	0.498705	−	0.866772i	$-0.333809\pi$
$647$	25.3703	0.997411	0.498705	+	0.866772i	$0.333809\pi$
$648$	0	0
$649$	0.314057	0.0123278
$650$	0	0
$651$	18.9113	0.741191
$652$	0	0
$653$	−0.617766	−0.0241750	−0.0120875	−	0.999927i	$-0.503848\pi$
$653$	−0.617766	−0.0241750	−0.0120875	+	0.999927i	$0.503848\pi$
$654$	0	0
$655$	−39.4546	−1.54162
$656$	0	0
$657$	8.00066	0.312136
$658$	0	0
$659$	35.0890	1.36687	0.683436	−	0.730010i	$-0.260485\pi$
$659$	35.0890	1.36687	0.683436	+	0.730010i	$0.260485\pi$
$660$	0	0
$661$	20.1003	0.781813	0.390906	−	0.920430i	$-0.372162\pi$
$661$	20.1003	0.781813	0.390906	+	0.920430i	$0.372162\pi$
$662$	0	0
$663$	−9.21156	−0.357747
$664$	0	0
$665$	11.8485	0.459464
$666$	0	0
$667$	48.0946	1.86223
$668$	0	0
$669$	1.85254	0.0716234
$670$	0	0
$671$	5.24601	0.202520
$672$	0	0
$673$	−19.3181	−0.744658	−0.372329	−	0.928101i	$-0.621441\pi$
$673$	−19.3181	−0.744658	−0.372329	+	0.928101i	$0.621441\pi$
$674$	0	0
$675$	12.8192	0.493411
$676$	0	0
$677$	−33.9200	−1.30365	−0.651825	−	0.758369i	$-0.725996\pi$
$677$	−33.9200	−1.30365	−0.651825	+	0.758369i	$0.725996\pi$
$678$	0	0
$679$	−5.32147	−0.204219
$680$	0	0
$681$	49.5982	1.90061
$682$	0	0
$683$	16.4959	0.631199	0.315599	−	0.948893i	$-0.397794\pi$
$683$	16.4959	0.631199	0.315599	+	0.948893i	$0.397794\pi$
$684$	0	0
$685$	−27.1258	−1.03642
$686$	0	0
$687$	0.622118	0.0237353
$688$	0	0
$689$	−13.7527	−0.523937
$690$	0	0
$691$	−51.1166	−1.94457	−0.972283	−	0.233805i	$-0.924882\pi$
$691$	−51.1166	−1.94457	−0.972283	+	0.233805i	$0.924882\pi$
$692$	0	0
$693$	1.11151	0.0422227
$694$	0	0
$695$	−18.1302	−0.687718
$696$	0	0
$697$	11.5388	0.437064
$698$	0	0
$699$	−36.6622	−1.38669
$700$	0	0
$701$	−8.12610	−0.306919	−0.153459	−	0.988155i	$-0.549041\pi$
$701$	−8.12610	−0.306919	−0.153459	+	0.988155i	$0.549041\pi$
$702$	0	0
$703$	−14.7411	−0.555970
$704$	0	0
$705$	4.56699	0.172003
$706$	0	0
$707$	10.3417	0.388938
$708$	0	0
$709$	15.8773	0.596284	0.298142	−	0.954522i	$-0.403633\pi$
$709$	15.8773	0.596284	0.298142	+	0.954522i	$0.403633\pi$
$710$	0	0
$711$	−2.29184	−0.0859506
$712$	0	0
$713$	71.2763	2.66932
$714$	0	0
$715$	2.88924	0.108051
$716$	0	0
$717$	−49.2272	−1.83843
$718$	0	0
$719$	36.4369	1.35887	0.679434	−	0.733736i	$-0.262225\pi$
$719$	36.4369	1.35887	0.679434	+	0.733736i	$0.262225\pi$
$720$	0	0
$721$	−6.82941	−0.254341
$722$	0	0
$723$	−18.1181	−0.673821
$724$	0	0
$725$	−21.0676	−0.782432
$726$	0	0
$727$	18.3611	0.680974	0.340487	−	0.940249i	$-0.389408\pi$
$727$	18.3611	0.680974	0.340487	+	0.940249i	$0.389408\pi$
$728$	0	0
$729$	10.9569	0.405812
$730$	0	0
$731$	−45.2821	−1.67482
$732$	0	0
$733$	−34.0949	−1.25932	−0.629661	−	0.776870i	$-0.716806\pi$
$733$	−34.0949	−1.25932	−0.629661	+	0.776870i	$0.716806\pi$
$734$	0	0
$735$	−5.85846	−0.216093
$736$	0	0
$737$	−14.2025	−0.523154
$738$	0	0
$739$	15.9097	0.585249	0.292624	−	0.956227i	$-0.405471\pi$
$739$	15.9097	0.585249	0.292624	+	0.956227i	$0.405471\pi$
$740$	0	0
$741$	−8.31533	−0.305471
$742$	0	0
$743$	5.43763	0.199487	0.0997436	−	0.995013i	$-0.468198\pi$
$743$	5.43763	0.199487	0.0997436	+	0.995013i	$0.468198\pi$
$744$	0	0
$745$	34.8002	1.27498
$746$	0	0
$747$	−0.267033	−0.00977023
$748$	0	0
$749$	−2.01195	−0.0735152
$750$	0	0
$751$	48.8206	1.78149	0.890745	−	0.454504i	$-0.150183\pi$
$751$	48.8206	1.78149	0.890745	+	0.454504i	$0.150183\pi$
$752$	0	0
$753$	4.38963	0.159967
$754$	0	0
$755$	−45.6322	−1.66072
$756$	0	0
$757$	1.00833	0.0366485	0.0183243	−	0.999832i	$-0.494167\pi$
$757$	1.00833	0.0366485	0.0183243	+	0.999832i	$0.494167\pi$
$758$	0	0
$759$	15.4962	0.562478
$760$	0	0
$761$	−20.0337	−0.726220	−0.363110	−	0.931746i	$-0.618285\pi$
$761$	−20.0337	−0.726220	−0.363110	+	0.931746i	$0.618285\pi$
$762$	0	0
$763$	14.2438	0.515659
$764$	0	0
$765$	14.5891	0.527470
$766$	0	0
$767$	0.314057	0.0113399
$768$	0	0
$769$	45.3283	1.63458	0.817291	−	0.576225i	$-0.195475\pi$
$769$	45.3283	1.63458	0.817291	+	0.576225i	$0.195475\pi$
$770$	0	0
$771$	25.1951	0.907380
$772$	0	0
$773$	14.2869	0.513862	0.256931	−	0.966430i	$-0.417289\pi$
$773$	14.2869	0.513862	0.256931	+	0.966430i	$0.417289\pi$
$774$	0	0
$775$	−31.2223	−1.12154
$776$	0	0
$777$	7.28871	0.261481
$778$	0	0
$779$	10.4162	0.373198
$780$	0	0
$781$	−2.27460	−0.0813915
$782$	0	0
$783$	−24.0983	−0.861202
$784$	0	0
$785$	32.8501	1.17247
$786$	0	0
$787$	0.406258	0.0144815	0.00724077	−	0.999974i	$-0.497695\pi$
$787$	0.406258	0.0144815	0.00724077	+	0.999974i	$0.497695\pi$
$788$	0	0
$789$	−22.0755	−0.785910
$790$	0	0
$791$	−11.0445	−0.392698
$792$	0	0
$793$	5.24601	0.186291
$794$	0	0
$795$	80.5699	2.85752
$796$	0	0
$797$	23.7347	0.840727	0.420363	−	0.907356i	$-0.361903\pi$
$797$	23.7347	0.840727	0.420363	+	0.907356i	$0.361903\pi$
$798$	0	0
$799$	−3.54143	−0.125287
$800$	0	0
$801$	−11.4398	−0.404207
$802$	0	0
$803$	7.19802	0.254013
$804$	0	0
$805$	−22.0805	−0.778235
$806$	0	0
$807$	61.8481	2.17716
$808$	0	0
$809$	16.3393	0.574460	0.287230	−	0.957862i	$-0.407266\pi$
$809$	16.3393	0.574460	0.287230	+	0.957862i	$0.407266\pi$
$810$	0	0
$811$	4.72271	0.165837	0.0829184	−	0.996556i	$-0.473576\pi$
$811$	4.72271	0.165837	0.0829184	+	0.996556i	$0.473576\pi$
$812$	0	0
$813$	−2.26111	−0.0793007
$814$	0	0
$815$	58.2434	2.04018
$816$	0	0
$817$	−40.8764	−1.43008
$818$	0	0
$819$	1.11151	0.0388393
$820$	0	0
$821$	−18.5475	−0.647311	−0.323656	−	0.946175i	$-0.604912\pi$
$821$	−18.5475	−0.647311	−0.323656	+	0.946175i	$0.604912\pi$
$822$	0	0
$823$	12.8364	0.447447	0.223724	−	0.974653i	$-0.428179\pi$
$823$	12.8364	0.447447	0.223724	+	0.974653i	$0.428179\pi$
$824$	0	0
$825$	−6.78806	−0.236330
$826$	0	0
$827$	−20.0219	−0.696229	−0.348114	−	0.937452i	$-0.613178\pi$
$827$	−20.0219	−0.696229	−0.348114	+	0.937452i	$0.613178\pi$
$828$	0	0
$829$	−35.5205	−1.23368	−0.616839	−	0.787090i	$-0.711587\pi$
$829$	−35.5205	−1.23368	−0.616839	+	0.787090i	$0.711587\pi$
$830$	0	0
$831$	−24.6500	−0.855098
$832$	0	0
$833$	4.54289	0.157402
$834$	0	0
$835$	5.22229	0.180725
$836$	0	0
$837$	−35.7137	−1.23445
$838$	0	0
$839$	−3.94627	−0.136240	−0.0681202	−	0.997677i	$-0.521700\pi$
$839$	−3.94627	−0.136240	−0.0681202	+	0.997677i	$0.521700\pi$
$840$	0	0
$841$	10.6042	0.365661
$842$	0	0
$843$	−9.41662	−0.324326
$844$	0	0
$845$	2.88924	0.0993928
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−45.9365	−1.57654
$850$	0	0
$851$	27.4711	0.941696
$852$	0	0
$853$	−26.8815	−0.920404	−0.460202	−	0.887814i	$-0.652223\pi$
$853$	−26.8815	−0.920404	−0.460202	+	0.887814i	$0.652223\pi$
$854$	0	0
$855$	13.1697	0.450394
$856$	0	0
$857$	11.6870	0.399221	0.199611	−	0.979875i	$-0.436032\pi$
$857$	11.6870	0.399221	0.199611	+	0.979875i	$0.436032\pi$
$858$	0	0
$859$	−14.6585	−0.500142	−0.250071	−	0.968227i	$-0.580454\pi$
$859$	−14.6585	−0.500142	−0.250071	+	0.968227i	$0.580454\pi$
$860$	0	0
$861$	−5.15026	−0.175520
$862$	0	0
$863$	47.9368	1.63179	0.815894	−	0.578202i	$-0.196245\pi$
$863$	47.9368	1.63179	0.815894	+	0.578202i	$0.196245\pi$
$864$	0	0
$865$	33.2524	1.13062
$866$	0	0
$867$	−7.37650	−0.250519
$868$	0	0
$869$	−2.06191	−0.0699456
$870$	0	0
$871$	−14.2025	−0.481232
$872$	0	0
$873$	−5.91486	−0.200188
$874$	0	0
$875$	−4.77392	−0.161388
$876$	0	0
$877$	−53.3768	−1.80240	−0.901202	−	0.433398i	$-0.857314\pi$
$877$	−53.3768	−1.80240	−0.901202	+	0.433398i	$0.857314\pi$
$878$	0	0
$879$	−56.2000	−1.89558
$880$	0	0
$881$	−51.8065	−1.74541	−0.872703	−	0.488251i	$-0.837635\pi$
$881$	−51.8065	−1.74541	−0.872703	+	0.488251i	$0.837635\pi$
$882$	0	0
$883$	−0.983865	−0.0331097	−0.0165548	−	0.999863i	$-0.505270\pi$
$883$	−0.983865	−0.0331097	−0.0165548	+	0.999863i	$0.505270\pi$
$884$	0	0
$885$	−1.83989	−0.0618472
$886$	0	0
$887$	13.0788	0.439144	0.219572	−	0.975596i	$-0.429534\pi$
$887$	13.0788	0.439144	0.219572	+	0.975596i	$0.429534\pi$
$888$	0	0
$889$	−13.3162	−0.446610
$890$	0	0
$891$	−11.0991	−0.371833
$892$	0	0
$893$	−3.19687	−0.106979
$894$	0	0
$895$	13.0459	0.436077
$896$	0	0
$897$	15.4962	0.517404
$898$	0	0
$899$	58.6935	1.95754
$900$	0	0
$901$	−62.4772	−2.08142
$902$	0	0
$903$	20.2113	0.672590
$904$	0	0
$905$	−32.2508	−1.07205
$906$	0	0
$907$	−37.7072	−1.25205	−0.626023	−	0.779805i	$-0.715318\pi$
$907$	−37.7072	−1.25205	−0.626023	+	0.779805i	$0.715318\pi$
$908$	0	0
$909$	11.4948	0.381260
$910$	0	0
$911$	14.8197	0.491000	0.245500	−	0.969397i	$-0.421048\pi$
$911$	14.8197	0.491000	0.245500	+	0.969397i	$0.421048\pi$
$912$	0	0
$913$	−0.240244	−0.00795090
$914$	0	0
$915$	−30.7336	−1.01602
$916$	0	0
$917$	−13.6557	−0.450952
$918$	0	0
$919$	11.9950	0.395679	0.197839	−	0.980234i	$-0.436608\pi$
$919$	11.9950	0.395679	0.197839	+	0.980234i	$0.436608\pi$
$920$	0	0
$921$	−35.8796	−1.18227
$922$	0	0
$923$	−2.27460	−0.0748693
$924$	0	0
$925$	−12.0336	−0.395662
$926$	0	0
$927$	−7.59095	−0.249320
$928$	0	0
$929$	−15.4282	−0.506183	−0.253092	−	0.967442i	$-0.581447\pi$
$929$	−15.4282	−0.506183	−0.253092	+	0.967442i	$0.581447\pi$
$930$	0	0
$931$	4.10090	0.134402
$932$	0	0
$933$	51.9496	1.70075
$934$	0	0
$935$	13.1255	0.429250
$936$	0	0
$937$	47.2319	1.54300	0.771499	−	0.636230i	$-0.219507\pi$
$937$	47.2319	1.54300	0.771499	+	0.636230i	$0.219507\pi$
$938$	0	0
$939$	−11.2144	−0.365968
$940$	0	0
$941$	19.9921	0.651723	0.325862	−	0.945417i	$-0.394346\pi$
$941$	19.9921	0.651723	0.325862	+	0.945417i	$0.394346\pi$
$942$	0	0
$943$	−19.4113	−0.632118
$944$	0	0
$945$	11.0637	0.359901
$946$	0	0
$947$	−3.68774	−0.119835	−0.0599177	−	0.998203i	$-0.519084\pi$
$947$	−3.68774	−0.119835	−0.0599177	+	0.998203i	$0.519084\pi$
$948$	0	0
$949$	7.19802	0.233658
$950$	0	0
$951$	−38.1867	−1.23829
$952$	0	0
$953$	−11.0361	−0.357495	−0.178747	−	0.983895i	$-0.557204\pi$
$953$	−11.0361	−0.357495	−0.178747	+	0.983895i	$0.557204\pi$
$954$	0	0
$955$	41.1453	1.33143
$956$	0	0
$957$	12.7606	0.412492
$958$	0	0
$959$	−9.38856	−0.303172
$960$	0	0
$961$	55.9840	1.80594
$962$	0	0
$963$	−2.23630	−0.0720639
$964$	0	0
$965$	−1.70438	−0.0548660
$966$	0	0
$967$	49.0803	1.57831	0.789157	−	0.614191i	$-0.210518\pi$
$967$	49.0803	1.57831	0.789157	+	0.614191i	$0.210518\pi$
$968$	0	0
$969$	−37.7757	−1.21353
$970$	0	0
$971$	14.2122	0.456090	0.228045	−	0.973651i	$-0.426767\pi$
$971$	14.2122	0.456090	0.228045	+	0.973651i	$0.426767\pi$
$972$	0	0
$973$	−6.27509	−0.201170
$974$	0	0
$975$	−6.78806	−0.217392
$976$	0	0
$977$	−31.7926	−1.01714	−0.508568	−	0.861022i	$-0.669825\pi$
$977$	−31.7926	−1.01714	−0.508568	+	0.861022i	$0.669825\pi$
$978$	0	0
$979$	−10.2922	−0.328939
$980$	0	0
$981$	15.8321	0.505479
$982$	0	0
$983$	−16.3645	−0.521947	−0.260974	−	0.965346i	$-0.584044\pi$
$983$	−16.3645	−0.521947	−0.260974	+	0.965346i	$0.584044\pi$
$984$	0	0
$985$	4.51409	0.143831
$986$	0	0
$987$	1.58069	0.0503139
$988$	0	0
$989$	76.1762	2.42226
$990$	0	0
$991$	44.9590	1.42817	0.714085	−	0.700059i	$-0.246843\pi$
$991$	44.9590	1.42817	0.714085	+	0.700059i	$0.246843\pi$
$992$	0	0
$993$	36.0897	1.14527
$994$	0	0
$995$	24.9337	0.790451
$996$	0	0
$997$	−13.3228	−0.421936	−0.210968	−	0.977493i	$-0.567662\pi$
$997$	−13.3228	−0.421936	−0.210968	+	0.977493i	$0.567662\pi$
$998$	0	0
$999$	−13.7647	−0.435495

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.l.1.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.l.1.3	✓	8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.02769 q^{3} +2.88924 q^{5} +1.00000 q^{7} +1.11151 q^{9} +O(q^{10})\)	\(q-2.02769 q^{3} +2.88924 q^{5} +1.00000 q^{7} +1.11151 q^{9} +1.00000 q^{11} +1.00000 q^{13} -5.85846 q^{15} +4.54289 q^{17} +4.10090 q^{19} -2.02769 q^{21} -7.64232 q^{23} +3.34769 q^{25} +3.82927 q^{27} -6.29319 q^{29} -9.32652 q^{31} -2.02769 q^{33} +2.88924 q^{35} -3.59460 q^{37} -2.02769 q^{39} +2.53997 q^{41} -9.96767 q^{43} +3.21141 q^{45} -0.779554 q^{47} +1.00000 q^{49} -9.21156 q^{51} -13.7527 q^{53} +2.88924 q^{55} -8.31533 q^{57} +0.314057 q^{59} +5.24601 q^{61} +1.11151 q^{63} +2.88924 q^{65} -14.2025 q^{67} +15.4962 q^{69} -2.27460 q^{71} +7.19802 q^{73} -6.78806 q^{75} +1.00000 q^{77} -2.06191 q^{79} -11.0991 q^{81} -0.240244 q^{83} +13.1255 q^{85} +12.7606 q^{87} -10.2922 q^{89} +1.00000 q^{91} +18.9113 q^{93} +11.8485 q^{95} -5.32147 q^{97} +1.11151 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{3} - 7 q^{5} + 8 q^{7} + 7 q^{9}+O(q^{10}) \) 8 * q - 5 * q^3 - 7 * q^5 + 8 * q^7 + 7 * q^9	\( 8 q - 5 q^{3} - 7 q^{5} + 8 q^{7} + 7 q^{9} + 8 q^{11} + 8 q^{13} - 5 q^{15} + 3 q^{17} - 19 q^{19} - 5 q^{21} + q^{23} + 15 q^{25} - 11 q^{27} - 21 q^{29} + 2 q^{31} - 5 q^{33} - 7 q^{35} - 12 q^{37} - 5 q^{39} - 6 q^{41} - 19 q^{43} - 17 q^{45} - 7 q^{47} + 8 q^{49} - 19 q^{51} - 26 q^{53} - 7 q^{55} + 2 q^{57} - 17 q^{59} + 7 q^{63} - 7 q^{65} - 24 q^{67} + 20 q^{69} + 2 q^{71} + 2 q^{73} + 18 q^{75} + 8 q^{77} + 7 q^{79} - 4 q^{81} - 6 q^{83} + 19 q^{85} - 13 q^{87} + 13 q^{89} + 8 q^{91} + 13 q^{93} + 21 q^{95} + 18 q^{97} + 7 q^{99}+O(q^{100}) \) 8 * q - 5 * q^3 - 7 * q^5 + 8 * q^7 + 7 * q^9 + 8 * q^11 + 8 * q^13 - 5 * q^15 + 3 * q^17 - 19 * q^19 - 5 * q^21 + q^23 + 15 * q^25 - 11 * q^27 - 21 * q^29 + 2 * q^31 - 5 * q^33 - 7 * q^35 - 12 * q^37 - 5 * q^39 - 6 * q^41 - 19 * q^43 - 17 * q^45 - 7 * q^47 + 8 * q^49 - 19 * q^51 - 26 * q^53 - 7 * q^55 + 2 * q^57 - 17 * q^59 + 7 * q^63 - 7 * q^65 - 24 * q^67 + 20 * q^69 + 2 * q^71 + 2 * q^73 + 18 * q^75 + 8 * q^77 + 7 * q^79 - 4 * q^81 - 6 * q^83 + 19 * q^85 - 13 * q^87 + 13 * q^89 + 8 * q^91 + 13 * q^93 + 21 * q^95 + 18 * q^97 + 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more