LMFDB - Embedded newform 4008.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4008,2,Mod(1,4008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4008 = 2^{3} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4008.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:	$32.0040411301$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
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			1.1
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	4008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +0.324869 q^{5} -4.96239 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.324869 q^{5} -4.96239 q^{7} +1.00000 q^{9} +2.15633 q^{11} +2.00000 q^{13} -0.324869 q^{15} +5.02539 q^{17} +1.61213 q^{19} +4.96239 q^{21} -8.00000 q^{23} -4.89446 q^{25} -1.00000 q^{27} +2.31265 q^{29} -1.61213 q^{31} -2.15633 q^{33} -1.61213 q^{35} -6.96239 q^{37} -2.00000 q^{39} +9.86177 q^{41} -4.06300 q^{43} +0.324869 q^{45} +10.4690 q^{47} +17.6253 q^{49} -5.02539 q^{51} -4.89938 q^{53} +0.700523 q^{55} -1.61213 q^{57} +8.31265 q^{59} -0.0303172 q^{61} -4.96239 q^{63} +0.649738 q^{65} -6.71274 q^{67} +8.00000 q^{69} -8.12601 q^{71} +4.26187 q^{73} +4.89446 q^{75} -10.7005 q^{77} +15.3380 q^{79} +1.00000 q^{81} -5.79877 q^{83} +1.63259 q^{85} -2.31265 q^{87} +12.2374 q^{89} -9.92478 q^{91} +1.61213 q^{93} +0.523730 q^{95} -8.05079 q^{97} +2.15633 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 6 q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 6 * q^5 - 4 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} + 6 q^{5} - 4 q^{7} + 3 q^{9} - 4 q^{11} + 6 q^{13} - 6 q^{15} + 4 q^{19} + 4 q^{21} - 24 q^{23} + 5 q^{25} - 3 q^{27} - 14 q^{29} - 4 q^{31} + 4 q^{33} - 4 q^{35} - 10 q^{37} - 6 q^{39} + 12 q^{41} - 8 q^{43} + 6 q^{45} + 11 q^{49} - 8 q^{53} - 18 q^{55} - 4 q^{57} + 4 q^{59} + 2 q^{61} - 4 q^{63} + 12 q^{65} - 26 q^{67} + 24 q^{69} - 16 q^{71} + 22 q^{73} - 5 q^{75} - 12 q^{77} + 10 q^{79} + 3 q^{81} - 4 q^{83} - 24 q^{85} + 14 q^{87} - 6 q^{89} - 8 q^{91} + 4 q^{93} + 20 q^{95} + 6 q^{97} - 4 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 6 * q^5 - 4 * q^7 + 3 * q^9 - 4 * q^11 + 6 * q^13 - 6 * q^15 + 4 * q^19 + 4 * q^21 - 24 * q^23 + 5 * q^25 - 3 * q^27 - 14 * q^29 - 4 * q^31 + 4 * q^33 - 4 * q^35 - 10 * q^37 - 6 * q^39 + 12 * q^41 - 8 * q^43 + 6 * q^45 + 11 * q^49 - 8 * q^53 - 18 * q^55 - 4 * q^57 + 4 * q^59 + 2 * q^61 - 4 * q^63 + 12 * q^65 - 26 * q^67 + 24 * q^69 - 16 * q^71 + 22 * q^73 - 5 * q^75 - 12 * q^77 + 10 * q^79 + 3 * q^81 - 4 * q^83 - 24 * q^85 + 14 * q^87 - 6 * q^89 - 8 * q^91 + 4 * q^93 + 20 * q^95 + 6 * q^97 - 4 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0.324869	0.145286	0.0726429	−	0.997358i	$-0.476857\pi$
$5$	0.324869	0.145286	0.0726429	+	0.997358i	$0.476857\pi$
$6$	0	0
$7$	−4.96239	−1.87561	−0.937803	−	0.347167i	$-0.887144\pi$
$7$	−4.96239	−1.87561	−0.937803	+	0.347167i	$0.887144\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.15633	0.650157	0.325078	−	0.945687i	$-0.394609\pi$
$11$	2.15633	0.650157	0.325078	+	0.945687i	$0.394609\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	−0.324869	−0.0838808
$16$	0	0
$17$	5.02539	1.21884	0.609418	−	0.792849i	$-0.291403\pi$
$17$	5.02539	1.21884	0.609418	+	0.792849i	$0.291403\pi$
$18$	0	0
$19$	1.61213	0.369847	0.184924	−	0.982753i	$-0.440796\pi$
$19$	1.61213	0.369847	0.184924	+	0.982753i	$0.440796\pi$
$20$	0	0
$21$	4.96239	1.08288
$22$	0	0
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	−4.89446	−0.978892
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.31265	0.429448	0.214724	−	0.976675i	$-0.431115\pi$
$29$	2.31265	0.429448	0.214724	+	0.976675i	$0.431115\pi$
$30$	0	0
$31$	−1.61213	−0.289547	−0.144773	−	0.989465i	$-0.546245\pi$
$31$	−1.61213	−0.289547	−0.144773	+	0.989465i	$0.546245\pi$
$32$	0	0
$33$	−2.15633	−0.375368
$34$	0	0
$35$	−1.61213	−0.272499
$36$	0	0
$37$	−6.96239	−1.14461	−0.572305	−	0.820041i	$-0.693951\pi$
$37$	−6.96239	−1.14461	−0.572305	+	0.820041i	$0.693951\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	9.86177	1.54015	0.770075	−	0.637953i	$-0.220219\pi$
$41$	9.86177	1.54015	0.770075	+	0.637953i	$0.220219\pi$
$42$	0	0
$43$	−4.06300	−0.619602	−0.309801	−	0.950801i	$-0.600262\pi$
$43$	−4.06300	−0.619602	−0.309801	+	0.950801i	$0.600262\pi$
$44$	0	0
$45$	0.324869	0.0484286
$46$	0	0
$47$	10.4690	1.52706	0.763529	−	0.645774i	$-0.223465\pi$
$47$	10.4690	1.52706	0.763529	+	0.645774i	$0.223465\pi$
$48$	0	0
$49$	17.6253	2.51790
$50$	0	0
$51$	−5.02539	−0.703696
$52$	0	0
$53$	−4.89938	−0.672982	−0.336491	−	0.941687i	$-0.609240\pi$
$53$	−4.89938	−0.672982	−0.336491	+	0.941687i	$0.609240\pi$
$54$	0	0
$55$	0.700523	0.0944586
$56$	0	0
$57$	−1.61213	−0.213531
$58$	0	0
$59$	8.31265	1.08221	0.541107	−	0.840953i	$-0.318005\pi$
$59$	8.31265	1.08221	0.541107	+	0.840953i	$0.318005\pi$
$60$	0	0
$61$	−0.0303172	−0.00388172	−0.00194086	−	0.999998i	$-0.500618\pi$
$61$	−0.0303172	−0.00388172	−0.00194086	+	0.999998i	$0.500618\pi$
$62$	0	0
$63$	−4.96239	−0.625202
$64$	0	0
$65$	0.649738	0.0805901
$66$	0	0
$67$	−6.71274	−0.820092	−0.410046	−	0.912065i	$-0.634487\pi$
$67$	−6.71274	−0.820092	−0.410046	+	0.912065i	$0.634487\pi$
$68$	0	0
$69$	8.00000	0.963087
$70$	0	0
$71$	−8.12601	−0.964380	−0.482190	−	0.876067i	$-0.660158\pi$
$71$	−8.12601	−0.964380	−0.482190	+	0.876067i	$0.660158\pi$
$72$	0	0
$73$	4.26187	0.498814	0.249407	−	0.968399i	$-0.419764\pi$
$73$	4.26187	0.498814	0.249407	+	0.968399i	$0.419764\pi$
$74$	0	0
$75$	4.89446	0.565164
$76$	0	0
$77$	−10.7005	−1.21944
$78$	0	0
$79$	15.3380	1.72566	0.862832	−	0.505490i	$-0.168688\pi$
$79$	15.3380	1.72566	0.862832	+	0.505490i	$0.168688\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−5.79877	−0.636498	−0.318249	−	0.948007i	$-0.603095\pi$
$83$	−5.79877	−0.636498	−0.318249	+	0.948007i	$0.603095\pi$
$84$	0	0
$85$	1.63259	0.177080
$86$	0	0
$87$	−2.31265	−0.247942
$88$	0	0
$89$	12.2374	1.29716	0.648582	−	0.761144i	$-0.275362\pi$
$89$	12.2374	1.29716	0.648582	+	0.761144i	$0.275362\pi$
$90$	0	0
$91$	−9.92478	−1.04040
$92$	0	0
$93$	1.61213	0.167170
$94$	0	0
$95$	0.523730	0.0537336
$96$	0	0
$97$	−8.05079	−0.817433	−0.408717	−	0.912661i	$-0.634024\pi$
$97$	−8.05079	−0.817433	−0.408717	+	0.912661i	$0.634024\pi$
$98$	0	0
$99$	2.15633	0.216719
$100$	0	0
$101$	−1.28726	−0.128087	−0.0640435	−	0.997947i	$-0.520400\pi$
$101$	−1.28726	−0.128087	−0.0640435	+	0.997947i	$0.520400\pi$
$102$	0	0
$103$	−9.28726	−0.915101	−0.457550	−	0.889184i	$-0.651273\pi$
$103$	−9.28726	−0.915101	−0.457550	+	0.889184i	$0.651273\pi$
$104$	0	0
$105$	1.61213	0.157327
$106$	0	0
$107$	−3.76845	−0.364310	−0.182155	−	0.983270i	$-0.558307\pi$
$107$	−3.76845	−0.364310	−0.182155	+	0.983270i	$0.558307\pi$
$108$	0	0
$109$	−16.2374	−1.55526	−0.777632	−	0.628720i	$-0.783579\pi$
$109$	−16.2374	−1.55526	−0.777632	+	0.628720i	$0.783579\pi$
$110$	0	0
$111$	6.96239	0.660841
$112$	0	0
$113$	−6.24965	−0.587917	−0.293959	−	0.955818i	$-0.594973\pi$
$113$	−6.24965	−0.587917	−0.293959	+	0.955818i	$0.594973\pi$
$114$	0	0
$115$	−2.59895	−0.242354
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	−24.9380	−2.28606
$120$	0	0
$121$	−6.35026	−0.577297
$122$	0	0
$123$	−9.86177	−0.889206
$124$	0	0
$125$	−3.21440	−0.287505
$126$	0	0
$127$	−13.2750	−1.17797	−0.588985	−	0.808144i	$-0.700472\pi$
$127$	−13.2750	−1.17797	−0.588985	+	0.808144i	$0.700472\pi$
$128$	0	0
$129$	4.06300	0.357728
$130$	0	0
$131$	−11.6629	−1.01899	−0.509497	−	0.860473i	$-0.670168\pi$
$131$	−11.6629	−1.01899	−0.509497	+	0.860473i	$0.670168\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	0	0
$135$	−0.324869	−0.0279603
$136$	0	0
$137$	−13.6629	−1.16730	−0.583651	−	0.812005i	$-0.698376\pi$
$137$	−13.6629	−1.16730	−0.583651	+	0.812005i	$0.698376\pi$
$138$	0	0
$139$	10.3249	0.875744	0.437872	−	0.899037i	$-0.355732\pi$
$139$	10.3249	0.875744	0.437872	+	0.899037i	$0.355732\pi$
$140$	0	0
$141$	−10.4690	−0.881647
$142$	0	0
$143$	4.31265	0.360642
$144$	0	0
$145$	0.751309	0.0623928
$146$	0	0
$147$	−17.6253	−1.45371
$148$	0	0
$149$	−14.1138	−1.15625	−0.578123	−	0.815949i	$-0.696215\pi$
$149$	−14.1138	−1.15625	−0.578123	+	0.815949i	$0.696215\pi$
$150$	0	0
$151$	−10.9502	−0.891112	−0.445556	−	0.895254i	$-0.646994\pi$
$151$	−10.9502	−0.891112	−0.445556	+	0.895254i	$0.646994\pi$
$152$	0	0
$153$	5.02539	0.406279
$154$	0	0
$155$	−0.523730	−0.0420670
$156$	0	0
$157$	−15.2750	−1.21908	−0.609540	−	0.792755i	$-0.708646\pi$
$157$	−15.2750	−1.21908	−0.609540	+	0.792755i	$0.708646\pi$
$158$	0	0
$159$	4.89938	0.388546
$160$	0	0
$161$	39.6991	3.12873
$162$	0	0
$163$	−10.3249	−0.808706	−0.404353	−	0.914603i	$-0.632503\pi$
$163$	−10.3249	−0.808706	−0.404353	+	0.914603i	$0.632503\pi$
$164$	0	0
$165$	−0.700523	−0.0545357
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	1.61213	0.123282
$172$	0	0
$173$	10.7757	0.819265	0.409632	−	0.912251i	$-0.365657\pi$
$173$	10.7757	0.819265	0.409632	+	0.912251i	$0.365657\pi$
$174$	0	0
$175$	24.2882	1.83602
$176$	0	0
$177$	−8.31265	−0.624817
$178$	0	0
$179$	−18.7005	−1.39774	−0.698871	−	0.715247i	$-0.746314\pi$
$179$	−18.7005	−1.39774	−0.698871	+	0.715247i	$0.746314\pi$
$180$	0	0
$181$	−7.81924	−0.581199	−0.290600	−	0.956845i	$-0.593855\pi$
$181$	−7.81924	−0.581199	−0.290600	+	0.956845i	$0.593855\pi$
$182$	0	0
$183$	0.0303172	0.00224111
$184$	0	0
$185$	−2.26187	−0.166296
$186$	0	0
$187$	10.8364	0.792435
$188$	0	0
$189$	4.96239	0.360961
$190$	0	0
$191$	−16.5442	−1.19710	−0.598548	−	0.801087i	$-0.704255\pi$
$191$	−16.5442	−1.19710	−0.598548	+	0.801087i	$0.704255\pi$
$192$	0	0
$193$	−2.31265	−0.166468	−0.0832341	−	0.996530i	$-0.526525\pi$
$193$	−2.31265	−0.166468	−0.0832341	+	0.996530i	$0.526525\pi$
$194$	0	0
$195$	−0.649738	−0.0465287
$196$	0	0
$197$	0.763527	0.0543991	0.0271995	−	0.999630i	$-0.491341\pi$
$197$	0.763527	0.0543991	0.0271995	+	0.999630i	$0.491341\pi$
$198$	0	0
$199$	−4.12601	−0.292485	−0.146242	−	0.989249i	$-0.546718\pi$
$199$	−4.12601	−0.292485	−0.146242	+	0.989249i	$0.546718\pi$
$200$	0	0
$201$	6.71274	0.473480
$202$	0	0
$203$	−11.4763	−0.805476
$204$	0	0
$205$	3.20379	0.223762
$206$	0	0
$207$	−8.00000	−0.556038
$208$	0	0
$209$	3.47627	0.240459
$210$	0	0
$211$	22.5501	1.55241	0.776206	−	0.630480i	$-0.217142\pi$
$211$	22.5501	1.55241	0.776206	+	0.630480i	$0.217142\pi$
$212$	0	0
$213$	8.12601	0.556785
$214$	0	0
$215$	−1.31994	−0.0900195
$216$	0	0
$217$	8.00000	0.543075
$218$	0	0
$219$	−4.26187	−0.287990
$220$	0	0
$221$	10.0508	0.676089
$222$	0	0
$223$	−4.96239	−0.332306	−0.166153	−	0.986100i	$-0.553135\pi$
$223$	−4.96239	−0.332306	−0.166153	+	0.986100i	$0.553135\pi$
$224$	0	0
$225$	−4.89446	−0.326297
$226$	0	0
$227$	−4.25202	−0.282216	−0.141108	−	0.989994i	$-0.545067\pi$
$227$	−4.25202	−0.282216	−0.141108	+	0.989994i	$0.545067\pi$
$228$	0	0
$229$	10.6702	0.705107	0.352554	−	0.935792i	$-0.385313\pi$
$229$	10.6702	0.705107	0.352554	+	0.935792i	$0.385313\pi$
$230$	0	0
$231$	10.7005	0.704043
$232$	0	0
$233$	3.58769	0.235037	0.117519	−	0.993071i	$-0.462506\pi$
$233$	3.58769	0.235037	0.117519	+	0.993071i	$0.462506\pi$
$234$	0	0
$235$	3.40105	0.221860
$236$	0	0
$237$	−15.3380	−0.996313
$238$	0	0
$239$	−26.3185	−1.70240	−0.851202	−	0.524838i	$-0.824126\pi$
$239$	−26.3185	−1.70240	−0.851202	+	0.524838i	$0.824126\pi$
$240$	0	0
$241$	−20.1114	−1.29549	−0.647745	−	0.761857i	$-0.724288\pi$
$241$	−20.1114	−1.29549	−0.647745	+	0.761857i	$0.724288\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	5.72592	0.365815
$246$	0	0
$247$	3.22425	0.205154
$248$	0	0
$249$	5.79877	0.367482
$250$	0	0
$251$	−9.21440	−0.581608	−0.290804	−	0.956783i	$-0.593923\pi$
$251$	−9.21440	−0.581608	−0.290804	+	0.956783i	$0.593923\pi$
$252$	0	0
$253$	−17.2506	−1.08454
$254$	0	0
$255$	−1.63259	−0.102237
$256$	0	0
$257$	−20.8242	−1.29898	−0.649488	−	0.760372i	$-0.725017\pi$
$257$	−20.8242	−1.29898	−0.649488	+	0.760372i	$0.725017\pi$
$258$	0	0
$259$	34.5501	2.14684
$260$	0	0
$261$	2.31265	0.143149
$262$	0	0
$263$	3.28963	0.202847	0.101424	−	0.994843i	$-0.467660\pi$
$263$	3.28963	0.202847	0.101424	+	0.994843i	$0.467660\pi$
$264$	0	0
$265$	−1.59166	−0.0977748
$266$	0	0
$267$	−12.2374	−0.748918
$268$	0	0
$269$	−5.54912	−0.338336	−0.169168	−	0.985587i	$-0.554108\pi$
$269$	−5.54912	−0.338336	−0.169168	+	0.985587i	$0.554108\pi$
$270$	0	0
$271$	−2.89938	−0.176125	−0.0880625	−	0.996115i	$-0.528068\pi$
$271$	−2.89938	−0.176125	−0.0880625	+	0.996115i	$0.528068\pi$
$272$	0	0
$273$	9.92478	0.600675
$274$	0	0
$275$	−10.5540	−0.636433
$276$	0	0
$277$	31.5633	1.89645	0.948226	−	0.317596i	$-0.102876\pi$
$277$	31.5633	1.89645	0.948226	+	0.317596i	$0.102876\pi$
$278$	0	0
$279$	−1.61213	−0.0965155
$280$	0	0
$281$	15.5877	0.929884	0.464942	−	0.885341i	$-0.346075\pi$
$281$	15.5877	0.929884	0.464942	+	0.885341i	$0.346075\pi$
$282$	0	0
$283$	31.0132	1.84354	0.921771	−	0.387735i	$-0.126742\pi$
$283$	31.0132	1.84354	0.921771	+	0.387735i	$0.126742\pi$
$284$	0	0
$285$	−0.523730	−0.0310231
$286$	0	0
$287$	−48.9380	−2.88872
$288$	0	0
$289$	8.25457	0.485563
$290$	0	0
$291$	8.05079	0.471945
$292$	0	0
$293$	−16.2981	−0.952143	−0.476071	−	0.879407i	$-0.657940\pi$
$293$	−16.2981	−0.952143	−0.476071	+	0.879407i	$0.657940\pi$
$294$	0	0
$295$	2.70052	0.157231
$296$	0	0
$297$	−2.15633	−0.125123
$298$	0	0
$299$	−16.0000	−0.925304
$300$	0	0
$301$	20.1622	1.16213
$302$	0	0
$303$	1.28726	0.0739510
$304$	0	0
$305$	−0.00984911	−0.000563959 0
$306$	0	0
$307$	4.24965	0.242540	0.121270	−	0.992620i	$-0.461303\pi$
$307$	4.24965	0.242540	0.121270	+	0.992620i	$0.461303\pi$
$308$	0	0
$309$	9.28726	0.528334
$310$	0	0
$311$	18.1114	1.02700	0.513502	−	0.858088i	$-0.328348\pi$
$311$	18.1114	1.02700	0.513502	+	0.858088i	$0.328348\pi$
$312$	0	0
$313$	−2.77575	−0.156894	−0.0784472	−	0.996918i	$-0.524996\pi$
$313$	−2.77575	−0.156894	−0.0784472	+	0.996918i	$0.524996\pi$
$314$	0	0
$315$	−1.61213	−0.0908331
$316$	0	0
$317$	13.5369	0.760308	0.380154	−	0.924923i	$-0.375871\pi$
$317$	13.5369	0.760308	0.380154	+	0.924923i	$0.375871\pi$
$318$	0	0
$319$	4.98683	0.279209
$320$	0	0
$321$	3.76845	0.210334
$322$	0	0
$323$	8.10157	0.450783
$324$	0	0
$325$	−9.78892	−0.542992
$326$	0	0
$327$	16.2374	0.897932
$328$	0	0
$329$	−51.9511	−2.86416
$330$	0	0
$331$	33.8881	1.86266	0.931330	−	0.364177i	$-0.118650\pi$
$331$	33.8881	1.86266	0.931330	+	0.364177i	$0.118650\pi$
$332$	0	0
$333$	−6.96239	−0.381537
$334$	0	0
$335$	−2.18076	−0.119148
$336$	0	0
$337$	−4.20123	−0.228856	−0.114428	−	0.993432i	$-0.536503\pi$
$337$	−4.20123	−0.228856	−0.114428	+	0.993432i	$0.536503\pi$
$338$	0	0
$339$	6.24965	0.339434
$340$	0	0
$341$	−3.47627	−0.188251
$342$	0	0
$343$	−52.7269	−2.84698
$344$	0	0
$345$	2.59895	0.139923
$346$	0	0
$347$	18.0870	0.970960	0.485480	−	0.874248i	$-0.338645\pi$
$347$	18.0870	0.970960	0.485480	+	0.874248i	$0.338645\pi$
$348$	0	0
$349$	−21.6629	−1.15959	−0.579795	−	0.814763i	$-0.696867\pi$
$349$	−21.6629	−1.15959	−0.579795	+	0.814763i	$0.696867\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0	0
$353$	2.33709	0.124391	0.0621953	−	0.998064i	$-0.480190\pi$
$353$	2.33709	0.124391	0.0621953	+	0.998064i	$0.480190\pi$
$354$	0	0
$355$	−2.63989	−0.140111
$356$	0	0
$357$	24.9380	1.31986
$358$	0	0
$359$	−26.5501	−1.40126	−0.700630	−	0.713525i	$-0.747098\pi$
$359$	−26.5501	−1.40126	−0.700630	+	0.713525i	$0.747098\pi$
$360$	0	0
$361$	−16.4010	−0.863213
$362$	0	0
$363$	6.35026	0.333302
$364$	0	0
$365$	1.38455	0.0724706
$366$	0	0
$367$	19.3865	1.01196	0.505982	−	0.862544i	$-0.331130\pi$
$367$	19.3865	1.01196	0.505982	+	0.862544i	$0.331130\pi$
$368$	0	0
$369$	9.86177	0.513383
$370$	0	0
$371$	24.3127	1.26225
$372$	0	0
$373$	34.2276	1.77224	0.886118	−	0.463459i	$-0.153392\pi$
$373$	34.2276	1.77224	0.886118	+	0.463459i	$0.153392\pi$
$374$	0	0
$375$	3.21440	0.165991
$376$	0	0
$377$	4.62530	0.238215
$378$	0	0
$379$	−24.1646	−1.24125	−0.620625	−	0.784107i	$-0.713121\pi$
$379$	−24.1646	−1.24125	−0.620625	+	0.784107i	$0.713121\pi$
$380$	0	0
$381$	13.2750	0.680101
$382$	0	0
$383$	9.69323	0.495301	0.247650	−	0.968849i	$-0.420342\pi$
$383$	9.69323	0.495301	0.247650	+	0.968849i	$0.420342\pi$
$384$	0	0
$385$	−3.47627	−0.177167
$386$	0	0
$387$	−4.06300	−0.206534
$388$	0	0
$389$	−28.7635	−1.45837	−0.729184	−	0.684317i	$-0.760100\pi$
$389$	−28.7635	−1.45837	−0.729184	+	0.684317i	$0.760100\pi$
$390$	0	0
$391$	−40.2031	−2.03316
$392$	0	0
$393$	11.6629	0.588316
$394$	0	0
$395$	4.98286	0.250715
$396$	0	0
$397$	−13.4920	−0.677144	−0.338572	−	0.940940i	$-0.609944\pi$
$397$	−13.4920	−0.677144	−0.338572	+	0.940940i	$0.609944\pi$
$398$	0	0
$399$	8.00000	0.400501
$400$	0	0
$401$	−10.7127	−0.534969	−0.267484	−	0.963562i	$-0.586192\pi$
$401$	−10.7127	−0.534969	−0.267484	+	0.963562i	$0.586192\pi$
$402$	0	0
$403$	−3.22425	−0.160612
$404$	0	0
$405$	0.324869	0.0161429
$406$	0	0
$407$	−15.0132	−0.744175
$408$	0	0
$409$	1.79289	0.0886527	0.0443263	−	0.999017i	$-0.485886\pi$
$409$	1.79289	0.0886527	0.0443263	+	0.999017i	$0.485886\pi$
$410$	0	0
$411$	13.6629	0.673942
$412$	0	0
$413$	−41.2506	−2.02981
$414$	0	0
$415$	−1.88384	−0.0924741
$416$	0	0
$417$	−10.3249	−0.505611
$418$	0	0
$419$	−24.7064	−1.20699	−0.603493	−	0.797368i	$-0.706225\pi$
$419$	−24.7064	−1.20699	−0.603493	+	0.797368i	$0.706225\pi$
$420$	0	0
$421$	4.82653	0.235231	0.117615	−	0.993059i	$-0.462475\pi$
$421$	4.82653	0.235231	0.117615	+	0.993059i	$0.462475\pi$
$422$	0	0
$423$	10.4690	0.509019
$424$	0	0
$425$	−24.5966	−1.19311
$426$	0	0
$427$	0.150446	0.00728057
$428$	0	0
$429$	−4.31265	−0.208217
$430$	0	0
$431$	4.28821	0.206556	0.103278	−	0.994653i	$-0.467067\pi$
$431$	4.28821	0.206556	0.103278	+	0.994653i	$0.467067\pi$
$432$	0	0
$433$	33.2809	1.59938	0.799689	−	0.600414i	$-0.204997\pi$
$433$	33.2809	1.59938	0.799689	+	0.600414i	$0.204997\pi$
$434$	0	0
$435$	−0.751309	−0.0360225
$436$	0	0
$437$	−12.8970	−0.616948
$438$	0	0
$439$	−12.6229	−0.602460	−0.301230	−	0.953552i	$-0.597397\pi$
$439$	−12.6229	−0.602460	−0.301230	+	0.953552i	$0.597397\pi$
$440$	0	0
$441$	17.6253	0.839300
$442$	0	0
$443$	17.7137	0.841603	0.420802	−	0.907153i	$-0.361749\pi$
$443$	17.7137	0.841603	0.420802	+	0.907153i	$0.361749\pi$
$444$	0	0
$445$	3.97556	0.188460
$446$	0	0
$447$	14.1138	0.667559
$448$	0	0
$449$	−31.6483	−1.49358	−0.746788	−	0.665062i	$-0.768405\pi$
$449$	−31.6483	−1.49358	−0.746788	+	0.665062i	$0.768405\pi$
$450$	0	0
$451$	21.2652	1.00134
$452$	0	0
$453$	10.9502	0.514484
$454$	0	0
$455$	−3.22425	−0.151155
$456$	0	0
$457$	6.12601	0.286563	0.143281	−	0.989682i	$-0.454235\pi$
$457$	6.12601	0.286563	0.143281	+	0.989682i	$0.454235\pi$
$458$	0	0
$459$	−5.02539	−0.234565
$460$	0	0
$461$	26.6497	1.24120	0.620601	−	0.784126i	$-0.286889\pi$
$461$	26.6497	1.24120	0.620601	+	0.784126i	$0.286889\pi$
$462$	0	0
$463$	30.4363	1.41449	0.707247	−	0.706966i	$-0.249937\pi$
$463$	30.4363	1.41449	0.707247	+	0.706966i	$0.249937\pi$
$464$	0	0
$465$	0.523730	0.0242874
$466$	0	0
$467$	−37.8759	−1.75269	−0.876344	−	0.481686i	$-0.840025\pi$
$467$	−37.8759	−1.75269	−0.876344	+	0.481686i	$0.840025\pi$
$468$	0	0
$469$	33.3112	1.53817
$470$	0	0
$471$	15.2750	0.703837
$472$	0	0
$473$	−8.76116	−0.402838
$474$	0	0
$475$	−7.89049	−0.362041
$476$	0	0
$477$	−4.89938	−0.224327
$478$	0	0
$479$	14.0508	0.641997	0.320998	−	0.947080i	$-0.395982\pi$
$479$	14.0508	0.641997	0.320998	+	0.947080i	$0.395982\pi$
$480$	0	0
$481$	−13.9248	−0.634915
$482$	0	0
$483$	−39.6991	−1.80637
$484$	0	0
$485$	−2.61545	−0.118762
$486$	0	0
$487$	−22.7734	−1.03196	−0.515980	−	0.856601i	$-0.672572\pi$
$487$	−22.7734	−1.03196	−0.515980	+	0.856601i	$0.672572\pi$
$488$	0	0
$489$	10.3249	0.466907
$490$	0	0
$491$	0.917483	0.0414054	0.0207027	−	0.999786i	$-0.493410\pi$
$491$	0.917483	0.0414054	0.0207027	+	0.999786i	$0.493410\pi$
$492$	0	0
$493$	11.6220	0.523427
$494$	0	0
$495$	0.700523	0.0314862
$496$	0	0
$497$	40.3244	1.80880
$498$	0	0
$499$	10.8388	0.485209	0.242605	−	0.970125i	$-0.421998\pi$
$499$	10.8388	0.485209	0.242605	+	0.970125i	$0.421998\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	−21.3199	−0.950609	−0.475305	−	0.879821i	$-0.657662\pi$
$503$	−21.3199	−0.950609	−0.475305	+	0.879821i	$0.657662\pi$
$504$	0	0
$505$	−0.418190	−0.0186092
$506$	0	0
$507$	9.00000	0.399704
$508$	0	0
$509$	33.0494	1.46489	0.732444	−	0.680827i	$-0.238380\pi$
$509$	33.0494	1.46489	0.732444	+	0.680827i	$0.238380\pi$
$510$	0	0
$511$	−21.1490	−0.935578
$512$	0	0
$513$	−1.61213	−0.0711771
$514$	0	0
$515$	−3.01714	−0.132951
$516$	0	0
$517$	22.5745	0.992826
$518$	0	0
$519$	−10.7757	−0.473003
$520$	0	0
$521$	−29.5002	−1.29243	−0.646215	−	0.763156i	$-0.723649\pi$
$521$	−29.5002	−1.29243	−0.646215	+	0.763156i	$0.723649\pi$
$522$	0	0
$523$	−17.6728	−0.772776	−0.386388	−	0.922336i	$-0.626277\pi$
$523$	−17.6728	−0.772776	−0.386388	+	0.922336i	$0.626277\pi$
$524$	0	0
$525$	−24.2882	−1.06002
$526$	0	0
$527$	−8.10157	−0.352910
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	8.31265	0.360738
$532$	0	0
$533$	19.7235	0.854322
$534$	0	0
$535$	−1.22425	−0.0529291
$536$	0	0
$537$	18.7005	0.806987
$538$	0	0
$539$	38.0059	1.63703
$540$	0	0
$541$	28.8021	1.23830	0.619149	−	0.785273i	$-0.287478\pi$
$541$	28.8021	1.23830	0.619149	+	0.785273i	$0.287478\pi$
$542$	0	0
$543$	7.81924	0.335556
$544$	0	0
$545$	−5.27504	−0.225958
$546$	0	0
$547$	0.349307	0.0149353	0.00746764	−	0.999972i	$-0.497623\pi$
$547$	0.349307	0.0149353	0.00746764	+	0.999972i	$0.497623\pi$
$548$	0	0
$549$	−0.0303172	−0.00129391
$550$	0	0
$551$	3.72829	0.158830
$552$	0	0
$553$	−76.1133	−3.23667
$554$	0	0
$555$	2.26187	0.0960108
$556$	0	0
$557$	−8.51388	−0.360745	−0.180372	−	0.983598i	$-0.557730\pi$
$557$	−8.51388	−0.360745	−0.180372	+	0.983598i	$0.557730\pi$
$558$	0	0
$559$	−8.12601	−0.343694
$560$	0	0
$561$	−10.8364	−0.457512
$562$	0	0
$563$	−11.5125	−0.485193	−0.242596	−	0.970127i	$-0.577999\pi$
$563$	−11.5125	−0.485193	−0.242596	+	0.970127i	$0.577999\pi$
$564$	0	0
$565$	−2.03032	−0.0854161
$566$	0	0
$567$	−4.96239	−0.208401
$568$	0	0
$569$	22.1598	0.928989	0.464494	−	0.885576i	$-0.346236\pi$
$569$	22.1598	0.928989	0.464494	+	0.885576i	$0.346236\pi$
$570$	0	0
$571$	−17.2774	−0.723037	−0.361519	−	0.932365i	$-0.617742\pi$
$571$	−17.2774	−0.723037	−0.361519	+	0.932365i	$0.617742\pi$
$572$	0	0
$573$	16.5442	0.691144
$574$	0	0
$575$	39.1557	1.63290
$576$	0	0
$577$	−14.0157	−0.583482	−0.291741	−	0.956497i	$-0.594235\pi$
$577$	−14.0157	−0.583482	−0.291741	+	0.956497i	$0.594235\pi$
$578$	0	0
$579$	2.31265	0.0961105
$580$	0	0
$581$	28.7757	1.19382
$582$	0	0
$583$	−10.5647	−0.437544
$584$	0	0
$585$	0.649738	0.0268634
$586$	0	0
$587$	−26.9525	−1.11245	−0.556225	−	0.831032i	$-0.687751\pi$
$587$	−26.9525	−1.11245	−0.556225	+	0.831032i	$0.687751\pi$
$588$	0	0
$589$	−2.59895	−0.107088
$590$	0	0
$591$	−0.763527	−0.0314073
$592$	0	0
$593$	13.5393	0.555991	0.277996	−	0.960582i	$-0.410330\pi$
$593$	13.5393	0.555991	0.277996	+	0.960582i	$0.410330\pi$
$594$	0	0
$595$	−8.10157	−0.332132
$596$	0	0
$597$	4.12601	0.168866
$598$	0	0
$599$	−36.4749	−1.49032	−0.745161	−	0.666885i	$-0.767627\pi$
$599$	−36.4749	−1.49032	−0.745161	+	0.666885i	$0.767627\pi$
$600$	0	0
$601$	12.1768	0.496702	0.248351	−	0.968670i	$-0.420111\pi$
$601$	12.1768	0.496702	0.248351	+	0.968670i	$0.420111\pi$
$602$	0	0
$603$	−6.71274	−0.273364
$604$	0	0
$605$	−2.06300	−0.0838730
$606$	0	0
$607$	18.0122	0.731093	0.365547	−	0.930793i	$-0.380882\pi$
$607$	18.0122	0.731093	0.365547	+	0.930793i	$0.380882\pi$
$608$	0	0
$609$	11.4763	0.465042
$610$	0	0
$611$	20.9380	0.847059
$612$	0	0
$613$	−9.11871	−0.368301	−0.184151	−	0.982898i	$-0.558953\pi$
$613$	−9.11871	−0.368301	−0.184151	+	0.982898i	$0.558953\pi$
$614$	0	0
$615$	−3.20379	−0.129189
$616$	0	0
$617$	7.92478	0.319040	0.159520	−	0.987195i	$-0.449005\pi$
$617$	7.92478	0.319040	0.159520	+	0.987195i	$0.449005\pi$
$618$	0	0
$619$	40.6375	1.63336	0.816680	−	0.577091i	$-0.195812\pi$
$619$	40.6375	1.63336	0.816680	+	0.577091i	$0.195812\pi$
$620$	0	0
$621$	8.00000	0.321029
$622$	0	0
$623$	−60.7269	−2.43297
$624$	0	0
$625$	23.4280	0.937122
$626$	0	0
$627$	−3.47627	−0.138829
$628$	0	0
$629$	−34.9887	−1.39509
$630$	0	0
$631$	−13.1246	−0.522482	−0.261241	−	0.965274i	$-0.584132\pi$
$631$	−13.1246	−0.522482	−0.261241	+	0.965274i	$0.584132\pi$
$632$	0	0
$633$	−22.5501	−0.896285
$634$	0	0
$635$	−4.31265	−0.171142
$636$	0	0
$637$	35.2506	1.39668
$638$	0	0
$639$	−8.12601	−0.321460
$640$	0	0
$641$	−23.8011	−0.940088	−0.470044	−	0.882643i	$-0.655762\pi$
$641$	−23.8011	−0.940088	−0.470044	+	0.882643i	$0.655762\pi$
$642$	0	0
$643$	−46.7001	−1.84167	−0.920835	−	0.389952i	$-0.872492\pi$
$643$	−46.7001	−1.84167	−0.920835	+	0.389952i	$0.872492\pi$
$644$	0	0
$645$	1.31994	0.0519728
$646$	0	0
$647$	40.1984	1.58036	0.790181	−	0.612873i	$-0.209986\pi$
$647$	40.1984	1.58036	0.790181	+	0.612873i	$0.209986\pi$
$648$	0	0
$649$	17.9248	0.703609
$650$	0	0
$651$	−8.00000	−0.313545
$652$	0	0
$653$	−1.47627	−0.0577709	−0.0288854	−	0.999583i	$-0.509196\pi$
$653$	−1.47627	−0.0577709	−0.0288854	+	0.999583i	$0.509196\pi$
$654$	0	0
$655$	−3.78892	−0.148045
$656$	0	0
$657$	4.26187	0.166271
$658$	0	0
$659$	−33.8350	−1.31802	−0.659012	−	0.752133i	$-0.729025\pi$
$659$	−33.8350	−1.31802	−0.659012	+	0.752133i	$0.729025\pi$
$660$	0	0
$661$	−34.6907	−1.34931	−0.674655	−	0.738133i	$-0.735708\pi$
$661$	−34.6907	−1.34931	−0.674655	+	0.738133i	$0.735708\pi$
$662$	0	0
$663$	−10.0508	−0.390340
$664$	0	0
$665$	−2.59895	−0.100783
$666$	0	0
$667$	−18.5012	−0.716369
$668$	0	0
$669$	4.96239	0.191857
$670$	0	0
$671$	−0.0653737	−0.00252372
$672$	0	0
$673$	−12.0263	−0.463582	−0.231791	−	0.972766i	$-0.574458\pi$
$673$	−12.0263	−0.463582	−0.231791	+	0.972766i	$0.574458\pi$
$674$	0	0
$675$	4.89446	0.188388
$676$	0	0
$677$	26.2228	1.00783	0.503913	−	0.863755i	$-0.331893\pi$
$677$	26.2228	1.00783	0.503913	+	0.863755i	$0.331893\pi$
$678$	0	0
$679$	39.9511	1.53318
$680$	0	0
$681$	4.25202	0.162938
$682$	0	0
$683$	−8.49929	−0.325216	−0.162608	−	0.986691i	$-0.551991\pi$
$683$	−8.49929	−0.325216	−0.162608	+	0.986691i	$0.551991\pi$
$684$	0	0
$685$	−4.43866	−0.169592
$686$	0	0
$687$	−10.6702	−0.407094
$688$	0	0
$689$	−9.79877	−0.373303
$690$	0	0
$691$	30.1236	1.14596	0.572979	−	0.819570i	$-0.305788\pi$
$691$	30.1236	1.14596	0.572979	+	0.819570i	$0.305788\pi$
$692$	0	0
$693$	−10.7005	−0.406479
$694$	0	0
$695$	3.35423	0.127233
$696$	0	0
$697$	49.5593	1.87719
$698$	0	0
$699$	−3.58769	−0.135699
$700$	0	0
$701$	−13.8251	−0.522167	−0.261084	−	0.965316i	$-0.584080\pi$
$701$	−13.8251	−0.522167	−0.261084	+	0.965316i	$0.584080\pi$
$702$	0	0
$703$	−11.2243	−0.423331
$704$	0	0
$705$	−3.40105	−0.128091
$706$	0	0
$707$	6.38787	0.240241
$708$	0	0
$709$	−17.0738	−0.641220	−0.320610	−	0.947211i	$-0.603888\pi$
$709$	−17.0738	−0.641220	−0.320610	+	0.947211i	$0.603888\pi$
$710$	0	0
$711$	15.3380	0.575222
$712$	0	0
$713$	12.8970	0.482997
$714$	0	0
$715$	1.40105	0.0523962
$716$	0	0
$717$	26.3185	0.982884
$718$	0	0
$719$	32.7513	1.22142	0.610709	−	0.791855i	$-0.290885\pi$
$719$	32.7513	1.22142	0.610709	+	0.791855i	$0.290885\pi$
$720$	0	0
$721$	46.0870	1.71637
$722$	0	0
$723$	20.1114	0.747952
$724$	0	0
$725$	−11.3192	−0.420384
$726$	0	0
$727$	23.1006	0.856754	0.428377	−	0.903600i	$-0.359085\pi$
$727$	23.1006	0.856754	0.428377	+	0.903600i	$0.359085\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−20.4182	−0.755194
$732$	0	0
$733$	42.9946	1.58804	0.794021	−	0.607890i	$-0.207984\pi$
$733$	42.9946	1.58804	0.794021	+	0.607890i	$0.207984\pi$
$734$	0	0
$735$	−5.72592	−0.211204
$736$	0	0
$737$	−14.4749	−0.533188
$738$	0	0
$739$	−10.6229	−0.390771	−0.195385	−	0.980727i	$-0.562596\pi$
$739$	−10.6229	−0.390771	−0.195385	+	0.980727i	$0.562596\pi$
$740$	0	0
$741$	−3.22425	−0.118446
$742$	0	0
$743$	0.373285	0.0136945	0.00684724	−	0.999977i	$-0.497820\pi$
$743$	0.373285	0.0136945	0.00684724	+	0.999977i	$0.497820\pi$
$744$	0	0
$745$	−4.58513	−0.167986
$746$	0	0
$747$	−5.79877	−0.212166
$748$	0	0
$749$	18.7005	0.683302
$750$	0	0
$751$	49.2120	1.79577	0.897886	−	0.440227i	$-0.145102\pi$
$751$	49.2120	1.79577	0.897886	+	0.440227i	$0.145102\pi$
$752$	0	0
$753$	9.21440	0.335792
$754$	0	0
$755$	−3.55737	−0.129466
$756$	0	0
$757$	−32.6516	−1.18674	−0.593372	−	0.804928i	$-0.702204\pi$
$757$	−32.6516	−1.18674	−0.593372	+	0.804928i	$0.702204\pi$
$758$	0	0
$759$	17.2506	0.626157
$760$	0	0
$761$	8.44851	0.306258	0.153129	−	0.988206i	$-0.451065\pi$
$761$	8.44851	0.306258	0.153129	+	0.988206i	$0.451065\pi$
$762$	0	0
$763$	80.5764	2.91706
$764$	0	0
$765$	1.63259	0.0590266
$766$	0	0
$767$	16.6253	0.600305
$768$	0	0
$769$	14.3733	0.518314	0.259157	−	0.965835i	$-0.416555\pi$
$769$	14.3733	0.518314	0.259157	+	0.965835i	$0.416555\pi$
$770$	0	0
$771$	20.8242	0.749964
$772$	0	0
$773$	24.5115	0.881618	0.440809	−	0.897601i	$-0.354692\pi$
$773$	24.5115	0.881618	0.440809	+	0.897601i	$0.354692\pi$
$774$	0	0
$775$	7.89049	0.283435
$776$	0	0
$777$	−34.5501	−1.23948
$778$	0	0
$779$	15.8984	0.569620
$780$	0	0
$781$	−17.5223	−0.626998
$782$	0	0
$783$	−2.31265	−0.0826474
$784$	0	0
$785$	−4.96239	−0.177115
$786$	0	0
$787$	−7.59991	−0.270907	−0.135454	−	0.990784i	$-0.543249\pi$
$787$	−7.59991	−0.270907	−0.135454	+	0.990784i	$0.543249\pi$
$788$	0	0
$789$	−3.28963	−0.117114
$790$	0	0
$791$	31.0132	1.10270
$792$	0	0
$793$	−0.0606343	−0.00215319
$794$	0	0
$795$	1.59166	0.0564503
$796$	0	0
$797$	31.5999	1.11933	0.559663	−	0.828720i	$-0.310931\pi$
$797$	31.5999	1.11933	0.559663	+	0.828720i	$0.310931\pi$
$798$	0	0
$799$	52.6107	1.86123
$800$	0	0
$801$	12.2374	0.432388
$802$	0	0
$803$	9.18997	0.324307
$804$	0	0
$805$	12.8970	0.454560
$806$	0	0
$807$	5.54912	0.195338
$808$	0	0
$809$	32.6155	1.14670	0.573349	−	0.819311i	$-0.305644\pi$
$809$	32.6155	1.14670	0.573349	+	0.819311i	$0.305644\pi$
$810$	0	0
$811$	21.7358	0.763246	0.381623	−	0.924318i	$-0.375365\pi$
$811$	21.7358	0.763246	0.381623	+	0.924318i	$0.375365\pi$
$812$	0	0
$813$	2.89938	0.101686
$814$	0	0
$815$	−3.35423	−0.117494
$816$	0	0
$817$	−6.55008	−0.229158
$818$	0	0
$819$	−9.92478	−0.346800
$820$	0	0
$821$	−30.7997	−1.07492	−0.537459	−	0.843290i	$-0.680616\pi$
$821$	−30.7997	−1.07492	−0.537459	+	0.843290i	$0.680616\pi$
$822$	0	0
$823$	−0.324869	−0.0113242	−0.00566211	−	0.999984i	$-0.501802\pi$
$823$	−0.324869	−0.0113242	−0.00566211	+	0.999984i	$0.501802\pi$
$824$	0	0
$825$	10.5540	0.367445
$826$	0	0
$827$	17.2506	0.599862	0.299931	−	0.953961i	$-0.403036\pi$
$827$	17.2506	0.599862	0.299931	+	0.953961i	$0.403036\pi$
$828$	0	0
$829$	−7.76257	−0.269605	−0.134803	−	0.990872i	$-0.543040\pi$
$829$	−7.76257	−0.269605	−0.134803	+	0.990872i	$0.543040\pi$
$830$	0	0
$831$	−31.5633	−1.09492
$832$	0	0
$833$	88.5741	3.06891
$834$	0	0
$835$	0.324869	0.0112426
$836$	0	0
$837$	1.61213	0.0557233
$838$	0	0
$839$	21.3317	0.736452	0.368226	−	0.929736i	$-0.379965\pi$
$839$	21.3317	0.736452	0.368226	+	0.929736i	$0.379965\pi$
$840$	0	0
$841$	−23.6516	−0.815574
$842$	0	0
$843$	−15.5877	−0.536869
$844$	0	0
$845$	−2.92382	−0.100583
$846$	0	0
$847$	31.5125	1.08278
$848$	0	0
$849$	−31.0132	−1.06437
$850$	0	0
$851$	55.6991	1.90934
$852$	0	0
$853$	0.0303172	0.00103804	0.000519020	−	1.00000i	$-0.499835\pi$
$853$	0.0303172	0.00103804	0.000519020		1.00000i	$0.499835\pi$
$854$	0	0
$855$	0.523730	0.0179112
$856$	0	0
$857$	18.6253	0.636228	0.318114	−	0.948052i	$-0.396950\pi$
$857$	18.6253	0.636228	0.318114	+	0.948052i	$0.396950\pi$
$858$	0	0
$859$	−6.57452	−0.224320	−0.112160	−	0.993690i	$-0.535777\pi$
$859$	−6.57452	−0.224320	−0.112160	+	0.993690i	$0.535777\pi$
$860$	0	0
$861$	48.9380	1.66780
$862$	0	0
$863$	−47.7645	−1.62592	−0.812961	−	0.582318i	$-0.802146\pi$
$863$	−47.7645	−1.62592	−0.812961	+	0.582318i	$0.802146\pi$
$864$	0	0
$865$	3.50071	0.119028
$866$	0	0
$867$	−8.25457	−0.280340
$868$	0	0
$869$	33.0738	1.12195
$870$	0	0
$871$	−13.4255	−0.454905
$872$	0	0
$873$	−8.05079	−0.272478
$874$	0	0
$875$	15.9511	0.539246
$876$	0	0
$877$	36.2736	1.22487	0.612437	−	0.790520i	$-0.290189\pi$
$877$	36.2736	1.22487	0.612437	+	0.790520i	$0.290189\pi$
$878$	0	0
$879$	16.2981	0.549720
$880$	0	0
$881$	−3.33804	−0.112462	−0.0562308	−	0.998418i	$-0.517908\pi$
$881$	−3.33804	−0.112462	−0.0562308	+	0.998418i	$0.517908\pi$
$882$	0	0
$883$	49.0395	1.65031	0.825156	−	0.564905i	$-0.191087\pi$
$883$	49.0395	1.65031	0.825156	+	0.564905i	$0.191087\pi$
$884$	0	0
$885$	−2.70052	−0.0907771
$886$	0	0
$887$	−30.8383	−1.03545	−0.517724	−	0.855548i	$-0.673221\pi$
$887$	−30.8383	−1.03545	−0.517724	+	0.855548i	$0.673221\pi$
$888$	0	0
$889$	65.8759	2.20941
$890$	0	0
$891$	2.15633	0.0722396
$892$	0	0
$893$	16.8773	0.564778
$894$	0	0
$895$	−6.07522	−0.203072
$896$	0	0
$897$	16.0000	0.534224
$898$	0	0
$899$	−3.72829	−0.124345
$900$	0	0
$901$	−24.6213	−0.820255
$902$	0	0
$903$	−20.1622	−0.670956
$904$	0	0
$905$	−2.54023	−0.0844401
$906$	0	0
$907$	−39.7235	−1.31900	−0.659499	−	0.751705i	$-0.729232\pi$
$907$	−39.7235	−1.31900	−0.659499	+	0.751705i	$0.729232\pi$
$908$	0	0
$909$	−1.28726	−0.0426956
$910$	0	0
$911$	56.2031	1.86209	0.931047	−	0.364900i	$-0.118897\pi$
$911$	56.2031	1.86209	0.931047	+	0.364900i	$0.118897\pi$
$912$	0	0
$913$	−12.5040	−0.413823
$914$	0	0
$915$	0.00984911	0.000325602 0
$916$	0	0
$917$	57.8759	1.91123
$918$	0	0
$919$	−42.7974	−1.41175	−0.705877	−	0.708334i	$-0.749447\pi$
$919$	−42.7974	−1.41175	−0.705877	+	0.708334i	$0.749447\pi$
$920$	0	0
$921$	−4.24965	−0.140031
$922$	0	0
$923$	−16.2520	−0.534942
$924$	0	0
$925$	34.0771	1.12045
$926$	0	0
$927$	−9.28726	−0.305034
$928$	0	0
$929$	−60.6516	−1.98992	−0.994958	−	0.100292i	$-0.968022\pi$
$929$	−60.6516	−1.98992	−0.994958	+	0.100292i	$0.968022\pi$
$930$	0	0
$931$	28.4142	0.931238
$932$	0	0
$933$	−18.1114	−0.592941
$934$	0	0
$935$	3.52041	0.115130
$936$	0	0
$937$	−6.21108	−0.202907	−0.101454	−	0.994840i	$-0.532349\pi$
$937$	−6.21108	−0.202907	−0.101454	+	0.994840i	$0.532349\pi$
$938$	0	0
$939$	2.77575	0.0905831
$940$	0	0
$941$	31.5901	1.02981	0.514903	−	0.857248i	$-0.327828\pi$
$941$	31.5901	1.02981	0.514903	+	0.857248i	$0.327828\pi$
$942$	0	0
$943$	−78.8942	−2.56915
$944$	0	0
$945$	1.61213	0.0524425
$946$	0	0
$947$	−15.9795	−0.519265	−0.259633	−	0.965707i	$-0.583601\pi$
$947$	−15.9795	−0.519265	−0.259633	+	0.965707i	$0.583601\pi$
$948$	0	0
$949$	8.52373	0.276692
$950$	0	0
$951$	−13.5369	−0.438964
$952$	0	0
$953$	−10.2741	−0.332810	−0.166405	−	0.986057i	$-0.553216\pi$
$953$	−10.2741	−0.332810	−0.166405	+	0.986057i	$0.553216\pi$
$954$	0	0
$955$	−5.37470	−0.173921
$956$	0	0
$957$	−4.98683	−0.161201
$958$	0	0
$959$	67.8007	2.18940
$960$	0	0
$961$	−28.4010	−0.916163
$962$	0	0
$963$	−3.76845	−0.121437
$964$	0	0
$965$	−0.751309	−0.0241855
$966$	0	0
$967$	−40.9741	−1.31764	−0.658820	−	0.752301i	$-0.728944\pi$
$967$	−40.9741	−1.31764	−0.658820	+	0.752301i	$0.728944\pi$
$968$	0	0
$969$	−8.10157	−0.260260
$970$	0	0
$971$	42.6155	1.36759	0.683797	−	0.729672i	$-0.260327\pi$
$971$	42.6155	1.36759	0.683797	+	0.729672i	$0.260327\pi$
$972$	0	0
$973$	−51.2360	−1.64255
$974$	0	0
$975$	9.78892	0.313496
$976$	0	0
$977$	−19.5148	−0.624335	−0.312167	−	0.950027i	$-0.601055\pi$
$977$	−19.5148	−0.624335	−0.312167	+	0.950027i	$0.601055\pi$
$978$	0	0
$979$	26.3879	0.843360
$980$	0	0
$981$	−16.2374	−0.518421
$982$	0	0
$983$	−15.5125	−0.494771	−0.247385	−	0.968917i	$-0.579571\pi$
$983$	−15.5125	−0.494771	−0.247385	+	0.968917i	$0.579571\pi$
$984$	0	0
$985$	0.248047	0.00790342
$986$	0	0
$987$	51.9511	1.65362
$988$	0	0
$989$	32.5040	1.03357
$990$	0	0
$991$	28.2351	0.896916	0.448458	−	0.893804i	$-0.351973\pi$
$991$	28.2351	0.896916	0.448458	+	0.893804i	$0.351973\pi$
$992$	0	0
$993$	−33.8881	−1.07541
$994$	0	0
$995$	−1.34041	−0.0424939
$996$	0	0
$997$	−28.2823	−0.895710	−0.447855	−	0.894106i	$-0.647812\pi$
$997$	−28.2823	−0.895710	−0.447855	+	0.894106i	$0.647812\pi$
$998$	0	0
$999$	6.96239	0.220280

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.e.1.1	✓	3
4.3	odd	2		8016.2.a.n.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.e.1.1	✓	3	1.1	even	1	trivial
8016.2.a.n.1.1		3	4.3	odd	2

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:	\( N \)	\(=\)	\( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4008.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.324869 q^{5} -4.96239 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.324869 q^{5} -4.96239 q^{7} +1.00000 q^{9} +2.15633 q^{11} +2.00000 q^{13} -0.324869 q^{15} +5.02539 q^{17} +1.61213 q^{19} +4.96239 q^{21} -8.00000 q^{23} -4.89446 q^{25} -1.00000 q^{27} +2.31265 q^{29} -1.61213 q^{31} -2.15633 q^{33} -1.61213 q^{35} -6.96239 q^{37} -2.00000 q^{39} +9.86177 q^{41} -4.06300 q^{43} +0.324869 q^{45} +10.4690 q^{47} +17.6253 q^{49} -5.02539 q^{51} -4.89938 q^{53} +0.700523 q^{55} -1.61213 q^{57} +8.31265 q^{59} -0.0303172 q^{61} -4.96239 q^{63} +0.649738 q^{65} -6.71274 q^{67} +8.00000 q^{69} -8.12601 q^{71} +4.26187 q^{73} +4.89446 q^{75} -10.7005 q^{77} +15.3380 q^{79} +1.00000 q^{81} -5.79877 q^{83} +1.63259 q^{85} -2.31265 q^{87} +12.2374 q^{89} -9.92478 q^{91} +1.61213 q^{93} +0.523730 q^{95} -8.05079 q^{97} +2.15633 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 6 q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 6 * q^5 - 4 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 6 q^{5} - 4 q^{7} + 3 q^{9} - 4 q^{11} + 6 q^{13} - 6 q^{15} + 4 q^{19} + 4 q^{21} - 24 q^{23} + 5 q^{25} - 3 q^{27} - 14 q^{29} - 4 q^{31} + 4 q^{33} - 4 q^{35} - 10 q^{37} - 6 q^{39} + 12 q^{41} - 8 q^{43} + 6 q^{45} + 11 q^{49} - 8 q^{53} - 18 q^{55} - 4 q^{57} + 4 q^{59} + 2 q^{61} - 4 q^{63} + 12 q^{65} - 26 q^{67} + 24 q^{69} - 16 q^{71} + 22 q^{73} - 5 q^{75} - 12 q^{77} + 10 q^{79} + 3 q^{81} - 4 q^{83} - 24 q^{85} + 14 q^{87} - 6 q^{89} - 8 q^{91} + 4 q^{93} + 20 q^{95} + 6 q^{97} - 4 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 6 * q^5 - 4 * q^7 + 3 * q^9 - 4 * q^11 + 6 * q^13 - 6 * q^15 + 4 * q^19 + 4 * q^21 - 24 * q^23 + 5 * q^25 - 3 * q^27 - 14 * q^29 - 4 * q^31 + 4 * q^33 - 4 * q^35 - 10 * q^37 - 6 * q^39 + 12 * q^41 - 8 * q^43 + 6 * q^45 + 11 * q^49 - 8 * q^53 - 18 * q^55 - 4 * q^57 + 4 * q^59 + 2 * q^61 - 4 * q^63 + 12 * q^65 - 26 * q^67 + 24 * q^69 - 16 * q^71 + 22 * q^73 - 5 * q^75 - 12 * q^77 + 10 * q^79 + 3 * q^81 - 4 * q^83 - 24 * q^85 + 14 * q^87 - 6 * q^89 - 8 * q^91 + 4 * q^93 + 20 * q^95 + 6 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

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