LMFDB - Embedded newform 4008.2.a.g.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:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 7x^{5} + 11x^{4} + 12x^{3} - 14x^{2} - 6x + 3 $ x^7 - 2x^6 - 7x^5 + 11x^4 + 12x^3 - 14x^2 - 6x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.80982$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -3.20729 q^{5} +3.68779 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.20729 q^{5} +3.68779 q^{7} +1.00000 q^{9} +2.41235 q^{11} -0.292389 q^{13} +3.20729 q^{15} +7.19061 q^{17} -2.96484 q^{19} -3.68779 q^{21} +5.50281 q^{23} +5.28672 q^{25} -1.00000 q^{27} +3.32252 q^{29} -3.32412 q^{31} -2.41235 q^{33} -11.8278 q^{35} -0.357486 q^{37} +0.292389 q^{39} +3.49808 q^{41} -11.1409 q^{43} -3.20729 q^{45} -0.406706 q^{47} +6.59982 q^{49} -7.19061 q^{51} -10.6914 q^{53} -7.73710 q^{55} +2.96484 q^{57} -3.65193 q^{59} +9.48361 q^{61} +3.68779 q^{63} +0.937777 q^{65} +4.18677 q^{67} -5.50281 q^{69} +2.68125 q^{71} -6.27452 q^{73} -5.28672 q^{75} +8.89624 q^{77} +6.66323 q^{79} +1.00000 q^{81} +10.2090 q^{83} -23.0624 q^{85} -3.32252 q^{87} -4.51995 q^{89} -1.07827 q^{91} +3.32412 q^{93} +9.50909 q^{95} +1.77054 q^{97} +2.41235 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{3} - 3 q^{5} + 8 q^{7} + 7 q^{9}+O(q^{10}) $ 7 * q - 7 * q^3 - 3 * q^5 + 8 * q^7 + 7 * q^9	$ 7 q - 7 q^{3} - 3 q^{5} + 8 q^{7} + 7 q^{9} + q^{11} - 2 q^{13} + 3 q^{15} + 11 q^{17} + 2 q^{19} - 8 q^{21} + 17 q^{23} + 4 q^{25} - 7 q^{27} - 7 q^{29} + 10 q^{31} - q^{33} + 10 q^{35} - 21 q^{37} + 2 q^{39} + 8 q^{41} - 12 q^{43} - 3 q^{45} + 25 q^{47} - 7 q^{49} - 11 q^{51} - 7 q^{53} + 15 q^{55} - 2 q^{57} + 3 q^{59} - 14 q^{61} + 8 q^{63} + 4 q^{65} + 4 q^{67} - 17 q^{69} + 27 q^{71} - 12 q^{73} - 4 q^{75} + 16 q^{77} + 8 q^{79} + 7 q^{81} + 15 q^{83} - 3 q^{85} + 7 q^{87} + 14 q^{89} - 3 q^{91} - 10 q^{93} + 37 q^{95} + 3 q^{97} + q^{99}+O(q^{100}) $ 7 * q - 7 * q^3 - 3 * q^5 + 8 * q^7 + 7 * q^9 + q^11 - 2 * q^13 + 3 * q^15 + 11 * q^17 + 2 * q^19 - 8 * q^21 + 17 * q^23 + 4 * q^25 - 7 * q^27 - 7 * q^29 + 10 * q^31 - q^33 + 10 * q^35 - 21 * q^37 + 2 * q^39 + 8 * q^41 - 12 * q^43 - 3 * q^45 + 25 * q^47 - 7 * q^49 - 11 * q^51 - 7 * q^53 + 15 * q^55 - 2 * q^57 + 3 * q^59 - 14 * q^61 + 8 * q^63 + 4 * q^65 + 4 * q^67 - 17 * q^69 + 27 * q^71 - 12 * q^73 - 4 * q^75 + 16 * q^77 + 8 * q^79 + 7 * q^81 + 15 * q^83 - 3 * q^85 + 7 * q^87 + 14 * q^89 - 3 * q^91 - 10 * q^93 + 37 * q^95 + 3 * q^97 + 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
$5$	−3.20729	−1.43434	−0.717172	−	0.696896i	$-0.754564\pi$
$5$	−3.20729	−1.43434	−0.717172	+	0.696896i	$0.754564\pi$
$6$	0	0
$7$	3.68779	1.39385	0.696927	−	0.717142i	$-0.254550\pi$
$7$	3.68779	1.39385	0.696927	+	0.717142i	$0.254550\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.41235	0.727350	0.363675	−	0.931526i	$-0.381522\pi$
$11$	2.41235	0.727350	0.363675	+	0.931526i	$0.381522\pi$
$12$	0	0
$13$	−0.292389	−0.0810942	−0.0405471	−	0.999178i	$-0.512910\pi$
$13$	−0.292389	−0.0810942	−0.0405471	+	0.999178i	$0.512910\pi$
$14$	0	0
$15$	3.20729	0.828119
$16$	0	0
$17$	7.19061	1.74398	0.871989	−	0.489525i	$-0.162830\pi$
$17$	7.19061	1.74398	0.871989	+	0.489525i	$0.162830\pi$
$18$	0	0
$19$	−2.96484	−0.680180	−0.340090	−	0.940393i	$-0.610458\pi$
$19$	−2.96484	−0.680180	−0.340090	+	0.940393i	$0.610458\pi$
$20$	0	0
$21$	−3.68779	−0.804743
$22$	0	0
$23$	5.50281	1.14742	0.573708	−	0.819060i	$-0.305504\pi$
$23$	5.50281	1.14742	0.573708	+	0.819060i	$0.305504\pi$
$24$	0	0
$25$	5.28672	1.05734
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.32252	0.616976	0.308488	−	0.951228i	$-0.400177\pi$
$29$	3.32252	0.616976	0.308488	+	0.951228i	$0.400177\pi$
$30$	0	0
$31$	−3.32412	−0.597029	−0.298514	−	0.954405i	$-0.596491\pi$
$31$	−3.32412	−0.597029	−0.298514	+	0.954405i	$0.596491\pi$
$32$	0	0
$33$	−2.41235	−0.419936
$34$	0	0
$35$	−11.8278	−1.99927
$36$	0	0
$37$	−0.357486	−0.0587703	−0.0293852	−	0.999568i	$-0.509355\pi$
$37$	−0.357486	−0.0587703	−0.0293852	+	0.999568i	$0.509355\pi$
$38$	0	0
$39$	0.292389	0.0468197
$40$	0	0
$41$	3.49808	0.546309	0.273154	−	0.961970i	$-0.411933\pi$
$41$	3.49808	0.546309	0.273154	+	0.961970i	$0.411933\pi$
$42$	0	0
$43$	−11.1409	−1.69898	−0.849488	−	0.527608i	$-0.823089\pi$
$43$	−11.1409	−1.69898	−0.849488	+	0.527608i	$0.823089\pi$
$44$	0	0
$45$	−3.20729	−0.478115
$46$	0	0
$47$	−0.406706	−0.0593241	−0.0296621	−	0.999560i	$-0.509443\pi$
$47$	−0.406706	−0.0593241	−0.0296621	+	0.999560i	$0.509443\pi$
$48$	0	0
$49$	6.59982	0.942832
$50$	0	0
$51$	−7.19061	−1.00689
$52$	0	0
$53$	−10.6914	−1.46858	−0.734291	−	0.678834i	$-0.762485\pi$
$53$	−10.6914	−1.46858	−0.734291	+	0.678834i	$0.762485\pi$
$54$	0	0
$55$	−7.73710	−1.04327
$56$	0	0
$57$	2.96484	0.392702
$58$	0	0
$59$	−3.65193	−0.475441	−0.237720	−	0.971334i	$-0.576400\pi$
$59$	−3.65193	−0.475441	−0.237720	+	0.971334i	$0.576400\pi$
$60$	0	0
$61$	9.48361	1.21425	0.607126	−	0.794606i	$-0.292322\pi$
$61$	9.48361	1.21425	0.607126	+	0.794606i	$0.292322\pi$
$62$	0	0
$63$	3.68779	0.464618
$64$	0	0
$65$	0.937777	0.116317
$66$	0	0
$67$	4.18677	0.511496	0.255748	−	0.966744i	$-0.417678\pi$
$67$	4.18677	0.511496	0.255748	+	0.966744i	$0.417678\pi$
$68$	0	0
$69$	−5.50281	−0.662461
$70$	0	0
$71$	2.68125	0.318206	0.159103	−	0.987262i	$-0.449140\pi$
$71$	2.68125	0.318206	0.159103	+	0.987262i	$0.449140\pi$
$72$	0	0
$73$	−6.27452	−0.734377	−0.367189	−	0.930147i	$-0.619680\pi$
$73$	−6.27452	−0.734377	−0.367189	+	0.930147i	$0.619680\pi$
$74$	0	0
$75$	−5.28672	−0.610457
$76$	0	0
$77$	8.89624	1.01382
$78$	0	0
$79$	6.66323	0.749672	0.374836	−	0.927091i	$-0.377699\pi$
$79$	6.66323	0.749672	0.374836	+	0.927091i	$0.377699\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.2090	1.12059	0.560293	−	0.828295i	$-0.310689\pi$
$83$	10.2090	1.12059	0.560293	+	0.828295i	$0.310689\pi$
$84$	0	0
$85$	−23.0624	−2.50147
$86$	0	0
$87$	−3.32252	−0.356211
$88$	0	0
$89$	−4.51995	−0.479114	−0.239557	−	0.970882i	$-0.577002\pi$
$89$	−4.51995	−0.479114	−0.239557	+	0.970882i	$0.577002\pi$
$90$	0	0
$91$	−1.07827	−0.113034
$92$	0	0
$93$	3.32412	0.344695
$94$	0	0
$95$	9.50909	0.975613
$96$	0	0
$97$	1.77054	0.179771	0.0898853	−	0.995952i	$-0.471350\pi$
$97$	1.77054	0.179771	0.0898853	+	0.995952i	$0.471350\pi$
$98$	0	0
$99$	2.41235	0.242450
$100$	0	0
$101$	18.8099	1.87165	0.935826	−	0.352463i	$-0.114656\pi$
$101$	18.8099	1.87165	0.935826	+	0.352463i	$0.114656\pi$
$102$	0	0
$103$	5.82075	0.573536	0.286768	−	0.958000i	$-0.407419\pi$
$103$	5.82075	0.573536	0.286768	+	0.958000i	$0.407419\pi$
$104$	0	0
$105$	11.8278	1.15428
$106$	0	0
$107$	9.11502	0.881182	0.440591	−	0.897708i	$-0.354769\pi$
$107$	9.11502	0.881182	0.440591	+	0.897708i	$0.354769\pi$
$108$	0	0
$109$	−18.8744	−1.80784	−0.903922	−	0.427697i	$-0.859325\pi$
$109$	−18.8744	−1.80784	−0.903922	+	0.427697i	$0.859325\pi$
$110$	0	0
$111$	0.357486	0.0339311
$112$	0	0
$113$	12.5304	1.17876	0.589381	−	0.807855i	$-0.299372\pi$
$113$	12.5304	1.17876	0.589381	+	0.807855i	$0.299372\pi$
$114$	0	0
$115$	−17.6491	−1.64579
$116$	0	0
$117$	−0.292389	−0.0270314
$118$	0	0
$119$	26.5175	2.43085
$120$	0	0
$121$	−5.18058	−0.470962
$122$	0	0
$123$	−3.49808	−0.315412
$124$	0	0
$125$	−0.919585	−0.0822502
$126$	0	0
$127$	−1.13671	−0.100867	−0.0504333	−	0.998727i	$-0.516060\pi$
$127$	−1.13671	−0.100867	−0.0504333	+	0.998727i	$0.516060\pi$
$128$	0	0
$129$	11.1409	0.980904
$130$	0	0
$131$	12.1630	1.06269	0.531343	−	0.847157i	$-0.321688\pi$
$131$	12.1630	1.06269	0.531343	+	0.847157i	$0.321688\pi$
$132$	0	0
$133$	−10.9337	−0.948072
$134$	0	0
$135$	3.20729	0.276040
$136$	0	0
$137$	−14.4346	−1.23323	−0.616616	−	0.787264i	$-0.711497\pi$
$137$	−14.4346	−1.23323	−0.616616	+	0.787264i	$0.711497\pi$
$138$	0	0
$139$	−0.433633	−0.0367803	−0.0183902	−	0.999831i	$-0.505854\pi$
$139$	−0.433633	−0.0367803	−0.0183902	+	0.999831i	$0.505854\pi$
$140$	0	0
$141$	0.406706	0.0342508
$142$	0	0
$143$	−0.705344	−0.0589839
$144$	0	0
$145$	−10.6563	−0.884956
$146$	0	0
$147$	−6.59982	−0.544344
$148$	0	0
$149$	−19.0992	−1.56466	−0.782332	−	0.622862i	$-0.785970\pi$
$149$	−19.0992	−1.56466	−0.782332	+	0.622862i	$0.785970\pi$
$150$	0	0
$151$	8.70546	0.708440	0.354220	−	0.935162i	$-0.384746\pi$
$151$	8.70546	0.708440	0.354220	+	0.935162i	$0.384746\pi$
$152$	0	0
$153$	7.19061	0.581326
$154$	0	0
$155$	10.6614	0.856345
$156$	0	0
$157$	22.6106	1.80452	0.902261	−	0.431191i	$-0.141907\pi$
$157$	22.6106	1.80452	0.902261	+	0.431191i	$0.141907\pi$
$158$	0	0
$159$	10.6914	0.847887
$160$	0	0
$161$	20.2932	1.59933
$162$	0	0
$163$	−9.50323	−0.744350	−0.372175	−	0.928162i	$-0.621388\pi$
$163$	−9.50323	−0.744350	−0.372175	+	0.928162i	$0.621388\pi$
$164$	0	0
$165$	7.73710	0.602332
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−12.9145	−0.993424
$170$	0	0
$171$	−2.96484	−0.226727
$172$	0	0
$173$	3.88163	0.295115	0.147557	−	0.989053i	$-0.452859\pi$
$173$	3.88163	0.295115	0.147557	+	0.989053i	$0.452859\pi$
$174$	0	0
$175$	19.4963	1.47378
$176$	0	0
$177$	3.65193	0.274496
$178$	0	0
$179$	−7.65909	−0.572467	−0.286234	−	0.958160i	$-0.592403\pi$
$179$	−7.65909	−0.572467	−0.286234	+	0.958160i	$0.592403\pi$
$180$	0	0
$181$	9.78926	0.727630	0.363815	−	0.931471i	$-0.381474\pi$
$181$	9.78926	0.727630	0.363815	+	0.931471i	$0.381474\pi$
$182$	0	0
$183$	−9.48361	−0.701048
$184$	0	0
$185$	1.14656	0.0842969
$186$	0	0
$187$	17.3462	1.26848
$188$	0	0
$189$	−3.68779	−0.268248
$190$	0	0
$191$	14.9580	1.08232	0.541162	−	0.840918i	$-0.317985\pi$
$191$	14.9580	1.08232	0.541162	+	0.840918i	$0.317985\pi$
$192$	0	0
$193$	−0.569004	−0.0409578	−0.0204789	−	0.999790i	$-0.506519\pi$
$193$	−0.569004	−0.0409578	−0.0204789	+	0.999790i	$0.506519\pi$
$194$	0	0
$195$	−0.937777	−0.0671556
$196$	0	0
$197$	24.1326	1.71938	0.859688	−	0.510820i	$-0.170658\pi$
$197$	24.1326	1.71938	0.859688	+	0.510820i	$0.170658\pi$
$198$	0	0
$199$	10.6151	0.752482	0.376241	−	0.926522i	$-0.377216\pi$
$199$	10.6151	0.752482	0.376241	+	0.926522i	$0.377216\pi$
$200$	0	0
$201$	−4.18677	−0.295312
$202$	0	0
$203$	12.2528	0.859975
$204$	0	0
$205$	−11.2194	−0.783595
$206$	0	0
$207$	5.50281	0.382472
$208$	0	0
$209$	−7.15222	−0.494729
$210$	0	0
$211$	14.9203	1.02716	0.513579	−	0.858042i	$-0.328319\pi$
$211$	14.9203	1.02716	0.513579	+	0.858042i	$0.328319\pi$
$212$	0	0
$213$	−2.68125	−0.183716
$214$	0	0
$215$	35.7322	2.43692
$216$	0	0
$217$	−12.2587	−0.832171
$218$	0	0
$219$	6.27452	0.423993
$220$	0	0
$221$	−2.10246	−0.141427
$222$	0	0
$223$	−14.5962	−0.977431	−0.488715	−	0.872443i	$-0.662534\pi$
$223$	−14.5962	−0.977431	−0.488715	+	0.872443i	$0.662534\pi$
$224$	0	0
$225$	5.28672	0.352448
$226$	0	0
$227$	7.67692	0.509535	0.254767	−	0.967002i	$-0.418001\pi$
$227$	7.67692	0.509535	0.254767	+	0.967002i	$0.418001\pi$
$228$	0	0
$229$	−7.74042	−0.511501	−0.255751	−	0.966743i	$-0.582323\pi$
$229$	−7.74042	−0.511501	−0.255751	+	0.966743i	$0.582323\pi$
$230$	0	0
$231$	−8.89624	−0.585330
$232$	0	0
$233$	−15.6688	−1.02650	−0.513250	−	0.858239i	$-0.671558\pi$
$233$	−15.6688	−1.02650	−0.513250	+	0.858239i	$0.671558\pi$
$234$	0	0
$235$	1.30442	0.0850912
$236$	0	0
$237$	−6.66323	−0.432823
$238$	0	0
$239$	20.3027	1.31327	0.656636	−	0.754208i	$-0.271979\pi$
$239$	20.3027	1.31327	0.656636	+	0.754208i	$0.271979\pi$
$240$	0	0
$241$	7.43764	0.479100	0.239550	−	0.970884i	$-0.423000\pi$
$241$	7.43764	0.479100	0.239550	+	0.970884i	$0.423000\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−21.1675	−1.35235
$246$	0	0
$247$	0.866886	0.0551586
$248$	0	0
$249$	−10.2090	−0.646970
$250$	0	0
$251$	28.3028	1.78646	0.893228	−	0.449604i	$-0.148435\pi$
$251$	28.3028	1.78646	0.893228	+	0.449604i	$0.148435\pi$
$252$	0	0
$253$	13.2747	0.834573
$254$	0	0
$255$	23.0624	1.44422
$256$	0	0
$257$	−7.82120	−0.487873	−0.243936	−	0.969791i	$-0.578439\pi$
$257$	−7.82120	−0.487873	−0.243936	+	0.969791i	$0.578439\pi$
$258$	0	0
$259$	−1.31833	−0.0819173
$260$	0	0
$261$	3.32252	0.205659
$262$	0	0
$263$	−6.44120	−0.397182	−0.198591	−	0.980082i	$-0.563636\pi$
$263$	−6.44120	−0.397182	−0.198591	+	0.980082i	$0.563636\pi$
$264$	0	0
$265$	34.2906	2.10645
$266$	0	0
$267$	4.51995	0.276617
$268$	0	0
$269$	21.7914	1.32865	0.664323	−	0.747446i	$-0.268720\pi$
$269$	21.7914	1.32865	0.664323	+	0.747446i	$0.268720\pi$
$270$	0	0
$271$	−10.1627	−0.617340	−0.308670	−	0.951169i	$-0.599884\pi$
$271$	−10.1627	−0.617340	−0.308670	+	0.951169i	$0.599884\pi$
$272$	0	0
$273$	1.07827	0.0652599
$274$	0	0
$275$	12.7534	0.769059
$276$	0	0
$277$	5.38295	0.323430	0.161715	−	0.986838i	$-0.448297\pi$
$277$	5.38295	0.323430	0.161715	+	0.986838i	$0.448297\pi$
$278$	0	0
$279$	−3.32412	−0.199010
$280$	0	0
$281$	−17.2566	−1.02945	−0.514723	−	0.857357i	$-0.672105\pi$
$281$	−17.2566	−1.02945	−0.514723	+	0.857357i	$0.672105\pi$
$282$	0	0
$283$	−6.13015	−0.364399	−0.182200	−	0.983262i	$-0.558322\pi$
$283$	−6.13015	−0.364399	−0.182200	+	0.983262i	$0.558322\pi$
$284$	0	0
$285$	−9.50909	−0.563270
$286$	0	0
$287$	12.9002	0.761475
$288$	0	0
$289$	34.7048	2.04146
$290$	0	0
$291$	−1.77054	−0.103791
$292$	0	0
$293$	17.6848	1.03316	0.516578	−	0.856240i	$-0.327206\pi$
$293$	17.6848	1.03316	0.516578	+	0.856240i	$0.327206\pi$
$294$	0	0
$295$	11.7128	0.681945
$296$	0	0
$297$	−2.41235	−0.139979
$298$	0	0
$299$	−1.60896	−0.0930488
$300$	0	0
$301$	−41.0854	−2.36813
$302$	0	0
$303$	−18.8099	−1.08060
$304$	0	0
$305$	−30.4167	−1.74165
$306$	0	0
$307$	11.1101	0.634085	0.317042	−	0.948411i	$-0.397310\pi$
$307$	11.1101	0.634085	0.317042	+	0.948411i	$0.397310\pi$
$308$	0	0
$309$	−5.82075	−0.331131
$310$	0	0
$311$	28.3481	1.60747	0.803737	−	0.594985i	$-0.202842\pi$
$311$	28.3481	1.60747	0.803737	+	0.594985i	$0.202842\pi$
$312$	0	0
$313$	31.4209	1.77601	0.888006	−	0.459831i	$-0.152090\pi$
$313$	31.4209	1.77601	0.888006	+	0.459831i	$0.152090\pi$
$314$	0	0
$315$	−11.8278	−0.666423
$316$	0	0
$317$	−7.57882	−0.425669	−0.212834	−	0.977088i	$-0.568269\pi$
$317$	−7.57882	−0.425669	−0.212834	+	0.977088i	$0.568269\pi$
$318$	0	0
$319$	8.01507	0.448758
$320$	0	0
$321$	−9.11502	−0.508751
$322$	0	0
$323$	−21.3190	−1.18622
$324$	0	0
$325$	−1.54578	−0.0857444
$326$	0	0
$327$	18.8744	1.04376
$328$	0	0
$329$	−1.49985	−0.0826892
$330$	0	0
$331$	29.3509	1.61327	0.806636	−	0.591049i	$-0.201286\pi$
$331$	29.3509	1.61327	0.806636	+	0.591049i	$0.201286\pi$
$332$	0	0
$333$	−0.357486	−0.0195901
$334$	0	0
$335$	−13.4282	−0.733661
$336$	0	0
$337$	−24.2626	−1.32167	−0.660833	−	0.750533i	$-0.729797\pi$
$337$	−24.2626	−1.32167	−0.660833	+	0.750533i	$0.729797\pi$
$338$	0	0
$339$	−12.5304	−0.680559
$340$	0	0
$341$	−8.01892	−0.434249
$342$	0	0
$343$	−1.47578	−0.0796844
$344$	0	0
$345$	17.6491	0.950197
$346$	0	0
$347$	16.9314	0.908924	0.454462	−	0.890766i	$-0.349831\pi$
$347$	16.9314	0.908924	0.454462	+	0.890766i	$0.349831\pi$
$348$	0	0
$349$	2.41024	0.129017	0.0645087	−	0.997917i	$-0.479452\pi$
$349$	2.41024	0.129017	0.0645087	+	0.997917i	$0.479452\pi$
$350$	0	0
$351$	0.292389	0.0156066
$352$	0	0
$353$	−12.2607	−0.652570	−0.326285	−	0.945271i	$-0.605797\pi$
$353$	−12.2607	−0.652570	−0.326285	+	0.945271i	$0.605797\pi$
$354$	0	0
$355$	−8.59956	−0.456417
$356$	0	0
$357$	−26.5175	−1.40345
$358$	0	0
$359$	12.1673	0.642166	0.321083	−	0.947051i	$-0.395953\pi$
$359$	12.1673	0.642166	0.321083	+	0.947051i	$0.395953\pi$
$360$	0	0
$361$	−10.2097	−0.537355
$362$	0	0
$363$	5.18058	0.271910
$364$	0	0
$365$	20.1242	1.05335
$366$	0	0
$367$	7.82364	0.408391	0.204195	−	0.978930i	$-0.434542\pi$
$367$	7.82364	0.408391	0.204195	+	0.978930i	$0.434542\pi$
$368$	0	0
$369$	3.49808	0.182103
$370$	0	0
$371$	−39.4278	−2.04699
$372$	0	0
$373$	−18.1180	−0.938114	−0.469057	−	0.883168i	$-0.655406\pi$
$373$	−18.1180	−0.938114	−0.469057	+	0.883168i	$0.655406\pi$
$374$	0	0
$375$	0.919585	0.0474872
$376$	0	0
$377$	−0.971469	−0.0500332
$378$	0	0
$379$	0.116915	0.00600553	0.00300276	−	0.999995i	$-0.499044\pi$
$379$	0.116915	0.00600553	0.00300276	+	0.999995i	$0.499044\pi$
$380$	0	0
$381$	1.13671	0.0582354
$382$	0	0
$383$	−0.723537	−0.0369710	−0.0184855	−	0.999829i	$-0.505884\pi$
$383$	−0.723537	−0.0369710	−0.0184855	+	0.999829i	$0.505884\pi$
$384$	0	0
$385$	−28.5328	−1.45417
$386$	0	0
$387$	−11.1409	−0.566325
$388$	0	0
$389$	−24.8520	−1.26004	−0.630022	−	0.776577i	$-0.716954\pi$
$389$	−24.8520	−1.26004	−0.630022	+	0.776577i	$0.716954\pi$
$390$	0	0
$391$	39.5686	2.00107
$392$	0	0
$393$	−12.1630	−0.613542
$394$	0	0
$395$	−21.3709	−1.07529
$396$	0	0
$397$	14.6468	0.735102	0.367551	−	0.930003i	$-0.380196\pi$
$397$	14.6468	0.735102	0.367551	+	0.930003i	$0.380196\pi$
$398$	0	0
$399$	10.9337	0.547370
$400$	0	0
$401$	−5.34823	−0.267078	−0.133539	−	0.991044i	$-0.542634\pi$
$401$	−5.34823	−0.267078	−0.133539	+	0.991044i	$0.542634\pi$
$402$	0	0
$403$	0.971935	0.0484156
$404$	0	0
$405$	−3.20729	−0.159372
$406$	0	0
$407$	−0.862380	−0.0427466
$408$	0	0
$409$	16.7435	0.827913	0.413957	−	0.910297i	$-0.364146\pi$
$409$	16.7435	0.827913	0.413957	+	0.910297i	$0.364146\pi$
$410$	0	0
$411$	14.4346	0.712007
$412$	0	0
$413$	−13.4676	−0.662695
$414$	0	0
$415$	−32.7433	−1.60731
$416$	0	0
$417$	0.433633	0.0212351
$418$	0	0
$419$	5.99429	0.292840	0.146420	−	0.989222i	$-0.453225\pi$
$419$	5.99429	0.292840	0.146420	+	0.989222i	$0.453225\pi$
$420$	0	0
$421$	2.60581	0.126999	0.0634996	−	0.997982i	$-0.479774\pi$
$421$	2.60581	0.126999	0.0634996	+	0.997982i	$0.479774\pi$
$422$	0	0
$423$	−0.406706	−0.0197747
$424$	0	0
$425$	38.0147	1.84398
$426$	0	0
$427$	34.9736	1.69249
$428$	0	0
$429$	0.705344	0.0340543
$430$	0	0
$431$	3.60689	0.173738	0.0868690	−	0.996220i	$-0.472314\pi$
$431$	3.60689	0.173738	0.0868690	+	0.996220i	$0.472314\pi$
$432$	0	0
$433$	11.1030	0.533576	0.266788	−	0.963755i	$-0.414038\pi$
$433$	11.1030	0.533576	0.266788	+	0.963755i	$0.414038\pi$
$434$	0	0
$435$	10.6563	0.510930
$436$	0	0
$437$	−16.3149	−0.780450
$438$	0	0
$439$	−37.7654	−1.80245	−0.901223	−	0.433356i	$-0.857329\pi$
$439$	−37.7654	−1.80245	−0.901223	+	0.433356i	$0.857329\pi$
$440$	0	0
$441$	6.59982	0.314277
$442$	0	0
$443$	−21.3294	−1.01339	−0.506695	−	0.862125i	$-0.669133\pi$
$443$	−21.3294	−1.01339	−0.506695	+	0.862125i	$0.669133\pi$
$444$	0	0
$445$	14.4968	0.687215
$446$	0	0
$447$	19.0992	0.903359
$448$	0	0
$449$	28.7910	1.35873	0.679365	−	0.733800i	$-0.262255\pi$
$449$	28.7910	1.35873	0.679365	+	0.733800i	$0.262255\pi$
$450$	0	0
$451$	8.43859	0.397358
$452$	0	0
$453$	−8.70546	−0.409018
$454$	0	0
$455$	3.45833	0.162129
$456$	0	0
$457$	18.7766	0.878332	0.439166	−	0.898406i	$-0.355274\pi$
$457$	18.7766	0.878332	0.439166	+	0.898406i	$0.355274\pi$
$458$	0	0
$459$	−7.19061	−0.335629
$460$	0	0
$461$	−12.0426	−0.560878	−0.280439	−	0.959872i	$-0.590480\pi$
$461$	−12.0426	−0.560878	−0.280439	+	0.959872i	$0.590480\pi$
$462$	0	0
$463$	21.0030	0.976094	0.488047	−	0.872817i	$-0.337709\pi$
$463$	21.0030	0.976094	0.488047	+	0.872817i	$0.337709\pi$
$464$	0	0
$465$	−10.6614	−0.494411
$466$	0	0
$467$	−36.3609	−1.68258	−0.841291	−	0.540582i	$-0.818204\pi$
$467$	−36.3609	−1.68258	−0.841291	+	0.540582i	$0.818204\pi$
$468$	0	0
$469$	15.4400	0.712951
$470$	0	0
$471$	−22.6106	−1.04184
$472$	0	0
$473$	−26.8758	−1.23575
$474$	0	0
$475$	−15.6743	−0.719184
$476$	0	0
$477$	−10.6914	−0.489528
$478$	0	0
$479$	10.2203	0.466976	0.233488	−	0.972360i	$-0.424986\pi$
$479$	10.2203	0.466976	0.233488	+	0.972360i	$0.424986\pi$
$480$	0	0
$481$	0.104525	0.00476593
$482$	0	0
$483$	−20.2932	−0.923375
$484$	0	0
$485$	−5.67862	−0.257853
$486$	0	0
$487$	−27.8144	−1.26039	−0.630195	−	0.776437i	$-0.717025\pi$
$487$	−27.8144	−1.26039	−0.630195	+	0.776437i	$0.717025\pi$
$488$	0	0
$489$	9.50323	0.429751
$490$	0	0
$491$	6.01324	0.271374	0.135687	−	0.990752i	$-0.456676\pi$
$491$	6.01324	0.271374	0.135687	+	0.990752i	$0.456676\pi$
$492$	0	0
$493$	23.8909	1.07599
$494$	0	0
$495$	−7.73710	−0.347757
$496$	0	0
$497$	9.88791	0.443533
$498$	0	0
$499$	6.63868	0.297188	0.148594	−	0.988898i	$-0.452525\pi$
$499$	6.63868	0.297188	0.148594	+	0.988898i	$0.452525\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	42.5316	1.89639	0.948195	−	0.317689i	$-0.102907\pi$
$503$	42.5316	1.89639	0.948195	+	0.317689i	$0.102907\pi$
$504$	0	0
$505$	−60.3287	−2.68459
$506$	0	0
$507$	12.9145	0.573553
$508$	0	0
$509$	−35.1257	−1.55692	−0.778461	−	0.627693i	$-0.783999\pi$
$509$	−35.1257	−1.55692	−0.778461	+	0.627693i	$0.783999\pi$
$510$	0	0
$511$	−23.1391	−1.02362
$512$	0	0
$513$	2.96484	0.130901
$514$	0	0
$515$	−18.6689	−0.822648
$516$	0	0
$517$	−0.981115	−0.0431494
$518$	0	0
$519$	−3.88163	−0.170385
$520$	0	0
$521$	−37.8196	−1.65691	−0.828454	−	0.560057i	$-0.810779\pi$
$521$	−37.8196	−1.65691	−0.828454	+	0.560057i	$0.810779\pi$
$522$	0	0
$523$	−2.84372	−0.124347	−0.0621737	−	0.998065i	$-0.519803\pi$
$523$	−2.84372	−0.124347	−0.0621737	+	0.998065i	$0.519803\pi$
$524$	0	0
$525$	−19.4963	−0.850889
$526$	0	0
$527$	−23.9024	−1.04121
$528$	0	0
$529$	7.28097	0.316564
$530$	0	0
$531$	−3.65193	−0.158480
$532$	0	0
$533$	−1.02280	−0.0443025
$534$	0	0
$535$	−29.2345	−1.26392
$536$	0	0
$537$	7.65909	0.330514
$538$	0	0
$539$	15.9211	0.685769
$540$	0	0
$541$	37.0033	1.59090	0.795448	−	0.606022i	$-0.207236\pi$
$541$	37.0033	1.59090	0.795448	+	0.606022i	$0.207236\pi$
$542$	0	0
$543$	−9.78926	−0.420097
$544$	0	0
$545$	60.5358	2.59307
$546$	0	0
$547$	−33.2638	−1.42226	−0.711128	−	0.703063i	$-0.751815\pi$
$547$	−33.2638	−1.42226	−0.711128	+	0.703063i	$0.751815\pi$
$548$	0	0
$549$	9.48361	0.404750
$550$	0	0
$551$	−9.85072	−0.419655
$552$	0	0
$553$	24.5726	1.04493
$554$	0	0
$555$	−1.14656	−0.0486688
$556$	0	0
$557$	−38.8570	−1.64642	−0.823212	−	0.567734i	$-0.807820\pi$
$557$	−38.8570	−1.64642	−0.823212	+	0.567734i	$0.807820\pi$
$558$	0	0
$559$	3.25749	0.137777
$560$	0	0
$561$	−17.3462	−0.732359
$562$	0	0
$563$	15.4700	0.651984	0.325992	−	0.945373i	$-0.394302\pi$
$563$	15.4700	0.651984	0.325992	+	0.945373i	$0.394302\pi$
$564$	0	0
$565$	−40.1887	−1.69075
$566$	0	0
$567$	3.68779	0.154873
$568$	0	0
$569$	−21.7094	−0.910107	−0.455053	−	0.890464i	$-0.650380\pi$
$569$	−21.7094	−0.910107	−0.455053	+	0.890464i	$0.650380\pi$
$570$	0	0
$571$	22.9576	0.960745	0.480372	−	0.877065i	$-0.340502\pi$
$571$	22.9576	0.960745	0.480372	+	0.877065i	$0.340502\pi$
$572$	0	0
$573$	−14.9580	−0.624880
$574$	0	0
$575$	29.0918	1.21321
$576$	0	0
$577$	−23.8858	−0.994381	−0.497190	−	0.867641i	$-0.665635\pi$
$577$	−23.8858	−0.994381	−0.497190	+	0.867641i	$0.665635\pi$
$578$	0	0
$579$	0.569004	0.0236470
$580$	0	0
$581$	37.6488	1.56193
$582$	0	0
$583$	−25.7915	−1.06817
$584$	0	0
$585$	0.937777	0.0387723
$586$	0	0
$587$	−18.2904	−0.754926	−0.377463	−	0.926025i	$-0.623203\pi$
$587$	−18.2904	−0.754926	−0.377463	+	0.926025i	$0.623203\pi$
$588$	0	0
$589$	9.85546	0.406087
$590$	0	0
$591$	−24.1326	−0.992682
$592$	0	0
$593$	−33.7008	−1.38393	−0.691964	−	0.721932i	$-0.743254\pi$
$593$	−33.7008	−1.38393	−0.691964	+	0.721932i	$0.743254\pi$
$594$	0	0
$595$	−85.0493	−3.48668
$596$	0	0
$597$	−10.6151	−0.434446
$598$	0	0
$599$	−22.1340	−0.904372	−0.452186	−	0.891924i	$-0.649356\pi$
$599$	−22.1340	−0.904372	−0.452186	+	0.891924i	$0.649356\pi$
$600$	0	0
$601$	6.70740	0.273601	0.136800	−	0.990599i	$-0.456318\pi$
$601$	6.70740	0.273601	0.136800	+	0.990599i	$0.456318\pi$
$602$	0	0
$603$	4.18677	0.170499
$604$	0	0
$605$	16.6156	0.675521
$606$	0	0
$607$	30.2730	1.22874	0.614371	−	0.789017i	$-0.289410\pi$
$607$	30.2730	1.22874	0.614371	+	0.789017i	$0.289410\pi$
$608$	0	0
$609$	−12.2528	−0.496507
$610$	0	0
$611$	0.118916	0.00481084
$612$	0	0
$613$	2.52415	0.101949	0.0509747	−	0.998700i	$-0.483767\pi$
$613$	2.52415	0.101949	0.0509747	+	0.998700i	$0.483767\pi$
$614$	0	0
$615$	11.2194	0.452409
$616$	0	0
$617$	12.6084	0.507593	0.253797	−	0.967258i	$-0.418321\pi$
$617$	12.6084	0.507593	0.253797	+	0.967258i	$0.418321\pi$
$618$	0	0
$619$	−3.97122	−0.159617	−0.0798085	−	0.996810i	$-0.525431\pi$
$619$	−3.97122	−0.159617	−0.0798085	+	0.996810i	$0.525431\pi$
$620$	0	0
$621$	−5.50281	−0.220820
$622$	0	0
$623$	−16.6687	−0.667816
$624$	0	0
$625$	−23.4842	−0.939368
$626$	0	0
$627$	7.15222	0.285632
$628$	0	0
$629$	−2.57054	−0.102494
$630$	0	0
$631$	−48.8417	−1.94436	−0.972179	−	0.234238i	$-0.924741\pi$
$631$	−48.8417	−1.94436	−0.972179	+	0.234238i	$0.924741\pi$
$632$	0	0
$633$	−14.9203	−0.593030
$634$	0	0
$635$	3.64576	0.144677
$636$	0	0
$637$	−1.92972	−0.0764582
$638$	0	0
$639$	2.68125	0.106069
$640$	0	0
$641$	−11.7538	−0.464246	−0.232123	−	0.972686i	$-0.574567\pi$
$641$	−11.7538	−0.464246	−0.232123	+	0.972686i	$0.574567\pi$
$642$	0	0
$643$	−18.9663	−0.747958	−0.373979	−	0.927437i	$-0.622007\pi$
$643$	−18.9663	−0.747958	−0.373979	+	0.927437i	$0.622007\pi$
$644$	0	0
$645$	−35.7322	−1.40695
$646$	0	0
$647$	37.2745	1.46541	0.732705	−	0.680546i	$-0.238257\pi$
$647$	37.2745	1.46541	0.732705	+	0.680546i	$0.238257\pi$
$648$	0	0
$649$	−8.80972	−0.345812
$650$	0	0
$651$	12.2587	0.480454
$652$	0	0
$653$	−9.70587	−0.379820	−0.189910	−	0.981802i	$-0.560820\pi$
$653$	−9.70587	−0.379820	−0.189910	+	0.981802i	$0.560820\pi$
$654$	0	0
$655$	−39.0103	−1.52426
$656$	0	0
$657$	−6.27452	−0.244792
$658$	0	0
$659$	21.9798	0.856211	0.428105	−	0.903729i	$-0.359181\pi$
$659$	21.9798	0.856211	0.428105	+	0.903729i	$0.359181\pi$
$660$	0	0
$661$	−24.2779	−0.944300	−0.472150	−	0.881518i	$-0.656522\pi$
$661$	−24.2779	−0.944300	−0.472150	+	0.881518i	$0.656522\pi$
$662$	0	0
$663$	2.10246	0.0816526
$664$	0	0
$665$	35.0676	1.35986
$666$	0	0
$667$	18.2832	0.707928
$668$	0	0
$669$	14.5962	0.564320
$670$	0	0
$671$	22.8778	0.883186
$672$	0	0
$673$	27.1769	1.04759	0.523795	−	0.851844i	$-0.324516\pi$
$673$	27.1769	1.04759	0.523795	+	0.851844i	$0.324516\pi$
$674$	0	0
$675$	−5.28672	−0.203486
$676$	0	0
$677$	31.2103	1.19951	0.599755	−	0.800184i	$-0.295265\pi$
$677$	31.2103	1.19951	0.599755	+	0.800184i	$0.295265\pi$
$678$	0	0
$679$	6.52937	0.250574
$680$	0	0
$681$	−7.67692	−0.294180
$682$	0	0
$683$	8.68394	0.332282	0.166141	−	0.986102i	$-0.446869\pi$
$683$	8.68394	0.332282	0.166141	+	0.986102i	$0.446869\pi$
$684$	0	0
$685$	46.2960	1.76888
$686$	0	0
$687$	7.74042	0.295316
$688$	0	0
$689$	3.12606	0.119094
$690$	0	0
$691$	−10.3984	−0.395573	−0.197787	−	0.980245i	$-0.563375\pi$
$691$	−10.3984	−0.395573	−0.197787	+	0.980245i	$0.563375\pi$
$692$	0	0
$693$	8.89624	0.337940
$694$	0	0
$695$	1.39079	0.0527556
$696$	0	0
$697$	25.1533	0.952751
$698$	0	0
$699$	15.6688	0.592650
$700$	0	0
$701$	33.3918	1.26119	0.630596	−	0.776111i	$-0.282811\pi$
$701$	33.3918	1.26119	0.630596	+	0.776111i	$0.282811\pi$
$702$	0	0
$703$	1.05989	0.0399744
$704$	0	0
$705$	−1.30442	−0.0491274
$706$	0	0
$707$	69.3669	2.60881
$708$	0	0
$709$	−5.92848	−0.222649	−0.111324	−	0.993784i	$-0.535509\pi$
$709$	−5.92848	−0.222649	−0.111324	+	0.993784i	$0.535509\pi$
$710$	0	0
$711$	6.66323	0.249891
$712$	0	0
$713$	−18.2920	−0.685040
$714$	0	0
$715$	2.26224	0.0846032
$716$	0	0
$717$	−20.3027	−0.758218
$718$	0	0
$719$	−4.67728	−0.174433	−0.0872166	−	0.996189i	$-0.527797\pi$
$719$	−4.67728	−0.174433	−0.0872166	+	0.996189i	$0.527797\pi$
$720$	0	0
$721$	21.4657	0.799426
$722$	0	0
$723$	−7.43764	−0.276609
$724$	0	0
$725$	17.5652	0.652356
$726$	0	0
$727$	27.8553	1.03310	0.516548	−	0.856258i	$-0.327217\pi$
$727$	27.8553	1.03310	0.516548	+	0.856258i	$0.327217\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−80.1101	−2.96298
$732$	0	0
$733$	−23.6613	−0.873949	−0.436974	−	0.899474i	$-0.643950\pi$
$733$	−23.6613	−0.873949	−0.436974	+	0.899474i	$0.643950\pi$
$734$	0	0
$735$	21.1675	0.780777
$736$	0	0
$737$	10.0999	0.372036
$738$	0	0
$739$	2.47093	0.0908945	0.0454473	−	0.998967i	$-0.485529\pi$
$739$	2.47093	0.0908945	0.0454473	+	0.998967i	$0.485529\pi$
$740$	0	0
$741$	−0.866886	−0.0318459
$742$	0	0
$743$	18.8973	0.693276	0.346638	−	0.937999i	$-0.387323\pi$
$743$	18.8973	0.693276	0.346638	+	0.937999i	$0.387323\pi$
$744$	0	0
$745$	61.2565	2.24427
$746$	0	0
$747$	10.2090	0.373529
$748$	0	0
$749$	33.6143	1.22824
$750$	0	0
$751$	39.2273	1.43142	0.715712	−	0.698396i	$-0.246102\pi$
$751$	39.2273	1.43142	0.715712	+	0.698396i	$0.246102\pi$
$752$	0	0
$753$	−28.3028	−1.03141
$754$	0	0
$755$	−27.9210	−1.01615
$756$	0	0
$757$	34.1963	1.24288	0.621442	−	0.783460i	$-0.286547\pi$
$757$	34.1963	1.24288	0.621442	+	0.783460i	$0.286547\pi$
$758$	0	0
$759$	−13.2747	−0.481841
$760$	0	0
$761$	−11.0455	−0.400398	−0.200199	−	0.979755i	$-0.564159\pi$
$761$	−11.0455	−0.400398	−0.200199	+	0.979755i	$0.564159\pi$
$762$	0	0
$763$	−69.6051	−2.51987
$764$	0	0
$765$	−23.0624	−0.833822
$766$	0	0
$767$	1.06778	0.0385555
$768$	0	0
$769$	8.17138	0.294668	0.147334	−	0.989087i	$-0.452931\pi$
$769$	8.17138	0.294668	0.147334	+	0.989087i	$0.452931\pi$
$770$	0	0
$771$	7.82120	0.281673
$772$	0	0
$773$	−47.9579	−1.72493	−0.862463	−	0.506120i	$-0.831079\pi$
$773$	−47.9579	−1.72493	−0.862463	+	0.506120i	$0.831079\pi$
$774$	0	0
$775$	−17.5737	−0.631264
$776$	0	0
$777$	1.31833	0.0472950
$778$	0	0
$779$	−10.3712	−0.371588
$780$	0	0
$781$	6.46812	0.231447
$782$	0	0
$783$	−3.32252	−0.118737
$784$	0	0
$785$	−72.5187	−2.58830
$786$	0	0
$787$	53.9997	1.92488	0.962440	−	0.271495i	$-0.0875180\pi$
$787$	53.9997	1.92488	0.962440	+	0.271495i	$0.0875180\pi$
$788$	0	0
$789$	6.44120	0.229313
$790$	0	0
$791$	46.2096	1.64302
$792$	0	0
$793$	−2.77290	−0.0984687
$794$	0	0
$795$	−34.2906	−1.21616
$796$	0	0
$797$	5.87813	0.208214	0.104107	−	0.994566i	$-0.466802\pi$
$797$	5.87813	0.208214	0.104107	+	0.994566i	$0.466802\pi$
$798$	0	0
$799$	−2.92446	−0.103460
$800$	0	0
$801$	−4.51995	−0.159705
$802$	0	0
$803$	−15.1363	−0.534149
$804$	0	0
$805$	−65.0863	−2.29399
$806$	0	0
$807$	−21.7914	−0.767094
$808$	0	0
$809$	−3.91670	−0.137704	−0.0688519	−	0.997627i	$-0.521934\pi$
$809$	−3.91670	−0.137704	−0.0688519	+	0.997627i	$0.521934\pi$
$810$	0	0
$811$	13.4761	0.473211	0.236605	−	0.971606i	$-0.423965\pi$
$811$	13.4761	0.473211	0.236605	+	0.971606i	$0.423965\pi$
$812$	0	0
$813$	10.1627	0.356421
$814$	0	0
$815$	30.4796	1.06765
$816$	0	0
$817$	33.0310	1.15561
$818$	0	0
$819$	−1.07827	−0.0376778
$820$	0	0
$821$	22.0639	0.770035	0.385018	−	0.922909i	$-0.374195\pi$
$821$	22.0639	0.770035	0.385018	+	0.922909i	$0.374195\pi$
$822$	0	0
$823$	−12.8677	−0.448540	−0.224270	−	0.974527i	$-0.572000\pi$
$823$	−12.8677	−0.448540	−0.224270	+	0.974527i	$0.572000\pi$
$824$	0	0
$825$	−12.7534	−0.444016
$826$	0	0
$827$	−23.6784	−0.823380	−0.411690	−	0.911324i	$-0.635061\pi$
$827$	−23.6784	−0.823380	−0.411690	+	0.911324i	$0.635061\pi$
$828$	0	0
$829$	32.1173	1.11548	0.557741	−	0.830015i	$-0.311668\pi$
$829$	32.1173	1.11548	0.557741	+	0.830015i	$0.311668\pi$
$830$	0	0
$831$	−5.38295	−0.186732
$832$	0	0
$833$	47.4567	1.64428
$834$	0	0
$835$	3.20729	0.110993
$836$	0	0
$837$	3.32412	0.114898
$838$	0	0
$839$	4.63931	0.160167	0.0800835	−	0.996788i	$-0.474481\pi$
$839$	4.63931	0.160167	0.0800835	+	0.996788i	$0.474481\pi$
$840$	0	0
$841$	−17.9609	−0.619340
$842$	0	0
$843$	17.2566	0.594351
$844$	0	0
$845$	41.4206	1.42491
$846$	0	0
$847$	−19.1049	−0.656452
$848$	0	0
$849$	6.13015	0.210386
$850$	0	0
$851$	−1.96718	−0.0674340
$852$	0	0
$853$	−4.45984	−0.152702	−0.0763510	−	0.997081i	$-0.524327\pi$
$853$	−4.45984	−0.152702	−0.0763510	+	0.997081i	$0.524327\pi$
$854$	0	0
$855$	9.50909	0.325204
$856$	0	0
$857$	48.4705	1.65572	0.827860	−	0.560935i	$-0.189558\pi$
$857$	48.4705	1.65572	0.827860	+	0.560935i	$0.189558\pi$
$858$	0	0
$859$	−11.3642	−0.387741	−0.193871	−	0.981027i	$-0.562104\pi$
$859$	−11.3642	−0.387741	−0.193871	+	0.981027i	$0.562104\pi$
$860$	0	0
$861$	−12.9002	−0.439638
$862$	0	0
$863$	31.2403	1.06343	0.531717	−	0.846922i	$-0.321547\pi$
$863$	31.2403	1.06343	0.531717	+	0.846922i	$0.321547\pi$
$864$	0	0
$865$	−12.4495	−0.423296
$866$	0	0
$867$	−34.7048	−1.17864
$868$	0	0
$869$	16.0740	0.545274
$870$	0	0
$871$	−1.22417	−0.0414793
$872$	0	0
$873$	1.77054	0.0599236
$874$	0	0
$875$	−3.39124	−0.114645
$876$	0	0
$877$	−3.45334	−0.116611	−0.0583055	−	0.998299i	$-0.518570\pi$
$877$	−3.45334	−0.116611	−0.0583055	+	0.998299i	$0.518570\pi$
$878$	0	0
$879$	−17.6848	−0.596493
$880$	0	0
$881$	−18.0397	−0.607772	−0.303886	−	0.952708i	$-0.598284\pi$
$881$	−18.0397	−0.607772	−0.303886	+	0.952708i	$0.598284\pi$
$882$	0	0
$883$	−19.7645	−0.665129	−0.332565	−	0.943081i	$-0.607914\pi$
$883$	−19.7645	−0.665129	−0.332565	+	0.943081i	$0.607914\pi$
$884$	0	0
$885$	−11.7128	−0.393721
$886$	0	0
$887$	14.0636	0.472209	0.236105	−	0.971728i	$-0.424129\pi$
$887$	14.0636	0.472209	0.236105	+	0.971728i	$0.424129\pi$
$888$	0	0
$889$	−4.19195	−0.140593
$890$	0	0
$891$	2.41235	0.0808167
$892$	0	0
$893$	1.20582	0.0403511
$894$	0	0
$895$	24.5649	0.821115
$896$	0	0
$897$	1.60896	0.0537217
$898$	0	0
$899$	−11.0444	−0.368353
$900$	0	0
$901$	−76.8780	−2.56118
$902$	0	0
$903$	41.0854	1.36724
$904$	0	0
$905$	−31.3970	−1.04367
$906$	0	0
$907$	−13.8587	−0.460170	−0.230085	−	0.973171i	$-0.573900\pi$
$907$	−13.8587	−0.460170	−0.230085	+	0.973171i	$0.573900\pi$
$908$	0	0
$909$	18.8099	0.623884
$910$	0	0
$911$	−37.7036	−1.24918	−0.624588	−	0.780954i	$-0.714733\pi$
$911$	−37.7036	−1.24918	−0.624588	+	0.780954i	$0.714733\pi$
$912$	0	0
$913$	24.6277	0.815058
$914$	0	0
$915$	30.4167	1.00554
$916$	0	0
$917$	44.8546	1.48123
$918$	0	0
$919$	−24.6184	−0.812086	−0.406043	−	0.913854i	$-0.633092\pi$
$919$	−24.6184	−0.812086	−0.406043	+	0.913854i	$0.633092\pi$
$920$	0	0
$921$	−11.1101	−0.366089
$922$	0	0
$923$	−0.783970	−0.0258047
$924$	0	0
$925$	−1.88993	−0.0621404
$926$	0	0
$927$	5.82075	0.191179
$928$	0	0
$929$	28.1432	0.923349	0.461675	−	0.887049i	$-0.347249\pi$
$929$	28.1432	0.923349	0.461675	+	0.887049i	$0.347249\pi$
$930$	0	0
$931$	−19.5674	−0.641295
$932$	0	0
$933$	−28.3481	−0.928076
$934$	0	0
$935$	−55.6345	−1.81944
$936$	0	0
$937$	14.4087	0.470713	0.235357	−	0.971909i	$-0.424374\pi$
$937$	14.4087	0.470713	0.235357	+	0.971909i	$0.424374\pi$
$938$	0	0
$939$	−31.4209	−1.02538
$940$	0	0
$941$	31.8231	1.03740	0.518702	−	0.854955i	$-0.326415\pi$
$941$	31.8231	1.03740	0.518702	+	0.854955i	$0.326415\pi$
$942$	0	0
$943$	19.2493	0.626844
$944$	0	0
$945$	11.8278	0.384759
$946$	0	0
$947$	7.53847	0.244967	0.122484	−	0.992471i	$-0.460914\pi$
$947$	7.53847	0.244967	0.122484	+	0.992471i	$0.460914\pi$
$948$	0	0
$949$	1.83460	0.0595537
$950$	0	0
$951$	7.57882	0.245760
$952$	0	0
$953$	9.31282	0.301672	0.150836	−	0.988559i	$-0.451803\pi$
$953$	9.31282	0.301672	0.150836	+	0.988559i	$0.451803\pi$
$954$	0	0
$955$	−47.9747	−1.55242
$956$	0	0
$957$	−8.01507	−0.259090
$958$	0	0
$959$	−53.2319	−1.71895
$960$	0	0
$961$	−19.9503	−0.643557
$962$	0	0
$963$	9.11502	0.293727
$964$	0	0
$965$	1.82496	0.0587476
$966$	0	0
$967$	8.56361	0.275387	0.137693	−	0.990475i	$-0.456031\pi$
$967$	8.56361	0.275387	0.137693	+	0.990475i	$0.456031\pi$
$968$	0	0
$969$	21.3190	0.684864
$970$	0	0
$971$	−17.1699	−0.551010	−0.275505	−	0.961300i	$-0.588845\pi$
$971$	−17.1699	−0.551010	−0.275505	+	0.961300i	$0.588845\pi$
$972$	0	0
$973$	−1.59915	−0.0512664
$974$	0	0
$975$	1.54578	0.0495045
$976$	0	0
$977$	16.8510	0.539112	0.269556	−	0.962985i	$-0.413123\pi$
$977$	16.8510	0.539112	0.269556	+	0.962985i	$0.413123\pi$
$978$	0	0
$979$	−10.9037	−0.348484
$980$	0	0
$981$	−18.8744	−0.602615
$982$	0	0
$983$	23.9668	0.764424	0.382212	−	0.924075i	$-0.375162\pi$
$983$	23.9668	0.764424	0.382212	+	0.924075i	$0.375162\pi$
$984$	0	0
$985$	−77.4002	−2.46618
$986$	0	0
$987$	1.49985	0.0477406
$988$	0	0
$989$	−61.3065	−1.94943
$990$	0	0
$991$	−17.7821	−0.564867	−0.282434	−	0.959287i	$-0.591142\pi$
$991$	−17.7821	−0.564867	−0.282434	+	0.959287i	$0.591142\pi$
$992$	0	0
$993$	−29.3509	−0.931423
$994$	0	0
$995$	−34.0456	−1.07932
$996$	0	0
$997$	−34.0948	−1.07979	−0.539896	−	0.841732i	$-0.681536\pi$
$997$	−34.0948	−1.07979	−0.539896	+	0.841732i	$0.681536\pi$
$998$	0	0
$999$	0.357486	0.0113104

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.g.1.1	✓	7
4.3	odd	2		8016.2.a.w.1.1		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.g.1.1	✓	7	1.1	even	1	trivial
8016.2.a.w.1.1		7	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.20729 q^{5} +3.68779 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.20729 q^{5} +3.68779 q^{7} +1.00000 q^{9} +2.41235 q^{11} -0.292389 q^{13} +3.20729 q^{15} +7.19061 q^{17} -2.96484 q^{19} -3.68779 q^{21} +5.50281 q^{23} +5.28672 q^{25} -1.00000 q^{27} +3.32252 q^{29} -3.32412 q^{31} -2.41235 q^{33} -11.8278 q^{35} -0.357486 q^{37} +0.292389 q^{39} +3.49808 q^{41} -11.1409 q^{43} -3.20729 q^{45} -0.406706 q^{47} +6.59982 q^{49} -7.19061 q^{51} -10.6914 q^{53} -7.73710 q^{55} +2.96484 q^{57} -3.65193 q^{59} +9.48361 q^{61} +3.68779 q^{63} +0.937777 q^{65} +4.18677 q^{67} -5.50281 q^{69} +2.68125 q^{71} -6.27452 q^{73} -5.28672 q^{75} +8.89624 q^{77} +6.66323 q^{79} +1.00000 q^{81} +10.2090 q^{83} -23.0624 q^{85} -3.32252 q^{87} -4.51995 q^{89} -1.07827 q^{91} +3.32412 q^{93} +9.50909 q^{95} +1.77054 q^{97} +2.41235 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{3} - 3 q^{5} + 8 q^{7} + 7 q^{9}+O(q^{10}) \) 7 * q - 7 * q^3 - 3 * q^5 + 8 * q^7 + 7 * q^9	\( 7 q - 7 q^{3} - 3 q^{5} + 8 q^{7} + 7 q^{9} + q^{11} - 2 q^{13} + 3 q^{15} + 11 q^{17} + 2 q^{19} - 8 q^{21} + 17 q^{23} + 4 q^{25} - 7 q^{27} - 7 q^{29} + 10 q^{31} - q^{33} + 10 q^{35} - 21 q^{37} + 2 q^{39} + 8 q^{41} - 12 q^{43} - 3 q^{45} + 25 q^{47} - 7 q^{49} - 11 q^{51} - 7 q^{53} + 15 q^{55} - 2 q^{57} + 3 q^{59} - 14 q^{61} + 8 q^{63} + 4 q^{65} + 4 q^{67} - 17 q^{69} + 27 q^{71} - 12 q^{73} - 4 q^{75} + 16 q^{77} + 8 q^{79} + 7 q^{81} + 15 q^{83} - 3 q^{85} + 7 q^{87} + 14 q^{89} - 3 q^{91} - 10 q^{93} + 37 q^{95} + 3 q^{97} + q^{99}+O(q^{100}) \) 7 * q - 7 * q^3 - 3 * q^5 + 8 * q^7 + 7 * q^9 + q^11 - 2 * q^13 + 3 * q^15 + 11 * q^17 + 2 * q^19 - 8 * q^21 + 17 * q^23 + 4 * q^25 - 7 * q^27 - 7 * q^29 + 10 * q^31 - q^33 + 10 * q^35 - 21 * q^37 + 2 * q^39 + 8 * q^41 - 12 * q^43 - 3 * q^45 + 25 * q^47 - 7 * q^49 - 11 * q^51 - 7 * q^53 + 15 * q^55 - 2 * q^57 + 3 * q^59 - 14 * q^61 + 8 * q^63 + 4 * q^65 + 4 * q^67 - 17 * q^69 + 27 * q^71 - 12 * q^73 - 4 * q^75 + 16 * q^77 + 8 * q^79 + 7 * q^81 + 15 * q^83 - 3 * q^85 + 7 * q^87 + 14 * q^89 - 3 * q^91 - 10 * q^93 + 37 * q^95 + 3 * q^97 + q^99

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