LMFDB - Embedded newform 8016.2.a.z.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, 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("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.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:	$64.0080822603$
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} - 8x^{6} + 15x^{5} + 19x^{4} - 31x^{3} - 13x^{2} + 14x + 1 $ x^8 - 2x^7 - 8x^6 + 15x^5 + 19x^4 - 31x^3 - 13x^2 + 14*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 501)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.0678707$ of defining polynomial
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.71255 q^{5} +5.14356 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.71255 q^{5} +5.14356 q^{7} +1.00000 q^{9} -1.40535 q^{11} +2.82421 q^{13} -2.71255 q^{15} +0.388999 q^{17} -2.47626 q^{19} +5.14356 q^{21} -2.05940 q^{23} +2.35794 q^{25} +1.00000 q^{27} -9.76448 q^{29} -5.79229 q^{31} -1.40535 q^{33} -13.9522 q^{35} -6.17693 q^{37} +2.82421 q^{39} -4.11791 q^{41} -8.58328 q^{43} -2.71255 q^{45} +8.78477 q^{47} +19.4563 q^{49} +0.388999 q^{51} -2.59514 q^{53} +3.81208 q^{55} -2.47626 q^{57} -13.6308 q^{59} -4.03421 q^{61} +5.14356 q^{63} -7.66082 q^{65} -11.5624 q^{67} -2.05940 q^{69} -16.1859 q^{71} +10.4689 q^{73} +2.35794 q^{75} -7.22851 q^{77} -7.27177 q^{79} +1.00000 q^{81} +11.5589 q^{83} -1.05518 q^{85} -9.76448 q^{87} +1.50198 q^{89} +14.5265 q^{91} -5.79229 q^{93} +6.71698 q^{95} +6.01313 q^{97} -1.40535 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} + q^{5} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^3 + q^5 + 8 * q^9	$ 8 q + 8 q^{3} + q^{5} + 8 q^{9} - 5 q^{11} + q^{15} - 7 q^{17} - 24 q^{19} - q^{23} + 3 q^{25} + 8 q^{27} - 11 q^{29} - 30 q^{31} - 5 q^{33} - 26 q^{35} + 11 q^{37} + 10 q^{41} - 24 q^{43} + q^{45} + 3 q^{47} + 6 q^{49} - 7 q^{51} - 25 q^{53} - 25 q^{55} - 24 q^{57} - 45 q^{59} + 16 q^{61} - 10 q^{65} - 18 q^{67} - q^{69} - 21 q^{71} - 8 q^{73} + 3 q^{75} - 18 q^{77} - 10 q^{79} + 8 q^{81} - 7 q^{83} - 11 q^{85} - 11 q^{87} + 26 q^{89} - 15 q^{91} - 30 q^{93} - q^{95} - 3 q^{97} - 5 q^{99}+O(q^{100}) $ 8 * q + 8 * q^3 + q^5 + 8 * q^9 - 5 * q^11 + q^15 - 7 * q^17 - 24 * q^19 - q^23 + 3 * q^25 + 8 * q^27 - 11 * q^29 - 30 * q^31 - 5 * q^33 - 26 * q^35 + 11 * q^37 + 10 * q^41 - 24 * q^43 + q^45 + 3 * q^47 + 6 * q^49 - 7 * q^51 - 25 * q^53 - 25 * q^55 - 24 * q^57 - 45 * q^59 + 16 * q^61 - 10 * q^65 - 18 * q^67 - q^69 - 21 * q^71 - 8 * q^73 + 3 * q^75 - 18 * q^77 - 10 * q^79 + 8 * q^81 - 7 * q^83 - 11 * q^85 - 11 * q^87 + 26 * q^89 - 15 * q^91 - 30 * q^93 - q^95 - 3 * q^97 - 5 * 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
$5$	−2.71255	−1.21309	−0.606545	−	0.795049i	$-0.707445\pi$
$5$	−2.71255	−1.21309	−0.606545	+	0.795049i	$0.707445\pi$
$6$	0	0
$7$	5.14356	1.94408	0.972042	−	0.234806i	$-0.0754455\pi$
$7$	5.14356	1.94408	0.972042	+	0.234806i	$0.0754455\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.40535	−0.423729	−0.211864	−	0.977299i	$-0.567954\pi$
$11$	−1.40535	−0.423729	−0.211864	+	0.977299i	$0.567954\pi$
$12$	0	0
$13$	2.82421	0.783295	0.391648	−	0.920115i	$-0.371905\pi$
$13$	2.82421	0.783295	0.391648	+	0.920115i	$0.371905\pi$
$14$	0	0
$15$	−2.71255	−0.700378
$16$	0	0
$17$	0.388999	0.0943462	0.0471731	−	0.998887i	$-0.484979\pi$
$17$	0.388999	0.0943462	0.0471731	+	0.998887i	$0.484979\pi$
$18$	0	0
$19$	−2.47626	−0.568092	−0.284046	−	0.958811i	$-0.591677\pi$
$19$	−2.47626	−0.568092	−0.284046	+	0.958811i	$0.591677\pi$
$20$	0	0
$21$	5.14356	1.12242
$22$	0	0
$23$	−2.05940	−0.429414	−0.214707	−	0.976679i	$-0.568880\pi$
$23$	−2.05940	−0.429414	−0.214707	+	0.976679i	$0.568880\pi$
$24$	0	0
$25$	2.35794	0.471587
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−9.76448	−1.81322	−0.906609	−	0.421972i	$-0.861338\pi$
$29$	−9.76448	−1.81322	−0.906609	+	0.421972i	$0.861338\pi$
$30$	0	0
$31$	−5.79229	−1.04033	−0.520163	−	0.854067i	$-0.674129\pi$
$31$	−5.79229	−1.04033	−0.520163	+	0.854067i	$0.674129\pi$
$32$	0	0
$33$	−1.40535	−0.244640
$34$	0	0
$35$	−13.9522	−2.35835
$36$	0	0
$37$	−6.17693	−1.01548	−0.507741	−	0.861510i	$-0.669519\pi$
$37$	−6.17693	−1.01548	−0.507741	+	0.861510i	$0.669519\pi$
$38$	0	0
$39$	2.82421	0.452236
$40$	0	0
$41$	−4.11791	−0.643110	−0.321555	−	0.946891i	$-0.604206\pi$
$41$	−4.11791	−0.643110	−0.321555	+	0.946891i	$0.604206\pi$
$42$	0	0
$43$	−8.58328	−1.30894	−0.654469	−	0.756089i	$-0.727108\pi$
$43$	−8.58328	−1.30894	−0.654469	+	0.756089i	$0.727108\pi$
$44$	0	0
$45$	−2.71255	−0.404363
$46$	0	0
$47$	8.78477	1.28139	0.640695	−	0.767796i	$-0.278646\pi$
$47$	8.78477	1.28139	0.640695	+	0.767796i	$0.278646\pi$
$48$	0	0
$49$	19.4563	2.77946
$50$	0	0
$51$	0.388999	0.0544708
$52$	0	0
$53$	−2.59514	−0.356470	−0.178235	−	0.983988i	$-0.557039\pi$
$53$	−2.59514	−0.356470	−0.178235	+	0.983988i	$0.557039\pi$
$54$	0	0
$55$	3.81208	0.514021
$56$	0	0
$57$	−2.47626	−0.327988
$58$	0	0
$59$	−13.6308	−1.77458	−0.887288	−	0.461216i	$-0.847413\pi$
$59$	−13.6308	−1.77458	−0.887288	+	0.461216i	$0.847413\pi$
$60$	0	0
$61$	−4.03421	−0.516527	−0.258264	−	0.966074i	$-0.583150\pi$
$61$	−4.03421	−0.516527	−0.258264	+	0.966074i	$0.583150\pi$
$62$	0	0
$63$	5.14356	0.648028
$64$	0	0
$65$	−7.66082	−0.950208
$66$	0	0
$67$	−11.5624	−1.41257	−0.706285	−	0.707928i	$-0.749630\pi$
$67$	−11.5624	−1.41257	−0.706285	+	0.707928i	$0.749630\pi$
$68$	0	0
$69$	−2.05940	−0.247922
$70$	0	0
$71$	−16.1859	−1.92091	−0.960457	−	0.278428i	$-0.910187\pi$
$71$	−16.1859	−1.92091	−0.960457	+	0.278428i	$0.910187\pi$
$72$	0	0
$73$	10.4689	1.22529	0.612646	−	0.790358i	$-0.290105\pi$
$73$	10.4689	1.22529	0.612646	+	0.790358i	$0.290105\pi$
$74$	0	0
$75$	2.35794	0.272271
$76$	0	0
$77$	−7.22851	−0.823765
$78$	0	0
$79$	−7.27177	−0.818138	−0.409069	−	0.912503i	$-0.634147\pi$
$79$	−7.27177	−0.818138	−0.409069	+	0.912503i	$0.634147\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.5589	1.26875	0.634375	−	0.773026i	$-0.281258\pi$
$83$	11.5589	1.26875	0.634375	+	0.773026i	$0.281258\pi$
$84$	0	0
$85$	−1.05518	−0.114450
$86$	0	0
$87$	−9.76448	−1.04686
$88$	0	0
$89$	1.50198	0.159209	0.0796047	−	0.996827i	$-0.474634\pi$
$89$	1.50198	0.159209	0.0796047	+	0.996827i	$0.474634\pi$
$90$	0	0
$91$	14.5265	1.52279
$92$	0	0
$93$	−5.79229	−0.600633
$94$	0	0
$95$	6.71698	0.689147
$96$	0	0
$97$	6.01313	0.610541	0.305271	−	0.952266i	$-0.401253\pi$
$97$	6.01313	0.610541	0.305271	+	0.952266i	$0.401253\pi$
$98$	0	0
$99$	−1.40535	−0.141243
$100$	0	0
$101$	11.8180	1.17594	0.587968	−	0.808884i	$-0.299928\pi$
$101$	11.8180	1.17594	0.587968	+	0.808884i	$0.299928\pi$
$102$	0	0
$103$	5.58380	0.550188	0.275094	−	0.961417i	$-0.411291\pi$
$103$	5.58380	0.550188	0.275094	+	0.961417i	$0.411291\pi$
$104$	0	0
$105$	−13.9522	−1.36159
$106$	0	0
$107$	−0.921043	−0.0890406	−0.0445203	−	0.999008i	$-0.514176\pi$
$107$	−0.921043	−0.0890406	−0.0445203	+	0.999008i	$0.514176\pi$
$108$	0	0
$109$	5.72379	0.548239	0.274120	−	0.961696i	$-0.411614\pi$
$109$	5.72379	0.548239	0.274120	+	0.961696i	$0.411614\pi$
$110$	0	0
$111$	−6.17693	−0.586288
$112$	0	0
$113$	−5.32663	−0.501087	−0.250544	−	0.968105i	$-0.580609\pi$
$113$	−5.32663	−0.501087	−0.250544	+	0.968105i	$0.580609\pi$
$114$	0	0
$115$	5.58622	0.520918
$116$	0	0
$117$	2.82421	0.261098
$118$	0	0
$119$	2.00084	0.183417
$120$	0	0
$121$	−9.02499	−0.820454
$122$	0	0
$123$	−4.11791	−0.371300
$124$	0	0
$125$	7.16673	0.641012
$126$	0	0
$127$	1.08499	0.0962770	0.0481385	−	0.998841i	$-0.484671\pi$
$127$	1.08499	0.0962770	0.0481385	+	0.998841i	$0.484671\pi$
$128$	0	0
$129$	−8.58328	−0.755716
$130$	0	0
$131$	2.10877	0.184244	0.0921222	−	0.995748i	$-0.470635\pi$
$131$	2.10877	0.184244	0.0921222	+	0.995748i	$0.470635\pi$
$132$	0	0
$133$	−12.7368	−1.10442
$134$	0	0
$135$	−2.71255	−0.233459
$136$	0	0
$137$	−9.56847	−0.817490	−0.408745	−	0.912649i	$-0.634033\pi$
$137$	−9.56847	−0.817490	−0.408745	+	0.912649i	$0.634033\pi$
$138$	0	0
$139$	−10.6906	−0.906763	−0.453382	−	0.891316i	$-0.649783\pi$
$139$	−10.6906	−0.906763	−0.453382	+	0.891316i	$0.649783\pi$
$140$	0	0
$141$	8.78477	0.739811
$142$	0	0
$143$	−3.96900	−0.331905
$144$	0	0
$145$	26.4866	2.19960
$146$	0	0
$147$	19.4563	1.60472
$148$	0	0
$149$	−5.28320	−0.432816	−0.216408	−	0.976303i	$-0.569434\pi$
$149$	−5.28320	−0.432816	−0.216408	+	0.976303i	$0.569434\pi$
$150$	0	0
$151$	2.03474	0.165585	0.0827923	−	0.996567i	$-0.473616\pi$
$151$	2.03474	0.165585	0.0827923	+	0.996567i	$0.473616\pi$
$152$	0	0
$153$	0.388999	0.0314487
$154$	0	0
$155$	15.7119	1.26201
$156$	0	0
$157$	20.3613	1.62501	0.812506	−	0.582953i	$-0.198103\pi$
$157$	20.3613	1.62501	0.812506	+	0.582953i	$0.198103\pi$
$158$	0	0
$159$	−2.59514	−0.205808
$160$	0	0
$161$	−10.5926	−0.834817
$162$	0	0
$163$	−1.61102	−0.126185	−0.0630925	−	0.998008i	$-0.520096\pi$
$163$	−1.61102	−0.126185	−0.0630925	+	0.998008i	$0.520096\pi$
$164$	0	0
$165$	3.81208	0.296770
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−5.02383	−0.386448
$170$	0	0
$171$	−2.47626	−0.189364
$172$	0	0
$173$	−25.1622	−1.91305	−0.956524	−	0.291652i	$-0.905795\pi$
$173$	−25.1622	−1.91305	−0.956524	+	0.291652i	$0.905795\pi$
$174$	0	0
$175$	12.1282	0.916806
$176$	0	0
$177$	−13.6308	−1.02455
$178$	0	0
$179$	−0.906147	−0.0677286	−0.0338643	−	0.999426i	$-0.510781\pi$
$179$	−0.906147	−0.0677286	−0.0338643	+	0.999426i	$0.510781\pi$
$180$	0	0
$181$	25.6772	1.90857	0.954287	−	0.298893i	$-0.0966173\pi$
$181$	25.6772	1.90857	0.954287	+	0.298893i	$0.0966173\pi$
$182$	0	0
$183$	−4.03421	−0.298217
$184$	0	0
$185$	16.7552	1.23187
$186$	0	0
$187$	−0.546680	−0.0399772
$188$	0	0
$189$	5.14356	0.374139
$190$	0	0
$191$	0.440700	0.0318879	0.0159440	−	0.999873i	$-0.494925\pi$
$191$	0.440700	0.0318879	0.0159440	+	0.999873i	$0.494925\pi$
$192$	0	0
$193$	−2.48672	−0.178998	−0.0894990	−	0.995987i	$-0.528527\pi$
$193$	−2.48672	−0.178998	−0.0894990	+	0.995987i	$0.528527\pi$
$194$	0	0
$195$	−7.66082	−0.548603
$196$	0	0
$197$	−14.7487	−1.05080	−0.525401	−	0.850855i	$-0.676085\pi$
$197$	−14.7487	−1.05080	−0.525401	+	0.850855i	$0.676085\pi$
$198$	0	0
$199$	14.3359	1.01624	0.508122	−	0.861285i	$-0.330340\pi$
$199$	14.3359	1.01624	0.508122	+	0.861285i	$0.330340\pi$
$200$	0	0
$201$	−11.5624	−0.815547
$202$	0	0
$203$	−50.2242	−3.52505
$204$	0	0
$205$	11.1701	0.780150
$206$	0	0
$207$	−2.05940	−0.143138
$208$	0	0
$209$	3.48001	0.240717
$210$	0	0
$211$	25.0127	1.72195	0.860974	−	0.508649i	$-0.169855\pi$
$211$	25.0127	1.72195	0.860974	+	0.508649i	$0.169855\pi$
$212$	0	0
$213$	−16.1859	−1.10904
$214$	0	0
$215$	23.2826	1.58786
$216$	0	0
$217$	−29.7930	−2.02248
$218$	0	0
$219$	10.4689	0.707422
$220$	0	0
$221$	1.09862	0.0739010
$222$	0	0
$223$	−9.45803	−0.633357	−0.316678	−	0.948533i	$-0.602568\pi$
$223$	−9.45803	−0.633357	−0.316678	+	0.948533i	$0.602568\pi$
$224$	0	0
$225$	2.35794	0.157196
$226$	0	0
$227$	−19.3657	−1.28535	−0.642673	−	0.766141i	$-0.722174\pi$
$227$	−19.3657	−1.28535	−0.642673	+	0.766141i	$0.722174\pi$
$228$	0	0
$229$	14.6602	0.968774	0.484387	−	0.874854i	$-0.339043\pi$
$229$	14.6602	0.968774	0.484387	+	0.874854i	$0.339043\pi$
$230$	0	0
$231$	−7.22851	−0.475601
$232$	0	0
$233$	9.62634	0.630642	0.315321	−	0.948985i	$-0.397888\pi$
$233$	9.62634	0.630642	0.315321	+	0.948985i	$0.397888\pi$
$234$	0	0
$235$	−23.8291	−1.55444
$236$	0	0
$237$	−7.27177	−0.472352
$238$	0	0
$239$	−15.4588	−0.999949	−0.499974	−	0.866040i	$-0.666657\pi$
$239$	−15.4588	−0.999949	−0.499974	+	0.866040i	$0.666657\pi$
$240$	0	0
$241$	−8.16505	−0.525957	−0.262979	−	0.964802i	$-0.584705\pi$
$241$	−8.16505	−0.525957	−0.262979	+	0.964802i	$0.584705\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−52.7761	−3.37174
$246$	0	0
$247$	−6.99348	−0.444984
$248$	0	0
$249$	11.5589	0.732513
$250$	0	0
$251$	−17.3741	−1.09664	−0.548322	−	0.836267i	$-0.684733\pi$
$251$	−17.3741	−1.09664	−0.548322	+	0.836267i	$0.684733\pi$
$252$	0	0
$253$	2.89417	0.181955
$254$	0	0
$255$	−1.05518	−0.0660780
$256$	0	0
$257$	7.07802	0.441515	0.220758	−	0.975329i	$-0.429147\pi$
$257$	7.07802	0.441515	0.220758	+	0.975329i	$0.429147\pi$
$258$	0	0
$259$	−31.7714	−1.97418
$260$	0	0
$261$	−9.76448	−0.604406
$262$	0	0
$263$	−19.6386	−1.21097	−0.605485	−	0.795857i	$-0.707021\pi$
$263$	−19.6386	−1.21097	−0.605485	+	0.795857i	$0.707021\pi$
$264$	0	0
$265$	7.03946	0.432431
$266$	0	0
$267$	1.50198	0.0919196
$268$	0	0
$269$	7.31320	0.445893	0.222947	−	0.974831i	$-0.428432\pi$
$269$	7.31320	0.445893	0.222947	+	0.974831i	$0.428432\pi$
$270$	0	0
$271$	−3.36585	−0.204461	−0.102231	−	0.994761i	$-0.532598\pi$
$271$	−3.36585	−0.204461	−0.102231	+	0.994761i	$0.532598\pi$
$272$	0	0
$273$	14.5265	0.879185
$274$	0	0
$275$	−3.31372	−0.199825
$276$	0	0
$277$	−6.15719	−0.369950	−0.184975	−	0.982743i	$-0.559220\pi$
$277$	−6.15719	−0.369950	−0.184975	+	0.982743i	$0.559220\pi$
$278$	0	0
$279$	−5.79229	−0.346776
$280$	0	0
$281$	21.5407	1.28501	0.642505	−	0.766281i	$-0.277895\pi$
$281$	21.5407	1.28501	0.642505	+	0.766281i	$0.277895\pi$
$282$	0	0
$283$	3.51558	0.208979	0.104490	−	0.994526i	$-0.466679\pi$
$283$	3.51558	0.208979	0.104490	+	0.994526i	$0.466679\pi$
$284$	0	0
$285$	6.71698	0.397879
$286$	0	0
$287$	−21.1808	−1.25026
$288$	0	0
$289$	−16.8487	−0.991099
$290$	0	0
$291$	6.01313	0.352496
$292$	0	0
$293$	30.5584	1.78524	0.892619	−	0.450812i	$-0.148865\pi$
$293$	30.5584	1.78524	0.892619	+	0.450812i	$0.148865\pi$
$294$	0	0
$295$	36.9742	2.15272
$296$	0	0
$297$	−1.40535	−0.0815466
$298$	0	0
$299$	−5.81617	−0.336358
$300$	0	0
$301$	−44.1486	−2.54469
$302$	0	0
$303$	11.8180	0.678927
$304$	0	0
$305$	10.9430	0.626594
$306$	0	0
$307$	−1.45782	−0.0832021	−0.0416011	−	0.999134i	$-0.513246\pi$
$307$	−1.45782	−0.0832021	−0.0416011	+	0.999134i	$0.513246\pi$
$308$	0	0
$309$	5.58380	0.317651
$310$	0	0
$311$	−3.58336	−0.203194	−0.101597	−	0.994826i	$-0.532395\pi$
$311$	−3.58336	−0.203194	−0.101597	+	0.994826i	$0.532395\pi$
$312$	0	0
$313$	24.2575	1.37112	0.685558	−	0.728018i	$-0.259558\pi$
$313$	24.2575	1.37112	0.685558	+	0.728018i	$0.259558\pi$
$314$	0	0
$315$	−13.9522	−0.786116
$316$	0	0
$317$	−26.6137	−1.49478	−0.747388	−	0.664388i	$-0.768692\pi$
$317$	−26.6137	−1.49478	−0.747388	+	0.664388i	$0.768692\pi$
$318$	0	0
$319$	13.7225	0.768313
$320$	0	0
$321$	−0.921043	−0.0514076
$322$	0	0
$323$	−0.963263	−0.0535974
$324$	0	0
$325$	6.65931	0.369392
$326$	0	0
$327$	5.72379	0.316526
$328$	0	0
$329$	45.1850	2.49113
$330$	0	0
$331$	−18.6333	−1.02418	−0.512090	−	0.858932i	$-0.671128\pi$
$331$	−18.6333	−1.02418	−0.512090	+	0.858932i	$0.671128\pi$
$332$	0	0
$333$	−6.17693	−0.338494
$334$	0	0
$335$	31.3635	1.71357
$336$	0	0
$337$	6.86178	0.373785	0.186893	−	0.982380i	$-0.440158\pi$
$337$	6.86178	0.373785	0.186893	+	0.982380i	$0.440158\pi$
$338$	0	0
$339$	−5.32663	−0.289303
$340$	0	0
$341$	8.14020	0.440816
$342$	0	0
$343$	64.0695	3.45943
$344$	0	0
$345$	5.58622	0.300752
$346$	0	0
$347$	19.8523	1.06573	0.532865	−	0.846200i	$-0.321115\pi$
$347$	19.8523	1.06573	0.532865	+	0.846200i	$0.321115\pi$
$348$	0	0
$349$	23.1045	1.23676	0.618379	−	0.785880i	$-0.287790\pi$
$349$	23.1045	1.23676	0.618379	+	0.785880i	$0.287790\pi$
$350$	0	0
$351$	2.82421	0.150745
$352$	0	0
$353$	−36.4319	−1.93907	−0.969536	−	0.244949i	$-0.921229\pi$
$353$	−36.4319	−1.93907	−0.969536	+	0.244949i	$0.921229\pi$
$354$	0	0
$355$	43.9051	2.33024
$356$	0	0
$357$	2.00084	0.105896
$358$	0	0
$359$	15.1264	0.798339	0.399169	−	0.916877i	$-0.369299\pi$
$359$	15.1264	0.798339	0.399169	+	0.916877i	$0.369299\pi$
$360$	0	0
$361$	−12.8681	−0.677271
$362$	0	0
$363$	−9.02499	−0.473689
$364$	0	0
$365$	−28.3974	−1.48639
$366$	0	0
$367$	−2.85609	−0.149087	−0.0745434	−	0.997218i	$-0.523750\pi$
$367$	−2.85609	−0.149087	−0.0745434	+	0.997218i	$0.523750\pi$
$368$	0	0
$369$	−4.11791	−0.214370
$370$	0	0
$371$	−13.3483	−0.693008
$372$	0	0
$373$	−17.9849	−0.931223	−0.465611	−	0.884989i	$-0.654166\pi$
$373$	−17.9849	−0.931223	−0.465611	+	0.884989i	$0.654166\pi$
$374$	0	0
$375$	7.16673	0.370089
$376$	0	0
$377$	−27.5770	−1.42029
$378$	0	0
$379$	−26.8332	−1.37833	−0.689165	−	0.724604i	$-0.742023\pi$
$379$	−26.8332	−1.37833	−0.689165	+	0.724604i	$0.742023\pi$
$380$	0	0
$381$	1.08499	0.0555855
$382$	0	0
$383$	9.99636	0.510790	0.255395	−	0.966837i	$-0.417794\pi$
$383$	9.99636	0.510790	0.255395	+	0.966837i	$0.417794\pi$
$384$	0	0
$385$	19.6077	0.999301
$386$	0	0
$387$	−8.58328	−0.436313
$388$	0	0
$389$	26.0150	1.31902	0.659508	−	0.751698i	$-0.270765\pi$
$389$	26.0150	1.31902	0.659508	+	0.751698i	$0.270765\pi$
$390$	0	0
$391$	−0.801104	−0.0405136
$392$	0	0
$393$	2.10877	0.106374
$394$	0	0
$395$	19.7251	0.992475
$396$	0	0
$397$	5.04844	0.253374	0.126687	−	0.991943i	$-0.459566\pi$
$397$	5.04844	0.253374	0.126687	+	0.991943i	$0.459566\pi$
$398$	0	0
$399$	−12.7368	−0.637637
$400$	0	0
$401$	4.62207	0.230815	0.115408	−	0.993318i	$-0.463183\pi$
$401$	4.62207	0.230815	0.115408	+	0.993318i	$0.463183\pi$
$402$	0	0
$403$	−16.3587	−0.814883
$404$	0	0
$405$	−2.71255	−0.134788
$406$	0	0
$407$	8.68075	0.430289
$408$	0	0
$409$	−20.8508	−1.03101	−0.515503	−	0.856888i	$-0.672395\pi$
$409$	−20.8508	−1.03101	−0.515503	+	0.856888i	$0.672395\pi$
$410$	0	0
$411$	−9.56847	−0.471978
$412$	0	0
$413$	−70.1108	−3.44993
$414$	0	0
$415$	−31.3540	−1.53911
$416$	0	0
$417$	−10.6906	−0.523520
$418$	0	0
$419$	4.38115	0.214033	0.107017	−	0.994257i	$-0.465870\pi$
$419$	4.38115	0.214033	0.107017	+	0.994257i	$0.465870\pi$
$420$	0	0
$421$	−7.10263	−0.346161	−0.173081	−	0.984908i	$-0.555372\pi$
$421$	−7.10263	−0.346161	−0.173081	+	0.984908i	$0.555372\pi$
$422$	0	0
$423$	8.78477	0.427130
$424$	0	0
$425$	0.917236	0.0444925
$426$	0	0
$427$	−20.7502	−1.00417
$428$	0	0
$429$	−3.96900	−0.191625
$430$	0	0
$431$	−23.2622	−1.12050	−0.560250	−	0.828324i	$-0.689295\pi$
$431$	−23.2622	−1.12050	−0.560250	+	0.828324i	$0.689295\pi$
$432$	0	0
$433$	10.9540	0.526416	0.263208	−	0.964739i	$-0.415219\pi$
$433$	10.9540	0.526416	0.263208	+	0.964739i	$0.415219\pi$
$434$	0	0
$435$	26.4866	1.26994
$436$	0	0
$437$	5.09960	0.243947
$438$	0	0
$439$	3.79855	0.181295	0.0906475	−	0.995883i	$-0.471106\pi$
$439$	3.79855	0.181295	0.0906475	+	0.995883i	$0.471106\pi$
$440$	0	0
$441$	19.4563	0.926488
$442$	0	0
$443$	−18.8716	−0.896619	−0.448309	−	0.893879i	$-0.647974\pi$
$443$	−18.8716	−0.896619	−0.448309	+	0.893879i	$0.647974\pi$
$444$	0	0
$445$	−4.07419	−0.193135
$446$	0	0
$447$	−5.28320	−0.249887
$448$	0	0
$449$	7.03407	0.331958	0.165979	−	0.986129i	$-0.446922\pi$
$449$	7.03407	0.331958	0.165979	+	0.986129i	$0.446922\pi$
$450$	0	0
$451$	5.78711	0.272504
$452$	0	0
$453$	2.03474	0.0956003
$454$	0	0
$455$	−39.4039	−1.84728
$456$	0	0
$457$	−7.16863	−0.335334	−0.167667	−	0.985844i	$-0.553623\pi$
$457$	−7.16863	−0.335334	−0.167667	+	0.985844i	$0.553623\pi$
$458$	0	0
$459$	0.388999	0.0181569
$460$	0	0
$461$	−15.2697	−0.711182	−0.355591	−	0.934642i	$-0.615720\pi$
$461$	−15.2697	−0.711182	−0.355591	+	0.934642i	$0.615720\pi$
$462$	0	0
$463$	−4.81091	−0.223582	−0.111791	−	0.993732i	$-0.535659\pi$
$463$	−4.81091	−0.223582	−0.111791	+	0.993732i	$0.535659\pi$
$464$	0	0
$465$	15.7119	0.728622
$466$	0	0
$467$	8.72861	0.403912	0.201956	−	0.979395i	$-0.435270\pi$
$467$	8.72861	0.403912	0.201956	+	0.979395i	$0.435270\pi$
$468$	0	0
$469$	−59.4718	−2.74615
$470$	0	0
$471$	20.3613	0.938201
$472$	0	0
$473$	12.0625	0.554635
$474$	0	0
$475$	−5.83886	−0.267905
$476$	0	0
$477$	−2.59514	−0.118823
$478$	0	0
$479$	−26.5577	−1.21345	−0.606726	−	0.794911i	$-0.707517\pi$
$479$	−26.5577	−1.21345	−0.606726	+	0.794911i	$0.707517\pi$
$480$	0	0
$481$	−17.4450	−0.795422
$482$	0	0
$483$	−10.5926	−0.481982
$484$	0	0
$485$	−16.3109	−0.740641
$486$	0	0
$487$	36.4048	1.64966	0.824830	−	0.565381i	$-0.191271\pi$
$487$	36.4048	1.64966	0.824830	+	0.565381i	$0.191271\pi$
$488$	0	0
$489$	−1.61102	−0.0728529
$490$	0	0
$491$	27.7254	1.25123	0.625615	−	0.780132i	$-0.284848\pi$
$491$	27.7254	1.25123	0.625615	+	0.780132i	$0.284848\pi$
$492$	0	0
$493$	−3.79838	−0.171070
$494$	0	0
$495$	3.81208	0.171340
$496$	0	0
$497$	−83.2533	−3.73442
$498$	0	0
$499$	−38.4599	−1.72170	−0.860850	−	0.508859i	$-0.830067\pi$
$499$	−38.4599	−1.72170	−0.860850	+	0.508859i	$0.830067\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	14.0245	0.625322	0.312661	−	0.949865i	$-0.398780\pi$
$503$	14.0245	0.625322	0.312661	+	0.949865i	$0.398780\pi$
$504$	0	0
$505$	−32.0570	−1.42652
$506$	0	0
$507$	−5.02383	−0.223116
$508$	0	0
$509$	2.79569	0.123917	0.0619584	−	0.998079i	$-0.480265\pi$
$509$	2.79569	0.123917	0.0619584	+	0.998079i	$0.480265\pi$
$510$	0	0
$511$	53.8474	2.38207
$512$	0	0
$513$	−2.47626	−0.109329
$514$	0	0
$515$	−15.1463	−0.667427
$516$	0	0
$517$	−12.3457	−0.542962
$518$	0	0
$519$	−25.1622	−1.10450
$520$	0	0
$521$	−29.2905	−1.28324	−0.641620	−	0.767023i	$-0.721737\pi$
$521$	−29.2905	−1.28324	−0.641620	+	0.767023i	$0.721737\pi$
$522$	0	0
$523$	−44.5439	−1.94777	−0.973885	−	0.227040i	$-0.927095\pi$
$523$	−44.5439	−1.94777	−0.973885	+	0.227040i	$0.927095\pi$
$524$	0	0
$525$	12.1282	0.529318
$526$	0	0
$527$	−2.25320	−0.0981509
$528$	0	0
$529$	−18.7589	−0.815604
$530$	0	0
$531$	−13.6308	−0.591525
$532$	0	0
$533$	−11.6299	−0.503745
$534$	0	0
$535$	2.49838	0.108014
$536$	0	0
$537$	−0.906147	−0.0391031
$538$	0	0
$539$	−27.3428	−1.17774
$540$	0	0
$541$	21.3219	0.916701	0.458350	−	0.888772i	$-0.348440\pi$
$541$	21.3219	0.916701	0.458350	+	0.888772i	$0.348440\pi$
$542$	0	0
$543$	25.6772	1.10192
$544$	0	0
$545$	−15.5261	−0.665064
$546$	0	0
$547$	19.8718	0.849656	0.424828	−	0.905274i	$-0.360334\pi$
$547$	19.8718	0.849656	0.424828	+	0.905274i	$0.360334\pi$
$548$	0	0
$549$	−4.03421	−0.172176
$550$	0	0
$551$	24.1794	1.03008
$552$	0	0
$553$	−37.4028	−1.59053
$554$	0	0
$555$	16.7552	0.711221
$556$	0	0
$557$	23.4045	0.991680	0.495840	−	0.868414i	$-0.334860\pi$
$557$	23.4045	0.991680	0.495840	+	0.868414i	$0.334860\pi$
$558$	0	0
$559$	−24.2410	−1.02529
$560$	0	0
$561$	−0.546680	−0.0230809
$562$	0	0
$563$	−13.0340	−0.549317	−0.274659	−	0.961542i	$-0.588565\pi$
$563$	−13.0340	−0.549317	−0.274659	+	0.961542i	$0.588565\pi$
$564$	0	0
$565$	14.4488	0.607864
$566$	0	0
$567$	5.14356	0.216009
$568$	0	0
$569$	−35.6943	−1.49638	−0.748191	−	0.663483i	$-0.769077\pi$
$569$	−35.6943	−1.49638	−0.748191	+	0.663483i	$0.769077\pi$
$570$	0	0
$571$	−3.74464	−0.156708	−0.0783542	−	0.996926i	$-0.524966\pi$
$571$	−3.74464	−0.156708	−0.0783542	+	0.996926i	$0.524966\pi$
$572$	0	0
$573$	0.440700	0.0184105
$574$	0	0
$575$	−4.85593	−0.202506
$576$	0	0
$577$	−4.82085	−0.200695	−0.100347	−	0.994952i	$-0.531995\pi$
$577$	−4.82085	−0.200695	−0.100347	+	0.994952i	$0.531995\pi$
$578$	0	0
$579$	−2.48672	−0.103345
$580$	0	0
$581$	59.4537	2.46656
$582$	0	0
$583$	3.64708	0.151047
$584$	0	0
$585$	−7.66082	−0.316736
$586$	0	0
$587$	9.18744	0.379206	0.189603	−	0.981861i	$-0.439280\pi$
$587$	9.18744	0.379206	0.189603	+	0.981861i	$0.439280\pi$
$588$	0	0
$589$	14.3432	0.591002
$590$	0	0
$591$	−14.7487	−0.606680
$592$	0	0
$593$	37.5374	1.54147	0.770737	−	0.637153i	$-0.219888\pi$
$593$	37.5374	1.54147	0.770737	+	0.637153i	$0.219888\pi$
$594$	0	0
$595$	−5.42739	−0.222501
$596$	0	0
$597$	14.3359	0.586729
$598$	0	0
$599$	6.72390	0.274731	0.137366	−	0.990520i	$-0.456136\pi$
$599$	6.72390	0.274731	0.137366	+	0.990520i	$0.456136\pi$
$600$	0	0
$601$	18.4937	0.754373	0.377186	−	0.926137i	$-0.376892\pi$
$601$	18.4937	0.754373	0.377186	+	0.926137i	$0.376892\pi$
$602$	0	0
$603$	−11.5624	−0.470856
$604$	0	0
$605$	24.4808	0.995284
$606$	0	0
$607$	−13.9398	−0.565799	−0.282900	−	0.959150i	$-0.591296\pi$
$607$	−13.9398	−0.565799	−0.282900	+	0.959150i	$0.591296\pi$
$608$	0	0
$609$	−50.2242	−2.03519
$610$	0	0
$611$	24.8100	1.00371
$612$	0	0
$613$	31.9354	1.28986	0.644929	−	0.764242i	$-0.276887\pi$
$613$	31.9354	1.28986	0.644929	+	0.764242i	$0.276887\pi$
$614$	0	0
$615$	11.1701	0.450420
$616$	0	0
$617$	−2.30028	−0.0926059	−0.0463029	−	0.998927i	$-0.514744\pi$
$617$	−2.30028	−0.0926059	−0.0463029	+	0.998927i	$0.514744\pi$
$618$	0	0
$619$	−34.5858	−1.39012	−0.695060	−	0.718951i	$-0.744622\pi$
$619$	−34.5858	−1.39012	−0.695060	+	0.718951i	$0.744622\pi$
$620$	0	0
$621$	−2.05940	−0.0826407
$622$	0	0
$623$	7.72552	0.309517
$624$	0	0
$625$	−31.2298	−1.24919
$626$	0	0
$627$	3.48001	0.138978
$628$	0	0
$629$	−2.40282	−0.0958068
$630$	0	0
$631$	7.33109	0.291846	0.145923	−	0.989296i	$-0.453385\pi$
$631$	7.33109	0.291846	0.145923	+	0.989296i	$0.453385\pi$
$632$	0	0
$633$	25.0127	0.994167
$634$	0	0
$635$	−2.94308	−0.116793
$636$	0	0
$637$	54.9486	2.17714
$638$	0	0
$639$	−16.1859	−0.640305
$640$	0	0
$641$	19.6501	0.776130	0.388065	−	0.921632i	$-0.373144\pi$
$641$	19.6501	0.776130	0.388065	+	0.921632i	$0.373144\pi$
$642$	0	0
$643$	−3.43164	−0.135331	−0.0676654	−	0.997708i	$-0.521555\pi$
$643$	−3.43164	−0.135331	−0.0676654	+	0.997708i	$0.521555\pi$
$644$	0	0
$645$	23.2826	0.916751
$646$	0	0
$647$	−18.0148	−0.708234	−0.354117	−	0.935201i	$-0.615219\pi$
$647$	−18.0148	−0.708234	−0.354117	+	0.935201i	$0.615219\pi$
$648$	0	0
$649$	19.1560	0.751939
$650$	0	0
$651$	−29.7930	−1.16768
$652$	0	0
$653$	−33.7933	−1.32243	−0.661216	−	0.750195i	$-0.729959\pi$
$653$	−33.7933	−1.32243	−0.661216	+	0.750195i	$0.729959\pi$
$654$	0	0
$655$	−5.72016	−0.223505
$656$	0	0
$657$	10.4689	0.408430
$658$	0	0
$659$	−19.8505	−0.773265	−0.386632	−	0.922234i	$-0.626362\pi$
$659$	−19.8505	−0.773265	−0.386632	+	0.922234i	$0.626362\pi$
$660$	0	0
$661$	31.8630	1.23933	0.619664	−	0.784868i	$-0.287269\pi$
$661$	31.8630	1.23933	0.619664	+	0.784868i	$0.287269\pi$
$662$	0	0
$663$	1.09862	0.0426667
$664$	0	0
$665$	34.5492	1.33976
$666$	0	0
$667$	20.1089	0.778621
$668$	0	0
$669$	−9.45803	−0.365669
$670$	0	0
$671$	5.66947	0.218867
$672$	0	0
$673$	−18.0848	−0.697119	−0.348560	−	0.937287i	$-0.613329\pi$
$673$	−18.0848	−0.697119	−0.348560	+	0.937287i	$0.613329\pi$
$674$	0	0
$675$	2.35794	0.0907570
$676$	0	0
$677$	4.29057	0.164900	0.0824500	−	0.996595i	$-0.473726\pi$
$677$	4.29057	0.164900	0.0824500	+	0.996595i	$0.473726\pi$
$678$	0	0
$679$	30.9289	1.18694
$680$	0	0
$681$	−19.3657	−0.742094
$682$	0	0
$683$	−20.6870	−0.791565	−0.395782	−	0.918344i	$-0.629527\pi$
$683$	−20.6870	−0.791565	−0.395782	+	0.918344i	$0.629527\pi$
$684$	0	0
$685$	25.9550	0.991689
$686$	0	0
$687$	14.6602	0.559322
$688$	0	0
$689$	−7.32923	−0.279222
$690$	0	0
$691$	−18.4431	−0.701608	−0.350804	−	0.936449i	$-0.614092\pi$
$691$	−18.4431	−0.701608	−0.350804	+	0.936449i	$0.614092\pi$
$692$	0	0
$693$	−7.22851	−0.274588
$694$	0	0
$695$	28.9988	1.09999
$696$	0	0
$697$	−1.60187	−0.0606750
$698$	0	0
$699$	9.62634	0.364101
$700$	0	0
$701$	29.2094	1.10322	0.551612	−	0.834101i	$-0.314013\pi$
$701$	29.2094	1.10322	0.551612	+	0.834101i	$0.314013\pi$
$702$	0	0
$703$	15.2957	0.576887
$704$	0	0
$705$	−23.8291	−0.897457
$706$	0	0
$707$	60.7867	2.28612
$708$	0	0
$709$	32.3140	1.21358	0.606790	−	0.794862i	$-0.292457\pi$
$709$	32.3140	1.21358	0.606790	+	0.794862i	$0.292457\pi$
$710$	0	0
$711$	−7.27177	−0.272713
$712$	0	0
$713$	11.9286	0.446731
$714$	0	0
$715$	10.7661	0.402630
$716$	0	0
$717$	−15.4588	−0.577321
$718$	0	0
$719$	41.2415	1.53805	0.769023	−	0.639221i	$-0.220743\pi$
$719$	41.2415	1.53805	0.769023	+	0.639221i	$0.220743\pi$
$720$	0	0
$721$	28.7206	1.06961
$722$	0	0
$723$	−8.16505	−0.303662
$724$	0	0
$725$	−23.0240	−0.855090
$726$	0	0
$727$	−51.6838	−1.91685	−0.958423	−	0.285351i	$-0.907890\pi$
$727$	−51.6838	−1.91685	−0.958423	+	0.285351i	$0.907890\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−3.33889	−0.123493
$732$	0	0
$733$	−2.07201	−0.0765314	−0.0382657	−	0.999268i	$-0.512183\pi$
$733$	−2.07201	−0.0765314	−0.0382657	+	0.999268i	$0.512183\pi$
$734$	0	0
$735$	−52.7761	−1.94668
$736$	0	0
$737$	16.2492	0.598546
$738$	0	0
$739$	−32.7990	−1.20653	−0.603266	−	0.797540i	$-0.706134\pi$
$739$	−32.7990	−1.20653	−0.603266	+	0.797540i	$0.706134\pi$
$740$	0	0
$741$	−6.99348	−0.256912
$742$	0	0
$743$	33.2360	1.21931	0.609655	−	0.792667i	$-0.291308\pi$
$743$	33.2360	1.21931	0.609655	+	0.792667i	$0.291308\pi$
$744$	0	0
$745$	14.3309	0.525045
$746$	0	0
$747$	11.5589	0.422917
$748$	0	0
$749$	−4.73745	−0.173102
$750$	0	0
$751$	36.9105	1.34688	0.673441	−	0.739241i	$-0.264816\pi$
$751$	36.9105	1.34688	0.673441	+	0.739241i	$0.264816\pi$
$752$	0	0
$753$	−17.3741	−0.633148
$754$	0	0
$755$	−5.51933	−0.200869
$756$	0	0
$757$	−30.8390	−1.12086	−0.560432	−	0.828201i	$-0.689365\pi$
$757$	−30.8390	−1.12086	−0.560432	+	0.828201i	$0.689365\pi$
$758$	0	0
$759$	2.89417	0.105052
$760$	0	0
$761$	19.4664	0.705656	0.352828	−	0.935688i	$-0.385220\pi$
$761$	19.4664	0.705656	0.352828	+	0.935688i	$0.385220\pi$
$762$	0	0
$763$	29.4407	1.06582
$764$	0	0
$765$	−1.05518	−0.0381502
$766$	0	0
$767$	−38.4962	−1.39002
$768$	0	0
$769$	−30.6488	−1.10522	−0.552611	−	0.833439i	$-0.686369\pi$
$769$	−30.6488	−1.10522	−0.552611	+	0.833439i	$0.686369\pi$
$770$	0	0
$771$	7.07802	0.254909
$772$	0	0
$773$	52.1823	1.87687	0.938433	−	0.345461i	$-0.112278\pi$
$773$	52.1823	1.87687	0.938433	+	0.345461i	$0.112278\pi$
$774$	0	0
$775$	−13.6579	−0.490605
$776$	0	0
$777$	−31.7714	−1.13979
$778$	0	0
$779$	10.1970	0.365346
$780$	0	0
$781$	22.7469	0.813947
$782$	0	0
$783$	−9.76448	−0.348954
$784$	0	0
$785$	−55.2312	−1.97129
$786$	0	0
$787$	−39.2188	−1.39800	−0.699000	−	0.715122i	$-0.746371\pi$
$787$	−39.2188	−1.39800	−0.699000	+	0.715122i	$0.746371\pi$
$788$	0	0
$789$	−19.6386	−0.699154
$790$	0	0
$791$	−27.3979	−0.974156
$792$	0	0
$793$	−11.3935	−0.404593
$794$	0	0
$795$	7.03946	0.249664
$796$	0	0
$797$	−23.4569	−0.830885	−0.415443	−	0.909619i	$-0.636373\pi$
$797$	−23.4569	−0.830885	−0.415443	+	0.909619i	$0.636373\pi$
$798$	0	0
$799$	3.41727	0.120894
$800$	0	0
$801$	1.50198	0.0530698
$802$	0	0
$803$	−14.7125	−0.519191
$804$	0	0
$805$	28.7331	1.01271
$806$	0	0
$807$	7.31320	0.257437
$808$	0	0
$809$	41.2822	1.45141	0.725703	−	0.688008i	$-0.241515\pi$
$809$	41.2822	1.45141	0.725703	+	0.688008i	$0.241515\pi$
$810$	0	0
$811$	−9.53786	−0.334920	−0.167460	−	0.985879i	$-0.553556\pi$
$811$	−9.53786	−0.334920	−0.167460	+	0.985879i	$0.553556\pi$
$812$	0	0
$813$	−3.36585	−0.118046
$814$	0	0
$815$	4.36998	0.153074
$816$	0	0
$817$	21.2544	0.743598
$818$	0	0
$819$	14.5265	0.507598
$820$	0	0
$821$	41.2716	1.44039	0.720194	−	0.693773i	$-0.244053\pi$
$821$	41.2716	1.44039	0.720194	+	0.693773i	$0.244053\pi$
$822$	0	0
$823$	−24.1633	−0.842280	−0.421140	−	0.906996i	$-0.638370\pi$
$823$	−24.1633	−0.842280	−0.421140	+	0.906996i	$0.638370\pi$
$824$	0	0
$825$	−3.31372	−0.115369
$826$	0	0
$827$	36.7132	1.27664	0.638321	−	0.769770i	$-0.279629\pi$
$827$	36.7132	1.27664	0.638321	+	0.769770i	$0.279629\pi$
$828$	0	0
$829$	−22.5652	−0.783723	−0.391862	−	0.920024i	$-0.628169\pi$
$829$	−22.5652	−0.783723	−0.391862	+	0.920024i	$0.628169\pi$
$830$	0	0
$831$	−6.15719	−0.213591
$832$	0	0
$833$	7.56847	0.262232
$834$	0	0
$835$	2.71255	0.0938717
$836$	0	0
$837$	−5.79229	−0.200211
$838$	0	0
$839$	39.3178	1.35740	0.678701	−	0.734414i	$-0.262543\pi$
$839$	39.3178	1.35740	0.678701	+	0.734414i	$0.262543\pi$
$840$	0	0
$841$	66.3450	2.28776
$842$	0	0
$843$	21.5407	0.741901
$844$	0	0
$845$	13.6274	0.468796
$846$	0	0
$847$	−46.4206	−1.59503
$848$	0	0
$849$	3.51558	0.120654
$850$	0	0
$851$	12.7208	0.436062
$852$	0	0
$853$	−9.05470	−0.310027	−0.155013	−	0.987912i	$-0.549542\pi$
$853$	−9.05470	−0.310027	−0.155013	+	0.987912i	$0.549542\pi$
$854$	0	0
$855$	6.71698	0.229716
$856$	0	0
$857$	31.9208	1.09039	0.545196	−	0.838308i	$-0.316455\pi$
$857$	31.9208	1.09039	0.545196	+	0.838308i	$0.316455\pi$
$858$	0	0
$859$	−42.7955	−1.46016	−0.730081	−	0.683361i	$-0.760518\pi$
$859$	−42.7955	−1.46016	−0.730081	+	0.683361i	$0.760518\pi$
$860$	0	0
$861$	−21.1808	−0.721838
$862$	0	0
$863$	−20.5577	−0.699792	−0.349896	−	0.936788i	$-0.613783\pi$
$863$	−20.5577	−0.699792	−0.349896	+	0.936788i	$0.613783\pi$
$864$	0	0
$865$	68.2538	2.32070
$866$	0	0
$867$	−16.8487	−0.572211
$868$	0	0
$869$	10.2194	0.346669
$870$	0	0
$871$	−32.6546	−1.10646
$872$	0	0
$873$	6.01313	0.203514
$874$	0	0
$875$	36.8626	1.24618
$876$	0	0
$877$	−19.4547	−0.656937	−0.328469	−	0.944515i	$-0.606533\pi$
$877$	−19.4547	−0.656937	−0.328469	+	0.944515i	$0.606533\pi$
$878$	0	0
$879$	30.5584	1.03071
$880$	0	0
$881$	18.1949	0.613001	0.306500	−	0.951871i	$-0.400842\pi$
$881$	18.1949	0.613001	0.306500	+	0.951871i	$0.400842\pi$
$882$	0	0
$883$	56.7006	1.90813	0.954064	−	0.299603i	$-0.0968542\pi$
$883$	56.7006	1.90813	0.954064	+	0.299603i	$0.0968542\pi$
$884$	0	0
$885$	36.9742	1.24287
$886$	0	0
$887$	25.2605	0.848163	0.424082	−	0.905624i	$-0.360597\pi$
$887$	25.2605	0.848163	0.424082	+	0.905624i	$0.360597\pi$
$888$	0	0
$889$	5.58070	0.187171
$890$	0	0
$891$	−1.40535	−0.0470810
$892$	0	0
$893$	−21.7533	−0.727948
$894$	0	0
$895$	2.45797	0.0821608
$896$	0	0
$897$	−5.81617	−0.194196
$898$	0	0
$899$	56.5587	1.88634
$900$	0	0
$901$	−1.00951	−0.0336316
$902$	0	0
$903$	−44.1486	−1.46917
$904$	0	0
$905$	−69.6508	−2.31527
$906$	0	0
$907$	−35.2144	−1.16927	−0.584637	−	0.811295i	$-0.698763\pi$
$907$	−35.2144	−1.16927	−0.584637	+	0.811295i	$0.698763\pi$
$908$	0	0
$909$	11.8180	0.391979
$910$	0	0
$911$	−24.5943	−0.814845	−0.407423	−	0.913240i	$-0.633572\pi$
$911$	−24.5943	−0.814845	−0.407423	+	0.913240i	$0.633572\pi$
$912$	0	0
$913$	−16.2442	−0.537606
$914$	0	0
$915$	10.9430	0.361764
$916$	0	0
$917$	10.8466	0.358187
$918$	0	0
$919$	41.2339	1.36018	0.680090	−	0.733128i	$-0.261941\pi$
$919$	41.2339	1.36018	0.680090	+	0.733128i	$0.261941\pi$
$920$	0	0
$921$	−1.45782	−0.0480368
$922$	0	0
$923$	−45.7124	−1.50464
$924$	0	0
$925$	−14.5648	−0.478888
$926$	0	0
$927$	5.58380	0.183396
$928$	0	0
$929$	−26.8601	−0.881251	−0.440625	−	0.897691i	$-0.645243\pi$
$929$	−26.8601	−0.881251	−0.440625	+	0.897691i	$0.645243\pi$
$930$	0	0
$931$	−48.1787	−1.57899
$932$	0	0
$933$	−3.58336	−0.117314
$934$	0	0
$935$	1.48290	0.0484960
$936$	0	0
$937$	−9.15808	−0.299181	−0.149591	−	0.988748i	$-0.547796\pi$
$937$	−9.15808	−0.299181	−0.149591	+	0.988748i	$0.547796\pi$
$938$	0	0
$939$	24.2575	0.791615
$940$	0	0
$941$	48.5212	1.58174	0.790872	−	0.611981i	$-0.209627\pi$
$941$	48.5212	1.58174	0.790872	+	0.611981i	$0.209627\pi$
$942$	0	0
$943$	8.48042	0.276160
$944$	0	0
$945$	−13.9522	−0.453865
$946$	0	0
$947$	41.0699	1.33459	0.667296	−	0.744792i	$-0.267452\pi$
$947$	41.0699	1.33459	0.667296	+	0.744792i	$0.267452\pi$
$948$	0	0
$949$	29.5664	0.959765
$950$	0	0
$951$	−26.6137	−0.863009
$952$	0	0
$953$	−12.7472	−0.412924	−0.206462	−	0.978455i	$-0.566195\pi$
$953$	−12.7472	−0.412924	−0.206462	+	0.978455i	$0.566195\pi$
$954$	0	0
$955$	−1.19542	−0.0386829
$956$	0	0
$957$	13.7225	0.443586
$958$	0	0
$959$	−49.2161	−1.58927
$960$	0	0
$961$	2.55068	0.0822799
$962$	0	0
$963$	−0.921043	−0.0296802
$964$	0	0
$965$	6.74536	0.217141
$966$	0	0
$967$	23.7599	0.764067	0.382034	−	0.924148i	$-0.375224\pi$
$967$	23.7599	0.764067	0.382034	+	0.924148i	$0.375224\pi$
$968$	0	0
$969$	−0.963263	−0.0309445
$970$	0	0
$971$	36.1431	1.15989	0.579944	−	0.814656i	$-0.303074\pi$
$971$	36.1431	1.15989	0.579944	+	0.814656i	$0.303074\pi$
$972$	0	0
$973$	−54.9877	−1.76282
$974$	0	0
$975$	6.65931	0.213269
$976$	0	0
$977$	−24.4262	−0.781464	−0.390732	−	0.920504i	$-0.627778\pi$
$977$	−24.4262	−0.781464	−0.390732	+	0.920504i	$0.627778\pi$
$978$	0	0
$979$	−2.11080	−0.0674616
$980$	0	0
$981$	5.72379	0.182746
$982$	0	0
$983$	22.2434	0.709454	0.354727	−	0.934970i	$-0.384574\pi$
$983$	22.2434	0.709454	0.354727	+	0.934970i	$0.384574\pi$
$984$	0	0
$985$	40.0066	1.27472
$986$	0	0
$987$	45.1850	1.43825
$988$	0	0
$989$	17.6764	0.562076
$990$	0	0
$991$	47.9472	1.52309	0.761546	−	0.648111i	$-0.224441\pi$
$991$	47.9472	1.52309	0.761546	+	0.648111i	$0.224441\pi$
$992$	0	0
$993$	−18.6333	−0.591310
$994$	0	0
$995$	−38.8868	−1.23280
$996$	0	0
$997$	−28.1624	−0.891913	−0.445957	−	0.895055i	$-0.647136\pi$
$997$	−28.1624	−0.891913	−0.445957	+	0.895055i	$0.647136\pi$
$998$	0	0
$999$	−6.17693	−0.195429

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.z.1.2		8
4.3	odd	2		501.2.a.d.1.5	✓	8
12.11	even	2		1503.2.a.f.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
501.2.a.d.1.5	✓	8	4.3	odd	2
1503.2.a.f.1.4		8	12.11	even	2
8016.2.a.z.1.2		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:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.71255 q^{5} +5.14356 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.71255 q^{5} +5.14356 q^{7} +1.00000 q^{9} -1.40535 q^{11} +2.82421 q^{13} -2.71255 q^{15} +0.388999 q^{17} -2.47626 q^{19} +5.14356 q^{21} -2.05940 q^{23} +2.35794 q^{25} +1.00000 q^{27} -9.76448 q^{29} -5.79229 q^{31} -1.40535 q^{33} -13.9522 q^{35} -6.17693 q^{37} +2.82421 q^{39} -4.11791 q^{41} -8.58328 q^{43} -2.71255 q^{45} +8.78477 q^{47} +19.4563 q^{49} +0.388999 q^{51} -2.59514 q^{53} +3.81208 q^{55} -2.47626 q^{57} -13.6308 q^{59} -4.03421 q^{61} +5.14356 q^{63} -7.66082 q^{65} -11.5624 q^{67} -2.05940 q^{69} -16.1859 q^{71} +10.4689 q^{73} +2.35794 q^{75} -7.22851 q^{77} -7.27177 q^{79} +1.00000 q^{81} +11.5589 q^{83} -1.05518 q^{85} -9.76448 q^{87} +1.50198 q^{89} +14.5265 q^{91} -5.79229 q^{93} +6.71698 q^{95} +6.01313 q^{97} -1.40535 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} + q^{5} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^3 + q^5 + 8 * q^9	\( 8 q + 8 q^{3} + q^{5} + 8 q^{9} - 5 q^{11} + q^{15} - 7 q^{17} - 24 q^{19} - q^{23} + 3 q^{25} + 8 q^{27} - 11 q^{29} - 30 q^{31} - 5 q^{33} - 26 q^{35} + 11 q^{37} + 10 q^{41} - 24 q^{43} + q^{45} + 3 q^{47} + 6 q^{49} - 7 q^{51} - 25 q^{53} - 25 q^{55} - 24 q^{57} - 45 q^{59} + 16 q^{61} - 10 q^{65} - 18 q^{67} - q^{69} - 21 q^{71} - 8 q^{73} + 3 q^{75} - 18 q^{77} - 10 q^{79} + 8 q^{81} - 7 q^{83} - 11 q^{85} - 11 q^{87} + 26 q^{89} - 15 q^{91} - 30 q^{93} - q^{95} - 3 q^{97} - 5 q^{99}+O(q^{100}) \) 8 * q + 8 * q^3 + q^5 + 8 * q^9 - 5 * q^11 + q^15 - 7 * q^17 - 24 * q^19 - q^23 + 3 * q^25 + 8 * q^27 - 11 * q^29 - 30 * q^31 - 5 * q^33 - 26 * q^35 + 11 * q^37 + 10 * q^41 - 24 * q^43 + q^45 + 3 * q^47 + 6 * q^49 - 7 * q^51 - 25 * q^53 - 25 * q^55 - 24 * q^57 - 45 * q^59 + 16 * q^61 - 10 * q^65 - 18 * q^67 - q^69 - 21 * q^71 - 8 * q^73 + 3 * q^75 - 18 * q^77 - 10 * q^79 + 8 * q^81 - 7 * q^83 - 11 * q^85 - 11 * q^87 + 26 * q^89 - 15 * q^91 - 30 * q^93 - q^95 - 3 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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