LMFDB - Embedded newform 8004.2.a.g.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8004,2,Mod(1,8004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8004, 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("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$63.9122617778$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 31 x^{10} + 80 x^{9} + 347 x^{8} - 697 x^{7} - 1714 x^{6} + 2146 x^{5} + 3304 x^{4} + \cdots + 16 $ x^12 - 3x^11 - 31x^10 + 80x^9 + 347x^8 - 697x^7 - 1714x^6 + 2146x^5 + 3304x^4 - 1156x^3 - 616x^2 + 136*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-2.48147$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.48147 q^{5} -0.0448560 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.48147 q^{5} -0.0448560 q^{7} +1.00000 q^{9} -0.683409 q^{11} -6.72642 q^{13} -2.48147 q^{15} -1.19277 q^{17} +1.53130 q^{19} +0.0448560 q^{21} +1.00000 q^{23} +1.15770 q^{25} -1.00000 q^{27} -1.00000 q^{29} +1.40385 q^{31} +0.683409 q^{33} -0.111309 q^{35} +8.68984 q^{37} +6.72642 q^{39} +11.5300 q^{41} -6.28573 q^{43} +2.48147 q^{45} +5.00128 q^{47} -6.99799 q^{49} +1.19277 q^{51} -1.85106 q^{53} -1.69586 q^{55} -1.53130 q^{57} -2.14821 q^{59} -2.75349 q^{61} -0.0448560 q^{63} -16.6914 q^{65} -0.0568657 q^{67} -1.00000 q^{69} -1.36240 q^{71} -1.19218 q^{73} -1.15770 q^{75} +0.0306550 q^{77} +0.0825347 q^{79} +1.00000 q^{81} +7.83176 q^{83} -2.95982 q^{85} +1.00000 q^{87} -0.473510 q^{89} +0.301720 q^{91} -1.40385 q^{93} +3.79989 q^{95} -5.20158 q^{97} -0.683409 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9	$ 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9} - 5 q^{11} - 6 q^{13} + 3 q^{15} - 7 q^{17} - 3 q^{19} - 4 q^{21} + 12 q^{23} + 11 q^{25} - 12 q^{27} - 12 q^{29} + 2 q^{31} + 5 q^{33} - 9 q^{35} - 20 q^{37} + 6 q^{39} - 3 q^{41} + 5 q^{43} - 3 q^{45} - 2 q^{49} + 7 q^{51} - 3 q^{53} + 19 q^{55} + 3 q^{57} - 20 q^{59} - 17 q^{61} + 4 q^{63} - 4 q^{65} - 9 q^{67} - 12 q^{69} + 7 q^{71} - 9 q^{73} - 11 q^{75} - 34 q^{77} + 14 q^{79} + 12 q^{81} + 5 q^{83} - 12 q^{85} + 12 q^{87} - 22 q^{89} - 3 q^{91} - 2 q^{93} - 27 q^{95} + 17 q^{97} - 5 q^{99}+O(q^{100}) $ 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9 - 5 * q^11 - 6 * q^13 + 3 * q^15 - 7 * q^17 - 3 * q^19 - 4 * q^21 + 12 * q^23 + 11 * q^25 - 12 * q^27 - 12 * q^29 + 2 * q^31 + 5 * q^33 - 9 * q^35 - 20 * q^37 + 6 * q^39 - 3 * q^41 + 5 * q^43 - 3 * q^45 - 2 * q^49 + 7 * q^51 - 3 * q^53 + 19 * q^55 + 3 * q^57 - 20 * q^59 - 17 * q^61 + 4 * q^63 - 4 * q^65 - 9 * q^67 - 12 * q^69 + 7 * q^71 - 9 * q^73 - 11 * q^75 - 34 * q^77 + 14 * q^79 + 12 * q^81 + 5 * q^83 - 12 * q^85 + 12 * q^87 - 22 * q^89 - 3 * q^91 - 2 * q^93 - 27 * q^95 + 17 * 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
$5$	2.48147	1.10975	0.554874	−	0.831934i	$-0.312766\pi$
$5$	2.48147	1.10975	0.554874	+	0.831934i	$0.312766\pi$
$6$	0	0
$7$	−0.0448560	−0.0169540	−0.00847699	−	0.999964i	$-0.502698\pi$
$7$	−0.0448560	−0.0169540	−0.00847699	+	0.999964i	$0.502698\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.683409	−0.206056	−0.103028	−	0.994678i	$-0.532853\pi$
$11$	−0.683409	−0.206056	−0.103028	+	0.994678i	$0.532853\pi$
$12$	0	0
$13$	−6.72642	−1.86557	−0.932787	−	0.360429i	$-0.882630\pi$
$13$	−6.72642	−1.86557	−0.932787	+	0.360429i	$0.882630\pi$
$14$	0	0
$15$	−2.48147	−0.640713
$16$	0	0
$17$	−1.19277	−0.289289	−0.144645	−	0.989484i	$-0.546204\pi$
$17$	−1.19277	−0.289289	−0.144645	+	0.989484i	$0.546204\pi$
$18$	0	0
$19$	1.53130	0.351305	0.175653	−	0.984452i	$-0.443796\pi$
$19$	1.53130	0.351305	0.175653	+	0.984452i	$0.443796\pi$
$20$	0	0
$21$	0.0448560	0.00978839
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.15770	0.231541
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	1.40385	0.252139	0.126069	−	0.992021i	$-0.459764\pi$
$31$	1.40385	0.252139	0.126069	+	0.992021i	$0.459764\pi$
$32$	0	0
$33$	0.683409	0.118966
$34$	0	0
$35$	−0.111309	−0.0188147
$36$	0	0
$37$	8.68984	1.42860	0.714301	−	0.699839i	$-0.246745\pi$
$37$	8.68984	1.42860	0.714301	+	0.699839i	$0.246745\pi$
$38$	0	0
$39$	6.72642	1.07709
$40$	0	0
$41$	11.5300	1.80069	0.900345	−	0.435178i	$-0.143314\pi$
$41$	11.5300	1.80069	0.900345	+	0.435178i	$0.143314\pi$
$42$	0	0
$43$	−6.28573	−0.958565	−0.479283	−	0.877661i	$-0.659103\pi$
$43$	−6.28573	−0.958565	−0.479283	+	0.877661i	$0.659103\pi$
$44$	0	0
$45$	2.48147	0.369916
$46$	0	0
$47$	5.00128	0.729512	0.364756	−	0.931103i	$-0.381152\pi$
$47$	5.00128	0.729512	0.364756	+	0.931103i	$0.381152\pi$
$48$	0	0
$49$	−6.99799	−0.999713
$50$	0	0
$51$	1.19277	0.167021
$52$	0	0
$53$	−1.85106	−0.254263	−0.127131	−	0.991886i	$-0.540577\pi$
$53$	−1.85106	−0.254263	−0.127131	+	0.991886i	$0.540577\pi$
$54$	0	0
$55$	−1.69586	−0.228670
$56$	0	0
$57$	−1.53130	−0.202826
$58$	0	0
$59$	−2.14821	−0.279673	−0.139837	−	0.990175i	$-0.544658\pi$
$59$	−2.14821	−0.279673	−0.139837	+	0.990175i	$0.544658\pi$
$60$	0	0
$61$	−2.75349	−0.352548	−0.176274	−	0.984341i	$-0.556404\pi$
$61$	−2.75349	−0.352548	−0.176274	+	0.984341i	$0.556404\pi$
$62$	0	0
$63$	−0.0448560	−0.00565133
$64$	0	0
$65$	−16.6914	−2.07032
$66$	0	0
$67$	−0.0568657	−0.00694725	−0.00347362	−	0.999994i	$-0.501106\pi$
$67$	−0.0568657	−0.00694725	−0.00347362	+	0.999994i	$0.501106\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−1.36240	−0.161687	−0.0808436	−	0.996727i	$-0.525761\pi$
$71$	−1.36240	−0.161687	−0.0808436	+	0.996727i	$0.525761\pi$
$72$	0	0
$73$	−1.19218	−0.139534	−0.0697672	−	0.997563i	$-0.522226\pi$
$73$	−1.19218	−0.139534	−0.0697672	+	0.997563i	$0.522226\pi$
$74$	0	0
$75$	−1.15770	−0.133680
$76$	0	0
$77$	0.0306550	0.00349346
$78$	0	0
$79$	0.0825347	0.00928587	0.00464294	−	0.999989i	$-0.498522\pi$
$79$	0.0825347	0.00928587	0.00464294	+	0.999989i	$0.498522\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.83176	0.859647	0.429824	−	0.902913i	$-0.358576\pi$
$83$	7.83176	0.859647	0.429824	+	0.902913i	$0.358576\pi$
$84$	0	0
$85$	−2.95982	−0.321038
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	−0.473510	−0.0501920	−0.0250960	−	0.999685i	$-0.507989\pi$
$89$	−0.473510	−0.0501920	−0.0250960	+	0.999685i	$0.507989\pi$
$90$	0	0
$91$	0.301720	0.0316289
$92$	0	0
$93$	−1.40385	−0.145572
$94$	0	0
$95$	3.79989	0.389860
$96$	0	0
$97$	−5.20158	−0.528141	−0.264070	−	0.964503i	$-0.585065\pi$
$97$	−5.20158	−0.528141	−0.264070	+	0.964503i	$0.585065\pi$
$98$	0	0
$99$	−0.683409	−0.0686852
$100$	0	0
$101$	−13.4401	−1.33734	−0.668669	−	0.743560i	$-0.733136\pi$
$101$	−13.4401	−1.33734	−0.668669	+	0.743560i	$0.733136\pi$
$102$	0	0
$103$	3.12497	0.307912	0.153956	−	0.988078i	$-0.450799\pi$
$103$	3.12497	0.307912	0.153956	+	0.988078i	$0.450799\pi$
$104$	0	0
$105$	0.111309	0.0108626
$106$	0	0
$107$	2.76781	0.267574	0.133787	−	0.991010i	$-0.457286\pi$
$107$	2.76781	0.267574	0.133787	+	0.991010i	$0.457286\pi$
$108$	0	0
$109$	−20.7736	−1.98975	−0.994877	−	0.101095i	$-0.967766\pi$
$109$	−20.7736	−1.98975	−0.994877	+	0.101095i	$0.967766\pi$
$110$	0	0
$111$	−8.68984	−0.824803
$112$	0	0
$113$	−8.18272	−0.769765	−0.384883	−	0.922965i	$-0.625758\pi$
$113$	−8.18272	−0.769765	−0.384883	+	0.922965i	$0.625758\pi$
$114$	0	0
$115$	2.48147	0.231398
$116$	0	0
$117$	−6.72642	−0.621858
$118$	0	0
$119$	0.0535029	0.00490460
$120$	0	0
$121$	−10.5330	−0.957541
$122$	0	0
$123$	−11.5300	−1.03963
$124$	0	0
$125$	−9.53455	−0.852796
$126$	0	0
$127$	−7.44035	−0.660225	−0.330112	−	0.943942i	$-0.607087\pi$
$127$	−7.44035	−0.660225	−0.330112	+	0.943942i	$0.607087\pi$
$128$	0	0
$129$	6.28573	0.553428
$130$	0	0
$131$	−17.7398	−1.54993	−0.774965	−	0.632004i	$-0.782233\pi$
$131$	−17.7398	−1.54993	−0.774965	+	0.632004i	$0.782233\pi$
$132$	0	0
$133$	−0.0686882	−0.00595603
$134$	0	0
$135$	−2.48147	−0.213571
$136$	0	0
$137$	1.55498	0.132851	0.0664255	−	0.997791i	$-0.478841\pi$
$137$	1.55498	0.132851	0.0664255	+	0.997791i	$0.478841\pi$
$138$	0	0
$139$	4.65605	0.394921	0.197461	−	0.980311i	$-0.436731\pi$
$139$	4.65605	0.394921	0.197461	+	0.980311i	$0.436731\pi$
$140$	0	0
$141$	−5.00128	−0.421184
$142$	0	0
$143$	4.59690	0.384412
$144$	0	0
$145$	−2.48147	−0.206075
$146$	0	0
$147$	6.99799	0.577184
$148$	0	0
$149$	8.44818	0.692102	0.346051	−	0.938216i	$-0.387522\pi$
$149$	8.44818	0.692102	0.346051	+	0.938216i	$0.387522\pi$
$150$	0	0
$151$	−19.0547	−1.55065	−0.775325	−	0.631562i	$-0.782414\pi$
$151$	−19.0547	−1.55065	−0.775325	+	0.631562i	$0.782414\pi$
$152$	0	0
$153$	−1.19277	−0.0964297
$154$	0	0
$155$	3.48361	0.279810
$156$	0	0
$157$	−2.91226	−0.232423	−0.116212	−	0.993224i	$-0.537075\pi$
$157$	−2.91226	−0.232423	−0.116212	+	0.993224i	$0.537075\pi$
$158$	0	0
$159$	1.85106	0.146799
$160$	0	0
$161$	−0.0448560	−0.00353515
$162$	0	0
$163$	−13.1417	−1.02934	−0.514670	−	0.857389i	$-0.672085\pi$
$163$	−13.1417	−1.02934	−0.514670	+	0.857389i	$0.672085\pi$
$164$	0	0
$165$	1.69586	0.132023
$166$	0	0
$167$	−16.0649	−1.24314	−0.621570	−	0.783359i	$-0.713505\pi$
$167$	−16.0649	−1.24314	−0.621570	+	0.783359i	$0.713505\pi$
$168$	0	0
$169$	32.2447	2.48036
$170$	0	0
$171$	1.53130	0.117102
$172$	0	0
$173$	−10.3946	−0.790286	−0.395143	−	0.918620i	$-0.629305\pi$
$173$	−10.3946	−0.790286	−0.395143	+	0.918620i	$0.629305\pi$
$174$	0	0
$175$	−0.0519300	−0.00392554
$176$	0	0
$177$	2.14821	0.161469
$178$	0	0
$179$	0.127749	0.00954841	0.00477421	−	0.999989i	$-0.498480\pi$
$179$	0.127749	0.00954841	0.00477421	+	0.999989i	$0.498480\pi$
$180$	0	0
$181$	−7.28371	−0.541394	−0.270697	−	0.962665i	$-0.587254\pi$
$181$	−7.28371	−0.541394	−0.270697	+	0.962665i	$0.587254\pi$
$182$	0	0
$183$	2.75349	0.203544
$184$	0	0
$185$	21.5636	1.58539
$186$	0	0
$187$	0.815150	0.0596097
$188$	0	0
$189$	0.0448560	0.00326280
$190$	0	0
$191$	1.58690	0.114824	0.0574119	−	0.998351i	$-0.481715\pi$
$191$	1.58690	0.114824	0.0574119	+	0.998351i	$0.481715\pi$
$192$	0	0
$193$	7.87850	0.567107	0.283553	−	0.958956i	$-0.408487\pi$
$193$	7.87850	0.567107	0.283553	+	0.958956i	$0.408487\pi$
$194$	0	0
$195$	16.6914	1.19530
$196$	0	0
$197$	9.56121	0.681208	0.340604	−	0.940207i	$-0.389368\pi$
$197$	9.56121	0.681208	0.340604	+	0.940207i	$0.389368\pi$
$198$	0	0
$199$	−10.9842	−0.778647	−0.389323	−	0.921101i	$-0.627291\pi$
$199$	−10.9842	−0.778647	−0.389323	+	0.921101i	$0.627291\pi$
$200$	0	0
$201$	0.0568657	0.00401100
$202$	0	0
$203$	0.0448560	0.00314828
$204$	0	0
$205$	28.6115	1.99831
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−1.04651	−0.0723885
$210$	0	0
$211$	−8.04284	−0.553692	−0.276846	−	0.960914i	$-0.589289\pi$
$211$	−8.04284	−0.553692	−0.276846	+	0.960914i	$0.589289\pi$
$212$	0	0
$213$	1.36240	0.0933501
$214$	0	0
$215$	−15.5979	−1.06377
$216$	0	0
$217$	−0.0629711	−0.00427476
$218$	0	0
$219$	1.19218	0.0805602
$220$	0	0
$221$	8.02307	0.539690
$222$	0	0
$223$	26.7333	1.79019	0.895097	−	0.445871i	$-0.147106\pi$
$223$	26.7333	1.79019	0.895097	+	0.445871i	$0.147106\pi$
$224$	0	0
$225$	1.15770	0.0771802
$226$	0	0
$227$	24.0072	1.59342	0.796708	−	0.604365i	$-0.206573\pi$
$227$	24.0072	1.59342	0.796708	+	0.604365i	$0.206573\pi$
$228$	0	0
$229$	−21.2022	−1.40108	−0.700539	−	0.713614i	$-0.747057\pi$
$229$	−21.2022	−1.40108	−0.700539	+	0.713614i	$0.747057\pi$
$230$	0	0
$231$	−0.0306550	−0.00201695
$232$	0	0
$233$	−5.50352	−0.360548	−0.180274	−	0.983616i	$-0.557698\pi$
$233$	−5.50352	−0.360548	−0.180274	+	0.983616i	$0.557698\pi$
$234$	0	0
$235$	12.4105	0.809575
$236$	0	0
$237$	−0.0825347	−0.00536120
$238$	0	0
$239$	26.0700	1.68633	0.843163	−	0.537658i	$-0.180691\pi$
$239$	26.0700	1.68633	0.843163	+	0.537658i	$0.180691\pi$
$240$	0	0
$241$	−23.5744	−1.51856	−0.759279	−	0.650765i	$-0.774448\pi$
$241$	−23.5744	−1.51856	−0.759279	+	0.650765i	$0.774448\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−17.3653	−1.10943
$246$	0	0
$247$	−10.3002	−0.655386
$248$	0	0
$249$	−7.83176	−0.496318
$250$	0	0
$251$	−14.8586	−0.937869	−0.468934	−	0.883233i	$-0.655362\pi$
$251$	−14.8586	−0.937869	−0.468934	+	0.883233i	$0.655362\pi$
$252$	0	0
$253$	−0.683409	−0.0429656
$254$	0	0
$255$	2.95982	0.185351
$256$	0	0
$257$	−6.87765	−0.429016	−0.214508	−	0.976722i	$-0.568815\pi$
$257$	−6.87765	−0.429016	−0.214508	+	0.976722i	$0.568815\pi$
$258$	0	0
$259$	−0.389792	−0.0242205
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	21.3741	1.31798	0.658992	−	0.752150i	$-0.270983\pi$
$263$	21.3741	1.31798	0.658992	+	0.752150i	$0.270983\pi$
$264$	0	0
$265$	−4.59335	−0.282167
$266$	0	0
$267$	0.473510	0.0289784
$268$	0	0
$269$	−12.5934	−0.767836	−0.383918	−	0.923367i	$-0.625425\pi$
$269$	−12.5934	−0.767836	−0.383918	+	0.923367i	$0.625425\pi$
$270$	0	0
$271$	4.59472	0.279109	0.139555	−	0.990214i	$-0.455433\pi$
$271$	4.59472	0.279109	0.139555	+	0.990214i	$0.455433\pi$
$272$	0	0
$273$	−0.301720	−0.0182610
$274$	0	0
$275$	−0.791185	−0.0477103
$276$	0	0
$277$	−2.28953	−0.137565	−0.0687823	−	0.997632i	$-0.521911\pi$
$277$	−2.28953	−0.137565	−0.0687823	+	0.997632i	$0.521911\pi$
$278$	0	0
$279$	1.40385	0.0840462
$280$	0	0
$281$	−20.3041	−1.21124	−0.605620	−	0.795754i	$-0.707075\pi$
$281$	−20.3041	−1.21124	−0.605620	+	0.795754i	$0.707075\pi$
$282$	0	0
$283$	−2.01310	−0.119666	−0.0598332	−	0.998208i	$-0.519057\pi$
$283$	−2.01310	−0.119666	−0.0598332	+	0.998208i	$0.519057\pi$
$284$	0	0
$285$	−3.79989	−0.225086
$286$	0	0
$287$	−0.517192	−0.0305289
$288$	0	0
$289$	−15.5773	−0.916312
$290$	0	0
$291$	5.20158	0.304922
$292$	0	0
$293$	14.6955	0.858518	0.429259	−	0.903181i	$-0.358775\pi$
$293$	14.6955	0.858518	0.429259	+	0.903181i	$0.358775\pi$
$294$	0	0
$295$	−5.33072	−0.310367
$296$	0	0
$297$	0.683409	0.0396554
$298$	0	0
$299$	−6.72642	−0.388999
$300$	0	0
$301$	0.281953	0.0162515
$302$	0	0
$303$	13.4401	0.772112
$304$	0	0
$305$	−6.83271	−0.391240
$306$	0	0
$307$	−24.8416	−1.41779	−0.708893	−	0.705316i	$-0.750805\pi$
$307$	−24.8416	−1.41779	−0.708893	+	0.705316i	$0.750805\pi$
$308$	0	0
$309$	−3.12497	−0.177773
$310$	0	0
$311$	−0.957541	−0.0542972	−0.0271486	−	0.999631i	$-0.508643\pi$
$311$	−0.957541	−0.0542972	−0.0271486	+	0.999631i	$0.508643\pi$
$312$	0	0
$313$	17.6885	0.999813	0.499907	−	0.866079i	$-0.333368\pi$
$313$	17.6885	0.999813	0.499907	+	0.866079i	$0.333368\pi$
$314$	0	0
$315$	−0.111309	−0.00627155
$316$	0	0
$317$	−30.3391	−1.70401	−0.852006	−	0.523533i	$-0.824614\pi$
$317$	−30.3391	−1.70401	−0.852006	+	0.523533i	$0.824614\pi$
$318$	0	0
$319$	0.683409	0.0382636
$320$	0	0
$321$	−2.76781	−0.154484
$322$	0	0
$323$	−1.82649	−0.101629
$324$	0	0
$325$	−7.78720	−0.431956
$326$	0	0
$327$	20.7736	1.14878
$328$	0	0
$329$	−0.224338	−0.0123681
$330$	0	0
$331$	14.8890	0.818371	0.409186	−	0.912451i	$-0.365813\pi$
$331$	14.8890	0.818371	0.409186	+	0.912451i	$0.365813\pi$
$332$	0	0
$333$	8.68984	0.476200
$334$	0	0
$335$	−0.141111	−0.00770970
$336$	0	0
$337$	18.4693	1.00608	0.503042	−	0.864262i	$-0.332214\pi$
$337$	18.4693	1.00608	0.503042	+	0.864262i	$0.332214\pi$
$338$	0	0
$339$	8.18272	0.444424
$340$	0	0
$341$	−0.959404	−0.0519546
$342$	0	0
$343$	0.627894	0.0339031
$344$	0	0
$345$	−2.48147	−0.133598
$346$	0	0
$347$	36.2946	1.94840	0.974198	−	0.225697i	$-0.0724658\pi$
$347$	36.2946	1.94840	0.974198	+	0.225697i	$0.0724658\pi$
$348$	0	0
$349$	5.14729	0.275528	0.137764	−	0.990465i	$-0.456008\pi$
$349$	5.14729	0.275528	0.137764	+	0.990465i	$0.456008\pi$
$350$	0	0
$351$	6.72642	0.359030
$352$	0	0
$353$	−23.6393	−1.25819	−0.629096	−	0.777327i	$-0.716575\pi$
$353$	−23.6393	−1.25819	−0.629096	+	0.777327i	$0.716575\pi$
$354$	0	0
$355$	−3.38076	−0.179432
$356$	0	0
$357$	−0.0535029	−0.00283167
$358$	0	0
$359$	24.7411	1.30578	0.652891	−	0.757451i	$-0.273556\pi$
$359$	24.7411	1.30578	0.652891	+	0.757451i	$0.273556\pi$
$360$	0	0
$361$	−16.6551	−0.876585
$362$	0	0
$363$	10.5330	0.552837
$364$	0	0
$365$	−2.95837	−0.154848
$366$	0	0
$367$	−12.6328	−0.659426	−0.329713	−	0.944081i	$-0.606952\pi$
$367$	−12.6328	−0.659426	−0.329713	+	0.944081i	$0.606952\pi$
$368$	0	0
$369$	11.5300	0.600230
$370$	0	0
$371$	0.0830312	0.00431076
$372$	0	0
$373$	−11.6850	−0.605024	−0.302512	−	0.953146i	$-0.597825\pi$
$373$	−11.6850	−0.605024	−0.302512	+	0.953146i	$0.597825\pi$
$374$	0	0
$375$	9.53455	0.492362
$376$	0	0
$377$	6.72642	0.346428
$378$	0	0
$379$	18.8719	0.969384	0.484692	−	0.874685i	$-0.338932\pi$
$379$	18.8719	0.969384	0.484692	+	0.874685i	$0.338932\pi$
$380$	0	0
$381$	7.44035	0.381181
$382$	0	0
$383$	7.17604	0.366678	0.183339	−	0.983050i	$-0.441309\pi$
$383$	7.17604	0.366678	0.183339	+	0.983050i	$0.441309\pi$
$384$	0	0
$385$	0.0760696	0.00387687
$386$	0	0
$387$	−6.28573	−0.319522
$388$	0	0
$389$	−23.5320	−1.19312	−0.596560	−	0.802568i	$-0.703466\pi$
$389$	−23.5320	−1.19312	−0.596560	+	0.802568i	$0.703466\pi$
$390$	0	0
$391$	−1.19277	−0.0603210
$392$	0	0
$393$	17.7398	0.894853
$394$	0	0
$395$	0.204807	0.0103050
$396$	0	0
$397$	−5.90830	−0.296529	−0.148264	−	0.988948i	$-0.547369\pi$
$397$	−5.90830	−0.296529	−0.148264	+	0.988948i	$0.547369\pi$
$398$	0	0
$399$	0.0686882	0.00343871
$400$	0	0
$401$	−33.3654	−1.66619	−0.833095	−	0.553130i	$-0.813433\pi$
$401$	−33.3654	−1.66619	−0.833095	+	0.553130i	$0.813433\pi$
$402$	0	0
$403$	−9.44288	−0.470383
$404$	0	0
$405$	2.48147	0.123305
$406$	0	0
$407$	−5.93872	−0.294371
$408$	0	0
$409$	16.6673	0.824143	0.412071	−	0.911152i	$-0.364805\pi$
$409$	16.6673	0.824143	0.412071	+	0.911152i	$0.364805\pi$
$410$	0	0
$411$	−1.55498	−0.0767016
$412$	0	0
$413$	0.0963602	0.00474157
$414$	0	0
$415$	19.4343	0.953992
$416$	0	0
$417$	−4.65605	−0.228008
$418$	0	0
$419$	−30.9304	−1.51105	−0.755523	−	0.655122i	$-0.772617\pi$
$419$	−30.9304	−1.51105	−0.755523	+	0.655122i	$0.772617\pi$
$420$	0	0
$421$	28.5383	1.39087	0.695435	−	0.718589i	$-0.255212\pi$
$421$	28.5383	1.39087	0.695435	+	0.718589i	$0.255212\pi$
$422$	0	0
$423$	5.00128	0.243171
$424$	0	0
$425$	−1.38087	−0.0669822
$426$	0	0
$427$	0.123511	0.00597710
$428$	0	0
$429$	−4.59690	−0.221940
$430$	0	0
$431$	−30.6926	−1.47841	−0.739206	−	0.673480i	$-0.764799\pi$
$431$	−30.6926	−1.47841	−0.739206	+	0.673480i	$0.764799\pi$
$432$	0	0
$433$	−18.3757	−0.883078	−0.441539	−	0.897242i	$-0.645567\pi$
$433$	−18.3757	−0.883078	−0.441539	+	0.897242i	$0.645567\pi$
$434$	0	0
$435$	2.48147	0.118977
$436$	0	0
$437$	1.53130	0.0732522
$438$	0	0
$439$	39.5696	1.88855	0.944277	−	0.329151i	$-0.106762\pi$
$439$	39.5696	1.88855	0.944277	+	0.329151i	$0.106762\pi$
$440$	0	0
$441$	−6.99799	−0.333238
$442$	0	0
$443$	−30.7869	−1.46273	−0.731365	−	0.681986i	$-0.761116\pi$
$443$	−30.7869	−1.46273	−0.731365	+	0.681986i	$0.761116\pi$
$444$	0	0
$445$	−1.17500	−0.0557005
$446$	0	0
$447$	−8.44818	−0.399585
$448$	0	0
$449$	−15.8707	−0.748986	−0.374493	−	0.927230i	$-0.622183\pi$
$449$	−15.8707	−0.748986	−0.374493	+	0.927230i	$0.622183\pi$
$450$	0	0
$451$	−7.87973	−0.371042
$452$	0	0
$453$	19.0547	0.895269
$454$	0	0
$455$	0.748711	0.0351001
$456$	0	0
$457$	11.4522	0.535710	0.267855	−	0.963459i	$-0.413685\pi$
$457$	11.4522	0.535710	0.267855	+	0.963459i	$0.413685\pi$
$458$	0	0
$459$	1.19277	0.0556737
$460$	0	0
$461$	−0.768140	−0.0357759	−0.0178879	−	0.999840i	$-0.505694\pi$
$461$	−0.768140	−0.0357759	−0.0178879	+	0.999840i	$0.505694\pi$
$462$	0	0
$463$	4.41811	0.205327	0.102663	−	0.994716i	$-0.467264\pi$
$463$	4.41811	0.205327	0.102663	+	0.994716i	$0.467264\pi$
$464$	0	0
$465$	−3.48361	−0.161549
$466$	0	0
$467$	−14.1128	−0.653060	−0.326530	−	0.945187i	$-0.605879\pi$
$467$	−14.1128	−0.653060	−0.326530	+	0.945187i	$0.605879\pi$
$468$	0	0
$469$	0.00255077	0.000117784 0
$470$	0	0
$471$	2.91226	0.134190
$472$	0	0
$473$	4.29573	0.197518
$474$	0	0
$475$	1.77280	0.0813415
$476$	0	0
$477$	−1.85106	−0.0847542
$478$	0	0
$479$	−11.8727	−0.542480	−0.271240	−	0.962512i	$-0.587434\pi$
$479$	−11.8727	−0.542480	−0.271240	+	0.962512i	$0.587434\pi$
$480$	0	0
$481$	−58.4515	−2.66516
$482$	0	0
$483$	0.0448560	0.00204102
$484$	0	0
$485$	−12.9076	−0.586103
$486$	0	0
$487$	17.7138	0.802687	0.401343	−	0.915928i	$-0.368543\pi$
$487$	17.7138	0.802687	0.401343	+	0.915928i	$0.368543\pi$
$488$	0	0
$489$	13.1417	0.594289
$490$	0	0
$491$	39.0560	1.76257	0.881287	−	0.472581i	$-0.156678\pi$
$491$	39.0560	1.76257	0.881287	+	0.472581i	$0.156678\pi$
$492$	0	0
$493$	1.19277	0.0537196
$494$	0	0
$495$	−1.69586	−0.0762233
$496$	0	0
$497$	0.0611119	0.00274124
$498$	0	0
$499$	−35.9240	−1.60818	−0.804091	−	0.594507i	$-0.797347\pi$
$499$	−35.9240	−1.60818	−0.804091	+	0.594507i	$0.797347\pi$
$500$	0	0
$501$	16.0649	0.717727
$502$	0	0
$503$	33.3600	1.48745	0.743724	−	0.668487i	$-0.233058\pi$
$503$	33.3600	1.48745	0.743724	+	0.668487i	$0.233058\pi$
$504$	0	0
$505$	−33.3512	−1.48411
$506$	0	0
$507$	−32.2447	−1.43204
$508$	0	0
$509$	−20.7899	−0.921497	−0.460748	−	0.887531i	$-0.652419\pi$
$509$	−20.7899	−0.921497	−0.460748	+	0.887531i	$0.652419\pi$
$510$	0	0
$511$	0.0534766	0.00236566
$512$	0	0
$513$	−1.53130	−0.0676087
$514$	0	0
$515$	7.75452	0.341705
$516$	0	0
$517$	−3.41792	−0.150320
$518$	0	0
$519$	10.3946	0.456272
$520$	0	0
$521$	−27.7417	−1.21539	−0.607693	−	0.794172i	$-0.707905\pi$
$521$	−27.7417	−1.21539	−0.607693	+	0.794172i	$0.707905\pi$
$522$	0	0
$523$	24.1577	1.05634	0.528170	−	0.849139i	$-0.322878\pi$
$523$	24.1577	1.05634	0.528170	+	0.849139i	$0.322878\pi$
$524$	0	0
$525$	0.0519300	0.00226641
$526$	0	0
$527$	−1.67447	−0.0729410
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−2.14821	−0.0932244
$532$	0	0
$533$	−77.5559	−3.35932
$534$	0	0
$535$	6.86823	0.296940
$536$	0	0
$537$	−0.127749	−0.00551278
$538$	0	0
$539$	4.78249	0.205996
$540$	0	0
$541$	−27.5006	−1.18234	−0.591171	−	0.806546i	$-0.701334\pi$
$541$	−27.5006	−1.18234	−0.591171	+	0.806546i	$0.701334\pi$
$542$	0	0
$543$	7.28371	0.312574
$544$	0	0
$545$	−51.5492	−2.20813
$546$	0	0
$547$	29.6516	1.26781	0.633905	−	0.773411i	$-0.281451\pi$
$547$	29.6516	1.26781	0.633905	+	0.773411i	$0.281451\pi$
$548$	0	0
$549$	−2.75349	−0.117516
$550$	0	0
$551$	−1.53130	−0.0652358
$552$	0	0
$553$	−0.00370218	−0.000157433 0
$554$	0	0
$555$	−21.5636	−0.915324
$556$	0	0
$557$	14.8913	0.630965	0.315483	−	0.948931i	$-0.397834\pi$
$557$	14.8913	0.630965	0.315483	+	0.948931i	$0.397834\pi$
$558$	0	0
$559$	42.2805	1.78827
$560$	0	0
$561$	−0.815150	−0.0344157
$562$	0	0
$563$	−33.4014	−1.40770	−0.703850	−	0.710348i	$-0.748537\pi$
$563$	−33.4014	−1.40770	−0.703850	+	0.710348i	$0.748537\pi$
$564$	0	0
$565$	−20.3052	−0.854246
$566$	0	0
$567$	−0.0448560	−0.00188378
$568$	0	0
$569$	−33.3289	−1.39722	−0.698611	−	0.715502i	$-0.746198\pi$
$569$	−33.3289	−1.39722	−0.698611	+	0.715502i	$0.746198\pi$
$570$	0	0
$571$	−2.86311	−0.119818	−0.0599088	−	0.998204i	$-0.519081\pi$
$571$	−2.86311	−0.119818	−0.0599088	+	0.998204i	$0.519081\pi$
$572$	0	0
$573$	−1.58690	−0.0662936
$574$	0	0
$575$	1.15770	0.0482796
$576$	0	0
$577$	35.6396	1.48370	0.741848	−	0.670568i	$-0.233949\pi$
$577$	35.6396	1.48370	0.741848	+	0.670568i	$0.233949\pi$
$578$	0	0
$579$	−7.87850	−0.327419
$580$	0	0
$581$	−0.351302	−0.0145744
$582$	0	0
$583$	1.26503	0.0523922
$584$	0	0
$585$	−16.6914	−0.690105
$586$	0	0
$587$	6.66341	0.275029	0.137514	−	0.990500i	$-0.456089\pi$
$587$	6.66341	0.275029	0.137514	+	0.990500i	$0.456089\pi$
$588$	0	0
$589$	2.14972	0.0885777
$590$	0	0
$591$	−9.56121	−0.393296
$592$	0	0
$593$	−35.6118	−1.46240	−0.731201	−	0.682162i	$-0.761040\pi$
$593$	−35.6118	−1.46240	−0.731201	+	0.682162i	$0.761040\pi$
$594$	0	0
$595$	0.132766	0.00544287
$596$	0	0
$597$	10.9842	0.449552
$598$	0	0
$599$	−9.38122	−0.383306	−0.191653	−	0.981463i	$-0.561385\pi$
$599$	−9.38122	−0.383306	−0.191653	+	0.981463i	$0.561385\pi$
$600$	0	0
$601$	33.4516	1.36452	0.682259	−	0.731111i	$-0.260998\pi$
$601$	33.4516	1.36452	0.682259	+	0.731111i	$0.260998\pi$
$602$	0	0
$603$	−0.0568657	−0.00231575
$604$	0	0
$605$	−26.1372	−1.06263
$606$	0	0
$607$	−15.7731	−0.640208	−0.320104	−	0.947382i	$-0.603718\pi$
$607$	−15.7731	−0.640208	−0.320104	+	0.947382i	$0.603718\pi$
$608$	0	0
$609$	−0.0448560	−0.00181766
$610$	0	0
$611$	−33.6407	−1.36096
$612$	0	0
$613$	−4.26903	−0.172424	−0.0862122	−	0.996277i	$-0.527476\pi$
$613$	−4.26903	−0.172424	−0.0862122	+	0.996277i	$0.527476\pi$
$614$	0	0
$615$	−28.6115	−1.15373
$616$	0	0
$617$	24.0080	0.966527	0.483263	−	0.875475i	$-0.339451\pi$
$617$	24.0080	0.966527	0.483263	+	0.875475i	$0.339451\pi$
$618$	0	0
$619$	−30.9394	−1.24356	−0.621780	−	0.783192i	$-0.713590\pi$
$619$	−30.9394	−1.24356	−0.621780	+	0.783192i	$0.713590\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	0.0212398	0.000850954 0
$624$	0	0
$625$	−29.4482	−1.17793
$626$	0	0
$627$	1.04651	0.0417935
$628$	0	0
$629$	−10.3650	−0.413279
$630$	0	0
$631$	−23.3669	−0.930222	−0.465111	−	0.885252i	$-0.653986\pi$
$631$	−23.3669	−0.930222	−0.465111	+	0.885252i	$0.653986\pi$
$632$	0	0
$633$	8.04284	0.319674
$634$	0	0
$635$	−18.4630	−0.732683
$636$	0	0
$637$	47.0714	1.86504
$638$	0	0
$639$	−1.36240	−0.0538957
$640$	0	0
$641$	−17.1059	−0.675641	−0.337820	−	0.941211i	$-0.609690\pi$
$641$	−17.1059	−0.675641	−0.337820	+	0.941211i	$0.609690\pi$
$642$	0	0
$643$	−12.0493	−0.475179	−0.237589	−	0.971366i	$-0.576357\pi$
$643$	−12.0493	−0.475179	−0.237589	+	0.971366i	$0.576357\pi$
$644$	0	0
$645$	15.5979	0.614166
$646$	0	0
$647$	18.7704	0.737940	0.368970	−	0.929441i	$-0.379711\pi$
$647$	18.7704	0.737940	0.368970	+	0.929441i	$0.379711\pi$
$648$	0	0
$649$	1.46811	0.0576282
$650$	0	0
$651$	0.0629711	0.00246803
$652$	0	0
$653$	−15.6034	−0.610607	−0.305303	−	0.952255i	$-0.598758\pi$
$653$	−15.6034	−0.610607	−0.305303	+	0.952255i	$0.598758\pi$
$654$	0	0
$655$	−44.0207	−1.72003
$656$	0	0
$657$	−1.19218	−0.0465115
$658$	0	0
$659$	14.8795	0.579623	0.289812	−	0.957084i	$-0.406407\pi$
$659$	14.8795	0.579623	0.289812	+	0.957084i	$0.406407\pi$
$660$	0	0
$661$	−4.34822	−0.169126	−0.0845630	−	0.996418i	$-0.526949\pi$
$661$	−4.34822	−0.169126	−0.0845630	+	0.996418i	$0.526949\pi$
$662$	0	0
$663$	−8.02307	−0.311590
$664$	0	0
$665$	−0.170448	−0.00660969
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	−26.7333	−1.03357
$670$	0	0
$671$	1.88176	0.0726445
$672$	0	0
$673$	21.5468	0.830567	0.415284	−	0.909692i	$-0.363682\pi$
$673$	21.5468	0.830567	0.415284	+	0.909692i	$0.363682\pi$
$674$	0	0
$675$	−1.15770	−0.0445600
$676$	0	0
$677$	32.0704	1.23257	0.616283	−	0.787525i	$-0.288638\pi$
$677$	32.0704	1.23257	0.616283	+	0.787525i	$0.288638\pi$
$678$	0	0
$679$	0.233322	0.00895409
$680$	0	0
$681$	−24.0072	−0.919959
$682$	0	0
$683$	18.8385	0.720836	0.360418	−	0.932791i	$-0.382634\pi$
$683$	18.8385	0.720836	0.360418	+	0.932791i	$0.382634\pi$
$684$	0	0
$685$	3.85864	0.147431
$686$	0	0
$687$	21.2022	0.808913
$688$	0	0
$689$	12.4510	0.474345
$690$	0	0
$691$	−11.6594	−0.443544	−0.221772	−	0.975099i	$-0.571184\pi$
$691$	−11.6594	−0.443544	−0.221772	+	0.975099i	$0.571184\pi$
$692$	0	0
$693$	0.0306550	0.00116449
$694$	0	0
$695$	11.5539	0.438263
$696$	0	0
$697$	−13.7527	−0.520920
$698$	0	0
$699$	5.50352	0.208162
$700$	0	0
$701$	29.2820	1.10597	0.552983	−	0.833192i	$-0.313489\pi$
$701$	29.2820	1.10597	0.552983	+	0.833192i	$0.313489\pi$
$702$	0	0
$703$	13.3068	0.501875
$704$	0	0
$705$	−12.4105	−0.467408
$706$	0	0
$707$	0.602869	0.0226732
$708$	0	0
$709$	−48.8523	−1.83469	−0.917343	−	0.398097i	$-0.869671\pi$
$709$	−48.8523	−1.83469	−0.917343	+	0.398097i	$0.869671\pi$
$710$	0	0
$711$	0.0825347	0.00309529
$712$	0	0
$713$	1.40385	0.0525746
$714$	0	0
$715$	11.4071	0.426600
$716$	0	0
$717$	−26.0700	−0.973601
$718$	0	0
$719$	−21.3601	−0.796597	−0.398299	−	0.917256i	$-0.630399\pi$
$719$	−21.3601	−0.796597	−0.398299	+	0.917256i	$0.630399\pi$
$720$	0	0
$721$	−0.140174	−0.00522034
$722$	0	0
$723$	23.5744	0.876740
$724$	0	0
$725$	−1.15770	−0.0429960
$726$	0	0
$727$	18.2219	0.675814	0.337907	−	0.941180i	$-0.390281\pi$
$727$	18.2219	0.675814	0.337907	+	0.941180i	$0.390281\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	7.49743	0.277303
$732$	0	0
$733$	−11.7818	−0.435169	−0.217584	−	0.976042i	$-0.569818\pi$
$733$	−11.7818	−0.435169	−0.217584	+	0.976042i	$0.569818\pi$
$734$	0	0
$735$	17.3653	0.640529
$736$	0	0
$737$	0.0388625	0.00143152
$738$	0	0
$739$	22.3717	0.822958	0.411479	−	0.911419i	$-0.365012\pi$
$739$	22.3717	0.822958	0.411479	+	0.911419i	$0.365012\pi$
$740$	0	0
$741$	10.3002	0.378387
$742$	0	0
$743$	−39.4351	−1.44673	−0.723367	−	0.690464i	$-0.757407\pi$
$743$	−39.4351	−1.44673	−0.723367	+	0.690464i	$0.757407\pi$
$744$	0	0
$745$	20.9639	0.768059
$746$	0	0
$747$	7.83176	0.286549
$748$	0	0
$749$	−0.124153	−0.00453644
$750$	0	0
$751$	−27.4160	−1.00042	−0.500211	−	0.865903i	$-0.666744\pi$
$751$	−27.4160	−1.00042	−0.500211	+	0.865903i	$0.666744\pi$
$752$	0	0
$753$	14.8586	0.541479
$754$	0	0
$755$	−47.2837	−1.72083
$756$	0	0
$757$	−22.2228	−0.807700	−0.403850	−	0.914825i	$-0.632328\pi$
$757$	−22.2228	−0.807700	−0.403850	+	0.914825i	$0.632328\pi$
$758$	0	0
$759$	0.683409	0.0248062
$760$	0	0
$761$	38.5659	1.39801	0.699006	−	0.715116i	$-0.253626\pi$
$761$	38.5659	1.39801	0.699006	+	0.715116i	$0.253626\pi$
$762$	0	0
$763$	0.931823	0.0337343
$764$	0	0
$765$	−2.95982	−0.107013
$766$	0	0
$767$	14.4498	0.521751
$768$	0	0
$769$	−10.9131	−0.393535	−0.196768	−	0.980450i	$-0.563044\pi$
$769$	−10.9131	−0.393535	−0.196768	+	0.980450i	$0.563044\pi$
$770$	0	0
$771$	6.87765	0.247692
$772$	0	0
$773$	−11.0493	−0.397416	−0.198708	−	0.980059i	$-0.563675\pi$
$773$	−11.0493	−0.397416	−0.198708	+	0.980059i	$0.563675\pi$
$774$	0	0
$775$	1.62524	0.0583804
$776$	0	0
$777$	0.389792	0.0139837
$778$	0	0
$779$	17.6560	0.632592
$780$	0	0
$781$	0.931077	0.0333166
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	−7.22668	−0.257931
$786$	0	0
$787$	43.2832	1.54288	0.771440	−	0.636301i	$-0.219537\pi$
$787$	43.2832	1.54288	0.771440	+	0.636301i	$0.219537\pi$
$788$	0	0
$789$	−21.3741	−0.760939
$790$	0	0
$791$	0.367044	0.0130506
$792$	0	0
$793$	18.5211	0.657704
$794$	0	0
$795$	4.59335	0.162909
$796$	0	0
$797$	−12.8858	−0.456437	−0.228218	−	0.973610i	$-0.573290\pi$
$797$	−12.8858	−0.456437	−0.228218	+	0.973610i	$0.573290\pi$
$798$	0	0
$799$	−5.96538	−0.211040
$800$	0	0
$801$	−0.473510	−0.0167307
$802$	0	0
$803$	0.814748	0.0287518
$804$	0	0
$805$	−0.111309	−0.00392313
$806$	0	0
$807$	12.5934	0.443310
$808$	0	0
$809$	−3.36974	−0.118474	−0.0592368	−	0.998244i	$-0.518867\pi$
$809$	−3.36974	−0.118474	−0.0592368	+	0.998244i	$0.518867\pi$
$810$	0	0
$811$	−12.5650	−0.441218	−0.220609	−	0.975362i	$-0.570805\pi$
$811$	−12.5650	−0.441218	−0.220609	+	0.975362i	$0.570805\pi$
$812$	0	0
$813$	−4.59472	−0.161144
$814$	0	0
$815$	−32.6108	−1.14231
$816$	0	0
$817$	−9.62537	−0.336749
$818$	0	0
$819$	0.301720	0.0105430
$820$	0	0
$821$	36.1970	1.26328	0.631642	−	0.775261i	$-0.282381\pi$
$821$	36.1970	1.26328	0.631642	+	0.775261i	$0.282381\pi$
$822$	0	0
$823$	12.1322	0.422903	0.211452	−	0.977388i	$-0.432181\pi$
$823$	12.1322	0.422903	0.211452	+	0.977388i	$0.432181\pi$
$824$	0	0
$825$	0.791185	0.0275455
$826$	0	0
$827$	−0.718101	−0.0249708	−0.0124854	−	0.999922i	$-0.503974\pi$
$827$	−0.718101	−0.0249708	−0.0124854	+	0.999922i	$0.503974\pi$
$828$	0	0
$829$	12.3297	0.428230	0.214115	−	0.976808i	$-0.431313\pi$
$829$	12.3297	0.428230	0.214115	+	0.976808i	$0.431313\pi$
$830$	0	0
$831$	2.28953	0.0794230
$832$	0	0
$833$	8.34699	0.289206
$834$	0	0
$835$	−39.8646	−1.37957
$836$	0	0
$837$	−1.40385	−0.0485241
$838$	0	0
$839$	29.8107	1.02918	0.514590	−	0.857437i	$-0.327944\pi$
$839$	29.8107	1.02918	0.514590	+	0.857437i	$0.327944\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	20.3041	0.699310
$844$	0	0
$845$	80.0144	2.75258
$846$	0	0
$847$	0.472466	0.0162341
$848$	0	0
$849$	2.01310	0.0690894
$850$	0	0
$851$	8.68984	0.297884
$852$	0	0
$853$	16.8950	0.578474	0.289237	−	0.957258i	$-0.406598\pi$
$853$	16.8950	0.578474	0.289237	+	0.957258i	$0.406598\pi$
$854$	0	0
$855$	3.79989	0.129953
$856$	0	0
$857$	13.3776	0.456970	0.228485	−	0.973547i	$-0.426623\pi$
$857$	13.3776	0.456970	0.228485	+	0.973547i	$0.426623\pi$
$858$	0	0
$859$	29.9155	1.02070	0.510352	−	0.859966i	$-0.329515\pi$
$859$	29.9155	1.02070	0.510352	+	0.859966i	$0.329515\pi$
$860$	0	0
$861$	0.517192	0.0176258
$862$	0	0
$863$	−23.9042	−0.813709	−0.406854	−	0.913493i	$-0.633374\pi$
$863$	−23.9042	−0.813709	−0.406854	+	0.913493i	$0.633374\pi$
$864$	0	0
$865$	−25.7939	−0.877018
$866$	0	0
$867$	15.5773	0.529033
$868$	0	0
$869$	−0.0564050	−0.00191341
$870$	0	0
$871$	0.382502	0.0129606
$872$	0	0
$873$	−5.20158	−0.176047
$874$	0	0
$875$	0.427682	0.0144583
$876$	0	0
$877$	22.7542	0.768356	0.384178	−	0.923259i	$-0.374485\pi$
$877$	22.7542	0.768356	0.384178	+	0.923259i	$0.374485\pi$
$878$	0	0
$879$	−14.6955	−0.495666
$880$	0	0
$881$	35.2527	1.18769	0.593846	−	0.804579i	$-0.297609\pi$
$881$	35.2527	1.18769	0.593846	+	0.804579i	$0.297609\pi$
$882$	0	0
$883$	20.1303	0.677437	0.338719	−	0.940888i	$-0.390007\pi$
$883$	20.1303	0.677437	0.338719	+	0.940888i	$0.390007\pi$
$884$	0	0
$885$	5.33072	0.179190
$886$	0	0
$887$	28.5084	0.957219	0.478610	−	0.878028i	$-0.341141\pi$
$887$	28.5084	0.957219	0.478610	+	0.878028i	$0.341141\pi$
$888$	0	0
$889$	0.333745	0.0111934
$890$	0	0
$891$	−0.683409	−0.0228951
$892$	0	0
$893$	7.65849	0.256282
$894$	0	0
$895$	0.317006	0.0105963
$896$	0	0
$897$	6.72642	0.224589
$898$	0	0
$899$	−1.40385	−0.0468210
$900$	0	0
$901$	2.20789	0.0735554
$902$	0	0
$903$	−0.281953	−0.00938281
$904$	0	0
$905$	−18.0743	−0.600811
$906$	0	0
$907$	−3.24488	−0.107744	−0.0538722	−	0.998548i	$-0.517156\pi$
$907$	−3.24488	−0.107744	−0.0538722	+	0.998548i	$0.517156\pi$
$908$	0	0
$909$	−13.4401	−0.445779
$910$	0	0
$911$	−50.9154	−1.68690	−0.843451	−	0.537207i	$-0.819479\pi$
$911$	−50.9154	−1.68690	−0.843451	+	0.537207i	$0.819479\pi$
$912$	0	0
$913$	−5.35230	−0.177135
$914$	0	0
$915$	6.83271	0.225882
$916$	0	0
$917$	0.795735	0.0262775
$918$	0	0
$919$	−25.6095	−0.844778	−0.422389	−	0.906415i	$-0.638808\pi$
$919$	−25.6095	−0.844778	−0.422389	+	0.906415i	$0.638808\pi$
$920$	0	0
$921$	24.8416	0.818559
$922$	0	0
$923$	9.16407	0.301639
$924$	0	0
$925$	10.0603	0.330779
$926$	0	0
$927$	3.12497	0.102637
$928$	0	0
$929$	−4.01347	−0.131678	−0.0658388	−	0.997830i	$-0.520972\pi$
$929$	−4.01347	−0.131678	−0.0658388	+	0.997830i	$0.520972\pi$
$930$	0	0
$931$	−10.7161	−0.351204
$932$	0	0
$933$	0.957541	0.0313485
$934$	0	0
$935$	2.02277	0.0661517
$936$	0	0
$937$	46.3819	1.51523	0.757615	−	0.652702i	$-0.226365\pi$
$937$	46.3819	1.51523	0.757615	+	0.652702i	$0.226365\pi$
$938$	0	0
$939$	−17.6885	−0.577242
$940$	0	0
$941$	11.1442	0.363292	0.181646	−	0.983364i	$-0.441858\pi$
$941$	11.1442	0.363292	0.181646	+	0.983364i	$0.441858\pi$
$942$	0	0
$943$	11.5300	0.375470
$944$	0	0
$945$	0.111309	0.00362088
$946$	0	0
$947$	16.6491	0.541022	0.270511	−	0.962717i	$-0.412807\pi$
$947$	16.6491	0.541022	0.270511	+	0.962717i	$0.412807\pi$
$948$	0	0
$949$	8.01912	0.260312
$950$	0	0
$951$	30.3391	0.983811
$952$	0	0
$953$	−47.7628	−1.54719	−0.773594	−	0.633681i	$-0.781543\pi$
$953$	−47.7628	−1.54719	−0.773594	+	0.633681i	$0.781543\pi$
$954$	0	0
$955$	3.93784	0.127426
$956$	0	0
$957$	−0.683409	−0.0220915
$958$	0	0
$959$	−0.0697503	−0.00225235
$960$	0	0
$961$	−29.0292	−0.936426
$962$	0	0
$963$	2.76781	0.0891913
$964$	0	0
$965$	19.5503	0.629345
$966$	0	0
$967$	−45.5564	−1.46499	−0.732497	−	0.680771i	$-0.761645\pi$
$967$	−45.5564	−1.46499	−0.732497	+	0.680771i	$0.761645\pi$
$968$	0	0
$969$	1.82649	0.0586754
$970$	0	0
$971$	−41.9958	−1.34771	−0.673854	−	0.738864i	$-0.735362\pi$
$971$	−41.9958	−1.34771	−0.673854	+	0.738864i	$0.735362\pi$
$972$	0	0
$973$	−0.208852	−0.00669549
$974$	0	0
$975$	7.78720	0.249390
$976$	0	0
$977$	−44.1202	−1.41153	−0.705765	−	0.708446i	$-0.749397\pi$
$977$	−44.1202	−1.41153	−0.705765	+	0.708446i	$0.749397\pi$
$978$	0	0
$979$	0.323601	0.0103423
$980$	0	0
$981$	−20.7736	−0.663251
$982$	0	0
$983$	−16.0584	−0.512185	−0.256092	−	0.966652i	$-0.582435\pi$
$983$	−16.0584	−0.512185	−0.256092	+	0.966652i	$0.582435\pi$
$984$	0	0
$985$	23.7259	0.755969
$986$	0	0
$987$	0.224338	0.00714075
$988$	0	0
$989$	−6.28573	−0.199875
$990$	0	0
$991$	50.4668	1.60313	0.801565	−	0.597907i	$-0.204001\pi$
$991$	50.4668	1.60313	0.801565	+	0.597907i	$0.204001\pi$
$992$	0	0
$993$	−14.8890	−0.472487
$994$	0	0
$995$	−27.2569	−0.864102
$996$	0	0
$997$	19.0699	0.603950	0.301975	−	0.953316i	$-0.402354\pi$
$997$	19.0699	0.603950	0.301975	+	0.953316i	$0.402354\pi$
$998$	0	0
$999$	−8.68984	−0.274934

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.g.1.10	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.g.1.10	✓	12	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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.48147 q^{5} -0.0448560 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.48147 q^{5} -0.0448560 q^{7} +1.00000 q^{9} -0.683409 q^{11} -6.72642 q^{13} -2.48147 q^{15} -1.19277 q^{17} +1.53130 q^{19} +0.0448560 q^{21} +1.00000 q^{23} +1.15770 q^{25} -1.00000 q^{27} -1.00000 q^{29} +1.40385 q^{31} +0.683409 q^{33} -0.111309 q^{35} +8.68984 q^{37} +6.72642 q^{39} +11.5300 q^{41} -6.28573 q^{43} +2.48147 q^{45} +5.00128 q^{47} -6.99799 q^{49} +1.19277 q^{51} -1.85106 q^{53} -1.69586 q^{55} -1.53130 q^{57} -2.14821 q^{59} -2.75349 q^{61} -0.0448560 q^{63} -16.6914 q^{65} -0.0568657 q^{67} -1.00000 q^{69} -1.36240 q^{71} -1.19218 q^{73} -1.15770 q^{75} +0.0306550 q^{77} +0.0825347 q^{79} +1.00000 q^{81} +7.83176 q^{83} -2.95982 q^{85} +1.00000 q^{87} -0.473510 q^{89} +0.301720 q^{91} -1.40385 q^{93} +3.79989 q^{95} -5.20158 q^{97} -0.683409 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9	\( 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9} - 5 q^{11} - 6 q^{13} + 3 q^{15} - 7 q^{17} - 3 q^{19} - 4 q^{21} + 12 q^{23} + 11 q^{25} - 12 q^{27} - 12 q^{29} + 2 q^{31} + 5 q^{33} - 9 q^{35} - 20 q^{37} + 6 q^{39} - 3 q^{41} + 5 q^{43} - 3 q^{45} - 2 q^{49} + 7 q^{51} - 3 q^{53} + 19 q^{55} + 3 q^{57} - 20 q^{59} - 17 q^{61} + 4 q^{63} - 4 q^{65} - 9 q^{67} - 12 q^{69} + 7 q^{71} - 9 q^{73} - 11 q^{75} - 34 q^{77} + 14 q^{79} + 12 q^{81} + 5 q^{83} - 12 q^{85} + 12 q^{87} - 22 q^{89} - 3 q^{91} - 2 q^{93} - 27 q^{95} + 17 q^{97} - 5 q^{99}+O(q^{100}) \) 12 * q - 12 * q^3 - 3 * q^5 + 4 * q^7 + 12 * q^9 - 5 * q^11 - 6 * q^13 + 3 * q^15 - 7 * q^17 - 3 * q^19 - 4 * q^21 + 12 * q^23 + 11 * q^25 - 12 * q^27 - 12 * q^29 + 2 * q^31 + 5 * q^33 - 9 * q^35 - 20 * q^37 + 6 * q^39 - 3 * q^41 + 5 * q^43 - 3 * q^45 - 2 * q^49 + 7 * q^51 - 3 * q^53 + 19 * q^55 + 3 * q^57 - 20 * q^59 - 17 * q^61 + 4 * q^63 - 4 * q^65 - 9 * q^67 - 12 * q^69 + 7 * q^71 - 9 * q^73 - 11 * q^75 - 34 * q^77 + 14 * q^79 + 12 * q^81 + 5 * q^83 - 12 * q^85 + 12 * q^87 - 22 * q^89 - 3 * q^91 - 2 * q^93 - 27 * q^95 + 17 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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