LMFDB - Embedded newform 8004.2.a.g.1.4

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			1.4
Root			$2.52462$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -2.52462 q^{5} -2.76852 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.52462 q^{5} -2.76852 q^{7} +1.00000 q^{9} +0.526219 q^{11} -1.93131 q^{13} +2.52462 q^{15} +0.739721 q^{17} -1.99953 q^{19} +2.76852 q^{21} +1.00000 q^{23} +1.37371 q^{25} -1.00000 q^{27} -1.00000 q^{29} +3.82478 q^{31} -0.526219 q^{33} +6.98946 q^{35} +0.747593 q^{37} +1.93131 q^{39} +3.52897 q^{41} +2.13846 q^{43} -2.52462 q^{45} +7.82942 q^{47} +0.664694 q^{49} -0.739721 q^{51} +7.39345 q^{53} -1.32850 q^{55} +1.99953 q^{57} +12.9991 q^{59} +10.0143 q^{61} -2.76852 q^{63} +4.87583 q^{65} -3.29209 q^{67} -1.00000 q^{69} -14.3864 q^{71} -4.34510 q^{73} -1.37371 q^{75} -1.45685 q^{77} -0.963327 q^{79} +1.00000 q^{81} -13.7019 q^{83} -1.86752 q^{85} +1.00000 q^{87} -1.29398 q^{89} +5.34687 q^{91} -3.82478 q^{93} +5.04806 q^{95} +3.48364 q^{97} +0.526219 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−2.52462	−1.12904	−0.564522	−	0.825418i	$-0.690940\pi$
$5$	−2.52462	−1.12904	−0.564522	+	0.825418i	$0.690940\pi$
$6$	0	0
$7$	−2.76852	−1.04640	−0.523201	−	0.852209i	$-0.675262\pi$
$7$	−2.76852	−1.04640	−0.523201	+	0.852209i	$0.675262\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.526219	0.158661	0.0793305	−	0.996848i	$-0.474722\pi$
$11$	0.526219	0.158661	0.0793305	+	0.996848i	$0.474722\pi$
$12$	0	0
$13$	−1.93131	−0.535649	−0.267825	−	0.963468i	$-0.586305\pi$
$13$	−1.93131	−0.535649	−0.267825	+	0.963468i	$0.586305\pi$
$14$	0	0
$15$	2.52462	0.651854
$16$	0	0
$17$	0.739721	0.179409	0.0897044	−	0.995968i	$-0.471408\pi$
$17$	0.739721	0.179409	0.0897044	+	0.995968i	$0.471408\pi$
$18$	0	0
$19$	−1.99953	−0.458724	−0.229362	−	0.973341i	$-0.573664\pi$
$19$	−1.99953	−0.458724	−0.229362	+	0.973341i	$0.573664\pi$
$20$	0	0
$21$	2.76852	0.604140
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.37371	0.274742
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	3.82478	0.686950	0.343475	−	0.939162i	$-0.388396\pi$
$31$	3.82478	0.686950	0.343475	+	0.939162i	$0.388396\pi$
$32$	0	0
$33$	−0.526219	−0.0916030
$34$	0	0
$35$	6.98946	1.18143
$36$	0	0
$37$	0.747593	0.122904	0.0614518	−	0.998110i	$-0.480427\pi$
$37$	0.747593	0.122904	0.0614518	+	0.998110i	$0.480427\pi$
$38$	0	0
$39$	1.93131	0.309257
$40$	0	0
$41$	3.52897	0.551133	0.275566	−	0.961282i	$-0.411135\pi$
$41$	3.52897	0.551133	0.275566	+	0.961282i	$0.411135\pi$
$42$	0	0
$43$	2.13846	0.326113	0.163056	−	0.986617i	$-0.447865\pi$
$43$	2.13846	0.326113	0.163056	+	0.986617i	$0.447865\pi$
$44$	0	0
$45$	−2.52462	−0.376348
$46$	0	0
$47$	7.82942	1.14204	0.571019	−	0.820937i	$-0.306548\pi$
$47$	7.82942	1.14204	0.571019	+	0.820937i	$0.306548\pi$
$48$	0	0
$49$	0.664694	0.0949562
$50$	0	0
$51$	−0.739721	−0.103582
$52$	0	0
$53$	7.39345	1.01557	0.507785	−	0.861484i	$-0.330465\pi$
$53$	7.39345	1.01557	0.507785	+	0.861484i	$0.330465\pi$
$54$	0	0
$55$	−1.32850	−0.179135
$56$	0	0
$57$	1.99953	0.264844
$58$	0	0
$59$	12.9991	1.69234	0.846172	−	0.532909i	$-0.178901\pi$
$59$	12.9991	1.69234	0.846172	+	0.532909i	$0.178901\pi$
$60$	0	0
$61$	10.0143	1.28220	0.641098	−	0.767459i	$-0.278479\pi$
$61$	10.0143	1.28220	0.641098	+	0.767459i	$0.278479\pi$
$62$	0	0
$63$	−2.76852	−0.348801
$64$	0	0
$65$	4.87583	0.604772
$66$	0	0
$67$	−3.29209	−0.402192	−0.201096	−	0.979572i	$-0.564450\pi$
$67$	−3.29209	−0.402192	−0.201096	+	0.979572i	$0.564450\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−14.3864	−1.70735	−0.853677	−	0.520803i	$-0.825633\pi$
$71$	−14.3864	−1.70735	−0.853677	+	0.520803i	$0.825633\pi$
$72$	0	0
$73$	−4.34510	−0.508556	−0.254278	−	0.967131i	$-0.581838\pi$
$73$	−4.34510	−0.508556	−0.254278	+	0.967131i	$0.581838\pi$
$74$	0	0
$75$	−1.37371	−0.158622
$76$	0	0
$77$	−1.45685	−0.166023
$78$	0	0
$79$	−0.963327	−0.108383	−0.0541914	−	0.998531i	$-0.517258\pi$
$79$	−0.963327	−0.108383	−0.0541914	+	0.998531i	$0.517258\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−13.7019	−1.50398	−0.751990	−	0.659175i	$-0.770906\pi$
$83$	−13.7019	−1.50398	−0.751990	+	0.659175i	$0.770906\pi$
$84$	0	0
$85$	−1.86752	−0.202560
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	−1.29398	−0.137161	−0.0685806	−	0.997646i	$-0.521847\pi$
$89$	−1.29398	−0.137161	−0.0685806	+	0.997646i	$0.521847\pi$
$90$	0	0
$91$	5.34687	0.560504
$92$	0	0
$93$	−3.82478	−0.396611
$94$	0	0
$95$	5.04806	0.517920
$96$	0	0
$97$	3.48364	0.353710	0.176855	−	0.984237i	$-0.443408\pi$
$97$	3.48364	0.353710	0.176855	+	0.984237i	$0.443408\pi$
$98$	0	0
$99$	0.526219	0.0528870
$100$	0	0
$101$	−10.3219	−1.02707	−0.513535	−	0.858068i	$-0.671664\pi$
$101$	−10.3219	−1.02707	−0.513535	+	0.858068i	$0.671664\pi$
$102$	0	0
$103$	15.2741	1.50500	0.752499	−	0.658594i	$-0.228848\pi$
$103$	15.2741	1.50500	0.752499	+	0.658594i	$0.228848\pi$
$104$	0	0
$105$	−6.98946	−0.682101
$106$	0	0
$107$	−13.8318	−1.33717	−0.668583	−	0.743637i	$-0.733099\pi$
$107$	−13.8318	−1.33717	−0.668583	+	0.743637i	$0.733099\pi$
$108$	0	0
$109$	−0.610729	−0.0584972	−0.0292486	−	0.999572i	$-0.509311\pi$
$109$	−0.610729	−0.0584972	−0.0292486	+	0.999572i	$0.509311\pi$
$110$	0	0
$111$	−0.747593	−0.0709584
$112$	0	0
$113$	10.9256	1.02780	0.513898	−	0.857851i	$-0.328201\pi$
$113$	10.9256	1.02780	0.513898	+	0.857851i	$0.328201\pi$
$114$	0	0
$115$	−2.52462	−0.235422
$116$	0	0
$117$	−1.93131	−0.178550
$118$	0	0
$119$	−2.04793	−0.187734
$120$	0	0
$121$	−10.7231	−0.974827
$122$	0	0
$123$	−3.52897	−0.318197
$124$	0	0
$125$	9.15501	0.818849
$126$	0	0
$127$	20.6821	1.83524	0.917621	−	0.397457i	$-0.130107\pi$
$127$	20.6821	1.83524	0.917621	+	0.397457i	$0.130107\pi$
$128$	0	0
$129$	−2.13846	−0.188281
$130$	0	0
$131$	−1.82772	−0.159688	−0.0798441	−	0.996807i	$-0.525442\pi$
$131$	−1.82772	−0.159688	−0.0798441	+	0.996807i	$0.525442\pi$
$132$	0	0
$133$	5.53574	0.480009
$134$	0	0
$135$	2.52462	0.217285
$136$	0	0
$137$	20.1737	1.72356	0.861779	−	0.507285i	$-0.169351\pi$
$137$	20.1737	1.72356	0.861779	+	0.507285i	$0.169351\pi$
$138$	0	0
$139$	−6.98799	−0.592714	−0.296357	−	0.955077i	$-0.595772\pi$
$139$	−6.98799	−0.592714	−0.296357	+	0.955077i	$0.595772\pi$
$140$	0	0
$141$	−7.82942	−0.659356
$142$	0	0
$143$	−1.01629	−0.0849867
$144$	0	0
$145$	2.52462	0.209658
$146$	0	0
$147$	−0.664694	−0.0548230
$148$	0	0
$149$	−7.38340	−0.604872	−0.302436	−	0.953170i	$-0.597800\pi$
$149$	−7.38340	−0.604872	−0.302436	+	0.953170i	$0.597800\pi$
$150$	0	0
$151$	15.4647	1.25850	0.629248	−	0.777204i	$-0.283363\pi$
$151$	15.4647	1.25850	0.629248	+	0.777204i	$0.283363\pi$
$152$	0	0
$153$	0.739721	0.0598029
$154$	0	0
$155$	−9.65611	−0.775598
$156$	0	0
$157$	−0.360889	−0.0288020	−0.0144010	−	0.999896i	$-0.504584\pi$
$157$	−0.360889	−0.0288020	−0.0144010	+	0.999896i	$0.504584\pi$
$158$	0	0
$159$	−7.39345	−0.586339
$160$	0	0
$161$	−2.76852	−0.218190
$162$	0	0
$163$	10.3594	0.811409	0.405704	−	0.914004i	$-0.367026\pi$
$163$	10.3594	0.811409	0.405704	+	0.914004i	$0.367026\pi$
$164$	0	0
$165$	1.32850	0.103424
$166$	0	0
$167$	5.80202	0.448973	0.224487	−	0.974477i	$-0.427929\pi$
$167$	5.80202	0.448973	0.224487	+	0.974477i	$0.427929\pi$
$168$	0	0
$169$	−9.27004	−0.713080
$170$	0	0
$171$	−1.99953	−0.152908
$172$	0	0
$173$	−11.2699	−0.856835	−0.428418	−	0.903581i	$-0.640929\pi$
$173$	−11.2699	−0.856835	−0.428418	+	0.903581i	$0.640929\pi$
$174$	0	0
$175$	−3.80314	−0.287491
$176$	0	0
$177$	−12.9991	−0.977076
$178$	0	0
$179$	24.2983	1.81614	0.908071	−	0.418816i	$-0.137555\pi$
$179$	24.2983	1.81614	0.908071	+	0.418816i	$0.137555\pi$
$180$	0	0
$181$	−0.434698	−0.0323108	−0.0161554	−	0.999869i	$-0.505143\pi$
$181$	−0.434698	−0.0323108	−0.0161554	+	0.999869i	$0.505143\pi$
$182$	0	0
$183$	−10.0143	−0.740277
$184$	0	0
$185$	−1.88739	−0.138764
$186$	0	0
$187$	0.389255	0.0284652
$188$	0	0
$189$	2.76852	0.201380
$190$	0	0
$191$	−26.2461	−1.89910	−0.949549	−	0.313619i	$-0.898459\pi$
$191$	−26.2461	−1.89910	−0.949549	+	0.313619i	$0.898459\pi$
$192$	0	0
$193$	−8.48991	−0.611117	−0.305559	−	0.952173i	$-0.598843\pi$
$193$	−8.48991	−0.611117	−0.305559	+	0.952173i	$0.598843\pi$
$194$	0	0
$195$	−4.87583	−0.349165
$196$	0	0
$197$	−13.1347	−0.935809	−0.467904	−	0.883779i	$-0.654991\pi$
$197$	−13.1347	−0.935809	−0.467904	+	0.883779i	$0.654991\pi$
$198$	0	0
$199$	7.99311	0.566617	0.283308	−	0.959029i	$-0.408568\pi$
$199$	7.99311	0.566617	0.283308	+	0.959029i	$0.408568\pi$
$200$	0	0
$201$	3.29209	0.232206
$202$	0	0
$203$	2.76852	0.194312
$204$	0	0
$205$	−8.90931	−0.622253
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−1.05219	−0.0727816
$210$	0	0
$211$	0.868521	0.0597914	0.0298957	−	0.999553i	$-0.490482\pi$
$211$	0.868521	0.0597914	0.0298957	+	0.999553i	$0.490482\pi$
$212$	0	0
$213$	14.3864	0.985741
$214$	0	0
$215$	−5.39881	−0.368196
$216$	0	0
$217$	−10.5890	−0.718826
$218$	0	0
$219$	4.34510	0.293615
$220$	0	0
$221$	−1.42863	−0.0961002
$222$	0	0
$223$	−2.31567	−0.155068	−0.0775342	−	0.996990i	$-0.524705\pi$
$223$	−2.31567	−0.155068	−0.0775342	+	0.996990i	$0.524705\pi$
$224$	0	0
$225$	1.37371	0.0915807
$226$	0	0
$227$	3.21958	0.213691	0.106845	−	0.994276i	$-0.465925\pi$
$227$	3.21958	0.213691	0.106845	+	0.994276i	$0.465925\pi$
$228$	0	0
$229$	−9.23186	−0.610059	−0.305029	−	0.952343i	$-0.598666\pi$
$229$	−9.23186	−0.610059	−0.305029	+	0.952343i	$0.598666\pi$
$230$	0	0
$231$	1.45685	0.0958535
$232$	0	0
$233$	−22.9381	−1.50273	−0.751363	−	0.659889i	$-0.770603\pi$
$233$	−22.9381	−1.50273	−0.751363	+	0.659889i	$0.770603\pi$
$234$	0	0
$235$	−19.7663	−1.28941
$236$	0	0
$237$	0.963327	0.0625748
$238$	0	0
$239$	−15.9646	−1.03266	−0.516331	−	0.856389i	$-0.672702\pi$
$239$	−15.9646	−1.03266	−0.516331	+	0.856389i	$0.672702\pi$
$240$	0	0
$241$	−25.1441	−1.61967	−0.809837	−	0.586655i	$-0.800445\pi$
$241$	−25.1441	−1.61967	−0.809837	+	0.586655i	$0.800445\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−1.67810	−0.107210
$246$	0	0
$247$	3.86172	0.245715
$248$	0	0
$249$	13.7019	0.868323
$250$	0	0
$251$	−21.4340	−1.35290	−0.676452	−	0.736487i	$-0.736483\pi$
$251$	−21.4340	−1.35290	−0.676452	+	0.736487i	$0.736483\pi$
$252$	0	0
$253$	0.526219	0.0330831
$254$	0	0
$255$	1.86752	0.116948
$256$	0	0
$257$	−18.8959	−1.17869	−0.589346	−	0.807881i	$-0.700615\pi$
$257$	−18.8959	−1.17869	−0.589346	+	0.807881i	$0.700615\pi$
$258$	0	0
$259$	−2.06973	−0.128606
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	26.6795	1.64513	0.822563	−	0.568673i	$-0.192543\pi$
$263$	26.6795	1.64513	0.822563	+	0.568673i	$0.192543\pi$
$264$	0	0
$265$	−18.6657	−1.14662
$266$	0	0
$267$	1.29398	0.0791900
$268$	0	0
$269$	13.7876	0.840647	0.420323	−	0.907374i	$-0.361917\pi$
$269$	13.7876	0.840647	0.420323	+	0.907374i	$0.361917\pi$
$270$	0	0
$271$	12.4101	0.753860	0.376930	−	0.926242i	$-0.376980\pi$
$271$	12.4101	0.753860	0.376930	+	0.926242i	$0.376980\pi$
$272$	0	0
$273$	−5.34687	−0.323607
$274$	0	0
$275$	0.722873	0.0435909
$276$	0	0
$277$	−5.28861	−0.317762	−0.158881	−	0.987298i	$-0.550789\pi$
$277$	−5.28861	−0.317762	−0.158881	+	0.987298i	$0.550789\pi$
$278$	0	0
$279$	3.82478	0.228983
$280$	0	0
$281$	19.8312	1.18303	0.591516	−	0.806293i	$-0.298530\pi$
$281$	19.8312	1.18303	0.591516	+	0.806293i	$0.298530\pi$
$282$	0	0
$283$	23.5538	1.40013	0.700064	−	0.714080i	$-0.253155\pi$
$283$	23.5538	1.40013	0.700064	+	0.714080i	$0.253155\pi$
$284$	0	0
$285$	−5.04806	−0.299021
$286$	0	0
$287$	−9.77002	−0.576706
$288$	0	0
$289$	−16.4528	−0.967813
$290$	0	0
$291$	−3.48364	−0.204215
$292$	0	0
$293$	7.00209	0.409066	0.204533	−	0.978860i	$-0.434432\pi$
$293$	7.00209	0.409066	0.204533	+	0.978860i	$0.434432\pi$
$294$	0	0
$295$	−32.8179	−1.91073
$296$	0	0
$297$	−0.526219	−0.0305343
$298$	0	0
$299$	−1.93131	−0.111691
$300$	0	0
$301$	−5.92037	−0.341245
$302$	0	0
$303$	10.3219	0.592980
$304$	0	0
$305$	−25.2822	−1.44766
$306$	0	0
$307$	−22.4598	−1.28185	−0.640925	−	0.767604i	$-0.721449\pi$
$307$	−22.4598	−1.28185	−0.640925	+	0.767604i	$0.721449\pi$
$308$	0	0
$309$	−15.2741	−0.868911
$310$	0	0
$311$	−27.7082	−1.57119	−0.785595	−	0.618741i	$-0.787643\pi$
$311$	−27.7082	−1.57119	−0.785595	+	0.618741i	$0.787643\pi$
$312$	0	0
$313$	−0.503230	−0.0284442	−0.0142221	−	0.999899i	$-0.504527\pi$
$313$	−0.503230	−0.0284442	−0.0142221	+	0.999899i	$0.504527\pi$
$314$	0	0
$315$	6.98946	0.393811
$316$	0	0
$317$	−14.6482	−0.822724	−0.411362	−	0.911472i	$-0.634947\pi$
$317$	−14.6482	−0.822724	−0.411362	+	0.911472i	$0.634947\pi$
$318$	0	0
$319$	−0.526219	−0.0294626
$320$	0	0
$321$	13.8318	0.772014
$322$	0	0
$323$	−1.47909	−0.0822990
$324$	0	0
$325$	−2.65306	−0.147165
$326$	0	0
$327$	0.610729	0.0337734
$328$	0	0
$329$	−21.6759	−1.19503
$330$	0	0
$331$	22.3670	1.22940	0.614702	−	0.788760i	$-0.289276\pi$
$331$	22.3670	1.22940	0.614702	+	0.788760i	$0.289276\pi$
$332$	0	0
$333$	0.747593	0.0409679
$334$	0	0
$335$	8.31127	0.454093
$336$	0	0
$337$	29.7405	1.62007	0.810034	−	0.586383i	$-0.199449\pi$
$337$	29.7405	1.62007	0.810034	+	0.586383i	$0.199449\pi$
$338$	0	0
$339$	−10.9256	−0.593398
$340$	0	0
$341$	2.01267	0.108992
$342$	0	0
$343$	17.5394	0.947039
$344$	0	0
$345$	2.52462	0.135921
$346$	0	0
$347$	−33.6176	−1.80469	−0.902343	−	0.431019i	$-0.858154\pi$
$347$	−33.6176	−1.80469	−0.902343	+	0.431019i	$0.858154\pi$
$348$	0	0
$349$	21.0513	1.12685	0.563424	−	0.826168i	$-0.309484\pi$
$349$	21.0513	1.12685	0.563424	+	0.826168i	$0.309484\pi$
$350$	0	0
$351$	1.93131	0.103086
$352$	0	0
$353$	−9.93922	−0.529011	−0.264506	−	0.964384i	$-0.585209\pi$
$353$	−9.93922	−0.529011	−0.264506	+	0.964384i	$0.585209\pi$
$354$	0	0
$355$	36.3202	1.92768
$356$	0	0
$357$	2.04793	0.108388
$358$	0	0
$359$	5.50696	0.290646	0.145323	−	0.989384i	$-0.453578\pi$
$359$	5.50696	0.290646	0.145323	+	0.989384i	$0.453578\pi$
$360$	0	0
$361$	−15.0019	−0.789573
$362$	0	0
$363$	10.7231	0.562816
$364$	0	0
$365$	10.9697	0.574182
$366$	0	0
$367$	−13.1984	−0.688953	−0.344477	−	0.938795i	$-0.611944\pi$
$367$	−13.1984	−0.688953	−0.344477	+	0.938795i	$0.611944\pi$
$368$	0	0
$369$	3.52897	0.183711
$370$	0	0
$371$	−20.4689	−1.06269
$372$	0	0
$373$	3.43321	0.177765	0.0888825	−	0.996042i	$-0.471670\pi$
$373$	3.43321	0.177765	0.0888825	+	0.996042i	$0.471670\pi$
$374$	0	0
$375$	−9.15501	−0.472763
$376$	0	0
$377$	1.93131	0.0994676
$378$	0	0
$379$	−32.5768	−1.67336	−0.836680	−	0.547692i	$-0.815507\pi$
$379$	−32.5768	−1.67336	−0.836680	+	0.547692i	$0.815507\pi$
$380$	0	0
$381$	−20.6821	−1.05958
$382$	0	0
$383$	−22.4133	−1.14527	−0.572633	−	0.819812i	$-0.694078\pi$
$383$	−22.4133	−1.14527	−0.572633	+	0.819812i	$0.694078\pi$
$384$	0	0
$385$	3.67799	0.187448
$386$	0	0
$387$	2.13846	0.108704
$388$	0	0
$389$	−27.6519	−1.40201	−0.701003	−	0.713159i	$-0.747264\pi$
$389$	−27.6519	−1.40201	−0.701003	+	0.713159i	$0.747264\pi$
$390$	0	0
$391$	0.739721	0.0374093
$392$	0	0
$393$	1.82772	0.0921961
$394$	0	0
$395$	2.43204	0.122369
$396$	0	0
$397$	−3.62435	−0.181901	−0.0909505	−	0.995855i	$-0.528991\pi$
$397$	−3.62435	−0.181901	−0.0909505	+	0.995855i	$0.528991\pi$
$398$	0	0
$399$	−5.53574	−0.277133
$400$	0	0
$401$	−14.4203	−0.720115	−0.360057	−	0.932930i	$-0.617243\pi$
$401$	−14.4203	−0.720115	−0.360057	+	0.932930i	$0.617243\pi$
$402$	0	0
$403$	−7.38684	−0.367965
$404$	0	0
$405$	−2.52462	−0.125449
$406$	0	0
$407$	0.393398	0.0195000
$408$	0	0
$409$	−17.4810	−0.864381	−0.432190	−	0.901782i	$-0.642259\pi$
$409$	−17.4810	−0.864381	−0.432190	+	0.901782i	$0.642259\pi$
$410$	0	0
$411$	−20.1737	−0.995096
$412$	0	0
$413$	−35.9884	−1.77087
$414$	0	0
$415$	34.5921	1.69806
$416$	0	0
$417$	6.98799	0.342203
$418$	0	0
$419$	−26.6543	−1.30215	−0.651074	−	0.759014i	$-0.725681\pi$
$419$	−26.6543	−1.30215	−0.651074	+	0.759014i	$0.725681\pi$
$420$	0	0
$421$	−28.6872	−1.39813	−0.699065	−	0.715058i	$-0.746400\pi$
$421$	−28.6872	−1.39813	−0.699065	+	0.715058i	$0.746400\pi$
$422$	0	0
$423$	7.82942	0.380679
$424$	0	0
$425$	1.01616	0.0492911
$426$	0	0
$427$	−27.7247	−1.34169
$428$	0	0
$429$	1.01629	0.0490671
$430$	0	0
$431$	24.2099	1.16615	0.583076	−	0.812418i	$-0.301849\pi$
$431$	24.2099	1.16615	0.583076	+	0.812418i	$0.301849\pi$
$432$	0	0
$433$	32.0026	1.53795	0.768974	−	0.639279i	$-0.220767\pi$
$433$	32.0026	1.53795	0.768974	+	0.639279i	$0.220767\pi$
$434$	0	0
$435$	−2.52462	−0.121046
$436$	0	0
$437$	−1.99953	−0.0956505
$438$	0	0
$439$	−29.8370	−1.42404	−0.712020	−	0.702159i	$-0.752220\pi$
$439$	−29.8370	−1.42404	−0.712020	+	0.702159i	$0.752220\pi$
$440$	0	0
$441$	0.664694	0.0316521
$442$	0	0
$443$	−29.4781	−1.40054	−0.700272	−	0.713876i	$-0.746938\pi$
$443$	−29.4781	−1.40054	−0.700272	+	0.713876i	$0.746938\pi$
$444$	0	0
$445$	3.26680	0.154861
$446$	0	0
$447$	7.38340	0.349223
$448$	0	0
$449$	−13.5353	−0.638772	−0.319386	−	0.947625i	$-0.603477\pi$
$449$	−13.5353	−0.638772	−0.319386	+	0.947625i	$0.603477\pi$
$450$	0	0
$451$	1.85701	0.0874433
$452$	0	0
$453$	−15.4647	−0.726593
$454$	0	0
$455$	−13.4988	−0.632835
$456$	0	0
$457$	33.2067	1.55334	0.776672	−	0.629906i	$-0.216906\pi$
$457$	33.2067	1.55334	0.776672	+	0.629906i	$0.216906\pi$
$458$	0	0
$459$	−0.739721	−0.0345272
$460$	0	0
$461$	10.0870	0.469797	0.234899	−	0.972020i	$-0.424524\pi$
$461$	10.0870	0.469797	0.234899	+	0.972020i	$0.424524\pi$
$462$	0	0
$463$	41.6137	1.93395	0.966977	−	0.254864i	$-0.0820306\pi$
$463$	41.6137	1.93395	0.966977	+	0.254864i	$0.0820306\pi$
$464$	0	0
$465$	9.65611	0.447791
$466$	0	0
$467$	33.9829	1.57254	0.786270	−	0.617883i	$-0.212009\pi$
$467$	33.9829	1.57254	0.786270	+	0.617883i	$0.212009\pi$
$468$	0	0
$469$	9.11420	0.420855
$470$	0	0
$471$	0.360889	0.0166289
$472$	0	0
$473$	1.12530	0.0517414
$474$	0	0
$475$	−2.74678	−0.126031
$476$	0	0
$477$	7.39345	0.338523
$478$	0	0
$479$	2.40384	0.109834	0.0549170	−	0.998491i	$-0.482511\pi$
$479$	2.40384	0.109834	0.0549170	+	0.998491i	$0.482511\pi$
$480$	0	0
$481$	−1.44384	−0.0658332
$482$	0	0
$483$	2.76852	0.125972
$484$	0	0
$485$	−8.79487	−0.399355
$486$	0	0
$487$	−22.0887	−1.00093	−0.500467	−	0.865756i	$-0.666838\pi$
$487$	−22.0887	−1.00093	−0.500467	+	0.865756i	$0.666838\pi$
$488$	0	0
$489$	−10.3594	−0.468467
$490$	0	0
$491$	−31.9350	−1.44121	−0.720603	−	0.693348i	$-0.756135\pi$
$491$	−31.9350	−1.44121	−0.720603	+	0.693348i	$0.756135\pi$
$492$	0	0
$493$	−0.739721	−0.0333154
$494$	0	0
$495$	−1.32850	−0.0597118
$496$	0	0
$497$	39.8291	1.78658
$498$	0	0
$499$	−20.2643	−0.907155	−0.453578	−	0.891217i	$-0.649853\pi$
$499$	−20.2643	−0.907155	−0.453578	+	0.891217i	$0.649853\pi$
$500$	0	0
$501$	−5.80202	−0.259215
$502$	0	0
$503$	8.20416	0.365805	0.182903	−	0.983131i	$-0.441451\pi$
$503$	8.20416	0.365805	0.182903	+	0.983131i	$0.441451\pi$
$504$	0	0
$505$	26.0590	1.15961
$506$	0	0
$507$	9.27004	0.411697
$508$	0	0
$509$	−19.2432	−0.852938	−0.426469	−	0.904502i	$-0.640243\pi$
$509$	−19.2432	−0.852938	−0.426469	+	0.904502i	$0.640243\pi$
$510$	0	0
$511$	12.0295	0.532154
$512$	0	0
$513$	1.99953	0.0882814
$514$	0	0
$515$	−38.5612	−1.69921
$516$	0	0
$517$	4.11999	0.181197
$518$	0	0
$519$	11.2699	0.494694
$520$	0	0
$521$	−42.9060	−1.87975	−0.939873	−	0.341524i	$-0.889057\pi$
$521$	−42.9060	−1.87975	−0.939873	+	0.341524i	$0.889057\pi$
$522$	0	0
$523$	−9.21145	−0.402789	−0.201394	−	0.979510i	$-0.564547\pi$
$523$	−9.21145	−0.402789	−0.201394	+	0.979510i	$0.564547\pi$
$524$	0	0
$525$	3.80314	0.165983
$526$	0	0
$527$	2.82927	0.123245
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	12.9991	0.564115
$532$	0	0
$533$	−6.81554	−0.295214
$534$	0	0
$535$	34.9200	1.50972
$536$	0	0
$537$	−24.2983	−1.04855
$538$	0	0
$539$	0.349775	0.0150659
$540$	0	0
$541$	19.0764	0.820161	0.410080	−	0.912049i	$-0.365501\pi$
$541$	19.0764	0.820161	0.410080	+	0.912049i	$0.365501\pi$
$542$	0	0
$543$	0.434698	0.0186547
$544$	0	0
$545$	1.54186	0.0660460
$546$	0	0
$547$	34.9038	1.49238	0.746189	−	0.665734i	$-0.231882\pi$
$547$	34.9038	1.49238	0.746189	+	0.665734i	$0.231882\pi$
$548$	0	0
$549$	10.0143	0.427399
$550$	0	0
$551$	1.99953	0.0851828
$552$	0	0
$553$	2.66699	0.113412
$554$	0	0
$555$	1.88739	0.0801152
$556$	0	0
$557$	−18.3979	−0.779545	−0.389772	−	0.920911i	$-0.627446\pi$
$557$	−18.3979	−0.779545	−0.389772	+	0.920911i	$0.627446\pi$
$558$	0	0
$559$	−4.13004	−0.174682
$560$	0	0
$561$	−0.389255	−0.0164344
$562$	0	0
$563$	21.6776	0.913601	0.456801	−	0.889569i	$-0.348995\pi$
$563$	21.6776	0.913601	0.456801	+	0.889569i	$0.348995\pi$
$564$	0	0
$565$	−27.5830	−1.16043
$566$	0	0
$567$	−2.76852	−0.116267
$568$	0	0
$569$	35.3588	1.48232	0.741160	−	0.671329i	$-0.234276\pi$
$569$	35.3588	1.48232	0.741160	+	0.671329i	$0.234276\pi$
$570$	0	0
$571$	−37.9152	−1.58670	−0.793350	−	0.608765i	$-0.791665\pi$
$571$	−37.9152	−1.58670	−0.793350	+	0.608765i	$0.791665\pi$
$572$	0	0
$573$	26.2461	1.09644
$574$	0	0
$575$	1.37371	0.0572877
$576$	0	0
$577$	−25.2547	−1.05137	−0.525684	−	0.850680i	$-0.676191\pi$
$577$	−25.2547	−1.05137	−0.525684	+	0.850680i	$0.676191\pi$
$578$	0	0
$579$	8.48991	0.352829
$580$	0	0
$581$	37.9340	1.57377
$582$	0	0
$583$	3.89058	0.161131
$584$	0	0
$585$	4.87583	0.201591
$586$	0	0
$587$	−25.7576	−1.06313	−0.531566	−	0.847017i	$-0.678396\pi$
$587$	−25.7576	−1.06313	−0.531566	+	0.847017i	$0.678396\pi$
$588$	0	0
$589$	−7.64776	−0.315120
$590$	0	0
$591$	13.1347	0.540289
$592$	0	0
$593$	24.6936	1.01404	0.507022	−	0.861933i	$-0.330746\pi$
$593$	24.6936	1.01404	0.507022	+	0.861933i	$0.330746\pi$
$594$	0	0
$595$	5.17025	0.211960
$596$	0	0
$597$	−7.99311	−0.327136
$598$	0	0
$599$	31.6646	1.29378	0.646889	−	0.762584i	$-0.276070\pi$
$599$	31.6646	1.29378	0.646889	+	0.762584i	$0.276070\pi$
$600$	0	0
$601$	18.6588	0.761109	0.380554	−	0.924759i	$-0.375733\pi$
$601$	18.6588	0.761109	0.380554	+	0.924759i	$0.375733\pi$
$602$	0	0
$603$	−3.29209	−0.134064
$604$	0	0
$605$	27.0717	1.10062
$606$	0	0
$607$	8.67966	0.352296	0.176148	−	0.984364i	$-0.443636\pi$
$607$	8.67966	0.352296	0.176148	+	0.984364i	$0.443636\pi$
$608$	0	0
$609$	−2.76852	−0.112186
$610$	0	0
$611$	−15.1210	−0.611732
$612$	0	0
$613$	−46.5278	−1.87924	−0.939620	−	0.342220i	$-0.888821\pi$
$613$	−46.5278	−1.87924	−0.939620	+	0.342220i	$0.888821\pi$
$614$	0	0
$615$	8.90931	0.359258
$616$	0	0
$617$	−43.1480	−1.73707	−0.868537	−	0.495625i	$-0.834939\pi$
$617$	−43.1480	−1.73707	−0.868537	+	0.495625i	$0.834939\pi$
$618$	0	0
$619$	17.5317	0.704658	0.352329	−	0.935876i	$-0.385390\pi$
$619$	17.5317	0.704658	0.352329	+	0.935876i	$0.385390\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	3.58239	0.143526
$624$	0	0
$625$	−29.9815	−1.19926
$626$	0	0
$627$	1.05219	0.0420205
$628$	0	0
$629$	0.553010	0.0220500
$630$	0	0
$631$	−22.9146	−0.912216	−0.456108	−	0.889925i	$-0.650757\pi$
$631$	−22.9146	−0.912216	−0.456108	+	0.889925i	$0.650757\pi$
$632$	0	0
$633$	−0.868521	−0.0345206
$634$	0	0
$635$	−52.2145	−2.07207
$636$	0	0
$637$	−1.28373	−0.0508633
$638$	0	0
$639$	−14.3864	−0.569118
$640$	0	0
$641$	−7.61469	−0.300762	−0.150381	−	0.988628i	$-0.548050\pi$
$641$	−7.61469	−0.300762	−0.150381	+	0.988628i	$0.548050\pi$
$642$	0	0
$643$	−5.73510	−0.226170	−0.113085	−	0.993585i	$-0.536073\pi$
$643$	−5.73510	−0.226170	−0.113085	+	0.993585i	$0.536073\pi$
$644$	0	0
$645$	5.39881	0.212578
$646$	0	0
$647$	−40.0251	−1.57355	−0.786775	−	0.617239i	$-0.788251\pi$
$647$	−40.0251	−1.57355	−0.786775	+	0.617239i	$0.788251\pi$
$648$	0	0
$649$	6.84040	0.268509
$650$	0	0
$651$	10.5890	0.415014
$652$	0	0
$653$	−46.5044	−1.81986	−0.909929	−	0.414764i	$-0.863864\pi$
$653$	−46.5044	−1.81986	−0.909929	+	0.414764i	$0.863864\pi$
$654$	0	0
$655$	4.61429	0.180295
$656$	0	0
$657$	−4.34510	−0.169519
$658$	0	0
$659$	29.8314	1.16207	0.581034	−	0.813879i	$-0.302648\pi$
$659$	29.8314	1.16207	0.581034	+	0.813879i	$0.302648\pi$
$660$	0	0
$661$	−19.3456	−0.752455	−0.376227	−	0.926527i	$-0.622779\pi$
$661$	−19.3456	−0.752455	−0.376227	+	0.926527i	$0.622779\pi$
$662$	0	0
$663$	1.42863	0.0554835
$664$	0	0
$665$	−13.9756	−0.541952
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	2.31567	0.0895288
$670$	0	0
$671$	5.26970	0.203435
$672$	0	0
$673$	5.04602	0.194510	0.0972550	−	0.995260i	$-0.468994\pi$
$673$	5.04602	0.194510	0.0972550	+	0.995260i	$0.468994\pi$
$674$	0	0
$675$	−1.37371	−0.0528741
$676$	0	0
$677$	−13.3579	−0.513387	−0.256693	−	0.966493i	$-0.582633\pi$
$677$	−13.3579	−0.513387	−0.256693	+	0.966493i	$0.582633\pi$
$678$	0	0
$679$	−9.64452	−0.370123
$680$	0	0
$681$	−3.21958	−0.123374
$682$	0	0
$683$	14.7302	0.563635	0.281817	−	0.959468i	$-0.409063\pi$
$683$	14.7302	0.563635	0.281817	+	0.959468i	$0.409063\pi$
$684$	0	0
$685$	−50.9310	−1.94597
$686$	0	0
$687$	9.23186	0.352218
$688$	0	0
$689$	−14.2791	−0.543989
$690$	0	0
$691$	33.0276	1.25643	0.628214	−	0.778040i	$-0.283786\pi$
$691$	33.0276	1.25643	0.628214	+	0.778040i	$0.283786\pi$
$692$	0	0
$693$	−1.45685	−0.0553411
$694$	0	0
$695$	17.6420	0.669200
$696$	0	0
$697$	2.61045	0.0988780
$698$	0	0
$699$	22.9381	0.867600
$700$	0	0
$701$	−38.1941	−1.44257	−0.721285	−	0.692638i	$-0.756448\pi$
$701$	−38.1941	−1.44257	−0.721285	+	0.692638i	$0.756448\pi$
$702$	0	0
$703$	−1.49484	−0.0563788
$704$	0	0
$705$	19.7663	0.744443
$706$	0	0
$707$	28.5765	1.07473
$708$	0	0
$709$	30.4902	1.14508	0.572542	−	0.819875i	$-0.305957\pi$
$709$	30.4902	1.14508	0.572542	+	0.819875i	$0.305957\pi$
$710$	0	0
$711$	−0.963327	−0.0361276
$712$	0	0
$713$	3.82478	0.143239
$714$	0	0
$715$	2.56576	0.0959538
$716$	0	0
$717$	15.9646	0.596207
$718$	0	0
$719$	19.3654	0.722209	0.361104	−	0.932525i	$-0.382400\pi$
$719$	19.3654	0.722209	0.361104	+	0.932525i	$0.382400\pi$
$720$	0	0
$721$	−42.2865	−1.57483
$722$	0	0
$723$	25.1441	0.935119
$724$	0	0
$725$	−1.37371	−0.0510183
$726$	0	0
$727$	41.2825	1.53108	0.765541	−	0.643387i	$-0.222472\pi$
$727$	41.2825	1.53108	0.765541	+	0.643387i	$0.222472\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.58187	0.0585074
$732$	0	0
$733$	−25.6687	−0.948097	−0.474048	−	0.880499i	$-0.657208\pi$
$733$	−25.6687	−0.948097	−0.474048	+	0.880499i	$0.657208\pi$
$734$	0	0
$735$	1.67810	0.0618976
$736$	0	0
$737$	−1.73236	−0.0638123
$738$	0	0
$739$	30.9111	1.13708	0.568541	−	0.822655i	$-0.307508\pi$
$739$	30.9111	1.13708	0.568541	+	0.822655i	$0.307508\pi$
$740$	0	0
$741$	−3.86172	−0.141864
$742$	0	0
$743$	16.4627	0.603959	0.301979	−	0.953314i	$-0.402353\pi$
$743$	16.4627	0.603959	0.301979	+	0.953314i	$0.402353\pi$
$744$	0	0
$745$	18.6403	0.682928
$746$	0	0
$747$	−13.7019	−0.501327
$748$	0	0
$749$	38.2935	1.39921
$750$	0	0
$751$	50.3553	1.83749	0.918746	−	0.394850i	$-0.129203\pi$
$751$	50.3553	1.83749	0.918746	+	0.394850i	$0.129203\pi$
$752$	0	0
$753$	21.4340	0.781099
$754$	0	0
$755$	−39.0424	−1.42090
$756$	0	0
$757$	−37.5477	−1.36469	−0.682347	−	0.731029i	$-0.739041\pi$
$757$	−37.5477	−1.36469	−0.682347	+	0.731029i	$0.739041\pi$
$758$	0	0
$759$	−0.526219	−0.0191005
$760$	0	0
$761$	16.6558	0.603771	0.301886	−	0.953344i	$-0.402384\pi$
$761$	16.6558	0.603771	0.301886	+	0.953344i	$0.402384\pi$
$762$	0	0
$763$	1.69081	0.0612116
$764$	0	0
$765$	−1.86752	−0.0675202
$766$	0	0
$767$	−25.1054	−0.906504
$768$	0	0
$769$	11.1202	0.401005	0.200502	−	0.979693i	$-0.435743\pi$
$769$	11.1202	0.401005	0.200502	+	0.979693i	$0.435743\pi$
$770$	0	0
$771$	18.8959	0.680519
$772$	0	0
$773$	28.0186	1.00776	0.503879	−	0.863774i	$-0.331906\pi$
$773$	28.0186	1.00776	0.503879	+	0.863774i	$0.331906\pi$
$774$	0	0
$775$	5.25414	0.188734
$776$	0	0
$777$	2.06973	0.0742510
$778$	0	0
$779$	−7.05628	−0.252818
$780$	0	0
$781$	−7.57041	−0.270891
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	0.911107	0.0325188
$786$	0	0
$787$	44.0485	1.57016	0.785080	−	0.619394i	$-0.212622\pi$
$787$	44.0485	1.57016	0.785080	+	0.619394i	$0.212622\pi$
$788$	0	0
$789$	−26.6795	−0.949814
$790$	0	0
$791$	−30.2478	−1.07549
$792$	0	0
$793$	−19.3407	−0.686808
$794$	0	0
$795$	18.6657	0.662003
$796$	0	0
$797$	6.11994	0.216779	0.108390	−	0.994108i	$-0.465431\pi$
$797$	6.11994	0.216779	0.108390	+	0.994108i	$0.465431\pi$
$798$	0	0
$799$	5.79159	0.204892
$800$	0	0
$801$	−1.29398	−0.0457204
$802$	0	0
$803$	−2.28648	−0.0806880
$804$	0	0
$805$	6.98946	0.246346
$806$	0	0
$807$	−13.7876	−0.485348
$808$	0	0
$809$	−11.5618	−0.406490	−0.203245	−	0.979128i	$-0.565149\pi$
$809$	−11.5618	−0.406490	−0.203245	+	0.979128i	$0.565149\pi$
$810$	0	0
$811$	−11.0513	−0.388065	−0.194032	−	0.980995i	$-0.562157\pi$
$811$	−11.0513	−0.388065	−0.194032	+	0.980995i	$0.562157\pi$
$812$	0	0
$813$	−12.4101	−0.435241
$814$	0	0
$815$	−26.1535	−0.916117
$816$	0	0
$817$	−4.27592	−0.149596
$818$	0	0
$819$	5.34687	0.186835
$820$	0	0
$821$	12.6542	0.441636	0.220818	−	0.975315i	$-0.429127\pi$
$821$	12.6542	0.441636	0.220818	+	0.975315i	$0.429127\pi$
$822$	0	0
$823$	49.0282	1.70902	0.854509	−	0.519437i	$-0.173858\pi$
$823$	49.0282	1.70902	0.854509	+	0.519437i	$0.173858\pi$
$824$	0	0
$825$	−0.722873	−0.0251672
$826$	0	0
$827$	−13.9071	−0.483599	−0.241799	−	0.970326i	$-0.577738\pi$
$827$	−13.9071	−0.483599	−0.241799	+	0.970326i	$0.577738\pi$
$828$	0	0
$829$	22.9863	0.798347	0.399174	−	0.916875i	$-0.369297\pi$
$829$	22.9863	0.798347	0.399174	+	0.916875i	$0.369297\pi$
$830$	0	0
$831$	5.28861	0.183460
$832$	0	0
$833$	0.491688	0.0170360
$834$	0	0
$835$	−14.6479	−0.506911
$836$	0	0
$837$	−3.82478	−0.132204
$838$	0	0
$839$	43.7697	1.51110	0.755549	−	0.655092i	$-0.227370\pi$
$839$	43.7697	1.51110	0.755549	+	0.655092i	$0.227370\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−19.8312	−0.683024
$844$	0	0
$845$	23.4033	0.805099
$846$	0	0
$847$	29.6871	1.02006
$848$	0	0
$849$	−23.5538	−0.808364
$850$	0	0
$851$	0.747593	0.0256272
$852$	0	0
$853$	29.6457	1.01505	0.507524	−	0.861637i	$-0.330561\pi$
$853$	29.6457	1.01505	0.507524	+	0.861637i	$0.330561\pi$
$854$	0	0
$855$	5.04806	0.172640
$856$	0	0
$857$	25.3516	0.865995	0.432997	−	0.901395i	$-0.357456\pi$
$857$	25.3516	0.865995	0.432997	+	0.901395i	$0.357456\pi$
$858$	0	0
$859$	−28.6168	−0.976394	−0.488197	−	0.872734i	$-0.662345\pi$
$859$	−28.6168	−0.976394	−0.488197	+	0.872734i	$0.662345\pi$
$860$	0	0
$861$	9.77002	0.332961
$862$	0	0
$863$	42.6297	1.45113	0.725565	−	0.688153i	$-0.241578\pi$
$863$	42.6297	1.45113	0.725565	+	0.688153i	$0.241578\pi$
$864$	0	0
$865$	28.4522	0.967405
$866$	0	0
$867$	16.4528	0.558767
$868$	0	0
$869$	−0.506921	−0.0171961
$870$	0	0
$871$	6.35804	0.215434
$872$	0	0
$873$	3.48364	0.117903
$874$	0	0
$875$	−25.3458	−0.856845
$876$	0	0
$877$	−6.42553	−0.216975	−0.108487	−	0.994098i	$-0.534601\pi$
$877$	−6.42553	−0.216975	−0.108487	+	0.994098i	$0.534601\pi$
$878$	0	0
$879$	−7.00209	−0.236174
$880$	0	0
$881$	−14.8442	−0.500113	−0.250056	−	0.968231i	$-0.580449\pi$
$881$	−14.8442	−0.500113	−0.250056	+	0.968231i	$0.580449\pi$
$882$	0	0
$883$	−43.4782	−1.46316	−0.731579	−	0.681757i	$-0.761216\pi$
$883$	−43.4782	−1.46316	−0.731579	+	0.681757i	$0.761216\pi$
$884$	0	0
$885$	32.8179	1.10316
$886$	0	0
$887$	4.64900	0.156098	0.0780490	−	0.996950i	$-0.475131\pi$
$887$	4.64900	0.156098	0.0780490	+	0.996950i	$0.475131\pi$
$888$	0	0
$889$	−57.2588	−1.92040
$890$	0	0
$891$	0.526219	0.0176290
$892$	0	0
$893$	−15.6552	−0.523880
$894$	0	0
$895$	−61.3441	−2.05051
$896$	0	0
$897$	1.93131	0.0644846
$898$	0	0
$899$	−3.82478	−0.127563
$900$	0	0
$901$	5.46909	0.182202
$902$	0	0
$903$	5.92037	0.197018
$904$	0	0
$905$	1.09745	0.0364804
$906$	0	0
$907$	−15.5997	−0.517981	−0.258990	−	0.965880i	$-0.583390\pi$
$907$	−15.5997	−0.517981	−0.258990	+	0.965880i	$0.583390\pi$
$908$	0	0
$909$	−10.3219	−0.342357
$910$	0	0
$911$	42.0282	1.39245	0.696227	−	0.717821i	$-0.254861\pi$
$911$	42.0282	1.39245	0.696227	+	0.717821i	$0.254861\pi$
$912$	0	0
$913$	−7.21021	−0.238623
$914$	0	0
$915$	25.2822	0.835805
$916$	0	0
$917$	5.06006	0.167098
$918$	0	0
$919$	20.0000	0.659738	0.329869	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659738	0.329869	+	0.944027i	$0.392995\pi$
$920$	0	0
$921$	22.4598	0.740076
$922$	0	0
$923$	27.7847	0.914543
$924$	0	0
$925$	1.02698	0.0337668
$926$	0	0
$927$	15.2741	0.501666
$928$	0	0
$929$	−22.5550	−0.740007	−0.370004	−	0.929030i	$-0.620644\pi$
$929$	−22.5550	−0.740007	−0.370004	+	0.929030i	$0.620644\pi$
$930$	0	0
$931$	−1.32907	−0.0435587
$932$	0	0
$933$	27.7082	0.907127
$934$	0	0
$935$	−0.982722	−0.0321385
$936$	0	0
$937$	−33.3680	−1.09008	−0.545042	−	0.838409i	$-0.683486\pi$
$937$	−33.3680	−1.09008	−0.545042	+	0.838409i	$0.683486\pi$
$938$	0	0
$939$	0.503230	0.0164223
$940$	0	0
$941$	30.2255	0.985324	0.492662	−	0.870221i	$-0.336024\pi$
$941$	30.2255	0.985324	0.492662	+	0.870221i	$0.336024\pi$
$942$	0	0
$943$	3.52897	0.114919
$944$	0	0
$945$	−6.98946	−0.227367
$946$	0	0
$947$	−19.5974	−0.636830	−0.318415	−	0.947951i	$-0.603151\pi$
$947$	−19.5974	−0.636830	−0.318415	+	0.947951i	$0.603151\pi$
$948$	0	0
$949$	8.39175	0.272408
$950$	0	0
$951$	14.6482	0.475000
$952$	0	0
$953$	−1.14762	−0.0371749	−0.0185875	−	0.999827i	$-0.505917\pi$
$953$	−1.14762	−0.0371749	−0.0185875	+	0.999827i	$0.505917\pi$
$954$	0	0
$955$	66.2613	2.14417
$956$	0	0
$957$	0.526219	0.0170103
$958$	0	0
$959$	−55.8513	−1.80353
$960$	0	0
$961$	−16.3711	−0.528099
$962$	0	0
$963$	−13.8318	−0.445722
$964$	0	0
$965$	21.4338	0.689979
$966$	0	0
$967$	6.81919	0.219290	0.109645	−	0.993971i	$-0.465029\pi$
$967$	6.81919	0.219290	0.109645	+	0.993971i	$0.465029\pi$
$968$	0	0
$969$	1.47909	0.0475154
$970$	0	0
$971$	−20.5144	−0.658337	−0.329169	−	0.944271i	$-0.606768\pi$
$971$	−20.5144	−0.658337	−0.329169	+	0.944271i	$0.606768\pi$
$972$	0	0
$973$	19.3464	0.620216
$974$	0	0
$975$	2.65306	0.0849660
$976$	0	0
$977$	−25.1984	−0.806168	−0.403084	−	0.915163i	$-0.632062\pi$
$977$	−25.1984	−0.806168	−0.403084	+	0.915163i	$0.632062\pi$
$978$	0	0
$979$	−0.680915	−0.0217621
$980$	0	0
$981$	−0.610729	−0.0194991
$982$	0	0
$983$	−17.2132	−0.549017	−0.274509	−	0.961585i	$-0.588515\pi$
$983$	−17.2132	−0.549017	−0.274509	+	0.961585i	$0.588515\pi$
$984$	0	0
$985$	33.1601	1.05657
$986$	0	0
$987$	21.6759	0.689951
$988$	0	0
$989$	2.13846	0.0679992
$990$	0	0
$991$	7.10587	0.225725	0.112863	−	0.993611i	$-0.463998\pi$
$991$	7.10587	0.225725	0.112863	+	0.993611i	$0.463998\pi$
$992$	0	0
$993$	−22.3670	−0.709796
$994$	0	0
$995$	−20.1796	−0.639735
$996$	0	0
$997$	20.5726	0.651539	0.325770	−	0.945449i	$-0.394377\pi$
$997$	20.5726	0.651539	0.325770	+	0.945449i	$0.394377\pi$
$998$	0	0
$999$	−0.747593	−0.0236528

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.g.1.4	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.g.1.4	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.52462 q^{5} -2.76852 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.52462 q^{5} -2.76852 q^{7} +1.00000 q^{9} +0.526219 q^{11} -1.93131 q^{13} +2.52462 q^{15} +0.739721 q^{17} -1.99953 q^{19} +2.76852 q^{21} +1.00000 q^{23} +1.37371 q^{25} -1.00000 q^{27} -1.00000 q^{29} +3.82478 q^{31} -0.526219 q^{33} +6.98946 q^{35} +0.747593 q^{37} +1.93131 q^{39} +3.52897 q^{41} +2.13846 q^{43} -2.52462 q^{45} +7.82942 q^{47} +0.664694 q^{49} -0.739721 q^{51} +7.39345 q^{53} -1.32850 q^{55} +1.99953 q^{57} +12.9991 q^{59} +10.0143 q^{61} -2.76852 q^{63} +4.87583 q^{65} -3.29209 q^{67} -1.00000 q^{69} -14.3864 q^{71} -4.34510 q^{73} -1.37371 q^{75} -1.45685 q^{77} -0.963327 q^{79} +1.00000 q^{81} -13.7019 q^{83} -1.86752 q^{85} +1.00000 q^{87} -1.29398 q^{89} +5.34687 q^{91} -3.82478 q^{93} +5.04806 q^{95} +3.48364 q^{97} +0.526219 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{3} - 3 q^{5} + 4 q^{7} + 12 q^{9}+O(q^{10}) \) 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

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