LMFDB - Embedded newform 8004.2.a.h.1.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8004,2,Mod(1,8004)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [13,0,-13,0,5,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$63.9122617778$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 5 x^{12} - 27 x^{11} + 158 x^{10} + 180 x^{9} - 1652 x^{8} + 65 x^{7} + 7388 x^{6} + \cdots - 5832 $ x^13 - 5x^12 - 27x^11 + 158x^10 + 180x^9 - 1652x^8 + 65x^7 + 7388x^6 - 3259x^5 - 15445x^4 + 7832x^3 + 15102x^2 - 5184x - 5832
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$2.93518$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +2.93518 q^{5} +2.71487 q^{7} +1.00000 q^{9} -3.58887 q^{11} -3.17111 q^{13} -2.93518 q^{15} +4.47862 q^{17} -2.55694 q^{19} -2.71487 q^{21} -1.00000 q^{23} +3.61530 q^{25} -1.00000 q^{27} -1.00000 q^{29} -5.38333 q^{31} +3.58887 q^{33} +7.96863 q^{35} +5.07603 q^{37} +3.17111 q^{39} +0.235962 q^{41} +8.18346 q^{43} +2.93518 q^{45} -6.76932 q^{47} +0.370499 q^{49} -4.47862 q^{51} -2.07077 q^{53} -10.5340 q^{55} +2.55694 q^{57} +5.67608 q^{59} +3.60240 q^{61} +2.71487 q^{63} -9.30780 q^{65} +2.30488 q^{67} +1.00000 q^{69} +5.54897 q^{71} +11.4518 q^{73} -3.61530 q^{75} -9.74330 q^{77} +4.16775 q^{79} +1.00000 q^{81} +6.64646 q^{83} +13.1456 q^{85} +1.00000 q^{87} +8.84427 q^{89} -8.60915 q^{91} +5.38333 q^{93} -7.50509 q^{95} +10.2089 q^{97} -3.58887 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 13 q^{3} + 5 q^{5} - 8 q^{7} + 13 q^{9} + q^{11} + q^{13} - 5 q^{15} - 2 q^{17} - 10 q^{19} + 8 q^{21} - 13 q^{23} + 14 q^{25} - 13 q^{27} - 13 q^{29} - 26 q^{31} - q^{33} + 19 q^{35} + 15 q^{37}+ \cdots + q^{99}+O(q^{100}) $ 13 * q - 13 * q^3 + 5 * q^5 - 8 * q^7 + 13 * q^9 + q^11 + q^13 - 5 * q^15 - 2 * q^17 - 10 * q^19 + 8 * q^21 - 13 * q^23 + 14 * q^25 - 13 * q^27 - 13 * q^29 - 26 * q^31 - q^33 + 19 * q^35 + 15 * q^37 - q^39 + 21 * q^41 - 6 * q^43 + 5 * q^45 + 16 * q^47 + 19 * q^49 + 2 * q^51 + 7 * q^53 + 15 * q^55 + 10 * q^57 - 11 * q^59 + 19 * q^61 - 8 * q^63 + 6 * q^65 - 13 * q^67 + 13 * q^69 + 9 * q^73 - 14 * q^75 + 10 * q^77 - 25 * q^79 + 13 * q^81 + 3 * q^83 + 14 * q^85 + 13 * q^87 + 23 * q^89 + 19 * q^91 + 26 * q^93 + 7 * q^95 + 25 * q^97 + q^99

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
$5$	2.93518	1.31265	0.656327	−	0.754477i	$-0.272109\pi$
$5$	2.93518	1.31265	0.656327	+	0.754477i	$0.272109\pi$
$6$	0	0
$7$	2.71487	1.02612	0.513062	−	0.858352i	$-0.328511\pi$
$7$	2.71487	1.02612	0.513062	+	0.858352i	$0.328511\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.58887	−1.08208	−0.541042	−	0.840996i	$-0.681970\pi$
$11$	−3.58887	−1.08208	−0.541042	+	0.840996i	$0.681970\pi$
$12$	0	0
$13$	−3.17111	−0.879509	−0.439754	−	0.898118i	$-0.644935\pi$
$13$	−3.17111	−0.879509	−0.439754	+	0.898118i	$0.644935\pi$
$14$	0	0
$15$	−2.93518	−0.757861
$16$	0	0
$17$	4.47862	1.08622	0.543112	−	0.839660i	$-0.317246\pi$
$17$	4.47862	1.08622	0.543112	+	0.839660i	$0.317246\pi$
$18$	0	0
$19$	−2.55694	−0.586603	−0.293301	−	0.956020i	$-0.594754\pi$
$19$	−2.55694	−0.586603	−0.293301	+	0.956020i	$0.594754\pi$
$20$	0	0
$21$	−2.71487	−0.592432
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	3.61530	0.723059
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−5.38333	−0.966874	−0.483437	−	0.875379i	$-0.660612\pi$
$31$	−5.38333	−0.966874	−0.483437	+	0.875379i	$0.660612\pi$
$32$	0	0
$33$	3.58887	0.624742
$34$	0	0
$35$	7.96863	1.34694
$36$	0	0
$37$	5.07603	0.834494	0.417247	−	0.908793i	$-0.362995\pi$
$37$	5.07603	0.834494	0.417247	+	0.908793i	$0.362995\pi$
$38$	0	0
$39$	3.17111	0.507785
$40$	0	0
$41$	0.235962	0.0368511	0.0184255	−	0.999830i	$-0.494135\pi$
$41$	0.235962	0.0368511	0.0184255	+	0.999830i	$0.494135\pi$
$42$	0	0
$43$	8.18346	1.24797	0.623983	−	0.781438i	$-0.285514\pi$
$43$	8.18346	1.24797	0.623983	+	0.781438i	$0.285514\pi$
$44$	0	0
$45$	2.93518	0.437551
$46$	0	0
$47$	−6.76932	−0.987407	−0.493703	−	0.869630i	$-0.664357\pi$
$47$	−6.76932	−0.987407	−0.493703	+	0.869630i	$0.664357\pi$
$48$	0	0
$49$	0.370499	0.0529285
$50$	0	0
$51$	−4.47862	−0.627132
$52$	0	0
$53$	−2.07077	−0.284442	−0.142221	−	0.989835i	$-0.545424\pi$
$53$	−2.07077	−0.284442	−0.142221	+	0.989835i	$0.545424\pi$
$54$	0	0
$55$	−10.5340	−1.42040
$56$	0	0
$57$	2.55694	0.338675
$58$	0	0
$59$	5.67608	0.738963	0.369481	−	0.929238i	$-0.379535\pi$
$59$	5.67608	0.738963	0.369481	+	0.929238i	$0.379535\pi$
$60$	0	0
$61$	3.60240	0.461240	0.230620	−	0.973044i	$-0.425925\pi$
$61$	3.60240	0.461240	0.230620	+	0.973044i	$0.425925\pi$
$62$	0	0
$63$	2.71487	0.342041
$64$	0	0
$65$	−9.30780	−1.15449
$66$	0	0
$67$	2.30488	0.281587	0.140793	−	0.990039i	$-0.455035\pi$
$67$	2.30488	0.281587	0.140793	+	0.990039i	$0.455035\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	5.54897	0.658542	0.329271	−	0.944235i	$-0.393197\pi$
$71$	5.54897	0.658542	0.329271	+	0.944235i	$0.393197\pi$
$72$	0	0
$73$	11.4518	1.34033	0.670163	−	0.742214i	$-0.266224\pi$
$73$	11.4518	1.34033	0.670163	+	0.742214i	$0.266224\pi$
$74$	0	0
$75$	−3.61530	−0.417458
$76$	0	0
$77$	−9.74330	−1.11035
$78$	0	0
$79$	4.16775	0.468908	0.234454	−	0.972127i	$-0.424670\pi$
$79$	4.16775	0.468908	0.234454	+	0.972127i	$0.424670\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.64646	0.729543	0.364772	−	0.931097i	$-0.381147\pi$
$83$	6.64646	0.729543	0.364772	+	0.931097i	$0.381147\pi$
$84$	0	0
$85$	13.1456	1.42584
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	8.84427	0.937491	0.468745	−	0.883333i	$-0.344706\pi$
$89$	8.84427	0.937491	0.468745	+	0.883333i	$0.344706\pi$
$90$	0	0
$91$	−8.60915	−0.902484
$92$	0	0
$93$	5.38333	0.558225
$94$	0	0
$95$	−7.50509	−0.770006
$96$	0	0
$97$	10.2089	1.03655	0.518277	−	0.855213i	$-0.326574\pi$
$97$	10.2089	1.03655	0.518277	+	0.855213i	$0.326574\pi$
$98$	0	0
$99$	−3.58887	−0.360695
$100$	0	0
$101$	14.0772	1.40073	0.700366	−	0.713784i	$-0.253020\pi$
$101$	14.0772	1.40073	0.700366	+	0.713784i	$0.253020\pi$
$102$	0	0
$103$	0.536603	0.0528730	0.0264365	−	0.999650i	$-0.491584\pi$
$103$	0.536603	0.0528730	0.0264365	+	0.999650i	$0.491584\pi$
$104$	0	0
$105$	−7.96863	−0.777658
$106$	0	0
$107$	6.01387	0.581383	0.290691	−	0.956817i	$-0.406115\pi$
$107$	6.01387	0.581383	0.290691	+	0.956817i	$0.406115\pi$
$108$	0	0
$109$	11.0166	1.05520	0.527600	−	0.849493i	$-0.323092\pi$
$109$	11.0166	1.05520	0.527600	+	0.849493i	$0.323092\pi$
$110$	0	0
$111$	−5.07603	−0.481795
$112$	0	0
$113$	19.0729	1.79422	0.897112	−	0.441803i	$-0.145661\pi$
$113$	19.0729	1.79422	0.897112	+	0.441803i	$0.145661\pi$
$114$	0	0
$115$	−2.93518	−0.273707
$116$	0	0
$117$	−3.17111	−0.293170
$118$	0	0
$119$	12.1588	1.11460
$120$	0	0
$121$	1.87997	0.170906
$122$	0	0
$123$	−0.235962	−0.0212760
$124$	0	0
$125$	−4.06436	−0.363527
$126$	0	0
$127$	16.5571	1.46920	0.734602	−	0.678498i	$-0.237369\pi$
$127$	16.5571	1.46920	0.734602	+	0.678498i	$0.237369\pi$
$128$	0	0
$129$	−8.18346	−0.720513
$130$	0	0
$131$	6.58954	0.575731	0.287865	−	0.957671i	$-0.407054\pi$
$131$	6.58954	0.575731	0.287865	+	0.957671i	$0.407054\pi$
$132$	0	0
$133$	−6.94176	−0.601927
$134$	0	0
$135$	−2.93518	−0.252620
$136$	0	0
$137$	2.66517	0.227701	0.113850	−	0.993498i	$-0.463682\pi$
$137$	2.66517	0.227701	0.113850	+	0.993498i	$0.463682\pi$
$138$	0	0
$139$	15.6879	1.33063	0.665314	−	0.746564i	$-0.268298\pi$
$139$	15.6879	1.33063	0.665314	+	0.746564i	$0.268298\pi$
$140$	0	0
$141$	6.76932	0.570079
$142$	0	0
$143$	11.3807	0.951703
$144$	0	0
$145$	−2.93518	−0.243754
$146$	0	0
$147$	−0.370499	−0.0305583
$148$	0	0
$149$	5.07527	0.415782	0.207891	−	0.978152i	$-0.433340\pi$
$149$	5.07527	0.415782	0.207891	+	0.978152i	$0.433340\pi$
$150$	0	0
$151$	−9.51640	−0.774434	−0.387217	−	0.921989i	$-0.626564\pi$
$151$	−9.51640	−0.774434	−0.387217	+	0.921989i	$0.626564\pi$
$152$	0	0
$153$	4.47862	0.362075
$154$	0	0
$155$	−15.8010	−1.26917
$156$	0	0
$157$	2.74605	0.219158	0.109579	−	0.993978i	$-0.465050\pi$
$157$	2.74605	0.219158	0.109579	+	0.993978i	$0.465050\pi$
$158$	0	0
$159$	2.07077	0.164223
$160$	0	0
$161$	−2.71487	−0.213961
$162$	0	0
$163$	3.29661	0.258210	0.129105	−	0.991631i	$-0.458790\pi$
$163$	3.29661	0.258210	0.129105	+	0.991631i	$0.458790\pi$
$164$	0	0
$165$	10.5340	0.820069
$166$	0	0
$167$	−21.0440	−1.62843	−0.814215	−	0.580563i	$-0.802832\pi$
$167$	−21.0440	−1.62843	−0.814215	+	0.580563i	$0.802832\pi$
$168$	0	0
$169$	−2.94404	−0.226464
$170$	0	0
$171$	−2.55694	−0.195534
$172$	0	0
$173$	14.9462	1.13634	0.568168	−	0.822913i	$-0.307652\pi$
$173$	14.9462	1.13634	0.568168	+	0.822913i	$0.307652\pi$
$174$	0	0
$175$	9.81505	0.741948
$176$	0	0
$177$	−5.67608	−0.426640
$178$	0	0
$179$	−5.34025	−0.399149	−0.199574	−	0.979883i	$-0.563956\pi$
$179$	−5.34025	−0.399149	−0.199574	+	0.979883i	$0.563956\pi$
$180$	0	0
$181$	−20.4669	−1.52129	−0.760645	−	0.649168i	$-0.775117\pi$
$181$	−20.4669	−1.52129	−0.760645	+	0.649168i	$0.775117\pi$
$182$	0	0
$183$	−3.60240	−0.266297
$184$	0	0
$185$	14.8991	1.09540
$186$	0	0
$187$	−16.0732	−1.17539
$188$	0	0
$189$	−2.71487	−0.197477
$190$	0	0
$191$	−23.0846	−1.67034	−0.835172	−	0.549989i	$-0.814632\pi$
$191$	−23.0846	−1.67034	−0.835172	+	0.549989i	$0.814632\pi$
$192$	0	0
$193$	18.8353	1.35580	0.677899	−	0.735155i	$-0.262891\pi$
$193$	18.8353	1.35580	0.677899	+	0.735155i	$0.262891\pi$
$194$	0	0
$195$	9.30780	0.666545
$196$	0	0
$197$	12.0193	0.856336	0.428168	−	0.903699i	$-0.359159\pi$
$197$	12.0193	0.856336	0.428168	+	0.903699i	$0.359159\pi$
$198$	0	0
$199$	−24.0112	−1.70211	−0.851055	−	0.525076i	$-0.824037\pi$
$199$	−24.0112	−1.70211	−0.851055	+	0.525076i	$0.824037\pi$
$200$	0	0
$201$	−2.30488	−0.162574
$202$	0	0
$203$	−2.71487	−0.190546
$204$	0	0
$205$	0.692592	0.0483727
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	9.17653	0.634754
$210$	0	0
$211$	26.2057	1.80408	0.902038	−	0.431656i	$-0.142071\pi$
$211$	26.2057	1.80408	0.902038	+	0.431656i	$0.142071\pi$
$212$	0	0
$213$	−5.54897	−0.380209
$214$	0	0
$215$	24.0199	1.63815
$216$	0	0
$217$	−14.6150	−0.992132
$218$	0	0
$219$	−11.4518	−0.773838
$220$	0	0
$221$	−14.2022	−0.955344
$222$	0	0
$223$	−25.8799	−1.73304	−0.866522	−	0.499138i	$-0.833650\pi$
$223$	−25.8799	−1.73304	−0.866522	+	0.499138i	$0.833650\pi$
$224$	0	0
$225$	3.61530	0.241020
$226$	0	0
$227$	3.84644	0.255297	0.127649	−	0.991819i	$-0.459257\pi$
$227$	3.84644	0.255297	0.127649	+	0.991819i	$0.459257\pi$
$228$	0	0
$229$	−28.1470	−1.86001	−0.930004	−	0.367550i	$-0.880197\pi$
$229$	−28.1470	−1.86001	−0.930004	+	0.367550i	$0.880197\pi$
$230$	0	0
$231$	9.74330	0.641062
$232$	0	0
$233$	−0.131560	−0.00861881	−0.00430941	−	0.999991i	$-0.501372\pi$
$233$	−0.131560	−0.00861881	−0.00430941	+	0.999991i	$0.501372\pi$
$234$	0	0
$235$	−19.8692	−1.29612
$236$	0	0
$237$	−4.16775	−0.270724
$238$	0	0
$239$	−10.7006	−0.692167	−0.346084	−	0.938204i	$-0.612489\pi$
$239$	−10.7006	−0.692167	−0.346084	+	0.938204i	$0.612489\pi$
$240$	0	0
$241$	−7.51815	−0.484286	−0.242143	−	0.970241i	$-0.577850\pi$
$241$	−7.51815	−0.484286	−0.242143	+	0.970241i	$0.577850\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	1.08748	0.0694768
$246$	0	0
$247$	8.10836	0.515923
$248$	0	0
$249$	−6.64646	−0.421202
$250$	0	0
$251$	16.0258	1.01154	0.505769	−	0.862669i	$-0.331209\pi$
$251$	16.0258	1.01154	0.505769	+	0.862669i	$0.331209\pi$
$252$	0	0
$253$	3.58887	0.225630
$254$	0	0
$255$	−13.1456	−0.823207
$256$	0	0
$257$	14.6440	0.913466	0.456733	−	0.889604i	$-0.349020\pi$
$257$	14.6440	0.913466	0.456733	+	0.889604i	$0.349020\pi$
$258$	0	0
$259$	13.7807	0.856294
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	−17.7426	−1.09406	−0.547029	−	0.837114i	$-0.684241\pi$
$263$	−17.7426	−1.09406	−0.547029	+	0.837114i	$0.684241\pi$
$264$	0	0
$265$	−6.07808	−0.373373
$266$	0	0
$267$	−8.84427	−0.541261
$268$	0	0
$269$	7.41764	0.452261	0.226131	−	0.974097i	$-0.427392\pi$
$269$	7.41764	0.452261	0.226131	+	0.974097i	$0.427392\pi$
$270$	0	0
$271$	−4.08324	−0.248039	−0.124020	−	0.992280i	$-0.539579\pi$
$271$	−4.08324	−0.248039	−0.124020	+	0.992280i	$0.539579\pi$
$272$	0	0
$273$	8.60915	0.521050
$274$	0	0
$275$	−12.9748	−0.782411
$276$	0	0
$277$	−14.8936	−0.894868	−0.447434	−	0.894317i	$-0.647662\pi$
$277$	−14.8936	−0.894868	−0.447434	+	0.894317i	$0.647662\pi$
$278$	0	0
$279$	−5.38333	−0.322291
$280$	0	0
$281$	9.09591	0.542617	0.271308	−	0.962492i	$-0.412544\pi$
$281$	9.09591	0.542617	0.271308	+	0.962492i	$0.412544\pi$
$282$	0	0
$283$	24.1814	1.43744	0.718719	−	0.695301i	$-0.244729\pi$
$283$	24.1814	1.43744	0.718719	+	0.695301i	$0.244729\pi$
$284$	0	0
$285$	7.50509	0.444563
$286$	0	0
$287$	0.640605	0.0378137
$288$	0	0
$289$	3.05800	0.179883
$290$	0	0
$291$	−10.2089	−0.598455
$292$	0	0
$293$	17.0939	0.998639	0.499319	−	0.866418i	$-0.333583\pi$
$293$	17.0939	0.998639	0.499319	+	0.866418i	$0.333583\pi$
$294$	0	0
$295$	16.6603	0.970002
$296$	0	0
$297$	3.58887	0.208247
$298$	0	0
$299$	3.17111	0.183390
$300$	0	0
$301$	22.2170	1.28057
$302$	0	0
$303$	−14.0772	−0.808713
$304$	0	0
$305$	10.5737	0.605448
$306$	0	0
$307$	24.2673	1.38501	0.692504	−	0.721414i	$-0.256507\pi$
$307$	24.2673	1.38501	0.692504	+	0.721414i	$0.256507\pi$
$308$	0	0
$309$	−0.536603	−0.0305263
$310$	0	0
$311$	−11.6000	−0.657774	−0.328887	−	0.944369i	$-0.606674\pi$
$311$	−11.6000	−0.657774	−0.328887	+	0.944369i	$0.606674\pi$
$312$	0	0
$313$	3.77413	0.213326	0.106663	−	0.994295i	$-0.465983\pi$
$313$	3.77413	0.213326	0.106663	+	0.994295i	$0.465983\pi$
$314$	0	0
$315$	7.96863	0.448981
$316$	0	0
$317$	−20.3142	−1.14096	−0.570479	−	0.821312i	$-0.693242\pi$
$317$	−20.3142	−1.14096	−0.570479	+	0.821312i	$0.693242\pi$
$318$	0	0
$319$	3.58887	0.200938
$320$	0	0
$321$	−6.01387	−0.335662
$322$	0	0
$323$	−11.4516	−0.637182
$324$	0	0
$325$	−11.4645	−0.635937
$326$	0	0
$327$	−11.0166	−0.609220
$328$	0	0
$329$	−18.3778	−1.01320
$330$	0	0
$331$	34.4015	1.89088	0.945439	−	0.325798i	$-0.105633\pi$
$331$	34.4015	1.89088	0.945439	+	0.325798i	$0.105633\pi$
$332$	0	0
$333$	5.07603	0.278165
$334$	0	0
$335$	6.76526	0.369626
$336$	0	0
$337$	−3.68666	−0.200825	−0.100413	−	0.994946i	$-0.532016\pi$
$337$	−3.68666	−0.200825	−0.100413	+	0.994946i	$0.532016\pi$
$338$	0	0
$339$	−19.0729	−1.03590
$340$	0	0
$341$	19.3200	1.04624
$342$	0	0
$343$	−17.9982	−0.971812
$344$	0	0
$345$	2.93518	0.158025
$346$	0	0
$347$	3.11968	0.167473	0.0837365	−	0.996488i	$-0.473315\pi$
$347$	3.11968	0.167473	0.0837365	+	0.996488i	$0.473315\pi$
$348$	0	0
$349$	−19.5424	−1.04608	−0.523039	−	0.852309i	$-0.675202\pi$
$349$	−19.5424	−1.04608	−0.523039	+	0.852309i	$0.675202\pi$
$350$	0	0
$351$	3.17111	0.169262
$352$	0	0
$353$	−16.0982	−0.856821	−0.428410	−	0.903584i	$-0.640926\pi$
$353$	−16.0982	−0.856821	−0.428410	+	0.903584i	$0.640926\pi$
$354$	0	0
$355$	16.2872	0.864437
$356$	0	0
$357$	−12.1588	−0.643514
$358$	0	0
$359$	11.0469	0.583034	0.291517	−	0.956566i	$-0.405840\pi$
$359$	11.0469	0.583034	0.291517	+	0.956566i	$0.405840\pi$
$360$	0	0
$361$	−12.4620	−0.655897
$362$	0	0
$363$	−1.87997	−0.0986728
$364$	0	0
$365$	33.6130	1.75938
$366$	0	0
$367$	17.0291	0.888913	0.444456	−	0.895801i	$-0.353397\pi$
$367$	17.0291	0.888913	0.444456	+	0.895801i	$0.353397\pi$
$368$	0	0
$369$	0.235962	0.0122837
$370$	0	0
$371$	−5.62186	−0.291872
$372$	0	0
$373$	23.7067	1.22749	0.613744	−	0.789505i	$-0.289663\pi$
$373$	23.7067	1.22749	0.613744	+	0.789505i	$0.289663\pi$
$374$	0	0
$375$	4.06436	0.209883
$376$	0	0
$377$	3.17111	0.163321
$378$	0	0
$379$	−28.2242	−1.44978	−0.724890	−	0.688865i	$-0.758109\pi$
$379$	−28.2242	−1.44978	−0.724890	+	0.688865i	$0.758109\pi$
$380$	0	0
$381$	−16.5571	−0.848246
$382$	0	0
$383$	5.55185	0.283686	0.141843	−	0.989889i	$-0.454697\pi$
$383$	5.55185	0.283686	0.141843	+	0.989889i	$0.454697\pi$
$384$	0	0
$385$	−28.5983	−1.45751
$386$	0	0
$387$	8.18346	0.415989
$388$	0	0
$389$	26.5610	1.34669	0.673347	−	0.739327i	$-0.264856\pi$
$389$	26.5610	1.34669	0.673347	+	0.739327i	$0.264856\pi$
$390$	0	0
$391$	−4.47862	−0.226493
$392$	0	0
$393$	−6.58954	−0.332398
$394$	0	0
$395$	12.2331	0.615514
$396$	0	0
$397$	−18.0667	−0.906740	−0.453370	−	0.891322i	$-0.649778\pi$
$397$	−18.0667	−0.906740	−0.453370	+	0.891322i	$0.649778\pi$
$398$	0	0
$399$	6.94176	0.347523
$400$	0	0
$401$	7.62011	0.380530	0.190265	−	0.981733i	$-0.439065\pi$
$401$	7.62011	0.380530	0.190265	+	0.981733i	$0.439065\pi$
$402$	0	0
$403$	17.0711	0.850374
$404$	0	0
$405$	2.93518	0.145850
$406$	0	0
$407$	−18.2172	−0.902993
$408$	0	0
$409$	−15.6778	−0.775218	−0.387609	−	0.921824i	$-0.626699\pi$
$409$	−15.6778	−0.775218	−0.387609	+	0.921824i	$0.626699\pi$
$410$	0	0
$411$	−2.66517	−0.131463
$412$	0	0
$413$	15.4098	0.758267
$414$	0	0
$415$	19.5086	0.957638
$416$	0	0
$417$	−15.6879	−0.768239
$418$	0	0
$419$	−32.3822	−1.58197	−0.790987	−	0.611833i	$-0.790432\pi$
$419$	−32.3822	−1.58197	−0.790987	+	0.611833i	$0.790432\pi$
$420$	0	0
$421$	7.08743	0.345420	0.172710	−	0.984973i	$-0.444748\pi$
$421$	7.08743	0.345420	0.172710	+	0.984973i	$0.444748\pi$
$422$	0	0
$423$	−6.76932	−0.329136
$424$	0	0
$425$	16.1915	0.785404
$426$	0	0
$427$	9.78003	0.473289
$428$	0	0
$429$	−11.3807	−0.549466
$430$	0	0
$431$	5.12783	0.246999	0.123499	−	0.992345i	$-0.460588\pi$
$431$	5.12783	0.246999	0.123499	+	0.992345i	$0.460588\pi$
$432$	0	0
$433$	−16.8665	−0.810550	−0.405275	−	0.914195i	$-0.632824\pi$
$433$	−16.8665	−0.810550	−0.405275	+	0.914195i	$0.632824\pi$
$434$	0	0
$435$	2.93518	0.140731
$436$	0	0
$437$	2.55694	0.122315
$438$	0	0
$439$	−25.1420	−1.19996	−0.599981	−	0.800014i	$-0.704825\pi$
$439$	−25.1420	−1.19996	−0.599981	+	0.800014i	$0.704825\pi$
$440$	0	0
$441$	0.370499	0.0176428
$442$	0	0
$443$	4.20154	0.199621	0.0998105	−	0.995006i	$-0.468176\pi$
$443$	4.20154	0.199621	0.0998105	+	0.995006i	$0.468176\pi$
$444$	0	0
$445$	25.9595	1.23060
$446$	0	0
$447$	−5.07527	−0.240052
$448$	0	0
$449$	13.0873	0.617629	0.308815	−	0.951122i	$-0.400068\pi$
$449$	13.0873	0.617629	0.308815	+	0.951122i	$0.400068\pi$
$450$	0	0
$451$	−0.846836	−0.0398760
$452$	0	0
$453$	9.51640	0.447120
$454$	0	0
$455$	−25.2694	−1.18465
$456$	0	0
$457$	−2.16189	−0.101129	−0.0505645	−	0.998721i	$-0.516102\pi$
$457$	−2.16189	−0.101129	−0.0505645	+	0.998721i	$0.516102\pi$
$458$	0	0
$459$	−4.47862	−0.209044
$460$	0	0
$461$	3.71289	0.172926	0.0864632	−	0.996255i	$-0.472443\pi$
$461$	3.71289	0.172926	0.0864632	+	0.996255i	$0.472443\pi$
$462$	0	0
$463$	−3.73581	−0.173618	−0.0868089	−	0.996225i	$-0.527667\pi$
$463$	−3.73581	−0.173618	−0.0868089	+	0.996225i	$0.527667\pi$
$464$	0	0
$465$	15.8010	0.732756
$466$	0	0
$467$	2.01144	0.0930785	0.0465393	−	0.998916i	$-0.485181\pi$
$467$	2.01144	0.0930785	0.0465393	+	0.998916i	$0.485181\pi$
$468$	0	0
$469$	6.25745	0.288942
$470$	0	0
$471$	−2.74605	−0.126531
$472$	0	0
$473$	−29.3693	−1.35040
$474$	0	0
$475$	−9.24411	−0.424149
$476$	0	0
$477$	−2.07077	−0.0948139
$478$	0	0
$479$	16.2844	0.744053	0.372026	−	0.928222i	$-0.378663\pi$
$479$	16.2844	0.744053	0.372026	+	0.928222i	$0.378663\pi$
$480$	0	0
$481$	−16.0967	−0.733945
$482$	0	0
$483$	2.71487	0.123531
$484$	0	0
$485$	29.9649	1.36064
$486$	0	0
$487$	−14.4120	−0.653069	−0.326534	−	0.945185i	$-0.605881\pi$
$487$	−14.4120	−0.653069	−0.326534	+	0.945185i	$0.605881\pi$
$488$	0	0
$489$	−3.29661	−0.149078
$490$	0	0
$491$	−11.8871	−0.536458	−0.268229	−	0.963355i	$-0.586438\pi$
$491$	−11.8871	−0.536458	−0.268229	+	0.963355i	$0.586438\pi$
$492$	0	0
$493$	−4.47862	−0.201707
$494$	0	0
$495$	−10.5340	−0.473467
$496$	0	0
$497$	15.0647	0.675745
$498$	0	0
$499$	−9.17119	−0.410559	−0.205279	−	0.978703i	$-0.565810\pi$
$499$	−9.17119	−0.410559	−0.205279	+	0.978703i	$0.565810\pi$
$500$	0	0
$501$	21.0440	0.940175
$502$	0	0
$503$	25.2300	1.12495	0.562474	−	0.826815i	$-0.309849\pi$
$503$	25.2300	1.12495	0.562474	+	0.826815i	$0.309849\pi$
$504$	0	0
$505$	41.3191	1.83868
$506$	0	0
$507$	2.94404	0.130749
$508$	0	0
$509$	2.94323	0.130456	0.0652282	−	0.997870i	$-0.479222\pi$
$509$	2.94323	0.130456	0.0652282	+	0.997870i	$0.479222\pi$
$510$	0	0
$511$	31.0900	1.37534
$512$	0	0
$513$	2.55694	0.112892
$514$	0	0
$515$	1.57503	0.0694040
$516$	0	0
$517$	24.2942	1.06846
$518$	0	0
$519$	−14.9462	−0.656064
$520$	0	0
$521$	−13.2602	−0.580939	−0.290469	−	0.956884i	$-0.593811\pi$
$521$	−13.2602	−0.580939	−0.290469	+	0.956884i	$0.593811\pi$
$522$	0	0
$523$	−1.65072	−0.0721809	−0.0360904	−	0.999349i	$-0.511490\pi$
$523$	−1.65072	−0.0721809	−0.0360904	+	0.999349i	$0.511490\pi$
$524$	0	0
$525$	−9.81505	−0.428364
$526$	0	0
$527$	−24.1099	−1.05024
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	5.67608	0.246321
$532$	0	0
$533$	−0.748262	−0.0324108
$534$	0	0
$535$	17.6518	0.763154
$536$	0	0
$537$	5.34025	0.230449
$538$	0	0
$539$	−1.32967	−0.0572731
$540$	0	0
$541$	−0.351096	−0.0150948	−0.00754740	−	0.999972i	$-0.502402\pi$
$541$	−0.351096	−0.0150948	−0.00754740	+	0.999972i	$0.502402\pi$
$542$	0	0
$543$	20.4669	0.878317
$544$	0	0
$545$	32.3357	1.38511
$546$	0	0
$547$	−0.518654	−0.0221760	−0.0110880	−	0.999939i	$-0.503530\pi$
$547$	−0.518654	−0.0221760	−0.0110880	+	0.999939i	$0.503530\pi$
$548$	0	0
$549$	3.60240	0.153747
$550$	0	0
$551$	2.55694	0.108929
$552$	0	0
$553$	11.3149	0.481157
$554$	0	0
$555$	−14.8991	−0.632430
$556$	0	0
$557$	23.0357	0.976056	0.488028	−	0.872828i	$-0.337716\pi$
$557$	23.0357	0.976056	0.488028	+	0.872828i	$0.337716\pi$
$558$	0	0
$559$	−25.9507	−1.09760
$560$	0	0
$561$	16.0732	0.678609
$562$	0	0
$563$	25.4451	1.07238	0.536191	−	0.844097i	$-0.319863\pi$
$563$	25.4451	1.07238	0.536191	+	0.844097i	$0.319863\pi$
$564$	0	0
$565$	55.9823	2.35519
$566$	0	0
$567$	2.71487	0.114014
$568$	0	0
$569$	−0.429499	−0.0180055	−0.00900276	−	0.999959i	$-0.502866\pi$
$569$	−0.429499	−0.0180055	−0.00900276	+	0.999959i	$0.502866\pi$
$570$	0	0
$571$	−40.2698	−1.68524	−0.842619	−	0.538510i	$-0.818988\pi$
$571$	−40.2698	−1.68524	−0.842619	+	0.538510i	$0.818988\pi$
$572$	0	0
$573$	23.0846	0.964373
$574$	0	0
$575$	−3.61530	−0.150768
$576$	0	0
$577$	29.7628	1.23904	0.619520	−	0.784981i	$-0.287327\pi$
$577$	29.7628	1.23904	0.619520	+	0.784981i	$0.287327\pi$
$578$	0	0
$579$	−18.8353	−0.782770
$580$	0	0
$581$	18.0442	0.748601
$582$	0	0
$583$	7.43171	0.307790
$584$	0	0
$585$	−9.30780	−0.384830
$586$	0	0
$587$	−3.66570	−0.151300	−0.0756498	−	0.997134i	$-0.524103\pi$
$587$	−3.66570	−0.151300	−0.0756498	+	0.997134i	$0.524103\pi$
$588$	0	0
$589$	13.7649	0.567171
$590$	0	0
$591$	−12.0193	−0.494406
$592$	0	0
$593$	41.1114	1.68824	0.844122	−	0.536151i	$-0.180122\pi$
$593$	41.1114	1.68824	0.844122	+	0.536151i	$0.180122\pi$
$594$	0	0
$595$	35.6884	1.46308
$596$	0	0
$597$	24.0112	0.982714
$598$	0	0
$599$	2.21185	0.0903736	0.0451868	−	0.998979i	$-0.485612\pi$
$599$	2.21185	0.0903736	0.0451868	+	0.998979i	$0.485612\pi$
$600$	0	0
$601$	−26.1361	−1.06612	−0.533058	−	0.846079i	$-0.678957\pi$
$601$	−26.1361	−1.06612	−0.533058	+	0.846079i	$0.678957\pi$
$602$	0	0
$603$	2.30488	0.0938622
$604$	0	0
$605$	5.51805	0.224341
$606$	0	0
$607$	39.4865	1.60271	0.801353	−	0.598192i	$-0.204114\pi$
$607$	39.4865	1.60271	0.801353	+	0.598192i	$0.204114\pi$
$608$	0	0
$609$	2.71487	0.110012
$610$	0	0
$611$	21.4663	0.868433
$612$	0	0
$613$	43.6230	1.76192	0.880958	−	0.473194i	$-0.156899\pi$
$613$	43.6230	1.76192	0.880958	+	0.473194i	$0.156899\pi$
$614$	0	0
$615$	−0.692592	−0.0279280
$616$	0	0
$617$	−41.5910	−1.67439	−0.837195	−	0.546904i	$-0.815806\pi$
$617$	−41.5910	−1.67439	−0.837195	+	0.546904i	$0.815806\pi$
$618$	0	0
$619$	−27.2209	−1.09410	−0.547050	−	0.837100i	$-0.684249\pi$
$619$	−27.2209	−1.09410	−0.547050	+	0.837100i	$0.684249\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	24.0110	0.961981
$624$	0	0
$625$	−30.0061	−1.20024
$626$	0	0
$627$	−9.17653	−0.366475
$628$	0	0
$629$	22.7336	0.906448
$630$	0	0
$631$	38.7594	1.54299	0.771493	−	0.636238i	$-0.219510\pi$
$631$	38.7594	1.54299	0.771493	+	0.636238i	$0.219510\pi$
$632$	0	0
$633$	−26.2057	−1.04158
$634$	0	0
$635$	48.5981	1.92856
$636$	0	0
$637$	−1.17490	−0.0465511
$638$	0	0
$639$	5.54897	0.219514
$640$	0	0
$641$	7.90994	0.312424	0.156212	−	0.987724i	$-0.450072\pi$
$641$	7.90994	0.312424	0.156212	+	0.987724i	$0.450072\pi$
$642$	0	0
$643$	16.2425	0.640543	0.320272	−	0.947326i	$-0.396226\pi$
$643$	16.2425	0.640543	0.320272	+	0.947326i	$0.396226\pi$
$644$	0	0
$645$	−24.0199	−0.945784
$646$	0	0
$647$	−25.9612	−1.02064	−0.510319	−	0.859985i	$-0.670473\pi$
$647$	−25.9612	−1.02064	−0.510319	+	0.859985i	$0.670473\pi$
$648$	0	0
$649$	−20.3707	−0.799620
$650$	0	0
$651$	14.6150	0.572808
$652$	0	0
$653$	17.4622	0.683348	0.341674	−	0.939818i	$-0.389006\pi$
$653$	17.4622	0.683348	0.341674	+	0.939818i	$0.389006\pi$
$654$	0	0
$655$	19.3415	0.755735
$656$	0	0
$657$	11.4518	0.446775
$658$	0	0
$659$	−10.6803	−0.416045	−0.208023	−	0.978124i	$-0.566703\pi$
$659$	−10.6803	−0.416045	−0.208023	+	0.978124i	$0.566703\pi$
$660$	0	0
$661$	17.2518	0.671016	0.335508	−	0.942037i	$-0.391092\pi$
$661$	17.2518	0.671016	0.335508	+	0.942037i	$0.391092\pi$
$662$	0	0
$663$	14.2022	0.551568
$664$	0	0
$665$	−20.3753	−0.790121
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	25.8799	1.00057
$670$	0	0
$671$	−12.9285	−0.499100
$672$	0	0
$673$	−21.3349	−0.822402	−0.411201	−	0.911545i	$-0.634891\pi$
$673$	−21.3349	−0.822402	−0.411201	+	0.911545i	$0.634891\pi$
$674$	0	0
$675$	−3.61530	−0.139153
$676$	0	0
$677$	−50.7006	−1.94858	−0.974291	−	0.225294i	$-0.927666\pi$
$677$	−50.7006	−1.94858	−0.974291	+	0.225294i	$0.927666\pi$
$678$	0	0
$679$	27.7157	1.06363
$680$	0	0
$681$	−3.84644	−0.147396
$682$	0	0
$683$	10.1622	0.388847	0.194424	−	0.980918i	$-0.437716\pi$
$683$	10.1622	0.388847	0.194424	+	0.980918i	$0.437716\pi$
$684$	0	0
$685$	7.82276	0.298892
$686$	0	0
$687$	28.1470	1.07388
$688$	0	0
$689$	6.56664	0.250169
$690$	0	0
$691$	17.4204	0.662703	0.331351	−	0.943507i	$-0.392495\pi$
$691$	17.4204	0.662703	0.331351	+	0.943507i	$0.392495\pi$
$692$	0	0
$693$	−9.74330	−0.370117
$694$	0	0
$695$	46.0468	1.74665
$696$	0	0
$697$	1.05678	0.0400285
$698$	0	0
$699$	0.131560	0.00497607
$700$	0	0
$701$	2.85603	0.107871	0.0539354	−	0.998544i	$-0.482824\pi$
$701$	2.85603	0.107871	0.0539354	+	0.998544i	$0.482824\pi$
$702$	0	0
$703$	−12.9791	−0.489517
$704$	0	0
$705$	19.8692	0.748317
$706$	0	0
$707$	38.2177	1.43732
$708$	0	0
$709$	25.5780	0.960604	0.480302	−	0.877103i	$-0.340527\pi$
$709$	25.5780	0.960604	0.480302	+	0.877103i	$0.340527\pi$
$710$	0	0
$711$	4.16775	0.156303
$712$	0	0
$713$	5.38333	0.201607
$714$	0	0
$715$	33.4045	1.24926
$716$	0	0
$717$	10.7006	0.399623
$718$	0	0
$719$	−43.6869	−1.62925	−0.814624	−	0.579990i	$-0.803057\pi$
$719$	−43.6869	−1.62925	−0.814624	+	0.579990i	$0.803057\pi$
$720$	0	0
$721$	1.45680	0.0542542
$722$	0	0
$723$	7.51815	0.279603
$724$	0	0
$725$	−3.61530	−0.134269
$726$	0	0
$727$	−29.7216	−1.10231	−0.551156	−	0.834402i	$-0.685813\pi$
$727$	−29.7216	−1.10231	−0.551156	+	0.834402i	$0.685813\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	36.6506	1.35557
$732$	0	0
$733$	−10.8570	−0.401012	−0.200506	−	0.979692i	$-0.564259\pi$
$733$	−10.8570	−0.401012	−0.200506	+	0.979692i	$0.564259\pi$
$734$	0	0
$735$	−1.08748	−0.0401124
$736$	0	0
$737$	−8.27193	−0.304700
$738$	0	0
$739$	1.35269	0.0497594	0.0248797	−	0.999690i	$-0.492080\pi$
$739$	1.35269	0.0497594	0.0248797	+	0.999690i	$0.492080\pi$
$740$	0	0
$741$	−8.10836	−0.297868
$742$	0	0
$743$	−32.4723	−1.19129	−0.595647	−	0.803246i	$-0.703104\pi$
$743$	−32.4723	−1.19129	−0.595647	+	0.803246i	$0.703104\pi$
$744$	0	0
$745$	14.8968	0.545778
$746$	0	0
$747$	6.64646	0.243181
$748$	0	0
$749$	16.3269	0.596570
$750$	0	0
$751$	−9.79509	−0.357428	−0.178714	−	0.983901i	$-0.557194\pi$
$751$	−9.79509	−0.357428	−0.178714	+	0.983901i	$0.557194\pi$
$752$	0	0
$753$	−16.0258	−0.584012
$754$	0	0
$755$	−27.9324	−1.01656
$756$	0	0
$757$	−28.0716	−1.02028	−0.510140	−	0.860092i	$-0.670406\pi$
$757$	−28.0716	−1.02028	−0.510140	+	0.860092i	$0.670406\pi$
$758$	0	0
$759$	−3.58887	−0.130268
$760$	0	0
$761$	18.6002	0.674257	0.337129	−	0.941459i	$-0.390544\pi$
$761$	18.6002	0.674257	0.337129	+	0.941459i	$0.390544\pi$
$762$	0	0
$763$	29.9086	1.08276
$764$	0	0
$765$	13.1456	0.475279
$766$	0	0
$767$	−17.9995	−0.649924
$768$	0	0
$769$	−36.7281	−1.32445	−0.662224	−	0.749306i	$-0.730387\pi$
$769$	−36.7281	−1.32445	−0.662224	+	0.749306i	$0.730387\pi$
$770$	0	0
$771$	−14.6440	−0.527390
$772$	0	0
$773$	22.6706	0.815406	0.407703	−	0.913115i	$-0.366330\pi$
$773$	22.6706	0.815406	0.407703	+	0.913115i	$0.366330\pi$
$774$	0	0
$775$	−19.4623	−0.699107
$776$	0	0
$777$	−13.7807	−0.494381
$778$	0	0
$779$	−0.603341	−0.0216169
$780$	0	0
$781$	−19.9145	−0.712598
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	8.06015	0.287679
$786$	0	0
$787$	−6.45810	−0.230206	−0.115103	−	0.993354i	$-0.536720\pi$
$787$	−6.45810	−0.230206	−0.115103	+	0.993354i	$0.536720\pi$
$788$	0	0
$789$	17.7426	0.631654
$790$	0	0
$791$	51.7803	1.84109
$792$	0	0
$793$	−11.4236	−0.405664
$794$	0	0
$795$	6.07808	0.215567
$796$	0	0
$797$	7.31648	0.259163	0.129581	−	0.991569i	$-0.458637\pi$
$797$	7.31648	0.259163	0.129581	+	0.991569i	$0.458637\pi$
$798$	0	0
$799$	−30.3172	−1.07254
$800$	0	0
$801$	8.84427	0.312497
$802$	0	0
$803$	−41.0988	−1.45035
$804$	0	0
$805$	−7.96863	−0.280857
$806$	0	0
$807$	−7.41764	−0.261113
$808$	0	0
$809$	−18.0091	−0.633167	−0.316584	−	0.948565i	$-0.602536\pi$
$809$	−18.0091	−0.633167	−0.316584	+	0.948565i	$0.602536\pi$
$810$	0	0
$811$	−34.1144	−1.19792	−0.598960	−	0.800779i	$-0.704419\pi$
$811$	−34.1144	−1.19792	−0.598960	+	0.800779i	$0.704419\pi$
$812$	0	0
$813$	4.08324	0.143205
$814$	0	0
$815$	9.67614	0.338941
$816$	0	0
$817$	−20.9246	−0.732060
$818$	0	0
$819$	−8.60915	−0.300828
$820$	0	0
$821$	22.2717	0.777288	0.388644	−	0.921388i	$-0.372944\pi$
$821$	22.2717	0.777288	0.388644	+	0.921388i	$0.372944\pi$
$822$	0	0
$823$	−15.0073	−0.523121	−0.261561	−	0.965187i	$-0.584237\pi$
$823$	−15.0073	−0.523121	−0.261561	+	0.965187i	$0.584237\pi$
$824$	0	0
$825$	12.9748	0.451725
$826$	0	0
$827$	−1.35082	−0.0469724	−0.0234862	−	0.999724i	$-0.507477\pi$
$827$	−1.35082	−0.0469724	−0.0234862	+	0.999724i	$0.507477\pi$
$828$	0	0
$829$	33.0917	1.14932	0.574662	−	0.818391i	$-0.305134\pi$
$829$	33.0917	1.14932	0.574662	+	0.818391i	$0.305134\pi$
$830$	0	0
$831$	14.8936	0.516652
$832$	0	0
$833$	1.65932	0.0574922
$834$	0	0
$835$	−61.7678	−2.13756
$836$	0	0
$837$	5.38333	0.186075
$838$	0	0
$839$	−46.9173	−1.61977	−0.809883	−	0.586592i	$-0.800469\pi$
$839$	−46.9173	−1.61977	−0.809883	+	0.586592i	$0.800469\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−9.09591	−0.313280
$844$	0	0
$845$	−8.64128	−0.297269
$846$	0	0
$847$	5.10386	0.175371
$848$	0	0
$849$	−24.1814	−0.829905
$850$	0	0
$851$	−5.07603	−0.174004
$852$	0	0
$853$	29.2457	1.00135	0.500677	−	0.865634i	$-0.333085\pi$
$853$	29.2457	1.00135	0.500677	+	0.865634i	$0.333085\pi$
$854$	0	0
$855$	−7.50509	−0.256669
$856$	0	0
$857$	50.3916	1.72134	0.860672	−	0.509160i	$-0.170044\pi$
$857$	50.3916	1.72134	0.860672	+	0.509160i	$0.170044\pi$
$858$	0	0
$859$	44.4056	1.51510	0.757550	−	0.652778i	$-0.226396\pi$
$859$	44.4056	1.51510	0.757550	+	0.652778i	$0.226396\pi$
$860$	0	0
$861$	−0.640605	−0.0218318
$862$	0	0
$863$	41.1402	1.40043	0.700214	−	0.713933i	$-0.253088\pi$
$863$	41.1402	1.40043	0.700214	+	0.713933i	$0.253088\pi$
$864$	0	0
$865$	43.8697	1.49162
$866$	0	0
$867$	−3.05800	−0.103855
$868$	0	0
$869$	−14.9575	−0.507398
$870$	0	0
$871$	−7.30905	−0.247658
$872$	0	0
$873$	10.2089	0.345518
$874$	0	0
$875$	−11.0342	−0.373024
$876$	0	0
$877$	−29.9193	−1.01030	−0.505152	−	0.863030i	$-0.668564\pi$
$877$	−29.9193	−1.01030	−0.505152	+	0.863030i	$0.668564\pi$
$878$	0	0
$879$	−17.0939	−0.576564
$880$	0	0
$881$	24.6463	0.830355	0.415178	−	0.909740i	$-0.363719\pi$
$881$	24.6463	0.830355	0.415178	+	0.909740i	$0.363719\pi$
$882$	0	0
$883$	36.3717	1.22401	0.612003	−	0.790856i	$-0.290364\pi$
$883$	36.3717	1.22401	0.612003	+	0.790856i	$0.290364\pi$
$884$	0	0
$885$	−16.6603	−0.560031
$886$	0	0
$887$	−30.0299	−1.00831	−0.504153	−	0.863614i	$-0.668195\pi$
$887$	−30.0299	−1.00831	−0.504153	+	0.863614i	$0.668195\pi$
$888$	0	0
$889$	44.9503	1.50758
$890$	0	0
$891$	−3.58887	−0.120232
$892$	0	0
$893$	17.3088	0.579216
$894$	0	0
$895$	−15.6746	−0.523944
$896$	0	0
$897$	−3.17111	−0.105880
$898$	0	0
$899$	5.38333	0.179544
$900$	0	0
$901$	−9.27417	−0.308967
$902$	0	0
$903$	−22.2170	−0.739335
$904$	0	0
$905$	−60.0740	−1.99693
$906$	0	0
$907$	−49.2448	−1.63515	−0.817574	−	0.575824i	$-0.804681\pi$
$907$	−49.2448	−1.63515	−0.817574	+	0.575824i	$0.804681\pi$
$908$	0	0
$909$	14.0772	0.466911
$910$	0	0
$911$	28.2619	0.936358	0.468179	−	0.883634i	$-0.344910\pi$
$911$	28.2619	0.936358	0.468179	+	0.883634i	$0.344910\pi$
$912$	0	0
$913$	−23.8532	−0.789427
$914$	0	0
$915$	−10.5737	−0.349555
$916$	0	0
$917$	17.8897	0.590770
$918$	0	0
$919$	−42.1139	−1.38921	−0.694604	−	0.719392i	$-0.744421\pi$
$919$	−42.1139	−1.38921	−0.694604	+	0.719392i	$0.744421\pi$
$920$	0	0
$921$	−24.2673	−0.799635
$922$	0	0
$923$	−17.5964	−0.579193
$924$	0	0
$925$	18.3513	0.603389
$926$	0	0
$927$	0.536603	0.0176243
$928$	0	0
$929$	45.4642	1.49163	0.745817	−	0.666151i	$-0.232059\pi$
$929$	45.4642	1.49163	0.745817	+	0.666151i	$0.232059\pi$
$930$	0	0
$931$	−0.947346	−0.0310480
$932$	0	0
$933$	11.6000	0.379766
$934$	0	0
$935$	−47.1777	−1.54287
$936$	0	0
$937$	−30.3802	−0.992478	−0.496239	−	0.868186i	$-0.665286\pi$
$937$	−30.3802	−0.992478	−0.496239	+	0.868186i	$0.665286\pi$
$938$	0	0
$939$	−3.77413	−0.123164
$940$	0	0
$941$	16.4577	0.536505	0.268253	−	0.963349i	$-0.413554\pi$
$941$	16.4577	0.536505	0.268253	+	0.963349i	$0.413554\pi$
$942$	0	0
$943$	−0.235962	−0.00768398
$944$	0	0
$945$	−7.96863	−0.259219
$946$	0	0
$947$	59.6861	1.93954	0.969768	−	0.244027i	$-0.0784686\pi$
$947$	59.6861	1.93954	0.969768	+	0.244027i	$0.0784686\pi$
$948$	0	0
$949$	−36.3148	−1.17883
$950$	0	0
$951$	20.3142	0.658732
$952$	0	0
$953$	18.7179	0.606331	0.303165	−	0.952938i	$-0.401957\pi$
$953$	18.7179	0.606331	0.303165	+	0.952938i	$0.401957\pi$
$954$	0	0
$955$	−67.7575	−2.19258
$956$	0	0
$957$	−3.58887	−0.116012
$958$	0	0
$959$	7.23558	0.233649
$960$	0	0
$961$	−2.01979	−0.0651546
$962$	0	0
$963$	6.01387	0.193794
$964$	0	0
$965$	55.2852	1.77969
$966$	0	0
$967$	−11.1343	−0.358054	−0.179027	−	0.983844i	$-0.557295\pi$
$967$	−11.1343	−0.358054	−0.179027	+	0.983844i	$0.557295\pi$
$968$	0	0
$969$	11.4516	0.367877
$970$	0	0
$971$	−54.6824	−1.75484	−0.877420	−	0.479723i	$-0.840737\pi$
$971$	−54.6824	−1.75484	−0.877420	+	0.479723i	$0.840737\pi$
$972$	0	0
$973$	42.5905	1.36539
$974$	0	0
$975$	11.4645	0.367158
$976$	0	0
$977$	4.47111	0.143043	0.0715217	−	0.997439i	$-0.477214\pi$
$977$	4.47111	0.143043	0.0715217	+	0.997439i	$0.477214\pi$
$978$	0	0
$979$	−31.7409	−1.01444
$980$	0	0
$981$	11.0166	0.351733
$982$	0	0
$983$	−31.4118	−1.00188	−0.500940	−	0.865482i	$-0.667012\pi$
$983$	−31.4118	−1.00188	−0.500940	+	0.865482i	$0.667012\pi$
$984$	0	0
$985$	35.2787	1.12407
$986$	0	0
$987$	18.3778	0.584972
$988$	0	0
$989$	−8.18346	−0.260219
$990$	0	0
$991$	39.0390	1.24011	0.620057	−	0.784557i	$-0.287110\pi$
$991$	39.0390	1.24011	0.620057	+	0.784557i	$0.287110\pi$
$992$	0	0
$993$	−34.4015	−1.09170
$994$	0	0
$995$	−70.4773	−2.23428
$996$	0	0
$997$	12.9729	0.410855	0.205428	−	0.978672i	$-0.434141\pi$
$997$	12.9729	0.410855	0.205428	+	0.978672i	$0.434141\pi$
$998$	0	0
$999$	−5.07603	−0.160598

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.h.1.12	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.h.1.12	✓	13	1.1	even	1	trivial

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.93518 q^{5} +2.71487 q^{7} +1.00000 q^{9} -3.58887 q^{11} -3.17111 q^{13} -2.93518 q^{15} +4.47862 q^{17} -2.55694 q^{19} -2.71487 q^{21} -1.00000 q^{23} +3.61530 q^{25} -1.00000 q^{27} -1.00000 q^{29} -5.38333 q^{31} +3.58887 q^{33} +7.96863 q^{35} +5.07603 q^{37} +3.17111 q^{39} +0.235962 q^{41} +8.18346 q^{43} +2.93518 q^{45} -6.76932 q^{47} +0.370499 q^{49} -4.47862 q^{51} -2.07077 q^{53} -10.5340 q^{55} +2.55694 q^{57} +5.67608 q^{59} +3.60240 q^{61} +2.71487 q^{63} -9.30780 q^{65} +2.30488 q^{67} +1.00000 q^{69} +5.54897 q^{71} +11.4518 q^{73} -3.61530 q^{75} -9.74330 q^{77} +4.16775 q^{79} +1.00000 q^{81} +6.64646 q^{83} +13.1456 q^{85} +1.00000 q^{87} +8.84427 q^{89} -8.60915 q^{91} +5.38333 q^{93} -7.50509 q^{95} +10.2089 q^{97} -3.58887 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 13 q^{3} + 5 q^{5} - 8 q^{7} + 13 q^{9} + q^{11} + q^{13} - 5 q^{15} - 2 q^{17} - 10 q^{19} + 8 q^{21} - 13 q^{23} + 14 q^{25} - 13 q^{27} - 13 q^{29} - 26 q^{31} - q^{33} + 19 q^{35} + 15 q^{37}+ \cdots + q^{99}+O(q^{100}) \) 13 * q - 13 * q^3 + 5 * q^5 - 8 * q^7 + 13 * q^9 + q^11 + q^13 - 5 * q^15 - 2 * q^17 - 10 * q^19 + 8 * q^21 - 13 * q^23 + 14 * q^25 - 13 * q^27 - 13 * q^29 - 26 * q^31 - q^33 + 19 * q^35 + 15 * q^37 - q^39 + 21 * q^41 - 6 * q^43 + 5 * q^45 + 16 * q^47 + 19 * q^49 + 2 * q^51 + 7 * q^53 + 15 * q^55 + 10 * q^57 - 11 * q^59 + 19 * q^61 - 8 * q^63 + 6 * q^65 - 13 * q^67 + 13 * q^69 + 9 * q^73 - 14 * q^75 + 10 * q^77 - 25 * q^79 + 13 * q^81 + 3 * q^83 + 14 * q^85 + 13 * q^87 + 23 * q^89 + 19 * q^91 + 26 * q^93 + 7 * q^95 + 25 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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