LMFDB - Embedded newform 3808.2.a.h.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3808,2,Mod(1,3808)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3808, 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("3808.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3808 = 2^{5} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3808.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-4,0,4,0,-6] 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:	$30.4070330897$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.109859312.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 7x^{4} + 15x^{3} + 13x^{2} - 9x - 2 $ x^6 - 3x^5 - 7x^4 + 15x^3 + 13x^2 - 9*x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.04046$ of defining polynomial
Character	$\chi$	$=$	3808.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73670 q^{3} -1.54368 q^{5} -1.00000 q^{7} +0.0161353 q^{9} -3.07643 q^{11} +5.13506 q^{13} -2.68091 q^{15} -1.00000 q^{17} +2.95679 q^{19} -1.73670 q^{21} +0.956789 q^{23} -2.61706 q^{25} -5.18209 q^{27} +4.83715 q^{29} -3.98836 q^{31} -5.34284 q^{33} +1.54368 q^{35} -5.56431 q^{37} +8.91807 q^{39} -3.34872 q^{41} -1.96538 q^{43} -0.0249077 q^{45} -4.46891 q^{47} +1.00000 q^{49} -1.73670 q^{51} -7.31576 q^{53} +4.74901 q^{55} +5.13506 q^{57} -0.494325 q^{59} +2.23788 q^{61} -0.0161353 q^{63} -7.92687 q^{65} +0.329470 q^{67} +1.66166 q^{69} +2.42120 q^{71} -14.2955 q^{73} -4.54506 q^{75} +3.07643 q^{77} +8.89622 q^{79} -9.04815 q^{81} -17.6958 q^{83} +1.54368 q^{85} +8.40069 q^{87} +9.59753 q^{89} -5.13506 q^{91} -6.92660 q^{93} -4.56433 q^{95} -3.61561 q^{97} -0.0496391 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{3} + 4 q^{5} - 6 q^{7} + 8 q^{9} - 8 q^{11} + 8 q^{13} - 6 q^{17} - 6 q^{19} + 4 q^{21} - 18 q^{23} + 12 q^{25} - 22 q^{27} - 8 q^{29} + 4 q^{31} - 6 q^{33} - 4 q^{35} - 8 q^{37} - 14 q^{39}+ \cdots - 28 q^{99}+O(q^{100}) $ 6 * q - 4 * q^3 + 4 * q^5 - 6 * q^7 + 8 * q^9 - 8 * q^11 + 8 * q^13 - 6 * q^17 - 6 * q^19 + 4 * q^21 - 18 * q^23 + 12 * q^25 - 22 * q^27 - 8 * q^29 + 4 * q^31 - 6 * q^33 - 4 * q^35 - 8 * q^37 - 14 * q^39 + 16 * q^41 - 12 * q^43 + 4 * q^45 - 18 * q^47 + 6 * q^49 + 4 * q^51 - 2 * q^53 + 8 * q^55 + 8 * q^57 - 16 * q^59 + 6 * q^61 - 8 * q^63 - 22 * q^65 - 12 * q^67 + 16 * q^69 + 2 * q^71 + 8 * q^73 - 24 * q^75 + 8 * q^77 + 18 * q^79 + 18 * q^81 - 28 * q^83 - 4 * q^85 + 2 * q^87 - 2 * q^89 - 8 * q^91 - 32 * q^93 - 26 * q^95 - 18 * q^97 - 28 * 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
$3$	1.73670	1.00269	0.501343	−	0.865249i	$-0.332840\pi$
$3$	1.73670	1.00269	0.501343	+	0.865249i	$0.332840\pi$
$4$	0	0
$5$	−1.54368	−0.690353	−0.345177	−	0.938538i	$-0.612181\pi$
$5$	−1.54368	−0.690353	−0.345177	+	0.938538i	$0.612181\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0.0161353	0.00537844
$10$	0	0
$11$	−3.07643	−0.927578	−0.463789	−	0.885946i	$-0.653510\pi$
$11$	−3.07643	−0.927578	−0.463789	+	0.885946i	$0.653510\pi$
$12$	0	0
$13$	5.13506	1.42421	0.712105	−	0.702073i	$-0.247742\pi$
$13$	5.13506	1.42421	0.712105	+	0.702073i	$0.247742\pi$
$14$	0	0
$15$	−2.68091	−0.692207
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	2.95679	0.678334	0.339167	−	0.940726i	$-0.389855\pi$
$19$	2.95679	0.678334	0.339167	+	0.940726i	$0.389855\pi$
$20$	0	0
$21$	−1.73670	−0.378980
$22$	0	0
$23$	0.956789	0.199504	0.0997521	−	0.995012i	$-0.468195\pi$
$23$	0.956789	0.199504	0.0997521	+	0.995012i	$0.468195\pi$
$24$	0	0
$25$	−2.61706	−0.523413
$26$	0	0
$27$	−5.18209	−0.997293
$28$	0	0
$29$	4.83715	0.898236	0.449118	−	0.893472i	$-0.351738\pi$
$29$	4.83715	0.898236	0.449118	+	0.893472i	$0.351738\pi$
$30$	0	0
$31$	−3.98836	−0.716331	−0.358165	−	0.933658i	$-0.616598\pi$
$31$	−3.98836	−0.716331	−0.358165	+	0.933658i	$0.616598\pi$
$32$	0	0
$33$	−5.34284	−0.930069
$34$	0	0
$35$	1.54368	0.260929
$36$	0	0
$37$	−5.56431	−0.914767	−0.457383	−	0.889270i	$-0.651213\pi$
$37$	−5.56431	−0.914767	−0.457383	+	0.889270i	$0.651213\pi$
$38$	0	0
$39$	8.91807	1.42803
$40$	0	0
$41$	−3.34872	−0.522982	−0.261491	−	0.965206i	$-0.584214\pi$
$41$	−3.34872	−0.522982	−0.261491	+	0.965206i	$0.584214\pi$
$42$	0	0
$43$	−1.96538	−0.299718	−0.149859	−	0.988707i	$-0.547882\pi$
$43$	−1.96538	−0.299718	−0.149859	+	0.988707i	$0.547882\pi$
$44$	0	0
$45$	−0.0249077	−0.00371302
$46$	0	0
$47$	−4.46891	−0.651857	−0.325929	−	0.945394i	$-0.605677\pi$
$47$	−4.46891	−0.651857	−0.325929	+	0.945394i	$0.605677\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.73670	−0.243187
$52$	0	0
$53$	−7.31576	−1.00490	−0.502449	−	0.864607i	$-0.667567\pi$
$53$	−7.31576	−1.00490	−0.502449	+	0.864607i	$0.667567\pi$
$54$	0	0
$55$	4.74901	0.640356
$56$	0	0
$57$	5.13506	0.680156
$58$	0	0
$59$	−0.494325	−0.0643556	−0.0321778	−	0.999482i	$-0.510244\pi$
$59$	−0.494325	−0.0643556	−0.0321778	+	0.999482i	$0.510244\pi$
$60$	0	0
$61$	2.23788	0.286531	0.143266	−	0.989684i	$-0.454240\pi$
$61$	2.23788	0.286531	0.143266	+	0.989684i	$0.454240\pi$
$62$	0	0
$63$	−0.0161353	−0.00203286
$64$	0	0
$65$	−7.92687	−0.983208
$66$	0	0
$67$	0.329470	0.0402512	0.0201256	−	0.999797i	$-0.493593\pi$
$67$	0.329470	0.0402512	0.0201256	+	0.999797i	$0.493593\pi$
$68$	0	0
$69$	1.66166	0.200040
$70$	0	0
$71$	2.42120	0.287343	0.143672	−	0.989625i	$-0.454109\pi$
$71$	2.42120	0.287343	0.143672	+	0.989625i	$0.454109\pi$
$72$	0	0
$73$	−14.2955	−1.67317	−0.836583	−	0.547841i	$-0.815450\pi$
$73$	−14.2955	−1.67317	−0.836583	+	0.547841i	$0.815450\pi$
$74$	0	0
$75$	−4.54506	−0.524818
$76$	0	0
$77$	3.07643	0.350592
$78$	0	0
$79$	8.89622	1.00090	0.500452	−	0.865764i	$-0.333167\pi$
$79$	8.89622	1.00090	0.500452	+	0.865764i	$0.333167\pi$
$80$	0	0
$81$	−9.04815	−1.00535
$82$	0	0
$83$	−17.6958	−1.94237	−0.971184	−	0.238330i	$-0.923400\pi$
$83$	−17.6958	−1.94237	−0.971184	+	0.238330i	$0.923400\pi$
$84$	0	0
$85$	1.54368	0.167435
$86$	0	0
$87$	8.40069	0.900648
$88$	0	0
$89$	9.59753	1.01734	0.508668	−	0.860963i	$-0.330138\pi$
$89$	9.59753	1.01734	0.508668	+	0.860963i	$0.330138\pi$
$90$	0	0
$91$	−5.13506	−0.538301
$92$	0	0
$93$	−6.92660	−0.718255
$94$	0	0
$95$	−4.56433	−0.468290
$96$	0	0
$97$	−3.61561	−0.367109	−0.183555	−	0.983010i	$-0.558760\pi$
$97$	−3.61561	−0.367109	−0.183555	+	0.983010i	$0.558760\pi$
$98$	0	0
$99$	−0.0496391	−0.00498892
$100$	0	0
$101$	−10.0531	−1.00032	−0.500162	−	0.865932i	$-0.666726\pi$
$101$	−10.0531	−1.00032	−0.500162	+	0.865932i	$0.666726\pi$
$102$	0	0
$103$	−7.02679	−0.692370	−0.346185	−	0.938166i	$-0.612523\pi$
$103$	−7.02679	−0.692370	−0.346185	+	0.938166i	$0.612523\pi$
$104$	0	0
$105$	2.68091	0.261630
$106$	0	0
$107$	−15.8632	−1.53355	−0.766775	−	0.641916i	$-0.778140\pi$
$107$	−15.8632	−1.53355	−0.766775	+	0.641916i	$0.778140\pi$
$108$	0	0
$109$	8.20151	0.785562	0.392781	−	0.919632i	$-0.371513\pi$
$109$	8.20151	0.785562	0.392781	+	0.919632i	$0.371513\pi$
$110$	0	0
$111$	−9.66355	−0.917223
$112$	0	0
$113$	−6.80081	−0.639766	−0.319883	−	0.947457i	$-0.603644\pi$
$113$	−6.80081	−0.639766	−0.319883	+	0.947457i	$0.603644\pi$
$114$	0	0
$115$	−1.47697	−0.137728
$116$	0	0
$117$	0.0828558	0.00766003
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	−1.53559	−0.139599
$122$	0	0
$123$	−5.81573	−0.524387
$124$	0	0
$125$	11.7583	1.05169
$126$	0	0
$127$	6.48233	0.575213	0.287607	−	0.957749i	$-0.407140\pi$
$127$	6.48233	0.575213	0.287607	+	0.957749i	$0.407140\pi$
$128$	0	0
$129$	−3.41329	−0.300523
$130$	0	0
$131$	−3.17648	−0.277530	−0.138765	−	0.990325i	$-0.544313\pi$
$131$	−3.17648	−0.277530	−0.138765	+	0.990325i	$0.544313\pi$
$132$	0	0
$133$	−2.95679	−0.256386
$134$	0	0
$135$	7.99946	0.688484
$136$	0	0
$137$	6.72383	0.574456	0.287228	−	0.957862i	$-0.407266\pi$
$137$	6.72383	0.574456	0.287228	+	0.957862i	$0.407266\pi$
$138$	0	0
$139$	−21.0484	−1.78530	−0.892650	−	0.450751i	$-0.851156\pi$
$139$	−21.0484	−1.78530	−0.892650	+	0.450751i	$0.851156\pi$
$140$	0	0
$141$	−7.76116	−0.653608
$142$	0	0
$143$	−15.7976	−1.32107
$144$	0	0
$145$	−7.46699	−0.620100
$146$	0	0
$147$	1.73670	0.143241
$148$	0	0
$149$	−4.15874	−0.340697	−0.170348	−	0.985384i	$-0.554489\pi$
$149$	−4.15874	−0.340697	−0.170348	+	0.985384i	$0.554489\pi$
$150$	0	0
$151$	−1.69992	−0.138337	−0.0691687	−	0.997605i	$-0.522035\pi$
$151$	−1.69992	−0.138337	−0.0691687	+	0.997605i	$0.522035\pi$
$152$	0	0
$153$	−0.0161353	−0.00130446
$154$	0	0
$155$	6.15674	0.494521
$156$	0	0
$157$	11.6963	0.933463	0.466732	−	0.884399i	$-0.345431\pi$
$157$	11.6963	0.933463	0.466732	+	0.884399i	$0.345431\pi$
$158$	0	0
$159$	−12.7053	−1.00760
$160$	0	0
$161$	−0.956789	−0.0754055
$162$	0	0
$163$	−1.99905	−0.156578	−0.0782890	−	0.996931i	$-0.524946\pi$
$163$	−1.99905	−0.156578	−0.0782890	+	0.996931i	$0.524946\pi$
$164$	0	0
$165$	8.24762	0.642076
$166$	0	0
$167$	−14.4867	−1.12101	−0.560506	−	0.828151i	$-0.689393\pi$
$167$	−14.4867	−1.12101	−0.560506	+	0.828151i	$0.689393\pi$
$168$	0	0
$169$	13.3689	1.02837
$170$	0	0
$171$	0.0477087	0.00364838
$172$	0	0
$173$	−6.91125	−0.525452	−0.262726	−	0.964870i	$-0.584622\pi$
$173$	−6.91125	−0.525452	−0.262726	+	0.964870i	$0.584622\pi$
$174$	0	0
$175$	2.61706	0.197831
$176$	0	0
$177$	−0.858495	−0.0645284
$178$	0	0
$179$	−24.3328	−1.81872	−0.909360	−	0.416010i	$-0.863428\pi$
$179$	−24.3328	−1.81872	−0.909360	+	0.416010i	$0.863428\pi$
$180$	0	0
$181$	18.0268	1.33992	0.669960	−	0.742397i	$-0.266311\pi$
$181$	18.0268	1.33992	0.669960	+	0.742397i	$0.266311\pi$
$182$	0	0
$183$	3.88653	0.287301
$184$	0	0
$185$	8.58949	0.631512
$186$	0	0
$187$	3.07643	0.224971
$188$	0	0
$189$	5.18209	0.376941
$190$	0	0
$191$	−0.499519	−0.0361439	−0.0180720	−	0.999837i	$-0.505753\pi$
$191$	−0.499519	−0.0361439	−0.0180720	+	0.999837i	$0.505753\pi$
$192$	0	0
$193$	−1.24333	−0.0894972	−0.0447486	−	0.998998i	$-0.514249\pi$
$193$	−1.24333	−0.0894972	−0.0447486	+	0.998998i	$0.514249\pi$
$194$	0	0
$195$	−13.7666	−0.985848
$196$	0	0
$197$	20.8997	1.48904	0.744521	−	0.667600i	$-0.232678\pi$
$197$	20.8997	1.48904	0.744521	+	0.667600i	$0.232678\pi$
$198$	0	0
$199$	−4.67889	−0.331678	−0.165839	−	0.986153i	$-0.553033\pi$
$199$	−4.67889	−0.331678	−0.165839	+	0.986153i	$0.553033\pi$
$200$	0	0
$201$	0.572192	0.0403593
$202$	0	0
$203$	−4.83715	−0.339501
$204$	0	0
$205$	5.16934	0.361042
$206$	0	0
$207$	0.0154381	0.00107302
$208$	0	0
$209$	−9.09635	−0.629208
$210$	0	0
$211$	2.13228	0.146792	0.0733961	−	0.997303i	$-0.476616\pi$
$211$	2.13228	0.146792	0.0733961	+	0.997303i	$0.476616\pi$
$212$	0	0
$213$	4.20490	0.288115
$214$	0	0
$215$	3.03392	0.206911
$216$	0	0
$217$	3.98836	0.270748
$218$	0	0
$219$	−24.8271	−1.67766
$220$	0	0
$221$	−5.13506	−0.345422
$222$	0	0
$223$	−1.58212	−0.105947	−0.0529733	−	0.998596i	$-0.516870\pi$
$223$	−1.58212	−0.105947	−0.0529733	+	0.998596i	$0.516870\pi$
$224$	0	0
$225$	−0.0422271	−0.00281514
$226$	0	0
$227$	−25.8874	−1.71821	−0.859105	−	0.511799i	$-0.828979\pi$
$227$	−25.8874	−1.71821	−0.859105	+	0.511799i	$0.828979\pi$
$228$	0	0
$229$	11.9860	0.792058	0.396029	−	0.918238i	$-0.370388\pi$
$229$	11.9860	0.792058	0.396029	+	0.918238i	$0.370388\pi$
$230$	0	0
$231$	5.34284	0.351533
$232$	0	0
$233$	3.89460	0.255144	0.127572	−	0.991829i	$-0.459282\pi$
$233$	3.89460	0.255144	0.127572	+	0.991829i	$0.459282\pi$
$234$	0	0
$235$	6.89855	0.450012
$236$	0	0
$237$	15.4501	1.00359
$238$	0	0
$239$	7.03555	0.455092	0.227546	−	0.973767i	$-0.426930\pi$
$239$	7.03555	0.455092	0.227546	+	0.973767i	$0.426930\pi$
$240$	0	0
$241$	14.1017	0.908373	0.454186	−	0.890907i	$-0.349930\pi$
$241$	14.1017	0.908373	0.454186	+	0.890907i	$0.349930\pi$
$242$	0	0
$243$	−0.167681	−0.0107568
$244$	0	0
$245$	−1.54368	−0.0986219
$246$	0	0
$247$	15.1833	0.966090
$248$	0	0
$249$	−30.7324	−1.94759
$250$	0	0
$251$	−1.80629	−0.114012	−0.0570060	−	0.998374i	$-0.518155\pi$
$251$	−1.80629	−0.114012	−0.0570060	+	0.998374i	$0.518155\pi$
$252$	0	0
$253$	−2.94349	−0.185056
$254$	0	0
$255$	2.68091	0.167885
$256$	0	0
$257$	20.7900	1.29685	0.648423	−	0.761280i	$-0.275429\pi$
$257$	20.7900	1.29685	0.648423	+	0.761280i	$0.275429\pi$
$258$	0	0
$259$	5.56431	0.345749
$260$	0	0
$261$	0.0780489	0.00483111
$262$	0	0
$263$	−4.38605	−0.270456	−0.135228	−	0.990815i	$-0.543177\pi$
$263$	−4.38605	−0.270456	−0.135228	+	0.990815i	$0.543177\pi$
$264$	0	0
$265$	11.2932	0.693734
$266$	0	0
$267$	16.6680	1.02007
$268$	0	0
$269$	−19.5885	−1.19433	−0.597166	−	0.802118i	$-0.703707\pi$
$269$	−19.5885	−1.19433	−0.597166	+	0.802118i	$0.703707\pi$
$270$	0	0
$271$	14.6852	0.892065	0.446032	−	0.895017i	$-0.352837\pi$
$271$	14.6852	0.892065	0.446032	+	0.895017i	$0.352837\pi$
$272$	0	0
$273$	−8.91807	−0.539746
$274$	0	0
$275$	8.05121	0.485506
$276$	0	0
$277$	−11.3853	−0.684074	−0.342037	−	0.939686i	$-0.611117\pi$
$277$	−11.3853	−0.684074	−0.342037	+	0.939686i	$0.611117\pi$
$278$	0	0
$279$	−0.0643535	−0.00385274
$280$	0	0
$281$	8.69046	0.518430	0.259215	−	0.965820i	$-0.416536\pi$
$281$	8.69046	0.518430	0.259215	+	0.965820i	$0.416536\pi$
$282$	0	0
$283$	4.03978	0.240140	0.120070	−	0.992765i	$-0.461688\pi$
$283$	4.03978	0.240140	0.120070	+	0.992765i	$0.461688\pi$
$284$	0	0
$285$	−7.92687	−0.469548
$286$	0	0
$287$	3.34872	0.197669
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−6.27923	−0.368095
$292$	0	0
$293$	−10.3337	−0.603700	−0.301850	−	0.953355i	$-0.597604\pi$
$293$	−10.3337	−0.603700	−0.301850	+	0.953355i	$0.597604\pi$
$294$	0	0
$295$	0.763077	0.0444281
$296$	0	0
$297$	15.9423	0.925067
$298$	0	0
$299$	4.91317	0.284136
$300$	0	0
$301$	1.96538	0.113283
$302$	0	0
$303$	−17.4593	−1.00301
$304$	0	0
$305$	−3.45456	−0.197808
$306$	0	0
$307$	33.4587	1.90959	0.954795	−	0.297264i	$-0.0960741\pi$
$307$	33.4587	1.90959	0.954795	+	0.297264i	$0.0960741\pi$
$308$	0	0
$309$	−12.2034	−0.694230
$310$	0	0
$311$	−14.5026	−0.822368	−0.411184	−	0.911552i	$-0.634885\pi$
$311$	−14.5026	−0.822368	−0.411184	+	0.911552i	$0.634885\pi$
$312$	0	0
$313$	−0.665479	−0.0376151	−0.0188076	−	0.999823i	$-0.505987\pi$
$313$	−0.665479	−0.0376151	−0.0188076	+	0.999823i	$0.505987\pi$
$314$	0	0
$315$	0.0249077	0.00140339
$316$	0	0
$317$	−2.52540	−0.141841	−0.0709203	−	0.997482i	$-0.522594\pi$
$317$	−2.52540	−0.141841	−0.0709203	+	0.997482i	$0.522594\pi$
$318$	0	0
$319$	−14.8811	−0.833184
$320$	0	0
$321$	−27.5496	−1.53767
$322$	0	0
$323$	−2.95679	−0.164520
$324$	0	0
$325$	−13.4388	−0.745449
$326$	0	0
$327$	14.2436	0.787672
$328$	0	0
$329$	4.46891	0.246379
$330$	0	0
$331$	29.0108	1.59458	0.797288	−	0.603599i	$-0.206267\pi$
$331$	29.0108	1.59458	0.797288	+	0.603599i	$0.206267\pi$
$332$	0	0
$333$	−0.0897819	−0.00492002
$334$	0	0
$335$	−0.508596	−0.0277876
$336$	0	0
$337$	5.79023	0.315414	0.157707	−	0.987486i	$-0.449590\pi$
$337$	5.79023	0.315414	0.157707	+	0.987486i	$0.449590\pi$
$338$	0	0
$339$	−11.8110	−0.641484
$340$	0	0
$341$	12.2699	0.664453
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−2.56506	−0.138098
$346$	0	0
$347$	−15.5324	−0.833826	−0.416913	−	0.908947i	$-0.636888\pi$
$347$	−15.5324	−0.833826	−0.416913	+	0.908947i	$0.636888\pi$
$348$	0	0
$349$	12.9869	0.695171	0.347585	−	0.937648i	$-0.387002\pi$
$349$	12.9869	0.695171	0.347585	+	0.937648i	$0.387002\pi$
$350$	0	0
$351$	−26.6103	−1.42035
$352$	0	0
$353$	26.5500	1.41311	0.706557	−	0.707656i	$-0.250247\pi$
$353$	26.5500	1.41311	0.706557	+	0.707656i	$0.250247\pi$
$354$	0	0
$355$	−3.73755	−0.198368
$356$	0	0
$357$	1.73670	0.0919160
$358$	0	0
$359$	12.2568	0.646888	0.323444	−	0.946247i	$-0.395159\pi$
$359$	12.2568	0.646888	0.323444	+	0.946247i	$0.395159\pi$
$360$	0	0
$361$	−10.2574	−0.539863
$362$	0	0
$363$	−2.66686	−0.139974
$364$	0	0
$365$	22.0677	1.15507
$366$	0	0
$367$	3.27195	0.170794	0.0853972	−	0.996347i	$-0.472784\pi$
$367$	3.27195	0.170794	0.0853972	+	0.996347i	$0.472784\pi$
$368$	0	0
$369$	−0.0540327	−0.00281283
$370$	0	0
$371$	7.31576	0.379816
$372$	0	0
$373$	9.66109	0.500232	0.250116	−	0.968216i	$-0.419531\pi$
$373$	9.66109	0.500232	0.250116	+	0.968216i	$0.419531\pi$
$374$	0	0
$375$	20.4206	1.05452
$376$	0	0
$377$	24.8391	1.27928
$378$	0	0
$379$	−0.596009	−0.0306149	−0.0153075	−	0.999883i	$-0.504873\pi$
$379$	−0.596009	−0.0306149	−0.0153075	+	0.999883i	$0.504873\pi$
$380$	0	0
$381$	11.2579	0.576758
$382$	0	0
$383$	4.43661	0.226700	0.113350	−	0.993555i	$-0.463842\pi$
$383$	4.43661	0.226700	0.113350	+	0.993555i	$0.463842\pi$
$384$	0	0
$385$	−4.74901	−0.242032
$386$	0	0
$387$	−0.0317121	−0.00161202
$388$	0	0
$389$	−29.7065	−1.50618	−0.753089	−	0.657919i	$-0.771437\pi$
$389$	−29.7065	−1.50618	−0.753089	+	0.657919i	$0.771437\pi$
$390$	0	0
$391$	−0.956789	−0.0483869
$392$	0	0
$393$	−5.51659	−0.278275
$394$	0	0
$395$	−13.7329	−0.690977
$396$	0	0
$397$	19.7774	0.992599	0.496299	−	0.868151i	$-0.334692\pi$
$397$	19.7774	0.992599	0.496299	+	0.868151i	$0.334692\pi$
$398$	0	0
$399$	−5.13506	−0.257075
$400$	0	0
$401$	36.5174	1.82359	0.911795	−	0.410645i	$-0.134696\pi$
$401$	36.5174	1.82359	0.911795	+	0.410645i	$0.134696\pi$
$402$	0	0
$403$	−20.4805	−1.02021
$404$	0	0
$405$	13.9674	0.694046
$406$	0	0
$407$	17.1182	0.848518
$408$	0	0
$409$	38.9254	1.92474	0.962368	−	0.271748i	$-0.0876017\pi$
$409$	38.9254	1.92474	0.962368	+	0.271748i	$0.0876017\pi$
$410$	0	0
$411$	11.6773	0.575998
$412$	0	0
$413$	0.494325	0.0243241
$414$	0	0
$415$	27.3166	1.34092
$416$	0	0
$417$	−36.5548	−1.79009
$418$	0	0
$419$	−29.2450	−1.42871	−0.714355	−	0.699784i	$-0.753280\pi$
$419$	−29.2450	−1.42871	−0.714355	+	0.699784i	$0.753280\pi$
$420$	0	0
$421$	−37.0885	−1.80758	−0.903791	−	0.427975i	$-0.859227\pi$
$421$	−37.0885	−1.80758	−0.903791	+	0.427975i	$0.859227\pi$
$422$	0	0
$423$	−0.0721072	−0.00350597
$424$	0	0
$425$	2.61706	0.126946
$426$	0	0
$427$	−2.23788	−0.108299
$428$	0	0
$429$	−27.4358	−1.32461
$430$	0	0
$431$	−27.7206	−1.33525	−0.667627	−	0.744496i	$-0.732690\pi$
$431$	−27.7206	−1.33525	−0.667627	+	0.744496i	$0.732690\pi$
$432$	0	0
$433$	15.2795	0.734285	0.367142	−	0.930165i	$-0.380336\pi$
$433$	15.2795	0.734285	0.367142	+	0.930165i	$0.380336\pi$
$434$	0	0
$435$	−12.9679	−0.621765
$436$	0	0
$437$	2.82902	0.135330
$438$	0	0
$439$	−11.5974	−0.553515	−0.276758	−	0.960940i	$-0.589260\pi$
$439$	−11.5974	−0.553515	−0.276758	+	0.960940i	$0.589260\pi$
$440$	0	0
$441$	0.0161353	0.000768348 0
$442$	0	0
$443$	−13.8718	−0.659068	−0.329534	−	0.944144i	$-0.606892\pi$
$443$	−13.8718	−0.659068	−0.329534	+	0.944144i	$0.606892\pi$
$444$	0	0
$445$	−14.8155	−0.702321
$446$	0	0
$447$	−7.22249	−0.341612
$448$	0	0
$449$	35.5894	1.67957	0.839785	−	0.542920i	$-0.182681\pi$
$449$	35.5894	1.67957	0.839785	+	0.542920i	$0.182681\pi$
$450$	0	0
$451$	10.3021	0.485107
$452$	0	0
$453$	−2.95225	−0.138709
$454$	0	0
$455$	7.92687	0.371618
$456$	0	0
$457$	−5.53234	−0.258792	−0.129396	−	0.991593i	$-0.541304\pi$
$457$	−5.53234	−0.258792	−0.129396	+	0.991593i	$0.541304\pi$
$458$	0	0
$459$	5.18209	0.241879
$460$	0	0
$461$	−15.3059	−0.712866	−0.356433	−	0.934321i	$-0.616007\pi$
$461$	−15.3059	−0.712866	−0.356433	+	0.934321i	$0.616007\pi$
$462$	0	0
$463$	5.67202	0.263601	0.131800	−	0.991276i	$-0.457924\pi$
$463$	5.67202	0.263601	0.131800	+	0.991276i	$0.457924\pi$
$464$	0	0
$465$	10.6924	0.495849
$466$	0	0
$467$	17.5647	0.812796	0.406398	−	0.913696i	$-0.366785\pi$
$467$	17.5647	0.812796	0.406398	+	0.913696i	$0.366785\pi$
$468$	0	0
$469$	−0.329470	−0.0152135
$470$	0	0
$471$	20.3129	0.935970
$472$	0	0
$473$	6.04636	0.278012
$474$	0	0
$475$	−7.73810	−0.355048
$476$	0	0
$477$	−0.118042	−0.00540478
$478$	0	0
$479$	39.6215	1.81035	0.905176	−	0.425038i	$-0.139739\pi$
$479$	39.6215	1.81035	0.905176	+	0.425038i	$0.139739\pi$
$480$	0	0
$481$	−28.5731	−1.30282
$482$	0	0
$483$	−1.66166	−0.0756080
$484$	0	0
$485$	5.58133	0.253435
$486$	0	0
$487$	−27.4853	−1.24548	−0.622740	−	0.782429i	$-0.713980\pi$
$487$	−27.4853	−1.24548	−0.622740	+	0.782429i	$0.713980\pi$
$488$	0	0
$489$	−3.47176	−0.156998
$490$	0	0
$491$	−8.80191	−0.397225	−0.198612	−	0.980078i	$-0.563643\pi$
$491$	−8.80191	−0.397225	−0.198612	+	0.980078i	$0.563643\pi$
$492$	0	0
$493$	−4.83715	−0.217854
$494$	0	0
$495$	0.0766268	0.00344412
$496$	0	0
$497$	−2.42120	−0.108606
$498$	0	0
$499$	−19.2935	−0.863697	−0.431848	−	0.901946i	$-0.642138\pi$
$499$	−19.2935	−0.863697	−0.431848	+	0.901946i	$0.642138\pi$
$500$	0	0
$501$	−25.1590	−1.12402
$502$	0	0
$503$	3.87021	0.172564	0.0862821	−	0.996271i	$-0.472501\pi$
$503$	3.87021	0.172564	0.0862821	+	0.996271i	$0.472501\pi$
$504$	0	0
$505$	15.5188	0.690577
$506$	0	0
$507$	23.2177	1.03114
$508$	0	0
$509$	−4.76533	−0.211219	−0.105610	−	0.994408i	$-0.533679\pi$
$509$	−4.76533	−0.211219	−0.105610	+	0.994408i	$0.533679\pi$
$510$	0	0
$511$	14.2955	0.632397
$512$	0	0
$513$	−15.3223	−0.676497
$514$	0	0
$515$	10.8471	0.477980
$516$	0	0
$517$	13.7483	0.604648
$518$	0	0
$519$	−12.0028	−0.526864
$520$	0	0
$521$	−5.59408	−0.245081	−0.122541	−	0.992463i	$-0.539104\pi$
$521$	−5.59408	−0.245081	−0.122541	+	0.992463i	$0.539104\pi$
$522$	0	0
$523$	39.8251	1.74143	0.870714	−	0.491789i	$-0.163657\pi$
$523$	39.8251	1.74143	0.870714	+	0.491789i	$0.163657\pi$
$524$	0	0
$525$	4.54506	0.198363
$526$	0	0
$527$	3.98836	0.173736
$528$	0	0
$529$	−22.0846	−0.960198
$530$	0	0
$531$	−0.00797608	−0.000346132 0
$532$	0	0
$533$	−17.1959	−0.744836
$534$	0	0
$535$	24.4876	1.05869
$536$	0	0
$537$	−42.2589	−1.82360
$538$	0	0
$539$	−3.07643	−0.132511
$540$	0	0
$541$	−25.3977	−1.09193	−0.545966	−	0.837807i	$-0.683837\pi$
$541$	−25.3977	−1.09193	−0.545966	+	0.837807i	$0.683837\pi$
$542$	0	0
$543$	31.3071	1.34352
$544$	0	0
$545$	−12.6605	−0.542315
$546$	0	0
$547$	−8.97660	−0.383812	−0.191906	−	0.981413i	$-0.561467\pi$
$547$	−8.97660	−0.383812	−0.191906	+	0.981413i	$0.561467\pi$
$548$	0	0
$549$	0.0361089	0.00154109
$550$	0	0
$551$	14.3024	0.609304
$552$	0	0
$553$	−8.89622	−0.378306
$554$	0	0
$555$	14.9174	0.633208
$556$	0	0
$557$	0.716380	0.0303540	0.0151770	−	0.999885i	$-0.495169\pi$
$557$	0.716380	0.0303540	0.0151770	+	0.999885i	$0.495169\pi$
$558$	0	0
$559$	−10.0924	−0.426862
$560$	0	0
$561$	5.34284	0.225575
$562$	0	0
$563$	−20.3312	−0.856860	−0.428430	−	0.903575i	$-0.640933\pi$
$563$	−20.3312	−0.856860	−0.428430	+	0.903575i	$0.640933\pi$
$564$	0	0
$565$	10.4982	0.441665
$566$	0	0
$567$	9.04815	0.379986
$568$	0	0
$569$	−3.83795	−0.160895	−0.0804475	−	0.996759i	$-0.525635\pi$
$569$	−3.83795	−0.160895	−0.0804475	+	0.996759i	$0.525635\pi$
$570$	0	0
$571$	−32.4717	−1.35890	−0.679449	−	0.733722i	$-0.737781\pi$
$571$	−32.4717	−1.35890	−0.679449	+	0.733722i	$0.737781\pi$
$572$	0	0
$573$	−0.867516	−0.0362410
$574$	0	0
$575$	−2.50398	−0.104423
$576$	0	0
$577$	25.6252	1.06679	0.533395	−	0.845866i	$-0.320916\pi$
$577$	25.6252	1.06679	0.533395	+	0.845866i	$0.320916\pi$
$578$	0	0
$579$	−2.15930	−0.0897375
$580$	0	0
$581$	17.6958	0.734146
$582$	0	0
$583$	22.5064	0.932121
$584$	0	0
$585$	−0.127903	−0.00528812
$586$	0	0
$587$	31.0885	1.28316	0.641579	−	0.767057i	$-0.278280\pi$
$587$	31.0885	1.28316	0.641579	+	0.767057i	$0.278280\pi$
$588$	0	0
$589$	−11.7927	−0.485911
$590$	0	0
$591$	36.2965	1.49304
$592$	0	0
$593$	−4.95402	−0.203437	−0.101719	−	0.994813i	$-0.532434\pi$
$593$	−4.95402	−0.203437	−0.101719	+	0.994813i	$0.532434\pi$
$594$	0	0
$595$	−1.54368	−0.0632846
$596$	0	0
$597$	−8.12584	−0.332568
$598$	0	0
$599$	43.5216	1.77825	0.889123	−	0.457668i	$-0.151315\pi$
$599$	43.5216	1.77825	0.889123	+	0.457668i	$0.151315\pi$
$600$	0	0
$601$	26.2441	1.07052	0.535259	−	0.844688i	$-0.320214\pi$
$601$	26.2441	1.07052	0.535259	+	0.844688i	$0.320214\pi$
$602$	0	0
$603$	0.00531611	0.000216489 0
$604$	0	0
$605$	2.37045	0.0963726
$606$	0	0
$607$	−26.9179	−1.09257	−0.546283	−	0.837601i	$-0.683958\pi$
$607$	−26.9179	−1.09257	−0.546283	+	0.837601i	$0.683958\pi$
$608$	0	0
$609$	−8.40069	−0.340413
$610$	0	0
$611$	−22.9481	−0.928381
$612$	0	0
$613$	−22.9795	−0.928134	−0.464067	−	0.885800i	$-0.653610\pi$
$613$	−22.9795	−0.928134	−0.464067	+	0.885800i	$0.653610\pi$
$614$	0	0
$615$	8.97761	0.362012
$616$	0	0
$617$	7.30604	0.294130	0.147065	−	0.989127i	$-0.453017\pi$
$617$	7.30604	0.294130	0.147065	+	0.989127i	$0.453017\pi$
$618$	0	0
$619$	−20.0699	−0.806679	−0.403339	−	0.915050i	$-0.632151\pi$
$619$	−20.0699	−0.806679	−0.403339	+	0.915050i	$0.632151\pi$
$620$	0	0
$621$	−4.95816	−0.198964
$622$	0	0
$623$	−9.59753	−0.384517
$624$	0	0
$625$	−5.06567	−0.202627
$626$	0	0
$627$	−15.7976	−0.630897
$628$	0	0
$629$	5.56431	0.221864
$630$	0	0
$631$	29.4824	1.17368	0.586838	−	0.809704i	$-0.300373\pi$
$631$	29.4824	1.17368	0.586838	+	0.809704i	$0.300373\pi$
$632$	0	0
$633$	3.70314	0.147186
$634$	0	0
$635$	−10.0066	−0.397100
$636$	0	0
$637$	5.13506	0.203459
$638$	0	0
$639$	0.0390668	0.00154546
$640$	0	0
$641$	−8.15820	−0.322230	−0.161115	−	0.986936i	$-0.551509\pi$
$641$	−8.15820	−0.322230	−0.161115	+	0.986936i	$0.551509\pi$
$642$	0	0
$643$	−20.0748	−0.791673	−0.395836	−	0.918321i	$-0.629545\pi$
$643$	−20.0748	−0.791673	−0.395836	+	0.918321i	$0.629545\pi$
$644$	0	0
$645$	5.26901	0.207467
$646$	0	0
$647$	−27.1506	−1.06740	−0.533699	−	0.845674i	$-0.679199\pi$
$647$	−27.1506	−1.06740	−0.533699	+	0.845674i	$0.679199\pi$
$648$	0	0
$649$	1.52075	0.0596948
$650$	0	0
$651$	6.92660	0.271475
$652$	0	0
$653$	50.4969	1.97610	0.988049	−	0.154143i	$-0.0492617\pi$
$653$	50.4969	1.97610	0.988049	+	0.154143i	$0.0492617\pi$
$654$	0	0
$655$	4.90345	0.191594
$656$	0	0
$657$	−0.230663	−0.00899902
$658$	0	0
$659$	−2.22940	−0.0868449	−0.0434225	−	0.999057i	$-0.513826\pi$
$659$	−2.22940	−0.0868449	−0.0434225	+	0.999057i	$0.513826\pi$
$660$	0	0
$661$	23.2104	0.902780	0.451390	−	0.892327i	$-0.350928\pi$
$661$	23.2104	0.902780	0.451390	+	0.892327i	$0.350928\pi$
$662$	0	0
$663$	−8.91807	−0.346349
$664$	0	0
$665$	4.56433	0.176997
$666$	0	0
$667$	4.62813	0.179202
$668$	0	0
$669$	−2.74767	−0.106231
$670$	0	0
$671$	−6.88468	−0.265780
$672$	0	0
$673$	−27.6117	−1.06435	−0.532177	−	0.846633i	$-0.678626\pi$
$673$	−27.6117	−1.06435	−0.532177	+	0.846633i	$0.678626\pi$
$674$	0	0
$675$	13.5618	0.521996
$676$	0	0
$677$	−34.5960	−1.32963	−0.664816	−	0.747007i	$-0.731490\pi$
$677$	−34.5960	−1.32963	−0.664816	+	0.747007i	$0.731490\pi$
$678$	0	0
$679$	3.61561	0.138754
$680$	0	0
$681$	−44.9588	−1.72282
$682$	0	0
$683$	35.8086	1.37018	0.685089	−	0.728459i	$-0.259763\pi$
$683$	35.8086	1.37018	0.685089	+	0.728459i	$0.259763\pi$
$684$	0	0
$685$	−10.3794	−0.396577
$686$	0	0
$687$	20.8161	0.794185
$688$	0	0
$689$	−37.5669	−1.43118
$690$	0	0
$691$	−7.50019	−0.285321	−0.142660	−	0.989772i	$-0.545566\pi$
$691$	−7.50019	−0.285321	−0.142660	+	0.989772i	$0.545566\pi$
$692$	0	0
$693$	0.0496391	0.00188563
$694$	0	0
$695$	32.4919	1.23249
$696$	0	0
$697$	3.34872	0.126842
$698$	0	0
$699$	6.76377	0.255829
$700$	0	0
$701$	11.2108	0.423427	0.211713	−	0.977332i	$-0.432096\pi$
$701$	11.2108	0.423427	0.211713	+	0.977332i	$0.432096\pi$
$702$	0	0
$703$	−16.4525	−0.620517
$704$	0	0
$705$	11.9807	0.451220
$706$	0	0
$707$	10.0531	0.378087
$708$	0	0
$709$	−48.4671	−1.82022	−0.910110	−	0.414367i	$-0.864003\pi$
$709$	−48.4671	−1.82022	−0.910110	+	0.414367i	$0.864003\pi$
$710$	0	0
$711$	0.143543	0.00538330
$712$	0	0
$713$	−3.81602	−0.142911
$714$	0	0
$715$	24.3865	0.912002
$716$	0	0
$717$	12.2186	0.456314
$718$	0	0
$719$	−3.47062	−0.129432	−0.0647161	−	0.997904i	$-0.520614\pi$
$719$	−3.47062	−0.129432	−0.0647161	+	0.997904i	$0.520614\pi$
$720$	0	0
$721$	7.02679	0.261691
$722$	0	0
$723$	24.4905	0.910812
$724$	0	0
$725$	−12.6591	−0.470148
$726$	0	0
$727$	−7.61839	−0.282550	−0.141275	−	0.989970i	$-0.545120\pi$
$727$	−7.61839	−0.282550	−0.141275	+	0.989970i	$0.545120\pi$
$728$	0	0
$729$	26.8532	0.994564
$730$	0	0
$731$	1.96538	0.0726924
$732$	0	0
$733$	−34.8340	−1.28662	−0.643311	−	0.765605i	$-0.722440\pi$
$733$	−34.8340	−1.28662	−0.643311	+	0.765605i	$0.722440\pi$
$734$	0	0
$735$	−2.68091	−0.0988867
$736$	0	0
$737$	−1.01359	−0.0373361
$738$	0	0
$739$	−18.1732	−0.668512	−0.334256	−	0.942482i	$-0.608485\pi$
$739$	−18.1732	−0.668512	−0.334256	+	0.942482i	$0.608485\pi$
$740$	0	0
$741$	26.3689	0.968684
$742$	0	0
$743$	1.78017	0.0653080	0.0326540	−	0.999467i	$-0.489604\pi$
$743$	1.78017	0.0653080	0.0326540	+	0.999467i	$0.489604\pi$
$744$	0	0
$745$	6.41974	0.235201
$746$	0	0
$747$	−0.285528	−0.0104469
$748$	0	0
$749$	15.8632	0.579628
$750$	0	0
$751$	33.0713	1.20679	0.603394	−	0.797443i	$-0.293815\pi$
$751$	33.0713	1.20679	0.603394	+	0.797443i	$0.293815\pi$
$752$	0	0
$753$	−3.13699	−0.114318
$754$	0	0
$755$	2.62412	0.0955017
$756$	0	0
$757$	−1.16353	−0.0422891	−0.0211446	−	0.999776i	$-0.506731\pi$
$757$	−1.16353	−0.0422891	−0.0211446	+	0.999776i	$0.506731\pi$
$758$	0	0
$759$	−5.11197	−0.185553
$760$	0	0
$761$	−25.5778	−0.927194	−0.463597	−	0.886046i	$-0.653441\pi$
$761$	−25.5778	−0.927194	−0.463597	+	0.886046i	$0.653441\pi$
$762$	0	0
$763$	−8.20151	−0.296915
$764$	0	0
$765$	0.0249077	0.000900540 0
$766$	0	0
$767$	−2.53839	−0.0916558
$768$	0	0
$769$	−7.58033	−0.273354	−0.136677	−	0.990616i	$-0.543642\pi$
$769$	−7.58033	−0.273354	−0.136677	+	0.990616i	$0.543642\pi$
$770$	0	0
$771$	36.1061	1.30033
$772$	0	0
$773$	19.2432	0.692131	0.346066	−	0.938210i	$-0.387517\pi$
$773$	19.2432	0.692131	0.346066	+	0.938210i	$0.387517\pi$
$774$	0	0
$775$	10.4378	0.374937
$776$	0	0
$777$	9.66355	0.346678
$778$	0	0
$779$	−9.90146	−0.354757
$780$	0	0
$781$	−7.44864	−0.266533
$782$	0	0
$783$	−25.0665	−0.895804
$784$	0	0
$785$	−18.0552	−0.644419
$786$	0	0
$787$	−46.0869	−1.64282	−0.821411	−	0.570337i	$-0.806813\pi$
$787$	−46.0869	−1.64282	−0.821411	+	0.570337i	$0.806813\pi$
$788$	0	0
$789$	−7.61727	−0.271182
$790$	0	0
$791$	6.80081	0.241809
$792$	0	0
$793$	11.4917	0.408081
$794$	0	0
$795$	19.6129	0.695597
$796$	0	0
$797$	−7.05749	−0.249989	−0.124995	−	0.992157i	$-0.539891\pi$
$797$	−7.05749	−0.249989	−0.124995	+	0.992157i	$0.539891\pi$
$798$	0	0
$799$	4.46891	0.158099
$800$	0	0
$801$	0.154859	0.00547168
$802$	0	0
$803$	43.9792	1.55199
$804$	0	0
$805$	1.47697	0.0520564
$806$	0	0
$807$	−34.0194	−1.19754
$808$	0	0
$809$	−24.8762	−0.874599	−0.437300	−	0.899316i	$-0.644065\pi$
$809$	−24.8762	−0.874599	−0.437300	+	0.899316i	$0.644065\pi$
$810$	0	0
$811$	12.6176	0.443064	0.221532	−	0.975153i	$-0.428894\pi$
$811$	12.6176	0.443064	0.221532	+	0.975153i	$0.428894\pi$
$812$	0	0
$813$	25.5039	0.894460
$814$	0	0
$815$	3.08589	0.108094
$816$	0	0
$817$	−5.81123	−0.203309
$818$	0	0
$819$	−0.0828558	−0.00289522
$820$	0	0
$821$	28.3528	0.989521	0.494760	−	0.869029i	$-0.335256\pi$
$821$	28.3528	0.989521	0.494760	+	0.869029i	$0.335256\pi$
$822$	0	0
$823$	44.1976	1.54063	0.770315	−	0.637663i	$-0.220099\pi$
$823$	44.1976	1.54063	0.770315	+	0.637663i	$0.220099\pi$
$824$	0	0
$825$	13.9825	0.486810
$826$	0	0
$827$	−4.51243	−0.156913	−0.0784563	−	0.996918i	$-0.524999\pi$
$827$	−4.51243	−0.156913	−0.0784563	+	0.996918i	$0.524999\pi$
$828$	0	0
$829$	−23.6425	−0.821138	−0.410569	−	0.911830i	$-0.634670\pi$
$829$	−23.6425	−0.821138	−0.410569	+	0.911830i	$0.634670\pi$
$830$	0	0
$831$	−19.7728	−0.685911
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	22.3627	0.773894
$836$	0	0
$837$	20.6680	0.714392
$838$	0	0
$839$	−22.3277	−0.770838	−0.385419	−	0.922742i	$-0.625943\pi$
$839$	−22.3277	−0.770838	−0.385419	+	0.922742i	$0.625943\pi$
$840$	0	0
$841$	−5.60199	−0.193172
$842$	0	0
$843$	15.0928	0.519822
$844$	0	0
$845$	−20.6372	−0.709941
$846$	0	0
$847$	1.53559	0.0527635
$848$	0	0
$849$	7.01590	0.240785
$850$	0	0
$851$	−5.32387	−0.182500
$852$	0	0
$853$	−12.1099	−0.414634	−0.207317	−	0.978274i	$-0.566473\pi$
$853$	−12.1099	−0.414634	−0.207317	+	0.978274i	$0.566473\pi$
$854$	0	0
$855$	−0.0736468	−0.00251867
$856$	0	0
$857$	28.0762	0.959065	0.479533	−	0.877524i	$-0.340806\pi$
$857$	28.0762	0.959065	0.479533	+	0.877524i	$0.340806\pi$
$858$	0	0
$859$	34.1333	1.16461	0.582306	−	0.812970i	$-0.302151\pi$
$859$	34.1333	1.16461	0.582306	+	0.812970i	$0.302151\pi$
$860$	0	0
$861$	5.81573	0.198200
$862$	0	0
$863$	−29.7597	−1.01303	−0.506515	−	0.862231i	$-0.669067\pi$
$863$	−29.7597	−1.01303	−0.506515	+	0.862231i	$0.669067\pi$
$864$	0	0
$865$	10.6687	0.362748
$866$	0	0
$867$	1.73670	0.0589815
$868$	0	0
$869$	−27.3686	−0.928416
$870$	0	0
$871$	1.69185	0.0573262
$872$	0	0
$873$	−0.0583390	−0.00197447
$874$	0	0
$875$	−11.7583	−0.397502
$876$	0	0
$877$	48.0126	1.62127	0.810636	−	0.585551i	$-0.199122\pi$
$877$	48.0126	1.62127	0.810636	+	0.585551i	$0.199122\pi$
$878$	0	0
$879$	−17.9465	−0.605321
$880$	0	0
$881$	−1.73011	−0.0582889	−0.0291444	−	0.999575i	$-0.509278\pi$
$881$	−1.73011	−0.0582889	−0.0291444	+	0.999575i	$0.509278\pi$
$882$	0	0
$883$	−13.4867	−0.453865	−0.226932	−	0.973910i	$-0.572870\pi$
$883$	−13.4867	−0.453865	−0.226932	+	0.973910i	$0.572870\pi$
$884$	0	0
$885$	1.32524	0.0445474
$886$	0	0
$887$	−18.7271	−0.628793	−0.314397	−	0.949292i	$-0.601802\pi$
$887$	−18.7271	−0.628793	−0.314397	+	0.949292i	$0.601802\pi$
$888$	0	0
$889$	−6.48233	−0.217410
$890$	0	0
$891$	27.8360	0.932540
$892$	0	0
$893$	−13.2136	−0.442177
$894$	0	0
$895$	37.5620	1.25556
$896$	0	0
$897$	8.53271	0.284899
$898$	0	0
$899$	−19.2923	−0.643434
$900$	0	0
$901$	7.31576	0.243723
$902$	0	0
$903$	3.41329	0.113587
$904$	0	0
$905$	−27.8275	−0.925018
$906$	0	0
$907$	11.5187	0.382474	0.191237	−	0.981544i	$-0.438750\pi$
$907$	11.5187	0.382474	0.191237	+	0.981544i	$0.438750\pi$
$908$	0	0
$909$	−0.162211	−0.00538018
$910$	0	0
$911$	−40.2975	−1.33512	−0.667559	−	0.744557i	$-0.732661\pi$
$911$	−40.2975	−1.33512	−0.667559	+	0.744557i	$0.732661\pi$
$912$	0	0
$913$	54.4399	1.80170
$914$	0	0
$915$	−5.99955	−0.198339
$916$	0	0
$917$	3.17648	0.104896
$918$	0	0
$919$	46.4359	1.53178	0.765889	−	0.642973i	$-0.222299\pi$
$919$	46.4359	1.53178	0.765889	+	0.642973i	$0.222299\pi$
$920$	0	0
$921$	58.1079	1.91472
$922$	0	0
$923$	12.4330	0.409237
$924$	0	0
$925$	14.5621	0.478800
$926$	0	0
$927$	−0.113379	−0.00372387
$928$	0	0
$929$	23.4205	0.768402	0.384201	−	0.923250i	$-0.374477\pi$
$929$	23.4205	0.768402	0.384201	+	0.923250i	$0.374477\pi$
$930$	0	0
$931$	2.95679	0.0969048
$932$	0	0
$933$	−25.1867	−0.824577
$934$	0	0
$935$	−4.74901	−0.155309
$936$	0	0
$937$	−15.0807	−0.492664	−0.246332	−	0.969186i	$-0.579225\pi$
$937$	−15.0807	−0.492664	−0.246332	+	0.969186i	$0.579225\pi$
$938$	0	0
$939$	−1.15574	−0.0377161
$940$	0	0
$941$	−43.8498	−1.42946	−0.714732	−	0.699399i	$-0.753451\pi$
$941$	−43.8498	−1.42946	−0.714732	+	0.699399i	$0.753451\pi$
$942$	0	0
$943$	−3.20402	−0.104337
$944$	0	0
$945$	−7.99946	−0.260223
$946$	0	0
$947$	−35.7859	−1.16288	−0.581442	−	0.813588i	$-0.697511\pi$
$947$	−35.7859	−1.16288	−0.581442	+	0.813588i	$0.697511\pi$
$948$	0	0
$949$	−73.4084	−2.38294
$950$	0	0
$951$	−4.38587	−0.142222
$952$	0	0
$953$	2.73924	0.0887327	0.0443663	−	0.999015i	$-0.485873\pi$
$953$	2.73924	0.0887327	0.0443663	+	0.999015i	$0.485873\pi$
$954$	0	0
$955$	0.771096	0.0249521
$956$	0	0
$957$	−25.8441	−0.835422
$958$	0	0
$959$	−6.72383	−0.217124
$960$	0	0
$961$	−15.0930	−0.486870
$962$	0	0
$963$	−0.255957	−0.00824811
$964$	0	0
$965$	1.91931	0.0617847
$966$	0	0
$967$	−3.24324	−0.104296	−0.0521478	−	0.998639i	$-0.516607\pi$
$967$	−3.24324	−0.104296	−0.0521478	+	0.998639i	$0.516607\pi$
$968$	0	0
$969$	−5.13506	−0.164962
$970$	0	0
$971$	11.8203	0.379333	0.189666	−	0.981849i	$-0.439259\pi$
$971$	11.8203	0.379333	0.189666	+	0.981849i	$0.439259\pi$
$972$	0	0
$973$	21.0484	0.674780
$974$	0	0
$975$	−23.3392	−0.747451
$976$	0	0
$977$	25.5359	0.816965	0.408483	−	0.912766i	$-0.366058\pi$
$977$	25.5359	0.816965	0.408483	+	0.912766i	$0.366058\pi$
$978$	0	0
$979$	−29.5261	−0.943658
$980$	0	0
$981$	0.132334	0.00422510
$982$	0	0
$983$	4.22087	0.134625	0.0673124	−	0.997732i	$-0.478558\pi$
$983$	4.22087	0.134625	0.0673124	+	0.997732i	$0.478558\pi$
$984$	0	0
$985$	−32.2624	−1.02796
$986$	0	0
$987$	7.76116	0.247041
$988$	0	0
$989$	−1.88046	−0.0597951
$990$	0	0
$991$	39.3864	1.25115	0.625575	−	0.780164i	$-0.284864\pi$
$991$	39.3864	1.25115	0.625575	+	0.780164i	$0.284864\pi$
$992$	0	0
$993$	50.3831	1.59886
$994$	0	0
$995$	7.22269	0.228975
$996$	0	0
$997$	9.15850	0.290053	0.145026	−	0.989428i	$-0.453673\pi$
$997$	9.15850	0.290053	0.145026	+	0.989428i	$0.453673\pi$
$998$	0	0
$999$	28.8347	0.912290

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3808.2.a.h.1.5	✓	6
4.3	odd	2		3808.2.a.p.1.2	yes	6
8.3	odd	2		7616.2.a.bu.1.5		6
8.5	even	2		7616.2.a.cc.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.h.1.5	✓	6	1.1	even	1	trivial
3808.2.a.p.1.2	yes	6	4.3	odd	2
7616.2.a.bu.1.5		6	8.3	odd	2
7616.2.a.cc.1.2		6	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3808 = 2^{5} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3808.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73670 q^{3} -1.54368 q^{5} -1.00000 q^{7} +0.0161353 q^{9} -3.07643 q^{11} +5.13506 q^{13} -2.68091 q^{15} -1.00000 q^{17} +2.95679 q^{19} -1.73670 q^{21} +0.956789 q^{23} -2.61706 q^{25} -5.18209 q^{27} +4.83715 q^{29} -3.98836 q^{31} -5.34284 q^{33} +1.54368 q^{35} -5.56431 q^{37} +8.91807 q^{39} -3.34872 q^{41} -1.96538 q^{43} -0.0249077 q^{45} -4.46891 q^{47} +1.00000 q^{49} -1.73670 q^{51} -7.31576 q^{53} +4.74901 q^{55} +5.13506 q^{57} -0.494325 q^{59} +2.23788 q^{61} -0.0161353 q^{63} -7.92687 q^{65} +0.329470 q^{67} +1.66166 q^{69} +2.42120 q^{71} -14.2955 q^{73} -4.54506 q^{75} +3.07643 q^{77} +8.89622 q^{79} -9.04815 q^{81} -17.6958 q^{83} +1.54368 q^{85} +8.40069 q^{87} +9.59753 q^{89} -5.13506 q^{91} -6.92660 q^{93} -4.56433 q^{95} -3.61561 q^{97} -0.0496391 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} + 4 q^{5} - 6 q^{7} + 8 q^{9} - 8 q^{11} + 8 q^{13} - 6 q^{17} - 6 q^{19} + 4 q^{21} - 18 q^{23} + 12 q^{25} - 22 q^{27} - 8 q^{29} + 4 q^{31} - 6 q^{33} - 4 q^{35} - 8 q^{37} - 14 q^{39}+ \cdots - 28 q^{99}+O(q^{100}) \) 6 * q - 4 * q^3 + 4 * q^5 - 6 * q^7 + 8 * q^9 - 8 * q^11 + 8 * q^13 - 6 * q^17 - 6 * q^19 + 4 * q^21 - 18 * q^23 + 12 * q^25 - 22 * q^27 - 8 * q^29 + 4 * q^31 - 6 * q^33 - 4 * q^35 - 8 * q^37 - 14 * q^39 + 16 * q^41 - 12 * q^43 + 4 * q^45 - 18 * q^47 + 6 * q^49 + 4 * q^51 - 2 * q^53 + 8 * q^55 + 8 * q^57 - 16 * q^59 + 6 * q^61 - 8 * q^63 - 22 * q^65 - 12 * q^67 + 16 * q^69 + 2 * q^71 + 8 * q^73 - 24 * q^75 + 8 * q^77 + 18 * q^79 + 18 * q^81 - 28 * q^83 - 4 * q^85 + 2 * q^87 - 2 * q^89 - 8 * q^91 - 32 * q^93 - 26 * q^95 - 18 * q^97 - 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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