LMFDB - Embedded newform 3808.2.a.i.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3808,2,Mod(1,3808)]

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

lf = mfeigenbasis(mf)

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")

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

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

Self dual:	yes
Analytic conductor:	$30.4070330897$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.80686992.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 10x^{4} + 15x^{3} + 8x^{2} - 15x + 3 $ x^6 - x^5 - 10x^4 + 15x^3 + 8x^2 - 15x + 3
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			1.6
Root			$-1.06298$ 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)

$f(q)$	$=$	$q+2.39573 q^{3} -1.16130 q^{5} +1.00000 q^{7} +2.73954 q^{9} -3.38533 q^{11} -2.57825 q^{13} -2.78216 q^{15} -1.00000 q^{17} -6.30698 q^{19} +2.39573 q^{21} +8.41214 q^{23} -3.65139 q^{25} -0.623981 q^{27} -5.38533 q^{29} -2.05025 q^{31} -8.11035 q^{33} -1.16130 q^{35} +3.25564 q^{37} -6.17680 q^{39} +7.90703 q^{41} -8.38012 q^{43} -3.18142 q^{45} -4.48077 q^{47} +1.00000 q^{49} -2.39573 q^{51} -1.33508 q^{53} +3.93137 q^{55} -15.1098 q^{57} -11.6695 q^{59} -3.78325 q^{61} +2.73954 q^{63} +2.99411 q^{65} +3.57716 q^{67} +20.1533 q^{69} +10.4329 q^{71} +1.13637 q^{73} -8.74777 q^{75} -3.38533 q^{77} -13.1994 q^{79} -9.71353 q^{81} -17.5584 q^{83} +1.16130 q^{85} -12.9018 q^{87} -7.74736 q^{89} -2.57825 q^{91} -4.91185 q^{93} +7.32427 q^{95} -8.44376 q^{97} -9.27427 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 6 q^{5} + 6 q^{7} + 4 q^{9} - 2 q^{11} - 4 q^{13} + 2 q^{15} - 6 q^{17} - 10 q^{19} - 2 q^{21} + 4 q^{23} + 4 q^{25} - 8 q^{27} - 14 q^{29} + 8 q^{31} - 8 q^{33} - 6 q^{35} - 4 q^{37} + 14 q^{39}+ \cdots - 4 q^{99}+O(q^{100}) $ 6 * q - 2 * q^3 - 6 * q^5 + 6 * q^7 + 4 * q^9 - 2 * q^11 - 4 * q^13 + 2 * q^15 - 6 * q^17 - 10 * q^19 - 2 * q^21 + 4 * q^23 + 4 * q^25 - 8 * q^27 - 14 * q^29 + 8 * q^31 - 8 * q^33 - 6 * q^35 - 4 * q^37 + 14 * q^39 - 2 * q^41 - 8 * q^43 - 8 * q^45 - 2 * q^47 + 6 * q^49 + 2 * q^51 - 10 * q^53 + 2 * q^55 - 18 * q^57 - 14 * q^59 - 20 * q^61 + 4 * q^63 - 20 * q^65 - 6 * q^67 - 14 * q^69 + 20 * q^71 - 6 * q^73 - 10 * q^75 - 2 * q^77 + 38 * q^79 - 18 * q^81 - 16 * q^83 + 6 * q^85 - 4 * q^87 - 2 * q^89 - 4 * q^91 - 16 * q^93 + 42 * q^95 - 12 * q^97 - 4 * 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
$3$	2.39573	1.38318	0.691589	−	0.722291i	$-0.256911\pi$
$3$	2.39573	1.38318	0.691589	+	0.722291i	$0.256911\pi$
$4$	0	0
$5$	−1.16130	−0.519347	−0.259674	−	0.965696i	$-0.583615\pi$
$5$	−1.16130	−0.519347	−0.259674	+	0.965696i	$0.583615\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	2.73954	0.913182
$10$	0	0
$11$	−3.38533	−1.02072	−0.510358	−	0.859962i	$-0.670487\pi$
$11$	−3.38533	−1.02072	−0.510358	+	0.859962i	$0.670487\pi$
$12$	0	0
$13$	−2.57825	−0.715078	−0.357539	−	0.933898i	$-0.616384\pi$
$13$	−2.57825	−0.715078	−0.357539	+	0.933898i	$0.616384\pi$
$14$	0	0
$15$	−2.78216	−0.718350
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−6.30698	−1.44692	−0.723460	−	0.690366i	$-0.757449\pi$
$19$	−6.30698	−1.44692	−0.723460	+	0.690366i	$0.757449\pi$
$20$	0	0
$21$	2.39573	0.522792
$22$	0	0
$23$	8.41214	1.75405	0.877026	−	0.480443i	$-0.159524\pi$
$23$	8.41214	1.75405	0.877026	+	0.480443i	$0.159524\pi$
$24$	0	0
$25$	−3.65139	−0.730278
$26$	0	0
$27$	−0.623981	−0.120085
$28$	0	0
$29$	−5.38533	−1.00003	−0.500015	−	0.866017i	$-0.666672\pi$
$29$	−5.38533	−1.00003	−0.500015	+	0.866017i	$0.666672\pi$
$30$	0	0
$31$	−2.05025	−0.368235	−0.184118	−	0.982904i	$-0.558943\pi$
$31$	−2.05025	−0.368235	−0.184118	+	0.982904i	$0.558943\pi$
$32$	0	0
$33$	−8.11035	−1.41183
$34$	0	0
$35$	−1.16130	−0.196295
$36$	0	0
$37$	3.25564	0.535224	0.267612	−	0.963527i	$-0.413765\pi$
$37$	3.25564	0.535224	0.267612	+	0.963527i	$0.413765\pi$
$38$	0	0
$39$	−6.17680	−0.989080
$40$	0	0
$41$	7.90703	1.23487	0.617436	−	0.786621i	$-0.288172\pi$
$41$	7.90703	1.23487	0.617436	+	0.786621i	$0.288172\pi$
$42$	0	0
$43$	−8.38012	−1.27796	−0.638978	−	0.769225i	$-0.720643\pi$
$43$	−8.38012	−1.27796	−0.638978	+	0.769225i	$0.720643\pi$
$44$	0	0
$45$	−3.18142	−0.474258
$46$	0	0
$47$	−4.48077	−0.653587	−0.326794	−	0.945096i	$-0.605968\pi$
$47$	−4.48077	−0.653587	−0.326794	+	0.945096i	$0.605968\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.39573	−0.335470
$52$	0	0
$53$	−1.33508	−0.183388	−0.0916940	−	0.995787i	$-0.529228\pi$
$53$	−1.33508	−0.183388	−0.0916940	+	0.995787i	$0.529228\pi$
$54$	0	0
$55$	3.93137	0.530106
$56$	0	0
$57$	−15.1098	−2.00135
$58$	0	0
$59$	−11.6695	−1.51924	−0.759618	−	0.650369i	$-0.774614\pi$
$59$	−11.6695	−1.51924	−0.759618	+	0.650369i	$0.774614\pi$
$60$	0	0
$61$	−3.78325	−0.484395	−0.242198	−	0.970227i	$-0.577868\pi$
$61$	−3.78325	−0.484395	−0.242198	+	0.970227i	$0.577868\pi$
$62$	0	0
$63$	2.73954	0.345150
$64$	0	0
$65$	2.99411	0.371374
$66$	0	0
$67$	3.57716	0.437019	0.218510	−	0.975835i	$-0.429880\pi$
$67$	3.57716	0.437019	0.218510	+	0.975835i	$0.429880\pi$
$68$	0	0
$69$	20.1533	2.42617
$70$	0	0
$71$	10.4329	1.23816	0.619081	−	0.785327i	$-0.287505\pi$
$71$	10.4329	1.23816	0.619081	+	0.785327i	$0.287505\pi$
$72$	0	0
$73$	1.13637	0.133002	0.0665011	−	0.997786i	$-0.478816\pi$
$73$	1.13637	0.133002	0.0665011	+	0.997786i	$0.478816\pi$
$74$	0	0
$75$	−8.74777	−1.01010
$76$	0	0
$77$	−3.38533	−0.385794
$78$	0	0
$79$	−13.1994	−1.48505	−0.742524	−	0.669819i	$-0.766372\pi$
$79$	−13.1994	−1.48505	−0.742524	+	0.669819i	$0.766372\pi$
$80$	0	0
$81$	−9.71353	−1.07928
$82$	0	0
$83$	−17.5584	−1.92729	−0.963643	−	0.267194i	$-0.913904\pi$
$83$	−17.5584	−1.92729	−0.963643	+	0.267194i	$0.913904\pi$
$84$	0	0
$85$	1.16130	0.125960
$86$	0	0
$87$	−12.9018	−1.38322
$88$	0	0
$89$	−7.74736	−0.821218	−0.410609	−	0.911812i	$-0.634684\pi$
$89$	−7.74736	−0.821218	−0.410609	+	0.911812i	$0.634684\pi$
$90$	0	0
$91$	−2.57825	−0.270274
$92$	0	0
$93$	−4.91185	−0.509335
$94$	0	0
$95$	7.32427	0.751454
$96$	0	0
$97$	−8.44376	−0.857334	−0.428667	−	0.903463i	$-0.641017\pi$
$97$	−8.44376	−0.857334	−0.428667	+	0.903463i	$0.641017\pi$
$98$	0	0
$99$	−9.27427	−0.932099
$100$	0	0
$101$	3.59855	0.358069	0.179035	−	0.983843i	$-0.442703\pi$
$101$	3.59855	0.358069	0.179035	+	0.983843i	$0.442703\pi$
$102$	0	0
$103$	3.44207	0.339157	0.169579	−	0.985517i	$-0.445759\pi$
$103$	3.44207	0.339157	0.169579	+	0.985517i	$0.445759\pi$
$104$	0	0
$105$	−2.78216	−0.271511
$106$	0	0
$107$	15.6200	1.51004	0.755019	−	0.655703i	$-0.227627\pi$
$107$	15.6200	1.51004	0.755019	+	0.655703i	$0.227627\pi$
$108$	0	0
$109$	13.4271	1.28608	0.643039	−	0.765833i	$-0.277673\pi$
$109$	13.4271	1.28608	0.643039	+	0.765833i	$0.277673\pi$
$110$	0	0
$111$	7.79965	0.740310
$112$	0	0
$113$	−5.15420	−0.484867	−0.242433	−	0.970168i	$-0.577946\pi$
$113$	−5.15420	−0.484867	−0.242433	+	0.970168i	$0.577946\pi$
$114$	0	0
$115$	−9.76898	−0.910962
$116$	0	0
$117$	−7.06323	−0.652996
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	0.460465	0.0418605
$122$	0	0
$123$	18.9432	1.70805
$124$	0	0
$125$	10.0468	0.898615
$126$	0	0
$127$	16.2555	1.44245	0.721223	−	0.692703i	$-0.243580\pi$
$127$	16.2555	1.44245	0.721223	+	0.692703i	$0.243580\pi$
$128$	0	0
$129$	−20.0765	−1.76764
$130$	0	0
$131$	8.84678	0.772947	0.386473	−	0.922301i	$-0.373693\pi$
$131$	8.84678	0.772947	0.386473	+	0.922301i	$0.373693\pi$
$132$	0	0
$133$	−6.30698	−0.546884
$134$	0	0
$135$	0.724627	0.0623660
$136$	0	0
$137$	−12.8215	−1.09541	−0.547706	−	0.836671i	$-0.684499\pi$
$137$	−12.8215	−1.09541	−0.547706	+	0.836671i	$0.684499\pi$
$138$	0	0
$139$	6.94688	0.589226	0.294613	−	0.955617i	$-0.404809\pi$
$139$	6.94688	0.589226	0.294613	+	0.955617i	$0.404809\pi$
$140$	0	0
$141$	−10.7347	−0.904027
$142$	0	0
$143$	8.72823	0.729891
$144$	0	0
$145$	6.25396	0.519363
$146$	0	0
$147$	2.39573	0.197597
$148$	0	0
$149$	−15.9355	−1.30549	−0.652744	−	0.757578i	$-0.726382\pi$
$149$	−15.9355	−1.30549	−0.652744	+	0.757578i	$0.726382\pi$
$150$	0	0
$151$	5.31591	0.432603	0.216301	−	0.976327i	$-0.430601\pi$
$151$	5.31591	0.432603	0.216301	+	0.976327i	$0.430601\pi$
$152$	0	0
$153$	−2.73954	−0.221479
$154$	0	0
$155$	2.38094	0.191242
$156$	0	0
$157$	1.66382	0.132788	0.0663938	−	0.997793i	$-0.478851\pi$
$157$	1.66382	0.132788	0.0663938	+	0.997793i	$0.478851\pi$
$158$	0	0
$159$	−3.19851	−0.253658
$160$	0	0
$161$	8.41214	0.662969
$162$	0	0
$163$	19.9624	1.56357	0.781787	−	0.623545i	$-0.214308\pi$
$163$	19.9624	1.56357	0.781787	+	0.623545i	$0.214308\pi$
$164$	0	0
$165$	9.41852	0.733231
$166$	0	0
$167$	−0.251143	−0.0194340	−0.00971701	−	0.999953i	$-0.503093\pi$
$167$	−0.251143	−0.0194340	−0.00971701	+	0.999953i	$0.503093\pi$
$168$	0	0
$169$	−6.35263	−0.488664
$170$	0	0
$171$	−17.2783	−1.32130
$172$	0	0
$173$	−14.1833	−1.07834	−0.539170	−	0.842197i	$-0.681262\pi$
$173$	−14.1833	−1.07834	−0.539170	+	0.842197i	$0.681262\pi$
$174$	0	0
$175$	−3.65139	−0.276019
$176$	0	0
$177$	−27.9570	−2.10137
$178$	0	0
$179$	−2.76596	−0.206737	−0.103369	−	0.994643i	$-0.532962\pi$
$179$	−2.76596	−0.206737	−0.103369	+	0.994643i	$0.532962\pi$
$180$	0	0
$181$	−8.53340	−0.634283	−0.317141	−	0.948378i	$-0.602723\pi$
$181$	−8.53340	−0.634283	−0.317141	+	0.948378i	$0.602723\pi$
$182$	0	0
$183$	−9.06366	−0.670005
$184$	0	0
$185$	−3.78076	−0.277967
$186$	0	0
$187$	3.38533	0.247560
$188$	0	0
$189$	−0.623981	−0.0453880
$190$	0	0
$191$	−17.8415	−1.29096	−0.645482	−	0.763776i	$-0.723343\pi$
$191$	−17.8415	−1.29096	−0.645482	+	0.763776i	$0.723343\pi$
$192$	0	0
$193$	24.1012	1.73484	0.867422	−	0.497573i	$-0.165775\pi$
$193$	24.1012	1.73484	0.867422	+	0.497573i	$0.165775\pi$
$194$	0	0
$195$	7.17309	0.513676
$196$	0	0
$197$	0.686765	0.0489300	0.0244650	−	0.999701i	$-0.492212\pi$
$197$	0.686765	0.0489300	0.0244650	+	0.999701i	$0.492212\pi$
$198$	0	0
$199$	6.64718	0.471206	0.235603	−	0.971849i	$-0.424293\pi$
$199$	6.64718	0.471206	0.235603	+	0.971849i	$0.424293\pi$
$200$	0	0
$201$	8.56992	0.604476
$202$	0	0
$203$	−5.38533	−0.377976
$204$	0	0
$205$	−9.18241	−0.641327
$206$	0	0
$207$	23.0454	1.60177
$208$	0	0
$209$	21.3512	1.47689
$210$	0	0
$211$	−10.1514	−0.698854	−0.349427	−	0.936964i	$-0.613624\pi$
$211$	−10.1514	−0.698854	−0.349427	+	0.936964i	$0.613624\pi$
$212$	0	0
$213$	24.9946	1.71260
$214$	0	0
$215$	9.73180	0.663703
$216$	0	0
$217$	−2.05025	−0.139180
$218$	0	0
$219$	2.72244	0.183966
$220$	0	0
$221$	2.57825	0.173432
$222$	0	0
$223$	−4.11035	−0.275250	−0.137625	−	0.990484i	$-0.543947\pi$
$223$	−4.11035	−0.275250	−0.137625	+	0.990484i	$0.543947\pi$
$224$	0	0
$225$	−10.0032	−0.666877
$226$	0	0
$227$	20.3981	1.35387	0.676935	−	0.736043i	$-0.263308\pi$
$227$	20.3981	1.35387	0.676935	+	0.736043i	$0.263308\pi$
$228$	0	0
$229$	11.1147	0.734478	0.367239	−	0.930127i	$-0.380303\pi$
$229$	11.1147	0.734478	0.367239	+	0.930127i	$0.380303\pi$
$230$	0	0
$231$	−8.11035	−0.533622
$232$	0	0
$233$	−17.1600	−1.12419	−0.562094	−	0.827073i	$-0.690004\pi$
$233$	−17.1600	−1.12419	−0.562094	+	0.827073i	$0.690004\pi$
$234$	0	0
$235$	5.20350	0.339439
$236$	0	0
$237$	−31.6223	−2.05409
$238$	0	0
$239$	15.8180	1.02318	0.511590	−	0.859230i	$-0.329057\pi$
$239$	15.8180	1.02318	0.511590	+	0.859230i	$0.329057\pi$
$240$	0	0
$241$	−2.02068	−0.130163	−0.0650816	−	0.997880i	$-0.520731\pi$
$241$	−2.02068	−0.130163	−0.0650816	+	0.997880i	$0.520731\pi$
$242$	0	0
$243$	−21.3991	−1.37275
$244$	0	0
$245$	−1.16130	−0.0741925
$246$	0	0
$247$	16.2610	1.03466
$248$	0	0
$249$	−42.0653	−2.66578
$250$	0	0
$251$	−25.5890	−1.61516	−0.807580	−	0.589758i	$-0.799223\pi$
$251$	−25.5890	−1.61516	−0.807580	+	0.589758i	$0.799223\pi$
$252$	0	0
$253$	−28.4779	−1.79039
$254$	0	0
$255$	2.78216	0.174225
$256$	0	0
$257$	−9.46517	−0.590421	−0.295211	−	0.955432i	$-0.595390\pi$
$257$	−9.46517	−0.590421	−0.295211	+	0.955432i	$0.595390\pi$
$258$	0	0
$259$	3.25564	0.202296
$260$	0	0
$261$	−14.7534	−0.913210
$262$	0	0
$263$	−16.4809	−1.01625	−0.508127	−	0.861282i	$-0.669662\pi$
$263$	−16.4809	−1.01625	−0.508127	+	0.861282i	$0.669662\pi$
$264$	0	0
$265$	1.55043	0.0952420
$266$	0	0
$267$	−18.5606	−1.13589
$268$	0	0
$269$	−20.6355	−1.25817	−0.629083	−	0.777338i	$-0.716569\pi$
$269$	−20.6355	−1.25817	−0.629083	+	0.777338i	$0.716569\pi$
$270$	0	0
$271$	25.2867	1.53605	0.768027	−	0.640417i	$-0.221238\pi$
$271$	25.2867	1.53605	0.768027	+	0.640417i	$0.221238\pi$
$272$	0	0
$273$	−6.17680	−0.373837
$274$	0	0
$275$	12.3612	0.745407
$276$	0	0
$277$	10.1964	0.612640	0.306320	−	0.951929i	$-0.400902\pi$
$277$	10.1964	0.612640	0.306320	+	0.951929i	$0.400902\pi$
$278$	0	0
$279$	−5.61674	−0.336266
$280$	0	0
$281$	−27.3535	−1.63177	−0.815886	−	0.578212i	$-0.803750\pi$
$281$	−27.3535	−1.63177	−0.815886	+	0.578212i	$0.803750\pi$
$282$	0	0
$283$	16.1874	0.962241	0.481120	−	0.876655i	$-0.340230\pi$
$283$	16.1874	0.962241	0.481120	+	0.876655i	$0.340230\pi$
$284$	0	0
$285$	17.5470	1.03940
$286$	0	0
$287$	7.90703	0.466737
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−20.2290	−1.18585
$292$	0	0
$293$	14.8361	0.866734	0.433367	−	0.901218i	$-0.357325\pi$
$293$	14.8361	0.866734	0.433367	+	0.901218i	$0.357325\pi$
$294$	0	0
$295$	13.5517	0.789011
$296$	0	0
$297$	2.11238	0.122573
$298$	0	0
$299$	−21.6886	−1.25428
$300$	0	0
$301$	−8.38012	−0.483022
$302$	0	0
$303$	8.62117	0.495274
$304$	0	0
$305$	4.39347	0.251569
$306$	0	0
$307$	17.0617	0.973761	0.486880	−	0.873469i	$-0.338135\pi$
$307$	17.0617	0.973761	0.486880	+	0.873469i	$0.338135\pi$
$308$	0	0
$309$	8.24628	0.469115
$310$	0	0
$311$	−24.8146	−1.40711	−0.703554	−	0.710641i	$-0.748405\pi$
$311$	−24.8146	−1.40711	−0.703554	+	0.710641i	$0.748405\pi$
$312$	0	0
$313$	−20.1543	−1.13919	−0.569595	−	0.821925i	$-0.692900\pi$
$313$	−20.1543	−1.13919	−0.569595	+	0.821925i	$0.692900\pi$
$314$	0	0
$315$	−3.18142	−0.179253
$316$	0	0
$317$	−4.01313	−0.225400	−0.112700	−	0.993629i	$-0.535950\pi$
$317$	−4.01313	−0.225400	−0.112700	+	0.993629i	$0.535950\pi$
$318$	0	0
$319$	18.2311	1.02075
$320$	0	0
$321$	37.4213	2.08865
$322$	0	0
$323$	6.30698	0.350930
$324$	0	0
$325$	9.41420	0.522206
$326$	0	0
$327$	32.1677	1.77888
$328$	0	0
$329$	−4.48077	−0.247033
$330$	0	0
$331$	33.0083	1.81430	0.907151	−	0.420805i	$-0.138252\pi$
$331$	33.0083	1.81430	0.907151	+	0.420805i	$0.138252\pi$
$332$	0	0
$333$	8.91898	0.488757
$334$	0	0
$335$	−4.15414	−0.226965
$336$	0	0
$337$	21.9800	1.19732	0.598662	−	0.801002i	$-0.295699\pi$
$337$	21.9800	1.19732	0.598662	+	0.801002i	$0.295699\pi$
$338$	0	0
$339$	−12.3481	−0.670657
$340$	0	0
$341$	6.94076	0.375863
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−23.4039	−1.26002
$346$	0	0
$347$	34.7488	1.86541	0.932707	−	0.360635i	$-0.117440\pi$
$347$	34.7488	1.86541	0.932707	+	0.360635i	$0.117440\pi$
$348$	0	0
$349$	13.4233	0.718535	0.359267	−	0.933235i	$-0.383027\pi$
$349$	13.4233	0.718535	0.359267	+	0.933235i	$0.383027\pi$
$350$	0	0
$351$	1.60878	0.0858703
$352$	0	0
$353$	18.0159	0.958887	0.479443	−	0.877573i	$-0.340839\pi$
$353$	18.0159	0.958887	0.479443	+	0.877573i	$0.340839\pi$
$354$	0	0
$355$	−12.1157	−0.643037
$356$	0	0
$357$	−2.39573	−0.126796
$358$	0	0
$359$	−20.0356	−1.05744	−0.528720	−	0.848796i	$-0.677328\pi$
$359$	−20.0356	−1.05744	−0.528720	+	0.848796i	$0.677328\pi$
$360$	0	0
$361$	20.7780	1.09358
$362$	0	0
$363$	1.10315	0.0579005
$364$	0	0
$365$	−1.31966	−0.0690743
$366$	0	0
$367$	33.6842	1.75830	0.879151	−	0.476543i	$-0.158110\pi$
$367$	33.6842	1.75830	0.879151	+	0.476543i	$0.158110\pi$
$368$	0	0
$369$	21.6617	1.12766
$370$	0	0
$371$	−1.33508	−0.0693141
$372$	0	0
$373$	3.98420	0.206294	0.103147	−	0.994666i	$-0.467109\pi$
$373$	3.98420	0.206294	0.103147	+	0.994666i	$0.467109\pi$
$374$	0	0
$375$	24.0695	1.24295
$376$	0	0
$377$	13.8847	0.715100
$378$	0	0
$379$	−24.3603	−1.25131	−0.625653	−	0.780101i	$-0.715167\pi$
$379$	−24.3603	−1.25131	−0.625653	+	0.780101i	$0.715167\pi$
$380$	0	0
$381$	38.9439	1.99516
$382$	0	0
$383$	−6.21921	−0.317787	−0.158893	−	0.987296i	$-0.550793\pi$
$383$	−6.21921	−0.317787	−0.158893	+	0.987296i	$0.550793\pi$
$384$	0	0
$385$	3.93137	0.200361
$386$	0	0
$387$	−22.9577	−1.16701
$388$	0	0
$389$	−37.8774	−1.92046	−0.960230	−	0.279212i	$-0.909927\pi$
$389$	−37.8774	−1.92046	−0.960230	+	0.279212i	$0.909927\pi$
$390$	0	0
$391$	−8.41214	−0.425420
$392$	0	0
$393$	21.1945	1.06912
$394$	0	0
$395$	15.3284	0.771256
$396$	0	0
$397$	−18.3294	−0.919924	−0.459962	−	0.887939i	$-0.652137\pi$
$397$	−18.3294	−0.919924	−0.459962	+	0.887939i	$0.652137\pi$
$398$	0	0
$399$	−15.1098	−0.756439
$400$	0	0
$401$	−11.6643	−0.582486	−0.291243	−	0.956649i	$-0.594069\pi$
$401$	−11.6643	−0.582486	−0.291243	+	0.956649i	$0.594069\pi$
$402$	0	0
$403$	5.28605	0.263317
$404$	0	0
$405$	11.2803	0.560522
$406$	0	0
$407$	−11.0214	−0.546312
$408$	0	0
$409$	−0.332982	−0.0164649	−0.00823246	−	0.999966i	$-0.502621\pi$
$409$	−0.332982	−0.0164649	−0.00823246	+	0.999966i	$0.502621\pi$
$410$	0	0
$411$	−30.7168	−1.51515
$412$	0	0
$413$	−11.6695	−0.574217
$414$	0	0
$415$	20.3905	1.00093
$416$	0	0
$417$	16.6429	0.815005
$418$	0	0
$419$	−27.1772	−1.32769	−0.663846	−	0.747869i	$-0.731077\pi$
$419$	−27.1772	−1.32769	−0.663846	+	0.747869i	$0.731077\pi$
$420$	0	0
$421$	−2.40440	−0.117183	−0.0585916	−	0.998282i	$-0.518661\pi$
$421$	−2.40440	−0.117183	−0.0585916	+	0.998282i	$0.518661\pi$
$422$	0	0
$423$	−12.2753	−0.596844
$424$	0	0
$425$	3.65139	0.177119
$426$	0	0
$427$	−3.78325	−0.183084
$428$	0	0
$429$	20.9105	1.00957
$430$	0	0
$431$	39.8697	1.92046	0.960228	−	0.279216i	$-0.0900747\pi$
$431$	39.8697	1.92046	0.960228	+	0.279216i	$0.0900747\pi$
$432$	0	0
$433$	1.06816	0.0513324	0.0256662	−	0.999671i	$-0.491829\pi$
$433$	1.06816	0.0513324	0.0256662	+	0.999671i	$0.491829\pi$
$434$	0	0
$435$	14.9828	0.718372
$436$	0	0
$437$	−53.0552	−2.53797
$438$	0	0
$439$	10.9720	0.523663	0.261831	−	0.965114i	$-0.415674\pi$
$439$	10.9720	0.523663	0.261831	+	0.965114i	$0.415674\pi$
$440$	0	0
$441$	2.73954	0.130455
$442$	0	0
$443$	6.81228	0.323661	0.161831	−	0.986819i	$-0.448260\pi$
$443$	6.81228	0.323661	0.161831	+	0.986819i	$0.448260\pi$
$444$	0	0
$445$	8.99697	0.426497
$446$	0	0
$447$	−38.1773	−1.80572
$448$	0	0
$449$	16.4833	0.777895	0.388947	−	0.921260i	$-0.372839\pi$
$449$	16.4833	0.777895	0.388947	+	0.921260i	$0.372839\pi$
$450$	0	0
$451$	−26.7679	−1.26045
$452$	0	0
$453$	12.7355	0.598367
$454$	0	0
$455$	2.99411	0.140366
$456$	0	0
$457$	−6.04469	−0.282758	−0.141379	−	0.989956i	$-0.545154\pi$
$457$	−6.04469	−0.282758	−0.141379	+	0.989956i	$0.545154\pi$
$458$	0	0
$459$	0.623981	0.0291250
$460$	0	0
$461$	−5.69584	−0.265282	−0.132641	−	0.991164i	$-0.542346\pi$
$461$	−5.69584	−0.265282	−0.132641	+	0.991164i	$0.542346\pi$
$462$	0	0
$463$	−20.8144	−0.967328	−0.483664	−	0.875254i	$-0.660694\pi$
$463$	−20.8144	−0.967328	−0.483664	+	0.875254i	$0.660694\pi$
$464$	0	0
$465$	5.70411	0.264522
$466$	0	0
$467$	28.1763	1.30384	0.651922	−	0.758286i	$-0.273963\pi$
$467$	28.1763	1.30384	0.651922	+	0.758286i	$0.273963\pi$
$468$	0	0
$469$	3.57716	0.165178
$470$	0	0
$471$	3.98608	0.183669
$472$	0	0
$473$	28.3695	1.30443
$474$	0	0
$475$	23.0293	1.05665
$476$	0	0
$477$	−3.65752	−0.167467
$478$	0	0
$479$	23.0546	1.05339	0.526696	−	0.850054i	$-0.323431\pi$
$479$	23.0546	1.05339	0.526696	+	0.850054i	$0.323431\pi$
$480$	0	0
$481$	−8.39385	−0.382727
$482$	0	0
$483$	20.1533	0.917005
$484$	0	0
$485$	9.80571	0.445254
$486$	0	0
$487$	10.8207	0.490331	0.245166	−	0.969481i	$-0.421158\pi$
$487$	10.8207	0.490331	0.245166	+	0.969481i	$0.421158\pi$
$488$	0	0
$489$	47.8246	2.16270
$490$	0	0
$491$	−23.5001	−1.06055	−0.530273	−	0.847827i	$-0.677911\pi$
$491$	−23.5001	−1.06055	−0.530273	+	0.847827i	$0.677911\pi$
$492$	0	0
$493$	5.38533	0.242543
$494$	0	0
$495$	10.7702	0.484083
$496$	0	0
$497$	10.4329	0.467982
$498$	0	0
$499$	−26.6838	−1.19453	−0.597265	−	0.802044i	$-0.703746\pi$
$499$	−26.6838	−1.19453	−0.597265	+	0.802044i	$0.703746\pi$
$500$	0	0
$501$	−0.601672	−0.0268807
$502$	0	0
$503$	24.8091	1.10618	0.553091	−	0.833121i	$-0.313448\pi$
$503$	24.8091	1.10618	0.553091	+	0.833121i	$0.313448\pi$
$504$	0	0
$505$	−4.17898	−0.185962
$506$	0	0
$507$	−15.2192	−0.675909
$508$	0	0
$509$	−40.6166	−1.80030	−0.900151	−	0.435579i	$-0.856544\pi$
$509$	−40.6166	−1.80030	−0.900151	+	0.435579i	$0.856544\pi$
$510$	0	0
$511$	1.13637	0.0502701
$512$	0	0
$513$	3.93544	0.173754
$514$	0	0
$515$	−3.99726	−0.176140
$516$	0	0
$517$	15.1689	0.667127
$518$	0	0
$519$	−33.9795	−1.49154
$520$	0	0
$521$	−22.1578	−0.970749	−0.485375	−	0.874306i	$-0.661317\pi$
$521$	−22.1578	−0.970749	−0.485375	+	0.874306i	$0.661317\pi$
$522$	0	0
$523$	−33.3672	−1.45905	−0.729523	−	0.683956i	$-0.760258\pi$
$523$	−33.3672	−1.45905	−0.729523	+	0.683956i	$0.760258\pi$
$524$	0	0
$525$	−8.74777	−0.381784
$526$	0	0
$527$	2.05025	0.0893101
$528$	0	0
$529$	47.7641	2.07670
$530$	0	0
$531$	−31.9690	−1.38734
$532$	0	0
$533$	−20.3863	−0.883029
$534$	0	0
$535$	−18.1394	−0.784235
$536$	0	0
$537$	−6.62650	−0.285955
$538$	0	0
$539$	−3.38533	−0.145817
$540$	0	0
$541$	−45.8423	−1.97092	−0.985458	−	0.169919i	$-0.945649\pi$
$541$	−45.8423	−1.97092	−0.985458	+	0.169919i	$0.945649\pi$
$542$	0	0
$543$	−20.4438	−0.877326
$544$	0	0
$545$	−15.5928	−0.667922
$546$	0	0
$547$	−21.6713	−0.926597	−0.463298	−	0.886202i	$-0.653334\pi$
$547$	−21.6713	−0.926597	−0.463298	+	0.886202i	$0.653334\pi$
$548$	0	0
$549$	−10.3644	−0.442341
$550$	0	0
$551$	33.9652	1.44696
$552$	0	0
$553$	−13.1994	−0.561295
$554$	0	0
$555$	−9.05771	−0.384478
$556$	0	0
$557$	37.8659	1.60443	0.802214	−	0.597036i	$-0.203655\pi$
$557$	37.8659	1.60443	0.802214	+	0.597036i	$0.203655\pi$
$558$	0	0
$559$	21.6060	0.913838
$560$	0	0
$561$	8.11035	0.342419
$562$	0	0
$563$	9.58873	0.404117	0.202058	−	0.979373i	$-0.435237\pi$
$563$	9.58873	0.404117	0.202058	+	0.979373i	$0.435237\pi$
$564$	0	0
$565$	5.98556	0.251814
$566$	0	0
$567$	−9.71353	−0.407930
$568$	0	0
$569$	23.6665	0.992153	0.496077	−	0.868279i	$-0.334774\pi$
$569$	23.6665	0.992153	0.496077	+	0.868279i	$0.334774\pi$
$570$	0	0
$571$	5.53501	0.231633	0.115816	−	0.993271i	$-0.463052\pi$
$571$	5.53501	0.231633	0.115816	+	0.993271i	$0.463052\pi$
$572$	0	0
$573$	−42.7434	−1.78563
$574$	0	0
$575$	−30.7160	−1.28095
$576$	0	0
$577$	37.1491	1.54654	0.773268	−	0.634079i	$-0.218621\pi$
$577$	37.1491	1.54654	0.773268	+	0.634079i	$0.218621\pi$
$578$	0	0
$579$	57.7401	2.39960
$580$	0	0
$581$	−17.5584	−0.728446
$582$	0	0
$583$	4.51970	0.187187
$584$	0	0
$585$	8.20250	0.339132
$586$	0	0
$587$	−21.7498	−0.897712	−0.448856	−	0.893604i	$-0.648168\pi$
$587$	−21.7498	−0.897712	−0.448856	+	0.893604i	$0.648168\pi$
$588$	0	0
$589$	12.9309	0.532807
$590$	0	0
$591$	1.64531	0.0676789
$592$	0	0
$593$	−37.2945	−1.53150	−0.765750	−	0.643138i	$-0.777632\pi$
$593$	−37.2945	−1.53150	−0.765750	+	0.643138i	$0.777632\pi$
$594$	0	0
$595$	1.16130	0.0476085
$596$	0	0
$597$	15.9249	0.651762
$598$	0	0
$599$	−36.8856	−1.50711	−0.753553	−	0.657387i	$-0.771662\pi$
$599$	−36.8856	−1.50711	−0.753553	+	0.657387i	$0.771662\pi$
$600$	0	0
$601$	−3.86531	−0.157669	−0.0788346	−	0.996888i	$-0.525120\pi$
$601$	−3.86531	−0.157669	−0.0788346	+	0.996888i	$0.525120\pi$
$602$	0	0
$603$	9.79978	0.399078
$604$	0	0
$605$	−0.534736	−0.0217401
$606$	0	0
$607$	6.04946	0.245540	0.122770	−	0.992435i	$-0.460822\pi$
$607$	6.04946	0.245540	0.122770	+	0.992435i	$0.460822\pi$
$608$	0	0
$609$	−12.9018	−0.522808
$610$	0	0
$611$	11.5525	0.467366
$612$	0	0
$613$	9.26251	0.374109	0.187055	−	0.982350i	$-0.440106\pi$
$613$	9.26251	0.374109	0.187055	+	0.982350i	$0.440106\pi$
$614$	0	0
$615$	−21.9986	−0.887070
$616$	0	0
$617$	10.9184	0.439558	0.219779	−	0.975550i	$-0.429466\pi$
$617$	10.9184	0.439558	0.219779	+	0.975550i	$0.429466\pi$
$618$	0	0
$619$	1.96825	0.0791108	0.0395554	−	0.999217i	$-0.487406\pi$
$619$	1.96825	0.0791108	0.0395554	+	0.999217i	$0.487406\pi$
$620$	0	0
$621$	−5.24902	−0.210636
$622$	0	0
$623$	−7.74736	−0.310391
$624$	0	0
$625$	6.58962	0.263585
$626$	0	0
$627$	51.1518	2.04281
$628$	0	0
$629$	−3.25564	−0.129811
$630$	0	0
$631$	−3.34802	−0.133283	−0.0666414	−	0.997777i	$-0.521228\pi$
$631$	−3.34802	−0.133283	−0.0666414	+	0.997777i	$0.521228\pi$
$632$	0	0
$633$	−24.3202	−0.966640
$634$	0	0
$635$	−18.8775	−0.749130
$636$	0	0
$637$	−2.57825	−0.102154
$638$	0	0
$639$	28.5815	1.13067
$640$	0	0
$641$	−8.23207	−0.325147	−0.162574	−	0.986696i	$-0.551980\pi$
$641$	−8.23207	−0.325147	−0.162574	+	0.986696i	$0.551980\pi$
$642$	0	0
$643$	−5.24865	−0.206986	−0.103493	−	0.994630i	$-0.533002\pi$
$643$	−5.24865	−0.206986	−0.103493	+	0.994630i	$0.533002\pi$
$644$	0	0
$645$	23.3148	0.918020
$646$	0	0
$647$	17.9286	0.704846	0.352423	−	0.935841i	$-0.385358\pi$
$647$	17.9286	0.704846	0.352423	+	0.935841i	$0.385358\pi$
$648$	0	0
$649$	39.5050	1.55071
$650$	0	0
$651$	−4.91185	−0.192510
$652$	0	0
$653$	−10.2360	−0.400565	−0.200282	−	0.979738i	$-0.564186\pi$
$653$	−10.2360	−0.400565	−0.200282	+	0.979738i	$0.564186\pi$
$654$	0	0
$655$	−10.2737	−0.401428
$656$	0	0
$657$	3.11314	0.121455
$658$	0	0
$659$	36.9011	1.43746	0.718732	−	0.695287i	$-0.244723\pi$
$659$	36.9011	1.43746	0.718732	+	0.695287i	$0.244723\pi$
$660$	0	0
$661$	42.7374	1.66229	0.831146	−	0.556055i	$-0.187685\pi$
$661$	42.7374	1.66229	0.831146	+	0.556055i	$0.187685\pi$
$662$	0	0
$663$	6.17680	0.239887
$664$	0	0
$665$	7.32427	0.284023
$666$	0	0
$667$	−45.3022	−1.75411
$668$	0	0
$669$	−9.84732	−0.380719
$670$	0	0
$671$	12.8075	0.494430
$672$	0	0
$673$	−28.3933	−1.09448	−0.547241	−	0.836975i	$-0.684322\pi$
$673$	−28.3933	−1.09448	−0.547241	+	0.836975i	$0.684322\pi$
$674$	0	0
$675$	2.27840	0.0876957
$676$	0	0
$677$	−42.9223	−1.64964	−0.824818	−	0.565398i	$-0.808723\pi$
$677$	−42.9223	−1.64964	−0.824818	+	0.565398i	$0.808723\pi$
$678$	0	0
$679$	−8.44376	−0.324042
$680$	0	0
$681$	48.8684	1.87264
$682$	0	0
$683$	13.2145	0.505640	0.252820	−	0.967513i	$-0.418642\pi$
$683$	13.2145	0.505640	0.252820	+	0.967513i	$0.418642\pi$
$684$	0	0
$685$	14.8895	0.568899
$686$	0	0
$687$	26.6278	1.01591
$688$	0	0
$689$	3.44218	0.131137
$690$	0	0
$691$	−13.5646	−0.516022	−0.258011	−	0.966142i	$-0.583067\pi$
$691$	−13.5646	−0.516022	−0.258011	+	0.966142i	$0.583067\pi$
$692$	0	0
$693$	−9.27427	−0.352300
$694$	0	0
$695$	−8.06738	−0.306013
$696$	0	0
$697$	−7.90703	−0.299500
$698$	0	0
$699$	−41.1108	−1.55495
$700$	0	0
$701$	18.3008	0.691212	0.345606	−	0.938380i	$-0.387673\pi$
$701$	18.3008	0.691212	0.345606	+	0.938380i	$0.387673\pi$
$702$	0	0
$703$	−20.5333	−0.774427
$704$	0	0
$705$	12.4662	0.469504
$706$	0	0
$707$	3.59855	0.135337
$708$	0	0
$709$	18.8780	0.708980	0.354490	−	0.935060i	$-0.384654\pi$
$709$	18.8780	0.708980	0.354490	+	0.935060i	$0.384654\pi$
$710$	0	0
$711$	−36.1603	−1.35612
$712$	0	0
$713$	−17.2470	−0.645904
$714$	0	0
$715$	−10.1361	−0.379067
$716$	0	0
$717$	37.8957	1.41524
$718$	0	0
$719$	38.8925	1.45044	0.725222	−	0.688515i	$-0.241737\pi$
$719$	38.8925	1.45044	0.725222	+	0.688515i	$0.241737\pi$
$720$	0	0
$721$	3.44207	0.128189
$722$	0	0
$723$	−4.84100	−0.180039
$724$	0	0
$725$	19.6640	0.730301
$726$	0	0
$727$	−33.3780	−1.23792	−0.618961	−	0.785421i	$-0.712446\pi$
$727$	−33.3780	−1.23792	−0.618961	+	0.785421i	$0.712446\pi$
$728$	0	0
$729$	−22.1260	−0.819480
$730$	0	0
$731$	8.38012	0.309950
$732$	0	0
$733$	−23.0143	−0.850053	−0.425026	−	0.905181i	$-0.639735\pi$
$733$	−23.0143	−0.850053	−0.425026	+	0.905181i	$0.639735\pi$
$734$	0	0
$735$	−2.78216	−0.102621
$736$	0	0
$737$	−12.1099	−0.446072
$738$	0	0
$739$	14.9546	0.550116	0.275058	−	0.961428i	$-0.411303\pi$
$739$	14.9546	0.550116	0.275058	+	0.961428i	$0.411303\pi$
$740$	0	0
$741$	38.9570	1.43112
$742$	0	0
$743$	−13.2351	−0.485549	−0.242774	−	0.970083i	$-0.578057\pi$
$743$	−13.2351	−0.485549	−0.242774	+	0.970083i	$0.578057\pi$
$744$	0	0
$745$	18.5059	0.678002
$746$	0	0
$747$	−48.1020	−1.75996
$748$	0	0
$749$	15.6200	0.570741
$750$	0	0
$751$	−38.9998	−1.42312	−0.711562	−	0.702624i	$-0.752012\pi$
$751$	−38.9998	−1.42312	−0.711562	+	0.702624i	$0.752012\pi$
$752$	0	0
$753$	−61.3043	−2.23405
$754$	0	0
$755$	−6.17335	−0.224671
$756$	0	0
$757$	−10.1665	−0.369509	−0.184754	−	0.982785i	$-0.559149\pi$
$757$	−10.1665	−0.369509	−0.184754	+	0.982785i	$0.559149\pi$
$758$	0	0
$759$	−68.2254	−2.47643
$760$	0	0
$761$	−34.7369	−1.25921	−0.629606	−	0.776914i	$-0.716784\pi$
$761$	−34.7369	−1.25921	−0.629606	+	0.776914i	$0.716784\pi$
$762$	0	0
$763$	13.4271	0.486092
$764$	0	0
$765$	3.18142	0.115025
$766$	0	0
$767$	30.0868	1.08637
$768$	0	0
$769$	11.8679	0.427967	0.213984	−	0.976837i	$-0.431356\pi$
$769$	11.8679	0.427967	0.213984	+	0.976837i	$0.431356\pi$
$770$	0	0
$771$	−22.6760	−0.816658
$772$	0	0
$773$	6.55388	0.235727	0.117863	−	0.993030i	$-0.462396\pi$
$773$	6.55388	0.235727	0.117863	+	0.993030i	$0.462396\pi$
$774$	0	0
$775$	7.48625	0.268914
$776$	0	0
$777$	7.79965	0.279811
$778$	0	0
$779$	−49.8695	−1.78676
$780$	0	0
$781$	−35.3190	−1.26381
$782$	0	0
$783$	3.36035	0.120089
$784$	0	0
$785$	−1.93219	−0.0689629
$786$	0	0
$787$	−21.4732	−0.765438	−0.382719	−	0.923865i	$-0.625012\pi$
$787$	−21.4732	−0.765438	−0.382719	+	0.923865i	$0.625012\pi$
$788$	0	0
$789$	−39.4838	−1.40566
$790$	0	0
$791$	−5.15420	−0.183262
$792$	0	0
$793$	9.75416	0.346380
$794$	0	0
$795$	3.71441	0.131737
$796$	0	0
$797$	9.66267	0.342269	0.171135	−	0.985248i	$-0.445257\pi$
$797$	9.66267	0.342269	0.171135	+	0.985248i	$0.445257\pi$
$798$	0	0
$799$	4.48077	0.158518
$800$	0	0
$801$	−21.2242	−0.749921
$802$	0	0
$803$	−3.84699	−0.135757
$804$	0	0
$805$	−9.76898	−0.344311
$806$	0	0
$807$	−49.4371	−1.74027
$808$	0	0
$809$	17.9528	0.631185	0.315593	−	0.948895i	$-0.397797\pi$
$809$	17.9528	0.631185	0.315593	+	0.948895i	$0.397797\pi$
$810$	0	0
$811$	−19.2609	−0.676342	−0.338171	−	0.941085i	$-0.609808\pi$
$811$	−19.2609	−0.676342	−0.338171	+	0.941085i	$0.609808\pi$
$812$	0	0
$813$	60.5801	2.12464
$814$	0	0
$815$	−23.1822	−0.812038
$816$	0	0
$817$	52.8533	1.84910
$818$	0	0
$819$	−7.06323	−0.246809
$820$	0	0
$821$	15.3663	0.536286	0.268143	−	0.963379i	$-0.413590\pi$
$821$	15.3663	0.536286	0.268143	+	0.963379i	$0.413590\pi$
$822$	0	0
$823$	29.8971	1.04215	0.521074	−	0.853511i	$-0.325531\pi$
$823$	29.8971	1.04215	0.521074	+	0.853511i	$0.325531\pi$
$824$	0	0
$825$	29.6141	1.03103
$826$	0	0
$827$	21.6766	0.753769	0.376884	−	0.926260i	$-0.376995\pi$
$827$	21.6766	0.753769	0.376884	+	0.926260i	$0.376995\pi$
$828$	0	0
$829$	−15.9927	−0.555450	−0.277725	−	0.960661i	$-0.589580\pi$
$829$	−15.9927	−0.555450	−0.277725	+	0.960661i	$0.589580\pi$
$830$	0	0
$831$	24.4278	0.847391
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	0.291651	0.0100930
$836$	0	0
$837$	1.27932	0.0442196
$838$	0	0
$839$	−12.7481	−0.440113	−0.220056	−	0.975487i	$-0.570624\pi$
$839$	−12.7481	−0.440113	−0.220056	+	0.975487i	$0.570624\pi$
$840$	0	0
$841$	0.00178882	6.16836e−5 0
$842$	0	0
$843$	−65.5317	−2.25703
$844$	0	0
$845$	7.37729	0.253786
$846$	0	0
$847$	0.460465	0.0158218
$848$	0	0
$849$	38.7807	1.33095
$850$	0	0
$851$	27.3869	0.938811
$852$	0	0
$853$	36.3795	1.24561	0.622805	−	0.782377i	$-0.285993\pi$
$853$	36.3795	1.24561	0.622805	+	0.782377i	$0.285993\pi$
$854$	0	0
$855$	20.0652	0.686214
$856$	0	0
$857$	−4.17301	−0.142547	−0.0712737	−	0.997457i	$-0.522706\pi$
$857$	−4.17301	−0.142547	−0.0712737	+	0.997457i	$0.522706\pi$
$858$	0	0
$859$	13.4228	0.457981	0.228991	−	0.973429i	$-0.426457\pi$
$859$	13.4228	0.457981	0.228991	+	0.973429i	$0.426457\pi$
$860$	0	0
$861$	18.9432	0.645581
$862$	0	0
$863$	11.6858	0.397790	0.198895	−	0.980021i	$-0.436265\pi$
$863$	11.6858	0.397790	0.198895	+	0.980021i	$0.436265\pi$
$864$	0	0
$865$	16.4711	0.560033
$866$	0	0
$867$	2.39573	0.0813634
$868$	0	0
$869$	44.6843	1.51581
$870$	0	0
$871$	−9.22280	−0.312503
$872$	0	0
$873$	−23.1321	−0.782902
$874$	0	0
$875$	10.0468	0.339645
$876$	0	0
$877$	−1.47281	−0.0497334	−0.0248667	−	0.999691i	$-0.507916\pi$
$877$	−1.47281	−0.0497334	−0.0248667	+	0.999691i	$0.507916\pi$
$878$	0	0
$879$	35.5434	1.19885
$880$	0	0
$881$	−11.1792	−0.376637	−0.188319	−	0.982108i	$-0.560304\pi$
$881$	−11.1792	−0.376637	−0.188319	+	0.982108i	$0.560304\pi$
$882$	0	0
$883$	−30.1071	−1.01318	−0.506592	−	0.862186i	$-0.669095\pi$
$883$	−30.1071	−1.01318	−0.506592	+	0.862186i	$0.669095\pi$
$884$	0	0
$885$	32.4663	1.09134
$886$	0	0
$887$	52.6410	1.76751	0.883756	−	0.467948i	$-0.155007\pi$
$887$	52.6410	1.76751	0.883756	+	0.467948i	$0.155007\pi$
$888$	0	0
$889$	16.2555	0.545193
$890$	0	0
$891$	32.8835	1.10164
$892$	0	0
$893$	28.2601	0.945689
$894$	0	0
$895$	3.21210	0.107369
$896$	0	0
$897$	−51.9601	−1.73490
$898$	0	0
$899$	11.0413	0.368246
$900$	0	0
$901$	1.33508	0.0444781
$902$	0	0
$903$	−20.0765	−0.668106
$904$	0	0
$905$	9.90980	0.329413
$906$	0	0
$907$	10.3695	0.344313	0.172156	−	0.985070i	$-0.444927\pi$
$907$	10.3695	0.344313	0.172156	+	0.985070i	$0.444927\pi$
$908$	0	0
$909$	9.85839	0.326982
$910$	0	0
$911$	6.60928	0.218975	0.109488	−	0.993988i	$-0.465079\pi$
$911$	6.60928	0.218975	0.109488	+	0.993988i	$0.465079\pi$
$912$	0	0
$913$	59.4410	1.96721
$914$	0	0
$915$	10.5256	0.347965
$916$	0	0
$917$	8.84678	0.292146
$918$	0	0
$919$	−18.1994	−0.600342	−0.300171	−	0.953885i	$-0.597044\pi$
$919$	−18.1994	−0.600342	−0.300171	+	0.953885i	$0.597044\pi$
$920$	0	0
$921$	40.8752	1.34688
$922$	0	0
$923$	−26.8987	−0.885383
$924$	0	0
$925$	−11.8876	−0.390863
$926$	0	0
$927$	9.42970	0.309712
$928$	0	0
$929$	3.90926	0.128259	0.0641294	−	0.997942i	$-0.479573\pi$
$929$	3.90926	0.128259	0.0641294	+	0.997942i	$0.479573\pi$
$930$	0	0
$931$	−6.30698	−0.206703
$932$	0	0
$933$	−59.4493	−1.94628
$934$	0	0
$935$	−3.93137	−0.128570
$936$	0	0
$937$	34.6920	1.13334	0.566669	−	0.823946i	$-0.308232\pi$
$937$	34.6920	1.13334	0.566669	+	0.823946i	$0.308232\pi$
$938$	0	0
$939$	−48.2844	−1.57570
$940$	0	0
$941$	30.1987	0.984449	0.492225	−	0.870468i	$-0.336184\pi$
$941$	30.1987	0.984449	0.492225	+	0.870468i	$0.336184\pi$
$942$	0	0
$943$	66.5151	2.16603
$944$	0	0
$945$	0.724627	0.0235721
$946$	0	0
$947$	48.6183	1.57988	0.789941	−	0.613183i	$-0.210111\pi$
$947$	48.6183	1.57988	0.789941	+	0.613183i	$0.210111\pi$
$948$	0	0
$949$	−2.92985	−0.0951069
$950$	0	0
$951$	−9.61439	−0.311768
$952$	0	0
$953$	−41.6805	−1.35016	−0.675082	−	0.737742i	$-0.735892\pi$
$953$	−41.6805	−1.35016	−0.675082	+	0.737742i	$0.735892\pi$
$954$	0	0
$955$	20.7192	0.670459
$956$	0	0
$957$	43.6769	1.41188
$958$	0	0
$959$	−12.8215	−0.414026
$960$	0	0
$961$	−26.7965	−0.864403
$962$	0	0
$963$	42.7916	1.37894
$964$	0	0
$965$	−27.9887	−0.900987
$966$	0	0
$967$	11.5094	0.370118	0.185059	−	0.982727i	$-0.440752\pi$
$967$	11.5094	0.370118	0.185059	+	0.982727i	$0.440752\pi$
$968$	0	0
$969$	15.1098	0.485398
$970$	0	0
$971$	−5.38443	−0.172795	−0.0863973	−	0.996261i	$-0.527535\pi$
$971$	−5.38443	−0.172795	−0.0863973	+	0.996261i	$0.527535\pi$
$972$	0	0
$973$	6.94688	0.222707
$974$	0	0
$975$	22.5539	0.722303
$976$	0	0
$977$	−12.1466	−0.388603	−0.194302	−	0.980942i	$-0.562244\pi$
$977$	−12.1466	−0.388603	−0.194302	+	0.980942i	$0.562244\pi$
$978$	0	0
$979$	26.2274	0.838230
$980$	0	0
$981$	36.7840	1.17442
$982$	0	0
$983$	−50.2908	−1.60403	−0.802013	−	0.597307i	$-0.796238\pi$
$983$	−50.2908	−1.60403	−0.802013	+	0.597307i	$0.796238\pi$
$984$	0	0
$985$	−0.797537	−0.0254116
$986$	0	0
$987$	−10.7347	−0.341690
$988$	0	0
$989$	−70.4948	−2.24160
$990$	0	0
$991$	−51.7997	−1.64547	−0.822735	−	0.568425i	$-0.807553\pi$
$991$	−51.7997	−1.64547	−0.822735	+	0.568425i	$0.807553\pi$
$992$	0	0
$993$	79.0792	2.50950
$994$	0	0
$995$	−7.71934	−0.244720
$996$	0	0
$997$	−14.9119	−0.472264	−0.236132	−	0.971721i	$-0.575880\pi$
$997$	−14.9119	−0.472264	−0.236132	+	0.971721i	$0.575880\pi$
$998$	0	0
$999$	−2.03146	−0.0642725

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3808.2.a.i.1.6	✓	6
4.3	odd	2		3808.2.a.m.1.1	yes	6
8.3	odd	2		7616.2.a.bx.1.6		6
8.5	even	2		7616.2.a.cb.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.i.1.6	✓	6	1.1	even	1	trivial
3808.2.a.m.1.1	yes	6	4.3	odd	2
7616.2.a.bx.1.6		6	8.3	odd	2
7616.2.a.cb.1.1		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)

\(f(q)\)	\(=\)	\(q+2.39573 q^{3} -1.16130 q^{5} +1.00000 q^{7} +2.73954 q^{9} -3.38533 q^{11} -2.57825 q^{13} -2.78216 q^{15} -1.00000 q^{17} -6.30698 q^{19} +2.39573 q^{21} +8.41214 q^{23} -3.65139 q^{25} -0.623981 q^{27} -5.38533 q^{29} -2.05025 q^{31} -8.11035 q^{33} -1.16130 q^{35} +3.25564 q^{37} -6.17680 q^{39} +7.90703 q^{41} -8.38012 q^{43} -3.18142 q^{45} -4.48077 q^{47} +1.00000 q^{49} -2.39573 q^{51} -1.33508 q^{53} +3.93137 q^{55} -15.1098 q^{57} -11.6695 q^{59} -3.78325 q^{61} +2.73954 q^{63} +2.99411 q^{65} +3.57716 q^{67} +20.1533 q^{69} +10.4329 q^{71} +1.13637 q^{73} -8.74777 q^{75} -3.38533 q^{77} -13.1994 q^{79} -9.71353 q^{81} -17.5584 q^{83} +1.16130 q^{85} -12.9018 q^{87} -7.74736 q^{89} -2.57825 q^{91} -4.91185 q^{93} +7.32427 q^{95} -8.44376 q^{97} -9.27427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 6 q^{5} + 6 q^{7} + 4 q^{9} - 2 q^{11} - 4 q^{13} + 2 q^{15} - 6 q^{17} - 10 q^{19} - 2 q^{21} + 4 q^{23} + 4 q^{25} - 8 q^{27} - 14 q^{29} + 8 q^{31} - 8 q^{33} - 6 q^{35} - 4 q^{37} + 14 q^{39}+ \cdots - 4 q^{99}+O(q^{100}) \) 6 * q - 2 * q^3 - 6 * q^5 + 6 * q^7 + 4 * q^9 - 2 * q^11 - 4 * q^13 + 2 * q^15 - 6 * q^17 - 10 * q^19 - 2 * q^21 + 4 * q^23 + 4 * q^25 - 8 * q^27 - 14 * q^29 + 8 * q^31 - 8 * q^33 - 6 * q^35 - 4 * q^37 + 14 * q^39 - 2 * q^41 - 8 * q^43 - 8 * q^45 - 2 * q^47 + 6 * q^49 + 2 * q^51 - 10 * q^53 + 2 * q^55 - 18 * q^57 - 14 * q^59 - 20 * q^61 + 4 * q^63 - 20 * q^65 - 6 * q^67 - 14 * q^69 + 20 * q^71 - 6 * q^73 - 10 * q^75 - 2 * q^77 + 38 * q^79 - 18 * q^81 - 16 * q^83 + 6 * q^85 - 4 * q^87 - 2 * q^89 - 4 * q^91 - 16 * q^93 + 42 * q^95 - 12 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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