LMFDB - Embedded newform 3808.2.a.i.1.2

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			1.2
Root			$-3.19746$ 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)

$f(q)$	$=$	$q-2.52160 q^{3} +0.600745 q^{5} +1.00000 q^{7} +3.35847 q^{9} +5.46196 q^{11} -4.95922 q^{13} -1.51484 q^{15} -1.00000 q^{17} -4.17649 q^{19} -2.52160 q^{21} +2.69068 q^{23} -4.63911 q^{25} -0.903920 q^{27} +3.46196 q^{29} +2.09416 q^{31} -13.7729 q^{33} +0.600745 q^{35} -7.22776 q^{37} +12.5052 q^{39} -1.58865 q^{41} -4.85637 q^{43} +2.01758 q^{45} +0.590571 q^{47} +1.00000 q^{49} +2.52160 q^{51} +3.36780 q^{53} +3.28125 q^{55} +10.5314 q^{57} -5.54387 q^{59} -15.0132 q^{61} +3.35847 q^{63} -2.97922 q^{65} -6.53919 q^{67} -6.78481 q^{69} +12.5714 q^{71} +9.33527 q^{73} +11.6980 q^{75} +5.46196 q^{77} +14.6393 q^{79} -7.79609 q^{81} +7.33801 q^{83} -0.600745 q^{85} -8.72969 q^{87} +13.8988 q^{89} -4.95922 q^{91} -5.28063 q^{93} -2.50900 q^{95} -15.7126 q^{97} +18.3438 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.52160	−1.45585	−0.727923	−	0.685658i	$-0.759514\pi$
$3$	−2.52160	−1.45585	−0.727923	+	0.685658i	$0.759514\pi$
$4$	0	0
$5$	0.600745	0.268661	0.134331	−	0.990937i	$-0.457112\pi$
$5$	0.600745	0.268661	0.134331	+	0.990937i	$0.457112\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	3.35847	1.11949
$10$	0	0
$11$	5.46196	1.64684	0.823422	−	0.567430i	$-0.192062\pi$
$11$	5.46196	1.64684	0.823422	+	0.567430i	$0.192062\pi$
$12$	0	0
$13$	−4.95922	−1.37544	−0.687720	−	0.725976i	$-0.741388\pi$
$13$	−4.95922	−1.37544	−0.687720	+	0.725976i	$0.741388\pi$
$14$	0	0
$15$	−1.51484	−0.391130
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−4.17649	−0.958152	−0.479076	−	0.877774i	$-0.659028\pi$
$19$	−4.17649	−0.958152	−0.479076	+	0.877774i	$0.659028\pi$
$20$	0	0
$21$	−2.52160	−0.550258
$22$	0	0
$23$	2.69068	0.561045	0.280522	−	0.959848i	$-0.409492\pi$
$23$	2.69068	0.561045	0.280522	+	0.959848i	$0.409492\pi$
$24$	0	0
$25$	−4.63911	−0.927821
$26$	0	0
$27$	−0.903920	−0.173960
$28$	0	0
$29$	3.46196	0.642870	0.321435	−	0.946932i	$-0.395835\pi$
$29$	3.46196	0.642870	0.321435	+	0.946932i	$0.395835\pi$
$30$	0	0
$31$	2.09416	0.376122	0.188061	−	0.982157i	$-0.439780\pi$
$31$	2.09416	0.376122	0.188061	+	0.982157i	$0.439780\pi$
$32$	0	0
$33$	−13.7729	−2.39755
$34$	0	0
$35$	0.600745	0.101544
$36$	0	0
$37$	−7.22776	−1.18824	−0.594118	−	0.804378i	$-0.702499\pi$
$37$	−7.22776	−1.18824	−0.594118	+	0.804378i	$0.702499\pi$
$38$	0	0
$39$	12.5052	2.00243
$40$	0	0
$41$	−1.58865	−0.248106	−0.124053	−	0.992276i	$-0.539589\pi$
$41$	−1.58865	−0.248106	−0.124053	+	0.992276i	$0.539589\pi$
$42$	0	0
$43$	−4.85637	−0.740590	−0.370295	−	0.928914i	$-0.620743\pi$
$43$	−4.85637	−0.740590	−0.370295	+	0.928914i	$0.620743\pi$
$44$	0	0
$45$	2.01758	0.300764
$46$	0	0
$47$	0.590571	0.0861437	0.0430718	−	0.999072i	$-0.486286\pi$
$47$	0.590571	0.0861437	0.0430718	+	0.999072i	$0.486286\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.52160	0.353095
$52$	0	0
$53$	3.36780	0.462603	0.231302	−	0.972882i	$-0.425702\pi$
$53$	3.36780	0.462603	0.231302	+	0.972882i	$0.425702\pi$
$54$	0	0
$55$	3.28125	0.442443
$56$	0	0
$57$	10.5314	1.39492
$58$	0	0
$59$	−5.54387	−0.721750	−0.360875	−	0.932614i	$-0.617522\pi$
$59$	−5.54387	−0.721750	−0.360875	+	0.932614i	$0.617522\pi$
$60$	0	0
$61$	−15.0132	−1.92225	−0.961124	−	0.276116i	$-0.910953\pi$
$61$	−15.0132	−1.92225	−0.961124	+	0.276116i	$0.910953\pi$
$62$	0	0
$63$	3.35847	0.423128
$64$	0	0
$65$	−2.97922	−0.369527
$66$	0	0
$67$	−6.53919	−0.798889	−0.399444	−	0.916757i	$-0.630797\pi$
$67$	−6.53919	−0.798889	−0.399444	+	0.916757i	$0.630797\pi$
$68$	0	0
$69$	−6.78481	−0.816795
$70$	0	0
$71$	12.5714	1.49195	0.745975	−	0.665974i	$-0.231984\pi$
$71$	12.5714	1.49195	0.745975	+	0.665974i	$0.231984\pi$
$72$	0	0
$73$	9.33527	1.09261	0.546305	−	0.837586i	$-0.316034\pi$
$73$	9.33527	1.09261	0.546305	+	0.837586i	$0.316034\pi$
$74$	0	0
$75$	11.6980	1.35077
$76$	0	0
$77$	5.46196	0.622448
$78$	0	0
$79$	14.6393	1.64705	0.823523	−	0.567283i	$-0.192005\pi$
$79$	14.6393	1.64705	0.823523	+	0.567283i	$0.192005\pi$
$80$	0	0
$81$	−7.79609	−0.866232
$82$	0	0
$83$	7.33801	0.805451	0.402726	−	0.915321i	$-0.368063\pi$
$83$	7.33801	0.805451	0.402726	+	0.915321i	$0.368063\pi$
$84$	0	0
$85$	−0.600745	−0.0651600
$86$	0	0
$87$	−8.72969	−0.935920
$88$	0	0
$89$	13.8988	1.47327	0.736636	−	0.676290i	$-0.236413\pi$
$89$	13.8988	1.47327	0.736636	+	0.676290i	$0.236413\pi$
$90$	0	0
$91$	−4.95922	−0.519867
$92$	0	0
$93$	−5.28063	−0.547576
$94$	0	0
$95$	−2.50900	−0.257418
$96$	0	0
$97$	−15.7126	−1.59537	−0.797686	−	0.603073i	$-0.793943\pi$
$97$	−15.7126	−1.59537	−0.797686	+	0.603073i	$0.793943\pi$
$98$	0	0
$99$	18.3438	1.84363
$100$	0	0
$101$	−17.4644	−1.73777	−0.868885	−	0.495013i	$-0.835163\pi$
$101$	−17.4644	−1.73777	−0.868885	+	0.495013i	$0.835163\pi$
$102$	0	0
$103$	−5.99435	−0.590641	−0.295320	−	0.955398i	$-0.595426\pi$
$103$	−5.99435	−0.590641	−0.295320	+	0.955398i	$0.595426\pi$
$104$	0	0
$105$	−1.51484	−0.147833
$106$	0	0
$107$	6.62489	0.640452	0.320226	−	0.947341i	$-0.396241\pi$
$107$	6.62489	0.640452	0.320226	+	0.947341i	$0.396241\pi$
$108$	0	0
$109$	9.59217	0.918763	0.459382	−	0.888239i	$-0.348071\pi$
$109$	9.59217	0.918763	0.459382	+	0.888239i	$0.348071\pi$
$110$	0	0
$111$	18.2255	1.72989
$112$	0	0
$113$	4.37106	0.411195	0.205597	−	0.978637i	$-0.434086\pi$
$113$	4.37106	0.411195	0.205597	+	0.978637i	$0.434086\pi$
$114$	0	0
$115$	1.61641	0.150731
$116$	0	0
$117$	−16.6554	−1.53979
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	18.8330	1.71209
$122$	0	0
$123$	4.00594	0.361204
$124$	0	0
$125$	−5.79065	−0.517931
$126$	0	0
$127$	−9.75420	−0.865545	−0.432773	−	0.901503i	$-0.642465\pi$
$127$	−9.75420	−0.865545	−0.432773	+	0.901503i	$0.642465\pi$
$128$	0	0
$129$	12.2458	1.07819
$130$	0	0
$131$	15.3430	1.34053	0.670263	−	0.742124i	$-0.266181\pi$
$131$	15.3430	1.34053	0.670263	+	0.742124i	$0.266181\pi$
$132$	0	0
$133$	−4.17649	−0.362147
$134$	0	0
$135$	−0.543026	−0.0467362
$136$	0	0
$137$	−13.5012	−1.15348	−0.576741	−	0.816927i	$-0.695676\pi$
$137$	−13.5012	−1.15348	−0.576741	+	0.816927i	$0.695676\pi$
$138$	0	0
$139$	−10.6232	−0.901046	−0.450523	−	0.892765i	$-0.648762\pi$
$139$	−10.6232	−0.901046	−0.450523	+	0.892765i	$0.648762\pi$
$140$	0	0
$141$	−1.48919	−0.125412
$142$	0	0
$143$	−27.0870	−2.26513
$144$	0	0
$145$	2.07976	0.172714
$146$	0	0
$147$	−2.52160	−0.207978
$148$	0	0
$149$	−3.25647	−0.266781	−0.133390	−	0.991064i	$-0.542586\pi$
$149$	−3.25647	−0.266781	−0.133390	+	0.991064i	$0.542586\pi$
$150$	0	0
$151$	−4.77097	−0.388256	−0.194128	−	0.980976i	$-0.562188\pi$
$151$	−4.77097	−0.388256	−0.194128	+	0.980976i	$0.562188\pi$
$152$	0	0
$153$	−3.35847	−0.271516
$154$	0	0
$155$	1.25806	0.101049
$156$	0	0
$157$	−6.13060	−0.489275	−0.244638	−	0.969615i	$-0.578669\pi$
$157$	−6.13060	−0.489275	−0.244638	+	0.969615i	$0.578669\pi$
$158$	0	0
$159$	−8.49225	−0.673479
$160$	0	0
$161$	2.69068	0.212055
$162$	0	0
$163$	−14.5276	−1.13789	−0.568944	−	0.822376i	$-0.692648\pi$
$163$	−14.5276	−1.13789	−0.568944	+	0.822376i	$0.692648\pi$
$164$	0	0
$165$	−8.27399	−0.644130
$166$	0	0
$167$	−8.19957	−0.634502	−0.317251	−	0.948342i	$-0.602760\pi$
$167$	−8.19957	−0.634502	−0.317251	+	0.948342i	$0.602760\pi$
$168$	0	0
$169$	11.5938	0.891833
$170$	0	0
$171$	−14.0266	−1.07264
$172$	0	0
$173$	19.2541	1.46386	0.731931	−	0.681379i	$-0.238619\pi$
$173$	19.2541	1.46386	0.731931	+	0.681379i	$0.238619\pi$
$174$	0	0
$175$	−4.63911	−0.350683
$176$	0	0
$177$	13.9794	1.05076
$178$	0	0
$179$	−21.6987	−1.62184	−0.810920	−	0.585158i	$-0.801033\pi$
$179$	−21.6987	−1.62184	−0.810920	+	0.585158i	$0.801033\pi$
$180$	0	0
$181$	−12.7427	−0.947161	−0.473580	−	0.880751i	$-0.657039\pi$
$181$	−12.7427	−0.947161	−0.473580	+	0.880751i	$0.657039\pi$
$182$	0	0
$183$	37.8574	2.79850
$184$	0	0
$185$	−4.34204	−0.319233
$186$	0	0
$187$	−5.46196	−0.399418
$188$	0	0
$189$	−0.903920	−0.0657505
$190$	0	0
$191$	0.442118	0.0319905	0.0159953	−	0.999872i	$-0.494908\pi$
$191$	0.442118	0.0319905	0.0159953	+	0.999872i	$0.494908\pi$
$192$	0	0
$193$	−7.90958	−0.569344	−0.284672	−	0.958625i	$-0.591885\pi$
$193$	−7.90958	−0.569344	−0.284672	+	0.958625i	$0.591885\pi$
$194$	0	0
$195$	7.51242	0.537975
$196$	0	0
$197$	−13.1416	−0.936302	−0.468151	−	0.883649i	$-0.655080\pi$
$197$	−13.1416	−0.936302	−0.468151	+	0.883649i	$0.655080\pi$
$198$	0	0
$199$	−4.64763	−0.329462	−0.164731	−	0.986339i	$-0.552676\pi$
$199$	−4.64763	−0.329462	−0.164731	+	0.986339i	$0.552676\pi$
$200$	0	0
$201$	16.4892	1.16306
$202$	0	0
$203$	3.46196	0.242982
$204$	0	0
$205$	−0.954374	−0.0666564
$206$	0	0
$207$	9.03655	0.628084
$208$	0	0
$209$	−22.8118	−1.57793
$210$	0	0
$211$	−26.9781	−1.85725	−0.928626	−	0.371017i	$-0.879009\pi$
$211$	−26.9781	−1.85725	−0.928626	+	0.371017i	$0.879009\pi$
$212$	0	0
$213$	−31.7000	−2.17205
$214$	0	0
$215$	−2.91744	−0.198968
$216$	0	0
$217$	2.09416	0.142161
$218$	0	0
$219$	−23.5398	−1.59067
$220$	0	0
$221$	4.95922	0.333593
$222$	0	0
$223$	−9.77289	−0.654441	−0.327220	−	0.944948i	$-0.606112\pi$
$223$	−9.77289	−0.654441	−0.327220	+	0.944948i	$0.606112\pi$
$224$	0	0
$225$	−15.5803	−1.03869
$226$	0	0
$227$	−5.76530	−0.382657	−0.191328	−	0.981526i	$-0.561280\pi$
$227$	−5.76530	−0.382657	−0.191328	+	0.981526i	$0.561280\pi$
$228$	0	0
$229$	1.96385	0.129775	0.0648875	−	0.997893i	$-0.479331\pi$
$229$	1.96385	0.129775	0.0648875	+	0.997893i	$0.479331\pi$
$230$	0	0
$231$	−13.7729	−0.906189
$232$	0	0
$233$	−21.0962	−1.38206	−0.691028	−	0.722828i	$-0.742842\pi$
$233$	−21.0962	−1.38206	−0.691028	+	0.722828i	$0.742842\pi$
$234$	0	0
$235$	0.354783	0.0231435
$236$	0	0
$237$	−36.9144	−2.39785
$238$	0	0
$239$	−9.42471	−0.609634	−0.304817	−	0.952411i	$-0.598595\pi$
$239$	−9.42471	−0.609634	−0.304817	+	0.952411i	$0.598595\pi$
$240$	0	0
$241$	−21.1024	−1.35933	−0.679663	−	0.733525i	$-0.737874\pi$
$241$	−21.1024	−1.35933	−0.679663	+	0.733525i	$0.737874\pi$
$242$	0	0
$243$	22.3704	1.43506
$244$	0	0
$245$	0.600745	0.0383802
$246$	0	0
$247$	20.7121	1.31788
$248$	0	0
$249$	−18.5035	−1.17261
$250$	0	0
$251$	−17.5740	−1.10926	−0.554630	−	0.832097i	$-0.687140\pi$
$251$	−17.5740	−1.10926	−0.554630	+	0.832097i	$0.687140\pi$
$252$	0	0
$253$	14.6964	0.923953
$254$	0	0
$255$	1.51484	0.0948629
$256$	0	0
$257$	−22.1759	−1.38329	−0.691646	−	0.722236i	$-0.743114\pi$
$257$	−22.1759	−1.38329	−0.691646	+	0.722236i	$0.743114\pi$
$258$	0	0
$259$	−7.22776	−0.449111
$260$	0	0
$261$	11.6269	0.719687
$262$	0	0
$263$	−0.702119	−0.0432945	−0.0216473	−	0.999766i	$-0.506891\pi$
$263$	−0.702119	−0.0432945	−0.0216473	+	0.999766i	$0.506891\pi$
$264$	0	0
$265$	2.02319	0.124284
$266$	0	0
$267$	−35.0473	−2.14486
$268$	0	0
$269$	28.2940	1.72512	0.862558	−	0.505959i	$-0.168861\pi$
$269$	28.2940	1.72512	0.862558	+	0.505959i	$0.168861\pi$
$270$	0	0
$271$	−19.0366	−1.15639	−0.578195	−	0.815898i	$-0.696243\pi$
$271$	−19.0366	−1.15639	−0.578195	+	0.815898i	$0.696243\pi$
$272$	0	0
$273$	12.5052	0.756847
$274$	0	0
$275$	−25.3386	−1.52798
$276$	0	0
$277$	−18.2896	−1.09892	−0.549458	−	0.835521i	$-0.685166\pi$
$277$	−18.2896	−1.09892	−0.549458	+	0.835521i	$0.685166\pi$
$278$	0	0
$279$	7.03317	0.421065
$280$	0	0
$281$	25.5368	1.52340	0.761698	−	0.647932i	$-0.224366\pi$
$281$	25.5368	1.52340	0.761698	+	0.647932i	$0.224366\pi$
$282$	0	0
$283$	4.44290	0.264103	0.132052	−	0.991243i	$-0.457844\pi$
$283$	4.44290	0.264103	0.132052	+	0.991243i	$0.457844\pi$
$284$	0	0
$285$	6.32670	0.374762
$286$	0	0
$287$	−1.58865	−0.0937751
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	39.6209	2.32262
$292$	0	0
$293$	4.98013	0.290943	0.145471	−	0.989362i	$-0.453530\pi$
$293$	4.98013	0.290943	0.145471	+	0.989362i	$0.453530\pi$
$294$	0	0
$295$	−3.33045	−0.193906
$296$	0	0
$297$	−4.93718	−0.286484
$298$	0	0
$299$	−13.3436	−0.771683
$300$	0	0
$301$	−4.85637	−0.279917
$302$	0	0
$303$	44.0382	2.52993
$304$	0	0
$305$	−9.01913	−0.516434
$306$	0	0
$307$	−11.3554	−0.648085	−0.324043	−	0.946042i	$-0.605042\pi$
$307$	−11.3554	−0.648085	−0.324043	+	0.946042i	$0.605042\pi$
$308$	0	0
$309$	15.1154	0.859882
$310$	0	0
$311$	29.3717	1.66552	0.832758	−	0.553638i	$-0.186761\pi$
$311$	29.3717	1.66552	0.832758	+	0.553638i	$0.186761\pi$
$312$	0	0
$313$	10.6749	0.603379	0.301690	−	0.953406i	$-0.402449\pi$
$313$	10.6749	0.603379	0.301690	+	0.953406i	$0.402449\pi$
$314$	0	0
$315$	2.01758	0.113678
$316$	0	0
$317$	−22.9163	−1.28711	−0.643554	−	0.765400i	$-0.722541\pi$
$317$	−22.9163	−1.28711	−0.643554	+	0.765400i	$0.722541\pi$
$318$	0	0
$319$	18.9091	1.05871
$320$	0	0
$321$	−16.7053	−0.932400
$322$	0	0
$323$	4.17649	0.232386
$324$	0	0
$325$	23.0063	1.27616
$326$	0	0
$327$	−24.1876	−1.33758
$328$	0	0
$329$	0.590571	0.0325593
$330$	0	0
$331$	5.21678	0.286740	0.143370	−	0.989669i	$-0.454206\pi$
$331$	5.21678	0.286740	0.143370	+	0.989669i	$0.454206\pi$
$332$	0	0
$333$	−24.2742	−1.33022
$334$	0	0
$335$	−3.92838	−0.214631
$336$	0	0
$337$	12.8981	0.702604	0.351302	−	0.936262i	$-0.385739\pi$
$337$	12.8981	0.702604	0.351302	+	0.936262i	$0.385739\pi$
$338$	0	0
$339$	−11.0221	−0.598637
$340$	0	0
$341$	11.4382	0.619414
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−4.07594	−0.219441
$346$	0	0
$347$	−24.2592	−1.30230	−0.651151	−	0.758948i	$-0.725714\pi$
$347$	−24.2592	−1.30230	−0.651151	+	0.758948i	$0.725714\pi$
$348$	0	0
$349$	12.2213	0.654191	0.327095	−	0.944991i	$-0.393930\pi$
$349$	12.2213	0.654191	0.327095	+	0.944991i	$0.393930\pi$
$350$	0	0
$351$	4.48274	0.239271
$352$	0	0
$353$	−22.0151	−1.17174	−0.585872	−	0.810403i	$-0.699248\pi$
$353$	−22.0151	−1.17174	−0.585872	+	0.810403i	$0.699248\pi$
$354$	0	0
$355$	7.55220	0.400829
$356$	0	0
$357$	2.52160	0.133457
$358$	0	0
$359$	−14.2558	−0.752391	−0.376196	−	0.926540i	$-0.622768\pi$
$359$	−14.2558	−0.752391	−0.376196	+	0.926540i	$0.622768\pi$
$360$	0	0
$361$	−1.55697	−0.0819457
$362$	0	0
$363$	−47.4894	−2.49255
$364$	0	0
$365$	5.60812	0.293542
$366$	0	0
$367$	6.31553	0.329668	0.164834	−	0.986321i	$-0.447291\pi$
$367$	6.31553	0.329668	0.164834	+	0.986321i	$0.447291\pi$
$368$	0	0
$369$	−5.33544	−0.277752
$370$	0	0
$371$	3.36780	0.174848
$372$	0	0
$373$	8.99627	0.465809	0.232905	−	0.972500i	$-0.425177\pi$
$373$	8.99627	0.465809	0.232905	+	0.972500i	$0.425177\pi$
$374$	0	0
$375$	14.6017	0.754028
$376$	0	0
$377$	−17.1686	−0.884229
$378$	0	0
$379$	31.3775	1.61176	0.805878	−	0.592081i	$-0.201694\pi$
$379$	31.3775	1.61176	0.805878	+	0.592081i	$0.201694\pi$
$380$	0	0
$381$	24.5962	1.26010
$382$	0	0
$383$	27.1190	1.38571	0.692857	−	0.721075i	$-0.256352\pi$
$383$	27.1190	1.38571	0.692857	+	0.721075i	$0.256352\pi$
$384$	0	0
$385$	3.28125	0.167228
$386$	0	0
$387$	−16.3100	−0.829084
$388$	0	0
$389$	21.6322	1.09679	0.548397	−	0.836218i	$-0.315238\pi$
$389$	21.6322	1.09679	0.548397	+	0.836218i	$0.315238\pi$
$390$	0	0
$391$	−2.69068	−0.136073
$392$	0	0
$393$	−38.6890	−1.95160
$394$	0	0
$395$	8.79447	0.442498
$396$	0	0
$397$	−2.22325	−0.111582	−0.0557909	−	0.998442i	$-0.517768\pi$
$397$	−2.22325	−0.111582	−0.0557909	+	0.998442i	$0.517768\pi$
$398$	0	0
$399$	10.5314	0.527231
$400$	0	0
$401$	3.71483	0.185510	0.0927548	−	0.995689i	$-0.470433\pi$
$401$	3.71483	0.185510	0.0927548	+	0.995689i	$0.470433\pi$
$402$	0	0
$403$	−10.3854	−0.517333
$404$	0	0
$405$	−4.68346	−0.232723
$406$	0	0
$407$	−39.4777	−1.95684
$408$	0	0
$409$	−15.3159	−0.757323	−0.378661	−	0.925535i	$-0.623615\pi$
$409$	−15.3159	−0.757323	−0.378661	+	0.925535i	$0.623615\pi$
$410$	0	0
$411$	34.0445	1.67929
$412$	0	0
$413$	−5.54387	−0.272796
$414$	0	0
$415$	4.40827	0.216394
$416$	0	0
$417$	26.7874	1.31178
$418$	0	0
$419$	−7.88425	−0.385171	−0.192585	−	0.981280i	$-0.561687\pi$
$419$	−7.88425	−0.385171	−0.192585	+	0.981280i	$0.561687\pi$
$420$	0	0
$421$	16.5852	0.808315	0.404158	−	0.914689i	$-0.367565\pi$
$421$	16.5852	0.808315	0.404158	+	0.914689i	$0.367565\pi$
$422$	0	0
$423$	1.98342	0.0964370
$424$	0	0
$425$	4.63911	0.225030
$426$	0	0
$427$	−15.0132	−0.726542
$428$	0	0
$429$	68.3027	3.29769
$430$	0	0
$431$	14.4179	0.694486	0.347243	−	0.937775i	$-0.387118\pi$
$431$	14.4179	0.694486	0.347243	+	0.937775i	$0.387118\pi$
$432$	0	0
$433$	−7.08548	−0.340506	−0.170253	−	0.985400i	$-0.554459\pi$
$433$	−7.08548	−0.340506	−0.170253	+	0.985400i	$0.554459\pi$
$434$	0	0
$435$	−5.24432	−0.251446
$436$	0	0
$437$	−11.2376	−0.537566
$438$	0	0
$439$	21.1477	1.00932	0.504662	−	0.863317i	$-0.331617\pi$
$439$	21.1477	1.00932	0.504662	+	0.863317i	$0.331617\pi$
$440$	0	0
$441$	3.35847	0.159927
$442$	0	0
$443$	4.83752	0.229837	0.114919	−	0.993375i	$-0.463339\pi$
$443$	4.83752	0.229837	0.114919	+	0.993375i	$0.463339\pi$
$444$	0	0
$445$	8.34964	0.395811
$446$	0	0
$447$	8.21152	0.388392
$448$	0	0
$449$	−19.2445	−0.908206	−0.454103	−	0.890949i	$-0.650040\pi$
$449$	−19.2445	−0.908206	−0.454103	+	0.890949i	$0.650040\pi$
$450$	0	0
$451$	−8.67715	−0.408591
$452$	0	0
$453$	12.0305	0.565241
$454$	0	0
$455$	−2.97922	−0.139668
$456$	0	0
$457$	17.3388	0.811073	0.405537	−	0.914079i	$-0.367085\pi$
$457$	17.3388	0.811073	0.405537	+	0.914079i	$0.367085\pi$
$458$	0	0
$459$	0.903920	0.0421914
$460$	0	0
$461$	−26.6692	−1.24211	−0.621054	−	0.783768i	$-0.713295\pi$
$461$	−26.6692	−1.24211	−0.621054	+	0.783768i	$0.713295\pi$
$462$	0	0
$463$	−26.3856	−1.22624	−0.613122	−	0.789988i	$-0.710087\pi$
$463$	−26.3856	−1.22624	−0.613122	+	0.789988i	$0.710087\pi$
$464$	0	0
$465$	−3.17232	−0.147113
$466$	0	0
$467$	−5.42102	−0.250855	−0.125427	−	0.992103i	$-0.540030\pi$
$467$	−5.42102	−0.250855	−0.125427	+	0.992103i	$0.540030\pi$
$468$	0	0
$469$	−6.53919	−0.301952
$470$	0	0
$471$	15.4589	0.712310
$472$	0	0
$473$	−26.5253	−1.21964
$474$	0	0
$475$	19.3752	0.888993
$476$	0	0
$477$	11.3107	0.517880
$478$	0	0
$479$	11.7002	0.534596	0.267298	−	0.963614i	$-0.413869\pi$
$479$	11.7002	0.534596	0.267298	+	0.963614i	$0.413869\pi$
$480$	0	0
$481$	35.8440	1.63435
$482$	0	0
$483$	−6.78481	−0.308720
$484$	0	0
$485$	−9.43927	−0.428615
$486$	0	0
$487$	10.9109	0.494421	0.247210	−	0.968962i	$-0.420486\pi$
$487$	10.9109	0.494421	0.247210	+	0.968962i	$0.420486\pi$
$488$	0	0
$489$	36.6328	1.65659
$490$	0	0
$491$	−10.6687	−0.481470	−0.240735	−	0.970591i	$-0.577389\pi$
$491$	−10.6687	−0.481470	−0.240735	+	0.970591i	$0.577389\pi$
$492$	0	0
$493$	−3.46196	−0.155919
$494$	0	0
$495$	11.0200	0.495311
$496$	0	0
$497$	12.5714	0.563904
$498$	0	0
$499$	−31.4888	−1.40963	−0.704816	−	0.709390i	$-0.748971\pi$
$499$	−31.4888	−1.40963	−0.704816	+	0.709390i	$0.748971\pi$
$500$	0	0
$501$	20.6760	0.923737
$502$	0	0
$503$	−41.7535	−1.86169	−0.930847	−	0.365408i	$-0.880929\pi$
$503$	−41.7535	−1.86169	−0.930847	+	0.365408i	$0.880929\pi$
$504$	0	0
$505$	−10.4916	−0.466872
$506$	0	0
$507$	−29.2350	−1.29837
$508$	0	0
$509$	−0.269289	−0.0119360	−0.00596802	−	0.999982i	$-0.501900\pi$
$509$	−0.269289	−0.0119360	−0.00596802	+	0.999982i	$0.501900\pi$
$510$	0	0
$511$	9.33527	0.412968
$512$	0	0
$513$	3.77521	0.166680
$514$	0	0
$515$	−3.60108	−0.158682
$516$	0	0
$517$	3.22568	0.141865
$518$	0	0
$519$	−48.5511	−2.13116
$520$	0	0
$521$	−26.0361	−1.14066	−0.570331	−	0.821415i	$-0.693185\pi$
$521$	−26.0361	−1.14066	−0.570331	+	0.821415i	$0.693185\pi$
$522$	0	0
$523$	19.4264	0.849456	0.424728	−	0.905321i	$-0.360370\pi$
$523$	19.4264	0.849456	0.424728	+	0.905321i	$0.360370\pi$
$524$	0	0
$525$	11.6980	0.510541
$526$	0	0
$527$	−2.09416	−0.0912230
$528$	0	0
$529$	−15.7603	−0.685229
$530$	0	0
$531$	−18.6189	−0.807992
$532$	0	0
$533$	7.87846	0.341254
$534$	0	0
$535$	3.97987	0.172065
$536$	0	0
$537$	54.7155	2.36115
$538$	0	0
$539$	5.46196	0.235263
$540$	0	0
$541$	7.38506	0.317509	0.158754	−	0.987318i	$-0.449252\pi$
$541$	7.38506	0.317509	0.158754	+	0.987318i	$0.449252\pi$
$542$	0	0
$543$	32.1321	1.37892
$544$	0	0
$545$	5.76245	0.246836
$546$	0	0
$547$	−14.9175	−0.637827	−0.318913	−	0.947784i	$-0.603318\pi$
$547$	−14.9175	−0.637827	−0.318913	+	0.947784i	$0.603318\pi$
$548$	0	0
$549$	−50.4215	−2.15194
$550$	0	0
$551$	−14.4588	−0.615967
$552$	0	0
$553$	14.6393	0.622525
$554$	0	0
$555$	10.9489	0.464755
$556$	0	0
$557$	−7.10542	−0.301066	−0.150533	−	0.988605i	$-0.548099\pi$
$557$	−7.10542	−0.301066	−0.150533	+	0.988605i	$0.548099\pi$
$558$	0	0
$559$	24.0838	1.01864
$560$	0	0
$561$	13.7729	0.581492
$562$	0	0
$563$	46.5174	1.96048	0.980238	−	0.197822i	$-0.0633868\pi$
$563$	46.5174	1.96048	0.980238	+	0.197822i	$0.0633868\pi$
$564$	0	0
$565$	2.62589	0.110472
$566$	0	0
$567$	−7.79609	−0.327405
$568$	0	0
$569$	20.7632	0.870440	0.435220	−	0.900324i	$-0.356671\pi$
$569$	20.7632	0.870440	0.435220	+	0.900324i	$0.356671\pi$
$570$	0	0
$571$	3.78287	0.158308	0.0791542	−	0.996862i	$-0.474778\pi$
$571$	3.78287	0.158308	0.0791542	+	0.996862i	$0.474778\pi$
$572$	0	0
$573$	−1.11484	−0.0465733
$574$	0	0
$575$	−12.4823	−0.520549
$576$	0	0
$577$	−28.8197	−1.19978	−0.599889	−	0.800083i	$-0.704789\pi$
$577$	−28.8197	−1.19978	−0.599889	+	0.800083i	$0.704789\pi$
$578$	0	0
$579$	19.9448	0.828878
$580$	0	0
$581$	7.33801	0.304432
$582$	0	0
$583$	18.3948	0.761835
$584$	0	0
$585$	−10.0056	−0.413682
$586$	0	0
$587$	20.1931	0.833459	0.416729	−	0.909031i	$-0.363176\pi$
$587$	20.1931	0.833459	0.416729	+	0.909031i	$0.363176\pi$
$588$	0	0
$589$	−8.74623	−0.360382
$590$	0	0
$591$	33.1379	1.36311
$592$	0	0
$593$	6.27942	0.257865	0.128932	−	0.991653i	$-0.458845\pi$
$593$	6.27942	0.257865	0.128932	+	0.991653i	$0.458845\pi$
$594$	0	0
$595$	−0.600745	−0.0246281
$596$	0	0
$597$	11.7195	0.479646
$598$	0	0
$599$	31.0338	1.26801	0.634003	−	0.773330i	$-0.281411\pi$
$599$	31.0338	1.26801	0.634003	+	0.773330i	$0.281411\pi$
$600$	0	0
$601$	−20.3227	−0.828981	−0.414490	−	0.910054i	$-0.636040\pi$
$601$	−20.3227	−0.828981	−0.414490	+	0.910054i	$0.636040\pi$
$602$	0	0
$603$	−21.9617	−0.894348
$604$	0	0
$605$	11.3138	0.459973
$606$	0	0
$607$	32.2811	1.31025	0.655124	−	0.755521i	$-0.272616\pi$
$607$	32.2811	1.31025	0.655124	+	0.755521i	$0.272616\pi$
$608$	0	0
$609$	−8.72969	−0.353745
$610$	0	0
$611$	−2.92877	−0.118485
$612$	0	0
$613$	−30.8167	−1.24467	−0.622337	−	0.782750i	$-0.713817\pi$
$613$	−30.8167	−1.24467	−0.622337	+	0.782750i	$0.713817\pi$
$614$	0	0
$615$	2.40655	0.0970415
$616$	0	0
$617$	−30.4533	−1.22600	−0.613001	−	0.790082i	$-0.710038\pi$
$617$	−30.4533	−1.22600	−0.613001	+	0.790082i	$0.710038\pi$
$618$	0	0
$619$	−42.9368	−1.72578	−0.862888	−	0.505394i	$-0.831347\pi$
$619$	−42.9368	−1.72578	−0.862888	+	0.505394i	$0.831347\pi$
$620$	0	0
$621$	−2.43216	−0.0975991
$622$	0	0
$623$	13.8988	0.556844
$624$	0	0
$625$	19.7168	0.788673
$626$	0	0
$627$	57.5223	2.29722
$628$	0	0
$629$	7.22776	0.288190
$630$	0	0
$631$	42.6584	1.69821	0.849103	−	0.528228i	$-0.177143\pi$
$631$	42.6584	1.69821	0.849103	+	0.528228i	$0.177143\pi$
$632$	0	0
$633$	68.0281	2.70387
$634$	0	0
$635$	−5.85979	−0.232539
$636$	0	0
$637$	−4.95922	−0.196491
$638$	0	0
$639$	42.2207	1.67022
$640$	0	0
$641$	−16.2407	−0.641470	−0.320735	−	0.947169i	$-0.603930\pi$
$641$	−16.2407	−0.641470	−0.320735	+	0.947169i	$0.603930\pi$
$642$	0	0
$643$	15.7543	0.621291	0.310645	−	0.950526i	$-0.399455\pi$
$643$	15.7543	0.621291	0.310645	+	0.950526i	$0.399455\pi$
$644$	0	0
$645$	7.35663	0.289667
$646$	0	0
$647$	−15.8946	−0.624881	−0.312441	−	0.949937i	$-0.601147\pi$
$647$	−15.8946	−0.624881	−0.312441	+	0.949937i	$0.601147\pi$
$648$	0	0
$649$	−30.2804	−1.18861
$650$	0	0
$651$	−5.28063	−0.206964
$652$	0	0
$653$	−17.0447	−0.667011	−0.333506	−	0.942748i	$-0.608232\pi$
$653$	−17.0447	−0.667011	−0.333506	+	0.942748i	$0.608232\pi$
$654$	0	0
$655$	9.21725	0.360148
$656$	0	0
$657$	31.3522	1.22317
$658$	0	0
$659$	42.6352	1.66083	0.830415	−	0.557145i	$-0.188103\pi$
$659$	42.6352	1.66083	0.830415	+	0.557145i	$0.188103\pi$
$660$	0	0
$661$	−25.0946	−0.976066	−0.488033	−	0.872825i	$-0.662285\pi$
$661$	−25.0946	−0.976066	−0.488033	+	0.872825i	$0.662285\pi$
$662$	0	0
$663$	−12.5052	−0.485660
$664$	0	0
$665$	−2.50900	−0.0972950
$666$	0	0
$667$	9.31501	0.360679
$668$	0	0
$669$	24.6433	0.952766
$670$	0	0
$671$	−82.0017	−3.16564
$672$	0	0
$673$	−31.2540	−1.20475	−0.602376	−	0.798213i	$-0.705779\pi$
$673$	−31.2540	−1.20475	−0.602376	+	0.798213i	$0.705779\pi$
$674$	0	0
$675$	4.19338	0.161403
$676$	0	0
$677$	−10.7104	−0.411634	−0.205817	−	0.978590i	$-0.565985\pi$
$677$	−10.7104	−0.411634	−0.205817	+	0.978590i	$0.565985\pi$
$678$	0	0
$679$	−15.7126	−0.602994
$680$	0	0
$681$	14.5378	0.557090
$682$	0	0
$683$	−30.9331	−1.18362	−0.591811	−	0.806077i	$-0.701587\pi$
$683$	−30.9331	−1.18362	−0.591811	+	0.806077i	$0.701587\pi$
$684$	0	0
$685$	−8.11076	−0.309896
$686$	0	0
$687$	−4.95205	−0.188933
$688$	0	0
$689$	−16.7017	−0.636282
$690$	0	0
$691$	3.29592	0.125383	0.0626914	−	0.998033i	$-0.480032\pi$
$691$	3.29592	0.125383	0.0626914	+	0.998033i	$0.480032\pi$
$692$	0	0
$693$	18.3438	0.696825
$694$	0	0
$695$	−6.38182	−0.242076
$696$	0	0
$697$	1.58865	0.0601745
$698$	0	0
$699$	53.1961	2.01206
$700$	0	0
$701$	−22.9139	−0.865447	−0.432723	−	0.901527i	$-0.642447\pi$
$701$	−22.9139	−0.865447	−0.432723	+	0.901527i	$0.642447\pi$
$702$	0	0
$703$	30.1866	1.13851
$704$	0	0
$705$	−0.894621	−0.0336934
$706$	0	0
$707$	−17.4644	−0.656816
$708$	0	0
$709$	51.3935	1.93012	0.965062	−	0.262023i	$-0.0843897\pi$
$709$	51.3935	1.93012	0.965062	+	0.262023i	$0.0843897\pi$
$710$	0	0
$711$	49.1655	1.84385
$712$	0	0
$713$	5.63470	0.211021
$714$	0	0
$715$	−16.2724	−0.608554
$716$	0	0
$717$	23.7654	0.887534
$718$	0	0
$719$	11.0252	0.411170	0.205585	−	0.978639i	$-0.434090\pi$
$719$	11.0252	0.411170	0.205585	+	0.978639i	$0.434090\pi$
$720$	0	0
$721$	−5.99435	−0.223241
$722$	0	0
$723$	53.2118	1.97897
$724$	0	0
$725$	−16.0604	−0.596468
$726$	0	0
$727$	−41.5890	−1.54245	−0.771226	−	0.636561i	$-0.780356\pi$
$727$	−41.5890	−1.54245	−0.771226	+	0.636561i	$0.780356\pi$
$728$	0	0
$729$	−33.0209	−1.22300
$730$	0	0
$731$	4.85637	0.179620
$732$	0	0
$733$	48.9142	1.80669	0.903344	−	0.428917i	$-0.141105\pi$
$733$	48.9142	1.80669	0.903344	+	0.428917i	$0.141105\pi$
$734$	0	0
$735$	−1.51484	−0.0558757
$736$	0	0
$737$	−35.7168	−1.31564
$738$	0	0
$739$	16.4447	0.604928	0.302464	−	0.953161i	$-0.402191\pi$
$739$	16.4447	0.604928	0.302464	+	0.953161i	$0.402191\pi$
$740$	0	0
$741$	−52.2276	−1.91863
$742$	0	0
$743$	−21.8911	−0.803106	−0.401553	−	0.915836i	$-0.631529\pi$
$743$	−21.8911	−0.803106	−0.401553	+	0.915836i	$0.631529\pi$
$744$	0	0
$745$	−1.95631	−0.0716736
$746$	0	0
$747$	24.6445	0.901695
$748$	0	0
$749$	6.62489	0.242068
$750$	0	0
$751$	−20.3651	−0.743132	−0.371566	−	0.928407i	$-0.621179\pi$
$751$	−20.3651	−0.743132	−0.371566	+	0.928407i	$0.621179\pi$
$752$	0	0
$753$	44.3146	1.61491
$754$	0	0
$755$	−2.86613	−0.104309
$756$	0	0
$757$	17.0412	0.619373	0.309687	−	0.950839i	$-0.399776\pi$
$757$	17.0412	0.619373	0.309687	+	0.950839i	$0.399776\pi$
$758$	0	0
$759$	−37.0584	−1.34513
$760$	0	0
$761$	−13.6557	−0.495020	−0.247510	−	0.968885i	$-0.579612\pi$
$761$	−13.6557	−0.495020	−0.247510	+	0.968885i	$0.579612\pi$
$762$	0	0
$763$	9.59217	0.347260
$764$	0	0
$765$	−2.01758	−0.0729459
$766$	0	0
$767$	27.4932	0.992723
$768$	0	0
$769$	−8.98523	−0.324016	−0.162008	−	0.986789i	$-0.551797\pi$
$769$	−8.98523	−0.324016	−0.162008	+	0.986789i	$0.551797\pi$
$770$	0	0
$771$	55.9187	2.01386
$772$	0	0
$773$	−1.72211	−0.0619400	−0.0309700	−	0.999520i	$-0.509860\pi$
$773$	−1.72211	−0.0619400	−0.0309700	+	0.999520i	$0.509860\pi$
$774$	0	0
$775$	−9.71503	−0.348974
$776$	0	0
$777$	18.2255	0.653837
$778$	0	0
$779$	6.63498	0.237723
$780$	0	0
$781$	68.6645	2.45701
$782$	0	0
$783$	−3.12934	−0.111833
$784$	0	0
$785$	−3.68293	−0.131449
$786$	0	0
$787$	−22.8493	−0.814491	−0.407245	−	0.913319i	$-0.633511\pi$
$787$	−22.8493	−0.814491	−0.407245	+	0.913319i	$0.633511\pi$
$788$	0	0
$789$	1.77047	0.0630302
$790$	0	0
$791$	4.37106	0.155417
$792$	0	0
$793$	74.4539	2.64394
$794$	0	0
$795$	−5.10168	−0.180938
$796$	0	0
$797$	2.65098	0.0939025	0.0469513	−	0.998897i	$-0.485049\pi$
$797$	2.65098	0.0939025	0.0469513	+	0.998897i	$0.485049\pi$
$798$	0	0
$799$	−0.590571	−0.0208929
$800$	0	0
$801$	46.6788	1.64931
$802$	0	0
$803$	50.9889	1.79936
$804$	0	0
$805$	1.61641	0.0569710
$806$	0	0
$807$	−71.3462	−2.51150
$808$	0	0
$809$	−36.2662	−1.27505	−0.637525	−	0.770430i	$-0.720042\pi$
$809$	−36.2662	−1.27505	−0.637525	+	0.770430i	$0.720042\pi$
$810$	0	0
$811$	−52.6193	−1.84771	−0.923857	−	0.382739i	$-0.874981\pi$
$811$	−52.6193	−1.84771	−0.923857	+	0.382739i	$0.874981\pi$
$812$	0	0
$813$	48.0027	1.68353
$814$	0	0
$815$	−8.72738	−0.305707
$816$	0	0
$817$	20.2826	0.709598
$818$	0	0
$819$	−16.6554	−0.581986
$820$	0	0
$821$	37.5447	1.31032	0.655159	−	0.755491i	$-0.272601\pi$
$821$	37.5447	1.31032	0.655159	+	0.755491i	$0.272601\pi$
$822$	0	0
$823$	14.9156	0.519924	0.259962	−	0.965619i	$-0.416290\pi$
$823$	14.9156	0.519924	0.259962	+	0.965619i	$0.416290\pi$
$824$	0	0
$825$	63.8939	2.22450
$826$	0	0
$827$	46.3286	1.61100	0.805502	−	0.592592i	$-0.201896\pi$
$827$	46.3286	1.61100	0.805502	+	0.592592i	$0.201896\pi$
$828$	0	0
$829$	−0.191257	−0.00664265	−0.00332132	−	0.999994i	$-0.501057\pi$
$829$	−0.191257	−0.00664265	−0.00332132	+	0.999994i	$0.501057\pi$
$830$	0	0
$831$	46.1191	1.59985
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	−4.92585	−0.170466
$836$	0	0
$837$	−1.89295	−0.0654301
$838$	0	0
$839$	12.2611	0.423300	0.211650	−	0.977345i	$-0.432116\pi$
$839$	12.2611	0.423300	0.211650	+	0.977345i	$0.432116\pi$
$840$	0	0
$841$	−17.0148	−0.586718
$842$	0	0
$843$	−64.3935	−2.21783
$844$	0	0
$845$	6.96493	0.239601
$846$	0	0
$847$	18.8330	0.647110
$848$	0	0
$849$	−11.2032	−0.384494
$850$	0	0
$851$	−19.4475	−0.666653
$852$	0	0
$853$	50.3652	1.72447	0.862236	−	0.506506i	$-0.169063\pi$
$853$	50.3652	1.72447	0.862236	+	0.506506i	$0.169063\pi$
$854$	0	0
$855$	−8.42641	−0.288177
$856$	0	0
$857$	40.1681	1.37212	0.686058	−	0.727547i	$-0.259340\pi$
$857$	40.1681	1.37212	0.686058	+	0.727547i	$0.259340\pi$
$858$	0	0
$859$	40.4420	1.37986	0.689932	−	0.723874i	$-0.257640\pi$
$859$	40.4420	1.37986	0.689932	+	0.723874i	$0.257640\pi$
$860$	0	0
$861$	4.00594	0.136522
$862$	0	0
$863$	40.2893	1.37147	0.685733	−	0.727854i	$-0.259482\pi$
$863$	40.2893	1.37147	0.685733	+	0.727854i	$0.259482\pi$
$864$	0	0
$865$	11.5668	0.393283
$866$	0	0
$867$	−2.52160	−0.0856381
$868$	0	0
$869$	79.9591	2.71243
$870$	0	0
$871$	32.4292	1.09882
$872$	0	0
$873$	−52.7703	−1.78600
$874$	0	0
$875$	−5.79065	−0.195760
$876$	0	0
$877$	19.3598	0.653734	0.326867	−	0.945070i	$-0.394007\pi$
$877$	19.3598	0.653734	0.326867	+	0.945070i	$0.394007\pi$
$878$	0	0
$879$	−12.5579	−0.423568
$880$	0	0
$881$	−12.0339	−0.405433	−0.202717	−	0.979237i	$-0.564977\pi$
$881$	−12.0339	−0.405433	−0.202717	+	0.979237i	$0.564977\pi$
$882$	0	0
$883$	18.1108	0.609476	0.304738	−	0.952436i	$-0.401431\pi$
$883$	18.1108	0.609476	0.304738	+	0.952436i	$0.401431\pi$
$884$	0	0
$885$	8.39807	0.282298
$886$	0	0
$887$	−19.6716	−0.660508	−0.330254	−	0.943892i	$-0.607134\pi$
$887$	−19.6716	−0.660508	−0.330254	+	0.943892i	$0.607134\pi$
$888$	0	0
$889$	−9.75420	−0.327145
$890$	0	0
$891$	−42.5819	−1.42655
$892$	0	0
$893$	−2.46651	−0.0825387
$894$	0	0
$895$	−13.0354	−0.435726
$896$	0	0
$897$	33.6473	1.12345
$898$	0	0
$899$	7.24990	0.241798
$900$	0	0
$901$	−3.36780	−0.112198
$902$	0	0
$903$	12.2458	0.407516
$904$	0	0
$905$	−7.65514	−0.254465
$906$	0	0
$907$	−7.69508	−0.255511	−0.127756	−	0.991806i	$-0.540777\pi$
$907$	−7.69508	−0.255511	−0.127756	+	0.991806i	$0.540777\pi$
$908$	0	0
$909$	−58.6536	−1.94542
$910$	0	0
$911$	40.7871	1.35134	0.675668	−	0.737206i	$-0.263855\pi$
$911$	40.7871	1.35134	0.675668	+	0.737206i	$0.263855\pi$
$912$	0	0
$913$	40.0799	1.32645
$914$	0	0
$915$	22.7426	0.751849
$916$	0	0
$917$	15.3430	0.506671
$918$	0	0
$919$	−48.0744	−1.58583	−0.792913	−	0.609334i	$-0.791437\pi$
$919$	−48.0744	−1.58583	−0.792913	+	0.609334i	$0.791437\pi$
$920$	0	0
$921$	28.6337	0.943513
$922$	0	0
$923$	−62.3443	−2.05209
$924$	0	0
$925$	33.5303	1.10247
$926$	0	0
$927$	−20.1318	−0.661217
$928$	0	0
$929$	33.9572	1.11410	0.557050	−	0.830479i	$-0.311933\pi$
$929$	33.9572	1.11410	0.557050	+	0.830479i	$0.311933\pi$
$930$	0	0
$931$	−4.17649	−0.136879
$932$	0	0
$933$	−74.0637	−2.42474
$934$	0	0
$935$	−3.28125	−0.107308
$936$	0	0
$937$	−2.74267	−0.0895992	−0.0447996	−	0.998996i	$-0.514265\pi$
$937$	−2.74267	−0.0895992	−0.0447996	+	0.998996i	$0.514265\pi$
$938$	0	0
$939$	−26.9178	−0.878428
$940$	0	0
$941$	−25.3253	−0.825581	−0.412790	−	0.910826i	$-0.635446\pi$
$941$	−25.3253	−0.825581	−0.412790	+	0.910826i	$0.635446\pi$
$942$	0	0
$943$	−4.27454	−0.139198
$944$	0	0
$945$	−0.543026	−0.0176646
$946$	0	0
$947$	−30.0639	−0.976947	−0.488473	−	0.872579i	$-0.662446\pi$
$947$	−30.0639	−0.976947	−0.488473	+	0.872579i	$0.662446\pi$
$948$	0	0
$949$	−46.2956	−1.50282
$950$	0	0
$951$	57.7858	1.87383
$952$	0	0
$953$	47.3950	1.53527	0.767637	−	0.640885i	$-0.221432\pi$
$953$	47.3950	1.53527	0.767637	+	0.640885i	$0.221432\pi$
$954$	0	0
$955$	0.265600	0.00859462
$956$	0	0
$957$	−47.6812	−1.54131
$958$	0	0
$959$	−13.5012	−0.435975
$960$	0	0
$961$	−26.6145	−0.858532
$962$	0	0
$963$	22.2495	0.716980
$964$	0	0
$965$	−4.75164	−0.152961
$966$	0	0
$967$	−42.6663	−1.37206	−0.686028	−	0.727575i	$-0.740647\pi$
$967$	−42.6663	−1.37206	−0.686028	+	0.727575i	$0.740647\pi$
$968$	0	0
$969$	−10.5314	−0.338318
$970$	0	0
$971$	−18.0128	−0.578059	−0.289030	−	0.957320i	$-0.593333\pi$
$971$	−18.0128	−0.578059	−0.289030	+	0.957320i	$0.593333\pi$
$972$	0	0
$973$	−10.6232	−0.340563
$974$	0	0
$975$	−58.0128	−1.85790
$976$	0	0
$977$	−35.8450	−1.14678	−0.573391	−	0.819282i	$-0.694372\pi$
$977$	−35.8450	−1.14678	−0.573391	+	0.819282i	$0.694372\pi$
$978$	0	0
$979$	75.9148	2.42625
$980$	0	0
$981$	32.2150	1.02855
$982$	0	0
$983$	−2.72370	−0.0868726	−0.0434363	−	0.999056i	$-0.513831\pi$
$983$	−2.72370	−0.0868726	−0.0434363	+	0.999056i	$0.513831\pi$
$984$	0	0
$985$	−7.89476	−0.251548
$986$	0	0
$987$	−1.48919	−0.0474013
$988$	0	0
$989$	−13.0669	−0.415504
$990$	0	0
$991$	0.518829	0.0164811	0.00824057	−	0.999966i	$-0.497377\pi$
$991$	0.518829	0.0164811	0.00824057	+	0.999966i	$0.497377\pi$
$992$	0	0
$993$	−13.1546	−0.417450
$994$	0	0
$995$	−2.79204	−0.0885137
$996$	0	0
$997$	40.8584	1.29400	0.646999	−	0.762491i	$-0.276024\pi$
$997$	40.8584	1.29400	0.646999	+	0.762491i	$0.276024\pi$
$998$	0	0
$999$	6.53332	0.206705

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3808.2.a.i.1.2	✓	6
4.3	odd	2		3808.2.a.m.1.5	yes	6
8.3	odd	2		7616.2.a.bx.1.2		6
8.5	even	2		7616.2.a.cb.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.i.1.2	✓	6	1.1	even	1	trivial
3808.2.a.m.1.5	yes	6	4.3	odd	2
7616.2.a.bx.1.2		6	8.3	odd	2
7616.2.a.cb.1.5		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)

\(f(q)\)	\(=\)	\(q-2.52160 q^{3} +0.600745 q^{5} +1.00000 q^{7} +3.35847 q^{9} +5.46196 q^{11} -4.95922 q^{13} -1.51484 q^{15} -1.00000 q^{17} -4.17649 q^{19} -2.52160 q^{21} +2.69068 q^{23} -4.63911 q^{25} -0.903920 q^{27} +3.46196 q^{29} +2.09416 q^{31} -13.7729 q^{33} +0.600745 q^{35} -7.22776 q^{37} +12.5052 q^{39} -1.58865 q^{41} -4.85637 q^{43} +2.01758 q^{45} +0.590571 q^{47} +1.00000 q^{49} +2.52160 q^{51} +3.36780 q^{53} +3.28125 q^{55} +10.5314 q^{57} -5.54387 q^{59} -15.0132 q^{61} +3.35847 q^{63} -2.97922 q^{65} -6.53919 q^{67} -6.78481 q^{69} +12.5714 q^{71} +9.33527 q^{73} +11.6980 q^{75} +5.46196 q^{77} +14.6393 q^{79} -7.79609 q^{81} +7.33801 q^{83} -0.600745 q^{85} -8.72969 q^{87} +13.8988 q^{89} -4.95922 q^{91} -5.28063 q^{93} -2.50900 q^{95} -15.7126 q^{97} +18.3438 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 6 q^{5} + 6 q^{7} + 4 q^{9} - 2 q^{11} - 4 q^{13} + 2 q^{15} - 6 q^{17} - 10 q^{19} - 2 q^{21} + 4 q^{23} + 4 q^{25} - 8 q^{27} - 14 q^{29} + 8 q^{31} - 8 q^{33} - 6 q^{35} - 4 q^{37} + 14 q^{39}+ \cdots - 4 q^{99}+O(q^{100}) \) 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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