LMFDB - Embedded newform 3808.2.a.g.1.4

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:	6.6.147697840.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 10x^{4} + 10x^{3} + 18x^{2} - 16x + 3 $ x^6 - 2x^5 - 10x^4 + 10x^3 + 18x^2 - 16*x + 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			1.4
Root			$0.478948$ 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)

$f(q)$	$=$	$q-0.521052 q^{3} -3.59354 q^{5} +1.00000 q^{7} -2.72850 q^{9} -1.77740 q^{11} +0.242172 q^{13} +1.87242 q^{15} +1.00000 q^{17} +4.14155 q^{19} -0.521052 q^{21} +7.07110 q^{23} +7.91355 q^{25} +2.98485 q^{27} -0.706946 q^{29} +4.08666 q^{31} +0.926117 q^{33} -3.59354 q^{35} +3.76442 q^{37} -0.126184 q^{39} +0.270103 q^{41} -3.01417 q^{43} +9.80500 q^{45} -2.59974 q^{47} +1.00000 q^{49} -0.521052 q^{51} -2.32540 q^{53} +6.38715 q^{55} -2.15796 q^{57} -10.7135 q^{59} -7.86267 q^{61} -2.72850 q^{63} -0.870255 q^{65} -0.657406 q^{67} -3.68441 q^{69} -7.68555 q^{71} +0.326796 q^{73} -4.12337 q^{75} -1.77740 q^{77} -0.628083 q^{79} +6.63025 q^{81} -10.2569 q^{83} -3.59354 q^{85} +0.368356 q^{87} -0.329813 q^{89} +0.242172 q^{91} -2.12937 q^{93} -14.8828 q^{95} -12.7273 q^{97} +4.84964 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{3} - 4 q^{5} + 6 q^{7} + 8 q^{9} - 8 q^{11} - 4 q^{13} + 6 q^{17} - 18 q^{19} - 4 q^{21} - 6 q^{23} + 12 q^{25} - 10 q^{27} + 4 q^{29} + 8 q^{31} - 6 q^{33} - 4 q^{35} - 8 q^{37} + 18 q^{39}+ \cdots - 32 q^{99}+O(q^{100}) $ 6 * q - 4 * q^3 - 4 * q^5 + 6 * q^7 + 8 * q^9 - 8 * q^11 - 4 * q^13 + 6 * q^17 - 18 * q^19 - 4 * q^21 - 6 * q^23 + 12 * q^25 - 10 * q^27 + 4 * q^29 + 8 * q^31 - 6 * q^33 - 4 * q^35 - 8 * q^37 + 18 * q^39 - 4 * q^41 - 20 * q^43 - 24 * q^45 - 6 * q^47 + 6 * q^49 - 4 * q^51 + 14 * q^53 - 4 * q^55 - 4 * q^57 - 20 * q^59 + 2 * q^61 + 8 * q^63 + 6 * q^65 - 28 * q^67 - 4 * q^69 - 18 * q^71 + 4 * q^73 - 20 * q^75 - 8 * q^77 + 2 * q^79 + 34 * q^81 - 20 * q^83 - 4 * q^85 - 22 * q^87 - 6 * q^89 - 4 * q^91 + 40 * q^93 + 6 * q^95 + 30 * q^97 - 32 * 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
$3$	−0.521052	−0.300830	−0.150415	−	0.988623i	$-0.548061\pi$
$3$	−0.521052	−0.300830	−0.150415	+	0.988623i	$0.548061\pi$
$4$	0	0
$5$	−3.59354	−1.60708	−0.803541	−	0.595250i	$-0.797053\pi$
$5$	−3.59354	−1.60708	−0.803541	+	0.595250i	$0.797053\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.72850	−0.909502
$10$	0	0
$11$	−1.77740	−0.535905	−0.267953	−	0.963432i	$-0.586347\pi$
$11$	−1.77740	−0.535905	−0.267953	+	0.963432i	$0.586347\pi$
$12$	0	0
$13$	0.242172	0.0671664	0.0335832	−	0.999436i	$-0.489308\pi$
$13$	0.242172	0.0671664	0.0335832	+	0.999436i	$0.489308\pi$
$14$	0	0
$15$	1.87242	0.483458
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	4.14155	0.950137	0.475068	−	0.879949i	$-0.342423\pi$
$19$	4.14155	0.950137	0.475068	+	0.879949i	$0.342423\pi$
$20$	0	0
$21$	−0.521052	−0.113703
$22$	0	0
$23$	7.07110	1.47443	0.737213	−	0.675660i	$-0.236141\pi$
$23$	7.07110	1.47443	0.737213	+	0.675660i	$0.236141\pi$
$24$	0	0
$25$	7.91355	1.58271
$26$	0	0
$27$	2.98485	0.574435
$28$	0	0
$29$	−0.706946	−0.131277	−0.0656383	−	0.997843i	$-0.520908\pi$
$29$	−0.706946	−0.131277	−0.0656383	+	0.997843i	$0.520908\pi$
$30$	0	0
$31$	4.08666	0.733987	0.366993	−	0.930224i	$-0.380387\pi$
$31$	4.08666	0.733987	0.366993	+	0.930224i	$0.380387\pi$
$32$	0	0
$33$	0.926117	0.161216
$34$	0	0
$35$	−3.59354	−0.607420
$36$	0	0
$37$	3.76442	0.618866	0.309433	−	0.950921i	$-0.399861\pi$
$37$	3.76442	0.618866	0.309433	+	0.950921i	$0.399861\pi$
$38$	0	0
$39$	−0.126184	−0.0202056
$40$	0	0
$41$	0.270103	0.0421830	0.0210915	−	0.999778i	$-0.493286\pi$
$41$	0.270103	0.0421830	0.0210915	+	0.999778i	$0.493286\pi$
$42$	0	0
$43$	−3.01417	−0.459657	−0.229829	−	0.973231i	$-0.573817\pi$
$43$	−3.01417	−0.459657	−0.229829	+	0.973231i	$0.573817\pi$
$44$	0	0
$45$	9.80500	1.46164
$46$	0	0
$47$	−2.59974	−0.379211	−0.189605	−	0.981860i	$-0.560721\pi$
$47$	−2.59974	−0.379211	−0.189605	+	0.981860i	$0.560721\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.521052	−0.0729619
$52$	0	0
$53$	−2.32540	−0.319419	−0.159709	−	0.987164i	$-0.551056\pi$
$53$	−2.32540	−0.319419	−0.159709	+	0.987164i	$0.551056\pi$
$54$	0	0
$55$	6.38715	0.861244
$56$	0	0
$57$	−2.15796	−0.285829
$58$	0	0
$59$	−10.7135	−1.39478	−0.697392	−	0.716690i	$-0.745656\pi$
$59$	−10.7135	−1.39478	−0.697392	+	0.716690i	$0.745656\pi$
$60$	0	0
$61$	−7.86267	−1.00671	−0.503356	−	0.864079i	$-0.667901\pi$
$61$	−7.86267	−1.00671	−0.503356	+	0.864079i	$0.667901\pi$
$62$	0	0
$63$	−2.72850	−0.343759
$64$	0	0
$65$	−0.870255	−0.107942
$66$	0	0
$67$	−0.657406	−0.0803150	−0.0401575	−	0.999193i	$-0.512786\pi$
$67$	−0.657406	−0.0803150	−0.0401575	+	0.999193i	$0.512786\pi$
$68$	0	0
$69$	−3.68441	−0.443551
$70$	0	0
$71$	−7.68555	−0.912107	−0.456054	−	0.889952i	$-0.650737\pi$
$71$	−7.68555	−0.912107	−0.456054	+	0.889952i	$0.650737\pi$
$72$	0	0
$73$	0.326796	0.0382486	0.0191243	−	0.999817i	$-0.493912\pi$
$73$	0.326796	0.0382486	0.0191243	+	0.999817i	$0.493912\pi$
$74$	0	0
$75$	−4.12337	−0.476126
$76$	0	0
$77$	−1.77740	−0.202553
$78$	0	0
$79$	−0.628083	−0.0706648	−0.0353324	−	0.999376i	$-0.511249\pi$
$79$	−0.628083	−0.0706648	−0.0353324	+	0.999376i	$0.511249\pi$
$80$	0	0
$81$	6.63025	0.736695
$82$	0	0
$83$	−10.2569	−1.12585	−0.562923	−	0.826509i	$-0.690323\pi$
$83$	−10.2569	−1.12585	−0.562923	+	0.826509i	$0.690323\pi$
$84$	0	0
$85$	−3.59354	−0.389774
$86$	0	0
$87$	0.368356	0.0394919
$88$	0	0
$89$	−0.329813	−0.0349601	−0.0174800	−	0.999847i	$-0.505564\pi$
$89$	−0.329813	−0.0349601	−0.0174800	+	0.999847i	$0.505564\pi$
$90$	0	0
$91$	0.242172	0.0253865
$92$	0	0
$93$	−2.12937	−0.220805
$94$	0	0
$95$	−14.8828	−1.52695
$96$	0	0
$97$	−12.7273	−1.29226	−0.646132	−	0.763226i	$-0.723615\pi$
$97$	−12.7273	−1.29226	−0.646132	+	0.763226i	$0.723615\pi$
$98$	0	0
$99$	4.84964	0.487407
$100$	0	0
$101$	−2.50926	−0.249680	−0.124840	−	0.992177i	$-0.539842\pi$
$101$	−2.50926	−0.249680	−0.124840	+	0.992177i	$0.539842\pi$
$102$	0	0
$103$	10.6270	1.04711	0.523556	−	0.851991i	$-0.324605\pi$
$103$	10.6270	1.04711	0.523556	+	0.851991i	$0.324605\pi$
$104$	0	0
$105$	1.87242	0.182730
$106$	0	0
$107$	−9.41312	−0.910001	−0.455000	−	0.890491i	$-0.650361\pi$
$107$	−9.41312	−0.910001	−0.455000	+	0.890491i	$0.650361\pi$
$108$	0	0
$109$	9.67677	0.926867	0.463433	−	0.886132i	$-0.346617\pi$
$109$	9.67677	0.926867	0.463433	+	0.886132i	$0.346617\pi$
$110$	0	0
$111$	−1.96146	−0.186173
$112$	0	0
$113$	1.19664	0.112570	0.0562850	−	0.998415i	$-0.482074\pi$
$113$	1.19664	0.112570	0.0562850	+	0.998415i	$0.482074\pi$
$114$	0	0
$115$	−25.4103	−2.36952
$116$	0	0
$117$	−0.660767	−0.0610879
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	−7.84086	−0.712805
$122$	0	0
$123$	−0.140738	−0.0126899
$124$	0	0
$125$	−10.4700	−0.936463
$126$	0	0
$127$	−8.67058	−0.769390	−0.384695	−	0.923044i	$-0.625693\pi$
$127$	−8.67058	−0.769390	−0.384695	+	0.923044i	$0.625693\pi$
$128$	0	0
$129$	1.57054	0.138278
$130$	0	0
$131$	−10.9781	−0.959162	−0.479581	−	0.877498i	$-0.659211\pi$
$131$	−10.9781	−0.959162	−0.479581	+	0.877498i	$0.659211\pi$
$132$	0	0
$133$	4.14155	0.359118
$134$	0	0
$135$	−10.7262	−0.923163
$136$	0	0
$137$	20.9432	1.78930	0.894649	−	0.446770i	$-0.147426\pi$
$137$	20.9432	1.78930	0.894649	+	0.446770i	$0.147426\pi$
$138$	0	0
$139$	−21.1638	−1.79509	−0.897543	−	0.440927i	$-0.854650\pi$
$139$	−21.1638	−1.79509	−0.897543	+	0.440927i	$0.854650\pi$
$140$	0	0
$141$	1.35460	0.114078
$142$	0	0
$143$	−0.430435	−0.0359948
$144$	0	0
$145$	2.54044	0.210972
$146$	0	0
$147$	−0.521052	−0.0429757
$148$	0	0
$149$	14.5850	1.19485	0.597424	−	0.801925i	$-0.296191\pi$
$149$	14.5850	1.19485	0.597424	+	0.801925i	$0.296191\pi$
$150$	0	0
$151$	18.1426	1.47642	0.738212	−	0.674568i	$-0.235670\pi$
$151$	18.1426	1.47642	0.738212	+	0.674568i	$0.235670\pi$
$152$	0	0
$153$	−2.72850	−0.220587
$154$	0	0
$155$	−14.6856	−1.17958
$156$	0	0
$157$	18.6110	1.48532	0.742658	−	0.669671i	$-0.233565\pi$
$157$	18.6110	1.48532	0.742658	+	0.669671i	$0.233565\pi$
$158$	0	0
$159$	1.21166	0.0960906
$160$	0	0
$161$	7.07110	0.557281
$162$	0	0
$163$	−0.692723	−0.0542582	−0.0271291	−	0.999632i	$-0.508637\pi$
$163$	−0.692723	−0.0542582	−0.0271291	+	0.999632i	$0.508637\pi$
$164$	0	0
$165$	−3.32804	−0.259088
$166$	0	0
$167$	5.43251	0.420380	0.210190	−	0.977661i	$-0.432592\pi$
$167$	5.43251	0.420380	0.210190	+	0.977661i	$0.432592\pi$
$168$	0	0
$169$	−12.9414	−0.995489
$170$	0	0
$171$	−11.3002	−0.864151
$172$	0	0
$173$	−6.64724	−0.505380	−0.252690	−	0.967547i	$-0.581315\pi$
$173$	−6.64724	−0.505380	−0.252690	+	0.967547i	$0.581315\pi$
$174$	0	0
$175$	7.91355	0.598208
$176$	0	0
$177$	5.58231	0.419592
$178$	0	0
$179$	−23.7425	−1.77460	−0.887298	−	0.461196i	$-0.847420\pi$
$179$	−23.7425	−1.77460	−0.887298	+	0.461196i	$0.847420\pi$
$180$	0	0
$181$	−0.807569	−0.0600261	−0.0300131	−	0.999550i	$-0.509555\pi$
$181$	−0.807569	−0.0600261	−0.0300131	+	0.999550i	$0.509555\pi$
$182$	0	0
$183$	4.09686	0.302849
$184$	0	0
$185$	−13.5276	−0.994568
$186$	0	0
$187$	−1.77740	−0.129976
$188$	0	0
$189$	2.98485	0.217116
$190$	0	0
$191$	1.49827	0.108411	0.0542053	−	0.998530i	$-0.482737\pi$
$191$	1.49827	0.108411	0.0542053	+	0.998530i	$0.482737\pi$
$192$	0	0
$193$	16.8296	1.21142	0.605710	−	0.795686i	$-0.292889\pi$
$193$	16.8296	1.21142	0.605710	+	0.795686i	$0.292889\pi$
$194$	0	0
$195$	0.453448	0.0324721
$196$	0	0
$197$	−14.6500	−1.04377	−0.521886	−	0.853015i	$-0.674771\pi$
$197$	−14.6500	−1.04377	−0.521886	+	0.853015i	$0.674771\pi$
$198$	0	0
$199$	2.62826	0.186313	0.0931563	−	0.995651i	$-0.470304\pi$
$199$	2.62826	0.186313	0.0931563	+	0.995651i	$0.470304\pi$
$200$	0	0
$201$	0.342543	0.0241611
$202$	0	0
$203$	−0.706946	−0.0496179
$204$	0	0
$205$	−0.970626	−0.0677915
$206$	0	0
$207$	−19.2935	−1.34099
$208$	0	0
$209$	−7.36118	−0.509183
$210$	0	0
$211$	1.86285	0.128244	0.0641219	−	0.997942i	$-0.479575\pi$
$211$	1.86285	0.128244	0.0641219	+	0.997942i	$0.479575\pi$
$212$	0	0
$213$	4.00457	0.274389
$214$	0	0
$215$	10.8316	0.738706
$216$	0	0
$217$	4.08666	0.277421
$218$	0	0
$219$	−0.170278	−0.0115063
$220$	0	0
$221$	0.242172	0.0162902
$222$	0	0
$223$	−19.8238	−1.32750	−0.663749	−	0.747955i	$-0.731036\pi$
$223$	−19.8238	−1.32750	−0.663749	+	0.747955i	$0.731036\pi$
$224$	0	0
$225$	−21.5922	−1.43948
$226$	0	0
$227$	4.40949	0.292668	0.146334	−	0.989235i	$-0.453253\pi$
$227$	4.40949	0.292668	0.146334	+	0.989235i	$0.453253\pi$
$228$	0	0
$229$	−0.133049	−0.00879212	−0.00439606	−	0.999990i	$-0.501399\pi$
$229$	−0.133049	−0.00879212	−0.00439606	+	0.999990i	$0.501399\pi$
$230$	0	0
$231$	0.926117	0.0609340
$232$	0	0
$233$	15.9579	1.04543	0.522717	−	0.852506i	$-0.324918\pi$
$233$	15.9579	1.04543	0.522717	+	0.852506i	$0.324918\pi$
$234$	0	0
$235$	9.34226	0.609422
$236$	0	0
$237$	0.327264	0.0212581
$238$	0	0
$239$	−20.3435	−1.31591	−0.657954	−	0.753058i	$-0.728578\pi$
$239$	−20.3435	−1.31591	−0.657954	+	0.753058i	$0.728578\pi$
$240$	0	0
$241$	−8.65865	−0.557753	−0.278876	−	0.960327i	$-0.589962\pi$
$241$	−8.65865	−0.557753	−0.278876	+	0.960327i	$0.589962\pi$
$242$	0	0
$243$	−12.4093	−0.796054
$244$	0	0
$245$	−3.59354	−0.229583
$246$	0	0
$247$	1.00297	0.0638172
$248$	0	0
$249$	5.34440	0.338688
$250$	0	0
$251$	−13.2464	−0.836104	−0.418052	−	0.908423i	$-0.637287\pi$
$251$	−13.2464	−0.836104	−0.418052	+	0.908423i	$0.637287\pi$
$252$	0	0
$253$	−12.5682	−0.790153
$254$	0	0
$255$	1.87242	0.117256
$256$	0	0
$257$	−28.6363	−1.78628	−0.893141	−	0.449777i	$-0.851504\pi$
$257$	−28.6363	−1.78628	−0.893141	+	0.449777i	$0.851504\pi$
$258$	0	0
$259$	3.76442	0.233909
$260$	0	0
$261$	1.92891	0.119396
$262$	0	0
$263$	8.76904	0.540722	0.270361	−	0.962759i	$-0.412857\pi$
$263$	8.76904	0.540722	0.270361	+	0.962759i	$0.412857\pi$
$264$	0	0
$265$	8.35644	0.513332
$266$	0	0
$267$	0.171850	0.0105170
$268$	0	0
$269$	4.50063	0.274408	0.137204	−	0.990543i	$-0.456188\pi$
$269$	4.50063	0.274408	0.137204	+	0.990543i	$0.456188\pi$
$270$	0	0
$271$	−31.7715	−1.92998	−0.964990	−	0.262286i	$-0.915524\pi$
$271$	−31.7715	−1.92998	−0.964990	+	0.262286i	$0.915524\pi$
$272$	0	0
$273$	−0.126184	−0.00763701
$274$	0	0
$275$	−14.0655	−0.848183
$276$	0	0
$277$	13.5501	0.814146	0.407073	−	0.913396i	$-0.366549\pi$
$277$	13.5501	0.814146	0.407073	+	0.913396i	$0.366549\pi$
$278$	0	0
$279$	−11.1505	−0.667562
$280$	0	0
$281$	−26.1800	−1.56177	−0.780885	−	0.624675i	$-0.785231\pi$
$281$	−26.1800	−1.56177	−0.780885	+	0.624675i	$0.785231\pi$
$282$	0	0
$283$	29.6230	1.76090	0.880451	−	0.474137i	$-0.157240\pi$
$283$	29.6230	1.76090	0.880451	+	0.474137i	$0.157240\pi$
$284$	0	0
$285$	7.75473	0.459351
$286$	0	0
$287$	0.270103	0.0159437
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	6.63160	0.388751
$292$	0	0
$293$	−11.3031	−0.660333	−0.330166	−	0.943923i	$-0.607105\pi$
$293$	−11.3031	−0.660333	−0.330166	+	0.943923i	$0.607105\pi$
$294$	0	0
$295$	38.4995	2.24153
$296$	0	0
$297$	−5.30526	−0.307843
$298$	0	0
$299$	1.71242	0.0990318
$300$	0	0
$301$	−3.01417	−0.173734
$302$	0	0
$303$	1.30745	0.0751113
$304$	0	0
$305$	28.2548	1.61787
$306$	0	0
$307$	0.132000	0.00753363	0.00376682	−	0.999993i	$-0.498801\pi$
$307$	0.132000	0.00753363	0.00376682	+	0.999993i	$0.498801\pi$
$308$	0	0
$309$	−5.53724	−0.315003
$310$	0	0
$311$	15.5261	0.880405	0.440203	−	0.897898i	$-0.354907\pi$
$311$	15.5261	0.880405	0.440203	+	0.897898i	$0.354907\pi$
$312$	0	0
$313$	−5.24435	−0.296428	−0.148214	−	0.988955i	$-0.547353\pi$
$313$	−5.24435	−0.296428	−0.148214	+	0.988955i	$0.547353\pi$
$314$	0	0
$315$	9.80500	0.552449
$316$	0	0
$317$	−4.82970	−0.271263	−0.135631	−	0.990759i	$-0.543306\pi$
$317$	−4.82970	−0.271263	−0.135631	+	0.990759i	$0.543306\pi$
$318$	0	0
$319$	1.25652	0.0703519
$320$	0	0
$321$	4.90473	0.273755
$322$	0	0
$323$	4.14155	0.230442
$324$	0	0
$325$	1.91644	0.106305
$326$	0	0
$327$	−5.04210	−0.278829
$328$	0	0
$329$	−2.59974	−0.143328
$330$	0	0
$331$	13.9415	0.766295	0.383148	−	0.923687i	$-0.374840\pi$
$331$	13.9415	0.766295	0.383148	+	0.923687i	$0.374840\pi$
$332$	0	0
$333$	−10.2712	−0.562860
$334$	0	0
$335$	2.36242	0.129073
$336$	0	0
$337$	7.73051	0.421107	0.210554	−	0.977582i	$-0.432473\pi$
$337$	7.73051	0.421107	0.210554	+	0.977582i	$0.432473\pi$
$338$	0	0
$339$	−0.623509	−0.0338644
$340$	0	0
$341$	−7.26363	−0.393347
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	13.2401	0.712822
$346$	0	0
$347$	−11.1765	−0.599984	−0.299992	−	0.953942i	$-0.596984\pi$
$347$	−11.1765	−0.599984	−0.299992	+	0.953942i	$0.596984\pi$
$348$	0	0
$349$	−16.2102	−0.867711	−0.433855	−	0.900983i	$-0.642847\pi$
$349$	−16.2102	−0.867711	−0.433855	+	0.900983i	$0.642847\pi$
$350$	0	0
$351$	0.722846	0.0385827
$352$	0	0
$353$	−24.0699	−1.28111	−0.640556	−	0.767911i	$-0.721296\pi$
$353$	−24.0699	−1.28111	−0.640556	+	0.767911i	$0.721296\pi$
$354$	0	0
$355$	27.6184	1.46583
$356$	0	0
$357$	−0.521052	−0.0275770
$358$	0	0
$359$	−20.0758	−1.05956	−0.529780	−	0.848135i	$-0.677726\pi$
$359$	−20.0758	−1.05956	−0.529780	+	0.848135i	$0.677726\pi$
$360$	0	0
$361$	−1.84757	−0.0972406
$362$	0	0
$363$	4.08550	0.214433
$364$	0	0
$365$	−1.17436	−0.0614686
$366$	0	0
$367$	−21.9057	−1.14347	−0.571733	−	0.820440i	$-0.693729\pi$
$367$	−21.9057	−1.14347	−0.571733	+	0.820440i	$0.693729\pi$
$368$	0	0
$369$	−0.736977	−0.0383655
$370$	0	0
$371$	−2.32540	−0.120729
$372$	0	0
$373$	5.14722	0.266513	0.133257	−	0.991082i	$-0.457457\pi$
$373$	5.14722	0.266513	0.133257	+	0.991082i	$0.457457\pi$
$374$	0	0
$375$	5.45540	0.281716
$376$	0	0
$377$	−0.171202	−0.00881737
$378$	0	0
$379$	−23.0656	−1.18480	−0.592400	−	0.805644i	$-0.701819\pi$
$379$	−23.0656	−1.18480	−0.592400	+	0.805644i	$0.701819\pi$
$380$	0	0
$381$	4.51783	0.231455
$382$	0	0
$383$	16.7575	0.856270	0.428135	−	0.903715i	$-0.359171\pi$
$383$	16.7575	0.856270	0.428135	+	0.903715i	$0.359171\pi$
$384$	0	0
$385$	6.38715	0.325519
$386$	0	0
$387$	8.22419	0.418059
$388$	0	0
$389$	21.8542	1.10805	0.554027	−	0.832499i	$-0.313090\pi$
$389$	21.8542	1.10805	0.554027	+	0.832499i	$0.313090\pi$
$390$	0	0
$391$	7.07110	0.357601
$392$	0	0
$393$	5.72017	0.288544
$394$	0	0
$395$	2.25704	0.113564
$396$	0	0
$397$	5.50933	0.276506	0.138253	−	0.990397i	$-0.455851\pi$
$397$	5.50933	0.276506	0.138253	+	0.990397i	$0.455851\pi$
$398$	0	0
$399$	−2.15796	−0.108033
$400$	0	0
$401$	11.9283	0.595672	0.297836	−	0.954617i	$-0.403735\pi$
$401$	11.9283	0.595672	0.297836	+	0.954617i	$0.403735\pi$
$402$	0	0
$403$	0.989675	0.0492992
$404$	0	0
$405$	−23.8261	−1.18393
$406$	0	0
$407$	−6.69086	−0.331654
$408$	0	0
$409$	−34.2245	−1.69229	−0.846145	−	0.532953i	$-0.821082\pi$
$409$	−34.2245	−1.69229	−0.846145	+	0.532953i	$0.821082\pi$
$410$	0	0
$411$	−10.9125	−0.538274
$412$	0	0
$413$	−10.7135	−0.527179
$414$	0	0
$415$	36.8588	1.80933
$416$	0	0
$417$	11.0274	0.540015
$418$	0	0
$419$	16.1257	0.787793	0.393897	−	0.919155i	$-0.371127\pi$
$419$	16.1257	0.787793	0.393897	+	0.919155i	$0.371127\pi$
$420$	0	0
$421$	−5.97331	−0.291121	−0.145561	−	0.989349i	$-0.546499\pi$
$421$	−5.97331	−0.291121	−0.145561	+	0.989349i	$0.546499\pi$
$422$	0	0
$423$	7.09339	0.344893
$424$	0	0
$425$	7.91355	0.383864
$426$	0	0
$427$	−7.86267	−0.380501
$428$	0	0
$429$	0.224279	0.0108283
$430$	0	0
$431$	2.34394	0.112903	0.0564517	−	0.998405i	$-0.482021\pi$
$431$	2.34394	0.112903	0.0564517	+	0.998405i	$0.482021\pi$
$432$	0	0
$433$	9.69611	0.465965	0.232983	−	0.972481i	$-0.425151\pi$
$433$	9.69611	0.465965	0.232983	+	0.972481i	$0.425151\pi$
$434$	0	0
$435$	−1.32370	−0.0634667
$436$	0	0
$437$	29.2853	1.40091
$438$	0	0
$439$	24.2470	1.15725	0.578623	−	0.815595i	$-0.303590\pi$
$439$	24.2470	1.15725	0.578623	+	0.815595i	$0.303590\pi$
$440$	0	0
$441$	−2.72850	−0.129929
$442$	0	0
$443$	−24.9603	−1.18590	−0.592949	−	0.805240i	$-0.702037\pi$
$443$	−24.9603	−1.18590	−0.592949	+	0.805240i	$0.702037\pi$
$444$	0	0
$445$	1.18520	0.0561837
$446$	0	0
$447$	−7.59954	−0.359446
$448$	0	0
$449$	29.2044	1.37824	0.689121	−	0.724646i	$-0.257997\pi$
$449$	29.2044	1.37824	0.689121	+	0.724646i	$0.257997\pi$
$450$	0	0
$451$	−0.480080	−0.0226061
$452$	0	0
$453$	−9.45325	−0.444152
$454$	0	0
$455$	−0.870255	−0.0407982
$456$	0	0
$457$	−19.4825	−0.911355	−0.455678	−	0.890145i	$-0.650603\pi$
$457$	−19.4825	−0.911355	−0.455678	+	0.890145i	$0.650603\pi$
$458$	0	0
$459$	2.98485	0.139321
$460$	0	0
$461$	−9.11030	−0.424309	−0.212155	−	0.977236i	$-0.568048\pi$
$461$	−9.11030	−0.424309	−0.212155	+	0.977236i	$0.568048\pi$
$462$	0	0
$463$	−14.5969	−0.678376	−0.339188	−	0.940719i	$-0.610152\pi$
$463$	−14.5969	−0.678376	−0.339188	+	0.940719i	$0.610152\pi$
$464$	0	0
$465$	7.65197	0.354851
$466$	0	0
$467$	−29.5801	−1.36880	−0.684401	−	0.729106i	$-0.739936\pi$
$467$	−29.5801	−1.36880	−0.684401	+	0.729106i	$0.739936\pi$
$468$	0	0
$469$	−0.657406	−0.0303562
$470$	0	0
$471$	−9.69728	−0.446827
$472$	0	0
$473$	5.35738	0.246333
$474$	0	0
$475$	32.7744	1.50379
$476$	0	0
$477$	6.34487	0.290512
$478$	0	0
$479$	−13.4317	−0.613709	−0.306855	−	0.951756i	$-0.599277\pi$
$479$	−13.4317	−0.613709	−0.306855	+	0.951756i	$0.599277\pi$
$480$	0	0
$481$	0.911635	0.0415670
$482$	0	0
$483$	−3.68441	−0.167647
$484$	0	0
$485$	45.7362	2.07677
$486$	0	0
$487$	40.5453	1.83728	0.918642	−	0.395092i	$-0.129287\pi$
$487$	40.5453	1.83728	0.918642	+	0.395092i	$0.129287\pi$
$488$	0	0
$489$	0.360945	0.0163225
$490$	0	0
$491$	34.2949	1.54771	0.773854	−	0.633364i	$-0.218326\pi$
$491$	34.2949	1.54771	0.773854	+	0.633364i	$0.218326\pi$
$492$	0	0
$493$	−0.706946	−0.0318393
$494$	0	0
$495$	−17.4274	−0.783302
$496$	0	0
$497$	−7.68555	−0.344744
$498$	0	0
$499$	18.6425	0.834553	0.417276	−	0.908780i	$-0.362985\pi$
$499$	18.6425	0.834553	0.417276	+	0.908780i	$0.362985\pi$
$500$	0	0
$501$	−2.83062	−0.126463
$502$	0	0
$503$	−2.38529	−0.106355	−0.0531775	−	0.998585i	$-0.516935\pi$
$503$	−2.38529	−0.106355	−0.0531775	+	0.998585i	$0.516935\pi$
$504$	0	0
$505$	9.01713	0.401257
$506$	0	0
$507$	6.74312	0.299472
$508$	0	0
$509$	−18.9969	−0.842021	−0.421011	−	0.907056i	$-0.638325\pi$
$509$	−18.9969	−0.842021	−0.421011	+	0.907056i	$0.638325\pi$
$510$	0	0
$511$	0.326796	0.0144566
$512$	0	0
$513$	12.3619	0.545791
$514$	0	0
$515$	−38.1887	−1.68280
$516$	0	0
$517$	4.62076	0.203221
$518$	0	0
$519$	3.46356	0.152033
$520$	0	0
$521$	−41.8609	−1.83396	−0.916980	−	0.398934i	$-0.869380\pi$
$521$	−41.8609	−1.83396	−0.916980	+	0.398934i	$0.869380\pi$
$522$	0	0
$523$	−18.2454	−0.797817	−0.398909	−	0.916991i	$-0.630611\pi$
$523$	−18.2454	−0.797817	−0.398909	+	0.916991i	$0.630611\pi$
$524$	0	0
$525$	−4.12337	−0.179959
$526$	0	0
$527$	4.08666	0.178018
$528$	0	0
$529$	27.0004	1.17393
$530$	0	0
$531$	29.2319	1.26856
$532$	0	0
$533$	0.0654113	0.00283328
$534$	0	0
$535$	33.8265	1.46245
$536$	0	0
$537$	12.3711	0.533851
$538$	0	0
$539$	−1.77740	−0.0765579
$540$	0	0
$541$	−1.17618	−0.0505680	−0.0252840	−	0.999680i	$-0.508049\pi$
$541$	−1.17618	−0.0505680	−0.0252840	+	0.999680i	$0.508049\pi$
$542$	0	0
$543$	0.420786	0.0180576
$544$	0	0
$545$	−34.7739	−1.48955
$546$	0	0
$547$	33.9132	1.45002	0.725011	−	0.688737i	$-0.241835\pi$
$547$	33.9132	1.45002	0.725011	+	0.688737i	$0.241835\pi$
$548$	0	0
$549$	21.4533	0.915606
$550$	0	0
$551$	−2.92785	−0.124731
$552$	0	0
$553$	−0.628083	−0.0267088
$554$	0	0
$555$	7.04858	0.299196
$556$	0	0
$557$	4.32264	0.183156	0.0915780	−	0.995798i	$-0.470809\pi$
$557$	4.32264	0.183156	0.0915780	+	0.995798i	$0.470809\pi$
$558$	0	0
$559$	−0.729948	−0.0308735
$560$	0	0
$561$	0.926117	0.0391007
$562$	0	0
$563$	−1.53402	−0.0646514	−0.0323257	−	0.999477i	$-0.510291\pi$
$563$	−1.53402	−0.0646514	−0.0323257	+	0.999477i	$0.510291\pi$
$564$	0	0
$565$	−4.30016	−0.180909
$566$	0	0
$567$	6.63025	0.278444
$568$	0	0
$569$	25.6759	1.07639	0.538194	−	0.842821i	$-0.319107\pi$
$569$	25.6759	1.07639	0.538194	+	0.842821i	$0.319107\pi$
$570$	0	0
$571$	9.20825	0.385353	0.192677	−	0.981262i	$-0.438283\pi$
$571$	9.20825	0.385353	0.192677	+	0.981262i	$0.438283\pi$
$572$	0	0
$573$	−0.780674	−0.0326131
$574$	0	0
$575$	55.9575	2.33359
$576$	0	0
$577$	−28.6499	−1.19271	−0.596356	−	0.802720i	$-0.703385\pi$
$577$	−28.6499	−1.19271	−0.596356	+	0.802720i	$0.703385\pi$
$578$	0	0
$579$	−8.76909	−0.364431
$580$	0	0
$581$	−10.2569	−0.425530
$582$	0	0
$583$	4.13317	0.171178
$584$	0	0
$585$	2.37449	0.0981732
$586$	0	0
$587$	−19.2263	−0.793553	−0.396777	−	0.917915i	$-0.629871\pi$
$587$	−19.2263	−0.793553	−0.396777	+	0.917915i	$0.629871\pi$
$588$	0	0
$589$	16.9251	0.697387
$590$	0	0
$591$	7.63343	0.313997
$592$	0	0
$593$	3.01321	0.123738	0.0618689	−	0.998084i	$-0.480294\pi$
$593$	3.01321	0.123738	0.0618689	+	0.998084i	$0.480294\pi$
$594$	0	0
$595$	−3.59354	−0.147321
$596$	0	0
$597$	−1.36946	−0.0560483
$598$	0	0
$599$	−31.9070	−1.30368	−0.651842	−	0.758355i	$-0.726003\pi$
$599$	−31.9070	−1.30368	−0.651842	+	0.758355i	$0.726003\pi$
$600$	0	0
$601$	−3.85064	−0.157071	−0.0785354	−	0.996911i	$-0.525024\pi$
$601$	−3.85064	−0.157071	−0.0785354	+	0.996911i	$0.525024\pi$
$602$	0	0
$603$	1.79374	0.0730466
$604$	0	0
$605$	28.1765	1.14554
$606$	0	0
$607$	12.4976	0.507262	0.253631	−	0.967301i	$-0.418375\pi$
$607$	12.4976	0.507262	0.253631	+	0.967301i	$0.418375\pi$
$608$	0	0
$609$	0.368356	0.0149265
$610$	0	0
$611$	−0.629583	−0.0254702
$612$	0	0
$613$	34.3960	1.38924	0.694621	−	0.719376i	$-0.255572\pi$
$613$	34.3960	1.38924	0.694621	+	0.719376i	$0.255572\pi$
$614$	0	0
$615$	0.505747	0.0203937
$616$	0	0
$617$	−3.87366	−0.155948	−0.0779739	−	0.996955i	$-0.524845\pi$
$617$	−3.87366	−0.155948	−0.0779739	+	0.996955i	$0.524845\pi$
$618$	0	0
$619$	−10.4694	−0.420801	−0.210400	−	0.977615i	$-0.567477\pi$
$619$	−10.4694	−0.420801	−0.210400	+	0.977615i	$0.567477\pi$
$620$	0	0
$621$	21.1062	0.846961
$622$	0	0
$623$	−0.329813	−0.0132137
$624$	0	0
$625$	−1.94347	−0.0777388
$626$	0	0
$627$	3.83556	0.153177
$628$	0	0
$629$	3.76442	0.150097
$630$	0	0
$631$	2.23285	0.0888884	0.0444442	−	0.999012i	$-0.485848\pi$
$631$	2.23285	0.0888884	0.0444442	+	0.999012i	$0.485848\pi$
$632$	0	0
$633$	−0.970641	−0.0385795
$634$	0	0
$635$	31.1581	1.23647
$636$	0	0
$637$	0.242172	0.00959519
$638$	0	0
$639$	20.9701	0.829563
$640$	0	0
$641$	3.17089	0.125243	0.0626213	−	0.998037i	$-0.480054\pi$
$641$	3.17089	0.125243	0.0626213	+	0.998037i	$0.480054\pi$
$642$	0	0
$643$	−28.5119	−1.12440	−0.562200	−	0.827001i	$-0.690045\pi$
$643$	−28.5119	−1.12440	−0.562200	+	0.827001i	$0.690045\pi$
$644$	0	0
$645$	−5.64381	−0.222225
$646$	0	0
$647$	−46.3718	−1.82306	−0.911531	−	0.411231i	$-0.865099\pi$
$647$	−46.3718	−1.82306	−0.911531	+	0.411231i	$0.865099\pi$
$648$	0	0
$649$	19.0422	0.747472
$650$	0	0
$651$	−2.12937	−0.0834564
$652$	0	0
$653$	−31.2361	−1.22236	−0.611181	−	0.791491i	$-0.709305\pi$
$653$	−31.2361	−1.22236	−0.611181	+	0.791491i	$0.709305\pi$
$654$	0	0
$655$	39.4503	1.54145
$656$	0	0
$657$	−0.891664	−0.0347871
$658$	0	0
$659$	−40.2223	−1.56684	−0.783420	−	0.621493i	$-0.786527\pi$
$659$	−40.2223	−1.56684	−0.783420	+	0.621493i	$0.786527\pi$
$660$	0	0
$661$	−32.2903	−1.25595	−0.627973	−	0.778235i	$-0.716115\pi$
$661$	−32.2903	−1.25595	−0.627973	+	0.778235i	$0.716115\pi$
$662$	0	0
$663$	−0.126184	−0.00490059
$664$	0	0
$665$	−14.8828	−0.577132
$666$	0	0
$667$	−4.99889	−0.193558
$668$	0	0
$669$	10.3292	0.399351
$670$	0	0
$671$	13.9751	0.539502
$672$	0	0
$673$	20.5674	0.792816	0.396408	−	0.918074i	$-0.370256\pi$
$673$	20.5674	0.792816	0.396408	+	0.918074i	$0.370256\pi$
$674$	0	0
$675$	23.6208	0.909163
$676$	0	0
$677$	−8.63622	−0.331917	−0.165958	−	0.986133i	$-0.553072\pi$
$677$	−8.63622	−0.331917	−0.165958	+	0.986133i	$0.553072\pi$
$678$	0	0
$679$	−12.7273	−0.488430
$680$	0	0
$681$	−2.29757	−0.0880432
$682$	0	0
$683$	−36.8513	−1.41008	−0.705038	−	0.709169i	$-0.749070\pi$
$683$	−36.8513	−1.41008	−0.705038	+	0.709169i	$0.749070\pi$
$684$	0	0
$685$	−75.2603	−2.87555
$686$	0	0
$687$	0.0693254	0.00264493
$688$	0	0
$689$	−0.563147	−0.0214542
$690$	0	0
$691$	39.1003	1.48745	0.743723	−	0.668488i	$-0.233058\pi$
$691$	39.1003	1.48745	0.743723	+	0.668488i	$0.233058\pi$
$692$	0	0
$693$	4.84964	0.184222
$694$	0	0
$695$	76.0529	2.88485
$696$	0	0
$697$	0.270103	0.0102309
$698$	0	0
$699$	−8.31488	−0.314498
$700$	0	0
$701$	24.0707	0.909139	0.454570	−	0.890711i	$-0.349793\pi$
$701$	24.0707	0.909139	0.454570	+	0.890711i	$0.349793\pi$
$702$	0	0
$703$	15.5905	0.588007
$704$	0	0
$705$	−4.86781	−0.183332
$706$	0	0
$707$	−2.50926	−0.0943704
$708$	0	0
$709$	19.0724	0.716280	0.358140	−	0.933668i	$-0.383411\pi$
$709$	19.0724	0.716280	0.358140	+	0.933668i	$0.383411\pi$
$710$	0	0
$711$	1.71373	0.0642698
$712$	0	0
$713$	28.8972	1.08221
$714$	0	0
$715$	1.54679	0.0578466
$716$	0	0
$717$	10.6000	0.395864
$718$	0	0
$719$	−13.3961	−0.499590	−0.249795	−	0.968299i	$-0.580363\pi$
$719$	−13.3961	−0.499590	−0.249795	+	0.968299i	$0.580363\pi$
$720$	0	0
$721$	10.6270	0.395771
$722$	0	0
$723$	4.51161	0.167788
$724$	0	0
$725$	−5.59446	−0.207773
$726$	0	0
$727$	−16.1554	−0.599172	−0.299586	−	0.954069i	$-0.596849\pi$
$727$	−16.1554	−0.599172	−0.299586	+	0.954069i	$0.596849\pi$
$728$	0	0
$729$	−13.4249	−0.497218
$730$	0	0
$731$	−3.01417	−0.111483
$732$	0	0
$733$	14.5144	0.536100	0.268050	−	0.963405i	$-0.413621\pi$
$733$	14.5144	0.536100	0.268050	+	0.963405i	$0.413621\pi$
$734$	0	0
$735$	1.87242	0.0690654
$736$	0	0
$737$	1.16847	0.0430412
$738$	0	0
$739$	−41.9965	−1.54486	−0.772432	−	0.635097i	$-0.780960\pi$
$739$	−41.9965	−1.54486	−0.772432	+	0.635097i	$0.780960\pi$
$740$	0	0
$741$	−0.522598	−0.0191981
$742$	0	0
$743$	5.81233	0.213234	0.106617	−	0.994300i	$-0.465998\pi$
$743$	5.81233	0.213234	0.106617	+	0.994300i	$0.465998\pi$
$744$	0	0
$745$	−52.4118	−1.92022
$746$	0	0
$747$	27.9861	1.02396
$748$	0	0
$749$	−9.41312	−0.343948
$750$	0	0
$751$	−35.0156	−1.27774	−0.638869	−	0.769315i	$-0.720597\pi$
$751$	−35.0156	−1.27774	−0.638869	+	0.769315i	$0.720597\pi$
$752$	0	0
$753$	6.90206	0.251525
$754$	0	0
$755$	−65.1963	−2.37273
$756$	0	0
$757$	−33.6148	−1.22175	−0.610875	−	0.791727i	$-0.709182\pi$
$757$	−33.6148	−1.22175	−0.610875	+	0.791727i	$0.709182\pi$
$758$	0	0
$759$	6.54866	0.237701
$760$	0	0
$761$	32.5823	1.18111	0.590554	−	0.806998i	$-0.298909\pi$
$761$	32.5823	1.18111	0.590554	+	0.806998i	$0.298909\pi$
$762$	0	0
$763$	9.67677	0.350323
$764$	0	0
$765$	9.80500	0.354500
$766$	0	0
$767$	−2.59452	−0.0936825
$768$	0	0
$769$	14.2206	0.512807	0.256404	−	0.966570i	$-0.417462\pi$
$769$	14.2206	0.512807	0.256404	+	0.966570i	$0.417462\pi$
$770$	0	0
$771$	14.9210	0.537367
$772$	0	0
$773$	17.5552	0.631417	0.315708	−	0.948856i	$-0.397758\pi$
$773$	17.5552	0.631417	0.315708	+	0.948856i	$0.397758\pi$
$774$	0	0
$775$	32.3400	1.16169
$776$	0	0
$777$	−1.96146	−0.0703669
$778$	0	0
$779$	1.11864	0.0400796
$780$	0	0
$781$	13.6603	0.488803
$782$	0	0
$783$	−2.11013	−0.0754098
$784$	0	0
$785$	−66.8793	−2.38702
$786$	0	0
$787$	−42.1505	−1.50250	−0.751251	−	0.660017i	$-0.770549\pi$
$787$	−42.1505	−1.50250	−0.751251	+	0.660017i	$0.770549\pi$
$788$	0	0
$789$	−4.56913	−0.162665
$790$	0	0
$791$	1.19664	0.0425474
$792$	0	0
$793$	−1.90412	−0.0676172
$794$	0	0
$795$	−4.35414	−0.154425
$796$	0	0
$797$	25.4660	0.902053	0.451027	−	0.892510i	$-0.351058\pi$
$797$	25.4660	0.902053	0.451027	+	0.892510i	$0.351058\pi$
$798$	0	0
$799$	−2.59974	−0.0919721
$800$	0	0
$801$	0.899896	0.0317963
$802$	0	0
$803$	−0.580846	−0.0204976
$804$	0	0
$805$	−25.4103	−0.895595
$806$	0	0
$807$	−2.34506	−0.0825502
$808$	0	0
$809$	−14.8300	−0.521395	−0.260698	−	0.965421i	$-0.583953\pi$
$809$	−14.8300	−0.521395	−0.260698	+	0.965421i	$0.583953\pi$
$810$	0	0
$811$	17.1020	0.600532	0.300266	−	0.953855i	$-0.402925\pi$
$811$	17.1020	0.600532	0.300266	+	0.953855i	$0.402925\pi$
$812$	0	0
$813$	16.5546	0.580595
$814$	0	0
$815$	2.48933	0.0871974
$816$	0	0
$817$	−12.4833	−0.436737
$818$	0	0
$819$	−0.660767	−0.0230891
$820$	0	0
$821$	20.1654	0.703777	0.351888	−	0.936042i	$-0.385540\pi$
$821$	20.1654	0.703777	0.351888	+	0.936042i	$0.385540\pi$
$822$	0	0
$823$	0.105743	0.00368595	0.00184298	−	0.999998i	$-0.499413\pi$
$823$	0.105743	0.00368595	0.00184298	+	0.999998i	$0.499413\pi$
$824$	0	0
$825$	7.32887	0.255159
$826$	0	0
$827$	−35.5296	−1.23549	−0.617743	−	0.786380i	$-0.711953\pi$
$827$	−35.5296	−1.23549	−0.617743	+	0.786380i	$0.711953\pi$
$828$	0	0
$829$	−43.2001	−1.50040	−0.750201	−	0.661210i	$-0.770043\pi$
$829$	−43.2001	−1.50040	−0.750201	+	0.661210i	$0.770043\pi$
$830$	0	0
$831$	−7.06031	−0.244919
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	−19.5220	−0.675585
$836$	0	0
$837$	12.1981	0.421627
$838$	0	0
$839$	19.8052	0.683752	0.341876	−	0.939745i	$-0.388938\pi$
$839$	19.8052	0.683752	0.341876	+	0.939745i	$0.388938\pi$
$840$	0	0
$841$	−28.5002	−0.982766
$842$	0	0
$843$	13.6412	0.469826
$844$	0	0
$845$	46.5053	1.59983
$846$	0	0
$847$	−7.84086	−0.269415
$848$	0	0
$849$	−15.4351	−0.529732
$850$	0	0
$851$	26.6186	0.912472
$852$	0	0
$853$	4.18275	0.143215	0.0716074	−	0.997433i	$-0.477187\pi$
$853$	4.18275	0.143215	0.0716074	+	0.997433i	$0.477187\pi$
$854$	0	0
$855$	40.6079	1.38876
$856$	0	0
$857$	−41.2979	−1.41071	−0.705355	−	0.708854i	$-0.749212\pi$
$857$	−41.2979	−1.41071	−0.705355	+	0.708854i	$0.749212\pi$
$858$	0	0
$859$	7.12069	0.242955	0.121477	−	0.992594i	$-0.461237\pi$
$859$	7.12069	0.242955	0.121477	+	0.992594i	$0.461237\pi$
$860$	0	0
$861$	−0.140738	−0.00479633
$862$	0	0
$863$	13.1066	0.446154	0.223077	−	0.974801i	$-0.428390\pi$
$863$	13.1066	0.446154	0.223077	+	0.974801i	$0.428390\pi$
$864$	0	0
$865$	23.8871	0.812187
$866$	0	0
$867$	−0.521052	−0.0176959
$868$	0	0
$869$	1.11635	0.0378697
$870$	0	0
$871$	−0.159205	−0.00539447
$872$	0	0
$873$	34.7266	1.17532
$874$	0	0
$875$	−10.4700	−0.353950
$876$	0	0
$877$	−19.8172	−0.669180	−0.334590	−	0.942364i	$-0.608598\pi$
$877$	−19.8172	−0.669180	−0.334590	+	0.942364i	$0.608598\pi$
$878$	0	0
$879$	5.88949	0.198648
$880$	0	0
$881$	16.4737	0.555014	0.277507	−	0.960724i	$-0.410492\pi$
$881$	16.4737	0.555014	0.277507	+	0.960724i	$0.410492\pi$
$882$	0	0
$883$	34.4713	1.16005	0.580026	−	0.814598i	$-0.303042\pi$
$883$	34.4713	1.16005	0.580026	+	0.814598i	$0.303042\pi$
$884$	0	0
$885$	−20.0603	−0.674319
$886$	0	0
$887$	−6.59459	−0.221425	−0.110712	−	0.993852i	$-0.535313\pi$
$887$	−6.59459	−0.221425	−0.110712	+	0.993852i	$0.535313\pi$
$888$	0	0
$889$	−8.67058	−0.290802
$890$	0	0
$891$	−11.7846	−0.394799
$892$	0	0
$893$	−10.7669	−0.360302
$894$	0	0
$895$	85.3196	2.85192
$896$	0	0
$897$	−0.892260	−0.0297917
$898$	0	0
$899$	−2.88905	−0.0963553
$900$	0	0
$901$	−2.32540	−0.0774704
$902$	0	0
$903$	1.57054	0.0522643
$904$	0	0
$905$	2.90203	0.0964669
$906$	0	0
$907$	45.8349	1.52192	0.760962	−	0.648797i	$-0.224727\pi$
$907$	45.8349	1.52192	0.760962	+	0.648797i	$0.224727\pi$
$908$	0	0
$909$	6.84652	0.227085
$910$	0	0
$911$	−39.2464	−1.30029	−0.650145	−	0.759810i	$-0.725292\pi$
$911$	−39.2464	−1.30029	−0.650145	+	0.759810i	$0.725292\pi$
$912$	0	0
$913$	18.2307	0.603347
$914$	0	0
$915$	−14.7222	−0.486702
$916$	0	0
$917$	−10.9781	−0.362529
$918$	0	0
$919$	−25.8827	−0.853792	−0.426896	−	0.904301i	$-0.640393\pi$
$919$	−25.8827	−0.853792	−0.426896	+	0.904301i	$0.640393\pi$
$920$	0	0
$921$	−0.0687788	−0.00226634
$922$	0	0
$923$	−1.86122	−0.0612629
$924$	0	0
$925$	29.7899	0.979486
$926$	0	0
$927$	−28.9959	−0.952351
$928$	0	0
$929$	36.2319	1.18873	0.594366	−	0.804195i	$-0.297403\pi$
$929$	36.2319	1.18873	0.594366	+	0.804195i	$0.297403\pi$
$930$	0	0
$931$	4.14155	0.135734
$932$	0	0
$933$	−8.08992	−0.264852
$934$	0	0
$935$	6.38715	0.208882
$936$	0	0
$937$	48.8281	1.59515	0.797573	−	0.603223i	$-0.206117\pi$
$937$	48.8281	1.59515	0.797573	+	0.603223i	$0.206117\pi$
$938$	0	0
$939$	2.73258	0.0891744
$940$	0	0
$941$	5.32917	0.173726	0.0868631	−	0.996220i	$-0.472316\pi$
$941$	5.32917	0.173726	0.0868631	+	0.996220i	$0.472316\pi$
$942$	0	0
$943$	1.90992	0.0621957
$944$	0	0
$945$	−10.7262	−0.348923
$946$	0	0
$947$	−9.61529	−0.312455	−0.156227	−	0.987721i	$-0.549933\pi$
$947$	−9.61529	−0.312455	−0.156227	+	0.987721i	$0.549933\pi$
$948$	0	0
$949$	0.0791408	0.00256902
$950$	0	0
$951$	2.51652	0.0816039
$952$	0	0
$953$	39.2917	1.27278	0.636392	−	0.771366i	$-0.280426\pi$
$953$	39.2917	1.27278	0.636392	+	0.771366i	$0.280426\pi$
$954$	0	0
$955$	−5.38408	−0.174225
$956$	0	0
$957$	−0.654715	−0.0211639
$958$	0	0
$959$	20.9432	0.676291
$960$	0	0
$961$	−14.2992	−0.461264
$962$	0	0
$963$	25.6837	0.827647
$964$	0	0
$965$	−60.4778	−1.94685
$966$	0	0
$967$	23.7753	0.764562	0.382281	−	0.924046i	$-0.375139\pi$
$967$	23.7753	0.764562	0.382281	+	0.924046i	$0.375139\pi$
$968$	0	0
$969$	−2.15796	−0.0693238
$970$	0	0
$971$	5.56247	0.178508	0.0892541	−	0.996009i	$-0.471552\pi$
$971$	5.56247	0.178508	0.0892541	+	0.996009i	$0.471552\pi$
$972$	0	0
$973$	−21.1638	−0.678479
$974$	0	0
$975$	−0.998564	−0.0319797
$976$	0	0
$977$	−22.9348	−0.733750	−0.366875	−	0.930270i	$-0.619572\pi$
$977$	−22.9348	−0.733750	−0.366875	+	0.930270i	$0.619572\pi$
$978$	0	0
$979$	0.586208	0.0187353
$980$	0	0
$981$	−26.4031	−0.842987
$982$	0	0
$983$	50.7404	1.61837	0.809183	−	0.587556i	$-0.199910\pi$
$983$	50.7404	1.61837	0.809183	+	0.587556i	$0.199910\pi$
$984$	0	0
$985$	52.6455	1.67743
$986$	0	0
$987$	1.35460	0.0431173
$988$	0	0
$989$	−21.3135	−0.677730
$990$	0	0
$991$	39.3792	1.25092	0.625461	−	0.780255i	$-0.284911\pi$
$991$	39.3792	1.25092	0.625461	+	0.780255i	$0.284911\pi$
$992$	0	0
$993$	−7.26426	−0.230524
$994$	0	0
$995$	−9.44477	−0.299419
$996$	0	0
$997$	−60.4604	−1.91480	−0.957400	−	0.288764i	$-0.906756\pi$
$997$	−60.4604	−1.91480	−0.957400	+	0.288764i	$0.906756\pi$
$998$	0	0
$999$	11.2362	0.355498

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3808.2.a.g.1.4	✓	6
4.3	odd	2		3808.2.a.o.1.3	yes	6
8.3	odd	2		7616.2.a.bv.1.4		6
8.5	even	2		7616.2.a.cd.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.g.1.4	✓	6	1.1	even	1	trivial
3808.2.a.o.1.3	yes	6	4.3	odd	2
7616.2.a.bv.1.4		6	8.3	odd	2
7616.2.a.cd.1.3		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)

\(f(q)\)	\(=\)	\(q-0.521052 q^{3} -3.59354 q^{5} +1.00000 q^{7} -2.72850 q^{9} -1.77740 q^{11} +0.242172 q^{13} +1.87242 q^{15} +1.00000 q^{17} +4.14155 q^{19} -0.521052 q^{21} +7.07110 q^{23} +7.91355 q^{25} +2.98485 q^{27} -0.706946 q^{29} +4.08666 q^{31} +0.926117 q^{33} -3.59354 q^{35} +3.76442 q^{37} -0.126184 q^{39} +0.270103 q^{41} -3.01417 q^{43} +9.80500 q^{45} -2.59974 q^{47} +1.00000 q^{49} -0.521052 q^{51} -2.32540 q^{53} +6.38715 q^{55} -2.15796 q^{57} -10.7135 q^{59} -7.86267 q^{61} -2.72850 q^{63} -0.870255 q^{65} -0.657406 q^{67} -3.68441 q^{69} -7.68555 q^{71} +0.326796 q^{73} -4.12337 q^{75} -1.77740 q^{77} -0.628083 q^{79} +6.63025 q^{81} -10.2569 q^{83} -3.59354 q^{85} +0.368356 q^{87} -0.329813 q^{89} +0.242172 q^{91} -2.12937 q^{93} -14.8828 q^{95} -12.7273 q^{97} +4.84964 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} - 4 q^{5} + 6 q^{7} + 8 q^{9} - 8 q^{11} - 4 q^{13} + 6 q^{17} - 18 q^{19} - 4 q^{21} - 6 q^{23} + 12 q^{25} - 10 q^{27} + 4 q^{29} + 8 q^{31} - 6 q^{33} - 4 q^{35} - 8 q^{37} + 18 q^{39}+ \cdots - 32 q^{99}+O(q^{100}) \) 6 * q - 4 * q^3 - 4 * q^5 + 6 * q^7 + 8 * q^9 - 8 * q^11 - 4 * q^13 + 6 * q^17 - 18 * q^19 - 4 * q^21 - 6 * q^23 + 12 * q^25 - 10 * q^27 + 4 * q^29 + 8 * q^31 - 6 * q^33 - 4 * q^35 - 8 * q^37 + 18 * q^39 - 4 * q^41 - 20 * q^43 - 24 * q^45 - 6 * q^47 + 6 * q^49 - 4 * q^51 + 14 * q^53 - 4 * q^55 - 4 * q^57 - 20 * q^59 + 2 * q^61 + 8 * q^63 + 6 * q^65 - 28 * q^67 - 4 * q^69 - 18 * q^71 + 4 * q^73 - 20 * q^75 - 8 * q^77 + 2 * q^79 + 34 * q^81 - 20 * q^83 - 4 * q^85 - 22 * q^87 - 6 * q^89 - 4 * q^91 + 40 * q^93 + 6 * q^95 + 30 * q^97 - 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more