LMFDB - Embedded newform 3808.2.a.h.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

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

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3808, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3808.1"); S:= CuspForms(chi, 2); N := Newforms(S);

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

Newform invariants

comment:select newform

sage:traces = [6,0,-4,0,4,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

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

Embedding invariants

Embedding label			1.4
Root			$3.55174$ 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)

$f(q)$	$=$	$q-0.531974 q^{3} +0.110568 q^{5} -1.00000 q^{7} -2.71700 q^{9} +3.70563 q^{11} -0.433088 q^{13} -0.0588193 q^{15} -1.00000 q^{17} +0.814115 q^{19} +0.531974 q^{21} -1.18589 q^{23} -4.98777 q^{25} +3.04130 q^{27} +7.33386 q^{29} -2.31917 q^{31} -1.97130 q^{33} -0.110568 q^{35} +3.22541 q^{37} +0.230392 q^{39} +9.96934 q^{41} -12.2989 q^{43} -0.300414 q^{45} -4.33389 q^{47} +1.00000 q^{49} +0.531974 q^{51} +0.931587 q^{53} +0.409725 q^{55} -0.433088 q^{57} -10.4980 q^{59} -5.63209 q^{61} +2.71700 q^{63} -0.0478857 q^{65} -11.3973 q^{67} +0.630861 q^{69} +4.54584 q^{71} +8.09723 q^{73} +2.65337 q^{75} -3.70563 q^{77} -6.73861 q^{79} +6.53312 q^{81} -4.28183 q^{83} -0.110568 q^{85} -3.90143 q^{87} -8.11693 q^{89} +0.433088 q^{91} +1.23374 q^{93} +0.0900150 q^{95} +0.418365 q^{97} -10.0682 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{3} + 4 q^{5} - 6 q^{7} + 8 q^{9} - 8 q^{11} + 8 q^{13} - 6 q^{17} - 6 q^{19} + 4 q^{21} - 18 q^{23} + 12 q^{25} - 22 q^{27} - 8 q^{29} + 4 q^{31} - 6 q^{33} - 4 q^{35} - 8 q^{37} - 14 q^{39}+ \cdots - 28 q^{99}+O(q^{100}) $ 6 * q - 4 * q^3 + 4 * q^5 - 6 * q^7 + 8 * q^9 - 8 * q^11 + 8 * q^13 - 6 * q^17 - 6 * q^19 + 4 * q^21 - 18 * q^23 + 12 * q^25 - 22 * q^27 - 8 * q^29 + 4 * q^31 - 6 * q^33 - 4 * q^35 - 8 * q^37 - 14 * q^39 + 16 * q^41 - 12 * q^43 + 4 * q^45 - 18 * q^47 + 6 * q^49 + 4 * q^51 - 2 * q^53 + 8 * q^55 + 8 * q^57 - 16 * q^59 + 6 * q^61 - 8 * q^63 - 22 * q^65 - 12 * q^67 + 16 * q^69 + 2 * q^71 + 8 * q^73 - 24 * q^75 + 8 * q^77 + 18 * q^79 + 18 * q^81 - 28 * q^83 - 4 * q^85 + 2 * q^87 - 2 * q^89 - 8 * q^91 - 32 * q^93 - 26 * q^95 - 18 * q^97 - 28 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−0.531974	−0.307135	−0.153568	−	0.988138i	$-0.549076\pi$
$3$	−0.531974	−0.307135	−0.153568	+	0.988138i	$0.549076\pi$
$4$	0	0
$5$	0.110568	0.0494475	0.0247238	−	0.999694i	$-0.492129\pi$
$5$	0.110568	0.0494475	0.0247238	+	0.999694i	$0.492129\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.71700	−0.905668
$10$	0	0
$11$	3.70563	1.11729	0.558645	−	0.829407i	$-0.311321\pi$
$11$	3.70563	1.11729	0.558645	+	0.829407i	$0.311321\pi$
$12$	0	0
$13$	−0.433088	−0.120117	−0.0600585	−	0.998195i	$-0.519129\pi$
$13$	−0.433088	−0.120117	−0.0600585	+	0.998195i	$0.519129\pi$
$14$	0	0
$15$	−0.0588193	−0.0151871
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	0.814115	0.186771	0.0933853	−	0.995630i	$-0.470231\pi$
$19$	0.814115	0.186771	0.0933853	+	0.995630i	$0.470231\pi$
$20$	0	0
$21$	0.531974	0.116086
$22$	0	0
$23$	−1.18589	−0.247274	−0.123637	−	0.992327i	$-0.539456\pi$
$23$	−1.18589	−0.247274	−0.123637	+	0.992327i	$0.539456\pi$
$24$	0	0
$25$	−4.98777	−0.997555
$26$	0	0
$27$	3.04130	0.585298
$28$	0	0
$29$	7.33386	1.36186	0.680932	−	0.732347i	$-0.261575\pi$
$29$	7.33386	1.36186	0.680932	+	0.732347i	$0.261575\pi$
$30$	0	0
$31$	−2.31917	−0.416535	−0.208267	−	0.978072i	$-0.566782\pi$
$31$	−2.31917	−0.416535	−0.208267	+	0.978072i	$0.566782\pi$
$32$	0	0
$33$	−1.97130	−0.343160
$34$	0	0
$35$	−0.110568	−0.0186894
$36$	0	0
$37$	3.22541	0.530254	0.265127	−	0.964214i	$-0.414586\pi$
$37$	3.22541	0.530254	0.265127	+	0.964214i	$0.414586\pi$
$38$	0	0
$39$	0.230392	0.0368922
$40$	0	0
$41$	9.96934	1.55695	0.778474	−	0.627677i	$-0.215994\pi$
$41$	9.96934	1.55695	0.778474	+	0.627677i	$0.215994\pi$
$42$	0	0
$43$	−12.2989	−1.87557	−0.937784	−	0.347218i	$-0.887126\pi$
$43$	−12.2989	−1.87557	−0.937784	+	0.347218i	$0.887126\pi$
$44$	0	0
$45$	−0.300414	−0.0447830
$46$	0	0
$47$	−4.33389	−0.632163	−0.316081	−	0.948732i	$-0.602367\pi$
$47$	−4.33389	−0.632163	−0.316081	+	0.948732i	$0.602367\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.531974	0.0744913
$52$	0	0
$53$	0.931587	0.127963	0.0639816	−	0.997951i	$-0.479620\pi$
$53$	0.931587	0.127963	0.0639816	+	0.997951i	$0.479620\pi$
$54$	0	0
$55$	0.409725	0.0552472
$56$	0	0
$57$	−0.433088	−0.0573639
$58$	0	0
$59$	−10.4980	−1.36672	−0.683359	−	0.730083i	$-0.739481\pi$
$59$	−10.4980	−1.36672	−0.683359	+	0.730083i	$0.739481\pi$
$60$	0	0
$61$	−5.63209	−0.721115	−0.360558	−	0.932737i	$-0.617414\pi$
$61$	−5.63209	−0.721115	−0.360558	+	0.932737i	$0.617414\pi$
$62$	0	0
$63$	2.71700	0.342310
$64$	0	0
$65$	−0.0478857	−0.00593949
$66$	0	0
$67$	−11.3973	−1.39240	−0.696201	−	0.717847i	$-0.745128\pi$
$67$	−11.3973	−1.39240	−0.696201	+	0.717847i	$0.745128\pi$
$68$	0	0
$69$	0.630861	0.0759467
$70$	0	0
$71$	4.54584	0.539492	0.269746	−	0.962932i	$-0.413060\pi$
$71$	4.54584	0.539492	0.269746	+	0.962932i	$0.413060\pi$
$72$	0	0
$73$	8.09723	0.947709	0.473855	−	0.880603i	$-0.342862\pi$
$73$	8.09723	0.947709	0.473855	+	0.880603i	$0.342862\pi$
$74$	0	0
$75$	2.65337	0.306385
$76$	0	0
$77$	−3.70563	−0.422296
$78$	0	0
$79$	−6.73861	−0.758152	−0.379076	−	0.925365i	$-0.623758\pi$
$79$	−6.73861	−0.758152	−0.379076	+	0.925365i	$0.623758\pi$
$80$	0	0
$81$	6.53312	0.725902
$82$	0	0
$83$	−4.28183	−0.469992	−0.234996	−	0.971996i	$-0.575508\pi$
$83$	−4.28183	−0.469992	−0.234996	+	0.971996i	$0.575508\pi$
$84$	0	0
$85$	−0.110568	−0.0119928
$86$	0	0
$87$	−3.90143	−0.418277
$88$	0	0
$89$	−8.11693	−0.860393	−0.430196	−	0.902735i	$-0.641556\pi$
$89$	−8.11693	−0.860393	−0.430196	+	0.902735i	$0.641556\pi$
$90$	0	0
$91$	0.433088	0.0454000
$92$	0	0
$93$	1.23374	0.127933
$94$	0	0
$95$	0.0900150	0.00923534
$96$	0	0
$97$	0.418365	0.0424785	0.0212393	−	0.999774i	$-0.493239\pi$
$97$	0.418365	0.0424785	0.0212393	+	0.999774i	$0.493239\pi$
$98$	0	0
$99$	−10.0682	−1.01189
$100$	0	0
$101$	4.20270	0.418184	0.209092	−	0.977896i	$-0.432949\pi$
$101$	4.20270	0.418184	0.209092	+	0.977896i	$0.432949\pi$
$102$	0	0
$103$	9.77386	0.963047	0.481523	−	0.876433i	$-0.340084\pi$
$103$	9.77386	0.963047	0.481523	+	0.876433i	$0.340084\pi$
$104$	0	0
$105$	0.0588193	0.00574018
$106$	0	0
$107$	4.44987	0.430185	0.215093	−	0.976594i	$-0.430995\pi$
$107$	4.44987	0.430185	0.215093	+	0.976594i	$0.430995\pi$
$108$	0	0
$109$	−11.0807	−1.06134	−0.530671	−	0.847578i	$-0.678060\pi$
$109$	−11.0807	−1.06134	−0.530671	+	0.847578i	$0.678060\pi$
$110$	0	0
$111$	−1.71583	−0.162860
$112$	0	0
$113$	4.31470	0.405893	0.202946	−	0.979190i	$-0.434948\pi$
$113$	4.31470	0.405893	0.202946	+	0.979190i	$0.434948\pi$
$114$	0	0
$115$	−0.131121	−0.0122271
$116$	0	0
$117$	1.17670	0.108786
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	2.73173	0.248339
$122$	0	0
$123$	−5.30343	−0.478194
$124$	0	0
$125$	−1.10433	−0.0987741
$126$	0	0
$127$	−18.8086	−1.66899	−0.834495	−	0.551015i	$-0.814241\pi$
$127$	−18.8086	−1.66899	−0.834495	+	0.551015i	$0.814241\pi$
$128$	0	0
$129$	6.54271	0.576054
$130$	0	0
$131$	−22.4407	−1.96065	−0.980326	−	0.197385i	$-0.936755\pi$
$131$	−22.4407	−1.96065	−0.980326	+	0.197385i	$0.936755\pi$
$132$	0	0
$133$	−0.814115	−0.0705927
$134$	0	0
$135$	0.336270	0.0289415
$136$	0	0
$137$	−15.4320	−1.31845	−0.659224	−	0.751947i	$-0.729115\pi$
$137$	−15.4320	−1.31845	−0.659224	+	0.751947i	$0.729115\pi$
$138$	0	0
$139$	6.34621	0.538278	0.269139	−	0.963101i	$-0.413261\pi$
$139$	6.34621	0.538278	0.269139	+	0.963101i	$0.413261\pi$
$140$	0	0
$141$	2.30552	0.194160
$142$	0	0
$143$	−1.60487	−0.134206
$144$	0	0
$145$	0.810891	0.0673408
$146$	0	0
$147$	−0.531974	−0.0438765
$148$	0	0
$149$	19.3519	1.58537	0.792685	−	0.609631i	$-0.208683\pi$
$149$	19.3519	1.58537	0.792685	+	0.609631i	$0.208683\pi$
$150$	0	0
$151$	−5.16448	−0.420279	−0.210140	−	0.977671i	$-0.567392\pi$
$151$	−5.16448	−0.420279	−0.210140	+	0.977671i	$0.567392\pi$
$152$	0	0
$153$	2.71700	0.219657
$154$	0	0
$155$	−0.256426	−0.0205966
$156$	0	0
$157$	−21.0632	−1.68103	−0.840513	−	0.541792i	$-0.817746\pi$
$157$	−21.0632	−1.68103	−0.840513	+	0.541792i	$0.817746\pi$
$158$	0	0
$159$	−0.495580	−0.0393021
$160$	0	0
$161$	1.18589	0.0934609
$162$	0	0
$163$	−5.45751	−0.427465	−0.213733	−	0.976892i	$-0.568562\pi$
$163$	−5.45751	−0.427465	−0.213733	+	0.976892i	$0.568562\pi$
$164$	0	0
$165$	−0.217963	−0.0169684
$166$	0	0
$167$	−4.42024	−0.342048	−0.171024	−	0.985267i	$-0.554708\pi$
$167$	−4.42024	−0.342048	−0.171024	+	0.985267i	$0.554708\pi$
$168$	0	0
$169$	−12.8124	−0.985572
$170$	0	0
$171$	−2.21195	−0.169152
$172$	0	0
$173$	−11.6285	−0.884102	−0.442051	−	0.896990i	$-0.645749\pi$
$173$	−11.6285	−0.884102	−0.442051	+	0.896990i	$0.645749\pi$
$174$	0	0
$175$	4.98777	0.377040
$176$	0	0
$177$	5.58464	0.419767
$178$	0	0
$179$	−12.0938	−0.903935	−0.451967	−	0.892035i	$-0.649278\pi$
$179$	−12.0938	−0.903935	−0.451967	+	0.892035i	$0.649278\pi$
$180$	0	0
$181$	−2.90925	−0.216243	−0.108121	−	0.994138i	$-0.534484\pi$
$181$	−2.90925	−0.216243	−0.108121	+	0.994138i	$0.534484\pi$
$182$	0	0
$183$	2.99613	0.221480
$184$	0	0
$185$	0.356627	0.0262197
$186$	0	0
$187$	−3.70563	−0.270983
$188$	0	0
$189$	−3.04130	−0.221222
$190$	0	0
$191$	−0.161060	−0.0116539	−0.00582693	−	0.999983i	$-0.501855\pi$
$191$	−0.161060	−0.0116539	−0.00582693	+	0.999983i	$0.501855\pi$
$192$	0	0
$193$	−6.90768	−0.497226	−0.248613	−	0.968603i	$-0.579975\pi$
$193$	−6.90768	−0.497226	−0.248613	+	0.968603i	$0.579975\pi$
$194$	0	0
$195$	0.0254739	0.00182423
$196$	0	0
$197$	−22.9991	−1.63862	−0.819311	−	0.573350i	$-0.805644\pi$
$197$	−22.9991	−1.63862	−0.819311	+	0.573350i	$0.805644\pi$
$198$	0	0
$199$	26.3057	1.86476	0.932380	−	0.361479i	$-0.117728\pi$
$199$	26.3057	1.86476	0.932380	+	0.361479i	$0.117728\pi$
$200$	0	0
$201$	6.06307	0.427656
$202$	0	0
$203$	−7.33386	−0.514736
$204$	0	0
$205$	1.10229	0.0769872
$206$	0	0
$207$	3.22205	0.223948
$208$	0	0
$209$	3.01681	0.208677
$210$	0	0
$211$	3.51712	0.242128	0.121064	−	0.992645i	$-0.461369\pi$
$211$	3.51712	0.242128	0.121064	+	0.992645i	$0.461369\pi$
$212$	0	0
$213$	−2.41827	−0.165697
$214$	0	0
$215$	−1.35987	−0.0927422
$216$	0	0
$217$	2.31917	0.157435
$218$	0	0
$219$	−4.30752	−0.291075
$220$	0	0
$221$	0.433088	0.0291326
$222$	0	0
$223$	11.0682	0.741181	0.370591	−	0.928796i	$-0.379155\pi$
$223$	11.0682	0.741181	0.370591	+	0.928796i	$0.379155\pi$
$224$	0	0
$225$	13.5518	0.903453
$226$	0	0
$227$	−13.2678	−0.880617	−0.440308	−	0.897847i	$-0.645131\pi$
$227$	−13.2678	−0.880617	−0.440308	+	0.897847i	$0.645131\pi$
$228$	0	0
$229$	21.2079	1.40146	0.700730	−	0.713427i	$-0.252858\pi$
$229$	21.2079	1.40146	0.700730	+	0.713427i	$0.252858\pi$
$230$	0	0
$231$	1.97130	0.129702
$232$	0	0
$233$	1.48189	0.0970820	0.0485410	−	0.998821i	$-0.484543\pi$
$233$	1.48189	0.0970820	0.0485410	+	0.998821i	$0.484543\pi$
$234$	0	0
$235$	−0.479189	−0.0312589
$236$	0	0
$237$	3.58476	0.232855
$238$	0	0
$239$	−10.8918	−0.704534	−0.352267	−	0.935900i	$-0.614589\pi$
$239$	−10.8918	−0.704534	−0.352267	+	0.935900i	$0.614589\pi$
$240$	0	0
$241$	−13.6857	−0.881574	−0.440787	−	0.897612i	$-0.645301\pi$
$241$	−13.6857	−0.881574	−0.440787	+	0.897612i	$0.645301\pi$
$242$	0	0
$243$	−12.5993	−0.808248
$244$	0	0
$245$	0.110568	0.00706393
$246$	0	0
$247$	−0.352583	−0.0224343
$248$	0	0
$249$	2.27782	0.144351
$250$	0	0
$251$	−2.02515	−0.127826	−0.0639131	−	0.997955i	$-0.520358\pi$
$251$	−2.02515	−0.127826	−0.0639131	+	0.997955i	$0.520358\pi$
$252$	0	0
$253$	−4.39446	−0.276277
$254$	0	0
$255$	0.0588193	0.00368341
$256$	0	0
$257$	−11.5374	−0.719682	−0.359841	−	0.933014i	$-0.617169\pi$
$257$	−11.5374	−0.719682	−0.359841	+	0.933014i	$0.617169\pi$
$258$	0	0
$259$	−3.22541	−0.200417
$260$	0	0
$261$	−19.9261	−1.23340
$262$	0	0
$263$	−3.15719	−0.194681	−0.0973403	−	0.995251i	$-0.531034\pi$
$263$	−3.15719	−0.194681	−0.0973403	+	0.995251i	$0.531034\pi$
$264$	0	0
$265$	0.103004	0.00632747
$266$	0	0
$267$	4.31800	0.264257
$268$	0	0
$269$	8.51645	0.519257	0.259629	−	0.965709i	$-0.416400\pi$
$269$	8.51645	0.519257	0.259629	+	0.965709i	$0.416400\pi$
$270$	0	0
$271$	27.2876	1.65760	0.828802	−	0.559541i	$-0.189023\pi$
$271$	27.2876	1.65760	0.828802	+	0.559541i	$0.189023\pi$
$272$	0	0
$273$	−0.230392	−0.0139439
$274$	0	0
$275$	−18.4829	−1.11456
$276$	0	0
$277$	−0.0119126	−0.000715757 0	−0.000357878	−	1.00000i	$-0.500114\pi$
$277$	−0.0119126	−0.000715757 0	−0.000357878		1.00000i	$0.500114\pi$
$278$	0	0
$279$	6.30118	0.377242
$280$	0	0
$281$	19.2215	1.14666	0.573330	−	0.819325i	$-0.305651\pi$
$281$	19.2215	1.14666	0.573330	+	0.819325i	$0.305651\pi$
$282$	0	0
$283$	−14.4359	−0.858125	−0.429063	−	0.903275i	$-0.641156\pi$
$283$	−14.4359	−0.858125	−0.429063	+	0.903275i	$0.641156\pi$
$284$	0	0
$285$	−0.0478857	−0.00283650
$286$	0	0
$287$	−9.96934	−0.588471
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−0.222559	−0.0130467
$292$	0	0
$293$	27.4719	1.60492	0.802462	−	0.596703i	$-0.203523\pi$
$293$	27.4719	1.60492	0.802462	+	0.596703i	$0.203523\pi$
$294$	0	0
$295$	−1.16074	−0.0675808
$296$	0	0
$297$	11.2699	0.653948
$298$	0	0
$299$	0.513593	0.0297018
$300$	0	0
$301$	12.2989	0.708898
$302$	0	0
$303$	−2.23573	−0.128439
$304$	0	0
$305$	−0.622729	−0.0356574
$306$	0	0
$307$	−25.1178	−1.43355	−0.716773	−	0.697306i	$-0.754382\pi$
$307$	−25.1178	−1.43355	−0.716773	+	0.697306i	$0.754382\pi$
$308$	0	0
$309$	−5.19944	−0.295786
$310$	0	0
$311$	6.08910	0.345281	0.172641	−	0.984985i	$-0.444770\pi$
$311$	6.08910	0.345281	0.172641	+	0.984985i	$0.444770\pi$
$312$	0	0
$313$	−16.2319	−0.917480	−0.458740	−	0.888570i	$-0.651699\pi$
$313$	−16.2319	−0.917480	−0.458740	+	0.888570i	$0.651699\pi$
$314$	0	0
$315$	0.300414	0.0169264
$316$	0	0
$317$	3.19596	0.179503	0.0897516	−	0.995964i	$-0.471393\pi$
$317$	3.19596	0.179503	0.0897516	+	0.995964i	$0.471393\pi$
$318$	0	0
$319$	27.1766	1.52160
$320$	0	0
$321$	−2.36722	−0.132125
$322$	0	0
$323$	−0.814115	−0.0452985
$324$	0	0
$325$	2.16015	0.119823
$326$	0	0
$327$	5.89466	0.325976
$328$	0	0
$329$	4.33389	0.238935
$330$	0	0
$331$	−6.93043	−0.380931	−0.190465	−	0.981694i	$-0.561000\pi$
$331$	−6.93043	−0.380931	−0.190465	+	0.981694i	$0.561000\pi$
$332$	0	0
$333$	−8.76345	−0.480234
$334$	0	0
$335$	−1.26018	−0.0688508
$336$	0	0
$337$	20.6911	1.12712	0.563558	−	0.826077i	$-0.309432\pi$
$337$	20.6911	1.12712	0.563558	+	0.826077i	$0.309432\pi$
$338$	0	0
$339$	−2.29531	−0.124664
$340$	0	0
$341$	−8.59398	−0.465390
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0.0697530	0.00375537
$346$	0	0
$347$	15.4072	0.827102	0.413551	−	0.910481i	$-0.364288\pi$
$347$	15.4072	0.827102	0.413551	+	0.910481i	$0.364288\pi$
$348$	0	0
$349$	2.97756	0.159385	0.0796924	−	0.996819i	$-0.474606\pi$
$349$	2.97756	0.159385	0.0796924	+	0.996819i	$0.474606\pi$
$350$	0	0
$351$	−1.31715	−0.0703043
$352$	0	0
$353$	1.48185	0.0788711	0.0394356	−	0.999222i	$-0.487444\pi$
$353$	1.48185	0.0788711	0.0394356	+	0.999222i	$0.487444\pi$
$354$	0	0
$355$	0.502624	0.0266765
$356$	0	0
$357$	−0.531974	−0.0281551
$358$	0	0
$359$	−26.6944	−1.40888	−0.704439	−	0.709765i	$-0.748801\pi$
$359$	−26.6944	−1.40888	−0.704439	+	0.709765i	$0.748801\pi$
$360$	0	0
$361$	−18.3372	−0.965117
$362$	0	0
$363$	−1.45321	−0.0762736
$364$	0	0
$365$	0.895295	0.0468619
$366$	0	0
$367$	−32.7540	−1.70974	−0.854872	−	0.518839i	$-0.826365\pi$
$367$	−32.7540	−1.70974	−0.854872	+	0.518839i	$0.826365\pi$
$368$	0	0
$369$	−27.0867	−1.41008
$370$	0	0
$371$	−0.931587	−0.0483656
$372$	0	0
$373$	−13.3275	−0.690072	−0.345036	−	0.938589i	$-0.612133\pi$
$373$	−13.3275	−0.690072	−0.345036	+	0.938589i	$0.612133\pi$
$374$	0	0
$375$	0.587474	0.0303370
$376$	0	0
$377$	−3.17621	−0.163583
$378$	0	0
$379$	−27.5592	−1.41562	−0.707810	−	0.706403i	$-0.750317\pi$
$379$	−27.5592	−1.41562	−0.707810	+	0.706403i	$0.750317\pi$
$380$	0	0
$381$	10.0057	0.512606
$382$	0	0
$383$	1.49711	0.0764987	0.0382493	−	0.999268i	$-0.487822\pi$
$383$	1.49711	0.0764987	0.0382493	+	0.999268i	$0.487822\pi$
$384$	0	0
$385$	−0.409725	−0.0208815
$386$	0	0
$387$	33.4162	1.69864
$388$	0	0
$389$	−2.22498	−0.112811	−0.0564054	−	0.998408i	$-0.517964\pi$
$389$	−2.22498	−0.112811	−0.0564054	+	0.998408i	$0.517964\pi$
$390$	0	0
$391$	1.18589	0.0599728
$392$	0	0
$393$	11.9379	0.602186
$394$	0	0
$395$	−0.745074	−0.0374887
$396$	0	0
$397$	22.7503	1.14181	0.570904	−	0.821017i	$-0.306593\pi$
$397$	22.7503	1.14181	0.570904	+	0.821017i	$0.306593\pi$
$398$	0	0
$399$	0.433088	0.0216815
$400$	0	0
$401$	−16.7098	−0.834449	−0.417224	−	0.908804i	$-0.636997\pi$
$401$	−16.7098	−0.834449	−0.417224	+	0.908804i	$0.636997\pi$
$402$	0	0
$403$	1.00440	0.0500329
$404$	0	0
$405$	0.722354	0.0358940
$406$	0	0
$407$	11.9522	0.592448
$408$	0	0
$409$	11.3246	0.559966	0.279983	−	0.960005i	$-0.409671\pi$
$409$	11.3246	0.559966	0.279983	+	0.960005i	$0.409671\pi$
$410$	0	0
$411$	8.20945	0.404942
$412$	0	0
$413$	10.4980	0.516571
$414$	0	0
$415$	−0.473433	−0.0232399
$416$	0	0
$417$	−3.37602	−0.165324
$418$	0	0
$419$	−10.4513	−0.510580	−0.255290	−	0.966865i	$-0.582171\pi$
$419$	−10.4513	−0.510580	−0.255290	+	0.966865i	$0.582171\pi$
$420$	0	0
$421$	4.82233	0.235026	0.117513	−	0.993071i	$-0.462508\pi$
$421$	4.82233	0.235026	0.117513	+	0.993071i	$0.462508\pi$
$422$	0	0
$423$	11.7752	0.572529
$424$	0	0
$425$	4.98777	0.241943
$426$	0	0
$427$	5.63209	0.272556
$428$	0	0
$429$	0.853747	0.0412193
$430$	0	0
$431$	−34.0631	−1.64076	−0.820381	−	0.571817i	$-0.806239\pi$
$431$	−34.0631	−1.64076	−0.820381	+	0.571817i	$0.806239\pi$
$432$	0	0
$433$	1.23339	0.0592731	0.0296366	−	0.999561i	$-0.490565\pi$
$433$	1.23339	0.0592731	0.0296366	+	0.999561i	$0.490565\pi$
$434$	0	0
$435$	−0.431373	−0.0206827
$436$	0	0
$437$	−0.965447	−0.0461836
$438$	0	0
$439$	−25.4586	−1.21507	−0.607537	−	0.794291i	$-0.707843\pi$
$439$	−25.4586	−1.21507	−0.607537	+	0.794291i	$0.707843\pi$
$440$	0	0
$441$	−2.71700	−0.129381
$442$	0	0
$443$	11.2248	0.533306	0.266653	−	0.963793i	$-0.414082\pi$
$443$	11.2248	0.533306	0.266653	+	0.963793i	$0.414082\pi$
$444$	0	0
$445$	−0.897472	−0.0425443
$446$	0	0
$447$	−10.2947	−0.486923
$448$	0	0
$449$	4.02603	0.190000	0.0950001	−	0.995477i	$-0.469715\pi$
$449$	4.02603	0.190000	0.0950001	+	0.995477i	$0.469715\pi$
$450$	0	0
$451$	36.9427	1.73956
$452$	0	0
$453$	2.74737	0.129083
$454$	0	0
$455$	0.0478857	0.00224491
$456$	0	0
$457$	−28.0207	−1.31075	−0.655377	−	0.755302i	$-0.727490\pi$
$457$	−28.0207	−1.31075	−0.655377	+	0.755302i	$0.727490\pi$
$458$	0	0
$459$	−3.04130	−0.141956
$460$	0	0
$461$	25.4253	1.18418	0.592088	−	0.805874i	$-0.298304\pi$
$461$	25.4253	1.18418	0.592088	+	0.805874i	$0.298304\pi$
$462$	0	0
$463$	32.0742	1.49062	0.745308	−	0.666721i	$-0.232303\pi$
$463$	32.0742	1.49062	0.745308	+	0.666721i	$0.232303\pi$
$464$	0	0
$465$	0.136412	0.00632594
$466$	0	0
$467$	−38.0364	−1.76012	−0.880058	−	0.474867i	$-0.842496\pi$
$467$	−38.0364	−1.76012	−0.880058	+	0.474867i	$0.842496\pi$
$468$	0	0
$469$	11.3973	0.526278
$470$	0	0
$471$	11.2051	0.516303
$472$	0	0
$473$	−45.5753	−2.09556
$474$	0	0
$475$	−4.06062	−0.186314
$476$	0	0
$477$	−2.53112	−0.115892
$478$	0	0
$479$	5.98856	0.273624	0.136812	−	0.990597i	$-0.456314\pi$
$479$	5.98856	0.273624	0.136812	+	0.990597i	$0.456314\pi$
$480$	0	0
$481$	−1.39689	−0.0636925
$482$	0	0
$483$	−0.630861	−0.0287051
$484$	0	0
$485$	0.0462578	0.00210046
$486$	0	0
$487$	38.2784	1.73456	0.867281	−	0.497819i	$-0.165866\pi$
$487$	38.2784	1.73456	0.867281	+	0.497819i	$0.165866\pi$
$488$	0	0
$489$	2.90326	0.131290
$490$	0	0
$491$	8.13912	0.367313	0.183657	−	0.982990i	$-0.441207\pi$
$491$	8.13912	0.367313	0.183657	+	0.982990i	$0.441207\pi$
$492$	0	0
$493$	−7.33386	−0.330301
$494$	0	0
$495$	−1.11322	−0.0500357
$496$	0	0
$497$	−4.54584	−0.203909
$498$	0	0
$499$	−31.2580	−1.39930	−0.699650	−	0.714486i	$-0.746661\pi$
$499$	−31.2580	−1.39930	−0.699650	+	0.714486i	$0.746661\pi$
$500$	0	0
$501$	2.35145	0.105055
$502$	0	0
$503$	38.4641	1.71503	0.857515	−	0.514459i	$-0.172007\pi$
$503$	38.4641	1.71503	0.857515	+	0.514459i	$0.172007\pi$
$504$	0	0
$505$	0.464684	0.0206782
$506$	0	0
$507$	6.81589	0.302704
$508$	0	0
$509$	−29.5499	−1.30978	−0.654889	−	0.755725i	$-0.727284\pi$
$509$	−29.5499	−1.30978	−0.654889	+	0.755725i	$0.727284\pi$
$510$	0	0
$511$	−8.09723	−0.358200
$512$	0	0
$513$	2.47597	0.109317
$514$	0	0
$515$	1.08068	0.0476203
$516$	0	0
$517$	−16.0598	−0.706310
$518$	0	0
$519$	6.18608	0.271539
$520$	0	0
$521$	38.6168	1.69183	0.845916	−	0.533316i	$-0.179054\pi$
$521$	38.6168	1.69183	0.845916	+	0.533316i	$0.179054\pi$
$522$	0	0
$523$	−20.1272	−0.880100	−0.440050	−	0.897973i	$-0.645039\pi$
$523$	−20.1272	−0.880100	−0.440050	+	0.897973i	$0.645039\pi$
$524$	0	0
$525$	−2.65337	−0.115802
$526$	0	0
$527$	2.31917	0.101024
$528$	0	0
$529$	−21.5937	−0.938855
$530$	0	0
$531$	28.5230	1.23779
$532$	0	0
$533$	−4.31760	−0.187016
$534$	0	0
$535$	0.492014	0.0212716
$536$	0	0
$537$	6.43360	0.277630
$538$	0	0
$539$	3.70563	0.159613
$540$	0	0
$541$	−10.3210	−0.443733	−0.221867	−	0.975077i	$-0.571215\pi$
$541$	−10.3210	−0.443733	−0.221867	+	0.975077i	$0.571215\pi$
$542$	0	0
$543$	1.54765	0.0664158
$544$	0	0
$545$	−1.22517	−0.0524807
$546$	0	0
$547$	18.4053	0.786955	0.393477	−	0.919334i	$-0.371272\pi$
$547$	18.4053	0.786955	0.393477	+	0.919334i	$0.371272\pi$
$548$	0	0
$549$	15.3024	0.653091
$550$	0	0
$551$	5.97060	0.254356
$552$	0	0
$553$	6.73861	0.286555
$554$	0	0
$555$	−0.189716	−0.00805301
$556$	0	0
$557$	−31.7820	−1.34664	−0.673322	−	0.739349i	$-0.735133\pi$
$557$	−31.7820	−1.34664	−0.673322	+	0.739349i	$0.735133\pi$
$558$	0	0
$559$	5.32652	0.225288
$560$	0	0
$561$	1.97130	0.0832284
$562$	0	0
$563$	−18.9686	−0.799432	−0.399716	−	0.916639i	$-0.630891\pi$
$563$	−18.9686	−0.799432	−0.399716	+	0.916639i	$0.630891\pi$
$564$	0	0
$565$	0.477068	0.0200704
$566$	0	0
$567$	−6.53312	−0.274365
$568$	0	0
$569$	−37.8743	−1.58777	−0.793886	−	0.608067i	$-0.791945\pi$
$569$	−37.8743	−1.58777	−0.793886	+	0.608067i	$0.791945\pi$
$570$	0	0
$571$	−13.6906	−0.572932	−0.286466	−	0.958090i	$-0.592481\pi$
$571$	−13.6906	−0.572932	−0.286466	+	0.958090i	$0.592481\pi$
$572$	0	0
$573$	0.0856796	0.00357932
$574$	0	0
$575$	5.91493	0.246670
$576$	0	0
$577$	36.4870	1.51897	0.759487	−	0.650522i	$-0.225450\pi$
$577$	36.4870	1.51897	0.759487	+	0.650522i	$0.225450\pi$
$578$	0	0
$579$	3.67471	0.152716
$580$	0	0
$581$	4.28183	0.177640
$582$	0	0
$583$	3.45212	0.142972
$584$	0	0
$585$	0.130106	0.00537920
$586$	0	0
$587$	14.8540	0.613088	0.306544	−	0.951856i	$-0.400827\pi$
$587$	14.8540	0.613088	0.306544	+	0.951856i	$0.400827\pi$
$588$	0	0
$589$	−1.88807	−0.0777964
$590$	0	0
$591$	12.2350	0.503279
$592$	0	0
$593$	−13.8997	−0.570793	−0.285396	−	0.958410i	$-0.592125\pi$
$593$	−13.8997	−0.570793	−0.285396	+	0.958410i	$0.592125\pi$
$594$	0	0
$595$	0.110568	0.00453285
$596$	0	0
$597$	−13.9939	−0.572734
$598$	0	0
$599$	−6.42996	−0.262721	−0.131361	−	0.991335i	$-0.541935\pi$
$599$	−6.42996	−0.262721	−0.131361	+	0.991335i	$0.541935\pi$
$600$	0	0
$601$	22.8578	0.932387	0.466194	−	0.884683i	$-0.345625\pi$
$601$	22.8578	0.932387	0.466194	+	0.884683i	$0.345625\pi$
$602$	0	0
$603$	30.9665	1.26105
$604$	0	0
$605$	0.302041	0.0122797
$606$	0	0
$607$	17.0491	0.692003	0.346002	−	0.938234i	$-0.387539\pi$
$607$	17.0491	0.692003	0.346002	+	0.938234i	$0.387539\pi$
$608$	0	0
$609$	3.90143	0.158094
$610$	0	0
$611$	1.87696	0.0759335
$612$	0	0
$613$	−16.5121	−0.666916	−0.333458	−	0.942765i	$-0.608215\pi$
$613$	−16.5121	−0.666916	−0.333458	+	0.942765i	$0.608215\pi$
$614$	0	0
$615$	−0.586390	−0.0236455
$616$	0	0
$617$	33.2628	1.33911	0.669555	−	0.742763i	$-0.266485\pi$
$617$	33.2628	1.33911	0.669555	+	0.742763i	$0.266485\pi$
$618$	0	0
$619$	22.0540	0.886423	0.443212	−	0.896417i	$-0.353839\pi$
$619$	22.0540	0.886423	0.443212	+	0.896417i	$0.353839\pi$
$620$	0	0
$621$	−3.60663	−0.144729
$622$	0	0
$623$	8.11693	0.325198
$624$	0	0
$625$	24.8168	0.992671
$626$	0	0
$627$	−1.60487	−0.0640922
$628$	0	0
$629$	−3.22541	−0.128605
$630$	0	0
$631$	43.5593	1.73407	0.867035	−	0.498247i	$-0.166023\pi$
$631$	43.5593	1.73407	0.867035	+	0.498247i	$0.166023\pi$
$632$	0	0
$633$	−1.87102	−0.0743662
$634$	0	0
$635$	−2.07962	−0.0825274
$636$	0	0
$637$	−0.433088	−0.0171596
$638$	0	0
$639$	−12.3511	−0.488601
$640$	0	0
$641$	29.0395	1.14699	0.573495	−	0.819209i	$-0.305587\pi$
$641$	29.0395	1.14699	0.573495	+	0.819209i	$0.305587\pi$
$642$	0	0
$643$	−0.838572	−0.0330701	−0.0165350	−	0.999863i	$-0.505264\pi$
$643$	−0.838572	−0.0330701	−0.0165350	+	0.999863i	$0.505264\pi$
$644$	0	0
$645$	0.723414	0.0284844
$646$	0	0
$647$	36.4525	1.43310	0.716548	−	0.697538i	$-0.245721\pi$
$647$	36.4525	1.43310	0.716548	+	0.697538i	$0.245721\pi$
$648$	0	0
$649$	−38.9016	−1.52702
$650$	0	0
$651$	−1.23374	−0.0483539
$652$	0	0
$653$	32.1268	1.25722	0.628610	−	0.777721i	$-0.283624\pi$
$653$	32.1268	1.25722	0.628610	+	0.777721i	$0.283624\pi$
$654$	0	0
$655$	−2.48122	−0.0969494
$656$	0	0
$657$	−22.0002	−0.858310
$658$	0	0
$659$	−13.0272	−0.507469	−0.253735	−	0.967274i	$-0.581659\pi$
$659$	−13.0272	−0.507469	−0.253735	+	0.967274i	$0.581659\pi$
$660$	0	0
$661$	43.9589	1.70980	0.854901	−	0.518790i	$-0.173618\pi$
$661$	43.9589	1.70980	0.854901	+	0.518790i	$0.173618\pi$
$662$	0	0
$663$	−0.230392	−0.00894767
$664$	0	0
$665$	−0.0900150	−0.00349063
$666$	0	0
$667$	−8.69712	−0.336754
$668$	0	0
$669$	−5.88800	−0.227643
$670$	0	0
$671$	−20.8705	−0.805696
$672$	0	0
$673$	16.3500	0.630246	0.315123	−	0.949051i	$-0.397954\pi$
$673$	16.3500	0.630246	0.315123	+	0.949051i	$0.397954\pi$
$674$	0	0
$675$	−15.1693	−0.583867
$676$	0	0
$677$	38.0727	1.46325	0.731627	−	0.681706i	$-0.238762\pi$
$677$	38.0727	1.46325	0.731627	+	0.681706i	$0.238762\pi$
$678$	0	0
$679$	−0.418365	−0.0160554
$680$	0	0
$681$	7.05814	0.270469
$682$	0	0
$683$	29.0790	1.11268	0.556338	−	0.830956i	$-0.312206\pi$
$683$	29.0790	1.11268	0.556338	+	0.830956i	$0.312206\pi$
$684$	0	0
$685$	−1.70629	−0.0651940
$686$	0	0
$687$	−11.2821	−0.430438
$688$	0	0
$689$	−0.403459	−0.0153706
$690$	0	0
$691$	−3.42192	−0.130176	−0.0650881	−	0.997880i	$-0.520733\pi$
$691$	−3.42192	−0.130176	−0.0650881	+	0.997880i	$0.520733\pi$
$692$	0	0
$693$	10.0682	0.382460
$694$	0	0
$695$	0.701688	0.0266165
$696$	0	0
$697$	−9.96934	−0.377615
$698$	0	0
$699$	−0.788329	−0.0298173
$700$	0	0
$701$	4.34309	0.164036	0.0820182	−	0.996631i	$-0.473863\pi$
$701$	4.34309	0.164036	0.0820182	+	0.996631i	$0.473863\pi$
$702$	0	0
$703$	2.62585	0.0990359
$704$	0	0
$705$	0.254916	0.00960071
$706$	0	0
$707$	−4.20270	−0.158059
$708$	0	0
$709$	0.879666	0.0330365	0.0165183	−	0.999864i	$-0.494742\pi$
$709$	0.879666	0.0330365	0.0165183	+	0.999864i	$0.494742\pi$
$710$	0	0
$711$	18.3088	0.686634
$712$	0	0
$713$	2.75027	0.102998
$714$	0	0
$715$	−0.177447	−0.00663613
$716$	0	0
$717$	5.79417	0.216387
$718$	0	0
$719$	−23.1460	−0.863200	−0.431600	−	0.902065i	$-0.642051\pi$
$719$	−23.1460	−0.863200	−0.431600	+	0.902065i	$0.642051\pi$
$720$	0	0
$721$	−9.77386	−0.363997
$722$	0	0
$723$	7.28044	0.270763
$724$	0	0
$725$	−36.5797	−1.35853
$726$	0	0
$727$	18.8856	0.700429	0.350214	−	0.936670i	$-0.386109\pi$
$727$	18.8856	0.700429	0.350214	+	0.936670i	$0.386109\pi$
$728$	0	0
$729$	−12.8968	−0.477660
$730$	0	0
$731$	12.2989	0.454892
$732$	0	0
$733$	8.39104	0.309930	0.154965	−	0.987920i	$-0.450474\pi$
$733$	8.39104	0.309930	0.154965	+	0.987920i	$0.450474\pi$
$734$	0	0
$735$	−0.0588193	−0.00216958
$736$	0	0
$737$	−42.2342	−1.55572
$738$	0	0
$739$	17.8675	0.657268	0.328634	−	0.944457i	$-0.393412\pi$
$739$	17.8675	0.657268	0.328634	+	0.944457i	$0.393412\pi$
$740$	0	0
$741$	0.187565	0.00689038
$742$	0	0
$743$	−1.88839	−0.0692784	−0.0346392	−	0.999400i	$-0.511028\pi$
$743$	−1.88839	−0.0692784	−0.0346392	+	0.999400i	$0.511028\pi$
$744$	0	0
$745$	2.13970	0.0783926
$746$	0	0
$747$	11.6337	0.425656
$748$	0	0
$749$	−4.44987	−0.162595
$750$	0	0
$751$	36.1740	1.32001	0.660004	−	0.751262i	$-0.270554\pi$
$751$	36.1740	1.32001	0.660004	+	0.751262i	$0.270554\pi$
$752$	0	0
$753$	1.07733	0.0392600
$754$	0	0
$755$	−0.571026	−0.0207818
$756$	0	0
$757$	5.38447	0.195702	0.0978509	−	0.995201i	$-0.468803\pi$
$757$	5.38447	0.195702	0.0978509	+	0.995201i	$0.468803\pi$
$758$	0	0
$759$	2.33774	0.0848545
$760$	0	0
$761$	0.554315	0.0200939	0.0100470	−	0.999950i	$-0.496802\pi$
$761$	0.554315	0.0200939	0.0100470	+	0.999950i	$0.496802\pi$
$762$	0	0
$763$	11.0807	0.401150
$764$	0	0
$765$	0.300414	0.0108615
$766$	0	0
$767$	4.54654	0.164166
$768$	0	0
$769$	35.0866	1.26525	0.632627	−	0.774457i	$-0.281977\pi$
$769$	35.0866	1.26525	0.632627	+	0.774457i	$0.281977\pi$
$770$	0	0
$771$	6.13759	0.221040
$772$	0	0
$773$	−29.5200	−1.06176	−0.530881	−	0.847446i	$-0.678139\pi$
$773$	−29.5200	−1.06176	−0.530881	+	0.847446i	$0.678139\pi$
$774$	0	0
$775$	11.5675	0.415516
$776$	0	0
$777$	1.71583	0.0615552
$778$	0	0
$779$	8.11618	0.290792
$780$	0	0
$781$	16.8452	0.602770
$782$	0	0
$783$	22.3045	0.797097
$784$	0	0
$785$	−2.32891	−0.0831225
$786$	0	0
$787$	25.1018	0.894783	0.447392	−	0.894338i	$-0.352353\pi$
$787$	25.1018	0.894783	0.447392	+	0.894338i	$0.352353\pi$
$788$	0	0
$789$	1.67954	0.0597933
$790$	0	0
$791$	−4.31470	−0.153413
$792$	0	0
$793$	2.43919	0.0866182
$794$	0	0
$795$	−0.0547953	−0.00194339
$796$	0	0
$797$	35.4984	1.25742	0.628709	−	0.777641i	$-0.283584\pi$
$797$	35.4984	1.25742	0.628709	+	0.777641i	$0.283584\pi$
$798$	0	0
$799$	4.33389	0.153322
$800$	0	0
$801$	22.0537	0.779230
$802$	0	0
$803$	30.0054	1.05887
$804$	0	0
$805$	0.131121	0.00462141
$806$	0	0
$807$	−4.53053	−0.159482
$808$	0	0
$809$	24.5309	0.862460	0.431230	−	0.902242i	$-0.358080\pi$
$809$	24.5309	0.862460	0.431230	+	0.902242i	$0.358080\pi$
$810$	0	0
$811$	11.4156	0.400857	0.200428	−	0.979708i	$-0.435767\pi$
$811$	11.4156	0.400857	0.200428	+	0.979708i	$0.435767\pi$
$812$	0	0
$813$	−14.5163	−0.509109
$814$	0	0
$815$	−0.603426	−0.0211371
$816$	0	0
$817$	−10.0127	−0.350301
$818$	0	0
$819$	−1.17670	−0.0411173
$820$	0	0
$821$	−20.7929	−0.725678	−0.362839	−	0.931852i	$-0.618193\pi$
$821$	−20.7929	−0.725678	−0.362839	+	0.931852i	$0.618193\pi$
$822$	0	0
$823$	29.7346	1.03648	0.518241	−	0.855235i	$-0.326587\pi$
$823$	29.7346	1.03648	0.518241	+	0.855235i	$0.326587\pi$
$824$	0	0
$825$	9.83241	0.342321
$826$	0	0
$827$	−27.7393	−0.964589	−0.482295	−	0.876009i	$-0.660197\pi$
$827$	−27.7393	−0.964589	−0.482295	+	0.876009i	$0.660197\pi$
$828$	0	0
$829$	33.8141	1.17441	0.587207	−	0.809437i	$-0.300228\pi$
$829$	33.8141	1.17441	0.587207	+	0.809437i	$0.300228\pi$
$830$	0	0
$831$	0.00633718	0.000219834 0
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	−0.488737	−0.0169134
$836$	0	0
$837$	−7.05328	−0.243797
$838$	0	0
$839$	−26.0443	−0.899150	−0.449575	−	0.893242i	$-0.648425\pi$
$839$	−26.0443	−0.899150	−0.449575	+	0.893242i	$0.648425\pi$
$840$	0	0
$841$	24.7856	0.854674
$842$	0	0
$843$	−10.2253	−0.352180
$844$	0	0
$845$	−1.41665	−0.0487341
$846$	0	0
$847$	−2.73173	−0.0938632
$848$	0	0
$849$	7.67953	0.263561
$850$	0	0
$851$	−3.82497	−0.131118
$852$	0	0
$853$	36.8054	1.26019	0.630096	−	0.776517i	$-0.283016\pi$
$853$	36.8054	1.26019	0.630096	+	0.776517i	$0.283016\pi$
$854$	0	0
$855$	−0.244571	−0.00836415
$856$	0	0
$857$	0.330681	0.0112959	0.00564793	−	0.999984i	$-0.498202\pi$
$857$	0.330681	0.0112959	0.00564793	+	0.999984i	$0.498202\pi$
$858$	0	0
$859$	36.0492	1.22998	0.614992	−	0.788533i	$-0.289159\pi$
$859$	36.0492	1.22998	0.614992	+	0.788533i	$0.289159\pi$
$860$	0	0
$861$	5.30343	0.180740
$862$	0	0
$863$	16.3776	0.557499	0.278749	−	0.960364i	$-0.410080\pi$
$863$	16.3776	0.557499	0.278749	+	0.960364i	$0.410080\pi$
$864$	0	0
$865$	−1.28574	−0.0437166
$866$	0	0
$867$	−0.531974	−0.0180668
$868$	0	0
$869$	−24.9708	−0.847077
$870$	0	0
$871$	4.93603	0.167251
$872$	0	0
$873$	−1.13670	−0.0384714
$874$	0	0
$875$	1.10433	0.0373331
$876$	0	0
$877$	−13.7292	−0.463602	−0.231801	−	0.972763i	$-0.574462\pi$
$877$	−13.7292	−0.463602	−0.231801	+	0.972763i	$0.574462\pi$
$878$	0	0
$879$	−14.6143	−0.492929
$880$	0	0
$881$	−11.2123	−0.377753	−0.188876	−	0.982001i	$-0.560485\pi$
$881$	−11.2123	−0.377753	−0.188876	+	0.982001i	$0.560485\pi$
$882$	0	0
$883$	−36.7643	−1.23722	−0.618609	−	0.785699i	$-0.712303\pi$
$883$	−36.7643	−1.23722	−0.618609	+	0.785699i	$0.712303\pi$
$884$	0	0
$885$	0.617483	0.0207565
$886$	0	0
$887$	23.8492	0.800777	0.400389	−	0.916345i	$-0.368875\pi$
$887$	23.8492	0.800777	0.400389	+	0.916345i	$0.368875\pi$
$888$	0	0
$889$	18.8086	0.630819
$890$	0	0
$891$	24.2093	0.811044
$892$	0	0
$893$	−3.52828	−0.118069
$894$	0	0
$895$	−1.33719	−0.0446973
$896$	0	0
$897$	−0.273218	−0.00912249
$898$	0	0
$899$	−17.0084	−0.567263
$900$	0	0
$901$	−0.931587	−0.0310357
$902$	0	0
$903$	−6.54271	−0.217728
$904$	0	0
$905$	−0.321670	−0.0106927
$906$	0	0
$907$	11.7889	0.391445	0.195723	−	0.980659i	$-0.437295\pi$
$907$	11.7889	0.391445	0.195723	+	0.980659i	$0.437295\pi$
$908$	0	0
$909$	−11.4187	−0.378736
$910$	0	0
$911$	16.5982	0.549923	0.274962	−	0.961455i	$-0.411335\pi$
$911$	16.5982	0.549923	0.274962	+	0.961455i	$0.411335\pi$
$912$	0	0
$913$	−15.8669	−0.525117
$914$	0	0
$915$	0.331276	0.0109516
$916$	0	0
$917$	22.4407	0.741057
$918$	0	0
$919$	20.0817	0.662435	0.331217	−	0.943554i	$-0.392541\pi$
$919$	20.0817	0.662435	0.331217	+	0.943554i	$0.392541\pi$
$920$	0	0
$921$	13.3620	0.440293
$922$	0	0
$923$	−1.96875	−0.0648022
$924$	0	0
$925$	−16.0876	−0.528957
$926$	0	0
$927$	−26.5556	−0.872200
$928$	0	0
$929$	27.2037	0.892525	0.446263	−	0.894902i	$-0.352755\pi$
$929$	27.2037	0.892525	0.446263	+	0.894902i	$0.352755\pi$
$930$	0	0
$931$	0.814115	0.0266815
$932$	0	0
$933$	−3.23924	−0.106048
$934$	0	0
$935$	−0.409725	−0.0133994
$936$	0	0
$937$	18.2375	0.595792	0.297896	−	0.954598i	$-0.403715\pi$
$937$	18.2375	0.595792	0.297896	+	0.954598i	$0.403715\pi$
$938$	0	0
$939$	8.63495	0.281791
$940$	0	0
$941$	2.53544	0.0826530	0.0413265	−	0.999146i	$-0.486842\pi$
$941$	2.53544	0.0826530	0.0413265	+	0.999146i	$0.486842\pi$
$942$	0	0
$943$	−11.8225	−0.384993
$944$	0	0
$945$	−0.336270	−0.0109389
$946$	0	0
$947$	−30.5617	−0.993123	−0.496561	−	0.868002i	$-0.665404\pi$
$947$	−30.5617	−0.993123	−0.496561	+	0.868002i	$0.665404\pi$
$948$	0	0
$949$	−3.50681	−0.113836
$950$	0	0
$951$	−1.70017	−0.0551318
$952$	0	0
$953$	−26.2235	−0.849461	−0.424730	−	0.905320i	$-0.639631\pi$
$953$	−26.2235	−0.849461	−0.424730	+	0.905320i	$0.639631\pi$
$954$	0	0
$955$	−0.0178080	−0.000576255 0
$956$	0	0
$957$	−14.4573	−0.467337
$958$	0	0
$959$	15.4320	0.498327
$960$	0	0
$961$	−25.6215	−0.826499
$962$	0	0
$963$	−12.0903	−0.389605
$964$	0	0
$965$	−0.763768	−0.0245866
$966$	0	0
$967$	−40.4833	−1.30185	−0.650927	−	0.759140i	$-0.725620\pi$
$967$	−40.4833	−1.30185	−0.650927	+	0.759140i	$0.725620\pi$
$968$	0	0
$969$	0.433088	0.0139128
$970$	0	0
$971$	27.1253	0.870493	0.435246	−	0.900311i	$-0.356661\pi$
$971$	27.1253	0.870493	0.435246	+	0.900311i	$0.356661\pi$
$972$	0	0
$973$	−6.34621	−0.203450
$974$	0	0
$975$	−1.14914	−0.0368020
$976$	0	0
$977$	−48.4161	−1.54897	−0.774484	−	0.632594i	$-0.781990\pi$
$977$	−48.4161	−1.54897	−0.774484	+	0.632594i	$0.781990\pi$
$978$	0	0
$979$	−30.0784	−0.961309
$980$	0	0
$981$	30.1064	0.961223
$982$	0	0
$983$	−44.0930	−1.40635	−0.703173	−	0.711018i	$-0.748234\pi$
$983$	−44.0930	−1.40635	−0.703173	+	0.711018i	$0.748234\pi$
$984$	0	0
$985$	−2.54297	−0.0810257
$986$	0	0
$987$	−2.30552	−0.0733854
$988$	0	0
$989$	14.5851	0.463780
$990$	0	0
$991$	−12.7892	−0.406263	−0.203131	−	0.979152i	$-0.565112\pi$
$991$	−12.7892	−0.406263	−0.203131	+	0.979152i	$0.565112\pi$
$992$	0	0
$993$	3.68681	0.116997
$994$	0	0
$995$	2.90857	0.0922078
$996$	0	0
$997$	−23.9339	−0.757995	−0.378998	−	0.925398i	$-0.623731\pi$
$997$	−23.9339	−0.757995	−0.378998	+	0.925398i	$0.623731\pi$
$998$	0	0
$999$	9.80943	0.310357

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3808.2.a.h.1.4	✓	6
4.3	odd	2		3808.2.a.p.1.3	yes	6
8.3	odd	2		7616.2.a.bu.1.4		6
8.5	even	2		7616.2.a.cc.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.h.1.4	✓	6	1.1	even	1	trivial
3808.2.a.p.1.3	yes	6	4.3	odd	2
7616.2.a.bu.1.4		6	8.3	odd	2
7616.2.a.cc.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)

\(f(q)\)	\(=\)	\(q-0.531974 q^{3} +0.110568 q^{5} -1.00000 q^{7} -2.71700 q^{9} +3.70563 q^{11} -0.433088 q^{13} -0.0588193 q^{15} -1.00000 q^{17} +0.814115 q^{19} +0.531974 q^{21} -1.18589 q^{23} -4.98777 q^{25} +3.04130 q^{27} +7.33386 q^{29} -2.31917 q^{31} -1.97130 q^{33} -0.110568 q^{35} +3.22541 q^{37} +0.230392 q^{39} +9.96934 q^{41} -12.2989 q^{43} -0.300414 q^{45} -4.33389 q^{47} +1.00000 q^{49} +0.531974 q^{51} +0.931587 q^{53} +0.409725 q^{55} -0.433088 q^{57} -10.4980 q^{59} -5.63209 q^{61} +2.71700 q^{63} -0.0478857 q^{65} -11.3973 q^{67} +0.630861 q^{69} +4.54584 q^{71} +8.09723 q^{73} +2.65337 q^{75} -3.70563 q^{77} -6.73861 q^{79} +6.53312 q^{81} -4.28183 q^{83} -0.110568 q^{85} -3.90143 q^{87} -8.11693 q^{89} +0.433088 q^{91} +1.23374 q^{93} +0.0900150 q^{95} +0.418365 q^{97} -10.0682 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} + 4 q^{5} - 6 q^{7} + 8 q^{9} - 8 q^{11} + 8 q^{13} - 6 q^{17} - 6 q^{19} + 4 q^{21} - 18 q^{23} + 12 q^{25} - 22 q^{27} - 8 q^{29} + 4 q^{31} - 6 q^{33} - 4 q^{35} - 8 q^{37} - 14 q^{39}+ \cdots - 28 q^{99}+O(q^{100}) \) 6 * q - 4 * q^3 + 4 * q^5 - 6 * q^7 + 8 * q^9 - 8 * q^11 + 8 * q^13 - 6 * q^17 - 6 * q^19 + 4 * q^21 - 18 * q^23 + 12 * q^25 - 22 * q^27 - 8 * q^29 + 4 * q^31 - 6 * q^33 - 4 * q^35 - 8 * q^37 - 14 * q^39 + 16 * q^41 - 12 * q^43 + 4 * q^45 - 18 * q^47 + 6 * q^49 + 4 * q^51 - 2 * q^53 + 8 * q^55 + 8 * q^57 - 16 * q^59 + 6 * q^61 - 8 * q^63 - 22 * q^65 - 12 * q^67 + 16 * q^69 + 2 * q^71 + 8 * q^73 - 24 * q^75 + 8 * q^77 + 18 * q^79 + 18 * q^81 - 28 * q^83 - 4 * q^85 + 2 * q^87 - 2 * q^89 - 8 * q^91 - 32 * q^93 - 26 * q^95 - 18 * q^97 - 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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