LMFDB - Embedded newform 3856.2.a.j.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3856,2,Mod(1,3856)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3856, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3856.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3856 = 2^{4} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3856.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:	$30.7903150194$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	7.7.31056073.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 3x^{5} + 11x^{4} + x^{3} - 9x^{2} + 1 $ x^7 - 3x^6 - 3x^5 + 11x^4 + x^3 - 9x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 241)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.73684$ of defining polynomial
Character	$\chi$	$=$	3856.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.37146 q^{3} -2.63180 q^{5} +2.01025 q^{7} +2.62382 q^{9} +O(q^{10})$	$q+2.37146 q^{3} -2.63180 q^{5} +2.01025 q^{7} +2.62382 q^{9} +3.39618 q^{11} +5.63669 q^{13} -6.24122 q^{15} +0.866432 q^{17} -2.46437 q^{19} +4.76723 q^{21} +6.37847 q^{23} +1.92640 q^{25} -0.892104 q^{27} -4.52212 q^{29} +3.51511 q^{31} +8.05390 q^{33} -5.29060 q^{35} -5.19315 q^{37} +13.3672 q^{39} +1.35422 q^{41} -8.49015 q^{43} -6.90537 q^{45} +9.44537 q^{47} -2.95888 q^{49} +2.05471 q^{51} +9.71877 q^{53} -8.93808 q^{55} -5.84415 q^{57} -6.03110 q^{59} +4.45402 q^{61} +5.27454 q^{63} -14.8347 q^{65} +10.9216 q^{67} +15.1263 q^{69} +3.01063 q^{71} -0.255916 q^{73} +4.56837 q^{75} +6.82718 q^{77} +10.3262 q^{79} -9.98704 q^{81} +16.8148 q^{83} -2.28028 q^{85} -10.7240 q^{87} -17.4574 q^{89} +11.3312 q^{91} +8.33594 q^{93} +6.48574 q^{95} +0.273223 q^{97} +8.91095 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 3 q^{3} - 8 q^{5} + 7 q^{7} - 2 q^{9}+O(q^{10}) $ 7 * q + 3 * q^3 - 8 * q^5 + 7 * q^7 - 2 * q^9	$ 7 q + 3 q^{3} - 8 q^{5} + 7 q^{7} - 2 q^{9} + 18 q^{11} - q^{13} + 11 q^{15} - 2 q^{17} + 6 q^{19} - 2 q^{21} + 22 q^{23} + 5 q^{25} - 3 q^{27} - 16 q^{29} + 18 q^{31} + 4 q^{33} - 7 q^{35} + 8 q^{37} + 9 q^{39} - 15 q^{41} - 14 q^{43} + 3 q^{45} + 10 q^{47} + 6 q^{49} - 13 q^{51} + 15 q^{53} - 29 q^{55} + 14 q^{57} + 18 q^{59} + 4 q^{61} + 16 q^{63} - 7 q^{65} - 18 q^{67} + 26 q^{69} + 50 q^{71} - 16 q^{75} + 17 q^{77} + 15 q^{79} - 9 q^{81} + 24 q^{83} - 2 q^{85} - 12 q^{87} - 13 q^{89} + 12 q^{91} + 14 q^{93} + 41 q^{95} + q^{97} + 20 q^{99}+O(q^{100}) $ 7 * q + 3 * q^3 - 8 * q^5 + 7 * q^7 - 2 * q^9 + 18 * q^11 - q^13 + 11 * q^15 - 2 * q^17 + 6 * q^19 - 2 * q^21 + 22 * q^23 + 5 * q^25 - 3 * q^27 - 16 * q^29 + 18 * q^31 + 4 * q^33 - 7 * q^35 + 8 * q^37 + 9 * q^39 - 15 * q^41 - 14 * q^43 + 3 * q^45 + 10 * q^47 + 6 * q^49 - 13 * q^51 + 15 * q^53 - 29 * q^55 + 14 * q^57 + 18 * q^59 + 4 * q^61 + 16 * q^63 - 7 * q^65 - 18 * q^67 + 26 * q^69 + 50 * q^71 - 16 * q^75 + 17 * q^77 + 15 * q^79 - 9 * q^81 + 24 * q^83 - 2 * q^85 - 12 * q^87 - 13 * q^89 + 12 * q^91 + 14 * q^93 + 41 * q^95 + q^97 + 20 * q^99

Display 100 coefficients 10 coefficients

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
$3$	2.37146	1.36916	0.684581	−	0.728937i	$-0.259985\pi$
$3$	2.37146	1.36916	0.684581	+	0.728937i	$0.259985\pi$
$4$	0	0
$5$	−2.63180	−1.17698	−0.588489	−	0.808505i	$-0.700277\pi$
$5$	−2.63180	−1.17698	−0.588489	+	0.808505i	$0.700277\pi$
$6$	0	0
$7$	2.01025	0.759805	0.379902	−	0.925027i	$-0.375958\pi$
$7$	2.01025	0.759805	0.379902	+	0.925027i	$0.375958\pi$
$8$	0	0
$9$	2.62382	0.874605
$10$	0	0
$11$	3.39618	1.02399	0.511993	−	0.858989i	$-0.328907\pi$
$11$	3.39618	1.02399	0.511993	+	0.858989i	$0.328907\pi$
$12$	0	0
$13$	5.63669	1.56334	0.781669	−	0.623694i	$-0.214369\pi$
$13$	5.63669	1.56334	0.781669	+	0.623694i	$0.214369\pi$
$14$	0	0
$15$	−6.24122	−1.61148
$16$	0	0
$17$	0.866432	0.210141	0.105070	−	0.994465i	$-0.466493\pi$
$17$	0.866432	0.210141	0.105070	+	0.994465i	$0.466493\pi$
$18$	0	0
$19$	−2.46437	−0.565365	−0.282682	−	0.959214i	$-0.591224\pi$
$19$	−2.46437	−0.565365	−0.282682	+	0.959214i	$0.591224\pi$
$20$	0	0
$21$	4.76723	1.04030
$22$	0	0
$23$	6.37847	1.33000	0.665001	−	0.746842i	$-0.268431\pi$
$23$	6.37847	1.33000	0.665001	+	0.746842i	$0.268431\pi$
$24$	0	0
$25$	1.92640	0.385279
$26$	0	0
$27$	−0.892104	−0.171686
$28$	0	0
$29$	−4.52212	−0.839737	−0.419868	−	0.907585i	$-0.637924\pi$
$29$	−4.52212	−0.839737	−0.419868	+	0.907585i	$0.637924\pi$
$30$	0	0
$31$	3.51511	0.631332	0.315666	−	0.948870i	$-0.397772\pi$
$31$	3.51511	0.631332	0.315666	+	0.948870i	$0.397772\pi$
$32$	0	0
$33$	8.05390	1.40200
$34$	0	0
$35$	−5.29060	−0.894274
$36$	0	0
$37$	−5.19315	−0.853748	−0.426874	−	0.904311i	$-0.640385\pi$
$37$	−5.19315	−0.853748	−0.426874	+	0.904311i	$0.640385\pi$
$38$	0	0
$39$	13.3672	2.14046
$40$	0	0
$41$	1.35422	0.211494	0.105747	−	0.994393i	$-0.466277\pi$
$41$	1.35422	0.211494	0.105747	+	0.994393i	$0.466277\pi$
$42$	0	0
$43$	−8.49015	−1.29474	−0.647368	−	0.762178i	$-0.724130\pi$
$43$	−8.49015	−1.29474	−0.647368	+	0.762178i	$0.724130\pi$
$44$	0	0
$45$	−6.90537	−1.02939
$46$	0	0
$47$	9.44537	1.37775	0.688875	−	0.724880i	$-0.258105\pi$
$47$	9.44537	1.37775	0.688875	+	0.724880i	$0.258105\pi$
$48$	0	0
$49$	−2.95888	−0.422697
$50$	0	0
$51$	2.05471	0.287717
$52$	0	0
$53$	9.71877	1.33498	0.667488	−	0.744620i	$-0.267369\pi$
$53$	9.71877	1.33498	0.667488	+	0.744620i	$0.267369\pi$
$54$	0	0
$55$	−8.93808	−1.20521
$56$	0	0
$57$	−5.84415	−0.774076
$58$	0	0
$59$	−6.03110	−0.785182	−0.392591	−	0.919713i	$-0.628421\pi$
$59$	−6.03110	−0.785182	−0.392591	+	0.919713i	$0.628421\pi$
$60$	0	0
$61$	4.45402	0.570279	0.285140	−	0.958486i	$-0.407960\pi$
$61$	4.45402	0.570279	0.285140	+	0.958486i	$0.407960\pi$
$62$	0	0
$63$	5.27454	0.664529
$64$	0	0
$65$	−14.8347	−1.84002
$66$	0	0
$67$	10.9216	1.33428	0.667141	−	0.744932i	$-0.267518\pi$
$67$	10.9216	1.33428	0.667141	+	0.744932i	$0.267518\pi$
$68$	0	0
$69$	15.1263	1.82099
$70$	0	0
$71$	3.01063	0.357296	0.178648	−	0.983913i	$-0.442828\pi$
$71$	3.01063	0.357296	0.178648	+	0.983913i	$0.442828\pi$
$72$	0	0
$73$	−0.255916	−0.0299527	−0.0149764	−	0.999888i	$-0.504767\pi$
$73$	−0.255916	−0.0299527	−0.0149764	+	0.999888i	$0.504767\pi$
$74$	0	0
$75$	4.56837	0.527510
$76$	0	0
$77$	6.82718	0.778030
$78$	0	0
$79$	10.3262	1.16179	0.580893	−	0.813980i	$-0.302703\pi$
$79$	10.3262	1.16179	0.580893	+	0.813980i	$0.302703\pi$
$80$	0	0
$81$	−9.98704	−1.10967
$82$	0	0
$83$	16.8148	1.84567	0.922833	−	0.385201i	$-0.125868\pi$
$83$	16.8148	1.84567	0.922833	+	0.385201i	$0.125868\pi$
$84$	0	0
$85$	−2.28028	−0.247331
$86$	0	0
$87$	−10.7240	−1.14974
$88$	0	0
$89$	−17.4574	−1.85048	−0.925241	−	0.379380i	$-0.876137\pi$
$89$	−17.4574	−1.85048	−0.925241	+	0.379380i	$0.876137\pi$
$90$	0	0
$91$	11.3312	1.18783
$92$	0	0
$93$	8.33594	0.864397
$94$	0	0
$95$	6.48574	0.665423
$96$	0	0
$97$	0.273223	0.0277416	0.0138708	−	0.999904i	$-0.495585\pi$
$97$	0.273223	0.0277416	0.0138708	+	0.999904i	$0.495585\pi$
$98$	0	0
$99$	8.91095	0.895584
$100$	0	0
$101$	7.47149	0.743441	0.371721	−	0.928345i	$-0.378768\pi$
$101$	7.47149	0.743441	0.371721	+	0.928345i	$0.378768\pi$
$102$	0	0
$103$	−14.5145	−1.43015	−0.715077	−	0.699046i	$-0.753608\pi$
$103$	−14.5145	−1.43015	−0.715077	+	0.699046i	$0.753608\pi$
$104$	0	0
$105$	−12.5464	−1.22441
$106$	0	0
$107$	−16.7492	−1.61921	−0.809605	−	0.586976i	$-0.800318\pi$
$107$	−16.7492	−1.61921	−0.809605	+	0.586976i	$0.800318\pi$
$108$	0	0
$109$	−6.01965	−0.576578	−0.288289	−	0.957543i	$-0.593086\pi$
$109$	−6.01965	−0.576578	−0.288289	+	0.957543i	$0.593086\pi$
$110$	0	0
$111$	−12.3153	−1.16892
$112$	0	0
$113$	18.9246	1.78028	0.890139	−	0.455689i	$-0.150607\pi$
$113$	18.9246	1.78028	0.890139	+	0.455689i	$0.150607\pi$
$114$	0	0
$115$	−16.7869	−1.56539
$116$	0	0
$117$	14.7896	1.36730
$118$	0	0
$119$	1.74175	0.159666
$120$	0	0
$121$	0.534035	0.0485487
$122$	0	0
$123$	3.21149	0.289570
$124$	0	0
$125$	8.08912	0.723513
$126$	0	0
$127$	13.7300	1.21834	0.609172	−	0.793038i	$-0.291502\pi$
$127$	13.7300	1.21834	0.609172	+	0.793038i	$0.291502\pi$
$128$	0	0
$129$	−20.1340	−1.77270
$130$	0	0
$131$	14.3193	1.25108	0.625542	−	0.780190i	$-0.284878\pi$
$131$	14.3193	1.25108	0.625542	+	0.780190i	$0.284878\pi$
$132$	0	0
$133$	−4.95401	−0.429567
$134$	0	0
$135$	2.34784	0.202070
$136$	0	0
$137$	10.1743	0.869252	0.434626	−	0.900611i	$-0.356881\pi$
$137$	10.1743	0.869252	0.434626	+	0.900611i	$0.356881\pi$
$138$	0	0
$139$	−8.00345	−0.678844	−0.339422	−	0.940634i	$-0.610231\pi$
$139$	−8.00345	−0.678844	−0.339422	+	0.940634i	$0.610231\pi$
$140$	0	0
$141$	22.3993	1.88636
$142$	0	0
$143$	19.1432	1.60084
$144$	0	0
$145$	11.9013	0.988352
$146$	0	0
$147$	−7.01686	−0.578741
$148$	0	0
$149$	−15.5640	−1.27505	−0.637527	−	0.770428i	$-0.720042\pi$
$149$	−15.5640	−1.27505	−0.637527	+	0.770428i	$0.720042\pi$
$150$	0	0
$151$	2.43764	0.198372	0.0991862	−	0.995069i	$-0.468376\pi$
$151$	2.43764	0.198372	0.0991862	+	0.995069i	$0.468376\pi$
$152$	0	0
$153$	2.27336	0.183790
$154$	0	0
$155$	−9.25108	−0.743065
$156$	0	0
$157$	12.1633	0.970735	0.485367	−	0.874310i	$-0.338686\pi$
$157$	12.1633	0.970735	0.485367	+	0.874310i	$0.338686\pi$
$158$	0	0
$159$	23.0477	1.82780
$160$	0	0
$161$	12.8223	1.01054
$162$	0	0
$163$	−4.23026	−0.331340	−0.165670	−	0.986181i	$-0.552979\pi$
$163$	−4.23026	−0.331340	−0.165670	+	0.986181i	$0.552979\pi$
$164$	0	0
$165$	−21.1963	−1.65013
$166$	0	0
$167$	−14.3677	−1.11181	−0.555903	−	0.831247i	$-0.687628\pi$
$167$	−14.3677	−1.11181	−0.555903	+	0.831247i	$0.687628\pi$
$168$	0	0
$169$	18.7723	1.44402
$170$	0	0
$171$	−6.46605	−0.494471
$172$	0	0
$173$	1.08253	0.0823030	0.0411515	−	0.999153i	$-0.486897\pi$
$173$	1.08253	0.0823030	0.0411515	+	0.999153i	$0.486897\pi$
$174$	0	0
$175$	3.87255	0.292737
$176$	0	0
$177$	−14.3025	−1.07504
$178$	0	0
$179$	24.0238	1.79562	0.897810	−	0.440382i	$-0.145157\pi$
$179$	24.0238	1.79562	0.897810	+	0.440382i	$0.145157\pi$
$180$	0	0
$181$	−4.72624	−0.351298	−0.175649	−	0.984453i	$-0.556202\pi$
$181$	−4.72624	−0.351298	−0.175649	+	0.984453i	$0.556202\pi$
$182$	0	0
$183$	10.5625	0.780805
$184$	0	0
$185$	13.6674	1.00484
$186$	0	0
$187$	2.94256	0.215181
$188$	0	0
$189$	−1.79336	−0.130447
$190$	0	0
$191$	24.3499	1.76190	0.880948	−	0.473213i	$-0.156906\pi$
$191$	24.3499	1.76190	0.880948	+	0.473213i	$0.156906\pi$
$192$	0	0
$193$	3.22480	0.232126	0.116063	−	0.993242i	$-0.462973\pi$
$193$	3.22480	0.232126	0.116063	+	0.993242i	$0.462973\pi$
$194$	0	0
$195$	−35.1798	−2.51928
$196$	0	0
$197$	−3.60952	−0.257167	−0.128584	−	0.991699i	$-0.541043\pi$
$197$	−3.60952	−0.257167	−0.128584	+	0.991699i	$0.541043\pi$
$198$	0	0
$199$	−11.6028	−0.822499	−0.411249	−	0.911523i	$-0.634907\pi$
$199$	−11.6028	−0.822499	−0.411249	+	0.911523i	$0.634907\pi$
$200$	0	0
$201$	25.9000	1.82685
$202$	0	0
$203$	−9.09061	−0.638036
$204$	0	0
$205$	−3.56405	−0.248924
$206$	0	0
$207$	16.7359	1.16323
$208$	0	0
$209$	−8.36944	−0.578926
$210$	0	0
$211$	−25.9439	−1.78605	−0.893025	−	0.450007i	$-0.851421\pi$
$211$	−25.9439	−1.78605	−0.893025	+	0.450007i	$0.851421\pi$
$212$	0	0
$213$	7.13958	0.489196
$214$	0	0
$215$	22.3444	1.52388
$216$	0	0
$217$	7.06626	0.479689
$218$	0	0
$219$	−0.606895	−0.0410102
$220$	0	0
$221$	4.88381	0.328521
$222$	0	0
$223$	4.40090	0.294706	0.147353	−	0.989084i	$-0.452925\pi$
$223$	4.40090	0.294706	0.147353	+	0.989084i	$0.452925\pi$
$224$	0	0
$225$	5.05451	0.336967
$226$	0	0
$227$	23.0781	1.53175	0.765873	−	0.642991i	$-0.222307\pi$
$227$	23.0781	1.53175	0.765873	+	0.642991i	$0.222307\pi$
$228$	0	0
$229$	3.21172	0.212236	0.106118	−	0.994354i	$-0.466158\pi$
$229$	3.21172	0.212236	0.106118	+	0.994354i	$0.466158\pi$
$230$	0	0
$231$	16.1904	1.06525
$232$	0	0
$233$	7.13210	0.467239	0.233620	−	0.972328i	$-0.424943\pi$
$233$	7.13210	0.467239	0.233620	+	0.972328i	$0.424943\pi$
$234$	0	0
$235$	−24.8584	−1.62158
$236$	0	0
$237$	24.4881	1.59067
$238$	0	0
$239$	−3.66241	−0.236902	−0.118451	−	0.992960i	$-0.537793\pi$
$239$	−3.66241	−0.236902	−0.118451	+	0.992960i	$0.537793\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	0	0
$243$	−21.0075	−1.34763
$244$	0	0
$245$	7.78719	0.497506
$246$	0	0
$247$	−13.8909	−0.883856
$248$	0	0
$249$	39.8756	2.52702
$250$	0	0
$251$	−1.41613	−0.0893854	−0.0446927	−	0.999001i	$-0.514231\pi$
$251$	−1.41613	−0.0893854	−0.0446927	+	0.999001i	$0.514231\pi$
$252$	0	0
$253$	21.6624	1.36190
$254$	0	0
$255$	−5.40759	−0.338636
$256$	0	0
$257$	−11.0690	−0.690466	−0.345233	−	0.938517i	$-0.612200\pi$
$257$	−11.0690	−0.690466	−0.345233	+	0.938517i	$0.612200\pi$
$258$	0	0
$259$	−10.4395	−0.648682
$260$	0	0
$261$	−11.8652	−0.734438
$262$	0	0
$263$	3.31569	0.204454	0.102227	−	0.994761i	$-0.467403\pi$
$263$	3.31569	0.204454	0.102227	+	0.994761i	$0.467403\pi$
$264$	0	0
$265$	−25.5779	−1.57124
$266$	0	0
$267$	−41.3995	−2.53361
$268$	0	0
$269$	−14.3455	−0.874663	−0.437332	−	0.899300i	$-0.644076\pi$
$269$	−14.3455	−0.874663	−0.437332	+	0.899300i	$0.644076\pi$
$270$	0	0
$271$	1.55240	0.0943014	0.0471507	−	0.998888i	$-0.484986\pi$
$271$	1.55240	0.0943014	0.0471507	+	0.998888i	$0.484986\pi$
$272$	0	0
$273$	26.8714	1.62633
$274$	0	0
$275$	6.54239	0.394521
$276$	0	0
$277$	18.0089	1.08205	0.541024	−	0.841007i	$-0.318037\pi$
$277$	18.0089	1.08205	0.541024	+	0.841007i	$0.318037\pi$
$278$	0	0
$279$	9.22300	0.552167
$280$	0	0
$281$	−11.5476	−0.688873	−0.344436	−	0.938810i	$-0.611930\pi$
$281$	−11.5476	−0.688873	−0.344436	+	0.938810i	$0.611930\pi$
$282$	0	0
$283$	1.84755	0.109825	0.0549126	−	0.998491i	$-0.482512\pi$
$283$	1.84755	0.109825	0.0549126	+	0.998491i	$0.482512\pi$
$284$	0	0
$285$	15.3807	0.911071
$286$	0	0
$287$	2.72233	0.160694
$288$	0	0
$289$	−16.2493	−0.955841
$290$	0	0
$291$	0.647937	0.0379827
$292$	0	0
$293$	−3.03213	−0.177139	−0.0885694	−	0.996070i	$-0.528229\pi$
$293$	−3.03213	−0.177139	−0.0885694	+	0.996070i	$0.528229\pi$
$294$	0	0
$295$	15.8727	0.924143
$296$	0	0
$297$	−3.02975	−0.175804
$298$	0	0
$299$	35.9535	2.07924
$300$	0	0
$301$	−17.0674	−0.983746
$302$	0	0
$303$	17.7183	1.01789
$304$	0	0
$305$	−11.7221	−0.671207
$306$	0	0
$307$	−8.40693	−0.479809	−0.239905	−	0.970796i	$-0.577116\pi$
$307$	−8.40693	−0.479809	−0.239905	+	0.970796i	$0.577116\pi$
$308$	0	0
$309$	−34.4205	−1.95811
$310$	0	0
$311$	−2.94070	−0.166752	−0.0833759	−	0.996518i	$-0.526570\pi$
$311$	−2.94070	−0.166752	−0.0833759	+	0.996518i	$0.526570\pi$
$312$	0	0
$313$	16.5110	0.933258	0.466629	−	0.884453i	$-0.345468\pi$
$313$	16.5110	0.933258	0.466629	+	0.884453i	$0.345468\pi$
$314$	0	0
$315$	−13.8816	−0.782137
$316$	0	0
$317$	−20.3799	−1.14465	−0.572325	−	0.820027i	$-0.693959\pi$
$317$	−20.3799	−1.14465	−0.572325	+	0.820027i	$0.693959\pi$
$318$	0	0
$319$	−15.3579	−0.859879
$320$	0	0
$321$	−39.7201	−2.21696
$322$	0	0
$323$	−2.13521	−0.118806
$324$	0	0
$325$	10.8585	0.602322
$326$	0	0
$327$	−14.2754	−0.789429
$328$	0	0
$329$	18.9876	1.04682
$330$	0	0
$331$	−9.08522	−0.499369	−0.249684	−	0.968327i	$-0.580327\pi$
$331$	−9.08522	−0.499369	−0.249684	+	0.968327i	$0.580327\pi$
$332$	0	0
$333$	−13.6259	−0.746693
$334$	0	0
$335$	−28.7434	−1.57042
$336$	0	0
$337$	−34.1864	−1.86225	−0.931125	−	0.364700i	$-0.881171\pi$
$337$	−34.1864	−1.86225	−0.931125	+	0.364700i	$0.881171\pi$
$338$	0	0
$339$	44.8789	2.43749
$340$	0	0
$341$	11.9379	0.646476
$342$	0	0
$343$	−20.0199	−1.08097
$344$	0	0
$345$	−39.8094	−2.14327
$346$	0	0
$347$	9.13414	0.490346	0.245173	−	0.969479i	$-0.421155\pi$
$347$	9.13414	0.490346	0.245173	+	0.969479i	$0.421155\pi$
$348$	0	0
$349$	−20.3980	−1.09188	−0.545940	−	0.837824i	$-0.683827\pi$
$349$	−20.3980	−1.09188	−0.545940	+	0.837824i	$0.683827\pi$
$350$	0	0
$351$	−5.02852	−0.268402
$352$	0	0
$353$	13.2353	0.704444	0.352222	−	0.935916i	$-0.385426\pi$
$353$	13.2353	0.704444	0.352222	+	0.935916i	$0.385426\pi$
$354$	0	0
$355$	−7.92339	−0.420530
$356$	0	0
$357$	4.13048	0.218608
$358$	0	0
$359$	−35.7474	−1.88668	−0.943339	−	0.331831i	$-0.892334\pi$
$359$	−35.7474	−1.88668	−0.943339	+	0.331831i	$0.892334\pi$
$360$	0	0
$361$	−12.9269	−0.680363
$362$	0	0
$363$	1.26644	0.0664710
$364$	0	0
$365$	0.673522	0.0352537
$366$	0	0
$367$	34.7219	1.81247	0.906234	−	0.422777i	$-0.138945\pi$
$367$	34.7219	1.81247	0.906234	+	0.422777i	$0.138945\pi$
$368$	0	0
$369$	3.55323	0.184974
$370$	0	0
$371$	19.5372	1.01432
$372$	0	0
$373$	−34.9987	−1.81217	−0.906083	−	0.423100i	$-0.860942\pi$
$373$	−34.9987	−1.81217	−0.906083	+	0.423100i	$0.860942\pi$
$374$	0	0
$375$	19.1830	0.990607
$376$	0	0
$377$	−25.4898	−1.31279
$378$	0	0
$379$	−17.4032	−0.893943	−0.446972	−	0.894548i	$-0.647498\pi$
$379$	−17.4032	−0.893943	−0.446972	+	0.894548i	$0.647498\pi$
$380$	0	0
$381$	32.5602	1.66811
$382$	0	0
$383$	−7.58514	−0.387582	−0.193791	−	0.981043i	$-0.562078\pi$
$383$	−7.58514	−0.387582	−0.193791	+	0.981043i	$0.562078\pi$
$384$	0	0
$385$	−17.9678	−0.915725
$386$	0	0
$387$	−22.2766	−1.13238
$388$	0	0
$389$	−9.68216	−0.490905	−0.245452	−	0.969409i	$-0.578937\pi$
$389$	−9.68216	−0.490905	−0.245452	+	0.969409i	$0.578937\pi$
$390$	0	0
$391$	5.52651	0.279488
$392$	0	0
$393$	33.9577	1.71294
$394$	0	0
$395$	−27.1765	−1.36740
$396$	0	0
$397$	−8.54563	−0.428893	−0.214446	−	0.976736i	$-0.568795\pi$
$397$	−8.54563	−0.428893	−0.214446	+	0.976736i	$0.568795\pi$
$398$	0	0
$399$	−11.7482	−0.588147
$400$	0	0
$401$	23.0124	1.14918	0.574592	−	0.818440i	$-0.305161\pi$
$401$	23.0124	1.14918	0.574592	+	0.818440i	$0.305161\pi$
$402$	0	0
$403$	19.8136	0.986986
$404$	0	0
$405$	26.2839	1.30606
$406$	0	0
$407$	−17.6369	−0.874227
$408$	0	0
$409$	−24.2236	−1.19778	−0.598891	−	0.800831i	$-0.704392\pi$
$409$	−24.2236	−1.19778	−0.598891	+	0.800831i	$0.704392\pi$
$410$	0	0
$411$	24.1280	1.19015
$412$	0	0
$413$	−12.1240	−0.596585
$414$	0	0
$415$	−44.2533	−2.17231
$416$	0	0
$417$	−18.9799	−0.929448
$418$	0	0
$419$	−27.7565	−1.35600	−0.677998	−	0.735064i	$-0.737152\pi$
$419$	−27.7565	−1.35600	−0.677998	+	0.735064i	$0.737152\pi$
$420$	0	0
$421$	−33.1205	−1.61420	−0.807098	−	0.590418i	$-0.798963\pi$
$421$	−33.1205	−1.61420	−0.807098	+	0.590418i	$0.798963\pi$
$422$	0	0
$423$	24.7829	1.20499
$424$	0	0
$425$	1.66909	0.0809628
$426$	0	0
$427$	8.95372	0.433301
$428$	0	0
$429$	45.3974	2.19181
$430$	0	0
$431$	5.07371	0.244392	0.122196	−	0.992506i	$-0.461006\pi$
$431$	5.07371	0.244392	0.122196	+	0.992506i	$0.461006\pi$
$432$	0	0
$433$	11.6416	0.559457	0.279729	−	0.960079i	$-0.409755\pi$
$433$	11.6416	0.559457	0.279729	+	0.960079i	$0.409755\pi$
$434$	0	0
$435$	28.2235	1.35321
$436$	0	0
$437$	−15.7189	−0.751937
$438$	0	0
$439$	14.7945	0.706105	0.353052	−	0.935604i	$-0.385144\pi$
$439$	14.7945	0.706105	0.353052	+	0.935604i	$0.385144\pi$
$440$	0	0
$441$	−7.76356	−0.369693
$442$	0	0
$443$	2.54544	0.120937	0.0604687	−	0.998170i	$-0.480740\pi$
$443$	2.54544	0.120937	0.0604687	+	0.998170i	$0.480740\pi$
$444$	0	0
$445$	45.9445	2.17798
$446$	0	0
$447$	−36.9094	−1.74575
$448$	0	0
$449$	−13.1558	−0.620860	−0.310430	−	0.950596i	$-0.600473\pi$
$449$	−13.1558	−0.620860	−0.310430	+	0.950596i	$0.600473\pi$
$450$	0	0
$451$	4.59919	0.216567
$452$	0	0
$453$	5.78077	0.271604
$454$	0	0
$455$	−29.8215	−1.39805
$456$	0	0
$457$	5.54809	0.259528	0.129764	−	0.991545i	$-0.458578\pi$
$457$	5.54809	0.259528	0.129764	+	0.991545i	$0.458578\pi$
$458$	0	0
$459$	−0.772948	−0.0360781
$460$	0	0
$461$	−32.2282	−1.50102	−0.750508	−	0.660861i	$-0.770191\pi$
$461$	−32.2282	−1.50102	−0.750508	+	0.660861i	$0.770191\pi$
$462$	0	0
$463$	18.5042	0.859965	0.429983	−	0.902837i	$-0.358520\pi$
$463$	18.5042	0.859965	0.429983	+	0.902837i	$0.358520\pi$
$464$	0	0
$465$	−21.9386	−1.01738
$466$	0	0
$467$	−20.9192	−0.968026	−0.484013	−	0.875061i	$-0.660821\pi$
$467$	−20.9192	−0.968026	−0.484013	+	0.875061i	$0.660821\pi$
$468$	0	0
$469$	21.9551	1.01379
$470$	0	0
$471$	28.8447	1.32909
$472$	0	0
$473$	−28.8341	−1.32579
$474$	0	0
$475$	−4.74735	−0.217823
$476$	0	0
$477$	25.5003	1.16758
$478$	0	0
$479$	9.86001	0.450515	0.225258	−	0.974299i	$-0.427678\pi$
$479$	9.86001	0.450515	0.225258	+	0.974299i	$0.427678\pi$
$480$	0	0
$481$	−29.2722	−1.33470
$482$	0	0
$483$	30.4077	1.38360
$484$	0	0
$485$	−0.719069	−0.0326512
$486$	0	0
$487$	25.9290	1.17496	0.587478	−	0.809240i	$-0.300121\pi$
$487$	25.9290	1.17496	0.587478	+	0.809240i	$0.300121\pi$
$488$	0	0
$489$	−10.0319	−0.453658
$490$	0	0
$491$	22.5594	1.01809	0.509046	−	0.860739i	$-0.329998\pi$
$491$	22.5594	1.01809	0.509046	+	0.860739i	$0.329998\pi$
$492$	0	0
$493$	−3.91811	−0.176463
$494$	0	0
$495$	−23.4519	−1.05408
$496$	0	0
$497$	6.05213	0.271475
$498$	0	0
$499$	−39.8682	−1.78475	−0.892373	−	0.451298i	$-0.850961\pi$
$499$	−39.8682	−1.78475	−0.892373	+	0.451298i	$0.850961\pi$
$500$	0	0
$501$	−34.0724	−1.52224
$502$	0	0
$503$	19.7008	0.878416	0.439208	−	0.898386i	$-0.355259\pi$
$503$	19.7008	0.878416	0.439208	+	0.898386i	$0.355259\pi$
$504$	0	0
$505$	−19.6635	−0.875015
$506$	0	0
$507$	44.5178	1.97710
$508$	0	0
$509$	−23.4312	−1.03857	−0.519284	−	0.854602i	$-0.673801\pi$
$509$	−23.4312	−1.03857	−0.519284	+	0.854602i	$0.673801\pi$
$510$	0	0
$511$	−0.514457	−0.0227582
$512$	0	0
$513$	2.19847	0.0970650
$514$	0	0
$515$	38.1993	1.68326
$516$	0	0
$517$	32.0782	1.41080
$518$	0	0
$519$	2.56717	0.112686
$520$	0	0
$521$	−5.76181	−0.252430	−0.126215	−	0.992003i	$-0.540283\pi$
$521$	−5.76181	−0.252430	−0.126215	+	0.992003i	$0.540283\pi$
$522$	0	0
$523$	−42.0766	−1.83988	−0.919942	−	0.392056i	$-0.871764\pi$
$523$	−42.0766	−1.83988	−0.919942	+	0.392056i	$0.871764\pi$
$524$	0	0
$525$	9.18358	0.400804
$526$	0	0
$527$	3.04560	0.132669
$528$	0	0
$529$	17.6849	0.768907
$530$	0	0
$531$	−15.8245	−0.686724
$532$	0	0
$533$	7.63334	0.330637
$534$	0	0
$535$	44.0807	1.90578
$536$	0	0
$537$	56.9714	2.45850
$538$	0	0
$539$	−10.0489	−0.432836
$540$	0	0
$541$	7.25557	0.311941	0.155971	−	0.987762i	$-0.450149\pi$
$541$	7.25557	0.311941	0.155971	+	0.987762i	$0.450149\pi$
$542$	0	0
$543$	−11.2081	−0.480984
$544$	0	0
$545$	15.8426	0.678620
$546$	0	0
$547$	27.6574	1.18254	0.591272	−	0.806472i	$-0.298626\pi$
$547$	27.6574	1.18254	0.591272	+	0.806472i	$0.298626\pi$
$548$	0	0
$549$	11.6865	0.498769
$550$	0	0
$551$	11.1442	0.474758
$552$	0	0
$553$	20.7582	0.882730
$554$	0	0
$555$	32.4116	1.37579
$556$	0	0
$557$	26.7523	1.13353	0.566766	−	0.823878i	$-0.308194\pi$
$557$	26.7523	1.13353	0.566766	+	0.823878i	$0.308194\pi$
$558$	0	0
$559$	−47.8564	−2.02411
$560$	0	0
$561$	6.97816	0.294618
$562$	0	0
$563$	−20.2482	−0.853362	−0.426681	−	0.904402i	$-0.640317\pi$
$563$	−20.2482	−0.853362	−0.426681	+	0.904402i	$0.640317\pi$
$564$	0	0
$565$	−49.8059	−2.09535
$566$	0	0
$567$	−20.0765	−0.843133
$568$	0	0
$569$	41.1274	1.72415	0.862076	−	0.506779i	$-0.169164\pi$
$569$	41.1274	1.72415	0.862076	+	0.506779i	$0.169164\pi$
$570$	0	0
$571$	−25.2419	−1.05634	−0.528170	−	0.849139i	$-0.677122\pi$
$571$	−25.2419	−1.05634	−0.528170	+	0.849139i	$0.677122\pi$
$572$	0	0
$573$	57.7448	2.41232
$574$	0	0
$575$	12.2875	0.512423
$576$	0	0
$577$	4.90936	0.204379	0.102190	−	0.994765i	$-0.467415\pi$
$577$	4.90936	0.204379	0.102190	+	0.994765i	$0.467415\pi$
$578$	0	0
$579$	7.64748	0.317818
$580$	0	0
$581$	33.8020	1.40234
$582$	0	0
$583$	33.0067	1.36700
$584$	0	0
$585$	−38.9235	−1.60929
$586$	0	0
$587$	−0.733495	−0.0302746	−0.0151373	−	0.999885i	$-0.504819\pi$
$587$	−0.733495	−0.0302746	−0.0151373	+	0.999885i	$0.504819\pi$
$588$	0	0
$589$	−8.66253	−0.356933
$590$	0	0
$591$	−8.55982	−0.352104
$592$	0	0
$593$	43.2540	1.77623	0.888115	−	0.459621i	$-0.152015\pi$
$593$	43.2540	1.77623	0.888115	+	0.459621i	$0.152015\pi$
$594$	0	0
$595$	−4.58394	−0.187923
$596$	0	0
$597$	−27.5155	−1.12613
$598$	0	0
$599$	21.1031	0.862251	0.431126	−	0.902292i	$-0.358117\pi$
$599$	21.1031	0.862251	0.431126	+	0.902292i	$0.358117\pi$
$600$	0	0
$601$	−16.1436	−0.658512	−0.329256	−	0.944241i	$-0.606798\pi$
$601$	−16.1436	−0.658512	−0.329256	+	0.944241i	$0.606798\pi$
$602$	0	0
$603$	28.6562	1.16697
$604$	0	0
$605$	−1.40548	−0.0571408
$606$	0	0
$607$	32.2628	1.30951	0.654753	−	0.755843i	$-0.272773\pi$
$607$	32.2628	1.30951	0.654753	+	0.755843i	$0.272773\pi$
$608$	0	0
$609$	−21.5580	−0.873574
$610$	0	0
$611$	53.2407	2.15389
$612$	0	0
$613$	−44.6321	−1.80267	−0.901337	−	0.433119i	$-0.857413\pi$
$613$	−44.6321	−1.80267	−0.901337	+	0.433119i	$0.857413\pi$
$614$	0	0
$615$	−8.45200	−0.340818
$616$	0	0
$617$	20.2532	0.815363	0.407682	−	0.913124i	$-0.366337\pi$
$617$	20.2532	0.815363	0.407682	+	0.913124i	$0.366337\pi$
$618$	0	0
$619$	−23.9533	−0.962766	−0.481383	−	0.876510i	$-0.659865\pi$
$619$	−23.9533	−0.962766	−0.481383	+	0.876510i	$0.659865\pi$
$620$	0	0
$621$	−5.69026	−0.228342
$622$	0	0
$623$	−35.0938	−1.40600
$624$	0	0
$625$	−30.9210	−1.23684
$626$	0	0
$627$	−19.8478	−0.792644
$628$	0	0
$629$	−4.49951	−0.179407
$630$	0	0
$631$	3.46481	0.137932	0.0689659	−	0.997619i	$-0.478030\pi$
$631$	3.46481	0.137932	0.0689659	+	0.997619i	$0.478030\pi$
$632$	0	0
$633$	−61.5248	−2.44539
$634$	0	0
$635$	−36.1348	−1.43396
$636$	0	0
$637$	−16.6783	−0.660818
$638$	0	0
$639$	7.89934	0.312493
$640$	0	0
$641$	−28.9401	−1.14307	−0.571533	−	0.820579i	$-0.693651\pi$
$641$	−28.9401	−1.14307	−0.571533	+	0.820579i	$0.693651\pi$
$642$	0	0
$643$	−18.6657	−0.736103	−0.368052	−	0.929805i	$-0.619975\pi$
$643$	−18.6657	−0.736103	−0.368052	+	0.929805i	$0.619975\pi$
$644$	0	0
$645$	52.9888	2.08643
$646$	0	0
$647$	−24.6633	−0.969615	−0.484808	−	0.874621i	$-0.661110\pi$
$647$	−24.6633	−0.969615	−0.484808	+	0.874621i	$0.661110\pi$
$648$	0	0
$649$	−20.4827	−0.804016
$650$	0	0
$651$	16.7574	0.656772
$652$	0	0
$653$	12.4942	0.488937	0.244469	−	0.969657i	$-0.421386\pi$
$653$	12.4942	0.488937	0.244469	+	0.969657i	$0.421386\pi$
$654$	0	0
$655$	−37.6856	−1.47250
$656$	0	0
$657$	−0.671477	−0.0261968
$658$	0	0
$659$	−34.3869	−1.33952	−0.669762	−	0.742576i	$-0.733604\pi$
$659$	−34.3869	−1.33952	−0.669762	+	0.742576i	$0.733604\pi$
$660$	0	0
$661$	19.4957	0.758295	0.379147	−	0.925336i	$-0.376217\pi$
$661$	19.4957	0.758295	0.379147	+	0.925336i	$0.376217\pi$
$662$	0	0
$663$	11.5818	0.449798
$664$	0	0
$665$	13.0380	0.505591
$666$	0	0
$667$	−28.8442	−1.11685
$668$	0	0
$669$	10.4365	0.403500
$670$	0	0
$671$	15.1267	0.583958
$672$	0	0
$673$	−26.1090	−1.00643	−0.503215	−	0.864161i	$-0.667850\pi$
$673$	−26.1090	−1.00643	−0.503215	+	0.864161i	$0.667850\pi$
$674$	0	0
$675$	−1.71855	−0.0661469
$676$	0	0
$677$	−20.5707	−0.790597	−0.395298	−	0.918553i	$-0.629359\pi$
$677$	−20.5707	−0.790597	−0.395298	+	0.918553i	$0.629359\pi$
$678$	0	0
$679$	0.549247	0.0210782
$680$	0	0
$681$	54.7287	2.09721
$682$	0	0
$683$	−23.7002	−0.906864	−0.453432	−	0.891291i	$-0.649801\pi$
$683$	−23.7002	−0.906864	−0.453432	+	0.891291i	$0.649801\pi$
$684$	0	0
$685$	−26.7768	−1.02309
$686$	0	0
$687$	7.61646	0.290586
$688$	0	0
$689$	54.7818	2.08702
$690$	0	0
$691$	7.36209	0.280067	0.140034	−	0.990147i	$-0.455279\pi$
$691$	7.36209	0.280067	0.140034	+	0.990147i	$0.455279\pi$
$692$	0	0
$693$	17.9133	0.680469
$694$	0	0
$695$	21.0635	0.798985
$696$	0	0
$697$	1.17334	0.0444435
$698$	0	0
$699$	16.9135	0.639727
$700$	0	0
$701$	−28.4805	−1.07569	−0.537846	−	0.843043i	$-0.680762\pi$
$701$	−28.4805	−1.07569	−0.537846	+	0.843043i	$0.680762\pi$
$702$	0	0
$703$	12.7978	0.482679
$704$	0	0
$705$	−58.9506	−2.22021
$706$	0	0
$707$	15.0196	0.564870
$708$	0	0
$709$	32.8736	1.23459	0.617297	−	0.786730i	$-0.288228\pi$
$709$	32.8736	1.23459	0.617297	+	0.786730i	$0.288228\pi$
$710$	0	0
$711$	27.0940	1.01610
$712$	0	0
$713$	22.4210	0.839674
$714$	0	0
$715$	−50.3812	−1.88415
$716$	0	0
$717$	−8.68526	−0.324357
$718$	0	0
$719$	13.3988	0.499690	0.249845	−	0.968286i	$-0.419620\pi$
$719$	13.3988	0.499690	0.249845	+	0.968286i	$0.419620\pi$
$720$	0	0
$721$	−29.1778	−1.08664
$722$	0	0
$723$	−2.37146	−0.0881955
$724$	0	0
$725$	−8.71140	−0.323533
$726$	0	0
$727$	−24.5791	−0.911587	−0.455793	−	0.890086i	$-0.650644\pi$
$727$	−24.5791	−0.911587	−0.455793	+	0.890086i	$0.650644\pi$
$728$	0	0
$729$	−19.8574	−0.735459
$730$	0	0
$731$	−7.35613	−0.272076
$732$	0	0
$733$	2.08207	0.0769029	0.0384515	−	0.999260i	$-0.487758\pi$
$733$	2.08207	0.0769029	0.0384515	+	0.999260i	$0.487758\pi$
$734$	0	0
$735$	18.4670	0.681166
$736$	0	0
$737$	37.0916	1.36629
$738$	0	0
$739$	46.1104	1.69620	0.848099	−	0.529838i	$-0.177747\pi$
$739$	46.1104	1.69620	0.848099	+	0.529838i	$0.177747\pi$
$740$	0	0
$741$	−32.9417	−1.21014
$742$	0	0
$743$	−16.5456	−0.606998	−0.303499	−	0.952832i	$-0.598155\pi$
$743$	−16.5456	−0.606998	−0.303499	+	0.952832i	$0.598155\pi$
$744$	0	0
$745$	40.9614	1.50071
$746$	0	0
$747$	44.1190	1.61423
$748$	0	0
$749$	−33.6702	−1.23028
$750$	0	0
$751$	−36.4121	−1.32870	−0.664349	−	0.747423i	$-0.731291\pi$
$751$	−36.4121	−1.32870	−0.664349	+	0.747423i	$0.731291\pi$
$752$	0	0
$753$	−3.35830	−0.122383
$754$	0	0
$755$	−6.41540	−0.233480
$756$	0	0
$757$	16.3953	0.595898	0.297949	−	0.954582i	$-0.403697\pi$
$757$	16.3953	0.595898	0.297949	+	0.954582i	$0.403697\pi$
$758$	0	0
$759$	51.3715	1.86467
$760$	0	0
$761$	44.2450	1.60388	0.801940	−	0.597404i	$-0.203801\pi$
$761$	44.2450	1.60388	0.801940	+	0.597404i	$0.203801\pi$
$762$	0	0
$763$	−12.1010	−0.438087
$764$	0	0
$765$	−5.98303	−0.216317
$766$	0	0
$767$	−33.9954	−1.22750
$768$	0	0
$769$	−21.3439	−0.769680	−0.384840	−	0.922983i	$-0.625743\pi$
$769$	−21.3439	−0.769680	−0.384840	+	0.922983i	$0.625743\pi$
$770$	0	0
$771$	−26.2497	−0.945360
$772$	0	0
$773$	20.7861	0.747623	0.373812	−	0.927505i	$-0.378051\pi$
$773$	20.7861	0.747623	0.373812	+	0.927505i	$0.378051\pi$
$774$	0	0
$775$	6.77150	0.243239
$776$	0	0
$777$	−24.7570	−0.888151
$778$	0	0
$779$	−3.33731	−0.119571
$780$	0	0
$781$	10.2246	0.365866
$782$	0	0
$783$	4.03420	0.144171
$784$	0	0
$785$	−32.0114	−1.14253
$786$	0	0
$787$	28.9072	1.03043	0.515215	−	0.857061i	$-0.327712\pi$
$787$	28.9072	1.03043	0.515215	+	0.857061i	$0.327712\pi$
$788$	0	0
$789$	7.86302	0.279931
$790$	0	0
$791$	38.0433	1.35266
$792$	0	0
$793$	25.1060	0.891539
$794$	0	0
$795$	−60.6570	−2.15128
$796$	0	0
$797$	21.6563	0.767106	0.383553	−	0.923519i	$-0.374700\pi$
$797$	21.6563	0.767106	0.383553	+	0.923519i	$0.374700\pi$
$798$	0	0
$799$	8.18377	0.289521
$800$	0	0
$801$	−45.8050	−1.61844
$802$	0	0
$803$	−0.869138	−0.0306712
$804$	0	0
$805$	−33.7459	−1.18939
$806$	0	0
$807$	−34.0199	−1.19756
$808$	0	0
$809$	4.12546	0.145043	0.0725217	−	0.997367i	$-0.476895\pi$
$809$	4.12546	0.145043	0.0725217	+	0.997367i	$0.476895\pi$
$810$	0	0
$811$	27.4598	0.964243	0.482121	−	0.876104i	$-0.339866\pi$
$811$	27.4598	0.964243	0.482121	+	0.876104i	$0.339866\pi$
$812$	0	0
$813$	3.68145	0.129114
$814$	0	0
$815$	11.1332	0.389980
$816$	0	0
$817$	20.9228	0.731998
$818$	0	0
$819$	29.7309	1.03888
$820$	0	0
$821$	26.7943	0.935127	0.467564	−	0.883959i	$-0.345132\pi$
$821$	26.7943	0.935127	0.467564	+	0.883959i	$0.345132\pi$
$822$	0	0
$823$	−29.5890	−1.03141	−0.515704	−	0.856767i	$-0.672469\pi$
$823$	−29.5890	−1.03141	−0.515704	+	0.856767i	$0.672469\pi$
$824$	0	0
$825$	15.5150	0.540163
$826$	0	0
$827$	41.2969	1.43603	0.718017	−	0.696025i	$-0.245050\pi$
$827$	41.2969	1.43603	0.718017	+	0.696025i	$0.245050\pi$
$828$	0	0
$829$	−3.90138	−0.135501	−0.0677503	−	0.997702i	$-0.521582\pi$
$829$	−3.90138	−0.135501	−0.0677503	+	0.997702i	$0.521582\pi$
$830$	0	0
$831$	42.7073	1.48150
$832$	0	0
$833$	−2.56367	−0.0888258
$834$	0	0
$835$	37.8130	1.30857
$836$	0	0
$837$	−3.13584	−0.108391
$838$	0	0
$839$	−18.4300	−0.636274	−0.318137	−	0.948045i	$-0.603057\pi$
$839$	−18.4300	−0.636274	−0.318137	+	0.948045i	$0.603057\pi$
$840$	0	0
$841$	−8.55043	−0.294842
$842$	0	0
$843$	−27.3847	−0.943179
$844$	0	0
$845$	−49.4051	−1.69959
$846$	0	0
$847$	1.07355	0.0368875
$848$	0	0
$849$	4.38138	0.150368
$850$	0	0
$851$	−33.1243	−1.13549
$852$	0	0
$853$	37.7897	1.29389	0.646947	−	0.762535i	$-0.276046\pi$
$853$	37.7897	1.29389	0.646947	+	0.762535i	$0.276046\pi$
$854$	0	0
$855$	17.0174	0.581982
$856$	0	0
$857$	−58.1926	−1.98782	−0.993910	−	0.110194i	$-0.964853\pi$
$857$	−58.1926	−1.98782	−0.993910	+	0.110194i	$0.964853\pi$
$858$	0	0
$859$	−35.3869	−1.20738	−0.603692	−	0.797218i	$-0.706304\pi$
$859$	−35.3869	−1.20738	−0.603692	+	0.797218i	$0.706304\pi$
$860$	0	0
$861$	6.45590	0.220016
$862$	0	0
$863$	−27.4547	−0.934568	−0.467284	−	0.884107i	$-0.654767\pi$
$863$	−27.4547	−0.934568	−0.467284	+	0.884107i	$0.654767\pi$
$864$	0	0
$865$	−2.84900	−0.0968689
$866$	0	0
$867$	−38.5345	−1.30870
$868$	0	0
$869$	35.0696	1.18965
$870$	0	0
$871$	61.5615	2.08593
$872$	0	0
$873$	0.716886	0.0242629
$874$	0	0
$875$	16.2612	0.549729
$876$	0	0
$877$	−14.8517	−0.501505	−0.250752	−	0.968051i	$-0.580678\pi$
$877$	−14.8517	−0.501505	−0.250752	+	0.968051i	$0.580678\pi$
$878$	0	0
$879$	−7.19056	−0.242532
$880$	0	0
$881$	37.8714	1.27592	0.637960	−	0.770070i	$-0.279779\pi$
$881$	37.8714	1.27592	0.637960	+	0.770070i	$0.279779\pi$
$882$	0	0
$883$	15.3679	0.517171	0.258586	−	0.965988i	$-0.416744\pi$
$883$	15.3679	0.517171	0.258586	+	0.965988i	$0.416744\pi$
$884$	0	0
$885$	37.6414	1.26530
$886$	0	0
$887$	−2.72841	−0.0916111	−0.0458056	−	0.998950i	$-0.514585\pi$
$887$	−2.72841	−0.0916111	−0.0458056	+	0.998950i	$0.514585\pi$
$888$	0	0
$889$	27.6008	0.925703
$890$	0	0
$891$	−33.9178	−1.13629
$892$	0	0
$893$	−23.2769	−0.778931
$894$	0	0
$895$	−63.2259	−2.11341
$896$	0	0
$897$	85.2622	2.84682
$898$	0	0
$899$	−15.8958	−0.530153
$900$	0	0
$901$	8.42066	0.280533
$902$	0	0
$903$	−40.4745	−1.34691
$904$	0	0
$905$	12.4385	0.413471
$906$	0	0
$907$	6.49334	0.215608	0.107804	−	0.994172i	$-0.465618\pi$
$907$	6.49334	0.215608	0.107804	+	0.994172i	$0.465618\pi$
$908$	0	0
$909$	19.6038	0.650218
$910$	0	0
$911$	32.2049	1.06699	0.533497	−	0.845802i	$-0.320877\pi$
$911$	32.2049	1.06699	0.533497	+	0.845802i	$0.320877\pi$
$912$	0	0
$913$	57.1061	1.88994
$914$	0	0
$915$	−27.7985	−0.918991
$916$	0	0
$917$	28.7855	0.950580
$918$	0	0
$919$	25.5609	0.843177	0.421588	−	0.906787i	$-0.361473\pi$
$919$	25.5609	0.843177	0.421588	+	0.906787i	$0.361473\pi$
$920$	0	0
$921$	−19.9367	−0.656936
$922$	0	0
$923$	16.9700	0.558574
$924$	0	0
$925$	−10.0041	−0.328932
$926$	0	0
$927$	−38.0833	−1.25082
$928$	0	0
$929$	−43.9109	−1.44067	−0.720336	−	0.693626i	$-0.756012\pi$
$929$	−43.9109	−1.44067	−0.720336	+	0.693626i	$0.756012\pi$
$930$	0	0
$931$	7.29177	0.238978
$932$	0	0
$933$	−6.97375	−0.228310
$934$	0	0
$935$	−7.74424	−0.253264
$936$	0	0
$937$	37.5543	1.22684	0.613422	−	0.789755i	$-0.289792\pi$
$937$	37.5543	1.22684	0.613422	+	0.789755i	$0.289792\pi$
$938$	0	0
$939$	39.1552	1.27778
$940$	0	0
$941$	−20.6958	−0.674663	−0.337332	−	0.941386i	$-0.609524\pi$
$941$	−20.6958	−0.674663	−0.337332	+	0.941386i	$0.609524\pi$
$942$	0	0
$943$	8.63787	0.281288
$944$	0	0
$945$	4.71976	0.153534
$946$	0	0
$947$	14.0412	0.456276	0.228138	−	0.973629i	$-0.426736\pi$
$947$	14.0412	0.456276	0.228138	+	0.973629i	$0.426736\pi$
$948$	0	0
$949$	−1.44252	−0.0468262
$950$	0	0
$951$	−48.3302	−1.56721
$952$	0	0
$953$	−32.4866	−1.05235	−0.526173	−	0.850378i	$-0.676373\pi$
$953$	−32.4866	−1.05235	−0.526173	+	0.850378i	$0.676373\pi$
$954$	0	0
$955$	−64.0842	−2.07371
$956$	0	0
$957$	−36.4207	−1.17731
$958$	0	0
$959$	20.4530	0.660461
$960$	0	0
$961$	−18.6440	−0.601419
$962$	0	0
$963$	−43.9469	−1.41617
$964$	0	0
$965$	−8.48704	−0.273208
$966$	0	0
$967$	53.7501	1.72849	0.864244	−	0.503073i	$-0.167797\pi$
$967$	53.7501	1.72849	0.864244	+	0.503073i	$0.167797\pi$
$968$	0	0
$969$	−5.06356	−0.162665
$970$	0	0
$971$	−34.1747	−1.09672	−0.548359	−	0.836243i	$-0.684747\pi$
$971$	−34.1747	−1.09672	−0.548359	+	0.836243i	$0.684747\pi$
$972$	0	0
$973$	−16.0890	−0.515789
$974$	0	0
$975$	25.7505	0.824676
$976$	0	0
$977$	−8.90298	−0.284832	−0.142416	−	0.989807i	$-0.545487\pi$
$977$	−8.90298	−0.284832	−0.142416	+	0.989807i	$0.545487\pi$
$978$	0	0
$979$	−59.2885	−1.89487
$980$	0	0
$981$	−15.7945	−0.504278
$982$	0	0
$983$	−22.5713	−0.719912	−0.359956	−	0.932969i	$-0.617208\pi$
$983$	−22.5713	−0.719912	−0.359956	+	0.932969i	$0.617208\pi$
$984$	0	0
$985$	9.49955	0.302681
$986$	0	0
$987$	45.0283	1.43327
$988$	0	0
$989$	−54.1541	−1.72200
$990$	0	0
$991$	−46.7740	−1.48583	−0.742913	−	0.669388i	$-0.766556\pi$
$991$	−46.7740	−1.48583	−0.742913	+	0.669388i	$0.766556\pi$
$992$	0	0
$993$	−21.5452	−0.683717
$994$	0	0
$995$	30.5362	0.968063
$996$	0	0
$997$	−20.7833	−0.658214	−0.329107	−	0.944293i	$-0.606748\pi$
$997$	−20.7833	−0.658214	−0.329107	+	0.944293i	$0.606748\pi$
$998$	0	0
$999$	4.63283	0.146576

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3856.2.a.j.1.6		7
4.3	odd	2		241.2.a.a.1.7	✓	7
12.11	even	2		2169.2.a.e.1.1		7
20.19	odd	2		6025.2.a.f.1.1		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
241.2.a.a.1.7	✓	7	4.3	odd	2
2169.2.a.e.1.1		7	12.11	even	2
3856.2.a.j.1.6		7	1.1	even	1	trivial
6025.2.a.f.1.1		7	20.19	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3856 = 2^{4} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3856.a (trivial)

\(f(q)\)	\(=\)	\(q+2.37146 q^{3} -2.63180 q^{5} +2.01025 q^{7} +2.62382 q^{9} +O(q^{10})\)	\(q+2.37146 q^{3} -2.63180 q^{5} +2.01025 q^{7} +2.62382 q^{9} +3.39618 q^{11} +5.63669 q^{13} -6.24122 q^{15} +0.866432 q^{17} -2.46437 q^{19} +4.76723 q^{21} +6.37847 q^{23} +1.92640 q^{25} -0.892104 q^{27} -4.52212 q^{29} +3.51511 q^{31} +8.05390 q^{33} -5.29060 q^{35} -5.19315 q^{37} +13.3672 q^{39} +1.35422 q^{41} -8.49015 q^{43} -6.90537 q^{45} +9.44537 q^{47} -2.95888 q^{49} +2.05471 q^{51} +9.71877 q^{53} -8.93808 q^{55} -5.84415 q^{57} -6.03110 q^{59} +4.45402 q^{61} +5.27454 q^{63} -14.8347 q^{65} +10.9216 q^{67} +15.1263 q^{69} +3.01063 q^{71} -0.255916 q^{73} +4.56837 q^{75} +6.82718 q^{77} +10.3262 q^{79} -9.98704 q^{81} +16.8148 q^{83} -2.28028 q^{85} -10.7240 q^{87} -17.4574 q^{89} +11.3312 q^{91} +8.33594 q^{93} +6.48574 q^{95} +0.273223 q^{97} +8.91095 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{3} - 8 q^{5} + 7 q^{7} - 2 q^{9}+O(q^{10}) \) 7 * q + 3 * q^3 - 8 * q^5 + 7 * q^7 - 2 * q^9	\( 7 q + 3 q^{3} - 8 q^{5} + 7 q^{7} - 2 q^{9} + 18 q^{11} - q^{13} + 11 q^{15} - 2 q^{17} + 6 q^{19} - 2 q^{21} + 22 q^{23} + 5 q^{25} - 3 q^{27} - 16 q^{29} + 18 q^{31} + 4 q^{33} - 7 q^{35} + 8 q^{37} + 9 q^{39} - 15 q^{41} - 14 q^{43} + 3 q^{45} + 10 q^{47} + 6 q^{49} - 13 q^{51} + 15 q^{53} - 29 q^{55} + 14 q^{57} + 18 q^{59} + 4 q^{61} + 16 q^{63} - 7 q^{65} - 18 q^{67} + 26 q^{69} + 50 q^{71} - 16 q^{75} + 17 q^{77} + 15 q^{79} - 9 q^{81} + 24 q^{83} - 2 q^{85} - 12 q^{87} - 13 q^{89} + 12 q^{91} + 14 q^{93} + 41 q^{95} + q^{97} + 20 q^{99}+O(q^{100}) \) 7 * q + 3 * q^3 - 8 * q^5 + 7 * q^7 - 2 * q^9 + 18 * q^11 - q^13 + 11 * q^15 - 2 * q^17 + 6 * q^19 - 2 * q^21 + 22 * q^23 + 5 * q^25 - 3 * q^27 - 16 * q^29 + 18 * q^31 + 4 * q^33 - 7 * q^35 + 8 * q^37 + 9 * q^39 - 15 * q^41 - 14 * q^43 + 3 * q^45 + 10 * q^47 + 6 * q^49 - 13 * q^51 + 15 * q^53 - 29 * q^55 + 14 * q^57 + 18 * q^59 + 4 * q^61 + 16 * q^63 - 7 * q^65 - 18 * q^67 + 26 * q^69 + 50 * q^71 - 16 * q^75 + 17 * q^77 + 15 * q^79 - 9 * q^81 + 24 * q^83 - 2 * q^85 - 12 * q^87 - 13 * q^89 + 12 * q^91 + 14 * q^93 + 41 * q^95 + q^97 + 20 * q^99

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