LMFDB - Embedded newform 3856.2.a.j.1.3

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			1.3
Root			$-0.911223$ 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)

$f(q)$	$=$	$q+0.186202 q^{3} -2.25110 q^{5} -3.52970 q^{7} -2.96533 q^{9} +O(q^{10})$	$q+0.186202 q^{3} -2.25110 q^{5} -3.52970 q^{7} -2.96533 q^{9} -0.515564 q^{11} -5.38098 q^{13} -0.419160 q^{15} -4.16566 q^{17} -4.92935 q^{19} -0.657237 q^{21} +7.69193 q^{23} +0.0674429 q^{25} -1.11076 q^{27} -8.93755 q^{29} +4.43182 q^{31} -0.0959992 q^{33} +7.94569 q^{35} +5.99816 q^{37} -1.00195 q^{39} -8.99946 q^{41} +1.66336 q^{43} +6.67525 q^{45} -8.55484 q^{47} +5.45876 q^{49} -0.775656 q^{51} +13.1736 q^{53} +1.16059 q^{55} -0.917857 q^{57} -9.25521 q^{59} -10.4203 q^{61} +10.4667 q^{63} +12.1131 q^{65} +3.91715 q^{67} +1.43226 q^{69} +13.6724 q^{71} +11.5529 q^{73} +0.0125580 q^{75} +1.81979 q^{77} +1.43448 q^{79} +8.68916 q^{81} -1.73047 q^{83} +9.37732 q^{85} -1.66419 q^{87} -1.07999 q^{89} +18.9932 q^{91} +0.825214 q^{93} +11.0965 q^{95} -16.0883 q^{97} +1.52882 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.186202	0.107504	0.0537520	−	0.998554i	$-0.482882\pi$
$3$	0.186202	0.107504	0.0537520	+	0.998554i	$0.482882\pi$
$4$	0	0
$5$	−2.25110	−1.00672	−0.503361	−	0.864076i	$-0.667903\pi$
$5$	−2.25110	−1.00672	−0.503361	+	0.864076i	$0.667903\pi$
$6$	0	0
$7$	−3.52970	−1.33410	−0.667050	−	0.745013i	$-0.732443\pi$
$7$	−3.52970	−1.33410	−0.667050	+	0.745013i	$0.732443\pi$
$8$	0	0
$9$	−2.96533	−0.988443
$10$	0	0
$11$	−0.515564	−0.155448	−0.0777242	−	0.996975i	$-0.524765\pi$
$11$	−0.515564	−0.155448	−0.0777242	+	0.996975i	$0.524765\pi$
$12$	0	0
$13$	−5.38098	−1.49242	−0.746208	−	0.665712i	$-0.768128\pi$
$13$	−5.38098	−1.49242	−0.746208	+	0.665712i	$0.768128\pi$
$14$	0	0
$15$	−0.419160	−0.108227
$16$	0	0
$17$	−4.16566	−1.01032	−0.505161	−	0.863025i	$-0.668567\pi$
$17$	−4.16566	−1.01032	−0.505161	+	0.863025i	$0.668567\pi$
$18$	0	0
$19$	−4.92935	−1.13087	−0.565436	−	0.824792i	$-0.691292\pi$
$19$	−4.92935	−1.13087	−0.565436	+	0.824792i	$0.691292\pi$
$20$	0	0
$21$	−0.657237	−0.143421
$22$	0	0
$23$	7.69193	1.60388	0.801940	−	0.597405i	$-0.203802\pi$
$23$	7.69193	1.60388	0.801940	+	0.597405i	$0.203802\pi$
$24$	0	0
$25$	0.0674429	0.0134886
$26$	0	0
$27$	−1.11076	−0.213765
$28$	0	0
$29$	−8.93755	−1.65966	−0.829831	−	0.558015i	$-0.811563\pi$
$29$	−8.93755	−1.65966	−0.829831	+	0.558015i	$0.811563\pi$
$30$	0	0
$31$	4.43182	0.795978	0.397989	−	0.917390i	$-0.369708\pi$
$31$	4.43182	0.795978	0.397989	+	0.917390i	$0.369708\pi$
$32$	0	0
$33$	−0.0959992	−0.0167113
$34$	0	0
$35$	7.94569	1.34307
$36$	0	0
$37$	5.99816	0.986091	0.493046	−	0.870003i	$-0.335884\pi$
$37$	5.99816	0.986091	0.493046	+	0.870003i	$0.335884\pi$
$38$	0	0
$39$	−1.00195	−0.160441
$40$	0	0
$41$	−8.99946	−1.40548	−0.702740	−	0.711447i	$-0.748040\pi$
$41$	−8.99946	−1.40548	−0.702740	+	0.711447i	$0.748040\pi$
$42$	0	0
$43$	1.66336	0.253660	0.126830	−	0.991924i	$-0.459520\pi$
$43$	1.66336	0.253660	0.126830	+	0.991924i	$0.459520\pi$
$44$	0	0
$45$	6.67525	0.995087
$46$	0	0
$47$	−8.55484	−1.24785	−0.623926	−	0.781483i	$-0.714463\pi$
$47$	−8.55484	−1.24785	−0.623926	+	0.781483i	$0.714463\pi$
$48$	0	0
$49$	5.45876	0.779823
$50$	0	0
$51$	−0.775656	−0.108614
$52$	0	0
$53$	13.1736	1.80954	0.904769	−	0.425903i	$-0.140043\pi$
$53$	13.1736	1.80954	0.904769	+	0.425903i	$0.140043\pi$
$54$	0	0
$55$	1.16059	0.156493
$56$	0	0
$57$	−0.917857	−0.121573
$58$	0	0
$59$	−9.25521	−1.20493	−0.602463	−	0.798147i	$-0.705814\pi$
$59$	−9.25521	−1.20493	−0.602463	+	0.798147i	$0.705814\pi$
$60$	0	0
$61$	−10.4203	−1.33418	−0.667090	−	0.744978i	$-0.732460\pi$
$61$	−10.4203	−1.33418	−0.667090	+	0.744978i	$0.732460\pi$
$62$	0	0
$63$	10.4667	1.31868
$64$	0	0
$65$	12.1131	1.50245
$66$	0	0
$67$	3.91715	0.478555	0.239278	−	0.970951i	$-0.423089\pi$
$67$	3.91715	0.478555	0.239278	+	0.970951i	$0.423089\pi$
$68$	0	0
$69$	1.43226	0.172423
$70$	0	0
$71$	13.6724	1.62262	0.811308	−	0.584619i	$-0.198756\pi$
$71$	13.6724	1.62262	0.811308	+	0.584619i	$0.198756\pi$
$72$	0	0
$73$	11.5529	1.35217	0.676083	−	0.736825i	$-0.263676\pi$
$73$	11.5529	1.35217	0.676083	+	0.736825i	$0.263676\pi$
$74$	0	0
$75$	0.0125580	0.00145007
$76$	0	0
$77$	1.81979	0.207384
$78$	0	0
$79$	1.43448	0.161391	0.0806956	−	0.996739i	$-0.474286\pi$
$79$	1.43448	0.161391	0.0806956	+	0.996739i	$0.474286\pi$
$80$	0	0
$81$	8.68916	0.965462
$82$	0	0
$83$	−1.73047	−0.189944	−0.0949720	−	0.995480i	$-0.530276\pi$
$83$	−1.73047	−0.189944	−0.0949720	+	0.995480i	$0.530276\pi$
$84$	0	0
$85$	9.37732	1.01711
$86$	0	0
$87$	−1.66419	−0.178420
$88$	0	0
$89$	−1.07999	−0.114478	−0.0572392	−	0.998360i	$-0.518230\pi$
$89$	−1.07999	−0.114478	−0.0572392	+	0.998360i	$0.518230\pi$
$90$	0	0
$91$	18.9932	1.99103
$92$	0	0
$93$	0.825214	0.0855707
$94$	0	0
$95$	11.0965	1.13847
$96$	0	0
$97$	−16.0883	−1.63352	−0.816759	−	0.576979i	$-0.804231\pi$
$97$	−16.0883	−1.63352	−0.816759	+	0.576979i	$0.804231\pi$
$98$	0	0
$99$	1.52882	0.153652
$100$	0	0
$101$	−1.23922	−0.123307	−0.0616536	−	0.998098i	$-0.519637\pi$
$101$	−1.23922	−0.123307	−0.0616536	+	0.998098i	$0.519637\pi$
$102$	0	0
$103$	−5.27633	−0.519892	−0.259946	−	0.965623i	$-0.583705\pi$
$103$	−5.27633	−0.519892	−0.259946	+	0.965623i	$0.583705\pi$
$104$	0	0
$105$	1.47951	0.144385
$106$	0	0
$107$	−10.7822	−1.04236	−0.521178	−	0.853448i	$-0.674507\pi$
$107$	−10.7822	−1.04236	−0.521178	+	0.853448i	$0.674507\pi$
$108$	0	0
$109$	−1.15255	−0.110394	−0.0551972	−	0.998475i	$-0.517579\pi$
$109$	−1.15255	−0.110394	−0.0551972	+	0.998475i	$0.517579\pi$
$110$	0	0
$111$	1.11687	0.106009
$112$	0	0
$113$	−8.70999	−0.819367	−0.409684	−	0.912228i	$-0.634361\pi$
$113$	−8.70999	−0.819367	−0.409684	+	0.912228i	$0.634361\pi$
$114$	0	0
$115$	−17.3153	−1.61466
$116$	0	0
$117$	15.9564	1.47517
$118$	0	0
$119$	14.7035	1.34787
$120$	0	0
$121$	−10.7342	−0.975836
$122$	0	0
$123$	−1.67572	−0.151095
$124$	0	0
$125$	11.1037	0.993142
$126$	0	0
$127$	8.22063	0.729463	0.364732	−	0.931113i	$-0.381161\pi$
$127$	8.22063	0.729463	0.364732	+	0.931113i	$0.381161\pi$
$128$	0	0
$129$	0.309722	0.0272695
$130$	0	0
$131$	7.30323	0.638086	0.319043	−	0.947740i	$-0.396638\pi$
$131$	7.30323	0.638086	0.319043	+	0.947740i	$0.396638\pi$
$132$	0	0
$133$	17.3991	1.50870
$134$	0	0
$135$	2.50042	0.215202
$136$	0	0
$137$	5.95317	0.508614	0.254307	−	0.967124i	$-0.418153\pi$
$137$	5.95317	0.508614	0.254307	+	0.967124i	$0.418153\pi$
$138$	0	0
$139$	−14.9650	−1.26931	−0.634656	−	0.772795i	$-0.718858\pi$
$139$	−14.9650	−1.26931	−0.634656	+	0.772795i	$0.718858\pi$
$140$	0	0
$141$	−1.59293	−0.134149
$142$	0	0
$143$	2.77424	0.231994
$144$	0	0
$145$	20.1193	1.67082
$146$	0	0
$147$	1.01643	0.0838340
$148$	0	0
$149$	−0.130576	−0.0106972	−0.00534860	−	0.999986i	$-0.501703\pi$
$149$	−0.130576	−0.0106972	−0.00534860	+	0.999986i	$0.501703\pi$
$150$	0	0
$151$	−0.276102	−0.0224688	−0.0112344	−	0.999937i	$-0.503576\pi$
$151$	−0.276102	−0.0224688	−0.0112344	+	0.999937i	$0.503576\pi$
$152$	0	0
$153$	12.3526	0.998645
$154$	0	0
$155$	−9.97646	−0.801328
$156$	0	0
$157$	16.1044	1.28527	0.642636	−	0.766172i	$-0.277841\pi$
$157$	16.1044	1.28527	0.642636	+	0.766172i	$0.277841\pi$
$158$	0	0
$159$	2.45296	0.194532
$160$	0	0
$161$	−27.1502	−2.13973
$162$	0	0
$163$	18.5215	1.45071	0.725357	−	0.688373i	$-0.241675\pi$
$163$	18.5215	1.45071	0.725357	+	0.688373i	$0.241675\pi$
$164$	0	0
$165$	0.216104	0.0168236
$166$	0	0
$167$	−15.5017	−1.19956	−0.599779	−	0.800166i	$-0.704745\pi$
$167$	−15.5017	−1.19956	−0.599779	+	0.800166i	$0.704745\pi$
$168$	0	0
$169$	15.9550	1.22731
$170$	0	0
$171$	14.6172	1.11780
$172$	0	0
$173$	−0.366438	−0.0278597	−0.0139299	−	0.999903i	$-0.504434\pi$
$173$	−0.366438	−0.0278597	−0.0139299	+	0.999903i	$0.504434\pi$
$174$	0	0
$175$	−0.238053	−0.0179951
$176$	0	0
$177$	−1.72334	−0.129534
$178$	0	0
$179$	−12.8719	−0.962090	−0.481045	−	0.876696i	$-0.659743\pi$
$179$	−12.8719	−0.962090	−0.481045	+	0.876696i	$0.659743\pi$
$180$	0	0
$181$	−12.2125	−0.907746	−0.453873	−	0.891066i	$-0.649958\pi$
$181$	−12.2125	−0.907746	−0.453873	+	0.891066i	$0.649958\pi$
$182$	0	0
$183$	−1.94028	−0.143429
$184$	0	0
$185$	−13.5024	−0.992719
$186$	0	0
$187$	2.14767	0.157053
$188$	0	0
$189$	3.92064	0.285184
$190$	0	0
$191$	3.92400	0.283930	0.141965	−	0.989872i	$-0.454658\pi$
$191$	3.92400	0.283930	0.141965	+	0.989872i	$0.454658\pi$
$192$	0	0
$193$	−1.69319	−0.121879	−0.0609393	−	0.998141i	$-0.519410\pi$
$193$	−1.69319	−0.121879	−0.0609393	+	0.998141i	$0.519410\pi$
$194$	0	0
$195$	2.25549	0.161519
$196$	0	0
$197$	−2.20382	−0.157016	−0.0785080	−	0.996913i	$-0.525016\pi$
$197$	−2.20382	−0.157016	−0.0785080	+	0.996913i	$0.525016\pi$
$198$	0	0
$199$	15.5533	1.10255	0.551274	−	0.834324i	$-0.314142\pi$
$199$	15.5533	1.10255	0.551274	+	0.834324i	$0.314142\pi$
$200$	0	0
$201$	0.729381	0.0514466
$202$	0	0
$203$	31.5468	2.21415
$204$	0	0
$205$	20.2587	1.41493
$206$	0	0
$207$	−22.8091	−1.58534
$208$	0	0
$209$	2.54140	0.175792
$210$	0	0
$211$	−6.55996	−0.451606	−0.225803	−	0.974173i	$-0.572501\pi$
$211$	−6.55996	−0.451606	−0.225803	+	0.974173i	$0.572501\pi$
$212$	0	0
$213$	2.54583	0.174438
$214$	0	0
$215$	−3.74439	−0.255365
$216$	0	0
$217$	−15.6430	−1.06191
$218$	0	0
$219$	2.15118	0.145363
$220$	0	0
$221$	22.4154	1.50782
$222$	0	0
$223$	−6.53800	−0.437817	−0.218908	−	0.975745i	$-0.570250\pi$
$223$	−6.53800	−0.437817	−0.218908	+	0.975745i	$0.570250\pi$
$224$	0	0
$225$	−0.199990	−0.0133327
$226$	0	0
$227$	−16.1673	−1.07306	−0.536530	−	0.843881i	$-0.680265\pi$
$227$	−16.1673	−1.07306	−0.536530	+	0.843881i	$0.680265\pi$
$228$	0	0
$229$	0.0208412	0.00137722	0.000688612	−	1.00000i	$-0.499781\pi$
$229$	0.0208412	0.00137722	0.000688612		1.00000i	$0.499781\pi$
$230$	0	0
$231$	0.338848	0.0222946
$232$	0	0
$233$	−0.651845	−0.0427038	−0.0213519	−	0.999772i	$-0.506797\pi$
$233$	−0.651845	−0.0427038	−0.0213519	+	0.999772i	$0.506797\pi$
$234$	0	0
$235$	19.2578	1.25624
$236$	0	0
$237$	0.267103	0.0173502
$238$	0	0
$239$	8.99997	0.582160	0.291080	−	0.956699i	$-0.405985\pi$
$239$	8.99997	0.582160	0.291080	+	0.956699i	$0.405985\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	0	0
$243$	4.95021	0.317556
$244$	0	0
$245$	−12.2882	−0.785064
$246$	0	0
$247$	26.5248	1.68773
$248$	0	0
$249$	−0.322218	−0.0204197
$250$	0	0
$251$	−16.4986	−1.04138	−0.520691	−	0.853745i	$-0.674326\pi$
$251$	−16.4986	−1.04138	−0.520691	+	0.853745i	$0.674326\pi$
$252$	0	0
$253$	−3.96569	−0.249321
$254$	0	0
$255$	1.74608	0.109344
$256$	0	0
$257$	7.99029	0.498420	0.249210	−	0.968449i	$-0.419829\pi$
$257$	7.99029	0.498420	0.249210	+	0.968449i	$0.419829\pi$
$258$	0	0
$259$	−21.1717	−1.31554
$260$	0	0
$261$	26.5028	1.64048
$262$	0	0
$263$	5.47488	0.337595	0.168798	−	0.985651i	$-0.446012\pi$
$263$	5.47488	0.337595	0.168798	+	0.985651i	$0.446012\pi$
$264$	0	0
$265$	−29.6551	−1.82170
$266$	0	0
$267$	−0.201096	−0.0123069
$268$	0	0
$269$	−2.34697	−0.143097	−0.0715486	−	0.997437i	$-0.522794\pi$
$269$	−2.34697	−0.143097	−0.0715486	+	0.997437i	$0.522794\pi$
$270$	0	0
$271$	−1.60301	−0.0973758	−0.0486879	−	0.998814i	$-0.515504\pi$
$271$	−1.60301	−0.0973758	−0.0486879	+	0.998814i	$0.515504\pi$
$272$	0	0
$273$	3.53658	0.214044
$274$	0	0
$275$	−0.0347711	−0.00209678
$276$	0	0
$277$	−21.6775	−1.30247	−0.651237	−	0.758875i	$-0.725750\pi$
$277$	−21.6775	−1.30247	−0.651237	+	0.758875i	$0.725750\pi$
$278$	0	0
$279$	−13.1418	−0.786779
$280$	0	0
$281$	26.9010	1.60478	0.802388	−	0.596802i	$-0.203562\pi$
$281$	26.9010	1.60478	0.802388	+	0.596802i	$0.203562\pi$
$282$	0	0
$283$	−3.89230	−0.231373	−0.115687	−	0.993286i	$-0.536907\pi$
$283$	−3.89230	−0.231373	−0.115687	+	0.993286i	$0.536907\pi$
$284$	0	0
$285$	2.06619	0.122390
$286$	0	0
$287$	31.7654	1.87505
$288$	0	0
$289$	0.352752	0.0207501
$290$	0	0
$291$	−2.99567	−0.175610
$292$	0	0
$293$	−5.57389	−0.325630	−0.162815	−	0.986657i	$-0.552057\pi$
$293$	−5.57389	−0.325630	−0.162815	+	0.986657i	$0.552057\pi$
$294$	0	0
$295$	20.8344	1.21302
$296$	0	0
$297$	0.572667	0.0332295
$298$	0	0
$299$	−41.3902	−2.39366
$300$	0	0
$301$	−5.87116	−0.338408
$302$	0	0
$303$	−0.230746	−0.0132560
$304$	0	0
$305$	23.4571	1.34315
$306$	0	0
$307$	−23.0795	−1.31722	−0.658608	−	0.752486i	$-0.728854\pi$
$307$	−23.0795	−1.31722	−0.658608	+	0.752486i	$0.728854\pi$
$308$	0	0
$309$	−0.982465	−0.0558905
$310$	0	0
$311$	14.9884	0.849916	0.424958	−	0.905213i	$-0.360289\pi$
$311$	14.9884	0.849916	0.424958	+	0.905213i	$0.360289\pi$
$312$	0	0
$313$	−2.31082	−0.130615	−0.0653076	−	0.997865i	$-0.520803\pi$
$313$	−2.31082	−0.130615	−0.0653076	+	0.997865i	$0.520803\pi$
$314$	0	0
$315$	−23.5616	−1.32755
$316$	0	0
$317$	−18.3630	−1.03137	−0.515684	−	0.856779i	$-0.672462\pi$
$317$	−18.3630	−1.03137	−0.515684	+	0.856779i	$0.672462\pi$
$318$	0	0
$319$	4.60788	0.257992
$320$	0	0
$321$	−2.00767	−0.112057
$322$	0	0
$323$	20.5340	1.14254
$324$	0	0
$325$	−0.362909	−0.0201306
$326$	0	0
$327$	−0.214608	−0.0118678
$328$	0	0
$329$	30.1960	1.66476
$330$	0	0
$331$	6.47933	0.356136	0.178068	−	0.984018i	$-0.443015\pi$
$331$	6.47933	0.356136	0.178068	+	0.984018i	$0.443015\pi$
$332$	0	0
$333$	−17.7865	−0.974695
$334$	0	0
$335$	−8.81788	−0.481772
$336$	0	0
$337$	−3.54257	−0.192976	−0.0964880	−	0.995334i	$-0.530761\pi$
$337$	−3.54257	−0.192976	−0.0964880	+	0.995334i	$0.530761\pi$
$338$	0	0
$339$	−1.62182	−0.0880852
$340$	0	0
$341$	−2.28489	−0.123734
$342$	0	0
$343$	5.44011	0.293739
$344$	0	0
$345$	−3.22415	−0.173582
$346$	0	0
$347$	14.4844	0.777566	0.388783	−	0.921329i	$-0.372896\pi$
$347$	14.4844	0.777566	0.388783	+	0.921329i	$0.372896\pi$
$348$	0	0
$349$	−23.6287	−1.26482	−0.632408	−	0.774636i	$-0.717933\pi$
$349$	−23.6287	−1.26482	−0.632408	+	0.774636i	$0.717933\pi$
$350$	0	0
$351$	5.97697	0.319027
$352$	0	0
$353$	−18.2662	−0.972209	−0.486105	−	0.873901i	$-0.661583\pi$
$353$	−18.2662	−0.972209	−0.486105	+	0.873901i	$0.661583\pi$
$354$	0	0
$355$	−30.7779	−1.63352
$356$	0	0
$357$	2.73783	0.144901
$358$	0	0
$359$	20.3758	1.07539	0.537697	−	0.843138i	$-0.319294\pi$
$359$	20.3758	1.07539	0.537697	+	0.843138i	$0.319294\pi$
$360$	0	0
$361$	5.29852	0.278870
$362$	0	0
$363$	−1.99873	−0.104906
$364$	0	0
$365$	−26.0067	−1.36125
$366$	0	0
$367$	1.76412	0.0920862	0.0460431	−	0.998939i	$-0.485339\pi$
$367$	1.76412	0.0920862	0.0460431	+	0.998939i	$0.485339\pi$
$368$	0	0
$369$	26.6864	1.38924
$370$	0	0
$371$	−46.4989	−2.41410
$372$	0	0
$373$	−32.1736	−1.66589	−0.832943	−	0.553359i	$-0.813346\pi$
$373$	−32.1736	−1.66589	−0.832943	+	0.553359i	$0.813346\pi$
$374$	0	0
$375$	2.06753	0.106767
$376$	0	0
$377$	48.0928	2.47691
$378$	0	0
$379$	31.3680	1.61126	0.805632	−	0.592416i	$-0.201826\pi$
$379$	31.3680	1.61126	0.805632	+	0.592416i	$0.201826\pi$
$380$	0	0
$381$	1.53070	0.0784201
$382$	0	0
$383$	−16.8623	−0.861624	−0.430812	−	0.902442i	$-0.641773\pi$
$383$	−16.8623	−0.861624	−0.430812	+	0.902442i	$0.641773\pi$
$384$	0	0
$385$	−4.09651	−0.208778
$386$	0	0
$387$	−4.93242	−0.250729
$388$	0	0
$389$	−0.816001	−0.0413729	−0.0206864	−	0.999786i	$-0.506585\pi$
$389$	−0.816001	−0.0413729	−0.0206864	+	0.999786i	$0.506585\pi$
$390$	0	0
$391$	−32.0420	−1.62043
$392$	0	0
$393$	1.35988	0.0685968
$394$	0	0
$395$	−3.22915	−0.162476
$396$	0	0
$397$	−21.8279	−1.09551	−0.547755	−	0.836639i	$-0.684517\pi$
$397$	−21.8279	−1.09551	−0.547755	+	0.836639i	$0.684517\pi$
$398$	0	0
$399$	3.23976	0.162191
$400$	0	0
$401$	−15.4857	−0.773321	−0.386661	−	0.922222i	$-0.626372\pi$
$401$	−15.4857	−0.773321	−0.386661	+	0.922222i	$0.626372\pi$
$402$	0	0
$403$	−23.8475	−1.18793
$404$	0	0
$405$	−19.5602	−0.971952
$406$	0	0
$407$	−3.09244	−0.153286
$408$	0	0
$409$	26.0377	1.28748	0.643740	−	0.765245i	$-0.277382\pi$
$409$	26.0377	1.28748	0.643740	+	0.765245i	$0.277382\pi$
$410$	0	0
$411$	1.10849	0.0546780
$412$	0	0
$413$	32.6681	1.60749
$414$	0	0
$415$	3.89546	0.191221
$416$	0	0
$417$	−2.78651	−0.136456
$418$	0	0
$419$	33.2204	1.62292	0.811462	−	0.584406i	$-0.198672\pi$
$419$	33.2204	1.62292	0.811462	+	0.584406i	$0.198672\pi$
$420$	0	0
$421$	2.90586	0.141623	0.0708115	−	0.997490i	$-0.477441\pi$
$421$	2.90586	0.141623	0.0708115	+	0.997490i	$0.477441\pi$
$422$	0	0
$423$	25.3679	1.23343
$424$	0	0
$425$	−0.280944	−0.0136278
$426$	0	0
$427$	36.7804	1.77993
$428$	0	0
$429$	0.516570	0.0249402
$430$	0	0
$431$	28.4892	1.37228	0.686139	−	0.727471i	$-0.259304\pi$
$431$	28.4892	1.37228	0.686139	+	0.727471i	$0.259304\pi$
$432$	0	0
$433$	−28.9311	−1.39034	−0.695169	−	0.718846i	$-0.744671\pi$
$433$	−28.9311	−1.39034	−0.695169	+	0.718846i	$0.744671\pi$
$434$	0	0
$435$	3.74626	0.179619
$436$	0	0
$437$	−37.9163	−1.81378
$438$	0	0
$439$	8.92156	0.425803	0.212901	−	0.977074i	$-0.431709\pi$
$439$	8.92156	0.425803	0.212901	+	0.977074i	$0.431709\pi$
$440$	0	0
$441$	−16.1870	−0.770810
$442$	0	0
$443$	−24.8216	−1.17931	−0.589656	−	0.807655i	$-0.700736\pi$
$443$	−24.8216	−1.17931	−0.589656	+	0.807655i	$0.700736\pi$
$444$	0	0
$445$	2.43116	0.115248
$446$	0	0
$447$	−0.0243136	−0.00114999
$448$	0	0
$449$	11.9254	0.562793	0.281397	−	0.959592i	$-0.409202\pi$
$449$	11.9254	0.562793	0.281397	+	0.959592i	$0.409202\pi$
$450$	0	0
$451$	4.63980	0.218480
$452$	0	0
$453$	−0.0514107	−0.00241549
$454$	0	0
$455$	−42.7557	−2.00442
$456$	0	0
$457$	8.44205	0.394902	0.197451	−	0.980313i	$-0.436734\pi$
$457$	8.44205	0.394902	0.197451	+	0.980313i	$0.436734\pi$
$458$	0	0
$459$	4.62704	0.215972
$460$	0	0
$461$	23.2925	1.08484	0.542419	−	0.840108i	$-0.317509\pi$
$461$	23.2925	1.08484	0.542419	+	0.840108i	$0.317509\pi$
$462$	0	0
$463$	−1.66578	−0.0774152	−0.0387076	−	0.999251i	$-0.512324\pi$
$463$	−1.66578	−0.0774152	−0.0387076	+	0.999251i	$0.512324\pi$
$464$	0	0
$465$	−1.85764	−0.0861459
$466$	0	0
$467$	6.66428	0.308386	0.154193	−	0.988041i	$-0.450722\pi$
$467$	6.66428	0.308386	0.154193	+	0.988041i	$0.450722\pi$
$468$	0	0
$469$	−13.8263	−0.638441
$470$	0	0
$471$	2.99868	0.138172
$472$	0	0
$473$	−0.857570	−0.0394311
$474$	0	0
$475$	−0.332450	−0.0152538
$476$	0	0
$477$	−39.0642	−1.78863
$478$	0	0
$479$	13.6902	0.625523	0.312762	−	0.949832i	$-0.398746\pi$
$479$	13.6902	0.625523	0.312762	+	0.949832i	$0.398746\pi$
$480$	0	0
$481$	−32.2760	−1.47166
$482$	0	0
$483$	−5.05543	−0.230030
$484$	0	0
$485$	36.2163	1.64450
$486$	0	0
$487$	−31.9634	−1.44840	−0.724200	−	0.689590i	$-0.757791\pi$
$487$	−31.9634	−1.44840	−0.724200	+	0.689590i	$0.757791\pi$
$488$	0	0
$489$	3.44874	0.155957
$490$	0	0
$491$	10.0550	0.453777	0.226889	−	0.973921i	$-0.427145\pi$
$491$	10.0550	0.453777	0.226889	+	0.973921i	$0.427145\pi$
$492$	0	0
$493$	37.2308	1.67679
$494$	0	0
$495$	−3.44152	−0.154685
$496$	0	0
$497$	−48.2594	−2.16473
$498$	0	0
$499$	−9.14295	−0.409295	−0.204647	−	0.978836i	$-0.565605\pi$
$499$	−9.14295	−0.409295	−0.204647	+	0.978836i	$0.565605\pi$
$500$	0	0
$501$	−2.88645	−0.128957
$502$	0	0
$503$	27.2050	1.21301	0.606506	−	0.795079i	$-0.292571\pi$
$503$	27.2050	1.21301	0.606506	+	0.795079i	$0.292571\pi$
$504$	0	0
$505$	2.78961	0.124136
$506$	0	0
$507$	2.97086	0.131940
$508$	0	0
$509$	7.79598	0.345551	0.172775	−	0.984961i	$-0.444726\pi$
$509$	7.79598	0.345551	0.172775	+	0.984961i	$0.444726\pi$
$510$	0	0
$511$	−40.7783	−1.80392
$512$	0	0
$513$	5.47532	0.241741
$514$	0	0
$515$	11.8775	0.523387
$516$	0	0
$517$	4.41057	0.193977
$518$	0	0
$519$	−0.0682315	−0.00299503
$520$	0	0
$521$	32.6959	1.43243	0.716216	−	0.697878i	$-0.245872\pi$
$521$	32.6959	1.43243	0.716216	+	0.697878i	$0.245872\pi$
$522$	0	0
$523$	7.80136	0.341129	0.170565	−	0.985346i	$-0.445441\pi$
$523$	7.80136	0.341129	0.170565	+	0.985346i	$0.445441\pi$
$524$	0	0
$525$	−0.0443260	−0.00193454
$526$	0	0
$527$	−18.4615	−0.804194
$528$	0	0
$529$	36.1658	1.57243
$530$	0	0
$531$	27.4447	1.19100
$532$	0	0
$533$	48.4260	2.09756
$534$	0	0
$535$	24.2718	1.04936
$536$	0	0
$537$	−2.39677	−0.103428
$538$	0	0
$539$	−2.81434	−0.121222
$540$	0	0
$541$	22.3971	0.962926	0.481463	−	0.876467i	$-0.340106\pi$
$541$	22.3971	0.962926	0.481463	+	0.876467i	$0.340106\pi$
$542$	0	0
$543$	−2.27399	−0.0975863
$544$	0	0
$545$	2.59451	0.111136
$546$	0	0
$547$	−37.2322	−1.59193	−0.795966	−	0.605341i	$-0.793037\pi$
$547$	−37.2322	−1.59193	−0.795966	+	0.605341i	$0.793037\pi$
$548$	0	0
$549$	30.8995	1.31876
$550$	0	0
$551$	44.0563	1.87686
$552$	0	0
$553$	−5.06326	−0.215312
$554$	0	0
$555$	−2.51419	−0.106721
$556$	0	0
$557$	−27.6457	−1.17139	−0.585693	−	0.810533i	$-0.699177\pi$
$557$	−27.6457	−1.17139	−0.585693	+	0.810533i	$0.699177\pi$
$558$	0	0
$559$	−8.95053	−0.378567
$560$	0	0
$561$	0.399900	0.0168838
$562$	0	0
$563$	23.3454	0.983891	0.491945	−	0.870626i	$-0.336286\pi$
$563$	23.3454	0.983891	0.491945	+	0.870626i	$0.336286\pi$
$564$	0	0
$565$	19.6070	0.824875
$566$	0	0
$567$	−30.6701	−1.28802
$568$	0	0
$569$	2.61346	0.109562	0.0547811	−	0.998498i	$-0.482554\pi$
$569$	2.61346	0.109562	0.0547811	+	0.998498i	$0.482554\pi$
$570$	0	0
$571$	−16.9259	−0.708326	−0.354163	−	0.935184i	$-0.615234\pi$
$571$	−16.9259	−0.708326	−0.354163	+	0.935184i	$0.615234\pi$
$572$	0	0
$573$	0.730657	0.0305236
$574$	0	0
$575$	0.518766	0.0216340
$576$	0	0
$577$	4.67534	0.194637	0.0973185	−	0.995253i	$-0.468973\pi$
$577$	4.67534	0.194637	0.0973185	+	0.995253i	$0.468973\pi$
$578$	0	0
$579$	−0.315276	−0.0131024
$580$	0	0
$581$	6.10804	0.253404
$582$	0	0
$583$	−6.79185	−0.281290
$584$	0	0
$585$	−35.9194	−1.48508
$586$	0	0
$587$	3.78940	0.156405	0.0782027	−	0.996937i	$-0.475082\pi$
$587$	3.78940	0.156405	0.0782027	+	0.996937i	$0.475082\pi$
$588$	0	0
$589$	−21.8460	−0.900148
$590$	0	0
$591$	−0.410357	−0.0168798
$592$	0	0
$593$	−6.81434	−0.279831	−0.139916	−	0.990163i	$-0.544683\pi$
$593$	−6.81434	−0.279831	−0.139916	+	0.990163i	$0.544683\pi$
$594$	0	0
$595$	−33.0991	−1.35693
$596$	0	0
$597$	2.89607	0.118528
$598$	0	0
$599$	39.7739	1.62512	0.812559	−	0.582879i	$-0.198074\pi$
$599$	39.7739	1.62512	0.812559	+	0.582879i	$0.198074\pi$
$600$	0	0
$601$	27.4256	1.11871	0.559356	−	0.828927i	$-0.311048\pi$
$601$	27.4256	1.11871	0.559356	+	0.828927i	$0.311048\pi$
$602$	0	0
$603$	−11.6156	−0.473025
$604$	0	0
$605$	24.1637	0.982395
$606$	0	0
$607$	−20.6168	−0.836811	−0.418405	−	0.908260i	$-0.637411\pi$
$607$	−20.6168	−0.836811	−0.418405	+	0.908260i	$0.637411\pi$
$608$	0	0
$609$	5.87409	0.238030
$610$	0	0
$611$	46.0335	1.86232
$612$	0	0
$613$	−12.9127	−0.521540	−0.260770	−	0.965401i	$-0.583976\pi$
$613$	−12.9127	−0.521540	−0.260770	+	0.965401i	$0.583976\pi$
$614$	0	0
$615$	3.77221	0.152110
$616$	0	0
$617$	26.3856	1.06225	0.531123	−	0.847295i	$-0.321770\pi$
$617$	26.3856	1.06225	0.531123	+	0.847295i	$0.321770\pi$
$618$	0	0
$619$	2.30850	0.0927866	0.0463933	−	0.998923i	$-0.485227\pi$
$619$	2.30850	0.0927866	0.0463933	+	0.998923i	$0.485227\pi$
$620$	0	0
$621$	−8.54387	−0.342854
$622$	0	0
$623$	3.81203	0.152726
$624$	0	0
$625$	−25.3327	−1.01331
$626$	0	0
$627$	0.473214	0.0188983
$628$	0	0
$629$	−24.9863	−0.996270
$630$	0	0
$631$	8.60820	0.342687	0.171343	−	0.985211i	$-0.445189\pi$
$631$	8.60820	0.342687	0.171343	+	0.985211i	$0.445189\pi$
$632$	0	0
$633$	−1.22148	−0.0485494
$634$	0	0
$635$	−18.5054	−0.734366
$636$	0	0
$637$	−29.3735	−1.16382
$638$	0	0
$639$	−40.5432	−1.60386
$640$	0	0
$641$	−11.7479	−0.464014	−0.232007	−	0.972714i	$-0.574529\pi$
$641$	−11.7479	−0.464014	−0.232007	+	0.972714i	$0.574529\pi$
$642$	0	0
$643$	−15.0808	−0.594729	−0.297364	−	0.954764i	$-0.596108\pi$
$643$	−15.0808	−0.594729	−0.297364	+	0.954764i	$0.596108\pi$
$644$	0	0
$645$	−0.697214	−0.0274528
$646$	0	0
$647$	5.68014	0.223309	0.111655	−	0.993747i	$-0.464385\pi$
$647$	5.68014	0.223309	0.111655	+	0.993747i	$0.464385\pi$
$648$	0	0
$649$	4.77166	0.187304
$650$	0	0
$651$	−2.91276	−0.114160
$652$	0	0
$653$	−48.9941	−1.91729	−0.958644	−	0.284608i	$-0.908137\pi$
$653$	−48.9941	−1.91729	−0.958644	+	0.284608i	$0.908137\pi$
$654$	0	0
$655$	−16.4403	−0.642375
$656$	0	0
$657$	−34.2582	−1.33654
$658$	0	0
$659$	23.5789	0.918505	0.459252	−	0.888306i	$-0.348117\pi$
$659$	23.5789	0.918505	0.459252	+	0.888306i	$0.348117\pi$
$660$	0	0
$661$	−38.6322	−1.50262	−0.751310	−	0.659949i	$-0.770578\pi$
$661$	−38.6322	−1.50262	−0.751310	+	0.659949i	$0.770578\pi$
$662$	0	0
$663$	4.17379	0.162097
$664$	0	0
$665$	−39.1671	−1.51884
$666$	0	0
$667$	−68.7470	−2.66190
$668$	0	0
$669$	−1.21739	−0.0470670
$670$	0	0
$671$	5.37232	0.207396
$672$	0	0
$673$	−37.4530	−1.44371	−0.721853	−	0.692047i	$-0.756709\pi$
$673$	−37.4530	−1.44371	−0.721853	+	0.692047i	$0.756709\pi$
$674$	0	0
$675$	−0.0749127	−0.00288339
$676$	0	0
$677$	16.0616	0.617297	0.308649	−	0.951176i	$-0.400123\pi$
$677$	16.0616	0.617297	0.308649	+	0.951176i	$0.400123\pi$
$678$	0	0
$679$	56.7868	2.17928
$680$	0	0
$681$	−3.01038	−0.115358
$682$	0	0
$683$	−13.5819	−0.519696	−0.259848	−	0.965649i	$-0.583673\pi$
$683$	−13.5819	−0.519696	−0.259848	+	0.965649i	$0.583673\pi$
$684$	0	0
$685$	−13.4012	−0.512032
$686$	0	0
$687$	0.00388067	0.000148057 0
$688$	0	0
$689$	−70.8871	−2.70058
$690$	0	0
$691$	5.72328	0.217724	0.108862	−	0.994057i	$-0.465279\pi$
$691$	5.72328	0.217724	0.108862	+	0.994057i	$0.465279\pi$
$692$	0	0
$693$	−5.39626	−0.204987
$694$	0	0
$695$	33.6876	1.27784
$696$	0	0
$697$	37.4887	1.41999
$698$	0	0
$699$	−0.121375	−0.00459083
$700$	0	0
$701$	−20.2921	−0.766421	−0.383211	−	0.923661i	$-0.625182\pi$
$701$	−20.2921	−0.766421	−0.383211	+	0.923661i	$0.625182\pi$
$702$	0	0
$703$	−29.5670	−1.11514
$704$	0	0
$705$	3.58584	0.135051
$706$	0	0
$707$	4.37408	0.164504
$708$	0	0
$709$	−31.5398	−1.18450	−0.592251	−	0.805754i	$-0.701761\pi$
$709$	−31.5398	−1.18450	−0.592251	+	0.805754i	$0.701761\pi$
$710$	0	0
$711$	−4.25369	−0.159526
$712$	0	0
$713$	34.0892	1.27665
$714$	0	0
$715$	−6.24509	−0.233553
$716$	0	0
$717$	1.67581	0.0625844
$718$	0	0
$719$	11.0676	0.412753	0.206376	−	0.978473i	$-0.433833\pi$
$719$	11.0676	0.412753	0.206376	+	0.978473i	$0.433833\pi$
$720$	0	0
$721$	18.6238	0.693588
$722$	0	0
$723$	−0.186202	−0.00692494
$724$	0	0
$725$	−0.602774	−0.0223865
$726$	0	0
$727$	49.5631	1.83819	0.919096	−	0.394033i	$-0.128921\pi$
$727$	49.5631	1.83819	0.919096	+	0.394033i	$0.128921\pi$
$728$	0	0
$729$	−25.1457	−0.931324
$730$	0	0
$731$	−6.92901	−0.256279
$732$	0	0
$733$	−2.44508	−0.0903110	−0.0451555	−	0.998980i	$-0.514378\pi$
$733$	−2.44508	−0.0903110	−0.0451555	+	0.998980i	$0.514378\pi$
$734$	0	0
$735$	−2.28809	−0.0843975
$736$	0	0
$737$	−2.01954	−0.0743907
$738$	0	0
$739$	−36.9067	−1.35764	−0.678818	−	0.734307i	$-0.737508\pi$
$739$	−36.9067	−1.35764	−0.678818	+	0.734307i	$0.737508\pi$
$740$	0	0
$741$	4.93897	0.181438
$742$	0	0
$743$	−40.6188	−1.49016	−0.745079	−	0.666976i	$-0.767588\pi$
$743$	−40.6188	−1.49016	−0.745079	+	0.666976i	$0.767588\pi$
$744$	0	0
$745$	0.293940	0.0107691
$746$	0	0
$747$	5.13142	0.187749
$748$	0	0
$749$	38.0579	1.39061
$750$	0	0
$751$	14.0406	0.512349	0.256174	−	0.966631i	$-0.417538\pi$
$751$	14.0406	0.512349	0.256174	+	0.966631i	$0.417538\pi$
$752$	0	0
$753$	−3.07207	−0.111953
$754$	0	0
$755$	0.621532	0.0226199
$756$	0	0
$757$	24.6936	0.897505	0.448752	−	0.893656i	$-0.351869\pi$
$757$	24.6936	0.897505	0.448752	+	0.893656i	$0.351869\pi$
$758$	0	0
$759$	−0.738419	−0.0268029
$760$	0	0
$761$	12.6083	0.457051	0.228525	−	0.973538i	$-0.426610\pi$
$761$	12.6083	0.457051	0.228525	+	0.973538i	$0.426610\pi$
$762$	0	0
$763$	4.06816	0.147277
$764$	0	0
$765$	−27.8068	−1.00536
$766$	0	0
$767$	49.8021	1.79825
$768$	0	0
$769$	42.1413	1.51965	0.759827	−	0.650125i	$-0.225283\pi$
$769$	42.1413	1.51965	0.759827	+	0.650125i	$0.225283\pi$
$770$	0	0
$771$	1.48781	0.0535821
$772$	0	0
$773$	−9.62659	−0.346244	−0.173122	−	0.984900i	$-0.555386\pi$
$773$	−9.62659	−0.346244	−0.173122	+	0.984900i	$0.555386\pi$
$774$	0	0
$775$	0.298895	0.0107366
$776$	0	0
$777$	−3.94221	−0.141426
$778$	0	0
$779$	44.3615	1.58942
$780$	0	0
$781$	−7.04900	−0.252233
$782$	0	0
$783$	9.92745	0.354778
$784$	0	0
$785$	−36.2526	−1.29391
$786$	0	0
$787$	−50.4413	−1.79804	−0.899019	−	0.437909i	$-0.855719\pi$
$787$	−50.4413	−1.79804	−0.899019	+	0.437909i	$0.855719\pi$
$788$	0	0
$789$	1.01943	0.0362928
$790$	0	0
$791$	30.7436	1.09312
$792$	0	0
$793$	56.0713	1.99115
$794$	0	0
$795$	−5.52185	−0.195840
$796$	0	0
$797$	34.3043	1.21512	0.607561	−	0.794273i	$-0.292148\pi$
$797$	34.3043	1.21512	0.607561	+	0.794273i	$0.292148\pi$
$798$	0	0
$799$	35.6366	1.26073
$800$	0	0
$801$	3.20252	0.113155
$802$	0	0
$803$	−5.95627	−0.210192
$804$	0	0
$805$	61.1177	2.15412
$806$	0	0
$807$	−0.437011	−0.0153835
$808$	0	0
$809$	−43.9886	−1.54656	−0.773279	−	0.634066i	$-0.781385\pi$
$809$	−43.9886	−1.54656	−0.773279	+	0.634066i	$0.781385\pi$
$810$	0	0
$811$	43.0270	1.51088	0.755440	−	0.655217i	$-0.227423\pi$
$811$	43.0270	1.51088	0.755440	+	0.655217i	$0.227423\pi$
$812$	0	0
$813$	−0.298484	−0.0104683
$814$	0	0
$815$	−41.6937	−1.46047
$816$	0	0
$817$	−8.19930	−0.286857
$818$	0	0
$819$	−56.3212	−1.96802
$820$	0	0
$821$	18.7704	0.655090	0.327545	−	0.944836i	$-0.393779\pi$
$821$	18.7704	0.655090	0.327545	+	0.944836i	$0.393779\pi$
$822$	0	0
$823$	−27.1588	−0.946695	−0.473348	−	0.880876i	$-0.656955\pi$
$823$	−27.1588	−0.946695	−0.473348	+	0.880876i	$0.656955\pi$
$824$	0	0
$825$	−0.00647446	−0.000225412 0
$826$	0	0
$827$	−41.5157	−1.44364	−0.721821	−	0.692079i	$-0.756695\pi$
$827$	−41.5157	−1.44364	−0.721821	+	0.692079i	$0.756695\pi$
$828$	0	0
$829$	−30.6349	−1.06399	−0.531997	−	0.846746i	$-0.678558\pi$
$829$	−30.6349	−1.06399	−0.531997	+	0.846746i	$0.678558\pi$
$830$	0	0
$831$	−4.03640	−0.140021
$832$	0	0
$833$	−22.7394	−0.787872
$834$	0	0
$835$	34.8959	1.20762
$836$	0	0
$837$	−4.92267	−0.170153
$838$	0	0
$839$	−30.2456	−1.04419	−0.522097	−	0.852886i	$-0.674850\pi$
$839$	−30.2456	−1.04419	−0.522097	+	0.852886i	$0.674850\pi$
$840$	0	0
$841$	50.8798	1.75447
$842$	0	0
$843$	5.00902	0.172520
$844$	0	0
$845$	−35.9163	−1.23556
$846$	0	0
$847$	37.8884	1.30186
$848$	0	0
$849$	−0.724755	−0.0248735
$850$	0	0
$851$	46.1374	1.58157
$852$	0	0
$853$	42.7734	1.46453	0.732267	−	0.681018i	$-0.238462\pi$
$853$	42.7734	1.46453	0.732267	+	0.681018i	$0.238462\pi$
$854$	0	0
$855$	−32.9046	−1.12532
$856$	0	0
$857$	−18.8048	−0.642361	−0.321181	−	0.947018i	$-0.604080\pi$
$857$	−18.8048	−0.642361	−0.321181	+	0.947018i	$0.604080\pi$
$858$	0	0
$859$	8.22676	0.280693	0.140347	−	0.990102i	$-0.455178\pi$
$859$	8.22676	0.280693	0.140347	+	0.990102i	$0.455178\pi$
$860$	0	0
$861$	5.91478	0.201575
$862$	0	0
$863$	41.8464	1.42447	0.712235	−	0.701941i	$-0.247683\pi$
$863$	41.8464	1.42447	0.712235	+	0.701941i	$0.247683\pi$
$864$	0	0
$865$	0.824887	0.0280470
$866$	0	0
$867$	0.0656832	0.00223072
$868$	0	0
$869$	−0.739564	−0.0250880
$870$	0	0
$871$	−21.0781	−0.714204
$872$	0	0
$873$	47.7071	1.61464
$874$	0	0
$875$	−39.1926	−1.32495
$876$	0	0
$877$	−6.11483	−0.206483	−0.103242	−	0.994656i	$-0.532921\pi$
$877$	−6.11483	−0.206483	−0.103242	+	0.994656i	$0.532921\pi$
$878$	0	0
$879$	−1.03787	−0.0350065
$880$	0	0
$881$	−13.5157	−0.455355	−0.227678	−	0.973737i	$-0.573113\pi$
$881$	−13.5157	−0.455355	−0.227678	+	0.973737i	$0.573113\pi$
$882$	0	0
$883$	−38.5489	−1.29727	−0.648637	−	0.761098i	$-0.724661\pi$
$883$	−38.5489	−1.29727	−0.648637	+	0.761098i	$0.724661\pi$
$884$	0	0
$885$	3.87941	0.130405
$886$	0	0
$887$	−40.8386	−1.37123	−0.685613	−	0.727966i	$-0.740466\pi$
$887$	−40.8386	−1.37123	−0.685613	+	0.727966i	$0.740466\pi$
$888$	0	0
$889$	−29.0163	−0.973177
$890$	0	0
$891$	−4.47982	−0.150080
$892$	0	0
$893$	42.1698	1.41116
$894$	0	0
$895$	28.9759	0.968557
$896$	0	0
$897$	−7.70694	−0.257327
$898$	0	0
$899$	−39.6096	−1.32105
$900$	0	0
$901$	−54.8769	−1.82822
$902$	0	0
$903$	−1.09322	−0.0363802
$904$	0	0
$905$	27.4915	0.913848
$906$	0	0
$907$	34.3321	1.13998	0.569989	−	0.821652i	$-0.306947\pi$
$907$	34.3321	1.13998	0.569989	+	0.821652i	$0.306947\pi$
$908$	0	0
$909$	3.67470	0.121882
$910$	0	0
$911$	−49.0410	−1.62480	−0.812400	−	0.583100i	$-0.801840\pi$
$911$	−49.0410	−1.62480	−0.812400	+	0.583100i	$0.801840\pi$
$912$	0	0
$913$	0.892169	0.0295265
$914$	0	0
$915$	4.36776	0.144394
$916$	0	0
$917$	−25.7782	−0.851271
$918$	0	0
$919$	−48.7329	−1.60755	−0.803775	−	0.594934i	$-0.797178\pi$
$919$	−48.7329	−1.60755	−0.803775	+	0.594934i	$0.797178\pi$
$920$	0	0
$921$	−4.29745	−0.141606
$922$	0	0
$923$	−73.5710	−2.42162
$924$	0	0
$925$	0.404533	0.0133010
$926$	0	0
$927$	15.6461	0.513884
$928$	0	0
$929$	52.9031	1.73569	0.867847	−	0.496832i	$-0.165504\pi$
$929$	52.9031	1.73569	0.867847	+	0.496832i	$0.165504\pi$
$930$	0	0
$931$	−26.9081	−0.881879
$932$	0	0
$933$	2.79088	0.0913693
$934$	0	0
$935$	−4.83461	−0.158109
$936$	0	0
$937$	−16.1310	−0.526976	−0.263488	−	0.964663i	$-0.584873\pi$
$937$	−16.1310	−0.526976	−0.263488	+	0.964663i	$0.584873\pi$
$938$	0	0
$939$	−0.430280	−0.0140416
$940$	0	0
$941$	10.7225	0.349544	0.174772	−	0.984609i	$-0.444081\pi$
$941$	10.7225	0.349544	0.174772	+	0.984609i	$0.444081\pi$
$942$	0	0
$943$	−69.2233	−2.25422
$944$	0	0
$945$	−8.82574	−0.287101
$946$	0	0
$947$	21.9984	0.714851	0.357426	−	0.933942i	$-0.383655\pi$
$947$	21.9984	0.714851	0.357426	+	0.933942i	$0.383655\pi$
$948$	0	0
$949$	−62.1660	−2.01800
$950$	0	0
$951$	−3.41923	−0.110876
$952$	0	0
$953$	−40.5214	−1.31262	−0.656309	−	0.754492i	$-0.727883\pi$
$953$	−40.5214	−1.31262	−0.656309	+	0.754492i	$0.727883\pi$
$954$	0	0
$955$	−8.83330	−0.285839
$956$	0	0
$957$	0.857998	0.0277351
$958$	0	0
$959$	−21.0129	−0.678542
$960$	0	0
$961$	−11.3590	−0.366419
$962$	0	0
$963$	31.9728	1.03031
$964$	0	0
$965$	3.81154	0.122698
$966$	0	0
$967$	46.5645	1.49741	0.748706	−	0.662902i	$-0.230676\pi$
$967$	46.5645	1.49741	0.748706	+	0.662902i	$0.230676\pi$
$968$	0	0
$969$	3.82348	0.122828
$970$	0	0
$971$	−2.03099	−0.0651775	−0.0325887	−	0.999469i	$-0.510375\pi$
$971$	−2.03099	−0.0651775	−0.0325887	+	0.999469i	$0.510375\pi$
$972$	0	0
$973$	52.8218	1.69339
$974$	0	0
$975$	−0.0675745	−0.00216412
$976$	0	0
$977$	−45.7170	−1.46262	−0.731308	−	0.682047i	$-0.761090\pi$
$977$	−45.7170	−1.46262	−0.731308	+	0.682047i	$0.761090\pi$
$978$	0	0
$979$	0.556803	0.0177955
$980$	0	0
$981$	3.41770	0.109119
$982$	0	0
$983$	47.0190	1.49967	0.749837	−	0.661623i	$-0.230132\pi$
$983$	47.0190	1.49967	0.749837	+	0.661623i	$0.230132\pi$
$984$	0	0
$985$	4.96103	0.158071
$986$	0	0
$987$	5.62256	0.178968
$988$	0	0
$989$	12.7945	0.406841
$990$	0	0
$991$	−54.4626	−1.73006	−0.865031	−	0.501718i	$-0.832702\pi$
$991$	−54.4626	−1.73006	−0.865031	+	0.501718i	$0.832702\pi$
$992$	0	0
$993$	1.20647	0.0382860
$994$	0	0
$995$	−35.0121	−1.10996
$996$	0	0
$997$	13.5868	0.430297	0.215149	−	0.976581i	$-0.430976\pi$
$997$	13.5868	0.430297	0.215149	+	0.976581i	$0.430976\pi$
$998$	0	0
$999$	−6.66250	−0.210792

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3856.2.a.j.1.3		7
4.3	odd	2		241.2.a.a.1.2	✓	7
12.11	even	2		2169.2.a.e.1.6		7
20.19	odd	2		6025.2.a.f.1.6		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
241.2.a.a.1.2	✓	7	4.3	odd	2
2169.2.a.e.1.6		7	12.11	even	2
3856.2.a.j.1.3		7	1.1	even	1	trivial
6025.2.a.f.1.6		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)

\(f(q)\)	\(=\)	\(q+0.186202 q^{3} -2.25110 q^{5} -3.52970 q^{7} -2.96533 q^{9} +O(q^{10})\)	\(q+0.186202 q^{3} -2.25110 q^{5} -3.52970 q^{7} -2.96533 q^{9} -0.515564 q^{11} -5.38098 q^{13} -0.419160 q^{15} -4.16566 q^{17} -4.92935 q^{19} -0.657237 q^{21} +7.69193 q^{23} +0.0674429 q^{25} -1.11076 q^{27} -8.93755 q^{29} +4.43182 q^{31} -0.0959992 q^{33} +7.94569 q^{35} +5.99816 q^{37} -1.00195 q^{39} -8.99946 q^{41} +1.66336 q^{43} +6.67525 q^{45} -8.55484 q^{47} +5.45876 q^{49} -0.775656 q^{51} +13.1736 q^{53} +1.16059 q^{55} -0.917857 q^{57} -9.25521 q^{59} -10.4203 q^{61} +10.4667 q^{63} +12.1131 q^{65} +3.91715 q^{67} +1.43226 q^{69} +13.6724 q^{71} +11.5529 q^{73} +0.0125580 q^{75} +1.81979 q^{77} +1.43448 q^{79} +8.68916 q^{81} -1.73047 q^{83} +9.37732 q^{85} -1.66419 q^{87} -1.07999 q^{89} +18.9932 q^{91} +0.825214 q^{93} +11.0965 q^{95} -16.0883 q^{97} +1.52882 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{3} - 8 q^{5} + 7 q^{7} - 2 q^{9}+O(q^{10}) \) 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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