LMFDB - Embedded newform 3856.2.a.j.1.1

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			1.1
Root			$0.369356$ 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)

$f(q)$	$=$	$q-2.33806 q^{3} -3.89634 q^{5} +3.68231 q^{7} +2.46654 q^{9} +O(q^{10})$	$q-2.33806 q^{3} -3.89634 q^{5} +3.68231 q^{7} +2.46654 q^{9} +4.96431 q^{11} -1.69048 q^{13} +9.10989 q^{15} +5.52260 q^{17} -4.21489 q^{19} -8.60948 q^{21} +2.77495 q^{23} +10.1815 q^{25} +1.24727 q^{27} -2.31253 q^{29} +0.199515 q^{31} -11.6069 q^{33} -14.3476 q^{35} +1.79089 q^{37} +3.95245 q^{39} -12.1960 q^{41} +5.34523 q^{43} -9.61047 q^{45} +10.7345 q^{47} +6.55944 q^{49} -12.9122 q^{51} +1.32229 q^{53} -19.3426 q^{55} +9.85467 q^{57} -5.78578 q^{59} -0.0766435 q^{61} +9.08256 q^{63} +6.58670 q^{65} -12.6166 q^{67} -6.48800 q^{69} +5.20391 q^{71} -14.0733 q^{73} -23.8050 q^{75} +18.2801 q^{77} -7.82844 q^{79} -10.3158 q^{81} -2.55372 q^{83} -21.5180 q^{85} +5.40683 q^{87} -4.35327 q^{89} -6.22489 q^{91} -0.466479 q^{93} +16.4226 q^{95} +9.02693 q^{97} +12.2446 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.33806	−1.34988	−0.674940	−	0.737872i	$-0.735831\pi$
$3$	−2.33806	−1.34988	−0.674940	+	0.737872i	$0.735831\pi$
$4$	0	0
$5$	−3.89634	−1.74250	−0.871249	−	0.490842i	$-0.836689\pi$
$5$	−3.89634	−1.74250	−0.871249	+	0.490842i	$0.836689\pi$
$6$	0	0
$7$	3.68231	1.39178	0.695892	−	0.718146i	$-0.255009\pi$
$7$	3.68231	1.39178	0.695892	+	0.718146i	$0.255009\pi$
$8$	0	0
$9$	2.46654	0.822178
$10$	0	0
$11$	4.96431	1.49679	0.748397	−	0.663250i	$-0.230824\pi$
$11$	4.96431	1.49679	0.748397	+	0.663250i	$0.230824\pi$
$12$	0	0
$13$	−1.69048	−0.468855	−0.234428	−	0.972134i	$-0.575322\pi$
$13$	−1.69048	−0.468855	−0.234428	+	0.972134i	$0.575322\pi$
$14$	0	0
$15$	9.10989	2.35216
$16$	0	0
$17$	5.52260	1.33943	0.669714	−	0.742619i	$-0.266417\pi$
$17$	5.52260	1.33943	0.669714	+	0.742619i	$0.266417\pi$
$18$	0	0
$19$	−4.21489	−0.966961	−0.483481	−	0.875355i	$-0.660628\pi$
$19$	−4.21489	−0.966961	−0.483481	+	0.875355i	$0.660628\pi$
$20$	0	0
$21$	−8.60948	−1.87874
$22$	0	0
$23$	2.77495	0.578617	0.289308	−	0.957236i	$-0.406575\pi$
$23$	2.77495	0.578617	0.289308	+	0.957236i	$0.406575\pi$
$24$	0	0
$25$	10.1815	2.03630
$26$	0	0
$27$	1.24727	0.240038
$28$	0	0
$29$	−2.31253	−0.429425	−0.214713	−	0.976677i	$-0.568881\pi$
$29$	−2.31253	−0.429425	−0.214713	+	0.976677i	$0.568881\pi$
$30$	0	0
$31$	0.199515	0.0358340	0.0179170	−	0.999839i	$-0.494297\pi$
$31$	0.199515	0.0358340	0.0179170	+	0.999839i	$0.494297\pi$
$32$	0	0
$33$	−11.6069	−2.02049
$34$	0	0
$35$	−14.3476	−2.42518
$36$	0	0
$37$	1.79089	0.294420	0.147210	−	0.989105i	$-0.452971\pi$
$37$	1.79089	0.294420	0.147210	+	0.989105i	$0.452971\pi$
$38$	0	0
$39$	3.95245	0.632899
$40$	0	0
$41$	−12.1960	−1.90470	−0.952351	−	0.305003i	$-0.901343\pi$
$41$	−12.1960	−1.90470	−0.952351	+	0.305003i	$0.901343\pi$
$42$	0	0
$43$	5.34523	0.815139	0.407570	−	0.913174i	$-0.366376\pi$
$43$	5.34523	0.815139	0.407570	+	0.913174i	$0.366376\pi$
$44$	0	0
$45$	−9.61047	−1.43264
$46$	0	0
$47$	10.7345	1.56579	0.782893	−	0.622157i	$-0.213743\pi$
$47$	10.7345	1.56579	0.782893	+	0.622157i	$0.213743\pi$
$48$	0	0
$49$	6.55944	0.937063
$50$	0	0
$51$	−12.9122	−1.80807
$52$	0	0
$53$	1.32229	0.181631	0.0908154	−	0.995868i	$-0.471053\pi$
$53$	1.32229	0.181631	0.0908154	+	0.995868i	$0.471053\pi$
$54$	0	0
$55$	−19.3426	−2.60816
$56$	0	0
$57$	9.85467	1.30528
$58$	0	0
$59$	−5.78578	−0.753244	−0.376622	−	0.926367i	$-0.622914\pi$
$59$	−5.78578	−0.753244	−0.376622	+	0.926367i	$0.622914\pi$
$60$	0	0
$61$	−0.0766435	−0.00981320	−0.00490660	−	0.999988i	$-0.501562\pi$
$61$	−0.0766435	−0.00981320	−0.00490660	+	0.999988i	$0.501562\pi$
$62$	0	0
$63$	9.08256	1.14429
$64$	0	0
$65$	6.58670	0.816979
$66$	0	0
$67$	−12.6166	−1.54136	−0.770681	−	0.637221i	$-0.780084\pi$
$67$	−12.6166	−1.54136	−0.770681	+	0.637221i	$0.780084\pi$
$68$	0	0
$69$	−6.48800	−0.781064
$70$	0	0
$71$	5.20391	0.617591	0.308796	−	0.951128i	$-0.400074\pi$
$71$	5.20391	0.617591	0.308796	+	0.951128i	$0.400074\pi$
$72$	0	0
$73$	−14.0733	−1.64715	−0.823577	−	0.567204i	$-0.808025\pi$
$73$	−14.0733	−1.64715	−0.823577	+	0.567204i	$0.808025\pi$
$74$	0	0
$75$	−23.8050	−2.74876
$76$	0	0
$77$	18.2801	2.08322
$78$	0	0
$79$	−7.82844	−0.880769	−0.440384	−	0.897809i	$-0.645158\pi$
$79$	−7.82844	−0.880769	−0.440384	+	0.897809i	$0.645158\pi$
$80$	0	0
$81$	−10.3158	−1.14620
$82$	0	0
$83$	−2.55372	−0.280307	−0.140153	−	0.990130i	$-0.544760\pi$
$83$	−2.55372	−0.280307	−0.140153	+	0.990130i	$0.544760\pi$
$84$	0	0
$85$	−21.5180	−2.33395
$86$	0	0
$87$	5.40683	0.579673
$88$	0	0
$89$	−4.35327	−0.461446	−0.230723	−	0.973019i	$-0.574109\pi$
$89$	−4.35327	−0.461446	−0.230723	+	0.973019i	$0.574109\pi$
$90$	0	0
$91$	−6.22489	−0.652546
$92$	0	0
$93$	−0.466479	−0.0483716
$94$	0	0
$95$	16.4226	1.68493
$96$	0	0
$97$	9.02693	0.916546	0.458273	−	0.888811i	$-0.348468\pi$
$97$	9.02693	0.916546	0.458273	+	0.888811i	$0.348468\pi$
$98$	0	0
$99$	12.2446	1.23063
$100$	0	0
$101$	6.94323	0.690877	0.345439	−	0.938441i	$-0.387730\pi$
$101$	6.94323	0.690877	0.345439	+	0.938441i	$0.387730\pi$
$102$	0	0
$103$	0.105009	0.0103469	0.00517343	−	0.999987i	$-0.498353\pi$
$103$	0.105009	0.0103469	0.00517343	+	0.999987i	$0.498353\pi$
$104$	0	0
$105$	33.5455	3.27371
$106$	0	0
$107$	18.1842	1.75793	0.878966	−	0.476885i	$-0.158234\pi$
$107$	18.1842	1.75793	0.878966	+	0.476885i	$0.158234\pi$
$108$	0	0
$109$	14.6576	1.40394	0.701972	−	0.712205i	$-0.252303\pi$
$109$	14.6576	1.40394	0.701972	+	0.712205i	$0.252303\pi$
$110$	0	0
$111$	−4.18720	−0.397432
$112$	0	0
$113$	1.08836	0.102384	0.0511922	−	0.998689i	$-0.483698\pi$
$113$	1.08836	0.102384	0.0511922	+	0.998689i	$0.483698\pi$
$114$	0	0
$115$	−10.8122	−1.00824
$116$	0	0
$117$	−4.16963	−0.385483
$118$	0	0
$119$	20.3360	1.86420
$120$	0	0
$121$	13.6443	1.24040
$122$	0	0
$123$	28.5151	2.57112
$124$	0	0
$125$	−20.1889	−1.80575
$126$	0	0
$127$	−0.364538	−0.0323475	−0.0161737	−	0.999869i	$-0.505148\pi$
$127$	−0.364538	−0.0323475	−0.0161737	+	0.999869i	$0.505148\pi$
$128$	0	0
$129$	−12.4975	−1.10034
$130$	0	0
$131$	−5.48865	−0.479546	−0.239773	−	0.970829i	$-0.577073\pi$
$131$	−5.48865	−0.479546	−0.239773	+	0.970829i	$0.577073\pi$
$132$	0	0
$133$	−15.5205	−1.34580
$134$	0	0
$135$	−4.85981	−0.418266
$136$	0	0
$137$	−17.1533	−1.46551	−0.732754	−	0.680493i	$-0.761766\pi$
$137$	−17.1533	−1.46551	−0.732754	+	0.680493i	$0.761766\pi$
$138$	0	0
$139$	19.6074	1.66308	0.831539	−	0.555467i	$-0.187460\pi$
$139$	19.6074	1.66308	0.831539	+	0.555467i	$0.187460\pi$
$140$	0	0
$141$	−25.0979	−2.11362
$142$	0	0
$143$	−8.39207	−0.701780
$144$	0	0
$145$	9.01039	0.748272
$146$	0	0
$147$	−15.3364	−1.26492
$148$	0	0
$149$	19.0508	1.56070	0.780350	−	0.625343i	$-0.215041\pi$
$149$	19.0508	1.56070	0.780350	+	0.625343i	$0.215041\pi$
$150$	0	0
$151$	4.76802	0.388016	0.194008	−	0.981000i	$-0.437851\pi$
$151$	4.76802	0.388016	0.194008	+	0.981000i	$0.437851\pi$
$152$	0	0
$153$	13.6217	1.10125
$154$	0	0
$155$	−0.777379	−0.0624406
$156$	0	0
$157$	6.12724	0.489007	0.244503	−	0.969648i	$-0.421375\pi$
$157$	6.12724	0.489007	0.244503	+	0.969648i	$0.421375\pi$
$158$	0	0
$159$	−3.09160	−0.245180
$160$	0	0
$161$	10.2182	0.805310
$162$	0	0
$163$	1.20414	0.0943159	0.0471579	−	0.998887i	$-0.484984\pi$
$163$	1.20414	0.0943159	0.0471579	+	0.998887i	$0.484984\pi$
$164$	0	0
$165$	45.2243	3.52071
$166$	0	0
$167$	0.900391	0.0696744	0.0348372	−	0.999393i	$-0.488909\pi$
$167$	0.900391	0.0696744	0.0348372	+	0.999393i	$0.488909\pi$
$168$	0	0
$169$	−10.1423	−0.780175
$170$	0	0
$171$	−10.3962	−0.795015
$172$	0	0
$173$	12.3564	0.939436	0.469718	−	0.882817i	$-0.344356\pi$
$173$	12.3564	0.939436	0.469718	+	0.882817i	$0.344356\pi$
$174$	0	0
$175$	37.4915	2.83409
$176$	0	0
$177$	13.5275	1.01679
$178$	0	0
$179$	10.0258	0.749367	0.374683	−	0.927153i	$-0.377751\pi$
$179$	10.0258	0.749367	0.374683	+	0.927153i	$0.377751\pi$
$180$	0	0
$181$	−17.0491	−1.26725	−0.633623	−	0.773642i	$-0.718433\pi$
$181$	−17.0491	−1.26725	−0.633623	+	0.773642i	$0.718433\pi$
$182$	0	0
$183$	0.179197	0.0132467
$184$	0	0
$185$	−6.97790	−0.513026
$186$	0	0
$187$	27.4159	2.00485
$188$	0	0
$189$	4.59286	0.334081
$190$	0	0
$191$	23.7744	1.72026	0.860129	−	0.510077i	$-0.170383\pi$
$191$	23.7744	1.72026	0.860129	+	0.510077i	$0.170383\pi$
$192$	0	0
$193$	7.49373	0.539411	0.269705	−	0.962943i	$-0.413074\pi$
$193$	7.49373	0.539411	0.269705	+	0.962943i	$0.413074\pi$
$194$	0	0
$195$	−15.4001	−1.10282
$196$	0	0
$197$	1.85647	0.132268	0.0661339	−	0.997811i	$-0.478934\pi$
$197$	1.85647	0.132268	0.0661339	+	0.997811i	$0.478934\pi$
$198$	0	0
$199$	−3.46667	−0.245746	−0.122873	−	0.992422i	$-0.539211\pi$
$199$	−3.46667	−0.245746	−0.122873	+	0.992422i	$0.539211\pi$
$200$	0	0
$201$	29.4984	2.08066
$202$	0	0
$203$	−8.51545	−0.597667
$204$	0	0
$205$	47.5200	3.31894
$206$	0	0
$207$	6.84451	0.475726
$208$	0	0
$209$	−20.9240	−1.44734
$210$	0	0
$211$	2.33201	0.160542	0.0802711	−	0.996773i	$-0.474421\pi$
$211$	2.33201	0.160542	0.0802711	+	0.996773i	$0.474421\pi$
$212$	0	0
$213$	−12.1671	−0.833674
$214$	0	0
$215$	−20.8268	−1.42038
$216$	0	0
$217$	0.734678	0.0498732
$218$	0	0
$219$	32.9042	2.22346
$220$	0	0
$221$	−9.33587	−0.627998
$222$	0	0
$223$	1.75535	0.117547	0.0587734	−	0.998271i	$-0.481281\pi$
$223$	1.75535	0.117547	0.0587734	+	0.998271i	$0.481281\pi$
$224$	0	0
$225$	25.1130	1.67420
$226$	0	0
$227$	−4.35631	−0.289139	−0.144569	−	0.989495i	$-0.546180\pi$
$227$	−4.35631	−0.289139	−0.144569	+	0.989495i	$0.546180\pi$
$228$	0	0
$229$	6.24160	0.412457	0.206228	−	0.978504i	$-0.433881\pi$
$229$	6.24160	0.412457	0.206228	+	0.978504i	$0.433881\pi$
$230$	0	0
$231$	−42.7401	−2.81209
$232$	0	0
$233$	13.8315	0.906135	0.453067	−	0.891476i	$-0.350330\pi$
$233$	13.8315	0.906135	0.453067	+	0.891476i	$0.350330\pi$
$234$	0	0
$235$	−41.8252	−2.72838
$236$	0	0
$237$	18.3034	1.18893
$238$	0	0
$239$	19.3924	1.25439	0.627194	−	0.778863i	$-0.284203\pi$
$239$	19.3924	1.25439	0.627194	+	0.778863i	$0.284203\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	0	0
$243$	20.3772	1.30720
$244$	0	0
$245$	−25.5578	−1.63283
$246$	0	0
$247$	7.12519	0.453365
$248$	0	0
$249$	5.97075	0.378381
$250$	0	0
$251$	14.4441	0.911705	0.455852	−	0.890055i	$-0.349334\pi$
$251$	14.4441	0.911705	0.455852	+	0.890055i	$0.349334\pi$
$252$	0	0
$253$	13.7757	0.866071
$254$	0	0
$255$	50.3103	3.15056
$256$	0	0
$257$	28.9004	1.80276	0.901379	−	0.433031i	$-0.142556\pi$
$257$	28.9004	1.80276	0.901379	+	0.433031i	$0.142556\pi$
$258$	0	0
$259$	6.59460	0.409769
$260$	0	0
$261$	−5.70393	−0.353064
$262$	0	0
$263$	−26.3078	−1.62221	−0.811103	−	0.584903i	$-0.801133\pi$
$263$	−26.3078	−1.62221	−0.811103	+	0.584903i	$0.801133\pi$
$264$	0	0
$265$	−5.15211	−0.316491
$266$	0	0
$267$	10.1782	0.622897
$268$	0	0
$269$	−9.42450	−0.574622	−0.287311	−	0.957837i	$-0.592761\pi$
$269$	−9.42450	−0.574622	−0.287311	+	0.957837i	$0.592761\pi$
$270$	0	0
$271$	5.86519	0.356285	0.178142	−	0.984005i	$-0.442991\pi$
$271$	5.86519	0.356285	0.178142	+	0.984005i	$0.442991\pi$
$272$	0	0
$273$	14.5542	0.880859
$274$	0	0
$275$	50.5440	3.04792
$276$	0	0
$277$	5.60395	0.336708	0.168354	−	0.985727i	$-0.446155\pi$
$277$	5.60395	0.336708	0.168354	+	0.985727i	$0.446155\pi$
$278$	0	0
$279$	0.492111	0.0294619
$280$	0	0
$281$	−4.71669	−0.281374	−0.140687	−	0.990054i	$-0.544931\pi$
$281$	−4.71669	−0.281374	−0.140687	+	0.990054i	$0.544931\pi$
$282$	0	0
$283$	−19.2068	−1.14173	−0.570864	−	0.821044i	$-0.693392\pi$
$283$	−19.2068	−1.14173	−0.570864	+	0.821044i	$0.693392\pi$
$284$	0	0
$285$	−38.3972	−2.27445
$286$	0	0
$287$	−44.9097	−2.65094
$288$	0	0
$289$	13.4992	0.794069
$290$	0	0
$291$	−21.1055	−1.23723
$292$	0	0
$293$	13.0140	0.760289	0.380144	−	0.924927i	$-0.375874\pi$
$293$	13.0140	0.760289	0.380144	+	0.924927i	$0.375874\pi$
$294$	0	0
$295$	22.5434	1.31253
$296$	0	0
$297$	6.19185	0.359288
$298$	0	0
$299$	−4.69100	−0.271288
$300$	0	0
$301$	19.6828	1.13450
$302$	0	0
$303$	−16.2337	−0.932602
$304$	0	0
$305$	0.298630	0.0170995
$306$	0	0
$307$	11.5372	0.658463	0.329232	−	0.944249i	$-0.393210\pi$
$307$	11.5372	0.658463	0.329232	+	0.944249i	$0.393210\pi$
$308$	0	0
$309$	−0.245518	−0.0139670
$310$	0	0
$311$	23.1542	1.31296	0.656478	−	0.754345i	$-0.272045\pi$
$311$	23.1542	1.31296	0.656478	+	0.754345i	$0.272045\pi$
$312$	0	0
$313$	−7.88902	−0.445914	−0.222957	−	0.974828i	$-0.571571\pi$
$313$	−7.88902	−0.445914	−0.222957	+	0.974828i	$0.571571\pi$
$314$	0	0
$315$	−35.3888	−1.99393
$316$	0	0
$317$	−2.38325	−0.133857	−0.0669283	−	0.997758i	$-0.521320\pi$
$317$	−2.38325	−0.133857	−0.0669283	+	0.997758i	$0.521320\pi$
$318$	0	0
$319$	−11.4801	−0.642761
$320$	0	0
$321$	−42.5157	−2.37300
$322$	0	0
$323$	−23.2772	−1.29518
$324$	0	0
$325$	−17.2116	−0.954729
$326$	0	0
$327$	−34.2704	−1.89516
$328$	0	0
$329$	39.5278	2.17924
$330$	0	0
$331$	3.05598	0.167972	0.0839860	−	0.996467i	$-0.473235\pi$
$331$	3.05598	0.167972	0.0839860	+	0.996467i	$0.473235\pi$
$332$	0	0
$333$	4.41728	0.242066
$334$	0	0
$335$	49.1586	2.68582
$336$	0	0
$337$	−16.1755	−0.881134	−0.440567	−	0.897720i	$-0.645223\pi$
$337$	−16.1755	−0.881134	−0.440567	+	0.897720i	$0.645223\pi$
$338$	0	0
$339$	−2.54466	−0.138207
$340$	0	0
$341$	0.990454	0.0536361
$342$	0	0
$343$	−1.62227	−0.0875943
$344$	0	0
$345$	25.2795	1.36100
$346$	0	0
$347$	−24.4716	−1.31370	−0.656851	−	0.754020i	$-0.728112\pi$
$347$	−24.4716	−1.31370	−0.656851	+	0.754020i	$0.728112\pi$
$348$	0	0
$349$	−13.0468	−0.698380	−0.349190	−	0.937052i	$-0.613543\pi$
$349$	−13.0468	−0.698380	−0.349190	+	0.937052i	$0.613543\pi$
$350$	0	0
$351$	−2.10849	−0.112543
$352$	0	0
$353$	23.6788	1.26029	0.630147	−	0.776476i	$-0.282995\pi$
$353$	23.6788	1.26029	0.630147	+	0.776476i	$0.282995\pi$
$354$	0	0
$355$	−20.2762	−1.07615
$356$	0	0
$357$	−47.5468	−2.51644
$358$	0	0
$359$	25.5499	1.34847	0.674236	−	0.738516i	$-0.264473\pi$
$359$	25.5499	1.34847	0.674236	+	0.738516i	$0.264473\pi$
$360$	0	0
$361$	−1.23473	−0.0649857
$362$	0	0
$363$	−31.9013	−1.67439
$364$	0	0
$365$	54.8344	2.87016
$366$	0	0
$367$	−8.35766	−0.436266	−0.218133	−	0.975919i	$-0.569997\pi$
$367$	−8.35766	−0.436266	−0.218133	+	0.975919i	$0.569997\pi$
$368$	0	0
$369$	−30.0820	−1.56601
$370$	0	0
$371$	4.86910	0.252791
$372$	0	0
$373$	−24.6934	−1.27858	−0.639288	−	0.768968i	$-0.720771\pi$
$373$	−24.6934	−1.27858	−0.639288	+	0.768968i	$0.720771\pi$
$374$	0	0
$375$	47.2028	2.43754
$376$	0	0
$377$	3.90928	0.201338
$378$	0	0
$379$	−18.6593	−0.958465	−0.479232	−	0.877688i	$-0.659085\pi$
$379$	−18.6593	−0.958465	−0.479232	+	0.877688i	$0.659085\pi$
$380$	0	0
$381$	0.852312	0.0436652
$382$	0	0
$383$	−11.9780	−0.612049	−0.306025	−	0.952024i	$-0.598999\pi$
$383$	−11.9780	−0.612049	−0.306025	+	0.952024i	$0.598999\pi$
$384$	0	0
$385$	−71.2257	−3.63000
$386$	0	0
$387$	13.1842	0.670190
$388$	0	0
$389$	−7.62670	−0.386689	−0.193344	−	0.981131i	$-0.561933\pi$
$389$	−7.62670	−0.386689	−0.193344	+	0.981131i	$0.561933\pi$
$390$	0	0
$391$	15.3249	0.775016
$392$	0	0
$393$	12.8328	0.647330
$394$	0	0
$395$	30.5023	1.53474
$396$	0	0
$397$	−33.4013	−1.67636	−0.838181	−	0.545392i	$-0.816381\pi$
$397$	−33.4013	−1.67636	−0.838181	+	0.545392i	$0.816381\pi$
$398$	0	0
$399$	36.2880	1.81667
$400$	0	0
$401$	−1.73530	−0.0866566	−0.0433283	−	0.999061i	$-0.513796\pi$
$401$	−1.73530	−0.0866566	−0.0433283	+	0.999061i	$0.513796\pi$
$402$	0	0
$403$	−0.337277	−0.0168010
$404$	0	0
$405$	40.1939	1.99725
$406$	0	0
$407$	8.89050	0.440686
$408$	0	0
$409$	−3.81908	−0.188841	−0.0944207	−	0.995532i	$-0.530100\pi$
$409$	−3.81908	−0.188841	−0.0944207	+	0.995532i	$0.530100\pi$
$410$	0	0
$411$	40.1056	1.97826
$412$	0	0
$413$	−21.3050	−1.04835
$414$	0	0
$415$	9.95015	0.488434
$416$	0	0
$417$	−45.8433	−2.24496
$418$	0	0
$419$	−0.640088	−0.0312703	−0.0156352	−	0.999878i	$-0.504977\pi$
$419$	−0.640088	−0.0312703	−0.0156352	+	0.999878i	$0.504977\pi$
$420$	0	0
$421$	13.2765	0.647057	0.323528	−	0.946218i	$-0.395131\pi$
$421$	13.2765	0.647057	0.323528	+	0.946218i	$0.395131\pi$
$422$	0	0
$423$	26.4770	1.28735
$424$	0	0
$425$	56.2283	2.72748
$426$	0	0
$427$	−0.282226	−0.0136579
$428$	0	0
$429$	19.6212	0.947320
$430$	0	0
$431$	26.8180	1.29178	0.645888	−	0.763432i	$-0.276487\pi$
$431$	26.8180	1.29178	0.645888	+	0.763432i	$0.276487\pi$
$432$	0	0
$433$	−20.7392	−0.996664	−0.498332	−	0.866986i	$-0.666054\pi$
$433$	−20.7392	−0.996664	−0.498332	+	0.866986i	$0.666054\pi$
$434$	0	0
$435$	−21.0669	−1.01008
$436$	0	0
$437$	−11.6961	−0.559500
$438$	0	0
$439$	12.7382	0.607960	0.303980	−	0.952678i	$-0.401684\pi$
$439$	12.7382	0.607960	0.303980	+	0.952678i	$0.401684\pi$
$440$	0	0
$441$	16.1791	0.770433
$442$	0	0
$443$	−23.4058	−1.11204	−0.556022	−	0.831167i	$-0.687673\pi$
$443$	−23.4058	−1.11204	−0.556022	+	0.831167i	$0.687673\pi$
$444$	0	0
$445$	16.9618	0.804068
$446$	0	0
$447$	−44.5419	−2.10676
$448$	0	0
$449$	35.8887	1.69369	0.846846	−	0.531838i	$-0.178498\pi$
$449$	35.8887	1.69369	0.846846	+	0.531838i	$0.178498\pi$
$450$	0	0
$451$	−60.5449	−2.85095
$452$	0	0
$453$	−11.1479	−0.523775
$454$	0	0
$455$	24.2543	1.13706
$456$	0	0
$457$	−23.4014	−1.09467	−0.547336	−	0.836913i	$-0.684358\pi$
$457$	−23.4014	−1.09467	−0.547336	+	0.836913i	$0.684358\pi$
$458$	0	0
$459$	6.88820	0.321514
$460$	0	0
$461$	17.9146	0.834366	0.417183	−	0.908822i	$-0.363017\pi$
$461$	17.9146	0.834366	0.417183	+	0.908822i	$0.363017\pi$
$462$	0	0
$463$	32.0159	1.48790	0.743952	−	0.668233i	$-0.232949\pi$
$463$	32.0159	1.48790	0.743952	+	0.668233i	$0.232949\pi$
$464$	0	0
$465$	1.81756	0.0842874
$466$	0	0
$467$	12.2949	0.568940	0.284470	−	0.958685i	$-0.408182\pi$
$467$	12.2949	0.568940	0.284470	+	0.958685i	$0.408182\pi$
$468$	0	0
$469$	−46.4583	−2.14524
$470$	0	0
$471$	−14.3259	−0.660101
$472$	0	0
$473$	26.5353	1.22010
$474$	0	0
$475$	−42.9138	−1.96902
$476$	0	0
$477$	3.26148	0.149333
$478$	0	0
$479$	−15.1555	−0.692472	−0.346236	−	0.938148i	$-0.612540\pi$
$479$	−15.1555	−0.692472	−0.346236	+	0.938148i	$0.612540\pi$
$480$	0	0
$481$	−3.02746	−0.138040
$482$	0	0
$483$	−23.8909	−1.08707
$484$	0	0
$485$	−35.1720	−1.59708
$486$	0	0
$487$	15.5008	0.702407	0.351204	−	0.936299i	$-0.385773\pi$
$487$	15.5008	0.702407	0.351204	+	0.936299i	$0.385773\pi$
$488$	0	0
$489$	−2.81536	−0.127315
$490$	0	0
$491$	−12.5838	−0.567901	−0.283950	−	0.958839i	$-0.591645\pi$
$491$	−12.5838	−0.567901	−0.283950	+	0.958839i	$0.591645\pi$
$492$	0	0
$493$	−12.7712	−0.575184
$494$	0	0
$495$	−47.7093	−2.14437
$496$	0	0
$497$	19.1625	0.859553
$498$	0	0
$499$	41.9638	1.87856	0.939280	−	0.343152i	$-0.111495\pi$
$499$	41.9638	1.87856	0.939280	+	0.343152i	$0.111495\pi$
$500$	0	0
$501$	−2.10517	−0.0940521
$502$	0	0
$503$	−7.23458	−0.322574	−0.161287	−	0.986908i	$-0.551564\pi$
$503$	−7.23458	−0.322574	−0.161287	+	0.986908i	$0.551564\pi$
$504$	0	0
$505$	−27.0532	−1.20385
$506$	0	0
$507$	23.7133	1.05314
$508$	0	0
$509$	43.2945	1.91899	0.959497	−	0.281718i	$-0.0909041\pi$
$509$	43.2945	1.91899	0.959497	+	0.281718i	$0.0909041\pi$
$510$	0	0
$511$	−51.8223	−2.29248
$512$	0	0
$513$	−5.25712	−0.232108
$514$	0	0
$515$	−0.409152	−0.0180294
$516$	0	0
$517$	53.2893	2.34366
$518$	0	0
$519$	−28.8899	−1.26813
$520$	0	0
$521$	−22.7560	−0.996961	−0.498480	−	0.866901i	$-0.666108\pi$
$521$	−22.7560	−0.996961	−0.498480	+	0.866901i	$0.666108\pi$
$522$	0	0
$523$	−36.7529	−1.60709	−0.803547	−	0.595241i	$-0.797056\pi$
$523$	−36.7529	−1.60709	−0.803547	+	0.595241i	$0.797056\pi$
$524$	0	0
$525$	−87.6573	−3.82568
$526$	0	0
$527$	1.10184	0.0479970
$528$	0	0
$529$	−15.2997	−0.665203
$530$	0	0
$531$	−14.2708	−0.619301
$532$	0	0
$533$	20.6172	0.893030
$534$	0	0
$535$	−70.8518	−3.06319
$536$	0	0
$537$	−23.4410	−1.01156
$538$	0	0
$539$	32.5631	1.40259
$540$	0	0
$541$	−19.7726	−0.850089	−0.425045	−	0.905172i	$-0.639742\pi$
$541$	−19.7726	−0.850089	−0.425045	+	0.905172i	$0.639742\pi$
$542$	0	0
$543$	39.8617	1.71063
$544$	0	0
$545$	−57.1110	−2.44637
$546$	0	0
$547$	9.94319	0.425140	0.212570	−	0.977146i	$-0.431817\pi$
$547$	9.94319	0.425140	0.212570	+	0.977146i	$0.431817\pi$
$548$	0	0
$549$	−0.189044	−0.00806820
$550$	0	0
$551$	9.74703	0.415238
$552$	0	0
$553$	−28.8268	−1.22584
$554$	0	0
$555$	16.3148	0.692524
$556$	0	0
$557$	−7.84163	−0.332260	−0.166130	−	0.986104i	$-0.553127\pi$
$557$	−7.84163	−0.332260	−0.166130	+	0.986104i	$0.553127\pi$
$558$	0	0
$559$	−9.03601	−0.382182
$560$	0	0
$561$	−64.1001	−2.70631
$562$	0	0
$563$	19.2105	0.809625	0.404813	−	0.914400i	$-0.367337\pi$
$563$	19.2105	0.809625	0.404813	+	0.914400i	$0.367337\pi$
$564$	0	0
$565$	−4.24063	−0.178405
$566$	0	0
$567$	−37.9861	−1.59526
$568$	0	0
$569$	25.1338	1.05366	0.526831	−	0.849970i	$-0.323380\pi$
$569$	25.1338	1.05366	0.526831	+	0.849970i	$0.323380\pi$
$570$	0	0
$571$	18.1912	0.761277	0.380638	−	0.924724i	$-0.375704\pi$
$571$	18.1912	0.761277	0.380638	+	0.924724i	$0.375704\pi$
$572$	0	0
$573$	−55.5861	−2.32214
$574$	0	0
$575$	28.2531	1.17824
$576$	0	0
$577$	28.9417	1.20486	0.602429	−	0.798172i	$-0.294200\pi$
$577$	28.9417	1.20486	0.602429	+	0.798172i	$0.294200\pi$
$578$	0	0
$579$	−17.5208	−0.728140
$580$	0	0
$581$	−9.40358	−0.390126
$582$	0	0
$583$	6.56427	0.271864
$584$	0	0
$585$	16.2463	0.671703
$586$	0	0
$587$	21.9772	0.907098	0.453549	−	0.891231i	$-0.350158\pi$
$587$	21.9772	0.907098	0.453549	+	0.891231i	$0.350158\pi$
$588$	0	0
$589$	−0.840934	−0.0346501
$590$	0	0
$591$	−4.34054	−0.178546
$592$	0	0
$593$	15.0138	0.616544	0.308272	−	0.951298i	$-0.400249\pi$
$593$	15.0138	0.616544	0.308272	+	0.951298i	$0.400249\pi$
$594$	0	0
$595$	−79.2359	−3.24836
$596$	0	0
$597$	8.10529	0.331728
$598$	0	0
$599$	31.7086	1.29558	0.647789	−	0.761820i	$-0.275694\pi$
$599$	31.7086	1.29558	0.647789	+	0.761820i	$0.275694\pi$
$600$	0	0
$601$	5.63682	0.229931	0.114965	−	0.993369i	$-0.463324\pi$
$601$	5.63682	0.229931	0.114965	+	0.993369i	$0.463324\pi$
$602$	0	0
$603$	−31.1193	−1.26727
$604$	0	0
$605$	−53.1631	−2.16139
$606$	0	0
$607$	−39.7534	−1.61354	−0.806770	−	0.590866i	$-0.798786\pi$
$607$	−39.7534	−1.61354	−0.806770	+	0.590866i	$0.798786\pi$
$608$	0	0
$609$	19.9096	0.806780
$610$	0	0
$611$	−18.1465	−0.734127
$612$	0	0
$613$	30.3066	1.22407	0.612035	−	0.790831i	$-0.290351\pi$
$613$	30.3066	1.22407	0.612035	+	0.790831i	$0.290351\pi$
$614$	0	0
$615$	−111.105	−4.48017
$616$	0	0
$617$	18.2240	0.733670	0.366835	−	0.930286i	$-0.380441\pi$
$617$	18.2240	0.733670	0.366835	+	0.930286i	$0.380441\pi$
$618$	0	0
$619$	26.3431	1.05882	0.529409	−	0.848366i	$-0.322414\pi$
$619$	26.3431	1.05882	0.529409	+	0.848366i	$0.322414\pi$
$620$	0	0
$621$	3.46112	0.138890
$622$	0	0
$623$	−16.0301	−0.642233
$624$	0	0
$625$	27.7553	1.11021
$626$	0	0
$627$	48.9216	1.95374
$628$	0	0
$629$	9.89035	0.394354
$630$	0	0
$631$	−18.4187	−0.733236	−0.366618	−	0.930372i	$-0.619484\pi$
$631$	−18.4187	−0.733236	−0.366618	+	0.930372i	$0.619484\pi$
$632$	0	0
$633$	−5.45238	−0.216713
$634$	0	0
$635$	1.42036	0.0563654
$636$	0	0
$637$	−11.0886	−0.439347
$638$	0	0
$639$	12.8356	0.507770
$640$	0	0
$641$	−12.6956	−0.501446	−0.250723	−	0.968059i	$-0.580668\pi$
$641$	−12.6956	−0.501446	−0.250723	+	0.968059i	$0.580668\pi$
$642$	0	0
$643$	35.1591	1.38654	0.693270	−	0.720678i	$-0.256169\pi$
$643$	35.1591	1.38654	0.693270	+	0.720678i	$0.256169\pi$
$644$	0	0
$645$	48.6944	1.91734
$646$	0	0
$647$	−19.7535	−0.776588	−0.388294	−	0.921535i	$-0.626936\pi$
$647$	−19.7535	−0.776588	−0.388294	+	0.921535i	$0.626936\pi$
$648$	0	0
$649$	−28.7224	−1.12745
$650$	0	0
$651$	−1.71772	−0.0673228
$652$	0	0
$653$	−44.0129	−1.72236	−0.861178	−	0.508303i	$-0.830273\pi$
$653$	−44.0129	−1.72236	−0.861178	+	0.508303i	$0.830273\pi$
$654$	0	0
$655$	21.3857	0.835608
$656$	0	0
$657$	−34.7123	−1.35426
$658$	0	0
$659$	32.2032	1.25446	0.627229	−	0.778835i	$-0.284189\pi$
$659$	32.2032	1.25446	0.627229	+	0.778835i	$0.284189\pi$
$660$	0	0
$661$	29.5485	1.14930	0.574651	−	0.818398i	$-0.305138\pi$
$661$	29.5485	1.14930	0.574651	+	0.818398i	$0.305138\pi$
$662$	0	0
$663$	21.8278	0.847723
$664$	0	0
$665$	60.4734	2.34506
$666$	0	0
$667$	−6.41714	−0.248473
$668$	0	0
$669$	−4.10412	−0.158674
$670$	0	0
$671$	−0.380482	−0.0146883
$672$	0	0
$673$	28.4424	1.09637	0.548186	−	0.836356i	$-0.315319\pi$
$673$	28.4424	1.09637	0.548186	+	0.836356i	$0.315319\pi$
$674$	0	0
$675$	12.6991	0.488789
$676$	0	0
$677$	16.9259	0.650514	0.325257	−	0.945626i	$-0.394549\pi$
$677$	16.9259	0.650514	0.325257	+	0.945626i	$0.394549\pi$
$678$	0	0
$679$	33.2400	1.27563
$680$	0	0
$681$	10.1853	0.390303
$682$	0	0
$683$	29.0324	1.11090	0.555448	−	0.831551i	$-0.312547\pi$
$683$	29.0324	1.11090	0.555448	+	0.831551i	$0.312547\pi$
$684$	0	0
$685$	66.8353	2.55365
$686$	0	0
$687$	−14.5933	−0.556767
$688$	0	0
$689$	−2.23531	−0.0851586
$690$	0	0
$691$	21.5798	0.820935	0.410467	−	0.911875i	$-0.365366\pi$
$691$	21.5798	0.820935	0.410467	+	0.911875i	$0.365366\pi$
$692$	0	0
$693$	45.0886	1.71277
$694$	0	0
$695$	−76.3972	−2.89791
$696$	0	0
$697$	−67.3540	−2.55121
$698$	0	0
$699$	−32.3390	−1.22317
$700$	0	0
$701$	−13.2247	−0.499489	−0.249744	−	0.968312i	$-0.580347\pi$
$701$	−13.2247	−0.499489	−0.249744	+	0.968312i	$0.580347\pi$
$702$	0	0
$703$	−7.54838	−0.284692
$704$	0	0
$705$	97.7900	3.68298
$706$	0	0
$707$	25.5672	0.961552
$708$	0	0
$709$	18.8314	0.707228	0.353614	−	0.935391i	$-0.384953\pi$
$709$	18.8314	0.707228	0.353614	+	0.935391i	$0.384953\pi$
$710$	0	0
$711$	−19.3091	−0.724149
$712$	0	0
$713$	0.553644	0.0207341
$714$	0	0
$715$	32.6984	1.22285
$716$	0	0
$717$	−45.3406	−1.69328
$718$	0	0
$719$	37.7936	1.40946	0.704731	−	0.709475i	$-0.251068\pi$
$719$	37.7936	1.40946	0.704731	+	0.709475i	$0.251068\pi$
$720$	0	0
$721$	0.386677	0.0144006
$722$	0	0
$723$	2.33806	0.0869535
$724$	0	0
$725$	−23.5450	−0.874438
$726$	0	0
$727$	9.96692	0.369653	0.184826	−	0.982771i	$-0.440828\pi$
$727$	9.96692	0.369653	0.184826	+	0.982771i	$0.440828\pi$
$728$	0	0
$729$	−16.6957	−0.618359
$730$	0	0
$731$	29.5196	1.09182
$732$	0	0
$733$	46.8282	1.72964	0.864819	−	0.502083i	$-0.167433\pi$
$733$	46.8282	1.72964	0.864819	+	0.502083i	$0.167433\pi$
$734$	0	0
$735$	59.7558	2.20413
$736$	0	0
$737$	−62.6326	−2.30710
$738$	0	0
$739$	−31.1213	−1.14482	−0.572408	−	0.819969i	$-0.693991\pi$
$739$	−31.1213	−1.14482	−0.572408	+	0.819969i	$0.693991\pi$
$740$	0	0
$741$	−16.6591	−0.611989
$742$	0	0
$743$	22.3465	0.819815	0.409907	−	0.912127i	$-0.365561\pi$
$743$	22.3465	0.819815	0.409907	+	0.912127i	$0.365561\pi$
$744$	0	0
$745$	−74.2283	−2.71952
$746$	0	0
$747$	−6.29883	−0.230462
$748$	0	0
$749$	66.9599	2.44666
$750$	0	0
$751$	41.3430	1.50863	0.754313	−	0.656515i	$-0.227970\pi$
$751$	41.3430	1.50863	0.754313	+	0.656515i	$0.227970\pi$
$752$	0	0
$753$	−33.7713	−1.23069
$754$	0	0
$755$	−18.5778	−0.676116
$756$	0	0
$757$	−5.70532	−0.207364	−0.103682	−	0.994611i	$-0.533062\pi$
$757$	−5.70532	−0.207364	−0.103682	+	0.994611i	$0.533062\pi$
$758$	0	0
$759$	−32.2084	−1.16909
$760$	0	0
$761$	15.1656	0.549754	0.274877	−	0.961479i	$-0.411363\pi$
$761$	15.1656	0.549754	0.274877	+	0.961479i	$0.411363\pi$
$762$	0	0
$763$	53.9739	1.95399
$764$	0	0
$765$	−53.0748	−1.91892
$766$	0	0
$767$	9.78075	0.353162
$768$	0	0
$769$	5.23538	0.188793	0.0943964	−	0.995535i	$-0.469908\pi$
$769$	5.23538	0.188793	0.0943964	+	0.995535i	$0.469908\pi$
$770$	0	0
$771$	−67.5710	−2.43351
$772$	0	0
$773$	−27.0422	−0.972641	−0.486321	−	0.873780i	$-0.661661\pi$
$773$	−27.0422	−0.972641	−0.486321	+	0.873780i	$0.661661\pi$
$774$	0	0
$775$	2.03136	0.0729687
$776$	0	0
$777$	−15.4186	−0.553139
$778$	0	0
$779$	51.4050	1.84177
$780$	0	0
$781$	25.8338	0.924407
$782$	0	0
$783$	−2.88435	−0.103078
$784$	0	0
$785$	−23.8738	−0.852093
$786$	0	0
$787$	25.6221	0.913329	0.456665	−	0.889639i	$-0.349044\pi$
$787$	25.6221	0.913329	0.456665	+	0.889639i	$0.349044\pi$
$788$	0	0
$789$	61.5092	2.18979
$790$	0	0
$791$	4.00769	0.142497
$792$	0	0
$793$	0.129565	0.00460097
$794$	0	0
$795$	12.0459	0.427226
$796$	0	0
$797$	28.8151	1.02068	0.510341	−	0.859972i	$-0.329519\pi$
$797$	28.8151	1.02068	0.510341	+	0.859972i	$0.329519\pi$
$798$	0	0
$799$	59.2823	2.09726
$800$	0	0
$801$	−10.7375	−0.379391
$802$	0	0
$803$	−69.8642	−2.46545
$804$	0	0
$805$	−39.8137	−1.40325
$806$	0	0
$807$	22.0351	0.775671
$808$	0	0
$809$	20.6265	0.725188	0.362594	−	0.931947i	$-0.381891\pi$
$809$	20.6265	0.725188	0.362594	+	0.931947i	$0.381891\pi$
$810$	0	0
$811$	−45.2088	−1.58749	−0.793747	−	0.608248i	$-0.791873\pi$
$811$	−45.2088	−1.58749	−0.793747	+	0.608248i	$0.791873\pi$
$812$	0	0
$813$	−13.7132	−0.480942
$814$	0	0
$815$	−4.69176	−0.164345
$816$	0	0
$817$	−22.5295	−0.788208
$818$	0	0
$819$	−15.3539	−0.536509
$820$	0	0
$821$	23.2674	0.812039	0.406019	−	0.913864i	$-0.366917\pi$
$821$	23.2674	0.812039	0.406019	+	0.913864i	$0.366917\pi$
$822$	0	0
$823$	−2.12031	−0.0739095	−0.0369547	−	0.999317i	$-0.511766\pi$
$823$	−2.12031	−0.0739095	−0.0369547	+	0.999317i	$0.511766\pi$
$824$	0	0
$825$	−118.175	−4.11433
$826$	0	0
$827$	16.5417	0.575213	0.287606	−	0.957749i	$-0.407141\pi$
$827$	16.5417	0.575213	0.287606	+	0.957749i	$0.407141\pi$
$828$	0	0
$829$	2.61675	0.0908835	0.0454418	−	0.998967i	$-0.485530\pi$
$829$	2.61675	0.0908835	0.0454418	+	0.998967i	$0.485530\pi$
$830$	0	0
$831$	−13.1024	−0.454516
$832$	0	0
$833$	36.2252	1.25513
$834$	0	0
$835$	−3.50823	−0.121407
$836$	0	0
$837$	0.248850	0.00860152
$838$	0	0
$839$	−40.9480	−1.41368	−0.706841	−	0.707373i	$-0.749880\pi$
$839$	−40.9480	−1.41368	−0.706841	+	0.707373i	$0.749880\pi$
$840$	0	0
$841$	−23.6522	−0.815594
$842$	0	0
$843$	11.0279	0.379821
$844$	0	0
$845$	39.5178	1.35945
$846$	0	0
$847$	50.2428	1.72636
$848$	0	0
$849$	44.9068	1.54120
$850$	0	0
$851$	4.96961	0.170356
$852$	0	0
$853$	17.3933	0.595537	0.297768	−	0.954638i	$-0.403758\pi$
$853$	17.3933	0.595537	0.297768	+	0.954638i	$0.403758\pi$
$854$	0	0
$855$	40.5070	1.38531
$856$	0	0
$857$	16.7101	0.570805	0.285402	−	0.958408i	$-0.407873\pi$
$857$	16.7101	0.570805	0.285402	+	0.958408i	$0.407873\pi$
$858$	0	0
$859$	−13.9877	−0.477255	−0.238628	−	0.971111i	$-0.576698\pi$
$859$	−13.9877	−0.477255	−0.238628	+	0.971111i	$0.576698\pi$
$860$	0	0
$861$	105.002	3.57845
$862$	0	0
$863$	23.1575	0.788290	0.394145	−	0.919048i	$-0.371041\pi$
$863$	23.1575	0.788290	0.394145	+	0.919048i	$0.371041\pi$
$864$	0	0
$865$	−48.1446	−1.63696
$866$	0	0
$867$	−31.5619	−1.07190
$868$	0	0
$869$	−38.8628	−1.31833
$870$	0	0
$871$	21.3281	0.722676
$872$	0	0
$873$	22.2652	0.753564
$874$	0	0
$875$	−74.3418	−2.51321
$876$	0	0
$877$	40.3104	1.36119	0.680594	−	0.732661i	$-0.261722\pi$
$877$	40.3104	1.36119	0.680594	+	0.732661i	$0.261722\pi$
$878$	0	0
$879$	−30.4276	−1.02630
$880$	0	0
$881$	−23.2156	−0.782152	−0.391076	−	0.920358i	$-0.627897\pi$
$881$	−23.2156	−0.782152	−0.391076	+	0.920358i	$0.627897\pi$
$882$	0	0
$883$	0.920055	0.0309623	0.0154812	−	0.999880i	$-0.495072\pi$
$883$	0.920055	0.0309623	0.0154812	+	0.999880i	$0.495072\pi$
$884$	0	0
$885$	−52.7078	−1.77175
$886$	0	0
$887$	28.9082	0.970643	0.485321	−	0.874336i	$-0.338703\pi$
$887$	28.9082	0.970643	0.485321	+	0.874336i	$0.338703\pi$
$888$	0	0
$889$	−1.34234	−0.0450207
$890$	0	0
$891$	−51.2109	−1.71563
$892$	0	0
$893$	−45.2446	−1.51405
$894$	0	0
$895$	−39.0641	−1.30577
$896$	0	0
$897$	10.9679	0.366206
$898$	0	0
$899$	−0.461384	−0.0153880
$900$	0	0
$901$	7.30250	0.243282
$902$	0	0
$903$	−46.0196	−1.53144
$904$	0	0
$905$	66.4290	2.20817
$906$	0	0
$907$	−36.3884	−1.20826	−0.604128	−	0.796887i	$-0.706478\pi$
$907$	−36.3884	−1.20826	−0.604128	+	0.796887i	$0.706478\pi$
$908$	0	0
$909$	17.1257	0.568024
$910$	0	0
$911$	10.8289	0.358776	0.179388	−	0.983778i	$-0.442588\pi$
$911$	10.8289	0.358776	0.179388	+	0.983778i	$0.442588\pi$
$912$	0	0
$913$	−12.6774	−0.419562
$914$	0	0
$915$	−0.698214	−0.0230823
$916$	0	0
$917$	−20.2110	−0.667424
$918$	0	0
$919$	−28.3998	−0.936823	−0.468412	−	0.883510i	$-0.655174\pi$
$919$	−28.3998	−0.936823	−0.468412	+	0.883510i	$0.655174\pi$
$920$	0	0
$921$	−26.9747	−0.888847
$922$	0	0
$923$	−8.79712	−0.289561
$924$	0	0
$925$	18.2339	0.599526
$926$	0	0
$927$	0.259009	0.00850696
$928$	0	0
$929$	−32.8431	−1.07755	−0.538774	−	0.842451i	$-0.681112\pi$
$929$	−32.8431	−1.07755	−0.538774	+	0.842451i	$0.681112\pi$
$930$	0	0
$931$	−27.6473	−0.906104
$932$	0	0
$933$	−54.1361	−1.77233
$934$	0	0
$935$	−106.822	−3.49345
$936$	0	0
$937$	4.11911	0.134566	0.0672828	−	0.997734i	$-0.478567\pi$
$937$	4.11911	0.134566	0.0672828	+	0.997734i	$0.478567\pi$
$938$	0	0
$939$	18.4450	0.601930
$940$	0	0
$941$	−8.26409	−0.269402	−0.134701	−	0.990886i	$-0.543007\pi$
$941$	−8.26409	−0.269402	−0.134701	+	0.990886i	$0.543007\pi$
$942$	0	0
$943$	−33.8434	−1.10209
$944$	0	0
$945$	−17.8953	−0.582136
$946$	0	0
$947$	−43.3860	−1.40986	−0.704928	−	0.709279i	$-0.749021\pi$
$947$	−43.3860	−1.40986	−0.704928	+	0.709279i	$0.749021\pi$
$948$	0	0
$949$	23.7907	0.772277
$950$	0	0
$951$	5.57218	0.180690
$952$	0	0
$953$	−26.7790	−0.867457	−0.433728	−	0.901044i	$-0.642802\pi$
$953$	−26.7790	−0.867457	−0.433728	+	0.901044i	$0.642802\pi$
$954$	0	0
$955$	−92.6333	−2.99754
$956$	0	0
$957$	26.8412	0.867651
$958$	0	0
$959$	−63.1640	−2.03967
$960$	0	0
$961$	−30.9602	−0.998716
$962$	0	0
$963$	44.8519	1.44533
$964$	0	0
$965$	−29.1981	−0.939922
$966$	0	0
$967$	−7.46107	−0.239932	−0.119966	−	0.992778i	$-0.538279\pi$
$967$	−7.46107	−0.239932	−0.119966	+	0.992778i	$0.538279\pi$
$968$	0	0
$969$	54.4234	1.74833
$970$	0	0
$971$	21.4365	0.687931	0.343966	−	0.938982i	$-0.388230\pi$
$971$	21.4365	0.687931	0.343966	+	0.938982i	$0.388230\pi$
$972$	0	0
$973$	72.2006	2.31465
$974$	0	0
$975$	40.2419	1.28877
$976$	0	0
$977$	−30.9584	−0.990446	−0.495223	−	0.868766i	$-0.664914\pi$
$977$	−30.9584	−0.990446	−0.495223	+	0.868766i	$0.664914\pi$
$978$	0	0
$979$	−21.6110	−0.690690
$980$	0	0
$981$	36.1535	1.15429
$982$	0	0
$983$	−13.6397	−0.435040	−0.217520	−	0.976056i	$-0.569797\pi$
$983$	−13.6397	−0.435040	−0.217520	+	0.976056i	$0.569797\pi$
$984$	0	0
$985$	−7.23343	−0.230476
$986$	0	0
$987$	−92.4183	−2.94171
$988$	0	0
$989$	14.8327	0.471653
$990$	0	0
$991$	46.0458	1.46269	0.731347	−	0.682006i	$-0.238892\pi$
$991$	46.0458	1.46269	0.731347	+	0.682006i	$0.238892\pi$
$992$	0	0
$993$	−7.14508	−0.226742
$994$	0	0
$995$	13.5073	0.428211
$996$	0	0
$997$	57.1775	1.81083	0.905415	−	0.424527i	$-0.139560\pi$
$997$	57.1775	1.81083	0.905415	+	0.424527i	$0.139560\pi$
$998$	0	0
$999$	2.23373	0.0706719

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3856.2.a.j.1.1		7
4.3	odd	2		241.2.a.a.1.4	✓	7
12.11	even	2		2169.2.a.e.1.4		7
20.19	odd	2		6025.2.a.f.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
241.2.a.a.1.4	✓	7	4.3	odd	2
2169.2.a.e.1.4		7	12.11	even	2
3856.2.a.j.1.1		7	1.1	even	1	trivial
6025.2.a.f.1.4		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)

\(f(q)\)	\(=\)	\(q-2.33806 q^{3} -3.89634 q^{5} +3.68231 q^{7} +2.46654 q^{9} +O(q^{10})\)	\(q-2.33806 q^{3} -3.89634 q^{5} +3.68231 q^{7} +2.46654 q^{9} +4.96431 q^{11} -1.69048 q^{13} +9.10989 q^{15} +5.52260 q^{17} -4.21489 q^{19} -8.60948 q^{21} +2.77495 q^{23} +10.1815 q^{25} +1.24727 q^{27} -2.31253 q^{29} +0.199515 q^{31} -11.6069 q^{33} -14.3476 q^{35} +1.79089 q^{37} +3.95245 q^{39} -12.1960 q^{41} +5.34523 q^{43} -9.61047 q^{45} +10.7345 q^{47} +6.55944 q^{49} -12.9122 q^{51} +1.32229 q^{53} -19.3426 q^{55} +9.85467 q^{57} -5.78578 q^{59} -0.0766435 q^{61} +9.08256 q^{63} +6.58670 q^{65} -12.6166 q^{67} -6.48800 q^{69} +5.20391 q^{71} -14.0733 q^{73} -23.8050 q^{75} +18.2801 q^{77} -7.82844 q^{79} -10.3158 q^{81} -2.55372 q^{83} -21.5180 q^{85} +5.40683 q^{87} -4.35327 q^{89} -6.22489 q^{91} -0.466479 q^{93} +16.4226 q^{95} +9.02693 q^{97} +12.2446 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{3} - 8 q^{5} + 7 q^{7} - 2 q^{9}+O(q^{10}) \) 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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