LMFDB - Embedded newform 4864.2.a.bl.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4864,2,Mod(1,4864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4864, 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("4864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4864 = 2^{8} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4864.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:	$38.8392355432$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 4 $ x^4 - 7*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1216)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.792287$ of defining polynomial
Character	$\chi$	$=$	4864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.37228 q^{3} -0.792287 q^{5} +3.31662 q^{7} +2.62772 q^{9} +O(q^{10})$	$q-2.37228 q^{3} -0.792287 q^{5} +3.31662 q^{7} +2.62772 q^{9} -3.37228 q^{11} +4.10891 q^{13} +1.87953 q^{15} +5.00000 q^{17} -1.00000 q^{19} -7.86797 q^{21} +2.52434 q^{23} -4.37228 q^{25} +0.883156 q^{27} +5.98844 q^{29} +3.46410 q^{31} +8.00000 q^{33} -2.62772 q^{35} -3.16915 q^{37} -9.74749 q^{39} -9.37228 q^{43} -2.08191 q^{45} +9.30506 q^{47} +4.00000 q^{49} -11.8614 q^{51} +9.45254 q^{53} +2.67181 q^{55} +2.37228 q^{57} -4.37228 q^{59} +4.55134 q^{61} +8.71516 q^{63} -3.25544 q^{65} -8.37228 q^{67} -5.98844 q^{69} -6.63325 q^{71} +14.4891 q^{73} +10.3723 q^{75} -11.1846 q^{77} -13.5615 q^{79} -9.97825 q^{81} +8.00000 q^{83} -3.96143 q^{85} -14.2063 q^{87} -7.48913 q^{89} +13.6277 q^{91} -8.21782 q^{93} +0.792287 q^{95} +8.74456 q^{97} -8.86141 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 22 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 + 22 * q^9	$ 4 q + 2 q^{3} + 22 q^{9} - 2 q^{11} + 20 q^{17} - 4 q^{19} - 6 q^{25} + 38 q^{27} + 32 q^{33} - 22 q^{35} - 26 q^{43} + 16 q^{49} + 10 q^{51} - 2 q^{57} - 6 q^{59} - 36 q^{65} - 22 q^{67} + 12 q^{73} + 30 q^{75} + 52 q^{81} + 32 q^{83} + 16 q^{89} + 66 q^{91} + 12 q^{97} + 22 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 + 22 * q^9 - 2 * q^11 + 20 * q^17 - 4 * q^19 - 6 * q^25 + 38 * q^27 + 32 * q^33 - 22 * q^35 - 26 * q^43 + 16 * q^49 + 10 * q^51 - 2 * q^57 - 6 * q^59 - 36 * q^65 - 22 * q^67 + 12 * q^73 + 30 * q^75 + 52 * q^81 + 32 * q^83 + 16 * q^89 + 66 * q^91 + 12 * q^97 + 22 * 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
$3$	−2.37228	−1.36964	−0.684819	−	0.728714i	$-0.740119\pi$
$3$	−2.37228	−1.36964	−0.684819	+	0.728714i	$0.740119\pi$
$4$	0	0
$5$	−0.792287	−0.354322	−0.177161	−	0.984182i	$-0.556691\pi$
$5$	−0.792287	−0.354322	−0.177161	+	0.984182i	$0.556691\pi$
$6$	0	0
$7$	3.31662	1.25357	0.626783	−	0.779194i	$-0.284371\pi$
$7$	3.31662	1.25357	0.626783	+	0.779194i	$0.284371\pi$
$8$	0	0
$9$	2.62772	0.875906
$10$	0	0
$11$	−3.37228	−1.01678	−0.508391	−	0.861127i	$-0.669759\pi$
$11$	−3.37228	−1.01678	−0.508391	+	0.861127i	$0.669759\pi$
$12$	0	0
$13$	4.10891	1.13961	0.569804	−	0.821781i	$-0.307019\pi$
$13$	4.10891	1.13961	0.569804	+	0.821781i	$0.307019\pi$
$14$	0	0
$15$	1.87953	0.485292
$16$	0	0
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−7.86797	−1.71693
$22$	0	0
$23$	2.52434	0.526361	0.263180	−	0.964747i	$-0.415229\pi$
$23$	2.52434	0.526361	0.263180	+	0.964747i	$0.415229\pi$
$24$	0	0
$25$	−4.37228	−0.874456
$26$	0	0
$27$	0.883156	0.169963
$28$	0	0
$29$	5.98844	1.11203	0.556013	−	0.831174i	$-0.312331\pi$
$29$	5.98844	1.11203	0.556013	+	0.831174i	$0.312331\pi$
$30$	0	0
$31$	3.46410	0.622171	0.311086	−	0.950382i	$-0.399307\pi$
$31$	3.46410	0.622171	0.311086	+	0.950382i	$0.399307\pi$
$32$	0	0
$33$	8.00000	1.39262
$34$	0	0
$35$	−2.62772	−0.444166
$36$	0	0
$37$	−3.16915	−0.521005	−0.260502	−	0.965473i	$-0.583888\pi$
$37$	−3.16915	−0.521005	−0.260502	+	0.965473i	$0.583888\pi$
$38$	0	0
$39$	−9.74749	−1.56085
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−9.37228	−1.42926	−0.714630	−	0.699503i	$-0.753405\pi$
$43$	−9.37228	−1.42926	−0.714630	+	0.699503i	$0.753405\pi$
$44$	0	0
$45$	−2.08191	−0.310352
$46$	0	0
$47$	9.30506	1.35728	0.678642	−	0.734470i	$-0.262569\pi$
$47$	9.30506	1.35728	0.678642	+	0.734470i	$0.262569\pi$
$48$	0	0
$49$	4.00000	0.571429
$50$	0	0
$51$	−11.8614	−1.66093
$52$	0	0
$53$	9.45254	1.29841	0.649203	−	0.760615i	$-0.275102\pi$
$53$	9.45254	1.29841	0.649203	+	0.760615i	$0.275102\pi$
$54$	0	0
$55$	2.67181	0.360267
$56$	0	0
$57$	2.37228	0.314216
$58$	0	0
$59$	−4.37228	−0.569223	−0.284611	−	0.958643i	$-0.591865\pi$
$59$	−4.37228	−0.569223	−0.284611	+	0.958643i	$0.591865\pi$
$60$	0	0
$61$	4.55134	0.582740	0.291370	−	0.956610i	$-0.405889\pi$
$61$	4.55134	0.582740	0.291370	+	0.956610i	$0.405889\pi$
$62$	0	0
$63$	8.71516	1.09801
$64$	0	0
$65$	−3.25544	−0.403787
$66$	0	0
$67$	−8.37228	−1.02284	−0.511418	−	0.859332i	$-0.670880\pi$
$67$	−8.37228	−1.02284	−0.511418	+	0.859332i	$0.670880\pi$
$68$	0	0
$69$	−5.98844	−0.720923
$70$	0	0
$71$	−6.63325	−0.787222	−0.393611	−	0.919277i	$-0.628774\pi$
$71$	−6.63325	−0.787222	−0.393611	+	0.919277i	$0.628774\pi$
$72$	0	0
$73$	14.4891	1.69582	0.847912	−	0.530137i	$-0.177860\pi$
$73$	14.4891	1.69582	0.847912	+	0.530137i	$0.177860\pi$
$74$	0	0
$75$	10.3723	1.19769
$76$	0	0
$77$	−11.1846	−1.27460
$78$	0	0
$79$	−13.5615	−1.52578	−0.762891	−	0.646527i	$-0.776221\pi$
$79$	−13.5615	−1.52578	−0.762891	+	0.646527i	$0.776221\pi$
$80$	0	0
$81$	−9.97825	−1.10869
$82$	0	0
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	0	0
$85$	−3.96143	−0.429678
$86$	0	0
$87$	−14.2063	−1.52307
$88$	0	0
$89$	−7.48913	−0.793846	−0.396923	−	0.917852i	$-0.629922\pi$
$89$	−7.48913	−0.793846	−0.396923	+	0.917852i	$0.629922\pi$
$90$	0	0
$91$	13.6277	1.42857
$92$	0	0
$93$	−8.21782	−0.852149
$94$	0	0
$95$	0.792287	0.0812869
$96$	0	0
$97$	8.74456	0.887876	0.443938	−	0.896058i	$-0.353581\pi$
$97$	8.74456	0.887876	0.443938	+	0.896058i	$0.353581\pi$
$98$	0	0
$99$	−8.86141	−0.890605
$100$	0	0
$101$	−13.8564	−1.37876	−0.689382	−	0.724398i	$-0.742118\pi$
$101$	−13.8564	−1.37876	−0.689382	+	0.724398i	$0.742118\pi$
$102$	0	0
$103$	−5.34363	−0.526523	−0.263262	−	0.964724i	$-0.584798\pi$
$103$	−5.34363	−0.526523	−0.263262	+	0.964724i	$0.584798\pi$
$104$	0	0
$105$	6.23369	0.608346
$106$	0	0
$107$	−7.62772	−0.737399	−0.368700	−	0.929549i	$-0.620197\pi$
$107$	−7.62772	−0.737399	−0.368700	+	0.929549i	$0.620197\pi$
$108$	0	0
$109$	12.6217	1.20894	0.604469	−	0.796628i	$-0.293385\pi$
$109$	12.6217	1.20894	0.604469	+	0.796628i	$0.293385\pi$
$110$	0	0
$111$	7.51811	0.713587
$112$	0	0
$113$	−3.25544	−0.306246	−0.153123	−	0.988207i	$-0.548933\pi$
$113$	−3.25544	−0.306246	−0.153123	+	0.988207i	$0.548933\pi$
$114$	0	0
$115$	−2.00000	−0.186501
$116$	0	0
$117$	10.7971	0.998189
$118$	0	0
$119$	16.5831	1.52017
$120$	0	0
$121$	0.372281	0.0338438
$122$	0	0
$123$	0	0
$124$	0	0
$125$	7.42554	0.664160
$126$	0	0
$127$	18.9051	1.67755	0.838777	−	0.544475i	$-0.183271\pi$
$127$	18.9051	1.67755	0.838777	+	0.544475i	$0.183271\pi$
$128$	0	0
$129$	22.2337	1.95757
$130$	0	0
$131$	−1.37228	−0.119897	−0.0599484	−	0.998201i	$-0.519094\pi$
$131$	−1.37228	−0.119897	−0.0599484	+	0.998201i	$0.519094\pi$
$132$	0	0
$133$	−3.31662	−0.287588
$134$	0	0
$135$	−0.699713	−0.0602217
$136$	0	0
$137$	−5.00000	−0.427179	−0.213589	−	0.976924i	$-0.568515\pi$
$137$	−5.00000	−0.427179	−0.213589	+	0.976924i	$0.568515\pi$
$138$	0	0
$139$	−9.37228	−0.794947	−0.397473	−	0.917614i	$-0.630113\pi$
$139$	−9.37228	−0.794947	−0.397473	+	0.917614i	$0.630113\pi$
$140$	0	0
$141$	−22.0742	−1.85899
$142$	0	0
$143$	−13.8564	−1.15873
$144$	0	0
$145$	−4.74456	−0.394014
$146$	0	0
$147$	−9.48913	−0.782650
$148$	0	0
$149$	21.2819	1.74348	0.871742	−	0.489965i	$-0.162990\pi$
$149$	21.2819	1.74348	0.871742	+	0.489965i	$0.162990\pi$
$150$	0	0
$151$	−22.3692	−1.82038	−0.910189	−	0.414193i	$-0.864064\pi$
$151$	−22.3692	−1.82038	−0.910189	+	0.414193i	$0.864064\pi$
$152$	0	0
$153$	13.1386	1.06219
$154$	0	0
$155$	−2.74456	−0.220449
$156$	0	0
$157$	−1.28962	−0.102923	−0.0514615	−	0.998675i	$-0.516388\pi$
$157$	−1.28962	−0.102923	−0.0514615	+	0.998675i	$0.516388\pi$
$158$	0	0
$159$	−22.4241	−1.77835
$160$	0	0
$161$	8.37228	0.659828
$162$	0	0
$163$	0.744563	0.0583186	0.0291593	−	0.999575i	$-0.490717\pi$
$163$	0.744563	0.0583186	0.0291593	+	0.999575i	$0.490717\pi$
$164$	0	0
$165$	−6.33830	−0.493436
$166$	0	0
$167$	20.1947	1.56271	0.781356	−	0.624085i	$-0.214528\pi$
$167$	20.1947	1.56271	0.781356	+	0.624085i	$0.214528\pi$
$168$	0	0
$169$	3.88316	0.298704
$170$	0	0
$171$	−2.62772	−0.200947
$172$	0	0
$173$	4.75372	0.361419	0.180709	−	0.983537i	$-0.442161\pi$
$173$	4.75372	0.361419	0.180709	+	0.983537i	$0.442161\pi$
$174$	0	0
$175$	−14.5012	−1.09619
$176$	0	0
$177$	10.3723	0.779628
$178$	0	0
$179$	22.9783	1.71748	0.858738	−	0.512416i	$-0.171249\pi$
$179$	22.9783	1.71748	0.858738	+	0.512416i	$0.171249\pi$
$180$	0	0
$181$	−11.6819	−0.868311	−0.434155	−	0.900838i	$-0.642953\pi$
$181$	−11.6819	−0.868311	−0.434155	+	0.900838i	$0.642953\pi$
$182$	0	0
$183$	−10.7971	−0.798142
$184$	0	0
$185$	2.51087	0.184603
$186$	0	0
$187$	−16.8614	−1.23303
$188$	0	0
$189$	2.92910	0.213060
$190$	0	0
$191$	−13.7089	−0.991943	−0.495972	−	0.868339i	$-0.665188\pi$
$191$	−13.7089	−0.991943	−0.495972	+	0.868339i	$0.665188\pi$
$192$	0	0
$193$	14.7446	1.06134	0.530668	−	0.847580i	$-0.321941\pi$
$193$	14.7446	1.06134	0.530668	+	0.847580i	$0.321941\pi$
$194$	0	0
$195$	7.72281	0.553042
$196$	0	0
$197$	20.1947	1.43881	0.719406	−	0.694589i	$-0.244414\pi$
$197$	20.1947	1.43881	0.719406	+	0.694589i	$0.244414\pi$
$198$	0	0
$199$	6.78073	0.480673	0.240336	−	0.970690i	$-0.422742\pi$
$199$	6.78073	0.480673	0.240336	+	0.970690i	$0.422742\pi$
$200$	0	0
$201$	19.8614	1.40092
$202$	0	0
$203$	19.8614	1.39400
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.63325	0.461043
$208$	0	0
$209$	3.37228	0.233266
$210$	0	0
$211$	8.37228	0.576372	0.288186	−	0.957575i	$-0.406948\pi$
$211$	8.37228	0.576372	0.288186	+	0.957575i	$0.406948\pi$
$212$	0	0
$213$	15.7359	1.07821
$214$	0	0
$215$	7.42554	0.506417
$216$	0	0
$217$	11.4891	0.779933
$218$	0	0
$219$	−34.3723	−2.32266
$220$	0	0
$221$	20.5446	1.38198
$222$	0	0
$223$	−6.92820	−0.463947	−0.231973	−	0.972722i	$-0.574518\pi$
$223$	−6.92820	−0.463947	−0.231973	+	0.972722i	$0.574518\pi$
$224$	0	0
$225$	−11.4891	−0.765942
$226$	0	0
$227$	−3.11684	−0.206872	−0.103436	−	0.994636i	$-0.532984\pi$
$227$	−3.11684	−0.206872	−0.103436	+	0.994636i	$0.532984\pi$
$228$	0	0
$229$	−3.96143	−0.261779	−0.130889	−	0.991397i	$-0.541783\pi$
$229$	−3.96143	−0.261779	−0.130889	+	0.991397i	$0.541783\pi$
$230$	0	0
$231$	26.5330	1.74574
$232$	0	0
$233$	28.1168	1.84200	0.920998	−	0.389568i	$-0.127376\pi$
$233$	28.1168	1.84200	0.920998	+	0.389568i	$0.127376\pi$
$234$	0	0
$235$	−7.37228	−0.480915
$236$	0	0
$237$	32.1716	2.08977
$238$	0	0
$239$	−24.1012	−1.55898	−0.779490	−	0.626415i	$-0.784522\pi$
$239$	−24.1012	−1.55898	−0.779490	+	0.626415i	$0.784522\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	0	0
$243$	21.0217	1.34855
$244$	0	0
$245$	−3.16915	−0.202469
$246$	0	0
$247$	−4.10891	−0.261444
$248$	0	0
$249$	−18.9783	−1.20270
$250$	0	0
$251$	10.8614	0.685566	0.342783	−	0.939415i	$-0.388630\pi$
$251$	10.8614	0.685566	0.342783	+	0.939415i	$0.388630\pi$
$252$	0	0
$253$	−8.51278	−0.535194
$254$	0	0
$255$	9.39764	0.588503
$256$	0	0
$257$	18.2337	1.13739	0.568693	−	0.822550i	$-0.307449\pi$
$257$	18.2337	1.13739	0.568693	+	0.822550i	$0.307449\pi$
$258$	0	0
$259$	−10.5109	−0.653114
$260$	0	0
$261$	15.7359	0.974030
$262$	0	0
$263$	24.1561	1.48953	0.744766	−	0.667326i	$-0.232561\pi$
$263$	24.1561	1.48953	0.744766	+	0.667326i	$0.232561\pi$
$264$	0	0
$265$	−7.48913	−0.460053
$266$	0	0
$267$	17.7663	1.08728
$268$	0	0
$269$	−8.51278	−0.519033	−0.259517	−	0.965739i	$-0.583563\pi$
$269$	−8.51278	−0.519033	−0.259517	+	0.965739i	$0.583563\pi$
$270$	0	0
$271$	−7.57301	−0.460028	−0.230014	−	0.973187i	$-0.573877\pi$
$271$	−7.57301	−0.460028	−0.230014	+	0.973187i	$0.573877\pi$
$272$	0	0
$273$	−32.3288	−1.95663
$274$	0	0
$275$	14.7446	0.889131
$276$	0	0
$277$	−13.0641	−0.784947	−0.392473	−	0.919763i	$-0.628381\pi$
$277$	−13.0641	−0.784947	−0.392473	+	0.919763i	$0.628381\pi$
$278$	0	0
$279$	9.10268	0.544963
$280$	0	0
$281$	28.2337	1.68428	0.842140	−	0.539258i	$-0.181295\pi$
$281$	28.2337	1.68428	0.842140	+	0.539258i	$0.181295\pi$
$282$	0	0
$283$	23.3723	1.38934	0.694669	−	0.719330i	$-0.255551\pi$
$283$	23.3723	1.38934	0.694669	+	0.719330i	$0.255551\pi$
$284$	0	0
$285$	−1.87953	−0.111334
$286$	0	0
$287$	0	0
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	−20.7446	−1.21607
$292$	0	0
$293$	28.0627	1.63944	0.819719	−	0.572765i	$-0.194129\pi$
$293$	28.0627	1.63944	0.819719	+	0.572765i	$0.194129\pi$
$294$	0	0
$295$	3.46410	0.201688
$296$	0	0
$297$	−2.97825	−0.172816
$298$	0	0
$299$	10.3723	0.599845
$300$	0	0
$301$	−31.0843	−1.79167
$302$	0	0
$303$	32.8713	1.88841
$304$	0	0
$305$	−3.60597	−0.206477
$306$	0	0
$307$	26.7446	1.52639	0.763196	−	0.646167i	$-0.223629\pi$
$307$	26.7446	1.52639	0.763196	+	0.646167i	$0.223629\pi$
$308$	0	0
$309$	12.6766	0.721146
$310$	0	0
$311$	23.2164	1.31648	0.658240	−	0.752808i	$-0.271301\pi$
$311$	23.2164	1.31648	0.658240	+	0.752808i	$0.271301\pi$
$312$	0	0
$313$	−21.1168	−1.19359	−0.596797	−	0.802392i	$-0.703560\pi$
$313$	−21.1168	−1.19359	−0.596797	+	0.802392i	$0.703560\pi$
$314$	0	0
$315$	−6.90491	−0.389047
$316$	0	0
$317$	−18.2603	−1.02560	−0.512800	−	0.858508i	$-0.671392\pi$
$317$	−18.2603	−1.02560	−0.512800	+	0.858508i	$0.671392\pi$
$318$	0	0
$319$	−20.1947	−1.13069
$320$	0	0
$321$	18.0951	1.00997
$322$	0	0
$323$	−5.00000	−0.278207
$324$	0	0
$325$	−17.9653	−0.996537
$326$	0	0
$327$	−29.9422	−1.65581
$328$	0	0
$329$	30.8614	1.70144
$330$	0	0
$331$	17.8614	0.981752	0.490876	−	0.871230i	$-0.336677\pi$
$331$	17.8614	0.981752	0.490876	+	0.871230i	$0.336677\pi$
$332$	0	0
$333$	−8.32763	−0.456351
$334$	0	0
$335$	6.63325	0.362413
$336$	0	0
$337$	−24.7446	−1.34792	−0.673961	−	0.738767i	$-0.735408\pi$
$337$	−24.7446	−1.34792	−0.673961	+	0.738767i	$0.735408\pi$
$338$	0	0
$339$	7.72281	0.419446
$340$	0	0
$341$	−11.6819	−0.632612
$342$	0	0
$343$	−9.94987	−0.537243
$344$	0	0
$345$	4.74456	0.255439
$346$	0	0
$347$	−19.8832	−1.06738	−0.533692	−	0.845679i	$-0.679196\pi$
$347$	−19.8832	−1.06738	−0.533692	+	0.845679i	$0.679196\pi$
$348$	0	0
$349$	−23.1615	−1.23981	−0.619903	−	0.784679i	$-0.712828\pi$
$349$	−23.1615	−1.23981	−0.619903	+	0.784679i	$0.712828\pi$
$350$	0	0
$351$	3.62881	0.193692
$352$	0	0
$353$	21.1168	1.12394	0.561968	−	0.827159i	$-0.310044\pi$
$353$	21.1168	1.12394	0.561968	+	0.827159i	$0.310044\pi$
$354$	0	0
$355$	5.25544	0.278930
$356$	0	0
$357$	−39.3398	−2.08208
$358$	0	0
$359$	−16.5831	−0.875224	−0.437612	−	0.899164i	$-0.644176\pi$
$359$	−16.5831	−0.875224	−0.437612	+	0.899164i	$0.644176\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−0.883156	−0.0463537
$364$	0	0
$365$	−11.4795	−0.600867
$366$	0	0
$367$	30.5870	1.59663	0.798314	−	0.602241i	$-0.205725\pi$
$367$	30.5870	1.59663	0.798314	+	0.602241i	$0.205725\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	31.3505	1.62764
$372$	0	0
$373$	24.8935	1.28894	0.644469	−	0.764631i	$-0.277079\pi$
$373$	24.8935	1.28894	0.644469	+	0.764631i	$0.277079\pi$
$374$	0	0
$375$	−17.6155	−0.909659
$376$	0	0
$377$	24.6060	1.26727
$378$	0	0
$379$	15.3505	0.788504	0.394252	−	0.919002i	$-0.371004\pi$
$379$	15.3505	0.788504	0.394252	+	0.919002i	$0.371004\pi$
$380$	0	0
$381$	−44.8482	−2.29764
$382$	0	0
$383$	7.92287	0.404840	0.202420	−	0.979299i	$-0.435119\pi$
$383$	7.92287	0.404840	0.202420	+	0.979299i	$0.435119\pi$
$384$	0	0
$385$	8.86141	0.451619
$386$	0	0
$387$	−24.6277	−1.25190
$388$	0	0
$389$	−12.1793	−0.617513	−0.308756	−	0.951141i	$-0.599913\pi$
$389$	−12.1793	−0.617513	−0.308756	+	0.951141i	$0.599913\pi$
$390$	0	0
$391$	12.6217	0.638306
$392$	0	0
$393$	3.25544	0.164215
$394$	0	0
$395$	10.7446	0.540618
$396$	0	0
$397$	9.30506	0.467008	0.233504	−	0.972356i	$-0.424981\pi$
$397$	9.30506	0.467008	0.233504	+	0.972356i	$0.424981\pi$
$398$	0	0
$399$	7.86797	0.393891
$400$	0	0
$401$	−21.7228	−1.08479	−0.542393	−	0.840125i	$-0.682482\pi$
$401$	−21.7228	−1.08479	−0.542393	+	0.840125i	$0.682482\pi$
$402$	0	0
$403$	14.2337	0.709030
$404$	0	0
$405$	7.90564	0.392834
$406$	0	0
$407$	10.6873	0.529748
$408$	0	0
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	0	0
$411$	11.8614	0.585080
$412$	0	0
$413$	−14.5012	−0.713558
$414$	0	0
$415$	−6.33830	−0.311135
$416$	0	0
$417$	22.2337	1.08879
$418$	0	0
$419$	34.2337	1.67243	0.836213	−	0.548405i	$-0.184765\pi$
$419$	34.2337	1.67243	0.836213	+	0.548405i	$0.184765\pi$
$420$	0	0
$421$	35.9855	1.75383	0.876914	−	0.480647i	$-0.159598\pi$
$421$	35.9855	1.75383	0.876914	+	0.480647i	$0.159598\pi$
$422$	0	0
$423$	24.4511	1.18885
$424$	0	0
$425$	−21.8614	−1.06043
$426$	0	0
$427$	15.0951	0.730503
$428$	0	0
$429$	32.8713	1.58704
$430$	0	0
$431$	11.3870	0.548491	0.274246	−	0.961660i	$-0.411572\pi$
$431$	11.3870	0.548491	0.274246	+	0.961660i	$0.411572\pi$
$432$	0	0
$433$	21.4891	1.03270	0.516351	−	0.856377i	$-0.327290\pi$
$433$	21.4891	1.03270	0.516351	+	0.856377i	$0.327290\pi$
$434$	0	0
$435$	11.2554	0.539657
$436$	0	0
$437$	−2.52434	−0.120755
$438$	0	0
$439$	6.92820	0.330665	0.165333	−	0.986238i	$-0.447130\pi$
$439$	6.92820	0.330665	0.165333	+	0.986238i	$0.447130\pi$
$440$	0	0
$441$	10.5109	0.500518
$442$	0	0
$443$	10.3505	0.491769	0.245884	−	0.969299i	$-0.420922\pi$
$443$	10.3505	0.491769	0.245884	+	0.969299i	$0.420922\pi$
$444$	0	0
$445$	5.93354	0.281277
$446$	0	0
$447$	−50.4868	−2.38794
$448$	0	0
$449$	14.5109	0.684811	0.342405	−	0.939552i	$-0.388758\pi$
$449$	14.5109	0.684811	0.342405	+	0.939552i	$0.388758\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	53.0660	2.49326
$454$	0	0
$455$	−10.7971	−0.506174
$456$	0	0
$457$	−31.0000	−1.45012	−0.725059	−	0.688686i	$-0.758188\pi$
$457$	−31.0000	−1.45012	−0.725059	+	0.688686i	$0.758188\pi$
$458$	0	0
$459$	4.41578	0.206111
$460$	0	0
$461$	−11.1846	−0.520918	−0.260459	−	0.965485i	$-0.583874\pi$
$461$	−11.1846	−0.520918	−0.260459	+	0.965485i	$0.583874\pi$
$462$	0	0
$463$	8.31040	0.386217	0.193108	−	0.981177i	$-0.438143\pi$
$463$	8.31040	0.386217	0.193108	+	0.981177i	$0.438143\pi$
$464$	0	0
$465$	6.51087	0.301935
$466$	0	0
$467$	−28.1168	−1.30109	−0.650546	−	0.759467i	$-0.725460\pi$
$467$	−28.1168	−1.30109	−0.650546	+	0.759467i	$0.725460\pi$
$468$	0	0
$469$	−27.7677	−1.28219
$470$	0	0
$471$	3.05934	0.140967
$472$	0	0
$473$	31.6060	1.45324
$474$	0	0
$475$	4.37228	0.200614
$476$	0	0
$477$	24.8386	1.13728
$478$	0	0
$479$	14.1514	0.646592	0.323296	−	0.946298i	$-0.395209\pi$
$479$	14.1514	0.646592	0.323296	+	0.946298i	$0.395209\pi$
$480$	0	0
$481$	−13.0217	−0.593741
$482$	0	0
$483$	−19.8614	−0.903725
$484$	0	0
$485$	−6.92820	−0.314594
$486$	0	0
$487$	−29.2974	−1.32759	−0.663796	−	0.747914i	$-0.731056\pi$
$487$	−29.2974	−1.32759	−0.663796	+	0.747914i	$0.731056\pi$
$488$	0	0
$489$	−1.76631	−0.0798754
$490$	0	0
$491$	14.9783	0.675959	0.337979	−	0.941153i	$-0.390257\pi$
$491$	14.9783	0.675959	0.337979	+	0.941153i	$0.390257\pi$
$492$	0	0
$493$	29.9422	1.34853
$494$	0	0
$495$	7.02078	0.315560
$496$	0	0
$497$	−22.0000	−0.986835
$498$	0	0
$499$	−1.37228	−0.0614317	−0.0307159	−	0.999528i	$-0.509779\pi$
$499$	−1.37228	−0.0614317	−0.0307159	+	0.999528i	$0.509779\pi$
$500$	0	0
$501$	−47.9075	−2.14035
$502$	0	0
$503$	21.4294	0.955491	0.477745	−	0.878498i	$-0.341454\pi$
$503$	21.4294	0.955491	0.477745	+	0.878498i	$0.341454\pi$
$504$	0	0
$505$	10.9783	0.488526
$506$	0	0
$507$	−9.21194	−0.409117
$508$	0	0
$509$	24.5437	1.08788	0.543939	−	0.839124i	$-0.316932\pi$
$509$	24.5437	1.08788	0.543939	+	0.839124i	$0.316932\pi$
$510$	0	0
$511$	48.0550	2.12583
$512$	0	0
$513$	−0.883156	−0.0389923
$514$	0	0
$515$	4.23369	0.186559
$516$	0	0
$517$	−31.3793	−1.38006
$518$	0	0
$519$	−11.2772	−0.495013
$520$	0	0
$521$	40.2337	1.76267	0.881335	−	0.472492i	$-0.156645\pi$
$521$	40.2337	1.76267	0.881335	+	0.472492i	$0.156645\pi$
$522$	0	0
$523$	19.6277	0.858260	0.429130	−	0.903243i	$-0.358820\pi$
$523$	19.6277	0.858260	0.429130	+	0.903243i	$0.358820\pi$
$524$	0	0
$525$	34.4010	1.50138
$526$	0	0
$527$	17.3205	0.754493
$528$	0	0
$529$	−16.6277	−0.722944
$530$	0	0
$531$	−11.4891	−0.498586
$532$	0	0
$533$	0	0
$534$	0	0
$535$	6.04334	0.261276
$536$	0	0
$537$	−54.5109	−2.35232
$538$	0	0
$539$	−13.4891	−0.581018
$540$	0	0
$541$	33.5538	1.44259	0.721295	−	0.692628i	$-0.243547\pi$
$541$	33.5538	1.44259	0.721295	+	0.692628i	$0.243547\pi$
$542$	0	0
$543$	27.7128	1.18927
$544$	0	0
$545$	−10.0000	−0.428353
$546$	0	0
$547$	25.4891	1.08984	0.544918	−	0.838489i	$-0.316561\pi$
$547$	25.4891	1.08984	0.544918	+	0.838489i	$0.316561\pi$
$548$	0	0
$549$	11.9596	0.510425
$550$	0	0
$551$	−5.98844	−0.255116
$552$	0	0
$553$	−44.9783	−1.91267
$554$	0	0
$555$	−5.95650	−0.252839
$556$	0	0
$557$	−28.5051	−1.20780	−0.603900	−	0.797060i	$-0.706387\pi$
$557$	−28.5051	−1.20780	−0.603900	+	0.797060i	$0.706387\pi$
$558$	0	0
$559$	−38.5099	−1.62879
$560$	0	0
$561$	40.0000	1.68880
$562$	0	0
$563$	−9.48913	−0.399919	−0.199959	−	0.979804i	$-0.564081\pi$
$563$	−9.48913	−0.399919	−0.199959	+	0.979804i	$0.564081\pi$
$564$	0	0
$565$	2.57924	0.108509
$566$	0	0
$567$	−33.0941	−1.38982
$568$	0	0
$569$	−18.9783	−0.795610	−0.397805	−	0.917470i	$-0.630228\pi$
$569$	−18.9783	−0.795610	−0.397805	+	0.917470i	$0.630228\pi$
$570$	0	0
$571$	2.97825	0.124636	0.0623180	−	0.998056i	$-0.480151\pi$
$571$	2.97825	0.124636	0.0623180	+	0.998056i	$0.480151\pi$
$572$	0	0
$573$	32.5214	1.35860
$574$	0	0
$575$	−11.0371	−0.460280
$576$	0	0
$577$	−35.4674	−1.47653	−0.738263	−	0.674513i	$-0.764354\pi$
$577$	−35.4674	−1.47653	−0.738263	+	0.674513i	$0.764354\pi$
$578$	0	0
$579$	−34.9783	−1.45365
$580$	0	0
$581$	26.5330	1.10077
$582$	0	0
$583$	−31.8766	−1.32020
$584$	0	0
$585$	−8.55437	−0.353680
$586$	0	0
$587$	3.13859	0.129544	0.0647718	−	0.997900i	$-0.479368\pi$
$587$	3.13859	0.129544	0.0647718	+	0.997900i	$0.479368\pi$
$588$	0	0
$589$	−3.46410	−0.142736
$590$	0	0
$591$	−47.9075	−1.97065
$592$	0	0
$593$	−40.9783	−1.68278	−0.841388	−	0.540432i	$-0.818261\pi$
$593$	−40.9783	−1.68278	−0.841388	+	0.540432i	$0.818261\pi$
$594$	0	0
$595$	−13.1386	−0.538630
$596$	0	0
$597$	−16.0858	−0.658348
$598$	0	0
$599$	33.0564	1.35065	0.675325	−	0.737520i	$-0.264003\pi$
$599$	33.0564	1.35065	0.675325	+	0.737520i	$0.264003\pi$
$600$	0	0
$601$	−46.4674	−1.89544	−0.947722	−	0.319097i	$-0.896620\pi$
$601$	−46.4674	−1.89544	−0.947722	+	0.319097i	$0.896620\pi$
$602$	0	0
$603$	−22.0000	−0.895909
$604$	0	0
$605$	−0.294954	−0.0119916
$606$	0	0
$607$	7.62792	0.309608	0.154804	−	0.987945i	$-0.450525\pi$
$607$	7.62792	0.309608	0.154804	+	0.987945i	$0.450525\pi$
$608$	0	0
$609$	−47.1168	−1.90927
$610$	0	0
$611$	38.2337	1.54677
$612$	0	0
$613$	−35.1383	−1.41922	−0.709612	−	0.704592i	$-0.751130\pi$
$613$	−35.1383	−1.41922	−0.709612	+	0.704592i	$0.751130\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−33.8397	−1.36233	−0.681167	−	0.732128i	$-0.738527\pi$
$617$	−33.8397	−1.36233	−0.681167	+	0.732128i	$0.738527\pi$
$618$	0	0
$619$	−18.2337	−0.732874	−0.366437	−	0.930443i	$-0.619422\pi$
$619$	−18.2337	−0.732874	−0.366437	+	0.930443i	$0.619422\pi$
$620$	0	0
$621$	2.22938	0.0894621
$622$	0	0
$623$	−24.8386	−0.995138
$624$	0	0
$625$	15.9783	0.639130
$626$	0	0
$627$	−8.00000	−0.319489
$628$	0	0
$629$	−15.8457	−0.631811
$630$	0	0
$631$	−11.8843	−0.473107	−0.236553	−	0.971619i	$-0.576018\pi$
$631$	−11.8843	−0.473107	−0.236553	+	0.971619i	$0.576018\pi$
$632$	0	0
$633$	−19.8614	−0.789420
$634$	0	0
$635$	−14.9783	−0.594394
$636$	0	0
$637$	16.4356	0.651204
$638$	0	0
$639$	−17.4303	−0.689533
$640$	0	0
$641$	34.9783	1.38156	0.690779	−	0.723066i	$-0.257268\pi$
$641$	34.9783	1.38156	0.690779	+	0.723066i	$0.257268\pi$
$642$	0	0
$643$	−30.6277	−1.20784	−0.603920	−	0.797045i	$-0.706395\pi$
$643$	−30.6277	−1.20784	−0.603920	+	0.797045i	$0.706395\pi$
$644$	0	0
$645$	−17.6155	−0.693608
$646$	0	0
$647$	−16.5831	−0.651950	−0.325975	−	0.945378i	$-0.605693\pi$
$647$	−16.5831	−0.651950	−0.325975	+	0.945378i	$0.605693\pi$
$648$	0	0
$649$	14.7446	0.578775
$650$	0	0
$651$	−27.2554	−1.06822
$652$	0	0
$653$	−47.5200	−1.85960	−0.929800	−	0.368064i	$-0.880021\pi$
$653$	−47.5200	−1.85960	−0.929800	+	0.368064i	$0.880021\pi$
$654$	0	0
$655$	1.08724	0.0424820
$656$	0	0
$657$	38.0733	1.48538
$658$	0	0
$659$	18.1386	0.706579	0.353290	−	0.935514i	$-0.385063\pi$
$659$	18.1386	0.706579	0.353290	+	0.935514i	$0.385063\pi$
$660$	0	0
$661$	−19.2549	−0.748930	−0.374465	−	0.927241i	$-0.622174\pi$
$661$	−19.2549	−0.748930	−0.374465	+	0.927241i	$0.622174\pi$
$662$	0	0
$663$	−48.7375	−1.89281
$664$	0	0
$665$	2.62772	0.101899
$666$	0	0
$667$	15.1168	0.585327
$668$	0	0
$669$	16.4356	0.635439
$670$	0	0
$671$	−15.3484	−0.592519
$672$	0	0
$673$	5.25544	0.202582	0.101291	−	0.994857i	$-0.467703\pi$
$673$	5.25544	0.202582	0.101291	+	0.994857i	$0.467703\pi$
$674$	0	0
$675$	−3.86141	−0.148626
$676$	0	0
$677$	4.10891	0.157918	0.0789592	−	0.996878i	$-0.474840\pi$
$677$	4.10891	0.157918	0.0789592	+	0.996878i	$0.474840\pi$
$678$	0	0
$679$	29.0024	1.11301
$680$	0	0
$681$	7.39403	0.283340
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	0	0
$685$	3.96143	0.151359
$686$	0	0
$687$	9.39764	0.358542
$688$	0	0
$689$	38.8397	1.47967
$690$	0	0
$691$	10.8614	0.413187	0.206594	−	0.978427i	$-0.433762\pi$
$691$	10.8614	0.413187	0.206594	+	0.978427i	$0.433762\pi$
$692$	0	0
$693$	−29.3900	−1.11643
$694$	0	0
$695$	7.42554	0.281667
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−66.7011	−2.52287
$700$	0	0
$701$	−27.7128	−1.04670	−0.523349	−	0.852118i	$-0.675318\pi$
$701$	−27.7128	−1.04670	−0.523349	+	0.852118i	$0.675318\pi$
$702$	0	0
$703$	3.16915	0.119527
$704$	0	0
$705$	17.4891	0.658679
$706$	0	0
$707$	−45.9565	−1.72837
$708$	0	0
$709$	0.699713	0.0262783	0.0131391	−	0.999914i	$-0.495818\pi$
$709$	0.699713	0.0262783	0.0131391	+	0.999914i	$0.495818\pi$
$710$	0	0
$711$	−35.6357	−1.33644
$712$	0	0
$713$	8.74456	0.327486
$714$	0	0
$715$	10.9783	0.410563
$716$	0	0
$717$	57.1749	2.13524
$718$	0	0
$719$	5.89587	0.219879	0.109939	−	0.993938i	$-0.464934\pi$
$719$	5.89587	0.219879	0.109939	+	0.993938i	$0.464934\pi$
$720$	0	0
$721$	−17.7228	−0.660032
$722$	0	0
$723$	−18.9783	−0.705809
$724$	0	0
$725$	−26.1831	−0.972417
$726$	0	0
$727$	20.6371	0.765389	0.382694	−	0.923875i	$-0.374996\pi$
$727$	20.6371	0.765389	0.382694	+	0.923875i	$0.374996\pi$
$728$	0	0
$729$	−19.9348	−0.738324
$730$	0	0
$731$	−46.8614	−1.73323
$732$	0	0
$733$	−38.5099	−1.42239	−0.711197	−	0.702992i	$-0.751847\pi$
$733$	−38.5099	−1.42239	−0.711197	+	0.702992i	$0.751847\pi$
$734$	0	0
$735$	7.51811	0.277310
$736$	0	0
$737$	28.2337	1.04000
$738$	0	0
$739$	−27.6060	−1.01550	−0.507751	−	0.861504i	$-0.669523\pi$
$739$	−27.6060	−1.01550	−0.507751	+	0.861504i	$0.669523\pi$
$740$	0	0
$741$	9.74749	0.358083
$742$	0	0
$743$	−6.63325	−0.243350	−0.121675	−	0.992570i	$-0.538827\pi$
$743$	−6.63325	−0.243350	−0.121675	+	0.992570i	$0.538827\pi$
$744$	0	0
$745$	−16.8614	−0.617754
$746$	0	0
$747$	21.0217	0.769146
$748$	0	0
$749$	−25.2983	−0.924379
$750$	0	0
$751$	30.2921	1.10537	0.552686	−	0.833389i	$-0.313603\pi$
$751$	30.2921	1.10537	0.552686	+	0.833389i	$0.313603\pi$
$752$	0	0
$753$	−25.7663	−0.938977
$754$	0	0
$755$	17.7228	0.644999
$756$	0	0
$757$	48.6998	1.77002	0.885012	−	0.465568i	$-0.154150\pi$
$757$	48.6998	1.77002	0.885012	+	0.465568i	$0.154150\pi$
$758$	0	0
$759$	20.1947	0.733021
$760$	0	0
$761$	11.0000	0.398750	0.199375	−	0.979923i	$-0.436109\pi$
$761$	11.0000	0.398750	0.199375	+	0.979923i	$0.436109\pi$
$762$	0	0
$763$	41.8614	1.51548
$764$	0	0
$765$	−10.4095	−0.376358
$766$	0	0
$767$	−17.9653	−0.648690
$768$	0	0
$769$	−31.4674	−1.13474	−0.567371	−	0.823462i	$-0.692040\pi$
$769$	−31.4674	−1.13474	−0.567371	+	0.823462i	$0.692040\pi$
$770$	0	0
$771$	−43.2554	−1.55781
$772$	0	0
$773$	18.9600	0.681943	0.340972	−	0.940074i	$-0.389244\pi$
$773$	18.9600	0.681943	0.340972	+	0.940074i	$0.389244\pi$
$774$	0	0
$775$	−15.1460	−0.544061
$776$	0	0
$777$	24.9348	0.894529
$778$	0	0
$779$	0	0
$780$	0	0
$781$	22.3692	0.800432
$782$	0	0
$783$	5.28873	0.189004
$784$	0	0
$785$	1.02175	0.0364678
$786$	0	0
$787$	−20.6060	−0.734523	−0.367262	−	0.930118i	$-0.619705\pi$
$787$	−20.6060	−0.734523	−0.367262	+	0.930118i	$0.619705\pi$
$788$	0	0
$789$	−57.3052	−2.04012
$790$	0	0
$791$	−10.7971	−0.383899
$792$	0	0
$793$	18.7011	0.664094
$794$	0	0
$795$	17.7663	0.630106
$796$	0	0
$797$	11.3321	0.401402	0.200701	−	0.979652i	$-0.435678\pi$
$797$	11.3321	0.401402	0.200701	+	0.979652i	$0.435678\pi$
$798$	0	0
$799$	46.5253	1.64595
$800$	0	0
$801$	−19.6793	−0.695334
$802$	0	0
$803$	−48.8614	−1.72428
$804$	0	0
$805$	−6.63325	−0.233791
$806$	0	0
$807$	20.1947	0.710887
$808$	0	0
$809$	−7.97825	−0.280500	−0.140250	−	0.990116i	$-0.544791\pi$
$809$	−7.97825	−0.280500	−0.140250	+	0.990116i	$0.544791\pi$
$810$	0	0
$811$	−0.138593	−0.00486667	−0.00243334	−	0.999997i	$-0.500775\pi$
$811$	−0.138593	−0.00486667	−0.00243334	+	0.999997i	$0.500775\pi$
$812$	0	0
$813$	17.9653	0.630071
$814$	0	0
$815$	−0.589907	−0.0206636
$816$	0	0
$817$	9.37228	0.327895
$818$	0	0
$819$	35.8098	1.25130
$820$	0	0
$821$	37.0179	1.29193	0.645966	−	0.763366i	$-0.276455\pi$
$821$	37.0179	1.29193	0.645966	+	0.763366i	$0.276455\pi$
$822$	0	0
$823$	−27.2704	−0.950586	−0.475293	−	0.879828i	$-0.657658\pi$
$823$	−27.2704	−0.950586	−0.475293	+	0.879828i	$0.657658\pi$
$824$	0	0
$825$	−34.9783	−1.21779
$826$	0	0
$827$	11.3505	0.394697	0.197348	−	0.980333i	$-0.436767\pi$
$827$	11.3505	0.394697	0.197348	+	0.980333i	$0.436767\pi$
$828$	0	0
$829$	−33.1113	−1.15000	−0.575002	−	0.818152i	$-0.694999\pi$
$829$	−33.1113	−1.15000	−0.575002	+	0.818152i	$0.694999\pi$
$830$	0	0
$831$	30.9918	1.07509
$832$	0	0
$833$	20.0000	0.692959
$834$	0	0
$835$	−16.0000	−0.553703
$836$	0	0
$837$	3.05934	0.105746
$838$	0	0
$839$	−16.7306	−0.577604	−0.288802	−	0.957389i	$-0.593257\pi$
$839$	−16.7306	−0.577604	−0.288802	+	0.957389i	$0.593257\pi$
$840$	0	0
$841$	6.86141	0.236600
$842$	0	0
$843$	−66.9783	−2.30685
$844$	0	0
$845$	−3.07657	−0.105837
$846$	0	0
$847$	1.23472	0.0424254
$848$	0	0
$849$	−55.4456	−1.90289
$850$	0	0
$851$	−8.00000	−0.274236
$852$	0	0
$853$	12.6766	0.434038	0.217019	−	0.976167i	$-0.430367\pi$
$853$	12.6766	0.434038	0.217019	+	0.976167i	$0.430367\pi$
$854$	0	0
$855$	2.08191	0.0711997
$856$	0	0
$857$	12.2337	0.417895	0.208947	−	0.977927i	$-0.432996\pi$
$857$	12.2337	0.417895	0.208947	+	0.977927i	$0.432996\pi$
$858$	0	0
$859$	−31.6060	−1.07838	−0.539191	−	0.842184i	$-0.681270\pi$
$859$	−31.6060	−1.07838	−0.539191	+	0.842184i	$0.681270\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	49.1971	1.67469	0.837345	−	0.546675i	$-0.184107\pi$
$863$	49.1971	1.67469	0.837345	+	0.546675i	$0.184107\pi$
$864$	0	0
$865$	−3.76631	−0.128058
$866$	0	0
$867$	−18.9783	−0.644535
$868$	0	0
$869$	45.7330	1.55139
$870$	0	0
$871$	−34.4010	−1.16563
$872$	0	0
$873$	22.9783	0.777696
$874$	0	0
$875$	24.6277	0.832569
$876$	0	0
$877$	24.5986	0.830635	0.415317	−	0.909677i	$-0.363671\pi$
$877$	24.5986	0.830635	0.415317	+	0.909677i	$0.363671\pi$
$878$	0	0
$879$	−66.5725	−2.24544
$880$	0	0
$881$	1.37228	0.0462333	0.0231167	−	0.999733i	$-0.492641\pi$
$881$	1.37228	0.0462333	0.0231167	+	0.999733i	$0.492641\pi$
$882$	0	0
$883$	36.1168	1.21543	0.607714	−	0.794156i	$-0.292087\pi$
$883$	36.1168	1.21543	0.607714	+	0.794156i	$0.292087\pi$
$884$	0	0
$885$	−8.21782	−0.276239
$886$	0	0
$887$	−45.0333	−1.51207	−0.756035	−	0.654531i	$-0.772866\pi$
$887$	−45.0333	−1.51207	−0.756035	+	0.654531i	$0.772866\pi$
$888$	0	0
$889$	62.7011	2.10293
$890$	0	0
$891$	33.6495	1.12730
$892$	0	0
$893$	−9.30506	−0.311382
$894$	0	0
$895$	−18.2054	−0.608538
$896$	0	0
$897$	−24.6060	−0.821569
$898$	0	0
$899$	20.7446	0.691870
$900$	0	0
$901$	47.2627	1.57455
$902$	0	0
$903$	73.7408	2.45394
$904$	0	0
$905$	9.25544	0.307661
$906$	0	0
$907$	18.1386	0.602282	0.301141	−	0.953580i	$-0.402632\pi$
$907$	18.1386	0.602282	0.301141	+	0.953580i	$0.402632\pi$
$908$	0	0
$909$	−36.4107	−1.20767
$910$	0	0
$911$	7.81306	0.258858	0.129429	−	0.991589i	$-0.458686\pi$
$911$	7.81306	0.258858	0.129429	+	0.991589i	$0.458686\pi$
$912$	0	0
$913$	−26.9783	−0.892850
$914$	0	0
$915$	8.55437	0.282799
$916$	0	0
$917$	−4.55134	−0.150299
$918$	0	0
$919$	−27.7677	−0.915972	−0.457986	−	0.888959i	$-0.651429\pi$
$919$	−27.7677	−0.915972	−0.457986	+	0.888959i	$0.651429\pi$
$920$	0	0
$921$	−63.4456	−2.09060
$922$	0	0
$923$	−27.2554	−0.897124
$924$	0	0
$925$	13.8564	0.455596
$926$	0	0
$927$	−14.0416	−0.461185
$928$	0	0
$929$	−45.1168	−1.48024	−0.740118	−	0.672477i	$-0.765230\pi$
$929$	−45.1168	−1.48024	−0.740118	+	0.672477i	$0.765230\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	0	0
$933$	−55.0758	−1.80310
$934$	0	0
$935$	13.3591	0.436888
$936$	0	0
$937$	−23.9783	−0.783335	−0.391668	−	0.920107i	$-0.628102\pi$
$937$	−23.9783	−0.783335	−0.391668	+	0.920107i	$0.628102\pi$
$938$	0	0
$939$	50.0951	1.63479
$940$	0	0
$941$	5.69349	0.185602	0.0928012	−	0.995685i	$-0.470418\pi$
$941$	5.69349	0.185602	0.0928012	+	0.995685i	$0.470418\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−2.32069	−0.0754919
$946$	0	0
$947$	−37.2119	−1.20923	−0.604613	−	0.796520i	$-0.706672\pi$
$947$	−37.2119	−1.20923	−0.604613	+	0.796520i	$0.706672\pi$
$948$	0	0
$949$	59.5345	1.93257
$950$	0	0
$951$	43.3185	1.40470
$952$	0	0
$953$	8.23369	0.266715	0.133358	−	0.991068i	$-0.457424\pi$
$953$	8.23369	0.266715	0.133358	+	0.991068i	$0.457424\pi$
$954$	0	0
$955$	10.8614	0.351467
$956$	0	0
$957$	47.9075	1.54863
$958$	0	0
$959$	−16.5831	−0.535497
$960$	0	0
$961$	−19.0000	−0.612903
$962$	0	0
$963$	−20.0435	−0.645893
$964$	0	0
$965$	−11.6819	−0.376054
$966$	0	0
$967$	41.8642	1.34626	0.673131	−	0.739524i	$-0.264949\pi$
$967$	41.8642	1.34626	0.673131	+	0.739524i	$0.264949\pi$
$968$	0	0
$969$	11.8614	0.381043
$970$	0	0
$971$	−4.00000	−0.128366	−0.0641831	−	0.997938i	$-0.520444\pi$
$971$	−4.00000	−0.128366	−0.0641831	+	0.997938i	$0.520444\pi$
$972$	0	0
$973$	−31.0843	−0.996518
$974$	0	0
$975$	42.6188	1.36489
$976$	0	0
$977$	4.51087	0.144316	0.0721578	−	0.997393i	$-0.477011\pi$
$977$	4.51087	0.144316	0.0721578	+	0.997393i	$0.477011\pi$
$978$	0	0
$979$	25.2554	0.807167
$980$	0	0
$981$	33.1662	1.05892
$982$	0	0
$983$	6.22849	0.198658	0.0993290	−	0.995055i	$-0.468330\pi$
$983$	6.22849	0.198658	0.0993290	+	0.995055i	$0.468330\pi$
$984$	0	0
$985$	−16.0000	−0.509802
$986$	0	0
$987$	−73.2119	−2.33036
$988$	0	0
$989$	−23.6588	−0.752306
$990$	0	0
$991$	−25.2434	−0.801882	−0.400941	−	0.916104i	$-0.631317\pi$
$991$	−25.2434	−0.801882	−0.400941	+	0.916104i	$0.631317\pi$
$992$	0	0
$993$	−42.3723	−1.34464
$994$	0	0
$995$	−5.37228	−0.170313
$996$	0	0
$997$	10.2997	0.326196	0.163098	−	0.986610i	$-0.447851\pi$
$997$	10.2997	0.326196	0.163098	+	0.986610i	$0.447851\pi$
$998$	0	0
$999$	−2.79885	−0.0885518

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4864.2.a.bl.1.1		4
4.3	odd	2		4864.2.a.bi.1.3		4
8.3	odd	2	inner	4864.2.a.bl.1.2		4
8.5	even	2		4864.2.a.bi.1.4		4
16.3	odd	4		1216.2.c.h.609.6	yes	8
16.5	even	4		1216.2.c.h.609.5	yes	8
16.11	odd	4		1216.2.c.h.609.3	✓	8
16.13	even	4		1216.2.c.h.609.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.2.c.h.609.3	✓	8	16.11	odd	4
1216.2.c.h.609.4	yes	8	16.13	even	4
1216.2.c.h.609.5	yes	8	16.5	even	4
1216.2.c.h.609.6	yes	8	16.3	odd	4
4864.2.a.bi.1.3		4	4.3	odd	2
4864.2.a.bi.1.4		4	8.5	even	2
4864.2.a.bl.1.1		4	1.1	even	1	trivial
4864.2.a.bl.1.2		4	8.3	odd	2	inner

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:	\( N \)	\(=\)	\( 4864 = 2^{8} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4864.a (trivial)

\(f(q)\)	\(=\)	\(q-2.37228 q^{3} -0.792287 q^{5} +3.31662 q^{7} +2.62772 q^{9} +O(q^{10})\)	\(q-2.37228 q^{3} -0.792287 q^{5} +3.31662 q^{7} +2.62772 q^{9} -3.37228 q^{11} +4.10891 q^{13} +1.87953 q^{15} +5.00000 q^{17} -1.00000 q^{19} -7.86797 q^{21} +2.52434 q^{23} -4.37228 q^{25} +0.883156 q^{27} +5.98844 q^{29} +3.46410 q^{31} +8.00000 q^{33} -2.62772 q^{35} -3.16915 q^{37} -9.74749 q^{39} -9.37228 q^{43} -2.08191 q^{45} +9.30506 q^{47} +4.00000 q^{49} -11.8614 q^{51} +9.45254 q^{53} +2.67181 q^{55} +2.37228 q^{57} -4.37228 q^{59} +4.55134 q^{61} +8.71516 q^{63} -3.25544 q^{65} -8.37228 q^{67} -5.98844 q^{69} -6.63325 q^{71} +14.4891 q^{73} +10.3723 q^{75} -11.1846 q^{77} -13.5615 q^{79} -9.97825 q^{81} +8.00000 q^{83} -3.96143 q^{85} -14.2063 q^{87} -7.48913 q^{89} +13.6277 q^{91} -8.21782 q^{93} +0.792287 q^{95} +8.74456 q^{97} -8.86141 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 22 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 + 22 * q^9	\( 4 q + 2 q^{3} + 22 q^{9} - 2 q^{11} + 20 q^{17} - 4 q^{19} - 6 q^{25} + 38 q^{27} + 32 q^{33} - 22 q^{35} - 26 q^{43} + 16 q^{49} + 10 q^{51} - 2 q^{57} - 6 q^{59} - 36 q^{65} - 22 q^{67} + 12 q^{73} + 30 q^{75} + 52 q^{81} + 32 q^{83} + 16 q^{89} + 66 q^{91} + 12 q^{97} + 22 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 + 22 * q^9 - 2 * q^11 + 20 * q^17 - 4 * q^19 - 6 * q^25 + 38 * q^27 + 32 * q^33 - 22 * q^35 - 26 * q^43 + 16 * q^49 + 10 * q^51 - 2 * q^57 - 6 * q^59 - 36 * q^65 - 22 * q^67 + 12 * q^73 + 30 * q^75 + 52 * q^81 + 32 * q^83 + 16 * q^89 + 66 * q^91 + 12 * q^97 + 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more