LMFDB - Embedded newform 4864.2.a.bn.1.2

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:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 13x^{6} + 24x^{5} + 48x^{4} - 68x^{3} - 62x^{2} + 32x + 16 $ x^8 - 2x^7 - 13x^6 + 24x^5 + 48x^4 - 68x^3 - 62x^2 + 32*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 152)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.77833$ 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)

$f(q)$	$=$	$q-2.32921 q^{3} -3.13887 q^{5} -0.535658 q^{7} +2.42523 q^{9} +O(q^{10})$	$q-2.32921 q^{3} -3.13887 q^{5} -0.535658 q^{7} +2.42523 q^{9} -0.425227 q^{11} -6.65786 q^{13} +7.31110 q^{15} +7.33239 q^{17} +1.00000 q^{19} +1.24766 q^{21} -5.90033 q^{23} +4.85252 q^{25} +1.33877 q^{27} -0.837037 q^{29} +3.16999 q^{31} +0.990444 q^{33} +1.68136 q^{35} -3.49490 q^{37} +15.5076 q^{39} +0.123059 q^{41} +5.39744 q^{43} -7.61248 q^{45} +2.02928 q^{47} -6.71307 q^{49} -17.0787 q^{51} +5.82785 q^{53} +1.33473 q^{55} -2.32921 q^{57} +5.56633 q^{59} +6.99606 q^{61} -1.29909 q^{63} +20.8982 q^{65} +12.3777 q^{67} +13.7431 q^{69} +12.1786 q^{71} +6.99382 q^{73} -11.3025 q^{75} +0.227776 q^{77} -2.07996 q^{79} -10.3940 q^{81} +11.8227 q^{83} -23.0154 q^{85} +1.94964 q^{87} -13.7684 q^{89} +3.56633 q^{91} -7.38358 q^{93} -3.13887 q^{95} +0.801240 q^{97} -1.03127 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{5} - 4 q^{7} + 12 q^{9}+O(q^{10}) $ 8 * q - 8 * q^5 - 4 * q^7 + 12 * q^9	$ 8 q - 8 q^{5} - 4 q^{7} + 12 q^{9} + 4 q^{11} - 8 q^{13} - 4 q^{17} + 8 q^{19} - 16 q^{21} + 12 q^{25} - 28 q^{29} - 8 q^{31} - 12 q^{35} - 4 q^{37} + 4 q^{39} - 8 q^{41} - 4 q^{43} - 24 q^{45} - 12 q^{47} + 12 q^{49} + 12 q^{51} - 32 q^{53} + 8 q^{55} + 12 q^{59} - 8 q^{61} + 16 q^{63} + 8 q^{65} - 4 q^{67} - 28 q^{69} + 24 q^{71} - 24 q^{77} + 24 q^{79} - 8 q^{81} + 40 q^{83} - 24 q^{85} - 24 q^{87} + 8 q^{89} - 4 q^{91} - 32 q^{93} - 8 q^{95} + 16 q^{97} - 76 q^{99}+O(q^{100}) $ 8 * q - 8 * q^5 - 4 * q^7 + 12 * q^9 + 4 * q^11 - 8 * q^13 - 4 * q^17 + 8 * q^19 - 16 * q^21 + 12 * q^25 - 28 * q^29 - 8 * q^31 - 12 * q^35 - 4 * q^37 + 4 * q^39 - 8 * q^41 - 4 * q^43 - 24 * q^45 - 12 * q^47 + 12 * q^49 + 12 * q^51 - 32 * q^53 + 8 * q^55 + 12 * q^59 - 8 * q^61 + 16 * q^63 + 8 * q^65 - 4 * q^67 - 28 * q^69 + 24 * q^71 - 24 * q^77 + 24 * q^79 - 8 * q^81 + 40 * q^83 - 24 * q^85 - 24 * q^87 + 8 * q^89 - 4 * q^91 - 32 * q^93 - 8 * q^95 + 16 * q^97 - 76 * 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
$3$	−2.32921	−1.34477	−0.672385	−	0.740201i	$-0.734730\pi$
$3$	−2.32921	−1.34477	−0.672385	+	0.740201i	$0.734730\pi$
$4$	0	0
$5$	−3.13887	−1.40375	−0.701873	−	0.712302i	$-0.747653\pi$
$5$	−3.13887	−1.40375	−0.701873	+	0.712302i	$0.747653\pi$
$6$	0	0
$7$	−0.535658	−0.202460	−0.101230	−	0.994863i	$-0.532278\pi$
$7$	−0.535658	−0.202460	−0.101230	+	0.994863i	$0.532278\pi$
$8$	0	0
$9$	2.42523	0.808409
$10$	0	0
$11$	−0.425227	−0.128211	−0.0641054	−	0.997943i	$-0.520419\pi$
$11$	−0.425227	−0.128211	−0.0641054	+	0.997943i	$0.520419\pi$
$12$	0	0
$13$	−6.65786	−1.84656	−0.923279	−	0.384129i	$-0.874502\pi$
$13$	−6.65786	−1.84656	−0.923279	+	0.384129i	$0.874502\pi$
$14$	0	0
$15$	7.31110	1.88772
$16$	0	0
$17$	7.33239	1.77837	0.889183	−	0.457552i	$-0.151273\pi$
$17$	7.33239	1.77837	0.889183	+	0.457552i	$0.151273\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	1.24766	0.272262
$22$	0	0
$23$	−5.90033	−1.23030	−0.615152	−	0.788408i	$-0.710905\pi$
$23$	−5.90033	−1.23030	−0.615152	+	0.788408i	$0.710905\pi$
$24$	0	0
$25$	4.85252	0.970503
$26$	0	0
$27$	1.33877	0.257646
$28$	0	0
$29$	−0.837037	−0.155434	−0.0777170	−	0.996975i	$-0.524763\pi$
$29$	−0.837037	−0.155434	−0.0777170	+	0.996975i	$0.524763\pi$
$30$	0	0
$31$	3.16999	0.569347	0.284674	−	0.958625i	$-0.408115\pi$
$31$	3.16999	0.569347	0.284674	+	0.958625i	$0.408115\pi$
$32$	0	0
$33$	0.990444	0.172414
$34$	0	0
$35$	1.68136	0.284202
$36$	0	0
$37$	−3.49490	−0.574558	−0.287279	−	0.957847i	$-0.592751\pi$
$37$	−3.49490	−0.574558	−0.287279	+	0.957847i	$0.592751\pi$
$38$	0	0
$39$	15.5076	2.48320
$40$	0	0
$41$	0.123059	0.0192186	0.00960932	−	0.999954i	$-0.496941\pi$
$41$	0.123059	0.0192186	0.00960932	+	0.999954i	$0.496941\pi$
$42$	0	0
$43$	5.39744	0.823101	0.411551	−	0.911387i	$-0.364987\pi$
$43$	5.39744	0.823101	0.411551	+	0.911387i	$0.364987\pi$
$44$	0	0
$45$	−7.61248	−1.13480
$46$	0	0
$47$	2.02928	0.296001	0.148000	−	0.988987i	$-0.452716\pi$
$47$	2.02928	0.296001	0.148000	+	0.988987i	$0.452716\pi$
$48$	0	0
$49$	−6.71307	−0.959010
$50$	0	0
$51$	−17.0787	−2.39150
$52$	0	0
$53$	5.82785	0.800517	0.400259	−	0.916402i	$-0.368920\pi$
$53$	5.82785	0.800517	0.400259	+	0.916402i	$0.368920\pi$
$54$	0	0
$55$	1.33473	0.179975
$56$	0	0
$57$	−2.32921	−0.308512
$58$	0	0
$59$	5.56633	0.724675	0.362337	−	0.932047i	$-0.381979\pi$
$59$	5.56633	0.724675	0.362337	+	0.932047i	$0.381979\pi$
$60$	0	0
$61$	6.99606	0.895753	0.447877	−	0.894095i	$-0.352180\pi$
$61$	6.99606	0.895753	0.447877	+	0.894095i	$0.352180\pi$
$62$	0	0
$63$	−1.29909	−0.163670
$64$	0	0
$65$	20.8982	2.59210
$66$	0	0
$67$	12.3777	1.51217	0.756087	−	0.654471i	$-0.227109\pi$
$67$	12.3777	1.51217	0.756087	+	0.654471i	$0.227109\pi$
$68$	0	0
$69$	13.7431	1.65448
$70$	0	0
$71$	12.1786	1.44534	0.722669	−	0.691194i	$-0.242915\pi$
$71$	12.1786	1.44534	0.722669	+	0.691194i	$0.242915\pi$
$72$	0	0
$73$	6.99382	0.818565	0.409282	−	0.912408i	$-0.365779\pi$
$73$	6.99382	0.818565	0.409282	+	0.912408i	$0.365779\pi$
$74$	0	0
$75$	−11.3025	−1.30510
$76$	0	0
$77$	0.227776	0.0259575
$78$	0	0
$79$	−2.07996	−0.234014	−0.117007	−	0.993131i	$-0.537330\pi$
$79$	−2.07996	−0.234014	−0.117007	+	0.993131i	$0.537330\pi$
$80$	0	0
$81$	−10.3940	−1.15488
$82$	0	0
$83$	11.8227	1.29771	0.648853	−	0.760914i	$-0.275249\pi$
$83$	11.8227	1.29771	0.648853	+	0.760914i	$0.275249\pi$
$84$	0	0
$85$	−23.0154	−2.49637
$86$	0	0
$87$	1.94964	0.209023
$88$	0	0
$89$	−13.7684	−1.45945	−0.729723	−	0.683743i	$-0.760351\pi$
$89$	−13.7684	−1.45945	−0.729723	+	0.683743i	$0.760351\pi$
$90$	0	0
$91$	3.56633	0.373853
$92$	0	0
$93$	−7.38358	−0.765641
$94$	0	0
$95$	−3.13887	−0.322041
$96$	0	0
$97$	0.801240	0.0813536	0.0406768	−	0.999172i	$-0.487049\pi$
$97$	0.801240	0.0813536	0.0406768	+	0.999172i	$0.487049\pi$
$98$	0	0
$99$	−1.03127	−0.103647
$100$	0	0
$101$	−12.4171	−1.23555	−0.617773	−	0.786356i	$-0.711965\pi$
$101$	−12.4171	−1.23555	−0.617773	+	0.786356i	$0.711965\pi$
$102$	0	0
$103$	−2.84869	−0.280690	−0.140345	−	0.990103i	$-0.544821\pi$
$103$	−2.84869	−0.280690	−0.140345	+	0.990103i	$0.544821\pi$
$104$	0	0
$105$	−3.91624	−0.382186
$106$	0	0
$107$	−0.288142	−0.0278557	−0.0139278	−	0.999903i	$-0.504434\pi$
$107$	−0.288142	−0.0278557	−0.0139278	+	0.999903i	$0.504434\pi$
$108$	0	0
$109$	1.91873	0.183781	0.0918905	−	0.995769i	$-0.470709\pi$
$109$	1.91873	0.183781	0.0918905	+	0.995769i	$0.470709\pi$
$110$	0	0
$111$	8.14036	0.772649
$112$	0	0
$113$	−13.4370	−1.26405	−0.632024	−	0.774949i	$-0.717776\pi$
$113$	−13.4370	−1.26405	−0.632024	+	0.774949i	$0.717776\pi$
$114$	0	0
$115$	18.5204	1.72704
$116$	0	0
$117$	−16.1468	−1.49277
$118$	0	0
$119$	−3.92765	−0.360047
$120$	0	0
$121$	−10.8192	−0.983562
$122$	0	0
$123$	−0.286631	−0.0258447
$124$	0	0
$125$	0.462931	0.0414058
$126$	0	0
$127$	18.9992	1.68591	0.842953	−	0.537987i	$-0.180815\pi$
$127$	18.9992	1.68591	0.842953	+	0.537987i	$0.180815\pi$
$128$	0	0
$129$	−12.5718	−1.10688
$130$	0	0
$131$	−10.1330	−0.885322	−0.442661	−	0.896689i	$-0.645965\pi$
$131$	−10.1330	−0.885322	−0.442661	+	0.896689i	$0.645965\pi$
$132$	0	0
$133$	−0.535658	−0.0464474
$134$	0	0
$135$	−4.20222	−0.361670
$136$	0	0
$137$	3.28221	0.280418	0.140209	−	0.990122i	$-0.455222\pi$
$137$	3.28221	0.280418	0.140209	+	0.990122i	$0.455222\pi$
$138$	0	0
$139$	−10.5345	−0.893527	−0.446763	−	0.894652i	$-0.647423\pi$
$139$	−10.5345	−0.893527	−0.446763	+	0.894652i	$0.647423\pi$
$140$	0	0
$141$	−4.72662	−0.398053
$142$	0	0
$143$	2.83110	0.236749
$144$	0	0
$145$	2.62735	0.218190
$146$	0	0
$147$	15.6362	1.28965
$148$	0	0
$149$	−5.93244	−0.486005	−0.243002	−	0.970026i	$-0.578132\pi$
$149$	−5.93244	−0.486005	−0.243002	+	0.970026i	$0.578132\pi$
$150$	0	0
$151$	4.57907	0.372639	0.186320	−	0.982489i	$-0.440344\pi$
$151$	4.57907	0.372639	0.186320	+	0.982489i	$0.440344\pi$
$152$	0	0
$153$	17.7827	1.43765
$154$	0	0
$155$	−9.95019	−0.799219
$156$	0	0
$157$	−3.80861	−0.303960	−0.151980	−	0.988384i	$-0.548565\pi$
$157$	−3.80861	−0.303960	−0.151980	+	0.988384i	$0.548565\pi$
$158$	0	0
$159$	−13.5743	−1.07651
$160$	0	0
$161$	3.16056	0.249087
$162$	0	0
$163$	−17.9293	−1.40433	−0.702164	−	0.712016i	$-0.747783\pi$
$163$	−17.9293	−1.40433	−0.702164	+	0.712016i	$0.747783\pi$
$164$	0	0
$165$	−3.10888	−0.242026
$166$	0	0
$167$	18.8189	1.45625	0.728123	−	0.685446i	$-0.240393\pi$
$167$	18.8189	1.45625	0.728123	+	0.685446i	$0.240393\pi$
$168$	0	0
$169$	31.3271	2.40978
$170$	0	0
$171$	2.42523	0.185462
$172$	0	0
$173$	−5.87750	−0.446858	−0.223429	−	0.974720i	$-0.571725\pi$
$173$	−5.87750	−0.446858	−0.223429	+	0.974720i	$0.571725\pi$
$174$	0	0
$175$	−2.59929	−0.196488
$176$	0	0
$177$	−12.9652	−0.974522
$178$	0	0
$179$	−18.9245	−1.41448	−0.707241	−	0.706973i	$-0.750060\pi$
$179$	−18.9245	−1.41448	−0.707241	+	0.706973i	$0.750060\pi$
$180$	0	0
$181$	10.5330	0.782914	0.391457	−	0.920196i	$-0.371971\pi$
$181$	10.5330	0.782914	0.391457	+	0.920196i	$0.371971\pi$
$182$	0	0
$183$	−16.2953	−1.20458
$184$	0	0
$185$	10.9700	0.806534
$186$	0	0
$187$	−3.11793	−0.228006
$188$	0	0
$189$	−0.717121	−0.0521629
$190$	0	0
$191$	−1.60481	−0.116120	−0.0580600	−	0.998313i	$-0.518491\pi$
$191$	−1.60481	−0.116120	−0.0580600	+	0.998313i	$0.518491\pi$
$192$	0	0
$193$	−13.2889	−0.956554	−0.478277	−	0.878209i	$-0.658739\pi$
$193$	−13.2889	−0.956554	−0.478277	+	0.878209i	$0.658739\pi$
$194$	0	0
$195$	−48.6763	−3.48578
$196$	0	0
$197$	−4.06689	−0.289754	−0.144877	−	0.989450i	$-0.546279\pi$
$197$	−4.06689	−0.289754	−0.144877	+	0.989450i	$0.546279\pi$
$198$	0	0
$199$	12.8273	0.909307	0.454653	−	0.890668i	$-0.349763\pi$
$199$	12.8273	0.909307	0.454653	+	0.890668i	$0.349763\pi$
$200$	0	0
$201$	−28.8302	−2.03353
$202$	0	0
$203$	0.448365	0.0314691
$204$	0	0
$205$	−0.386268	−0.0269781
$206$	0	0
$207$	−14.3096	−0.994589
$208$	0	0
$209$	−0.425227	−0.0294136
$210$	0	0
$211$	15.9906	1.10084	0.550420	−	0.834888i	$-0.314468\pi$
$211$	15.9906	1.10084	0.550420	+	0.834888i	$0.314468\pi$
$212$	0	0
$213$	−28.3666	−1.94365
$214$	0	0
$215$	−16.9419	−1.15543
$216$	0	0
$217$	−1.69803	−0.115270
$218$	0	0
$219$	−16.2901	−1.10078
$220$	0	0
$221$	−48.8181	−3.28386
$222$	0	0
$223$	−21.1424	−1.41580	−0.707899	−	0.706314i	$-0.750357\pi$
$223$	−21.1424	−1.41580	−0.707899	+	0.706314i	$0.750357\pi$
$224$	0	0
$225$	11.7685	0.784564
$226$	0	0
$227$	22.2056	1.47384	0.736918	−	0.675982i	$-0.236280\pi$
$227$	22.2056	1.47384	0.736918	+	0.675982i	$0.236280\pi$
$228$	0	0
$229$	−12.6331	−0.834816	−0.417408	−	0.908719i	$-0.637061\pi$
$229$	−12.6331	−0.834816	−0.417408	+	0.908719i	$0.637061\pi$
$230$	0	0
$231$	−0.530539	−0.0349069
$232$	0	0
$233$	−3.01852	−0.197750	−0.0988749	−	0.995100i	$-0.531524\pi$
$233$	−3.01852	−0.197750	−0.0988749	+	0.995100i	$0.531524\pi$
$234$	0	0
$235$	−6.36965	−0.415510
$236$	0	0
$237$	4.84468	0.314695
$238$	0	0
$239$	−12.7156	−0.822504	−0.411252	−	0.911522i	$-0.634908\pi$
$239$	−12.7156	−0.822504	−0.411252	+	0.911522i	$0.634908\pi$
$240$	0	0
$241$	−1.05734	−0.0681091	−0.0340545	−	0.999420i	$-0.510842\pi$
$241$	−1.05734	−0.0681091	−0.0340545	+	0.999420i	$0.510842\pi$
$242$	0	0
$243$	20.1934	1.29541
$244$	0	0
$245$	21.0715	1.34621
$246$	0	0
$247$	−6.65786	−0.423630
$248$	0	0
$249$	−27.5375	−1.74512
$250$	0	0
$251$	13.4689	0.850151	0.425076	−	0.905158i	$-0.360248\pi$
$251$	13.4689	0.850151	0.425076	+	0.905158i	$0.360248\pi$
$252$	0	0
$253$	2.50898	0.157738
$254$	0	0
$255$	53.6078	3.35705
$256$	0	0
$257$	19.1631	1.19536	0.597680	−	0.801734i	$-0.296089\pi$
$257$	19.1631	1.19536	0.597680	+	0.801734i	$0.296089\pi$
$258$	0	0
$259$	1.87207	0.116325
$260$	0	0
$261$	−2.03001	−0.125654
$262$	0	0
$263$	−2.55940	−0.157819	−0.0789097	−	0.996882i	$-0.525144\pi$
$263$	−2.55940	−0.157819	−0.0789097	+	0.996882i	$0.525144\pi$
$264$	0	0
$265$	−18.2929	−1.12372
$266$	0	0
$267$	32.0695	1.96262
$268$	0	0
$269$	4.88596	0.297902	0.148951	−	0.988845i	$-0.452410\pi$
$269$	4.88596	0.297902	0.148951	+	0.988845i	$0.452410\pi$
$270$	0	0
$271$	−32.1299	−1.95175	−0.975875	−	0.218329i	$-0.929940\pi$
$271$	−32.1299	−1.95175	−0.975875	+	0.218329i	$0.929940\pi$
$272$	0	0
$273$	−8.30675	−0.502747
$274$	0	0
$275$	−2.06342	−0.124429
$276$	0	0
$277$	7.46639	0.448612	0.224306	−	0.974519i	$-0.427988\pi$
$277$	7.46639	0.448612	0.224306	+	0.974519i	$0.427988\pi$
$278$	0	0
$279$	7.68795	0.460265
$280$	0	0
$281$	−18.9770	−1.13208	−0.566038	−	0.824379i	$-0.691524\pi$
$281$	−18.9770	−1.13208	−0.566038	+	0.824379i	$0.691524\pi$
$282$	0	0
$283$	1.24321	0.0739012	0.0369506	−	0.999317i	$-0.488236\pi$
$283$	1.24321	0.0739012	0.0369506	+	0.999317i	$0.488236\pi$
$284$	0	0
$285$	7.31110	0.433072
$286$	0	0
$287$	−0.0659177	−0.00389100
$288$	0	0
$289$	36.7640	2.16259
$290$	0	0
$291$	−1.86626	−0.109402
$292$	0	0
$293$	−5.35018	−0.312561	−0.156280	−	0.987713i	$-0.549950\pi$
$293$	−5.35018	−0.312561	−0.156280	+	0.987713i	$0.549950\pi$
$294$	0	0
$295$	−17.4720	−1.01726
$296$	0	0
$297$	−0.569280	−0.0330330
$298$	0	0
$299$	39.2836	2.27183
$300$	0	0
$301$	−2.89118	−0.166645
$302$	0	0
$303$	28.9220	1.66153
$304$	0	0
$305$	−21.9597	−1.25741
$306$	0	0
$307$	10.4800	0.598123	0.299062	−	0.954234i	$-0.403326\pi$
$307$	10.4800	0.598123	0.299062	+	0.954234i	$0.403326\pi$
$308$	0	0
$309$	6.63521	0.377464
$310$	0	0
$311$	−27.3720	−1.55212	−0.776061	−	0.630658i	$-0.782785\pi$
$311$	−27.3720	−1.55212	−0.776061	+	0.630658i	$0.782785\pi$
$312$	0	0
$313$	16.9214	0.956451	0.478226	−	0.878237i	$-0.341280\pi$
$313$	16.9214	0.956451	0.478226	+	0.878237i	$0.341280\pi$
$314$	0	0
$315$	4.07768	0.229751
$316$	0	0
$317$	−3.82118	−0.214619	−0.107309	−	0.994226i	$-0.534224\pi$
$317$	−3.82118	−0.214619	−0.107309	+	0.994226i	$0.534224\pi$
$318$	0	0
$319$	0.355931	0.0199283
$320$	0	0
$321$	0.671143	0.0374595
$322$	0	0
$323$	7.33239	0.407985
$324$	0	0
$325$	−32.3074	−1.79209
$326$	0	0
$327$	−4.46913	−0.247143
$328$	0	0
$329$	−1.08700	−0.0599282
$330$	0	0
$331$	−26.6082	−1.46252	−0.731258	−	0.682101i	$-0.761067\pi$
$331$	−26.6082	−1.46252	−0.731258	+	0.682101i	$0.761067\pi$
$332$	0	0
$333$	−8.47592	−0.464478
$334$	0	0
$335$	−38.8520	−2.12271
$336$	0	0
$337$	−11.6884	−0.636707	−0.318354	−	0.947972i	$-0.603130\pi$
$337$	−11.6884	−0.636707	−0.318354	+	0.947972i	$0.603130\pi$
$338$	0	0
$339$	31.2976	1.69985
$340$	0	0
$341$	−1.34797	−0.0729964
$342$	0	0
$343$	7.34551	0.396620
$344$	0	0
$345$	−43.1379	−2.32247
$346$	0	0
$347$	−16.2573	−0.872739	−0.436370	−	0.899767i	$-0.643736\pi$
$347$	−16.2573	−0.872739	−0.436370	+	0.899767i	$0.643736\pi$
$348$	0	0
$349$	−22.5885	−1.20913	−0.604567	−	0.796554i	$-0.706654\pi$
$349$	−22.5885	−1.20913	−0.604567	+	0.796554i	$0.706654\pi$
$350$	0	0
$351$	−8.91333	−0.475758
$352$	0	0
$353$	−10.2804	−0.547169	−0.273584	−	0.961848i	$-0.588209\pi$
$353$	−10.2804	−0.547169	−0.273584	+	0.961848i	$0.588209\pi$
$354$	0	0
$355$	−38.2272	−2.02889
$356$	0	0
$357$	9.14833	0.484181
$358$	0	0
$359$	−16.1694	−0.853386	−0.426693	−	0.904397i	$-0.640321\pi$
$359$	−16.1694	−0.853386	−0.426693	+	0.904397i	$0.640321\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	25.2002	1.32267
$364$	0	0
$365$	−21.9527	−1.14906
$366$	0	0
$367$	−19.5371	−1.01983	−0.509914	−	0.860225i	$-0.670323\pi$
$367$	−19.5371	−1.01983	−0.509914	+	0.860225i	$0.670323\pi$
$368$	0	0
$369$	0.298447	0.0155365
$370$	0	0
$371$	−3.12173	−0.162072
$372$	0	0
$373$	−12.9672	−0.671418	−0.335709	−	0.941966i	$-0.608976\pi$
$373$	−12.9672	−0.671418	−0.335709	+	0.941966i	$0.608976\pi$
$374$	0	0
$375$	−1.07826	−0.0556813
$376$	0	0
$377$	5.57288	0.287018
$378$	0	0
$379$	−0.229262	−0.0117764	−0.00588821	−	0.999983i	$-0.501874\pi$
$379$	−0.229262	−0.0117764	−0.00588821	+	0.999983i	$0.501874\pi$
$380$	0	0
$381$	−44.2532	−2.26716
$382$	0	0
$383$	18.9168	0.966601	0.483301	−	0.875455i	$-0.339438\pi$
$383$	18.9168	0.966601	0.483301	+	0.875455i	$0.339438\pi$
$384$	0	0
$385$	−0.714960	−0.0364377
$386$	0	0
$387$	13.0900	0.665403
$388$	0	0
$389$	−9.38404	−0.475790	−0.237895	−	0.971291i	$-0.576457\pi$
$389$	−9.38404	−0.475790	−0.237895	+	0.971291i	$0.576457\pi$
$390$	0	0
$391$	−43.2636	−2.18793
$392$	0	0
$393$	23.6018	1.19055
$394$	0	0
$395$	6.52874	0.328496
$396$	0	0
$397$	18.3307	0.919990	0.459995	−	0.887921i	$-0.347851\pi$
$397$	18.3307	0.919990	0.459995	+	0.887921i	$0.347851\pi$
$398$	0	0
$399$	1.24766	0.0624611
$400$	0	0
$401$	−22.7518	−1.13617	−0.568085	−	0.822970i	$-0.692316\pi$
$401$	−22.7518	−1.13617	−0.568085	+	0.822970i	$0.692316\pi$
$402$	0	0
$403$	−21.1054	−1.05133
$404$	0	0
$405$	32.6253	1.62116
$406$	0	0
$407$	1.48613	0.0736645
$408$	0	0
$409$	19.3401	0.956305	0.478152	−	0.878277i	$-0.341307\pi$
$409$	19.3401	0.956305	0.478152	+	0.878277i	$0.341307\pi$
$410$	0	0
$411$	−7.64497	−0.377099
$412$	0	0
$413$	−2.98165	−0.146717
$414$	0	0
$415$	−37.1098	−1.82165
$416$	0	0
$417$	24.5371	1.20159
$418$	0	0
$419$	−28.0225	−1.36899	−0.684495	−	0.729017i	$-0.739977\pi$
$419$	−28.0225	−1.36899	−0.684495	+	0.729017i	$0.739977\pi$
$420$	0	0
$421$	15.5487	0.757798	0.378899	−	0.925438i	$-0.376303\pi$
$421$	15.5487	0.757798	0.378899	+	0.925438i	$0.376303\pi$
$422$	0	0
$423$	4.92146	0.239290
$424$	0	0
$425$	35.5806	1.72591
$426$	0	0
$427$	−3.74749	−0.181354
$428$	0	0
$429$	−6.59424	−0.318373
$430$	0	0
$431$	−15.5573	−0.749368	−0.374684	−	0.927153i	$-0.622249\pi$
$431$	−15.5573	−0.749368	−0.374684	+	0.927153i	$0.622249\pi$
$432$	0	0
$433$	4.16219	0.200022	0.100011	−	0.994986i	$-0.468112\pi$
$433$	4.16219	0.200022	0.100011	+	0.994986i	$0.468112\pi$
$434$	0	0
$435$	−6.11966	−0.293415
$436$	0	0
$437$	−5.90033	−0.282251
$438$	0	0
$439$	24.6600	1.17696	0.588478	−	0.808513i	$-0.299727\pi$
$439$	24.6600	1.17696	0.588478	+	0.808513i	$0.299727\pi$
$440$	0	0
$441$	−16.2807	−0.775272
$442$	0	0
$443$	−19.7595	−0.938801	−0.469401	−	0.882985i	$-0.655530\pi$
$443$	−19.7595	−0.938801	−0.469401	+	0.882985i	$0.655530\pi$
$444$	0	0
$445$	43.2172	2.04869
$446$	0	0
$447$	13.8179	0.653565
$448$	0	0
$449$	−16.4826	−0.777862	−0.388931	−	0.921267i	$-0.627156\pi$
$449$	−16.4826	−0.777862	−0.388931	+	0.921267i	$0.627156\pi$
$450$	0	0
$451$	−0.0523282	−0.00246404
$452$	0	0
$453$	−10.6656	−0.501115
$454$	0	0
$455$	−11.1943	−0.524795
$456$	0	0
$457$	−7.44504	−0.348264	−0.174132	−	0.984722i	$-0.555712\pi$
$457$	−7.44504	−0.348264	−0.174132	+	0.984722i	$0.555712\pi$
$458$	0	0
$459$	9.81637	0.458189
$460$	0	0
$461$	14.9337	0.695532	0.347766	−	0.937581i	$-0.386940\pi$
$461$	14.9337	0.695532	0.347766	+	0.937581i	$0.386940\pi$
$462$	0	0
$463$	23.6042	1.09698	0.548491	−	0.836156i	$-0.315202\pi$
$463$	23.6042	1.09698	0.548491	+	0.836156i	$0.315202\pi$
$464$	0	0
$465$	23.1761	1.07477
$466$	0	0
$467$	−0.808996	−0.0374359	−0.0187179	−	0.999825i	$-0.505958\pi$
$467$	−0.808996	−0.0374359	−0.0187179	+	0.999825i	$0.505958\pi$
$468$	0	0
$469$	−6.63020	−0.306154
$470$	0	0
$471$	8.87106	0.408757
$472$	0	0
$473$	−2.29514	−0.105530
$474$	0	0
$475$	4.85252	0.222649
$476$	0	0
$477$	14.1339	0.647145
$478$	0	0
$479$	12.2547	0.559933	0.279966	−	0.960010i	$-0.409677\pi$
$479$	12.2547	0.559933	0.279966	+	0.960010i	$0.409677\pi$
$480$	0	0
$481$	23.2686	1.06095
$482$	0	0
$483$	−7.36161	−0.334965
$484$	0	0
$485$	−2.51499	−0.114200
$486$	0	0
$487$	23.1692	1.04990	0.524948	−	0.851134i	$-0.324085\pi$
$487$	23.1692	1.04990	0.524948	+	0.851134i	$0.324085\pi$
$488$	0	0
$489$	41.7610	1.88850
$490$	0	0
$491$	20.6669	0.932684	0.466342	−	0.884605i	$-0.345572\pi$
$491$	20.6669	0.932684	0.466342	+	0.884605i	$0.345572\pi$
$492$	0	0
$493$	−6.13749	−0.276418
$494$	0	0
$495$	3.23703	0.145494
$496$	0	0
$497$	−6.52358	−0.292623
$498$	0	0
$499$	5.36134	0.240007	0.120003	−	0.992773i	$-0.461709\pi$
$499$	5.36134	0.240007	0.120003	+	0.992773i	$0.461709\pi$
$500$	0	0
$501$	−43.8331	−1.95832
$502$	0	0
$503$	−25.6878	−1.14536	−0.572681	−	0.819778i	$-0.694097\pi$
$503$	−25.6878	−1.14536	−0.572681	+	0.819778i	$0.694097\pi$
$504$	0	0
$505$	38.9757	1.73439
$506$	0	0
$507$	−72.9675	−3.24060
$508$	0	0
$509$	16.2273	0.719261	0.359631	−	0.933095i	$-0.382903\pi$
$509$	16.2273	0.719261	0.359631	+	0.933095i	$0.382903\pi$
$510$	0	0
$511$	−3.74629	−0.165726
$512$	0	0
$513$	1.33877	0.0591080
$514$	0	0
$515$	8.94168	0.394017
$516$	0	0
$517$	−0.862904	−0.0379505
$518$	0	0
$519$	13.6899	0.600922
$520$	0	0
$521$	−32.4863	−1.42325	−0.711626	−	0.702559i	$-0.752041\pi$
$521$	−32.4863	−1.42325	−0.711626	+	0.702559i	$0.752041\pi$
$522$	0	0
$523$	−11.1406	−0.487143	−0.243571	−	0.969883i	$-0.578319\pi$
$523$	−11.1406	−0.487143	−0.243571	+	0.969883i	$0.578319\pi$
$524$	0	0
$525$	6.05429	0.264231
$526$	0	0
$527$	23.2436	1.01251
$528$	0	0
$529$	11.8139	0.513649
$530$	0	0
$531$	13.4996	0.585834
$532$	0	0
$533$	−0.819312	−0.0354884
$534$	0	0
$535$	0.904439	0.0391023
$536$	0	0
$537$	44.0791	1.90215
$538$	0	0
$539$	2.85458	0.122955
$540$	0	0
$541$	1.33136	0.0572394	0.0286197	−	0.999590i	$-0.490889\pi$
$541$	1.33136	0.0572394	0.0286197	+	0.999590i	$0.490889\pi$
$542$	0	0
$543$	−24.5336	−1.05284
$544$	0	0
$545$	−6.02264	−0.257982
$546$	0	0
$547$	−21.1345	−0.903644	−0.451822	−	0.892108i	$-0.649226\pi$
$547$	−21.1345	−0.903644	−0.451822	+	0.892108i	$0.649226\pi$
$548$	0	0
$549$	16.9670	0.724135
$550$	0	0
$551$	−0.837037	−0.0356590
$552$	0	0
$553$	1.11415	0.0473784
$554$	0	0
$555$	−25.5515	−1.08460
$556$	0	0
$557$	−6.60875	−0.280022	−0.140011	−	0.990150i	$-0.544714\pi$
$557$	−6.60875	−0.280022	−0.140011	+	0.990150i	$0.544714\pi$
$558$	0	0
$559$	−35.9354	−1.51991
$560$	0	0
$561$	7.26232	0.306615
$562$	0	0
$563$	4.42073	0.186312	0.0931559	−	0.995652i	$-0.470305\pi$
$563$	4.42073	0.186312	0.0931559	+	0.995652i	$0.470305\pi$
$564$	0	0
$565$	42.1771	1.77440
$566$	0	0
$567$	5.56760	0.233817
$568$	0	0
$569$	−8.59442	−0.360297	−0.180149	−	0.983639i	$-0.557658\pi$
$569$	−8.59442	−0.360297	−0.180149	+	0.983639i	$0.557658\pi$
$570$	0	0
$571$	34.6802	1.45132	0.725661	−	0.688052i	$-0.241534\pi$
$571$	34.6802	1.45132	0.725661	+	0.688052i	$0.241534\pi$
$572$	0	0
$573$	3.73794	0.156155
$574$	0	0
$575$	−28.6315	−1.19401
$576$	0	0
$577$	26.2877	1.09437	0.547185	−	0.837012i	$-0.315699\pi$
$577$	26.2877	1.09437	0.547185	+	0.837012i	$0.315699\pi$
$578$	0	0
$579$	30.9526	1.28635
$580$	0	0
$581$	−6.33290	−0.262733
$582$	0	0
$583$	−2.47816	−0.102635
$584$	0	0
$585$	50.6828	2.09548
$586$	0	0
$587$	−7.01067	−0.289361	−0.144681	−	0.989478i	$-0.546215\pi$
$587$	−7.01067	−0.289361	−0.144681	+	0.989478i	$0.546215\pi$
$588$	0	0
$589$	3.16999	0.130617
$590$	0	0
$591$	9.47265	0.389653
$592$	0	0
$593$	34.7115	1.42543	0.712716	−	0.701453i	$-0.247465\pi$
$593$	34.7115	1.42543	0.712716	+	0.701453i	$0.247465\pi$
$594$	0	0
$595$	12.3284	0.505415
$596$	0	0
$597$	−29.8776	−1.22281
$598$	0	0
$599$	−32.4456	−1.32569	−0.662845	−	0.748757i	$-0.730651\pi$
$599$	−32.4456	−1.32569	−0.662845	+	0.748757i	$0.730651\pi$
$600$	0	0
$601$	36.2901	1.48030	0.740152	−	0.672440i	$-0.234754\pi$
$601$	36.2901	1.48030	0.740152	+	0.672440i	$0.234754\pi$
$602$	0	0
$603$	30.0187	1.22246
$604$	0	0
$605$	33.9600	1.38067
$606$	0	0
$607$	−5.54708	−0.225149	−0.112575	−	0.993643i	$-0.535910\pi$
$607$	−5.54708	−0.225149	−0.112575	+	0.993643i	$0.535910\pi$
$608$	0	0
$609$	−1.04434	−0.0423187
$610$	0	0
$611$	−13.5107	−0.546583
$612$	0	0
$613$	41.7239	1.68521	0.842606	−	0.538530i	$-0.181020\pi$
$613$	41.7239	1.68521	0.842606	+	0.538530i	$0.181020\pi$
$614$	0	0
$615$	0.899699	0.0362794
$616$	0	0
$617$	−16.0859	−0.647593	−0.323797	−	0.946127i	$-0.604959\pi$
$617$	−16.0859	−0.647593	−0.323797	+	0.946127i	$0.604959\pi$
$618$	0	0
$619$	2.93257	0.117870	0.0589349	−	0.998262i	$-0.481230\pi$
$619$	2.93257	0.117870	0.0589349	+	0.998262i	$0.481230\pi$
$620$	0	0
$621$	−7.89918	−0.316983
$622$	0	0
$623$	7.37514	0.295479
$624$	0	0
$625$	−25.7157	−1.02863
$626$	0	0
$627$	0.990444	0.0395545
$628$	0	0
$629$	−25.6260	−1.02177
$630$	0	0
$631$	−23.0095	−0.915994	−0.457997	−	0.888954i	$-0.651433\pi$
$631$	−23.0095	−0.915994	−0.457997	+	0.888954i	$0.651433\pi$
$632$	0	0
$633$	−37.2455	−1.48038
$634$	0	0
$635$	−59.6361	−2.36658
$636$	0	0
$637$	44.6947	1.77087
$638$	0	0
$639$	29.5360	1.16842
$640$	0	0
$641$	−0.00476234	−0.000188101 0	−9.40506e−5	−	1.00000i	$-0.500030\pi$
$641$	−0.00476234	−0.000188101 0	−9.40506e−5		1.00000i	$0.500030\pi$
$642$	0	0
$643$	37.6587	1.48511	0.742557	−	0.669783i	$-0.233613\pi$
$643$	37.6587	1.48511	0.742557	+	0.669783i	$0.233613\pi$
$644$	0	0
$645$	39.4612	1.55378
$646$	0	0
$647$	25.3744	0.997570	0.498785	−	0.866726i	$-0.333780\pi$
$647$	25.3744	0.997570	0.498785	+	0.866726i	$0.333780\pi$
$648$	0	0
$649$	−2.36696	−0.0929111
$650$	0	0
$651$	3.95507	0.155011
$652$	0	0
$653$	−22.5548	−0.882637	−0.441319	−	0.897350i	$-0.645489\pi$
$653$	−22.5548	−0.882637	−0.441319	+	0.897350i	$0.645489\pi$
$654$	0	0
$655$	31.8061	1.24277
$656$	0	0
$657$	16.9616	0.661735
$658$	0	0
$659$	18.3990	0.716724	0.358362	−	0.933583i	$-0.383335\pi$
$659$	18.3990	0.716724	0.358362	+	0.933583i	$0.383335\pi$
$660$	0	0
$661$	39.9383	1.55342	0.776710	−	0.629859i	$-0.216887\pi$
$661$	39.9383	1.55342	0.776710	+	0.629859i	$0.216887\pi$
$662$	0	0
$663$	113.708	4.41604
$664$	0	0
$665$	1.68136	0.0652004
$666$	0	0
$667$	4.93880	0.191231
$668$	0	0
$669$	49.2451	1.90392
$670$	0	0
$671$	−2.97491	−0.114845
$672$	0	0
$673$	−4.04663	−0.155986	−0.0779931	−	0.996954i	$-0.524851\pi$
$673$	−4.04663	−0.155986	−0.0779931	+	0.996954i	$0.524851\pi$
$674$	0	0
$675$	6.49639	0.250046
$676$	0	0
$677$	−42.8041	−1.64509	−0.822547	−	0.568697i	$-0.807448\pi$
$677$	−42.8041	−1.64509	−0.822547	+	0.568697i	$0.807448\pi$
$678$	0	0
$679$	−0.429190	−0.0164708
$680$	0	0
$681$	−51.7215	−1.98197
$682$	0	0
$683$	6.33556	0.242423	0.121212	−	0.992627i	$-0.461322\pi$
$683$	6.33556	0.242423	0.121212	+	0.992627i	$0.461322\pi$
$684$	0	0
$685$	−10.3024	−0.393636
$686$	0	0
$687$	29.4251	1.12264
$688$	0	0
$689$	−38.8010	−1.47820
$690$	0	0
$691$	15.3700	0.584702	0.292351	−	0.956311i	$-0.405562\pi$
$691$	15.3700	0.584702	0.292351	+	0.956311i	$0.405562\pi$
$692$	0	0
$693$	0.552409	0.0209843
$694$	0	0
$695$	33.0665	1.25428
$696$	0	0
$697$	0.902320	0.0341778
$698$	0	0
$699$	7.03077	0.265928
$700$	0	0
$701$	−36.0602	−1.36197	−0.680987	−	0.732296i	$-0.738449\pi$
$701$	−36.0602	−1.36197	−0.680987	+	0.732296i	$0.738449\pi$
$702$	0	0
$703$	−3.49490	−0.131813
$704$	0	0
$705$	14.8363	0.558766
$706$	0	0
$707$	6.65131	0.250148
$708$	0	0
$709$	18.1426	0.681360	0.340680	−	0.940179i	$-0.389343\pi$
$709$	18.1426	0.681360	0.340680	+	0.940179i	$0.389343\pi$
$710$	0	0
$711$	−5.04438	−0.189179
$712$	0	0
$713$	−18.7040	−0.700470
$714$	0	0
$715$	−8.88647	−0.332335
$716$	0	0
$717$	29.6173	1.10608
$718$	0	0
$719$	38.2801	1.42761	0.713804	−	0.700345i	$-0.246971\pi$
$719$	38.2801	1.42761	0.713804	+	0.700345i	$0.246971\pi$
$720$	0	0
$721$	1.52592	0.0568284
$722$	0	0
$723$	2.46276	0.0915911
$724$	0	0
$725$	−4.06174	−0.150849
$726$	0	0
$727$	23.4550	0.869897	0.434949	−	0.900455i	$-0.356767\pi$
$727$	23.4550	0.869897	0.434949	+	0.900455i	$0.356767\pi$
$728$	0	0
$729$	−15.8529	−0.587144
$730$	0	0
$731$	39.5761	1.46378
$732$	0	0
$733$	19.3814	0.715870	0.357935	−	0.933747i	$-0.383481\pi$
$733$	19.3814	0.715870	0.357935	+	0.933747i	$0.383481\pi$
$734$	0	0
$735$	−49.0799	−1.81034
$736$	0	0
$737$	−5.26332	−0.193877
$738$	0	0
$739$	−12.1161	−0.445697	−0.222848	−	0.974853i	$-0.571536\pi$
$739$	−12.1161	−0.445697	−0.222848	+	0.974853i	$0.571536\pi$
$740$	0	0
$741$	15.5076	0.569685
$742$	0	0
$743$	−29.1148	−1.06812	−0.534060	−	0.845447i	$-0.679334\pi$
$743$	−29.1148	−1.06812	−0.534060	+	0.845447i	$0.679334\pi$
$744$	0	0
$745$	18.6212	0.682227
$746$	0	0
$747$	28.6726	1.04908
$748$	0	0
$749$	0.154345	0.00563965
$750$	0	0
$751$	2.09618	0.0764906	0.0382453	−	0.999268i	$-0.487823\pi$
$751$	2.09618	0.0764906	0.0382453	+	0.999268i	$0.487823\pi$
$752$	0	0
$753$	−31.3720	−1.14326
$754$	0	0
$755$	−14.3731	−0.523091
$756$	0	0
$757$	36.7044	1.33404	0.667022	−	0.745038i	$-0.267569\pi$
$757$	36.7044	1.33404	0.667022	+	0.745038i	$0.267569\pi$
$758$	0	0
$759$	−5.84395	−0.212122
$760$	0	0
$761$	−26.8362	−0.972812	−0.486406	−	0.873733i	$-0.661692\pi$
$761$	−26.8362	−0.972812	−0.486406	+	0.873733i	$0.661692\pi$
$762$	0	0
$763$	−1.02778	−0.0372082
$764$	0	0
$765$	−55.8177	−2.01809
$766$	0	0
$767$	−37.0599	−1.33815
$768$	0	0
$769$	15.3434	0.553299	0.276649	−	0.960971i	$-0.410776\pi$
$769$	15.3434	0.553299	0.276649	+	0.960971i	$0.410776\pi$
$770$	0	0
$771$	−44.6349	−1.60749
$772$	0	0
$773$	−54.9160	−1.97519	−0.987596	−	0.157017i	$-0.949812\pi$
$773$	−54.9160	−1.97519	−0.987596	+	0.157017i	$0.949812\pi$
$774$	0	0
$775$	15.3824	0.552553
$776$	0	0
$777$	−4.36045	−0.156430
$778$	0	0
$779$	0.123059	0.00440906
$780$	0	0
$781$	−5.17869	−0.185308
$782$	0	0
$783$	−1.12060	−0.0400469
$784$	0	0
$785$	11.9547	0.426683
$786$	0	0
$787$	16.2826	0.580413	0.290206	−	0.956964i	$-0.406276\pi$
$787$	16.2826	0.580413	0.290206	+	0.956964i	$0.406276\pi$
$788$	0	0
$789$	5.96139	0.212231
$790$	0	0
$791$	7.19764	0.255918
$792$	0	0
$793$	−46.5788	−1.65406
$794$	0	0
$795$	42.6080	1.51115
$796$	0	0
$797$	20.2945	0.718867	0.359434	−	0.933171i	$-0.382970\pi$
$797$	20.2945	0.718867	0.359434	+	0.933171i	$0.382970\pi$
$798$	0	0
$799$	14.8795	0.526398
$800$	0	0
$801$	−33.3915	−1.17983
$802$	0	0
$803$	−2.97396	−0.104949
$804$	0	0
$805$	−9.92059	−0.349655
$806$	0	0
$807$	−11.3804	−0.400610
$808$	0	0
$809$	5.57944	0.196163	0.0980813	−	0.995178i	$-0.468729\pi$
$809$	5.57944	0.196163	0.0980813	+	0.995178i	$0.468729\pi$
$810$	0	0
$811$	−28.4695	−0.999699	−0.499849	−	0.866112i	$-0.666611\pi$
$811$	−28.4695	−0.999699	−0.499849	+	0.866112i	$0.666611\pi$
$812$	0	0
$813$	74.8373	2.62466
$814$	0	0
$815$	56.2776	1.97132
$816$	0	0
$817$	5.39744	0.188832
$818$	0	0
$819$	8.64917	0.302226
$820$	0	0
$821$	3.41291	0.119111	0.0595556	−	0.998225i	$-0.481032\pi$
$821$	3.41291	0.119111	0.0595556	+	0.998225i	$0.481032\pi$
$822$	0	0
$823$	−12.8444	−0.447729	−0.223865	−	0.974620i	$-0.571867\pi$
$823$	−12.8444	−0.447729	−0.223865	+	0.974620i	$0.571867\pi$
$824$	0	0
$825$	4.80615	0.167328
$826$	0	0
$827$	11.4599	0.398501	0.199251	−	0.979949i	$-0.436149\pi$
$827$	11.4599	0.398501	0.199251	+	0.979949i	$0.436149\pi$
$828$	0	0
$829$	15.6953	0.545119	0.272559	−	0.962139i	$-0.412130\pi$
$829$	15.6953	0.545119	0.272559	+	0.962139i	$0.412130\pi$
$830$	0	0
$831$	−17.3908	−0.603280
$832$	0	0
$833$	−49.2229	−1.70547
$834$	0	0
$835$	−59.0700	−2.04420
$836$	0	0
$837$	4.24388	0.146690
$838$	0	0
$839$	40.4629	1.39693	0.698467	−	0.715642i	$-0.253866\pi$
$839$	40.4629	1.39693	0.698467	+	0.715642i	$0.253866\pi$
$840$	0	0
$841$	−28.2994	−0.975840
$842$	0	0
$843$	44.2016	1.52238
$844$	0	0
$845$	−98.3318	−3.38272
$846$	0	0
$847$	5.79538	0.199131
$848$	0	0
$849$	−2.89570	−0.0993802
$850$	0	0
$851$	20.6211	0.706881
$852$	0	0
$853$	−1.22071	−0.0417962	−0.0208981	−	0.999782i	$-0.506653\pi$
$853$	−1.22071	−0.0417962	−0.0208981	+	0.999782i	$0.506653\pi$
$854$	0	0
$855$	−7.61248	−0.260341
$856$	0	0
$857$	49.3646	1.68626	0.843131	−	0.537709i	$-0.180710\pi$
$857$	49.3646	1.68626	0.843131	+	0.537709i	$0.180710\pi$
$858$	0	0
$859$	−41.6828	−1.42220	−0.711099	−	0.703092i	$-0.751802\pi$
$859$	−41.6828	−1.42220	−0.711099	+	0.703092i	$0.751802\pi$
$860$	0	0
$861$	0.153536	0.00523250
$862$	0	0
$863$	22.0429	0.750348	0.375174	−	0.926954i	$-0.377583\pi$
$863$	22.0429	0.750348	0.375174	+	0.926954i	$0.377583\pi$
$864$	0	0
$865$	18.4487	0.627275
$866$	0	0
$867$	−85.6311	−2.90818
$868$	0	0
$869$	0.884457	0.0300031
$870$	0	0
$871$	−82.4089	−2.79232
$872$	0	0
$873$	1.94319	0.0657670
$874$	0	0
$875$	−0.247973	−0.00838300
$876$	0	0
$877$	−33.7237	−1.13877	−0.569385	−	0.822071i	$-0.692819\pi$
$877$	−33.7237	−1.13877	−0.569385	+	0.822071i	$0.692819\pi$
$878$	0	0
$879$	12.4617	0.420323
$880$	0	0
$881$	7.24781	0.244185	0.122092	−	0.992519i	$-0.461040\pi$
$881$	7.24781	0.244185	0.122092	+	0.992519i	$0.461040\pi$
$882$	0	0
$883$	24.1525	0.812797	0.406398	−	0.913696i	$-0.366785\pi$
$883$	24.1525	0.812797	0.406398	+	0.913696i	$0.366785\pi$
$884$	0	0
$885$	40.6960	1.36798
$886$	0	0
$887$	20.3957	0.684821	0.342411	−	0.939550i	$-0.388757\pi$
$887$	20.3957	0.684821	0.342411	+	0.939550i	$0.388757\pi$
$888$	0	0
$889$	−10.1771	−0.341328
$890$	0	0
$891$	4.41979	0.148069
$892$	0	0
$893$	2.02928	0.0679072
$894$	0	0
$895$	59.4015	1.98557
$896$	0	0
$897$	−91.4998	−3.05509
$898$	0	0
$899$	−2.65340	−0.0884959
$900$	0	0
$901$	42.7321	1.42361
$902$	0	0
$903$	6.73416	0.224099
$904$	0	0
$905$	−33.0618	−1.09901
$906$	0	0
$907$	6.56562	0.218008	0.109004	−	0.994041i	$-0.465234\pi$
$907$	6.56562	0.218008	0.109004	+	0.994041i	$0.465234\pi$
$908$	0	0
$909$	−30.1143	−0.998827
$910$	0	0
$911$	−6.98339	−0.231370	−0.115685	−	0.993286i	$-0.536906\pi$
$911$	−6.98339	−0.231370	−0.115685	+	0.993286i	$0.536906\pi$
$912$	0	0
$913$	−5.02732	−0.166380
$914$	0	0
$915$	51.1488	1.69093
$916$	0	0
$917$	5.42780	0.179242
$918$	0	0
$919$	12.4237	0.409819	0.204910	−	0.978781i	$-0.434310\pi$
$919$	12.4237	0.409819	0.204910	+	0.978781i	$0.434310\pi$
$920$	0	0
$921$	−24.4101	−0.804339
$922$	0	0
$923$	−81.0837	−2.66890
$924$	0	0
$925$	−16.9591	−0.557610
$926$	0	0
$927$	−6.90873	−0.226912
$928$	0	0
$929$	−23.0649	−0.756735	−0.378367	−	0.925655i	$-0.623514\pi$
$929$	−23.0649	−0.756735	−0.378367	+	0.925655i	$0.623514\pi$
$930$	0	0
$931$	−6.71307	−0.220012
$932$	0	0
$933$	63.7551	2.08725
$934$	0	0
$935$	9.78679	0.320062
$936$	0	0
$937$	20.2747	0.662347	0.331173	−	0.943570i	$-0.392555\pi$
$937$	20.2747	0.662347	0.331173	+	0.943570i	$0.392555\pi$
$938$	0	0
$939$	−39.4134	−1.28621
$940$	0	0
$941$	33.8390	1.10312	0.551559	−	0.834136i	$-0.314033\pi$
$941$	33.8390	1.10312	0.551559	+	0.834136i	$0.314033\pi$
$942$	0	0
$943$	−0.726091	−0.0236448
$944$	0	0
$945$	2.25095	0.0732234
$946$	0	0
$947$	−55.4208	−1.80093	−0.900466	−	0.434926i	$-0.856775\pi$
$947$	−55.4208	−1.80093	−0.900466	+	0.434926i	$0.856775\pi$
$948$	0	0
$949$	−46.5639	−1.51153
$950$	0	0
$951$	8.90033	0.288613
$952$	0	0
$953$	19.8530	0.643102	0.321551	−	0.946892i	$-0.395796\pi$
$953$	19.8530	0.643102	0.321551	+	0.946892i	$0.395796\pi$
$954$	0	0
$955$	5.03729	0.163003
$956$	0	0
$957$	−0.829039	−0.0267990
$958$	0	0
$959$	−1.75814	−0.0567734
$960$	0	0
$961$	−20.9512	−0.675844
$962$	0	0
$963$	−0.698809	−0.0225188
$964$	0	0
$965$	41.7121	1.34276
$966$	0	0
$967$	−5.95336	−0.191447	−0.0957236	−	0.995408i	$-0.530516\pi$
$967$	−5.95336	−0.191447	−0.0957236	+	0.995408i	$0.530516\pi$
$968$	0	0
$969$	−17.0787	−0.548647
$970$	0	0
$971$	−5.33267	−0.171134	−0.0855668	−	0.996332i	$-0.527270\pi$
$971$	−5.33267	−0.171134	−0.0855668	+	0.996332i	$0.527270\pi$
$972$	0	0
$973$	5.64290	0.180903
$974$	0	0
$975$	75.2507	2.40995
$976$	0	0
$977$	−12.6063	−0.403312	−0.201656	−	0.979456i	$-0.564632\pi$
$977$	−12.6063	−0.403312	−0.201656	+	0.979456i	$0.564632\pi$
$978$	0	0
$979$	5.85469	0.187117
$980$	0	0
$981$	4.65335	0.148570
$982$	0	0
$983$	−19.8397	−0.632788	−0.316394	−	0.948628i	$-0.602472\pi$
$983$	−19.8397	−0.632788	−0.316394	+	0.948628i	$0.602472\pi$
$984$	0	0
$985$	12.7655	0.406741
$986$	0	0
$987$	2.53185	0.0805897
$988$	0	0
$989$	−31.8467	−1.01267
$990$	0	0
$991$	53.9425	1.71354	0.856770	−	0.515699i	$-0.172468\pi$
$991$	53.9425	1.71354	0.856770	+	0.515699i	$0.172468\pi$
$992$	0	0
$993$	61.9760	1.96675
$994$	0	0
$995$	−40.2634	−1.27644
$996$	0	0
$997$	−20.6213	−0.653084	−0.326542	−	0.945183i	$-0.605884\pi$
$997$	−20.6213	−0.653084	−0.326542	+	0.945183i	$0.605884\pi$
$998$	0	0
$999$	−4.67886	−0.148033

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4864.2.a.bn.1.2		8
4.3	odd	2		4864.2.a.bo.1.7		8
8.3	odd	2		4864.2.a.bq.1.2		8
8.5	even	2		4864.2.a.bp.1.7		8
16.3	odd	4		152.2.c.b.77.13	✓	16
16.5	even	4		608.2.c.b.305.14		16
16.11	odd	4		152.2.c.b.77.14	yes	16
16.13	even	4		608.2.c.b.305.3		16
48.5	odd	4		5472.2.g.b.2737.14		16
48.11	even	4		1368.2.g.b.685.3		16
48.29	odd	4		5472.2.g.b.2737.3		16
48.35	even	4		1368.2.g.b.685.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.c.b.77.13	✓	16	16.3	odd	4
152.2.c.b.77.14	yes	16	16.11	odd	4
608.2.c.b.305.3		16	16.13	even	4
608.2.c.b.305.14		16	16.5	even	4
1368.2.g.b.685.3		16	48.11	even	4
1368.2.g.b.685.4		16	48.35	even	4
4864.2.a.bn.1.2		8	1.1	even	1	trivial
4864.2.a.bo.1.7		8	4.3	odd	2
4864.2.a.bp.1.7		8	8.5	even	2
4864.2.a.bq.1.2		8	8.3	odd	2
5472.2.g.b.2737.3		16	48.29	odd	4
5472.2.g.b.2737.14		16	48.5	odd	4

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)

\(f(q)\)	\(=\)	\(q-2.32921 q^{3} -3.13887 q^{5} -0.535658 q^{7} +2.42523 q^{9} +O(q^{10})\)	\(q-2.32921 q^{3} -3.13887 q^{5} -0.535658 q^{7} +2.42523 q^{9} -0.425227 q^{11} -6.65786 q^{13} +7.31110 q^{15} +7.33239 q^{17} +1.00000 q^{19} +1.24766 q^{21} -5.90033 q^{23} +4.85252 q^{25} +1.33877 q^{27} -0.837037 q^{29} +3.16999 q^{31} +0.990444 q^{33} +1.68136 q^{35} -3.49490 q^{37} +15.5076 q^{39} +0.123059 q^{41} +5.39744 q^{43} -7.61248 q^{45} +2.02928 q^{47} -6.71307 q^{49} -17.0787 q^{51} +5.82785 q^{53} +1.33473 q^{55} -2.32921 q^{57} +5.56633 q^{59} +6.99606 q^{61} -1.29909 q^{63} +20.8982 q^{65} +12.3777 q^{67} +13.7431 q^{69} +12.1786 q^{71} +6.99382 q^{73} -11.3025 q^{75} +0.227776 q^{77} -2.07996 q^{79} -10.3940 q^{81} +11.8227 q^{83} -23.0154 q^{85} +1.94964 q^{87} -13.7684 q^{89} +3.56633 q^{91} -7.38358 q^{93} -3.13887 q^{95} +0.801240 q^{97} -1.03127 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5} - 4 q^{7} + 12 q^{9}+O(q^{10}) \) 8 * q - 8 * q^5 - 4 * q^7 + 12 * q^9	\( 8 q - 8 q^{5} - 4 q^{7} + 12 q^{9} + 4 q^{11} - 8 q^{13} - 4 q^{17} + 8 q^{19} - 16 q^{21} + 12 q^{25} - 28 q^{29} - 8 q^{31} - 12 q^{35} - 4 q^{37} + 4 q^{39} - 8 q^{41} - 4 q^{43} - 24 q^{45} - 12 q^{47} + 12 q^{49} + 12 q^{51} - 32 q^{53} + 8 q^{55} + 12 q^{59} - 8 q^{61} + 16 q^{63} + 8 q^{65} - 4 q^{67} - 28 q^{69} + 24 q^{71} - 24 q^{77} + 24 q^{79} - 8 q^{81} + 40 q^{83} - 24 q^{85} - 24 q^{87} + 8 q^{89} - 4 q^{91} - 32 q^{93} - 8 q^{95} + 16 q^{97} - 76 q^{99}+O(q^{100}) \) 8 * q - 8 * q^5 - 4 * q^7 + 12 * q^9 + 4 * q^11 - 8 * q^13 - 4 * q^17 + 8 * q^19 - 16 * q^21 + 12 * q^25 - 28 * q^29 - 8 * q^31 - 12 * q^35 - 4 * q^37 + 4 * q^39 - 8 * q^41 - 4 * q^43 - 24 * q^45 - 12 * q^47 + 12 * q^49 + 12 * q^51 - 32 * q^53 + 8 * q^55 + 12 * q^59 - 8 * q^61 + 16 * q^63 + 8 * q^65 - 4 * q^67 - 28 * q^69 + 24 * q^71 - 24 * q^77 + 24 * q^79 - 8 * q^81 + 40 * q^83 - 24 * q^85 - 24 * q^87 + 8 * q^89 - 4 * q^91 - 32 * q^93 - 8 * q^95 + 16 * q^97 - 76 * q^99

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