LMFDB - Embedded newform 3072.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3072,2,Mod(1,3072)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3072, 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("3072.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3072 = 2^{10} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3072.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:	$24.5300435009$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{16})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 2 $ x^4 - 4*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1536)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.84776$ of defining polynomial
Character	$\chi$	$=$	3072.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.49661 q^{5} -0.917608 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.49661 q^{5} -0.917608 q^{7} +1.00000 q^{9} +3.69552 q^{11} -5.81204 q^{13} -2.49661 q^{15} -0.867091 q^{17} +6.52395 q^{19} -0.917608 q^{21} -4.00000 q^{23} +1.23304 q^{25} +1.00000 q^{27} +7.72286 q^{29} +2.14386 q^{31} +3.69552 q^{33} +2.29090 q^{35} +2.47568 q^{37} -5.81204 q^{39} -9.58541 q^{41} -9.58541 q^{43} -2.49661 q^{45} +1.65685 q^{47} -6.15800 q^{49} -0.867091 q^{51} -3.39329 q^{53} -9.22625 q^{55} +6.52395 q^{57} -12.7183 q^{59} -0.0231773 q^{61} -0.917608 q^{63} +14.5104 q^{65} +5.32729 q^{67} -4.00000 q^{69} -11.8216 q^{71} -15.2809 q^{73} +1.23304 q^{75} -3.39104 q^{77} -8.40968 q^{79} +1.00000 q^{81} +1.96134 q^{83} +2.16478 q^{85} +7.72286 q^{87} +2.79565 q^{89} +5.33317 q^{91} +2.14386 q^{93} -16.2877 q^{95} -2.26582 q^{97} +3.69552 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 8 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 8 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} - 8 q^{7} + 4 q^{9} - 8 q^{13} - 8 q^{21} - 16 q^{23} + 4 q^{25} + 4 q^{27} - 8 q^{31} - 16 q^{35} - 8 q^{37} - 8 q^{39} - 16 q^{47} + 4 q^{49} - 16 q^{55} - 16 q^{59} - 24 q^{61} - 8 q^{63} - 8 q^{65} + 16 q^{67} - 16 q^{69} - 16 q^{71} - 8 q^{73} + 4 q^{75} + 16 q^{77} - 24 q^{79} + 4 q^{81} - 8 q^{89} + 16 q^{91} - 8 q^{93} - 32 q^{95} - 16 q^{97}+O(q^{100}) $ 4 * q + 4 * q^3 - 8 * q^7 + 4 * q^9 - 8 * q^13 - 8 * q^21 - 16 * q^23 + 4 * q^25 + 4 * q^27 - 8 * q^31 - 16 * q^35 - 8 * q^37 - 8 * q^39 - 16 * q^47 + 4 * q^49 - 16 * q^55 - 16 * q^59 - 24 * q^61 - 8 * q^63 - 8 * q^65 + 16 * q^67 - 16 * q^69 - 16 * q^71 - 8 * q^73 + 4 * q^75 + 16 * q^77 - 24 * q^79 + 4 * q^81 - 8 * q^89 + 16 * q^91 - 8 * q^93 - 32 * q^95 - 16 * q^97

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−2.49661	−1.11652	−0.558258	−	0.829667i	$-0.688530\pi$
$5$	−2.49661	−1.11652	−0.558258	+	0.829667i	$0.688530\pi$
$6$	0	0
$7$	−0.917608	−0.346823	−0.173412	−	0.984849i	$-0.555479\pi$
$7$	−0.917608	−0.346823	−0.173412	+	0.984849i	$0.555479\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.69552	1.11424	0.557120	−	0.830432i	$-0.311906\pi$
$11$	3.69552	1.11424	0.557120	+	0.830432i	$0.311906\pi$
$12$	0	0
$13$	−5.81204	−1.61197	−0.805985	−	0.591936i	$-0.798364\pi$
$13$	−5.81204	−1.61197	−0.805985	+	0.591936i	$0.798364\pi$
$14$	0	0
$15$	−2.49661	−0.644621
$16$	0	0
$17$	−0.867091	−0.210300	−0.105150	−	0.994456i	$-0.533532\pi$
$17$	−0.867091	−0.210300	−0.105150	+	0.994456i	$0.533532\pi$
$18$	0	0
$19$	6.52395	1.49670	0.748348	−	0.663306i	$-0.230847\pi$
$19$	6.52395	1.49670	0.748348	+	0.663306i	$0.230847\pi$
$20$	0	0
$21$	−0.917608	−0.200238
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	1.23304	0.246608
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	7.72286	1.43410	0.717049	−	0.697022i	$-0.245492\pi$
$29$	7.72286	1.43410	0.717049	+	0.697022i	$0.245492\pi$
$30$	0	0
$31$	2.14386	0.385049	0.192524	−	0.981292i	$-0.438333\pi$
$31$	2.14386	0.385049	0.192524	+	0.981292i	$0.438333\pi$
$32$	0	0
$33$	3.69552	0.643307
$34$	0	0
$35$	2.29090	0.387234
$36$	0	0
$37$	2.47568	0.406999	0.203500	−	0.979075i	$-0.434768\pi$
$37$	2.47568	0.406999	0.203500	+	0.979075i	$0.434768\pi$
$38$	0	0
$39$	−5.81204	−0.930671
$40$	0	0
$41$	−9.58541	−1.49699	−0.748495	−	0.663140i	$-0.769223\pi$
$41$	−9.58541	−1.49699	−0.748495	+	0.663140i	$0.769223\pi$
$42$	0	0
$43$	−9.58541	−1.46176	−0.730881	−	0.682505i	$-0.760891\pi$
$43$	−9.58541	−1.46176	−0.730881	+	0.682505i	$0.760891\pi$
$44$	0	0
$45$	−2.49661	−0.372172
$46$	0	0
$47$	1.65685	0.241677	0.120839	−	0.992672i	$-0.461442\pi$
$47$	1.65685	0.241677	0.120839	+	0.992672i	$0.461442\pi$
$48$	0	0
$49$	−6.15800	−0.879714
$50$	0	0
$51$	−0.867091	−0.121417
$52$	0	0
$53$	−3.39329	−0.466104	−0.233052	−	0.972464i	$-0.574871\pi$
$53$	−3.39329	−0.466104	−0.233052	+	0.972464i	$0.574871\pi$
$54$	0	0
$55$	−9.22625	−1.24407
$56$	0	0
$57$	6.52395	0.864118
$58$	0	0
$59$	−12.7183	−1.65578	−0.827892	−	0.560887i	$-0.810460\pi$
$59$	−12.7183	−1.65578	−0.827892	+	0.560887i	$0.810460\pi$
$60$	0	0
$61$	−0.0231773	−0.00296755	−0.00148377	−	0.999999i	$-0.500472\pi$
$61$	−0.0231773	−0.00296755	−0.00148377	+	0.999999i	$0.500472\pi$
$62$	0	0
$63$	−0.917608	−0.115608
$64$	0	0
$65$	14.5104	1.79979
$66$	0	0
$67$	5.32729	0.650832	0.325416	−	0.945571i	$-0.394496\pi$
$67$	5.32729	0.650832	0.325416	+	0.945571i	$0.394496\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−11.8216	−1.40297	−0.701485	−	0.712684i	$-0.747479\pi$
$71$	−11.8216	−1.40297	−0.701485	+	0.712684i	$0.747479\pi$
$72$	0	0
$73$	−15.2809	−1.78850	−0.894249	−	0.447570i	$-0.852289\pi$
$73$	−15.2809	−1.78850	−0.894249	+	0.447570i	$0.852289\pi$
$74$	0	0
$75$	1.23304	0.142379
$76$	0	0
$77$	−3.39104	−0.386444
$78$	0	0
$79$	−8.40968	−0.946163	−0.473081	−	0.881019i	$-0.656858\pi$
$79$	−8.40968	−0.946163	−0.473081	+	0.881019i	$0.656858\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.96134	0.215285	0.107642	−	0.994190i	$-0.465670\pi$
$83$	1.96134	0.215285	0.107642	+	0.994190i	$0.465670\pi$
$84$	0	0
$85$	2.16478	0.234804
$86$	0	0
$87$	7.72286	0.827977
$88$	0	0
$89$	2.79565	0.296338	0.148169	−	0.988962i	$-0.452662\pi$
$89$	2.79565	0.296338	0.148169	+	0.988962i	$0.452662\pi$
$90$	0	0
$91$	5.33317	0.559068
$92$	0	0
$93$	2.14386	0.222308
$94$	0	0
$95$	−16.2877	−1.67108
$96$	0	0
$97$	−2.26582	−0.230059	−0.115029	−	0.993362i	$-0.536696\pi$
$97$	−2.26582	−0.230059	−0.115029	+	0.993362i	$0.536696\pi$
$98$	0	0
$99$	3.69552	0.371414
$100$	0	0
$101$	9.98868	0.993910	0.496955	−	0.867776i	$-0.334451\pi$
$101$	9.98868	0.993910	0.496955	+	0.867776i	$0.334451\pi$
$102$	0	0
$103$	16.4180	1.61771	0.808857	−	0.588006i	$-0.200087\pi$
$103$	16.4180	1.61771	0.808857	+	0.588006i	$0.200087\pi$
$104$	0	0
$105$	2.29090	0.223569
$106$	0	0
$107$	−12.2522	−1.18447	−0.592234	−	0.805766i	$-0.701754\pi$
$107$	−12.2522	−1.18447	−0.592234	+	0.805766i	$0.701754\pi$
$108$	0	0
$109$	−18.2973	−1.75257	−0.876283	−	0.481797i	$-0.839984\pi$
$109$	−18.2973	−1.75257	−0.876283	+	0.481797i	$0.839984\pi$
$110$	0	0
$111$	2.47568	0.234981
$112$	0	0
$113$	3.65685	0.344008	0.172004	−	0.985096i	$-0.444976\pi$
$113$	3.65685	0.344008	0.172004	+	0.985096i	$0.444976\pi$
$114$	0	0
$115$	9.98642	0.931239
$116$	0	0
$117$	−5.81204	−0.537323
$118$	0	0
$119$	0.795649	0.0729371
$120$	0	0
$121$	2.65685	0.241532
$122$	0	0
$123$	−9.58541	−0.864288
$124$	0	0
$125$	9.40461	0.841174
$126$	0	0
$127$	−20.6382	−1.83135	−0.915673	−	0.401925i	$-0.868341\pi$
$127$	−20.6382	−1.83135	−0.915673	+	0.401925i	$0.868341\pi$
$128$	0	0
$129$	−9.58541	−0.843949
$130$	0	0
$131$	17.5140	1.53020	0.765101	−	0.643910i	$-0.222689\pi$
$131$	17.5140	1.53020	0.765101	+	0.643910i	$0.222689\pi$
$132$	0	0
$133$	−5.98642	−0.519089
$134$	0	0
$135$	−2.49661	−0.214874
$136$	0	0
$137$	8.25813	0.705539	0.352770	−	0.935710i	$-0.385240\pi$
$137$	8.25813	0.705539	0.352770	+	0.935710i	$0.385240\pi$
$138$	0	0
$139$	−14.7047	−1.24724	−0.623620	−	0.781728i	$-0.714339\pi$
$139$	−14.7047	−1.24724	−0.623620	+	0.781728i	$0.714339\pi$
$140$	0	0
$141$	1.65685	0.139532
$142$	0	0
$143$	−21.4785	−1.79612
$144$	0	0
$145$	−19.2809	−1.60119
$146$	0	0
$147$	−6.15800	−0.507903
$148$	0	0
$149$	16.1535	1.32334	0.661672	−	0.749794i	$-0.269847\pi$
$149$	16.1535	1.32334	0.661672	+	0.749794i	$0.269847\pi$
$150$	0	0
$151$	−19.2691	−1.56810	−0.784048	−	0.620701i	$-0.786848\pi$
$151$	−19.2691	−1.56810	−0.784048	+	0.620701i	$0.786848\pi$
$152$	0	0
$153$	−0.867091	−0.0701002
$154$	0	0
$155$	−5.35237	−0.429913
$156$	0	0
$157$	−0.818827	−0.0653495	−0.0326747	−	0.999466i	$-0.510403\pi$
$157$	−0.818827	−0.0653495	−0.0326747	+	0.999466i	$0.510403\pi$
$158$	0	0
$159$	−3.39329	−0.269105
$160$	0	0
$161$	3.67043	0.289271
$162$	0	0
$163$	−23.6492	−1.85235	−0.926173	−	0.377100i	$-0.876921\pi$
$163$	−23.6492	−1.85235	−0.926173	+	0.377100i	$0.876921\pi$
$164$	0	0
$165$	−9.22625	−0.718263
$166$	0	0
$167$	−9.30358	−0.719933	−0.359966	−	0.932965i	$-0.617212\pi$
$167$	−9.30358	−0.719933	−0.359966	+	0.932965i	$0.617212\pi$
$168$	0	0
$169$	20.7798	1.59845
$170$	0	0
$171$	6.52395	0.498899
$172$	0	0
$173$	7.10557	0.540226	0.270113	−	0.962829i	$-0.412939\pi$
$173$	7.10557	0.540226	0.270113	+	0.962829i	$0.412939\pi$
$174$	0	0
$175$	−1.13145	−0.0855294
$176$	0	0
$177$	−12.7183	−0.955968
$178$	0	0
$179$	−2.67271	−0.199768	−0.0998840	−	0.994999i	$-0.531847\pi$
$179$	−2.67271	−0.199768	−0.0998840	+	0.994999i	$0.531847\pi$
$180$	0	0
$181$	0.640465	0.0476054	0.0238027	−	0.999717i	$-0.492423\pi$
$181$	0.640465	0.0476054	0.0238027	+	0.999717i	$0.492423\pi$
$182$	0	0
$183$	−0.0231773	−0.00171332
$184$	0	0
$185$	−6.18080	−0.454421
$186$	0	0
$187$	−3.20435	−0.234325
$188$	0	0
$189$	−0.917608	−0.0667461
$190$	0	0
$191$	14.6037	1.05669	0.528344	−	0.849031i	$-0.322813\pi$
$191$	14.6037	1.05669	0.528344	+	0.849031i	$0.322813\pi$
$192$	0	0
$193$	−8.81485	−0.634507	−0.317253	−	0.948341i	$-0.602761\pi$
$193$	−8.81485	−0.634507	−0.317253	+	0.948341i	$0.602761\pi$
$194$	0	0
$195$	14.5104	1.03911
$196$	0	0
$197$	3.74551	0.266856	0.133428	−	0.991058i	$-0.457401\pi$
$197$	3.74551	0.266856	0.133428	+	0.991058i	$0.457401\pi$
$198$	0	0
$199$	−14.8267	−1.05104	−0.525519	−	0.850782i	$-0.676129\pi$
$199$	−14.8267	−1.05104	−0.525519	+	0.850782i	$0.676129\pi$
$200$	0	0
$201$	5.32729	0.375758
$202$	0	0
$203$	−7.08655	−0.497379
$204$	0	0
$205$	23.9310	1.67141
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	24.1094	1.66768
$210$	0	0
$211$	11.3910	0.784191	0.392096	−	0.919924i	$-0.371750\pi$
$211$	11.3910	0.784191	0.392096	+	0.919924i	$0.371750\pi$
$212$	0	0
$213$	−11.8216	−0.810005
$214$	0	0
$215$	23.9310	1.63208
$216$	0	0
$217$	−1.96722	−0.133544
$218$	0	0
$219$	−15.2809	−1.03259
$220$	0	0
$221$	5.03957	0.338998
$222$	0	0
$223$	−4.78110	−0.320166	−0.160083	−	0.987104i	$-0.551176\pi$
$223$	−4.78110	−0.320166	−0.160083	+	0.987104i	$0.551176\pi$
$224$	0	0
$225$	1.23304	0.0822027
$226$	0	0
$227$	3.83840	0.254764	0.127382	−	0.991854i	$-0.459343\pi$
$227$	3.83840	0.254764	0.127382	+	0.991854i	$0.459343\pi$
$228$	0	0
$229$	1.34596	0.0889434	0.0444717	−	0.999011i	$-0.485840\pi$
$229$	1.34596	0.0889434	0.0444717	+	0.999011i	$0.485840\pi$
$230$	0	0
$231$	−3.39104	−0.223114
$232$	0	0
$233$	−17.6433	−1.15585	−0.577925	−	0.816090i	$-0.696137\pi$
$233$	−17.6433	−1.15585	−0.577925	+	0.816090i	$0.696137\pi$
$234$	0	0
$235$	−4.13651	−0.269836
$236$	0	0
$237$	−8.40968	−0.546267
$238$	0	0
$239$	9.61500	0.621943	0.310971	−	0.950419i	$-0.399346\pi$
$239$	9.61500	0.621943	0.310971	+	0.950419i	$0.399346\pi$
$240$	0	0
$241$	21.3583	1.37581	0.687903	−	0.725802i	$-0.258531\pi$
$241$	21.3583	1.37581	0.687903	+	0.725802i	$0.258531\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	15.3741	0.982214
$246$	0	0
$247$	−37.9174	−2.41263
$248$	0	0
$249$	1.96134	0.124295
$250$	0	0
$251$	4.30448	0.271696	0.135848	−	0.990730i	$-0.456624\pi$
$251$	4.30448	0.271696	0.135848	+	0.990730i	$0.456624\pi$
$252$	0	0
$253$	−14.7821	−0.929341
$254$	0	0
$255$	2.16478	0.135564
$256$	0	0
$257$	−4.67271	−0.291476	−0.145738	−	0.989323i	$-0.546556\pi$
$257$	−4.67271	−0.291476	−0.145738	+	0.989323i	$0.546556\pi$
$258$	0	0
$259$	−2.27170	−0.141157
$260$	0	0
$261$	7.72286	0.478033
$262$	0	0
$263$	−15.4502	−0.952701	−0.476351	−	0.879255i	$-0.658041\pi$
$263$	−15.4502	−0.952701	−0.476351	+	0.879255i	$0.658041\pi$
$264$	0	0
$265$	8.47170	0.520413
$266$	0	0
$267$	2.79565	0.171091
$268$	0	0
$269$	31.2459	1.90510	0.952548	−	0.304388i	$-0.0984520\pi$
$269$	31.2459	1.90510	0.952548	+	0.304388i	$0.0984520\pi$
$270$	0	0
$271$	−9.88817	−0.600664	−0.300332	−	0.953835i	$-0.597097\pi$
$271$	−9.88817	−0.600664	−0.300332	+	0.953835i	$0.597097\pi$
$272$	0	0
$273$	5.33317	0.322778
$274$	0	0
$275$	4.55672	0.274781
$276$	0	0
$277$	−22.9565	−1.37932	−0.689660	−	0.724133i	$-0.742240\pi$
$277$	−22.9565	−1.37932	−0.689660	+	0.724133i	$0.742240\pi$
$278$	0	0
$279$	2.14386	0.128350
$280$	0	0
$281$	−14.5754	−0.869498	−0.434749	−	0.900552i	$-0.643163\pi$
$281$	−14.5754	−0.869498	−0.434749	+	0.900552i	$0.643163\pi$
$282$	0	0
$283$	4.71832	0.280475	0.140238	−	0.990118i	$-0.455213\pi$
$283$	4.71832	0.280475	0.140238	+	0.990118i	$0.455213\pi$
$284$	0	0
$285$	−16.2877	−0.964801
$286$	0	0
$287$	8.79565	0.519191
$288$	0	0
$289$	−16.2482	−0.955774
$290$	0	0
$291$	−2.26582	−0.132825
$292$	0	0
$293$	23.9204	1.39745	0.698723	−	0.715392i	$-0.253752\pi$
$293$	23.9204	1.39745	0.698723	+	0.715392i	$0.253752\pi$
$294$	0	0
$295$	31.7526	1.84871
$296$	0	0
$297$	3.69552	0.214436
$298$	0	0
$299$	23.2482	1.34448
$300$	0	0
$301$	8.79565	0.506973
$302$	0	0
$303$	9.98868	0.573834
$304$	0	0
$305$	0.0578646	0.00331332
$306$	0	0
$307$	20.7685	1.18532	0.592660	−	0.805453i	$-0.298078\pi$
$307$	20.7685	1.18532	0.592660	+	0.805453i	$0.298078\pi$
$308$	0	0
$309$	16.4180	0.933988
$310$	0	0
$311$	−11.7798	−0.667971	−0.333985	−	0.942578i	$-0.608394\pi$
$311$	−11.7798	−0.667971	−0.333985	+	0.942578i	$0.608394\pi$
$312$	0	0
$313$	11.1580	0.630687	0.315344	−	0.948978i	$-0.397880\pi$
$313$	11.1580	0.630687	0.315344	+	0.948978i	$0.397880\pi$
$314$	0	0
$315$	2.29090	0.129078
$316$	0	0
$317$	−15.9140	−0.893822	−0.446911	−	0.894578i	$-0.647476\pi$
$317$	−15.9140	−0.893822	−0.446911	+	0.894578i	$0.647476\pi$
$318$	0	0
$319$	28.5400	1.59793
$320$	0	0
$321$	−12.2522	−0.683853
$322$	0	0
$323$	−5.65685	−0.314756
$324$	0	0
$325$	−7.16648	−0.397525
$326$	0	0
$327$	−18.2973	−1.01184
$328$	0	0
$329$	−1.52034	−0.0838192
$330$	0	0
$331$	−2.73190	−0.150159	−0.0750794	−	0.997178i	$-0.523921\pi$
$331$	−2.73190	−0.150159	−0.0750794	+	0.997178i	$0.523921\pi$
$332$	0	0
$333$	2.47568	0.135666
$334$	0	0
$335$	−13.3001	−0.726664
$336$	0	0
$337$	−1.49657	−0.0815236	−0.0407618	−	0.999169i	$-0.512978\pi$
$337$	−1.49657	−0.0815236	−0.0407618	+	0.999169i	$0.512978\pi$
$338$	0	0
$339$	3.65685	0.198613
$340$	0	0
$341$	7.92267	0.429037
$342$	0	0
$343$	12.0739	0.651928
$344$	0	0
$345$	9.98642	0.537651
$346$	0	0
$347$	3.22944	0.173365	0.0866826	−	0.996236i	$-0.472373\pi$
$347$	3.22944	0.173365	0.0866826	+	0.996236i	$0.472373\pi$
$348$	0	0
$349$	−7.51074	−0.402041	−0.201020	−	0.979587i	$-0.564426\pi$
$349$	−7.51074	−0.402041	−0.201020	+	0.979587i	$0.564426\pi$
$350$	0	0
$351$	−5.81204	−0.310224
$352$	0	0
$353$	2.17491	0.115759	0.0578795	−	0.998324i	$-0.481566\pi$
$353$	2.17491	0.115759	0.0578795	+	0.998324i	$0.481566\pi$
$354$	0	0
$355$	29.5140	1.56644
$356$	0	0
$357$	0.795649	0.0421102
$358$	0	0
$359$	12.6828	0.669375	0.334687	−	0.942329i	$-0.391369\pi$
$359$	12.6828	0.669375	0.334687	+	0.942329i	$0.391369\pi$
$360$	0	0
$361$	23.5619	1.24010
$362$	0	0
$363$	2.65685	0.139449
$364$	0	0
$365$	38.1505	1.99689
$366$	0	0
$367$	21.4576	1.12008	0.560038	−	0.828467i	$-0.310787\pi$
$367$	21.4576	1.12008	0.560038	+	0.828467i	$0.310787\pi$
$368$	0	0
$369$	−9.58541	−0.498997
$370$	0	0
$371$	3.11371	0.161656
$372$	0	0
$373$	0.508459	0.0263270	0.0131635	−	0.999913i	$-0.495810\pi$
$373$	0.508459	0.0263270	0.0131635	+	0.999913i	$0.495810\pi$
$374$	0	0
$375$	9.40461	0.485652
$376$	0	0
$377$	−44.8855	−2.31172
$378$	0	0
$379$	19.9286	1.02366	0.511831	−	0.859086i	$-0.328968\pi$
$379$	19.9286	1.02366	0.511831	+	0.859086i	$0.328968\pi$
$380$	0	0
$381$	−20.6382	−1.05733
$382$	0	0
$383$	−29.3858	−1.50154	−0.750772	−	0.660562i	$-0.770318\pi$
$383$	−29.3858	−1.50154	−0.750772	+	0.660562i	$0.770318\pi$
$384$	0	0
$385$	8.46608	0.431471
$386$	0	0
$387$	−9.58541	−0.487254
$388$	0	0
$389$	−1.00379	−0.0508941	−0.0254470	−	0.999676i	$-0.508101\pi$
$389$	−1.00379	−0.0508941	−0.0254470	+	0.999676i	$0.508101\pi$
$390$	0	0
$391$	3.46836	0.175403
$392$	0	0
$393$	17.5140	0.883463
$394$	0	0
$395$	20.9956	1.05641
$396$	0	0
$397$	−7.02546	−0.352598	−0.176299	−	0.984337i	$-0.556413\pi$
$397$	−7.02546	−0.352598	−0.176299	+	0.984337i	$0.556413\pi$
$398$	0	0
$399$	−5.98642	−0.299696
$400$	0	0
$401$	4.18080	0.208779	0.104390	−	0.994536i	$-0.466711\pi$
$401$	4.18080	0.208779	0.104390	+	0.994536i	$0.466711\pi$
$402$	0	0
$403$	−12.4602	−0.620686
$404$	0	0
$405$	−2.49661	−0.124057
$406$	0	0
$407$	9.14892	0.453495
$408$	0	0
$409$	16.5818	0.819918	0.409959	−	0.912104i	$-0.365543\pi$
$409$	16.5818	0.819918	0.409959	+	0.912104i	$0.365543\pi$
$410$	0	0
$411$	8.25813	0.407343
$412$	0	0
$413$	11.6704	0.574264
$414$	0	0
$415$	−4.89668	−0.240369
$416$	0	0
$417$	−14.7047	−0.720094
$418$	0	0
$419$	22.2574	1.08734	0.543672	−	0.839298i	$-0.317034\pi$
$419$	22.2574	1.08734	0.543672	+	0.839298i	$0.317034\pi$
$420$	0	0
$421$	−28.4203	−1.38512	−0.692559	−	0.721361i	$-0.743517\pi$
$421$	−28.4203	−1.38512	−0.692559	+	0.721361i	$0.743517\pi$
$422$	0	0
$423$	1.65685	0.0805590
$424$	0	0
$425$	−1.06916	−0.0518618
$426$	0	0
$427$	0.0212677	0.00102921
$428$	0	0
$429$	−21.4785	−1.03699
$430$	0	0
$431$	36.6138	1.76363	0.881813	−	0.471599i	$-0.156323\pi$
$431$	36.6138	1.76363	0.881813	+	0.471599i	$0.156323\pi$
$432$	0	0
$433$	27.7526	1.33371	0.666853	−	0.745189i	$-0.267641\pi$
$433$	27.7526	1.33371	0.666853	+	0.745189i	$0.267641\pi$
$434$	0	0
$435$	−19.2809	−0.924450
$436$	0	0
$437$	−26.0958	−1.24833
$438$	0	0
$439$	21.1461	1.00925	0.504625	−	0.863339i	$-0.331631\pi$
$439$	21.1461	1.00925	0.504625	+	0.863339i	$0.331631\pi$
$440$	0	0
$441$	−6.15800	−0.293238
$442$	0	0
$443$	−7.30677	−0.347155	−0.173577	−	0.984820i	$-0.555533\pi$
$443$	−7.30677	−0.347155	−0.173577	+	0.984820i	$0.555533\pi$
$444$	0	0
$445$	−6.97963	−0.330866
$446$	0	0
$447$	16.1535	0.764032
$448$	0	0
$449$	−17.6356	−0.832275	−0.416137	−	0.909302i	$-0.636616\pi$
$449$	−17.6356	−0.832275	−0.416137	+	0.909302i	$0.636616\pi$
$450$	0	0
$451$	−35.4231	−1.66801
$452$	0	0
$453$	−19.2691	−0.905340
$454$	0	0
$455$	−13.3148	−0.624209
$456$	0	0
$457$	17.7070	0.828300	0.414150	−	0.910209i	$-0.364079\pi$
$457$	17.7070	0.828300	0.414150	+	0.910209i	$0.364079\pi$
$458$	0	0
$459$	−0.867091	−0.0404723
$460$	0	0
$461$	−36.9811	−1.72238	−0.861192	−	0.508280i	$-0.830281\pi$
$461$	−36.9811	−1.72238	−0.861192	+	0.508280i	$0.830281\pi$
$462$	0	0
$463$	−10.9632	−0.509504	−0.254752	−	0.967006i	$-0.581994\pi$
$463$	−10.9632	−0.509504	−0.254752	+	0.967006i	$0.581994\pi$
$464$	0	0
$465$	−5.35237	−0.248210
$466$	0	0
$467$	−12.6661	−0.586116	−0.293058	−	0.956095i	$-0.594673\pi$
$467$	−12.6661	−0.586116	−0.293058	+	0.956095i	$0.594673\pi$
$468$	0	0
$469$	−4.88836	−0.225723
$470$	0	0
$471$	−0.818827	−0.0377295
$472$	0	0
$473$	−35.4231	−1.62875
$474$	0	0
$475$	8.04429	0.369097
$476$	0	0
$477$	−3.39329	−0.155368
$478$	0	0
$479$	13.3036	0.607856	0.303928	−	0.952695i	$-0.401702\pi$
$479$	13.3036	0.607856	0.303928	+	0.952695i	$0.401702\pi$
$480$	0	0
$481$	−14.3888	−0.656071
$482$	0	0
$483$	3.67043	0.167010
$484$	0	0
$485$	5.65685	0.256865
$486$	0	0
$487$	5.44924	0.246929	0.123464	−	0.992349i	$-0.460600\pi$
$487$	5.44924	0.246929	0.123464	+	0.992349i	$0.460600\pi$
$488$	0	0
$489$	−23.6492	−1.06945
$490$	0	0
$491$	−21.7844	−0.983114	−0.491557	−	0.870845i	$-0.663572\pi$
$491$	−21.7844	−0.983114	−0.491557	+	0.870845i	$0.663572\pi$
$492$	0	0
$493$	−6.69642	−0.301592
$494$	0	0
$495$	−9.22625	−0.414689
$496$	0	0
$497$	10.8476	0.486583
$498$	0	0
$499$	−20.1595	−0.902465	−0.451232	−	0.892407i	$-0.649015\pi$
$499$	−20.1595	−0.902465	−0.451232	+	0.892407i	$0.649015\pi$
$500$	0	0
$501$	−9.30358	−0.415653
$502$	0	0
$503$	−33.5879	−1.49761	−0.748804	−	0.662791i	$-0.769372\pi$
$503$	−33.5879	−1.49761	−0.748804	+	0.662791i	$0.769372\pi$
$504$	0	0
$505$	−24.9378	−1.10972
$506$	0	0
$507$	20.7798	0.922863
$508$	0	0
$509$	11.1957	0.496242	0.248121	−	0.968729i	$-0.420187\pi$
$509$	11.1957	0.496242	0.248121	+	0.968729i	$0.420187\pi$
$510$	0	0
$511$	14.0219	0.620292
$512$	0	0
$513$	6.52395	0.288039
$514$	0	0
$515$	−40.9893	−1.80620
$516$	0	0
$517$	6.12293	0.269286
$518$	0	0
$519$	7.10557	0.311900
$520$	0	0
$521$	−35.2151	−1.54280	−0.771401	−	0.636349i	$-0.780444\pi$
$521$	−35.2151	−1.54280	−0.771401	+	0.636349i	$0.780444\pi$
$522$	0	0
$523$	−12.9444	−0.566020	−0.283010	−	0.959117i	$-0.591333\pi$
$523$	−12.9444	−0.566020	−0.283010	+	0.959117i	$0.591333\pi$
$524$	0	0
$525$	−1.13145	−0.0493804
$526$	0	0
$527$	−1.85892	−0.0809759
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−12.7183	−0.551928
$532$	0	0
$533$	55.7108	2.41310
$534$	0	0
$535$	30.5890	1.32248
$536$	0	0
$537$	−2.67271	−0.115336
$538$	0	0
$539$	−22.7570	−0.980213
$540$	0	0
$541$	−17.8584	−0.767792	−0.383896	−	0.923376i	$-0.625418\pi$
$541$	−17.8584	−0.767792	−0.383896	+	0.923376i	$0.625418\pi$
$542$	0	0
$543$	0.640465	0.0274850
$544$	0	0
$545$	45.6812	1.95677
$546$	0	0
$547$	−14.9191	−0.637893	−0.318947	−	0.947773i	$-0.603329\pi$
$547$	−14.9191	−0.637893	−0.318947	+	0.947773i	$0.603329\pi$
$548$	0	0
$549$	−0.0231773	−0.000989183 0
$550$	0	0
$551$	50.3835	2.14641
$552$	0	0
$553$	7.71679	0.328151
$554$	0	0
$555$	−6.18080	−0.262360
$556$	0	0
$557$	24.4649	1.03661	0.518305	−	0.855196i	$-0.326563\pi$
$557$	24.4649	1.03661	0.518305	+	0.855196i	$0.326563\pi$
$558$	0	0
$559$	55.7108	2.35632
$560$	0	0
$561$	−3.20435	−0.135288
$562$	0	0
$563$	−40.4459	−1.70459	−0.852295	−	0.523061i	$-0.824790\pi$
$563$	−40.4459	−1.70459	−0.852295	+	0.523061i	$0.824790\pi$
$564$	0	0
$565$	−9.12972	−0.384090
$566$	0	0
$567$	−0.917608	−0.0385359
$568$	0	0
$569$	42.1017	1.76499	0.882497	−	0.470318i	$-0.155861\pi$
$569$	42.1017	1.76499	0.882497	+	0.470318i	$0.155861\pi$
$570$	0	0
$571$	1.24996	0.0523091	0.0261546	−	0.999658i	$-0.491674\pi$
$571$	1.24996	0.0523091	0.0261546	+	0.999658i	$0.491674\pi$
$572$	0	0
$573$	14.6037	0.610079
$574$	0	0
$575$	−4.93216	−0.205685
$576$	0	0
$577$	29.3767	1.22297	0.611484	−	0.791257i	$-0.290573\pi$
$577$	29.3767	1.22297	0.611484	+	0.791257i	$0.290573\pi$
$578$	0	0
$579$	−8.81485	−0.366333
$580$	0	0
$581$	−1.79974	−0.0746657
$582$	0	0
$583$	−12.5400	−0.519352
$584$	0	0
$585$	14.5104	0.599930
$586$	0	0
$587$	17.7070	0.730847	0.365424	−	0.930841i	$-0.380924\pi$
$587$	17.7070	0.730847	0.365424	+	0.930841i	$0.380924\pi$
$588$	0	0
$589$	13.9864	0.576301
$590$	0	0
$591$	3.74551	0.154070
$592$	0	0
$593$	17.1207	0.703061	0.351530	−	0.936176i	$-0.385661\pi$
$593$	17.1207	0.703061	0.351530	+	0.936176i	$0.385661\pi$
$594$	0	0
$595$	−1.98642	−0.0814354
$596$	0	0
$597$	−14.8267	−0.606817
$598$	0	0
$599$	25.1908	1.02927	0.514634	−	0.857410i	$-0.327928\pi$
$599$	25.1908	1.02927	0.514634	+	0.857410i	$0.327928\pi$
$600$	0	0
$601$	22.5946	0.921655	0.460827	−	0.887490i	$-0.347553\pi$
$601$	22.5946	0.921655	0.460827	+	0.887490i	$0.347553\pi$
$602$	0	0
$603$	5.32729	0.216944
$604$	0	0
$605$	−6.63312	−0.269675
$606$	0	0
$607$	−8.88408	−0.360594	−0.180297	−	0.983612i	$-0.557706\pi$
$607$	−8.88408	−0.360594	−0.180297	+	0.983612i	$0.557706\pi$
$608$	0	0
$609$	−7.08655	−0.287162
$610$	0	0
$611$	−9.62970	−0.389576
$612$	0	0
$613$	−15.9960	−0.646073	−0.323037	−	0.946386i	$-0.604704\pi$
$613$	−15.9960	−0.646073	−0.323037	+	0.946386i	$0.604704\pi$
$614$	0	0
$615$	23.9310	0.964991
$616$	0	0
$617$	−2.99772	−0.120684	−0.0603418	−	0.998178i	$-0.519219\pi$
$617$	−2.99772	−0.120684	−0.0603418	+	0.998178i	$0.519219\pi$
$618$	0	0
$619$	15.0279	0.604024	0.302012	−	0.953304i	$-0.402342\pi$
$619$	15.0279	0.604024	0.302012	+	0.953304i	$0.402342\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	0	0
$623$	−2.56531	−0.102777
$624$	0	0
$625$	−29.6448	−1.18579
$626$	0	0
$627$	24.1094	0.962835
$628$	0	0
$629$	−2.14664	−0.0855922
$630$	0	0
$631$	19.3701	0.771112	0.385556	−	0.922684i	$-0.374010\pi$
$631$	19.3701	0.771112	0.385556	+	0.922684i	$0.374010\pi$
$632$	0	0
$633$	11.3910	0.452753
$634$	0	0
$635$	51.5255	2.04473
$636$	0	0
$637$	35.7905	1.41807
$638$	0	0
$639$	−11.8216	−0.467657
$640$	0	0
$641$	13.4489	0.531200	0.265600	−	0.964083i	$-0.414430\pi$
$641$	13.4489	0.531200	0.265600	+	0.964083i	$0.414430\pi$
$642$	0	0
$643$	−2.44662	−0.0964852	−0.0482426	−	0.998836i	$-0.515362\pi$
$643$	−2.44662	−0.0964852	−0.0482426	+	0.998836i	$0.515362\pi$
$644$	0	0
$645$	23.9310	0.942282
$646$	0	0
$647$	16.1275	0.634038	0.317019	−	0.948419i	$-0.397318\pi$
$647$	16.1275	0.634038	0.317019	+	0.948419i	$0.397318\pi$
$648$	0	0
$649$	−47.0008	−1.84494
$650$	0	0
$651$	−1.96722	−0.0771015
$652$	0	0
$653$	−28.6244	−1.12016	−0.560080	−	0.828439i	$-0.689230\pi$
$653$	−28.6244	−1.12016	−0.560080	+	0.828439i	$0.689230\pi$
$654$	0	0
$655$	−43.7255	−1.70850
$656$	0	0
$657$	−15.2809	−0.596166
$658$	0	0
$659$	−4.75185	−0.185106	−0.0925528	−	0.995708i	$-0.529503\pi$
$659$	−4.75185	−0.185106	−0.0925528	+	0.995708i	$0.529503\pi$
$660$	0	0
$661$	−21.8086	−0.848256	−0.424128	−	0.905602i	$-0.639419\pi$
$661$	−21.8086	−0.848256	−0.424128	+	0.905602i	$0.639419\pi$
$662$	0	0
$663$	5.03957	0.195721
$664$	0	0
$665$	14.9457	0.579571
$666$	0	0
$667$	−30.8914	−1.19612
$668$	0	0
$669$	−4.78110	−0.184848
$670$	0	0
$671$	−0.0856521	−0.00330656
$672$	0	0
$673$	6.34315	0.244510	0.122255	−	0.992499i	$-0.460987\pi$
$673$	6.34315	0.244510	0.122255	+	0.992499i	$0.460987\pi$
$674$	0	0
$675$	1.23304	0.0474597
$676$	0	0
$677$	45.6339	1.75385	0.876927	−	0.480624i	$-0.159590\pi$
$677$	45.6339	1.75385	0.876927	+	0.480624i	$0.159590\pi$
$678$	0	0
$679$	2.07913	0.0797898
$680$	0	0
$681$	3.83840	0.147088
$682$	0	0
$683$	37.9342	1.45151	0.725756	−	0.687953i	$-0.241490\pi$
$683$	37.9342	1.45151	0.725756	+	0.687953i	$0.241490\pi$
$684$	0	0
$685$	−20.6173	−0.787746
$686$	0	0
$687$	1.34596	0.0513515
$688$	0	0
$689$	19.7219	0.751345
$690$	0	0
$691$	−33.3716	−1.26951	−0.634757	−	0.772712i	$-0.718900\pi$
$691$	−33.3716	−1.26951	−0.634757	+	0.772712i	$0.718900\pi$
$692$	0	0
$693$	−3.39104	−0.128815
$694$	0	0
$695$	36.7120	1.39256
$696$	0	0
$697$	8.31143	0.314818
$698$	0	0
$699$	−17.6433	−0.667330
$700$	0	0
$701$	23.4480	0.885618	0.442809	−	0.896616i	$-0.353982\pi$
$701$	23.4480	0.885618	0.442809	+	0.896616i	$0.353982\pi$
$702$	0	0
$703$	16.1512	0.609154
$704$	0	0
$705$	−4.13651	−0.155790
$706$	0	0
$707$	−9.16569	−0.344711
$708$	0	0
$709$	12.1562	0.456537	0.228269	−	0.973598i	$-0.426694\pi$
$709$	12.1562	0.456537	0.228269	+	0.973598i	$0.426694\pi$
$710$	0	0
$711$	−8.40968	−0.315388
$712$	0	0
$713$	−8.57544	−0.321153
$714$	0	0
$715$	53.6233	2.00540
$716$	0	0
$717$	9.61500	0.359079
$718$	0	0
$719$	4.06900	0.151748	0.0758741	−	0.997117i	$-0.475825\pi$
$719$	4.06900	0.151748	0.0758741	+	0.997117i	$0.475825\pi$
$720$	0	0
$721$	−15.0653	−0.561061
$722$	0	0
$723$	21.3583	0.794322
$724$	0	0
$725$	9.52259	0.353660
$726$	0	0
$727$	17.3211	0.642402	0.321201	−	0.947011i	$-0.395913\pi$
$727$	17.3211	0.642402	0.321201	+	0.947011i	$0.395913\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.31143	0.307409
$732$	0	0
$733$	21.4361	0.791761	0.395880	−	0.918302i	$-0.370439\pi$
$733$	21.4361	0.791761	0.395880	+	0.918302i	$0.370439\pi$
$734$	0	0
$735$	15.3741	0.567082
$736$	0	0
$737$	19.6871	0.725183
$738$	0	0
$739$	31.2545	1.14972	0.574858	−	0.818253i	$-0.305057\pi$
$739$	31.2545	1.14972	0.574858	+	0.818253i	$0.305057\pi$
$740$	0	0
$741$	−37.9174	−1.39293
$742$	0	0
$743$	17.7244	0.650244	0.325122	−	0.945672i	$-0.394595\pi$
$743$	17.7244	0.650244	0.325122	+	0.945672i	$0.394595\pi$
$744$	0	0
$745$	−40.3288	−1.47753
$746$	0	0
$747$	1.96134	0.0717615
$748$	0	0
$749$	11.2428	0.410801
$750$	0	0
$751$	−7.38826	−0.269601	−0.134801	−	0.990873i	$-0.543039\pi$
$751$	−7.38826	−0.269601	−0.134801	+	0.990873i	$0.543039\pi$
$752$	0	0
$753$	4.30448	0.156864
$754$	0	0
$755$	48.1073	1.75080
$756$	0	0
$757$	−13.2804	−0.482684	−0.241342	−	0.970440i	$-0.577588\pi$
$757$	−13.2804	−0.482684	−0.241342	+	0.970440i	$0.577588\pi$
$758$	0	0
$759$	−14.7821	−0.536555
$760$	0	0
$761$	28.1473	1.02034	0.510169	−	0.860074i	$-0.329583\pi$
$761$	28.1473	1.02034	0.510169	+	0.860074i	$0.329583\pi$
$762$	0	0
$763$	16.7898	0.607830
$764$	0	0
$765$	2.16478	0.0782679
$766$	0	0
$767$	73.9194	2.66907
$768$	0	0
$769$	−11.9355	−0.430405	−0.215203	−	0.976569i	$-0.569041\pi$
$769$	−11.9355	−0.430405	−0.215203	+	0.976569i	$0.569041\pi$
$770$	0	0
$771$	−4.67271	−0.168284
$772$	0	0
$773$	−14.5868	−0.524649	−0.262325	−	0.964980i	$-0.584489\pi$
$773$	−14.5868	−0.524649	−0.262325	+	0.964980i	$0.584489\pi$
$774$	0	0
$775$	2.64347	0.0949561
$776$	0	0
$777$	−2.27170	−0.0814969
$778$	0	0
$779$	−62.5347	−2.24054
$780$	0	0
$781$	−43.6871	−1.56325
$782$	0	0
$783$	7.72286	0.275992
$784$	0	0
$785$	2.04429	0.0729638
$786$	0	0
$787$	−38.6835	−1.37892	−0.689459	−	0.724325i	$-0.742151\pi$
$787$	−38.6835	−1.37892	−0.689459	+	0.724325i	$0.742151\pi$
$788$	0	0
$789$	−15.4502	−0.550042
$790$	0	0
$791$	−3.35556	−0.119310
$792$	0	0
$793$	0.134707	0.00478360
$794$	0	0
$795$	8.47170	0.300460
$796$	0	0
$797$	29.4107	1.04178	0.520890	−	0.853624i	$-0.325600\pi$
$797$	29.4107	1.04178	0.520890	+	0.853624i	$0.325600\pi$
$798$	0	0
$799$	−1.43664	−0.0508248
$800$	0	0
$801$	2.79565	0.0987794
$802$	0	0
$803$	−56.4710	−1.99282
$804$	0	0
$805$	−9.16362	−0.322975
$806$	0	0
$807$	31.2459	1.09991
$808$	0	0
$809$	2.80334	0.0985602	0.0492801	−	0.998785i	$-0.484307\pi$
$809$	2.80334	0.0985602	0.0492801	+	0.998785i	$0.484307\pi$
$810$	0	0
$811$	51.0722	1.79339	0.896694	−	0.442650i	$-0.145962\pi$
$811$	51.0722	1.79339	0.896694	+	0.442650i	$0.145962\pi$
$812$	0	0
$813$	−9.88817	−0.346793
$814$	0	0
$815$	59.0426	2.06817
$816$	0	0
$817$	−62.5347	−2.18781
$818$	0	0
$819$	5.33317	0.186356
$820$	0	0
$821$	−17.4389	−0.608622	−0.304311	−	0.952573i	$-0.598426\pi$
$821$	−17.4389	−0.608622	−0.304311	+	0.952573i	$0.598426\pi$
$822$	0	0
$823$	−22.0665	−0.769191	−0.384595	−	0.923085i	$-0.625659\pi$
$823$	−22.0665	−0.769191	−0.384595	+	0.923085i	$0.625659\pi$
$824$	0	0
$825$	4.55672	0.158645
$826$	0	0
$827$	1.76621	0.0614172	0.0307086	−	0.999528i	$-0.490224\pi$
$827$	1.76621	0.0614172	0.0307086	+	0.999528i	$0.490224\pi$
$828$	0	0
$829$	−22.0514	−0.765879	−0.382939	−	0.923774i	$-0.625088\pi$
$829$	−22.0514	−0.765879	−0.382939	+	0.923774i	$0.625088\pi$
$830$	0	0
$831$	−22.9565	−0.796351
$832$	0	0
$833$	5.33954	0.185004
$834$	0	0
$835$	23.2274	0.803816
$836$	0	0
$837$	2.14386	0.0741026
$838$	0	0
$839$	−22.1904	−0.766099	−0.383050	−	0.923728i	$-0.625126\pi$
$839$	−22.1904	−0.766099	−0.383050	+	0.923728i	$0.625126\pi$
$840$	0	0
$841$	30.6425	1.05664
$842$	0	0
$843$	−14.5754	−0.502005
$844$	0	0
$845$	−51.8789	−1.78469
$846$	0	0
$847$	−2.43795	−0.0837690
$848$	0	0
$849$	4.71832	0.161932
$850$	0	0
$851$	−9.90272	−0.339461
$852$	0	0
$853$	37.4972	1.28388	0.641939	−	0.766756i	$-0.278130\pi$
$853$	37.4972	1.28388	0.641939	+	0.766756i	$0.278130\pi$
$854$	0	0
$855$	−16.2877	−0.557028
$856$	0	0
$857$	−17.4761	−0.596971	−0.298485	−	0.954414i	$-0.596481\pi$
$857$	−17.4761	−0.596971	−0.298485	+	0.954414i	$0.596481\pi$
$858$	0	0
$859$	10.7442	0.366586	0.183293	−	0.983058i	$-0.441324\pi$
$859$	10.7442	0.366586	0.183293	+	0.983058i	$0.441324\pi$
$860$	0	0
$861$	8.79565	0.299755
$862$	0	0
$863$	23.1791	0.789027	0.394514	−	0.918890i	$-0.370913\pi$
$863$	23.1791	0.789027	0.394514	+	0.918890i	$0.370913\pi$
$864$	0	0
$865$	−17.7398	−0.603171
$866$	0	0
$867$	−16.2482	−0.551816
$868$	0	0
$869$	−31.0781	−1.05425
$870$	0	0
$871$	−30.9624	−1.04912
$872$	0	0
$873$	−2.26582	−0.0766863
$874$	0	0
$875$	−8.62975	−0.291739
$876$	0	0
$877$	−28.3694	−0.957966	−0.478983	−	0.877824i	$-0.658995\pi$
$877$	−28.3694	−0.957966	−0.478983	+	0.877824i	$0.658995\pi$
$878$	0	0
$879$	23.9204	0.806816
$880$	0	0
$881$	−58.0822	−1.95684	−0.978420	−	0.206628i	$-0.933751\pi$
$881$	−58.0822	−1.95684	−0.978420	+	0.206628i	$0.933751\pi$
$882$	0	0
$883$	−5.55826	−0.187050	−0.0935252	−	0.995617i	$-0.529814\pi$
$883$	−5.55826	−0.187050	−0.0935252	+	0.995617i	$0.529814\pi$
$884$	0	0
$885$	31.7526	1.06735
$886$	0	0
$887$	−9.24359	−0.310369	−0.155185	−	0.987885i	$-0.549597\pi$
$887$	−9.24359	−0.310369	−0.155185	+	0.987885i	$0.549597\pi$
$888$	0	0
$889$	18.9378	0.635153
$890$	0	0
$891$	3.69552	0.123805
$892$	0	0
$893$	10.8092	0.361717
$894$	0	0
$895$	6.67271	0.223044
$896$	0	0
$897$	23.2482	0.776233
$898$	0	0
$899$	16.5567	0.552198
$900$	0	0
$901$	2.94229	0.0980219
$902$	0	0
$903$	8.79565	0.292701
$904$	0	0
$905$	−1.59899	−0.0531522
$906$	0	0
$907$	15.4945	0.514487	0.257243	−	0.966347i	$-0.417186\pi$
$907$	15.4945	0.514487	0.257243	+	0.966347i	$0.417186\pi$
$908$	0	0
$909$	9.98868	0.331303
$910$	0	0
$911$	−43.7108	−1.44820	−0.724101	−	0.689693i	$-0.757745\pi$
$911$	−43.7108	−1.44820	−0.724101	+	0.689693i	$0.757745\pi$
$912$	0	0
$913$	7.24815	0.239879
$914$	0	0
$915$	0.0578646	0.00191294
$916$	0	0
$917$	−16.0710	−0.530710
$918$	0	0
$919$	41.7955	1.37871	0.689353	−	0.724426i	$-0.257895\pi$
$919$	41.7955	1.37871	0.689353	+	0.724426i	$0.257895\pi$
$920$	0	0
$921$	20.7685	0.684345
$922$	0	0
$923$	68.7078	2.26155
$924$	0	0
$925$	3.05261	0.100369
$926$	0	0
$927$	16.4180	0.539238
$928$	0	0
$929$	24.1989	0.793942	0.396971	−	0.917831i	$-0.370061\pi$
$929$	24.1989	0.793942	0.396971	+	0.917831i	$0.370061\pi$
$930$	0	0
$931$	−40.1744	−1.31666
$932$	0	0
$933$	−11.7798	−0.385653
$934$	0	0
$935$	8.00000	0.261628
$936$	0	0
$937$	15.3183	0.500426	0.250213	−	0.968191i	$-0.419499\pi$
$937$	15.3183	0.500426	0.250213	+	0.968191i	$0.419499\pi$
$938$	0	0
$939$	11.1580	0.364127
$940$	0	0
$941$	35.3016	1.15080	0.575400	−	0.817872i	$-0.304846\pi$
$941$	35.3016	1.15080	0.575400	+	0.817872i	$0.304846\pi$
$942$	0	0
$943$	38.3417	1.24858
$944$	0	0
$945$	2.29090	0.0745231
$946$	0	0
$947$	14.2051	0.461605	0.230802	−	0.973001i	$-0.425865\pi$
$947$	14.2051	0.461605	0.230802	+	0.973001i	$0.425865\pi$
$948$	0	0
$949$	88.8134	2.88300
$950$	0	0
$951$	−15.9140	−0.516048
$952$	0	0
$953$	−19.6428	−0.636292	−0.318146	−	0.948042i	$-0.603060\pi$
$953$	−19.6428	−0.636292	−0.318146	+	0.948042i	$0.603060\pi$
$954$	0	0
$955$	−36.4597	−1.17981
$956$	0	0
$957$	28.5400	0.922566
$958$	0	0
$959$	−7.57772	−0.244697
$960$	0	0
$961$	−26.4039	−0.851738
$962$	0	0
$963$	−12.2522	−0.394823
$964$	0	0
$965$	22.0072	0.708437
$966$	0	0
$967$	−44.5337	−1.43211	−0.716054	−	0.698045i	$-0.754054\pi$
$967$	−44.5337	−1.43211	−0.716054	+	0.698045i	$0.754054\pi$
$968$	0	0
$969$	−5.65685	−0.181724
$970$	0	0
$971$	−1.40255	−0.0450099	−0.0225049	−	0.999747i	$-0.507164\pi$
$971$	−1.40255	−0.0450099	−0.0225049	+	0.999747i	$0.507164\pi$
$972$	0	0
$973$	13.4932	0.432572
$974$	0	0
$975$	−7.16648	−0.229511
$976$	0	0
$977$	−37.0794	−1.18628	−0.593138	−	0.805101i	$-0.702111\pi$
$977$	−37.0794	−1.18628	−0.593138	+	0.805101i	$0.702111\pi$
$978$	0	0
$979$	10.3314	0.330192
$980$	0	0
$981$	−18.2973	−0.584188
$982$	0	0
$983$	−46.6894	−1.48916	−0.744580	−	0.667534i	$-0.767350\pi$
$983$	−46.6894	−1.48916	−0.744580	+	0.667534i	$0.767350\pi$
$984$	0	0
$985$	−9.35105	−0.297949
$986$	0	0
$987$	−1.52034	−0.0483930
$988$	0	0
$989$	38.3417	1.21919
$990$	0	0
$991$	11.8780	0.377318	0.188659	−	0.982043i	$-0.439586\pi$
$991$	11.8780	0.377318	0.188659	+	0.982043i	$0.439586\pi$
$992$	0	0
$993$	−2.73190	−0.0866942
$994$	0	0
$995$	37.0164	1.17350
$996$	0	0
$997$	51.1515	1.61998	0.809992	−	0.586441i	$-0.199472\pi$
$997$	51.1515	1.61998	0.809992	+	0.586441i	$0.199472\pi$
$998$	0	0
$999$	2.47568	0.0783271

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3072.2.a.p.1.1		4
3.2	odd	2		9216.2.a.z.1.4		4
4.3	odd	2		3072.2.a.m.1.1		4
8.3	odd	2		3072.2.a.s.1.4		4
8.5	even	2		3072.2.a.j.1.4		4
12.11	even	2		9216.2.a.bl.1.4		4
16.3	odd	4		3072.2.d.e.1537.4		8
16.5	even	4		3072.2.d.j.1537.1		8
16.11	odd	4		3072.2.d.e.1537.5		8
16.13	even	4		3072.2.d.j.1537.8		8
24.5	odd	2		9216.2.a.ba.1.1		4
24.11	even	2		9216.2.a.bm.1.1		4
32.3	odd	8		1536.2.j.j.1153.2	yes	8
32.5	even	8		1536.2.j.f.385.1	yes	8
32.11	odd	8		1536.2.j.j.385.2	yes	8
32.13	even	8		1536.2.j.f.1153.1	yes	8
32.19	odd	8		1536.2.j.e.1153.3	yes	8
32.21	even	8		1536.2.j.i.385.4	yes	8
32.27	odd	8		1536.2.j.e.385.3	✓	8
32.29	even	8		1536.2.j.i.1153.4	yes	8
96.5	odd	8		4608.2.k.bj.3457.3		8
96.11	even	8		4608.2.k.be.3457.2		8
96.29	odd	8		4608.2.k.bc.1153.2		8
96.35	even	8		4608.2.k.be.1153.2		8
96.53	odd	8		4608.2.k.bc.3457.2		8
96.59	even	8		4608.2.k.bh.3457.3		8
96.77	odd	8		4608.2.k.bj.1153.3		8
96.83	even	8		4608.2.k.bh.1153.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.j.e.385.3	✓	8	32.27	odd	8
1536.2.j.e.1153.3	yes	8	32.19	odd	8
1536.2.j.f.385.1	yes	8	32.5	even	8
1536.2.j.f.1153.1	yes	8	32.13	even	8
1536.2.j.i.385.4	yes	8	32.21	even	8
1536.2.j.i.1153.4	yes	8	32.29	even	8
1536.2.j.j.385.2	yes	8	32.11	odd	8
1536.2.j.j.1153.2	yes	8	32.3	odd	8
3072.2.a.j.1.4		4	8.5	even	2
3072.2.a.m.1.1		4	4.3	odd	2
3072.2.a.p.1.1		4	1.1	even	1	trivial
3072.2.a.s.1.4		4	8.3	odd	2
3072.2.d.e.1537.4		8	16.3	odd	4
3072.2.d.e.1537.5		8	16.11	odd	4
3072.2.d.j.1537.1		8	16.5	even	4
3072.2.d.j.1537.8		8	16.13	even	4
4608.2.k.bc.1153.2		8	96.29	odd	8
4608.2.k.bc.3457.2		8	96.53	odd	8
4608.2.k.be.1153.2		8	96.35	even	8
4608.2.k.be.3457.2		8	96.11	even	8
4608.2.k.bh.1153.3		8	96.83	even	8
4608.2.k.bh.3457.3		8	96.59	even	8
4608.2.k.bj.1153.3		8	96.77	odd	8
4608.2.k.bj.3457.3		8	96.5	odd	8
9216.2.a.z.1.4		4	3.2	odd	2
9216.2.a.ba.1.1		4	24.5	odd	2
9216.2.a.bl.1.4		4	12.11	even	2
9216.2.a.bm.1.1		4	24.11	even	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:	\( N \)	\(=\)	\( 3072 = 2^{10} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3072.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.49661 q^{5} -0.917608 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.49661 q^{5} -0.917608 q^{7} +1.00000 q^{9} +3.69552 q^{11} -5.81204 q^{13} -2.49661 q^{15} -0.867091 q^{17} +6.52395 q^{19} -0.917608 q^{21} -4.00000 q^{23} +1.23304 q^{25} +1.00000 q^{27} +7.72286 q^{29} +2.14386 q^{31} +3.69552 q^{33} +2.29090 q^{35} +2.47568 q^{37} -5.81204 q^{39} -9.58541 q^{41} -9.58541 q^{43} -2.49661 q^{45} +1.65685 q^{47} -6.15800 q^{49} -0.867091 q^{51} -3.39329 q^{53} -9.22625 q^{55} +6.52395 q^{57} -12.7183 q^{59} -0.0231773 q^{61} -0.917608 q^{63} +14.5104 q^{65} +5.32729 q^{67} -4.00000 q^{69} -11.8216 q^{71} -15.2809 q^{73} +1.23304 q^{75} -3.39104 q^{77} -8.40968 q^{79} +1.00000 q^{81} +1.96134 q^{83} +2.16478 q^{85} +7.72286 q^{87} +2.79565 q^{89} +5.33317 q^{91} +2.14386 q^{93} -16.2877 q^{95} -2.26582 q^{97} +3.69552 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 8 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 8 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} - 8 q^{7} + 4 q^{9} - 8 q^{13} - 8 q^{21} - 16 q^{23} + 4 q^{25} + 4 q^{27} - 8 q^{31} - 16 q^{35} - 8 q^{37} - 8 q^{39} - 16 q^{47} + 4 q^{49} - 16 q^{55} - 16 q^{59} - 24 q^{61} - 8 q^{63} - 8 q^{65} + 16 q^{67} - 16 q^{69} - 16 q^{71} - 8 q^{73} + 4 q^{75} + 16 q^{77} - 24 q^{79} + 4 q^{81} - 8 q^{89} + 16 q^{91} - 8 q^{93} - 32 q^{95} - 16 q^{97}+O(q^{100}) \) 4 * q + 4 * q^3 - 8 * q^7 + 4 * q^9 - 8 * q^13 - 8 * q^21 - 16 * q^23 + 4 * q^25 + 4 * q^27 - 8 * q^31 - 16 * q^35 - 8 * q^37 - 8 * q^39 - 16 * q^47 + 4 * q^49 - 16 * q^55 - 16 * q^59 - 24 * q^61 - 8 * q^63 - 8 * q^65 + 16 * q^67 - 16 * q^69 - 16 * q^71 - 8 * q^73 + 4 * q^75 + 16 * q^77 - 24 * q^79 + 4 * q^81 - 8 * q^89 + 16 * q^91 - 8 * q^93 - 32 * q^95 - 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more