LMFDB - Embedded newform 3072.2.a.p.1.3

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			1.3
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)

$f(q)$	$=$	$q+1.00000 q^{3} -0.331821 q^{5} -3.08239 q^{7} +1.00000 q^{9} -3.69552 q^{11} +4.64047 q^{13} -0.331821 q^{15} +6.52395 q^{17} -0.867091 q^{19} -3.08239 q^{21} -4.00000 q^{23} -4.88989 q^{25} +1.00000 q^{27} -4.89443 q^{29} -6.14386 q^{31} -3.69552 q^{33} +1.02280 q^{35} -3.64725 q^{37} +4.64047 q^{39} +3.92856 q^{41} +3.92856 q^{43} -0.331821 q^{45} +1.65685 q^{47} +2.50114 q^{49} +6.52395 q^{51} +0.564862 q^{53} +1.22625 q^{55} -0.867091 q^{57} -6.59539 q^{59} -14.8052 q^{61} -3.08239 q^{63} -1.53981 q^{65} +13.9864 q^{67} -4.00000 q^{69} -7.49207 q^{71} +5.62408 q^{73} -4.88989 q^{75} +11.3910 q^{77} -14.9040 q^{79} +1.00000 q^{81} +9.35237 q^{83} -2.16478 q^{85} -4.89443 q^{87} -18.1094 q^{89} -14.3037 q^{91} -6.14386 q^{93} +0.287719 q^{95} -17.0479 q^{97} -3.69552 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 4 q^{3} - 8 q^{7} + 4 q^{9} - 8 q^{13} - 8 q^{21} - 16 q^{23} + 4 q^{25} + 4 q^{27} - 8 q^{31} - 16 q^{35} - 8 q^{37} - 8 q^{39} - 16 q^{47} + 4 q^{49} - 16 q^{55} - 16 q^{59} - 24 q^{61} - 8 q^{63}+ \cdots - 16 q^{97}+O(q^{100})$ 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

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
$5$	−0.331821	−0.148395	−0.0741975	−	0.997244i	$-0.523640\pi$
$5$	−0.331821	−0.148395	−0.0741975	+	0.997244i	$0.523640\pi$
$6$	0	0
$7$	−3.08239	−1.16503	−0.582517	−	0.812818i	$-0.697932\pi$
$7$	−3.08239	−1.16503	−0.582517	+	0.812818i	$0.697932\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.69552	−1.11424	−0.557120	−	0.830432i	$-0.688094\pi$
$11$	−3.69552	−1.11424	−0.557120	+	0.830432i	$0.688094\pi$
$12$	0	0
$13$	4.64047	1.28703	0.643517	−	0.765432i	$-0.277475\pi$
$13$	4.64047	1.28703	0.643517	+	0.765432i	$0.277475\pi$
$14$	0	0
$15$	−0.331821	−0.0856759
$16$	0	0
$17$	6.52395	1.58229	0.791145	−	0.611629i	$-0.209486\pi$
$17$	6.52395	1.58229	0.791145	+	0.611629i	$0.209486\pi$
$18$	0	0
$19$	−0.867091	−0.198924	−0.0994622	−	0.995041i	$-0.531712\pi$
$19$	−0.867091	−0.198924	−0.0994622	+	0.995041i	$0.531712\pi$
$20$	0	0
$21$	−3.08239	−0.672633
$22$	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
$25$	−4.88989	−0.977979
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.89443	−0.908873	−0.454436	−	0.890779i	$-0.650159\pi$
$29$	−4.89443	−0.908873	−0.454436	+	0.890779i	$0.650159\pi$
$30$	0	0
$31$	−6.14386	−1.10347	−0.551735	−	0.834020i	$-0.686034\pi$
$31$	−6.14386	−1.10347	−0.551735	+	0.834020i	$0.686034\pi$
$32$	0	0
$33$	−3.69552	−0.643307
$34$	0	0
$35$	1.02280	0.172885
$36$	0	0
$37$	−3.64725	−0.599605	−0.299802	−	0.954001i	$-0.596921\pi$
$37$	−3.64725	−0.599605	−0.299802	+	0.954001i	$0.596921\pi$
$38$	0	0
$39$	4.64047	0.743069
$40$	0	0
$41$	3.92856	0.613538	0.306769	−	0.951784i	$-0.400752\pi$
$41$	3.92856	0.613538	0.306769	+	0.951784i	$0.400752\pi$
$42$	0	0
$43$	3.92856	0.599100	0.299550	−	0.954081i	$-0.403164\pi$
$43$	3.92856	0.599100	0.299550	+	0.954081i	$0.403164\pi$
$44$	0	0
$45$	−0.331821	−0.0494650
$46$	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
$49$	2.50114	0.357306
$50$	0	0
$51$	6.52395	0.913535
$52$	0	0
$53$	0.564862	0.0775897	0.0387949	−	0.999247i	$-0.487648\pi$
$53$	0.564862	0.0775897	0.0387949	+	0.999247i	$0.487648\pi$
$54$	0	0
$55$	1.22625	0.165348
$56$	0	0
$57$	−0.867091	−0.114849
$58$	0	0
$59$	−6.59539	−0.858646	−0.429323	−	0.903151i	$-0.641248\pi$
$59$	−6.59539	−0.858646	−0.429323	+	0.903151i	$0.641248\pi$
$60$	0	0
$61$	−14.8052	−1.89562	−0.947809	−	0.318839i	$-0.896707\pi$
$61$	−14.8052	−1.89562	−0.947809	+	0.318839i	$0.896707\pi$
$62$	0	0
$63$	−3.08239	−0.388345
$64$	0	0
$65$	−1.53981	−0.190989
$66$	0	0
$67$	13.9864	1.70871	0.854357	−	0.519687i	$-0.173951\pi$
$67$	13.9864	1.70871	0.854357	+	0.519687i	$0.173951\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−7.49207	−0.889145	−0.444573	−	0.895743i	$-0.646644\pi$
$71$	−7.49207	−0.889145	−0.444573	+	0.895743i	$0.646644\pi$
$72$	0	0
$73$	5.62408	0.658248	0.329124	−	0.944287i	$-0.393247\pi$
$73$	5.62408	0.658248	0.329124	+	0.944287i	$0.393247\pi$
$74$	0	0
$75$	−4.88989	−0.564636
$76$	0	0
$77$	11.3910	1.29813
$78$	0	0
$79$	−14.9040	−1.67683	−0.838417	−	0.545029i	$-0.816519\pi$
$79$	−14.9040	−1.67683	−0.838417	+	0.545029i	$0.816519\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.35237	1.02656	0.513278	−	0.858222i	$-0.328431\pi$
$83$	9.35237	1.02656	0.513278	+	0.858222i	$0.328431\pi$
$84$	0	0
$85$	−2.16478	−0.234804
$86$	0	0
$87$	−4.89443	−0.524738
$88$	0	0
$89$	−18.1094	−1.91959	−0.959794	−	0.280705i	$-0.909432\pi$
$89$	−18.1094	−1.91959	−0.959794	+	0.280705i	$0.909432\pi$
$90$	0	0
$91$	−14.3037	−1.49944
$92$	0	0
$93$	−6.14386	−0.637089
$94$	0	0
$95$	0.287719	0.0295194
$96$	0	0
$97$	−17.0479	−1.73095	−0.865476	−	0.500951i	$-0.832984\pi$
$97$	−17.0479	−1.73095	−0.865476	+	0.500951i	$0.832984\pi$
$98$	0	0
$99$	−3.69552	−0.371414
$100$	0	0
$101$	12.1535	1.20931	0.604657	−	0.796486i	$-0.293310\pi$
$101$	12.1535	1.20931	0.604657	+	0.796486i	$0.293310\pi$
$102$	0	0
$103$	−17.1043	−1.68534	−0.842668	−	0.538433i	$-0.819016\pi$
$103$	−17.1043	−1.68534	−0.842668	+	0.538433i	$0.819016\pi$
$104$	0	0
$105$	1.02280	0.0998154
$106$	0	0
$107$	−18.3752	−1.77640	−0.888198	−	0.459462i	$-0.848042\pi$
$107$	−18.3752	−1.77640	−0.888198	+	0.459462i	$0.848042\pi$
$108$	0	0
$109$	−7.84482	−0.751397	−0.375699	−	0.926742i	$-0.622597\pi$
$109$	−7.84482	−0.751397	−0.375699	+	0.926742i	$0.622597\pi$
$110$	0	0
$111$	−3.64725	−0.346182
$112$	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
$115$	1.32729	0.123770
$116$	0	0
$117$	4.64047	0.429011
$118$	0	0
$119$	−20.1094	−1.84342
$120$	0	0
$121$	2.65685	0.241532
$122$	0	0
$123$	3.92856	0.354226
$124$	0	0
$125$	3.28168	0.293522
$126$	0	0
$127$	0.638213	0.0566322	0.0283161	−	0.999599i	$-0.490985\pi$
$127$	0.638213	0.0566322	0.0283161	+	0.999599i	$0.490985\pi$
$128$	0	0
$129$	3.92856	0.345890
$130$	0	0
$131$	−9.51397	−0.831240	−0.415620	−	0.909538i	$-0.636435\pi$
$131$	−9.51397	−0.831240	−0.415620	+	0.909538i	$0.636435\pi$
$132$	0	0
$133$	2.67271	0.231754
$134$	0	0
$135$	−0.331821	−0.0285586
$136$	0	0
$137$	−13.9150	−1.18884	−0.594419	−	0.804156i	$-0.702618\pi$
$137$	−13.9150	−1.18884	−0.594419	+	0.804156i	$0.702618\pi$
$138$	0	0
$139$	0.0773278	0.00655886	0.00327943	−	0.999995i	$-0.498956\pi$
$139$	0.0773278	0.00655886	0.00327943	+	0.999995i	$0.498956\pi$
$140$	0	0
$141$	1.65685	0.139532
$142$	0	0
$143$	−17.1489	−1.43407
$144$	0	0
$145$	1.62408	0.134872
$146$	0	0
$147$	2.50114	0.206291
$148$	0	0
$149$	13.9887	1.14600	0.572998	−	0.819556i	$-0.305780\pi$
$149$	13.9887	1.14600	0.572998	+	0.819556i	$0.305780\pi$
$150$	0	0
$151$	18.5828	1.51225	0.756123	−	0.654430i	$-0.227091\pi$
$151$	18.5828	1.51225	0.756123	+	0.654430i	$0.227091\pi$
$152$	0	0
$153$	6.52395	0.527430
$154$	0	0
$155$	2.03866	0.163749
$156$	0	0
$157$	5.30411	0.423314	0.211657	−	0.977344i	$-0.432114\pi$
$157$	5.30411	0.423314	0.211657	+	0.977344i	$0.432114\pi$
$158$	0	0
$159$	0.564862	0.0447964
$160$	0	0
$161$	12.3296	0.971706
$162$	0	0
$163$	13.3060	1.04221	0.521104	−	0.853493i	$-0.325520\pi$
$163$	13.3060	1.04221	0.521104	+	0.853493i	$0.325520\pi$
$164$	0	0
$165$	1.22625	0.0954636
$166$	0	0
$167$	15.9310	1.23278	0.616389	−	0.787442i	$-0.288595\pi$
$167$	15.9310	1.23278	0.616389	+	0.787442i	$0.288595\pi$
$168$	0	0
$169$	8.53392	0.656455
$170$	0	0
$171$	−0.867091	−0.0663081
$172$	0	0
$173$	19.7229	1.49950	0.749751	−	0.661721i	$-0.230173\pi$
$173$	19.7229	1.49950	0.749751	+	0.661721i	$0.230173\pi$
$174$	0	0
$175$	15.0726	1.13938
$176$	0	0
$177$	−6.59539	−0.495740
$178$	0	0
$179$	5.98642	0.447446	0.223723	−	0.974653i	$-0.428179\pi$
$179$	5.98642	0.447446	0.223723	+	0.974653i	$0.428179\pi$
$180$	0	0
$181$	−9.81204	−0.729323	−0.364662	−	0.931140i	$-0.618815\pi$
$181$	−9.81204	−0.729323	−0.364662	+	0.931140i	$0.618815\pi$
$182$	0	0
$183$	−14.8052	−1.09444
$184$	0	0
$185$	1.21024	0.0889784
$186$	0	0
$187$	−24.1094	−1.76305
$188$	0	0
$189$	−3.08239	−0.224211
$190$	0	0
$191$	−19.2900	−1.39578	−0.697888	−	0.716207i	$-0.745877\pi$
$191$	−19.2900	−1.39578	−0.697888	+	0.716207i	$0.745877\pi$
$192$	0	0
$193$	−0.155713	−0.0112084	−0.00560422	−	0.999984i	$-0.501784\pi$
$193$	−0.155713	−0.0112084	−0.00560422	+	0.999984i	$0.501784\pi$
$194$	0	0
$195$	−1.53981	−0.110268
$196$	0	0
$197$	−13.2014	−0.940557	−0.470279	−	0.882518i	$-0.655847\pi$
$197$	−13.2014	−0.940557	−0.470279	+	0.882518i	$0.655847\pi$
$198$	0	0
$199$	−23.1144	−1.63854	−0.819269	−	0.573409i	$-0.805620\pi$
$199$	−23.1144	−1.63854	−0.819269	+	0.573409i	$0.805620\pi$
$200$	0	0
$201$	13.9864	0.986526
$202$	0	0
$203$	15.0866	1.05887
$204$	0	0
$205$	−1.30358	−0.0910459
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	3.20435	0.221650
$210$	0	0
$211$	−3.39104	−0.233449	−0.116724	−	0.993164i	$-0.537239\pi$
$211$	−3.39104	−0.233449	−0.116724	+	0.993164i	$0.537239\pi$
$212$	0	0
$213$	−7.49207	−0.513348
$214$	0	0
$215$	−1.30358	−0.0889034
$216$	0	0
$217$	18.9378	1.28558
$218$	0	0
$219$	5.62408	0.380040
$220$	0	0
$221$	30.2741	2.03646
$222$	0	0
$223$	−10.5326	−0.705316	−0.352658	−	0.935752i	$-0.614722\pi$
$223$	−10.5326	−0.705316	−0.352658	+	0.935752i	$0.614722\pi$
$224$	0	0
$225$	−4.88989	−0.325993
$226$	0	0
$227$	23.4753	1.55811	0.779055	−	0.626955i	$-0.215699\pi$
$227$	23.4753	1.55811	0.779055	+	0.626955i	$0.215699\pi$
$228$	0	0
$229$	3.13932	0.207452	0.103726	−	0.994606i	$-0.466923\pi$
$229$	3.13932	0.207452	0.103726	+	0.994606i	$0.466923\pi$
$230$	0	0
$231$	11.3910	0.749475
$232$	0	0
$233$	−8.98414	−0.588571	−0.294285	−	0.955718i	$-0.595082\pi$
$233$	−8.98414	−0.588571	−0.294285	+	0.955718i	$0.595082\pi$
$234$	0	0
$235$	−0.549780	−0.0358637
$236$	0	0
$237$	−14.9040	−0.968121
$238$	0	0
$239$	1.69870	0.109880	0.0549400	−	0.998490i	$-0.482503\pi$
$239$	1.69870	0.109880	0.0549400	+	0.998490i	$0.482503\pi$
$240$	0	0
$241$	−14.3288	−0.923001	−0.461500	−	0.887140i	$-0.652689\pi$
$241$	−14.3288	−0.923001	−0.461500	+	0.887140i	$0.652689\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−0.829932	−0.0530224
$246$	0	0
$247$	−4.02371	−0.256022
$248$	0	0
$249$	9.35237	0.592683
$250$	0	0
$251$	11.6955	0.738215	0.369107	−	0.929387i	$-0.379663\pi$
$251$	11.6955	0.738215	0.369107	+	0.929387i	$0.379663\pi$
$252$	0	0
$253$	14.7821	0.929341
$254$	0	0
$255$	−2.16478	−0.135564
$256$	0	0
$257$	3.98642	0.248666	0.124333	−	0.992241i	$-0.460321\pi$
$257$	3.98642	0.248666	0.124333	+	0.992241i	$0.460321\pi$
$258$	0	0
$259$	11.2423	0.698560
$260$	0	0
$261$	−4.89443	−0.302958
$262$	0	0
$263$	−11.8635	−0.731534	−0.365767	−	0.930706i	$-0.619193\pi$
$263$	−11.8635	−0.731534	−0.365767	+	0.930706i	$0.619193\pi$
$264$	0	0
$265$	−0.187433	−0.0115139
$266$	0	0
$267$	−18.1094	−1.10827
$268$	0	0
$269$	−21.3880	−1.30405	−0.652026	−	0.758197i	$-0.726081\pi$
$269$	−21.3880	−1.30405	−0.652026	+	0.758197i	$0.726081\pi$
$270$	0	0
$271$	−12.0530	−0.732165	−0.366082	−	0.930582i	$-0.619301\pi$
$271$	−12.0530	−0.732165	−0.366082	+	0.930582i	$0.619301\pi$
$272$	0	0
$273$	−14.3037	−0.865701
$274$	0	0
$275$	18.0707	1.08970
$276$	0	0
$277$	4.81432	0.289265	0.144632	−	0.989485i	$-0.453800\pi$
$277$	4.81432	0.289265	0.144632	+	0.989485i	$0.453800\pi$
$278$	0	0
$279$	−6.14386	−0.367823
$280$	0	0
$281$	18.5754	1.10812	0.554059	−	0.832477i	$-0.313078\pi$
$281$	18.5754	1.10812	0.554059	+	0.832477i	$0.313078\pi$
$282$	0	0
$283$	−1.40461	−0.0834956	−0.0417478	−	0.999128i	$-0.513293\pi$
$283$	−1.40461	−0.0834956	−0.0417478	+	0.999128i	$0.513293\pi$
$284$	0	0
$285$	0.287719	0.0170430
$286$	0	0
$287$	−12.1094	−0.714793
$288$	0	0
$289$	25.5619	1.50364
$290$	0	0
$291$	−17.0479	−0.999365
$292$	0	0
$293$	27.8786	1.62868	0.814342	−	0.580386i	$-0.197098\pi$
$293$	27.8786	1.62868	0.814342	+	0.580386i	$0.197098\pi$
$294$	0	0
$295$	2.18849	0.127419
$296$	0	0
$297$	−3.69552	−0.214436
$298$	0	0
$299$	−18.5619	−1.07346
$300$	0	0
$301$	−12.1094	−0.697972
$302$	0	0
$303$	12.1535	0.698198
$304$	0	0
$305$	4.91270	0.281300
$306$	0	0
$307$	−17.4548	−0.996197	−0.498099	−	0.867120i	$-0.665968\pi$
$307$	−17.4548	−0.996197	−0.498099	+	0.867120i	$0.665968\pi$
$308$	0	0
$309$	−17.1043	−0.973029
$310$	0	0
$311$	0.466081	0.0264290	0.0132145	−	0.999913i	$-0.495794\pi$
$311$	0.466081	0.0264290	0.0132145	+	0.999913i	$0.495794\pi$
$312$	0	0
$313$	2.49886	0.141244	0.0706219	−	0.997503i	$-0.477502\pi$
$313$	2.49886	0.141244	0.0706219	+	0.997503i	$0.477502\pi$
$314$	0	0
$315$	1.02280	0.0576285
$316$	0	0
$317$	10.7425	0.603358	0.301679	−	0.953410i	$-0.402453\pi$
$317$	10.7425	0.603358	0.301679	+	0.953410i	$0.402453\pi$
$318$	0	0
$319$	18.0875	1.01270
$320$	0	0
$321$	−18.3752	−1.02560
$322$	0	0
$323$	−5.65685	−0.314756
$324$	0	0
$325$	−22.6914	−1.25869
$326$	0	0
$327$	−7.84482	−0.433819
$328$	0	0
$329$	−5.10707	−0.281562
$330$	0	0
$331$	−5.26810	−0.289561	−0.144781	−	0.989464i	$-0.546248\pi$
$331$	−5.26810	−0.289561	−0.144781	+	0.989464i	$0.546248\pi$
$332$	0	0
$333$	−3.64725	−0.199868
$334$	0	0
$335$	−4.64099	−0.253565
$336$	0	0
$337$	−27.4740	−1.49660	−0.748302	−	0.663359i	$-0.769130\pi$
$337$	−27.4740	−1.49660	−0.748302	+	0.663359i	$0.769130\pi$
$338$	0	0
$339$	3.65685	0.198613
$340$	0	0
$341$	22.7047	1.22953
$342$	0	0
$343$	13.8672	0.748761
$344$	0	0
$345$	1.32729	0.0714586
$346$	0	0
$347$	8.08427	0.433986	0.216993	−	0.976173i	$-0.430375\pi$
$347$	8.08427	0.433986	0.216993	+	0.976173i	$0.430375\pi$
$348$	0	0
$349$	−4.97454	−0.266281	−0.133140	−	0.991097i	$-0.542506\pi$
$349$	−4.97454	−0.266281	−0.133140	+	0.991097i	$0.542506\pi$
$350$	0	0
$351$	4.64047	0.247690
$352$	0	0
$353$	23.0799	1.22842	0.614210	−	0.789143i	$-0.289475\pi$
$353$	23.0799	1.22842	0.614210	+	0.789143i	$0.289475\pi$
$354$	0	0
$355$	2.48603	0.131945
$356$	0	0
$357$	−20.1094	−1.06430
$358$	0	0
$359$	29.2583	1.54419	0.772097	−	0.635505i	$-0.219208\pi$
$359$	29.2583	1.54419	0.772097	+	0.635505i	$0.219208\pi$
$360$	0	0
$361$	−18.2482	−0.960429
$362$	0	0
$363$	2.65685	0.139449
$364$	0	0
$365$	−1.86619	−0.0976808
$366$	0	0
$367$	13.1698	0.687461	0.343730	−	0.939068i	$-0.388309\pi$
$367$	13.1698	0.687461	0.343730	+	0.939068i	$0.388309\pi$
$368$	0	0
$369$	3.92856	0.204513
$370$	0	0
$371$	−1.74113	−0.0903947
$372$	0	0
$373$	15.2905	0.791714	0.395857	−	0.918312i	$-0.370448\pi$
$373$	15.2905	0.791714	0.395857	+	0.918312i	$0.370448\pi$
$374$	0	0
$375$	3.28168	0.169465
$376$	0	0
$377$	−22.7124	−1.16975
$378$	0	0
$379$	6.41459	0.329495	0.164748	−	0.986336i	$-0.447319\pi$
$379$	6.41459	0.329495	0.164748	+	0.986336i	$0.447319\pi$
$380$	0	0
$381$	0.638213	0.0326966
$382$	0	0
$383$	34.0721	1.74100	0.870501	−	0.492167i	$-0.163795\pi$
$383$	34.0721	1.74100	0.870501	+	0.492167i	$0.163795\pi$
$384$	0	0
$385$	−3.77979	−0.192636
$386$	0	0
$387$	3.92856	0.199700
$388$	0	0
$389$	32.5185	1.64875	0.824377	−	0.566041i	$-0.191526\pi$
$389$	32.5185	1.64875	0.824377	+	0.566041i	$0.191526\pi$
$390$	0	0
$391$	−26.0958	−1.31972
$392$	0	0
$393$	−9.51397	−0.479916
$394$	0	0
$395$	4.94548	0.248834
$396$	0	0
$397$	−4.48926	−0.225309	−0.112655	−	0.993634i	$-0.535935\pi$
$397$	−4.48926	−0.225309	−0.112655	+	0.993634i	$0.535935\pi$
$398$	0	0
$399$	2.67271	0.133803
$400$	0	0
$401$	−3.21024	−0.160312	−0.0801558	−	0.996782i	$-0.525542\pi$
$401$	−3.21024	−0.160312	−0.0801558	+	0.996782i	$0.525542\pi$
$402$	0	0
$403$	−28.5104	−1.42020
$404$	0	0
$405$	−0.331821	−0.0164883
$406$	0	0
$407$	13.4785	0.668104
$408$	0	0
$409$	14.0456	0.694511	0.347255	−	0.937771i	$-0.387114\pi$
$409$	14.0456	0.694511	0.347255	+	0.937771i	$0.387114\pi$
$410$	0	0
$411$	−13.9150	−0.686375
$412$	0	0
$413$	20.3296	1.00035
$414$	0	0
$415$	−3.10332	−0.152336
$416$	0	0
$417$	0.0773278	0.00378676
$418$	0	0
$419$	−26.9437	−1.31628	−0.658142	−	0.752894i	$-0.728657\pi$
$419$	−26.9437	−1.31628	−0.658142	+	0.752894i	$0.728657\pi$
$420$	0	0
$421$	−5.72188	−0.278867	−0.139434	−	0.990231i	$-0.544528\pi$
$421$	−5.72188	−0.278867	−0.139434	+	0.990231i	$0.544528\pi$
$422$	0	0
$423$	1.65685	0.0805590
$424$	0	0
$425$	−31.9014	−1.54745
$426$	0	0
$427$	45.6356	2.20846
$428$	0	0
$429$	−17.1489	−0.827958
$430$	0	0
$431$	27.9547	1.34653	0.673265	−	0.739401i	$-0.264891\pi$
$431$	27.9547	1.34653	0.673265	+	0.739401i	$0.264891\pi$
$432$	0	0
$433$	−1.81151	−0.0870556	−0.0435278	−	0.999052i	$-0.513860\pi$
$433$	−1.81151	−0.0870556	−0.0435278	+	0.999052i	$0.513860\pi$
$434$	0	0
$435$	1.62408	0.0778685
$436$	0	0
$437$	3.46836	0.165914
$438$	0	0
$439$	−4.45985	−0.212857	−0.106429	−	0.994320i	$-0.533942\pi$
$439$	−4.45985	−0.212857	−0.106429	+	0.994320i	$0.533942\pi$
$440$	0	0
$441$	2.50114	0.119102
$442$	0	0
$443$	2.62047	0.124502	0.0622512	−	0.998061i	$-0.480172\pi$
$443$	2.62047	0.124502	0.0622512	+	0.998061i	$0.480172\pi$
$444$	0	0
$445$	6.00907	0.284857
$446$	0	0
$447$	13.9887	0.661642
$448$	0	0
$449$	27.9787	1.32040	0.660199	−	0.751091i	$-0.270472\pi$
$449$	27.9787	1.32040	0.660199	+	0.751091i	$0.270472\pi$
$450$	0	0
$451$	−14.5181	−0.683629
$452$	0	0
$453$	18.5828	0.873095
$454$	0	0
$455$	4.74628	0.222509
$456$	0	0
$457$	−14.3933	−0.673291	−0.336646	−	0.941631i	$-0.609292\pi$
$457$	−14.3933	−0.673291	−0.336646	+	0.941631i	$0.609292\pi$
$458$	0	0
$459$	6.52395	0.304512
$460$	0	0
$461$	−7.78841	−0.362743	−0.181371	−	0.983415i	$-0.558054\pi$
$461$	−7.78841	−0.362743	−0.181371	+	0.983415i	$0.558054\pi$
$462$	0	0
$463$	−15.6642	−0.727977	−0.363989	−	0.931403i	$-0.618585\pi$
$463$	−15.6642	−0.727977	−0.363989	+	0.931403i	$0.618585\pi$
$464$	0	0
$465$	2.03866	0.0945408
$466$	0	0
$467$	−5.27504	−0.244100	−0.122050	−	0.992524i	$-0.538947\pi$
$467$	−5.27504	−0.244100	−0.122050	+	0.992524i	$0.538947\pi$
$468$	0	0
$469$	−43.1116	−1.99071
$470$	0	0
$471$	5.30411	0.244400
$472$	0	0
$473$	−14.5181	−0.667541
$474$	0	0
$475$	4.23998	0.194544
$476$	0	0
$477$	0.564862	0.0258632
$478$	0	0
$479$	−11.9310	−0.545141	−0.272571	−	0.962136i	$-0.587874\pi$
$479$	−11.9310	−0.545141	−0.272571	+	0.962136i	$0.587874\pi$
$480$	0	0
$481$	−16.9250	−0.771712
$482$	0	0
$483$	12.3296	0.561015
$484$	0	0
$485$	5.65685	0.256865
$486$	0	0
$487$	37.1782	1.68470	0.842352	−	0.538928i	$-0.181170\pi$
$487$	37.1782	1.68470	0.842352	+	0.538928i	$0.181170\pi$
$488$	0	0
$489$	13.3060	0.601719
$490$	0	0
$491$	25.0981	1.13266	0.566330	−	0.824179i	$-0.308363\pi$
$491$	25.0981	1.13266	0.566330	+	0.824179i	$0.308363\pi$
$492$	0	0
$493$	−31.9310	−1.43810
$494$	0	0
$495$	1.22625	0.0551159
$496$	0	0
$497$	23.0935	1.03588
$498$	0	0
$499$	32.8458	1.47038	0.735190	−	0.677861i	$-0.237093\pi$
$499$	32.8458	1.47038	0.735190	+	0.677861i	$0.237093\pi$
$500$	0	0
$501$	15.9310	0.711744
$502$	0	0
$503$	−8.35327	−0.372454	−0.186227	−	0.982507i	$-0.559626\pi$
$503$	−8.35327	−0.372454	−0.186227	+	0.982507i	$0.559626\pi$
$504$	0	0
$505$	−4.03278	−0.179456
$506$	0	0
$507$	8.53392	0.379005
$508$	0	0
$509$	−9.33786	−0.413893	−0.206947	−	0.978352i	$-0.566353\pi$
$509$	−9.33786	−0.413893	−0.206947	+	0.978352i	$0.566353\pi$
$510$	0	0
$511$	−17.3356	−0.766882
$512$	0	0
$513$	−0.867091	−0.0382830
$514$	0	0
$515$	5.67557	0.250095
$516$	0	0
$517$	−6.12293	−0.269286
$518$	0	0
$519$	19.7229	0.865737
$520$	0	0
$521$	−4.38287	−0.192017	−0.0960084	−	0.995381i	$-0.530608\pi$
$521$	−4.38287	−0.192017	−0.0960084	+	0.995381i	$0.530608\pi$
$522$	0	0
$523$	9.22869	0.403542	0.201771	−	0.979433i	$-0.435330\pi$
$523$	9.22869	0.403542	0.201771	+	0.979433i	$0.435330\pi$
$524$	0	0
$525$	15.0726	0.657821
$526$	0	0
$527$	−40.0822	−1.74601
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−6.59539	−0.286215
$532$	0	0
$533$	18.2303	0.789644
$534$	0	0
$535$	6.09728	0.263608
$536$	0	0
$537$	5.98642	0.258333
$538$	0	0
$539$	−9.24301	−0.398125
$540$	0	0
$541$	−36.9700	−1.58947	−0.794733	−	0.606959i	$-0.792389\pi$
$541$	−36.9700	−1.58947	−0.794733	+	0.606959i	$0.792389\pi$
$542$	0	0
$543$	−9.81204	−0.421075
$544$	0	0
$545$	2.60308	0.111504
$546$	0	0
$547$	−40.6789	−1.73930	−0.869652	−	0.493665i	$-0.835657\pi$
$547$	−40.6789	−1.73930	−0.869652	+	0.493665i	$0.835657\pi$
$548$	0	0
$549$	−14.8052	−0.631873
$550$	0	0
$551$	4.24392	0.180797
$552$	0	0
$553$	45.9401	1.95357
$554$	0	0
$555$	1.21024	0.0513717
$556$	0	0
$557$	39.6184	1.67868	0.839342	−	0.543603i	$-0.182940\pi$
$557$	39.6184	1.67868	0.839342	+	0.543603i	$0.182940\pi$
$558$	0	0
$559$	18.2303	0.771061
$560$	0	0
$561$	−24.1094	−1.01790
$562$	0	0
$563$	−20.8090	−0.876993	−0.438497	−	0.898733i	$-0.644489\pi$
$563$	−20.8090	−0.876993	−0.438497	+	0.898733i	$0.644489\pi$
$564$	0	0
$565$	−1.21342	−0.0510491
$566$	0	0
$567$	−3.08239	−0.129448
$568$	0	0
$569$	−15.7585	−0.660632	−0.330316	−	0.943870i	$-0.607155\pi$
$569$	−15.7585	−0.660632	−0.330316	+	0.943870i	$0.607155\pi$
$570$	0	0
$571$	24.6912	1.03329	0.516647	−	0.856199i	$-0.327180\pi$
$571$	24.6912	1.03329	0.516647	+	0.856199i	$0.327180\pi$
$572$	0	0
$573$	−19.2900	−0.805851
$574$	0	0
$575$	19.5596	0.815691
$576$	0	0
$577$	−21.0924	−0.878090	−0.439045	−	0.898465i	$-0.644683\pi$
$577$	−21.0924	−0.878090	−0.439045	+	0.898465i	$0.644683\pi$
$578$	0	0
$579$	−0.155713	−0.00647119
$580$	0	0
$581$	−28.8277	−1.19597
$582$	0	0
$583$	−2.08746	−0.0864536
$584$	0	0
$585$	−1.53981	−0.0636631
$586$	0	0
$587$	−14.3933	−0.594076	−0.297038	−	0.954866i	$-0.595999\pi$
$587$	−14.3933	−0.594076	−0.297038	+	0.954866i	$0.595999\pi$
$588$	0	0
$589$	5.32729	0.219507
$590$	0	0
$591$	−13.2014	−0.543031
$592$	0	0
$593$	22.1931	0.911360	0.455680	−	0.890144i	$-0.349396\pi$
$593$	22.1931	0.911360	0.455680	+	0.890144i	$0.349396\pi$
$594$	0	0
$595$	6.67271	0.273555
$596$	0	0
$597$	−23.1144	−0.946010
$598$	0	0
$599$	37.4366	1.52962	0.764810	−	0.644256i	$-0.222833\pi$
$599$	37.4366	1.52962	0.764810	+	0.644256i	$0.222833\pi$
$600$	0	0
$601$	1.68963	0.0689215	0.0344608	−	0.999406i	$-0.489029\pi$
$601$	1.68963	0.0689215	0.0344608	+	0.999406i	$0.489029\pi$
$602$	0	0
$603$	13.9864	0.569571
$604$	0	0
$605$	−0.881601	−0.0358422
$606$	0	0
$607$	36.8841	1.49708	0.748539	−	0.663090i	$-0.230755\pi$
$607$	36.8841	1.49708	0.748539	+	0.663090i	$0.230755\pi$
$608$	0	0
$609$	15.0866	0.611338
$610$	0	0
$611$	7.68857	0.311046
$612$	0	0
$613$	−13.4598	−0.543637	−0.271819	−	0.962349i	$-0.587625\pi$
$613$	−13.4598	−0.543637	−0.271819	+	0.962349i	$0.587625\pi$
$614$	0	0
$615$	−1.30358	−0.0525654
$616$	0	0
$617$	−20.3160	−0.817891	−0.408946	−	0.912559i	$-0.634103\pi$
$617$	−20.3160	−0.817891	−0.408946	+	0.912559i	$0.634103\pi$
$618$	0	0
$619$	−39.0279	−1.56867	−0.784333	−	0.620340i	$-0.786994\pi$
$619$	−39.0279	−1.56867	−0.784333	+	0.620340i	$0.786994\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	0	0
$623$	55.8201	2.23639
$624$	0	0
$625$	23.3605	0.934422
$626$	0	0
$627$	3.20435	0.127969
$628$	0	0
$629$	−23.7945	−0.948748
$630$	0	0
$631$	0.629888	0.0250755	0.0125377	−	0.999921i	$-0.496009\pi$
$631$	0.629888	0.0250755	0.0125377	+	0.999921i	$0.496009\pi$
$632$	0	0
$633$	−3.39104	−0.134782
$634$	0	0
$635$	−0.211773	−0.00840394
$636$	0	0
$637$	11.6065	0.459865
$638$	0	0
$639$	−7.49207	−0.296382
$640$	0	0
$641$	3.52166	0.139097	0.0695486	−	0.997579i	$-0.477844\pi$
$641$	3.52166	0.139097	0.0695486	+	0.997579i	$0.477844\pi$
$642$	0	0
$643$	−9.83765	−0.387959	−0.193980	−	0.981006i	$-0.562140\pi$
$643$	−9.83765	−0.387959	−0.193980	+	0.981006i	$0.562140\pi$
$644$	0	0
$645$	−1.30358	−0.0513284
$646$	0	0
$647$	−30.7549	−1.20910	−0.604550	−	0.796567i	$-0.706647\pi$
$647$	−30.7549	−1.20910	−0.604550	+	0.796567i	$0.706647\pi$
$648$	0	0
$649$	24.3734	0.956739
$650$	0	0
$651$	18.9378	0.742230
$652$	0	0
$653$	9.22745	0.361098	0.180549	−	0.983566i	$-0.442213\pi$
$653$	9.22745	0.361098	0.180549	+	0.983566i	$0.442213\pi$
$654$	0	0
$655$	3.15694	0.123352
$656$	0	0
$657$	5.62408	0.219416
$658$	0	0
$659$	−46.5619	−1.81379	−0.906896	−	0.421354i	$-0.861555\pi$
$659$	−46.5619	−1.81379	−0.906896	+	0.421354i	$0.861555\pi$
$660$	0	0
$661$	−27.9315	−1.08641	−0.543205	−	0.839600i	$-0.682789\pi$
$661$	−27.9315	−1.08641	−0.543205	+	0.839600i	$0.682789\pi$
$662$	0	0
$663$	30.2741	1.17575
$664$	0	0
$665$	−0.886864	−0.0343911
$666$	0	0
$667$	19.5777	0.758052
$668$	0	0
$669$	−10.5326	−0.407214
$670$	0	0
$671$	54.7131	2.11217
$672$	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
$675$	−4.88989	−0.188212
$676$	0	0
$677$	−31.4918	−1.21033	−0.605164	−	0.796101i	$-0.706892\pi$
$677$	−31.4918	−1.21033	−0.605164	+	0.796101i	$0.706892\pi$
$678$	0	0
$679$	52.5483	2.01662
$680$	0	0
$681$	23.4753	0.899576
$682$	0	0
$683$	28.0069	1.07166	0.535828	−	0.844327i	$-0.320000\pi$
$683$	28.0069	1.07166	0.535828	+	0.844327i	$0.320000\pi$
$684$	0	0
$685$	4.61729	0.176418
$686$	0	0
$687$	3.13932	0.119773
$688$	0	0
$689$	2.62122	0.0998606
$690$	0	0
$691$	−38.2264	−1.45420	−0.727101	−	0.686531i	$-0.759133\pi$
$691$	−38.2264	−1.45420	−0.727101	+	0.686531i	$0.759133\pi$
$692$	0	0
$693$	11.3910	0.432710
$694$	0	0
$695$	−0.0256590	−0.000973301 0
$696$	0	0
$697$	25.6297	0.970794
$698$	0	0
$699$	−8.98414	−0.339811
$700$	0	0
$701$	9.03731	0.341335	0.170667	−	0.985329i	$-0.445408\pi$
$701$	9.03731	0.341335	0.170667	+	0.985329i	$0.445408\pi$
$702$	0	0
$703$	3.16250	0.119276
$704$	0	0
$705$	−0.549780	−0.0207059
$706$	0	0
$707$	−37.4617	−1.40889
$708$	0	0
$709$	39.9270	1.49949	0.749745	−	0.661726i	$-0.230176\pi$
$709$	39.9270	1.49949	0.749745	+	0.661726i	$0.230176\pi$
$710$	0	0
$711$	−14.9040	−0.558945
$712$	0	0
$713$	24.5754	0.920357
$714$	0	0
$715$	5.69038	0.212808
$716$	0	0
$717$	1.69870	0.0634393
$718$	0	0
$719$	29.3036	1.09284	0.546420	−	0.837512i	$-0.315990\pi$
$719$	29.3036	1.09284	0.546420	+	0.837512i	$0.315990\pi$
$720$	0	0
$721$	52.7221	1.96348
$722$	0	0
$723$	−14.3288	−0.532895
$724$	0	0
$725$	23.9332	0.888859
$726$	0	0
$727$	12.6201	0.468052	0.234026	−	0.972230i	$-0.424810\pi$
$727$	12.6201	0.468052	0.234026	+	0.972230i	$0.424810\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	25.6297	0.947949
$732$	0	0
$733$	−9.92140	−0.366455	−0.183228	−	0.983071i	$-0.558655\pi$
$733$	−9.92140	−0.366455	−0.183228	+	0.983071i	$0.558655\pi$
$734$	0	0
$735$	−0.829932	−0.0306125
$736$	0	0
$737$	−51.6871	−1.90392
$738$	0	0
$739$	20.0592	0.737889	0.368945	−	0.929451i	$-0.379719\pi$
$739$	20.0592	0.737889	0.368945	+	0.929451i	$0.379719\pi$
$740$	0	0
$741$	−4.02371	−0.147815
$742$	0	0
$743$	−11.0969	−0.407107	−0.203554	−	0.979064i	$-0.565249\pi$
$743$	−11.0969	−0.407107	−0.203554	+	0.979064i	$0.565249\pi$
$744$	0	0
$745$	−4.64174	−0.170060
$746$	0	0
$747$	9.35237	0.342185
$748$	0	0
$749$	56.6395	2.06956
$750$	0	0
$751$	37.3294	1.36217	0.681084	−	0.732205i	$-0.261509\pi$
$751$	37.3294	1.36217	0.681084	+	0.732205i	$0.261509\pi$
$752$	0	0
$753$	11.6955	0.426208
$754$	0	0
$755$	−6.16617	−0.224410
$756$	0	0
$757$	26.7362	0.971745	0.485873	−	0.874030i	$-0.338502\pi$
$757$	26.7362	0.971745	0.485873	+	0.874030i	$0.338502\pi$
$758$	0	0
$759$	14.7821	0.536555
$760$	0	0
$761$	−27.1767	−0.985155	−0.492578	−	0.870269i	$-0.663945\pi$
$761$	−27.1767	−0.985155	−0.492578	+	0.870269i	$0.663945\pi$
$762$	0	0
$763$	24.1808	0.875404
$764$	0	0
$765$	−2.16478	−0.0782679
$766$	0	0
$767$	−30.6057	−1.10511
$768$	0	0
$769$	−8.34877	−0.301064	−0.150532	−	0.988605i	$-0.548099\pi$
$769$	−8.34877	−0.301064	−0.150532	+	0.988605i	$0.548099\pi$
$770$	0	0
$771$	3.98642	0.143568
$772$	0	0
$773$	20.7289	0.745567	0.372783	−	0.927918i	$-0.378403\pi$
$773$	20.7289	0.745567	0.372783	+	0.927918i	$0.378403\pi$
$774$	0	0
$775$	30.0428	1.07917
$776$	0	0
$777$	11.2423	0.403314
$778$	0	0
$779$	−3.40642	−0.122048
$780$	0	0
$781$	27.6871	0.990722
$782$	0	0
$783$	−4.89443	−0.174913
$784$	0	0
$785$	−1.76002	−0.0628177
$786$	0	0
$787$	21.7129	0.773982	0.386991	−	0.922084i	$-0.373514\pi$
$787$	21.7129	0.773982	0.386991	+	0.922084i	$0.373514\pi$
$788$	0	0
$789$	−11.8635	−0.422351
$790$	0	0
$791$	−11.2719	−0.400781
$792$	0	0
$793$	−68.7032	−2.43972
$794$	0	0
$795$	−0.187433	−0.00664757
$796$	0	0
$797$	−27.5528	−0.975971	−0.487985	−	0.872852i	$-0.662268\pi$
$797$	−27.5528	−0.975971	−0.487985	+	0.872852i	$0.662268\pi$
$798$	0	0
$799$	10.8092	0.382403
$800$	0	0
$801$	−18.1094	−0.639863
$802$	0	0
$803$	−20.7839	−0.733447
$804$	0	0
$805$	−4.09121	−0.144196
$806$	0	0
$807$	−21.3880	−0.752895
$808$	0	0
$809$	18.8535	0.662854	0.331427	−	0.943481i	$-0.392470\pi$
$809$	18.8535	0.662854	0.331427	+	0.943481i	$0.392470\pi$
$810$	0	0
$811$	−6.78796	−0.238357	−0.119179	−	0.992873i	$-0.538026\pi$
$811$	−6.78796	−0.238357	−0.119179	+	0.992873i	$0.538026\pi$
$812$	0	0
$813$	−12.0530	−0.422716
$814$	0	0
$815$	−4.41522	−0.154658
$816$	0	0
$817$	−3.40642	−0.119175
$818$	0	0
$819$	−14.3037	−0.499813
$820$	0	0
$821$	−16.0169	−0.558995	−0.279498	−	0.960146i	$-0.590168\pi$
$821$	−16.0169	−0.558995	−0.279498	+	0.960146i	$0.590168\pi$
$822$	0	0
$823$	−28.5609	−0.995570	−0.497785	−	0.867301i	$-0.665853\pi$
$823$	−28.5609	−0.995570	−0.497785	+	0.867301i	$0.665853\pi$
$824$	0	0
$825$	18.0707	0.629141
$826$	0	0
$827$	−19.1388	−0.665521	−0.332761	−	0.943011i	$-0.607980\pi$
$827$	−19.1388	−0.665521	−0.332761	+	0.943011i	$0.607980\pi$
$828$	0	0
$829$	−36.0907	−1.25348	−0.626741	−	0.779228i	$-0.715611\pi$
$829$	−36.0907	−1.25348	−0.626741	+	0.779228i	$0.715611\pi$
$830$	0	0
$831$	4.81432	0.167007
$832$	0	0
$833$	16.3173	0.565361
$834$	0	0
$835$	−5.28624	−0.182938
$836$	0	0
$837$	−6.14386	−0.212363
$838$	0	0
$839$	18.8767	0.651697	0.325849	−	0.945422i	$-0.394350\pi$
$839$	18.8767	0.651697	0.325849	+	0.945422i	$0.394350\pi$
$840$	0	0
$841$	−5.04455	−0.173950
$842$	0	0
$843$	18.5754	0.639772
$844$	0	0
$845$	−2.83174	−0.0974147
$846$	0	0
$847$	−8.18947	−0.281393
$848$	0	0
$849$	−1.40461	−0.0482062
$850$	0	0
$851$	14.5890	0.500105
$852$	0	0
$853$	26.3018	0.900557	0.450279	−	0.892888i	$-0.351325\pi$
$853$	26.3018	0.900557	0.450279	+	0.892888i	$0.351325\pi$
$854$	0	0
$855$	0.287719	0.00983979
$856$	0	0
$857$	−24.8671	−0.849444	−0.424722	−	0.905324i	$-0.639628\pi$
$857$	−24.8671	−0.849444	−0.424722	+	0.905324i	$0.639628\pi$
$858$	0	0
$859$	15.5990	0.532231	0.266115	−	0.963941i	$-0.414260\pi$
$859$	15.5990	0.532231	0.266115	+	0.963941i	$0.414260\pi$
$860$	0	0
$861$	−12.1094	−0.412686
$862$	0	0
$863$	−43.8654	−1.49320	−0.746598	−	0.665275i	$-0.768314\pi$
$863$	−43.8654	−1.49320	−0.746598	+	0.665275i	$0.768314\pi$
$864$	0	0
$865$	−6.54447	−0.222519
$866$	0	0
$867$	25.5619	0.868126
$868$	0	0
$869$	55.0781	1.86840
$870$	0	0
$871$	64.9035	2.19917
$872$	0	0
$873$	−17.0479	−0.576984
$874$	0	0
$875$	−10.1154	−0.341964
$876$	0	0
$877$	45.5410	1.53781	0.768905	−	0.639364i	$-0.220802\pi$
$877$	45.5410	1.53781	0.768905	+	0.639364i	$0.220802\pi$
$878$	0	0
$879$	27.8786	0.940321
$880$	0	0
$881$	−19.8589	−0.669064	−0.334532	−	0.942384i	$-0.608578\pi$
$881$	−19.8589	−0.669064	−0.334532	+	0.942384i	$0.608578\pi$
$882$	0	0
$883$	25.2740	0.850537	0.425269	−	0.905067i	$-0.360180\pi$
$883$	25.2740	0.850537	0.425269	+	0.905067i	$0.360180\pi$
$884$	0	0
$885$	2.18849	0.0735653
$886$	0	0
$887$	−2.07012	−0.0695079	−0.0347539	−	0.999396i	$-0.511065\pi$
$887$	−2.07012	−0.0695079	−0.0347539	+	0.999396i	$0.511065\pi$
$888$	0	0
$889$	−1.96722	−0.0659785
$890$	0	0
$891$	−3.69552	−0.123805
$892$	0	0
$893$	−1.43664	−0.0480754
$894$	0	0
$895$	−1.98642	−0.0663988
$896$	0	0
$897$	−18.5619	−0.619763
$898$	0	0
$899$	30.0707	1.00291
$900$	0	0
$901$	3.68513	0.122769
$902$	0	0
$903$	−12.1094	−0.402974
$904$	0	0
$905$	3.25584	0.108228
$906$	0	0
$907$	8.10347	0.269071	0.134536	−	0.990909i	$-0.457046\pi$
$907$	8.10347	0.269071	0.134536	+	0.990909i	$0.457046\pi$
$908$	0	0
$909$	12.1535	0.403105
$910$	0	0
$911$	−6.23034	−0.206420	−0.103210	−	0.994660i	$-0.532911\pi$
$911$	−6.23034	−0.206420	−0.103210	+	0.994660i	$0.532911\pi$
$912$	0	0
$913$	−34.5619	−1.14383
$914$	0	0
$915$	4.91270	0.162409
$916$	0	0
$917$	29.3258	0.968423
$918$	0	0
$919$	−15.1680	−0.500348	−0.250174	−	0.968201i	$-0.580488\pi$
$919$	−15.1680	−0.500348	−0.250174	+	0.968201i	$0.580488\pi$
$920$	0	0
$921$	−17.4548	−0.575155
$922$	0	0
$923$	−34.7667	−1.14436
$924$	0	0
$925$	17.8347	0.586401
$926$	0	0
$927$	−17.1043	−0.561779
$928$	0	0
$929$	−9.16951	−0.300842	−0.150421	−	0.988622i	$-0.548063\pi$
$929$	−9.16951	−0.300842	−0.150421	+	0.988622i	$0.548063\pi$
$930$	0	0
$931$	−2.16872	−0.0710768
$932$	0	0
$933$	0.466081	0.0152588
$934$	0	0
$935$	8.00000	0.261628
$936$	0	0
$937$	−19.3183	−0.631101	−0.315550	−	0.948909i	$-0.602189\pi$
$937$	−19.3183	−0.631101	−0.315550	+	0.948909i	$0.602189\pi$
$938$	0	0
$939$	2.49886	0.0815472
$940$	0	0
$941$	10.4385	0.340285	0.170142	−	0.985419i	$-0.445577\pi$
$941$	10.4385	0.340285	0.170142	+	0.985419i	$0.445577\pi$
$942$	0	0
$943$	−15.7142	−0.511726
$944$	0	0
$945$	1.02280	0.0332718
$946$	0	0
$947$	−36.2640	−1.17842	−0.589211	−	0.807979i	$-0.700561\pi$
$947$	−36.2640	−1.17842	−0.589211	+	0.807979i	$0.700561\pi$
$948$	0	0
$949$	26.0983	0.847188
$950$	0	0
$951$	10.7425	0.348349
$952$	0	0
$953$	47.9271	1.55251	0.776255	−	0.630419i	$-0.217117\pi$
$953$	47.9271	1.55251	0.776255	+	0.630419i	$0.217117\pi$
$954$	0	0
$955$	6.40083	0.207126
$956$	0	0
$957$	18.0875	0.584684
$958$	0	0
$959$	42.8914	1.38504
$960$	0	0
$961$	6.74701	0.217646
$962$	0	0
$963$	−18.3752	−0.592132
$964$	0	0
$965$	0.0516688	0.00166328
$966$	0	0
$967$	−20.7211	−0.666346	−0.333173	−	0.942866i	$-0.608119\pi$
$967$	−20.7211	−0.666346	−0.333173	+	0.942866i	$0.608119\pi$
$968$	0	0
$969$	−5.65685	−0.181724
$970$	0	0
$971$	38.0888	1.22233	0.611164	−	0.791504i	$-0.290701\pi$
$971$	38.0888	1.22233	0.611164	+	0.791504i	$0.290701\pi$
$972$	0	0
$973$	−0.238354	−0.00764129
$974$	0	0
$975$	−22.6914	−0.726706
$976$	0	0
$977$	42.7363	1.36726	0.683628	−	0.729831i	$-0.260401\pi$
$977$	42.7363	1.36726	0.683628	+	0.729831i	$0.260401\pi$
$978$	0	0
$979$	66.9235	2.13888
$980$	0	0
$981$	−7.84482	−0.250466
$982$	0	0
$983$	42.0031	1.33969	0.669845	−	0.742501i	$-0.266361\pi$
$983$	42.0031	1.33969	0.669845	+	0.742501i	$0.266361\pi$
$984$	0	0
$985$	4.38049	0.139574
$986$	0	0
$987$	−5.10707	−0.162560
$988$	0	0
$989$	−15.7142	−0.499684
$990$	0	0
$991$	−11.1918	−0.355518	−0.177759	−	0.984074i	$-0.556885\pi$
$991$	−11.1918	−0.355518	−0.177759	+	0.984074i	$0.556885\pi$
$992$	0	0
$993$	−5.26810	−0.167178
$994$	0	0
$995$	7.66986	0.243151
$996$	0	0
$997$	−52.3230	−1.65709	−0.828544	−	0.559924i	$-0.810830\pi$
$997$	−52.3230	−1.65709	−0.828544	+	0.559924i	$0.810830\pi$
$998$	0	0
$999$	−3.64725	−0.115394

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3072.2.a.p.1.3		4
3.2	odd	2		9216.2.a.z.1.2		4
4.3	odd	2		3072.2.a.m.1.3		4
8.3	odd	2		3072.2.a.s.1.2		4
8.5	even	2		3072.2.a.j.1.2		4
12.11	even	2		9216.2.a.bl.1.2		4
16.3	odd	4		3072.2.d.e.1537.2		8
16.5	even	4		3072.2.d.j.1537.3		8
16.11	odd	4		3072.2.d.e.1537.7		8
16.13	even	4		3072.2.d.j.1537.6		8
24.5	odd	2		9216.2.a.ba.1.3		4
24.11	even	2		9216.2.a.bm.1.3		4
32.3	odd	8		1536.2.j.j.1153.1	yes	8
32.5	even	8		1536.2.j.f.385.2	yes	8
32.11	odd	8		1536.2.j.j.385.1	yes	8
32.13	even	8		1536.2.j.f.1153.2	yes	8
32.19	odd	8		1536.2.j.e.1153.4	yes	8
32.21	even	8		1536.2.j.i.385.3	yes	8
32.27	odd	8		1536.2.j.e.385.4	✓	8
32.29	even	8		1536.2.j.i.1153.3	yes	8
96.5	odd	8		4608.2.k.bj.3457.2		8
96.11	even	8		4608.2.k.be.3457.3		8
96.29	odd	8		4608.2.k.bc.1153.3		8
96.35	even	8		4608.2.k.be.1153.3		8
96.53	odd	8		4608.2.k.bc.3457.3		8
96.59	even	8		4608.2.k.bh.3457.2		8
96.77	odd	8		4608.2.k.bj.1153.2		8
96.83	even	8		4608.2.k.bh.1153.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.j.e.385.4	✓	8	32.27	odd	8
1536.2.j.e.1153.4	yes	8	32.19	odd	8
1536.2.j.f.385.2	yes	8	32.5	even	8
1536.2.j.f.1153.2	yes	8	32.13	even	8
1536.2.j.i.385.3	yes	8	32.21	even	8
1536.2.j.i.1153.3	yes	8	32.29	even	8
1536.2.j.j.385.1	yes	8	32.11	odd	8
1536.2.j.j.1153.1	yes	8	32.3	odd	8
3072.2.a.j.1.2		4	8.5	even	2
3072.2.a.m.1.3		4	4.3	odd	2
3072.2.a.p.1.3		4	1.1	even	1	trivial
3072.2.a.s.1.2		4	8.3	odd	2
3072.2.d.e.1537.2		8	16.3	odd	4
3072.2.d.e.1537.7		8	16.11	odd	4
3072.2.d.j.1537.3		8	16.5	even	4
3072.2.d.j.1537.6		8	16.13	even	4
4608.2.k.bc.1153.3		8	96.29	odd	8
4608.2.k.bc.3457.3		8	96.53	odd	8
4608.2.k.be.1153.3		8	96.35	even	8
4608.2.k.be.3457.3		8	96.11	even	8
4608.2.k.bh.1153.2		8	96.83	even	8
4608.2.k.bh.3457.2		8	96.59	even	8
4608.2.k.bj.1153.2		8	96.77	odd	8
4608.2.k.bj.3457.2		8	96.5	odd	8
9216.2.a.z.1.2		4	3.2	odd	2
9216.2.a.ba.1.3		4	24.5	odd	2
9216.2.a.bl.1.2		4	12.11	even	2
9216.2.a.bm.1.3		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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	3072.2.a.p.1.3

Level	$3072$
Weight	$2$
Character	3072.1
Self dual	yes
Analytic conductor	$24.530$
Analytic rank	$1$
Dimension	$4$
CM	no
Inner twists	$1$