LMFDB - Embedded newform 9216.2.a.bl.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9216,2,Mod(1,9216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9216, 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("9216.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$9216 = 2^{10} \cdot 3^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	9216.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:	$73.5901305028$
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			1.2
Root			$1.84776$ of defining polynomial
Character	$\chi$	$=$	9216.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.331821 q^{5} +3.08239 q^{7} -3.69552 q^{11} +4.64047 q^{13} -6.52395 q^{17} +0.867091 q^{19} -4.00000 q^{23} -4.88989 q^{25} +4.89443 q^{29} +6.14386 q^{31} +1.02280 q^{35} -3.64725 q^{37} -3.92856 q^{41} -3.92856 q^{43} +1.65685 q^{47} +2.50114 q^{49} -0.564862 q^{53} -1.22625 q^{55} -6.59539 q^{59} -14.8052 q^{61} +1.53981 q^{65} -13.9864 q^{67} -7.49207 q^{71} +5.62408 q^{73} -11.3910 q^{77} +14.9040 q^{79} +9.35237 q^{83} -2.16478 q^{85} +18.1094 q^{89} +14.3037 q^{91} +0.287719 q^{95} -17.0479 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 8 q^{7} - 8 q^{13} - 16 q^{23} + 4 q^{25} + 8 q^{31} - 16 q^{35} - 8 q^{37} - 16 q^{47} + 4 q^{49} + 16 q^{55} - 16 q^{59} - 24 q^{61} + 8 q^{65} - 16 q^{67} - 16 q^{71} - 8 q^{73} - 16 q^{77} + 24 q^{79}+ \cdots - 16 q^{97}+O(q^{100})$ 4 * q + 8 * q^7 - 8 * q^13 - 16 * q^23 + 4 * q^25 + 8 * q^31 - 16 * q^35 - 8 * q^37 - 16 * q^47 + 4 * q^49 + 16 * q^55 - 16 * q^59 - 24 * q^61 + 8 * q^65 - 16 * q^67 - 16 * q^71 - 8 * q^73 - 16 * q^77 + 24 * q^79 + 8 * q^89 - 16 * q^91 - 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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0.331821	0.148395	0.0741975	−	0.997244i	$-0.476360\pi$
$5$	0.331821	0.148395	0.0741975	+	0.997244i	$0.476360\pi$
$6$	0	0
$7$	3.08239	1.16503	0.582517	−	0.812818i	$-0.302068\pi$
$7$	3.08239	1.16503	0.582517	+	0.812818i	$0.302068\pi$
$8$	0	0
$9$	0	0
$10$	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	0
$16$	0	0
$17$	−6.52395	−1.58229	−0.791145	−	0.611629i	$-0.790514\pi$
$17$	−6.52395	−1.58229	−0.791145	+	0.611629i	$0.790514\pi$
$18$	0	0
$19$	0.867091	0.198924	0.0994622	−	0.995041i	$-0.468288\pi$
$19$	0.867091	0.198924	0.0994622	+	0.995041i	$0.468288\pi$
$20$	0	0
$21$	0	0
$22$	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
$27$	0	0
$28$	0	0
$29$	4.89443	0.908873	0.454436	−	0.890779i	$-0.349841\pi$
$29$	4.89443	0.908873	0.454436	+	0.890779i	$0.349841\pi$
$30$	0	0
$31$	6.14386	1.10347	0.551735	−	0.834020i	$-0.313966\pi$
$31$	6.14386	1.10347	0.551735	+	0.834020i	$0.313966\pi$
$32$	0	0
$33$	0	0
$34$	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
$39$	0	0
$40$	0	0
$41$	−3.92856	−0.613538	−0.306769	−	0.951784i	$-0.599248\pi$
$41$	−3.92856	−0.613538	−0.306769	+	0.951784i	$0.599248\pi$
$42$	0	0
$43$	−3.92856	−0.599100	−0.299550	−	0.954081i	$-0.596836\pi$
$43$	−3.92856	−0.599100	−0.299550	+	0.954081i	$0.596836\pi$
$44$	0	0
$45$	0	0
$46$	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
$51$	0	0
$52$	0	0
$53$	−0.564862	−0.0775897	−0.0387949	−	0.999247i	$-0.512352\pi$
$53$	−0.564862	−0.0775897	−0.0387949	+	0.999247i	$0.512352\pi$
$54$	0	0
$55$	−1.22625	−0.165348
$56$	0	0
$57$	0	0
$58$	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
$63$	0	0
$64$	0	0
$65$	1.53981	0.190989
$66$	0	0
$67$	−13.9864	−1.70871	−0.854357	−	0.519687i	$-0.826049\pi$
$67$	−13.9864	−1.70871	−0.854357	+	0.519687i	$0.826049\pi$
$68$	0	0
$69$	0	0
$70$	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
$75$	0	0
$76$	0	0
$77$	−11.3910	−1.29813
$78$	0	0
$79$	14.9040	1.67683	0.838417	−	0.545029i	$-0.183481\pi$
$79$	14.9040	1.67683	0.838417	+	0.545029i	$0.183481\pi$
$80$	0	0
$81$	0	0
$82$	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
$87$	0	0
$88$	0	0
$89$	18.1094	1.91959	0.959794	−	0.280705i	$-0.0905683\pi$
$89$	18.1094	1.91959	0.959794	+	0.280705i	$0.0905683\pi$
$90$	0	0
$91$	14.3037	1.49944
$92$	0	0
$93$	0	0
$94$	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
$99$	0	0
$100$	0	0
$101$	−12.1535	−1.20931	−0.604657	−	0.796486i	$-0.706690\pi$
$101$	−12.1535	−1.20931	−0.604657	+	0.796486i	$0.706690\pi$
$102$	0	0
$103$	17.1043	1.68534	0.842668	−	0.538433i	$-0.180984\pi$
$103$	17.1043	1.68534	0.842668	+	0.538433i	$0.180984\pi$
$104$	0	0
$105$	0	0
$106$	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
$111$	0	0
$112$	0	0
$113$	−3.65685	−0.344008	−0.172004	−	0.985096i	$-0.555024\pi$
$113$	−3.65685	−0.344008	−0.172004	+	0.985096i	$0.555024\pi$
$114$	0	0
$115$	−1.32729	−0.123770
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−20.1094	−1.84342
$120$	0	0
$121$	2.65685	0.241532
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−3.28168	−0.293522
$126$	0	0
$127$	−0.638213	−0.0566322	−0.0283161	−	0.999599i	$-0.509015\pi$
$127$	−0.638213	−0.0566322	−0.0283161	+	0.999599i	$0.509015\pi$
$128$	0	0
$129$	0	0
$130$	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	0
$136$	0	0
$137$	13.9150	1.18884	0.594419	−	0.804156i	$-0.297382\pi$
$137$	13.9150	1.18884	0.594419	+	0.804156i	$0.297382\pi$
$138$	0	0
$139$	−0.0773278	−0.00655886	−0.00327943	−	0.999995i	$-0.501044\pi$
$139$	−0.0773278	−0.00655886	−0.00327943	+	0.999995i	$0.501044\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−17.1489	−1.43407
$144$	0	0
$145$	1.62408	0.134872
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−13.9887	−1.14600	−0.572998	−	0.819556i	$-0.694220\pi$
$149$	−13.9887	−1.14600	−0.572998	+	0.819556i	$0.694220\pi$
$150$	0	0
$151$	−18.5828	−1.51225	−0.756123	−	0.654430i	$-0.772909\pi$
$151$	−18.5828	−1.51225	−0.756123	+	0.654430i	$0.772909\pi$
$152$	0	0
$153$	0	0
$154$	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	0
$160$	0	0
$161$	−12.3296	−0.971706
$162$	0	0
$163$	−13.3060	−1.04221	−0.521104	−	0.853493i	$-0.674480\pi$
$163$	−13.3060	−1.04221	−0.521104	+	0.853493i	$0.674480\pi$
$164$	0	0
$165$	0	0
$166$	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	0
$172$	0	0
$173$	−19.7229	−1.49950	−0.749751	−	0.661721i	$-0.769827\pi$
$173$	−19.7229	−1.49950	−0.749751	+	0.661721i	$0.769827\pi$
$174$	0	0
$175$	−15.0726	−1.13938
$176$	0	0
$177$	0	0
$178$	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
$183$	0	0
$184$	0	0
$185$	−1.21024	−0.0889784
$186$	0	0
$187$	24.1094	1.76305
$188$	0	0
$189$	0	0
$190$	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
$195$	0	0
$196$	0	0
$197$	13.2014	0.940557	0.470279	−	0.882518i	$-0.344153\pi$
$197$	13.2014	0.940557	0.470279	+	0.882518i	$0.344153\pi$
$198$	0	0
$199$	23.1144	1.63854	0.819269	−	0.573409i	$-0.194380\pi$
$199$	23.1144	1.63854	0.819269	+	0.573409i	$0.194380\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	15.0866	1.05887
$204$	0	0
$205$	−1.30358	−0.0910459
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.20435	−0.221650
$210$	0	0
$211$	3.39104	0.233449	0.116724	−	0.993164i	$-0.462761\pi$
$211$	3.39104	0.233449	0.116724	+	0.993164i	$0.462761\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.30358	−0.0889034
$216$	0	0
$217$	18.9378	1.28558
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−30.2741	−2.03646
$222$	0	0
$223$	10.5326	0.705316	0.352658	−	0.935752i	$-0.385278\pi$
$223$	10.5326	0.705316	0.352658	+	0.935752i	$0.385278\pi$
$224$	0	0
$225$	0	0
$226$	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
$231$	0	0
$232$	0	0
$233$	8.98414	0.588571	0.294285	−	0.955718i	$-0.404918\pi$
$233$	8.98414	0.588571	0.294285	+	0.955718i	$0.404918\pi$
$234$	0	0
$235$	0.549780	0.0358637
$236$	0	0
$237$	0	0
$238$	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
$243$	0	0
$244$	0	0
$245$	0.829932	0.0530224
$246$	0	0
$247$	4.02371	0.256022
$248$	0	0
$249$	0	0
$250$	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
$255$	0	0
$256$	0	0
$257$	−3.98642	−0.248666	−0.124333	−	0.992241i	$-0.539679\pi$
$257$	−3.98642	−0.248666	−0.124333	+	0.992241i	$0.539679\pi$
$258$	0	0
$259$	−11.2423	−0.698560
$260$	0	0
$261$	0	0
$262$	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
$267$	0	0
$268$	0	0
$269$	21.3880	1.30405	0.652026	−	0.758197i	$-0.273919\pi$
$269$	21.3880	1.30405	0.652026	+	0.758197i	$0.273919\pi$
$270$	0	0
$271$	12.0530	0.732165	0.366082	−	0.930582i	$-0.380699\pi$
$271$	12.0530	0.732165	0.366082	+	0.930582i	$0.380699\pi$
$272$	0	0
$273$	0	0
$274$	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
$279$	0	0
$280$	0	0
$281$	−18.5754	−1.10812	−0.554059	−	0.832477i	$-0.686922\pi$
$281$	−18.5754	−1.10812	−0.554059	+	0.832477i	$0.686922\pi$
$282$	0	0
$283$	1.40461	0.0834956	0.0417478	−	0.999128i	$-0.486707\pi$
$283$	1.40461	0.0834956	0.0417478	+	0.999128i	$0.486707\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−12.1094	−0.714793
$288$	0	0
$289$	25.5619	1.50364
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−27.8786	−1.62868	−0.814342	−	0.580386i	$-0.802902\pi$
$293$	−27.8786	−1.62868	−0.814342	+	0.580386i	$0.802902\pi$
$294$	0	0
$295$	−2.18849	−0.127419
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−18.5619	−1.07346
$300$	0	0
$301$	−12.1094	−0.697972
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−4.91270	−0.281300
$306$	0	0
$307$	17.4548	0.996197	0.498099	−	0.867120i	$-0.334032\pi$
$307$	17.4548	0.996197	0.498099	+	0.867120i	$0.334032\pi$
$308$	0	0
$309$	0	0
$310$	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
$315$	0	0
$316$	0	0
$317$	−10.7425	−0.603358	−0.301679	−	0.953410i	$-0.597547\pi$
$317$	−10.7425	−0.603358	−0.301679	+	0.953410i	$0.597547\pi$
$318$	0	0
$319$	−18.0875	−1.01270
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−5.65685	−0.314756
$324$	0	0
$325$	−22.6914	−1.25869
$326$	0	0
$327$	0	0
$328$	0	0
$329$	5.10707	0.281562
$330$	0	0
$331$	5.26810	0.289561	0.144781	−	0.989464i	$-0.453752\pi$
$331$	5.26810	0.289561	0.144781	+	0.989464i	$0.453752\pi$
$332$	0	0
$333$	0	0
$334$	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
$339$	0	0
$340$	0	0
$341$	−22.7047	−1.22953
$342$	0	0
$343$	−13.8672	−0.748761
$344$	0	0
$345$	0	0
$346$	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
$351$	0	0
$352$	0	0
$353$	−23.0799	−1.22842	−0.614210	−	0.789143i	$-0.710525\pi$
$353$	−23.0799	−1.22842	−0.614210	+	0.789143i	$0.710525\pi$
$354$	0	0
$355$	−2.48603	−0.131945
$356$	0	0
$357$	0	0
$358$	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
$363$	0	0
$364$	0	0
$365$	1.86619	0.0976808
$366$	0	0
$367$	−13.1698	−0.687461	−0.343730	−	0.939068i	$-0.611691\pi$
$367$	−13.1698	−0.687461	−0.343730	+	0.939068i	$0.611691\pi$
$368$	0	0
$369$	0	0
$370$	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
$375$	0	0
$376$	0	0
$377$	22.7124	1.16975
$378$	0	0
$379$	−6.41459	−0.329495	−0.164748	−	0.986336i	$-0.552681\pi$
$379$	−6.41459	−0.329495	−0.164748	+	0.986336i	$0.552681\pi$
$380$	0	0
$381$	0	0
$382$	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
$387$	0	0
$388$	0	0
$389$	−32.5185	−1.64875	−0.824377	−	0.566041i	$-0.808474\pi$
$389$	−32.5185	−1.64875	−0.824377	+	0.566041i	$0.808474\pi$
$390$	0	0
$391$	26.0958	1.31972
$392$	0	0
$393$	0	0
$394$	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
$399$	0	0
$400$	0	0
$401$	3.21024	0.160312	0.0801558	−	0.996782i	$-0.474458\pi$
$401$	3.21024	0.160312	0.0801558	+	0.996782i	$0.474458\pi$
$402$	0	0
$403$	28.5104	1.42020
$404$	0	0
$405$	0	0
$406$	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
$411$	0	0
$412$	0	0
$413$	−20.3296	−1.00035
$414$	0	0
$415$	3.10332	0.152336
$416$	0	0
$417$	0	0
$418$	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
$423$	0	0
$424$	0	0
$425$	31.9014	1.54745
$426$	0	0
$427$	−45.6356	−2.20846
$428$	0	0
$429$	0	0
$430$	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
$435$	0	0
$436$	0	0
$437$	−3.46836	−0.165914
$438$	0	0
$439$	4.45985	0.212857	0.106429	−	0.994320i	$-0.466058\pi$
$439$	4.45985	0.212857	0.106429	+	0.994320i	$0.466058\pi$
$440$	0	0
$441$	0	0
$442$	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
$447$	0	0
$448$	0	0
$449$	−27.9787	−1.32040	−0.660199	−	0.751091i	$-0.729528\pi$
$449$	−27.9787	−1.32040	−0.660199	+	0.751091i	$0.729528\pi$
$450$	0	0
$451$	14.5181	0.683629
$452$	0	0
$453$	0	0
$454$	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
$459$	0	0
$460$	0	0
$461$	7.78841	0.362743	0.181371	−	0.983415i	$-0.441946\pi$
$461$	7.78841	0.362743	0.181371	+	0.983415i	$0.441946\pi$
$462$	0	0
$463$	15.6642	0.727977	0.363989	−	0.931403i	$-0.381415\pi$
$463$	15.6642	0.727977	0.363989	+	0.931403i	$0.381415\pi$
$464$	0	0
$465$	0	0
$466$	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
$471$	0	0
$472$	0	0
$473$	14.5181	0.667541
$474$	0	0
$475$	−4.23998	−0.194544
$476$	0	0
$477$	0	0
$478$	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
$483$	0	0
$484$	0	0
$485$	−5.65685	−0.256865
$486$	0	0
$487$	−37.1782	−1.68470	−0.842352	−	0.538928i	$-0.818830\pi$
$487$	−37.1782	−1.68470	−0.842352	+	0.538928i	$0.818830\pi$
$488$	0	0
$489$	0	0
$490$	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
$495$	0	0
$496$	0	0
$497$	−23.0935	−1.03588
$498$	0	0
$499$	−32.8458	−1.47038	−0.735190	−	0.677861i	$-0.762907\pi$
$499$	−32.8458	−1.47038	−0.735190	+	0.677861i	$0.762907\pi$
$500$	0	0
$501$	0	0
$502$	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
$507$	0	0
$508$	0	0
$509$	9.33786	0.413893	0.206947	−	0.978352i	$-0.433647\pi$
$509$	9.33786	0.413893	0.206947	+	0.978352i	$0.433647\pi$
$510$	0	0
$511$	17.3356	0.766882
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5.67557	0.250095
$516$	0	0
$517$	−6.12293	−0.269286
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4.38287	0.192017	0.0960084	−	0.995381i	$-0.469392\pi$
$521$	4.38287	0.192017	0.0960084	+	0.995381i	$0.469392\pi$
$522$	0	0
$523$	−9.22869	−0.403542	−0.201771	−	0.979433i	$-0.564670\pi$
$523$	−9.22869	−0.403542	−0.201771	+	0.979433i	$0.564670\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−40.0822	−1.74601
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−18.2303	−0.789644
$534$	0	0
$535$	−6.09728	−0.263608
$536$	0	0
$537$	0	0
$538$	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
$543$	0	0
$544$	0	0
$545$	−2.60308	−0.111504
$546$	0	0
$547$	40.6789	1.73930	0.869652	−	0.493665i	$-0.164343\pi$
$547$	40.6789	1.73930	0.869652	+	0.493665i	$0.164343\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.24392	0.180797
$552$	0	0
$553$	45.9401	1.95357
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−39.6184	−1.67868	−0.839342	−	0.543603i	$-0.817060\pi$
$557$	−39.6184	−1.67868	−0.839342	+	0.543603i	$0.817060\pi$
$558$	0	0
$559$	−18.2303	−0.771061
$560$	0	0
$561$	0	0
$562$	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
$567$	0	0
$568$	0	0
$569$	15.7585	0.660632	0.330316	−	0.943870i	$-0.392845\pi$
$569$	15.7585	0.660632	0.330316	+	0.943870i	$0.392845\pi$
$570$	0	0
$571$	−24.6912	−1.03329	−0.516647	−	0.856199i	$-0.672820\pi$
$571$	−24.6912	−1.03329	−0.516647	+	0.856199i	$0.672820\pi$
$572$	0	0
$573$	0	0
$574$	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	0
$580$	0	0
$581$	28.8277	1.19597
$582$	0	0
$583$	2.08746	0.0864536
$584$	0	0
$585$	0	0
$586$	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
$591$	0	0
$592$	0	0
$593$	−22.1931	−0.911360	−0.455680	−	0.890144i	$-0.650604\pi$
$593$	−22.1931	−0.911360	−0.455680	+	0.890144i	$0.650604\pi$
$594$	0	0
$595$	−6.67271	−0.273555
$596$	0	0
$597$	0	0
$598$	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
$603$	0	0
$604$	0	0
$605$	0.881601	0.0358422
$606$	0	0
$607$	−36.8841	−1.49708	−0.748539	−	0.663090i	$-0.769245\pi$
$607$	−36.8841	−1.49708	−0.748539	+	0.663090i	$0.769245\pi$
$608$	0	0
$609$	0	0
$610$	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
$615$	0	0
$616$	0	0
$617$	20.3160	0.817891	0.408946	−	0.912559i	$-0.365897\pi$
$617$	20.3160	0.817891	0.408946	+	0.912559i	$0.365897\pi$
$618$	0	0
$619$	39.0279	1.56867	0.784333	−	0.620340i	$-0.213006\pi$
$619$	39.0279	1.56867	0.784333	+	0.620340i	$0.213006\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	55.8201	2.23639
$624$	0	0
$625$	23.3605	0.934422
$626$	0	0
$627$	0	0
$628$	0	0
$629$	23.7945	0.948748
$630$	0	0
$631$	−0.629888	−0.0250755	−0.0125377	−	0.999921i	$-0.503991\pi$
$631$	−0.629888	−0.0250755	−0.0125377	+	0.999921i	$0.503991\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−0.211773	−0.00840394
$636$	0	0
$637$	11.6065	0.459865
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−3.52166	−0.139097	−0.0695486	−	0.997579i	$-0.522156\pi$
$641$	−3.52166	−0.139097	−0.0695486	+	0.997579i	$0.522156\pi$
$642$	0	0
$643$	9.83765	0.387959	0.193980	−	0.981006i	$-0.437860\pi$
$643$	9.83765	0.387959	0.193980	+	0.981006i	$0.437860\pi$
$644$	0	0
$645$	0	0
$646$	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
$651$	0	0
$652$	0	0
$653$	−9.22745	−0.361098	−0.180549	−	0.983566i	$-0.557787\pi$
$653$	−9.22745	−0.361098	−0.180549	+	0.983566i	$0.557787\pi$
$654$	0	0
$655$	−3.15694	−0.123352
$656$	0	0
$657$	0	0
$658$	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
$663$	0	0
$664$	0	0
$665$	0.886864	0.0343911
$666$	0	0
$667$	−19.5777	−0.758052
$668$	0	0
$669$	0	0
$670$	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
$675$	0	0
$676$	0	0
$677$	31.4918	1.21033	0.605164	−	0.796101i	$-0.293108\pi$
$677$	31.4918	1.21033	0.605164	+	0.796101i	$0.293108\pi$
$678$	0	0
$679$	−52.5483	−2.01662
$680$	0	0
$681$	0	0
$682$	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
$687$	0	0
$688$	0	0
$689$	−2.62122	−0.0998606
$690$	0	0
$691$	38.2264	1.45420	0.727101	−	0.686531i	$-0.240867\pi$
$691$	38.2264	1.45420	0.727101	+	0.686531i	$0.240867\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−0.0256590	−0.000973301 0
$696$	0	0
$697$	25.6297	0.970794
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−9.03731	−0.341335	−0.170667	−	0.985329i	$-0.554592\pi$
$701$	−9.03731	−0.341335	−0.170667	+	0.985329i	$0.554592\pi$
$702$	0	0
$703$	−3.16250	−0.119276
$704$	0	0
$705$	0	0
$706$	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
$711$	0	0
$712$	0	0
$713$	−24.5754	−0.920357
$714$	0	0
$715$	−5.69038	−0.212808
$716$	0	0
$717$	0	0
$718$	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
$723$	0	0
$724$	0	0
$725$	−23.9332	−0.888859
$726$	0	0
$727$	−12.6201	−0.468052	−0.234026	−	0.972230i	$-0.575190\pi$
$727$	−12.6201	−0.468052	−0.234026	+	0.972230i	$0.575190\pi$
$728$	0	0
$729$	0	0
$730$	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	0
$736$	0	0
$737$	51.6871	1.90392
$738$	0	0
$739$	−20.0592	−0.737889	−0.368945	−	0.929451i	$-0.620281\pi$
$739$	−20.0592	−0.737889	−0.368945	+	0.929451i	$0.620281\pi$
$740$	0	0
$741$	0	0
$742$	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
$747$	0	0
$748$	0	0
$749$	−56.6395	−2.06956
$750$	0	0
$751$	−37.3294	−1.36217	−0.681084	−	0.732205i	$-0.738491\pi$
$751$	−37.3294	−1.36217	−0.681084	+	0.732205i	$0.738491\pi$
$752$	0	0
$753$	0	0
$754$	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
$759$	0	0
$760$	0	0
$761$	27.1767	0.985155	0.492578	−	0.870269i	$-0.336055\pi$
$761$	27.1767	0.985155	0.492578	+	0.870269i	$0.336055\pi$
$762$	0	0
$763$	−24.1808	−0.875404
$764$	0	0
$765$	0	0
$766$	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
$771$	0	0
$772$	0	0
$773$	−20.7289	−0.745567	−0.372783	−	0.927918i	$-0.621597\pi$
$773$	−20.7289	−0.745567	−0.372783	+	0.927918i	$0.621597\pi$
$774$	0	0
$775$	−30.0428	−1.07917
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.40642	−0.122048
$780$	0	0
$781$	27.6871	0.990722
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.76002	0.0628177
$786$	0	0
$787$	−21.7129	−0.773982	−0.386991	−	0.922084i	$-0.626486\pi$
$787$	−21.7129	−0.773982	−0.386991	+	0.922084i	$0.626486\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−11.2719	−0.400781
$792$	0	0
$793$	−68.7032	−2.43972
$794$	0	0
$795$	0	0
$796$	0	0
$797$	27.5528	0.975971	0.487985	−	0.872852i	$-0.337732\pi$
$797$	27.5528	0.975971	0.487985	+	0.872852i	$0.337732\pi$
$798$	0	0
$799$	−10.8092	−0.382403
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−20.7839	−0.733447
$804$	0	0
$805$	−4.09121	−0.144196
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−18.8535	−0.662854	−0.331427	−	0.943481i	$-0.607530\pi$
$809$	−18.8535	−0.662854	−0.331427	+	0.943481i	$0.607530\pi$
$810$	0	0
$811$	6.78796	0.238357	0.119179	−	0.992873i	$-0.461974\pi$
$811$	6.78796	0.238357	0.119179	+	0.992873i	$0.461974\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−4.41522	−0.154658
$816$	0	0
$817$	−3.40642	−0.119175
$818$	0	0
$819$	0	0
$820$	0	0
$821$	16.0169	0.558995	0.279498	−	0.960146i	$-0.409832\pi$
$821$	16.0169	0.558995	0.279498	+	0.960146i	$0.409832\pi$
$822$	0	0
$823$	28.5609	0.995570	0.497785	−	0.867301i	$-0.334147\pi$
$823$	28.5609	0.995570	0.497785	+	0.867301i	$0.334147\pi$
$824$	0	0
$825$	0	0
$826$	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
$831$	0	0
$832$	0	0
$833$	−16.3173	−0.565361
$834$	0	0
$835$	5.28624	0.182938
$836$	0	0
$837$	0	0
$838$	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
$843$	0	0
$844$	0	0
$845$	2.83174	0.0974147
$846$	0	0
$847$	8.18947	0.281393
$848$	0	0
$849$	0	0
$850$	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	0
$856$	0	0
$857$	24.8671	0.849444	0.424722	−	0.905324i	$-0.360372\pi$
$857$	24.8671	0.849444	0.424722	+	0.905324i	$0.360372\pi$
$858$	0	0
$859$	−15.5990	−0.532231	−0.266115	−	0.963941i	$-0.585740\pi$
$859$	−15.5990	−0.532231	−0.266115	+	0.963941i	$0.585740\pi$
$860$	0	0
$861$	0	0
$862$	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
$867$	0	0
$868$	0	0
$869$	−55.0781	−1.86840
$870$	0	0
$871$	−64.9035	−2.19917
$872$	0	0
$873$	0	0
$874$	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
$879$	0	0
$880$	0	0
$881$	19.8589	0.669064	0.334532	−	0.942384i	$-0.391422\pi$
$881$	19.8589	0.669064	0.334532	+	0.942384i	$0.391422\pi$
$882$	0	0
$883$	−25.2740	−0.850537	−0.425269	−	0.905067i	$-0.639820\pi$
$883$	−25.2740	−0.850537	−0.425269	+	0.905067i	$0.639820\pi$
$884$	0	0
$885$	0	0
$886$	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
$891$	0	0
$892$	0	0
$893$	1.43664	0.0480754
$894$	0	0
$895$	1.98642	0.0663988
$896$	0	0
$897$	0	0
$898$	0	0
$899$	30.0707	1.00291
$900$	0	0
$901$	3.68513	0.122769
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−3.25584	−0.108228
$906$	0	0
$907$	−8.10347	−0.269071	−0.134536	−	0.990909i	$-0.542954\pi$
$907$	−8.10347	−0.269071	−0.134536	+	0.990909i	$0.542954\pi$
$908$	0	0
$909$	0	0
$910$	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
$915$	0	0
$916$	0	0
$917$	−29.3258	−0.968423
$918$	0	0
$919$	15.1680	0.500348	0.250174	−	0.968201i	$-0.419512\pi$
$919$	15.1680	0.500348	0.250174	+	0.968201i	$0.419512\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−34.7667	−1.14436
$924$	0	0
$925$	17.8347	0.586401
$926$	0	0
$927$	0	0
$928$	0	0
$929$	9.16951	0.300842	0.150421	−	0.988622i	$-0.451937\pi$
$929$	9.16951	0.300842	0.150421	+	0.988622i	$0.451937\pi$
$930$	0	0
$931$	2.16872	0.0710768
$932$	0	0
$933$	0	0
$934$	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
$939$	0	0
$940$	0	0
$941$	−10.4385	−0.340285	−0.170142	−	0.985419i	$-0.554423\pi$
$941$	−10.4385	−0.340285	−0.170142	+	0.985419i	$0.554423\pi$
$942$	0	0
$943$	15.7142	0.511726
$944$	0	0
$945$	0	0
$946$	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
$951$	0	0
$952$	0	0
$953$	−47.9271	−1.55251	−0.776255	−	0.630419i	$-0.782883\pi$
$953$	−47.9271	−1.55251	−0.776255	+	0.630419i	$0.782883\pi$
$954$	0	0
$955$	−6.40083	−0.207126
$956$	0	0
$957$	0	0
$958$	0	0
$959$	42.8914	1.38504
$960$	0	0
$961$	6.74701	0.217646
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−0.0516688	−0.00166328
$966$	0	0
$967$	20.7211	0.666346	0.333173	−	0.942866i	$-0.391881\pi$
$967$	20.7211	0.666346	0.333173	+	0.942866i	$0.391881\pi$
$968$	0	0
$969$	0	0
$970$	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
$975$	0	0
$976$	0	0
$977$	−42.7363	−1.36726	−0.683628	−	0.729831i	$-0.739599\pi$
$977$	−42.7363	−1.36726	−0.683628	+	0.729831i	$0.739599\pi$
$978$	0	0
$979$	−66.9235	−2.13888
$980$	0	0
$981$	0	0
$982$	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
$987$	0	0
$988$	0	0
$989$	15.7142	0.499684
$990$	0	0
$991$	11.1918	0.355518	0.177759	−	0.984074i	$-0.443115\pi$
$991$	11.1918	0.355518	0.177759	+	0.984074i	$0.443115\pi$
$992$	0	0
$993$	0	0
$994$	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
$999$	0	0

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	9216.2.a.bl.1.2		4
3.2	odd	2		3072.2.a.m.1.3		4
4.3	odd	2		9216.2.a.z.1.2		4
8.3	odd	2		9216.2.a.ba.1.3		4
8.5	even	2		9216.2.a.bm.1.3		4
12.11	even	2		3072.2.a.p.1.3		4
24.5	odd	2		3072.2.a.s.1.2		4
24.11	even	2		3072.2.a.j.1.2		4
32.3	odd	8		4608.2.k.bc.1153.3		8
32.5	even	8		4608.2.k.bh.3457.2		8
32.11	odd	8		4608.2.k.bc.3457.3		8
32.13	even	8		4608.2.k.bh.1153.2		8
32.19	odd	8		4608.2.k.bj.1153.2		8
32.21	even	8		4608.2.k.be.3457.3		8
32.27	odd	8		4608.2.k.bj.3457.2		8
32.29	even	8		4608.2.k.be.1153.3		8
48.5	odd	4		3072.2.d.e.1537.7		8
48.11	even	4		3072.2.d.j.1537.3		8
48.29	odd	4		3072.2.d.e.1537.2		8
48.35	even	4		3072.2.d.j.1537.6		8
96.5	odd	8		1536.2.j.e.385.4	✓	8
96.11	even	8		1536.2.j.i.385.3	yes	8
96.29	odd	8		1536.2.j.j.1153.1	yes	8
96.35	even	8		1536.2.j.i.1153.3	yes	8
96.53	odd	8		1536.2.j.j.385.1	yes	8
96.59	even	8		1536.2.j.f.385.2	yes	8
96.77	odd	8		1536.2.j.e.1153.4	yes	8
96.83	even	8		1536.2.j.f.1153.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.j.e.385.4	✓	8	96.5	odd	8
1536.2.j.e.1153.4	yes	8	96.77	odd	8
1536.2.j.f.385.2	yes	8	96.59	even	8
1536.2.j.f.1153.2	yes	8	96.83	even	8
1536.2.j.i.385.3	yes	8	96.11	even	8
1536.2.j.i.1153.3	yes	8	96.35	even	8
1536.2.j.j.385.1	yes	8	96.53	odd	8
1536.2.j.j.1153.1	yes	8	96.29	odd	8
3072.2.a.j.1.2		4	24.11	even	2
3072.2.a.m.1.3		4	3.2	odd	2
3072.2.a.p.1.3		4	12.11	even	2
3072.2.a.s.1.2		4	24.5	odd	2
3072.2.d.e.1537.2		8	48.29	odd	4
3072.2.d.e.1537.7		8	48.5	odd	4
3072.2.d.j.1537.3		8	48.11	even	4
3072.2.d.j.1537.6		8	48.35	even	4
4608.2.k.bc.1153.3		8	32.3	odd	8
4608.2.k.bc.3457.3		8	32.11	odd	8
4608.2.k.be.1153.3		8	32.29	even	8
4608.2.k.be.3457.3		8	32.21	even	8
4608.2.k.bh.1153.2		8	32.13	even	8
4608.2.k.bh.3457.2		8	32.5	even	8
4608.2.k.bj.1153.2		8	32.19	odd	8
4608.2.k.bj.3457.2		8	32.27	odd	8
9216.2.a.z.1.2		4	4.3	odd	2
9216.2.a.ba.1.3		4	8.3	odd	2
9216.2.a.bl.1.2		4	1.1	even	1	trivial
9216.2.a.bm.1.3		4	8.5	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

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Modular forms

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Fields

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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	9216.2.a.bl.1.2

Level	$9216$
Weight	$2$
Character	9216.1
Self dual	yes
Analytic conductor	$73.590$
Analytic rank	$1$
Dimension	$4$
CM	no
Inner twists	$1$