LMFDB - Embedded newform 976.2.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [976,2,Mod(1,976)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(976, 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("976.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 976 = 2^{4} \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	976.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:	$7.79339923728$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.13676.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 7x + 1 $ x^4 - x^3 - 6x^2 + 7x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 488)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.18363$ of defining polynomial
Character	$\chi$	$=$	976.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.49391 q^{3} +2.26214 q^{5} +0.183629 q^{7} -0.768234 q^{9} +O(q^{10})$	$q-1.49391 q^{3} +2.26214 q^{5} +0.183629 q^{7} -0.768234 q^{9} -3.95186 q^{11} +0.725675 q^{13} -3.37944 q^{15} -4.36726 q^{17} -3.76823 q^{19} -0.274325 q^{21} -2.92148 q^{23} +0.117294 q^{25} +5.62940 q^{27} +4.13549 q^{29} +7.76489 q^{31} +5.90373 q^{33} +0.415394 q^{35} +0.506090 q^{37} -1.08409 q^{39} -9.96628 q^{41} -7.90373 q^{43} -1.73786 q^{45} -9.28316 q^{47} -6.96628 q^{49} +6.52429 q^{51} -1.12665 q^{53} -8.93968 q^{55} +5.62940 q^{57} -2.39103 q^{59} -1.00000 q^{61} -0.141070 q^{63} +1.64158 q^{65} -0.964044 q^{67} +4.36443 q^{69} +3.14767 q^{71} -6.18741 q^{73} -0.175227 q^{75} -0.725675 q^{77} -3.79483 q^{79} -6.10511 q^{81} +11.5149 q^{83} -9.87936 q^{85} -6.17805 q^{87} -10.0608 q^{89} +0.133255 q^{91} -11.6000 q^{93} -8.52429 q^{95} +0.897710 q^{97} +3.03596 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} - 2 q^{5} - 7 q^{7} + 3 q^{9}+O(q^{10}) $ 4 * q - q^3 - 2 * q^5 - 7 * q^7 + 3 * q^9	$ 4 q - q^{3} - 2 q^{5} - 7 q^{7} + 3 q^{9} - 2 q^{11} + 4 q^{13} - 8 q^{15} - 2 q^{17} - 9 q^{19} - 15 q^{23} + 6 q^{25} - 4 q^{27} - 5 q^{29} - 17 q^{31} - 4 q^{33} + 7 q^{37} - 22 q^{39} - 15 q^{41} - 4 q^{43} - 18 q^{45} - 4 q^{47} - 3 q^{49} + 4 q^{51} - 15 q^{53} - 12 q^{55} - 4 q^{57} + 12 q^{59} - 4 q^{61} - 10 q^{65} - 3 q^{69} + q^{71} - q^{73} + 33 q^{75} - 4 q^{77} - 8 q^{79} - 20 q^{81} + 19 q^{83} + 8 q^{85} + 4 q^{87} - 6 q^{89} - 5 q^{93} - 12 q^{95} + 13 q^{97} + 16 q^{99}+O(q^{100}) $ 4 * q - q^3 - 2 * q^5 - 7 * q^7 + 3 * q^9 - 2 * q^11 + 4 * q^13 - 8 * q^15 - 2 * q^17 - 9 * q^19 - 15 * q^23 + 6 * q^25 - 4 * q^27 - 5 * q^29 - 17 * q^31 - 4 * q^33 + 7 * q^37 - 22 * q^39 - 15 * q^41 - 4 * q^43 - 18 * q^45 - 4 * q^47 - 3 * q^49 + 4 * q^51 - 15 * q^53 - 12 * q^55 - 4 * q^57 + 12 * q^59 - 4 * q^61 - 10 * q^65 - 3 * q^69 + q^71 - q^73 + 33 * q^75 - 4 * q^77 - 8 * q^79 - 20 * q^81 + 19 * q^83 + 8 * q^85 + 4 * q^87 - 6 * q^89 - 5 * q^93 - 12 * q^95 + 13 * q^97 + 16 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.49391	−0.862509	−0.431255	−	0.902230i	$-0.641929\pi$
$3$	−1.49391	−0.862509	−0.431255	+	0.902230i	$0.641929\pi$
$4$	0	0
$5$	2.26214	1.01166	0.505831	−	0.862633i	$-0.331186\pi$
$5$	2.26214	1.01166	0.505831	+	0.862633i	$0.331186\pi$
$6$	0	0
$7$	0.183629	0.0694051	0.0347026	−	0.999398i	$-0.488952\pi$
$7$	0.183629	0.0694051	0.0347026	+	0.999398i	$0.488952\pi$
$8$	0	0
$9$	−0.768234	−0.256078
$10$	0	0
$11$	−3.95186	−1.19153	−0.595766	−	0.803158i	$-0.703151\pi$
$11$	−3.95186	−1.19153	−0.595766	+	0.803158i	$0.703151\pi$
$12$	0	0
$13$	0.725675	0.201266	0.100633	−	0.994924i	$-0.467913\pi$
$13$	0.725675	0.201266	0.100633	+	0.994924i	$0.467913\pi$
$14$	0	0
$15$	−3.37944	−0.872567
$16$	0	0
$17$	−4.36726	−1.05922	−0.529608	−	0.848243i	$-0.677661\pi$
$17$	−4.36726	−1.05922	−0.529608	+	0.848243i	$0.677661\pi$
$18$	0	0
$19$	−3.76823	−0.864492	−0.432246	−	0.901756i	$-0.642279\pi$
$19$	−3.76823	−0.864492	−0.432246	+	0.901756i	$0.642279\pi$
$20$	0	0
$21$	−0.274325	−0.0598625
$22$	0	0
$23$	−2.92148	−0.609172	−0.304586	−	0.952485i	$-0.598518\pi$
$23$	−2.92148	−0.609172	−0.304586	+	0.952485i	$0.598518\pi$
$24$	0	0
$25$	0.117294	0.0234589
$26$	0	0
$27$	5.62940	1.08338
$28$	0	0
$29$	4.13549	0.767942	0.383971	−	0.923345i	$-0.374556\pi$
$29$	4.13549	0.767942	0.383971	+	0.923345i	$0.374556\pi$
$30$	0	0
$31$	7.76489	1.39462	0.697308	−	0.716772i	$-0.254381\pi$
$31$	7.76489	1.39462	0.697308	+	0.716772i	$0.254381\pi$
$32$	0	0
$33$	5.90373	1.02771
$34$	0	0
$35$	0.415394	0.0702145
$36$	0	0
$37$	0.506090	0.0832008	0.0416004	−	0.999134i	$-0.486754\pi$
$37$	0.506090	0.0832008	0.0416004	+	0.999134i	$0.486754\pi$
$38$	0	0
$39$	−1.08409	−0.173594
$40$	0	0
$41$	−9.96628	−1.55647	−0.778236	−	0.627972i	$-0.783885\pi$
$41$	−9.96628	−1.55647	−0.778236	+	0.627972i	$0.783885\pi$
$42$	0	0
$43$	−7.90373	−1.20531	−0.602653	−	0.798003i	$-0.705890\pi$
$43$	−7.90373	−1.20531	−0.602653	+	0.798003i	$0.705890\pi$
$44$	0	0
$45$	−1.73786	−0.259064
$46$	0	0
$47$	−9.28316	−1.35409	−0.677044	−	0.735942i	$-0.736739\pi$
$47$	−9.28316	−1.35409	−0.677044	+	0.735942i	$0.736739\pi$
$48$	0	0
$49$	−6.96628	−0.995183
$50$	0	0
$51$	6.52429	0.913583
$52$	0	0
$53$	−1.12665	−0.154758	−0.0773788	−	0.997002i	$-0.524655\pi$
$53$	−1.12665	−0.154758	−0.0773788	+	0.997002i	$0.524655\pi$
$54$	0	0
$55$	−8.93968	−1.20543
$56$	0	0
$57$	5.62940	0.745632
$58$	0	0
$59$	−2.39103	−0.311286	−0.155643	−	0.987813i	$-0.549745\pi$
$59$	−2.39103	−0.311286	−0.155643	+	0.987813i	$0.549745\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	0	0
$63$	−0.141070	−0.0177731
$64$	0	0
$65$	1.64158	0.203613
$66$	0	0
$67$	−0.964044	−0.117777	−0.0588883	−	0.998265i	$-0.518756\pi$
$67$	−0.964044	−0.117777	−0.0588883	+	0.998265i	$0.518756\pi$
$68$	0	0
$69$	4.36443	0.525416
$70$	0	0
$71$	3.14767	0.373560	0.186780	−	0.982402i	$-0.440195\pi$
$71$	3.14767	0.373560	0.186780	+	0.982402i	$0.440195\pi$
$72$	0	0
$73$	−6.18741	−0.724181	−0.362091	−	0.932143i	$-0.617937\pi$
$73$	−6.18741	−0.724181	−0.362091	+	0.932143i	$0.617937\pi$
$74$	0	0
$75$	−0.175227	−0.0202335
$76$	0	0
$77$	−0.725675	−0.0826984
$78$	0	0
$79$	−3.79483	−0.426952	−0.213476	−	0.976948i	$-0.568479\pi$
$79$	−3.79483	−0.426952	−0.213476	+	0.976948i	$0.568479\pi$
$80$	0	0
$81$	−6.10511	−0.678346
$82$	0	0
$83$	11.5149	1.26393	0.631964	−	0.774998i	$-0.282249\pi$
$83$	11.5149	1.26393	0.631964	+	0.774998i	$0.282249\pi$
$84$	0	0
$85$	−9.87936	−1.07157
$86$	0	0
$87$	−6.17805	−0.662357
$88$	0	0
$89$	−10.0608	−1.06644	−0.533219	−	0.845977i	$-0.679018\pi$
$89$	−10.0608	−1.06644	−0.533219	+	0.845977i	$0.679018\pi$
$90$	0	0
$91$	0.133255	0.0139689
$92$	0	0
$93$	−11.6000	−1.20287
$94$	0	0
$95$	−8.52429	−0.874573
$96$	0	0
$97$	0.897710	0.0911486	0.0455743	−	0.998961i	$-0.485488\pi$
$97$	0.897710	0.0911486	0.0455743	+	0.998961i	$0.485488\pi$
$98$	0	0
$99$	3.03596	0.305125
$100$	0	0
$101$	−12.5425	−1.24802	−0.624012	−	0.781415i	$-0.714498\pi$
$101$	−12.5425	−1.24802	−0.624012	+	0.781415i	$0.714498\pi$
$102$	0	0
$103$	3.74670	0.369173	0.184586	−	0.982816i	$-0.440905\pi$
$103$	3.74670	0.369173	0.184586	+	0.982816i	$0.440905\pi$
$104$	0	0
$105$	−0.620562	−0.0605606
$106$	0	0
$107$	7.46173	0.721353	0.360676	−	0.932691i	$-0.382546\pi$
$107$	7.46173	0.721353	0.360676	+	0.932691i	$0.382546\pi$
$108$	0	0
$109$	15.0818	1.44457	0.722286	−	0.691594i	$-0.243091\pi$
$109$	15.0818	1.44457	0.722286	+	0.691594i	$0.243091\pi$
$110$	0	0
$111$	−0.756053	−0.0717614
$112$	0	0
$113$	3.38545	0.318477	0.159238	−	0.987240i	$-0.449096\pi$
$113$	3.38545	0.318477	0.159238	+	0.987240i	$0.449096\pi$
$114$	0	0
$115$	−6.60882	−0.616276
$116$	0	0
$117$	−0.557489	−0.0515398
$118$	0	0
$119$	−0.801954	−0.0735150
$120$	0	0
$121$	4.61722	0.419747
$122$	0	0
$123$	14.8887	1.34247
$124$	0	0
$125$	−11.0454	−0.987929
$126$	0	0
$127$	−0.705679	−0.0626189	−0.0313095	−	0.999510i	$-0.509968\pi$
$127$	−0.705679	−0.0626189	−0.0313095	+	0.999510i	$0.509968\pi$
$128$	0	0
$129$	11.8075	1.03959
$130$	0	0
$131$	10.4999	0.917383	0.458691	−	0.888596i	$-0.348318\pi$
$131$	10.4999	0.917383	0.458691	+	0.888596i	$0.348318\pi$
$132$	0	0
$133$	−0.691956	−0.0600002
$134$	0	0
$135$	12.7345	1.09601
$136$	0	0
$137$	−16.4662	−1.40680	−0.703401	−	0.710793i	$-0.748336\pi$
$137$	−16.4662	−1.40680	−0.703401	+	0.710793i	$0.748336\pi$
$138$	0	0
$139$	−8.13773	−0.690233	−0.345117	−	0.938560i	$-0.612161\pi$
$139$	−8.13773	−0.690233	−0.345117	+	0.938560i	$0.612161\pi$
$140$	0	0
$141$	13.8682	1.16791
$142$	0	0
$143$	−2.86777	−0.239815
$144$	0	0
$145$	9.35508	0.776897
$146$	0	0
$147$	10.4070	0.858354
$148$	0	0
$149$	2.03320	0.166566	0.0832832	−	0.996526i	$-0.473459\pi$
$149$	2.03320	0.166566	0.0832832	+	0.996526i	$0.473459\pi$
$150$	0	0
$151$	−6.74009	−0.548501	−0.274250	−	0.961658i	$-0.588430\pi$
$151$	−6.74009	−0.548501	−0.274250	+	0.961658i	$0.588430\pi$
$152$	0	0
$153$	3.35508	0.271242
$154$	0	0
$155$	17.5653	1.41088
$156$	0	0
$157$	7.61120	0.607440	0.303720	−	0.952761i	$-0.401771\pi$
$157$	7.61120	0.607440	0.303720	+	0.952761i	$0.401771\pi$
$158$	0	0
$159$	1.68312	0.133480
$160$	0	0
$161$	−0.536468	−0.0422796
$162$	0	0
$163$	3.46955	0.271756	0.135878	−	0.990726i	$-0.456614\pi$
$163$	3.46955	0.271756	0.135878	+	0.990726i	$0.456614\pi$
$164$	0	0
$165$	13.3551	1.03969
$166$	0	0
$167$	8.06076	0.623760	0.311880	−	0.950121i	$-0.399041\pi$
$167$	8.06076	0.623760	0.311880	+	0.950121i	$0.399041\pi$
$168$	0	0
$169$	−12.4734	−0.959492
$170$	0	0
$171$	2.89489	0.221377
$172$	0	0
$173$	0.545826	0.0414984	0.0207492	−	0.999785i	$-0.493395\pi$
$173$	0.545826	0.0414984	0.0207492	+	0.999785i	$0.493395\pi$
$174$	0	0
$175$	0.0215386	0.00162817
$176$	0	0
$177$	3.57199	0.268487
$178$	0	0
$179$	14.8378	1.10903	0.554516	−	0.832173i	$-0.312904\pi$
$179$	14.8378	1.10903	0.554516	+	0.832173i	$0.312904\pi$
$180$	0	0
$181$	18.2251	1.35466	0.677330	−	0.735679i	$-0.263137\pi$
$181$	18.2251	1.35466	0.677330	+	0.735679i	$0.263137\pi$
$182$	0	0
$183$	1.49391	0.110433
$184$	0	0
$185$	1.14485	0.0841710
$186$	0	0
$187$	17.2588	1.26209
$188$	0	0
$189$	1.03372	0.0751920
$190$	0	0
$191$	−3.10288	−0.224516	−0.112258	−	0.993679i	$-0.535808\pi$
$191$	−3.10288	−0.224516	−0.112258	+	0.993679i	$0.535808\pi$
$192$	0	0
$193$	18.7532	1.34989	0.674944	−	0.737869i	$-0.264168\pi$
$193$	18.7532	1.34989	0.674944	+	0.737869i	$0.264168\pi$
$194$	0	0
$195$	−2.45238	−0.175618
$196$	0	0
$197$	5.02102	0.357733	0.178866	−	0.983873i	$-0.442757\pi$
$197$	5.02102	0.357733	0.178866	+	0.983873i	$0.442757\pi$
$198$	0	0
$199$	−27.3915	−1.94173	−0.970865	−	0.239627i	$-0.922975\pi$
$199$	−27.3915	−1.94173	−0.970865	+	0.239627i	$0.922975\pi$
$200$	0	0
$201$	1.44019	0.101583
$202$	0	0
$203$	0.759395	0.0532991
$204$	0	0
$205$	−22.5452	−1.57462
$206$	0	0
$207$	2.24438	0.155996
$208$	0	0
$209$	14.8915	1.03007
$210$	0	0
$211$	23.0774	1.58871	0.794357	−	0.607451i	$-0.207808\pi$
$211$	23.0774	1.58871	0.794357	+	0.607451i	$0.207808\pi$
$212$	0	0
$213$	−4.70234	−0.322199
$214$	0	0
$215$	−17.8794	−1.21936
$216$	0	0
$217$	1.42586	0.0967935
$218$	0	0
$219$	9.24343	0.624613
$220$	0	0
$221$	−3.16921	−0.213184
$222$	0	0
$223$	4.01262	0.268705	0.134352	−	0.990934i	$-0.457105\pi$
$223$	4.01262	0.268705	0.134352	+	0.990934i	$0.457105\pi$
$224$	0	0
$225$	−0.0901096	−0.00600731
$226$	0	0
$227$	−6.99288	−0.464134	−0.232067	−	0.972700i	$-0.574549\pi$
$227$	−6.99288	−0.464134	−0.232067	+	0.972700i	$0.574549\pi$
$228$	0	0
$229$	21.2500	1.40424	0.702119	−	0.712060i	$-0.252238\pi$
$229$	21.2500	1.40424	0.702119	+	0.712060i	$0.252238\pi$
$230$	0	0
$231$	1.08409	0.0713281
$232$	0	0
$233$	23.6868	1.55177	0.775887	−	0.630871i	$-0.217302\pi$
$233$	23.6868	1.55177	0.775887	+	0.630871i	$0.217302\pi$
$234$	0	0
$235$	−20.9999	−1.36988
$236$	0	0
$237$	5.66914	0.368250
$238$	0	0
$239$	−11.2943	−0.730569	−0.365284	−	0.930896i	$-0.619028\pi$
$239$	−11.2943	−0.730569	−0.365284	+	0.930896i	$0.619028\pi$
$240$	0	0
$241$	13.3003	0.856750	0.428375	−	0.903601i	$-0.359086\pi$
$241$	13.3003	0.856750	0.428375	+	0.903601i	$0.359086\pi$
$242$	0	0
$243$	−7.76772	−0.498299
$244$	0	0
$245$	−15.7587	−1.00679
$246$	0	0
$247$	−2.73451	−0.173993
$248$	0	0
$249$	−17.2023	−1.09015
$250$	0	0
$251$	−14.6990	−0.927792	−0.463896	−	0.885890i	$-0.653549\pi$
$251$	−14.6990	−0.927792	−0.463896	+	0.885890i	$0.653549\pi$
$252$	0	0
$253$	11.5453	0.725847
$254$	0	0
$255$	14.7589	0.924237
$256$	0	0
$257$	28.4540	1.77491	0.887457	−	0.460891i	$-0.152470\pi$
$257$	28.4540	1.77491	0.887457	+	0.460891i	$0.152470\pi$
$258$	0	0
$259$	0.0929327	0.00577456
$260$	0	0
$261$	−3.17703	−0.196653
$262$	0	0
$263$	−28.4756	−1.75588	−0.877939	−	0.478772i	$-0.841082\pi$
$263$	−28.4756	−1.75588	−0.877939	+	0.478772i	$0.841082\pi$
$264$	0	0
$265$	−2.54865	−0.156562
$266$	0	0
$267$	15.0299	0.919812
$268$	0	0
$269$	−29.5729	−1.80309	−0.901545	−	0.432686i	$-0.857566\pi$
$269$	−29.5729	−1.80309	−0.901545	+	0.432686i	$0.857566\pi$
$270$	0	0
$271$	16.6139	1.00922	0.504611	−	0.863347i	$-0.331636\pi$
$271$	16.6139	1.00922	0.504611	+	0.863347i	$0.331636\pi$
$272$	0	0
$273$	−0.199071	−0.0120483
$274$	0	0
$275$	−0.463532	−0.0279520
$276$	0	0
$277$	29.1841	1.75350	0.876750	−	0.480946i	$-0.159707\pi$
$277$	29.1841	1.75350	0.876750	+	0.480946i	$0.159707\pi$
$278$	0	0
$279$	−5.96526	−0.357131
$280$	0	0
$281$	−3.12166	−0.186223	−0.0931113	−	0.995656i	$-0.529681\pi$
$281$	−3.12166	−0.186223	−0.0931113	+	0.995656i	$0.529681\pi$
$282$	0	0
$283$	−20.5668	−1.22257	−0.611286	−	0.791410i	$-0.709348\pi$
$283$	−20.5668	−1.22257	−0.611286	+	0.791410i	$0.709348\pi$
$284$	0	0
$285$	12.7345	0.754328
$286$	0	0
$287$	−1.83009	−0.108027
$288$	0	0
$289$	2.07294	0.121937
$290$	0	0
$291$	−1.34110	−0.0786165
$292$	0	0
$293$	17.3008	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$293$	17.3008	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$294$	0	0
$295$	−5.40886	−0.314916
$296$	0	0
$297$	−22.2466	−1.29088
$298$	0	0
$299$	−2.12005	−0.122606
$300$	0	0
$301$	−1.45135	−0.0836545
$302$	0	0
$303$	18.7373	1.07643
$304$	0	0
$305$	−2.26214	−0.129530
$306$	0	0
$307$	−2.98723	−0.170490	−0.0852452	−	0.996360i	$-0.527167\pi$
$307$	−2.98723	−0.170490	−0.0852452	+	0.996360i	$0.527167\pi$
$308$	0	0
$309$	−5.59722	−0.318415
$310$	0	0
$311$	−2.52205	−0.143012	−0.0715062	−	0.997440i	$-0.522781\pi$
$311$	−2.52205	−0.143012	−0.0715062	+	0.997440i	$0.522781\pi$
$312$	0	0
$313$	18.9048	1.06856	0.534280	−	0.845308i	$-0.320583\pi$
$313$	18.9048	1.06856	0.534280	+	0.845308i	$0.320583\pi$
$314$	0	0
$315$	−0.319120	−0.0179804
$316$	0	0
$317$	−25.2101	−1.41594	−0.707970	−	0.706243i	$-0.750389\pi$
$317$	−25.2101	−1.41594	−0.707970	+	0.706243i	$0.750389\pi$
$318$	0	0
$319$	−16.3429	−0.915026
$320$	0	0
$321$	−11.1472	−0.622173
$322$	0	0
$323$	16.4568	0.915684
$324$	0	0
$325$	0.0851177	0.00472148
$326$	0	0
$327$	−22.5308	−1.24596
$328$	0	0
$329$	−1.70466	−0.0939807
$330$	0	0
$331$	2.85456	0.156901	0.0784505	−	0.996918i	$-0.475003\pi$
$331$	2.85456	0.156901	0.0784505	+	0.996918i	$0.475003\pi$
$332$	0	0
$333$	−0.388796	−0.0213059
$334$	0	0
$335$	−2.18081	−0.119150
$336$	0	0
$337$	21.4934	1.17082	0.585410	−	0.810738i	$-0.300934\pi$
$337$	21.4934	1.17082	0.585410	+	0.810738i	$0.300934\pi$
$338$	0	0
$339$	−5.05756	−0.274689
$340$	0	0
$341$	−30.6858	−1.66173
$342$	0	0
$343$	−2.56461	−0.138476
$344$	0	0
$345$	9.87298	0.531543
$346$	0	0
$347$	−13.1373	−0.705247	−0.352623	−	0.935765i	$-0.614710\pi$
$347$	−13.1373	−0.705247	−0.352623	+	0.935765i	$0.614710\pi$
$348$	0	0
$349$	−2.42801	−0.129969	−0.0649843	−	0.997886i	$-0.520700\pi$
$349$	−2.42801	−0.129969	−0.0649843	+	0.997886i	$0.520700\pi$
$350$	0	0
$351$	4.08512	0.218047
$352$	0	0
$353$	−32.1376	−1.71051	−0.855257	−	0.518204i	$-0.826601\pi$
$353$	−32.1376	−1.71051	−0.855257	+	0.518204i	$0.826601\pi$
$354$	0	0
$355$	7.12049	0.377916
$356$	0	0
$357$	1.19805	0.0634073
$358$	0	0
$359$	−15.5138	−0.818785	−0.409392	−	0.912358i	$-0.634259\pi$
$359$	−15.5138	−0.818785	−0.409392	+	0.912358i	$0.634259\pi$
$360$	0	0
$361$	−4.80041	−0.252653
$362$	0	0
$363$	−6.89771	−0.362036
$364$	0	0
$365$	−13.9968	−0.732626
$366$	0	0
$367$	29.4522	1.53739	0.768697	−	0.639613i	$-0.220905\pi$
$367$	29.4522	1.53739	0.768697	+	0.639613i	$0.220905\pi$
$368$	0	0
$369$	7.65644	0.398578
$370$	0	0
$371$	−0.206886	−0.0107410
$372$	0	0
$373$	−20.2461	−1.04830	−0.524152	−	0.851625i	$-0.675618\pi$
$373$	−20.2461	−1.04830	−0.524152	+	0.851625i	$0.675618\pi$
$374$	0	0
$375$	16.5008	0.852098
$376$	0	0
$377$	3.00102	0.154561
$378$	0	0
$379$	−14.1111	−0.724840	−0.362420	−	0.932015i	$-0.618049\pi$
$379$	−14.1111	−0.724840	−0.362420	+	0.932015i	$0.618049\pi$
$380$	0	0
$381$	1.05422	0.0540094
$382$	0	0
$383$	27.8461	1.42287	0.711434	−	0.702753i	$-0.248046\pi$
$383$	27.8461	1.42287	0.711434	+	0.702753i	$0.248046\pi$
$384$	0	0
$385$	−1.64158	−0.0836628
$386$	0	0
$387$	6.07191	0.308653
$388$	0	0
$389$	−14.4840	−0.734370	−0.367185	−	0.930148i	$-0.619678\pi$
$389$	−14.4840	−0.734370	−0.367185	+	0.930148i	$0.619678\pi$
$390$	0	0
$391$	12.7589	0.645244
$392$	0	0
$393$	−15.6859	−0.791251
$394$	0	0
$395$	−8.58446	−0.431931
$396$	0	0
$397$	−7.46738	−0.374777	−0.187389	−	0.982286i	$-0.560002\pi$
$397$	−7.46738	−0.374777	−0.187389	+	0.982286i	$0.560002\pi$
$398$	0	0
$399$	1.03372	0.0517507
$400$	0	0
$401$	−17.1337	−0.855616	−0.427808	−	0.903870i	$-0.640714\pi$
$401$	−17.1337	−0.855616	−0.427808	+	0.903870i	$0.640714\pi$
$402$	0	0
$403$	5.63479	0.280689
$404$	0	0
$405$	−13.8106	−0.686256
$406$	0	0
$407$	−2.00000	−0.0991363
$408$	0	0
$409$	−5.12166	−0.253250	−0.126625	−	0.991951i	$-0.540414\pi$
$409$	−5.12166	−0.253250	−0.126625	+	0.991951i	$0.540414\pi$
$410$	0	0
$411$	24.5990	1.21338
$412$	0	0
$413$	−0.439062	−0.0216048
$414$	0	0
$415$	26.0484	1.27867
$416$	0	0
$417$	12.1570	0.595332
$418$	0	0
$419$	14.3429	0.700696	0.350348	−	0.936620i	$-0.386063\pi$
$419$	14.3429	0.700696	0.350348	+	0.936620i	$0.386063\pi$
$420$	0	0
$421$	10.3644	0.505132	0.252566	−	0.967580i	$-0.418726\pi$
$421$	10.3644	0.505132	0.252566	+	0.967580i	$0.418726\pi$
$422$	0	0
$423$	7.13164	0.346752
$424$	0	0
$425$	−0.512255	−0.0248480
$426$	0	0
$427$	−0.183629	−0.00888641
$428$	0	0
$429$	4.28419	0.206843
$430$	0	0
$431$	−7.67478	−0.369681	−0.184841	−	0.982769i	$-0.559177\pi$
$431$	−7.67478	−0.369681	−0.184841	+	0.982769i	$0.559177\pi$
$432$	0	0
$433$	23.6736	1.13768	0.568841	−	0.822448i	$-0.307392\pi$
$433$	23.6736	1.13768	0.568841	+	0.822448i	$0.307392\pi$
$434$	0	0
$435$	−13.9756	−0.670081
$436$	0	0
$437$	11.0088	0.526624
$438$	0	0
$439$	3.97564	0.189747	0.0948734	−	0.995489i	$-0.469755\pi$
$439$	3.97564	0.189747	0.0948734	+	0.995489i	$0.469755\pi$
$440$	0	0
$441$	5.35173	0.254845
$442$	0	0
$443$	−30.0070	−1.42568	−0.712839	−	0.701328i	$-0.752591\pi$
$443$	−30.0070	−1.42568	−0.712839	+	0.701328i	$0.752591\pi$
$444$	0	0
$445$	−22.7589	−1.07887
$446$	0	0
$447$	−3.03742	−0.143665
$448$	0	0
$449$	−19.7317	−0.931196	−0.465598	−	0.884996i	$-0.654161\pi$
$449$	−19.7317	−0.931196	−0.465598	+	0.884996i	$0.654161\pi$
$450$	0	0
$451$	39.3854	1.85458
$452$	0	0
$453$	10.0691	0.473087
$454$	0	0
$455$	0.301442	0.0141318
$456$	0	0
$457$	35.5186	1.66149	0.830746	−	0.556652i	$-0.187914\pi$
$457$	35.5186	1.66149	0.830746	+	0.556652i	$0.187914\pi$
$458$	0	0
$459$	−24.5850	−1.14753
$460$	0	0
$461$	36.2399	1.68786	0.843931	−	0.536452i	$-0.180236\pi$
$461$	36.2399	1.68786	0.843931	+	0.536452i	$0.180236\pi$
$462$	0	0
$463$	−0.0420404	−0.00195379	−0.000976893	−	1.00000i	$-0.500311\pi$
$463$	−0.0420404	−0.00195379	−0.000976893		1.00000i	$0.500311\pi$
$464$	0	0
$465$	−26.2410	−1.21690
$466$	0	0
$467$	−33.4892	−1.54970	−0.774848	−	0.632148i	$-0.782174\pi$
$467$	−33.4892	−1.54970	−0.774848	+	0.632148i	$0.782174\pi$
$468$	0	0
$469$	−0.177026	−0.00817430
$470$	0	0
$471$	−11.3705	−0.523923
$472$	0	0
$473$	31.2344	1.43616
$474$	0	0
$475$	−0.441993	−0.0202800
$476$	0	0
$477$	0.865533	0.0396300
$478$	0	0
$479$	−42.6934	−1.95071	−0.975354	−	0.220644i	$-0.929184\pi$
$479$	−42.6934	−1.95071	−0.975354	+	0.220644i	$0.929184\pi$
$480$	0	0
$481$	0.367257	0.0167455
$482$	0	0
$483$	0.801435	0.0364666
$484$	0	0
$485$	2.03075	0.0922115
$486$	0	0
$487$	32.3738	1.46700	0.733498	−	0.679692i	$-0.237886\pi$
$487$	32.3738	1.46700	0.733498	+	0.679692i	$0.237886\pi$
$488$	0	0
$489$	−5.18319	−0.234392
$490$	0	0
$491$	15.7195	0.709412	0.354706	−	0.934978i	$-0.384581\pi$
$491$	15.7195	0.709412	0.354706	+	0.934978i	$0.384581\pi$
$492$	0	0
$493$	−18.0608	−0.813416
$494$	0	0
$495$	6.86777	0.308683
$496$	0	0
$497$	0.578003	0.0259270
$498$	0	0
$499$	−26.2349	−1.17443	−0.587217	−	0.809429i	$-0.699777\pi$
$499$	−26.2349	−1.17443	−0.587217	+	0.809429i	$0.699777\pi$
$500$	0	0
$501$	−12.0420	−0.537999
$502$	0	0
$503$	39.4091	1.75717	0.878584	−	0.477589i	$-0.158489\pi$
$503$	39.4091	1.75717	0.878584	+	0.477589i	$0.158489\pi$
$504$	0	0
$505$	−28.3729	−1.26258
$506$	0	0
$507$	18.6341	0.827571
$508$	0	0
$509$	−10.9022	−0.483231	−0.241615	−	0.970372i	$-0.577677\pi$
$509$	−10.9022	−0.483231	−0.241615	+	0.970372i	$0.577677\pi$
$510$	0	0
$511$	−1.13619	−0.0502619
$512$	0	0
$513$	−21.2129	−0.936572
$514$	0	0
$515$	8.47556	0.373478
$516$	0	0
$517$	36.6858	1.61344
$518$	0	0
$519$	−0.815415	−0.0357927
$520$	0	0
$521$	−37.7355	−1.65322	−0.826612	−	0.562772i	$-0.809735\pi$
$521$	−37.7355	−1.65322	−0.826612	+	0.562772i	$0.809735\pi$
$522$	0	0
$523$	−34.0977	−1.49099	−0.745495	−	0.666511i	$-0.767787\pi$
$523$	−34.0977	−1.49099	−0.745495	+	0.666511i	$0.767787\pi$
$524$	0	0
$525$	−0.0321768	−0.00140431
$526$	0	0
$527$	−33.9113	−1.47720
$528$	0	0
$529$	−14.4649	−0.628910
$530$	0	0
$531$	1.83687	0.0797135
$532$	0	0
$533$	−7.23228	−0.313265
$534$	0	0
$535$	16.8795	0.729765
$536$	0	0
$537$	−22.1664	−0.956550
$538$	0	0
$539$	27.5298	1.18579
$540$	0	0
$541$	−34.6995	−1.49185	−0.745924	−	0.666031i	$-0.767992\pi$
$541$	−34.6995	−1.49185	−0.745924	+	0.666031i	$0.767992\pi$
$542$	0	0
$543$	−27.2266	−1.16841
$544$	0	0
$545$	34.1171	1.46142
$546$	0	0
$547$	15.9994	0.684086	0.342043	−	0.939684i	$-0.388881\pi$
$547$	15.9994	0.684086	0.342043	+	0.939684i	$0.388881\pi$
$548$	0	0
$549$	0.768234	0.0327874
$550$	0	0
$551$	−15.5835	−0.663879
$552$	0	0
$553$	−0.696840	−0.0296327
$554$	0	0
$555$	−1.71030	−0.0725982
$556$	0	0
$557$	18.2102	0.771592	0.385796	−	0.922584i	$-0.373927\pi$
$557$	18.2102	0.771592	0.385796	+	0.922584i	$0.373927\pi$
$558$	0	0
$559$	−5.73554	−0.242587
$560$	0	0
$561$	−25.7831	−1.08856
$562$	0	0
$563$	−13.3064	−0.560796	−0.280398	−	0.959884i	$-0.590466\pi$
$563$	−13.3064	−0.560796	−0.280398	+	0.959884i	$0.590466\pi$
$564$	0	0
$565$	7.65838	0.322191
$566$	0	0
$567$	−1.12107	−0.0470807
$568$	0	0
$569$	−16.8340	−0.705717	−0.352859	−	0.935677i	$-0.614790\pi$
$569$	−16.8340	−0.705717	−0.352859	+	0.935677i	$0.614790\pi$
$570$	0	0
$571$	−10.5238	−0.440406	−0.220203	−	0.975454i	$-0.570672\pi$
$571$	−10.5238	−0.440406	−0.220203	+	0.975454i	$0.570672\pi$
$572$	0	0
$573$	4.63542	0.193647
$574$	0	0
$575$	−0.342674	−0.0142905
$576$	0	0
$577$	25.6681	1.06858	0.534288	−	0.845302i	$-0.320580\pi$
$577$	25.6681	1.06858	0.534288	+	0.845302i	$0.320580\pi$
$578$	0	0
$579$	−28.0156	−1.16429
$580$	0	0
$581$	2.11447	0.0877231
$582$	0	0
$583$	4.45238	0.184399
$584$	0	0
$585$	−1.26112	−0.0521409
$586$	0	0
$587$	21.6017	0.891597	0.445799	−	0.895133i	$-0.352920\pi$
$587$	21.6017	0.891597	0.445799	+	0.895133i	$0.352920\pi$
$588$	0	0
$589$	−29.2599	−1.20563
$590$	0	0
$591$	−7.50095	−0.308548
$592$	0	0
$593$	−17.8119	−0.731448	−0.365724	−	0.930723i	$-0.619179\pi$
$593$	−17.8119	−0.731448	−0.365724	+	0.930723i	$0.619179\pi$
$594$	0	0
$595$	−1.81413	−0.0743723
$596$	0	0
$597$	40.9204	1.67476
$598$	0	0
$599$	−7.25287	−0.296344	−0.148172	−	0.988962i	$-0.547339\pi$
$599$	−7.25287	−0.296344	−0.148172	+	0.988962i	$0.547339\pi$
$600$	0	0
$601$	−29.1243	−1.18801	−0.594003	−	0.804463i	$-0.702453\pi$
$601$	−29.1243	−1.18801	−0.594003	+	0.804463i	$0.702453\pi$
$602$	0	0
$603$	0.740611	0.0301600
$604$	0	0
$605$	10.4448	0.424642
$606$	0	0
$607$	5.79425	0.235181	0.117591	−	0.993062i	$-0.462483\pi$
$607$	5.79425	0.235181	0.117591	+	0.993062i	$0.462483\pi$
$608$	0	0
$609$	−1.13447	−0.0459709
$610$	0	0
$611$	−6.73656	−0.272532
$612$	0	0
$613$	10.3074	0.416311	0.208156	−	0.978096i	$-0.433254\pi$
$613$	10.3074	0.416311	0.208156	+	0.978096i	$0.433254\pi$
$614$	0	0
$615$	33.6804	1.35813
$616$	0	0
$617$	−36.2830	−1.46070	−0.730350	−	0.683074i	$-0.760643\pi$
$617$	−36.2830	−1.46070	−0.730350	+	0.683074i	$0.760643\pi$
$618$	0	0
$619$	−17.4651	−0.701981	−0.350990	−	0.936379i	$-0.614155\pi$
$619$	−17.4651	−0.701981	−0.350990	+	0.936379i	$0.614155\pi$
$620$	0	0
$621$	−16.4462	−0.659964
$622$	0	0
$623$	−1.84744	−0.0740163
$624$	0	0
$625$	−25.5727	−1.02291
$626$	0	0
$627$	−22.2466	−0.888444
$628$	0	0
$629$	−2.21023	−0.0881275
$630$	0	0
$631$	28.8219	1.14738	0.573690	−	0.819072i	$-0.305511\pi$
$631$	28.8219	1.14738	0.573690	+	0.819072i	$0.305511\pi$
$632$	0	0
$633$	−34.4756	−1.37028
$634$	0	0
$635$	−1.59635	−0.0633492
$636$	0	0
$637$	−5.05526	−0.200297
$638$	0	0
$639$	−2.41815	−0.0956605
$640$	0	0
$641$	25.3372	1.00076	0.500380	−	0.865806i	$-0.333194\pi$
$641$	25.3372	1.00076	0.500380	+	0.865806i	$0.333194\pi$
$642$	0	0
$643$	13.0597	0.515026	0.257513	−	0.966275i	$-0.417097\pi$
$643$	13.0597	0.515026	0.257513	+	0.966275i	$0.417097\pi$
$644$	0	0
$645$	26.7102	1.05171
$646$	0	0
$647$	−21.7957	−0.856878	−0.428439	−	0.903571i	$-0.640936\pi$
$647$	−21.7957	−0.856878	−0.428439	+	0.903571i	$0.640936\pi$
$648$	0	0
$649$	9.44903	0.370907
$650$	0	0
$651$	−2.13010	−0.0834853
$652$	0	0
$653$	22.9906	0.899693	0.449847	−	0.893106i	$-0.351479\pi$
$653$	22.9906	0.899693	0.449847	+	0.893106i	$0.351479\pi$
$654$	0	0
$655$	23.7523	0.928081
$656$	0	0
$657$	4.75338	0.185447
$658$	0	0
$659$	−31.6398	−1.23251	−0.616255	−	0.787546i	$-0.711351\pi$
$659$	−31.6398	−1.23251	−0.616255	+	0.787546i	$0.711351\pi$
$660$	0	0
$661$	14.7771	0.574762	0.287381	−	0.957816i	$-0.407215\pi$
$661$	14.7771	0.574762	0.287381	+	0.957816i	$0.407215\pi$
$662$	0	0
$663$	4.73451	0.183873
$664$	0	0
$665$	−1.56530	−0.0606999
$666$	0	0
$667$	−12.0818	−0.467808
$668$	0	0
$669$	−5.99449	−0.231760
$670$	0	0
$671$	3.95186	0.152560
$672$	0	0
$673$	−6.19907	−0.238957	−0.119478	−	0.992837i	$-0.538122\pi$
$673$	−6.19907	−0.238957	−0.119478	+	0.992837i	$0.538122\pi$
$674$	0	0
$675$	0.660298	0.0254149
$676$	0	0
$677$	41.2832	1.58664	0.793320	−	0.608804i	$-0.208351\pi$
$677$	41.2832	1.58664	0.793320	+	0.608804i	$0.208351\pi$
$678$	0	0
$679$	0.164845	0.00632618
$680$	0	0
$681$	10.4467	0.400320
$682$	0	0
$683$	−7.55045	−0.288910	−0.144455	−	0.989511i	$-0.546143\pi$
$683$	−7.55045	−0.288910	−0.144455	+	0.989511i	$0.546143\pi$
$684$	0	0
$685$	−37.2489	−1.42321
$686$	0	0
$687$	−31.7455	−1.21117
$688$	0	0
$689$	−0.817584	−0.0311475
$690$	0	0
$691$	37.5591	1.42882	0.714408	−	0.699729i	$-0.246696\pi$
$691$	37.5591	1.42882	0.714408	+	0.699729i	$0.246696\pi$
$692$	0	0
$693$	0.557489	0.0211772
$694$	0	0
$695$	−18.4087	−0.698282
$696$	0	0
$697$	43.5253	1.64864
$698$	0	0
$699$	−35.3860	−1.33842
$700$	0	0
$701$	−22.2329	−0.839725	−0.419862	−	0.907588i	$-0.637922\pi$
$701$	−22.2329	−0.839725	−0.419862	+	0.907588i	$0.637922\pi$
$702$	0	0
$703$	−1.90707	−0.0719264
$704$	0	0
$705$	31.3719	1.18153
$706$	0	0
$707$	−2.30316	−0.0866192
$708$	0	0
$709$	−8.80643	−0.330732	−0.165366	−	0.986232i	$-0.552881\pi$
$709$	−8.80643	−0.330732	−0.165366	+	0.986232i	$0.552881\pi$
$710$	0	0
$711$	2.91532	0.109333
$712$	0	0
$713$	−22.6850	−0.849561
$714$	0	0
$715$	−6.48731	−0.242612
$716$	0	0
$717$	16.8727	0.630122
$718$	0	0
$719$	23.4522	0.874620	0.437310	−	0.899311i	$-0.355931\pi$
$719$	23.4522	0.874620	0.437310	+	0.899311i	$0.355931\pi$
$720$	0	0
$721$	0.688001	0.0256225
$722$	0	0
$723$	−19.8695	−0.738955
$724$	0	0
$725$	0.485070	0.0180151
$726$	0	0
$727$	0.975787	0.0361899	0.0180950	−	0.999836i	$-0.494240\pi$
$727$	0.975787	0.0361899	0.0180950	+	0.999836i	$0.494240\pi$
$728$	0	0
$729$	29.9196	1.10813
$730$	0	0
$731$	34.5176	1.27668
$732$	0	0
$733$	−1.35071	−0.0498896	−0.0249448	−	0.999689i	$-0.507941\pi$
$733$	−1.35071	−0.0498896	−0.0249448	+	0.999689i	$0.507941\pi$
$734$	0	0
$735$	23.5421	0.868364
$736$	0	0
$737$	3.80977	0.140335
$738$	0	0
$739$	−35.5846	−1.30900	−0.654500	−	0.756062i	$-0.727121\pi$
$739$	−35.5846	−1.30900	−0.654500	+	0.756062i	$0.727121\pi$
$740$	0	0
$741$	4.08512	0.150071
$742$	0	0
$743$	5.22773	0.191787	0.0958934	−	0.995392i	$-0.469429\pi$
$743$	5.22773	0.191787	0.0958934	+	0.995392i	$0.469429\pi$
$744$	0	0
$745$	4.59939	0.168509
$746$	0	0
$747$	−8.84616	−0.323664
$748$	0	0
$749$	1.37019	0.0500656
$750$	0	0
$751$	−34.1503	−1.24616	−0.623082	−	0.782157i	$-0.714120\pi$
$751$	−34.1503	−1.24616	−0.623082	+	0.782157i	$0.714120\pi$
$752$	0	0
$753$	21.9590	0.800229
$754$	0	0
$755$	−15.2471	−0.554897
$756$	0	0
$757$	21.6824	0.788062	0.394031	−	0.919097i	$-0.371080\pi$
$757$	21.6824	0.788062	0.394031	+	0.919097i	$0.371080\pi$
$758$	0	0
$759$	−17.2476	−0.626050
$760$	0	0
$761$	43.4646	1.57559	0.787794	−	0.615938i	$-0.211223\pi$
$761$	43.4646	1.57559	0.787794	+	0.615938i	$0.211223\pi$
$762$	0	0
$763$	2.76945	0.100261
$764$	0	0
$765$	7.58967	0.274405
$766$	0	0
$767$	−1.73511	−0.0626513
$768$	0	0
$769$	−10.3009	−0.371458	−0.185729	−	0.982601i	$-0.559465\pi$
$769$	−10.3009	−0.371458	−0.185729	+	0.982601i	$0.559465\pi$
$770$	0	0
$771$	−42.5077	−1.53088
$772$	0	0
$773$	−7.71030	−0.277320	−0.138660	−	0.990340i	$-0.544280\pi$
$773$	−7.71030	−0.277320	−0.138660	+	0.990340i	$0.544280\pi$
$774$	0	0
$775$	0.910779	0.0327161
$776$	0	0
$777$	−0.138833	−0.00498061
$778$	0	0
$779$	37.5553	1.34556
$780$	0	0
$781$	−12.4392	−0.445108
$782$	0	0
$783$	23.2803	0.831972
$784$	0	0
$785$	17.2176	0.614524
$786$	0	0
$787$	−52.9612	−1.88786	−0.943932	−	0.330139i	$-0.892904\pi$
$787$	−52.9612	−1.88786	−0.943932	+	0.330139i	$0.892904\pi$
$788$	0	0
$789$	42.5399	1.51446
$790$	0	0
$791$	0.621667	0.0221039
$792$	0	0
$793$	−0.725675	−0.0257695
$794$	0	0
$795$	3.80745	0.135036
$796$	0	0
$797$	−9.65170	−0.341881	−0.170940	−	0.985281i	$-0.554681\pi$
$797$	−9.65170	−0.341881	−0.170940	+	0.985281i	$0.554681\pi$
$798$	0	0
$799$	40.5420	1.43427
$800$	0	0
$801$	7.72902	0.273091
$802$	0	0
$803$	24.4518	0.862885
$804$	0	0
$805$	−1.21357	−0.0427727
$806$	0	0
$807$	44.1792	1.55518
$808$	0	0
$809$	16.4797	0.579394	0.289697	−	0.957118i	$-0.406445\pi$
$809$	16.4797	0.579394	0.289697	+	0.957118i	$0.406445\pi$
$810$	0	0
$811$	−15.9620	−0.560501	−0.280251	−	0.959927i	$-0.590418\pi$
$811$	−15.9620	−0.560501	−0.280251	+	0.959927i	$0.590418\pi$
$812$	0	0
$813$	−24.8196	−0.870463
$814$	0	0
$815$	7.84862	0.274925
$816$	0	0
$817$	29.7831	1.04198
$818$	0	0
$819$	−0.102371	−0.00357713
$820$	0	0
$821$	−47.3090	−1.65110	−0.825548	−	0.564332i	$-0.809134\pi$
$821$	−47.3090	−1.65110	−0.825548	+	0.564332i	$0.809134\pi$
$822$	0	0
$823$	−41.3663	−1.44194	−0.720970	−	0.692966i	$-0.756303\pi$
$823$	−41.3663	−1.44194	−0.720970	+	0.692966i	$0.756303\pi$
$824$	0	0
$825$	0.692474	0.0241089
$826$	0	0
$827$	35.6863	1.24093	0.620467	−	0.784233i	$-0.286943\pi$
$827$	35.6863	1.24093	0.620467	+	0.784233i	$0.286943\pi$
$828$	0	0
$829$	−46.4444	−1.61308	−0.806541	−	0.591179i	$-0.798663\pi$
$829$	−46.4444	−1.61308	−0.806541	+	0.591179i	$0.798663\pi$
$830$	0	0
$831$	−43.5984	−1.51241
$832$	0	0
$833$	30.4235	1.05411
$834$	0	0
$835$	18.2346	0.631034
$836$	0	0
$837$	43.7117	1.51090
$838$	0	0
$839$	−14.7964	−0.510830	−0.255415	−	0.966832i	$-0.582212\pi$
$839$	−14.7964	−0.510830	−0.255415	+	0.966832i	$0.582212\pi$
$840$	0	0
$841$	−11.8977	−0.410266
$842$	0	0
$843$	4.66348	0.160619
$844$	0	0
$845$	−28.2166	−0.970681
$846$	0	0
$847$	0.847854	0.0291326
$848$	0	0
$849$	30.7250	1.05448
$850$	0	0
$851$	−1.47854	−0.0506835
$852$	0	0
$853$	19.3627	0.662968	0.331484	−	0.943461i	$-0.392451\pi$
$853$	19.3627	0.662968	0.331484	+	0.943461i	$0.392451\pi$
$854$	0	0
$855$	6.54865	0.223959
$856$	0	0
$857$	−10.2969	−0.351735	−0.175867	−	0.984414i	$-0.556273\pi$
$857$	−10.2969	−0.351735	−0.175867	+	0.984414i	$0.556273\pi$
$858$	0	0
$859$	24.8809	0.848926	0.424463	−	0.905445i	$-0.360463\pi$
$859$	24.8809	0.848926	0.424463	+	0.905445i	$0.360463\pi$
$860$	0	0
$861$	2.73400	0.0931743
$862$	0	0
$863$	−42.3894	−1.44295	−0.721476	−	0.692439i	$-0.756536\pi$
$863$	−42.3894	−1.44295	−0.721476	+	0.692439i	$0.756536\pi$
$864$	0	0
$865$	1.23474	0.0419823
$866$	0	0
$867$	−3.09678	−0.105172
$868$	0	0
$869$	14.9967	0.508727
$870$	0	0
$871$	−0.699583	−0.0237045
$872$	0	0
$873$	−0.689651	−0.0233412
$874$	0	0
$875$	−2.02825	−0.0685673
$876$	0	0
$877$	33.9838	1.14755	0.573776	−	0.819012i	$-0.305478\pi$
$877$	33.9838	1.14755	0.573776	+	0.819012i	$0.305478\pi$
$878$	0	0
$879$	−25.8459	−0.871760
$880$	0	0
$881$	−29.1509	−0.982118	−0.491059	−	0.871126i	$-0.663390\pi$
$881$	−29.1509	−0.982118	−0.491059	+	0.871126i	$0.663390\pi$
$882$	0	0
$883$	−10.6323	−0.357806	−0.178903	−	0.983867i	$-0.557255\pi$
$883$	−10.6323	−0.357806	−0.178903	+	0.983867i	$0.557255\pi$
$884$	0	0
$885$	8.08035	0.271618
$886$	0	0
$887$	30.3149	1.01788	0.508938	−	0.860803i	$-0.330038\pi$
$887$	30.3149	1.01788	0.508938	+	0.860803i	$0.330038\pi$
$888$	0	0
$889$	−0.129583	−0.00434607
$890$	0	0
$891$	24.1266	0.808271
$892$	0	0
$893$	34.9811	1.17060
$894$	0	0
$895$	33.5653	1.12196
$896$	0	0
$897$	3.16716	0.105748
$898$	0	0
$899$	32.1116	1.07098
$900$	0	0
$901$	4.92038	0.163922
$902$	0	0
$903$	2.16819	0.0721527
$904$	0	0
$905$	41.2278	1.37046
$906$	0	0
$907$	49.5586	1.64557	0.822783	−	0.568355i	$-0.192420\pi$
$907$	49.5586	1.64557	0.822783	+	0.568355i	$0.192420\pi$
$908$	0	0
$909$	9.63557	0.319592
$910$	0	0
$911$	55.2322	1.82993	0.914963	−	0.403538i	$-0.132220\pi$
$911$	55.2322	1.82993	0.914963	+	0.403538i	$0.132220\pi$
$912$	0	0
$913$	−45.5054	−1.50601
$914$	0	0
$915$	3.37944	0.111721
$916$	0	0
$917$	1.92809	0.0636711
$918$	0	0
$919$	47.1670	1.55590	0.777948	−	0.628329i	$-0.216261\pi$
$919$	47.1670	1.55590	0.777948	+	0.628329i	$0.216261\pi$
$920$	0	0
$921$	4.46266	0.147050
$922$	0	0
$923$	2.28419	0.0751850
$924$	0	0
$925$	0.0593616	0.00195180
$926$	0	0
$927$	−2.87834	−0.0945371
$928$	0	0
$929$	−55.8536	−1.83250	−0.916248	−	0.400612i	$-0.868798\pi$
$929$	−55.8536	−1.83250	−0.916248	+	0.400612i	$0.868798\pi$
$930$	0	0
$931$	26.2506	0.860328
$932$	0	0
$933$	3.76772	0.123349
$934$	0	0
$935$	39.0419	1.27681
$936$	0	0
$937$	−57.9031	−1.89161	−0.945805	−	0.324734i	$-0.894725\pi$
$937$	−57.9031	−1.89161	−0.945805	+	0.324734i	$0.894725\pi$
$938$	0	0
$939$	−28.2420	−0.921642
$940$	0	0
$941$	12.5785	0.410048	0.205024	−	0.978757i	$-0.434273\pi$
$941$	12.5785	0.410048	0.205024	+	0.978757i	$0.434273\pi$
$942$	0	0
$943$	29.1163	0.948158
$944$	0	0
$945$	2.33842	0.0760689
$946$	0	0
$947$	−30.3312	−0.985630	−0.492815	−	0.870134i	$-0.664032\pi$
$947$	−30.3312	−0.985630	−0.492815	+	0.870134i	$0.664032\pi$
$948$	0	0
$949$	−4.49005	−0.145753
$950$	0	0
$951$	37.6616	1.22126
$952$	0	0
$953$	−32.7046	−1.05941	−0.529704	−	0.848183i	$-0.677697\pi$
$953$	−32.7046	−1.05941	−0.529704	+	0.848183i	$0.677697\pi$
$954$	0	0
$955$	−7.01915	−0.227134
$956$	0	0
$957$	24.4148	0.789219
$958$	0	0
$959$	−3.02367	−0.0976393
$960$	0	0
$961$	29.2936	0.944954
$962$	0	0
$963$	−5.73236	−0.184723
$964$	0	0
$965$	42.4225	1.36563
$966$	0	0
$967$	−3.84297	−0.123582	−0.0617908	−	0.998089i	$-0.519681\pi$
$967$	−3.84297	−0.123582	−0.0617908	+	0.998089i	$0.519681\pi$
$968$	0	0
$969$	−24.5850	−0.789785
$970$	0	0
$971$	5.24729	0.168393	0.0841967	−	0.996449i	$-0.473168\pi$
$971$	5.24729	0.168393	0.0841967	+	0.996449i	$0.473168\pi$
$972$	0	0
$973$	−1.49432	−0.0479057
$974$	0	0
$975$	−0.127158	−0.00407232
$976$	0	0
$977$	32.6077	1.04321	0.521607	−	0.853186i	$-0.325333\pi$
$977$	32.6077	1.04321	0.521607	+	0.853186i	$0.325333\pi$
$978$	0	0
$979$	39.7587	1.27069
$980$	0	0
$981$	−11.5863	−0.369923
$982$	0	0
$983$	0.549168	0.0175157	0.00875786	−	0.999962i	$-0.497212\pi$
$983$	0.549168	0.0175157	0.00875786	+	0.999962i	$0.497212\pi$
$984$	0	0
$985$	11.3583	0.361905
$986$	0	0
$987$	2.54660	0.0810592
$988$	0	0
$989$	23.0906	0.734239
$990$	0	0
$991$	−59.3669	−1.88585	−0.942927	−	0.333001i	$-0.891939\pi$
$991$	−59.3669	−1.88585	−0.942927	+	0.333001i	$0.891939\pi$
$992$	0	0
$993$	−4.26446	−0.135329
$994$	0	0
$995$	−61.9634	−1.96437
$996$	0	0
$997$	−43.4318	−1.37550	−0.687750	−	0.725948i	$-0.741401\pi$
$997$	−43.4318	−1.37550	−0.687750	+	0.725948i	$0.741401\pi$
$998$	0	0
$999$	2.84899	0.0901379

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	976.2.a.i.1.2		4
3.2	odd	2		8784.2.a.bv.1.1		4
4.3	odd	2		488.2.a.c.1.3	✓	4
8.3	odd	2		3904.2.a.y.1.2		4
8.5	even	2		3904.2.a.bf.1.3		4
12.11	even	2		4392.2.a.n.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
488.2.a.c.1.3	✓	4	4.3	odd	2
976.2.a.i.1.2		4	1.1	even	1	trivial
3904.2.a.y.1.2		4	8.3	odd	2
3904.2.a.bf.1.3		4	8.5	even	2
4392.2.a.n.1.1		4	12.11	even	2
8784.2.a.bv.1.1		4	3.2	odd	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:	\( N \)	\(=\)	\( 976 = 2^{4} \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	976.a (trivial)

\(f(q)\)	\(=\)	\(q-1.49391 q^{3} +2.26214 q^{5} +0.183629 q^{7} -0.768234 q^{9} +O(q^{10})\)	\(q-1.49391 q^{3} +2.26214 q^{5} +0.183629 q^{7} -0.768234 q^{9} -3.95186 q^{11} +0.725675 q^{13} -3.37944 q^{15} -4.36726 q^{17} -3.76823 q^{19} -0.274325 q^{21} -2.92148 q^{23} +0.117294 q^{25} +5.62940 q^{27} +4.13549 q^{29} +7.76489 q^{31} +5.90373 q^{33} +0.415394 q^{35} +0.506090 q^{37} -1.08409 q^{39} -9.96628 q^{41} -7.90373 q^{43} -1.73786 q^{45} -9.28316 q^{47} -6.96628 q^{49} +6.52429 q^{51} -1.12665 q^{53} -8.93968 q^{55} +5.62940 q^{57} -2.39103 q^{59} -1.00000 q^{61} -0.141070 q^{63} +1.64158 q^{65} -0.964044 q^{67} +4.36443 q^{69} +3.14767 q^{71} -6.18741 q^{73} -0.175227 q^{75} -0.725675 q^{77} -3.79483 q^{79} -6.10511 q^{81} +11.5149 q^{83} -9.87936 q^{85} -6.17805 q^{87} -10.0608 q^{89} +0.133255 q^{91} -11.6000 q^{93} -8.52429 q^{95} +0.897710 q^{97} +3.03596 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} - 2 q^{5} - 7 q^{7} + 3 q^{9}+O(q^{10}) \) 4 * q - q^3 - 2 * q^5 - 7 * q^7 + 3 * q^9	\( 4 q - q^{3} - 2 q^{5} - 7 q^{7} + 3 q^{9} - 2 q^{11} + 4 q^{13} - 8 q^{15} - 2 q^{17} - 9 q^{19} - 15 q^{23} + 6 q^{25} - 4 q^{27} - 5 q^{29} - 17 q^{31} - 4 q^{33} + 7 q^{37} - 22 q^{39} - 15 q^{41} - 4 q^{43} - 18 q^{45} - 4 q^{47} - 3 q^{49} + 4 q^{51} - 15 q^{53} - 12 q^{55} - 4 q^{57} + 12 q^{59} - 4 q^{61} - 10 q^{65} - 3 q^{69} + q^{71} - q^{73} + 33 q^{75} - 4 q^{77} - 8 q^{79} - 20 q^{81} + 19 q^{83} + 8 q^{85} + 4 q^{87} - 6 q^{89} - 5 q^{93} - 12 q^{95} + 13 q^{97} + 16 q^{99}+O(q^{100}) \) 4 * q - q^3 - 2 * q^5 - 7 * q^7 + 3 * q^9 - 2 * q^11 + 4 * q^13 - 8 * q^15 - 2 * q^17 - 9 * q^19 - 15 * q^23 + 6 * q^25 - 4 * q^27 - 5 * q^29 - 17 * q^31 - 4 * q^33 + 7 * q^37 - 22 * q^39 - 15 * q^41 - 4 * q^43 - 18 * q^45 - 4 * q^47 - 3 * q^49 + 4 * q^51 - 15 * q^53 - 12 * q^55 - 4 * q^57 + 12 * q^59 - 4 * q^61 - 10 * q^65 - 3 * q^69 + q^71 - q^73 + 33 * q^75 - 4 * q^77 - 8 * q^79 - 20 * q^81 + 19 * q^83 + 8 * q^85 + 4 * q^87 - 6 * q^89 - 5 * q^93 - 12 * q^95 + 13 * q^97 + 16 * q^99

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