LMFDB - Embedded newform 9576.2.a.be.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9576,2,Mod(1,9576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9576, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("9576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9576.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:	$76.4647449756$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	9576.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.23607 q^{5} -1.00000 q^{7} +O(q^{10})$	$q+1.23607 q^{5} -1.00000 q^{7} -4.47214 q^{11} -4.47214 q^{13} -1.23607 q^{17} +1.00000 q^{19} -0.472136 q^{23} -3.47214 q^{25} +7.23607 q^{29} +6.47214 q^{31} -1.23607 q^{35} -4.47214 q^{37} -2.00000 q^{41} +8.00000 q^{43} -3.23607 q^{47} +1.00000 q^{49} -3.23607 q^{53} -5.52786 q^{55} -4.00000 q^{59} -3.52786 q^{61} -5.52786 q^{65} -1.52786 q^{67} +5.23607 q^{71} +4.47214 q^{73} +4.47214 q^{77} -12.0000 q^{79} -5.70820 q^{83} -1.52786 q^{85} +10.9443 q^{89} +4.47214 q^{91} +1.23607 q^{95} +3.52786 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 - 2 * q^7	$ 2 q - 2 q^{5} - 2 q^{7} + 2 q^{17} + 2 q^{19} + 8 q^{23} + 2 q^{25} + 10 q^{29} + 4 q^{31} + 2 q^{35} - 4 q^{41} + 16 q^{43} - 2 q^{47} + 2 q^{49} - 2 q^{53} - 20 q^{55} - 8 q^{59} - 16 q^{61} - 20 q^{65} - 12 q^{67} + 6 q^{71} - 24 q^{79} + 2 q^{83} - 12 q^{85} + 4 q^{89} - 2 q^{95} + 16 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 2 * q^7 + 2 * q^17 + 2 * q^19 + 8 * q^23 + 2 * q^25 + 10 * q^29 + 4 * q^31 + 2 * q^35 - 4 * q^41 + 16 * q^43 - 2 * q^47 + 2 * q^49 - 2 * q^53 - 20 * q^55 - 8 * q^59 - 16 * q^61 - 20 * q^65 - 12 * q^67 + 6 * q^71 - 24 * q^79 + 2 * q^83 - 12 * q^85 + 4 * q^89 - 2 * q^95 + 16 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.23607	0.552786	0.276393	−	0.961045i	$-0.410861\pi$
$5$	1.23607	0.552786	0.276393	+	0.961045i	$0.410861\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.47214	−1.34840	−0.674200	−	0.738549i	$-0.735511\pi$
$11$	−4.47214	−1.34840	−0.674200	+	0.738549i	$0.735511\pi$
$12$	0	0
$13$	−4.47214	−1.24035	−0.620174	−	0.784465i	$-0.712938\pi$
$13$	−4.47214	−1.24035	−0.620174	+	0.784465i	$0.712938\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.23607	−0.299791	−0.149895	−	0.988702i	$-0.547894\pi$
$17$	−1.23607	−0.299791	−0.149895	+	0.988702i	$0.547894\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.472136	−0.0984472	−0.0492236	−	0.998788i	$-0.515675\pi$
$23$	−0.472136	−0.0984472	−0.0492236	+	0.998788i	$0.515675\pi$
$24$	0	0
$25$	−3.47214	−0.694427
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.23607	1.34370	0.671852	−	0.740685i	$-0.265499\pi$
$29$	7.23607	1.34370	0.671852	+	0.740685i	$0.265499\pi$
$30$	0	0
$31$	6.47214	1.16243	0.581215	−	0.813750i	$-0.302578\pi$
$31$	6.47214	1.16243	0.581215	+	0.813750i	$0.302578\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.23607	−0.208934
$36$	0	0
$37$	−4.47214	−0.735215	−0.367607	−	0.929981i	$-0.619823\pi$
$37$	−4.47214	−0.735215	−0.367607	+	0.929981i	$0.619823\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.23607	−0.472029	−0.236015	−	0.971750i	$-0.575841\pi$
$47$	−3.23607	−0.472029	−0.236015	+	0.971750i	$0.575841\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−3.23607	−0.444508	−0.222254	−	0.974989i	$-0.571341\pi$
$53$	−3.23607	−0.444508	−0.222254	+	0.974989i	$0.571341\pi$
$54$	0	0
$55$	−5.52786	−0.745377
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−3.52786	−0.451697	−0.225848	−	0.974162i	$-0.572515\pi$
$61$	−3.52786	−0.451697	−0.225848	+	0.974162i	$0.572515\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5.52786	−0.685647
$66$	0	0
$67$	−1.52786	−0.186658	−0.0933292	−	0.995635i	$-0.529751\pi$
$67$	−1.52786	−0.186658	−0.0933292	+	0.995635i	$0.529751\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.23607	0.621407	0.310703	−	0.950507i	$-0.399435\pi$
$71$	5.23607	0.621407	0.310703	+	0.950507i	$0.399435\pi$
$72$	0	0
$73$	4.47214	0.523424	0.261712	−	0.965146i	$-0.415713\pi$
$73$	4.47214	0.523424	0.261712	+	0.965146i	$0.415713\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.47214	0.509647
$78$	0	0
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−5.70820	−0.626557	−0.313278	−	0.949661i	$-0.601427\pi$
$83$	−5.70820	−0.626557	−0.313278	+	0.949661i	$0.601427\pi$
$84$	0	0
$85$	−1.52786	−0.165720
$86$	0	0
$87$	0	0
$88$	0	0
$89$	10.9443	1.16009	0.580045	−	0.814584i	$-0.303035\pi$
$89$	10.9443	1.16009	0.580045	+	0.814584i	$0.303035\pi$
$90$	0	0
$91$	4.47214	0.468807
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.23607	0.126818
$96$	0	0
$97$	3.52786	0.358200	0.179100	−	0.983831i	$-0.442681\pi$
$97$	3.52786	0.358200	0.179100	+	0.983831i	$0.442681\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	18.1803	1.80901	0.904506	−	0.426461i	$-0.140240\pi$
$101$	18.1803	1.80901	0.904506	+	0.426461i	$0.140240\pi$
$102$	0	0
$103$	−8.94427	−0.881305	−0.440653	−	0.897678i	$-0.645253\pi$
$103$	−8.94427	−0.881305	−0.440653	+	0.897678i	$0.645253\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.70820	−0.745180	−0.372590	−	0.927996i	$-0.621530\pi$
$107$	−7.70820	−0.745180	−0.372590	+	0.927996i	$0.621530\pi$
$108$	0	0
$109$	14.9443	1.43140	0.715701	−	0.698407i	$-0.246107\pi$
$109$	14.9443	1.43140	0.715701	+	0.698407i	$0.246107\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	15.2361	1.43329	0.716644	−	0.697439i	$-0.245677\pi$
$113$	15.2361	1.43329	0.716644	+	0.697439i	$0.245677\pi$
$114$	0	0
$115$	−0.583592	−0.0544202
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.23607	0.113310
$120$	0	0
$121$	9.00000	0.818182
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−10.4721	−0.936656
$126$	0	0
$127$	−10.4721	−0.929252	−0.464626	−	0.885507i	$-0.653811\pi$
$127$	−10.4721	−0.929252	−0.464626	+	0.885507i	$0.653811\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	8.76393	0.765708	0.382854	−	0.923809i	$-0.374941\pi$
$131$	8.76393	0.765708	0.382854	+	0.923809i	$0.374941\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−0.944272	−0.0806746	−0.0403373	−	0.999186i	$-0.512843\pi$
$137$	−0.944272	−0.0806746	−0.0403373	+	0.999186i	$0.512843\pi$
$138$	0	0
$139$	−1.52786	−0.129592	−0.0647959	−	0.997899i	$-0.520640\pi$
$139$	−1.52786	−0.129592	−0.0647959	+	0.997899i	$0.520640\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	20.0000	1.67248
$144$	0	0
$145$	8.94427	0.742781
$146$	0	0
$147$	0	0
$148$	0	0
$149$	10.4721	0.857911	0.428955	−	0.903326i	$-0.358882\pi$
$149$	10.4721	0.857911	0.428955	+	0.903326i	$0.358882\pi$
$150$	0	0
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	8.00000	0.642575
$156$	0	0
$157$	22.3607	1.78458	0.892288	−	0.451466i	$-0.149099\pi$
$157$	22.3607	1.78458	0.892288	+	0.451466i	$0.149099\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0.472136	0.0372095
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.47214	−0.191300	−0.0956498	−	0.995415i	$-0.530493\pi$
$167$	−2.47214	−0.191300	−0.0956498	+	0.995415i	$0.530493\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	0	0
$171$	0	0
$172$	0	0
$173$	6.94427	0.527963	0.263982	−	0.964528i	$-0.414964\pi$
$173$	6.94427	0.527963	0.263982	+	0.964528i	$0.414964\pi$
$174$	0	0
$175$	3.47214	0.262469
$176$	0	0
$177$	0	0
$178$	0	0
$179$	7.70820	0.576138	0.288069	−	0.957610i	$-0.406987\pi$
$179$	7.70820	0.576138	0.288069	+	0.957610i	$0.406987\pi$
$180$	0	0
$181$	4.47214	0.332411	0.166206	−	0.986091i	$-0.446848\pi$
$181$	4.47214	0.332411	0.166206	+	0.986091i	$0.446848\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−5.52786	−0.406417
$186$	0	0
$187$	5.52786	0.404237
$188$	0	0
$189$	0	0
$190$	0	0
$191$	24.4721	1.77074	0.885371	−	0.464886i	$-0.153905\pi$
$191$	24.4721	1.77074	0.885371	+	0.464886i	$0.153905\pi$
$192$	0	0
$193$	−0.472136	−0.0339851	−0.0169925	−	0.999856i	$-0.505409\pi$
$193$	−0.472136	−0.0339851	−0.0169925	+	0.999856i	$0.505409\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.00000	0.284988	0.142494	−	0.989796i	$-0.454488\pi$
$197$	4.00000	0.284988	0.142494	+	0.989796i	$0.454488\pi$
$198$	0	0
$199$	4.94427	0.350490	0.175245	−	0.984525i	$-0.443928\pi$
$199$	4.94427	0.350490	0.175245	+	0.984525i	$0.443928\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−7.23607	−0.507872
$204$	0	0
$205$	−2.47214	−0.172661
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.47214	−0.309344
$210$	0	0
$211$	−6.47214	−0.445560	−0.222780	−	0.974869i	$-0.571513\pi$
$211$	−6.47214	−0.445560	−0.222780	+	0.974869i	$0.571513\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	9.88854	0.674393
$216$	0	0
$217$	−6.47214	−0.439357
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5.52786	0.371844
$222$	0	0
$223$	−23.4164	−1.56808	−0.784039	−	0.620711i	$-0.786844\pi$
$223$	−23.4164	−1.56808	−0.784039	+	0.620711i	$0.786844\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−15.4164	−1.02322	−0.511611	−	0.859217i	$-0.670951\pi$
$227$	−15.4164	−1.02322	−0.511611	+	0.859217i	$0.670951\pi$
$228$	0	0
$229$	22.9443	1.51620	0.758100	−	0.652138i	$-0.226128\pi$
$229$	22.9443	1.51620	0.758100	+	0.652138i	$0.226128\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.47214	−0.161955	−0.0809775	−	0.996716i	$-0.525804\pi$
$233$	−2.47214	−0.161955	−0.0809775	+	0.996716i	$0.525804\pi$
$234$	0	0
$235$	−4.00000	−0.260931
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−5.05573	−0.327028	−0.163514	−	0.986541i	$-0.552283\pi$
$239$	−5.05573	−0.327028	−0.163514	+	0.986541i	$0.552283\pi$
$240$	0	0
$241$	0.472136	0.0304130	0.0152065	−	0.999884i	$-0.495159\pi$
$241$	0.472136	0.0304130	0.0152065	+	0.999884i	$0.495159\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.23607	0.0789695
$246$	0	0
$247$	−4.47214	−0.284555
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−0.180340	−0.0113830	−0.00569148	−	0.999984i	$-0.501812\pi$
$251$	−0.180340	−0.0113830	−0.00569148	+	0.999984i	$0.501812\pi$
$252$	0	0
$253$	2.11146	0.132746
$254$	0	0
$255$	0	0
$256$	0	0
$257$	12.4721	0.777990	0.388995	−	0.921240i	$-0.372822\pi$
$257$	12.4721	0.777990	0.388995	+	0.921240i	$0.372822\pi$
$258$	0	0
$259$	4.47214	0.277885
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2.94427	0.181552	0.0907758	−	0.995871i	$-0.471065\pi$
$263$	2.94427	0.181552	0.0907758	+	0.995871i	$0.471065\pi$
$264$	0	0
$265$	−4.00000	−0.245718
$266$	0	0
$267$	0	0
$268$	0	0
$269$	15.5279	0.946751	0.473375	−	0.880861i	$-0.343035\pi$
$269$	15.5279	0.946751	0.473375	+	0.880861i	$0.343035\pi$
$270$	0	0
$271$	−5.52786	−0.335794	−0.167897	−	0.985805i	$-0.553698\pi$
$271$	−5.52786	−0.335794	−0.167897	+	0.985805i	$0.553698\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	15.5279	0.936365
$276$	0	0
$277$	−4.47214	−0.268705	−0.134352	−	0.990934i	$-0.542895\pi$
$277$	−4.47214	−0.268705	−0.134352	+	0.990934i	$0.542895\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	8.18034	0.487998	0.243999	−	0.969775i	$-0.421541\pi$
$281$	8.18034	0.487998	0.243999	+	0.969775i	$0.421541\pi$
$282$	0	0
$283$	−7.05573	−0.419419	−0.209710	−	0.977764i	$-0.567252\pi$
$283$	−7.05573	−0.419419	−0.209710	+	0.977764i	$0.567252\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.00000	0.118056
$288$	0	0
$289$	−15.4721	−0.910126
$290$	0	0
$291$	0	0
$292$	0	0
$293$	26.0000	1.51894	0.759468	−	0.650545i	$-0.225459\pi$
$293$	26.0000	1.51894	0.759468	+	0.650545i	$0.225459\pi$
$294$	0	0
$295$	−4.94427	−0.287867
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.11146	0.122109
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−4.36068	−0.249692
$306$	0	0
$307$	17.5279	1.00037	0.500184	−	0.865919i	$-0.333266\pi$
$307$	17.5279	1.00037	0.500184	+	0.865919i	$0.333266\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−16.1803	−0.917503	−0.458751	−	0.888565i	$-0.651703\pi$
$311$	−16.1803	−0.917503	−0.458751	+	0.888565i	$0.651703\pi$
$312$	0	0
$313$	3.52786	0.199407	0.0997033	−	0.995017i	$-0.468211\pi$
$313$	3.52786	0.199407	0.0997033	+	0.995017i	$0.468211\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.1246	0.737152	0.368576	−	0.929598i	$-0.379845\pi$
$317$	13.1246	0.737152	0.368576	+	0.929598i	$0.379845\pi$
$318$	0	0
$319$	−32.3607	−1.81185
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.23607	−0.0687767
$324$	0	0
$325$	15.5279	0.861331
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.23607	0.178410
$330$	0	0
$331$	−9.88854	−0.543524	−0.271762	−	0.962365i	$-0.587606\pi$
$331$	−9.88854	−0.543524	−0.271762	+	0.962365i	$0.587606\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.88854	−0.103182
$336$	0	0
$337$	2.94427	0.160385	0.0801924	−	0.996779i	$-0.474447\pi$
$337$	2.94427	0.160385	0.0801924	+	0.996779i	$0.474447\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−28.9443	−1.56742
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	30.0000	1.61048	0.805242	−	0.592946i	$-0.202035\pi$
$347$	30.0000	1.61048	0.805242	+	0.592946i	$0.202035\pi$
$348$	0	0
$349$	−24.8328	−1.32927	−0.664635	−	0.747168i	$-0.731413\pi$
$349$	−24.8328	−1.32927	−0.664635	+	0.747168i	$0.731413\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	27.7082	1.47476	0.737379	−	0.675479i	$-0.236063\pi$
$353$	27.7082	1.47476	0.737379	+	0.675479i	$0.236063\pi$
$354$	0	0
$355$	6.47214	0.343505
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.58359	−0.136357	−0.0681784	−	0.997673i	$-0.521719\pi$
$359$	−2.58359	−0.136357	−0.0681784	+	0.997673i	$0.521719\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	5.52786	0.289342
$366$	0	0
$367$	4.94427	0.258089	0.129044	−	0.991639i	$-0.458809\pi$
$367$	4.94427	0.258089	0.129044	+	0.991639i	$0.458809\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.23607	0.168008
$372$	0	0
$373$	−14.9443	−0.773785	−0.386893	−	0.922125i	$-0.626452\pi$
$373$	−14.9443	−0.773785	−0.386893	+	0.922125i	$0.626452\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−32.3607	−1.66666
$378$	0	0
$379$	−20.9443	−1.07583	−0.537917	−	0.842997i	$-0.680789\pi$
$379$	−20.9443	−1.07583	−0.537917	+	0.842997i	$0.680789\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.52786	−0.0780702	−0.0390351	−	0.999238i	$-0.512428\pi$
$383$	−1.52786	−0.0780702	−0.0390351	+	0.999238i	$0.512428\pi$
$384$	0	0
$385$	5.52786	0.281726
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−4.58359	−0.232397	−0.116199	−	0.993226i	$-0.537071\pi$
$389$	−4.58359	−0.232397	−0.116199	+	0.993226i	$0.537071\pi$
$390$	0	0
$391$	0.583592	0.0295135
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−14.8328	−0.746320
$396$	0	0
$397$	23.8885	1.19893	0.599466	−	0.800400i	$-0.295380\pi$
$397$	23.8885	1.19893	0.599466	+	0.800400i	$0.295380\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7.81966	0.390495	0.195248	−	0.980754i	$-0.437449\pi$
$401$	7.81966	0.390495	0.195248	+	0.980754i	$0.437449\pi$
$402$	0	0
$403$	−28.9443	−1.44182
$404$	0	0
$405$	0	0
$406$	0	0
$407$	20.0000	0.991363
$408$	0	0
$409$	22.3607	1.10566	0.552832	−	0.833293i	$-0.313547\pi$
$409$	22.3607	1.10566	0.552832	+	0.833293i	$0.313547\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.00000	0.196827
$414$	0	0
$415$	−7.05573	−0.346352
$416$	0	0
$417$	0	0
$418$	0	0
$419$	21.1246	1.03200	0.516002	−	0.856587i	$-0.327420\pi$
$419$	21.1246	1.03200	0.516002	+	0.856587i	$0.327420\pi$
$420$	0	0
$421$	−39.3050	−1.91561	−0.957803	−	0.287425i	$-0.907201\pi$
$421$	−39.3050	−1.91561	−0.957803	+	0.287425i	$0.907201\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.29180	0.208183
$426$	0	0
$427$	3.52786	0.170725
$428$	0	0
$429$	0	0
$430$	0	0
$431$	36.0689	1.73738	0.868688	−	0.495359i	$-0.164963\pi$
$431$	36.0689	1.73738	0.868688	+	0.495359i	$0.164963\pi$
$432$	0	0
$433$	−0.472136	−0.0226894	−0.0113447	−	0.999936i	$-0.503611\pi$
$433$	−0.472136	−0.0226894	−0.0113447	+	0.999936i	$0.503611\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−0.472136	−0.0225853
$438$	0	0
$439$	−39.4164	−1.88124	−0.940621	−	0.339458i	$-0.889756\pi$
$439$	−39.4164	−1.88124	−0.940621	+	0.339458i	$0.889756\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	15.8885	0.754887	0.377444	−	0.926033i	$-0.376803\pi$
$443$	15.8885	0.754887	0.377444	+	0.926033i	$0.376803\pi$
$444$	0	0
$445$	13.5279	0.641282
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−15.2361	−0.719035	−0.359517	−	0.933138i	$-0.617059\pi$
$449$	−15.2361	−0.719035	−0.359517	+	0.933138i	$0.617059\pi$
$450$	0	0
$451$	8.94427	0.421169
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5.52786	0.259150
$456$	0	0
$457$	21.4164	1.00182	0.500909	−	0.865500i	$-0.332999\pi$
$457$	21.4164	1.00182	0.500909	+	0.865500i	$0.332999\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	5.23607	0.243868	0.121934	−	0.992538i	$-0.461090\pi$
$461$	5.23607	0.243868	0.121934	+	0.992538i	$0.461090\pi$
$462$	0	0
$463$	−25.8885	−1.20314	−0.601571	−	0.798819i	$-0.705458\pi$
$463$	−25.8885	−1.20314	−0.601571	+	0.798819i	$0.705458\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−24.1803	−1.11893	−0.559466	−	0.828853i	$-0.688994\pi$
$467$	−24.1803	−1.11893	−0.559466	+	0.828853i	$0.688994\pi$
$468$	0	0
$469$	1.52786	0.0705502
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−35.7771	−1.64503
$474$	0	0
$475$	−3.47214	−0.159313
$476$	0	0
$477$	0	0
$478$	0	0
$479$	21.1246	0.965208	0.482604	−	0.875839i	$-0.339691\pi$
$479$	21.1246	0.965208	0.482604	+	0.875839i	$0.339691\pi$
$480$	0	0
$481$	20.0000	0.911922
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.36068	0.198008
$486$	0	0
$487$	23.4164	1.06110	0.530549	−	0.847654i	$-0.321986\pi$
$487$	23.4164	1.06110	0.530549	+	0.847654i	$0.321986\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	30.0000	1.35388	0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000	1.35388	0.676941	+	0.736038i	$0.263305\pi$
$492$	0	0
$493$	−8.94427	−0.402830
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−5.23607	−0.234870
$498$	0	0
$499$	37.8885	1.69612	0.848062	−	0.529897i	$-0.177769\pi$
$499$	37.8885	1.69612	0.848062	+	0.529897i	$0.177769\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−28.1803	−1.25650	−0.628250	−	0.778012i	$-0.716228\pi$
$503$	−28.1803	−1.25650	−0.628250	+	0.778012i	$0.716228\pi$
$504$	0	0
$505$	22.4721	0.999997
$506$	0	0
$507$	0	0
$508$	0	0
$509$	31.5279	1.39745	0.698724	−	0.715391i	$-0.253752\pi$
$509$	31.5279	1.39745	0.698724	+	0.715391i	$0.253752\pi$
$510$	0	0
$511$	−4.47214	−0.197836
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−11.0557	−0.487174
$516$	0	0
$517$	14.4721	0.636484
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.41641	−0.0620540	−0.0310270	−	0.999519i	$-0.509878\pi$
$521$	−1.41641	−0.0620540	−0.0310270	+	0.999519i	$0.509878\pi$
$522$	0	0
$523$	−21.5279	−0.941348	−0.470674	−	0.882307i	$-0.655989\pi$
$523$	−21.5279	−0.941348	−0.470674	+	0.882307i	$0.655989\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.00000	−0.348485
$528$	0	0
$529$	−22.7771	−0.990308
$530$	0	0
$531$	0	0
$532$	0	0
$533$	8.94427	0.387419
$534$	0	0
$535$	−9.52786	−0.411925
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−4.47214	−0.192629
$540$	0	0
$541$	−35.8885	−1.54297	−0.771485	−	0.636248i	$-0.780485\pi$
$541$	−35.8885	−1.54297	−0.771485	+	0.636248i	$0.780485\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	18.4721	0.791259
$546$	0	0
$547$	−14.8328	−0.634205	−0.317103	−	0.948391i	$-0.602710\pi$
$547$	−14.8328	−0.634205	−0.317103	+	0.948391i	$0.602710\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	7.23607	0.308267
$552$	0	0
$553$	12.0000	0.510292
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0.583592	0.0247276	0.0123638	−	0.999924i	$-0.496064\pi$
$557$	0.583592	0.0247276	0.0123638	+	0.999924i	$0.496064\pi$
$558$	0	0
$559$	−35.7771	−1.51321
$560$	0	0
$561$	0	0
$562$	0	0
$563$	23.4164	0.986884	0.493442	−	0.869779i	$-0.335739\pi$
$563$	23.4164	0.986884	0.493442	+	0.869779i	$0.335739\pi$
$564$	0	0
$565$	18.8328	0.792303
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−25.7082	−1.07774	−0.538872	−	0.842388i	$-0.681149\pi$
$569$	−25.7082	−1.07774	−0.538872	+	0.842388i	$0.681149\pi$
$570$	0	0
$571$	5.88854	0.246428	0.123214	−	0.992380i	$-0.460680\pi$
$571$	5.88854	0.246428	0.123214	+	0.992380i	$0.460680\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.63932	0.0683644
$576$	0	0
$577$	−25.4164	−1.05810	−0.529049	−	0.848591i	$-0.677451\pi$
$577$	−25.4164	−1.05810	−0.529049	+	0.848591i	$0.677451\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	5.70820	0.236816
$582$	0	0
$583$	14.4721	0.599375
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−22.2918	−0.920081	−0.460040	−	0.887898i	$-0.652165\pi$
$587$	−22.2918	−0.920081	−0.460040	+	0.887898i	$0.652165\pi$
$588$	0	0
$589$	6.47214	0.266680
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−39.7082	−1.63062	−0.815310	−	0.579024i	$-0.803434\pi$
$593$	−39.7082	−1.63062	−0.815310	+	0.579024i	$0.803434\pi$
$594$	0	0
$595$	1.52786	0.0626363
$596$	0	0
$597$	0	0
$598$	0	0
$599$	21.5967	0.882419	0.441210	−	0.897404i	$-0.354550\pi$
$599$	21.5967	0.882419	0.441210	+	0.897404i	$0.354550\pi$
$600$	0	0
$601$	18.5836	0.758041	0.379020	−	0.925388i	$-0.376261\pi$
$601$	18.5836	0.758041	0.379020	+	0.925388i	$0.376261\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	11.1246	0.452280
$606$	0	0
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	14.4721	0.585480
$612$	0	0
$613$	10.3607	0.418464	0.209232	−	0.977866i	$-0.432904\pi$
$613$	10.3607	0.418464	0.209232	+	0.977866i	$0.432904\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	48.9443	1.97042	0.985211	−	0.171345i	$-0.0548113\pi$
$617$	48.9443	1.97042	0.985211	+	0.171345i	$0.0548113\pi$
$618$	0	0
$619$	4.58359	0.184230	0.0921151	−	0.995748i	$-0.470637\pi$
$619$	4.58359	0.184230	0.0921151	+	0.995748i	$0.470637\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−10.9443	−0.438473
$624$	0	0
$625$	4.41641	0.176656
$626$	0	0
$627$	0	0
$628$	0	0
$629$	5.52786	0.220410
$630$	0	0
$631$	−16.9443	−0.674541	−0.337270	−	0.941408i	$-0.609504\pi$
$631$	−16.9443	−0.674541	−0.337270	+	0.941408i	$0.609504\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−12.9443	−0.513678
$636$	0	0
$637$	−4.47214	−0.177192
$638$	0	0
$639$	0	0
$640$	0	0
$641$	33.7082	1.33139	0.665697	−	0.746222i	$-0.268134\pi$
$641$	33.7082	1.33139	0.665697	+	0.746222i	$0.268134\pi$
$642$	0	0
$643$	15.0557	0.593740	0.296870	−	0.954918i	$-0.404057\pi$
$643$	15.0557	0.593740	0.296870	+	0.954918i	$0.404057\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.65248	0.261536	0.130768	−	0.991413i	$-0.458256\pi$
$647$	6.65248	0.261536	0.130768	+	0.991413i	$0.458256\pi$
$648$	0	0
$649$	17.8885	0.702187
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.11146	0.239160	0.119580	−	0.992825i	$-0.461845\pi$
$653$	6.11146	0.239160	0.119580	+	0.992825i	$0.461845\pi$
$654$	0	0
$655$	10.8328	0.423273
$656$	0	0
$657$	0	0
$658$	0	0
$659$	43.7082	1.70263	0.851315	−	0.524655i	$-0.175806\pi$
$659$	43.7082	1.70263	0.851315	+	0.524655i	$0.175806\pi$
$660$	0	0
$661$	25.4164	0.988584	0.494292	−	0.869296i	$-0.335427\pi$
$661$	25.4164	0.988584	0.494292	+	0.869296i	$0.335427\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.23607	−0.0479327
$666$	0	0
$667$	−3.41641	−0.132284
$668$	0	0
$669$	0	0
$670$	0	0
$671$	15.7771	0.609068
$672$	0	0
$673$	−36.2492	−1.39730	−0.698652	−	0.715461i	$-0.746217\pi$
$673$	−36.2492	−1.39730	−0.698652	+	0.715461i	$0.746217\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−9.41641	−0.361902	−0.180951	−	0.983492i	$-0.557918\pi$
$677$	−9.41641	−0.361902	−0.180951	+	0.983492i	$0.557918\pi$
$678$	0	0
$679$	−3.52786	−0.135387
$680$	0	0
$681$	0	0
$682$	0	0
$683$	18.7639	0.717982	0.358991	−	0.933341i	$-0.383121\pi$
$683$	18.7639	0.717982	0.358991	+	0.933341i	$0.383121\pi$
$684$	0	0
$685$	−1.16718	−0.0445958
$686$	0	0
$687$	0	0
$688$	0	0
$689$	14.4721	0.551344
$690$	0	0
$691$	22.4721	0.854880	0.427440	−	0.904044i	$-0.359415\pi$
$691$	22.4721	0.854880	0.427440	+	0.904044i	$0.359415\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.88854	−0.0716366
$696$	0	0
$697$	2.47214	0.0936388
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−14.8328	−0.560228	−0.280114	−	0.959967i	$-0.590372\pi$
$701$	−14.8328	−0.560228	−0.280114	+	0.959967i	$0.590372\pi$
$702$	0	0
$703$	−4.47214	−0.168670
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−18.1803	−0.683742
$708$	0	0
$709$	−36.2492	−1.36137	−0.680684	−	0.732577i	$-0.738317\pi$
$709$	−36.2492	−1.36137	−0.680684	+	0.732577i	$0.738317\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−3.05573	−0.114438
$714$	0	0
$715$	24.7214	0.924526
$716$	0	0
$717$	0	0
$718$	0	0
$719$	41.1246	1.53369	0.766845	−	0.641833i	$-0.221826\pi$
$719$	41.1246	1.53369	0.766845	+	0.641833i	$0.221826\pi$
$720$	0	0
$721$	8.94427	0.333102
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−25.1246	−0.933105
$726$	0	0
$727$	4.36068	0.161729	0.0808643	−	0.996725i	$-0.474232\pi$
$727$	4.36068	0.161729	0.0808643	+	0.996725i	$0.474232\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−9.88854	−0.365741
$732$	0	0
$733$	11.5279	0.425791	0.212896	−	0.977075i	$-0.431711\pi$
$733$	11.5279	0.425791	0.212896	+	0.977075i	$0.431711\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.83282	0.251690
$738$	0	0
$739$	17.8885	0.658041	0.329020	−	0.944323i	$-0.393282\pi$
$739$	17.8885	0.658041	0.329020	+	0.944323i	$0.393282\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−4.65248	−0.170683	−0.0853414	−	0.996352i	$-0.527198\pi$
$743$	−4.65248	−0.170683	−0.0853414	+	0.996352i	$0.527198\pi$
$744$	0	0
$745$	12.9443	0.474241
$746$	0	0
$747$	0	0
$748$	0	0
$749$	7.70820	0.281652
$750$	0	0
$751$	18.8328	0.687219	0.343610	−	0.939113i	$-0.388350\pi$
$751$	18.8328	0.687219	0.343610	+	0.939113i	$0.388350\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	24.7214	0.899702
$756$	0	0
$757$	19.8885	0.722861	0.361431	−	0.932399i	$-0.382288\pi$
$757$	19.8885	0.722861	0.361431	+	0.932399i	$0.382288\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−38.7639	−1.40519	−0.702596	−	0.711589i	$-0.747976\pi$
$761$	−38.7639	−1.40519	−0.702596	+	0.711589i	$0.747976\pi$
$762$	0	0
$763$	−14.9443	−0.541019
$764$	0	0
$765$	0	0
$766$	0	0
$767$	17.8885	0.645918
$768$	0	0
$769$	−14.3607	−0.517859	−0.258930	−	0.965896i	$-0.583370\pi$
$769$	−14.3607	−0.517859	−0.258930	+	0.965896i	$0.583370\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−31.8885	−1.14695	−0.573476	−	0.819223i	$-0.694405\pi$
$773$	−31.8885	−1.14695	−0.573476	+	0.819223i	$0.694405\pi$
$774$	0	0
$775$	−22.4721	−0.807223
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.00000	−0.0716574
$780$	0	0
$781$	−23.4164	−0.837905
$782$	0	0
$783$	0	0
$784$	0	0
$785$	27.6393	0.986490
$786$	0	0
$787$	−16.5836	−0.591141	−0.295571	−	0.955321i	$-0.595510\pi$
$787$	−16.5836	−0.591141	−0.295571	+	0.955321i	$0.595510\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−15.2361	−0.541732
$792$	0	0
$793$	15.7771	0.560261
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−25.0557	−0.887519	−0.443760	−	0.896146i	$-0.646356\pi$
$797$	−25.0557	−0.887519	−0.443760	+	0.896146i	$0.646356\pi$
$798$	0	0
$799$	4.00000	0.141510
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−20.0000	−0.705785
$804$	0	0
$805$	0.583592	0.0205689
$806$	0	0
$807$	0	0
$808$	0	0
$809$	24.0000	0.843795	0.421898	−	0.906644i	$-0.361364\pi$
$809$	24.0000	0.843795	0.421898	+	0.906644i	$0.361364\pi$
$810$	0	0
$811$	6.83282	0.239933	0.119966	−	0.992778i	$-0.461721\pi$
$811$	6.83282	0.239933	0.119966	+	0.992778i	$0.461721\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	8.00000	0.279885
$818$	0	0
$819$	0	0
$820$	0	0
$821$	28.3607	0.989795	0.494897	−	0.868951i	$-0.335206\pi$
$821$	28.3607	0.989795	0.494897	+	0.868951i	$0.335206\pi$
$822$	0	0
$823$	−42.8328	−1.49306	−0.746529	−	0.665353i	$-0.768281\pi$
$823$	−42.8328	−1.49306	−0.746529	+	0.665353i	$0.768281\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−26.7639	−0.930673	−0.465337	−	0.885134i	$-0.654067\pi$
$827$	−26.7639	−0.930673	−0.465337	+	0.885134i	$0.654067\pi$
$828$	0	0
$829$	−16.4721	−0.572101	−0.286050	−	0.958215i	$-0.592343\pi$
$829$	−16.4721	−0.572101	−0.286050	+	0.958215i	$0.592343\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.23607	−0.0428272
$834$	0	0
$835$	−3.05573	−0.105748
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−4.00000	−0.138095	−0.0690477	−	0.997613i	$-0.521996\pi$
$839$	−4.00000	−0.138095	−0.0690477	+	0.997613i	$0.521996\pi$
$840$	0	0
$841$	23.3607	0.805541
$842$	0	0
$843$	0	0
$844$	0	0
$845$	8.65248	0.297654
$846$	0	0
$847$	−9.00000	−0.309244
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.11146	0.0723798
$852$	0	0
$853$	33.7771	1.15651	0.578253	−	0.815858i	$-0.303735\pi$
$853$	33.7771	1.15651	0.578253	+	0.815858i	$0.303735\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	28.8328	0.984910	0.492455	−	0.870338i	$-0.336100\pi$
$857$	28.8328	0.984910	0.492455	+	0.870338i	$0.336100\pi$
$858$	0	0
$859$	0.944272	0.0322181	0.0161091	−	0.999870i	$-0.494872\pi$
$859$	0.944272	0.0322181	0.0161091	+	0.999870i	$0.494872\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−34.1803	−1.16351	−0.581756	−	0.813363i	$-0.697634\pi$
$863$	−34.1803	−1.16351	−0.581756	+	0.813363i	$0.697634\pi$
$864$	0	0
$865$	8.58359	0.291851
$866$	0	0
$867$	0	0
$868$	0	0
$869$	53.6656	1.82048
$870$	0	0
$871$	6.83282	0.231521
$872$	0	0
$873$	0	0
$874$	0	0
$875$	10.4721	0.354023
$876$	0	0
$877$	39.8885	1.34694	0.673470	−	0.739214i	$-0.264803\pi$
$877$	39.8885	1.34694	0.673470	+	0.739214i	$0.264803\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	11.7082	0.394459	0.197230	−	0.980357i	$-0.436806\pi$
$881$	11.7082	0.394459	0.197230	+	0.980357i	$0.436806\pi$
$882$	0	0
$883$	18.1115	0.609499	0.304750	−	0.952433i	$-0.401427\pi$
$883$	18.1115	0.609499	0.304750	+	0.952433i	$0.401427\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−3.63932	−0.122196	−0.0610982	−	0.998132i	$-0.519460\pi$
$887$	−3.63932	−0.122196	−0.0610982	+	0.998132i	$0.519460\pi$
$888$	0	0
$889$	10.4721	0.351224
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−3.23607	−0.108291
$894$	0	0
$895$	9.52786	0.318481
$896$	0	0
$897$	0	0
$898$	0	0
$899$	46.8328	1.56196
$900$	0	0
$901$	4.00000	0.133259
$902$	0	0
$903$	0	0
$904$	0	0
$905$	5.52786	0.183752
$906$	0	0
$907$	12.5836	0.417831	0.208916	−	0.977934i	$-0.433007\pi$
$907$	12.5836	0.417831	0.208916	+	0.977934i	$0.433007\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−35.1246	−1.16373	−0.581865	−	0.813285i	$-0.697677\pi$
$911$	−35.1246	−1.16373	−0.581865	+	0.813285i	$0.697677\pi$
$912$	0	0
$913$	25.5279	0.844849
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−8.76393	−0.289411
$918$	0	0
$919$	25.8885	0.853984	0.426992	−	0.904255i	$-0.359573\pi$
$919$	25.8885	0.853984	0.426992	+	0.904255i	$0.359573\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−23.4164	−0.770760
$924$	0	0
$925$	15.5279	0.510553
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−33.0132	−1.08313	−0.541563	−	0.840660i	$-0.682167\pi$
$929$	−33.0132	−1.08313	−0.541563	+	0.840660i	$0.682167\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	6.83282	0.223457
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−15.3050	−0.498927	−0.249464	−	0.968384i	$-0.580254\pi$
$941$	−15.3050	−0.498927	−0.249464	+	0.968384i	$0.580254\pi$
$942$	0	0
$943$	0.944272	0.0307497
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−28.4721	−0.925220	−0.462610	−	0.886562i	$-0.653087\pi$
$947$	−28.4721	−0.925220	−0.462610	+	0.886562i	$0.653087\pi$
$948$	0	0
$949$	−20.0000	−0.649227
$950$	0	0
$951$	0	0
$952$	0	0
$953$	52.5410	1.70197	0.850985	−	0.525190i	$-0.176006\pi$
$953$	52.5410	1.70197	0.850985	+	0.525190i	$0.176006\pi$
$954$	0	0
$955$	30.2492	0.978842
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0.944272	0.0304921
$960$	0	0
$961$	10.8885	0.351243
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−0.583592	−0.0187865
$966$	0	0
$967$	34.8328	1.12015	0.560074	−	0.828443i	$-0.310773\pi$
$967$	34.8328	1.12015	0.560074	+	0.828443i	$0.310773\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−14.8328	−0.476008	−0.238004	−	0.971264i	$-0.576493\pi$
$971$	−14.8328	−0.476008	−0.238004	+	0.971264i	$0.576493\pi$
$972$	0	0
$973$	1.52786	0.0489811
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−5.12461	−0.163951	−0.0819754	−	0.996634i	$-0.526123\pi$
$977$	−5.12461	−0.163951	−0.0819754	+	0.996634i	$0.526123\pi$
$978$	0	0
$979$	−48.9443	−1.56427
$980$	0	0
$981$	0	0
$982$	0	0
$983$	19.0557	0.607783	0.303892	−	0.952707i	$-0.401714\pi$
$983$	19.0557	0.607783	0.303892	+	0.952707i	$0.401714\pi$
$984$	0	0
$985$	4.94427	0.157538
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3.77709	−0.120104
$990$	0	0
$991$	−0.583592	−0.0185384	−0.00926921	−	0.999957i	$-0.502951\pi$
$991$	−0.583592	−0.0185384	−0.00926921	+	0.999957i	$0.502951\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	6.11146	0.193746
$996$	0	0
$997$	−10.3607	−0.328126	−0.164063	−	0.986450i	$-0.552460\pi$
$997$	−10.3607	−0.328126	−0.164063	+	0.986450i	$0.552460\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9576.2.a.be.1.2	✓	2
3.2	odd	2		9576.2.a.bq.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9576.2.a.be.1.2	✓	2	1.1	even	1	trivial
9576.2.a.bq.1.1	yes	2	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:	\( N \)	\(=\)	\( 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9576.a (trivial)

\(f(q)\)	\(=\)	\(q+1.23607 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+1.23607 q^{5} -1.00000 q^{7} -4.47214 q^{11} -4.47214 q^{13} -1.23607 q^{17} +1.00000 q^{19} -0.472136 q^{23} -3.47214 q^{25} +7.23607 q^{29} +6.47214 q^{31} -1.23607 q^{35} -4.47214 q^{37} -2.00000 q^{41} +8.00000 q^{43} -3.23607 q^{47} +1.00000 q^{49} -3.23607 q^{53} -5.52786 q^{55} -4.00000 q^{59} -3.52786 q^{61} -5.52786 q^{65} -1.52786 q^{67} +5.23607 q^{71} +4.47214 q^{73} +4.47214 q^{77} -12.0000 q^{79} -5.70820 q^{83} -1.52786 q^{85} +10.9443 q^{89} +4.47214 q^{91} +1.23607 q^{95} +3.52786 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 - 2 * q^7	\( 2 q - 2 q^{5} - 2 q^{7} + 2 q^{17} + 2 q^{19} + 8 q^{23} + 2 q^{25} + 10 q^{29} + 4 q^{31} + 2 q^{35} - 4 q^{41} + 16 q^{43} - 2 q^{47} + 2 q^{49} - 2 q^{53} - 20 q^{55} - 8 q^{59} - 16 q^{61} - 20 q^{65} - 12 q^{67} + 6 q^{71} - 24 q^{79} + 2 q^{83} - 12 q^{85} + 4 q^{89} - 2 q^{95} + 16 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 2 * q^7 + 2 * q^17 + 2 * q^19 + 8 * q^23 + 2 * q^25 + 10 * q^29 + 4 * q^31 + 2 * q^35 - 4 * q^41 + 16 * q^43 - 2 * q^47 + 2 * q^49 - 2 * q^53 - 20 * q^55 - 8 * q^59 - 16 * q^61 - 20 * q^65 - 12 * q^67 + 6 * q^71 - 24 * q^79 + 2 * q^83 - 12 * q^85 + 4 * q^89 - 2 * q^95 + 16 * q^97

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