LMFDB - Embedded newform 6468.2.a.be.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6468,2,Mod(1,6468)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6468, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6468.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6468 = 2^{2} \cdot 3 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6468.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:	$51.6472400274$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.126956032.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 18x^{4} - 18x^{3} + 44x^{2} + 56x + 7 $ x^6 - 18x^4 - 18x^3 + 44x^2 + 56x + 7
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.75500$ of defining polynomial
Character	$\chi$	$=$	6468.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -4.38468 q^{5} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -4.38468 q^{5} +1.00000 q^{9} -1.00000 q^{11} -3.17941 q^{13} -4.38468 q^{15} -5.71894 q^{17} +6.24116 q^{19} +8.93655 q^{23} +14.2254 q^{25} +1.00000 q^{27} +8.29526 q^{29} -0.636783 q^{31} -1.00000 q^{33} -8.79450 q^{37} -3.17941 q^{39} +2.64912 q^{41} -4.11021 q^{43} -4.38468 q^{45} +5.85587 q^{47} -5.71894 q^{51} -0.386160 q^{53} +4.38468 q^{55} +6.24116 q^{57} -14.1943 q^{59} +11.7029 q^{61} +13.9407 q^{65} +1.90274 q^{67} +8.93655 q^{69} -6.54402 q^{71} -13.4617 q^{73} +14.2254 q^{75} +5.74642 q^{79} +1.00000 q^{81} -10.5777 q^{83} +25.0757 q^{85} +8.29526 q^{87} +3.28512 q^{89} -0.636783 q^{93} -27.3655 q^{95} +8.13178 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} - 4 q^{5} + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^3 - 4 * q^5 + 6 * q^9	$ 6 q + 6 q^{3} - 4 q^{5} + 6 q^{9} - 6 q^{11} - 8 q^{13} - 4 q^{15} - 16 q^{17} - 4 q^{19} + 22 q^{25} + 6 q^{27} - 4 q^{29} - 6 q^{33} + 4 q^{37} - 8 q^{39} - 12 q^{41} - 8 q^{43} - 4 q^{45} + 12 q^{47} - 16 q^{51} + 4 q^{53} + 4 q^{55} - 4 q^{57} - 24 q^{59} - 16 q^{61} - 12 q^{65} + 8 q^{67} - 36 q^{71} - 36 q^{73} + 22 q^{75} + 6 q^{81} - 32 q^{83} - 20 q^{85} - 4 q^{87} - 12 q^{89} - 44 q^{95} - 16 q^{97} - 6 q^{99}+O(q^{100}) $ 6 * q + 6 * q^3 - 4 * q^5 + 6 * q^9 - 6 * q^11 - 8 * q^13 - 4 * q^15 - 16 * q^17 - 4 * q^19 + 22 * q^25 + 6 * q^27 - 4 * q^29 - 6 * q^33 + 4 * q^37 - 8 * q^39 - 12 * q^41 - 8 * q^43 - 4 * q^45 + 12 * q^47 - 16 * q^51 + 4 * q^53 + 4 * q^55 - 4 * q^57 - 24 * q^59 - 16 * q^61 - 12 * q^65 + 8 * q^67 - 36 * q^71 - 36 * q^73 + 22 * q^75 + 6 * q^81 - 32 * q^83 - 20 * q^85 - 4 * q^87 - 12 * q^89 - 44 * q^95 - 16 * q^97 - 6 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−4.38468	−1.96089	−0.980445	−	0.196793i	$-0.936947\pi$
$5$	−4.38468	−1.96089	−0.980445	+	0.196793i	$0.936947\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−3.17941	−0.881810	−0.440905	−	0.897554i	$-0.645342\pi$
$13$	−3.17941	−0.881810	−0.440905	+	0.897554i	$0.645342\pi$
$14$	0	0
$15$	−4.38468	−1.13212
$16$	0	0
$17$	−5.71894	−1.38705	−0.693523	−	0.720434i	$-0.743942\pi$
$17$	−5.71894	−1.38705	−0.693523	+	0.720434i	$0.743942\pi$
$18$	0	0
$19$	6.24116	1.43182	0.715911	−	0.698192i	$-0.246012\pi$
$19$	6.24116	1.43182	0.715911	+	0.698192i	$0.246012\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.93655	1.86340	0.931699	−	0.363230i	$-0.118326\pi$
$23$	8.93655	1.86340	0.931699	+	0.363230i	$0.118326\pi$
$24$	0	0
$25$	14.2254	2.84509
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	8.29526	1.54039	0.770196	−	0.637807i	$-0.220158\pi$
$29$	8.29526	1.54039	0.770196	+	0.637807i	$0.220158\pi$
$30$	0	0
$31$	−0.636783	−0.114370	−0.0571848	−	0.998364i	$-0.518212\pi$
$31$	−0.636783	−0.114370	−0.0571848	+	0.998364i	$0.518212\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.79450	−1.44581	−0.722904	−	0.690949i	$-0.757193\pi$
$37$	−8.79450	−1.44581	−0.722904	+	0.690949i	$0.757193\pi$
$38$	0	0
$39$	−3.17941	−0.509113
$40$	0	0
$41$	2.64912	0.413723	0.206862	−	0.978370i	$-0.433675\pi$
$41$	2.64912	0.413723	0.206862	+	0.978370i	$0.433675\pi$
$42$	0	0
$43$	−4.11021	−0.626801	−0.313400	−	0.949621i	$-0.601468\pi$
$43$	−4.11021	−0.626801	−0.313400	+	0.949621i	$0.601468\pi$
$44$	0	0
$45$	−4.38468	−0.653630
$46$	0	0
$47$	5.85587	0.854166	0.427083	−	0.904212i	$-0.359541\pi$
$47$	5.85587	0.854166	0.427083	+	0.904212i	$0.359541\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−5.71894	−0.800811
$52$	0	0
$53$	−0.386160	−0.0530431	−0.0265216	−	0.999648i	$-0.508443\pi$
$53$	−0.386160	−0.0530431	−0.0265216	+	0.999648i	$0.508443\pi$
$54$	0	0
$55$	4.38468	0.591231
$56$	0	0
$57$	6.24116	0.826662
$58$	0	0
$59$	−14.1943	−1.84794	−0.923970	−	0.382466i	$-0.875075\pi$
$59$	−14.1943	−1.84794	−0.923970	+	0.382466i	$0.875075\pi$
$60$	0	0
$61$	11.7029	1.49840	0.749201	−	0.662343i	$-0.230438\pi$
$61$	11.7029	1.49840	0.749201	+	0.662343i	$0.230438\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	13.9407	1.72913
$66$	0	0
$67$	1.90274	0.232457	0.116228	−	0.993223i	$-0.462920\pi$
$67$	1.90274	0.232457	0.116228	+	0.993223i	$0.462920\pi$
$68$	0	0
$69$	8.93655	1.07583
$70$	0	0
$71$	−6.54402	−0.776633	−0.388316	−	0.921526i	$-0.626943\pi$
$71$	−6.54402	−0.776633	−0.388316	+	0.921526i	$0.626943\pi$
$72$	0	0
$73$	−13.4617	−1.57558	−0.787788	−	0.615947i	$-0.788774\pi$
$73$	−13.4617	−1.57558	−0.787788	+	0.615947i	$0.788774\pi$
$74$	0	0
$75$	14.2254	1.64261
$76$	0	0
$77$	0	0
$78$	0	0
$79$	5.74642	0.646523	0.323262	−	0.946310i	$-0.395221\pi$
$79$	5.74642	0.646523	0.323262	+	0.946310i	$0.395221\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.5777	−1.16106	−0.580528	−	0.814240i	$-0.697154\pi$
$83$	−10.5777	−1.16106	−0.580528	+	0.814240i	$0.697154\pi$
$84$	0	0
$85$	25.0757	2.71984
$86$	0	0
$87$	8.29526	0.889346
$88$	0	0
$89$	3.28512	0.348222	0.174111	−	0.984726i	$-0.444295\pi$
$89$	3.28512	0.348222	0.174111	+	0.984726i	$0.444295\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−0.636783	−0.0660313
$94$	0	0
$95$	−27.3655	−2.80764
$96$	0	0
$97$	8.13178	0.825657	0.412829	−	0.910809i	$-0.364541\pi$
$97$	8.13178	0.825657	0.412829	+	0.910809i	$0.364541\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−18.2938	−1.82030	−0.910150	−	0.414279i	$-0.864034\pi$
$101$	−18.2938	−1.82030	−0.910150	+	0.414279i	$0.864034\pi$
$102$	0	0
$103$	12.9283	1.27387	0.636933	−	0.770919i	$-0.280203\pi$
$103$	12.9283	1.27387	0.636933	+	0.770919i	$0.280203\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−16.2566	−1.57159	−0.785793	−	0.618490i	$-0.787745\pi$
$107$	−16.2566	−1.57159	−0.785793	+	0.618490i	$0.787745\pi$
$108$	0	0
$109$	−1.91656	−0.183573	−0.0917864	−	0.995779i	$-0.529258\pi$
$109$	−1.91656	−0.183573	−0.0917864	+	0.995779i	$0.529258\pi$
$110$	0	0
$111$	−8.79450	−0.834737
$112$	0	0
$113$	−10.4213	−0.980353	−0.490176	−	0.871623i	$-0.663068\pi$
$113$	−10.4213	−0.980353	−0.490176	+	0.871623i	$0.663068\pi$
$114$	0	0
$115$	−39.1839	−3.65392
$116$	0	0
$117$	−3.17941	−0.293937
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	2.64912	0.238863
$124$	0	0
$125$	−40.4507	−3.61802
$126$	0	0
$127$	−5.44170	−0.482873	−0.241436	−	0.970417i	$-0.577619\pi$
$127$	−5.44170	−0.482873	−0.241436	+	0.970417i	$0.577619\pi$
$128$	0	0
$129$	−4.11021	−0.361884
$130$	0	0
$131$	−20.5006	−1.79115	−0.895575	−	0.444911i	$-0.853235\pi$
$131$	−20.5006	−1.79115	−0.895575	+	0.444911i	$0.853235\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−4.38468	−0.377373
$136$	0	0
$137$	−0.525791	−0.0449214	−0.0224607	−	0.999748i	$-0.507150\pi$
$137$	−0.525791	−0.0449214	−0.0224607	+	0.999748i	$0.507150\pi$
$138$	0	0
$139$	5.38583	0.456820	0.228410	−	0.973565i	$-0.426647\pi$
$139$	5.38583	0.456820	0.228410	+	0.973565i	$0.426647\pi$
$140$	0	0
$141$	5.85587	0.493153
$142$	0	0
$143$	3.17941	0.265876
$144$	0	0
$145$	−36.3721	−3.02054
$146$	0	0
$147$	0	0
$148$	0	0
$149$	14.6296	1.19851	0.599253	−	0.800560i	$-0.295464\pi$
$149$	14.6296	1.19851	0.599253	+	0.800560i	$0.295464\pi$
$150$	0	0
$151$	−2.37716	−0.193451	−0.0967253	−	0.995311i	$-0.530837\pi$
$151$	−2.37716	−0.193451	−0.0967253	+	0.995311i	$0.530837\pi$
$152$	0	0
$153$	−5.71894	−0.462349
$154$	0	0
$155$	2.79209	0.224266
$156$	0	0
$157$	7.98698	0.637430	0.318715	−	0.947851i	$-0.396749\pi$
$157$	7.98698	0.637430	0.318715	+	0.947851i	$0.396749\pi$
$158$	0	0
$159$	−0.386160	−0.0306245
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−0.922732	−0.0722740	−0.0361370	−	0.999347i	$-0.511505\pi$
$163$	−0.922732	−0.0722740	−0.0361370	+	0.999347i	$0.511505\pi$
$164$	0	0
$165$	4.38468	0.341347
$166$	0	0
$167$	−8.60219	−0.665657	−0.332829	−	0.942987i	$-0.608003\pi$
$167$	−8.60219	−0.665657	−0.332829	+	0.942987i	$0.608003\pi$
$168$	0	0
$169$	−2.89134	−0.222411
$170$	0	0
$171$	6.24116	0.477274
$172$	0	0
$173$	−16.7819	−1.27591	−0.637954	−	0.770075i	$-0.720219\pi$
$173$	−16.7819	−1.27591	−0.637954	+	0.770075i	$0.720219\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−14.1943	−1.06691
$178$	0	0
$179$	−15.5846	−1.16485	−0.582425	−	0.812885i	$-0.697896\pi$
$179$	−15.5846	−1.16485	−0.582425	+	0.812885i	$0.697896\pi$
$180$	0	0
$181$	6.73405	0.500538	0.250269	−	0.968176i	$-0.419481\pi$
$181$	6.73405	0.500538	0.250269	+	0.968176i	$0.419481\pi$
$182$	0	0
$183$	11.7029	0.865103
$184$	0	0
$185$	38.5611	2.83507
$186$	0	0
$187$	5.71894	0.418210
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−20.6085	−1.49118	−0.745591	−	0.666404i	$-0.767833\pi$
$191$	−20.6085	−1.49118	−0.745591	+	0.666404i	$0.767833\pi$
$192$	0	0
$193$	−19.6134	−1.41180	−0.705901	−	0.708310i	$-0.749458\pi$
$193$	−19.6134	−1.41180	−0.705901	+	0.708310i	$0.749458\pi$
$194$	0	0
$195$	13.9407	0.998315
$196$	0	0
$197$	−4.61668	−0.328925	−0.164462	−	0.986383i	$-0.552589\pi$
$197$	−4.61668	−0.328925	−0.164462	+	0.986383i	$0.552589\pi$
$198$	0	0
$199$	6.32501	0.448368	0.224184	−	0.974547i	$-0.428028\pi$
$199$	6.32501	0.448368	0.224184	+	0.974547i	$0.428028\pi$
$200$	0	0
$201$	1.90274	0.134209
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−11.6156	−0.811266
$206$	0	0
$207$	8.93655	0.621133
$208$	0	0
$209$	−6.24116	−0.431710
$210$	0	0
$211$	15.1192	1.04085	0.520424	−	0.853908i	$-0.325774\pi$
$211$	15.1192	1.04085	0.520424	+	0.853908i	$0.325774\pi$
$212$	0	0
$213$	−6.54402	−0.448389
$214$	0	0
$215$	18.0220	1.22909
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−13.4617	−0.909659
$220$	0	0
$221$	18.1829	1.22311
$222$	0	0
$223$	−5.76785	−0.386244	−0.193122	−	0.981175i	$-0.561861\pi$
$223$	−5.76785	−0.386244	−0.193122	+	0.981175i	$0.561861\pi$
$224$	0	0
$225$	14.2254	0.948363
$226$	0	0
$227$	−8.56849	−0.568711	−0.284355	−	0.958719i	$-0.591780\pi$
$227$	−8.56849	−0.568711	−0.284355	+	0.958719i	$0.591780\pi$
$228$	0	0
$229$	5.49278	0.362973	0.181486	−	0.983393i	$-0.441909\pi$
$229$	5.49278	0.362973	0.181486	+	0.983393i	$0.441909\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.17585	0.470105	0.235053	−	0.971983i	$-0.424474\pi$
$233$	7.17585	0.470105	0.235053	+	0.971983i	$0.424474\pi$
$234$	0	0
$235$	−25.6761	−1.67493
$236$	0	0
$237$	5.74642	0.373270
$238$	0	0
$239$	13.5212	0.874612	0.437306	−	0.899313i	$-0.355933\pi$
$239$	13.5212	0.874612	0.437306	+	0.899313i	$0.355933\pi$
$240$	0	0
$241$	16.2988	1.04990	0.524950	−	0.851133i	$-0.324084\pi$
$241$	16.2988	1.04990	0.524950	+	0.851133i	$0.324084\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−19.8432	−1.26259
$248$	0	0
$249$	−10.5777	−0.670336
$250$	0	0
$251$	−16.4421	−1.03782	−0.518909	−	0.854830i	$-0.673662\pi$
$251$	−16.4421	−1.03782	−0.518909	+	0.854830i	$0.673662\pi$
$252$	0	0
$253$	−8.93655	−0.561836
$254$	0	0
$255$	25.0757	1.57030
$256$	0	0
$257$	14.6468	0.913643	0.456821	−	0.889558i	$-0.348988\pi$
$257$	14.6468	0.913643	0.456821	+	0.889558i	$0.348988\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	8.29526	0.513464
$262$	0	0
$263$	7.60525	0.468960	0.234480	−	0.972121i	$-0.424661\pi$
$263$	7.60525	0.468960	0.234480	+	0.972121i	$0.424661\pi$
$264$	0	0
$265$	1.69319	0.104012
$266$	0	0
$267$	3.28512	0.201046
$268$	0	0
$269$	22.1608	1.35117	0.675583	−	0.737284i	$-0.263892\pi$
$269$	22.1608	1.35117	0.675583	+	0.737284i	$0.263892\pi$
$270$	0	0
$271$	15.0990	0.917198	0.458599	−	0.888643i	$-0.348351\pi$
$271$	15.0990	0.917198	0.458599	+	0.888643i	$0.348351\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−14.2254	−0.857827
$276$	0	0
$277$	15.8964	0.955122	0.477561	−	0.878599i	$-0.341521\pi$
$277$	15.8964	0.955122	0.477561	+	0.878599i	$0.341521\pi$
$278$	0	0
$279$	−0.636783	−0.0381232
$280$	0	0
$281$	−6.75346	−0.402878	−0.201439	−	0.979501i	$-0.564562\pi$
$281$	−6.75346	−0.402878	−0.201439	+	0.979501i	$0.564562\pi$
$282$	0	0
$283$	−4.03215	−0.239687	−0.119843	−	0.992793i	$-0.538239\pi$
$283$	−4.03215	−0.239687	−0.119843	+	0.992793i	$0.538239\pi$
$284$	0	0
$285$	−27.3655	−1.62099
$286$	0	0
$287$	0	0
$288$	0	0
$289$	15.7062	0.923897
$290$	0	0
$291$	8.13178	0.476693
$292$	0	0
$293$	21.1652	1.23649	0.618243	−	0.785987i	$-0.287845\pi$
$293$	21.1652	1.23649	0.618243	+	0.785987i	$0.287845\pi$
$294$	0	0
$295$	62.2375	3.62361
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−28.4130	−1.64316
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−18.2938	−1.05095
$304$	0	0
$305$	−51.3135	−2.93820
$306$	0	0
$307$	−14.2449	−0.812998	−0.406499	−	0.913651i	$-0.633251\pi$
$307$	−14.2449	−0.812998	−0.406499	+	0.913651i	$0.633251\pi$
$308$	0	0
$309$	12.9283	0.735467
$310$	0	0
$311$	−7.15443	−0.405690	−0.202845	−	0.979211i	$-0.565019\pi$
$311$	−7.15443	−0.405690	−0.202845	+	0.979211i	$0.565019\pi$
$312$	0	0
$313$	−7.32552	−0.414063	−0.207031	−	0.978334i	$-0.566380\pi$
$313$	−7.32552	−0.414063	−0.207031	+	0.978334i	$0.566380\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−11.0384	−0.619977	−0.309989	−	0.950740i	$-0.600325\pi$
$317$	−11.0384	−0.619977	−0.309989	+	0.950740i	$0.600325\pi$
$318$	0	0
$319$	−8.29526	−0.464446
$320$	0	0
$321$	−16.2566	−0.907355
$322$	0	0
$323$	−35.6928	−1.98600
$324$	0	0
$325$	−45.2286	−2.50883
$326$	0	0
$327$	−1.91656	−0.105986
$328$	0	0
$329$	0	0
$330$	0	0
$331$	5.42659	0.298272	0.149136	−	0.988817i	$-0.452351\pi$
$331$	5.42659	0.298272	0.149136	+	0.988817i	$0.452351\pi$
$332$	0	0
$333$	−8.79450	−0.481936
$334$	0	0
$335$	−8.34292	−0.455823
$336$	0	0
$337$	−0.0878322	−0.00478453	−0.00239226	−	0.999997i	$-0.500761\pi$
$337$	−0.0878322	−0.00478453	−0.00239226	+	0.999997i	$0.500761\pi$
$338$	0	0
$339$	−10.4213	−0.566007
$340$	0	0
$341$	0.636783	0.0344837
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−39.1839	−2.10959
$346$	0	0
$347$	−1.09506	−0.0587861	−0.0293930	−	0.999568i	$-0.509357\pi$
$347$	−1.09506	−0.0587861	−0.0293930	+	0.999568i	$0.509357\pi$
$348$	0	0
$349$	−22.4605	−1.20228	−0.601142	−	0.799142i	$-0.705287\pi$
$349$	−22.4605	−1.20228	−0.601142	+	0.799142i	$0.705287\pi$
$350$	0	0
$351$	−3.17941	−0.169704
$352$	0	0
$353$	−27.5088	−1.46415	−0.732073	−	0.681226i	$-0.761447\pi$
$353$	−27.5088	−1.46415	−0.732073	+	0.681226i	$0.761447\pi$
$354$	0	0
$355$	28.6935	1.52289
$356$	0	0
$357$	0	0
$358$	0	0
$359$	11.9741	0.631967	0.315984	−	0.948765i	$-0.397666\pi$
$359$	11.9741	0.631967	0.315984	+	0.948765i	$0.397666\pi$
$360$	0	0
$361$	19.9521	1.05011
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	59.0254	3.08953
$366$	0	0
$367$	−13.2258	−0.690381	−0.345190	−	0.938533i	$-0.612186\pi$
$367$	−13.2258	−0.690381	−0.345190	+	0.938533i	$0.612186\pi$
$368$	0	0
$369$	2.64912	0.137908
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−15.6186	−0.808703	−0.404351	−	0.914604i	$-0.632503\pi$
$373$	−15.6186	−0.808703	−0.404351	+	0.914604i	$0.632503\pi$
$374$	0	0
$375$	−40.4507	−2.08886
$376$	0	0
$377$	−26.3741	−1.35833
$378$	0	0
$379$	−8.99227	−0.461902	−0.230951	−	0.972965i	$-0.574184\pi$
$379$	−8.99227	−0.461902	−0.230951	+	0.972965i	$0.574184\pi$
$380$	0	0
$381$	−5.44170	−0.278787
$382$	0	0
$383$	−34.0484	−1.73979	−0.869895	−	0.493237i	$-0.835813\pi$
$383$	−34.0484	−1.73979	−0.869895	+	0.493237i	$0.835813\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.11021	−0.208934
$388$	0	0
$389$	15.7855	0.800357	0.400178	−	0.916437i	$-0.368948\pi$
$389$	15.7855	0.800357	0.400178	+	0.916437i	$0.368948\pi$
$390$	0	0
$391$	−51.1076	−2.58462
$392$	0	0
$393$	−20.5006	−1.03412
$394$	0	0
$395$	−25.1962	−1.26776
$396$	0	0
$397$	−7.80017	−0.391479	−0.195740	−	0.980656i	$-0.562711\pi$
$397$	−7.80017	−0.391479	−0.195740	+	0.980656i	$0.562711\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.77429	0.188479	0.0942395	−	0.995550i	$-0.469958\pi$
$401$	3.77429	0.188479	0.0942395	+	0.995550i	$0.469958\pi$
$402$	0	0
$403$	2.02460	0.100852
$404$	0	0
$405$	−4.38468	−0.217877
$406$	0	0
$407$	8.79450	0.435927
$408$	0	0
$409$	3.58768	0.177399	0.0886996	−	0.996058i	$-0.471729\pi$
$409$	3.58768	0.177399	0.0886996	+	0.996058i	$0.471729\pi$
$410$	0	0
$411$	−0.525791	−0.0259354
$412$	0	0
$413$	0	0
$414$	0	0
$415$	46.3800	2.27670
$416$	0	0
$417$	5.38583	0.263745
$418$	0	0
$419$	−6.96399	−0.340213	−0.170107	−	0.985426i	$-0.554411\pi$
$419$	−6.96399	−0.340213	−0.170107	+	0.985426i	$0.554411\pi$
$420$	0	0
$421$	10.5236	0.512889	0.256444	−	0.966559i	$-0.417449\pi$
$421$	10.5236	0.512889	0.256444	+	0.966559i	$0.417449\pi$
$422$	0	0
$423$	5.85587	0.284722
$424$	0	0
$425$	−81.3545	−3.94627
$426$	0	0
$427$	0	0
$428$	0	0
$429$	3.17941	0.153503
$430$	0	0
$431$	3.75937	0.181083	0.0905413	−	0.995893i	$-0.471140\pi$
$431$	3.75937	0.181083	0.0905413	+	0.995893i	$0.471140\pi$
$432$	0	0
$433$	8.09935	0.389230	0.194615	−	0.980880i	$-0.437654\pi$
$433$	8.09935	0.389230	0.194615	+	0.980880i	$0.437654\pi$
$434$	0	0
$435$	−36.3721	−1.74391
$436$	0	0
$437$	55.7745	2.66805
$438$	0	0
$439$	−21.7714	−1.03909	−0.519547	−	0.854442i	$-0.673899\pi$
$439$	−21.7714	−1.03909	−0.519547	+	0.854442i	$0.673899\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4.79186	0.227668	0.113834	−	0.993500i	$-0.463687\pi$
$443$	4.79186	0.227668	0.113834	+	0.993500i	$0.463687\pi$
$444$	0	0
$445$	−14.4042	−0.682825
$446$	0	0
$447$	14.6296	0.691958
$448$	0	0
$449$	−7.31656	−0.345290	−0.172645	−	0.984984i	$-0.555231\pi$
$449$	−7.31656	−0.345290	−0.172645	+	0.984984i	$0.555231\pi$
$450$	0	0
$451$	−2.64912	−0.124742
$452$	0	0
$453$	−2.37716	−0.111689
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−39.5813	−1.85154	−0.925768	−	0.378093i	$-0.876580\pi$
$457$	−39.5813	−1.85154	−0.925768	+	0.378093i	$0.876580\pi$
$458$	0	0
$459$	−5.71894	−0.266937
$460$	0	0
$461$	20.5998	0.959430	0.479715	−	0.877424i	$-0.340740\pi$
$461$	20.5998	0.959430	0.479715	+	0.877424i	$0.340740\pi$
$462$	0	0
$463$	−17.9500	−0.834205	−0.417102	−	0.908859i	$-0.636954\pi$
$463$	−17.9500	−0.834205	−0.417102	+	0.908859i	$0.636954\pi$
$464$	0	0
$465$	2.79209	0.129480
$466$	0	0
$467$	28.6494	1.32573	0.662867	−	0.748737i	$-0.269339\pi$
$467$	28.6494	1.32573	0.662867	+	0.748737i	$0.269339\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	7.98698	0.368020
$472$	0	0
$473$	4.11021	0.188988
$474$	0	0
$475$	88.7834	4.07366
$476$	0	0
$477$	−0.386160	−0.0176810
$478$	0	0
$479$	−21.8274	−0.997321	−0.498661	−	0.866797i	$-0.666175\pi$
$479$	−21.8274	−0.997321	−0.498661	+	0.866797i	$0.666175\pi$
$480$	0	0
$481$	27.9614	1.27493
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−35.6553	−1.61902
$486$	0	0
$487$	24.7405	1.12110	0.560550	−	0.828121i	$-0.310590\pi$
$487$	24.7405	1.12110	0.560550	+	0.828121i	$0.310590\pi$
$488$	0	0
$489$	−0.922732	−0.0417274
$490$	0	0
$491$	3.60310	0.162606	0.0813028	−	0.996689i	$-0.474092\pi$
$491$	3.60310	0.162606	0.0813028	+	0.996689i	$0.474092\pi$
$492$	0	0
$493$	−47.4401	−2.13659
$494$	0	0
$495$	4.38468	0.197077
$496$	0	0
$497$	0	0
$498$	0	0
$499$	10.6171	0.475288	0.237644	−	0.971352i	$-0.423625\pi$
$499$	10.6171	0.475288	0.237644	+	0.971352i	$0.423625\pi$
$500$	0	0
$501$	−8.60219	−0.384317
$502$	0	0
$503$	1.76461	0.0786799	0.0393399	−	0.999226i	$-0.487474\pi$
$503$	1.76461	0.0786799	0.0393399	+	0.999226i	$0.487474\pi$
$504$	0	0
$505$	80.2125	3.56941
$506$	0	0
$507$	−2.89134	−0.128409
$508$	0	0
$509$	−31.6511	−1.40291	−0.701454	−	0.712714i	$-0.747466\pi$
$509$	−31.6511	−1.40291	−0.701454	+	0.712714i	$0.747466\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	6.24116	0.275554
$514$	0	0
$515$	−56.6866	−2.49791
$516$	0	0
$517$	−5.85587	−0.257541
$518$	0	0
$519$	−16.7819	−0.736645
$520$	0	0
$521$	−4.25641	−0.186477	−0.0932383	−	0.995644i	$-0.529722\pi$
$521$	−4.25641	−0.186477	−0.0932383	+	0.995644i	$0.529722\pi$
$522$	0	0
$523$	−1.82094	−0.0796240	−0.0398120	−	0.999207i	$-0.512676\pi$
$523$	−1.82094	−0.0796240	−0.0398120	+	0.999207i	$0.512676\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.64172	0.158636
$528$	0	0
$529$	56.8619	2.47226
$530$	0	0
$531$	−14.1943	−0.615980
$532$	0	0
$533$	−8.42265	−0.364825
$534$	0	0
$535$	71.2801	3.08171
$536$	0	0
$537$	−15.5846	−0.672526
$538$	0	0
$539$	0	0
$540$	0	0
$541$	9.04281	0.388781	0.194390	−	0.980924i	$-0.437727\pi$
$541$	9.04281	0.388781	0.194390	+	0.980924i	$0.437727\pi$
$542$	0	0
$543$	6.73405	0.288986
$544$	0	0
$545$	8.40350	0.359966
$546$	0	0
$547$	2.38990	0.102185	0.0510923	−	0.998694i	$-0.483730\pi$
$547$	2.38990	0.102185	0.0510923	+	0.998694i	$0.483730\pi$
$548$	0	0
$549$	11.7029	0.499467
$550$	0	0
$551$	51.7721	2.20557
$552$	0	0
$553$	0	0
$554$	0	0
$555$	38.5611	1.63683
$556$	0	0
$557$	31.4920	1.33436	0.667179	−	0.744897i	$-0.267502\pi$
$557$	31.4920	1.33436	0.667179	+	0.744897i	$0.267502\pi$
$558$	0	0
$559$	13.0680	0.552719
$560$	0	0
$561$	5.71894	0.241454
$562$	0	0
$563$	−17.7938	−0.749918	−0.374959	−	0.927041i	$-0.622343\pi$
$563$	−17.7938	−0.749918	−0.374959	+	0.927041i	$0.622343\pi$
$564$	0	0
$565$	45.6941	1.92236
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−11.4827	−0.481382	−0.240691	−	0.970602i	$-0.577374\pi$
$569$	−11.4827	−0.481382	−0.240691	+	0.970602i	$0.577374\pi$
$570$	0	0
$571$	−33.6109	−1.40657	−0.703287	−	0.710906i	$-0.748285\pi$
$571$	−33.6109	−1.40657	−0.703287	+	0.710906i	$0.748285\pi$
$572$	0	0
$573$	−20.6085	−0.860935
$574$	0	0
$575$	127.126	5.30154
$576$	0	0
$577$	−40.9300	−1.70394	−0.851970	−	0.523591i	$-0.824592\pi$
$577$	−40.9300	−1.70394	−0.851970	+	0.523591i	$0.824592\pi$
$578$	0	0
$579$	−19.6134	−0.815105
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0.386160	0.0159931
$584$	0	0
$585$	13.9407	0.576378
$586$	0	0
$587$	−1.92954	−0.0796406	−0.0398203	−	0.999207i	$-0.512679\pi$
$587$	−1.92954	−0.0796406	−0.0398203	+	0.999207i	$0.512679\pi$
$588$	0	0
$589$	−3.97427	−0.163757
$590$	0	0
$591$	−4.61668	−0.189905
$592$	0	0
$593$	15.8909	0.652561	0.326280	−	0.945273i	$-0.394205\pi$
$593$	15.8909	0.652561	0.326280	+	0.945273i	$0.394205\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6.32501	0.258865
$598$	0	0
$599$	−13.8197	−0.564659	−0.282329	−	0.959318i	$-0.591107\pi$
$599$	−13.8197	−0.564659	−0.282329	+	0.959318i	$0.591107\pi$
$600$	0	0
$601$	34.9295	1.42480	0.712401	−	0.701772i	$-0.247608\pi$
$601$	34.9295	1.42480	0.712401	+	0.701772i	$0.247608\pi$
$602$	0	0
$603$	1.90274	0.0774856
$604$	0	0
$605$	−4.38468	−0.178263
$606$	0	0
$607$	−27.6226	−1.12117	−0.560583	−	0.828098i	$-0.689423\pi$
$607$	−27.6226	−1.12117	−0.560583	+	0.828098i	$0.689423\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−18.6182	−0.753213
$612$	0	0
$613$	−22.4123	−0.905223	−0.452611	−	0.891708i	$-0.649508\pi$
$613$	−22.4123	−0.905223	−0.452611	+	0.891708i	$0.649508\pi$
$614$	0	0
$615$	−11.6156	−0.468385
$616$	0	0
$617$	−13.1912	−0.531058	−0.265529	−	0.964103i	$-0.585547\pi$
$617$	−13.1912	−0.531058	−0.265529	+	0.964103i	$0.585547\pi$
$618$	0	0
$619$	−14.7932	−0.594588	−0.297294	−	0.954786i	$-0.596084\pi$
$619$	−14.7932	−0.594588	−0.297294	+	0.954786i	$0.596084\pi$
$620$	0	0
$621$	8.93655	0.358611
$622$	0	0
$623$	0	0
$624$	0	0
$625$	106.236	4.24945
$626$	0	0
$627$	−6.24116	−0.249248
$628$	0	0
$629$	50.2952	2.00540
$630$	0	0
$631$	−44.4777	−1.77063	−0.885315	−	0.464992i	$-0.846057\pi$
$631$	−44.4777	−1.77063	−0.885315	+	0.464992i	$0.846057\pi$
$632$	0	0
$633$	15.1192	0.600934
$634$	0	0
$635$	23.8601	0.946860
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−6.54402	−0.258878
$640$	0	0
$641$	−32.8630	−1.29801	−0.649004	−	0.760785i	$-0.724814\pi$
$641$	−32.8630	−1.29801	−0.649004	+	0.760785i	$0.724814\pi$
$642$	0	0
$643$	23.5098	0.927135	0.463568	−	0.886062i	$-0.346569\pi$
$643$	23.5098	0.927135	0.463568	+	0.886062i	$0.346569\pi$
$644$	0	0
$645$	18.0220	0.709614
$646$	0	0
$647$	28.2743	1.11158	0.555789	−	0.831323i	$-0.312416\pi$
$647$	28.2743	1.11158	0.555789	+	0.831323i	$0.312416\pi$
$648$	0	0
$649$	14.1943	0.557175
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−10.3715	−0.405867	−0.202933	−	0.979193i	$-0.565048\pi$
$653$	−10.3715	−0.405867	−0.202933	+	0.979193i	$0.565048\pi$
$654$	0	0
$655$	89.8888	3.51225
$656$	0	0
$657$	−13.4617	−0.525192
$658$	0	0
$659$	30.7881	1.19933	0.599667	−	0.800250i	$-0.295300\pi$
$659$	30.7881	1.19933	0.599667	+	0.800250i	$0.295300\pi$
$660$	0	0
$661$	−34.1518	−1.32835	−0.664175	−	0.747578i	$-0.731217\pi$
$661$	−34.1518	−1.32835	−0.664175	+	0.747578i	$0.731217\pi$
$662$	0	0
$663$	18.1829	0.706164
$664$	0	0
$665$	0	0
$666$	0	0
$667$	74.1310	2.87036
$668$	0	0
$669$	−5.76785	−0.222998
$670$	0	0
$671$	−11.7029	−0.451785
$672$	0	0
$673$	−28.7754	−1.10921	−0.554605	−	0.832114i	$-0.687131\pi$
$673$	−28.7754	−1.10921	−0.554605	+	0.832114i	$0.687131\pi$
$674$	0	0
$675$	14.2254	0.547538
$676$	0	0
$677$	7.16187	0.275253	0.137627	−	0.990484i	$-0.456053\pi$
$677$	7.16187	0.275253	0.137627	+	0.990484i	$0.456053\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−8.56849	−0.328345
$682$	0	0
$683$	−10.1533	−0.388504	−0.194252	−	0.980952i	$-0.562228\pi$
$683$	−10.1533	−0.388504	−0.194252	+	0.980952i	$0.562228\pi$
$684$	0	0
$685$	2.30543	0.0880859
$686$	0	0
$687$	5.49278	0.209563
$688$	0	0
$689$	1.22776	0.0467740
$690$	0	0
$691$	−24.1163	−0.917427	−0.458714	−	0.888584i	$-0.651690\pi$
$691$	−24.1163	−0.917427	−0.458714	+	0.888584i	$0.651690\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−23.6152	−0.895774
$696$	0	0
$697$	−15.1502	−0.573853
$698$	0	0
$699$	7.17585	0.271415
$700$	0	0
$701$	−45.7772	−1.72898	−0.864490	−	0.502650i	$-0.832358\pi$
$701$	−45.7772	−1.72898	−0.864490	+	0.502650i	$0.832358\pi$
$702$	0	0
$703$	−54.8879	−2.07014
$704$	0	0
$705$	−25.6761	−0.967019
$706$	0	0
$707$	0	0
$708$	0	0
$709$	20.3063	0.762618	0.381309	−	0.924448i	$-0.375473\pi$
$709$	20.3063	0.762618	0.381309	+	0.924448i	$0.375473\pi$
$710$	0	0
$711$	5.74642	0.215508
$712$	0	0
$713$	−5.69064	−0.213116
$714$	0	0
$715$	−13.9407	−0.521353
$716$	0	0
$717$	13.5212	0.504958
$718$	0	0
$719$	45.0074	1.67849	0.839246	−	0.543752i	$-0.182997\pi$
$719$	45.0074	1.67849	0.839246	+	0.543752i	$0.182997\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	16.2988	0.606160
$724$	0	0
$725$	118.004	4.38255
$726$	0	0
$727$	−2.82251	−0.104681	−0.0523406	−	0.998629i	$-0.516668\pi$
$727$	−2.82251	−0.104681	−0.0523406	+	0.998629i	$0.516668\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	23.5060	0.869402
$732$	0	0
$733$	12.6620	0.467680	0.233840	−	0.972275i	$-0.424871\pi$
$733$	12.6620	0.467680	0.233840	+	0.972275i	$0.424871\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.90274	−0.0700884
$738$	0	0
$739$	−14.7919	−0.544130	−0.272065	−	0.962279i	$-0.587707\pi$
$739$	−14.7919	−0.544130	−0.272065	+	0.962279i	$0.587707\pi$
$740$	0	0
$741$	−19.8432	−0.728959
$742$	0	0
$743$	−29.9648	−1.09930	−0.549652	−	0.835394i	$-0.685240\pi$
$743$	−29.9648	−1.09930	−0.549652	+	0.835394i	$0.685240\pi$
$744$	0	0
$745$	−64.1463	−2.35014
$746$	0	0
$747$	−10.5777	−0.387019
$748$	0	0
$749$	0	0
$750$	0	0
$751$	6.07537	0.221693	0.110847	−	0.993838i	$-0.464644\pi$
$751$	6.07537	0.221693	0.110847	+	0.993838i	$0.464644\pi$
$752$	0	0
$753$	−16.4421	−0.599184
$754$	0	0
$755$	10.4231	0.379335
$756$	0	0
$757$	10.8378	0.393906	0.196953	−	0.980413i	$-0.436895\pi$
$757$	10.8378	0.393906	0.196953	+	0.980413i	$0.436895\pi$
$758$	0	0
$759$	−8.93655	−0.324376
$760$	0	0
$761$	4.93239	0.178799	0.0893995	−	0.995996i	$-0.471505\pi$
$761$	4.93239	0.178799	0.0893995	+	0.995996i	$0.471505\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	25.0757	0.906615
$766$	0	0
$767$	45.1295	1.62953
$768$	0	0
$769$	−33.6085	−1.21195	−0.605976	−	0.795483i	$-0.707217\pi$
$769$	−33.6085	−1.21195	−0.605976	+	0.795483i	$0.707217\pi$
$770$	0	0
$771$	14.6468	0.527492
$772$	0	0
$773$	−15.5922	−0.560812	−0.280406	−	0.959882i	$-0.590469\pi$
$773$	−15.5922	−0.560812	−0.280406	+	0.959882i	$0.590469\pi$
$774$	0	0
$775$	−9.05853	−0.325392
$776$	0	0
$777$	0	0
$778$	0	0
$779$	16.5336	0.592378
$780$	0	0
$781$	6.54402	0.234164
$782$	0	0
$783$	8.29526	0.296449
$784$	0	0
$785$	−35.0204	−1.24993
$786$	0	0
$787$	−31.8054	−1.13374	−0.566870	−	0.823807i	$-0.691846\pi$
$787$	−31.8054	−1.13374	−0.566870	+	0.823807i	$0.691846\pi$
$788$	0	0
$789$	7.60525	0.270754
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−37.2083	−1.32131
$794$	0	0
$795$	1.69319	0.0600512
$796$	0	0
$797$	−27.9876	−0.991372	−0.495686	−	0.868502i	$-0.665083\pi$
$797$	−27.9876	−0.991372	−0.495686	+	0.868502i	$0.665083\pi$
$798$	0	0
$799$	−33.4894	−1.18477
$800$	0	0
$801$	3.28512	0.116074
$802$	0	0
$803$	13.4617	0.475054
$804$	0	0
$805$	0	0
$806$	0	0
$807$	22.1608	0.780096
$808$	0	0
$809$	12.6246	0.443857	0.221929	−	0.975063i	$-0.428765\pi$
$809$	12.6246	0.443857	0.221929	+	0.975063i	$0.428765\pi$
$810$	0	0
$811$	−3.43480	−0.120612	−0.0603060	−	0.998180i	$-0.519208\pi$
$811$	−3.43480	−0.120612	−0.0603060	+	0.998180i	$0.519208\pi$
$812$	0	0
$813$	15.0990	0.529545
$814$	0	0
$815$	4.04589	0.141721
$816$	0	0
$817$	−25.6525	−0.897467
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−0.567311	−0.0197993	−0.00989964	−	0.999951i	$-0.503151\pi$
$821$	−0.567311	−0.0197993	−0.00989964	+	0.999951i	$0.503151\pi$
$822$	0	0
$823$	−46.4880	−1.62047	−0.810235	−	0.586105i	$-0.800661\pi$
$823$	−46.4880	−1.62047	−0.810235	+	0.586105i	$0.800661\pi$
$824$	0	0
$825$	−14.2254	−0.495267
$826$	0	0
$827$	−4.56180	−0.158629	−0.0793147	−	0.996850i	$-0.525273\pi$
$827$	−4.56180	−0.158629	−0.0793147	+	0.996850i	$0.525273\pi$
$828$	0	0
$829$	−30.8634	−1.07193	−0.535964	−	0.844241i	$-0.680052\pi$
$829$	−30.8634	−1.07193	−0.535964	+	0.844241i	$0.680052\pi$
$830$	0	0
$831$	15.8964	0.551440
$832$	0	0
$833$	0	0
$834$	0	0
$835$	37.7179	1.30528
$836$	0	0
$837$	−0.636783	−0.0220104
$838$	0	0
$839$	−11.2504	−0.388406	−0.194203	−	0.980961i	$-0.562212\pi$
$839$	−11.2504	−0.388406	−0.194203	+	0.980961i	$0.562212\pi$
$840$	0	0
$841$	39.8114	1.37281
$842$	0	0
$843$	−6.75346	−0.232602
$844$	0	0
$845$	12.6776	0.436123
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−4.03215	−0.138383
$850$	0	0
$851$	−78.5925	−2.69412
$852$	0	0
$853$	−27.3626	−0.936878	−0.468439	−	0.883496i	$-0.655184\pi$
$853$	−27.3626	−0.936878	−0.468439	+	0.883496i	$0.655184\pi$
$854$	0	0
$855$	−27.3655	−0.935881
$856$	0	0
$857$	47.4470	1.62076	0.810379	−	0.585906i	$-0.199261\pi$
$857$	47.4470	1.62076	0.810379	+	0.585906i	$0.199261\pi$
$858$	0	0
$859$	24.0759	0.821461	0.410730	−	0.911757i	$-0.365274\pi$
$859$	24.0759	0.821461	0.410730	+	0.911757i	$0.365274\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−14.0500	−0.478269	−0.239134	−	0.970986i	$-0.576864\pi$
$863$	−14.0500	−0.478269	−0.239134	+	0.970986i	$0.576864\pi$
$864$	0	0
$865$	73.5835	2.50191
$866$	0	0
$867$	15.7062	0.533412
$868$	0	0
$869$	−5.74642	−0.194934
$870$	0	0
$871$	−6.04960	−0.204983
$872$	0	0
$873$	8.13178	0.275219
$874$	0	0
$875$	0	0
$876$	0	0
$877$	11.4812	0.387692	0.193846	−	0.981032i	$-0.437904\pi$
$877$	11.4812	0.387692	0.193846	+	0.981032i	$0.437904\pi$
$878$	0	0
$879$	21.1652	0.713885
$880$	0	0
$881$	34.2651	1.15442	0.577210	−	0.816596i	$-0.304141\pi$
$881$	34.2651	1.15442	0.577210	+	0.816596i	$0.304141\pi$
$882$	0	0
$883$	−6.37653	−0.214587	−0.107294	−	0.994227i	$-0.534219\pi$
$883$	−6.37653	−0.214587	−0.107294	+	0.994227i	$0.534219\pi$
$884$	0	0
$885$	62.2375	2.09209
$886$	0	0
$887$	33.7670	1.13379	0.566893	−	0.823791i	$-0.308145\pi$
$887$	33.7670	1.13379	0.566893	+	0.823791i	$0.308145\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	36.5474	1.22301
$894$	0	0
$895$	68.3336	2.28414
$896$	0	0
$897$	−28.4130	−0.948681
$898$	0	0
$899$	−5.28229	−0.176174
$900$	0	0
$901$	2.20842	0.0735733
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−29.5267	−0.981500
$906$	0	0
$907$	−37.5947	−1.24831	−0.624155	−	0.781301i	$-0.714557\pi$
$907$	−37.5947	−1.24831	−0.624155	+	0.781301i	$0.714557\pi$
$908$	0	0
$909$	−18.2938	−0.606767
$910$	0	0
$911$	38.4219	1.27297	0.636487	−	0.771287i	$-0.280387\pi$
$911$	38.4219	1.27297	0.636487	+	0.771287i	$0.280387\pi$
$912$	0	0
$913$	10.5777	0.350072
$914$	0	0
$915$	−51.3135	−1.69637
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−39.1967	−1.29298	−0.646489	−	0.762923i	$-0.723763\pi$
$919$	−39.1967	−1.29298	−0.646489	+	0.762923i	$0.723763\pi$
$920$	0	0
$921$	−14.2449	−0.469385
$922$	0	0
$923$	20.8062	0.684843
$924$	0	0
$925$	−125.106	−4.11345
$926$	0	0
$927$	12.9283	0.424622
$928$	0	0
$929$	37.5627	1.23239	0.616195	−	0.787593i	$-0.288673\pi$
$929$	37.5627	1.23239	0.616195	+	0.787593i	$0.288673\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−7.15443	−0.234225
$934$	0	0
$935$	−25.0757	−0.820064
$936$	0	0
$937$	−52.9460	−1.72967	−0.864835	−	0.502055i	$-0.832577\pi$
$937$	−52.9460	−1.72967	−0.864835	+	0.502055i	$0.832577\pi$
$938$	0	0
$939$	−7.32552	−0.239059
$940$	0	0
$941$	−7.74649	−0.252528	−0.126264	−	0.991997i	$-0.540299\pi$
$941$	−7.74649	−0.252528	−0.126264	+	0.991997i	$0.540299\pi$
$942$	0	0
$943$	23.6740	0.770932
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−47.1091	−1.53084	−0.765420	−	0.643531i	$-0.777469\pi$
$947$	−47.1091	−1.53084	−0.765420	+	0.643531i	$0.777469\pi$
$948$	0	0
$949$	42.8004	1.38936
$950$	0	0
$951$	−11.0384	−0.357944
$952$	0	0
$953$	54.0344	1.75035	0.875174	−	0.483809i	$-0.160747\pi$
$953$	54.0344	1.75035	0.875174	+	0.483809i	$0.160747\pi$
$954$	0	0
$955$	90.3620	2.92404
$956$	0	0
$957$	−8.29526	−0.268148
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.5945	−0.986920
$962$	0	0
$963$	−16.2566	−0.523862
$964$	0	0
$965$	85.9985	2.76839
$966$	0	0
$967$	45.0018	1.44716	0.723580	−	0.690241i	$-0.242495\pi$
$967$	45.0018	1.44716	0.723580	+	0.690241i	$0.242495\pi$
$968$	0	0
$969$	−35.6928	−1.14662
$970$	0	0
$971$	−10.9247	−0.350589	−0.175295	−	0.984516i	$-0.556088\pi$
$971$	−10.9247	−0.350589	−0.175295	+	0.984516i	$0.556088\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−45.2286	−1.44847
$976$	0	0
$977$	1.85311	0.0592862	0.0296431	−	0.999561i	$-0.490563\pi$
$977$	1.85311	0.0592862	0.0296431	+	0.999561i	$0.490563\pi$
$978$	0	0
$979$	−3.28512	−0.104993
$980$	0	0
$981$	−1.91656	−0.0611910
$982$	0	0
$983$	−9.01543	−0.287548	−0.143774	−	0.989611i	$-0.545924\pi$
$983$	−9.01543	−0.287548	−0.143774	+	0.989611i	$0.545924\pi$
$984$	0	0
$985$	20.2427	0.644985
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−36.7311	−1.16798
$990$	0	0
$991$	27.7923	0.882852	0.441426	−	0.897298i	$-0.354473\pi$
$991$	27.7923	0.882852	0.441426	+	0.897298i	$0.354473\pi$
$992$	0	0
$993$	5.42659	0.172207
$994$	0	0
$995$	−27.7332	−0.879201
$996$	0	0
$997$	−29.9125	−0.947340	−0.473670	−	0.880703i	$-0.657071\pi$
$997$	−29.9125	−0.947340	−0.473670	+	0.880703i	$0.657071\pi$
$998$	0	0
$999$	−8.79450	−0.278246

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6468.2.a.be.1.1	yes	6
7.6	odd	2		6468.2.a.bd.1.6	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6468.2.a.bd.1.6	✓	6	7.6	odd	2
6468.2.a.be.1.1	yes	6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6468 = 2^{2} \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6468.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -4.38468 q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -4.38468 q^{5} +1.00000 q^{9} -1.00000 q^{11} -3.17941 q^{13} -4.38468 q^{15} -5.71894 q^{17} +6.24116 q^{19} +8.93655 q^{23} +14.2254 q^{25} +1.00000 q^{27} +8.29526 q^{29} -0.636783 q^{31} -1.00000 q^{33} -8.79450 q^{37} -3.17941 q^{39} +2.64912 q^{41} -4.11021 q^{43} -4.38468 q^{45} +5.85587 q^{47} -5.71894 q^{51} -0.386160 q^{53} +4.38468 q^{55} +6.24116 q^{57} -14.1943 q^{59} +11.7029 q^{61} +13.9407 q^{65} +1.90274 q^{67} +8.93655 q^{69} -6.54402 q^{71} -13.4617 q^{73} +14.2254 q^{75} +5.74642 q^{79} +1.00000 q^{81} -10.5777 q^{83} +25.0757 q^{85} +8.29526 q^{87} +3.28512 q^{89} -0.636783 q^{93} -27.3655 q^{95} +8.13178 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} - 4 q^{5} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 - 4 * q^5 + 6 * q^9	\( 6 q + 6 q^{3} - 4 q^{5} + 6 q^{9} - 6 q^{11} - 8 q^{13} - 4 q^{15} - 16 q^{17} - 4 q^{19} + 22 q^{25} + 6 q^{27} - 4 q^{29} - 6 q^{33} + 4 q^{37} - 8 q^{39} - 12 q^{41} - 8 q^{43} - 4 q^{45} + 12 q^{47} - 16 q^{51} + 4 q^{53} + 4 q^{55} - 4 q^{57} - 24 q^{59} - 16 q^{61} - 12 q^{65} + 8 q^{67} - 36 q^{71} - 36 q^{73} + 22 q^{75} + 6 q^{81} - 32 q^{83} - 20 q^{85} - 4 q^{87} - 12 q^{89} - 44 q^{95} - 16 q^{97} - 6 q^{99}+O(q^{100}) \) 6 * q + 6 * q^3 - 4 * q^5 + 6 * q^9 - 6 * q^11 - 8 * q^13 - 4 * q^15 - 16 * q^17 - 4 * q^19 + 22 * q^25 + 6 * q^27 - 4 * q^29 - 6 * q^33 + 4 * q^37 - 8 * q^39 - 12 * q^41 - 8 * q^43 - 4 * q^45 + 12 * q^47 - 16 * q^51 + 4 * q^53 + 4 * q^55 - 4 * q^57 - 24 * q^59 - 16 * q^61 - 12 * q^65 + 8 * q^67 - 36 * q^71 - 36 * q^73 + 22 * q^75 + 6 * q^81 - 32 * q^83 - 20 * q^85 - 4 * q^87 - 12 * q^89 - 44 * q^95 - 16 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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