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Properties

Label 5054.2.a.bk
Level $5054$
Weight $2$
Character orbit 5054.a
Self dual yes
Analytic conductor $40.356$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5054,2,Mod(1,5054)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5054, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("5054.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5054.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: \(40.3563931816\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 15x^{7} + 40x^{6} + 81x^{5} - 162x^{4} - 205x^{3} + 204x^{2} + 210x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 266)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta_1 q^{3} + q^{4} + \beta_{6} q^{5} + \beta_1 q^{6} - q^{7} + q^{8} + (\beta_{4} + \beta_{3} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta_1 q^{3} + q^{4} + \beta_{6} q^{5} + \beta_1 q^{6} - q^{7} + q^{8} + (\beta_{4} + \beta_{3} + \beta_1 + 1) q^{9} + \beta_{6} q^{10} + (\beta_{8} + \beta_{6} - 2 \beta_{3} - 1) q^{11} + \beta_1 q^{12} + (\beta_{8} + \beta_{2} + \beta_1 + 1) q^{13} - q^{14} + (\beta_{8} + \beta_{7} + \beta_{6} + \cdots - 2) q^{15}+ \cdots + (\beta_{8} + \beta_{6} + 2 \beta_{5} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 9 q^{2} + 3 q^{3} + 9 q^{4} + 3 q^{5} + 3 q^{6} - 9 q^{7} + 9 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 9 q^{2} + 3 q^{3} + 9 q^{4} + 3 q^{5} + 3 q^{6} - 9 q^{7} + 9 q^{8} + 12 q^{9} + 3 q^{10} - 3 q^{11} + 3 q^{12} + 12 q^{13} - 9 q^{14} - 6 q^{15} + 9 q^{16} + 12 q^{17} + 12 q^{18} + 3 q^{20} - 3 q^{21} - 3 q^{22} + 6 q^{23} + 3 q^{24} + 12 q^{26} + 24 q^{27} - 9 q^{28} - 6 q^{29} - 6 q^{30} + 9 q^{31} + 9 q^{32} - 3 q^{33} + 12 q^{34} - 3 q^{35} + 12 q^{36} + 9 q^{37} + 33 q^{39} + 3 q^{40} + 3 q^{41} - 3 q^{42} + 21 q^{43} - 3 q^{44} + 18 q^{45} + 6 q^{46} + 21 q^{47} + 3 q^{48} + 9 q^{49} + 39 q^{51} + 12 q^{52} + 24 q^{54} + 24 q^{55} - 9 q^{56} - 6 q^{58} - 9 q^{59} - 6 q^{60} + 6 q^{61} + 9 q^{62} - 12 q^{63} + 9 q^{64} - 3 q^{65} - 3 q^{66} + 27 q^{67} + 12 q^{68} + 6 q^{69} - 3 q^{70} + 9 q^{71} + 12 q^{72} + 51 q^{73} + 9 q^{74} - 3 q^{75} + 3 q^{77} + 33 q^{78} + 24 q^{79} + 3 q^{80} - 3 q^{81} + 3 q^{82} + 9 q^{83} - 3 q^{84} + 6 q^{85} + 21 q^{86} - 3 q^{87} - 3 q^{88} + 9 q^{89} + 18 q^{90} - 12 q^{91} + 6 q^{92} + 3 q^{93} + 21 q^{94} + 3 q^{96} - 12 q^{97} + 9 q^{98} - 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3x^{8} - 15x^{7} + 40x^{6} + 81x^{5} - 162x^{4} - 205x^{3} + 204x^{2} + 210x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{8} + 9\nu^{7} - 280\nu^{6} + 463\nu^{5} + 2443\nu^{4} - 4613\nu^{3} - 4884\nu^{2} + 7433\nu + 1878 ) / 1156 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9\nu^{8} + 27\nu^{7} - 262\nu^{6} - 345\nu^{5} + 2127\nu^{4} + 1189\nu^{3} - 5404\nu^{2} - 1399\nu + 2166 ) / 1156 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -9\nu^{8} - 27\nu^{7} + 262\nu^{6} + 345\nu^{5} - 2127\nu^{4} - 1189\nu^{3} + 6560\nu^{2} + 243\nu - 6790 ) / 1156 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 32 \nu^{8} + 193 \nu^{7} + 193 \nu^{6} - 2434 \nu^{5} + 433 \nu^{4} + 9227 \nu^{3} - 1369 \nu^{2} + \cdots - 1825 ) / 1156 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 43 \nu^{8} + 160 \nu^{7} + 449 \nu^{6} - 1627 \nu^{5} - 1396 \nu^{4} + 3214 \nu^{3} + 2667 \nu^{2} + \cdots - 1197 ) / 1156 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -97\nu^{8} + 287\nu^{7} + 1154\nu^{6} - 3025\nu^{5} - 4043\nu^{4} + 7929\nu^{3} + 4746\nu^{2} - 4895\nu - 32 ) / 1156 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 151 \nu^{8} - 414 \nu^{7} - 1859 \nu^{6} + 4423 \nu^{5} + 6690 \nu^{4} - 11488 \nu^{3} - 7981 \nu^{2} + \cdots + 1179 ) / 1156 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + 2\beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} + 6\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} + 3\beta_{7} - \beta_{6} - 2\beta_{5} + 9\beta_{4} + 13\beta_{3} - \beta_{2} + 10\beta _1 + 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 14\beta_{8} + 27\beta_{7} - 8\beta_{6} - 4\beta_{5} + 14\beta_{4} + 18\beta_{3} - \beta_{2} + 46\beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 25\beta_{8} + 56\beta_{7} - 7\beta_{6} - 30\beta_{5} + 81\beta_{4} + 131\beta_{3} - 18\beta_{2} + 97\beta _1 + 181 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 167 \beta_{8} + 309 \beta_{7} - 45 \beta_{6} - 68 \beta_{5} + 160 \beta_{4} + 242 \beta_{3} - 31 \beta_{2} + \cdots + 221 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 395 \beta_{8} + 765 \beta_{7} - 7 \beta_{6} - 350 \beta_{5} + 756 \beta_{4} + 1302 \beta_{3} + \cdots + 1438 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.64845
−1.81276
−1.47653
−0.906887
−0.00478425
1.68026
2.18756
2.67565
3.30594
1.00000 −2.64845 1.00000 2.31395 −2.64845 −1.00000 1.00000 4.01430 2.31395
1.2 1.00000 −1.81276 1.00000 3.21562 −1.81276 −1.00000 1.00000 0.286090 3.21562
1.3 1.00000 −1.47653 1.00000 −1.23411 −1.47653 −1.00000 1.00000 −0.819871 −1.23411
1.4 1.00000 −0.906887 1.00000 −1.94678 −0.906887 −1.00000 1.00000 −2.17756 −1.94678
1.5 1.00000 −0.00478425 1.00000 −1.04022 −0.00478425 −1.00000 1.00000 −2.99998 −1.04022
1.6 1.00000 1.68026 1.00000 3.48998 1.68026 −1.00000 1.00000 −0.176713 3.48998
1.7 1.00000 2.18756 1.00000 −0.616139 2.18756 −1.00000 1.00000 1.78540 −0.616139
1.8 1.00000 2.67565 1.00000 −2.78796 2.67565 −1.00000 1.00000 4.15909 −2.78796
1.9 1.00000 3.30594 1.00000 1.60566 3.30594 −1.00000 1.00000 7.92924 1.60566
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)
\(19\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5054.2.a.bk 9
19.b odd 2 1 5054.2.a.bj 9
19.e even 9 2 266.2.u.c ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
266.2.u.c ✓ 18 19.e even 9 2
5054.2.a.bj 9 19.b odd 2 1
5054.2.a.bk 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5054))\):

\( T_{3}^{9} - 3T_{3}^{8} - 15T_{3}^{7} + 40T_{3}^{6} + 81T_{3}^{5} - 162T_{3}^{4} - 205T_{3}^{3} + 204T_{3}^{2} + 210T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{9} - 3T_{5}^{8} - 18T_{5}^{7} + 43T_{5}^{6} + 123T_{5}^{5} - 171T_{5}^{4} - 393T_{5}^{3} + 132T_{5}^{2} + 468T_{5} + 179 \) Copy content Toggle raw display
\( T_{13}^{9} - 12 T_{13}^{8} + 3 T_{13}^{7} + 451 T_{13}^{6} - 1491 T_{13}^{5} - 2796 T_{13}^{4} + \cdots + 10099 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{9} \) Copy content Toggle raw display
$3$ \( T^{9} - 3 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{9} - 3 T^{8} + \cdots + 179 \) Copy content Toggle raw display
$7$ \( (T + 1)^{9} \) Copy content Toggle raw display
$11$ \( T^{9} + 3 T^{8} + \cdots + 8704 \) Copy content Toggle raw display
$13$ \( T^{9} - 12 T^{8} + \cdots + 10099 \) Copy content Toggle raw display
$17$ \( T^{9} - 12 T^{8} + \cdots - 1216 \) Copy content Toggle raw display
$19$ \( T^{9} \) Copy content Toggle raw display
$23$ \( T^{9} - 6 T^{8} + \cdots + 49096 \) Copy content Toggle raw display
$29$ \( T^{9} + 6 T^{8} + \cdots - 23104 \) Copy content Toggle raw display
$31$ \( T^{9} - 9 T^{8} + \cdots + 239104 \) Copy content Toggle raw display
$37$ \( T^{9} - 9 T^{8} + \cdots + 2078144 \) Copy content Toggle raw display
$41$ \( T^{9} - 3 T^{8} + \cdots + 12718144 \) Copy content Toggle raw display
$43$ \( T^{9} - 21 T^{8} + \cdots + 17294912 \) Copy content Toggle raw display
$47$ \( T^{9} - 21 T^{8} + \cdots + 5202496 \) Copy content Toggle raw display
$53$ \( T^{9} - 207 T^{7} + \cdots + 56512 \) Copy content Toggle raw display
$59$ \( T^{9} + 9 T^{8} + \cdots + 83156571 \) Copy content Toggle raw display
$61$ \( T^{9} - 6 T^{8} + \cdots + 4158872 \) Copy content Toggle raw display
$67$ \( T^{9} - 27 T^{8} + \cdots - 2331584 \) Copy content Toggle raw display
$71$ \( T^{9} - 9 T^{8} + \cdots - 703 \) Copy content Toggle raw display
$73$ \( T^{9} - 51 T^{8} + \cdots - 212405696 \) Copy content Toggle raw display
$79$ \( T^{9} - 24 T^{8} + \cdots - 243181 \) Copy content Toggle raw display
$83$ \( T^{9} - 9 T^{8} + \cdots + 9838979 \) Copy content Toggle raw display
$89$ \( T^{9} - 9 T^{8} + \cdots + 43964992 \) Copy content Toggle raw display
$97$ \( T^{9} + 12 T^{8} + \cdots - 11829248 \) Copy content Toggle raw display
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