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Properties

Label 5054.2.a.bl
Level $5054$
Weight $2$
Character orbit 5054.a
Self dual yes
Analytic conductor $40.356$
Analytic rank $0$
Dimension $12$
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} - 24 x^{10} + 68 x^{9} + 213 x^{8} - 555 x^{7} - 814 x^{6} + 1911 x^{5} + 1005 x^{4} + \cdots + 1 \) 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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - \beta_1 q^{3} + q^{4} + ( - \beta_{5} + \beta_{2}) q^{5} + \beta_1 q^{6} + q^{7} - q^{8} + (\beta_{9} + \beta_{7} - \beta_{6} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - \beta_1 q^{3} + q^{4} + ( - \beta_{5} + \beta_{2}) q^{5} + \beta_1 q^{6} + q^{7} - q^{8} + (\beta_{9} + \beta_{7} - \beta_{6} + \cdots + 2) q^{9}+ \cdots + (2 \beta_{11} + \beta_{10} + \beta_{9} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} - 3 q^{3} + 12 q^{4} + 3 q^{5} + 3 q^{6} + 12 q^{7} - 12 q^{8} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} - 3 q^{3} + 12 q^{4} + 3 q^{5} + 3 q^{6} + 12 q^{7} - 12 q^{8} + 21 q^{9} - 3 q^{10} + 3 q^{11} - 3 q^{12} + 6 q^{13} - 12 q^{14} + 6 q^{15} + 12 q^{16} + 3 q^{17} - 21 q^{18} + 3 q^{20} - 3 q^{21} - 3 q^{22} + 12 q^{23} + 3 q^{24} + 33 q^{25} - 6 q^{26} - 21 q^{27} + 12 q^{28} - 6 q^{29} - 6 q^{30} - 3 q^{31} - 12 q^{32} + 6 q^{33} - 3 q^{34} + 3 q^{35} + 21 q^{36} - 27 q^{37} + 27 q^{39} - 3 q^{40} + 27 q^{41} + 3 q^{42} + 30 q^{43} + 3 q^{44} + 30 q^{45} - 12 q^{46} + 21 q^{47} - 3 q^{48} + 12 q^{49} - 33 q^{50} - 3 q^{51} + 6 q^{52} - 6 q^{53} + 21 q^{54} + 48 q^{55} - 12 q^{56} + 6 q^{58} + 27 q^{59} + 6 q^{60} + 18 q^{61} + 3 q^{62} + 21 q^{63} + 12 q^{64} + 9 q^{65} - 6 q^{66} - 9 q^{67} + 3 q^{68} - 18 q^{69} - 3 q^{70} - 27 q^{71} - 21 q^{72} - 9 q^{73} + 27 q^{74} - 33 q^{75} + 3 q^{77} - 27 q^{78} + 12 q^{79} + 3 q^{80} + 24 q^{81} - 27 q^{82} + 9 q^{83} - 3 q^{84} + 78 q^{85} - 30 q^{86} - 45 q^{87} - 3 q^{88} + 24 q^{89} - 30 q^{90} + 6 q^{91} + 12 q^{92} + 3 q^{93} - 21 q^{94} + 3 q^{96} + 18 q^{97} - 12 q^{98} + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} - 24 x^{10} + 68 x^{9} + 213 x^{8} - 555 x^{7} - 814 x^{6} + 1911 x^{5} + 1005 x^{4} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 447 \nu^{11} + 1225 \nu^{10} + 13139 \nu^{9} - 20343 \nu^{8} - 158096 \nu^{7} + 67730 \nu^{6} + \cdots - 530962 ) / 162716 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 839 \nu^{11} + 3252 \nu^{10} - 17654 \nu^{9} - 82762 \nu^{8} + 104083 \nu^{7} + 693188 \nu^{6} + \cdots - 2181 ) / 162716 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 895 \nu^{11} + 7731 \nu^{10} + 18299 \nu^{9} - 193073 \nu^{8} - 119612 \nu^{7} + 1731830 \nu^{6} + \cdots - 293578 ) / 162716 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1286 \nu^{11} - 2027 \nu^{10} + 30793 \nu^{9} + 62419 \nu^{8} - 262179 \nu^{7} - 625458 \nu^{6} + \cdots - 40633 ) / 162716 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5046 \nu^{11} - 3181 \nu^{10} - 132213 \nu^{9} + 71023 \nu^{8} + 1235105 \nu^{7} - 596946 \nu^{6} + \cdots + 895 ) / 162716 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 5411 \nu^{11} + 18833 \nu^{10} + 108087 \nu^{9} - 409517 \nu^{8} - 712334 \nu^{7} + \cdots - 768532 ) / 162716 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 4331 \nu^{11} + 10413 \nu^{10} + 101459 \nu^{9} - 215396 \nu^{8} - 877749 \nu^{7} + 1540074 \nu^{6} + \cdots + 205005 ) / 81358 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10457 \nu^{11} - 22014 \nu^{10} - 240300 \nu^{9} + 480540 \nu^{8} + 1947439 \nu^{7} - 3867760 \nu^{6} + \cdots + 443995 ) / 162716 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 15220 \nu^{11} - 35795 \nu^{10} - 393225 \nu^{9} + 830535 \nu^{8} + 3775997 \nu^{7} - 6913710 \nu^{6} + \cdots + 701627 ) / 162716 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 25258 \nu^{11} - 82415 \nu^{10} - 600369 \nu^{9} + 1855779 \nu^{8} + 5308691 \nu^{7} + \cdots + 1189669 ) / 162716 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} + \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{7} - \beta_{6} + 7\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{9} + 10\beta_{7} - 10\beta_{6} - 14\beta_{5} - 4\beta_{4} - 9\beta_{3} + 8\beta_{2} + 2\beta _1 + 38 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} - \beta_{10} + 12 \beta_{9} + 15 \beta_{7} - 13 \beta_{6} - 6 \beta_{5} - 4 \beta_{4} + \cdots + 33 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{11} + \beta_{10} + 80 \beta_{9} - 3 \beta_{8} + 96 \beta_{7} - 97 \beta_{6} - 151 \beta_{5} + \cdots + 311 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 9 \beta_{11} - 6 \beta_{10} + 132 \beta_{9} - 5 \beta_{8} + 177 \beta_{7} - 164 \beta_{6} - 130 \beta_{5} + \cdots + 401 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 31 \beta_{11} + 36 \beta_{10} + 728 \beta_{9} - 65 \beta_{8} + 911 \beta_{7} - 960 \beta_{6} + \cdots + 2668 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 23 \beta_{11} + 42 \beta_{10} + 1409 \beta_{9} - 131 \beta_{8} + 1918 \beta_{7} - 1979 \beta_{6} + \cdots + 4401 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 561 \beta_{11} + 692 \beta_{10} + 6765 \beta_{9} - 981 \beta_{8} + 8642 \beta_{7} - 9667 \beta_{6} + \cdots + 23676 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 679 \beta_{11} + 1660 \beta_{10} + 14781 \beta_{9} - 2267 \beta_{8} + 20005 \beta_{7} - 22914 \beta_{6} + \cdots + 46188 \) 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
3.20919
2.98161
2.93688
2.22802
0.767178
0.572050
−0.0101006
−0.113874
−1.95625
−2.25339
−2.59018
−2.77114
−1.00000 −3.20919 1.00000 2.74348 3.20919 1.00000 −1.00000 7.29887 −2.74348
1.2 −1.00000 −2.98161 1.00000 −4.31337 2.98161 1.00000 −1.00000 5.89001 4.31337
1.3 −1.00000 −2.93688 1.00000 −0.167883 2.93688 1.00000 −1.00000 5.62525 0.167883
1.4 −1.00000 −2.22802 1.00000 3.08661 2.22802 1.00000 −1.00000 1.96409 −3.08661
1.5 −1.00000 −0.767178 1.00000 2.24084 0.767178 1.00000 −1.00000 −2.41144 −2.24084
1.6 −1.00000 −0.572050 1.00000 2.21950 0.572050 1.00000 −1.00000 −2.67276 −2.21950
1.7 −1.00000 0.0101006 1.00000 −3.08958 −0.0101006 1.00000 −1.00000 −2.99990 3.08958
1.8 −1.00000 0.113874 1.00000 −3.74313 −0.113874 1.00000 −1.00000 −2.98703 3.74313
1.9 −1.00000 1.95625 1.00000 −2.38328 −1.95625 1.00000 −1.00000 0.826915 2.38328
1.10 −1.00000 2.25339 1.00000 1.19857 −2.25339 1.00000 −1.00000 2.07775 −1.19857
1.11 −1.00000 2.59018 1.00000 3.85497 −2.59018 1.00000 −1.00000 3.70904 −3.85497
1.12 −1.00000 2.77114 1.00000 1.35329 −2.77114 1.00000 −1.00000 4.67919 −1.35329
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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.bl 12
19.b odd 2 1 5054.2.a.bm 12
19.e even 9 2 266.2.u.d ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
266.2.u.d ✓ 24 19.e even 9 2
5054.2.a.bl 12 1.a even 1 1 trivial
5054.2.a.bm 12 19.b odd 2 1

Hecke kernels

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

\( T_{3}^{12} + 3 T_{3}^{11} - 24 T_{3}^{10} - 68 T_{3}^{9} + 213 T_{3}^{8} + 555 T_{3}^{7} - 814 T_{3}^{6} + \cdots + 1 \) Copy content Toggle raw display
\( T_{5}^{12} - 3 T_{5}^{11} - 42 T_{5}^{10} + 149 T_{5}^{9} + 561 T_{5}^{8} - 2571 T_{5}^{7} - 1847 T_{5}^{6} + \cdots - 5256 \) Copy content Toggle raw display
\( T_{13}^{12} - 6 T_{13}^{11} - 87 T_{13}^{10} + 545 T_{13}^{9} + 2613 T_{13}^{8} - 17346 T_{13}^{7} + \cdots + 2483704 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 3 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{12} - 3 T^{11} + \cdots - 5256 \) Copy content Toggle raw display
$7$ \( (T - 1)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} - 3 T^{11} + \cdots + 56832 \) Copy content Toggle raw display
$13$ \( T^{12} - 6 T^{11} + \cdots + 2483704 \) Copy content Toggle raw display
$17$ \( T^{12} - 3 T^{11} + \cdots + 4198848 \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( T^{12} - 12 T^{11} + \cdots - 20544 \) Copy content Toggle raw display
$29$ \( T^{12} + 6 T^{11} + \cdots - 3319296 \) Copy content Toggle raw display
$31$ \( T^{12} + 3 T^{11} + \cdots - 32997376 \) Copy content Toggle raw display
$37$ \( T^{12} + 27 T^{11} + \cdots - 7091712 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 1993061184 \) Copy content Toggle raw display
$43$ \( T^{12} - 30 T^{11} + \cdots - 21992896 \) Copy content Toggle raw display
$47$ \( T^{12} - 21 T^{11} + \cdots + 6744576 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 332453376 \) Copy content Toggle raw display
$59$ \( T^{12} - 27 T^{11} + \cdots - 11454423 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 531001664 \) Copy content Toggle raw display
$67$ \( T^{12} + 9 T^{11} + \cdots + 7338176 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 127877837592 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 3241283264 \) Copy content Toggle raw display
$79$ \( T^{12} - 12 T^{11} + \cdots - 979192 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots - 6892045047 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots - 6220575552 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots - 6948749312 \) Copy content Toggle raw display
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