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Properties

Label 9054.2.a.bj
Level $9054$
Weight $2$
Character orbit 9054.a
Self dual yes
Analytic conductor $72.297$
Analytic rank $1$
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] = [9054,2,Mod(1,9054)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9054, 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("9054.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9054 = 2 \cdot 3^{2} \cdot 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9054.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: \(72.2965539901\)
Analytic rank: \(1\)
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: \( x^{12} - 4 x^{11} - 22 x^{10} + 67 x^{9} + 180 x^{8} - 383 x^{7} - 674 x^{6} + 875 x^{5} + 1077 x^{4} + \cdots + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1006)
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + (\beta_{10} - \beta_{7}) q^{5} + ( - \beta_{10} + \beta_{7} + \cdots + \beta_{2}) q^{7}+ \cdots + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + (\beta_{10} - \beta_{7}) q^{5} + ( - \beta_{10} + \beta_{7} + \cdots + \beta_{2}) q^{7}+ \cdots + (2 \beta_{11} + 2 \beta_{10} + \cdots + 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{2} + 12 q^{4} - 7 q^{5} - 2 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{2} + 12 q^{4} - 7 q^{5} - 2 q^{7} + 12 q^{8} - 7 q^{10} - 18 q^{11} - 4 q^{13} - 2 q^{14} + 12 q^{16} - 12 q^{17} - 7 q^{20} - 18 q^{22} + 9 q^{23} + 25 q^{25} - 4 q^{26} - 2 q^{28} - 34 q^{29} - 11 q^{31} + 12 q^{32} - 12 q^{34} - 21 q^{35} - 22 q^{37} - 7 q^{40} - 32 q^{41} - 8 q^{43} - 18 q^{44} + 9 q^{46} - 24 q^{47} + 36 q^{49} + 25 q^{50} - 4 q^{52} + 2 q^{53} - 12 q^{55} - 2 q^{56} - 34 q^{58} - 26 q^{59} + 12 q^{61} - 11 q^{62} + 12 q^{64} - 66 q^{65} - 21 q^{67} - 12 q^{68} - 21 q^{70} - 50 q^{71} + 17 q^{73} - 22 q^{74} - 25 q^{77} - 9 q^{79} - 7 q^{80} - 32 q^{82} - 25 q^{83} + 24 q^{85} - 8 q^{86} - 18 q^{88} - 21 q^{89} - 9 q^{91} + 9 q^{92} - 24 q^{94} - 22 q^{95} + 18 q^{97} + 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} - 22 x^{10} + 67 x^{9} + 180 x^{8} - 383 x^{7} - 674 x^{6} + 875 x^{5} + 1077 x^{4} + \cdots + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 75810347 \nu^{11} - 750391815 \nu^{10} + 1111655415 \nu^{9} + 8946386707 \nu^{8} + \cdots - 21112206658 ) / 7685328593 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 85937175 \nu^{11} - 183905264 \nu^{10} - 2549240479 \nu^{9} + 2711003021 \nu^{8} + \cdots - 7917809651 ) / 7685328593 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 106132083 \nu^{11} - 842730380 \nu^{10} + 89329233 \nu^{9} + 11763798682 \nu^{8} + \cdots - 34213508272 ) / 7685328593 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 146823889 \nu^{11} + 471833772 \nu^{10} + 3883703902 \nu^{9} - 8276421068 \nu^{8} + \cdots + 8817559953 ) / 7685328593 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 159236033 \nu^{11} + 320476659 \nu^{10} + 4957239696 \nu^{9} - 5371489680 \nu^{8} + \cdots + 3243266922 ) / 7685328593 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 194027177 \nu^{11} + 1001308266 \nu^{10} + 2733553715 \nu^{9} - 14135059823 \nu^{8} + \cdots - 485767656 ) / 7685328593 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 206991323 \nu^{11} - 573051401 \nu^{10} - 5843045880 \nu^{9} + 9800401503 \nu^{8} + \cdots + 11406893253 ) / 7685328593 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 210621704 \nu^{11} + 1410256576 \nu^{10} + 1550724992 \nu^{9} - 21641699784 \nu^{8} + \cdots - 34320821610 ) / 7685328593 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 246499891 \nu^{11} - 1029238772 \nu^{10} - 4971149628 \nu^{9} + 16303798652 \nu^{8} + \cdots + 24583412642 ) / 7685328593 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 566115680 \nu^{11} - 2117638831 \nu^{10} - 12926378732 \nu^{9} + 34046046658 \nu^{8} + \cdots + 37969929527 ) / 7685328593 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 678429148 \nu^{11} + 3136864086 \nu^{10} + 11829253654 \nu^{9} - 46947000260 \nu^{8} + \cdots - 915307019 ) / 7685328593 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - \beta_{8} - \beta_{5} - \beta_{4} + \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{9} - 2\beta_{8} - 3\beta_{7} + \beta_{6} - 2\beta_{5} - 4\beta_{4} - \beta_{3} + 3\beta_{2} + \beta _1 + 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6 \beta_{11} - 2 \beta_{10} - 13 \beta_{9} - 18 \beta_{8} - 5 \beta_{7} + 5 \beta_{6} - 12 \beta_{5} + \cdots + 28 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4 \beta_{11} - 14 \beta_{10} - 83 \beta_{9} - 78 \beta_{8} - 49 \beta_{7} + 21 \beta_{6} - 50 \beta_{5} + \cdots + 184 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 54 \beta_{11} - 80 \beta_{10} - 413 \beta_{9} - 442 \beta_{8} - 173 \beta_{7} + 121 \beta_{6} + \cdots + 744 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 114 \beta_{11} - 454 \beta_{10} - 2185 \beta_{9} - 2162 \beta_{8} - 1033 \beta_{7} + 533 \beta_{6} + \cdots + 4060 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 804 \beta_{11} - 2316 \beta_{10} - 11057 \beta_{9} - 11262 \beta_{8} - 4717 \beta_{7} + 2873 \beta_{6} + \cdots + 19366 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 3012 \beta_{11} - 12194 \beta_{10} - 56555 \beta_{9} - 56556 \beta_{8} - 24987 \beta_{7} + 13785 \beta_{6} + \cdots + 100286 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 16918 \beta_{11} - 61660 \beta_{10} - 286719 \beta_{9} - 288812 \beta_{8} - 123013 \beta_{7} + \cdots + 500102 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 78302 \beta_{11} - 316360 \beta_{10} - 1456547 \beta_{9} - 1460328 \beta_{8} - 631427 \beta_{7} + \cdots + 2551408 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 409448 \beta_{11} - 1601440 \beta_{10} - 7385239 \beta_{9} - 7418958 \beta_{8} - 3174907 \beta_{7} + \cdots + 12876656 ) / 2 \) 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
0.396940
−1.33502
1.84513
0.753640
−0.875563
−1.97018
−2.01674
2.29094
−2.75875
2.60861
5.07033
−0.00933287
1.00000 0 1.00000 −4.23481 0 −1.79853 1.00000 0 −4.23481
1.2 1.00000 0 1.00000 −4.07967 0 −0.827612 1.00000 0 −4.07967
1.3 1.00000 0 1.00000 −2.93695 0 3.10313 1.00000 0 −2.93695
1.4 1.00000 0 1.00000 −2.79920 0 1.31469 1.00000 0 −2.79920
1.5 1.00000 0 1.00000 −2.23842 0 4.41277 1.00000 0 −2.23842
1.6 1.00000 0 1.00000 −1.77374 0 −3.87316 1.00000 0 −1.77374
1.7 1.00000 0 1.00000 0.847598 0 −2.65658 1.00000 0 0.847598
1.8 1.00000 0 1.00000 1.07414 0 4.58087 1.00000 0 1.07414
1.9 1.00000 0 1.00000 1.48375 0 2.26909 1.00000 0 1.48375
1.10 1.00000 0 1.00000 1.53308 0 −4.18455 1.00000 0 1.53308
1.11 1.00000 0 1.00000 2.50185 0 0.0804128 1.00000 0 2.50185
1.12 1.00000 0 1.00000 3.62237 0 −4.42054 1.00000 0 3.62237
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(503\) \(1\)

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 9054.2.a.bj 12
3.b odd 2 1 1006.2.a.i ✓ 12
12.b even 2 1 8048.2.a.r 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1006.2.a.i ✓ 12 3.b odd 2 1
8048.2.a.r 12 12.b even 2 1
9054.2.a.bj 12 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9054))\):

\( T_{5}^{12} + 7 T_{5}^{11} - 18 T_{5}^{10} - 195 T_{5}^{9} + 31 T_{5}^{8} + 1922 T_{5}^{7} + 584 T_{5}^{6} + \cdots + 10584 \) Copy content Toggle raw display
\( T_{7}^{12} + 2 T_{7}^{11} - 58 T_{7}^{10} - 113 T_{7}^{9} + 1213 T_{7}^{8} + 2224 T_{7}^{7} + \cdots + 4263 \) Copy content Toggle raw display
\( T_{11}^{12} + 18 T_{11}^{11} + 74 T_{11}^{10} - 427 T_{11}^{9} - 3693 T_{11}^{8} - 1417 T_{11}^{7} + \cdots + 32144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 7 T^{11} + \cdots + 10584 \) Copy content Toggle raw display
$7$ \( T^{12} + 2 T^{11} + \cdots + 4263 \) Copy content Toggle raw display
$11$ \( T^{12} + 18 T^{11} + \cdots + 32144 \) Copy content Toggle raw display
$13$ \( T^{12} + 4 T^{11} + \cdots + 238928 \) Copy content Toggle raw display
$17$ \( T^{12} + 12 T^{11} + \cdots + 1179136 \) Copy content Toggle raw display
$19$ \( T^{12} - 169 T^{10} + \cdots - 14590592 \) Copy content Toggle raw display
$23$ \( T^{12} - 9 T^{11} + \cdots - 3244067 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 231936768 \) Copy content Toggle raw display
$31$ \( T^{12} + 11 T^{11} + \cdots - 23817472 \) Copy content Toggle raw display
$37$ \( T^{12} + 22 T^{11} + \cdots + 4886848 \) Copy content Toggle raw display
$41$ \( T^{12} + 32 T^{11} + \cdots - 417024 \) Copy content Toggle raw display
$43$ \( T^{12} + 8 T^{11} + \cdots + 3607792 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 107614057 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 395038848 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots - 26353645568 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 903709552 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots - 455670064 \) Copy content Toggle raw display
$71$ \( T^{12} + 50 T^{11} + \cdots - 257024 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots - 2309284344 \) Copy content Toggle raw display
$79$ \( T^{12} + 9 T^{11} + \cdots - 57869696 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 4260038928 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 117284352 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots - 26594882712 \) Copy content Toggle raw display
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