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Properties

Label 2254.4.a.u
Level $2254$
Weight $4$
Character orbit 2254.a
Self dual yes
Analytic conductor $132.990$
Analytic rank $0$
Dimension $11$
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] = [2254,4,Mod(1,2254)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2254, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2254.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2254 = 2 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2254.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: \(132.990305153\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 234 x^{9} - 105 x^{8} + 18997 x^{7} + 16513 x^{6} - 621598 x^{5} - 743169 x^{4} + \cdots - 12103441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 7 \)
Twist minimal: no (minimal twist has level 322)
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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} + \beta_1 q^{3} + 4 q^{4} + (\beta_{2} + 2) q^{5} - 2 \beta_1 q^{6} - 8 q^{8} + (\beta_{8} + \beta_{7} + \beta_1 + 16) q^{9} + ( - 2 \beta_{2} - 4) q^{10} + ( - \beta_{7} - \beta_{4} - \beta_{3} - 5) q^{11}+ \cdots + (8 \beta_{10} - 19 \beta_{8} + \cdots - 254) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q - 22 q^{2} + 44 q^{4} + 23 q^{5} - 88 q^{8} + 171 q^{9} - 46 q^{10} - 48 q^{11} + 77 q^{13} + 104 q^{15} + 176 q^{16} + 97 q^{17} - 342 q^{18} + 138 q^{19} + 92 q^{20} + 96 q^{22} + 253 q^{23} + 30 q^{25}+ \cdots - 2545 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - 234 x^{9} - 105 x^{8} + 18997 x^{7} + 16513 x^{6} - 621598 x^{5} - 743169 x^{4} + \cdots - 12103441 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 124970979065318 \nu^{10} - 315523500145021 \nu^{9} + \cdots + 32\!\cdots\!58 ) / 36\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 175883848495615 \nu^{10} - 186335656266977 \nu^{9} + \cdots + 54\!\cdots\!54 ) / 36\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 9311891222740 \nu^{10} + 25085860949219 \nu^{9} + \cdots + 60\!\cdots\!48 ) / 13\!\cdots\!93 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 258931730435996 \nu^{10} + \cdots + 37\!\cdots\!84 ) / 36\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 372009683931775 \nu^{10} + \cdots + 88\!\cdots\!41 ) / 36\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 58800937298822 \nu^{10} - 206018849405971 \nu^{9} + \cdots - 46\!\cdots\!73 ) / 51\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 58800937298822 \nu^{10} + 206018849405971 \nu^{9} + \cdots + 24\!\cdots\!34 ) / 51\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 147048562972945 \nu^{10} - 509547597313958 \nu^{9} + \cdots - 69\!\cdots\!48 ) / 12\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 59711715036880 \nu^{10} + 266000450260034 \nu^{9} + \cdots + 26\!\cdots\!02 ) / 40\!\cdots\!79 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + \beta_{7} + \beta _1 + 43 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{10} - 4\beta_{9} - \beta_{8} + 2\beta_{6} + 2\beta_{5} - 3\beta_{4} - 2\beta_{2} + 74\beta _1 + 32 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 14 \beta_{10} - 15 \beta_{9} + 99 \beta_{8} + 95 \beta_{7} + 4 \beta_{6} - 4 \beta_{5} + 9 \beta_{4} + \cdots + 3091 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 214 \beta_{10} - 553 \beta_{9} - 99 \beta_{8} + 111 \beta_{7} + 171 \beta_{6} + 187 \beta_{5} + \cdots + 4100 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2165 \beta_{10} - 2747 \beta_{9} + 8801 \beta_{8} + 8681 \beta_{7} + 480 \beta_{6} - 952 \beta_{5} + \cdots + 251033 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 20555 \beta_{10} - 62505 \beta_{9} - 7770 \beta_{8} + 20003 \beta_{7} + 13432 \beta_{6} + \cdots + 450339 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 250324 \beta_{10} - 362434 \beta_{9} + 774955 \beta_{8} + 791590 \beta_{7} + 40797 \beta_{6} + \cdots + 21375735 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1922848 \beta_{10} - 6596013 \beta_{9} - 539016 \beta_{8} + 2652487 \beta_{7} + 1070084 \beta_{6} + \cdots + 48605475 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 26171915 \beta_{10} - 42327131 \beta_{9} + 68470741 \beta_{8} + 72681124 \beta_{7} + \cdots + 1871359727 \) 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
−9.31831
−8.73782
−5.89611
−3.73826
−3.39536
−0.423740
2.27413
3.24819
7.30492
8.71952
9.96286
−2.00000 −9.31831 4.00000 10.7136 18.6366 0 −8.00000 59.8309 −21.4272
1.2 −2.00000 −8.73782 4.00000 1.29770 17.4756 0 −8.00000 49.3496 −2.59539
1.3 −2.00000 −5.89611 4.00000 −12.9445 11.7922 0 −8.00000 7.76415 25.8891
1.4 −2.00000 −3.73826 4.00000 −15.3436 7.47652 0 −8.00000 −13.0254 30.6872
1.5 −2.00000 −3.39536 4.00000 2.94232 6.79072 0 −8.00000 −15.4715 −5.88465
1.6 −2.00000 −0.423740 4.00000 18.0271 0.847481 0 −8.00000 −26.8204 −36.0542
1.7 −2.00000 2.27413 4.00000 14.8127 −4.54825 0 −8.00000 −21.8284 −29.6254
1.8 −2.00000 3.24819 4.00000 −5.66055 −6.49637 0 −8.00000 −16.4493 11.3211
1.9 −2.00000 7.30492 4.00000 7.55338 −14.6098 0 −8.00000 26.3619 −15.1068
1.10 −2.00000 8.71952 4.00000 −10.2018 −17.4390 0 −8.00000 49.0300 20.4036
1.11 −2.00000 9.96286 4.00000 11.8037 −19.9257 0 −8.00000 72.2585 −23.6074
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(7\) \( -1 \)
\(23\) \( -1 \)

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 2254.4.a.u 11
7.b odd 2 1 2254.4.a.r 11
7.d odd 6 2 322.4.e.d ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.4.e.d ✓ 22 7.d odd 6 2
2254.4.a.r 11 7.b odd 2 1
2254.4.a.u 11 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{11} - 234 T_{3}^{9} - 105 T_{3}^{8} + 18997 T_{3}^{7} + 16513 T_{3}^{6} - 621598 T_{3}^{5} + \cdots - 12103441 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2254))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{11} \) Copy content Toggle raw display
$3$ \( T^{11} - 234 T^{9} + \cdots - 12103441 \) Copy content Toggle raw display
$5$ \( T^{11} + \cdots - 11170323063 \) Copy content Toggle raw display
$7$ \( T^{11} \) Copy content Toggle raw display
$11$ \( T^{11} + \cdots - 94646114017749 \) Copy content Toggle raw display
$13$ \( T^{11} + \cdots + 94\!\cdots\!61 \) Copy content Toggle raw display
$17$ \( T^{11} + \cdots + 49\!\cdots\!93 \) Copy content Toggle raw display
$19$ \( T^{11} + \cdots - 63\!\cdots\!97 \) Copy content Toggle raw display
$23$ \( (T - 23)^{11} \) Copy content Toggle raw display
$29$ \( T^{11} + \cdots + 71\!\cdots\!71 \) Copy content Toggle raw display
$31$ \( T^{11} + \cdots + 24\!\cdots\!39 \) Copy content Toggle raw display
$37$ \( T^{11} + \cdots - 45\!\cdots\!57 \) Copy content Toggle raw display
$41$ \( T^{11} + \cdots + 76\!\cdots\!17 \) Copy content Toggle raw display
$43$ \( T^{11} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$47$ \( T^{11} + \cdots - 43\!\cdots\!23 \) Copy content Toggle raw display
$53$ \( T^{11} + \cdots + 76\!\cdots\!89 \) Copy content Toggle raw display
$59$ \( T^{11} + \cdots + 18\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{11} + \cdots + 14\!\cdots\!57 \) Copy content Toggle raw display
$67$ \( T^{11} + \cdots - 34\!\cdots\!73 \) Copy content Toggle raw display
$71$ \( T^{11} + \cdots + 36\!\cdots\!93 \) Copy content Toggle raw display
$73$ \( T^{11} + \cdots - 43\!\cdots\!81 \) Copy content Toggle raw display
$79$ \( T^{11} + \cdots + 10\!\cdots\!07 \) Copy content Toggle raw display
$83$ \( T^{11} + \cdots - 17\!\cdots\!47 \) Copy content Toggle raw display
$89$ \( T^{11} + \cdots - 29\!\cdots\!79 \) Copy content Toggle raw display
$97$ \( T^{11} + \cdots - 17\!\cdots\!01 \) Copy content Toggle raw display
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