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Properties

Label 3.37.b.b
Level $3$
Weight $37$
Character orbit 3.b
Analytic conductor $24.627$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3,37,Mod(2,3)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 37, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3.2");
 
S:= CuspForms(chi, 37);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 37 \)
Character orbit: \([\chi]\) \(=\) 3.b (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(24.6273775978\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 1732091138 x^{8} + \cdots + 87\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{44}\cdot 3^{71}\cdot 5^{4}\cdot 7^{2}\cdot 13 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} - 69 \beta_1 - 55215675) q^{3} + (\beta_{3} + 26 \beta_{2} + \cdots - 43520029006) q^{4}+ \cdots + ( - 3 \beta_{9} + \cdots + 69\!\cdots\!75) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} - 69 \beta_1 - 55215675) q^{3} + (\beta_{3} + 26 \beta_{2} + \cdots - 43520029006) q^{4}+ \cdots + (27\!\cdots\!36 \beta_{9} + \cdots + 29\!\cdots\!51) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 552156750 q^{3} - 435200290064 q^{4} + 77238909590832 q^{6} - 12\!\cdots\!00 q^{7}+ \cdots + 69\!\cdots\!78 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 552156750 q^{3} - 435200290064 q^{4} + 77238909590832 q^{6} - 12\!\cdots\!00 q^{7}+ \cdots + 29\!\cdots\!20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 1732091138 x^{8} + \cdots + 87\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 18\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 11\!\cdots\!25 \nu^{9} + \cdots + 98\!\cdots\!80 ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15\!\cdots\!25 \nu^{9} + \cdots + 71\!\cdots\!60 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 18\!\cdots\!25 \nu^{9} + \cdots - 62\!\cdots\!80 ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15\!\cdots\!99 \nu^{9} + \cdots + 50\!\cdots\!80 ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 35\!\cdots\!75 \nu^{9} + \cdots - 28\!\cdots\!60 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 44\!\cdots\!49 \nu^{9} + \cdots - 29\!\cdots\!60 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 98\!\cdots\!23 \nu^{9} + \cdots - 60\!\cdots\!60 ) / 89\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 19\!\cdots\!05 \nu^{9} + \cdots - 23\!\cdots\!40 ) / 35\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 26\beta_{2} + 5\beta _1 - 112239505742 ) / 324 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{9} - 18 \beta_{8} + 13 \beta_{7} - 186 \beta_{6} + 318 \beta_{5} + 2 \beta_{4} + 6231 \beta_{3} + \cdots + 2456 ) / 5832 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2805467 \beta_{8} + 2805467 \beta_{7} - 9264038 \beta_{6} - 1376724 \beta_{4} + \cdots + 24\!\cdots\!04 ) / 13122 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 33128433137 \beta_{9} + 1292482381266 \beta_{8} - 961846945181 \beta_{7} + \cdots - 129463204740568 ) / 236196 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 42\!\cdots\!89 \beta_{8} + \cdots - 19\!\cdots\!92 ) / 177147 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 35\!\cdots\!79 \beta_{9} + \cdots + 17\!\cdots\!72 ) / 354294 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 36\!\cdots\!86 \beta_{8} + \cdots + 12\!\cdots\!80 ) / 177147 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 12\!\cdots\!95 \beta_{9} + \cdots - 68\!\cdots\!96 ) / 177147 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
− 26735.5i
− 23576.4i
− 16809.5i
− 10236.4i
− 8608.85i
8608.85i
10236.4i
16809.5i
23576.4i
26735.5i
− 481239.i −3.00483e8 + 2.44550e8i −1.62872e11 − 1.10872e12i 1.17687e14 + 1.44604e14i 2.90621e15 4.53097e16i 3.04857e16 − 1.46966e17i −5.33560e17
2.2 − 424376.i 3.83451e8 − 5.53148e7i −1.11375e11 3.68903e12i −2.34743e13 − 1.62727e14i −2.28183e15 1.81020e16i 1.43975e17 − 4.24211e16i 1.56554e18
2.3 − 302571.i −5.37826e7 − 3.83669e8i −2.28297e10 − 4.03107e12i −1.16087e14 + 1.62730e13i 4.01364e14 − 1.38849e16i −1.44310e17 + 4.12694e16i −1.21969e18
2.4 − 184256.i 7.73223e7 + 3.79626e8i 3.47692e10 − 1.78906e12i 6.99484e13 − 1.42471e13i −1.12077e15 − 1.90684e16i −1.38137e17 + 5.87071e16i −3.29645e17
2.5 − 154959.i −3.82586e8 − 6.10115e7i 4.47071e10 6.48901e12i −9.45430e12 + 5.92853e13i −5.18677e14 − 1.75765e16i 1.42650e17 + 4.66843e16i 1.00553e18
2.6 154959.i −3.82586e8 + 6.10115e7i 4.47071e10 − 6.48901e12i −9.45430e12 − 5.92853e13i −5.18677e14 1.75765e16i 1.42650e17 − 4.66843e16i 1.00553e18
2.7 184256.i 7.73223e7 − 3.79626e8i 3.47692e10 1.78906e12i 6.99484e13 + 1.42471e13i −1.12077e15 1.90684e16i −1.38137e17 − 5.87071e16i −3.29645e17
2.8 302571.i −5.37826e7 + 3.83669e8i −2.28297e10 4.03107e12i −1.16087e14 − 1.62730e13i 4.01364e14 1.38849e16i −1.44310e17 − 4.12694e16i −1.21969e18
2.9 424376.i 3.83451e8 + 5.53148e7i −1.11375e11 − 3.68903e12i −2.34743e13 + 1.62727e14i −2.28183e15 − 1.81020e16i 1.43975e17 + 4.24211e16i 1.56554e18
2.10 481239.i −3.00483e8 − 2.44550e8i −1.62872e11 1.10872e12i 1.17687e14 − 1.44604e14i 2.90621e15 − 4.53097e16i 3.04857e16 + 1.46966e17i −5.33560e17
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.37.b.b ✓ 10
3.b odd 2 1 inner 3.37.b.b ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.37.b.b ✓ 10 1.a even 1 1 trivial
3.37.b.b ✓ 10 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 561197528712 T_{2}^{8} + \cdots + 31\!\cdots\!00 \) acting on \(S_{37}^{\mathrm{new}}(3, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 76\!\cdots\!01 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{5} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{5} + \cdots + 44\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots + 29\!\cdots\!12)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 79\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots + 26\!\cdots\!88)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 93\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots - 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots - 22\!\cdots\!32)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 21\!\cdots\!92)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots - 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
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