Properties

Label 49.22.a.e
Level $49$
Weight $22$
Character orbit 49.a
Self dual yes
Analytic conductor $136.944$
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] = [49,22,Mod(1,49)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(49, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("49.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 49.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: \(136.943898701\)
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: \( x^{10} - 1979868308 x^{8} + \cdots - 34\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{13}\cdot 5\cdot 7^{10} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - 46) q^{2} - \beta_1 q^{3} + (\beta_{3} - 404 \beta_{2} + 524612) q^{4} + ( - \beta_{5} - 18 \beta_1) q^{5} + (\beta_{7} - 2 \beta_{5} - 513 \beta_1) q^{6} + ( - 4 \beta_{4} - 1034 \beta_{3} - 249304 \beta_{2} + \cdots - 984935016) q^{8}+ \cdots + (17 \beta_{6} - 21 \beta_{4} + 1801 \beta_{3} + 3040860 \beta_{2} + \cdots + 3794698618) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 46) q^{2} - \beta_1 q^{3} + (\beta_{3} - 404 \beta_{2} + 524612) q^{4} + ( - \beta_{5} - 18 \beta_1) q^{5} + (\beta_{7} - 2 \beta_{5} - 513 \beta_1) q^{6} + ( - 4 \beta_{4} - 1034 \beta_{3} - 249304 \beta_{2} + \cdots - 984935016) q^{8}+ \cdots + (147936608123 \beta_{6} - 281587903494 \beta_{4} + \cdots + 45\!\cdots\!19) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 460 q^{2} + 5246120 q^{4} - 9849350160 q^{8} + 37946986146 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 460 q^{2} + 5246120 q^{4} - 9849350160 q^{8} + 37946986146 q^{9} - 31261752856 q^{11} + 2576031066144 q^{15} - 17090753001952 q^{16} + 77933284833060 q^{18} - 229993408699760 q^{22} - 462501482091920 q^{23} + 331660226812942 q^{25} - 30\!\cdots\!24 q^{29}+ \cdots + 45\!\cdots\!44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 1979868308 x^{8} + \cdots - 34\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 6\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 24\!\cdots\!09 \nu^{8} + \cdots - 36\!\cdots\!00 ) / 77\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 23\!\cdots\!27 \nu^{8} + \cdots + 37\!\cdots\!00 ) / 38\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 52\!\cdots\!17 \nu^{8} + \cdots - 83\!\cdots\!00 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 68\!\cdots\!61 \nu^{9} + \cdots - 17\!\cdots\!00 \nu ) / 83\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 16\!\cdots\!63 \nu^{8} + \cdots - 30\!\cdots\!00 ) / 38\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12\!\cdots\!81 \nu^{9} + \cdots + 18\!\cdots\!00 \nu ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 28\!\cdots\!17 \nu^{9} + \cdots - 35\!\cdots\!00 \nu ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 62\!\cdots\!33 \nu^{9} + \cdots + 10\!\cdots\!00 \nu ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 17\beta_{6} - 21\beta_{4} + 1801\beta_{3} + 3040860\beta_{2} + 14255051821 ) / 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 1957\beta_{9} - 5574\beta_{8} - 3926122\beta_{7} + 18596423\beta_{5} + 13942363211\beta_1 ) / 108 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 19324118198 \beta_{6} - 22058369679 \beta_{4} + 3215684993929 \beta_{3} + \cdots + 11\!\cdots\!14 ) / 36 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5345669235701 \beta_{9} - 16878946365582 \beta_{8} + \cdots + 27\!\cdots\!13 \beta_1 ) / 216 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 11\!\cdots\!26 \beta_{6} + \cdots + 55\!\cdots\!58 ) / 18 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 30\!\cdots\!77 \beta_{9} + \cdots + 15\!\cdots\!41 \beta_1 ) / 108 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 62\!\cdots\!62 \beta_{6} + \cdots + 30\!\cdots\!26 ) / 9 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 17\!\cdots\!79 \beta_{9} + \cdots + 84\!\cdots\!87 \beta_1 ) / 54 \) Copy content Toggle raw display

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
14743.6
−14743.6
19997.9
−19997.9
14665.0
−14665.0
4022.09
−4022.09
33635.3
−33635.3
−2387.05 −88461.8 3.60085e6 2.49047e7 2.11163e8 0 −3.58939e9 −2.63486e9 −5.94487e10
1.2 −2387.05 88461.8 3.60085e6 −2.49047e7 −2.11163e8 0 −3.58939e9 −2.63486e9 5.94487e10
1.3 −1282.25 −119988. −452999. −2.26317e7 1.53853e8 0 3.26992e9 3.93665e9 2.90193e10
1.4 −1282.25 119988. −452999. 2.26317e7 −1.53853e8 0 3.26992e9 3.93665e9 −2.90193e10
1.5 47.8215 −87990.2 −2.09487e6 1.61037e7 −4.20782e6 0 −2.00469e8 −2.71808e9 7.70103e8
1.6 47.8215 87990.2 −2.09487e6 −1.61037e7 4.20782e6 0 −2.00469e8 −2.71808e9 −7.70103e8
1.7 1614.05 −24132.5 508019. 3.07690e7 −3.89512e7 0 −2.56495e9 −9.87797e9 4.96628e10
1.8 1614.05 24132.5 508019. −3.07690e7 3.89512e7 0 −2.56495e9 −9.87797e9 −4.96628e10
1.9 1777.42 −201812. 1.06206e6 −1.45438e7 −3.58704e8 0 −1.83979e9 3.02678e10 −2.58504e10
1.10 1777.42 201812. 1.06206e6 1.45438e7 3.58704e8 0 −1.83979e9 3.02678e10 2.58504e10
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.22.a.e 10
7.b odd 2 1 inner 49.22.a.e 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.22.a.e 10 1.a even 1 1 trivial
49.22.a.e 10 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(49))\):

\( T_{2}^{5} + 230T_{2}^{4} - 6527960T_{2}^{3} + 457635200T_{2}^{2} + 8773926866944T_{2} - 419916764938240 \) Copy content Toggle raw display
\( T_{3}^{10} - 71275259088 T_{3}^{8} + \cdots - 20\!\cdots\!00 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{5} + 230 T^{4} + \cdots - 419916764938240)^{2} \) Copy content Toggle raw display
$3$ \( T^{10} - 71275259088 T^{8} + \cdots - 20\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( (T^{5} + 15630876428 T^{4} + \cdots - 50\!\cdots\!92)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots - 46\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots - 75\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots - 34\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{5} + 231250741045960 T^{4} + \cdots + 12\!\cdots\!20)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots - 60\!\cdots\!44)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots - 40\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots + 22\!\cdots\!40)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots - 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 45\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 20\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{5} + \cdots - 53\!\cdots\!80)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots - 74\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots - 86\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 35\!\cdots\!56)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 31\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 84\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 29\!\cdots\!00 \) Copy content Toggle raw display
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