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Properties

Label 49.10.c.f
Level $49$
Weight $10$
Character orbit 49.c
Analytic conductor $25.237$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [49,10,Mod(18,49)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(49, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("49.18");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 49.c (of order \(3\), degree \(2\), not 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: \(25.2367559720\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 674x^{6} + 399556x^{4} + 36881280x^{2} + 2994278400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{4}\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} - \beta_{3} - 3 \beta_1 + 3) q^{2} + \beta_{2} q^{3} + (6 \beta_{5} - 698 \beta_1) q^{4} + (\beta_{7} + \beta_{6} - \beta_{2}) q^{5} + ( - 5 \beta_{7} + 12 \beta_{6} + 5 \beta_{4}) q^{6}+ \cdots + (17954244 \beta_{3} + 657848592) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{2} - 2792 q^{4} - 62112 q^{8} - 128196 q^{9} - 163200 q^{11} + 259488 q^{15} + 356320 q^{16} + 2632860 q^{18} + 10127648 q^{22} - 1813968 q^{23} - 1232572 q^{25} + 8599824 q^{29} - 43019712 q^{30}+ \cdots + 5262788736 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 674x^{6} + 399556x^{4} + 36881280x^{2} + 2994278400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 337\nu^{6} + 199778\nu^{4} + 134650372\nu^{2} + 12428991360 ) / 10931852160 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 150910040\nu ) / 5593784 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - 97769092 ) / 2796892 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 385848968\nu ) / 2796892 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -86209\nu^{6} - 67325186\nu^{4} - 34445323204\nu^{2} - 3179498267520 ) / 76522965120 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -15839\nu^{7} - 9389566\nu^{5} - 5235382268\nu^{3} - 70365542400\nu ) / 15304593024 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 164593\nu^{7} + 105682562\nu^{5} + 65764120708\nu^{3} + 6070400519040\nu ) / 38261482560 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - 2\beta_{2} ) / 84 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 7\beta_{5} - 7\beta_{3} + 337\beta _1 - 337 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 235\beta_{7} + 1058\beta_{6} - 235\beta_{4} ) / 42 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -4718\beta_{5} - 172418\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -65515\beta_{7} - 329186\beta_{6} + 329186\beta_{2} ) / 21 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2796892\beta_{3} + 97769092 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 37727510\beta_{4} - 192924484\beta_{2} ) / 21 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1 + \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1
12.0373 − 20.8492i
−12.0373 + 20.8492i
−4.85829 + 8.41480i
4.85829 − 8.41480i
12.0373 + 20.8492i
−12.0373 − 20.8492i
−4.85829 − 8.41480i
4.85829 + 8.41480i
−15.8277 − 27.4144i −94.2248 + 163.202i −245.034 + 424.411i −916.910 − 1588.13i 5965.46 0 −694.289 −7915.13 − 13709.4i −29025.2 + 50273.1i
18.2 −15.8277 − 27.4144i 94.2248 − 163.202i −245.034 + 424.411i 916.910 + 1588.13i −5965.46 0 −694.289 −7915.13 − 13709.4i 29025.2 − 50273.1i
18.3 18.8277 + 32.6106i −130.337 + 225.750i −452.966 + 784.561i 538.433 + 932.593i −9815.78 0 −14833.7 −24133.9 − 41801.1i −20274.9 + 35117.2i
18.4 18.8277 + 32.6106i 130.337 − 225.750i −452.966 + 784.561i −538.433 − 932.593i 9815.78 0 −14833.7 −24133.9 − 41801.1i 20274.9 − 35117.2i
30.1 −15.8277 + 27.4144i −94.2248 − 163.202i −245.034 − 424.411i −916.910 + 1588.13i 5965.46 0 −694.289 −7915.13 + 13709.4i −29025.2 − 50273.1i
30.2 −15.8277 + 27.4144i 94.2248 + 163.202i −245.034 − 424.411i 916.910 − 1588.13i −5965.46 0 −694.289 −7915.13 + 13709.4i 29025.2 + 50273.1i
30.3 18.8277 − 32.6106i −130.337 − 225.750i −452.966 − 784.561i 538.433 − 932.593i −9815.78 0 −14833.7 −24133.9 + 41801.1i −20274.9 − 35117.2i
30.4 18.8277 − 32.6106i 130.337 + 225.750i −452.966 − 784.561i −538.433 + 932.593i 9815.78 0 −14833.7 −24133.9 + 41801.1i 20274.9 + 35117.2i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 18.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.10.c.f 8
7.b odd 2 1 inner 49.10.c.f 8
7.c even 3 1 49.10.a.d ✓ 4
7.c even 3 1 inner 49.10.c.f 8
7.d odd 6 1 49.10.a.d ✓ 4
7.d odd 6 1 inner 49.10.c.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.10.a.d ✓ 4 7.c even 3 1
49.10.a.d ✓ 4 7.d odd 6 1
49.10.c.f 8 1.a even 1 1 trivial
49.10.c.f 8 7.b odd 2 1 inner
49.10.c.f 8 7.c even 3 1 inner
49.10.c.f 8 7.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(49, [\chi])\):

\( T_{2}^{4} - 6T_{2}^{3} + 1228T_{2}^{2} + 7152T_{2} + 1420864 \) Copy content Toggle raw display
\( T_{3}^{8} + 103464T_{3}^{6} + 8291647296T_{3}^{4} + 249674358528000T_{3}^{2} + 5823302575104000000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 6 T^{3} + \cdots + 1420864)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 35\!\cdots\!96)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 55\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 21\!\cdots\!04)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + \cdots + 1085915549828)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 59\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 74\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + \cdots + 232157250991152)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 89\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 39\!\cdots\!76)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + \cdots + 697944613332992)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 27\!\cdots\!24)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
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