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Properties

Label 6900.2.a.bd
Level $6900$
Weight $2$
Character orbit 6900.a
Self dual yes
Analytic conductor $55.097$
Analytic rank $0$
Dimension $7$
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] = [6900,2,Mod(1,6900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6900, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("6900.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6900.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: \(55.0967773947\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 13x^{5} + 11x^{4} + 46x^{3} - 32x^{2} - 30x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1380)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta_1 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + \beta_1 q^{7} + q^{9} + ( - \beta_{4} - \beta_{2} + \beta_1) q^{11} + ( - \beta_{5} - \beta_{2}) q^{13} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{17} + ( - \beta_{6} - \beta_{4} + \beta_{3} - 1) q^{19} + \beta_1 q^{21} - q^{23} + q^{27} + (\beta_{5} + \beta_{3} + \beta_{2} + \cdots + 2) q^{29}+ \cdots + ( - \beta_{4} - \beta_{2} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 7 q^{3} - q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 7 q^{3} - q^{7} + 7 q^{9} - 2 q^{13} + q^{17} - 2 q^{19} - q^{21} - 7 q^{23} + 7 q^{27} + 15 q^{29} + 3 q^{31} + 19 q^{37} - 2 q^{39} + 23 q^{41} - 4 q^{43} + 10 q^{47} + 10 q^{49} + q^{51} + 11 q^{53} - 2 q^{57} + 5 q^{59} + 32 q^{61} - q^{63} + 15 q^{67} - 7 q^{69} + 21 q^{71} - 18 q^{73} + 28 q^{77} + 16 q^{79} + 7 q^{81} + 25 q^{83} + 15 q^{87} + 26 q^{89} + 14 q^{91} + 3 q^{93} + 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 13x^{5} + 11x^{4} + 46x^{3} - 32x^{2} - 30x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{4} + 2\nu^{3} + 7\nu^{2} - 10\nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 2\nu^{3} - 7\nu^{2} + 12\nu + 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} - 11\nu^{4} - 2\nu^{3} + 28\nu^{2} + 4\nu - 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} - 2\nu^{5} - 9\nu^{4} + 14\nu^{3} + 18\nu^{2} - 16\nu - 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\nu^{5} + 2\nu^{4} + 8\nu^{3} - 12\nu^{2} - 10\nu + 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{6} - \beta_{5} + \beta_{4} + 5\beta_{3} + 6\beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} - \beta_{5} + \beta_{4} + 7\beta _1 + 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 5\beta_{6} - 6\beta_{5} + 6\beta_{4} + 15\beta_{3} + 19\beta_{2} + 2\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12\beta_{6} - 12\beta_{5} + 14\beta_{4} + 3\beta_{3} + 4\beta_{2} + 49\beta _1 + 170 \) 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
−0.0729221
−0.511427
1.52925
2.04690
−2.39625
−2.47928
2.88373
0 1.00000 0 0 0 −3.99468 0 1.00000 0
1.2 0 1.00000 0 0 0 −3.73844 0 1.00000 0
1.3 0 1.00000 0 0 0 −1.66138 0 1.00000 0
1.4 0 1.00000 0 0 0 0.189781 0 1.00000 0
1.5 0 1.00000 0 0 0 1.74202 0 1.00000 0
1.6 0 1.00000 0 0 0 2.14682 0 1.00000 0
1.7 0 1.00000 0 0 0 4.31589 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(23\) \(1\)

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 6900.2.a.bd 7
5.b even 2 1 6900.2.a.bc 7
5.c odd 4 2 1380.2.f.b ✓ 14
15.e even 4 2 4140.2.f.c 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.f.b ✓ 14 5.c odd 4 2
4140.2.f.c 14 15.e even 4 2
6900.2.a.bc 7 5.b even 2 1
6900.2.a.bd 7 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6900))\):

\( T_{7}^{7} + T_{7}^{6} - 29T_{7}^{5} - 21T_{7}^{4} + 220T_{7}^{3} + 20T_{7}^{2} - 412T_{7} + 76 \) Copy content Toggle raw display
\( T_{11}^{7} - 64T_{11}^{5} + 62T_{11}^{4} + 1200T_{11}^{3} - 2112T_{11}^{2} - 4608T_{11} + 9216 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \) Copy content Toggle raw display
$3$ \( (T - 1)^{7} \) Copy content Toggle raw display
$5$ \( T^{7} \) Copy content Toggle raw display
$7$ \( T^{7} + T^{6} + \cdots + 76 \) Copy content Toggle raw display
$11$ \( T^{7} - 64 T^{5} + \cdots + 9216 \) Copy content Toggle raw display
$13$ \( T^{7} + 2 T^{6} + \cdots - 2464 \) Copy content Toggle raw display
$17$ \( T^{7} - T^{6} + \cdots + 100 \) Copy content Toggle raw display
$19$ \( T^{7} + 2 T^{6} + \cdots + 36680 \) Copy content Toggle raw display
$23$ \( (T + 1)^{7} \) Copy content Toggle raw display
$29$ \( T^{7} - 15 T^{6} + \cdots + 12272 \) Copy content Toggle raw display
$31$ \( T^{7} - 3 T^{6} + \cdots + 54720 \) Copy content Toggle raw display
$37$ \( T^{7} - 19 T^{6} + \cdots - 126992 \) Copy content Toggle raw display
$41$ \( T^{7} - 23 T^{6} + \cdots + 15248 \) Copy content Toggle raw display
$43$ \( T^{7} + 4 T^{6} + \cdots + 19728 \) Copy content Toggle raw display
$47$ \( T^{7} - 10 T^{6} + \cdots + 1404640 \) Copy content Toggle raw display
$53$ \( T^{7} - 11 T^{6} + \cdots + 19348 \) Copy content Toggle raw display
$59$ \( T^{7} - 5 T^{6} + \cdots + 1483200 \) Copy content Toggle raw display
$61$ \( T^{7} - 32 T^{6} + \cdots + 377248 \) Copy content Toggle raw display
$67$ \( T^{7} - 15 T^{6} + \cdots - 548 \) Copy content Toggle raw display
$71$ \( T^{7} - 21 T^{6} + \cdots - 784 \) Copy content Toggle raw display
$73$ \( T^{7} + 18 T^{6} + \cdots - 217088 \) Copy content Toggle raw display
$79$ \( T^{7} - 16 T^{6} + \cdots + 4755584 \) Copy content Toggle raw display
$83$ \( T^{7} - 25 T^{6} + \cdots - 1280 \) Copy content Toggle raw display
$89$ \( T^{7} - 26 T^{6} + \cdots + 526112 \) Copy content Toggle raw display
$97$ \( T^{7} - 6 T^{6} + \cdots + 7488 \) Copy content Toggle raw display
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