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Properties

Label 784.2.m.d
Level $784$
Weight $2$
Character orbit 784.m
Analytic conductor $6.260$
Analytic rank $1$
Dimension $4$
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] = [784,2,Mod(197,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("784.197");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.m (of order \(4\), degree \(2\), 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: \(6.26027151847\)
Analytic rank: \(1\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - \beta_{2}) q^{2} + \beta_{3} q^{3} + ( - \beta_{3} - \beta_1) q^{4} + ( - 2 \beta_{2} + \beta_1 - 2) q^{5} + ( - \beta_{3} - \beta_{2}) q^{6} + (2 \beta_{2} - 2) q^{8} + ( - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - \beta_{2}) q^{2} + \beta_{3} q^{3} + ( - \beta_{3} - \beta_1) q^{4} + ( - 2 \beta_{2} + \beta_1 - 2) q^{5} + ( - \beta_{3} - \beta_{2}) q^{6} + (2 \beta_{2} - 2) q^{8} + ( - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{9} + ( - \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 2) q^{10} + (2 \beta_{2} - 3 \beta_1 + 2) q^{11} + (\beta_{3} + 2 \beta_{2} - \beta_1) q^{12} + ( - 2 \beta_{3} + 3 \beta_{2} - 3) q^{13} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots + 3) q^{15}+ \cdots + (2 \beta_{3} - 5 \beta_{2} + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{3} - 6 q^{5} + 2 q^{6} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 2 q^{3} - 6 q^{5} + 2 q^{6} - 8 q^{8} + 6 q^{10} + 2 q^{11} - 4 q^{12} - 8 q^{13} + 12 q^{15} + 8 q^{16} - 12 q^{17} + 8 q^{18} + 2 q^{19} - 12 q^{20} - 18 q^{22} + 8 q^{26} - 2 q^{27} - 8 q^{29} - 18 q^{30} - 8 q^{31} + 8 q^{32} - 20 q^{33} - 6 q^{34} + 4 q^{36} - 14 q^{37} - 2 q^{38} + 24 q^{40} + 12 q^{43} + 20 q^{44} - 14 q^{46} - 24 q^{47} + 8 q^{48} - 16 q^{50} - 6 q^{51} + 20 q^{52} - 22 q^{53} + 18 q^{54} + 8 q^{58} + 2 q^{59} + 24 q^{60} - 10 q^{61} - 2 q^{62} + 12 q^{65} + 22 q^{66} + 18 q^{67} + 24 q^{68} - 10 q^{69} - 8 q^{72} + 30 q^{74} - 20 q^{75} + 16 q^{76} + 8 q^{78} - 32 q^{79} - 24 q^{80} + 8 q^{81} - 20 q^{82} - 20 q^{83} + 6 q^{85} + 24 q^{86} - 8 q^{88} - 4 q^{92} - 2 q^{93} + 18 q^{94} - 24 q^{95} - 8 q^{96} - 16 q^{97} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{2} + \zeta_{12} \) Copy content Toggle raw display

Display \(\zeta_{12}^j\) in terms of \(\beta_i\)

\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( -\beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{12}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 − 0.500000i
0.866025 − 0.500000i
−1.36603 + 0.366025i −1.36603 + 1.36603i 1.73205 − 1.00000i −2.36603 − 2.36603i 1.36603 − 2.36603i 0 −2.00000 + 2.00000i − 0.732051i 4.09808 + 2.36603i
197.2 0.366025 − 1.36603i 0.366025 − 0.366025i −1.73205 − 1.00000i −0.633975 − 0.633975i −0.366025 − 0.633975i 0 −2.00000 + 2.00000i 2.73205i −1.09808 + 0.633975i
589.1 −1.36603 − 0.366025i −1.36603 − 1.36603i 1.73205 + 1.00000i −2.36603 + 2.36603i 1.36603 + 2.36603i 0 −2.00000 − 2.00000i 0.732051i 4.09808 − 2.36603i
589.2 0.366025 + 1.36603i 0.366025 + 0.366025i −1.73205 + 1.00000i −0.633975 + 0.633975i −0.366025 + 0.633975i 0 −2.00000 − 2.00000i − 2.73205i −1.09808 − 0.633975i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.m.d 4
7.b odd 2 1 784.2.m.e 4
7.c even 3 1 784.2.x.a 4
7.c even 3 1 784.2.x.h 4
7.d odd 6 1 112.2.w.a ✓ 4
7.d odd 6 1 112.2.w.b yes 4
16.e even 4 1 inner 784.2.m.d 4
28.f even 6 1 448.2.ba.a 4
28.f even 6 1 448.2.ba.b 4
56.j odd 6 1 896.2.ba.b 4
56.j odd 6 1 896.2.ba.d 4
56.m even 6 1 896.2.ba.a 4
56.m even 6 1 896.2.ba.c 4
112.l odd 4 1 784.2.m.e 4
112.v even 12 1 448.2.ba.a 4
112.v even 12 1 448.2.ba.b 4
112.v even 12 1 896.2.ba.a 4
112.v even 12 1 896.2.ba.c 4
112.w even 12 1 784.2.x.a 4
112.w even 12 1 784.2.x.h 4
112.x odd 12 1 112.2.w.a ✓ 4
112.x odd 12 1 112.2.w.b yes 4
112.x odd 12 1 896.2.ba.b 4
112.x odd 12 1 896.2.ba.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.w.a ✓ 4 7.d odd 6 1
112.2.w.a ✓ 4 112.x odd 12 1
112.2.w.b yes 4 7.d odd 6 1
112.2.w.b yes 4 112.x odd 12 1
448.2.ba.a 4 28.f even 6 1
448.2.ba.a 4 112.v even 12 1
448.2.ba.b 4 28.f even 6 1
448.2.ba.b 4 112.v even 12 1
784.2.m.d 4 1.a even 1 1 trivial
784.2.m.d 4 16.e even 4 1 inner
784.2.m.e 4 7.b odd 2 1
784.2.m.e 4 112.l odd 4 1
784.2.x.a 4 7.c even 3 1
784.2.x.a 4 112.w even 12 1
784.2.x.h 4 7.c even 3 1
784.2.x.h 4 112.w even 12 1
896.2.ba.a 4 56.m even 6 1
896.2.ba.a 4 112.v even 12 1
896.2.ba.b 4 56.j odd 6 1
896.2.ba.b 4 112.x odd 12 1
896.2.ba.c 4 56.m even 6 1
896.2.ba.c 4 112.v even 12 1
896.2.ba.d 4 56.j odd 6 1
896.2.ba.d 4 112.x odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(784, [\chi])\):

\( T_{3}^{4} + 2T_{3}^{3} + 2T_{3}^{2} - 2T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{4} + 6T_{5}^{3} + 18T_{5}^{2} + 18T_{5} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{4} + 6 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 2 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$13$ \( T^{4} + 8 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$17$ \( (T^{2} + 6 T - 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 2 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$23$ \( T^{4} + 26T^{2} + 121 \) Copy content Toggle raw display
$29$ \( T^{4} + 8 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$31$ \( (T^{2} + 4 T + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 14 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$41$ \( T^{4} + 104T^{2} + 1936 \) Copy content Toggle raw display
$43$ \( T^{4} - 12 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$47$ \( (T^{2} + 12 T + 33)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 22 T^{3} + \cdots + 2209 \) Copy content Toggle raw display
$59$ \( T^{4} - 2 T^{3} + \cdots + 14641 \) Copy content Toggle raw display
$61$ \( T^{4} + 10 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{4} - 18 T^{3} + \cdots + 1521 \) Copy content Toggle raw display
$71$ \( T^{4} + 56T^{2} + 16 \) Copy content Toggle raw display
$73$ \( T^{4} + 62T^{2} + 529 \) Copy content Toggle raw display
$79$ \( (T^{2} + 16 T - 11)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 20 T^{3} + \cdots + 676 \) Copy content Toggle raw display
$89$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 8 T - 32)^{2} \) Copy content Toggle raw display
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