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Properties

Label 378.2.g.h
Level $378$
Weight $2$
Character orbit 378.g
Analytic conductor $3.018$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [378,2,Mod(109,378)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(378, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("378.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 378.g (of order \(3\), 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: \(3.01834519640\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{2} + \beta_{2} q^{4} + ( - \beta_{2} + \beta_1 - 1) q^{5} - \beta_{3} q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{2} + \beta_{2} q^{4} + ( - \beta_{2} + \beta_1 - 1) q^{5} - \beta_{3} q^{7} - q^{8} + (\beta_{3} - \beta_{2} + \beta_1) q^{10} + (\beta_{3} - \beta_{2} + \beta_1) q^{11} + ( - \beta_{3} - 2) q^{13} + \beta_1 q^{14} + ( - \beta_{2} - 1) q^{16} + (\beta_{3} - \beta_{2} + \beta_1) q^{17} + ( - 2 \beta_{2} - 2) q^{19} + (\beta_{3} + 1) q^{20} + (\beta_{3} + 1) q^{22} + (4 \beta_{2} + 2 \beta_1 + 4) q^{23} + ( - 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{25} + ( - 2 \beta_{2} + \beta_1 - 2) q^{26} + (\beta_{3} + \beta_1) q^{28} + (\beta_{3} - 5) q^{29} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{31} - \beta_{2} q^{32} + (\beta_{3} + 1) q^{34} + (7 \beta_{2} - \beta_1 + 7) q^{35} + (4 \beta_{2} - 3 \beta_1 + 4) q^{37} - 2 \beta_{2} q^{38} + (\beta_{2} - \beta_1 + 1) q^{40} + ( - 3 \beta_{3} - 3) q^{41} + 5 q^{43} + (\beta_{2} - \beta_1 + 1) q^{44} + (2 \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{46} + ( - 3 \beta_{2} - 3 \beta_1 - 3) q^{47} + 7 q^{49} + ( - 2 \beta_{3} - 3) q^{50} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{52} - 6 \beta_{2} q^{53} + ( - 2 \beta_{3} - 8) q^{55} + \beta_{3} q^{56} + ( - 5 \beta_{2} - \beta_1 - 5) q^{58} + ( - \beta_{3} - 11 \beta_{2} - \beta_1) q^{59} + ( - 10 \beta_{2} - \beta_1 - 10) q^{61} + (\beta_{3} + 2) q^{62} + q^{64} + (9 \beta_{2} - 3 \beta_1 + 9) q^{65} + (2 \beta_{3} + 3 \beta_{2} + 2 \beta_1) q^{67} + (\beta_{2} - \beta_1 + 1) q^{68} + ( - \beta_{3} + 7 \beta_{2} - \beta_1) q^{70} + ( - \beta_{3} - 13) q^{71} + ( - 4 \beta_{3} - 4 \beta_1) q^{73} + ( - 3 \beta_{3} + 4 \beta_{2} - 3 \beta_1) q^{74} + 2 q^{76} + ( - \beta_{3} + 7 \beta_{2} - \beta_1) q^{77} + (2 \beta_{2} + 5 \beta_1 + 2) q^{79} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{80} + ( - 3 \beta_{2} + 3 \beta_1 - 3) q^{82} + (2 \beta_{3} + 8) q^{83} + ( - 2 \beta_{3} - 8) q^{85} + (5 \beta_{2} + 5) q^{86} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{88} + ( - 3 \beta_{2} - 3 \beta_1 - 3) q^{89} + (2 \beta_{3} + 7) q^{91} + (2 \beta_{3} - 4) q^{92} + ( - 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{94} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{95} + (4 \beta_{3} + 3) q^{97} + (7 \beta_{2} + 7) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{4} - 2 q^{5} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{4} - 2 q^{5} - 4 q^{8} + 2 q^{10} + 2 q^{11} - 8 q^{13} - 2 q^{16} + 2 q^{17} - 4 q^{19} + 4 q^{20} + 4 q^{22} + 8 q^{23} - 6 q^{25} - 4 q^{26} - 20 q^{29} + 4 q^{31} + 2 q^{32} + 4 q^{34} + 14 q^{35} + 8 q^{37} + 4 q^{38} + 2 q^{40} - 12 q^{41} + 20 q^{43} + 2 q^{44} - 8 q^{46} - 6 q^{47} + 28 q^{49} - 12 q^{50} + 4 q^{52} + 12 q^{53} - 32 q^{55} - 10 q^{58} + 22 q^{59} - 20 q^{61} + 8 q^{62} + 4 q^{64} + 18 q^{65} - 6 q^{67} + 2 q^{68} - 14 q^{70} - 52 q^{71} - 8 q^{74} + 8 q^{76} - 14 q^{77} + 4 q^{79} - 2 q^{80} - 6 q^{82} + 32 q^{83} - 32 q^{85} + 10 q^{86} - 2 q^{88} - 6 q^{89} + 28 q^{91} - 16 q^{92} + 6 q^{94} - 4 q^{95} + 12 q^{97} + 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 7 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 7\beta_{3} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\)
\(\chi(n)\) \(1\) \(-1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
−1.32288 − 2.29129i
1.32288 + 2.29129i
−1.32288 + 2.29129i
1.32288 − 2.29129i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.82288 − 3.15731i 0 −2.64575 −1.00000 0 1.82288 − 3.15731i
109.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.822876 + 1.42526i 0 2.64575 −1.00000 0 −0.822876 + 1.42526i
163.1 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −1.82288 + 3.15731i 0 −2.64575 −1.00000 0 1.82288 + 3.15731i
163.2 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0.822876 − 1.42526i 0 2.64575 −1.00000 0 −0.822876 − 1.42526i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.2.g.h yes 4
3.b odd 2 1 378.2.g.g ✓ 4
7.c even 3 1 inner 378.2.g.h yes 4
7.c even 3 1 2646.2.a.bi 2
7.d odd 6 1 2646.2.a.bf 2
9.c even 3 1 1134.2.e.q 4
9.c even 3 1 1134.2.h.t 4
9.d odd 6 1 1134.2.e.t 4
9.d odd 6 1 1134.2.h.q 4
21.g even 6 1 2646.2.a.bo 2
21.h odd 6 1 378.2.g.g ✓ 4
21.h odd 6 1 2646.2.a.bl 2
63.g even 3 1 1134.2.e.q 4
63.h even 3 1 1134.2.h.t 4
63.j odd 6 1 1134.2.h.q 4
63.n odd 6 1 1134.2.e.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.g.g ✓ 4 3.b odd 2 1
378.2.g.g ✓ 4 21.h odd 6 1
378.2.g.h yes 4 1.a even 1 1 trivial
378.2.g.h yes 4 7.c even 3 1 inner
1134.2.e.q 4 9.c even 3 1
1134.2.e.q 4 63.g even 3 1
1134.2.e.t 4 9.d odd 6 1
1134.2.e.t 4 63.n odd 6 1
1134.2.h.q 4 9.d odd 6 1
1134.2.h.q 4 63.j odd 6 1
1134.2.h.t 4 9.c even 3 1
1134.2.h.t 4 63.h even 3 1
2646.2.a.bf 2 7.d odd 6 1
2646.2.a.bi 2 7.c even 3 1
2646.2.a.bl 2 21.h odd 6 1
2646.2.a.bo 2 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\):

\( T_{5}^{4} + 2T_{5}^{3} + 10T_{5}^{2} - 12T_{5} + 36 \) Copy content Toggle raw display
\( T_{11}^{4} - 2T_{11}^{3} + 10T_{11}^{2} + 12T_{11} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 2 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$7$ \( (T^{2} - 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 2 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$13$ \( (T^{2} + 4 T - 3)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 2 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$19$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 8 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$29$ \( (T^{2} + 10 T + 18)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 4 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$37$ \( T^{4} - 8 T^{3} + \cdots + 2209 \) Copy content Toggle raw display
$41$ \( (T^{2} + 6 T - 54)^{2} \) Copy content Toggle raw display
$43$ \( (T - 5)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 6 T^{3} + \cdots + 2916 \) Copy content Toggle raw display
$53$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 22 T^{3} + \cdots + 12996 \) Copy content Toggle raw display
$61$ \( T^{4} + 20 T^{3} + \cdots + 8649 \) Copy content Toggle raw display
$67$ \( T^{4} + 6 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$71$ \( (T^{2} + 26 T + 162)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 112 T^{2} + 12544 \) Copy content Toggle raw display
$79$ \( T^{4} - 4 T^{3} + \cdots + 29241 \) Copy content Toggle raw display
$83$ \( (T^{2} - 16 T + 36)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 6 T^{3} + \cdots + 2916 \) Copy content Toggle raw display
$97$ \( (T^{2} - 6 T - 103)^{2} \) Copy content Toggle raw display
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