Properties

Label 378.4.d.b
Level $378$
Weight $4$
Character orbit 378.d
Analytic conductor $22.303$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [378,4,Mod(377,378)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(378, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("378.377");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 378.d (of order \(2\), degree \(1\), 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: \(22.3027219822\)
Analytic rank: \(0\)
Dimension: \(4\)
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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

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

\(f(q)\) \(=\) \( q + 2 \beta_1 q^{2} - 4 q^{4} + 2 \beta_{3} q^{5} + ( - 7 \beta_{2} + 14) q^{7} - 8 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_1 q^{2} - 4 q^{4} + 2 \beta_{3} q^{5} + ( - 7 \beta_{2} + 14) q^{7} - 8 \beta_1 q^{8} + 4 \beta_{2} q^{10} - 30 \beta_1 q^{11} + 25 \beta_{2} q^{13} + (14 \beta_{3} + 28 \beta_1) q^{14} + 16 q^{16} - 61 \beta_{3} q^{17} - 52 \beta_{2} q^{19} - 8 \beta_{3} q^{20} + 60 q^{22} - 201 \beta_1 q^{23} - 113 q^{25} - 50 \beta_{3} q^{26} + (28 \beta_{2} - 56) q^{28} + 111 \beta_1 q^{29} - 117 \beta_{2} q^{31} + 32 \beta_1 q^{32} - 122 \beta_{2} q^{34} + (28 \beta_{3} - 42 \beta_1) q^{35} + 430 q^{37} + 104 \beta_{3} q^{38} - 16 \beta_{2} q^{40} - 68 \beta_{3} q^{41} + 355 q^{43} + 120 \beta_1 q^{44} + 402 q^{46} - 340 \beta_{3} q^{47} + ( - 196 \beta_{2} + 49) q^{49} - 226 \beta_1 q^{50} - 100 \beta_{2} q^{52} - 33 \beta_1 q^{53} - 60 \beta_{2} q^{55} + ( - 56 \beta_{3} - 112 \beta_1) q^{56} - 222 q^{58} + 337 \beta_{3} q^{59} - 400 \beta_{2} q^{61} + 234 \beta_{3} q^{62} - 64 q^{64} + 150 \beta_1 q^{65} + 575 q^{67} + 244 \beta_{3} q^{68} + (56 \beta_{2} + 84) q^{70} - 159 \beta_1 q^{71} - 28 \beta_{2} q^{73} + 860 \beta_1 q^{74} + 208 \beta_{2} q^{76} + ( - 210 \beta_{3} - 420 \beta_1) q^{77} - 376 q^{79} + 32 \beta_{3} q^{80} - 136 \beta_{2} q^{82} - 134 \beta_{3} q^{83} - 366 q^{85} + 710 \beta_1 q^{86} - 240 q^{88} + 327 \beta_{3} q^{89} + (350 \beta_{2} + 525) q^{91} + 804 \beta_1 q^{92} - 680 \beta_{2} q^{94} - 312 \beta_1 q^{95} + 364 \beta_{2} q^{97} + (392 \beta_{3} + 98 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{4} + 56 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{4} + 56 q^{7} + 64 q^{16} + 240 q^{22} - 452 q^{25} - 224 q^{28} + 1720 q^{37} + 1420 q^{43} + 1608 q^{46} + 196 q^{49} - 888 q^{58} - 256 q^{64} + 2300 q^{67} + 336 q^{70} - 1504 q^{79} - 1464 q^{85} - 960 q^{88} + 2100 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(325\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
377.1
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
2.00000i 0 −4.00000 −3.46410 0 14.0000 12.1244i 8.00000i 0 6.92820i
377.2 2.00000i 0 −4.00000 3.46410 0 14.0000 + 12.1244i 8.00000i 0 6.92820i
377.3 2.00000i 0 −4.00000 −3.46410 0 14.0000 + 12.1244i 8.00000i 0 6.92820i
377.4 2.00000i 0 −4.00000 3.46410 0 14.0000 12.1244i 8.00000i 0 6.92820i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.4.d.b 4
3.b odd 2 1 inner 378.4.d.b 4
7.b odd 2 1 inner 378.4.d.b 4
21.c even 2 1 inner 378.4.d.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.4.d.b 4 1.a even 1 1 trivial
378.4.d.b 4 3.b odd 2 1 inner
378.4.d.b 4 7.b odd 2 1 inner
378.4.d.b 4 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 12 \) acting on \(S_{4}^{\mathrm{new}}(378, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 28 T + 343)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 900)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1875)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 11163)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 8112)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 40401)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 12321)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 41067)^{2} \) Copy content Toggle raw display
$37$ \( (T - 430)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 13872)^{2} \) Copy content Toggle raw display
$43$ \( (T - 355)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 346800)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 1089)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 340707)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 480000)^{2} \) Copy content Toggle raw display
$67$ \( (T - 575)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 25281)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 2352)^{2} \) Copy content Toggle raw display
$79$ \( (T + 376)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 53868)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 320787)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 397488)^{2} \) Copy content Toggle raw display
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