Properties

Label 378.4.f.a
Level $378$
Weight $4$
Character orbit 378.f
Analytic conductor $22.303$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [378,4,Mod(127,378)]
 
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([4, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("378.127");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 378.f (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: \(22.3027219822\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: no (minimal twist has level 126)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_{3} q^{2} + (4 \beta_{3} - 4) q^{4} + ( - \beta_{5} + \beta_{4} - 3 \beta_{3} + \cdots + 3) q^{5}+ \cdots + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_{3} q^{2} + (4 \beta_{3} - 4) q^{4} + ( - \beta_{5} + \beta_{4} - 3 \beta_{3} + \cdots + 3) q^{5}+ \cdots + 98 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} - 12 q^{4} + 9 q^{5} - 21 q^{7} + 48 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{2} - 12 q^{4} + 9 q^{5} - 21 q^{7} + 48 q^{8} - 36 q^{10} + 48 q^{11} + 57 q^{13} - 42 q^{14} - 48 q^{16} - 48 q^{17} - 282 q^{19} + 36 q^{20} + 96 q^{22} - 30 q^{23} + 210 q^{25} - 228 q^{26} + 168 q^{28} + 117 q^{29} + 165 q^{31} - 96 q^{32} + 48 q^{34} - 126 q^{35} - 798 q^{37} + 282 q^{38} + 72 q^{40} - 384 q^{41} + 771 q^{43} - 384 q^{44} + 120 q^{46} + 81 q^{47} - 147 q^{49} + 420 q^{50} + 228 q^{52} - 1044 q^{53} + 156 q^{55} - 168 q^{56} + 234 q^{58} + 21 q^{59} + 780 q^{61} - 660 q^{62} + 384 q^{64} + 531 q^{65} + 384 q^{67} + 96 q^{68} + 126 q^{70} - 114 q^{71} - 234 q^{73} + 798 q^{74} + 564 q^{76} + 336 q^{77} + 1383 q^{79} - 288 q^{80} + 1536 q^{82} + 12 q^{83} + 1095 q^{85} + 1542 q^{86} + 384 q^{88} + 1296 q^{89} - 798 q^{91} - 120 q^{92} + 162 q^{94} - 1314 q^{95} - 741 q^{97} + 588 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} - \nu^{4} + 5\nu^{3} - 2\nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} - 11\nu^{3} + 17\nu^{2} - 12\nu + 3 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{5} - 5\nu^{4} + 16\nu^{3} - 19\nu^{2} + 21\nu - 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 2\nu^{3} + 8\nu^{2} - 7\nu + 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7\nu^{5} - 16\nu^{4} + 62\nu^{3} - 68\nu^{2} + 99\nu - 30 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + \beta_{3} - 2\beta _1 - 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} + \beta_{4} - 6\beta_{3} - 4\beta_{2} + \beta _1 - 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{5} - 3\beta_{4} - 13\beta_{3} - \beta_{2} + 11\beta _1 + 19 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3\beta_{5} - 6\beta_{4} + 19\beta_{3} + 19\beta_{2} + 11\beta _1 + 37 ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(325\)
\(\chi(n)\) \(-1 + \beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
0.500000 2.05195i
0.500000 + 1.41036i
0.500000 0.224437i
0.500000 + 2.05195i
0.500000 1.41036i
0.500000 + 0.224437i
−1.00000 + 1.73205i 0 −2.00000 3.46410i −3.21780 5.57339i 0 −3.50000 + 6.06218i 8.00000 0 12.8712
127.2 −1.00000 + 1.73205i 0 −2.00000 3.46410i 3.11273 + 5.39140i 0 −3.50000 + 6.06218i 8.00000 0 −12.4509
127.3 −1.00000 + 1.73205i 0 −2.00000 3.46410i 4.60507 + 7.97622i 0 −3.50000 + 6.06218i 8.00000 0 −18.4203
253.1 −1.00000 1.73205i 0 −2.00000 + 3.46410i −3.21780 + 5.57339i 0 −3.50000 6.06218i 8.00000 0 12.8712
253.2 −1.00000 1.73205i 0 −2.00000 + 3.46410i 3.11273 5.39140i 0 −3.50000 6.06218i 8.00000 0 −12.4509
253.3 −1.00000 1.73205i 0 −2.00000 + 3.46410i 4.60507 7.97622i 0 −3.50000 6.06218i 8.00000 0 −18.4203
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.3
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.4.f.a 6
3.b odd 2 1 126.4.f.a 6
9.c even 3 1 inner 378.4.f.a 6
9.c even 3 1 1134.4.a.h 3
9.d odd 6 1 126.4.f.a 6
9.d odd 6 1 1134.4.a.g 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.f.a 6 3.b odd 2 1
126.4.f.a 6 9.d odd 6 1
378.4.f.a 6 1.a even 1 1 trivial
378.4.f.a 6 9.c even 3 1 inner
1134.4.a.g 3 9.d odd 6 1
1134.4.a.h 3 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 9T_{5}^{5} + 123T_{5}^{4} - 360T_{5}^{3} + 5085T_{5}^{2} - 15498T_{5} + 136161 \) acting on \(S_{4}^{\mathrm{new}}(378, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 4)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 9 T^{5} + \cdots + 136161 \) Copy content Toggle raw display
$7$ \( (T^{2} + 7 T + 49)^{3} \) Copy content Toggle raw display
$11$ \( T^{6} - 48 T^{5} + \cdots + 79548561 \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 1824400369 \) Copy content Toggle raw display
$17$ \( (T^{3} + 24 T^{2} + \cdots - 157239)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + 141 T^{2} + \cdots - 106169)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 104760326889 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 78328896129 \) Copy content Toggle raw display
$31$ \( T^{6} - 165 T^{5} + \cdots + 559369801 \) Copy content Toggle raw display
$37$ \( (T^{3} + 399 T^{2} + \cdots - 18438443)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 264917119401 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 40798917084409 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 57\!\cdots\!61 \) Copy content Toggle raw display
$53$ \( (T^{3} + 522 T^{2} + \cdots - 106645401)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 71\!\cdots\!61 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 837556418281489 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 16\!\cdots\!69 \) Copy content Toggle raw display
$71$ \( (T^{3} + 57 T^{2} + \cdots + 141832269)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 117 T^{2} + \cdots - 225440003)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 32\!\cdots\!89 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 43\!\cdots\!89 \) Copy content Toggle raw display
$89$ \( (T^{3} - 648 T^{2} + \cdots + 21052413)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 66\!\cdots\!09 \) Copy content Toggle raw display
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