Properties

Label 378.3.s.d
Level $378$
Weight $3$
Character orbit 378.s
Analytic conductor $10.300$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [378,3,Mod(53,378)]
 
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([3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("378.53");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 378.s (of order \(6\), 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: \(10.2997539928\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.621801639936.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 34x^{6} + 116x^{5} + 413x^{4} - 1024x^{3} - 1664x^{2} + 2196x + 4467 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + (2 \beta_1 + 2) q^{4} - \beta_{2} q^{5} + ( - \beta_{7} + 2 \beta_1 - 2) q^{7} + (2 \beta_{5} + 2 \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + (2 \beta_1 + 2) q^{4} - \beta_{2} q^{5} + ( - \beta_{7} + 2 \beta_1 - 2) q^{7} + (2 \beta_{5} + 2 \beta_{3}) q^{8} + ( - \beta_{7} + \beta_1 + 1) q^{10} + (2 \beta_{5} + 2 \beta_{4} + \cdots + 2 \beta_{2}) q^{11}+ \cdots + ( - 8 \beta_{4} + 21 \beta_{3} + 8 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4} - 24 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{4} - 24 q^{7} + 4 q^{10} + 56 q^{13} - 16 q^{16} + 52 q^{19} + 80 q^{22} - 24 q^{25} - 48 q^{28} - 28 q^{31} + 56 q^{34} + 4 q^{37} - 8 q^{40} - 176 q^{43} + 92 q^{46} - 100 q^{49} + 56 q^{52} - 256 q^{55} - 68 q^{58} + 152 q^{61} - 64 q^{64} + 180 q^{67} - 172 q^{70} - 264 q^{73} + 208 q^{76} + 392 q^{79} + 36 q^{82} - 712 q^{85} + 80 q^{88} - 316 q^{91} + 84 q^{94} + 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 34x^{6} + 116x^{5} + 413x^{4} - 1024x^{3} - 1664x^{2} + 2196x + 4467 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{6} - 6\nu^{5} - 91\nu^{4} + 192\nu^{3} + 1187\nu^{2} - 1284\nu - 4425 ) / 2418 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 919 \nu^{7} - 16681 \nu^{6} + 82598 \nu^{5} + 496373 \nu^{4} - 1354204 \nu^{3} - 4328389 \nu^{2} + \cdots + 2711814 ) / 2749266 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 724 \nu^{7} - 2534 \nu^{6} - 27020 \nu^{5} + 73885 \nu^{4} + 387688 \nu^{3} - 656684 \nu^{2} + \cdots + 872256 ) / 1374633 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1643 \nu^{7} + 27922 \nu^{6} - 16589 \nu^{5} - 804335 \nu^{4} + 922099 \nu^{3} + \cdots - 20444505 ) / 2749266 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1838 \nu^{7} - 6433 \nu^{6} - 45811 \nu^{5} + 130610 \nu^{4} + 262721 \nu^{3} - 527908 \nu^{2} + \cdots - 542487 ) / 2749266 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1448 \nu^{7} + 5068 \nu^{6} + 54040 \nu^{5} - 147770 \nu^{4} - 775376 \nu^{3} + 1313368 \nu^{2} + \cdots - 3119145 ) / 1374633 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 8224 \nu^{7} + 28784 \nu^{6} + 291734 \nu^{5} - 801295 \nu^{4} - 3006376 \nu^{3} + \cdots - 1196985 ) / 2749266 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} + 2\beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + 4\beta_{5} + 4\beta_{4} + 4\beta_{3} + 4\beta_{2} - 4\beta _1 + 19 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{7} + 8\beta_{6} + 5\beta_{5} + 3\beta_{4} + 32\beta_{3} + 3\beta_{2} - 3\beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 12\beta_{7} + 31\beta_{6} + 96\beta_{5} + 96\beta_{4} + 168\beta_{3} + 80\beta_{2} - 236\beta _1 + 191 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 220\beta_{7} + 309\beta_{6} + 598\beta_{5} + 230\beta_{4} + 1490\beta_{3} + 190\beta_{2} - 580\beta _1 + 431 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 315\beta_{7} + 425\beta_{6} + 1414\beta_{5} + 1054\beta_{4} + 2440\beta_{3} + 630\beta_{2} - 3555\beta _1 + 543 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 5978 \beta_{7} + 5783 \beta_{6} + 20692 \beta_{5} + 6580 \beta_{4} + 32898 \beta_{3} + 3752 \beta_{2} + \cdots + 2325 ) / 2 \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
4.76613 + 0.707107i
−1.31664 + 0.707107i
−3.76613 0.707107i
2.31664 0.707107i
4.76613 0.707107i
−1.31664 0.707107i
−3.76613 + 0.707107i
2.31664 + 0.707107i
−1.22474 + 0.707107i 0 1.00000 1.73205i −4.33729 + 2.50413i 0 0.0413813 6.99988i 2.82843i 0 3.54138 6.13385i
53.2 −1.22474 + 0.707107i 0 1.00000 1.73205i 3.11254 1.79703i 0 −6.04138 + 3.53578i 2.82843i 0 −2.54138 + 4.40180i
53.3 1.22474 0.707107i 0 1.00000 1.73205i −3.11254 + 1.79703i 0 −6.04138 + 3.53578i 2.82843i 0 −2.54138 + 4.40180i
53.4 1.22474 0.707107i 0 1.00000 1.73205i 4.33729 2.50413i 0 0.0413813 6.99988i 2.82843i 0 3.54138 6.13385i
107.1 −1.22474 0.707107i 0 1.00000 + 1.73205i −4.33729 2.50413i 0 0.0413813 + 6.99988i 2.82843i 0 3.54138 + 6.13385i
107.2 −1.22474 0.707107i 0 1.00000 + 1.73205i 3.11254 + 1.79703i 0 −6.04138 3.53578i 2.82843i 0 −2.54138 4.40180i
107.3 1.22474 + 0.707107i 0 1.00000 + 1.73205i −3.11254 1.79703i 0 −6.04138 3.53578i 2.82843i 0 −2.54138 4.40180i
107.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i 4.33729 + 2.50413i 0 0.0413813 + 6.99988i 2.82843i 0 3.54138 + 6.13385i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.3.s.d 8
3.b odd 2 1 inner 378.3.s.d 8
7.c even 3 1 inner 378.3.s.d 8
21.h odd 6 1 inner 378.3.s.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.3.s.d 8 1.a even 1 1 trivial
378.3.s.d 8 3.b odd 2 1 inner
378.3.s.d 8 7.c even 3 1 inner
378.3.s.d 8 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 38T_{5}^{6} + 1120T_{5}^{4} - 12312T_{5}^{2} + 104976 \) acting on \(S_{3}^{\mathrm{new}}(378, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 38 T^{6} + \cdots + 104976 \) Copy content Toggle raw display
$7$ \( (T^{4} + 12 T^{3} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 248 T^{6} + \cdots + 331776 \) Copy content Toggle raw display
$13$ \( (T^{2} - 14 T + 12)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 36804120336 \) Copy content Toggle raw display
$19$ \( (T^{2} - 13 T + 169)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 3662186256 \) Copy content Toggle raw display
$29$ \( (T^{4} + 1214 T^{2} + 101124)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 14 T^{3} + \cdots + 294849)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 2 T^{3} + \cdots + 8976016)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 414 T^{2} + 15876)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 44 T - 2513)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 2667616624656 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 208074970438416 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 240005038473216 \) Copy content Toggle raw display
$61$ \( (T^{4} - 76 T^{3} + \cdots + 136161)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 90 T^{3} + \cdots + 44944)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 17048 T^{2} + 70157376)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 132 T^{3} + \cdots + 18653761)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 196 T^{3} + \cdots + 81216144)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 33344 T^{2} + 204604416)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 22\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( (T^{2} - 12 T - 1)^{4} \) Copy content Toggle raw display
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