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Properties

Label 405.3.l.i
Level $405$
Weight $3$
Character orbit 405.l
Analytic conductor $11.035$
Analytic rank $0$
Dimension $8$
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] = [405,3,Mod(28,405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(405, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([4, 9]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("405.28");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 405.l (of order \(12\), degree \(4\), 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: \(11.0354507066\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: 8.0.49787136.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_{4} + 1) q^{2} + (\beta_{7} - \beta_{6} - \beta_{5} + \cdots - \beta_1) q^{4}+ \cdots + ( - 2 \beta_{4} + 4 \beta_{3} + \cdots - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_{4} + 1) q^{2} + (\beta_{7} - \beta_{6} - \beta_{5} + \cdots - \beta_1) q^{4}+ \cdots + ( - 85 \beta_{6} - 111 \beta_{4} + \cdots - 111) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{2} + 6 q^{5} + 26 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{2} + 6 q^{5} + 26 q^{7} + 44 q^{10} - 28 q^{13} + 120 q^{14} + 8 q^{16} + 24 q^{20} + 46 q^{22} + 60 q^{23} + 32 q^{25} + 136 q^{28} + 120 q^{29} + 148 q^{31} - 168 q^{32} + 20 q^{37} - 54 q^{38} - 48 q^{40} + 2 q^{43} + 232 q^{46} + 6 q^{47} + 42 q^{50} - 112 q^{52} + 52 q^{55} - 66 q^{58} - 528 q^{59} + 112 q^{61} - 84 q^{65} - 76 q^{67} + 156 q^{68} + 396 q^{70} + 260 q^{73} - 852 q^{74} - 204 q^{76} - 318 q^{77} + 340 q^{82} + 582 q^{83} + 194 q^{85} - 72 q^{88} - 896 q^{91} + 228 q^{92} + 342 q^{95} - 274 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{6} + 5\nu^{4} - 5\nu^{2} + 20\nu - 12 ) / 20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 5\nu^{5} - 5\nu^{3} - 12\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{6} + 5\nu^{4} + 15\nu^{2} + 36 ) / 20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{7} - 5\nu^{5} + 5\nu^{3} - 16\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{7} + \nu^{6} - 5\nu^{4} + 5\nu^{2} + 6\nu + 12 ) / 20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{7} - 4\nu^{6} - 5\nu^{5} - 15\nu^{3} - 16\nu - 28 ) / 20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9\nu^{7} - 8\nu^{6} + 15\nu^{5} + 25\nu^{3} + 48\nu - 56 ) / 40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} - \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} + 2\beta_{3} - \beta_{2} - \beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - \beta_{6} + 5\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -3\beta_{5} + 3\beta_{4} + 2\beta_{3} + 3\beta_{2} + 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{7} - \beta_{6} + \beta_{5} + 11\beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -5\beta_{7} - 5\beta_{6} - 5\beta_{4} - 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -7\beta_{5} - 13\beta_{4} - 13\beta_{2} - 7\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).

\(n\) \(82\) \(326\)
\(\chi(n)\) \(-\beta_{4}\) \(-1 + \beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1
1.09445 + 0.895644i
−0.228425 − 1.39564i
1.09445 − 0.895644i
−0.228425 + 1.39564i
−1.09445 + 0.895644i
0.228425 − 1.39564i
−1.09445 − 0.895644i
0.228425 + 1.39564i
−0.624069 − 0.167219i 0 −3.10260 − 1.79129i −4.64741 + 1.84434i 0 5.74921 + 1.54050i 3.46410 + 3.46410i 0 3.20871 − 0.373864i
28.2 2.99009 + 0.801193i 0 4.83465 + 2.79129i 3.54933 + 3.52168i 0 12.0091 + 3.21783i 3.46410 + 3.46410i 0 7.79129 + 13.3739i
217.1 −0.624069 + 0.167219i 0 −3.10260 + 1.79129i −4.64741 − 1.84434i 0 5.74921 − 1.54050i 3.46410 − 3.46410i 0 3.20871 + 0.373864i
217.2 2.99009 − 0.801193i 0 4.83465 − 2.79129i 3.54933 − 3.52168i 0 12.0091 − 3.21783i 3.46410 − 3.46410i 0 7.79129 − 13.3739i
298.1 −0.167219 − 0.624069i 0 3.10260 − 1.79129i −0.726456 + 4.94694i 0 −1.54050 − 5.74921i −3.46410 − 3.46410i 0 3.20871 − 0.373864i
298.2 0.801193 + 2.99009i 0 −4.83465 + 2.79129i 4.82453 − 1.31297i 0 −3.21783 − 12.0091i −3.46410 − 3.46410i 0 7.79129 + 13.3739i
352.1 −0.167219 + 0.624069i 0 3.10260 + 1.79129i −0.726456 − 4.94694i 0 −1.54050 + 5.74921i −3.46410 + 3.46410i 0 3.20871 + 0.373864i
352.2 0.801193 − 2.99009i 0 −4.83465 − 2.79129i 4.82453 + 1.31297i 0 −3.21783 + 12.0091i −3.46410 + 3.46410i 0 7.79129 − 13.3739i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 28.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
15.e even 4 1 inner
45.k odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.3.l.i 8
3.b odd 2 1 405.3.l.e 8
5.c odd 4 1 405.3.l.e 8
9.c even 3 1 405.3.g.d ✓ 8
9.c even 3 1 405.3.l.e 8
9.d odd 6 1 405.3.g.d ✓ 8
9.d odd 6 1 inner 405.3.l.i 8
15.e even 4 1 inner 405.3.l.i 8
45.k odd 12 1 405.3.g.d ✓ 8
45.k odd 12 1 inner 405.3.l.i 8
45.l even 12 1 405.3.g.d ✓ 8
45.l even 12 1 405.3.l.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.3.g.d ✓ 8 9.c even 3 1
405.3.g.d ✓ 8 9.d odd 6 1
405.3.g.d ✓ 8 45.k odd 12 1
405.3.g.d ✓ 8 45.l even 12 1
405.3.l.e 8 3.b odd 2 1
405.3.l.e 8 5.c odd 4 1
405.3.l.e 8 9.c even 3 1
405.3.l.e 8 45.l even 12 1
405.3.l.i 8 1.a even 1 1 trivial
405.3.l.i 8 9.d odd 6 1 inner
405.3.l.i 8 15.e even 4 1 inner
405.3.l.i 8 45.k odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 6T_{2}^{7} + 18T_{2}^{6} - 36T_{2}^{5} + 8T_{2}^{4} + 72T_{2}^{3} + 72T_{2}^{2} + 48T_{2} + 16 \) acting on \(S_{3}^{\mathrm{new}}(405, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 6 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 6 T^{7} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{8} - 26 T^{7} + \cdots + 29986576 \) Copy content Toggle raw display
$11$ \( T^{8} + 110 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{8} + 28 T^{7} + \cdots + 9834496 \) Copy content Toggle raw display
$17$ \( T^{8} + 172412 T^{4} + 1336336 \) Copy content Toggle raw display
$19$ \( (T^{4} + 666 T^{2} + 2025)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 60 T^{7} + \cdots + 78074896 \) Copy content Toggle raw display
$29$ \( (T^{4} - 60 T^{3} + \cdots + 71289)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 37 T + 1369)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 10 T^{3} + \cdots + 21325924)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 1550 T^{6} + \cdots + 62742241 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 2601446410000 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 121173610000 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{4} + 264 T^{3} + \cdots + 27468081)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 56 T^{3} + \cdots + 784)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 76 T^{7} + \cdots + 6250000 \) Copy content Toggle raw display
$71$ \( (T^{4} - 5270 T^{2} + 5851561)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 130 T^{3} + \cdots + 4072324)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 38\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{4} + 9630 T^{2} + 18584721)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 77\!\cdots\!76 \) Copy content Toggle raw display
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