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Properties

Label 405.3.l.h
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: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 15)
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.

\(f(q)\) \(=\) \( q + (\beta_{7} - \beta_{4} + \beta_{2} + \beta_1) q^{2} + (2 \beta_{6} + 2 \beta_{5} + \beta_1) q^{4} + ( - \beta_{6} + 2 \beta_{5} - \beta_{2} + \cdots + 1) q^{5}+ \cdots + (3 \beta_{4} + \beta_{3} + 3) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} - \beta_{4} + \beta_{2} + \beta_1) q^{2} + (2 \beta_{6} + 2 \beta_{5} + \beta_1) q^{4} + ( - \beta_{6} + 2 \beta_{5} - \beta_{2} + \cdots + 1) q^{5}+ \cdots + ( - 47 \beta_{4} - 43 \beta_{3} - 47) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} + 4 q^{5} - 4 q^{7} + 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} + 4 q^{5} - 4 q^{7} + 24 q^{8} + 8 q^{10} - 16 q^{11} + 32 q^{13} + 20 q^{16} - 80 q^{17} + 36 q^{20} - 20 q^{22} - 56 q^{23} - 16 q^{25} + 176 q^{26} + 88 q^{28} + 16 q^{31} + 76 q^{32} - 80 q^{35} + 128 q^{37} + 96 q^{38} - 48 q^{40} + 56 q^{41} + 8 q^{43} - 272 q^{46} - 128 q^{47} - 164 q^{50} + 80 q^{52} + 112 q^{53} - 248 q^{55} + 12 q^{58} - 200 q^{61} + 176 q^{62} + 112 q^{65} + 200 q^{67} + 104 q^{68} + 60 q^{70} - 544 q^{71} + 152 q^{73} - 312 q^{76} - 88 q^{77} + 328 q^{80} + 256 q^{82} + 16 q^{83} - 232 q^{85} + 224 q^{86} - 12 q^{88} - 32 q^{91} - 104 q^{92} - 144 q^{95} + 20 q^{97} - 376 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{5} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \zeta_{24}^{7} + \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{5} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{24}^{7} + 2\zeta_{24}^{3} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{6} + \beta_{3} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} + \beta_{5} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{6} + 2\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{4} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{7} + 2\beta_{5} ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

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

\(n\) \(82\) \(326\)
\(\chi(n)\) \(-\beta_{4}\) \(-1 + \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1
−0.965926 − 0.258819i
0.965926 + 0.258819i
−0.965926 + 0.258819i
0.965926 − 0.258819i
0.258819 + 0.965926i
−0.258819 − 0.965926i
0.258819 − 0.965926i
−0.258819 + 0.965926i
−0.307007 − 0.0822623i 0 −3.37662 − 1.94949i 3.87453 − 3.16038i 0 −4.71209 − 1.26260i 1.77526 + 1.77526i 0 −1.44949 + 0.651531i
28.2 3.03906 + 0.814313i 0 5.10867 + 2.94949i 2.32162 + 4.42833i 0 1.98004 + 0.530550i 4.22474 + 4.22474i 0 3.44949 + 15.3485i
217.1 −0.307007 + 0.0822623i 0 −3.37662 + 1.94949i 3.87453 + 3.16038i 0 −4.71209 + 1.26260i 1.77526 − 1.77526i 0 −1.44949 − 0.651531i
217.2 3.03906 − 0.814313i 0 5.10867 − 2.94949i 2.32162 − 4.42833i 0 1.98004 − 0.530550i 4.22474 − 4.22474i 0 3.44949 − 15.3485i
298.1 −0.814313 − 3.03906i 0 −5.10867 + 2.94949i −4.99585 − 0.203583i 0 −0.530550 − 1.98004i 4.22474 + 4.22474i 0 3.44949 + 15.3485i
298.2 0.0822623 + 0.307007i 0 3.37662 − 1.94949i 0.799701 + 4.93563i 0 1.26260 + 4.71209i 1.77526 + 1.77526i 0 −1.44949 + 0.651531i
352.1 −0.814313 + 3.03906i 0 −5.10867 − 2.94949i −4.99585 + 0.203583i 0 −0.530550 + 1.98004i 4.22474 − 4.22474i 0 3.44949 − 15.3485i
352.2 0.0822623 − 0.307007i 0 3.37662 + 1.94949i 0.799701 − 4.93563i 0 1.26260 − 4.71209i 1.77526 − 1.77526i 0 −1.44949 − 0.651531i
\(n\): 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
5.c odd 4 1 inner
9.c even 3 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.h 8
3.b odd 2 1 405.3.l.f 8
5.c odd 4 1 inner 405.3.l.h 8
9.c even 3 1 15.3.f.a ✓ 4
9.c even 3 1 inner 405.3.l.h 8
9.d odd 6 1 45.3.g.b 4
9.d odd 6 1 405.3.l.f 8
15.e even 4 1 405.3.l.f 8
36.f odd 6 1 240.3.bg.a 4
36.h even 6 1 720.3.bh.k 4
45.h odd 6 1 225.3.g.a 4
45.j even 6 1 75.3.f.c 4
45.k odd 12 1 15.3.f.a ✓ 4
45.k odd 12 1 75.3.f.c 4
45.k odd 12 1 inner 405.3.l.h 8
45.l even 12 1 45.3.g.b 4
45.l even 12 1 225.3.g.a 4
45.l even 12 1 405.3.l.f 8
72.n even 6 1 960.3.bg.i 4
72.p odd 6 1 960.3.bg.h 4
180.p odd 6 1 1200.3.bg.k 4
180.v odd 12 1 720.3.bh.k 4
180.x even 12 1 240.3.bg.a 4
180.x even 12 1 1200.3.bg.k 4
360.bo even 12 1 960.3.bg.h 4
360.bu odd 12 1 960.3.bg.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.3.f.a ✓ 4 9.c even 3 1
15.3.f.a ✓ 4 45.k odd 12 1
45.3.g.b 4 9.d odd 6 1
45.3.g.b 4 45.l even 12 1
75.3.f.c 4 45.j even 6 1
75.3.f.c 4 45.k odd 12 1
225.3.g.a 4 45.h odd 6 1
225.3.g.a 4 45.l even 12 1
240.3.bg.a 4 36.f odd 6 1
240.3.bg.a 4 180.x even 12 1
405.3.l.f 8 3.b odd 2 1
405.3.l.f 8 9.d odd 6 1
405.3.l.f 8 15.e even 4 1
405.3.l.f 8 45.l even 12 1
405.3.l.h 8 1.a even 1 1 trivial
405.3.l.h 8 5.c odd 4 1 inner
405.3.l.h 8 9.c even 3 1 inner
405.3.l.h 8 45.k odd 12 1 inner
720.3.bh.k 4 36.h even 6 1
720.3.bh.k 4 180.v odd 12 1
960.3.bg.h 4 72.p odd 6 1
960.3.bg.h 4 360.bo even 12 1
960.3.bg.i 4 72.n even 6 1
960.3.bg.i 4 360.bu odd 12 1
1200.3.bg.k 4 180.p odd 6 1
1200.3.bg.k 4 180.x even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 4T_{2}^{7} + 8T_{2}^{6} - 40T_{2}^{5} + 79T_{2}^{4} + 40T_{2}^{3} + 8T_{2}^{2} + 4T_{2} + 1 \) acting on \(S_{3}^{\mathrm{new}}(405, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 4 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 4 T^{7} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{8} + 4 T^{7} + \cdots + 10000 \) Copy content Toggle raw display
$11$ \( (T^{4} + 8 T^{3} + \cdots + 1444)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 32 T^{7} + \cdots + 181063936 \) Copy content Toggle raw display
$17$ \( (T^{4} + 40 T^{3} + \cdots + 8464)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 504 T^{2} + 32400)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 20851360000 \) Copy content Toggle raw display
$29$ \( T^{8} - 1236 T^{6} + \cdots + 810000 \) Copy content Toggle raw display
$31$ \( (T^{4} - 8 T^{3} + \cdots + 40000)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 64 T^{3} + \cdots + 211600)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 28 T^{3} + \cdots + 400)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 2018854506496 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 9336104694016 \) Copy content Toggle raw display
$53$ \( (T^{4} - 56 T^{3} + \cdots + 1600)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 399236364810000 \) Copy content Toggle raw display
$61$ \( (T^{4} + 100 T^{3} + \cdots + 309136)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 601343393468416 \) Copy content Toggle raw display
$71$ \( (T + 68)^{8} \) Copy content Toggle raw display
$73$ \( (T^{4} - 76 T^{3} + \cdots + 38316100)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 600 T^{2} + 360000)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 95565066496 \) Copy content Toggle raw display
$89$ \( (T^{4} + 15624 T^{2} + 59907600)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 265764994576 \) Copy content Toggle raw display
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