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Properties

Label 405.3.l.g
Level $405$
Weight $3$
Character orbit 405.l
Analytic conductor $11.035$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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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: 8.0.3317760000.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 25x^{4} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
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.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} - \beta_{2} q^{4} + ( - 2 \beta_{7} + \beta_{5} + 2 \beta_{3}) q^{5} + (5 \beta_{6} + 5 \beta_{4} - 5 \beta_{2}) q^{7} + (3 \beta_{5} - 3 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{2} - \beta_{2} q^{4} + ( - 2 \beta_{7} + \beta_{5} + 2 \beta_{3}) q^{5} + (5 \beta_{6} + 5 \beta_{4} - 5 \beta_{2}) q^{7} + (3 \beta_{5} - 3 \beta_1) q^{8} + (10 \beta_{6} - 5) q^{10} + ( - 5 \beta_{7} - 5 \beta_1) q^{11} + (10 \beta_{4} + 10 \beta_{2} - 10) q^{13} + (5 \beta_{7} - 5 \beta_{5} - 5 \beta_{3}) q^{14} - 19 \beta_{4} q^{16} + 2 \beta_{3} q^{17} + 18 \beta_{6} q^{19} + ( - \beta_{7} - 2 \beta_1) q^{20} + ( - 25 \beta_{4} + 25 \beta_{2} + 25) q^{22} + 2 \beta_{5} q^{23} + ( - 15 \beta_{6} + 20 \beta_{4} + 15 \beta_{2}) q^{25} + (10 \beta_{5} - 10 \beta_{3} - 10 \beta_1) q^{26} + ( - 5 \beta_{6} + 5) q^{28} + (15 \beta_{7} - 15 \beta_1) q^{29} + (8 \beta_{4} - 8) q^{31} + ( - 7 \beta_{7} + 7 \beta_{3}) q^{32} + (10 \beta_{6} - 10 \beta_{2}) q^{34} + (15 \beta_{5} + 5 \beta_{3} - 15 \beta_1) q^{35} + (10 \beta_{6} + 10) q^{37} - 18 \beta_1 q^{38} + (30 \beta_{4} - 15 \beta_{2} - 30) q^{40} + ( - 10 \beta_{7} - 10 \beta_{5} + 10 \beta_{3}) q^{41} + (10 \beta_{6} - 10 \beta_{4} - 10 \beta_{2}) q^{43} + (5 \beta_{5} + 5 \beta_{3} - 5 \beta_1) q^{44} - 10 q^{46} - 26 \beta_{7} q^{47} - \beta_{2} q^{49} + (20 \beta_{7} + 15 \beta_{5} - 20 \beta_{3}) q^{50} + ( - 10 \beta_{6} - 10 \beta_{4} + 10 \beta_{2}) q^{52} + (16 \beta_{5} - 16 \beta_1) q^{53} + ( - 75 \beta_{6} - 25) q^{55} + ( - 15 \beta_{7} - 15 \beta_1) q^{56} + ( - 75 \beta_{4} - 75 \beta_{2} + 75) q^{58} + (15 \beta_{7} - 15 \beta_{5} - 15 \beta_{3}) q^{59} + 58 \beta_{4} q^{61} - 8 \beta_{3} q^{62} - 41 \beta_{6} q^{64} + (30 \beta_{7} + 10 \beta_1) q^{65} + (70 \beta_{4} - 70 \beta_{2} - 70) q^{67} - 2 \beta_{5} q^{68} + (25 \beta_{6} - 75 \beta_{4} - 25 \beta_{2}) q^{70} + ( - 20 \beta_{5} + 20 \beta_{3} + 20 \beta_1) q^{71} + ( - 55 \beta_{6} + 55) q^{73} + (10 \beta_{7} - 10 \beta_1) q^{74} + ( - 18 \beta_{4} + 18) q^{76} + ( - 50 \beta_{7} + 50 \beta_{3}) q^{77} + ( - 12 \beta_{6} + 12 \beta_{2}) q^{79} + ( - 19 \beta_{5} - 38 \beta_{3} + 19 \beta_1) q^{80} + (50 \beta_{6} + 50) q^{82} + 34 \beta_1 q^{83} + (10 \beta_{4} + 20 \beta_{2} - 10) q^{85} + ( - 10 \beta_{7} - 10 \beta_{5} + 10 \beta_{3}) q^{86} + ( - 75 \beta_{6} + 75 \beta_{4} + 75 \beta_{2}) q^{88} - 100 q^{91} - 2 \beta_{7} q^{92} + 130 \beta_{2} q^{94} + (18 \beta_{7} + 36 \beta_{5} - 18 \beta_{3}) q^{95} + (5 \beta_{6} + 5 \beta_{4} - 5 \beta_{2}) q^{97} + ( - \beta_{5} + \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 20 q^{7} - 40 q^{10} - 40 q^{13} - 76 q^{16} + 100 q^{22} + 80 q^{25} + 40 q^{28} - 32 q^{31} + 80 q^{37} - 120 q^{40} - 40 q^{43} - 80 q^{46} - 40 q^{52} - 200 q^{55} + 300 q^{58} + 232 q^{61} - 280 q^{67} - 300 q^{70} + 440 q^{73} + 72 q^{76} + 400 q^{82} - 40 q^{85} + 300 q^{88} - 800 q^{91} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 25x^{4} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 25 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 125 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 125 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 25\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 25\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 125\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 125\beta_{7} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(82\) \(326\)
\(\chi(n)\) \(\beta_{6}\) \(-1 + \beta_{4}\)

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
0.578737 − 2.15988i
−0.578737 + 2.15988i
0.578737 + 2.15988i
−0.578737 − 2.15988i
2.15988 − 0.578737i
−2.15988 + 0.578737i
2.15988 + 0.578737i
−2.15988 − 0.578737i
−2.15988 − 0.578737i 0 0.866025 + 0.500000i 3.31735 + 3.74101i 0 6.83013 + 1.83013i 4.74342 + 4.74342i 0 −5.00000 − 10.0000i
28.2 2.15988 + 0.578737i 0 0.866025 + 0.500000i −3.31735 − 3.74101i 0 6.83013 + 1.83013i −4.74342 − 4.74342i 0 −5.00000 − 10.0000i
217.1 −2.15988 + 0.578737i 0 0.866025 − 0.500000i 3.31735 − 3.74101i 0 6.83013 − 1.83013i 4.74342 − 4.74342i 0 −5.00000 + 10.0000i
217.2 2.15988 − 0.578737i 0 0.866025 − 0.500000i −3.31735 + 3.74101i 0 6.83013 − 1.83013i −4.74342 + 4.74342i 0 −5.00000 + 10.0000i
298.1 −0.578737 − 2.15988i 0 −0.866025 + 0.500000i 4.89849 − 1.00240i 0 −1.83013 − 6.83013i −4.74342 − 4.74342i 0 −5.00000 − 10.0000i
298.2 0.578737 + 2.15988i 0 −0.866025 + 0.500000i −4.89849 + 1.00240i 0 −1.83013 − 6.83013i 4.74342 + 4.74342i 0 −5.00000 − 10.0000i
352.1 −0.578737 + 2.15988i 0 −0.866025 − 0.500000i 4.89849 + 1.00240i 0 −1.83013 + 6.83013i −4.74342 + 4.74342i 0 −5.00000 + 10.0000i
352.2 0.578737 − 2.15988i 0 −0.866025 − 0.500000i −4.89849 − 1.00240i 0 −1.83013 + 6.83013i 4.74342 − 4.74342i 0 −5.00000 + 10.0000i
\(n\): 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
3.b odd 2 1 inner
5.c odd 4 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
15.e even 4 1 inner
45.k odd 12 1 inner
45.l even 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.g 8
3.b odd 2 1 inner 405.3.l.g 8
5.c odd 4 1 inner 405.3.l.g 8
9.c even 3 1 45.3.g.a ✓ 4
9.c even 3 1 inner 405.3.l.g 8
9.d odd 6 1 45.3.g.a ✓ 4
9.d odd 6 1 inner 405.3.l.g 8
15.e even 4 1 inner 405.3.l.g 8
36.f odd 6 1 720.3.bh.j 4
36.h even 6 1 720.3.bh.j 4
45.h odd 6 1 225.3.g.g 4
45.j even 6 1 225.3.g.g 4
45.k odd 12 1 45.3.g.a ✓ 4
45.k odd 12 1 225.3.g.g 4
45.k odd 12 1 inner 405.3.l.g 8
45.l even 12 1 45.3.g.a ✓ 4
45.l even 12 1 225.3.g.g 4
45.l even 12 1 inner 405.3.l.g 8
180.v odd 12 1 720.3.bh.j 4
180.x even 12 1 720.3.bh.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.3.g.a ✓ 4 9.c even 3 1
45.3.g.a ✓ 4 9.d odd 6 1
45.3.g.a ✓ 4 45.k odd 12 1
45.3.g.a ✓ 4 45.l even 12 1
225.3.g.g 4 45.h odd 6 1
225.3.g.g 4 45.j even 6 1
225.3.g.g 4 45.k odd 12 1
225.3.g.g 4 45.l even 12 1
405.3.l.g 8 1.a even 1 1 trivial
405.3.l.g 8 3.b odd 2 1 inner
405.3.l.g 8 5.c odd 4 1 inner
405.3.l.g 8 9.c even 3 1 inner
405.3.l.g 8 9.d odd 6 1 inner
405.3.l.g 8 15.e even 4 1 inner
405.3.l.g 8 45.k odd 12 1 inner
405.3.l.g 8 45.l even 12 1 inner
720.3.bh.j 4 36.f odd 6 1
720.3.bh.j 4 36.h even 6 1
720.3.bh.j 4 180.v odd 12 1
720.3.bh.j 4 180.x even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 25T_{2}^{4} + 625 \) acting on \(S_{3}^{\mathrm{new}}(405, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 25T^{4} + 625 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 40 T^{6} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( (T^{4} - 10 T^{3} + \cdots + 2500)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 250 T^{2} + 62500)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 20 T^{3} + \cdots + 40000)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 400)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 324)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} - 400 T^{4} + 160000 \) Copy content Toggle raw display
$29$ \( (T^{4} - 2250 T^{2} + 5062500)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T + 64)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 20 T + 200)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 1000 T^{2} + 1000000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 20 T^{3} + \cdots + 40000)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 130516915360000 \) Copy content Toggle raw display
$53$ \( (T^{4} + 1638400)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 2250 T^{2} + 5062500)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 58 T + 3364)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 140 T^{3} + \cdots + 96040000)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 4000)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 110 T + 6050)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 144 T^{2} + 20736)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( (T^{4} - 10 T^{3} + \cdots + 2500)^{2} \) Copy content Toggle raw display
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