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Properties

Label 324.2.b.b
Level $324$
Weight $2$
Character orbit 324.b
Analytic conductor $2.587$
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] = [324,2,Mod(323,324)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(324, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("324.323");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 324.b (of order \(2\), degree \(1\), 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: \(2.58715302549\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.170772624.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - \beta_{2} q^{4} + (\beta_{7} - \beta_{4} + \beta_{3}) q^{5} + \beta_{6} q^{7} + (\beta_{7} + \beta_{3} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} - \beta_{2} q^{4} + (\beta_{7} - \beta_{4} + \beta_{3}) q^{5} + \beta_{6} q^{7} + (\beta_{7} + \beta_{3} - \beta_1) q^{8} + ( - \beta_{6} + \beta_{5} - \beta_{2} - 1) q^{10} - \beta_1 q^{11} + (\beta_{5} + \beta_{2} + 1) q^{13} + ( - \beta_{7} - \beta_1) q^{14} + ( - 2 \beta_{6} - \beta_{2}) q^{16} + ( - \beta_{4} + \beta_{3}) q^{17} + (2 \beta_{6} - \beta_{5} + \beta_{2}) q^{19} + (2 \beta_{7} - 2 \beta_{4}) q^{20} + ( - \beta_{6} - \beta_{5} - 1) q^{22} + (\beta_{4} + \beta_{3} - \beta_1) q^{23} + (\beta_{5} + \beta_{2} + 2) q^{25} + ( - \beta_{7} - 2 \beta_{4} + \beta_1) q^{26} + ( - 2 \beta_{5} - 2) q^{28} + (\beta_{7} - \beta_{4} + \beta_{3}) q^{29} + (3 \beta_{6} - 2 \beta_{5} + 2 \beta_{2}) q^{31} + (3 \beta_{7} + \beta_{3} + \beta_1) q^{32} + ( - \beta_{2} - 2) q^{34} + (\beta_{4} + \beta_{3} - \beta_1) q^{35} + (2 \beta_{5} + 2 \beta_{2}) q^{37} + ( - 3 \beta_{7} + 2 \beta_{4} + \cdots - \beta_1) q^{38}+ \cdots + (\beta_{7} + 2 \beta_{4} + \cdots - \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{4} - 8 q^{10} + 4 q^{13} + 2 q^{16} - 6 q^{22} + 12 q^{25} - 12 q^{28} - 14 q^{34} - 8 q^{37} - 20 q^{40} + 12 q^{46} + 20 q^{49} - 32 q^{52} - 8 q^{58} + 4 q^{61} + 26 q^{64} + 12 q^{70} + 4 q^{73} + 6 q^{76} + 10 q^{82} - 16 q^{85} + 42 q^{88} + 36 q^{94} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + \nu^{6} - \nu^{5} + 2\nu^{4} + 8\nu^{2} - 4\nu + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 2\nu^{5} - \nu^{4} + 2\nu^{3} - 6\nu^{2} + 4\nu - 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{7} + 3\nu^{6} - 5\nu^{5} + 5\nu^{4} - 6\nu^{3} + 14\nu^{2} - 16\nu + 20 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + 3\nu^{5} - 2\nu^{4} + 3\nu^{3} - 8\nu^{2} + 12\nu - 12 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} - 4\nu^{5} + 3\nu^{4} - 4\nu^{3} + 10\nu^{2} - 12\nu + 16 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{7} + 4\nu^{6} - 8\nu^{5} + 5\nu^{4} - 8\nu^{3} + 22\nu^{2} - 20\nu + 32 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{7} - 5\nu^{6} + 7\nu^{5} - 6\nu^{4} + 10\nu^{3} - 24\nu^{2} + 28\nu - 28 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + \beta_{4} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + \beta_{5} + \beta_{4} - 2\beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{4} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{7} - 2\beta_{6} + \beta_{5} + \beta_{4} + 4\beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3\beta_{7} - 2\beta_{6} - \beta_{5} + \beta_{4} + \beta _1 + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{6} - \beta_{5} + 4\beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -3\beta_{7} - 2\beta_{6} + 9\beta_{5} + 5\beta_{4} - 8\beta_{3} + 4\beta_{2} + \beta _1 + 4 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).

\(n\) \(163\) \(245\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
323.1
−1.02187 − 0.977642i
−1.02187 + 0.977642i
0.774115 + 1.18353i
0.774115 − 1.18353i
1.41203 + 0.0786378i
1.41203 − 0.0786378i
0.335728 + 1.37379i
0.335728 − 1.37379i
−1.35760 − 0.396143i 0 1.68614 + 1.07561i − 2.52434i 0 − 1.27582i −1.86301 − 2.12819i 0 −1.00000 + 3.42703i
323.2 −1.35760 + 0.396143i 0 1.68614 − 1.07561i 2.52434i 0 1.27582i −1.86301 + 2.12819i 0 −1.00000 − 3.42703i
323.3 −0.637910 − 1.26217i 0 −1.18614 + 1.61030i − 0.792287i 0 2.71519i 2.78912 + 0.469882i 0 −1.00000 + 0.505408i
323.4 −0.637910 + 1.26217i 0 −1.18614 − 1.61030i 0.792287i 0 − 2.71519i 2.78912 − 0.469882i 0 −1.00000 − 0.505408i
323.5 0.637910 − 1.26217i 0 −1.18614 − 1.61030i − 0.792287i 0 − 2.71519i −2.78912 + 0.469882i 0 −1.00000 − 0.505408i
323.6 0.637910 + 1.26217i 0 −1.18614 + 1.61030i 0.792287i 0 2.71519i −2.78912 − 0.469882i 0 −1.00000 + 0.505408i
323.7 1.35760 − 0.396143i 0 1.68614 − 1.07561i − 2.52434i 0 1.27582i 1.86301 − 2.12819i 0 −1.00000 − 3.42703i
323.8 1.35760 + 0.396143i 0 1.68614 + 1.07561i 2.52434i 0 − 1.27582i 1.86301 + 2.12819i 0 −1.00000 + 3.42703i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 323.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.2.b.b 8
3.b odd 2 1 inner 324.2.b.b 8
4.b odd 2 1 inner 324.2.b.b 8
8.b even 2 1 5184.2.c.j 8
8.d odd 2 1 5184.2.c.j 8
9.c even 3 1 36.2.h.a ✓ 8
9.c even 3 1 108.2.h.a 8
9.d odd 6 1 36.2.h.a ✓ 8
9.d odd 6 1 108.2.h.a 8
12.b even 2 1 inner 324.2.b.b 8
24.f even 2 1 5184.2.c.j 8
24.h odd 2 1 5184.2.c.j 8
36.f odd 6 1 36.2.h.a ✓ 8
36.f odd 6 1 108.2.h.a 8
36.h even 6 1 36.2.h.a ✓ 8
36.h even 6 1 108.2.h.a 8
45.h odd 6 1 900.2.r.c 8
45.j even 6 1 900.2.r.c 8
45.k odd 12 2 900.2.o.a 16
45.l even 12 2 900.2.o.a 16
72.j odd 6 1 576.2.s.f 8
72.j odd 6 1 1728.2.s.f 8
72.l even 6 1 576.2.s.f 8
72.l even 6 1 1728.2.s.f 8
72.n even 6 1 576.2.s.f 8
72.n even 6 1 1728.2.s.f 8
72.p odd 6 1 576.2.s.f 8
72.p odd 6 1 1728.2.s.f 8
180.n even 6 1 900.2.r.c 8
180.p odd 6 1 900.2.r.c 8
180.v odd 12 2 900.2.o.a 16
180.x even 12 2 900.2.o.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.h.a ✓ 8 9.c even 3 1
36.2.h.a ✓ 8 9.d odd 6 1
36.2.h.a ✓ 8 36.f odd 6 1
36.2.h.a ✓ 8 36.h even 6 1
108.2.h.a 8 9.c even 3 1
108.2.h.a 8 9.d odd 6 1
108.2.h.a 8 36.f odd 6 1
108.2.h.a 8 36.h even 6 1
324.2.b.b 8 1.a even 1 1 trivial
324.2.b.b 8 3.b odd 2 1 inner
324.2.b.b 8 4.b odd 2 1 inner
324.2.b.b 8 12.b even 2 1 inner
576.2.s.f 8 72.j odd 6 1
576.2.s.f 8 72.l even 6 1
576.2.s.f 8 72.n even 6 1
576.2.s.f 8 72.p odd 6 1
900.2.o.a 16 45.k odd 12 2
900.2.o.a 16 45.l even 12 2
900.2.o.a 16 180.v odd 12 2
900.2.o.a 16 180.x even 12 2
900.2.r.c 8 45.h odd 6 1
900.2.r.c 8 45.j even 6 1
900.2.r.c 8 180.n even 6 1
900.2.r.c 8 180.p odd 6 1
1728.2.s.f 8 72.j odd 6 1
1728.2.s.f 8 72.l even 6 1
1728.2.s.f 8 72.n even 6 1
1728.2.s.f 8 72.p odd 6 1
5184.2.c.j 8 8.b even 2 1
5184.2.c.j 8 8.d odd 2 1
5184.2.c.j 8 24.f even 2 1
5184.2.c.j 8 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 7T_{5}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(324, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{6} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 7 T^{2} + 4)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 9 T^{2} + 12)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 12 T^{2} + 3)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - T - 8)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 7 T^{2} + 4)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 27 T^{2} + 108)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 15 T^{2} + 48)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 7 T^{2} + 4)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 69 T^{2} + 192)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 2 T - 32)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 46 T^{2} + 1)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 108 T^{2} + 2883)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 135 T^{2} + 192)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 76 T^{2} + 256)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 180 T^{2} + 4107)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - T - 8)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 108 T^{2} + 2883)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 144 T^{2} + 432)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - T - 8)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 201 T^{2} + 10092)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 111 T^{2} + 3072)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 172 T^{2} + 4096)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 2 T - 131)^{4} \) Copy content Toggle raw display
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