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Properties

Label 324.3.d.i
Level $324$
Weight $3$
Character orbit 324.d
Analytic conductor $8.828$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [324,3,Mod(163,324)]
 
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, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("324.163");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 324.d (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: \(8.82836056527\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1919698923024.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 2x^{6} + 12x^{5} - 36x^{4} + 48x^{3} + 32x^{2} - 192x + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + (\beta_{6} + \beta_{4} + \beta_{2} - \beta_1) q^{4} + ( - \beta_{7} - \beta_{3} - 1) q^{5} + (\beta_{7} - \beta_{5} - \beta_{3}) q^{7} + (\beta_{7} + 2 \beta_{6} - \beta_{5} + \cdots - 4) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + (\beta_{6} + \beta_{4} + \beta_{2} - \beta_1) q^{4} + ( - \beta_{7} - \beta_{3} - 1) q^{5} + (\beta_{7} - \beta_{5} - \beta_{3}) q^{7} + (\beta_{7} + 2 \beta_{6} - \beta_{5} + \cdots - 4) q^{8}+ \cdots + (\beta_{7} - 2 \beta_{6} - \beta_{5} + \cdots + 70) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 3 q^{2} + 5 q^{4} - 6 q^{5} - 27 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 3 q^{2} + 5 q^{4} - 6 q^{5} - 27 q^{8} + 10 q^{10} + 46 q^{13} + 12 q^{14} + 17 q^{16} + 6 q^{17} - 36 q^{20} - 33 q^{22} + 30 q^{25} + 36 q^{26} + 6 q^{28} - 42 q^{29} - 87 q^{32} - 11 q^{34} + 28 q^{37} + 99 q^{38} - 68 q^{40} - 84 q^{41} - 111 q^{44} - 132 q^{46} - 58 q^{49} + 219 q^{50} - 110 q^{52} - 36 q^{53} - 270 q^{56} + 16 q^{58} + 34 q^{61} + 258 q^{62} - 127 q^{64} + 30 q^{65} - 375 q^{68} - 150 q^{70} + 58 q^{73} + 372 q^{74} + 15 q^{76} + 330 q^{77} - 360 q^{80} + 127 q^{82} + 140 q^{85} + 273 q^{86} - 75 q^{88} - 192 q^{89} - 258 q^{92} - 36 q^{94} + 148 q^{97} + 585 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 2x^{6} + 12x^{5} - 36x^{4} + 48x^{3} + 32x^{2} - 192x + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - \nu^{6} + 10\nu^{5} - 20\nu^{4} - 12\nu^{3} + 32\nu^{2} - 96\nu + 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} - 5\nu^{6} + 6\nu^{5} - 12\nu^{4} + 4\nu^{3} + 48\nu^{2} - 160\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 3\nu^{6} - 2\nu^{5} - 12\nu^{4} + 36\nu^{3} + 16\nu^{2} - 32\nu + 128 ) / 64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + \nu^{5} - 2\nu^{4} + 12\nu^{3} - 4\nu^{2} - 16\nu + 80 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 3\nu^{6} - 2\nu^{5} - 12\nu^{4} + 36\nu^{3} - 48\nu^{2} - 32\nu + 192 ) / 64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} + \nu^{5} - 6\nu^{4} + 16\nu^{3} - 12\nu^{2} - 32\nu + 80 ) / 16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + \beta_{4} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{3} - \beta_{2} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{7} - 4\beta_{6} + \beta_{5} - 2\beta_{4} + \beta_{3} - 3\beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{7} - 4\beta_{6} + 5\beta_{5} - 6\beta_{4} + \beta_{3} + \beta_{2} + 4\beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{7} - 8\beta_{6} + \beta_{5} + 6\beta_{4} - 11\beta_{3} + 5\beta_{2} - 8\beta _1 - 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -22\beta_{7} + 16\beta_{6} + 17\beta_{5} + 6\beta_{4} - 11\beta_{3} + 13\beta_{2} - 12\beta _1 - 60 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
−1.97436 + 0.319229i
−1.97436 − 0.319229i
0.247102 + 1.98468i
0.247102 − 1.98468i
1.40960 + 1.41881i
1.40960 − 1.41881i
1.81766 + 0.834343i
1.81766 − 0.834343i
−1.97436 − 0.319229i 0 3.79619 + 1.26055i −2.71218 0 − 11.5967i −7.09263 − 3.70062i 0 5.35481 + 0.865806i
163.2 −1.97436 + 0.319229i 0 3.79619 − 1.26055i −2.71218 0 11.5967i −7.09263 + 3.70062i 0 5.35481 − 0.865806i
163.3 0.247102 − 1.98468i 0 −3.87788 − 0.980835i −2.20185 0 8.35824i −2.90487 + 7.45397i 0 −0.544081 + 4.36996i
163.4 0.247102 + 1.98468i 0 −3.87788 + 0.980835i −2.20185 0 − 8.35824i −2.90487 − 7.45397i 0 −0.544081 − 4.36996i
163.5 1.40960 − 1.41881i 0 −0.0260491 − 3.99992i 8.06209 0 − 4.50627i −5.71184 − 5.60133i 0 11.3643 − 11.4386i
163.6 1.40960 + 1.41881i 0 −0.0260491 + 3.99992i 8.06209 0 4.50627i −5.71184 + 5.60133i 0 11.3643 + 11.4386i
163.7 1.81766 − 0.834343i 0 2.60775 − 3.03310i −6.14806 0 − 0.590679i 2.20934 − 7.68888i 0 −11.1751 + 5.12959i
163.8 1.81766 + 0.834343i 0 2.60775 + 3.03310i −6.14806 0 0.590679i 2.20934 + 7.68888i 0 −11.1751 − 5.12959i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.3.d.i 8
3.b odd 2 1 324.3.d.g 8
4.b odd 2 1 inner 324.3.d.i 8
9.c even 3 2 36.3.f.c ✓ 16
9.d odd 6 2 108.3.f.c 16
12.b even 2 1 324.3.d.g 8
36.f odd 6 2 36.3.f.c ✓ 16
36.h even 6 2 108.3.f.c 16
72.j odd 6 2 1728.3.o.g 16
72.l even 6 2 1728.3.o.g 16
72.n even 6 2 576.3.o.g 16
72.p odd 6 2 576.3.o.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.3.f.c ✓ 16 9.c even 3 2
36.3.f.c ✓ 16 36.f odd 6 2
108.3.f.c 16 9.d odd 6 2
108.3.f.c 16 36.h even 6 2
324.3.d.g 8 3.b odd 2 1
324.3.d.g 8 12.b even 2 1
324.3.d.i 8 1.a even 1 1 trivial
324.3.d.i 8 4.b odd 2 1 inner
576.3.o.g 16 72.n even 6 2
576.3.o.g 16 72.p odd 6 2
1728.3.o.g 16 72.j odd 6 2
1728.3.o.g 16 72.l even 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 3T_{5}^{3} - 53T_{5}^{2} - 255T_{5} - 296 \) acting on \(S_{3}^{\mathrm{new}}(324, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 3 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 3 T^{3} + \cdots - 296)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 225 T^{6} + \cdots + 66564 \) Copy content Toggle raw display
$11$ \( T^{8} + 444 T^{6} + \cdots + 11594025 \) Copy content Toggle raw display
$13$ \( (T^{4} - 23 T^{3} + \cdots - 110)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 3 T^{3} + \cdots + 2200)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 7464960000 \) Copy content Toggle raw display
$23$ \( T^{8} + 1545 T^{6} + \cdots + 383298084 \) Copy content Toggle raw display
$29$ \( (T^{4} + 21 T^{3} + \cdots - 60482)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 93159248400 \) Copy content Toggle raw display
$37$ \( (T^{4} - 14 T^{3} + \cdots + 5920)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 42 T^{3} + \cdots + 111319)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 1197323973729 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 1334733156 \) Copy content Toggle raw display
$53$ \( (T^{4} + 18 T^{3} + \cdots - 16160)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 481518027225 \) Copy content Toggle raw display
$61$ \( (T^{4} - 17 T^{3} + \cdots + 30547408)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 1226398990041 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{4} - 29 T^{3} + \cdots + 20112040)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 10483996410000 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 60\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{4} + 96 T^{3} + \cdots - 4957424)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 74 T^{3} + \cdots + 64675)^{2} \) Copy content Toggle raw display
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