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Properties

Label 324.3.f.q
Level $324$
Weight $3$
Character orbit 324.f
Analytic conductor $8.828$
Analytic rank $0$
Dimension $12$
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] = [324,3,Mod(55,324)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(324, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("324.55");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 324.f (of order \(6\), degree \(2\), 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: \(8.82836056527\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.119023932416481.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} - 3 x^{10} + 11 x^{9} - 5 x^{8} - 14 x^{7} + 29 x^{6} - 28 x^{5} - 20 x^{4} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} - \beta_{3}) q^{2} + (\beta_{4} + \beta_{2} + 1) q^{4} + ( - \beta_{7} - \beta_{5} - \beta_{4} + \cdots - 1) q^{5}+ \cdots + (\beta_{11} + \beta_{9} - \beta_{7} + \cdots - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} - \beta_{3}) q^{2} + (\beta_{4} + \beta_{2} + 1) q^{4} + ( - \beta_{7} - \beta_{5} - \beta_{4} + \cdots - 1) q^{5}+ \cdots + ( - 2 \beta_{11} - 2 \beta_{9} + \cdots - 56) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{2} + 3 q^{4} - 2 q^{5} + 14 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - q^{2} + 3 q^{4} - 2 q^{5} + 14 q^{8} + 18 q^{10} - 6 q^{13} + 15 q^{16} - 20 q^{17} + 67 q^{20} - 48 q^{22} - 146 q^{26} - 96 q^{28} + 22 q^{29} - 31 q^{32} - 81 q^{34} + 108 q^{37} - 168 q^{38} + 81 q^{40} - 92 q^{41} + 336 q^{44} + 240 q^{46} + 66 q^{49} + 48 q^{50} + 117 q^{52} + 232 q^{53} + 312 q^{56} - 201 q^{58} - 54 q^{61} - 624 q^{62} - 510 q^{64} - 82 q^{65} - 53 q^{68} - 264 q^{70} - 156 q^{73} - 383 q^{74} + 192 q^{76} + 168 q^{77} + 754 q^{80} + 300 q^{82} - 66 q^{85} + 144 q^{86} + 336 q^{88} - 500 q^{89} + 504 q^{92} - 216 q^{94} + 204 q^{97} - 814 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} - 3 x^{10} + 11 x^{9} - 5 x^{8} - 14 x^{7} + 29 x^{6} - 28 x^{5} - 20 x^{4} + \cdots + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{10} - 5\nu^{9} + \nu^{8} + 12\nu^{7} - 16\nu^{6} + 7\nu^{5} + 13\nu^{4} - 51\nu^{3} + 38\nu^{2} + 40\nu - 64 ) / 24 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} - 3\nu^{9} + 5\nu^{8} - 3\nu^{7} - 4\nu^{6} + 21\nu^{5} - 18\nu^{4} - 8\nu^{3} + 16\nu^{2} - 32\nu + 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3 \nu^{11} - 2 \nu^{10} - 11 \nu^{9} + 13 \nu^{8} + 3 \nu^{7} - 16 \nu^{6} + 37 \nu^{5} - 32 \nu^{4} + \cdots - 64 ) / 24 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{10} + \nu^{9} + 4\nu^{8} - 3\nu^{7} - \nu^{6} + \nu^{5} - 14\nu^{4} + 3\nu^{3} + 32\nu^{2} - 8\nu - 40 ) / 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9 \nu^{11} - 10 \nu^{10} - 13 \nu^{9} + 35 \nu^{8} - 63 \nu^{7} + 40 \nu^{6} + 131 \nu^{5} - 244 \nu^{4} + \cdots + 400 ) / 48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{11} + \nu^{10} - 7 \nu^{9} + 2 \nu^{8} + 14 \nu^{7} - 19 \nu^{6} + 13 \nu^{5} + 19 \nu^{4} + \cdots - 80 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{11} - 4 \nu^{10} - \nu^{9} + 13 \nu^{8} - 13 \nu^{7} - 2 \nu^{6} + 19 \nu^{5} - 42 \nu^{4} + \cdots - 40 ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{11} - 2\nu^{10} + 5\nu^{8} - 10\nu^{7} + 3\nu^{6} + 10\nu^{5} - 26\nu^{4} + 23\nu^{3} + 12\nu^{2} - 48\nu + 16 ) / 4 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3 \nu^{11} + 22 \nu^{10} - 65 \nu^{9} + 13 \nu^{8} + 165 \nu^{7} - 262 \nu^{6} + 79 \nu^{5} + \cdots - 928 ) / 48 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 15 \nu^{11} - 8 \nu^{10} - 89 \nu^{9} + 139 \nu^{8} + 87 \nu^{7} - 280 \nu^{6} + 343 \nu^{5} + \cdots - 1120 ) / 48 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 33 \nu^{11} + 22 \nu^{10} + 133 \nu^{9} - 167 \nu^{8} - 141 \nu^{7} + 332 \nu^{6} - 371 \nu^{5} + \cdots + 1568 ) / 48 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + 3\beta_{5} - 3\beta_{3} - 2\beta_{2} - 5\beta _1 - 2 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + 2\beta_{10} - \beta_{9} + \beta_{8} + 3\beta_{6} - 2\beta_{4} - 4\beta_{2} - 5\beta _1 + 6 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + 3 \beta_{5} - 4 \beta_{4} + \cdots - 8 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{9} + 2\beta_{7} + 3\beta_{6} - \beta_{5} - 2\beta_{4} - 3\beta_{3} - \beta _1 + 5 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - \beta_{10} - 2 \beta_{9} - 3 \beta_{8} + 4 \beta_{7} + 16 \beta_{6} + 9 \beta_{5} + 4 \beta_{4} + \cdots - 19 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2 \beta_{10} - 9 \beta_{9} - 5 \beta_{8} + 3 \beta_{7} + 3 \beta_{6} - 9 \beta_{5} - 8 \beta_{4} + \cdots - 20 ) / 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 9 \beta_{11} - 11 \beta_{9} - \beta_{8} - 14 \beta_{7} - 11 \beta_{6} - 6 \beta_{5} + 6 \beta_{4} + \cdots - 2 ) / 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 3 \beta_{11} - \beta_{10} - \beta_{9} - 5 \beta_{8} + 3 \beta_{7} - 11 \beta_{6} - 9 \beta_{5} + \cdots + 10 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 4 \beta_{11} + 2 \beta_{10} + 5 \beta_{9} - 6 \beta_{8} - 26 \beta_{7} + 13 \beta_{6} + 3 \beta_{5} + \cdots - 3 ) / 12 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 8 \beta_{11} + 27 \beta_{10} + 16 \beta_{9} - 9 \beta_{8} - 18 \beta_{7} - 54 \beta_{6} - 21 \beta_{5} + \cdots - 191 ) / 12 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 18 \beta_{11} + 26 \beta_{10} - 11 \beta_{9} + 57 \beta_{8} - 89 \beta_{7} + \beta_{6} - 33 \beta_{5} + \cdots - 88 ) / 12 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(163\) \(245\)
\(\chi(n)\) \(-1\) \(\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
−1.40311 + 0.176844i
1.08837 + 0.903022i
1.25131 − 0.658947i
1.38685 + 0.276848i
0.0389494 + 1.41368i
−1.36237 − 0.379393i
−1.40311 − 0.176844i
1.08837 − 0.903022i
1.25131 + 0.658947i
1.38685 − 0.276848i
0.0389494 − 1.41368i
−1.36237 + 0.379393i
−1.82144 − 0.826041i 0 2.63531 + 3.00917i −3.61903 − 6.26834i 0 5.95847 + 3.44013i −2.31438 − 7.65792i 0 1.41395 + 14.4069i
55.2 −1.70333 + 1.04817i 0 1.80268 − 3.57076i 0.931627 + 1.61363i 0 −9.80189 − 5.65913i 0.672219 + 7.97171i 0 −3.27823 − 1.77203i
55.3 −0.461217 − 1.94609i 0 −3.57456 + 1.79514i 2.18740 + 3.78869i 0 −1.84918 − 1.06762i 5.14216 + 6.12848i 0 6.36427 − 6.00429i
55.4 −0.0560770 + 1.99921i 0 −3.99371 − 0.224220i 0.931627 + 1.61363i 0 9.80189 + 5.65913i 0.672219 − 7.97171i 0 −3.27823 + 1.77203i
55.5 1.62609 + 1.16440i 0 1.28836 + 3.78684i −3.61903 − 6.26834i 0 −5.95847 − 3.44013i −2.31438 + 7.65792i 0 1.41395 − 14.4069i
55.6 1.91597 − 0.573621i 0 3.34192 − 2.19809i 2.18740 + 3.78869i 0 1.84918 + 1.06762i 5.14216 − 6.12848i 0 6.36427 + 6.00429i
271.1 −1.82144 + 0.826041i 0 2.63531 − 3.00917i −3.61903 + 6.26834i 0 5.95847 − 3.44013i −2.31438 + 7.65792i 0 1.41395 − 14.4069i
271.2 −1.70333 − 1.04817i 0 1.80268 + 3.57076i 0.931627 − 1.61363i 0 −9.80189 + 5.65913i 0.672219 − 7.97171i 0 −3.27823 + 1.77203i
271.3 −0.461217 + 1.94609i 0 −3.57456 − 1.79514i 2.18740 − 3.78869i 0 −1.84918 + 1.06762i 5.14216 − 6.12848i 0 6.36427 + 6.00429i
271.4 −0.0560770 − 1.99921i 0 −3.99371 + 0.224220i 0.931627 − 1.61363i 0 9.80189 − 5.65913i 0.672219 + 7.97171i 0 −3.27823 − 1.77203i
271.5 1.62609 − 1.16440i 0 1.28836 − 3.78684i −3.61903 + 6.26834i 0 −5.95847 + 3.44013i −2.31438 − 7.65792i 0 1.41395 + 14.4069i
271.6 1.91597 + 0.573621i 0 3.34192 + 2.19809i 2.18740 − 3.78869i 0 1.84918 − 1.06762i 5.14216 + 6.12848i 0 6.36427 − 6.00429i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.3.f.q 12
3.b odd 2 1 324.3.f.r 12
4.b odd 2 1 inner 324.3.f.q 12
9.c even 3 1 324.3.d.f yes 6
9.c even 3 1 inner 324.3.f.q 12
9.d odd 6 1 324.3.d.e ✓ 6
9.d odd 6 1 324.3.f.r 12
12.b even 2 1 324.3.f.r 12
36.f odd 6 1 324.3.d.f yes 6
36.f odd 6 1 inner 324.3.f.q 12
36.h even 6 1 324.3.d.e ✓ 6
36.h even 6 1 324.3.f.r 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.3.d.e ✓ 6 9.d odd 6 1
324.3.d.e ✓ 6 36.h even 6 1
324.3.d.f yes 6 9.c even 3 1
324.3.d.f yes 6 36.f odd 6 1
324.3.f.q 12 1.a even 1 1 trivial
324.3.f.q 12 4.b odd 2 1 inner
324.3.f.q 12 9.c even 3 1 inner
324.3.f.q 12 36.f odd 6 1 inner
324.3.f.r 12 3.b odd 2 1
324.3.f.r 12 9.d odd 6 1
324.3.f.r 12 12.b even 2 1
324.3.f.r 12 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(324, [\chi])\):

\( T_{5}^{6} + T_{5}^{5} + 38T_{5}^{4} - 155T_{5}^{3} + 1310T_{5}^{2} - 2183T_{5} + 3481 \) Copy content Toggle raw display
\( T_{7}^{12} - 180T_{7}^{10} + 25536T_{7}^{8} - 1180224T_{7}^{6} + 42137856T_{7}^{4} - 189775872T_{7}^{2} + 764411904 \) Copy content Toggle raw display
\( T_{11}^{12} - 516 T_{11}^{10} + 198336 T_{11}^{8} - 31507776 T_{11}^{6} + 3700078848 T_{11}^{4} + \cdots + 3131031158784 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + T^{11} + \cdots + 4096 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} + T^{5} + 38 T^{4} + \cdots + 3481)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 764411904 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 3131031158784 \) Copy content Toggle raw display
$13$ \( (T^{6} + 3 T^{5} + \cdots + 982081)^{2} \) Copy content Toggle raw display
$17$ \( (T^{3} + 5 T^{2} + \cdots - 233)^{4} \) Copy content Toggle raw display
$19$ \( (T^{6} + 1476 T^{4} + \cdots + 20155392)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 18\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( (T^{6} - 11 T^{5} + \cdots + 1057485361)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 50096498540544 \) Copy content Toggle raw display
$37$ \( (T^{3} - 27 T^{2} + \cdots + 144607)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + 46 T^{5} + \cdots + 482944576)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 41\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 55\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( (T^{3} - 58 T^{2} + \cdots + 5128)^{4} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 24\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( (T^{6} + 27 T^{5} + \cdots + 10239618481)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 16\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( (T^{6} + 8532 T^{4} + \cdots + 808455168)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 39 T^{2} + \cdots - 447851)^{4} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 12\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 81\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{3} + 125 T^{2} + \cdots - 107777)^{4} \) Copy content Toggle raw display
$97$ \( (T^{6} - 102 T^{5} + \cdots + 136361809984)^{2} \) Copy content Toggle raw display
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