Properties

Label 324.4.e.i
Level $324$
Weight $4$
Character orbit 324.e
Analytic conductor $19.117$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 324.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.1166188419\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.49787136.1
Defining polynomial: \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{5} + ( - \beta_{7} + \beta_{4} - 4 \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{5} + ( - \beta_{7} + \beta_{4} - 4 \beta_1) q^{7} + (\beta_{6} - \beta_{5} - \beta_{2}) q^{11} + ( - \beta_{7} - 29 \beta_1 - 29) q^{13} + (5 \beta_{6} + 2 \beta_{5} + 2 \beta_{3}) q^{17} + (\beta_{4} + 56) q^{19} + (7 \beta_{3} + 5 \beta_{2}) q^{23} + (6 \beta_{7} - 6 \beta_{4} + 154 \beta_1) q^{25} + (12 \beta_{6} + 5 \beta_{5} - 12 \beta_{2}) q^{29} + (6 \beta_{7} - 92 \beta_1 - 92) q^{31} + (25 \beta_{6} - 13 \beta_{5} - 13 \beta_{3}) q^{35} + ( - 7 \beta_{4} + 167) q^{37} + ( - 16 \beta_{3} + 4 \beta_{2}) q^{41} + ( - 9 \beta_{7} + 9 \beta_{4} + 164 \beta_1) q^{43} + (20 \beta_{6} - 20 \beta_{2}) q^{47} + ( - 8 \beta_{7} - 429 \beta_1 - 429) q^{49} + (30 \beta_{6} + 22 \beta_{5} + 22 \beta_{3}) q^{53} + (15 \beta_{4} + 360) q^{55} + (4 \beta_{3} + 28 \beta_{2}) q^{59} + ( - 9 \beta_{7} + 9 \beta_{4} + 251 \beta_1) q^{61} + (25 \beta_{6} - 38 \beta_{5} - 25 \beta_{2}) q^{65} + ( - 15 \beta_{7} - 80 \beta_1 - 80) q^{67} + (41 \beta_{6} + 23 \beta_{5} + 23 \beta_{3}) q^{71} + 305 q^{73} + (40 \beta_{3} - 20 \beta_{2}) q^{77} + (25 \beta_{7} - 25 \beta_{4} - 16 \beta_1) q^{79} + ( - 10 \beta_{6} + 54 \beta_{5} + 10 \beta_{2}) q^{83} + (33 \beta_{7} - 153 \beta_1 - 153) q^{85} + ( - 43 \beta_{6} - 88 \beta_{5} - 88 \beta_{3}) q^{89} + ( - 33 \beta_{4} - 872) q^{91} + ( - 65 \beta_{3} + 25 \beta_{2}) q^{95} + (4 \beta_{7} - 4 \beta_{4} + 506 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{7} - 116 q^{13} + 448 q^{19} - 616 q^{25} - 368 q^{31} + 1336 q^{37} - 656 q^{43} - 1716 q^{49} + 2880 q^{55} - 1004 q^{61} - 320 q^{67} + 2440 q^{73} + 64 q^{79} - 612 q^{85} - 6976 q^{91} - 2024 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( 3\nu^{6} + 5\nu^{4} + 15\nu^{2} + 16 ) / 20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -9\nu^{7} - 45\nu^{5} + 45\nu^{3} - 18\nu ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{7} + 3\nu^{5} - 3\nu^{3} - 30\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -3\nu^{6} - 9\nu^{4} - 3\nu^{2} - 18 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9\nu^{7} + 27\nu^{5} + 69\nu^{3} + 132\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9\nu^{7} - 9\nu^{5} + 9\nu^{3} + 72\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -51\nu^{6} - 45\nu^{4} - 15\nu^{2} - 252 ) / 10 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 4\beta_{6} - 9\beta_{3} - 5\beta_{2} ) / 108 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{4} + 54\beta_1 ) / 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{6} + 9\beta_{5} + 9\beta_{3} + 20\beta_{2} ) / 108 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{7} - 2\beta_{4} - 6\beta _1 - 6 ) / 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -19\beta_{6} + 9\beta_{5} - 25\beta_{2} ) / 108 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -10\beta_{7} + 5\beta_{4} - 162 ) / 36 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 52\beta_{6} + 63\beta_{3} - 5\beta_{2} ) / 108 \) Copy content Toggle raw display

Character values

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

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

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
−1.09445 + 0.895644i
−0.228425 + 1.39564i
0.228425 1.39564i
1.09445 0.895644i
−1.09445 0.895644i
−0.228425 1.39564i
0.228425 + 1.39564i
1.09445 + 0.895644i
0 0 0 −10.5353 + 18.2477i 0 15.7477 + 27.2759i 0 0 0
109.2 0 0 0 −5.33918 + 9.24773i 0 −11.7477 20.3477i 0 0 0
109.3 0 0 0 5.33918 9.24773i 0 −11.7477 20.3477i 0 0 0
109.4 0 0 0 10.5353 18.2477i 0 15.7477 + 27.2759i 0 0 0
217.1 0 0 0 −10.5353 18.2477i 0 15.7477 27.2759i 0 0 0
217.2 0 0 0 −5.33918 9.24773i 0 −11.7477 + 20.3477i 0 0 0
217.3 0 0 0 5.33918 + 9.24773i 0 −11.7477 + 20.3477i 0 0 0
217.4 0 0 0 10.5353 + 18.2477i 0 15.7477 27.2759i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 217.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.4.e.i 8
3.b odd 2 1 inner 324.4.e.i 8
9.c even 3 1 324.4.a.e 4
9.c even 3 1 inner 324.4.e.i 8
9.d odd 6 1 324.4.a.e 4
9.d odd 6 1 inner 324.4.e.i 8
36.f odd 6 1 1296.4.a.z 4
36.h even 6 1 1296.4.a.z 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.4.a.e 4 9.c even 3 1
324.4.a.e 4 9.d odd 6 1
324.4.e.i 8 1.a even 1 1 trivial
324.4.e.i 8 3.b odd 2 1 inner
324.4.e.i 8 9.c even 3 1 inner
324.4.e.i 8 9.d odd 6 1 inner
1296.4.a.z 4 36.f odd 6 1
1296.4.a.z 4 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 \(S_{4}^{\mathrm{new}}(324, [\chi])\):

\( T_{5}^{8} + 558T_{5}^{6} + 260739T_{5}^{4} + 28248750T_{5}^{2} + 2562890625 \) Copy content Toggle raw display
\( T_{7}^{4} - 8T_{7}^{3} + 804T_{7}^{2} + 5920T_{7} + 547600 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 558 T^{6} + \cdots + 2562890625 \) Copy content Toggle raw display
$7$ \( (T^{4} - 8 T^{3} + 804 T^{2} + \cdots + 547600)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 1368 T^{6} + \cdots + 1049760000 \) Copy content Toggle raw display
$13$ \( (T^{4} + 58 T^{3} + 3279 T^{2} + \cdots + 7225)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 11142 T^{2} + 12638025)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 112 T + 2380)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} + 28152 T^{6} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{8} + 64494 T^{6} + \cdots + 14\!\cdots\!81 \) Copy content Toggle raw display
$31$ \( (T^{4} + 184 T^{3} + 52608 T^{2} + \cdots + 351637504)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 334 T - 9155)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} + 171360 T^{6} + \cdots + 35\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( (T^{4} + 328 T^{3} + 141924 T^{2} + \cdots + 1179235600)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 97200 T^{2} + \cdots + 9447840000)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 493632 T^{2} + 8294400)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 353664 T^{6} + \cdots + 81\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( (T^{4} + 502 T^{3} + 250239 T^{2} + \cdots + 3115225)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 160 T^{3} + \cdots + 26797690000)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 806616 T^{2} + \cdots + 18684702864)^{2} \) Copy content Toggle raw display
$73$ \( (T - 305)^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} - 32 T^{3} + \cdots + 223014395536)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 1850688 T^{6} + \cdots + 87\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{4} - 3993750 T^{2} + \cdots + 3633221022201)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 1012 T^{3} + \cdots + 59506723600)^{2} \) Copy content Toggle raw display
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