Properties

Label 1728.3.e.v
Level $1728$
Weight $3$
Character orbit 1728.e
Analytic conductor $47.085$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.2441150464.4
Defining polynomial: \( x^{8} - 14x^{6} + 77x^{4} - 188x^{2} + 196 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 864)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{5} - \beta_{4} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{5} - \beta_{4} q^{7} + ( - \beta_{2} - \beta_1) q^{11} + ( - \beta_{5} + 2) q^{13} + (\beta_{7} + \beta_{6}) q^{17} - \beta_{3} q^{19} + ( - 4 \beta_{2} - 3 \beta_1) q^{23} + (3 \beta_{5} - 4) q^{25} + (\beta_{7} - 3 \beta_{6}) q^{29} + (\beta_{4} - 3 \beta_{3}) q^{31} + ( - 5 \beta_{2} + 2 \beta_1) q^{35} + ( - 5 \beta_{5} - 4) q^{37} + ( - \beta_{7} - 3 \beta_{6}) q^{41} + ( - 6 \beta_{4} - 2 \beta_{3}) q^{43} + ( - 6 \beta_{2} + 14 \beta_1) q^{47} + (7 \beta_{5} + 12) q^{49} + ( - \beta_{7} + 12 \beta_{6}) q^{53} + (\beta_{4} + 2 \beta_{3}) q^{55} + ( - 18 \beta_{2} + 10 \beta_1) q^{59} + ( - 8 \beta_{5} - 12) q^{61} + ( - \beta_{7} - 9 \beta_{6}) q^{65} + (6 \beta_{4} + 4 \beta_{3}) q^{67} + ( - 26 \beta_{2} - 13 \beta_1) q^{71} + (11 \beta_{5} - 27) q^{73} - \beta_{6} q^{77} + ( - 10 \beta_{4} + \beta_{3}) q^{79} + ( - 19 \beta_{2} - 33 \beta_1) q^{83} + ( - 11 \beta_{5} + 42) q^{85} + ( - 4 \beta_{7} + 22 \beta_{6}) q^{89} + (6 \beta_{4} - \beta_{3}) q^{91} + ( - 24 \beta_{2} - 29 \beta_1) q^{95} + (8 \beta_{5} - 5) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{13} - 32 q^{25} - 32 q^{37} + 96 q^{49} - 96 q^{61} - 216 q^{73} + 336 q^{85} - 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 14x^{6} + 77x^{4} - 188x^{2} + 196 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} - 14\nu^{5} + 63\nu^{3} - 90\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5\nu^{7} - 56\nu^{5} + 245\nu^{3} - 338\nu ) / 56 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 7\nu^{5} - 106\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -9\nu^{7} + 84\nu^{5} - 245\nu^{3} + 250\nu ) / 56 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + 10\nu^{4} - 31\nu^{2} + 22 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} - 14\nu^{4} + 67\nu^{2} - 106 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{6} - 26\nu^{4} + 105\nu^{2} - 150 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{4} - \beta_{3} + 2\beta_{2} ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} + 2\beta_{5} + 42 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 10\beta_{4} - 2\beta_{3} + 22\beta_{2} - 9\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{7} - \beta_{6} + 4\beta_{5} + 42 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 34\beta_{4} - 2\beta_{3} + 142\beta_{2} - 105\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 59\beta_{7} - 61\beta_{6} + 34\beta_{5} + 222 ) / 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 26\beta_{4} + 8\beta_{3} + 782\beta_{2} - 735\beta_1 ) / 12 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1025.1
−1.52833 0.500000i
1.52833 + 0.500000i
2.27249 0.500000i
−2.27249 + 0.500000i
2.27249 + 0.500000i
−2.27249 0.500000i
−1.52833 + 0.500000i
1.52833 0.500000i
0 0 0 7.37942i 0 −1.26611 0 0 0
1025.2 0 0 0 7.37942i 0 1.26611 0 0 0
1025.3 0 0 0 1.88259i 0 −10.9726 0 0 0
1025.4 0 0 0 1.88259i 0 10.9726 0 0 0
1025.5 0 0 0 1.88259i 0 −10.9726 0 0 0
1025.6 0 0 0 1.88259i 0 10.9726 0 0 0
1025.7 0 0 0 7.37942i 0 −1.26611 0 0 0
1025.8 0 0 0 7.37942i 0 1.26611 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1025.8
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 1728.3.e.v 8
3.b odd 2 1 inner 1728.3.e.v 8
4.b odd 2 1 inner 1728.3.e.v 8
8.b even 2 1 864.3.e.e 8
8.d odd 2 1 864.3.e.e 8
12.b even 2 1 inner 1728.3.e.v 8
24.f even 2 1 864.3.e.e 8
24.h odd 2 1 864.3.e.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.3.e.e 8 8.b even 2 1
864.3.e.e 8 8.d odd 2 1
864.3.e.e 8 24.f even 2 1
864.3.e.e 8 24.h odd 2 1
1728.3.e.v 8 1.a even 1 1 trivial
1728.3.e.v 8 3.b odd 2 1 inner
1728.3.e.v 8 4.b odd 2 1 inner
1728.3.e.v 8 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{4} + 58T_{5}^{2} + 193 \) Copy content Toggle raw display
\( T_{7}^{4} - 122T_{7}^{2} + 193 \) 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
$5$ \( (T^{4} + 58 T^{2} + 193)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 122 T^{2} + 193)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 34 T^{2} + 1)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 4 T - 68)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 1080 T^{2} + 250128)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 464 T^{2} + 12352)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 432 T^{2} + 5184)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 1336 T^{2} + 151312)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 4490 T^{2} + 2733073)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 8 T - 1784)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 1648 T^{2} + 605248)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 5480 T^{2} + 7414288)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 3784 T^{2} + 1547536)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 8698 T^{2} + 16119553)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 7432 T^{2} + 4477456)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 24 T - 4464)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 10280 T^{2} + 16455952)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 14872 T^{2} + 22391824)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 54 T - 7983)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 13304 T^{2} + 892432)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 23922 T^{2} + 29844369)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 39016 T^{2} + \cdots + 350701072)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 10 T - 4583)^{4} \) Copy content Toggle raw display
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