Properties

Label 1728.4.f.g
Level $1728$
Weight $4$
Character orbit 1728.f
Analytic conductor $101.955$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1728.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(101.955300490\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.58594980096.3
Defining polynomial: \( x^{8} - 21x^{6} + 341x^{4} - 2100x^{2} + 10000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{4} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{5} + (\beta_{7} - 2 \beta_{3}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{5} + (\beta_{7} - 2 \beta_{3}) q^{7} + ( - \beta_{5} - 8 \beta_1) q^{11} + (3 \beta_{5} + \beta_1) q^{13} + ( - 3 \beta_{7} + 63 \beta_{3}) q^{17} + ( - 6 \beta_{6} + 5 \beta_{2}) q^{19} + ( - 3 \beta_{4} + 153) q^{23} + ( - 2 \beta_{4} + 1) q^{25} - 16 \beta_{2} q^{29} + (2 \beta_{7} + 2 \beta_{3}) q^{31} + ( - \beta_{5} + 40 \beta_1) q^{35} + (21 \beta_{5} + \beta_1) q^{37} + ( - 18 \beta_{7} - 54 \beta_{3}) q^{41} + ( - 30 \beta_{6} + 34 \beta_{2}) q^{43} + ( - 21 \beta_{4} - 225) q^{47} + ( - 4 \beta_{4} - 30) q^{49} + ( - 26 \beta_{6} + 80 \beta_{2}) q^{53} + ( - 22 \beta_{7} + 54 \beta_{3}) q^{55} + (27 \beta_{5} - 40 \beta_1) q^{59} + (9 \beta_{5} - 45 \beta_1) q^{61} + ( - 3 \beta_{7} - 369 \beta_{3}) q^{65} + (36 \beta_{6} + 39 \beta_{2}) q^{67} + ( - 18 \beta_{4} + 54) q^{71} + (44 \beta_{4} + 43) q^{73} + (71 \beta_{6} - 80 \beta_{2}) q^{77} + (27 \beta_{7} - 196 \beta_{3}) q^{79} + ( - 44 \beta_{5} - 80 \beta_1) q^{83} + ( - 54 \beta_{5} - 120 \beta_1) q^{85} + ( - 21 \beta_{7} - 423 \beta_{3}) q^{89} + ( - 6 \beta_{6} + 125 \beta_{2}) q^{91} + ( - 3 \beta_{4} - 711) q^{95} + ( - 50 \beta_{4} + 155) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 1224 q^{23} + 8 q^{25} - 1800 q^{47} - 240 q^{49} + 432 q^{71} + 344 q^{73} - 5688 q^{95} + 1240 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 21x^{6} + 341x^{4} - 2100x^{2} + 10000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -63\nu^{6} + 1023\nu^{4} - 21483\nu^{2} + 81150 ) / 17050 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{7} - 363\nu^{5} + 4323\nu^{3} - 42600\nu ) / 11000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 21\nu^{7} - 341\nu^{5} + 4061\nu^{3} - 10000\nu ) / 31000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -6\nu^{6} - 8883 ) / 341 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 181\nu^{6} - 3751\nu^{4} + 44671\nu^{2} - 192550 ) / 8525 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 241\nu^{7} - 7161\nu^{5} + 116281\nu^{3} - 1222200\nu ) / 170500 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 63\nu^{7} - 1023\nu^{5} + 18183\nu^{3} - 30000\nu ) / 11000 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} - 3\beta_{6} - 3\beta_{3} + 2\beta_{2} ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{5} + \beta_{4} - 22\beta _1 + 63 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 11\beta_{7} - 93\beta_{3} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -63\beta_{5} - 21\beta_{4} - 262\beta _1 - 723 ) / 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 131\beta_{7} + 393\beta_{6} - 1653\beta_{3} - 682\beta_{2} ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -341\beta_{4} - 8883 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -1651\beta_{7} + 4953\beta_{6} + 25413\beta_{3} - 10122\beta_{2} ) / 12 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
863.1
2.33962 1.35078i
2.33962 + 1.35078i
3.20565 1.85078i
3.20565 + 1.85078i
−3.20565 1.85078i
−3.20565 + 1.85078i
−2.33962 1.35078i
−2.33962 + 1.35078i
0 0 0 −12.8226 0 17.2094i 0 0 0
863.2 0 0 0 −12.8226 0 17.2094i 0 0 0
863.3 0 0 0 −9.35849 0 21.2094i 0 0 0
863.4 0 0 0 −9.35849 0 21.2094i 0 0 0
863.5 0 0 0 9.35849 0 21.2094i 0 0 0
863.6 0 0 0 9.35849 0 21.2094i 0 0 0
863.7 0 0 0 12.8226 0 17.2094i 0 0 0
863.8 0 0 0 12.8226 0 17.2094i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 863.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.4.f.g yes 8
3.b odd 2 1 1728.4.f.c 8
4.b odd 2 1 1728.4.f.c 8
8.b even 2 1 inner 1728.4.f.g yes 8
8.d odd 2 1 1728.4.f.c 8
12.b even 2 1 inner 1728.4.f.g yes 8
24.f even 2 1 inner 1728.4.f.g yes 8
24.h odd 2 1 1728.4.f.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1728.4.f.c 8 3.b odd 2 1
1728.4.f.c 8 4.b odd 2 1
1728.4.f.c 8 8.d odd 2 1
1728.4.f.c 8 24.h odd 2 1
1728.4.f.g yes 8 1.a even 1 1 trivial
1728.4.f.g yes 8 8.b even 2 1 inner
1728.4.f.g yes 8 12.b even 2 1 inner
1728.4.f.g yes 8 24.f 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 \(S_{4}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{4} - 252T_{5}^{2} + 14400 \) Copy content Toggle raw display
\( T_{7}^{4} + 746T_{7}^{2} + 133225 \) Copy content Toggle raw display
\( T_{23}^{2} - 306T_{23} + 20088 \) 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
$5$ \( (T^{4} - 252 T^{2} + 14400)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 746 T^{2} + 133225)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 3420 T^{2} + 2143296)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1107)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 14580 T^{2} + 419904)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 9342 T^{2} + 17514225)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 306 T + 20088)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 6912)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2960 T^{2} + 2166784)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 110430 T^{2} + \cdots + 2837799441)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 244944 T^{2} + \cdots + 13604889600)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 252504 T^{2} + \cdots + 9053141904)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 450 T - 112104)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} - 441072 T^{2} + \cdots + 2941977600)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 308988 T^{2} + \cdots + 617025600)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 144342 T^{2} + \cdots + 2729540025)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 459270 T^{2} + \cdots + 7953250761)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 108 T - 116640)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 86 T - 712535)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 614834 T^{2} + \cdots + 53169442225)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 706752 T^{2} + \cdots + 15099494400)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 683316 T^{2} + \cdots + 262440000)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 310 T - 898475)^{4} \) Copy content Toggle raw display
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