Properties

Label 192.3.i.a
Level $192$
Weight $3$
Character orbit 192.i
Analytic conductor $5.232$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 192.i (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.23162107572\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.629407744.1
Defining polynomial: \( x^{8} - 2x^{6} + 2x^{4} - 8x^{2} + 16 \) 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 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{7} - \beta_{6} - \beta_{3} - \beta_{2} - \beta_1) q^{3} + ( - \beta_{7} - \beta_{6} + 3 \beta_{4}) q^{5} + ( - \beta_{6} - \beta_{3} - \beta_{2} + 1) q^{7} + (2 \beta_{6} + 3 \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} - \beta_{6} - \beta_{3} - \beta_{2} - \beta_1) q^{3} + ( - \beta_{7} - \beta_{6} + 3 \beta_{4}) q^{5} + ( - \beta_{6} - \beta_{3} - \beta_{2} + 1) q^{7} + (2 \beta_{6} + 3 \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{9} + 2 \beta_{4} q^{11} + ( - \beta_{7} - \beta_{6} - 13 \beta_{3} - 2 \beta_{2} - 11) q^{13} + (5 \beta_{7} + \beta_{6} + 3 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 14) q^{15} + (8 \beta_{7} + 8 \beta_{6} + 9 \beta_{5} + \beta_{4} - \beta_1) q^{17} + ( - 4 \beta_{7} - 4 \beta_{6} - 6 \beta_{3} - 8 \beta_{2} + 2) q^{19} + ( - 2 \beta_{7} - \beta_{4} + 3 \beta_{3} - 5) q^{21} + (10 \beta_{5} - 2 \beta_{4} - 2 \beta_1) q^{23} + ( - 14 \beta_{6} + 17 \beta_{3} - 14 \beta_{2} + 14) q^{25} + (2 \beta_{7} + 9 \beta_{6} + \beta_{4} - 12 \beta_{3} + 5) q^{27} + ( - 7 \beta_{7} - 7 \beta_{6} - 7 \beta_{5} - 7 \beta_1) q^{29} + ( - \beta_{7} - \beta_{3} - \beta_{2} - 9) q^{31} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 8) q^{33} + (\beta_{7} + \beta_{6} + 6 \beta_{5} - 4 \beta_1) q^{35} + ( - 5 \beta_{7} + 5 \beta_{6} - 19 \beta_{3} + 9) q^{37} + (13 \beta_{6} + 3 \beta_{5} + 10 \beta_{4} + 19 \beta_{3} + 13 \beta_{2} + \cdots - 13) q^{39}+ \cdots + (8 \beta_{7} + 8 \beta_{6} - 6 \beta_{5} - 16 \beta_{3} + 8 \beta_{2} + 14 \beta_1 - 24) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{3} - 96 q^{13} - 112 q^{15} - 16 q^{19} - 32 q^{21} + 68 q^{27} - 72 q^{31} - 64 q^{33} + 112 q^{37} + 240 q^{43} - 112 q^{45} + 328 q^{49} + 32 q^{51} + 208 q^{61} + 104 q^{63} + 232 q^{67} - 324 q^{75} + 136 q^{79} + 184 q^{81} - 112 q^{85} - 152 q^{91} + 64 q^{93} - 480 q^{97} - 160 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 2\nu^{5} + 10\nu^{3} + 24\nu^{2} + 8\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 2\nu^{4} - 2\nu^{2} - 4 ) / 12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 4\nu^{5} - 10\nu^{3} + 4\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + \nu^{5} + 2\nu^{3} + 10\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 8\nu^{6} - 2\nu^{5} - 8\nu^{4} - 10\nu^{3} + 8\nu^{2} - 8\nu - 32 ) / 24 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} - 8\nu^{6} - 2\nu^{5} + 8\nu^{4} - 10\nu^{3} - 8\nu^{2} - 8\nu + 32 ) / 24 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} + 2\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + 4\beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{7} - 2\beta_{6} + 2\beta_{4} - \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{7} + \beta_{6} + 4\beta_{3} + 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -3\beta_{7} - 3\beta_{6} - 5\beta_{5} + \beta_{4} + 3\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/192\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(133\)
\(\chi(n)\) \(-1\) \(1\) \(-\beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
1.38255 + 0.297594i
0.767178 1.18804i
−1.38255 0.297594i
−0.767178 + 1.18804i
1.38255 0.297594i
0.767178 + 1.18804i
−1.38255 + 0.297594i
−0.767178 1.18804i
0 −2.90783 0.737922i 0 1.57472 + 1.57472i 0 3.64575i 0 7.91094 + 4.29150i 0
17.2 0 −1.13234 + 2.77809i 0 6.28651 + 6.28651i 0 1.64575i 0 −6.43560 6.29150i 0
17.3 0 −0.737922 2.90783i 0 −1.57472 1.57472i 0 3.64575i 0 −7.91094 + 4.29150i 0
17.4 0 2.77809 1.13234i 0 −6.28651 6.28651i 0 1.64575i 0 6.43560 6.29150i 0
113.1 0 −2.90783 + 0.737922i 0 1.57472 1.57472i 0 3.64575i 0 7.91094 4.29150i 0
113.2 0 −1.13234 2.77809i 0 6.28651 6.28651i 0 1.64575i 0 −6.43560 + 6.29150i 0
113.3 0 −0.737922 + 2.90783i 0 −1.57472 + 1.57472i 0 3.64575i 0 −7.91094 4.29150i 0
113.4 0 2.77809 + 1.13234i 0 −6.28651 + 6.28651i 0 1.64575i 0 6.43560 + 6.29150i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.e even 4 1 inner
48.i odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.3.i.a 8
3.b odd 2 1 inner 192.3.i.a 8
4.b odd 2 1 48.3.i.a 8
8.b even 2 1 384.3.i.b 8
8.d odd 2 1 384.3.i.a 8
12.b even 2 1 48.3.i.a 8
16.e even 4 1 inner 192.3.i.a 8
16.e even 4 1 384.3.i.b 8
16.f odd 4 1 48.3.i.a 8
16.f odd 4 1 384.3.i.a 8
24.f even 2 1 384.3.i.a 8
24.h odd 2 1 384.3.i.b 8
48.i odd 4 1 inner 192.3.i.a 8
48.i odd 4 1 384.3.i.b 8
48.k even 4 1 48.3.i.a 8
48.k even 4 1 384.3.i.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.3.i.a 8 4.b odd 2 1
48.3.i.a 8 12.b even 2 1
48.3.i.a 8 16.f odd 4 1
48.3.i.a 8 48.k even 4 1
192.3.i.a 8 1.a even 1 1 trivial
192.3.i.a 8 3.b odd 2 1 inner
192.3.i.a 8 16.e even 4 1 inner
192.3.i.a 8 48.i odd 4 1 inner
384.3.i.a 8 8.d odd 2 1
384.3.i.a 8 16.f odd 4 1
384.3.i.a 8 24.f even 2 1
384.3.i.a 8 48.k even 4 1
384.3.i.b 8 8.b even 2 1
384.3.i.b 8 16.e even 4 1
384.3.i.b 8 24.h odd 2 1
384.3.i.b 8 48.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 6272T_{5}^{4} + 153664 \) acting on \(S_{3}^{\mathrm{new}}(192, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 4 T^{7} + 8 T^{6} - 12 T^{5} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{8} + 6272 T^{4} + 153664 \) Copy content Toggle raw display
$7$ \( (T^{4} + 16 T^{2} + 36)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 2048 T^{4} + 16384 \) Copy content Toggle raw display
$13$ \( (T^{4} + 48 T^{3} + 1152 T^{2} + \cdots + 75076)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 920 T^{2} + 103968)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 8 T^{3} + 32 T^{2} - 1728 T + 46656)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 1120 T^{2} + 225792)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 614656 T^{4} + \cdots + 29884728384 \) Copy content Toggle raw display
$31$ \( (T^{2} + 18 T + 74)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 56 T^{3} + 1568 T^{2} + \cdots + 1764)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 664 T^{2} + 2592)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 120 T^{3} + 7200 T^{2} + \cdots + 2483776)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 8144 T^{2} + 14193792)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 18284288 T^{4} + \cdots + 18847994944 \) Copy content Toggle raw display
$59$ \( T^{8} + 3520000 T^{4} + \cdots + 2624400000000 \) Copy content Toggle raw display
$61$ \( (T^{4} - 104 T^{3} + 5408 T^{2} + \cdots + 1004004)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 116 T^{3} + 6728 T^{2} + \cdots + 2155024)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 16560 T^{2} + 24668288)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 3712 T^{2} + 788544)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 34 T - 894)^{4} \) Copy content Toggle raw display
$83$ \( T^{8} + 135735296 T^{4} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{4} - 6240 T^{2} + 184832)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 120 T + 3152)^{4} \) Copy content Toggle raw display
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