Properties

Label 192.2.k.a
Level $192$
Weight $2$
Character orbit 192.k
Analytic conductor $1.533$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(1.53312771881\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.163368480538624.2
Defining polynomial: \( x^{12} - 2x^{10} - 2x^{8} + 16x^{6} - 8x^{4} - 32x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{10} q^{3} + \beta_{7} q^{5} + ( - \beta_{11} + 1) q^{7} + ( - \beta_{9} - \beta_{8} - \beta_{5} + \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{10} q^{3} + \beta_{7} q^{5} + ( - \beta_{11} + 1) q^{7} + ( - \beta_{9} - \beta_{8} - \beta_{5} + \beta_{3}) q^{9} + ( - \beta_{9} - \beta_{8} - \beta_{4} - \beta_{2}) q^{11} + ( - \beta_{11} + \beta_{5}) q^{13} + ( - \beta_{9} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} - \beta_1) q^{15} + ( - \beta_{10} - \beta_{9} - \beta_{7} + \beta_{6} - \beta_{4} - \beta_{3}) q^{17} + (\beta_{11} + \beta_{10} + \beta_{5} - \beta_{3} + \beta_1 + 1) q^{19} + ( - 2 \beta_{10} + \beta_{8} - \beta_{7} - \beta_{2} - \beta_1 - 1) q^{21} + (\beta_{10} + \beta_{9} + \beta_{4} + \beta_{3} + 2 \beta_{2}) q^{23} + (\beta_{10} + \beta_{9} - \beta_{4} - \beta_{3} + \beta_1) q^{25} + (\beta_{11} + \beta_{9} + \beta_{8} - 2 \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{2} + \beta_1 - 1) q^{27} + (2 \beta_{9} - \beta_{6} + 2 \beta_{4}) q^{29} + (\beta_{10} + \beta_{9} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_1) q^{31} + (\beta_{11} + \beta_{10} + 2 \beta_{9} + \beta_{7} - \beta_{6} + \beta_{4} + 2 \beta_{3} + \beta_{2} - 1) q^{33} + (\beta_{8} - \beta_{2}) q^{35} + (\beta_{11} + 2 \beta_{10} + \beta_{5} - 2 \beta_{3}) q^{37} + ( - \beta_{10} - \beta_{7} + \beta_{6} - \beta_{4} - 2 \beta_{2} - 2) q^{39} + (2 \beta_{8} - \beta_{7} - \beta_{6}) q^{41} + ( - \beta_{11} + \beta_{9} + \beta_{5} - \beta_{4} + \beta_1 - 1) q^{43} + (\beta_{11} - 2 \beta_{9} + \beta_{8} + \beta_{6} - \beta_{5} + \beta_{2} + \beta_1 - 1) q^{45} + ( - 2 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{4} - 2 \beta_{3}) q^{47} + (\beta_{10} - \beta_{9} + \beta_{4} - \beta_{3} - 1) q^{49} + ( - \beta_{8} - 2 \beta_{7} + \beta_{2} - 2 \beta_1 - 2) q^{51} + (4 \beta_{10} - 2 \beta_{8} + \beta_{7} + 4 \beta_{3} + 2 \beta_{2}) q^{53} + (2 \beta_{11} + \beta_{10} - \beta_{9} + \beta_{4} - \beta_{3} - 2) q^{55} + (\beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} - \beta_1) q^{57} + ( - \beta_{9} + 4 \beta_{6} - \beta_{4}) q^{59} + (\beta_{11} + 2 \beta_{9} - \beta_{5} - 2 \beta_{4}) q^{61} + (2 \beta_{10} - 2 \beta_{9} - 2 \beta_{8} + \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 3 \beta_1) q^{63} + ( - \beta_{7} + \beta_{6} - 2 \beta_{2}) q^{65} + ( - 2 \beta_{11} - \beta_{10} - 2 \beta_{5} + \beta_{3} - 2 \beta_1 - 2) q^{67} + ( - \beta_{11} - 2 \beta_{10} + \beta_{8} - 2 \beta_{7} - \beta_{5} - 4 \beta_{3} - \beta_{2} + \cdots + 1) q^{69}+ \cdots + ( - \beta_{11} + \beta_{10} + 2 \beta_{7} - \beta_{5} - \beta_{3} + 5 \beta_1 + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{3} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{3} + 8 q^{7} - 4 q^{13} + 12 q^{19} - 8 q^{21} - 10 q^{27} - 4 q^{33} - 4 q^{37} - 20 q^{39} - 12 q^{43} - 12 q^{45} - 20 q^{49} - 24 q^{51} - 24 q^{55} + 12 q^{61} - 28 q^{67} + 4 q^{69} + 34 q^{75} - 4 q^{81} + 32 q^{85} + 60 q^{87} + 56 q^{91} + 28 q^{93} - 8 q^{97} + 52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2x^{10} - 2x^{8} + 16x^{6} - 8x^{4} - 32x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{8} - 2\nu^{4} + 4\nu^{2} + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{9} - 2\nu^{7} + 2\nu^{5} - 24\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{11} - \nu^{10} - 8\nu^{7} + 2\nu^{6} + 16\nu^{5} + 4\nu^{4} + 24\nu^{3} - 16\nu^{2} - 32\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{11} + \nu^{10} + 6\nu^{7} - 10\nu^{6} - 4\nu^{5} + 12\nu^{4} - 8\nu^{3} + 32\nu^{2} + 32\nu - 64 ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{10} - 2\nu^{6} + 12\nu^{4} + 16\nu^{2} - 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{11} + 2\nu^{9} + 2\nu^{7} - 8\nu^{5} + 8\nu^{3} + 16\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{11} + 2\nu^{7} - 4\nu^{5} - 16\nu^{3} + 16\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{11} - 6\nu^{7} + 12\nu^{5} + 8\nu^{3} - 48\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{11} - \nu^{10} + 6\nu^{7} + 10\nu^{6} - 4\nu^{5} - 12\nu^{4} - 8\nu^{3} - 32\nu^{2} + 32\nu + 64 ) / 32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2\nu^{11} + \nu^{10} - 8\nu^{7} - 2\nu^{6} + 16\nu^{5} - 4\nu^{4} + 24\nu^{3} + 16\nu^{2} - 32\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -3\nu^{10} + 2\nu^{8} + 14\nu^{6} - 24\nu^{4} - 24\nu^{2} + 80 ) / 16 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{10} - \beta_{8} + \beta_{7} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + \beta_{10} - \beta_{9} + \beta_{5} + \beta_{4} - \beta_{3} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{10} + \beta_{5} + \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{10} + 2\beta_{9} + \beta_{8} + \beta_{7} + 2\beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{11} + \beta_{10} + \beta_{9} + 3\beta_{5} - \beta_{4} - \beta_{3} - \beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -3\beta_{10} + 3\beta_{9} + 3\beta_{8} - 5\beta_{7} - \beta_{6} + 3\beta_{4} - 3\beta_{3} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -2\beta_{11} - 4\beta_{10} + 2\beta_{9} - 2\beta_{4} + 4\beta_{3} + 10\beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -4\beta_{10} - 2\beta_{9} + 8\beta_{8} + 2\beta_{6} - 2\beta_{4} - 4\beta_{3} - 6\beta_{2} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -6\beta_{11} + 6\beta_{10} + 10\beta_{9} + 2\beta_{5} - 10\beta_{4} - 6\beta_{3} + 6\beta _1 - 14 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -10\beta_{10} - 10\beta_{9} + 2\beta_{8} - 14\beta_{7} - 10\beta_{6} - 10\beta_{4} - 10\beta_{3} - 10\beta_{2} \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
1.27715 + 0.607364i
−1.35164 + 0.416001i
−1.27715 0.607364i
0.204810 + 1.39930i
1.35164 0.416001i
−0.204810 1.39930i
1.27715 0.607364i
−1.35164 0.416001i
−1.27715 + 0.607364i
0.204810 1.39930i
1.35164 + 0.416001i
−0.204810 + 1.39930i
0 −1.73003 + 0.0835731i 0 −0.431733 0.431733i 0 3.10278 0 2.98603 0.289169i 0
47.2 0 −0.966579 1.43726i 0 −1.57184 1.57184i 0 −2.24914 0 −1.13145 + 2.77846i 0
47.3 0 −0.0835731 + 1.73003i 0 0.431733 + 0.431733i 0 3.10278 0 −2.98603 0.289169i 0
47.4 0 0.814141 1.52878i 0 2.08397 + 2.08397i 0 1.14637 0 −1.67435 2.48929i 0
47.5 0 1.43726 + 0.966579i 0 1.57184 + 1.57184i 0 −2.24914 0 1.13145 + 2.77846i 0
47.6 0 1.52878 0.814141i 0 −2.08397 2.08397i 0 1.14637 0 1.67435 2.48929i 0
143.1 0 −1.73003 0.0835731i 0 −0.431733 + 0.431733i 0 3.10278 0 2.98603 + 0.289169i 0
143.2 0 −0.966579 + 1.43726i 0 −1.57184 + 1.57184i 0 −2.24914 0 −1.13145 2.77846i 0
143.3 0 −0.0835731 1.73003i 0 0.431733 0.431733i 0 3.10278 0 −2.98603 + 0.289169i 0
143.4 0 0.814141 + 1.52878i 0 2.08397 2.08397i 0 1.14637 0 −1.67435 + 2.48929i 0
143.5 0 1.43726 0.966579i 0 1.57184 1.57184i 0 −2.24914 0 1.13145 2.77846i 0
143.6 0 1.52878 + 0.814141i 0 −2.08397 + 2.08397i 0 1.14637 0 1.67435 + 2.48929i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 143.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.2.k.a 12
3.b odd 2 1 inner 192.2.k.a 12
4.b odd 2 1 48.2.k.a 12
8.b even 2 1 384.2.k.a 12
8.d odd 2 1 384.2.k.b 12
12.b even 2 1 48.2.k.a 12
16.e even 4 1 48.2.k.a 12
16.e even 4 1 384.2.k.b 12
16.f odd 4 1 inner 192.2.k.a 12
16.f odd 4 1 384.2.k.a 12
24.f even 2 1 384.2.k.b 12
24.h odd 2 1 384.2.k.a 12
48.i odd 4 1 48.2.k.a 12
48.i odd 4 1 384.2.k.b 12
48.k even 4 1 inner 192.2.k.a 12
48.k even 4 1 384.2.k.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.2.k.a 12 4.b odd 2 1
48.2.k.a 12 12.b even 2 1
48.2.k.a 12 16.e even 4 1
48.2.k.a 12 48.i odd 4 1
192.2.k.a 12 1.a even 1 1 trivial
192.2.k.a 12 3.b odd 2 1 inner
192.2.k.a 12 16.f odd 4 1 inner
192.2.k.a 12 48.k even 4 1 inner
384.2.k.a 12 8.b even 2 1
384.2.k.a 12 16.f odd 4 1
384.2.k.a 12 24.h odd 2 1
384.2.k.a 12 48.k even 4 1
384.2.k.b 12 8.d odd 2 1
384.2.k.b 12 16.e even 4 1
384.2.k.b 12 24.f even 2 1
384.2.k.b 12 48.i odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(192, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 2 T^{11} + 2 T^{10} + 2 T^{9} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( T^{12} + 100 T^{8} + 1856 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( (T^{3} - 2 T^{2} - 6 T + 8)^{4} \) Copy content Toggle raw display
$11$ \( T^{12} + 356 T^{8} + 12288 T^{4} + \cdots + 65536 \) Copy content Toggle raw display
$13$ \( (T^{6} + 2 T^{5} + 2 T^{4} - 32 T^{3} + 196 T^{2} + \cdots + 8)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 40 T^{4} + 288 T^{2} + 512)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 6 T^{5} + 18 T^{4} + 32 T^{3} + \cdots + 128)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 52 T^{4} + 640 T^{2} + 2048)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 1316 T^{8} + \cdots + 71639296 \) Copy content Toggle raw display
$31$ \( (T^{6} + 36 T^{4} + 68 T^{2} + 16)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 2 T^{5} + 2 T^{4} - 128 T^{3} + \cdots + 2312)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 108 T^{4} + 384 T^{2} - 128)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 6 T^{5} + 18 T^{4} - 32 T^{3} + \cdots + 128)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 112 T^{4} + 3712 T^{2} + \cdots - 32768)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + 24356 T^{8} + 87360 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$59$ \( T^{12} + 27748 T^{8} + 24418304 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$61$ \( (T^{6} - 6 T^{5} + 18 T^{4} - 64 T^{3} + \cdots + 95048)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 14 T^{5} + 98 T^{4} - 232 T^{3} + \cdots + 32)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 196 T^{4} + 10176 T^{2} + \cdots + 147968)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 272 T^{4} + 18496 T^{2} + \cdots + 369664)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 116 T^{4} + 1956 T^{2} + \cdots + 8464)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 19044 T^{8} + \cdots + 5473632256 \) Copy content Toggle raw display
$89$ \( (T^{6} - 212 T^{4} + 8576 T^{2} + \cdots - 2048)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 2 T^{2} - 128 T - 608)^{4} \) Copy content Toggle raw display
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