Properties

Label 7200.2.k.u
Level $7200$
Weight $2$
Character orbit 7200.k
Analytic conductor $57.492$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7200.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(57.4922894553\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.180227832610816.1
Defining polynomial: \( x^{12} + x^{10} - 8x^{6} + 16x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{53}]\)
Coefficient ring index: \( 2^{17} \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{7} + \beta_{8} q^{11} + \beta_{9} q^{13} + (\beta_{5} + \beta_{4}) q^{17} + (\beta_{7} - \beta_{2}) q^{19} + ( - 2 \beta_{5} - \beta_{4} + \beta_{3}) q^{23} + ( - 2 \beta_{8} - \beta_{7} - 2 \beta_{2}) q^{29} + ( - \beta_{11} + 3) q^{31} + (\beta_{9} + \beta_{6}) q^{37} + 2 \beta_{11} q^{41} + 2 \beta_1 q^{43} + (\beta_{4} - \beta_{3}) q^{47} + ( - \beta_{11} + \beta_{10} + 1) q^{49} + ( - \beta_{6} + \beta_1) q^{53} + (\beta_{8} - 2 \beta_{2}) q^{59} + (2 \beta_{7} + 2 \beta_{2}) q^{61} + 2 \beta_1 q^{67} + (2 \beta_{11} + 2) q^{71} + ( - 2 \beta_{5} - 4 \beta_{4}) q^{73} + \beta_{6} q^{77} + ( - \beta_{11} + 3) q^{79} + 2 \beta_{9} q^{83} + (2 \beta_{10} - 4) q^{89} + (2 \beta_{8} + 4 \beta_{7} + 4 \beta_{2}) q^{91} + (2 \beta_{5} - 2 \beta_{4}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 32 q^{31} + 8 q^{41} + 12 q^{49} + 32 q^{71} + 32 q^{79} - 40 q^{89}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{9} + \nu^{7} + 8\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{8} + \nu^{6} + 4\nu^{4} - 4\nu^{2} ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{11} - 3\nu^{9} + 2\nu^{7} + 4\nu^{5} + 24\nu^{3} ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{11} + \nu^{9} - 2\nu^{7} + 12\nu^{5} - 24\nu^{3} + 32\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{11} - \nu^{9} - 6\nu^{7} - 4\nu^{5} + 40\nu^{3} + 32\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{11} - \nu^{9} + 2\nu^{7} + 4\nu^{5} + 8\nu^{3} ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{8} + 3\nu^{6} + 4\nu^{2} - 16 ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{10} + \nu^{8} - 4\nu^{6} - 12\nu^{4} + 16\nu^{2} + 32 ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{11} - \nu^{9} - 6\nu^{7} - 20\nu^{5} - 8\nu^{3} + 64\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -\nu^{10} - \nu^{8} + 8\nu^{4} + 16\nu^{2} - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -\nu^{10} + \nu^{8} + 2\nu^{6} - 8\nu^{2} - 8 ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{9} + \beta_{6} + 2\beta_{4} + 2\beta_{3} + 2\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{10} + 2\beta_{8} + \beta_{7} + \beta_{2} - 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{9} - \beta_{6} + 4\beta_{5} - 2\beta_{4} + 2\beta_{3} + 2\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{11} + \beta_{10} - 2\beta_{8} - \beta_{7} + 3\beta_{2} + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -6\beta_{9} + 5\beta_{6} + 4\beta_{5} + 10\beta_{4} - 2\beta_{3} - 2\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2\beta_{11} - \beta_{10} + 2\beta_{8} + 9\beta_{7} + 5\beta_{2} + 15 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 6\beta_{9} + 11\beta_{6} - 20\beta_{5} - 10\beta_{4} + 18\beta_{3} + 18\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 6\beta_{11} + \beta_{10} + 14\beta_{8} - \beta_{7} + 19\beta_{2} - 23 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -22\beta_{9} - 19\beta_{6} + 20\beta_{5} - 6\beta_{4} - 34\beta_{3} + 30\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -22\beta_{11} - 9\beta_{10} + 2\beta_{8} + 9\beta_{7} + 21\beta_{2} - 17 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 6\beta_{9} + 75\beta_{6} + 12\beta_{5} - 10\beta_{4} - 78\beta_{3} - 14\beta_1 ) / 8 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7200\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(6401\) \(6751\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3601.1
1.37729 + 0.321037i
1.37729 0.321037i
−0.450129 + 1.34067i
−0.450129 1.34067i
0.806504 1.16170i
0.806504 + 1.16170i
−0.806504 + 1.16170i
−0.806504 1.16170i
0.450129 1.34067i
0.450129 + 1.34067i
−1.37729 0.321037i
−1.37729 + 0.321037i
0 0 0 0 0 −4.05705 0 0 0
3601.2 0 0 0 0 0 −4.05705 0 0 0
3601.3 0 0 0 0 0 −2.64265 0 0 0
3601.4 0 0 0 0 0 −2.64265 0 0 0
3601.5 0 0 0 0 0 −0.746175 0 0 0
3601.6 0 0 0 0 0 −0.746175 0 0 0
3601.7 0 0 0 0 0 0.746175 0 0 0
3601.8 0 0 0 0 0 0.746175 0 0 0
3601.9 0 0 0 0 0 2.64265 0 0 0
3601.10 0 0 0 0 0 2.64265 0 0 0
3601.11 0 0 0 0 0 4.05705 0 0 0
3601.12 0 0 0 0 0 4.05705 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3601.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.b even 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7200.2.k.u 12
3.b odd 2 1 2400.2.k.f 12
4.b odd 2 1 1800.2.k.u 12
5.b even 2 1 inner 7200.2.k.u 12
5.c odd 4 1 1440.2.d.e 6
5.c odd 4 1 1440.2.d.f 6
8.b even 2 1 inner 7200.2.k.u 12
8.d odd 2 1 1800.2.k.u 12
12.b even 2 1 600.2.k.f 12
15.d odd 2 1 2400.2.k.f 12
15.e even 4 1 480.2.d.a 6
15.e even 4 1 480.2.d.b 6
20.d odd 2 1 1800.2.k.u 12
20.e even 4 1 360.2.d.e 6
20.e even 4 1 360.2.d.f 6
24.f even 2 1 600.2.k.f 12
24.h odd 2 1 2400.2.k.f 12
40.e odd 2 1 1800.2.k.u 12
40.f even 2 1 inner 7200.2.k.u 12
40.i odd 4 1 1440.2.d.e 6
40.i odd 4 1 1440.2.d.f 6
40.k even 4 1 360.2.d.e 6
40.k even 4 1 360.2.d.f 6
60.h even 2 1 600.2.k.f 12
60.l odd 4 1 120.2.d.a 6
60.l odd 4 1 120.2.d.b yes 6
120.i odd 2 1 2400.2.k.f 12
120.m even 2 1 600.2.k.f 12
120.q odd 4 1 120.2.d.a 6
120.q odd 4 1 120.2.d.b yes 6
120.w even 4 1 480.2.d.a 6
120.w even 4 1 480.2.d.b 6
240.z odd 4 2 3840.2.f.l 12
240.bb even 4 2 3840.2.f.m 12
240.bd odd 4 2 3840.2.f.l 12
240.bf even 4 2 3840.2.f.m 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.d.a 6 60.l odd 4 1
120.2.d.a 6 120.q odd 4 1
120.2.d.b yes 6 60.l odd 4 1
120.2.d.b yes 6 120.q odd 4 1
360.2.d.e 6 20.e even 4 1
360.2.d.e 6 40.k even 4 1
360.2.d.f 6 20.e even 4 1
360.2.d.f 6 40.k even 4 1
480.2.d.a 6 15.e even 4 1
480.2.d.a 6 120.w even 4 1
480.2.d.b 6 15.e even 4 1
480.2.d.b 6 120.w even 4 1
600.2.k.f 12 12.b even 2 1
600.2.k.f 12 24.f even 2 1
600.2.k.f 12 60.h even 2 1
600.2.k.f 12 120.m even 2 1
1440.2.d.e 6 5.c odd 4 1
1440.2.d.e 6 40.i odd 4 1
1440.2.d.f 6 5.c odd 4 1
1440.2.d.f 6 40.i odd 4 1
1800.2.k.u 12 4.b odd 2 1
1800.2.k.u 12 8.d odd 2 1
1800.2.k.u 12 20.d odd 2 1
1800.2.k.u 12 40.e odd 2 1
2400.2.k.f 12 3.b odd 2 1
2400.2.k.f 12 15.d odd 2 1
2400.2.k.f 12 24.h odd 2 1
2400.2.k.f 12 120.i odd 2 1
3840.2.f.l 12 240.z odd 4 2
3840.2.f.l 12 240.bd odd 4 2
3840.2.f.m 12 240.bb even 4 2
3840.2.f.m 12 240.bf even 4 2
7200.2.k.u 12 1.a even 1 1 trivial
7200.2.k.u 12 5.b even 2 1 inner
7200.2.k.u 12 8.b even 2 1 inner
7200.2.k.u 12 40.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 \(S_{2}^{\mathrm{new}}(7200, [\chi])\):

\( T_{7}^{6} - 24T_{7}^{4} + 128T_{7}^{2} - 64 \) Copy content Toggle raw display
\( T_{11}^{6} + 32T_{11}^{4} + 96T_{11}^{2} + 64 \) Copy content Toggle raw display
\( T_{17}^{6} - 36T_{17}^{4} + 368T_{17}^{2} - 1024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} - 24 T^{4} + 128 T^{2} - 64)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 32 T^{4} + 96 T^{2} + 64)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 48 T^{4} + 704 T^{2} + 3136)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 36 T^{4} + 368 T^{2} - 1024)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 60 T^{4} + 512 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 92 T^{4} + 2304 T^{2} + \cdots - 16384)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 108 T^{4} + 3120 T^{2} + \cdots + 12544)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 8 T^{2} - 4 T + 64)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + 64 T^{4} + 128 T^{2} + 64)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} - 2 T^{2} - 100 T - 56)^{4} \) Copy content Toggle raw display
$43$ \( (T^{6} + 128 T^{4} + 4096 T^{2} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 60 T^{4} + 512 T^{2} - 1024)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 80 T^{4} + 1216 T^{2} + 64)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 176 T^{4} + 9888 T^{2} + \cdots + 179776)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 176 T^{4} + 7168 T^{2} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 128 T^{4} + 4096 T^{2} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} - 8 T^{2} - 80 T + 128)^{4} \) Copy content Toggle raw display
$73$ \( (T^{6} - 384 T^{4} + 34560 T^{2} + \cdots - 16384)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 8 T^{2} - 4 T + 64)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + 192 T^{4} + 11264 T^{2} + \cdots + 200704)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} + 10 T^{2} - 164 T - 1384)^{4} \) Copy content Toggle raw display
$97$ \( (T^{6} - 336 T^{4} + 28416 T^{2} + \cdots - 262144)^{2} \) Copy content Toggle raw display
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