Properties

Label 2400.2.k.f
Level $2400$
Weight $2$
Character orbit 2400.k
Analytic conductor $19.164$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2400.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.1640964851\)
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: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 2^{14} \)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} - \beta_{5} q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} - \beta_{5} q^{7} - q^{9} + \beta_{7} q^{11} + ( - \beta_{11} + \beta_{3}) q^{13} + ( - \beta_{9} - \beta_{8}) q^{17} + ( - \beta_{6} + \beta_{2}) q^{19} - \beta_{2} q^{21} + (2 \beta_{9} + \beta_{8} - \beta_{5}) q^{23} - \beta_{3} q^{27} + ( - 2 \beta_{7} - \beta_{6} - 2 \beta_{2}) q^{29} + ( - \beta_{10} + \beta_{4} + 3) q^{31} + ( - \beta_{9} + \beta_{5}) q^{33} + ( - \beta_{11} - 3 \beta_{3}) q^{37} + ( - \beta_{10} - 1) q^{39} + ( - 2 \beta_{10} + 2 \beta_{4}) q^{41} - 2 \beta_1 q^{43} + ( - \beta_{8} + \beta_{5}) q^{47} + (2 \beta_{4} + 1) q^{49} + ( - \beta_{7} - \beta_{6} - \beta_{2}) q^{51} + ( - 4 \beta_{3} + \beta_1) q^{53} + (\beta_{8} - \beta_{5}) q^{57} + (\beta_{7} - 2 \beta_{2}) q^{59} + ( - 2 \beta_{6} - 2 \beta_{2}) q^{61} + \beta_{5} q^{63} - 2 \beta_1 q^{67} + (2 \beta_{7} + \beta_{6} + \beta_{2}) q^{69} + ( - 2 \beta_{10} + 2 \beta_{4} - 2) q^{71} + ( - 2 \beta_{9} - 4 \beta_{8}) q^{73} + 4 \beta_{3} q^{77} + ( - \beta_{10} + \beta_{4} + 3) q^{79} + q^{81} + (2 \beta_{11} - 2 \beta_{3}) q^{83} + (2 \beta_{9} + \beta_{8}) q^{87} + ( - 2 \beta_{10} - 2 \beta_{4} + 4) q^{89} + ( - 2 \beta_{7} - 4 \beta_{6} - 4 \beta_{2}) q^{91} + (\beta_{11} + 3 \beta_{3} + \beta_1) q^{93} + (2 \beta_{9} - 2 \beta_{8}) q^{97} - \beta_{7} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{9} + 32 q^{31} - 16 q^{39} - 8 q^{41} + 12 q^{49} - 32 q^{71} + 32 q^{79} + 12 q^{81} + 40 q^{89}+O(q^{100}) \) 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
\(\beta_{3}\)\(=\) \( ( \nu^{11} - \nu^{9} + 2\nu^{7} + 4\nu^{5} + 8\nu^{3} ) / 64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{8} - \nu^{6} + 4\nu^{4} + 12\nu^{2} ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{11} - 3\nu^{9} + 2\nu^{7} + 4\nu^{5} + 24\nu^{3} ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{8} + 3\nu^{6} + 4\nu^{2} - 16 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{10} + \nu^{8} - 4\nu^{6} - 12\nu^{4} + 16\nu^{2} + 32 ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{11} + \nu^{9} - 2\nu^{7} + 12\nu^{5} - 24\nu^{3} + 32\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{11} - \nu^{9} - 6\nu^{7} - 4\nu^{5} + 40\nu^{3} + 32\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -\nu^{10} + \nu^{6} + 4\nu^{4} + 4\nu^{2} - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3\nu^{11} - 3\nu^{9} - 10\nu^{7} - 36\nu^{5} - 8\nu^{3} + 128\nu ) / 64 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{11} + \beta_{8} + \beta_{5} + \beta_{3} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{10} + 2\beta_{7} + \beta_{6} + \beta_{4} + \beta_{2} - 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{11} + 2\beta_{9} - \beta_{8} + \beta_{5} - \beta_{3} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{10} - 2\beta_{7} - \beta_{6} + 3\beta_{4} + 3\beta_{2} + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3\beta_{11} + 2\beta_{9} + 5\beta_{8} - \beta_{5} + 13\beta_{3} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( \beta_{10} + 2\beta_{7} + 9\beta_{6} - 3\beta_{4} + 5\beta_{2} + 15 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3\beta_{11} - 10\beta_{9} - 5\beta_{8} + 9\beta_{5} + 19\beta_{3} + 9\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 7\beta_{10} + 14\beta_{7} - \beta_{6} - 5\beta_{4} + 19\beta_{2} - 23 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -11\beta_{11} + 10\beta_{9} - 3\beta_{8} - 17\beta_{5} - 27\beta_{3} + 15\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -31\beta_{10} + 2\beta_{7} + 9\beta_{6} + 13\beta_{4} + 21\beta_{2} - 17 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 3\beta_{11} + 6\beta_{9} - 5\beta_{8} - 39\beta_{5} + 147\beta_{3} - 7\beta_1 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1601\) \(1951\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1201.1
1.37729 0.321037i
−0.450129 1.34067i
0.806504 + 1.16170i
−0.806504 + 1.16170i
0.450129 1.34067i
−1.37729 0.321037i
1.37729 + 0.321037i
−0.450129 + 1.34067i
0.806504 1.16170i
−0.806504 1.16170i
0.450129 + 1.34067i
−1.37729 + 0.321037i
0 1.00000i 0 0 0 −4.05705 0 −1.00000 0
1201.2 0 1.00000i 0 0 0 −2.64265 0 −1.00000 0
1201.3 0 1.00000i 0 0 0 −0.746175 0 −1.00000 0
1201.4 0 1.00000i 0 0 0 0.746175 0 −1.00000 0
1201.5 0 1.00000i 0 0 0 2.64265 0 −1.00000 0
1201.6 0 1.00000i 0 0 0 4.05705 0 −1.00000 0
1201.7 0 1.00000i 0 0 0 −4.05705 0 −1.00000 0
1201.8 0 1.00000i 0 0 0 −2.64265 0 −1.00000 0
1201.9 0 1.00000i 0 0 0 −0.746175 0 −1.00000 0
1201.10 0 1.00000i 0 0 0 0.746175 0 −1.00000 0
1201.11 0 1.00000i 0 0 0 2.64265 0 −1.00000 0
1201.12 0 1.00000i 0 0 0 4.05705 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1201.12
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 2400.2.k.f 12
3.b odd 2 1 7200.2.k.u 12
4.b odd 2 1 600.2.k.f 12
5.b even 2 1 inner 2400.2.k.f 12
5.c odd 4 1 480.2.d.a 6
5.c odd 4 1 480.2.d.b 6
8.b even 2 1 inner 2400.2.k.f 12
8.d odd 2 1 600.2.k.f 12
12.b even 2 1 1800.2.k.u 12
15.d odd 2 1 7200.2.k.u 12
15.e even 4 1 1440.2.d.e 6
15.e even 4 1 1440.2.d.f 6
20.d odd 2 1 600.2.k.f 12
20.e even 4 1 120.2.d.a 6
20.e even 4 1 120.2.d.b yes 6
24.f even 2 1 1800.2.k.u 12
24.h odd 2 1 7200.2.k.u 12
40.e odd 2 1 600.2.k.f 12
40.f even 2 1 inner 2400.2.k.f 12
40.i odd 4 1 480.2.d.a 6
40.i odd 4 1 480.2.d.b 6
40.k even 4 1 120.2.d.a 6
40.k even 4 1 120.2.d.b yes 6
60.h even 2 1 1800.2.k.u 12
60.l odd 4 1 360.2.d.e 6
60.l odd 4 1 360.2.d.f 6
80.i odd 4 2 3840.2.f.m 12
80.j even 4 2 3840.2.f.l 12
80.s even 4 2 3840.2.f.l 12
80.t odd 4 2 3840.2.f.m 12
120.i odd 2 1 7200.2.k.u 12
120.m even 2 1 1800.2.k.u 12
120.q odd 4 1 360.2.d.e 6
120.q odd 4 1 360.2.d.f 6
120.w even 4 1 1440.2.d.e 6
120.w even 4 1 1440.2.d.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.d.a 6 20.e even 4 1
120.2.d.a 6 40.k even 4 1
120.2.d.b yes 6 20.e even 4 1
120.2.d.b yes 6 40.k even 4 1
360.2.d.e 6 60.l odd 4 1
360.2.d.e 6 120.q odd 4 1
360.2.d.f 6 60.l odd 4 1
360.2.d.f 6 120.q odd 4 1
480.2.d.a 6 5.c odd 4 1
480.2.d.a 6 40.i odd 4 1
480.2.d.b 6 5.c odd 4 1
480.2.d.b 6 40.i odd 4 1
600.2.k.f 12 4.b odd 2 1
600.2.k.f 12 8.d odd 2 1
600.2.k.f 12 20.d odd 2 1
600.2.k.f 12 40.e odd 2 1
1440.2.d.e 6 15.e even 4 1
1440.2.d.e 6 120.w even 4 1
1440.2.d.f 6 15.e even 4 1
1440.2.d.f 6 120.w even 4 1
1800.2.k.u 12 12.b even 2 1
1800.2.k.u 12 24.f even 2 1
1800.2.k.u 12 60.h even 2 1
1800.2.k.u 12 120.m even 2 1
2400.2.k.f 12 1.a even 1 1 trivial
2400.2.k.f 12 5.b even 2 1 inner
2400.2.k.f 12 8.b even 2 1 inner
2400.2.k.f 12 40.f even 2 1 inner
3840.2.f.l 12 80.j even 4 2
3840.2.f.l 12 80.s even 4 2
3840.2.f.m 12 80.i odd 4 2
3840.2.f.m 12 80.t odd 4 2
7200.2.k.u 12 3.b odd 2 1
7200.2.k.u 12 15.d odd 2 1
7200.2.k.u 12 24.h odd 2 1
7200.2.k.u 12 120.i odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} - 24T_{7}^{4} + 128T_{7}^{2} - 64 \) acting on \(S_{2}^{\mathrm{new}}(2400, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{6} \) 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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