Properties

Label 3840.2.f.m
Level $3840$
Weight $2$
Character orbit 3840.f
Analytic conductor $30.663$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3840.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.6625543762\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12} \)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} - \beta_{2} q^{5} - \beta_1 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} - \beta_{2} q^{5} - \beta_1 q^{7} - q^{9} + (\beta_{9} - \beta_{8}) q^{11} + (\beta_{11} - \beta_{3}) q^{13} + \beta_{7} q^{15} + ( - \beta_{10} - \beta_{7} - \beta_{4}) q^{17} + (\beta_{8} + \beta_{6} + \beta_{2}) q^{19} - \beta_{8} q^{21} + (2 \beta_{10} + \beta_{7} + \beta_{4} + \beta_1) q^{23} + (\beta_{10} - \beta_{5}) q^{25} + \beta_{3} q^{27} + ( - 2 \beta_{9} + \beta_{6} + \beta_{2}) q^{29} + ( - \beta_{7} - \beta_{5} + \beta_{4} + 3) q^{31} + (\beta_{10} + \beta_1) q^{33} + (\beta_{11} - \beta_{9} - \beta_{8} + \beta_{6} + \beta_{3} + \beta_{2}) q^{35} + (\beta_{11} + 3 \beta_{3}) q^{37} + ( - \beta_{5} - 1) q^{39} + (2 \beta_{7} + 2 \beta_{5} - 2 \beta_{4}) q^{41} + (2 \beta_{6} - 2 \beta_{2}) q^{43} + \beta_{2} q^{45} + ( - \beta_{7} - \beta_{4} - \beta_1) q^{47} + (2 \beta_{7} - 2 \beta_{4} - 1) q^{49} + (\beta_{9} - \beta_{6} - \beta_{2}) q^{51} + ( - \beta_{6} - 4 \beta_{3} + \beta_{2}) q^{53} + ( - 2 \beta_{10} - \beta_{7} - \beta_{5} + \beta_{4} - \beta_1 - 1) q^{55} + ( - \beta_{7} - \beta_{4} - \beta_1) q^{57} + ( - \beta_{9} + 3 \beta_{8}) q^{59} + (2 \beta_{8} - 2 \beta_{6} - 2 \beta_{2}) q^{61} + \beta_1 q^{63} + ( - \beta_{10} + \beta_{7} - 3 \beta_{4} - 2 \beta_1 - 2) q^{65} + (2 \beta_{6} - 2 \beta_{2}) q^{67} + ( - 2 \beta_{9} + \beta_{8} + \beta_{6} + \beta_{2}) q^{69} + (2 \beta_{7} + 2 \beta_{5} - 2 \beta_{4} + 2) q^{71} + ( - 2 \beta_{10} - 4 \beta_{7} - 4 \beta_{4}) q^{73} + ( - \beta_{11} - \beta_{9}) q^{75} + 4 \beta_{3} q^{77} + (\beta_{7} + \beta_{5} - \beta_{4} - 3) q^{79} + q^{81} + ( - 2 \beta_{11} + 2 \beta_{3}) q^{83} + (\beta_{11} + 2 \beta_{8} - 2 \beta_{6} + 3 \beta_{3}) q^{85} + ( - 2 \beta_{10} - \beta_{7} - \beta_{4}) q^{87} + (2 \beta_{7} - 2 \beta_{5} - 2 \beta_{4} + 4) q^{89} + ( - 2 \beta_{9} - 2 \beta_{8} + 4 \beta_{6} + 4 \beta_{2}) q^{91} + ( - \beta_{11} + \beta_{6} - 3 \beta_{3} - \beta_{2}) q^{93} + (\beta_{7} + 2 \beta_{5} + \beta_{4} - \beta_1 - 6) q^{95} + (2 \beta_{10} - 2 \beta_{7} - 2 \beta_{4}) q^{97} + ( - \beta_{9} + \beta_{8}) 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 \( 12 q - 12 q^{9} - 4 q^{25} + 32 q^{31} - 16 q^{39} + 8 q^{41} - 12 q^{49} - 16 q^{55} - 24 q^{65} + 32 q^{71} - 32 q^{79} + 12 q^{81} + 40 q^{89} - 64 q^{95}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( \nu^{8} + \nu^{6} + 4\nu^{4} - 4\nu^{2} ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} + 3\nu^{9} + 6\nu^{7} - 12\nu^{5} + 24\nu^{3} ) / 64 \) 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
\(\beta_{4}\)\(=\) \( ( -\nu^{6} + \nu^{4} + 2\nu^{2} + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{10} + \nu^{6} + 4\nu^{4} + 4\nu^{2} - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{11} - 5\nu^{9} - 2\nu^{7} - 12\nu^{5} + 24\nu^{3} - 64\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{8} - \nu^{6} - 2\nu^{4} - 8\nu^{2} + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{11} - 3\nu^{9} + 2\nu^{7} + 4\nu^{5} + 24\nu^{3} ) / 32 \) 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
\(\beta_{10}\)\(=\) \( ( -\nu^{10} - 3\nu^{8} + 2\nu^{6} + 4\nu^{4} - 8\nu^{2} - 32 ) / 16 \) 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
\(\nu\)\(=\) \( ( \beta_{11} + \beta_{8} - 2\beta_{6} + \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{10} - 2\beta_{7} + \beta_{5} - \beta _1 - 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{11} + 2\beta_{9} + \beta_{8} - \beta_{3} + 2\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{10} - 2\beta_{7} - \beta_{5} + 4\beta_{4} + 5\beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3\beta_{11} + 2\beta_{9} - \beta_{8} - 4\beta_{6} + 13\beta_{3} - 6\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -2\beta_{10} - 6\beta_{7} + \beta_{5} - 12\beta_{4} + 3\beta _1 + 15 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3\beta_{11} - 10\beta_{9} + 9\beta_{8} - 4\beta_{6} + 19\beta_{3} + 14\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -14\beta_{10} + 6\beta_{7} + 7\beta_{5} - 4\beta_{4} + 5\beta _1 - 23 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -11\beta_{11} + 10\beta_{9} - 17\beta_{8} - 12\beta_{6} - 27\beta_{3} + 18\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -2\beta_{10} - 22\beta_{7} - 31\beta_{5} + 4\beta_{4} + 19\beta _1 - 17 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 3\beta_{11} + 6\beta_{9} - 39\beta_{8} + 12\beta_{6} + 147\beta_{3} - 2\beta_{2} ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(1537\) \(2561\) \(2821\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
769.1
−0.450129 + 1.34067i
−0.806504 1.16170i
−1.37729 + 0.321037i
1.37729 + 0.321037i
0.806504 1.16170i
0.450129 + 1.34067i
−0.450129 1.34067i
−0.806504 + 1.16170i
−1.37729 0.321037i
1.37729 0.321037i
0.806504 + 1.16170i
0.450129 1.34067i
0 1.00000i 0 −2.22158 + 0.254102i 0 2.64265i 0 −1.00000 0
769.2 0 1.00000i 0 −1.23992 + 1.86081i 0 0.746175i 0 −1.00000 0
769.3 0 1.00000i 0 −0.726062 2.11491i 0 4.05705i 0 −1.00000 0
769.4 0 1.00000i 0 0.726062 2.11491i 0 4.05705i 0 −1.00000 0
769.5 0 1.00000i 0 1.23992 + 1.86081i 0 0.746175i 0 −1.00000 0
769.6 0 1.00000i 0 2.22158 + 0.254102i 0 2.64265i 0 −1.00000 0
769.7 0 1.00000i 0 −2.22158 0.254102i 0 2.64265i 0 −1.00000 0
769.8 0 1.00000i 0 −1.23992 1.86081i 0 0.746175i 0 −1.00000 0
769.9 0 1.00000i 0 −0.726062 + 2.11491i 0 4.05705i 0 −1.00000 0
769.10 0 1.00000i 0 0.726062 + 2.11491i 0 4.05705i 0 −1.00000 0
769.11 0 1.00000i 0 1.23992 1.86081i 0 0.746175i 0 −1.00000 0
769.12 0 1.00000i 0 2.22158 0.254102i 0 2.64265i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 769.12
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 3840.2.f.m 12
4.b odd 2 1 3840.2.f.l 12
5.b even 2 1 inner 3840.2.f.m 12
8.b even 2 1 inner 3840.2.f.m 12
8.d odd 2 1 3840.2.f.l 12
16.e even 4 1 480.2.d.a 6
16.e even 4 1 480.2.d.b 6
16.f odd 4 1 120.2.d.a 6
16.f odd 4 1 120.2.d.b yes 6
20.d odd 2 1 3840.2.f.l 12
40.e odd 2 1 3840.2.f.l 12
40.f even 2 1 inner 3840.2.f.m 12
48.i odd 4 1 1440.2.d.e 6
48.i odd 4 1 1440.2.d.f 6
48.k even 4 1 360.2.d.e 6
48.k even 4 1 360.2.d.f 6
80.i odd 4 2 2400.2.k.f 12
80.j even 4 2 600.2.k.f 12
80.k odd 4 1 120.2.d.a 6
80.k odd 4 1 120.2.d.b yes 6
80.q even 4 1 480.2.d.a 6
80.q even 4 1 480.2.d.b 6
80.s even 4 2 600.2.k.f 12
80.t odd 4 2 2400.2.k.f 12
240.t even 4 1 360.2.d.e 6
240.t even 4 1 360.2.d.f 6
240.z odd 4 2 1800.2.k.u 12
240.bb even 4 2 7200.2.k.u 12
240.bd odd 4 2 1800.2.k.u 12
240.bf even 4 2 7200.2.k.u 12
240.bm odd 4 1 1440.2.d.e 6
240.bm odd 4 1 1440.2.d.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.d.a 6 16.f odd 4 1
120.2.d.a 6 80.k odd 4 1
120.2.d.b yes 6 16.f odd 4 1
120.2.d.b yes 6 80.k odd 4 1
360.2.d.e 6 48.k even 4 1
360.2.d.e 6 240.t even 4 1
360.2.d.f 6 48.k even 4 1
360.2.d.f 6 240.t even 4 1
480.2.d.a 6 16.e even 4 1
480.2.d.a 6 80.q even 4 1
480.2.d.b 6 16.e even 4 1
480.2.d.b 6 80.q even 4 1
600.2.k.f 12 80.j even 4 2
600.2.k.f 12 80.s even 4 2
1440.2.d.e 6 48.i odd 4 1
1440.2.d.e 6 240.bm odd 4 1
1440.2.d.f 6 48.i odd 4 1
1440.2.d.f 6 240.bm odd 4 1
1800.2.k.u 12 240.z odd 4 2
1800.2.k.u 12 240.bd odd 4 2
2400.2.k.f 12 80.i odd 4 2
2400.2.k.f 12 80.t odd 4 2
3840.2.f.l 12 4.b odd 2 1
3840.2.f.l 12 8.d odd 2 1
3840.2.f.l 12 20.d odd 2 1
3840.2.f.l 12 40.e odd 2 1
3840.2.f.m 12 1.a even 1 1 trivial
3840.2.f.m 12 5.b even 2 1 inner
3840.2.f.m 12 8.b even 2 1 inner
3840.2.f.m 12 40.f even 2 1 inner
7200.2.k.u 12 240.bb even 4 2
7200.2.k.u 12 240.bf even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3840, [\chi])\):

\( T_{7}^{6} + 24T_{7}^{4} + 128T_{7}^{2} + 64 \) Copy content Toggle raw display
\( T_{11}^{6} - 32T_{11}^{4} + 96T_{11}^{2} - 64 \) Copy content Toggle raw display
\( T_{29}^{6} - 108T_{29}^{4} + 3120T_{29}^{2} - 12544 \) Copy content Toggle raw display
\( T_{31}^{3} - 8T_{31}^{2} - 4T_{31} + 64 \) 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
$5$ \( T^{12} + 2 T^{10} - 9 T^{8} + \cdots + 15625 \) 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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