Properties

Label 4140.2.f.b
Level $4140$
Weight $2$
Character orbit 4140.f
Analytic conductor $33.058$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4140.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(33.0580664368\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \( x^{12} + 24x^{10} + 188x^{8} + 530x^{6} + 508x^{4} + 80x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 460)
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.

\(f(q)\) \(=\) \( q + \beta_{8} q^{5} + ( - \beta_{10} + \beta_{6} + \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{5} + ( - \beta_{10} + \beta_{6} + \beta_1) q^{7} + ( - \beta_{9} + \beta_{8} + \beta_{4} + \beta_{3} - \beta_{2} - 1) q^{11} + ( - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5}) q^{13} + ( - \beta_{11} + \beta_{5}) q^{17} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{19} + \beta_{6} q^{23} + (\beta_{11} + \beta_{10} - \beta_{9} + \beta_{7} - \beta_{6} - \beta_{2} - \beta_1 + 1) q^{25} + ( - \beta_{4} - 2 \beta_{2} + 2) q^{29} + ( - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{4} + 2 \beta_{3}) q^{31} + ( - \beta_{11} - \beta_{10} - \beta_{9} + \beta_{7} + 4 \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} - \beta_1) q^{35} + ( - 2 \beta_{11} - \beta_{10} + 2 \beta_{7} + 5 \beta_{6} - \beta_1) q^{37} + ( - \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} - \beta_{4} + 2 \beta_{2}) q^{41} + (\beta_{11} - \beta_{10} - 2 \beta_{7} - 5 \beta_{6} - \beta_{5} + 3 \beta_1) q^{43} + ( - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} - 3 \beta_{6} - \beta_{5}) q^{47} + (\beta_{10} + \beta_{7} - \beta_{4} + 2 \beta_{3} + \beta_{2} - 4) q^{49} + (\beta_{11} - 2 \beta_{10} + \beta_{9} + \beta_{8} - 2 \beta_{6} - \beta_{5} + 2 \beta_1) q^{53} + ( - \beta_{11} + \beta_{10} - \beta_{8} + \beta_{7} - \beta_{5} - 2 \beta_{3} - 2 \beta_{2} - \beta_1 + 3) q^{55} + (2 \beta_{10} - 3 \beta_{9} + 3 \beta_{8} + 2 \beta_{7} + \beta_{4} + 2 \beta_{3} - \beta_{2} + \cdots - 3) q^{59}+ \cdots + (\beta_{11} - \beta_{10} + 3 \beta_{9} + 3 \beta_{8} + 2 \beta_{7} + \beta_{6} + \beta_{5} - \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{11} - 8 q^{19} + 8 q^{25} + 10 q^{29} + 18 q^{31} + 10 q^{35} + 2 q^{41} - 38 q^{49} + 16 q^{55} - 22 q^{59} - 8 q^{61} - 38 q^{65} + 34 q^{71} - 20 q^{79} + 6 q^{85} - 48 q^{89} - 8 q^{91} - 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 24x^{10} + 188x^{8} + 530x^{6} + 508x^{4} + 80x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{10} - 23\nu^{8} - 8\nu^{7} - 157\nu^{6} - 144\nu^{5} - 261\nu^{4} - 752\nu^{3} + 121\nu^{2} - 1064\nu + 159 ) / 96 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{10} + 69\nu^{8} + 503\nu^{6} + 1215\nu^{4} + 869\nu^{2} + 83 ) / 48 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{10} + 169\nu^{8} + 1331\nu^{6} + 3779\nu^{4} + 3689\nu^{2} + 415 ) / 96 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9\nu^{10} + 215\nu^{8} + 1661\nu^{6} + 4493\nu^{4} + 3895\nu^{2} + 497 ) / 96 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -7\nu^{11} - 169\nu^{9} - 1339\nu^{7} - 3875\nu^{5} - 3913\nu^{3} - 663\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{11} + 24\nu^{9} + 188\nu^{7} + 529\nu^{5} + 496\nu^{3} + 58\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 27 \nu^{11} - \nu^{10} - 645 \nu^{9} - 23 \nu^{8} - 5007 \nu^{7} - 157 \nu^{6} - 13815 \nu^{5} - 261 \nu^{4} - 12597 \nu^{3} + 121 \nu^{2} - 1419 \nu + 159 ) / 96 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 27 \nu^{11} + \nu^{10} + 645 \nu^{9} + 23 \nu^{8} + 5007 \nu^{7} + 173 \nu^{6} + 13815 \nu^{5} + 453 \nu^{4} + 12597 \nu^{3} + 231 \nu^{2} + 1515 \nu - 143 ) / 96 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 27 \nu^{11} - \nu^{10} + 645 \nu^{9} - 23 \nu^{8} + 5007 \nu^{7} - 173 \nu^{6} + 13815 \nu^{5} - 453 \nu^{4} + 12597 \nu^{3} - 231 \nu^{2} + 1515 \nu + 143 ) / 96 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 27 \nu^{11} - \nu^{10} + 645 \nu^{9} - 23 \nu^{8} + 5007 \nu^{7} - 157 \nu^{6} + 13815 \nu^{5} - 261 \nu^{4} + 12597 \nu^{3} + 121 \nu^{2} + 1419 \nu + 159 ) / 96 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -37\nu^{11} - 891\nu^{9} - 7017\nu^{7} - 19993\nu^{5} - 19547\nu^{3} - 3245\nu ) / 96 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} + 2\beta_{4} - 2\beta_{3} - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + 4\beta_{10} - 4\beta_{9} - 4\beta_{8} - 3\beta_{7} + 2\beta_{6} + 3\beta_{5} - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -9\beta_{10} - 7\beta_{9} + 7\beta_{8} - 9\beta_{7} - 22\beta_{4} + 22\beta_{3} + 2\beta_{2} + 66 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 22\beta_{11} - 73\beta_{10} + 75\beta_{9} + 75\beta_{8} + 51\beta_{7} - 56\beta_{6} - 72\beta_{5} + 22\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 46\beta_{10} + 28\beta_{9} - 28\beta_{8} + 46\beta_{7} + 110\beta_{4} - 110\beta_{3} - 12\beta_{2} - 309 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 208 \beta_{11} + 707 \beta_{10} - 731 \beta_{9} - 731 \beta_{8} - 475 \beta_{7} + 632 \beta_{6} + 732 \beta_{5} - 232 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 955 \beta_{10} - 483 \beta_{9} + 483 \beta_{8} - 955 \beta_{7} - 2146 \beta_{4} + 2170 \beta_{3} + 208 \beta_{2} + 5972 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 941 \beta_{11} - 3488 \beta_{10} + 3592 \beta_{9} + 3592 \beta_{8} + 2271 \beta_{7} - 3376 \beta_{6} - 3597 \beta_{5} + 1217 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 9895 \beta_{10} + 4265 \beta_{9} - 4265 \beta_{8} + 9895 \beta_{7} + 20802 \beta_{4} - 21354 \beta_{3} - 1538 \beta_{2} - 58206 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 16710 \beta_{11} + 69215 \beta_{10} - 70753 \beta_{9} - 70753 \beta_{8} - 43769 \beta_{7} + 70884 \beta_{6} + 70152 \beta_{5} - 25446 \beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times\).

\(n\) \(461\) \(1657\) \(2071\) \(3961\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
829.1
0.420790i
0.420790i
3.16223i
3.16223i
0.116918i
0.116918i
1.26443i
1.26443i
3.08006i
3.08006i
1.65047i
1.65047i
0 0 0 −1.77747 1.35668i 0 3.32224i 0 0 0
829.2 0 0 0 −1.77747 + 1.35668i 0 3.32224i 0 0 0
829.3 0 0 0 −1.59013 1.57210i 0 2.43185i 0 0 0
829.4 0 0 0 −1.59013 + 1.57210i 0 2.43185i 0 0 0
829.5 0 0 0 −1.52160 1.63852i 0 1.80495i 0 0 0
829.6 0 0 0 −1.52160 + 1.63852i 0 1.80495i 0 0 0
829.7 0 0 0 0.817027 2.08146i 0 4.41307i 0 0 0
829.8 0 0 0 0.817027 + 2.08146i 0 4.41307i 0 0 0
829.9 0 0 0 1.89824 1.18182i 0 0.992530i 0 0 0
829.10 0 0 0 1.89824 + 1.18182i 0 0.992530i 0 0 0
829.11 0 0 0 2.17393 0.523461i 0 4.50896i 0 0 0
829.12 0 0 0 2.17393 + 0.523461i 0 4.50896i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 829.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4140.2.f.b 12
3.b odd 2 1 460.2.c.a 12
5.b even 2 1 inner 4140.2.f.b 12
12.b even 2 1 1840.2.e.f 12
15.d odd 2 1 460.2.c.a 12
15.e even 4 1 2300.2.a.n 6
15.e even 4 1 2300.2.a.o 6
60.h even 2 1 1840.2.e.f 12
60.l odd 4 1 9200.2.a.cx 6
60.l odd 4 1 9200.2.a.cy 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.c.a 12 3.b odd 2 1
460.2.c.a 12 15.d odd 2 1
1840.2.e.f 12 12.b even 2 1
1840.2.e.f 12 60.h even 2 1
2300.2.a.n 6 15.e even 4 1
2300.2.a.o 6 15.e even 4 1
4140.2.f.b 12 1.a even 1 1 trivial
4140.2.f.b 12 5.b even 2 1 inner
9200.2.a.cx 6 60.l odd 4 1
9200.2.a.cy 6 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} + 61T_{7}^{10} + 1380T_{7}^{8} + 14312T_{7}^{6} + 68992T_{7}^{4} + 139536T_{7}^{2} + 82944 \) acting on \(S_{2}^{\mathrm{new}}(4140, [\chi])\). 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
$5$ \( T^{12} - 4 T^{10} - 12 T^{9} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( T^{12} + 61 T^{10} + 1380 T^{8} + \cdots + 82944 \) Copy content Toggle raw display
$11$ \( (T^{6} + 2 T^{5} - 32 T^{4} - 28 T^{3} + \cdots - 256)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 108 T^{10} + 4126 T^{8} + \cdots + 1401856 \) Copy content Toggle raw display
$17$ \( T^{12} + 57 T^{10} + 628 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( (T^{6} + 4 T^{5} - 42 T^{4} - 140 T^{3} + \cdots + 256)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$29$ \( (T^{6} - 5 T^{5} - 84 T^{4} + 450 T^{3} + \cdots + 11862)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 9 T^{5} - 58 T^{4} + 838 T^{3} + \cdots + 916)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 357 T^{10} + \cdots + 3996262656 \) Copy content Toggle raw display
$41$ \( (T^{6} - T^{5} - 64 T^{4} + 158 T^{3} + 257 T^{2} + \cdots - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 340 T^{10} + \cdots + 8399355904 \) Copy content Toggle raw display
$47$ \( T^{12} + 228 T^{10} + 17974 T^{8} + \cdots + 215296 \) Copy content Toggle raw display
$53$ \( T^{12} + 273 T^{10} + \cdots + 149426176 \) Copy content Toggle raw display
$59$ \( (T^{6} + 11 T^{5} - 188 T^{4} + \cdots - 360576)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 4 T^{5} - 198 T^{4} + \cdots - 171088)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 341 T^{10} + \cdots + 55115776 \) Copy content Toggle raw display
$71$ \( (T^{6} - 17 T^{5} - 62 T^{4} + 1570 T^{3} + \cdots - 16108)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 376 T^{10} + \cdots + 30294016 \) Copy content Toggle raw display
$79$ \( (T^{6} + 10 T^{5} - 10 T^{4} - 208 T^{3} + \cdots - 128)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 777 T^{10} + \cdots + 401124622336 \) Copy content Toggle raw display
$89$ \( (T^{6} + 24 T^{5} - 142 T^{4} + \cdots + 180432)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 704 T^{10} + \cdots + 2091049984 \) Copy content Toggle raw display
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