Properties

Label 4140.2.f.c
Level $4140$
Weight $2$
Character orbit 4140.f
Analytic conductor $33.058$
Analytic rank $0$
Dimension $14$
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: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
Defining polynomial: \( x^{14} + 27x^{12} + 283x^{10} + 1441x^{8} + 3596x^{6} + 3740x^{4} + 772x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 1380)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{10} q^{5} + \beta_{13} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{10} q^{5} + \beta_{13} q^{7} + (\beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{5} - \beta_{4} + \beta_{2} - 1) q^{11} + (\beta_{12} + \beta_{10} + \beta_{7} - \beta_1) q^{13} + (\beta_{13} + \beta_{12} - \beta_{9} - \beta_{8} - \beta_1) q^{17} + ( - \beta_{4} + \beta_{3}) q^{19} - \beta_{6} q^{23} + (\beta_{13} - \beta_{6} - \beta_{5} + \beta_{2} - 1) q^{25} + (\beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{4} - \beta_{3} - \beta_{2} + 2) q^{29} + ( - 2 \beta_{10} + 2 \beta_{7} + \beta_{5} - \beta_{3} + 1) q^{31} + ( - \beta_{12} - 2 \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + 1) q^{35} + (\beta_{11} + 3 \beta_{6} + \beta_1) q^{37} + ( - \beta_{9} + \beta_{8} + \beta_{5} + \beta_{4} - 3) q^{41} + ( - \beta_{13} - 2 \beta_{12} + \beta_{11} + \beta_{6} + \beta_1) q^{43} + ( - \beta_{13} + \beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6}) q^{47} + ( - \beta_{10} + \beta_{7} - 2 \beta_{3} - \beta_{2} - 1) q^{49} + (\beta_{12} - \beta_{11} - 2 \beta_{9} - 2 \beta_{8} + \beta_{6}) q^{53} + ( - \beta_{12} + 2 \beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} - \beta_1 - 1) q^{55} + (\beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} + 2 \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 2) q^{59} + ( - \beta_{9} + \beta_{8} + \beta_{4} + \beta_{3} + 2 \beta_{2} + 4) q^{61} + ( - \beta_{13} + \beta_{11} - \beta_{10} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} + \beta_1 + 3) q^{65} + (2 \beta_{12} - \beta_{11} + 2 \beta_{10} - \beta_{9} - \beta_{8} + 2 \beta_{7} + \beta_{6} - \beta_1) q^{67} + ( - \beta_{9} + \beta_{8} + \beta_{5} - \beta_{4} - 3) q^{71} + ( - 2 \beta_{13} + \beta_{12} + \beta_{10} + \beta_{7} + 2 \beta_{6} - \beta_1) q^{73} + (\beta_{13} - 3 \beta_{12} + \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} - 3 \beta_{6} + 2 \beta_1) q^{77} + ( - \beta_{9} + \beta_{8} + 2 \beta_{4} - 2 \beta_{2} - 2) q^{79} + (\beta_{13} + \beta_{12} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} + 4 \beta_{6} - \beta_1) q^{83} + (\beta_{13} - \beta_{12} + 2 \beta_{11} - \beta_{10} - \beta_{7} - 3 \beta_{6} + 2 \beta_{4} + \beta_{3} + \cdots - 3) q^{85}+ \cdots + ( - \beta_{13} - 2 \beta_{12} + \beta_{11} - \beta_{9} - \beta_{8} + \beta_{6} + 3 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 4 q^{19} - 6 q^{25} + 30 q^{29} + 6 q^{31} + 14 q^{35} - 46 q^{41} - 20 q^{49} - 16 q^{55} + 10 q^{59} + 64 q^{61} + 36 q^{65} - 42 q^{71} - 32 q^{79} - 42 q^{85} + 52 q^{89} + 28 q^{91} + 44 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 27x^{12} + 283x^{10} + 1441x^{8} + 3596x^{6} + 3740x^{4} + 772x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 6\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{12} + 6\nu^{10} - 153\nu^{8} - 1732\nu^{6} - 5726\nu^{4} - 5036\nu^{2} + 136 ) / 372 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 15\nu^{12} + 338\nu^{10} + 2851\nu^{8} + 11282\nu^{6} + 21122\nu^{4} + 15228\nu^{2} + 800 ) / 372 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4\nu^{12} - 86\nu^{10} - 659\nu^{8} - 2124\nu^{6} - 2423\nu^{4} - 6\nu^{2} + 200 ) / 62 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6\nu^{13} + 160\nu^{11} + 1655\nu^{9} + 8332\nu^{7} + 20731\nu^{5} + 22050\nu^{3} + 5218\nu ) / 372 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 21 \nu^{13} + 2 \nu^{12} - 591 \nu^{11} + 12 \nu^{10} - 6459 \nu^{9} - 306 \nu^{8} - 34215 \nu^{7} - 3464 \nu^{6} - 88260 \nu^{5} - 11824 \nu^{4} - 93078 \nu^{3} - 12676 \nu^{2} + \cdots - 1216 ) / 744 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 21 \nu^{13} - 2 \nu^{12} - 591 \nu^{11} - 74 \nu^{10} - 6459 \nu^{9} - 934 \nu^{8} - 34215 \nu^{7} - 5154 \nu^{6} - 88260 \nu^{5} - 12604 \nu^{4} - 93078 \nu^{3} - 11504 \nu^{2} + \cdots - 1264 ) / 744 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 21 \nu^{13} + 2 \nu^{12} - 591 \nu^{11} + 74 \nu^{10} - 6459 \nu^{9} + 934 \nu^{8} - 34215 \nu^{7} + 5154 \nu^{6} - 88260 \nu^{5} + 12604 \nu^{4} - 93078 \nu^{3} + 11504 \nu^{2} + \cdots + 1264 ) / 744 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 21 \nu^{13} - 2 \nu^{12} - 591 \nu^{11} - 12 \nu^{10} - 6459 \nu^{9} + 306 \nu^{8} - 34215 \nu^{7} + 3464 \nu^{6} - 88260 \nu^{5} + 11824 \nu^{4} - 93078 \nu^{3} + 12676 \nu^{2} + \cdots + 1216 ) / 744 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 16\nu^{13} + 468\nu^{11} + 5271\nu^{9} + 28460\nu^{7} + 74017\nu^{5} + 78826\nu^{3} + 16870\nu ) / 372 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -27\nu^{13} - 689\nu^{11} - 6781\nu^{9} - 32255\nu^{7} - 74798\nu^{5} - 71046\nu^{3} - 10244\nu ) / 372 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -22\nu^{13} - 597\nu^{11} - 6306\nu^{9} - 32483\nu^{7} - 82534\nu^{5} - 88786\nu^{3} - 20848\nu ) / 372 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{10} - 3\beta_{9} - 3\beta_{8} + 3\beta_{7} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} - \beta_{7} + 2\beta_{3} - 7\beta_{2} + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{13} + \beta_{11} - 19\beta_{10} + 21\beta_{9} + 21\beta_{8} - 19\beta_{7} - 3\beta_{6} - 11\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -17\beta_{10} + 4\beta_{9} - 4\beta_{8} + 17\beta_{7} + 2\beta_{5} + 2\beta_{4} - 24\beta_{3} + 49\beta_{2} - 156 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 30 \beta_{13} + 2 \beta_{12} - 15 \beta_{11} + 128 \beta_{10} - 156 \beta_{9} - 156 \beta_{8} + 128 \beta_{7} + 61 \beta_{6} + 99 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 197 \beta_{10} - 68 \beta_{9} + 68 \beta_{8} - 197 \beta_{7} - 30 \beta_{5} - 28 \beta_{4} + 230 \beta_{3} - 353 \beta_{2} + 1072 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 338 \beta_{13} - 32 \beta_{12} + 157 \beta_{11} - 904 \beta_{10} + 1194 \beta_{9} + 1194 \beta_{8} - 904 \beta_{7} - 787 \beta_{6} - 841 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1965 \beta_{10} + 810 \beta_{9} - 810 \beta_{8} + 1965 \beta_{7} + 322 \beta_{5} + 282 \beta_{4} - 2052 \beta_{3} + 2617 \beta_{2} - 7692 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3390 \beta_{13} + 362 \beta_{12} - 1437 \beta_{11} + 6624 \beta_{10} - 9320 \beta_{9} - 9320 \beta_{8} + 6624 \beta_{7} + 8455 \beta_{6} + 7003 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 18213 \beta_{10} - 8336 \beta_{9} + 8336 \beta_{8} - 18213 \beta_{7} - 3058 \beta_{5} - 2512 \beta_{4} + 17758 \beta_{3} - 19889 \beta_{2} + 57120 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 31918 \beta_{13} - 3604 \beta_{12} + 12389 \beta_{11} - 49978 \beta_{10} + 73852 \beta_{9} + 73852 \beta_{8} - 49978 \beta_{7} - 82667 \beta_{6} - 57917 \beta_1 \) 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
1.52925i
1.52925i
0.0729221i
0.0729221i
2.88373i
2.88373i
2.04690i
2.04690i
2.47928i
2.47928i
0.511427i
0.511427i
2.39625i
2.39625i
0 0 0 −2.14859 0.619333i 0 1.66138i 0 0 0
829.2 0 0 0 −2.14859 + 0.619333i 0 1.66138i 0 0 0
829.3 0 0 0 −1.54426 1.61718i 0 3.99468i 0 0 0
829.4 0 0 0 −1.54426 + 1.61718i 0 3.99468i 0 0 0
829.5 0 0 0 −0.792997 2.09073i 0 4.31589i 0 0 0
829.6 0 0 0 −0.792997 + 2.09073i 0 4.31589i 0 0 0
829.7 0 0 0 0.181772 2.22867i 0 0.189781i 0 0 0
829.8 0 0 0 0.181772 + 2.22867i 0 0.189781i 0 0 0
829.9 0 0 0 0.258163 2.22111i 0 2.14682i 0 0 0
829.10 0 0 0 0.258163 + 2.22111i 0 2.14682i 0 0 0
829.11 0 0 0 1.81604 1.30461i 0 3.73844i 0 0 0
829.12 0 0 0 1.81604 + 1.30461i 0 3.73844i 0 0 0
829.13 0 0 0 2.22987 0.166381i 0 1.74202i 0 0 0
829.14 0 0 0 2.22987 + 0.166381i 0 1.74202i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 829.14
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.c 14
3.b odd 2 1 1380.2.f.b 14
5.b even 2 1 inner 4140.2.f.c 14
15.d odd 2 1 1380.2.f.b 14
15.e even 4 1 6900.2.a.bc 7
15.e even 4 1 6900.2.a.bd 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.f.b 14 3.b odd 2 1
1380.2.f.b 14 15.d odd 2 1
4140.2.f.c 14 1.a even 1 1 trivial
4140.2.f.c 14 5.b even 2 1 inner
6900.2.a.bc 7 15.e even 4 1
6900.2.a.bd 7 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{14} + 59T_{7}^{12} + 1323T_{7}^{10} + 14065T_{7}^{8} + 72984T_{7}^{6} + 178488T_{7}^{4} + 166704T_{7}^{2} + 5776 \) acting on \(S_{2}^{\mathrm{new}}(4140, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + 3 T^{12} - 8 T^{11} + \cdots + 78125 \) Copy content Toggle raw display
$7$ \( T^{14} + 59 T^{12} + 1323 T^{10} + \cdots + 5776 \) Copy content Toggle raw display
$11$ \( (T^{7} - 64 T^{5} - 62 T^{4} + 1200 T^{3} + \cdots - 9216)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + 104 T^{12} + 4268 T^{10} + \cdots + 6071296 \) Copy content Toggle raw display
$17$ \( T^{14} + 167 T^{12} + 10683 T^{10} + \cdots + 10000 \) Copy content Toggle raw display
$19$ \( (T^{7} - 2 T^{6} - 118 T^{5} + 158 T^{4} + \cdots - 36680)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{7} \) Copy content Toggle raw display
$29$ \( (T^{7} - 15 T^{6} + 17 T^{5} + 595 T^{4} + \cdots + 12272)^{2} \) Copy content Toggle raw display
$31$ \( (T^{7} - 3 T^{6} - 141 T^{5} + 499 T^{4} + \cdots + 54720)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + 243 T^{12} + \cdots + 16126968064 \) Copy content Toggle raw display
$41$ \( (T^{7} + 23 T^{6} + 103 T^{5} + \cdots - 15248)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + 336 T^{12} + \cdots + 389193984 \) Copy content Toggle raw display
$47$ \( T^{14} + 492 T^{12} + \cdots + 1973013529600 \) Copy content Toggle raw display
$53$ \( T^{14} + 251 T^{12} + \cdots + 374345104 \) Copy content Toggle raw display
$59$ \( (T^{7} - 5 T^{6} - 255 T^{5} + \cdots + 1483200)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} - 32 T^{6} + 274 T^{5} + \cdots + 377248)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + 315 T^{12} + 37075 T^{10} + \cdots + 300304 \) Copy content Toggle raw display
$71$ \( (T^{7} + 21 T^{6} + 55 T^{5} - 875 T^{4} + \cdots + 784)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + 440 T^{12} + \cdots + 47127199744 \) Copy content Toggle raw display
$79$ \( (T^{7} + 16 T^{6} - 264 T^{5} + \cdots - 4755584)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + 415 T^{12} + 47443 T^{10} + \cdots + 1638400 \) Copy content Toggle raw display
$89$ \( (T^{7} - 26 T^{6} + 12 T^{5} + \cdots + 526112)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + 804 T^{12} + \cdots + 56070144 \) Copy content Toggle raw display
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