Properties

Label 4140.3.d.a
Level $4140$
Weight $3$
Character orbit 4140.d
Analytic conductor $112.807$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(112.806829445\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \( x^{16} - 64 x^{14} - 16 x^{13} + 2252 x^{12} + 648 x^{11} - 30106 x^{10} + 12360 x^{9} + 374528 x^{8} + 196544 x^{7} + 1261236 x^{6} - 4237944 x^{5} + \cdots + 1535848276 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{5} - \beta_{7} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{5} - \beta_{7} q^{7} + (\beta_{9} - 2 \beta_{4}) q^{11} + ( - \beta_{11} - 1) q^{13} + ( - \beta_{10} + \beta_1) q^{17} + ( - \beta_{9} - \beta_{8} - \beta_{7}) q^{19} + (\beta_{15} + \beta_{7} - \beta_{5} + \beta_{3} + \beta_1 + 1) q^{23} - 5 q^{25} + ( - \beta_{13} + \beta_{11} - 2 \beta_{5} - 4 \beta_{3} - 5) q^{29} + ( - \beta_{13} - \beta_{12} + \beta_{11} - \beta_{5} - 2 \beta_{3} + 1) q^{31} + (\beta_{13} - 2) q^{35} + ( - \beta_{15} + \beta_{10} + 2 \beta_{9} + 2 \beta_{7} - 2 \beta_{4}) q^{37} + ( - \beta_{14} + \beta_{13} + \beta_{12} - \beta_{5} - 2 \beta_{3} - \beta_{2} - 12) q^{41} + ( - 2 \beta_{15} + 2 \beta_{10} - \beta_{9} - \beta_{8} - \beta_{6} - 4 \beta_{4} - 2 \beta_1) q^{43} + ( - \beta_{14} - \beta_{13} + 2 \beta_{12} - \beta_{5} + \beta_{3} + \beta_{2} + 20) q^{47} + ( - \beta_{14} + \beta_{13} + 2 \beta_{11}) q^{49} + (\beta_{15} - \beta_{10} + 3 \beta_{9} - \beta_{8} + \beta_{6} - 8 \beta_{4} + 2 \beta_1) q^{53} + (\beta_{13} - \beta_{11} + \beta_{5} - \beta_{3} - \beta_{2} - 8) q^{55} + (\beta_{13} + \beta_{12} + \beta_{11} - \beta_{5} + 6 \beta_{3} + \beta_{2} + 6) q^{59} + ( - 2 \beta_{15} - 2 \beta_{10} + 3 \beta_{9} + 2 \beta_{6} - 6 \beta_{4} - 2 \beta_1) q^{61} + ( - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} + 2 \beta_{4}) q^{65} + ( - 3 \beta_{15} - \beta_{10} + 4 \beta_{9} + 2 \beta_{7} + 2 \beta_{6} - 6 \beta_{4} + \cdots + 4 \beta_1) q^{67}+ \cdots + (\beta_{15} - 4 \beta_{10} - 2 \beta_{9} + 2 \beta_{6} - 8 \beta_{4} + 3 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 12 q^{13} + 14 q^{23} - 80 q^{25} - 90 q^{29} + 10 q^{31} - 30 q^{35} - 186 q^{41} + 320 q^{47} + 2 q^{49} - 120 q^{55} + 90 q^{59} + 238 q^{71} - 280 q^{73} - 324 q^{77} - 30 q^{85} - 80 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 64 x^{14} - 16 x^{13} + 2252 x^{12} + 648 x^{11} - 30106 x^{10} + 12360 x^{9} + 374528 x^{8} + 196544 x^{7} + 1261236 x^{6} - 4237944 x^{5} + \cdots + 1535848276 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 23\!\cdots\!29 \nu^{15} + \cdots - 21\!\cdots\!52 ) / 97\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 23\!\cdots\!29 \nu^{15} + \cdots - 60\!\cdots\!12 ) / 48\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 34\!\cdots\!21 \nu^{15} + \cdots - 12\!\cdots\!88 ) / 19\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 34\!\cdots\!21 \nu^{15} + \cdots - 12\!\cdots\!88 ) / 19\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 14\!\cdots\!29 \nu^{15} + \cdots - 33\!\cdots\!12 ) / 34\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 18\!\cdots\!44 \nu^{15} + \cdots + 93\!\cdots\!56 ) / 34\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 20\!\cdots\!08 \nu^{15} + \cdots + 72\!\cdots\!04 ) / 34\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 31\!\cdots\!80 \nu^{15} + \cdots - 54\!\cdots\!00 ) / 34\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 16\!\cdots\!59 \nu^{15} + \cdots + 32\!\cdots\!44 ) / 17\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 60\!\cdots\!66 \nu^{15} + \cdots + 41\!\cdots\!16 ) / 34\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 32\!\cdots\!82 \nu^{15} + \cdots + 87\!\cdots\!64 ) / 17\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 66\!\cdots\!45 \nu^{15} + \cdots + 10\!\cdots\!68 ) / 34\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 67\!\cdots\!87 \nu^{15} + \cdots - 20\!\cdots\!12 ) / 34\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 59\!\cdots\!01 \nu^{15} + \cdots + 15\!\cdots\!28 ) / 17\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 64\!\cdots\!75 \nu^{15} + \cdots + 16\!\cdots\!20 ) / 17\!\cdots\!40 \) Copy content Toggle raw display
\(\nu\)\(=\) \( -\beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 3 \beta_{15} - \beta_{14} - \beta_{13} + \beta_{11} + 6 \beta_{9} + 3 \beta_{7} + 3 \beta_{6} - \beta_{5} - 37 \beta_{4} + 9 \beta_{3} + \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 12 \beta_{15} - 3 \beta_{14} + 2 \beta_{12} + 3 \beta_{11} + 12 \beta_{10} + 16 \beta_{9} + 8 \beta_{8} + 4 \beta_{7} - 4 \beta_{5} - 24 \beta_{4} + 14 \beta_{3} + 2 \beta_{2} - 80 \beta _1 - 51 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 105 \beta_{15} + 12 \beta_{14} + 14 \beta_{13} + \beta_{11} + 50 \beta_{10} + 260 \beta_{9} + 25 \beta_{8} + 180 \beta_{7} + 105 \beta_{6} + 8 \beta_{5} - 1125 \beta_{4} - 403 \beta_{3} + 10 \beta_{2} - 70 \beta _1 + 121 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 492 \beta_{15} + 76 \beta_{14} + 2 \beta_{13} - 67 \beta_{12} - 88 \beta_{11} + 438 \beta_{10} + 902 \beta_{9} + 202 \beta_{8} + 470 \beta_{7} + 54 \beta_{6} + 105 \beta_{5} - 1992 \beta_{4} - 96 \beta_{3} - 1029 \beta_{2} + \cdots - 9948 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2093 \beta_{15} + 1833 \beta_{14} + 1717 \beta_{13} + 6 \beta_{12} - 2546 \beta_{11} + 1554 \beta_{10} + 5040 \beta_{9} + 1036 \beta_{8} + 4452 \beta_{7} + 1694 \beta_{6} + 2005 \beta_{5} - 17921 \beta_{4} + \cdots - 10828 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 7032 \beta_{15} + 9718 \beta_{14} - 476 \beta_{13} - 5252 \beta_{12} - 8558 \beta_{11} + 3808 \beta_{10} + 17352 \beta_{9} + 1272 \beta_{8} + 11040 \beta_{7} + 3464 \beta_{6} + 12708 \beta_{5} + \cdots - 467398 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 28053 \beta_{15} + 78623 \beta_{14} + 72687 \beta_{13} - 80 \beta_{12} - 130637 \beta_{11} - 1824 \beta_{10} - 75510 \beta_{9} + 1314 \beta_{8} - 47247 \beta_{7} - 39969 \beta_{6} + 100283 \beta_{5} + \cdots - 1205919 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 401930 \beta_{15} + 404837 \beta_{14} + 32552 \beta_{13} - 167844 \beta_{12} - 363817 \beta_{11} - 385430 \beta_{10} - 679140 \beta_{9} - 191720 \beta_{8} - 294010 \beta_{7} + 15500 \beta_{6} + \cdots - 13138755 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 4719737 \beta_{15} + 1871644 \beta_{14} + 1472966 \beta_{13} - 152960 \beta_{12} - 3126813 \beta_{11} - 2668270 \beta_{10} - 11292204 \beta_{9} - 1695265 \beta_{8} - 8956222 \beta_{7} + \cdots - 49274749 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 35357024 \beta_{15} + 7306256 \beta_{14} + 3282590 \beta_{13} - 1523959 \beta_{12} - 8191892 \beta_{11} - 27317252 \beta_{10} - 69695660 \beta_{9} - 12986388 \beta_{8} + \cdots - 121656184 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 238683133 \beta_{15} - 5803841 \beta_{14} - 19627253 \beta_{13} - 7811938 \beta_{12} + 17346326 \beta_{11} - 155666914 \beta_{10} - 544563500 \beta_{9} - 97260826 \beta_{8} + \cdots - 546243096 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 1439013442 \beta_{15} - 228382132 \beta_{14} + 68953924 \beta_{13} + 144852348 \beta_{12} + 139890736 \beta_{11} - 1015933716 \beta_{10} - 2988873050 \beta_{9} + \cdots + 11047082428 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 6968471215 \beta_{15} - 3180242887 \beta_{14} - 2976448991 \beta_{13} + 10488064 \beta_{12} + 5374132165 \beta_{11} - 4918684580 \beta_{10} - 15147940310 \beta_{9} + \cdots + 57576702111 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2161.1
3.41677 2.23607i
−0.886481 2.23607i
−4.53300 2.23607i
−1.83366 2.23607i
5.89296 2.23607i
0.613622 2.23607i
−5.14043 2.23607i
2.47022 2.23607i
2.47022 + 2.23607i
−5.14043 + 2.23607i
0.613622 + 2.23607i
5.89296 + 2.23607i
−1.83366 + 2.23607i
−4.53300 + 2.23607i
−0.886481 + 2.23607i
3.41677 + 2.23607i
0 0 0 2.23607i 0 11.7428i 0 0 0
2161.2 0 0 0 2.23607i 0 9.10808i 0 0 0
2161.3 0 0 0 2.23607i 0 5.73038i 0 0 0
2161.4 0 0 0 2.23607i 0 0.167846i 0 0 0
2161.5 0 0 0 2.23607i 0 1.21480i 0 0 0
2161.6 0 0 0 2.23607i 0 2.35348i 0 0 0
2161.7 0 0 0 2.23607i 0 7.88014i 0 0 0
2161.8 0 0 0 2.23607i 0 8.25674i 0 0 0
2161.9 0 0 0 2.23607i 0 8.25674i 0 0 0
2161.10 0 0 0 2.23607i 0 7.88014i 0 0 0
2161.11 0 0 0 2.23607i 0 2.35348i 0 0 0
2161.12 0 0 0 2.23607i 0 1.21480i 0 0 0
2161.13 0 0 0 2.23607i 0 0.167846i 0 0 0
2161.14 0 0 0 2.23607i 0 5.73038i 0 0 0
2161.15 0 0 0 2.23607i 0 9.10808i 0 0 0
2161.16 0 0 0 2.23607i 0 11.7428i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2161.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4140.3.d.a 16
3.b odd 2 1 460.3.f.a 16
12.b even 2 1 1840.3.k.c 16
15.d odd 2 1 2300.3.f.e 16
15.e even 4 2 2300.3.d.b 32
23.b odd 2 1 inner 4140.3.d.a 16
69.c even 2 1 460.3.f.a 16
276.h odd 2 1 1840.3.k.c 16
345.h even 2 1 2300.3.f.e 16
345.l odd 4 2 2300.3.d.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.3.f.a 16 3.b odd 2 1
460.3.f.a 16 69.c even 2 1
1840.3.k.c 16 12.b even 2 1
1840.3.k.c 16 276.h odd 2 1
2300.3.d.b 32 15.e even 4 2
2300.3.d.b 32 345.l odd 4 2
2300.3.f.e 16 15.d odd 2 1
2300.3.f.e 16 345.h even 2 1
4140.3.d.a 16 1.a even 1 1 trivial
4140.3.d.a 16 23.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} + 391 T_{7}^{14} + 58685 T_{7}^{12} + 4281913 T_{7}^{10} + 155886747 T_{7}^{8} + 2524614569 T_{7}^{6} + 12272252524 T_{7}^{4} + 13341794000 T_{7}^{2} + 366186496 \) acting on \(S_{3}^{\mathrm{new}}(4140, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} + 391 T^{14} + \cdots + 366186496 \) Copy content Toggle raw display
$11$ \( T^{16} + 842 T^{14} + \cdots + 538982656 \) Copy content Toggle raw display
$13$ \( (T^{8} + 6 T^{7} - 832 T^{6} + \cdots + 12532480)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 645469713961216 \) Copy content Toggle raw display
$19$ \( T^{16} + 3578 T^{14} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{16} - 14 T^{15} + \cdots + 61\!\cdots\!61 \) Copy content Toggle raw display
$29$ \( (T^{8} + 45 T^{7} - 2411 T^{6} + \cdots - 519358880)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 5 T^{7} - 4394 T^{6} + \cdots - 162152849534)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 216591737307136 \) Copy content Toggle raw display
$41$ \( (T^{8} + 93 T^{7} + \cdots + 580968030770)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + 16556 T^{14} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( (T^{8} - 160 T^{7} + \cdots - 66361863793216)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + 18049 T^{14} + \cdots + 55\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( (T^{8} - 45 T^{7} + \cdots + 199085136256)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + 31266 T^{14} + \cdots + 21\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{16} + 40765 T^{14} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{8} - 119 T^{7} + \cdots - 318523111508570)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 140 T^{7} + \cdots - 71365498673344)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + 56440 T^{14} + \cdots + 17\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{16} + 51025 T^{14} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{16} + 93608 T^{14} + \cdots + 43\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( T^{16} + 51438 T^{14} + \cdots + 31\!\cdots\!56 \) Copy content Toggle raw display
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