Properties

Label 7942.2.a.bs
Level $7942$
Weight $2$
Character orbit 7942.a
Self dual yes
Analytic conductor $63.417$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7942.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.4171892853\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 3 x^{11} - 21 x^{10} + 59 x^{9} + 162 x^{8} - 408 x^{7} - 581 x^{6} + 1236 x^{5} + 972 x^{4} - 1596 x^{3} - 621 x^{2} + 648 x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 418)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q - q^{2} + \beta_1 q^{3} + q^{4} + (\beta_{9} - 1) q^{5} - \beta_1 q^{6} + ( - \beta_{9} - \beta_{6} - \beta_{4} - \beta_{2}) q^{7} - q^{8} + (\beta_{10} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta_1 q^{3} + q^{4} + (\beta_{9} - 1) q^{5} - \beta_1 q^{6} + ( - \beta_{9} - \beta_{6} - \beta_{4} - \beta_{2}) q^{7} - q^{8} + (\beta_{10} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{9} + ( - \beta_{9} + 1) q^{10} + q^{11} + \beta_1 q^{12} + (\beta_{11} - \beta_{7} + \beta_{6} - \beta_{4}) q^{13} + (\beta_{9} + \beta_{6} + \beta_{4} + \beta_{2}) q^{14} + (\beta_{10} + \beta_{9} + \beta_{7} + \beta_{3} - \beta_1 + 2) q^{15} + q^{16} + (\beta_{11} + \beta_{9} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} + \beta_{2} - 4) q^{17} + ( - \beta_{10} - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{18} + (\beta_{9} - 1) q^{20} + ( - 2 \beta_{11} - \beta_{9} - \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_1) q^{21} - q^{22} + ( - \beta_{11} - 2 \beta_{10} - \beta_{9} - 2 \beta_{8} - \beta_{7} - 2 \beta_{6} - \beta_{4} + \cdots + 1) q^{23}+ \cdots + (\beta_{10} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} + 3 q^{3} + 12 q^{4} - 9 q^{5} - 3 q^{6} - 12 q^{7} - 12 q^{8} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} + 3 q^{3} + 12 q^{4} - 9 q^{5} - 3 q^{6} - 12 q^{7} - 12 q^{8} + 15 q^{9} + 9 q^{10} + 12 q^{11} + 3 q^{12} + 12 q^{14} + 21 q^{15} + 12 q^{16} - 33 q^{17} - 15 q^{18} - 9 q^{20} - 9 q^{21} - 12 q^{22} - 18 q^{23} - 3 q^{24} + 15 q^{25} + 21 q^{27} - 12 q^{28} - 15 q^{29} - 21 q^{30} + 9 q^{31} - 12 q^{32} + 3 q^{33} + 33 q^{34} - 30 q^{35} + 15 q^{36} + 9 q^{37} - 6 q^{39} + 9 q^{40} - 15 q^{41} + 9 q^{42} - 21 q^{43} + 12 q^{44} + 18 q^{46} - 27 q^{47} + 3 q^{48} + 18 q^{49} - 15 q^{50} - 6 q^{51} + 6 q^{53} - 21 q^{54} - 9 q^{55} + 12 q^{56} + 15 q^{58} + 48 q^{59} + 21 q^{60} + 12 q^{61} - 9 q^{62} - 66 q^{63} + 12 q^{64} - 36 q^{65} - 3 q^{66} - 3 q^{67} - 33 q^{68} - 24 q^{69} + 30 q^{70} - 3 q^{71} - 15 q^{72} - 30 q^{73} - 9 q^{74} + 21 q^{75} - 12 q^{77} + 6 q^{78} + 12 q^{79} - 9 q^{80} + 12 q^{81} + 15 q^{82} - 66 q^{83} - 9 q^{84} + 15 q^{85} + 21 q^{86} - 51 q^{87} - 12 q^{88} + 30 q^{89} + 54 q^{91} - 18 q^{92} - 66 q^{93} + 27 q^{94} - 3 q^{96} + 36 q^{97} - 18 q^{98} + 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} - 21 x^{10} + 59 x^{9} + 162 x^{8} - 408 x^{7} - 581 x^{6} + 1236 x^{5} + 972 x^{4} - 1596 x^{3} - 621 x^{2} + 648 x + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 36106 \nu^{11} + 1662627 \nu^{10} - 4612281 \nu^{9} - 27776825 \nu^{8} + 84133296 \nu^{7} + 144702246 \nu^{6} - 451891531 \nu^{5} + \cdots + 40812606 ) / 32651937 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 128564 \nu^{11} - 1747173 \nu^{10} + 2182854 \nu^{9} + 29652583 \nu^{8} - 62468688 \nu^{7} - 156938793 \nu^{6} + 377717441 \nu^{5} + 282218643 \nu^{4} + \cdots - 29700081 ) / 32651937 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 405461 \nu^{11} + 4432509 \nu^{10} - 3745263 \nu^{9} - 75920590 \nu^{8} + 145872735 \nu^{7} + 421020771 \nu^{6} - 924159005 \nu^{5} + \cdots - 373530798 ) / 97955811 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 411575 \nu^{11} + 1512099 \nu^{10} + 8389437 \nu^{9} - 31571206 \nu^{8} - 61047876 \nu^{7} + 232916424 \nu^{6} + 202415434 \nu^{5} + \cdots - 176546574 ) / 97955811 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 557519 \nu^{11} + 832122 \nu^{10} + 13443351 \nu^{9} - 14428411 \nu^{8} - 116721489 \nu^{7} + 72284757 \nu^{6} + 430643887 \nu^{5} + \cdots + 177627060 ) / 97955811 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 269641 \nu^{11} - 1494300 \nu^{10} - 3903051 \nu^{9} + 27509099 \nu^{8} + 18010035 \nu^{7} - 171119802 \nu^{6} - 55678748 \nu^{5} + 437072313 \nu^{4} + \cdots + 61834725 ) / 32651937 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 13107 \nu^{11} - 60119 \nu^{10} - 163248 \nu^{9} + 1027749 \nu^{8} + 124142 \nu^{7} - 5635923 \nu^{6} + 4114755 \nu^{5} + 11684083 \nu^{4} - 13793571 \nu^{3} + \cdots + 80775 ) / 572841 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2304427 \nu^{11} + 3438855 \nu^{10} + 51859659 \nu^{9} - 52047371 \nu^{8} - 427430148 \nu^{7} + 209996652 \nu^{6} + 1585199306 \nu^{5} + \cdots + 224961003 ) / 97955811 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2364614 \nu^{11} + 9752256 \nu^{10} + 37214754 \nu^{9} - 173550649 \nu^{8} - 169403457 \nu^{7} + 1020506352 \nu^{6} + 179450527 \nu^{5} + \cdots - 330617970 ) / 97955811 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1072042 \nu^{11} - 4086648 \nu^{10} - 17332797 \nu^{9} + 70519139 \nu^{8} + 85197561 \nu^{7} - 386577282 \nu^{6} - 141787421 \nu^{5} + \cdots - 54356427 ) / 32651937 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + 2\beta_{5} + 2\beta_{2} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9 \beta_{10} + 9 \beta_{8} + 10 \beta_{7} + 11 \beta_{6} + 2 \beta_{5} + 8 \beta_{4} + 12 \beta_{3} + 13 \beta_{2} + 11 \beta _1 + 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 13 \beta_{11} + 10 \beta_{10} + 14 \beta_{9} + 13 \beta_{8} + 12 \beta_{7} + 16 \beta_{6} + 30 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + 28 \beta_{2} + 60 \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11 \beta_{11} + 81 \beta_{10} + 8 \beta_{9} + 80 \beta_{8} + 90 \beta_{7} + 116 \beta_{6} + 38 \beta_{5} + 70 \beta_{4} + 121 \beta_{3} + 143 \beta_{2} + 119 \beta _1 + 148 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 157 \beta_{11} + 95 \beta_{10} + 163 \beta_{9} + 141 \beta_{8} + 123 \beta_{7} + 230 \beta_{6} + 357 \beta_{5} + 60 \beta_{4} + 39 \beta_{3} + 330 \beta_{2} + 571 \beta _1 + 33 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 250 \beta_{11} + 742 \beta_{10} + 185 \beta_{9} + 740 \beta_{8} + 796 \beta_{7} + 1280 \beta_{6} + 543 \beta_{5} + 668 \beta_{4} + 1160 \beta_{3} + 1524 \beta_{2} + 1292 \beta _1 + 1239 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1876 \beta_{11} + 935 \beta_{10} + 1825 \beta_{9} + 1480 \beta_{8} + 1181 \beta_{7} + 3076 \beta_{6} + 3966 \beta_{5} + 874 \beta_{4} + 562 \beta_{3} + 3735 \beta_{2} + 5713 \beta _1 + 388 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3973 \beta_{11} + 6887 \beta_{10} + 2932 \beta_{9} + 7071 \beta_{8} + 7001 \beta_{7} + 14520 \beta_{6} + 7057 \beta_{5} + 6715 \beta_{4} + 10950 \beta_{3} + 16142 \beta_{2} + 14066 \beta _1 + 10649 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 22200 \beta_{11} + 9432 \beta_{10} + 20226 \beta_{9} + 15457 \beta_{8} + 10870 \beta_{7} + 39028 \beta_{6} + 43050 \beta_{5} + 11324 \beta_{4} + 7161 \beta_{3} + 41708 \beta_{2} + 58616 \beta _1 + 3958 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.94662
−2.15917
−1.75685
−1.63824
−0.909909
−0.116160
0.763444
1.31121
1.87195
2.19083
3.08389
3.30563
−1.00000 −2.94662 1.00000 −2.95323 2.94662 −2.07806 −1.00000 5.68257 2.95323
1.2 −1.00000 −2.15917 1.00000 −0.973672 2.15917 −2.13909 −1.00000 1.66201 0.973672
1.3 −1.00000 −1.75685 1.00000 −0.498553 1.75685 −5.22967 −1.00000 0.0865324 0.498553
1.4 −1.00000 −1.63824 1.00000 −1.31219 1.63824 2.28188 −1.00000 −0.316160 1.31219
1.5 −1.00000 −0.909909 1.00000 −4.17433 0.909909 4.37485 −1.00000 −2.17207 4.17433
1.6 −1.00000 −0.116160 1.00000 −0.498551 0.116160 −2.29128 −1.00000 −2.98651 0.498551
1.7 −1.00000 0.763444 1.00000 2.22399 −0.763444 0.0798976 −1.00000 −2.41715 −2.22399
1.8 −1.00000 1.31121 1.00000 −3.25078 −1.31121 0.670809 −1.00000 −1.28073 3.25078
1.9 −1.00000 1.87195 1.00000 3.81859 −1.87195 −2.96733 −1.00000 0.504212 −3.81859
1.10 −1.00000 2.19083 1.00000 −3.36681 −2.19083 1.45728 −1.00000 1.79972 3.36681
1.11 −1.00000 3.08389 1.00000 2.06152 −3.08389 −1.37573 −1.00000 6.51036 −2.06152
1.12 −1.00000 3.30563 1.00000 −0.0759854 −3.30563 −4.78357 −1.00000 7.92721 0.0759854
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(11\) \(-1\)
\(19\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7942.2.a.bs 12
19.b odd 2 1 7942.2.a.bw 12
19.f odd 18 2 418.2.j.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.j.b 24 19.f odd 18 2
7942.2.a.bs 12 1.a even 1 1 trivial
7942.2.a.bw 12 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7942))\):

\( T_{3}^{12} - 3 T_{3}^{11} - 21 T_{3}^{10} + 59 T_{3}^{9} + 162 T_{3}^{8} - 408 T_{3}^{7} - 581 T_{3}^{6} + 1236 T_{3}^{5} + 972 T_{3}^{4} - 1596 T_{3}^{3} - 621 T_{3}^{2} + 648 T_{3} + 81 \) Copy content Toggle raw display
\( T_{5}^{12} + 9 T_{5}^{11} + 3 T_{5}^{10} - 183 T_{5}^{9} - 465 T_{5}^{8} + 663 T_{5}^{7} + 3449 T_{5}^{6} + 1968 T_{5}^{5} - 5604 T_{5}^{4} - 9042 T_{5}^{3} - 4995 T_{5}^{2} - 1080 T_{5} - 57 \) Copy content Toggle raw display
\( T_{13}^{12} - 75 T_{13}^{10} + 100 T_{13}^{9} + 1857 T_{13}^{8} - 4080 T_{13}^{7} - 17028 T_{13}^{6} + 49008 T_{13}^{5} + 45258 T_{13}^{4} - 190063 T_{13}^{3} + 17967 T_{13}^{2} + 174321 T_{13} - 46567 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 3 T^{11} - 21 T^{10} + 59 T^{9} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{12} + 9 T^{11} + 3 T^{10} - 183 T^{9} + \cdots - 57 \) Copy content Toggle raw display
$7$ \( T^{12} + 12 T^{11} + 21 T^{10} + \cdots - 811 \) Copy content Toggle raw display
$11$ \( (T - 1)^{12} \) Copy content Toggle raw display
$13$ \( T^{12} - 75 T^{10} + 100 T^{9} + \cdots - 46567 \) Copy content Toggle raw display
$17$ \( T^{12} + 33 T^{11} + 411 T^{10} + \cdots - 40797 \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( T^{12} + 18 T^{11} - 33 T^{10} + \cdots + 3314007 \) Copy content Toggle raw display
$29$ \( T^{12} + 15 T^{11} - 72 T^{10} + \cdots + 1782657 \) Copy content Toggle raw display
$31$ \( T^{12} - 9 T^{11} - 198 T^{10} + \cdots - 12062793 \) Copy content Toggle raw display
$37$ \( T^{12} - 9 T^{11} - 216 T^{10} + \cdots + 131055759 \) Copy content Toggle raw display
$41$ \( T^{12} + 15 T^{11} - 153 T^{10} + \cdots + 33656391 \) Copy content Toggle raw display
$43$ \( T^{12} + 21 T^{11} + \cdots - 19585826369 \) Copy content Toggle raw display
$47$ \( T^{12} + 27 T^{11} + 57 T^{10} + \cdots - 57774789 \) Copy content Toggle raw display
$53$ \( T^{12} - 6 T^{11} - 318 T^{10} + \cdots + 341949087 \) Copy content Toggle raw display
$59$ \( T^{12} - 48 T^{11} + 693 T^{10} + \cdots - 1718523 \) Copy content Toggle raw display
$61$ \( T^{12} - 12 T^{11} + \cdots + 512283909 \) Copy content Toggle raw display
$67$ \( T^{12} + 3 T^{11} - 225 T^{10} + \cdots + 2277123 \) Copy content Toggle raw display
$71$ \( T^{12} + 3 T^{11} - 225 T^{10} + \cdots - 3265191 \) Copy content Toggle raw display
$73$ \( T^{12} + 30 T^{11} + \cdots + 3142939797 \) Copy content Toggle raw display
$79$ \( T^{12} - 12 T^{11} + \cdots - 106048227 \) Copy content Toggle raw display
$83$ \( T^{12} + 66 T^{11} + \cdots - 188944699113 \) Copy content Toggle raw display
$89$ \( T^{12} - 30 T^{11} + 174 T^{10} + \cdots - 88047537 \) Copy content Toggle raw display
$97$ \( T^{12} - 36 T^{11} + \cdots + 2928630491 \) Copy content Toggle raw display
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