Properties

Label 7942.2.a.bm
Level $7942$
Weight $2$
Character orbit 7942.a
Self dual yes
Analytic conductor $63.417$
Analytic rank $1$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 14x^{6} - 4x^{5} + 38x^{4} + 3x^{3} - 29x^{2} + 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + ( - \beta_{7} + \beta_1 - 1) q^{3} + q^{4} + ( - \beta_{7} + \beta_{2}) q^{5} + (\beta_{7} - \beta_1 + 1) q^{6} + \beta_{6} q^{7} - q^{8} + (\beta_{7} + \beta_{5} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + ( - \beta_{7} + \beta_1 - 1) q^{3} + q^{4} + ( - \beta_{7} + \beta_{2}) q^{5} + (\beta_{7} - \beta_1 + 1) q^{6} + \beta_{6} q^{7} - q^{8} + (\beta_{7} + \beta_{5} - \beta_1 + 1) q^{9} + (\beta_{7} - \beta_{2}) q^{10} + q^{11} + ( - \beta_{7} + \beta_1 - 1) q^{12} + ( - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - 2) q^{13} - \beta_{6} q^{14} + (\beta_{7} + \beta_{5} + \beta_{3} - \beta_{2} + 1) q^{15} + q^{16} + (\beta_{5} + \beta_{4} - \beta_{3} + 1) q^{17} + ( - \beta_{7} - \beta_{5} + \beta_1 - 1) q^{18} + ( - \beta_{7} + \beta_{2}) q^{20} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \beta_1 - 1) q^{21} - q^{22} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} - 2 \beta_1 - 1) q^{23} + (\beta_{7} - \beta_1 + 1) q^{24} + (3 \beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_1 + 2) q^{25} + (\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 2) q^{26} + ( - \beta_{6} - 2 \beta_{5} + \beta_{4} + \beta_{2} - 1) q^{27} + \beta_{6} q^{28} + (2 \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{29} + ( - \beta_{7} - \beta_{5} - \beta_{3} + \beta_{2} - 1) q^{30} + ( - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} + \beta_1 - 3) q^{31} - q^{32} + ( - \beta_{7} + \beta_1 - 1) q^{33} + ( - \beta_{5} - \beta_{4} + \beta_{3} - 1) q^{34} + (\beta_{6} - 2 \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 + 1) q^{35} + (\beta_{7} + \beta_{5} - \beta_1 + 1) q^{36} + ( - \beta_{7} - \beta_{4} + \beta_{3} + \beta_{2} - 2) q^{37} + (\beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} - 3 \beta_1) q^{39} + (\beta_{7} - \beta_{2}) q^{40} + ( - 2 \beta_{7} - \beta_{6} + 2 \beta_{4} - 2) q^{41} + (\beta_{5} - \beta_{4} + \beta_{3} - \beta_1 + 1) q^{42} + ( - \beta_{6} + \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1) q^{43} + q^{44} + ( - \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_1 - 1) q^{45} + (\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{2} + 2 \beta_1 + 1) q^{46} + (3 \beta_{7} - \beta_{6} - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_1 + 5) q^{47} + ( - \beta_{7} + \beta_1 - 1) q^{48} + ( - 2 \beta_{7} + \beta_{6} - \beta_{4} + 3 \beta_{3} + \beta_{2} - 1) q^{49} + ( - 3 \beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_1 - 2) q^{50} + (\beta_{6} - \beta_{5} + 4 \beta_1 - 2) q^{51} + ( - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - 2) q^{52} + (\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 2) q^{53} + (\beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{2} + 1) q^{54} + ( - \beta_{7} + \beta_{2}) q^{55} - \beta_{6} q^{56} + ( - 2 \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{58} + (3 \beta_{7} - 2 \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1) q^{59} + (\beta_{7} + \beta_{5} + \beta_{3} - \beta_{2} + 1) q^{60} + ( - 2 \beta_{7} - \beta_{6} - \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{61} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} - \beta_1 + 3) q^{62} + (6 \beta_{7} + 2 \beta_{5} - \beta_{4} - 2 \beta_{2} - 2 \beta_1 + 3) q^{63} + q^{64} + (2 \beta_{7} + 2 \beta_{5} - \beta_{3} - 2 \beta_1 - 4) q^{65} + (\beta_{7} - \beta_1 + 1) q^{66} + (5 \beta_{7} - \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_1 + 2) q^{67} + (\beta_{5} + \beta_{4} - \beta_{3} + 1) q^{68} + ( - 3 \beta_{7} - 2 \beta_{6} - \beta_{5} - \beta_{4} + 2 \beta_{3} - 3) q^{69} + ( - \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{3} - \beta_1 - 1) q^{70} + ( - 2 \beta_{7} - 3 \beta_{4} - \beta_{3} + 2 \beta_{2} - 4 \beta_1 - 1) q^{71} + ( - \beta_{7} - \beta_{5} + \beta_1 - 1) q^{72} + (2 \beta_{6} - 2 \beta_{5} + \beta_{4} - 4 \beta_{2} + \beta_1 - 1) q^{73} + (\beta_{7} + \beta_{4} - \beta_{3} - \beta_{2} + 2) q^{74} + ( - 3 \beta_{7} - \beta_{6} + 2 \beta_{4} - \beta_1 - 4) q^{75} + \beta_{6} q^{77} + ( - \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + 3 \beta_1) q^{78} + (\beta_{6} + \beta_{5} + 3 \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{79} + ( - \beta_{7} + \beta_{2}) q^{80} + (6 \beta_{7} + 3 \beta_{6} + \beta_{5} - 3 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{81} + (2 \beta_{7} + \beta_{6} - 2 \beta_{4} + 2) q^{82} + ( - 2 \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 - 3) q^{83} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \beta_1 - 1) q^{84} + (4 \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{85} + (\beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1) q^{86} + ( - 4 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 3 \beta_{2} + \beta_1 - 3) q^{87} - q^{88} + (\beta_{7} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{89} + (\beta_{7} + 2 \beta_{6} + 2 \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_1 + 1) q^{90} + ( - 4 \beta_{7} - 3 \beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{3} - 5) q^{91} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} - 2 \beta_1 - 1) q^{92} + ( - 3 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \cdots + 4) q^{93}+ \cdots + (\beta_{7} + \beta_{5} - \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + q^{5} + 4 q^{6} + 4 q^{7} - 8 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + q^{5} + 4 q^{6} + 4 q^{7} - 8 q^{8} + 6 q^{9} - q^{10} + 8 q^{11} - 4 q^{12} - 12 q^{13} - 4 q^{14} + 9 q^{15} + 8 q^{16} + 7 q^{17} - 6 q^{18} + q^{20} - 13 q^{21} - 8 q^{22} - 6 q^{23} + 4 q^{24} + 5 q^{25} + 12 q^{26} - 22 q^{27} + 4 q^{28} + q^{29} - 9 q^{30} - 8 q^{31} - 8 q^{32} - 4 q^{33} - 7 q^{34} + 5 q^{35} + 6 q^{36} - 12 q^{37} + 10 q^{39} - q^{40} - 18 q^{41} + 13 q^{42} - 4 q^{43} + 8 q^{44} - 19 q^{45} + 6 q^{46} + 13 q^{47} - 4 q^{48} + 4 q^{49} - 5 q^{50} - 14 q^{51} - 12 q^{52} + 17 q^{53} + 22 q^{54} + q^{55} - 4 q^{56} - q^{58} - 26 q^{59} + 9 q^{60} - 7 q^{61} + 8 q^{62} + 13 q^{63} + 8 q^{64} - 36 q^{65} + 4 q^{66} - 6 q^{67} + 7 q^{68} - 19 q^{69} - 5 q^{70} + 3 q^{71} - 6 q^{72} + 5 q^{73} + 12 q^{74} - 30 q^{75} + 4 q^{77} - 10 q^{78} - 16 q^{79} + q^{80} + 8 q^{81} + 18 q^{82} - 21 q^{83} - 13 q^{84} - 10 q^{85} + 4 q^{86} - 29 q^{87} - 8 q^{88} - 16 q^{89} + 19 q^{90} - 31 q^{91} - 6 q^{92} + 20 q^{93} - 13 q^{94} + 4 q^{96} + 15 q^{97} - 4 q^{98} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 14x^{6} - 4x^{5} + 38x^{4} + 3x^{3} - 29x^{2} + 2x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 6\nu^{7} - 9\nu^{6} - 101\nu^{5} + 97\nu^{4} + 479\nu^{3} - 182\nu^{2} - 511\nu + 77 ) / 61 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5\nu^{7} + 23\nu^{6} - 74\nu^{5} - 336\nu^{4} + 145\nu^{3} + 804\nu^{2} - 131\nu - 312 ) / 61 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9\nu^{7} + 17\nu^{6} - 121\nu^{5} - 251\nu^{4} + 200\nu^{3} + 398\nu^{2} - 126\nu - 159 ) / 61 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -18\nu^{7} - 34\nu^{6} + 242\nu^{5} + 502\nu^{4} - 400\nu^{3} - 735\nu^{2} + 191\nu + 74 ) / 61 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -23\nu^{7} + 4\nu^{6} + 316\nu^{5} + 45\nu^{4} - 789\nu^{3} - 14\nu^{2} + 261\nu + 20 ) / 61 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 49\nu^{7} + 18\nu^{6} - 652\nu^{5} - 438\nu^{4} + 1360\nu^{3} + 547\nu^{2} - 625\nu - 154 ) / 122 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{4} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} + 2\beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} + 9\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + 10\beta_{5} + 23\beta_{4} - 2\beta_{3} + \beta_{2} + 16\beta _1 + 36 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 26\beta_{7} + 25\beta_{6} + 17\beta_{5} + 21\beta_{4} - \beta_{3} + 10\beta_{2} + 98\beta _1 + 41 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{7} + 22\beta_{6} + 108\beta_{5} + 253\beta_{4} - 25\beta_{3} + 17\beta_{2} + 220\beta _1 + 382 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 290\beta_{7} + 278\beta_{6} + 237\beta_{5} + 342\beta_{4} - 22\beta_{3} + 108\beta_{2} + 1118\beta _1 + 630 \) 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
−1.78054
−2.97448
−0.360811
0.893598
−1.03493
0.517227
3.48382
1.25611
−1.00000 −3.39858 1.00000 −2.89336 3.39858 3.18396 −1.00000 8.55033 2.89336
1.2 −1.00000 −2.35644 1.00000 −0.0660228 2.35644 −0.317755 −1.00000 2.55283 0.0660228
1.3 −1.00000 −1.97884 1.00000 2.94620 1.97884 −0.656982 −1.00000 0.915826 −2.94620
1.4 −1.00000 −0.724436 1.00000 −3.58061 0.724436 −1.97748 −1.00000 −2.47519 3.58061
1.5 −1.00000 −0.416894 1.00000 3.13364 0.416894 5.24744 −1.00000 −2.82620 −3.13364
1.6 −1.00000 1.13526 1.00000 −1.11348 −1.13526 0.931861 −1.00000 −1.71118 1.11348
1.7 −1.00000 1.86579 1.00000 0.673675 −1.86579 1.45050 −1.00000 0.481174 −0.673675
1.8 −1.00000 1.87414 1.00000 1.89996 −1.87414 −3.86154 −1.00000 0.512419 −1.89996
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.bm 8
19.b odd 2 1 7942.2.a.br yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7942.2.a.bm 8 1.a even 1 1 trivial
7942.2.a.br yes 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7942))\):

\( T_{3}^{8} + 4T_{3}^{7} - 7T_{3}^{6} - 34T_{3}^{5} + 13T_{3}^{4} + 87T_{3}^{3} + 2T_{3}^{2} - 58T_{3} - 19 \) Copy content Toggle raw display
\( T_{5}^{8} - T_{5}^{7} - 22T_{5}^{6} + 25T_{5}^{5} + 134T_{5}^{4} - 154T_{5}^{3} - 167T_{5}^{2} + 126T_{5} + 9 \) Copy content Toggle raw display
\( T_{13}^{8} + 12T_{13}^{7} + 3T_{13}^{6} - 486T_{13}^{5} - 1903T_{13}^{4} + 1281T_{13}^{3} + 19105T_{13}^{2} + 31494T_{13} + 10604 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 4 T^{7} - 7 T^{6} - 34 T^{5} + \cdots - 19 \) Copy content Toggle raw display
$5$ \( T^{8} - T^{7} - 22 T^{6} + 25 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{8} - 4 T^{7} - 22 T^{6} + 71 T^{5} + \cdots + 36 \) Copy content Toggle raw display
$11$ \( (T - 1)^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + 12 T^{7} + 3 T^{6} + \cdots + 10604 \) Copy content Toggle raw display
$17$ \( T^{8} - 7 T^{7} - 44 T^{6} + \cdots - 9424 \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} + 6 T^{7} - 96 T^{6} + \cdots + 119291 \) Copy content Toggle raw display
$29$ \( T^{8} - T^{7} - 110 T^{6} + \cdots + 128304 \) Copy content Toggle raw display
$31$ \( T^{8} + 8 T^{7} - 72 T^{6} + \cdots + 25821 \) Copy content Toggle raw display
$37$ \( T^{8} + 12 T^{7} - T^{6} - 482 T^{5} + \cdots + 12889 \) Copy content Toggle raw display
$41$ \( T^{8} + 18 T^{7} + 36 T^{6} + \cdots - 21796 \) Copy content Toggle raw display
$43$ \( T^{8} + 4 T^{7} - 140 T^{6} + \cdots - 22900 \) Copy content Toggle raw display
$47$ \( T^{8} - 13 T^{7} - 198 T^{6} + \cdots - 1709159 \) Copy content Toggle raw display
$53$ \( T^{8} - 17 T^{7} + 33 T^{6} + \cdots - 1025 \) Copy content Toggle raw display
$59$ \( T^{8} + 26 T^{7} + 87 T^{6} + \cdots + 442775 \) Copy content Toggle raw display
$61$ \( T^{8} + 7 T^{7} - 93 T^{6} + \cdots + 22716 \) Copy content Toggle raw display
$67$ \( T^{8} + 6 T^{7} - 243 T^{6} + \cdots + 13275 \) Copy content Toggle raw display
$71$ \( T^{8} - 3 T^{7} - 306 T^{6} + \cdots + 283376 \) Copy content Toggle raw display
$73$ \( T^{8} - 5 T^{7} - 479 T^{6} + \cdots - 2105476 \) Copy content Toggle raw display
$79$ \( T^{8} + 16 T^{7} - 278 T^{6} + \cdots - 27251316 \) Copy content Toggle raw display
$83$ \( T^{8} + 21 T^{7} + 61 T^{6} + \cdots - 32644 \) Copy content Toggle raw display
$89$ \( T^{8} + 16 T^{7} - 169 T^{6} + \cdots - 251299 \) Copy content Toggle raw display
$97$ \( T^{8} - 15 T^{7} - 350 T^{6} + \cdots + 68364 \) Copy content Toggle raw display
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