Properties

Label 7942.2.a.bo
Level $7942$
Weight $2$
Character orbit 7942.a
Self dual yes
Analytic conductor $63.417$
Analytic rank $0$
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: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 3x^{7} - 14x^{6} + 29x^{5} + 64x^{4} - 50x^{3} - 36x^{2} + 30x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} - 14x^{6} + 29x^{5} + 64x^{4} - 50x^{3} - 36x^{2} + 30x - 5 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{7} + 10\nu^{6} + 36\nu^{5} - 83\nu^{4} - 159\nu^{3} + 91\nu^{2} + 91\nu - 35 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{7} - 18\nu^{6} - 108\nu^{5} + 167\nu^{4} + 531\nu^{3} - 191\nu^{2} - 343\nu + 111 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -9\nu^{7} + 22\nu^{6} + 140\nu^{5} - 193\nu^{4} - 685\nu^{3} + 129\nu^{2} + 369\nu - 97 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{7} - 8\nu^{6} - 44\nu^{5} + 69\nu^{4} + 213\nu^{3} - 57\nu^{2} - 133\nu + 33 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} - 8\nu^{6} - 44\nu^{5} + 69\nu^{4} + 213\nu^{3} - 59\nu^{2} - 129\nu + 39 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 39\nu^{7} - 106\nu^{6} - 572\nu^{5} + 951\nu^{4} + 2747\nu^{3} - 1047\nu^{2} - 1695\nu + 583 ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + \beta_{5} + 2\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} + 2\beta_{5} + \beta_{4} + \beta_{2} + 11\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - 10\beta_{6} + 11\beta_{5} + 2\beta_{4} - 3\beta_{3} + 4\beta_{2} + 32\beta _1 + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{7} - 20\beta_{6} + 32\beta_{5} + 14\beta_{4} - 15\beta_{3} + 23\beta_{2} + 135\beta _1 + 49 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 23\beta_{7} - 106\beta_{6} + 135\beta_{5} + 43\beta_{4} - 81\beta_{3} + 97\beta_{2} + 454\beta _1 + 233 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 97\beta_{7} - 294\beta_{6} + 454\beta_{5} + 203\beta_{4} - 367\beta_{3} + 433\beta_{2} + 1756\beta _1 + 621 \) 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
3.74399
3.33202
0.725465
0.370513
0.304617
−0.960117
−2.22193
−2.29456
−1.00000 −2.74399 1.00000 −2.12596 2.74399 −0.877808 −1.00000 4.52948 2.12596
1.2 −1.00000 −2.33202 1.00000 −3.95006 2.33202 −0.471110 −1.00000 2.43832 3.95006
1.3 −1.00000 0.274535 1.00000 0.892569 −0.274535 2.55547 −1.00000 −2.92463 −0.892569
1.4 −1.00000 0.629487 1.00000 1.24752 −0.629487 −2.54316 −1.00000 −2.60375 −1.24752
1.5 −1.00000 0.695383 1.00000 −0.922651 −0.695383 0.361679 −1.00000 −2.51644 0.922651
1.6 −1.00000 1.96012 1.00000 0.342083 −1.96012 2.97928 −1.00000 0.842060 −0.342083
1.7 −1.00000 3.22193 1.00000 3.83997 −3.22193 −1.13450 −1.00000 7.38086 −3.83997
1.8 −1.00000 3.29456 1.00000 1.67652 −3.29456 −4.86985 −1.00000 7.85409 −1.67652
\(n\): 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.bo 8
19.b odd 2 1 7942.2.a.bp yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7942.2.a.bo 8 1.a even 1 1 trivial
7942.2.a.bp 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7942))\):

\( T_{3}^{8} - 5T_{3}^{7} - 7T_{3}^{6} + 62T_{3}^{5} - 36T_{3}^{4} - 167T_{3}^{3} + 243T_{3}^{2} - 112T_{3} + 16 \) Copy content Toggle raw display
\( T_{5}^{8} - T_{5}^{7} - 20T_{5}^{6} + 23T_{5}^{5} + 74T_{5}^{4} - 106T_{5}^{3} - 27T_{5}^{2} + 74T_{5} - 19 \) Copy content Toggle raw display
\( T_{13}^{8} - 3T_{13}^{7} - 72T_{13}^{6} + 241T_{13}^{5} + 1342T_{13}^{4} - 4374T_{13}^{3} - 7689T_{13}^{2} + 23004T_{13} + 2911 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 5 T^{7} - 7 T^{6} + 62 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{8} - T^{7} - 20 T^{6} + 23 T^{5} + \cdots - 19 \) Copy content Toggle raw display
$7$ \( T^{8} + 4 T^{7} - 16 T^{6} - 55 T^{5} + \cdots - 16 \) Copy content Toggle raw display
$11$ \( (T + 1)^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - 3 T^{7} - 72 T^{6} + \cdots + 2911 \) Copy content Toggle raw display
$17$ \( T^{8} + 6 T^{7} - 20 T^{6} - 127 T^{5} + \cdots + 241 \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} - 9 T^{7} - 62 T^{6} + \cdots + 27920 \) Copy content Toggle raw display
$29$ \( T^{8} - 4 T^{7} - 100 T^{6} + \cdots - 87169 \) Copy content Toggle raw display
$31$ \( T^{8} - 11 T^{7} - 26 T^{6} + \cdots + 8720 \) Copy content Toggle raw display
$37$ \( T^{8} + 10 T^{7} - 75 T^{6} + \cdots + 12269 \) Copy content Toggle raw display
$41$ \( T^{8} - 29 T^{7} + 263 T^{6} + \cdots - 1301 \) Copy content Toggle raw display
$43$ \( T^{8} - 2 T^{7} - 108 T^{6} + \cdots + 16624 \) Copy content Toggle raw display
$47$ \( T^{8} - 259 T^{6} + 136 T^{5} + \cdots - 15280 \) Copy content Toggle raw display
$53$ \( T^{8} - 59 T^{7} + 1349 T^{6} + \cdots + 10843831 \) Copy content Toggle raw display
$59$ \( T^{8} - 23 T^{7} - 81 T^{6} + \cdots - 5134144 \) Copy content Toggle raw display
$61$ \( T^{8} + 2 T^{7} - 429 T^{6} + \cdots + 37902131 \) Copy content Toggle raw display
$67$ \( T^{8} + 3 T^{7} - 197 T^{6} + \cdots + 281264 \) Copy content Toggle raw display
$71$ \( T^{8} - T^{7} - 368 T^{6} + \cdots + 8315920 \) Copy content Toggle raw display
$73$ \( T^{8} + 6 T^{7} - 429 T^{6} + \cdots + 58388219 \) Copy content Toggle raw display
$79$ \( T^{8} - 410 T^{6} + 1008 T^{5} + \cdots - 9246464 \) Copy content Toggle raw display
$83$ \( T^{8} - 15 T^{7} - 239 T^{6} + \cdots + 5294656 \) Copy content Toggle raw display
$89$ \( T^{8} - 4 T^{7} - 97 T^{6} + \cdots - 24631 \) Copy content Toggle raw display
$97$ \( T^{8} - 62 T^{7} + 1374 T^{6} + \cdots + 53342671 \) Copy content Toggle raw display
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