Properties

Label 83.2.a.b
Level $83$
Weight $2$
Character orbit 83.a
Self dual yes
Analytic conductor $0.663$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 83 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 83.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.662758336777\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.9059636.1
Defining polynomial: \( x^{6} - x^{5} - 8x^{4} + 11x^{3} + 4x^{2} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - \beta_1) q^{2} - \beta_{3} q^{3} + ( - \beta_{5} + 1) q^{4} + (\beta_{5} + \beta_{3} + \beta_{2} + 1) q^{5} + (\beta_{5} + \beta_{3} + \beta_1 - 1) q^{6} + ( - \beta_{4} + \beta_1) q^{7} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_1 + 1) q^{8} + (\beta_{4} + \beta_{2} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - \beta_1) q^{2} - \beta_{3} q^{3} + ( - \beta_{5} + 1) q^{4} + (\beta_{5} + \beta_{3} + \beta_{2} + 1) q^{5} + (\beta_{5} + \beta_{3} + \beta_1 - 1) q^{6} + ( - \beta_{4} + \beta_1) q^{7} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_1 + 1) q^{8} + (\beta_{4} + \beta_{2} + \beta_1 + 1) q^{9} + (\beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{10} + ( - \beta_{5} - \beta_{4} - \beta_{3} - \beta_1 - 1) q^{11} + ( - 2 \beta_{4} - \beta_{3} + \beta_{2} - 2) q^{12} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 + 3) q^{13} + (\beta_{5} - \beta_{3} - 2 \beta_{2} - \beta_1 - 3) q^{14} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} - 1) q^{15} + ( - 2 \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2}) q^{16} + (\beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 1) q^{17} - 2 q^{18} + (\beta_{5} + \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{19} + (3 \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} - 3) q^{20} + ( - \beta_{5} - 3 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 2) q^{21} + (\beta_{5} + 2 \beta_{4} + 3 \beta_{3} + \beta_1 + 1) q^{22} + ( - 2 \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{23} + (2 \beta_{5} + \beta_{2} - \beta_1 - 2) q^{24} + (2 \beta_{3} - 2 \beta_1 + 3) q^{25} + ( - 3 \beta_{5} + 2 \beta_{4} - \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 1) q^{26} + (2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 3) q^{27} + ( - \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 2) q^{28} + (\beta_{5} - 3 \beta_{2} - 1) q^{29} + ( - 3 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} + 4 \beta_1 + 3) q^{30} + (\beta_{5} + \beta_{4} - \beta_{3} - \beta_1 + 1) q^{31} + ( - 2 \beta_{5} - \beta_{2} + \beta_1 + 2) q^{32} + ( - \beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{33} + ( - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{34} + ( - \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + \beta_{2} + 4 \beta_1 - 1) q^{35} + ( - 2 \beta_{4} - 2) q^{36} + ( - \beta_{4} - 2 \beta_{3} - \beta_{2} - 3 \beta_1 + 6) q^{37} + ( - \beta_{5} - 2 \beta_{4} - 5 \beta_{3} - \beta_{2} - 3) q^{38} + (\beta_{5} - 3 \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{39} + (3 \beta_{5} - 2 \beta_{4} - \beta_{3} + 3 \beta_{2} + 4 \beta_1 - 3) q^{40} + (2 \beta_{4} + \beta_{3} + \beta_{2} + 1) q^{41} + (2 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} - \beta_{2} + \beta_1 + 4) q^{42} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{2} - 2 \beta_1 - 2) q^{43} + ( - 2 \beta_{5} - 3 \beta_{3} + \beta_{2} + 4) q^{44} + (2 \beta_{4} + 2 \beta_1 + 2) q^{45} + (4 \beta_{5} + 4 \beta_{3} + 2 \beta_{2}) q^{46} + ( - \beta_{5} + 4 \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{47} + (3 \beta_{5} + 2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 3) q^{48} + (\beta_{5} + \beta_{2} + 2 \beta_1 + 2) q^{49} + ( - 2 \beta_{5} - \beta_{2} - 5 \beta_1 + 6) q^{50} + ( - \beta_{4} + \beta_{3} - 3 \beta_{2} - \beta_1 - 5) q^{51} + ( - 5 \beta_{5} + 2 \beta_{4} + \beta_{3} - 3 \beta_{2} + 5) q^{52} + ( - 2 \beta_{5} + 2 \beta_{2} + 4 \beta_1 + 2) q^{53} + ( - 3 \beta_{5} - \beta_{3} - 3 \beta_1 + 3) q^{54} + (\beta_{5} - 2 \beta_{4} - \beta_{3} - 3 \beta_{2} - 4 \beta_1 - 5) q^{55} + (2 \beta_{5} + 4 \beta_{3} + 3 \beta_{2} + \beta_1 - 2) q^{56} + (\beta_{5} - 2 \beta_{4} - \beta_{3} - 5 \beta_{2} - 6 \beta_1 - 3) q^{57} + ( - 2 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} - \beta_{2} + \beta_1 + 2) q^{58} + (2 \beta_{5} - \beta_{4} + 2 \beta_{2} - \beta_1 - 2) q^{59} + ( - 3 \beta_{5} + 2 \beta_{4} + \beta_{3} - 5 \beta_{2} - 2 \beta_1 - 1) q^{60} + ( - \beta_{5} - 4 \beta_{3} - 5 \beta_{2} - 2 \beta_1 - 3) q^{61} + (\beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{62} + ( - \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + \beta_{2} - 2) q^{63} + (\beta_{5} + 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 1) q^{64} + (4 \beta_{5} + 2 \beta_{3} + 4 \beta_{2} + 6 \beta_1) q^{65} + (2 \beta_{5} + 2 \beta_{4} - 3 \beta_{2} - 3 \beta_1 - 6) q^{66} + ( - 2 \beta_{5} + 2 \beta_{3} + 4 \beta_{2} + 4) q^{67} + (3 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 1) q^{68} + ( - 3 \beta_{5} + \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 1) q^{69} + ( - 5 \beta_{5} + 2 \beta_{4} - 5 \beta_{3} - \beta_{2} - 3) q^{70} + ( - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 6) q^{71} + (2 \beta_{5} + 2) q^{72} + ( - 3 \beta_{5} + 4 \beta_{4} - 3 \beta_{3} - \beta_{2} - 1) q^{73} + (2 \beta_{5} + 4 \beta_{3} - 5 \beta_{2} - 5 \beta_1 + 4) q^{74} + ( - \beta_{3} + 2 \beta_1 - 8) q^{75} + (3 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + \beta_{2} + 2 \beta_1 - 3) q^{76} + ( - 2 \beta_{5} + \beta_{4} - 5 \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{77} + (5 \beta_{5} - 2 \beta_{4} + 5 \beta_{3} + 5 \beta_{2} + 4 \beta_1 - 1) q^{78} + (\beta_{5} + \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 1) q^{79} + (3 \beta_{5} - 2 \beta_{4} - 5 \beta_{3} + \beta_{2} + 2 \beta_1 - 11) q^{80} + (\beta_{5} - \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 6) q^{81} + ( - 2 \beta_{5} + 2 \beta_{2} + 2) q^{82} + q^{83} + ( - 3 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} + 7) q^{84} + ( - \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 3) q^{85} + (2 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 2) q^{86} + (3 \beta_{5} - \beta_{4} + \beta_{3} - 4 \beta_{2} - 3 \beta_1 - 1) q^{87} + ( - 5 \beta_{2} - 3 \beta_1 - 4) q^{88} + (2 \beta_{3} + 6 \beta_{2} + 4 \beta_1 - 2) q^{89} + ( - 2 \beta_{5} - 2 \beta_{3} - 2 \beta_{2} - 2) q^{90} + (\beta_{5} - 2 \beta_{4} + 5 \beta_{3} + 5 \beta_{2} + 4 \beta_1 + 1) q^{91} + (2 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 4 \beta_{2}) q^{92} + (\beta_{5} + 6 \beta_{4} + \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 8) q^{93} + ( - 3 \beta_{5} + 2 \beta_{4} + \beta_{3} + 3 \beta_{2} + 6 \beta_1 - 1) q^{94} + ( - 4 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + 10 \beta_1) q^{95} + (\beta_{5} - 6 \beta_{4} - 3 \beta_{3} - 3 \beta_1 - 5) q^{96} + (2 \beta_{5} - 4 \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{97} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 6) q^{98} + ( - \beta_{5} - 4 \beta_{4} - \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} + q^{3} + 7 q^{4} + 2 q^{5} - 7 q^{6} + 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{2} + q^{3} + 7 q^{4} + 2 q^{5} - 7 q^{6} + 3 q^{7} + 3 q^{8} + 3 q^{9} - 6 q^{10} - 3 q^{11} - 9 q^{12} + 14 q^{13} - 15 q^{14} - 6 q^{15} + q^{16} - 5 q^{17} - 12 q^{18} - 4 q^{19} - 20 q^{20} - 2 q^{21} - q^{22} - 5 q^{23} - 17 q^{24} + 14 q^{25} - 2 q^{26} + 10 q^{27} + 9 q^{28} - q^{29} + 18 q^{30} + 3 q^{31} + 17 q^{32} + 2 q^{33} + 15 q^{34} - 10 q^{35} - 8 q^{36} + 39 q^{37} - 6 q^{38} - 8 q^{39} - 18 q^{40} - q^{41} + 17 q^{42} - 8 q^{43} + 27 q^{44} + 10 q^{45} - 12 q^{46} - 12 q^{47} + 7 q^{48} + 11 q^{49} + 35 q^{50} - 24 q^{51} + 36 q^{52} + 14 q^{53} + 19 q^{54} - 24 q^{55} - 23 q^{56} - 10 q^{57} + 25 q^{58} - 17 q^{59} - 5 q^{61} + 5 q^{62} - 11 q^{63} + 9 q^{64} - 8 q^{65} - 39 q^{66} + 16 q^{67} - 15 q^{68} + 8 q^{69} - 10 q^{70} - 26 q^{71} + 10 q^{72} - 6 q^{73} + 23 q^{74} - 45 q^{75} - 28 q^{76} + 6 q^{77} - 18 q^{78} - 12 q^{79} - 60 q^{80} - 34 q^{81} + 10 q^{82} + 6 q^{83} + 41 q^{84} + 18 q^{85} + 16 q^{86} - 3 q^{87} - 17 q^{88} - 22 q^{89} - 4 q^{90} - 2 q^{91} + 2 q^{92} + 34 q^{93} - 8 q^{94} - 4 q^{95} - 19 q^{96} + 6 q^{97} - 30 q^{98} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 8x^{4} + 11x^{3} + 4x^{2} - 4x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{4} - 7\nu^{2} + 4\nu + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{5} - \nu^{4} - 7\nu^{3} + 12\nu^{2} - 2\nu - 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{5} + \nu^{4} + 8\nu^{3} - 11\nu^{2} - 4\nu + 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -2\nu^{5} + 2\nu^{4} + 15\nu^{3} - 22\nu^{2} - \nu + 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} + \beta_{3} - \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + 2\beta_{4} + 7\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{5} - 7\beta_{4} + 7\beta_{3} + \beta_{2} - 11\beta _1 + 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -12\beta_{5} + 19\beta_{4} - 4\beta_{3} + \beta_{2} + 52\beta _1 - 27 \) 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
0.705771
2.14357
−0.236470
1.80570
−2.88130
−0.537266
−2.29018 1.96807 3.24494 −1.62860 −4.50724 3.12266 −2.85114 0.873293 3.72978
1.2 −1.66658 −2.04941 0.777479 2.79494 3.41549 3.61008 2.03743 1.20006 −4.65798
1.3 −0.429349 2.76735 −1.81566 1.71413 −1.18816 −3.46533 1.63825 4.65821 −0.735961
1.4 1.16417 1.13227 −0.644717 −1.45742 1.31815 3.35950 −3.07889 −1.71797 −1.69668
1.5 1.59835 −1.32239 0.554733 4.05060 −2.11364 −2.22837 −2.31005 −1.25129 6.47429
1.6 2.62359 −1.49589 4.88322 −3.47366 −3.92460 −1.39854 7.56440 −0.762314 −9.11346
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(83\) \(-1\)

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 83.2.a.b 6
3.b odd 2 1 747.2.a.j 6
4.b odd 2 1 1328.2.a.l 6
5.b even 2 1 2075.2.a.g 6
7.b odd 2 1 4067.2.a.d 6
8.b even 2 1 5312.2.a.bn 6
8.d odd 2 1 5312.2.a.bo 6
83.b odd 2 1 6889.2.a.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
83.2.a.b 6 1.a even 1 1 trivial
747.2.a.j 6 3.b odd 2 1
1328.2.a.l 6 4.b odd 2 1
2075.2.a.g 6 5.b even 2 1
4067.2.a.d 6 7.b odd 2 1
5312.2.a.bn 6 8.b even 2 1
5312.2.a.bo 6 8.d odd 2 1
6889.2.a.e 6 83.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - T_{2}^{5} - 9T_{2}^{4} + 7T_{2}^{3} + 20T_{2}^{2} - 12T_{2} - 8 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(83))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - T^{5} - 9 T^{4} + 7 T^{3} + 20 T^{2} + \cdots - 8 \) Copy content Toggle raw display
$3$ \( T^{6} - T^{5} - 10 T^{4} + 5 T^{3} + \cdots - 25 \) Copy content Toggle raw display
$5$ \( T^{6} - 2 T^{5} - 20 T^{4} + 28 T^{3} + \cdots - 160 \) Copy content Toggle raw display
$7$ \( T^{6} - 3 T^{5} - 22 T^{4} + 55 T^{3} + \cdots - 409 \) Copy content Toggle raw display
$11$ \( T^{6} + 3 T^{5} - 26 T^{4} - 83 T^{3} + \cdots - 113 \) Copy content Toggle raw display
$13$ \( T^{6} - 14 T^{5} + 44 T^{4} + \cdots + 992 \) Copy content Toggle raw display
$17$ \( T^{6} + 5 T^{5} - 20 T^{4} - 77 T^{3} + \cdots - 275 \) Copy content Toggle raw display
$19$ \( T^{6} + 4 T^{5} - 68 T^{4} + \cdots + 6176 \) Copy content Toggle raw display
$23$ \( T^{6} + 5 T^{5} - 61 T^{4} + \cdots + 10912 \) Copy content Toggle raw display
$29$ \( T^{6} + T^{5} - 88 T^{4} - 181 T^{3} + \cdots - 55 \) Copy content Toggle raw display
$31$ \( T^{6} - 3 T^{5} - 66 T^{4} - 93 T^{3} + \cdots - 313 \) Copy content Toggle raw display
$37$ \( T^{6} - 39 T^{5} + 576 T^{4} + \cdots - 91499 \) Copy content Toggle raw display
$41$ \( T^{6} + T^{5} - 47 T^{4} - T^{3} + \cdots - 248 \) Copy content Toggle raw display
$43$ \( T^{6} + 8 T^{5} - 44 T^{4} + \cdots + 6400 \) Copy content Toggle raw display
$47$ \( T^{6} + 12 T^{5} - 96 T^{4} + \cdots + 25952 \) Copy content Toggle raw display
$53$ \( T^{6} - 14 T^{5} - 64 T^{4} + 1064 T^{3} + \cdots - 64 \) Copy content Toggle raw display
$59$ \( T^{6} + 17 T^{5} + 10 T^{4} + \cdots + 3527 \) Copy content Toggle raw display
$61$ \( T^{6} + 5 T^{5} - 208 T^{4} + \cdots - 47347 \) Copy content Toggle raw display
$67$ \( T^{6} - 16 T^{5} - 128 T^{4} + \cdots + 264256 \) Copy content Toggle raw display
$71$ \( T^{6} + 26 T^{5} + 168 T^{4} + \cdots + 7232 \) Copy content Toggle raw display
$73$ \( T^{6} + 6 T^{5} - 268 T^{4} + \cdots - 39136 \) Copy content Toggle raw display
$79$ \( T^{6} + 12 T^{5} - 12 T^{4} + \cdots - 160 \) Copy content Toggle raw display
$83$ \( (T - 1)^{6} \) Copy content Toggle raw display
$89$ \( T^{6} + 22 T^{5} - 28 T^{4} + \cdots + 144896 \) Copy content Toggle raw display
$97$ \( T^{6} - 6 T^{5} - 300 T^{4} + \cdots - 101120 \) Copy content Toggle raw display
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