Properties

Label 966.4.a.n
Level $966$
Weight $4$
Character orbit 966.a
Self dual yes
Analytic conductor $56.996$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 966.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(56.9958450655\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \( x^{5} - 2x^{4} - 267x^{3} + 1502x^{2} + 1857x + 198 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} + 3 q^{3} + 4 q^{4} + \beta_{2} q^{5} - 6 q^{6} - 7 q^{7} - 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + 3 q^{3} + 4 q^{4} + \beta_{2} q^{5} - 6 q^{6} - 7 q^{7} - 8 q^{8} + 9 q^{9} - 2 \beta_{2} q^{10} + (3 \beta_{2} + \beta_1 + 4) q^{11} + 12 q^{12} + (\beta_{4} + 2 \beta_{2} - 3 \beta_1 - 6) q^{13} + 14 q^{14} + 3 \beta_{2} q^{15} + 16 q^{16} + ( - \beta_{4} + 5 \beta_{2} - 3 \beta_1 + 8) q^{17} - 18 q^{18} + (3 \beta_{4} - \beta_{2} + 5 \beta_1 - 2) q^{19} + 4 \beta_{2} q^{20} - 21 q^{21} + ( - 6 \beta_{2} - 2 \beta_1 - 8) q^{22} + 23 q^{23} - 24 q^{24} + ( - 4 \beta_{4} - 4 \beta_{2} - 3 \beta_1 + 9) q^{25} + ( - 2 \beta_{4} - 4 \beta_{2} + 6 \beta_1 + 12) q^{26} + 27 q^{27} - 28 q^{28} + ( - 2 \beta_{4} - \beta_{3} + 3 \beta_{2} + 6 \beta_1 - 5) q^{29} - 6 \beta_{2} q^{30} + (\beta_{4} + 2 \beta_{3} + 7 \beta_{2} - \beta_1 - 20) q^{31} - 32 q^{32} + (9 \beta_{2} + 3 \beta_1 + 12) q^{33} + (2 \beta_{4} - 10 \beta_{2} + 6 \beta_1 - 16) q^{34} - 7 \beta_{2} q^{35} + 36 q^{36} + (2 \beta_{4} - 3 \beta_{3} - 7 \beta_{2} + 49) q^{37} + ( - 6 \beta_{4} + 2 \beta_{2} - 10 \beta_1 + 4) q^{38} + (3 \beta_{4} + 6 \beta_{2} - 9 \beta_1 - 18) q^{39} - 8 \beta_{2} q^{40} + ( - 2 \beta_{4} + 3 \beta_{3} + 13 \beta_{2} - 2 \beta_1 + 15) q^{41} + 42 q^{42} + ( - 4 \beta_{4} - 2 \beta_{2} + 5 \beta_1 + 186) q^{43} + (12 \beta_{2} + 4 \beta_1 + 16) q^{44} + 9 \beta_{2} q^{45} - 46 q^{46} + (15 \beta_{4} - \beta_{3} + 10 \beta_{2} + 13 \beta_1 - 65) q^{47} + 48 q^{48} + 49 q^{49} + (8 \beta_{4} + 8 \beta_{2} + 6 \beta_1 - 18) q^{50} + ( - 3 \beta_{4} + 15 \beta_{2} - 9 \beta_1 + 24) q^{51} + (4 \beta_{4} + 8 \beta_{2} - 12 \beta_1 - 24) q^{52} + ( - 14 \beta_{4} - \beta_{3} - 5 \beta_{2} - 13 \beta_1 + 149) q^{53} - 54 q^{54} + ( - 14 \beta_{4} - \beta_{3} - 15 \beta_{2} - 5 \beta_1 + 443) q^{55} + 56 q^{56} + (9 \beta_{4} - 3 \beta_{2} + 15 \beta_1 - 6) q^{57} + (4 \beta_{4} + 2 \beta_{3} - 6 \beta_{2} - 12 \beta_1 + 10) q^{58} + (18 \beta_{4} + \beta_{3} + 15 \beta_{2} + 7 \beta_1 - 67) q^{59} + 12 \beta_{2} q^{60} + ( - 2 \beta_{4} + 3 \beta_{3} + 10 \beta_{2} - 18 \beta_1 + 213) q^{61} + ( - 2 \beta_{4} - 4 \beta_{3} - 14 \beta_{2} + 2 \beta_1 + 40) q^{62} - 63 q^{63} + 64 q^{64} + (\beta_{3} - 9 \beta_{2} - 20 \beta_1 + 99) q^{65} + ( - 18 \beta_{2} - 6 \beta_1 - 24) q^{66} + ( - 2 \beta_{4} - \beta_{3} - 13 \beta_{2} + 21 \beta_1 + 207) q^{67} + ( - 4 \beta_{4} + 20 \beta_{2} - 12 \beta_1 + 32) q^{68} + 69 q^{69} + 14 \beta_{2} q^{70} + ( - 14 \beta_{4} + \beta_{3} + 26 \beta_1 + 201) q^{71} - 72 q^{72} + (4 \beta_{4} - 8 \beta_{3} - 9 \beta_{2} - 5 \beta_1 + 222) q^{73} + ( - 4 \beta_{4} + 6 \beta_{3} + 14 \beta_{2} - 98) q^{74} + ( - 12 \beta_{4} - 12 \beta_{2} - 9 \beta_1 + 27) q^{75} + (12 \beta_{4} - 4 \beta_{2} + 20 \beta_1 - 8) q^{76} + ( - 21 \beta_{2} - 7 \beta_1 - 28) q^{77} + ( - 6 \beta_{4} - 12 \beta_{2} + 18 \beta_1 + 36) q^{78} + (12 \beta_{4} + 4 \beta_{3} - 21 \beta_{2} + 33 \beta_1 - 12) q^{79} + 16 \beta_{2} q^{80} + 81 q^{81} + (4 \beta_{4} - 6 \beta_{3} - 26 \beta_{2} + 4 \beta_1 - 30) q^{82} + (7 \beta_{4} + 4 \beta_{3} - 31 \beta_{2} - \beta_1 + 42) q^{83} - 84 q^{84} + ( - 16 \beta_{4} + 5 \beta_{3} + 25 \beta_{2} - 25 \beta_1 + 593) q^{85} + (8 \beta_{4} + 4 \beta_{2} - 10 \beta_1 - 372) q^{86} + ( - 6 \beta_{4} - 3 \beta_{3} + 9 \beta_{2} + 18 \beta_1 - 15) q^{87} + ( - 24 \beta_{2} - 8 \beta_1 - 32) q^{88} + ( - \beta_{4} - 5 \beta_{3} - 28 \beta_{2} - 12 \beta_1 - 9) q^{89} - 18 \beta_{2} q^{90} + ( - 7 \beta_{4} - 14 \beta_{2} + 21 \beta_1 + 42) q^{91} + 92 q^{92} + (3 \beta_{4} + 6 \beta_{3} + 21 \beta_{2} - 3 \beta_1 - 60) q^{93} + ( - 30 \beta_{4} + 2 \beta_{3} - 20 \beta_{2} - 26 \beta_1 + 130) q^{94} + ( - 11 \beta_{3} - 81 \beta_{2} + 17 \beta_1 - 67) q^{95} - 96 q^{96} + (13 \beta_{4} - \beta_{3} - 31 \beta_{2} - 54 \beta_1 + 125) q^{97} - 98 q^{98} + (27 \beta_{2} + 9 \beta_1 + 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 10 q^{2} + 15 q^{3} + 20 q^{4} - 30 q^{6} - 35 q^{7} - 40 q^{8} + 45 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 10 q^{2} + 15 q^{3} + 20 q^{4} - 30 q^{6} - 35 q^{7} - 40 q^{8} + 45 q^{9} + 18 q^{11} + 60 q^{12} - 22 q^{13} + 70 q^{14} + 80 q^{16} + 44 q^{17} - 90 q^{18} - 14 q^{19} - 105 q^{21} - 36 q^{22} + 115 q^{23} - 120 q^{24} + 43 q^{25} + 44 q^{26} + 135 q^{27} - 140 q^{28} - 39 q^{29} - 100 q^{31} - 160 q^{32} + 54 q^{33} - 88 q^{34} + 180 q^{36} + 255 q^{37} + 28 q^{38} - 66 q^{39} + 69 q^{41} + 210 q^{42} + 912 q^{43} + 72 q^{44} - 230 q^{46} - 319 q^{47} + 240 q^{48} + 245 q^{49} - 86 q^{50} + 132 q^{51} - 88 q^{52} + 745 q^{53} - 270 q^{54} + 2199 q^{55} + 280 q^{56} - 42 q^{57} + 78 q^{58} - 315 q^{59} + 1091 q^{61} + 200 q^{62} - 315 q^{63} + 320 q^{64} + 533 q^{65} - 108 q^{66} + 991 q^{67} + 176 q^{68} + 345 q^{69} + 923 q^{71} - 360 q^{72} + 1144 q^{73} - 510 q^{74} + 129 q^{75} - 56 q^{76} - 126 q^{77} + 132 q^{78} - 110 q^{79} + 405 q^{81} - 138 q^{82} + 218 q^{83} - 420 q^{84} + 2973 q^{85} - 1824 q^{86} - 117 q^{87} - 144 q^{88} - 13 q^{89} + 154 q^{91} + 460 q^{92} - 300 q^{93} + 638 q^{94} - 347 q^{95} - 480 q^{96} + 761 q^{97} - 490 q^{98} + 162 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 267x^{3} + 1502x^{2} + 1857x + 198 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -8\nu^{4} - 11\nu^{3} + 2031\nu^{2} - 5365\nu - 19320 ) / 1086 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13\nu^{4} - 50\nu^{3} - 3504\nu^{2} + 24533\nu + 6960 ) / 1086 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{4} + 133\nu^{3} + 2001\nu^{2} - 25369\nu - 77592 ) / 1086 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7\nu^{4} - 13\nu^{3} - 1845\nu^{2} + 10690\nu + 8398 ) / 181 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} - 2\beta_{2} + 2\beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -7\beta_{4} + 4\beta_{3} + 22\beta_{2} - 2\beta _1 + 434 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 127\beta_{4} - 6\beta_{3} - 298\beta_{2} + 184\beta _1 - 1138 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2797\beta_{4} + 1032\beta_{3} + 7746\beta_{2} - 2898\beta _1 + 102310 ) / 4 \) 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
−0.940566
12.7990
−0.118152
7.89405
−17.6344
−2.00000 3.00000 4.00000 −17.6455 −6.00000 −7.00000 −8.00000 9.00000 35.2910
1.2 −2.00000 3.00000 4.00000 −8.30883 −6.00000 −7.00000 −8.00000 9.00000 16.6177
1.3 −2.00000 3.00000 4.00000 3.69480 −6.00000 −7.00000 −8.00000 9.00000 −7.38960
1.4 −2.00000 3.00000 4.00000 7.50994 −6.00000 −7.00000 −8.00000 9.00000 −15.0199
1.5 −2.00000 3.00000 4.00000 14.7496 −6.00000 −7.00000 −8.00000 9.00000 −29.4991
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(7\) \(1\)
\(23\) \(-1\)

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 966.4.a.n 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.4.a.n 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} - 334T_{5}^{3} + 795T_{5}^{2} + 17676T_{5} - 60004 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(966))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{5} \) Copy content Toggle raw display
$3$ \( (T - 3)^{5} \) Copy content Toggle raw display
$5$ \( T^{5} - 334 T^{3} + 795 T^{2} + \cdots - 60004 \) Copy content Toggle raw display
$7$ \( (T + 7)^{5} \) Copy content Toggle raw display
$11$ \( T^{5} - 18 T^{4} - 4055 T^{3} + \cdots + 4096576 \) Copy content Toggle raw display
$13$ \( T^{5} + 22 T^{4} - 7596 T^{3} + \cdots + 39550252 \) Copy content Toggle raw display
$17$ \( T^{5} - 44 T^{4} + \cdots + 256399952 \) Copy content Toggle raw display
$19$ \( T^{5} + 14 T^{4} + \cdots - 3376969856 \) Copy content Toggle raw display
$23$ \( (T - 23)^{5} \) Copy content Toggle raw display
$29$ \( T^{5} + 39 T^{4} + \cdots + 499949716 \) Copy content Toggle raw display
$31$ \( T^{5} + 100 T^{4} + \cdots + 129027454688 \) Copy content Toggle raw display
$37$ \( T^{5} - 255 T^{4} + \cdots - 1752755460964 \) Copy content Toggle raw display
$41$ \( T^{5} - 69 T^{4} + \cdots + 1347225491444 \) Copy content Toggle raw display
$43$ \( T^{5} - 912 T^{4} + \cdots + 198454918768 \) Copy content Toggle raw display
$47$ \( T^{5} + 319 T^{4} + \cdots - 3919588628032 \) Copy content Toggle raw display
$53$ \( T^{5} - 745 T^{4} + \cdots + 460114266352 \) Copy content Toggle raw display
$59$ \( T^{5} + 315 T^{4} + \cdots - 88518485664 \) Copy content Toggle raw display
$61$ \( T^{5} - 1091 T^{4} + \cdots + 4528088303864 \) Copy content Toggle raw display
$67$ \( T^{5} - 991 T^{4} + \cdots - 1352835082304 \) Copy content Toggle raw display
$71$ \( T^{5} - 923 T^{4} + \cdots + 10575799339392 \) Copy content Toggle raw display
$73$ \( T^{5} + \cdots - 202562391842272 \) Copy content Toggle raw display
$79$ \( T^{5} + 110 T^{4} + \cdots + 36500687712256 \) Copy content Toggle raw display
$83$ \( T^{5} - 218 T^{4} + \cdots + 3555275628608 \) Copy content Toggle raw display
$89$ \( T^{5} + 13 T^{4} + \cdots + 10886725290664 \) Copy content Toggle raw display
$97$ \( T^{5} + \cdots - 190187620864468 \) Copy content Toggle raw display
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