Properties

Label 966.4.a.i
Level $966$
Weight $4$
Character orbit 966.a
Self dual yes
Analytic conductor $56.996$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - 2x^{3} - 264x^{2} - 1037x + 3708 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} - 3 q^{3} + 4 q^{4} + (\beta_1 + 2) q^{5} + 6 q^{6} + 7 q^{7} - 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} - 3 q^{3} + 4 q^{4} + (\beta_1 + 2) q^{5} + 6 q^{6} + 7 q^{7} - 8 q^{8} + 9 q^{9} + ( - 2 \beta_1 - 4) q^{10} + (\beta_{3} + 2 \beta_{2} - \beta_1 + 7) q^{11} - 12 q^{12} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 - 11) q^{13} - 14 q^{14} + ( - 3 \beta_1 - 6) q^{15} + 16 q^{16} + ( - \beta_{3} - 2 \beta_{2} - 5 \beta_1 + 23) q^{17} - 18 q^{18} + (\beta_{3} + 4 \beta_{2} + \beta_1 + 19) q^{19} + (4 \beta_1 + 8) q^{20} - 21 q^{21} + ( - 2 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 14) q^{22} + 23 q^{23} + 24 q^{24} + (3 \beta_{3} + 4 \beta_{2} + 14 \beta_1 + 8) q^{25} + ( - 2 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 22) q^{26} - 27 q^{27} + 28 q^{28} + (3 \beta_{3} - 7 \beta_{2} + 5 \beta_1 + 109) q^{29} + (6 \beta_1 + 12) q^{30} + ( - 5 \beta_{3} + 6 \beta_{2} + 11 \beta_1 + 57) q^{31} - 32 q^{32} + ( - 3 \beta_{3} - 6 \beta_{2} + 3 \beta_1 - 21) q^{33} + (2 \beta_{3} + 4 \beta_{2} + 10 \beta_1 - 46) q^{34} + (7 \beta_1 + 14) q^{35} + 36 q^{36} + ( - 5 \beta_{3} + \beta_{2} - 3 \beta_1 - 43) q^{37} + ( - 2 \beta_{3} - 8 \beta_{2} - 2 \beta_1 - 38) q^{38} + ( - 3 \beta_{3} + 6 \beta_{2} + 6 \beta_1 + 33) q^{39} + ( - 8 \beta_1 - 16) q^{40} + (7 \beta_{3} - \beta_{2} + 15 \beta_1 + 57) q^{41} + 42 q^{42} + ( - 3 \beta_{3} - 6 \beta_{2} - 8 \beta_1 - 185) q^{43} + (4 \beta_{3} + 8 \beta_{2} - 4 \beta_1 + 28) q^{44} + (9 \beta_1 + 18) q^{45} - 46 q^{46} + (6 \beta_{3} - 3 \beta_{2} - 24 \beta_1 + 84) q^{47} - 48 q^{48} + 49 q^{49} + ( - 6 \beta_{3} - 8 \beta_{2} - 28 \beta_1 - 16) q^{50} + (3 \beta_{3} + 6 \beta_{2} + 15 \beta_1 - 69) q^{51} + (4 \beta_{3} - 8 \beta_{2} - 8 \beta_1 - 44) q^{52} + ( - 10 \beta_{3} - 11 \beta_{2} - \beta_1 + 46) q^{53} + 54 q^{54} + (3 \beta_{2} + 13 \beta_1 - 214) q^{55} - 56 q^{56} + ( - 3 \beta_{3} - 12 \beta_{2} - 3 \beta_1 - 57) q^{57} + ( - 6 \beta_{3} + 14 \beta_{2} - 10 \beta_1 - 218) q^{58} + ( - 8 \beta_{3} + 13 \beta_{2} - 3 \beta_1 - 54) q^{59} + ( - 12 \beta_1 - 24) q^{60} + (17 \beta_{3} + \beta_{2} + 40 \beta_1 - 53) q^{61} + (10 \beta_{3} - 12 \beta_{2} - 22 \beta_1 - 114) q^{62} + 63 q^{63} + 64 q^{64} + ( - 19 \beta_{3} - 5 \beta_{2} - 41 \beta_1 - 259) q^{65} + (6 \beta_{3} + 12 \beta_{2} - 6 \beta_1 + 42) q^{66} + (14 \beta_{3} + 9 \beta_{2} + 21 \beta_1 + 20) q^{67} + ( - 4 \beta_{3} - 8 \beta_{2} - 20 \beta_1 + 92) q^{68} - 69 q^{69} + ( - 14 \beta_1 - 28) q^{70} + ( - 11 \beta_{3} - 5 \beta_{2} - 12 \beta_1 - 3) q^{71} - 72 q^{72} + (11 \beta_{3} - 12 \beta_{2} + 41 \beta_1 - 177) q^{73} + (10 \beta_{3} - 2 \beta_{2} + 6 \beta_1 + 86) q^{74} + ( - 9 \beta_{3} - 12 \beta_{2} - 42 \beta_1 - 24) q^{75} + (4 \beta_{3} + 16 \beta_{2} + 4 \beta_1 + 76) q^{76} + (7 \beta_{3} + 14 \beta_{2} - 7 \beta_1 + 49) q^{77} + (6 \beta_{3} - 12 \beta_{2} - 12 \beta_1 - 66) q^{78} + (31 \beta_{3} + 16 \beta_{2} + 23 \beta_1 - 15) q^{79} + (16 \beta_1 + 32) q^{80} + 81 q^{81} + ( - 14 \beta_{3} + 2 \beta_{2} - 30 \beta_1 - 114) q^{82} + ( - 3 \beta_{3} + 30 \beta_{2} + 11 \beta_1 + 619) q^{83} - 84 q^{84} + ( - 18 \beta_{3} - 27 \beta_{2} - 55 \beta_1 - 500) q^{85} + (6 \beta_{3} + 12 \beta_{2} + 16 \beta_1 + 370) q^{86} + ( - 9 \beta_{3} + 21 \beta_{2} - 15 \beta_1 - 327) q^{87} + ( - 8 \beta_{3} - 16 \beta_{2} + 8 \beta_1 - 56) q^{88} + (37 \beta_{3} + 35 \beta_{2} - 18 \beta_1 + 159) q^{89} + ( - 18 \beta_1 - 36) q^{90} + (7 \beta_{3} - 14 \beta_{2} - 14 \beta_1 - 77) q^{91} + 92 q^{92} + (15 \beta_{3} - 18 \beta_{2} - 33 \beta_1 - 171) q^{93} + ( - 12 \beta_{3} + 6 \beta_{2} + 48 \beta_1 - 168) q^{94} + (14 \beta_{3} + 13 \beta_{2} + 61 \beta_1 + 8) q^{95} + 96 q^{96} + ( - 43 \beta_{3} - 35 \beta_{2} - 37 \beta_1 - 105) q^{97} - 98 q^{98} + (9 \beta_{3} + 18 \beta_{2} - 9 \beta_1 + 63) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{2} - 12 q^{3} + 16 q^{4} + 10 q^{5} + 24 q^{6} + 28 q^{7} - 32 q^{8} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{2} - 12 q^{3} + 16 q^{4} + 10 q^{5} + 24 q^{6} + 28 q^{7} - 32 q^{8} + 36 q^{9} - 20 q^{10} + 24 q^{11} - 48 q^{12} - 46 q^{13} - 56 q^{14} - 30 q^{15} + 64 q^{16} + 84 q^{17} - 72 q^{18} + 74 q^{19} + 40 q^{20} - 84 q^{21} - 48 q^{22} + 92 q^{23} + 96 q^{24} + 56 q^{25} + 92 q^{26} - 108 q^{27} + 112 q^{28} + 453 q^{29} + 60 q^{30} + 244 q^{31} - 128 q^{32} - 72 q^{33} - 168 q^{34} + 70 q^{35} + 144 q^{36} - 179 q^{37} - 148 q^{38} + 138 q^{39} - 80 q^{40} + 259 q^{41} + 168 q^{42} - 750 q^{43} + 96 q^{44} + 90 q^{45} - 184 q^{46} + 291 q^{47} - 192 q^{48} + 196 q^{49} - 112 q^{50} - 252 q^{51} - 184 q^{52} + 193 q^{53} + 216 q^{54} - 833 q^{55} - 224 q^{56} - 222 q^{57} - 906 q^{58} - 235 q^{59} - 120 q^{60} - 133 q^{61} - 488 q^{62} + 252 q^{63} + 256 q^{64} - 1113 q^{65} + 144 q^{66} + 113 q^{67} + 336 q^{68} - 276 q^{69} - 140 q^{70} - 31 q^{71} - 288 q^{72} - 614 q^{73} + 358 q^{74} - 168 q^{75} + 296 q^{76} + 168 q^{77} - 276 q^{78} - 30 q^{79} + 160 q^{80} + 324 q^{81} - 518 q^{82} + 2468 q^{83} - 336 q^{84} - 2083 q^{85} + 1500 q^{86} - 1359 q^{87} - 192 q^{88} + 565 q^{89} - 180 q^{90} - 322 q^{91} + 368 q^{92} - 732 q^{93} - 582 q^{94} + 141 q^{95} + 384 q^{96} - 459 q^{97} - 392 q^{98} + 216 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 264x^{2} - 1037x + 3708 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{3} - 25\nu^{2} - 563\nu + 66 ) / 53 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{3} + 51\nu^{2} + 574\nu - 2367 ) / 53 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{3} + 4\beta_{2} + 10\beta _1 + 129 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 25\beta_{3} + 51\beta_{2} + 271\beta _1 + 1053 \) 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
−10.5485
−8.32744
2.26856
18.6074
−2.00000 −3.00000 4.00000 −8.54853 6.00000 7.00000 −8.00000 9.00000 17.0971
1.2 −2.00000 −3.00000 4.00000 −6.32744 6.00000 7.00000 −8.00000 9.00000 12.6549
1.3 −2.00000 −3.00000 4.00000 4.26856 6.00000 7.00000 −8.00000 9.00000 −8.53712
1.4 −2.00000 −3.00000 4.00000 20.6074 6.00000 7.00000 −8.00000 9.00000 −41.2148
\(n\): e.g. 2-40 or 990-1000
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.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.4.a.i 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 10T_{5}^{3} - 228T_{5}^{2} - 37T_{5} + 4758 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(966))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{4} \) Copy content Toggle raw display
$3$ \( (T + 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 10 T^{3} - 228 T^{2} + \cdots + 4758 \) Copy content Toggle raw display
$7$ \( (T - 7)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 24 T^{3} - 3249 T^{2} + \cdots - 212832 \) Copy content Toggle raw display
$13$ \( T^{4} + 46 T^{3} - 4538 T^{2} + \cdots + 3764782 \) Copy content Toggle raw display
$17$ \( T^{4} - 84 T^{3} - 5019 T^{2} + \cdots - 13121592 \) Copy content Toggle raw display
$19$ \( T^{4} - 74 T^{3} - 7211 T^{2} + \cdots - 10790864 \) Copy content Toggle raw display
$23$ \( (T - 23)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} - 453 T^{3} + \cdots - 1589154186 \) Copy content Toggle raw display
$31$ \( T^{4} - 244 T^{3} + \cdots - 252612336 \) Copy content Toggle raw display
$37$ \( T^{4} + 179 T^{3} + \cdots + 16592278 \) Copy content Toggle raw display
$41$ \( T^{4} - 259 T^{3} + \cdots - 53519922 \) Copy content Toggle raw display
$43$ \( T^{4} + 750 T^{3} + \cdots + 93284468 \) Copy content Toggle raw display
$47$ \( T^{4} - 291 T^{3} + \cdots - 94654656 \) Copy content Toggle raw display
$53$ \( T^{4} - 193 T^{3} + \cdots + 248176368 \) Copy content Toggle raw display
$59$ \( T^{4} + 235 T^{3} + \cdots - 3616212024 \) Copy content Toggle raw display
$61$ \( T^{4} + 133 T^{3} + \cdots + 22161880364 \) Copy content Toggle raw display
$67$ \( T^{4} - 113 T^{3} + \cdots + 14109282416 \) Copy content Toggle raw display
$71$ \( T^{4} + 31 T^{3} + \cdots + 4881628944 \) Copy content Toggle raw display
$73$ \( T^{4} + 614 T^{3} + \cdots + 2976713544 \) Copy content Toggle raw display
$79$ \( T^{4} + 30 T^{3} + \cdots + 238252459968 \) Copy content Toggle raw display
$83$ \( T^{4} - 2468 T^{3} + \cdots - 13321543632 \) Copy content Toggle raw display
$89$ \( T^{4} - 565 T^{3} + \cdots - 69999776532 \) Copy content Toggle raw display
$97$ \( T^{4} + 459 T^{3} + \cdots + 688298945778 \) Copy content Toggle raw display
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