Properties

Label 966.4.a.f
Level $966$
Weight $4$
Character orbit 966.a
Self dual yes
Analytic conductor $56.996$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.12197.1
Defining polynomial: \( x^{3} - x^{2} - 15x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} + ( - 3 \beta_1 + 4) q^{5} - 6 q^{6} + 7 q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} + ( - 3 \beta_1 + 4) q^{5} - 6 q^{6} + 7 q^{7} + 8 q^{8} + 9 q^{9} + ( - 6 \beta_1 + 8) q^{10} + ( - 2 \beta_{2} - 12 \beta_1 + 21) q^{11} - 12 q^{12} + ( - 3 \beta_{2} - 2 \beta_1 + 20) q^{13} + 14 q^{14} + (9 \beta_1 - 12) q^{15} + 16 q^{16} + (3 \beta_{2} - 5 \beta_1 + 2) q^{17} + 18 q^{18} + ( - 4 \beta_{2} - 10 \beta_1 + 23) q^{19} + ( - 12 \beta_1 + 16) q^{20} - 21 q^{21} + ( - 4 \beta_{2} - 24 \beta_1 + 42) q^{22} - 23 q^{23} - 24 q^{24} + (9 \beta_{2} - 6 \beta_1 - 19) q^{25} + ( - 6 \beta_{2} - 4 \beta_1 + 40) q^{26} - 27 q^{27} + 28 q^{28} + (9 \beta_{2} + 45 \beta_1 + 92) q^{29} + (18 \beta_1 - 24) q^{30} + (15 \beta_{2} + 43 \beta_1 + 64) q^{31} + 32 q^{32} + (6 \beta_{2} + 36 \beta_1 - 63) q^{33} + (6 \beta_{2} - 10 \beta_1 + 4) q^{34} + ( - 21 \beta_1 + 28) q^{35} + 36 q^{36} + (21 \beta_{2} - 59 \beta_1 + 12) q^{37} + ( - 8 \beta_{2} - 20 \beta_1 + 46) q^{38} + (9 \beta_{2} + 6 \beta_1 - 60) q^{39} + ( - 24 \beta_1 + 32) q^{40} + ( - 48 \beta_{2} + 42 \beta_1 + 27) q^{41} - 42 q^{42} + ( - 49 \beta_{2} - 34 \beta_1 - 214) q^{43} + ( - 8 \beta_{2} - 48 \beta_1 + 84) q^{44} + ( - 27 \beta_1 + 36) q^{45} - 46 q^{46} + (69 \beta_{2} + 11 \beta_1 + 137) q^{47} - 48 q^{48} + 49 q^{49} + (18 \beta_{2} - 12 \beta_1 - 38) q^{50} + ( - 9 \beta_{2} + 15 \beta_1 - 6) q^{51} + ( - 12 \beta_{2} - 8 \beta_1 + 80) q^{52} + ( - 65 \beta_{2} - 14 \beta_1 - 185) q^{53} - 54 q^{54} + (22 \beta_{2} - 21 \beta_1 + 408) q^{55} + 56 q^{56} + (12 \beta_{2} + 30 \beta_1 - 69) q^{57} + (18 \beta_{2} + 90 \beta_1 + 184) q^{58} + ( - 7 \beta_{2} + 138 \beta_1 + 131) q^{59} + (36 \beta_1 - 48) q^{60} + (20 \beta_{2} - 111 \beta_1 + 129) q^{61} + (30 \beta_{2} + 86 \beta_1 + 128) q^{62} + 63 q^{63} + 64 q^{64} + ( - 15 \beta_{2} - 29 \beta_1 + 86) q^{65} + (12 \beta_{2} + 72 \beta_1 - 126) q^{66} + ( - 50 \beta_{2} + 43 \beta_1 + 340) q^{67} + (12 \beta_{2} - 20 \beta_1 + 8) q^{68} + 69 q^{69} + ( - 42 \beta_1 + 56) q^{70} + (75 \beta_{2} + 194 \beta_1 - 22) q^{71} + 72 q^{72} + ( - 53 \beta_{2} + 121 \beta_1 + 62) q^{73} + (42 \beta_{2} - 118 \beta_1 + 24) q^{74} + ( - 27 \beta_{2} + 18 \beta_1 + 57) q^{75} + ( - 16 \beta_{2} - 40 \beta_1 + 92) q^{76} + ( - 14 \beta_{2} - 84 \beta_1 + 147) q^{77} + (18 \beta_{2} + 12 \beta_1 - 120) q^{78} + (27 \beta_{2} - 151 \beta_1 + 104) q^{79} + ( - 48 \beta_1 + 64) q^{80} + 81 q^{81} + ( - 96 \beta_{2} + 84 \beta_1 + 54) q^{82} + (13 \beta_{2} - 41 \beta_1 + 520) q^{83} - 84 q^{84} + (36 \beta_{2} - 23 \beta_1 + 212) q^{85} + ( - 98 \beta_{2} - 68 \beta_1 - 428) q^{86} + ( - 27 \beta_{2} - 135 \beta_1 - 276) q^{87} + ( - 16 \beta_{2} - 96 \beta_1 + 168) q^{88} + ( - 23 \beta_{2} + 186 \beta_1 + 724) q^{89} + ( - 54 \beta_1 + 72) q^{90} + ( - 21 \beta_{2} - 14 \beta_1 + 140) q^{91} - 92 q^{92} + ( - 45 \beta_{2} - 129 \beta_1 - 192) q^{93} + (138 \beta_{2} + 22 \beta_1 + 274) q^{94} + (2 \beta_{2} - 13 \beta_1 + 320) q^{95} - 96 q^{96} + (29 \beta_{2} - 59 \beta_1) q^{97} + 98 q^{98} + ( - 18 \beta_{2} - 108 \beta_1 + 189) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 6 q^{2} - 9 q^{3} + 12 q^{4} + 9 q^{5} - 18 q^{6} + 21 q^{7} + 24 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 6 q^{2} - 9 q^{3} + 12 q^{4} + 9 q^{5} - 18 q^{6} + 21 q^{7} + 24 q^{8} + 27 q^{9} + 18 q^{10} + 53 q^{11} - 36 q^{12} + 61 q^{13} + 42 q^{14} - 27 q^{15} + 48 q^{16} - 2 q^{17} + 54 q^{18} + 63 q^{19} + 36 q^{20} - 63 q^{21} + 106 q^{22} - 69 q^{23} - 72 q^{24} - 72 q^{25} + 122 q^{26} - 81 q^{27} + 84 q^{28} + 312 q^{29} - 54 q^{30} + 220 q^{31} + 96 q^{32} - 159 q^{33} - 4 q^{34} + 63 q^{35} + 108 q^{36} - 44 q^{37} + 126 q^{38} - 183 q^{39} + 72 q^{40} + 171 q^{41} - 126 q^{42} - 627 q^{43} + 212 q^{44} + 81 q^{45} - 138 q^{46} + 353 q^{47} - 144 q^{48} + 147 q^{49} - 144 q^{50} + 6 q^{51} + 244 q^{52} - 504 q^{53} - 162 q^{54} + 1181 q^{55} + 168 q^{56} - 189 q^{57} + 624 q^{58} + 538 q^{59} - 108 q^{60} + 256 q^{61} + 440 q^{62} + 189 q^{63} + 192 q^{64} + 244 q^{65} - 318 q^{66} + 1113 q^{67} - 8 q^{68} + 207 q^{69} + 126 q^{70} + 53 q^{71} + 216 q^{72} + 360 q^{73} - 88 q^{74} + 216 q^{75} + 252 q^{76} + 371 q^{77} - 366 q^{78} + 134 q^{79} + 144 q^{80} + 243 q^{81} + 342 q^{82} + 1506 q^{83} - 252 q^{84} + 577 q^{85} - 1254 q^{86} - 936 q^{87} + 424 q^{88} + 2381 q^{89} + 162 q^{90} + 427 q^{91} - 276 q^{92} - 660 q^{93} + 706 q^{94} + 945 q^{95} - 288 q^{96} - 88 q^{97} + 294 q^{98} + 477 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 15x - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 10 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 10 \) 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
4.51691
−0.272991
−3.24392
2.00000 −3.00000 4.00000 −9.55073 −6.00000 7.00000 8.00000 9.00000 −19.1015
1.2 2.00000 −3.00000 4.00000 4.81897 −6.00000 7.00000 8.00000 9.00000 9.63795
1.3 2.00000 −3.00000 4.00000 13.7318 −6.00000 7.00000 8.00000 9.00000 27.4635
\(n\): 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.f 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.4.a.f 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} - 9T_{5}^{2} - 111T_{5} + 632 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(966))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{3} \) Copy content Toggle raw display
$3$ \( (T + 3)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 9 T^{2} - 111 T + 632 \) Copy content Toggle raw display
$7$ \( (T - 7)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - 53 T^{2} - 1221 T + 71001 \) Copy content Toggle raw display
$13$ \( T^{3} - 61 T^{2} + 637 T - 1822 \) Copy content Toggle raw display
$17$ \( T^{3} + 2 T^{2} - 1211 T - 16024 \) Copy content Toggle raw display
$19$ \( T^{3} - 63 T^{2} - 773 T + 47899 \) Copy content Toggle raw display
$23$ \( (T + 23)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} - 312 T^{2} + 1317 T + 13186 \) Copy content Toggle raw display
$31$ \( T^{3} - 220 T^{2} - 19003 T + 731490 \) Copy content Toggle raw display
$37$ \( T^{3} + 44 T^{2} - 100239 T - 13366222 \) Copy content Toggle raw display
$41$ \( T^{3} - 171 T^{2} + \cdots + 31352967 \) Copy content Toggle raw display
$43$ \( T^{3} + 627 T^{2} + \cdots - 49544756 \) Copy content Toggle raw display
$47$ \( T^{3} - 353 T^{2} + \cdots + 84406692 \) Copy content Toggle raw display
$53$ \( T^{3} + 504 T^{2} + \cdots - 86011099 \) Copy content Toggle raw display
$59$ \( T^{3} - 538 T^{2} + \cdots + 43304511 \) Copy content Toggle raw display
$61$ \( T^{3} - 256 T^{2} - 225150 T - 6141519 \) Copy content Toggle raw display
$67$ \( T^{3} - 1113 T^{2} + \cdots + 55713572 \) Copy content Toggle raw display
$71$ \( T^{3} - 53 T^{2} - 767299 T - 93489564 \) Copy content Toggle raw display
$73$ \( T^{3} - 360 T^{2} + \cdots + 197977404 \) Copy content Toggle raw display
$79$ \( T^{3} - 134 T^{2} + \cdots - 45775266 \) Copy content Toggle raw display
$83$ \( T^{3} - 1506 T^{2} + \cdots - 107382958 \) Copy content Toggle raw display
$89$ \( T^{3} - 2381 T^{2} + \cdots + 55340850 \) Copy content Toggle raw display
$97$ \( T^{3} + 88 T^{2} - 132487 T - 22908546 \) Copy content Toggle raw display
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