Properties

Label 966.4.a.g
Level $966$
Weight $4$
Character orbit 966.a
Self dual yes
Analytic conductor $56.996$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.2981.1
Defining polynomial: \( x^{3} - x^{2} - 11x + 14 \) 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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + 3 q^{3} + 4 q^{4} + (\beta_{2} - 3 \beta_1 - 4) q^{5} + 6 q^{6} - 7 q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 3 q^{3} + 4 q^{4} + (\beta_{2} - 3 \beta_1 - 4) q^{5} + 6 q^{6} - 7 q^{7} + 8 q^{8} + 9 q^{9} + (2 \beta_{2} - 6 \beta_1 - 8) q^{10} + ( - 5 \beta_{2} + 8 \beta_1 - 24) q^{11} + 12 q^{12} + (4 \beta_{2} + 19 \beta_1 - 8) q^{13} - 14 q^{14} + (3 \beta_{2} - 9 \beta_1 - 12) q^{15} + 16 q^{16} + ( - 15 \beta_{2} + 2 \beta_1 - 44) q^{17} + 18 q^{18} + (\beta_{2} - 12 \beta_1 - 24) q^{19} + (4 \beta_{2} - 12 \beta_1 - 16) q^{20} - 21 q^{21} + ( - 10 \beta_{2} + 16 \beta_1 - 48) q^{22} + 23 q^{23} + 24 q^{24} + ( - 12 \beta_{2} + 13 \beta_1 - 35) q^{25} + (8 \beta_{2} + 38 \beta_1 - 16) q^{26} + 27 q^{27} - 28 q^{28} + (29 \beta_{2} + 4 \beta_1 - 104) q^{29} + (6 \beta_{2} - 18 \beta_1 - 24) q^{30} + ( - 13 \beta_{2} - 12 \beta_1 - 62) q^{31} + 32 q^{32} + ( - 15 \beta_{2} + 24 \beta_1 - 72) q^{33} + ( - 30 \beta_{2} + 4 \beta_1 - 88) q^{34} + ( - 7 \beta_{2} + 21 \beta_1 + 28) q^{35} + 36 q^{36} + ( - 27 \beta_{2} + 6 \beta_1 - 234) q^{37} + (2 \beta_{2} - 24 \beta_1 - 48) q^{38} + (12 \beta_{2} + 57 \beta_1 - 24) q^{39} + (8 \beta_{2} - 24 \beta_1 - 32) q^{40} + (71 \beta_{2} + 82 \beta_1 - 148) q^{41} - 42 q^{42} + (8 \beta_{2} - 61 \beta_1 - 152) q^{43} + ( - 20 \beta_{2} + 32 \beta_1 - 96) q^{44} + (9 \beta_{2} - 27 \beta_1 - 36) q^{45} + 46 q^{46} + ( - 68 \beta_{2} - 18 \beta_1 + 84) q^{47} + 48 q^{48} + 49 q^{49} + ( - 24 \beta_{2} + 26 \beta_1 - 70) q^{50} + ( - 45 \beta_{2} + 6 \beta_1 - 132) q^{51} + (16 \beta_{2} + 76 \beta_1 - 32) q^{52} + (87 \beta_{2} - 39 \beta_1 - 12) q^{53} + 54 q^{54} + (23 \beta_{2} + 67 \beta_1 - 120) q^{55} - 56 q^{56} + (3 \beta_{2} - 36 \beta_1 - 72) q^{57} + (58 \beta_{2} + 8 \beta_1 - 208) q^{58} + (141 \beta_{2} - 113 \beta_1 + 84) q^{59} + (12 \beta_{2} - 36 \beta_1 - 48) q^{60} + ( - 156 \beta_{2} - 9 \beta_1 - 46) q^{61} + ( - 26 \beta_{2} - 24 \beta_1 - 124) q^{62} - 63 q^{63} + 64 q^{64} + ( - 71 \beta_{2} + 28 \beta_1 - 354) q^{65} + ( - 30 \beta_{2} + 48 \beta_1 - 144) q^{66} + (99 \beta_{2} - 129 \beta_1 - 50) q^{67} + ( - 60 \beta_{2} + 8 \beta_1 - 176) q^{68} + 69 q^{69} + ( - 14 \beta_{2} + 42 \beta_1 + 56) q^{70} + ( - 28 \beta_{2} - 145 \beta_1 - 106) q^{71} + 72 q^{72} + (13 \beta_{2} - 190 \beta_1 + 120) q^{73} + ( - 54 \beta_{2} + 12 \beta_1 - 468) q^{74} + ( - 36 \beta_{2} + 39 \beta_1 - 105) q^{75} + (4 \beta_{2} - 48 \beta_1 - 96) q^{76} + (35 \beta_{2} - 56 \beta_1 + 168) q^{77} + (24 \beta_{2} + 114 \beta_1 - 48) q^{78} + (65 \beta_{2} - 268 \beta_1 - 546) q^{79} + (16 \beta_{2} - 48 \beta_1 - 64) q^{80} + 81 q^{81} + (142 \beta_{2} + 164 \beta_1 - 296) q^{82} + ( - 237 \beta_{2} + 92 \beta_1 - 376) q^{83} - 84 q^{84} + (119 \beta_{2} + 117 \beta_1 + 12) q^{85} + (16 \beta_{2} - 122 \beta_1 - 304) q^{86} + (87 \beta_{2} + 12 \beta_1 - 312) q^{87} + ( - 40 \beta_{2} + 64 \beta_1 - 192) q^{88} + (134 \beta_{2} - 133 \beta_1 - 104) q^{89} + (18 \beta_{2} - 54 \beta_1 - 72) q^{90} + ( - 28 \beta_{2} - 133 \beta_1 + 56) q^{91} + 92 q^{92} + ( - 39 \beta_{2} - 36 \beta_1 - 186) q^{93} + ( - 136 \beta_{2} - 36 \beta_1 + 168) q^{94} + ( - 23 \beta_{2} + 73 \beta_1 + 368) q^{95} + 96 q^{96} + (97 \beta_{2} + 22 \beta_1 + 144) q^{97} + 98 q^{98} + ( - 45 \beta_{2} + 72 \beta_1 - 216) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 6 q^{2} + 9 q^{3} + 12 q^{4} - 15 q^{5} + 18 q^{6} - 21 q^{7} + 24 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 6 q^{2} + 9 q^{3} + 12 q^{4} - 15 q^{5} + 18 q^{6} - 21 q^{7} + 24 q^{8} + 27 q^{9} - 30 q^{10} - 64 q^{11} + 36 q^{12} - 5 q^{13} - 42 q^{14} - 45 q^{15} + 48 q^{16} - 130 q^{17} + 54 q^{18} - 84 q^{19} - 60 q^{20} - 63 q^{21} - 128 q^{22} + 69 q^{23} + 72 q^{24} - 92 q^{25} - 10 q^{26} + 81 q^{27} - 84 q^{28} - 308 q^{29} - 90 q^{30} - 198 q^{31} + 96 q^{32} - 192 q^{33} - 260 q^{34} + 105 q^{35} + 108 q^{36} - 696 q^{37} - 168 q^{38} - 15 q^{39} - 120 q^{40} - 362 q^{41} - 126 q^{42} - 517 q^{43} - 256 q^{44} - 135 q^{45} + 138 q^{46} + 234 q^{47} + 144 q^{48} + 147 q^{49} - 184 q^{50} - 390 q^{51} - 20 q^{52} - 75 q^{53} + 162 q^{54} - 293 q^{55} - 168 q^{56} - 252 q^{57} - 616 q^{58} + 139 q^{59} - 180 q^{60} - 147 q^{61} - 396 q^{62} - 189 q^{63} + 192 q^{64} - 1034 q^{65} - 384 q^{66} - 279 q^{67} - 520 q^{68} + 207 q^{69} + 210 q^{70} - 463 q^{71} + 216 q^{72} + 170 q^{73} - 1392 q^{74} - 276 q^{75} - 336 q^{76} + 448 q^{77} - 30 q^{78} - 1906 q^{79} - 240 q^{80} + 243 q^{81} - 724 q^{82} - 1036 q^{83} - 252 q^{84} + 153 q^{85} - 1034 q^{86} - 924 q^{87} - 512 q^{88} - 445 q^{89} - 270 q^{90} + 35 q^{91} + 276 q^{92} - 594 q^{93} + 468 q^{94} + 1177 q^{95} + 288 q^{96} + 454 q^{97} + 294 q^{98} - 576 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 11x + 14 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + \nu - 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - \beta _1 + 8 \) 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
1.32447
3.09300
−3.41747
2.00000 3.00000 4.00000 −12.8947 6.00000 −7.00000 8.00000 9.00000 −25.7894
1.2 2.00000 3.00000 4.00000 −8.61935 6.00000 −7.00000 8.00000 9.00000 −17.2387
1.3 2.00000 3.00000 4.00000 6.51407 6.00000 −7.00000 8.00000 9.00000 13.0281
\(n\): 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.g 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.4.a.g 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} + 15T_{5}^{2} - 29T_{5} - 724 \) 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
$5$ \( T^{3} + 15 T^{2} - 29 T - 724 \) Copy content Toggle raw display
$7$ \( (T + 7)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 64 T^{2} + 345 T - 13302 \) Copy content Toggle raw display
$13$ \( T^{3} + 5 T^{2} - 4983 T - 12574 \) Copy content Toggle raw display
$17$ \( T^{3} + 130 T^{2} + 623 T - 191502 \) Copy content Toggle raw display
$19$ \( T^{3} + 84 T^{2} + 781 T - 43698 \) Copy content Toggle raw display
$23$ \( (T - 23)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} + 308 T^{2} + 11285 T - 1156106 \) Copy content Toggle raw display
$31$ \( T^{3} + 198 T^{2} + 6457 T + 54208 \) Copy content Toggle raw display
$37$ \( T^{3} + 696 T^{2} + 145431 T + 8317188 \) Copy content Toggle raw display
$41$ \( T^{3} + 362 T^{2} + \cdots - 69517986 \) Copy content Toggle raw display
$43$ \( T^{3} + 517 T^{2} + 48869 T - 4835406 \) Copy content Toggle raw display
$47$ \( T^{3} - 234 T^{2} + \cdots + 14549376 \) Copy content Toggle raw display
$53$ \( T^{3} + 75 T^{2} - 165699 T + 19323144 \) Copy content Toggle raw display
$59$ \( T^{3} - 139 T^{2} + \cdots + 150787736 \) Copy content Toggle raw display
$61$ \( T^{3} + 147 T^{2} + \cdots - 31862708 \) Copy content Toggle raw display
$67$ \( T^{3} + 279 T^{2} - 298677 T + 3632114 \) Copy content Toggle raw display
$71$ \( T^{3} + 463 T^{2} + \cdots - 41953208 \) Copy content Toggle raw display
$73$ \( T^{3} - 170 T^{2} + \cdots - 61538418 \) Copy content Toggle raw display
$79$ \( T^{3} + 1906 T^{2} + \cdots - 506361848 \) Copy content Toggle raw display
$83$ \( T^{3} + 1036 T^{2} + \cdots - 820731062 \) Copy content Toggle raw display
$89$ \( T^{3} + 445 T^{2} + \cdots + 39499362 \) Copy content Toggle raw display
$97$ \( T^{3} - 454 T^{2} + \cdots + 19029086 \) Copy content Toggle raw display
show more
show less