Properties

Label 966.4.a.e
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: 3.3.65101.1
Defining polynomial: \( x^{3} - 28x - 29 \) 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.

\(f(q)\) \(=\) \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} + (\beta_{2} + 2) q^{5} - 6 q^{6} - 7 q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} + (\beta_{2} + 2) q^{5} - 6 q^{6} - 7 q^{7} + 8 q^{8} + 9 q^{9} + (2 \beta_{2} + 4) q^{10} + ( - \beta_{2} - \beta_1 - 19) q^{11} - 12 q^{12} + ( - 2 \beta_{2} - 5 \beta_1 + 9) q^{13} - 14 q^{14} + ( - 3 \beta_{2} - 6) q^{15} + 16 q^{16} + ( - \beta_{2} + 19 \beta_1 + 9) q^{17} + 18 q^{18} + ( - 9 \beta_{2} - \beta_1 - 9) q^{19} + (4 \beta_{2} + 8) q^{20} + 21 q^{21} + ( - 2 \beta_{2} - 2 \beta_1 - 38) q^{22} - 23 q^{23} - 24 q^{24} + ( - 2 \beta_{2} - 19 \beta_1 + 10) q^{25} + ( - 4 \beta_{2} - 10 \beta_1 + 18) q^{26} - 27 q^{27} - 28 q^{28} + (5 \beta_{2} + 37 \beta_1 - 79) q^{29} + ( - 6 \beta_{2} - 12) q^{30} + ( - 7 \beta_{2} - 35 \beta_1 - 31) q^{31} + 32 q^{32} + (3 \beta_{2} + 3 \beta_1 + 57) q^{33} + ( - 2 \beta_{2} + 38 \beta_1 + 18) q^{34} + ( - 7 \beta_{2} - 14) q^{35} + 36 q^{36} + ( - 5 \beta_{2} - \beta_1 - 25) q^{37} + ( - 18 \beta_{2} - 2 \beta_1 - 18) q^{38} + (6 \beta_{2} + 15 \beta_1 - 27) q^{39} + (8 \beta_{2} + 16) q^{40} + (9 \beta_{2} - 35 \beta_1 - 131) q^{41} + 42 q^{42} + (4 \beta_{2} + 19 \beta_1 - 183) q^{43} + ( - 4 \beta_{2} - 4 \beta_1 - 76) q^{44} + (9 \beta_{2} + 18) q^{45} - 46 q^{46} + (14 \beta_{2} + 4 \beta_1 - 118) q^{47} - 48 q^{48} + 49 q^{49} + ( - 4 \beta_{2} - 38 \beta_1 + 20) q^{50} + (3 \beta_{2} - 57 \beta_1 - 27) q^{51} + ( - 8 \beta_{2} - 20 \beta_1 + 36) q^{52} + ( - 5 \beta_{2} - 28 \beta_1 + 6) q^{53} - 54 q^{54} + ( - 13 \beta_{2} + 12 \beta_1 - 160) q^{55} - 56 q^{56} + (27 \beta_{2} + 3 \beta_1 + 27) q^{57} + (10 \beta_{2} + 74 \beta_1 - 158) q^{58} + ( - 3 \beta_{2} + 48 \beta_1 - 288) q^{59} + ( - 12 \beta_{2} - 24) q^{60} + (38 \beta_{2} - 35 \beta_1 - 91) q^{61} + ( - 14 \beta_{2} - 70 \beta_1 - 62) q^{62} - 63 q^{63} + 64 q^{64} + (27 \beta_{2} + 3 \beta_1 - 199) q^{65} + (6 \beta_{2} + 6 \beta_1 + 114) q^{66} + (23 \beta_{2} - 88 \beta_1 - 64) q^{67} + ( - 4 \beta_{2} + 76 \beta_1 + 36) q^{68} + 69 q^{69} + ( - 14 \beta_{2} - 28) q^{70} + (8 \beta_{2} - 23 \beta_1 + 43) q^{71} + 72 q^{72} + (9 \beta_{2} + 187 \beta_1 - 193) q^{73} + ( - 10 \beta_{2} - 2 \beta_1 - 50) q^{74} + (6 \beta_{2} + 57 \beta_1 - 30) q^{75} + ( - 36 \beta_{2} - 4 \beta_1 - 36) q^{76} + (7 \beta_{2} + 7 \beta_1 + 133) q^{77} + (12 \beta_{2} + 30 \beta_1 - 54) q^{78} + ( - 19 \beta_{2} + 59 \beta_1 - 479) q^{79} + (16 \beta_{2} + 32) q^{80} + 81 q^{81} + (18 \beta_{2} - 70 \beta_1 - 262) q^{82} + ( - 29 \beta_{2} - 143 \beta_1 - 127) q^{83} + 84 q^{84} + ( - 25 \beta_{2} + 152 \beta_1 - 284) q^{85} + (8 \beta_{2} + 38 \beta_1 - 366) q^{86} + ( - 15 \beta_{2} - 111 \beta_1 + 237) q^{87} + ( - 8 \beta_{2} - 8 \beta_1 - 152) q^{88} + ( - 30 \beta_{2} + 125 \beta_1 + 415) q^{89} + (18 \beta_{2} + 36) q^{90} + (14 \beta_{2} + 35 \beta_1 - 63) q^{91} - 92 q^{92} + (21 \beta_{2} + 105 \beta_1 + 93) q^{93} + (28 \beta_{2} + 8 \beta_1 - 236) q^{94} + (29 \beta_{2} + 164 \beta_1 - 1188) q^{95} - 96 q^{96} + ( - 79 \beta_{2} - 229 \beta_1 - 227) q^{97} + 98 q^{98} + ( - 9 \beta_{2} - 9 \beta_1 - 171) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 6 q^{2} - 9 q^{3} + 12 q^{4} + 5 q^{5} - 18 q^{6} - 21 q^{7} + 24 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 6 q^{2} - 9 q^{3} + 12 q^{4} + 5 q^{5} - 18 q^{6} - 21 q^{7} + 24 q^{8} + 27 q^{9} + 10 q^{10} - 56 q^{11} - 36 q^{12} + 29 q^{13} - 42 q^{14} - 15 q^{15} + 48 q^{16} + 28 q^{17} + 54 q^{18} - 18 q^{19} + 20 q^{20} + 63 q^{21} - 112 q^{22} - 69 q^{23} - 72 q^{24} + 32 q^{25} + 58 q^{26} - 81 q^{27} - 84 q^{28} - 242 q^{29} - 30 q^{30} - 86 q^{31} + 96 q^{32} + 168 q^{33} + 56 q^{34} - 35 q^{35} + 108 q^{36} - 70 q^{37} - 36 q^{38} - 87 q^{39} + 40 q^{40} - 402 q^{41} + 126 q^{42} - 553 q^{43} - 224 q^{44} + 45 q^{45} - 138 q^{46} - 368 q^{47} - 144 q^{48} + 147 q^{49} + 64 q^{50} - 84 q^{51} + 116 q^{52} + 23 q^{53} - 162 q^{54} - 467 q^{55} - 168 q^{56} + 54 q^{57} - 484 q^{58} - 861 q^{59} - 60 q^{60} - 311 q^{61} - 172 q^{62} - 189 q^{63} + 192 q^{64} - 624 q^{65} + 336 q^{66} - 215 q^{67} + 112 q^{68} + 207 q^{69} - 70 q^{70} + 121 q^{71} + 216 q^{72} - 588 q^{73} - 140 q^{74} - 96 q^{75} - 72 q^{76} + 392 q^{77} - 174 q^{78} - 1418 q^{79} + 80 q^{80} + 243 q^{81} - 804 q^{82} - 352 q^{83} + 252 q^{84} - 827 q^{85} - 1106 q^{86} + 726 q^{87} - 448 q^{88} + 1275 q^{89} + 90 q^{90} - 203 q^{91} - 276 q^{92} + 258 q^{93} - 736 q^{94} - 3593 q^{95} - 288 q^{96} - 602 q^{97} + 294 q^{98} - 504 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 28x - 29 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 19 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 19 \) 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
−1.08080
5.74846
−4.66766
2.00000 −3.00000 4.00000 −13.6703 −6.00000 −7.00000 8.00000 9.00000 −27.3405
1.2 2.00000 −3.00000 4.00000 4.54790 −6.00000 −7.00000 8.00000 9.00000 9.09580
1.3 2.00000 −3.00000 4.00000 14.1224 −6.00000 −7.00000 8.00000 9.00000 28.2447
\(n\): 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.e 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.4.a.e 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} - 5T_{5}^{2} - 191T_{5} + 878 \) 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
$5$ \( T^{3} - 5 T^{2} - 191 T + 878 \) Copy content Toggle raw display
$7$ \( (T + 7)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 56 T^{2} + 843 T + 1624 \) Copy content Toggle raw display
$13$ \( T^{3} - 29 T^{2} - 967 T + 9196 \) Copy content Toggle raw display
$17$ \( T^{3} - 28 T^{2} - 10521 T + 43912 \) Copy content Toggle raw display
$19$ \( T^{3} + 18 T^{2} - 15841 T - 568942 \) Copy content Toggle raw display
$23$ \( (T + 23)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} + 242 T^{2} - 19169 T - 5522006 \) Copy content Toggle raw display
$31$ \( T^{3} + 86 T^{2} - 35477 T + 1384186 \) Copy content Toggle raw display
$37$ \( T^{3} + 70 T^{2} - 3253 T - 191606 \) Copy content Toggle raw display
$41$ \( T^{3} + 402 T^{2} + \cdots - 10246778 \) Copy content Toggle raw display
$43$ \( T^{3} + 553 T^{2} + 90539 T + 3778276 \) Copy content Toggle raw display
$47$ \( T^{3} + 368 T^{2} + 7024 T - 669944 \) Copy content Toggle raw display
$53$ \( T^{3} - 23 T^{2} - 23259 T + 1462334 \) Copy content Toggle raw display
$59$ \( T^{3} + 861 T^{2} + 177201 T + 3166884 \) Copy content Toggle raw display
$61$ \( T^{3} + 311 T^{2} + \cdots - 67548734 \) Copy content Toggle raw display
$67$ \( T^{3} + 215 T^{2} + \cdots - 105321548 \) Copy content Toggle raw display
$71$ \( T^{3} - 121 T^{2} - 27289 T - 978964 \) Copy content Toggle raw display
$73$ \( T^{3} + 588 T^{2} + \cdots - 464169806 \) Copy content Toggle raw display
$79$ \( T^{3} + 1418 T^{2} + \cdots + 45422608 \) Copy content Toggle raw display
$83$ \( T^{3} + 352 T^{2} + \cdots + 93148418 \) Copy content Toggle raw display
$89$ \( T^{3} - 1275 T^{2} + \cdots + 421896500 \) Copy content Toggle raw display
$97$ \( T^{3} + 602 T^{2} + \cdots - 254189276 \) Copy content Toggle raw display
show more
show less