Properties

Label 966.4.a.j
Level $966$
Weight $4$
Character orbit 966.a
Self dual yes
Analytic conductor $56.996$
Analytic rank $1$
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: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - x^{3} - 171x^{2} + 17x + 1050 \) 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.

\(f(q)\) \(=\) \( q - 2 q^{2} + 3 q^{3} + 4 q^{4} + ( - \beta_{2} + 1) q^{5} - 6 q^{6} - 7 q^{7} - 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + 3 q^{3} + 4 q^{4} + ( - \beta_{2} + 1) q^{5} - 6 q^{6} - 7 q^{7} - 8 q^{8} + 9 q^{9} + (2 \beta_{2} - 2) q^{10} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 + 6) q^{11} + 12 q^{12} + (\beta_{3} - 2 \beta_1 - 18) q^{13} + 14 q^{14} + ( - 3 \beta_{2} + 3) q^{15} + 16 q^{16} + (\beta_{3} + 5 \beta_{2} + 10 \beta_1 - 5) q^{17} - 18 q^{18} + (\beta_{3} + \beta_{2} + 5 \beta_1 - 14) q^{19} + ( - 4 \beta_{2} + 4) q^{20} - 21 q^{21} + (2 \beta_{3} - 2 \beta_{2} + 6 \beta_1 - 12) q^{22} - 23 q^{23} - 24 q^{24} + ( - 11 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 51) q^{25} + ( - 2 \beta_{3} + 4 \beta_1 + 36) q^{26} + 27 q^{27} - 28 q^{28} + (11 \beta_{3} + 11 \beta_{2} - 4 \beta_1 - 57) q^{29} + (6 \beta_{2} - 6) q^{30} + (9 \beta_{3} + 11 \beta_{2} + 6 \beta_1 + 19) q^{31} - 32 q^{32} + ( - 3 \beta_{3} + 3 \beta_{2} - 9 \beta_1 + 18) q^{33} + ( - 2 \beta_{3} - 10 \beta_{2} - 20 \beta_1 + 10) q^{34} + (7 \beta_{2} - 7) q^{35} + 36 q^{36} + (3 \beta_{3} + 9 \beta_{2} + 12 \beta_1 - 39) q^{37} + ( - 2 \beta_{3} - 2 \beta_{2} - 10 \beta_1 + 28) q^{38} + (3 \beta_{3} - 6 \beta_1 - 54) q^{39} + (8 \beta_{2} - 8) q^{40} + ( - 13 \beta_{3} - 7 \beta_{2} - 13 \beta_1 - 118) q^{41} + 42 q^{42} + ( - 11 \beta_{3} - 6 \beta_1 - 116) q^{43} + ( - 4 \beta_{3} + 4 \beta_{2} - 12 \beta_1 + 24) q^{44} + ( - 9 \beta_{2} + 9) q^{45} + 46 q^{46} + ( - 8 \beta_{3} + 26 \beta_{2} - 19 \beta_1 - 59) q^{47} + 48 q^{48} + 49 q^{49} + (22 \beta_{3} + 8 \beta_{2} + 4 \beta_1 - 102) q^{50} + (3 \beta_{3} + 15 \beta_{2} + 30 \beta_1 - 15) q^{51} + (4 \beta_{3} - 8 \beta_1 - 72) q^{52} + (28 \beta_{3} + 9 \beta_{2} + 5 \beta_1 - 50) q^{53} - 54 q^{54} + (2 \beta_{3} - 17 \beta_{2} - 18 \beta_1 - 267) q^{55} + 56 q^{56} + (3 \beta_{3} + 3 \beta_{2} + 15 \beta_1 - 42) q^{57} + ( - 22 \beta_{3} - 22 \beta_{2} + 8 \beta_1 + 114) q^{58} + ( - 12 \beta_{3} - 11 \beta_{2} + 37 \beta_1 + 44) q^{59} + ( - 12 \beta_{2} + 12) q^{60} + (5 \beta_{3} + 6 \beta_{2} - 29 \beta_1 - 509) q^{61} + ( - 18 \beta_{3} - 22 \beta_{2} - 12 \beta_1 - 38) q^{62} - 63 q^{63} + 64 q^{64} + ( - 11 \beta_{3} + 27 \beta_{2} + 45) q^{65} + (6 \beta_{3} - 6 \beta_{2} + 18 \beta_1 - 36) q^{66} + ( - 22 \beta_{3} - 27 \beta_{2} - 2 \beta_1 - 93) q^{67} + (4 \beta_{3} + 20 \beta_{2} + 40 \beta_1 - 20) q^{68} - 69 q^{69} + ( - 14 \beta_{2} + 14) q^{70} + (3 \beta_{3} - 6 \beta_{2} + 50 \beta_1 - 406) q^{71} - 72 q^{72} + ( - 15 \beta_{3} - 55 \beta_{2} - 38 \beta_1 - 121) q^{73} + ( - 6 \beta_{3} - 18 \beta_{2} - 24 \beta_1 + 78) q^{74} + ( - 33 \beta_{3} - 12 \beta_{2} - 6 \beta_1 + 153) q^{75} + (4 \beta_{3} + 4 \beta_{2} + 20 \beta_1 - 56) q^{76} + (7 \beta_{3} - 7 \beta_{2} + 21 \beta_1 - 42) q^{77} + ( - 6 \beta_{3} + 12 \beta_1 + 108) q^{78} + ( - \beta_{3} - 33 \beta_{2} + 2 \beta_1 - 545) q^{79} + ( - 16 \beta_{2} + 16) q^{80} + 81 q^{81} + (26 \beta_{3} + 14 \beta_{2} + 26 \beta_1 + 236) q^{82} + (63 \beta_{3} + 5 \beta_{2} + 2 \beta_1 + 245) q^{83} - 84 q^{84} + (92 \beta_{3} + 41 \beta_{2} + 58 \beta_1 - 733) q^{85} + (22 \beta_{3} + 12 \beta_1 + 232) q^{86} + (33 \beta_{3} + 33 \beta_{2} - 12 \beta_1 - 171) q^{87} + (8 \beta_{3} - 8 \beta_{2} + 24 \beta_1 - 48) q^{88} + ( - 61 \beta_{3} - 18 \beta_{2} - 90 \beta_1 + 524) q^{89} + (18 \beta_{2} - 18) q^{90} + ( - 7 \beta_{3} + 14 \beta_1 + 126) q^{91} - 92 q^{92} + (27 \beta_{3} + 33 \beta_{2} + 18 \beta_1 + 57) q^{93} + (16 \beta_{3} - 52 \beta_{2} + 38 \beta_1 + 118) q^{94} + (28 \beta_{3} + 33 \beta_{2} + 30 \beta_1 - 77) q^{95} - 96 q^{96} + ( - 41 \beta_{3} - 75 \beta_{2} - 56 \beta_1 - 199) q^{97} - 98 q^{98} + ( - 9 \beta_{3} + 9 \beta_{2} - 27 \beta_1 + 54) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{2} + 12 q^{3} + 16 q^{4} + 5 q^{5} - 24 q^{6} - 28 q^{7} - 32 q^{8} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{2} + 12 q^{3} + 16 q^{4} + 5 q^{5} - 24 q^{6} - 28 q^{7} - 32 q^{8} + 36 q^{9} - 10 q^{10} + 19 q^{11} + 48 q^{12} - 73 q^{13} + 56 q^{14} + 15 q^{15} + 64 q^{16} - 14 q^{17} - 72 q^{18} - 51 q^{19} + 20 q^{20} - 84 q^{21} - 38 q^{22} - 92 q^{23} - 96 q^{24} + 195 q^{25} + 146 q^{26} + 108 q^{27} - 112 q^{28} - 232 q^{29} - 30 q^{30} + 80 q^{31} - 128 q^{32} + 57 q^{33} + 28 q^{34} - 35 q^{35} + 144 q^{36} - 150 q^{37} + 102 q^{38} - 219 q^{39} - 40 q^{40} - 491 q^{41} + 168 q^{42} - 481 q^{43} + 76 q^{44} + 45 q^{45} + 184 q^{46} - 289 q^{47} + 192 q^{48} + 196 q^{49} - 390 q^{50} - 42 q^{51} - 292 q^{52} - 176 q^{53} - 216 q^{54} - 1067 q^{55} + 224 q^{56} - 153 q^{57} + 464 q^{58} + 212 q^{59} + 60 q^{60} - 2066 q^{61} - 160 q^{62} - 252 q^{63} + 256 q^{64} + 142 q^{65} - 114 q^{66} - 369 q^{67} - 56 q^{68} - 276 q^{69} + 70 q^{70} - 1565 q^{71} - 288 q^{72} - 482 q^{73} + 300 q^{74} + 585 q^{75} - 204 q^{76} - 133 q^{77} + 438 q^{78} - 2146 q^{79} + 80 q^{80} + 324 q^{81} + 982 q^{82} + 1040 q^{83} - 336 q^{84} - 2823 q^{85} + 962 q^{86} - 696 q^{87} - 152 q^{88} + 1963 q^{89} - 90 q^{90} + 511 q^{91} - 368 q^{92} + 240 q^{93} + 578 q^{94} - 283 q^{95} - 384 q^{96} - 818 q^{97} - 392 q^{98} + 171 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 171x^{2} + 17x + 1050 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{3} + 2\nu^{2} + 471\nu + 42 ) / 64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 10\nu^{2} - 189\nu - 910 ) / 64 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 6\beta_{3} + 2\beta_{2} + 3\beta _1 + 84 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{3} - 20\beta_{2} + 159\beta _1 + 70 \) 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
2.55880
−12.3751
13.3080
−2.49169
−2.00000 3.00000 4.00000 −17.9067 −6.00000 −7.00000 −8.00000 9.00000 35.8134
1.2 −2.00000 3.00000 4.00000 −2.20416 −6.00000 −7.00000 −8.00000 9.00000 4.40833
1.3 −2.00000 3.00000 4.00000 7.34899 −6.00000 −7.00000 −8.00000 9.00000 −14.6980
1.4 −2.00000 3.00000 4.00000 17.7619 −6.00000 −7.00000 −8.00000 9.00000 −35.5237
\(n\): 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.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.4.a.j 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 5T_{5}^{3} - 335T_{5}^{2} + 1634T_{5} + 5152 \) 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
$5$ \( T^{4} - 5 T^{3} - 335 T^{2} + \cdots + 5152 \) Copy content Toggle raw display
$7$ \( (T + 7)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 19 T^{3} - 2289 T^{2} + \cdots - 163296 \) Copy content Toggle raw display
$13$ \( T^{4} + 73 T^{3} + 691 T^{2} + \cdots - 657068 \) Copy content Toggle raw display
$17$ \( T^{4} + 14 T^{3} + \cdots + 116955720 \) Copy content Toggle raw display
$19$ \( T^{4} + 51 T^{3} - 3125 T^{2} + \cdots - 441000 \) Copy content Toggle raw display
$23$ \( (T + 23)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 232 T^{3} + \cdots - 316294572 \) Copy content Toggle raw display
$31$ \( T^{4} - 80 T^{3} + \cdots - 226272896 \) Copy content Toggle raw display
$37$ \( T^{4} + 150 T^{3} + \cdots + 250856676 \) Copy content Toggle raw display
$41$ \( T^{4} + 491 T^{3} + \cdots - 146499174 \) Copy content Toggle raw display
$43$ \( T^{4} + 481 T^{3} + \cdots - 224904384 \) Copy content Toggle raw display
$47$ \( T^{4} + 289 T^{3} + \cdots + 15568412832 \) Copy content Toggle raw display
$53$ \( T^{4} + 176 T^{3} + \cdots + 17341218234 \) Copy content Toggle raw display
$59$ \( T^{4} - 212 T^{3} + \cdots - 11404237476 \) Copy content Toggle raw display
$61$ \( T^{4} + 2066 T^{3} + \cdots + 13608223298 \) Copy content Toggle raw display
$67$ \( T^{4} + 369 T^{3} + \cdots - 7790300448 \) Copy content Toggle raw display
$71$ \( T^{4} + 1565 T^{3} + \cdots - 66716380416 \) Copy content Toggle raw display
$73$ \( T^{4} + 482 T^{3} + \cdots - 31813792344 \) Copy content Toggle raw display
$79$ \( T^{4} + 2146 T^{3} + \cdots - 2351442528 \) Copy content Toggle raw display
$83$ \( T^{4} - 1040 T^{3} + \cdots + 260613132640 \) Copy content Toggle raw display
$89$ \( T^{4} - 1963 T^{3} + \cdots - 1189692588420 \) Copy content Toggle raw display
$97$ \( T^{4} + 818 T^{3} + \cdots - 524872461924 \) Copy content Toggle raw display
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