Properties

Label 966.4.a.k
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: 4.4.9814581.1
Defining polynomial: \( x^{4} - x^{3} - 66x^{2} + 271x - 241 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} + ( - \beta_{2} - \beta_1 - 3) q^{5} - 6 q^{6} + 7 q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} - 3 q^{3} + 4 q^{4} + ( - \beta_{2} - \beta_1 - 3) q^{5} - 6 q^{6} + 7 q^{7} + 8 q^{8} + 9 q^{9} + ( - 2 \beta_{2} - 2 \beta_1 - 6) q^{10} + (5 \beta_{2} - 4 \beta_1 - 8) q^{11} - 12 q^{12} + (3 \beta_{3} - 5 \beta_{2} + 8 \beta_1 - 7) q^{13} + 14 q^{14} + (3 \beta_{2} + 3 \beta_1 + 9) q^{15} + 16 q^{16} + ( - 6 \beta_{3} + \beta_{2} + 8 \beta_1 - 46) q^{17} + 18 q^{18} + ( - 3 \beta_{3} + 6 \beta_{2} + 7 \beta_1 - 40) q^{19} + ( - 4 \beta_{2} - 4 \beta_1 - 12) q^{20} - 21 q^{21} + (10 \beta_{2} - 8 \beta_1 - 16) q^{22} + 23 q^{23} - 24 q^{24} + (2 \beta_{3} + 24 \beta_{2} + \beta_1 - 22) q^{25} + (6 \beta_{3} - 10 \beta_{2} + 16 \beta_1 - 14) q^{26} - 27 q^{27} + 28 q^{28} + ( - 3 \beta_{3} + 12 \beta_{2} - 15 \beta_1 + 10) q^{29} + (6 \beta_{2} + 6 \beta_1 + 18) q^{30} + (10 \beta_{3} - 7 \beta_{2} - 56) q^{31} + 32 q^{32} + ( - 15 \beta_{2} + 12 \beta_1 + 24) q^{33} + ( - 12 \beta_{3} + 2 \beta_{2} + 16 \beta_1 - 92) q^{34} + ( - 7 \beta_{2} - 7 \beta_1 - 21) q^{35} + 36 q^{36} + (4 \beta_{3} - 45 \beta_{2} + 24 \beta_1 + 74) q^{37} + ( - 6 \beta_{3} + 12 \beta_{2} + 14 \beta_1 - 80) q^{38} + ( - 9 \beta_{3} + 15 \beta_{2} - 24 \beta_1 + 21) q^{39} + ( - 8 \beta_{2} - 8 \beta_1 - 24) q^{40} + ( - 9 \beta_{3} - 24 \beta_{2} + 21 \beta_1 - 64) q^{41} - 42 q^{42} + (15 \beta_{3} + 3 \beta_{2} + 10 \beta_1 + 67) q^{43} + (20 \beta_{2} - 16 \beta_1 - 32) q^{44} + ( - 9 \beta_{2} - 9 \beta_1 - 27) q^{45} + 46 q^{46} + (17 \beta_{3} + 31 \beta_{2} - 21 \beta_1 - 108) q^{47} - 48 q^{48} + 49 q^{49} + (4 \beta_{3} + 48 \beta_{2} + 2 \beta_1 - 44) q^{50} + (18 \beta_{3} - 3 \beta_{2} - 24 \beta_1 + 138) q^{51} + (12 \beta_{3} - 20 \beta_{2} + 32 \beta_1 - 28) q^{52} + (4 \beta_{3} + 21 \beta_{2} - 69 \beta_1 - 203) q^{53} - 54 q^{54} + ( - \beta_{3} - 34 \beta_{2} - 18 \beta_1 - 41) q^{55} + 56 q^{56} + (9 \beta_{3} - 18 \beta_{2} - 21 \beta_1 + 120) q^{57} + ( - 6 \beta_{3} + 24 \beta_{2} - 30 \beta_1 + 20) q^{58} + ( - 3 \beta_{3} + 44 \beta_{2} - 52 \beta_1 - 173) q^{59} + (12 \beta_{2} + 12 \beta_1 + 36) q^{60} + ( - 15 \beta_{3} - 3 \beta_{2} - 243) q^{61} + (20 \beta_{3} - 14 \beta_{2} - 112) q^{62} + 63 q^{63} + 64 q^{64} + ( - 9 \beta_{3} + 43 \beta_1 - 160) q^{65} + ( - 30 \beta_{2} + 24 \beta_1 + 48) q^{66} + ( - 42 \beta_{3} + 53 \beta_{2} + 19 \beta_1 - 239) q^{67} + ( - 24 \beta_{3} + 4 \beta_{2} + 32 \beta_1 - 184) q^{68} - 69 q^{69} + ( - 14 \beta_{2} - 14 \beta_1 - 42) q^{70} + ( - 40 \beta_{3} - 8 \beta_{2} - 5 \beta_1 - 429) q^{71} + 72 q^{72} + (17 \beta_{3} - 28 \beta_{2} - 7 \beta_1 - 430) q^{73} + (8 \beta_{3} - 90 \beta_{2} + 48 \beta_1 + 148) q^{74} + ( - 6 \beta_{3} - 72 \beta_{2} - 3 \beta_1 + 66) q^{75} + ( - 12 \beta_{3} + 24 \beta_{2} + 28 \beta_1 - 160) q^{76} + (35 \beta_{2} - 28 \beta_1 - 56) q^{77} + ( - 18 \beta_{3} + 30 \beta_{2} - 48 \beta_1 + 42) q^{78} + (11 \beta_{3} - 16 \beta_{2} + 5 \beta_1 + 78) q^{79} + ( - 16 \beta_{2} - 16 \beta_1 - 48) q^{80} + 81 q^{81} + ( - 18 \beta_{3} - 48 \beta_{2} + 42 \beta_1 - 128) q^{82} + ( - 73 \beta_{3} - 14 \beta_{2} - 29 \beta_1 - 378) q^{83} - 84 q^{84} + (3 \beta_{3} + 18 \beta_{2} + 88 \beta_1 - 139) q^{85} + (30 \beta_{3} + 6 \beta_{2} + 20 \beta_1 + 134) q^{86} + (9 \beta_{3} - 36 \beta_{2} + 45 \beta_1 - 30) q^{87} + (40 \beta_{2} - 32 \beta_1 - 64) q^{88} + (71 \beta_{3} - 89 \beta_{2} - 16 \beta_1 - 391) q^{89} + ( - 18 \beta_{2} - 18 \beta_1 - 54) q^{90} + (21 \beta_{3} - 35 \beta_{2} + 56 \beta_1 - 49) q^{91} + 92 q^{92} + ( - 30 \beta_{3} + 21 \beta_{2} + 168) q^{93} + (34 \beta_{3} + 62 \beta_{2} - 42 \beta_1 - 216) q^{94} + ( - 7 \beta_{3} - 72 \beta_{2} + 62 \beta_1 - 423) q^{95} - 96 q^{96} + ( - 34 \beta_{3} - 3 \beta_{2} - 146 \beta_1 - 448) q^{97} + 98 q^{98} + (45 \beta_{2} - 36 \beta_1 - 72) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{2} - 12 q^{3} + 16 q^{4} - 15 q^{5} - 24 q^{6} + 28 q^{7} + 32 q^{8} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{2} - 12 q^{3} + 16 q^{4} - 15 q^{5} - 24 q^{6} + 28 q^{7} + 32 q^{8} + 36 q^{9} - 30 q^{10} - 26 q^{11} - 48 q^{12} - 33 q^{13} + 56 q^{14} + 45 q^{15} + 64 q^{16} - 168 q^{17} + 72 q^{18} - 138 q^{19} - 60 q^{20} - 84 q^{21} - 52 q^{22} + 92 q^{23} - 96 q^{24} - 41 q^{25} - 66 q^{26} - 108 q^{27} + 112 q^{28} + 52 q^{29} + 90 q^{30} - 248 q^{31} + 128 q^{32} + 78 q^{33} - 336 q^{34} - 105 q^{35} + 144 q^{36} + 226 q^{37} - 276 q^{38} + 99 q^{39} - 120 q^{40} - 274 q^{41} - 168 q^{42} + 269 q^{43} - 104 q^{44} - 135 q^{45} + 184 q^{46} - 408 q^{47} - 192 q^{48} + 196 q^{49} - 82 q^{50} + 504 q^{51} - 132 q^{52} - 843 q^{53} - 216 q^{54} - 249 q^{55} + 224 q^{56} + 414 q^{57} + 104 q^{58} - 653 q^{59} + 180 q^{60} - 963 q^{61} - 496 q^{62} + 252 q^{63} + 256 q^{64} - 588 q^{65} + 156 q^{66} - 789 q^{67} - 672 q^{68} - 276 q^{69} - 210 q^{70} - 1697 q^{71} + 288 q^{72} - 1800 q^{73} + 452 q^{74} + 123 q^{75} - 552 q^{76} - 182 q^{77} + 198 q^{78} + 274 q^{79} - 240 q^{80} + 324 q^{81} - 548 q^{82} - 1496 q^{83} - 336 q^{84} - 435 q^{85} + 538 q^{86} - 156 q^{87} - 208 q^{88} - 1829 q^{89} - 270 q^{90} - 231 q^{91} + 368 q^{92} + 744 q^{93} - 816 q^{94} - 1767 q^{95} - 384 q^{96} - 1910 q^{97} + 392 q^{98} - 234 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 66x^{2} + 271x - 241 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 4\nu^{2} - 46\nu + 35 ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} - \nu^{2} + 64\nu - 137 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 2\beta_{2} - 6\beta _1 + 34 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{3} - 2\beta_{2} + 70\beta _1 - 171 \) 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
5.48574
1.29595
3.60967
−9.39135
2.00000 −3.00000 4.00000 −19.8380 −6.00000 7.00000 8.00000 9.00000 −39.6760
1.2 2.00000 −3.00000 4.00000 −1.67610 −6.00000 7.00000 8.00000 9.00000 −3.35220
1.3 2.00000 −3.00000 4.00000 −1.29416 −6.00000 7.00000 8.00000 9.00000 −2.58831
1.4 2.00000 −3.00000 4.00000 7.80827 −6.00000 7.00000 8.00000 9.00000 15.6165
\(n\): 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.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.4.a.k 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 15T_{5}^{3} - 117T_{5}^{2} - 434T_{5} - 336 \) 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
$5$ \( T^{4} + 15 T^{3} - 117 T^{2} + \cdots - 336 \) Copy content Toggle raw display
$7$ \( (T - 7)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 26 T^{3} - 1821 T^{2} + \cdots + 776712 \) Copy content Toggle raw display
$13$ \( T^{4} + 33 T^{3} - 6375 T^{2} + \cdots - 240792 \) Copy content Toggle raw display
$17$ \( T^{4} + 168 T^{3} + \cdots - 25371672 \) Copy content Toggle raw display
$19$ \( T^{4} + 138 T^{3} - 1683 T^{2} + \cdots + 5026572 \) Copy content Toggle raw display
$23$ \( (T - 23)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} - 52 T^{3} - 18561 T^{2} + \cdots - 14350100 \) Copy content Toggle raw display
$31$ \( T^{4} + 248 T^{3} + \cdots - 61915040 \) Copy content Toggle raw display
$37$ \( T^{4} - 226 T^{3} + \cdots + 1542318928 \) Copy content Toggle raw display
$41$ \( T^{4} + 274 T^{3} + \cdots + 486223156 \) Copy content Toggle raw display
$43$ \( T^{4} - 269 T^{3} + \cdots + 292721328 \) Copy content Toggle raw display
$47$ \( T^{4} + 408 T^{3} + \cdots + 1141661952 \) Copy content Toggle raw display
$53$ \( T^{4} + 843 T^{3} + \cdots - 29253184632 \) Copy content Toggle raw display
$59$ \( T^{4} + 653 T^{3} + \cdots + 1048823328 \) Copy content Toggle raw display
$61$ \( T^{4} + 963 T^{3} + \cdots - 1998848016 \) Copy content Toggle raw display
$67$ \( T^{4} + 789 T^{3} + \cdots + 44722780272 \) Copy content Toggle raw display
$71$ \( T^{4} + 1697 T^{3} + \cdots - 97565087680 \) Copy content Toggle raw display
$73$ \( T^{4} + 1800 T^{3} + \cdots + 16815427644 \) Copy content Toggle raw display
$79$ \( T^{4} - 274 T^{3} + \cdots + 11053248 \) Copy content Toggle raw display
$83$ \( T^{4} + 1496 T^{3} + \cdots - 199551125836 \) Copy content Toggle raw display
$89$ \( T^{4} + 1829 T^{3} + \cdots - 88345748592 \) Copy content Toggle raw display
$97$ \( T^{4} + 1910 T^{3} + \cdots + 49171636784 \) Copy content Toggle raw display
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