Properties

Label 966.4.a.h
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: \( x^{4} - x^{3} - 356x^{2} + 245x + 16751 \) 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.

\(f(q)\) \(=\) \( q - 2 q^{2} - 3 q^{3} + 4 q^{4} + ( - \beta_1 - 1) q^{5} + 6 q^{6} - 7 q^{7} - 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} - 3 q^{3} + 4 q^{4} + ( - \beta_1 - 1) q^{5} + 6 q^{6} - 7 q^{7} - 8 q^{8} + 9 q^{9} + (2 \beta_1 + 2) q^{10} + (\beta_{2} - 2 \beta_1 - 10) q^{11} - 12 q^{12} + ( - \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 5) q^{13} + 14 q^{14} + (3 \beta_1 + 3) q^{15} + 16 q^{16} + (3 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 6) q^{17} - 18 q^{18} + ( - \beta_{2} + 4) q^{19} + ( - 4 \beta_1 - 4) q^{20} + 21 q^{21} + ( - 2 \beta_{2} + 4 \beta_1 + 20) q^{22} + 23 q^{23} + 24 q^{24} + ( - 7 \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 50) q^{25} + (2 \beta_{3} + 4 \beta_{2} - 6 \beta_1 - 10) q^{26} - 27 q^{27} - 28 q^{28} + (\beta_{3} - 4 \beta_{2} + 8 \beta_1 - 12) q^{29} + ( - 6 \beta_1 - 6) q^{30} + (5 \beta_{3} - 2 \beta_{2} + 10 \beta_1 + 36) q^{31} - 32 q^{32} + ( - 3 \beta_{2} + 6 \beta_1 + 30) q^{33} + ( - 6 \beta_{3} - 4 \beta_{2} + 4 \beta_1 - 12) q^{34} + (7 \beta_1 + 7) q^{35} + 36 q^{36} + (3 \beta_{3} + 8 \beta_{2} + 16 \beta_1 - 40) q^{37} + (2 \beta_{2} - 8) q^{38} + (3 \beta_{3} + 6 \beta_{2} - 9 \beta_1 - 15) q^{39} + (8 \beta_1 + 8) q^{40} + (4 \beta_{3} - \beta_{2} + 4 \beta_1 - 154) q^{41} - 42 q^{42} + (11 \beta_{3} - 14 \beta_{2} + 9 \beta_1 + 53) q^{43} + (4 \beta_{2} - 8 \beta_1 - 40) q^{44} + ( - 9 \beta_1 - 9) q^{45} - 46 q^{46} + ( - 19 \beta_{3} + 17 \beta_{2} + 6 \beta_1 + 152) q^{47} - 48 q^{48} + 49 q^{49} + (14 \beta_{3} - 4 \beta_{2} - 6 \beta_1 - 100) q^{50} + ( - 9 \beta_{3} - 6 \beta_{2} + 6 \beta_1 - 18) q^{51} + ( - 4 \beta_{3} - 8 \beta_{2} + 12 \beta_1 + 20) q^{52} + ( - 13 \beta_{3} - 11 \beta_{2} - 29 \beta_1 + 109) q^{53} + 54 q^{54} + ( - 22 \beta_{3} + 6 \beta_{2} + 13 \beta_1 + 343) q^{55} + 56 q^{56} + (3 \beta_{2} - 12) q^{57} + ( - 2 \beta_{3} + 8 \beta_{2} - 16 \beta_1 + 24) q^{58} + ( - 15 \beta_{3} - 15 \beta_{2} + 3 \beta_1 + 63) q^{59} + (12 \beta_1 + 12) q^{60} + (12 \beta_{3} + 13 \beta_{2} - \beta_1 + 71) q^{61} + ( - 10 \beta_{3} + 4 \beta_{2} - 20 \beta_1 - 72) q^{62} - 63 q^{63} + 64 q^{64} + (41 \beta_{3} - 22 \beta_1 - 472) q^{65} + (6 \beta_{2} - 12 \beta_1 - 60) q^{66} + (36 \beta_{3} - 22 \beta_{2} - 25 \beta_1 + 13) q^{67} + (12 \beta_{3} + 8 \beta_{2} - 8 \beta_1 + 24) q^{68} - 69 q^{69} + ( - 14 \beta_1 - 14) q^{70} + ( - 15 \beta_{3} + 38 \beta_{2} + 19 \beta_1 + 131) q^{71} - 72 q^{72} + (5 \beta_{3} + 46 \beta_{2} - 16 \beta_1 - 64) q^{73} + ( - 6 \beta_{3} - 16 \beta_{2} - 32 \beta_1 + 80) q^{74} + (21 \beta_{3} - 6 \beta_{2} - 9 \beta_1 - 150) q^{75} + ( - 4 \beta_{2} + 16) q^{76} + ( - 7 \beta_{2} + 14 \beta_1 + 70) q^{77} + ( - 6 \beta_{3} - 12 \beta_{2} + 18 \beta_1 + 30) q^{78} + (17 \beta_{3} - 10 \beta_{2} + 10 \beta_1 + 148) q^{79} + ( - 16 \beta_1 - 16) q^{80} + 81 q^{81} + ( - 8 \beta_{3} + 2 \beta_{2} - 8 \beta_1 + 308) q^{82} + ( - 7 \beta_{3} - 46 \beta_{2} + 28 \beta_1 + 118) q^{83} + 84 q^{84} + ( - 42 \beta_{3} - 22 \beta_{2} + 35 \beta_1 + 237) q^{85} + ( - 22 \beta_{3} + 28 \beta_{2} - 18 \beta_1 - 106) q^{86} + ( - 3 \beta_{3} + 12 \beta_{2} - 24 \beta_1 + 36) q^{87} + ( - 8 \beta_{2} + 16 \beta_1 + 80) q^{88} + ( - 3 \beta_{3} - 50 \beta_{2} + 33 \beta_1 - 377) q^{89} + (18 \beta_1 + 18) q^{90} + (7 \beta_{3} + 14 \beta_{2} - 21 \beta_1 - 35) q^{91} + 92 q^{92} + ( - 15 \beta_{3} + 6 \beta_{2} - 30 \beta_1 - 108) q^{93} + (38 \beta_{3} - 34 \beta_{2} - 12 \beta_1 - 304) q^{94} + (8 \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 11) q^{95} + 96 q^{96} + (23 \beta_{3} + 32 \beta_{2} - 50 \beta_1 + 166) q^{97} - 98 q^{98} + (9 \beta_{2} - 18 \beta_1 - 90) 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 \( 4 q - 8 q^{2} - 12 q^{3} + 16 q^{4} - 5 q^{5} + 24 q^{6} - 28 q^{7} - 32 q^{8} + 36 q^{9} + 10 q^{10} - 41 q^{11} - 48 q^{12} + 23 q^{13} + 56 q^{14} + 15 q^{15} + 64 q^{16} + 18 q^{17} - 72 q^{18} + 15 q^{19} - 20 q^{20} + 84 q^{21} + 82 q^{22} + 92 q^{23} + 96 q^{24} + 219 q^{25} - 46 q^{26} - 108 q^{27} - 112 q^{28} - 46 q^{29} - 30 q^{30} + 142 q^{31} - 128 q^{32} + 123 q^{33} - 36 q^{34} + 35 q^{35} + 144 q^{36} - 142 q^{37} - 30 q^{38} - 69 q^{39} + 40 q^{40} - 621 q^{41} - 168 q^{42} + 185 q^{43} - 164 q^{44} - 45 q^{45} - 184 q^{46} + 669 q^{47} - 192 q^{48} + 196 q^{49} - 438 q^{50} - 54 q^{51} + 92 q^{52} + 422 q^{53} + 216 q^{54} + 1435 q^{55} + 224 q^{56} - 45 q^{57} + 92 q^{58} + 270 q^{59} + 60 q^{60} + 272 q^{61} - 284 q^{62} - 252 q^{63} + 256 q^{64} - 1992 q^{65} - 246 q^{66} - 67 q^{67} + 72 q^{68} - 276 q^{69} - 70 q^{70} + 611 q^{71} - 288 q^{72} - 236 q^{73} + 284 q^{74} - 657 q^{75} + 60 q^{76} + 287 q^{77} + 138 q^{78} + 558 q^{79} - 80 q^{80} + 324 q^{81} + 1242 q^{82} + 468 q^{83} + 336 q^{84} + 1045 q^{85} - 370 q^{86} + 138 q^{87} + 328 q^{88} - 1519 q^{89} + 90 q^{90} - 161 q^{91} + 368 q^{92} - 426 q^{93} - 1338 q^{94} + 23 q^{95} + 384 q^{96} + 600 q^{97} - 392 q^{98} - 369 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 356x^{2} + 245x + 16751 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -7\nu^{3} + 12\nu^{2} + 1864\nu - 1885 ) / 542 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} - 37\nu^{2} + 305\nu + 6467 ) / 271 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -7\beta_{3} + 2\beta_{2} + \beta _1 + 174 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -12\beta_{3} - 74\beta_{2} + 268\beta _1 + 29 \) 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
17.4461
7.86398
−7.09048
−17.2196
−2.00000 −3.00000 4.00000 −18.4461 6.00000 −7.00000 −8.00000 9.00000 36.8923
1.2 −2.00000 −3.00000 4.00000 −8.86398 6.00000 −7.00000 −8.00000 9.00000 17.7280
1.3 −2.00000 −3.00000 4.00000 6.09048 6.00000 −7.00000 −8.00000 9.00000 −12.1810
1.4 −2.00000 −3.00000 4.00000 16.2196 6.00000 −7.00000 −8.00000 9.00000 −32.4393
\(n\): 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.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.4.a.h 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 5T_{5}^{3} - 347T_{5}^{2} - 950T_{5} + 16152 \) 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
$5$ \( T^{4} + 5 T^{3} - 347 T^{2} + \cdots + 16152 \) Copy content Toggle raw display
$7$ \( (T + 7)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 41 T^{3} - 1193 T^{2} + \cdots - 218500 \) Copy content Toggle raw display
$13$ \( T^{4} - 23 T^{3} - 5251 T^{2} + \cdots + 2320500 \) Copy content Toggle raw display
$17$ \( T^{4} - 18 T^{3} - 8689 T^{2} + \cdots - 519800 \) Copy content Toggle raw display
$19$ \( T^{4} - 15 T^{3} - 397 T^{2} + \cdots + 20748 \) Copy content Toggle raw display
$23$ \( (T - 23)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 46 T^{3} - 29255 T^{2} + \cdots + 1007820 \) Copy content Toggle raw display
$31$ \( T^{4} - 142 T^{3} + \cdots - 371797776 \) Copy content Toggle raw display
$37$ \( T^{4} + 142 T^{3} + \cdots + 3755562180 \) Copy content Toggle raw display
$41$ \( T^{4} + 621 T^{3} + \cdots + 274699398 \) Copy content Toggle raw display
$43$ \( T^{4} - 185 T^{3} + \cdots - 1753530768 \) Copy content Toggle raw display
$47$ \( T^{4} - 669 T^{3} + \cdots - 11237142080 \) Copy content Toggle raw display
$53$ \( T^{4} - 422 T^{3} + \cdots + 20598425582 \) Copy content Toggle raw display
$59$ \( T^{4} - 270 T^{3} + \cdots - 6491744028 \) Copy content Toggle raw display
$61$ \( T^{4} - 272 T^{3} + \cdots + 110431466 \) Copy content Toggle raw display
$67$ \( T^{4} + 67 T^{3} + \cdots + 82920565088 \) Copy content Toggle raw display
$71$ \( T^{4} - 611 T^{3} + \cdots - 148669378176 \) Copy content Toggle raw display
$73$ \( T^{4} + 236 T^{3} + \cdots + 279450763592 \) Copy content Toggle raw display
$79$ \( T^{4} - 558 T^{3} + \cdots - 6264993920 \) Copy content Toggle raw display
$83$ \( T^{4} - 468 T^{3} + \cdots + 397443049096 \) Copy content Toggle raw display
$89$ \( T^{4} + 1519 T^{3} + \cdots + 372666895804 \) Copy content Toggle raw display
$97$ \( T^{4} - 600 T^{3} + \cdots - 53326747724 \) Copy content Toggle raw display
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