Properties

Label 966.4.a.q
Level $966$
Weight $4$
Character orbit 966.a
Self dual yes
Analytic conductor $56.996$
Analytic rank $0$
Dimension $5$
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: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \( x^{5} - 570x^{3} - 189x^{2} + 63838x + 254320 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + 3 q^{3} + 4 q^{4} + ( - \beta_1 - 2) q^{5} + 6 q^{6} - 7 q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 3 q^{3} + 4 q^{4} + ( - \beta_1 - 2) q^{5} + 6 q^{6} - 7 q^{7} + 8 q^{8} + 9 q^{9} + ( - 2 \beta_1 - 4) q^{10} + ( - \beta_{4} - \beta_{3} + \beta_1 + 9) q^{11} + 12 q^{12} + (\beta_{3} - 2 \beta_1 + 15) q^{13} - 14 q^{14} + ( - 3 \beta_1 - 6) q^{15} + 16 q^{16} + (\beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{17} + 18 q^{18} + (\beta_{4} - \beta_{3} - \beta_1 + 43) q^{19} + ( - 4 \beta_1 - 8) q^{20} - 21 q^{21} + ( - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_1 + 18) q^{22} - 23 q^{23} + 24 q^{24} + (2 \beta_{4} - \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 108) q^{25} + (2 \beta_{3} - 4 \beta_1 + 30) q^{26} + 27 q^{27} - 28 q^{28} + (\beta_{3} + \beta_{2} + 5 \beta_1 + 55) q^{29} + ( - 6 \beta_1 - 12) q^{30} + (\beta_{3} + 4 \beta_{2} + 5 \beta_1 + 133) q^{31} + 32 q^{32} + ( - 3 \beta_{4} - 3 \beta_{3} + 3 \beta_1 + 27) q^{33} + (2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 2) q^{34} + (7 \beta_1 + 14) q^{35} + 36 q^{36} + (4 \beta_{4} + 5 \beta_{3} - 3 \beta_{2} - 5 \beta_1 - 13) q^{37} + (2 \beta_{4} - 2 \beta_{3} - 2 \beta_1 + 86) q^{38} + (3 \beta_{3} - 6 \beta_1 + 45) q^{39} + ( - 8 \beta_1 - 16) q^{40} + (\beta_{4} - 5 \beta_{3} + \beta_{2} - \beta_1 + 85) q^{41} - 42 q^{42} + (2 \beta_{4} - 5 \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 121) q^{43} + ( - 4 \beta_{4} - 4 \beta_{3} + 4 \beta_1 + 36) q^{44} + ( - 9 \beta_1 - 18) q^{45} - 46 q^{46} + (5 \beta_{4} + 4 \beta_{3} - 3 \beta_{2} - 10 \beta_1 + 104) q^{47} + 48 q^{48} + 49 q^{49} + (4 \beta_{4} - 2 \beta_{3} + 8 \beta_{2} + 8 \beta_1 + 216) q^{50} + (3 \beta_{3} - 6 \beta_{2} - 3 \beta_1 - 3) q^{51} + (4 \beta_{3} - 8 \beta_1 + 60) q^{52} + ( - 9 \beta_{4} - 10 \beta_{3} - 7 \beta_{2} + 19 \beta_1 + 70) q^{53} + 54 q^{54} + (10 \beta_{4} + 18 \beta_{3} - 13 \beta_{2} - 15 \beta_1 - 76) q^{55} - 56 q^{56} + (3 \beta_{4} - 3 \beta_{3} - 3 \beta_1 + 129) q^{57} + (2 \beta_{3} + 2 \beta_{2} + 10 \beta_1 + 110) q^{58} + ( - 5 \beta_{4} + 6 \beta_{3} - 3 \beta_{2} - 5 \beta_1 + 212) q^{59} + ( - 12 \beta_1 - 24) q^{60} + ( - 7 \beta_{4} + \beta_{3} + \beta_{2} - 8 \beta_1 + 11) q^{61} + (2 \beta_{3} + 8 \beta_{2} + 10 \beta_1 + 266) q^{62} - 63 q^{63} + 64 q^{64} + ( - 4 \beta_{4} - 9 \beta_{3} + 13 \beta_{2} + \beta_1 + 351) q^{65} + ( - 6 \beta_{4} - 6 \beta_{3} + 6 \beta_1 + 54) q^{66} + ( - 10 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + 19 \beta_1 + 132) q^{67} + (4 \beta_{3} - 8 \beta_{2} - 4 \beta_1 - 4) q^{68} - 69 q^{69} + (14 \beta_1 + 28) q^{70} + ( - 2 \beta_{4} + \beta_{3} + \beta_{2} + 12 \beta_1 + 377) q^{71} + 72 q^{72} + (6 \beta_{4} - 5 \beta_{3} + 4 \beta_{2} + 25 \beta_1 + 159) q^{73} + (8 \beta_{4} + 10 \beta_{3} - 6 \beta_{2} - 10 \beta_1 - 26) q^{74} + (6 \beta_{4} - 3 \beta_{3} + 12 \beta_{2} + 12 \beta_1 + 324) q^{75} + (4 \beta_{4} - 4 \beta_{3} - 4 \beta_1 + 172) q^{76} + (7 \beta_{4} + 7 \beta_{3} - 7 \beta_1 - 63) q^{77} + (6 \beta_{3} - 12 \beta_1 + 90) q^{78} + ( - 6 \beta_{4} + 17 \beta_{3} + 6 \beta_{2} + 11 \beta_1 + 151) q^{79} + ( - 16 \beta_1 - 32) q^{80} + 81 q^{81} + (2 \beta_{4} - 10 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 170) q^{82} + ( - 8 \beta_{4} - 17 \beta_{3} + 6 \beta_{2} + 11 \beta_1 + 151) q^{83} - 84 q^{84} + ( - 22 \beta_{4} - 28 \beta_{3} - \beta_{2} + 71 \beta_1 + 146) q^{85} + (4 \beta_{4} - 10 \beta_{3} + 4 \beta_{2} + 8 \beta_1 + 242) q^{86} + (3 \beta_{3} + 3 \beta_{2} + 15 \beta_1 + 165) q^{87} + ( - 8 \beta_{4} - 8 \beta_{3} + 8 \beta_1 + 72) q^{88} + ( - \beta_{3} + 5 \beta_{2} - 22 \beta_1 + 27) q^{89} + ( - 18 \beta_1 - 36) q^{90} + ( - 7 \beta_{3} + 14 \beta_1 - 105) q^{91} - 92 q^{92} + (3 \beta_{3} + 12 \beta_{2} + 15 \beta_1 + 399) q^{93} + (10 \beta_{4} + 8 \beta_{3} - 6 \beta_{2} - 20 \beta_1 + 208) q^{94} + (6 \beta_{4} - 4 \beta_{3} + 3 \beta_{2} - 61 \beta_1 + 126) q^{95} + 96 q^{96} + ( - 6 \beta_{4} - 13 \beta_{3} - 15 \beta_{2} + 29 \beta_1 - 17) q^{97} + 98 q^{98} + ( - 9 \beta_{4} - 9 \beta_{3} + 9 \beta_1 + 81) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 10 q^{2} + 15 q^{3} + 20 q^{4} - 10 q^{5} + 30 q^{6} - 35 q^{7} + 40 q^{8} + 45 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 10 q^{2} + 15 q^{3} + 20 q^{4} - 10 q^{5} + 30 q^{6} - 35 q^{7} + 40 q^{8} + 45 q^{9} - 20 q^{10} + 47 q^{11} + 60 q^{12} + 74 q^{13} - 70 q^{14} - 30 q^{15} + 80 q^{16} - 4 q^{17} + 90 q^{18} + 215 q^{19} - 40 q^{20} - 105 q^{21} + 94 q^{22} - 115 q^{23} + 120 q^{24} + 535 q^{25} + 148 q^{26} + 135 q^{27} - 140 q^{28} + 273 q^{29} - 60 q^{30} + 660 q^{31} + 160 q^{32} + 141 q^{33} - 8 q^{34} + 70 q^{35} + 180 q^{36} - 71 q^{37} + 430 q^{38} + 222 q^{39} - 80 q^{40} + 428 q^{41} - 210 q^{42} + 606 q^{43} + 188 q^{44} - 90 q^{45} - 230 q^{46} + 514 q^{47} + 240 q^{48} + 245 q^{49} + 1070 q^{50} - 12 q^{51} + 296 q^{52} + 376 q^{53} + 270 q^{54} - 395 q^{55} - 280 q^{56} + 645 q^{57} + 546 q^{58} + 1062 q^{59} - 120 q^{60} + 60 q^{61} + 1320 q^{62} - 315 q^{63} + 320 q^{64} + 1755 q^{65} + 282 q^{66} + 671 q^{67} - 16 q^{68} - 345 q^{69} + 140 q^{70} + 1885 q^{71} + 360 q^{72} + 790 q^{73} - 142 q^{74} + 1605 q^{75} + 860 q^{76} - 329 q^{77} + 444 q^{78} + 738 q^{79} - 160 q^{80} + 405 q^{81} + 856 q^{82} + 774 q^{83} - 420 q^{84} + 781 q^{85} + 1212 q^{86} + 819 q^{87} + 376 q^{88} + 131 q^{89} - 180 q^{90} - 518 q^{91} - 460 q^{92} + 1980 q^{93} + 1028 q^{94} + 625 q^{95} + 480 q^{96} - 51 q^{97} + 490 q^{98} + 423 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 570x^{3} - 189x^{2} + 63838x + 254320 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 43\nu^{4} - 187\nu^{3} - 18717\nu^{2} + 101726\nu + 893370 ) / 15295 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 41\nu^{4} - 534\nu^{3} - 24249\nu^{2} + 228247\nu + 2351795 ) / 15295 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -131\nu^{4} + 214\nu^{3} + 65914\nu^{2} - 178657\nu - 4724240 ) / 30590 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{4} - \beta_{3} + 4\beta_{2} + 229 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -36\beta_{4} - 25\beta_{3} - 31\beta_{2} + 369\beta _1 + 95 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 714\beta_{4} - 544\beta_{3} + 1962\beta_{2} - 761\beta _1 + 79316 \) 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
19.3342
15.0684
−4.94452
−8.37318
−21.0849
2.00000 3.00000 4.00000 −21.3342 6.00000 −7.00000 8.00000 9.00000 −42.6683
1.2 2.00000 3.00000 4.00000 −17.0684 6.00000 −7.00000 8.00000 9.00000 −34.1369
1.3 2.00000 3.00000 4.00000 2.94452 6.00000 −7.00000 8.00000 9.00000 5.88904
1.4 2.00000 3.00000 4.00000 6.37318 6.00000 −7.00000 8.00000 9.00000 12.7464
1.5 2.00000 3.00000 4.00000 19.0849 6.00000 −7.00000 8.00000 9.00000 38.1698
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.q 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.4.a.q 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} + 10T_{5}^{4} - 530T_{5}^{3} - 3151T_{5}^{2} + 57834T_{5} - 130416 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(966))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{5} \) Copy content Toggle raw display
$3$ \( (T - 3)^{5} \) Copy content Toggle raw display
$5$ \( T^{5} + 10 T^{4} - 530 T^{3} + \cdots - 130416 \) Copy content Toggle raw display
$7$ \( (T + 7)^{5} \) Copy content Toggle raw display
$11$ \( T^{5} - 47 T^{4} - 4225 T^{3} + \cdots - 73078208 \) Copy content Toggle raw display
$13$ \( T^{5} - 74 T^{4} - 2758 T^{3} + \cdots - 68093400 \) Copy content Toggle raw display
$17$ \( T^{5} + 4 T^{4} + \cdots - 2138415104 \) Copy content Toggle raw display
$19$ \( T^{5} - 215 T^{4} + \cdots + 1351264904 \) Copy content Toggle raw display
$23$ \( (T + 23)^{5} \) Copy content Toggle raw display
$29$ \( T^{5} - 273 T^{4} + \cdots - 2602801916 \) Copy content Toggle raw display
$31$ \( T^{5} - 660 T^{4} + \cdots + 372375172984 \) Copy content Toggle raw display
$37$ \( T^{5} + 71 T^{4} + \cdots + 9965603956 \) Copy content Toggle raw display
$41$ \( T^{5} - 428 T^{4} + \cdots - 262649729210 \) Copy content Toggle raw display
$43$ \( T^{5} - 606 T^{4} + \cdots + 340242583264 \) Copy content Toggle raw display
$47$ \( T^{5} - 514 T^{4} + \cdots - 316596333000 \) Copy content Toggle raw display
$53$ \( T^{5} - 376 T^{4} + \cdots - 18023680374544 \) Copy content Toggle raw display
$59$ \( T^{5} - 1062 T^{4} + \cdots + 42010142616 \) Copy content Toggle raw display
$61$ \( T^{5} - 60 T^{4} + \cdots + 380306833900 \) Copy content Toggle raw display
$67$ \( T^{5} - 671 T^{4} + \cdots - 1322680877888 \) Copy content Toggle raw display
$71$ \( T^{5} - 1885 T^{4} + \cdots - 2557042300608 \) Copy content Toggle raw display
$73$ \( T^{5} - 790 T^{4} + \cdots - 1322849220128 \) Copy content Toggle raw display
$79$ \( T^{5} - 738 T^{4} + \cdots - 6937624173056 \) Copy content Toggle raw display
$83$ \( T^{5} - 774 T^{4} + \cdots - 6921450883600 \) Copy content Toggle raw display
$89$ \( T^{5} - 131 T^{4} + \cdots + 254330006480 \) Copy content Toggle raw display
$97$ \( T^{5} + 51 T^{4} + \cdots + 25172472305992 \) Copy content Toggle raw display
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