Properties

Label 966.4.a.o
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: \( x^{5} - x^{4} - 560x^{3} + 2247x^{2} + 58113x - 197784 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
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.

\(f(q)\) \(=\) \( q - 2 q^{2} + 3 q^{3} + 4 q^{4} + (\beta_1 + 1) q^{5} - 6 q^{6} + 7 q^{7} - 8 q^{8} + 9 q^{9}+O(q^{10}) \) 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} + \beta_1 - 10) q^{11} + 12 q^{12} + ( - \beta_{4} + \beta_1 + 13) q^{13} - 14 q^{14} + (3 \beta_1 + 3) q^{15} + 16 q^{16} + (\beta_{4} + 2 \beta_1 - 40) q^{17} - 18 q^{18} + ( - \beta_{4} + 24) q^{19} + (4 \beta_1 + 4) q^{20} + 21 q^{21} + (2 \beta_{2} - 2 \beta_1 + 20) q^{22} - 23 q^{23} - 24 q^{24} + ( - 2 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 102) q^{25} + (2 \beta_{4} - 2 \beta_1 - 26) q^{26} + 27 q^{27} + 28 q^{28} + (3 \beta_{3} + 3 \beta_1 + 59) q^{29} + ( - 6 \beta_1 - 6) q^{30} + (\beta_{4} + 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 62) q^{31} - 32 q^{32} + ( - 3 \beta_{2} + 3 \beta_1 - 30) q^{33} + ( - 2 \beta_{4} - 4 \beta_1 + 80) q^{34} + (7 \beta_1 + 7) q^{35} + 36 q^{36} + (2 \beta_{4} - \beta_{3} + 5 \beta_1 + 53) q^{37} + (2 \beta_{4} - 48) q^{38} + ( - 3 \beta_{4} + 3 \beta_1 + 39) q^{39} + ( - 8 \beta_1 - 8) q^{40} + ( - 3 \beta_{3} + 2 \beta_{2} + 15 \beta_1 + 93) q^{41} - 42 q^{42} + (2 \beta_{4} + 9 \beta_{2} + 6 \beta_1 - 61) q^{43} + ( - 4 \beta_{2} + 4 \beta_1 - 40) q^{44} + (9 \beta_1 + 9) q^{45} + 46 q^{46} + (\beta_{4} - 7 \beta_{3} + \beta_1 + 15) q^{47} + 48 q^{48} + 49 q^{49} + (4 \beta_{3} + 6 \beta_{2} + 4 \beta_1 - 204) q^{50} + (3 \beta_{4} + 6 \beta_1 - 120) q^{51} + ( - 4 \beta_{4} + 4 \beta_1 + 52) q^{52} + ( - 2 \beta_{4} - 5 \beta_{3} + 7 \beta_{2} + 7 \beta_1 + 66) q^{53} - 54 q^{54} + (7 \beta_{3} - 9 \beta_{2} + 9 \beta_1 + 190) q^{55} - 56 q^{56} + ( - 3 \beta_{4} + 72) q^{57} + ( - 6 \beta_{3} - 6 \beta_1 - 118) q^{58} + (3 \beta_{3} - 7 \beta_{2} - 11 \beta_1 + 416) q^{59} + (12 \beta_1 + 12) q^{60} + (9 \beta_{3} - 4 \beta_{2} - 24 \beta_1 + 110) q^{61} + ( - 2 \beta_{4} - 4 \beta_{3} - 8 \beta_{2} + 4 \beta_1 - 124) q^{62} + 63 q^{63} + 64 q^{64} + ( - 2 \beta_{4} + 15 \beta_{3} - 2 \beta_{2} + 9 \beta_1 + 241) q^{65} + (6 \beta_{2} - 6 \beta_1 + 60) q^{66} + ( - 2 \beta_{4} + 9 \beta_{3} + 3 \beta_{2} - 7 \beta_1 + 170) q^{67} + (4 \beta_{4} + 8 \beta_1 - 160) q^{68} - 69 q^{69} + ( - 14 \beta_1 - 14) q^{70} + ( - 10 \beta_{4} - 7 \beta_{3} - 26 \beta_1 - 166) q^{71} - 72 q^{72} + (10 \beta_{4} - 14 \beta_{3} + 3 \beta_{2} - 21 \beta_1 + 130) q^{73} + ( - 4 \beta_{4} + 2 \beta_{3} - 10 \beta_1 - 106) q^{74} + ( - 6 \beta_{3} - 9 \beta_{2} - 6 \beta_1 + 306) q^{75} + ( - 4 \beta_{4} + 96) q^{76} + ( - 7 \beta_{2} + 7 \beta_1 - 70) q^{77} + (6 \beta_{4} - 6 \beta_1 - 78) q^{78} + (10 \beta_{4} - 8 \beta_{3} + 7 \beta_{2} - 15 \beta_1 + 54) q^{79} + (16 \beta_1 + 16) q^{80} + 81 q^{81} + (6 \beta_{3} - 4 \beta_{2} - 30 \beta_1 - 186) q^{82} + ( - \beta_{4} - 4 \beta_{3} + 4 \beta_1 - 340) q^{83} + 84 q^{84} + (2 \beta_{4} - 21 \beta_{3} - 7 \beta_{2} - 45 \beta_1 + 410) q^{85} + ( - 4 \beta_{4} - 18 \beta_{2} - 12 \beta_1 + 122) q^{86} + (9 \beta_{3} + 9 \beta_1 + 177) q^{87} + (8 \beta_{2} - 8 \beta_1 + 80) q^{88} + (3 \beta_{4} + 5 \beta_{3} + 15 \beta_{2} - 35 \beta_1 - 52) q^{89} + ( - 18 \beta_1 - 18) q^{90} + ( - 7 \beta_{4} + 7 \beta_1 + 91) q^{91} - 92 q^{92} + (3 \beta_{4} + 6 \beta_{3} + 12 \beta_{2} - 6 \beta_1 + 186) q^{93} + ( - 2 \beta_{4} + 14 \beta_{3} - 2 \beta_1 - 30) q^{94} + ( - 2 \beta_{4} + 17 \beta_{3} + \beta_{2} + 23 \beta_1 + 26) q^{95} - 96 q^{96} + ( - 9 \beta_{4} + \beta_{3} + 11 \beta_{2} + 4 \beta_1 - 199) q^{97} - 98 q^{98} + ( - 9 \beta_{2} + 9 \beta_1 - 90) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 10 q^{2} + 15 q^{3} + 20 q^{4} + 6 q^{5} - 30 q^{6} + 35 q^{7} - 40 q^{8} + 45 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 10 q^{2} + 15 q^{3} + 20 q^{4} + 6 q^{5} - 30 q^{6} + 35 q^{7} - 40 q^{8} + 45 q^{9} - 12 q^{10} - 50 q^{11} + 60 q^{12} + 66 q^{13} - 70 q^{14} + 18 q^{15} + 80 q^{16} - 198 q^{17} - 90 q^{18} + 120 q^{19} + 24 q^{20} + 105 q^{21} + 100 q^{22} - 115 q^{23} - 120 q^{24} + 503 q^{25} - 132 q^{26} + 135 q^{27} + 140 q^{28} + 301 q^{29} - 36 q^{30} + 314 q^{31} - 160 q^{32} - 150 q^{33} + 396 q^{34} + 42 q^{35} + 180 q^{36} + 269 q^{37} - 240 q^{38} + 198 q^{39} - 48 q^{40} + 479 q^{41} - 210 q^{42} - 290 q^{43} - 200 q^{44} + 54 q^{45} + 230 q^{46} + 69 q^{47} + 240 q^{48} + 245 q^{49} - 1006 q^{50} - 594 q^{51} + 264 q^{52} + 339 q^{53} - 270 q^{54} + 957 q^{55} - 280 q^{56} + 360 q^{57} - 602 q^{58} + 2065 q^{59} + 72 q^{60} + 531 q^{61} - 628 q^{62} + 315 q^{63} + 320 q^{64} + 1227 q^{65} + 300 q^{66} + 855 q^{67} - 792 q^{68} - 345 q^{69} - 84 q^{70} - 863 q^{71} - 360 q^{72} + 618 q^{73} - 538 q^{74} + 1509 q^{75} + 480 q^{76} - 350 q^{77} - 396 q^{78} + 254 q^{79} + 96 q^{80} + 405 q^{81} - 958 q^{82} - 1700 q^{83} + 420 q^{84} + 1977 q^{85} + 580 q^{86} + 903 q^{87} + 400 q^{88} - 275 q^{89} - 108 q^{90} + 462 q^{91} - 460 q^{92} + 942 q^{93} - 138 q^{94} + 171 q^{95} - 480 q^{96} - 979 q^{97} - 490 q^{98} - 450 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 560x^{3} + 2247x^{2} + 58113x - 197784 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{4} - 70\nu^{3} + 945\nu^{2} + 14196\nu + 25106 ) / 1267 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6\nu^{4} + 105\nu^{3} - 2051\nu^{2} - 23828\nu + 105512 ) / 1267 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -81\nu^{4} - 784\nu^{3} + 30856\nu^{2} + 120225\nu - 1449752 ) / 10136 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{3} - 3\beta_{2} - 4\beta _1 + 226 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 16\beta_{4} + 37\beta_{3} + 15\beta_{2} + 338\beta _1 - 1090 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -280\beta_{4} - 1120\beta_{3} - 1288\beta_{2} - 3311\beta _1 + 78744 \) 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
−22.4726
−10.9609
3.32532
14.8741
16.2340
−2.00000 3.00000 4.00000 −21.4726 −6.00000 7.00000 −8.00000 9.00000 42.9452
1.2 −2.00000 3.00000 4.00000 −9.96090 −6.00000 7.00000 −8.00000 9.00000 19.9218
1.3 −2.00000 3.00000 4.00000 4.32532 −6.00000 7.00000 −8.00000 9.00000 −8.65064
1.4 −2.00000 3.00000 4.00000 15.8741 −6.00000 7.00000 −8.00000 9.00000 −31.7483
1.5 −2.00000 3.00000 4.00000 17.2340 −6.00000 7.00000 −8.00000 9.00000 −34.4681
\(n\): 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.o 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.4.a.o 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} - 6T_{5}^{4} - 546T_{5}^{3} + 3911T_{5}^{2} + 51948T_{5} - 253092 \) 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
$5$ \( T^{5} - 6 T^{4} - 546 T^{3} + \cdots - 253092 \) Copy content Toggle raw display
$7$ \( (T - 7)^{5} \) Copy content Toggle raw display
$11$ \( T^{5} + 50 T^{4} - 3851 T^{3} + \cdots + 1580544 \) Copy content Toggle raw display
$13$ \( T^{5} - 66 T^{4} + \cdots - 338313228 \) Copy content Toggle raw display
$17$ \( T^{5} + 198 T^{4} + \cdots + 461848128 \) Copy content Toggle raw display
$19$ \( T^{5} - 120 T^{4} + \cdots - 659836416 \) Copy content Toggle raw display
$23$ \( (T + 23)^{5} \) Copy content Toggle raw display
$29$ \( T^{5} - 301 T^{4} + \cdots - 23330640716 \) Copy content Toggle raw display
$31$ \( T^{5} - 314 T^{4} + \cdots - 105545477824 \) Copy content Toggle raw display
$37$ \( T^{5} - 269 T^{4} + \cdots - 6433347004 \) Copy content Toggle raw display
$41$ \( T^{5} - 479 T^{4} + \cdots - 533133774388 \) Copy content Toggle raw display
$43$ \( T^{5} + 290 T^{4} + \cdots - 5204122290432 \) Copy content Toggle raw display
$47$ \( T^{5} - 69 T^{4} + \cdots + 877810784256 \) Copy content Toggle raw display
$53$ \( T^{5} - 339 T^{4} + \cdots + 1138765942800 \) Copy content Toggle raw display
$59$ \( T^{5} - 2065 T^{4} + \cdots - 55391020704 \) Copy content Toggle raw display
$61$ \( T^{5} - 531 T^{4} + \cdots - 10079254220328 \) Copy content Toggle raw display
$67$ \( T^{5} - 855 T^{4} + \cdots - 4400681714688 \) Copy content Toggle raw display
$71$ \( T^{5} + \cdots + 194629543039232 \) Copy content Toggle raw display
$73$ \( T^{5} + \cdots - 286748840756976 \) Copy content Toggle raw display
$79$ \( T^{5} - 254 T^{4} + \cdots - 49849731412992 \) Copy content Toggle raw display
$83$ \( T^{5} + 1700 T^{4} + \cdots + 2240300109824 \) Copy content Toggle raw display
$89$ \( T^{5} + 275 T^{4} + \cdots + 30964164369768 \) Copy content Toggle raw display
$97$ \( T^{5} + 979 T^{4} + \cdots - 87530849611844 \) Copy content Toggle raw display
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