Properties

Label 966.4.a.l
Level $966$
Weight $4$
Character orbit 966.a
Self dual yes
Analytic conductor $56.996$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \( x^{5} - 319x^{3} - 666x^{2} + 23460x + 101568 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} - 3 q^{3} + 4 q^{4} + (\beta_{2} - 3) q^{5} + 6 q^{6} + 7 q^{7} - 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} - 3 q^{3} + 4 q^{4} + (\beta_{2} - 3) q^{5} + 6 q^{6} + 7 q^{7} - 8 q^{8} + 9 q^{9} + ( - 2 \beta_{2} + 6) q^{10} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 4) q^{11} - 12 q^{12} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 4) q^{13} - 14 q^{14} + ( - 3 \beta_{2} + 9) q^{15} + 16 q^{16} + (\beta_{4} - \beta_{2} - 5 \beta_1 - 8) q^{17} - 18 q^{18} + (3 \beta_{4} - \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 4) q^{19} + (4 \beta_{2} - 12) q^{20} - 21 q^{21} + (2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 8) q^{22} - 23 q^{23} + 24 q^{24} + ( - \beta_{3} - 11 \beta_{2} - 6 \beta_1 + 41) q^{25} + (2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 8) q^{26} - 27 q^{27} + 28 q^{28} + ( - \beta_{4} + 4 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 58) q^{29} + (6 \beta_{2} - 18) q^{30} + ( - 3 \beta_{4} - 4 \beta_{3} - 5 \beta_{2} + 7 \beta_1 - 14) q^{31} - 32 q^{32} + (3 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} + 12) q^{33} + ( - 2 \beta_{4} + 2 \beta_{2} + 10 \beta_1 + 16) q^{34} + (7 \beta_{2} - 21) q^{35} + 36 q^{36} + ( - 2 \beta_{4} + 5 \beta_{3} + 2 \beta_{2} + 14 \beta_1 + 53) q^{37} + ( - 6 \beta_{4} + 2 \beta_{3} + 6 \beta_{2} - 4 \beta_1 + 8) q^{38} + (3 \beta_{3} - 3 \beta_{2} - 6 \beta_1 + 12) q^{39} + ( - 8 \beta_{2} + 24) q^{40} + ( - 2 \beta_{4} - 4 \beta_{3} - 18 \beta_{2} - 9 \beta_1 + 43) q^{41} + 42 q^{42} + ( - \beta_{4} + 6 \beta_{3} - 12 \beta_{2} - 19 \beta_1 + 133) q^{43} + ( - 4 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} - 16) q^{44} + (9 \beta_{2} - 27) q^{45} + 46 q^{46} + (\beta_{4} - 8 \beta_{3} - 5 \beta_{2} + 16 \beta_1 - 57) q^{47} - 48 q^{48} + 49 q^{49} + (2 \beta_{3} + 22 \beta_{2} + 12 \beta_1 - 82) q^{50} + ( - 3 \beta_{4} + 3 \beta_{2} + 15 \beta_1 + 24) q^{51} + ( - 4 \beta_{3} + 4 \beta_{2} + 8 \beta_1 - 16) q^{52} + (\beta_{4} + 2 \beta_{3} - 20 \beta_{2} + 22 \beta_1 - 64) q^{53} + 54 q^{54} + (9 \beta_{4} + 5 \beta_{3} - 24 \beta_{2} - 45 \beta_1 + 220) q^{55} - 56 q^{56} + ( - 9 \beta_{4} + 3 \beta_{3} + 9 \beta_{2} - 6 \beta_1 + 12) q^{57} + (2 \beta_{4} - 8 \beta_{3} + 6 \beta_{2} - 6 \beta_1 + 116) q^{58} + (2 \beta_{4} - 5 \beta_{3} + 13 \beta_{2} + 9 \beta_1 + 5) q^{59} + ( - 12 \beta_{2} + 36) q^{60} + (4 \beta_{4} + 2 \beta_{3} + 27 \beta_{2} + 21 \beta_1 - 90) q^{61} + (6 \beta_{4} + 8 \beta_{3} + 10 \beta_{2} - 14 \beta_1 + 28) q^{62} + 63 q^{63} + 64 q^{64} + (7 \beta_{4} - 12 \beta_{3} - 9 \beta_{2} - 5 \beta_1 + 106) q^{65} + ( - 6 \beta_{4} + 6 \beta_{3} + 6 \beta_{2} - 24) q^{66} + ( - 4 \beta_{4} + 2 \beta_{3} - 31 \beta_{2} - 38 \beta_1 + 307) q^{67} + (4 \beta_{4} - 4 \beta_{2} - 20 \beta_1 - 32) q^{68} + 69 q^{69} + ( - 14 \beta_{2} + 42) q^{70} + (10 \beta_{4} - \beta_{3} + 13 \beta_{2} - 40 \beta_1 - 60) q^{71} - 72 q^{72} + ( - \beta_{4} + 10 \beta_{3} + 3 \beta_{2} + 25 \beta_1 + 308) q^{73} + (4 \beta_{4} - 10 \beta_{3} - 4 \beta_{2} - 28 \beta_1 - 106) q^{74} + (3 \beta_{3} + 33 \beta_{2} + 18 \beta_1 - 123) q^{75} + (12 \beta_{4} - 4 \beta_{3} - 12 \beta_{2} + 8 \beta_1 - 16) q^{76} + ( - 7 \beta_{4} + 7 \beta_{3} + 7 \beta_{2} - 28) q^{77} + ( - 6 \beta_{3} + 6 \beta_{2} + 12 \beta_1 - 24) q^{78} + (3 \beta_{4} - 18 \beta_{3} + 3 \beta_{2} + \beta_1 + 246) q^{79} + (16 \beta_{2} - 48) q^{80} + 81 q^{81} + (4 \beta_{4} + 8 \beta_{3} + 36 \beta_{2} + 18 \beta_1 - 86) q^{82} + (10 \beta_{4} + 5 \beta_{3} + 52 \beta_{2} + 20 \beta_1 - 273) q^{83} - 84 q^{84} + ( - 25 \beta_{4} + 9 \beta_{3} + 18 \beta_{2} + 53 \beta_1 - 16) q^{85} + (2 \beta_{4} - 12 \beta_{3} + 24 \beta_{2} + 38 \beta_1 - 266) q^{86} + (3 \beta_{4} - 12 \beta_{3} + 9 \beta_{2} - 9 \beta_1 + 174) q^{87} + (8 \beta_{4} - 8 \beta_{3} - 8 \beta_{2} + 32) q^{88} + (14 \beta_{4} - 7 \beta_{3} - 5 \beta_{2} - 60 \beta_1 - 230) q^{89} + ( - 18 \beta_{2} + 54) q^{90} + ( - 7 \beta_{3} + 7 \beta_{2} + 14 \beta_1 - 28) q^{91} - 92 q^{92} + (9 \beta_{4} + 12 \beta_{3} + 15 \beta_{2} - 21 \beta_1 + 42) q^{93} + ( - 2 \beta_{4} + 16 \beta_{3} + 10 \beta_{2} - 32 \beta_1 + 114) q^{94} + ( - 23 \beta_{4} + \beta_{3} + 32 \beta_{2} + 115 \beta_1 - 696) q^{95} + 96 q^{96} + ( - 13 \beta_{4} - 24 \beta_{3} - 47 \beta_{2} + 15 \beta_1 - 370) q^{97} - 98 q^{98} + ( - 9 \beta_{4} + 9 \beta_{3} + 9 \beta_{2} - 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 10 q^{2} - 15 q^{3} + 20 q^{4} - 15 q^{5} + 30 q^{6} + 35 q^{7} - 40 q^{8} + 45 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 10 q^{2} - 15 q^{3} + 20 q^{4} - 15 q^{5} + 30 q^{6} + 35 q^{7} - 40 q^{8} + 45 q^{9} + 30 q^{10} - 19 q^{11} - 60 q^{12} - 19 q^{13} - 70 q^{14} + 45 q^{15} + 80 q^{16} - 42 q^{17} - 90 q^{18} - 25 q^{19} - 60 q^{20} - 105 q^{21} + 38 q^{22} - 115 q^{23} + 120 q^{24} + 206 q^{25} + 38 q^{26} - 135 q^{27} + 140 q^{28} - 292 q^{29} - 90 q^{30} - 60 q^{31} - 160 q^{32} + 57 q^{33} + 84 q^{34} - 105 q^{35} + 180 q^{36} + 264 q^{37} + 50 q^{38} + 57 q^{39} + 120 q^{40} + 223 q^{41} + 210 q^{42} + 661 q^{43} - 76 q^{44} - 135 q^{45} + 230 q^{46} - 279 q^{47} - 240 q^{48} + 245 q^{49} - 412 q^{50} + 126 q^{51} - 76 q^{52} - 324 q^{53} + 270 q^{54} + 1077 q^{55} - 280 q^{56} + 75 q^{57} + 584 q^{58} + 26 q^{59} + 180 q^{60} - 460 q^{61} + 120 q^{62} + 315 q^{63} + 320 q^{64} + 528 q^{65} - 114 q^{66} + 1541 q^{67} - 168 q^{68} + 345 q^{69} + 210 q^{70} - 319 q^{71} - 360 q^{72} + 1532 q^{73} - 528 q^{74} - 618 q^{75} - 100 q^{76} - 133 q^{77} - 114 q^{78} + 1242 q^{79} - 240 q^{80} + 405 q^{81} - 446 q^{82} - 1390 q^{83} - 420 q^{84} - 39 q^{85} - 1322 q^{86} + 876 q^{87} + 152 q^{88} - 1171 q^{89} + 270 q^{90} - 133 q^{91} - 460 q^{92} + 180 q^{93} + 558 q^{94} - 3435 q^{95} + 480 q^{96} - 1800 q^{97} - 490 q^{98} - 171 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 319x^{3} - 666x^{2} + 23460x + 101568 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{4} - 23\nu^{3} - 1023\nu^{2} + 3500\nu + 51980 ) / 644 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{4} + 23\nu^{3} + 1345\nu^{2} - 4466\nu - 93196 ) / 644 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{4} + 69\nu^{3} + 1480\nu^{2} - 11942\nu - 106996 ) / 644 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{3} + 2\beta_{2} + 3\beta _1 + 128 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 16\beta_{4} - 10\beta_{3} + 10\beta_{2} + 173\beta _1 + 404 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 92\beta_{4} + 454\beta_{3} + 730\beta_{2} + 887\beta _1 + 22064 \) 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
−7.69907
11.4760
15.2907
−13.4981
−5.56964
−2.00000 −3.00000 4.00000 −20.1660 6.00000 7.00000 −8.00000 9.00000 40.3320
1.2 −2.00000 −3.00000 4.00000 −15.3686 6.00000 7.00000 −8.00000 9.00000 30.7372
1.3 −2.00000 −3.00000 4.00000 1.26807 6.00000 7.00000 −8.00000 9.00000 −2.53613
1.4 −2.00000 −3.00000 4.00000 8.95143 6.00000 7.00000 −8.00000 9.00000 −17.9029
1.5 −2.00000 −3.00000 4.00000 10.3151 6.00000 7.00000 −8.00000 9.00000 −20.6302
\(n\): 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.l 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.4.a.l 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} + 15T_{5}^{4} - 303T_{5}^{3} - 2332T_{5}^{2} + 32028T_{5} - 36288 \) 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
$5$ \( T^{5} + 15 T^{4} - 303 T^{3} + \cdots - 36288 \) Copy content Toggle raw display
$7$ \( (T - 7)^{5} \) Copy content Toggle raw display
$11$ \( T^{5} + 19 T^{4} - 4737 T^{3} + \cdots + 62634096 \) Copy content Toggle raw display
$13$ \( T^{5} + 19 T^{4} - 3171 T^{3} + \cdots - 4287000 \) Copy content Toggle raw display
$17$ \( T^{5} + 42 T^{4} + \cdots + 117524736 \) Copy content Toggle raw display
$19$ \( T^{5} + 25 T^{4} + \cdots + 14087905104 \) Copy content Toggle raw display
$23$ \( (T + 23)^{5} \) Copy content Toggle raw display
$29$ \( T^{5} + 292 T^{4} + \cdots - 1129841688 \) Copy content Toggle raw display
$31$ \( T^{5} + 60 T^{4} + \cdots + 126870808576 \) Copy content Toggle raw display
$37$ \( T^{5} - 264 T^{4} + \cdots - 378516115880 \) Copy content Toggle raw display
$41$ \( T^{5} - 223 T^{4} + \cdots - 589161509700 \) Copy content Toggle raw display
$43$ \( T^{5} - 661 T^{4} + \cdots - 421735977792 \) Copy content Toggle raw display
$47$ \( T^{5} + 279 T^{4} + \cdots - 1783648512 \) Copy content Toggle raw display
$53$ \( T^{5} + 324 T^{4} + \cdots - 4637528572932 \) Copy content Toggle raw display
$59$ \( T^{5} - 26 T^{4} + \cdots + 257121804240 \) Copy content Toggle raw display
$61$ \( T^{5} + 460 T^{4} + \cdots - 170741533284 \) Copy content Toggle raw display
$67$ \( T^{5} - 1541 T^{4} + \cdots + 9598713835584 \) Copy content Toggle raw display
$71$ \( T^{5} + 319 T^{4} + \cdots - 7396227885312 \) Copy content Toggle raw display
$73$ \( T^{5} - 1532 T^{4} + \cdots - 1792003628016 \) Copy content Toggle raw display
$79$ \( T^{5} - 1242 T^{4} + \cdots - 1271307835392 \) Copy content Toggle raw display
$83$ \( T^{5} + 1390 T^{4} + \cdots + 9587344906272 \) Copy content Toggle raw display
$89$ \( T^{5} + 1171 T^{4} + \cdots - 2368861030872 \) Copy content Toggle raw display
$97$ \( T^{5} + 1800 T^{4} + \cdots + 25515582030104 \) Copy content Toggle raw display
show more
show less