Properties

Label 124.6.a.b
Level $124$
Weight $6$
Character orbit 124.a
Self dual yes
Analytic conductor $19.888$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 124 = 2^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 124.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.8875936568\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - 2x^{5} - 847x^{4} + 1184x^{3} + 199815x^{2} - 13326x - 12452553 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 3) q^{3} + ( - \beta_{5} + \beta_1 + 4) q^{5} + ( - \beta_{5} - \beta_{4} + 2 \beta_{3} + 6) q^{7} + ( - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 5 \beta_1 + 48) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 3) q^{3} + ( - \beta_{5} + \beta_1 + 4) q^{5} + ( - \beta_{5} - \beta_{4} + 2 \beta_{3} + 6) q^{7} + ( - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 5 \beta_1 + 48) q^{9} + (\beta_{5} - \beta_{4} - 6 \beta_{3} - \beta_{2} + 17 \beta_1 + 44) q^{11} + (3 \beta_{5} - 4 \beta_{4} - 11 \beta_{3} - 2 \beta_{2} + 22 \beta_1 + 201) q^{13} + (\beta_{5} + 11 \beta_{4} - 5 \beta_{2} + 10 \beta_1 + 311) q^{15} + (10 \beta_{5} - 7 \beta_{4} + 9 \beta_{3} + 3 \beta_{2} + 13 \beta_1 + 125) q^{17} + ( - 3 \beta_{5} + 3 \beta_{4} + 9 \beta_{3} + 29 \beta_{2} + 4 \beta_1 + 523) q^{19} + (14 \beta_{5} + 7 \beta_{4} + 39 \beta_{3} - 17 \beta_{2} - 29 \beta_1 + 163) q^{21} + ( - 15 \beta_{5} - 7 \beta_{4} - 16 \beta_{3} + 3 \beta_{2} - 58 \beta_1 + 1553) q^{23} + (4 \beta_{5} - 22 \beta_{4} - 33 \beta_{3} - 42 \beta_{2} - 26 \beta_1 + 1105) q^{25} + ( - 16 \beta_{5} + 22 \beta_{4} - 36 \beta_{2} - 94 \beta_1 + 1240) q^{27} + ( - 9 \beta_{5} - 42 \beta_{4} + \beta_{3} + 92 \beta_{2} - 26 \beta_1 + 2209) q^{29} - 961 q^{31} + ( - 20 \beta_{5} + 22 \beta_{4} - 29 \beta_{3} + 87 \beta_{2} + \cdots + 4149) q^{33}+ \cdots + ( - 47 \beta_{5} - 67 \beta_{4} - 1098 \beta_{3} - \beta_{2} + 2813 \beta_1 + 5318) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 20 q^{3} + 25 q^{5} + 39 q^{7} + 306 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 20 q^{3} + 25 q^{5} + 39 q^{7} + 306 q^{9} + 280 q^{11} + 1214 q^{13} + 1914 q^{15} + 796 q^{17} + 3147 q^{19} + 1082 q^{21} + 9122 q^{23} + 6481 q^{25} + 7316 q^{27} + 13020 q^{29} - 5766 q^{31} + 24804 q^{33} + 20059 q^{35} + 21678 q^{37} + 30680 q^{39} + 3227 q^{41} + 37882 q^{43} + 26169 q^{45} + 29708 q^{47} + 34849 q^{49} + 24432 q^{51} - 9976 q^{53} + 23758 q^{55} + 17318 q^{57} + 20573 q^{59} + 21610 q^{61} - 17697 q^{63} + 3894 q^{65} - 17024 q^{67} - 83692 q^{69} + 44509 q^{71} - 161864 q^{73} - 49430 q^{75} - 144202 q^{77} - 24420 q^{79} - 181158 q^{81} - 114160 q^{83} - 228882 q^{85} - 56180 q^{87} - 199742 q^{89} - 186774 q^{91} - 19220 q^{93} - 12793 q^{95} - 282951 q^{97} + 34060 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 847x^{4} + 1184x^{3} + 199815x^{2} - 13326x - 12452553 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -527\nu^{5} - 4605\nu^{4} + 205184\nu^{3} + 2426160\nu^{2} + 6013815\nu - 230250843 ) / 5613600 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 667\nu^{5} + 14705\nu^{4} - 376864\nu^{3} - 7771760\nu^{2} + 40226685\nu + 691203303 ) / 5613600 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 303\nu^{5} - 11555\nu^{4} - 211176\nu^{3} + 8091560\nu^{2} + 26940465\nu - 1024004973 ) / 2806800 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2019\nu^{5} - 21415\nu^{4} - 1393248\nu^{3} + 13635280\nu^{2} + 188615445\nu - 1360828929 ) / 5613600 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + 2\beta_{4} + 2\beta_{3} + \beta_{2} - \beta _1 + 282 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -7\beta_{5} + 4\beta_{4} - 18\beta_{3} - 45\beta_{2} + 374\beta _1 + 133 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -622\beta_{5} + 1112\beta_{4} + 1454\beta_{3} + 736\beta_{2} - 982\beta _1 + 106065 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -1894\beta_{5} + 1048\beta_{4} - 10506\beta_{3} - 30000\beta_{2} + 161003\beta _1 - 13688 \) 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
−22.5479
−13.5834
−11.0199
9.40536
18.2423
21.5036
0 −19.5479 0 −16.3712 0 −21.6934 0 139.120 0
1.2 0 −10.5834 0 −84.7805 0 −158.720 0 −130.991 0
1.3 0 −8.01991 0 93.2607 0 174.764 0 −178.681 0
1.4 0 12.4054 0 −65.1867 0 116.704 0 −89.1069 0
1.5 0 21.2423 0 45.9749 0 −213.890 0 208.235 0
1.6 0 24.5036 0 52.1028 0 141.835 0 357.424 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(31\) \(1\)

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 124.6.a.b 6
4.b odd 2 1 496.6.a.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.6.a.b 6 1.a even 1 1 trivial
496.6.a.f 6 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 20T_{3}^{5} - 682T_{3}^{4} + 10628T_{3}^{3} + 145176T_{3}^{2} - 1091040T_{3} - 10713600 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(124))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 20 T^{5} - 682 T^{4} + \cdots - 10713600 \) Copy content Toggle raw display
$5$ \( T^{6} - 25 T^{5} + \cdots - 20212306536 \) Copy content Toggle raw display
$7$ \( T^{6} - 39 T^{5} + \cdots - 2130457113216 \) Copy content Toggle raw display
$11$ \( T^{6} - 280 T^{5} + \cdots - 1676525942016 \) Copy content Toggle raw display
$13$ \( T^{6} - 1214 T^{5} + \cdots + 15\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( T^{6} - 796 T^{5} + \cdots - 58\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{6} - 3147 T^{5} + \cdots - 23\!\cdots\!20 \) Copy content Toggle raw display
$23$ \( T^{6} - 9122 T^{5} + \cdots + 43\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{6} - 13020 T^{5} + \cdots + 20\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( (T + 961)^{6} \) Copy content Toggle raw display
$37$ \( T^{6} - 21678 T^{5} + \cdots + 98\!\cdots\!88 \) Copy content Toggle raw display
$41$ \( T^{6} - 3227 T^{5} + \cdots + 28\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( T^{6} - 37882 T^{5} + \cdots + 22\!\cdots\!40 \) Copy content Toggle raw display
$47$ \( T^{6} - 29708 T^{5} + \cdots - 97\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{6} + 9976 T^{5} + \cdots - 36\!\cdots\!92 \) Copy content Toggle raw display
$59$ \( T^{6} - 20573 T^{5} + \cdots - 63\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{6} - 21610 T^{5} + \cdots - 21\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{6} + 17024 T^{5} + \cdots + 49\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{6} - 44509 T^{5} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{6} + 161864 T^{5} + \cdots - 11\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{6} + 24420 T^{5} + \cdots + 44\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T^{6} + 114160 T^{5} + \cdots + 12\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{6} + 199742 T^{5} + \cdots + 65\!\cdots\!28 \) Copy content Toggle raw display
$97$ \( T^{6} + 282951 T^{5} + \cdots - 27\!\cdots\!20 \) Copy content Toggle raw display
show more
show less