Properties

Label 120.6.f.a
Level $120$
Weight $6$
Character orbit 120.f
Analytic conductor $19.246$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 120.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.2460583776\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.25787221056.1
Defining polynomial: \( x^{6} + 61x^{4} + 852x^{2} + 576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 9 \beta_1 q^{3} + ( - 2 \beta_{5} + \beta_{3} - 9 \beta_1 + 8) q^{5} + ( - 3 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 38 \beta_1) q^{7} - 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 \beta_1 q^{3} + ( - 2 \beta_{5} + \beta_{3} - 9 \beta_1 + 8) q^{5} + ( - 3 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 38 \beta_1) q^{7} - 81 q^{9} + (9 \beta_{5} + 9 \beta_{4} + 4 \beta_{3} - 4 \beta_{2} - 102) q^{11} + ( - 2 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} - 30 \beta_1) q^{13} + ( - 9 \beta_{5} - 18 \beta_{3} + 72 \beta_1 + 81) q^{15} + (16 \beta_{5} - 16 \beta_{4} - 23 \beta_{3} - 23 \beta_{2} - 398 \beta_1) q^{17} + (64 \beta_{5} + 64 \beta_{4} + 20 \beta_{3} - 20 \beta_{2} + 104) q^{19} + ( - 18 \beta_{5} - 18 \beta_{4} - 27 \beta_{3} + 27 \beta_{2} - 342) q^{21} + ( - 80 \beta_{5} + 80 \beta_{4} + 52 \beta_{3} + 52 \beta_{2} - 868 \beta_1) q^{23} + ( - 20 \beta_{5} + 100 \beta_{4} + 85 \beta_{3} + 75 \beta_{2} + \cdots + 655) q^{25}+ \cdots + ( - 729 \beta_{5} - 729 \beta_{4} - 324 \beta_{3} + 324 \beta_{2} + \cdots + 8262) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 50 q^{5} - 486 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 50 q^{5} - 486 q^{9} - 664 q^{11} + 540 q^{15} + 288 q^{19} - 1872 q^{21} + 3750 q^{25} - 1892 q^{29} - 10248 q^{31} - 18880 q^{35} + 1584 q^{39} - 1324 q^{41} - 4050 q^{45} + 28410 q^{49} + 20088 q^{51} - 47160 q^{55} - 10296 q^{59} - 52116 q^{61} - 13480 q^{65} + 51624 q^{69} - 28288 q^{71} - 41400 q^{75} + 220200 q^{79} + 39366 q^{81} + 95880 q^{85} + 351420 q^{89} - 17088 q^{91} - 349120 q^{95} + 53784 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 61x^{4} + 852x^{2} + 576 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 85\nu^{3} + 1596\nu ) / 1296 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -7\nu^{5} - 28\nu^{4} - 379\nu^{3} - 1516\nu^{2} - 2748\nu - 11856 ) / 432 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{5} + 28\nu^{4} - 379\nu^{3} + 1516\nu^{2} - 2748\nu + 11856 ) / 432 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 65\nu^{5} - 108\nu^{4} + 3581\nu^{3} - 3996\nu^{2} + 40884\nu - 9072 ) / 1296 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -65\nu^{5} - 108\nu^{4} - 3581\nu^{3} - 3996\nu^{2} - 40884\nu - 9072 ) / 1296 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{5} + \beta_{4} + 3\beta_{3} + 3\beta_{2} - 4\beta_1 ) / 20 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 7\beta_{5} + 7\beta_{4} + 9\beta_{3} - 9\beta_{2} - 396 ) / 20 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 39\beta_{5} - 39\beta_{4} - 97\beta_{3} - 97\beta_{2} + 996\beta_1 ) / 20 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -379\beta_{5} - 379\beta_{4} - 333\beta_{3} + 333\beta_{2} + 12972 ) / 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -1719\beta_{5} + 1719\beta_{4} + 3457\beta_{3} + 3457\beta_{2} - 52356\beta_1 ) / 20 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/120\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(41\) \(61\) \(97\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
4.49053i
6.33429i
0.843753i
4.49053i
6.33429i
0.843753i
0 9.00000i 0 −51.5439 + 21.6386i 0 21.3742i 0 −81.0000 0
49.2 0 9.00000i 0 33.8706 + 44.4723i 0 85.8595i 0 −81.0000 0
49.3 0 9.00000i 0 42.6733 36.1108i 0 168.485i 0 −81.0000 0
49.4 0 9.00000i 0 −51.5439 21.6386i 0 21.3742i 0 −81.0000 0
49.5 0 9.00000i 0 33.8706 44.4723i 0 85.8595i 0 −81.0000 0
49.6 0 9.00000i 0 42.6733 + 36.1108i 0 168.485i 0 −81.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 120.6.f.a 6
3.b odd 2 1 360.6.f.a 6
4.b odd 2 1 240.6.f.e 6
5.b even 2 1 inner 120.6.f.a 6
5.c odd 4 1 600.6.a.r 3
5.c odd 4 1 600.6.a.s 3
12.b even 2 1 720.6.f.l 6
15.d odd 2 1 360.6.f.a 6
20.d odd 2 1 240.6.f.e 6
60.h even 2 1 720.6.f.l 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.6.f.a 6 1.a even 1 1 trivial
120.6.f.a 6 5.b even 2 1 inner
240.6.f.e 6 4.b odd 2 1
240.6.f.e 6 20.d odd 2 1
360.6.f.a 6 3.b odd 2 1
360.6.f.a 6 15.d odd 2 1
600.6.a.r 3 5.c odd 4 1
600.6.a.s 3 5.c odd 4 1
720.6.f.l 6 12.b even 2 1
720.6.f.l 6 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} + 36216T_{7}^{4} + 225603600T_{7}^{2} + 95604640000 \) acting on \(S_{6}^{\mathrm{new}}(120, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} + 81)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} - 50 T^{5} + \cdots + 30517578125 \) Copy content Toggle raw display
$7$ \( T^{6} + 36216 T^{4} + \cdots + 95604640000 \) Copy content Toggle raw display
$11$ \( (T^{3} + 332 T^{2} - 74644 T + 751600)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 62184 T^{4} + \cdots + 743071584256 \) Copy content Toggle raw display
$17$ \( T^{6} + 2125752 T^{4} + \cdots + 31\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( (T^{3} - 144 T^{2} - 5516736 T + 2584131584)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 25770672 T^{4} + \cdots + 21\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( (T^{3} + 946 T^{2} + \cdots + 37303134464)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 5124 T^{2} + \cdots - 71486323200)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 125310888 T^{4} + \cdots + 54\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{3} + 662 T^{2} + \cdots + 289547160040)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 379478448 T^{4} + \cdots + 40\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{6} + 1087347984 T^{4} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{6} + 545716248 T^{4} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{3} + 5148 T^{2} + \cdots - 6990217203600)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 26058 T^{2} + \cdots - 30927698302664)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 593605008 T^{4} + \cdots + 34\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{3} + 14144 T^{2} + \cdots - 37186025753600)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 10300131936 T^{4} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{3} - 110100 T^{2} + \cdots + 4303737800704)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 17756024496 T^{4} + \cdots + 33\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{3} - 175710 T^{2} + \cdots + 11306161037592)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 33401863872 T^{4} + \cdots + 34\!\cdots\!24 \) Copy content Toggle raw display
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