Properties

Label 120.2.k.b
Level $120$
Weight $2$
Character orbit 120.k
Analytic conductor $0.958$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 120.k (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.958204824255\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.399424.1
Defining polynomial: \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_1 q^{3} + (\beta_{5} + \beta_1) q^{4} + \beta_1 q^{5} - \beta_{3} q^{6} + ( - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{7} + (\beta_{5} - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + \beta_1 q^{3} + (\beta_{5} + \beta_1) q^{4} + \beta_1 q^{5} - \beta_{3} q^{6} + ( - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{7} + (\beta_{5} - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{8} - q^{9} - \beta_{3} q^{10} + ( - 2 \beta_{4} + 2 \beta_{2} - 2 \beta_1) q^{11} + (\beta_{4} - 1) q^{12} + (\beta_{5} + \beta_{4} + \beta_{3} - 3 \beta_{2} + \beta_1 + 1) q^{13} + ( - 2 \beta_{5} + 2 \beta_{2} - 2 \beta_1 - 4) q^{14} - q^{15} + (\beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{16} + ( - \beta_{5} + \beta_{4} + 3 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{17} - \beta_{2} q^{18} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{19} + (\beta_{4} - 1) q^{20} + (\beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 1) q^{21} + (2 \beta_{5} - 2 \beta_{4} - 2 \beta_1 - 2) q^{22} + ( - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{23} + (\beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{24} - q^{25} + ( - 2 \beta_{5} - 2 \beta_1 + 4) q^{26} - \beta_1 q^{27} + (2 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 4) q^{28} - 2 \beta_1 q^{29} - \beta_{2} q^{30} + ( - \beta_{5} + \beta_{4} + 3 \beta_{3} + \beta_{2} + \beta_1 - 3) q^{31} + (\beta_{5} + 3 \beta_{3} + \beta_{2} + 3 \beta_1 + 2) q^{32} + (2 \beta_{5} - 2 \beta_{3} + 2) q^{33} + ( - 2 \beta_{4} + 2 \beta_{2} + 4 \beta_1 + 2) q^{34} + (\beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 1) q^{35} + ( - \beta_{5} - \beta_1) q^{36} + ( - \beta_{5} + 3 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{37} + (2 \beta_{4} + 4 \beta_1 - 2) q^{38} + ( - \beta_{5} + \beta_{4} + 3 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{39} + (\beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{40} + (2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 4) q^{41} + ( - 2 \beta_{4} - 2 \beta_{3} - 4 \beta_1 + 2) q^{42} + ( - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 2) q^{43} + (2 \beta_{5} - 2 \beta_{4} - 4 \beta_{2} - 6 \beta_1 + 2) q^{44} - \beta_1 q^{45} + ( - 2 \beta_{5} - 2 \beta_1 - 4) q^{46} + (3 \beta_{5} - \beta_{4} - 5 \beta_{3} - \beta_{2} - \beta_1 + 3) q^{47} + ( - \beta_{5} - \beta_{3} + \beta_{2} + \beta_1 + 2) q^{48} + (2 \beta_{5} - 2 \beta_{4} - 6 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 7) q^{49} - \beta_{2} q^{50} + ( - \beta_{5} - \beta_{4} - \beta_{3} + 3 \beta_{2} + \beta_1 - 1) q^{51} + ( - 2 \beta_{5} + 2 \beta_{3} + 6 \beta_{2} + 2 \beta_1 - 4) q^{52} + 2 \beta_1 q^{53} + \beta_{3} q^{54} + (2 \beta_{5} - 2 \beta_{3} + 2) q^{55} + ( - 2 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 2) q^{56} + ( - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{57} + 2 \beta_{3} q^{58} + (2 \beta_{4} - 2 \beta_{2} - 2 \beta_1) q^{59} + ( - \beta_{5} - \beta_1) q^{60} + ( - 4 \beta_{4} + 4 \beta_{2} - 4 \beta_1) q^{61} + ( - 2 \beta_{4} - 2 \beta_{2} + 4 \beta_1 + 2) q^{62} + (\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{63} + (2 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + \beta_{2} + 5) q^{64} + ( - \beta_{5} + \beta_{4} + 3 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{65} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_1 + 2) q^{66} - 4 \beta_1 q^{67} + (2 \beta_{5} - 2 \beta_{4} - 6 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 2) q^{68} + (\beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{69} + ( - 2 \beta_{4} - 2 \beta_{3} - 4 \beta_1 + 2) q^{70} + (4 \beta_{5} - 4 \beta_{3}) q^{71} + ( - \beta_{5} + \beta_{3} + \beta_{2} + \beta_1 - 2) q^{72} - 6 q^{73} + ( - 2 \beta_{5} + 4 \beta_{4} + 4 \beta_{3} + 6 \beta_1) q^{74} - \beta_1 q^{75} + (2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 4 \beta_1 + 2) q^{76} + ( - 4 \beta_{5} + 4 \beta_{4} - 4 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 4) q^{77} + ( - 2 \beta_{4} + 4 \beta_1 + 2) q^{78} + ( - \beta_{5} + \beta_{4} + 3 \beta_{3} + \beta_{2} + \beta_1 + 5) q^{79} + ( - \beta_{5} - \beta_{3} + \beta_{2} + \beta_1 + 2) q^{80} + q^{81} + (4 \beta_{5} - 6 \beta_{2} + 4 \beta_1 + 8) q^{82} + (2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 6 \beta_1 + 2) q^{83} + (2 \beta_{3} + 2 \beta_{2} - 4 \beta_1 - 4) q^{84} + ( - \beta_{5} - \beta_{4} - \beta_{3} + 3 \beta_{2} + \beta_1 - 1) q^{85} + (4 \beta_{4} + 4 \beta_{3} + 8 \beta_1 - 4) q^{86} + 2 q^{87} + ( - 2 \beta_{5} - 2 \beta_{4} + 4 \beta_{3} - 10 \beta_1 + 2) q^{88} + ( - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 4) q^{89} + \beta_{3} q^{90} + (4 \beta_{5} + 4 \beta_{3} - 8 \beta_{2} + 8 \beta_1 + 4) q^{91} + ( - 2 \beta_{5} + 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 4) q^{92} + ( - \beta_{5} - \beta_{4} - \beta_{3} + 3 \beta_{2} - 3 \beta_1 - 1) q^{93} + (2 \beta_{5} + 4 \beta_{4} - 6 \beta_1) q^{94} + ( - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{95} + (\beta_{4} - \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 3) q^{96} + ( - 2 \beta_{5} + 2 \beta_{4} + 6 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 4) q^{97} + (4 \beta_{4} + 5 \beta_{2} - 8 \beta_1 - 4) q^{98} + (2 \beta_{4} - 2 \beta_{2} + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{2} - 2 q^{4} + 4 q^{7} + 8 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{2} - 2 q^{4} + 4 q^{7} + 8 q^{8} - 6 q^{9} - 4 q^{12} - 16 q^{14} - 6 q^{15} + 10 q^{16} + 12 q^{17} - 2 q^{18} - 4 q^{20} - 20 q^{22} - 8 q^{23} + 6 q^{24} - 6 q^{25} + 28 q^{26} - 28 q^{28} - 2 q^{30} - 12 q^{31} + 12 q^{32} + 8 q^{33} + 12 q^{34} + 2 q^{36} - 8 q^{38} + 6 q^{40} - 20 q^{41} + 8 q^{42} - 4 q^{44} - 20 q^{46} + 8 q^{47} + 16 q^{48} + 30 q^{49} - 2 q^{50} - 8 q^{52} + 8 q^{55} + 4 q^{56} - 8 q^{57} + 2 q^{60} + 4 q^{62} - 4 q^{63} + 22 q^{64} + 12 q^{66} - 16 q^{68} + 8 q^{70} - 8 q^{71} - 8 q^{72} - 36 q^{73} + 12 q^{74} + 12 q^{76} + 8 q^{78} + 36 q^{79} + 16 q^{80} + 6 q^{81} + 28 q^{82} - 20 q^{84} - 16 q^{86} + 12 q^{87} + 12 q^{88} - 28 q^{89} - 24 q^{92} + 4 q^{94} - 8 q^{95} - 10 q^{96} + 36 q^{97} - 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 3\nu^{3} - 4\nu^{2} + 2\nu - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 2\nu^{4} - 3\nu^{3} + 6\nu^{2} - 6\nu + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 3\nu^{3} + 3\nu^{2} - 2\nu + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{5} - 2\nu^{4} + 5\nu^{3} - 6\nu^{2} + 6\nu - 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{5} + 2\nu^{4} - 5\nu^{3} + 10\nu^{2} - 2\nu + 12 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} - \beta_{4} - 3\beta_{3} + \beta_{2} + \beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{5} + \beta_{4} + 3\beta_{3} + \beta_{2} + 7\beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{5} + 5\beta_{4} + 3\beta_{3} + 3\beta_{2} - \beta _1 + 1 ) / 2 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1
−0.671462 + 1.24464i
−0.671462 1.24464i
0.264658 + 1.38923i
0.264658 1.38923i
1.40680 + 0.144584i
1.40680 0.144584i
−0.671462 1.24464i 1.00000i −1.09828 + 1.67146i 1.00000i 1.24464 0.671462i 4.68585 2.81783 + 0.244644i −1.00000 1.24464 0.671462i
61.2 −0.671462 + 1.24464i 1.00000i −1.09828 1.67146i 1.00000i 1.24464 + 0.671462i 4.68585 2.81783 0.244644i −1.00000 1.24464 + 0.671462i
61.3 0.264658 1.38923i 1.00000i −1.85991 0.735342i 1.00000i −1.38923 0.264658i 0.941367 −1.51380 + 2.38923i −1.00000 −1.38923 0.264658i
61.4 0.264658 + 1.38923i 1.00000i −1.85991 + 0.735342i 1.00000i −1.38923 + 0.264658i 0.941367 −1.51380 2.38923i −1.00000 −1.38923 + 0.264658i
61.5 1.40680 0.144584i 1.00000i 1.95819 0.406803i 1.00000i 0.144584 + 1.40680i −3.62721 2.69597 0.855416i −1.00000 0.144584 + 1.40680i
61.6 1.40680 + 0.144584i 1.00000i 1.95819 + 0.406803i 1.00000i 0.144584 1.40680i −3.62721 2.69597 + 0.855416i −1.00000 0.144584 1.40680i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 120.2.k.b 6
3.b odd 2 1 360.2.k.f 6
4.b odd 2 1 480.2.k.b 6
5.b even 2 1 600.2.k.c 6
5.c odd 4 1 600.2.d.e 6
5.c odd 4 1 600.2.d.f 6
8.b even 2 1 inner 120.2.k.b 6
8.d odd 2 1 480.2.k.b 6
12.b even 2 1 1440.2.k.f 6
15.d odd 2 1 1800.2.k.p 6
15.e even 4 1 1800.2.d.q 6
15.e even 4 1 1800.2.d.r 6
16.e even 4 1 3840.2.a.bp 3
16.e even 4 1 3840.2.a.bq 3
16.f odd 4 1 3840.2.a.bo 3
16.f odd 4 1 3840.2.a.br 3
20.d odd 2 1 2400.2.k.c 6
20.e even 4 1 2400.2.d.e 6
20.e even 4 1 2400.2.d.f 6
24.f even 2 1 1440.2.k.f 6
24.h odd 2 1 360.2.k.f 6
40.e odd 2 1 2400.2.k.c 6
40.f even 2 1 600.2.k.c 6
40.i odd 4 1 600.2.d.e 6
40.i odd 4 1 600.2.d.f 6
40.k even 4 1 2400.2.d.e 6
40.k even 4 1 2400.2.d.f 6
60.h even 2 1 7200.2.k.p 6
60.l odd 4 1 7200.2.d.q 6
60.l odd 4 1 7200.2.d.r 6
120.i odd 2 1 1800.2.k.p 6
120.m even 2 1 7200.2.k.p 6
120.q odd 4 1 7200.2.d.q 6
120.q odd 4 1 7200.2.d.r 6
120.w even 4 1 1800.2.d.q 6
120.w even 4 1 1800.2.d.r 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.k.b 6 1.a even 1 1 trivial
120.2.k.b 6 8.b even 2 1 inner
360.2.k.f 6 3.b odd 2 1
360.2.k.f 6 24.h odd 2 1
480.2.k.b 6 4.b odd 2 1
480.2.k.b 6 8.d odd 2 1
600.2.d.e 6 5.c odd 4 1
600.2.d.e 6 40.i odd 4 1
600.2.d.f 6 5.c odd 4 1
600.2.d.f 6 40.i odd 4 1
600.2.k.c 6 5.b even 2 1
600.2.k.c 6 40.f even 2 1
1440.2.k.f 6 12.b even 2 1
1440.2.k.f 6 24.f even 2 1
1800.2.d.q 6 15.e even 4 1
1800.2.d.q 6 120.w even 4 1
1800.2.d.r 6 15.e even 4 1
1800.2.d.r 6 120.w even 4 1
1800.2.k.p 6 15.d odd 2 1
1800.2.k.p 6 120.i odd 2 1
2400.2.d.e 6 20.e even 4 1
2400.2.d.e 6 40.k even 4 1
2400.2.d.f 6 20.e even 4 1
2400.2.d.f 6 40.k even 4 1
2400.2.k.c 6 20.d odd 2 1
2400.2.k.c 6 40.e odd 2 1
3840.2.a.bo 3 16.f odd 4 1
3840.2.a.bp 3 16.e even 4 1
3840.2.a.bq 3 16.e even 4 1
3840.2.a.br 3 16.f odd 4 1
7200.2.d.q 6 60.l odd 4 1
7200.2.d.q 6 120.q odd 4 1
7200.2.d.r 6 60.l odd 4 1
7200.2.d.r 6 120.q odd 4 1
7200.2.k.p 6 60.h even 2 1
7200.2.k.p 6 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{3} - 2T_{7}^{2} - 16T_{7} + 16 \) acting on \(S_{2}^{\mathrm{new}}(120, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 2 T^{5} + 3 T^{4} - 6 T^{3} + \cdots + 8 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$7$ \( (T^{3} - 2 T^{2} - 16 T + 16)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} + 64 T^{4} + 1088 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$13$ \( T^{6} + 56 T^{4} + 784 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( (T^{3} - 6 T^{2} - 16 T + 32)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 40 T^{4} + 272 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( (T^{3} + 4 T^{2} - 12 T - 16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 4)^{3} \) Copy content Toggle raw display
$31$ \( (T^{3} + 6 T^{2} - 16 T - 64)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 136 T^{4} + 5648 T^{2} + \cdots + 65536 \) Copy content Toggle raw display
$41$ \( (T^{3} + 10 T^{2} - 36 T - 232)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 144 T^{4} + 5120 T^{2} + \cdots + 16384 \) Copy content Toggle raw display
$47$ \( (T^{3} - 4 T^{2} - 92 T + 496)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 4)^{3} \) Copy content Toggle raw display
$59$ \( T^{6} + 80 T^{4} + 576 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$61$ \( T^{6} + 256 T^{4} + 17408 T^{2} + \cdots + 262144 \) Copy content Toggle raw display
$67$ \( (T^{2} + 16)^{3} \) Copy content Toggle raw display
$71$ \( (T^{3} + 4 T^{2} - 112 T + 64)^{2} \) Copy content Toggle raw display
$73$ \( (T + 6)^{6} \) Copy content Toggle raw display
$79$ \( (T^{3} - 18 T^{2} + 80 T - 64)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 224 T^{4} + 8448 T^{2} + \cdots + 65536 \) Copy content Toggle raw display
$89$ \( (T^{3} + 14 T^{2} - 4 T - 184)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 18 T^{2} - 4 T + 328)^{2} \) Copy content Toggle raw display
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