Properties

Label 120.2.d.b
Level $120$
Weight $2$
Character orbit 120.d
Analytic conductor $0.958$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.958204824255\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.839056.1
Defining polynomial: \( x^{6} + 6x^{4} + 8x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 6x^{4} + 8x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + \nu^{4} + 5\nu^{3} + 5\nu^{2} + 5\nu + 3 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 5\nu^{3} + 3\nu^{2} - 3\nu - 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} - \nu^{4} - 7\nu^{3} - 5\nu^{2} - 9\nu - 3 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{5} + \nu^{4} - 17\nu^{3} + 5\nu^{2} - 19\nu + 3 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} + \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 2\beta_{3} + \beta_{2} + \beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{4} - 2\beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4\beta_{5} - \beta_{4} + 10\beta_{3} - 3\beta_{2} - 5\beta _1 + 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{5} + 16\beta_{4} + 17\beta_{2} + 5\beta_1 ) / 2 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
0.373087i
0.373087i
2.02852i
2.02852i
1.32132i
1.32132i
−1.16170 0.806504i −1.00000 0.699104 + 1.87383i 1.86081 1.23992i 1.16170 + 0.806504i 0.746175i 0.699104 2.74067i 1.00000 −3.16170 0.0603290i
109.2 −1.16170 + 0.806504i −1.00000 0.699104 1.87383i 1.86081 + 1.23992i 1.16170 0.806504i 0.746175i 0.699104 + 2.74067i 1.00000 −3.16170 + 0.0603290i
109.3 0.321037 1.37729i −1.00000 −1.79387 0.884323i −2.11491 0.726062i −0.321037 + 1.37729i 4.05705i −1.79387 + 2.18678i 1.00000 −1.67896 + 2.67975i
109.4 0.321037 + 1.37729i −1.00000 −1.79387 + 0.884323i −2.11491 + 0.726062i −0.321037 1.37729i 4.05705i −1.79387 2.18678i 1.00000 −1.67896 2.67975i
109.5 1.34067 0.450129i −1.00000 1.59477 1.20695i 0.254102 2.22158i −1.34067 + 0.450129i 2.64265i 1.59477 2.33596i 1.00000 −0.659335 3.09278i
109.6 1.34067 + 0.450129i −1.00000 1.59477 + 1.20695i 0.254102 + 2.22158i −1.34067 0.450129i 2.64265i 1.59477 + 2.33596i 1.00000 −0.659335 + 3.09278i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.f 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.d.b yes 6
3.b odd 2 1 360.2.d.e 6
4.b odd 2 1 480.2.d.b 6
5.b even 2 1 120.2.d.a 6
5.c odd 4 2 600.2.k.f 12
8.b even 2 1 120.2.d.a 6
8.d odd 2 1 480.2.d.a 6
12.b even 2 1 1440.2.d.f 6
15.d odd 2 1 360.2.d.f 6
15.e even 4 2 1800.2.k.u 12
16.e even 4 2 3840.2.f.l 12
16.f odd 4 2 3840.2.f.m 12
20.d odd 2 1 480.2.d.a 6
20.e even 4 2 2400.2.k.f 12
24.f even 2 1 1440.2.d.e 6
24.h odd 2 1 360.2.d.f 6
40.e odd 2 1 480.2.d.b 6
40.f even 2 1 inner 120.2.d.b yes 6
40.i odd 4 2 600.2.k.f 12
40.k even 4 2 2400.2.k.f 12
60.h even 2 1 1440.2.d.e 6
60.l odd 4 2 7200.2.k.u 12
80.k odd 4 2 3840.2.f.m 12
80.q even 4 2 3840.2.f.l 12
120.i odd 2 1 360.2.d.e 6
120.m even 2 1 1440.2.d.f 6
120.q odd 4 2 7200.2.k.u 12
120.w even 4 2 1800.2.k.u 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.d.a 6 5.b even 2 1
120.2.d.a 6 8.b even 2 1
120.2.d.b yes 6 1.a even 1 1 trivial
120.2.d.b yes 6 40.f even 2 1 inner
360.2.d.e 6 3.b odd 2 1
360.2.d.e 6 120.i odd 2 1
360.2.d.f 6 15.d odd 2 1
360.2.d.f 6 24.h odd 2 1
480.2.d.a 6 8.d odd 2 1
480.2.d.a 6 20.d odd 2 1
480.2.d.b 6 4.b odd 2 1
480.2.d.b 6 40.e odd 2 1
600.2.k.f 12 5.c odd 4 2
600.2.k.f 12 40.i odd 4 2
1440.2.d.e 6 24.f even 2 1
1440.2.d.e 6 60.h even 2 1
1440.2.d.f 6 12.b even 2 1
1440.2.d.f 6 120.m even 2 1
1800.2.k.u 12 15.e even 4 2
1800.2.k.u 12 120.w even 4 2
2400.2.k.f 12 20.e even 4 2
2400.2.k.f 12 40.k even 4 2
3840.2.f.l 12 16.e even 4 2
3840.2.f.l 12 80.q even 4 2
3840.2.f.m 12 16.f odd 4 2
3840.2.f.m 12 80.k odd 4 2
7200.2.k.u 12 60.l odd 4 2
7200.2.k.u 12 120.q odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{3} - 4T_{13}^{2} - 16T_{13} + 56 \) acting on \(S_{2}^{\mathrm{new}}(120, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - T^{5} - 4T + 8 \) Copy content Toggle raw display
$3$ \( (T + 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - T^{4} + 8 T^{3} - 5 T^{2} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( T^{6} + 24 T^{4} + 128 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( T^{6} + 32 T^{4} + 96 T^{2} + 64 \) Copy content Toggle raw display
$13$ \( (T^{3} - 4 T^{2} - 16 T + 56)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + 36 T^{4} + 368 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$19$ \( T^{6} + 60 T^{4} + 512 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$23$ \( T^{6} + 92 T^{4} + 2304 T^{2} + \cdots + 16384 \) Copy content Toggle raw display
$29$ \( T^{6} + 108 T^{4} + 3120 T^{2} + \cdots + 12544 \) Copy content Toggle raw display
$31$ \( (T^{3} + 8 T^{2} - 4 T - 64)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 8 T^{2} + 8)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} + 2 T^{2} - 100 T + 56)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 64 T - 64)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 60 T^{4} + 512 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$53$ \( (T^{3} + 12 T^{2} + 32 T - 8)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 176 T^{4} + 9888 T^{2} + \cdots + 179776 \) Copy content Toggle raw display
$61$ \( T^{6} + 176 T^{4} + 7168 T^{2} + \cdots + 65536 \) Copy content Toggle raw display
$67$ \( (T^{3} - 64 T + 64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} - 8 T^{2} - 80 T + 128)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 384 T^{4} + 34560 T^{2} + \cdots + 16384 \) Copy content Toggle raw display
$79$ \( (T^{3} - 8 T^{2} - 4 T + 64)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} - 8 T^{2} - 64 T + 448)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} + 10 T^{2} - 164 T - 1384)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 336 T^{4} + 28416 T^{2} + \cdots + 262144 \) Copy content Toggle raw display
show more
show less