Properties

Label 120.3.l.a
Level $120$
Weight $3$
Character orbit 120.l
Analytic conductor $3.270$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(3.26976317232\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.681615360000.5
Defining polynomial: \(x^{8} - 4 x^{7} - 2 x^{6} + 20 x^{5} + 49 x^{4} - 136 x^{3} + 168 x^{2} - 96 x + 864\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + \beta_{6} q^{5} + ( 1 + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{7} + ( 2 - \beta_{2} - \beta_{3} + \beta_{5} + 2 \beta_{6} ) q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} + \beta_{6} q^{5} + ( 1 + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{7} + ( 2 - \beta_{2} - \beta_{3} + \beta_{5} + 2 \beta_{6} ) q^{9} + ( 2 + \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{11} + ( -4 + \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{13} + \beta_{1} q^{15} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{7} ) q^{17} + ( -2 - \beta_{1} + \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 3 \beta_{6} - 2 \beta_{7} ) q^{19} + ( 9 - \beta_{1} + 3 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + 5 \beta_{6} + \beta_{7} ) q^{21} + ( -3 + \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + 4 \beta_{6} ) q^{23} -5 q^{25} + ( 5 + 2 \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{5} - 7 \beta_{6} ) q^{27} + ( \beta_{1} + 3 \beta_{2} + \beta_{3} - 9 \beta_{6} + 2 \beta_{7} ) q^{29} + ( 16 - \beta_{1} - \beta_{2} + \beta_{3} - 6 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{31} + ( -16 + \beta_{1} - \beta_{2} - \beta_{3} + 6 \beta_{4} + 4 \beta_{5} + 11 \beta_{6} + 6 \beta_{7} ) q^{33} + ( -1 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{35} + ( -\beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 6 \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{37} + ( -2 - 3 \beta_{1} - \beta_{2} - 5 \beta_{3} + 6 \beta_{4} - 4 \beta_{5} + \beta_{6} ) q^{39} + ( -5 \beta_{1} - 9 \beta_{2} - 5 \beta_{3} - 13 \beta_{6} - 4 \beta_{7} ) q^{41} + ( -45 + \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{43} + ( -9 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{45} + ( 1 - \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - 4 \beta_{7} ) q^{47} + ( 5 + 3 \beta_{1} - 7 \beta_{2} - 3 \beta_{3} - 2 \beta_{5} + 3 \beta_{6} + 4 \beta_{7} ) q^{49} + ( 10 + 3 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - 16 \beta_{6} + 5 \beta_{7} ) q^{51} + ( -4 + 6 \beta_{1} + 10 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} ) q^{53} + ( -12 + 3 \beta_{1} - 6 \beta_{2} + \beta_{3} - 4 \beta_{4} + 4 \beta_{5} - \beta_{6} - \beta_{7} ) q^{55} + ( 16 + \beta_{1} + 4 \beta_{2} - 7 \beta_{3} - 2 \beta_{4} - 10 \beta_{5} + 17 \beta_{6} + 5 \beta_{7} ) q^{57} + ( -3 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 19 \beta_{6} + 5 \beta_{7} ) q^{59} + ( 8 - 7 \beta_{1} + 15 \beta_{2} + 7 \beta_{3} + 2 \beta_{5} - 7 \beta_{6} - 8 \beta_{7} ) q^{61} + ( 1 + 5 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} - \beta_{4} + 11 \beta_{5} - 6 \beta_{6} + 4 \beta_{7} ) q^{63} + ( -4 - \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{65} + ( 31 - 3 \beta_{1} + 11 \beta_{2} - 6 \beta_{3} + 9 \beta_{4} + \beta_{5} + 6 \beta_{6} + \beta_{7} ) q^{67} + ( 13 + 4 \beta_{1} - 2 \beta_{2} + 7 \beta_{3} - 6 \beta_{4} - \beta_{5} + 4 \beta_{6} - 15 \beta_{7} ) q^{69} + ( 6 - 4 \beta_{1} + 4 \beta_{2} - 10 \beta_{3} - 6 \beta_{4} - 6 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{71} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + 8 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} ) q^{73} -5 \beta_{2} q^{75} + ( 12 + 12 \beta_{1} + 20 \beta_{2} - 12 \beta_{4} - 12 \beta_{5} - 20 \beta_{6} - 4 \beta_{7} ) q^{77} + ( 14 - \beta_{1} + 5 \beta_{2} - \beta_{3} + 2 \beta_{4} + 4 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{79} + ( 23 - 9 \beta_{1} - \beta_{2} + 5 \beta_{3} + 10 \beta_{5} - \beta_{6} - 6 \beta_{7} ) q^{81} + ( -3 - 3 \beta_{1} - 13 \beta_{2} + 3 \beta_{4} + 3 \beta_{5} + 40 \beta_{6} - 7 \beta_{7} ) q^{83} + ( 4 - 2 \beta_{1} + 7 \beta_{2} + 2 \beta_{4} + 4 \beta_{5} - 3 \beta_{7} ) q^{85} + ( -20 - 9 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} - 5 \beta_{7} ) q^{87} + ( 8 + 2 \beta_{1} + 26 \beta_{2} - 6 \beta_{3} - 8 \beta_{4} - 8 \beta_{5} + 2 \beta_{6} + 16 \beta_{7} ) q^{89} + ( 54 + 10 \beta_{1} - 22 \beta_{2} - 4 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} + 6 \beta_{7} ) q^{91} + ( -46 + 7 \beta_{1} + 20 \beta_{2} + 9 \beta_{3} - 14 \beta_{4} - 8 \beta_{5} + 3 \beta_{6} + 5 \beta_{7} ) q^{93} + ( 2 + 3 \beta_{1} + 11 \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + 6 \beta_{7} ) q^{95} + ( 6 + 8 \beta_{1} - 16 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} ) q^{97} + ( -10 + 5 \beta_{1} - 24 \beta_{2} + 7 \beta_{3} - 10 \beta_{4} + 10 \beta_{5} - 35 \beta_{6} - 23 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} + 16 q^{7} + 20 q^{9} + O(q^{10}) \) \( 8 q - 4 q^{3} + 16 q^{7} + 20 q^{9} - 8 q^{13} - 8 q^{19} + 28 q^{21} - 40 q^{25} + 20 q^{27} + 120 q^{31} - 112 q^{33} + 8 q^{37} - 72 q^{39} - 328 q^{43} - 60 q^{45} + 64 q^{49} + 64 q^{51} - 40 q^{55} + 72 q^{57} + 8 q^{61} + 88 q^{63} + 152 q^{67} + 100 q^{69} + 32 q^{73} + 20 q^{75} + 88 q^{79} + 224 q^{81} - 152 q^{87} + 560 q^{91} - 368 q^{93} + 144 q^{97} + 32 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} - 2 x^{6} + 20 x^{5} + 49 x^{4} - 136 x^{3} + 168 x^{2} - 96 x + 864\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{7} - 43 \nu^{6} + 27 \nu^{5} + 691 \nu^{4} - 866 \nu^{3} - 6528 \nu^{2} + 3264 \nu + 8424 \)\()/3720\)
\(\beta_{2}\)\(=\)\((\)\( -2 \nu^{7} + 7 \nu^{6} + 85 \nu^{5} - 230 \nu^{4} - 523 \nu^{3} + 243 \nu^{2} + 4668 \nu - 2124 \)\()/3100\)
\(\beta_{3}\)\(=\)\((\)\( 17 \nu^{7} - 447 \nu^{6} + 1215 \nu^{5} + 1955 \nu^{4} - 3692 \nu^{3} - 18728 \nu^{2} + 11472 \nu - 33096 \)\()/18600\)
\(\beta_{4}\)\(=\)\((\)\( 21 \nu^{7} - 151 \nu^{6} + 115 \nu^{5} + 2725 \nu^{4} - 4816 \nu^{3} - 16734 \nu^{2} + 2136 \nu + 56712 \)\()/18600\)
\(\beta_{5}\)\(=\)\((\)\( -69 \nu^{7} + 319 \nu^{6} + 375 \nu^{5} - 2045 \nu^{4} - 6186 \nu^{3} + 16366 \nu^{2} + 35496 \nu + 3912 \)\()/18600\)
\(\beta_{6}\)\(=\)\((\)\( -8 \nu^{7} + 28 \nu^{6} + 30 \nu^{5} - 145 \nu^{4} - 232 \nu^{3} + 507 \nu^{2} - 1788 \nu + 804 \)\()/1860\)
\(\beta_{7}\)\(=\)\((\)\( 82 \nu^{7} - 287 \nu^{6} - 385 \nu^{5} + 1680 \nu^{4} + 5943 \nu^{3} - 8413 \nu^{2} + 13212 \nu - 5916 \)\()/9300\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + 2 \beta_{5} - \beta_{4} + \beta_{1} + 1\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{6} + 2 \beta_{5} + \beta_{3} - 3 \beta_{2} - \beta_{1} + 2\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(6 \beta_{7} + 11 \beta_{6} + 2 \beta_{5} + \beta_{3} - \beta_{2} - \beta_{1} + 2\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(16 \beta_{7} + 27 \beta_{6} + 18 \beta_{5} + 20 \beta_{4} + 5 \beta_{3} - 19 \beta_{2} - 9 \beta_{1} - 50\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(54 \beta_{7} + 123 \beta_{6} + 10 \beta_{5} + 72 \beta_{4} + 17 \beta_{3} + 31 \beta_{2} - 49 \beta_{1} - 78\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(188 \beta_{7} + 427 \beta_{6} + 10 \beta_{5} + 316 \beta_{4} - 75 \beta_{3} + 169 \beta_{2} - 137 \beta_{1} - 738\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(322 \beta_{7} + 619 \beta_{6} - 334 \beta_{5} + 1088 \beta_{4} - 255 \beta_{3} + 891 \beta_{2} - 609 \beta_{1} - 1774\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1
3.22255 + 1.41421i
3.22255 1.41421i
−0.542939 1.41421i
−0.542939 + 1.41421i
−2.22255 1.41421i
−2.22255 + 1.41421i
1.54294 1.41421i
1.54294 + 1.41421i
0 −2.87275 0.864473i 0 2.23607i 0 9.02416 0 7.50537 + 4.96683i 0
41.2 0 −2.87275 + 0.864473i 0 2.23607i 0 9.02416 0 7.50537 4.96683i 0
41.3 0 −2.40140 1.79813i 0 2.23607i 0 −10.2132 0 2.53346 + 8.63606i 0
41.4 0 −2.40140 + 1.79813i 0 2.23607i 0 −10.2132 0 2.53346 8.63606i 0
41.5 0 0.291610 2.98579i 0 2.23607i 0 4.46268 0 −8.82993 1.74137i 0
41.6 0 0.291610 + 2.98579i 0 2.23607i 0 4.46268 0 −8.82993 + 1.74137i 0
41.7 0 2.98254 0.323191i 0 2.23607i 0 4.72640 0 8.79110 1.92786i 0
41.8 0 2.98254 + 0.323191i 0 2.23607i 0 4.72640 0 8.79110 + 1.92786i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 120.3.l.a 8
3.b odd 2 1 inner 120.3.l.a 8
4.b odd 2 1 240.3.l.d 8
5.b even 2 1 600.3.l.f 8
5.c odd 4 2 600.3.c.d 16
8.b even 2 1 960.3.l.h 8
8.d odd 2 1 960.3.l.g 8
12.b even 2 1 240.3.l.d 8
15.d odd 2 1 600.3.l.f 8
15.e even 4 2 600.3.c.d 16
20.d odd 2 1 1200.3.l.x 8
20.e even 4 2 1200.3.c.m 16
24.f even 2 1 960.3.l.g 8
24.h odd 2 1 960.3.l.h 8
60.h even 2 1 1200.3.l.x 8
60.l odd 4 2 1200.3.c.m 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.3.l.a 8 1.a even 1 1 trivial
120.3.l.a 8 3.b odd 2 1 inner
240.3.l.d 8 4.b odd 2 1
240.3.l.d 8 12.b even 2 1
600.3.c.d 16 5.c odd 4 2
600.3.c.d 16 15.e even 4 2
600.3.l.f 8 5.b even 2 1
600.3.l.f 8 15.d odd 2 1
960.3.l.g 8 8.d odd 2 1
960.3.l.g 8 24.f even 2 1
960.3.l.h 8 8.b even 2 1
960.3.l.h 8 24.h odd 2 1
1200.3.c.m 16 20.e even 4 2
1200.3.c.m 16 60.l odd 4 2
1200.3.l.x 8 20.d odd 2 1
1200.3.l.x 8 60.h even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(120, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 6561 + 2916 T - 162 T^{2} - 324 T^{3} - 102 T^{4} - 36 T^{5} - 2 T^{6} + 4 T^{7} + T^{8} \)
$5$ \( ( 5 + T^{2} )^{4} \)
$7$ \( ( -1944 + 872 T - 82 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$11$ \( 232989696 + 14951168 T^{2} + 226128 T^{4} + 888 T^{6} + T^{8} \)
$13$ \( ( -3456 - 3648 T - 380 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$17$ \( 15872256 + 25369856 T^{2} + 312672 T^{4} + 1104 T^{6} + T^{8} \)
$19$ \( ( 16736 - 3424 T - 708 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$23$ \( 93650688576 + 885144416 T^{2} + 2697012 T^{4} + 2964 T^{6} + T^{8} \)
$29$ \( 3474395136 + 145293824 T^{2} + 1160592 T^{4} + 2376 T^{6} + T^{8} \)
$31$ \( ( -151296 + 71360 T - 924 T^{2} - 60 T^{3} + T^{4} )^{2} \)
$37$ \( ( -31104 + 39744 T - 3228 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$41$ \( 43961355472896 + 77260718592 T^{2} + 46587024 T^{4} + 11528 T^{6} + T^{8} \)
$43$ \( ( 1582656 + 220576 T + 9482 T^{2} + 164 T^{3} + T^{4} )^{2} \)
$47$ \( 13517317696 + 266492768 T^{2} + 1541364 T^{4} + 2612 T^{6} + T^{8} \)
$53$ \( 6801580544256 + 88193961216 T^{2} + 82087776 T^{4} + 16976 T^{6} + T^{8} \)
$59$ \( 15563214360576 + 64836527616 T^{2} + 49411216 T^{4} + 12616 T^{6} + T^{8} \)
$61$ \( ( 30631296 + 12544 T - 12508 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$67$ \( ( -668224 + 236224 T - 4854 T^{2} - 76 T^{3} + T^{4} )^{2} \)
$71$ \( 35499479924736 + 145459224576 T^{2} + 83806272 T^{4} + 16304 T^{6} + T^{8} \)
$73$ \( ( 2938896 + 29376 T - 3528 T^{2} - 16 T^{3} + T^{4} )^{2} \)
$79$ \( ( -12384 - 8256 T - 1268 T^{2} - 44 T^{3} + T^{4} )^{2} \)
$83$ \( 336130569170496 + 2057217800544 T^{2} + 532895092 T^{4} + 41716 T^{6} + T^{8} \)
$89$ \( 1334603390386176 + 2220134105088 T^{2} + 649648384 T^{4} + 49312 T^{6} + T^{8} \)
$97$ \( ( 10270096 + 535392 T - 10376 T^{2} - 72 T^{3} + T^{4} )^{2} \)
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