Properties

Label 140.2.m.a
Level $140$
Weight $2$
Character orbit 140.m
Analytic conductor $1.118$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 140.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.11790562830\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.11574317056.3
Defining polynomial: \(x^{8} + 45 x^{4} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( -\beta_{2} + \beta_{6} ) q^{5} + ( \beta_{4} - \beta_{7} ) q^{7} + ( -\beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( -\beta_{2} + \beta_{6} ) q^{5} + ( \beta_{4} - \beta_{7} ) q^{7} + ( -\beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{9} + ( -1 + \beta_{4} + \beta_{5} ) q^{11} + \beta_{1} q^{13} + ( 2 - \beta_{3} - \beta_{4} - \beta_{5} ) q^{15} + ( \beta_{2} - 2 \beta_{6} + 2 \beta_{7} ) q^{17} + ( -\beta_{1} + \beta_{2} - 2 \beta_{7} ) q^{19} + ( -1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{6} ) q^{21} + ( -2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{23} + ( -2 + \beta_{3} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{25} + ( \beta_{2} - \beta_{6} + \beta_{7} ) q^{27} + ( 3 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{29} + ( 3 \beta_{1} + 3 \beta_{2} - 4 \beta_{6} ) q^{31} + ( -5 \beta_{1} + \beta_{6} + \beta_{7} ) q^{33} + ( -2 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{6} ) q^{35} + ( 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{37} + ( -3 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{39} + ( -\beta_{1} - \beta_{2} - 2 \beta_{6} ) q^{41} + ( -5 - 5 \beta_{3} ) q^{43} + ( 3 \beta_{1} + \beta_{2} - \beta_{6} + 2 \beta_{7} ) q^{45} + ( -5 \beta_{2} + 2 \beta_{6} - 2 \beta_{7} ) q^{47} + ( \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{49} + ( 1 + \beta_{4} + \beta_{5} ) q^{51} + ( 5 + 5 \beta_{3} ) q^{53} + ( -\beta_{1} + 4 \beta_{2} + \beta_{6} + \beta_{7} ) q^{55} + ( -5 + 5 \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{57} + ( \beta_{1} - \beta_{2} + 2 \beta_{7} ) q^{59} + ( 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{6} ) q^{61} + ( 5 - \beta_{1} + 5 \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{63} + ( 2 - \beta_{3} - \beta_{4} - \beta_{5} ) q^{65} + ( 5 - 5 \beta_{3} ) q^{67} + ( 4 \beta_{1} - 4 \beta_{2} - 2 \beta_{7} ) q^{69} + ( 2 + 2 \beta_{4} + 2 \beta_{5} ) q^{71} + ( 2 \beta_{1} - \beta_{6} - \beta_{7} ) q^{73} + ( 2 \beta_{1} - 5 \beta_{2} - 2 \beta_{7} ) q^{75} + ( 5 - \beta_{2} - 5 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} ) q^{77} + ( 3 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{79} + ( -1 - 2 \beta_{4} - 2 \beta_{5} ) q^{81} + ( 4 \beta_{1} + \beta_{6} + \beta_{7} ) q^{83} + ( 7 - 6 \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{85} + ( \beta_{2} - \beta_{6} + \beta_{7} ) q^{87} + ( -4 \beta_{1} + 4 \beta_{2} + 2 \beta_{7} ) q^{89} + ( -1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{6} ) q^{91} + ( -5 - 5 \beta_{3} + 6 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{93} + ( -5 + 5 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{95} + 3 \beta_{2} q^{97} + ( 10 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 2q^{7} + O(q^{10}) \) \( 8q - 2q^{7} - 12q^{11} + 20q^{15} - 8q^{21} + 4q^{23} - 12q^{25} - 14q^{35} - 4q^{37} - 40q^{43} + 4q^{51} + 40q^{53} - 44q^{57} + 42q^{63} + 20q^{65} + 40q^{67} + 8q^{71} + 44q^{77} + 48q^{85} - 8q^{91} - 52q^{93} - 44q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 45 x^{4} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} + 47 \nu^{3} \)\()/14\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{6} - 47 \nu^{2} \)\()/14\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{6} + 2 \nu^{5} + 2 \nu^{4} + 127 \nu^{2} + 94 \nu + 38 \)\()/28\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{6} - 2 \nu^{5} + 2 \nu^{4} - 127 \nu^{2} - 94 \nu + 38 \)\()/28\)
\(\beta_{6}\)\(=\)\((\)\( 7 \nu^{7} + 2 \nu^{5} + 315 \nu^{3} + 94 \nu \)\()/28\)
\(\beta_{7}\)\(=\)\((\)\( -7 \nu^{7} + 2 \nu^{5} - 315 \nu^{3} + 94 \nu \)\()/28\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - 3 \beta_{3}\)
\(\nu^{3}\)\(=\)\(\beta_{7} - \beta_{6} + 7 \beta_{2}\)
\(\nu^{4}\)\(=\)\(7 \beta_{5} + 7 \beta_{4} - 19\)
\(\nu^{5}\)\(=\)\(7 \beta_{7} + 7 \beta_{6} - 47 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-47 \beta_{7} - 47 \beta_{6} - 47 \beta_{5} + 47 \beta_{4} + 127 \beta_{3}\)
\(\nu^{7}\)\(=\)\(-47 \beta_{7} + 47 \beta_{6} - 315 \beta_{2}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/140\mathbb{Z}\right)^\times\).

\(n\) \(57\) \(71\) \(101\)
\(\chi(n)\) \(-\beta_{3}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
−1.83051 + 1.83051i
−0.386289 + 0.386289i
0.386289 0.386289i
1.83051 1.83051i
−1.83051 1.83051i
−0.386289 0.386289i
0.386289 + 0.386289i
1.83051 + 1.83051i
0 −1.83051 + 1.83051i 0 −1.83051 1.28422i 0 −1.57763 + 2.12393i 0 3.70156i 0
13.2 0 −0.386289 + 0.386289i 0 −0.386289 + 2.20245i 0 2.64515 0.0564123i 0 2.70156i 0
13.3 0 0.386289 0.386289i 0 0.386289 2.20245i 0 0.0564123 2.64515i 0 2.70156i 0
13.4 0 1.83051 1.83051i 0 1.83051 + 1.28422i 0 −2.12393 + 1.57763i 0 3.70156i 0
97.1 0 −1.83051 1.83051i 0 −1.83051 + 1.28422i 0 −1.57763 2.12393i 0 3.70156i 0
97.2 0 −0.386289 0.386289i 0 −0.386289 2.20245i 0 2.64515 + 0.0564123i 0 2.70156i 0
97.3 0 0.386289 + 0.386289i 0 0.386289 + 2.20245i 0 0.0564123 + 2.64515i 0 2.70156i 0
97.4 0 1.83051 + 1.83051i 0 1.83051 1.28422i 0 −2.12393 1.57763i 0 3.70156i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
35.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 140.2.m.a 8
3.b odd 2 1 1260.2.ba.a 8
4.b odd 2 1 560.2.bj.b 8
5.b even 2 1 700.2.m.c 8
5.c odd 4 1 inner 140.2.m.a 8
5.c odd 4 1 700.2.m.c 8
7.b odd 2 1 inner 140.2.m.a 8
7.c even 3 2 980.2.v.b 16
7.d odd 6 2 980.2.v.b 16
15.e even 4 1 1260.2.ba.a 8
20.e even 4 1 560.2.bj.b 8
21.c even 2 1 1260.2.ba.a 8
28.d even 2 1 560.2.bj.b 8
35.c odd 2 1 700.2.m.c 8
35.f even 4 1 inner 140.2.m.a 8
35.f even 4 1 700.2.m.c 8
35.k even 12 2 980.2.v.b 16
35.l odd 12 2 980.2.v.b 16
105.k odd 4 1 1260.2.ba.a 8
140.j odd 4 1 560.2.bj.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.m.a 8 1.a even 1 1 trivial
140.2.m.a 8 5.c odd 4 1 inner
140.2.m.a 8 7.b odd 2 1 inner
140.2.m.a 8 35.f even 4 1 inner
560.2.bj.b 8 4.b odd 2 1
560.2.bj.b 8 20.e even 4 1
560.2.bj.b 8 28.d even 2 1
560.2.bj.b 8 140.j odd 4 1
700.2.m.c 8 5.b even 2 1
700.2.m.c 8 5.c odd 4 1
700.2.m.c 8 35.c odd 2 1
700.2.m.c 8 35.f even 4 1
980.2.v.b 16 7.c even 3 2
980.2.v.b 16 7.d odd 6 2
980.2.v.b 16 35.k even 12 2
980.2.v.b 16 35.l odd 12 2
1260.2.ba.a 8 3.b odd 2 1
1260.2.ba.a 8 15.e even 4 1
1260.2.ba.a 8 21.c even 2 1
1260.2.ba.a 8 105.k odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(140, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 4 + 45 T^{4} + T^{8} \)
$5$ \( 625 + 150 T^{2} + 18 T^{4} + 6 T^{6} + T^{8} \)
$7$ \( 2401 + 686 T + 98 T^{2} - 182 T^{3} - 62 T^{4} - 26 T^{5} + 2 T^{6} + 2 T^{7} + T^{8} \)
$11$ \( ( -8 + 3 T + T^{2} )^{4} \)
$13$ \( 4 + 45 T^{4} + T^{8} \)
$17$ \( 2500 + 2109 T^{4} + T^{8} \)
$19$ \( ( 800 - 58 T^{2} + T^{4} )^{2} \)
$23$ \( ( 400 + 40 T + 2 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$29$ \( ( 16 + 33 T^{2} + T^{4} )^{2} \)
$31$ \( ( 5000 + 142 T^{2} + T^{4} )^{2} \)
$37$ \( ( 400 - 40 T + 2 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$41$ \( ( 800 + 58 T^{2} + T^{4} )^{2} \)
$43$ \( ( 50 + 10 T + T^{2} )^{4} \)
$47$ \( 7496644 + 17325 T^{4} + T^{8} \)
$53$ \( ( 50 - 10 T + T^{2} )^{4} \)
$59$ \( ( 800 - 58 T^{2} + T^{4} )^{2} \)
$61$ \( ( 12800 + 232 T^{2} + T^{4} )^{2} \)
$67$ \( ( 50 - 10 T + T^{2} )^{4} \)
$71$ \( ( -40 - 2 T + T^{2} )^{4} \)
$73$ \( 16384 + 420 T^{4} + T^{8} \)
$79$ \( ( 4 + 45 T^{2} + T^{4} )^{2} \)
$83$ \( 17909824 + 16500 T^{4} + T^{8} \)
$89$ \( ( 800 - 188 T^{2} + T^{4} )^{2} \)
$97$ \( 26244 + 3645 T^{4} + T^{8} \)
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