Properties

Label 124.2.f.a
Level $124$
Weight $2$
Character orbit 124.f
Analytic conductor $0.990$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 124 = 2^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 124.f (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.990144985064\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{10}^{3} - 1) q^{3} + 2 q^{5} + (\zeta_{10}^{3} + \zeta_{10}^{2} + \zeta_{10}) q^{7} + (\zeta_{10}^{3} - \zeta_{10} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{10}^{3} - 1) q^{3} + 2 q^{5} + (\zeta_{10}^{3} + \zeta_{10}^{2} + \zeta_{10}) q^{7} + (\zeta_{10}^{3} - \zeta_{10} + 1) q^{9} + (\zeta_{10}^{3} + \zeta_{10}^{2} + \zeta_{10}) q^{11} + ( - \zeta_{10}^{3} + 1) q^{13} + (2 \zeta_{10}^{3} - 2) q^{15} + ( - 4 \zeta_{10}^{3} - \zeta_{10} + 1) q^{17} + ( - 3 \zeta_{10}^{2} - 3) q^{19} + ( - 2 \zeta_{10}^{2} - \zeta_{10} - 2) q^{21} + ( - 6 \zeta_{10}^{3} + \zeta_{10} - 1) q^{23} - q^{25} + ( - 4 \zeta_{10}^{3} + \zeta_{10}^{2} - 4 \zeta_{10}) q^{27} + ( - 3 \zeta_{10}^{2} + 4 \zeta_{10} - 3) q^{29} + (6 \zeta_{10}^{2} - 2 \zeta_{10} + 3) q^{31} + ( - 2 \zeta_{10}^{2} - \zeta_{10} - 2) q^{33} + (2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{35} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} - 2) q^{37} + (2 \zeta_{10}^{3} + \zeta_{10} - 1) q^{39} + (5 \zeta_{10}^{2} + 4 \zeta_{10} + 5) q^{41} + ( - 3 \zeta_{10}^{2} - 4 \zeta_{10} - 3) q^{43} + (2 \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{45} + (\zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 1) q^{47} + ( - 2 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 5 \zeta_{10} + 2) q^{49} + (4 \zeta_{10}^{3} + \zeta_{10}^{2} + 4 \zeta_{10}) q^{51} + ( - 8 \zeta_{10}^{3} - \zeta_{10} + 1) q^{53} + (2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{55} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} + 6) q^{57} + ( - 7 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} + 7) q^{59} + (12 \zeta_{10}^{3} - 12 \zeta_{10}^{2} - 6) q^{61} - q^{63} + ( - 2 \zeta_{10}^{3} + 2) q^{65} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4) q^{67} + (6 \zeta_{10}^{3} - \zeta_{10}^{2} + 6 \zeta_{10}) q^{69} + (6 \zeta_{10}^{3} - 3 \zeta_{10} + 3) q^{71} + (\zeta_{10}^{3} + 11 \zeta_{10}^{2} + \zeta_{10}) q^{73} + ( - \zeta_{10}^{3} + 1) q^{75} + (5 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 5) q^{77} + ( - 2 \zeta_{10}^{3} - 3 \zeta_{10} + 3) q^{79} + (6 \zeta_{10}^{2} - 2 \zeta_{10} + 6) q^{81} + ( - 3 \zeta_{10}^{2} + 8 \zeta_{10} - 3) q^{83} + ( - 8 \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{85} + (\zeta_{10}^{3} - \zeta_{10}^{2} + 2) q^{87} + ( - 7 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 7 \zeta_{10}) q^{89} + (2 \zeta_{10}^{2} + \zeta_{10} + 2) q^{91} + (\zeta_{10}^{3} - 4 \zeta_{10}^{2} - 7) q^{93} + ( - 6 \zeta_{10}^{2} - 6) q^{95} + (\zeta_{10}^{3} + 15 \zeta_{10}^{2} + \zeta_{10}) q^{97} - q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{3} + 8 q^{5} + q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 3 q^{3} + 8 q^{5} + q^{7} + 4 q^{9} + q^{11} + 3 q^{13} - 6 q^{15} - q^{17} - 9 q^{19} - 7 q^{21} - 9 q^{23} - 4 q^{25} - 9 q^{27} - 5 q^{29} + 4 q^{31} - 7 q^{33} + 2 q^{35} - q^{39} + 19 q^{41} - 13 q^{43} + 8 q^{45} + 5 q^{47} - 4 q^{49} + 7 q^{51} - 5 q^{53} + 2 q^{55} + 18 q^{57} + 13 q^{59} - 4 q^{63} + 6 q^{65} + 24 q^{67} + 13 q^{69} + 15 q^{71} - 9 q^{73} + 3 q^{75} - 11 q^{77} + 7 q^{79} + 16 q^{81} - q^{83} - 2 q^{85} + 10 q^{87} - 17 q^{89} + 7 q^{91} - 23 q^{93} - 18 q^{95} - 13 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/124\mathbb{Z}\right)^\times\).

\(n\) \(63\) \(65\)
\(\chi(n)\) \(1\) \(-\zeta_{10}^{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
0.809017 + 0.587785i
−0.309017 0.951057i
−0.309017 + 0.951057i
0.809017 0.587785i
0 −1.30902 + 0.951057i 0 2.00000 0 0.809017 + 2.48990i 0 −0.118034 + 0.363271i 0
97.1 0 −0.190983 + 0.587785i 0 2.00000 0 −0.309017 + 0.224514i 0 2.11803 + 1.53884i 0
101.1 0 −0.190983 0.587785i 0 2.00000 0 −0.309017 0.224514i 0 2.11803 1.53884i 0
109.1 0 −1.30902 0.951057i 0 2.00000 0 0.809017 2.48990i 0 −0.118034 0.363271i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 124.2.f.a 4
3.b odd 2 1 1116.2.m.a 4
4.b odd 2 1 496.2.n.c 4
31.d even 5 1 inner 124.2.f.a 4
31.d even 5 1 3844.2.a.h 2
31.f odd 10 1 3844.2.a.e 2
93.l odd 10 1 1116.2.m.a 4
124.l odd 10 1 496.2.n.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.2.f.a 4 1.a even 1 1 trivial
124.2.f.a 4 31.d even 5 1 inner
496.2.n.c 4 4.b odd 2 1
496.2.n.c 4 124.l odd 10 1
1116.2.m.a 4 3.b odd 2 1
1116.2.m.a 4 93.l odd 10 1
3844.2.a.e 2 31.f odd 10 1
3844.2.a.h 2 31.d even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 3T_{3}^{3} + 4T_{3}^{2} + 2T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(124, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 3 T^{3} + 4 T^{2} + 2 T + 1 \) Copy content Toggle raw display
$5$ \( (T - 2)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - T^{3} + 6 T^{2} + 4 T + 1 \) Copy content Toggle raw display
$11$ \( T^{4} - T^{3} + 6 T^{2} + 4 T + 1 \) Copy content Toggle raw display
$13$ \( T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$17$ \( T^{4} + T^{3} + 16 T^{2} + 66 T + 121 \) Copy content Toggle raw display
$19$ \( T^{4} + 9 T^{3} + 36 T^{2} + 54 T + 81 \) Copy content Toggle raw display
$23$ \( T^{4} + 9 T^{3} + 46 T^{2} + \cdots + 1681 \) Copy content Toggle raw display
$29$ \( T^{4} + 5 T^{3} + 40 T^{2} + 50 T + 25 \) Copy content Toggle raw display
$31$ \( T^{4} - 4 T^{3} + 46 T^{2} - 124 T + 961 \) Copy content Toggle raw display
$37$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 19 T^{3} + 136 T^{2} + \cdots + 121 \) Copy content Toggle raw display
$43$ \( T^{4} + 13 T^{3} + 64 T^{2} + \cdots + 361 \) Copy content Toggle raw display
$47$ \( T^{4} - 5 T^{3} + 40 T^{2} - 50 T + 25 \) Copy content Toggle raw display
$53$ \( T^{4} + 5 T^{3} + 60 T^{2} + \cdots + 3025 \) Copy content Toggle raw display
$59$ \( T^{4} - 13 T^{3} + 64 T^{2} + \cdots + 361 \) Copy content Toggle raw display
$61$ \( (T^{2} - 180)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 12 T + 16)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 15 T^{3} + 90 T^{2} + \cdots + 2025 \) Copy content Toggle raw display
$73$ \( T^{4} + 9 T^{3} + 136 T^{2} + \cdots + 17161 \) Copy content Toggle raw display
$79$ \( T^{4} - 7 T^{3} + 34 T^{2} - 88 T + 121 \) Copy content Toggle raw display
$83$ \( T^{4} + T^{3} + 76 T^{2} - 434 T + 961 \) Copy content Toggle raw display
$89$ \( T^{4} + 17 T^{3} + 184 T^{2} + \cdots + 3721 \) Copy content Toggle raw display
$97$ \( T^{4} + 13 T^{3} + 244 T^{2} + \cdots + 57121 \) Copy content Toggle raw display
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