Properties

Label 124.2.f.b
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: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} + 1) q^{3} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 1) q^{5} + (2 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 2 \zeta_{10}) q^{7} - 2 \zeta_{10}^{3} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} + 1) q^{3} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 1) q^{5} + (2 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 2 \zeta_{10}) q^{7} - 2 \zeta_{10}^{3} q^{9} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10}) q^{11} + (4 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 4) q^{13} + ( - \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} + 1) q^{15} + ( - \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{17} + (6 \zeta_{10}^{2} - 3 \zeta_{10} + 6) q^{19} + ( - 4 \zeta_{10}^{2} + 7 \zeta_{10} - 4) q^{21} + (3 \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{23} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} - 3) q^{25} + (2 \zeta_{10}^{3} - \zeta_{10}^{2} + 2 \zeta_{10}) q^{27} + ( - \zeta_{10}^{2} - 4 \zeta_{10} - 1) q^{29} + (2 \zeta_{10}^{3} + 3 \zeta_{10} - 6) q^{31} + (2 \zeta_{10}^{2} - 6 \zeta_{10} + 2) q^{33} + ( - 5 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 5 \zeta_{10}) q^{35} + ( - 6 \zeta_{10}^{3} + 6 \zeta_{10}^{2} + 1) q^{37} + (6 \zeta_{10}^{3} + 7 \zeta_{10} - 7) q^{39} + ( - 4 \zeta_{10}^{2} - 4 \zeta_{10} - 4) q^{41} + (3 \zeta_{10}^{2} - 6 \zeta_{10} + 3) q^{43} + (2 \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{45} + ( - 3 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} + 3) q^{47} + ( - 2 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} + 2) q^{49} - 5 \zeta_{10}^{2} q^{51} + (3 \zeta_{10}^{3} + 10 \zeta_{10} - 10) q^{53} + (4 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 4 \zeta_{10}) q^{55} + 15 q^{57} + ( - 9 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10} + 9) q^{59} + 2 q^{61} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 2) q^{63} + ( - 3 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 5 \zeta_{10} + 3) q^{65} + 3 q^{67} + (8 \zeta_{10}^{3} - 9 \zeta_{10}^{2} + 8 \zeta_{10}) q^{69} + ( - 10 \zeta_{10}^{3} - 5 \zeta_{10} + 5) q^{71} + ( - 3 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 3 \zeta_{10}) q^{73} + (9 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} - 9) q^{75} + ( - 4 \zeta_{10}^{3} + 10 \zeta_{10}^{2} - 10 \zeta_{10} + 4) q^{77} + 11 \zeta_{10} q^{81} + ( - 3 \zeta_{10}^{2} - 3 \zeta_{10} - 3) q^{83} + ( - \zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{85} + (9 \zeta_{10}^{3} - 9 \zeta_{10}^{2} - 7) q^{87} + ( - 13 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 13 \zeta_{10}) q^{89} + ( - 5 \zeta_{10}^{2} + 7 \zeta_{10} - 5) q^{91} + (4 \zeta_{10}^{3} + 10 \zeta_{10}^{2} - 2 \zeta_{10} - 3) q^{93} + ( - 9 \zeta_{10}^{2} + 12 \zeta_{10} - 9) q^{95} + (6 \zeta_{10}^{3} - 5 \zeta_{10}^{2} + 6 \zeta_{10}) q^{97} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 5 q^{3} - 6 q^{5} + 7 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 5 q^{3} - 6 q^{5} + 7 q^{7} - 2 q^{9} - 6 q^{11} - 6 q^{13} - 5 q^{15} + 5 q^{17} + 15 q^{19} - 5 q^{21} + 9 q^{23} - 6 q^{25} + 5 q^{27} - 7 q^{29} - 19 q^{31} - 18 q^{35} - 8 q^{37} - 15 q^{39} - 16 q^{41} + 3 q^{43} + 8 q^{45} + 21 q^{47} + 18 q^{49} + 5 q^{51} - 27 q^{53} + 14 q^{55} + 60 q^{57} + 21 q^{59} + 8 q^{61} - 16 q^{63} - q^{65} + 12 q^{67} + 25 q^{69} + 5 q^{71} - 11 q^{73} - 15 q^{75} - 8 q^{77} + 11 q^{81} - 12 q^{83} - 10 q^{85} - 10 q^{87} - 31 q^{89} - 8 q^{91} - 20 q^{93} - 15 q^{95} + 17 q^{97} + 8 q^{99}+O(q^{100}) \) 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 1.80902 1.31433i 0 −0.381966 0 0.0729490 + 0.224514i 0 0.618034 1.90211i 0
97.1 0 0.690983 2.12663i 0 −2.61803 0 3.42705 2.48990i 0 −1.61803 1.17557i 0
101.1 0 0.690983 + 2.12663i 0 −2.61803 0 3.42705 + 2.48990i 0 −1.61803 + 1.17557i 0
109.1 0 1.80902 + 1.31433i 0 −0.381966 0 0.0729490 0.224514i 0 0.618034 + 1.90211i 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.b 4
3.b odd 2 1 1116.2.m.c 4
4.b odd 2 1 496.2.n.a 4
31.d even 5 1 inner 124.2.f.b 4
31.d even 5 1 3844.2.a.g 2
31.f odd 10 1 3844.2.a.f 2
93.l odd 10 1 1116.2.m.c 4
124.l odd 10 1 496.2.n.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.2.f.b 4 1.a even 1 1 trivial
124.2.f.b 4 31.d even 5 1 inner
496.2.n.a 4 4.b odd 2 1
496.2.n.a 4 124.l odd 10 1
1116.2.m.c 4 3.b odd 2 1
1116.2.m.c 4 93.l odd 10 1
3844.2.a.f 2 31.f odd 10 1
3844.2.a.g 2 31.d even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 5T_{3}^{3} + 15T_{3}^{2} - 25T_{3} + 25 \) 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
$3$ \( T^{4} - 5 T^{3} + 15 T^{2} - 25 T + 25 \) Copy content Toggle raw display
$5$ \( (T^{2} + 3 T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 7 T^{3} + 19 T^{2} - 3 T + 1 \) Copy content Toggle raw display
$11$ \( T^{4} + 6 T^{3} + 16 T^{2} + 16 T + 16 \) Copy content Toggle raw display
$13$ \( T^{4} + 6 T^{3} + 16 T^{2} + 11 T + 121 \) Copy content Toggle raw display
$17$ \( T^{4} - 5 T^{3} + 15 T^{2} - 25 T + 25 \) Copy content Toggle raw display
$19$ \( T^{4} - 15 T^{3} + 135 T^{2} + \cdots + 2025 \) Copy content Toggle raw display
$23$ \( T^{4} - 9 T^{3} + 31 T^{2} + 11 T + 121 \) Copy content Toggle raw display
$29$ \( T^{4} + 7 T^{3} + 24 T^{2} + 38 T + 361 \) Copy content Toggle raw display
$31$ \( T^{4} + 19 T^{3} + 151 T^{2} + \cdots + 961 \) Copy content Toggle raw display
$37$ \( (T^{2} + 4 T - 41)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 16 T^{3} + 96 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$43$ \( T^{4} - 3 T^{3} + 54 T^{2} + 108 T + 81 \) Copy content Toggle raw display
$47$ \( T^{4} - 21 T^{3} + 306 T^{2} + \cdots + 9801 \) Copy content Toggle raw display
$53$ \( T^{4} + 27 T^{3} + 379 T^{2} + \cdots + 14641 \) Copy content Toggle raw display
$59$ \( T^{4} - 21 T^{3} + 171 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$61$ \( (T - 2)^{4} \) Copy content Toggle raw display
$67$ \( (T - 3)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} - 5 T^{3} + 150 T^{2} + \cdots + 625 \) Copy content Toggle raw display
$73$ \( T^{4} + 11 T^{3} + 46 T^{2} - 4 T + 1 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 12 T^{3} + 54 T^{2} - 27 T + 81 \) Copy content Toggle raw display
$89$ \( T^{4} + 31 T^{3} + 636 T^{2} + \cdots + 43681 \) Copy content Toggle raw display
$97$ \( T^{4} - 17 T^{3} + 139 T^{2} + \cdots + 1681 \) Copy content Toggle raw display
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