Properties

Label 124.2.j.a
Level $124$
Weight $2$
Character orbit 124.j
Analytic conductor $0.990$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 124 = 2^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 124.j (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.990144985064\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{20}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{20}^{6} - \zeta_{20}^{4} - \zeta_{20}^{3} + \zeta_{20}^{2} - 1) q^{2} + (\zeta_{20}^{7} - 2 \zeta_{20}^{5} - 2 \zeta_{20}) q^{3} + 2 \zeta_{20} q^{4} + (\zeta_{20}^{6} - \zeta_{20}^{4} - 1) q^{5} + ( - 2 \zeta_{20}^{7} + 2 \zeta_{20}^{6} + \zeta_{20}^{5} + 2 \zeta_{20}^{2} + 2 \zeta_{20} - 1) q^{6} + ( - 2 \zeta_{20}^{7} + 2 \zeta_{20}^{5} - \zeta_{20}) q^{7} + (2 \zeta_{20}^{7} - 2 \zeta_{20}^{5} - 2 \zeta_{20}^{4} + 2 \zeta_{20}^{3} - 2 \zeta_{20}) q^{8} + (4 \zeta_{20}^{6} + 4 \zeta_{20}^{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{20}^{6} - \zeta_{20}^{4} - \zeta_{20}^{3} + \zeta_{20}^{2} - 1) q^{2} + (\zeta_{20}^{7} - 2 \zeta_{20}^{5} - 2 \zeta_{20}) q^{3} + 2 \zeta_{20} q^{4} + (\zeta_{20}^{6} - \zeta_{20}^{4} - 1) q^{5} + ( - 2 \zeta_{20}^{7} + 2 \zeta_{20}^{6} + \zeta_{20}^{5} + 2 \zeta_{20}^{2} + 2 \zeta_{20} - 1) q^{6} + ( - 2 \zeta_{20}^{7} + 2 \zeta_{20}^{5} - \zeta_{20}) q^{7} + (2 \zeta_{20}^{7} - 2 \zeta_{20}^{5} - 2 \zeta_{20}^{4} + 2 \zeta_{20}^{3} - 2 \zeta_{20}) q^{8} + (4 \zeta_{20}^{6} + 4 \zeta_{20}^{2}) q^{9} + ( - \zeta_{20}^{6} + \zeta_{20}^{5} + \zeta_{20} + 1) q^{10} + (4 \zeta_{20}^{7} + 4 \zeta_{20}^{3} - 2 \zeta_{20}) q^{11} + ( - 2 \zeta_{20}^{6} - 2 \zeta_{20}^{4} - 2 \zeta_{20}^{2} - 2) q^{12} + ( - 2 \zeta_{20}^{6} + 3 \zeta_{20}^{4} - \zeta_{20}^{2} - 1) q^{13} + ( - \zeta_{20}^{7} - 2 \zeta_{20}^{6} + 3 \zeta_{20}^{5} + 3 \zeta_{20}^{4} - 3 \zeta_{20}^{3} - 2 \zeta_{20}^{2} + \zeta_{20}) q^{14} + ( - \zeta_{20}^{7} + 2 \zeta_{20}^{5} + \zeta_{20}^{3} + 3 \zeta_{20}) q^{15} + 4 \zeta_{20}^{2} q^{16} + (2 \zeta_{20}^{6} + \zeta_{20}^{4} - 4 \zeta_{20}^{2} + 2) q^{17} + ( - 4 \zeta_{20}^{7} - 4 \zeta_{20}^{4} - 4 \zeta_{20}^{3} + 4 \zeta_{20} - 4) q^{18} - 5 \zeta_{20}^{3} q^{19} + (2 \zeta_{20}^{7} - 2 \zeta_{20}^{5} - 2 \zeta_{20}) q^{20} + (\zeta_{20}^{6} - \zeta_{20}^{4} - \zeta_{20}^{2} + 1) q^{21} + ( - 2 \zeta_{20}^{7} - 4 \zeta_{20}^{6} - 2 \zeta_{20}^{5} + 2 \zeta_{20}^{4} - 2 \zeta_{20}^{3} - 2 \zeta_{20} + 4) q^{22} + ( - \zeta_{20}^{7} + \zeta_{20}^{5} + \zeta_{20}^{3} - \zeta_{20}) q^{23} + (4 \zeta_{20}^{7} - 2 \zeta_{20}^{6} + 4 \zeta_{20}^{4} + 4 \zeta_{20}^{3} - 2 \zeta_{20} + 4) q^{24} + ( - \zeta_{20}^{6} + \zeta_{20}^{4} - 3) q^{25} + ( - \zeta_{20}^{7} - \zeta_{20}^{6} - \zeta_{20}^{5} + 3 \zeta_{20}^{4} + 3 \zeta_{20}^{3} - 4 \zeta_{20}^{2} - 2 \zeta_{20} + 2) q^{26} + ( - 6 \zeta_{20}^{7} - 4 \zeta_{20}^{5} - 2 \zeta_{20}^{3} + \zeta_{20}) q^{27} + (4 \zeta_{20}^{4} - 6 \zeta_{20}^{2} + 4) q^{28} + ( - 3 \zeta_{20}^{6} - 2 \zeta_{20}^{4} - 2 \zeta_{20}^{2} - 3) q^{29} + (3 \zeta_{20}^{7} - 3 \zeta_{20}^{6} - 2 \zeta_{20}^{5} - \zeta_{20}^{4} + \zeta_{20}^{3} - 2 \zeta_{20}^{2} - 4 \zeta_{20} + 1) q^{30} + (6 \zeta_{20}^{7} - 2 \zeta_{20}^{5} + 3 \zeta_{20}^{3}) q^{31} + ( - 4 \zeta_{20}^{5} - 4) q^{32} + ( - 14 \zeta_{20}^{6} + 6 \zeta_{20}^{4} - 6 \zeta_{20}^{2} + 14) q^{33} + ( - 3 \zeta_{20}^{7} + 2 \zeta_{20}^{6} + 6 \zeta_{20}^{5} - 4 \zeta_{20}^{4} - 4 \zeta_{20}^{3} + \zeta_{20}^{2} + \cdots + 2) q^{34} + \cdots + (24 \zeta_{20}^{7} - 16 \zeta_{20}^{5} + 8 \zeta_{20}^{3} - 32 \zeta_{20}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} - 4 q^{5} + 4 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} - 4 q^{5} + 4 q^{8} + 16 q^{9} + 6 q^{10} - 20 q^{12} - 20 q^{13} - 14 q^{14} + 8 q^{16} + 10 q^{17} - 24 q^{18} + 10 q^{21} + 20 q^{22} + 20 q^{24} - 28 q^{25} + 12 q^{28} - 30 q^{29} - 32 q^{32} + 60 q^{33} + 30 q^{34} + 10 q^{38} + 8 q^{40} + 16 q^{41} + 10 q^{42} - 40 q^{44} - 8 q^{45} - 10 q^{46} - 36 q^{49} + 2 q^{50} - 30 q^{53} - 50 q^{54} + 32 q^{56} + 20 q^{58} + 20 q^{60} + 38 q^{62} - 10 q^{65} - 20 q^{66} + 10 q^{69} - 8 q^{70} + 48 q^{72} - 10 q^{73} - 30 q^{74} + 20 q^{76} + 20 q^{77} - 24 q^{80} - 38 q^{81} - 4 q^{82} + 20 q^{85} + 20 q^{86} - 50 q^{89} + 32 q^{90} + 40 q^{93} + 36 q^{94} - 40 q^{96} + 26 q^{97} - 16 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(63\) \(65\)
\(\chi(n)\) \(-1\) \(-\zeta_{20}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1
0.951057 + 0.309017i
−0.951057 0.309017i
−0.587785 + 0.809017i
0.587785 0.809017i
−0.587785 0.809017i
0.587785 + 0.809017i
0.951057 0.309017i
−0.951057 + 0.309017i
−1.39680 0.221232i −2.48990 1.80902i 1.90211 + 0.618034i −1.61803 3.07768 + 3.07768i 0.224514 + 0.0729490i −2.52015 1.28408i 2.00000 + 6.15537i 2.26007 + 0.357960i
15.2 −0.221232 + 1.39680i 2.48990 + 1.80902i −1.90211 0.618034i −1.61803 −3.07768 + 3.07768i −0.224514 0.0729490i 1.28408 2.52015i 2.00000 + 6.15537i 0.357960 2.26007i
23.1 −0.642040 1.26007i 0.224514 + 0.690983i −1.17557 + 1.61803i 0.618034 0.726543 0.726543i 2.48990 3.42705i 2.79360 + 0.442463i 2.00000 1.45309i −0.396802 0.778768i
23.2 1.26007 0.642040i −0.224514 0.690983i 1.17557 1.61803i 0.618034 −0.726543 0.726543i −2.48990 + 3.42705i 0.442463 2.79360i 2.00000 1.45309i 0.778768 0.396802i
27.1 −0.642040 + 1.26007i 0.224514 0.690983i −1.17557 1.61803i 0.618034 0.726543 + 0.726543i 2.48990 + 3.42705i 2.79360 0.442463i 2.00000 + 1.45309i −0.396802 + 0.778768i
27.2 1.26007 + 0.642040i −0.224514 + 0.690983i 1.17557 + 1.61803i 0.618034 −0.726543 + 0.726543i −2.48990 3.42705i 0.442463 + 2.79360i 2.00000 + 1.45309i 0.778768 + 0.396802i
91.1 −1.39680 + 0.221232i −2.48990 + 1.80902i 1.90211 0.618034i −1.61803 3.07768 3.07768i 0.224514 0.0729490i −2.52015 + 1.28408i 2.00000 6.15537i 2.26007 0.357960i
91.2 −0.221232 1.39680i 2.48990 1.80902i −1.90211 + 0.618034i −1.61803 −3.07768 3.07768i −0.224514 + 0.0729490i 1.28408 + 2.52015i 2.00000 6.15537i 0.357960 + 2.26007i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
31.f odd 10 1 inner
124.j even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 124.2.j.a 8
4.b odd 2 1 inner 124.2.j.a 8
31.f odd 10 1 inner 124.2.j.a 8
124.j even 10 1 inner 124.2.j.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.2.j.a 8 1.a even 1 1 trivial
124.2.j.a 8 4.b odd 2 1 inner
124.2.j.a 8 31.f odd 10 1 inner
124.2.j.a 8 124.j even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 5T_{3}^{6} + 85T_{3}^{4} + 75T_{3}^{2} + 25 \) acting on \(S_{2}^{\mathrm{new}}(124, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 2 T^{7} + 2 T^{6} - 4 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{8} - 5 T^{6} + 85 T^{4} + 75 T^{2} + \cdots + 25 \) Copy content Toggle raw display
$5$ \( (T^{2} + T - 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} + 11 T^{6} + 321 T^{4} - 29 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{8} + 60 T^{6} + 1360 T^{4} + \cdots + 6400 \) Copy content Toggle raw display
$13$ \( (T^{4} + 10 T^{3} + 30 T^{2} + 15 T + 5)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 5 T^{3} + 25 T^{2} + 125 T + 125)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} - 25 T^{6} + 625 T^{4} + \cdots + 390625 \) Copy content Toggle raw display
$23$ \( T^{8} - 5 T^{6} + 85 T^{4} + 75 T^{2} + \cdots + 25 \) Copy content Toggle raw display
$29$ \( (T^{4} + 15 T^{3} + 120 T^{2} + 440 T + 605)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 79 T^{6} + 3381 T^{4} + \cdots + 923521 \) Copy content Toggle raw display
$37$ \( (T^{4} + 90 T^{2} + 405)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 8 T^{3} + 24 T^{2} + 8 T + 16)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 25 T^{6} + 250 T^{4} + \cdots + 15625 \) Copy content Toggle raw display
$47$ \( T^{8} + T^{6} + 766 T^{4} + \cdots + 130321 \) Copy content Toggle raw display
$53$ \( (T^{4} + 15 T^{3} + 45 T^{2} - 135 T + 405)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - T^{6} + 681 T^{4} + \cdots + 130321 \) Copy content Toggle raw display
$61$ \( (T^{4} + 100 T^{2} + 80)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 138 T^{2} + 3481)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 59 T^{6} + 1446 T^{4} + \cdots + 707281 \) Copy content Toggle raw display
$73$ \( (T^{4} + 5 T^{3} + 10 T^{2} + 10 T + 5)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} - 60 T^{6} + 38560 T^{4} + \cdots + 93702400 \) Copy content Toggle raw display
$83$ \( T^{8} + 40 T^{6} + 1810 T^{4} + \cdots + 3258025 \) Copy content Toggle raw display
$89$ \( (T^{4} + 25 T^{3} + 400 T^{2} + 1520 T + 1805)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 13 T^{3} + 69 T^{2} - 77 T + 121)^{2} \) Copy content Toggle raw display
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