Properties

Label 124.4.e.a
Level $124$
Weight $4$
Character orbit 124.e
Analytic conductor $7.316$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 124 = 2^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 124.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.31623684071\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 5 \zeta_{6} + 5) q^{3} + 15 \zeta_{6} q^{5} + (29 \zeta_{6} - 29) q^{7} + 2 \zeta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 5 \zeta_{6} + 5) q^{3} + 15 \zeta_{6} q^{5} + (29 \zeta_{6} - 29) q^{7} + 2 \zeta_{6} q^{9} + 51 \zeta_{6} q^{11} - 53 \zeta_{6} q^{13} + 75 q^{15} + (51 \zeta_{6} - 51) q^{17} + ( - 145 \zeta_{6} + 145) q^{19} + 145 \zeta_{6} q^{21} + 108 q^{23} + (100 \zeta_{6} - 100) q^{25} + 145 q^{27} - 198 q^{29} + (186 \zeta_{6} - 155) q^{31} + 255 q^{33} - 435 q^{35} + ( - 55 \zeta_{6} + 55) q^{37} - 265 q^{39} - 39 \zeta_{6} q^{41} + ( - 217 \zeta_{6} + 217) q^{43} + (30 \zeta_{6} - 30) q^{45} + 504 q^{47} - 498 \zeta_{6} q^{49} + 255 \zeta_{6} q^{51} - 237 \zeta_{6} q^{53} + (765 \zeta_{6} - 765) q^{55} - 725 \zeta_{6} q^{57} + ( - 597 \zeta_{6} + 597) q^{59} - 466 q^{61} - 58 q^{63} + ( - 795 \zeta_{6} + 795) q^{65} - 749 \zeta_{6} q^{67} + ( - 540 \zeta_{6} + 540) q^{69} + 321 \zeta_{6} q^{71} + 793 \zeta_{6} q^{73} + 500 \zeta_{6} q^{75} - 1479 q^{77} + (1217 \zeta_{6} - 1217) q^{79} + ( - 671 \zeta_{6} + 671) q^{81} + 555 \zeta_{6} q^{83} - 765 q^{85} + (990 \zeta_{6} - 990) q^{87} - 378 q^{89} + 1537 q^{91} + (775 \zeta_{6} + 155) q^{93} + 2175 q^{95} + 1070 q^{97} + (102 \zeta_{6} - 102) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{3} + 15 q^{5} - 29 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 5 q^{3} + 15 q^{5} - 29 q^{7} + 2 q^{9} + 51 q^{11} - 53 q^{13} + 150 q^{15} - 51 q^{17} + 145 q^{19} + 145 q^{21} + 216 q^{23} - 100 q^{25} + 290 q^{27} - 396 q^{29} - 124 q^{31} + 510 q^{33} - 870 q^{35} + 55 q^{37} - 530 q^{39} - 39 q^{41} + 217 q^{43} - 30 q^{45} + 1008 q^{47} - 498 q^{49} + 255 q^{51} - 237 q^{53} - 765 q^{55} - 725 q^{57} + 597 q^{59} - 932 q^{61} - 116 q^{63} + 795 q^{65} - 749 q^{67} + 540 q^{69} + 321 q^{71} + 793 q^{73} + 500 q^{75} - 2958 q^{77} - 1217 q^{79} + 671 q^{81} + 555 q^{83} - 1530 q^{85} - 990 q^{87} - 756 q^{89} + 3074 q^{91} + 1085 q^{93} + 4350 q^{95} + 2140 q^{97} - 102 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(63\) \(65\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0.500000 + 0.866025i
0.500000 0.866025i
0 2.50000 4.33013i 0 7.50000 + 12.9904i 0 −14.5000 + 25.1147i 0 1.00000 + 1.73205i 0
25.1 0 2.50000 + 4.33013i 0 7.50000 12.9904i 0 −14.5000 25.1147i 0 1.00000 1.73205i 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.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 124.4.e.a 2
3.b odd 2 1 1116.4.i.a 2
4.b odd 2 1 496.4.i.a 2
31.c even 3 1 inner 124.4.e.a 2
93.h odd 6 1 1116.4.i.a 2
124.i odd 6 1 496.4.i.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.4.e.a 2 1.a even 1 1 trivial
124.4.e.a 2 31.c even 3 1 inner
496.4.i.a 2 4.b odd 2 1
496.4.i.a 2 124.i odd 6 1
1116.4.i.a 2 3.b odd 2 1
1116.4.i.a 2 93.h odd 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$5$ \( T^{2} - 15T + 225 \) Copy content Toggle raw display
$7$ \( T^{2} + 29T + 841 \) Copy content Toggle raw display
$11$ \( T^{2} - 51T + 2601 \) Copy content Toggle raw display
$13$ \( T^{2} + 53T + 2809 \) Copy content Toggle raw display
$17$ \( T^{2} + 51T + 2601 \) Copy content Toggle raw display
$19$ \( T^{2} - 145T + 21025 \) Copy content Toggle raw display
$23$ \( (T - 108)^{2} \) Copy content Toggle raw display
$29$ \( (T + 198)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 124T + 29791 \) Copy content Toggle raw display
$37$ \( T^{2} - 55T + 3025 \) Copy content Toggle raw display
$41$ \( T^{2} + 39T + 1521 \) Copy content Toggle raw display
$43$ \( T^{2} - 217T + 47089 \) Copy content Toggle raw display
$47$ \( (T - 504)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 237T + 56169 \) Copy content Toggle raw display
$59$ \( T^{2} - 597T + 356409 \) Copy content Toggle raw display
$61$ \( (T + 466)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 749T + 561001 \) Copy content Toggle raw display
$71$ \( T^{2} - 321T + 103041 \) Copy content Toggle raw display
$73$ \( T^{2} - 793T + 628849 \) Copy content Toggle raw display
$79$ \( T^{2} + 1217 T + 1481089 \) Copy content Toggle raw display
$83$ \( T^{2} - 555T + 308025 \) Copy content Toggle raw display
$89$ \( (T + 378)^{2} \) Copy content Toggle raw display
$97$ \( (T - 1070)^{2} \) Copy content Toggle raw display
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