Properties

Label 124.4.d.c
Level $124$
Weight $4$
Character orbit 124.d
Analytic conductor $7.316$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 124 = 2^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 124.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.31623684071\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 2 q^{2} + 10 q^{4} - 4 q^{5} + 94 q^{8} + 536 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 2 q^{2} + 10 q^{4} - 4 q^{5} + 94 q^{8} + 536 q^{9} + 228 q^{10} - 104 q^{14} - 78 q^{16} + 114 q^{18} - 44 q^{20} + 28 q^{25} + 48 q^{28} - 602 q^{32} - 136 q^{33} - 482 q^{36} + 420 q^{38} - 516 q^{40} - 4 q^{41} - 1596 q^{45} + 1876 q^{49} - 662 q^{50} + 1576 q^{56} - 838 q^{62} - 302 q^{64} - 3900 q^{66} - 872 q^{69} - 912 q^{70} - 2166 q^{72} + 3220 q^{76} - 476 q^{78} + 572 q^{80} - 2056 q^{81} + 3096 q^{82} - 6220 q^{90} - 2904 q^{93} + 6408 q^{94} - 1836 q^{97} - 1358 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
123.1 −2.73824 0.708563i −3.81956 6.99588 + 3.88043i −9.46257 10.4589 + 2.70640i 9.26693i −16.4068 15.5825i −12.4110 25.9108 + 6.70483i
123.2 −2.73824 0.708563i 3.81956 6.99588 + 3.88043i −9.46257 −10.4589 2.70640i 9.26693i −16.4068 15.5825i −12.4110 25.9108 + 6.70483i
123.3 −2.73824 + 0.708563i −3.81956 6.99588 3.88043i −9.46257 10.4589 2.70640i 9.26693i −16.4068 + 15.5825i −12.4110 25.9108 6.70483i
123.4 −2.73824 + 0.708563i 3.81956 6.99588 3.88043i −9.46257 −10.4589 + 2.70640i 9.26693i −16.4068 + 15.5825i −12.4110 25.9108 6.70483i
123.5 −2.59794 1.11835i −8.21784 5.49858 + 5.81081i 6.37189 21.3495 + 9.19042i 24.3788i −7.78647 21.2455i 40.5329 −16.5538 7.12601i
123.6 −2.59794 1.11835i 8.21784 5.49858 + 5.81081i 6.37189 −21.3495 9.19042i 24.3788i −7.78647 21.2455i 40.5329 −16.5538 7.12601i
123.7 −2.59794 + 1.11835i −8.21784 5.49858 5.81081i 6.37189 21.3495 9.19042i 24.3788i −7.78647 + 21.2455i 40.5329 −16.5538 + 7.12601i
123.8 −2.59794 + 1.11835i 8.21784 5.49858 5.81081i 6.37189 −21.3495 + 9.19042i 24.3788i −7.78647 + 21.2455i 40.5329 −16.5538 + 7.12601i
123.9 −1.77211 2.20446i −7.49955 −1.71925 + 7.81308i −18.0979 13.2900 + 16.5324i 21.7591i 20.2703 10.0556i 29.2432 32.0715 + 39.8960i
123.10 −1.77211 2.20446i 7.49955 −1.71925 + 7.81308i −18.0979 −13.2900 16.5324i 21.7591i 20.2703 10.0556i 29.2432 32.0715 + 39.8960i
123.11 −1.77211 + 2.20446i −7.49955 −1.71925 7.81308i −18.0979 13.2900 16.5324i 21.7591i 20.2703 + 10.0556i 29.2432 32.0715 39.8960i
123.12 −1.77211 + 2.20446i 7.49955 −1.71925 7.81308i −18.0979 −13.2900 + 16.5324i 21.7591i 20.2703 + 10.0556i 29.2432 32.0715 39.8960i
123.13 −1.63773 2.30605i −0.254294 −2.63571 + 7.55335i 3.31209 0.416464 + 0.586414i 7.54711i 21.7349 6.29225i −26.9353 −5.42430 7.63784i
123.14 −1.63773 2.30605i 0.254294 −2.63571 + 7.55335i 3.31209 −0.416464 0.586414i 7.54711i 21.7349 6.29225i −26.9353 −5.42430 7.63784i
123.15 −1.63773 + 2.30605i −0.254294 −2.63571 7.55335i 3.31209 0.416464 0.586414i 7.54711i 21.7349 + 6.29225i −26.9353 −5.42430 + 7.63784i
123.16 −1.63773 + 2.30605i 0.254294 −2.63571 7.55335i 3.31209 −0.416464 + 0.586414i 7.54711i 21.7349 + 6.29225i −26.9353 −5.42430 + 7.63784i
123.17 −0.546214 2.77518i −7.92016 −7.40330 + 3.03169i 15.6847 4.32610 + 21.9799i 9.09573i 12.4573 + 18.8896i 35.7289 −8.56720 43.5279i
123.18 −0.546214 2.77518i 7.92016 −7.40330 + 3.03169i 15.6847 −4.32610 21.9799i 9.09573i 12.4573 + 18.8896i 35.7289 −8.56720 43.5279i
123.19 −0.546214 + 2.77518i −7.92016 −7.40330 3.03169i 15.6847 4.32610 21.9799i 9.09573i 12.4573 18.8896i 35.7289 −8.56720 + 43.5279i
123.20 −0.546214 + 2.77518i 7.92016 −7.40330 3.03169i 15.6847 −4.32610 + 21.9799i 9.09573i 12.4573 18.8896i 35.7289 −8.56720 + 43.5279i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 123.40
Significant digits:
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Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 124.4.d.c 40
4.b odd 2 1 inner 124.4.d.c 40
31.b odd 2 1 inner 124.4.d.c 40
124.d even 2 1 inner 124.4.d.c 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.4.d.c 40 1.a even 1 1 trivial
124.4.d.c 40 4.b odd 2 1 inner
124.4.d.c 40 31.b odd 2 1 inner
124.4.d.c 40 124.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(124, [\chi])\):

\( T_{3}^{20} - 404 T_{3}^{18} + 69148 T_{3}^{16} - 6576608 T_{3}^{14} + 382699024 T_{3}^{12} - 14122671680 T_{3}^{10} + 330615016768 T_{3}^{8} - 4740799173632 T_{3}^{6} + \cdots + 8218725875712 \) Copy content Toggle raw display
\( T_{5}^{10} + T_{5}^{9} - 628 T_{5}^{8} - 466 T_{5}^{7} + 128573 T_{5}^{6} + 104761 T_{5}^{5} - 10043754 T_{5}^{4} - 7181688 T_{5}^{3} + 293435776 T_{5}^{2} + 170815696 T_{5} - 2515695776 \) Copy content Toggle raw display