Properties

Label 105.4.g.b
Level $105$
Weight $4$
Character orbit 105.g
Analytic conductor $6.195$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 105.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.19520055060\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40q + 184q^{4} + 4q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 40q + 184q^{4} + 4q^{9} - 188q^{15} + 184q^{16} + 148q^{21} + 712q^{25} - 336q^{30} - 1520q^{36} + 644q^{39} - 1488q^{46} - 1496q^{49} - 220q^{51} + 1984q^{60} + 40q^{64} - 3000q^{70} - 1192q^{79} + 4636q^{81} - 2192q^{84} + 4808q^{85} - 4408q^{91} + 5276q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
104.1 −5.11138 −1.51783 4.96953i 18.1262 −10.9509 + 2.25323i 7.75821 + 25.4011i −11.4695 14.5413i −51.7589 −22.3924 + 15.0858i 55.9744 11.5171i
104.2 −5.11138 −1.51783 + 4.96953i 18.1262 −10.9509 2.25323i 7.75821 25.4011i −11.4695 + 14.5413i −51.7589 −22.3924 15.0858i 55.9744 + 11.5171i
104.3 −5.11138 1.51783 4.96953i 18.1262 10.9509 + 2.25323i −7.75821 + 25.4011i 11.4695 + 14.5413i −51.7589 −22.3924 15.0858i −55.9744 11.5171i
104.4 −5.11138 1.51783 + 4.96953i 18.1262 10.9509 2.25323i −7.75821 25.4011i 11.4695 14.5413i −51.7589 −22.3924 + 15.0858i −55.9744 + 11.5171i
104.5 −4.45958 −5.00290 1.40393i 11.8879 −0.892950 11.1446i 22.3108 + 6.26096i 12.5955 + 13.5777i −17.3382 23.0579 + 14.0475i 3.98218 + 49.7003i
104.6 −4.45958 −5.00290 + 1.40393i 11.8879 −0.892950 + 11.1446i 22.3108 6.26096i 12.5955 13.5777i −17.3382 23.0579 14.0475i 3.98218 49.7003i
104.7 −4.45958 5.00290 1.40393i 11.8879 0.892950 11.1446i −22.3108 + 6.26096i −12.5955 13.5777i −17.3382 23.0579 14.0475i −3.98218 + 49.7003i
104.8 −4.45958 5.00290 + 1.40393i 11.8879 0.892950 + 11.1446i −22.3108 6.26096i −12.5955 + 13.5777i −17.3382 23.0579 + 14.0475i −3.98218 49.7003i
104.9 −3.23327 −3.88311 3.45274i 2.45405 8.12252 + 7.68276i 12.5552 + 11.1637i −17.9067 + 4.72745i 17.9316 3.15713 + 26.8148i −26.2623 24.8404i
104.10 −3.23327 −3.88311 + 3.45274i 2.45405 8.12252 7.68276i 12.5552 11.1637i −17.9067 4.72745i 17.9316 3.15713 26.8148i −26.2623 + 24.8404i
104.11 −3.23327 3.88311 3.45274i 2.45405 −8.12252 + 7.68276i −12.5552 + 11.1637i 17.9067 4.72745i 17.9316 3.15713 26.8148i 26.2623 24.8404i
104.12 −3.23327 3.88311 + 3.45274i 2.45405 −8.12252 7.68276i −12.5552 11.1637i 17.9067 + 4.72745i 17.9316 3.15713 + 26.8148i 26.2623 + 24.8404i
104.13 −2.24438 −1.06702 5.08542i −2.96275 8.51335 7.24727i 2.39479 + 11.4136i 12.2768 13.8665i 24.6046 −24.7230 + 10.8524i −19.1072 + 16.2656i
104.14 −2.24438 −1.06702 + 5.08542i −2.96275 8.51335 + 7.24727i 2.39479 11.4136i 12.2768 + 13.8665i 24.6046 −24.7230 10.8524i −19.1072 16.2656i
104.15 −2.24438 1.06702 5.08542i −2.96275 −8.51335 7.24727i −2.39479 + 11.4136i −12.2768 + 13.8665i 24.6046 −24.7230 10.8524i 19.1072 + 16.2656i
104.16 −2.24438 1.06702 + 5.08542i −2.96275 −8.51335 + 7.24727i −2.39479 11.4136i −12.2768 13.8665i 24.6046 −24.7230 + 10.8524i 19.1072 16.2656i
104.17 −1.22256 −4.91936 1.67328i −6.50535 −9.89077 5.21274i 6.01420 + 2.04568i −1.55936 18.4545i 17.7336 21.4003 + 16.4630i 12.0920 + 6.37288i
104.18 −1.22256 −4.91936 + 1.67328i −6.50535 −9.89077 + 5.21274i 6.01420 2.04568i −1.55936 + 18.4545i 17.7336 21.4003 16.4630i 12.0920 6.37288i
104.19 −1.22256 4.91936 1.67328i −6.50535 9.89077 5.21274i −6.01420 + 2.04568i 1.55936 + 18.4545i 17.7336 21.4003 16.4630i −12.0920 + 6.37288i
104.20 −1.22256 4.91936 + 1.67328i −6.50535 9.89077 + 5.21274i −6.01420 2.04568i 1.55936 18.4545i 17.7336 21.4003 + 16.4630i −12.0920 6.37288i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 104.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
15.d odd 2 1 inner
21.c even 2 1 inner
35.c odd 2 1 inner
105.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.4.g.b 40
3.b odd 2 1 inner 105.4.g.b 40
5.b even 2 1 inner 105.4.g.b 40
7.b odd 2 1 inner 105.4.g.b 40
15.d odd 2 1 inner 105.4.g.b 40
21.c even 2 1 inner 105.4.g.b 40
35.c odd 2 1 inner 105.4.g.b 40
105.g even 2 1 inner 105.4.g.b 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.g.b 40 1.a even 1 1 trivial
105.4.g.b 40 3.b odd 2 1 inner
105.4.g.b 40 5.b even 2 1 inner
105.4.g.b 40 7.b odd 2 1 inner
105.4.g.b 40 15.d odd 2 1 inner
105.4.g.b 40 21.c even 2 1 inner
105.4.g.b 40 35.c odd 2 1 inner
105.4.g.b 40 105.g even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - 63 T_{2}^{8} + 1377 T_{2}^{6} - 12393 T_{2}^{4} + 43014 T_{2}^{2} - 40896 \) acting on \(S_{4}^{\mathrm{new}}(105, [\chi])\).