Properties

Label 768.2.o.a
Level 768
Weight 2
Character orbit 768.o
Analytic conductor 6.133
Analytic rank 0
Dimension 56
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 768.o (of order \(8\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.13251087523\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{8})\)
Coefficient ring index: multiple of None
Twist minimal: no (minimal twist has level 96)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56q - 4q^{3} + 8q^{7} - 4q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 56q - 4q^{3} + 8q^{7} - 4q^{9} + 8q^{13} + 8q^{15} - 8q^{19} + 4q^{21} - 8q^{25} - 28q^{27} - 8q^{33} + 8q^{37} + 28q^{39} - 8q^{43} + 4q^{45} - 16q^{51} - 24q^{55} - 4q^{57} + 40q^{61} + 56q^{67} + 4q^{69} - 8q^{73} + 16q^{75} - 16q^{79} + 48q^{85} - 52q^{87} + 40q^{91} - 8q^{93} - 16q^{97} + 60q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
95.1 0 −1.70463 + 0.306981i 0 2.70066 1.11865i 0 3.28204 + 3.28204i 0 2.81152 1.04658i 0
95.2 0 −1.68370 0.406387i 0 −2.81491 + 1.16597i 0 0.543879 + 0.543879i 0 2.66970 + 1.36847i 0
95.3 0 −1.57410 + 0.722645i 0 0.378520 0.156788i 0 −2.01144 2.01144i 0 1.95557 2.27503i 0
95.4 0 −1.29202 1.15356i 0 −0.180206 + 0.0746437i 0 0.289055 + 0.289055i 0 0.338620 + 2.98083i 0
95.5 0 −1.02188 + 1.39848i 0 −3.14689 + 1.30348i 0 −0.663471 0.663471i 0 −0.911503 2.85817i 0
95.6 0 −0.266294 + 1.71146i 0 3.14689 1.30348i 0 −0.663471 0.663471i 0 −2.85817 0.911503i 0
95.7 0 −0.0380372 1.73163i 0 −2.18808 + 0.906333i 0 1.93241 + 1.93241i 0 −2.99711 + 0.131733i 0
95.8 0 0.0499748 1.73133i 0 1.06973 0.443098i 0 −2.37247 2.37247i 0 −2.99501 0.173046i 0
95.9 0 0.602068 + 1.62404i 0 −0.378520 + 0.156788i 0 −2.01144 2.01144i 0 −2.27503 + 1.95557i 0
95.10 0 0.988287 + 1.42242i 0 −2.70066 + 1.11865i 0 3.28204 + 3.28204i 0 −1.04658 + 2.81152i 0
95.11 0 1.18890 1.25957i 0 −1.06973 + 0.443098i 0 −2.37247 2.37247i 0 −0.173046 2.99501i 0
95.12 0 1.25135 1.19755i 0 2.18808 0.906333i 0 1.93241 + 1.93241i 0 0.131733 2.99711i 0
95.13 0 1.47792 + 0.903198i 0 2.81491 1.16597i 0 0.543879 + 0.543879i 0 1.36847 + 2.66970i 0
95.14 0 1.72928 + 0.0979076i 0 0.180206 0.0746437i 0 0.289055 + 0.289055i 0 2.98083 + 0.338620i 0
287.1 0 −1.71185 + 0.263778i 0 1.20190 2.90164i 0 −2.49510 + 2.49510i 0 2.86084 0.903094i 0
287.2 0 −1.67029 0.458417i 0 0.632159 1.52617i 0 2.65940 2.65940i 0 2.57971 + 1.53137i 0
287.3 0 −1.54863 0.775716i 0 −0.584611 + 1.41138i 0 0.696730 0.696730i 0 1.79653 + 2.40260i 0
287.4 0 −1.39698 + 1.02394i 0 −1.20190 + 2.90164i 0 −2.49510 + 2.49510i 0 0.903094 2.86084i 0
287.5 0 −0.856920 + 1.50522i 0 −0.632159 + 1.52617i 0 2.65940 2.65940i 0 −1.53137 2.57971i 0
287.6 0 −0.641495 1.60888i 0 −0.296199 + 0.715088i 0 −2.77714 + 2.77714i 0 −2.17697 + 2.06417i 0
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 671.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
32.h odd 8 1 inner
96.o even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 768.2.o.a 56
3.b odd 2 1 inner 768.2.o.a 56
4.b odd 2 1 768.2.o.b 56
8.b even 2 1 384.2.o.a 56
8.d odd 2 1 96.2.o.a 56
12.b even 2 1 768.2.o.b 56
24.f even 2 1 96.2.o.a 56
24.h odd 2 1 384.2.o.a 56
32.g even 8 1 96.2.o.a 56
32.g even 8 1 768.2.o.b 56
32.h odd 8 1 384.2.o.a 56
32.h odd 8 1 inner 768.2.o.a 56
96.o even 8 1 384.2.o.a 56
96.o even 8 1 inner 768.2.o.a 56
96.p odd 8 1 96.2.o.a 56
96.p odd 8 1 768.2.o.b 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.2.o.a 56 8.d odd 2 1
96.2.o.a 56 24.f even 2 1
96.2.o.a 56 32.g even 8 1
96.2.o.a 56 96.p odd 8 1
384.2.o.a 56 8.b even 2 1
384.2.o.a 56 24.h odd 2 1
384.2.o.a 56 32.h odd 8 1
384.2.o.a 56 96.o even 8 1
768.2.o.a 56 1.a even 1 1 trivial
768.2.o.a 56 3.b odd 2 1 inner
768.2.o.a 56 32.h odd 8 1 inner
768.2.o.a 56 96.o even 8 1 inner
768.2.o.b 56 4.b odd 2 1
768.2.o.b 56 12.b even 2 1
768.2.o.b 56 32.g even 8 1
768.2.o.b 56 96.p odd 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{7}^{28} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(768, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database