Properties

Label 6012.2.h.a
Level 6012
Weight 2
Character orbit 6012.h
Analytic conductor 48.006
Analytic rank 0
Dimension 56
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6012.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.0060616952\)
Analytic rank: \(0\)
Dimension: \(56\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 56q + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 56q + 8q^{19} + 64q^{25} - 8q^{31} + 56q^{49} - 8q^{61} + 32q^{85} - 48q^{97} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3005.1 0 0 0 −4.09297 0 2.29241 0 0 0
3005.2 0 0 0 −4.09297 0 2.29241 0 0 0
3005.3 0 0 0 −3.84494 0 −2.14597 0 0 0
3005.4 0 0 0 −3.84494 0 −2.14597 0 0 0
3005.5 0 0 0 −3.47368 0 3.93299 0 0 0
3005.6 0 0 0 −3.47368 0 3.93299 0 0 0
3005.7 0 0 0 −3.30258 0 −4.47594 0 0 0
3005.8 0 0 0 −3.30258 0 −4.47594 0 0 0
3005.9 0 0 0 −3.04368 0 0.233711 0 0 0
3005.10 0 0 0 −3.04368 0 0.233711 0 0 0
3005.11 0 0 0 −2.76473 0 0.618008 0 0 0
3005.12 0 0 0 −2.76473 0 0.618008 0 0 0
3005.13 0 0 0 −2.21423 0 −1.10661 0 0 0
3005.14 0 0 0 −2.21423 0 −1.10661 0 0 0
3005.15 0 0 0 −2.08738 0 3.97701 0 0 0
3005.16 0 0 0 −2.08738 0 3.97701 0 0 0
3005.17 0 0 0 −1.39469 0 −4.57475 0 0 0
3005.18 0 0 0 −1.39469 0 −4.57475 0 0 0
3005.19 0 0 0 −1.15018 0 −0.123775 0 0 0
3005.20 0 0 0 −1.15018 0 −0.123775 0 0 0
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3005.56
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
167.b odd 2 1 inner
501.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6012.2.h.a 56
3.b odd 2 1 inner 6012.2.h.a 56
167.b odd 2 1 inner 6012.2.h.a 56
501.c even 2 1 inner 6012.2.h.a 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6012.2.h.a 56 1.a even 1 1 trivial
6012.2.h.a 56 3.b odd 2 1 inner
6012.2.h.a 56 167.b odd 2 1 inner
6012.2.h.a 56 501.c even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(6012, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database