Properties

Label 276.2.k.a
Level $276$
Weight $2$
Character orbit 276.k
Analytic conductor $2.204$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 276 = 2^{2} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 276.k (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.20387109579\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(8\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80q - 2q^{3} + 6q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 80q - 2q^{3} + 6q^{9} - 4q^{13} + 11q^{15} + 33q^{21} + 25q^{27} + 20q^{31} + 11q^{33} - 44q^{37} - 18q^{39} - 44q^{43} - 100q^{49} - 98q^{55} - 33q^{57} - 44q^{61} - 55q^{63} - 22q^{67} - 41q^{69} - 26q^{73} - 65q^{75} - 44q^{79} - 42q^{81} + 2q^{85} - 64q^{87} - 46q^{93} + 66q^{97} - 66q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 0 −1.56788 0.736032i 0 0.185142 1.28769i 0 −4.35485 + 1.98879i 0 1.91651 + 2.30802i 0
5.2 0 −1.25009 + 1.19887i 0 −0.395237 + 2.74894i 0 0.714786 0.326432i 0 0.125437 2.99738i 0
5.3 0 −1.09629 1.34095i 0 −0.333236 + 2.31771i 0 2.57171 1.17446i 0 −0.596294 + 2.94014i 0
5.4 0 −1.00876 + 1.40798i 0 0.395237 2.74894i 0 0.714786 0.326432i 0 −0.964814 2.84062i 0
5.5 0 0.297274 1.70635i 0 0.377490 2.62550i 0 1.06836 0.487902i 0 −2.82326 1.01451i 0
5.6 0 0.951673 + 1.44718i 0 −0.185142 + 1.28769i 0 −4.35485 + 1.98879i 0 −1.18864 + 2.75448i 0
5.7 0 1.48332 + 0.894295i 0 0.333236 2.31771i 0 2.57171 1.17446i 0 1.40047 + 2.65305i 0
5.8 0 1.64667 0.537087i 0 −0.377490 + 2.62550i 0 1.06836 0.487902i 0 2.42308 1.76881i 0
17.1 0 −1.72769 + 0.122800i 0 1.67238 1.07478i 0 −0.700880 0.100771i 0 2.96984 0.424320i 0
17.2 0 −1.38704 1.03737i 0 −1.67238 + 1.07478i 0 −0.700880 0.100771i 0 0.847741 + 2.87773i 0
17.3 0 −1.27466 + 1.17271i 0 −2.95319 + 1.89790i 0 0.486747 + 0.0699836i 0 0.249516 2.98961i 0
17.4 0 −0.438299 1.67568i 0 2.95319 1.89790i 0 0.486747 + 0.0699836i 0 −2.61579 + 1.46889i 0
17.5 0 0.604048 + 1.62331i 0 0.998768 0.641869i 0 3.17327 + 0.456247i 0 −2.27025 + 1.96111i 0
17.6 0 1.38367 + 1.04185i 0 −2.86447 + 1.84088i 0 −2.95914 0.425460i 0 0.829091 + 2.88316i 0
17.7 0 1.38578 1.03904i 0 −0.998768 + 0.641869i 0 3.17327 + 0.456247i 0 0.840791 2.87977i 0
17.8 0 1.72729 0.128392i 0 2.86447 1.84088i 0 −2.95914 0.425460i 0 2.96703 0.443540i 0
53.1 0 −1.73158 + 0.0405044i 0 −0.551710 1.20808i 0 0.335628 + 1.14304i 0 2.99672 0.140273i 0
53.2 0 −1.67064 0.457117i 0 1.12657 + 2.46685i 0 −1.33973 4.56271i 0 2.58209 + 1.52736i 0
53.3 0 −0.756167 + 1.55827i 0 0.551710 + 1.20808i 0 0.335628 + 1.14304i 0 −1.85642 2.35663i 0
53.4 0 −0.751560 1.56050i 0 1.12484 + 2.46306i 0 1.26778 + 4.31765i 0 −1.87032 + 2.34562i 0
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 245.8
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
23.d odd 22 1 inner
69.g even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 276.2.k.a 80
3.b odd 2 1 inner 276.2.k.a 80
23.d odd 22 1 inner 276.2.k.a 80
69.g even 22 1 inner 276.2.k.a 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
276.2.k.a 80 1.a even 1 1 trivial
276.2.k.a 80 3.b odd 2 1 inner
276.2.k.a 80 23.d odd 22 1 inner
276.2.k.a 80 69.g even 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(276, [\chi])\).