Properties

Label 1045.2.f.a
Level $1045$
Weight $2$
Character orbit 1045.f
Analytic conductor $8.344$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1045.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.34436701122\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40q + 40q^{4} - 40q^{5} - 28q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 40q + 40q^{4} - 40q^{5} - 28q^{9} - 4q^{11} + 32q^{16} - 40q^{20} - 16q^{23} + 40q^{25} + 8q^{26} + 8q^{36} + 28q^{38} - 84q^{42} - 48q^{44} + 28q^{45} + 32q^{47} - 20q^{49} + 4q^{55} - 20q^{58} + 72q^{64} + 36q^{66} + 16q^{77} - 32q^{80} + 16q^{81} + 16q^{82} - 20q^{92} - 8q^{93} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
626.1 −2.71965 2.28311i 5.39648 −1.00000 6.20925i 2.59337i −9.23723 −2.21259 2.71965
626.2 −2.71965 2.28311i 5.39648 −1.00000 6.20925i 2.59337i −9.23723 −2.21259 2.71965
626.3 −2.52972 0.384887i 4.39948 −1.00000 0.973657i 1.35297i −6.07000 2.85186 2.52972
626.4 −2.52972 0.384887i 4.39948 −1.00000 0.973657i 1.35297i −6.07000 2.85186 2.52972
626.5 −2.08652 2.80799i 2.35358 −1.00000 5.85893i 0.950248i −0.737755 −4.88478 2.08652
626.6 −2.08652 2.80799i 2.35358 −1.00000 5.85893i 0.950248i −0.737755 −4.88478 2.08652
626.7 −2.04337 0.838315i 2.17535 −1.00000 1.71299i 4.78057i −0.358306 2.29723 2.04337
626.8 −2.04337 0.838315i 2.17535 −1.00000 1.71299i 4.78057i −0.358306 2.29723 2.04337
626.9 −1.55769 2.37450i 0.426394 −1.00000 3.69874i 4.79166i 2.45119 −2.63826 1.55769
626.10 −1.55769 2.37450i 0.426394 −1.00000 3.69874i 4.79166i 2.45119 −2.63826 1.55769
626.11 −1.53521 1.36588i 0.356884 −1.00000 2.09692i 1.36858i 2.52254 1.13438 1.53521
626.12 −1.53521 1.36588i 0.356884 −1.00000 2.09692i 1.36858i 2.52254 1.13438 1.53521
626.13 −1.42582 1.05301i 0.0329672 −1.00000 1.50140i 0.901840i 2.80464 1.89118 1.42582
626.14 −1.42582 1.05301i 0.0329672 −1.00000 1.50140i 0.901840i 2.80464 1.89118 1.42582
626.15 −0.743263 1.99663i −1.44756 −1.00000 1.48402i 2.79593i 2.56244 −0.986525 0.743263
626.16 −0.743263 1.99663i −1.44756 −1.00000 1.48402i 2.79593i 2.56244 −0.986525 0.743263
626.17 −0.512926 3.19882i −1.73691 −1.00000 1.64076i 1.77223i 1.91676 −7.23242 0.512926
626.18 −0.512926 3.19882i −1.73691 −1.00000 1.64076i 1.77223i 1.91676 −7.23242 0.512926
626.19 −0.208163 0.469109i −1.95667 −1.00000 0.0976512i 2.46634i 0.823632 2.77994 0.208163
626.20 −0.208163 0.469109i −1.95667 −1.00000 0.0976512i 2.46634i 0.823632 2.77994 0.208163
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 626.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
19.b odd 2 1 inner
209.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1045.2.f.a 40
11.b odd 2 1 inner 1045.2.f.a 40
19.b odd 2 1 inner 1045.2.f.a 40
209.d even 2 1 inner 1045.2.f.a 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.f.a 40 1.a even 1 1 trivial
1045.2.f.a 40 11.b odd 2 1 inner
1045.2.f.a 40 19.b odd 2 1 inner
1045.2.f.a 40 209.d even 2 1 inner