Properties

Label 207.7.d.e
Level $207$
Weight $7$
Character orbit 207.d
Analytic conductor $47.621$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 207.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(47.6211953093\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 69)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24q + 20q^{2} + 816q^{4} + 940q^{8} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 24q + 20q^{2} + 816q^{4} + 940q^{8} + 384q^{13} + 29544q^{16} - 29336q^{23} - 61272q^{25} - 10088q^{26} - 64672q^{29} + 9696q^{31} + 319620q^{32} + 225744q^{35} - 135280q^{41} + 233232q^{46} + 74336q^{47} - 722136q^{49} - 619324q^{50} + 1059720q^{52} - 1019328q^{55} - 694344q^{58} - 1057648q^{59} + 488776q^{62} - 273888q^{64} + 2785512q^{70} + 255392q^{71} - 322560q^{73} + 1002960q^{77} - 5732712q^{82} - 2704704q^{85} + 1611444q^{92} - 147720q^{94} + 1672656q^{95} - 9104212q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1 −13.5405 0 119.346 141.090i 0 238.919i −749.409 0 1910.43i
91.2 −13.5405 0 119.346 141.090i 0 238.919i −749.409 0 1910.43i
91.3 −13.3624 0 114.554 40.7784i 0 469.379i −675.521 0 544.897i
91.4 −13.3624 0 114.554 40.7784i 0 469.379i −675.521 0 544.897i
91.5 −10.8996 0 54.8005 162.476i 0 219.782i 100.271 0 1770.91i
91.6 −10.8996 0 54.8005 162.476i 0 219.782i 100.271 0 1770.91i
91.7 −6.36312 0 −23.5107 60.2242i 0 233.613i 556.841 0 383.214i
91.8 −6.36312 0 −23.5107 60.2242i 0 233.613i 556.841 0 383.214i
91.9 −2.88321 0 −55.6871 133.948i 0 59.3150i 345.083 0 386.201i
91.10 −2.88321 0 −55.6871 133.948i 0 59.3150i 345.083 0 386.201i
91.11 0.368531 0 −63.8642 113.928i 0 569.901i −47.1219 0 41.9859i
91.12 0.368531 0 −63.8642 113.928i 0 569.901i −47.1219 0 41.9859i
91.13 3.69366 0 −50.3569 218.943i 0 261.444i −422.396 0 808.702i
91.14 3.69366 0 −50.3569 218.943i 0 261.444i −422.396 0 808.702i
91.15 4.73591 0 −41.5712 38.4146i 0 655.803i −499.975 0 181.928i
91.16 4.73591 0 −41.5712 38.4146i 0 655.803i −499.975 0 181.928i
91.17 8.53823 0 8.90142 233.104i 0 155.980i −470.445 0 1990.29i
91.18 8.53823 0 8.90142 233.104i 0 155.980i −470.445 0 1990.29i
91.19 10.6305 0 49.0083 66.6032i 0 306.844i −159.370 0 708.028i
91.20 10.6305 0 49.0083 66.6032i 0 306.844i −159.370 0 708.028i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 207.7.d.e 24
3.b odd 2 1 69.7.d.a 24
23.b odd 2 1 inner 207.7.d.e 24
69.c even 2 1 69.7.d.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.7.d.a 24 3.b odd 2 1
69.7.d.a 24 69.c even 2 1
207.7.d.e 24 1.a even 1 1 trivial
207.7.d.e 24 23.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{12} - \cdots\) acting on \(S_{7}^{\mathrm{new}}(207, [\chi])\).