Properties

Label 531.5.c.d
Level $531$
Weight $5$
Character orbit 531.c
Analytic conductor $54.889$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 531.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(54.8894503975\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: no (minimal twist has level 177)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40q - 320q^{4} + 80q^{7} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 40q - 320q^{4} + 80q^{7} + 3944q^{16} + 528q^{17} + 444q^{19} - 444q^{20} + 1304q^{22} + 4880q^{25} + 1452q^{26} - 1160q^{28} + 996q^{29} - 10320q^{35} + 5196q^{41} - 10476q^{46} + 5104q^{49} + 2184q^{53} + 11736q^{59} - 15240q^{62} - 81012q^{64} - 29568q^{68} + 5964q^{71} - 14376q^{74} + 3480q^{76} + 19020q^{79} - 33096q^{80} + 20220q^{85} + 65880q^{86} - 14932q^{88} - 17864q^{94} - 11004q^{95} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
235.1 7.81880i 0 −45.1337 −21.3172 0 −30.7266 227.790i 0 166.675i
235.2 7.77798i 0 −44.4970 −28.5836 0 72.7001 221.650i 0 222.323i
235.3 7.65331i 0 −42.5731 38.1687 0 −35.1454 203.372i 0 292.117i
235.4 6.76718i 0 −29.7948 −6.77685 0 45.8846 93.3518i 0 45.8602i
235.5 6.70986i 0 −29.0222 41.0870 0 −6.70931 87.3773i 0 275.688i
235.6 6.10324i 0 −21.2495 12.8499 0 4.61608 32.0389i 0 78.4262i
235.7 5.77154i 0 −17.3107 −34.3452 0 −39.2727 7.56484i 0 198.225i
235.8 4.96663i 0 −8.66741 −41.3974 0 1.08721 36.4182i 0 205.606i
235.9 4.85825i 0 −7.60257 12.5087 0 50.4755 40.7968i 0 60.7705i
235.10 4.64719i 0 −5.59638 0.691812 0 −76.1022 48.3476i 0 3.21498i
235.11 4.10319i 0 −0.836195 39.6276 0 −85.8400 62.2200i 0 162.600i
235.12 4.05068i 0 −0.408022 16.4107 0 47.3137 63.1581i 0 66.4744i
235.13 3.44422i 0 4.13738 16.2205 0 92.6602 69.3575i 0 55.8670i
235.14 2.43119i 0 10.0893 −25.9319 0 −17.4595 63.4281i 0 63.0454i
235.15 2.33963i 0 10.5261 −30.0439 0 −41.7287 62.0613i 0 70.2916i
235.16 2.31914i 0 10.6216 38.6371 0 9.48964 61.7392i 0 89.6048i
235.17 2.15560i 0 11.3534 −30.9385 0 73.3307 58.9629i 0 66.6909i
235.18 1.07792i 0 14.8381 −26.1400 0 49.1298 33.2410i 0 28.1769i
235.19 0.850217i 0 15.2771 11.2738 0 −26.4494 26.5923i 0 9.58520i
235.20 0.389117i 0 15.8486 17.9988 0 −47.2538 12.3928i 0 7.00365i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 235.40
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 531.5.c.d 40
3.b odd 2 1 177.5.c.a 40
59.b odd 2 1 inner 531.5.c.d 40
177.d even 2 1 177.5.c.a 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.5.c.a 40 3.b odd 2 1
177.5.c.a 40 177.d even 2 1
531.5.c.d 40 1.a even 1 1 trivial
531.5.c.d 40 59.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(39\!\cdots\!65\)\( T_{2}^{28} + \)\(14\!\cdots\!94\)\( T_{2}^{26} + \)\(39\!\cdots\!14\)\( T_{2}^{24} + \)\(84\!\cdots\!60\)\( T_{2}^{22} + \)\(13\!\cdots\!73\)\( T_{2}^{20} + \)\(17\!\cdots\!52\)\( T_{2}^{18} + \)\(16\!\cdots\!68\)\( T_{2}^{16} + \)\(11\!\cdots\!56\)\( T_{2}^{14} + \)\(60\!\cdots\!52\)\( T_{2}^{12} + \)\(21\!\cdots\!48\)\( T_{2}^{10} + \)\(53\!\cdots\!80\)\( T_{2}^{8} + \)\(79\!\cdots\!16\)\( T_{2}^{6} + \)\(64\!\cdots\!00\)\( T_{2}^{4} + \)\(23\!\cdots\!92\)\( T_{2}^{2} + \)\(23\!\cdots\!36\)\( \)">\(T_{2}^{40} + \cdots\) acting on \(S_{5}^{\mathrm{new}}(531, [\chi])\).