Properties

Label 27.8.a.d
Level 27
Weight 8
Character orbit 27.a
Self dual Yes
Analytic conductor 8.434
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 27 = 3^{3} \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 27.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(8.43439568807\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{42}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{42}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 250 q^{4} + 20 \beta q^{5} -1261 q^{7} + 122 \beta q^{8} +O(q^{10})\) \( q + \beta q^{2} + 250 q^{4} + 20 \beta q^{5} -1261 q^{7} + 122 \beta q^{8} + 7560 q^{10} + 76 \beta q^{11} + 9581 q^{13} -1261 \beta q^{14} + 14116 q^{16} -1092 \beta q^{17} -21931 q^{19} + 5000 \beta q^{20} + 28728 q^{22} -4420 \beta q^{23} + 73075 q^{25} + 9581 \beta q^{26} -315250 q^{28} -1664 \beta q^{29} -50908 q^{31} -1500 \beta q^{32} -412776 q^{34} -25220 \beta q^{35} + 246467 q^{37} -21931 \beta q^{38} + 922320 q^{40} + 31424 \beta q^{41} + 315512 q^{43} + 19000 \beta q^{44} -1670760 q^{46} + 21868 \beta q^{47} + 766578 q^{49} + 73075 \beta q^{50} + 2395250 q^{52} + 6552 \beta q^{53} + 574560 q^{55} -153842 \beta q^{56} -628992 q^{58} -49588 \beta q^{59} -497953 q^{61} -50908 \beta q^{62} -2373848 q^{64} + 191620 \beta q^{65} + 1336361 q^{67} -273000 \beta q^{68} -9533160 q^{70} + 46392 \beta q^{71} + 3250793 q^{73} + 246467 \beta q^{74} -5482750 q^{76} -95836 \beta q^{77} + 6075485 q^{79} + 282320 \beta q^{80} + 11878272 q^{82} + 421336 \beta q^{83} -8255520 q^{85} + 315512 \beta q^{86} + 3504816 q^{88} -670740 \beta q^{89} -12081641 q^{91} -1105000 \beta q^{92} + 8266104 q^{94} -438620 \beta q^{95} + 6570629 q^{97} + 766578 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 500q^{4} - 2522q^{7} + O(q^{10}) \) \( 2q + 500q^{4} - 2522q^{7} + 15120q^{10} + 19162q^{13} + 28232q^{16} - 43862q^{19} + 57456q^{22} + 146150q^{25} - 630500q^{28} - 101816q^{31} - 825552q^{34} + 492934q^{37} + 1844640q^{40} + 631024q^{43} - 3341520q^{46} + 1533156q^{49} + 4790500q^{52} + 1149120q^{55} - 1257984q^{58} - 995906q^{61} - 4747696q^{64} + 2672722q^{67} - 19066320q^{70} + 6501586q^{73} - 10965500q^{76} + 12150970q^{79} + 23756544q^{82} - 16511040q^{85} + 7009632q^{88} - 24163282q^{91} + 16532208q^{94} + 13141258q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−6.48074
6.48074
−19.4422 0 250.000 −388.844 0 −1261.00 −2371.95 0 7560.00
1.2 19.4422 0 250.000 388.844 0 −1261.00 2371.95 0 7560.00
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2}^{2} - 378 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(27))\).