Properties

Label 27.8.a.c
Level 27
Weight 8
Character orbit 27.a
Self dual Yes
Analytic conductor 8.434
Analytic rank 1
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 6\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta q^{2} \) \( -20 q^{4} \) \( -34 \beta q^{5} \) \( -559 q^{7} \) \( -148 \beta q^{8} \) \(+O(q^{10})\) \( q\) \( + \beta q^{2} \) \( -20 q^{4} \) \( -34 \beta q^{5} \) \( -559 q^{7} \) \( -148 \beta q^{8} \) \( -3672 q^{10} \) \( + 454 \beta q^{11} \) \( -8671 q^{13} \) \( -559 \beta q^{14} \) \( -13424 q^{16} \) \( + 2418 \beta q^{17} \) \( -32461 q^{19} \) \( + 680 \beta q^{20} \) \( + 49032 q^{22} \) \( -7930 \beta q^{23} \) \( + 46723 q^{25} \) \( -8671 \beta q^{26} \) \( + 11180 q^{28} \) \( + 15184 \beta q^{29} \) \( + 229892 q^{31} \) \( + 5520 \beta q^{32} \) \( + 261144 q^{34} \) \( + 19006 \beta q^{35} \) \( -541177 q^{37} \) \( -32461 \beta q^{38} \) \( + 543456 q^{40} \) \( + 34016 \beta q^{41} \) \( -465112 q^{43} \) \( -9080 \beta q^{44} \) \( -856440 q^{46} \) \( -79922 \beta q^{47} \) \( -511062 q^{49} \) \( + 46723 \beta q^{50} \) \( + 173420 q^{52} \) \( -98748 \beta q^{53} \) \( -1667088 q^{55} \) \( + 82732 \beta q^{56} \) \( + 1639872 q^{58} \) \( -75562 \beta q^{59} \) \( -137773 q^{61} \) \( + 229892 \beta q^{62} \) \( + 2314432 q^{64} \) \( + 294814 \beta q^{65} \) \( -314041 q^{67} \) \( -48360 \beta q^{68} \) \( + 2052648 q^{70} \) \( -270372 \beta q^{71} \) \( + 2669537 q^{73} \) \( -541177 \beta q^{74} \) \( + 649220 q^{76} \) \( -253786 \beta q^{77} \) \( + 1101815 q^{79} \) \( + 456416 \beta q^{80} \) \( + 3673728 q^{82} \) \( + 584956 \beta q^{83} \) \( -8878896 q^{85} \) \( -465112 \beta q^{86} \) \( -7256736 q^{88} \) \( -316230 \beta q^{89} \) \( + 4847089 q^{91} \) \( + 158600 \beta q^{92} \) \( -8631576 q^{94} \) \( + 1103674 \beta q^{95} \) \( -2979379 q^{97} \) \( -511062 \beta q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 40q^{4} \) \(\mathstrut -\mathstrut 1118q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 40q^{4} \) \(\mathstrut -\mathstrut 1118q^{7} \) \(\mathstrut -\mathstrut 7344q^{10} \) \(\mathstrut -\mathstrut 17342q^{13} \) \(\mathstrut -\mathstrut 26848q^{16} \) \(\mathstrut -\mathstrut 64922q^{19} \) \(\mathstrut +\mathstrut 98064q^{22} \) \(\mathstrut +\mathstrut 93446q^{25} \) \(\mathstrut +\mathstrut 22360q^{28} \) \(\mathstrut +\mathstrut 459784q^{31} \) \(\mathstrut +\mathstrut 522288q^{34} \) \(\mathstrut -\mathstrut 1082354q^{37} \) \(\mathstrut +\mathstrut 1086912q^{40} \) \(\mathstrut -\mathstrut 930224q^{43} \) \(\mathstrut -\mathstrut 1712880q^{46} \) \(\mathstrut -\mathstrut 1022124q^{49} \) \(\mathstrut +\mathstrut 346840q^{52} \) \(\mathstrut -\mathstrut 3334176q^{55} \) \(\mathstrut +\mathstrut 3279744q^{58} \) \(\mathstrut -\mathstrut 275546q^{61} \) \(\mathstrut +\mathstrut 4628864q^{64} \) \(\mathstrut -\mathstrut 628082q^{67} \) \(\mathstrut +\mathstrut 4105296q^{70} \) \(\mathstrut +\mathstrut 5339074q^{73} \) \(\mathstrut +\mathstrut 1298440q^{76} \) \(\mathstrut +\mathstrut 2203630q^{79} \) \(\mathstrut +\mathstrut 7347456q^{82} \) \(\mathstrut -\mathstrut 17757792q^{85} \) \(\mathstrut -\mathstrut 14513472q^{88} \) \(\mathstrut +\mathstrut 9694178q^{91} \) \(\mathstrut -\mathstrut 17263152q^{94} \) \(\mathstrut -\mathstrut 5958758q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−1.73205
1.73205
−10.3923 0 −20.0000 353.338 0 −559.000 1538.06 0 −3672.00
1.2 10.3923 0 −20.0000 −353.338 0 −559.000 −1538.06 0 −3672.00
\(n\): 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 \(T_{2}^{2} \) \(\mathstrut -\mathstrut 108 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(27))\).