Properties

Label 27.7.b.a
Level 27
Weight 7
Character orbit 27.b
Analytic conductor 6.211
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 27 = 3^{3} \)
Weight: \( k \) = \( 7 \)
Character orbit: \([\chi]\) = 27.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(6.21146025774\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-10}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{-10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta q^{2} \) \( -26 q^{4} \) \( + 14 \beta q^{5} \) \( -403 q^{7} \) \( + 38 \beta q^{8} \) \(+O(q^{10})\) \( q\) \( + \beta q^{2} \) \( -26 q^{4} \) \( + 14 \beta q^{5} \) \( -403 q^{7} \) \( + 38 \beta q^{8} \) \( -1260 q^{10} \) \( -158 \beta q^{11} \) \( -961 q^{13} \) \( -403 \beta q^{14} \) \( -5084 q^{16} \) \( + 1014 \beta q^{17} \) \( + 8021 q^{19} \) \( -364 \beta q^{20} \) \( + 14220 q^{22} \) \( + 1118 \beta q^{23} \) \( -2015 q^{25} \) \( -961 \beta q^{26} \) \( + 10478 q^{28} \) \( -260 \beta q^{29} \) \( + 48854 q^{31} \) \( -2652 \beta q^{32} \) \( -91260 q^{34} \) \( -5642 \beta q^{35} \) \( + 24167 q^{37} \) \( + 8021 \beta q^{38} \) \( -47880 q^{40} \) \( + 7460 \beta q^{41} \) \( -60802 q^{43} \) \( + 4108 \beta q^{44} \) \( -100620 q^{46} \) \( -2846 \beta q^{47} \) \( + 44760 q^{49} \) \( -2015 \beta q^{50} \) \( + 24986 q^{52} \) \( -14508 \beta q^{53} \) \( + 199080 q^{55} \) \( -15314 \beta q^{56} \) \( + 23400 q^{58} \) \( -10282 \beta q^{59} \) \( + 272999 q^{61} \) \( + 48854 \beta q^{62} \) \( -86696 q^{64} \) \( -13454 \beta q^{65} \) \( -85579 q^{67} \) \( -26364 \beta q^{68} \) \( + 507780 q^{70} \) \( + 36024 \beta q^{71} \) \( -152737 q^{73} \) \( + 24167 \beta q^{74} \) \( -208546 q^{76} \) \( + 63674 \beta q^{77} \) \( -74059 q^{79} \) \( -71176 \beta q^{80} \) \( -671400 q^{82} \) \( + 10180 \beta q^{83} \) \( -1277640 q^{85} \) \( -60802 \beta q^{86} \) \( + 540360 q^{88} \) \( -125826 \beta q^{89} \) \( + 387283 q^{91} \) \( -29068 \beta q^{92} \) \( + 256140 q^{94} \) \( + 112294 \beta q^{95} \) \( -1197313 q^{97} \) \( + 44760 \beta q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 52q^{4} \) \(\mathstrut -\mathstrut 806q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 52q^{4} \) \(\mathstrut -\mathstrut 806q^{7} \) \(\mathstrut -\mathstrut 2520q^{10} \) \(\mathstrut -\mathstrut 1922q^{13} \) \(\mathstrut -\mathstrut 10168q^{16} \) \(\mathstrut +\mathstrut 16042q^{19} \) \(\mathstrut +\mathstrut 28440q^{22} \) \(\mathstrut -\mathstrut 4030q^{25} \) \(\mathstrut +\mathstrut 20956q^{28} \) \(\mathstrut +\mathstrut 97708q^{31} \) \(\mathstrut -\mathstrut 182520q^{34} \) \(\mathstrut +\mathstrut 48334q^{37} \) \(\mathstrut -\mathstrut 95760q^{40} \) \(\mathstrut -\mathstrut 121604q^{43} \) \(\mathstrut -\mathstrut 201240q^{46} \) \(\mathstrut +\mathstrut 89520q^{49} \) \(\mathstrut +\mathstrut 49972q^{52} \) \(\mathstrut +\mathstrut 398160q^{55} \) \(\mathstrut +\mathstrut 46800q^{58} \) \(\mathstrut +\mathstrut 545998q^{61} \) \(\mathstrut -\mathstrut 173392q^{64} \) \(\mathstrut -\mathstrut 171158q^{67} \) \(\mathstrut +\mathstrut 1015560q^{70} \) \(\mathstrut -\mathstrut 305474q^{73} \) \(\mathstrut -\mathstrut 417092q^{76} \) \(\mathstrut -\mathstrut 148118q^{79} \) \(\mathstrut -\mathstrut 1342800q^{82} \) \(\mathstrut -\mathstrut 2555280q^{85} \) \(\mathstrut +\mathstrut 1080720q^{88} \) \(\mathstrut +\mathstrut 774566q^{91} \) \(\mathstrut +\mathstrut 512280q^{94} \) \(\mathstrut -\mathstrut 2394626q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
3.16228i
3.16228i
9.48683i 0 −26.0000 132.816i 0 −403.000 360.500i 0 −1260.00
26.2 9.48683i 0 −26.0000 132.816i 0 −403.000 360.500i 0 −1260.00
\(n\): 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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{2} \) \(\mathstrut +\mathstrut 90 \) acting on \(S_{7}^{\mathrm{new}}(27, [\chi])\).