Properties

Label 27.11.b.b
Level 27
Weight 11
Character orbit 27.b
Analytic conductor 17.155
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 27 = 3^{3} \)
Weight: \( k \) = \( 11 \)
Character orbit: \([\chi]\) = 27.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(17.1546458222\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 5 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 30\sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta q^{2} \) \( -1676 q^{4} \) \( + 16 \beta q^{5} \) \( -16093 q^{7} \) \( + 652 \beta q^{8} \) \(+O(q^{10})\) \( q\) \( -\beta q^{2} \) \( -1676 q^{4} \) \( + 16 \beta q^{5} \) \( -16093 q^{7} \) \( + 652 \beta q^{8} \) \( + 43200 q^{10} \) \( + 5648 \beta q^{11} \) \( + 608639 q^{13} \) \( + 16093 \beta q^{14} \) \( + 44176 q^{16} \) \( + 336 \beta q^{17} \) \( -2104549 q^{19} \) \( -26816 \beta q^{20} \) \( + 15249600 q^{22} \) \( -88208 \beta q^{23} \) \( + 9074425 q^{25} \) \( -608639 \beta q^{26} \) \( + 26971868 q^{28} \) \( + 605600 \beta q^{29} \) \( -26014126 q^{31} \) \( + 623472 \beta q^{32} \) \( + 907200 q^{34} \) \( -257488 \beta q^{35} \) \( + 51946607 q^{37} \) \( + 2104549 \beta q^{38} \) \( -28166400 q^{40} \) \( + 3457120 \beta q^{41} \) \( -71964982 q^{43} \) \( -9466048 \beta q^{44} \) \( -238161600 q^{46} \) \( + 2882576 \beta q^{47} \) \( -23490600 q^{49} \) \( -9074425 \beta q^{50} \) \( -1020078964 q^{52} \) \( + 2024928 \beta q^{53} \) \( -243993600 q^{55} \) \( -10492636 \beta q^{56} \) \( + 1635120000 q^{58} \) \( + 11668912 \beta q^{59} \) \( -356794201 q^{61} \) \( + 26014126 \beta q^{62} \) \( + 1728610624 q^{64} \) \( + 9738224 \beta q^{65} \) \( -591548989 q^{67} \) \( -563136 \beta q^{68} \) \( -695217600 q^{70} \) \( + 59468736 \beta q^{71} \) \( + 413274143 q^{73} \) \( -51946607 \beta q^{74} \) \( + 3527224124 q^{76} \) \( -90893264 \beta q^{77} \) \( -1080519949 q^{79} \) \( + 706816 \beta q^{80} \) \( + 9334224000 q^{82} \) \( -109721440 \beta q^{83} \) \( -14515200 q^{85} \) \( + 71964982 \beta q^{86} \) \( -9942739200 q^{88} \) \( -54807024 \beta q^{89} \) \( -9794827427 q^{91} \) \( + 147836608 \beta q^{92} \) \( + 7782955200 q^{94} \) \( -33672784 \beta q^{95} \) \( -13694177593 q^{97} \) \( + 23490600 \beta q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 3352q^{4} \) \(\mathstrut -\mathstrut 32186q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 3352q^{4} \) \(\mathstrut -\mathstrut 32186q^{7} \) \(\mathstrut +\mathstrut 86400q^{10} \) \(\mathstrut +\mathstrut 1217278q^{13} \) \(\mathstrut +\mathstrut 88352q^{16} \) \(\mathstrut -\mathstrut 4209098q^{19} \) \(\mathstrut +\mathstrut 30499200q^{22} \) \(\mathstrut +\mathstrut 18148850q^{25} \) \(\mathstrut +\mathstrut 53943736q^{28} \) \(\mathstrut -\mathstrut 52028252q^{31} \) \(\mathstrut +\mathstrut 1814400q^{34} \) \(\mathstrut +\mathstrut 103893214q^{37} \) \(\mathstrut -\mathstrut 56332800q^{40} \) \(\mathstrut -\mathstrut 143929964q^{43} \) \(\mathstrut -\mathstrut 476323200q^{46} \) \(\mathstrut -\mathstrut 46981200q^{49} \) \(\mathstrut -\mathstrut 2040157928q^{52} \) \(\mathstrut -\mathstrut 487987200q^{55} \) \(\mathstrut +\mathstrut 3270240000q^{58} \) \(\mathstrut -\mathstrut 713588402q^{61} \) \(\mathstrut +\mathstrut 3457221248q^{64} \) \(\mathstrut -\mathstrut 1183097978q^{67} \) \(\mathstrut -\mathstrut 1390435200q^{70} \) \(\mathstrut +\mathstrut 826548286q^{73} \) \(\mathstrut +\mathstrut 7054448248q^{76} \) \(\mathstrut -\mathstrut 2161039898q^{79} \) \(\mathstrut +\mathstrut 18668448000q^{82} \) \(\mathstrut -\mathstrut 29030400q^{85} \) \(\mathstrut -\mathstrut 19885478400q^{88} \) \(\mathstrut -\mathstrut 19589654854q^{91} \) \(\mathstrut +\mathstrut 15565910400q^{94} \) \(\mathstrut -\mathstrut 27388355186q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
0.500000 + 0.866025i
0.500000 0.866025i
51.9615i 0 −1676.00 831.384i 0 −16093.0 33878.9i 0 43200.0
26.2 51.9615i 0 −1676.00 831.384i 0 −16093.0 33878.9i 0 43200.0
\(n\): 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 \(T_{2}^{2} \) \(\mathstrut +\mathstrut 2700 \) acting on \(S_{11}^{\mathrm{new}}(27, [\chi])\).