Properties

Label 27.7.b.b
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: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 28 q^{4} -40 \beta q^{5} + 299 q^{7} + 92 \beta q^{8} +O(q^{10})\) \( q + \beta q^{2} + 28 q^{4} -40 \beta q^{5} + 299 q^{7} + 92 \beta q^{8} + 1440 q^{10} -104 \beta q^{11} + 2495 q^{13} + 299 \beta q^{14} -1520 q^{16} + 312 \beta q^{17} -2509 q^{19} -1120 \beta q^{20} + 3744 q^{22} -2392 \beta q^{23} -41975 q^{25} + 2495 \beta q^{26} + 8372 q^{28} + 3952 \beta q^{29} + 5330 q^{31} + 4368 \beta q^{32} -11232 q^{34} -11960 \beta q^{35} + 32591 q^{37} -2509 \beta q^{38} + 132480 q^{40} + 11024 \beta q^{41} -70630 q^{43} -2912 \beta q^{44} + 86112 q^{46} + 664 \beta q^{47} -28248 q^{49} -41975 \beta q^{50} + 69860 q^{52} + 31824 \beta q^{53} -149760 q^{55} + 27508 \beta q^{56} -142272 q^{58} + 39560 \beta q^{59} -61801 q^{61} + 5330 \beta q^{62} -254528 q^{64} -99800 \beta q^{65} -430261 q^{67} + 8736 \beta q^{68} + 430560 q^{70} -41952 \beta q^{71} + 251615 q^{73} + 32591 \beta q^{74} -70252 q^{76} -31096 \beta q^{77} + 660827 q^{79} + 60800 \beta q^{80} -396864 q^{82} + 132976 \beta q^{83} + 449280 q^{85} -70630 \beta q^{86} + 344448 q^{88} -45096 \beta q^{89} + 746005 q^{91} -66976 \beta q^{92} -23904 q^{94} + 100360 \beta q^{95} + 220727 q^{97} -28248 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 56q^{4} + 598q^{7} + O(q^{10}) \) \( 2q + 56q^{4} + 598q^{7} + 2880q^{10} + 4990q^{13} - 3040q^{16} - 5018q^{19} + 7488q^{22} - 83950q^{25} + 16744q^{28} + 10660q^{31} - 22464q^{34} + 65182q^{37} + 264960q^{40} - 141260q^{43} + 172224q^{46} - 56496q^{49} + 139720q^{52} - 299520q^{55} - 284544q^{58} - 123602q^{61} - 509056q^{64} - 860522q^{67} + 861120q^{70} + 503230q^{73} - 140504q^{76} + 1321654q^{79} - 793728q^{82} + 898560q^{85} + 688896q^{88} + 1492010q^{91} - 47808q^{94} + 441454q^{97} + O(q^{100}) \)

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
1.00000i
1.00000i
6.00000i 0 28.0000 240.000i 0 299.000 552.000i 0 1440.00
26.2 6.00000i 0 28.0000 240.000i 0 299.000 552.000i 0 1440.00
\(n\): 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 \( T_{2}^{2} + 36 \) acting on \(S_{7}^{\mathrm{new}}(27, [\chi])\).