Properties

Label 27.7.b.c
Level 27
Weight 7
Character orbit 27.b
Analytic conductor 6.211
Analytic rank 0
Dimension 4
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{41})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{8} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{2} \) \( + ( -49 + \beta_{3} ) q^{4} \) \( + ( 7 \beta_{1} + \beta_{2} ) q^{5} \) \( + ( -171 + 4 \beta_{3} ) q^{7} \) \( + ( 49 \beta_{1} + 8 \beta_{2} ) q^{8} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{2} \) \( + ( -49 + \beta_{3} ) q^{4} \) \( + ( 7 \beta_{1} + \beta_{2} ) q^{5} \) \( + ( -171 + 4 \beta_{3} ) q^{7} \) \( + ( 49 \beta_{1} + 8 \beta_{2} ) q^{8} \) \( + ( 821 - 13 \beta_{3} ) q^{10} \) \( + ( -133 \beta_{1} + 17 \beta_{2} ) q^{11} \) \( + ( -858 - 8 \beta_{3} ) q^{13} \) \( + ( 427 \beta_{1} + 32 \beta_{2} ) q^{14} \) \( + ( 2641 - 33 \beta_{3} ) q^{16} \) \( + ( 150 \beta_{1} - 14 \beta_{2} ) q^{17} \) \( + ( -976 + 120 \beta_{3} ) q^{19} \) \( + ( -1205 \beta_{1} - 40 \beta_{2} ) q^{20} \) \( + ( -14519 + 31 \beta_{3} ) q^{22} \) \( + ( -212 \beta_{1} - 196 \beta_{2} ) q^{23} \) \( + ( 4304 + 88 \beta_{3} ) q^{25} \) \( + ( 346 \beta_{1} - 64 \beta_{2} ) q^{26} \) \( + ( 38267 - 363 \beta_{3} ) q^{28} \) \( + ( 1322 \beta_{1} - 258 \beta_{2} ) q^{29} \) \( + ( -38241 - 140 \beta_{3} ) q^{31} \) \( + ( -1617 \beta_{1} + 248 \beta_{2} ) q^{32} \) \( + ( 16530 - 66 \beta_{3} ) q^{34} \) \( + ( -4645 \beta_{1} - 135 \beta_{2} ) q^{35} \) \( + ( 32180 + 120 \beta_{3} ) q^{37} \) \( + ( 8656 \beta_{1} + 960 \beta_{2} ) q^{38} \) \( + ( -84821 + 613 \beta_{3} ) q^{40} \) \( + ( -3068 \beta_{1} + 132 \beta_{2} ) q^{41} \) \( + ( 31458 + 88 \beta_{3} ) q^{43} \) \( + ( 7991 \beta_{1} + 1336 \beta_{2} ) q^{44} \) \( + ( -29836 + 1388 \beta_{3} ) q^{46} \) \( + ( 3182 \beta_{1} - 470 \beta_{2} ) q^{47} \) \( + ( 31144 - 1352 \beta_{3} ) q^{49} \) \( + ( 1328 \beta_{1} + 704 \beta_{2} ) q^{50} \) \( + ( -17734 - 474 \beta_{3} ) q^{52} \) \( + ( -11313 \beta_{1} - 1407 \beta_{2} ) q^{53} \) \( + ( 14435 - 1780 \beta_{3} ) q^{55} \) \( + ( -34171 \beta_{1} - 856 \beta_{2} ) q^{56} \) \( + ( 141646 + 226 \beta_{3} ) q^{58} \) \( + ( 18358 \beta_{1} - 2462 \beta_{2} ) q^{59} \) \( + ( 42324 - 800 \beta_{3} ) q^{61} \) \( + ( 29281 \beta_{1} - 1120 \beta_{2} ) q^{62} \) \( + ( -6257 - 1983 \beta_{3} ) q^{64} \) \( + ( 890 \beta_{1} - 930 \beta_{2} ) q^{65} \) \( + ( -2202 + 4408 \beta_{3} ) q^{67} \) \( + ( -11154 \beta_{1} - 1424 \beta_{2} ) q^{68} \) \( + ( -528935 + 5455 \beta_{3} ) q^{70} \) \( + ( -44958 \beta_{1} - 1058 \beta_{2} ) q^{71} \) \( + ( -472111 + 3384 \beta_{3} ) q^{73} \) \( + ( -24500 \beta_{1} + 960 \beta_{2} ) q^{74} \) \( + ( 944464 - 6736 \beta_{3} ) q^{76} \) \( + ( 28639 \beta_{1} + 5769 \beta_{2} ) q^{77} \) \( + ( 298446 - 2528 \beta_{3} ) q^{79} \) \( + ( 46933 \beta_{1} + 2344 \beta_{2} ) q^{80} \) \( + ( -342724 + 2276 \beta_{3} ) q^{82} \) \( + ( 53009 \beta_{1} + 4035 \beta_{2} ) q^{83} \) \( + ( -45114 + 1992 \beta_{3} ) q^{85} \) \( + ( -25826 \beta_{1} + 704 \beta_{2} ) q^{86} \) \( + ( 13847 - 14023 \beta_{3} ) q^{88} \) \( + ( 11838 \beta_{1} + 12338 \beta_{2} ) q^{89} \) \( + ( -92386 - 2096 \beta_{3} ) q^{91} \) \( + ( 105100 \beta_{1} - 1440 \beta_{2} ) q^{92} \) \( + ( 345466 - 362 \beta_{3} ) q^{94} \) \( + ( -110272 \beta_{1} + 104 \beta_{2} ) q^{95} \) \( + ( 238827 + 9952 \beta_{3} ) q^{97} \) \( + ( -117672 \beta_{1} - 10816 \beta_{2} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 194q^{4} \) \(\mathstrut -\mathstrut 676q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 194q^{4} \) \(\mathstrut -\mathstrut 676q^{7} \) \(\mathstrut +\mathstrut 3258q^{10} \) \(\mathstrut -\mathstrut 3448q^{13} \) \(\mathstrut +\mathstrut 10498q^{16} \) \(\mathstrut -\mathstrut 3664q^{19} \) \(\mathstrut -\mathstrut 58014q^{22} \) \(\mathstrut +\mathstrut 17392q^{25} \) \(\mathstrut +\mathstrut 152342q^{28} \) \(\mathstrut -\mathstrut 153244q^{31} \) \(\mathstrut +\mathstrut 65988q^{34} \) \(\mathstrut +\mathstrut 128960q^{37} \) \(\mathstrut -\mathstrut 338058q^{40} \) \(\mathstrut +\mathstrut 126008q^{43} \) \(\mathstrut -\mathstrut 116568q^{46} \) \(\mathstrut +\mathstrut 121872q^{49} \) \(\mathstrut -\mathstrut 71884q^{52} \) \(\mathstrut +\mathstrut 54180q^{55} \) \(\mathstrut +\mathstrut 567036q^{58} \) \(\mathstrut +\mathstrut 167696q^{61} \) \(\mathstrut -\mathstrut 28994q^{64} \) \(\mathstrut +\mathstrut 8q^{67} \) \(\mathstrut -\mathstrut 2104830q^{70} \) \(\mathstrut -\mathstrut 1881676q^{73} \) \(\mathstrut +\mathstrut 3764384q^{76} \) \(\mathstrut +\mathstrut 1188728q^{79} \) \(\mathstrut -\mathstrut 1366344q^{82} \) \(\mathstrut -\mathstrut 176472q^{85} \) \(\mathstrut +\mathstrut 27342q^{88} \) \(\mathstrut -\mathstrut 373736q^{91} \) \(\mathstrut +\mathstrut 1381140q^{94} \) \(\mathstrut +\mathstrut 975212q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut +\mathstrut \) \(21\) \(x^{2}\mathstrut +\mathstrut \) \(100\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 3 \nu^{3} + 3 \nu \)\()/10\)
\(\beta_{2}\)\(=\)\((\)\( 36 \nu^{3} + 441 \nu \)\()/5\)
\(\beta_{3}\)\(=\)\( 27 \nu^{2} + 284 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2}\mathstrut -\mathstrut \) \(24\) \(\beta_{1}\)\()/81\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(284\)\()/27\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(\beta_{2}\mathstrut +\mathstrut \) \(294\) \(\beta_{1}\)\()/81\)

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
3.70156i
2.70156i
2.70156i
3.70156i
14.1047i 0 −134.942 137.419i 0 −514.769 1000.62i 0 1938.25
26.2 5.10469i 0 37.9422 60.5813i 0 176.769 520.383i 0 −309.248
26.3 5.10469i 0 37.9422 60.5813i 0 176.769 520.383i 0 −309.248
26.4 14.1047i 0 −134.942 137.419i 0 −514.769 1000.62i 0 1938.25
\(n\): 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 \(T_{2}^{4} \) \(\mathstrut +\mathstrut 225 T_{2}^{2} \) \(\mathstrut +\mathstrut 5184 \) acting on \(S_{7}^{\mathrm{new}}(27, [\chi])\).