Properties

Label 27.6.a.d
Level 27
Weight 6
Character orbit 27.a
Self dual Yes
Analytic conductor 4.330
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 27 = 3^{3} \)
Weight: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 27.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(4.33036313495\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + 3\sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 4 + \beta ) q^{2} \) \( + ( 22 + 9 \beta ) q^{4} \) \( + ( 41 - 10 \beta ) q^{5} \) \( + ( 5 - 18 \beta ) q^{7} \) \( + ( 302 + 35 \beta ) q^{8} \) \(+O(q^{10})\) \( q\) \( + ( 4 + \beta ) q^{2} \) \( + ( 22 + 9 \beta ) q^{4} \) \( + ( 41 - 10 \beta ) q^{5} \) \( + ( 5 - 18 \beta ) q^{7} \) \( + ( 302 + 35 \beta ) q^{8} \) \( + ( -216 - 9 \beta ) q^{10} \) \( + ( 277 - 32 \beta ) q^{11} \) \( + ( -316 - 72 \beta ) q^{13} \) \( + ( -664 - 85 \beta ) q^{14} \) \( + ( 1834 + 189 \beta ) q^{16} \) \( + ( 24 + 168 \beta ) q^{17} \) \( + ( -1474 + 108 \beta ) q^{19} \) \( + ( -2518 + 59 \beta ) q^{20} \) \( + ( -108 + 117 \beta ) q^{22} \) \( + ( 2 - 40 \beta ) q^{23} \) \( + ( 2356 - 720 \beta ) q^{25} \) \( + ( -4000 - 676 \beta ) q^{26} \) \( + ( -6046 - 513 \beta ) q^{28} \) \( + ( 6034 + 172 \beta ) q^{29} \) \( + ( -793 + 522 \beta ) q^{31} \) \( + ( 4854 + 1659 \beta ) q^{32} \) \( + ( 6480 + 864 \beta ) q^{34} \) \( + ( 7045 - 608 \beta ) q^{35} \) \( + ( 3962 + 1080 \beta ) q^{37} \) \( + ( -1792 - 934 \beta ) q^{38} \) \( + ( -918 - 1935 \beta ) q^{40} \) \( + ( 3878 - 2068 \beta ) q^{41} \) \( + ( -382 + 1548 \beta ) q^{43} \) \( + ( -4850 + 1501 \beta ) q^{44} \) \( + ( -1512 - 198 \beta ) q^{46} \) \( + ( -1034 + 3184 \beta ) q^{47} \) \( + ( -4470 + 144 \beta ) q^{49} \) \( + ( -17936 - 1244 \beta ) q^{50} \) \( + ( -31576 - 5076 \beta ) q^{52} \) \( + ( 1179 + 2178 \beta ) q^{53} \) \( + ( 23517 - 3762 \beta ) q^{55} \) \( + ( -22430 - 5891 \beta ) q^{56} \) \( + ( 30672 + 6894 \beta ) q^{58} \) \( + ( -33124 - 1072 \beta ) q^{59} \) \( + ( -26104 + 2304 \beta ) q^{61} \) \( + ( 16664 + 1817 \beta ) q^{62} \) \( + ( 23770 + 7101 \beta ) q^{64} \) \( + ( 14404 + 928 \beta ) q^{65} \) \( + ( -18406 - 5364 \beta ) q^{67} \) \( + ( 57984 + 5424 \beta ) q^{68} \) \( + ( 5076 + 4005 \beta ) q^{70} \) \( + ( -19560 - 4728 \beta ) q^{71} \) \( + ( 24905 - 2592 \beta ) q^{73} \) \( + ( 56888 + 9362 \beta ) q^{74} \) \( + ( 4508 - 9918 \beta ) q^{76} \) \( + ( 23273 - 4570 \beta ) q^{77} \) \( + ( -24664 - 288 \beta ) q^{79} \) \( + ( 3374 - 12481 \beta ) q^{80} \) \( + ( -63072 - 6462 \beta ) q^{82} \) \( + ( -47867 - 6560 \beta ) q^{83} \) \( + ( -62856 + 4968 \beta ) q^{85} \) \( + ( 57296 + 7358 \beta ) q^{86} \) \( + ( 41094 - 1089 \beta ) q^{88} \) \( + ( -23046 + 10236 \beta ) q^{89} \) \( + ( 47668 + 6624 \beta ) q^{91} \) \( + ( -13636 - 1222 \beta ) q^{92} \) \( + ( 116856 + 14886 \beta ) q^{94} \) \( + ( -101474 + 18088 \beta ) q^{95} \) \( + ( 91823 - 13680 \beta ) q^{97} \) \( + ( -12408 - 3750 \beta ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 9q^{2} \) \(\mathstrut +\mathstrut 53q^{4} \) \(\mathstrut +\mathstrut 72q^{5} \) \(\mathstrut -\mathstrut 8q^{7} \) \(\mathstrut +\mathstrut 639q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 9q^{2} \) \(\mathstrut +\mathstrut 53q^{4} \) \(\mathstrut +\mathstrut 72q^{5} \) \(\mathstrut -\mathstrut 8q^{7} \) \(\mathstrut +\mathstrut 639q^{8} \) \(\mathstrut -\mathstrut 441q^{10} \) \(\mathstrut +\mathstrut 522q^{11} \) \(\mathstrut -\mathstrut 704q^{13} \) \(\mathstrut -\mathstrut 1413q^{14} \) \(\mathstrut +\mathstrut 3857q^{16} \) \(\mathstrut +\mathstrut 216q^{17} \) \(\mathstrut -\mathstrut 2840q^{19} \) \(\mathstrut -\mathstrut 4977q^{20} \) \(\mathstrut -\mathstrut 99q^{22} \) \(\mathstrut -\mathstrut 36q^{23} \) \(\mathstrut +\mathstrut 3992q^{25} \) \(\mathstrut -\mathstrut 8676q^{26} \) \(\mathstrut -\mathstrut 12605q^{28} \) \(\mathstrut +\mathstrut 12240q^{29} \) \(\mathstrut -\mathstrut 1064q^{31} \) \(\mathstrut +\mathstrut 11367q^{32} \) \(\mathstrut +\mathstrut 13824q^{34} \) \(\mathstrut +\mathstrut 13482q^{35} \) \(\mathstrut +\mathstrut 9004q^{37} \) \(\mathstrut -\mathstrut 4518q^{38} \) \(\mathstrut -\mathstrut 3771q^{40} \) \(\mathstrut +\mathstrut 5688q^{41} \) \(\mathstrut +\mathstrut 784q^{43} \) \(\mathstrut -\mathstrut 8199q^{44} \) \(\mathstrut -\mathstrut 3222q^{46} \) \(\mathstrut +\mathstrut 1116q^{47} \) \(\mathstrut -\mathstrut 8796q^{49} \) \(\mathstrut -\mathstrut 37116q^{50} \) \(\mathstrut -\mathstrut 68228q^{52} \) \(\mathstrut +\mathstrut 4536q^{53} \) \(\mathstrut +\mathstrut 43272q^{55} \) \(\mathstrut -\mathstrut 50751q^{56} \) \(\mathstrut +\mathstrut 68238q^{58} \) \(\mathstrut -\mathstrut 67320q^{59} \) \(\mathstrut -\mathstrut 49904q^{61} \) \(\mathstrut +\mathstrut 35145q^{62} \) \(\mathstrut +\mathstrut 54641q^{64} \) \(\mathstrut +\mathstrut 29736q^{65} \) \(\mathstrut -\mathstrut 42176q^{67} \) \(\mathstrut +\mathstrut 121392q^{68} \) \(\mathstrut +\mathstrut 14157q^{70} \) \(\mathstrut -\mathstrut 43848q^{71} \) \(\mathstrut +\mathstrut 47218q^{73} \) \(\mathstrut +\mathstrut 123138q^{74} \) \(\mathstrut -\mathstrut 902q^{76} \) \(\mathstrut +\mathstrut 41976q^{77} \) \(\mathstrut -\mathstrut 49616q^{79} \) \(\mathstrut -\mathstrut 5733q^{80} \) \(\mathstrut -\mathstrut 132606q^{82} \) \(\mathstrut -\mathstrut 102294q^{83} \) \(\mathstrut -\mathstrut 120744q^{85} \) \(\mathstrut +\mathstrut 121950q^{86} \) \(\mathstrut +\mathstrut 81099q^{88} \) \(\mathstrut -\mathstrut 35856q^{89} \) \(\mathstrut +\mathstrut 101960q^{91} \) \(\mathstrut -\mathstrut 28494q^{92} \) \(\mathstrut +\mathstrut 248598q^{94} \) \(\mathstrut -\mathstrut 184860q^{95} \) \(\mathstrut +\mathstrut 169966q^{97} \) \(\mathstrut -\mathstrut 28566q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−1.56155
2.56155
−1.68466 0 −29.1619 97.8466 0 107.324 103.037 0 −164.838
1.2 10.6847 0 82.1619 −25.8466 0 −115.324 535.963 0 −276.162
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{2} \) \(\mathstrut -\mathstrut 9 T_{2} \) \(\mathstrut -\mathstrut 18 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(27))\).