Properties

Label 27.10.a.a
Level 27
Weight 10
Character orbit 27.a
Self dual Yes
Analytic conductor 13.906
Analytic rank 1
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 27 = 3^{3} \)
Weight: \( k \) = \( 10 \)
Character orbit: \([\chi]\) = 27.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(13.9059675764\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{14}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 6\sqrt{14}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta q^{2} \) \( -8 q^{4} \) \( -22 \beta q^{5} \) \( -763 q^{7} \) \( -520 \beta q^{8} \) \(+O(q^{10})\) \( q\) \( + \beta q^{2} \) \( -8 q^{4} \) \( -22 \beta q^{5} \) \( -763 q^{7} \) \( -520 \beta q^{8} \) \( -11088 q^{10} \) \( -2534 \beta q^{11} \) \( -73015 q^{13} \) \( -763 \beta q^{14} \) \( -257984 q^{16} \) \( + 7494 \beta q^{17} \) \( -598129 q^{19} \) \( + 176 \beta q^{20} \) \( -1277136 q^{22} \) \( + 106922 \beta q^{23} \) \( -1709189 q^{25} \) \( -73015 \beta q^{26} \) \( + 6104 q^{28} \) \( -207296 \beta q^{29} \) \( -1824100 q^{31} \) \( + 8256 \beta q^{32} \) \( + 3776976 q^{34} \) \( + 16786 \beta q^{35} \) \( + 14272175 q^{37} \) \( -598129 \beta q^{38} \) \( + 5765760 q^{40} \) \( + 1328480 \beta q^{41} \) \( + 7756904 q^{43} \) \( + 20272 \beta q^{44} \) \( + 53888688 q^{46} \) \( + 1380418 \beta q^{47} \) \( -39771438 q^{49} \) \( -1709189 \beta q^{50} \) \( + 584120 q^{52} \) \( + 441036 \beta q^{53} \) \( + 28096992 q^{55} \) \( + 396760 \beta q^{56} \) \( -104477184 q^{58} \) \( -5577286 \beta q^{59} \) \( -152766493 q^{61} \) \( -1824100 \beta q^{62} \) \( + 136248832 q^{64} \) \( + 1606330 \beta q^{65} \) \( -320752069 q^{67} \) \( -59952 \beta q^{68} \) \( + 8460144 q^{70} \) \( + 4862004 \beta q^{71} \) \( + 66463985 q^{73} \) \( + 14272175 \beta q^{74} \) \( + 4785032 q^{76} \) \( + 1933442 \beta q^{77} \) \( -115357453 q^{79} \) \( + 5675648 \beta q^{80} \) \( + 669553920 q^{82} \) \( -27378188 \beta q^{83} \) \( -83093472 q^{85} \) \( + 7756904 \beta q^{86} \) \( + 664110720 q^{88} \) \( -15046674 \beta q^{89} \) \( + 55710445 q^{91} \) \( -855376 \beta q^{92} \) \( + 695730672 q^{94} \) \( + 13158838 \beta q^{95} \) \( -235306843 q^{97} \) \( -39771438 \beta q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 16q^{4} \) \(\mathstrut -\mathstrut 1526q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 16q^{4} \) \(\mathstrut -\mathstrut 1526q^{7} \) \(\mathstrut -\mathstrut 22176q^{10} \) \(\mathstrut -\mathstrut 146030q^{13} \) \(\mathstrut -\mathstrut 515968q^{16} \) \(\mathstrut -\mathstrut 1196258q^{19} \) \(\mathstrut -\mathstrut 2554272q^{22} \) \(\mathstrut -\mathstrut 3418378q^{25} \) \(\mathstrut +\mathstrut 12208q^{28} \) \(\mathstrut -\mathstrut 3648200q^{31} \) \(\mathstrut +\mathstrut 7553952q^{34} \) \(\mathstrut +\mathstrut 28544350q^{37} \) \(\mathstrut +\mathstrut 11531520q^{40} \) \(\mathstrut +\mathstrut 15513808q^{43} \) \(\mathstrut +\mathstrut 107777376q^{46} \) \(\mathstrut -\mathstrut 79542876q^{49} \) \(\mathstrut +\mathstrut 1168240q^{52} \) \(\mathstrut +\mathstrut 56193984q^{55} \) \(\mathstrut -\mathstrut 208954368q^{58} \) \(\mathstrut -\mathstrut 305532986q^{61} \) \(\mathstrut +\mathstrut 272497664q^{64} \) \(\mathstrut -\mathstrut 641504138q^{67} \) \(\mathstrut +\mathstrut 16920288q^{70} \) \(\mathstrut +\mathstrut 132927970q^{73} \) \(\mathstrut +\mathstrut 9570064q^{76} \) \(\mathstrut -\mathstrut 230714906q^{79} \) \(\mathstrut +\mathstrut 1339107840q^{82} \) \(\mathstrut -\mathstrut 166186944q^{85} \) \(\mathstrut +\mathstrut 1328221440q^{88} \) \(\mathstrut +\mathstrut 111420890q^{91} \) \(\mathstrut +\mathstrut 1391461344q^{94} \) \(\mathstrut -\mathstrut 470613686q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−3.74166
3.74166
−22.4499 0 −8.00000 493.899 0 −763.000 11674.0 0 −11088.0
1.2 22.4499 0 −8.00000 −493.899 0 −763.000 −11674.0 0 −11088.0
\(n\): 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

Atkin-Lehner signs

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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{2} \) \(\mathstrut -\mathstrut 504 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(27))\).