Properties

Label 27.8.a.e
Level 27
Weight 8
Character orbit 27.a
Self dual Yes
Analytic conductor 8.434
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 27 = 3^{3} \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 27.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(8.43439568807\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{65}) \)
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 \(\beta = \frac{1}{2}(-1 + 3\sqrt{65})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 5 + \beta ) q^{2} \) \( + ( 43 + 9 \beta ) q^{4} \) \( + ( 91 + 2 \beta ) q^{5} \) \( + ( 395 + 90 \beta ) q^{7} \) \( + ( 889 - 49 \beta ) q^{8} \) \(+O(q^{10})\) \( q\) \( + ( 5 + \beta ) q^{2} \) \( + ( 43 + 9 \beta ) q^{4} \) \( + ( 91 + 2 \beta ) q^{5} \) \( + ( 395 + 90 \beta ) q^{7} \) \( + ( 889 - 49 \beta ) q^{8} \) \( + ( 747 + 99 \beta ) q^{10} \) \( + ( 5465 + 40 \beta ) q^{11} \) \( + ( -3100 - 720 \beta ) q^{13} \) \( + ( 15115 + 755 \beta ) q^{14} \) \( + ( -8213 - 459 \beta ) q^{16} \) \( + ( 7032 - 2352 \beta ) q^{17} \) \( + ( 7418 - 1188 \beta ) q^{19} \) \( + ( 6541 + 887 \beta ) q^{20} \) \( + ( 33165 + 5625 \beta ) q^{22} \) \( + ( 14818 + 5264 \beta ) q^{23} \) \( + ( -69260 + 360 \beta ) q^{25} \) \( + ( -120620 - 5980 \beta ) q^{26} \) \( + ( 135245 + 6615 \beta ) q^{28} \) \( + ( -71770 - 260 \beta ) q^{29} \) \( + ( -26383 - 14058 \beta ) q^{31} \) \( + ( -221871 - 3777 \beta ) q^{32} \) \( + ( -308232 - 2376 \beta ) q^{34} \) \( + ( 62225 + 8800 \beta ) q^{35} \) \( + ( 217010 - 21600 \beta ) q^{37} \) \( + ( -136358 + 2666 \beta ) q^{38} \) \( + ( 66591 - 2583 \beta ) q^{40} \) \( + ( 356650 - 18580 \beta ) q^{41} \) \( + ( -532090 + 24660 \beta ) q^{43} \) \( + ( 287555 + 50545 \beta ) q^{44} \) \( + ( 842634 + 35874 \beta ) q^{46} \) \( + ( 762326 - 36848 \beta ) q^{47} \) \( + ( 515082 + 63000 \beta ) q^{49} \) \( + ( -293740 - 67820 \beta ) q^{50} \) \( + ( -1079380 - 52380 \beta ) q^{52} \) \( + ( 1319553 + 28638 \beta ) q^{53} \) \( + ( 508995 + 14490 \beta ) q^{55} \) \( + ( -292705 + 65065 \beta ) q^{56} \) \( + ( -396810 - 72810 \beta ) q^{58} \) \( + ( 914860 + 97760 \beta ) q^{59} \) \( + ( -321688 - 23184 \beta ) q^{61} \) \( + ( -2184383 - 82615 \beta ) q^{62} \) \( + ( -609533 - 178227 \beta ) q^{64} \) \( + ( -492340 - 70280 \beta ) q^{65} \) \( + ( 168350 - 9900 \beta ) q^{67} \) \( + ( -2788152 - 16680 \beta ) q^{68} \) \( + ( 1595925 + 97425 \beta ) q^{70} \) \( + ( -2238360 - 234480 \beta ) q^{71} \) \( + ( -1473775 + 197640 \beta ) q^{73} \) \( + ( -2068550 + 130610 \beta ) q^{74} \) \( + ( -1242058 + 26370 \beta ) q^{76} \) \( + ( 2684275 + 504050 \beta ) q^{77} \) \( + ( 5159384 + 208152 \beta ) q^{79} \) \( + ( -881411 - 57277 \beta ) q^{80} \) \( + ( -929430 + 282330 \beta ) q^{82} \) \( + ( 266657 - 110888 \beta ) q^{83} \) \( + ( -46872 - 195264 \beta ) q^{85} \) \( + ( 939910 - 433450 \beta ) q^{86} \) \( + ( 4572225 - 230265 \beta ) q^{88} \) \( + ( 2942790 - 135420 \beta ) q^{89} \) \( + ( -10685300 - 498600 \beta ) q^{91} \) \( + ( 7554070 + 312338 \beta ) q^{92} \) \( + ( -1568178 + 614934 \beta ) q^{94} \) \( + ( 328142 - 90896 \beta ) q^{95} \) \( + ( 1526255 - 1046160 \beta ) q^{97} \) \( + ( 11773410 + 767082 \beta ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 9q^{2} \) \(\mathstrut +\mathstrut 77q^{4} \) \(\mathstrut +\mathstrut 180q^{5} \) \(\mathstrut +\mathstrut 700q^{7} \) \(\mathstrut +\mathstrut 1827q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 9q^{2} \) \(\mathstrut +\mathstrut 77q^{4} \) \(\mathstrut +\mathstrut 180q^{5} \) \(\mathstrut +\mathstrut 700q^{7} \) \(\mathstrut +\mathstrut 1827q^{8} \) \(\mathstrut +\mathstrut 1395q^{10} \) \(\mathstrut +\mathstrut 10890q^{11} \) \(\mathstrut -\mathstrut 5480q^{13} \) \(\mathstrut +\mathstrut 29475q^{14} \) \(\mathstrut -\mathstrut 15967q^{16} \) \(\mathstrut +\mathstrut 16416q^{17} \) \(\mathstrut +\mathstrut 16024q^{19} \) \(\mathstrut +\mathstrut 12195q^{20} \) \(\mathstrut +\mathstrut 60705q^{22} \) \(\mathstrut +\mathstrut 24372q^{23} \) \(\mathstrut -\mathstrut 138880q^{25} \) \(\mathstrut -\mathstrut 235260q^{26} \) \(\mathstrut +\mathstrut 263875q^{28} \) \(\mathstrut -\mathstrut 143280q^{29} \) \(\mathstrut -\mathstrut 38708q^{31} \) \(\mathstrut -\mathstrut 439965q^{32} \) \(\mathstrut -\mathstrut 614088q^{34} \) \(\mathstrut +\mathstrut 115650q^{35} \) \(\mathstrut +\mathstrut 455620q^{37} \) \(\mathstrut -\mathstrut 275382q^{38} \) \(\mathstrut +\mathstrut 135765q^{40} \) \(\mathstrut +\mathstrut 731880q^{41} \) \(\mathstrut -\mathstrut 1088840q^{43} \) \(\mathstrut +\mathstrut 524565q^{44} \) \(\mathstrut +\mathstrut 1649394q^{46} \) \(\mathstrut +\mathstrut 1561500q^{47} \) \(\mathstrut +\mathstrut 967164q^{49} \) \(\mathstrut -\mathstrut 519660q^{50} \) \(\mathstrut -\mathstrut 2106380q^{52} \) \(\mathstrut +\mathstrut 2610468q^{53} \) \(\mathstrut +\mathstrut 1003500q^{55} \) \(\mathstrut -\mathstrut 650475q^{56} \) \(\mathstrut -\mathstrut 720810q^{58} \) \(\mathstrut +\mathstrut 1731960q^{59} \) \(\mathstrut -\mathstrut 620192q^{61} \) \(\mathstrut -\mathstrut 4286151q^{62} \) \(\mathstrut -\mathstrut 1040839q^{64} \) \(\mathstrut -\mathstrut 914400q^{65} \) \(\mathstrut +\mathstrut 346600q^{67} \) \(\mathstrut -\mathstrut 5559624q^{68} \) \(\mathstrut +\mathstrut 3094425q^{70} \) \(\mathstrut -\mathstrut 4242240q^{71} \) \(\mathstrut -\mathstrut 3145190q^{73} \) \(\mathstrut -\mathstrut 4267710q^{74} \) \(\mathstrut -\mathstrut 2510486q^{76} \) \(\mathstrut +\mathstrut 4864500q^{77} \) \(\mathstrut +\mathstrut 10110616q^{79} \) \(\mathstrut -\mathstrut 1705545q^{80} \) \(\mathstrut -\mathstrut 2141190q^{82} \) \(\mathstrut +\mathstrut 644202q^{83} \) \(\mathstrut +\mathstrut 101520q^{85} \) \(\mathstrut +\mathstrut 2313270q^{86} \) \(\mathstrut +\mathstrut 9374715q^{88} \) \(\mathstrut +\mathstrut 6021000q^{89} \) \(\mathstrut -\mathstrut 20872000q^{91} \) \(\mathstrut +\mathstrut 14795802q^{92} \) \(\mathstrut -\mathstrut 3751290q^{94} \) \(\mathstrut +\mathstrut 747180q^{95} \) \(\mathstrut +\mathstrut 4098670q^{97} \) \(\mathstrut +\mathstrut 22779738q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−3.53113
4.53113
−7.59339 0 −70.3405 65.8132 0 −738.405 1506.08 0 −499.745
1.2 16.5934 0 147.340 114.187 0 1438.40 320.924 0 1894.75
\(n\): 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 \(T_{2}^{2} \) \(\mathstrut -\mathstrut 9 T_{2} \) \(\mathstrut -\mathstrut 126 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(27))\).