Properties

Label 27.10.a.b
Level 27
Weight 10
Character orbit 27.a
Self dual Yes
Analytic conductor 13.906
Analytic rank 0
Dimension 3
CM No
Inner twists 1

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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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.177113.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3^{4} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + ( -1 - \beta_{1} ) q^{2} \) \( + ( 199 - 2 \beta_{1} + \beta_{2} ) q^{4} \) \( + ( 661 + 7 \beta_{1} + 3 \beta_{2} ) q^{5} \) \( + ( -1231 - 233 \beta_{1} - 5 \beta_{2} ) q^{7} \) \( + ( 1501 - 32 \beta_{1} - 3 \beta_{2} ) q^{8} \) \(+O(q^{10})\) \( q\) \( + ( -1 - \beta_{1} ) q^{2} \) \( + ( 199 - 2 \beta_{1} + \beta_{2} ) q^{4} \) \( + ( 661 + 7 \beta_{1} + 3 \beta_{2} ) q^{5} \) \( + ( -1231 - 233 \beta_{1} - 5 \beta_{2} ) q^{7} \) \( + ( 1501 - 32 \beta_{1} - 3 \beta_{2} ) q^{8} \) \( + ( -6327 - 1657 \beta_{1} - 22 \beta_{2} ) q^{10} \) \( + ( 5621 - 124 \beta_{1} - 84 \beta_{2} ) q^{11} \) \( + ( 38972 - 2360 \beta_{1} + 208 \beta_{2} ) q^{13} \) \( + ( 167821 + 2227 \beta_{1} + 258 \beta_{2} ) q^{14} \) \( + ( -79973 + 444 \beta_{1} - 465 \beta_{2} ) q^{16} \) \( + ( 338016 - 8472 \beta_{1} - 96 \beta_{2} ) q^{17} \) \( + ( -5074 + 23682 \beta_{1} + 66 \beta_{2} ) q^{19} \) \( + ( 849469 + 5230 \beta_{1} + 231 \beta_{2} ) q^{20} \) \( + ( 101907 + 22483 \beta_{1} + 544 \beta_{2} ) q^{22} \) \( + ( 975706 + 44248 \beta_{1} - 2208 \beta_{2} ) q^{23} \) \( + ( 711244 + 37588 \beta_{1} + 1132 \beta_{2} ) q^{25} \) \( + ( 1588372 - 116564 \beta_{1} + 1320 \beta_{2} ) q^{26} \) \( + ( -1178575 - 129306 \beta_{1} - 957 \beta_{2} ) q^{28} \) \( + ( 1922930 + 44306 \beta_{1} + 3762 \beta_{2} ) q^{29} \) \( + ( -2191741 - 72215 \beta_{1} - 8483 \beta_{2} ) q^{31} \) \( + ( -895899 + 255324 \beta_{1} + 3417 \beta_{2} ) q^{32} \) \( + ( 5699376 - 330888 \beta_{1} + 8952 \beta_{2} ) q^{34} \) \( + ( -5780179 - 428752 \beta_{1} - 6480 \beta_{2} ) q^{35} \) \( + ( -3895342 + 303456 \beta_{1} + 9624 \beta_{2} ) q^{37} \) \( + ( -16824458 + 53746 \beta_{1} - 24012 \beta_{2} ) q^{38} \) \( + ( -1376937 - 63704 \beta_{1} + 4879 \beta_{2} ) q^{40} \) \( + ( -7404506 + 420274 \beta_{1} + 16434 \beta_{2} ) q^{41} \) \( + ( 15128138 + 1100902 \beta_{1} - 30674 \beta_{2} ) q^{43} \) \( + ( -19068997 - 155386 \beta_{1} + 17805 \beta_{2} ) q^{44} \) \( + ( -31879530 - 94450 \beta_{1} - 33208 \beta_{2} ) q^{46} \) \( + ( 4130678 - 60952 \beta_{1} + 44328 \beta_{2} ) q^{47} \) \( + ( 6311154 + 1186060 \beta_{1} + 67444 \beta_{2} ) q^{49} \) \( + ( -27661348 - 982228 \beta_{1} - 43248 \beta_{2} ) q^{50} \) \( + ( 60912164 - 1177224 \beta_{1} + 3468 \beta_{2} ) q^{52} \) \( + ( 26859879 + 1534761 \beta_{1} - 42939 \beta_{2} ) q^{53} \) \( + ( -58245111 - 764761 \beta_{1} + 42059 \beta_{2} ) q^{55} \) \( + ( 7283507 - 25144 \beta_{1} + 1995 \beta_{2} ) q^{56} \) \( + ( -34252974 - 3065330 \beta_{1} - 63116 \beta_{2} ) q^{58} \) \( + ( 81342220 - 2420336 \beta_{1} - 85632 \beta_{2} ) q^{59} \) \( + ( 123243320 - 2098280 \beta_{1} - 32696 \beta_{2} ) q^{61} \) \( + ( 55432447 + 4850833 \beta_{1} + 114630 \beta_{2} ) q^{62} \) \( + ( -140230709 + 276180 \beta_{1} - 34329 \beta_{2} ) q^{64} \) \( + ( 164075228 - 2136700 \beta_{1} + 3852 \beta_{2} ) q^{65} \) \( + ( -84204862 - 235562 \beta_{1} - 227522 \beta_{2} ) q^{67} \) \( + ( 54090048 - 5389104 \beta_{1} + 335280 \beta_{2} ) q^{68} \) \( + ( 311697459 + 6690643 \beta_{1} + 461152 \beta_{2} ) q^{70} \) \( + ( 134397696 + 3010488 \beta_{1} - 188880 \beta_{2} ) q^{71} \) \( + ( -135542239 + 2361348 \beta_{1} + 194220 \beta_{2} ) q^{73} \) \( + ( -213791186 + 1543174 \beta_{1} - 351576 \beta_{2} ) q^{74} \) \( + ( -13166530 + 13000580 \beta_{1} + 32522 \beta_{2} ) q^{76} \) \( + ( 120147733 + 6239431 \beta_{1} + 42579 \beta_{2} ) q^{77} \) \( + ( 88483952 - 4693196 \beta_{1} - 673892 \beta_{2} ) q^{79} \) \( + ( -389453279 - 3145916 \beta_{1} - 78963 \beta_{2} ) q^{80} \) \( + ( -294802722 + 3094202 \beta_{1} - 502444 \beta_{2} ) q^{82} \) \( + ( 40541957 - 3456100 \beta_{1} + 296820 \beta_{2} ) q^{83} \) \( + ( 105417072 - 12297840 \beta_{1} + 884568 \beta_{2} ) q^{85} \) \( + ( -789652190 - 1426946 \beta_{1} - 947532 \beta_{2} ) q^{86} \) \( + ( 73085913 + 1055648 \beta_{1} - 212167 \beta_{2} ) q^{88} \) \( + ( 125968002 - 20852538 \beta_{1} + 927582 \beta_{2} ) q^{89} \) \( + ( 81686380 - 22455452 \beta_{1} + 304876 \beta_{2} ) q^{91} \) \( + ( -392918186 + 20198716 \beta_{1} + 1390986 \beta_{2} ) q^{92} \) \( + ( 28861146 - 19340726 \beta_{1} - 160688 \beta_{2} ) q^{94} \) \( + ( 178959590 + 39510824 \beta_{1} + 413088 \beta_{2} ) q^{95} \) \( + ( -146302513 - 7785896 \beta_{1} + 606616 \beta_{2} ) q^{97} \) \( + ( -864060762 - 25616490 \beta_{1} - 1523280 \beta_{2} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(3q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut +\mathstrut 597q^{4} \) \(\mathstrut +\mathstrut 1983q^{5} \) \(\mathstrut -\mathstrut 3693q^{7} \) \(\mathstrut +\mathstrut 4503q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut +\mathstrut 597q^{4} \) \(\mathstrut +\mathstrut 1983q^{5} \) \(\mathstrut -\mathstrut 3693q^{7} \) \(\mathstrut +\mathstrut 4503q^{8} \) \(\mathstrut -\mathstrut 18981q^{10} \) \(\mathstrut +\mathstrut 16863q^{11} \) \(\mathstrut +\mathstrut 116916q^{13} \) \(\mathstrut +\mathstrut 503463q^{14} \) \(\mathstrut -\mathstrut 239919q^{16} \) \(\mathstrut +\mathstrut 1014048q^{17} \) \(\mathstrut -\mathstrut 15222q^{19} \) \(\mathstrut +\mathstrut 2548407q^{20} \) \(\mathstrut +\mathstrut 305721q^{22} \) \(\mathstrut +\mathstrut 2927118q^{23} \) \(\mathstrut +\mathstrut 2133732q^{25} \) \(\mathstrut +\mathstrut 4765116q^{26} \) \(\mathstrut -\mathstrut 3535725q^{28} \) \(\mathstrut +\mathstrut 5768790q^{29} \) \(\mathstrut -\mathstrut 6575223q^{31} \) \(\mathstrut -\mathstrut 2687697q^{32} \) \(\mathstrut +\mathstrut 17098128q^{34} \) \(\mathstrut -\mathstrut 17340537q^{35} \) \(\mathstrut -\mathstrut 11686026q^{37} \) \(\mathstrut -\mathstrut 50473374q^{38} \) \(\mathstrut -\mathstrut 4130811q^{40} \) \(\mathstrut -\mathstrut 22213518q^{41} \) \(\mathstrut +\mathstrut 45384414q^{43} \) \(\mathstrut -\mathstrut 57206991q^{44} \) \(\mathstrut -\mathstrut 95638590q^{46} \) \(\mathstrut +\mathstrut 12392034q^{47} \) \(\mathstrut +\mathstrut 18933462q^{49} \) \(\mathstrut -\mathstrut 82984044q^{50} \) \(\mathstrut +\mathstrut 182736492q^{52} \) \(\mathstrut +\mathstrut 80579637q^{53} \) \(\mathstrut -\mathstrut 174735333q^{55} \) \(\mathstrut +\mathstrut 21850521q^{56} \) \(\mathstrut -\mathstrut 102758922q^{58} \) \(\mathstrut +\mathstrut 244026660q^{59} \) \(\mathstrut +\mathstrut 369729960q^{61} \) \(\mathstrut +\mathstrut 166297341q^{62} \) \(\mathstrut -\mathstrut 420692127q^{64} \) \(\mathstrut +\mathstrut 492225684q^{65} \) \(\mathstrut -\mathstrut 252614586q^{67} \) \(\mathstrut +\mathstrut 162270144q^{68} \) \(\mathstrut +\mathstrut 935092377q^{70} \) \(\mathstrut +\mathstrut 403193088q^{71} \) \(\mathstrut -\mathstrut 406626717q^{73} \) \(\mathstrut -\mathstrut 641373558q^{74} \) \(\mathstrut -\mathstrut 39499590q^{76} \) \(\mathstrut +\mathstrut 360443199q^{77} \) \(\mathstrut +\mathstrut 265451856q^{79} \) \(\mathstrut -\mathstrut 1168359837q^{80} \) \(\mathstrut -\mathstrut 884408166q^{82} \) \(\mathstrut +\mathstrut 121625871q^{83} \) \(\mathstrut +\mathstrut 316251216q^{85} \) \(\mathstrut -\mathstrut 2368956570q^{86} \) \(\mathstrut +\mathstrut 219257739q^{88} \) \(\mathstrut +\mathstrut 377904006q^{89} \) \(\mathstrut +\mathstrut 245059140q^{91} \) \(\mathstrut -\mathstrut 1178754558q^{92} \) \(\mathstrut +\mathstrut 86583438q^{94} \) \(\mathstrut +\mathstrut 536878770q^{95} \) \(\mathstrut -\mathstrut 438907539q^{97} \) \(\mathstrut -\mathstrut 2592182286q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3}\mathstrut -\mathstrut \) \(x^{2}\mathstrut -\mathstrut \) \(118\) \(x\mathstrut +\mathstrut \) \(136\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 3 \nu - 1 \)
\(\beta_{2}\)\(=\)\( 9 \nu^{2} + 6 \nu - 713 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(1\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2}\mathstrut -\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(711\)\()/9\)

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
10.7777
1.15428
−10.9320
−32.3331 0 533.432 2071.63 0 −10517.1 −692.954 0 −66982.2
1.2 −3.46285 0 −500.009 −1404.01 0 1665.57 3504.44 0 4861.88
1.3 32.7960 0 563.577 1315.38 0 5158.54 1691.52 0 43139.3
\(n\): 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 \(T_{2}^{3} \) \(\mathstrut +\mathstrut 3 T_{2}^{2} \) \(\mathstrut -\mathstrut 1062 T_{2} \) \(\mathstrut -\mathstrut 3672 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(27))\).