Properties

Label 27.11.b.d
Level 27
Weight 11
Character orbit 27.b
Analytic conductor 17.155
Analytic rank 0
Dimension 6
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 27 = 3^{3} \)
Weight: \( k \) = \( 11 \)
Character orbit: \([\chi]\) = 27.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(17.1546458222\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{21} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( -443 + \beta_{2} ) q^{4} \) \( + ( -26 \beta_{1} - \beta_{4} - \beta_{5} ) q^{5} \) \( + ( 773 - \beta_{2} + \beta_{3} ) q^{7} \) \( + ( -128 \beta_{1} - 4 \beta_{4} - 5 \beta_{5} ) q^{8} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( + ( -443 + \beta_{2} ) q^{4} \) \( + ( -26 \beta_{1} - \beta_{4} - \beta_{5} ) q^{5} \) \( + ( 773 - \beta_{2} + \beta_{3} ) q^{7} \) \( + ( -128 \beta_{1} - 4 \beta_{4} - 5 \beta_{5} ) q^{8} \) \( + ( 38943 - 78 \beta_{2} + 13 \beta_{3} ) q^{10} \) \( + ( 1291 \beta_{1} + 51 \beta_{4} + 34 \beta_{5} ) q^{11} \) \( + ( 21974 - 18 \beta_{2} + 50 \beta_{3} ) q^{13} \) \( + ( 1193 \beta_{1} + 192 \beta_{4} + 76 \beta_{5} ) q^{14} \) \( + ( -262517 + 647 \beta_{2} + 56 \beta_{3} ) q^{16} \) \( + ( -13522 \beta_{1} - 10 \beta_{4} - 484 \beta_{5} ) q^{17} \) \( + ( 1284464 - 2810 \beta_{2} - 38 \beta_{3} ) q^{19} \) \( + ( 63864 \beta_{1} + 1732 \beta_{4} + 289 \beta_{5} ) q^{20} \) \( + ( -1932453 + 3246 \beta_{2} - 595 \beta_{3} ) q^{22} \) \( + ( -67754 \beta_{1} + 3116 \beta_{4} - 74 \beta_{5} ) q^{23} \) \( + ( -1778288 + 4238 \beta_{2} - 686 \beta_{3} ) q^{25} \) \( + ( 20286 \beta_{1} + 9472 \beta_{4} + 3640 \beta_{5} ) q^{26} \) \( + ( -1096711 + 5397 \beta_{2} - 1008 \beta_{3} ) q^{28} \) \( + ( 624110 \beta_{1} + 3418 \beta_{4} - 4152 \beta_{5} ) q^{29} \) \( + ( 1931999 - 15865 \beta_{2} + 2521 \beta_{3} ) q^{31} \) \( + ( -868496 \beta_{1} + 3844 \beta_{4} - 4379 \beta_{5} ) q^{32} \) \( + ( 19908774 - 33476 \beta_{2} + 2026 \beta_{3} ) q^{34} \) \( + ( -784359 \beta_{1} - 2557 \beta_{4} - 3924 \beta_{5} ) q^{35} \) \( + ( 1195220 + 41230 \beta_{2} + 8722 \beta_{3} ) q^{37} \) \( + ( 3287736 \beta_{1} + 4096 \beta_{4} + 11352 \beta_{5} ) q^{38} \) \( + ( -55003383 + 14893 \beta_{2} - 3432 \beta_{3} ) q^{40} \) \( + ( -2191746 \beta_{1} - 8060 \beta_{4} + 26742 \beta_{5} ) q^{41} \) \( + ( 33835346 - 6542 \beta_{2} + 5838 \beta_{3} ) q^{43} \) \( + ( -2739928 \beta_{1} - 72620 \beta_{4} - 23659 \beta_{5} ) q^{44} \) \( + ( 97329852 - 36512 \beta_{2} - 27748 \beta_{3} ) q^{46} \) \( + ( 4053414 \beta_{1} - 92722 \beta_{4} + 5036 \beta_{5} ) q^{47} \) \( + ( -199281048 - 47882 \beta_{2} - 2070 \beta_{3} ) q^{49} \) \( + ( -4584776 \beta_{1} - 145920 \beta_{4} - 69896 \beta_{5} ) q^{50} \) \( + ( -14057938 + 255286 \beta_{2} - 48608 \beta_{3} ) q^{52} \) \( + ( -103204 \beta_{1} - 115033 \beta_{4} + 15177 \beta_{5} ) q^{53} \) \( + ( 453193875 - 280235 \beta_{2} + 28139 \beta_{3} ) q^{55} \) \( + ( -3410240 \beta_{1} - 14484 \beta_{4} - 20729 \beta_{5} ) q^{56} \) \( + ( -917285238 + 491476 \beta_{2} - 14154 \beta_{3} ) q^{58} \) \( + ( 8805526 \beta_{1} + 57830 \beta_{4} + 115268 \beta_{5} ) q^{59} \) \( + ( 387693044 - 43640 \beta_{2} + 92280 \beta_{3} ) q^{61} \) \( + ( 12451715 \beta_{1} + 537408 \beta_{4} + 258316 \beta_{5} ) q^{62} \) \( + ( 1003297285 - 343223 \beta_{2} + 40264 \beta_{3} ) q^{64} \) \( + ( -37109302 \beta_{1} - 69926 \beta_{4} - 184452 \beta_{5} ) q^{65} \) \( + ( -1094909674 - 1212862 \beta_{2} + 110782 \beta_{3} ) q^{67} \) \( + ( 29211216 \beta_{1} + 504552 \beta_{4} - 184390 \beta_{5} ) q^{68} \) \( + ( 1152887355 - 973370 \beta_{2} + 38709 \beta_{3} ) q^{70} \) \( + ( 15070956 \beta_{1} - 306654 \beta_{4} - 525526 \beta_{5} ) q^{71} \) \( + ( 436470113 + 2732694 \beta_{2} - 82614 \beta_{3} ) q^{73} \) \( + ( -30557508 \beta_{1} + 1474816 \beta_{4} + 413112 \beta_{5} ) q^{74} \) \( + ( -3512078032 + 920784 \beta_{2} - 121184 \beta_{3} ) q^{76} \) \( + ( 35809630 \beta_{1} + 129671 \beta_{4} + 142911 \beta_{5} ) q^{77} \) \( + ( 939722318 + 619212 \beta_{2} - 304108 \beta_{3} ) q^{79} \) \( + ( 826064 \beta_{1} + 1068780 \beta_{4} - 22201 \beta_{5} ) q^{80} \) \( + ( 3217049172 - 1183984 \beta_{2} - 34428 \beta_{3} ) q^{82} \) \( + ( -50163915 \beta_{1} - 384343 \beta_{4} + 524274 \beta_{5} ) q^{83} \) \( + ( -4373230122 - 786478 \beta_{2} - 357490 \beta_{3} ) q^{85} \) \( + ( 36786442 \beta_{1} + 1123712 \beta_{4} + 447208 \beta_{5} ) q^{86} \) \( + ( 2092201389 - 1184863 \beta_{2} + 138936 \beta_{3} ) q^{88} \) \( + ( 35475428 \beta_{1} - 2089474 \beta_{4} + 294790 \beta_{5} ) q^{89} \) \( + ( 4122682798 - 2218896 \beta_{2} - 138256 \beta_{3} ) q^{91} \) \( + ( 61855936 \beta_{1} - 1879792 \beta_{4} - 1863324 \beta_{5} ) q^{92} \) \( + ( -5885285346 + 3239948 \beta_{2} + 814354 \beta_{3} ) q^{94} \) \( + ( -153435554 \beta_{1} - 4887744 \beta_{4} - 726134 \beta_{5} ) q^{95} \) \( + ( -778793941 + 6214552 \beta_{2} + 119784 \beta_{3} ) q^{97} \) \( + ( -164734480 \beta_{1} - 197632 \beta_{4} + 92440 \beta_{5} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(6q \) \(\mathstrut -\mathstrut 2658q^{4} \) \(\mathstrut +\mathstrut 4638q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut -\mathstrut 2658q^{4} \) \(\mathstrut +\mathstrut 4638q^{7} \) \(\mathstrut +\mathstrut 233658q^{10} \) \(\mathstrut +\mathstrut 131844q^{13} \) \(\mathstrut -\mathstrut 1575102q^{16} \) \(\mathstrut +\mathstrut 7706784q^{19} \) \(\mathstrut -\mathstrut 11594718q^{22} \) \(\mathstrut -\mathstrut 10669728q^{25} \) \(\mathstrut -\mathstrut 6580266q^{28} \) \(\mathstrut +\mathstrut 11591994q^{31} \) \(\mathstrut +\mathstrut 119452644q^{34} \) \(\mathstrut +\mathstrut 7171320q^{37} \) \(\mathstrut -\mathstrut 330020298q^{40} \) \(\mathstrut +\mathstrut 203012076q^{43} \) \(\mathstrut +\mathstrut 583979112q^{46} \) \(\mathstrut -\mathstrut 1195686288q^{49} \) \(\mathstrut -\mathstrut 84347628q^{52} \) \(\mathstrut +\mathstrut 2719163250q^{55} \) \(\mathstrut -\mathstrut 5503711428q^{58} \) \(\mathstrut +\mathstrut 2326158264q^{61} \) \(\mathstrut +\mathstrut 6019783710q^{64} \) \(\mathstrut -\mathstrut 6569458044q^{67} \) \(\mathstrut +\mathstrut 6917324130q^{70} \) \(\mathstrut +\mathstrut 2618820678q^{73} \) \(\mathstrut -\mathstrut 21072468192q^{76} \) \(\mathstrut +\mathstrut 5638333908q^{79} \) \(\mathstrut +\mathstrut 19302295032q^{82} \) \(\mathstrut -\mathstrut 26239380732q^{85} \) \(\mathstrut +\mathstrut 12553208334q^{88} \) \(\mathstrut +\mathstrut 24736096788q^{91} \) \(\mathstrut -\mathstrut 35311712076q^{94} \) \(\mathstrut -\mathstrut 4672763646q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6}\mathstrut -\mathstrut \) \(196\) \(x^{3}\mathstrut +\mathstrut \) \(11881\) \(x^{2}\mathstrut -\mathstrut \) \(21364\) \(x\mathstrut +\mathstrut \) \(19208\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -39349 \nu^{5} - 35378 \nu^{4} - 124264 \nu^{3} + 8895068 \nu^{2} - 474116157 \nu + 432503890 \)\()/8398110\)
\(\beta_{2}\)\(=\)\((\)\( -1092 \nu^{5} - 22209 \nu^{4} - 119028 \nu^{3} + 107016 \nu^{2} - 144804102 \)\()/142825\)
\(\beta_{3}\)\(=\)\((\)\( -4128 \nu^{5} + 144879 \nu^{4} - 449952 \nu^{3} + 404544 \nu^{2} + 1265125722 \)\()/142825\)
\(\beta_{4}\)\(=\)\((\)\( -3642017 \nu^{5} - 3274474 \nu^{4} - 1326038 \nu^{3} + 268737511 \nu^{2} - 42773545215 \nu + 39033977318 \)\()/20995275\)
\(\beta_{5}\)\(=\)\((\)\( -2429719 \nu^{5} - 2184518 \nu^{4} + 29008784 \nu^{3} + 817582052 \nu^{2} - 25277368455 \nu + 23111397526 \)\()/13996850\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(27\) \(\beta_{5}\mathstrut +\mathstrut \) \(16\) \(\beta_{4}\mathstrut +\mathstrut \) \(9\) \(\beta_{3}\mathstrut +\mathstrut \) \(63\) \(\beta_{2}\mathstrut -\mathstrut \) \(1751\) \(\beta_{1}\)\()/17496\)
\(\nu^{2}\)\(=\)\((\)\(27\) \(\beta_{5}\mathstrut -\mathstrut \) \(146\) \(\beta_{4}\mathstrut +\mathstrut \) \(4405\) \(\beta_{1}\)\()/4374\)
\(\nu^{3}\)\(=\)\((\)\(2943\) \(\beta_{5}\mathstrut +\mathstrut \) \(2528\) \(\beta_{4}\mathstrut -\mathstrut \) \(981\) \(\beta_{3}\mathstrut -\mathstrut \) \(6867\) \(\beta_{2}\mathstrut -\mathstrut \) \(185371\) \(\beta_{1}\mathstrut +\mathstrut \) \(1714608\)\()/17496\)
\(\nu^{4}\)\(=\)\((\)\(455\) \(\beta_{3}\mathstrut -\mathstrut \) \(1720\) \(\beta_{2}\mathstrut -\mathstrut \) \(5774166\)\()/729\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(103401\) \(\beta_{5}\mathstrut -\mathstrut \) \(110928\) \(\beta_{4}\mathstrut -\mathstrut \) \(38387\) \(\beta_{3}\mathstrut -\mathstrut \) \(233429\) \(\beta_{2}\mathstrut +\mathstrut \) \(7310733\) \(\beta_{1}\mathstrut +\mathstrut \) \(103829040\)\()/5832\)

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
0.913049 + 0.913049i
−7.79647 + 7.79647i
6.88342 + 6.88342i
6.88342 6.88342i
−7.79647 7.79647i
0.913049 0.913049i
49.7909i 0 −1455.14 4680.95i 0 10645.1 21466.7i 0 233069.
26.2 43.0534i 0 −829.595 2154.41i 0 −11290.6 8369.79i 0 −92754.7
26.3 8.26247i 0 955.732 2842.36i 0 2964.51 16357.5i 0 −23484.9
26.4 8.26247i 0 955.732 2842.36i 0 2964.51 16357.5i 0 −23484.9
26.5 43.0534i 0 −829.595 2154.41i 0 −11290.6 8369.79i 0 −92754.7
26.6 49.7909i 0 −1455.14 4680.95i 0 10645.1 21466.7i 0 233069.
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.6
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 \(T_{2}^{6} \) \(\mathstrut +\mathstrut 4401 T_{2}^{4} \) \(\mathstrut +\mathstrut 4891104 T_{2}^{2} \) \(\mathstrut +\mathstrut 313714944 \) acting on \(S_{11}^{\mathrm{new}}(27, [\chi])\).