Properties

Label 27.6.c.a
Level 27
Weight 6
Character orbit 27.c
Analytic conductor 4.330
Analytic rank 0
Dimension 8
CM No
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 27 = 3^{3} \)
Weight: \( k \) = \( 6 \)
Character orbit: \([\chi]\) = 27.c (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(4.33036313495\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{8} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{2} + ( -13 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{4} + ( -1 - 19 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{5} + ( 7 - 7 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} + 2 \beta_{5} + 3 \beta_{6} ) q^{7} + ( 95 + 8 \beta_{3} - 5 \beta_{4} + 5 \beta_{5} + 4 \beta_{7} ) q^{8} +O(q^{10})\) \( q + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{2} + ( -13 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{4} + ( -1 - 19 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{5} + ( 7 - 7 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} + 2 \beta_{5} + 3 \beta_{6} ) q^{7} + ( 95 + 8 \beta_{3} - 5 \beta_{4} + 5 \beta_{5} + 4 \beta_{7} ) q^{8} + ( 18 + 55 \beta_{3} + \beta_{4} - \beta_{5} + 6 \beta_{7} ) q^{10} + ( -107 + 107 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} + 7 \beta_{5} + 2 \beta_{6} ) q^{11} + ( 3 - 65 \beta_{1} + 71 \beta_{2} + \beta_{4} + 3 \beta_{6} - 3 \beta_{7} ) q^{13} + ( -2 - 342 \beta_{1} - 13 \beta_{2} - 7 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{14} + ( -109 + 109 \beta_{1} - 120 \beta_{2} - 120 \beta_{3} - 3 \beta_{5} - 12 \beta_{6} ) q^{16} + ( 531 - 57 \beta_{3} + 35 \beta_{4} - 35 \beta_{5} - 19 \beta_{7} ) q^{17} + ( 119 - 81 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} - 39 \beta_{7} ) q^{19} + ( -1692 + 1692 \beta_{1} - 14 \beta_{2} - 14 \beta_{3} - 38 \beta_{5} - 16 \beta_{6} ) q^{20} + ( -24 + 285 \beta_{1} - 49 \beta_{2} - 32 \beta_{4} - 24 \beta_{6} + 24 \beta_{7} ) q^{22} + ( 49 - 2265 \beta_{1} + 104 \beta_{2} + 14 \beta_{4} + 49 \beta_{6} - 49 \beta_{7} ) q^{23} + ( -346 + 346 \beta_{1} + 355 \beta_{2} + 355 \beta_{3} - 55 \beta_{5} - 15 \beta_{6} ) q^{25} + ( 3164 + 125 \beta_{3} - 83 \beta_{4} + 83 \beta_{5} - 2 \beta_{7} ) q^{26} + ( -250 - 78 \beta_{3} - 24 \beta_{4} + 24 \beta_{5} + 72 \beta_{7} ) q^{28} + ( -2947 + 2947 \beta_{1} + 107 \beta_{2} + 107 \beta_{3} + 21 \beta_{5} + 51 \beta_{6} ) q^{29} + ( 69 + 409 \beta_{1} - 682 \beta_{2} + 40 \beta_{4} + 69 \beta_{6} - 69 \beta_{7} ) q^{31} + ( -140 - 2083 \beta_{1} - 96 \beta_{2} + 5 \beta_{4} - 140 \beta_{6} + 140 \beta_{7} ) q^{32} + ( 2685 - 2685 \beta_{1} + 192 \beta_{2} + 192 \beta_{3} + 219 \beta_{5} + 102 \beta_{6} ) q^{34} + ( 1966 + 40 \beta_{3} + 66 \beta_{4} - 66 \beta_{5} + 111 \beta_{7} ) q^{35} + ( -2140 - 810 \beta_{3} + 150 \beta_{4} - 150 \beta_{5} + 6 \beta_{7} ) q^{37} + ( 2653 - 2653 \beta_{1} - 542 \beta_{2} - 542 \beta_{3} + 189 \beta_{5} - 90 \beta_{6} ) q^{38} + ( -72 - 2376 \beta_{1} + 704 \beta_{2} + 208 \beta_{4} - 72 \beta_{6} + 72 \beta_{7} ) q^{40} + ( 54 - 173 \beta_{1} - 646 \beta_{2} - 18 \beta_{4} + 54 \beta_{6} - 54 \beta_{7} ) q^{41} + ( -1517 + 1517 \beta_{1} + 205 \beta_{2} + 205 \beta_{3} - 181 \beta_{5} ) q^{43} + ( -6353 - 686 \beta_{3} - 7 \beta_{4} + 7 \beta_{5} - 16 \beta_{7} ) q^{44} + ( 6372 + 2131 \beta_{3} - 293 \beta_{4} + 293 \beta_{5} - 42 \beta_{7} ) q^{46} + ( 297 - 297 \beta_{1} + 1100 \beta_{2} + 1100 \beta_{3} - 190 \beta_{5} + 109 \beta_{6} ) q^{47} + ( -9 + 2364 \beta_{1} + 77 \beta_{2} - 425 \beta_{4} - 9 \beta_{6} + 9 \beta_{7} ) q^{49} + ( 190 + 13909 \beta_{1} + 1186 \beta_{2} - 145 \beta_{4} + 190 \beta_{6} - 190 \beta_{7} ) q^{50} + ( -9110 + 9110 \beta_{1} - 3114 \beta_{2} - 3114 \beta_{3} - 336 \beta_{5} - 240 \beta_{6} ) q^{52} + ( -2052 + 1458 \beta_{3} - 30 \beta_{4} + 30 \beta_{5} - 390 \beta_{7} ) q^{53} + ( -4482 - 182 \beta_{3} + 76 \beta_{4} - 76 \beta_{5} - 195 \beta_{7} ) q^{55} + ( 15202 - 15202 \beta_{1} + 104 \beta_{2} + 104 \beta_{3} - 434 \beta_{5} - 16 \beta_{6} ) q^{56} + ( 18 + 1422 \beta_{1} + 2927 \beta_{2} - 323 \beta_{4} + 18 \beta_{6} - 18 \beta_{7} ) q^{58} + ( 14 - 1033 \beta_{1} + 1637 \beta_{2} + 391 \beta_{4} + 14 \beta_{6} - 14 \beta_{7} ) q^{59} + ( 12271 - 12271 \beta_{1} + 919 \beta_{2} + 919 \beta_{3} + 545 \beta_{5} - 261 \beta_{6} ) q^{61} + ( -30550 - 919 \beta_{3} + 355 \beta_{4} - 355 \beta_{5} + 22 \beta_{7} ) q^{62} + ( -2735 - 48 \beta_{3} + 597 \beta_{4} - 597 \beta_{5} - 84 \beta_{7} ) q^{64} + ( 2015 - 2015 \beta_{1} - 4411 \beta_{2} - 4411 \beta_{3} + 63 \beta_{5} - 369 \beta_{6} ) q^{65} + ( 24 - 1679 \beta_{1} - 1459 \beta_{2} + 979 \beta_{4} + 24 \beta_{6} - 24 \beta_{7} ) q^{67} + ( -64 + 32149 \beta_{1} - 5304 \beta_{2} - 35 \beta_{4} - 64 \beta_{6} + 64 \beta_{7} ) q^{68} + ( 252 - 252 \beta_{1} + 865 \beta_{2} + 865 \beta_{3} - 175 \beta_{5} + 486 \beta_{6} ) q^{70} + ( 7956 - 3288 \beta_{3} - 616 \beta_{4} + 616 \beta_{5} + 716 \beta_{7} ) q^{71} + ( 14699 - 1701 \beta_{3} - 1017 \beta_{4} + 1017 \beta_{5} + 873 \beta_{7} ) q^{73} + ( 42388 - 42388 \beta_{1} + 6766 \beta_{2} + 6766 \beta_{3} + 1242 \beta_{5} + 612 \beta_{6} ) q^{74} + ( 312 - 11255 \beta_{1} - 5324 \beta_{2} + 149 \beta_{4} + 312 \beta_{6} - 312 \beta_{7} ) q^{76} + ( -969 - 9689 \beta_{1} + 1955 \beta_{2} - 795 \beta_{4} - 969 \beta_{6} + 969 \beta_{7} ) q^{77} + ( -12761 + 12761 \beta_{1} + 5494 \beta_{2} + 5494 \beta_{3} + 392 \beta_{5} + 1293 \beta_{6} ) q^{79} + ( -13648 + 8696 \beta_{3} + 104 \beta_{4} - 104 \beta_{5} + 464 \beta_{7} ) q^{80} + ( -29637 - 1571 \beta_{3} + 538 \beta_{4} - 538 \beta_{5} - 180 \beta_{7} ) q^{82} + ( -28999 + 28999 \beta_{1} + 1016 \beta_{2} + 1016 \beta_{3} + 846 \beta_{5} - 81 \beta_{6} ) q^{83} + ( -942 + 8862 \beta_{1} - 4434 \beta_{2} - 294 \beta_{4} - 942 \beta_{6} + 942 \beta_{7} ) q^{85} + ( 724 + 2073 \beta_{1} + 5837 \beta_{2} + 338 \beta_{4} + 724 \beta_{6} - 724 \beta_{7} ) q^{86} + ( 26919 - 26919 \beta_{1} + 5120 \beta_{2} + 5120 \beta_{3} - 311 \beta_{5} - 828 \beta_{6} ) q^{88} + ( 56034 - 3696 \beta_{3} - 536 \beta_{4} + 536 \beta_{5} - 716 \beta_{7} ) q^{89} + ( 13702 - 1132 \beta_{3} + 146 \beta_{4} - 146 \beta_{5} - 195 \beta_{7} ) q^{91} + ( -37202 + 37202 \beta_{1} - 12962 \beta_{2} - 12962 \beta_{3} - 2436 \beta_{5} + 312 \beta_{6} ) q^{92} + ( 978 + 40926 \beta_{1} + 4661 \beta_{2} - 857 \beta_{4} + 978 \beta_{6} - 978 \beta_{7} ) q^{94} + ( 412 - 70940 \beta_{1} - 12476 \beta_{2} + 524 \beta_{4} + 412 \beta_{6} - 412 \beta_{7} ) q^{95} + ( 4549 - 4549 \beta_{1} - 11342 \beta_{2} - 11342 \beta_{3} - 154 \beta_{5} - 2142 \beta_{6} ) q^{97} + ( -9873 - 12804 \beta_{3} + 1225 \beta_{4} - 1225 \beta_{5} - 1682 \beta_{7} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 3q^{2} - 49q^{4} - 78q^{5} + 28q^{7} + 750q^{8} + O(q^{10}) \) \( 8q - 3q^{2} - 49q^{4} - 78q^{5} + 28q^{7} + 750q^{8} + 60q^{10} - 444q^{11} - 182q^{13} - 1392q^{14} - 289q^{16} + 4356q^{17} + 952q^{19} - 6684q^{20} + 1011q^{22} - 8844q^{23} - 1654q^{25} + 24888q^{26} - 1604q^{28} - 12018q^{29} + 1132q^{31} - 8703q^{32} + 10125q^{34} + 16224q^{35} - 15176q^{37} + 11145q^{38} - 8736q^{40} - 1248q^{41} - 6092q^{43} - 49530q^{44} + 45960q^{46} + 60q^{47} + 9090q^{49} + 57057q^{50} - 32510q^{52} - 20952q^{53} - 36120q^{55} + 61170q^{56} + 8328q^{58} - 2076q^{59} + 48142q^{61} - 241764q^{62} - 20926q^{64} + 13146q^{65} - 7148q^{67} + 123129q^{68} - 654q^{70} + 71856q^{71} + 122452q^{73} + 160320q^{74} - 49571q^{76} - 39534q^{77} - 59516q^{79} - 124512q^{80} - 233598q^{82} - 117696q^{83} + 28836q^{85} + 15915q^{86} + 104523q^{88} + 451728q^{89} + 111392q^{91} - 134034q^{92} + 169464q^{94} - 294888q^{95} + 33976q^{97} - 57654q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 40 x^{6} + 568 x^{4} + 3363 x^{2} + 7056\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{7} + 40 \nu^{5} + 484 \nu^{3} + 1683 \nu + 84 \)\()/168\)
\(\beta_{2}\)\(=\)\((\)\( -9 \nu^{7} + 28 \nu^{6} - 276 \nu^{5} + 868 \nu^{4} - 2592 \nu^{3} + 8428 \nu^{2} - 7167 \nu + 25116 \)\()/168\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{6} - 31 \nu^{4} - 301 \nu^{2} - 897 \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( -9 \nu^{7} + 28 \nu^{6} - 276 \nu^{5} + 1372 \nu^{4} - 2592 \nu^{3} + 18508 \nu^{2} - 5655 \nu + 67452 \)\()/168\)
\(\beta_{5}\)\(=\)\((\)\( -9 \nu^{7} - 28 \nu^{6} - 276 \nu^{5} - 1372 \nu^{4} - 2592 \nu^{3} - 18508 \nu^{2} - 5655 \nu - 67452 \)\()/168\)
\(\beta_{6}\)\(=\)\((\)\( -71 \nu^{7} + 56 \nu^{6} - 2252 \nu^{5} + 1988 \nu^{4} - 22772 \nu^{3} + 22652 \nu^{2} - 72705 \nu + 78876 \)\()/168\)
\(\beta_{7}\)\(=\)\((\)\( 2 \nu^{6} + 71 \nu^{4} + 809 \nu^{2} + 2820 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{2}\)\()/18\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{7} + \beta_{5} - \beta_{4} + 3 \beta_{3} - 180\)\()/18\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{7} - 2 \beta_{6} - 6 \beta_{5} - 6 \beta_{4} + 13 \beta_{3} + 26 \beta_{2} - 16 \beta_{1} + 7\)\()/9\)
\(\nu^{4}\)\(=\)\((\)\(-40 \beta_{7} - 23 \beta_{5} + 23 \beta_{4} - 57 \beta_{3} + 2088\)\()/18\)
\(\nu^{5}\)\(=\)\((\)\(-42 \beta_{7} + 84 \beta_{6} + 157 \beta_{5} + 157 \beta_{4} - 433 \beta_{3} - 866 \beta_{2} + 996 \beta_{1} - 456\)\()/18\)
\(\nu^{6}\)\(=\)\((\)\(319 \beta_{7} + 206 \beta_{5} - 206 \beta_{4} + 405 \beta_{3} - 13347\)\()/9\)
\(\nu^{7}\)\(=\)\((\)\(712 \beta_{7} - 1424 \beta_{6} - 2155 \beta_{5} - 2155 \beta_{4} + 6419 \beta_{3} + 12838 \beta_{2} - 21328 \beta_{1} + 9952\)\()/18\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
2.56934i
3.84183i
2.34949i
3.62198i
2.56934i
3.84183i
2.34949i
3.62198i
−4.96163 8.59380i 0 −33.2356 + 57.5657i −23.6560 + 40.9735i 0 1.01546 + 1.75882i 342.066 0 469.490
10.2 −1.78978 3.09998i 0 9.59340 16.6162i −4.05388 + 7.02152i 0 −87.7139 151.925i −183.226 0 29.0221
10.3 1.47673 + 2.55778i 0 11.6385 20.1585i 32.4255 56.1625i 0 80.0952 + 138.729i 163.259 0 191.535
10.4 3.77467 + 6.53793i 0 −12.4963 + 21.6443i −43.7155 + 75.7175i 0 20.6033 + 35.6859i 52.9007 0 −660.048
19.1 −4.96163 + 8.59380i 0 −33.2356 57.5657i −23.6560 40.9735i 0 1.01546 1.75882i 342.066 0 469.490
19.2 −1.78978 + 3.09998i 0 9.59340 + 16.6162i −4.05388 7.02152i 0 −87.7139 + 151.925i −183.226 0 29.0221
19.3 1.47673 2.55778i 0 11.6385 + 20.1585i 32.4255 + 56.1625i 0 80.0952 138.729i 163.259 0 191.535
19.4 3.77467 6.53793i 0 −12.4963 21.6443i −43.7155 75.7175i 0 20.6033 35.6859i 52.9007 0 −660.048
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.4
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
9.c Even 1 yes

Hecke kernels

There are no other newforms in \(S_{6}^{\mathrm{new}}(27, [\chi])\).