Properties

Label 189.2.s.b
Level 189
Weight 2
Character orbit 189.s
Analytic conductor 1.509
Analytic rank 0
Dimension 10
CM no
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 189.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.50917259820\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: 10.0.288778218147.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} ) q^{2} + ( \beta_{2} + \beta_{4} - \beta_{6} + \beta_{8} - \beta_{9} ) q^{4} + ( \beta_{2} + \beta_{8} + \beta_{9} ) q^{5} + ( 1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{7} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} ) q^{8} +O(q^{10})\) \( q + ( -\beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} ) q^{2} + ( \beta_{2} + \beta_{4} - \beta_{6} + \beta_{8} - \beta_{9} ) q^{4} + ( \beta_{2} + \beta_{8} + \beta_{9} ) q^{5} + ( 1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{7} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} ) q^{8} + ( -1 - \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{10} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + 2 \beta_{6} ) q^{11} + ( -\beta_{1} + \beta_{3} + \beta_{7} ) q^{13} + ( 1 + \beta_{2} - \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{14} + ( -1 - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{16} + ( 3 + \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{6} - \beta_{7} + \beta_{9} ) q^{17} + ( 2 \beta_{4} + \beta_{5} + 3 \beta_{7} + \beta_{8} - \beta_{9} ) q^{19} + ( -\beta_{2} + 4 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} + \beta_{8} - \beta_{9} ) q^{20} + ( 3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{22} + ( -1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} - \beta_{8} + \beta_{9} ) q^{23} + ( -2 - 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{7} - 2 \beta_{9} ) q^{25} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{26} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} - 4 \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{28} + ( 2 - \beta_{1} + 2 \beta_{2} + 3 \beta_{4} + 4 \beta_{5} + \beta_{6} + 4 \beta_{7} - \beta_{8} - \beta_{9} ) q^{29} + ( -2 \beta_{1} + \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{31} + ( -6 - \beta_{1} - \beta_{2} - 3 \beta_{6} ) q^{32} + ( \beta_{1} + \beta_{2} - 3 \beta_{4} - 3 \beta_{7} - 3 \beta_{8} ) q^{34} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - 3 \beta_{6} - \beta_{7} - \beta_{8} ) q^{35} + ( 2 \beta_{2} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{8} ) q^{37} + ( -3 - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{7} - 2 \beta_{8} ) q^{38} + ( 1 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{40} + ( 5 \beta_{1} - 4 \beta_{2} - 4 \beta_{5} + 4 \beta_{8} - 4 \beta_{9} ) q^{41} + ( 6 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} + 3 \beta_{8} - 3 \beta_{9} ) q^{43} + ( -2 - \beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - 3 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{44} + ( -3 - \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{46} + ( -3 - 2 \beta_{1} - \beta_{2} - \beta_{4} - 3 \beta_{6} - \beta_{9} ) q^{47} + ( -2 - 2 \beta_{1} - 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} + 4 \beta_{9} ) q^{49} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} ) q^{50} + ( 1 - 2 \beta_{1} + \beta_{2} + 3 \beta_{3} + 3 \beta_{5} + 2 \beta_{6} ) q^{52} + ( 1 + 2 \beta_{1} - 4 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{53} + ( -1 + 4 \beta_{3} + 4 \beta_{4} - 2 \beta_{6} + \beta_{8} - \beta_{9} ) q^{55} + ( 6 + 3 \beta_{1} + 3 \beta_{2} + \beta_{5} - 4 \beta_{7} - \beta_{8} ) q^{56} + ( -3 \beta_{3} - 3 \beta_{4} - 6 \beta_{7} - 7 \beta_{8} - \beta_{9} ) q^{58} + ( -2 \beta_{1} - 3 \beta_{4} + \beta_{5} - 3 \beta_{6} - 2 \beta_{8} + 3 \beta_{9} ) q^{59} + ( 1 + 2 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - \beta_{6} - 4 \beta_{7} - 3 \beta_{8} ) q^{61} + ( -3 + \beta_{2} + 3 \beta_{3} + 4 \beta_{4} + \beta_{5} + 6 \beta_{7} + 4 \beta_{8} - 2 \beta_{9} ) q^{62} + ( -1 - \beta_{2} + 3 \beta_{3} + 4 \beta_{4} + \beta_{5} + 6 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} ) q^{64} + ( -2 \beta_{1} + 4 \beta_{2} - \beta_{4} + \beta_{5} + \beta_{8} + 2 \beta_{9} ) q^{65} + ( -\beta_{2} + 2 \beta_{3} - \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + \beta_{7} + 2 \beta_{9} ) q^{67} + ( 6 + 3 \beta_{2} + 3 \beta_{4} + 3 \beta_{5} - 3 \beta_{8} - 3 \beta_{9} ) q^{68} + ( -1 + 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + 3 \beta_{8} - \beta_{9} ) q^{70} + ( 2 \beta_{1} - \beta_{2} - 5 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} + 2 \beta_{8} - 2 \beta_{9} ) q^{71} + ( -\beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{7} - 3 \beta_{8} ) q^{73} + ( 2 - 4 \beta_{3} - 4 \beta_{5} + 4 \beta_{6} ) q^{74} + ( -3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{5} + 3 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} ) q^{76} + ( -3 + \beta_{2} - \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 3 \beta_{6} - 4 \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{77} + ( 4 - 3 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - \beta_{4} + 3 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} - 3 \beta_{8} + 2 \beta_{9} ) q^{79} + ( 6 - 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + 6 \beta_{6} + \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{80} + ( 2 - \beta_{1} - 3 \beta_{2} - \beta_{4} - 3 \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{82} + ( 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{4} - 3 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} ) q^{83} + ( 3 - 2 \beta_{2} - \beta_{3} - 2 \beta_{5} + 3 \beta_{6} + \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{85} + ( 5 - 6 \beta_{1} + 3 \beta_{2} + 5 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + 10 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} ) q^{86} + ( 3 + 3 \beta_{2} + 3 \beta_{4} + 3 \beta_{5} + \beta_{8} + \beta_{9} ) q^{88} + ( -3 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} + 5 \beta_{4} + \beta_{5} + 6 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 3 \beta_{9} ) q^{89} + ( -4 - 3 \beta_{1} - 2 \beta_{3} - 6 \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{91} + ( -6 + 2 \beta_{1} - 4 \beta_{2} - 6 \beta_{5} - 3 \beta_{6} + 6 \beta_{8} ) q^{92} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + 6 \beta_{4} - 2 \beta_{5} - \beta_{6} + 8 \beta_{7} + 8 \beta_{8} - 2 \beta_{9} ) q^{94} + ( \beta_{1} + 4 \beta_{2} + 4 \beta_{4} + 2 \beta_{5} + 3 \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{95} + ( 3 \beta_{1} - 2 \beta_{2} - \beta_{4} - 6 \beta_{5} - 2 \beta_{7} + 5 \beta_{8} + \beta_{9} ) q^{97} + ( -9 - 2 \beta_{1} - \beta_{2} + 5 \beta_{3} + \beta_{4} + 3 \beta_{5} - 6 \beta_{6} + 5 \beta_{7} + 4 \beta_{8} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 4q^{4} + 3q^{7} + O(q^{10}) \) \( 10q + 4q^{4} + 3q^{7} - 15q^{10} + 6q^{13} + 6q^{14} - 6q^{16} + 12q^{17} + 3q^{19} + 3q^{20} + 5q^{22} - 14q^{25} - 3q^{26} + 2q^{28} + 15q^{29} - 9q^{31} - 48q^{32} + 3q^{34} + 15q^{35} + 6q^{37} - 36q^{38} + 9q^{41} + 3q^{43} - 24q^{44} - 13q^{46} - 15q^{47} - 23q^{49} - 3q^{50} + 9q^{53} + 51q^{56} - 16q^{58} + 18q^{59} + 12q^{61} - 12q^{62} + 6q^{64} + 3q^{65} - 10q^{67} + 54q^{68} + 9q^{70} + 3q^{73} + 9q^{76} - 45q^{77} + 20q^{79} + 30q^{80} + 9q^{82} + 15q^{83} + 18q^{85} + 16q^{88} - 24q^{89} - 24q^{91} - 39q^{92} - 3q^{94} + 6q^{97} - 45q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - x^{9} + 7 x^{8} - 4 x^{7} + 34 x^{6} - 19 x^{5} + 64 x^{4} - x^{3} + 64 x^{2} - 21 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -339 \nu^{9} + 1348 \nu^{8} - 4381 \nu^{7} + 7882 \nu^{6} - 19883 \nu^{5} + 36059 \nu^{4} - 75410 \nu^{3} + 44484 \nu^{2} - 15165 \nu + 29709 \)\()/72795\)
\(\beta_{3}\)\(=\)\((\)\( 658 \nu^{9} + 2394 \nu^{8} + 4352 \nu^{7} + 10326 \nu^{6} + 25351 \nu^{5} + 51907 \nu^{4} + 47450 \nu^{3} + 30472 \nu^{2} + 130790 \nu + 98232 \)\()/72795\)
\(\beta_{4}\)\(=\)\((\)\( -4192 \nu^{9} - 796 \nu^{8} - 21678 \nu^{7} - 20279 \nu^{6} - 85319 \nu^{5} - 118353 \nu^{4} - 2560 \nu^{3} - 414508 \nu^{2} + 81750 \nu - 398583 \)\()/218385\)
\(\beta_{5}\)\(=\)\((\)\( 8236 \nu^{9} - 9272 \nu^{8} + 54399 \nu^{7} - 28438 \nu^{6} + 233822 \nu^{5} - 150966 \nu^{4} + 361225 \nu^{3} + 82264 \nu^{2} + 31515 \nu - 336546 \)\()/218385\)
\(\beta_{6}\)\(=\)\((\)\( 3301 \nu^{9} - 2962 \nu^{8} + 21759 \nu^{7} - 8823 \nu^{6} + 104352 \nu^{5} - 42836 \nu^{4} + 175205 \nu^{3} + 72109 \nu^{2} + 166780 \nu - 54156 \)\()/72795\)
\(\beta_{7}\)\(=\)\((\)\( -840 \nu^{9} + 248 \nu^{8} - 5659 \nu^{7} - 998 \nu^{6} - 27923 \nu^{5} - 3072 \nu^{4} - 51488 \nu^{3} - 30640 \nu^{2} - 51320 \nu + 11514 \)\()/14559\)
\(\beta_{8}\)\(=\)\((\)\( 3085 \nu^{9} - 1373 \nu^{8} + 17808 \nu^{7} + 1181 \nu^{6} + 84554 \nu^{5} + 5736 \nu^{4} + 111910 \nu^{3} + 124546 \nu^{2} + 106440 \nu + 17856 \)\()/43677\)
\(\beta_{9}\)\(=\)\((\)\( -18476 \nu^{9} + 18997 \nu^{8} - 128469 \nu^{7} + 65033 \nu^{6} - 601717 \nu^{5} + 295851 \nu^{4} - 1019855 \nu^{3} - 222374 \nu^{2} - 668685 \nu + 178101 \)\()/218385\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{9} - \beta_{8} + 3 \beta_{6} - \beta_{4} - \beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{9} + \beta_{8} - 3 \beta_{2}\)
\(\nu^{4}\)\(=\)\(-5 \beta_{9} + 5 \beta_{8} + \beta_{7} - 12 \beta_{6} - 5 \beta_{5} - \beta_{3} - 5 \beta_{2} + 5 \beta_{1} - 12\)
\(\nu^{5}\)\(=\)\(-5 \beta_{9} - \beta_{8} - \beta_{7} - 7 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} + 11 \beta_{2} - 11 \beta_{1}\)
\(\nu^{6}\)\(=\)\(6 \beta_{9} - 8 \beta_{8} - 14 \beta_{7} + 16 \beta_{5} + 9 \beta_{4} - 7 \beta_{3} + 22 \beta_{2} + 51\)
\(\nu^{7}\)\(=\)\(\beta_{9} - 31 \beta_{8} - 8 \beta_{7} + 31 \beta_{5} - 30 \beta_{4} + 8 \beta_{3} + \beta_{2} + 43 \beta_{1}\)
\(\nu^{8}\)\(=\)\(75 \beta_{9} - 66 \beta_{8} + 38 \beta_{7} + 222 \beta_{6} + 47 \beta_{5} - 37 \beta_{4} + 76 \beta_{3} + 8 \beta_{2} - 112 \beta_{1}\)
\(\nu^{9}\)\(=\)\(95 \beta_{9} + 189 \beta_{8} + 94 \beta_{7} + 37 \beta_{5} + 84 \beta_{4} + 47 \beta_{3} - 194 \beta_{2} - 3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/189\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(136\)
\(\chi(n)\) \(1 + \beta_{6}\) \(-\beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
0.827154 1.43267i
−1.04536 + 1.81062i
−0.539982 + 0.935277i
0.187540 0.324828i
1.07065 1.85442i
0.827154 + 1.43267i
−1.04536 1.81062i
−0.539982 0.935277i
0.187540 + 0.324828i
1.07065 + 1.85442i
−1.81474 1.04774i 0 1.19552 + 2.07070i 2.08983 0 −0.879217 + 2.49539i 0.819421i 0 −3.79250 2.18960i
17.2 −1.30778 0.755047i 0 0.140193 + 0.242822i 0.775876 0 2.05881 1.66171i 2.59678i 0 −1.01468 0.585823i
17.3 0.254498 + 0.146935i 0 −0.956820 1.65726i −3.06027 0 −1.22581 2.34465i 1.15010i 0 −0.778834 0.449660i
17.4 0.621951 + 0.359083i 0 −0.742118 1.28539i 1.44755 0 2.19442 + 1.47801i 2.50226i 0 0.900304 + 0.519791i
17.5 2.24607 + 1.29677i 0 2.36322 + 4.09323i −1.25299 0 −0.648211 2.56512i 7.07116i 0 −2.81429 1.62483i
89.1 −1.81474 + 1.04774i 0 1.19552 2.07070i 2.08983 0 −0.879217 2.49539i 0.819421i 0 −3.79250 + 2.18960i
89.2 −1.30778 + 0.755047i 0 0.140193 0.242822i 0.775876 0 2.05881 + 1.66171i 2.59678i 0 −1.01468 + 0.585823i
89.3 0.254498 0.146935i 0 −0.956820 + 1.65726i −3.06027 0 −1.22581 + 2.34465i 1.15010i 0 −0.778834 + 0.449660i
89.4 0.621951 0.359083i 0 −0.742118 + 1.28539i 1.44755 0 2.19442 1.47801i 2.50226i 0 0.900304 0.519791i
89.5 2.24607 1.29677i 0 2.36322 4.09323i −1.25299 0 −0.648211 + 2.56512i 7.07116i 0 −2.81429 + 1.62483i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 89.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.2.s.b 10
3.b odd 2 1 63.2.s.b yes 10
4.b odd 2 1 3024.2.df.b 10
7.b odd 2 1 1323.2.s.b 10
7.c even 3 1 1323.2.i.b 10
7.c even 3 1 1323.2.o.d 10
7.d odd 6 1 189.2.i.b 10
7.d odd 6 1 1323.2.o.c 10
9.c even 3 1 63.2.i.b 10
9.c even 3 1 567.2.p.d 10
9.d odd 6 1 189.2.i.b 10
9.d odd 6 1 567.2.p.c 10
12.b even 2 1 1008.2.df.b 10
21.c even 2 1 441.2.s.b 10
21.g even 6 1 63.2.i.b 10
21.g even 6 1 441.2.o.d 10
21.h odd 6 1 441.2.i.b 10
21.h odd 6 1 441.2.o.c 10
28.f even 6 1 3024.2.ca.b 10
36.f odd 6 1 1008.2.ca.b 10
36.h even 6 1 3024.2.ca.b 10
63.g even 3 1 441.2.s.b 10
63.h even 3 1 441.2.o.d 10
63.i even 6 1 567.2.p.d 10
63.i even 6 1 1323.2.o.d 10
63.j odd 6 1 1323.2.o.c 10
63.k odd 6 1 63.2.s.b yes 10
63.l odd 6 1 441.2.i.b 10
63.n odd 6 1 1323.2.s.b 10
63.o even 6 1 1323.2.i.b 10
63.s even 6 1 inner 189.2.s.b 10
63.t odd 6 1 441.2.o.c 10
63.t odd 6 1 567.2.p.c 10
84.j odd 6 1 1008.2.ca.b 10
252.n even 6 1 1008.2.df.b 10
252.bn odd 6 1 3024.2.df.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.i.b 10 9.c even 3 1
63.2.i.b 10 21.g even 6 1
63.2.s.b yes 10 3.b odd 2 1
63.2.s.b yes 10 63.k odd 6 1
189.2.i.b 10 7.d odd 6 1
189.2.i.b 10 9.d odd 6 1
189.2.s.b 10 1.a even 1 1 trivial
189.2.s.b 10 63.s even 6 1 inner
441.2.i.b 10 21.h odd 6 1
441.2.i.b 10 63.l odd 6 1
441.2.o.c 10 21.h odd 6 1
441.2.o.c 10 63.t odd 6 1
441.2.o.d 10 21.g even 6 1
441.2.o.d 10 63.h even 3 1
441.2.s.b 10 21.c even 2 1
441.2.s.b 10 63.g even 3 1
567.2.p.c 10 9.d odd 6 1
567.2.p.c 10 63.t odd 6 1
567.2.p.d 10 9.c even 3 1
567.2.p.d 10 63.i even 6 1
1008.2.ca.b 10 36.f odd 6 1
1008.2.ca.b 10 84.j odd 6 1
1008.2.df.b 10 12.b even 2 1
1008.2.df.b 10 252.n even 6 1
1323.2.i.b 10 7.c even 3 1
1323.2.i.b 10 63.o even 6 1
1323.2.o.c 10 7.d odd 6 1
1323.2.o.c 10 63.j odd 6 1
1323.2.o.d 10 7.c even 3 1
1323.2.o.d 10 63.i even 6 1
1323.2.s.b 10 7.b odd 2 1
1323.2.s.b 10 63.n odd 6 1
3024.2.ca.b 10 28.f even 6 1
3024.2.ca.b 10 36.h even 6 1
3024.2.df.b 10 4.b odd 2 1
3024.2.df.b 10 252.bn odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{10} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(189, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T^{2} + 4 T^{4} + 6 T^{5} - 2 T^{6} - 3 T^{7} - 14 T^{8} - 39 T^{9} + T^{10} - 78 T^{11} - 56 T^{12} - 24 T^{13} - 32 T^{14} + 192 T^{15} + 256 T^{16} + 768 T^{18} + 1024 T^{20} \)
$3$ 1
$5$ \( ( 1 + 16 T^{2} + 6 T^{3} + 127 T^{4} + 51 T^{5} + 635 T^{6} + 150 T^{7} + 2000 T^{8} + 3125 T^{10} )^{2} \)
$7$ \( 1 - 3 T + 16 T^{2} - 62 T^{3} + 220 T^{4} - 473 T^{5} + 1540 T^{6} - 3038 T^{7} + 5488 T^{8} - 7203 T^{9} + 16807 T^{10} \)
$11$ \( 1 - 66 T^{2} + 2257 T^{4} - 51461 T^{6} + 855052 T^{8} - 10752323 T^{10} + 103461292 T^{12} - 753440501 T^{14} + 3998413177 T^{16} - 14147686146 T^{18} + 25937424601 T^{20} \)
$13$ \( 1 - 6 T + 68 T^{2} - 336 T^{3} + 2292 T^{4} - 9399 T^{5} + 51837 T^{6} - 187401 T^{7} + 909867 T^{8} - 3004662 T^{9} + 13054461 T^{10} - 39060606 T^{11} + 153767523 T^{12} - 411719997 T^{13} + 1480516557 T^{14} - 3489782907 T^{15} + 11063046228 T^{16} - 21083501712 T^{17} + 55469689028 T^{18} - 63626996238 T^{19} + 137858491849 T^{20} \)
$17$ \( 1 - 12 T + 26 T^{2} + 36 T^{3} + 1143 T^{4} - 5247 T^{5} - 21540 T^{6} + 73476 T^{7} + 337539 T^{8} + 599625 T^{9} - 13374333 T^{10} + 10193625 T^{11} + 97548771 T^{12} + 360987588 T^{13} - 1799042340 T^{14} - 7449989679 T^{15} + 27589241367 T^{16} + 14772192228 T^{17} + 181369693466 T^{18} - 1423054517964 T^{19} + 2015993900449 T^{20} \)
$19$ \( 1 - 3 T + 71 T^{2} - 204 T^{3} + 2646 T^{4} - 8547 T^{5} + 74607 T^{6} - 258954 T^{7} + 1743726 T^{8} - 5989488 T^{9} + 35261703 T^{10} - 113800272 T^{11} + 629485086 T^{12} - 1776165486 T^{13} + 9722858847 T^{14} - 21163218153 T^{15} + 124483401126 T^{16} - 182349834756 T^{17} + 1205832975911 T^{18} - 968063093337 T^{19} + 6131066257801 T^{20} \)
$23$ \( 1 - 165 T^{2} + 13144 T^{4} - 668429 T^{6} + 24050461 T^{8} - 639494549 T^{10} + 12722693869 T^{12} - 187053839789 T^{14} + 1945783725016 T^{16} - 12921312571365 T^{18} + 41426511213649 T^{20} \)
$29$ \( 1 - 15 T + 150 T^{2} - 1125 T^{3} + 6691 T^{4} - 30108 T^{5} + 81631 T^{6} + 232971 T^{7} - 5137202 T^{8} + 44535417 T^{9} - 275752187 T^{10} + 1291527093 T^{11} - 4320386882 T^{12} + 5681929719 T^{13} + 57736055311 T^{14} - 617549674092 T^{15} + 3979962840811 T^{16} - 19406110847625 T^{17} + 75036961944150 T^{18} - 217607189638035 T^{19} + 420707233300201 T^{20} \)
$31$ \( 1 + 9 T + 149 T^{2} + 1098 T^{3} + 10878 T^{4} + 60723 T^{5} + 461409 T^{6} + 2027286 T^{7} + 13421802 T^{8} + 50078664 T^{9} + 374531595 T^{10} + 1552438584 T^{11} + 12898351722 T^{12} + 60394877226 T^{13} + 426120901089 T^{14} + 1738447936173 T^{15} + 9654265041918 T^{16} + 30208850293878 T^{17} + 127080764578709 T^{18} + 237956599446039 T^{19} + 819628286980801 T^{20} \)
$37$ \( 1 - 6 T - 97 T^{2} + 194 T^{3} + 7179 T^{4} + 3556 T^{5} - 323794 T^{6} - 533292 T^{7} + 10739317 T^{8} + 10946526 T^{9} - 345629139 T^{10} + 405021462 T^{11} + 14702124973 T^{12} - 27012839676 T^{13} - 606842086834 T^{14} + 246587111092 T^{15} + 18419349890211 T^{16} + 18416784163802 T^{17} - 340710507030337 T^{18} - 779770438770462 T^{19} + 4808584372417849 T^{20} \)
$41$ \( 1 - 9 T - 34 T^{2} + 747 T^{3} - 2085 T^{4} - 20394 T^{5} + 110775 T^{6} + 623979 T^{7} - 5992218 T^{8} - 18494757 T^{9} + 381591615 T^{10} - 758285037 T^{11} - 10072918458 T^{12} + 43005256659 T^{13} + 313023674775 T^{14} - 2362771363194 T^{15} - 9903967342485 T^{16} + 145481442589107 T^{17} - 271487457790114 T^{18} - 2946437409545649 T^{19} + 13422659310152401 T^{20} \)
$43$ \( 1 - 3 T - 79 T^{2} + 1100 T^{3} + 1674 T^{4} - 79931 T^{5} + 324899 T^{6} + 3229902 T^{7} - 28512986 T^{8} - 52724394 T^{9} + 1438527201 T^{10} - 2267148942 T^{11} - 52720511114 T^{12} + 256799818314 T^{13} + 1110765026099 T^{14} - 11750531857433 T^{15} + 10581961744026 T^{16} + 299000472217700 T^{17} - 923367821930479 T^{18} - 1507777835810529 T^{19} + 21611482313284249 T^{20} \)
$47$ \( 1 + 15 T - 49 T^{2} - 1068 T^{3} + 9486 T^{4} + 83265 T^{5} - 760914 T^{6} - 2887821 T^{7} + 57371082 T^{8} + 59131839 T^{9} - 3026317959 T^{10} + 2779196433 T^{11} + 126732720138 T^{12} - 299822239683 T^{13} - 3713017588434 T^{14} + 19096412007855 T^{15} + 102251636610894 T^{16} - 541073492654484 T^{17} - 1166753046426289 T^{18} + 16786957096541505 T^{19} + 52599132235830049 T^{20} \)
$53$ \( 1 - 9 T + 237 T^{2} - 1890 T^{3} + 28720 T^{4} - 208689 T^{5} + 2523571 T^{6} - 16852668 T^{7} + 178203742 T^{8} - 1088604978 T^{9} + 10361882797 T^{10} - 57696063834 T^{11} + 500574311278 T^{12} - 2508974653836 T^{13} + 19912189027651 T^{14} - 87272799238677 T^{15} + 636560451624880 T^{16} - 2220204054291930 T^{17} + 14755546627492557 T^{18} - 29697872326219197 T^{19} + 174887470365513049 T^{20} \)
$59$ \( 1 - 18 T - 55 T^{2} + 1536 T^{3} + 19971 T^{4} - 205494 T^{5} - 1764945 T^{6} + 8798931 T^{7} + 181121100 T^{8} - 308804295 T^{9} - 11121159681 T^{10} - 18219453405 T^{11} + 630482549100 T^{12} + 1807115649849 T^{13} - 21386475710145 T^{14} - 146912653898706 T^{15} + 842387437344411 T^{16} + 3822568680681984 T^{17} - 8075674068237655 T^{18} - 155933924735788902 T^{19} + 511116753300641401 T^{20} \)
$61$ \( 1 - 12 T + 251 T^{2} - 2436 T^{3} + 34779 T^{4} - 347730 T^{5} + 3716049 T^{6} - 34819755 T^{7} + 301038894 T^{8} - 2709237273 T^{9} + 20488848807 T^{10} - 165263473653 T^{11} + 1120165724574 T^{12} - 7903422809655 T^{13} + 51451823602209 T^{14} - 293691471746730 T^{15} + 1791827099901219 T^{16} - 7655721548547156 T^{17} + 48118535562317531 T^{18} - 140329753114009692 T^{19} + 713342911662882601 T^{20} \)
$67$ \( 1 + 10 T - 182 T^{2} - 2448 T^{3} + 18867 T^{4} + 319605 T^{5} - 772530 T^{6} - 23213154 T^{7} - 12219093 T^{8} + 697872289 T^{9} + 3674653819 T^{10} + 46757443363 T^{11} - 54851508477 T^{12} - 6981657836502 T^{13} - 15567345506130 T^{14} + 431506734822735 T^{15} + 1706678296382523 T^{16} - 14836622009830704 T^{17} - 73904317315308662 T^{18} + 272065343962949470 T^{19} + 1822837804551761449 T^{20} \)
$71$ \( 1 - 351 T^{2} + 63037 T^{4} - 7800935 T^{6} + 747809113 T^{8} - 58386380555 T^{10} + 3769705738633 T^{12} - 198234871721735 T^{14} + 8075057597528077 T^{16} - 226659489467262111 T^{18} + 3255243551009881201 T^{20} \)
$73$ \( 1 - 3 T + 260 T^{2} - 771 T^{3} + 36639 T^{4} - 155724 T^{5} + 3663555 T^{6} - 21043473 T^{7} + 293486934 T^{8} - 2074196103 T^{9} + 21799556757 T^{10} - 151416315519 T^{11} + 1563991871286 T^{12} - 8186268736041 T^{13} + 104038517806755 T^{14} - 322827000748332 T^{15} + 5544734717002671 T^{16} - 8517544258223787 T^{17} + 209679623892461060 T^{18} - 176614760124803739 T^{19} + 4297625829703557649 T^{20} \)
$79$ \( 1 - 20 T + 46 T^{2} - 144 T^{3} + 21153 T^{4} - 101181 T^{5} - 106944 T^{6} - 9264000 T^{7} + 7962453 T^{8} + 230795113 T^{9} + 4723714795 T^{10} + 18232813927 T^{11} + 49693669173 T^{12} - 4567513296000 T^{13} - 4165477462464 T^{14} - 311339643507219 T^{15} + 5142028946635713 T^{16} - 2765362894006896 T^{17} + 69787005255701806 T^{18} - 2397031919652366380 T^{19} + 9468276082626847201 T^{20} \)
$83$ \( 1 - 15 T - 136 T^{2} + 1773 T^{3} + 22674 T^{4} - 93717 T^{5} - 3687774 T^{6} + 10067337 T^{7} + 346135869 T^{8} - 496605294 T^{9} - 27460905396 T^{10} - 41218239402 T^{11} + 2384530001541 T^{12} + 5756372421219 T^{13} - 175015562267454 T^{14} - 369155071940031 T^{15} + 7413046025768706 T^{16} + 48112218404608671 T^{17} - 306311743570909576 T^{18} - 2804103829013106045 T^{19} + 15516041187205853449 T^{20} \)
$89$ \( 1 + 24 T + 44 T^{2} - 2592 T^{3} - 1287 T^{4} + 278721 T^{5} - 235110 T^{6} - 13705920 T^{7} + 186157425 T^{8} + 904992183 T^{9} - 14602879521 T^{10} + 80544304287 T^{11} + 1474552963425 T^{12} - 9662248716480 T^{13} - 14751328281510 T^{14} + 1556394633684729 T^{15} - 639614921466807 T^{16} - 114647620049211168 T^{17} + 173209907450891564 T^{18} + 8408553688979645016 T^{19} + 31181719929966183601 T^{20} \)
$97$ \( 1 - 6 T + 311 T^{2} - 1794 T^{3} + 51903 T^{4} - 385032 T^{5} + 6010353 T^{6} - 63309837 T^{7} + 574248354 T^{8} - 8264282925 T^{9} + 54719955099 T^{10} - 801635443725 T^{11} + 5403102762786 T^{12} - 57781178864301 T^{13} + 532092229646193 T^{14} - 3306400793833224 T^{15} + 43233745971829887 T^{16} - 144952122353734722 T^{17} + 2437441847851234871 T^{18} - 4561386351927391302 T^{19} + 73742412689492826049 T^{20} \)
show more
show less