Properties

Label 190.2.b.b
Level $190$
Weight $2$
Character orbit 190.b
Analytic conductor $1.517$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 190 = 2 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 190.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.51715763840\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.5161984.1
Defining polynomial: \(x^{6} - 4 x^{3} + 25 x^{2} - 20 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q -\beta_{4} q^{2} + ( \beta_{4} + \beta_{5} ) q^{3} - q^{4} + ( -\beta_{1} - \beta_{3} ) q^{5} + ( 1 + \beta_{3} ) q^{6} + ( -\beta_{4} + \beta_{5} ) q^{7} + \beta_{4} q^{8} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})\) \( q -\beta_{4} q^{2} + ( \beta_{4} + \beta_{5} ) q^{3} - q^{4} + ( -\beta_{1} - \beta_{3} ) q^{5} + ( 1 + \beta_{3} ) q^{6} + ( -\beta_{4} + \beta_{5} ) q^{7} + \beta_{4} q^{8} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{9} + ( \beta_{2} + \beta_{5} ) q^{10} + ( -\beta_{1} - \beta_{2} ) q^{11} + ( -\beta_{4} - \beta_{5} ) q^{12} + ( -\beta_{1} + \beta_{2} + 3 \beta_{4} + \beta_{5} ) q^{13} + ( -1 + \beta_{3} ) q^{14} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{15} + q^{16} + ( -\beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{17} + ( -\beta_{1} + \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{18} - q^{19} + ( \beta_{1} + \beta_{3} ) q^{20} + ( -3 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{21} + ( -\beta_{1} + \beta_{2} ) q^{22} + ( 2 \beta_{1} - 2 \beta_{2} - \beta_{4} - 3 \beta_{5} ) q^{23} + ( -1 - \beta_{3} ) q^{24} + ( 2 + 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{25} + ( 3 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{26} + ( 2 \beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{5} ) q^{27} + ( \beta_{4} - \beta_{5} ) q^{28} + ( 3 + 3 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{29} + ( -3 - 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{30} + ( 2 + 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{31} -\beta_{4} q^{32} + ( \beta_{1} - \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{33} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{34} + ( 1 + \beta_{1} - \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{35} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{36} + ( -4 \beta_{4} + 2 \beta_{5} ) q^{37} + \beta_{4} q^{38} + ( -5 - 2 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} ) q^{39} + ( -\beta_{2} - \beta_{5} ) q^{40} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{41} + ( -\beta_{1} + \beta_{2} + 3 \beta_{4} - \beta_{5} ) q^{42} + ( -\beta_{1} + \beta_{2} + 6 \beta_{4} ) q^{43} + ( \beta_{1} + \beta_{2} ) q^{44} + ( 4 + \beta_{1} + \beta_{2} + 3 \beta_{3} - 2 \beta_{4} ) q^{45} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{46} + ( 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{4} - 2 \beta_{5} ) q^{47} + ( \beta_{4} + \beta_{5} ) q^{48} + ( 2 - \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{49} + ( -1 + 2 \beta_{1} - \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{50} + ( -3 - 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{51} + ( \beta_{1} - \beta_{2} - 3 \beta_{4} - \beta_{5} ) q^{52} + ( \beta_{1} - \beta_{2} + 5 \beta_{4} - \beta_{5} ) q^{53} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{54} + ( 1 - 2 \beta_{1} + \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{55} + ( 1 - \beta_{3} ) q^{56} + ( -\beta_{4} - \beta_{5} ) q^{57} + ( 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{4} - \beta_{5} ) q^{58} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{59} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{60} + ( -10 - \beta_{1} - \beta_{2} ) q^{61} + ( 3 \beta_{1} - 3 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} ) q^{62} -2 \beta_{5} q^{63} - q^{64} + ( -2 + \beta_{1} - 2 \beta_{2} - 4 \beta_{4} - 4 \beta_{5} ) q^{65} + ( 2 - \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{66} + ( -2 \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} ) q^{67} + ( \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} ) q^{68} + ( 9 + 5 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} ) q^{69} + ( -3 - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{70} + ( -4 + \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{71} + ( \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{72} + ( -3 \beta_{1} + 3 \beta_{2} + 5 \beta_{4} + 5 \beta_{5} ) q^{73} + ( -4 + 2 \beta_{3} ) q^{74} + ( 7 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 4 \beta_{4} + 4 \beta_{5} ) q^{75} + q^{76} + ( -\beta_{1} + \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{77} + ( -2 \beta_{1} + 2 \beta_{2} + 5 \beta_{4} + 5 \beta_{5} ) q^{78} + ( 2 + 6 \beta_{3} ) q^{79} + ( -\beta_{1} - \beta_{3} ) q^{80} + ( -5 + 2 \beta_{3} ) q^{81} + ( \beta_{1} - \beta_{2} + 2 \beta_{5} ) q^{82} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} ) q^{83} + ( 3 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{84} + ( -2 + \beta_{1} - 4 \beta_{4} - 2 \beta_{5} ) q^{85} + ( 6 + \beta_{1} + \beta_{2} ) q^{86} + ( -4 \beta_{1} + 4 \beta_{2} + \beta_{4} + 9 \beta_{5} ) q^{87} + ( \beta_{1} - \beta_{2} ) q^{88} + ( 4 - \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{89} + ( -2 + \beta_{1} - \beta_{2} - 4 \beta_{4} - 3 \beta_{5} ) q^{90} + ( 1 - 3 \beta_{3} ) q^{91} + ( -2 \beta_{1} + 2 \beta_{2} + \beta_{4} + 3 \beta_{5} ) q^{92} + ( -5 \beta_{1} + 5 \beta_{2} + 4 \beta_{4} + 8 \beta_{5} ) q^{93} + ( 4 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{94} + ( \beta_{1} + \beta_{3} ) q^{95} + ( 1 + \beta_{3} ) q^{96} + ( -4 \beta_{1} + 4 \beta_{2} - 2 \beta_{4} + 4 \beta_{5} ) q^{97} + ( -\beta_{1} + \beta_{2} - 2 \beta_{4} - 3 \beta_{5} ) q^{98} + 4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{4} + 2q^{5} + 4q^{6} - 10q^{9} + O(q^{10}) \) \( 6q - 6q^{4} + 2q^{5} + 4q^{6} - 10q^{9} - 8q^{14} + 8q^{15} + 6q^{16} - 6q^{19} - 2q^{20} - 20q^{21} - 4q^{24} + 10q^{25} + 16q^{26} + 16q^{29} - 16q^{30} + 8q^{31} + 4q^{34} + 8q^{35} + 10q^{36} - 20q^{39} + 4q^{41} + 18q^{45} + 6q^{49} - 4q^{50} - 12q^{51} - 4q^{54} + 4q^{55} + 8q^{56} - 4q^{59} - 8q^{60} - 60q^{61} - 6q^{64} - 12q^{65} + 16q^{66} + 44q^{69} - 20q^{70} - 16q^{71} - 28q^{74} + 44q^{75} + 6q^{76} + 2q^{80} - 34q^{81} + 20q^{84} - 12q^{85} + 36q^{86} + 28q^{89} - 12q^{90} + 12q^{91} + 28q^{94} - 2q^{95} + 4q^{96} + 24q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 4 x^{3} + 25 x^{2} - 20 x + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -5 \nu^{5} - 2 \nu^{4} - 25 \nu^{3} + 10 \nu^{2} - 121 \nu + 100 \)\()/121\)
\(\beta_{3}\)\(=\)\((\)\( 7 \nu^{5} + 27 \nu^{4} + 35 \nu^{3} - 14 \nu^{2} + 223 \)\()/121\)
\(\beta_{4}\)\(=\)\((\)\( -25 \nu^{5} - 10 \nu^{4} - 4 \nu^{3} + 50 \nu^{2} - 605 \nu + 258 \)\()/242\)
\(\beta_{5}\)\(=\)\((\)\( -65 \nu^{5} - 26 \nu^{4} + 38 \nu^{3} + 372 \nu^{2} - 1331 \nu + 574 \)\()/242\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - 3 \beta_{4} + \beta_{2} - \beta_{1}\)
\(\nu^{3}\)\(=\)\(2 \beta_{4} - 5 \beta_{2} + 2\)
\(\nu^{4}\)\(=\)\(5 \beta_{3} + 7 \beta_{2} + 7 \beta_{1} - 15\)
\(\nu^{5}\)\(=\)\(2 \beta_{5} - 16 \beta_{4} - 2 \beta_{3} - 29 \beta_{1} + 16\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/190\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
39.1
1.32001 1.32001i
−1.75233 + 1.75233i
0.432320 0.432320i
0.432320 + 0.432320i
−1.75233 1.75233i
1.32001 + 1.32001i
1.00000i 2.12489i −1.00000 1.80487 + 1.32001i −2.12489 4.12489i 1.00000i −1.51514 1.32001 1.80487i
39.2 1.00000i 1.36333i −1.00000 1.38900 1.75233i 1.36333 0.636672i 1.00000i 1.14134 −1.75233 1.38900i
39.3 1.00000i 2.76156i −1.00000 −2.19388 + 0.432320i 2.76156 0.761557i 1.00000i −4.62620 0.432320 + 2.19388i
39.4 1.00000i 2.76156i −1.00000 −2.19388 0.432320i 2.76156 0.761557i 1.00000i −4.62620 0.432320 2.19388i
39.5 1.00000i 1.36333i −1.00000 1.38900 + 1.75233i 1.36333 0.636672i 1.00000i 1.14134 −1.75233 + 1.38900i
39.6 1.00000i 2.12489i −1.00000 1.80487 1.32001i −2.12489 4.12489i 1.00000i −1.51514 1.32001 + 1.80487i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 39.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 190.2.b.b 6
3.b odd 2 1 1710.2.d.d 6
4.b odd 2 1 1520.2.d.j 6
5.b even 2 1 inner 190.2.b.b 6
5.c odd 4 1 950.2.a.i 3
5.c odd 4 1 950.2.a.n 3
15.d odd 2 1 1710.2.d.d 6
15.e even 4 1 8550.2.a.ck 3
15.e even 4 1 8550.2.a.cl 3
20.d odd 2 1 1520.2.d.j 6
20.e even 4 1 7600.2.a.bi 3
20.e even 4 1 7600.2.a.cd 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.b.b 6 1.a even 1 1 trivial
190.2.b.b 6 5.b even 2 1 inner
950.2.a.i 3 5.c odd 4 1
950.2.a.n 3 5.c odd 4 1
1520.2.d.j 6 4.b odd 2 1
1520.2.d.j 6 20.d odd 2 1
1710.2.d.d 6 3.b odd 2 1
1710.2.d.d 6 15.d odd 2 1
7600.2.a.bi 3 20.e even 4 1
7600.2.a.cd 3 20.e even 4 1
8550.2.a.ck 3 15.e even 4 1
8550.2.a.cl 3 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 14 T_{3}^{4} + 57 T_{3}^{2} + 64 \) acting on \(S_{2}^{\mathrm{new}}(190, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{3} \)
$3$ \( 64 + 57 T^{2} + 14 T^{4} + T^{6} \)
$5$ \( 125 - 50 T - 15 T^{2} + 24 T^{3} - 3 T^{4} - 2 T^{5} + T^{6} \)
$7$ \( 4 + 17 T^{2} + 18 T^{4} + T^{6} \)
$11$ \( ( -8 - 10 T + T^{3} )^{2} \)
$13$ \( 4 + 201 T^{2} + 38 T^{4} + T^{6} \)
$17$ \( 16 + 65 T^{2} + 18 T^{4} + T^{6} \)
$19$ \( ( 1 + T )^{6} \)
$23$ \( 14884 + 2401 T^{2} + 98 T^{4} + T^{6} \)
$29$ \( ( 410 - 51 T - 8 T^{2} + T^{3} )^{2} \)
$31$ \( ( 232 - 62 T - 4 T^{2} + T^{3} )^{2} \)
$37$ \( 256 + 1152 T^{2} + 116 T^{4} + T^{6} \)
$41$ \( ( -100 - 50 T - 2 T^{2} + T^{3} )^{2} \)
$43$ \( 21904 + 4276 T^{2} + 128 T^{4} + T^{6} \)
$47$ \( 4096 + 2816 T^{2} + 132 T^{4} + T^{6} \)
$53$ \( 11236 + 2537 T^{2} + 102 T^{4} + T^{6} \)
$59$ \( ( -80 - 29 T + 2 T^{2} + T^{3} )^{2} \)
$61$ \( ( 892 + 290 T + 30 T^{2} + T^{3} )^{2} \)
$67$ \( 4096 + 3977 T^{2} + 126 T^{4} + T^{6} \)
$71$ \( ( -1016 - 122 T + 8 T^{2} + T^{3} )^{2} \)
$73$ \( 26896 + 5745 T^{2} + 290 T^{4} + T^{6} \)
$79$ \( ( 880 - 228 T + T^{3} )^{2} \)
$83$ \( 64 + 880 T^{2} + 92 T^{4} + T^{6} \)
$89$ \( ( 20 + 46 T - 14 T^{2} + T^{3} )^{2} \)
$97$ \( 238144 + 19760 T^{2} + 300 T^{4} + T^{6} \)
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