Properties

Label 190.2.b.a
Level $190$
Weight $2$
Character orbit 190.b
Analytic conductor $1.517$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−0.707107 0.707107i
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
1.00000i 2.41421i −1.00000 −0.707107 2.12132i −2.41421 1.58579i 1.00000i −2.82843 −2.12132 + 0.707107i
39.2 1.00000i 0.414214i −1.00000 0.707107 + 2.12132i 0.414214 4.41421i 1.00000i 2.82843 2.12132 0.707107i
39.3 1.00000i 0.414214i −1.00000 0.707107 2.12132i 0.414214 4.41421i 1.00000i 2.82843 2.12132 + 0.707107i
39.4 1.00000i 2.41421i −1.00000 −0.707107 + 2.12132i −2.41421 1.58579i 1.00000i −2.82843 −2.12132 0.707107i
\(n\): e.g. 2-40 or 990-1000
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.a 4
3.b odd 2 1 1710.2.d.c 4
4.b odd 2 1 1520.2.d.e 4
5.b even 2 1 inner 190.2.b.a 4
5.c odd 4 1 950.2.a.f 2
5.c odd 4 1 950.2.a.g 2
15.d odd 2 1 1710.2.d.c 4
15.e even 4 1 8550.2.a.bn 2
15.e even 4 1 8550.2.a.cb 2
20.d odd 2 1 1520.2.d.e 4
20.e even 4 1 7600.2.a.v 2
20.e even 4 1 7600.2.a.bg 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.b.a 4 1.a even 1 1 trivial
190.2.b.a 4 5.b even 2 1 inner
950.2.a.f 2 5.c odd 4 1
950.2.a.g 2 5.c odd 4 1
1520.2.d.e 4 4.b odd 2 1
1520.2.d.e 4 20.d odd 2 1
1710.2.d.c 4 3.b odd 2 1
1710.2.d.c 4 15.d odd 2 1
7600.2.a.v 2 20.e even 4 1
7600.2.a.bg 2 20.e even 4 1
8550.2.a.bn 2 15.e even 4 1
8550.2.a.cb 2 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 6 T_{3}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(190, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( 1 + 6 T^{2} + T^{4} \)
$5$ \( 25 + 8 T^{2} + T^{4} \)
$7$ \( 49 + 22 T^{2} + T^{4} \)
$11$ \( ( -2 + T^{2} )^{2} \)
$13$ \( 1 + 34 T^{2} + T^{4} \)
$17$ \( ( 1 + T^{2} )^{2} \)
$19$ \( ( -1 + T )^{4} \)
$23$ \( 49 + 86 T^{2} + T^{4} \)
$29$ \( ( 1 - 6 T + T^{2} )^{2} \)
$31$ \( ( -14 - 4 T + T^{2} )^{2} \)
$37$ \( ( 72 + T^{2} )^{2} \)
$41$ \( ( -18 + T^{2} )^{2} \)
$43$ \( 324 + 108 T^{2} + T^{4} \)
$47$ \( T^{4} \)
$53$ \( 3969 + 162 T^{2} + T^{4} \)
$59$ \( ( -89 - 6 T + T^{2} )^{2} \)
$61$ \( ( 82 - 20 T + T^{2} )^{2} \)
$67$ \( 3969 + 198 T^{2} + T^{4} \)
$71$ \( ( 142 + 24 T + T^{2} )^{2} \)
$73$ \( 3969 + 162 T^{2} + T^{4} \)
$79$ \( ( -68 + 4 T + T^{2} )^{2} \)
$83$ \( 1296 + 216 T^{2} + T^{4} \)
$89$ \( ( -50 + T^{2} )^{2} \)
$97$ \( 16 + 136 T^{2} + T^{4} \)
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