Properties

Label 230.2.b.a
Level $230$
Weight $2$
Character orbit 230.b
Analytic conductor $1.837$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 230.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.83655924649\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 2 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 1\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 2 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
139.1
1.61803i
0.618034i
0.618034i
1.61803i
1.00000i 1.61803i −1.00000 −2.23607 −1.61803 1.85410i 1.00000i 0.381966 2.23607i
139.2 1.00000i 0.618034i −1.00000 2.23607 0.618034 4.85410i 1.00000i 2.61803 2.23607i
139.3 1.00000i 0.618034i −1.00000 2.23607 0.618034 4.85410i 1.00000i 2.61803 2.23607i
139.4 1.00000i 1.61803i −1.00000 −2.23607 −1.61803 1.85410i 1.00000i 0.381966 2.23607i
\(n\): 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 230.2.b.a 4
3.b odd 2 1 2070.2.d.c 4
4.b odd 2 1 1840.2.e.c 4
5.b even 2 1 inner 230.2.b.a 4
5.c odd 4 1 1150.2.a.l 2
5.c odd 4 1 1150.2.a.n 2
15.d odd 2 1 2070.2.d.c 4
20.d odd 2 1 1840.2.e.c 4
20.e even 4 1 9200.2.a.bo 2
20.e even 4 1 9200.2.a.by 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.b.a 4 1.a even 1 1 trivial
230.2.b.a 4 5.b even 2 1 inner
1150.2.a.l 2 5.c odd 4 1
1150.2.a.n 2 5.c odd 4 1
1840.2.e.c 4 4.b odd 2 1
1840.2.e.c 4 20.d odd 2 1
2070.2.d.c 4 3.b odd 2 1
2070.2.d.c 4 15.d odd 2 1
9200.2.a.bo 2 20.e even 4 1
9200.2.a.by 2 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 3 T_{3}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(230, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( 1 + 3 T^{2} + T^{4} \)
$5$ \( ( -5 + T^{2} )^{2} \)
$7$ \( 81 + 27 T^{2} + T^{4} \)
$11$ \( ( 19 + 9 T + T^{2} )^{2} \)
$13$ \( 1 + 7 T^{2} + T^{4} \)
$17$ \( 25 + 35 T^{2} + T^{4} \)
$19$ \( ( 1 - 7 T + T^{2} )^{2} \)
$23$ \( ( 1 + T^{2} )^{2} \)
$29$ \( ( -36 - 6 T + T^{2} )^{2} \)
$31$ \( ( 19 + 11 T + T^{2} )^{2} \)
$37$ \( 1296 + 108 T^{2} + T^{4} \)
$41$ \( ( 19 + 9 T + T^{2} )^{2} \)
$43$ \( 5776 + 172 T^{2} + T^{4} \)
$47$ \( 400 + 140 T^{2} + T^{4} \)
$53$ \( ( 4 + T^{2} )^{2} \)
$59$ \( ( -6 + T )^{4} \)
$61$ \( ( -11 + T + T^{2} )^{2} \)
$67$ \( 16 + 28 T^{2} + T^{4} \)
$71$ \( ( 1 - 3 T + T^{2} )^{2} \)
$73$ \( 15376 + 328 T^{2} + T^{4} \)
$79$ \( ( -44 - 2 T + T^{2} )^{2} \)
$83$ \( 1936 + 92 T^{2} + T^{4} \)
$89$ \( ( -36 - 6 T + T^{2} )^{2} \)
$97$ \( 43681 + 427 T^{2} + T^{4} \)
show more
show less