Properties

Label 605.2.b.b
Level $605$
Weight $2$
Character orbit 605.b
Analytic conductor $4.831$
Analytic rank $0$
Dimension $4$
CM discriminant -55
Inner twists $4$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.4400.1
Defining polynomial: \(x^{4} + 7 x^{2} + 11\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} + ( -2 - \beta_{3} ) q^{4} + \beta_{3} q^{5} -\beta_{1} q^{7} + ( \beta_{1} + \beta_{2} ) q^{8} + 3 q^{9} +O(q^{10})\) \( q -\beta_{2} q^{2} + ( -2 - \beta_{3} ) q^{4} + \beta_{3} q^{5} -\beta_{1} q^{7} + ( \beta_{1} + \beta_{2} ) q^{8} + 3 q^{9} + ( -\beta_{1} - \beta_{2} ) q^{10} + ( -\beta_{1} + 2 \beta_{2} ) q^{13} + ( -1 - 3 \beta_{3} ) q^{14} + ( 1 + 2 \beta_{3} ) q^{16} + ( -\beta_{1} - 2 \beta_{2} ) q^{17} -3 \beta_{2} q^{18} + ( -5 - 2 \beta_{3} ) q^{20} + 5 q^{25} + ( 7 - \beta_{3} ) q^{26} + ( \beta_{1} + 4 \beta_{2} ) q^{28} -4 \beta_{3} q^{31} -\beta_{2} q^{32} + ( -9 - 5 \beta_{3} ) q^{34} + ( \beta_{1} - 4 \beta_{2} ) q^{35} + ( -6 - 3 \beta_{3} ) q^{36} + 5 \beta_{2} q^{40} + ( \beta_{1} + 4 \beta_{2} ) q^{43} + 3 \beta_{3} q^{45} + ( -7 + 2 \beta_{3} ) q^{49} -5 \beta_{2} q^{50} + ( -\beta_{1} - 2 \beta_{2} ) q^{52} + ( 15 + \beta_{3} ) q^{56} -4 q^{59} + ( 4 \beta_{1} + 4 \beta_{2} ) q^{62} -3 \beta_{1} q^{63} + ( -2 + 3 \beta_{3} ) q^{64} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{65} + ( 3 \beta_{1} + 10 \beta_{2} ) q^{68} + ( -15 - \beta_{3} ) q^{70} + 8 q^{71} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{72} + ( \beta_{1} - 6 \beta_{2} ) q^{73} + ( 10 + \beta_{3} ) q^{80} + 9 q^{81} + ( -3 \beta_{1} + 4 \beta_{2} ) q^{83} + ( -\beta_{1} - 6 \beta_{2} ) q^{85} + ( 17 + 7 \beta_{3} ) q^{86} + 6 \beta_{3} q^{89} + ( -3 \beta_{1} - 3 \beta_{2} ) q^{90} + ( -12 + 8 \beta_{3} ) q^{91} + ( -2 \beta_{1} + 5 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{4} + 12q^{9} + O(q^{10}) \) \( 4q - 8q^{4} + 12q^{9} - 4q^{14} + 4q^{16} - 20q^{20} + 20q^{25} + 28q^{26} - 36q^{34} - 24q^{36} - 28q^{49} + 60q^{56} - 16q^{59} - 8q^{64} - 60q^{70} + 32q^{71} + 40q^{80} + 36q^{81} + 68q^{86} - 48q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 7 x^{2} + 11\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} + 4 \nu \)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} + 7 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 7\)\()/2\)
\(\nu^{3}\)\(=\)\(\beta_{2} - 2 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(486\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
364.1
1.54336i
2.14896i
2.14896i
1.54336i
2.49721i 0 −4.23607 2.23607 0 3.08672i 5.58394i 3.00000 5.58394i
364.2 1.32813i 0 0.236068 −2.23607 0 4.29792i 2.96979i 3.00000 2.96979i
364.3 1.32813i 0 0.236068 −2.23607 0 4.29792i 2.96979i 3.00000 2.96979i
364.4 2.49721i 0 −4.23607 2.23607 0 3.08672i 5.58394i 3.00000 5.58394i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
55.d odd 2 1 CM by \(\Q(\sqrt{-55}) \)
5.b even 2 1 inner
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.b.b 4
5.b even 2 1 inner 605.2.b.b 4
5.c odd 4 2 3025.2.a.bb 4
11.b odd 2 1 inner 605.2.b.b 4
11.c even 5 2 605.2.j.a 8
11.c even 5 2 605.2.j.c 8
11.d odd 10 2 605.2.j.a 8
11.d odd 10 2 605.2.j.c 8
55.d odd 2 1 CM 605.2.b.b 4
55.e even 4 2 3025.2.a.bb 4
55.h odd 10 2 605.2.j.a 8
55.h odd 10 2 605.2.j.c 8
55.j even 10 2 605.2.j.a 8
55.j even 10 2 605.2.j.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.b.b 4 1.a even 1 1 trivial
605.2.b.b 4 5.b even 2 1 inner
605.2.b.b 4 11.b odd 2 1 inner
605.2.b.b 4 55.d odd 2 1 CM
605.2.j.a 8 11.c even 5 2
605.2.j.a 8 11.d odd 10 2
605.2.j.a 8 55.h odd 10 2
605.2.j.a 8 55.j even 10 2
605.2.j.c 8 11.c even 5 2
605.2.j.c 8 11.d odd 10 2
605.2.j.c 8 55.h odd 10 2
605.2.j.c 8 55.j even 10 2
3025.2.a.bb 4 5.c odd 4 2
3025.2.a.bb 4 55.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(605, [\chi])\):

\( T_{2}^{4} + 8 T_{2}^{2} + 11 \)
\( T_{19} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 11 + 8 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( -5 + T^{2} )^{2} \)
$7$ \( 176 + 28 T^{2} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( 176 + 52 T^{2} + T^{4} \)
$17$ \( 176 + 68 T^{2} + T^{4} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( ( -80 + T^{2} )^{2} \)
$37$ \( T^{4} \)
$41$ \( T^{4} \)
$43$ \( 176 + 172 T^{2} + T^{4} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( ( 4 + T )^{4} \)
$61$ \( T^{4} \)
$67$ \( T^{4} \)
$71$ \( ( -8 + T )^{4} \)
$73$ \( 21296 + 292 T^{2} + T^{4} \)
$79$ \( T^{4} \)
$83$ \( 176 + 332 T^{2} + T^{4} \)
$89$ \( ( -180 + T^{2} )^{2} \)
$97$ \( T^{4} \)
show more
show less