Properties

Label 605.2.b.a
Level $605$
Weight $2$
Character orbit 605.b
Analytic conductor $4.831$
Analytic rank $0$
Dimension $2$
CM discriminant -11
Inner twists $4$

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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: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
Defining polynomial: \(x^{2} - x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{-11})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + 2 \beta ) q^{3} + 2 q^{4} + ( 1 + \beta ) q^{5} -8 q^{9} +O(q^{10})\) \( q + ( -1 + 2 \beta ) q^{3} + 2 q^{4} + ( 1 + \beta ) q^{5} -8 q^{9} + ( -2 + 4 \beta ) q^{12} + ( -7 + 3 \beta ) q^{15} + 4 q^{16} + ( 2 + 2 \beta ) q^{20} + ( 1 - 2 \beta ) q^{23} + ( -2 + 3 \beta ) q^{25} + ( 5 - 10 \beta ) q^{27} + 5 q^{31} -16 q^{36} + ( 3 - 6 \beta ) q^{37} + ( -8 - 8 \beta ) q^{45} + ( 2 - 4 \beta ) q^{47} + ( -4 + 8 \beta ) q^{48} + 7 q^{49} + ( -4 + 8 \beta ) q^{53} -15 q^{59} + ( -14 + 6 \beta ) q^{60} + 8 q^{64} + ( -3 + 6 \beta ) q^{67} + 11 q^{69} -3 q^{71} + ( -16 - \beta ) q^{75} + ( 4 + 4 \beta ) q^{80} + 31 q^{81} + 9 q^{89} + ( 2 - 4 \beta ) q^{92} + ( -5 + 10 \beta ) q^{93} + ( 3 - 6 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{4} + 3q^{5} - 16q^{9} + O(q^{10}) \) \( 2q + 4q^{4} + 3q^{5} - 16q^{9} - 11q^{15} + 8q^{16} + 6q^{20} - q^{25} + 10q^{31} - 32q^{36} - 24q^{45} + 14q^{49} - 30q^{59} - 22q^{60} + 16q^{64} + 22q^{69} - 6q^{71} - 33q^{75} + 12q^{80} + 62q^{81} + 18q^{89} + O(q^{100}) \)

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
0.500000 1.65831i
0.500000 + 1.65831i
0 3.31662i 2.00000 1.50000 1.65831i 0 0 0 −8.00000 0
364.2 0 3.31662i 2.00000 1.50000 + 1.65831i 0 0 0 −8.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
5.b even 2 1 inner
55.d 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.a 2
5.b even 2 1 inner 605.2.b.a 2
5.c odd 4 2 3025.2.a.k 2
11.b odd 2 1 CM 605.2.b.a 2
11.c even 5 4 605.2.j.b 8
11.d odd 10 4 605.2.j.b 8
55.d odd 2 1 inner 605.2.b.a 2
55.e even 4 2 3025.2.a.k 2
55.h odd 10 4 605.2.j.b 8
55.j even 10 4 605.2.j.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.b.a 2 1.a even 1 1 trivial
605.2.b.a 2 5.b even 2 1 inner
605.2.b.a 2 11.b odd 2 1 CM
605.2.b.a 2 55.d odd 2 1 inner
605.2.j.b 8 11.c even 5 4
605.2.j.b 8 11.d odd 10 4
605.2.j.b 8 55.h odd 10 4
605.2.j.b 8 55.j even 10 4
3025.2.a.k 2 5.c odd 4 2
3025.2.a.k 2 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 \(S_{2}^{\mathrm{new}}(605, [\chi])\):

\( T_{2} \)
\( T_{19} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 11 + T^{2} \)
$5$ \( 5 - 3 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( T^{2} \)
$17$ \( T^{2} \)
$19$ \( T^{2} \)
$23$ \( 11 + T^{2} \)
$29$ \( T^{2} \)
$31$ \( ( -5 + T )^{2} \)
$37$ \( 99 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( T^{2} \)
$47$ \( 44 + T^{2} \)
$53$ \( 176 + T^{2} \)
$59$ \( ( 15 + T )^{2} \)
$61$ \( T^{2} \)
$67$ \( 99 + T^{2} \)
$71$ \( ( 3 + T )^{2} \)
$73$ \( T^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( ( -9 + T )^{2} \)
$97$ \( 99 + T^{2} \)
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