Properties

Label 605.2.j.b
Level $605$
Weight $2$
Character orbit 605.j
Analytic conductor $4.831$
Analytic rank $0$
Dimension $8$
CM discriminant -11
Inner twists $16$

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Newspace parameters

Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.j (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.228765625.1
Defining polynomial: \(x^{8} - x^{7} - 2 x^{6} + 5 x^{5} + x^{4} + 15 x^{3} - 18 x^{2} - 27 x + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{10}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{4} - 2 \beta_{5} ) q^{3} + ( -2 + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{7} ) q^{4} + ( -\beta_{2} + \beta_{7} ) q^{5} + 8 \beta_{3} q^{9} +O(q^{10})\) \( q + ( -\beta_{4} - 2 \beta_{5} ) q^{3} + ( -2 + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{7} ) q^{4} + ( -\beta_{2} + \beta_{7} ) q^{5} + 8 \beta_{3} q^{9} + ( -2 + 4 \beta_{6} ) q^{12} + ( 7 - 3 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + 7 \beta_{7} ) q^{15} + 4 \beta_{7} q^{16} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{20} + ( 1 - 2 \beta_{6} ) q^{23} + ( -\beta_{4} - 3 \beta_{5} ) q^{25} + ( 10 \beta_{2} + 5 \beta_{7} ) q^{27} -5 \beta_{3} q^{31} + 16 \beta_{4} q^{36} + ( -3 + 6 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 6 \beta_{5} + 6 \beta_{6} - 3 \beta_{7} ) q^{37} + ( -8 - 8 \beta_{6} ) q^{45} + ( 2 \beta_{4} + 4 \beta_{5} ) q^{47} + ( 4 - 8 \beta_{1} + 8 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} + 8 \beta_{5} - 8 \beta_{6} + 4 \beta_{7} ) q^{48} + 7 \beta_{7} q^{49} + ( 8 \beta_{1} - 4 \beta_{3} ) q^{53} + ( 15 - 15 \beta_{3} - 15 \beta_{4} + 15 \beta_{7} ) q^{59} + ( -6 \beta_{2} - 14 \beta_{7} ) q^{60} -8 \beta_{3} q^{64} + ( -3 + 6 \beta_{6} ) q^{67} -11 \beta_{4} q^{69} -3 \beta_{7} q^{71} + ( -\beta_{1} + 17 \beta_{3} ) q^{75} + ( -8 \beta_{4} - 4 \beta_{5} ) q^{80} + ( -31 + 31 \beta_{3} + 31 \beta_{4} - 31 \beta_{7} ) q^{81} + 9 q^{89} + ( -2 + 4 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} ) q^{92} + ( -10 \beta_{2} - 5 \beta_{7} ) q^{93} + ( -6 \beta_{1} + 3 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{4} - 3q^{5} + 16q^{9} + O(q^{10}) \) \( 8q - 4q^{4} - 3q^{5} + 16q^{9} + 11q^{15} - 8q^{16} - 6q^{20} + q^{25} - 10q^{31} + 32q^{36} - 96q^{45} - 14q^{49} + 30q^{59} + 22q^{60} - 16q^{64} - 22q^{69} + 6q^{71} + 33q^{75} - 12q^{80} - 62q^{81} + 72q^{89} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} - 2 x^{6} + 5 x^{5} + x^{4} + 15 x^{3} - 18 x^{2} - 27 x + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} + \nu^{6} + 2 \nu^{5} - 5 \nu^{4} - \nu^{3} - 15 \nu^{2} + 18 \nu + 27 \)\()/27\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} + 16 \nu \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{7} - 4 \nu^{6} + 10 \nu^{5} + 2 \nu^{4} - 5 \nu^{3} - 36 \nu^{2} - 54 \nu + 162 \)\()/27\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} - 2 \nu^{6} + 5 \nu^{5} + \nu^{4} + 2 \nu^{3} - 18 \nu^{2} - 27 \nu + 81 \)\()/9\)
\(\beta_{6}\)\(=\)\( \nu^{5} + 16 \)
\(\beta_{7}\)\(=\)\((\)\( -5 \nu^{7} + 5 \nu^{6} + 10 \nu^{5} + 2 \nu^{4} - 5 \nu^{3} - 75 \nu^{2} + 90 \nu + 135 \)\()/27\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{7} + \beta_{6} - \beta_{5} - 3 \beta_{4} - 3 \beta_{3} - \beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(2 \beta_{5} - 3 \beta_{4}\)
\(\nu^{4}\)\(=\)\(\beta_{7} - 5 \beta_{2}\)
\(\nu^{5}\)\(=\)\(\beta_{6} - 16\)
\(\nu^{6}\)\(=\)\(3 \beta_{3} - 16 \beta_{1}\)
\(\nu^{7}\)\(=\)\(-35 \beta_{7} - 13 \beta_{6} + 13 \beta_{5} + 48 \beta_{4} + 48 \beta_{3} + 13 \beta_{2} - 13 \beta_{1} - 35\)

Character values

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

\(n\) \(122\) \(486\)
\(\chi(n)\) \(-1\) \(-1 + \beta_{3} + \beta_{4} - \beta_{7}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
1.42264 + 0.987975i
−1.73166 0.0369185i
−0.570223 1.63550i
1.37924 + 1.04771i
1.42264 0.987975i
−1.73166 + 0.0369185i
−0.570223 + 1.63550i
1.37924 1.04771i
0 −1.94946 2.68321i −1.61803 1.17557i −1.11362 + 1.93903i 0 0 0 −2.47214 + 7.60845i 0
9.2 0 1.94946 + 2.68321i −1.61803 1.17557i 2.04067 + 0.914138i 0 0 0 −2.47214 + 7.60845i 0
124.1 0 −3.15430 + 1.02489i 0.618034 1.90211i −0.238794 2.22328i 0 0 0 6.47214 4.70228i 0
124.2 0 3.15430 1.02489i 0.618034 1.90211i −2.18826 + 0.459925i 0 0 0 6.47214 4.70228i 0
269.1 0 −1.94946 + 2.68321i −1.61803 + 1.17557i −1.11362 1.93903i 0 0 0 −2.47214 7.60845i 0
269.2 0 1.94946 2.68321i −1.61803 + 1.17557i 2.04067 0.914138i 0 0 0 −2.47214 7.60845i 0
444.1 0 −3.15430 1.02489i 0.618034 + 1.90211i −0.238794 + 2.22328i 0 0 0 6.47214 + 4.70228i 0
444.2 0 3.15430 + 1.02489i 0.618034 + 1.90211i −2.18826 0.459925i 0 0 0 6.47214 + 4.70228i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 444.2
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
11.c even 5 3 inner
11.d odd 10 3 inner
55.d odd 2 1 inner
55.h odd 10 3 inner
55.j even 10 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.j.b 8
5.b even 2 1 inner 605.2.j.b 8
11.b odd 2 1 CM 605.2.j.b 8
11.c even 5 1 605.2.b.a 2
11.c even 5 3 inner 605.2.j.b 8
11.d odd 10 1 605.2.b.a 2
11.d odd 10 3 inner 605.2.j.b 8
55.d odd 2 1 inner 605.2.j.b 8
55.h odd 10 1 605.2.b.a 2
55.h odd 10 3 inner 605.2.j.b 8
55.j even 10 1 605.2.b.a 2
55.j even 10 3 inner 605.2.j.b 8
55.k odd 20 2 3025.2.a.k 2
55.l even 20 2 3025.2.a.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.b.a 2 11.c even 5 1
605.2.b.a 2 11.d odd 10 1
605.2.b.a 2 55.h odd 10 1
605.2.b.a 2 55.j even 10 1
605.2.j.b 8 1.a even 1 1 trivial
605.2.j.b 8 5.b even 2 1 inner
605.2.j.b 8 11.b odd 2 1 CM
605.2.j.b 8 11.c even 5 3 inner
605.2.j.b 8 11.d odd 10 3 inner
605.2.j.b 8 55.d odd 2 1 inner
605.2.j.b 8 55.h odd 10 3 inner
605.2.j.b 8 55.j even 10 3 inner
3025.2.a.k 2 55.k odd 20 2
3025.2.a.k 2 55.l even 20 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

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 14641 - 1331 T^{2} + 121 T^{4} - 11 T^{6} + T^{8} \)
$5$ \( 625 + 375 T + 100 T^{2} - 15 T^{3} - 29 T^{4} - 3 T^{5} + 4 T^{6} + 3 T^{7} + T^{8} \)
$7$ \( T^{8} \)
$11$ \( T^{8} \)
$13$ \( T^{8} \)
$17$ \( T^{8} \)
$19$ \( T^{8} \)
$23$ \( ( 11 + T^{2} )^{4} \)
$29$ \( T^{8} \)
$31$ \( ( 625 + 125 T + 25 T^{2} + 5 T^{3} + T^{4} )^{2} \)
$37$ \( 96059601 - 970299 T^{2} + 9801 T^{4} - 99 T^{6} + T^{8} \)
$41$ \( T^{8} \)
$43$ \( T^{8} \)
$47$ \( 3748096 - 85184 T^{2} + 1936 T^{4} - 44 T^{6} + T^{8} \)
$53$ \( 959512576 - 5451776 T^{2} + 30976 T^{4} - 176 T^{6} + T^{8} \)
$59$ \( ( 50625 - 3375 T + 225 T^{2} - 15 T^{3} + T^{4} )^{2} \)
$61$ \( T^{8} \)
$67$ \( ( 99 + T^{2} )^{4} \)
$71$ \( ( 81 - 27 T + 9 T^{2} - 3 T^{3} + T^{4} )^{2} \)
$73$ \( T^{8} \)
$79$ \( T^{8} \)
$83$ \( T^{8} \)
$89$ \( ( -9 + T )^{8} \)
$97$ \( 96059601 - 970299 T^{2} + 9801 T^{4} - 99 T^{6} + T^{8} \)
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