Properties

Label 605.2.j.a
Level $605$
Weight $2$
Character orbit 605.j
Analytic conductor $4.831$
Analytic rank $0$
Dimension $8$
CM discriminant -55
Inner twists $8$

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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: 8.0.484000000.6
Defining polynomial: \(x^{8} - x^{6} + 16 x^{4} - 66 x^{2} + 121\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
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.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{4} ) q^{2} + ( 1 + \beta_{2} - 3 \beta_{3} + 3 \beta_{5} ) q^{4} + ( -2 - \beta_{2} + 2 \beta_{3} ) q^{5} -2 \beta_{1} q^{7} + ( -\beta_{1} + 2 \beta_{4} + \beta_{6} + 2 \beta_{7} ) q^{8} + 3 \beta_{5} q^{9} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{4} ) q^{2} + ( 1 + \beta_{2} - 3 \beta_{3} + 3 \beta_{5} ) q^{4} + ( -2 - \beta_{2} + 2 \beta_{3} ) q^{5} -2 \beta_{1} q^{7} + ( -\beta_{1} + 2 \beta_{4} + \beta_{6} + 2 \beta_{7} ) q^{8} + 3 \beta_{5} q^{9} + ( -\beta_{1} + 2 \beta_{4} - \beta_{7} ) q^{10} + ( 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{13} + ( -6 + 4 \beta_{3} - 6 \beta_{5} ) q^{14} + ( -4 - \beta_{2} + 4 \beta_{3} ) q^{16} + ( 2 \beta_{6} + 2 \beta_{7} ) q^{17} + ( -3 \beta_{1} + 3 \beta_{4} + 3 \beta_{6} ) q^{18} + ( -4 \beta_{2} + 4 \beta_{3} - 7 \beta_{5} ) q^{20} -5 \beta_{3} q^{25} + ( -8 - 8 \beta_{2} + 6 \beta_{3} - 6 \beta_{5} ) q^{26} + ( -2 \beta_{6} - 4 \beta_{7} ) q^{28} + ( -8 \beta_{2} + 8 \beta_{3} - 4 \beta_{5} ) q^{31} + ( -\beta_{1} - \beta_{7} ) q^{32} + ( -4 + 10 \beta_{2} + 10 \beta_{5} ) q^{34} + ( 2 \beta_{1} - 4 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} ) q^{35} + ( -6 + 9 \beta_{3} - 6 \beta_{5} ) q^{36} + ( 5 \beta_{1} - 5 \beta_{4} - 5 \beta_{6} ) q^{40} + ( 4 \beta_{1} - 2 \beta_{4} + 4 \beta_{7} ) q^{43} + ( -3 - 6 \beta_{2} - 6 \beta_{5} ) q^{45} + ( -4 - 9 \beta_{2} + 4 \beta_{3} ) q^{49} + 5 \beta_{7} q^{50} + ( 2 \beta_{1} - 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{52} + ( 14 - 2 \beta_{2} - 2 \beta_{5} ) q^{56} + ( 4 + 4 \beta_{2} - 4 \beta_{3} + 4 \beta_{5} ) q^{59} + ( 12 \beta_{1} - 4 \beta_{4} - 4 \beta_{6} ) q^{62} + ( -6 \beta_{4} - 6 \beta_{7} ) q^{63} + ( 6 \beta_{2} - 6 \beta_{3} + \beta_{5} ) q^{64} + ( -2 \beta_{1} - 6 \beta_{4} - 2 \beta_{7} ) q^{65} + ( -16 \beta_{1} + 10 \beta_{4} + 6 \beta_{6} - 6 \beta_{7} ) q^{68} + ( 14 + 14 \beta_{2} - 16 \beta_{3} + 16 \beta_{5} ) q^{70} + 8 \beta_{2} q^{71} + ( -6 \beta_{6} - 3 \beta_{7} ) q^{72} + ( -4 \beta_{1} + 6 \beta_{4} + 6 \beta_{6} ) q^{73} + ( 2 - 11 \beta_{3} + 2 \beta_{5} ) q^{80} + ( -9 - 9 \beta_{2} + 9 \beta_{3} - 9 \beta_{5} ) q^{81} + ( 6 \beta_{6} - 4 \beta_{7} ) q^{83} + ( 6 \beta_{1} - 2 \beta_{4} - 6 \beta_{6} - 2 \beta_{7} ) q^{85} + ( 14 \beta_{2} - 14 \beta_{3} + 24 \beta_{5} ) q^{86} + ( -6 - 12 \beta_{2} - 12 \beta_{5} ) q^{89} + ( 9 \beta_{1} - 3 \beta_{4} - 6 \beta_{6} + 6 \beta_{7} ) q^{90} + ( 16 + 4 \beta_{3} + 16 \beta_{5} ) q^{91} + ( 5 \beta_{1} + 4 \beta_{4} + 5 \beta_{7} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 6q^{4} - 10q^{5} - 6q^{9} + O(q^{10}) \) \( 8q - 6q^{4} - 10q^{5} - 6q^{9} - 28q^{14} - 22q^{16} + 30q^{20} - 10q^{25} - 24q^{26} + 40q^{31} - 72q^{34} - 18q^{36} - 6q^{49} + 120q^{56} + 8q^{59} - 26q^{64} + 20q^{70} - 16q^{71} - 10q^{80} - 18q^{81} - 104q^{86} + 104q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{6} + 16 x^{4} - 66 x^{2} + 121\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -7 \nu^{6} - 37 \nu^{4} - 629 \nu^{2} - 363 \)\()/1991\)
\(\beta_{3}\)\(=\)\((\)\( -28 \nu^{6} - 148 \nu^{4} - 525 \nu^{2} + 539 \)\()/1991\)
\(\beta_{4}\)\(=\)\((\)\( -28 \nu^{7} - 148 \nu^{5} - 525 \nu^{3} + 539 \nu \)\()/1991\)
\(\beta_{5}\)\(=\)\((\)\( 40 \nu^{6} - 73 \nu^{4} + 750 \nu^{2} - 2761 \)\()/1991\)
\(\beta_{6}\)\(=\)\((\)\( 61 \nu^{7} + 38 \nu^{5} + 646 \nu^{3} - 1672 \nu \)\()/1991\)
\(\beta_{7}\)\(=\)\((\)\( 68 \nu^{7} + 75 \nu^{5} + 1275 \nu^{3} - 3300 \nu \)\()/1991\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 4 \beta_{2} - 1\)
\(\nu^{3}\)\(=\)\(4 \beta_{7} - 4 \beta_{6} + \beta_{4} + 3 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-7 \beta_{5} - 10 \beta_{3} - 7\)
\(\nu^{5}\)\(=\)\(-7 \beta_{7} - 17 \beta_{4} - 7 \beta_{1}\)
\(\nu^{6}\)\(=\)\(37 \beta_{5} - 37 \beta_{3} + 75 \beta_{2} + 75\)
\(\nu^{7}\)\(=\)\(-38 \beta_{7} + 75 \beta_{6}\)

Character values

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

\(n\) \(122\) \(486\)
\(\chi(n)\) \(-1\) \(-1 - \beta_{2} + \beta_{3} - \beta_{5}\)

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
−1.26313 + 1.73855i
1.26313 1.73855i
−1.46782 0.476925i
1.46782 + 0.476925i
−1.26313 1.73855i
1.26313 + 1.73855i
−1.46782 + 0.476925i
1.46782 0.476925i
−1.26313 0.410415i 0 −0.190983 0.138757i −0.690983 2.12663i 0 2.52626 3.47709i 1.74560 + 2.40261i 0.927051 2.85317i 2.96979i
9.2 1.26313 + 0.410415i 0 −0.190983 0.138757i −0.690983 2.12663i 0 −2.52626 + 3.47709i −1.74560 2.40261i 0.927051 2.85317i 2.96979i
124.1 −1.46782 2.02029i 0 −1.30902 + 4.02874i −1.80902 1.31433i 0 2.93565 + 0.953850i 5.31064 1.72553i −2.42705 + 1.76336i 5.58394i
124.2 1.46782 + 2.02029i 0 −1.30902 + 4.02874i −1.80902 1.31433i 0 −2.93565 0.953850i −5.31064 + 1.72553i −2.42705 + 1.76336i 5.58394i
269.1 −1.26313 + 0.410415i 0 −0.190983 + 0.138757i −0.690983 + 2.12663i 0 2.52626 + 3.47709i 1.74560 2.40261i 0.927051 + 2.85317i 2.96979i
269.2 1.26313 0.410415i 0 −0.190983 + 0.138757i −0.690983 + 2.12663i 0 −2.52626 3.47709i −1.74560 + 2.40261i 0.927051 + 2.85317i 2.96979i
444.1 −1.46782 + 2.02029i 0 −1.30902 4.02874i −1.80902 + 1.31433i 0 2.93565 0.953850i 5.31064 + 1.72553i −2.42705 1.76336i 5.58394i
444.2 1.46782 2.02029i 0 −1.30902 4.02874i −1.80902 + 1.31433i 0 −2.93565 + 0.953850i −5.31064 1.72553i −2.42705 1.76336i 5.58394i
\(n\): 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
55.d odd 2 1 CM by \(\Q(\sqrt{-55}) \)
5.b even 2 1 inner
11.b odd 2 1 inner
11.c even 5 1 inner
11.d odd 10 1 inner
55.h odd 10 1 inner
55.j even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.j.a 8
5.b even 2 1 inner 605.2.j.a 8
11.b odd 2 1 inner 605.2.j.a 8
11.c even 5 1 605.2.b.b 4
11.c even 5 1 inner 605.2.j.a 8
11.c even 5 2 605.2.j.c 8
11.d odd 10 1 605.2.b.b 4
11.d odd 10 1 inner 605.2.j.a 8
11.d odd 10 2 605.2.j.c 8
55.d odd 2 1 CM 605.2.j.a 8
55.h odd 10 1 605.2.b.b 4
55.h odd 10 1 inner 605.2.j.a 8
55.h odd 10 2 605.2.j.c 8
55.j even 10 1 605.2.b.b 4
55.j even 10 1 inner 605.2.j.a 8
55.j even 10 2 605.2.j.c 8
55.k odd 20 2 3025.2.a.bb 4
55.l even 20 2 3025.2.a.bb 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.b.b 4 11.c even 5 1
605.2.b.b 4 11.d odd 10 1
605.2.b.b 4 55.h odd 10 1
605.2.b.b 4 55.j even 10 1
605.2.j.a 8 1.a even 1 1 trivial
605.2.j.a 8 5.b even 2 1 inner
605.2.j.a 8 11.b odd 2 1 inner
605.2.j.a 8 11.c even 5 1 inner
605.2.j.a 8 11.d odd 10 1 inner
605.2.j.a 8 55.d odd 2 1 CM
605.2.j.a 8 55.h odd 10 1 inner
605.2.j.a 8 55.j even 10 1 inner
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 55.k odd 20 2
3025.2.a.bb 4 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 \(S_{2}^{\mathrm{new}}(605, [\chi])\):

\( T_{2}^{8} + T_{2}^{6} + 31 T_{2}^{4} - 99 T_{2}^{2} + 121 \)
\( T_{19} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 121 - 99 T^{2} + 31 T^{4} + T^{6} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 25 + 25 T + 15 T^{2} + 5 T^{3} + T^{4} )^{2} \)
$7$ \( 30976 - 4224 T^{2} + 256 T^{4} - 4 T^{6} + T^{8} \)
$11$ \( T^{8} \)
$13$ \( 30976 + 4224 T^{2} + 2176 T^{4} - 76 T^{6} + T^{8} \)
$17$ \( 30976 - 18304 T^{2} + 4096 T^{4} + 36 T^{6} + T^{8} \)
$19$ \( T^{8} \)
$23$ \( T^{8} \)
$29$ \( T^{8} \)
$31$ \( ( 6400 - 1600 T + 240 T^{2} - 20 T^{3} + T^{4} )^{2} \)
$37$ \( T^{8} \)
$41$ \( T^{8} \)
$43$ \( ( 176 + 172 T^{2} + T^{4} )^{2} \)
$47$ \( T^{8} \)
$53$ \( T^{8} \)
$59$ \( ( 256 - 64 T + 16 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$61$ \( T^{8} \)
$67$ \( T^{8} \)
$71$ \( ( 4096 + 512 T + 64 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$73$ \( 453519616 - 3322176 T^{2} + 21376 T^{4} - 136 T^{6} + T^{8} \)
$79$ \( T^{8} \)
$83$ \( 30976 + 35904 T^{2} + 109696 T^{4} - 536 T^{6} + T^{8} \)
$89$ \( ( -180 + T^{2} )^{4} \)
$97$ \( T^{8} \)
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