Properties

Label 65.2.l.a
Level $65$
Weight $2$
Character orbit 65.l
Analytic conductor $0.519$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 65 = 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 65.l (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.519027613138\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.49787136.1
Defining polynomial: \(x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{6} + 5 \nu^{4} - 5 \nu^{2} - 12 \)\()/20\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} + 5 \nu^{5} - 5 \nu^{3} - 12 \nu \)\()/40\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{6} + 5 \nu^{4} + 15 \nu^{2} + 36 \)\()/20\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{7} - 5 \nu^{5} + 5 \nu^{3} - 16 \nu \)\()/40\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{6} - 7 \)\()/5\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} + 3 \nu^{5} + 5 \nu^{3} + 12 \nu \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} + \beta_{4} - \beta_{2} - 1\)
\(\nu^{3}\)\(=\)\(\beta_{7} + 2 \beta_{5} - \beta_{3} - \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4} + 3 \beta_{2}\)
\(\nu^{5}\)\(=\)\(\beta_{7} + 5 \beta_{3}\)
\(\nu^{6}\)\(=\)\(-5 \beta_{6} - 7\)
\(\nu^{7}\)\(=\)\(-10 \beta_{5} - 10 \beta_{3} - 7 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/65\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(41\)
\(\chi(n)\) \(-1\) \(1 - \beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
−1.09445 + 0.895644i
−0.228425 + 1.39564i
0.228425 1.39564i
1.09445 0.895644i
−1.09445 0.895644i
−0.228425 1.39564i
0.228425 + 1.39564i
1.09445 + 0.895644i
−1.09445 + 1.89564i 0.866025 + 0.500000i −1.39564 2.41733i 0.456850 + 2.18890i −1.89564 + 1.09445i −0.866025 1.50000i 1.73205 −1.00000 1.73205i −4.64938 1.52962i
4.2 −0.228425 + 0.395644i −0.866025 0.500000i 0.895644 + 1.55130i 2.18890 0.456850i 0.395644 0.228425i 0.866025 + 1.50000i −1.73205 −1.00000 1.73205i −0.319250 + 0.970381i
4.3 0.228425 0.395644i 0.866025 + 0.500000i 0.895644 + 1.55130i −2.18890 0.456850i 0.395644 0.228425i −0.866025 1.50000i 1.73205 −1.00000 1.73205i −0.680750 + 0.761669i
4.4 1.09445 1.89564i −0.866025 0.500000i −1.39564 2.41733i −0.456850 + 2.18890i −1.89564 + 1.09445i 0.866025 + 1.50000i −1.73205 −1.00000 1.73205i 3.64938 + 3.26167i
49.1 −1.09445 1.89564i 0.866025 0.500000i −1.39564 + 2.41733i 0.456850 2.18890i −1.89564 1.09445i −0.866025 + 1.50000i 1.73205 −1.00000 + 1.73205i −4.64938 + 1.52962i
49.2 −0.228425 0.395644i −0.866025 + 0.500000i 0.895644 1.55130i 2.18890 + 0.456850i 0.395644 + 0.228425i 0.866025 1.50000i −1.73205 −1.00000 + 1.73205i −0.319250 0.970381i
49.3 0.228425 + 0.395644i 0.866025 0.500000i 0.895644 1.55130i −2.18890 + 0.456850i 0.395644 + 0.228425i −0.866025 + 1.50000i 1.73205 −1.00000 + 1.73205i −0.680750 0.761669i
49.4 1.09445 + 1.89564i −0.866025 + 0.500000i −1.39564 + 2.41733i −0.456850 2.18890i −1.89564 1.09445i 0.866025 1.50000i −1.73205 −1.00000 + 1.73205i 3.64938 3.26167i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.e even 6 1 inner
65.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 65.2.l.a 8
3.b odd 2 1 585.2.bf.a 8
4.b odd 2 1 1040.2.df.b 8
5.b even 2 1 inner 65.2.l.a 8
5.c odd 4 1 325.2.n.b 4
5.c odd 4 1 325.2.n.c 4
13.b even 2 1 845.2.l.c 8
13.c even 3 1 845.2.d.c 8
13.c even 3 1 845.2.l.c 8
13.d odd 4 1 845.2.n.c 8
13.d odd 4 1 845.2.n.d 8
13.e even 6 1 inner 65.2.l.a 8
13.e even 6 1 845.2.d.c 8
13.f odd 12 2 845.2.b.f 8
13.f odd 12 1 845.2.n.c 8
13.f odd 12 1 845.2.n.d 8
15.d odd 2 1 585.2.bf.a 8
20.d odd 2 1 1040.2.df.b 8
39.h odd 6 1 585.2.bf.a 8
52.i odd 6 1 1040.2.df.b 8
65.d even 2 1 845.2.l.c 8
65.g odd 4 1 845.2.n.c 8
65.g odd 4 1 845.2.n.d 8
65.l even 6 1 inner 65.2.l.a 8
65.l even 6 1 845.2.d.c 8
65.n even 6 1 845.2.d.c 8
65.n even 6 1 845.2.l.c 8
65.o even 12 1 4225.2.a.bj 4
65.o even 12 1 4225.2.a.bk 4
65.r odd 12 1 325.2.n.b 4
65.r odd 12 1 325.2.n.c 4
65.s odd 12 2 845.2.b.f 8
65.s odd 12 1 845.2.n.c 8
65.s odd 12 1 845.2.n.d 8
65.t even 12 1 4225.2.a.bj 4
65.t even 12 1 4225.2.a.bk 4
195.y odd 6 1 585.2.bf.a 8
260.w odd 6 1 1040.2.df.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.l.a 8 1.a even 1 1 trivial
65.2.l.a 8 5.b even 2 1 inner
65.2.l.a 8 13.e even 6 1 inner
65.2.l.a 8 65.l even 6 1 inner
325.2.n.b 4 5.c odd 4 1
325.2.n.b 4 65.r odd 12 1
325.2.n.c 4 5.c odd 4 1
325.2.n.c 4 65.r odd 12 1
585.2.bf.a 8 3.b odd 2 1
585.2.bf.a 8 15.d odd 2 1
585.2.bf.a 8 39.h odd 6 1
585.2.bf.a 8 195.y odd 6 1
845.2.b.f 8 13.f odd 12 2
845.2.b.f 8 65.s odd 12 2
845.2.d.c 8 13.c even 3 1
845.2.d.c 8 13.e even 6 1
845.2.d.c 8 65.l even 6 1
845.2.d.c 8 65.n even 6 1
845.2.l.c 8 13.b even 2 1
845.2.l.c 8 13.c even 3 1
845.2.l.c 8 65.d even 2 1
845.2.l.c 8 65.n even 6 1
845.2.n.c 8 13.d odd 4 1
845.2.n.c 8 13.f odd 12 1
845.2.n.c 8 65.g odd 4 1
845.2.n.c 8 65.s odd 12 1
845.2.n.d 8 13.d odd 4 1
845.2.n.d 8 13.f odd 12 1
845.2.n.d 8 65.g odd 4 1
845.2.n.d 8 65.s odd 12 1
1040.2.df.b 8 4.b odd 2 1
1040.2.df.b 8 20.d odd 2 1
1040.2.df.b 8 52.i odd 6 1
1040.2.df.b 8 260.w odd 6 1
4225.2.a.bj 4 65.o even 12 1
4225.2.a.bj 4 65.t even 12 1
4225.2.a.bk 4 65.o even 12 1
4225.2.a.bk 4 65.t even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(65, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 5 T^{2} + 24 T^{4} + 5 T^{6} + T^{8} \)
$3$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$5$ \( 625 - 34 T^{4} + T^{8} \)
$7$ \( ( 9 + 3 T^{2} + T^{4} )^{2} \)
$11$ \( ( 49 - 7 T^{2} + T^{4} )^{2} \)
$13$ \( ( 169 - 22 T^{2} + T^{4} )^{2} \)
$17$ \( ( 441 - 21 T^{2} + T^{4} )^{2} \)
$19$ \( ( 3 + 3 T + T^{2} )^{4} \)
$23$ \( ( 441 - 21 T^{2} + T^{4} )^{2} \)
$29$ \( ( 441 + 21 T^{2} + T^{4} )^{2} \)
$31$ \( ( 3600 + 132 T^{2} + T^{4} )^{2} \)
$37$ \( ( 3969 + 63 T^{2} + T^{4} )^{2} \)
$41$ \( ( 49 - 7 T^{2} + T^{4} )^{2} \)
$43$ \( 50625 - 25650 T^{2} + 12771 T^{4} - 114 T^{6} + T^{8} \)
$47$ \( ( 256 - 80 T^{2} + T^{4} )^{2} \)
$53$ \( ( 144 + 60 T^{2} + T^{4} )^{2} \)
$59$ \( ( 2209 + 1410 T + 347 T^{2} + 30 T^{3} + T^{4} )^{2} \)
$61$ \( ( 225 - 180 T + 129 T^{2} - 12 T^{3} + T^{4} )^{2} \)
$67$ \( 50625 + 49950 T^{2} + 49059 T^{4} + 222 T^{6} + T^{8} \)
$71$ \( ( 625 + 150 T - 13 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$73$ \( T^{8} \)
$79$ \( ( -6 + T )^{8} \)
$83$ \( ( 4624 - 164 T^{2} + T^{4} )^{2} \)
$89$ \( ( 1681 - 984 T + 233 T^{2} - 24 T^{3} + T^{4} )^{2} \)
$97$ \( 6765201 + 390150 T^{2} + 19899 T^{4} + 150 T^{6} + T^{8} \)
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