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Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 65 = 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 65.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.519027613138\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + i q^{2} + ( 1 - i ) q^{3} + q^{4} + ( -2 - i ) q^{5} + ( 1 + i ) q^{6} -2 q^{7} + 3 i q^{8} + i q^{9} +O(q^{10})\) \( q + i q^{2} + ( 1 - i ) q^{3} + q^{4} + ( -2 - i ) q^{5} + ( 1 + i ) q^{6} -2 q^{7} + 3 i q^{8} + i q^{9} + ( 1 - 2 i ) q^{10} + ( -1 + i ) q^{11} + ( 1 - i ) q^{12} + ( -2 - 3 i ) q^{13} -2 i q^{14} + ( -3 + i ) q^{15} - q^{16} + ( -1 + i ) q^{17} - q^{18} + ( 5 - 5 i ) q^{19} + ( -2 - i ) q^{20} + ( -2 + 2 i ) q^{21} + ( -1 - i ) q^{22} + ( -3 - 3 i ) q^{23} + ( 3 + 3 i ) q^{24} + ( 3 + 4 i ) q^{25} + ( 3 - 2 i ) q^{26} + ( 4 + 4 i ) q^{27} -2 q^{28} + ( -1 - 3 i ) q^{30} + ( 5 + 5 i ) q^{31} + 5 i q^{32} + 2 i q^{33} + ( -1 - i ) q^{34} + ( 4 + 2 i ) q^{35} + i q^{36} + ( 5 + 5 i ) q^{38} + ( -5 - i ) q^{39} + ( 3 - 6 i ) q^{40} + ( -7 - 7 i ) q^{41} + ( -2 - 2 i ) q^{42} + ( 1 + i ) q^{43} + ( -1 + i ) q^{44} + ( 1 - 2 i ) q^{45} + ( 3 - 3 i ) q^{46} + 6 q^{47} + ( -1 + i ) q^{48} -3 q^{49} + ( -4 + 3 i ) q^{50} + 2 i q^{51} + ( -2 - 3 i ) q^{52} + ( 5 - 5 i ) q^{53} + ( -4 + 4 i ) q^{54} + ( 3 - i ) q^{55} -6 i q^{56} -10 i q^{57} + ( 7 + 7 i ) q^{59} + ( -3 + i ) q^{60} -14 q^{61} + ( -5 + 5 i ) q^{62} -2 i q^{63} -7 q^{64} + ( 1 + 8 i ) q^{65} -2 q^{66} -4 i q^{67} + ( -1 + i ) q^{68} -6 q^{69} + ( -2 + 4 i ) q^{70} + ( 1 + i ) q^{71} -3 q^{72} + 10 i q^{73} + ( 7 + i ) q^{75} + ( 5 - 5 i ) q^{76} + ( 2 - 2 i ) q^{77} + ( 1 - 5 i ) q^{78} -2 i q^{79} + ( 2 + i ) q^{80} + 5 q^{81} + ( 7 - 7 i ) q^{82} + 6 q^{83} + ( -2 + 2 i ) q^{84} + ( 3 - i ) q^{85} + ( -1 + i ) q^{86} + ( -3 - 3 i ) q^{88} + ( -5 - 5 i ) q^{89} + ( 2 + i ) q^{90} + ( 4 + 6 i ) q^{91} + ( -3 - 3 i ) q^{92} + 10 q^{93} + 6 i q^{94} + ( -15 + 5 i ) q^{95} + ( 5 + 5 i ) q^{96} + 2 i q^{97} -3 i q^{98} + ( -1 - i ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + 2q^{4} - 4q^{5} + 2q^{6} - 4q^{7} + O(q^{10}) \) \( 2q + 2q^{3} + 2q^{4} - 4q^{5} + 2q^{6} - 4q^{7} + 2q^{10} - 2q^{11} + 2q^{12} - 4q^{13} - 6q^{15} - 2q^{16} - 2q^{17} - 2q^{18} + 10q^{19} - 4q^{20} - 4q^{21} - 2q^{22} - 6q^{23} + 6q^{24} + 6q^{25} + 6q^{26} + 8q^{27} - 4q^{28} - 2q^{30} + 10q^{31} - 2q^{34} + 8q^{35} + 10q^{38} - 10q^{39} + 6q^{40} - 14q^{41} - 4q^{42} + 2q^{43} - 2q^{44} + 2q^{45} + 6q^{46} + 12q^{47} - 2q^{48} - 6q^{49} - 8q^{50} - 4q^{52} + 10q^{53} - 8q^{54} + 6q^{55} + 14q^{59} - 6q^{60} - 28q^{61} - 10q^{62} - 14q^{64} + 2q^{65} - 4q^{66} - 2q^{68} - 12q^{69} - 4q^{70} + 2q^{71} - 6q^{72} + 14q^{75} + 10q^{76} + 4q^{77} + 2q^{78} + 4q^{80} + 10q^{81} + 14q^{82} + 12q^{83} - 4q^{84} + 6q^{85} - 2q^{86} - 6q^{88} - 10q^{89} + 4q^{90} + 8q^{91} - 6q^{92} + 20q^{93} - 30q^{95} + 10q^{96} - 2q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(27\) \(41\)
\(\chi(n)\) \(i\) \(i\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1
− 1.00000i
1.00000i
− 1.00000i 1.00000 + 1.00000i 1.00000 −2.00000 + 1.00000i 1.00000 − 1.00000i −2.00000 − 3.00000i − 1.00000i 1.00000 + 2.00000i
47.1 1.00000i 1.00000 − 1.00000i 1.00000 −2.00000 − 1.00000i 1.00000 + 1.00000i −2.00000 3.00000i 1.00000i 1.00000 − 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 65.2.f.a ✓ 2
3.b odd 2 1 585.2.n.c 2
4.b odd 2 1 1040.2.cd.b 2
5.b even 2 1 325.2.f.a 2
5.c odd 4 1 65.2.k.a yes 2
5.c odd 4 1 325.2.k.a 2
13.b even 2 1 845.2.f.a 2
13.c even 3 2 845.2.t.a 4
13.d odd 4 1 65.2.k.a yes 2
13.d odd 4 1 845.2.k.a 2
13.e even 6 2 845.2.t.b 4
13.f odd 12 2 845.2.o.a 4
13.f odd 12 2 845.2.o.b 4
15.e even 4 1 585.2.w.b 2
20.e even 4 1 1040.2.bg.a 2
39.f even 4 1 585.2.w.b 2
52.f even 4 1 1040.2.bg.a 2
65.f even 4 1 inner 65.2.f.a ✓ 2
65.g odd 4 1 325.2.k.a 2
65.h odd 4 1 845.2.k.a 2
65.k even 4 1 325.2.f.a 2
65.k even 4 1 845.2.f.a 2
65.o even 12 2 845.2.t.b 4
65.q odd 12 2 845.2.o.a 4
65.r odd 12 2 845.2.o.b 4
65.t even 12 2 845.2.t.a 4
195.u odd 4 1 585.2.n.c 2
260.l odd 4 1 1040.2.cd.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.f.a ✓ 2 1.a even 1 1 trivial
65.2.f.a ✓ 2 65.f even 4 1 inner
65.2.k.a yes 2 5.c odd 4 1
65.2.k.a yes 2 13.d odd 4 1
325.2.f.a 2 5.b even 2 1
325.2.f.a 2 65.k even 4 1
325.2.k.a 2 5.c odd 4 1
325.2.k.a 2 65.g odd 4 1
585.2.n.c 2 3.b odd 2 1
585.2.n.c 2 195.u odd 4 1
585.2.w.b 2 15.e even 4 1
585.2.w.b 2 39.f even 4 1
845.2.f.a 2 13.b even 2 1
845.2.f.a 2 65.k even 4 1
845.2.k.a 2 13.d odd 4 1
845.2.k.a 2 65.h odd 4 1
845.2.o.a 4 13.f odd 12 2
845.2.o.a 4 65.q odd 12 2
845.2.o.b 4 13.f odd 12 2
845.2.o.b 4 65.r odd 12 2
845.2.t.a 4 13.c even 3 2
845.2.t.a 4 65.t even 12 2
845.2.t.b 4 13.e even 6 2
845.2.t.b 4 65.o even 12 2
1040.2.bg.a 2 20.e even 4 1
1040.2.bg.a 2 52.f even 4 1
1040.2.cd.b 2 4.b odd 2 1
1040.2.cd.b 2 260.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(65, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 2 - 2 T + T^{2} \)
$5$ \( 5 + 4 T + T^{2} \)
$7$ \( ( 2 + T )^{2} \)
$11$ \( 2 + 2 T + T^{2} \)
$13$ \( 13 + 4 T + T^{2} \)
$17$ \( 2 + 2 T + T^{2} \)
$19$ \( 50 - 10 T + T^{2} \)
$23$ \( 18 + 6 T + T^{2} \)
$29$ \( T^{2} \)
$31$ \( 50 - 10 T + T^{2} \)
$37$ \( T^{2} \)
$41$ \( 98 + 14 T + T^{2} \)
$43$ \( 2 - 2 T + T^{2} \)
$47$ \( ( -6 + T )^{2} \)
$53$ \( 50 - 10 T + T^{2} \)
$59$ \( 98 - 14 T + T^{2} \)
$61$ \( ( 14 + T )^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( 2 - 2 T + T^{2} \)
$73$ \( 100 + T^{2} \)
$79$ \( 4 + T^{2} \)
$83$ \( ( -6 + T )^{2} \)
$89$ \( 50 + 10 T + T^{2} \)
$97$ \( 4 + T^{2} \)
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