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Properties

Label 585.2.h.d
Level $585$
Weight $2$
Character orbit 585.h
Analytic conductor $4.671$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 2 q^{2} + 2 q^{4} + ( 2 - i ) q^{5} + 3 q^{7} +O(q^{10})\) \( q + 2 q^{2} + 2 q^{4} + ( 2 - i ) q^{5} + 3 q^{7} + ( 4 - 2 i ) q^{10} + 5 i q^{11} + ( -3 - 2 i ) q^{13} + 6 q^{14} -4 q^{16} -3 i q^{17} -6 i q^{19} + ( 4 - 2 i ) q^{20} + 10 i q^{22} + 9 i q^{23} + ( 3 - 4 i ) q^{25} + ( -6 - 4 i ) q^{26} + 6 q^{28} -8 q^{32} -6 i q^{34} + ( 6 - 3 i ) q^{35} + 3 q^{37} -12 i q^{38} -5 i q^{41} + 6 i q^{43} + 10 i q^{44} + 18 i q^{46} -8 q^{47} + 2 q^{49} + ( 6 - 8 i ) q^{50} + ( -6 - 4 i ) q^{52} + 9 i q^{53} + ( 5 + 10 i ) q^{55} -4 i q^{59} -3 q^{61} -8 q^{64} + ( -8 - i ) q^{65} -12 q^{67} -6 i q^{68} + ( 12 - 6 i ) q^{70} -5 i q^{71} -6 q^{73} + 6 q^{74} -12 i q^{76} + 15 i q^{77} + 15 q^{79} + ( -8 + 4 i ) q^{80} -10 i q^{82} -4 q^{83} + ( -3 - 6 i ) q^{85} + 12 i q^{86} + i q^{89} + ( -9 - 6 i ) q^{91} + 18 i q^{92} -16 q^{94} + ( -6 - 12 i ) q^{95} + 3 q^{97} + 4 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 6 q^{7} + O(q^{10}) \) \( 2 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 6 q^{7} + 8 q^{10} - 6 q^{13} + 12 q^{14} - 8 q^{16} + 8 q^{20} + 6 q^{25} - 12 q^{26} + 12 q^{28} - 16 q^{32} + 12 q^{35} + 6 q^{37} - 16 q^{47} + 4 q^{49} + 12 q^{50} - 12 q^{52} + 10 q^{55} - 6 q^{61} - 16 q^{64} - 16 q^{65} - 24 q^{67} + 24 q^{70} - 12 q^{73} + 12 q^{74} + 30 q^{79} - 16 q^{80} - 8 q^{83} - 6 q^{85} - 18 q^{91} - 32 q^{94} - 12 q^{95} + 6 q^{97} + 8 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
1.00000i
− 1.00000i
2.00000 0 2.00000 2.00000 − 1.00000i 0 3.00000 0 0 4.00000 − 2.00000i
64.2 2.00000 0 2.00000 2.00000 + 1.00000i 0 3.00000 0 0 4.00000 + 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.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.h.d 2
3.b odd 2 1 195.2.h.a ✓ 2
5.b even 2 1 585.2.h.a 2
12.b even 2 1 3120.2.r.a 2
13.b even 2 1 585.2.h.a 2
15.d odd 2 1 195.2.h.b yes 2
15.e even 4 1 975.2.b.a 2
15.e even 4 1 975.2.b.c 2
39.d odd 2 1 195.2.h.b yes 2
60.h even 2 1 3120.2.r.f 2
65.d even 2 1 inner 585.2.h.d 2
156.h even 2 1 3120.2.r.f 2
195.e odd 2 1 195.2.h.a ✓ 2
195.s even 4 1 975.2.b.a 2
195.s even 4 1 975.2.b.c 2
780.d even 2 1 3120.2.r.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.h.a ✓ 2 3.b odd 2 1
195.2.h.a ✓ 2 195.e odd 2 1
195.2.h.b yes 2 15.d odd 2 1
195.2.h.b yes 2 39.d odd 2 1
585.2.h.a 2 5.b even 2 1
585.2.h.a 2 13.b even 2 1
585.2.h.d 2 1.a even 1 1 trivial
585.2.h.d 2 65.d even 2 1 inner
975.2.b.a 2 15.e even 4 1
975.2.b.a 2 195.s even 4 1
975.2.b.c 2 15.e even 4 1
975.2.b.c 2 195.s even 4 1
3120.2.r.a 2 12.b even 2 1
3120.2.r.a 2 780.d even 2 1
3120.2.r.f 2 60.h even 2 1
3120.2.r.f 2 156.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 2 \) acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( 5 - 4 T + T^{2} \)
$7$ \( ( -3 + T )^{2} \)
$11$ \( 25 + T^{2} \)
$13$ \( 13 + 6 T + T^{2} \)
$17$ \( 9 + T^{2} \)
$19$ \( 36 + T^{2} \)
$23$ \( 81 + T^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( ( -3 + T )^{2} \)
$41$ \( 25 + T^{2} \)
$43$ \( 36 + T^{2} \)
$47$ \( ( 8 + T )^{2} \)
$53$ \( 81 + T^{2} \)
$59$ \( 16 + T^{2} \)
$61$ \( ( 3 + T )^{2} \)
$67$ \( ( 12 + T )^{2} \)
$71$ \( 25 + T^{2} \)
$73$ \( ( 6 + T )^{2} \)
$79$ \( ( -15 + T )^{2} \)
$83$ \( ( 4 + T )^{2} \)
$89$ \( 1 + T^{2} \)
$97$ \( ( -3 + T )^{2} \)
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