Properties

Label 3264.2.c.d
Level $3264$
Weight $2$
Character orbit 3264.c
Analytic conductor $26.063$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3264 = 2^{6} \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3264.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.0631712197\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 51)
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.

\(f(q)\) \(=\) \( q -i q^{3} -4 i q^{7} - q^{9} +O(q^{10})\) \( q -i q^{3} -4 i q^{7} - q^{9} -4 i q^{11} -2 q^{13} + ( 1 - 4 i ) q^{17} -4 q^{19} -4 q^{21} -4 i q^{23} + 5 q^{25} + i q^{27} -4 i q^{31} -4 q^{33} + 8 i q^{37} + 2 i q^{39} + 8 i q^{41} -4 q^{43} + 8 q^{47} -9 q^{49} + ( -4 - i ) q^{51} -6 q^{53} + 4 i q^{57} -12 q^{59} -8 i q^{61} + 4 i q^{63} + 12 q^{67} -4 q^{69} + 12 i q^{71} -5 i q^{75} -16 q^{77} -4 i q^{79} + q^{81} + 12 q^{83} -10 q^{89} + 8 i q^{91} -4 q^{93} -16 i q^{97} + 4 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} - 4q^{13} + 2q^{17} - 8q^{19} - 8q^{21} + 10q^{25} - 8q^{33} - 8q^{43} + 16q^{47} - 18q^{49} - 8q^{51} - 12q^{53} - 24q^{59} + 24q^{67} - 8q^{69} - 32q^{77} + 2q^{81} + 24q^{83} - 20q^{89} - 8q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3264\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(2177\) \(2245\) \(2689\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
1.00000i
1.00000i
0 1.00000i 0 0 0 4.00000i 0 −1.00000 0
577.2 0 1.00000i 0 0 0 4.00000i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3264.2.c.d 2
4.b odd 2 1 3264.2.c.e 2
8.b even 2 1 816.2.c.c 2
8.d odd 2 1 51.2.d.b 2
17.b even 2 1 inner 3264.2.c.d 2
24.f even 2 1 153.2.d.a 2
24.h odd 2 1 2448.2.c.j 2
40.e odd 2 1 1275.2.g.a 2
40.k even 4 1 1275.2.d.b 2
40.k even 4 1 1275.2.d.d 2
68.d odd 2 1 3264.2.c.e 2
136.e odd 2 1 51.2.d.b 2
136.h even 2 1 816.2.c.c 2
136.j odd 4 1 867.2.a.a 1
136.j odd 4 1 867.2.a.b 1
136.p odd 8 4 867.2.e.d 4
136.s even 16 8 867.2.h.d 8
408.b odd 2 1 2448.2.c.j 2
408.h even 2 1 153.2.d.a 2
408.q even 4 1 2601.2.a.i 1
408.q even 4 1 2601.2.a.j 1
680.k odd 2 1 1275.2.g.a 2
680.u even 4 1 1275.2.d.b 2
680.u even 4 1 1275.2.d.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.d.b 2 8.d odd 2 1
51.2.d.b 2 136.e odd 2 1
153.2.d.a 2 24.f even 2 1
153.2.d.a 2 408.h even 2 1
816.2.c.c 2 8.b even 2 1
816.2.c.c 2 136.h even 2 1
867.2.a.a 1 136.j odd 4 1
867.2.a.b 1 136.j odd 4 1
867.2.e.d 4 136.p odd 8 4
867.2.h.d 8 136.s even 16 8
1275.2.d.b 2 40.k even 4 1
1275.2.d.b 2 680.u even 4 1
1275.2.d.d 2 40.k even 4 1
1275.2.d.d 2 680.u even 4 1
1275.2.g.a 2 40.e odd 2 1
1275.2.g.a 2 680.k odd 2 1
2448.2.c.j 2 24.h odd 2 1
2448.2.c.j 2 408.b odd 2 1
2601.2.a.i 1 408.q even 4 1
2601.2.a.j 1 408.q even 4 1
3264.2.c.d 2 1.a even 1 1 trivial
3264.2.c.d 2 17.b even 2 1 inner
3264.2.c.e 2 4.b odd 2 1
3264.2.c.e 2 68.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3264, [\chi])\):

\( T_{5} \)
\( T_{13} + 2 \)
\( T_{19} + 4 \)
\( T_{43} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 16 + T^{2} \)
$11$ \( 16 + T^{2} \)
$13$ \( ( 2 + T )^{2} \)
$17$ \( 17 - 2 T + T^{2} \)
$19$ \( ( 4 + T )^{2} \)
$23$ \( 16 + T^{2} \)
$29$ \( T^{2} \)
$31$ \( 16 + T^{2} \)
$37$ \( 64 + T^{2} \)
$41$ \( 64 + T^{2} \)
$43$ \( ( 4 + T )^{2} \)
$47$ \( ( -8 + T )^{2} \)
$53$ \( ( 6 + T )^{2} \)
$59$ \( ( 12 + T )^{2} \)
$61$ \( 64 + T^{2} \)
$67$ \( ( -12 + T )^{2} \)
$71$ \( 144 + T^{2} \)
$73$ \( T^{2} \)
$79$ \( 16 + T^{2} \)
$83$ \( ( -12 + T )^{2} \)
$89$ \( ( 10 + T )^{2} \)
$97$ \( 256 + T^{2} \)
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