Properties

Label 4056.2.c.e
Level $4056$
Weight $2$
Character orbit 4056.c
Analytic conductor $32.387$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4056.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.3873230598\)
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: \( 2 \)
Twist minimal: no (minimal twist has level 24)
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.

\(f(q)\) \(=\) \( q - q^{3} + 2 i q^{5} + q^{9} +O(q^{10})\) \( q - q^{3} + 2 i q^{5} + q^{9} + 4 i q^{11} -2 i q^{15} -2 q^{17} + 4 i q^{19} + 8 q^{23} + q^{25} - q^{27} + 6 q^{29} -8 i q^{31} -4 i q^{33} + 6 i q^{37} + 6 i q^{41} -4 q^{43} + 2 i q^{45} + 7 q^{49} + 2 q^{51} -2 q^{53} -8 q^{55} -4 i q^{57} + 4 i q^{59} -2 q^{61} + 4 i q^{67} -8 q^{69} -8 i q^{71} + 10 i q^{73} - q^{75} -8 q^{79} + q^{81} + 4 i q^{83} -4 i q^{85} -6 q^{87} -6 i q^{89} + 8 i q^{93} -8 q^{95} -2 i q^{97} + 4 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{9} - 4q^{17} + 16q^{23} + 2q^{25} - 2q^{27} + 12q^{29} - 8q^{43} + 14q^{49} + 4q^{51} - 4q^{53} - 16q^{55} - 4q^{61} - 16q^{69} - 2q^{75} - 16q^{79} + 2q^{81} - 12q^{87} - 16q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4056\mathbb{Z}\right)^\times\).

\(n\) \(1015\) \(2029\) \(2705\) \(3889\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
1.00000i
1.00000i
0 −1.00000 0 2.00000i 0 0 0 1.00000 0
337.2 0 −1.00000 0 2.00000i 0 0 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
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4056.2.c.e 2
13.b even 2 1 inner 4056.2.c.e 2
13.d odd 4 1 24.2.a.a 1
13.d odd 4 1 4056.2.a.i 1
39.f even 4 1 72.2.a.a 1
52.f even 4 1 48.2.a.a 1
52.f even 4 1 8112.2.a.be 1
65.f even 4 1 600.2.f.e 2
65.g odd 4 1 600.2.a.h 1
65.k even 4 1 600.2.f.e 2
91.i even 4 1 1176.2.a.i 1
91.z odd 12 2 1176.2.q.i 2
91.bb even 12 2 1176.2.q.a 2
104.j odd 4 1 192.2.a.d 1
104.m even 4 1 192.2.a.b 1
117.y odd 12 2 648.2.i.g 2
117.z even 12 2 648.2.i.b 2
143.g even 4 1 2904.2.a.c 1
156.l odd 4 1 144.2.a.b 1
195.j odd 4 1 1800.2.f.c 2
195.n even 4 1 1800.2.a.m 1
195.u odd 4 1 1800.2.f.c 2
208.l even 4 1 768.2.d.d 2
208.m odd 4 1 768.2.d.e 2
208.r odd 4 1 768.2.d.e 2
208.s even 4 1 768.2.d.d 2
221.g odd 4 1 6936.2.a.p 1
247.i even 4 1 8664.2.a.j 1
260.l odd 4 1 1200.2.f.b 2
260.s odd 4 1 1200.2.f.b 2
260.u even 4 1 1200.2.a.d 1
273.o odd 4 1 3528.2.a.d 1
273.cb odd 12 2 3528.2.s.y 2
273.cd even 12 2 3528.2.s.j 2
312.w odd 4 1 576.2.a.b 1
312.y even 4 1 576.2.a.d 1
364.p odd 4 1 2352.2.a.i 1
364.bw odd 12 2 2352.2.q.r 2
364.ce even 12 2 2352.2.q.l 2
429.l odd 4 1 8712.2.a.u 1
468.bs even 12 2 1296.2.i.m 2
468.ch odd 12 2 1296.2.i.e 2
520.t even 4 1 4800.2.a.cc 1
520.x odd 4 1 4800.2.f.bg 2
520.y even 4 1 4800.2.f.d 2
520.bj even 4 1 4800.2.f.d 2
520.bk odd 4 1 4800.2.f.bg 2
520.bo odd 4 1 4800.2.a.q 1
572.k odd 4 1 5808.2.a.s 1
624.s odd 4 1 2304.2.d.k 2
624.u even 4 1 2304.2.d.i 2
624.bm even 4 1 2304.2.d.i 2
624.bo odd 4 1 2304.2.d.k 2
728.x odd 4 1 9408.2.a.cc 1
728.ba even 4 1 9408.2.a.h 1
780.u even 4 1 3600.2.f.r 2
780.bb odd 4 1 3600.2.a.v 1
780.bn even 4 1 3600.2.f.r 2
1092.u even 4 1 7056.2.a.q 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.a.a 1 13.d odd 4 1
48.2.a.a 1 52.f even 4 1
72.2.a.a 1 39.f even 4 1
144.2.a.b 1 156.l odd 4 1
192.2.a.b 1 104.m even 4 1
192.2.a.d 1 104.j odd 4 1
576.2.a.b 1 312.w odd 4 1
576.2.a.d 1 312.y even 4 1
600.2.a.h 1 65.g odd 4 1
600.2.f.e 2 65.f even 4 1
600.2.f.e 2 65.k even 4 1
648.2.i.b 2 117.z even 12 2
648.2.i.g 2 117.y odd 12 2
768.2.d.d 2 208.l even 4 1
768.2.d.d 2 208.s even 4 1
768.2.d.e 2 208.m odd 4 1
768.2.d.e 2 208.r odd 4 1
1176.2.a.i 1 91.i even 4 1
1176.2.q.a 2 91.bb even 12 2
1176.2.q.i 2 91.z odd 12 2
1200.2.a.d 1 260.u even 4 1
1200.2.f.b 2 260.l odd 4 1
1200.2.f.b 2 260.s odd 4 1
1296.2.i.e 2 468.ch odd 12 2
1296.2.i.m 2 468.bs even 12 2
1800.2.a.m 1 195.n even 4 1
1800.2.f.c 2 195.j odd 4 1
1800.2.f.c 2 195.u odd 4 1
2304.2.d.i 2 624.u even 4 1
2304.2.d.i 2 624.bm even 4 1
2304.2.d.k 2 624.s odd 4 1
2304.2.d.k 2 624.bo odd 4 1
2352.2.a.i 1 364.p odd 4 1
2352.2.q.l 2 364.ce even 12 2
2352.2.q.r 2 364.bw odd 12 2
2904.2.a.c 1 143.g even 4 1
3528.2.a.d 1 273.o odd 4 1
3528.2.s.j 2 273.cd even 12 2
3528.2.s.y 2 273.cb odd 12 2
3600.2.a.v 1 780.bb odd 4 1
3600.2.f.r 2 780.u even 4 1
3600.2.f.r 2 780.bn even 4 1
4056.2.a.i 1 13.d odd 4 1
4056.2.c.e 2 1.a even 1 1 trivial
4056.2.c.e 2 13.b even 2 1 inner
4800.2.a.q 1 520.bo odd 4 1
4800.2.a.cc 1 520.t even 4 1
4800.2.f.d 2 520.y even 4 1
4800.2.f.d 2 520.bj even 4 1
4800.2.f.bg 2 520.x odd 4 1
4800.2.f.bg 2 520.bk odd 4 1
5808.2.a.s 1 572.k odd 4 1
6936.2.a.p 1 221.g odd 4 1
7056.2.a.q 1 1092.u even 4 1
8112.2.a.be 1 52.f even 4 1
8664.2.a.j 1 247.i even 4 1
8712.2.a.u 1 429.l odd 4 1
9408.2.a.h 1 728.ba even 4 1
9408.2.a.cc 1 728.x odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4056, [\chi])\):

\( T_{5}^{2} + 4 \)
\( T_{7} \)
\( T_{11}^{2} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \)
$7$ \( ( 1 - 7 T^{2} )^{2} \)
$11$ \( 1 - 6 T^{2} + 121 T^{4} \)
$13$ 1
$17$ \( ( 1 + 2 T + 17 T^{2} )^{2} \)
$19$ \( 1 - 22 T^{2} + 361 T^{4} \)
$23$ \( ( 1 - 8 T + 23 T^{2} )^{2} \)
$29$ \( ( 1 - 6 T + 29 T^{2} )^{2} \)
$31$ \( 1 + 2 T^{2} + 961 T^{4} \)
$37$ \( 1 - 38 T^{2} + 1369 T^{4} \)
$41$ \( 1 - 46 T^{2} + 1681 T^{4} \)
$43$ \( ( 1 + 4 T + 43 T^{2} )^{2} \)
$47$ \( ( 1 - 47 T^{2} )^{2} \)
$53$ \( ( 1 + 2 T + 53 T^{2} )^{2} \)
$59$ \( 1 - 102 T^{2} + 3481 T^{4} \)
$61$ \( ( 1 + 2 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 118 T^{2} + 4489 T^{4} \)
$71$ \( 1 - 78 T^{2} + 5041 T^{4} \)
$73$ \( 1 - 46 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 + 8 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 150 T^{2} + 6889 T^{4} \)
$89$ \( 1 - 142 T^{2} + 7921 T^{4} \)
$97$ \( 1 - 190 T^{2} + 9409 T^{4} \)
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