Properties

Label 832.4.f.e
Level $832$
Weight $4$
Character orbit 832.f
Analytic conductor $49.090$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 832 = 2^{6} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 832.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(49.0895891248\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 13)
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.

\(f(q)\) \(=\) \( q + q^{3} + 9 i q^{5} -15 i q^{7} -26 q^{9} +O(q^{10})\) \( q + q^{3} + 9 i q^{5} -15 i q^{7} -26 q^{9} -48 i q^{11} + ( -26 + 39 i ) q^{13} + 9 i q^{15} -45 q^{17} -6 i q^{19} -15 i q^{21} + 162 q^{23} + 44 q^{25} -53 q^{27} + 144 q^{29} + 264 i q^{31} -48 i q^{33} + 135 q^{35} + 303 i q^{37} + ( -26 + 39 i ) q^{39} -192 i q^{41} + 97 q^{43} -234 i q^{45} -111 i q^{47} + 118 q^{49} -45 q^{51} + 414 q^{53} + 432 q^{55} -6 i q^{57} + 522 i q^{59} -376 q^{61} + 390 i q^{63} + ( -351 - 234 i ) q^{65} + 36 i q^{67} + 162 q^{69} + 357 i q^{71} + 1098 i q^{73} + 44 q^{75} -720 q^{77} -830 q^{79} + 649 q^{81} + 438 i q^{83} -405 i q^{85} + 144 q^{87} + 438 i q^{89} + ( 585 + 390 i ) q^{91} + 264 i q^{93} + 54 q^{95} -852 i q^{97} + 1248 i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 52 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} - 52 q^{9} - 52 q^{13} - 90 q^{17} + 324 q^{23} + 88 q^{25} - 106 q^{27} + 288 q^{29} + 270 q^{35} - 52 q^{39} + 194 q^{43} + 236 q^{49} - 90 q^{51} + 828 q^{53} + 864 q^{55} - 752 q^{61} - 702 q^{65} + 324 q^{69} + 88 q^{75} - 1440 q^{77} - 1660 q^{79} + 1298 q^{81} + 288 q^{87} + 1170 q^{91} + 108 q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/832\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(703\) \(769\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
1.00000i
1.00000i
0 1.00000 0 9.00000i 0 15.0000i 0 −26.0000 0
129.2 0 1.00000 0 9.00000i 0 15.0000i 0 −26.0000 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 832.4.f.e 2
4.b odd 2 1 832.4.f.c 2
8.b even 2 1 13.4.b.a 2
8.d odd 2 1 208.4.f.b 2
13.b even 2 1 inner 832.4.f.e 2
24.h odd 2 1 117.4.b.a 2
40.f even 2 1 325.4.c.b 2
40.i odd 4 1 325.4.d.a 2
40.i odd 4 1 325.4.d.b 2
52.b odd 2 1 832.4.f.c 2
104.e even 2 1 13.4.b.a 2
104.h odd 2 1 208.4.f.b 2
104.j odd 4 1 169.4.a.b 1
104.j odd 4 1 169.4.a.c 1
104.r even 6 2 169.4.e.d 4
104.s even 6 2 169.4.e.d 4
104.x odd 12 2 169.4.c.b 2
104.x odd 12 2 169.4.c.c 2
312.b odd 2 1 117.4.b.a 2
312.y even 4 1 1521.4.a.d 1
312.y even 4 1 1521.4.a.i 1
520.p even 2 1 325.4.c.b 2
520.bg odd 4 1 325.4.d.a 2
520.bg odd 4 1 325.4.d.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.b.a 2 8.b even 2 1
13.4.b.a 2 104.e even 2 1
117.4.b.a 2 24.h odd 2 1
117.4.b.a 2 312.b odd 2 1
169.4.a.b 1 104.j odd 4 1
169.4.a.c 1 104.j odd 4 1
169.4.c.b 2 104.x odd 12 2
169.4.c.c 2 104.x odd 12 2
169.4.e.d 4 104.r even 6 2
169.4.e.d 4 104.s even 6 2
208.4.f.b 2 8.d odd 2 1
208.4.f.b 2 104.h odd 2 1
325.4.c.b 2 40.f even 2 1
325.4.c.b 2 520.p even 2 1
325.4.d.a 2 40.i odd 4 1
325.4.d.a 2 520.bg odd 4 1
325.4.d.b 2 40.i odd 4 1
325.4.d.b 2 520.bg odd 4 1
832.4.f.c 2 4.b odd 2 1
832.4.f.c 2 52.b odd 2 1
832.4.f.e 2 1.a even 1 1 trivial
832.4.f.e 2 13.b even 2 1 inner
1521.4.a.d 1 312.y even 4 1
1521.4.a.i 1 312.y even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(832, [\chi])\):

\( T_{3} - 1 \)
\( T_{5}^{2} + 81 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( 81 + T^{2} \)
$7$ \( 225 + T^{2} \)
$11$ \( 2304 + T^{2} \)
$13$ \( 2197 + 52 T + T^{2} \)
$17$ \( ( 45 + T )^{2} \)
$19$ \( 36 + T^{2} \)
$23$ \( ( -162 + T )^{2} \)
$29$ \( ( -144 + T )^{2} \)
$31$ \( 69696 + T^{2} \)
$37$ \( 91809 + T^{2} \)
$41$ \( 36864 + T^{2} \)
$43$ \( ( -97 + T )^{2} \)
$47$ \( 12321 + T^{2} \)
$53$ \( ( -414 + T )^{2} \)
$59$ \( 272484 + T^{2} \)
$61$ \( ( 376 + T )^{2} \)
$67$ \( 1296 + T^{2} \)
$71$ \( 127449 + T^{2} \)
$73$ \( 1205604 + T^{2} \)
$79$ \( ( 830 + T )^{2} \)
$83$ \( 191844 + T^{2} \)
$89$ \( 191844 + T^{2} \)
$97$ \( 725904 + T^{2} \)
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