Properties

Label 848.1.x.a
Level $848$
Weight $1$
Character orbit 848.x
Analytic conductor $0.423$
Analytic rank $0$
Dimension $12$
Projective image $D_{26}$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 848 = 2^{4} \cdot 53 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 848.x (of order \(26\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.423207130713\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{26})\)
Defining polynomial: \(x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{26}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{26} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + ( -\zeta_{26} + \zeta_{26}^{11} ) q^{5} -\zeta_{26}^{6} q^{9} +O(q^{10})\) \( q + ( -\zeta_{26} + \zeta_{26}^{11} ) q^{5} -\zeta_{26}^{6} q^{9} + ( -\zeta_{26}^{3} - \zeta_{26}^{7} ) q^{13} + ( -\zeta_{26}^{5} + \zeta_{26}^{10} ) q^{17} + ( \zeta_{26}^{2} - \zeta_{26}^{9} - \zeta_{26}^{12} ) q^{25} + ( -\zeta_{26}^{8} + \zeta_{26}^{9} ) q^{29} + ( -\zeta_{26}^{2} - \zeta_{26}^{4} ) q^{37} + ( -\zeta_{26}^{10} + \zeta_{26}^{12} ) q^{41} + ( \zeta_{26}^{4} + \zeta_{26}^{7} ) q^{45} + \zeta_{26}^{8} q^{49} + \zeta_{26}^{3} q^{53} + ( \zeta_{26}^{5} + \zeta_{26}^{6} ) q^{61} + ( \zeta_{26} + \zeta_{26}^{4} + \zeta_{26}^{5} + \zeta_{26}^{8} ) q^{65} + ( -\zeta_{26}^{4} - \zeta_{26}^{11} ) q^{73} + \zeta_{26}^{12} q^{81} + ( \zeta_{26}^{3} + \zeta_{26}^{6} - \zeta_{26}^{8} - \zeta_{26}^{11} ) q^{85} + ( -1 - \zeta_{26}^{2} ) q^{89} + ( -1 - \zeta_{26}^{12} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + q^{9} + O(q^{10}) \) \( 12q + q^{9} - 2q^{13} - 2q^{17} - q^{25} + 2q^{29} + 2q^{37} - q^{49} + q^{53} - q^{81} - 11q^{89} - 11q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/848\mathbb{Z}\right)^\times\).

\(n\) \(161\) \(213\) \(319\)
\(\chi(n)\) \(-\zeta_{26}^{8}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1
0.748511 0.663123i
−0.885456 0.464723i
0.748511 + 0.663123i
−0.120537 0.992709i
−0.568065 0.822984i
0.970942 0.239316i
−0.568065 + 0.822984i
−0.885456 + 0.464723i
0.354605 0.935016i
−0.120537 + 0.992709i
0.970942 + 0.239316i
0.354605 + 0.935016i
0 0 0 −0.869047 0.329586i 0 0 0 0.354605 0.935016i 0
223.1 0 0 0 0.317391 + 1.28771i 0 0 0 0.970942 0.239316i 0
255.1 0 0 0 −0.869047 + 0.329586i 0 0 0 0.354605 + 0.935016i 0
271.1 0 0 0 1.09148 + 1.23202i 0 0 0 0.748511 0.663123i 0
303.1 0 0 0 0.922670 + 1.75800i 0 0 0 −0.885456 + 0.464723i 0
335.1 0 0 0 −1.85640 0.225408i 0 0 0 −0.120537 + 0.992709i 0
431.1 0 0 0 0.922670 1.75800i 0 0 0 −0.885456 0.464723i 0
559.1 0 0 0 0.317391 1.28771i 0 0 0 0.970942 + 0.239316i 0
623.1 0 0 0 0.393906 + 0.271894i 0 0 0 −0.568065 + 0.822984i 0
751.1 0 0 0 1.09148 1.23202i 0 0 0 0.748511 + 0.663123i 0
767.1 0 0 0 −1.85640 + 0.225408i 0 0 0 −0.120537 0.992709i 0
799.1 0 0 0 0.393906 0.271894i 0 0 0 −0.568065 0.822984i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 799.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
53.e even 26 1 inner
212.h odd 26 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 848.1.x.a 12
4.b odd 2 1 CM 848.1.x.a 12
8.b even 2 1 3392.1.bn.a 12
8.d odd 2 1 3392.1.bn.a 12
53.e even 26 1 inner 848.1.x.a 12
212.h odd 26 1 inner 848.1.x.a 12
424.n even 26 1 3392.1.bn.a 12
424.q odd 26 1 3392.1.bn.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
848.1.x.a 12 1.a even 1 1 trivial
848.1.x.a 12 4.b odd 2 1 CM
848.1.x.a 12 53.e even 26 1 inner
848.1.x.a 12 212.h odd 26 1 inner
3392.1.bn.a 12 8.b even 2 1
3392.1.bn.a 12 8.d odd 2 1
3392.1.bn.a 12 424.n even 26 1
3392.1.bn.a 12 424.q odd 26 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(848, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( T^{12} \)
$5$ \( 13 - 26 T + 65 T^{3} + 13 T^{4} + 52 T^{6} + 13 T^{9} + T^{12} \)
$7$ \( T^{12} \)
$11$ \( T^{12} \)
$13$ \( 1 - 6 T + 10 T^{2} + 5 T^{3} + 35 T^{4} + 24 T^{5} + 12 T^{6} - 20 T^{7} - 10 T^{8} - 5 T^{9} + 4 T^{10} + 2 T^{11} + T^{12} \)
$17$ \( 1 + 7 T + 36 T^{2} + 96 T^{3} + 139 T^{4} + 115 T^{5} + 64 T^{6} + 32 T^{7} + 16 T^{8} + 8 T^{9} + 4 T^{10} + 2 T^{11} + T^{12} \)
$19$ \( T^{12} \)
$23$ \( T^{12} \)
$29$ \( 1 + 6 T + 10 T^{2} - 5 T^{3} + 35 T^{4} - 24 T^{5} + 12 T^{6} + 20 T^{7} - 10 T^{8} + 5 T^{9} + 4 T^{10} - 2 T^{11} + T^{12} \)
$31$ \( T^{12} \)
$37$ \( 1 - 7 T + 36 T^{2} - 96 T^{3} + 139 T^{4} - 115 T^{5} + 64 T^{6} - 32 T^{7} + 16 T^{8} - 8 T^{9} + 4 T^{10} - 2 T^{11} + T^{12} \)
$41$ \( 13 + 13 T + 26 T^{2} - 52 T^{3} + 65 T^{5} - 13 T^{7} + T^{12} \)
$43$ \( T^{12} \)
$47$ \( T^{12} \)
$53$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} \)
$59$ \( T^{12} \)
$61$ \( 13 + 13 T + 26 T^{2} - 52 T^{3} + 65 T^{5} - 13 T^{7} + T^{12} \)
$67$ \( T^{12} \)
$71$ \( T^{12} \)
$73$ \( 13 - 26 T + 65 T^{3} + 13 T^{4} + 52 T^{6} + 13 T^{9} + T^{12} \)
$79$ \( T^{12} \)
$83$ \( T^{12} \)
$89$ \( 1 + 6 T + 36 T^{2} + 125 T^{3} + 295 T^{4} + 496 T^{5} + 610 T^{6} + 553 T^{7} + 367 T^{8} + 174 T^{9} + 56 T^{10} + 11 T^{11} + T^{12} \)
$97$ \( 1 + 6 T + 36 T^{2} + 125 T^{3} + 295 T^{4} + 496 T^{5} + 610 T^{6} + 553 T^{7} + 367 T^{8} + 174 T^{9} + 56 T^{10} + 11 T^{11} + T^{12} \)
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