Properties

Label 1600.2.o.k
Level $1600$
Weight $2$
Character orbit 1600.o
Analytic conductor $12.776$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.o (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.12745506816.1
Defining polynomial: \(x^{8} + 23 x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{3} -2 \beta_{6} q^{7} -4 \beta_{3} q^{9} +O(q^{10})\) \( q + \beta_{7} q^{3} -2 \beta_{6} q^{7} -4 \beta_{3} q^{9} -\beta_{4} q^{11} -2 \beta_{2} q^{13} + \beta_{1} q^{17} + \beta_{5} q^{19} + 2 \beta_{5} q^{21} + 4 \beta_{1} q^{23} + \beta_{2} q^{27} + 2 \beta_{4} q^{29} -4 \beta_{3} q^{31} + 7 \beta_{6} q^{33} + 2 \beta_{7} q^{37} + 14 q^{39} -9 q^{41} + 2 \beta_{7} q^{43} + 2 \beta_{6} q^{47} -5 \beta_{3} q^{49} -\beta_{4} q^{51} + 7 \beta_{1} q^{57} -2 \beta_{5} q^{61} + 8 \beta_{1} q^{63} -\beta_{2} q^{67} -4 \beta_{4} q^{69} + 6 \beta_{3} q^{71} + 5 \beta_{6} q^{73} -6 \beta_{7} q^{77} + 10 q^{79} + 5 q^{81} -3 \beta_{7} q^{83} -14 \beta_{6} q^{87} -15 \beta_{3} q^{89} + 4 \beta_{4} q^{91} + 4 \beta_{2} q^{93} + 4 \beta_{1} q^{97} -4 \beta_{5} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 112q^{39} - 72q^{41} + 80q^{79} + 40q^{81} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 23 x^{4} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{5} + 19 \nu \)\()/5\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} + 29 \nu \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} + 24 \nu^{2} \)\()/5\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{4} + 23 \)\()/5\)
\(\beta_{5}\)\(=\)\( -\nu^{6} - 22 \nu^{2} \)
\(\beta_{6}\)\(=\)\((\)\( -4 \nu^{7} - 91 \nu^{3} \)\()/5\)
\(\beta_{7}\)\(=\)\((\)\( 6 \nu^{7} + 139 \nu^{3} \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} + 5 \beta_{3}\)\()/2\)
\(\nu^{3}\)\(=\)\(2 \beta_{7} + 3 \beta_{6}\)
\(\nu^{4}\)\(=\)\((\)\(5 \beta_{4} - 23\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-19 \beta_{2} + 29 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\(-12 \beta_{5} - 55 \beta_{3}\)
\(\nu^{7}\)\(=\)\((\)\(-91 \beta_{7} - 139 \beta_{6}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(\beta_{3}\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
543.1
0.323042 0.323042i
1.54779 1.54779i
−1.54779 + 1.54779i
−0.323042 + 0.323042i
0.323042 + 0.323042i
1.54779 + 1.54779i
−1.54779 1.54779i
−0.323042 0.323042i
0 −1.87083 1.87083i 0 0 0 −2.44949 2.44949i 0 4.00000i 0
543.2 0 −1.87083 1.87083i 0 0 0 2.44949 + 2.44949i 0 4.00000i 0
543.3 0 1.87083 + 1.87083i 0 0 0 −2.44949 2.44949i 0 4.00000i 0
543.4 0 1.87083 + 1.87083i 0 0 0 2.44949 + 2.44949i 0 4.00000i 0
607.1 0 −1.87083 + 1.87083i 0 0 0 −2.44949 + 2.44949i 0 4.00000i 0
607.2 0 −1.87083 + 1.87083i 0 0 0 2.44949 2.44949i 0 4.00000i 0
607.3 0 1.87083 1.87083i 0 0 0 −2.44949 + 2.44949i 0 4.00000i 0
607.4 0 1.87083 1.87083i 0 0 0 2.44949 2.44949i 0 4.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 607.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.b even 2 1 inner
20.e even 4 2 inner
40.f even 2 1 inner
40.k even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.o.k yes 8
4.b odd 2 1 1600.2.o.f 8
5.b even 2 1 inner 1600.2.o.k yes 8
5.c odd 4 2 1600.2.o.f 8
8.b even 2 1 inner 1600.2.o.k yes 8
8.d odd 2 1 1600.2.o.f 8
20.d odd 2 1 1600.2.o.f 8
20.e even 4 2 inner 1600.2.o.k yes 8
40.e odd 2 1 1600.2.o.f 8
40.f even 2 1 inner 1600.2.o.k yes 8
40.i odd 4 2 1600.2.o.f 8
40.k even 4 2 inner 1600.2.o.k yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1600.2.o.f 8 4.b odd 2 1
1600.2.o.f 8 5.c odd 4 2
1600.2.o.f 8 8.d odd 2 1
1600.2.o.f 8 20.d odd 2 1
1600.2.o.f 8 40.e odd 2 1
1600.2.o.f 8 40.i odd 4 2
1600.2.o.k yes 8 1.a even 1 1 trivial
1600.2.o.k yes 8 5.b even 2 1 inner
1600.2.o.k yes 8 8.b even 2 1 inner
1600.2.o.k yes 8 20.e even 4 2 inner
1600.2.o.k yes 8 40.f even 2 1 inner
1600.2.o.k yes 8 40.k even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1600, [\chi])\):

\( T_{3}^{4} + 49 \)
\( T_{7}^{4} + 144 \)
\( T_{79} - 10 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 49 + T^{4} )^{2} \)
$5$ \( T^{8} \)
$7$ \( ( 144 + T^{4} )^{2} \)
$11$ \( ( -21 + T^{2} )^{4} \)
$13$ \( ( 784 + T^{4} )^{2} \)
$17$ \( ( 9 + T^{4} )^{2} \)
$19$ \( ( 21 + T^{2} )^{4} \)
$23$ \( ( 2304 + T^{4} )^{2} \)
$29$ \( ( -84 + T^{2} )^{4} \)
$31$ \( ( 16 + T^{2} )^{4} \)
$37$ \( ( 784 + T^{4} )^{2} \)
$41$ \( ( 9 + T )^{8} \)
$43$ \( ( 784 + T^{4} )^{2} \)
$47$ \( ( 144 + T^{4} )^{2} \)
$53$ \( T^{8} \)
$59$ \( T^{8} \)
$61$ \( ( 84 + T^{2} )^{4} \)
$67$ \( ( 49 + T^{4} )^{2} \)
$71$ \( ( 36 + T^{2} )^{4} \)
$73$ \( ( 5625 + T^{4} )^{2} \)
$79$ \( ( -10 + T )^{8} \)
$83$ \( ( 3969 + T^{4} )^{2} \)
$89$ \( ( 225 + T^{2} )^{4} \)
$97$ \( ( 2304 + T^{4} )^{2} \)
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