Properties

Label 1600.2.l.e
Level $1600$
Weight $2$
Character orbit 1600.l
Analytic conductor $12.776$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.l (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{11})\)
Defining polynomial: \(x^{4} - 5 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 400)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{2} ) q^{3} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + ( -1 + 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{2} ) q^{3} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + ( -1 + 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{9} + ( -2 + \beta_{1} + \beta_{3} ) q^{11} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{13} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{17} + ( -2 - \beta_{1} - \beta_{2} ) q^{19} + ( 5 - 5 \beta_{1} ) q^{21} + ( 1 + 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{23} + ( -6 + 5 \beta_{1} + \beta_{3} ) q^{27} + ( 2 + 2 \beta_{1} ) q^{29} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{31} + ( -7 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{33} + ( -3 + 5 \beta_{1} - 2 \beta_{3} ) q^{37} + ( -2 + 12 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{39} -5 \beta_{1} q^{41} + ( -3 - \beta_{1} + 4 \beta_{3} ) q^{43} -8 q^{47} + ( -5 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{49} + ( -5 - 4 \beta_{1} - \beta_{2} ) q^{51} + ( 1 + \beta_{1} - 2 \beta_{3} ) q^{53} + ( 1 + 4 \beta_{1} + \beta_{2} - \beta_{3} ) q^{57} + ( -1 + 3 \beta_{1} - 2 \beta_{3} ) q^{59} + ( 1 + 5 \beta_{1} - 4 \beta_{2} ) q^{61} + ( 8 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{63} + ( 7 + 8 \beta_{1} - \beta_{2} ) q^{67} + ( 3 - 5 \beta_{1} + 2 \beta_{3} ) q^{69} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{71} + ( 1 + 10 \beta_{1} + \beta_{2} - \beta_{3} ) q^{73} + ( 3 + 5 \beta_{1} - 2 \beta_{2} ) q^{77} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{79} + ( -2 - 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{81} + ( -5 - 6 \beta_{1} + \beta_{2} ) q^{83} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{87} + ( 3 + 6 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{89} + ( 10 - 10 \beta_{1} ) q^{91} + ( 5 + 5 \beta_{1} ) q^{93} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{97} + ( -7 - 13 \beta_{1} + 6 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} + O(q^{10}) \) \( 4q + 2q^{3} - 6q^{11} + 4q^{13} + 8q^{17} - 6q^{19} + 20q^{21} - 22q^{27} + 8q^{29} - 4q^{31} - 28q^{33} - 16q^{37} - 4q^{43} - 32q^{47} - 20q^{49} - 18q^{51} - 8q^{59} + 12q^{61} + 32q^{63} + 30q^{67} + 16q^{69} + 16q^{77} - 4q^{79} - 8q^{81} - 22q^{83} + 40q^{91} + 20q^{93} - 40q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 5 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - 2 \nu \)\()/3\)
\(\beta_{2}\)\(=\)\( \nu^{2} + \nu - 3 \)
\(\beta_{3}\)\(=\)\( -\nu^{2} + \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + \beta_{2} + 6\)\()/2\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 3 \beta_{1}\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
1.65831 + 0.500000i
−1.65831 + 0.500000i
1.65831 0.500000i
−1.65831 0.500000i
0 −1.15831 1.15831i 0 0 0 4.31662i 0 0.316625i 0
401.2 0 2.15831 + 2.15831i 0 0 0 2.31662i 0 6.31662i 0
1201.1 0 −1.15831 + 1.15831i 0 0 0 4.31662i 0 0.316625i 0
1201.2 0 2.15831 2.15831i 0 0 0 2.31662i 0 6.31662i 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
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.l.e 4
4.b odd 2 1 400.2.l.d 4
5.b even 2 1 1600.2.l.d 4
5.c odd 4 1 1600.2.q.c 4
5.c odd 4 1 1600.2.q.d 4
16.e even 4 1 inner 1600.2.l.e 4
16.f odd 4 1 400.2.l.d 4
20.d odd 2 1 400.2.l.e yes 4
20.e even 4 1 400.2.q.c 4
20.e even 4 1 400.2.q.d 4
80.i odd 4 1 1600.2.q.c 4
80.j even 4 1 400.2.q.d 4
80.k odd 4 1 400.2.l.e yes 4
80.q even 4 1 1600.2.l.d 4
80.s even 4 1 400.2.q.c 4
80.t odd 4 1 1600.2.q.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
400.2.l.d 4 4.b odd 2 1
400.2.l.d 4 16.f odd 4 1
400.2.l.e yes 4 20.d odd 2 1
400.2.l.e yes 4 80.k odd 4 1
400.2.q.c 4 20.e even 4 1
400.2.q.c 4 80.s even 4 1
400.2.q.d 4 20.e even 4 1
400.2.q.d 4 80.j even 4 1
1600.2.l.d 4 5.b even 2 1
1600.2.l.d 4 80.q even 4 1
1600.2.l.e 4 1.a even 1 1 trivial
1600.2.l.e 4 16.e even 4 1 inner
1600.2.q.c 4 5.c odd 4 1
1600.2.q.c 4 80.i odd 4 1
1600.2.q.d 4 5.c odd 4 1
1600.2.q.d 4 80.t odd 4 1

Hecke kernels

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

\( T_{3}^{4} - 2 T_{3}^{3} + 2 T_{3}^{2} + 10 T_{3} + 25 \)
\( T_{7}^{4} + 24 T_{7}^{2} + 100 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 25 + 10 T + 2 T^{2} - 2 T^{3} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 100 + 24 T^{2} + T^{4} \)
$11$ \( 1 - 6 T + 18 T^{2} + 6 T^{3} + T^{4} \)
$13$ \( 400 + 80 T + 8 T^{2} - 4 T^{3} + T^{4} \)
$17$ \( ( -7 - 4 T + T^{2} )^{2} \)
$19$ \( 1 - 6 T + 18 T^{2} + 6 T^{3} + T^{4} \)
$23$ \( 4 + 40 T^{2} + T^{4} \)
$29$ \( ( 8 - 4 T + T^{2} )^{2} \)
$31$ \( ( -10 + 2 T + T^{2} )^{2} \)
$37$ \( 100 + 160 T + 128 T^{2} + 16 T^{3} + T^{4} \)
$41$ \( ( 25 + T^{2} )^{2} \)
$43$ \( 7396 - 344 T + 8 T^{2} + 4 T^{3} + T^{4} \)
$47$ \( ( 8 + T )^{4} \)
$53$ \( 484 + T^{4} \)
$59$ \( 196 - 112 T + 32 T^{2} + 8 T^{3} + T^{4} \)
$61$ \( 4900 + 840 T + 72 T^{2} - 12 T^{3} + T^{4} \)
$67$ \( 11449 - 3210 T + 450 T^{2} - 30 T^{3} + T^{4} \)
$71$ \( 1600 + 96 T^{2} + T^{4} \)
$73$ \( 7921 + 222 T^{2} + T^{4} \)
$79$ \( ( -10 + 2 T + T^{2} )^{2} \)
$83$ \( 3025 + 1210 T + 242 T^{2} + 22 T^{3} + T^{4} \)
$89$ \( 3969 + 270 T^{2} + T^{4} \)
$97$ \( ( -44 + T^{2} )^{2} \)
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