Properties

Label 1600.2.n.t
Level $1600$
Weight $2$
Character orbit 1600.n
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.n (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: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 800)
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.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{1} + \beta_{2} ) q^{3} + ( 2 - 2 \beta_{2} ) q^{7} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{1} + \beta_{2} ) q^{3} + ( 2 - 2 \beta_{2} ) q^{7} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{9} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{11} + ( 2 - 2 \beta_{2} - 2 \beta_{3} ) q^{13} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{17} + ( 5 - \beta_{1} + \beta_{3} ) q^{19} + ( 4 + 2 \beta_{1} - 2 \beta_{3} ) q^{21} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{23} + ( -5 + 5 \beta_{2} + 3 \beta_{3} ) q^{27} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{29} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{31} + ( 4 - 4 \beta_{2} - 3 \beta_{3} ) q^{33} + ( -6 - 6 \beta_{2} ) q^{37} + ( 10 + 4 \beta_{1} - 4 \beta_{3} ) q^{39} + 5 q^{41} + ( -2 + 4 \beta_{1} - 2 \beta_{2} ) q^{43} + 6 \beta_{3} q^{47} -\beta_{2} q^{49} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{51} + ( -4 + 4 \beta_{2} + 2 \beta_{3} ) q^{53} + ( 2 + 3 \beta_{1} + 2 \beta_{2} ) q^{57} + ( 2 - 4 \beta_{1} + 4 \beta_{3} ) q^{59} + ( -4 + 2 \beta_{1} - 2 \beta_{3} ) q^{61} + ( 4 + 8 \beta_{1} + 4 \beta_{2} ) q^{63} + ( 3 - 3 \beta_{2} - \beta_{3} ) q^{67} + 2 \beta_{2} q^{69} + 2 \beta_{2} q^{71} + ( -2 + 2 \beta_{2} + 3 \beta_{3} ) q^{73} + ( -2 - 4 \beta_{1} - 2 \beta_{2} ) q^{77} + ( 4 - 2 \beta_{1} + 2 \beta_{3} ) q^{79} + ( -13 - 2 \beta_{1} + 2 \beta_{3} ) q^{81} + ( 1 + 3 \beta_{1} + \beta_{2} ) q^{83} + ( 4 - 4 \beta_{2} - 2 \beta_{3} ) q^{87} + ( -2 \beta_{1} - 7 \beta_{2} - 2 \beta_{3} ) q^{89} + ( -4 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} ) q^{91} + ( -4 + 4 \beta_{2} + 2 \beta_{3} ) q^{93} -4 \beta_{1} q^{97} + ( 14 + 4 \beta_{1} - 4 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 8q^{7} + O(q^{10}) \) \( 4q + 4q^{3} + 8q^{7} + 8q^{13} + 8q^{17} + 20q^{19} + 16q^{21} - 8q^{23} - 20q^{27} + 16q^{33} - 24q^{37} + 40q^{39} + 20q^{41} - 8q^{43} - 16q^{53} + 8q^{57} + 8q^{59} - 16q^{61} + 16q^{63} + 12q^{67} - 8q^{73} - 8q^{77} + 16q^{79} - 52q^{81} + 4q^{83} + 16q^{87} - 16q^{93} + 56q^{99} + O(q^{100}) \)

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

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

Character values

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

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1343.1
−1.22474 + 1.22474i
1.22474 1.22474i
−1.22474 1.22474i
1.22474 + 1.22474i
0 −0.224745 + 0.224745i 0 0 0 2.00000 + 2.00000i 0 2.89898i 0
1343.2 0 2.22474 2.22474i 0 0 0 2.00000 + 2.00000i 0 6.89898i 0
1407.1 0 −0.224745 0.224745i 0 0 0 2.00000 2.00000i 0 2.89898i 0
1407.2 0 2.22474 + 2.22474i 0 0 0 2.00000 2.00000i 0 6.89898i 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
20.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.n.t 4
4.b odd 2 1 1600.2.n.p 4
5.b even 2 1 1600.2.n.o 4
5.c odd 4 1 1600.2.n.p 4
5.c odd 4 1 1600.2.n.s 4
8.b even 2 1 800.2.n.k 4
8.d odd 2 1 800.2.n.m yes 4
20.d odd 2 1 1600.2.n.s 4
20.e even 4 1 1600.2.n.o 4
20.e even 4 1 inner 1600.2.n.t 4
40.e odd 2 1 800.2.n.l yes 4
40.f even 2 1 800.2.n.n yes 4
40.i odd 4 1 800.2.n.l yes 4
40.i odd 4 1 800.2.n.m yes 4
40.k even 4 1 800.2.n.k 4
40.k even 4 1 800.2.n.n yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.2.n.k 4 8.b even 2 1
800.2.n.k 4 40.k even 4 1
800.2.n.l yes 4 40.e odd 2 1
800.2.n.l yes 4 40.i odd 4 1
800.2.n.m yes 4 8.d odd 2 1
800.2.n.m yes 4 40.i odd 4 1
800.2.n.n yes 4 40.f even 2 1
800.2.n.n yes 4 40.k even 4 1
1600.2.n.o 4 5.b even 2 1
1600.2.n.o 4 20.e even 4 1
1600.2.n.p 4 4.b odd 2 1
1600.2.n.p 4 5.c odd 4 1
1600.2.n.s 4 5.c odd 4 1
1600.2.n.s 4 20.d odd 2 1
1600.2.n.t 4 1.a even 1 1 trivial
1600.2.n.t 4 20.e even 4 1 inner

Hecke kernels

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

\( T_{3}^{4} - 4 T_{3}^{3} + 8 T_{3}^{2} + 4 T_{3} + 1 \)
\( T_{7}^{2} - 4 T_{7} + 8 \)
\( T_{11}^{4} + 14 T_{11}^{2} + 25 \)
\( T_{13}^{4} - 8 T_{13}^{3} + 32 T_{13}^{2} + 32 T_{13} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 1 + 4 T + 8 T^{2} - 4 T^{3} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 8 - 4 T + T^{2} )^{2} \)
$11$ \( 25 + 14 T^{2} + T^{4} \)
$13$ \( 16 + 32 T + 32 T^{2} - 8 T^{3} + T^{4} \)
$17$ \( 25 - 40 T + 32 T^{2} - 8 T^{3} + T^{4} \)
$19$ \( ( 19 - 10 T + T^{2} )^{2} \)
$23$ \( 16 - 32 T + 32 T^{2} + 8 T^{3} + T^{4} \)
$29$ \( 400 + 56 T^{2} + T^{4} \)
$31$ \( 400 + 56 T^{2} + T^{4} \)
$37$ \( ( 72 + 12 T + T^{2} )^{2} \)
$41$ \( ( -5 + T )^{4} \)
$43$ \( 1600 - 320 T + 32 T^{2} + 8 T^{3} + T^{4} \)
$47$ \( 11664 + T^{4} \)
$53$ \( 400 + 320 T + 128 T^{2} + 16 T^{3} + T^{4} \)
$59$ \( ( -92 - 4 T + T^{2} )^{2} \)
$61$ \( ( -8 + 8 T + T^{2} )^{2} \)
$67$ \( 225 - 180 T + 72 T^{2} - 12 T^{3} + T^{4} \)
$71$ \( ( 4 + T^{2} )^{2} \)
$73$ \( 361 - 152 T + 32 T^{2} + 8 T^{3} + T^{4} \)
$79$ \( ( -8 - 8 T + T^{2} )^{2} \)
$83$ \( 625 + 100 T + 8 T^{2} - 4 T^{3} + T^{4} \)
$89$ \( 625 + 146 T^{2} + T^{4} \)
$97$ \( 2304 + T^{4} \)
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