Properties

Label 1600.2.n.s
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: \( x^{4} + 9 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1 + 1) q^{3} + ( - 2 \beta_{2} + 2) q^{7} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1 + 1) q^{3} + ( - 2 \beta_{2} + 2) q^{7} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{9} + (\beta_{3} + \beta_{2} + \beta_1) q^{11} + (2 \beta_{3} + 2 \beta_{2} - 2) q^{13} + ( - 2 \beta_{2} + \beta_1 - 2) q^{17} + ( - \beta_{3} + \beta_1 - 5) q^{19} + ( - 2 \beta_{3} + 2 \beta_1 + 4) q^{21} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{23} + (3 \beta_{3} + 5 \beta_{2} - 5) q^{27} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{29} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{31} + (3 \beta_{3} + 4 \beta_{2} - 4) q^{33} + (6 \beta_{2} + 6) q^{37} + (4 \beta_{3} - 4 \beta_1 - 10) q^{39} + 5 q^{41} + ( - 2 \beta_{2} + 4 \beta_1 - 2) q^{43} + 6 \beta_{3} q^{47} - \beta_{2} q^{49} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{51} + ( - 2 \beta_{3} - 4 \beta_{2} + 4) q^{53} + ( - 2 \beta_{2} - 3 \beta_1 - 2) q^{57} + ( - 4 \beta_{3} + 4 \beta_1 - 2) q^{59} + ( - 2 \beta_{3} + 2 \beta_1 - 4) q^{61} + (4 \beta_{2} + 8 \beta_1 + 4) q^{63} + ( - \beta_{3} - 3 \beta_{2} + 3) q^{67} + 2 \beta_{2} q^{69} - 2 \beta_{2} q^{71} + ( - 3 \beta_{3} - 2 \beta_{2} + 2) q^{73} + (2 \beta_{2} + 4 \beta_1 + 2) q^{77} + ( - 2 \beta_{3} + 2 \beta_1 - 4) q^{79} + (2 \beta_{3} - 2 \beta_1 - 13) q^{81} + (\beta_{2} + 3 \beta_1 + 1) q^{83} + ( - 2 \beta_{3} - 4 \beta_{2} + 4) q^{87} + ( - 2 \beta_{3} - 7 \beta_{2} - 2 \beta_1) q^{89} + (4 \beta_{3} + 8 \beta_{2} + 4 \beta_1) q^{91} + ( - 2 \beta_{3} - 4 \beta_{2} + 4) q^{93} + 4 \beta_1 q^{97} + (4 \beta_{3} - 4 \beta_1 - 14) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 8 q^{7} - 8 q^{13} - 8 q^{17} - 20 q^{19} + 16 q^{21} - 8 q^{23} - 20 q^{27} - 16 q^{33} + 24 q^{37} - 40 q^{39} + 20 q^{41} - 8 q^{43} + 16 q^{53} - 8 q^{57} - 8 q^{59} - 16 q^{61} + 16 q^{63} + 12 q^{67} + 8 q^{73} + 8 q^{77} - 16 q^{79} - 52 q^{81} + 4 q^{83} + 16 q^{87} + 16 q^{93} - 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} \) Copy content Toggle raw display

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.s 4
4.b odd 2 1 1600.2.n.o 4
5.b even 2 1 1600.2.n.p 4
5.c odd 4 1 1600.2.n.o 4
5.c odd 4 1 1600.2.n.t 4
8.b even 2 1 800.2.n.l yes 4
8.d odd 2 1 800.2.n.n yes 4
20.d odd 2 1 1600.2.n.t 4
20.e even 4 1 1600.2.n.p 4
20.e even 4 1 inner 1600.2.n.s 4
40.e odd 2 1 800.2.n.k 4
40.f even 2 1 800.2.n.m yes 4
40.i odd 4 1 800.2.n.k 4
40.i odd 4 1 800.2.n.n yes 4
40.k even 4 1 800.2.n.l yes 4
40.k even 4 1 800.2.n.m yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.2.n.k 4 40.e odd 2 1
800.2.n.k 4 40.i odd 4 1
800.2.n.l yes 4 8.b even 2 1
800.2.n.l yes 4 40.k even 4 1
800.2.n.m yes 4 40.f even 2 1
800.2.n.m yes 4 40.k even 4 1
800.2.n.n yes 4 8.d odd 2 1
800.2.n.n yes 4 40.i odd 4 1
1600.2.n.o 4 4.b odd 2 1
1600.2.n.o 4 5.c odd 4 1
1600.2.n.p 4 5.b even 2 1
1600.2.n.p 4 20.e even 4 1
1600.2.n.s 4 1.a even 1 1 trivial
1600.2.n.s 4 20.e even 4 1 inner
1600.2.n.t 4 5.c odd 4 1
1600.2.n.t 4 20.d odd 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

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