Properties

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

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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: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 80)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} - \beta_{3} q^{7} + 3 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} - \beta_{3} q^{7} + 3 \beta_1 q^{9} + (\beta_{3} - \beta_{2}) q^{11} + ( - \beta_1 + 1) q^{13} + ( - \beta_1 - 1) q^{17} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{19} + 6 q^{21} + \beta_{2} q^{23} + 4 \beta_1 q^{29} + (\beta_{3} - \beta_{2}) q^{31} + (6 \beta_1 - 6) q^{33} + (5 \beta_1 + 5) q^{37} + ( - \beta_{3} - \beta_{2}) q^{39} + 2 q^{41} - \beta_{2} q^{43} - \beta_{3} q^{47} + \beta_1 q^{49} + ( - \beta_{3} + \beta_{2}) q^{51} + (7 \beta_1 - 7) q^{53} + (12 \beta_1 + 12) q^{57} + (2 \beta_{3} + 2 \beta_{2}) q^{59} - 6 q^{61} - 3 \beta_{2} q^{63} - 3 \beta_{3} q^{67} - 6 \beta_1 q^{69} + ( - 3 \beta_{3} + 3 \beta_{2}) q^{71} + ( - 7 \beta_1 + 7) q^{73} + (6 \beta_1 + 6) q^{77} + 9 q^{81} + 7 \beta_{2} q^{83} + 4 \beta_{3} q^{87} - 8 \beta_1 q^{89} + ( - \beta_{3} + \beta_{2}) q^{91} + (6 \beta_1 - 6) q^{93} + (7 \beta_1 + 7) q^{97} + (3 \beta_{3} + 3 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{13} - 4 q^{17} + 24 q^{21} - 24 q^{33} + 20 q^{37} + 8 q^{41} - 28 q^{53} + 48 q^{57} - 24 q^{61} + 28 q^{73} + 24 q^{77} + 36 q^{81} - 24 q^{93} + 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12}^{2} + 2\zeta_{12} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} - 2\zeta_{12}^{2} + 2\zeta_{12} + 1 \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_{2} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( -\beta_{3} + \beta_{2} + 2 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

Character values

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

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

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
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0 −1.73205 + 1.73205i 0 0 0 −1.73205 1.73205i 0 3.00000i 0
1343.2 0 1.73205 1.73205i 0 0 0 1.73205 + 1.73205i 0 3.00000i 0
1407.1 0 −1.73205 1.73205i 0 0 0 −1.73205 + 1.73205i 0 3.00000i 0
1407.2 0 1.73205 + 1.73205i 0 0 0 1.73205 1.73205i 0 3.00000i 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
4.b odd 2 1 inner
5.c odd 4 1 inner
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.r 4
4.b odd 2 1 inner 1600.2.n.r 4
5.b even 2 1 320.2.n.i 4
5.c odd 4 1 320.2.n.i 4
5.c odd 4 1 inner 1600.2.n.r 4
8.b even 2 1 400.2.n.b 4
8.d odd 2 1 400.2.n.b 4
20.d odd 2 1 320.2.n.i 4
20.e even 4 1 320.2.n.i 4
20.e even 4 1 inner 1600.2.n.r 4
24.f even 2 1 3600.2.x.e 4
24.h odd 2 1 3600.2.x.e 4
40.e odd 2 1 80.2.n.b 4
40.f even 2 1 80.2.n.b 4
40.i odd 4 1 80.2.n.b 4
40.i odd 4 1 400.2.n.b 4
40.k even 4 1 80.2.n.b 4
40.k even 4 1 400.2.n.b 4
80.i odd 4 1 1280.2.o.r 4
80.j even 4 1 1280.2.o.q 4
80.k odd 4 1 1280.2.o.q 4
80.k odd 4 1 1280.2.o.r 4
80.q even 4 1 1280.2.o.q 4
80.q even 4 1 1280.2.o.r 4
80.s even 4 1 1280.2.o.r 4
80.t odd 4 1 1280.2.o.q 4
120.i odd 2 1 720.2.x.d 4
120.m even 2 1 720.2.x.d 4
120.q odd 4 1 720.2.x.d 4
120.q odd 4 1 3600.2.x.e 4
120.w even 4 1 720.2.x.d 4
120.w even 4 1 3600.2.x.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.n.b 4 40.e odd 2 1
80.2.n.b 4 40.f even 2 1
80.2.n.b 4 40.i odd 4 1
80.2.n.b 4 40.k even 4 1
320.2.n.i 4 5.b even 2 1
320.2.n.i 4 5.c odd 4 1
320.2.n.i 4 20.d odd 2 1
320.2.n.i 4 20.e even 4 1
400.2.n.b 4 8.b even 2 1
400.2.n.b 4 8.d odd 2 1
400.2.n.b 4 40.i odd 4 1
400.2.n.b 4 40.k even 4 1
720.2.x.d 4 120.i odd 2 1
720.2.x.d 4 120.m even 2 1
720.2.x.d 4 120.q odd 4 1
720.2.x.d 4 120.w even 4 1
1280.2.o.q 4 80.j even 4 1
1280.2.o.q 4 80.k odd 4 1
1280.2.o.q 4 80.q even 4 1
1280.2.o.q 4 80.t odd 4 1
1280.2.o.r 4 80.i odd 4 1
1280.2.o.r 4 80.k odd 4 1
1280.2.o.r 4 80.q even 4 1
1280.2.o.r 4 80.s even 4 1
1600.2.n.r 4 1.a even 1 1 trivial
1600.2.n.r 4 4.b odd 2 1 inner
1600.2.n.r 4 5.c odd 4 1 inner
1600.2.n.r 4 20.e even 4 1 inner
3600.2.x.e 4 24.f even 2 1
3600.2.x.e 4 24.h odd 2 1
3600.2.x.e 4 120.q odd 4 1
3600.2.x.e 4 120.w even 4 1

Hecke kernels

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

\( T_{3}^{4} + 36 \) Copy content Toggle raw display
\( T_{7}^{4} + 36 \) Copy content Toggle raw display
\( T_{11}^{2} + 12 \) Copy content Toggle raw display
\( T_{13}^{2} - 2T_{13} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 36 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 36 \) Copy content Toggle raw display
$11$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 36 \) Copy content Toggle raw display
$29$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 10 T + 50)^{2} \) Copy content Toggle raw display
$41$ \( (T - 2)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 36 \) Copy content Toggle raw display
$47$ \( T^{4} + 36 \) Copy content Toggle raw display
$53$ \( (T^{2} + 14 T + 98)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$61$ \( (T + 6)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 2916 \) Copy content Toggle raw display
$71$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 14 T + 98)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 86436 \) Copy content Toggle raw display
$89$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 14 T + 98)^{2} \) Copy content Toggle raw display
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