Properties

Label 1600.2.n.u
Level $1600$
Weight $2$
Character orbit 1600.n
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.n (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} - 4 \beta_1 q^{7} + 2 \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{3} - 4 \beta_1 q^{7} + 2 \beta_{3} q^{9} + 3 \beta_{4} q^{11} + 2 \beta_{2} q^{13} - 3 \beta_{7} q^{17} - \beta_{6} q^{19} - 4 q^{21} - 6 \beta_{5} q^{23} + 5 \beta_1 q^{27} + 6 \beta_{3} q^{29} - 2 \beta_{4} q^{31} + 3 \beta_{2} q^{33} + 2 \beta_{6} q^{39} - 3 q^{41} - 4 \beta_{5} q^{43} + 6 \beta_1 q^{47} + 9 \beta_{3} q^{49} - 3 \beta_{4} q^{51} + 6 \beta_{2} q^{53} + \beta_{7} q^{57} + 6 \beta_{6} q^{59} + 4 q^{61} + 8 \beta_{5} q^{63} - 7 \beta_1 q^{67} + 6 \beta_{3} q^{69} + 6 \beta_{4} q^{71} + 9 \beta_{2} q^{73} - 12 \beta_{7} q^{77} - q^{81} + 3 \beta_{5} q^{83} + 6 \beta_1 q^{87} + 3 \beta_{3} q^{89} - 8 \beta_{4} q^{91} - 2 \beta_{2} q^{93} - 4 \beta_{7} q^{97} - 6 \beta_{6} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 32 q^{21} - 24 q^{41} + 32 q^{61} - 8 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{5} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 2\zeta_{24}^{4} - 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{5} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{6} + 2\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 2\zeta_{24}^{7} - \zeta_{24}^{3} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{5} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{6} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta_{4} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{5} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{7} + \beta_1 ) / 2 \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1343.1
0.258819 + 0.965926i
−0.965926 0.258819i
−0.258819 0.965926i
0.965926 + 0.258819i
−0.965926 + 0.258819i
0.258819 0.965926i
0.965926 0.258819i
−0.258819 + 0.965926i
0 −0.707107 + 0.707107i 0 0 0 2.82843 + 2.82843i 0 2.00000i 0
1343.2 0 −0.707107 + 0.707107i 0 0 0 2.82843 + 2.82843i 0 2.00000i 0
1343.3 0 0.707107 0.707107i 0 0 0 −2.82843 2.82843i 0 2.00000i 0
1343.4 0 0.707107 0.707107i 0 0 0 −2.82843 2.82843i 0 2.00000i 0
1407.1 0 −0.707107 0.707107i 0 0 0 2.82843 2.82843i 0 2.00000i 0
1407.2 0 −0.707107 0.707107i 0 0 0 2.82843 2.82843i 0 2.00000i 0
1407.3 0 0.707107 + 0.707107i 0 0 0 −2.82843 + 2.82843i 0 2.00000i 0
1407.4 0 0.707107 + 0.707107i 0 0 0 −2.82843 + 2.82843i 0 2.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1407.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
20.d odd 2 1 inner
20.e 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.n.u 8
4.b odd 2 1 inner 1600.2.n.u 8
5.b even 2 1 inner 1600.2.n.u 8
5.c odd 4 2 inner 1600.2.n.u 8
8.b even 2 1 400.2.n.d 8
8.d odd 2 1 400.2.n.d 8
20.d odd 2 1 inner 1600.2.n.u 8
20.e even 4 2 inner 1600.2.n.u 8
24.f even 2 1 3600.2.x.m 8
24.h odd 2 1 3600.2.x.m 8
40.e odd 2 1 400.2.n.d 8
40.f even 2 1 400.2.n.d 8
40.i odd 4 2 400.2.n.d 8
40.k even 4 2 400.2.n.d 8
120.i odd 2 1 3600.2.x.m 8
120.m even 2 1 3600.2.x.m 8
120.q odd 4 2 3600.2.x.m 8
120.w even 4 2 3600.2.x.m 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
400.2.n.d 8 8.b even 2 1
400.2.n.d 8 8.d odd 2 1
400.2.n.d 8 40.e odd 2 1
400.2.n.d 8 40.f even 2 1
400.2.n.d 8 40.i odd 4 2
400.2.n.d 8 40.k even 4 2
1600.2.n.u 8 1.a even 1 1 trivial
1600.2.n.u 8 4.b odd 2 1 inner
1600.2.n.u 8 5.b even 2 1 inner
1600.2.n.u 8 5.c odd 4 2 inner
1600.2.n.u 8 20.d odd 2 1 inner
1600.2.n.u 8 20.e even 4 2 inner
3600.2.x.m 8 24.f even 2 1
3600.2.x.m 8 24.h odd 2 1
3600.2.x.m 8 120.i odd 2 1
3600.2.x.m 8 120.m even 2 1
3600.2.x.m 8 120.q odd 4 2
3600.2.x.m 8 120.w even 4 2

Hecke kernels

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

\( T_{3}^{4} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} + 256 \) Copy content Toggle raw display
\( T_{11}^{2} + 27 \) Copy content Toggle raw display
\( T_{13}^{4} + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 256)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 27)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 144)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 729)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 3)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 1296)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 36)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 12)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T + 3)^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} + 256)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 1296)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 11664)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 108)^{4} \) Copy content Toggle raw display
$61$ \( (T - 4)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} + 2401)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 108)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 59049)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} + 81)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 9)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 2304)^{2} \) Copy content Toggle raw display
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