Properties

Label 400.6.n.a
Level $400$
Weight $6$
Character orbit 400.n
Analytic conductor $64.154$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 400.n (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(64.1535279252\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -243 i q^{9} +O(q^{10})\) \( q -243 i q^{9} + ( -475 + 475 i ) q^{13} + ( 1525 + 1525 i ) q^{17} -8564 i q^{29} + ( 475 + 475 i ) q^{37} -4952 q^{41} + 16807 i q^{49} + ( 16475 - 16475 i ) q^{53} + 54948 q^{61} + ( 54475 - 54475 i ) q^{73} -59049 q^{81} -140464 i q^{89} + ( 126475 + 126475 i ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q - 950q^{13} + 3050q^{17} + 950q^{37} - 9904q^{41} + 32950q^{53} + 109896q^{61} + 108950q^{73} - 118098q^{81} + 252950q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(i\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1
1.00000i
1.00000i
0 0 0 0 0 0 0 243.000i 0
207.1 0 0 0 0 0 0 0 243.000i 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 CM by \(\Q(\sqrt{-1}) \)
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 400.6.n.a 2
4.b odd 2 1 CM 400.6.n.a 2
5.b even 2 1 80.6.n.a 2
5.c odd 4 1 80.6.n.a 2
5.c odd 4 1 inner 400.6.n.a 2
20.d odd 2 1 80.6.n.a 2
20.e even 4 1 80.6.n.a 2
20.e even 4 1 inner 400.6.n.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.6.n.a 2 5.b even 2 1
80.6.n.a 2 5.c odd 4 1
80.6.n.a 2 20.d odd 2 1
80.6.n.a 2 20.e even 4 1
400.6.n.a 2 1.a even 1 1 trivial
400.6.n.a 2 4.b odd 2 1 CM
400.6.n.a 2 5.c odd 4 1 inner
400.6.n.a 2 20.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{6}^{\mathrm{new}}(400, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 59049 T^{4} \)
$5$ 1
$7$ \( 1 + 282475249 T^{4} \)
$11$ \( ( 1 - 161051 T^{2} )^{2} \)
$13$ \( ( 1 - 244 T + 371293 T^{2} )( 1 + 1194 T + 371293 T^{2} ) \)
$17$ \( ( 1 - 2242 T + 1419857 T^{2} )( 1 - 808 T + 1419857 T^{2} ) \)
$19$ \( ( 1 + 2476099 T^{2} )^{2} \)
$23$ \( 1 + 41426511213649 T^{4} \)
$29$ \( ( 1 - 2950 T + 20511149 T^{2} )( 1 + 2950 T + 20511149 T^{2} ) \)
$31$ \( ( 1 - 28629151 T^{2} )^{2} \)
$37$ \( ( 1 - 12242 T + 69343957 T^{2} )( 1 + 11292 T + 69343957 T^{2} ) \)
$41$ \( ( 1 + 4952 T + 115856201 T^{2} )^{2} \)
$43$ \( 1 + 21611482313284249 T^{4} \)
$47$ \( 1 + 52599132235830049 T^{4} \)
$53$ \( ( 1 - 40244 T + 418195493 T^{2} )( 1 + 7294 T + 418195493 T^{2} ) \)
$59$ \( ( 1 + 714924299 T^{2} )^{2} \)
$61$ \( ( 1 - 54948 T + 844596301 T^{2} )^{2} \)
$67$ \( 1 + 1822837804551761449 T^{4} \)
$71$ \( ( 1 - 1804229351 T^{2} )^{2} \)
$73$ \( ( 1 - 88806 T + 2073071593 T^{2} )( 1 - 20144 T + 2073071593 T^{2} ) \)
$79$ \( ( 1 + 3077056399 T^{2} )^{2} \)
$83$ \( 1 + 15516041187205853449 T^{4} \)
$89$ \( ( 1 - 51050 T + 5584059449 T^{2} )( 1 + 51050 T + 5584059449 T^{2} ) \)
$97$ \( ( 1 - 160808 T + 8587340257 T^{2} )( 1 - 92142 T + 8587340257 T^{2} ) \)
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