# 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 $4$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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