# Properties

 Label 400.6.a.t Level 400 Weight 6 Character orbit 400.a Self dual yes Analytic conductor 64.154 Analytic rank 1 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 400.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.1535279252$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{11})$$ Defining polynomial: $$x^{2} - 11$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 5) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{11}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 \beta q^{3} -9 \beta q^{7} + 153 q^{9} +O(q^{10})$$ $$q + 3 \beta q^{3} -9 \beta q^{7} + 153 q^{9} -252 q^{11} -18 \beta q^{13} + 104 \beta q^{17} -220 q^{19} -1188 q^{21} -367 \beta q^{23} -270 \beta q^{27} + 6930 q^{29} -6752 q^{31} -756 \beta q^{33} -2106 \beta q^{37} -2376 q^{39} -198 q^{41} + 63 \beta q^{43} -1589 \beta q^{47} -13243 q^{49} + 13728 q^{51} -878 \beta q^{53} -660 \beta q^{57} -24660 q^{59} -5698 q^{61} -1377 \beta q^{63} -6579 \beta q^{67} -48444 q^{69} -53352 q^{71} + 10692 \beta q^{73} + 2268 \beta q^{77} + 51920 q^{79} -72819 q^{81} + 9323 \beta q^{83} + 20790 \beta q^{87} + 9990 q^{89} + 7128 q^{91} -20256 \beta q^{93} + 15264 \beta q^{97} -38556 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 306q^{9} + O(q^{10})$$ $$2q + 306q^{9} - 504q^{11} - 440q^{19} - 2376q^{21} + 13860q^{29} - 13504q^{31} - 4752q^{39} - 396q^{41} - 26486q^{49} + 27456q^{51} - 49320q^{59} - 11396q^{61} - 96888q^{69} - 106704q^{71} + 103840q^{79} - 145638q^{81} + 19980q^{89} + 14256q^{91} - 77112q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −3.31662 3.31662
0 −19.8997 0 0 0 59.6992 0 153.000 0
1.2 0 19.8997 0 0 0 −59.6992 0 153.000 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.6.a.t 2
4.b odd 2 1 25.6.a.c 2
5.b even 2 1 inner 400.6.a.t 2
5.c odd 4 2 80.6.c.a 2
12.b even 2 1 225.6.a.n 2
15.e even 4 2 720.6.f.f 2
20.d odd 2 1 25.6.a.c 2
20.e even 4 2 5.6.b.a 2
40.i odd 4 2 320.6.c.g 2
40.k even 4 2 320.6.c.f 2
60.h even 2 1 225.6.a.n 2
60.l odd 4 2 45.6.b.b 2
140.j odd 4 2 245.6.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.6.b.a 2 20.e even 4 2
25.6.a.c 2 4.b odd 2 1
25.6.a.c 2 20.d odd 2 1
45.6.b.b 2 60.l odd 4 2
80.6.c.a 2 5.c odd 4 2
225.6.a.n 2 12.b even 2 1
225.6.a.n 2 60.h even 2 1
245.6.b.a 2 140.j odd 4 2
320.6.c.f 2 40.k even 4 2
320.6.c.g 2 40.i odd 4 2
400.6.a.t 2 1.a even 1 1 trivial
400.6.a.t 2 5.b even 2 1 inner
720.6.f.f 2 15.e even 4 2

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 396$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(400))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 90 T^{2} + 59049 T^{4}$$
$5$ 1
$7$ $$1 + 30050 T^{2} + 282475249 T^{4}$$
$11$ $$( 1 + 252 T + 161051 T^{2} )^{2}$$
$13$ $$1 + 728330 T^{2} + 137858491849 T^{4}$$
$17$ $$1 + 2363810 T^{2} + 2015993900449 T^{4}$$
$19$ $$( 1 + 220 T + 2476099 T^{2} )^{2}$$
$23$ $$1 + 6946370 T^{2} + 41426511213649 T^{4}$$
$29$ $$( 1 - 6930 T + 20511149 T^{2} )^{2}$$
$31$ $$( 1 + 6752 T + 28629151 T^{2} )^{2}$$
$37$ $$1 - 56462470 T^{2} + 4808584372417849 T^{4}$$
$41$ $$( 1 + 198 T + 115856201 T^{2} )^{2}$$
$43$ $$1 + 293842250 T^{2} + 21611482313284249 T^{4}$$
$47$ $$1 + 347593490 T^{2} + 52599132235830049 T^{4}$$
$53$ $$1 + 802472090 T^{2} + 174887470365513049 T^{4}$$
$59$ $$( 1 + 24660 T + 714924299 T^{2} )^{2}$$
$61$ $$( 1 + 5698 T + 844596301 T^{2} )^{2}$$
$67$ $$1 + 795787610 T^{2} + 1822837804551761449 T^{4}$$
$71$ $$( 1 + 53352 T + 1804229351 T^{2} )^{2}$$
$73$ $$1 - 883886830 T^{2} + 4297625829703557649 T^{4}$$
$79$ $$( 1 - 51920 T + 3077056399 T^{2} )^{2}$$
$83$ $$1 + 4053674810 T^{2} + 15516041187205853449 T^{4}$$
$89$ $$( 1 - 9990 T + 5584059449 T^{2} )^{2}$$
$97$ $$1 + 6923133890 T^{2} + 73742412689492826049 T^{4}$$