# Properties

 Label 400.6.c.k Level 400 Weight 6 Character orbit 400.c Analytic conductor 64.154 Analytic rank 0 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.c (of order $$2$$, degree $$1$$, not 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 40) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{3} -62 i q^{7} + 239 q^{9} +O(q^{10})$$ $$q + 2 i q^{3} -62 i q^{7} + 239 q^{9} + 144 q^{11} -654 i q^{13} + 1190 i q^{17} + 556 q^{19} + 124 q^{21} -2182 i q^{23} + 964 i q^{27} + 1578 q^{29} -9660 q^{31} + 288 i q^{33} + 3534 i q^{37} + 1308 q^{39} + 7462 q^{41} + 7114 i q^{43} -28294 i q^{47} + 12963 q^{49} -2380 q^{51} -13046 i q^{53} + 1112 i q^{57} -37092 q^{59} + 39570 q^{61} -14818 i q^{63} -56734 i q^{67} + 4364 q^{69} -45588 q^{71} + 11842 i q^{73} -8928 i q^{77} + 94216 q^{79} + 56149 q^{81} + 31482 i q^{83} + 3156 i q^{87} + 94054 q^{89} -40548 q^{91} -19320 i q^{93} -23714 i q^{97} + 34416 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 478q^{9} + O(q^{10})$$ $$2q + 478q^{9} + 288q^{11} + 1112q^{19} + 248q^{21} + 3156q^{29} - 19320q^{31} + 2616q^{39} + 14924q^{41} + 25926q^{49} - 4760q^{51} - 74184q^{59} + 79140q^{61} + 8728q^{69} - 91176q^{71} + 188432q^{79} + 112298q^{81} + 188108q^{89} - 81096q^{91} + 68832q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 − 1.00000i 1.00000i
0 2.00000i 0 0 0 62.0000i 0 239.000 0
49.2 0 2.00000i 0 0 0 62.0000i 0 239.000 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.c.k 2
4.b odd 2 1 200.6.c.d 2
5.b even 2 1 inner 400.6.c.k 2
5.c odd 4 1 80.6.a.d 1
5.c odd 4 1 400.6.a.h 1
15.e even 4 1 720.6.a.t 1
20.d odd 2 1 200.6.c.d 2
20.e even 4 1 40.6.a.c 1
20.e even 4 1 200.6.a.b 1
40.i odd 4 1 320.6.a.h 1
40.k even 4 1 320.6.a.i 1
60.l odd 4 1 360.6.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.a.c 1 20.e even 4 1
80.6.a.d 1 5.c odd 4 1
200.6.a.b 1 20.e even 4 1
200.6.c.d 2 4.b odd 2 1
200.6.c.d 2 20.d odd 2 1
320.6.a.h 1 40.i odd 4 1
320.6.a.i 1 40.k even 4 1
360.6.a.f 1 60.l odd 4 1
400.6.a.h 1 5.c odd 4 1
400.6.c.k 2 1.a even 1 1 trivial
400.6.c.k 2 5.b even 2 1 inner
720.6.a.t 1 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 4$$ acting on $$S_{6}^{\mathrm{new}}(400, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 482 T^{2} + 59049 T^{4}$$
$5$ 1
$7$ $$1 - 29770 T^{2} + 282475249 T^{4}$$
$11$ $$( 1 - 144 T + 161051 T^{2} )^{2}$$
$13$ $$1 - 314870 T^{2} + 137858491849 T^{4}$$
$17$ $$1 - 1423614 T^{2} + 2015993900449 T^{4}$$
$19$ $$( 1 - 556 T + 2476099 T^{2} )^{2}$$
$23$ $$1 - 8111562 T^{2} + 41426511213649 T^{4}$$
$29$ $$( 1 - 1578 T + 20511149 T^{2} )^{2}$$
$31$ $$( 1 + 9660 T + 28629151 T^{2} )^{2}$$
$37$ $$1 - 126198758 T^{2} + 4808584372417849 T^{4}$$
$41$ $$( 1 - 7462 T + 115856201 T^{2} )^{2}$$
$43$ $$1 - 243407890 T^{2} + 21611482313284249 T^{4}$$
$47$ $$1 + 341860422 T^{2} + 52599132235830049 T^{4}$$
$53$ $$1 - 666192870 T^{2} + 174887470365513049 T^{4}$$
$59$ $$( 1 + 37092 T + 714924299 T^{2} )^{2}$$
$61$ $$( 1 - 39570 T + 844596301 T^{2} )^{2}$$
$67$ $$1 + 518496542 T^{2} + 1822837804551761449 T^{4}$$
$71$ $$( 1 + 45588 T + 1804229351 T^{2} )^{2}$$
$73$ $$1 - 4005910222 T^{2} + 4297625829703557649 T^{4}$$
$79$ $$( 1 - 94216 T + 3077056399 T^{2} )^{2}$$
$83$ $$1 - 6886964962 T^{2} + 15516041187205853449 T^{4}$$
$89$ $$( 1 - 94054 T + 5584059449 T^{2} )^{2}$$
$97$ $$1 - 16612326718 T^{2} + 73742412689492826049 T^{4}$$