# Properties

 Label 400.6.c.j 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 5) 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.

 $$f(q)$$ $$=$$ $$q + 4 i q^{3} + 192 i q^{7} + 227 q^{9} +O(q^{10})$$ $$q + 4 i q^{3} + 192 i q^{7} + 227 q^{9} + 148 q^{11} + 286 i q^{13} + 1678 i q^{17} + 1060 q^{19} -768 q^{21} -2976 i q^{23} + 1880 i q^{27} + 3410 q^{29} + 2448 q^{31} + 592 i q^{33} -182 i q^{37} -1144 q^{39} -9398 q^{41} + 1244 i q^{43} -12088 i q^{47} -20057 q^{49} -6712 q^{51} + 23846 i q^{53} + 4240 i q^{57} -20020 q^{59} + 32302 q^{61} + 43584 i q^{63} + 60972 i q^{67} + 11904 q^{69} + 32648 q^{71} -38774 i q^{73} + 28416 i q^{77} -33360 q^{79} + 47641 q^{81} -16716 i q^{83} + 13640 i q^{87} -101370 q^{89} -54912 q^{91} + 9792 i q^{93} + 119038 i q^{97} + 33596 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 454 q^{9} + O(q^{10})$$ $$2 q + 454 q^{9} + 296 q^{11} + 2120 q^{19} - 1536 q^{21} + 6820 q^{29} + 4896 q^{31} - 2288 q^{39} - 18796 q^{41} - 40114 q^{49} - 13424 q^{51} - 40040 q^{59} + 64604 q^{61} + 23808 q^{69} + 65296 q^{71} - 66720 q^{79} + 95282 q^{81} - 202740 q^{89} - 109824 q^{91} + 67192 q^{99} + O(q^{100})$$

## 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 4.00000i 0 0 0 192.000i 0 227.000 0
49.2 0 4.00000i 0 0 0 192.000i 0 227.000 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.j 2
4.b odd 2 1 25.6.b.a 2
5.b even 2 1 inner 400.6.c.j 2
5.c odd 4 1 80.6.a.e 1
5.c odd 4 1 400.6.a.g 1
12.b even 2 1 225.6.b.e 2
15.e even 4 1 720.6.a.a 1
20.d odd 2 1 25.6.b.a 2
20.e even 4 1 5.6.a.a 1
20.e even 4 1 25.6.a.a 1
40.i odd 4 1 320.6.a.g 1
40.k even 4 1 320.6.a.j 1
60.h even 2 1 225.6.b.e 2
60.l odd 4 1 45.6.a.b 1
60.l odd 4 1 225.6.a.f 1
140.j odd 4 1 245.6.a.b 1
220.i odd 4 1 605.6.a.a 1
260.p even 4 1 845.6.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.6.a.a 1 20.e even 4 1
25.6.a.a 1 20.e even 4 1
25.6.b.a 2 4.b odd 2 1
25.6.b.a 2 20.d odd 2 1
45.6.a.b 1 60.l odd 4 1
80.6.a.e 1 5.c odd 4 1
225.6.a.f 1 60.l odd 4 1
225.6.b.e 2 12.b even 2 1
225.6.b.e 2 60.h even 2 1
245.6.a.b 1 140.j odd 4 1
320.6.a.g 1 40.i odd 4 1
320.6.a.j 1 40.k even 4 1
400.6.a.g 1 5.c odd 4 1
400.6.c.j 2 1.a even 1 1 trivial
400.6.c.j 2 5.b even 2 1 inner
605.6.a.a 1 220.i odd 4 1
720.6.a.a 1 15.e even 4 1
845.6.a.b 1 260.p even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 16$$ acting on $$S_{6}^{\mathrm{new}}(400, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$16 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$36864 + T^{2}$$
$11$ $$( -148 + T )^{2}$$
$13$ $$81796 + T^{2}$$
$17$ $$2815684 + T^{2}$$
$19$ $$( -1060 + T )^{2}$$
$23$ $$8856576 + T^{2}$$
$29$ $$( -3410 + T )^{2}$$
$31$ $$( -2448 + T )^{2}$$
$37$ $$33124 + T^{2}$$
$41$ $$( 9398 + T )^{2}$$
$43$ $$1547536 + T^{2}$$
$47$ $$146119744 + T^{2}$$
$53$ $$568631716 + T^{2}$$
$59$ $$( 20020 + T )^{2}$$
$61$ $$( -32302 + T )^{2}$$
$67$ $$3717584784 + T^{2}$$
$71$ $$( -32648 + T )^{2}$$
$73$ $$1503423076 + T^{2}$$
$79$ $$( 33360 + T )^{2}$$
$83$ $$279424656 + T^{2}$$
$89$ $$( 101370 + T )^{2}$$
$97$ $$14170045444 + T^{2}$$