# Properties

 Label 400.8.c.b Level $400$ Weight $8$ Character orbit 400.c Analytic conductor $124.954$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 400.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$124.954010194$$ 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 8) 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.

 $$f(q)$$ $$=$$ $$q + 84 i q^{3} -456 i q^{7} -4869 q^{9} +O(q^{10})$$ $$q + 84 i q^{3} -456 i q^{7} -4869 q^{9} + 2524 q^{11} -10778 i q^{13} + 11150 i q^{17} + 4124 q^{19} + 38304 q^{21} -81704 i q^{23} -225288 i q^{27} -99798 q^{29} + 40480 q^{31} + 212016 i q^{33} + 419442 i q^{37} + 905352 q^{39} + 141402 q^{41} + 690428 i q^{43} -682032 i q^{47} + 615607 q^{49} -936600 q^{51} + 1813118 i q^{53} + 346416 i q^{57} -966028 q^{59} + 1887670 q^{61} + 2220264 i q^{63} + 2965868 i q^{67} + 6863136 q^{69} + 2548232 q^{71} -1680326 i q^{73} -1150944 i q^{77} + 4038064 q^{79} + 8275689 q^{81} + 5385764 i q^{83} -8383032 i q^{87} + 6473046 q^{89} -4914768 q^{91} + 3400320 i q^{93} + 6065758 i q^{97} -12289356 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 9738q^{9} + O(q^{10})$$ $$2q - 9738q^{9} + 5048q^{11} + 8248q^{19} + 76608q^{21} - 199596q^{29} + 80960q^{31} + 1810704q^{39} + 282804q^{41} + 1231214q^{49} - 1873200q^{51} - 1932056q^{59} + 3775340q^{61} + 13726272q^{69} + 5096464q^{71} + 8076128q^{79} + 16551378q^{81} + 12946092q^{89} - 9829536q^{91} - 24578712q^{99} + O(q^{100})$$

## 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 84.0000i 0 0 0 456.000i 0 −4869.00 0
49.2 0 84.0000i 0 0 0 456.000i 0 −4869.00 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 400.8.c.b 2
4.b odd 2 1 200.8.c.a 2
5.b even 2 1 inner 400.8.c.b 2
5.c odd 4 1 16.8.a.c 1
5.c odd 4 1 400.8.a.b 1
15.e even 4 1 144.8.a.g 1
20.d odd 2 1 200.8.c.a 2
20.e even 4 1 8.8.a.a 1
20.e even 4 1 200.8.a.i 1
40.i odd 4 1 64.8.a.a 1
40.k even 4 1 64.8.a.g 1
60.l odd 4 1 72.8.a.d 1
80.i odd 4 1 256.8.b.c 2
80.j even 4 1 256.8.b.e 2
80.s even 4 1 256.8.b.e 2
80.t odd 4 1 256.8.b.c 2
120.q odd 4 1 576.8.a.j 1
120.w even 4 1 576.8.a.k 1
140.j odd 4 1 392.8.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.8.a.a 1 20.e even 4 1
16.8.a.c 1 5.c odd 4 1
64.8.a.a 1 40.i odd 4 1
64.8.a.g 1 40.k even 4 1
72.8.a.d 1 60.l odd 4 1
144.8.a.g 1 15.e even 4 1
200.8.a.i 1 20.e even 4 1
200.8.c.a 2 4.b odd 2 1
200.8.c.a 2 20.d odd 2 1
256.8.b.c 2 80.i odd 4 1
256.8.b.c 2 80.t odd 4 1
256.8.b.e 2 80.j even 4 1
256.8.b.e 2 80.s even 4 1
392.8.a.d 1 140.j odd 4 1
400.8.a.b 1 5.c odd 4 1
400.8.c.b 2 1.a even 1 1 trivial
400.8.c.b 2 5.b even 2 1 inner
576.8.a.j 1 120.q odd 4 1
576.8.a.k 1 120.w even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 7056$$ acting on $$S_{8}^{\mathrm{new}}(400, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$7056 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$207936 + T^{2}$$
$11$ $$( -2524 + T )^{2}$$
$13$ $$116165284 + T^{2}$$
$17$ $$124322500 + T^{2}$$
$19$ $$( -4124 + T )^{2}$$
$23$ $$6675543616 + T^{2}$$
$29$ $$( 99798 + T )^{2}$$
$31$ $$( -40480 + T )^{2}$$
$37$ $$175931591364 + T^{2}$$
$41$ $$( -141402 + T )^{2}$$
$43$ $$476690823184 + T^{2}$$
$47$ $$465167649024 + T^{2}$$
$53$ $$3287396881924 + T^{2}$$
$59$ $$( 966028 + T )^{2}$$
$61$ $$( -1887670 + T )^{2}$$
$67$ $$8796372993424 + T^{2}$$
$71$ $$( -2548232 + T )^{2}$$
$73$ $$2823495466276 + T^{2}$$
$79$ $$( -4038064 + T )^{2}$$
$83$ $$29006453863696 + T^{2}$$
$89$ $$( -6473046 + T )^{2}$$
$97$ $$36793420114564 + T^{2}$$