# Properties

 Label 400.2.q.b Level $400$ Weight $2$ Character orbit 400.q Analytic conductor $3.194$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.19401608085$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 16) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 - i ) q^{2} + ( 1 + i ) q^{3} -2 i q^{4} + 2 q^{6} + 2 q^{7} + ( -2 - 2 i ) q^{8} -i q^{9} +O(q^{10})$$ $$q + ( 1 - i ) q^{2} + ( 1 + i ) q^{3} -2 i q^{4} + 2 q^{6} + 2 q^{7} + ( -2 - 2 i ) q^{8} -i q^{9} + ( 1 + i ) q^{11} + ( 2 - 2 i ) q^{12} + ( 1 + i ) q^{13} + ( 2 - 2 i ) q^{14} -4 q^{16} -2 i q^{17} + ( -1 - i ) q^{18} + ( -3 + 3 i ) q^{19} + ( 2 + 2 i ) q^{21} + 2 q^{22} + 6 q^{23} -4 i q^{24} + 2 q^{26} + ( 4 - 4 i ) q^{27} -4 i q^{28} + ( -3 + 3 i ) q^{29} -8 q^{31} + ( -4 + 4 i ) q^{32} + 2 i q^{33} + ( -2 - 2 i ) q^{34} -2 q^{36} + ( -3 + 3 i ) q^{37} + 6 i q^{38} + 2 i q^{39} + 4 q^{42} + ( 5 - 5 i ) q^{43} + ( 2 - 2 i ) q^{44} + ( 6 - 6 i ) q^{46} + 8 i q^{47} + ( -4 - 4 i ) q^{48} -3 q^{49} + ( 2 - 2 i ) q^{51} + ( 2 - 2 i ) q^{52} + ( -5 + 5 i ) q^{53} -8 i q^{54} + ( -4 - 4 i ) q^{56} -6 q^{57} + 6 i q^{58} + ( 3 + 3 i ) q^{59} + ( -9 + 9 i ) q^{61} + ( -8 + 8 i ) q^{62} -2 i q^{63} + 8 i q^{64} + ( 2 + 2 i ) q^{66} + ( -5 - 5 i ) q^{67} -4 q^{68} + ( 6 + 6 i ) q^{69} -10 i q^{71} + ( -2 + 2 i ) q^{72} -4 q^{73} + 6 i q^{74} + ( 6 + 6 i ) q^{76} + ( 2 + 2 i ) q^{77} + ( 2 + 2 i ) q^{78} + 5 q^{81} + ( 1 + i ) q^{83} + ( 4 - 4 i ) q^{84} -10 i q^{86} -6 q^{87} -4 i q^{88} -4 i q^{89} + ( 2 + 2 i ) q^{91} -12 i q^{92} + ( -8 - 8 i ) q^{93} + ( 8 + 8 i ) q^{94} -8 q^{96} -2 i q^{97} + ( -3 + 3 i ) q^{98} + ( 1 - i ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + 2q^{3} + 4q^{6} + 4q^{7} - 4q^{8} + O(q^{10})$$ $$2q + 2q^{2} + 2q^{3} + 4q^{6} + 4q^{7} - 4q^{8} + 2q^{11} + 4q^{12} + 2q^{13} + 4q^{14} - 8q^{16} - 2q^{18} - 6q^{19} + 4q^{21} + 4q^{22} + 12q^{23} + 4q^{26} + 8q^{27} - 6q^{29} - 16q^{31} - 8q^{32} - 4q^{34} - 4q^{36} - 6q^{37} + 8q^{42} + 10q^{43} + 4q^{44} + 12q^{46} - 8q^{48} - 6q^{49} + 4q^{51} + 4q^{52} - 10q^{53} - 8q^{56} - 12q^{57} + 6q^{59} - 18q^{61} - 16q^{62} + 4q^{66} - 10q^{67} - 8q^{68} + 12q^{69} - 4q^{72} - 8q^{73} + 12q^{76} + 4q^{77} + 4q^{78} + 10q^{81} + 2q^{83} + 8q^{84} - 12q^{87} + 4q^{91} - 16q^{93} + 16q^{94} - 16q^{96} - 6q^{98} + 2q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1
 1.00000i − 1.00000i
1.00000 1.00000i 1.00000 + 1.00000i 2.00000i 0 2.00000 2.00000 −2.00000 2.00000i 1.00000i 0
349.1 1.00000 + 1.00000i 1.00000 1.00000i 2.00000i 0 2.00000 2.00000 −2.00000 + 2.00000i 1.00000i 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
80.q even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.2.q.b 2
4.b odd 2 1 1600.2.q.a 2
5.b even 2 1 400.2.q.a 2
5.c odd 4 1 16.2.e.a 2
5.c odd 4 1 400.2.l.c 2
15.e even 4 1 144.2.k.a 2
16.e even 4 1 400.2.q.a 2
16.f odd 4 1 1600.2.q.b 2
20.d odd 2 1 1600.2.q.b 2
20.e even 4 1 64.2.e.a 2
20.e even 4 1 1600.2.l.a 2
35.f even 4 1 784.2.m.b 2
35.k even 12 2 784.2.x.c 4
35.l odd 12 2 784.2.x.f 4
40.i odd 4 1 128.2.e.b 2
40.k even 4 1 128.2.e.a 2
60.l odd 4 1 576.2.k.a 2
80.i odd 4 1 16.2.e.a 2
80.j even 4 1 128.2.e.a 2
80.j even 4 1 1600.2.l.a 2
80.k odd 4 1 1600.2.q.a 2
80.q even 4 1 inner 400.2.q.b 2
80.s even 4 1 64.2.e.a 2
80.t odd 4 1 128.2.e.b 2
80.t odd 4 1 400.2.l.c 2
120.q odd 4 1 1152.2.k.a 2
120.w even 4 1 1152.2.k.b 2
160.u even 8 2 1024.2.b.b 2
160.v odd 8 2 1024.2.a.b 2
160.ba even 8 2 1024.2.a.e 2
160.bb odd 8 2 1024.2.b.e 2
240.z odd 4 1 576.2.k.a 2
240.bb even 4 1 144.2.k.a 2
240.bd odd 4 1 1152.2.k.a 2
240.bf even 4 1 1152.2.k.b 2
480.br even 8 2 9216.2.a.d 2
480.ca odd 8 2 9216.2.a.s 2
560.bn even 4 1 784.2.m.b 2
560.ch even 12 2 784.2.x.c 4
560.cy odd 12 2 784.2.x.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.2.e.a 2 5.c odd 4 1
16.2.e.a 2 80.i odd 4 1
64.2.e.a 2 20.e even 4 1
64.2.e.a 2 80.s even 4 1
128.2.e.a 2 40.k even 4 1
128.2.e.a 2 80.j even 4 1
128.2.e.b 2 40.i odd 4 1
128.2.e.b 2 80.t odd 4 1
144.2.k.a 2 15.e even 4 1
144.2.k.a 2 240.bb even 4 1
400.2.l.c 2 5.c odd 4 1
400.2.l.c 2 80.t odd 4 1
400.2.q.a 2 5.b even 2 1
400.2.q.a 2 16.e even 4 1
400.2.q.b 2 1.a even 1 1 trivial
400.2.q.b 2 80.q even 4 1 inner
576.2.k.a 2 60.l odd 4 1
576.2.k.a 2 240.z odd 4 1
784.2.m.b 2 35.f even 4 1
784.2.m.b 2 560.bn even 4 1
784.2.x.c 4 35.k even 12 2
784.2.x.c 4 560.ch even 12 2
784.2.x.f 4 35.l odd 12 2
784.2.x.f 4 560.cy odd 12 2
1024.2.a.b 2 160.v odd 8 2
1024.2.a.e 2 160.ba even 8 2
1024.2.b.b 2 160.u even 8 2
1024.2.b.e 2 160.bb odd 8 2
1152.2.k.a 2 120.q odd 4 1
1152.2.k.a 2 240.bd odd 4 1
1152.2.k.b 2 120.w even 4 1
1152.2.k.b 2 240.bf even 4 1
1600.2.l.a 2 20.e even 4 1
1600.2.l.a 2 80.j even 4 1
1600.2.q.a 2 4.b odd 2 1
1600.2.q.a 2 80.k odd 4 1
1600.2.q.b 2 16.f odd 4 1
1600.2.q.b 2 20.d odd 2 1
9216.2.a.d 2 480.br even 8 2
9216.2.a.s 2 480.ca odd 8 2

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 - 2 T + T^{2}$$
$3$ $$2 - 2 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$( -2 + T )^{2}$$
$11$ $$2 - 2 T + T^{2}$$
$13$ $$2 - 2 T + T^{2}$$
$17$ $$4 + T^{2}$$
$19$ $$18 + 6 T + T^{2}$$
$23$ $$( -6 + T )^{2}$$
$29$ $$18 + 6 T + T^{2}$$
$31$ $$( 8 + T )^{2}$$
$37$ $$18 + 6 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$50 - 10 T + T^{2}$$
$47$ $$64 + T^{2}$$
$53$ $$50 + 10 T + T^{2}$$
$59$ $$18 - 6 T + T^{2}$$
$61$ $$162 + 18 T + T^{2}$$
$67$ $$50 + 10 T + T^{2}$$
$71$ $$100 + T^{2}$$
$73$ $$( 4 + T )^{2}$$
$79$ $$T^{2}$$
$83$ $$2 - 2 T + T^{2}$$
$89$ $$16 + T^{2}$$
$97$ $$4 + T^{2}$$
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