# Properties

 Label 400.3.p.b Level 400 Weight 3 Character orbit 400.p Analytic conductor 10.899 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 400.p (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.8992105744$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 10) 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.

 $$f(q)$$ $$=$$ $$q + ( -2 + 2 i ) q^{3} + ( 2 + 2 i ) q^{7} + i q^{9} +O(q^{10})$$ $$q + ( -2 + 2 i ) q^{3} + ( 2 + 2 i ) q^{7} + i q^{9} + 8 q^{11} + ( -3 + 3 i ) q^{13} + ( -7 - 7 i ) q^{17} + 20 i q^{19} -8 q^{21} + ( -2 + 2 i ) q^{23} + ( -20 - 20 i ) q^{27} + 40 i q^{29} -52 q^{31} + ( -16 + 16 i ) q^{33} + ( 3 + 3 i ) q^{37} -12 i q^{39} -8 q^{41} + ( -42 + 42 i ) q^{43} + ( -18 - 18 i ) q^{47} -41 i q^{49} + 28 q^{51} + ( -53 + 53 i ) q^{53} + ( -40 - 40 i ) q^{57} + 20 i q^{59} -48 q^{61} + ( -2 + 2 i ) q^{63} + ( 62 + 62 i ) q^{67} -8 i q^{69} + 28 q^{71} + ( 47 - 47 i ) q^{73} + ( 16 + 16 i ) q^{77} + 71 q^{81} + ( 18 - 18 i ) q^{83} + ( -80 - 80 i ) q^{87} + 80 i q^{89} -12 q^{91} + ( 104 - 104 i ) q^{93} + ( 63 + 63 i ) q^{97} + 8 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{3} + 4q^{7} + O(q^{10})$$ $$2q - 4q^{3} + 4q^{7} + 16q^{11} - 6q^{13} - 14q^{17} - 16q^{21} - 4q^{23} - 40q^{27} - 104q^{31} - 32q^{33} + 6q^{37} - 16q^{41} - 84q^{43} - 36q^{47} + 56q^{51} - 106q^{53} - 80q^{57} - 96q^{61} - 4q^{63} + 124q^{67} + 56q^{71} + 94q^{73} + 32q^{77} + 142q^{81} + 36q^{83} - 160q^{87} - 24q^{91} + 208q^{93} + 126q^{97} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 − 1.00000i 1.00000i
0 −2.00000 2.00000i 0 0 0 2.00000 2.00000i 0 1.00000i 0
257.1 0 −2.00000 + 2.00000i 0 0 0 2.00000 + 2.00000i 0 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
5.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.3.p.b 2
4.b odd 2 1 50.3.c.c 2
5.b even 2 1 80.3.p.c 2
5.c odd 4 1 80.3.p.c 2
5.c odd 4 1 inner 400.3.p.b 2
12.b even 2 1 450.3.g.b 2
15.d odd 2 1 720.3.bh.c 2
15.e even 4 1 720.3.bh.c 2
20.d odd 2 1 10.3.c.a 2
20.e even 4 1 10.3.c.a 2
20.e even 4 1 50.3.c.c 2
40.e odd 2 1 320.3.p.h 2
40.f even 2 1 320.3.p.a 2
40.i odd 4 1 320.3.p.a 2
40.k even 4 1 320.3.p.h 2
60.h even 2 1 90.3.g.b 2
60.l odd 4 1 90.3.g.b 2
60.l odd 4 1 450.3.g.b 2
140.c even 2 1 490.3.f.b 2
140.j odd 4 1 490.3.f.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.3.c.a 2 20.d odd 2 1
10.3.c.a 2 20.e even 4 1
50.3.c.c 2 4.b odd 2 1
50.3.c.c 2 20.e even 4 1
80.3.p.c 2 5.b even 2 1
80.3.p.c 2 5.c odd 4 1
90.3.g.b 2 60.h even 2 1
90.3.g.b 2 60.l odd 4 1
320.3.p.a 2 40.f even 2 1
320.3.p.a 2 40.i odd 4 1
320.3.p.h 2 40.e odd 2 1
320.3.p.h 2 40.k even 4 1
400.3.p.b 2 1.a even 1 1 trivial
400.3.p.b 2 5.c odd 4 1 inner
450.3.g.b 2 12.b even 2 1
450.3.g.b 2 60.l odd 4 1
490.3.f.b 2 140.c even 2 1
490.3.f.b 2 140.j odd 4 1
720.3.bh.c 2 15.d odd 2 1
720.3.bh.c 2 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(400, [\chi])$$:

 $$T_{3}^{2} + 4 T_{3} + 8$$ $$T_{7}^{2} - 4 T_{7} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 4 T + 8 T^{2} + 36 T^{3} + 81 T^{4}$$
$5$ 1
$7$ $$1 - 4 T + 8 T^{2} - 196 T^{3} + 2401 T^{4}$$
$11$ $$( 1 - 8 T + 121 T^{2} )^{2}$$
$13$ $$1 + 6 T + 18 T^{2} + 1014 T^{3} + 28561 T^{4}$$
$17$ $$( 1 - 16 T + 289 T^{2} )( 1 + 30 T + 289 T^{2} )$$
$19$ $$1 - 322 T^{2} + 130321 T^{4}$$
$23$ $$1 + 4 T + 8 T^{2} + 2116 T^{3} + 279841 T^{4}$$
$29$ $$( 1 - 42 T + 841 T^{2} )( 1 + 42 T + 841 T^{2} )$$
$31$ $$( 1 + 52 T + 961 T^{2} )^{2}$$
$37$ $$1 - 6 T + 18 T^{2} - 8214 T^{3} + 1874161 T^{4}$$
$41$ $$( 1 + 8 T + 1681 T^{2} )^{2}$$
$43$ $$1 + 84 T + 3528 T^{2} + 155316 T^{3} + 3418801 T^{4}$$
$47$ $$1 + 36 T + 648 T^{2} + 79524 T^{3} + 4879681 T^{4}$$
$53$ $$( 1 + 53 T )^{2}( 1 + 2809 T^{2} )$$
$59$ $$1 - 6562 T^{2} + 12117361 T^{4}$$
$61$ $$( 1 + 48 T + 3721 T^{2} )^{2}$$
$67$ $$1 - 124 T + 7688 T^{2} - 556636 T^{3} + 20151121 T^{4}$$
$71$ $$( 1 - 28 T + 5041 T^{2} )^{2}$$
$73$ $$1 - 94 T + 4418 T^{2} - 500926 T^{3} + 28398241 T^{4}$$
$79$ $$( 1 - 79 T )^{2}( 1 + 79 T )^{2}$$
$83$ $$1 - 36 T + 648 T^{2} - 248004 T^{3} + 47458321 T^{4}$$
$89$ $$1 - 9442 T^{2} + 62742241 T^{4}$$
$97$ $$1 - 126 T + 7938 T^{2} - 1185534 T^{3} + 88529281 T^{4}$$