# Properties

 Label 400.4.c.i Level $400$ Weight $4$ Character orbit 400.c Analytic conductor $23.601$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$23.6007640023$$ 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.

 $$f(q)$$ $$=$$ $$q + 4 i q^{3} + 24 i q^{7} + 11 q^{9} +O(q^{10})$$ $$q + 4 i q^{3} + 24 i q^{7} + 11 q^{9} + 44 q^{11} + 22 i q^{13} -50 i q^{17} + 44 q^{19} -96 q^{21} + 56 i q^{23} + 152 i q^{27} -198 q^{29} + 160 q^{31} + 176 i q^{33} + 162 i q^{37} -88 q^{39} -198 q^{41} -52 i q^{43} + 528 i q^{47} -233 q^{49} + 200 q^{51} -242 i q^{53} + 176 i q^{57} -668 q^{59} + 550 q^{61} + 264 i q^{63} + 188 i q^{67} -224 q^{69} -728 q^{71} + 154 i q^{73} + 1056 i q^{77} -656 q^{79} -311 q^{81} -236 i q^{83} -792 i q^{87} -714 q^{89} -528 q^{91} + 640 i q^{93} + 478 i q^{97} + 484 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 22q^{9} + O(q^{10})$$ $$2q + 22q^{9} + 88q^{11} + 88q^{19} - 192q^{21} - 396q^{29} + 320q^{31} - 176q^{39} - 396q^{41} - 466q^{49} + 400q^{51} - 1336q^{59} + 1100q^{61} - 448q^{69} - 1456q^{71} - 1312q^{79} - 622q^{81} - 1428q^{89} - 1056q^{91} + 968q^{99} + O(q^{100})$$

## 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 24.0000i 0 11.0000 0
49.2 0 4.00000i 0 0 0 24.0000i 0 11.0000 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 400.4.c.i 2
4.b odd 2 1 200.4.c.e 2
5.b even 2 1 inner 400.4.c.i 2
5.c odd 4 1 16.4.a.a 1
5.c odd 4 1 400.4.a.g 1
12.b even 2 1 1800.4.f.u 2
15.e even 4 1 144.4.a.e 1
20.d odd 2 1 200.4.c.e 2
20.e even 4 1 8.4.a.a 1
20.e even 4 1 200.4.a.g 1
35.f even 4 1 784.4.a.e 1
40.i odd 4 1 64.4.a.b 1
40.i odd 4 1 1600.4.a.bm 1
40.k even 4 1 64.4.a.d 1
40.k even 4 1 1600.4.a.o 1
55.e even 4 1 1936.4.a.l 1
60.h even 2 1 1800.4.f.u 2
60.l odd 4 1 72.4.a.c 1
60.l odd 4 1 1800.4.a.d 1
80.i odd 4 1 256.4.b.g 2
80.j even 4 1 256.4.b.a 2
80.s even 4 1 256.4.b.a 2
80.t odd 4 1 256.4.b.g 2
120.q odd 4 1 576.4.a.k 1
120.w even 4 1 576.4.a.j 1
140.j odd 4 1 392.4.a.e 1
140.w even 12 2 392.4.i.g 2
140.x odd 12 2 392.4.i.b 2
180.v odd 12 2 648.4.i.e 2
180.x even 12 2 648.4.i.h 2
220.i odd 4 1 968.4.a.a 1
260.p even 4 1 1352.4.a.a 1
340.r even 4 1 2312.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.4.a.a 1 20.e even 4 1
16.4.a.a 1 5.c odd 4 1
64.4.a.b 1 40.i odd 4 1
64.4.a.d 1 40.k even 4 1
72.4.a.c 1 60.l odd 4 1
144.4.a.e 1 15.e even 4 1
200.4.a.g 1 20.e even 4 1
200.4.c.e 2 4.b odd 2 1
200.4.c.e 2 20.d odd 2 1
256.4.b.a 2 80.j even 4 1
256.4.b.a 2 80.s even 4 1
256.4.b.g 2 80.i odd 4 1
256.4.b.g 2 80.t odd 4 1
392.4.a.e 1 140.j odd 4 1
392.4.i.b 2 140.x odd 12 2
392.4.i.g 2 140.w even 12 2
400.4.a.g 1 5.c odd 4 1
400.4.c.i 2 1.a even 1 1 trivial
400.4.c.i 2 5.b even 2 1 inner
576.4.a.j 1 120.w even 4 1
576.4.a.k 1 120.q odd 4 1
648.4.i.e 2 180.v odd 12 2
648.4.i.h 2 180.x even 12 2
784.4.a.e 1 35.f even 4 1
968.4.a.a 1 220.i odd 4 1
1352.4.a.a 1 260.p even 4 1
1600.4.a.o 1 40.k even 4 1
1600.4.a.bm 1 40.i odd 4 1
1800.4.a.d 1 60.l odd 4 1
1800.4.f.u 2 12.b even 2 1
1800.4.f.u 2 60.h even 2 1
1936.4.a.l 1 55.e even 4 1
2312.4.a.a 1 340.r even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(400, [\chi])$$:

 $$T_{3}^{2} + 16$$ $$T_{7}^{2} + 576$$ $$T_{11} - 44$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 38 T^{2} + 729 T^{4}$$
$5$ 1
$7$ $$1 - 110 T^{2} + 117649 T^{4}$$
$11$ $$( 1 - 44 T + 1331 T^{2} )^{2}$$
$13$ $$1 - 3910 T^{2} + 4826809 T^{4}$$
$17$ $$1 - 7326 T^{2} + 24137569 T^{4}$$
$19$ $$( 1 - 44 T + 6859 T^{2} )^{2}$$
$23$ $$1 - 21198 T^{2} + 148035889 T^{4}$$
$29$ $$( 1 + 198 T + 24389 T^{2} )^{2}$$
$31$ $$( 1 - 160 T + 29791 T^{2} )^{2}$$
$37$ $$1 - 75062 T^{2} + 2565726409 T^{4}$$
$41$ $$( 1 + 198 T + 68921 T^{2} )^{2}$$
$43$ $$1 - 156310 T^{2} + 6321363049 T^{4}$$
$47$ $$1 + 71138 T^{2} + 10779215329 T^{4}$$
$53$ $$1 - 239190 T^{2} + 22164361129 T^{4}$$
$59$ $$( 1 + 668 T + 205379 T^{2} )^{2}$$
$61$ $$( 1 - 550 T + 226981 T^{2} )^{2}$$
$67$ $$1 - 566182 T^{2} + 90458382169 T^{4}$$
$71$ $$( 1 + 728 T + 357911 T^{2} )^{2}$$
$73$ $$1 - 754318 T^{2} + 151334226289 T^{4}$$
$79$ $$( 1 + 656 T + 493039 T^{2} )^{2}$$
$83$ $$1 - 1087878 T^{2} + 326940373369 T^{4}$$
$89$ $$( 1 + 714 T + 704969 T^{2} )^{2}$$
$97$ $$1 - 1596862 T^{2} + 832972004929 T^{4}$$