# Properties

 Label 400.6.c.c 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 20) 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.

 $$f(q)$$ $$=$$ $$q + 22 i q^{3} -218 i q^{7} -241 q^{9} +O(q^{10})$$ $$q + 22 i q^{3} -218 i q^{7} -241 q^{9} + 480 q^{11} + 622 i q^{13} + 186 i q^{17} -1204 q^{19} + 4796 q^{21} -3186 i q^{23} + 44 i q^{27} -5526 q^{29} -9356 q^{31} + 10560 i q^{33} + 5618 i q^{37} -13684 q^{39} -14394 q^{41} -370 i q^{43} -16146 i q^{47} -30717 q^{49} -4092 q^{51} + 4374 i q^{53} -26488 i q^{57} -11748 q^{59} + 13202 q^{61} + 52538 i q^{63} + 11542 i q^{67} + 70092 q^{69} + 29532 q^{71} -33698 i q^{73} -104640 i q^{77} + 31208 q^{79} -59531 q^{81} -38466 i q^{83} -121572 i q^{87} -119514 q^{89} + 135596 q^{91} -205832 i q^{93} + 94658 i q^{97} -115680 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 482q^{9} + O(q^{10})$$ $$2q - 482q^{9} + 960q^{11} - 2408q^{19} + 9592q^{21} - 11052q^{29} - 18712q^{31} - 27368q^{39} - 28788q^{41} - 61434q^{49} - 8184q^{51} - 23496q^{59} + 26404q^{61} + 140184q^{69} + 59064q^{71} + 62416q^{79} - 119062q^{81} - 239028q^{89} + 271192q^{91} - 231360q^{99} + O(q^{100})$$

## 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 22.0000i 0 0 0 218.000i 0 −241.000 0
49.2 0 22.0000i 0 0 0 218.000i 0 −241.000 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.c 2
4.b odd 2 1 100.6.c.a 2
5.b even 2 1 inner 400.6.c.c 2
5.c odd 4 1 80.6.a.b 1
5.c odd 4 1 400.6.a.m 1
12.b even 2 1 900.6.d.h 2
15.e even 4 1 720.6.a.l 1
20.d odd 2 1 100.6.c.a 2
20.e even 4 1 20.6.a.a 1
20.e even 4 1 100.6.a.a 1
40.i odd 4 1 320.6.a.n 1
40.k even 4 1 320.6.a.c 1
60.h even 2 1 900.6.d.h 2
60.l odd 4 1 180.6.a.e 1
60.l odd 4 1 900.6.a.b 1
140.j odd 4 1 980.6.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.6.a.a 1 20.e even 4 1
80.6.a.b 1 5.c odd 4 1
100.6.a.a 1 20.e even 4 1
100.6.c.a 2 4.b odd 2 1
100.6.c.a 2 20.d odd 2 1
180.6.a.e 1 60.l odd 4 1
320.6.a.c 1 40.k even 4 1
320.6.a.n 1 40.i odd 4 1
400.6.a.m 1 5.c odd 4 1
400.6.c.c 2 1.a even 1 1 trivial
400.6.c.c 2 5.b even 2 1 inner
720.6.a.l 1 15.e even 4 1
900.6.a.b 1 60.l odd 4 1
900.6.d.h 2 12.b even 2 1
900.6.d.h 2 60.h even 2 1
980.6.a.b 1 140.j odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 484$$ acting on $$S_{6}^{\mathrm{new}}(400, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 2 T^{2} + 59049 T^{4}$$
$5$ 1
$7$ $$1 + 13910 T^{2} + 282475249 T^{4}$$
$11$ $$( 1 - 480 T + 161051 T^{2} )^{2}$$
$13$ $$1 - 355702 T^{2} + 137858491849 T^{4}$$
$17$ $$1 - 2805118 T^{2} + 2015993900449 T^{4}$$
$19$ $$( 1 + 1204 T + 2476099 T^{2} )^{2}$$
$23$ $$1 - 2722090 T^{2} + 41426511213649 T^{4}$$
$29$ $$( 1 + 5526 T + 20511149 T^{2} )^{2}$$
$31$ $$( 1 + 9356 T + 28629151 T^{2} )^{2}$$
$37$ $$1 - 107125990 T^{2} + 4808584372417849 T^{4}$$
$41$ $$( 1 + 14394 T + 115856201 T^{2} )^{2}$$
$43$ $$1 - 293879986 T^{2} + 21611482313284249 T^{4}$$
$47$ $$1 - 197996698 T^{2} + 52599132235830049 T^{4}$$
$53$ $$1 - 817259110 T^{2} + 174887470365513049 T^{4}$$
$59$ $$( 1 + 11748 T + 714924299 T^{2} )^{2}$$
$61$ $$( 1 - 13202 T + 844596301 T^{2} )^{2}$$
$67$ $$1 - 2567032450 T^{2} + 1822837804551761449 T^{4}$$
$71$ $$( 1 - 29532 T + 1804229351 T^{2} )^{2}$$
$73$ $$1 - 3010587982 T^{2} + 4297625829703557649 T^{4}$$
$79$ $$( 1 - 31208 T + 3077056399 T^{2} )^{2}$$
$83$ $$1 - 6398448130 T^{2} + 15516041187205853449 T^{4}$$
$89$ $$( 1 + 119514 T + 5584059449 T^{2} )^{2}$$
$97$ $$1 - 8214543550 T^{2} + 73742412689492826049 T^{4}$$