# Properties

 Label 400.6.c.i 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 10) 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.

 $$f(q)$$ $$=$$ $$q + 6 i q^{3} + 118 i q^{7} + 207 q^{9} +O(q^{10})$$ $$q + 6 i q^{3} + 118 i q^{7} + 207 q^{9} -192 q^{11} -1106 i q^{13} + 762 i q^{17} -2740 q^{19} -708 q^{21} + 1566 i q^{23} + 2700 i q^{27} -5910 q^{29} + 6868 q^{31} -1152 i q^{33} -5518 i q^{37} + 6636 q^{39} -378 q^{41} -2434 i q^{43} -13122 i q^{47} + 2883 q^{49} -4572 q^{51} + 9174 i q^{53} -16440 i q^{57} -34980 q^{59} -9838 q^{61} + 24426 i q^{63} -33722 i q^{67} -9396 q^{69} -70212 q^{71} -21986 i q^{73} -22656 i q^{77} + 4520 q^{79} + 34101 q^{81} -109074 i q^{83} -35460 i q^{87} -38490 q^{89} + 130508 q^{91} + 41208 i q^{93} -1918 i q^{97} -39744 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 414q^{9} + O(q^{10})$$ $$2q + 414q^{9} - 384q^{11} - 5480q^{19} - 1416q^{21} - 11820q^{29} + 13736q^{31} + 13272q^{39} - 756q^{41} + 5766q^{49} - 9144q^{51} - 69960q^{59} - 19676q^{61} - 18792q^{69} - 140424q^{71} + 9040q^{79} + 68202q^{81} - 76980q^{89} + 261016q^{91} - 79488q^{99} + O(q^{100})$$

## 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 6.00000i 0 0 0 118.000i 0 207.000 0
49.2 0 6.00000i 0 0 0 118.000i 0 207.000 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.i 2
4.b odd 2 1 50.6.b.b 2
5.b even 2 1 inner 400.6.c.i 2
5.c odd 4 1 80.6.a.c 1
5.c odd 4 1 400.6.a.i 1
12.b even 2 1 450.6.c.f 2
15.e even 4 1 720.6.a.v 1
20.d odd 2 1 50.6.b.b 2
20.e even 4 1 10.6.a.c 1
20.e even 4 1 50.6.a.b 1
40.i odd 4 1 320.6.a.k 1
40.k even 4 1 320.6.a.f 1
60.h even 2 1 450.6.c.f 2
60.l odd 4 1 90.6.a.b 1
60.l odd 4 1 450.6.a.u 1
140.j odd 4 1 490.6.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.a.c 1 20.e even 4 1
50.6.a.b 1 20.e even 4 1
50.6.b.b 2 4.b odd 2 1
50.6.b.b 2 20.d odd 2 1
80.6.a.c 1 5.c odd 4 1
90.6.a.b 1 60.l odd 4 1
320.6.a.f 1 40.k even 4 1
320.6.a.k 1 40.i odd 4 1
400.6.a.i 1 5.c odd 4 1
400.6.c.i 2 1.a even 1 1 trivial
400.6.c.i 2 5.b even 2 1 inner
450.6.a.u 1 60.l odd 4 1
450.6.c.f 2 12.b even 2 1
450.6.c.f 2 60.h even 2 1
490.6.a.k 1 140.j odd 4 1
720.6.a.v 1 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 36$$ acting on $$S_{6}^{\mathrm{new}}(400, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 450 T^{2} + 59049 T^{4}$$
$5$ 1
$7$ $$1 - 19690 T^{2} + 282475249 T^{4}$$
$11$ $$( 1 + 192 T + 161051 T^{2} )^{2}$$
$13$ $$1 + 480650 T^{2} + 137858491849 T^{4}$$
$17$ $$1 - 2259070 T^{2} + 2015993900449 T^{4}$$
$19$ $$( 1 + 2740 T + 2476099 T^{2} )^{2}$$
$23$ $$1 - 10420330 T^{2} + 41426511213649 T^{4}$$
$29$ $$( 1 + 5910 T + 20511149 T^{2} )^{2}$$
$31$ $$( 1 - 6868 T + 28629151 T^{2} )^{2}$$
$37$ $$1 - 108239590 T^{2} + 4808584372417849 T^{4}$$
$41$ $$( 1 + 378 T + 115856201 T^{2} )^{2}$$
$43$ $$1 - 288092530 T^{2} + 21611482313284249 T^{4}$$
$47$ $$1 - 286503130 T^{2} + 52599132235830049 T^{4}$$
$53$ $$1 - 752228710 T^{2} + 174887470365513049 T^{4}$$
$59$ $$( 1 + 34980 T + 714924299 T^{2} )^{2}$$
$61$ $$( 1 + 9838 T + 844596301 T^{2} )^{2}$$
$67$ $$1 - 1563076930 T^{2} + 1822837804551761449 T^{4}$$
$71$ $$( 1 + 70212 T + 1804229351 T^{2} )^{2}$$
$73$ $$1 - 3662758990 T^{2} + 4297625829703557649 T^{4}$$
$79$ $$( 1 - 4520 T + 3077056399 T^{2} )^{2}$$
$83$ $$1 + 4019056190 T^{2} + 15516041187205853449 T^{4}$$
$89$ $$( 1 + 38490 T + 5584059449 T^{2} )^{2}$$
$97$ $$1 - 17171001790 T^{2} + 73742412689492826049 T^{4}$$