# Properties

 Label 200.6.c.c Level 200 Weight 6 Character orbit 200.c Analytic conductor 32.077 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$200 = 2^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 200.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.0767639626$$ 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 40) 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.

 $$f(q)$$ $$=$$ $$q + 8 i q^{3} -108 i q^{7} + 179 q^{9} +O(q^{10})$$ $$q + 8 i q^{3} -108 i q^{7} + 179 q^{9} -604 q^{11} + 306 i q^{13} + 930 i q^{17} + 1324 q^{19} + 864 q^{21} + 852 i q^{23} + 3376 i q^{27} -5902 q^{29} -3320 q^{31} -4832 i q^{33} + 10774 i q^{37} -2448 q^{39} -17958 q^{41} -9264 i q^{43} -9796 i q^{47} + 5143 q^{49} -7440 q^{51} + 31434 i q^{53} + 10592 i q^{57} -33228 q^{59} -40210 q^{61} -19332 i q^{63} + 58864 i q^{67} -6816 q^{69} -55312 q^{71} -27258 i q^{73} + 65232 i q^{77} -31456 q^{79} + 16489 q^{81} -24552 i q^{83} -47216 i q^{87} + 90854 q^{89} + 33048 q^{91} -26560 i q^{93} + 154706 i q^{97} -108116 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 358q^{9} + O(q^{10})$$ $$2q + 358q^{9} - 1208q^{11} + 2648q^{19} + 1728q^{21} - 11804q^{29} - 6640q^{31} - 4896q^{39} - 35916q^{41} + 10286q^{49} - 14880q^{51} - 66456q^{59} - 80420q^{61} - 13632q^{69} - 110624q^{71} - 62912q^{79} + 32978q^{81} + 181708q^{89} + 66096q^{91} - 216232q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/200\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$151$$ $$177$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 8.00000i 0 0 0 108.000i 0 179.000 0
49.2 0 8.00000i 0 0 0 108.000i 0 179.000 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 200.6.c.c 2
4.b odd 2 1 400.6.c.h 2
5.b even 2 1 inner 200.6.c.c 2
5.c odd 4 1 40.6.a.b 1
5.c odd 4 1 200.6.a.c 1
15.e even 4 1 360.6.a.b 1
20.d odd 2 1 400.6.c.h 2
20.e even 4 1 80.6.a.f 1
20.e even 4 1 400.6.a.f 1
40.i odd 4 1 320.6.a.l 1
40.k even 4 1 320.6.a.e 1
60.l odd 4 1 720.6.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.a.b 1 5.c odd 4 1
80.6.a.f 1 20.e even 4 1
200.6.a.c 1 5.c odd 4 1
200.6.c.c 2 1.a even 1 1 trivial
200.6.c.c 2 5.b even 2 1 inner
320.6.a.e 1 40.k even 4 1
320.6.a.l 1 40.i odd 4 1
360.6.a.b 1 15.e even 4 1
400.6.a.f 1 20.e even 4 1
400.6.c.h 2 4.b odd 2 1
400.6.c.h 2 20.d odd 2 1
720.6.a.h 1 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 64$$ acting on $$S_{6}^{\mathrm{new}}(200, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 422 T^{2} + 59049 T^{4}$$
$5$ 1
$7$ $$1 - 21950 T^{2} + 282475249 T^{4}$$
$11$ $$( 1 + 604 T + 161051 T^{2} )^{2}$$
$13$ $$1 - 648950 T^{2} + 137858491849 T^{4}$$
$17$ $$1 - 1974814 T^{2} + 2015993900449 T^{4}$$
$19$ $$( 1 - 1324 T + 2476099 T^{2} )^{2}$$
$23$ $$1 - 12146782 T^{2} + 41426511213649 T^{4}$$
$29$ $$( 1 + 5902 T + 20511149 T^{2} )^{2}$$
$31$ $$( 1 + 3320 T + 28629151 T^{2} )^{2}$$
$37$ $$1 - 22608838 T^{2} + 4808584372417849 T^{4}$$
$41$ $$( 1 + 17958 T + 115856201 T^{2} )^{2}$$
$43$ $$1 - 208195190 T^{2} + 21611482313284249 T^{4}$$
$47$ $$1 - 362728398 T^{2} + 52599132235830049 T^{4}$$
$53$ $$1 + 151705370 T^{2} + 174887470365513049 T^{4}$$
$59$ $$( 1 + 33228 T + 714924299 T^{2} )^{2}$$
$61$ $$( 1 + 40210 T + 844596301 T^{2} )^{2}$$
$67$ $$1 + 764720282 T^{2} + 1822837804551761449 T^{4}$$
$71$ $$( 1 + 55312 T + 1804229351 T^{2} )^{2}$$
$73$ $$1 - 3403144622 T^{2} + 4297625829703557649 T^{4}$$
$79$ $$( 1 + 31456 T + 3077056399 T^{2} )^{2}$$
$83$ $$1 - 7275280582 T^{2} + 15516041187205853449 T^{4}$$
$89$ $$( 1 - 90854 T + 5584059449 T^{2} )^{2}$$
$97$ $$1 + 6759265922 T^{2} + 73742412689492826049 T^{4}$$