# Properties

 Label 200.6.c.b 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: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + 18 i q^{3} + 242 i q^{7} -81 q^{9} +O(q^{10})$$ $$q + 18 i q^{3} + 242 i q^{7} -81 q^{9} + 656 q^{11} + 206 i q^{13} + 1690 i q^{17} + 1364 q^{19} -4356 q^{21} -2198 i q^{23} + 2916 i q^{27} + 2218 q^{29} -1700 q^{31} + 11808 i q^{33} -846 i q^{37} -3708 q^{39} -1818 q^{41} -10534 i q^{43} + 12074 i q^{47} -41757 q^{49} -30420 q^{51} -32586 i q^{53} + 24552 i q^{57} -8668 q^{59} -34670 q^{61} -19602 i q^{63} -47566 i q^{67} + 39564 q^{69} + 948 q^{71} + 63102 i q^{73} + 158752 i q^{77} -46536 q^{79} -72171 q^{81} + 88778 i q^{83} + 39924 i q^{87} + 104934 q^{89} -49852 q^{91} -30600 i q^{93} -36254 i q^{97} -53136 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 162q^{9} + O(q^{10})$$ $$2q - 162q^{9} + 1312q^{11} + 2728q^{19} - 8712q^{21} + 4436q^{29} - 3400q^{31} - 7416q^{39} - 3636q^{41} - 83514q^{49} - 60840q^{51} - 17336q^{59} - 69340q^{61} + 79128q^{69} + 1896q^{71} - 93072q^{79} - 144342q^{81} + 209868q^{89} - 99704q^{91} - 106272q^{99} + O(q^{100})$$

## 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 18.0000i 0 0 0 242.000i 0 −81.0000 0
49.2 0 18.0000i 0 0 0 242.000i 0 −81.0000 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 200.6.c.b 2
4.b odd 2 1 400.6.c.e 2
5.b even 2 1 inner 200.6.c.b 2
5.c odd 4 1 40.6.a.a 1
5.c odd 4 1 200.6.a.d 1
15.e even 4 1 360.6.a.i 1
20.d odd 2 1 400.6.c.e 2
20.e even 4 1 80.6.a.g 1
20.e even 4 1 400.6.a.b 1
40.i odd 4 1 320.6.a.m 1
40.k even 4 1 320.6.a.d 1
60.l odd 4 1 720.6.a.k 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 324$$ acting on $$S_{6}^{\mathrm{new}}(200, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 162 T^{2} + 59049 T^{4}$$
$5$ 1
$7$ $$1 + 24950 T^{2} + 282475249 T^{4}$$
$11$ $$( 1 - 656 T + 161051 T^{2} )^{2}$$
$13$ $$1 - 700150 T^{2} + 137858491849 T^{4}$$
$17$ $$1 + 16386 T^{2} + 2015993900449 T^{4}$$
$19$ $$( 1 - 1364 T + 2476099 T^{2} )^{2}$$
$23$ $$1 - 8041482 T^{2} + 41426511213649 T^{4}$$
$29$ $$( 1 - 2218 T + 20511149 T^{2} )^{2}$$
$31$ $$( 1 + 1700 T + 28629151 T^{2} )^{2}$$
$37$ $$1 - 137972198 T^{2} + 4808584372417849 T^{4}$$
$41$ $$( 1 + 1818 T + 115856201 T^{2} )^{2}$$
$43$ $$1 - 183051730 T^{2} + 21611482313284249 T^{4}$$
$47$ $$1 - 312908538 T^{2} + 52599132235830049 T^{4}$$
$53$ $$1 + 225456410 T^{2} + 174887470365513049 T^{4}$$
$59$ $$( 1 + 8668 T + 714924299 T^{2} )^{2}$$
$61$ $$( 1 + 34670 T + 844596301 T^{2} )^{2}$$
$67$ $$1 - 437725858 T^{2} + 1822837804551761449 T^{4}$$
$71$ $$( 1 - 948 T + 1804229351 T^{2} )^{2}$$
$73$ $$1 - 164280782 T^{2} + 4297625829703557649 T^{4}$$
$79$ $$( 1 + 46536 T + 3077056399 T^{2} )^{2}$$
$83$ $$1 + 3451998 T^{2} + 15516041187205853449 T^{4}$$
$89$ $$( 1 - 104934 T + 5584059449 T^{2} )^{2}$$
$97$ $$1 - 15860327998 T^{2} + 73742412689492826049 T^{4}$$