# Properties

 Label 200.2.f.e Level $200$ Weight $2$ Character orbit 200.f Analytic conductor $1.597$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$1.59700804043$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{2} + ( -1 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12} q^{4} + ( 2 - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{6} + ( 1 - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{7} + ( 2 + 2 \zeta_{12}^{3} ) q^{8} + ( 1 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{9} +O(q^{10})$$ $$q + ( \zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{2} + ( -1 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12} q^{4} + ( 2 - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{6} + ( 1 - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{7} + ( 2 + 2 \zeta_{12}^{3} ) q^{8} + ( 1 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{9} + 2 \zeta_{12}^{3} q^{11} + ( 2 - 2 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{12} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{13} + ( 2 - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{14} + 4 \zeta_{12}^{2} q^{16} + ( 2 - 4 \zeta_{12}^{2} ) q^{17} + ( -4 - \zeta_{12} + 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{18} + ( -2 + 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{19} -2 \zeta_{12}^{3} q^{21} + ( -2 \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{22} + ( 3 - 6 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{23} + ( -4 + 4 \zeta_{12} + 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{24} + ( -4 - 2 \zeta_{12} + 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{26} -4 q^{27} + ( 2 + 2 \zeta_{12} - 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{28} + ( 4 - 8 \zeta_{12}^{2} ) q^{29} + ( -2 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{31} + ( -4 + 4 \zeta_{12} + 4 \zeta_{12}^{2} ) q^{32} + ( -2 + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{33} + ( 4 - 2 \zeta_{12} - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{34} + ( -4 + 2 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{36} -2 q^{37} + ( -4 - 2 \zeta_{12} + 6 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{38} + ( -6 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{39} + ( -2 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{41} + ( 2 \zeta_{12} - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{42} + ( 7 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{43} + ( -4 + 4 \zeta_{12}^{2} ) q^{44} + ( 6 - 4 \zeta_{12} - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{46} + ( -1 + 2 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{47} + ( 4 \zeta_{12} - 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{48} + ( 3 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{49} + ( -2 + 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{51} + ( -4 - 4 \zeta_{12}^{2} ) q^{52} + ( -8 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{53} + ( -4 \zeta_{12} - 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{54} + ( 4 + 4 \zeta_{12} - 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{56} + ( -2 + 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{57} + ( 8 - 4 \zeta_{12} - 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{58} + ( -2 + 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{59} + ( 4 - 8 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{61} + ( 4 - 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{62} + ( -1 + 2 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{63} + 8 \zeta_{12}^{3} q^{64} + ( -4 + 4 \zeta_{12} ) q^{66} + ( 9 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{67} + ( 4 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{68} + ( -4 + 8 \zeta_{12}^{2} - 10 \zeta_{12}^{3} ) q^{69} + ( 2 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{71} + ( 6 - 8 \zeta_{12} - 8 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{72} + ( -2 + 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{73} + ( -2 \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{74} + ( -8 - 4 \zeta_{12} + 8 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{76} + ( 2 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{77} + ( 4 - 4 \zeta_{12} - 8 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{78} + ( 8 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{79} + ( 1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{81} + ( -4 - 4 \zeta_{12} ) q^{82} + ( -3 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{83} + ( 4 - 4 \zeta_{12}^{2} ) q^{84} + ( 2 + 8 \zeta_{12} + 6 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{86} + ( -4 + 8 \zeta_{12}^{2} - 12 \zeta_{12}^{3} ) q^{87} + ( -4 + 4 \zeta_{12}^{3} ) q^{88} + ( -2 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{89} + ( -2 + 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{91} + ( -2 + 6 \zeta_{12} + 2 \zeta_{12}^{2} - 12 \zeta_{12}^{3} ) q^{92} + ( 8 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{93} + ( -2 - 4 \zeta_{12} + 6 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{94} + ( 8 - 8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{96} + ( -6 + 12 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{97} + ( -4 + \zeta_{12} + 5 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{98} + ( 4 - 8 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 4 q^{3} + 4 q^{6} + 8 q^{8} + 4 q^{9} + O(q^{10})$$ $$4 q + 2 q^{2} - 4 q^{3} + 4 q^{6} + 8 q^{8} + 4 q^{9} + 12 q^{12} + 4 q^{14} + 8 q^{16} - 10 q^{18} + 4 q^{22} - 8 q^{24} - 12 q^{26} - 16 q^{27} + 4 q^{28} - 8 q^{31} - 8 q^{32} + 12 q^{34} - 24 q^{36} - 8 q^{37} - 4 q^{38} - 24 q^{39} - 8 q^{41} - 4 q^{42} + 28 q^{43} - 8 q^{44} + 20 q^{46} - 8 q^{48} + 12 q^{49} - 24 q^{52} - 32 q^{53} - 8 q^{54} + 8 q^{56} + 24 q^{58} + 8 q^{62} - 16 q^{66} + 36 q^{67} + 8 q^{71} + 8 q^{72} - 4 q^{74} - 16 q^{76} + 8 q^{77} + 32 q^{79} + 4 q^{81} - 16 q^{82} - 12 q^{83} + 8 q^{84} + 20 q^{86} - 16 q^{88} - 8 q^{89} - 4 q^{92} + 32 q^{93} + 4 q^{94} + 32 q^{96} - 6 q^{98} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1
 −0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i 0.866025 + 0.500000i
−0.366025 1.36603i −2.73205 −1.73205 + 1.00000i 0 1.00000 + 3.73205i 0.732051i 2.00000 + 2.00000i 4.46410 0
149.2 −0.366025 + 1.36603i −2.73205 −1.73205 1.00000i 0 1.00000 3.73205i 0.732051i 2.00000 2.00000i 4.46410 0
149.3 1.36603 0.366025i 0.732051 1.73205 1.00000i 0 1.00000 0.267949i 2.73205i 2.00000 2.00000i −2.46410 0
149.4 1.36603 + 0.366025i 0.732051 1.73205 + 1.00000i 0 1.00000 + 0.267949i 2.73205i 2.00000 + 2.00000i −2.46410 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
40.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 200.2.f.e 4
3.b odd 2 1 1800.2.d.l 4
4.b odd 2 1 800.2.f.e 4
5.b even 2 1 200.2.f.c 4
5.c odd 4 1 40.2.d.a 4
5.c odd 4 1 200.2.d.f 4
8.b even 2 1 200.2.f.c 4
8.d odd 2 1 800.2.f.c 4
12.b even 2 1 7200.2.d.n 4
15.d odd 2 1 1800.2.d.p 4
15.e even 4 1 360.2.k.e 4
15.e even 4 1 1800.2.k.j 4
20.d odd 2 1 800.2.f.c 4
20.e even 4 1 160.2.d.a 4
20.e even 4 1 800.2.d.e 4
24.f even 2 1 7200.2.d.o 4
24.h odd 2 1 1800.2.d.p 4
40.e odd 2 1 800.2.f.e 4
40.f even 2 1 inner 200.2.f.e 4
40.i odd 4 1 40.2.d.a 4
40.i odd 4 1 200.2.d.f 4
40.k even 4 1 160.2.d.a 4
40.k even 4 1 800.2.d.e 4
60.h even 2 1 7200.2.d.o 4
60.l odd 4 1 1440.2.k.e 4
60.l odd 4 1 7200.2.k.j 4
80.i odd 4 1 1280.2.a.o 2
80.i odd 4 1 6400.2.a.ce 2
80.j even 4 1 1280.2.a.n 2
80.j even 4 1 6400.2.a.cj 2
80.s even 4 1 1280.2.a.d 2
80.s even 4 1 6400.2.a.be 2
80.t odd 4 1 1280.2.a.a 2
80.t odd 4 1 6400.2.a.z 2
120.i odd 2 1 1800.2.d.l 4
120.m even 2 1 7200.2.d.n 4
120.q odd 4 1 1440.2.k.e 4
120.q odd 4 1 7200.2.k.j 4
120.w even 4 1 360.2.k.e 4
120.w even 4 1 1800.2.k.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.d.a 4 5.c odd 4 1
40.2.d.a 4 40.i odd 4 1
160.2.d.a 4 20.e even 4 1
160.2.d.a 4 40.k even 4 1
200.2.d.f 4 5.c odd 4 1
200.2.d.f 4 40.i odd 4 1
200.2.f.c 4 5.b even 2 1
200.2.f.c 4 8.b even 2 1
200.2.f.e 4 1.a even 1 1 trivial
200.2.f.e 4 40.f even 2 1 inner
360.2.k.e 4 15.e even 4 1
360.2.k.e 4 120.w even 4 1
800.2.d.e 4 20.e even 4 1
800.2.d.e 4 40.k even 4 1
800.2.f.c 4 8.d odd 2 1
800.2.f.c 4 20.d odd 2 1
800.2.f.e 4 4.b odd 2 1
800.2.f.e 4 40.e odd 2 1
1280.2.a.a 2 80.t odd 4 1
1280.2.a.d 2 80.s even 4 1
1280.2.a.n 2 80.j even 4 1
1280.2.a.o 2 80.i odd 4 1
1440.2.k.e 4 60.l odd 4 1
1440.2.k.e 4 120.q odd 4 1
1800.2.d.l 4 3.b odd 2 1
1800.2.d.l 4 120.i odd 2 1
1800.2.d.p 4 15.d odd 2 1
1800.2.d.p 4 24.h odd 2 1
1800.2.k.j 4 15.e even 4 1
1800.2.k.j 4 120.w even 4 1
6400.2.a.z 2 80.t odd 4 1
6400.2.a.be 2 80.s even 4 1
6400.2.a.ce 2 80.i odd 4 1
6400.2.a.cj 2 80.j even 4 1
7200.2.d.n 4 12.b even 2 1
7200.2.d.n 4 120.m even 2 1
7200.2.d.o 4 24.f even 2 1
7200.2.d.o 4 60.h even 2 1
7200.2.k.j 4 60.l odd 4 1
7200.2.k.j 4 120.q odd 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4}$$
$3$ $$( -2 + 2 T + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$4 + 8 T^{2} + T^{4}$$
$11$ $$( 4 + T^{2} )^{2}$$
$13$ $$( -12 + T^{2} )^{2}$$
$17$ $$( 12 + T^{2} )^{2}$$
$19$ $$16 + 56 T^{2} + T^{4}$$
$23$ $$676 + 56 T^{2} + T^{4}$$
$29$ $$( 48 + T^{2} )^{2}$$
$31$ $$( -8 + 4 T + T^{2} )^{2}$$
$37$ $$( 2 + T )^{4}$$
$41$ $$( -8 + 4 T + T^{2} )^{2}$$
$43$ $$( 46 - 14 T + T^{2} )^{2}$$
$47$ $$484 + 56 T^{2} + T^{4}$$
$53$ $$( 52 + 16 T + T^{2} )^{2}$$
$59$ $$16 + 56 T^{2} + T^{4}$$
$61$ $$1936 + 104 T^{2} + T^{4}$$
$67$ $$( 78 - 18 T + T^{2} )^{2}$$
$71$ $$( -8 - 4 T + T^{2} )^{2}$$
$73$ $$16 + 56 T^{2} + T^{4}$$
$79$ $$( 16 - 16 T + T^{2} )^{2}$$
$83$ $$( 6 + 6 T + T^{2} )^{2}$$
$89$ $$( -44 + 4 T + T^{2} )^{2}$$
$97$ $$8464 + 248 T^{2} + T^{4}$$