# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$200 = 2^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 200.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.59700804043$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-7})$$ Defining polynomial: $$x^{2} - x + 2$$ x^2 - x + 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{-7})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 0.5 − 1.32288i 0.5 + 1.32288i
0.500000 1.32288i 2.64575i −1.50000 1.32288i 0 −3.50000 1.32288i 4.00000 −2.50000 + 1.32288i −4.00000 0
101.2 0.500000 + 1.32288i 2.64575i −1.50000 + 1.32288i 0 −3.50000 + 1.32288i 4.00000 −2.50000 1.32288i −4.00000 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
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 200.2.d.c yes 2
3.b odd 2 1 1800.2.k.d 2
4.b odd 2 1 800.2.d.a 2
5.b even 2 1 200.2.d.b 2
5.c odd 4 2 200.2.f.d 4
8.b even 2 1 inner 200.2.d.c yes 2
8.d odd 2 1 800.2.d.a 2
12.b even 2 1 7200.2.k.b 2
15.d odd 2 1 1800.2.k.f 2
15.e even 4 2 1800.2.d.m 4
16.e even 4 2 6400.2.a.bg 2
16.f odd 4 2 6400.2.a.cb 2
20.d odd 2 1 800.2.d.d 2
20.e even 4 2 800.2.f.d 4
24.f even 2 1 7200.2.k.b 2
24.h odd 2 1 1800.2.k.d 2
40.e odd 2 1 800.2.d.d 2
40.f even 2 1 200.2.d.b 2
40.i odd 4 2 200.2.f.d 4
40.k even 4 2 800.2.f.d 4
60.h even 2 1 7200.2.k.i 2
60.l odd 4 2 7200.2.d.m 4
80.k odd 4 2 6400.2.a.bh 2
80.q even 4 2 6400.2.a.cc 2
120.i odd 2 1 1800.2.k.f 2
120.m even 2 1 7200.2.k.i 2
120.q odd 4 2 7200.2.d.m 4
120.w even 4 2 1800.2.d.m 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.d.b 2 5.b even 2 1
200.2.d.b 2 40.f even 2 1
200.2.d.c yes 2 1.a even 1 1 trivial
200.2.d.c yes 2 8.b even 2 1 inner
200.2.f.d 4 5.c odd 4 2
200.2.f.d 4 40.i odd 4 2
800.2.d.a 2 4.b odd 2 1
800.2.d.a 2 8.d odd 2 1
800.2.d.d 2 20.d odd 2 1
800.2.d.d 2 40.e odd 2 1
800.2.f.d 4 20.e even 4 2
800.2.f.d 4 40.k even 4 2
1800.2.d.m 4 15.e even 4 2
1800.2.d.m 4 120.w even 4 2
1800.2.k.d 2 3.b odd 2 1
1800.2.k.d 2 24.h odd 2 1
1800.2.k.f 2 15.d odd 2 1
1800.2.k.f 2 120.i odd 2 1
6400.2.a.bg 2 16.e even 4 2
6400.2.a.bh 2 80.k odd 4 2
6400.2.a.cb 2 16.f odd 4 2
6400.2.a.cc 2 80.q even 4 2
7200.2.d.m 4 60.l odd 4 2
7200.2.d.m 4 120.q odd 4 2
7200.2.k.b 2 12.b even 2 1
7200.2.k.b 2 24.f even 2 1
7200.2.k.i 2 60.h even 2 1
7200.2.k.i 2 120.m even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(200, [\chi])$$:

 $$T_{3}^{2} + 7$$ T3^2 + 7 $$T_{7} - 4$$ T7 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 2$$
$3$ $$T^{2} + 7$$
$5$ $$T^{2}$$
$7$ $$(T - 4)^{2}$$
$11$ $$T^{2} + 7$$
$13$ $$T^{2}$$
$17$ $$(T + 3)^{2}$$
$19$ $$T^{2} + 7$$
$23$ $$(T - 4)^{2}$$
$29$ $$T^{2}$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + 112$$
$41$ $$(T + 5)^{2}$$
$43$ $$T^{2} + 28$$
$47$ $$(T + 8)^{2}$$
$53$ $$T^{2} + 112$$
$59$ $$T^{2} + 28$$
$61$ $$T^{2} + 112$$
$67$ $$T^{2} + 63$$
$71$ $$(T - 8)^{2}$$
$73$ $$(T + 7)^{2}$$
$79$ $$(T - 4)^{2}$$
$83$ $$T^{2} + 63$$
$89$ $$(T + 1)^{2}$$
$97$ $$(T + 2)^{2}$$