# Properties

 Label 200.4.d.a Level $200$ Weight $4$ Character orbit 200.d Analytic conductor $11.800$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$200 = 2^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 200.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.8003820011$$ 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: $$2$$ Twist minimal: no (minimal twist has level 8) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} - 2 \beta q^{3} + (2 \beta - 6) q^{4} + ( - 2 \beta + 14) q^{6} + 8 q^{7} + ( - 4 \beta - 20) q^{8} - q^{9} +O(q^{10})$$ q + (b + 1) * q^2 - 2*b * q^3 + (2*b - 6) * q^4 + (-2*b + 14) * q^6 + 8 * q^7 + (-4*b - 20) * q^8 - q^9 $$q + (\beta + 1) q^{2} - 2 \beta q^{3} + (2 \beta - 6) q^{4} + ( - 2 \beta + 14) q^{6} + 8 q^{7} + ( - 4 \beta - 20) q^{8} - q^{9} - 6 \beta q^{11} + (12 \beta + 28) q^{12} - 20 \beta q^{13} + (8 \beta + 8) q^{14} + ( - 24 \beta + 8) q^{16} + 14 q^{17} + ( - \beta - 1) q^{18} - 14 \beta q^{19} - 16 \beta q^{21} + ( - 6 \beta + 42) q^{22} + 152 q^{23} + (40 \beta - 56) q^{24} + ( - 20 \beta + 140) q^{26} - 52 \beta q^{27} + (16 \beta - 48) q^{28} - 60 \beta q^{29} + 224 q^{31} + ( - 16 \beta + 176) q^{32} - 84 q^{33} + (14 \beta + 14) q^{34} + ( - 2 \beta + 6) q^{36} - 92 \beta q^{37} + ( - 14 \beta + 98) q^{38} - 280 q^{39} - 70 q^{41} + ( - 16 \beta + 112) q^{42} + 166 \beta q^{43} + (36 \beta + 84) q^{44} + (152 \beta + 152) q^{46} - 336 q^{47} + ( - 16 \beta - 336) q^{48} - 279 q^{49} - 28 \beta q^{51} + (120 \beta + 280) q^{52} - 12 \beta q^{53} + ( - 52 \beta + 364) q^{54} + ( - 32 \beta - 160) q^{56} - 196 q^{57} + ( - 60 \beta + 420) q^{58} + 202 \beta q^{59} + 36 \beta q^{61} + (224 \beta + 224) q^{62} - 8 q^{63} + (160 \beta + 288) q^{64} + ( - 84 \beta - 84) q^{66} - 66 \beta q^{67} + (28 \beta - 84) q^{68} - 304 \beta q^{69} - 72 q^{71} + (4 \beta + 20) q^{72} + 294 q^{73} + ( - 92 \beta + 644) q^{74} + (84 \beta + 196) q^{76} - 48 \beta q^{77} + ( - 280 \beta - 280) q^{78} - 464 q^{79} - 755 q^{81} + ( - 70 \beta - 70) q^{82} + 206 \beta q^{83} + (96 \beta + 224) q^{84} + (166 \beta - 1162) q^{86} - 840 q^{87} + (120 \beta - 168) q^{88} + 266 q^{89} - 160 \beta q^{91} + (304 \beta - 912) q^{92} - 448 \beta q^{93} + ( - 336 \beta - 336) q^{94} + ( - 352 \beta - 224) q^{96} - 994 q^{97} + ( - 279 \beta - 279) q^{98} + 6 \beta q^{99} +O(q^{100})$$ q + (b + 1) * q^2 - 2*b * q^3 + (2*b - 6) * q^4 + (-2*b + 14) * q^6 + 8 * q^7 + (-4*b - 20) * q^8 - q^9 - 6*b * q^11 + (12*b + 28) * q^12 - 20*b * q^13 + (8*b + 8) * q^14 + (-24*b + 8) * q^16 + 14 * q^17 + (-b - 1) * q^18 - 14*b * q^19 - 16*b * q^21 + (-6*b + 42) * q^22 + 152 * q^23 + (40*b - 56) * q^24 + (-20*b + 140) * q^26 - 52*b * q^27 + (16*b - 48) * q^28 - 60*b * q^29 + 224 * q^31 + (-16*b + 176) * q^32 - 84 * q^33 + (14*b + 14) * q^34 + (-2*b + 6) * q^36 - 92*b * q^37 + (-14*b + 98) * q^38 - 280 * q^39 - 70 * q^41 + (-16*b + 112) * q^42 + 166*b * q^43 + (36*b + 84) * q^44 + (152*b + 152) * q^46 - 336 * q^47 + (-16*b - 336) * q^48 - 279 * q^49 - 28*b * q^51 + (120*b + 280) * q^52 - 12*b * q^53 + (-52*b + 364) * q^54 + (-32*b - 160) * q^56 - 196 * q^57 + (-60*b + 420) * q^58 + 202*b * q^59 + 36*b * q^61 + (224*b + 224) * q^62 - 8 * q^63 + (160*b + 288) * q^64 + (-84*b - 84) * q^66 - 66*b * q^67 + (28*b - 84) * q^68 - 304*b * q^69 - 72 * q^71 + (4*b + 20) * q^72 + 294 * q^73 + (-92*b + 644) * q^74 + (84*b + 196) * q^76 - 48*b * q^77 + (-280*b - 280) * q^78 - 464 * q^79 - 755 * q^81 + (-70*b - 70) * q^82 + 206*b * q^83 + (96*b + 224) * q^84 + (166*b - 1162) * q^86 - 840 * q^87 + (120*b - 168) * q^88 + 266 * q^89 - 160*b * q^91 + (304*b - 912) * q^92 - 448*b * q^93 + (-336*b - 336) * q^94 + (-352*b - 224) * q^96 - 994 * q^97 + (-279*b - 279) * q^98 + 6*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 12 q^{4} + 28 q^{6} + 16 q^{7} - 40 q^{8} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 12 * q^4 + 28 * q^6 + 16 * q^7 - 40 * q^8 - 2 * q^9 $$2 q + 2 q^{2} - 12 q^{4} + 28 q^{6} + 16 q^{7} - 40 q^{8} - 2 q^{9} + 56 q^{12} + 16 q^{14} + 16 q^{16} + 28 q^{17} - 2 q^{18} + 84 q^{22} + 304 q^{23} - 112 q^{24} + 280 q^{26} - 96 q^{28} + 448 q^{31} + 352 q^{32} - 168 q^{33} + 28 q^{34} + 12 q^{36} + 196 q^{38} - 560 q^{39} - 140 q^{41} + 224 q^{42} + 168 q^{44} + 304 q^{46} - 672 q^{47} - 672 q^{48} - 558 q^{49} + 560 q^{52} + 728 q^{54} - 320 q^{56} - 392 q^{57} + 840 q^{58} + 448 q^{62} - 16 q^{63} + 576 q^{64} - 168 q^{66} - 168 q^{68} - 144 q^{71} + 40 q^{72} + 588 q^{73} + 1288 q^{74} + 392 q^{76} - 560 q^{78} - 928 q^{79} - 1510 q^{81} - 140 q^{82} + 448 q^{84} - 2324 q^{86} - 1680 q^{87} - 336 q^{88} + 532 q^{89} - 1824 q^{92} - 672 q^{94} - 448 q^{96} - 1988 q^{97} - 558 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 - 12 * q^4 + 28 * q^6 + 16 * q^7 - 40 * q^8 - 2 * q^9 + 56 * q^12 + 16 * q^14 + 16 * q^16 + 28 * q^17 - 2 * q^18 + 84 * q^22 + 304 * q^23 - 112 * q^24 + 280 * q^26 - 96 * q^28 + 448 * q^31 + 352 * q^32 - 168 * q^33 + 28 * q^34 + 12 * q^36 + 196 * q^38 - 560 * q^39 - 140 * q^41 + 224 * q^42 + 168 * q^44 + 304 * q^46 - 672 * q^47 - 672 * q^48 - 558 * q^49 + 560 * q^52 + 728 * q^54 - 320 * q^56 - 392 * q^57 + 840 * q^58 + 448 * q^62 - 16 * q^63 + 576 * q^64 - 168 * q^66 - 168 * q^68 - 144 * q^71 + 40 * q^72 + 588 * q^73 + 1288 * q^74 + 392 * q^76 - 560 * q^78 - 928 * q^79 - 1510 * q^81 - 140 * q^82 + 448 * q^84 - 2324 * q^86 - 1680 * q^87 - 336 * q^88 + 532 * q^89 - 1824 * q^92 - 672 * q^94 - 448 * q^96 - 1988 * q^97 - 558 * 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
1.00000 2.64575i 5.29150i −6.00000 5.29150i 0 14.0000 + 5.29150i 8.00000 −20.0000 + 10.5830i −1.00000 0
101.2 1.00000 + 2.64575i 5.29150i −6.00000 + 5.29150i 0 14.0000 5.29150i 8.00000 −20.0000 10.5830i −1.00000 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 200.4.d.a 2
4.b odd 2 1 800.4.d.a 2
5.b even 2 1 8.4.b.a 2
5.c odd 4 2 200.4.f.a 4
8.b even 2 1 inner 200.4.d.a 2
8.d odd 2 1 800.4.d.a 2
15.d odd 2 1 72.4.d.b 2
20.d odd 2 1 32.4.b.a 2
20.e even 4 2 800.4.f.a 4
40.e odd 2 1 32.4.b.a 2
40.f even 2 1 8.4.b.a 2
40.i odd 4 2 200.4.f.a 4
40.k even 4 2 800.4.f.a 4
60.h even 2 1 288.4.d.a 2
80.k odd 4 2 256.4.a.j 2
80.q even 4 2 256.4.a.l 2
120.i odd 2 1 72.4.d.b 2
120.m even 2 1 288.4.d.a 2
240.t even 4 2 2304.4.a.v 2
240.bm odd 4 2 2304.4.a.bn 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.4.b.a 2 5.b even 2 1
8.4.b.a 2 40.f even 2 1
32.4.b.a 2 20.d odd 2 1
32.4.b.a 2 40.e odd 2 1
72.4.d.b 2 15.d odd 2 1
72.4.d.b 2 120.i odd 2 1
200.4.d.a 2 1.a even 1 1 trivial
200.4.d.a 2 8.b even 2 1 inner
200.4.f.a 4 5.c odd 4 2
200.4.f.a 4 40.i odd 4 2
256.4.a.j 2 80.k odd 4 2
256.4.a.l 2 80.q even 4 2
288.4.d.a 2 60.h even 2 1
288.4.d.a 2 120.m even 2 1
800.4.d.a 2 4.b odd 2 1
800.4.d.a 2 8.d odd 2 1
800.4.f.a 4 20.e even 4 2
800.4.f.a 4 40.k even 4 2
2304.4.a.v 2 240.t even 4 2
2304.4.a.bn 2 240.bm odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(200, [\chi])$$:

 $$T_{3}^{2} + 28$$ T3^2 + 28 $$T_{7} - 8$$ T7 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 8$$
$3$ $$T^{2} + 28$$
$5$ $$T^{2}$$
$7$ $$(T - 8)^{2}$$
$11$ $$T^{2} + 252$$
$13$ $$T^{2} + 2800$$
$17$ $$(T - 14)^{2}$$
$19$ $$T^{2} + 1372$$
$23$ $$(T - 152)^{2}$$
$29$ $$T^{2} + 25200$$
$31$ $$(T - 224)^{2}$$
$37$ $$T^{2} + 59248$$
$41$ $$(T + 70)^{2}$$
$43$ $$T^{2} + 192892$$
$47$ $$(T + 336)^{2}$$
$53$ $$T^{2} + 1008$$
$59$ $$T^{2} + 285628$$
$61$ $$T^{2} + 9072$$
$67$ $$T^{2} + 30492$$
$71$ $$(T + 72)^{2}$$
$73$ $$(T - 294)^{2}$$
$79$ $$(T + 464)^{2}$$
$83$ $$T^{2} + 297052$$
$89$ $$(T - 266)^{2}$$
$97$ $$(T + 994)^{2}$$