# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$200 = 2^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 200.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.8003820011$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{7})$$ Defining polynomial: $$x^{4} - 3x^{2} + 4$$ x^4 - 3*x^2 + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{2} - 2 \beta_1) q^{3} + (\beta_{3} + 6) q^{4} + (\beta_{3} + 14) q^{6} - 4 \beta_{2} q^{7} + ( - 8 \beta_{2} - 4 \beta_1) q^{8} + q^{9}+O(q^{10})$$ q - b1 * q^2 + (b2 - 2*b1) * q^3 + (b3 + 6) * q^4 + (b3 + 14) * q^6 - 4*b2 * q^7 + (-8*b2 - 4*b1) * q^8 + q^9 $$q - \beta_1 q^{2} + (\beta_{2} - 2 \beta_1) q^{3} + (\beta_{3} + 6) q^{4} + (\beta_{3} + 14) q^{6} - 4 \beta_{2} q^{7} + ( - 8 \beta_{2} - 4 \beta_1) q^{8} + q^{9} + 3 \beta_{3} q^{11} + ( - 8 \beta_{2} - 12 \beta_1) q^{12} + (10 \beta_{2} - 20 \beta_1) q^{13} + (4 \beta_{3} - 8) q^{14} + (12 \beta_{3} + 8) q^{16} - 7 \beta_{2} q^{17} - \beta_1 q^{18} - 7 \beta_{3} q^{19} + 8 \beta_{3} q^{21} + ( - 24 \beta_{2} + 6 \beta_1) q^{22} + 76 \beta_{2} q^{23} + (20 \beta_{3} + 56) q^{24} + (10 \beta_{3} + 140) q^{26} + ( - 26 \beta_{2} + 52 \beta_1) q^{27} + ( - 32 \beta_{2} + 16 \beta_1) q^{28} - 30 \beta_{3} q^{29} + 224 q^{31} + ( - 96 \beta_{2} + 16 \beta_1) q^{32} - 42 \beta_{2} q^{33} + (7 \beta_{3} - 14) q^{34} + (\beta_{3} + 6) q^{36} + ( - 46 \beta_{2} + 92 \beta_1) q^{37} + (56 \beta_{2} - 14 \beta_1) q^{38} + 280 q^{39} - 70 q^{41} + ( - 64 \beta_{2} + 16 \beta_1) q^{42} + ( - 83 \beta_{2} + 166 \beta_1) q^{43} + (18 \beta_{3} - 84) q^{44} + ( - 76 \beta_{3} + 152) q^{46} + 168 \beta_{2} q^{47} + ( - 160 \beta_{2} - 16 \beta_1) q^{48} + 279 q^{49} + 14 \beta_{3} q^{51} + ( - 80 \beta_{2} - 120 \beta_1) q^{52} + (6 \beta_{2} - 12 \beta_1) q^{53} + ( - 26 \beta_{3} - 364) q^{54} + (16 \beta_{3} - 160) q^{56} + 98 \beta_{2} q^{57} + (240 \beta_{2} - 60 \beta_1) q^{58} + 101 \beta_{3} q^{59} - 18 \beta_{3} q^{61} - 224 \beta_1 q^{62} - 4 \beta_{2} q^{63} + (80 \beta_{3} - 288) q^{64} + (42 \beta_{3} - 84) q^{66} + ( - 33 \beta_{2} + 66 \beta_1) q^{67} + ( - 56 \beta_{2} + 28 \beta_1) q^{68} - 152 \beta_{3} q^{69} - 72 q^{71} + ( - 8 \beta_{2} - 4 \beta_1) q^{72} + 147 \beta_{2} q^{73} + ( - 46 \beta_{3} - 644) q^{74} + ( - 42 \beta_{3} + 196) q^{76} + ( - 24 \beta_{2} + 48 \beta_1) q^{77} - 280 \beta_1 q^{78} + 464 q^{79} - 755 q^{81} + 70 \beta_1 q^{82} + ( - 103 \beta_{2} + 206 \beta_1) q^{83} + (48 \beta_{3} - 224) q^{84} + ( - 83 \beta_{3} - 1162) q^{86} + 420 \beta_{2} q^{87} + ( - 144 \beta_{2} + 120 \beta_1) q^{88} - 266 q^{89} + 80 \beta_{3} q^{91} + (608 \beta_{2} - 304 \beta_1) q^{92} + (224 \beta_{2} - 448 \beta_1) q^{93} + ( - 168 \beta_{3} + 336) q^{94} + (176 \beta_{3} - 224) q^{96} + 497 \beta_{2} q^{97} - 279 \beta_1 q^{98} + 3 \beta_{3} q^{99}+O(q^{100})$$ q - b1 * q^2 + (b2 - 2*b1) * q^3 + (b3 + 6) * q^4 + (b3 + 14) * q^6 - 4*b2 * q^7 + (-8*b2 - 4*b1) * q^8 + q^9 + 3*b3 * q^11 + (-8*b2 - 12*b1) * q^12 + (10*b2 - 20*b1) * q^13 + (4*b3 - 8) * q^14 + (12*b3 + 8) * q^16 - 7*b2 * q^17 - b1 * q^18 - 7*b3 * q^19 + 8*b3 * q^21 + (-24*b2 + 6*b1) * q^22 + 76*b2 * q^23 + (20*b3 + 56) * q^24 + (10*b3 + 140) * q^26 + (-26*b2 + 52*b1) * q^27 + (-32*b2 + 16*b1) * q^28 - 30*b3 * q^29 + 224 * q^31 + (-96*b2 + 16*b1) * q^32 - 42*b2 * q^33 + (7*b3 - 14) * q^34 + (b3 + 6) * q^36 + (-46*b2 + 92*b1) * q^37 + (56*b2 - 14*b1) * q^38 + 280 * q^39 - 70 * q^41 + (-64*b2 + 16*b1) * q^42 + (-83*b2 + 166*b1) * q^43 + (18*b3 - 84) * q^44 + (-76*b3 + 152) * q^46 + 168*b2 * q^47 + (-160*b2 - 16*b1) * q^48 + 279 * q^49 + 14*b3 * q^51 + (-80*b2 - 120*b1) * q^52 + (6*b2 - 12*b1) * q^53 + (-26*b3 - 364) * q^54 + (16*b3 - 160) * q^56 + 98*b2 * q^57 + (240*b2 - 60*b1) * q^58 + 101*b3 * q^59 - 18*b3 * q^61 - 224*b1 * q^62 - 4*b2 * q^63 + (80*b3 - 288) * q^64 + (42*b3 - 84) * q^66 + (-33*b2 + 66*b1) * q^67 + (-56*b2 + 28*b1) * q^68 - 152*b3 * q^69 - 72 * q^71 + (-8*b2 - 4*b1) * q^72 + 147*b2 * q^73 + (-46*b3 - 644) * q^74 + (-42*b3 + 196) * q^76 + (-24*b2 + 48*b1) * q^77 - 280*b1 * q^78 + 464 * q^79 - 755 * q^81 + 70*b1 * q^82 + (-103*b2 + 206*b1) * q^83 + (48*b3 - 224) * q^84 + (-83*b3 - 1162) * q^86 + 420*b2 * q^87 + (-144*b2 + 120*b1) * q^88 - 266 * q^89 + 80*b3 * q^91 + (608*b2 - 304*b1) * q^92 + (224*b2 - 448*b1) * q^93 + (-168*b3 + 336) * q^94 + (176*b3 - 224) * q^96 + 497*b2 * q^97 - 279*b1 * q^98 + 3*b3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 24 q^{4} + 56 q^{6} + 4 q^{9}+O(q^{10})$$ 4 * q + 24 * q^4 + 56 * q^6 + 4 * q^9 $$4 q + 24 q^{4} + 56 q^{6} + 4 q^{9} - 32 q^{14} + 32 q^{16} + 224 q^{24} + 560 q^{26} + 896 q^{31} - 56 q^{34} + 24 q^{36} + 1120 q^{39} - 280 q^{41} - 336 q^{44} + 608 q^{46} + 1116 q^{49} - 1456 q^{54} - 640 q^{56} - 1152 q^{64} - 336 q^{66} - 288 q^{71} - 2576 q^{74} + 784 q^{76} + 1856 q^{79} - 3020 q^{81} - 896 q^{84} - 4648 q^{86} - 1064 q^{89} + 1344 q^{94} - 896 q^{96}+O(q^{100})$$ 4 * q + 24 * q^4 + 56 * q^6 + 4 * q^9 - 32 * q^14 + 32 * q^16 + 224 * q^24 + 560 * q^26 + 896 * q^31 - 56 * q^34 + 24 * q^36 + 1120 * q^39 - 280 * q^41 - 336 * q^44 + 608 * q^46 + 1116 * q^49 - 1456 * q^54 - 640 * q^56 - 1152 * q^64 - 336 * q^66 - 288 * q^71 - 2576 * q^74 + 784 * q^76 + 1856 * q^79 - 3020 * q^81 - 896 * q^84 - 4648 * q^86 - 1064 * q^89 + 1344 * q^94 - 896 * q^96

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 3x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu$$ 2*v $$\beta_{2}$$ $$=$$ $$\nu^{3} - \nu$$ v^3 - v $$\beta_{3}$$ $$=$$ $$4\nu^{2} - 6$$ 4*v^2 - 6
 $$\nu$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 6 ) / 4$$ (b3 + 6) / 4 $$\nu^{3}$$ $$=$$ $$( 2\beta_{2} + \beta_1 ) / 2$$ (2*b2 + b1) / 2

## 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
 1.32288 + 0.500000i 1.32288 − 0.500000i −1.32288 + 0.500000i −1.32288 − 0.500000i
−2.64575 1.00000i −5.29150 6.00000 + 5.29150i 0 14.0000 + 5.29150i 8.00000i −10.5830 20.0000i 1.00000 0
149.2 −2.64575 + 1.00000i −5.29150 6.00000 5.29150i 0 14.0000 5.29150i 8.00000i −10.5830 + 20.0000i 1.00000 0
149.3 2.64575 1.00000i 5.29150 6.00000 5.29150i 0 14.0000 5.29150i 8.00000i 10.5830 20.0000i 1.00000 0
149.4 2.64575 + 1.00000i 5.29150 6.00000 + 5.29150i 0 14.0000 + 5.29150i 8.00000i 10.5830 + 20.0000i 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
5.b even 2 1 inner
8.b even 2 1 inner
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 200.4.f.a 4
4.b odd 2 1 800.4.f.a 4
5.b even 2 1 inner 200.4.f.a 4
5.c odd 4 1 8.4.b.a 2
5.c odd 4 1 200.4.d.a 2
8.b even 2 1 inner 200.4.f.a 4
8.d odd 2 1 800.4.f.a 4
15.e even 4 1 72.4.d.b 2
20.d odd 2 1 800.4.f.a 4
20.e even 4 1 32.4.b.a 2
20.e even 4 1 800.4.d.a 2
40.e odd 2 1 800.4.f.a 4
40.f even 2 1 inner 200.4.f.a 4
40.i odd 4 1 8.4.b.a 2
40.i odd 4 1 200.4.d.a 2
40.k even 4 1 32.4.b.a 2
40.k even 4 1 800.4.d.a 2
60.l odd 4 1 288.4.d.a 2
80.i odd 4 1 256.4.a.l 2
80.j even 4 1 256.4.a.j 2
80.s even 4 1 256.4.a.j 2
80.t odd 4 1 256.4.a.l 2
120.q odd 4 1 288.4.d.a 2
120.w even 4 1 72.4.d.b 2
240.z odd 4 1 2304.4.a.v 2
240.bb even 4 1 2304.4.a.bn 2
240.bd odd 4 1 2304.4.a.v 2
240.bf even 4 1 2304.4.a.bn 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 28$$ acting on $$S_{4}^{\mathrm{new}}(200, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 12T^{2} + 64$$
$3$ $$(T^{2} - 28)^{2}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 64)^{2}$$
$11$ $$(T^{2} + 252)^{2}$$
$13$ $$(T^{2} - 2800)^{2}$$
$17$ $$(T^{2} + 196)^{2}$$
$19$ $$(T^{2} + 1372)^{2}$$
$23$ $$(T^{2} + 23104)^{2}$$
$29$ $$(T^{2} + 25200)^{2}$$
$31$ $$(T - 224)^{4}$$
$37$ $$(T^{2} - 59248)^{2}$$
$41$ $$(T + 70)^{4}$$
$43$ $$(T^{2} - 192892)^{2}$$
$47$ $$(T^{2} + 112896)^{2}$$
$53$ $$(T^{2} - 1008)^{2}$$
$59$ $$(T^{2} + 285628)^{2}$$
$61$ $$(T^{2} + 9072)^{2}$$
$67$ $$(T^{2} - 30492)^{2}$$
$71$ $$(T + 72)^{4}$$
$73$ $$(T^{2} + 86436)^{2}$$
$79$ $$(T - 464)^{4}$$
$83$ $$(T^{2} - 297052)^{2}$$
$89$ $$(T + 266)^{4}$$
$97$ $$(T^{2} + 988036)^{2}$$