# Properties

 Label 200.6.a.g Level $200$ Weight $6$ Character orbit 200.a Self dual yes Analytic conductor $32.077$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$200 = 2^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 200.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.0767639626$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{129})$$ Defining polynomial: $$x^{2} - x - 32$$ x^2 - x - 32 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 40) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{129}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 6) q^{3} + (3 \beta - 26) q^{7} + (12 \beta + 309) q^{9}+O(q^{10})$$ q + (b + 6) * q^3 + (3*b - 26) * q^7 + (12*b + 309) * q^9 $$q + (\beta + 6) q^{3} + (3 \beta - 26) q^{7} + (12 \beta + 309) q^{9} + (6 \beta + 280) q^{11} + ( - 12 \beta - 694) q^{13} + (84 \beta - 74) q^{17} + (36 \beta - 500) q^{19} + ( - 8 \beta + 1392) q^{21} + ( - 123 \beta + 1226) q^{23} + (138 \beta + 6588) q^{27} + (312 \beta + 670) q^{29} + (54 \beta - 1124) q^{31} + (316 \beta + 4776) q^{33} + ( - 216 \beta + 2970) q^{37} + ( - 766 \beta - 10356) q^{39} + ( - 156 \beta + 11538) q^{41} + ( - 339 \beta - 8842) q^{43} + ( - 885 \beta + 1454) q^{47} + ( - 156 \beta - 11487) q^{49} + (430 \beta + 42900) q^{51} + (540 \beta + 2706) q^{53} + ( - 284 \beta + 15576) q^{57} + ( - 504 \beta + 31292) q^{59} + ( - 1104 \beta + 7054) q^{61} + (615 \beta + 10542) q^{63} + (543 \beta + 42706) q^{67} + (488 \beta - 56112) q^{69} + ( - 546 \beta + 23604) q^{71} + ( - 1308 \beta + 33726) q^{73} + (684 \beta + 2008) q^{77} + (2508 \beta - 32952) q^{79} + (4500 \beta + 35649) q^{81} + ( - 711 \beta - 54362) q^{83} + (2542 \beta + 165012) q^{87} + ( - 1464 \beta - 27510) q^{89} + ( - 1770 \beta - 532) q^{91} + ( - 800 \beta + 21120) q^{93} + (4620 \beta - 73834) q^{97} + (5214 \beta + 123672) q^{99}+O(q^{100})$$ q + (b + 6) * q^3 + (3*b - 26) * q^7 + (12*b + 309) * q^9 + (6*b + 280) * q^11 + (-12*b - 694) * q^13 + (84*b - 74) * q^17 + (36*b - 500) * q^19 + (-8*b + 1392) * q^21 + (-123*b + 1226) * q^23 + (138*b + 6588) * q^27 + (312*b + 670) * q^29 + (54*b - 1124) * q^31 + (316*b + 4776) * q^33 + (-216*b + 2970) * q^37 + (-766*b - 10356) * q^39 + (-156*b + 11538) * q^41 + (-339*b - 8842) * q^43 + (-885*b + 1454) * q^47 + (-156*b - 11487) * q^49 + (430*b + 42900) * q^51 + (540*b + 2706) * q^53 + (-284*b + 15576) * q^57 + (-504*b + 31292) * q^59 + (-1104*b + 7054) * q^61 + (615*b + 10542) * q^63 + (543*b + 42706) * q^67 + (488*b - 56112) * q^69 + (-546*b + 23604) * q^71 + (-1308*b + 33726) * q^73 + (684*b + 2008) * q^77 + (2508*b - 32952) * q^79 + (4500*b + 35649) * q^81 + (-711*b - 54362) * q^83 + (2542*b + 165012) * q^87 + (-1464*b - 27510) * q^89 + (-1770*b - 532) * q^91 + (-800*b + 21120) * q^93 + (4620*b - 73834) * q^97 + (5214*b + 123672) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 12 q^{3} - 52 q^{7} + 618 q^{9}+O(q^{10})$$ 2 * q + 12 * q^3 - 52 * q^7 + 618 * q^9 $$2 q + 12 q^{3} - 52 q^{7} + 618 q^{9} + 560 q^{11} - 1388 q^{13} - 148 q^{17} - 1000 q^{19} + 2784 q^{21} + 2452 q^{23} + 13176 q^{27} + 1340 q^{29} - 2248 q^{31} + 9552 q^{33} + 5940 q^{37} - 20712 q^{39} + 23076 q^{41} - 17684 q^{43} + 2908 q^{47} - 22974 q^{49} + 85800 q^{51} + 5412 q^{53} + 31152 q^{57} + 62584 q^{59} + 14108 q^{61} + 21084 q^{63} + 85412 q^{67} - 112224 q^{69} + 47208 q^{71} + 67452 q^{73} + 4016 q^{77} - 65904 q^{79} + 71298 q^{81} - 108724 q^{83} + 330024 q^{87} - 55020 q^{89} - 1064 q^{91} + 42240 q^{93} - 147668 q^{97} + 247344 q^{99}+O(q^{100})$$ 2 * q + 12 * q^3 - 52 * q^7 + 618 * q^9 + 560 * q^11 - 1388 * q^13 - 148 * q^17 - 1000 * q^19 + 2784 * q^21 + 2452 * q^23 + 13176 * q^27 + 1340 * q^29 - 2248 * q^31 + 9552 * q^33 + 5940 * q^37 - 20712 * q^39 + 23076 * q^41 - 17684 * q^43 + 2908 * q^47 - 22974 * q^49 + 85800 * q^51 + 5412 * q^53 + 31152 * q^57 + 62584 * q^59 + 14108 * q^61 + 21084 * q^63 + 85412 * q^67 - 112224 * q^69 + 47208 * q^71 + 67452 * q^73 + 4016 * q^77 - 65904 * q^79 + 71298 * q^81 - 108724 * q^83 + 330024 * q^87 - 55020 * q^89 - 1064 * q^91 + 42240 * q^93 - 147668 * q^97 + 247344 * q^99

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −5.17891 6.17891
0 −16.7156 0 0 0 −94.1469 0 36.4124 0
1.2 0 28.7156 0 0 0 42.1469 0 581.588 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 200.6.a.g 2
4.b odd 2 1 400.6.a.q 2
5.b even 2 1 40.6.a.d 2
5.c odd 4 2 200.6.c.e 4
15.d odd 2 1 360.6.a.l 2
20.d odd 2 1 80.6.a.i 2
20.e even 4 2 400.6.c.l 4
40.e odd 2 1 320.6.a.q 2
40.f even 2 1 320.6.a.w 2
60.h even 2 1 720.6.a.z 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 12T_{3} - 480$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(200))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 12T - 480$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 52T - 3968$$
$11$ $$T^{2} - 560T + 59824$$
$13$ $$T^{2} + 1388 T + 407332$$
$17$ $$T^{2} + 148 T - 3635420$$
$19$ $$T^{2} + 1000 T - 418736$$
$23$ $$T^{2} - 2452 T - 6303488$$
$29$ $$T^{2} - 1340 T - 49780604$$
$31$ $$T^{2} + 2248 T - 241280$$
$37$ $$T^{2} - 5940 T - 15253596$$
$41$ $$T^{2} - 23076 T + 120568068$$
$43$ $$T^{2} + 17684 T + 18881728$$
$47$ $$T^{2} - 2908 T - 402029984$$
$53$ $$T^{2} - 5412 T - 143143164$$
$59$ $$T^{2} - 62584 T + 848117008$$
$61$ $$T^{2} - 14108 T - 579150140$$
$67$ $$T^{2} - 85412 T + 1671660352$$
$71$ $$T^{2} - 47208 T + 403320960$$
$73$ $$T^{2} - 67452 T + 254637252$$
$79$ $$T^{2} + 65904 T - 2159838720$$
$83$ $$T^{2} + 108724 T + 2694378208$$
$89$ $$T^{2} + 55020 T - 349140636$$
$97$ $$T^{2} + 147668 T - 5562250844$$