# Properties

 Label 6498.2.a.ca Level $6498$ Weight $2$ Character orbit 6498.a Self dual yes Analytic conductor $51.887$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6498 = 2 \cdot 3^{2} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6498.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$51.8867912334$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{20})^+$$ Defining polynomial: $$x^{4} - 5x^{2} + 5$$ x^4 - 5*x^2 + 5 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 722) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{5} + (\beta_{2} + \beta_1) q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + (b3 + b2 - b1 + 1) * q^5 + (b2 + b1) * q^7 + q^8 $$q + q^{2} + q^{4} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{5} + (\beta_{2} + \beta_1) q^{7} + q^{8} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{10} + ( - 3 \beta_{2} + \beta_1 - 2) q^{11} + ( - \beta_{3} + \beta_{2} - \beta_1 - 4) q^{13} + (\beta_{2} + \beta_1) q^{14} + q^{16} + (\beta_{2} + 2 \beta_1 - 1) q^{17} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{20} + ( - 3 \beta_{2} + \beta_1 - 2) q^{22} + ( - 3 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 1) q^{23} + ( - 2 \beta_{3} - 3 \beta_{2}) q^{25} + ( - \beta_{3} + \beta_{2} - \beta_1 - 4) q^{26} + (\beta_{2} + \beta_1) q^{28} + (\beta_{3} - 3 \beta_{2} - \beta_1 - 2) q^{29} + ( - \beta_{3} - \beta_{2} - 7) q^{31} + q^{32} + (\beta_{2} + 2 \beta_1 - 1) q^{34} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{35} + ( - \beta_{3} - 2 \beta_1 - 1) q^{37} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{40} + (2 \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{41} + (\beta_{3} - \beta_{2} - 2 \beta_1 - 3) q^{43} + ( - 3 \beta_{2} + \beta_1 - 2) q^{44} + ( - 3 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 1) q^{46} + (\beta_{3} + 4 \beta_{2} + \beta_1 + 5) q^{47} + (2 \beta_{3} - 3) q^{49} + ( - 2 \beta_{3} - 3 \beta_{2}) q^{50} + ( - \beta_{3} + \beta_{2} - \beta_1 - 4) q^{52} + ( - 3 \beta_{3} - 2 \beta_{2} + 1) q^{53} + (5 \beta_{3} - \beta_{2} - 7) q^{55} + (\beta_{2} + \beta_1) q^{56} + (\beta_{3} - 3 \beta_{2} - \beta_1 - 2) q^{58} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{59} + (2 \beta_{3} - 6 \beta_{2} + \beta_1 - 3) q^{61} + ( - \beta_{3} - \beta_{2} - 7) q^{62} + q^{64} + ( - 7 \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 2) q^{65} + ( - 3 \beta_{3} + 3 \beta_{2} - 1) q^{67} + (\beta_{2} + 2 \beta_1 - 1) q^{68} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{70} + ( - 3 \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 1) q^{71} + ( - 2 \beta_{3} - \beta_{2} - 4) q^{73} + ( - \beta_{3} - 2 \beta_1 - 1) q^{74} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{77} + (3 \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 7) q^{79} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{80} + (2 \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{82} + ( - 3 \beta_{3} + 8 \beta_{2} + 3 \beta_1 + 7) q^{83} + ( - \beta_{3} + \beta_{2} + 4 \beta_1 - 4) q^{85} + (\beta_{3} - \beta_{2} - 2 \beta_1 - 3) q^{86} + ( - 3 \beta_{2} + \beta_1 - 2) q^{88} + (4 \beta_{3} - 4 \beta_{2} + \beta_1 - 6) q^{89} + (\beta_{3} - 8 \beta_{2} - 5 \beta_1 - 3) q^{91} + ( - 3 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 1) q^{92} + (\beta_{3} + 4 \beta_{2} + \beta_1 + 5) q^{94} + (\beta_{3} + 2 \beta_{2} + 6 \beta_1 - 6) q^{97} + (2 \beta_{3} - 3) q^{98}+O(q^{100})$$ q + q^2 + q^4 + (b3 + b2 - b1 + 1) * q^5 + (b2 + b1) * q^7 + q^8 + (b3 + b2 - b1 + 1) * q^10 + (-3*b2 + b1 - 2) * q^11 + (-b3 + b2 - b1 - 4) * q^13 + (b2 + b1) * q^14 + q^16 + (b2 + 2*b1 - 1) * q^17 + (b3 + b2 - b1 + 1) * q^20 + (-3*b2 + b1 - 2) * q^22 + (-3*b3 - 3*b2 - 2*b1 + 1) * q^23 + (-2*b3 - 3*b2) * q^25 + (-b3 + b2 - b1 - 4) * q^26 + (b2 + b1) * q^28 + (b3 - 3*b2 - b1 - 2) * q^29 + (-b3 - b2 - 7) * q^31 + q^32 + (b2 + 2*b1 - 1) * q^34 + (-b3 + b2 + 2*b1 - 1) * q^35 + (-b3 - 2*b1 - 1) * q^37 + (b3 + b2 - b1 + 1) * q^40 + (2*b3 + 2*b2 + b1 - 2) * q^41 + (b3 - b2 - 2*b1 - 3) * q^43 + (-3*b2 + b1 - 2) * q^44 + (-3*b3 - 3*b2 - 2*b1 + 1) * q^46 + (b3 + 4*b2 + b1 + 5) * q^47 + (2*b3 - 3) * q^49 + (-2*b3 - 3*b2) * q^50 + (-b3 + b2 - b1 - 4) * q^52 + (-3*b3 - 2*b2 + 1) * q^53 + (5*b3 - b2 - 7) * q^55 + (b2 + b1) * q^56 + (b3 - 3*b2 - b1 - 2) * q^58 + (-b3 + 2*b2 - b1 - 1) * q^59 + (2*b3 - 6*b2 + b1 - 3) * q^61 + (-b3 - b2 - 7) * q^62 + q^64 + (-7*b3 - 2*b2 + 3*b1 - 2) * q^65 + (-3*b3 + 3*b2 - 1) * q^67 + (b2 + 2*b1 - 1) * q^68 + (-b3 + b2 + 2*b1 - 1) * q^70 + (-3*b3 - 2*b2 + 3*b1 - 1) * q^71 + (-2*b3 - b2 - 4) * q^73 + (-b3 - 2*b1 - 1) * q^74 + (-2*b3 + 2*b2 - 2*b1) * q^77 + (3*b3 - 3*b2 - 2*b1 - 7) * q^79 + (b3 + b2 - b1 + 1) * q^80 + (2*b3 + 2*b2 + b1 - 2) * q^82 + (-3*b3 + 8*b2 + 3*b1 + 7) * q^83 + (-b3 + b2 + 4*b1 - 4) * q^85 + (b3 - b2 - 2*b1 - 3) * q^86 + (-3*b2 + b1 - 2) * q^88 + (4*b3 - 4*b2 + b1 - 6) * q^89 + (b3 - 8*b2 - 5*b1 - 3) * q^91 + (-3*b3 - 3*b2 - 2*b1 + 1) * q^92 + (b3 + 4*b2 + b1 + 5) * q^94 + (b3 + 2*b2 + 6*b1 - 6) * q^97 + (2*b3 - 3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{2} + 4 q^{4} + 2 q^{5} - 2 q^{7} + 4 q^{8}+O(q^{10})$$ 4 * q + 4 * q^2 + 4 * q^4 + 2 * q^5 - 2 * q^7 + 4 * q^8 $$4 q + 4 q^{2} + 4 q^{4} + 2 q^{5} - 2 q^{7} + 4 q^{8} + 2 q^{10} - 2 q^{11} - 18 q^{13} - 2 q^{14} + 4 q^{16} - 6 q^{17} + 2 q^{20} - 2 q^{22} + 10 q^{23} + 6 q^{25} - 18 q^{26} - 2 q^{28} - 2 q^{29} - 26 q^{31} + 4 q^{32} - 6 q^{34} - 6 q^{35} - 4 q^{37} + 2 q^{40} - 12 q^{41} - 10 q^{43} - 2 q^{44} + 10 q^{46} + 12 q^{47} - 12 q^{49} + 6 q^{50} - 18 q^{52} + 8 q^{53} - 26 q^{55} - 2 q^{56} - 2 q^{58} - 8 q^{59} - 26 q^{62} + 4 q^{64} - 4 q^{65} - 10 q^{67} - 6 q^{68} - 6 q^{70} - 14 q^{73} - 4 q^{74} - 4 q^{77} - 22 q^{79} + 2 q^{80} - 12 q^{82} + 12 q^{83} - 18 q^{85} - 10 q^{86} - 2 q^{88} - 16 q^{89} + 4 q^{91} + 10 q^{92} + 12 q^{94} - 28 q^{97} - 12 q^{98}+O(q^{100})$$ 4 * q + 4 * q^2 + 4 * q^4 + 2 * q^5 - 2 * q^7 + 4 * q^8 + 2 * q^10 - 2 * q^11 - 18 * q^13 - 2 * q^14 + 4 * q^16 - 6 * q^17 + 2 * q^20 - 2 * q^22 + 10 * q^23 + 6 * q^25 - 18 * q^26 - 2 * q^28 - 2 * q^29 - 26 * q^31 + 4 * q^32 - 6 * q^34 - 6 * q^35 - 4 * q^37 + 2 * q^40 - 12 * q^41 - 10 * q^43 - 2 * q^44 + 10 * q^46 + 12 * q^47 - 12 * q^49 + 6 * q^50 - 18 * q^52 + 8 * q^53 - 26 * q^55 - 2 * q^56 - 2 * q^58 - 8 * q^59 - 26 * q^62 + 4 * q^64 - 4 * q^65 - 10 * q^67 - 6 * q^68 - 6 * q^70 - 14 * q^73 - 4 * q^74 - 4 * q^77 - 22 * q^79 + 2 * q^80 - 12 * q^82 + 12 * q^83 - 18 * q^85 - 10 * q^86 - 2 * q^88 - 16 * q^89 + 4 * q^91 + 10 * q^92 + 12 * q^94 - 28 * q^97 - 12 * q^98

Basis of coefficient ring in terms of $$\nu = \zeta_{20} + \zeta_{20}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 3\nu$$ v^3 - 3*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 3\beta_1$$ b3 + 3*b1

## 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
 1.17557 1.90211 −1.90211 −1.17557
1.00000 0 1.00000 −3.69572 0 −0.442463 1.00000 0 −3.69572
1.2 1.00000 0 1.00000 0.891491 0 2.52015 1.00000 0 0.891491
1.3 1.00000 0 1.00000 2.34458 0 −1.28408 1.00000 0 2.34458
1.4 1.00000 0 1.00000 2.45965 0 −2.79360 1.00000 0 2.45965
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$19$$ $$1$$

## 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 6498.2.a.ca 4
3.b odd 2 1 722.2.a.m 4
12.b even 2 1 5776.2.a.bv 4
19.b odd 2 1 6498.2.a.bx 4
57.d even 2 1 722.2.a.n yes 4
57.f even 6 2 722.2.c.m 8
57.h odd 6 2 722.2.c.n 8
57.j even 18 6 722.2.e.r 24
57.l odd 18 6 722.2.e.s 24
228.b odd 2 1 5776.2.a.bt 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
722.2.a.m 4 3.b odd 2 1
722.2.a.n yes 4 57.d even 2 1
722.2.c.m 8 57.f even 6 2
722.2.c.n 8 57.h odd 6 2
722.2.e.r 24 57.j even 18 6
722.2.e.s 24 57.l odd 18 6
5776.2.a.bt 4 228.b odd 2 1
5776.2.a.bv 4 12.b even 2 1
6498.2.a.bx 4 19.b odd 2 1
6498.2.a.ca 4 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6498))$$:

 $$T_{5}^{4} - 2T_{5}^{3} - 11T_{5}^{2} + 32T_{5} - 19$$ T5^4 - 2*T5^3 - 11*T5^2 + 32*T5 - 19 $$T_{7}^{4} + 2T_{7}^{3} - 6T_{7}^{2} - 12T_{7} - 4$$ T7^4 + 2*T7^3 - 6*T7^2 - 12*T7 - 4 $$T_{11}^{4} + 2T_{11}^{3} - 26T_{11}^{2} - 12T_{11} + 76$$ T11^4 + 2*T11^3 - 26*T11^2 - 12*T11 + 76 $$T_{13}^{4} + 18T_{13}^{3} + 109T_{13}^{2} + 232T_{13} + 61$$ T13^4 + 18*T13^3 + 109*T13^2 + 232*T13 + 61 $$T_{29}^{4} + 2T_{29}^{3} - 31T_{29}^{2} - 92T_{29} - 19$$ T29^4 + 2*T29^3 - 31*T29^2 - 92*T29 - 19

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 2 T^{3} - 11 T^{2} + 32 T - 19$$
$7$ $$T^{4} + 2 T^{3} - 6 T^{2} - 12 T - 4$$
$11$ $$T^{4} + 2 T^{3} - 26 T^{2} - 12 T + 76$$
$13$ $$T^{4} + 18 T^{3} + 109 T^{2} + \cdots + 61$$
$17$ $$T^{4} + 6 T^{3} - 9 T^{2} - 74 T - 19$$
$19$ $$T^{4}$$
$23$ $$T^{4} - 10 T^{3} - 50 T^{2} + \cdots - 1220$$
$29$ $$T^{4} + 2 T^{3} - 31 T^{2} - 92 T - 19$$
$31$ $$T^{4} + 26 T^{3} + 246 T^{2} + \cdots + 1436$$
$37$ $$T^{4} + 4 T^{3} - 19 T^{2} - 46 T - 19$$
$41$ $$T^{4} + 12 T^{3} + 19 T^{2} + \cdots - 359$$
$43$ $$T^{4} + 10 T^{3} + 10 T^{2} + \cdots - 100$$
$47$ $$T^{4} - 12 T^{3} + 4 T^{2} + 112 T + 76$$
$53$ $$T^{4} - 8 T^{3} - 31 T^{2} + 98 T + 181$$
$59$ $$T^{4} + 8 T^{3} + 4 T^{2} - 88 T - 164$$
$61$ $$T^{4} - 115 T^{2} + 150 T + 1025$$
$67$ $$T^{4} + 10 T^{3} - 30 T^{2} - 140 T - 20$$
$71$ $$T^{4} - 100 T^{2} - 360 T - 20$$
$73$ $$T^{4} + 14 T^{3} + 51 T^{2} + \cdots - 139$$
$79$ $$T^{4} + 22 T^{3} + 94 T^{2} + \cdots - 4724$$
$83$ $$T^{4} - 12 T^{3} - 196 T^{2} + \cdots - 6884$$
$89$ $$T^{4} + 16 T^{3} - 29 T^{2} + \cdots - 1159$$
$97$ $$T^{4} + 28 T^{3} + 99 T^{2} + \cdots - 7979$$