# Properties

 Label 4598.2.a.br Level $4598$ Weight $2$ Character orbit 4598.a Self dual yes Analytic conductor $36.715$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4598 = 2 \cdot 11^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4598.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$36.7152148494$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.33452.1 Defining polynomial: $$x^{4} - x^{3} - 8x^{2} + 7x - 1$$ x^4 - x^3 - 8*x^2 + 7*x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + \beta_1 q^{3} + q^{4} + (\beta_{3} + \beta_{2} + 1) q^{5} - \beta_1 q^{6} - \beta_{3} q^{7} - q^{8} + ( - \beta_{3} + \beta_{2} + 1) q^{9}+O(q^{10})$$ q - q^2 + b1 * q^3 + q^4 + (b3 + b2 + 1) * q^5 - b1 * q^6 - b3 * q^7 - q^8 + (-b3 + b2 + 1) * q^9 $$q - q^{2} + \beta_1 q^{3} + q^{4} + (\beta_{3} + \beta_{2} + 1) q^{5} - \beta_1 q^{6} - \beta_{3} q^{7} - q^{8} + ( - \beta_{3} + \beta_{2} + 1) q^{9} + ( - \beta_{3} - \beta_{2} - 1) q^{10} + \beta_1 q^{12} + ( - \beta_{3} - \beta_{2} + 2) q^{13} + \beta_{3} q^{14} + (\beta_{2} + \beta_1 + 1) q^{15} + q^{16} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{17} + (\beta_{3} - \beta_{2} - 1) q^{18} + q^{19} + (\beta_{3} + \beta_{2} + 1) q^{20} + (2 \beta_1 - 1) q^{21} + ( - \beta_{3} + \beta_1 - 1) q^{23} - \beta_1 q^{24} + (2 \beta_{3} - \beta_{2} + \beta_1 + 4) q^{25} + (\beta_{3} + \beta_{2} - 2) q^{26} + (\beta_{2} + 2 \beta_1 - 1) q^{27} - \beta_{3} q^{28} + (2 \beta_{3} + \beta_{2} - 3 \beta_1) q^{29} + ( - \beta_{2} - \beta_1 - 1) q^{30} + ( - \beta_{2} - \beta_1 + 7) q^{31} - q^{32} + ( - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{34} + ( - 3 \beta_{3} + \beta_1 - 4) q^{35} + ( - \beta_{3} + \beta_{2} + 1) q^{36} + (2 \beta_{2} - 2 \beta_1 + 2) q^{37} - q^{38} + ( - \beta_{2} + 2 \beta_1 - 1) q^{39} + ( - \beta_{3} - \beta_{2} - 1) q^{40} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{41} + ( - 2 \beta_1 + 1) q^{42} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{43} + ( - 4 \beta_{3} - \beta_{2} + 3 \beta_1 + 1) q^{45} + (\beta_{3} - \beta_1 + 1) q^{46} + (2 \beta_{3} + 2 \beta_{2} + 6) q^{47} + \beta_1 q^{48} + (2 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{49} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 - 4) q^{50} + (\beta_{3} + \beta_1 - 3) q^{51} + ( - \beta_{3} - \beta_{2} + 2) q^{52} + (\beta_{3} - \beta_{2} - 3 \beta_1 + 1) q^{53} + ( - \beta_{2} - 2 \beta_1 + 1) q^{54} + \beta_{3} q^{56} + \beta_1 q^{57} + ( - 2 \beta_{3} - \beta_{2} + 3 \beta_1) q^{58} + ( - \beta_{3} - \beta_{2} + 3 \beta_1 - 1) q^{59} + (\beta_{2} + \beta_1 + 1) q^{60} + ( - 2 \beta_{3} - 2) q^{61} + (\beta_{2} + \beta_1 - 7) q^{62} + (\beta_{3} + 2 \beta_{2} - \beta_1 + 8) q^{63} + q^{64} + (\beta_{3} + 4 \beta_{2} - \beta_1 - 6) q^{65} + ( - \beta_{3} - 2 \beta_{2} + 4) q^{67} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{68} + ( - \beta_{3} + \beta_{2} + \beta_1 + 3) q^{69} + (3 \beta_{3} - \beta_1 + 4) q^{70} + ( - \beta_{3} + \beta_{2} - 5) q^{71} + (\beta_{3} - \beta_{2} - 1) q^{72} + ( - \beta_{3} - \beta_{2} - \beta_1 - 5) q^{73} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{74} + ( - \beta_{3} - 2 \beta_1 + 6) q^{75} + q^{76} + (\beta_{2} - 2 \beta_1 + 1) q^{78} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 4) q^{79} + (\beta_{3} + \beta_{2} + 1) q^{80} + (\beta_{3} + \beta_1 + 5) q^{81} + (2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{82} + (3 \beta_{3} - \beta_{2} - 2 \beta_1 + 7) q^{83} + (2 \beta_1 - 1) q^{84} + (2 \beta_{3} - 2 \beta_{2} + 8) q^{85} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{86} + (3 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 10) q^{87} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 4) q^{89} + (4 \beta_{3} + \beta_{2} - 3 \beta_1 - 1) q^{90} + ( - \beta_1 + 4) q^{91} + ( - \beta_{3} + \beta_1 - 1) q^{92} + (\beta_{3} - 2 \beta_{2} + 5 \beta_1 - 4) q^{93} + ( - 2 \beta_{3} - 2 \beta_{2} - 6) q^{94} + (\beta_{3} + \beta_{2} + 1) q^{95} - \beta_1 q^{96} + (4 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 4) q^{97} + ( - 2 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{98}+O(q^{100})$$ q - q^2 + b1 * q^3 + q^4 + (b3 + b2 + 1) * q^5 - b1 * q^6 - b3 * q^7 - q^8 + (-b3 + b2 + 1) * q^9 + (-b3 - b2 - 1) * q^10 + b1 * q^12 + (-b3 - b2 + 2) * q^13 + b3 * q^14 + (b2 + b1 + 1) * q^15 + q^16 + (b3 + b2 - b1 + 1) * q^17 + (b3 - b2 - 1) * q^18 + q^19 + (b3 + b2 + 1) * q^20 + (2*b1 - 1) * q^21 + (-b3 + b1 - 1) * q^23 - b1 * q^24 + (2*b3 - b2 + b1 + 4) * q^25 + (b3 + b2 - 2) * q^26 + (b2 + 2*b1 - 1) * q^27 - b3 * q^28 + (2*b3 + b2 - 3*b1) * q^29 + (-b2 - b1 - 1) * q^30 + (-b2 - b1 + 7) * q^31 - q^32 + (-b3 - b2 + b1 - 1) * q^34 + (-3*b3 + b1 - 4) * q^35 + (-b3 + b2 + 1) * q^36 + (2*b2 - 2*b1 + 2) * q^37 - q^38 + (-b2 + 2*b1 - 1) * q^39 + (-b3 - b2 - 1) * q^40 + (-2*b3 + b2 - b1 + 1) * q^41 + (-2*b1 + 1) * q^42 + (-b3 + b2 + 2*b1 - 1) * q^43 + (-4*b3 - b2 + 3*b1 + 1) * q^45 + (b3 - b1 + 1) * q^46 + (2*b3 + 2*b2 + 6) * q^47 + b1 * q^48 + (2*b3 + b2 - b1 - 1) * q^49 + (-2*b3 + b2 - b1 - 4) * q^50 + (b3 + b1 - 3) * q^51 + (-b3 - b2 + 2) * q^52 + (b3 - b2 - 3*b1 + 1) * q^53 + (-b2 - 2*b1 + 1) * q^54 + b3 * q^56 + b1 * q^57 + (-2*b3 - b2 + 3*b1) * q^58 + (-b3 - b2 + 3*b1 - 1) * q^59 + (b2 + b1 + 1) * q^60 + (-2*b3 - 2) * q^61 + (b2 + b1 - 7) * q^62 + (b3 + 2*b2 - b1 + 8) * q^63 + q^64 + (b3 + 4*b2 - b1 - 6) * q^65 + (-b3 - 2*b2 + 4) * q^67 + (b3 + b2 - b1 + 1) * q^68 + (-b3 + b2 + b1 + 3) * q^69 + (3*b3 - b1 + 4) * q^70 + (-b3 + b2 - 5) * q^71 + (b3 - b2 - 1) * q^72 + (-b3 - b2 - b1 - 5) * q^73 + (-2*b2 + 2*b1 - 2) * q^74 + (-b3 - 2*b1 + 6) * q^75 + q^76 + (b2 - 2*b1 + 1) * q^78 + (-2*b3 + 2*b2 + 2*b1 - 4) * q^79 + (b3 + b2 + 1) * q^80 + (b3 + b1 + 5) * q^81 + (2*b3 - b2 + b1 - 1) * q^82 + (3*b3 - b2 - 2*b1 + 7) * q^83 + (2*b1 - 1) * q^84 + (2*b3 - 2*b2 + 8) * q^85 + (b3 - b2 - 2*b1 + 1) * q^86 + (3*b3 - 2*b2 - 2*b1 - 10) * q^87 + (2*b3 - 2*b2 + 2*b1 + 4) * q^89 + (4*b3 + b2 - 3*b1 - 1) * q^90 + (-b1 + 4) * q^91 + (-b3 + b1 - 1) * q^92 + (b3 - 2*b2 + 5*b1 - 4) * q^93 + (-2*b3 - 2*b2 - 6) * q^94 + (b3 + b2 + 1) * q^95 - b1 * q^96 + (4*b3 + 2*b2 - 4*b1 + 4) * q^97 + (-2*b3 - b2 + b1 + 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} + q^{3} + 4 q^{4} + 3 q^{5} - q^{6} + q^{7} - 4 q^{8} + 5 q^{9}+O(q^{10})$$ 4 * q - 4 * q^2 + q^3 + 4 * q^4 + 3 * q^5 - q^6 + q^7 - 4 * q^8 + 5 * q^9 $$4 q - 4 q^{2} + q^{3} + 4 q^{4} + 3 q^{5} - q^{6} + q^{7} - 4 q^{8} + 5 q^{9} - 3 q^{10} + q^{12} + 9 q^{13} - q^{14} + 5 q^{15} + 4 q^{16} + 2 q^{17} - 5 q^{18} + 4 q^{19} + 3 q^{20} - 2 q^{21} - 2 q^{23} - q^{24} + 15 q^{25} - 9 q^{26} - 2 q^{27} + q^{28} - 5 q^{29} - 5 q^{30} + 27 q^{31} - 4 q^{32} - 2 q^{34} - 12 q^{35} + 5 q^{36} + 6 q^{37} - 4 q^{38} - 2 q^{39} - 3 q^{40} + 5 q^{41} + 2 q^{42} - q^{43} + 11 q^{45} + 2 q^{46} + 22 q^{47} + q^{48} - 7 q^{49} - 15 q^{50} - 12 q^{51} + 9 q^{52} + 2 q^{54} - q^{56} + q^{57} + 5 q^{58} + 5 q^{60} - 6 q^{61} - 27 q^{62} + 30 q^{63} + 4 q^{64} - 26 q^{65} + 17 q^{67} + 2 q^{68} + 14 q^{69} + 12 q^{70} - 19 q^{71} - 5 q^{72} - 20 q^{73} - 6 q^{74} + 23 q^{75} + 4 q^{76} + 2 q^{78} - 12 q^{79} + 3 q^{80} + 20 q^{81} - 5 q^{82} + 23 q^{83} - 2 q^{84} + 30 q^{85} + q^{86} - 45 q^{87} + 16 q^{89} - 11 q^{90} + 15 q^{91} - 2 q^{92} - 12 q^{93} - 22 q^{94} + 3 q^{95} - q^{96} + 8 q^{97} + 7 q^{98}+O(q^{100})$$ 4 * q - 4 * q^2 + q^3 + 4 * q^4 + 3 * q^5 - q^6 + q^7 - 4 * q^8 + 5 * q^9 - 3 * q^10 + q^12 + 9 * q^13 - q^14 + 5 * q^15 + 4 * q^16 + 2 * q^17 - 5 * q^18 + 4 * q^19 + 3 * q^20 - 2 * q^21 - 2 * q^23 - q^24 + 15 * q^25 - 9 * q^26 - 2 * q^27 + q^28 - 5 * q^29 - 5 * q^30 + 27 * q^31 - 4 * q^32 - 2 * q^34 - 12 * q^35 + 5 * q^36 + 6 * q^37 - 4 * q^38 - 2 * q^39 - 3 * q^40 + 5 * q^41 + 2 * q^42 - q^43 + 11 * q^45 + 2 * q^46 + 22 * q^47 + q^48 - 7 * q^49 - 15 * q^50 - 12 * q^51 + 9 * q^52 + 2 * q^54 - q^56 + q^57 + 5 * q^58 + 5 * q^60 - 6 * q^61 - 27 * q^62 + 30 * q^63 + 4 * q^64 - 26 * q^65 + 17 * q^67 + 2 * q^68 + 14 * q^69 + 12 * q^70 - 19 * q^71 - 5 * q^72 - 20 * q^73 - 6 * q^74 + 23 * q^75 + 4 * q^76 + 2 * q^78 - 12 * q^79 + 3 * q^80 + 20 * q^81 - 5 * q^82 + 23 * q^83 - 2 * q^84 + 30 * q^85 + q^86 - 45 * q^87 + 16 * q^89 - 11 * q^90 + 15 * q^91 - 2 * q^92 - 12 * q^93 - 22 * q^94 + 3 * q^95 - q^96 + 8 * q^97 + 7 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 8x^{2} + 7x - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{3} - 8\nu + 1$$ v^3 - 8*v + 1 $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 8\nu + 5$$ v^3 - v^2 - 8*v + 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{3} + \beta_{2} + 4$$ -b3 + b2 + 4 $$\nu^{3}$$ $$=$$ $$\beta_{2} + 8\beta _1 - 1$$ b2 + 8*b1 - 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.79836 0.180986 0.669883 2.94749
−1.00000 −2.79836 1.00000 0.116104 2.79836 2.35735 −1.00000 4.83081 −0.116104
1.2 −1.00000 0.180986 1.00000 4.08332 −0.180986 −3.52528 −1.00000 −2.96724 −4.08332
1.3 −1.00000 0.669883 1.00000 −3.56566 −0.669883 0.507202 −1.00000 −2.55126 3.56566
1.4 −1.00000 2.94749 1.00000 2.36624 −2.94749 1.66073 −1.00000 5.68769 −2.36624
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$11$$ $$1$$
$$19$$ $$-1$$

## 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 4598.2.a.br 4
11.b odd 2 1 4598.2.a.bu yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4598.2.a.br 4 1.a even 1 1 trivial
4598.2.a.bu yes 4 11.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4598))$$:

 $$T_{3}^{4} - T_{3}^{3} - 8T_{3}^{2} + 7T_{3} - 1$$ T3^4 - T3^3 - 8*T3^2 + 7*T3 - 1 $$T_{5}^{4} - 3T_{5}^{3} - 13T_{5}^{2} + 36T_{5} - 4$$ T5^4 - 3*T5^3 - 13*T5^2 + 36*T5 - 4 $$T_{7}^{4} - T_{7}^{3} - 10T_{7}^{2} + 19T_{7} - 7$$ T7^4 - T7^3 - 10*T7^2 + 19*T7 - 7 $$T_{13}^{4} - 9T_{13}^{3} + 14T_{13}^{2} + 15T_{13} - 13$$ T13^4 - 9*T13^3 + 14*T13^2 + 15*T13 - 13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{4}$$
$3$ $$T^{4} - T^{3} - 8 T^{2} + 7 T - 1$$
$5$ $$T^{4} - 3 T^{3} - 13 T^{2} + 36 T - 4$$
$7$ $$T^{4} - T^{3} - 10 T^{2} + 19 T - 7$$
$11$ $$T^{4}$$
$13$ $$T^{4} - 9 T^{3} + 14 T^{2} + 15 T - 13$$
$17$ $$T^{4} - 2 T^{3} - 19 T^{2} + 38 T + 28$$
$19$ $$(T - 1)^{4}$$
$23$ $$T^{4} + 2 T^{3} - 15 T^{2} - 20 T + 4$$
$29$ $$T^{4} + 5 T^{3} - 86 T^{2} + \cdots + 2023$$
$31$ $$T^{4} - 27 T^{3} + 249 T^{2} + \cdots + 644$$
$37$ $$T^{4} - 6 T^{3} - 68 T^{2} + 224 T - 128$$
$41$ $$T^{4} - 5 T^{3} - 73 T^{2} + 152 T + 796$$
$43$ $$T^{4} + T^{3} - 65 T^{2} - 308 T - 392$$
$47$ $$T^{4} - 22 T^{3} + 116 T^{2} + \cdots - 1408$$
$53$ $$T^{4} - 107 T^{2} + 564 T - 784$$
$59$ $$T^{4} - 79 T^{2} + 72 T + 1088$$
$61$ $$T^{4} + 6 T^{3} - 28 T^{2} + 32$$
$67$ $$T^{4} - 17 T^{3} + 58 T^{2} - 33 T - 23$$
$71$ $$T^{4} + 19 T^{3} + 103 T^{2} + \cdots + 28$$
$73$ $$T^{4} + 20 T^{3} + 121 T^{2} + \cdots + 112$$
$79$ $$T^{4} + 12 T^{3} - 108 T^{2} + \cdots - 2464$$
$83$ $$T^{4} - 23 T^{3} + 43 T^{2} + \cdots - 4048$$
$89$ $$T^{4} - 16 T^{3} - 68 T^{2} + \cdots - 736$$
$97$ $$T^{4} - 8 T^{3} - 216 T^{2} + \cdots + 10624$$