# Properties

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

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## 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: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.7488.1 Defining polynomial: $$x^{4} - 2x^{3} - 4x^{2} + 2x + 1$$ x^4 - 2*x^3 - 4*x^2 + 2*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.

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

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2\nu - 2$$ v^2 - 2*v - 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 2\nu^{2} - 4\nu + 1$$ v^3 - 2*v^2 - 4*v + 1
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2\beta _1 + 2$$ b2 + 2*b1 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2\beta_{2} + 8\beta _1 + 3$$ b3 + 2*b2 + 8*b1 + 3

## 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
 −0.326909 −1.43091 3.05896 0.698857
−1.00000 −3.05896 1.00000 −1.32691 3.05896 1.00000 −1.00000 6.35723 1.32691
1.2 −1.00000 −0.698857 1.00000 −2.43091 0.698857 1.00000 −1.00000 −2.51160 2.43091
1.3 −1.00000 0.326909 1.00000 2.05896 −0.326909 1.00000 −1.00000 −2.89313 −2.05896
1.4 −1.00000 1.43091 1.00000 −0.301143 −1.43091 1.00000 −1.00000 −0.952503 0.301143
 $$n$$: 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.bq 4
11.b odd 2 1 4598.2.a.bt yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4598.2.a.bq 4 1.a even 1 1 trivial
4598.2.a.bt 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4598))$$:

 $$T_{3}^{4} + 2T_{3}^{3} - 4T_{3}^{2} - 2T_{3} + 1$$ T3^4 + 2*T3^3 - 4*T3^2 - 2*T3 + 1 $$T_{5}^{4} + 2T_{5}^{3} - 4T_{5}^{2} - 8T_{5} - 2$$ T5^4 + 2*T5^3 - 4*T5^2 - 8*T5 - 2 $$T_{7} - 1$$ T7 - 1 $$T_{13}^{4} - 4T_{13}^{3} - 16T_{13}^{2} + 16T_{13} + 16$$ T13^4 - 4*T13^3 - 16*T13^2 + 16*T13 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{4}$$
$3$ $$T^{4} + 2 T^{3} - 4 T^{2} - 2 T + 1$$
$5$ $$T^{4} + 2 T^{3} - 4 T^{2} - 8 T - 2$$
$7$ $$(T - 1)^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4} - 4 T^{3} - 16 T^{2} + 16 T + 16$$
$17$ $$T^{4} - 4 T^{3} - 4 T^{2} + 16 T + 4$$
$19$ $$(T + 1)^{4}$$
$23$ $$T^{4} + 8 T^{3} - 30 T^{2} - 328 T - 587$$
$29$ $$T^{4} + 2 T^{3} - 72 T^{2} + 62 T - 11$$
$31$ $$T^{4} - 88 T^{2} + 72 T + 1336$$
$37$ $$T^{4} + 10 T^{3} - 16 T^{2} + 2 T + 1$$
$41$ $$T^{4} - 2 T^{3} - 88 T^{2} + \cdots + 1606$$
$43$ $$T^{4} + 16 T^{3} + 80 T^{2} + \cdots - 104$$
$47$ $$T^{4} - 46 T^{2} + 96 T + 97$$
$53$ $$T^{4} + 6 T^{3} - 28 T^{2} - 42 T - 11$$
$59$ $$T^{4} - 22 T^{3} + 104 T^{2} + \cdots - 107$$
$61$ $$T^{4} - 2 T^{3} - 168 T^{2} + \cdots + 1126$$
$67$ $$T^{4} + 20 T^{3} + 32 T^{2} + \cdots - 6512$$
$71$ $$T^{4} + 10 T^{3} - 120 T^{2} + \cdots + 4174$$
$73$ $$T^{4} - 100 T^{2} + 24 T + 1924$$
$79$ $$T^{4} + 6 T^{3} - 208 T^{2} + \cdots - 2882$$
$83$ $$T^{4} - 34 T^{3} + 332 T^{2} + \cdots - 5402$$
$89$ $$T^{4} + 2 T^{3} - 160 T^{2} + \cdots + 1678$$
$97$ $$T^{4} + 8 T^{3} - 280 T^{2} + \cdots - 4232$$
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