# Properties

 Label 1850.2.a.bb Level $1850$ Weight $2$ Character orbit 1850.a Self dual yes Analytic conductor $14.772$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.7723243739$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + ( - \beta_{2} - 1) q^{3} + q^{4} + ( - \beta_{2} - 1) q^{6} + (\beta_{2} + 2 \beta_1 - 2) q^{7} + q^{8} + (\beta_{2} - \beta_1 + 1) q^{9}+O(q^{10})$$ q + q^2 + (-b2 - 1) * q^3 + q^4 + (-b2 - 1) * q^6 + (b2 + 2*b1 - 2) * q^7 + q^8 + (b2 - b1 + 1) * q^9 $$q + q^{2} + ( - \beta_{2} - 1) q^{3} + q^{4} + ( - \beta_{2} - 1) q^{6} + (\beta_{2} + 2 \beta_1 - 2) q^{7} + q^{8} + (\beta_{2} - \beta_1 + 1) q^{9} + (2 \beta_{2} - \beta_1) q^{11} + ( - \beta_{2} - 1) q^{12} + ( - \beta_{2} - 4) q^{13} + (\beta_{2} + 2 \beta_1 - 2) q^{14} + q^{16} + (\beta_{2} - 2 \beta_1 + 1) q^{17} + (\beta_{2} - \beta_1 + 1) q^{18} + ( - \beta_{2} - 3 \beta_1 + 2) q^{19} + (2 \beta_{2} - 3 \beta_1 + 1) q^{21} + (2 \beta_{2} - \beta_1) q^{22} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{23} + ( - \beta_{2} - 1) q^{24} + ( - \beta_{2} - 4) q^{26} + (2 \beta_{2} + 3 \beta_1 - 2) q^{27} + (\beta_{2} + 2 \beta_1 - 2) q^{28} + (4 \beta_1 - 2) q^{29} + (3 \beta_1 - 3) q^{31} + q^{32} + (4 \beta_1 - 7) q^{33} + (\beta_{2} - 2 \beta_1 + 1) q^{34} + (\beta_{2} - \beta_1 + 1) q^{36} - q^{37} + ( - \beta_{2} - 3 \beta_1 + 2) q^{38} + (4 \beta_{2} - \beta_1 + 7) q^{39} + ( - 2 \beta_{2} - 7) q^{41} + (2 \beta_{2} - 3 \beta_1 + 1) q^{42} + (3 \beta_{2} - 5 \beta_1 - 1) q^{43} + (2 \beta_{2} - \beta_1) q^{44} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{46} - 4 \beta_1 q^{47} + ( - \beta_{2} - 1) q^{48} + ( - \beta_{2} - \beta_1 + 4) q^{49} + ( - \beta_{2} + 5 \beta_1 - 6) q^{51} + ( - \beta_{2} - 4) q^{52} + (2 \beta_{2} + 6 \beta_1 - 4) q^{53} + (2 \beta_{2} + 3 \beta_1 - 2) q^{54} + (\beta_{2} + 2 \beta_1 - 2) q^{56} + ( - 2 \beta_{2} + 5 \beta_1 - 2) q^{57} + (4 \beta_1 - 2) q^{58} + ( - 3 \beta_{2} + 5 \beta_1 + 1) q^{59} + (3 \beta_{2} + 4 \beta_1 - 4) q^{61} + (3 \beta_1 - 3) q^{62} + ( - 4 \beta_{2} + 2 \beta_1 - 4) q^{63} + q^{64} + (4 \beta_1 - 7) q^{66} + ( - 3 \beta_{2} - 4 \beta_1 - 3) q^{67} + (\beta_{2} - 2 \beta_1 + 1) q^{68} + (2 \beta_{2} + 2 \beta_1 + 6) q^{69} + (3 \beta_{2} + 4 \beta_1 - 4) q^{71} + (\beta_{2} - \beta_1 + 1) q^{72} + ( - \beta_{2} - \beta_1 - 10) q^{73} - q^{74} + ( - \beta_{2} - 3 \beta_1 + 2) q^{76} + ( - 8 \beta_{2} + \beta_1 - 1) q^{77} + (4 \beta_{2} - \beta_1 + 7) q^{78} + ( - 2 \beta_{2} - 2 \beta_1 + 8) q^{79} + ( - \beta_{2} - \beta_1 - 4) q^{81} + ( - 2 \beta_{2} - 7) q^{82} + ( - 8 \beta_{2} - \beta_1 - 2) q^{83} + (2 \beta_{2} - 3 \beta_1 + 1) q^{84} + (3 \beta_{2} - 5 \beta_1 - 1) q^{86} + (2 \beta_{2} - 8 \beta_1 + 6) q^{87} + (2 \beta_{2} - \beta_1) q^{88} - 3 \beta_1 q^{89} + ( - \beta_{2} - 9 \beta_1 + 7) q^{91} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{92} + (3 \beta_{2} - 6 \beta_1 + 6) q^{93} - 4 \beta_1 q^{94} + ( - \beta_{2} - 1) q^{96} + ( - 2 \beta_{2} - 2) q^{97} + ( - \beta_{2} - \beta_1 + 4) q^{98} + (\beta_{2} - 5 \beta_1 + 11) q^{99}+O(q^{100})$$ q + q^2 + (-b2 - 1) * q^3 + q^4 + (-b2 - 1) * q^6 + (b2 + 2*b1 - 2) * q^7 + q^8 + (b2 - b1 + 1) * q^9 + (2*b2 - b1) * q^11 + (-b2 - 1) * q^12 + (-b2 - 4) * q^13 + (b2 + 2*b1 - 2) * q^14 + q^16 + (b2 - 2*b1 + 1) * q^17 + (b2 - b1 + 1) * q^18 + (-b2 - 3*b1 + 2) * q^19 + (2*b2 - 3*b1 + 1) * q^21 + (2*b2 - b1) * q^22 + (-2*b2 - 2*b1 - 2) * q^23 + (-b2 - 1) * q^24 + (-b2 - 4) * q^26 + (2*b2 + 3*b1 - 2) * q^27 + (b2 + 2*b1 - 2) * q^28 + (4*b1 - 2) * q^29 + (3*b1 - 3) * q^31 + q^32 + (4*b1 - 7) * q^33 + (b2 - 2*b1 + 1) * q^34 + (b2 - b1 + 1) * q^36 - q^37 + (-b2 - 3*b1 + 2) * q^38 + (4*b2 - b1 + 7) * q^39 + (-2*b2 - 7) * q^41 + (2*b2 - 3*b1 + 1) * q^42 + (3*b2 - 5*b1 - 1) * q^43 + (2*b2 - b1) * q^44 + (-2*b2 - 2*b1 - 2) * q^46 - 4*b1 * q^47 + (-b2 - 1) * q^48 + (-b2 - b1 + 4) * q^49 + (-b2 + 5*b1 - 6) * q^51 + (-b2 - 4) * q^52 + (2*b2 + 6*b1 - 4) * q^53 + (2*b2 + 3*b1 - 2) * q^54 + (b2 + 2*b1 - 2) * q^56 + (-2*b2 + 5*b1 - 2) * q^57 + (4*b1 - 2) * q^58 + (-3*b2 + 5*b1 + 1) * q^59 + (3*b2 + 4*b1 - 4) * q^61 + (3*b1 - 3) * q^62 + (-4*b2 + 2*b1 - 4) * q^63 + q^64 + (4*b1 - 7) * q^66 + (-3*b2 - 4*b1 - 3) * q^67 + (b2 - 2*b1 + 1) * q^68 + (2*b2 + 2*b1 + 6) * q^69 + (3*b2 + 4*b1 - 4) * q^71 + (b2 - b1 + 1) * q^72 + (-b2 - b1 - 10) * q^73 - q^74 + (-b2 - 3*b1 + 2) * q^76 + (-8*b2 + b1 - 1) * q^77 + (4*b2 - b1 + 7) * q^78 + (-2*b2 - 2*b1 + 8) * q^79 + (-b2 - b1 - 4) * q^81 + (-2*b2 - 7) * q^82 + (-8*b2 - b1 - 2) * q^83 + (2*b2 - 3*b1 + 1) * q^84 + (3*b2 - 5*b1 - 1) * q^86 + (2*b2 - 8*b1 + 6) * q^87 + (2*b2 - b1) * q^88 - 3*b1 * q^89 + (-b2 - 9*b1 + 7) * q^91 + (-2*b2 - 2*b1 - 2) * q^92 + (3*b2 - 6*b1 + 6) * q^93 - 4*b1 * q^94 + (-b2 - 1) * q^96 + (-2*b2 - 2) * q^97 + (-b2 - b1 + 4) * q^98 + (b2 - 5*b1 + 11) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} - 4 q^{7} + 3 q^{8} + 2 q^{9}+O(q^{10})$$ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 - 4 * q^7 + 3 * q^8 + 2 * q^9 $$3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} - 4 q^{7} + 3 q^{8} + 2 q^{9} - q^{11} - 3 q^{12} - 12 q^{13} - 4 q^{14} + 3 q^{16} + q^{17} + 2 q^{18} + 3 q^{19} - q^{22} - 8 q^{23} - 3 q^{24} - 12 q^{26} - 3 q^{27} - 4 q^{28} - 2 q^{29} - 6 q^{31} + 3 q^{32} - 17 q^{33} + q^{34} + 2 q^{36} - 3 q^{37} + 3 q^{38} + 20 q^{39} - 21 q^{41} - 8 q^{43} - q^{44} - 8 q^{46} - 4 q^{47} - 3 q^{48} + 11 q^{49} - 13 q^{51} - 12 q^{52} - 6 q^{53} - 3 q^{54} - 4 q^{56} - q^{57} - 2 q^{58} + 8 q^{59} - 8 q^{61} - 6 q^{62} - 10 q^{63} + 3 q^{64} - 17 q^{66} - 13 q^{67} + q^{68} + 20 q^{69} - 8 q^{71} + 2 q^{72} - 31 q^{73} - 3 q^{74} + 3 q^{76} - 2 q^{77} + 20 q^{78} + 22 q^{79} - 13 q^{81} - 21 q^{82} - 7 q^{83} - 8 q^{86} + 10 q^{87} - q^{88} - 3 q^{89} + 12 q^{91} - 8 q^{92} + 12 q^{93} - 4 q^{94} - 3 q^{96} - 6 q^{97} + 11 q^{98} + 28 q^{99}+O(q^{100})$$ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 - 4 * q^7 + 3 * q^8 + 2 * q^9 - q^11 - 3 * q^12 - 12 * q^13 - 4 * q^14 + 3 * q^16 + q^17 + 2 * q^18 + 3 * q^19 - q^22 - 8 * q^23 - 3 * q^24 - 12 * q^26 - 3 * q^27 - 4 * q^28 - 2 * q^29 - 6 * q^31 + 3 * q^32 - 17 * q^33 + q^34 + 2 * q^36 - 3 * q^37 + 3 * q^38 + 20 * q^39 - 21 * q^41 - 8 * q^43 - q^44 - 8 * q^46 - 4 * q^47 - 3 * q^48 + 11 * q^49 - 13 * q^51 - 12 * q^52 - 6 * q^53 - 3 * q^54 - 4 * q^56 - q^57 - 2 * q^58 + 8 * q^59 - 8 * q^61 - 6 * q^62 - 10 * q^63 + 3 * q^64 - 17 * q^66 - 13 * q^67 + q^68 + 20 * q^69 - 8 * q^71 + 2 * q^72 - 31 * q^73 - 3 * q^74 + 3 * q^76 - 2 * q^77 + 20 * q^78 + 22 * q^79 - 13 * q^81 - 21 * q^82 - 7 * q^83 - 8 * q^86 + 10 * q^87 - q^88 - 3 * q^89 + 12 * q^91 - 8 * q^92 + 12 * q^93 - 4 * q^94 - 3 * q^96 - 6 * q^97 + 11 * q^98 + 28 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ v^2 - v - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 2$$ b2 + b1 + 2

## 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
 −1.48119 2.17009 0.311108
1.00000 −2.67513 1.00000 0 −2.67513 −3.28726 1.00000 4.15633 0
1.2 1.00000 −1.53919 1.00000 0 −1.53919 2.87936 1.00000 −0.630898 0
1.3 1.00000 1.21432 1.00000 0 1.21432 −3.59210 1.00000 −1.52543 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
$$2$$ $$-1$$
$$5$$ $$-1$$
$$37$$ $$1$$

## 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 1850.2.a.bb yes 3
5.b even 2 1 1850.2.a.ba 3
5.c odd 4 2 1850.2.b.n 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1850.2.a.ba 3 5.b even 2 1
1850.2.a.bb yes 3 1.a even 1 1 trivial
1850.2.b.n 6 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1850))$$:

 $$T_{3}^{3} + 3T_{3}^{2} - T_{3} - 5$$ T3^3 + 3*T3^2 - T3 - 5 $$T_{7}^{3} + 4T_{7}^{2} - 8T_{7} - 34$$ T7^3 + 4*T7^2 - 8*T7 - 34

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{3}$$
$3$ $$T^{3} + 3T^{2} - T - 5$$
$5$ $$T^{3}$$
$7$ $$T^{3} + 4 T^{2} - 8 T - 34$$
$11$ $$T^{3} + T^{2} - 23 T - 25$$
$13$ $$T^{3} + 12 T^{2} + 44 T + 46$$
$17$ $$T^{3} - T^{2} - 21 T - 29$$
$19$ $$T^{3} - 3 T^{2} - 25 T + 79$$
$23$ $$T^{3} + 8T^{2} - 32$$
$29$ $$T^{3} + 2 T^{2} - 52 T - 40$$
$31$ $$T^{3} + 6 T^{2} - 18 T - 54$$
$37$ $$(T + 1)^{3}$$
$41$ $$T^{3} + 21 T^{2} + 131 T + 215$$
$43$ $$T^{3} + 8 T^{2} - 128 T - 1076$$
$47$ $$T^{3} + 4 T^{2} - 48 T - 64$$
$53$ $$T^{3} + 6 T^{2} - 100 T - 632$$
$59$ $$T^{3} - 8 T^{2} - 128 T + 1076$$
$61$ $$T^{3} + 8 T^{2} - 44 T - 290$$
$67$ $$T^{3} + 13 T^{2} - 9 T - 67$$
$71$ $$T^{3} + 8 T^{2} - 44 T - 290$$
$73$ $$T^{3} + 31 T^{2} + 315 T + 1049$$
$79$ $$T^{3} - 22 T^{2} + 140 T - 232$$
$83$ $$T^{3} + 7 T^{2} - 227 T - 1819$$
$89$ $$T^{3} + 3 T^{2} - 27 T - 27$$
$97$ $$T^{3} + 6 T^{2} - 4 T - 40$$