# Properties

 Label 1850.2.a.bd Level $1850$ Weight $2$ Character orbit 1850.a Self dual yes Analytic conductor $14.772$ Analytic rank $0$ Dimension $5$ 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: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.1791440.1 Defining polynomial: $$x^{5} - 9x^{3} + 13x - 4$$ x^5 - 9*x^3 + 13*x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 370) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + \beta_1 q^{3} + q^{4} - \beta_1 q^{6} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{7} - q^{8} + (\beta_{2} + 1) q^{9}+O(q^{10})$$ q - q^2 + b1 * q^3 + q^4 - b1 * q^6 + (-b4 + b3 + b2 + b1) * q^7 - q^8 + (b2 + 1) * q^9 $$q - q^{2} + \beta_1 q^{3} + q^{4} - \beta_1 q^{6} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{7} - q^{8} + (\beta_{2} + 1) q^{9} + ( - \beta_{4} + \beta_{3} - \beta_1) q^{11} + \beta_1 q^{12} + (\beta_{4} + \beta_{3} + \beta_1 + 1) q^{13} + (\beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{14} + q^{16} + ( - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 2) q^{17} + ( - \beta_{2} - 1) q^{18} + 2 \beta_{3} q^{19} + ( - 2 \beta_{3} + 2 \beta_1 + 4) q^{21} + (\beta_{4} - \beta_{3} + \beta_1) q^{22} + ( - \beta_{4} - \beta_{3} + \beta_1 - 1) q^{23} - \beta_1 q^{24} + ( - \beta_{4} - \beta_{3} - \beta_1 - 1) q^{26} + (\beta_{4} - 2 \beta_{3} + 1) q^{27} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{28} + ( - \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 + 2) q^{29} + (2 \beta_{4} + \beta_{3} + \beta_{2} + 5) q^{31} - q^{32} + ( - \beta_{4} - 2 \beta_{2} - 5) q^{33} + (\beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{34} + (\beta_{2} + 1) q^{36} - q^{37} - 2 \beta_{3} q^{38} + (\beta_{4} - 2 \beta_{3} + \beta_1 + 5) q^{39} + ( - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 3) q^{41} + (2 \beta_{3} - 2 \beta_1 - 4) q^{42} + ( - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 4) q^{43} + ( - \beta_{4} + \beta_{3} - \beta_1) q^{44} + (\beta_{4} + \beta_{3} - \beta_1 + 1) q^{46} + (2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 2) q^{47} + \beta_1 q^{48} + (\beta_{4} - 3 \beta_{3} - \beta_{2} + \beta_1 + 7) q^{49} + ( - 2 \beta_{3} - 2 \beta_{2} - 4) q^{51} + (\beta_{4} + \beta_{3} + \beta_1 + 1) q^{52} + (3 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 2) q^{53} + ( - \beta_{4} + 2 \beta_{3} - 1) q^{54} + (\beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{56} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{57} + (\beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{58} + ( - 2 \beta_{3} - 2 \beta_{2} + 4) q^{59} + ( - 2 \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 - 3) q^{61} + ( - 2 \beta_{4} - \beta_{3} - \beta_{2} - 5) q^{62} + (3 \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 8) q^{63} + q^{64} + (\beta_{4} + 2 \beta_{2} + 5) q^{66} + (\beta_{4} + 2 \beta_{3} + \beta_1 - 3) q^{67} + ( - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 2) q^{68} + ( - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 3) q^{69} + (4 \beta_{4} + 2) q^{71} + ( - \beta_{2} - 1) q^{72} + ( - 3 \beta_{4} + 4 \beta_{3} - \beta_1 - 1) q^{73} + q^{74} + 2 \beta_{3} q^{76} + ( - 3 \beta_{4} + 3 \beta_{3} - \beta_{2} - 3 \beta_1 - 2) q^{77} + ( - \beta_{4} + 2 \beta_{3} - \beta_1 - 5) q^{78} + ( - 2 \beta_{4} + 4 \beta_{3} + \beta_1 + 2) q^{79} + (\beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 2) q^{81} + (2 \beta_{4} - 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 3) q^{82} + (2 \beta_{4} - 4 \beta_1 - 2) q^{83} + ( - 2 \beta_{3} + 2 \beta_1 + 4) q^{84} + (\beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 4) q^{86} + ( - 2 \beta_{3} + \beta_{2} + 4 \beta_1 + 8) q^{87} + (\beta_{4} - \beta_{3} + \beta_1) q^{88} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 4) q^{89} + ( - 2 \beta_{4} - 6 \beta_{3} + 2 \beta_{2} + 6) q^{91} + ( - \beta_{4} - \beta_{3} + \beta_1 - 1) q^{92} + (3 \beta_{4} - 5 \beta_{3} - \beta_{2} + 7 \beta_1 + 3) q^{93} + ( - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 2) q^{94} - \beta_1 q^{96} + (\beta_{4} + \beta_{3} - 3 \beta_{2} - \beta_1 - 2) q^{97} + ( - \beta_{4} + 3 \beta_{3} + \beta_{2} - \beta_1 - 7) q^{98} + (2 \beta_{3} - 6 \beta_1 - 3) q^{99}+O(q^{100})$$ q - q^2 + b1 * q^3 + q^4 - b1 * q^6 + (-b4 + b3 + b2 + b1) * q^7 - q^8 + (b2 + 1) * q^9 + (-b4 + b3 - b1) * q^11 + b1 * q^12 + (b4 + b3 + b1 + 1) * q^13 + (b4 - b3 - b2 - b1) * q^14 + q^16 + (-b4 + b3 + b2 - b1 - 2) * q^17 + (-b2 - 1) * q^18 + 2*b3 * q^19 + (-2*b3 + 2*b1 + 4) * q^21 + (b4 - b3 + b1) * q^22 + (-b4 - b3 + b1 - 1) * q^23 - b1 * q^24 + (-b4 - b3 - b1 - 1) * q^26 + (b4 - 2*b3 + 1) * q^27 + (-b4 + b3 + b2 + b1) * q^28 + (-b4 + b3 + b2 + 2*b1 + 2) * q^29 + (2*b4 + b3 + b2 + 5) * q^31 - q^32 + (-b4 - 2*b2 - 5) * q^33 + (b4 - b3 - b2 + b1 + 2) * q^34 + (b2 + 1) * q^36 - q^37 - 2*b3 * q^38 + (b4 - 2*b3 + b1 + 5) * q^39 + (-2*b4 + 2*b3 - b2 + 2*b1 - 3) * q^41 + (2*b3 - 2*b1 - 4) * q^42 + (-b4 - b3 + b2 + b1 + 4) * q^43 + (-b4 + b3 - b1) * q^44 + (b4 + b3 - b1 + 1) * q^46 + (2*b4 - 2*b3 - 2*b2 - 2) * q^47 + b1 * q^48 + (b4 - 3*b3 - b2 + b1 + 7) * q^49 + (-2*b3 - 2*b2 - 4) * q^51 + (b4 + b3 + b1 + 1) * q^52 + (3*b4 + b3 + b2 - b1 + 2) * q^53 + (-b4 + 2*b3 - 1) * q^54 + (b4 - b3 - b2 - b1) * q^56 + (-2*b3 - 2*b2) * q^57 + (b4 - b3 - b2 - 2*b1 - 2) * q^58 + (-2*b3 - 2*b2 + 4) * q^59 + (-2*b4 + b3 - b2 + 2*b1 - 3) * q^61 + (-2*b4 - b3 - b2 - 5) * q^62 + (3*b4 - b3 + b2 + b1 + 8) * q^63 + q^64 + (b4 + 2*b2 + 5) * q^66 + (b4 + 2*b3 + b1 - 3) * q^67 + (-b4 + b3 + b2 - b1 - 2) * q^68 + (-b4 + 2*b3 + 2*b2 - b1 + 3) * q^69 + (4*b4 + 2) * q^71 + (-b2 - 1) * q^72 + (-3*b4 + 4*b3 - b1 - 1) * q^73 + q^74 + 2*b3 * q^76 + (-3*b4 + 3*b3 - b2 - 3*b1 - 2) * q^77 + (-b4 + 2*b3 - b1 - 5) * q^78 + (-2*b4 + 4*b3 + b1 + 2) * q^79 + (b4 + b3 - b2 + b1 - 2) * q^81 + (2*b4 - 2*b3 + b2 - 2*b1 + 3) * q^82 + (2*b4 - 4*b1 - 2) * q^83 + (-2*b3 + 2*b1 + 4) * q^84 + (b4 + b3 - b2 - b1 - 4) * q^86 + (-2*b3 + b2 + 4*b1 + 8) * q^87 + (b4 - b3 + b1) * q^88 + (-2*b3 + 2*b2 - 2*b1 + 4) * q^89 + (-2*b4 - 6*b3 + 2*b2 + 6) * q^91 + (-b4 - b3 + b1 - 1) * q^92 + (3*b4 - 5*b3 - b2 + 7*b1 + 3) * q^93 + (-2*b4 + 2*b3 + 2*b2 + 2) * q^94 - b1 * q^96 + (b4 + b3 - 3*b2 - b1 - 2) * q^97 + (-b4 + 3*b3 + b2 - b1 - 7) * q^98 + (2*b3 - 6*b1 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 5 q^{2} + 5 q^{4} + q^{7} - 5 q^{8} + 3 q^{9}+O(q^{10})$$ 5 * q - 5 * q^2 + 5 * q^4 + q^7 - 5 * q^8 + 3 * q^9 $$5 q - 5 q^{2} + 5 q^{4} + q^{7} - 5 q^{8} + 3 q^{9} + 3 q^{11} + 6 q^{13} - q^{14} + 5 q^{16} - 9 q^{17} - 3 q^{18} + 4 q^{19} + 16 q^{21} - 3 q^{22} - 6 q^{23} - 6 q^{26} + q^{28} + 11 q^{29} + 23 q^{31} - 5 q^{32} - 20 q^{33} + 9 q^{34} + 3 q^{36} - 5 q^{37} - 4 q^{38} + 20 q^{39} - 7 q^{41} - 16 q^{42} + 17 q^{43} + 3 q^{44} + 6 q^{46} - 12 q^{47} + 30 q^{49} - 20 q^{51} + 6 q^{52} + 7 q^{53} - q^{56} - 11 q^{58} + 20 q^{59} - 9 q^{61} - 23 q^{62} + 33 q^{63} + 5 q^{64} + 20 q^{66} - 12 q^{67} - 9 q^{68} + 16 q^{69} + 6 q^{71} - 3 q^{72} + 6 q^{73} + 5 q^{74} + 4 q^{76} + q^{77} - 20 q^{78} + 20 q^{79} - 7 q^{81} + 7 q^{82} - 12 q^{83} + 16 q^{84} - 17 q^{86} + 34 q^{87} - 3 q^{88} + 12 q^{89} + 16 q^{91} - 6 q^{92} + 4 q^{93} + 12 q^{94} - 3 q^{97} - 30 q^{98} - 11 q^{99}+O(q^{100})$$ 5 * q - 5 * q^2 + 5 * q^4 + q^7 - 5 * q^8 + 3 * q^9 + 3 * q^11 + 6 * q^13 - q^14 + 5 * q^16 - 9 * q^17 - 3 * q^18 + 4 * q^19 + 16 * q^21 - 3 * q^22 - 6 * q^23 - 6 * q^26 + q^28 + 11 * q^29 + 23 * q^31 - 5 * q^32 - 20 * q^33 + 9 * q^34 + 3 * q^36 - 5 * q^37 - 4 * q^38 + 20 * q^39 - 7 * q^41 - 16 * q^42 + 17 * q^43 + 3 * q^44 + 6 * q^46 - 12 * q^47 + 30 * q^49 - 20 * q^51 + 6 * q^52 + 7 * q^53 - q^56 - 11 * q^58 + 20 * q^59 - 9 * q^61 - 23 * q^62 + 33 * q^63 + 5 * q^64 + 20 * q^66 - 12 * q^67 - 9 * q^68 + 16 * q^69 + 6 * q^71 - 3 * q^72 + 6 * q^73 + 5 * q^74 + 4 * q^76 + q^77 - 20 * q^78 + 20 * q^79 - 7 * q^81 + 7 * q^82 - 12 * q^83 + 16 * q^84 - 17 * q^86 + 34 * q^87 - 3 * q^88 + 12 * q^89 + 16 * q^91 - 6 * q^92 + 4 * q^93 + 12 * q^94 - 3 * q^97 - 30 * q^98 - 11 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 9x^{3} + 13x - 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ v^2 - 4 $$\beta_{3}$$ $$=$$ $$( \nu^{4} - \nu^{3} - 8\nu^{2} + 5\nu + 8 ) / 3$$ (v^4 - v^3 - 8*v^2 + 5*v + 8) / 3 $$\beta_{4}$$ $$=$$ $$( 2\nu^{4} + \nu^{3} - 16\nu^{2} - 8\nu + 13 ) / 3$$ (2*v^4 + v^3 - 16*v^2 - 8*v + 13) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ b2 + 4 $$\nu^{3}$$ $$=$$ $$\beta_{4} - 2\beta_{3} + 6\beta _1 + 1$$ b4 - 2*b3 + 6*b1 + 1 $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{3} + 8\beta_{2} + \beta _1 + 25$$ b4 + b3 + 8*b2 + b1 + 25

## 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
 −2.62545 −1.53175 0.332924 1.09441 2.72987
−1.00000 −2.62545 1.00000 0 2.62545 1.83227 −1.00000 3.89300 0
1.2 −1.00000 −1.53175 1.00000 0 1.53175 −4.67211 −1.00000 −0.653743 0
1.3 −1.00000 0.332924 1.00000 0 −0.332924 −3.51336 −1.00000 −2.88916 0
1.4 −1.00000 1.09441 1.00000 0 −1.09441 3.20984 −1.00000 −1.80226 0
1.5 −1.00000 2.72987 1.00000 0 −2.72987 4.14336 −1.00000 4.45216 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.bd 5
5.b even 2 1 1850.2.a.be 5
5.c odd 4 2 370.2.b.d 10
15.e even 4 2 3330.2.d.p 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.b.d 10 5.c odd 4 2
1850.2.a.bd 5 1.a even 1 1 trivial
1850.2.a.be 5 5.b even 2 1
3330.2.d.p 10 15.e even 4 2

## Hecke kernels

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

 $$T_{3}^{5} - 9T_{3}^{3} + 13T_{3} - 4$$ T3^5 - 9*T3^3 + 13*T3 - 4 $$T_{7}^{5} - T_{7}^{4} - 32T_{7}^{3} + 44T_{7}^{2} + 240T_{7} - 400$$ T7^5 - T7^4 - 32*T7^3 + 44*T7^2 + 240*T7 - 400

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{5}$$
$3$ $$T^{5} - 9 T^{3} + 13 T - 4$$
$5$ $$T^{5}$$
$7$ $$T^{5} - T^{4} - 32 T^{3} + 44 T^{2} + \cdots - 400$$
$11$ $$T^{5} - 3 T^{4} - 23 T^{3} + 71 T^{2} + \cdots - 1$$
$13$ $$T^{5} - 6 T^{4} - 29 T^{3} + 154 T^{2} + \cdots - 8$$
$17$ $$T^{5} + 9 T^{4} - 4 T^{3} - 216 T^{2} + \cdots - 368$$
$19$ $$T^{5} - 4 T^{4} - 40 T^{3} + 176 T^{2} + \cdots - 640$$
$23$ $$T^{5} + 6 T^{4} - 25 T^{3} - 82 T^{2} + \cdots - 288$$
$29$ $$T^{5} - 11 T^{4} - 9 T^{3} + 251 T^{2} + \cdots - 335$$
$31$ $$T^{5} - 23 T^{4} + 107 T^{3} + \cdots - 10511$$
$37$ $$(T + 1)^{5}$$
$41$ $$T^{5} + 7 T^{4} - 119 T^{3} + \cdots + 5821$$
$43$ $$T^{5} - 17 T^{4} + 52 T^{3} + \cdots + 3056$$
$47$ $$T^{5} + 12 T^{4} - 44 T^{3} + \cdots + 8000$$
$53$ $$T^{5} - 7 T^{4} - 164 T^{3} + \cdots - 32816$$
$59$ $$T^{5} - 20 T^{4} + 52 T^{3} + \cdots - 3136$$
$61$ $$T^{5} + 9 T^{4} - 81 T^{3} + \cdots - 2323$$
$67$ $$T^{5} + 12 T^{4} - 29 T^{3} + \cdots - 392$$
$71$ $$T^{5} - 6 T^{4} - 184 T^{3} + \cdots - 13792$$
$73$ $$T^{5} - 6 T^{4} - 197 T^{3} + \cdots - 14726$$
$79$ $$T^{5} - 20 T^{4} - 13 T^{3} + \cdots - 8168$$
$83$ $$T^{5} + 12 T^{4} - 120 T^{3} + \cdots - 2752$$
$89$ $$T^{5} - 12 T^{4} - 144 T^{3} + \cdots - 55552$$
$97$ $$T^{5} + 3 T^{4} - 244 T^{3} + \cdots + 1008$$