# Properties

 Label 1850.2.a.q Level $1850$ Weight $2$ Character orbit 1850.a Self dual yes Analytic conductor $14.772$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 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 $$\beta = \frac{1}{2}(1 + \sqrt{33})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} - 2 q^{3} + q^{4} + 2 q^{6} + (\beta + 1) q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 - 2 * q^3 + q^4 + 2 * q^6 + (b + 1) * q^7 - q^8 + q^9 $$q - q^{2} - 2 q^{3} + q^{4} + 2 q^{6} + (\beta + 1) q^{7} - q^{8} + q^{9} + (\beta - 1) q^{11} - 2 q^{12} - 2 \beta q^{13} + ( - \beta - 1) q^{14} + q^{16} + ( - \beta + 3) q^{17} - q^{18} - 2 q^{19} + ( - 2 \beta - 2) q^{21} + ( - \beta + 1) q^{22} + (2 \beta - 2) q^{23} + 2 q^{24} + 2 \beta q^{26} + 4 q^{27} + (\beta + 1) q^{28} + ( - 3 \beta + 1) q^{29} + ( - \beta - 5) q^{31} - q^{32} + ( - 2 \beta + 2) q^{33} + (\beta - 3) q^{34} + q^{36} - q^{37} + 2 q^{38} + 4 \beta q^{39} + ( - \beta + 3) q^{41} + (2 \beta + 2) q^{42} + (\beta - 5) q^{43} + (\beta - 1) q^{44} + ( - 2 \beta + 2) q^{46} + ( - 2 \beta + 4) q^{47} - 2 q^{48} + (3 \beta + 2) q^{49} + (2 \beta - 6) q^{51} - 2 \beta q^{52} + (\beta + 1) q^{53} - 4 q^{54} + ( - \beta - 1) q^{56} + 4 q^{57} + (3 \beta - 1) q^{58} + ( - 2 \beta + 8) q^{59} + (\beta - 3) q^{61} + (\beta + 5) q^{62} + (\beta + 1) q^{63} + q^{64} + (2 \beta - 2) q^{66} + 2 \beta q^{67} + ( - \beta + 3) q^{68} + ( - 4 \beta + 4) q^{69} + (2 \beta - 2) q^{71} - q^{72} + (2 \beta - 4) q^{73} + q^{74} - 2 q^{76} + (\beta + 7) q^{77} - 4 \beta q^{78} + 2 \beta q^{79} - 11 q^{81} + (\beta - 3) q^{82} + ( - 2 \beta - 4) q^{83} + ( - 2 \beta - 2) q^{84} + ( - \beta + 5) q^{86} + (6 \beta - 2) q^{87} + ( - \beta + 1) q^{88} + 10 q^{89} + ( - 4 \beta - 16) q^{91} + (2 \beta - 2) q^{92} + (2 \beta + 10) q^{93} + (2 \beta - 4) q^{94} + 2 q^{96} + ( - 3 \beta - 7) q^{97} + ( - 3 \beta - 2) q^{98} + (\beta - 1) q^{99} +O(q^{100})$$ q - q^2 - 2 * q^3 + q^4 + 2 * q^6 + (b + 1) * q^7 - q^8 + q^9 + (b - 1) * q^11 - 2 * q^12 - 2*b * q^13 + (-b - 1) * q^14 + q^16 + (-b + 3) * q^17 - q^18 - 2 * q^19 + (-2*b - 2) * q^21 + (-b + 1) * q^22 + (2*b - 2) * q^23 + 2 * q^24 + 2*b * q^26 + 4 * q^27 + (b + 1) * q^28 + (-3*b + 1) * q^29 + (-b - 5) * q^31 - q^32 + (-2*b + 2) * q^33 + (b - 3) * q^34 + q^36 - q^37 + 2 * q^38 + 4*b * q^39 + (-b + 3) * q^41 + (2*b + 2) * q^42 + (b - 5) * q^43 + (b - 1) * q^44 + (-2*b + 2) * q^46 + (-2*b + 4) * q^47 - 2 * q^48 + (3*b + 2) * q^49 + (2*b - 6) * q^51 - 2*b * q^52 + (b + 1) * q^53 - 4 * q^54 + (-b - 1) * q^56 + 4 * q^57 + (3*b - 1) * q^58 + (-2*b + 8) * q^59 + (b - 3) * q^61 + (b + 5) * q^62 + (b + 1) * q^63 + q^64 + (2*b - 2) * q^66 + 2*b * q^67 + (-b + 3) * q^68 + (-4*b + 4) * q^69 + (2*b - 2) * q^71 - q^72 + (2*b - 4) * q^73 + q^74 - 2 * q^76 + (b + 7) * q^77 - 4*b * q^78 + 2*b * q^79 - 11 * q^81 + (b - 3) * q^82 + (-2*b - 4) * q^83 + (-2*b - 2) * q^84 + (-b + 5) * q^86 + (6*b - 2) * q^87 + (-b + 1) * q^88 + 10 * q^89 + (-4*b - 16) * q^91 + (2*b - 2) * q^92 + (2*b + 10) * q^93 + (2*b - 4) * q^94 + 2 * q^96 + (-3*b - 7) * q^97 + (-3*b - 2) * q^98 + (b - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} + 4 q^{6} + 3 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 4 * q^3 + 2 * q^4 + 4 * q^6 + 3 * q^7 - 2 * q^8 + 2 * q^9 $$2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} + 4 q^{6} + 3 q^{7} - 2 q^{8} + 2 q^{9} - q^{11} - 4 q^{12} - 2 q^{13} - 3 q^{14} + 2 q^{16} + 5 q^{17} - 2 q^{18} - 4 q^{19} - 6 q^{21} + q^{22} - 2 q^{23} + 4 q^{24} + 2 q^{26} + 8 q^{27} + 3 q^{28} - q^{29} - 11 q^{31} - 2 q^{32} + 2 q^{33} - 5 q^{34} + 2 q^{36} - 2 q^{37} + 4 q^{38} + 4 q^{39} + 5 q^{41} + 6 q^{42} - 9 q^{43} - q^{44} + 2 q^{46} + 6 q^{47} - 4 q^{48} + 7 q^{49} - 10 q^{51} - 2 q^{52} + 3 q^{53} - 8 q^{54} - 3 q^{56} + 8 q^{57} + q^{58} + 14 q^{59} - 5 q^{61} + 11 q^{62} + 3 q^{63} + 2 q^{64} - 2 q^{66} + 2 q^{67} + 5 q^{68} + 4 q^{69} - 2 q^{71} - 2 q^{72} - 6 q^{73} + 2 q^{74} - 4 q^{76} + 15 q^{77} - 4 q^{78} + 2 q^{79} - 22 q^{81} - 5 q^{82} - 10 q^{83} - 6 q^{84} + 9 q^{86} + 2 q^{87} + q^{88} + 20 q^{89} - 36 q^{91} - 2 q^{92} + 22 q^{93} - 6 q^{94} + 4 q^{96} - 17 q^{97} - 7 q^{98} - q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 4 * q^3 + 2 * q^4 + 4 * q^6 + 3 * q^7 - 2 * q^8 + 2 * q^9 - q^11 - 4 * q^12 - 2 * q^13 - 3 * q^14 + 2 * q^16 + 5 * q^17 - 2 * q^18 - 4 * q^19 - 6 * q^21 + q^22 - 2 * q^23 + 4 * q^24 + 2 * q^26 + 8 * q^27 + 3 * q^28 - q^29 - 11 * q^31 - 2 * q^32 + 2 * q^33 - 5 * q^34 + 2 * q^36 - 2 * q^37 + 4 * q^38 + 4 * q^39 + 5 * q^41 + 6 * q^42 - 9 * q^43 - q^44 + 2 * q^46 + 6 * q^47 - 4 * q^48 + 7 * q^49 - 10 * q^51 - 2 * q^52 + 3 * q^53 - 8 * q^54 - 3 * q^56 + 8 * q^57 + q^58 + 14 * q^59 - 5 * q^61 + 11 * q^62 + 3 * q^63 + 2 * q^64 - 2 * q^66 + 2 * q^67 + 5 * q^68 + 4 * q^69 - 2 * q^71 - 2 * q^72 - 6 * q^73 + 2 * q^74 - 4 * q^76 + 15 * q^77 - 4 * q^78 + 2 * q^79 - 22 * q^81 - 5 * q^82 - 10 * q^83 - 6 * q^84 + 9 * q^86 + 2 * q^87 + q^88 + 20 * q^89 - 36 * q^91 - 2 * q^92 + 22 * q^93 - 6 * q^94 + 4 * q^96 - 17 * q^97 - 7 * q^98 - q^99

## 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
 −2.37228 3.37228
−1.00000 −2.00000 1.00000 0 2.00000 −1.37228 −1.00000 1.00000 0
1.2 −1.00000 −2.00000 1.00000 0 2.00000 4.37228 −1.00000 1.00000 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.q 2
5.b even 2 1 370.2.a.f 2
5.c odd 4 2 1850.2.b.m 4
15.d odd 2 1 3330.2.a.bb 2
20.d odd 2 1 2960.2.a.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.a.f 2 5.b even 2 1
1850.2.a.q 2 1.a even 1 1 trivial
1850.2.b.m 4 5.c odd 4 2
2960.2.a.o 2 20.d odd 2 1
3330.2.a.bb 2 15.d odd 2 1

## Hecke kernels

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

 $$T_{3} + 2$$ T3 + 2 $$T_{7}^{2} - 3T_{7} - 6$$ T7^2 - 3*T7 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$(T + 2)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 3T - 6$$
$11$ $$T^{2} + T - 8$$
$13$ $$T^{2} + 2T - 32$$
$17$ $$T^{2} - 5T - 2$$
$19$ $$(T + 2)^{2}$$
$23$ $$T^{2} + 2T - 32$$
$29$ $$T^{2} + T - 74$$
$31$ $$T^{2} + 11T + 22$$
$37$ $$(T + 1)^{2}$$
$41$ $$T^{2} - 5T - 2$$
$43$ $$T^{2} + 9T + 12$$
$47$ $$T^{2} - 6T - 24$$
$53$ $$T^{2} - 3T - 6$$
$59$ $$T^{2} - 14T + 16$$
$61$ $$T^{2} + 5T - 2$$
$67$ $$T^{2} - 2T - 32$$
$71$ $$T^{2} + 2T - 32$$
$73$ $$T^{2} + 6T - 24$$
$79$ $$T^{2} - 2T - 32$$
$83$ $$T^{2} + 10T - 8$$
$89$ $$(T - 10)^{2}$$
$97$ $$T^{2} + 17T - 2$$