# Properties

 Label 1850.2.b.k Level $1850$ Weight $2$ Character orbit 1850.b Analytic conductor $14.772$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.7723243739$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{2} + \beta_1) q^{3} - q^{4} + (\beta_{3} + 1) q^{6} + \beta_{2} q^{7} + \beta_1 q^{8} + ( - 2 \beta_{3} - 4) q^{9}+O(q^{10})$$ q - b1 * q^2 + (b2 + b1) * q^3 - q^4 + (b3 + 1) * q^6 + b2 * q^7 + b1 * q^8 + (-2*b3 - 4) * q^9 $$q - \beta_1 q^{2} + (\beta_{2} + \beta_1) q^{3} - q^{4} + (\beta_{3} + 1) q^{6} + \beta_{2} q^{7} + \beta_1 q^{8} + ( - 2 \beta_{3} - 4) q^{9} + ( - \beta_{3} + 1) q^{11} + ( - \beta_{2} - \beta_1) q^{12} + (\beta_{2} + 2 \beta_1) q^{13} + \beta_{3} q^{14} + q^{16} + (\beta_{2} - \beta_1) q^{17} + (2 \beta_{2} + 4 \beta_1) q^{18} + 5 q^{19} + ( - \beta_{3} - 6) q^{21} + (\beta_{2} - \beta_1) q^{22} + 2 \beta_1 q^{23} + ( - \beta_{3} - 1) q^{24} + (\beta_{3} + 2) q^{26} + ( - 3 \beta_{2} - 13 \beta_1) q^{27} - \beta_{2} q^{28} + ( - 2 \beta_{3} - 4) q^{29} + ( - \beta_{3} + 2) q^{31} - \beta_1 q^{32} - 5 \beta_1 q^{33} + (\beta_{3} - 1) q^{34} + (2 \beta_{3} + 4) q^{36} + \beta_1 q^{37} - 5 \beta_1 q^{38} + ( - 3 \beta_{3} - 8) q^{39} + q^{41} + (\beta_{2} + 6 \beta_1) q^{42} + ( - 2 \beta_{2} - 6 \beta_1) q^{43} + (\beta_{3} - 1) q^{44} + 2 q^{46} + 4 \beta_{2} q^{47} + (\beta_{2} + \beta_1) q^{48} + q^{49} - 5 q^{51} + ( - \beta_{2} - 2 \beta_1) q^{52} + 6 \beta_1 q^{53} + ( - 3 \beta_{3} - 13) q^{54} - \beta_{3} q^{56} + (5 \beta_{2} + 5 \beta_1) q^{57} + (2 \beta_{2} + 4 \beta_1) q^{58} + 2 q^{59} + (\beta_{3} - 4) q^{61} + (\beta_{2} - 2 \beta_1) q^{62} + ( - 4 \beta_{2} - 12 \beta_1) q^{63} - q^{64} - 5 q^{66} + ( - \beta_{2} - 7 \beta_1) q^{67} + ( - \beta_{2} + \beta_1) q^{68} + ( - 2 \beta_{3} - 2) q^{69} + ( - \beta_{3} - 10) q^{71} + ( - 2 \beta_{2} - 4 \beta_1) q^{72} + ( - 4 \beta_{2} + 3 \beta_1) q^{73} + q^{74} - 5 q^{76} + (\beta_{2} - 6 \beta_1) q^{77} + (3 \beta_{2} + 8 \beta_1) q^{78} + (4 \beta_{3} + 2) q^{79} + (10 \beta_{3} + 19) q^{81} - \beta_1 q^{82} + ( - \beta_{2} + \beta_1) q^{83} + (\beta_{3} + 6) q^{84} + ( - 2 \beta_{3} - 6) q^{86} + ( - 6 \beta_{2} - 16 \beta_1) q^{87} + ( - \beta_{2} + \beta_1) q^{88} + (3 \beta_{3} - 7) q^{89} + ( - 2 \beta_{3} - 6) q^{91} - 2 \beta_1 q^{92} + (\beta_{2} - 4 \beta_1) q^{93} + 4 \beta_{3} q^{94} + (\beta_{3} + 1) q^{96} - 14 \beta_1 q^{97} - \beta_1 q^{98} + (2 \beta_{3} + 8) q^{99}+O(q^{100})$$ q - b1 * q^2 + (b2 + b1) * q^3 - q^4 + (b3 + 1) * q^6 + b2 * q^7 + b1 * q^8 + (-2*b3 - 4) * q^9 + (-b3 + 1) * q^11 + (-b2 - b1) * q^12 + (b2 + 2*b1) * q^13 + b3 * q^14 + q^16 + (b2 - b1) * q^17 + (2*b2 + 4*b1) * q^18 + 5 * q^19 + (-b3 - 6) * q^21 + (b2 - b1) * q^22 + 2*b1 * q^23 + (-b3 - 1) * q^24 + (b3 + 2) * q^26 + (-3*b2 - 13*b1) * q^27 - b2 * q^28 + (-2*b3 - 4) * q^29 + (-b3 + 2) * q^31 - b1 * q^32 - 5*b1 * q^33 + (b3 - 1) * q^34 + (2*b3 + 4) * q^36 + b1 * q^37 - 5*b1 * q^38 + (-3*b3 - 8) * q^39 + q^41 + (b2 + 6*b1) * q^42 + (-2*b2 - 6*b1) * q^43 + (b3 - 1) * q^44 + 2 * q^46 + 4*b2 * q^47 + (b2 + b1) * q^48 + q^49 - 5 * q^51 + (-b2 - 2*b1) * q^52 + 6*b1 * q^53 + (-3*b3 - 13) * q^54 - b3 * q^56 + (5*b2 + 5*b1) * q^57 + (2*b2 + 4*b1) * q^58 + 2 * q^59 + (b3 - 4) * q^61 + (b2 - 2*b1) * q^62 + (-4*b2 - 12*b1) * q^63 - q^64 - 5 * q^66 + (-b2 - 7*b1) * q^67 + (-b2 + b1) * q^68 + (-2*b3 - 2) * q^69 + (-b3 - 10) * q^71 + (-2*b2 - 4*b1) * q^72 + (-4*b2 + 3*b1) * q^73 + q^74 - 5 * q^76 + (b2 - 6*b1) * q^77 + (3*b2 + 8*b1) * q^78 + (4*b3 + 2) * q^79 + (10*b3 + 19) * q^81 - b1 * q^82 + (-b2 + b1) * q^83 + (b3 + 6) * q^84 + (-2*b3 - 6) * q^86 + (-6*b2 - 16*b1) * q^87 + (-b2 + b1) * q^88 + (3*b3 - 7) * q^89 + (-2*b3 - 6) * q^91 - 2*b1 * q^92 + (b2 - 4*b1) * q^93 + 4*b3 * q^94 + (b3 + 1) * q^96 - 14*b1 * q^97 - b1 * q^98 + (2*b3 + 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} + 4 q^{6} - 16 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 + 4 * q^6 - 16 * q^9 $$4 q - 4 q^{4} + 4 q^{6} - 16 q^{9} + 4 q^{11} + 4 q^{16} + 20 q^{19} - 24 q^{21} - 4 q^{24} + 8 q^{26} - 16 q^{29} + 8 q^{31} - 4 q^{34} + 16 q^{36} - 32 q^{39} + 4 q^{41} - 4 q^{44} + 8 q^{46} + 4 q^{49} - 20 q^{51} - 52 q^{54} + 8 q^{59} - 16 q^{61} - 4 q^{64} - 20 q^{66} - 8 q^{69} - 40 q^{71} + 4 q^{74} - 20 q^{76} + 8 q^{79} + 76 q^{81} + 24 q^{84} - 24 q^{86} - 28 q^{89} - 24 q^{91} + 4 q^{96} + 32 q^{99}+O(q^{100})$$ 4 * q - 4 * q^4 + 4 * q^6 - 16 * q^9 + 4 * q^11 + 4 * q^16 + 20 * q^19 - 24 * q^21 - 4 * q^24 + 8 * q^26 - 16 * q^29 + 8 * q^31 - 4 * q^34 + 16 * q^36 - 32 * q^39 + 4 * q^41 - 4 * q^44 + 8 * q^46 + 4 * q^49 - 20 * q^51 - 52 * q^54 + 8 * q^59 - 16 * q^61 - 4 * q^64 - 20 * q^66 - 8 * q^69 - 40 * q^71 + 4 * q^74 - 20 * q^76 + 8 * q^79 + 76 * q^81 + 24 * q^84 - 24 * q^86 - 28 * q^89 - 24 * q^91 + 4 * q^96 + 32 * q^99

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

 $$\beta_{1}$$ $$=$$ $$( \nu^{2} ) / 3$$ (v^2) / 3 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 3\nu ) / 3$$ (v^3 + 3*v) / 3 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 3\nu ) / 3$$ (-v^3 + 3*v) / 3
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\nu^{2}$$ $$=$$ $$3\beta_1$$ 3*b1 $$\nu^{3}$$ $$=$$ $$( -3\beta_{3} + 3\beta_{2} ) / 2$$ (-3*b3 + 3*b2) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1777$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1
 −1.22474 − 1.22474i 1.22474 + 1.22474i 1.22474 − 1.22474i −1.22474 + 1.22474i
1.00000i 1.44949i −1.00000 0 −1.44949 2.44949i 1.00000i 0.898979 0
149.2 1.00000i 3.44949i −1.00000 0 3.44949 2.44949i 1.00000i −8.89898 0
149.3 1.00000i 3.44949i −1.00000 0 3.44949 2.44949i 1.00000i −8.89898 0
149.4 1.00000i 1.44949i −1.00000 0 −1.44949 2.44949i 1.00000i 0.898979 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1850.2.b.k 4
5.b even 2 1 inner 1850.2.b.k 4
5.c odd 4 1 1850.2.a.r 2
5.c odd 4 1 1850.2.a.w yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1850.2.a.r 2 5.c odd 4 1
1850.2.a.w yes 2 5.c odd 4 1
1850.2.b.k 4 1.a even 1 1 trivial
1850.2.b.k 4 5.b even 2 1 inner

## Hecke kernels

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

 $$T_{3}^{4} + 14T_{3}^{2} + 25$$ T3^4 + 14*T3^2 + 25 $$T_{7}^{2} + 6$$ T7^2 + 6 $$T_{13}^{4} + 20T_{13}^{2} + 4$$ T13^4 + 20*T13^2 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$T^{4} + 14T^{2} + 25$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 6)^{2}$$
$11$ $$(T^{2} - 2 T - 5)^{2}$$
$13$ $$T^{4} + 20T^{2} + 4$$
$17$ $$T^{4} + 14T^{2} + 25$$
$19$ $$(T - 5)^{4}$$
$23$ $$(T^{2} + 4)^{2}$$
$29$ $$(T^{2} + 8 T - 8)^{2}$$
$31$ $$(T^{2} - 4 T - 2)^{2}$$
$37$ $$(T^{2} + 1)^{2}$$
$41$ $$(T - 1)^{4}$$
$43$ $$T^{4} + 120T^{2} + 144$$
$47$ $$(T^{2} + 96)^{2}$$
$53$ $$(T^{2} + 36)^{2}$$
$59$ $$(T - 2)^{4}$$
$61$ $$(T^{2} + 8 T + 10)^{2}$$
$67$ $$T^{4} + 110T^{2} + 1849$$
$71$ $$(T^{2} + 20 T + 94)^{2}$$
$73$ $$T^{4} + 210T^{2} + 7569$$
$79$ $$(T^{2} - 4 T - 92)^{2}$$
$83$ $$T^{4} + 14T^{2} + 25$$
$89$ $$(T^{2} + 14 T - 5)^{2}$$
$97$ $$(T^{2} + 196)^{2}$$