# Properties

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

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## 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: $$\Q(i, \sqrt{33})$$ Defining polynomial: $$x^{4} + 17x^{2} + 64$$ x^4 + 17*x^2 + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 370) 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 17x^{2} + 64$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 9\nu ) / 8$$ (v^3 + 9*v) / 8 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 9$$ v^2 + 9
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 9$$ b3 - 9 $$\nu^{3}$$ $$=$$ $$8\beta_{2} - 9\beta_1$$ 8*b2 - 9*b1

## 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
 − 3.37228i 2.37228i − 2.37228i 3.37228i
1.00000i 2.00000i −1.00000 0 2.00000 1.37228i 1.00000i −1.00000 0
149.2 1.00000i 2.00000i −1.00000 0 2.00000 4.37228i 1.00000i −1.00000 0
149.3 1.00000i 2.00000i −1.00000 0 2.00000 4.37228i 1.00000i −1.00000 0
149.4 1.00000i 2.00000i −1.00000 0 2.00000 1.37228i 1.00000i −1.00000 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.m 4
5.b even 2 1 inner 1850.2.b.m 4
5.c odd 4 1 370.2.a.f 2
5.c odd 4 1 1850.2.a.q 2
15.e even 4 1 3330.2.a.bb 2
20.e even 4 1 2960.2.a.o 2

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

## Hecke kernels

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

 $$T_{3}^{2} + 4$$ T3^2 + 4 $$T_{7}^{4} + 21T_{7}^{2} + 36$$ T7^4 + 21*T7^2 + 36 $$T_{13}^{4} + 68T_{13}^{2} + 1024$$ T13^4 + 68*T13^2 + 1024

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$(T^{2} + 4)^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 21T^{2} + 36$$
$11$ $$(T^{2} + T - 8)^{2}$$
$13$ $$T^{4} + 68T^{2} + 1024$$
$17$ $$T^{4} + 29T^{2} + 4$$
$19$ $$(T - 2)^{4}$$
$23$ $$T^{4} + 68T^{2} + 1024$$
$29$ $$(T^{2} - T - 74)^{2}$$
$31$ $$(T^{2} + 11 T + 22)^{2}$$
$37$ $$(T^{2} + 1)^{2}$$
$41$ $$(T^{2} - 5 T - 2)^{2}$$
$43$ $$T^{4} + 57T^{2} + 144$$
$47$ $$T^{4} + 84T^{2} + 576$$
$53$ $$T^{4} + 21T^{2} + 36$$
$59$ $$(T^{2} + 14 T + 16)^{2}$$
$61$ $$(T^{2} + 5 T - 2)^{2}$$
$67$ $$T^{4} + 68T^{2} + 1024$$
$71$ $$(T^{2} + 2 T - 32)^{2}$$
$73$ $$T^{4} + 84T^{2} + 576$$
$79$ $$(T^{2} + 2 T - 32)^{2}$$
$83$ $$T^{4} + 116T^{2} + 64$$
$89$ $$(T + 10)^{4}$$
$97$ $$T^{4} + 293T^{2} + 4$$
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