# Properties

 Label 1805.2.b.b Level $1805$ Weight $2$ Character orbit 1805.b Analytic conductor $14.413$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1805 = 5 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1805.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.4129975648$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 95) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - 2 q^{4} + ( - \beta - 1) q^{5} - 2 \beta q^{7} + 3 q^{9} +O(q^{10})$$ q + b * q^2 - 2 * q^4 + (-b - 1) * q^5 - 2*b * q^7 + 3 * q^9 $$q + \beta q^{2} - 2 q^{4} + ( - \beta - 1) q^{5} - 2 \beta q^{7} + 3 q^{9} + ( - \beta + 4) q^{10} - q^{11} + \beta q^{13} + 8 q^{14} - 4 q^{16} - \beta q^{17} + 3 \beta q^{18} + (2 \beta + 2) q^{20} - \beta q^{22} - 3 \beta q^{23} + (2 \beta - 3) q^{25} - 4 q^{26} + 4 \beta q^{28} - 9 q^{29} - 7 q^{31} - 4 \beta q^{32} + 4 q^{34} + (2 \beta - 8) q^{35} - 6 q^{36} + \beta q^{37} + 2 q^{41} - \beta q^{43} + 2 q^{44} + ( - 3 \beta - 3) q^{45} + 12 q^{46} + 3 \beta q^{47} - 9 q^{49} + ( - 3 \beta - 8) q^{50} - 2 \beta q^{52} - 2 \beta q^{53} + (\beta + 1) q^{55} - 9 \beta q^{58} - 9 q^{59} - 7 q^{61} - 7 \beta q^{62} - 6 \beta q^{63} + 8 q^{64} + ( - \beta + 4) q^{65} - 5 \beta q^{67} + 2 \beta q^{68} + ( - 8 \beta - 8) q^{70} + q^{71} - 5 \beta q^{73} - 4 q^{74} + 2 \beta q^{77} - q^{79} + (4 \beta + 4) q^{80} + 9 q^{81} + 2 \beta q^{82} + 3 \beta q^{83} + (\beta - 4) q^{85} + 4 q^{86} + 11 q^{89} + ( - 3 \beta + 12) q^{90} + 8 q^{91} + 6 \beta q^{92} - 12 q^{94} - 3 \beta q^{97} - 9 \beta q^{98} - 3 q^{99} +O(q^{100})$$ q + b * q^2 - 2 * q^4 + (-b - 1) * q^5 - 2*b * q^7 + 3 * q^9 + (-b + 4) * q^10 - q^11 + b * q^13 + 8 * q^14 - 4 * q^16 - b * q^17 + 3*b * q^18 + (2*b + 2) * q^20 - b * q^22 - 3*b * q^23 + (2*b - 3) * q^25 - 4 * q^26 + 4*b * q^28 - 9 * q^29 - 7 * q^31 - 4*b * q^32 + 4 * q^34 + (2*b - 8) * q^35 - 6 * q^36 + b * q^37 + 2 * q^41 - b * q^43 + 2 * q^44 + (-3*b - 3) * q^45 + 12 * q^46 + 3*b * q^47 - 9 * q^49 + (-3*b - 8) * q^50 - 2*b * q^52 - 2*b * q^53 + (b + 1) * q^55 - 9*b * q^58 - 9 * q^59 - 7 * q^61 - 7*b * q^62 - 6*b * q^63 + 8 * q^64 + (-b + 4) * q^65 - 5*b * q^67 + 2*b * q^68 + (-8*b - 8) * q^70 + q^71 - 5*b * q^73 - 4 * q^74 + 2*b * q^77 - q^79 + (4*b + 4) * q^80 + 9 * q^81 + 2*b * q^82 + 3*b * q^83 + (b - 4) * q^85 + 4 * q^86 + 11 * q^89 + (-3*b + 12) * q^90 + 8 * q^91 + 6*b * q^92 - 12 * q^94 - 3*b * q^97 - 9*b * q^98 - 3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{4} - 2 q^{5} + 6 q^{9}+O(q^{10})$$ 2 * q - 4 * q^4 - 2 * q^5 + 6 * q^9 $$2 q - 4 q^{4} - 2 q^{5} + 6 q^{9} + 8 q^{10} - 2 q^{11} + 16 q^{14} - 8 q^{16} + 4 q^{20} - 6 q^{25} - 8 q^{26} - 18 q^{29} - 14 q^{31} + 8 q^{34} - 16 q^{35} - 12 q^{36} + 4 q^{41} + 4 q^{44} - 6 q^{45} + 24 q^{46} - 18 q^{49} - 16 q^{50} + 2 q^{55} - 18 q^{59} - 14 q^{61} + 16 q^{64} + 8 q^{65} - 16 q^{70} + 2 q^{71} - 8 q^{74} - 2 q^{79} + 8 q^{80} + 18 q^{81} - 8 q^{85} + 8 q^{86} + 22 q^{89} + 24 q^{90} + 16 q^{91} - 24 q^{94} - 6 q^{99}+O(q^{100})$$ 2 * q - 4 * q^4 - 2 * q^5 + 6 * q^9 + 8 * q^10 - 2 * q^11 + 16 * q^14 - 8 * q^16 + 4 * q^20 - 6 * q^25 - 8 * q^26 - 18 * q^29 - 14 * q^31 + 8 * q^34 - 16 * q^35 - 12 * q^36 + 4 * q^41 + 4 * q^44 - 6 * q^45 + 24 * q^46 - 18 * q^49 - 16 * q^50 + 2 * q^55 - 18 * q^59 - 14 * q^61 + 16 * q^64 + 8 * q^65 - 16 * q^70 + 2 * q^71 - 8 * q^74 - 2 * q^79 + 8 * q^80 + 18 * q^81 - 8 * q^85 + 8 * q^86 + 22 * q^89 + 24 * q^90 + 16 * q^91 - 24 * q^94 - 6 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1805\mathbb{Z}\right)^\times$$.

 $$n$$ $$362$$ $$1446$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1084.1
 − 1.00000i 1.00000i
2.00000i 0 −2.00000 −1.00000 + 2.00000i 0 4.00000i 0 3.00000 4.00000 + 2.00000i
1084.2 2.00000i 0 −2.00000 −1.00000 2.00000i 0 4.00000i 0 3.00000 4.00000 2.00000i
 $$n$$: 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 1805.2.b.b 2
5.b even 2 1 inner 1805.2.b.b 2
5.c odd 4 1 9025.2.a.b 1
5.c odd 4 1 9025.2.a.i 1
19.b odd 2 1 1805.2.b.a 2
19.c even 3 2 95.2.i.a 4
57.h odd 6 2 855.2.be.a 4
95.d odd 2 1 1805.2.b.a 2
95.g even 4 1 9025.2.a.a 1
95.g even 4 1 9025.2.a.j 1
95.i even 6 2 95.2.i.a 4
95.m odd 12 2 475.2.e.a 2
95.m odd 12 2 475.2.e.c 2
285.n odd 6 2 855.2.be.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.i.a 4 19.c even 3 2
95.2.i.a 4 95.i even 6 2
475.2.e.a 2 95.m odd 12 2
475.2.e.c 2 95.m odd 12 2
855.2.be.a 4 57.h odd 6 2
855.2.be.a 4 285.n odd 6 2
1805.2.b.a 2 19.b odd 2 1
1805.2.b.a 2 95.d odd 2 1
1805.2.b.b 2 1.a even 1 1 trivial
1805.2.b.b 2 5.b even 2 1 inner
9025.2.a.a 1 95.g even 4 1
9025.2.a.b 1 5.c odd 4 1
9025.2.a.i 1 5.c odd 4 1
9025.2.a.j 1 95.g even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1805, [\chi])$$:

 $$T_{2}^{2} + 4$$ T2^2 + 4 $$T_{29} + 9$$ T29 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 2T + 5$$
$7$ $$T^{2} + 16$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 4$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 36$$
$29$ $$(T + 9)^{2}$$
$31$ $$(T + 7)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T - 2)^{2}$$
$43$ $$T^{2} + 4$$
$47$ $$T^{2} + 36$$
$53$ $$T^{2} + 16$$
$59$ $$(T + 9)^{2}$$
$61$ $$(T + 7)^{2}$$
$67$ $$T^{2} + 100$$
$71$ $$(T - 1)^{2}$$
$73$ $$T^{2} + 100$$
$79$ $$(T + 1)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$(T - 11)^{2}$$
$97$ $$T^{2} + 36$$
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