# Properties

 Label 1805.2.b.g Level $1805$ Weight $2$ Character orbit 1805.b Analytic conductor $14.413$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## 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: $$6$$ Coefficient field: 6.0.4227136.1 Defining polynomial: $$x^{6} + 6x^{4} + 7x^{2} + 1$$ x^6 + 6*x^4 + 7*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{5} + \beta_1) q^{2} + ( - \beta_{5} - \beta_{4} - \beta_1) q^{3} + ( - \beta_{3} - \beta_{2}) q^{4} + (\beta_{3} - \beta_1) q^{5} + (\beta_{2} + 2) q^{6} + ( - 2 \beta_{5} - \beta_1) q^{7} + ( - 2 \beta_{5} + \beta_{4}) q^{8} + (2 \beta_{3} - \beta_{2} - 2) q^{9}+O(q^{10})$$ q + (b5 + b1) * q^2 + (-b5 - b4 - b1) * q^3 + (-b3 - b2) * q^4 + (b3 - b1) * q^5 + (b2 + 2) * q^6 + (-2*b5 - b1) * q^7 + (-2*b5 + b4) * q^8 + (2*b3 - b2 - 2) * q^9 $$q + (\beta_{5} + \beta_1) q^{2} + ( - \beta_{5} - \beta_{4} - \beta_1) q^{3} + ( - \beta_{3} - \beta_{2}) q^{4} + (\beta_{3} - \beta_1) q^{5} + (\beta_{2} + 2) q^{6} + ( - 2 \beta_{5} - \beta_1) q^{7} + ( - 2 \beta_{5} + \beta_{4}) q^{8} + (2 \beta_{3} - \beta_{2} - 2) q^{9} + (2 \beta_{5} - \beta_{4} + \beta_1 + 1) q^{10} + \beta_{3} q^{11} + (2 \beta_{5} - 2 \beta_{4} + \beta_1) q^{12} + (\beta_{5} - \beta_{4} + 2 \beta_1) q^{13} + (2 \beta_{3} + 2 \beta_{2} + 3) q^{14} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{15} + (\beta_{3} + 2) q^{16} + ( - \beta_{5} - 2 \beta_{4} - \beta_1) q^{17} + ( - 2 \beta_{4} - \beta_1) q^{18} + (\beta_{5} - \beta_{3} - 2 \beta_{2} - \beta_1 - 3) q^{20} + ( - \beta_{3} - \beta_{2} - 3) q^{21} + (2 \beta_{5} - \beta_{4} + \beta_1) q^{22} + (\beta_{4} + \beta_1) q^{23} + ( - 4 \beta_{3} + 1) q^{24} + (2 \beta_{4} + 2 \beta_{2} + 1) q^{25} + ( - 2 \beta_{3} - \beta_{2} - 3) q^{26} + ( - 2 \beta_{5} + 3 \beta_{4} + 3 \beta_1) q^{27} + (7 \beta_{5} - 2 \beta_{4} + 5 \beta_1) q^{28} + (3 \beta_{3} - 3) q^{29} + ( - \beta_{5} - \beta_{4} + 3 \beta_{3} + \beta_{2} - \beta_1) q^{30} + ( - \beta_{2} - 5) q^{31} + (\beta_{4} + 3 \beta_1) q^{32} + ( - \beta_{5} + 2 \beta_{4} + \beta_1) q^{33} + ( - \beta_{3} + \beta_{2} + 2) q^{34} + ( - 4 \beta_{5} + \beta_{4} - \beta_{2} - 2 \beta_1) q^{35} + (2 \beta_{3} - 2 \beta_{2} - 3) q^{36} + (2 \beta_{4} - 3 \beta_1) q^{37} + (\beta_{3} + 2 \beta_{2}) q^{39} + ( - 5 \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - 4 \beta_1 + 2) q^{40} + (3 \beta_{3} - 2 \beta_{2} - 3) q^{41} + ( - 7 \beta_{5} + \beta_{4} - 5 \beta_1) q^{42} + (\beta_{5} - 4 \beta_1) q^{43} + ( - \beta_{3} - 2 \beta_{2} - 3) q^{44} + (\beta_{5} + 3 \beta_{4} - 3 \beta_{3} + \beta_{2} + \beta_1 + 6) q^{45} + (\beta_{3} - 1) q^{46} + ( - 4 \beta_{5} - 4 \beta_{4} - \beta_1) q^{47} + ( - 3 \beta_{5} - \beta_1) q^{48} + ( - 4 \beta_{3} - 3 \beta_{2} + 1) q^{49} + (5 \beta_{5} + 2 \beta_{3} + 3 \beta_1) q^{50} + (4 \beta_{3} - \beta_{2} - 8) q^{51} + ( - 7 \beta_{5} - 2 \beta_1) q^{52} + ( - \beta_{5} + 2 \beta_{4} + 3 \beta_1) q^{53} + (5 \beta_{3} + 2 \beta_{2} - 1) q^{54} + (\beta_{4} + \beta_{2} + 3) q^{55} + ( - 5 \beta_{3} - 3 \beta_{2} - 6) q^{56} + (3 \beta_{5} - 3 \beta_{4}) q^{58} + (\beta_{3} + 4 \beta_{2} + 3) q^{59} + (6 \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} + 6 \beta_1) q^{60} + (2 \beta_{3} + 2 \beta_{2} - 1) q^{61} + ( - 7 \beta_{5} - 6 \beta_1) q^{62} + ( - \beta_{5} + \beta_{4} + \beta_1) q^{63} + (3 \beta_{3} + 1) q^{64} + (3 \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 3) q^{65} + (3 \beta_{3} + \beta_{2}) q^{66} + (2 \beta_{5} + 4 \beta_1) q^{67} - 3 \beta_{4} q^{68} + ( - 3 \beta_{3} + \beta_{2} + 4) q^{69} + ( - 2 \beta_{5} + 5 \beta_{3} + 4 \beta_{2} - \beta_1 + 6) q^{70} + (\beta_{3} - 5 \beta_{2}) q^{71} + ( - 3 \beta_{5} - 6 \beta_{4} - 5 \beta_1) q^{72} + ( - 2 \beta_{5} + \beta_{4} + 5 \beta_1) q^{73} + (2 \beta_{3} + 3) q^{74} + ( - 3 \beta_{5} - \beta_{4} - 4 \beta_{3} - 5 \beta_1 + 6) q^{75} + ( - 4 \beta_{5} + \beta_{4} - 2 \beta_1) q^{77} + (6 \beta_{5} - \beta_{4} + 3 \beta_1) q^{78} + (4 \beta_{2} + 4) q^{79} + (\beta_{4} + 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 3) q^{80} + ( - 5 \beta_{3} + 4) q^{81} + ( - \beta_{5} - 3 \beta_{4} - 2 \beta_1) q^{82} + ( - 5 \beta_{5} + 2 \beta_{4} - 9 \beta_1) q^{83} + (6 \beta_{3} + 5 \beta_{2} + 6) q^{84} + (3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 1) q^{85} + ( - \beta_{3} - \beta_{2} + 3) q^{86} + (9 \beta_{4} + 6 \beta_1) q^{87} + ( - 5 \beta_{5} - \beta_{4} - 4 \beta_1) q^{88} + ( - 4 \beta_{2} + 6) q^{89} + (2 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} - \beta_{2} + 4 \beta_1 - 2) q^{90} + (3 \beta_{3} + 4 \beta_{2} + 3) q^{91} + (\beta_{5} + \beta_{4} + 2 \beta_1) q^{92} + (6 \beta_{5} + 5 \beta_{4} + 7 \beta_1) q^{93} + (4 \beta_{2} + 5) q^{94} + ( - 5 \beta_{3} + 3 \beta_{2} + 6) q^{96} + (5 \beta_{5} + 7 \beta_{4} + \beta_1) q^{97} + ( - 13 \beta_{5} + 4 \beta_{4} - 6 \beta_1) q^{98} + ( - 3 \beta_{3} + \beta_{2} + 6) q^{99}+O(q^{100})$$ q + (b5 + b1) * q^2 + (-b5 - b4 - b1) * q^3 + (-b3 - b2) * q^4 + (b3 - b1) * q^5 + (b2 + 2) * q^6 + (-2*b5 - b1) * q^7 + (-2*b5 + b4) * q^8 + (2*b3 - b2 - 2) * q^9 + (2*b5 - b4 + b1 + 1) * q^10 + b3 * q^11 + (2*b5 - 2*b4 + b1) * q^12 + (b5 - b4 + 2*b1) * q^13 + (2*b3 + 2*b2 + 3) * q^14 + (-b5 + 2*b4 + b3 - b2 + b1 - 1) * q^15 + (b3 + 2) * q^16 + (-b5 - 2*b4 - b1) * q^17 + (-2*b4 - b1) * q^18 + (b5 - b3 - 2*b2 - b1 - 3) * q^20 + (-b3 - b2 - 3) * q^21 + (2*b5 - b4 + b1) * q^22 + (b4 + b1) * q^23 + (-4*b3 + 1) * q^24 + (2*b4 + 2*b2 + 1) * q^25 + (-2*b3 - b2 - 3) * q^26 + (-2*b5 + 3*b4 + 3*b1) * q^27 + (7*b5 - 2*b4 + 5*b1) * q^28 + (3*b3 - 3) * q^29 + (-b5 - b4 + 3*b3 + b2 - b1) * q^30 + (-b2 - 5) * q^31 + (b4 + 3*b1) * q^32 + (-b5 + 2*b4 + b1) * q^33 + (-b3 + b2 + 2) * q^34 + (-4*b5 + b4 - b2 - 2*b1) * q^35 + (2*b3 - 2*b2 - 3) * q^36 + (2*b4 - 3*b1) * q^37 + (b3 + 2*b2) * q^39 + (-5*b5 - b4 - b3 - b2 - 4*b1 + 2) * q^40 + (3*b3 - 2*b2 - 3) * q^41 + (-7*b5 + b4 - 5*b1) * q^42 + (b5 - 4*b1) * q^43 + (-b3 - 2*b2 - 3) * q^44 + (b5 + 3*b4 - 3*b3 + b2 + b1 + 6) * q^45 + (b3 - 1) * q^46 + (-4*b5 - 4*b4 - b1) * q^47 + (-3*b5 - b1) * q^48 + (-4*b3 - 3*b2 + 1) * q^49 + (5*b5 + 2*b3 + 3*b1) * q^50 + (4*b3 - b2 - 8) * q^51 + (-7*b5 - 2*b1) * q^52 + (-b5 + 2*b4 + 3*b1) * q^53 + (5*b3 + 2*b2 - 1) * q^54 + (b4 + b2 + 3) * q^55 + (-5*b3 - 3*b2 - 6) * q^56 + (3*b5 - 3*b4) * q^58 + (b3 + 4*b2 + 3) * q^59 + (6*b5 + b4 + 2*b3 - b2 + 6*b1) * q^60 + (2*b3 + 2*b2 - 1) * q^61 + (-7*b5 - 6*b1) * q^62 + (-b5 + b4 + b1) * q^63 + (3*b3 + 1) * q^64 + (3*b5 - b4 + b3 - 2*b2 + 3*b1 + 3) * q^65 + (3*b3 + b2) * q^66 + (2*b5 + 4*b1) * q^67 - 3*b4 * q^68 + (-3*b3 + b2 + 4) * q^69 + (-2*b5 + 5*b3 + 4*b2 - b1 + 6) * q^70 + (b3 - 5*b2) * q^71 + (-3*b5 - 6*b4 - 5*b1) * q^72 + (-2*b5 + b4 + 5*b1) * q^73 + (2*b3 + 3) * q^74 + (-3*b5 - b4 - 4*b3 - 5*b1 + 6) * q^75 + (-4*b5 + b4 - 2*b1) * q^77 + (6*b5 - b4 + 3*b1) * q^78 + (4*b2 + 4) * q^79 + (b4 + 2*b3 + b2 - 2*b1 + 3) * q^80 + (-5*b3 + 4) * q^81 + (-b5 - 3*b4 - 2*b1) * q^82 + (-5*b5 + 2*b4 - 9*b1) * q^83 + (6*b3 + 5*b2 + 6) * q^84 + (3*b4 + 2*b3 - 2*b2 + 3*b1 - 1) * q^85 + (-b3 - b2 + 3) * q^86 + (9*b4 + 6*b1) * q^87 + (-5*b5 - b4 - 4*b1) * q^88 + (-4*b2 + 6) * q^89 + (2*b5 + 3*b4 + 2*b3 - b2 + 4*b1 - 2) * q^90 + (3*b3 + 4*b2 + 3) * q^91 + (b5 + b4 + 2*b1) * q^92 + (6*b5 + 5*b4 + 7*b1) * q^93 + (4*b2 + 5) * q^94 + (-5*b3 + 3*b2 + 6) * q^96 + (5*b5 + 7*b4 + b1) * q^97 + (-13*b5 + 4*b4 - 6*b1) * q^98 + (-3*b3 + b2 + 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 2 q^{4} + 2 q^{5} + 12 q^{6} - 8 q^{9}+O(q^{10})$$ 6 * q - 2 * q^4 + 2 * q^5 + 12 * q^6 - 8 * q^9 $$6 q - 2 q^{4} + 2 q^{5} + 12 q^{6} - 8 q^{9} + 6 q^{10} + 2 q^{11} + 22 q^{14} - 4 q^{15} + 14 q^{16} - 20 q^{20} - 20 q^{21} - 2 q^{24} + 6 q^{25} - 22 q^{26} - 12 q^{29} + 6 q^{30} - 30 q^{31} + 10 q^{34} - 14 q^{36} + 2 q^{39} + 10 q^{40} - 12 q^{41} - 20 q^{44} + 30 q^{45} - 4 q^{46} - 2 q^{49} + 4 q^{50} - 40 q^{51} + 4 q^{54} + 18 q^{55} - 46 q^{56} + 20 q^{59} + 4 q^{60} - 2 q^{61} + 12 q^{64} + 20 q^{65} + 6 q^{66} + 18 q^{69} + 46 q^{70} + 2 q^{71} + 22 q^{74} + 28 q^{75} + 24 q^{79} + 22 q^{80} + 14 q^{81} + 48 q^{84} - 2 q^{85} + 16 q^{86} + 36 q^{89} - 8 q^{90} + 24 q^{91} + 30 q^{94} + 26 q^{96} + 30 q^{99}+O(q^{100})$$ 6 * q - 2 * q^4 + 2 * q^5 + 12 * q^6 - 8 * q^9 + 6 * q^10 + 2 * q^11 + 22 * q^14 - 4 * q^15 + 14 * q^16 - 20 * q^20 - 20 * q^21 - 2 * q^24 + 6 * q^25 - 22 * q^26 - 12 * q^29 + 6 * q^30 - 30 * q^31 + 10 * q^34 - 14 * q^36 + 2 * q^39 + 10 * q^40 - 12 * q^41 - 20 * q^44 + 30 * q^45 - 4 * q^46 - 2 * q^49 + 4 * q^50 - 40 * q^51 + 4 * q^54 + 18 * q^55 - 46 * q^56 + 20 * q^59 + 4 * q^60 - 2 * q^61 + 12 * q^64 + 20 * q^65 + 6 * q^66 + 18 * q^69 + 46 * q^70 + 2 * q^71 + 22 * q^74 + 28 * q^75 + 24 * q^79 + 22 * q^80 + 14 * q^81 + 48 * q^84 - 2 * q^85 + 16 * q^86 + 36 * q^89 - 8 * q^90 + 24 * q^91 + 30 * q^94 + 26 * q^96 + 30 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 6x^{4} + 7x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2$$ v^2 + 2 $$\beta_{3}$$ $$=$$ $$\nu^{4} + 5\nu^{2} + 3$$ v^4 + 5*v^2 + 3 $$\beta_{4}$$ $$=$$ $$-\nu^{5} - 5\nu^{3} - 3\nu$$ -v^5 - 5*v^3 - 3*v $$\beta_{5}$$ $$=$$ $$\nu^{5} + 6\nu^{3} + 6\nu$$ v^5 + 6*v^3 + 6*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2$$ b2 - 2 $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} - 3\beta_1$$ b5 + b4 - 3*b1 $$\nu^{4}$$ $$=$$ $$\beta_{3} - 5\beta_{2} + 7$$ b3 - 5*b2 + 7 $$\nu^{5}$$ $$=$$ $$-5\beta_{5} - 6\beta_{4} + 12\beta_1$$ -5*b5 - 6*b4 + 12*b1

## 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
 − 0.407132i − 1.15904i − 2.11917i 2.11917i 1.15904i 0.407132i
2.45620i 1.56104i −4.03293 2.19869 + 0.407132i 3.83424 4.50527i 4.99330i 0.563139 1.00000 5.40043i
1084.2 0.862781i 3.07914i 1.25561 −1.91223 + 1.15904i 2.65662 0.566520i 2.80888i −6.48108 1.00000 + 1.64984i
1084.3 0.471884i 1.04022i 1.77733 0.713538 + 2.11917i −0.490864 1.17540i 1.78246i 1.91794 1.00000 0.336707i
1084.4 0.471884i 1.04022i 1.77733 0.713538 2.11917i −0.490864 1.17540i 1.78246i 1.91794 1.00000 + 0.336707i
1084.5 0.862781i 3.07914i 1.25561 −1.91223 1.15904i 2.65662 0.566520i 2.80888i −6.48108 1.00000 1.64984i
1084.6 2.45620i 1.56104i −4.03293 2.19869 0.407132i 3.83424 4.50527i 4.99330i 0.563139 1.00000 + 5.40043i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1084.6 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.g 6
5.b even 2 1 inner 1805.2.b.g 6
5.c odd 4 2 9025.2.a.bt 6
19.b odd 2 1 1805.2.b.f 6
19.d odd 6 2 95.2.i.b 12
57.f even 6 2 855.2.be.d 12
95.d odd 2 1 1805.2.b.f 6
95.g even 4 2 9025.2.a.bu 6
95.h odd 6 2 95.2.i.b 12
95.l even 12 4 475.2.e.g 12
285.q even 6 2 855.2.be.d 12

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

## Hecke kernels

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

 $$T_{2}^{6} + 7T_{2}^{4} + 6T_{2}^{2} + 1$$ T2^6 + 7*T2^4 + 6*T2^2 + 1 $$T_{29}^{3} + 6T_{29}^{2} - 27T_{29} - 27$$ T29^3 + 6*T29^2 - 27*T29 - 27

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 7 T^{4} + 6 T^{2} + 1$$
$3$ $$T^{6} + 13 T^{4} + 36 T^{2} + 25$$
$5$ $$T^{6} - 2 T^{5} - T^{4} + 4 T^{3} + \cdots + 125$$
$7$ $$T^{6} + 22 T^{4} + 35 T^{2} + 9$$
$11$ $$(T^{3} - T^{2} - 4 T + 3)^{2}$$
$13$ $$T^{6} + 31 T^{4} + 239 T^{2} + 9$$
$17$ $$T^{6} + 35 T^{4} + 198 T^{2} + \cdots + 81$$
$19$ $$T^{6}$$
$23$ $$T^{6} + 12 T^{4} + 7 T^{2} + 1$$
$29$ $$(T^{3} + 6 T^{2} - 27 T - 27)^{2}$$
$31$ $$(T^{3} + 15 T^{2} + 70 T + 97)^{2}$$
$37$ $$T^{6} + 98 T^{4} + 887 T^{2} + \cdots + 729$$
$41$ $$(T^{3} + 6 T^{2} - 41 T - 3)^{2}$$
$43$ $$T^{6} + 127 T^{4} + 2516 T^{2} + \cdots + 441$$
$47$ $$T^{6} + 214 T^{4} + 13407 T^{2} + \cdots + 218089$$
$53$ $$T^{6} + 99 T^{4} + 2302 T^{2} + \cdots + 11449$$
$59$ $$(T^{3} - 10 T^{2} - 55 T + 291)^{2}$$
$61$ $$(T^{3} + T^{2} - 41 T - 113)^{2}$$
$67$ $$T^{6} + 76 T^{4} + 1856 T^{2} + \cdots + 14400$$
$71$ $$(T^{3} - T^{2} - 124 T - 477)^{2}$$
$73$ $$T^{6} + 236 T^{4} + 13331 T^{2} + \cdots + 99225$$
$79$ $$(T^{3} - 12 T^{2} - 32 T + 448)^{2}$$
$83$ $$T^{6} + 459 T^{4} + 56302 T^{2} + \cdots + 966289$$
$89$ $$(T^{3} - 18 T^{2} + 28 T + 72)^{2}$$
$97$ $$T^{6} + 529 T^{4} + 74192 T^{2} + \cdots + 1238769$$