# Properties

 Label 1805.2.a.m Level $1805$ Weight $2$ Character orbit 1805.a Self dual yes Analytic conductor $14.413$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1805,2,Mod(1,1805)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1805, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1805.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1805 = 5 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1805.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$14.4129975648$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.7168.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 6x^{2} + 7$$ x^4 - 6*x^2 + 7 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

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

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 4\nu$$ v^3 - 4*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 4\beta_1$$ b3 + 4*b1

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.10100 −1.25928 1.25928 2.10100
−2.10100 2.97127 2.41421 1.00000 −6.24264 4.82843 −0.870264 5.82843 −2.10100
1.2 −1.25928 −1.78089 −0.414214 1.00000 2.24264 −0.828427 3.04017 0.171573 −1.25928
1.3 1.25928 1.78089 −0.414214 1.00000 2.24264 −0.828427 −3.04017 0.171573 1.25928
1.4 2.10100 −2.97127 2.41421 1.00000 −6.24264 4.82843 0.870264 5.82843 2.10100
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$19$$ $$1$$

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1805.2.a.m 4
5.b even 2 1 9025.2.a.bn 4
19.b odd 2 1 inner 1805.2.a.m 4
95.d odd 2 1 9025.2.a.bn 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1805.2.a.m 4 1.a even 1 1 trivial
1805.2.a.m 4 19.b odd 2 1 inner
9025.2.a.bn 4 5.b even 2 1
9025.2.a.bn 4 95.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1805))$$:

 $$T_{2}^{4} - 6T_{2}^{2} + 7$$ T2^4 - 6*T2^2 + 7 $$T_{3}^{4} - 12T_{3}^{2} + 28$$ T3^4 - 12*T3^2 + 28

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 6T^{2} + 7$$
$3$ $$T^{4} - 12T^{2} + 28$$
$5$ $$(T - 1)^{4}$$
$7$ $$(T^{2} - 4 T - 4)^{2}$$
$11$ $$(T - 2)^{4}$$
$13$ $$T^{4} - 20T^{2} + 28$$
$17$ $$(T^{2} - 4 T - 28)^{2}$$
$19$ $$T^{4}$$
$23$ $$(T^{2} - 4 T - 4)^{2}$$
$29$ $$T^{4} - 80T^{2} + 448$$
$31$ $$T^{4} - 48T^{2} + 448$$
$37$ $$T^{4} - 52T^{2} + 28$$
$41$ $$T^{4}$$
$43$ $$(T^{2} + 4 T - 4)^{2}$$
$47$ $$(T^{2} - 4 T - 68)^{2}$$
$53$ $$T^{4} - 84T^{2} + 1372$$
$59$ $$T^{4} - 192T^{2} + 7168$$
$61$ $$(T^{2} - 8)^{2}$$
$67$ $$T^{4} - 76T^{2} + 1372$$
$71$ $$T^{4} - 304 T^{2} + 21952$$
$73$ $$(T^{2} + 12 T + 4)^{2}$$
$79$ $$T^{4} - 192T^{2} + 7168$$
$83$ $$(T^{2} - 12 T + 28)^{2}$$
$89$ $$T^{4} - 336 T^{2} + 21952$$
$97$ $$T^{4} - 20T^{2} + 28$$