# Properties

 Label 1110.2.h.e Level $1110$ Weight $2$ Character orbit 1110.h Analytic conductor $8.863$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1110,2,Mod(961,1110)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1110, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("1110.961");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1110 = 2 \cdot 3 \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1110.h (of order $$2$$, degree $$1$$, minimal)

## 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: no Analytic conductor: $$8.86339462436$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{33})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field 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: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} - q^{3} - q^{4} + \beta_{2} q^{5} - \beta_{2} q^{6} + ( - \beta_{3} + 1) q^{7} - \beta_{2} q^{8} + q^{9}+O(q^{10})$$ q + b2 * q^2 - q^3 - q^4 + b2 * q^5 - b2 * q^6 + (-b3 + 1) * q^7 - b2 * q^8 + q^9 $$q + \beta_{2} q^{2} - q^{3} - q^{4} + \beta_{2} q^{5} - \beta_{2} q^{6} + ( - \beta_{3} + 1) q^{7} - \beta_{2} q^{8} + q^{9} - q^{10} + (\beta_{3} + 3) q^{11} + q^{12} + 2 \beta_1 q^{13} - \beta_1 q^{14} - \beta_{2} q^{15} + q^{16} + (2 \beta_{2} - \beta_1) q^{17} + \beta_{2} q^{18} - 6 \beta_{2} q^{19} - \beta_{2} q^{20} + (\beta_{3} - 1) q^{21} + (4 \beta_{2} + \beta_1) q^{22} + (4 \beta_{2} + 2 \beta_1) q^{23} + \beta_{2} q^{24} - q^{25} + ( - 2 \beta_{3} + 2) q^{26} - q^{27} + (\beta_{3} - 1) q^{28} + ( - 6 \beta_{2} + \beta_1) q^{29} + q^{30} + (6 \beta_{2} + \beta_1) q^{31} + \beta_{2} q^{32} + ( - \beta_{3} - 3) q^{33} + (\beta_{3} - 3) q^{34} - \beta_1 q^{35} - q^{36} + ( - 2 \beta_{3} - 2 \beta_{2} + 1) q^{37} + 6 q^{38} - 2 \beta_1 q^{39} + q^{40} + ( - 3 \beta_{3} + 1) q^{41} + \beta_1 q^{42} + (2 \beta_{2} + \beta_1) q^{43} + ( - \beta_{3} - 3) q^{44} + \beta_{2} q^{45} + ( - 2 \beta_{3} - 2) q^{46} - q^{48} + ( - \beta_{3} + 2) q^{49} - \beta_{2} q^{50} + ( - 2 \beta_{2} + \beta_1) q^{51} - 2 \beta_1 q^{52} + ( - \beta_{3} - 5) q^{53} - \beta_{2} q^{54} + (4 \beta_{2} + \beta_1) q^{55} + \beta_1 q^{56} + 6 \beta_{2} q^{57} + ( - \beta_{3} + 7) q^{58} + 8 \beta_{2} q^{59} + \beta_{2} q^{60} + (8 \beta_{2} - \beta_1) q^{61} + ( - \beta_{3} - 5) q^{62} + ( - \beta_{3} + 1) q^{63} - q^{64} + ( - 2 \beta_{3} + 2) q^{65} + ( - 4 \beta_{2} - \beta_1) q^{66} + (2 \beta_{3} + 2) q^{67} + ( - 2 \beta_{2} + \beta_1) q^{68} + ( - 4 \beta_{2} - 2 \beta_1) q^{69} + (\beta_{3} - 1) q^{70} + ( - 2 \beta_{3} - 2) q^{71} - \beta_{2} q^{72} + 14 q^{73} + ( - \beta_{2} - 2 \beta_1 + 2) q^{74} + q^{75} + 6 \beta_{2} q^{76} + ( - 3 \beta_{3} - 5) q^{77} + (2 \beta_{3} - 2) q^{78} + (2 \beta_{2} + 2 \beta_1) q^{79} + \beta_{2} q^{80} + q^{81} + ( - 2 \beta_{2} - 3 \beta_1) q^{82} + ( - 4 \beta_{3} + 8) q^{83} + ( - \beta_{3} + 1) q^{84} + (\beta_{3} - 3) q^{85} + ( - \beta_{3} - 1) q^{86} + (6 \beta_{2} - \beta_1) q^{87} + ( - 4 \beta_{2} - \beta_1) q^{88} + (2 \beta_{2} - 2 \beta_1) q^{89} - q^{90} + ( - 16 \beta_{2} + 2 \beta_1) q^{91} + ( - 4 \beta_{2} - 2 \beta_1) q^{92} + ( - 6 \beta_{2} - \beta_1) q^{93} + 6 q^{95} - \beta_{2} q^{96} + 3 \beta_1 q^{97} + (\beta_{2} - \beta_1) q^{98} + (\beta_{3} + 3) q^{99}+O(q^{100})$$ q + b2 * q^2 - q^3 - q^4 + b2 * q^5 - b2 * q^6 + (-b3 + 1) * q^7 - b2 * q^8 + q^9 - q^10 + (b3 + 3) * q^11 + q^12 + 2*b1 * q^13 - b1 * q^14 - b2 * q^15 + q^16 + (2*b2 - b1) * q^17 + b2 * q^18 - 6*b2 * q^19 - b2 * q^20 + (b3 - 1) * q^21 + (4*b2 + b1) * q^22 + (4*b2 + 2*b1) * q^23 + b2 * q^24 - q^25 + (-2*b3 + 2) * q^26 - q^27 + (b3 - 1) * q^28 + (-6*b2 + b1) * q^29 + q^30 + (6*b2 + b1) * q^31 + b2 * q^32 + (-b3 - 3) * q^33 + (b3 - 3) * q^34 - b1 * q^35 - q^36 + (-2*b3 - 2*b2 + 1) * q^37 + 6 * q^38 - 2*b1 * q^39 + q^40 + (-3*b3 + 1) * q^41 + b1 * q^42 + (2*b2 + b1) * q^43 + (-b3 - 3) * q^44 + b2 * q^45 + (-2*b3 - 2) * q^46 - q^48 + (-b3 + 2) * q^49 - b2 * q^50 + (-2*b2 + b1) * q^51 - 2*b1 * q^52 + (-b3 - 5) * q^53 - b2 * q^54 + (4*b2 + b1) * q^55 + b1 * q^56 + 6*b2 * q^57 + (-b3 + 7) * q^58 + 8*b2 * q^59 + b2 * q^60 + (8*b2 - b1) * q^61 + (-b3 - 5) * q^62 + (-b3 + 1) * q^63 - q^64 + (-2*b3 + 2) * q^65 + (-4*b2 - b1) * q^66 + (2*b3 + 2) * q^67 + (-2*b2 + b1) * q^68 + (-4*b2 - 2*b1) * q^69 + (b3 - 1) * q^70 + (-2*b3 - 2) * q^71 - b2 * q^72 + 14 * q^73 + (-b2 - 2*b1 + 2) * q^74 + q^75 + 6*b2 * q^76 + (-3*b3 - 5) * q^77 + (2*b3 - 2) * q^78 + (2*b2 + 2*b1) * q^79 + b2 * q^80 + q^81 + (-2*b2 - 3*b1) * q^82 + (-4*b3 + 8) * q^83 + (-b3 + 1) * q^84 + (b3 - 3) * q^85 + (-b3 - 1) * q^86 + (6*b2 - b1) * q^87 + (-4*b2 - b1) * q^88 + (2*b2 - 2*b1) * q^89 - q^90 + (-16*b2 + 2*b1) * q^91 + (-4*b2 - 2*b1) * q^92 + (-6*b2 - b1) * q^93 + 6 * q^95 - b2 * q^96 + 3*b1 * q^97 + (b2 - b1) * q^98 + (b3 + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} - 4 q^{4} + 2 q^{7} + 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^3 - 4 * q^4 + 2 * q^7 + 4 * q^9 $$4 q - 4 q^{3} - 4 q^{4} + 2 q^{7} + 4 q^{9} - 4 q^{10} + 14 q^{11} + 4 q^{12} + 4 q^{16} - 2 q^{21} - 4 q^{25} + 4 q^{26} - 4 q^{27} - 2 q^{28} + 4 q^{30} - 14 q^{33} - 10 q^{34} - 4 q^{36} + 24 q^{38} + 4 q^{40} - 2 q^{41} - 14 q^{44} - 12 q^{46} - 4 q^{48} + 6 q^{49} - 22 q^{53} + 26 q^{58} - 22 q^{62} + 2 q^{63} - 4 q^{64} + 4 q^{65} + 12 q^{67} - 2 q^{70} - 12 q^{71} + 56 q^{73} + 8 q^{74} + 4 q^{75} - 26 q^{77} - 4 q^{78} + 4 q^{81} + 24 q^{83} + 2 q^{84} - 10 q^{85} - 6 q^{86} - 4 q^{90} + 24 q^{95} + 14 q^{99}+O(q^{100})$$ 4 * q - 4 * q^3 - 4 * q^4 + 2 * q^7 + 4 * q^9 - 4 * q^10 + 14 * q^11 + 4 * q^12 + 4 * q^16 - 2 * q^21 - 4 * q^25 + 4 * q^26 - 4 * q^27 - 2 * q^28 + 4 * q^30 - 14 * q^33 - 10 * q^34 - 4 * q^36 + 24 * q^38 + 4 * q^40 - 2 * q^41 - 14 * q^44 - 12 * q^46 - 4 * q^48 + 6 * q^49 - 22 * q^53 + 26 * q^58 - 22 * q^62 + 2 * q^63 - 4 * q^64 + 4 * q^65 + 12 * q^67 - 2 * q^70 - 12 * q^71 + 56 * q^73 + 8 * q^74 + 4 * q^75 - 26 * q^77 - 4 * q^78 + 4 * q^81 + 24 * q^83 + 2 * q^84 - 10 * q^85 - 6 * q^86 - 4 * q^90 + 24 * q^95 + 14 * 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 $$\left(\mathbb{Z}/1110\mathbb{Z}\right)^\times$$.

 $$n$$ $$371$$ $$631$$ $$667$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
961.1
 − 2.37228i 3.37228i 2.37228i − 3.37228i
1.00000i −1.00000 −1.00000 1.00000i 1.00000i −2.37228 1.00000i 1.00000 −1.00000
961.2 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 3.37228 1.00000i 1.00000 −1.00000
961.3 1.00000i −1.00000 −1.00000 1.00000i 1.00000i −2.37228 1.00000i 1.00000 −1.00000
961.4 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 3.37228 1.00000i 1.00000 −1.00000
 $$n$$: 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
37.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1110.2.h.e 4
3.b odd 2 1 3330.2.h.k 4
37.b even 2 1 inner 1110.2.h.e 4
111.d odd 2 1 3330.2.h.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.h.e 4 1.a even 1 1 trivial
1110.2.h.e 4 37.b even 2 1 inner
3330.2.h.k 4 3.b odd 2 1
3330.2.h.k 4 111.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 $$S_{2}^{\mathrm{new}}(1110, [\chi])$$:

 $$T_{7}^{2} - T_{7} - 8$$ T7^2 - T7 - 8 $$T_{13}^{4} + 68T_{13}^{2} + 1024$$ T13^4 + 68*T13^2 + 1024

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$(T + 1)^{4}$$
$5$ $$(T^{2} + 1)^{2}$$
$7$ $$(T^{2} - T - 8)^{2}$$
$11$ $$(T^{2} - 7 T + 4)^{2}$$
$13$ $$T^{4} + 68T^{2} + 1024$$
$17$ $$T^{4} + 29T^{2} + 4$$
$19$ $$(T^{2} + 36)^{2}$$
$23$ $$T^{4} + 84T^{2} + 576$$
$29$ $$T^{4} + 101T^{2} + 1156$$
$31$ $$T^{4} + 77T^{2} + 484$$
$37$ $$T^{4} - 58T^{2} + 1369$$
$41$ $$(T^{2} + T - 74)^{2}$$
$43$ $$T^{4} + 21T^{2} + 36$$
$47$ $$T^{4}$$
$53$ $$(T^{2} + 11 T + 22)^{2}$$
$59$ $$(T^{2} + 64)^{2}$$
$61$ $$T^{4} + 161T^{2} + 4096$$
$67$ $$(T^{2} - 6 T - 24)^{2}$$
$71$ $$(T^{2} + 6 T - 24)^{2}$$
$73$ $$(T - 14)^{4}$$
$79$ $$T^{4} + 68T^{2} + 1024$$
$83$ $$(T^{2} - 12 T - 96)^{2}$$
$89$ $$T^{4} + 84T^{2} + 576$$
$97$ $$T^{4} + 153T^{2} + 5184$$