Properties

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

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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: $$\Q(i, \sqrt{73})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 37x^{2} + 324$$ x^4 + 37*x^2 + 324 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 37x^{2} + 324$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 19\nu ) / 18$$ (v^3 + 19*v) / 18 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 19$$ v^2 + 19
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 19$$ b3 - 19 $$\nu^{3}$$ $$=$$ $$18\beta_{2} - 19\beta_1$$ 18*b2 - 19*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
 − 4.77200i 3.77200i − 3.77200i 4.77200i
1.00000i −1.00000 −1.00000 1.00000i 1.00000i −1.00000 1.00000i 1.00000 1.00000
961.2 1.00000i −1.00000 −1.00000 1.00000i 1.00000i −1.00000 1.00000i 1.00000 1.00000
961.3 1.00000i −1.00000 −1.00000 1.00000i 1.00000i −1.00000 1.00000i 1.00000 1.00000
961.4 1.00000i −1.00000 −1.00000 1.00000i 1.00000i −1.00000 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.d 4
3.b odd 2 1 3330.2.h.j 4
37.b even 2 1 inner 1110.2.h.d 4
111.d odd 2 1 3330.2.h.j 4

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

 $$T_{7} + 1$$ T7 + 1 $$T_{13}^{4} + 37T_{13}^{2} + 324$$ T13^4 + 37*T13^2 + 324

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$(T + 1)^{4}$$
$5$ $$(T^{2} + 1)^{2}$$
$7$ $$(T + 1)^{4}$$
$11$ $$(T + 1)^{4}$$
$13$ $$T^{4} + 37T^{2} + 324$$
$17$ $$(T^{2} + 1)^{2}$$
$19$ $$T^{4} + 61T^{2} + 36$$
$23$ $$T^{4} + 61T^{2} + 36$$
$29$ $$T^{4} + 61T^{2} + 36$$
$31$ $$T^{4} + 61T^{2} + 36$$
$37$ $$T^{4} - 2 T^{3} + \cdots + 1369$$
$41$ $$(T^{2} - 7 T - 6)^{2}$$
$43$ $$T^{4} + 97T^{2} + 144$$
$47$ $$T^{4}$$
$53$ $$(T^{2} + 4 T - 69)^{2}$$
$59$ $$T^{4} + 148T^{2} + 5184$$
$61$ $$T^{4} + 181T^{2} + 2916$$
$67$ $$(T^{2} - 2 T - 72)^{2}$$
$71$ $$(T + 6)^{4}$$
$73$ $$(T^{2} - 9 T + 2)^{2}$$
$79$ $$T^{4} + 164T^{2} + 4096$$
$83$ $$(T^{2} - 5 T - 12)^{2}$$
$89$ $$T^{4} + 369 T^{2} + 20736$$
$97$ $$T^{4} + 121T^{2} + 576$$
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