Properties

 Label 1110.2.h.g Level $1110$ Weight $2$ Character orbit 1110.h Analytic conductor $8.863$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 20x^{6} + 132x^{4} + 297x^{2} + 64$$ x^8 + 20*x^6 + 132*x^4 + 297*x^2 + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + q^{3} - q^{4} + \beta_{2} q^{5} - \beta_{2} q^{6} - \beta_{5} q^{7} + \beta_{2} q^{8} + q^{9}+O(q^{10})$$ q - b2 * q^2 + q^3 - q^4 + b2 * q^5 - b2 * q^6 - b5 * q^7 + b2 * q^8 + q^9 $$q - \beta_{2} q^{2} + q^{3} - q^{4} + \beta_{2} q^{5} - \beta_{2} q^{6} - \beta_{5} q^{7} + \beta_{2} q^{8} + q^{9} + q^{10} + \beta_{7} q^{11} - q^{12} + \beta_{3} q^{13} + \beta_{4} q^{14} + \beta_{2} q^{15} + q^{16} + ( - \beta_{4} + 2 \beta_{2}) q^{17} - \beta_{2} q^{18} + (\beta_{6} - \beta_{4} + \cdots + 2 \beta_{2}) q^{19}+ \cdots + \beta_{7} q^{99}+O(q^{100})$$ q - b2 * q^2 + q^3 - q^4 + b2 * q^5 - b2 * q^6 - b5 * q^7 + b2 * q^8 + q^9 + q^10 + b7 * q^11 - q^12 + b3 * q^13 + b4 * q^14 + b2 * q^15 + q^16 + (-b4 + 2*b2) * q^17 - b2 * q^18 + (b6 - b4 + b3 + 2*b2) * q^19 - b2 * q^20 - b5 * q^21 - b6 * q^22 + b3 * q^23 + b2 * q^24 - q^25 - b1 * q^26 + q^27 + b5 * q^28 + (2*b6 - b4 + b3 + 2*b2) * q^29 + q^30 + (b4 + b3 - 2*b2) * q^31 - b2 * q^32 + b7 * q^33 + (-b5 + 2) * q^34 - b4 * q^35 - q^36 + (b7 - b5 - b3 - b1 + 3) * q^37 + (b7 - b5 - b1 + 2) * q^38 + b3 * q^39 - q^40 + (b5 - b1 - 2) * q^41 + b4 * q^42 + (-b4 + b3 + 2*b2) * q^43 - b7 * q^44 + b2 * q^45 - b1 * q^46 + (b7 + b5 + 2*b1 + 4) * q^47 + q^48 + (2*b7 - b5 + 5) * q^49 + b2 * q^50 + (-b4 + 2*b2) * q^51 - b3 * q^52 + (-b5 + 2) * q^53 - b2 * q^54 + b6 * q^55 - b4 * q^56 + (b6 - b4 + b3 + 2*b2) * q^57 + (2*b7 - b5 - b1 + 2) * q^58 + 2*b3 * q^59 - b2 * q^60 + (-b6 + 2*b4 - b3 - 4*b2) * q^61 + (b5 - b1 - 2) * q^62 - b5 * q^63 - q^64 + b1 * q^65 - b6 * q^66 + (2*b5 - 2*b1) * q^67 + (b4 - 2*b2) * q^68 + b3 * q^69 - b5 * q^70 + (-2*b5 + 4) * q^71 + b2 * q^72 + (-2*b5 - b1 + 2) * q^73 + (-b6 + b4 - b3 - 3*b2 + b1) * q^74 - q^75 + (-b6 + b4 - b3 - 2*b2) * q^76 + (-2*b7 - b5 + 2*b1 - 4) * q^77 - b1 * q^78 + (2*b4 + 2*b3 + 2*b2) * q^79 + b2 * q^80 + q^81 + (-b4 - b3 + 2*b2) * q^82 + (-2*b7 + b1) * q^83 + b5 * q^84 + (b5 - 2) * q^85 + (-b5 - b1 + 2) * q^86 + (2*b6 - b4 + b3 + 2*b2) * q^87 + b6 * q^88 + (-b3 + 6*b2) * q^89 + q^90 + (-2*b6 + 2*b4 - b3 - 4*b2) * q^91 - b3 * q^92 + (b4 + b3 - 2*b2) * q^93 + (-b6 - b4 + 2*b3 - 4*b2) * q^94 + (-b7 + b5 + b1 - 2) * q^95 - b2 * q^96 + (-3*b6 + 2*b4 - b3 - 4*b2) * q^97 + (-2*b6 + b4 - 5*b2) * q^98 + b7 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{3} - 8 q^{4} - 4 q^{7} + 8 q^{9}+O(q^{10})$$ 8 * q + 8 * q^3 - 8 * q^4 - 4 * q^7 + 8 * q^9 $$8 q + 8 q^{3} - 8 q^{4} - 4 q^{7} + 8 q^{9} + 8 q^{10} - 4 q^{11} - 8 q^{12} + 8 q^{16} - 4 q^{21} - 8 q^{25} + 2 q^{26} + 8 q^{27} + 4 q^{28} + 8 q^{30} - 4 q^{33} + 12 q^{34} - 8 q^{36} + 18 q^{37} + 10 q^{38} - 8 q^{40} - 10 q^{41} + 4 q^{44} + 2 q^{46} + 28 q^{47} + 8 q^{48} + 28 q^{49} + 12 q^{53} + 6 q^{58} - 10 q^{62} - 4 q^{63} - 8 q^{64} - 2 q^{65} + 12 q^{67} - 4 q^{70} + 24 q^{71} + 10 q^{73} - 2 q^{74} - 8 q^{75} - 32 q^{77} + 2 q^{78} + 8 q^{81} + 6 q^{83} + 4 q^{84} - 12 q^{85} + 14 q^{86} + 8 q^{90} - 10 q^{95} - 4 q^{99}+O(q^{100})$$ 8 * q + 8 * q^3 - 8 * q^4 - 4 * q^7 + 8 * q^9 + 8 * q^10 - 4 * q^11 - 8 * q^12 + 8 * q^16 - 4 * q^21 - 8 * q^25 + 2 * q^26 + 8 * q^27 + 4 * q^28 + 8 * q^30 - 4 * q^33 + 12 * q^34 - 8 * q^36 + 18 * q^37 + 10 * q^38 - 8 * q^40 - 10 * q^41 + 4 * q^44 + 2 * q^46 + 28 * q^47 + 8 * q^48 + 28 * q^49 + 12 * q^53 + 6 * q^58 - 10 * q^62 - 4 * q^63 - 8 * q^64 - 2 * q^65 + 12 * q^67 - 4 * q^70 + 24 * q^71 + 10 * q^73 - 2 * q^74 - 8 * q^75 - 32 * q^77 + 2 * q^78 + 8 * q^81 + 6 * q^83 + 4 * q^84 - 12 * q^85 + 14 * q^86 + 8 * q^90 - 10 * q^95 - 4 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 20x^{6} + 132x^{4} + 297x^{2} + 64$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{6} + 10\nu^{4} + 18\nu^{2} - 9 ) / 7$$ (v^6 + 10*v^4 + 18*v^2 - 9) / 7 $$\beta_{2}$$ $$=$$ $$( 3\nu^{7} + 44\nu^{5} + 180\nu^{3} + 155\nu ) / 56$$ (3*v^7 + 44*v^5 + 180*v^3 + 155*v) / 56 $$\beta_{3}$$ $$=$$ $$( -3\nu^{7} - 44\nu^{5} - 124\nu^{3} + 181\nu ) / 56$$ (-3*v^7 - 44*v^5 - 124*v^3 + 181*v) / 56 $$\beta_{4}$$ $$=$$ $$( -\nu^{7} + 4\nu^{5} + 164\nu^{3} + 527\nu ) / 56$$ (-v^7 + 4*v^5 + 164*v^3 + 527*v) / 56 $$\beta_{5}$$ $$=$$ $$( -2\nu^{6} - 27\nu^{4} - 85\nu^{2} + 11 ) / 7$$ (-2*v^6 - 27*v^4 - 85*v^2 + 11) / 7 $$\beta_{6}$$ $$=$$ $$( -\nu^{7} + 4\nu^{5} + 164\nu^{3} + 639\nu ) / 56$$ (-v^7 + 4*v^5 + 164*v^3 + 639*v) / 56 $$\beta_{7}$$ $$=$$ $$( 2\nu^{6} + 27\nu^{4} + 99\nu^{2} + 59 ) / 7$$ (2*v^6 + 27*v^4 + 99*v^2 + 59) / 7
 $$\nu$$ $$=$$ $$( \beta_{6} - \beta_{4} ) / 2$$ (b6 - b4) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{7} + \beta_{5} - 10 ) / 2$$ (b7 + b5 - 10) / 2 $$\nu^{3}$$ $$=$$ $$-3\beta_{6} + 3\beta_{4} + \beta_{3} + \beta_{2}$$ -3*b6 + 3*b4 + b3 + b2 $$\nu^{4}$$ $$=$$ $$( -7\beta_{7} - 9\beta_{5} - 4\beta _1 + 68 ) / 2$$ (-7*b7 - 9*b5 - 4*b1 + 68) / 2 $$\nu^{5}$$ $$=$$ $$( 41\beta_{6} - 35\beta_{4} - 24\beta_{3} - 22\beta_{2} ) / 2$$ (41*b6 - 35*b4 - 24*b3 - 22*b2) / 2 $$\nu^{6}$$ $$=$$ $$26\beta_{7} + 36\beta_{5} + 27\beta _1 - 241$$ 26*b7 + 36*b5 + 27*b1 - 241 $$\nu^{7}$$ $$=$$ $$( -293\beta_{6} + 205\beta_{4} + 232\beta_{3} + 240\beta_{2} ) / 2$$ (-293*b6 + 205*b4 + 232*b3 + 240*b2) / 2

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
 0.490113i − 2.20111i − 2.87151i 2.58251i − 0.490113i 2.20111i 2.87151i − 2.58251i
1.00000i 1.00000 −1.00000 1.00000i 1.00000i −4.26968 1.00000i 1.00000 1.00000
961.2 1.00000i 1.00000 −1.00000 1.00000i 1.00000i −2.35622 1.00000i 1.00000 1.00000
961.3 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 0.374052 1.00000i 1.00000 1.00000
961.4 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 4.25184 1.00000i 1.00000 1.00000
961.5 1.00000i 1.00000 −1.00000 1.00000i 1.00000i −4.26968 1.00000i 1.00000 1.00000
961.6 1.00000i 1.00000 −1.00000 1.00000i 1.00000i −2.35622 1.00000i 1.00000 1.00000
961.7 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 0.374052 1.00000i 1.00000 1.00000
961.8 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 4.25184 1.00000i 1.00000 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 961.8 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.g 8
3.b odd 2 1 3330.2.h.o 8
37.b even 2 1 inner 1110.2.h.g 8
111.d odd 2 1 3330.2.h.o 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.h.g 8 1.a even 1 1 trivial
1110.2.h.g 8 37.b even 2 1 inner
3330.2.h.o 8 3.b odd 2 1
3330.2.h.o 8 111.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 $$S_{2}^{\mathrm{new}}(1110, [\chi])$$:

 $$T_{7}^{4} + 2T_{7}^{3} - 19T_{7}^{2} - 36T_{7} + 16$$ T7^4 + 2*T7^3 - 19*T7^2 - 36*T7 + 16 $$T_{13}^{8} + 53T_{13}^{6} + 852T_{13}^{4} + 5056T_{13}^{2} + 9216$$ T13^8 + 53*T13^6 + 852*T13^4 + 5056*T13^2 + 9216

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{4}$$
$3$ $$(T - 1)^{8}$$
$5$ $$(T^{2} + 1)^{4}$$
$7$ $$(T^{4} + 2 T^{3} - 19 T^{2} + \cdots + 16)^{2}$$
$11$ $$(T^{4} + 2 T^{3} - 33 T^{2} + \cdots + 60)^{2}$$
$13$ $$T^{8} + 53 T^{6} + \cdots + 9216$$
$17$ $$T^{8} + 50 T^{6} + \cdots + 144$$
$19$ $$T^{8} + 81 T^{6} + \cdots + 46656$$
$23$ $$T^{8} + 53 T^{6} + \cdots + 9216$$
$29$ $$T^{8} + 213 T^{6} + \cdots + 3594816$$
$31$ $$T^{8} + 117 T^{6} + \cdots + 129600$$
$37$ $$T^{8} - 18 T^{7} + \cdots + 1874161$$
$41$ $$(T^{4} + 5 T^{3} + \cdots + 360)^{2}$$
$43$ $$T^{8} + 89 T^{6} + \cdots + 2304$$
$47$ $$(T^{4} - 14 T^{3} + \cdots - 9216)^{2}$$
$53$ $$(T^{4} - 6 T^{3} - 7 T^{2} + \cdots + 12)^{2}$$
$59$ $$T^{8} + 212 T^{6} + \cdots + 2359296$$
$61$ $$T^{8} + 177 T^{6} + \cdots + 484416$$
$67$ $$(T^{4} - 6 T^{3} + \cdots + 4096)^{2}$$
$71$ $$(T^{4} - 12 T^{3} + \cdots + 192)^{2}$$
$73$ $$(T^{4} - 5 T^{3} + \cdots + 1880)^{2}$$
$79$ $$T^{8} + 492 T^{6} + \cdots + 589824$$
$83$ $$(T^{4} - 3 T^{3} + \cdots - 576)^{2}$$
$89$ $$T^{8} + 185 T^{6} + \cdots + 36864$$
$97$ $$T^{8} + 501 T^{6} + \cdots + 191102976$$
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