Properties

 Label 740.2.l.a Level $740$ Weight $2$ Character orbit 740.l Analytic conductor $5.909$ Analytic rank $0$ Dimension $6$ Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [740,2,Mod(413,740)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(740, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 3]))

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

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

chi := DirichletCharacter("740.413");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$740 = 2^{2} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 740.l (of order $$4$$, degree $$2$$, 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: $$5.90892974957$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(i)$$ Coefficient field: 6.0.79423744.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} + 2x^{4} + 10x^{3} + 36x^{2} - 12x + 2$$ x^6 - 2*x^5 + 2*x^4 + 10*x^3 + 36*x^2 - 12*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + ( - 2 \beta_{4} - 1) q^{5} + ( - \beta_{4} + \beta_1 - 1) q^{7} + (\beta_{5} + \beta_{4} - \beta_{2} + \beta_1) q^{9}+O(q^{10})$$ q + b1 * q^3 + (-2*b4 - 1) * q^5 + (-b4 + b1 - 1) * q^7 + (b5 + b4 - b2 + b1) * q^9 $$q + \beta_1 q^{3} + ( - 2 \beta_{4} - 1) q^{5} + ( - \beta_{4} + \beta_1 - 1) q^{7} + (\beta_{5} + \beta_{4} - \beta_{2} + \beta_1) q^{9} + (\beta_{2} - \beta_1) q^{11} + ( - \beta_{5} - \beta_{2} + \beta_1) q^{13} + (2 \beta_{2} - \beta_1) q^{15} + (\beta_{3} + \beta_{2} + \beta_1) q^{17} + (\beta_{5} + \beta_{3} + \beta_1) q^{19} + (\beta_{5} + 4 \beta_{4}) q^{21} + (2 \beta_{4} - \beta_{2} + \beta_1) q^{23} + (4 \beta_{4} - 3) q^{25} + (\beta_{5} + 3 \beta_{4} - \beta_{3} + \cdots - 3) q^{27}+ \cdots + (2 \beta_{3} + 5 \beta_{2} + 5 \beta_1 + 6) q^{99}+O(q^{100})$$ q + b1 * q^3 + (-2*b4 - 1) * q^5 + (-b4 + b1 - 1) * q^7 + (b5 + b4 - b2 + b1) * q^9 + (b2 - b1) * q^11 + (-b5 - b2 + b1) * q^13 + (2*b2 - b1) * q^15 + (b3 + b2 + b1) * q^17 + (b5 + b3 + b1) * q^19 + (b5 + 4*b4) * q^21 + (2*b4 - b2 + b1) * q^23 + (4*b4 - 3) * q^25 + (b5 + 3*b4 - b3 - 2*b2 - 3) * q^27 + (3*b4 - 3) * q^29 + (b5 - 3*b4 + b3 - b1 - 3) * q^31 + (-b5 - 4*b4 + b3 + 2*b2 + 4) * q^33 + (3*b4 + 2*b2 - b1 - 1) * q^35 + (b5 - 2*b4 + b3 - b2 + b1 + 3) * q^37 + (b5 + 5*b4 - b3 - 5) * q^39 + (-2*b5 - 2*b4 + b2 - b1) * q^41 + (2*b5 + b2 - b1) * q^43 + (-b5 - b4 + 2*b3 + 3*b2 + b1 + 2) * q^45 + (-b5 - b4 - b3 - 3*b1 - 1) * q^47 + (b5 - b4 + b2 - b1) * q^49 + (b5 + 3*b4 + b3 + 4*b1 + 3) * q^51 + (b5 - 3*b4 - b3 + 2*b2 + 3) * q^53 + (-3*b2 - b1) * q^55 + (b5 + 2*b4 - 3*b2 + 3*b1) * q^57 + (b5 + b4 + b3 - 3*b1 + 1) * q^59 + (-2*b5 - 3*b4 - 2*b3 - 3) * q^61 + (2*b4 - 3*b2 - 2) * q^63 + (b5 - 2*b3 + 3*b2 + b1) * q^65 + (b5 + 2*b4 - b3 + b2 - 2) * q^67 + (b5 + 4*b4 - b3 - 4*b2 - 4) * q^69 + (-2*b3 - 2) * q^71 + (-b5 - b4 - b3 + 2*b1 - 1) * q^73 + (-4*b2 - 3*b1) * q^75 + (-b5 - 4*b4 + b3 + 4) * q^77 + (-5*b4 + 3*b1 - 5) * q^79 + (b3 - 4*b2 - 4*b1 - 3) * q^81 + (-3*b5 - 2*b4 + 3*b3 + b2 + 2) * q^83 + (-2*b5 - b3 + b2 - 3*b1) * q^85 + (-3*b2 - 3*b1) * q^87 + (-b5 + 3*b4 + b3 + 2*b2 - 3) * q^89 + (2*b5 + 5*b4 - 2*b3 + 2*b2 - 5) * q^91 + (-b5 - 6*b4 + 2*b2 - 2*b1) * q^93 + (-3*b5 + b3 + 2*b2 - b1) * q^95 + (-2*b3 + b2 + b1 + 4) * q^97 + (2*b3 + 5*b2 + 5*b1 + 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 2 q^{3} - 6 q^{5} - 4 q^{7}+O(q^{10})$$ 6 * q + 2 * q^3 - 6 * q^5 - 4 * q^7 $$6 q + 2 q^{3} - 6 q^{5} - 4 q^{7} + 2 q^{15} + 4 q^{17} + 2 q^{19} - 18 q^{25} - 22 q^{27} - 18 q^{29} - 20 q^{31} + 28 q^{33} - 4 q^{35} + 18 q^{37} - 30 q^{39} + 20 q^{45} - 12 q^{47} + 26 q^{51} + 22 q^{53} - 8 q^{55} - 18 q^{61} - 18 q^{63} + 8 q^{65} - 10 q^{67} - 32 q^{69} - 12 q^{71} - 2 q^{73} - 14 q^{75} + 24 q^{77} - 24 q^{79} - 34 q^{81} + 14 q^{83} - 4 q^{85} - 12 q^{87} - 14 q^{89} - 26 q^{91} + 2 q^{95} + 28 q^{97} + 56 q^{99}+O(q^{100})$$ 6 * q + 2 * q^3 - 6 * q^5 - 4 * q^7 + 2 * q^15 + 4 * q^17 + 2 * q^19 - 18 * q^25 - 22 * q^27 - 18 * q^29 - 20 * q^31 + 28 * q^33 - 4 * q^35 + 18 * q^37 - 30 * q^39 + 20 * q^45 - 12 * q^47 + 26 * q^51 + 22 * q^53 - 8 * q^55 - 18 * q^61 - 18 * q^63 + 8 * q^65 - 10 * q^67 - 32 * q^69 - 12 * q^71 - 2 * q^73 - 14 * q^75 + 24 * q^77 - 24 * q^79 - 34 * q^81 + 14 * q^83 - 4 * q^85 - 12 * q^87 - 14 * q^89 - 26 * q^91 + 2 * q^95 + 28 * q^97 + 56 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} + 2x^{4} + 10x^{3} + 36x^{2} - 12x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -8\nu^{5} + 21\nu^{4} - 64\nu^{3} - 40\nu^{2} - 263\nu + 86 ) / 279$$ (-8*v^5 + 21*v^4 - 64*v^3 - 40*v^2 - 263*v + 86) / 279 $$\beta_{3}$$ $$=$$ $$( 13\nu^{5} - 69\nu^{4} + 104\nu^{3} + 65\nu^{2} - 26\nu - 1186 ) / 279$$ (13*v^5 - 69*v^4 + 104*v^3 + 65*v^2 - 26*v - 1186) / 279 $$\beta_{4}$$ $$=$$ $$( 43\nu^{5} - 78\nu^{4} + 65\nu^{3} + 494\nu^{2} + 1588\nu - 253 ) / 279$$ (43*v^5 - 78*v^4 + 65*v^3 + 494*v^2 + 1588*v - 253) / 279 $$\beta_{5}$$ $$=$$ $$( -20\nu^{5} + 37\nu^{4} - 36\nu^{3} - 193\nu^{2} - 766\nu + 122 ) / 31$$ (-20*v^5 + 37*v^4 - 36*v^3 - 193*v^2 - 766*v + 122) / 31
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} + 4\beta_{4} - \beta_{2} + \beta_1$$ b5 + 4*b4 - b2 + b1 $$\nu^{3}$$ $$=$$ $$\beta_{5} + 3\beta_{4} - \beta_{3} - 8\beta_{2} - 3$$ b5 + 3*b4 - b3 - 8*b2 - 3 $$\nu^{4}$$ $$=$$ $$-8\beta_{3} - 13\beta_{2} - 13\beta _1 - 30$$ -8*b3 - 13*b2 - 13*b1 - 30 $$\nu^{5}$$ $$=$$ $$-13\beta_{5} - 44\beta_{4} - 13\beta_{3} - 72\beta _1 - 44$$ -13*b5 - 44*b4 - 13*b3 - 72*b1 - 44

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/740\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$297$$ $$371$$ $$\chi(n)$$ $$\beta_{4}$$ $$\beta_{4}$$ $$1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
413.1
 −1.39930 + 1.39930i 0.159536 − 0.159536i 2.23976 − 2.23976i −1.39930 − 1.39930i 0.159536 + 0.159536i 2.23976 + 2.23976i
0 −1.39930 + 1.39930i 0 −1.00000 + 2.00000i 0 −2.39930 + 2.39930i 0 0.916055i 0
413.2 0 0.159536 0.159536i 0 −1.00000 + 2.00000i 0 −0.840464 + 0.840464i 0 2.94910i 0
413.3 0 2.23976 2.23976i 0 −1.00000 + 2.00000i 0 1.23976 1.23976i 0 7.03304i 0
697.1 0 −1.39930 1.39930i 0 −1.00000 2.00000i 0 −2.39930 2.39930i 0 0.916055i 0
697.2 0 0.159536 + 0.159536i 0 −1.00000 2.00000i 0 −0.840464 0.840464i 0 2.94910i 0
697.3 0 2.23976 + 2.23976i 0 −1.00000 2.00000i 0 1.23976 + 1.23976i 0 7.03304i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 413.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.f even 4 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.l.a 6
5.c odd 4 1 740.2.o.a yes 6
37.d odd 4 1 740.2.o.a yes 6
185.f even 4 1 inner 740.2.l.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.l.a 6 1.a even 1 1 trivial
740.2.l.a 6 185.f even 4 1 inner
740.2.o.a yes 6 5.c odd 4 1
740.2.o.a yes 6 37.d odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} - 2T_{3}^{5} + 2T_{3}^{4} + 10T_{3}^{3} + 36T_{3}^{2} - 12T_{3} + 2$$ acting on $$S_{2}^{\mathrm{new}}(740, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} - 2 T^{5} + \cdots + 2$$
$5$ $$(T^{2} + 2 T + 5)^{3}$$
$7$ $$T^{6} + 4 T^{5} + \cdots + 50$$
$11$ $$T^{6} + 28 T^{4} + \cdots + 16$$
$13$ $$T^{6} + 60 T^{4} + \cdots + 5476$$
$17$ $$(T^{3} - 2 T^{2} - 24 T - 2)^{2}$$
$19$ $$T^{6} - 2 T^{5} + \cdots + 3362$$
$23$ $$T^{6} + 48 T^{4} + \cdots + 144$$
$29$ $$(T^{2} + 6 T + 18)^{3}$$
$31$ $$T^{6} + 20 T^{5} + \cdots + 7442$$
$37$ $$T^{6} - 18 T^{5} + \cdots + 50653$$
$41$ $$T^{6} + 152 T^{4} + \cdots + 76176$$
$43$ $$T^{6} + 148 T^{4} + \cdots + 10000$$
$47$ $$T^{6} + 12 T^{5} + \cdots + 1250$$
$53$ $$T^{6} - 22 T^{5} + \cdots + 102152$$
$59$ $$T^{6} + 494 T^{3} + \cdots + 122018$$
$61$ $$T^{6} + 18 T^{5} + \cdots + 649800$$
$67$ $$T^{6} + 10 T^{5} + \cdots + 3042$$
$71$ $$(T^{3} + 6 T^{2} + \cdots - 248)^{2}$$
$73$ $$T^{6} + 2 T^{5} + \cdots + 16200$$
$79$ $$T^{6} + 24 T^{5} + \cdots + 40898$$
$83$ $$T^{6} - 14 T^{5} + \cdots + 5491298$$
$89$ $$T^{6} + 14 T^{5} + \cdots + 33800$$
$97$ $$(T^{3} - 14 T^{2} + \cdots + 292)^{2}$$
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