# Properties

 Label 2960.2.a.w Level $2960$ Weight $2$ Character orbit 2960.a Self dual yes Analytic conductor $23.636$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2960,2,Mod(1,2960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2960, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("2960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2960 = 2^{4} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2960.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: $$23.6357189983$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: 5.5.973904.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - 2x^{4} - 8x^{3} + 6x^{2} + 19x + 6$$ x^5 - 2*x^4 - 8*x^3 + 6*x^2 + 19*x + 6 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 185) 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 1) q^{3} - q^{5} + (\beta_{4} - \beta_{2} - \beta_1 - 2) q^{7} + (\beta_{2} + 1) q^{9}+O(q^{10})$$ q + (b1 - 1) * q^3 - q^5 + (b4 - b2 - b1 - 2) * q^7 + (b2 + 1) * q^9 $$q + (\beta_1 - 1) q^{3} - q^{5} + (\beta_{4} - \beta_{2} - \beta_1 - 2) q^{7} + (\beta_{2} + 1) q^{9} + (\beta_{3} + 1) q^{11} + ( - \beta_{3} - \beta_{2} + 1) q^{13} + ( - \beta_1 + 1) q^{15} + ( - 2 \beta_{4} + \beta_{3} - \beta_{2} + 1) q^{17} + ( - \beta_{3} + 2 \beta_{2} + \beta_1) q^{19} + ( - 2 \beta_{4} + \beta_{2} - 2 \beta_1 + 2) q^{21} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots - 2) q^{23}+ \cdots + ( - 2 \beta_{4} + 2 \beta_{3} + \cdots + 2) q^{99}+O(q^{100})$$ q + (b1 - 1) * q^3 - q^5 + (b4 - b2 - b1 - 2) * q^7 + (b2 + 1) * q^9 + (b3 + 1) * q^11 + (-b3 - b2 + 1) * q^13 + (-b1 + 1) * q^15 + (-2*b4 + b3 - b2 + 1) * q^17 + (-b3 + 2*b2 + b1) * q^19 + (-2*b4 + b2 - 2*b1 + 2) * q^21 + (-2*b3 + 2*b2 + 2*b1 - 2) * q^23 + q^25 + (b4 + b3 - b2 - 2*b1) * q^27 - 2*b1 * q^29 + (-b4 + b3 + b2 - b1 - 1) * q^31 + (2*b4 - b3 + 2*b2 + 2*b1 - 1) * q^33 + (-b4 + b2 + b1 + 2) * q^35 + q^37 + (-3*b4 - b2 + 1) * q^39 + (-b3 - 1) * q^41 + (-2*b4 + 2*b3 - 2*b2 - 2*b1) * q^43 + (-b2 - 1) * q^45 + (b4 - 3*b2 - 3*b1) * q^47 + (-2*b4 + 2*b3 + b2 + 5) * q^49 + (3*b4 - 4*b3 + b2 - 1) * q^51 + (4*b4 - b3 - 2*b1 - 1) * q^53 + (-b3 - 1) * q^55 + (3*b3 - 3*b2 - 1) * q^57 + (b4 + b3 - b2 - 3*b1 + 7) * q^59 + (2*b4 + 2*b2 - 4) * q^61 + (-b3 - 2*b2 + b1 - 6) * q^63 + (b3 + b2 - 1) * q^65 + (-2*b4 + b3 + 2*b2 - b1 - 4) * q^67 + (-2*b4 + 4*b3 - 4*b2 - 2*b1 + 4) * q^69 + (2*b4 + b3 + 2*b2 - 2*b1 + 1) * q^71 + (-4*b4 - b3 + 4*b1 + 1) * q^73 + (b1 - 1) * q^75 + (2*b4 - 3*b3 - 2*b2 - 2*b1 - 3) * q^77 + (3*b4 + b3 - b2 + b1 - 7) * q^79 + (-b3 - b2 - 6) * q^81 + (-2*b4 - 2*b3 + b1 - 5) * q^83 + (2*b4 - b3 + b2 - 1) * q^85 + (-2*b2 - 2*b1 - 6) * q^87 + (2*b4 - 2*b3 + 4*b2 + 4*b1 - 2) * q^89 + (b4 + 4*b3 + b2 - 2*b1 + 3) * q^91 + (4*b4 - b3 - b2 - 2*b1 - 5) * q^93 + (b3 - 2*b2 - b1) * q^95 + (2*b4 - 2*b3 + 4*b2 - 2*b1 - 6) * q^97 + (-2*b4 + 2*b3 + 2*b1 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 3 q^{3} - 5 q^{5} - 11 q^{7} + 6 q^{9}+O(q^{10})$$ 5 * q - 3 * q^3 - 5 * q^5 - 11 * q^7 + 6 * q^9 $$5 q - 3 q^{3} - 5 q^{5} - 11 q^{7} + 6 q^{9} + 5 q^{11} + 4 q^{13} + 3 q^{15} + 4 q^{19} + 3 q^{21} - 4 q^{23} + 5 q^{25} - 3 q^{27} - 4 q^{29} - 8 q^{31} + 5 q^{33} + 11 q^{35} + 5 q^{37} - 2 q^{39} - 5 q^{41} - 10 q^{43} - 6 q^{45} - 7 q^{47} + 22 q^{49} + 2 q^{51} - q^{53} - 5 q^{55} - 8 q^{57} + 30 q^{59} - 14 q^{61} - 30 q^{63} - 4 q^{65} - 24 q^{67} + 8 q^{69} + 7 q^{71} + 5 q^{73} - 3 q^{75} - 17 q^{77} - 28 q^{79} - 31 q^{81} - 27 q^{83} - 36 q^{87} + 6 q^{89} + 14 q^{91} - 22 q^{93} - 4 q^{95} - 26 q^{97} + 10 q^{99}+O(q^{100})$$ 5 * q - 3 * q^3 - 5 * q^5 - 11 * q^7 + 6 * q^9 + 5 * q^11 + 4 * q^13 + 3 * q^15 + 4 * q^19 + 3 * q^21 - 4 * q^23 + 5 * q^25 - 3 * q^27 - 4 * q^29 - 8 * q^31 + 5 * q^33 + 11 * q^35 + 5 * q^37 - 2 * q^39 - 5 * q^41 - 10 * q^43 - 6 * q^45 - 7 * q^47 + 22 * q^49 + 2 * q^51 - q^53 - 5 * q^55 - 8 * q^57 + 30 * q^59 - 14 * q^61 - 30 * q^63 - 4 * q^65 - 24 * q^67 + 8 * q^69 + 7 * q^71 + 5 * q^73 - 3 * q^75 - 17 * q^77 - 28 * q^79 - 31 * q^81 - 27 * q^83 - 36 * q^87 + 6 * q^89 + 14 * q^91 - 22 * q^93 - 4 * q^95 - 26 * q^97 + 10 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2x^{4} - 8x^{3} + 6x^{2} + 19x + 6$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2\nu - 3$$ v^2 - 2*v - 3 $$\beta_{3}$$ $$=$$ $$-\nu^{4} + 4\nu^{3} + 2\nu^{2} - 12\nu - 4$$ -v^4 + 4*v^3 + 2*v^2 - 12*v - 4 $$\beta_{4}$$ $$=$$ $$\nu^{4} - 3\nu^{3} - 4\nu^{2} + 9\nu + 6$$ v^4 - 3*v^3 - 4*v^2 + 9*v + 6
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2\beta _1 + 3$$ b2 + 2*b1 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{4} + \beta_{3} + 2\beta_{2} + 7\beta _1 + 4$$ b4 + b3 + 2*b2 + 7*b1 + 4 $$\nu^{4}$$ $$=$$ $$4\beta_{4} + 3\beta_{3} + 10\beta_{2} + 20\beta _1 + 18$$ 4*b4 + 3*b3 + 10*b2 + 20*b1 + 18

## 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
 −1.62871 −1.38679 −0.383115 2.10563 3.29298
0 −2.62871 0 −1.00000 0 −2.55244 0 3.91009 0
1.2 0 −2.38679 0 −1.00000 0 −4.78404 0 2.69675 0
1.3 0 −1.38311 0 −1.00000 0 2.62521 0 −1.08699 0
1.4 0 1.10563 0 −1.00000 0 −2.46164 0 −1.77758 0
1.5 0 2.29298 0 −1.00000 0 −3.82710 0 2.25774 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$+1$$
$$37$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2960.2.a.w 5
4.b odd 2 1 185.2.a.e 5
12.b even 2 1 1665.2.a.p 5
20.d odd 2 1 925.2.a.f 5
20.e even 4 2 925.2.b.f 10
28.d even 2 1 9065.2.a.k 5
60.h even 2 1 8325.2.a.ch 5
148.b odd 2 1 6845.2.a.f 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.2.a.e 5 4.b odd 2 1
925.2.a.f 5 20.d odd 2 1
925.2.b.f 10 20.e even 4 2
1665.2.a.p 5 12.b even 2 1
2960.2.a.w 5 1.a even 1 1 trivial
6845.2.a.f 5 148.b odd 2 1
8325.2.a.ch 5 60.h even 2 1
9065.2.a.k 5 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2960))$$:

 $$T_{3}^{5} + 3T_{3}^{4} - 6T_{3}^{3} - 20T_{3}^{2} + 4T_{3} + 22$$ T3^5 + 3*T3^4 - 6*T3^3 - 20*T3^2 + 4*T3 + 22 $$T_{7}^{5} + 11T_{7}^{4} + 32T_{7}^{3} - 32T_{7}^{2} - 268T_{7} - 302$$ T7^5 + 11*T7^4 + 32*T7^3 - 32*T7^2 - 268*T7 - 302 $$T_{13}^{5} - 4T_{13}^{4} - 28T_{13}^{3} + 60T_{13}^{2} + 148T_{13} - 256$$ T13^5 - 4*T13^4 - 28*T13^3 + 60*T13^2 + 148*T13 - 256

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$T^{5} + 3 T^{4} + \cdots + 22$$
$5$ $$(T + 1)^{5}$$
$7$ $$T^{5} + 11 T^{4} + \cdots - 302$$
$11$ $$T^{5} - 5 T^{4} + \cdots - 96$$
$13$ $$T^{5} - 4 T^{4} + \cdots - 256$$
$17$ $$T^{5} - 52 T^{3} + \cdots + 192$$
$19$ $$T^{5} - 4 T^{4} + \cdots + 8$$
$23$ $$T^{5} + 4 T^{4} + \cdots - 192$$
$29$ $$T^{5} + 4 T^{4} + \cdots - 192$$
$31$ $$T^{5} + 8 T^{4} + \cdots - 324$$
$37$ $$(T - 1)^{5}$$
$41$ $$T^{5} + 5 T^{4} + \cdots + 96$$
$43$ $$T^{5} + 10 T^{4} + \cdots + 2528$$
$47$ $$T^{5} + 7 T^{4} + \cdots + 978$$
$53$ $$T^{5} + T^{4} + \cdots + 528$$
$59$ $$T^{5} - 30 T^{4} + \cdots + 576$$
$61$ $$T^{5} + 14 T^{4} + \cdots + 3296$$
$67$ $$T^{5} + 24 T^{4} + \cdots - 10952$$
$71$ $$T^{5} - 7 T^{4} + \cdots + 7104$$
$73$ $$T^{5} - 5 T^{4} + \cdots + 368$$
$79$ $$T^{5} + 28 T^{4} + \cdots + 19508$$
$83$ $$T^{5} + 27 T^{4} + \cdots - 4818$$
$89$ $$T^{5} - 6 T^{4} + \cdots + 22944$$
$97$ $$T^{5} + 26 T^{4} + \cdots - 166976$$