# Properties

 Label 2960.2.a.ba 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: 5.5.368464.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 6x - 4$$ x^5 - 2*x^4 - 6*x^3 + 6*x^2 + 6*x - 4 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} + q^{5} + ( - \beta_{4} - 1) q^{7} + (\beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{9}+O(q^{10})$$ q + b3 * q^3 + q^5 + (-b4 - 1) * q^7 + (b4 + b3 + b2 + 2*b1 - 1) * q^9 $$q + \beta_{3} q^{3} + q^{5} + ( - \beta_{4} - 1) q^{7} + (\beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{9} + (\beta_{4} - 2 \beta_{3} - \beta_1 - 1) q^{11} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{13} + \beta_{3} q^{15} + ( - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{17} + ( - \beta_{4} + \beta_{3} - 2 \beta_{2} + \beta_1 - 3) q^{19} + ( - \beta_{4} - 3 \beta_{3} - \beta_{2} - 2 \beta_1) q^{21} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{23} + q^{25} + (2 \beta_{4} + \beta_{3} + \beta_1) q^{27} + ( - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{29} + ( - 2 \beta_{3} - 3 \beta_1) q^{31} + ( - \beta_{4} - 2 \beta_{3} - \beta_1 - 3) q^{33} + ( - \beta_{4} - 1) q^{35} - q^{37} + ( - \beta_{4} - \beta_{3} - 2 \beta_1 - 1) q^{39} + (\beta_{4} - 3 \beta_1 - 1) q^{41} + (2 \beta_{4} - 2 \beta_{3} + 2 \beta_1 - 4) q^{43} + (\beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{45} + (\beta_{4} + 2 \beta_{2} + 4 \beta_1 - 1) q^{47} + (3 \beta_{4} + \beta_{3} - \beta_{2} - 1) q^{49} + ( - \beta_{4} - 3 \beta_{3} - 1) q^{51} + (\beta_{4} - \beta_1 - 3) q^{53} + (\beta_{4} - 2 \beta_{3} - \beta_1 - 1) q^{55} + ( - 3 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{57} + (2 \beta_{3} + 4 \beta_{2} + 3 \beta_1 - 4) q^{59} + ( - 2 \beta_{3} + 2 \beta_1 + 2) q^{61} + ( - \beta_{4} - 7 \beta_{3} - 2 \beta_{2} - 5 \beta_1 - 1) q^{63} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{65} + (\beta_{4} + 3 \beta_{3} + 4 \beta_{2} + \beta_1 - 1) q^{67} + ( - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 4) q^{69} + ( - \beta_{4} + 6 \beta_{3} + 4 \beta_{2} - \beta_1 - 3) q^{71} + (\beta_{4} + 2 \beta_{3} + 3 \beta_1 - 3) q^{73} + \beta_{3} q^{75} + (\beta_{4} + 6 \beta_{3} + 4 \beta_{2} + 5 \beta_1 - 5) q^{77} + (2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \beta_1 - 8) q^{79} + (3 \beta_{3} - \beta_{2} - \beta_1 + 4) q^{81} + (2 \beta_{4} + \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{83} + ( - \beta_{3} - \beta_{2} - \beta_1 - 1) q^{85} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{87} + ( - 4 \beta_{4} + 2 \beta_{2} - 4 \beta_1) q^{89} + (\beta_{4} + 3 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{91} + ( - 2 \beta_{4} - 5 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{93} + ( - \beta_{4} + \beta_{3} - 2 \beta_{2} + \beta_1 - 3) q^{95} + (6 \beta_{4} + 2 \beta_{2} + 4 \beta_1 - 4) q^{97} + ( - 6 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{99}+O(q^{100})$$ q + b3 * q^3 + q^5 + (-b4 - 1) * q^7 + (b4 + b3 + b2 + 2*b1 - 1) * q^9 + (b4 - 2*b3 - b1 - 1) * q^11 + (-b3 + b2 - b1 + 1) * q^13 + b3 * q^15 + (-b3 - b2 - b1 - 1) * q^17 + (-b4 + b3 - 2*b2 + b1 - 3) * q^19 + (-b4 - 3*b3 - b2 - 2*b1) * q^21 + (-2*b3 - 2*b2) * q^23 + q^25 + (2*b4 + b3 + b1) * q^27 + (-2*b4 + 2*b3 + 2*b1) * q^29 + (-2*b3 - 3*b1) * q^31 + (-b4 - 2*b3 - b1 - 3) * q^33 + (-b4 - 1) * q^35 - q^37 + (-b4 - b3 - 2*b1 - 1) * q^39 + (b4 - 3*b1 - 1) * q^41 + (2*b4 - 2*b3 + 2*b1 - 4) * q^43 + (b4 + b3 + b2 + 2*b1 - 1) * q^45 + (b4 + 2*b2 + 4*b1 - 1) * q^47 + (3*b4 + b3 - b2 - 1) * q^49 + (-b4 - 3*b3 - 1) * q^51 + (b4 - b1 - 3) * q^53 + (b4 - 2*b3 - b1 - 1) * q^55 + (-3*b3 - b2 + b1 + 1) * q^57 + (2*b3 + 4*b2 + 3*b1 - 4) * q^59 + (-2*b3 + 2*b1 + 2) * q^61 + (-b4 - 7*b3 - 2*b2 - 5*b1 - 1) * q^63 + (-b3 + b2 - b1 + 1) * q^65 + (b4 + 3*b3 + 4*b2 + b1 - 1) * q^67 + (-2*b4 - 2*b3 - 2*b2 - 2*b1 - 4) * q^69 + (-b4 + 6*b3 + 4*b2 - b1 - 3) * q^71 + (b4 + 2*b3 + 3*b1 - 3) * q^73 + b3 * q^75 + (b4 + 6*b3 + 4*b2 + 5*b1 - 5) * q^77 + (2*b4 - 2*b3 + 2*b2 + b1 - 8) * q^79 + (3*b3 - b2 - b1 + 4) * q^81 + (2*b4 + b3 - 2*b2 + 2*b1) * q^83 + (-b3 - b2 - b1 - 1) * q^85 + (-2*b2 - 2*b1 + 2) * q^87 + (-4*b4 + 2*b2 - 4*b1) * q^89 + (b4 + 3*b3 + 2*b2 + 2*b1 - 1) * q^91 + (-2*b4 - 5*b3 + b2 - b1 - 1) * q^93 + (-b4 + b3 - 2*b2 + b1 - 3) * q^95 + (6*b4 + 2*b2 + 4*b1 - 4) * q^97 + (-6*b4 - 2*b3 - 2*b2 - 2*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + q^{3} + 5 q^{5} - 7 q^{7} + 2 q^{9}+O(q^{10})$$ 5 * q + q^3 + 5 * q^5 - 7 * q^7 + 2 * q^9 $$5 q + q^{3} + 5 q^{5} - 7 q^{7} + 2 q^{9} - 7 q^{11} + 2 q^{13} + q^{15} - 8 q^{17} - 14 q^{19} - 9 q^{21} - 2 q^{23} + 5 q^{25} + 7 q^{27} + 2 q^{29} - 8 q^{31} - 21 q^{33} - 7 q^{35} - 5 q^{37} - 12 q^{39} - 9 q^{41} - 14 q^{43} + 2 q^{45} + 5 q^{47} + 2 q^{49} - 10 q^{51} - 15 q^{53} - 7 q^{55} + 4 q^{57} - 12 q^{59} + 12 q^{61} - 24 q^{63} + 2 q^{65} + 2 q^{67} - 30 q^{69} - 13 q^{71} - 5 q^{73} + q^{75} - 7 q^{77} - 36 q^{79} + 21 q^{81} + 9 q^{83} - 8 q^{85} + 6 q^{87} - 16 q^{89} + 4 q^{91} - 16 q^{93} - 14 q^{95} - 18 q^{99}+O(q^{100})$$ 5 * q + q^3 + 5 * q^5 - 7 * q^7 + 2 * q^9 - 7 * q^11 + 2 * q^13 + q^15 - 8 * q^17 - 14 * q^19 - 9 * q^21 - 2 * q^23 + 5 * q^25 + 7 * q^27 + 2 * q^29 - 8 * q^31 - 21 * q^33 - 7 * q^35 - 5 * q^37 - 12 * q^39 - 9 * q^41 - 14 * q^43 + 2 * q^45 + 5 * q^47 + 2 * q^49 - 10 * q^51 - 15 * q^53 - 7 * q^55 + 4 * q^57 - 12 * q^59 + 12 * q^61 - 24 * q^63 + 2 * q^65 + 2 * q^67 - 30 * q^69 - 13 * q^71 - 5 * q^73 + q^75 - 7 * q^77 - 36 * q^79 + 21 * q^81 + 9 * q^83 - 8 * q^85 + 6 * q^87 - 16 * q^89 + 4 * q^91 - 16 * q^93 - 14 * q^95 - 18 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{4} - 2\nu^{3} - 4\nu^{2} + 2\nu ) / 2$$ (v^4 - 2*v^3 - 4*v^2 + 2*v) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{4} - 2\nu^{3} - 6\nu^{2} + 4\nu + 6 ) / 2$$ (v^4 - 2*v^3 - 6*v^2 + 4*v + 6) / 2 $$\beta_{4}$$ $$=$$ $$( \nu^{4} - 10\nu^{2} - 4\nu + 12 ) / 2$$ (v^4 - 10*v^2 - 4*v + 12) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{3} + \beta_{2} + \beta _1 + 3$$ -b3 + b2 + b1 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{4} - 3\beta_{3} + 2\beta_{2} + 6\beta _1 + 3$$ b4 - 3*b3 + 2*b2 + 6*b1 + 3 $$\nu^{4}$$ $$=$$ $$2\beta_{4} - 10\beta_{3} + 10\beta_{2} + 14\beta _1 + 18$$ 2*b4 - 10*b3 + 10*b2 + 14*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
 3.14884 −1.09027 1.17837 −1.78948 0.552543
0 −2.51369 0 1.00000 0 −0.281955 0 3.31863 0
1.2 0 −0.744131 0 1.00000 0 −3.94357 0 −2.44627 0
1.3 0 0.518894 0 1.00000 0 1.33546 0 −2.73075 0
1.4 0 0.671838 0 1.00000 0 0.305070 0 −2.54863 0
1.5 0 3.06709 0 1.00000 0 −4.41500 0 6.40702 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.ba 5
4.b odd 2 1 185.2.a.d 5
12.b even 2 1 1665.2.a.q 5
20.d odd 2 1 925.2.a.h 5
20.e even 4 2 925.2.b.g 10
28.d even 2 1 9065.2.a.j 5
60.h even 2 1 8325.2.a.cc 5
148.b odd 2 1 6845.2.a.g 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.2.a.d 5 4.b odd 2 1
925.2.a.h 5 20.d odd 2 1
925.2.b.g 10 20.e even 4 2
1665.2.a.q 5 12.b even 2 1
2960.2.a.ba 5 1.a even 1 1 trivial
6845.2.a.g 5 148.b odd 2 1
8325.2.a.cc 5 60.h even 2 1
9065.2.a.j 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2960))$$:

 $$T_{3}^{5} - T_{3}^{4} - 8T_{3}^{3} + 4T_{3}^{2} + 4T_{3} - 2$$ T3^5 - T3^4 - 8*T3^3 + 4*T3^2 + 4*T3 - 2 $$T_{7}^{5} + 7T_{7}^{4} + 6T_{7}^{3} - 24T_{7}^{2} + 2$$ T7^5 + 7*T7^4 + 6*T7^3 - 24*T7^2 + 2 $$T_{13}^{5} - 2T_{13}^{4} - 20T_{13}^{3} + 20T_{13}^{2} + 76T_{13} - 88$$ T13^5 - 2*T13^4 - 20*T13^3 + 20*T13^2 + 76*T13 - 88

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$T^{5} - T^{4} - 8 T^{3} + 4 T^{2} + 4 T - 2$$
$5$ $$(T - 1)^{5}$$
$7$ $$T^{5} + 7 T^{4} + 6 T^{3} - 24 T^{2} + \cdots + 2$$
$11$ $$T^{5} + 7 T^{4} - 12 T^{3} - 144 T^{2} + \cdots + 32$$
$13$ $$T^{5} - 2 T^{4} - 20 T^{3} + 20 T^{2} + \cdots - 88$$
$17$ $$T^{5} + 8 T^{4} + 12 T^{3} - 36 T^{2} + \cdots - 32$$
$19$ $$T^{5} + 14 T^{4} + 26 T^{3} + \cdots - 2224$$
$23$ $$T^{5} + 2 T^{4} - 56 T^{3} + \cdots - 1504$$
$29$ $$T^{5} - 2 T^{4} - 80 T^{3} + 272 T^{2} + \cdots + 32$$
$31$ $$T^{5} + 8 T^{4} - 38 T^{3} + \cdots + 3016$$
$37$ $$(T + 1)^{5}$$
$41$ $$T^{5} + 9 T^{4} - 64 T^{3} - 304 T^{2} + \cdots - 928$$
$43$ $$T^{5} + 14 T^{4} - 16 T^{3} + \cdots + 1312$$
$47$ $$T^{5} - 5 T^{4} - 114 T^{3} + \cdots + 3994$$
$53$ $$T^{5} + 15 T^{4} + 64 T^{3} + 40 T^{2} + \cdots - 16$$
$59$ $$T^{5} + 12 T^{4} - 94 T^{3} + \cdots - 5456$$
$61$ $$T^{5} - 12 T^{4} - 32 T^{3} + \cdots - 512$$
$67$ $$T^{5} - 2 T^{4} - 174 T^{3} + \cdots + 7568$$
$71$ $$T^{5} + 13 T^{4} - 348 T^{3} + \cdots + 291136$$
$73$ $$T^{5} + 5 T^{4} - 68 T^{3} - 336 T^{2} + \cdots + 176$$
$79$ $$T^{5} + 36 T^{4} + 430 T^{3} + \cdots - 5912$$
$83$ $$T^{5} - 9 T^{4} - 104 T^{3} + \cdots - 314$$
$89$ $$T^{5} + 16 T^{4} - 240 T^{3} + \cdots + 1856$$
$97$ $$T^{5} - 464 T^{3} + 496 T^{2} + \cdots - 193408$$