# Properties

 Label 2960.2.a.bc Level $2960$ Weight $2$ Character orbit 2960.a Self dual yes Analytic conductor $23.636$ Analytic rank $0$ Dimension $6$ 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: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.693982032.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} - 16x^{4} + 12x^{3} + 60x^{2} - 18x - 44$$ x^6 - x^5 - 16*x^4 + 12*x^3 + 60*x^2 - 18*x - 44 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1480) 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} + q^{5} + ( - \beta_{2} - 1) q^{7} + (\beta_{5} - \beta_{4} + \beta_{3} + 2) q^{9}+O(q^{10})$$ q - b1 * q^3 + q^5 + (-b2 - 1) * q^7 + (b5 - b4 + b3 + 2) * q^9 $$q - \beta_1 q^{3} + q^{5} + ( - \beta_{2} - 1) q^{7} + (\beta_{5} - \beta_{4} + \beta_{3} + 2) q^{9} + ( - \beta_{4} - \beta_{2}) q^{11} + (\beta_{5} + \beta_{4} - \beta_1 + 2) q^{13} - \beta_1 q^{15} + ( - \beta_{5} - \beta_{3} - \beta_{2} + 1) q^{17} + (\beta_{3} + 1) q^{19} + ( - \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_1 + 1) q^{21} + 2 \beta_{4} q^{23} + q^{25} + ( - \beta_{4} - 2 \beta_{3} - 3 \beta_1 - 1) q^{27} + (\beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{29} + (2 \beta_{5} + \beta_{3} + \beta_{2} - \beta_1) q^{31} + ( - 2 \beta_{4} + \beta_{3} - \beta_1 + 1) q^{33} + ( - \beta_{2} - 1) q^{35} - q^{37} + ( - \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{39} + (\beta_{4} - 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{41} + (2 \beta_{5} + \beta_{4} + \beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{43} + (\beta_{5} - \beta_{4} + \beta_{3} + 2) q^{45} + ( - 2 \beta_{5} - \beta_{4} - \beta_{3} + \beta_1 - 4) q^{47} + ( - \beta_{5} + \beta_{2} + \beta_1 + 1) q^{49} + ( - \beta_{4} + 3 \beta_{3} + 2 \beta_1 + 2) q^{51} + (\beta_{4} + \beta_{2} - 2 \beta_1 + 2) q^{53} + ( - \beta_{4} - \beta_{2}) q^{55} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{57} + ( - 2 \beta_{5} - \beta_{4} - 2 \beta_{3} - 1) q^{59} + (\beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{61} + ( - 2 \beta_{5} + 3 \beta_{4} - 4 \beta_{3} - \beta_{2} - \beta_1 - 2) q^{63} + (\beta_{5} + \beta_{4} - \beta_1 + 2) q^{65} + (2 \beta_{4} + \beta_{3} + 2 \beta_1 + 1) q^{67} + ( - 2 \beta_{5} + 2 \beta_{4} + 4 \beta_1) q^{69} + (2 \beta_{4} - \beta_{3} + \beta_1 - 1) q^{71} + (2 \beta_{4} + \beta_{3} - 3 \beta_1 + 3) q^{73} - \beta_1 q^{75} + (3 \beta_{4} + \beta_{2} + 6) q^{77} + (\beta_{4} - 2 \beta_{3} - 2 \beta_1 + 3) q^{79} + (3 \beta_{5} - 3 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} - \beta_1 + 5) q^{81} + ( - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{2} + 3 \beta_1 - 2) q^{83} + ( - \beta_{5} - \beta_{3} - \beta_{2} + 1) q^{85} + ( - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 6) q^{87} + ( - 2 \beta_{5} - 4 \beta_{4} + 4) q^{89} + ( - 2 \beta_{5} - 3 \beta_{4} - \beta_{3} - 2 \beta_{2} + 4 \beta_1) q^{91} + (\beta_{5} - \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 5 \beta_1) q^{93} + (\beta_{3} + 1) q^{95} + ( - 2 \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 5) q^{97} + (2 \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - 5 \beta_1 + 7) q^{99}+O(q^{100})$$ q - b1 * q^3 + q^5 + (-b2 - 1) * q^7 + (b5 - b4 + b3 + 2) * q^9 + (-b4 - b2) * q^11 + (b5 + b4 - b1 + 2) * q^13 - b1 * q^15 + (-b5 - b3 - b2 + 1) * q^17 + (b3 + 1) * q^19 + (-b5 - b4 + b3 + 2*b1 + 1) * q^21 + 2*b4 * q^23 + q^25 + (-b4 - 2*b3 - 3*b1 - 1) * q^27 + (b4 - b3 - b2 + b1 + 1) * q^29 + (2*b5 + b3 + b2 - b1) * q^31 + (-2*b4 + b3 - b1 + 1) * q^33 + (-b2 - 1) * q^35 - q^37 + (-b4 + b3 + 2*b2 - 2*b1 + 2) * q^39 + (b4 - 2*b3 + b2 - 2*b1 + 2) * q^41 + (2*b5 + b4 + b3 + 3*b2 - b1 + 1) * q^43 + (b5 - b4 + b3 + 2) * q^45 + (-2*b5 - b4 - b3 + b1 - 4) * q^47 + (-b5 + b2 + b1 + 1) * q^49 + (-b4 + 3*b3 + 2*b1 + 2) * q^51 + (b4 + b2 - 2*b1 + 2) * q^53 + (-b4 - b2) * q^55 + (-b5 + b4 - 2*b3 - 2*b2 - b1 + 2) * q^57 + (-2*b5 - b4 - 2*b3 - 1) * q^59 + (b4 - b3 - b2 - b1 + 1) * q^61 + (-2*b5 + 3*b4 - 4*b3 - b2 - b1 - 2) * q^63 + (b5 + b4 - b1 + 2) * q^65 + (2*b4 + b3 + 2*b1 + 1) * q^67 + (-2*b5 + 2*b4 + 4*b1) * q^69 + (2*b4 - b3 + b1 - 1) * q^71 + (2*b4 + b3 - 3*b1 + 3) * q^73 - b1 * q^75 + (3*b4 + b2 + 6) * q^77 + (b4 - 2*b3 - 2*b1 + 3) * q^79 + (3*b5 - 3*b4 + 4*b3 + 4*b2 - b1 + 5) * q^81 + (-2*b5 - 2*b4 - 2*b2 + 3*b1 - 2) * q^83 + (-b5 - b3 - b2 + 1) * q^85 + (-2*b5 + 2*b3 + 2*b2 + 2*b1 - 6) * q^87 + (-2*b5 - 4*b4 + 4) * q^89 + (-2*b5 - 3*b4 - b3 - 2*b2 + 4*b1) * q^91 + (b5 - b4 - 2*b3 + 2*b2 - 5*b1) * q^93 + (b3 + 1) * q^95 + (-2*b5 + b4 + b3 + b2 + b1 + 5) * q^97 + (2*b5 + b4 - b3 + b2 - 5*b1 + 7) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - q^{3} + 6 q^{5} - 8 q^{7} + 15 q^{9}+O(q^{10})$$ 6 * q - q^3 + 6 * q^5 - 8 * q^7 + 15 * q^9 $$6 q - q^{3} + 6 q^{5} - 8 q^{7} + 15 q^{9} + 10 q^{13} - q^{15} + 3 q^{17} + 6 q^{19} + 9 q^{21} - 4 q^{23} + 6 q^{25} - 7 q^{27} + 3 q^{29} + 3 q^{31} + 9 q^{33} - 8 q^{35} - 6 q^{37} + 16 q^{39} + 10 q^{41} + 11 q^{43} + 15 q^{45} - 23 q^{47} + 8 q^{49} + 16 q^{51} + 10 q^{53} + 4 q^{57} - 6 q^{59} + q^{61} - 23 q^{63} + 10 q^{65} + 4 q^{67} - 2 q^{69} - 9 q^{71} + 11 q^{73} - q^{75} + 32 q^{77} + 14 q^{79} + 46 q^{81} - 11 q^{83} + 3 q^{85} - 32 q^{87} + 30 q^{89} + 4 q^{91} + 2 q^{93} + 6 q^{95} + 29 q^{97} + 39 q^{99}+O(q^{100})$$ 6 * q - q^3 + 6 * q^5 - 8 * q^7 + 15 * q^9 + 10 * q^13 - q^15 + 3 * q^17 + 6 * q^19 + 9 * q^21 - 4 * q^23 + 6 * q^25 - 7 * q^27 + 3 * q^29 + 3 * q^31 + 9 * q^33 - 8 * q^35 - 6 * q^37 + 16 * q^39 + 10 * q^41 + 11 * q^43 + 15 * q^45 - 23 * q^47 + 8 * q^49 + 16 * q^51 + 10 * q^53 + 4 * q^57 - 6 * q^59 + q^61 - 23 * q^63 + 10 * q^65 + 4 * q^67 - 2 * q^69 - 9 * q^71 + 11 * q^73 - q^75 + 32 * q^77 + 14 * q^79 + 46 * q^81 - 11 * q^83 + 3 * q^85 - 32 * q^87 + 30 * q^89 + 4 * q^91 + 2 * q^93 + 6 * q^95 + 29 * q^97 + 39 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 16x^{4} + 12x^{3} + 60x^{2} - 18x - 44$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{5} + 25\nu^{4} + 4\nu^{3} - 304\nu^{2} + 82\nu + 500 ) / 98$$ (-v^5 + 25*v^4 + 4*v^3 - 304*v^2 + 82*v + 500) / 98 $$\beta_{3}$$ $$=$$ $$( 2\nu^{5} - \nu^{4} - 8\nu^{3} + 20\nu^{2} - 115\nu - 69 ) / 49$$ (2*v^5 - v^4 - 8*v^3 + 20*v^2 - 115*v - 69) / 49 $$\beta_{4}$$ $$=$$ $$( -4\nu^{5} + 2\nu^{4} + 65\nu^{3} - 40\nu^{2} - 211\nu + 89 ) / 49$$ (-4*v^5 + 2*v^4 + 65*v^3 - 40*v^2 - 211*v + 89) / 49 $$\beta_{5}$$ $$=$$ $$( -6\nu^{5} + 3\nu^{4} + 73\nu^{3} - 11\nu^{2} - 96\nu - 87 ) / 49$$ (-6*v^5 + 3*v^4 + 73*v^3 - 11*v^2 - 96*v - 87) / 49
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} - \beta_{4} + \beta_{3} + 5$$ b5 - b4 + b3 + 5 $$\nu^{3}$$ $$=$$ $$\beta_{4} + 2\beta_{3} + 9\beta _1 + 1$$ b4 + 2*b3 + 9*b1 + 1 $$\nu^{4}$$ $$=$$ $$12\beta_{5} - 12\beta_{4} + 13\beta_{3} + 4\beta_{2} - \beta _1 + 41$$ 12*b5 - 12*b4 + 13*b3 + 4*b2 - b1 + 41 $$\nu^{5}$$ $$=$$ $$-4\beta_{5} + 8\beta_{4} + 29\beta_{3} + 2\beta_{2} + 93\beta _1 + 9$$ -4*b5 + 8*b4 + 29*b3 + 2*b2 + 93*b1 + 9

## 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
 3.36175 2.45040 1.07986 −0.854190 −1.77441 −3.26341
0 −3.36175 0 1.00000 0 −3.60894 0 8.30138 0
1.2 0 −2.45040 0 1.00000 0 1.57732 0 3.00447 0
1.3 0 −1.07986 0 1.00000 0 −3.77160 0 −1.83390 0
1.4 0 0.854190 0 1.00000 0 −3.23895 0 −2.27036 0
1.5 0 1.77441 0 1.00000 0 2.66921 0 0.148539 0
1.6 0 3.26341 0 1.00000 0 −1.62704 0 7.64987 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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.bc 6
4.b odd 2 1 1480.2.a.k 6
20.d odd 2 1 7400.2.a.s 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1480.2.a.k 6 4.b odd 2 1
2960.2.a.bc 6 1.a even 1 1 trivial
7400.2.a.s 6 20.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2960))$$:

 $$T_{3}^{6} + T_{3}^{5} - 16T_{3}^{4} - 12T_{3}^{3} + 60T_{3}^{2} + 18T_{3} - 44$$ T3^6 + T3^5 - 16*T3^4 - 12*T3^3 + 60*T3^2 + 18*T3 - 44 $$T_{7}^{6} + 8T_{7}^{5} + 7T_{7}^{4} - 76T_{7}^{3} - 144T_{7}^{2} + 138T_{7} + 302$$ T7^6 + 8*T7^5 + 7*T7^4 - 76*T7^3 - 144*T7^2 + 138*T7 + 302 $$T_{13}^{6} - 10T_{13}^{5} + 4T_{13}^{4} + 180T_{13}^{3} - 260T_{13}^{2} - 648T_{13} + 384$$ T13^6 - 10*T13^5 + 4*T13^4 + 180*T13^3 - 260*T13^2 - 648*T13 + 384

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} + T^{5} - 16 T^{4} - 12 T^{3} + \cdots - 44$$
$5$ $$(T - 1)^{6}$$
$7$ $$T^{6} + 8 T^{5} + 7 T^{4} - 76 T^{3} + \cdots + 302$$
$11$ $$T^{6} - 33 T^{4} + 64 T^{3} + 80 T^{2} + \cdots - 96$$
$13$ $$T^{6} - 10 T^{5} + 4 T^{4} + 180 T^{3} + \cdots + 384$$
$17$ $$T^{6} - 3 T^{5} - 44 T^{4} + 160 T^{3} + \cdots - 464$$
$19$ $$T^{6} - 6 T^{5} - 18 T^{4} + 110 T^{3} + \cdots - 64$$
$23$ $$T^{6} + 4 T^{5} - 76 T^{4} - 304 T^{3} + \cdots - 576$$
$29$ $$T^{6} - 3 T^{5} - 74 T^{4} - 64 T^{3} + \cdots + 32$$
$31$ $$T^{6} - 3 T^{5} - 92 T^{4} + \cdots + 5496$$
$37$ $$(T + 1)^{6}$$
$41$ $$T^{6} - 10 T^{5} - 123 T^{4} + \cdots + 81504$$
$43$ $$T^{6} - 11 T^{5} - 154 T^{4} + \cdots - 51488$$
$47$ $$T^{6} + 23 T^{5} + 128 T^{4} + \cdots - 37784$$
$53$ $$T^{6} - 10 T^{5} - 39 T^{4} + \cdots + 1296$$
$59$ $$T^{6} + 6 T^{5} - 134 T^{4} + \cdots - 128$$
$61$ $$T^{6} - T^{5} - 80 T^{4} + 368 T^{3} + \cdots + 576$$
$67$ $$T^{6} - 4 T^{5} - 202 T^{4} + \cdots + 44544$$
$71$ $$T^{6} + 9 T^{5} - 74 T^{4} + \cdots + 4224$$
$73$ $$T^{6} - 11 T^{5} - 252 T^{4} + \cdots - 300672$$
$79$ $$T^{6} - 14 T^{5} - 82 T^{4} + \cdots + 5632$$
$83$ $$T^{6} + 11 T^{5} - 146 T^{4} + \cdots - 9624$$
$89$ $$T^{6} - 30 T^{5} + 48 T^{4} + \cdots + 1147008$$
$97$ $$T^{6} - 29 T^{5} + 44 T^{4} + \cdots + 1463296$$