Properties

 Label 1300.2.ba.b Level $1300$ Weight $2$ Character orbit 1300.ba Analytic conductor $10.381$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1300,2,Mod(49,1300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1300, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 5]))

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

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

chi := DirichletCharacter("1300.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1300 = 2^{2} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1300.ba (of order $$6$$, degree $$2$$, not 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: $$10.3805522628$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.22581504.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16$$ x^8 - 4*x^7 + 5*x^6 + 2*x^5 - 11*x^4 + 4*x^3 + 20*x^2 - 32*x + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 260) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{6} + \beta_{2}) q^{3} + ( - \beta_{6} - \beta_{5} - \beta_{4} + 1) q^{7} + ( - 2 \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2}) q^{9}+O(q^{10})$$ q + (b6 + b2) * q^3 + (-b6 - b5 - b4 + 1) * q^7 + (-2*b6 - b5 - b4 - b3 - 2*b2) * q^9 $$q + (\beta_{6} + \beta_{2}) q^{3} + ( - \beta_{6} - \beta_{5} - \beta_{4} + 1) q^{7} + ( - 2 \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2}) q^{9} + ( - \beta_{6} + 1) q^{11} + (\beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1) q^{13} + ( - \beta_{6} + \beta_{5} - \beta_{4} - 3) q^{17} + (\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_1) q^{19} + ( - \beta_{7} + 2 \beta_{6} + 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{21} + ( - \beta_{7} + \beta_{6} + 2 \beta_{5} + \beta_{4} + 2 \beta_{2} - 1) q^{23} + ( - \beta_{7} + 2 \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_1) q^{27} + ( - \beta_{7} - \beta_{6} - \beta_{4} - 2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{29} + ( - \beta_{7} - 4 \beta_{6} - 4 \beta_{3} - 4 \beta_{2} - \beta_1 - 2) q^{31} + (3 \beta_{6} + \beta_{3} + 2 \beta_{2} + 1) q^{33} + ( - \beta_{7} + 3 \beta_{6} + 3 \beta_{4} - 2 \beta_1 + 3) q^{37} + ( - \beta_{7} - 5 \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{39} + ( - 3 \beta_{7} - \beta_{6} + 1) q^{41} + (\beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} + 2 \beta_1 + 3) q^{43} + ( - \beta_{7} + \beta_1) q^{47} + (4 \beta_{6} - 3 \beta_{4} + 2 \beta_{3} + \beta_{2} + 3) q^{49} + (\beta_{7} + 2 \beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{51} + (4 \beta_{7} - 3 \beta_{5} - 6 \beta_{4} + \beta_{3} + \beta_{2} + 4 \beta_1 + 3) q^{53} + (2 \beta_{7} - \beta_{5} - 2 \beta_1 - 4) q^{57} + (3 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 4) q^{59} + (4 \beta_{7} + \beta_{6} - 3 \beta_{5} - 3 \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{61} + (\beta_{7} - 4 \beta_{6} - \beta_{4} - 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 3) q^{63} + (2 \beta_{7} - 5 \beta_{6} + \beta_{4} + 4 \beta_1 - 5) q^{67} + (4 \beta_{7} - 7 \beta_{6} - 4 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + \cdots + 2) q^{69}+ \cdots + ( - 4 \beta_{6} - \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} - 1) q^{99}+O(q^{100})$$ q + (b6 + b2) * q^3 + (-b6 - b5 - b4 + 1) * q^7 + (-2*b6 - b5 - b4 - b3 - 2*b2) * q^9 + (-b6 + 1) * q^11 + (b4 + b3 + 2*b2 - b1) * q^13 + (-b6 + b5 - b4 - 3) * q^17 + (b6 + b5 - b4 + b3 + b1) * q^19 + (-b7 + 2*b6 + 2*b3 + 2*b2 - b1 + 1) * q^21 + (-b7 + b6 + 2*b5 + b4 + 2*b2 - 1) * q^23 + (-b7 + 2*b6 + b5 + 2*b4 - b1) * q^27 + (-b7 - b6 - b4 - 2*b3 - b2 - 2*b1) * q^29 + (-b7 - 4*b6 - 4*b3 - 4*b2 - b1 - 2) * q^31 + (3*b6 + b3 + 2*b2 + 1) * q^33 + (-b7 + 3*b6 + 3*b4 - 2*b1 + 3) * q^37 + (-b7 - 5*b6 + b5 - 2*b4 - b3 - b2 + 2*b1 - 2) * q^39 + (-3*b7 - b6 + 1) * q^41 + (b6 + 2*b5 - 2*b4 - 3*b3 + 2*b1 + 3) * q^43 + (-b7 + b1) * q^47 + (4*b6 - 3*b4 + 2*b3 + b2 + 3) * q^49 + (b7 + 2*b3 - 2*b2 - b1 - 1) * q^51 + (4*b7 - 3*b5 - 6*b4 + b3 + b2 + 4*b1 + 3) * q^53 + (2*b7 - b5 - 2*b1 - 4) * q^57 + (3*b6 + 2*b5 - 2*b4 + 4) * q^59 + (4*b7 + b6 - 3*b5 - 3*b4 - b3 - 2*b2 + 2*b1 + 2) * q^61 + (b7 - 4*b6 - b4 - 2*b3 - b2 + 2*b1 - 3) * q^63 + (2*b7 - 5*b6 + b4 + 4*b1 - 5) * q^67 + (4*b7 - 7*b6 - 4*b5 - 4*b4 - 2*b3 - 4*b2 + 2*b1 + 2) * q^69 + (-3*b6 + 3*b5 - 3*b4 - 3*b3 - 6) * q^71 + (-b7 + 3*b5 - b3 + b2 + b1 + 5) * q^73 + (-2*b6 - b5 - 2*b4) * q^77 + (-3*b7 + 3*b5 - b3 + b2 + 3*b1 + 1) * q^79 + (2*b7 + 3*b6 + b4 + 2*b3 + b2 + 4*b1 + 2) * q^81 + (2*b3 - 2*b2 + 8) * q^83 + (3*b6 + b5 - b4 + 2*b3 + 2*b1 + 3) * q^87 + (-b6 - 4*b5 - 2*b4 + 5) * q^89 + (b6 - b5 - 2*b4 + 3*b3 + b2 - 2*b1 + 1) * q^91 + (b7 + 14*b6 + 2*b4 + 4*b3 + 2*b2 + 2*b1 + 12) * q^93 + (4*b7 + b6 - 5*b5 - 5*b4 - 2*b3 - 4*b2 + 2*b1 + 3) * q^97 + (-4*b6 - b5 - 2*b4 - 3*b3 - 3*b2 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 6 q^{3} + 6 q^{7} + 4 q^{9}+O(q^{10})$$ 8 * q - 6 * q^3 + 6 * q^7 + 4 * q^9 $$8 q - 6 q^{3} + 6 q^{7} + 4 q^{9} + 12 q^{11} - 18 q^{17} - 6 q^{23} - 6 q^{33} + 18 q^{37} + 4 q^{39} + 12 q^{41} + 18 q^{43} + 4 q^{49} - 36 q^{57} + 24 q^{59} - 4 q^{61} - 12 q^{63} - 18 q^{67} + 24 q^{69} - 36 q^{71} + 48 q^{73} + 16 q^{79} + 8 q^{81} + 72 q^{83} + 18 q^{87} + 24 q^{89} + 48 q^{93} - 6 q^{97}+O(q^{100})$$ 8 * q - 6 * q^3 + 6 * q^7 + 4 * q^9 + 12 * q^11 - 18 * q^17 - 6 * q^23 - 6 * q^33 + 18 * q^37 + 4 * q^39 + 12 * q^41 + 18 * q^43 + 4 * q^49 - 36 * q^57 + 24 * q^59 - 4 * q^61 - 12 * q^63 - 18 * q^67 + 24 * q^69 - 36 * q^71 + 48 * q^73 + 16 * q^79 + 8 * q^81 + 72 * q^83 + 18 * q^87 + 24 * q^89 + 48 * q^93 - 6 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{7} - 2\nu^{6} + \nu^{5} + 4\nu^{4} - 3\nu^{3} - 2\nu^{2} + 8\nu - 8 ) / 4$$ (v^7 - 2*v^6 + v^5 + 4*v^4 - 3*v^3 - 2*v^2 + 8*v - 8) / 4 $$\beta_{2}$$ $$=$$ $$( \nu^{7} - 3\nu^{6} + 3\nu^{5} + 3\nu^{4} - 7\nu^{3} - 3\nu^{2} + 14\nu - 16 ) / 4$$ (v^7 - 3*v^6 + 3*v^5 + 3*v^4 - 7*v^3 - 3*v^2 + 14*v - 16) / 4 $$\beta_{3}$$ $$=$$ $$( 3\nu^{7} - 6\nu^{6} - \nu^{5} + 12\nu^{4} - 5\nu^{3} - 22\nu^{2} + 36\nu - 8 ) / 8$$ (3*v^7 - 6*v^6 - v^5 + 12*v^4 - 5*v^3 - 22*v^2 + 36*v - 8) / 8 $$\beta_{4}$$ $$=$$ $$( -\nu^{7} + 3\nu^{6} - 3\nu^{5} - 3\nu^{4} + 7\nu^{3} + 3\nu^{2} - 22\nu + 20 ) / 4$$ (-v^7 + 3*v^6 - 3*v^5 - 3*v^4 + 7*v^3 + 3*v^2 - 22*v + 20) / 4 $$\beta_{5}$$ $$=$$ $$( -3\nu^{7} + 8\nu^{6} - 3\nu^{5} - 10\nu^{4} + 13\nu^{3} + 8\nu^{2} - 32\nu + 32 ) / 8$$ (-3*v^7 + 8*v^6 - 3*v^5 - 10*v^4 + 13*v^3 + 8*v^2 - 32*v + 32) / 8 $$\beta_{6}$$ $$=$$ $$( 3\nu^{7} - 7\nu^{6} + 3\nu^{5} + 11\nu^{4} - 15\nu^{3} - 11\nu^{2} + 40\nu - 32 ) / 4$$ (3*v^7 - 7*v^6 + 3*v^5 + 11*v^4 - 15*v^3 - 11*v^2 + 40*v - 32) / 4 $$\beta_{7}$$ $$=$$ $$( 7\nu^{7} - 20\nu^{6} + 11\nu^{5} + 30\nu^{4} - 45\nu^{3} - 28\nu^{2} + 116\nu - 88 ) / 4$$ (7*v^7 - 20*v^6 + 11*v^5 + 30*v^4 - 45*v^3 - 28*v^2 + 116*v - 88) / 4
 $$\nu$$ $$=$$ $$( -\beta_{4} - \beta_{2} + 1 ) / 2$$ (-b4 - b2 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + \beta _1 + 2 ) / 2$$ (-b5 - b4 - b3 - 2*b2 + b1 + 2) / 2 $$\nu^{3}$$ $$=$$ $$( -2\beta_{6} - \beta_{5} - 2\beta_{4} + \beta_{3} - \beta_{2} + 2\beta _1 - 1 ) / 2$$ (-2*b6 - b5 - 2*b4 + b3 - b2 + 2*b1 - 1) / 2 $$\nu^{4}$$ $$=$$ $$( \beta_{7} - 2\beta_{6} + 2\beta_{5} + \beta_{4} - \beta_{2} + 4\beta _1 - 3 ) / 2$$ (b7 - 2*b6 + 2*b5 + b4 - b2 + 4*b1 - 3) / 2 $$\nu^{5}$$ $$=$$ $$( -2\beta_{6} + 3\beta_{5} - \beta_{4} + \beta_{3} + 4\beta_{2} + 4\beta _1 + 2 ) / 2$$ (-2*b6 + 3*b5 - b4 + b3 + 4*b2 + 4*b1 + 2) / 2 $$\nu^{6}$$ $$=$$ $$( -\beta_{7} + 6\beta_{6} + 9\beta_{5} - \beta_{3} + \beta_{2} + 3\beta _1 - 1 ) / 2$$ (-b7 + 6*b6 + 9*b5 - b3 + b2 + 3*b1 - 1) / 2 $$\nu^{7}$$ $$=$$ $$( -6\beta_{7} + 16\beta_{6} + 2\beta_{5} - 3\beta_{4} - 2\beta_{3} + 3\beta_{2} + 2\beta _1 + 17 ) / 2$$ (-6*b7 + 16*b6 + 2*b5 - 3*b4 - 2*b3 + 3*b2 + 2*b1 + 17) / 2

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$651$$ $$677$$ $$\chi(n)$$ $$-\beta_{6}$$ $$1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 1.20036 − 0.747754i 1.40994 + 0.109843i 0.665665 + 1.24775i −1.27597 − 0.609843i 1.20036 + 0.747754i 1.40994 − 0.109843i 0.665665 − 1.24775i −1.27597 + 0.609843i
0 −2.44811 + 1.41342i 0 0 0 1.04739 1.81414i 0 2.49551 4.32235i 0
49.2 0 −2.01978 + 1.16612i 0 0 0 0.199902 0.346241i 0 1.21969 2.11256i 0
49.3 0 0.0820885 0.0473938i 0 0 0 −0.413419 + 0.716063i 0 −1.49551 + 2.59030i 0
49.4 0 1.38581 0.800098i 0 0 0 2.16612 3.75184i 0 −0.219687 + 0.380509i 0
849.1 0 −2.44811 1.41342i 0 0 0 1.04739 + 1.81414i 0 2.49551 + 4.32235i 0
849.2 0 −2.01978 1.16612i 0 0 0 0.199902 + 0.346241i 0 1.21969 + 2.11256i 0
849.3 0 0.0820885 + 0.0473938i 0 0 0 −0.413419 0.716063i 0 −1.49551 2.59030i 0
849.4 0 1.38581 + 0.800098i 0 0 0 2.16612 + 3.75184i 0 −0.219687 0.380509i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 849.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.l even 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1300.2.ba.b 8
5.b even 2 1 1300.2.ba.c 8
5.c odd 4 1 260.2.x.a 8
5.c odd 4 1 1300.2.y.b 8
13.e even 6 1 1300.2.ba.c 8
15.e even 4 1 2340.2.dj.d 8
20.e even 4 1 1040.2.da.c 8
65.l even 6 1 inner 1300.2.ba.b 8
65.o even 12 1 3380.2.a.q 4
65.q odd 12 1 3380.2.f.i 8
65.r odd 12 1 260.2.x.a 8
65.r odd 12 1 1300.2.y.b 8
65.r odd 12 1 3380.2.f.i 8
65.t even 12 1 3380.2.a.p 4
195.bf even 12 1 2340.2.dj.d 8
260.bg even 12 1 1040.2.da.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.x.a 8 5.c odd 4 1
260.2.x.a 8 65.r odd 12 1
1040.2.da.c 8 20.e even 4 1
1040.2.da.c 8 260.bg even 12 1
1300.2.y.b 8 5.c odd 4 1
1300.2.y.b 8 65.r odd 12 1
1300.2.ba.b 8 1.a even 1 1 trivial
1300.2.ba.b 8 65.l even 6 1 inner
1300.2.ba.c 8 5.b even 2 1
1300.2.ba.c 8 13.e even 6 1
2340.2.dj.d 8 15.e even 4 1
2340.2.dj.d 8 195.bf even 12 1
3380.2.a.p 4 65.t even 12 1
3380.2.a.q 4 65.o even 12 1
3380.2.f.i 8 65.q odd 12 1
3380.2.f.i 8 65.r odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + 6T_{3}^{7} + 10T_{3}^{6} - 12T_{3}^{5} - 33T_{3}^{4} + 36T_{3}^{3} + 106T_{3}^{2} - 18T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(1300, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8} + 6 T^{7} + 10 T^{6} - 12 T^{5} + \cdots + 1$$
$5$ $$T^{8}$$
$7$ $$T^{8} - 6 T^{7} + 30 T^{6} - 48 T^{5} + \cdots + 9$$
$11$ $$(T^{2} - 3 T + 3)^{4}$$
$13$ $$T^{8} + 16 T^{6} + 96 T^{5} + \cdots + 28561$$
$17$ $$T^{8} + 18 T^{7} + 126 T^{6} + \cdots + 729$$
$19$ $$T^{8} - 30 T^{6} + 867 T^{4} + \cdots + 1089$$
$23$ $$T^{8} + 6 T^{7} - 18 T^{6} - 180 T^{5} + \cdots + 9$$
$29$ $$T^{8} + 42 T^{6} + 192 T^{5} + \cdots + 1521$$
$31$ $$(T^{4} + 96 T^{2} + 2112)^{2}$$
$37$ $$T^{8} - 18 T^{7} + 258 T^{6} + \cdots + 42849$$
$41$ $$(T^{4} - 6 T^{3} - 21 T^{2} + 198 T + 1089)^{2}$$
$43$ $$T^{8} - 18 T^{7} + 46 T^{6} + \cdots + 5031049$$
$47$ $$(T^{2} - 12)^{4}$$
$53$ $$T^{8} + 456 T^{6} + 69168 T^{4} + \cdots + 389376$$
$59$ $$T^{8} - 24 T^{7} + 174 T^{6} + \cdots + 558009$$
$61$ $$T^{8} + 4 T^{7} + 118 T^{6} + \cdots + 942841$$
$67$ $$T^{8} + 18 T^{7} + 330 T^{6} + \cdots + 1083681$$
$71$ $$T^{8} + 36 T^{7} + 414 T^{6} + \cdots + 45198729$$
$73$ $$(T^{4} - 24 T^{3} + 156 T^{2} - 1584)^{2}$$
$79$ $$(T^{4} - 8 T^{3} - 180 T^{2} + 1504 T - 368)^{2}$$
$83$ $$(T^{4} - 36 T^{3} + 408 T^{2} - 1440 T + 576)^{2}$$
$89$ $$T^{8} - 24 T^{7} + 174 T^{6} + \cdots + 13689$$
$97$ $$T^{8} + 6 T^{7} + 222 T^{6} + \cdots + 12981609$$