# Properties

 Label 1300.2.d.c Level $1300$ Weight $2$ Character orbit 1300.d Analytic conductor $10.381$ Analytic rank $0$ Dimension $6$ 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(649,1300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1300, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

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

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

chi := DirichletCharacter("1300.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1300 = 2^{2} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1300.d (of order $$2$$, degree $$1$$, 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: $$6$$ Coefficient field: 6.0.9144576.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 12x^{4} + 36x^{2} + 4$$ x^6 + 12*x^4 + 36*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 260) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + \beta_{3} q^{7} + (\beta_{3} - 1) q^{9}+O(q^{10})$$ q + b1 * q^3 + b3 * q^7 + (b3 - 1) * q^9 $$q + \beta_1 q^{3} + \beta_{3} q^{7} + (\beta_{3} - 1) q^{9} + ( - \beta_{5} + \beta_{4}) q^{11} + ( - \beta_{5} - \beta_{2} - 1) q^{13} + (\beta_{5} - 2 \beta_1) q^{19} + (\beta_{4} - 2 \beta_1) q^{21} + ( - 2 \beta_{5} + \beta_{4} - \beta_1) q^{23} + \beta_{4} q^{27} + (\beta_{3} + 2 \beta_{2} + 2) q^{29} + ( - \beta_{5} - \beta_{4} - 2 \beta_1) q^{31} + (\beta_{3} + 4 \beta_{2} - 2) q^{33} + ( - \beta_{3} + 2 \beta_{2} - 4) q^{37} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 2) q^{39} + (2 \beta_{5} + \beta_{4}) q^{41} + ( - \beta_{4} + 3 \beta_1) q^{43} + ( - \beta_{3} - 4 \beta_{2} - 4) q^{47} + (2 \beta_{3} + 2 \beta_{2} + 1) q^{49} + ( - 2 \beta_{5} + \beta_{4} - 4 \beta_1) q^{53} + ( - 3 \beta_{3} - 2 \beta_{2} + 10) q^{57} + ( - \beta_{5} + 4 \beta_{4}) q^{59} + (3 \beta_{3} - 2) q^{61} + (\beta_{3} + 2 \beta_{2} + 8) q^{63} + (3 \beta_{3} + 2 \beta_{2} - 4) q^{67} + (\beta_{3} + 6 \beta_{2}) q^{69} + (3 \beta_{5} + 6 \beta_1) q^{71} + ( - \beta_{3} + 2 \beta_{2} + 8) q^{73} - 3 \beta_{4} q^{77} + (2 \beta_{3} - 4 \beta_{2} + 4) q^{79} + (3 \beta_{3} + 2 \beta_{2} - 3) q^{81} + ( - \beta_{3} + 2 \beta_{2} - 4) q^{83} + (2 \beta_{5} - 3 \beta_{4} + 2 \beta_1) q^{87} + ( - 2 \beta_{5} - 4 \beta_{4} - 6 \beta_1) q^{89} + ( - 2 \beta_{5} - 3 \beta_{4} - \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{91} + ( - \beta_{3} + 6) q^{93} + (2 \beta_{3} + 6) q^{97} + (\beta_{5} - 4 \beta_{4}) q^{99}+O(q^{100})$$ q + b1 * q^3 + b3 * q^7 + (b3 - 1) * q^9 + (-b5 + b4) * q^11 + (-b5 - b2 - 1) * q^13 + (b5 - 2*b1) * q^19 + (b4 - 2*b1) * q^21 + (-2*b5 + b4 - b1) * q^23 + b4 * q^27 + (b3 + 2*b2 + 2) * q^29 + (-b5 - b4 - 2*b1) * q^31 + (b3 + 4*b2 - 2) * q^33 + (-b3 + 2*b2 - 4) * q^37 + (-b5 + 2*b4 + b3 + 2*b2 - 2*b1 - 2) * q^39 + (2*b5 + b4) * q^41 + (-b4 + 3*b1) * q^43 + (-b3 - 4*b2 - 4) * q^47 + (2*b3 + 2*b2 + 1) * q^49 + (-2*b5 + b4 - 4*b1) * q^53 + (-3*b3 - 2*b2 + 10) * q^57 + (-b5 + 4*b4) * q^59 + (3*b3 - 2) * q^61 + (b3 + 2*b2 + 8) * q^63 + (3*b3 + 2*b2 - 4) * q^67 + (b3 + 6*b2) * q^69 + (3*b5 + 6*b1) * q^71 + (-b3 + 2*b2 + 8) * q^73 - 3*b4 * q^77 + (2*b3 - 4*b2 + 4) * q^79 + (3*b3 + 2*b2 - 3) * q^81 + (-b3 + 2*b2 - 4) * q^83 + (2*b5 - 3*b4 + 2*b1) * q^87 + (-2*b5 - 4*b4 - 6*b1) * q^89 + (-2*b5 - 3*b4 - b3 + 2*b2 - 2*b1 + 2) * q^91 + (-b3 + 6) * q^93 + (2*b3 + 6) * q^97 + (b5 - 4*b4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{9}+O(q^{10})$$ 6 * q - 6 * q^9 $$6 q - 6 q^{9} - 6 q^{13} + 12 q^{29} - 12 q^{33} - 24 q^{37} - 12 q^{39} - 24 q^{47} + 6 q^{49} + 60 q^{57} - 12 q^{61} + 48 q^{63} - 24 q^{67} + 48 q^{73} + 24 q^{79} - 18 q^{81} - 24 q^{83} + 12 q^{91} + 36 q^{93} + 36 q^{97}+O(q^{100})$$ 6 * q - 6 * q^9 - 6 * q^13 + 12 * q^29 - 12 * q^33 - 24 * q^37 - 12 * q^39 - 24 * q^47 + 6 * q^49 + 60 * q^57 - 12 * q^61 + 48 * q^63 - 24 * q^67 + 48 * q^73 + 24 * q^79 - 18 * q^81 - 24 * q^83 + 12 * q^91 + 36 * q^93 + 36 * q^97

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{4} + 6\nu^{2} ) / 2$$ (v^4 + 6*v^2) / 2 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 4$$ v^2 + 4 $$\beta_{4}$$ $$=$$ $$\nu^{3} + 6\nu$$ v^3 + 6*v $$\beta_{5}$$ $$=$$ $$( \nu^{5} + 10\nu^{3} + 22\nu ) / 2$$ (v^5 + 10*v^3 + 22*v) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 4$$ b3 - 4 $$\nu^{3}$$ $$=$$ $$\beta_{4} - 6\beta_1$$ b4 - 6*b1 $$\nu^{4}$$ $$=$$ $$-6\beta_{3} + 2\beta_{2} + 24$$ -6*b3 + 2*b2 + 24 $$\nu^{5}$$ $$=$$ $$2\beta_{5} - 10\beta_{4} + 38\beta_1$$ 2*b5 - 10*b4 + 38*b1

## Character values

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

 $$n$$ $$301$$ $$651$$ $$677$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
649.1
 − 2.60168i − 2.26180i − 0.339877i 0.339877i 2.26180i 2.60168i
0 2.60168i 0 0 0 −2.76873 0 −3.76873 0
649.2 0 2.26180i 0 0 0 −1.11575 0 −2.11575 0
649.3 0 0.339877i 0 0 0 3.88448 0 2.88448 0
649.4 0 0.339877i 0 0 0 3.88448 0 2.88448 0
649.5 0 2.26180i 0 0 0 −1.11575 0 −2.11575 0
649.6 0 2.60168i 0 0 0 −2.76873 0 −3.76873 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 649.6 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.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1300.2.d.c 6
5.b even 2 1 1300.2.d.d 6
5.c odd 4 1 260.2.f.a 6
5.c odd 4 1 1300.2.f.e 6
13.b even 2 1 1300.2.d.d 6
15.e even 4 1 2340.2.c.d 6
20.e even 4 1 1040.2.k.c 6
65.d even 2 1 inner 1300.2.d.c 6
65.f even 4 1 3380.2.a.m 3
65.h odd 4 1 260.2.f.a 6
65.h odd 4 1 1300.2.f.e 6
65.k even 4 1 3380.2.a.n 3
195.s even 4 1 2340.2.c.d 6
260.p even 4 1 1040.2.k.c 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.f.a 6 5.c odd 4 1
260.2.f.a 6 65.h odd 4 1
1040.2.k.c 6 20.e even 4 1
1040.2.k.c 6 260.p even 4 1
1300.2.d.c 6 1.a even 1 1 trivial
1300.2.d.c 6 65.d even 2 1 inner
1300.2.d.d 6 5.b even 2 1
1300.2.d.d 6 13.b even 2 1
1300.2.f.e 6 5.c odd 4 1
1300.2.f.e 6 65.h odd 4 1
2340.2.c.d 6 15.e even 4 1
2340.2.c.d 6 195.s even 4 1
3380.2.a.m 3 65.f even 4 1
3380.2.a.n 3 65.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1300, [\chi])$$:

 $$T_{3}^{6} + 12T_{3}^{4} + 36T_{3}^{2} + 4$$ T3^6 + 12*T3^4 + 36*T3^2 + 4 $$T_{7}^{3} - 12T_{7} - 12$$ T7^3 - 12*T7 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} + 12 T^{4} + 36 T^{2} + 4$$
$5$ $$T^{6}$$
$7$ $$(T^{3} - 12 T - 12)^{2}$$
$11$ $$T^{6} + 36 T^{4} + 216 T^{2} + \cdots + 324$$
$13$ $$T^{6} + 6 T^{5} + 27 T^{4} + \cdots + 2197$$
$17$ $$T^{6}$$
$19$ $$T^{6} + 96 T^{4} + 2304 T^{2} + \cdots + 12996$$
$23$ $$T^{6} + 96 T^{4} + 2988 T^{2} + \cdots + 30276$$
$29$ $$(T^{3} - 6 T^{2} - 12 T + 84)^{2}$$
$31$ $$T^{6} + 60 T^{4} + 936 T^{2} + \cdots + 4356$$
$37$ $$(T^{3} + 12 T^{2} - 252)^{2}$$
$41$ $$T^{6} + 108 T^{4} + 2160 T^{2} + \cdots + 5184$$
$43$ $$T^{6} + 120 T^{4} + 2316 T^{2} + \cdots + 2116$$
$47$ $$(T^{3} + 12 T^{2} - 36 T - 468)^{2}$$
$53$ $$T^{6} + 204 T^{4} + 6768 T^{2} + \cdots + 576$$
$59$ $$T^{6} + 216 T^{4} + 12528 T^{2} + \cdots + 171396$$
$61$ $$(T^{3} + 6 T^{2} - 96 T - 532)^{2}$$
$67$ $$(T^{3} + 12 T^{2} - 48 T - 588)^{2}$$
$71$ $$T^{6} + 432 T^{4} + 46656 T^{2} + \cdots + 492804$$
$73$ $$(T^{3} - 24 T^{2} + 144 T - 252)^{2}$$
$79$ $$(T^{3} - 12 T^{2} - 144 T + 1696)^{2}$$
$83$ $$(T^{3} + 12 T^{2} - 252)^{2}$$
$89$ $$T^{6} + 576 T^{4} + 96768 T^{2} + \cdots + 4064256$$
$97$ $$(T^{3} - 18 T^{2} + 60 T - 24)^{2}$$