# Properties

 Label 1300.2.c.f Level $1300$ Weight $2$ Character orbit 1300.c Analytic conductor $10.381$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1300,2,Mod(1249,1300)]

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, 0]))

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

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

chi := DirichletCharacter("1300.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1300 = 2^{2} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1300.c (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: 6.0.5089536.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18$$ x^6 - 2*x^5 + 2*x^4 + 2*x^3 + 16*x^2 - 24*x + 18 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{3}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{4} - \beta_1) q^{3} + ( - \beta_{5} - \beta_{4}) q^{7} + (\beta_{2} - 4) q^{9}+O(q^{10})$$ q + (-b4 - b1) * q^3 + (-b5 - b4) * q^7 + (b2 - 4) * q^9 $$q + ( - \beta_{4} - \beta_1) q^{3} + ( - \beta_{5} - \beta_{4}) q^{7} + (\beta_{2} - 4) q^{9} + ( - \beta_{3} - \beta_{2}) q^{11} - \beta_{4} q^{13} - 2 \beta_1 q^{17} + (\beta_{3} - \beta_{2} - 2) q^{19} + ( - 2 \beta_{3} + 2 \beta_{2} - 4) q^{21} + (3 \beta_{4} - \beta_1) q^{23} + (2 \beta_{5} + 4 \beta_{4} + 2 \beta_1) q^{27} + ( - \beta_{2} - 3) q^{29} + ( - \beta_{3} - \beta_{2} - 4) q^{31} + ( - \beta_{5} + 3 \beta_{4} - 2 \beta_1) q^{33} + (\beta_{5} - \beta_{4} - 2 \beta_1) q^{37} + ( - \beta_{3} - 1) q^{39} + 2 \beta_{3} q^{41} + (\beta_{4} + \beta_1) q^{43} + (\beta_{5} - 3 \beta_{4}) q^{47} + ( - 4 \beta_{3} - 9) q^{49} + (2 \beta_{3} + 2 \beta_{2} - 12) q^{51} + (2 \beta_{5} + 6 \beta_{4} - 2 \beta_1) q^{53} + ( - 3 \beta_{5} - 7 \beta_{4} + 2 \beta_1) q^{57} + (3 \beta_{3} + \beta_{2} + 6) q^{59} + ( - \beta_{2} + 5) q^{61} + (3 \beta_{5} + 19 \beta_{4} + 4 \beta_1) q^{63} + ( - \beta_{5} + 5 \beta_{4} + 2 \beta_1) q^{67} + (4 \beta_{3} + \beta_{2} - 3) q^{69} + ( - \beta_{3} - \beta_{2}) q^{71} + ( - 3 \beta_{5} - 5 \beta_{4} + 2 \beta_1) q^{73} + (2 \beta_{5} - 12 \beta_{4} - 2 \beta_1) q^{77} + (2 \beta_{3} - 2) q^{79} + (4 \beta_{3} - 3 \beta_{2} + 10) q^{81} + ( - \beta_{5} + 3 \beta_{4} + 4 \beta_1) q^{83} + ( - 2 \beta_{5} + 2 \beta_1) q^{87} + ( - 4 \beta_{3} - 2 \beta_{2}) q^{89} + (\beta_{2} - 1) q^{91} + ( - \beta_{5} + 7 \beta_{4} + 2 \beta_1) q^{93} + ( - 2 \beta_{5} + 8 \beta_{4}) q^{97} + (\beta_{3} + \beta_{2} - 12) q^{99}+O(q^{100})$$ q + (-b4 - b1) * q^3 + (-b5 - b4) * q^7 + (b2 - 4) * q^9 + (-b3 - b2) * q^11 - b4 * q^13 - 2*b1 * q^17 + (b3 - b2 - 2) * q^19 + (-2*b3 + 2*b2 - 4) * q^21 + (3*b4 - b1) * q^23 + (2*b5 + 4*b4 + 2*b1) * q^27 + (-b2 - 3) * q^29 + (-b3 - b2 - 4) * q^31 + (-b5 + 3*b4 - 2*b1) * q^33 + (b5 - b4 - 2*b1) * q^37 + (-b3 - 1) * q^39 + 2*b3 * q^41 + (b4 + b1) * q^43 + (b5 - 3*b4) * q^47 + (-4*b3 - 9) * q^49 + (2*b3 + 2*b2 - 12) * q^51 + (2*b5 + 6*b4 - 2*b1) * q^53 + (-3*b5 - 7*b4 + 2*b1) * q^57 + (3*b3 + b2 + 6) * q^59 + (-b2 + 5) * q^61 + (3*b5 + 19*b4 + 4*b1) * q^63 + (-b5 + 5*b4 + 2*b1) * q^67 + (4*b3 + b2 - 3) * q^69 + (-b3 - b2) * q^71 + (-3*b5 - 5*b4 + 2*b1) * q^73 + (2*b5 - 12*b4 - 2*b1) * q^77 + (2*b3 - 2) * q^79 + (4*b3 - 3*b2 + 10) * q^81 + (-b5 + 3*b4 + 4*b1) * q^83 + (-2*b5 + 2*b1) * q^87 + (-4*b3 - 2*b2) * q^89 + (b2 - 1) * q^91 + (-b5 + 7*b4 + 2*b1) * q^93 + (-2*b5 + 8*b4) * q^97 + (b3 + b2 - 12) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 22 q^{9}+O(q^{10})$$ 6 * q - 22 * q^9 $$6 q - 22 q^{9} - 16 q^{19} - 16 q^{21} - 20 q^{29} - 24 q^{31} - 4 q^{39} - 4 q^{41} - 46 q^{49} - 72 q^{51} + 32 q^{59} + 28 q^{61} - 24 q^{69} - 16 q^{79} + 46 q^{81} + 4 q^{89} - 4 q^{91} - 72 q^{99}+O(q^{100})$$ 6 * q - 22 * q^9 - 16 * q^19 - 16 * q^21 - 20 * q^29 - 24 * q^31 - 4 * q^39 - 4 * q^41 - 46 * q^49 - 72 * q^51 + 32 * q^59 + 28 * q^61 - 24 * q^69 - 16 * q^79 + 46 * q^81 + 4 * q^89 - 4 * q^91 - 72 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18$$ :

 $$\beta_{1}$$ $$=$$ $$( -5\nu^{5} - 11\nu^{4} + 101\nu^{3} - 136\nu^{2} + 292\nu - 147 ) / 393$$ (-5*v^5 - 11*v^4 + 101*v^3 - 136*v^2 + 292*v - 147) / 393 $$\beta_{2}$$ $$=$$ $$( -2\nu^{5} + 48\nu^{4} - 12\nu^{3} - 2\nu^{2} + 12\nu + 701 ) / 131$$ (-2*v^5 + 48*v^4 - 12*v^3 - 2*v^2 + 12*v + 701) / 131 $$\beta_{3}$$ $$=$$ $$( -6\nu^{5} + 13\nu^{4} - 36\nu^{3} - 6\nu^{2} + 36\nu + 7 ) / 131$$ (-6*v^5 + 13*v^4 - 36*v^3 - 6*v^2 + 36*v + 7) / 131 $$\beta_{4}$$ $$=$$ $$( 23\nu^{5} - 28\nu^{4} + 7\nu^{3} + 154\nu^{2} + 386\nu - 267 ) / 393$$ (23*v^5 - 28*v^4 + 7*v^3 + 154*v^2 + 386*v - 267) / 393 $$\beta_{5}$$ $$=$$ $$( -161\nu^{5} + 196\nu^{4} - 49\nu^{3} - 292\nu^{2} - 2702\nu + 1869 ) / 393$$ (-161*v^5 + 196*v^4 - 49*v^3 - 292*v^2 - 2702*v + 1869) / 393
 $$\nu$$ $$=$$ $$( \beta_{4} + \beta_{3} + \beta _1 + 1 ) / 2$$ (b4 + b3 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{5} + 7\beta_{4} ) / 2$$ (b5 + 7*b4) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{5} + 5\beta_{4} - 4\beta_{3} + \beta_{2} + 4\beta _1 - 5 ) / 2$$ (b5 + 5*b4 - 4*b3 + b2 + 4*b1 - 5) / 2 $$\nu^{4}$$ $$=$$ $$-\beta_{3} + 3\beta_{2} - 16$$ -b3 + 3*b2 - 16 $$\nu^{5}$$ $$=$$ $$( -7\beta_{5} - 31\beta_{4} - 18\beta_{3} + 7\beta_{2} - 18\beta _1 - 31 ) / 2$$ (-7*b5 - 31*b4 - 18*b3 + 7*b2 - 18*b1 - 31) / 2

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1249.1
 1.66044 + 1.66044i −1.33641 + 1.33641i 0.675970 + 0.675970i 0.675970 − 0.675970i −1.33641 − 1.33641i 1.66044 − 1.66044i
0 3.32088i 0 0 0 5.02827i 0 −8.02827 0
1249.2 0 2.67282i 0 0 0 1.14399i 0 −4.14399 0
1249.3 0 1.35194i 0 0 0 4.17226i 0 1.17226 0
1249.4 0 1.35194i 0 0 0 4.17226i 0 1.17226 0
1249.5 0 2.67282i 0 0 0 1.14399i 0 −4.14399 0
1249.6 0 3.32088i 0 0 0 5.02827i 0 −8.02827 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1249.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b 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.c.f 6
5.b even 2 1 inner 1300.2.c.f 6
5.c odd 4 1 260.2.a.b 3
5.c odd 4 1 1300.2.a.i 3
15.e even 4 1 2340.2.a.n 3
20.e even 4 1 1040.2.a.o 3
20.e even 4 1 5200.2.a.ci 3
40.i odd 4 1 4160.2.a.bo 3
40.k even 4 1 4160.2.a.br 3
60.l odd 4 1 9360.2.a.da 3
65.f even 4 1 3380.2.f.h 6
65.h odd 4 1 3380.2.a.o 3
65.k even 4 1 3380.2.f.h 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.a.b 3 5.c odd 4 1
1040.2.a.o 3 20.e even 4 1
1300.2.a.i 3 5.c odd 4 1
1300.2.c.f 6 1.a even 1 1 trivial
1300.2.c.f 6 5.b even 2 1 inner
2340.2.a.n 3 15.e even 4 1
3380.2.a.o 3 65.h odd 4 1
3380.2.f.h 6 65.f even 4 1
3380.2.f.h 6 65.k even 4 1
4160.2.a.bo 3 40.i odd 4 1
4160.2.a.br 3 40.k even 4 1
5200.2.a.ci 3 20.e even 4 1
9360.2.a.da 3 60.l odd 4 1

## Hecke kernels

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

 $$T_{3}^{6} + 20T_{3}^{4} + 112T_{3}^{2} + 144$$ T3^6 + 20*T3^4 + 112*T3^2 + 144 $$T_{7}^{6} + 44T_{7}^{4} + 496T_{7}^{2} + 576$$ T7^6 + 44*T7^4 + 496*T7^2 + 576

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} + 20 T^{4} + \cdots + 144$$
$5$ $$T^{6}$$
$7$ $$T^{6} + 44 T^{4} + \cdots + 576$$
$11$ $$(T^{3} - 24 T + 36)^{2}$$
$13$ $$(T^{2} + 1)^{3}$$
$17$ $$T^{6} + 76 T^{4} + \cdots + 576$$
$19$ $$(T^{3} + 8 T^{2} + \cdots - 164)^{2}$$
$23$ $$T^{6} + 52 T^{4} + \cdots + 144$$
$29$ $$(T^{3} + 10 T^{2} + \cdots - 24)^{2}$$
$31$ $$(T^{3} + 12 T^{2} + 24 T + 4)^{2}$$
$37$ $$T^{6} + 92 T^{4} + \cdots + 5184$$
$41$ $$(T^{3} + 2 T^{2} - 36 T + 24)^{2}$$
$43$ $$T^{6} + 20 T^{4} + \cdots + 144$$
$47$ $$T^{6} + 76 T^{4} + \cdots + 576$$
$53$ $$T^{6} + 300 T^{4} + \cdots + 419904$$
$59$ $$(T^{3} - 16 T^{2} + 564)^{2}$$
$61$ $$(T^{3} - 14 T^{2} + 44 T + 8)^{2}$$
$67$ $$T^{6} + 156 T^{4} + \cdots + 23104$$
$71$ $$(T^{3} - 24 T + 36)^{2}$$
$73$ $$T^{6} + 444 T^{4} + \cdots + 3182656$$
$79$ $$(T^{3} + 8 T^{2} - 16 T - 32)^{2}$$
$83$ $$T^{6} + 300 T^{4} + \cdots + 876096$$
$89$ $$(T^{3} - 2 T^{2} + \cdots - 216)^{2}$$
$97$ $$T^{6} + 396 T^{4} + \cdots + 64$$