# Properties

 Label 2900.2.c.f Level $2900$ Weight $2$ Character orbit 2900.c Analytic conductor $23.157$ 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] = [2900,2,Mod(349,2900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2900, 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("2900.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2900 = 2^{2} \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2900.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: $$23.1566165862$$ 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: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 580) 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} + (\beta_{4} + \beta_{2}) q^{7} + (\beta_1 - 4) q^{9}+O(q^{10})$$ q - b2 * q^3 + (b4 + b2) * q^7 + (b1 - 4) * q^9 $$q - \beta_{2} q^{3} + (\beta_{4} + \beta_{2}) q^{7} + (\beta_1 - 4) q^{9} + ( - \beta_{3} + 3) q^{11} + \beta_{5} q^{13} + ( - \beta_{5} - \beta_{4} - \beta_{2}) q^{17} + ( - \beta_{3} - \beta_1 - 2) q^{19} + (2 \beta_{3} - \beta_1 + 5) q^{21} + (\beta_{5} - \beta_{2}) q^{23} + (2 \beta_{5} + 2 \beta_{2}) q^{27} + q^{29} + ( - \beta_{3} - \beta_1 + 2) q^{31} + ( - \beta_{5} + 3 \beta_{4} - 2 \beta_{2}) q^{33} + (\beta_{5} + \beta_{4} - 3 \beta_{2}) q^{37} + ( - 2 \beta_{3} - 2 \beta_1 + 4) q^{39} + (\beta_1 + 3) q^{41} + ( - 2 \beta_{5} + 2 \beta_{4} + \beta_{2}) q^{43} + (2 \beta_{4} + \beta_{2}) q^{47} + ( - 4 \beta_{3} + \beta_1) q^{49} + (3 \beta_1 - 9) q^{51} + ( - \beta_{5} + 2 \beta_{4} + 2 \beta_{2}) q^{53} + ( - 3 \beta_{5} + 3 \beta_{4} + 2 \beta_{2}) q^{57} + ( - 2 \beta_{3} + 3 \beta_1 - 3) q^{59} + ( - 2 \beta_{3} + \beta_1 - 7) q^{61} + ( - 3 \beta_{4} - 5 \beta_{2}) q^{63} + ( - \beta_{5} + 2 \beta_{4} + \beta_{2}) q^{67} + ( - 2 \beta_{3} - \beta_1 - 3) q^{69} + (2 \beta_{3} - \beta_1 - 3) q^{71} + (2 \beta_{5} - \beta_{2}) q^{73} + (\beta_{5} - \beta_{4}) q^{77} + ( - \beta_{3} - 2 \beta_1 + 1) q^{79} + ( - 4 \beta_{3} - 3 \beta_1 + 10) q^{81} + (3 \beta_{4} + 3 \beta_{2}) q^{83} - \beta_{2} q^{87} + (2 \beta_{3} - 2 \beta_1 - 6) q^{89} + (2 \beta_{3} - 2) q^{91} + ( - 3 \beta_{5} + 3 \beta_{4} - 2 \beta_{2}) q^{93} + (6 \beta_{4} - \beta_{2}) q^{97} + (5 \beta_{3} + 4 \beta_1 - 15) q^{99}+O(q^{100})$$ q - b2 * q^3 + (b4 + b2) * q^7 + (b1 - 4) * q^9 + (-b3 + 3) * q^11 + b5 * q^13 + (-b5 - b4 - b2) * q^17 + (-b3 - b1 - 2) * q^19 + (2*b3 - b1 + 5) * q^21 + (b5 - b2) * q^23 + (2*b5 + 2*b2) * q^27 + q^29 + (-b3 - b1 + 2) * q^31 + (-b5 + 3*b4 - 2*b2) * q^33 + (b5 + b4 - 3*b2) * q^37 + (-2*b3 - 2*b1 + 4) * q^39 + (b1 + 3) * q^41 + (-2*b5 + 2*b4 + b2) * q^43 + (2*b4 + b2) * q^47 + (-4*b3 + b1) * q^49 + (3*b1 - 9) * q^51 + (-b5 + 2*b4 + 2*b2) * q^53 + (-3*b5 + 3*b4 + 2*b2) * q^57 + (-2*b3 + 3*b1 - 3) * q^59 + (-2*b3 + b1 - 7) * q^61 + (-3*b4 - 5*b2) * q^63 + (-b5 + 2*b4 + b2) * q^67 + (-2*b3 - b1 - 3) * q^69 + (2*b3 - b1 - 3) * q^71 + (2*b5 - b2) * q^73 + (b5 - b4) * q^77 + (-b3 - 2*b1 + 1) * q^79 + (-4*b3 - 3*b1 + 10) * q^81 + (3*b4 + 3*b2) * q^83 - b2 * q^87 + (2*b3 - 2*b1 - 6) * q^89 + (2*b3 - 2) * q^91 + (-3*b5 + 3*b4 - 2*b2) * q^93 + (6*b4 - b2) * q^97 + (5*b3 + 4*b1 - 15) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 22 q^{9}+O(q^{10})$$ 6 * q - 22 * q^9 $$6 q - 22 q^{9} + 16 q^{11} - 16 q^{19} + 32 q^{21} + 6 q^{29} + 8 q^{31} + 16 q^{39} + 20 q^{41} - 6 q^{49} - 48 q^{51} - 16 q^{59} - 44 q^{61} - 24 q^{69} - 16 q^{71} + 46 q^{81} - 36 q^{89} - 8 q^{91} - 72 q^{99}+O(q^{100})$$ 6 * q - 22 * q^9 + 16 * q^11 - 16 * q^19 + 32 * q^21 + 6 * q^29 + 8 * q^31 + 16 * q^39 + 20 * q^41 - 6 * q^49 - 48 * q^51 - 16 * q^59 - 44 * q^61 - 24 * q^69 - 16 * q^71 + 46 * q^81 - 36 * q^89 - 8 * q^91 - 72 * q^99

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

 $$\beta_{1}$$ $$=$$ $$( -2\nu^{5} + 48\nu^{4} - 12\nu^{3} - 2\nu^{2} + 12\nu + 701 ) / 131$$ (-2*v^5 + 48*v^4 - 12*v^3 - 2*v^2 + 12*v + 701) / 131 $$\beta_{2}$$ $$=$$ $$( 6\nu^{5} - 13\nu^{4} + 36\nu^{3} + 6\nu^{2} + 226\nu - 138 ) / 131$$ (6*v^5 - 13*v^4 + 36*v^3 + 6*v^2 + 226*v - 138) / 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 $$\beta_{4}$$ $$=$$ $$( -46\nu^{5} + 56\nu^{4} - 14\nu^{3} - 308\nu^{2} - 772\nu + 534 ) / 393$$ (-46*v^5 + 56*v^4 - 14*v^3 - 308*v^2 - 772*v + 534) / 393 $$\beta_{5}$$ $$=$$ $$( -46\nu^{5} + 56\nu^{4} - 14\nu^{3} - 46\nu^{2} - 772\nu + 534 ) / 131$$ (-46*v^5 + 56*v^4 - 14*v^3 - 46*v^2 - 772*v + 534) / 131
 $$\nu$$ $$=$$ $$( -\beta_{3} + \beta_{2} + 1 ) / 2$$ (-b3 + b2 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{5} - 3\beta_{4} ) / 2$$ (b5 - 3*b4) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{5} + 4\beta_{3} + 4\beta_{2} + \beta _1 - 5 ) / 2$$ (b5 + 4*b3 + 4*b2 + b1 - 5) / 2 $$\nu^{4}$$ $$=$$ $$\beta_{3} + 3\beta _1 - 16$$ b3 + 3*b1 - 16 $$\nu^{5}$$ $$=$$ $$( -7\beta_{5} + 3\beta_{4} + 18\beta_{3} - 18\beta_{2} + 7\beta _1 - 31 ) / 2$$ (-7*b5 + 3*b4 + 18*b3 - 18*b2 + 7*b1 - 31) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2900\mathbb{Z}\right)^\times$$.

 $$n$$ $$901$$ $$1277$$ $$1451$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
349.1
 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 1.32088i 0 −8.02827 0
349.2 0 2.67282i 0 0 0 4.67282i 0 −4.14399 0
349.3 0 1.35194i 0 0 0 0.648061i 0 1.17226 0
349.4 0 1.35194i 0 0 0 0.648061i 0 1.17226 0
349.5 0 2.67282i 0 0 0 4.67282i 0 −4.14399 0
349.6 0 3.32088i 0 0 0 1.32088i 0 −8.02827 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 349.6 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 2900.2.c.f 6
5.b even 2 1 inner 2900.2.c.f 6
5.c odd 4 1 580.2.a.c 3
5.c odd 4 1 2900.2.a.g 3
15.e even 4 1 5220.2.a.x 3
20.e even 4 1 2320.2.a.m 3
40.i odd 4 1 9280.2.a.bk 3
40.k even 4 1 9280.2.a.bw 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
580.2.a.c 3 5.c odd 4 1
2320.2.a.m 3 20.e even 4 1
2900.2.a.g 3 5.c odd 4 1
2900.2.c.f 6 1.a even 1 1 trivial
2900.2.c.f 6 5.b even 2 1 inner
5220.2.a.x 3 15.e even 4 1
9280.2.a.bk 3 40.i odd 4 1
9280.2.a.bw 3 40.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 $$S_{2}^{\mathrm{new}}(2900, [\chi])$$:

 $$T_{3}^{6} + 20T_{3}^{4} + 112T_{3}^{2} + 144$$ T3^6 + 20*T3^4 + 112*T3^2 + 144 $$T_{7}^{6} + 24T_{7}^{4} + 48T_{7}^{2} + 16$$ T7^6 + 24*T7^4 + 48*T7^2 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} + 20 T^{4} + \cdots + 144$$
$5$ $$T^{6}$$
$7$ $$T^{6} + 24 T^{4} + \cdots + 16$$
$11$ $$(T^{3} - 8 T^{2} + 12 T + 12)^{2}$$
$13$ $$T^{6} + 44 T^{4} + \cdots + 576$$
$17$ $$T^{6} + 76 T^{4} + \cdots + 11664$$
$19$ $$(T^{3} + 8 T^{2} + \cdots - 164)^{2}$$
$23$ $$T^{6} + 48 T^{4} + \cdots + 1296$$
$29$ $$(T - 1)^{6}$$
$31$ $$(T^{3} - 4 T^{2} - 32 T - 36)^{2}$$
$37$ $$T^{6} + 204 T^{4} + \cdots + 258064$$
$41$ $$(T^{3} - 10 T^{2} + \cdots + 24)^{2}$$
$43$ $$T^{6} + 228 T^{4} + \cdots + 300304$$
$47$ $$T^{6} + 52 T^{4} + \cdots + 144$$
$53$ $$T^{6} + 124 T^{4} + \cdots + 5184$$
$59$ $$(T^{3} + 8 T^{2} + \cdots - 1488)^{2}$$
$61$ $$(T^{3} + 22 T^{2} + \cdots + 104)^{2}$$
$67$ $$T^{6} + 96 T^{4} + \cdots + 4624$$
$71$ $$(T^{3} + 8 T^{2} + \cdots - 144)^{2}$$
$73$ $$T^{6} + 164 T^{4} + \cdots + 104976$$
$79$ $$(T^{3} - 108 T - 244)^{2}$$
$83$ $$T^{6} + 216 T^{4} + \cdots + 11664$$
$89$ $$(T^{3} + 18 T^{2} + \cdots - 72)^{2}$$
$97$ $$T^{6} + 500 T^{4} + \cdots + 3640464$$