# Properties

 Label 2925.2.c.p Level $2925$ Weight $2$ Character orbit 2925.c Analytic conductor $23.356$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2925,2,Mod(2224,2925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2925, 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("2925.2224");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2925 = 3^{2} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2925.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: $$23.3562425912$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9x^{2} + 16$$ x^4 + 9*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 585) 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{3} - 3) q^{4} + (3 \beta_{2} + \beta_1) q^{7} + (4 \beta_{2} - \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b3 - 3) * q^4 + (3*b2 + b1) * q^7 + (4*b2 - b1) * q^8 $$q + \beta_1 q^{2} + (\beta_{3} - 3) q^{4} + (3 \beta_{2} + \beta_1) q^{7} + (4 \beta_{2} - \beta_1) q^{8} + \beta_{3} q^{11} - \beta_{2} q^{13} + ( - 2 \beta_{3} - 2) q^{14} + ( - 3 \beta_{3} + 3) q^{16} + (\beta_{2} + \beta_1) q^{17} + ( - 2 \beta_{3} + 2) q^{19} + 4 \beta_{2} q^{22} + (5 \beta_{2} + \beta_1) q^{23} + (\beta_{3} - 1) q^{26} - 2 \beta_{2} q^{28} + ( - 2 \beta_{3} + 4) q^{29} + 6 q^{31} + ( - 4 \beta_{2} + \beta_1) q^{32} - 4 q^{34} + (3 \beta_{2} - 3 \beta_1) q^{37} + ( - 8 \beta_{2} + 2 \beta_1) q^{38} + ( - \beta_{3} + 2) q^{41} + ( - 2 \beta_{2} - 2 \beta_1) q^{43} + ( - 2 \beta_{3} + 4) q^{44} - 4 \beta_{3} q^{46} + ( - 6 \beta_{2} + 2 \beta_1) q^{47} + ( - 5 \beta_{3} - 1) q^{49} + (2 \beta_{2} - \beta_1) q^{52} + (3 \beta_{2} + 3 \beta_1) q^{53} + ( - 2 \beta_{3} - 6) q^{56} + ( - 8 \beta_{2} + 4 \beta_1) q^{58} - 12 q^{59} + (3 \beta_{3} - 2) q^{61} + 6 \beta_1 q^{62} + ( - \beta_{3} - 3) q^{64} + ( - 4 \beta_{2} - 6 \beta_1) q^{67} + (2 \beta_{2} - 2 \beta_1) q^{68} + (\beta_{3} + 12) q^{71} - 6 \beta_{2} q^{73} + ( - 6 \beta_{3} + 18) q^{74} + (6 \beta_{3} - 14) q^{76} + (7 \beta_{2} + 3 \beta_1) q^{77} + 3 \beta_{3} q^{79} + ( - 4 \beta_{2} + 2 \beta_1) q^{82} + (4 \beta_{2} + 8 \beta_1) q^{83} + 8 q^{86} + 4 \beta_1 q^{88} + (3 \beta_{3} - 6) q^{89} + (\beta_{3} + 2) q^{91} + ( - 6 \beta_{2} + 2 \beta_1) q^{92} + (8 \beta_{3} - 16) q^{94} + (\beta_{2} + 7 \beta_1) q^{97} + ( - 20 \beta_{2} - \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b3 - 3) * q^4 + (3*b2 + b1) * q^7 + (4*b2 - b1) * q^8 + b3 * q^11 - b2 * q^13 + (-2*b3 - 2) * q^14 + (-3*b3 + 3) * q^16 + (b2 + b1) * q^17 + (-2*b3 + 2) * q^19 + 4*b2 * q^22 + (5*b2 + b1) * q^23 + (b3 - 1) * q^26 - 2*b2 * q^28 + (-2*b3 + 4) * q^29 + 6 * q^31 + (-4*b2 + b1) * q^32 - 4 * q^34 + (3*b2 - 3*b1) * q^37 + (-8*b2 + 2*b1) * q^38 + (-b3 + 2) * q^41 + (-2*b2 - 2*b1) * q^43 + (-2*b3 + 4) * q^44 - 4*b3 * q^46 + (-6*b2 + 2*b1) * q^47 + (-5*b3 - 1) * q^49 + (2*b2 - b1) * q^52 + (3*b2 + 3*b1) * q^53 + (-2*b3 - 6) * q^56 + (-8*b2 + 4*b1) * q^58 - 12 * q^59 + (3*b3 - 2) * q^61 + 6*b1 * q^62 + (-b3 - 3) * q^64 + (-4*b2 - 6*b1) * q^67 + (2*b2 - 2*b1) * q^68 + (b3 + 12) * q^71 - 6*b2 * q^73 + (-6*b3 + 18) * q^74 + (6*b3 - 14) * q^76 + (7*b2 + 3*b1) * q^77 + 3*b3 * q^79 + (-4*b2 + 2*b1) * q^82 + (4*b2 + 8*b1) * q^83 + 8 * q^86 + 4*b1 * q^88 + (3*b3 - 6) * q^89 + (b3 + 2) * q^91 + (-6*b2 + 2*b1) * q^92 + (8*b3 - 16) * q^94 + (b2 + 7*b1) * q^97 + (-20*b2 - b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 10 q^{4}+O(q^{10})$$ 4 * q - 10 * q^4 $$4 q - 10 q^{4} + 2 q^{11} - 12 q^{14} + 6 q^{16} + 4 q^{19} - 2 q^{26} + 12 q^{29} + 24 q^{31} - 16 q^{34} + 6 q^{41} + 12 q^{44} - 8 q^{46} - 14 q^{49} - 28 q^{56} - 48 q^{59} - 2 q^{61} - 14 q^{64} + 50 q^{71} + 60 q^{74} - 44 q^{76} + 6 q^{79} + 32 q^{86} - 18 q^{89} + 10 q^{91} - 48 q^{94}+O(q^{100})$$ 4 * q - 10 * q^4 + 2 * q^11 - 12 * q^14 + 6 * q^16 + 4 * q^19 - 2 * q^26 + 12 * q^29 + 24 * q^31 - 16 * q^34 + 6 * q^41 + 12 * q^44 - 8 * q^46 - 14 * q^49 - 28 * q^56 - 48 * q^59 - 2 * q^61 - 14 * q^64 + 50 * q^71 + 60 * q^74 - 44 * q^76 + 6 * q^79 + 32 * q^86 - 18 * q^89 + 10 * q^91 - 48 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9x^{2} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 5\nu ) / 4$$ (v^3 + 5*v) / 4 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 5$$ v^2 + 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 5$$ b3 - 5 $$\nu^{3}$$ $$=$$ $$4\beta_{2} - 5\beta_1$$ 4*b2 - 5*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2925\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$352$$ $$2251$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2224.1
 − 2.56155i − 1.56155i 1.56155i 2.56155i
2.56155i 0 −4.56155 0 0 0.438447i 6.56155i 0 0
2224.2 1.56155i 0 −0.438447 0 0 4.56155i 2.43845i 0 0
2224.3 1.56155i 0 −0.438447 0 0 4.56155i 2.43845i 0 0
2224.4 2.56155i 0 −4.56155 0 0 0.438447i 6.56155i 0 0
 $$n$$: e.g. 2-40 or 990-1000 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 2925.2.c.p 4
3.b odd 2 1 2925.2.c.o 4
5.b even 2 1 inner 2925.2.c.p 4
5.c odd 4 1 585.2.a.l yes 2
5.c odd 4 1 2925.2.a.x 2
15.d odd 2 1 2925.2.c.o 4
15.e even 4 1 585.2.a.j 2
15.e even 4 1 2925.2.a.bc 2
20.e even 4 1 9360.2.a.cw 2
60.l odd 4 1 9360.2.a.cl 2
65.h odd 4 1 7605.2.a.bd 2
195.s even 4 1 7605.2.a.bi 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.a.j 2 15.e even 4 1
585.2.a.l yes 2 5.c odd 4 1
2925.2.a.x 2 5.c odd 4 1
2925.2.a.bc 2 15.e even 4 1
2925.2.c.o 4 3.b odd 2 1
2925.2.c.o 4 15.d odd 2 1
2925.2.c.p 4 1.a even 1 1 trivial
2925.2.c.p 4 5.b even 2 1 inner
7605.2.a.bd 2 65.h odd 4 1
7605.2.a.bi 2 195.s even 4 1
9360.2.a.cl 2 60.l odd 4 1
9360.2.a.cw 2 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2925, [\chi])$$:

 $$T_{2}^{4} + 9T_{2}^{2} + 16$$ T2^4 + 9*T2^2 + 16 $$T_{7}^{4} + 21T_{7}^{2} + 4$$ T7^4 + 21*T7^2 + 4 $$T_{11}^{2} - T_{11} - 4$$ T11^2 - T11 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 9T^{2} + 16$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 21T^{2} + 4$$
$11$ $$(T^{2} - T - 4)^{2}$$
$13$ $$(T^{2} + 1)^{2}$$
$17$ $$T^{4} + 9T^{2} + 16$$
$19$ $$(T^{2} - 2 T - 16)^{2}$$
$23$ $$T^{4} + 49T^{2} + 256$$
$29$ $$(T^{2} - 6 T - 8)^{2}$$
$31$ $$(T - 6)^{4}$$
$37$ $$T^{4} + 117T^{2} + 324$$
$41$ $$(T^{2} - 3 T - 2)^{2}$$
$43$ $$T^{4} + 36T^{2} + 256$$
$47$ $$T^{4} + 132T^{2} + 1024$$
$53$ $$T^{4} + 81T^{2} + 1296$$
$59$ $$(T + 12)^{4}$$
$61$ $$(T^{2} + T - 38)^{2}$$
$67$ $$T^{4} + 308 T^{2} + 23104$$
$71$ $$(T^{2} - 25 T + 152)^{2}$$
$73$ $$(T^{2} + 36)^{2}$$
$79$ $$(T^{2} - 3 T - 36)^{2}$$
$83$ $$(T^{2} + 272)^{2}$$
$89$ $$(T^{2} + 9 T - 18)^{2}$$
$97$ $$T^{4} + 429 T^{2} + 40804$$