# Properties

 Label 2475.2.c.o Level $2475$ Weight $2$ Character orbit 2475.c Analytic conductor $19.763$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(199,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2475.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: $$19.7629745003$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 825) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + \beta_1) q^{2} + ( - 2 \beta_{3} - 1) q^{4} + ( - \beta_{2} + \beta_1) q^{7} + ( - \beta_{2} - 3 \beta_1) q^{8}+O(q^{10})$$ q + (b2 + b1) * q^2 + (-2*b3 - 1) * q^4 + (-b2 + b1) * q^7 + (-b2 - 3*b1) * q^8 $$q + (\beta_{2} + \beta_1) q^{2} + ( - 2 \beta_{3} - 1) q^{4} + ( - \beta_{2} + \beta_1) q^{7} + ( - \beta_{2} - 3 \beta_1) q^{8} + q^{11} - 2 \beta_{2} q^{13} + q^{14} + 3 q^{16} + (\beta_{2} + \beta_1) q^{17} + ( - \beta_{3} - 5) q^{19} + (\beta_{2} + \beta_1) q^{22} - \beta_1 q^{23} + (2 \beta_{3} + 4) q^{26} + ( - \beta_{2} + 3 \beta_1) q^{28} + ( - 2 \beta_{3} + 4) q^{29} - 6 \beta_{3} q^{31} + (\beta_{2} - 3 \beta_1) q^{32} + ( - 2 \beta_{3} - 3) q^{34} + ( - 2 \beta_{2} + 3 \beta_1) q^{37} + ( - 6 \beta_{2} - 7 \beta_1) q^{38} + (7 \beta_{3} + 1) q^{41} + ( - 4 \beta_{2} - 6 \beta_1) q^{43} + ( - 2 \beta_{3} - 1) q^{44} + (\beta_{3} + 1) q^{46} + ( - 6 \beta_{2} + \beta_1) q^{47} + (2 \beta_{3} + 4) q^{49} + (2 \beta_{2} + 8 \beta_1) q^{52} + ( - 4 \beta_{2} - 2 \beta_1) q^{53} + ( - 2 \beta_{3} + 1) q^{56} + 2 \beta_{2} q^{58} + 11 q^{59} + (2 \beta_{3} + 6) q^{61} + ( - 6 \beta_{2} - 12 \beta_1) q^{62} + (2 \beta_{3} + 7) q^{64} + ( - 4 \beta_{2} + 6 \beta_1) q^{67} + ( - 3 \beta_{2} - 5 \beta_1) q^{68} + ( - 2 \beta_{3} - 5) q^{71} + ( - 2 \beta_{2} - 6 \beta_1) q^{73} + ( - \beta_{3} + 1) q^{74} + (11 \beta_{3} + 9) q^{76} + ( - \beta_{2} + \beta_1) q^{77} + ( - 3 \beta_{3} - 9) q^{79} + (8 \beta_{2} + 15 \beta_1) q^{82} + (6 \beta_{2} - 4 \beta_1) q^{83} + (10 \beta_{3} + 14) q^{86} + ( - \beta_{2} - 3 \beta_1) q^{88} + (4 \beta_{3} - 2) q^{89} + (2 \beta_{3} - 4) q^{91} + (2 \beta_{2} + \beta_1) q^{92} + (5 \beta_{3} + 11) q^{94} + ( - 2 \beta_{2} - 3 \beta_1) q^{97} + (6 \beta_{2} + 8 \beta_1) q^{98}+O(q^{100})$$ q + (b2 + b1) * q^2 + (-2*b3 - 1) * q^4 + (-b2 + b1) * q^7 + (-b2 - 3*b1) * q^8 + q^11 - 2*b2 * q^13 + q^14 + 3 * q^16 + (b2 + b1) * q^17 + (-b3 - 5) * q^19 + (b2 + b1) * q^22 - b1 * q^23 + (2*b3 + 4) * q^26 + (-b2 + 3*b1) * q^28 + (-2*b3 + 4) * q^29 - 6*b3 * q^31 + (b2 - 3*b1) * q^32 + (-2*b3 - 3) * q^34 + (-2*b2 + 3*b1) * q^37 + (-6*b2 - 7*b1) * q^38 + (7*b3 + 1) * q^41 + (-4*b2 - 6*b1) * q^43 + (-2*b3 - 1) * q^44 + (b3 + 1) * q^46 + (-6*b2 + b1) * q^47 + (2*b3 + 4) * q^49 + (2*b2 + 8*b1) * q^52 + (-4*b2 - 2*b1) * q^53 + (-2*b3 + 1) * q^56 + 2*b2 * q^58 + 11 * q^59 + (2*b3 + 6) * q^61 + (-6*b2 - 12*b1) * q^62 + (2*b3 + 7) * q^64 + (-4*b2 + 6*b1) * q^67 + (-3*b2 - 5*b1) * q^68 + (-2*b3 - 5) * q^71 + (-2*b2 - 6*b1) * q^73 + (-b3 + 1) * q^74 + (11*b3 + 9) * q^76 + (-b2 + b1) * q^77 + (-3*b3 - 9) * q^79 + (8*b2 + 15*b1) * q^82 + (6*b2 - 4*b1) * q^83 + (10*b3 + 14) * q^86 + (-b2 - 3*b1) * q^88 + (4*b3 - 2) * q^89 + (2*b3 - 4) * q^91 + (2*b2 + b1) * q^92 + (5*b3 + 11) * q^94 + (-2*b2 - 3*b1) * q^97 + (6*b2 + 8*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4}+O(q^{10})$$ 4 * q - 4 * q^4 $$4 q - 4 q^{4} + 4 q^{11} + 4 q^{14} + 12 q^{16} - 20 q^{19} + 16 q^{26} + 16 q^{29} - 12 q^{34} + 4 q^{41} - 4 q^{44} + 4 q^{46} + 16 q^{49} + 4 q^{56} + 44 q^{59} + 24 q^{61} + 28 q^{64} - 20 q^{71} + 4 q^{74} + 36 q^{76} - 36 q^{79} + 56 q^{86} - 8 q^{89} - 16 q^{91} + 44 q^{94}+O(q^{100})$$ 4 * q - 4 * q^4 + 4 * q^11 + 4 * q^14 + 12 * q^16 - 20 * q^19 + 16 * q^26 + 16 * q^29 - 12 * q^34 + 4 * q^41 - 4 * q^44 + 4 * q^46 + 16 * q^49 + 4 * q^56 + 44 * q^59 + 24 * q^61 + 28 * q^64 - 20 * q^71 + 4 * q^74 + 36 * q^76 - 36 * q^79 + 56 * q^86 - 8 * q^89 - 16 * q^91 + 44 * q^94

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{8}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}$$ -v^3 + v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\zeta_{8}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 2$$ (-b3 + b2) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times$$.

 $$n$$ $$551$$ $$2026$$ $$2377$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i
2.41421i 0 −3.82843 0 0 0.414214i 4.41421i 0 0
199.2 0.414214i 0 1.82843 0 0 2.41421i 1.58579i 0 0
199.3 0.414214i 0 1.82843 0 0 2.41421i 1.58579i 0 0
199.4 2.41421i 0 −3.82843 0 0 0.414214i 4.41421i 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 2475.2.c.o 4
3.b odd 2 1 825.2.c.d 4
5.b even 2 1 inner 2475.2.c.o 4
5.c odd 4 1 2475.2.a.l 2
5.c odd 4 1 2475.2.a.w 2
15.d odd 2 1 825.2.c.d 4
15.e even 4 1 825.2.a.d 2
15.e even 4 1 825.2.a.f yes 2
165.l odd 4 1 9075.2.a.w 2
165.l odd 4 1 9075.2.a.ca 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.a.d 2 15.e even 4 1
825.2.a.f yes 2 15.e even 4 1
825.2.c.d 4 3.b odd 2 1
825.2.c.d 4 15.d odd 2 1
2475.2.a.l 2 5.c odd 4 1
2475.2.a.w 2 5.c odd 4 1
2475.2.c.o 4 1.a even 1 1 trivial
2475.2.c.o 4 5.b even 2 1 inner
9075.2.a.w 2 165.l odd 4 1
9075.2.a.ca 2 165.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 $$S_{2}^{\mathrm{new}}(2475, [\chi])$$:

 $$T_{2}^{4} + 6T_{2}^{2} + 1$$ T2^4 + 6*T2^2 + 1 $$T_{7}^{4} + 6T_{7}^{2} + 1$$ T7^4 + 6*T7^2 + 1 $$T_{29}^{2} - 8T_{29} + 8$$ T29^2 - 8*T29 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 6T^{2} + 1$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 6T^{2} + 1$$
$11$ $$(T - 1)^{4}$$
$13$ $$(T^{2} + 8)^{2}$$
$17$ $$T^{4} + 6T^{2} + 1$$
$19$ $$(T^{2} + 10 T + 23)^{2}$$
$23$ $$(T^{2} + 1)^{2}$$
$29$ $$(T^{2} - 8 T + 8)^{2}$$
$31$ $$(T^{2} - 72)^{2}$$
$37$ $$T^{4} + 34T^{2} + 1$$
$41$ $$(T^{2} - 2 T - 97)^{2}$$
$43$ $$T^{4} + 136T^{2} + 16$$
$47$ $$T^{4} + 146T^{2} + 5041$$
$53$ $$T^{4} + 72T^{2} + 784$$
$59$ $$(T - 11)^{4}$$
$61$ $$(T^{2} - 12 T + 28)^{2}$$
$67$ $$T^{4} + 136T^{2} + 16$$
$71$ $$(T^{2} + 10 T + 17)^{2}$$
$73$ $$T^{4} + 88T^{2} + 784$$
$79$ $$(T^{2} + 18 T + 63)^{2}$$
$83$ $$T^{4} + 176T^{2} + 3136$$
$89$ $$(T^{2} + 4 T - 28)^{2}$$
$97$ $$T^{4} + 34T^{2} + 1$$