# Properties

 Label 2475.2.c.l 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 55) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{3} - 1) q^{4} + (\beta_{2} + \beta_1) q^{7} + ( - \beta_{2} - 2 \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (-b3 - 1) * q^4 + (b2 + b1) * q^7 + (-b2 - 2*b1) * q^8 $$q + \beta_1 q^{2} + ( - \beta_{3} - 1) q^{4} + (\beta_{2} + \beta_1) q^{7} + ( - \beta_{2} - 2 \beta_1) q^{8} - q^{11} + ( - 3 \beta_{2} - \beta_1) q^{13} + ( - \beta_{3} - 2) q^{14} + 3 q^{16} + (\beta_{2} + 3 \beta_1) q^{17} - \beta_1 q^{22} + ( - \beta_{2} + \beta_1) q^{23} + \beta_{3} q^{26} + (\beta_{2} - 3 \beta_1) q^{28} + ( - 2 \beta_{3} + 2) q^{29} + ( - 2 \beta_{2} - \beta_1) q^{32} + ( - 3 \beta_{3} - 8) q^{34} + ( - \beta_{2} + 3 \beta_1) q^{37} - 6 q^{41} + ( - 3 \beta_{2} - 3 \beta_1) q^{43} + (\beta_{3} + 1) q^{44} + ( - \beta_{3} - 4) q^{46} + ( - \beta_{2} + \beta_1) q^{47} + 3 q^{49} + ( - 5 \beta_{2} + \beta_1) q^{52} + ( - \beta_{2} - 5 \beta_1) q^{53} + (\beta_{3} + 6) q^{56} + ( - 2 \beta_{2} - 4 \beta_1) q^{58} + (2 \beta_{3} - 4) q^{59} + ( - 4 \beta_{3} + 2) q^{61} + (\beta_{3} + 7) q^{64} + (\beta_{2} - 5 \beta_1) q^{67} + ( - \beta_{2} - 11 \beta_1) q^{68} - 4 \beta_{3} q^{71} + ( - 3 \beta_{2} - \beta_1) q^{73} + ( - 3 \beta_{3} - 10) q^{74} + ( - \beta_{2} - \beta_1) q^{77} - 4 q^{79} - 6 \beta_1 q^{82} + (3 \beta_{2} + 3 \beta_1) q^{83} + (3 \beta_{3} + 6) q^{86} + (\beta_{2} + 2 \beta_1) q^{88} + ( - 4 \beta_{3} - 2) q^{89} + ( - 2 \beta_{3} + 8) q^{91} + ( - 3 \beta_{2} - 5 \beta_1) q^{92} + ( - \beta_{3} - 4) q^{94} + (3 \beta_{2} - \beta_1) q^{97} + 3 \beta_1 q^{98}+O(q^{100})$$ q + b1 * q^2 + (-b3 - 1) * q^4 + (b2 + b1) * q^7 + (-b2 - 2*b1) * q^8 - q^11 + (-3*b2 - b1) * q^13 + (-b3 - 2) * q^14 + 3 * q^16 + (b2 + 3*b1) * q^17 - b1 * q^22 + (-b2 + b1) * q^23 + b3 * q^26 + (b2 - 3*b1) * q^28 + (-2*b3 + 2) * q^29 + (-2*b2 - b1) * q^32 + (-3*b3 - 8) * q^34 + (-b2 + 3*b1) * q^37 - 6 * q^41 + (-3*b2 - 3*b1) * q^43 + (b3 + 1) * q^44 + (-b3 - 4) * q^46 + (-b2 + b1) * q^47 + 3 * q^49 + (-5*b2 + b1) * q^52 + (-b2 - 5*b1) * q^53 + (b3 + 6) * q^56 + (-2*b2 - 4*b1) * q^58 + (2*b3 - 4) * q^59 + (-4*b3 + 2) * q^61 + (b3 + 7) * q^64 + (b2 - 5*b1) * q^67 + (-b2 - 11*b1) * q^68 - 4*b3 * q^71 + (-3*b2 - b1) * q^73 + (-3*b3 - 10) * q^74 + (-b2 - b1) * q^77 - 4 * q^79 - 6*b1 * q^82 + (3*b2 + 3*b1) * q^83 + (3*b3 + 6) * q^86 + (b2 + 2*b1) * q^88 + (-4*b3 - 2) * q^89 + (-2*b3 + 8) * q^91 + (-3*b2 - 5*b1) * q^92 + (-b3 - 4) * q^94 + (3*b2 - b1) * q^97 + 3*b1 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4}+O(q^{10})$$ 4 * q - 4 * q^4 $$4 q - 4 q^{4} - 4 q^{11} - 8 q^{14} + 12 q^{16} + 8 q^{29} - 32 q^{34} - 24 q^{41} + 4 q^{44} - 16 q^{46} + 12 q^{49} + 24 q^{56} - 16 q^{59} + 8 q^{61} + 28 q^{64} - 40 q^{74} - 16 q^{79} + 24 q^{86} - 8 q^{89} + 32 q^{91} - 16 q^{94}+O(q^{100})$$ 4 * q - 4 * q^4 - 4 * q^11 - 8 * q^14 + 12 * q^16 + 8 * q^29 - 32 * q^34 - 24 * q^41 + 4 * q^44 - 16 * q^46 + 12 * q^49 + 24 * q^56 - 16 * q^59 + 8 * q^61 + 28 * q^64 - 40 * q^74 - 16 * q^79 + 24 * q^86 - 8 * q^89 + 32 * q^91 - 16 * q^94

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}$$ v^3 + v^2 + v $$\beta_{2}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}^{2} - \zeta_{8}$$ -v^3 + v^2 - v $$\beta_{3}$$ $$=$$ $$-2\zeta_{8}^{3} + 2\zeta_{8}$$ -2*v^3 + 2*v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} - \beta_{2} + \beta_1 ) / 4$$ (b3 - b2 + b1) / 4 $$\zeta_{8}^{2}$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} - \beta_{2} + \beta_1 ) / 4$$ (-b3 - b2 + b1) / 4

## 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 2.00000i 4.41421i 0 0
199.2 0.414214i 0 1.82843 0 0 2.00000i 1.58579i 0 0
199.3 0.414214i 0 1.82843 0 0 2.00000i 1.58579i 0 0
199.4 2.41421i 0 −3.82843 0 0 2.00000i 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.l 4
3.b odd 2 1 275.2.b.d 4
5.b even 2 1 inner 2475.2.c.l 4
5.c odd 4 1 495.2.a.b 2
5.c odd 4 1 2475.2.a.x 2
12.b even 2 1 4400.2.b.q 4
15.d odd 2 1 275.2.b.d 4
15.e even 4 1 55.2.a.b 2
15.e even 4 1 275.2.a.c 2
20.e even 4 1 7920.2.a.ch 2
55.e even 4 1 5445.2.a.y 2
60.h even 2 1 4400.2.b.q 4
60.l odd 4 1 880.2.a.m 2
60.l odd 4 1 4400.2.a.bn 2
105.k odd 4 1 2695.2.a.f 2
120.q odd 4 1 3520.2.a.bo 2
120.w even 4 1 3520.2.a.bn 2
165.l odd 4 1 605.2.a.d 2
165.l odd 4 1 3025.2.a.o 2
165.u odd 20 4 605.2.g.l 8
165.v even 20 4 605.2.g.f 8
195.s even 4 1 9295.2.a.g 2
660.q even 4 1 9680.2.a.bn 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.b 2 15.e even 4 1
275.2.a.c 2 15.e even 4 1
275.2.b.d 4 3.b odd 2 1
275.2.b.d 4 15.d odd 2 1
495.2.a.b 2 5.c odd 4 1
605.2.a.d 2 165.l odd 4 1
605.2.g.f 8 165.v even 20 4
605.2.g.l 8 165.u odd 20 4
880.2.a.m 2 60.l odd 4 1
2475.2.a.x 2 5.c odd 4 1
2475.2.c.l 4 1.a even 1 1 trivial
2475.2.c.l 4 5.b even 2 1 inner
2695.2.a.f 2 105.k odd 4 1
3025.2.a.o 2 165.l odd 4 1
3520.2.a.bn 2 120.w even 4 1
3520.2.a.bo 2 120.q odd 4 1
4400.2.a.bn 2 60.l odd 4 1
4400.2.b.q 4 12.b even 2 1
4400.2.b.q 4 60.h even 2 1
5445.2.a.y 2 55.e even 4 1
7920.2.a.ch 2 20.e even 4 1
9295.2.a.g 2 195.s even 4 1
9680.2.a.bn 2 660.q even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2475, [\chi])$$:

 $$T_{2}^{4} + 6T_{2}^{2} + 1$$ T2^4 + 6*T2^2 + 1 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{29}^{2} - 4T_{29} - 28$$ T29^2 - 4*T29 - 28

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 6T^{2} + 1$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 4)^{2}$$
$11$ $$(T + 1)^{4}$$
$13$ $$T^{4} + 48T^{2} + 64$$
$17$ $$T^{4} + 48T^{2} + 64$$
$19$ $$T^{4}$$
$23$ $$(T^{2} + 8)^{2}$$
$29$ $$(T^{2} - 4 T - 28)^{2}$$
$31$ $$T^{4}$$
$37$ $$T^{4} + 72T^{2} + 784$$
$41$ $$(T + 6)^{4}$$
$43$ $$(T^{2} + 36)^{2}$$
$47$ $$(T^{2} + 8)^{2}$$
$53$ $$T^{4} + 136T^{2} + 16$$
$59$ $$(T^{2} + 8 T - 16)^{2}$$
$61$ $$(T^{2} - 4 T - 124)^{2}$$
$67$ $$T^{4} + 176T^{2} + 3136$$
$71$ $$(T^{2} - 128)^{2}$$
$73$ $$T^{4} + 48T^{2} + 64$$
$79$ $$(T + 4)^{4}$$
$83$ $$(T^{2} + 36)^{2}$$
$89$ $$(T^{2} + 4 T - 124)^{2}$$
$97$ $$T^{4} + 72T^{2} + 784$$