# Properties

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

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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 165) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{3} - 1) q^{4} - 2 \beta_{2} q^{7} + ( - \beta_{2} - 2 \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (-b3 - 1) * q^4 - 2*b2 * q^7 + (-b2 - 2*b1) * q^8 $$q + \beta_1 q^{2} + ( - \beta_{3} - 1) q^{4} - 2 \beta_{2} q^{7} + ( - \beta_{2} - 2 \beta_1) q^{8} + q^{11} + ( - 2 \beta_{2} + 2 \beta_1) q^{13} - 2 q^{14} + 3 q^{16} + (3 \beta_{2} + \beta_1) q^{17} + (\beta_{3} + 4) q^{19} + \beta_1 q^{22} + ( - 2 \beta_{2} - 2 \beta_1) q^{23} + ( - 2 \beta_{3} - 8) q^{26} + ( - 4 \beta_{2} - 2 \beta_1) q^{28} + ( - \beta_{3} - 2) q^{29} + ( - 2 \beta_{2} - \beta_1) q^{32} - \beta_{3} q^{34} + (\beta_{2} + 5 \beta_1) q^{37} + (\beta_{2} + 7 \beta_1) q^{38} + ( - \beta_{3} - 2) q^{41} + (2 \beta_{2} + 4 \beta_1) q^{43} + ( - \beta_{3} - 1) q^{44} + (2 \beta_{3} + 4) q^{46} + (2 \beta_{2} + 2 \beta_1) q^{47} + (4 \beta_{3} - 5) q^{49} + ( - 6 \beta_{2} - 10 \beta_1) q^{52} + ( - 5 \beta_{2} + 3 \beta_1) q^{53} + (2 \beta_{3} - 2) q^{56} + ( - \beta_{2} - 5 \beta_1) q^{58} - 4 q^{59} + ( - 2 \beta_{3} - 6) q^{61} + (\beta_{3} + 7) q^{64} + (2 \beta_{2} - 2 \beta_1) q^{67} + (5 \beta_{2} - \beta_1) q^{68} + (2 \beta_{3} - 8) q^{71} + (4 \beta_{2} - 4 \beta_1) q^{73} + ( - 5 \beta_{3} - 14) q^{74} + ( - 5 \beta_{3} - 12) q^{76} - 2 \beta_{2} q^{77} - 3 \beta_{3} q^{79} + ( - \beta_{2} - 5 \beta_1) q^{82} + ( - 5 \beta_{2} - 5 \beta_1) q^{83} + ( - 4 \beta_{3} - 10) q^{86} + ( - \beta_{2} - 2 \beta_1) q^{88} + (2 \beta_{3} - 2) q^{89} + (4 \beta_{3} - 16) q^{91} + ( - 2 \beta_{2} + 6 \beta_1) q^{92} + ( - 2 \beta_{3} - 4) q^{94} + (\beta_{2} + 5 \beta_1) q^{97} + (4 \beta_{2} + 7 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + (-b3 - 1) * q^4 - 2*b2 * q^7 + (-b2 - 2*b1) * q^8 + q^11 + (-2*b2 + 2*b1) * q^13 - 2 * q^14 + 3 * q^16 + (3*b2 + b1) * q^17 + (b3 + 4) * q^19 + b1 * q^22 + (-2*b2 - 2*b1) * q^23 + (-2*b3 - 8) * q^26 + (-4*b2 - 2*b1) * q^28 + (-b3 - 2) * q^29 + (-2*b2 - b1) * q^32 - b3 * q^34 + (b2 + 5*b1) * q^37 + (b2 + 7*b1) * q^38 + (-b3 - 2) * q^41 + (2*b2 + 4*b1) * q^43 + (-b3 - 1) * q^44 + (2*b3 + 4) * q^46 + (2*b2 + 2*b1) * q^47 + (4*b3 - 5) * q^49 + (-6*b2 - 10*b1) * q^52 + (-5*b2 + 3*b1) * q^53 + (2*b3 - 2) * q^56 + (-b2 - 5*b1) * q^58 - 4 * q^59 + (-2*b3 - 6) * q^61 + (b3 + 7) * q^64 + (2*b2 - 2*b1) * q^67 + (5*b2 - b1) * q^68 + (2*b3 - 8) * q^71 + (4*b2 - 4*b1) * q^73 + (-5*b3 - 14) * q^74 + (-5*b3 - 12) * q^76 - 2*b2 * q^77 - 3*b3 * q^79 + (-b2 - 5*b1) * q^82 + (-5*b2 - 5*b1) * q^83 + (-4*b3 - 10) * q^86 + (-b2 - 2*b1) * q^88 + (2*b3 - 2) * q^89 + (4*b3 - 16) * q^91 + (-2*b2 + 6*b1) * q^92 + (-2*b3 - 4) * q^94 + (b2 + 5*b1) * q^97 + (4*b2 + 7*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4}+O(q^{10})$$ 4 * q - 4 * q^4 $$4 q - 4 q^{4} + 4 q^{11} - 8 q^{14} + 12 q^{16} + 16 q^{19} - 32 q^{26} - 8 q^{29} - 8 q^{41} - 4 q^{44} + 16 q^{46} - 20 q^{49} - 8 q^{56} - 16 q^{59} - 24 q^{61} + 28 q^{64} - 32 q^{71} - 56 q^{74} - 48 q^{76} - 40 q^{86} - 8 q^{89} - 64 q^{91} - 16 q^{94}+O(q^{100})$$ 4 * q - 4 * q^4 + 4 * q^11 - 8 * q^14 + 12 * q^16 + 16 * q^19 - 32 * q^26 - 8 * q^29 - 8 * q^41 - 4 * q^44 + 16 * q^46 - 20 * q^49 - 8 * q^56 - 16 * q^59 - 24 * q^61 + 28 * q^64 - 32 * q^71 - 56 * q^74 - 48 * q^76 - 40 * q^86 - 8 * q^89 - 64 * 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 0.828427i 4.41421i 0 0
199.2 0.414214i 0 1.82843 0 0 4.82843i 1.58579i 0 0
199.3 0.414214i 0 1.82843 0 0 4.82843i 1.58579i 0 0
199.4 2.41421i 0 −3.82843 0 0 0.828427i 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.m 4
3.b odd 2 1 825.2.c.e 4
5.b even 2 1 inner 2475.2.c.m 4
5.c odd 4 1 495.2.a.d 2
5.c odd 4 1 2475.2.a.m 2
15.d odd 2 1 825.2.c.e 4
15.e even 4 1 165.2.a.a 2
15.e even 4 1 825.2.a.g 2
20.e even 4 1 7920.2.a.cg 2
55.e even 4 1 5445.2.a.m 2
60.l odd 4 1 2640.2.a.bb 2
105.k odd 4 1 8085.2.a.ba 2
165.l odd 4 1 1815.2.a.k 2
165.l odd 4 1 9075.2.a.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.a 2 15.e even 4 1
495.2.a.d 2 5.c odd 4 1
825.2.a.g 2 15.e even 4 1
825.2.c.e 4 3.b odd 2 1
825.2.c.e 4 15.d odd 2 1
1815.2.a.k 2 165.l odd 4 1
2475.2.a.m 2 5.c odd 4 1
2475.2.c.m 4 1.a even 1 1 trivial
2475.2.c.m 4 5.b even 2 1 inner
2640.2.a.bb 2 60.l odd 4 1
5445.2.a.m 2 55.e even 4 1
7920.2.a.cg 2 20.e even 4 1
8085.2.a.ba 2 105.k odd 4 1
9075.2.a.v 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 $$T_{7}^{4} + 24T_{7}^{2} + 16$$ T7^4 + 24*T7^2 + 16 $$T_{29}^{2} + 4T_{29} - 4$$ T29^2 + 4*T29 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 6T^{2} + 1$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 24T^{2} + 16$$
$11$ $$(T - 1)^{4}$$
$13$ $$(T^{2} + 32)^{2}$$
$17$ $$T^{4} + 48T^{2} + 64$$
$19$ $$(T^{2} - 8 T + 8)^{2}$$
$23$ $$(T^{2} + 16)^{2}$$
$29$ $$(T^{2} + 4 T - 4)^{2}$$
$31$ $$T^{4}$$
$37$ $$T^{4} + 136T^{2} + 16$$
$41$ $$(T^{2} + 4 T - 4)^{2}$$
$43$ $$T^{4} + 88T^{2} + 784$$
$47$ $$(T^{2} + 16)^{2}$$
$53$ $$T^{4} + 264 T^{2} + 15376$$
$59$ $$(T + 4)^{4}$$
$61$ $$(T^{2} + 12 T + 4)^{2}$$
$67$ $$(T^{2} + 32)^{2}$$
$71$ $$(T^{2} + 16 T + 32)^{2}$$
$73$ $$(T^{2} + 128)^{2}$$
$79$ $$(T^{2} - 72)^{2}$$
$83$ $$(T^{2} + 100)^{2}$$
$89$ $$(T^{2} + 4 T - 28)^{2}$$
$97$ $$T^{4} + 136T^{2} + 16$$
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