Properties

 Label 2475.2.c.s Level $2475$ Weight $2$ Character orbit 2475.c Analytic conductor $19.763$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Coefficient field: 8.0.9488318464.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 16x^{6} + 82x^{4} + 136x^{2} + 9$$ x^8 + 16*x^6 + 82*x^4 + 136*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{7}$$ Twist minimal: no (minimal twist has level 495) 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} + ( - \beta_{5} + \beta_{3} - \beta_1) q^{7} + (\beta_{3} - 2 \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b2 - 2) * q^4 + (-b5 + b3 - b1) * q^7 + (b3 - 2*b1) * q^8 $$q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} + ( - \beta_{5} + \beta_{3} - \beta_1) q^{7} + (\beta_{3} - 2 \beta_1) q^{8} - q^{11} + (\beta_{5} + \beta_{3} + \beta_1) q^{13} + (\beta_{6} + \beta_{4} - 2 \beta_{2} + 2) q^{14} + (\beta_{4} - \beta_{2} + 3) q^{16} + \beta_{7} q^{17} + (\beta_{6} - 2 \beta_{2} - 1) q^{19} - \beta_1 q^{22} + (\beta_{7} - \beta_{3} + \beta_1) q^{23} + ( - \beta_{6} + \beta_{4} - 4) q^{26} + (\beta_{5} - 3 \beta_{3} + 5 \beta_1) q^{28} + ( - \beta_{6} - \beta_{4} - \beta_{2} + 1) q^{29} + ( - \beta_{4} + \beta_{2}) q^{31} + (\beta_{7} - 3 \beta_{3} + 2 \beta_1) q^{32} + (2 \beta_{6} - \beta_{2} - 1) q^{34} + ( - \beta_{7} - \beta_{5} - \beta_{3} + \beta_1) q^{37} + ( - \beta_{7} + 3 \beta_{5} - \beta_{3} + 3 \beta_1) q^{38} + (\beta_{6} - \beta_{4} - \beta_{2} - 1) q^{41} + (\beta_{5} + \beta_{3} - \beta_1) q^{43} + ( - \beta_{2} + 2) q^{44} + (2 \beta_{6} - \beta_{4} + \beta_{2} - 4) q^{46} + ( - 2 \beta_{3} - 2 \beta_1) q^{47} + (2 \beta_{6} + \beta_{4} + \beta_{2} - 5) q^{49} + (2 \beta_{7} - \beta_{5} - 3 \beta_{3} - \beta_1) q^{52} + (\beta_{7} + 3 \beta_{5} - \beta_{3} + \beta_1) q^{53} + (\beta_{6} - \beta_{4} + 4 \beta_{2} - 12) q^{56} + ( - 3 \beta_{5} + 2 \beta_{3} + 2 \beta_1) q^{58} + ( - 2 \beta_{6} + 2 \beta_{2} + 6) q^{59} + (2 \beta_{6} + \beta_{4} - \beta_{2} + 2) q^{61} + ( - \beta_{7} + 5 \beta_{3} - 3 \beta_1) q^{62} + (2 \beta_{6} - \beta_{4} + 2 \beta_{2}) q^{64} + (\beta_{7} + 2 \beta_{5} - \beta_{3} + 5 \beta_1) q^{67} + (6 \beta_{5} + \beta_{3} + \beta_1) q^{68} + ( - 2 \beta_{2} - 4) q^{71} + (\beta_{5} + \beta_{3} + \beta_1) q^{73} + ( - \beta_{6} - \beta_{4} + 3 \beta_{2} - 3) q^{74} + ( - 3 \beta_{6} - \beta_{4} + \beta_{2} - 9) q^{76} + (\beta_{5} - \beta_{3} + \beta_1) q^{77} + ( - \beta_{6} + 2 \beta_{2} - 3) q^{79} + ( - 2 \beta_{7} + 3 \beta_{5} + 4 \beta_{3}) q^{82} + ( - \beta_{7} + 2 \beta_{3}) q^{83} + ( - \beta_{6} + \beta_{4} - 2 \beta_{2} + 4) q^{86} + ( - \beta_{3} + 2 \beta_1) q^{88} + (2 \beta_{6} + 4) q^{89} + (4 \beta_{6} + \beta_{4} - \beta_{2} - 4) q^{91} + ( - \beta_{7} + 6 \beta_{5} + 5 \beta_{3} - 5 \beta_1) q^{92} + ( - 2 \beta_{4} + 10) q^{94} + ( - \beta_{7} - \beta_{5} + \beta_{3} + 3 \beta_1) q^{97} + ( - \beta_{7} + 6 \beta_{5} - \beta_{3} - 6 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b2 - 2) * q^4 + (-b5 + b3 - b1) * q^7 + (b3 - 2*b1) * q^8 - q^11 + (b5 + b3 + b1) * q^13 + (b6 + b4 - 2*b2 + 2) * q^14 + (b4 - b2 + 3) * q^16 + b7 * q^17 + (b6 - 2*b2 - 1) * q^19 - b1 * q^22 + (b7 - b3 + b1) * q^23 + (-b6 + b4 - 4) * q^26 + (b5 - 3*b3 + 5*b1) * q^28 + (-b6 - b4 - b2 + 1) * q^29 + (-b4 + b2) * q^31 + (b7 - 3*b3 + 2*b1) * q^32 + (2*b6 - b2 - 1) * q^34 + (-b7 - b5 - b3 + b1) * q^37 + (-b7 + 3*b5 - b3 + 3*b1) * q^38 + (b6 - b4 - b2 - 1) * q^41 + (b5 + b3 - b1) * q^43 + (-b2 + 2) * q^44 + (2*b6 - b4 + b2 - 4) * q^46 + (-2*b3 - 2*b1) * q^47 + (2*b6 + b4 + b2 - 5) * q^49 + (2*b7 - b5 - 3*b3 - b1) * q^52 + (b7 + 3*b5 - b3 + b1) * q^53 + (b6 - b4 + 4*b2 - 12) * q^56 + (-3*b5 + 2*b3 + 2*b1) * q^58 + (-2*b6 + 2*b2 + 6) * q^59 + (2*b6 + b4 - b2 + 2) * q^61 + (-b7 + 5*b3 - 3*b1) * q^62 + (2*b6 - b4 + 2*b2) * q^64 + (b7 + 2*b5 - b3 + 5*b1) * q^67 + (6*b5 + b3 + b1) * q^68 + (-2*b2 - 4) * q^71 + (b5 + b3 + b1) * q^73 + (-b6 - b4 + 3*b2 - 3) * q^74 + (-3*b6 - b4 + b2 - 9) * q^76 + (b5 - b3 + b1) * q^77 + (-b6 + 2*b2 - 3) * q^79 + (-2*b7 + 3*b5 + 4*b3) * q^82 + (-b7 + 2*b3) * q^83 + (-b6 + b4 - 2*b2 + 4) * q^86 + (-b3 + 2*b1) * q^88 + (2*b6 + 4) * q^89 + (4*b6 + b4 - b2 - 4) * q^91 + (-b7 + 6*b5 + 5*b3 - 5*b1) * q^92 + (-2*b4 + 10) * q^94 + (-b7 - b5 + b3 + 3*b1) * q^97 + (-b7 + 6*b5 - b3 - 6*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 16 q^{4}+O(q^{10})$$ 8 * q - 16 * q^4 $$8 q - 16 q^{4} - 8 q^{11} + 16 q^{14} + 24 q^{16} - 8 q^{19} - 32 q^{26} + 8 q^{29} - 8 q^{34} - 8 q^{41} + 16 q^{44} - 32 q^{46} - 40 q^{49} - 96 q^{56} + 48 q^{59} + 16 q^{61} - 32 q^{71} - 24 q^{74} - 72 q^{76} - 24 q^{79} + 32 q^{86} + 32 q^{89} - 32 q^{91} + 80 q^{94}+O(q^{100})$$ 8 * q - 16 * q^4 - 8 * q^11 + 16 * q^14 + 24 * q^16 - 8 * q^19 - 32 * q^26 + 8 * q^29 - 8 * q^34 - 8 * q^41 + 16 * q^44 - 32 * q^46 - 40 * q^49 - 96 * q^56 + 48 * q^59 + 16 * q^61 - 32 * q^71 - 24 * q^74 - 72 * q^76 - 24 * q^79 + 32 * q^86 + 32 * q^89 - 32 * q^91 + 80 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 16x^{6} + 82x^{4} + 136x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 4$$ v^2 + 4 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 6\nu$$ v^3 + 6*v $$\beta_{4}$$ $$=$$ $$\nu^{4} + 7\nu^{2} + 5$$ v^4 + 7*v^2 + 5 $$\beta_{5}$$ $$=$$ $$( \nu^{7} + 13\nu^{5} + 49\nu^{3} + 49\nu ) / 6$$ (v^7 + 13*v^5 + 49*v^3 + 49*v) / 6 $$\beta_{6}$$ $$=$$ $$( \nu^{6} + 11\nu^{4} + 29\nu^{2} + 5 ) / 2$$ (v^6 + 11*v^4 + 29*v^2 + 5) / 2 $$\beta_{7}$$ $$=$$ $$\nu^{5} + 11\nu^{3} + 28\nu$$ v^5 + 11*v^3 + 28*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 4$$ b2 - 4 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 6\beta_1$$ b3 - 6*b1 $$\nu^{4}$$ $$=$$ $$\beta_{4} - 7\beta_{2} + 23$$ b4 - 7*b2 + 23 $$\nu^{5}$$ $$=$$ $$\beta_{7} - 11\beta_{3} + 38\beta_1$$ b7 - 11*b3 + 38*b1 $$\nu^{6}$$ $$=$$ $$2\beta_{6} - 11\beta_{4} + 48\beta_{2} - 142$$ 2*b6 - 11*b4 + 48*b2 - 142 $$\nu^{7}$$ $$=$$ $$-13\beta_{7} + 6\beta_{5} + 94\beta_{3} - 249\beta_1$$ -13*b7 + 6*b5 + 94*b3 - 249*b1

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
 − 2.69696i − 2.28632i − 1.85206i − 0.262696i 0.262696i 1.85206i 2.28632i 2.69696i
2.69696i 0 −5.27358 0 0 4.13176i 8.82872i 0 0
199.2 2.28632i 0 −3.22727 0 0 2.51962i 2.80595i 0 0
199.3 1.85206i 0 −1.43013 0 0 4.90749i 1.05543i 0 0
199.4 0.262696i 0 1.93099 0 0 0.704647i 1.03266i 0 0
199.5 0.262696i 0 1.93099 0 0 0.704647i 1.03266i 0 0
199.6 1.85206i 0 −1.43013 0 0 4.90749i 1.05543i 0 0
199.7 2.28632i 0 −3.22727 0 0 2.51962i 2.80595i 0 0
199.8 2.69696i 0 −5.27358 0 0 4.13176i 8.82872i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 199.8 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.s 8
3.b odd 2 1 2475.2.c.t 8
5.b even 2 1 inner 2475.2.c.s 8
5.c odd 4 1 495.2.a.g yes 4
5.c odd 4 1 2475.2.a.bf 4
15.d odd 2 1 2475.2.c.t 8
15.e even 4 1 495.2.a.f 4
15.e even 4 1 2475.2.a.bj 4
20.e even 4 1 7920.2.a.cn 4
55.e even 4 1 5445.2.a.bh 4
60.l odd 4 1 7920.2.a.cm 4
165.l odd 4 1 5445.2.a.bs 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.2.a.f 4 15.e even 4 1
495.2.a.g yes 4 5.c odd 4 1
2475.2.a.bf 4 5.c odd 4 1
2475.2.a.bj 4 15.e even 4 1
2475.2.c.s 8 1.a even 1 1 trivial
2475.2.c.s 8 5.b even 2 1 inner
2475.2.c.t 8 3.b odd 2 1
2475.2.c.t 8 15.d odd 2 1
5445.2.a.bh 4 55.e even 4 1
5445.2.a.bs 4 165.l odd 4 1
7920.2.a.cm 4 60.l odd 4 1
7920.2.a.cn 4 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 $$S_{2}^{\mathrm{new}}(2475, [\chi])$$:

 $$T_{2}^{8} + 16T_{2}^{6} + 82T_{2}^{4} + 136T_{2}^{2} + 9$$ T2^8 + 16*T2^6 + 82*T2^4 + 136*T2^2 + 9 $$T_{7}^{8} + 48T_{7}^{6} + 696T_{7}^{4} + 2944T_{7}^{2} + 1296$$ T7^8 + 48*T7^6 + 696*T7^4 + 2944*T7^2 + 1296 $$T_{29}^{4} - 4T_{29}^{3} - 88T_{29}^{2} + 240T_{29} - 144$$ T29^4 - 4*T29^3 - 88*T29^2 + 240*T29 - 144

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 16 T^{6} + 82 T^{4} + 136 T^{2} + \cdots + 9$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$T^{8} + 48 T^{6} + 696 T^{4} + \cdots + 1296$$
$11$ $$(T + 1)^{8}$$
$13$ $$T^{8} + 80 T^{6} + 2168 T^{4} + \cdots + 85264$$
$17$ $$T^{8} + 96 T^{6} + 2872 T^{4} + \cdots + 104976$$
$19$ $$(T^{4} + 4 T^{3} - 56 T^{2} - 192 T + 288)^{2}$$
$23$ $$T^{8} + 128 T^{6} + 4736 T^{4} + \cdots + 36864$$
$29$ $$(T^{4} - 4 T^{3} - 88 T^{2} + 240 T - 144)^{2}$$
$31$ $$(T^{4} - 88 T^{2} + 192 T + 144)^{2}$$
$37$ $$T^{8} + 176 T^{6} + 8672 T^{4} + \cdots + 952576$$
$41$ $$(T^{4} + 4 T^{3} - 120 T^{2} - 432 T + 2160)^{2}$$
$43$ $$T^{8} + 80 T^{6} + 952 T^{4} + \cdots + 1296$$
$47$ $$T^{8} + 256 T^{6} + 15232 T^{4} + \cdots + 331776$$
$53$ $$T^{8} + 176 T^{6} + 5216 T^{4} + \cdots + 57600$$
$59$ $$(T^{4} - 24 T^{3} + 88 T^{2} + 1376 T - 8496)^{2}$$
$61$ $$(T^{4} - 8 T^{3} - 104 T^{2} + 1056 T - 2224)^{2}$$
$67$ $$T^{8} + 448 T^{6} + \cdots + 38539264$$
$71$ $$(T^{4} + 16 T^{3} + 40 T^{2} - 128 T - 240)^{2}$$
$73$ $$T^{8} + 80 T^{6} + 2168 T^{4} + \cdots + 85264$$
$79$ $$(T^{4} + 12 T^{3} - 8 T^{2} - 192 T + 160)^{2}$$
$83$ $$T^{8} + 208 T^{6} + 14616 T^{4} + \cdots + 3370896$$
$89$ $$(T^{4} - 16 T^{3} - 24 T^{2} + 512 T + 720)^{2}$$
$97$ $$T^{8} + 272 T^{6} + 24032 T^{4} + \cdots + 8202496$$