Properties

 Label 675.4.b.d Level $675$ Weight $4$ Character orbit 675.b Analytic conductor $39.826$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,4,Mod(649,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("675.649");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$675 = 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 675.b (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: $$39.8262892539$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 135) 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 i q^{2} + 4 q^{4} + 24 i q^{8}+O(q^{10})$$ q + 2*i * q^2 + 4 * q^4 + 24*i * q^8 $$q + 2 i q^{2} + 4 q^{4} + 24 i q^{8} + 10 q^{11} - 80 i q^{13} - 16 q^{16} - 7 i q^{17} + 113 q^{19} + 20 i q^{22} - 81 i q^{23} + 160 q^{26} + 220 q^{29} - 189 q^{31} + 160 i q^{32} + 14 q^{34} - 170 i q^{37} + 226 i q^{38} - 130 q^{41} + 10 i q^{43} + 40 q^{44} + 162 q^{46} - 160 i q^{47} + 343 q^{49} - 320 i q^{52} + 631 i q^{53} + 440 i q^{58} + 560 q^{59} + 229 q^{61} - 378 i q^{62} - 448 q^{64} - 750 i q^{67} - 28 i q^{68} + 890 q^{71} - 890 i q^{73} + 340 q^{74} + 452 q^{76} + 27 q^{79} - 260 i q^{82} + 429 i q^{83} - 20 q^{86} + 240 i q^{88} + 750 q^{89} - 324 i q^{92} + 320 q^{94} + 1480 i q^{97} + 686 i q^{98} +O(q^{100})$$ q + 2*i * q^2 + 4 * q^4 + 24*i * q^8 + 10 * q^11 - 80*i * q^13 - 16 * q^16 - 7*i * q^17 + 113 * q^19 + 20*i * q^22 - 81*i * q^23 + 160 * q^26 + 220 * q^29 - 189 * q^31 + 160*i * q^32 + 14 * q^34 - 170*i * q^37 + 226*i * q^38 - 130 * q^41 + 10*i * q^43 + 40 * q^44 + 162 * q^46 - 160*i * q^47 + 343 * q^49 - 320*i * q^52 + 631*i * q^53 + 440*i * q^58 + 560 * q^59 + 229 * q^61 - 378*i * q^62 - 448 * q^64 - 750*i * q^67 - 28*i * q^68 + 890 * q^71 - 890*i * q^73 + 340 * q^74 + 452 * q^76 + 27 * q^79 - 260*i * q^82 + 429*i * q^83 - 20 * q^86 + 240*i * q^88 + 750 * q^89 - 324*i * q^92 + 320 * q^94 + 1480*i * q^97 + 686*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 8 q^{4}+O(q^{10})$$ 2 * q + 8 * q^4 $$2 q + 8 q^{4} + 20 q^{11} - 32 q^{16} + 226 q^{19} + 320 q^{26} + 440 q^{29} - 378 q^{31} + 28 q^{34} - 260 q^{41} + 80 q^{44} + 324 q^{46} + 686 q^{49} + 1120 q^{59} + 458 q^{61} - 896 q^{64} + 1780 q^{71} + 680 q^{74} + 904 q^{76} + 54 q^{79} - 40 q^{86} + 1500 q^{89} + 640 q^{94}+O(q^{100})$$ 2 * q + 8 * q^4 + 20 * q^11 - 32 * q^16 + 226 * q^19 + 320 * q^26 + 440 * q^29 - 378 * q^31 + 28 * q^34 - 260 * q^41 + 80 * q^44 + 324 * q^46 + 686 * q^49 + 1120 * q^59 + 458 * q^61 - 896 * q^64 + 1780 * q^71 + 680 * q^74 + 904 * q^76 + 54 * q^79 - 40 * q^86 + 1500 * q^89 + 640 * q^94

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$$.

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

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
649.1
 − 1.00000i 1.00000i
2.00000i 0 4.00000 0 0 0 24.0000i 0 0
649.2 2.00000i 0 4.00000 0 0 0 24.0000i 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 675.4.b.d 2
3.b odd 2 1 675.4.b.c 2
5.b even 2 1 inner 675.4.b.d 2
5.c odd 4 1 135.4.a.a 1
5.c odd 4 1 675.4.a.i 1
15.d odd 2 1 675.4.b.c 2
15.e even 4 1 135.4.a.d yes 1
15.e even 4 1 675.4.a.b 1
20.e even 4 1 2160.4.a.n 1
45.k odd 12 2 405.4.e.j 2
45.l even 12 2 405.4.e.e 2
60.l odd 4 1 2160.4.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.a.a 1 5.c odd 4 1
135.4.a.d yes 1 15.e even 4 1
405.4.e.e 2 45.l even 12 2
405.4.e.j 2 45.k odd 12 2
675.4.a.b 1 15.e even 4 1
675.4.a.i 1 5.c odd 4 1
675.4.b.c 2 3.b odd 2 1
675.4.b.c 2 15.d odd 2 1
675.4.b.d 2 1.a even 1 1 trivial
675.4.b.d 2 5.b even 2 1 inner
2160.4.a.d 1 60.l odd 4 1
2160.4.a.n 1 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 $$S_{4}^{\mathrm{new}}(675, [\chi])$$:

 $$T_{2}^{2} + 4$$ T2^2 + 4 $$T_{7}$$ T7 $$T_{11} - 10$$ T11 - 10

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$(T - 10)^{2}$$
$13$ $$T^{2} + 6400$$
$17$ $$T^{2} + 49$$
$19$ $$(T - 113)^{2}$$
$23$ $$T^{2} + 6561$$
$29$ $$(T - 220)^{2}$$
$31$ $$(T + 189)^{2}$$
$37$ $$T^{2} + 28900$$
$41$ $$(T + 130)^{2}$$
$43$ $$T^{2} + 100$$
$47$ $$T^{2} + 25600$$
$53$ $$T^{2} + 398161$$
$59$ $$(T - 560)^{2}$$
$61$ $$(T - 229)^{2}$$
$67$ $$T^{2} + 562500$$
$71$ $$(T - 890)^{2}$$
$73$ $$T^{2} + 792100$$
$79$ $$(T - 27)^{2}$$
$83$ $$T^{2} + 184041$$
$89$ $$(T - 750)^{2}$$
$97$ $$T^{2} + 2190400$$