# Properties

 Label 675.4.b.a Level $675$ Weight $4$ Character orbit 675.b Analytic conductor $39.826$ Analytic rank $0$ Dimension $2$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 27) 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.

 $$f(q)$$ $$=$$ $$q + 3 i q^{2} - q^{4} - 25 i q^{7} + 21 i q^{8} +O(q^{10})$$ q + 3*i * q^2 - q^4 - 25*i * q^7 + 21*i * q^8 $$q + 3 i q^{2} - q^{4} - 25 i q^{7} + 21 i q^{8} - 15 q^{11} - 20 i q^{13} + 75 q^{14} - 71 q^{16} + 72 i q^{17} - 2 q^{19} - 45 i q^{22} - 114 i q^{23} + 60 q^{26} + 25 i q^{28} - 30 q^{29} + 101 q^{31} - 45 i q^{32} - 216 q^{34} - 430 i q^{37} - 6 i q^{38} - 30 q^{41} - 110 i q^{43} + 15 q^{44} + 342 q^{46} - 330 i q^{47} - 282 q^{49} + 20 i q^{52} - 621 i q^{53} + 525 q^{56} - 90 i q^{58} + 660 q^{59} - 376 q^{61} + 303 i q^{62} - 433 q^{64} - 250 i q^{67} - 72 i q^{68} - 360 q^{71} - 785 i q^{73} + 1290 q^{74} + 2 q^{76} + 375 i q^{77} - 488 q^{79} - 90 i q^{82} - 489 i q^{83} + 330 q^{86} - 315 i q^{88} + 450 q^{89} - 500 q^{91} + 114 i q^{92} + 990 q^{94} - 1105 i q^{97} - 846 i q^{98} +O(q^{100})$$ q + 3*i * q^2 - q^4 - 25*i * q^7 + 21*i * q^8 - 15 * q^11 - 20*i * q^13 + 75 * q^14 - 71 * q^16 + 72*i * q^17 - 2 * q^19 - 45*i * q^22 - 114*i * q^23 + 60 * q^26 + 25*i * q^28 - 30 * q^29 + 101 * q^31 - 45*i * q^32 - 216 * q^34 - 430*i * q^37 - 6*i * q^38 - 30 * q^41 - 110*i * q^43 + 15 * q^44 + 342 * q^46 - 330*i * q^47 - 282 * q^49 + 20*i * q^52 - 621*i * q^53 + 525 * q^56 - 90*i * q^58 + 660 * q^59 - 376 * q^61 + 303*i * q^62 - 433 * q^64 - 250*i * q^67 - 72*i * q^68 - 360 * q^71 - 785*i * q^73 + 1290 * q^74 + 2 * q^76 + 375*i * q^77 - 488 * q^79 - 90*i * q^82 - 489*i * q^83 + 330 * q^86 - 315*i * q^88 + 450 * q^89 - 500 * q^91 + 114*i * q^92 + 990 * q^94 - 1105*i * q^97 - 846*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4}+O(q^{10})$$ 2 * q - 2 * q^4 $$2 q - 2 q^{4} - 30 q^{11} + 150 q^{14} - 142 q^{16} - 4 q^{19} + 120 q^{26} - 60 q^{29} + 202 q^{31} - 432 q^{34} - 60 q^{41} + 30 q^{44} + 684 q^{46} - 564 q^{49} + 1050 q^{56} + 1320 q^{59} - 752 q^{61} - 866 q^{64} - 720 q^{71} + 2580 q^{74} + 4 q^{76} - 976 q^{79} + 660 q^{86} + 900 q^{89} - 1000 q^{91} + 1980 q^{94}+O(q^{100})$$ 2 * q - 2 * q^4 - 30 * q^11 + 150 * q^14 - 142 * q^16 - 4 * q^19 + 120 * q^26 - 60 * q^29 + 202 * q^31 - 432 * q^34 - 60 * q^41 + 30 * q^44 + 684 * q^46 - 564 * q^49 + 1050 * q^56 + 1320 * q^59 - 752 * q^61 - 866 * q^64 - 720 * q^71 + 2580 * q^74 + 4 * q^76 - 976 * q^79 + 660 * q^86 + 900 * q^89 - 1000 * q^91 + 1980 * 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
3.00000i 0 −1.00000 0 0 25.0000i 21.0000i 0 0
649.2 3.00000i 0 −1.00000 0 0 25.0000i 21.0000i 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.a 2
3.b odd 2 1 675.4.b.b 2
5.b even 2 1 inner 675.4.b.a 2
5.c odd 4 1 27.4.a.b yes 1
5.c odd 4 1 675.4.a.a 1
15.d odd 2 1 675.4.b.b 2
15.e even 4 1 27.4.a.a 1
15.e even 4 1 675.4.a.j 1
20.e even 4 1 432.4.a.n 1
35.f even 4 1 1323.4.a.k 1
40.i odd 4 1 1728.4.a.c 1
40.k even 4 1 1728.4.a.d 1
45.k odd 12 2 81.4.c.a 2
45.l even 12 2 81.4.c.c 2
60.l odd 4 1 432.4.a.a 1
105.k odd 4 1 1323.4.a.d 1
120.q odd 4 1 1728.4.a.bd 1
120.w even 4 1 1728.4.a.bc 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.4.a.a 1 15.e even 4 1
27.4.a.b yes 1 5.c odd 4 1
81.4.c.a 2 45.k odd 12 2
81.4.c.c 2 45.l even 12 2
432.4.a.a 1 60.l odd 4 1
432.4.a.n 1 20.e even 4 1
675.4.a.a 1 5.c odd 4 1
675.4.a.j 1 15.e even 4 1
675.4.b.a 2 1.a even 1 1 trivial
675.4.b.a 2 5.b even 2 1 inner
675.4.b.b 2 3.b odd 2 1
675.4.b.b 2 15.d odd 2 1
1323.4.a.d 1 105.k odd 4 1
1323.4.a.k 1 35.f even 4 1
1728.4.a.c 1 40.i odd 4 1
1728.4.a.d 1 40.k even 4 1
1728.4.a.bc 1 120.w even 4 1
1728.4.a.bd 1 120.q odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(675, [\chi])$$:

 $$T_{2}^{2} + 9$$ T2^2 + 9 $$T_{7}^{2} + 625$$ T7^2 + 625 $$T_{11} + 15$$ T11 + 15

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 9$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 625$$
$11$ $$(T + 15)^{2}$$
$13$ $$T^{2} + 400$$
$17$ $$T^{2} + 5184$$
$19$ $$(T + 2)^{2}$$
$23$ $$T^{2} + 12996$$
$29$ $$(T + 30)^{2}$$
$31$ $$(T - 101)^{2}$$
$37$ $$T^{2} + 184900$$
$41$ $$(T + 30)^{2}$$
$43$ $$T^{2} + 12100$$
$47$ $$T^{2} + 108900$$
$53$ $$T^{2} + 385641$$
$59$ $$(T - 660)^{2}$$
$61$ $$(T + 376)^{2}$$
$67$ $$T^{2} + 62500$$
$71$ $$(T + 360)^{2}$$
$73$ $$T^{2} + 616225$$
$79$ $$(T + 488)^{2}$$
$83$ $$T^{2} + 239121$$
$89$ $$(T - 450)^{2}$$
$97$ $$T^{2} + 1221025$$