# Properties

 Label 675.2.b.i Level $675$ Weight $2$ Character orbit 675.b Analytic conductor $5.390$ 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] = [675,2,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, 2, names="a")

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

chi := DirichletCharacter("675.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$675 = 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$5.38990213644$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 7x^{2} + 9$$ x^4 + 7*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{3} - 2) q^{4} + (2 \beta_{2} + 2 \beta_1) q^{7} + 3 \beta_{2} q^{8}+O(q^{10})$$ q + b1 * q^2 + (b3 - 2) * q^4 + (2*b2 + 2*b1) * q^7 + 3*b2 * q^8 $$q + \beta_1 q^{2} + (\beta_{3} - 2) q^{4} + (2 \beta_{2} + 2 \beta_1) q^{7} + 3 \beta_{2} q^{8} + ( - 2 \beta_{3} + 2) q^{11} + ( - 2 \beta_{2} + 2 \beta_1) q^{13} - 6 q^{14} + ( - \beta_{3} - 1) q^{16} + ( - 3 \beta_{2} - 2 \beta_1) q^{17} + (2 \beta_{3} - 1) q^{19} + ( - 6 \beta_{2} + 2 \beta_1) q^{22} + 3 \beta_{2} q^{23} + (4 \beta_{3} - 10) q^{26} + (4 \beta_{2} - 2 \beta_1) q^{28} + ( - 2 \beta_{3} - 4) q^{29} + (2 \beta_{3} - 3) q^{31} + (3 \beta_{2} - \beta_1) q^{32} + (\beta_{3} + 5) q^{34} + 2 \beta_{2} q^{37} + (6 \beta_{2} - \beta_1) q^{38} + (2 \beta_{3} - 2) q^{41} + (4 \beta_{2} + 2 \beta_1) q^{43} + (4 \beta_{3} - 10) q^{44} + ( - 3 \beta_{3} + 3) q^{46} + 4 \beta_1 q^{47} + ( - 4 \beta_{3} - 5) q^{49} + (8 \beta_{2} - 6 \beta_1) q^{52} + (3 \beta_{2} + 2 \beta_1) q^{53} - 6 \beta_{3} q^{56} + ( - 6 \beta_{2} - 4 \beta_1) q^{58} + (2 \beta_{3} + 4) q^{59} + (4 \beta_{3} + 1) q^{61} + (6 \beta_{2} - 3 \beta_1) q^{62} + ( - 6 \beta_{3} + 5) q^{64} + ( - 10 \beta_{2} - 4 \beta_1) q^{67} + ( - 3 \beta_{2} + \beta_1) q^{68} + (2 \beta_{3} + 10) q^{71} + ( - 8 \beta_{2} + 2 \beta_1) q^{73} + ( - 2 \beta_{3} + 2) q^{74} + ( - 3 \beta_{3} + 8) q^{76} - 12 \beta_{2} q^{77} + ( - 2 \beta_{3} + 9) q^{79} + (6 \beta_{2} - 2 \beta_1) q^{82} - 3 \beta_{2} q^{83} + ( - 2 \beta_{3} - 4) q^{86} - 6 \beta_1 q^{88} + ( - 6 \beta_{3} + 6) q^{89} + (4 \beta_{3} - 12) q^{91} + ( - 3 \beta_{2} + 3 \beta_1) q^{92} + (4 \beta_{3} - 16) q^{94} + 8 \beta_{2} q^{97} + ( - 12 \beta_{2} - 5 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b3 - 2) * q^4 + (2*b2 + 2*b1) * q^7 + 3*b2 * q^8 + (-2*b3 + 2) * q^11 + (-2*b2 + 2*b1) * q^13 - 6 * q^14 + (-b3 - 1) * q^16 + (-3*b2 - 2*b1) * q^17 + (2*b3 - 1) * q^19 + (-6*b2 + 2*b1) * q^22 + 3*b2 * q^23 + (4*b3 - 10) * q^26 + (4*b2 - 2*b1) * q^28 + (-2*b3 - 4) * q^29 + (2*b3 - 3) * q^31 + (3*b2 - b1) * q^32 + (b3 + 5) * q^34 + 2*b2 * q^37 + (6*b2 - b1) * q^38 + (2*b3 - 2) * q^41 + (4*b2 + 2*b1) * q^43 + (4*b3 - 10) * q^44 + (-3*b3 + 3) * q^46 + 4*b1 * q^47 + (-4*b3 - 5) * q^49 + (8*b2 - 6*b1) * q^52 + (3*b2 + 2*b1) * q^53 - 6*b3 * q^56 + (-6*b2 - 4*b1) * q^58 + (2*b3 + 4) * q^59 + (4*b3 + 1) * q^61 + (6*b2 - 3*b1) * q^62 + (-6*b3 + 5) * q^64 + (-10*b2 - 4*b1) * q^67 + (-3*b2 + b1) * q^68 + (2*b3 + 10) * q^71 + (-8*b2 + 2*b1) * q^73 + (-2*b3 + 2) * q^74 + (-3*b3 + 8) * q^76 - 12*b2 * q^77 + (-2*b3 + 9) * q^79 + (6*b2 - 2*b1) * q^82 - 3*b2 * q^83 + (-2*b3 - 4) * q^86 - 6*b1 * q^88 + (-6*b3 + 6) * q^89 + (4*b3 - 12) * q^91 + (-3*b2 + 3*b1) * q^92 + (4*b3 - 16) * q^94 + 8*b2 * q^97 + (-12*b2 - 5*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{4}+O(q^{10})$$ 4 * q - 6 * q^4 $$4 q - 6 q^{4} + 4 q^{11} - 24 q^{14} - 6 q^{16} - 32 q^{26} - 20 q^{29} - 8 q^{31} + 22 q^{34} - 4 q^{41} - 32 q^{44} + 6 q^{46} - 28 q^{49} - 12 q^{56} + 20 q^{59} + 12 q^{61} + 8 q^{64} + 44 q^{71} + 4 q^{74} + 26 q^{76} + 32 q^{79} - 20 q^{86} + 12 q^{89} - 40 q^{91} - 56 q^{94}+O(q^{100})$$ 4 * q - 6 * q^4 + 4 * q^11 - 24 * q^14 - 6 * q^16 - 32 * q^26 - 20 * q^29 - 8 * q^31 + 22 * q^34 - 4 * q^41 - 32 * q^44 + 6 * q^46 - 28 * q^49 - 12 * q^56 + 20 * q^59 + 12 * q^61 + 8 * q^64 + 44 * q^71 + 4 * q^74 + 26 * q^76 + 32 * q^79 - 20 * q^86 + 12 * q^89 - 40 * q^91 - 56 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7x^{2} + 9$$ :

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

## 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
 − 2.30278i − 1.30278i 1.30278i 2.30278i
2.30278i 0 −3.30278 0 0 2.60555i 3.00000i 0 0
649.2 1.30278i 0 0.302776 0 0 4.60555i 3.00000i 0 0
649.3 1.30278i 0 0.302776 0 0 4.60555i 3.00000i 0 0
649.4 2.30278i 0 −3.30278 0 0 2.60555i 3.00000i 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 675.2.b.i 4
3.b odd 2 1 675.2.b.h 4
5.b even 2 1 inner 675.2.b.i 4
5.c odd 4 1 135.2.a.c 2
5.c odd 4 1 675.2.a.p 2
15.d odd 2 1 675.2.b.h 4
15.e even 4 1 135.2.a.d yes 2
15.e even 4 1 675.2.a.k 2
20.e even 4 1 2160.2.a.ba 2
35.f even 4 1 6615.2.a.p 2
40.i odd 4 1 8640.2.a.cr 2
40.k even 4 1 8640.2.a.ck 2
45.k odd 12 2 405.2.e.k 4
45.l even 12 2 405.2.e.j 4
60.l odd 4 1 2160.2.a.y 2
105.k odd 4 1 6615.2.a.v 2
120.q odd 4 1 8640.2.a.cy 2
120.w even 4 1 8640.2.a.df 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.a.c 2 5.c odd 4 1
135.2.a.d yes 2 15.e even 4 1
405.2.e.j 4 45.l even 12 2
405.2.e.k 4 45.k odd 12 2
675.2.a.k 2 15.e even 4 1
675.2.a.p 2 5.c odd 4 1
675.2.b.h 4 3.b odd 2 1
675.2.b.h 4 15.d odd 2 1
675.2.b.i 4 1.a even 1 1 trivial
675.2.b.i 4 5.b even 2 1 inner
2160.2.a.y 2 60.l odd 4 1
2160.2.a.ba 2 20.e even 4 1
6615.2.a.p 2 35.f even 4 1
6615.2.a.v 2 105.k odd 4 1
8640.2.a.ck 2 40.k even 4 1
8640.2.a.cr 2 40.i odd 4 1
8640.2.a.cy 2 120.q odd 4 1
8640.2.a.df 2 120.w even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(675, [\chi])$$:

 $$T_{2}^{4} + 7T_{2}^{2} + 9$$ T2^4 + 7*T2^2 + 9 $$T_{7}^{4} + 28T_{7}^{2} + 144$$ T7^4 + 28*T7^2 + 144 $$T_{11}^{2} - 2T_{11} - 12$$ T11^2 - 2*T11 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 7T^{2} + 9$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 28T^{2} + 144$$
$11$ $$(T^{2} - 2 T - 12)^{2}$$
$13$ $$T^{4} + 44T^{2} + 16$$
$17$ $$T^{4} + 34T^{2} + 81$$
$19$ $$(T^{2} - 13)^{2}$$
$23$ $$(T^{2} + 9)^{2}$$
$29$ $$(T^{2} + 10 T + 12)^{2}$$
$31$ $$(T^{2} + 4 T - 9)^{2}$$
$37$ $$(T^{2} + 4)^{2}$$
$41$ $$(T^{2} + 2 T - 12)^{2}$$
$43$ $$T^{4} + 44T^{2} + 16$$
$47$ $$T^{4} + 112T^{2} + 2304$$
$53$ $$T^{4} + 34T^{2} + 81$$
$59$ $$(T^{2} - 10 T + 12)^{2}$$
$61$ $$(T^{2} - 6 T - 43)^{2}$$
$67$ $$T^{4} + 232T^{2} + 144$$
$71$ $$(T^{2} - 22 T + 108)^{2}$$
$73$ $$T^{4} + 188T^{2} + 4624$$
$79$ $$(T^{2} - 16 T + 51)^{2}$$
$83$ $$(T^{2} + 9)^{2}$$
$89$ $$(T^{2} - 6 T - 108)^{2}$$
$97$ $$(T^{2} + 64)^{2}$$