# Properties

 Label 1575.2.b.c Level $1575$ Weight $2$ Character orbit 1575.b Analytic conductor $12.576$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,2,Mod(251,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("1575.251");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1575 = 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1575.b (of order $$2$$, degree $$1$$, 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: $$12.5764383184$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 4x^{2} + 1$$ x^4 + 4*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 315) 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 4x^{2} + 1$$ :

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$1226$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 − 1.93185i − 0.517638i 0.517638i 1.93185i
1.93185i 0 −1.73205 0 0 −1.00000 + 2.44949i 0.517638i 0 0
251.2 0.517638i 0 1.73205 0 0 −1.00000 + 2.44949i 1.93185i 0 0
251.3 0.517638i 0 1.73205 0 0 −1.00000 2.44949i 1.93185i 0 0
251.4 1.93185i 0 −1.73205 0 0 −1.00000 2.44949i 0.517638i 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
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1575.2.b.c 4
3.b odd 2 1 1575.2.b.b 4
5.b even 2 1 315.2.b.b yes 4
5.c odd 4 2 1575.2.g.a 8
7.b odd 2 1 1575.2.b.b 4
15.d odd 2 1 315.2.b.a 4
15.e even 4 2 1575.2.g.c 8
20.d odd 2 1 5040.2.f.c 4
21.c even 2 1 inner 1575.2.b.c 4
35.c odd 2 1 315.2.b.a 4
35.f even 4 2 1575.2.g.c 8
60.h even 2 1 5040.2.f.a 4
105.g even 2 1 315.2.b.b yes 4
105.k odd 4 2 1575.2.g.a 8
140.c even 2 1 5040.2.f.a 4
420.o odd 2 1 5040.2.f.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.b.a 4 15.d odd 2 1
315.2.b.a 4 35.c odd 2 1
315.2.b.b yes 4 5.b even 2 1
315.2.b.b yes 4 105.g even 2 1
1575.2.b.b 4 3.b odd 2 1
1575.2.b.b 4 7.b odd 2 1
1575.2.b.c 4 1.a even 1 1 trivial
1575.2.b.c 4 21.c even 2 1 inner
1575.2.g.a 8 5.c odd 4 2
1575.2.g.a 8 105.k odd 4 2
1575.2.g.c 8 15.e even 4 2
1575.2.g.c 8 35.f even 4 2
5040.2.f.a 4 60.h even 2 1
5040.2.f.a 4 140.c even 2 1
5040.2.f.c 4 20.d odd 2 1
5040.2.f.c 4 420.o odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1575, [\chi])$$:

 $$T_{2}^{4} + 4T_{2}^{2} + 1$$ T2^4 + 4*T2^2 + 1 $$T_{37}^{2} + 4T_{37} - 8$$ T37^2 + 4*T37 - 8 $$T_{47}^{2} - 12T_{47} + 24$$ T47^2 - 12*T47 + 24 $$T_{67}^{2} - 8T_{67} - 32$$ T67^2 - 8*T67 - 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 4T^{2} + 1$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 2 T + 7)^{2}$$
$11$ $$T^{4} + 28T^{2} + 4$$
$13$ $$T^{4} + 48T^{2} + 144$$
$17$ $$(T^{2} - 12)^{2}$$
$19$ $$T^{4} + 48T^{2} + 144$$
$23$ $$(T^{2} + 2)^{2}$$
$29$ $$(T^{2} + 2)^{2}$$
$31$ $$T^{4} + 48T^{2} + 144$$
$37$ $$(T^{2} + 4 T - 8)^{2}$$
$41$ $$(T^{2} - 108)^{2}$$
$43$ $$(T^{2} + 4 T - 44)^{2}$$
$47$ $$(T^{2} - 12 T + 24)^{2}$$
$53$ $$T^{4} + 124T^{2} + 2116$$
$59$ $$(T^{2} + 12 T + 24)^{2}$$
$61$ $$T^{4} + 192T^{2} + 2304$$
$67$ $$(T^{2} - 8 T - 32)^{2}$$
$71$ $$T^{4} + 316 T^{2} + 20164$$
$73$ $$T^{4} + 48T^{2} + 144$$
$79$ $$(T^{2} + 8 T - 32)^{2}$$
$83$ $$(T^{2} + 12 T + 24)^{2}$$
$89$ $$(T^{2} + 12 T - 12)^{2}$$
$97$ $$T^{4} + 336 T^{2} + 24336$$