# Properties

 Label 1700.1.n.a Level $1700$ Weight $1$ Character orbit 1700.n Analytic conductor $0.848$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -4 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1700,1,Mod(599,1700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1700, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 2, 3]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("1700.599");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1700 = 2^{2} \cdot 5^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1700.n (of order $$4$$, degree $$2$$, 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: $$0.848410521476$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 68) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.19652.1 Artin image: $C_4\times C_4\wr C_2$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{32} - \cdots)$$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - q^{8} + i q^{9}+O(q^{10})$$ q - q^2 + q^4 - q^8 + z * q^9 $$q - q^{2} + q^{4} - q^{8} + i q^{9} + q^{16} + i q^{17} - i q^{18} + (i - 1) q^{29} - q^{32} - i q^{34} + i q^{36} + (i + 1) q^{37} + (i + 1) q^{41} - i q^{49} + ( - i + 1) q^{58} + ( - i - 1) q^{61} + q^{64} + i q^{68} - i q^{72} + (i + 1) q^{73} + ( - i - 1) q^{74} - q^{81} + ( - i - 1) q^{82} + (i + 1) q^{97} + i q^{98} +O(q^{100})$$ q - q^2 + q^4 - q^8 + z * q^9 + q^16 + z * q^17 - z * q^18 + (z - 1) * q^29 - q^32 - z * q^34 + z * q^36 + (z + 1) * q^37 + (z + 1) * q^41 - z * q^49 + (-z + 1) * q^58 + (-z - 1) * q^61 + q^64 + z * q^68 - z * q^72 + (z + 1) * q^73 + (-z - 1) * q^74 - q^81 + (-z - 1) * q^82 + (z + 1) * q^97 + z * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 2 q^{16} - 2 q^{29} - 2 q^{32} + 2 q^{37} + 2 q^{41} + 2 q^{58} - 2 q^{61} + 2 q^{64} + 2 q^{73} - 2 q^{74} - 2 q^{81} - 2 q^{82} + 2 q^{97}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 2 * q^16 - 2 * q^29 - 2 * q^32 + 2 * q^37 + 2 * q^41 + 2 * q^58 - 2 * q^61 + 2 * q^64 + 2 * q^73 - 2 * q^74 - 2 * q^81 - 2 * q^82 + 2 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1700\mathbb{Z}\right)^\times$$.

 $$n$$ $$477$$ $$851$$ $$1601$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-i$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
599.1
 1.00000i − 1.00000i
−1.00000 0 1.00000 0 0 0 −1.00000 1.00000i 0
999.1 −1.00000 0 1.00000 0 0 0 −1.00000 1.00000i 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
85.j even 4 1 inner
340.n odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1700.1.n.a 2
4.b odd 2 1 CM 1700.1.n.a 2
5.b even 2 1 1700.1.n.b 2
5.c odd 4 1 68.1.f.a 2
5.c odd 4 1 1700.1.p.a 2
15.e even 4 1 612.1.l.a 2
17.c even 4 1 1700.1.n.b 2
20.d odd 2 1 1700.1.n.b 2
20.e even 4 1 68.1.f.a 2
20.e even 4 1 1700.1.p.a 2
35.f even 4 1 3332.1.m.b 2
35.k even 12 2 3332.1.bc.b 4
35.l odd 12 2 3332.1.bc.c 4
40.i odd 4 1 1088.1.p.a 2
40.k even 4 1 1088.1.p.a 2
60.l odd 4 1 612.1.l.a 2
68.f odd 4 1 1700.1.n.b 2
85.f odd 4 1 1156.1.f.b 2
85.f odd 4 1 1700.1.p.a 2
85.g odd 4 1 1156.1.f.b 2
85.i odd 4 1 68.1.f.a 2
85.j even 4 1 inner 1700.1.n.a 2
85.k odd 8 2 1156.1.d.a 2
85.n odd 8 2 1156.1.c.b 2
85.o even 16 4 1156.1.g.b 8
85.r even 16 4 1156.1.g.b 8
140.j odd 4 1 3332.1.m.b 2
140.w even 12 2 3332.1.bc.c 4
140.x odd 12 2 3332.1.bc.b 4
255.r even 4 1 612.1.l.a 2
340.i even 4 1 68.1.f.a 2
340.n odd 4 1 inner 1700.1.n.a 2
340.r even 4 1 1156.1.f.b 2
340.s even 4 1 1156.1.f.b 2
340.s even 4 1 1700.1.p.a 2
340.w even 8 2 1156.1.c.b 2
340.z even 8 2 1156.1.d.a 2
340.bc odd 16 4 1156.1.g.b 8
340.bj odd 16 4 1156.1.g.b 8
595.l even 4 1 3332.1.m.b 2
595.bm odd 12 2 3332.1.bc.c 4
595.bt even 12 2 3332.1.bc.b 4
680.s odd 4 1 1088.1.p.a 2
680.bl even 4 1 1088.1.p.a 2
1020.q odd 4 1 612.1.l.a 2
2380.bo odd 4 1 3332.1.m.b 2
2380.cu odd 12 2 3332.1.bc.b 4
2380.dq even 12 2 3332.1.bc.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.1.f.a 2 5.c odd 4 1
68.1.f.a 2 20.e even 4 1
68.1.f.a 2 85.i odd 4 1
68.1.f.a 2 340.i even 4 1
612.1.l.a 2 15.e even 4 1
612.1.l.a 2 60.l odd 4 1
612.1.l.a 2 255.r even 4 1
612.1.l.a 2 1020.q odd 4 1
1088.1.p.a 2 40.i odd 4 1
1088.1.p.a 2 40.k even 4 1
1088.1.p.a 2 680.s odd 4 1
1088.1.p.a 2 680.bl even 4 1
1156.1.c.b 2 85.n odd 8 2
1156.1.c.b 2 340.w even 8 2
1156.1.d.a 2 85.k odd 8 2
1156.1.d.a 2 340.z even 8 2
1156.1.f.b 2 85.f odd 4 1
1156.1.f.b 2 85.g odd 4 1
1156.1.f.b 2 340.r even 4 1
1156.1.f.b 2 340.s even 4 1
1156.1.g.b 8 85.o even 16 4
1156.1.g.b 8 85.r even 16 4
1156.1.g.b 8 340.bc odd 16 4
1156.1.g.b 8 340.bj odd 16 4
1700.1.n.a 2 1.a even 1 1 trivial
1700.1.n.a 2 4.b odd 2 1 CM
1700.1.n.a 2 85.j even 4 1 inner
1700.1.n.a 2 340.n odd 4 1 inner
1700.1.n.b 2 5.b even 2 1
1700.1.n.b 2 17.c even 4 1
1700.1.n.b 2 20.d odd 2 1
1700.1.n.b 2 68.f odd 4 1
1700.1.p.a 2 5.c odd 4 1
1700.1.p.a 2 20.e even 4 1
1700.1.p.a 2 85.f odd 4 1
1700.1.p.a 2 340.s even 4 1
3332.1.m.b 2 35.f even 4 1
3332.1.m.b 2 140.j odd 4 1
3332.1.m.b 2 595.l even 4 1
3332.1.m.b 2 2380.bo odd 4 1
3332.1.bc.b 4 35.k even 12 2
3332.1.bc.b 4 140.x odd 12 2
3332.1.bc.b 4 595.bt even 12 2
3332.1.bc.b 4 2380.cu odd 12 2
3332.1.bc.c 4 35.l odd 12 2
3332.1.bc.c 4 140.w even 12 2
3332.1.bc.c 4 595.bm odd 12 2
3332.1.bc.c 4 2380.dq even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{37}^{2} - 2T_{37} + 2$$ acting on $$S_{1}^{\mathrm{new}}(1700, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 1$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 2T + 2$$
$31$ $$T^{2}$$
$37$ $$T^{2} - 2T + 2$$
$41$ $$T^{2} - 2T + 2$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 2T + 2$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 2T + 2$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} - 2T + 2$$