# Properties

 Label 1700.1.d.b Level $1700$ Weight $1$ Character orbit 1700.d Analytic conductor $0.848$ Analytic rank $0$ Dimension $2$ Projective image $D_{2}$ CM/RM discs -4, -68, 17 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1700,1,Mod(1699,1700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1700, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1700.1699");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1700 = 2^{2} \cdot 5^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1700.d (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: $$0.848410521476$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 68) Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(i, \sqrt{17})$$ Artin image: $D_4:C_2$ Artin field: Galois closure of 8.0.1156000000.2

## $q$-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.

 $$f(q)$$ $$=$$ $$q - i q^{2} - q^{4} + i q^{8} + q^{9} +O(q^{10})$$ q - z * q^2 - q^4 + z * q^8 + q^9 $$q - i q^{2} - q^{4} + i q^{8} + q^{9} + 2 i q^{13} + q^{16} + i q^{17} - i q^{18} + 2 q^{26} - i q^{32} + q^{34} - q^{36} + q^{49} - 2 i q^{52} - 2 i q^{53} - q^{64} - i q^{68} + i q^{72} + q^{81} + 2 q^{89} - i q^{98} +O(q^{100})$$ q - z * q^2 - q^4 + z * q^8 + q^9 + 2*z * q^13 + q^16 + z * q^17 - z * q^18 + 2 * q^26 - z * q^32 + q^34 - q^36 + q^49 - 2*z * q^52 - 2*z * q^53 - q^64 - z * q^68 + z * q^72 + q^81 + 2 * q^89 - z * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 2 * q^9 $$2 q - 2 q^{4} + 2 q^{9} + 2 q^{16} + 4 q^{26} + 2 q^{34} - 2 q^{36} + 2 q^{49} - 2 q^{64} + 2 q^{81} + 4 q^{89}+O(q^{100})$$ 2 * q - 2 * q^4 + 2 * q^9 + 2 * q^16 + 4 * q^26 + 2 * q^34 - 2 * q^36 + 2 * q^49 - 2 * q^64 + 2 * q^81 + 4 * q^89

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1699.1
 1.00000i − 1.00000i
1.00000i 0 −1.00000 0 0 0 1.00000i 1.00000 0
1699.2 1.00000i 0 −1.00000 0 0 0 1.00000i 1.00000 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 $$\Q(\sqrt{-1})$$
17.b even 2 1 RM by $$\Q(\sqrt{17})$$
68.d odd 2 1 CM by $$\Q(\sqrt{-17})$$
5.b even 2 1 inner
20.d odd 2 1 inner
85.c even 2 1 inner
340.d odd 2 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1700, [\chi])$$:

 $$T_{3}$$ T3 $$T_{11}$$ T11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 1$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 4$$
$59$ $$T^{2}$$
$61$ $$T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$(T - 2)^{2}$$
$97$ $$T^{2}$$