# Properties

 Label 1700.2.g.a Level $1700$ Weight $2$ Character orbit 1700.g Analytic conductor $13.575$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1700,2,Mod(849,1700)]

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([0, 1, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1700.849");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1700 = 2^{2} \cdot 5^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1700.g (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: $$13.5745683436$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 68) 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.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{3} - 3 \beta_{3} q^{7} - q^{9}+O(q^{10})$$ q - b3 * q^3 - 3*b3 * q^7 - q^9 $$q - \beta_{3} q^{3} - 3 \beta_{3} q^{7} - q^{9} + \beta_{2} q^{11} + 4 \beta_1 q^{13} + ( - 2 \beta_{3} + 3 \beta_1) q^{17} + 4 q^{19} + 6 q^{21} + \beta_{3} q^{23} + 4 \beta_{3} q^{27} + 2 \beta_{2} q^{29} - 3 \beta_{2} q^{31} - 2 \beta_1 q^{33} - 6 \beta_{3} q^{37} - 4 \beta_{2} q^{39} - 8 \beta_{2} q^{41} - 8 \beta_1 q^{43} - 12 \beta_1 q^{47} + 11 q^{49} + ( - 3 \beta_{2} + 4) q^{51} + 6 \beta_1 q^{53} - 4 \beta_{3} q^{57} + 6 \beta_{2} q^{61} + 3 \beta_{3} q^{63} - 4 \beta_1 q^{67} - 2 q^{69} - 5 \beta_{2} q^{71} - 6 \beta_1 q^{77} + 3 \beta_{2} q^{79} - 5 q^{81} - 4 \beta_1 q^{87} - 12 q^{89} - 12 \beta_{2} q^{91} + 6 \beta_1 q^{93} - \beta_{2} q^{99}+O(q^{100})$$ q - b3 * q^3 - 3*b3 * q^7 - q^9 + b2 * q^11 + 4*b1 * q^13 + (-2*b3 + 3*b1) * q^17 + 4 * q^19 + 6 * q^21 + b3 * q^23 + 4*b3 * q^27 + 2*b2 * q^29 - 3*b2 * q^31 - 2*b1 * q^33 - 6*b3 * q^37 - 4*b2 * q^39 - 8*b2 * q^41 - 8*b1 * q^43 - 12*b1 * q^47 + 11 * q^49 + (-3*b2 + 4) * q^51 + 6*b1 * q^53 - 4*b3 * q^57 + 6*b2 * q^61 + 3*b3 * q^63 - 4*b1 * q^67 - 2 * q^69 - 5*b2 * q^71 - 6*b1 * q^77 + 3*b2 * q^79 - 5 * q^81 - 4*b1 * q^87 - 12 * q^89 - 12*b2 * q^91 + 6*b1 * q^93 - b2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^9 $$4 q - 4 q^{9} + 16 q^{19} + 24 q^{21} + 44 q^{49} + 16 q^{51} - 8 q^{69} - 20 q^{81} - 48 q^{89}+O(q^{100})$$ 4 * q - 4 * q^9 + 16 * q^19 + 24 * q^21 + 44 * q^49 + 16 * q^51 - 8 * q^69 - 20 * q^81 - 48 * q^89

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{8}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}$$ -v^3 + v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\zeta_{8}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 2$$ (-b3 + b2) / 2

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
849.1
 0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i
0 −1.41421 0 0 0 −4.24264 0 −1.00000 0
849.2 0 −1.41421 0 0 0 −4.24264 0 −1.00000 0
849.3 0 1.41421 0 0 0 4.24264 0 −1.00000 0
849.4 0 1.41421 0 0 0 4.24264 0 −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
5.b even 2 1 inner
17.b even 2 1 inner
85.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1700.2.g.a 4
5.b even 2 1 inner 1700.2.g.a 4
5.c odd 4 1 68.2.b.a 2
5.c odd 4 1 1700.2.c.a 2
15.e even 4 1 612.2.b.a 2
17.b even 2 1 inner 1700.2.g.a 4
20.e even 4 1 272.2.b.c 2
35.f even 4 1 3332.2.b.a 2
40.i odd 4 1 1088.2.b.e 2
40.k even 4 1 1088.2.b.f 2
60.l odd 4 1 2448.2.c.d 2
85.c even 2 1 inner 1700.2.g.a 4
85.f odd 4 1 1156.2.a.c 2
85.g odd 4 1 68.2.b.a 2
85.g odd 4 1 1700.2.c.a 2
85.i odd 4 1 1156.2.a.c 2
85.k odd 8 1 1156.2.e.a 2
85.k odd 8 1 1156.2.e.b 2
85.n odd 8 1 1156.2.e.a 2
85.n odd 8 1 1156.2.e.b 2
85.o even 16 4 1156.2.h.d 8
85.r even 16 4 1156.2.h.d 8
255.o even 4 1 612.2.b.a 2
340.i even 4 1 4624.2.a.n 2
340.r even 4 1 272.2.b.c 2
340.s even 4 1 4624.2.a.n 2
595.p even 4 1 3332.2.b.a 2
680.u even 4 1 1088.2.b.f 2
680.bi odd 4 1 1088.2.b.e 2
1020.x odd 4 1 2448.2.c.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.2.b.a 2 5.c odd 4 1
68.2.b.a 2 85.g odd 4 1
272.2.b.c 2 20.e even 4 1
272.2.b.c 2 340.r even 4 1
612.2.b.a 2 15.e even 4 1
612.2.b.a 2 255.o even 4 1
1088.2.b.e 2 40.i odd 4 1
1088.2.b.e 2 680.bi odd 4 1
1088.2.b.f 2 40.k even 4 1
1088.2.b.f 2 680.u even 4 1
1156.2.a.c 2 85.f odd 4 1
1156.2.a.c 2 85.i odd 4 1
1156.2.e.a 2 85.k odd 8 1
1156.2.e.a 2 85.n odd 8 1
1156.2.e.b 2 85.k odd 8 1
1156.2.e.b 2 85.n odd 8 1
1156.2.h.d 8 85.o even 16 4
1156.2.h.d 8 85.r even 16 4
1700.2.c.a 2 5.c odd 4 1
1700.2.c.a 2 85.g odd 4 1
1700.2.g.a 4 1.a even 1 1 trivial
1700.2.g.a 4 5.b even 2 1 inner
1700.2.g.a 4 17.b even 2 1 inner
1700.2.g.a 4 85.c even 2 1 inner
2448.2.c.d 2 60.l odd 4 1
2448.2.c.d 2 1020.x odd 4 1
3332.2.b.a 2 35.f even 4 1
3332.2.b.a 2 595.p even 4 1
4624.2.a.n 2 340.i even 4 1
4624.2.a.n 2 340.s even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 2$$ acting on $$S_{2}^{\mathrm{new}}(1700, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - 2)^{2}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} - 18)^{2}$$
$11$ $$(T^{2} + 2)^{2}$$
$13$ $$(T^{2} + 16)^{2}$$
$17$ $$T^{4} + 2T^{2} + 289$$
$19$ $$(T - 4)^{4}$$
$23$ $$(T^{2} - 2)^{2}$$
$29$ $$(T^{2} + 8)^{2}$$
$31$ $$(T^{2} + 18)^{2}$$
$37$ $$(T^{2} - 72)^{2}$$
$41$ $$(T^{2} + 128)^{2}$$
$43$ $$(T^{2} + 64)^{2}$$
$47$ $$(T^{2} + 144)^{2}$$
$53$ $$(T^{2} + 36)^{2}$$
$59$ $$T^{4}$$
$61$ $$(T^{2} + 72)^{2}$$
$67$ $$(T^{2} + 16)^{2}$$
$71$ $$(T^{2} + 50)^{2}$$
$73$ $$T^{4}$$
$79$ $$(T^{2} + 18)^{2}$$
$83$ $$T^{4}$$
$89$ $$(T + 12)^{4}$$
$97$ $$T^{4}$$