# Properties

 Label 34.2.c.b Level $34$ Weight $2$ Character orbit 34.c Analytic conductor $0.271$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [34,2,Mod(13,34)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(34, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("34.13");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$34 = 2 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 34.c (of order $$4$$, degree $$2$$, 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: $$0.271491366872$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ 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: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} - q^{4} + ( - i - 1) q^{5} - i q^{8} - 3 i q^{9} +O(q^{10})$$ q + i * q^2 - q^4 + (-i - 1) * q^5 - i * q^8 - 3*i * q^9 $$q + i q^{2} - q^{4} + ( - i - 1) q^{5} - i q^{8} - 3 i q^{9} + ( - i + 1) q^{10} + (4 i - 4) q^{11} + 4 q^{13} + q^{16} + (4 i - 1) q^{17} + 3 q^{18} + 4 i q^{19} + (i + 1) q^{20} + ( - 4 i - 4) q^{22} + ( - 4 i + 4) q^{23} - 3 i q^{25} + 4 i q^{26} + ( - 3 i - 3) q^{29} + ( - 4 i - 4) q^{31} + i q^{32} + ( - i - 4) q^{34} + 3 i q^{36} + (3 i + 3) q^{37} - 4 q^{38} + (i - 1) q^{40} + ( - i + 1) q^{41} - 4 i q^{43} + ( - 4 i + 4) q^{44} + (3 i - 3) q^{45} + (4 i + 4) q^{46} + 8 q^{47} + 7 i q^{49} + 3 q^{50} - 4 q^{52} - 4 i q^{53} + 8 q^{55} + ( - 3 i + 3) q^{58} + 4 i q^{59} + (9 i - 9) q^{61} + ( - 4 i + 4) q^{62} - q^{64} + ( - 4 i - 4) q^{65} - 12 q^{67} + ( - 4 i + 1) q^{68} + ( - 4 i - 4) q^{71} - 3 q^{72} + (5 i + 5) q^{73} + (3 i - 3) q^{74} - 4 i q^{76} + ( - 8 i + 8) q^{79} + ( - i - 1) q^{80} - 9 q^{81} + (i + 1) q^{82} - 4 i q^{83} + ( - 3 i + 5) q^{85} + 4 q^{86} + (4 i + 4) q^{88} + ( - 3 i - 3) q^{90} + (4 i - 4) q^{92} + 8 i q^{94} + ( - 4 i + 4) q^{95} + (3 i + 3) q^{97} - 7 q^{98} + (12 i + 12) q^{99} +O(q^{100})$$ q + i * q^2 - q^4 + (-i - 1) * q^5 - i * q^8 - 3*i * q^9 + (-i + 1) * q^10 + (4*i - 4) * q^11 + 4 * q^13 + q^16 + (4*i - 1) * q^17 + 3 * q^18 + 4*i * q^19 + (i + 1) * q^20 + (-4*i - 4) * q^22 + (-4*i + 4) * q^23 - 3*i * q^25 + 4*i * q^26 + (-3*i - 3) * q^29 + (-4*i - 4) * q^31 + i * q^32 + (-i - 4) * q^34 + 3*i * q^36 + (3*i + 3) * q^37 - 4 * q^38 + (i - 1) * q^40 + (-i + 1) * q^41 - 4*i * q^43 + (-4*i + 4) * q^44 + (3*i - 3) * q^45 + (4*i + 4) * q^46 + 8 * q^47 + 7*i * q^49 + 3 * q^50 - 4 * q^52 - 4*i * q^53 + 8 * q^55 + (-3*i + 3) * q^58 + 4*i * q^59 + (9*i - 9) * q^61 + (-4*i + 4) * q^62 - q^64 + (-4*i - 4) * q^65 - 12 * q^67 + (-4*i + 1) * q^68 + (-4*i - 4) * q^71 - 3 * q^72 + (5*i + 5) * q^73 + (3*i - 3) * q^74 - 4*i * q^76 + (-8*i + 8) * q^79 + (-i - 1) * q^80 - 9 * q^81 + (i + 1) * q^82 - 4*i * q^83 + (-3*i + 5) * q^85 + 4 * q^86 + (4*i + 4) * q^88 + (-3*i - 3) * q^90 + (4*i - 4) * q^92 + 8*i * q^94 + (-4*i + 4) * q^95 + (3*i + 3) * q^97 - 7 * q^98 + (12*i + 12) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 2 q^{5}+O(q^{10})$$ 2 * q - 2 * q^4 - 2 * q^5 $$2 q - 2 q^{4} - 2 q^{5} + 2 q^{10} - 8 q^{11} + 8 q^{13} + 2 q^{16} - 2 q^{17} + 6 q^{18} + 2 q^{20} - 8 q^{22} + 8 q^{23} - 6 q^{29} - 8 q^{31} - 8 q^{34} + 6 q^{37} - 8 q^{38} - 2 q^{40} + 2 q^{41} + 8 q^{44} - 6 q^{45} + 8 q^{46} + 16 q^{47} + 6 q^{50} - 8 q^{52} + 16 q^{55} + 6 q^{58} - 18 q^{61} + 8 q^{62} - 2 q^{64} - 8 q^{65} - 24 q^{67} + 2 q^{68} - 8 q^{71} - 6 q^{72} + 10 q^{73} - 6 q^{74} + 16 q^{79} - 2 q^{80} - 18 q^{81} + 2 q^{82} + 10 q^{85} + 8 q^{86} + 8 q^{88} - 6 q^{90} - 8 q^{92} + 8 q^{95} + 6 q^{97} - 14 q^{98} + 24 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 - 2 * q^5 + 2 * q^10 - 8 * q^11 + 8 * q^13 + 2 * q^16 - 2 * q^17 + 6 * q^18 + 2 * q^20 - 8 * q^22 + 8 * q^23 - 6 * q^29 - 8 * q^31 - 8 * q^34 + 6 * q^37 - 8 * q^38 - 2 * q^40 + 2 * q^41 + 8 * q^44 - 6 * q^45 + 8 * q^46 + 16 * q^47 + 6 * q^50 - 8 * q^52 + 16 * q^55 + 6 * q^58 - 18 * q^61 + 8 * q^62 - 2 * q^64 - 8 * q^65 - 24 * q^67 + 2 * q^68 - 8 * q^71 - 6 * q^72 + 10 * q^73 - 6 * q^74 + 16 * q^79 - 2 * q^80 - 18 * q^81 + 2 * q^82 + 10 * q^85 + 8 * q^86 + 8 * q^88 - 6 * q^90 - 8 * q^92 + 8 * q^95 + 6 * q^97 - 14 * q^98 + 24 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/34\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$i$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 1.00000i − 1.00000i
1.00000i 0 −1.00000 −1.00000 1.00000i 0 0 1.00000i 3.00000i 1.00000 1.00000i
21.1 1.00000i 0 −1.00000 −1.00000 + 1.00000i 0 0 1.00000i 3.00000i 1.00000 + 1.00000i
 $$n$$: 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
17.c even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 34.2.c.b 2
3.b odd 2 1 306.2.g.d 2
4.b odd 2 1 272.2.o.c 2
5.b even 2 1 850.2.h.c 2
5.c odd 4 1 850.2.g.a 2
5.c odd 4 1 850.2.g.d 2
8.b even 2 1 1088.2.o.k 2
8.d odd 2 1 1088.2.o.i 2
12.b even 2 1 2448.2.be.j 2
17.b even 2 1 578.2.c.b 2
17.c even 4 1 inner 34.2.c.b 2
17.c even 4 1 578.2.c.b 2
17.d even 8 2 578.2.a.b 2
17.d even 8 2 578.2.b.c 2
17.e odd 16 8 578.2.d.f 8
51.f odd 4 1 306.2.g.d 2
51.g odd 8 2 5202.2.a.bb 2
68.f odd 4 1 272.2.o.c 2
68.g odd 8 2 4624.2.a.l 2
85.f odd 4 1 850.2.g.d 2
85.i odd 4 1 850.2.g.a 2
85.j even 4 1 850.2.h.c 2
136.i even 4 1 1088.2.o.k 2
136.j odd 4 1 1088.2.o.i 2
204.l even 4 1 2448.2.be.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
34.2.c.b 2 1.a even 1 1 trivial
34.2.c.b 2 17.c even 4 1 inner
272.2.o.c 2 4.b odd 2 1
272.2.o.c 2 68.f odd 4 1
306.2.g.d 2 3.b odd 2 1
306.2.g.d 2 51.f odd 4 1
578.2.a.b 2 17.d even 8 2
578.2.b.c 2 17.d even 8 2
578.2.c.b 2 17.b even 2 1
578.2.c.b 2 17.c even 4 1
578.2.d.f 8 17.e odd 16 8
850.2.g.a 2 5.c odd 4 1
850.2.g.a 2 85.i odd 4 1
850.2.g.d 2 5.c odd 4 1
850.2.g.d 2 85.f odd 4 1
850.2.h.c 2 5.b even 2 1
850.2.h.c 2 85.j even 4 1
1088.2.o.i 2 8.d odd 2 1
1088.2.o.i 2 136.j odd 4 1
1088.2.o.k 2 8.b even 2 1
1088.2.o.k 2 136.i even 4 1
2448.2.be.j 2 12.b even 2 1
2448.2.be.j 2 204.l even 4 1
4624.2.a.l 2 68.g odd 8 2
5202.2.a.bb 2 51.g odd 8 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{2}^{\mathrm{new}}(34, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 2T + 2$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 8T + 32$$
$13$ $$(T - 4)^{2}$$
$17$ $$T^{2} + 2T + 17$$
$19$ $$T^{2} + 16$$
$23$ $$T^{2} - 8T + 32$$
$29$ $$T^{2} + 6T + 18$$
$31$ $$T^{2} + 8T + 32$$
$37$ $$T^{2} - 6T + 18$$
$41$ $$T^{2} - 2T + 2$$
$43$ $$T^{2} + 16$$
$47$ $$(T - 8)^{2}$$
$53$ $$T^{2} + 16$$
$59$ $$T^{2} + 16$$
$61$ $$T^{2} + 18T + 162$$
$67$ $$(T + 12)^{2}$$
$71$ $$T^{2} + 8T + 32$$
$73$ $$T^{2} - 10T + 50$$
$79$ $$T^{2} - 16T + 128$$
$83$ $$T^{2} + 16$$
$89$ $$T^{2}$$
$97$ $$T^{2} - 6T + 18$$