# Properties

 Label 34.2.b.a Level $34$ Weight $2$ Character orbit 34.b Analytic conductor $0.271$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [34,2,Mod(33,34)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(34, base_ring=CyclotomicField(2))

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("34.33");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$34 = 2 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 34.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: $$0.271491366872$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes 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 $$\beta = 2\sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + \beta q^{3} + q^{4} - \beta q^{5} - \beta q^{6} - q^{8} - 5 q^{9} +O(q^{10})$$ q - q^2 + b * q^3 + q^4 - b * q^5 - b * q^6 - q^8 - 5 * q^9 $$q - q^{2} + \beta q^{3} + q^{4} - \beta q^{5} - \beta q^{6} - q^{8} - 5 q^{9} + \beta q^{10} - \beta q^{11} + \beta q^{12} + 2 q^{13} + 8 q^{15} + q^{16} + (\beta - 3) q^{17} + 5 q^{18} - 4 q^{19} - \beta q^{20} + \beta q^{22} + 2 \beta q^{23} - \beta q^{24} - 3 q^{25} - 2 q^{26} - 2 \beta q^{27} - \beta q^{29} - 8 q^{30} - q^{32} + 8 q^{33} + ( - \beta + 3) q^{34} - 5 q^{36} - 3 \beta q^{37} + 4 q^{38} + 2 \beta q^{39} + \beta q^{40} + 2 \beta q^{41} - 4 q^{43} - \beta q^{44} + 5 \beta q^{45} - 2 \beta q^{46} + \beta q^{48} + 7 q^{49} + 3 q^{50} + ( - 3 \beta - 8) q^{51} + 2 q^{52} + 6 q^{53} + 2 \beta q^{54} - 8 q^{55} - 4 \beta q^{57} + \beta q^{58} + 12 q^{59} + 8 q^{60} + 3 \beta q^{61} + q^{64} - 2 \beta q^{65} - 8 q^{66} - 4 q^{67} + (\beta - 3) q^{68} - 16 q^{69} + 2 \beta q^{71} + 5 q^{72} + 3 \beta q^{74} - 3 \beta q^{75} - 4 q^{76} - 2 \beta q^{78} - 6 \beta q^{79} - \beta q^{80} + q^{81} - 2 \beta q^{82} - 12 q^{83} + (3 \beta + 8) q^{85} + 4 q^{86} + 8 q^{87} + \beta q^{88} + 6 q^{89} - 5 \beta q^{90} + 2 \beta q^{92} + 4 \beta q^{95} - \beta q^{96} + 6 \beta q^{97} - 7 q^{98} + 5 \beta q^{99} +O(q^{100})$$ q - q^2 + b * q^3 + q^4 - b * q^5 - b * q^6 - q^8 - 5 * q^9 + b * q^10 - b * q^11 + b * q^12 + 2 * q^13 + 8 * q^15 + q^16 + (b - 3) * q^17 + 5 * q^18 - 4 * q^19 - b * q^20 + b * q^22 + 2*b * q^23 - b * q^24 - 3 * q^25 - 2 * q^26 - 2*b * q^27 - b * q^29 - 8 * q^30 - q^32 + 8 * q^33 + (-b + 3) * q^34 - 5 * q^36 - 3*b * q^37 + 4 * q^38 + 2*b * q^39 + b * q^40 + 2*b * q^41 - 4 * q^43 - b * q^44 + 5*b * q^45 - 2*b * q^46 + b * q^48 + 7 * q^49 + 3 * q^50 + (-3*b - 8) * q^51 + 2 * q^52 + 6 * q^53 + 2*b * q^54 - 8 * q^55 - 4*b * q^57 + b * q^58 + 12 * q^59 + 8 * q^60 + 3*b * q^61 + q^64 - 2*b * q^65 - 8 * q^66 - 4 * q^67 + (b - 3) * q^68 - 16 * q^69 + 2*b * q^71 + 5 * q^72 + 3*b * q^74 - 3*b * q^75 - 4 * q^76 - 2*b * q^78 - 6*b * q^79 - b * q^80 + q^81 - 2*b * q^82 - 12 * q^83 + (3*b + 8) * q^85 + 4 * q^86 + 8 * q^87 + b * q^88 + 6 * q^89 - 5*b * q^90 + 2*b * q^92 + 4*b * q^95 - b * q^96 + 6*b * q^97 - 7 * q^98 + 5*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 10 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 10 * q^9 $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 10 q^{9} + 4 q^{13} + 16 q^{15} + 2 q^{16} - 6 q^{17} + 10 q^{18} - 8 q^{19} - 6 q^{25} - 4 q^{26} - 16 q^{30} - 2 q^{32} + 16 q^{33} + 6 q^{34} - 10 q^{36} + 8 q^{38} - 8 q^{43} + 14 q^{49} + 6 q^{50} - 16 q^{51} + 4 q^{52} + 12 q^{53} - 16 q^{55} + 24 q^{59} + 16 q^{60} + 2 q^{64} - 16 q^{66} - 8 q^{67} - 6 q^{68} - 32 q^{69} + 10 q^{72} - 8 q^{76} + 2 q^{81} - 24 q^{83} + 16 q^{85} + 8 q^{86} + 16 q^{87} + 12 q^{89} - 14 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 10 * q^9 + 4 * q^13 + 16 * q^15 + 2 * q^16 - 6 * q^17 + 10 * q^18 - 8 * q^19 - 6 * q^25 - 4 * q^26 - 16 * q^30 - 2 * q^32 + 16 * q^33 + 6 * q^34 - 10 * q^36 + 8 * q^38 - 8 * q^43 + 14 * q^49 + 6 * q^50 - 16 * q^51 + 4 * q^52 + 12 * q^53 - 16 * q^55 + 24 * q^59 + 16 * q^60 + 2 * q^64 - 16 * q^66 - 8 * q^67 - 6 * q^68 - 32 * q^69 + 10 * q^72 - 8 * q^76 + 2 * q^81 - 24 * q^83 + 16 * q^85 + 8 * q^86 + 16 * q^87 + 12 * q^89 - 14 * q^98

## Character values

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

 $$n$$ $$3$$ $$\chi(n)$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
33.1
 − 1.41421i 1.41421i
−1.00000 2.82843i 1.00000 2.82843i 2.82843i 0 −1.00000 −5.00000 2.82843i
33.2 −1.00000 2.82843i 1.00000 2.82843i 2.82843i 0 −1.00000 −5.00000 2.82843i
 $$n$$: 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 34.2.b.a 2
3.b odd 2 1 306.2.b.d 2
4.b odd 2 1 272.2.b.a 2
5.b even 2 1 850.2.b.f 2
5.c odd 4 2 850.2.d.i 4
7.b odd 2 1 1666.2.b.c 2
8.b even 2 1 1088.2.b.b 2
8.d odd 2 1 1088.2.b.a 2
12.b even 2 1 2448.2.c.n 2
17.b even 2 1 inner 34.2.b.a 2
17.c even 4 2 578.2.a.d 2
17.d even 8 2 578.2.c.a 2
17.d even 8 2 578.2.c.d 2
17.e odd 16 8 578.2.d.g 8
51.c odd 2 1 306.2.b.d 2
51.f odd 4 2 5202.2.a.u 2
68.d odd 2 1 272.2.b.a 2
68.f odd 4 2 4624.2.a.s 2
85.c even 2 1 850.2.b.f 2
85.g odd 4 2 850.2.d.i 4
119.d odd 2 1 1666.2.b.c 2
136.e odd 2 1 1088.2.b.a 2
136.h even 2 1 1088.2.b.b 2
204.h even 2 1 2448.2.c.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
34.2.b.a 2 1.a even 1 1 trivial
34.2.b.a 2 17.b even 2 1 inner
272.2.b.a 2 4.b odd 2 1
272.2.b.a 2 68.d odd 2 1
306.2.b.d 2 3.b odd 2 1
306.2.b.d 2 51.c odd 2 1
578.2.a.d 2 17.c even 4 2
578.2.c.a 2 17.d even 8 2
578.2.c.d 2 17.d even 8 2
578.2.d.g 8 17.e odd 16 8
850.2.b.f 2 5.b even 2 1
850.2.b.f 2 85.c even 2 1
850.2.d.i 4 5.c odd 4 2
850.2.d.i 4 85.g odd 4 2
1088.2.b.a 2 8.d odd 2 1
1088.2.b.a 2 136.e odd 2 1
1088.2.b.b 2 8.b even 2 1
1088.2.b.b 2 136.h even 2 1
1666.2.b.c 2 7.b odd 2 1
1666.2.b.c 2 119.d odd 2 1
2448.2.c.n 2 12.b even 2 1
2448.2.c.n 2 204.h even 2 1
4624.2.a.s 2 68.f odd 4 2
5202.2.a.u 2 51.f odd 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(34, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2} + 8$$
$5$ $$T^{2} + 8$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 8$$
$13$ $$(T - 2)^{2}$$
$17$ $$T^{2} + 6T + 17$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2} + 32$$
$29$ $$T^{2} + 8$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 72$$
$41$ $$T^{2} + 32$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2}$$
$53$ $$(T - 6)^{2}$$
$59$ $$(T - 12)^{2}$$
$61$ $$T^{2} + 72$$
$67$ $$(T + 4)^{2}$$
$71$ $$T^{2} + 32$$
$73$ $$T^{2}$$
$79$ $$T^{2} + 288$$
$83$ $$(T + 12)^{2}$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 288$$