# Properties

 Label 34.2.d.a Level $34$ Weight $2$ Character orbit 34.d Analytic conductor $0.271$ Analytic rank $0$ Dimension $4$ 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(9,34)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(34, base_ring=CyclotomicField(8))

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("34.9");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$34 = 2 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 34.d (of order $$8$$, degree $$4$$, 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: $$4$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-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 primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{8}^{3} q^{2} + ( - \zeta_{8}^{3} + \zeta_{8}^{2}) q^{3} - \zeta_{8}^{2} q^{4} + (2 \zeta_{8} - 2) q^{5} + (\zeta_{8}^{2} - \zeta_{8}) q^{6} + ( - 2 \zeta_{8}^{2} - 2 \zeta_{8}) q^{7} + \zeta_{8} q^{8} + ( - \zeta_{8}^{2} - \zeta_{8} - 1) q^{9} +O(q^{10})$$ q + z^3 * q^2 + (-z^3 + z^2) * q^3 - z^2 * q^4 + (2*z - 2) * q^5 + (z^2 - z) * q^6 + (-2*z^2 - 2*z) * q^7 + z * q^8 + (-z^2 - z - 1) * q^9 $$q + \zeta_{8}^{3} q^{2} + ( - \zeta_{8}^{3} + \zeta_{8}^{2}) q^{3} - \zeta_{8}^{2} q^{4} + (2 \zeta_{8} - 2) q^{5} + (\zeta_{8}^{2} - \zeta_{8}) q^{6} + ( - 2 \zeta_{8}^{2} - 2 \zeta_{8}) q^{7} + \zeta_{8} q^{8} + ( - \zeta_{8}^{2} - \zeta_{8} - 1) q^{9} + ( - 2 \zeta_{8}^{3} - 2) q^{10} + ( - \zeta_{8}^{3} + 2 \zeta_{8}^{2} + \cdots + 1) q^{11} + \cdots + (\zeta_{8}^{3} - \zeta_{8}^{2}) q^{99} +O(q^{100})$$ q + z^3 * q^2 + (-z^3 + z^2) * q^3 - z^2 * q^4 + (2*z - 2) * q^5 + (z^2 - z) * q^6 + (-2*z^2 - 2*z) * q^7 + z * q^8 + (-z^2 - z - 1) * q^9 + (-2*z^3 - 2) * q^10 + (-z^3 + 2*z^2 - 2*z + 1) * q^11 + (-z + 1) * q^12 + (2*z^3 + 2*z^2 + 2*z) * q^13 + (2*z + 2) * q^14 + (4*z^3 - 2*z^2 + 2) * q^15 - q^16 + (z^3 + 4*z) * q^17 + (-z^3 + z + 1) * q^18 + (-2*z^3 - 2*z^2 + 2) * q^19 + (-2*z^3 + 2*z^2) * q^20 + (-2*z^3 - 2*z) * q^21 + (z^3 + z^2 - 2*z + 2) * q^22 + (4*z^3 - 4) * q^23 + (z^3 + 1) * q^24 + (4*z^2 - 3*z + 4) * q^25 + (-2*z^2 - 2*z - 2) * q^26 + (-3*z^3 - z^2 - z - 3) * q^27 + (2*z^3 - 2) * q^28 + (-2*z^3 - 2*z^2) * q^29 + (2*z^3 - 4*z^2 + 2*z) * q^30 + (-2*z^3 + 2*z^2) * q^31 - z^3 * q^32 + (-3*z^3 + 3*z - 4) * q^33 + (-z^2 - 4) * q^34 + (-4*z^3 + 4*z) * q^35 + (z^3 + z^2 - 1) * q^36 + (2*z^3 - 2*z^2 - 2*z - 2) * q^37 + (2*z^3 + 2*z^2 + 2*z) * q^38 + (2*z^3 + 2*z^2) * q^39 + (2*z^2 - 2*z) * q^40 + (4*z^3 - 5*z^2 - 5*z + 4) * q^41 + (2*z^2 + 2) * q^42 + (3*z^2 + 6*z + 3) * q^43 + (2*z^3 - z^2 - z + 2) * q^44 + (-2*z^3 + 2) * q^45 + (-4*z^3 - 4*z^2) * q^46 + (-4*z^3 + 4*z^2 - 4*z) * q^47 + (z^3 - z^2) * q^48 + (z^3 + 4*z^2 - 4) * q^49 + (4*z^3 - 4*z + 3) * q^50 + (4*z^3 + z^2 - z + 4) * q^51 + (-2*z^3 + 2*z + 2) * q^52 + (-2*z^3 - 2*z^2 + 2) * q^53 + (-3*z^3 + 3*z^2 + z + 1) * q^54 + (6*z^3 - 8*z^2 + 6*z) * q^55 + (-2*z^3 - 2*z^2) * q^56 + (-2*z^3 + 2) * q^57 + (2*z^2 + 2*z) * q^58 + (-z^2 - 2*z - 1) * q^59 + (-2*z^2 + 4*z - 2) * q^60 + (4*z^3 + 4) * q^61 + (2*z^2 - 2*z) * q^62 + (4*z^3 + 4*z^2 + 2*z - 2) * q^63 + z^2 * q^64 + (-4*z - 4) * q^65 + (-4*z^3 + 3*z^2 - 3) * q^66 + (-z^3 + z - 6) * q^67 + (-4*z^3 + z) * q^68 + (4*z^3 - 4*z) * q^69 + (4*z^2 - 4) * q^70 + (2*z^3 - 2*z^2 - 2*z - 2) * q^71 + (-z^3 - z^2 - z) * q^72 + (-7*z^3 - 7*z^2) * q^73 + (-2*z^3 - 2*z^2 + 2*z + 2) * q^74 + (-7*z^3 + 4*z^2 + 4*z - 7) * q^75 + (-2*z^2 - 2*z - 2) * q^76 + (2*z^2 - 4*z + 2) * q^77 + (-2*z^2 - 2*z) * q^78 + (2*z^3 - 2*z^2 + 2*z - 2) * q^79 + (-2*z + 2) * q^80 + (5*z^3 - 3*z^2 + 5*z) * q^81 + (4*z^3 - 4*z^2 + 5*z + 5) * q^82 + (2*z^3 - 3*z^2 + 3) * q^83 + (2*z^3 - 2*z) * q^84 + (-2*z^3 + 8*z^2 - 8*z - 2) * q^85 + (3*z^3 - 3*z - 6) * q^86 + (-2*z^2 + 2) * q^87 + (2*z^3 - 2*z^2 + z + 1) * q^88 + (-5*z^3 + 8*z^2 - 5*z) * q^89 + (2*z^3 + 2*z^2) * q^90 + (-8*z^3 - 4*z^2 + 4*z + 8) * q^91 + (4*z^2 + 4*z) * q^92 + (-2*z^2 + 4*z - 2) * q^93 + (4*z^2 - 4*z + 4) * q^94 + (4*z^2 + 4*z) * q^95 + (-z^2 + z) * q^96 + (3*z - 3) * q^97 + (-4*z^3 - z^2 - 4*z) * q^98 + (z^3 - z^2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{5} - 4 q^{9}+O(q^{10})$$ 4 * q - 8 * q^5 - 4 * q^9 $$4 q - 8 q^{5} - 4 q^{9} - 8 q^{10} + 4 q^{11} + 4 q^{12} + 8 q^{14} + 8 q^{15} - 4 q^{16} + 4 q^{18} + 8 q^{19} + 8 q^{22} - 16 q^{23} + 4 q^{24} + 16 q^{25} - 8 q^{26} - 12 q^{27} - 8 q^{28} - 16 q^{33} - 16 q^{34} - 4 q^{36} - 8 q^{37} + 16 q^{41} + 8 q^{42} + 12 q^{43} + 8 q^{44} + 8 q^{45} - 16 q^{49} + 12 q^{50} + 16 q^{51} + 8 q^{52} + 8 q^{53} + 4 q^{54} + 8 q^{57} - 4 q^{59} - 8 q^{60} + 16 q^{61} - 8 q^{63} - 16 q^{65} - 12 q^{66} - 24 q^{67} - 16 q^{70} - 8 q^{71} + 8 q^{74} - 28 q^{75} - 8 q^{76} + 8 q^{77} - 8 q^{79} + 8 q^{80} + 20 q^{82} + 12 q^{83} - 8 q^{85} - 24 q^{86} + 8 q^{87} + 4 q^{88} + 32 q^{91} - 8 q^{93} + 16 q^{94} - 12 q^{97}+O(q^{100})$$ 4 * q - 8 * q^5 - 4 * q^9 - 8 * q^10 + 4 * q^11 + 4 * q^12 + 8 * q^14 + 8 * q^15 - 4 * q^16 + 4 * q^18 + 8 * q^19 + 8 * q^22 - 16 * q^23 + 4 * q^24 + 16 * q^25 - 8 * q^26 - 12 * q^27 - 8 * q^28 - 16 * q^33 - 16 * q^34 - 4 * q^36 - 8 * q^37 + 16 * q^41 + 8 * q^42 + 12 * q^43 + 8 * q^44 + 8 * q^45 - 16 * q^49 + 12 * q^50 + 16 * q^51 + 8 * q^52 + 8 * q^53 + 4 * q^54 + 8 * q^57 - 4 * q^59 - 8 * q^60 + 16 * q^61 - 8 * q^63 - 16 * q^65 - 12 * q^66 - 24 * q^67 - 16 * q^70 - 8 * q^71 + 8 * q^74 - 28 * q^75 - 8 * q^76 + 8 * q^77 - 8 * q^79 + 8 * q^80 + 20 * q^82 + 12 * q^83 - 8 * q^85 - 24 * q^86 + 8 * q^87 + 4 * q^88 + 32 * q^91 - 8 * q^93 + 16 * q^94 - 12 * q^97

## Character values

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

 $$n$$ $$3$$ $$\chi(n)$$ $$\zeta_{8}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1
 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i
−0.707107 + 0.707107i 0.707107 + 0.292893i 1.00000i −0.585786 + 1.41421i −0.707107 + 0.292893i −1.41421 3.41421i 0.707107 + 0.707107i −1.70711 1.70711i −0.585786 1.41421i
15.1 0.707107 + 0.707107i −0.707107 1.70711i 1.00000i −3.41421 + 1.41421i 0.707107 1.70711i 1.41421 + 0.585786i −0.707107 + 0.707107i −0.292893 + 0.292893i −3.41421 1.41421i
19.1 −0.707107 0.707107i 0.707107 0.292893i 1.00000i −0.585786 1.41421i −0.707107 0.292893i −1.41421 + 3.41421i 0.707107 0.707107i −1.70711 + 1.70711i −0.585786 + 1.41421i
25.1 0.707107 0.707107i −0.707107 + 1.70711i 1.00000i −3.41421 1.41421i 0.707107 + 1.70711i 1.41421 0.585786i −0.707107 0.707107i −0.292893 0.292893i −3.41421 + 1.41421i
 $$n$$: 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.d even 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 34.2.d.a 4
3.b odd 2 1 306.2.l.c 4
4.b odd 2 1 272.2.v.b 4
5.b even 2 1 850.2.l.a 4
5.c odd 4 1 850.2.o.a 4
5.c odd 4 1 850.2.o.b 4
17.b even 2 1 578.2.d.b 4
17.c even 4 1 578.2.d.a 4
17.c even 4 1 578.2.d.c 4
17.d even 8 1 inner 34.2.d.a 4
17.d even 8 1 578.2.d.a 4
17.d even 8 1 578.2.d.b 4
17.d even 8 1 578.2.d.c 4
17.e odd 16 2 578.2.a.i 4
17.e odd 16 2 578.2.b.d 4
17.e odd 16 4 578.2.c.f 8
51.g odd 8 1 306.2.l.c 4
51.i even 16 2 5202.2.a.bw 4
68.g odd 8 1 272.2.v.b 4
68.i even 16 2 4624.2.a.bn 4
85.k odd 8 1 850.2.o.a 4
85.m even 8 1 850.2.l.a 4
85.n odd 8 1 850.2.o.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
34.2.d.a 4 1.a even 1 1 trivial
34.2.d.a 4 17.d even 8 1 inner
272.2.v.b 4 4.b odd 2 1
272.2.v.b 4 68.g odd 8 1
306.2.l.c 4 3.b odd 2 1
306.2.l.c 4 51.g odd 8 1
578.2.a.i 4 17.e odd 16 2
578.2.b.d 4 17.e odd 16 2
578.2.c.f 8 17.e odd 16 4
578.2.d.a 4 17.c even 4 1
578.2.d.a 4 17.d even 8 1
578.2.d.b 4 17.b even 2 1
578.2.d.b 4 17.d even 8 1
578.2.d.c 4 17.c even 4 1
578.2.d.c 4 17.d even 8 1
850.2.l.a 4 5.b even 2 1
850.2.l.a 4 85.m even 8 1
850.2.o.a 4 5.c odd 4 1
850.2.o.a 4 85.k odd 8 1
850.2.o.b 4 5.c odd 4 1
850.2.o.b 4 85.n odd 8 1
4624.2.a.bn 4 68.i even 16 2
5202.2.a.bw 4 51.i even 16 2

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 1$$
$3$ $$T^{4} + 2 T^{2} + \cdots + 2$$
$5$ $$T^{4} + 8 T^{3} + \cdots + 32$$
$7$ $$T^{4} + 8 T^{2} + \cdots + 32$$
$11$ $$T^{4} - 4 T^{3} + \cdots + 2$$
$13$ $$T^{4} + 24T^{2} + 16$$
$17$ $$T^{4} + 16T^{2} + 289$$
$19$ $$T^{4} - 8 T^{3} + \cdots + 16$$
$23$ $$T^{4} + 16 T^{3} + \cdots + 512$$
$29$ $$T^{4} + 8 T^{2} + \cdots + 32$$
$31$ $$T^{4} + 8 T^{2} + \cdots + 32$$
$37$ $$T^{4} + 8 T^{3} + \cdots + 128$$
$41$ $$T^{4} - 16 T^{3} + \cdots + 4802$$
$43$ $$T^{4} - 12 T^{3} + \cdots + 324$$
$47$ $$T^{4} + 96T^{2} + 256$$
$53$ $$T^{4} - 8 T^{3} + \cdots + 16$$
$59$ $$T^{4} + 4 T^{3} + \cdots + 4$$
$61$ $$T^{4} - 16 T^{3} + \cdots + 512$$
$67$ $$(T^{2} + 12 T + 34)^{2}$$
$71$ $$T^{4} + 8 T^{3} + \cdots + 128$$
$73$ $$T^{4} + 98 T^{2} + \cdots + 4802$$
$79$ $$T^{4} + 8 T^{3} + \cdots + 128$$
$83$ $$T^{4} - 12 T^{3} + \cdots + 196$$
$89$ $$T^{4} + 228T^{2} + 196$$
$97$ $$T^{4} + 12 T^{3} + \cdots + 162$$