# Properties

 Label 34.2.a.a Level $34$ Weight $2$ Character orbit 34.a Self dual yes Analytic conductor $0.271$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [34,2,Mod(1,34)]

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([0]))

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

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

chi := DirichletCharacter("34.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$34 = 2 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 34.a (trivial)

## 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: yes Analytic conductor: $$0.271491366872$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{2} - 2 q^{3} + q^{4} - 2 q^{6} - 4 q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 - 2 * q^3 + q^4 - 2 * q^6 - 4 * q^7 + q^8 + q^9 $$q + q^{2} - 2 q^{3} + q^{4} - 2 q^{6} - 4 q^{7} + q^{8} + q^{9} + 6 q^{11} - 2 q^{12} + 2 q^{13} - 4 q^{14} + q^{16} - q^{17} + q^{18} - 4 q^{19} + 8 q^{21} + 6 q^{22} - 2 q^{24} - 5 q^{25} + 2 q^{26} + 4 q^{27} - 4 q^{28} - 4 q^{31} + q^{32} - 12 q^{33} - q^{34} + q^{36} - 4 q^{37} - 4 q^{38} - 4 q^{39} + 6 q^{41} + 8 q^{42} + 8 q^{43} + 6 q^{44} - 2 q^{48} + 9 q^{49} - 5 q^{50} + 2 q^{51} + 2 q^{52} - 6 q^{53} + 4 q^{54} - 4 q^{56} + 8 q^{57} - 4 q^{61} - 4 q^{62} - 4 q^{63} + q^{64} - 12 q^{66} + 8 q^{67} - q^{68} + q^{72} + 2 q^{73} - 4 q^{74} + 10 q^{75} - 4 q^{76} - 24 q^{77} - 4 q^{78} + 8 q^{79} - 11 q^{81} + 6 q^{82} + 8 q^{84} + 8 q^{86} + 6 q^{88} - 6 q^{89} - 8 q^{91} + 8 q^{93} - 2 q^{96} + 14 q^{97} + 9 q^{98} + 6 q^{99}+O(q^{100})$$ q + q^2 - 2 * q^3 + q^4 - 2 * q^6 - 4 * q^7 + q^8 + q^9 + 6 * q^11 - 2 * q^12 + 2 * q^13 - 4 * q^14 + q^16 - q^17 + q^18 - 4 * q^19 + 8 * q^21 + 6 * q^22 - 2 * q^24 - 5 * q^25 + 2 * q^26 + 4 * q^27 - 4 * q^28 - 4 * q^31 + q^32 - 12 * q^33 - q^34 + q^36 - 4 * q^37 - 4 * q^38 - 4 * q^39 + 6 * q^41 + 8 * q^42 + 8 * q^43 + 6 * q^44 - 2 * q^48 + 9 * q^49 - 5 * q^50 + 2 * q^51 + 2 * q^52 - 6 * q^53 + 4 * q^54 - 4 * q^56 + 8 * q^57 - 4 * q^61 - 4 * q^62 - 4 * q^63 + q^64 - 12 * q^66 + 8 * q^67 - q^68 + q^72 + 2 * q^73 - 4 * q^74 + 10 * q^75 - 4 * q^76 - 24 * q^77 - 4 * q^78 + 8 * q^79 - 11 * q^81 + 6 * q^82 + 8 * q^84 + 8 * q^86 + 6 * q^88 - 6 * q^89 - 8 * q^91 + 8 * q^93 - 2 * q^96 + 14 * q^97 + 9 * q^98 + 6 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
1.00000 −2.00000 1.00000 0 −2.00000 −4.00000 1.00000 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$17$$ $$+1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 34.2.a.a 1
3.b odd 2 1 306.2.a.a 1
4.b odd 2 1 272.2.a.d 1
5.b even 2 1 850.2.a.e 1
5.c odd 4 2 850.2.c.b 2
7.b odd 2 1 1666.2.a.m 1
8.b even 2 1 1088.2.a.l 1
8.d odd 2 1 1088.2.a.d 1
11.b odd 2 1 4114.2.a.a 1
12.b even 2 1 2448.2.a.k 1
13.b even 2 1 5746.2.a.b 1
15.d odd 2 1 7650.2.a.ci 1
17.b even 2 1 578.2.a.a 1
17.c even 4 2 578.2.b.a 2
17.d even 8 4 578.2.c.e 4
17.e odd 16 8 578.2.d.e 8
20.d odd 2 1 6800.2.a.b 1
24.f even 2 1 9792.2.a.bj 1
24.h odd 2 1 9792.2.a.y 1
51.c odd 2 1 5202.2.a.d 1
68.d odd 2 1 4624.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
34.2.a.a 1 1.a even 1 1 trivial
272.2.a.d 1 4.b odd 2 1
306.2.a.a 1 3.b odd 2 1
578.2.a.a 1 17.b even 2 1
578.2.b.a 2 17.c even 4 2
578.2.c.e 4 17.d even 8 4
578.2.d.e 8 17.e odd 16 8
850.2.a.e 1 5.b even 2 1
850.2.c.b 2 5.c odd 4 2
1088.2.a.d 1 8.d odd 2 1
1088.2.a.l 1 8.b even 2 1
1666.2.a.m 1 7.b odd 2 1
2448.2.a.k 1 12.b even 2 1
4114.2.a.a 1 11.b odd 2 1
4624.2.a.a 1 68.d odd 2 1
5202.2.a.d 1 51.c odd 2 1
5746.2.a.b 1 13.b even 2 1
6800.2.a.b 1 20.d odd 2 1
7650.2.a.ci 1 15.d odd 2 1
9792.2.a.y 1 24.h odd 2 1
9792.2.a.bj 1 24.f even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(\Gamma_0(34))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 2$$
$5$ $$T$$
$7$ $$T + 4$$
$11$ $$T - 6$$
$13$ $$T - 2$$
$17$ $$T + 1$$
$19$ $$T + 4$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T + 4$$
$37$ $$T + 4$$
$41$ $$T - 6$$
$43$ $$T - 8$$
$47$ $$T$$
$53$ $$T + 6$$
$59$ $$T$$
$61$ $$T + 4$$
$67$ $$T - 8$$
$71$ $$T$$
$73$ $$T - 2$$
$79$ $$T - 8$$
$83$ $$T$$
$89$ $$T + 6$$
$97$ $$T - 14$$