# Properties

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

# Learn more

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.

 $$f(q)$$ $$=$$ $$q - i q^{2} + ( - i - 1) q^{3} - q^{4} + (2 i + 2) q^{5} + (i - 1) q^{6} + (2 i - 2) q^{7} + i q^{8} - i q^{9} +O(q^{10})$$ q - i * q^2 + (-i - 1) * q^3 - q^4 + (2*i + 2) * q^5 + (i - 1) * q^6 + (2*i - 2) * q^7 + i * q^8 - i * q^9 $$q - i q^{2} + ( - i - 1) q^{3} - q^{4} + (2 i + 2) q^{5} + (i - 1) q^{6} + (2 i - 2) q^{7} + i q^{8} - i q^{9} + ( - 2 i + 2) q^{10} + ( - i + 1) q^{11} + (i + 1) q^{12} - 6 q^{13} + (2 i + 2) q^{14} - 4 i q^{15} + q^{16} + (i + 4) q^{17} - q^{18} - 4 i q^{19} + ( - 2 i - 2) q^{20} + 4 q^{21} + ( - i - 1) q^{22} + ( - i + 1) q^{24} + 3 i q^{25} + 6 i q^{26} + (4 i - 4) q^{27} + ( - 2 i + 2) q^{28} + ( - 2 i - 2) q^{29} - 4 q^{30} + (6 i + 6) q^{31} - i q^{32} - 2 q^{33} + ( - 4 i + 1) q^{34} - 8 q^{35} + i q^{36} - 4 q^{38} + (6 i + 6) q^{39} + (2 i - 2) q^{40} + ( - i + 1) q^{41} - 4 i q^{42} - 6 i q^{43} + (i - 1) q^{44} + ( - 2 i + 2) q^{45} + 8 q^{47} + ( - i - 1) q^{48} - i q^{49} + 3 q^{50} + ( - 5 i - 3) q^{51} + 6 q^{52} - 6 i q^{53} + (4 i + 4) q^{54} + 4 q^{55} + ( - 2 i - 2) q^{56} + (4 i - 4) q^{57} + (2 i - 2) q^{58} + 6 i q^{59} + 4 i q^{60} + (4 i - 4) q^{61} + ( - 6 i + 6) q^{62} + (2 i + 2) q^{63} - q^{64} + ( - 12 i - 12) q^{65} + 2 i q^{66} - 2 q^{67} + ( - i - 4) q^{68} + 8 i q^{70} + ( - 4 i - 4) q^{71} + q^{72} + ( - i - 1) q^{73} + ( - 3 i + 3) q^{75} + 4 i q^{76} + 4 i q^{77} + ( - 6 i + 6) q^{78} + (8 i - 8) q^{79} + (2 i + 2) q^{80} + 5 q^{81} + ( - i - 1) q^{82} + 14 i q^{83} - 4 q^{84} + (10 i + 6) q^{85} - 6 q^{86} + 4 i q^{87} + (i + 1) q^{88} + ( - 2 i - 2) q^{90} + ( - 12 i + 12) q^{91} - 12 i q^{93} - 8 i q^{94} + ( - 8 i + 8) q^{95} + (i - 1) q^{96} + ( - 5 i - 5) q^{97} - q^{98} + ( - i - 1) q^{99} +O(q^{100})$$ q - i * q^2 + (-i - 1) * q^3 - q^4 + (2*i + 2) * q^5 + (i - 1) * q^6 + (2*i - 2) * q^7 + i * q^8 - i * q^9 + (-2*i + 2) * q^10 + (-i + 1) * q^11 + (i + 1) * q^12 - 6 * q^13 + (2*i + 2) * q^14 - 4*i * q^15 + q^16 + (i + 4) * q^17 - q^18 - 4*i * q^19 + (-2*i - 2) * q^20 + 4 * q^21 + (-i - 1) * q^22 + (-i + 1) * q^24 + 3*i * q^25 + 6*i * q^26 + (4*i - 4) * q^27 + (-2*i + 2) * q^28 + (-2*i - 2) * q^29 - 4 * q^30 + (6*i + 6) * q^31 - i * q^32 - 2 * q^33 + (-4*i + 1) * q^34 - 8 * q^35 + i * q^36 - 4 * q^38 + (6*i + 6) * q^39 + (2*i - 2) * q^40 + (-i + 1) * q^41 - 4*i * q^42 - 6*i * q^43 + (i - 1) * q^44 + (-2*i + 2) * q^45 + 8 * q^47 + (-i - 1) * q^48 - i * q^49 + 3 * q^50 + (-5*i - 3) * q^51 + 6 * q^52 - 6*i * q^53 + (4*i + 4) * q^54 + 4 * q^55 + (-2*i - 2) * q^56 + (4*i - 4) * q^57 + (2*i - 2) * q^58 + 6*i * q^59 + 4*i * q^60 + (4*i - 4) * q^61 + (-6*i + 6) * q^62 + (2*i + 2) * q^63 - q^64 + (-12*i - 12) * q^65 + 2*i * q^66 - 2 * q^67 + (-i - 4) * q^68 + 8*i * q^70 + (-4*i - 4) * q^71 + q^72 + (-i - 1) * q^73 + (-3*i + 3) * q^75 + 4*i * q^76 + 4*i * q^77 + (-6*i + 6) * q^78 + (8*i - 8) * q^79 + (2*i + 2) * q^80 + 5 * q^81 + (-i - 1) * q^82 + 14*i * q^83 - 4 * q^84 + (10*i + 6) * q^85 - 6 * q^86 + 4*i * q^87 + (i + 1) * q^88 + (-2*i - 2) * q^90 + (-12*i + 12) * q^91 - 12*i * q^93 - 8*i * q^94 + (-8*i + 8) * q^95 + (i - 1) * q^96 + (-5*i - 5) * q^97 - q^98 + (-i - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 2 q^{4} + 4 q^{5} - 2 q^{6} - 4 q^{7}+O(q^{10})$$ 2 * q - 2 * q^3 - 2 * q^4 + 4 * q^5 - 2 * q^6 - 4 * q^7 $$2 q - 2 q^{3} - 2 q^{4} + 4 q^{5} - 2 q^{6} - 4 q^{7} + 4 q^{10} + 2 q^{11} + 2 q^{12} - 12 q^{13} + 4 q^{14} + 2 q^{16} + 8 q^{17} - 2 q^{18} - 4 q^{20} + 8 q^{21} - 2 q^{22} + 2 q^{24} - 8 q^{27} + 4 q^{28} - 4 q^{29} - 8 q^{30} + 12 q^{31} - 4 q^{33} + 2 q^{34} - 16 q^{35} - 8 q^{38} + 12 q^{39} - 4 q^{40} + 2 q^{41} - 2 q^{44} + 4 q^{45} + 16 q^{47} - 2 q^{48} + 6 q^{50} - 6 q^{51} + 12 q^{52} + 8 q^{54} + 8 q^{55} - 4 q^{56} - 8 q^{57} - 4 q^{58} - 8 q^{61} + 12 q^{62} + 4 q^{63} - 2 q^{64} - 24 q^{65} - 4 q^{67} - 8 q^{68} - 8 q^{71} + 2 q^{72} - 2 q^{73} + 6 q^{75} + 12 q^{78} - 16 q^{79} + 4 q^{80} + 10 q^{81} - 2 q^{82} - 8 q^{84} + 12 q^{85} - 12 q^{86} + 2 q^{88} - 4 q^{90} + 24 q^{91} + 16 q^{95} - 2 q^{96} - 10 q^{97} - 2 q^{98} - 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 - 2 * q^4 + 4 * q^5 - 2 * q^6 - 4 * q^7 + 4 * q^10 + 2 * q^11 + 2 * q^12 - 12 * q^13 + 4 * q^14 + 2 * q^16 + 8 * q^17 - 2 * q^18 - 4 * q^20 + 8 * q^21 - 2 * q^22 + 2 * q^24 - 8 * q^27 + 4 * q^28 - 4 * q^29 - 8 * q^30 + 12 * q^31 - 4 * q^33 + 2 * q^34 - 16 * q^35 - 8 * q^38 + 12 * q^39 - 4 * q^40 + 2 * q^41 - 2 * q^44 + 4 * q^45 + 16 * q^47 - 2 * q^48 + 6 * q^50 - 6 * q^51 + 12 * q^52 + 8 * q^54 + 8 * q^55 - 4 * q^56 - 8 * q^57 - 4 * q^58 - 8 * q^61 + 12 * q^62 + 4 * q^63 - 2 * q^64 - 24 * q^65 - 4 * q^67 - 8 * q^68 - 8 * q^71 + 2 * q^72 - 2 * q^73 + 6 * q^75 + 12 * q^78 - 16 * q^79 + 4 * q^80 + 10 * q^81 - 2 * q^82 - 8 * q^84 + 12 * q^85 - 12 * q^86 + 2 * q^88 - 4 * q^90 + 24 * q^91 + 16 * q^95 - 2 * q^96 - 10 * q^97 - 2 * q^98 - 2 * 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 −1.00000 1.00000i −1.00000 2.00000 + 2.00000i −1.00000 + 1.00000i −2.00000 + 2.00000i 1.00000i 1.00000i 2.00000 2.00000i
21.1 1.00000i −1.00000 + 1.00000i −1.00000 2.00000 2.00000i −1.00000 1.00000i −2.00000 2.00000i 1.00000i 1.00000i 2.00000 + 2.00000i
 $$n$$: 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.a 2
3.b odd 2 1 306.2.g.b 2
4.b odd 2 1 272.2.o.f 2
5.b even 2 1 850.2.h.f 2
5.c odd 4 1 850.2.g.b 2
5.c odd 4 1 850.2.g.c 2
8.b even 2 1 1088.2.o.l 2
8.d odd 2 1 1088.2.o.d 2
12.b even 2 1 2448.2.be.c 2
17.b even 2 1 578.2.c.c 2
17.c even 4 1 inner 34.2.c.a 2
17.c even 4 1 578.2.c.c 2
17.d even 8 2 578.2.a.c 2
17.d even 8 2 578.2.b.b 2
17.e odd 16 8 578.2.d.d 8
51.f odd 4 1 306.2.g.b 2
51.g odd 8 2 5202.2.a.v 2
68.f odd 4 1 272.2.o.f 2
68.g odd 8 2 4624.2.a.r 2
85.f odd 4 1 850.2.g.b 2
85.i odd 4 1 850.2.g.c 2
85.j even 4 1 850.2.h.f 2
136.i even 4 1 1088.2.o.l 2
136.j odd 4 1 1088.2.o.d 2
204.l even 4 1 2448.2.be.c 2

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 2T + 2$$
$5$ $$T^{2} - 4T + 8$$
$7$ $$T^{2} + 4T + 8$$
$11$ $$T^{2} - 2T + 2$$
$13$ $$(T + 6)^{2}$$
$17$ $$T^{2} - 8T + 17$$
$19$ $$T^{2} + 16$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 4T + 8$$
$31$ $$T^{2} - 12T + 72$$
$37$ $$T^{2}$$
$41$ $$T^{2} - 2T + 2$$
$43$ $$T^{2} + 36$$
$47$ $$(T - 8)^{2}$$
$53$ $$T^{2} + 36$$
$59$ $$T^{2} + 36$$
$61$ $$T^{2} + 8T + 32$$
$67$ $$(T + 2)^{2}$$
$71$ $$T^{2} + 8T + 32$$
$73$ $$T^{2} + 2T + 2$$
$79$ $$T^{2} + 16T + 128$$
$83$ $$T^{2} + 196$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 10T + 50$$
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