# Properties

 Label 35.2.b.a Level $35$ Weight $2$ Character orbit 35.b Analytic conductor $0.279$ 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] = [35,2,Mod(29,35)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(35, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("35.29");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$35 = 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 35.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: $$0.279476407074$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/35\mathbb{Z}\right)^\times$$.

 $$n$$ $$22$$ $$31$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1
 − 1.00000i 1.00000i
2.00000i 1.00000i −2.00000 −2.00000 + 1.00000i 2.00000 1.00000i 0 2.00000 2.00000 + 4.00000i
29.2 2.00000i 1.00000i −2.00000 −2.00000 1.00000i 2.00000 1.00000i 0 2.00000 2.00000 4.00000i
 $$n$$: 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.2.b.a 2
3.b odd 2 1 315.2.d.a 2
4.b odd 2 1 560.2.g.b 2
5.b even 2 1 inner 35.2.b.a 2
5.c odd 4 1 175.2.a.a 1
5.c odd 4 1 175.2.a.c 1
7.b odd 2 1 245.2.b.a 2
7.c even 3 2 245.2.j.e 4
7.d odd 6 2 245.2.j.d 4
8.b even 2 1 2240.2.g.h 2
8.d odd 2 1 2240.2.g.g 2
12.b even 2 1 5040.2.t.p 2
15.d odd 2 1 315.2.d.a 2
15.e even 4 1 1575.2.a.a 1
15.e even 4 1 1575.2.a.k 1
20.d odd 2 1 560.2.g.b 2
20.e even 4 1 2800.2.a.l 1
20.e even 4 1 2800.2.a.w 1
21.c even 2 1 2205.2.d.b 2
35.c odd 2 1 245.2.b.a 2
35.f even 4 1 1225.2.a.a 1
35.f even 4 1 1225.2.a.i 1
35.i odd 6 2 245.2.j.d 4
35.j even 6 2 245.2.j.e 4
40.e odd 2 1 2240.2.g.g 2
40.f even 2 1 2240.2.g.h 2
60.h even 2 1 5040.2.t.p 2
105.g even 2 1 2205.2.d.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.b.a 2 1.a even 1 1 trivial
35.2.b.a 2 5.b even 2 1 inner
175.2.a.a 1 5.c odd 4 1
175.2.a.c 1 5.c odd 4 1
245.2.b.a 2 7.b odd 2 1
245.2.b.a 2 35.c odd 2 1
245.2.j.d 4 7.d odd 6 2
245.2.j.d 4 35.i odd 6 2
245.2.j.e 4 7.c even 3 2
245.2.j.e 4 35.j even 6 2
315.2.d.a 2 3.b odd 2 1
315.2.d.a 2 15.d odd 2 1
560.2.g.b 2 4.b odd 2 1
560.2.g.b 2 20.d odd 2 1
1225.2.a.a 1 35.f even 4 1
1225.2.a.i 1 35.f even 4 1
1575.2.a.a 1 15.e even 4 1
1575.2.a.k 1 15.e even 4 1
2205.2.d.b 2 21.c even 2 1
2205.2.d.b 2 105.g even 2 1
2240.2.g.g 2 8.d odd 2 1
2240.2.g.g 2 40.e odd 2 1
2240.2.g.h 2 8.b even 2 1
2240.2.g.h 2 40.f even 2 1
2800.2.a.l 1 20.e even 4 1
2800.2.a.w 1 20.e even 4 1
5040.2.t.p 2 12.b even 2 1
5040.2.t.p 2 60.h even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(35, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2} + 4T + 5$$
$7$ $$T^{2} + 1$$
$11$ $$(T + 3)^{2}$$
$13$ $$T^{2} + 1$$
$17$ $$T^{2} + 49$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 36$$
$29$ $$(T - 5)^{2}$$
$31$ $$(T - 2)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T - 2)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 9$$
$53$ $$T^{2} + 36$$
$59$ $$(T + 10)^{2}$$
$61$ $$(T + 8)^{2}$$
$67$ $$T^{2} + 4$$
$71$ $$(T + 8)^{2}$$
$73$ $$T^{2} + 36$$
$79$ $$(T - 5)^{2}$$
$83$ $$T^{2} + 16$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 49$$