# Properties

 Label 35.3.d.b Level $35$ Weight $3$ Character orbit 35.d Analytic conductor $0.954$ 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] = [35,3,Mod(6,35)]

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([0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("35.6");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$35 = 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 35.d (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: $$0.953680925261$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 5$$ x^2 + 5 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 $$\beta = \sqrt{-5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + \beta q^{3} - \beta q^{5} + 2 \beta q^{6} + ( - 3 \beta - 2) q^{7} - 8 q^{8} + 4 q^{9} +O(q^{10})$$ q + 2 * q^2 + b * q^3 - b * q^5 + 2*b * q^6 + (-3*b - 2) * q^7 - 8 * q^8 + 4 * q^9 $$q + 2 q^{2} + \beta q^{3} - \beta q^{5} + 2 \beta q^{6} + ( - 3 \beta - 2) q^{7} - 8 q^{8} + 4 q^{9} - 2 \beta q^{10} - q^{11} + 9 \beta q^{13} + ( - 6 \beta - 4) q^{14} + 5 q^{15} - 16 q^{16} + 3 \beta q^{17} + 8 q^{18} - 6 \beta q^{19} + ( - 2 \beta + 15) q^{21} - 2 q^{22} + 8 q^{23} - 8 \beta q^{24} - 5 q^{25} + 18 \beta q^{26} + 13 \beta q^{27} + 41 q^{29} + 10 q^{30} - 18 \beta q^{31} - \beta q^{33} + 6 \beta q^{34} + (2 \beta - 15) q^{35} - 28 q^{37} - 12 \beta q^{38} - 45 q^{39} + 8 \beta q^{40} - 6 \beta q^{41} + ( - 4 \beta + 30) q^{42} - 82 q^{43} - 4 \beta q^{45} + 16 q^{46} - 9 \beta q^{47} - 16 \beta q^{48} + (12 \beta - 41) q^{49} - 10 q^{50} - 15 q^{51} + 74 q^{53} + 26 \beta q^{54} + \beta q^{55} + (24 \beta + 16) q^{56} + 30 q^{57} + 82 q^{58} + 42 \beta q^{59} - 36 \beta q^{61} - 36 \beta q^{62} + ( - 12 \beta - 8) q^{63} + 64 q^{64} + 45 q^{65} - 2 \beta q^{66} + 2 q^{67} + 8 \beta q^{69} + (4 \beta - 30) q^{70} + 14 q^{71} - 32 q^{72} + 30 \beta q^{73} - 56 q^{74} - 5 \beta q^{75} + (3 \beta + 2) q^{77} - 90 q^{78} - 19 q^{79} + 16 \beta q^{80} - 29 q^{81} - 12 \beta q^{82} - 42 \beta q^{83} + 15 q^{85} - 164 q^{86} + 41 \beta q^{87} + 8 q^{88} - 48 \beta q^{89} - 8 \beta q^{90} + ( - 18 \beta + 135) q^{91} + 90 q^{93} - 18 \beta q^{94} - 30 q^{95} + 27 \beta q^{97} + (24 \beta - 82) q^{98} - 4 q^{99} +O(q^{100})$$ q + 2 * q^2 + b * q^3 - b * q^5 + 2*b * q^6 + (-3*b - 2) * q^7 - 8 * q^8 + 4 * q^9 - 2*b * q^10 - q^11 + 9*b * q^13 + (-6*b - 4) * q^14 + 5 * q^15 - 16 * q^16 + 3*b * q^17 + 8 * q^18 - 6*b * q^19 + (-2*b + 15) * q^21 - 2 * q^22 + 8 * q^23 - 8*b * q^24 - 5 * q^25 + 18*b * q^26 + 13*b * q^27 + 41 * q^29 + 10 * q^30 - 18*b * q^31 - b * q^33 + 6*b * q^34 + (2*b - 15) * q^35 - 28 * q^37 - 12*b * q^38 - 45 * q^39 + 8*b * q^40 - 6*b * q^41 + (-4*b + 30) * q^42 - 82 * q^43 - 4*b * q^45 + 16 * q^46 - 9*b * q^47 - 16*b * q^48 + (12*b - 41) * q^49 - 10 * q^50 - 15 * q^51 + 74 * q^53 + 26*b * q^54 + b * q^55 + (24*b + 16) * q^56 + 30 * q^57 + 82 * q^58 + 42*b * q^59 - 36*b * q^61 - 36*b * q^62 + (-12*b - 8) * q^63 + 64 * q^64 + 45 * q^65 - 2*b * q^66 + 2 * q^67 + 8*b * q^69 + (4*b - 30) * q^70 + 14 * q^71 - 32 * q^72 + 30*b * q^73 - 56 * q^74 - 5*b * q^75 + (3*b + 2) * q^77 - 90 * q^78 - 19 * q^79 + 16*b * q^80 - 29 * q^81 - 12*b * q^82 - 42*b * q^83 + 15 * q^85 - 164 * q^86 + 41*b * q^87 + 8 * q^88 - 48*b * q^89 - 8*b * q^90 + (-18*b + 135) * q^91 + 90 * q^93 - 18*b * q^94 - 30 * q^95 + 27*b * q^97 + (24*b - 82) * q^98 - 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{2} - 4 q^{7} - 16 q^{8} + 8 q^{9}+O(q^{10})$$ 2 * q + 4 * q^2 - 4 * q^7 - 16 * q^8 + 8 * q^9 $$2 q + 4 q^{2} - 4 q^{7} - 16 q^{8} + 8 q^{9} - 2 q^{11} - 8 q^{14} + 10 q^{15} - 32 q^{16} + 16 q^{18} + 30 q^{21} - 4 q^{22} + 16 q^{23} - 10 q^{25} + 82 q^{29} + 20 q^{30} - 30 q^{35} - 56 q^{37} - 90 q^{39} + 60 q^{42} - 164 q^{43} + 32 q^{46} - 82 q^{49} - 20 q^{50} - 30 q^{51} + 148 q^{53} + 32 q^{56} + 60 q^{57} + 164 q^{58} - 16 q^{63} + 128 q^{64} + 90 q^{65} + 4 q^{67} - 60 q^{70} + 28 q^{71} - 64 q^{72} - 112 q^{74} + 4 q^{77} - 180 q^{78} - 38 q^{79} - 58 q^{81} + 30 q^{85} - 328 q^{86} + 16 q^{88} + 270 q^{91} + 180 q^{93} - 60 q^{95} - 164 q^{98} - 8 q^{99}+O(q^{100})$$ 2 * q + 4 * q^2 - 4 * q^7 - 16 * q^8 + 8 * q^9 - 2 * q^11 - 8 * q^14 + 10 * q^15 - 32 * q^16 + 16 * q^18 + 30 * q^21 - 4 * q^22 + 16 * q^23 - 10 * q^25 + 82 * q^29 + 20 * q^30 - 30 * q^35 - 56 * q^37 - 90 * q^39 + 60 * q^42 - 164 * q^43 + 32 * q^46 - 82 * q^49 - 20 * q^50 - 30 * q^51 + 148 * q^53 + 32 * q^56 + 60 * q^57 + 164 * q^58 - 16 * q^63 + 128 * q^64 + 90 * q^65 + 4 * q^67 - 60 * q^70 + 28 * q^71 - 64 * q^72 - 112 * q^74 + 4 * q^77 - 180 * q^78 - 38 * q^79 - 58 * q^81 + 30 * q^85 - 328 * q^86 + 16 * q^88 + 270 * q^91 + 180 * q^93 - 60 * q^95 - 164 * q^98 - 8 * 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
6.1
 − 2.23607i 2.23607i
2.00000 2.23607i 0 2.23607i 4.47214i −2.00000 + 6.70820i −8.00000 4.00000 4.47214i
6.2 2.00000 2.23607i 0 2.23607i 4.47214i −2.00000 6.70820i −8.00000 4.00000 4.47214i
 $$n$$: 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
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.3.d.b 2
3.b odd 2 1 315.3.h.a 2
4.b odd 2 1 560.3.f.b 2
5.b even 2 1 175.3.d.c 2
5.c odd 4 2 175.3.c.c 4
7.b odd 2 1 inner 35.3.d.b 2
7.c even 3 2 245.3.h.a 4
7.d odd 6 2 245.3.h.a 4
21.c even 2 1 315.3.h.a 2
28.d even 2 1 560.3.f.b 2
35.c odd 2 1 175.3.d.c 2
35.f even 4 2 175.3.c.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.d.b 2 1.a even 1 1 trivial
35.3.d.b 2 7.b odd 2 1 inner
175.3.c.c 4 5.c odd 4 2
175.3.c.c 4 35.f even 4 2
175.3.d.c 2 5.b even 2 1
175.3.d.c 2 35.c odd 2 1
245.3.h.a 4 7.c even 3 2
245.3.h.a 4 7.d odd 6 2
315.3.h.a 2 3.b odd 2 1
315.3.h.a 2 21.c even 2 1
560.3.f.b 2 4.b odd 2 1
560.3.f.b 2 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 2$$ acting on $$S_{3}^{\mathrm{new}}(35, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 2)^{2}$$
$3$ $$T^{2} + 5$$
$5$ $$T^{2} + 5$$
$7$ $$T^{2} + 4T + 49$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} + 405$$
$17$ $$T^{2} + 45$$
$19$ $$T^{2} + 180$$
$23$ $$(T - 8)^{2}$$
$29$ $$(T - 41)^{2}$$
$31$ $$T^{2} + 1620$$
$37$ $$(T + 28)^{2}$$
$41$ $$T^{2} + 180$$
$43$ $$(T + 82)^{2}$$
$47$ $$T^{2} + 405$$
$53$ $$(T - 74)^{2}$$
$59$ $$T^{2} + 8820$$
$61$ $$T^{2} + 6480$$
$67$ $$(T - 2)^{2}$$
$71$ $$(T - 14)^{2}$$
$73$ $$T^{2} + 4500$$
$79$ $$(T + 19)^{2}$$
$83$ $$T^{2} + 8820$$
$89$ $$T^{2} + 11520$$
$97$ $$T^{2} + 3645$$