# Properties

 Label 3610.2.a.b Level $3610$ Weight $2$ Character orbit 3610.a Self dual yes Analytic conductor $28.826$ Analytic rank $1$ 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] = [3610,2,Mod(1,3610)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3610, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3610.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3610 = 2 \cdot 5 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3610.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: $$28.8259951297$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 190) 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)

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

## 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 −1.00000 1.00000 1.00000 1.00000 −1.00000 −1.00000 −2.00000 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$5$$ $$-1$$
$$19$$ $$-1$$

## 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 3610.2.a.b 1
19.b odd 2 1 190.2.a.c 1
57.d even 2 1 1710.2.a.d 1
76.d even 2 1 1520.2.a.d 1
95.d odd 2 1 950.2.a.a 1
95.g even 4 2 950.2.b.e 2
133.c even 2 1 9310.2.a.o 1
152.b even 2 1 6080.2.a.p 1
152.g odd 2 1 6080.2.a.h 1
285.b even 2 1 8550.2.a.bd 1
380.d even 2 1 7600.2.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.a.c 1 19.b odd 2 1
950.2.a.a 1 95.d odd 2 1
950.2.b.e 2 95.g even 4 2
1520.2.a.d 1 76.d even 2 1
1710.2.a.d 1 57.d even 2 1
3610.2.a.b 1 1.a even 1 1 trivial
6080.2.a.h 1 152.g odd 2 1
6080.2.a.p 1 152.b even 2 1
7600.2.a.m 1 380.d even 2 1
8550.2.a.bd 1 285.b even 2 1
9310.2.a.o 1 133.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3610))$$:

 $$T_{3} + 1$$ T3 + 1 $$T_{7} + 1$$ T7 + 1 $$T_{13} - 1$$ T13 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T + 1$$
$5$ $$T - 1$$
$7$ $$T + 1$$
$11$ $$T$$
$13$ $$T - 1$$
$17$ $$T + 3$$
$19$ $$T$$
$23$ $$T - 3$$
$29$ $$T - 3$$
$31$ $$T + 2$$
$37$ $$T - 10$$
$41$ $$T + 6$$
$43$ $$T - 2$$
$47$ $$T$$
$53$ $$T + 3$$
$59$ $$T + 3$$
$61$ $$T - 8$$
$67$ $$T - 7$$
$71$ $$T + 12$$
$73$ $$T + 13$$
$79$ $$T + 14$$
$83$ $$T - 6$$
$89$ $$T + 6$$
$97$ $$T - 10$$