# Properties

 Label 7605.2.a.g Level $7605$ Weight $2$ Character orbit 7605.a Self dual yes Analytic conductor $60.726$ 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] = [7605,2,Mod(1,7605)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7605, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7605.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7605 = 3^{2} \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7605.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: $$60.7262307372$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{4} + q^{5} + 3 q^{8}+O(q^{10})$$ q - q^2 - q^4 + q^5 + 3 * q^8 $$q - q^{2} - q^{4} + q^{5} + 3 q^{8} - q^{10} - 4 q^{11} - q^{16} - 2 q^{17} - 4 q^{19} - q^{20} + 4 q^{22} + q^{25} + 2 q^{29} - 5 q^{32} + 2 q^{34} + 10 q^{37} + 4 q^{38} + 3 q^{40} + 10 q^{41} + 4 q^{43} + 4 q^{44} + 8 q^{47} - 7 q^{49} - q^{50} + 10 q^{53} - 4 q^{55} - 2 q^{58} - 4 q^{59} - 2 q^{61} + 7 q^{64} - 12 q^{67} + 2 q^{68} - 8 q^{71} - 10 q^{73} - 10 q^{74} + 4 q^{76} - q^{80} - 10 q^{82} + 12 q^{83} - 2 q^{85} - 4 q^{86} - 12 q^{88} - 6 q^{89} - 8 q^{94} - 4 q^{95} - 2 q^{97} + 7 q^{98}+O(q^{100})$$ q - q^2 - q^4 + q^5 + 3 * q^8 - q^10 - 4 * q^11 - q^16 - 2 * q^17 - 4 * q^19 - q^20 + 4 * q^22 + q^25 + 2 * q^29 - 5 * q^32 + 2 * q^34 + 10 * q^37 + 4 * q^38 + 3 * q^40 + 10 * q^41 + 4 * q^43 + 4 * q^44 + 8 * q^47 - 7 * q^49 - q^50 + 10 * q^53 - 4 * q^55 - 2 * q^58 - 4 * q^59 - 2 * q^61 + 7 * q^64 - 12 * q^67 + 2 * q^68 - 8 * q^71 - 10 * q^73 - 10 * q^74 + 4 * q^76 - q^80 - 10 * q^82 + 12 * q^83 - 2 * q^85 - 4 * q^86 - 12 * q^88 - 6 * q^89 - 8 * q^94 - 4 * q^95 - 2 * q^97 + 7 * 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 0 −1.00000 1.00000 0 0 3.00000 0 −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
$$3$$ $$-1$$
$$5$$ $$-1$$
$$13$$ $$+1$$

## 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 7605.2.a.g 1
3.b odd 2 1 2535.2.a.j 1
13.b even 2 1 45.2.a.a 1
39.d odd 2 1 15.2.a.a 1
52.b odd 2 1 720.2.a.c 1
65.d even 2 1 225.2.a.b 1
65.h odd 4 2 225.2.b.b 2
91.b odd 2 1 2205.2.a.i 1
104.e even 2 1 2880.2.a.y 1
104.h odd 2 1 2880.2.a.bc 1
117.n odd 6 2 405.2.e.f 2
117.t even 6 2 405.2.e.c 2
143.d odd 2 1 5445.2.a.c 1
156.h even 2 1 240.2.a.d 1
195.e odd 2 1 75.2.a.b 1
195.s even 4 2 75.2.b.b 2
260.g odd 2 1 3600.2.a.u 1
260.p even 4 2 3600.2.f.e 2
273.g even 2 1 735.2.a.c 1
273.w odd 6 2 735.2.i.e 2
273.ba even 6 2 735.2.i.d 2
312.b odd 2 1 960.2.a.l 1
312.h even 2 1 960.2.a.a 1
429.e even 2 1 1815.2.a.d 1
624.v even 4 2 3840.2.k.r 2
624.bi odd 4 2 3840.2.k.m 2
663.g odd 2 1 4335.2.a.c 1
741.d even 2 1 5415.2.a.j 1
780.d even 2 1 1200.2.a.e 1
780.w odd 4 2 1200.2.f.h 2
897.g even 2 1 7935.2.a.d 1
1365.g even 2 1 3675.2.a.j 1
1560.n even 2 1 4800.2.a.bz 1
1560.y odd 2 1 4800.2.a.t 1
1560.bq even 4 2 4800.2.f.bf 2
1560.cs odd 4 2 4800.2.f.c 2
2145.k even 2 1 9075.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.2.a.a 1 39.d odd 2 1
45.2.a.a 1 13.b even 2 1
75.2.a.b 1 195.e odd 2 1
75.2.b.b 2 195.s even 4 2
225.2.a.b 1 65.d even 2 1
225.2.b.b 2 65.h odd 4 2
240.2.a.d 1 156.h even 2 1
405.2.e.c 2 117.t even 6 2
405.2.e.f 2 117.n odd 6 2
720.2.a.c 1 52.b odd 2 1
735.2.a.c 1 273.g even 2 1
735.2.i.d 2 273.ba even 6 2
735.2.i.e 2 273.w odd 6 2
960.2.a.a 1 312.h even 2 1
960.2.a.l 1 312.b odd 2 1
1200.2.a.e 1 780.d even 2 1
1200.2.f.h 2 780.w odd 4 2
1815.2.a.d 1 429.e even 2 1
2205.2.a.i 1 91.b odd 2 1
2535.2.a.j 1 3.b odd 2 1
2880.2.a.y 1 104.e even 2 1
2880.2.a.bc 1 104.h odd 2 1
3600.2.a.u 1 260.g odd 2 1
3600.2.f.e 2 260.p even 4 2
3675.2.a.j 1 1365.g even 2 1
3840.2.k.m 2 624.bi odd 4 2
3840.2.k.r 2 624.v even 4 2
4335.2.a.c 1 663.g odd 2 1
4800.2.a.t 1 1560.y odd 2 1
4800.2.a.bz 1 1560.n even 2 1
4800.2.f.c 2 1560.cs odd 4 2
4800.2.f.bf 2 1560.bq even 4 2
5415.2.a.j 1 741.d even 2 1
5445.2.a.c 1 143.d odd 2 1
7605.2.a.g 1 1.a even 1 1 trivial
7935.2.a.d 1 897.g even 2 1
9075.2.a.g 1 2145.k even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7605))$$:

 $$T_{2} + 1$$ T2 + 1 $$T_{7}$$ T7 $$T_{11} + 4$$ T11 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T$$
$11$ $$T + 4$$
$13$ $$T$$
$17$ $$T + 2$$
$19$ $$T + 4$$
$23$ $$T$$
$29$ $$T - 2$$
$31$ $$T$$
$37$ $$T - 10$$
$41$ $$T - 10$$
$43$ $$T - 4$$
$47$ $$T - 8$$
$53$ $$T - 10$$
$59$ $$T + 4$$
$61$ $$T + 2$$
$67$ $$T + 12$$
$71$ $$T + 8$$
$73$ $$T + 10$$
$79$ $$T$$
$83$ $$T - 12$$
$89$ $$T + 6$$
$97$ $$T + 2$$