# Properties

 Label 175.2.a.a Level $175$ Weight $2$ Character orbit 175.a Self dual yes Analytic conductor $1.397$ 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] = [175,2,Mod(1,175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, 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("175.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 175.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: $$1.39738203537$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) 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)

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

## 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
−2.00000 −1.00000 2.00000 0 2.00000 1.00000 0 −2.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$7$$ $$-1$$

## 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 175.2.a.a 1
3.b odd 2 1 1575.2.a.k 1
4.b odd 2 1 2800.2.a.w 1
5.b even 2 1 175.2.a.c 1
5.c odd 4 2 35.2.b.a 2
7.b odd 2 1 1225.2.a.a 1
15.d odd 2 1 1575.2.a.a 1
15.e even 4 2 315.2.d.a 2
20.d odd 2 1 2800.2.a.l 1
20.e even 4 2 560.2.g.b 2
35.c odd 2 1 1225.2.a.i 1
35.f even 4 2 245.2.b.a 2
35.k even 12 4 245.2.j.d 4
35.l odd 12 4 245.2.j.e 4
40.i odd 4 2 2240.2.g.h 2
40.k even 4 2 2240.2.g.g 2
60.l odd 4 2 5040.2.t.p 2
105.k odd 4 2 2205.2.d.b 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 2$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(175))$$.

## Hecke characteristic polynomials

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