# Properties

 Label 175.3.d.f Level $175$ Weight $3$ Character orbit 175.d Analytic conductor $4.768$ 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] = [175,3,Mod(76,175)]

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, 1]))

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

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

chi := DirichletCharacter("175.76");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 175.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: $$4.76840462631$$ 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: $$2$$ Twist minimal: no (minimal twist has level 35) 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 $$\beta = 2\sqrt{-5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + \beta q^{3} - 3 q^{4} + \beta q^{6} - 7 q^{7} - 7 q^{8} - 11 q^{9}+O(q^{10})$$ q + q^2 + b * q^3 - 3 * q^4 + b * q^6 - 7 * q^7 - 7 * q^8 - 11 * q^9 $$q + q^{2} + \beta q^{3} - 3 q^{4} + \beta q^{6} - 7 q^{7} - 7 q^{8} - 11 q^{9} + 2 q^{11} - 3 \beta q^{12} - 3 \beta q^{13} - 7 q^{14} + 5 q^{16} + 6 \beta q^{17} - 11 q^{18} + 3 \beta q^{19} - 7 \beta q^{21} + 2 q^{22} - 26 q^{23} - 7 \beta q^{24} - 3 \beta q^{26} - 2 \beta q^{27} + 21 q^{28} - 22 q^{29} + 12 \beta q^{31} + 33 q^{32} + 2 \beta q^{33} + 6 \beta q^{34} + 33 q^{36} - 14 q^{37} + 3 \beta q^{38} + 60 q^{39} + 6 \beta q^{41} - 7 \beta q^{42} + 34 q^{43} - 6 q^{44} - 26 q^{46} - 6 \beta q^{47} + 5 \beta q^{48} + 49 q^{49} - 120 q^{51} + 9 \beta q^{52} + 34 q^{53} - 2 \beta q^{54} + 49 q^{56} - 60 q^{57} - 22 q^{58} + 9 \beta q^{59} - 21 \beta q^{61} + 12 \beta q^{62} + 77 q^{63} + 13 q^{64} + 2 \beta q^{66} - 14 q^{67} - 18 \beta q^{68} - 26 \beta q^{69} + 62 q^{71} + 77 q^{72} - 12 \beta q^{73} - 14 q^{74} - 9 \beta q^{76} - 14 q^{77} + 60 q^{78} + 38 q^{79} - 59 q^{81} + 6 \beta q^{82} + 9 \beta q^{83} + 21 \beta q^{84} + 34 q^{86} - 22 \beta q^{87} - 14 q^{88} + 6 \beta q^{89} + 21 \beta q^{91} + 78 q^{92} - 240 q^{93} - 6 \beta q^{94} + 33 \beta q^{96} - 6 \beta q^{97} + 49 q^{98} - 22 q^{99} +O(q^{100})$$ q + q^2 + b * q^3 - 3 * q^4 + b * q^6 - 7 * q^7 - 7 * q^8 - 11 * q^9 + 2 * q^11 - 3*b * q^12 - 3*b * q^13 - 7 * q^14 + 5 * q^16 + 6*b * q^17 - 11 * q^18 + 3*b * q^19 - 7*b * q^21 + 2 * q^22 - 26 * q^23 - 7*b * q^24 - 3*b * q^26 - 2*b * q^27 + 21 * q^28 - 22 * q^29 + 12*b * q^31 + 33 * q^32 + 2*b * q^33 + 6*b * q^34 + 33 * q^36 - 14 * q^37 + 3*b * q^38 + 60 * q^39 + 6*b * q^41 - 7*b * q^42 + 34 * q^43 - 6 * q^44 - 26 * q^46 - 6*b * q^47 + 5*b * q^48 + 49 * q^49 - 120 * q^51 + 9*b * q^52 + 34 * q^53 - 2*b * q^54 + 49 * q^56 - 60 * q^57 - 22 * q^58 + 9*b * q^59 - 21*b * q^61 + 12*b * q^62 + 77 * q^63 + 13 * q^64 + 2*b * q^66 - 14 * q^67 - 18*b * q^68 - 26*b * q^69 + 62 * q^71 + 77 * q^72 - 12*b * q^73 - 14 * q^74 - 9*b * q^76 - 14 * q^77 + 60 * q^78 + 38 * q^79 - 59 * q^81 + 6*b * q^82 + 9*b * q^83 + 21*b * q^84 + 34 * q^86 - 22*b * q^87 - 14 * q^88 + 6*b * q^89 + 21*b * q^91 + 78 * q^92 - 240 * q^93 - 6*b * q^94 + 33*b * q^96 - 6*b * q^97 + 49 * q^98 - 22 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 6 q^{4} - 14 q^{7} - 14 q^{8} - 22 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 6 * q^4 - 14 * q^7 - 14 * q^8 - 22 * q^9 $$2 q + 2 q^{2} - 6 q^{4} - 14 q^{7} - 14 q^{8} - 22 q^{9} + 4 q^{11} - 14 q^{14} + 10 q^{16} - 22 q^{18} + 4 q^{22} - 52 q^{23} + 42 q^{28} - 44 q^{29} + 66 q^{32} + 66 q^{36} - 28 q^{37} + 120 q^{39} + 68 q^{43} - 12 q^{44} - 52 q^{46} + 98 q^{49} - 240 q^{51} + 68 q^{53} + 98 q^{56} - 120 q^{57} - 44 q^{58} + 154 q^{63} + 26 q^{64} - 28 q^{67} + 124 q^{71} + 154 q^{72} - 28 q^{74} - 28 q^{77} + 120 q^{78} + 76 q^{79} - 118 q^{81} + 68 q^{86} - 28 q^{88} + 156 q^{92} - 480 q^{93} + 98 q^{98} - 44 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - 6 * q^4 - 14 * q^7 - 14 * q^8 - 22 * q^9 + 4 * q^11 - 14 * q^14 + 10 * q^16 - 22 * q^18 + 4 * q^22 - 52 * q^23 + 42 * q^28 - 44 * q^29 + 66 * q^32 + 66 * q^36 - 28 * q^37 + 120 * q^39 + 68 * q^43 - 12 * q^44 - 52 * q^46 + 98 * q^49 - 240 * q^51 + 68 * q^53 + 98 * q^56 - 120 * q^57 - 44 * q^58 + 154 * q^63 + 26 * q^64 - 28 * q^67 + 124 * q^71 + 154 * q^72 - 28 * q^74 - 28 * q^77 + 120 * q^78 + 76 * q^79 - 118 * q^81 + 68 * q^86 - 28 * q^88 + 156 * q^92 - 480 * q^93 + 98 * q^98 - 44 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/175\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
76.1
 − 2.23607i 2.23607i
1.00000 4.47214i −3.00000 0 4.47214i −7.00000 −7.00000 −11.0000 0
76.2 1.00000 4.47214i −3.00000 0 4.47214i −7.00000 −7.00000 −11.0000 0
 $$n$$: 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 175.3.d.f 2
5.b even 2 1 35.3.d.a 2
5.c odd 4 2 175.3.c.d 4
7.b odd 2 1 inner 175.3.d.f 2
15.d odd 2 1 315.3.h.b 2
20.d odd 2 1 560.3.f.a 2
35.c odd 2 1 35.3.d.a 2
35.f even 4 2 175.3.c.d 4
35.i odd 6 2 245.3.h.b 4
35.j even 6 2 245.3.h.b 4
105.g even 2 1 315.3.h.b 2
140.c even 2 1 560.3.f.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.d.a 2 5.b even 2 1
35.3.d.a 2 35.c odd 2 1
175.3.c.d 4 5.c odd 4 2
175.3.c.d 4 35.f even 4 2
175.3.d.f 2 1.a even 1 1 trivial
175.3.d.f 2 7.b odd 2 1 inner
245.3.h.b 4 35.i odd 6 2
245.3.h.b 4 35.j even 6 2
315.3.h.b 2 15.d odd 2 1
315.3.h.b 2 105.g even 2 1
560.3.f.a 2 20.d odd 2 1
560.3.f.a 2 140.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 $$S_{3}^{\mathrm{new}}(175, [\chi])$$:

 $$T_{2} - 1$$ T2 - 1 $$T_{3}^{2} + 20$$ T3^2 + 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2} + 20$$
$5$ $$T^{2}$$
$7$ $$(T + 7)^{2}$$
$11$ $$(T - 2)^{2}$$
$13$ $$T^{2} + 180$$
$17$ $$T^{2} + 720$$
$19$ $$T^{2} + 180$$
$23$ $$(T + 26)^{2}$$
$29$ $$(T + 22)^{2}$$
$31$ $$T^{2} + 2880$$
$37$ $$(T + 14)^{2}$$
$41$ $$T^{2} + 720$$
$43$ $$(T - 34)^{2}$$
$47$ $$T^{2} + 720$$
$53$ $$(T - 34)^{2}$$
$59$ $$T^{2} + 1620$$
$61$ $$T^{2} + 8820$$
$67$ $$(T + 14)^{2}$$
$71$ $$(T - 62)^{2}$$
$73$ $$T^{2} + 2880$$
$79$ $$(T - 38)^{2}$$
$83$ $$T^{2} + 1620$$
$89$ $$T^{2} + 720$$
$97$ $$T^{2} + 720$$