# Properties

 Label 175.2.e.c Level $175$ Weight $2$ Character orbit 175.e Analytic conductor $1.397$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,2,Mod(51,175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("175.51");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 175.e (of order $$3$$, degree $$2$$, 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: $$1.39738203537$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + \beta_1 + 1) q^{2} + (\beta_{3} - \beta_{2} + \beta_1) q^{3} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{4} - q^{6} + (\beta_{3} + \beta_{2} - \beta_1) q^{7} + (\beta_{3} - 3) q^{8} + 2 \beta_1 q^{9}+O(q^{10})$$ q + (b2 + b1 + 1) * q^2 + (b3 - b2 + b1) * q^3 + (2*b3 + b2 + 2*b1) * q^4 - q^6 + (b3 + b2 - b1) * q^7 + (b3 - 3) * q^8 + 2*b1 * q^9 $$q + (\beta_{2} + \beta_1 + 1) q^{2} + (\beta_{3} - \beta_{2} + \beta_1) q^{3} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{4} - q^{6} + (\beta_{3} + \beta_{2} - \beta_1) q^{7} + (\beta_{3} - 3) q^{8} + 2 \beta_1 q^{9} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{11} + ( - 3 \beta_{2} + \beta_1 - 3) q^{12} + ( - 2 \beta_{3} + 2) q^{13} + ( - 4 \beta_{2} - 2 \beta_1 - 3) q^{14} + ( - 3 \beta_{2} - 3) q^{16} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{17} + (2 \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{18} - 2 \beta_1 q^{19} + (2 \beta_{3} - \beta_{2} + 3) q^{21} + 2 q^{22} + ( - \beta_{2} + \beta_1 - 1) q^{23} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{24} + (6 \beta_{2} + 4 \beta_1 + 6) q^{26} + (\beta_{3} - 1) q^{27} + ( - 2 \beta_{3} - 5 \beta_{2} - 3 \beta_1 + 3) q^{28} - q^{29} - 6 \beta_{2} q^{31} + ( - \beta_{3} + 3 \beta_{2} - \beta_1) q^{32} + (6 \beta_{2} - 4 \beta_1 + 6) q^{33} + ( - 4 \beta_{3} + 6) q^{34} + (2 \beta_{3} - 8) q^{36} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{38} + 2 \beta_{2} q^{39} + (2 \beta_{3} - 5) q^{41} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{42} + ( - \beta_{3} - 5) q^{43} + (6 \beta_{2} - 2 \beta_1 + 6) q^{44} + \beta_{2} q^{46} + (2 \beta_{2} + 2) q^{47} + ( - 3 \beta_{3} - 3) q^{48} + ( - 4 \beta_{3} + 5 \beta_{2} - 2 \beta_1 + 5) q^{49} + (2 \beta_{2} + 2) q^{51} + (6 \beta_{3} + 10 \beta_{2} + 6 \beta_1) q^{52} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{53} + ( - 3 \beta_{2} - 2 \beta_1 - 3) q^{54} + ( - 4 \beta_{3} - \beta_{2} + 2 \beta_1 + 4) q^{56} + (2 \beta_{3} + 4) q^{57} + ( - \beta_{2} - \beta_1 - 1) q^{58} + (6 \beta_{3} - 4 \beta_{2} + 6 \beta_1) q^{59} + (3 \beta_{2} - 6 \beta_1 + 3) q^{61} + ( - 6 \beta_{3} + 6) q^{62} + (2 \beta_{3} - 8 \beta_{2} - 4) q^{63} + (2 \beta_{3} - 7) q^{64} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{66} + (\beta_{3} - 11 \beta_{2} + \beta_1) q^{67} + (10 \beta_{2} + 6 \beta_1 + 10) q^{68} + ( - 2 \beta_{3} - 3) q^{69} + ( - 6 \beta_{3} - 4) q^{71} + ( - 4 \beta_{2} - 6 \beta_1 - 4) q^{72} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{73} + ( - 2 \beta_{3} + 8) q^{76} + ( - 4 \beta_{3} + 2 \beta_{2} - 6) q^{77} + (2 \beta_{3} - 2) q^{78} + ( - 12 \beta_{2} - 2 \beta_1 - 12) q^{79} + (6 \beta_{3} - \beta_{2} + 6 \beta_1) q^{81} + ( - 9 \beta_{2} - 7 \beta_1 - 9) q^{82} + (9 \beta_{3} - 1) q^{83} + (4 \beta_{3} - 4 \beta_{2} + 6 \beta_1 + 1) q^{84} + ( - 3 \beta_{2} - 4 \beta_1 - 3) q^{86} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{87} + (4 \beta_{3} - 2 \beta_{2} + 4 \beta_1) q^{88} + (3 \beta_{2} + 4 \beta_1 + 3) q^{89} + (4 \beta_{3} - 2 \beta_{2} - 8) q^{91} + ( - \beta_{3} - 3) q^{92} + ( - 6 \beta_{2} + 6 \beta_1 - 6) q^{93} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{94} + (5 \beta_{2} - 4 \beta_1 + 5) q^{96} + (4 \beta_{3} - 6) q^{97} + (3 \beta_{3} + 9 \beta_{2} + 7 \beta_1 + 8) q^{98} + (4 \beta_{3} + 8) q^{99}+O(q^{100})$$ q + (b2 + b1 + 1) * q^2 + (b3 - b2 + b1) * q^3 + (2*b3 + b2 + 2*b1) * q^4 - q^6 + (b3 + b2 - b1) * q^7 + (b3 - 3) * q^8 + 2*b1 * q^9 + (-2*b3 + 2*b2 - 2*b1) * q^11 + (-3*b2 + b1 - 3) * q^12 + (-2*b3 + 2) * q^13 + (-4*b2 - 2*b1 - 3) * q^14 + (-3*b2 - 3) * q^16 + (-2*b3 - 2*b2 - 2*b1) * q^17 + (2*b3 + 4*b2 + 2*b1) * q^18 - 2*b1 * q^19 + (2*b3 - b2 + 3) * q^21 + 2 * q^22 + (-b2 + b1 - 1) * q^23 + (-2*b3 + b2 - 2*b1) * q^24 + (6*b2 + 4*b1 + 6) * q^26 + (b3 - 1) * q^27 + (-2*b3 - 5*b2 - 3*b1 + 3) * q^28 - q^29 - 6*b2 * q^31 + (-b3 + 3*b2 - b1) * q^32 + (6*b2 - 4*b1 + 6) * q^33 + (-4*b3 + 6) * q^34 + (2*b3 - 8) * q^36 + (-2*b3 - 4*b2 - 2*b1) * q^38 + 2*b2 * q^39 + (2*b3 - 5) * q^41 + (-b3 - b2 + b1) * q^42 + (-b3 - 5) * q^43 + (6*b2 - 2*b1 + 6) * q^44 + b2 * q^46 + (2*b2 + 2) * q^47 + (-3*b3 - 3) * q^48 + (-4*b3 + 5*b2 - 2*b1 + 5) * q^49 + (2*b2 + 2) * q^51 + (6*b3 + 10*b2 + 6*b1) * q^52 + (-2*b3 + 4*b2 - 2*b1) * q^53 + (-3*b2 - 2*b1 - 3) * q^54 + (-4*b3 - b2 + 2*b1 + 4) * q^56 + (2*b3 + 4) * q^57 + (-b2 - b1 - 1) * q^58 + (6*b3 - 4*b2 + 6*b1) * q^59 + (3*b2 - 6*b1 + 3) * q^61 + (-6*b3 + 6) * q^62 + (2*b3 - 8*b2 - 4) * q^63 + (2*b3 - 7) * q^64 + (2*b3 - 2*b2 + 2*b1) * q^66 + (b3 - 11*b2 + b1) * q^67 + (10*b2 + 6*b1 + 10) * q^68 + (-2*b3 - 3) * q^69 + (-6*b3 - 4) * q^71 + (-4*b2 - 6*b1 - 4) * q^72 + (2*b3 - 2*b2 + 2*b1) * q^73 + (-2*b3 + 8) * q^76 + (-4*b3 + 2*b2 - 6) * q^77 + (2*b3 - 2) * q^78 + (-12*b2 - 2*b1 - 12) * q^79 + (6*b3 - b2 + 6*b1) * q^81 + (-9*b2 - 7*b1 - 9) * q^82 + (9*b3 - 1) * q^83 + (4*b3 - 4*b2 + 6*b1 + 1) * q^84 + (-3*b2 - 4*b1 - 3) * q^86 + (-b3 + b2 - b1) * q^87 + (4*b3 - 2*b2 + 4*b1) * q^88 + (3*b2 + 4*b1 + 3) * q^89 + (4*b3 - 2*b2 - 8) * q^91 + (-b3 - 3) * q^92 + (-6*b2 + 6*b1 - 6) * q^93 + (2*b3 + 2*b2 + 2*b1) * q^94 + (5*b2 - 4*b1 + 5) * q^96 + (4*b3 - 6) * q^97 + (3*b3 + 9*b2 + 7*b1 + 8) * q^98 + (4*b3 + 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} + 2 q^{3} - 2 q^{4} - 4 q^{6} - 2 q^{7} - 12 q^{8}+O(q^{10})$$ 4 * q + 2 * q^2 + 2 * q^3 - 2 * q^4 - 4 * q^6 - 2 * q^7 - 12 * q^8 $$4 q + 2 q^{2} + 2 q^{3} - 2 q^{4} - 4 q^{6} - 2 q^{7} - 12 q^{8} - 4 q^{11} - 6 q^{12} + 8 q^{13} - 4 q^{14} - 6 q^{16} + 4 q^{17} - 8 q^{18} + 14 q^{21} + 8 q^{22} - 2 q^{23} - 2 q^{24} + 12 q^{26} - 4 q^{27} + 22 q^{28} - 4 q^{29} + 12 q^{31} - 6 q^{32} + 12 q^{33} + 24 q^{34} - 32 q^{36} + 8 q^{38} - 4 q^{39} - 20 q^{41} + 2 q^{42} - 20 q^{43} + 12 q^{44} - 2 q^{46} + 4 q^{47} - 12 q^{48} + 10 q^{49} + 4 q^{51} - 20 q^{52} - 8 q^{53} - 6 q^{54} + 18 q^{56} + 16 q^{57} - 2 q^{58} + 8 q^{59} + 6 q^{61} + 24 q^{62} - 28 q^{64} + 4 q^{66} + 22 q^{67} + 20 q^{68} - 12 q^{69} - 16 q^{71} - 8 q^{72} + 4 q^{73} + 32 q^{76} - 28 q^{77} - 8 q^{78} - 24 q^{79} + 2 q^{81} - 18 q^{82} - 4 q^{83} + 12 q^{84} - 6 q^{86} - 2 q^{87} + 4 q^{88} + 6 q^{89} - 28 q^{91} - 12 q^{92} - 12 q^{93} - 4 q^{94} + 10 q^{96} - 24 q^{97} + 14 q^{98} + 32 q^{99}+O(q^{100})$$ 4 * q + 2 * q^2 + 2 * q^3 - 2 * q^4 - 4 * q^6 - 2 * q^7 - 12 * q^8 - 4 * q^11 - 6 * q^12 + 8 * q^13 - 4 * q^14 - 6 * q^16 + 4 * q^17 - 8 * q^18 + 14 * q^21 + 8 * q^22 - 2 * q^23 - 2 * q^24 + 12 * q^26 - 4 * q^27 + 22 * q^28 - 4 * q^29 + 12 * q^31 - 6 * q^32 + 12 * q^33 + 24 * q^34 - 32 * q^36 + 8 * q^38 - 4 * q^39 - 20 * q^41 + 2 * q^42 - 20 * q^43 + 12 * q^44 - 2 * q^46 + 4 * q^47 - 12 * q^48 + 10 * q^49 + 4 * q^51 - 20 * q^52 - 8 * q^53 - 6 * q^54 + 18 * q^56 + 16 * q^57 - 2 * q^58 + 8 * q^59 + 6 * q^61 + 24 * q^62 - 28 * q^64 + 4 * q^66 + 22 * q^67 + 20 * q^68 - 12 * q^69 - 16 * q^71 - 8 * q^72 + 4 * q^73 + 32 * q^76 - 28 * q^77 - 8 * q^78 - 24 * q^79 + 2 * q^81 - 18 * q^82 - 4 * q^83 + 12 * q^84 - 6 * q^86 - 2 * q^87 + 4 * q^88 + 6 * q^89 - 28 * q^91 - 12 * q^92 - 12 * q^93 - 4 * q^94 + 10 * q^96 - 24 * q^97 + 14 * q^98 + 32 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-1 - \beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
51.1
 −0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
−0.207107 + 0.358719i 1.20711 + 2.09077i 0.914214 + 1.58346i 0 −1.00000 1.62132 2.09077i −1.58579 −1.41421 + 2.44949i 0
51.2 1.20711 2.09077i −0.207107 0.358719i −1.91421 3.31552i 0 −1.00000 −2.62132 + 0.358719i −4.41421 1.41421 2.44949i 0
151.1 −0.207107 0.358719i 1.20711 2.09077i 0.914214 1.58346i 0 −1.00000 1.62132 + 2.09077i −1.58579 −1.41421 2.44949i 0
151.2 1.20711 + 2.09077i −0.207107 + 0.358719i −1.91421 + 3.31552i 0 −1.00000 −2.62132 0.358719i −4.41421 1.41421 + 2.44949i 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.2.e.c 4
5.b even 2 1 35.2.e.a 4
5.c odd 4 2 175.2.k.a 8
7.c even 3 1 inner 175.2.e.c 4
7.c even 3 1 1225.2.a.k 2
7.d odd 6 1 1225.2.a.m 2
15.d odd 2 1 315.2.j.e 4
20.d odd 2 1 560.2.q.k 4
35.c odd 2 1 245.2.e.e 4
35.i odd 6 1 245.2.a.g 2
35.i odd 6 1 245.2.e.e 4
35.j even 6 1 35.2.e.a 4
35.j even 6 1 245.2.a.h 2
35.k even 12 2 1225.2.b.h 4
35.l odd 12 2 175.2.k.a 8
35.l odd 12 2 1225.2.b.g 4
105.o odd 6 1 315.2.j.e 4
105.o odd 6 1 2205.2.a.n 2
105.p even 6 1 2205.2.a.q 2
140.p odd 6 1 560.2.q.k 4
140.p odd 6 1 3920.2.a.bq 2
140.s even 6 1 3920.2.a.bv 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.e.a 4 5.b even 2 1
35.2.e.a 4 35.j even 6 1
175.2.e.c 4 1.a even 1 1 trivial
175.2.e.c 4 7.c even 3 1 inner
175.2.k.a 8 5.c odd 4 2
175.2.k.a 8 35.l odd 12 2
245.2.a.g 2 35.i odd 6 1
245.2.a.h 2 35.j even 6 1
245.2.e.e 4 35.c odd 2 1
245.2.e.e 4 35.i odd 6 1
315.2.j.e 4 15.d odd 2 1
315.2.j.e 4 105.o odd 6 1
560.2.q.k 4 20.d odd 2 1
560.2.q.k 4 140.p odd 6 1
1225.2.a.k 2 7.c even 3 1
1225.2.a.m 2 7.d odd 6 1
1225.2.b.g 4 35.l odd 12 2
1225.2.b.h 4 35.k even 12 2
2205.2.a.n 2 105.o odd 6 1
2205.2.a.q 2 105.p even 6 1
3920.2.a.bq 2 140.p odd 6 1
3920.2.a.bv 2 140.s even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 2T_{2}^{3} + 5T_{2}^{2} + 2T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(175, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} + 5 T^{2} + 2 T + 1$$
$3$ $$T^{4} - 2 T^{3} + 5 T^{2} + 2 T + 1$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 2 T^{3} - 3 T^{2} + 14 T + 49$$
$11$ $$T^{4} + 4 T^{3} + 20 T^{2} - 16 T + 16$$
$13$ $$(T^{2} - 4 T - 4)^{2}$$
$17$ $$T^{4} - 4 T^{3} + 20 T^{2} + 16 T + 16$$
$19$ $$T^{4} + 8T^{2} + 64$$
$23$ $$T^{4} + 2 T^{3} + 5 T^{2} - 2 T + 1$$
$29$ $$(T + 1)^{4}$$
$31$ $$(T^{2} - 6 T + 36)^{2}$$
$37$ $$T^{4}$$
$41$ $$(T^{2} + 10 T + 17)^{2}$$
$43$ $$(T^{2} + 10 T + 23)^{2}$$
$47$ $$(T^{2} - 2 T + 4)^{2}$$
$53$ $$T^{4} + 8 T^{3} + 56 T^{2} + 64 T + 64$$
$59$ $$T^{4} - 8 T^{3} + 120 T^{2} + \cdots + 3136$$
$61$ $$T^{4} - 6 T^{3} + 99 T^{2} + \cdots + 3969$$
$67$ $$T^{4} - 22 T^{3} + 365 T^{2} + \cdots + 14161$$
$71$ $$(T^{2} + 8 T - 56)^{2}$$
$73$ $$T^{4} - 4 T^{3} + 20 T^{2} + 16 T + 16$$
$79$ $$T^{4} + 24 T^{3} + 440 T^{2} + \cdots + 18496$$
$83$ $$(T^{2} + 2 T - 161)^{2}$$
$89$ $$T^{4} - 6 T^{3} + 59 T^{2} + 138 T + 529$$
$97$ $$(T^{2} + 12 T + 4)^{2}$$