# Properties

 Label 175.2.k.a Level $175$ Weight $2$ Character orbit 175.k Analytic conductor $1.397$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,2,Mod(74,175)]

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([3, 4]))

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

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

chi := DirichletCharacter("175.74");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 175.k (of order $$6$$, degree $$2$$, not 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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 35) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} - \beta_{4} q^{3} + ( - \beta_{6} - \beta_1 + 1) q^{4} - q^{6} + ( - \beta_{7} - \beta_{3} + \beta_{2}) q^{7} + ( - \beta_{7} + \beta_{4} + 2 \beta_{3} - 2 \beta_{2}) q^{8} + \beta_{5} q^{9}+O(q^{10})$$ q + b3 * q^2 - b4 * q^3 + (-b6 - b1 + 1) * q^4 - q^6 + (-b7 - b3 + b2) * q^7 + (-b7 + b4 + 2*b3 - 2*b2) * q^8 + b5 * q^9 $$q + \beta_{3} q^{2} - \beta_{4} q^{3} + ( - \beta_{6} - \beta_1 + 1) q^{4} - q^{6} + ( - \beta_{7} - \beta_{3} + \beta_{2}) q^{7} + ( - \beta_{7} + \beta_{4} + 2 \beta_{3} - 2 \beta_{2}) q^{8} + \beta_{5} q^{9} + ( - \beta_{6} + 2 \beta_1 - 2) q^{11} + (2 \beta_{7} - \beta_{3}) q^{12} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{13} + ( - \beta_{5} + 4 \beta_1 - 1) q^{14} - 3 \beta_1 q^{16} - 2 \beta_{2} q^{17} + (\beta_{4} - 3 \beta_{2}) q^{18} - \beta_{5} q^{19} + (\beta_{6} + \beta_{5} - \beta_1 + 4) q^{21} + ( - \beta_{7} + \beta_{4} + \beta_{3} - \beta_{2}) q^{22} - \beta_{7} q^{23} + (\beta_{6} - \beta_1 + 1) q^{24} + ( - 2 \beta_{5} + 6 \beta_1) q^{26} + ( - \beta_{3} + \beta_{2}) q^{27} + (2 \beta_{7} - 3 \beta_{4} - \beta_{3} + 5 \beta_{2}) q^{28} + q^{29} + ( - 6 \beta_1 + 6) q^{31} + ( - 2 \beta_{4} + \beta_{2}) q^{32} + (5 \beta_{7} - \beta_{3}) q^{33} + (2 \beta_{6} + 2 \beta_{5} - 6) q^{34} + (\beta_{6} + \beta_{5} - 8) q^{36} + ( - \beta_{4} + 3 \beta_{2}) q^{38} + ( - 2 \beta_1 + 2) q^{39} + (\beta_{6} + \beta_{5} - 5) q^{41} + (\beta_{7} + \beta_{3} - \beta_{2}) q^{42} + ( - 3 \beta_{7} + 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2}) q^{43} + ( - \beta_{5} - 6 \beta_1) q^{44} + (\beta_1 - 1) q^{46} + ( - \beta_{7} + \beta_{3}) q^{47} + ( - 3 \beta_{7} + 3 \beta_{4}) q^{48} + (2 \beta_{6} + \beta_{5} - 5 \beta_1) q^{49} + 2 \beta_1 q^{51} + ( - 2 \beta_{4} + 8 \beta_{2}) q^{52} + (3 \beta_{4} - \beta_{2}) q^{53} + ( - \beta_{5} + 3 \beta_1) q^{54} + ( - 2 \beta_{6} - 3 \beta_{5} - \beta_1 + 5) q^{56} + ( - 3 \beta_{7} + 3 \beta_{4} + \beta_{3} - \beta_{2}) q^{57} + \beta_{3} q^{58} + ( - 3 \beta_{6} + 4 \beta_1 - 4) q^{59} + (3 \beta_{5} + 3 \beta_1) q^{61} + (6 \beta_{3} - 6 \beta_{2}) q^{62} + ( - \beta_{7} - 3 \beta_{4} + 3 \beta_{3} + \beta_{2}) q^{63} + ( - \beta_{6} - \beta_{5} + 7) q^{64} + (\beta_{6} - 2 \beta_1 + 2) q^{66} + (6 \beta_{4} - 5 \beta_{2}) q^{67} + (2 \beta_{7} - 8 \beta_{3}) q^{68} + (\beta_{6} + \beta_{5} + 3) q^{69} + ( - 3 \beta_{6} - 3 \beta_{5} - 4) q^{71} + ( - \beta_{7} - 5 \beta_{3}) q^{72} - 2 \beta_{4} q^{73} + ( - \beta_{6} - \beta_{5} + 8) q^{76} + (5 \beta_{7} - 6 \beta_{4} - \beta_{3} + 2 \beta_{2}) q^{77} + (2 \beta_{3} - 2 \beta_{2}) q^{78} + ( - \beta_{5} + 12 \beta_1) q^{79} + (3 \beta_{6} - \beta_1 + 1) q^{81} + (\beta_{7} - 8 \beta_{3}) q^{82} + (4 \beta_{7} - 4 \beta_{4} + 5 \beta_{3} - 5 \beta_{2}) q^{83} + ( - 2 \beta_{6} + \beta_{5} + 4 \beta_1 - 5) q^{84} + (2 \beta_{5} - 3 \beta_1) q^{86} - \beta_{4} q^{87} + ( - 3 \beta_{4} - \beta_{2}) q^{88} + (2 \beta_{5} - 3 \beta_1) q^{89} + (2 \beta_{6} + 2 \beta_{5} - 2 \beta_1 - 6) q^{91} + (2 \beta_{7} - 2 \beta_{4} - \beta_{3} + \beta_{2}) q^{92} - 6 \beta_{7} q^{93} + ( - \beta_{6} - 2 \beta_1 + 2) q^{94} + (2 \beta_{5} + 5 \beta_1) q^{96} + (\beta_{7} - \beta_{4} - 5 \beta_{3} + 5 \beta_{2}) q^{97} + (2 \beta_{7} - \beta_{4} - 6 \beta_{3} - 2 \beta_{2}) q^{98} + ( - 2 \beta_{6} - 2 \beta_{5} - 8) q^{99}+O(q^{100})$$ q + b3 * q^2 - b4 * q^3 + (-b6 - b1 + 1) * q^4 - q^6 + (-b7 - b3 + b2) * q^7 + (-b7 + b4 + 2*b3 - 2*b2) * q^8 + b5 * q^9 + (-b6 + 2*b1 - 2) * q^11 + (2*b7 - b3) * q^12 + (-2*b3 + 2*b2) * q^13 + (-b5 + 4*b1 - 1) * q^14 - 3*b1 * q^16 - 2*b2 * q^17 + (b4 - 3*b2) * q^18 - b5 * q^19 + (b6 + b5 - b1 + 4) * q^21 + (-b7 + b4 + b3 - b2) * q^22 - b7 * q^23 + (b6 - b1 + 1) * q^24 + (-2*b5 + 6*b1) * q^26 + (-b3 + b2) * q^27 + (2*b7 - 3*b4 - b3 + 5*b2) * q^28 + q^29 + (-6*b1 + 6) * q^31 + (-2*b4 + b2) * q^32 + (5*b7 - b3) * q^33 + (2*b6 + 2*b5 - 6) * q^34 + (b6 + b5 - 8) * q^36 + (-b4 + 3*b2) * q^38 + (-2*b1 + 2) * q^39 + (b6 + b5 - 5) * q^41 + (b7 + b3 - b2) * q^42 + (-3*b7 + 3*b4 + 2*b3 - 2*b2) * q^43 + (-b5 - 6*b1) * q^44 + (b1 - 1) * q^46 + (-b7 + b3) * q^47 + (-3*b7 + 3*b4) * q^48 + (2*b6 + b5 - 5*b1) * q^49 + 2*b1 * q^51 + (-2*b4 + 8*b2) * q^52 + (3*b4 - b2) * q^53 + (-b5 + 3*b1) * q^54 + (-2*b6 - 3*b5 - b1 + 5) * q^56 + (-3*b7 + 3*b4 + b3 - b2) * q^57 + b3 * q^58 + (-3*b6 + 4*b1 - 4) * q^59 + (3*b5 + 3*b1) * q^61 + (6*b3 - 6*b2) * q^62 + (-b7 - 3*b4 + 3*b3 + b2) * q^63 + (-b6 - b5 + 7) * q^64 + (b6 - 2*b1 + 2) * q^66 + (6*b4 - 5*b2) * q^67 + (2*b7 - 8*b3) * q^68 + (b6 + b5 + 3) * q^69 + (-3*b6 - 3*b5 - 4) * q^71 + (-b7 - 5*b3) * q^72 - 2*b4 * q^73 + (-b6 - b5 + 8) * q^76 + (5*b7 - 6*b4 - b3 + 2*b2) * q^77 + (2*b3 - 2*b2) * q^78 + (-b5 + 12*b1) * q^79 + (3*b6 - b1 + 1) * q^81 + (b7 - 8*b3) * q^82 + (4*b7 - 4*b4 + 5*b3 - 5*b2) * q^83 + (-2*b6 + b5 + 4*b1 - 5) * q^84 + (2*b5 - 3*b1) * q^86 - b4 * q^87 + (-3*b4 - b2) * q^88 + (2*b5 - 3*b1) * q^89 + (2*b6 + 2*b5 - 2*b1 - 6) * q^91 + (2*b7 - 2*b4 - b3 + b2) * q^92 - 6*b7 * q^93 + (-b6 - 2*b1 + 2) * q^94 + (2*b5 + 5*b1) * q^96 + (b7 - b4 - 5*b3 + 5*b2) * q^97 + (2*b7 - b4 - 6*b3 - 2*b2) * q^98 + (-2*b6 - 2*b5 - 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 4 q^{4} - 8 q^{6}+O(q^{10})$$ 8 * q + 4 * q^4 - 8 * q^6 $$8 q + 4 q^{4} - 8 q^{6} - 8 q^{11} + 8 q^{14} - 12 q^{16} + 28 q^{21} + 4 q^{24} + 24 q^{26} + 8 q^{29} + 24 q^{31} - 48 q^{34} - 64 q^{36} + 8 q^{39} - 40 q^{41} - 24 q^{44} - 4 q^{46} - 20 q^{49} + 8 q^{51} + 12 q^{54} + 36 q^{56} - 16 q^{59} + 12 q^{61} + 56 q^{64} + 8 q^{66} + 24 q^{69} - 32 q^{71} + 64 q^{76} + 48 q^{79} + 4 q^{81} - 24 q^{84} - 12 q^{86} - 12 q^{89} - 56 q^{91} + 8 q^{94} + 20 q^{96} - 64 q^{99}+O(q^{100})$$ 8 * q + 4 * q^4 - 8 * q^6 - 8 * q^11 + 8 * q^14 - 12 * q^16 + 28 * q^21 + 4 * q^24 + 24 * q^26 + 8 * q^29 + 24 * q^31 - 48 * q^34 - 64 * q^36 + 8 * q^39 - 40 * q^41 - 24 * q^44 - 4 * q^46 - 20 * q^49 + 8 * q^51 + 12 * q^54 + 36 * q^56 - 16 * q^59 + 12 * q^61 + 56 * q^64 + 8 * q^66 + 24 * q^69 - 32 * q^71 + 64 * q^76 + 48 * q^79 + 4 * q^81 - 24 * q^84 - 12 * q^86 - 12 * q^89 - 56 * q^91 + 8 * q^94 + 20 * q^96 - 64 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{24}^{4}$$ v^4 $$\beta_{2}$$ $$=$$ $$-\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} - \zeta_{24}^{2}$$ -v^7 + v^5 + v^3 - v^2 $$\beta_{3}$$ $$=$$ $$-\zeta_{24}^{7} + \zeta_{24}^{6} - \zeta_{24}^{2} + \zeta_{24}$$ -v^7 + v^6 - v^2 + v $$\beta_{4}$$ $$=$$ $$-\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}^{2}$$ -v^7 + v^5 + v^3 + v^2 $$\beta_{5}$$ $$=$$ $$2\zeta_{24}^{7} + 2\zeta_{24}$$ 2*v^7 + 2*v $$\beta_{6}$$ $$=$$ $$-2\zeta_{24}^{7} - 2\zeta_{24}^{5} + 2\zeta_{24}^{3}$$ -2*v^7 - 2*v^5 + 2*v^3 $$\beta_{7}$$ $$=$$ $$-\zeta_{24}^{7} - \zeta_{24}^{6} + \zeta_{24}^{2} + \zeta_{24}$$ -v^7 - v^6 + v^2 + v
 $$\zeta_{24}$$ $$=$$ $$( \beta_{7} + \beta_{5} + \beta_{3} ) / 4$$ (b7 + b5 + b3) / 4 $$\zeta_{24}^{2}$$ $$=$$ $$( \beta_{4} - \beta_{2} ) / 2$$ (b4 - b2) / 2 $$\zeta_{24}^{3}$$ $$=$$ $$( -\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} ) / 4$$ (-b7 + b6 + b5 + b4 - b3 + b2) / 4 $$\zeta_{24}^{4}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{24}^{5}$$ $$=$$ $$( -\beta_{6} + \beta_{4} + \beta_{2} ) / 4$$ (-b6 + b4 + b2) / 4 $$\zeta_{24}^{6}$$ $$=$$ $$( -\beta_{7} + \beta_{4} + \beta_{3} - \beta_{2} ) / 2$$ (-b7 + b4 + b3 - b2) / 2 $$\zeta_{24}^{7}$$ $$=$$ $$( -\beta_{7} + \beta_{5} - \beta_{3} ) / 4$$ (-b7 + b5 - b3) / 4

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
74.1
 −0.965926 − 0.258819i −0.258819 + 0.965926i 0.965926 + 0.258819i 0.258819 − 0.965926i −0.965926 + 0.258819i −0.258819 − 0.965926i 0.965926 − 0.258819i 0.258819 + 0.965926i
−2.09077 + 1.20711i 0.358719 + 0.207107i 1.91421 3.31552i 0 −1.00000 0.358719 2.62132i 4.41421i −1.41421 2.44949i 0
74.2 −0.358719 + 0.207107i 2.09077 + 1.20711i −0.914214 + 1.58346i 0 −1.00000 2.09077 1.62132i 1.58579i 1.41421 + 2.44949i 0
74.3 0.358719 0.207107i −2.09077 1.20711i −0.914214 + 1.58346i 0 −1.00000 −2.09077 + 1.62132i 1.58579i 1.41421 + 2.44949i 0
74.4 2.09077 1.20711i −0.358719 0.207107i 1.91421 3.31552i 0 −1.00000 −0.358719 + 2.62132i 4.41421i −1.41421 2.44949i 0
149.1 −2.09077 1.20711i 0.358719 0.207107i 1.91421 + 3.31552i 0 −1.00000 0.358719 + 2.62132i 4.41421i −1.41421 + 2.44949i 0
149.2 −0.358719 0.207107i 2.09077 1.20711i −0.914214 1.58346i 0 −1.00000 2.09077 + 1.62132i 1.58579i 1.41421 2.44949i 0
149.3 0.358719 + 0.207107i −2.09077 + 1.20711i −0.914214 1.58346i 0 −1.00000 −2.09077 1.62132i 1.58579i 1.41421 2.44949i 0
149.4 2.09077 + 1.20711i −0.358719 + 0.207107i 1.91421 + 3.31552i 0 −1.00000 −0.358719 2.62132i 4.41421i −1.41421 + 2.44949i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 149.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - 6T_{2}^{6} + 35T_{2}^{4} - 6T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(175, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 6 T^{6} + 35 T^{4} - 6 T^{2} + \cdots + 1$$
$3$ $$T^{8} - 6 T^{6} + 35 T^{4} - 6 T^{2} + \cdots + 1$$
$5$ $$T^{8}$$
$7$ $$T^{8} + 10 T^{6} + 51 T^{4} + \cdots + 2401$$
$11$ $$(T^{4} + 4 T^{3} + 20 T^{2} - 16 T + 16)^{2}$$
$13$ $$(T^{4} + 24 T^{2} + 16)^{2}$$
$17$ $$T^{8} - 24 T^{6} + 560 T^{4} + \cdots + 256$$
$19$ $$(T^{4} + 8 T^{2} + 64)^{2}$$
$23$ $$T^{8} - 6 T^{6} + 35 T^{4} - 6 T^{2} + \cdots + 1$$
$29$ $$(T - 1)^{8}$$
$31$ $$(T^{2} - 6 T + 36)^{4}$$
$37$ $$T^{8}$$
$41$ $$(T^{2} + 10 T + 17)^{4}$$
$43$ $$(T^{4} + 54 T^{2} + 529)^{2}$$
$47$ $$(T^{4} - 4 T^{2} + 16)^{2}$$
$53$ $$T^{8} - 48 T^{6} + 2240 T^{4} + \cdots + 4096$$
$59$ $$(T^{4} + 8 T^{3} + 120 T^{2} - 448 T + 3136)^{2}$$
$61$ $$(T^{4} - 6 T^{3} + 99 T^{2} + 378 T + 3969)^{2}$$
$67$ $$T^{8} - 246 T^{6} + \cdots + 200533921$$
$71$ $$(T^{2} + 8 T - 56)^{4}$$
$73$ $$T^{8} - 24 T^{6} + 560 T^{4} + \cdots + 256$$
$79$ $$(T^{4} - 24 T^{3} + 440 T^{2} + \cdots + 18496)^{2}$$
$83$ $$(T^{4} + 326 T^{2} + 25921)^{2}$$
$89$ $$(T^{4} + 6 T^{3} + 59 T^{2} - 138 T + 529)^{2}$$
$97$ $$(T^{4} + 136 T^{2} + 16)^{2}$$