# Properties

 Label 175.3.c.c Level $175$ Weight $3$ Character orbit 175.c Analytic conductor $4.768$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,3,Mod(174,175)]

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([1, 1]))

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

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

chi := DirichletCharacter("175.174");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 175.c (of order $$2$$, degree $$1$$, 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: $$4.76840462631$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ 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 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + \beta_{3} q^{3} + \beta_{2} q^{6} + (3 \beta_{3} - \beta_1) q^{7} + 4 \beta_1 q^{8} - 4 q^{9}+O(q^{10})$$ q + b1 * q^2 + b3 * q^3 + b2 * q^6 + (3*b3 - b1) * q^7 + 4*b1 * q^8 - 4 * q^9 $$q + \beta_1 q^{2} + \beta_{3} q^{3} + \beta_{2} q^{6} + (3 \beta_{3} - \beta_1) q^{7} + 4 \beta_1 q^{8} - 4 q^{9} - q^{11} + 9 \beta_{3} q^{13} + (3 \beta_{2} + 4) q^{14} - 16 q^{16} - 3 \beta_{3} q^{17} - 4 \beta_1 q^{18} + 3 \beta_{2} q^{19} + ( - \beta_{2} + 15) q^{21} - \beta_1 q^{22} - 4 \beta_1 q^{23} + 4 \beta_{2} q^{24} + 9 \beta_{2} q^{26} - 13 \beta_{3} q^{27} - 41 q^{29} - 9 \beta_{2} q^{31} - \beta_{3} q^{33} - 3 \beta_{2} q^{34} - 14 \beta_1 q^{37} - 12 \beta_{3} q^{38} + 45 q^{39} - 3 \beta_{2} q^{41} + (4 \beta_{3} + 15 \beta_1) q^{42} + 41 \beta_1 q^{43} + 16 q^{46} + 9 \beta_{3} q^{47} - 16 \beta_{3} q^{48} + ( - 6 \beta_{2} + 41) q^{49} - 15 q^{51} - 37 \beta_1 q^{53} - 13 \beta_{2} q^{54} + (12 \beta_{2} + 16) q^{56} + 15 \beta_1 q^{57} - 41 \beta_1 q^{58} - 21 \beta_{2} q^{59} - 18 \beta_{2} q^{61} + 36 \beta_{3} q^{62} + ( - 12 \beta_{3} + 4 \beta_1) q^{63} - 64 q^{64} - \beta_{2} q^{66} + \beta_1 q^{67} - 4 \beta_{2} q^{69} + 14 q^{71} - 16 \beta_1 q^{72} + 30 \beta_{3} q^{73} + 56 q^{74} + ( - 3 \beta_{3} + \beta_1) q^{77} + 45 \beta_1 q^{78} + 19 q^{79} - 29 q^{81} + 12 \beta_{3} q^{82} - 42 \beta_{3} q^{83} - 164 q^{86} - 41 \beta_{3} q^{87} - 4 \beta_1 q^{88} + 24 \beta_{2} q^{89} + ( - 9 \beta_{2} + 135) q^{91} - 45 \beta_1 q^{93} + 9 \beta_{2} q^{94} - 27 \beta_{3} q^{97} + (24 \beta_{3} + 41 \beta_1) q^{98} + 4 q^{99}+O(q^{100})$$ q + b1 * q^2 + b3 * q^3 + b2 * q^6 + (3*b3 - b1) * q^7 + 4*b1 * q^8 - 4 * q^9 - q^11 + 9*b3 * q^13 + (3*b2 + 4) * q^14 - 16 * q^16 - 3*b3 * q^17 - 4*b1 * q^18 + 3*b2 * q^19 + (-b2 + 15) * q^21 - b1 * q^22 - 4*b1 * q^23 + 4*b2 * q^24 + 9*b2 * q^26 - 13*b3 * q^27 - 41 * q^29 - 9*b2 * q^31 - b3 * q^33 - 3*b2 * q^34 - 14*b1 * q^37 - 12*b3 * q^38 + 45 * q^39 - 3*b2 * q^41 + (4*b3 + 15*b1) * q^42 + 41*b1 * q^43 + 16 * q^46 + 9*b3 * q^47 - 16*b3 * q^48 + (-6*b2 + 41) * q^49 - 15 * q^51 - 37*b1 * q^53 - 13*b2 * q^54 + (12*b2 + 16) * q^56 + 15*b1 * q^57 - 41*b1 * q^58 - 21*b2 * q^59 - 18*b2 * q^61 + 36*b3 * q^62 + (-12*b3 + 4*b1) * q^63 - 64 * q^64 - b2 * q^66 + b1 * q^67 - 4*b2 * q^69 + 14 * q^71 - 16*b1 * q^72 + 30*b3 * q^73 + 56 * q^74 + (-3*b3 + b1) * q^77 + 45*b1 * q^78 + 19 * q^79 - 29 * q^81 + 12*b3 * q^82 - 42*b3 * q^83 - 164 * q^86 - 41*b3 * q^87 - 4*b1 * q^88 + 24*b2 * q^89 + (-9*b2 + 135) * q^91 - 45*b1 * q^93 + 9*b2 * q^94 - 27*b3 * q^97 + (24*b3 + 41*b1) * q^98 + 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 16 q^{9}+O(q^{10})$$ 4 * q - 16 * q^9 $$4 q - 16 q^{9} - 4 q^{11} + 16 q^{14} - 64 q^{16} + 60 q^{21} - 164 q^{29} + 180 q^{39} + 64 q^{46} + 164 q^{49} - 60 q^{51} + 64 q^{56} - 256 q^{64} + 56 q^{71} + 224 q^{74} + 76 q^{79} - 116 q^{81} - 656 q^{86} + 540 q^{91} + 16 q^{99}+O(q^{100})$$ 4 * q - 16 * q^9 - 4 * q^11 + 16 * q^14 - 64 * q^16 + 60 * q^21 - 164 * q^29 + 180 * q^39 + 64 * q^46 + 164 * q^49 - 60 * q^51 + 64 * q^56 - 256 * q^64 + 56 * q^71 + 224 * q^74 + 76 * q^79 - 116 * q^81 - 656 * q^86 + 540 * q^91 + 16 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu^{3} + 4\nu$$ 2*v^3 + 4*v $$\beta_{2}$$ $$=$$ $$2\nu^{3} + 8\nu$$ 2*v^3 + 8*v $$\beta_{3}$$ $$=$$ $$2\nu^{2} + 3$$ 2*v^2 + 3
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 4$$ (b2 - b1) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 3 ) / 2$$ (b3 - 3) / 2 $$\nu^{3}$$ $$=$$ $$( -\beta_{2} + 2\beta_1 ) / 2$$ (-b2 + 2*b1) / 2

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
174.1
 1.61803i − 0.618034i − 1.61803i 0.618034i
2.00000i −2.23607 0 0 4.47214i −6.70820 + 2.00000i 8.00000i −4.00000 0
174.2 2.00000i 2.23607 0 0 4.47214i 6.70820 + 2.00000i 8.00000i −4.00000 0
174.3 2.00000i −2.23607 0 0 4.47214i −6.70820 2.00000i 8.00000i −4.00000 0
174.4 2.00000i 2.23607 0 0 4.47214i 6.70820 2.00000i 8.00000i −4.00000 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
5.b even 2 1 inner
7.b odd 2 1 inner
35.c 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.c.c 4
5.b even 2 1 inner 175.3.c.c 4
5.c odd 4 1 35.3.d.b 2
5.c odd 4 1 175.3.d.c 2
7.b odd 2 1 inner 175.3.c.c 4
15.e even 4 1 315.3.h.a 2
20.e even 4 1 560.3.f.b 2
35.c odd 2 1 inner 175.3.c.c 4
35.f even 4 1 35.3.d.b 2
35.f even 4 1 175.3.d.c 2
35.k even 12 2 245.3.h.a 4
35.l odd 12 2 245.3.h.a 4
105.k odd 4 1 315.3.h.a 2
140.j odd 4 1 560.3.f.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.d.b 2 5.c odd 4 1
35.3.d.b 2 35.f even 4 1
175.3.c.c 4 1.a even 1 1 trivial
175.3.c.c 4 5.b even 2 1 inner
175.3.c.c 4 7.b odd 2 1 inner
175.3.c.c 4 35.c odd 2 1 inner
175.3.d.c 2 5.c odd 4 1
175.3.d.c 2 35.f even 4 1
245.3.h.a 4 35.k even 12 2
245.3.h.a 4 35.l odd 12 2
315.3.h.a 2 15.e even 4 1
315.3.h.a 2 105.k odd 4 1
560.3.f.b 2 20.e even 4 1
560.3.f.b 2 140.j odd 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 4)^{2}$$
$3$ $$(T^{2} - 5)^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 82T^{2} + 2401$$
$11$ $$(T + 1)^{4}$$
$13$ $$(T^{2} - 405)^{2}$$
$17$ $$(T^{2} - 45)^{2}$$
$19$ $$(T^{2} + 180)^{2}$$
$23$ $$(T^{2} + 64)^{2}$$
$29$ $$(T + 41)^{4}$$
$31$ $$(T^{2} + 1620)^{2}$$
$37$ $$(T^{2} + 784)^{2}$$
$41$ $$(T^{2} + 180)^{2}$$
$43$ $$(T^{2} + 6724)^{2}$$
$47$ $$(T^{2} - 405)^{2}$$
$53$ $$(T^{2} + 5476)^{2}$$
$59$ $$(T^{2} + 8820)^{2}$$
$61$ $$(T^{2} + 6480)^{2}$$
$67$ $$(T^{2} + 4)^{2}$$
$71$ $$(T - 14)^{4}$$
$73$ $$(T^{2} - 4500)^{2}$$
$79$ $$(T - 19)^{4}$$
$83$ $$(T^{2} - 8820)^{2}$$
$89$ $$(T^{2} + 11520)^{2}$$
$97$ $$(T^{2} - 3645)^{2}$$