# Properties

 Label 175.2.b.b Level $175$ Weight $2$ Character orbit 175.b 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(99,175)]

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, 0]))

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

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

chi := DirichletCharacter("175.99");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$175 = 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 175.b (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: $$1.39738203537$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9x^{2} + 16$$ x^4 + 9*x^2 + 16 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} + \beta_1) q^{3} + (\beta_{3} - 3) q^{4} - 4 q^{6} - \beta_{2} q^{7} + (4 \beta_{2} - \beta_1) q^{8} + ( - \beta_{3} - 1) q^{9}+O(q^{10})$$ q + b1 * q^2 + (b2 + b1) * q^3 + (b3 - 3) * q^4 - 4 * q^6 - b2 * q^7 + (4*b2 - b1) * q^8 + (-b3 - 1) * q^9 $$q + \beta_1 q^{2} + (\beta_{2} + \beta_1) q^{3} + (\beta_{3} - 3) q^{4} - 4 q^{6} - \beta_{2} q^{7} + (4 \beta_{2} - \beta_1) q^{8} + ( - \beta_{3} - 1) q^{9} + \beta_{3} q^{11} + (2 \beta_{2} - 2 \beta_1) q^{12} + ( - 3 \beta_{2} - \beta_1) q^{13} + (\beta_{3} - 1) q^{14} + ( - 3 \beta_{3} + 3) q^{16} + ( - 3 \beta_{2} - \beta_1) q^{17} + ( - 4 \beta_{2} - \beta_1) q^{18} + ( - 2 \beta_{3} + 4) q^{19} + \beta_{3} q^{21} + 4 \beta_{2} q^{22} + (2 \beta_{2} + 2 \beta_1) q^{23} + ( - 4 \beta_{3} + 4) q^{24} + (2 \beta_{3} + 2) q^{26} + ( - 3 \beta_{2} + \beta_1) q^{27} + (2 \beta_{2} - \beta_1) q^{28} + (3 \beta_{3} - 2) q^{29} + ( - 4 \beta_{2} + \beta_1) q^{32} + (5 \beta_{2} + \beta_1) q^{33} + (2 \beta_{3} + 2) q^{34} + (\beta_{3} - 1) q^{36} + 6 \beta_{2} q^{37} + ( - 8 \beta_{2} + 4 \beta_1) q^{38} + (3 \beta_{3} + 4) q^{39} + ( - 2 \beta_{3} + 2) q^{41} + 4 \beta_{2} q^{42} + ( - 6 \beta_{2} - 2 \beta_1) q^{43} + ( - 2 \beta_{3} + 4) q^{44} - 8 q^{46} + ( - \beta_{2} + 3 \beta_1) q^{47} - 12 \beta_{2} q^{48} - q^{49} + (3 \beta_{3} + 4) q^{51} + 2 \beta_{2} q^{52} - 2 \beta_1 q^{53} + (4 \beta_{3} - 8) q^{54} + ( - \beta_{3} + 5) q^{56} + ( - 6 \beta_{2} + 2 \beta_1) q^{57} + (12 \beta_{2} - 2 \beta_1) q^{58} + 4 q^{59} + ( - 6 \beta_{3} + 6) q^{61} + (2 \beta_{2} + \beta_1) q^{63} + ( - \beta_{3} - 3) q^{64} - 4 \beta_{3} q^{66} - 4 \beta_1 q^{67} + 2 \beta_{2} q^{68} + ( - 2 \beta_{3} - 8) q^{69} + 8 q^{71} + ( - 4 \beta_{2} - 3 \beta_1) q^{72} + (2 \beta_{2} - 4 \beta_1) q^{73} + ( - 6 \beta_{3} + 6) q^{74} + (8 \beta_{3} - 20) q^{76} + ( - \beta_{2} - \beta_1) q^{77} + (12 \beta_{2} + 4 \beta_1) q^{78} + (\beta_{3} + 4) q^{79} - 7 q^{81} + ( - 8 \beta_{2} + 2 \beta_1) q^{82} - 4 \beta_{2} q^{83} + ( - 2 \beta_{3} + 4) q^{84} + (4 \beta_{3} + 4) q^{86} + (13 \beta_{2} + \beta_1) q^{87} + 4 \beta_1 q^{88} + ( - 2 \beta_{3} - 2) q^{89} + ( - \beta_{3} - 2) q^{91} + (4 \beta_{2} - 4 \beta_1) q^{92} + (4 \beta_{3} - 16) q^{94} + (4 \beta_{3} - 4) q^{96} + ( - 7 \beta_{2} - 5 \beta_1) q^{97} - \beta_1 q^{98} + ( - 2 \beta_{3} - 4) q^{99}+O(q^{100})$$ q + b1 * q^2 + (b2 + b1) * q^3 + (b3 - 3) * q^4 - 4 * q^6 - b2 * q^7 + (4*b2 - b1) * q^8 + (-b3 - 1) * q^9 + b3 * q^11 + (2*b2 - 2*b1) * q^12 + (-3*b2 - b1) * q^13 + (b3 - 1) * q^14 + (-3*b3 + 3) * q^16 + (-3*b2 - b1) * q^17 + (-4*b2 - b1) * q^18 + (-2*b3 + 4) * q^19 + b3 * q^21 + 4*b2 * q^22 + (2*b2 + 2*b1) * q^23 + (-4*b3 + 4) * q^24 + (2*b3 + 2) * q^26 + (-3*b2 + b1) * q^27 + (2*b2 - b1) * q^28 + (3*b3 - 2) * q^29 + (-4*b2 + b1) * q^32 + (5*b2 + b1) * q^33 + (2*b3 + 2) * q^34 + (b3 - 1) * q^36 + 6*b2 * q^37 + (-8*b2 + 4*b1) * q^38 + (3*b3 + 4) * q^39 + (-2*b3 + 2) * q^41 + 4*b2 * q^42 + (-6*b2 - 2*b1) * q^43 + (-2*b3 + 4) * q^44 - 8 * q^46 + (-b2 + 3*b1) * q^47 - 12*b2 * q^48 - q^49 + (3*b3 + 4) * q^51 + 2*b2 * q^52 - 2*b1 * q^53 + (4*b3 - 8) * q^54 + (-b3 + 5) * q^56 + (-6*b2 + 2*b1) * q^57 + (12*b2 - 2*b1) * q^58 + 4 * q^59 + (-6*b3 + 6) * q^61 + (2*b2 + b1) * q^63 + (-b3 - 3) * q^64 - 4*b3 * q^66 - 4*b1 * q^67 + 2*b2 * q^68 + (-2*b3 - 8) * q^69 + 8 * q^71 + (-4*b2 - 3*b1) * q^72 + (2*b2 - 4*b1) * q^73 + (-6*b3 + 6) * q^74 + (8*b3 - 20) * q^76 + (-b2 - b1) * q^77 + (12*b2 + 4*b1) * q^78 + (b3 + 4) * q^79 - 7 * q^81 + (-8*b2 + 2*b1) * q^82 - 4*b2 * q^83 + (-2*b3 + 4) * q^84 + (4*b3 + 4) * q^86 + (13*b2 + b1) * q^87 + 4*b1 * q^88 + (-2*b3 - 2) * q^89 + (-b3 - 2) * q^91 + (4*b2 - 4*b1) * q^92 + (4*b3 - 16) * q^94 + (4*b3 - 4) * q^96 + (-7*b2 - 5*b1) * q^97 - b1 * q^98 + (-2*b3 - 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 10 q^{4} - 16 q^{6} - 6 q^{9}+O(q^{10})$$ 4 * q - 10 * q^4 - 16 * q^6 - 6 * q^9 $$4 q - 10 q^{4} - 16 q^{6} - 6 q^{9} + 2 q^{11} - 2 q^{14} + 6 q^{16} + 12 q^{19} + 2 q^{21} + 8 q^{24} + 12 q^{26} - 2 q^{29} + 12 q^{34} - 2 q^{36} + 22 q^{39} + 4 q^{41} + 12 q^{44} - 32 q^{46} - 4 q^{49} + 22 q^{51} - 24 q^{54} + 18 q^{56} + 16 q^{59} + 12 q^{61} - 14 q^{64} - 8 q^{66} - 36 q^{69} + 32 q^{71} + 12 q^{74} - 64 q^{76} + 18 q^{79} - 28 q^{81} + 12 q^{84} + 24 q^{86} - 12 q^{89} - 10 q^{91} - 56 q^{94} - 8 q^{96} - 20 q^{99}+O(q^{100})$$ 4 * q - 10 * q^4 - 16 * q^6 - 6 * q^9 + 2 * q^11 - 2 * q^14 + 6 * q^16 + 12 * q^19 + 2 * q^21 + 8 * q^24 + 12 * q^26 - 2 * q^29 + 12 * q^34 - 2 * q^36 + 22 * q^39 + 4 * q^41 + 12 * q^44 - 32 * q^46 - 4 * q^49 + 22 * q^51 - 24 * q^54 + 18 * q^56 + 16 * q^59 + 12 * q^61 - 14 * q^64 - 8 * q^66 - 36 * q^69 + 32 * q^71 + 12 * q^74 - 64 * q^76 + 18 * q^79 - 28 * q^81 + 12 * q^84 + 24 * q^86 - 12 * q^89 - 10 * q^91 - 56 * q^94 - 8 * q^96 - 20 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9x^{2} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 5\nu ) / 4$$ (v^3 + 5*v) / 4 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 5$$ v^2 + 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 5$$ b3 - 5 $$\nu^{3}$$ $$=$$ $$4\beta_{2} - 5\beta_1$$ 4*b2 - 5*b1

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 − 2.56155i − 1.56155i 1.56155i 2.56155i
2.56155i 1.56155i −4.56155 0 −4.00000 1.00000i 6.56155i 0.561553 0
99.2 1.56155i 2.56155i −0.438447 0 −4.00000 1.00000i 2.43845i −3.56155 0
99.3 1.56155i 2.56155i −0.438447 0 −4.00000 1.00000i 2.43845i −3.56155 0
99.4 2.56155i 1.56155i −4.56155 0 −4.00000 1.00000i 6.56155i 0.561553 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.2.b.b 4
3.b odd 2 1 1575.2.d.e 4
4.b odd 2 1 2800.2.g.t 4
5.b even 2 1 inner 175.2.b.b 4
5.c odd 4 1 35.2.a.b 2
5.c odd 4 1 175.2.a.f 2
7.b odd 2 1 1225.2.b.f 4
15.d odd 2 1 1575.2.d.e 4
15.e even 4 1 315.2.a.e 2
15.e even 4 1 1575.2.a.p 2
20.d odd 2 1 2800.2.g.t 4
20.e even 4 1 560.2.a.i 2
20.e even 4 1 2800.2.a.bi 2
35.c odd 2 1 1225.2.b.f 4
35.f even 4 1 245.2.a.d 2
35.f even 4 1 1225.2.a.s 2
35.k even 12 2 245.2.e.h 4
35.l odd 12 2 245.2.e.i 4
40.i odd 4 1 2240.2.a.bh 2
40.k even 4 1 2240.2.a.bd 2
55.e even 4 1 4235.2.a.m 2
60.l odd 4 1 5040.2.a.bt 2
65.h odd 4 1 5915.2.a.l 2
105.k odd 4 1 2205.2.a.x 2
140.j odd 4 1 3920.2.a.bs 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.a.b 2 5.c odd 4 1
175.2.a.f 2 5.c odd 4 1
175.2.b.b 4 1.a even 1 1 trivial
175.2.b.b 4 5.b even 2 1 inner
245.2.a.d 2 35.f even 4 1
245.2.e.h 4 35.k even 12 2
245.2.e.i 4 35.l odd 12 2
315.2.a.e 2 15.e even 4 1
560.2.a.i 2 20.e even 4 1
1225.2.a.s 2 35.f even 4 1
1225.2.b.f 4 7.b odd 2 1
1225.2.b.f 4 35.c odd 2 1
1575.2.a.p 2 15.e even 4 1
1575.2.d.e 4 3.b odd 2 1
1575.2.d.e 4 15.d odd 2 1
2205.2.a.x 2 105.k odd 4 1
2240.2.a.bd 2 40.k even 4 1
2240.2.a.bh 2 40.i odd 4 1
2800.2.a.bi 2 20.e even 4 1
2800.2.g.t 4 4.b odd 2 1
2800.2.g.t 4 20.d odd 2 1
3920.2.a.bs 2 140.j odd 4 1
4235.2.a.m 2 55.e even 4 1
5040.2.a.bt 2 60.l odd 4 1
5915.2.a.l 2 65.h odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 9T_{2}^{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(175, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 9T^{2} + 16$$
$3$ $$T^{4} + 9T^{2} + 16$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 1)^{2}$$
$11$ $$(T^{2} - T - 4)^{2}$$
$13$ $$T^{4} + 21T^{2} + 4$$
$17$ $$T^{4} + 21T^{2} + 4$$
$19$ $$(T^{2} - 6 T - 8)^{2}$$
$23$ $$T^{4} + 36T^{2} + 256$$
$29$ $$(T^{2} + T - 38)^{2}$$
$31$ $$T^{4}$$
$37$ $$(T^{2} + 36)^{2}$$
$41$ $$(T^{2} - 2 T - 16)^{2}$$
$43$ $$T^{4} + 84T^{2} + 64$$
$47$ $$T^{4} + 89T^{2} + 1024$$
$53$ $$T^{4} + 36T^{2} + 256$$
$59$ $$(T - 4)^{4}$$
$61$ $$(T^{2} - 6 T - 144)^{2}$$
$67$ $$T^{4} + 144T^{2} + 4096$$
$71$ $$(T - 8)^{4}$$
$73$ $$T^{4} + 168T^{2} + 2704$$
$79$ $$(T^{2} - 9 T + 16)^{2}$$
$83$ $$(T^{2} + 16)^{2}$$
$89$ $$(T^{2} + 6 T - 8)^{2}$$
$97$ $$T^{4} + 253T^{2} + 7396$$