# Properties

 Label 2175.2.d.b Level $2175$ Weight $2$ Character orbit 2175.d Analytic conductor $17.367$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2175,2,Mod(376,2175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("2175.376");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2175 = 3 \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2175.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: $$17.3674624396$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 435) 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 i q^{2} + i q^{3} - 2 q^{4} - 2 q^{6} + 4 q^{7} - q^{9}+O(q^{10})$$ q + 2*i * q^2 + i * q^3 - 2 * q^4 - 2 * q^6 + 4 * q^7 - q^9 $$q + 2 i q^{2} + i q^{3} - 2 q^{4} - 2 q^{6} + 4 q^{7} - q^{9} + i q^{11} - 2 i q^{12} - 2 q^{13} + 8 i q^{14} - 4 q^{16} + 6 i q^{17} - 2 i q^{18} - 4 i q^{19} + 4 i q^{21} - 2 q^{22} + 9 q^{23} - 4 i q^{26} - i q^{27} - 8 q^{28} + (5 i + 2) q^{29} + 2 i q^{31} - 8 i q^{32} - q^{33} - 12 q^{34} + 2 q^{36} + i q^{37} + 8 q^{38} - 2 i q^{39} + 9 i q^{41} - 8 q^{42} - i q^{43} - 2 i q^{44} + 18 i q^{46} - 8 i q^{47} - 4 i q^{48} + 9 q^{49} - 6 q^{51} + 4 q^{52} - 9 q^{53} + 2 q^{54} + 4 q^{57} + (4 i - 10) q^{58} - 8 q^{59} + 6 i q^{61} - 4 q^{62} - 4 q^{63} + 8 q^{64} - 2 i q^{66} - 12 q^{67} - 12 i q^{68} + 9 i q^{69} + 2 q^{71} + 15 i q^{73} - 2 q^{74} + 8 i q^{76} + 4 i q^{77} + 4 q^{78} + 4 i q^{79} + q^{81} - 18 q^{82} + 7 q^{83} - 8 i q^{84} + 2 q^{86} + (2 i - 5) q^{87} + 2 i q^{89} - 8 q^{91} - 18 q^{92} - 2 q^{93} + 16 q^{94} + 8 q^{96} - 11 i q^{97} + 18 i q^{98} - i q^{99} +O(q^{100})$$ q + 2*i * q^2 + i * q^3 - 2 * q^4 - 2 * q^6 + 4 * q^7 - q^9 + i * q^11 - 2*i * q^12 - 2 * q^13 + 8*i * q^14 - 4 * q^16 + 6*i * q^17 - 2*i * q^18 - 4*i * q^19 + 4*i * q^21 - 2 * q^22 + 9 * q^23 - 4*i * q^26 - i * q^27 - 8 * q^28 + (5*i + 2) * q^29 + 2*i * q^31 - 8*i * q^32 - q^33 - 12 * q^34 + 2 * q^36 + i * q^37 + 8 * q^38 - 2*i * q^39 + 9*i * q^41 - 8 * q^42 - i * q^43 - 2*i * q^44 + 18*i * q^46 - 8*i * q^47 - 4*i * q^48 + 9 * q^49 - 6 * q^51 + 4 * q^52 - 9 * q^53 + 2 * q^54 + 4 * q^57 + (4*i - 10) * q^58 - 8 * q^59 + 6*i * q^61 - 4 * q^62 - 4 * q^63 + 8 * q^64 - 2*i * q^66 - 12 * q^67 - 12*i * q^68 + 9*i * q^69 + 2 * q^71 + 15*i * q^73 - 2 * q^74 + 8*i * q^76 + 4*i * q^77 + 4 * q^78 + 4*i * q^79 + q^81 - 18 * q^82 + 7 * q^83 - 8*i * q^84 + 2 * q^86 + (2*i - 5) * q^87 + 2*i * q^89 - 8 * q^91 - 18 * q^92 - 2 * q^93 + 16 * q^94 + 8 * q^96 - 11*i * q^97 + 18*i * q^98 - i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{4} - 4 q^{6} + 8 q^{7} - 2 q^{9}+O(q^{10})$$ 2 * q - 4 * q^4 - 4 * q^6 + 8 * q^7 - 2 * q^9 $$2 q - 4 q^{4} - 4 q^{6} + 8 q^{7} - 2 q^{9} - 4 q^{13} - 8 q^{16} - 4 q^{22} + 18 q^{23} - 16 q^{28} + 4 q^{29} - 2 q^{33} - 24 q^{34} + 4 q^{36} + 16 q^{38} - 16 q^{42} + 18 q^{49} - 12 q^{51} + 8 q^{52} - 18 q^{53} + 4 q^{54} + 8 q^{57} - 20 q^{58} - 16 q^{59} - 8 q^{62} - 8 q^{63} + 16 q^{64} - 24 q^{67} + 4 q^{71} - 4 q^{74} + 8 q^{78} + 2 q^{81} - 36 q^{82} + 14 q^{83} + 4 q^{86} - 10 q^{87} - 16 q^{91} - 36 q^{92} - 4 q^{93} + 32 q^{94} + 16 q^{96}+O(q^{100})$$ 2 * q - 4 * q^4 - 4 * q^6 + 8 * q^7 - 2 * q^9 - 4 * q^13 - 8 * q^16 - 4 * q^22 + 18 * q^23 - 16 * q^28 + 4 * q^29 - 2 * q^33 - 24 * q^34 + 4 * q^36 + 16 * q^38 - 16 * q^42 + 18 * q^49 - 12 * q^51 + 8 * q^52 - 18 * q^53 + 4 * q^54 + 8 * q^57 - 20 * q^58 - 16 * q^59 - 8 * q^62 - 8 * q^63 + 16 * q^64 - 24 * q^67 + 4 * q^71 - 4 * q^74 + 8 * q^78 + 2 * q^81 - 36 * q^82 + 14 * q^83 + 4 * q^86 - 10 * q^87 - 16 * q^91 - 36 * q^92 - 4 * q^93 + 32 * q^94 + 16 * q^96

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times$$.

 $$n$$ $$901$$ $$1451$$ $$2002$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
376.1
 − 1.00000i 1.00000i
2.00000i 1.00000i −2.00000 0 −2.00000 4.00000 0 −1.00000 0
376.2 2.00000i 1.00000i −2.00000 0 −2.00000 4.00000 0 −1.00000 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
29.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2175.2.d.b 2
5.b even 2 1 2175.2.d.a 2
5.c odd 4 1 435.2.f.a 2
5.c odd 4 1 435.2.f.d yes 2
15.e even 4 1 1305.2.f.a 2
15.e even 4 1 1305.2.f.d 2
29.b even 2 1 inner 2175.2.d.b 2
145.d even 2 1 2175.2.d.a 2
145.h odd 4 1 435.2.f.a 2
145.h odd 4 1 435.2.f.d yes 2
435.p even 4 1 1305.2.f.a 2
435.p even 4 1 1305.2.f.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.f.a 2 5.c odd 4 1
435.2.f.a 2 145.h odd 4 1
435.2.f.d yes 2 5.c odd 4 1
435.2.f.d yes 2 145.h odd 4 1
1305.2.f.a 2 15.e even 4 1
1305.2.f.a 2 435.p even 4 1
1305.2.f.d 2 15.e even 4 1
1305.2.f.d 2 435.p even 4 1
2175.2.d.a 2 5.b even 2 1
2175.2.d.a 2 145.d even 2 1
2175.2.d.b 2 1.a even 1 1 trivial
2175.2.d.b 2 29.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2175, [\chi])$$:

 $$T_{2}^{2} + 4$$ T2^2 + 4 $$T_{7} - 4$$ T7 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$(T - 4)^{2}$$
$11$ $$T^{2} + 1$$
$13$ $$(T + 2)^{2}$$
$17$ $$T^{2} + 36$$
$19$ $$T^{2} + 16$$
$23$ $$(T - 9)^{2}$$
$29$ $$T^{2} - 4T + 29$$
$31$ $$T^{2} + 4$$
$37$ $$T^{2} + 1$$
$41$ $$T^{2} + 81$$
$43$ $$T^{2} + 1$$
$47$ $$T^{2} + 64$$
$53$ $$(T + 9)^{2}$$
$59$ $$(T + 8)^{2}$$
$61$ $$T^{2} + 36$$
$67$ $$(T + 12)^{2}$$
$71$ $$(T - 2)^{2}$$
$73$ $$T^{2} + 225$$
$79$ $$T^{2} + 16$$
$83$ $$(T - 7)^{2}$$
$89$ $$T^{2} + 4$$
$97$ $$T^{2} + 121$$