# Properties

 Label 2175.2.c.f Level $2175$ Weight $2$ Character orbit 2175.c Analytic conductor $17.367$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2175,2,Mod(349,2175)]

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, 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("2175.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2175 = 3 \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2175.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: $$17.3674624396$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{21})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 11x^{2} + 25$$ x^4 + 11*x^2 + 25 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 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - \beta_{2} q^{3} + (\beta_{3} - 4) q^{4} + (\beta_{3} - 1) q^{6} + \beta_{2} q^{7} + (5 \beta_{2} - 2 \beta_1) q^{8} - q^{9}+O(q^{10})$$ q + b1 * q^2 - b2 * q^3 + (b3 - 4) * q^4 + (b3 - 1) * q^6 + b2 * q^7 + (5*b2 - 2*b1) * q^8 - q^9 $$q + \beta_1 q^{2} - \beta_{2} q^{3} + (\beta_{3} - 4) q^{4} + (\beta_{3} - 1) q^{6} + \beta_{2} q^{7} + (5 \beta_{2} - 2 \beta_1) q^{8} - q^{9} + 5 q^{11} + (3 \beta_{2} - \beta_1) q^{12} + (\beta_{2} + 2 \beta_1) q^{13} + ( - \beta_{3} + 1) q^{14} + ( - 5 \beta_{3} + 9) q^{16} - 3 \beta_{2} q^{17} - \beta_1 q^{18} + ( - 2 \beta_{3} + 2) q^{19} + q^{21} + 5 \beta_1 q^{22} + 4 \beta_{2} q^{23} + ( - 2 \beta_{3} + 7) q^{24} + (\beta_{3} - 11) q^{26} + \beta_{2} q^{27} + ( - 3 \beta_{2} + \beta_1) q^{28} - q^{29} + 4 q^{31} + ( - 15 \beta_{2} + 5 \beta_1) q^{32} - 5 \beta_{2} q^{33} + (3 \beta_{3} - 3) q^{34} + ( - \beta_{3} + 4) q^{36} - 4 \beta_{2} q^{37} + ( - 10 \beta_{2} + 2 \beta_1) q^{38} + (2 \beta_{3} - 1) q^{39} + ( - 4 \beta_{3} + 2) q^{41} + \beta_1 q^{42} + (6 \beta_{2} + 2 \beta_1) q^{43} + (5 \beta_{3} - 20) q^{44} + ( - 4 \beta_{3} + 4) q^{46} + (7 \beta_{2} + 2 \beta_1) q^{47} + ( - 4 \beta_{2} + 5 \beta_1) q^{48} + 6 q^{49} - 3 q^{51} + (7 \beta_{2} - 7 \beta_1) q^{52} + ( - 4 \beta_{2} + 2 \beta_1) q^{53} + ( - \beta_{3} + 1) q^{54} + (2 \beta_{3} - 7) q^{56} + 2 \beta_1 q^{57} - \beta_1 q^{58} + (2 \beta_{3} + 2) q^{59} + (6 \beta_{3} - 4) q^{61} + 4 \beta_1 q^{62} - \beta_{2} q^{63} + (10 \beta_{3} - 27) q^{64} + (5 \beta_{3} - 5) q^{66} + (3 \beta_{2} - 4 \beta_1) q^{67} + (9 \beta_{2} - 3 \beta_1) q^{68} + 4 q^{69} + ( - 2 \beta_{3} - 4) q^{71} + ( - 5 \beta_{2} + 2 \beta_1) q^{72} - 4 \beta_{2} q^{73} + (4 \beta_{3} - 4) q^{74} + (8 \beta_{3} - 18) q^{76} + 5 \beta_{2} q^{77} + (10 \beta_{2} - \beta_1) q^{78} + ( - 2 \beta_{3} - 2) q^{79} + q^{81} + ( - 20 \beta_{2} + 2 \beta_1) q^{82} + (8 \beta_{2} + 2 \beta_1) q^{83} + (\beta_{3} - 4) q^{84} + ( - 4 \beta_{3} - 6) q^{86} + \beta_{2} q^{87} + (25 \beta_{2} - 10 \beta_1) q^{88} + (2 \beta_{3} - 7) q^{89} + ( - 2 \beta_{3} + 1) q^{91} + ( - 12 \beta_{2} + 4 \beta_1) q^{92} - 4 \beta_{2} q^{93} + ( - 5 \beta_{3} - 5) q^{94} + (5 \beta_{3} - 20) q^{96} + (8 \beta_{2} + 2 \beta_1) q^{97} + 6 \beta_1 q^{98} - 5 q^{99}+O(q^{100})$$ q + b1 * q^2 - b2 * q^3 + (b3 - 4) * q^4 + (b3 - 1) * q^6 + b2 * q^7 + (5*b2 - 2*b1) * q^8 - q^9 + 5 * q^11 + (3*b2 - b1) * q^12 + (b2 + 2*b1) * q^13 + (-b3 + 1) * q^14 + (-5*b3 + 9) * q^16 - 3*b2 * q^17 - b1 * q^18 + (-2*b3 + 2) * q^19 + q^21 + 5*b1 * q^22 + 4*b2 * q^23 + (-2*b3 + 7) * q^24 + (b3 - 11) * q^26 + b2 * q^27 + (-3*b2 + b1) * q^28 - q^29 + 4 * q^31 + (-15*b2 + 5*b1) * q^32 - 5*b2 * q^33 + (3*b3 - 3) * q^34 + (-b3 + 4) * q^36 - 4*b2 * q^37 + (-10*b2 + 2*b1) * q^38 + (2*b3 - 1) * q^39 + (-4*b3 + 2) * q^41 + b1 * q^42 + (6*b2 + 2*b1) * q^43 + (5*b3 - 20) * q^44 + (-4*b3 + 4) * q^46 + (7*b2 + 2*b1) * q^47 + (-4*b2 + 5*b1) * q^48 + 6 * q^49 - 3 * q^51 + (7*b2 - 7*b1) * q^52 + (-4*b2 + 2*b1) * q^53 + (-b3 + 1) * q^54 + (2*b3 - 7) * q^56 + 2*b1 * q^57 - b1 * q^58 + (2*b3 + 2) * q^59 + (6*b3 - 4) * q^61 + 4*b1 * q^62 - b2 * q^63 + (10*b3 - 27) * q^64 + (5*b3 - 5) * q^66 + (3*b2 - 4*b1) * q^67 + (9*b2 - 3*b1) * q^68 + 4 * q^69 + (-2*b3 - 4) * q^71 + (-5*b2 + 2*b1) * q^72 - 4*b2 * q^73 + (4*b3 - 4) * q^74 + (8*b3 - 18) * q^76 + 5*b2 * q^77 + (10*b2 - b1) * q^78 + (-2*b3 - 2) * q^79 + q^81 + (-20*b2 + 2*b1) * q^82 + (8*b2 + 2*b1) * q^83 + (b3 - 4) * q^84 + (-4*b3 - 6) * q^86 + b2 * q^87 + (25*b2 - 10*b1) * q^88 + (2*b3 - 7) * q^89 + (-2*b3 + 1) * q^91 + (-12*b2 + 4*b1) * q^92 - 4*b2 * q^93 + (-5*b3 - 5) * q^94 + (5*b3 - 20) * q^96 + (8*b2 + 2*b1) * q^97 + 6*b1 * q^98 - 5 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 14 q^{4} - 2 q^{6} - 4 q^{9}+O(q^{10})$$ 4 * q - 14 * q^4 - 2 * q^6 - 4 * q^9 $$4 q - 14 q^{4} - 2 q^{6} - 4 q^{9} + 20 q^{11} + 2 q^{14} + 26 q^{16} + 4 q^{19} + 4 q^{21} + 24 q^{24} - 42 q^{26} - 4 q^{29} + 16 q^{31} - 6 q^{34} + 14 q^{36} - 70 q^{44} + 8 q^{46} + 24 q^{49} - 12 q^{51} + 2 q^{54} - 24 q^{56} + 12 q^{59} - 4 q^{61} - 88 q^{64} - 10 q^{66} + 16 q^{69} - 20 q^{71} - 8 q^{74} - 56 q^{76} - 12 q^{79} + 4 q^{81} - 14 q^{84} - 32 q^{86} - 24 q^{89} - 30 q^{94} - 70 q^{96} - 20 q^{99}+O(q^{100})$$ 4 * q - 14 * q^4 - 2 * q^6 - 4 * q^9 + 20 * q^11 + 2 * q^14 + 26 * q^16 + 4 * q^19 + 4 * q^21 + 24 * q^24 - 42 * q^26 - 4 * q^29 + 16 * q^31 - 6 * q^34 + 14 * q^36 - 70 * q^44 + 8 * q^46 + 24 * q^49 - 12 * q^51 + 2 * q^54 - 24 * q^56 + 12 * q^59 - 4 * q^61 - 88 * q^64 - 10 * q^66 + 16 * q^69 - 20 * q^71 - 8 * q^74 - 56 * q^76 - 12 * q^79 + 4 * q^81 - 14 * q^84 - 32 * q^86 - 24 * q^89 - 30 * q^94 - 70 * q^96 - 20 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 11x^{2} + 25$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 6\nu ) / 5$$ (v^3 + 6*v) / 5 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 6$$ v^2 + 6
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 6$$ b3 - 6 $$\nu^{3}$$ $$=$$ $$5\beta_{2} - 6\beta_1$$ 5*b2 - 6*b1

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
349.1
 − 2.79129i − 1.79129i 1.79129i 2.79129i
2.79129i 1.00000i −5.79129 0 −2.79129 1.00000i 10.5826i −1.00000 0
349.2 1.79129i 1.00000i −1.20871 0 1.79129 1.00000i 1.41742i −1.00000 0
349.3 1.79129i 1.00000i −1.20871 0 1.79129 1.00000i 1.41742i −1.00000 0
349.4 2.79129i 1.00000i −5.79129 0 −2.79129 1.00000i 10.5826i −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
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 2175.2.c.f 4
5.b even 2 1 inner 2175.2.c.f 4
5.c odd 4 1 435.2.a.f 2
5.c odd 4 1 2175.2.a.r 2
15.e even 4 1 1305.2.a.m 2
15.e even 4 1 6525.2.a.t 2
20.e even 4 1 6960.2.a.bw 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.a.f 2 5.c odd 4 1
1305.2.a.m 2 15.e even 4 1
2175.2.a.r 2 5.c odd 4 1
2175.2.c.f 4 1.a even 1 1 trivial
2175.2.c.f 4 5.b even 2 1 inner
6525.2.a.t 2 15.e even 4 1
6960.2.a.bw 2 20.e even 4 1

## Hecke kernels

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

 $$T_{2}^{4} + 11T_{2}^{2} + 25$$ T2^4 + 11*T2^2 + 25 $$T_{7}^{2} + 1$$ T7^2 + 1 $$T_{11} - 5$$ T11 - 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 11T^{2} + 25$$
$3$ $$(T^{2} + 1)^{2}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 1)^{2}$$
$11$ $$(T - 5)^{4}$$
$13$ $$(T^{2} + 21)^{2}$$
$17$ $$(T^{2} + 9)^{2}$$
$19$ $$(T^{2} - 2 T - 20)^{2}$$
$23$ $$(T^{2} + 16)^{2}$$
$29$ $$(T + 1)^{4}$$
$31$ $$(T - 4)^{4}$$
$37$ $$(T^{2} + 16)^{2}$$
$41$ $$(T^{2} - 84)^{2}$$
$43$ $$T^{4} + 92T^{2} + 16$$
$47$ $$T^{4} + 114T^{2} + 225$$
$53$ $$T^{4} + 92T^{2} + 16$$
$59$ $$(T^{2} - 6 T - 12)^{2}$$
$61$ $$(T^{2} + 2 T - 188)^{2}$$
$67$ $$T^{4} + 218T^{2} + 3481$$
$71$ $$(T^{2} + 10 T + 4)^{2}$$
$73$ $$(T^{2} + 16)^{2}$$
$79$ $$(T^{2} + 6 T - 12)^{2}$$
$83$ $$T^{4} + 140T^{2} + 784$$
$89$ $$(T^{2} + 12 T + 15)^{2}$$
$97$ $$T^{4} + 140T^{2} + 784$$