# Properties

 Label 375.2.b.b Level $375$ Weight $2$ Character orbit 375.b Analytic conductor $2.994$ 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] = [375,2,Mod(124,375)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(375, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("375.124");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$375 = 3 \cdot 5^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 375.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: $$2.99439007580$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

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

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ v^2 + 1 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 2\nu$$ v^3 + 2*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ b2 - 1 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 2\beta_1$$ b3 - 2*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/375\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$251$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
124.1
 − 1.61803i − 0.618034i 0.618034i 1.61803i
1.61803i 1.00000i −0.618034 0 1.61803 1.61803i 2.23607i −1.00000 0
124.2 0.618034i 1.00000i 1.61803 0 −0.618034 0.618034i 2.23607i −1.00000 0
124.3 0.618034i 1.00000i 1.61803 0 −0.618034 0.618034i 2.23607i −1.00000 0
124.4 1.61803i 1.00000i −0.618034 0 1.61803 1.61803i 2.23607i −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 375.2.b.b 4
3.b odd 2 1 1125.2.b.e 4
4.b odd 2 1 6000.2.f.b 4
5.b even 2 1 inner 375.2.b.b 4
5.c odd 4 1 375.2.a.b 2
5.c odd 4 1 375.2.a.c yes 2
15.d odd 2 1 1125.2.b.e 4
15.e even 4 1 1125.2.a.b 2
15.e even 4 1 1125.2.a.e 2
20.d odd 2 1 6000.2.f.b 4
20.e even 4 1 6000.2.a.e 2
20.e even 4 1 6000.2.a.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
375.2.a.b 2 5.c odd 4 1
375.2.a.c yes 2 5.c odd 4 1
375.2.b.b 4 1.a even 1 1 trivial
375.2.b.b 4 5.b even 2 1 inner
1125.2.a.b 2 15.e even 4 1
1125.2.a.e 2 15.e even 4 1
1125.2.b.e 4 3.b odd 2 1
1125.2.b.e 4 15.d odd 2 1
6000.2.a.e 2 20.e even 4 1
6000.2.a.v 2 20.e even 4 1
6000.2.f.b 4 4.b odd 2 1
6000.2.f.b 4 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 3T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(375, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 3T^{2} + 1$$
$3$ $$(T^{2} + 1)^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 3T^{2} + 1$$
$11$ $$(T^{2} - 4 T - 1)^{2}$$
$13$ $$T^{4} + 42T^{2} + 121$$
$17$ $$T^{4} + 63T^{2} + 841$$
$19$ $$(T - 5)^{4}$$
$23$ $$T^{4} + 42T^{2} + 361$$
$29$ $$(T^{2} - 45)^{2}$$
$31$ $$(T^{2} + 11 T + 19)^{2}$$
$37$ $$T^{4} + 18T^{2} + 1$$
$41$ $$(T^{2} + T - 31)^{2}$$
$43$ $$T^{4} + 207T^{2} + 9801$$
$47$ $$(T^{2} + 9)^{2}$$
$53$ $$T^{4} + 67T^{2} + 841$$
$59$ $$(T^{2} + 15 T + 55)^{2}$$
$61$ $$(T^{2} + 11 T + 29)^{2}$$
$67$ $$T^{4} + 368 T^{2} + 30976$$
$71$ $$(T^{2} - 19 T + 89)^{2}$$
$73$ $$T^{4} + 27T^{2} + 81$$
$79$ $$(T^{2} - 10 T - 20)^{2}$$
$83$ $$T^{4} + 447 T^{2} + 39601$$
$89$ $$(T^{2} - 10 T - 55)^{2}$$
$97$ $$T^{4} + 503 T^{2} + 52441$$