# Properties

 Label 375.2.b.a 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: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} + \beta_1) q^{2} - \beta_{3} q^{3} + 3 \beta_{2} q^{4} + (\beta_{2} - 1) q^{6} + ( - 3 \beta_{3} - \beta_1) q^{7} + (\beta_{3} - 4 \beta_1) q^{8} - q^{9}+O(q^{10})$$ q + (-b3 + b1) * q^2 - b3 * q^3 + 3*b2 * q^4 + (b2 - 1) * q^6 + (-3*b3 - b1) * q^7 + (b3 - 4*b1) * q^8 - q^9 $$q + ( - \beta_{3} + \beta_1) q^{2} - \beta_{3} q^{3} + 3 \beta_{2} q^{4} + (\beta_{2} - 1) q^{6} + ( - 3 \beta_{3} - \beta_1) q^{7} + (\beta_{3} - 4 \beta_1) q^{8} - q^{9} + (2 \beta_{2} - 3) q^{11} - 3 \beta_1 q^{12} + 3 \beta_{3} q^{13} + (\beta_{2} - 2) q^{14} + ( - 3 \beta_{2} + 5) q^{16} + ( - 5 \beta_{3} - \beta_1) q^{17} + (\beta_{3} - \beta_1) q^{18} + q^{19} + ( - \beta_{2} - 3) q^{21} + (5 \beta_{3} - 7 \beta_1) q^{22} + (3 \beta_{3} + 6 \beta_1) q^{23} + ( - 4 \beta_{2} + 1) q^{24} + ( - 3 \beta_{2} + 3) q^{26} + \beta_{3} q^{27} + ( - 3 \beta_{3} - 6 \beta_1) q^{28} + ( - 2 \beta_{2} + 1) q^{29} + (5 \beta_{2} + 5) q^{31} + ( - 6 \beta_{3} + 3 \beta_1) q^{32} + (3 \beta_{3} - 2 \beta_1) q^{33} + (3 \beta_{2} - 4) q^{34} - 3 \beta_{2} q^{36} - 5 \beta_{3} q^{37} + ( - \beta_{3} + \beta_1) q^{38} + 3 q^{39} + ( - 3 \beta_{2} - 9) q^{41} + (2 \beta_{3} - \beta_1) q^{42} + ( - 4 \beta_{3} - \beta_1) q^{43} + ( - 15 \beta_{2} + 6) q^{44} + (9 \beta_{2} - 3) q^{46} + (7 \beta_{3} + 10 \beta_1) q^{47} + ( - 5 \beta_{3} + 3 \beta_1) q^{48} + ( - 5 \beta_{2} - 3) q^{49} + ( - \beta_{2} - 5) q^{51} + 9 \beta_1 q^{52} + ( - 2 \beta_{3} + \beta_1) q^{53} + ( - \beta_{2} + 1) q^{54} + ( - 7 \beta_{2} - 1) q^{56} - \beta_{3} q^{57} + ( - 3 \beta_{3} + 5 \beta_1) q^{58} + (11 \beta_{2} + 7) q^{59} + (5 \beta_{2} + 3) q^{61} - 5 \beta_1 q^{62} + (3 \beta_{3} + \beta_1) q^{63} + (6 \beta_{2} + 1) q^{64} + ( - 7 \beta_{2} + 5) q^{66} + 8 \beta_{3} q^{67} + ( - 3 \beta_{3} - 12 \beta_1) q^{68} + (6 \beta_{2} + 3) q^{69} + (3 \beta_{2} + 6) q^{71} + ( - \beta_{3} + 4 \beta_1) q^{72} + (14 \beta_{3} + 3 \beta_1) q^{73} + (5 \beta_{2} - 5) q^{74} + 3 \beta_{2} q^{76} + (7 \beta_{3} - \beta_1) q^{77} + ( - 3 \beta_{3} + 3 \beta_1) q^{78} + ( - 2 \beta_{2} - 6) q^{79} + q^{81} + (6 \beta_{3} - 3 \beta_1) q^{82} + (3 \beta_{3} + 7 \beta_1) q^{83} + ( - 6 \beta_{2} - 3) q^{84} + (2 \beta_{2} - 3) q^{86} + ( - \beta_{3} + 2 \beta_1) q^{87} + ( - 11 \beta_{3} + 22 \beta_1) q^{88} + ( - 4 \beta_{2} - 1) q^{89} + (3 \beta_{2} + 9) q^{91} + (18 \beta_{3} - 9 \beta_1) q^{92} + ( - 5 \beta_{3} - 5 \beta_1) q^{93} + (13 \beta_{2} - 3) q^{94} + (3 \beta_{2} - 6) q^{96} + (5 \beta_{3} + 9 \beta_1) q^{97} + ( - 2 \beta_{3} + 7 \beta_1) q^{98} + ( - 2 \beta_{2} + 3) q^{99}+O(q^{100})$$ q + (-b3 + b1) * q^2 - b3 * q^3 + 3*b2 * q^4 + (b2 - 1) * q^6 + (-3*b3 - b1) * q^7 + (b3 - 4*b1) * q^8 - q^9 + (2*b2 - 3) * q^11 - 3*b1 * q^12 + 3*b3 * q^13 + (b2 - 2) * q^14 + (-3*b2 + 5) * q^16 + (-5*b3 - b1) * q^17 + (b3 - b1) * q^18 + q^19 + (-b2 - 3) * q^21 + (5*b3 - 7*b1) * q^22 + (3*b3 + 6*b1) * q^23 + (-4*b2 + 1) * q^24 + (-3*b2 + 3) * q^26 + b3 * q^27 + (-3*b3 - 6*b1) * q^28 + (-2*b2 + 1) * q^29 + (5*b2 + 5) * q^31 + (-6*b3 + 3*b1) * q^32 + (3*b3 - 2*b1) * q^33 + (3*b2 - 4) * q^34 - 3*b2 * q^36 - 5*b3 * q^37 + (-b3 + b1) * q^38 + 3 * q^39 + (-3*b2 - 9) * q^41 + (2*b3 - b1) * q^42 + (-4*b3 - b1) * q^43 + (-15*b2 + 6) * q^44 + (9*b2 - 3) * q^46 + (7*b3 + 10*b1) * q^47 + (-5*b3 + 3*b1) * q^48 + (-5*b2 - 3) * q^49 + (-b2 - 5) * q^51 + 9*b1 * q^52 + (-2*b3 + b1) * q^53 + (-b2 + 1) * q^54 + (-7*b2 - 1) * q^56 - b3 * q^57 + (-3*b3 + 5*b1) * q^58 + (11*b2 + 7) * q^59 + (5*b2 + 3) * q^61 - 5*b1 * q^62 + (3*b3 + b1) * q^63 + (6*b2 + 1) * q^64 + (-7*b2 + 5) * q^66 + 8*b3 * q^67 + (-3*b3 - 12*b1) * q^68 + (6*b2 + 3) * q^69 + (3*b2 + 6) * q^71 + (-b3 + 4*b1) * q^72 + (14*b3 + 3*b1) * q^73 + (5*b2 - 5) * q^74 + 3*b2 * q^76 + (7*b3 - b1) * q^77 + (-3*b3 + 3*b1) * q^78 + (-2*b2 - 6) * q^79 + q^81 + (6*b3 - 3*b1) * q^82 + (3*b3 + 7*b1) * q^83 + (-6*b2 - 3) * q^84 + (2*b2 - 3) * q^86 + (-b3 + 2*b1) * q^87 + (-11*b3 + 22*b1) * q^88 + (-4*b2 - 1) * q^89 + (3*b2 + 9) * q^91 + (18*b3 - 9*b1) * q^92 + (-5*b3 - 5*b1) * q^93 + (13*b2 - 3) * q^94 + (3*b2 - 6) * q^96 + (5*b3 + 9*b1) * q^97 + (-2*b3 + 7*b1) * q^98 + (-2*b2 + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{4} - 6 q^{6} - 4 q^{9}+O(q^{10})$$ 4 * q - 6 * q^4 - 6 * q^6 - 4 * q^9 $$4 q - 6 q^{4} - 6 q^{6} - 4 q^{9} - 16 q^{11} - 10 q^{14} + 26 q^{16} + 4 q^{19} - 10 q^{21} + 12 q^{24} + 18 q^{26} + 8 q^{29} + 10 q^{31} - 22 q^{34} + 6 q^{36} + 12 q^{39} - 30 q^{41} + 54 q^{44} - 30 q^{46} - 2 q^{49} - 18 q^{51} + 6 q^{54} + 10 q^{56} + 6 q^{59} + 2 q^{61} - 8 q^{64} + 34 q^{66} + 18 q^{71} - 30 q^{74} - 6 q^{76} - 20 q^{79} + 4 q^{81} - 16 q^{86} + 4 q^{89} + 30 q^{91} - 38 q^{94} - 30 q^{96} + 16 q^{99}+O(q^{100})$$ 4 * q - 6 * q^4 - 6 * q^6 - 4 * q^9 - 16 * q^11 - 10 * q^14 + 26 * q^16 + 4 * q^19 - 10 * q^21 + 12 * q^24 + 18 * q^26 + 8 * q^29 + 10 * q^31 - 22 * q^34 + 6 * q^36 + 12 * q^39 - 30 * q^41 + 54 * q^44 - 30 * q^46 - 2 * q^49 - 18 * q^51 + 6 * q^54 + 10 * q^56 + 6 * q^59 + 2 * q^61 - 8 * q^64 + 34 * q^66 + 18 * q^71 - 30 * q^74 - 6 * q^76 - 20 * q^79 + 4 * q^81 - 16 * q^86 + 4 * q^89 + 30 * q^91 - 38 * q^94 - 30 * q^96 + 16 * 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
2.61803i 1.00000i −4.85410 0 −2.61803 1.38197i 7.47214i −1.00000 0
124.2 0.381966i 1.00000i 1.85410 0 −0.381966 3.61803i 1.47214i −1.00000 0
124.3 0.381966i 1.00000i 1.85410 0 −0.381966 3.61803i 1.47214i −1.00000 0
124.4 2.61803i 1.00000i −4.85410 0 −2.61803 1.38197i 7.47214i −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.a 4
3.b odd 2 1 1125.2.b.b 4
4.b odd 2 1 6000.2.f.n 4
5.b even 2 1 inner 375.2.b.a 4
5.c odd 4 1 375.2.a.a 2
5.c odd 4 1 375.2.a.d yes 2
15.d odd 2 1 1125.2.b.b 4
15.e even 4 1 1125.2.a.a 2
15.e even 4 1 1125.2.a.f 2
20.d odd 2 1 6000.2.f.n 4
20.e even 4 1 6000.2.a.m 2
20.e even 4 1 6000.2.a.q 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 7T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(375, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 7T^{2} + 1$$
$3$ $$(T^{2} + 1)^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 15T^{2} + 25$$
$11$ $$(T^{2} + 8 T + 11)^{2}$$
$13$ $$(T^{2} + 9)^{2}$$
$17$ $$T^{4} + 43T^{2} + 361$$
$19$ $$(T - 1)^{4}$$
$23$ $$(T^{2} + 45)^{2}$$
$29$ $$(T^{2} - 4 T - 1)^{2}$$
$31$ $$(T^{2} - 5 T - 25)^{2}$$
$37$ $$(T^{2} + 25)^{2}$$
$41$ $$(T^{2} + 15 T + 45)^{2}$$
$43$ $$T^{4} + 27T^{2} + 121$$
$47$ $$T^{4} + 258 T^{2} + 14641$$
$53$ $$T^{4} + 15T^{2} + 25$$
$59$ $$(T^{2} - 3 T - 149)^{2}$$
$61$ $$(T^{2} - T - 31)^{2}$$
$67$ $$(T^{2} + 64)^{2}$$
$71$ $$(T^{2} - 9 T + 9)^{2}$$
$73$ $$T^{4} + 335 T^{2} + 21025$$
$79$ $$(T^{2} + 10 T + 20)^{2}$$
$83$ $$T^{4} + 123T^{2} + 3721$$
$89$ $$(T^{2} - 2 T - 19)^{2}$$
$97$ $$T^{4} + 203 T^{2} + 10201$$