# Properties

 Label 150.3.f.a Level $150$ Weight $3$ Character orbit 150.f Analytic conductor $4.087$ 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] = [150,3,Mod(7,150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(150, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("150.7");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$150 = 2 \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 150.f (of order $$4$$, degree $$2$$, 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: $$4.08720396540$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} - 1) q^{2} + \beta_1 q^{3} - 2 \beta_{2} q^{4} + (\beta_{3} - \beta_1) q^{6} + (\beta_{3} + 6 \beta_{2} - 6) q^{7} + (2 \beta_{2} + 2) q^{8} + 3 \beta_{2} q^{9}+O(q^{10})$$ q + (b2 - 1) * q^2 + b1 * q^3 - 2*b2 * q^4 + (b3 - b1) * q^6 + (b3 + 6*b2 - 6) * q^7 + (2*b2 + 2) * q^8 + 3*b2 * q^9 $$q + (\beta_{2} - 1) q^{2} + \beta_1 q^{3} - 2 \beta_{2} q^{4} + (\beta_{3} - \beta_1) q^{6} + (\beta_{3} + 6 \beta_{2} - 6) q^{7} + (2 \beta_{2} + 2) q^{8} + 3 \beta_{2} q^{9} + (6 \beta_{3} - 6 \beta_1 + 6) q^{11} - 2 \beta_{3} q^{12} + (12 \beta_{2} + 3 \beta_1 + 12) q^{13} + ( - \beta_{3} - 12 \beta_{2} - \beta_1) q^{14} - 4 q^{16} + (6 \beta_{3} + 6 \beta_{2} - 6) q^{17} + ( - 3 \beta_{2} - 3) q^{18} + ( - 6 \beta_{3} + 19 \beta_{2} - 6 \beta_1) q^{19} + (6 \beta_{3} - 6 \beta_1 - 3) q^{21} + ( - 12 \beta_{3} + 6 \beta_{2} - 6) q^{22} + ( - 6 \beta_{2} - 18 \beta_1 - 6) q^{23} + (2 \beta_{3} + 2 \beta_1) q^{24} + (3 \beta_{3} - 3 \beta_1 - 24) q^{26} + 3 \beta_{3} q^{27} + (12 \beta_{2} + 2 \beta_1 + 12) q^{28} + (6 \beta_{3} + 6 \beta_{2} + 6 \beta_1) q^{29} + ( - 18 \beta_{3} + 18 \beta_1 - 5) q^{31} + ( - 4 \beta_{2} + 4) q^{32} + ( - 18 \beta_{2} + 6 \beta_1 - 18) q^{33} + ( - 6 \beta_{3} - 12 \beta_{2} - 6 \beta_1) q^{34} + 6 q^{36} + (20 \beta_{3} - 12 \beta_{2} + 12) q^{37} + ( - 19 \beta_{2} + 12 \beta_1 - 19) q^{38} + (12 \beta_{3} + 9 \beta_{2} + 12 \beta_1) q^{39} + ( - 6 \beta_{3} + 6 \beta_1 + 48) q^{41} + ( - 12 \beta_{3} - 3 \beta_{2} + 3) q^{42} + (18 \beta_{2} - 5 \beta_1 + 18) q^{43} + (12 \beta_{3} - 12 \beta_{2} + 12 \beta_1) q^{44} + ( - 18 \beta_{3} + 18 \beta_1 + 12) q^{46} + ( - 18 \beta_{3} - 36 \beta_{2} + 36) q^{47} - 4 \beta_1 q^{48} + ( - 12 \beta_{3} - 26 \beta_{2} - 12 \beta_1) q^{49} + (6 \beta_{3} - 6 \beta_1 - 18) q^{51} + ( - 6 \beta_{3} - 24 \beta_{2} + 24) q^{52} + ( - 30 \beta_{2} + 24 \beta_1 - 30) q^{53} + ( - 3 \beta_{3} - 3 \beta_1) q^{54} + (2 \beta_{3} - 2 \beta_1 - 24) q^{56} + (19 \beta_{3} - 18 \beta_{2} + 18) q^{57} + ( - 6 \beta_{2} - 12 \beta_1 - 6) q^{58} + 30 \beta_{2} q^{59} + ( - 24 \beta_{3} + 24 \beta_1 + 11) q^{61} + (36 \beta_{3} - 5 \beta_{2} + 5) q^{62} + ( - 18 \beta_{2} - 3 \beta_1 - 18) q^{63} + 8 \beta_{2} q^{64} + (6 \beta_{3} - 6 \beta_1 + 36) q^{66} + ( - 9 \beta_{3} + 6 \beta_{2} - 6) q^{67} + (12 \beta_{2} + 12 \beta_1 + 12) q^{68} + ( - 6 \beta_{3} - 54 \beta_{2} - 6 \beta_1) q^{69} + (6 \beta_{3} - 6 \beta_1 - 24) q^{71} + (6 \beta_{2} - 6) q^{72} + (12 \beta_{2} + 28 \beta_1 + 12) q^{73} + ( - 20 \beta_{3} + 24 \beta_{2} - 20 \beta_1) q^{74} + (12 \beta_{3} - 12 \beta_1 + 38) q^{76} + ( - 66 \beta_{3} + 18 \beta_{2} - 18) q^{77} + ( - 9 \beta_{2} - 24 \beta_1 - 9) q^{78} + (12 \beta_{3} + 2 \beta_{2} + 12 \beta_1) q^{79} - 9 q^{81} + (12 \beta_{3} + 48 \beta_{2} - 48) q^{82} + (12 \beta_{2} - 42 \beta_1 + 12) q^{83} + (12 \beta_{3} + 6 \beta_{2} + 12 \beta_1) q^{84} + ( - 5 \beta_{3} + 5 \beta_1 - 36) q^{86} + (6 \beta_{3} + 18 \beta_{2} - 18) q^{87} + (12 \beta_{2} - 24 \beta_1 + 12) q^{88} + (12 \beta_{3} + 12 \beta_{2} + 12 \beta_1) q^{89} + (30 \beta_{3} - 30 \beta_1 - 153) q^{91} + (36 \beta_{3} + 12 \beta_{2} - 12) q^{92} + (54 \beta_{2} - 5 \beta_1 + 54) q^{93} + (18 \beta_{3} + 72 \beta_{2} + 18 \beta_1) q^{94} + ( - 4 \beta_{3} + 4 \beta_1) q^{96} + (5 \beta_{3} + 48 \beta_{2} - 48) q^{97} + (26 \beta_{2} + 24 \beta_1 + 26) q^{98} + ( - 18 \beta_{3} + 18 \beta_{2} - 18 \beta_1) q^{99}+O(q^{100})$$ q + (b2 - 1) * q^2 + b1 * q^3 - 2*b2 * q^4 + (b3 - b1) * q^6 + (b3 + 6*b2 - 6) * q^7 + (2*b2 + 2) * q^8 + 3*b2 * q^9 + (6*b3 - 6*b1 + 6) * q^11 - 2*b3 * q^12 + (12*b2 + 3*b1 + 12) * q^13 + (-b3 - 12*b2 - b1) * q^14 - 4 * q^16 + (6*b3 + 6*b2 - 6) * q^17 + (-3*b2 - 3) * q^18 + (-6*b3 + 19*b2 - 6*b1) * q^19 + (6*b3 - 6*b1 - 3) * q^21 + (-12*b3 + 6*b2 - 6) * q^22 + (-6*b2 - 18*b1 - 6) * q^23 + (2*b3 + 2*b1) * q^24 + (3*b3 - 3*b1 - 24) * q^26 + 3*b3 * q^27 + (12*b2 + 2*b1 + 12) * q^28 + (6*b3 + 6*b2 + 6*b1) * q^29 + (-18*b3 + 18*b1 - 5) * q^31 + (-4*b2 + 4) * q^32 + (-18*b2 + 6*b1 - 18) * q^33 + (-6*b3 - 12*b2 - 6*b1) * q^34 + 6 * q^36 + (20*b3 - 12*b2 + 12) * q^37 + (-19*b2 + 12*b1 - 19) * q^38 + (12*b3 + 9*b2 + 12*b1) * q^39 + (-6*b3 + 6*b1 + 48) * q^41 + (-12*b3 - 3*b2 + 3) * q^42 + (18*b2 - 5*b1 + 18) * q^43 + (12*b3 - 12*b2 + 12*b1) * q^44 + (-18*b3 + 18*b1 + 12) * q^46 + (-18*b3 - 36*b2 + 36) * q^47 - 4*b1 * q^48 + (-12*b3 - 26*b2 - 12*b1) * q^49 + (6*b3 - 6*b1 - 18) * q^51 + (-6*b3 - 24*b2 + 24) * q^52 + (-30*b2 + 24*b1 - 30) * q^53 + (-3*b3 - 3*b1) * q^54 + (2*b3 - 2*b1 - 24) * q^56 + (19*b3 - 18*b2 + 18) * q^57 + (-6*b2 - 12*b1 - 6) * q^58 + 30*b2 * q^59 + (-24*b3 + 24*b1 + 11) * q^61 + (36*b3 - 5*b2 + 5) * q^62 + (-18*b2 - 3*b1 - 18) * q^63 + 8*b2 * q^64 + (6*b3 - 6*b1 + 36) * q^66 + (-9*b3 + 6*b2 - 6) * q^67 + (12*b2 + 12*b1 + 12) * q^68 + (-6*b3 - 54*b2 - 6*b1) * q^69 + (6*b3 - 6*b1 - 24) * q^71 + (6*b2 - 6) * q^72 + (12*b2 + 28*b1 + 12) * q^73 + (-20*b3 + 24*b2 - 20*b1) * q^74 + (12*b3 - 12*b1 + 38) * q^76 + (-66*b3 + 18*b2 - 18) * q^77 + (-9*b2 - 24*b1 - 9) * q^78 + (12*b3 + 2*b2 + 12*b1) * q^79 - 9 * q^81 + (12*b3 + 48*b2 - 48) * q^82 + (12*b2 - 42*b1 + 12) * q^83 + (12*b3 + 6*b2 + 12*b1) * q^84 + (-5*b3 + 5*b1 - 36) * q^86 + (6*b3 + 18*b2 - 18) * q^87 + (12*b2 - 24*b1 + 12) * q^88 + (12*b3 + 12*b2 + 12*b1) * q^89 + (30*b3 - 30*b1 - 153) * q^91 + (36*b3 + 12*b2 - 12) * q^92 + (54*b2 - 5*b1 + 54) * q^93 + (18*b3 + 72*b2 + 18*b1) * q^94 + (-4*b3 + 4*b1) * q^96 + (5*b3 + 48*b2 - 48) * q^97 + (26*b2 + 24*b1 + 26) * q^98 + (-18*b3 + 18*b2 - 18*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} - 24 q^{7} + 8 q^{8}+O(q^{10})$$ 4 * q - 4 * q^2 - 24 * q^7 + 8 * q^8 $$4 q - 4 q^{2} - 24 q^{7} + 8 q^{8} + 24 q^{11} + 48 q^{13} - 16 q^{16} - 24 q^{17} - 12 q^{18} - 12 q^{21} - 24 q^{22} - 24 q^{23} - 96 q^{26} + 48 q^{28} - 20 q^{31} + 16 q^{32} - 72 q^{33} + 24 q^{36} + 48 q^{37} - 76 q^{38} + 192 q^{41} + 12 q^{42} + 72 q^{43} + 48 q^{46} + 144 q^{47} - 72 q^{51} + 96 q^{52} - 120 q^{53} - 96 q^{56} + 72 q^{57} - 24 q^{58} + 44 q^{61} + 20 q^{62} - 72 q^{63} + 144 q^{66} - 24 q^{67} + 48 q^{68} - 96 q^{71} - 24 q^{72} + 48 q^{73} + 152 q^{76} - 72 q^{77} - 36 q^{78} - 36 q^{81} - 192 q^{82} + 48 q^{83} - 144 q^{86} - 72 q^{87} + 48 q^{88} - 612 q^{91} - 48 q^{92} + 216 q^{93} - 192 q^{97} + 104 q^{98}+O(q^{100})$$ 4 * q - 4 * q^2 - 24 * q^7 + 8 * q^8 + 24 * q^11 + 48 * q^13 - 16 * q^16 - 24 * q^17 - 12 * q^18 - 12 * q^21 - 24 * q^22 - 24 * q^23 - 96 * q^26 + 48 * q^28 - 20 * q^31 + 16 * q^32 - 72 * q^33 + 24 * q^36 + 48 * q^37 - 76 * q^38 + 192 * q^41 + 12 * q^42 + 72 * q^43 + 48 * q^46 + 144 * q^47 - 72 * q^51 + 96 * q^52 - 120 * q^53 - 96 * q^56 + 72 * q^57 - 24 * q^58 + 44 * q^61 + 20 * q^62 - 72 * q^63 + 144 * q^66 - 24 * q^67 + 48 * q^68 - 96 * q^71 - 24 * q^72 + 48 * q^73 + 152 * q^76 - 72 * q^77 - 36 * q^78 - 36 * q^81 - 192 * q^82 + 48 * q^83 - 144 * q^86 - 72 * q^87 + 48 * q^88 - 612 * q^91 - 48 * q^92 + 216 * q^93 - 192 * q^97 + 104 * q^98

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

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/150\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-\beta_{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 −1.22474 + 1.22474i 1.22474 − 1.22474i −1.22474 − 1.22474i 1.22474 + 1.22474i
−1.00000 1.00000i −1.22474 + 1.22474i 2.00000i 0 2.44949 −4.77526 4.77526i 2.00000 2.00000i 3.00000i 0
7.2 −1.00000 1.00000i 1.22474 1.22474i 2.00000i 0 −2.44949 −7.22474 7.22474i 2.00000 2.00000i 3.00000i 0
43.1 −1.00000 + 1.00000i −1.22474 1.22474i 2.00000i 0 2.44949 −4.77526 + 4.77526i 2.00000 + 2.00000i 3.00000i 0
43.2 −1.00000 + 1.00000i 1.22474 + 1.22474i 2.00000i 0 −2.44949 −7.22474 + 7.22474i 2.00000 + 2.00000i 3.00000i 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.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 150.3.f.a 4
3.b odd 2 1 450.3.g.h 4
4.b odd 2 1 1200.3.bg.p 4
5.b even 2 1 150.3.f.c yes 4
5.c odd 4 1 inner 150.3.f.a 4
5.c odd 4 1 150.3.f.c yes 4
15.d odd 2 1 450.3.g.g 4
15.e even 4 1 450.3.g.g 4
15.e even 4 1 450.3.g.h 4
20.d odd 2 1 1200.3.bg.a 4
20.e even 4 1 1200.3.bg.a 4
20.e even 4 1 1200.3.bg.p 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.3.f.a 4 1.a even 1 1 trivial
150.3.f.a 4 5.c odd 4 1 inner
150.3.f.c yes 4 5.b even 2 1
150.3.f.c yes 4 5.c odd 4 1
450.3.g.g 4 15.d odd 2 1
450.3.g.g 4 15.e even 4 1
450.3.g.h 4 3.b odd 2 1
450.3.g.h 4 15.e even 4 1
1200.3.bg.a 4 20.d odd 2 1
1200.3.bg.a 4 20.e even 4 1
1200.3.bg.p 4 4.b odd 2 1
1200.3.bg.p 4 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 24T_{7}^{3} + 288T_{7}^{2} + 1656T_{7} + 4761$$ acting on $$S_{3}^{\mathrm{new}}(150, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 2 T + 2)^{2}$$
$3$ $$T^{4} + 9$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 24 T^{3} + \cdots + 4761$$
$11$ $$(T^{2} - 12 T - 180)^{2}$$
$13$ $$T^{4} - 48 T^{3} + \cdots + 68121$$
$17$ $$T^{4} + 24 T^{3} + \cdots + 1296$$
$19$ $$T^{4} + 1154 T^{2} + 21025$$
$23$ $$T^{4} + 24 T^{3} + \cdots + 810000$$
$29$ $$T^{4} + 504 T^{2} + 32400$$
$31$ $$(T^{2} + 10 T - 1919)^{2}$$
$37$ $$T^{4} - 48 T^{3} + \cdots + 831744$$
$41$ $$(T^{2} - 96 T + 2088)^{2}$$
$43$ $$T^{4} - 72 T^{3} + \cdots + 328329$$
$47$ $$T^{4} - 144 T^{3} + \cdots + 2624400$$
$53$ $$T^{4} + 120 T^{3} + \cdots + 5184$$
$59$ $$(T^{2} + 900)^{2}$$
$61$ $$(T^{2} - 22 T - 3335)^{2}$$
$67$ $$T^{4} + 24 T^{3} + \cdots + 29241$$
$71$ $$(T^{2} + 48 T + 360)^{2}$$
$73$ $$T^{4} - 48 T^{3} + \cdots + 4260096$$
$79$ $$T^{4} + 1736 T^{2} + 739600$$
$83$ $$T^{4} - 48 T^{3} + \cdots + 25040016$$
$89$ $$T^{4} + 2016 T^{2} + 518400$$
$97$ $$T^{4} + 192 T^{3} + \cdots + 20548089$$