# Properties

 Label 150.3.d.c Level $150$ Weight $3$ Character orbit 150.d Analytic conductor $4.087$ 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] = [150,3,Mod(101,150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(150, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("150.101");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$150 = 2 \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 150.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: $$4.08720396540$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 4x^{2} + 9$$ x^4 - 4*x^2 + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 30) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{2} + \beta_1 - 1) q^{3} - 2 q^{4} + (\beta_{3} - \beta_1 - 1) q^{6} + (2 \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{7} - 2 \beta_1 q^{8} + (\beta_{3} + 2 \beta_{2} + 3 \beta_1 - 2) q^{9}+O(q^{10})$$ q + b1 * q^2 + (-b2 + b1 - 1) * q^3 - 2 * q^4 + (b3 - b1 - 1) * q^6 + (2*b3 + 2*b2 - b1 - 2) * q^7 - 2*b1 * q^8 + (b3 + 2*b2 + 3*b1 - 2) * q^9 $$q + \beta_1 q^{2} + ( - \beta_{2} + \beta_1 - 1) q^{3} - 2 q^{4} + (\beta_{3} - \beta_1 - 1) q^{6} + (2 \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{7} - 2 \beta_1 q^{8} + (\beta_{3} + 2 \beta_{2} + 3 \beta_1 - 2) q^{9} + 6 \beta_1 q^{11} + (2 \beta_{2} - 2 \beta_1 + 2) q^{12} + 10 q^{13} + ( - 2 \beta_{3} + 4 \beta_{2} - 4 \beta_1) q^{14} + 4 q^{16} + ( - 2 \beta_{3} + 4 \beta_{2} + 10 \beta_1) q^{17} + ( - 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 8) q^{18} + (4 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 8) q^{19} + ( - 3 \beta_{3} + 2 \beta_{2} - 17 \beta_1 - 13) q^{21} - 12 q^{22} + (2 \beta_{3} - 4 \beta_{2} + 5 \beta_1) q^{23} + ( - 2 \beta_{3} + 2 \beta_1 + 2) q^{24} + 10 \beta_1 q^{26} + (2 \beta_{3} + \beta_{2} - 18 \beta_1 - 7) q^{27} + ( - 4 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 4) q^{28} + (4 \beta_{3} - 8 \beta_{2} + 4 \beta_1) q^{29} + 8 q^{31} + 4 \beta_1 q^{32} + (6 \beta_{3} - 6 \beta_1 - 6) q^{33} + ( - 4 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 24) q^{34} + ( - 2 \beta_{3} - 4 \beta_{2} - 6 \beta_1 + 4) q^{36} + ( - 8 \beta_{3} - 8 \beta_{2} + 4 \beta_1 + 22) q^{37} + ( - 4 \beta_{3} + 8 \beta_{2} + 4 \beta_1) q^{38} + ( - 10 \beta_{2} + 10 \beta_1 - 10) q^{39} + ( - 2 \beta_{3} + 4 \beta_{2} + 22 \beta_1) q^{41} + ( - 2 \beta_{3} - 6 \beta_{2} - 10 \beta_1 + 32) q^{42} + (6 \beta_{3} + 6 \beta_{2} - 3 \beta_1 - 14) q^{43} - 12 \beta_1 q^{44} + (4 \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 6) q^{46} + (10 \beta_{3} - 20 \beta_{2} + 25 \beta_1) q^{47} + ( - 4 \beta_{2} + 4 \beta_1 - 4) q^{48} + ( - 8 \beta_{3} - 8 \beta_{2} + 4 \beta_1 + 45) q^{49} + (12 \beta_{3} - 6 \beta_{2} - 24 \beta_1 + 18) q^{51} - 20 q^{52} + ( - 2 \beta_{3} + 4 \beta_{2} + 10 \beta_1) q^{53} + ( - \beta_{3} + 4 \beta_{2} - 9 \beta_1 + 35) q^{54} + (4 \beta_{3} - 8 \beta_{2} + 8 \beta_1) q^{56} + ( - 6 \beta_{3} - 8 \beta_{2} - 22 \beta_1 - 38) q^{57} + (8 \beta_{3} + 8 \beta_{2} - 4 \beta_1) q^{58} + ( - 4 \beta_{3} + 8 \beta_{2} - 40 \beta_1) q^{59} + ( - 8 \beta_{3} - 8 \beta_{2} + 4 \beta_1 - 16) q^{61} + 8 \beta_1 q^{62} + ( - 14 \beta_{3} + 8 \beta_{2} + 3 \beta_1 + 64) q^{63} - 8 q^{64} + (12 \beta_{2} - 12 \beta_1 + 12) q^{66} + (6 \beta_{3} + 6 \beta_{2} - 3 \beta_1 + 82) q^{67} + (4 \beta_{3} - 8 \beta_{2} - 20 \beta_1) q^{68} + (3 \beta_{3} + 6 \beta_{2} + 9 \beta_1 - 33) q^{69} + (4 \beta_{3} - 8 \beta_{2} + 34 \beta_1) q^{71} + (4 \beta_{3} - 4 \beta_{2} + 6 \beta_1 + 16) q^{72} + ( - 8 \beta_{3} - 8 \beta_{2} + 4 \beta_1 - 50) q^{73} + (8 \beta_{3} - 16 \beta_{2} + 30 \beta_1) q^{74} + ( - 8 \beta_{3} - 8 \beta_{2} + 4 \beta_1 - 16) q^{76} + ( - 12 \beta_{3} + 24 \beta_{2} - 24 \beta_1) q^{77} + (10 \beta_{3} - 10 \beta_1 - 10) q^{78} + ( - 4 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 28) q^{79} + ( - 20 \beta_{3} + 8 \beta_{2} + 7) q^{81} + ( - 4 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 48) q^{82} + (2 \beta_{3} - 4 \beta_{2} - 7 \beta_1) q^{83} + (6 \beta_{3} - 4 \beta_{2} + 34 \beta_1 + 26) q^{84} + ( - 6 \beta_{3} + 12 \beta_{2} - 20 \beta_1) q^{86} + (12 \beta_{2} + 24 \beta_1 - 60) q^{87} + 24 q^{88} + ( - 4 \beta_{3} + 8 \beta_{2} + 20 \beta_1) q^{89} + (20 \beta_{3} + 20 \beta_{2} - 10 \beta_1 - 20) q^{91} + ( - 4 \beta_{3} + 8 \beta_{2} - 10 \beta_1) q^{92} + ( - 8 \beta_{2} + 8 \beta_1 - 8) q^{93} + (20 \beta_{3} + 20 \beta_{2} - 10 \beta_1 - 30) q^{94} + (4 \beta_{3} - 4 \beta_1 - 4) q^{96} + (8 \beta_{3} + 8 \beta_{2} - 4 \beta_1 - 74) q^{97} + (8 \beta_{3} - 16 \beta_{2} + 53 \beta_1) q^{98} + ( - 12 \beta_{3} + 12 \beta_{2} - 18 \beta_1 - 48) q^{99}+O(q^{100})$$ q + b1 * q^2 + (-b2 + b1 - 1) * q^3 - 2 * q^4 + (b3 - b1 - 1) * q^6 + (2*b3 + 2*b2 - b1 - 2) * q^7 - 2*b1 * q^8 + (b3 + 2*b2 + 3*b1 - 2) * q^9 + 6*b1 * q^11 + (2*b2 - 2*b1 + 2) * q^12 + 10 * q^13 + (-2*b3 + 4*b2 - 4*b1) * q^14 + 4 * q^16 + (-2*b3 + 4*b2 + 10*b1) * q^17 + (-2*b3 + 2*b2 - 3*b1 - 8) * q^18 + (4*b3 + 4*b2 - 2*b1 + 8) * q^19 + (-3*b3 + 2*b2 - 17*b1 - 13) * q^21 - 12 * q^22 + (2*b3 - 4*b2 + 5*b1) * q^23 + (-2*b3 + 2*b1 + 2) * q^24 + 10*b1 * q^26 + (2*b3 + b2 - 18*b1 - 7) * q^27 + (-4*b3 - 4*b2 + 2*b1 + 4) * q^28 + (4*b3 - 8*b2 + 4*b1) * q^29 + 8 * q^31 + 4*b1 * q^32 + (6*b3 - 6*b1 - 6) * q^33 + (-4*b3 - 4*b2 + 2*b1 - 24) * q^34 + (-2*b3 - 4*b2 - 6*b1 + 4) * q^36 + (-8*b3 - 8*b2 + 4*b1 + 22) * q^37 + (-4*b3 + 8*b2 + 4*b1) * q^38 + (-10*b2 + 10*b1 - 10) * q^39 + (-2*b3 + 4*b2 + 22*b1) * q^41 + (-2*b3 - 6*b2 - 10*b1 + 32) * q^42 + (6*b3 + 6*b2 - 3*b1 - 14) * q^43 - 12*b1 * q^44 + (4*b3 + 4*b2 - 2*b1 - 6) * q^46 + (10*b3 - 20*b2 + 25*b1) * q^47 + (-4*b2 + 4*b1 - 4) * q^48 + (-8*b3 - 8*b2 + 4*b1 + 45) * q^49 + (12*b3 - 6*b2 - 24*b1 + 18) * q^51 - 20 * q^52 + (-2*b3 + 4*b2 + 10*b1) * q^53 + (-b3 + 4*b2 - 9*b1 + 35) * q^54 + (4*b3 - 8*b2 + 8*b1) * q^56 + (-6*b3 - 8*b2 - 22*b1 - 38) * q^57 + (8*b3 + 8*b2 - 4*b1) * q^58 + (-4*b3 + 8*b2 - 40*b1) * q^59 + (-8*b3 - 8*b2 + 4*b1 - 16) * q^61 + 8*b1 * q^62 + (-14*b3 + 8*b2 + 3*b1 + 64) * q^63 - 8 * q^64 + (12*b2 - 12*b1 + 12) * q^66 + (6*b3 + 6*b2 - 3*b1 + 82) * q^67 + (4*b3 - 8*b2 - 20*b1) * q^68 + (3*b3 + 6*b2 + 9*b1 - 33) * q^69 + (4*b3 - 8*b2 + 34*b1) * q^71 + (4*b3 - 4*b2 + 6*b1 + 16) * q^72 + (-8*b3 - 8*b2 + 4*b1 - 50) * q^73 + (8*b3 - 16*b2 + 30*b1) * q^74 + (-8*b3 - 8*b2 + 4*b1 - 16) * q^76 + (-12*b3 + 24*b2 - 24*b1) * q^77 + (10*b3 - 10*b1 - 10) * q^78 + (-4*b3 - 4*b2 + 2*b1 - 28) * q^79 + (-20*b3 + 8*b2 + 7) * q^81 + (-4*b3 - 4*b2 + 2*b1 - 48) * q^82 + (2*b3 - 4*b2 - 7*b1) * q^83 + (6*b3 - 4*b2 + 34*b1 + 26) * q^84 + (-6*b3 + 12*b2 - 20*b1) * q^86 + (12*b2 + 24*b1 - 60) * q^87 + 24 * q^88 + (-4*b3 + 8*b2 + 20*b1) * q^89 + (20*b3 + 20*b2 - 10*b1 - 20) * q^91 + (-4*b3 + 8*b2 - 10*b1) * q^92 + (-8*b2 + 8*b1 - 8) * q^93 + (20*b3 + 20*b2 - 10*b1 - 30) * q^94 + (4*b3 - 4*b1 - 4) * q^96 + (8*b3 + 8*b2 - 4*b1 - 74) * q^97 + (8*b3 - 16*b2 + 53*b1) * q^98 + (-12*b3 + 12*b2 - 18*b1 - 48) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} - 8 q^{4} - 4 q^{6} - 8 q^{7} - 8 q^{9}+O(q^{10})$$ 4 * q - 4 * q^3 - 8 * q^4 - 4 * q^6 - 8 * q^7 - 8 * q^9 $$4 q - 4 q^{3} - 8 q^{4} - 4 q^{6} - 8 q^{7} - 8 q^{9} + 8 q^{12} + 40 q^{13} + 16 q^{16} - 32 q^{18} + 32 q^{19} - 52 q^{21} - 48 q^{22} + 8 q^{24} - 28 q^{27} + 16 q^{28} + 32 q^{31} - 24 q^{33} - 96 q^{34} + 16 q^{36} + 88 q^{37} - 40 q^{39} + 128 q^{42} - 56 q^{43} - 24 q^{46} - 16 q^{48} + 180 q^{49} + 72 q^{51} - 80 q^{52} + 140 q^{54} - 152 q^{57} - 64 q^{61} + 256 q^{63} - 32 q^{64} + 48 q^{66} + 328 q^{67} - 132 q^{69} + 64 q^{72} - 200 q^{73} - 64 q^{76} - 40 q^{78} - 112 q^{79} + 28 q^{81} - 192 q^{82} + 104 q^{84} - 240 q^{87} + 96 q^{88} - 80 q^{91} - 32 q^{93} - 120 q^{94} - 16 q^{96} - 296 q^{97} - 192 q^{99}+O(q^{100})$$ 4 * q - 4 * q^3 - 8 * q^4 - 4 * q^6 - 8 * q^7 - 8 * q^9 + 8 * q^12 + 40 * q^13 + 16 * q^16 - 32 * q^18 + 32 * q^19 - 52 * q^21 - 48 * q^22 + 8 * q^24 - 28 * q^27 + 16 * q^28 + 32 * q^31 - 24 * q^33 - 96 * q^34 + 16 * q^36 + 88 * q^37 - 40 * q^39 + 128 * q^42 - 56 * q^43 - 24 * q^46 - 16 * q^48 + 180 * q^49 + 72 * q^51 - 80 * q^52 + 140 * q^54 - 152 * q^57 - 64 * q^61 + 256 * q^63 - 32 * q^64 + 48 * q^66 + 328 * q^67 - 132 * q^69 + 64 * q^72 - 200 * q^73 - 64 * q^76 - 40 * q^78 - 112 * q^79 + 28 * q^81 - 192 * q^82 + 104 * q^84 - 240 * q^87 + 96 * q^88 - 80 * q^91 - 32 * q^93 - 120 * q^94 - 16 * q^96 - 296 * q^97 - 192 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 4x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} - \nu ) / 3$$ (v^3 - v) / 3 $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu - 2$$ v^2 + v - 2 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} - 3\nu^{2} + 7\nu + 6 ) / 3$$ (-v^3 - 3*v^2 + 7*v + 6) / 3
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} + \beta_1 ) / 3$$ (b3 + b2 + b1) / 3 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} - \beta _1 + 6 ) / 3$$ (-b3 + 2*b2 - b1 + 6) / 3 $$\nu^{3}$$ $$=$$ $$( \beta_{3} + \beta_{2} + 10\beta_1 ) / 3$$ (b3 + b2 + 10*b1) / 3

## Character values

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

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 1.58114 − 0.707107i −1.58114 − 0.707107i 1.58114 + 0.707107i −1.58114 + 0.707107i
1.41421i −2.58114 + 1.52896i −2.00000 0 2.16228 + 3.65028i 7.48683 2.82843i 4.32456 7.89292i 0
101.2 1.41421i 0.581139 2.94317i −2.00000 0 −4.16228 0.821854i −11.4868 2.82843i −8.32456 3.42079i 0
101.3 1.41421i −2.58114 1.52896i −2.00000 0 2.16228 3.65028i 7.48683 2.82843i 4.32456 + 7.89292i 0
101.4 1.41421i 0.581139 + 2.94317i −2.00000 0 −4.16228 + 0.821854i −11.4868 2.82843i −8.32456 + 3.42079i 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
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 150.3.d.c 4
3.b odd 2 1 inner 150.3.d.c 4
4.b odd 2 1 1200.3.l.u 4
5.b even 2 1 30.3.d.a 4
5.c odd 4 2 150.3.b.b 8
12.b even 2 1 1200.3.l.u 4
15.d odd 2 1 30.3.d.a 4
15.e even 4 2 150.3.b.b 8
20.d odd 2 1 240.3.l.c 4
20.e even 4 2 1200.3.c.k 8
40.e odd 2 1 960.3.l.f 4
40.f even 2 1 960.3.l.e 4
45.h odd 6 2 810.3.h.a 8
45.j even 6 2 810.3.h.a 8
60.h even 2 1 240.3.l.c 4
60.l odd 4 2 1200.3.c.k 8
120.i odd 2 1 960.3.l.e 4
120.m even 2 1 960.3.l.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.3.d.a 4 5.b even 2 1
30.3.d.a 4 15.d odd 2 1
150.3.b.b 8 5.c odd 4 2
150.3.b.b 8 15.e even 4 2
150.3.d.c 4 1.a even 1 1 trivial
150.3.d.c 4 3.b odd 2 1 inner
240.3.l.c 4 20.d odd 2 1
240.3.l.c 4 60.h even 2 1
810.3.h.a 8 45.h odd 6 2
810.3.h.a 8 45.j even 6 2
960.3.l.e 4 40.f even 2 1
960.3.l.e 4 120.i odd 2 1
960.3.l.f 4 40.e odd 2 1
960.3.l.f 4 120.m even 2 1
1200.3.c.k 8 20.e even 4 2
1200.3.c.k 8 60.l odd 4 2
1200.3.l.u 4 4.b odd 2 1
1200.3.l.u 4 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 4T_{7} - 86$$ acting on $$S_{3}^{\mathrm{new}}(150, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 2)^{2}$$
$3$ $$T^{4} + 4 T^{3} + 12 T^{2} + 36 T + 81$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 4 T - 86)^{2}$$
$11$ $$(T^{2} + 72)^{2}$$
$13$ $$(T - 10)^{4}$$
$17$ $$T^{4} + 936 T^{2} + 11664$$
$19$ $$(T^{2} - 16 T - 296)^{2}$$
$23$ $$T^{4} + 396 T^{2} + 26244$$
$29$ $$(T^{2} + 720)^{2}$$
$31$ $$(T - 8)^{4}$$
$37$ $$(T^{2} - 44 T - 956)^{2}$$
$41$ $$T^{4} + 2664 T^{2} + 944784$$
$43$ $$(T^{2} + 28 T - 614)^{2}$$
$47$ $$T^{4} + 9900 T^{2} + \cdots + 16402500$$
$53$ $$T^{4} + 936 T^{2} + 11664$$
$59$ $$T^{4} + 6624 T^{2} + \cdots + 3504384$$
$61$ $$(T^{2} + 32 T - 1184)^{2}$$
$67$ $$(T^{2} - 164 T + 5914)^{2}$$
$71$ $$T^{4} + 5040 T^{2} + \cdots + 1166400$$
$73$ $$(T^{2} + 100 T + 1060)^{2}$$
$79$ $$(T^{2} + 56 T + 424)^{2}$$
$83$ $$T^{4} + 684T^{2} + 324$$
$89$ $$T^{4} + 3744 T^{2} + 186624$$
$97$ $$(T^{2} + 148 T + 4036)^{2}$$