# Properties

 Label 150.3.d.a Level $150$ Weight $3$ Character orbit 150.d Analytic conductor $4.087$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 2$$ x^2 + 2 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 $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + ( - 2 \beta - 1) q^{3} - 2 q^{4} + ( - \beta + 4) q^{6} - 7 q^{7} - 2 \beta q^{8} + (4 \beta - 7) q^{9} +O(q^{10})$$ q + b * q^2 + (-2*b - 1) * q^3 - 2 * q^4 + (-b + 4) * q^6 - 7 * q^7 - 2*b * q^8 + (4*b - 7) * q^9 $$q + \beta q^{2} + ( - 2 \beta - 1) q^{3} - 2 q^{4} + ( - \beta + 4) q^{6} - 7 q^{7} - 2 \beta q^{8} + (4 \beta - 7) q^{9} + 6 \beta q^{11} + (4 \beta + 2) q^{12} - 25 q^{13} - 7 \beta q^{14} + 4 q^{16} - 18 \beta q^{17} + ( - 7 \beta - 8) q^{18} - 7 q^{19} + (14 \beta + 7) q^{21} - 12 q^{22} + 18 \beta q^{23} + (2 \beta - 8) q^{24} - 25 \beta q^{26} + (10 \beta + 23) q^{27} + 14 q^{28} - 30 \beta q^{29} - 7 q^{31} + 4 \beta q^{32} + ( - 6 \beta + 24) q^{33} + 36 q^{34} + ( - 8 \beta + 14) q^{36} + 2 q^{37} - 7 \beta q^{38} + (50 \beta + 25) q^{39} - 6 \beta q^{41} + (7 \beta - 28) q^{42} + 41 q^{43} - 12 \beta q^{44} - 36 q^{46} + ( - 8 \beta - 4) q^{48} + (18 \beta - 72) q^{51} + 50 q^{52} + 42 \beta q^{53} + (23 \beta - 20) q^{54} + 14 \beta q^{56} + (14 \beta + 7) q^{57} + 60 q^{58} + 24 \beta q^{59} - q^{61} - 7 \beta q^{62} + ( - 28 \beta + 49) q^{63} - 8 q^{64} + (24 \beta + 12) q^{66} + 17 q^{67} + 36 \beta q^{68} + ( - 18 \beta + 72) q^{69} - 30 \beta q^{71} + (14 \beta + 16) q^{72} - 70 q^{73} + 2 \beta q^{74} + 14 q^{76} - 42 \beta q^{77} + (25 \beta - 100) q^{78} - 58 q^{79} + ( - 56 \beta + 17) q^{81} + 12 q^{82} - 84 \beta q^{83} + ( - 28 \beta - 14) q^{84} + 41 \beta q^{86} + (30 \beta - 120) q^{87} + 24 q^{88} - 96 \beta q^{89} + 175 q^{91} - 36 \beta q^{92} + (14 \beta + 7) q^{93} + ( - 4 \beta + 16) q^{96} - 49 q^{97} + ( - 42 \beta - 48) q^{99} +O(q^{100})$$ q + b * q^2 + (-2*b - 1) * q^3 - 2 * q^4 + (-b + 4) * q^6 - 7 * q^7 - 2*b * q^8 + (4*b - 7) * q^9 + 6*b * q^11 + (4*b + 2) * q^12 - 25 * q^13 - 7*b * q^14 + 4 * q^16 - 18*b * q^17 + (-7*b - 8) * q^18 - 7 * q^19 + (14*b + 7) * q^21 - 12 * q^22 + 18*b * q^23 + (2*b - 8) * q^24 - 25*b * q^26 + (10*b + 23) * q^27 + 14 * q^28 - 30*b * q^29 - 7 * q^31 + 4*b * q^32 + (-6*b + 24) * q^33 + 36 * q^34 + (-8*b + 14) * q^36 + 2 * q^37 - 7*b * q^38 + (50*b + 25) * q^39 - 6*b * q^41 + (7*b - 28) * q^42 + 41 * q^43 - 12*b * q^44 - 36 * q^46 + (-8*b - 4) * q^48 + (18*b - 72) * q^51 + 50 * q^52 + 42*b * q^53 + (23*b - 20) * q^54 + 14*b * q^56 + (14*b + 7) * q^57 + 60 * q^58 + 24*b * q^59 - q^61 - 7*b * q^62 + (-28*b + 49) * q^63 - 8 * q^64 + (24*b + 12) * q^66 + 17 * q^67 + 36*b * q^68 + (-18*b + 72) * q^69 - 30*b * q^71 + (14*b + 16) * q^72 - 70 * q^73 + 2*b * q^74 + 14 * q^76 - 42*b * q^77 + (25*b - 100) * q^78 - 58 * q^79 + (-56*b + 17) * q^81 + 12 * q^82 - 84*b * q^83 + (-28*b - 14) * q^84 + 41*b * q^86 + (30*b - 120) * q^87 + 24 * q^88 - 96*b * q^89 + 175 * q^91 - 36*b * q^92 + (14*b + 7) * q^93 + (-4*b + 16) * q^96 - 49 * q^97 + (-42*b - 48) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 4 q^{4} + 8 q^{6} - 14 q^{7} - 14 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 4 * q^4 + 8 * q^6 - 14 * q^7 - 14 * q^9 $$2 q - 2 q^{3} - 4 q^{4} + 8 q^{6} - 14 q^{7} - 14 q^{9} + 4 q^{12} - 50 q^{13} + 8 q^{16} - 16 q^{18} - 14 q^{19} + 14 q^{21} - 24 q^{22} - 16 q^{24} + 46 q^{27} + 28 q^{28} - 14 q^{31} + 48 q^{33} + 72 q^{34} + 28 q^{36} + 4 q^{37} + 50 q^{39} - 56 q^{42} + 82 q^{43} - 72 q^{46} - 8 q^{48} - 144 q^{51} + 100 q^{52} - 40 q^{54} + 14 q^{57} + 120 q^{58} - 2 q^{61} + 98 q^{63} - 16 q^{64} + 24 q^{66} + 34 q^{67} + 144 q^{69} + 32 q^{72} - 140 q^{73} + 28 q^{76} - 200 q^{78} - 116 q^{79} + 34 q^{81} + 24 q^{82} - 28 q^{84} - 240 q^{87} + 48 q^{88} + 350 q^{91} + 14 q^{93} + 32 q^{96} - 98 q^{97} - 96 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 - 4 * q^4 + 8 * q^6 - 14 * q^7 - 14 * q^9 + 4 * q^12 - 50 * q^13 + 8 * q^16 - 16 * q^18 - 14 * q^19 + 14 * q^21 - 24 * q^22 - 16 * q^24 + 46 * q^27 + 28 * q^28 - 14 * q^31 + 48 * q^33 + 72 * q^34 + 28 * q^36 + 4 * q^37 + 50 * q^39 - 56 * q^42 + 82 * q^43 - 72 * q^46 - 8 * q^48 - 144 * q^51 + 100 * q^52 - 40 * q^54 + 14 * q^57 + 120 * q^58 - 2 * q^61 + 98 * q^63 - 16 * q^64 + 24 * q^66 + 34 * q^67 + 144 * q^69 + 32 * q^72 - 140 * q^73 + 28 * q^76 - 200 * q^78 - 116 * q^79 + 34 * q^81 + 24 * q^82 - 28 * q^84 - 240 * q^87 + 48 * q^88 + 350 * q^91 + 14 * q^93 + 32 * q^96 - 98 * q^97 - 96 * q^99

## 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
 − 1.41421i 1.41421i
1.41421i −1.00000 + 2.82843i −2.00000 0 4.00000 + 1.41421i −7.00000 2.82843i −7.00000 5.65685i 0
101.2 1.41421i −1.00000 2.82843i −2.00000 0 4.00000 1.41421i −7.00000 2.82843i −7.00000 + 5.65685i 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.a 2
3.b odd 2 1 inner 150.3.d.a 2
4.b odd 2 1 1200.3.l.p 2
5.b even 2 1 150.3.d.b yes 2
5.c odd 4 2 150.3.b.a 4
12.b even 2 1 1200.3.l.p 2
15.d odd 2 1 150.3.d.b yes 2
15.e even 4 2 150.3.b.a 4
20.d odd 2 1 1200.3.l.i 2
20.e even 4 2 1200.3.c.h 4
60.h even 2 1 1200.3.l.i 2
60.l odd 4 2 1200.3.c.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.3.b.a 4 5.c odd 4 2
150.3.b.a 4 15.e even 4 2
150.3.d.a 2 1.a even 1 1 trivial
150.3.d.a 2 3.b odd 2 1 inner
150.3.d.b yes 2 5.b even 2 1
150.3.d.b yes 2 15.d odd 2 1
1200.3.c.h 4 20.e even 4 2
1200.3.c.h 4 60.l odd 4 2
1200.3.l.i 2 20.d odd 2 1
1200.3.l.i 2 60.h even 2 1
1200.3.l.p 2 4.b odd 2 1
1200.3.l.p 2 12.b even 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2$$
$3$ $$T^{2} + 2T + 9$$
$5$ $$T^{2}$$
$7$ $$(T + 7)^{2}$$
$11$ $$T^{2} + 72$$
$13$ $$(T + 25)^{2}$$
$17$ $$T^{2} + 648$$
$19$ $$(T + 7)^{2}$$
$23$ $$T^{2} + 648$$
$29$ $$T^{2} + 1800$$
$31$ $$(T + 7)^{2}$$
$37$ $$(T - 2)^{2}$$
$41$ $$T^{2} + 72$$
$43$ $$(T - 41)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 3528$$
$59$ $$T^{2} + 1152$$
$61$ $$(T + 1)^{2}$$
$67$ $$(T - 17)^{2}$$
$71$ $$T^{2} + 1800$$
$73$ $$(T + 70)^{2}$$
$79$ $$(T + 58)^{2}$$
$83$ $$T^{2} + 14112$$
$89$ $$T^{2} + 18432$$
$97$ $$(T + 49)^{2}$$