# Properties

 Label 150.3.b.a Level $150$ Weight $3$ Character orbit 150.b Analytic conductor $4.087$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [150,3,Mod(149,150)]

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, 1]))

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

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

chi := DirichletCharacter("150.149");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$150 = 2 \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 150.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: $$4.08720396540$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{2} + ( - 2 \beta_{3} - \beta_1) q^{3} + 2 q^{4} + (\beta_{2} + 4) q^{6} + 7 \beta_1 q^{7} - 2 \beta_{3} q^{8} + (4 \beta_{2} + 7) q^{9}+O(q^{10})$$ q - b3 * q^2 + (-2*b3 - b1) * q^3 + 2 * q^4 + (b2 + 4) * q^6 + 7*b1 * q^7 - 2*b3 * q^8 + (4*b2 + 7) * q^9 $$q - \beta_{3} q^{2} + ( - 2 \beta_{3} - \beta_1) q^{3} + 2 q^{4} + (\beta_{2} + 4) q^{6} + 7 \beta_1 q^{7} - 2 \beta_{3} q^{8} + (4 \beta_{2} + 7) q^{9} - 6 \beta_{2} q^{11} + ( - 4 \beta_{3} - 2 \beta_1) q^{12} - 25 \beta_1 q^{13} - 7 \beta_{2} q^{14} + 4 q^{16} + 18 \beta_{3} q^{17} + ( - 7 \beta_{3} - 8 \beta_1) q^{18} + 7 q^{19} + ( - 14 \beta_{2} + 7) q^{21} + 12 \beta_1 q^{22} + 18 \beta_{3} q^{23} + (2 \beta_{2} + 8) q^{24} + 25 \beta_{2} q^{26} + ( - 10 \beta_{3} - 23 \beta_1) q^{27} + 14 \beta_1 q^{28} - 30 \beta_{2} q^{29} - 7 q^{31} - 4 \beta_{3} q^{32} + ( - 6 \beta_{3} + 24 \beta_1) q^{33} - 36 q^{34} + (8 \beta_{2} + 14) q^{36} - 2 \beta_1 q^{37} - 7 \beta_{3} q^{38} + (50 \beta_{2} - 25) q^{39} + 6 \beta_{2} q^{41} + ( - 7 \beta_{3} + 28 \beta_1) q^{42} + 41 \beta_1 q^{43} - 12 \beta_{2} q^{44} - 36 q^{46} + ( - 8 \beta_{3} - 4 \beta_1) q^{48} + ( - 18 \beta_{2} - 72) q^{51} - 50 \beta_1 q^{52} + 42 \beta_{3} q^{53} + (23 \beta_{2} + 20) q^{54} - 14 \beta_{2} q^{56} + ( - 14 \beta_{3} - 7 \beta_1) q^{57} + 60 \beta_1 q^{58} + 24 \beta_{2} q^{59} - q^{61} + 7 \beta_{3} q^{62} + ( - 28 \beta_{3} + 49 \beta_1) q^{63} + 8 q^{64} + ( - 24 \beta_{2} + 12) q^{66} - 17 \beta_1 q^{67} + 36 \beta_{3} q^{68} + ( - 18 \beta_{2} - 72) q^{69} + 30 \beta_{2} q^{71} + ( - 14 \beta_{3} - 16 \beta_1) q^{72} - 70 \beta_1 q^{73} + 2 \beta_{2} q^{74} + 14 q^{76} + 42 \beta_{3} q^{77} + (25 \beta_{3} - 100 \beta_1) q^{78} + 58 q^{79} + (56 \beta_{2} + 17) q^{81} - 12 \beta_1 q^{82} - 84 \beta_{3} q^{83} + ( - 28 \beta_{2} + 14) q^{84} - 41 \beta_{2} q^{86} + ( - 30 \beta_{3} + 120 \beta_1) q^{87} + 24 \beta_1 q^{88} - 96 \beta_{2} q^{89} + 175 q^{91} + 36 \beta_{3} q^{92} + (14 \beta_{3} + 7 \beta_1) q^{93} + (4 \beta_{2} + 16) q^{96} + 49 \beta_1 q^{97} + ( - 42 \beta_{2} + 48) q^{99}+O(q^{100})$$ q - b3 * q^2 + (-2*b3 - b1) * q^3 + 2 * q^4 + (b2 + 4) * q^6 + 7*b1 * q^7 - 2*b3 * q^8 + (4*b2 + 7) * q^9 - 6*b2 * q^11 + (-4*b3 - 2*b1) * q^12 - 25*b1 * q^13 - 7*b2 * q^14 + 4 * q^16 + 18*b3 * q^17 + (-7*b3 - 8*b1) * q^18 + 7 * q^19 + (-14*b2 + 7) * q^21 + 12*b1 * q^22 + 18*b3 * q^23 + (2*b2 + 8) * q^24 + 25*b2 * q^26 + (-10*b3 - 23*b1) * q^27 + 14*b1 * q^28 - 30*b2 * q^29 - 7 * q^31 - 4*b3 * q^32 + (-6*b3 + 24*b1) * q^33 - 36 * q^34 + (8*b2 + 14) * q^36 - 2*b1 * q^37 - 7*b3 * q^38 + (50*b2 - 25) * q^39 + 6*b2 * q^41 + (-7*b3 + 28*b1) * q^42 + 41*b1 * q^43 - 12*b2 * q^44 - 36 * q^46 + (-8*b3 - 4*b1) * q^48 + (-18*b2 - 72) * q^51 - 50*b1 * q^52 + 42*b3 * q^53 + (23*b2 + 20) * q^54 - 14*b2 * q^56 + (-14*b3 - 7*b1) * q^57 + 60*b1 * q^58 + 24*b2 * q^59 - q^61 + 7*b3 * q^62 + (-28*b3 + 49*b1) * q^63 + 8 * q^64 + (-24*b2 + 12) * q^66 - 17*b1 * q^67 + 36*b3 * q^68 + (-18*b2 - 72) * q^69 + 30*b2 * q^71 + (-14*b3 - 16*b1) * q^72 - 70*b1 * q^73 + 2*b2 * q^74 + 14 * q^76 + 42*b3 * q^77 + (25*b3 - 100*b1) * q^78 + 58 * q^79 + (56*b2 + 17) * q^81 - 12*b1 * q^82 - 84*b3 * q^83 + (-28*b2 + 14) * q^84 - 41*b2 * q^86 + (-30*b3 + 120*b1) * q^87 + 24*b1 * q^88 - 96*b2 * q^89 + 175 * q^91 + 36*b3 * q^92 + (14*b3 + 7*b1) * q^93 + (4*b2 + 16) * q^96 + 49*b1 * q^97 + (-42*b2 + 48) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{4} + 16 q^{6} + 28 q^{9}+O(q^{10})$$ 4 * q + 8 * q^4 + 16 * q^6 + 28 * q^9 $$4 q + 8 q^{4} + 16 q^{6} + 28 q^{9} + 16 q^{16} + 28 q^{19} + 28 q^{21} + 32 q^{24} - 28 q^{31} - 144 q^{34} + 56 q^{36} - 100 q^{39} - 144 q^{46} - 288 q^{51} + 80 q^{54} - 4 q^{61} + 32 q^{64} + 48 q^{66} - 288 q^{69} + 56 q^{76} + 232 q^{79} + 68 q^{81} + 56 q^{84} + 700 q^{91} + 64 q^{96} + 192 q^{99}+O(q^{100})$$ 4 * q + 8 * q^4 + 16 * q^6 + 28 * q^9 + 16 * q^16 + 28 * q^19 + 28 * q^21 + 32 * q^24 - 28 * q^31 - 144 * q^34 + 56 * q^36 - 100 * q^39 - 144 * q^46 - 288 * q^51 + 80 * q^54 - 4 * q^61 + 32 * q^64 + 48 * q^66 - 288 * q^69 + 56 * q^76 + 232 * q^79 + 68 * q^81 + 56 * q^84 + 700 * q^91 + 64 * q^96 + 192 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{8}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}$$ -v^3 + v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\zeta_{8}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 2$$ (-b3 + b2) / 2

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1
 0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i
−1.41421 −2.82843 1.00000i 2.00000 0 4.00000 + 1.41421i 7.00000i −2.82843 7.00000 + 5.65685i 0
149.2 −1.41421 −2.82843 + 1.00000i 2.00000 0 4.00000 1.41421i 7.00000i −2.82843 7.00000 5.65685i 0
149.3 1.41421 2.82843 1.00000i 2.00000 0 4.00000 1.41421i 7.00000i 2.82843 7.00000 5.65685i 0
149.4 1.41421 2.82843 + 1.00000i 2.00000 0 4.00000 + 1.41421i 7.00000i 2.82843 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
5.b even 2 1 inner
15.d 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.b.a 4
3.b odd 2 1 inner 150.3.b.a 4
4.b odd 2 1 1200.3.c.h 4
5.b even 2 1 inner 150.3.b.a 4
5.c odd 4 1 150.3.d.a 2
5.c odd 4 1 150.3.d.b yes 2
12.b even 2 1 1200.3.c.h 4
15.d odd 2 1 inner 150.3.b.a 4
15.e even 4 1 150.3.d.a 2
15.e even 4 1 150.3.d.b yes 2
20.d odd 2 1 1200.3.c.h 4
20.e even 4 1 1200.3.l.i 2
20.e even 4 1 1200.3.l.p 2
60.h even 2 1 1200.3.c.h 4
60.l odd 4 1 1200.3.l.i 2
60.l odd 4 1 1200.3.l.p 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 49$$ acting on $$S_{3}^{\mathrm{new}}(150, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2)^{2}$$
$3$ $$T^{4} - 14T^{2} + 81$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 49)^{2}$$
$11$ $$(T^{2} + 72)^{2}$$
$13$ $$(T^{2} + 625)^{2}$$
$17$ $$(T^{2} - 648)^{2}$$
$19$ $$(T - 7)^{4}$$
$23$ $$(T^{2} - 648)^{2}$$
$29$ $$(T^{2} + 1800)^{2}$$
$31$ $$(T + 7)^{4}$$
$37$ $$(T^{2} + 4)^{2}$$
$41$ $$(T^{2} + 72)^{2}$$
$43$ $$(T^{2} + 1681)^{2}$$
$47$ $$T^{4}$$
$53$ $$(T^{2} - 3528)^{2}$$
$59$ $$(T^{2} + 1152)^{2}$$
$61$ $$(T + 1)^{4}$$
$67$ $$(T^{2} + 289)^{2}$$
$71$ $$(T^{2} + 1800)^{2}$$
$73$ $$(T^{2} + 4900)^{2}$$
$79$ $$(T - 58)^{4}$$
$83$ $$(T^{2} - 14112)^{2}$$
$89$ $$(T^{2} + 18432)^{2}$$
$97$ $$(T^{2} + 2401)^{2}$$