# Properties

 Label 150.4.c.d Level $150$ Weight $4$ Character orbit 150.c Analytic conductor $8.850$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [150,4,Mod(49,150)]

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([0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("150.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$150 = 2 \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 150.c (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: $$8.85028650086$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 6) 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 i q^{2} - 3 i q^{3} - 4 q^{4} + 6 q^{6} + 16 i q^{7} - 8 i q^{8} - 9 q^{9} +O(q^{10})$$ q + 2*i * q^2 - 3*i * q^3 - 4 * q^4 + 6 * q^6 + 16*i * q^7 - 8*i * q^8 - 9 * q^9 $$q + 2 i q^{2} - 3 i q^{3} - 4 q^{4} + 6 q^{6} + 16 i q^{7} - 8 i q^{8} - 9 q^{9} + 12 q^{11} + 12 i q^{12} + 38 i q^{13} - 32 q^{14} + 16 q^{16} + 126 i q^{17} - 18 i q^{18} - 20 q^{19} + 48 q^{21} + 24 i q^{22} + 168 i q^{23} - 24 q^{24} - 76 q^{26} + 27 i q^{27} - 64 i q^{28} - 30 q^{29} - 88 q^{31} + 32 i q^{32} - 36 i q^{33} - 252 q^{34} + 36 q^{36} - 254 i q^{37} - 40 i q^{38} + 114 q^{39} + 42 q^{41} + 96 i q^{42} - 52 i q^{43} - 48 q^{44} - 336 q^{46} + 96 i q^{47} - 48 i q^{48} + 87 q^{49} + 378 q^{51} - 152 i q^{52} + 198 i q^{53} - 54 q^{54} + 128 q^{56} + 60 i q^{57} - 60 i q^{58} + 660 q^{59} - 538 q^{61} - 176 i q^{62} - 144 i q^{63} - 64 q^{64} + 72 q^{66} - 884 i q^{67} - 504 i q^{68} + 504 q^{69} + 792 q^{71} + 72 i q^{72} + 218 i q^{73} + 508 q^{74} + 80 q^{76} + 192 i q^{77} + 228 i q^{78} + 520 q^{79} + 81 q^{81} + 84 i q^{82} - 492 i q^{83} - 192 q^{84} + 104 q^{86} + 90 i q^{87} - 96 i q^{88} - 810 q^{89} - 608 q^{91} - 672 i q^{92} + 264 i q^{93} - 192 q^{94} + 96 q^{96} - 1154 i q^{97} + 174 i q^{98} - 108 q^{99} +O(q^{100})$$ q + 2*i * q^2 - 3*i * q^3 - 4 * q^4 + 6 * q^6 + 16*i * q^7 - 8*i * q^8 - 9 * q^9 + 12 * q^11 + 12*i * q^12 + 38*i * q^13 - 32 * q^14 + 16 * q^16 + 126*i * q^17 - 18*i * q^18 - 20 * q^19 + 48 * q^21 + 24*i * q^22 + 168*i * q^23 - 24 * q^24 - 76 * q^26 + 27*i * q^27 - 64*i * q^28 - 30 * q^29 - 88 * q^31 + 32*i * q^32 - 36*i * q^33 - 252 * q^34 + 36 * q^36 - 254*i * q^37 - 40*i * q^38 + 114 * q^39 + 42 * q^41 + 96*i * q^42 - 52*i * q^43 - 48 * q^44 - 336 * q^46 + 96*i * q^47 - 48*i * q^48 + 87 * q^49 + 378 * q^51 - 152*i * q^52 + 198*i * q^53 - 54 * q^54 + 128 * q^56 + 60*i * q^57 - 60*i * q^58 + 660 * q^59 - 538 * q^61 - 176*i * q^62 - 144*i * q^63 - 64 * q^64 + 72 * q^66 - 884*i * q^67 - 504*i * q^68 + 504 * q^69 + 792 * q^71 + 72*i * q^72 + 218*i * q^73 + 508 * q^74 + 80 * q^76 + 192*i * q^77 + 228*i * q^78 + 520 * q^79 + 81 * q^81 + 84*i * q^82 - 492*i * q^83 - 192 * q^84 + 104 * q^86 + 90*i * q^87 - 96*i * q^88 - 810 * q^89 - 608 * q^91 - 672*i * q^92 + 264*i * q^93 - 192 * q^94 + 96 * q^96 - 1154*i * q^97 + 174*i * q^98 - 108 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{4} + 12 q^{6} - 18 q^{9}+O(q^{10})$$ 2 * q - 8 * q^4 + 12 * q^6 - 18 * q^9 $$2 q - 8 q^{4} + 12 q^{6} - 18 q^{9} + 24 q^{11} - 64 q^{14} + 32 q^{16} - 40 q^{19} + 96 q^{21} - 48 q^{24} - 152 q^{26} - 60 q^{29} - 176 q^{31} - 504 q^{34} + 72 q^{36} + 228 q^{39} + 84 q^{41} - 96 q^{44} - 672 q^{46} + 174 q^{49} + 756 q^{51} - 108 q^{54} + 256 q^{56} + 1320 q^{59} - 1076 q^{61} - 128 q^{64} + 144 q^{66} + 1008 q^{69} + 1584 q^{71} + 1016 q^{74} + 160 q^{76} + 1040 q^{79} + 162 q^{81} - 384 q^{84} + 208 q^{86} - 1620 q^{89} - 1216 q^{91} - 384 q^{94} + 192 q^{96} - 216 q^{99}+O(q^{100})$$ 2 * q - 8 * q^4 + 12 * q^6 - 18 * q^9 + 24 * q^11 - 64 * q^14 + 32 * q^16 - 40 * q^19 + 96 * q^21 - 48 * q^24 - 152 * q^26 - 60 * q^29 - 176 * q^31 - 504 * q^34 + 72 * q^36 + 228 * q^39 + 84 * q^41 - 96 * q^44 - 672 * q^46 + 174 * q^49 + 756 * q^51 - 108 * q^54 + 256 * q^56 + 1320 * q^59 - 1076 * q^61 - 128 * q^64 + 144 * q^66 + 1008 * q^69 + 1584 * q^71 + 1016 * q^74 + 160 * q^76 + 1040 * q^79 + 162 * q^81 - 384 * q^84 + 208 * q^86 - 1620 * q^89 - 1216 * q^91 - 384 * q^94 + 192 * q^96 - 216 * 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 − 1.00000i 1.00000i
2.00000i 3.00000i −4.00000 0 6.00000 16.0000i 8.00000i −9.00000 0
49.2 2.00000i 3.00000i −4.00000 0 6.00000 16.0000i 8.00000i −9.00000 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 150.4.c.d 2
3.b odd 2 1 450.4.c.e 2
4.b odd 2 1 1200.4.f.j 2
5.b even 2 1 inner 150.4.c.d 2
5.c odd 4 1 6.4.a.a 1
5.c odd 4 1 150.4.a.i 1
15.d odd 2 1 450.4.c.e 2
15.e even 4 1 18.4.a.a 1
15.e even 4 1 450.4.a.h 1
20.d odd 2 1 1200.4.f.j 2
20.e even 4 1 48.4.a.c 1
20.e even 4 1 1200.4.a.b 1
35.f even 4 1 294.4.a.e 1
35.k even 12 2 294.4.e.g 2
35.l odd 12 2 294.4.e.h 2
40.i odd 4 1 192.4.a.i 1
40.k even 4 1 192.4.a.c 1
45.k odd 12 2 162.4.c.f 2
45.l even 12 2 162.4.c.c 2
55.e even 4 1 726.4.a.f 1
60.l odd 4 1 144.4.a.c 1
65.f even 4 1 1014.4.b.d 2
65.h odd 4 1 1014.4.a.g 1
65.k even 4 1 1014.4.b.d 2
80.i odd 4 1 768.4.d.n 2
80.j even 4 1 768.4.d.c 2
80.s even 4 1 768.4.d.c 2
80.t odd 4 1 768.4.d.n 2
85.g odd 4 1 1734.4.a.d 1
95.g even 4 1 2166.4.a.i 1
105.k odd 4 1 882.4.a.n 1
105.w odd 12 2 882.4.g.f 2
105.x even 12 2 882.4.g.i 2
120.q odd 4 1 576.4.a.r 1
120.w even 4 1 576.4.a.q 1
140.j odd 4 1 2352.4.a.e 1
165.l odd 4 1 2178.4.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 5.c odd 4 1
18.4.a.a 1 15.e even 4 1
48.4.a.c 1 20.e even 4 1
144.4.a.c 1 60.l odd 4 1
150.4.a.i 1 5.c odd 4 1
150.4.c.d 2 1.a even 1 1 trivial
150.4.c.d 2 5.b even 2 1 inner
162.4.c.c 2 45.l even 12 2
162.4.c.f 2 45.k odd 12 2
192.4.a.c 1 40.k even 4 1
192.4.a.i 1 40.i odd 4 1
294.4.a.e 1 35.f even 4 1
294.4.e.g 2 35.k even 12 2
294.4.e.h 2 35.l odd 12 2
450.4.a.h 1 15.e even 4 1
450.4.c.e 2 3.b odd 2 1
450.4.c.e 2 15.d odd 2 1
576.4.a.q 1 120.w even 4 1
576.4.a.r 1 120.q odd 4 1
726.4.a.f 1 55.e even 4 1
768.4.d.c 2 80.j even 4 1
768.4.d.c 2 80.s even 4 1
768.4.d.n 2 80.i odd 4 1
768.4.d.n 2 80.t odd 4 1
882.4.a.n 1 105.k odd 4 1
882.4.g.f 2 105.w odd 12 2
882.4.g.i 2 105.x even 12 2
1014.4.a.g 1 65.h odd 4 1
1014.4.b.d 2 65.f even 4 1
1014.4.b.d 2 65.k even 4 1
1200.4.a.b 1 20.e even 4 1
1200.4.f.j 2 4.b odd 2 1
1200.4.f.j 2 20.d odd 2 1
1734.4.a.d 1 85.g odd 4 1
2166.4.a.i 1 95.g even 4 1
2178.4.a.e 1 165.l odd 4 1
2352.4.a.e 1 140.j odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 256$$ acting on $$S_{4}^{\mathrm{new}}(150, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$T^{2} + 9$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 256$$
$11$ $$(T - 12)^{2}$$
$13$ $$T^{2} + 1444$$
$17$ $$T^{2} + 15876$$
$19$ $$(T + 20)^{2}$$
$23$ $$T^{2} + 28224$$
$29$ $$(T + 30)^{2}$$
$31$ $$(T + 88)^{2}$$
$37$ $$T^{2} + 64516$$
$41$ $$(T - 42)^{2}$$
$43$ $$T^{2} + 2704$$
$47$ $$T^{2} + 9216$$
$53$ $$T^{2} + 39204$$
$59$ $$(T - 660)^{2}$$
$61$ $$(T + 538)^{2}$$
$67$ $$T^{2} + 781456$$
$71$ $$(T - 792)^{2}$$
$73$ $$T^{2} + 47524$$
$79$ $$(T - 520)^{2}$$
$83$ $$T^{2} + 242064$$
$89$ $$(T + 810)^{2}$$
$97$ $$T^{2} + 1331716$$