LMFDB - Newform orbit 150.10.c.h

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [150,10,Mod(49,150)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("150.49"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$N$	$=$	$150 = 2 \cdot 3 \cdot 5^{2}$
Weight:	$k$	$=$	$10$
Character orbit:	$[\chi]$	$=$	150.c (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-512,0,2592,0,0,-13122,0,15504] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$77.2553754246$
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 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 $i = \sqrt{-1}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - 16 i q^{2} + 81 i q^{3} - 256 q^{4} + 1296 q^{6} + 6332 i q^{7} + 4096 i q^{8} - 6561 q^{9} + 7752 q^{11} - 20736 i q^{12} - 72626 i q^{13} + 101312 q^{14} + 65536 q^{16} - 334698 i q^{17} + 104976 i q^{18} + \cdots - 50860872 q^{99} +O(q^{100})$ q - 16i q^2 + 81i q^3 - 256 * q^4 + 1296 * q^6 + 6332i q^7 + 4096i q^8 - 6561 * q^9 + 7752 * q^11 - 20736i q^12 - 72626i q^13 + 101312 * q^14 + 65536 * q^16 - 334698i q^17 + 104976i q^18 + 934660 * q^19 - 512892 * q^21 - 124032i q^22 + 991704i q^23 - 331776 * q^24 - 1162016 * q^26 - 531441i q^27 - 1620992i q^28 + 3638790 * q^29 - 6063688 * q^31 - 1048576i q^32 + 627912i q^33 - 5355168 * q^34 + 1679616 * q^36 + 12489842i q^37 - 14954560i q^38 + 5882706 * q^39 - 5035398 * q^41 + 8206272i q^42 - 30163316i q^43 - 1984512 * q^44 + 15867264 * q^46 - 743928i q^47 + 5308416i q^48 + 259383 * q^49 + 27110538 * q^51 + 18592256i q^52 + 102388134i q^53 - 8503056 * q^54 - 25935872 * q^56 + 75707460i q^57 - 58220640i q^58 + 49464840 * q^59 - 130545898 * q^61 + 97019008i q^62 - 41544252i q^63 - 16777216 * q^64 + 10046592 * q^66 + 102905012i q^67 + 85682688i q^68 - 80328024 * q^69 - 190423008 * q^71 - 26873856i q^72 + 367621054i q^73 + 199837472 * q^74 - 239272960 * q^76 + 49085664i q^77 - 94123296i q^78 - 175880360 * q^79 + 43046721 * q^81 + 80566368i q^82 + 100482444i q^83 + 131300352 * q^84 - 482613056 * q^86 + 294741990i q^87 + 31752192i q^88 + 660904110 * q^89 + 459867832 * q^91 - 253876224i q^92 - 491158728i q^93 - 11902848 * q^94 + 84934656 * q^96 + 1321991522i q^97 - 4150128i q^98 - 50860872 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 512 q^{4} + 2592 q^{6} - 13122 q^{9} + 15504 q^{11} + 202624 q^{14} + 131072 q^{16} + 1869320 q^{19} - 1025784 q^{21} - 663552 q^{24} - 2324032 q^{26} + 7277580 q^{29} - 12127376 q^{31} - 10710336 q^{34}+ \cdots - 101721744 q^{99}+O(q^{100})$ 2 * q - 512 * q^4 + 2592 * q^6 - 13122 * q^9 + 15504 * q^11 + 202624 * q^14 + 131072 * q^16 + 1869320 * q^19 - 1025784 * q^21 - 663552 * q^24 - 2324032 * q^26 + 7277580 * q^29 - 12127376 * q^31 - 10710336 * q^34 + 3359232 * q^36 + 11765412 * q^39 - 10070796 * q^41 - 3969024 * q^44 + 31734528 * q^46 + 518766 * q^49 + 54221076 * q^51 - 17006112 * q^54 - 51871744 * q^56 + 98929680 * q^59 - 261091796 * q^61 - 33554432 * q^64 + 20093184 * q^66 - 160656048 * q^69 - 380846016 * q^71 + 399674944 * q^74 - 478545920 * q^76 - 351760720 * q^79 + 86093442 * q^81 + 262600704 * q^84 - 965226112 * q^86 + 1321808220 * q^89 + 919735664 * q^91 - 23805696 * q^94 + 169869312 * q^96 - 101721744 * 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

−

16.0000i

81.0000i

−256.000

1296.00

6332.00i

4096.00i

−6561.00

49.2

16.0000i

−

81.0000i

−256.000

1296.00

−

6332.00i

−

4096.00i

−6561.00

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	150.10.c.h		2
5.b	even	2	1	inner	150.10.c.h		2
5.c	odd	4	1		30.10.a.a	✓	1
5.c	odd	4	1		150.10.a.i		1
15.e	even	4	1		90.10.a.j		1
20.e	even	4	1		240.10.a.d		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
30.10.a.a	✓	1	5.c	odd	4	1
90.10.a.j		1	15.e	even	4	1
150.10.a.i		1	5.c	odd	4	1
150.10.c.h		2	1.a	even	1	1	trivial
150.10.c.h		2	5.b	even	2	1	inner
240.10.a.d		1	20.e	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{7}^{2} + 40094224$ T7^2 + 40094224 acting on $S_{10}^{\mathrm{new}}(150, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} + 256$ T^2 + 256
$3$	$T^{2} + 6561$ T^2 + 6561
$5$	$T^{2}$ T^2
$7$	$T^{2} + 40094224$ T^2 + 40094224
$11$	$(T - 7752)^{2}$ (T - 7752)^2
$13$	$T^{2} + 5274535876$ T^2 + 5274535876
$17$	$T^{2} + 112022751204$ T^2 + 112022751204
$19$	$(T - 934660)^{2}$ (T - 934660)^2
$23$	$T^{2} + 983476823616$ T^2 + 983476823616
$29$	$(T - 3638790)^{2}$ (T - 3638790)^2
$31$	$(T + 6063688)^{2}$ (T + 6063688)^2
$37$	$T^{2} + 155996153184964$ T^2 + 155996153184964
$41$	$(T + 5035398)^{2}$ (T + 5035398)^2
$43$	$T^{2} + 909825632115856$ T^2 + 909825632115856
$47$	$T^{2} + 553428869184$ T^2 + 553428869184
$53$	$T^{2} + 10\!\cdots\!56$ T^2 + 10483329984001956
$59$	$(T - 49464840)^{2}$ (T - 49464840)^2
$61$	$(T + 130545898)^{2}$ (T + 130545898)^2
$67$	$T^{2} + 10\!\cdots\!44$ T^2 + 10589441494720144
$71$	$(T + 190423008)^{2}$ (T + 190423008)^2
$73$	$T^{2} + 13\!\cdots\!16$ T^2 + 135145239344070916
$79$	$(T + 175880360)^{2}$ (T + 175880360)^2
$83$	$T^{2} + 10\!\cdots\!36$ T^2 + 10096721552213136
$89$	$(T - 660904110)^{2}$ (T - 660904110)^2
$97$	$T^{2} + 17\!\cdots\!84$ T^2 + 1747661584239876484
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