LMFDB - Newform orbit 150.10.c.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [150,10,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, 10, names="a")

//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: f = N[0] # Warning: the index may be different

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.

$f(q)$	$=$	$q + 16 i q^{2} - 81 i q^{3} - 256 q^{4} + 1296 q^{6} + 7168 i q^{7} - 4096 i q^{8} - 6561 q^{9} - 83748 q^{11} + 20736 i q^{12} + 128126 i q^{13} - 114688 q^{14} + 65536 q^{16} - 560802 i q^{17} - 104976 i q^{18} + \cdots + 549470628 q^{99} +O(q^{100})$ q + 16i q^2 - 81i q^3 - 256 * q^4 + 1296 * q^6 + 7168i q^7 - 4096i q^8 - 6561 * q^9 - 83748 * q^11 + 20736i q^12 + 128126i q^13 - 114688 * q^14 + 65536 * q^16 - 560802i q^17 - 104976i q^18 + 577660 * q^19 + 580608 * q^21 - 1339968i q^22 + 2431296i q^23 - 331776 * q^24 - 2050016 * q^26 + 531441i q^27 - 1835008i q^28 - 5791710 * q^29 + 4145312 * q^31 + 1048576i q^32 + 6783588i q^33 + 8972832 * q^34 + 1679616 * q^36 + 7011658i q^37 + 9242560i q^38 + 10378206 * q^39 - 8881398 * q^41 + 9289728i q^42 - 15730684i q^43 + 21439488 * q^44 - 38900736 * q^46 - 60552072i q^47 - 5308416i q^48 - 11026617 * q^49 - 45424962 * q^51 - 32800256i q^52 + 30273366i q^53 - 8503056 * q^54 + 29360128 * q^56 - 46790460i q^57 - 92667360i q^58 - 45957660 * q^59 + 37595102 * q^61 + 66324992i q^62 - 47029248i q^63 - 16777216 * q^64 - 108537408 * q^66 - 196784012i q^67 + 143565312i q^68 + 196934976 * q^69 + 56047992 * q^71 + 26873856i q^72 - 159688054i q^73 - 112186528 * q^74 - 147880960 * q^76 - 600305664i q^77 + 166051296i q^78 - 201923360 * q^79 + 43046721 * q^81 - 142102368i q^82 - 362955444i q^83 - 148635648 * q^84 + 251690944 * q^86 + 469128510i q^87 + 343031808i q^88 + 272479110 * q^89 - 918407168 * q^91 - 622411776i q^92 - 335770272i q^93 + 968833152 * q^94 + 84934656 * q^96 + 600852478i q^97 - 176425872i q^98 + 549470628 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 512 q^{4} + 2592 q^{6} - 13122 q^{9} - 167496 q^{11} - 229376 q^{14} + 131072 q^{16} + 1155320 q^{19} + 1161216 q^{21} - 663552 q^{24} - 4100032 q^{26} - 11583420 q^{29} + 8290624 q^{31} + 17945664 q^{34}+ \cdots + 1098941256 q^{99}+O(q^{100})$ 2 * q - 512 * q^4 + 2592 * q^6 - 13122 * q^9 - 167496 * q^11 - 229376 * q^14 + 131072 * q^16 + 1155320 * q^19 + 1161216 * q^21 - 663552 * q^24 - 4100032 * q^26 - 11583420 * q^29 + 8290624 * q^31 + 17945664 * q^34 + 3359232 * q^36 + 20756412 * q^39 - 17762796 * q^41 + 42878976 * q^44 - 77801472 * q^46 - 22053234 * q^49 - 90849924 * q^51 - 17006112 * q^54 + 58720256 * q^56 - 91915320 * q^59 + 75190204 * q^61 - 33554432 * q^64 - 217074816 * q^66 + 393869952 * q^69 + 112095984 * q^71 - 224373056 * q^74 - 295761920 * q^76 - 403846720 * q^79 + 86093442 * q^81 - 297271296 * q^84 + 503381888 * q^86 + 544958220 * q^89 - 1836814336 * q^91 + 1937666304 * q^94 + 169869312 * q^96 + 1098941256 * 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

−

7168.00i

4096.00i

−6561.00

49.2

16.0000i

−

81.0000i

−256.000

1296.00

7168.00i

−

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

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{7}^{2} + 51380224$ T7^2 + 51380224 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
$7$	$T^{2} + 51380224$ T^2 + 51380224
$11$	$(T + 83748)^{2}$ (T + 83748)^2
$13$	$T^{2} + 16416271876$ T^2 + 16416271876
$17$	$T^{2} + 314498883204$ T^2 + 314498883204
$19$	$(T - 577660)^{2}$ (T - 577660)^2
$23$	$T^{2} + 5911200239616$ T^2 + 5911200239616
$29$	$(T + 5791710)^{2}$ (T + 5791710)^2
$31$	$(T - 4145312)^{2}$ (T - 4145312)^2
$37$	$T^{2} + 49163347908964$ T^2 + 49163347908964
$41$	$(T + 8881398)^{2}$ (T + 8881398)^2
$43$	$T^{2} + 247454419107856$ T^2 + 247454419107856
$47$	$T^{2} + 36\!\cdots\!84$ T^2 + 3666553423493184
$53$	$T^{2} + 916476688969956$ T^2 + 916476688969956
$59$	$(T + 45957660)^{2}$ (T + 45957660)^2
$61$	$(T - 37595102)^{2}$ (T - 37595102)^2
$67$	$T^{2} + 38\!\cdots\!44$ T^2 + 38723947378816144
$71$	$(T - 56047992)^{2}$ (T - 56047992)^2
$73$	$T^{2} + 25\!\cdots\!16$ T^2 + 25500274590306916
$79$	$(T + 201923360)^{2}$ (T + 201923360)^2
$83$	$T^{2} + 13\!\cdots\!36$ T^2 + 131736654329237136
$89$	$(T - 272479110)^{2}$ (T - 272479110)^2
$97$	$T^{2} + 36\!\cdots\!84$ T^2 + 361023700318740484
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