LMFDB - Newform orbit 162.12.c.g

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [162,12,Mod(55,162)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("162.55"); S:= CuspForms(chi, 12); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 12, names="a")

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	162.c (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,32,0,-1024,-5766] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$124.471595251$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-3}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 6)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity $\zeta_{6}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - 32 \zeta_{6} + 32) q^{2} - 1024 \zeta_{6} q^{4} - 5766 \zeta_{6} q^{5} + (72464 \zeta_{6} - 72464) q^{7} - 32768 q^{8} - 184512 q^{10} + ( - 408948 \zeta_{6} + 408948) q^{11} - 1367558 \zeta_{6} q^{13} + \cdots - 104758545696 q^{98} +O(q^{100}) $ q + (-32z + 32) q^2 - 1024z q^4 - 5766z q^5 + (72464z - 72464) q^7 - 32768 * q^8 - 184512 * q^10 + (-408948z + 408948) q^11 - 1367558z q^13 + 2318848z q^14 + (1048576z - 1048576) q^16 + 5422914 * q^17 + 15166100 * q^19 + (5904384z - 5904384) q^20 - 13086336z q^22 + 52194072z q^23 + (-15581369z + 15581369) q^25 - 43761856 * q^26 + 74203136 * q^28 + (118581150z - 118581150) q^29 + 57652408z q^31 + 33554432z q^32 + (-173533248z + 173533248) q^34 + 417827424 * q^35 - 375985186 * q^37 + (-485315200z + 485315200) q^38 + 188940288z q^40 - 856316202z q^41 + (-1245189172z + 1245189172) q^43 - 418762752 * q^44 + 1670210304 * q^46 + (-1306762656z + 1306762656) q^47 - 3273704553z q^49 - 498603808z q^50 + (1400379392z - 1400379392) q^52 + 409556358 * q^53 - 2357994168 * q^55 + (-2374500352z + 2374500352) q^56 + 3794596800z q^58 + 2882866260z q^59 + (5731767302z - 5731767302) q^61 + 1844877056 * q^62 + 1073741824 * q^64 + (7885339428z - 7885339428) q^65 - 3893272244z q^67 - 5553063936z q^68 + (-13370477568z + 13370477568) q^70 - 9075890088 * q^71 - 15571822822 * q^73 + (12031525952z - 12031525952) q^74 - 15530086400z q^76 + 29634007872z q^77 + (-30196762600z + 30196762600) q^79 + 6046089216 * q^80 - 27402118464 * q^82 + (23135252628z - 23135252628) q^83 - 31268522124z q^85 - 39846053504z q^86 + (13400408064z - 13400408064) q^88 - 25614819990 * q^89 + 99098722912 * q^91 + (-53446729728z + 53446729728) q^92 - 41816404992z q^94 - 87447732600z q^95 + (-61937553406z + 61937553406) q^97 - 104758545696 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 32 q^{2} - 1024 q^{4} - 5766 q^{5} - 72464 q^{7} - 65536 q^{8} - 369024 q^{10} + 408948 q^{11} - 1367558 q^{13} + 2318848 q^{14} - 1048576 q^{16} + 10845828 q^{17} + 30332200 q^{19} - 5904384 q^{20}+ \cdots - 209517091392 q^{98}+O(q^{100}) $ 2 * q + 32 * q^2 - 1024 * q^4 - 5766 * q^5 - 72464 * q^7 - 65536 * q^8 - 369024 * q^10 + 408948 * q^11 - 1367558 * q^13 + 2318848 * q^14 - 1048576 * q^16 + 10845828 * q^17 + 30332200 * q^19 - 5904384 * q^20 - 13086336 * q^22 + 52194072 * q^23 + 15581369 * q^25 - 87523712 * q^26 + 148406272 * q^28 - 118581150 * q^29 + 57652408 * q^31 + 33554432 * q^32 + 173533248 * q^34 + 835654848 * q^35 - 751970372 * q^37 + 485315200 * q^38 + 188940288 * q^40 - 856316202 * q^41 + 1245189172 * q^43 - 837525504 * q^44 + 3340420608 * q^46 + 1306762656 * q^47 - 3273704553 * q^49 - 498603808 * q^50 - 1400379392 * q^52 + 819112716 * q^53 - 4715988336 * q^55 + 2374500352 * q^56 + 3794596800 * q^58 + 2882866260 * q^59 - 5731767302 * q^61 + 3689754112 * q^62 + 2147483648 * q^64 - 7885339428 * q^65 - 3893272244 * q^67 - 5553063936 * q^68 + 13370477568 * q^70 - 18151780176 * q^71 - 31143645644 * q^73 - 12031525952 * q^74 - 15530086400 * q^76 + 29634007872 * q^77 + 30196762600 * q^79 + 12092178432 * q^80 - 54804236928 * q^82 - 23135252628 * q^83 - 31268522124 * q^85 - 39846053504 * q^86 - 13400408064 * q^88 - 51229639980 * q^89 + 198197445824 * q^91 + 53446729728 * q^92 - 41816404992 * q^94 - 87447732600 * q^95 + 61937553406 * q^97 - 209517091392 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$.

$n$	$83$
$\chi(n)$	$-\zeta_{6}$

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} $

55.1

0.500000	+	0.866025i
0.500000	−	0.866025i

16.0000

−

27.7128i

−512.000

−

886.810i

−2883.00

−

4993.50i

−36232.0

62755.7i

−32768.0

−184512.

109.1

16.0000

27.7128i

−512.000

886.810i

−2883.00

4993.50i

−36232.0

−

62755.7i

−32768.0

−184512.

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	162.12.c.g		2
3.b	odd	2	1		162.12.c.d		2
9.c	even	3	1		6.12.a.a	✓	1
9.c	even	3	1	inner	162.12.c.g		2
9.d	odd	6	1		18.12.a.c		1
9.d	odd	6	1		162.12.c.d		2
36.f	odd	6	1		48.12.a.h		1
36.h	even	6	1		144.12.a.b		1
45.j	even	6	1		150.12.a.g		1
45.k	odd	12	2		150.12.c.f		2
72.n	even	6	1		192.12.a.l		1
72.p	odd	6	1		192.12.a.b		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
6.12.a.a	✓	1	9.c	even	3	1
18.12.a.c		1	9.d	odd	6	1
48.12.a.h		1	36.f	odd	6	1
144.12.a.b		1	36.h	even	6	1
150.12.a.g		1	45.j	even	6	1
150.12.c.f		2	45.k	odd	12	2
162.12.c.d		2	3.b	odd	2	1
162.12.c.d		2	9.d	odd	6	1
162.12.c.g		2	1.a	even	1	1	trivial
162.12.c.g		2	9.c	even	3	1	inner
192.12.a.b		1	72.p	odd	6	1
192.12.a.l		1	72.n	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{2} + 5766T_{5} + 33246756 $ T5^2 + 5766*T5 + 33246756 acting on $S_{12}^{\mathrm{new}}(162, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} - 32T + 1024 $ T^2 - 32*T + 1024
$3$	$ T^{2} $ T^2
$5$	$ T^{2} + 5766 T + 33246756 $ T^2 + 5766*T + 33246756
$7$	$ T^{2} + \cdots + 5251031296 $ T^2 + 72464*T + 5251031296
$11$	$ T^{2} + \cdots + 167238466704 $ T^2 - 408948*T + 167238466704
$13$	$ T^{2} + \cdots + 1870214883364 $ T^2 + 1367558*T + 1870214883364
$17$	$ (T - 5422914)^{2} $ (T - 5422914)^2
$19$	$ (T - 15166100)^{2} $ (T - 15166100)^2
$23$	$ T^{2} + \cdots + 27\!\cdots\!84 $ T^2 - 52194072*T + 2724221151941184
$29$	$ T^{2} + \cdots + 14\!\cdots\!00 $ T^2 + 118581150*T + 14061489135322500
$31$	$ T^{2} + \cdots + 33\!\cdots\!64 $ T^2 - 57652408*T + 3323800148198464
$37$	$ (T + 375985186)^{2} $ (T + 375985186)^2
$41$	$ T^{2} + \cdots + 73\!\cdots\!04 $ T^2 + 856316202*T + 733277437807704804
$43$	$ T^{2} + \cdots + 15\!\cdots\!84 $ T^2 - 1245189172*T + 1550496074066045584
$47$	$ T^{2} + \cdots + 17\!\cdots\!36 $ T^2 - 1306762656*T + 1707628639116174336
$53$	$ (T - 409556358)^{2} $ (T - 409556358)^2
$59$	$ T^{2} + \cdots + 83\!\cdots\!00 $ T^2 - 2882866260*T + 8310917873046387600
$61$	$ T^{2} + \cdots + 32\!\cdots\!04 $ T^2 + 5731767302*T + 32853156404276359204
$67$	$ T^{2} + \cdots + 15\!\cdots\!36 $ T^2 + 3893272244*T + 15157568765900795536
$71$	$ (T + 9075890088)^{2} $ (T + 9075890088)^2
$73$	$ (T + 15571822822)^{2} $ (T + 15571822822)^2
$79$	$ T^{2} + \cdots + 91\!\cdots\!00 $ T^2 - 30196762600*T + 911844471520758760000
$83$	$ T^{2} + \cdots + 53\!\cdots\!84 $ T^2 + 23135252628*T + 535239914161380906384
$89$	$ (T + 25614819990)^{2} $ (T + 25614819990)^2
$97$	$ T^{2} + \cdots + 38\!\cdots\!36 $ T^2 - 61937553406*T + 3836260521921102200836
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Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	162.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - 32 \zeta_{6} + 32) q^{2} - 1024 \zeta_{6} q^{4} - 5766 \zeta_{6} q^{5} + (72464 \zeta_{6} - 72464) q^{7} - 32768 q^{8} - 184512 q^{10} + ( - 408948 \zeta_{6} + 408948) q^{11} - 1367558 \zeta_{6} q^{13} + \cdots - 104758545696 q^{98} +O(q^{100}) \) q + (-32z + 32) q^2 - 1024z q^4 - 5766z q^5 + (72464z - 72464) q^7 - 32768 * q^8 - 184512 * q^10 + (-408948z + 408948) q^11 - 1367558z q^13 + 2318848z q^14 + (1048576z - 1048576) q^16 + 5422914 * q^17 + 15166100 * q^19 + (5904384z - 5904384) q^20 - 13086336z q^22 + 52194072z q^23 + (-15581369z + 15581369) q^25 - 43761856 * q^26 + 74203136 * q^28 + (118581150z - 118581150) q^29 + 57652408z q^31 + 33554432z q^32 + (-173533248z + 173533248) q^34 + 417827424 * q^35 - 375985186 * q^37 + (-485315200z + 485315200) q^38 + 188940288z q^40 - 856316202z q^41 + (-1245189172z + 1245189172) q^43 - 418762752 * q^44 + 1670210304 * q^46 + (-1306762656z + 1306762656) q^47 - 3273704553z q^49 - 498603808z q^50 + (1400379392z - 1400379392) q^52 + 409556358 * q^53 - 2357994168 * q^55 + (-2374500352z + 2374500352) q^56 + 3794596800z q^58 + 2882866260z q^59 + (5731767302z - 5731767302) q^61 + 1844877056 * q^62 + 1073741824 * q^64 + (7885339428z - 7885339428) q^65 - 3893272244z q^67 - 5553063936z q^68 + (-13370477568z + 13370477568) q^70 - 9075890088 * q^71 - 15571822822 * q^73 + (12031525952z - 12031525952) q^74 - 15530086400z q^76 + 29634007872z q^77 + (-30196762600z + 30196762600) q^79 + 6046089216 * q^80 - 27402118464 * q^82 + (23135252628z - 23135252628) q^83 - 31268522124z q^85 - 39846053504z q^86 + (13400408064z - 13400408064) q^88 - 25614819990 * q^89 + 99098722912 * q^91 + (-53446729728z + 53446729728) q^92 - 41816404992z q^94 - 87447732600z q^95 + (-61937553406z + 61937553406) q^97 - 104758545696 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 32 q^{2} - 1024 q^{4} - 5766 q^{5} - 72464 q^{7} - 65536 q^{8} - 369024 q^{10} + 408948 q^{11} - 1367558 q^{13} + 2318848 q^{14} - 1048576 q^{16} + 10845828 q^{17} + 30332200 q^{19} - 5904384 q^{20}+ \cdots - 209517091392 q^{98}+O(q^{100}) \) 2 * q + 32 * q^2 - 1024 * q^4 - 5766 * q^5 - 72464 * q^7 - 65536 * q^8 - 369024 * q^10 + 408948 * q^11 - 1367558 * q^13 + 2318848 * q^14 - 1048576 * q^16 + 10845828 * q^17 + 30332200 * q^19 - 5904384 * q^20 - 13086336 * q^22 + 52194072 * q^23 + 15581369 * q^25 - 87523712 * q^26 + 148406272 * q^28 - 118581150 * q^29 + 57652408 * q^31 + 33554432 * q^32 + 173533248 * q^34 + 835654848 * q^35 - 751970372 * q^37 + 485315200 * q^38 + 188940288 * q^40 - 856316202 * q^41 + 1245189172 * q^43 - 837525504 * q^44 + 3340420608 * q^46 + 1306762656 * q^47 - 3273704553 * q^49 - 498603808 * q^50 - 1400379392 * q^52 + 819112716 * q^53 - 4715988336 * q^55 + 2374500352 * q^56 + 3794596800 * q^58 + 2882866260 * q^59 - 5731767302 * q^61 + 3689754112 * q^62 + 2147483648 * q^64 - 7885339428 * q^65 - 3893272244 * q^67 - 5553063936 * q^68 + 13370477568 * q^70 - 18151780176 * q^71 - 31143645644 * q^73 - 12031525952 * q^74 - 15530086400 * q^76 + 29634007872 * q^77 + 30196762600 * q^79 + 12092178432 * q^80 - 54804236928 * q^82 - 23135252628 * q^83 - 31268522124 * q^85 - 39846053504 * q^86 - 13400408064 * q^88 - 51229639980 * q^89 + 198197445824 * q^91 + 53446729728 * q^92 - 41816404992 * q^94 - 87447732600 * q^95 + 61937553406 * q^97 - 209517091392 * q^98

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