LMFDB - Newform orbit 108.4.e.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,4,Mod(37,108)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("108.37");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	108.e (of order $3$, degree $2$, 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:	$6.37220628062$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.6831243.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 13x^{4} + 49x^{2} + 48 $ x^6 + 13x^4 + 49x^2 + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 36)
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 basis $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\beta_{5} + 2 \beta_1 - 2) q^{5} + ( - \beta_{4} - \beta_{2} - 2 \beta_1) q^{7}+O(q^{10}) $ q + (b5 + 2b1 - 2) q^5 + (-b4 - b2 - 2b1) q^7	$ q + (\beta_{5} + 2 \beta_1 - 2) q^{5} + ( - \beta_{4} - \beta_{2} - 2 \beta_1) q^{7} + (2 \beta_{4} - 3 \beta_{2} - 17 \beta_1) q^{11} + (5 \beta_{5} - 4 \beta_{3} - 4 \beta_{2} + \cdots + 4) q^{13}+ \cdots + (70 \beta_{4} - 56 \beta_{2} + 31 \beta_1) q^{97}+O(q^{100}) $ q + (b5 + 2b1 - 2) q^5 + (-b4 - b2 - 2b1) q^7 + (2b4 - 3b2 - 17b1) q^11 + (5b5 - 4b3 - 4b2 - 4b1 + 4) * q^13 + (b5 - b4 + 6b3 + 37) q^17 + (7b5 - 7b4 - 2b3 + 5) q^19 + (5b5 - 3b3 - 3b2 + 70b1 - 70) * q^23 + (3b4 - 6b2 - b1) * q^25 + (5b4 - 152b1) * q^29 + (-15b5 + 3b3 + 3b2 - 16b1 + 16) * q^31 + (b5 - b4 - 15b3 + 184) q^35 + (-10b5 + 10b4 + 8b3 - 16) q^37 + (-14b5 + 12b3 + 12b2 + 299b1 - 299) * q^41 + (27b2 + 43b1) * q^43 + (-39b4 + 21b2 - 174b1) q^47 + (13b5 + 22b3 + 22b2 + 75b1 - 75) * q^49 + (-22b5 + 22b4 + 368) * q^53 + (-33b5 + 33b4 - 15b3 - 36) q^55 + (-34b5 - 15b3 - 15b2 + 151b1 - 151) * q^59 + (-3b4 - 12b2 - 134b1) q^61 + (37b4 + 6b2 - 370b1) q^65 + (-24b5 + 3b3 + 3b2 + 71b1 - 71) * q^67 + (52b5 - 52b4 - 12b3 - 20) q^71 + (21b5 - 21b4 + 30b3 + 125) q^73 + (65b5 + 12b3 + 12b2 + 376b1 - 376) * q^77 + (-23b4 - 5b2 + 184b1) q^79 + (15b4 - 33b2 + 204b1) q^83 + (30b5 - 60b3 - 60b2 - 396b1 + 396) * q^85 + (-44b5 + 44b4 + 60b3 + 154) q^89 + (75b5 - 75b4 - 33b3 - 44) q^91 + (44b5 - 24b3 - 24b2 - 728b1 + 728) * q^95 + (70b4 - 56b2 + 31b1) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{5} - 6 q^{7}+O(q^{10}) $ 6 * q - 6 * q^5 - 6 * q^7	$ 6 q - 6 q^{5} - 6 q^{7} - 51 q^{11} + 12 q^{13} + 222 q^{17} + 30 q^{19} - 210 q^{23} - 3 q^{25} - 456 q^{29} + 48 q^{31} + 1104 q^{35} - 96 q^{37} - 897 q^{41} + 129 q^{43} - 522 q^{47} - 225 q^{49} + 2208 q^{53} - 216 q^{55} - 453 q^{59} - 402 q^{61} - 1110 q^{65} - 213 q^{67} - 120 q^{71} + 750 q^{73} - 1128 q^{77} + 552 q^{79} + 612 q^{83} + 1188 q^{85} + 924 q^{89} - 264 q^{91} + 2184 q^{95} + 93 q^{97}+O(q^{100}) $ 6 * q - 6 * q^5 - 6 * q^7 - 51 * q^11 + 12 * q^13 + 222 * q^17 + 30 * q^19 - 210 * q^23 - 3 * q^25 - 456 * q^29 + 48 * q^31 + 1104 * q^35 - 96 * q^37 - 897 * q^41 + 129 * q^43 - 522 * q^47 - 225 * q^49 + 2208 * q^53 - 216 * q^55 - 453 * q^59 - 402 * q^61 - 1110 * q^65 - 213 * q^67 - 120 * q^71 + 750 * q^73 - 1128 * q^77 + 552 * q^79 + 612 * q^83 + 1188 * q^85 + 924 * q^89 - 264 * q^91 + 2184 * q^95 + 93 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} + 13x^{4} + 49x^{2} + 48 $ x^6 + 13*x^4 + 49*x^2 + 48 :

$\beta_{1}$	$=$	$ ( \nu^{5} + 9\nu^{3} + 17\nu + 4 ) / 8 $ (v^5 + 9v^3 + 17v + 4) / 8
$\beta_{2}$	$=$	$ ( \nu^{5} + 21\nu^{3} + 12\nu^{2} + 77\nu + 52 ) / 4 $ (v^5 + 21v^3 + 12v^2 + 77*v + 52) / 4
$\beta_{3}$	$=$	$ -6\nu^{2} - 26 $ -6*v^2 - 26
$\beta_{4}$	$=$	$ ( -\nu^{5} + 6\nu^{4} - 3\nu^{3} + 54\nu^{2} + 31\nu + 92 ) / 4 $ (-v^5 + 6v^4 - 3v^3 + 54v^2 + 31v + 92) / 4
$\beta_{5}$	$=$	$ ( -\nu^{5} - 6\nu^{4} - 3\nu^{3} - 54\nu^{2} + 31\nu - 92 ) / 4 $ (-v^5 - 6v^4 - 3v^3 - 54v^2 + 31v - 92) / 4

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( 2\beta_{5} + 2\beta_{4} - \beta_{3} - 2\beta_{2} + 12\beta _1 - 6 ) / 18 $ (2b5 + 2b4 - b3 - 2b2 + 12b1 - 6) / 18
$\nu^{2}$	$=$	$ ( -\beta_{3} - 26 ) / 6 $ (-b3 - 26) / 6
$\nu^{3}$	$=$	$ ( -5\beta_{5} - 5\beta_{4} + 4\beta_{3} + 8\beta_{2} - 36\beta _1 + 18 ) / 9 $ (-5b5 - 5b4 + 4b3 + 8b2 - 36*b1 + 18) / 9
$\nu^{4}$	$=$	$ ( -2\beta_{5} + 2\beta_{4} + 9\beta_{3} + 142 ) / 6 $ (-2b5 + 2b4 + 9*b3 + 142) / 6
$\nu^{5}$	$=$	$ ( 56\beta_{5} + 56\beta_{4} - 55\beta_{3} - 110\beta_{2} + 588\beta _1 - 294 ) / 18 $ (56b5 + 56b4 - 55b3 - 110b2 + 588*b1 - 294) / 18

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$-1 + \beta_{1}$	$1$

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} $

37.1

		1.23396i
		2.63162i
	−	2.13353i
	−	1.23396i
	−	2.63162i
		2.13353i

−6.92194

11.9892i

−15.3540

−

26.5939i

37.2

−2.44901

4.24182i

5.32725

9.22708i

37.3

6.37096

−

11.0348i

7.02674

12.1707i

73.1

−6.92194

−

11.9892i

−15.3540

26.5939i

73.2

−2.44901

−

4.24182i

5.32725

−

9.22708i

73.3

6.37096

11.0348i

7.02674

−

12.1707i

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	108.4.e.a		6
3.b	odd	2	1		36.4.e.a	✓	6
4.b	odd	2	1		432.4.i.d		6
9.c	even	3	1	inner	108.4.e.a		6
9.c	even	3	1		324.4.a.d		3
9.d	odd	6	1		36.4.e.a	✓	6
9.d	odd	6	1		324.4.a.c		3
12.b	even	2	1		144.4.i.d		6
36.f	odd	6	1		432.4.i.d		6
36.f	odd	6	1		1296.4.a.w		3
36.h	even	6	1		144.4.i.d		6
36.h	even	6	1		1296.4.a.v		3

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
36.4.e.a	✓	6	3.b	odd	2	1
36.4.e.a	✓	6	9.d	odd	6	1
108.4.e.a		6	1.a	even	1	1	trivial
108.4.e.a		6	9.c	even	3	1	inner
144.4.i.d		6	12.b	even	2	1
144.4.i.d		6	36.h	even	6	1
324.4.a.c		3	9.d	odd	6	1
324.4.a.d		3	9.c	even	3	1
432.4.i.d		6	4.b	odd	2	1
432.4.i.d		6	36.f	odd	6	1
1296.4.a.v		3	36.h	even	6	1
1296.4.a.w		3	36.f	odd	6	1

Hecke kernels

This newform subspace is the entire newspace $S_{4}^{\mathrm{new}}(108, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} $ T^6
$5$	$ T^{6} + 6 T^{5} + \cdots + 746496 $ T^6 + 6T^5 + 207T^4 + 702T^3 + 34425T^2 + 147744*T + 746496
$7$	$ T^{6} + 6 T^{5} + \cdots + 21141604 $ T^6 + 6T^5 + 645T^4 - 12850T^3 + 343293T^2 - 2800182*T + 21141604
$11$	$ T^{6} + \cdots + 4386545361 $ T^6 + 51T^5 + 3798T^4 + 71415T^3 + 4810590T^2 + 79278507*T + 4386545361
$13$	$ T^{6} + \cdots + 4155865156 $ T^6 - 12T^5 + 5415T^4 - 65680T^3 + 28557033T^2 - 339800286*T + 4155865156
$17$	$ (T^{3} - 111 T^{2} + \cdots + 577476)^{2} $ (T^3 - 111T^2 - 6624T + 577476)^2
$19$	$ (T^{3} - 15 T^{2} + \cdots - 216368)^{2} $ (T^3 - 15T^2 - 7512T - 216368)^2
$23$	$ T^{6} + 210 T^{5} + \cdots + 5391684 $ T^6 + 210T^5 + 33741T^4 + 2170746T^3 + 106821261T^2 + 24053598*T + 5391684
$29$	$ T^{6} + \cdots + 8292430196964 $ T^6 + 456T^5 + 143199T^4 + 23760756T^3 + 2877755121T^2 + 186420419946*T + 8292430196964
$31$	$ T^{6} + \cdots + 9331729724944 $ T^6 - 48T^5 + 37257T^4 - 4431832T^3 + 1368342033T^2 - 106774004964*T + 9331729724944
$37$	$ (T^{3} + 48 T^{2} + \cdots - 682352)^{2} $ (T^3 + 48T^2 - 20508T - 682352)^2
$41$	$ T^{6} + \cdots + 139158426750849 $ T^6 + 897T^5 + 581058T^4 + 176932161T^3 + 39393550530T^2 + 2637128984193*T + 139158426750849
$43$	$ T^{6} + \cdots + 2031049672201 $ T^6 - 129T^5 + 203550T^4 + 21260963T^3 + 35118818502T^2 - 266373174441*T + 2031049672201
$47$	$ T^{6} + \cdots + 41\!\cdots\!24 $ T^6 + 522T^5 + 433917T^4 + 44849538T^3 + 59760297693T^2 + 10421917854606*T + 4167836333323524
$53$	$ (T^{3} - 1104 T^{2} + \cdots - 11853648)^{2} $ (T^3 - 1104T^2 + 317700T - 11853648)^2
$59$	$ T^{6} + \cdots + 93\!\cdots\!69 $ T^6 + 453T^5 + 496494T^4 + 61833321T^3 + 128739350214T^2 + 28223393906205*T + 9388197832500369
$61$	$ T^{6} + \cdots + 1463461189696 $ T^6 + 402T^5 + 153663T^4 + 5611754T^3 + 549373353T^2 - 9606513576*T + 1463461189696
$67$	$ T^{6} + \cdots + 9579414973969 $ T^6 + 213T^5 + 125502T^4 - 23258455T^3 + 5762049270T^2 - 248016683379*T + 9579414973969
$71$	$ (T^{3} + 60 T^{2} + \cdots - 113211648)^{2} $ (T^3 + 60T^2 - 423072T - 113211648)^2
$73$	$ (T^{3} - 375 T^{2} + \cdots + 158369284)^{2} $ (T^3 - 375T^2 - 381048T + 158369284)^2
$79$	$ T^{6} + \cdots + 318578661907984 $ T^6 - 552T^5 + 326553T^4 - 23636896T^3 + 10329900945T^2 - 389977819428*T + 318578661907984
$83$	$ T^{6} + \cdots + 12101965948944 $ T^6 - 612T^5 + 492237T^4 + 65070540T^3 + 15980660505T^2 - 409428996084*T + 12101965948944
$89$	$ (T^{3} - 462 T^{2} + \cdots - 170122248)^{2} $ (T^3 - 462T^2 - 774180T - 170122248)^2
$97$	$ T^{6} + \cdots + 74\!\cdots\!29 $ T^6 - 93T^5 + 1048290T^4 + 269487659T^3 + 1072818160242T^2 + 89825526132243*T + 7465050374673529
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	108.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{5} + 2 \beta_1 - 2) q^{5} + ( - \beta_{4} - \beta_{2} - 2 \beta_1) q^{7}+O(q^{10}) \) q + (b5 + 2b1 - 2) q^5 + (-b4 - b2 - 2b1) q^7	\( q + (\beta_{5} + 2 \beta_1 - 2) q^{5} + ( - \beta_{4} - \beta_{2} - 2 \beta_1) q^{7} + (2 \beta_{4} - 3 \beta_{2} - 17 \beta_1) q^{11} + (5 \beta_{5} - 4 \beta_{3} - 4 \beta_{2} + \cdots + 4) q^{13}+ \cdots + (70 \beta_{4} - 56 \beta_{2} + 31 \beta_1) q^{97}+O(q^{100}) \) q + (b5 + 2b1 - 2) q^5 + (-b4 - b2 - 2b1) q^7 + (2b4 - 3b2 - 17b1) q^11 + (5b5 - 4b3 - 4b2 - 4b1 + 4) * q^13 + (b5 - b4 + 6b3 + 37) q^17 + (7b5 - 7b4 - 2b3 + 5) q^19 + (5b5 - 3b3 - 3b2 + 70b1 - 70) * q^23 + (3b4 - 6b2 - b1) * q^25 + (5b4 - 152b1) * q^29 + (-15b5 + 3b3 + 3b2 - 16b1 + 16) * q^31 + (b5 - b4 - 15b3 + 184) q^35 + (-10b5 + 10b4 + 8b3 - 16) q^37 + (-14b5 + 12b3 + 12b2 + 299b1 - 299) * q^41 + (27b2 + 43b1) * q^43 + (-39b4 + 21b2 - 174b1) q^47 + (13b5 + 22b3 + 22b2 + 75b1 - 75) * q^49 + (-22b5 + 22b4 + 368) * q^53 + (-33b5 + 33b4 - 15b3 - 36) q^55 + (-34b5 - 15b3 - 15b2 + 151b1 - 151) * q^59 + (-3b4 - 12b2 - 134b1) q^61 + (37b4 + 6b2 - 370b1) q^65 + (-24b5 + 3b3 + 3b2 + 71b1 - 71) * q^67 + (52b5 - 52b4 - 12b3 - 20) q^71 + (21b5 - 21b4 + 30b3 + 125) q^73 + (65b5 + 12b3 + 12b2 + 376b1 - 376) * q^77 + (-23b4 - 5b2 + 184b1) q^79 + (15b4 - 33b2 + 204b1) q^83 + (30b5 - 60b3 - 60b2 - 396b1 + 396) * q^85 + (-44b5 + 44b4 + 60b3 + 154) q^89 + (75b5 - 75b4 - 33b3 - 44) q^91 + (44b5 - 24b3 - 24b2 - 728b1 + 728) * q^95 + (70b4 - 56b2 + 31b1) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{5} - 6 q^{7}+O(q^{10}) \) 6 * q - 6 * q^5 - 6 * q^7	\( 6 q - 6 q^{5} - 6 q^{7} - 51 q^{11} + 12 q^{13} + 222 q^{17} + 30 q^{19} - 210 q^{23} - 3 q^{25} - 456 q^{29} + 48 q^{31} + 1104 q^{35} - 96 q^{37} - 897 q^{41} + 129 q^{43} - 522 q^{47} - 225 q^{49} + 2208 q^{53} - 216 q^{55} - 453 q^{59} - 402 q^{61} - 1110 q^{65} - 213 q^{67} - 120 q^{71} + 750 q^{73} - 1128 q^{77} + 552 q^{79} + 612 q^{83} + 1188 q^{85} + 924 q^{89} - 264 q^{91} + 2184 q^{95} + 93 q^{97}+O(q^{100}) \) 6 * q - 6 * q^5 - 6 * q^7 - 51 * q^11 + 12 * q^13 + 222 * q^17 + 30 * q^19 - 210 * q^23 - 3 * q^25 - 456 * q^29 + 48 * q^31 + 1104 * q^35 - 96 * q^37 - 897 * q^41 + 129 * q^43 - 522 * q^47 - 225 * q^49 + 2208 * q^53 - 216 * q^55 - 453 * q^59 - 402 * q^61 - 1110 * q^65 - 213 * q^67 - 120 * q^71 + 750 * q^73 - 1128 * q^77 + 552 * q^79 + 612 * q^83 + 1188 * q^85 + 924 * q^89 - 264 * q^91 + 2184 * q^95 + 93 * q^97

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