LMFDB - Newform orbit 368.2.m.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [368,2,Mod(49,368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(368, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 0, 16]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("368.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 368 = 2^{4} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	368.m (of order $11$, degree $10$, 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:	$2.93849479438$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\Q(\zeta_{22})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 $ x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 23)
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/368\mathbb{Z}\right)^\times$.

$n$	$47$	$97$	$277$
$\chi(n)$	$1$	$\zeta_{22}^{4}$	$1$

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

−0.415415	−	0.909632i
0.142315	+	0.989821i
−0.841254	−	0.540641i
0.654861	−	0.755750i
0.142315	−	0.989821i
0.654861	+	0.755750i
0.959493	+	0.281733i
−0.841254	+	0.540641i
0.959493	−	0.281733i
−0.415415	+	0.909632i

1.04408

−

2.28621i

0.809721

−

0.934468i

1.99611

−

1.28282i

−2.17208

−

2.50672i

81.1

0.226900

−

1.57812i

1.41899

−

0.416652i

0.804632

0.928595i

0.439490

0.129046i

177.1

−0.198939

0.127850i

−1.33083

−

2.91411i

−0.874908

0.256896i

−1.22301

2.67803i

193.1

0.313607

0.361922i

−0.215370

−

1.49793i

1.05773

−

2.31611i

0.394306

−

2.74246i

209.1

0.226900

1.57812i

1.41899

0.416652i

0.804632

−

0.928595i

0.439490

−

0.129046i

225.1

0.313607

−

0.361922i

−0.215370

1.49793i

1.05773

2.31611i

0.394306

2.74246i

257.1

2.11435

−

0.620830i

−2.18251

−

1.40261i

−0.483568

−

3.36329i

1.56130

−

1.00339i

289.1

−0.198939

−

0.127850i

−1.33083

2.91411i

−0.874908

−

0.256896i

−1.22301

−

2.67803i

305.1

2.11435

0.620830i

−2.18251

1.40261i

−0.483568

3.36329i

1.56130

1.00339i

353.1

1.04408

2.28621i

0.809721

0.934468i

1.99611

1.28282i

−2.17208

2.50672i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
23.c	even	11	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	368.2.m.c		10
4.b	odd	2	1		23.2.c.a	✓	10
12.b	even	2	1		207.2.i.c		10
20.d	odd	2	1		575.2.k.b		10
20.e	even	4	2		575.2.p.b		20
23.c	even	11	1	inner	368.2.m.c		10
23.c	even	11	1		8464.2.a.bs		5
23.d	odd	22	1		8464.2.a.bt		5
92.b	even	2	1		529.2.c.a		10
92.g	odd	22	1		23.2.c.a	✓	10
92.g	odd	22	1		529.2.a.i		5
92.g	odd	22	2		529.2.c.b		10
92.g	odd	22	2		529.2.c.d		10
92.g	odd	22	2		529.2.c.g		10
92.g	odd	22	2		529.2.c.i		10
92.h	even	22	1		529.2.a.j		5
92.h	even	22	1		529.2.c.a		10
92.h	even	22	2		529.2.c.c		10
92.h	even	22	2		529.2.c.e		10
92.h	even	22	2		529.2.c.f		10
92.h	even	22	2		529.2.c.h		10
276.j	odd	22	1		4761.2.a.bn		5
276.o	even	22	1		207.2.i.c		10
276.o	even	22	1		4761.2.a.bo		5
460.n	odd	22	1		575.2.k.b		10
460.w	even	44	2		575.2.p.b		20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
23.2.c.a	✓	10	4.b	odd	2	1
23.2.c.a	✓	10	92.g	odd	22	1
207.2.i.c		10	12.b	even	2	1
207.2.i.c		10	276.o	even	22	1
368.2.m.c		10	1.a	even	1	1	trivial
368.2.m.c		10	23.c	even	11	1	inner
529.2.a.i		5	92.g	odd	22	1
529.2.a.j		5	92.h	even	22	1
529.2.c.a		10	92.b	even	2	1
529.2.c.a		10	92.h	even	22	1
529.2.c.b		10	92.g	odd	22	2
529.2.c.c		10	92.h	even	22	2
529.2.c.d		10	92.g	odd	22	2
529.2.c.e		10	92.h	even	22	2
529.2.c.f		10	92.h	even	22	2
529.2.c.g		10	92.g	odd	22	2
529.2.c.h		10	92.h	even	22	2
529.2.c.i		10	92.g	odd	22	2
575.2.k.b		10	20.d	odd	2	1
575.2.k.b		10	460.n	odd	22	1
575.2.p.b		20	20.e	even	4	2
575.2.p.b		20	460.w	even	44	2
4761.2.a.bn		5	276.j	odd	22	1
4761.2.a.bo		5	276.o	even	22	1
8464.2.a.bs		5	23.c	even	11	1
8464.2.a.bt		5	23.d	odd	22	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{10} - 7T_{3}^{9} + 27T_{3}^{8} - 68T_{3}^{7} + 113T_{3}^{6} - 131T_{3}^{5} + 103T_{3}^{4} - 17T_{3}^{3} - 2T_{3}^{2} + 3T_{3} + 1 $ T3^10 - 7*T3^9 + 27*T3^8 - 68*T3^7 + 113*T3^6 - 131*T3^5 + 103*T3^4 - 17*T3^3 - 2*T3^2 + 3*T3 + 1 acting on $S_{2}^{\mathrm{new}}(368, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{10} $ T^10
$3$	$ T^{10} - 7 T^{9} + \cdots + 1 $ T^10 - 7T^9 + 27T^8 - 68T^7 + 113T^6 - 131T^5 + 103T^4 - 17T^3 - 2T^2 + 3*T + 1
$5$	$ T^{10} + 3 T^{9} + \cdots + 529 $ T^10 + 3T^9 + 9T^8 - 6T^7 - 18T^6 - 32T^5 + 124T^4 - 233T^3 + 489T^2 - 667*T + 529
$7$	$ T^{10} - 5 T^{9} + \cdots + 529 $ T^10 - 5T^9 + 25T^8 - 81T^7 + 207T^6 - 298T^5 + 170T^4 + 448T^3 - 425T^2 + 46*T + 529
$11$	$ T^{10} + 7 T^{9} + \cdots + 529 $ T^10 + 7T^9 + 16T^8 - 31T^7 - 74T^6 + 813T^5 + 3282T^4 + 6111T^3 + 11317T^2 + 2208*T + 529
$13$	$ T^{10} + 3 T^{9} + \cdots + 1 $ T^10 + 3T^9 + 9T^8 + 71T^7 + 125T^6 + 166T^5 + 1290T^4 - 684T^3 + 159T^2 - 18*T + 1
$17$	$ T^{10} + 10 T^{9} + \cdots + 529 $ T^10 + 10T^9 + 56T^8 + 65T^7 - 274T^6 - 2003T^5 + 10880T^4 - 20197T^3 + 14521T^2 - 1541*T + 529
$19$	$ T^{10} + 2 T^{9} + \cdots + 541696 $ T^10 + 2T^9 - 40T^8 + 184T^7 + 544T^6 - 8416T^5 + 52864T^4 - 180096T^3 + 450816T^2 - 635904*T + 541696
$23$	$ T^{10} - 12 T^{9} + \cdots + 6436343 $ T^10 - 12T^9 - 10T^8 + 527T^7 - 32T^6 - 14103T^5 - 736T^4 + 278783T^3 - 121670T^2 - 3358092*T + 6436343
$29$	$ T^{10} - 14 T^{9} + \cdots + 4932841 $ T^10 - 14T^9 + 86T^8 + 6T^7 - 2394T^6 + 13100T^5 - 833T^4 - 184633T^3 + 1146117T^2 - 1896734*T + 4932841
$31$	$ T^{10} + 10 T^{9} + \cdots + 17161 $ T^10 + 10T^9 + 56T^8 + 164T^7 + 375T^6 + 505T^5 + 56T^4 - 3862T^3 - 1066T^2 - 2751*T + 17161
$37$	$ T^{10} + 19 T^{9} + \cdots + 529 $ T^10 + 19T^9 + 174T^8 + 1117T^7 + 6956T^6 + 38345T^5 + 148800T^4 + 346183T^3 + 370815T^2 + 4462*T + 529
$41$	$ T^{10} - 7 T^{9} + \cdots + 1849 $ T^10 - 7T^9 + 16T^8 + 9T^7 + 1235T^6 + 4181T^5 + 15173T^4 + 27978T^3 + 23747T^2 + 9331*T + 1849
$43$	$ T^{10} - 11 T^{9} + \cdots + 64009 $ T^10 - 11T^9 + 66T^8 - 341T^7 + 1826T^6 - 6875T^5 + 20449T^4 - 55297T^3 + 106843T^2 - 114103*T + 64009
$47$	$ (T^{5} - 9 T^{4} - 5 T^{3} + \cdots - 1)^{2} $ (T^5 - 9T^4 - 5T^3 + 97T^2 - 106T - 1)^2
$53$	$ T^{10} + \cdots + 517426009 $ T^10 - 29T^9 + 478T^8 - 5579T^7 + 55454T^6 - 448645T^5 + 2974063T^4 - 16104853T^3 + 83193185T^2 - 312339057*T + 517426009
$59$	$ T^{10} - 21 T^{9} + \cdots + 4489 $ T^10 - 21T^9 + 265T^8 - 1649T^7 + 3433T^6 + 22111T^5 - 98174T^4 + 33991T^3 + 1176352T^2 - 20569*T + 4489
$61$	$ T^{10} - 3 T^{9} + \cdots + 529 $ T^10 - 3T^9 + 218T^8 + 281T^7 + 7066T^6 + 46573T^5 + 215097T^4 + 359141T^3 + 236219T^2 - 22103*T + 529
$67$	$ T^{10} + \cdots + 1113757129 $ T^10 + 45T^9 + 1090T^8 + 17513T^7 + 203501T^6 + 1799216T^5 + 12627902T^4 + 70242899T^3 + 291205025T^2 + 798415652*T + 1113757129
$71$	$ T^{10} - 14 T^{9} + \cdots + 529 $ T^10 - 14T^9 + 130T^8 - 566T^7 + 785T^6 + 197T^5 + 1686T^4 - 1560T^3 + 1204T^2 - 851*T + 529
$73$	$ T^{10} - 19 T^{9} + \cdots + 982081 $ T^10 - 19T^9 + 251T^8 - 2415T^7 + 15063T^6 - 73963T^5 + 379118T^4 - 1529101T^3 + 3411556T^2 - 2470563*T + 982081
$79$	$ T^{10} + \cdots + 517426009 $ T^10 - 15T^9 + 115T^8 - 2715T^7 + 26304T^6 - 111816T^5 + 2966242T^4 - 21480574T^3 + 54168013T^2 + 24589507*T + 517426009
$83$	$ T^{10} + 18 T^{9} + \cdots + 529 $ T^10 + 18T^9 + 192T^8 + 321T^7 + 1455T^6 + 2914T^5 + 224T^4 + 17T^3 + 3804T^2 - 2346*T + 529
$89$	$ T^{10} + \cdots + 78310985281 $ T^10 - 25T^9 + 262T^8 - 5615T^7 + 111566T^6 - 1048455T^5 + 11505750T^4 - 150274617T^3 + 1491272689T^2 - 10362512230*T + 78310985281
$97$	$ T^{10} + 34 T^{9} + \cdots + 2374681 $ T^10 + 34T^9 + 430T^8 + 4423T^7 + 52812T^6 + 183261T^5 + 1071940T^4 + 3733313T^3 + 17156261T^2 + 10511161*T + 2374681
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 368 = 2^{4} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	368.m (of order \(11\), degree \(10\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{22}^{9} - \zeta_{22}^{5} + \cdots + 1) q^{3}+ \cdots + ( - \zeta_{22}^{6} - \zeta_{22}^{5} + \cdots + \zeta_{22}) q^{9}+O(q^{10}) \) q + (-z^9 - z^5 + z^4 + 1) * q^3 + (z^9 - z^6 - z^4 + z^3 - 2z^2 + z - 1) q^5 + (z^9 - z^8 - z^6 - z^5 - z^4 - z^2 + z) * q^7 + (-z^6 - z^5 + z^4 - z^3 + z^2 + z) * q^9	\( q + ( - \zeta_{22}^{9} - \zeta_{22}^{5} + \cdots + 1) q^{3}+ \cdots + ( - 5 \zeta_{22}^{8} + \cdots - 5 \zeta_{22}^{2}) q^{99}+O(q^{100}) \) q + (-z^9 - z^5 + z^4 + 1) * q^3 + (z^9 - z^6 - z^4 + z^3 - 2z^2 + z - 1) q^5 + (z^9 - z^8 - z^6 - z^5 - z^4 - z^2 + z) * q^7 + (-z^6 - z^5 + z^4 - z^3 + z^2 + z) * q^9 + (z^9 - 2z^7 + z^5 + z^3 - z^2 + z - 1) q^11 + (-2z^9 + z^8 - 2z^7 + 2z^6 - 2z^5 + z^4 - 2z^3 - z + 1) q^13 + (z^9 + 2z^7 - 2z^6 - z^4 - 2z^2 - 2) q^15 + (-z^6 - z^5 + z^4 + 3z^3 + z^2 - z - 1) q^17 + (-2z^9 + 4z^8 - 2z^7 - 2z^5 + 2z^4 - 4z^3 + 4z^2 - 2z + 2) * q^19 + (-z^8 + z^7 - 3z^6 - z^5 - z^4 - z^3 - 3z^2 + z - 1) * q^21 + (-2z^9 + z^8 + 3z^6 - 3z^5 + z^3 + z^2 + z + 2) q^23 + (z^8 - z^7 + 3z^6 - 2z^5 + z^4 - 2z^3 + 3z^2 - z + 1) * q^25 + (3z^8 - z^5 + z^4 - z + 1) q^27 + (-2z^9 + 2z^8 + 2z^6 + 2z^5 + z^4 - 2z^3 + z^2 + 2z + 2) * q^29 + (-z^9 - z^8 - z^7 + z^6 + z^5 + z^4 - z^2 + z - 1) * q^31 + (z^9 + z^8 - z^7 - z^5 + z^4 + z^3 - 2z + 2) q^33 + (4z^9 - 2z^8 + 4z^7 - 2z^6 + 4z^5 + 3z^3 - 2z^2 + 2z - 3) * q^35 + (z^9 + 4z^7 - 2z^6 + 3z^5 - 3z^4 + 3z^3 - 3z^2 + 2z - 4) q^37 + (-z^9 - z^8 - z^7 - z^6 - z^5 - z^4 - z^3 - z^2 - z) * q^39 + (-z^9 + z^8 - z^7 + z^6 + z^4 + 2z^3 + 2z^2 + 2z + 1) q^41 + (-z^9 - z^7 + 2z^5 - 2z^4 + z^2 + 1) * q^43 + (2z^9 - 2z^8 + 3z^7 - 5z^6 + 5z^5 - 3z^4 + 2z^3 - 2z^2 - 3) * q^45 + (z^9 - 3z^8 + 2z^7 - 3z^6 + 3z^5 - 2z^4 + 3z^3 - z^2) * q^47 + (3z^8 - z^7 + 2z^6 - 3z^5 + 3z^4 - 2z^3 + z^2 - 3z) * q^49 + (-3z^8 + 3z^7 - 2z^4 + 3z^3 + z^2 + 3z - 2) q^51 + (2z^9 - 7z^8 + z^7 - 3z^6 + 3z^5 - 3z^4 + z^3 - 7z^2 + 2z) q^53 + (-3z^7 + 4z^6 - 3z^5 + 6z^4 - 6z^3 + 3z^2 - 4z + 3) q^55 + (-4z^9 + 6z^8 - 8z^7 + 6z^6 - 4z^5 - 4z^3 + 8z^2 - 8z + 4) * q^57 + (-3z^9 + 3z^8 - 3z^7 + 6z^6 - 3z^5 + 3z^4 - 3z^3 - 5z + 5) * q^59 + (3z^9 - 4z^8 - 2z^7 + 2z^6 + 4z^5 - 3z^4 - z^2 + 2z - 1) q^61 + (z^9 - z^8 - 5z^6 + 2z^5 - 2z^4 + z^3 - 2z^2 + 2z - 5) q^63 + (4z^9 + 4z^7 + z^4 - 2z^3 + 2z^2 - z) * q^65 + (-4z^8 - 3z^7 + z^6 - 2z^5 - 3z^4 - 2z^3 + z^2 - 3z - 4) * q^67 + (-2z^9 - 5z^8 + 3z^7 - z^6 + z^5 + z^3 - z^2 + 6z + 1) * q^69 + (2z^8 + z^7 + 2z^6 + 2z^4 + 2z^2 + z + 2) * q^71 + (3z^9 - 4z^8 + 3z^7 - z^5 - z^4 - z^3 + z^2 + z + 1) q^73 + (2z^9 - 2z^8 + 2z^6 + z^5 - z^4 + 2z^3 - z^2 + z + 2) * q^75 + (-3z^8 - z^7 + z^6 + 3z^5 - 2z) q^77 + (6z^9 - 6z^8 + 8z^7 - 2z^6 + 8z^5 - 6z^4 + 6z^3 + 3z - 3) * q^79 + (2z^9 - z^8 + 5z^7 - z^6 + 2z^5 + 7z^3 - 4z^2 + 4z - 7) * q^81 + (-z^9 + 3z^7 - 4z^6 + z^5 - z^2 + 4z - 3) q^83 + (-5z^9 - z^7 + 2z^6 - 3z^5 + 2z^4 - z^3 - 5z) q^85 + (-5z^9 + 7z^8 - 7z^7 + 5z^6 + 7z^4 - 4z^3 + 8z^2 - 4z + 7) * q^87 + (-5z^9 + 4z^8 - 8z^7 - 3z^6 - z^5 + z^4 + 3z^3 + 8z^2 - 4z + 5) q^89 + (3z^9 + z^8 + 4z^7 - z^6 + z^5 - 4z^4 - z^3 - 3z^2 - 1) * q^91 + (-z^9 + z^8 - 2z^7 - z^6 + z^5 + 2z^4 - z^3 + z^2 + 1) * q^93 + (2z^9 + 2z^8 - 4z^6 + 4z^5 - 4z^4 + 4z^3 - 2z - 2) q^95 + (-5z^9 - 2z^8 + 2z^7 + 5z^6 - 2z^4 - 3z^3 + 4z^2 - 3z - 2) * q^97 + (-5z^8 + 4z^7 - 3z^6 + 6z^5 - 3z^4 + 4z^3 - 5z^2) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 7 q^{3} - 3 q^{5} + 5 q^{7} - 2 q^{9}+O(q^{10}) \) 10 * q + 7 * q^3 - 3 * q^5 + 5 * q^7 - 2 * q^9	\( 10 q + 7 q^{3} - 3 q^{5} + 5 q^{7} - 2 q^{9} - 7 q^{11} - 3 q^{13} - 12 q^{15} - 10 q^{17} - 2 q^{19} - 2 q^{21} + 12 q^{23} - 4 q^{25} + 4 q^{27} + 14 q^{29} - 10 q^{31} + 16 q^{33} - 7 q^{35} - 19 q^{37} - q^{39} + 7 q^{41} + 11 q^{43} - 6 q^{45} + 18 q^{47} - 18 q^{49} - 7 q^{51} + 29 q^{53} + q^{55} - 8 q^{57} + 21 q^{59} + 3 q^{61} - 34 q^{63} + 2 q^{65} - 45 q^{67} + 26 q^{69} + 14 q^{71} + 19 q^{73} + 28 q^{75} + 2 q^{77} + 15 q^{79} - 44 q^{81} - 18 q^{83} - 19 q^{85} + 23 q^{87} + 25 q^{89} + 4 q^{91} + 4 q^{93} - 6 q^{95} - 34 q^{97} + 30 q^{99}+O(q^{100}) \) 10 * q + 7 * q^3 - 3 * q^5 + 5 * q^7 - 2 * q^9 - 7 * q^11 - 3 * q^13 - 12 * q^15 - 10 * q^17 - 2 * q^19 - 2 * q^21 + 12 * q^23 - 4 * q^25 + 4 * q^27 + 14 * q^29 - 10 * q^31 + 16 * q^33 - 7 * q^35 - 19 * q^37 - q^39 + 7 * q^41 + 11 * q^43 - 6 * q^45 + 18 * q^47 - 18 * q^49 - 7 * q^51 + 29 * q^53 + q^55 - 8 * q^57 + 21 * q^59 + 3 * q^61 - 34 * q^63 + 2 * q^65 - 45 * q^67 + 26 * q^69 + 14 * q^71 + 19 * q^73 + 28 * q^75 + 2 * q^77 + 15 * q^79 - 44 * q^81 - 18 * q^83 - 19 * q^85 + 23 * q^87 + 25 * q^89 + 4 * q^91 + 4 * q^93 - 6 * q^95 - 34 * q^97 + 30 * q^99

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