LMFDB - Newform orbit 841.2.e.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [841,2,Mod(63,841)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(841, base_ring=CyclotomicField(14))

chi = DirichletCharacter(H, H._module([11]))

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

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

chi := DirichletCharacter("841.63");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 841 = 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	841.e (of order $14$, degree $6$, 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:	$6.71541880999$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{14})$
Coefficient field:	$\Q(\zeta_{28})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 29)
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/841\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$\zeta_{28}^{2}$

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} $

63.1

0.781831	−	0.623490i
−0.781831	+	0.623490i
0.974928	−	0.222521i
−0.974928	+	0.222521i
0.974928	+	0.222521i
−0.974928	−	0.222521i
0.781831	+	0.623490i
−0.781831	−	0.623490i
−0.433884	−	0.900969i
0.433884	+	0.900969i
−0.433884	+	0.900969i
0.433884	−	0.900969i

−0.347948

0.277479i

1.21572

−

0.277479i

−0.400969

1.75676i

−0.431468

−

0.541044i

−0.346011

0.433884i

−0.0794168

−

0.347948i

−0.734141

−

1.52446i

−1.30194

0.626980i

0.300257

0.0685317i

63.2

0.347948

−

0.277479i

−1.21572

0.277479i

−0.400969

1.75676i

−0.431468

−

0.541044i

−0.346011

0.433884i

−0.0794168

−

0.347948i

0.734141

1.52446i

−1.30194

0.626980i

−0.300257

−

0.0685317i

196.1

−1.75676

0.400969i

0.193096

−

0.400969i

1.12349

−

0.541044i

0.0794168

0.347948i

−0.178448

0.781831i

3.64795

1.75676i

1.06086

−

0.846011i

1.74698

2.19064i

−0.279032

−

0.579417i

196.2

1.75676

−

0.400969i

−0.193096

0.400969i

1.12349

−

0.541044i

0.0794168

0.347948i

−0.178448

0.781831i

3.64795

1.75676i

−1.06086

0.846011i

1.74698

2.19064i

0.279032

0.579417i

236.1

−1.75676

−

0.400969i

0.193096

0.400969i

1.12349

0.541044i

0.0794168

−

0.347948i

−0.178448

−

0.781831i

3.64795

−

1.75676i

1.06086

0.846011i

1.74698

−

2.19064i

−0.279032

0.579417i

236.2

1.75676

0.400969i

−0.193096

−

0.400969i

1.12349

0.541044i

0.0794168

−

0.347948i

−0.178448

−

0.781831i

3.64795

−

1.75676i

−1.06086

−

0.846011i

1.74698

−

2.19064i

0.279032

−

0.579417i

267.1

−0.347948

−

0.277479i

1.21572

0.277479i

−0.400969

−

1.75676i

−0.431468

0.541044i

−0.346011

−

0.433884i

−0.0794168

0.347948i

−0.734141

1.52446i

−1.30194

−

0.626980i

0.300257

−

0.0685317i

267.2

0.347948

0.277479i

−1.21572

−

0.277479i

−0.400969

−

1.75676i

−0.431468

0.541044i

−0.346011

−

0.433884i

−0.0794168

0.347948i

0.734141

−

1.52446i

−1.30194

−

0.626980i

−0.300257

0.0685317i

270.1

−0.541044

−

1.12349i

−1.40881

1.12349i

0.277479

−

0.347948i

−3.64795

1.75676i

2.02446

0.974928i

0.431468

0.541044i

−2.97247

−

0.678448i

0.0549581

−

0.240787i

3.94740

3.14795i

270.2

0.541044

1.12349i

1.40881

−

1.12349i

0.277479

−

0.347948i

−3.64795

1.75676i

2.02446

0.974928i

0.431468

0.541044i

2.97247

0.678448i

0.0549581

−

0.240787i

−3.94740

−

3.14795i

651.1

−0.541044

1.12349i

−1.40881

−

1.12349i

0.277479

0.347948i

−3.64795

−

1.75676i

2.02446

−

0.974928i

0.431468

−

0.541044i

−2.97247

0.678448i

0.0549581

0.240787i

3.94740

−

3.14795i

651.2

0.541044

−

1.12349i

1.40881

1.12349i

0.277479

0.347948i

−3.64795

−

1.75676i

2.02446

−

0.974928i

0.431468

−

0.541044i

2.97247

−

0.678448i

0.0549581

0.240787i

−3.94740

3.14795i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
29.b	even	2	1	inner
29.d	even	7	1	inner
29.e	even	14	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	841.2.e.c		12
29.b	even	2	1	inner	841.2.e.c		12
29.c	odd	4	1		841.2.d.b		6
29.c	odd	4	1		841.2.d.c		6
29.d	even	7	1		841.2.b.c		6
29.d	even	7	2		841.2.e.b		12
29.d	even	7	1	inner	841.2.e.c		12
29.d	even	7	2		841.2.e.d		12
29.e	even	14	1		841.2.b.c		6
29.e	even	14	2		841.2.e.b		12
29.e	even	14	1	inner	841.2.e.c		12
29.e	even	14	2		841.2.e.d		12
29.f	odd	28	2		29.2.d.a	✓	6
29.f	odd	28	1		841.2.a.e		3
29.f	odd	28	1		841.2.a.f		3
29.f	odd	28	2		841.2.d.a		6
29.f	odd	28	1		841.2.d.b		6
29.f	odd	28	1		841.2.d.c		6
29.f	odd	28	2		841.2.d.d		6
29.f	odd	28	2		841.2.d.e		6
87.k	even	28	2		261.2.k.a		6
87.k	even	28	1		7569.2.a.p		3
87.k	even	28	1		7569.2.a.r		3
116.l	even	28	2		464.2.u.f		6
145.o	even	28	2		725.2.r.b		12
145.s	odd	28	2		725.2.l.b		6
145.t	even	28	2		725.2.r.b		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
29.2.d.a	✓	6	29.f	odd	28	2
261.2.k.a		6	87.k	even	28	2
464.2.u.f		6	116.l	even	28	2
725.2.l.b		6	145.s	odd	28	2
725.2.r.b		12	145.o	even	28	2
725.2.r.b		12	145.t	even	28	2
841.2.a.e		3	29.f	odd	28	1
841.2.a.f		3	29.f	odd	28	1
841.2.b.c		6	29.d	even	7	1
841.2.b.c		6	29.e	even	14	1
841.2.d.a		6	29.f	odd	28	2
841.2.d.b		6	29.c	odd	4	1
841.2.d.b		6	29.f	odd	28	1
841.2.d.c		6	29.c	odd	4	1
841.2.d.c		6	29.f	odd	28	1
841.2.d.d		6	29.f	odd	28	2
841.2.d.e		6	29.f	odd	28	2
841.2.e.b		12	29.d	even	7	2
841.2.e.b		12	29.e	even	14	2
841.2.e.c		12	1.a	even	1	1	trivial
841.2.e.c		12	29.b	even	2	1	inner
841.2.e.c		12	29.d	even	7	1	inner
841.2.e.c		12	29.e	even	14	1	inner
841.2.e.d		12	29.d	even	7	2
841.2.e.d		12	29.e	even	14	2
7569.2.a.p		3	87.k	even	28	1
7569.2.a.r		3	87.k	even	28	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{12} - 4T_{2}^{10} + 2T_{2}^{8} + 6T_{2}^{6} + 25T_{2}^{4} - 2T_{2}^{2} + 1 $ T2^12 - 4*T2^10 + 2*T2^8 + 6*T2^6 + 25*T2^4 - 2*T2^2 + 1 acting on $S_{2}^{\mathrm{new}}(841, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} - 4 T^{10} + \cdots + 1 $ T^12 - 4T^10 + 2T^8 + 6T^6 + 25T^4 - 2*T^2 + 1
$3$	$ T^{12} - 4 T^{10} + \cdots + 1 $ T^12 - 4T^10 + 16T^8 - 29T^6 + 18T^4 + 5*T^2 + 1
$5$	$ (T^{6} + 8 T^{5} + 22 T^{4} + \cdots + 1)^{2} $ (T^6 + 8T^5 + 22T^4 + 15T^3 + 8T^2 + T + 1)^2
$7$	$ (T^{6} - 8 T^{5} + 22 T^{4} + \cdots + 1)^{2} $ (T^6 - 8T^5 + 22T^4 - 15T^3 + 8T^2 - T + 1)^2
$11$	$ T^{12} + 12 T^{10} + \cdots + 2825761 $ T^12 + 12T^10 + 228T^8 - 6385T^6 + 80110T^4 - 630375*T^2 + 2825761
$13$	$ (T^{6} + 16 T^{5} + \cdots + 1849)^{2} $ (T^6 + 16T^5 + 116T^4 + 435T^3 + 814T^2 + 473*T + 1849)^2
$17$	$ (T^{6} + 24 T^{4} + \cdots + 64)^{2} $ (T^6 + 24T^4 + 80T^2 + 64)^2
$19$	$ T^{12} - 8 T^{10} + \cdots + 28561 $ T^12 - 8T^10 + 64T^8 - 253T^6 + 386T^4 - 2535*T^2 + 28561
$23$	$ (T^{6} - 14 T^{5} + \cdots + 2401)^{2} $ (T^6 - 14T^5 + 98T^4 - 441T^3 + 1372T^2 - 2401*T + 2401)^2
$29$	$ T^{12} $ T^12
$31$	$ T^{12} - 88 T^{10} + \cdots + 47458321 $ T^12 - 88T^10 + 4664T^8 - 155765T^6 + 2501034T^4 + 12958209*T^2 + 47458321
$37$	$ T^{12} + 12 T^{10} + \cdots + 2825761 $ T^12 + 12T^10 + 228T^8 - 6385T^6 + 80110T^4 - 630375*T^2 + 2825761
$41$	$ (T^{6} + 52 T^{4} + \cdots + 64)^{2} $ (T^6 + 52T^4 + 416T^2 + 64)^2
$43$	$ T^{12} - 64 T^{10} + \cdots + 28561 $ T^12 - 64T^10 + 1492T^8 - 12769T^6 + 138958T^4 + 77909*T^2 + 28561
$47$	$ T^{12} + \cdots + 3262808641 $ T^12 - 16T^10 + 3532T^8 - 6161T^6 + 2538678T^4 - 186157339*T^2 + 3262808641
$53$	$ (T^{6} + 4 T^{5} + 30 T^{4} + \cdots + 1)^{2} $ (T^6 + 4T^5 + 30T^4 + 71T^3 + 424T^2 + 37*T + 1)^2
$59$	$ (T^{3} + 28 T^{2} + \cdots + 728)^{4} $ (T^3 + 28T^2 + 252T + 728)^4
$61$	$ T^{12} + 40 T^{10} + \cdots + 28561 $ T^12 + 40T^10 + 1096T^8 - 6917T^6 + 17754T^4 - 25519*T^2 + 28561
$67$	$ (T^{6} - 2 T^{5} + \cdots + 169)^{2} $ (T^6 - 2T^5 + 200T^4 + 993T^3 + 1654T^2 + 689*T + 169)^2
$71$	$ (T^{6} + 945 T^{3} + \cdots + 35721)^{2} $ (T^6 + 945T^3 + 7938T^2 + 11907*T + 35721)^2
$73$	$ T^{12} + \cdots + 30821664721 $ T^12 - 72T^10 + 564T^8 - 178865T^6 + 31468334T^4 - 572153299*T^2 + 30821664721
$79$	$ T^{12} - 144 T^{10} + \cdots + 531441 $ T^12 - 144T^10 + 4860T^8 + 162567T^6 + 4238406T^4 - 649539*T^2 + 531441
$83$	$ (T^{6} - 10 T^{5} + \cdots + 28561)^{2} $ (T^6 - 10T^5 + 16T^4 - 27T^3 + 2174T^2 + 6253*T + 28561)^2
$89$	$ T^{12} + 140 T^{10} + \cdots + 68574961 $ T^12 + 140T^10 + 14308T^8 - 119609T^6 + 17013486T^4 - 61271119*T^2 + 68574961
$97$	$ T^{12} + 300 T^{10} + \cdots + 28561 $ T^12 + 300T^10 + 59088T^8 - 440581T^6 + 26870866T^4 - 1578291*T^2 + 28561
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Elliptic curves over $\Q$
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Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 841 = 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	841.e (of order \(14\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{28}^{11} - \zeta_{28}^{9} + \cdots - \zeta_{28}) q^{2}+ \cdots + ( - 2 \zeta_{28}^{10} - \zeta_{28}^{6} + \cdots + 1) q^{9}+O(q^{10}) \) q + (z^11 - z^9 + z^7 - z^5 - z) * q^2 + (z^9 - z^7 + z^5 - z^3 + z) * q^3 + (-z^10 + z^8 + z^4 - z^2 + 1) * q^4 + (-z^10 + z^8 - z^6 + z^2 - 1) * q^5 + (-z^8 + z^6 - z^4) * q^6 + (-2z^10 - z^6 + z^4 + 2) q^7 + (z^11 - z^9 + z^5 + z) * q^8 + (-2z^10 - z^6 + z^4 - z^2 + 1) q^9	\( q + (\zeta_{28}^{11} - \zeta_{28}^{9} + \cdots - \zeta_{28}) q^{2}+ \cdots + (\zeta_{28}^{11} + \cdots + \zeta_{28}^{3}) q^{99}+O(q^{100}) \) q + (z^11 - z^9 + z^7 - z^5 - z) * q^2 + (z^9 - z^7 + z^5 - z^3 + z) * q^3 + (-z^10 + z^8 + z^4 - z^2 + 1) * q^4 + (-z^10 + z^8 - z^6 + z^2 - 1) * q^5 + (-z^8 + z^6 - z^4) * q^6 + (-2z^10 - z^6 + z^4 + 2) q^7 + (z^11 - z^9 + z^5 + z) * q^8 + (-2z^10 - z^6 + z^4 - z^2 + 1) q^9 + (-z^11 + 2z^9 - z^7 + z^3 - z) q^10 + (-3z^11 - z^7 + 3z^5 - 3z^3 + z) q^11 + (-z^11 + z^9 + z^5 - z^3) * q^12 + (-4z^8 + z^6 - 2z^4 + z^2 - 4) * q^13 + (4z^11 - z^9 + 3z^7 - 3z^5 + z^3 - 4z) * q^14 + (z^11 - 3z^9 + 4z^7 - 4z^5 + 3z^3 - z) * q^15 + (-3z^8 + z^6 - 3z^4 + z^2 - 3) * q^16 + (2z^11 - 2z^9 - 2z^5 + 2z^3) * q^17 + (3z^11 + 2z^7 - z^5 + z^3 - 2z) q^18 + (2z^11 - 2z^9 + 2z^7 + z^3 - z) q^19 + (-z^4 + z^2) * q^20 + (z^5 + z) * q^21 + (4z^10 - 3z^8 + 4z^6 - 4z^4 + 3z^2 - 4) q^22 + (3z^10 - 2z^8 + 4z^6 - 2z^4 + 3z^2) q^23 + (2z^10 - 3z^8 + 2z^6) q^24 + (3z^10 - 3z^8 + 4z^4 - z^2 + 4) q^25 + (3z^9 + 2z^7 + 2z^5 + 2z^3 + 3z) q^26 + (-z^11 - 3z^9 + 3z^7 + z^5) * q^27 + (-2z^10 + z^8 - z^6 + 2z^4 + 3) * q^28 + (-2z^10 + 4z^8 - 4z^6 + 2z^4 - 1) * q^30 + (z^11 - 2z^9 + 2z^7 - z^5 - 6z) q^31 + (4z^9 - z^7 + 3z^5 - z^3 + 4z) q^32 + (-2z^10 + 2z^8 + 3z^4 - 5z^2 + 3) * q^33 + 2z^8 q^34 + (-z^10 + z^8 - 2z^6 + z^4 - z^2) q^35 + (-z^10 - z^8 - z^6 + z^4 + z^2 + 1) * q^36 + (-2z^11 + 2z^9 - z^5 - 2z^3 - z) q^37 + (-2z^10 - z^6 - z^4 + z^2 + 1) q^38 + (z^11 - 5z^9 + z^7 + 2z^3 - 2z) q^39 + (-2z^11 + 3z^7 - 5z^5 + 5z^3 - 3z) q^40 + (-2z^9 + 4z^7 - 2z^5) q^41 + (-z^8 - 2z^4 - 1) q^42 + (2z^11 + z^9 + 2z^7 - 2z^5 - z^3 - 2z) * q^43 + (z^11 - 4z^9 - z^7 + z^5 + 4z^3 - z) * q^44 - z^4 * q^45 + (-z^11 - 4z^9 + z^7 - 4z^5 - z^3) * q^46 + (-7z^11 - z^7 + z^5 - z^3 + z) q^47 + (z^11 - 4z^9 + z^7) q^48 + (-2z^10 - 7z^6 + 3z^4 - 3z^2 + 7) * q^49 + (4z^11 - 4z^9 - 7z^3) q^50 + (2z^10 - 4z^8 + 2z^6 - 2z^4 + 4z^2 - 2) q^51 + (3z^10 - 4z^8 - 3z^6 - 4z^4 + 3z^2) q^52 + (-z^10 - 3z^8 - z^6 + z^2 - 1) q^53 + (z^10 - z^8 - 4z^4 + 3z^2 - 4) * q^54 + (-z^9 + 4z^7 - 7z^5 + 4z^3 - z) q^55 + (-2z^11 + 2z^5 + 3z) q^56 + (z^10 - 3z^8 + 3z^6 - z^4 - 1) * q^57 + (-2z^8 + 2z^6 - 10) * q^59 + (z^11 - z^9 + z^7 - z^5 + z) * q^60 + (-z^9 - z^7 - 3z^5 - z^3 - z) q^61 + (-z^10 + z^8 + 5z^4 + 2z^2 + 5) * q^62 + (-5z^10 - z^8 - 5z^6 - 3z^2 + 3) q^63 + (3z^10 - 5z^8 + 4z^6 - 5z^4 + 3z^2) q^64 + (-z^10 + 4z^8 - 3z^6 + 3z^4 - 4z^2 + 1) * q^65 + (3z^11 - 3z^9 - z^3) * q^66 + (z^10 - 3z^6 + 7z^4 - 7z^2 + 3) q^67 + (-2z^11 + 4z^9 - 2z^7 - 4z^3 + 4z) q^68 + (3z^11 - 2z^7 + 3z^5 - 3z^3 + 2z) q^69 + (2z^9 - z^7 + 2z^5) * q^70 + (3z^6 + 6z^4 + 3z^2) q^71 + (-3z^11 + 2z^9 - 2z^7 + 2z^5 - 2z^3 + 3z) * q^72 + (2z^11 + 5z^9 - 2z^7 + 2z^5 - 5z^3 - 2z) * q^73 + (z^8 + 4z^4 + 1) q^74 + (-z^11 + 8z^9 - 11z^7 + 8z^5 - z^3) q^75 + (3z^11 - z^5 + z^3) q^76 + (-5z^11 - z^9 - 5z^7 - 4z^3 + 4z) * q^77 + (3z^10 + 2z^6 - 3z^4 + 3z^2 - 2) * q^78 + (-3z^5 + 6z^3 - 3z) q^79 + (-z^10 + 5z^8 - 6z^6 + 6z^4 - 5z^2 + 1) * q^80 + (-z^10 - 5z^8 + 3z^6 - 5z^4 - z^2) q^81 + (-2z^10 + 2z^8 - 2z^6 + 2z^2 - 2) * q^82 + (-5z^10 + 5z^8 + 4z^4 + 4) q^83 + (z^9 + z^5 + z) * q^84 + (-2z^11 + 2z^9 - 2z^7 + 2z^5 - 2z) q^85 + (-5z^10 + 2z^8 - 2z^6 + 5z^4 + 5) * q^86 + (2z^10 - z^8 + z^6 - 2z^4 + 6) * q^88 + (z^11 - 6z^9 + 6z^7 - z^5) * q^89 + (z^7 + z^3) * q^90 + (4z^10 - 4z^8 - 9z^4 - 3z^2 - 9) * q^91 + (2z^10 + 2z^8 + 2z^6 - z^2 + 1) q^92 + (-5z^10 + 3z^8 - 2z^6 + 3z^4 - 5z^2) q^93 + (8z^10 - z^8 + 2z^6 - 2z^4 + z^2 - 8) q^94 + (-3z^11 + 3z^9 - 2z^5 + 3z^3 - 2z) q^95 + (4z^10 - z^6 + 1) q^96 + (-8z^11 - 8z^7 - 3z^3 + 3z) * q^97 + (9z^11 + 6z^7 + z^5 - z^3 - 6z) q^98 + (z^11 - 3z^9 - 4z^7 - 3z^5 + z^3) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{4} - 16 q^{5} + 6 q^{6} + 16 q^{7} + 2 q^{9}+O(q^{10}) \) 12 * q + 4 * q^4 - 16 * q^5 + 6 * q^6 + 16 * q^7 + 2 * q^9	\( 12 q + 4 q^{4} - 16 q^{5} + 6 q^{6} + 16 q^{7} + 2 q^{9} - 32 q^{13} - 20 q^{16} + 4 q^{20} - 12 q^{22} + 28 q^{23} + 14 q^{24} + 50 q^{25} + 24 q^{28} - 36 q^{30} + 12 q^{33} - 4 q^{34} - 12 q^{35} + 10 q^{36} + 10 q^{38} - 6 q^{42} + 2 q^{45} + 54 q^{49} + 4 q^{51} + 22 q^{52} - 8 q^{53} - 30 q^{54} + 4 q^{57} - 112 q^{59} + 50 q^{62} + 12 q^{63} + 40 q^{64} - 18 q^{65} + 4 q^{67} + 2 q^{74} - 2 q^{78} - 34 q^{80} + 22 q^{81} - 32 q^{82} + 20 q^{83} + 32 q^{86} + 84 q^{88} - 80 q^{91} + 14 q^{92} - 36 q^{93} - 68 q^{94} + 18 q^{96}+O(q^{100}) \) 12 * q + 4 * q^4 - 16 * q^5 + 6 * q^6 + 16 * q^7 + 2 * q^9 - 32 * q^13 - 20 * q^16 + 4 * q^20 - 12 * q^22 + 28 * q^23 + 14 * q^24 + 50 * q^25 + 24 * q^28 - 36 * q^30 + 12 * q^33 - 4 * q^34 - 12 * q^35 + 10 * q^36 + 10 * q^38 - 6 * q^42 + 2 * q^45 + 54 * q^49 + 4 * q^51 + 22 * q^52 - 8 * q^53 - 30 * q^54 + 4 * q^57 - 112 * q^59 + 50 * q^62 + 12 * q^63 + 40 * q^64 - 18 * q^65 + 4 * q^67 + 2 * q^74 - 2 * q^78 - 34 * q^80 + 22 * q^81 - 32 * q^82 + 20 * q^83 + 32 * q^86 + 84 * q^88 - 80 * q^91 + 14 * q^92 - 36 * q^93 - 68 * q^94 + 18 * q^96

Introduction

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	841.2.e.c

Level	$841$
Weight	$2$
Character orbit	841.e
Analytic conductor	$6.715$
Analytic rank	$0$
Dimension	$12$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(6.71541880999\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(2\) over \(\Q(\zeta_{14})\)
Coefficient field:	\(\Q(\zeta_{28})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 29)
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 63.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{12} - 4 T^{10} + \cdots + 1 \) T^12 - 4T^10 + 2T^8 + 6T^6 + 25T^4 - 2*T^2 + 1
$3$	\( T^{12} - 4 T^{10} + \cdots + 1 \) T^12 - 4T^10 + 16T^8 - 29T^6 + 18T^4 + 5*T^2 + 1
$5$	\( (T^{6} + 8 T^{5} + 22 T^{4} + \cdots + 1)^{2} \) (T^6 + 8T^5 + 22T^4 + 15T^3 + 8T^2 + T + 1)^2
$7$	\( (T^{6} - 8 T^{5} + 22 T^{4} + \cdots + 1)^{2} \) (T^6 - 8T^5 + 22T^4 - 15T^3 + 8T^2 - T + 1)^2
$11$	\( T^{12} + 12 T^{10} + \cdots + 2825761 \) T^12 + 12T^10 + 228T^8 - 6385T^6 + 80110T^4 - 630375*T^2 + 2825761
$13$	\( (T^{6} + 16 T^{5} + \cdots + 1849)^{2} \) (T^6 + 16T^5 + 116T^4 + 435T^3 + 814T^2 + 473*T + 1849)^2
$17$	\( (T^{6} + 24 T^{4} + \cdots + 64)^{2} \) (T^6 + 24T^4 + 80T^2 + 64)^2
$19$	\( T^{12} - 8 T^{10} + \cdots + 28561 \) T^12 - 8T^10 + 64T^8 - 253T^6 + 386T^4 - 2535*T^2 + 28561
$23$	\( (T^{6} - 14 T^{5} + \cdots + 2401)^{2} \) (T^6 - 14T^5 + 98T^4 - 441T^3 + 1372T^2 - 2401*T + 2401)^2
$29$	\( T^{12} \) T^12
$31$	\( T^{12} - 88 T^{10} + \cdots + 47458321 \) T^12 - 88T^10 + 4664T^8 - 155765T^6 + 2501034T^4 + 12958209*T^2 + 47458321
$37$	\( T^{12} + 12 T^{10} + \cdots + 2825761 \) T^12 + 12T^10 + 228T^8 - 6385T^6 + 80110T^4 - 630375*T^2 + 2825761
$41$	\( (T^{6} + 52 T^{4} + \cdots + 64)^{2} \) (T^6 + 52T^4 + 416T^2 + 64)^2
$43$	\( T^{12} - 64 T^{10} + \cdots + 28561 \) T^12 - 64T^10 + 1492T^8 - 12769T^6 + 138958T^4 + 77909*T^2 + 28561
$47$	\( T^{12} + \cdots + 3262808641 \) T^12 - 16T^10 + 3532T^8 - 6161T^6 + 2538678T^4 - 186157339*T^2 + 3262808641
$53$	\( (T^{6} + 4 T^{5} + 30 T^{4} + \cdots + 1)^{2} \) (T^6 + 4T^5 + 30T^4 + 71T^3 + 424T^2 + 37*T + 1)^2
$59$	\( (T^{3} + 28 T^{2} + \cdots + 728)^{4} \) (T^3 + 28T^2 + 252T + 728)^4
$61$	\( T^{12} + 40 T^{10} + \cdots + 28561 \) T^12 + 40T^10 + 1096T^8 - 6917T^6 + 17754T^4 - 25519*T^2 + 28561
$67$	\( (T^{6} - 2 T^{5} + \cdots + 169)^{2} \) (T^6 - 2T^5 + 200T^4 + 993T^3 + 1654T^2 + 689*T + 169)^2
$71$	\( (T^{6} + 945 T^{3} + \cdots + 35721)^{2} \) (T^6 + 945T^3 + 7938T^2 + 11907*T + 35721)^2
$73$	\( T^{12} + \cdots + 30821664721 \) T^12 - 72T^10 + 564T^8 - 178865T^6 + 31468334T^4 - 572153299*T^2 + 30821664721
$79$	\( T^{12} - 144 T^{10} + \cdots + 531441 \) T^12 - 144T^10 + 4860T^8 + 162567T^6 + 4238406T^4 - 649539*T^2 + 531441
$83$	\( (T^{6} - 10 T^{5} + \cdots + 28561)^{2} \) (T^6 - 10T^5 + 16T^4 - 27T^3 + 2174T^2 + 6253*T + 28561)^2
$89$	\( T^{12} + 140 T^{10} + \cdots + 68574961 \) T^12 + 140T^10 + 14308T^8 - 119609T^6 + 17013486T^4 - 61271119*T^2 + 68574961
$97$	\( T^{12} + 300 T^{10} + \cdots + 28561 \) T^12 + 300T^10 + 59088T^8 - 440581T^6 + 26870866T^4 - 1578291*T^2 + 28561
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\(n\)	\(2\)
\(\chi(n)\)	\(\zeta_{28}^{2}\)