LMFDB - Newform orbit 891.2.u.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [891,2,Mod(107,891)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(891, base_ring=CyclotomicField(30))

chi = DirichletCharacter(H, H._module([5, 9]))

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

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

chi := DirichletCharacter("891.107");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 891 = 3^{4} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	891.u (of order $30$, degree $8$, 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:	$7.11467082010$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$2$ over $\Q(\zeta_{30})$
Coefficient field:	$\Q(\zeta_{60})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + x^{14} - x^{10} - x^{8} - x^{6} + x^{2} + 1 $ x^16 + x^14 - x^10 - x^8 - x^6 + x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/891\mathbb{Z}\right)^\times$.

$n$	$244$	$650$
$\chi(n)$	$\zeta_{60}^{6}$	$\zeta_{60}^{10}$

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} $

107.1

−0.207912	−	0.978148i
0.207912	+	0.978148i
−0.406737	−	0.913545i
0.406737	+	0.913545i
−0.994522	+	0.104528i
0.994522	−	0.104528i
−0.994522	−	0.104528i
0.994522	+	0.104528i
−0.207912	+	0.978148i
0.207912	−	0.978148i
−0.406737	+	0.913545i
0.406737	−	0.913545i
0.743145	−	0.669131i
−0.743145	+	0.669131i
0.743145	+	0.669131i
−0.743145	−	0.669131i

−0.122881

−

1.16913i

0.604528

−

0.128496i

0.379874

0.0399263i

2.28716

−

2.05937i

−0.951057

−

2.92705i

−

0.449028i

107.2

0.122881

1.16913i

0.604528

−

0.128496i

−0.379874

−

0.0399263i

2.28716

−

2.05937i

0.951057

2.92705i

−

0.449028i

134.1

−1.86055

0.395472i

1.47815

−

0.658114i

0.544320

−

2.56082i

−0.722562

−

0.0759444i

0.587785

−

0.427051i

4.97980i

134.2

1.86055

−

0.395472i

1.47815

−

0.658114i

−0.544320

2.56082i

−0.722562

−

0.0759444i

−0.587785

0.427051i

4.97980i

215.1

−1.27276

1.41355i

−0.169131

−

1.60917i

−1.94558

1.75181i

0.295511

−

0.663730i

−0.587785

−

0.427051i

−

4.97980i

215.2

1.27276

−

1.41355i

−0.169131

−

1.60917i

1.94558

−

1.75181i

0.295511

−

0.663730i

0.587785

0.427051i

−

4.97980i

431.1

−1.27276

−

1.41355i

−0.169131

1.60917i

−1.94558

−

1.75181i

0.295511

0.663730i

−0.587785

0.427051i

4.97980i

431.2

1.27276

1.41355i

−0.169131

1.60917i

1.94558

1.75181i

0.295511

0.663730i

0.587785

−

0.427051i

4.97980i

458.1

−0.122881

1.16913i

0.604528

0.128496i

0.379874

−

0.0399263i

2.28716

2.05937i

−0.951057

2.92705i

0.449028i

458.2

0.122881

−

1.16913i

0.604528

0.128496i

−0.379874

0.0399263i

2.28716

2.05937i

0.951057

−

2.92705i

0.449028i

512.1

−1.86055

−

0.395472i

1.47815

0.658114i

0.544320

2.56082i

−0.722562

0.0759444i

0.587785

0.427051i

−

4.97980i

512.2

1.86055

0.395472i

1.47815

0.658114i

−0.544320

−

2.56082i

−0.722562

0.0759444i

−0.587785

−

0.427051i

−

4.97980i

701.1

−1.07394

−

0.478148i

−0.413545

−

0.459289i

0.155360

0.348943i

0.639886

3.01043i

0.951057

2.92705i

−

0.449028i

701.2

1.07394

0.478148i

−0.413545

−

0.459289i

−0.155360

−

0.348943i

0.639886

3.01043i

−0.951057

−

2.92705i

−

0.449028i

755.1

−1.07394

0.478148i

−0.413545

0.459289i

0.155360

−

0.348943i

0.639886

−

3.01043i

0.951057

−

2.92705i

0.449028i

755.2

1.07394

−

0.478148i

−0.413545

0.459289i

−0.155360

0.348943i

0.639886

−

3.01043i

−0.951057

2.92705i

0.449028i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
9.c	even	3	1	inner
9.d	odd	6	1	inner
11.d	odd	10	1	inner
33.f	even	10	1	inner
99.o	odd	30	1	inner
99.p	even	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	891.2.u.a		16
3.b	odd	2	1	inner	891.2.u.a		16
9.c	even	3	1		33.2.f.a	✓	8
9.c	even	3	1	inner	891.2.u.a		16
9.d	odd	6	1		33.2.f.a	✓	8
9.d	odd	6	1	inner	891.2.u.a		16
11.d	odd	10	1	inner	891.2.u.a		16
33.f	even	10	1	inner	891.2.u.a		16
36.f	odd	6	1		528.2.bn.c		8
36.h	even	6	1		528.2.bn.c		8
45.h	odd	6	1		825.2.bi.b		8
45.j	even	6	1		825.2.bi.b		8
45.k	odd	12	1		825.2.bs.a		8
45.k	odd	12	1		825.2.bs.d		8
45.l	even	12	1		825.2.bs.a		8
45.l	even	12	1		825.2.bs.d		8
99.g	even	6	1		363.2.f.b		8
99.h	odd	6	1		363.2.f.b		8
99.m	even	15	1		363.2.d.f		8
99.m	even	15	1		363.2.f.b		8
99.m	even	15	1		363.2.f.d		8
99.m	even	15	1		363.2.f.e		8
99.n	odd	30	1		363.2.d.f		8
99.n	odd	30	1		363.2.f.b		8
99.n	odd	30	1		363.2.f.d		8
99.n	odd	30	1		363.2.f.e		8
99.o	odd	30	1		33.2.f.a	✓	8
99.o	odd	30	1		363.2.d.f		8
99.o	odd	30	1		363.2.f.d		8
99.o	odd	30	1		363.2.f.e		8
99.o	odd	30	1	inner	891.2.u.a		16
99.p	even	30	1		33.2.f.a	✓	8
99.p	even	30	1		363.2.d.f		8
99.p	even	30	1		363.2.f.d		8
99.p	even	30	1		363.2.f.e		8
99.p	even	30	1	inner	891.2.u.a		16
396.bb	odd	30	1		528.2.bn.c		8
396.bf	even	30	1		528.2.bn.c		8
495.bo	even	30	1		825.2.bi.b		8
495.br	odd	30	1		825.2.bi.b		8
495.bs	even	60	1		825.2.bs.a		8
495.bs	even	60	1		825.2.bs.d		8
495.bu	odd	60	1		825.2.bs.a		8
495.bu	odd	60	1		825.2.bs.d		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
33.2.f.a	✓	8	9.c	even	3	1
33.2.f.a	✓	8	9.d	odd	6	1
33.2.f.a	✓	8	99.o	odd	30	1
33.2.f.a	✓	8	99.p	even	30	1
363.2.d.f		8	99.m	even	15	1
363.2.d.f		8	99.n	odd	30	1
363.2.d.f		8	99.o	odd	30	1
363.2.d.f		8	99.p	even	30	1
363.2.f.b		8	99.g	even	6	1
363.2.f.b		8	99.h	odd	6	1
363.2.f.b		8	99.m	even	15	1
363.2.f.b		8	99.n	odd	30	1
363.2.f.d		8	99.m	even	15	1
363.2.f.d		8	99.n	odd	30	1
363.2.f.d		8	99.o	odd	30	1
363.2.f.d		8	99.p	even	30	1
363.2.f.e		8	99.m	even	15	1
363.2.f.e		8	99.n	odd	30	1
363.2.f.e		8	99.o	odd	30	1
363.2.f.e		8	99.p	even	30	1
528.2.bn.c		8	36.f	odd	6	1
528.2.bn.c		8	36.h	even	6	1
528.2.bn.c		8	396.bb	odd	30	1
528.2.bn.c		8	396.bf	even	30	1
825.2.bi.b		8	45.h	odd	6	1
825.2.bi.b		8	45.j	even	6	1
825.2.bi.b		8	495.bo	even	30	1
825.2.bi.b		8	495.br	odd	30	1
825.2.bs.a		8	45.k	odd	12	1
825.2.bs.a		8	45.l	even	12	1
825.2.bs.a		8	495.bs	even	60	1
825.2.bs.a		8	495.bu	odd	60	1
825.2.bs.d		8	45.k	odd	12	1
825.2.bs.d		8	45.l	even	12	1
825.2.bs.d		8	495.bs	even	60	1
825.2.bs.d		8	495.bu	odd	60	1
891.2.u.a		16	1.a	even	1	1	trivial
891.2.u.a		16	3.b	odd	2	1	inner
891.2.u.a		16	9.c	even	3	1	inner
891.2.u.a		16	9.d	odd	6	1	inner
891.2.u.a		16	11.d	odd	10	1	inner
891.2.u.a		16	33.f	even	10	1	inner
891.2.u.a		16	99.o	odd	30	1	inner
891.2.u.a		16	99.p	even	30	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{16} - 5T_{2}^{14} + 15T_{2}^{12} - 50T_{2}^{10} + 75T_{2}^{8} + 250T_{2}^{6} - 250T_{2}^{4} + 625 $ T2^16 - 5*T2^14 + 15*T2^12 - 50*T2^10 + 75*T2^8 + 250*T2^6 - 250*T2^4 + 625 acting on $S_{2}^{\mathrm{new}}(891, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} - 5 T^{14} + \cdots + 625 $ T^16 - 5T^14 + 15T^12 - 50T^10 + 75T^8 + 250T^6 - 250T^4 + 625
$3$	$ T^{16} $ T^16
$5$	$ T^{16} + 11 T^{14} + \cdots + 1 $ T^16 + 11T^14 + 75T^12 + 514T^10 + 2159T^8 - 206T^6 - 30T^4 - 4*T^2 + 1
$7$	$ (T^{8} - 5 T^{7} + 20 T^{6} + \cdots + 25)^{2} $ (T^8 - 5T^7 + 20T^6 - 35T^5 + 45T^4 + 75T^3 + 25T + 25)^2
$11$	$ T^{16} + \cdots + 214358881 $ T^16 - 19T^14 + 60T^12 - 1121T^10 + 32279T^8 - 135641T^6 + 878460T^4 - 33659659*T^2 + 214358881
$13$	$ (T^{8} - 5 T^{7} + 20 T^{6} + \cdots + 25)^{2} $ (T^8 - 5T^7 + 20T^6 - 35T^5 + 45T^4 + 75T^3 + 25T + 25)^2
$17$	$ (T^{8} + 250 T^{4} + \cdots + 15625)^{2} $ (T^8 + 250T^4 + 3125T^2 + 15625)^2
$19$	$ (T^{4} - 10 T^{3} + \cdots + 125)^{4} $ (T^4 - 10T^3 + 50T^2 - 125*T + 125)^4
$23$	$ (T^{8} - 42 T^{6} + \cdots + 14641)^{2} $ (T^8 - 42T^6 + 1643T^4 - 5082*T^2 + 14641)^2
$29$	$ T^{16} - 160 T^{12} + \cdots + 40960000 $ T^16 - 160T^12 + 3200T^10 + 19200T^8 - 256000T^6 + 1536000T^4 - 10240000T^2 + 40960000
$31$	$ (T^{8} - 10 T^{7} + \cdots + 625)^{2} $ (T^8 - 10T^7 + 60T^6 - 350T^5 + 1325T^4 - 500T^3 - 375T^2 - 625*T + 625)^2
$37$	$ (T^{4} + 3 T^{3} + 9 T^{2} + \cdots + 81)^{4} $ (T^4 + 3T^3 + 9T^2 + 27*T + 81)^4
$41$	$ T^{16} + 5 T^{14} + \cdots + 625 $ T^16 + 5T^14 - 60T^12 + 575T^10 + 7575T^8 - 6625T^6 + 3500T^4 - 1875*T^2 + 625
$43$	$ (T^{8} - 50 T^{6} + \cdots + 15625)^{2} $ (T^8 - 50T^6 + 2375T^4 - 6250*T^2 + 15625)^2
$47$	$ T^{16} + \cdots + 191707312997281 $ T^16 + 79T^14 + 2275T^12 - 14134T^10 - 11050881T^8 - 1538313494T^6 - 28107057230T^4 + 2266896471884*T^2 + 191707312997281
$53$	$ (T^{8} - 36 T^{6} + \cdots + 6561)^{2} $ (T^8 - 36T^6 + 486T^4 + 729*T^2 + 6561)^2
$59$	$ T^{16} - 4 T^{14} + \cdots + 1 $ T^16 - 4T^14 - 30T^12 - 206T^10 + 2159T^8 + 514T^6 + 75T^4 + 11*T^2 + 1
$61$	$ (T^{8} - 5 T^{7} + \cdots + 15625)^{2} $ (T^8 - 5T^7 + 25T^6 - 125T^4 + 1250T^3 + 15625)^2
$67$	$ (T^{4} - T^{3} + 62 T^{2} + \cdots + 3721)^{4} $ (T^4 - T^3 + 62T^2 + 61T + 3721)^4
$71$	$ (T^{8} - 155 T^{6} + \cdots + 9150625)^{2} $ (T^8 - 155T^6 + 9150T^4 + 60500*T^2 + 9150625)^2
$73$	$ (T^{4} + 2560 T + 20480)^{4} $ (T^4 + 2560*T + 20480)^4
$79$	$ (T^{8} + 25 T^{7} + \cdots + 3258025)^{2} $ (T^8 + 25T^7 + 400T^6 + 3915T^5 + 27445T^4 + 102125T^3 + 324900T^2 + 1543275*T + 3258025)^2
$83$	$ T^{16} + \cdots + 49\!\cdots\!25 $ T^16 - 315T^14 + 61940T^12 - 12837425T^10 + 1491619575T^8 + 64875463375T^6 - 2335491466500T^4 + 38594596958125*T^2 + 4990578268200625
$89$	$ (T^{4} + 90 T^{2} + 25)^{4} $ (T^4 + 90*T^2 + 25)^4
$97$	$ (T^{8} + 3 T^{7} + \cdots + 707281)^{2} $ (T^8 + 3T^7 - 25T^6 - 362T^5 - 381T^4 + 2842T^3 + 25230T^2 + 195112*T + 707281)^2
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Elliptic curves over $\Q$
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Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 891 = 3^{4} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	891.u (of order \(30\), degree \(8\), not minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{60}^{11} + \zeta_{60}^{5}) q^{2} + (\zeta_{60}^{12} + \cdots - \zeta_{60}^{2}) q^{4}+ \cdots + (2 \zeta_{60}^{15} + \cdots + 2 \zeta_{60}) q^{8}+O(q^{10}) \) q + (z^11 + z^5) * q^2 + (z^12 + z^10 - z^2) * q^4 + (z^13 + z^7 + z) * q^5 + (z^14 - z^10 + z^8 - z^2 + 1) * q^7 + (2z^15 - 2z^11 - z^9 + 2z) q^8	\( q + (\zeta_{60}^{11} + \zeta_{60}^{5}) q^{2} + (\zeta_{60}^{12} + \cdots - \zeta_{60}^{2}) q^{4}+ \cdots + (4 \zeta_{60}^{15} + \cdots + 8 \zeta_{60}) q^{98}+O(q^{100}) \) q + (z^11 + z^5) * q^2 + (z^12 + z^10 - z^2) * q^4 + (z^13 + z^7 + z) * q^5 + (z^14 - z^10 + z^8 - z^2 + 1) * q^7 + (2z^15 - 2z^11 - z^9 + 2z) q^8 + (3z^14 + 4z^12 - z^6 - 3z^4 + 2) q^10 + (3z^15 + 4z^13 - z^11 - 3z^9 - 3z^7 - 2z^5 - z^3 + 3z) * q^11 + (-z^14 + z^12 + 2z^10 + z^8 + z^6 + z^4 - 2z^2 - 1) * q^13 + (2z^15 + 2z^13 - 3z^7 - 2z^5 + 3z) q^14 + (z^14 - z^10 + 4z^2 + 1) q^16 + (-2z^15 + z^11 - 2z^9 + z^3 - z) * q^17 + (2z^14 + 3z^12 - z^6 - 2z^4 + 4) q^19 + (3z^13 + 2z^11 - 2z^5 - 3z^3) * q^20 + (z^14 - 2z^10 - z^8 - z^6 - 5z^4 + z^2 + 1) * q^22 + (-3z^15 + 2z^13 - 2z^7 + 3z^5) * q^23 + (-3z^14 + 3z^10 + 3z^8 - 3) q^25 + (4z^11 + 3z^9 - 3z^3 - 4z) * q^26 + (z^14 + z^12 - 2z^6 - z^4 + 1) q^28 + (2z^15 + 4z^13 - 2z^9 - 2z^7 - 2z^5 - 2z^3 + 2z) q^29 + (4z^14 + z^12 - 4z^8 - 4z^6 - z^4 + 3z^2 + 4) * q^31 + (3z^15 + 5z^13 - z^7 - 3z^5 + 6z) * q^32 + (-5z^10 + 5) q^34 + (z^15 + z^11 - z) * q^35 - 3z^6 q^37 + (5z^13 + 5z^11 - 5z^3) q^38 + (-z^14 - z^12 + z^8 + z^6 + 2z^4 - 1) q^40 + (z^15 - z^13 - z^7 - z^5 - z) * q^41 + (6z^14 - 3z^10 - 4z^8 + 2z^2 + 3) * q^43 + (-z^15 - 3z^11 - 2z^9 - z^3 + 3z) q^44 + (2z^14 - 2z^12 + 3z^6 - 2z^4 + 3) * q^46 + (4z^15 - 3z^13 - 4z^9 - 4z^7 + 3z^5 + 7z^3 + 4z) q^47 + (-4z^12 - 4z^10 + 4z^4 + 4z^2) * q^49 + (3z^13 - 3z) * q^50 + (-2z^14 + 3z^10 + z^8 - 4z^2 - 3) q^52 + (-3z^15 + 3z^11 - 3z^3 - 3z) * q^53 + (-3z^12 - 3z^6 - 4) * q^55 + (4z^15 + 4z^13 - 4z^11 - 4z^9 - 4z^7 + 3z^5 + 4z) q^56 + (2z^14 + 2z^12 - 2z^10 - 2z^8 - 2z^6 - 6z^4 + 2) * q^58 + (z^15 - z^7 - z^5 - z) * q^59 + (z^14 - 3z^10 - 4z^8 + 2z^2 + 3) q^61 + (6z^15 - 3z^11 - 4z^9 + 2z^3 + 3z) q^62 + (z^14 + z^12 + 5z^6 - z^4 - 5) q^64 + (-z^15 + 5z^13 + 5z^11 + z^9 + z^7 - 2z^5 - 6z^3 - z) * q^65 + (-7z^14 + 4z^10 + 7z^8 + 7z^6 + 7z^4 - 7z^2 - 7) * q^67 + (-3z^15 - 2z^13 + 4z^7 + 3z^5 - z) * q^68 + (-2z^14 + 3z^10 + 2z^8 - 3z^2 - 3) * q^70 + (z^15 + 6z^11 - z^9 - 6z^3 - 6z) q^71 + (-16z^14 - 8z^12 + 8z^6 + 16z^4 - 8) * q^73 + (3z^15 - 6z^11 - 3z^9 - 3z^7 + 3z^3 + 3z) * q^74 + (2z^14 + 6z^12 + z^10 - 2z^8 - 2z^6 - 4z^4 - 4z^2 + 2) * q^76 + (3z^13 - 8z^7 + 4z) q^77 + (-4z^14 + 10z^10 + 5z^8 - 6z^2 - 10) * q^79 + (5z^15 + 4z^9 + 5z^3) q^80 + (-3z^14 - 4z^12 + 3z^4 - 3) q^82 + (7z^15 + 8z^13 - z^11 - 7z^9 - 7z^7 - 14z^5 - z^3 + 7z) * q^83 + (2z^14 - 4z^12 - 4z^10 - 2z^8 - 2z^6 + z^4 + 6z^2 + 2) * q^85 + (5z^15 - 5z^5 + 5z) q^86 + (-z^14 + 5z^10 - 3z^8 + 3z^2 - 5) q^88 + (-3z^15 - 4z^11 - 4z^9 + 4z) * q^89 + (z^14 + 2z^12 - 2z^6 - z^4) * q^91 + (-3z^15 + z^11 + 3z^9 + 3z^7 + 3z^5 - 3z^3 - 3z) * q^92 + (-14z^14 - 7z^12 + 10z^10 + 14z^8 + 14z^6 + 10z^4 - 7z^2 - 14) q^94 + (7z^15 + 8z^13 - z^7 - 7z^5 + 4z) * q^95 + (-z^14 - 5z^8 - z^2) q^97 + (4z^15 - 8z^11 + 8z^3 + 8z) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 6 q^{4} + 10 q^{7}+O(q^{10}) \) 16 * q + 6 * q^4 + 10 * q^7	\( 16 q + 6 q^{4} + 10 q^{7} + 10 q^{13} - 2 q^{16} + 40 q^{19} - 20 q^{22} - 12 q^{25} + 20 q^{31} + 40 q^{34} - 12 q^{37} + 60 q^{46} - 16 q^{49} - 10 q^{52} - 64 q^{55} - 20 q^{58} + 10 q^{61} - 68 q^{64} + 4 q^{67} - 10 q^{70} - 50 q^{79} - 20 q^{82} - 10 q^{85} - 50 q^{88} - 20 q^{91} + 30 q^{94} - 6 q^{97}+O(q^{100}) \) 16 * q + 6 * q^4 + 10 * q^7 + 10 * q^13 - 2 * q^16 + 40 * q^19 - 20 * q^22 - 12 * q^25 + 20 * q^31 + 40 * q^34 - 12 * q^37 + 60 * q^46 - 16 * q^49 - 10 * q^52 - 64 * q^55 - 20 * q^58 + 10 * q^61 - 68 * q^64 + 4 * q^67 - 10 * q^70 - 50 * q^79 - 20 * q^82 - 10 * q^85 - 50 * q^88 - 20 * q^91 + 30 * q^94 - 6 * q^97

Introduction

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Newspace parameters

Newform invariants

$q$-expansion

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Hecke characteristic polynomials

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	891.2.u.a

Level	$891$
Weight	$2$
Character orbit	891.u
Analytic conductor	$7.115$
Analytic rank	$0$
Dimension	$16$
CM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(7.11467082010\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(2\) over \(\Q(\zeta_{30})\)
Coefficient field:	\(\Q(\zeta_{60})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} + x^{14} - x^{10} - x^{8} - x^{6} + x^{2} + 1 \) x^16 + x^14 - x^10 - x^8 - x^6 + x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

\(n\)	\(244\)	\(650\)
\(\chi(n)\)	\(\zeta_{60}^{6}\)	\(\zeta_{60}^{10}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 107.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{16} - 5 T^{14} + \cdots + 625 \) T^16 - 5T^14 + 15T^12 - 50T^10 + 75T^8 + 250T^6 - 250T^4 + 625
$3$	\( T^{16} \) T^16
$5$	\( T^{16} + 11 T^{14} + \cdots + 1 \) T^16 + 11T^14 + 75T^12 + 514T^10 + 2159T^8 - 206T^6 - 30T^4 - 4*T^2 + 1
$7$	\( (T^{8} - 5 T^{7} + 20 T^{6} + \cdots + 25)^{2} \) (T^8 - 5T^7 + 20T^6 - 35T^5 + 45T^4 + 75T^3 + 25T + 25)^2
$11$	\( T^{16} + \cdots + 214358881 \) T^16 - 19T^14 + 60T^12 - 1121T^10 + 32279T^8 - 135641T^6 + 878460T^4 - 33659659*T^2 + 214358881
$13$	\( (T^{8} - 5 T^{7} + 20 T^{6} + \cdots + 25)^{2} \) (T^8 - 5T^7 + 20T^6 - 35T^5 + 45T^4 + 75T^3 + 25T + 25)^2
$17$	\( (T^{8} + 250 T^{4} + \cdots + 15625)^{2} \) (T^8 + 250T^4 + 3125T^2 + 15625)^2
$19$	\( (T^{4} - 10 T^{3} + \cdots + 125)^{4} \) (T^4 - 10T^3 + 50T^2 - 125*T + 125)^4
$23$	\( (T^{8} - 42 T^{6} + \cdots + 14641)^{2} \) (T^8 - 42T^6 + 1643T^4 - 5082*T^2 + 14641)^2
$29$	\( T^{16} - 160 T^{12} + \cdots + 40960000 \) T^16 - 160T^12 + 3200T^10 + 19200T^8 - 256000T^6 + 1536000T^4 - 10240000T^2 + 40960000
$31$	\( (T^{8} - 10 T^{7} + \cdots + 625)^{2} \) (T^8 - 10T^7 + 60T^6 - 350T^5 + 1325T^4 - 500T^3 - 375T^2 - 625*T + 625)^2
$37$	\( (T^{4} + 3 T^{3} + 9 T^{2} + \cdots + 81)^{4} \) (T^4 + 3T^3 + 9T^2 + 27*T + 81)^4
$41$	\( T^{16} + 5 T^{14} + \cdots + 625 \) T^16 + 5T^14 - 60T^12 + 575T^10 + 7575T^8 - 6625T^6 + 3500T^4 - 1875*T^2 + 625
$43$	\( (T^{8} - 50 T^{6} + \cdots + 15625)^{2} \) (T^8 - 50T^6 + 2375T^4 - 6250*T^2 + 15625)^2
$47$	\( T^{16} + \cdots + 191707312997281 \) T^16 + 79T^14 + 2275T^12 - 14134T^10 - 11050881T^8 - 1538313494T^6 - 28107057230T^4 + 2266896471884*T^2 + 191707312997281
$53$	\( (T^{8} - 36 T^{6} + \cdots + 6561)^{2} \) (T^8 - 36T^6 + 486T^4 + 729*T^2 + 6561)^2
$59$	\( T^{16} - 4 T^{14} + \cdots + 1 \) T^16 - 4T^14 - 30T^12 - 206T^10 + 2159T^8 + 514T^6 + 75T^4 + 11*T^2 + 1
$61$	\( (T^{8} - 5 T^{7} + \cdots + 15625)^{2} \) (T^8 - 5T^7 + 25T^6 - 125T^4 + 1250T^3 + 15625)^2
$67$	\( (T^{4} - T^{3} + 62 T^{2} + \cdots + 3721)^{4} \) (T^4 - T^3 + 62T^2 + 61T + 3721)^4
$71$	\( (T^{8} - 155 T^{6} + \cdots + 9150625)^{2} \) (T^8 - 155T^6 + 9150T^4 + 60500*T^2 + 9150625)^2
$73$	\( (T^{4} + 2560 T + 20480)^{4} \) (T^4 + 2560*T + 20480)^4
$79$	\( (T^{8} + 25 T^{7} + \cdots + 3258025)^{2} \) (T^8 + 25T^7 + 400T^6 + 3915T^5 + 27445T^4 + 102125T^3 + 324900T^2 + 1543275*T + 3258025)^2
$83$	\( T^{16} + \cdots + 49\!\cdots\!25 \) T^16 - 315T^14 + 61940T^12 - 12837425T^10 + 1491619575T^8 + 64875463375T^6 - 2335491466500T^4 + 38594596958125*T^2 + 4990578268200625
$89$	\( (T^{4} + 90 T^{2} + 25)^{4} \) (T^4 + 90*T^2 + 25)^4
$97$	\( (T^{8} + 3 T^{7} + \cdots + 707281)^{2} \) (T^8 + 3T^7 - 25T^6 - 362T^5 - 381T^4 + 2842T^3 + 25230T^2 + 195112*T + 707281)^2
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