LMFDB - Newform orbit 33.2.e.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [33,2,Mod(4,33)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(33, base_ring=CyclotomicField(10))

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

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

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

chi := DirichletCharacter("33.4");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 33 = 3 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	33.e (of order $5$, degree $4$, 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:	$0.263506326670$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{10})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + x^{2} - x + 1 $ x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/33\mathbb{Z}\right)^\times$.

$n$	$13$	$23$
$\chi(n)$	$\zeta_{10}^{2}$	$1$

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} $

4.1

0.809017	+	0.587785i
−0.309017	−	0.951057i
0.809017	−	0.587785i
−0.309017	+	0.951057i

−1.30902

−

0.951057i

0.309017

−

0.951057i

0.190983

0.587785i

0.309017

−

0.224514i

−1.30902

0.951057i

0.927051

2.85317i

−0.690983

2.12663i

−0.809017

−

0.587785i

−0.618034

16.1

−0.190983

−

0.587785i

−0.809017

−

0.587785i

1.30902

−

0.951057i

−0.809017

2.48990i

−0.190983

0.587785i

−2.42705

1.76336i

−1.80902

−

1.31433i

0.309017

0.951057i

1.61803

25.1

−1.30902

0.951057i

0.309017

0.951057i

0.190983

−

0.587785i

0.309017

0.224514i

−1.30902

−

0.951057i

0.927051

−

2.85317i

−0.690983

−

2.12663i

−0.809017

0.587785i

−0.618034

31.1

−0.190983

0.587785i

−0.809017

0.587785i

1.30902

0.951057i

−0.809017

−

2.48990i

−0.190983

−

0.587785i

−2.42705

−

1.76336i

−1.80902

1.31433i

0.309017

−

0.951057i

1.61803

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.c	even	5	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	33.2.e.a	✓	4
3.b	odd	2	1		99.2.f.b		4
4.b	odd	2	1		528.2.y.f		4
5.b	even	2	1		825.2.n.f		4
5.c	odd	4	2		825.2.bx.b		8
9.c	even	3	2		891.2.n.d		8
9.d	odd	6	2		891.2.n.a		8
11.b	odd	2	1		363.2.e.j		4
11.c	even	5	1	inner	33.2.e.a	✓	4
11.c	even	5	1		363.2.a.h		2
11.c	even	5	2		363.2.e.h		4
11.d	odd	10	1		363.2.a.e		2
11.d	odd	10	2		363.2.e.c		4
11.d	odd	10	1		363.2.e.j		4
33.f	even	10	1		1089.2.a.s		2
33.h	odd	10	1		99.2.f.b		4
33.h	odd	10	1		1089.2.a.m		2
44.g	even	10	1		5808.2.a.bm		2
44.h	odd	10	1		528.2.y.f		4
44.h	odd	10	1		5808.2.a.bl		2
55.h	odd	10	1		9075.2.a.bv		2
55.j	even	10	1		825.2.n.f		4
55.j	even	10	1		9075.2.a.x		2
55.k	odd	20	2		825.2.bx.b		8
99.m	even	15	2		891.2.n.d		8
99.n	odd	30	2		891.2.n.a		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
33.2.e.a	✓	4	1.a	even	1	1	trivial
33.2.e.a	✓	4	11.c	even	5	1	inner
99.2.f.b		4	3.b	odd	2	1
99.2.f.b		4	33.h	odd	10	1
363.2.a.e		2	11.d	odd	10	1
363.2.a.h		2	11.c	even	5	1
363.2.e.c		4	11.d	odd	10	2
363.2.e.h		4	11.c	even	5	2
363.2.e.j		4	11.b	odd	2	1
363.2.e.j		4	11.d	odd	10	1
528.2.y.f		4	4.b	odd	2	1
528.2.y.f		4	44.h	odd	10	1
825.2.n.f		4	5.b	even	2	1
825.2.n.f		4	55.j	even	10	1
825.2.bx.b		8	5.c	odd	4	2
825.2.bx.b		8	55.k	odd	20	2
891.2.n.a		8	9.d	odd	6	2
891.2.n.a		8	99.n	odd	30	2
891.2.n.d		8	9.c	even	3	2
891.2.n.d		8	99.m	even	15	2
1089.2.a.m		2	33.h	odd	10	1
1089.2.a.s		2	33.f	even	10	1
5808.2.a.bl		2	44.h	odd	10	1
5808.2.a.bm		2	44.g	even	10	1
9075.2.a.x		2	55.j	even	10	1
9075.2.a.bv		2	55.h	odd	10	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{4} + 3T_{2}^{3} + 4T_{2}^{2} + 2T_{2} + 1 $ T2^4 + 3*T2^3 + 4*T2^2 + 2*T2 + 1 acting on $S_{2}^{\mathrm{new}}(33, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} + 3 T^{3} + 4 T^{2} + 2 T + 1 $ T^4 + 3T^3 + 4T^2 + 2*T + 1
$3$	$ T^{4} + T^{3} + T^{2} + T + 1 $ T^4 + T^3 + T^2 + T + 1
$5$	$ T^{4} + T^{3} + 6 T^{2} - 4 T + 1 $ T^4 + T^3 + 6T^2 - 4T + 1
$7$	$ T^{4} + 3 T^{3} + 9 T^{2} + 27 T + 81 $ T^4 + 3T^3 + 9T^2 + 27*T + 81
$11$	$ T^{4} - 9 T^{3} + 41 T^{2} - 99 T + 121 $ T^4 - 9T^3 + 41T^2 - 99*T + 121
$13$	$ T^{4} + 9 T^{3} + 31 T^{2} - 11 T + 121 $ T^4 + 9T^3 + 31T^2 - 11*T + 121
$17$	$ T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1 $ T^4 - 2T^3 + 4T^2 - 3*T + 1
$19$	$ T^{4} + 10 T^{3} + 40 T^{2} + 25 T + 25 $ T^4 + 10T^3 + 40T^2 + 25*T + 25
$23$	$ (T^{2} + 2 T - 19)^{2} $ (T^2 + 2*T - 19)^2
$29$	$ T^{4} + 10 T^{3} + 60 T^{2} + \cdots + 400 $ T^4 + 10T^3 + 60T^2 + 200*T + 400
$31$	$ T^{4} - 8 T^{3} + 34 T^{2} - 77 T + 121 $ T^4 - 8T^3 + 34T^2 - 77*T + 121
$37$	$ T^{4} + 3 T^{3} + 19 T^{2} + 7 T + 1 $ T^4 + 3T^3 + 19T^2 + 7*T + 1
$41$	$ T^{4} - 23 T^{3} + 249 T^{2} + \cdots + 5041 $ T^4 - 23T^3 + 249T^2 - 1207*T + 5041
$43$	$ (T^{2} - 8 T + 11)^{2} $ (T^2 - 8*T + 11)^2
$47$	$ T^{4} + 3 T^{3} + 4 T^{2} + 2 T + 1 $ T^4 + 3T^3 + 4T^2 + 2*T + 1
$53$	$ T^{4} - 6 T^{3} + 76 T^{2} + \cdots + 5041 $ T^4 - 6T^3 + 76T^2 - 781*T + 5041
$59$	$ T^{4} + 20 T^{3} + 190 T^{2} + \cdots + 3025 $ T^4 + 20T^3 + 190T^2 + 825*T + 3025
$61$	$ T^{4} - 3 T^{3} + 54 T^{2} + 108 T + 81 $ T^4 - 3T^3 + 54T^2 + 108*T + 81
$67$	$ (T^{2} - T - 101)^{2} $ (T^2 - T - 101)^2
$71$	$ T^{4} + 27 T^{3} + 324 T^{2} + \cdots + 6561 $ T^4 + 27T^3 + 324T^2 + 1458*T + 6561
$73$	$ T^{4} - 6 T^{3} + 16 T^{2} - 16 T + 16 $ T^4 - 6T^3 + 16T^2 - 16*T + 16
$79$	$ T^{4} - 5 T^{3} + 85 T^{2} + 75 T + 25 $ T^4 - 5T^3 + 85T^2 + 75*T + 25
$83$	$ T^{4} - 21 T^{3} + 171 T^{2} + \cdots + 81 $ T^4 - 21T^3 + 171T^2 - 81*T + 81
$89$	$ (T^{2} - 10 T + 5)^{2} $ (T^2 - 10*T + 5)^2
$97$	$ T^{4} + 33 T^{3} + 634 T^{2} + \cdots + 44521 $ T^4 + 33T^3 + 634T^2 + 6752*T + 44521
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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:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	33.e (of order \(5\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{10}^{2} - 1) q^{2} - \zeta_{10}^{3} q^{3} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10}) q^{4} + ( - \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 1) q^{5} + (\zeta_{10}^{3} - 1) q^{6} + 3 \zeta_{10}^{2} q^{7} + (\zeta_{10}^{3} + 2 \zeta_{10} - 2) q^{8} - \zeta_{10} q^{9} +O(q^{10}) \) q + (-z^2 - 1) * q^2 - z^3 * q^3 + (z^3 - z^2 + z) * q^4 + (-z^3 + 2z^2 - 2z + 1) * q^5 + (z^3 - 1) * q^6 + 3z^2 q^7 + (z^3 + 2z - 2) q^8 - z * q^9	\( q + ( - \zeta_{10}^{2} - 1) q^{2} - \zeta_{10}^{3} q^{3} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10}) q^{4} + ( - \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 1) q^{5} + (\zeta_{10}^{3} - 1) q^{6} + 3 \zeta_{10}^{2} q^{7} + (\zeta_{10}^{3} + 2 \zeta_{10} - 2) q^{8} - \zeta_{10} q^{9} + (\zeta_{10}^{3} - \zeta_{10}^{2}) q^{10} + (\zeta_{10}^{3} - \zeta_{10}^{2} + 3 \zeta_{10} + 1) q^{11} + ( - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{12} + ( - 2 \zeta_{10}^{2} - 3 \zeta_{10} - 2) q^{13} + ( - 3 \zeta_{10}^{3} - 3 \zeta_{10} + 3) q^{14} + (\zeta_{10}^{3} - 2 \zeta_{10}^{2} + \zeta_{10}) q^{15} + ( - 3 \zeta_{10}^{3} + 3) q^{16} + ( - \zeta_{10}^{2} + \zeta_{10}) q^{17} + (\zeta_{10}^{3} + \zeta_{10}) q^{18} + ( - \zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{19} + (2 \zeta_{10}^{2} - 3 \zeta_{10} + 2) q^{20} + 3 q^{21} + ( - 3 \zeta_{10}^{3} - \zeta_{10}^{2} - 2 \zeta_{10} - 1) q^{22} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} - 3) q^{23} + (2 \zeta_{10}^{2} - \zeta_{10} + 2) q^{24} + (3 \zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{25} + (5 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 5 \zeta_{10}) q^{26} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{27} + (3 \zeta_{10}^{2} - 3 \zeta_{10}) q^{28} + ( - 4 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 4 \zeta_{10}) q^{29} + (\zeta_{10} - 1) q^{30} + (3 \zeta_{10}^{2} - \zeta_{10} + 3) q^{31} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 4) q^{32} + ( - 4 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{33} - q^{34} + ( - 3 \zeta_{10}^{2} + 6 \zeta_{10} - 3) q^{35} + ( - \zeta_{10} + 1) q^{36} + ( - 2 \zeta_{10}^{3} - \zeta_{10}^{2} - 2 \zeta_{10}) q^{37} + ( - 2 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10} + 2) q^{38} + (5 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 5) q^{39} + (3 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 3 \zeta_{10}) q^{40} + ( - \zeta_{10}^{3} - 8 \zeta_{10} + 8) q^{41} + ( - 3 \zeta_{10}^{2} - 3) q^{42} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 3) q^{43} + (2 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 5 \zeta_{10} - 3) q^{44} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 1) q^{45} + (3 \zeta_{10}^{2} + 4 \zeta_{10} + 3) q^{46} + (\zeta_{10} - 1) q^{47} + ( - 3 \zeta_{10}^{3} - 3 \zeta_{10}) q^{48} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 2) q^{49} + ( - 6 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10} + 6) q^{50} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10}) q^{51} + ( - 2 \zeta_{10}^{3} - 3 \zeta_{10} + 3) q^{52} + ( - \zeta_{10}^{2} + 9 \zeta_{10} - 1) q^{53} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + 1) q^{54} + (\zeta_{10}^{3} + 2 \zeta_{10}^{2} - 5 \zeta_{10} + 5) q^{55} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} - 3) q^{56} + (3 \zeta_{10}^{2} - 4 \zeta_{10} + 3) q^{57} + (6 \zeta_{10}^{3} + 2 \zeta_{10} - 2) q^{58} + ( - 7 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 7 \zeta_{10}) q^{59} + (\zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 1) q^{60} + (3 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} - 3) q^{61} + ( - 2 \zeta_{10}^{3} - 3 \zeta_{10}^{2} - 2 \zeta_{10}) q^{62} - 3 \zeta_{10}^{3} q^{63} + ( - 2 \zeta_{10}^{2} - \zeta_{10} - 2) q^{64} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 3) q^{65} + (3 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - \zeta_{10} - 3) q^{66} + ( - 9 \zeta_{10}^{3} + 9 \zeta_{10}^{2} + 5) q^{67} + ( - \zeta_{10}^{2} + 2 \zeta_{10} - 1) q^{68} + (3 \zeta_{10}^{3} + 4 \zeta_{10} - 4) q^{69} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10}) q^{70} + (9 \zeta_{10}^{3} - 9) q^{71} + ( - \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} + 1) q^{72} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{73} + (5 \zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{74} + (3 \zeta_{10}^{2} + 3) q^{75} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 3) q^{76} + (6 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 3 \zeta_{10}) q^{77} + ( - 5 \zeta_{10}^{3} + 5 \zeta_{10}^{2} + 7) q^{78} + (4 \zeta_{10}^{2} - 7 \zeta_{10} + 4) q^{79} + ( - 3 \zeta_{10} + 3) q^{80} + \zeta_{10}^{2} q^{81} + (9 \zeta_{10}^{3} - 8 \zeta_{10}^{2} + 8 \zeta_{10} - 9) q^{82} + ( - 9 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10} + 9) q^{83} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10}) q^{84} + (\zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{85} + ( - 3 \zeta_{10}^{2} + 2 \zeta_{10} - 3) q^{86} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} - 2) q^{87} + (2 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 6) q^{88} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} + 7) q^{89} + (\zeta_{10}^{2} - \zeta_{10} + 1) q^{90} + ( - 15 \zeta_{10}^{3} - 6 \zeta_{10} + 6) q^{91} + (\zeta_{10}^{3} - 5 \zeta_{10}^{2} + \zeta_{10}) q^{92} + ( - 2 \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} + 2) q^{93} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{94} + (7 \zeta_{10}^{3} - 11 \zeta_{10}^{2} + 7 \zeta_{10}) q^{95} + (4 \zeta_{10}^{3} - \zeta_{10} + 1) q^{96} + ( - 13 \zeta_{10}^{2} + 6 \zeta_{10} - 13) q^{97} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 2) q^{98} + ( - 2 \zeta_{10}^{2} - 2 \zeta_{10} + 1) q^{99} +O(q^{100}) \) q + (-z^2 - 1) * q^2 - z^3 * q^3 + (z^3 - z^2 + z) * q^4 + (-z^3 + 2z^2 - 2z + 1) * q^5 + (z^3 - 1) * q^6 + 3z^2 q^7 + (z^3 + 2z - 2) q^8 - z * q^9 + (z^3 - z^2) * q^10 + (z^3 - z^2 + 3z + 1) q^11 + (-z^3 + z^2) * q^12 + (-2z^2 - 3z - 2) * q^13 + (-3z^3 - 3z + 3) * q^14 + (z^3 - 2z^2 + z) q^15 + (-3z^3 + 3) q^16 + (-z^2 + z) * q^17 + (z^3 + z) * q^18 + (-z^3 + 3z - 3) q^19 + (2z^2 - 3z + 2) * q^20 + 3 * q^21 + (-3z^3 - z^2 - 2z - 1) * q^22 + (4z^3 - 4z^2 - 3) * q^23 + (2z^2 - z + 2) q^24 + (3z^3 + 3z - 3) * q^25 + (5z^3 + 2z^2 + 5z) q^26 + (z^3 - z^2 + z - 1) * q^27 + (3z^2 - 3z) * q^28 + (-4z^3 + 2z^2 - 4z) q^29 + (z - 1) * q^30 + (3z^2 - z + 3) q^31 + (-z^3 + z^2 - 4) * q^32 + (-4z^3 + 3z^2 - 2z + 2) q^33 - q^34 + (-3z^2 + 6z - 3) * q^35 + (-z + 1) * q^36 + (-2z^3 - z^2 - 2z) * q^37 + (-2z^3 + 3z^2 - 3z + 2) q^38 + (5z^3 - 3z^2 + 3z - 5) q^39 + (3z^3 - 4z^2 + 3z) q^40 + (-z^3 - 8z + 8) q^41 + (-3z^2 - 3) q^42 + (2z^3 - 2z^2 + 3) * q^43 + (2z^3 - 3z^2 + 5z - 3) q^44 + (-z^3 + z^2 - 1) * q^45 + (3z^2 + 4z + 3) * q^46 + (z - 1) * q^47 + (-3z^3 - 3z) * q^48 + (2z^3 - 2z^2 + 2z - 2) q^49 + (-6z^3 + 3z^2 - 3z + 6) q^50 + (-z^3 + z^2 - z) * q^51 + (-2z^3 - 3z + 3) * q^52 + (-z^2 + 9z - 1) q^53 + (-z^3 + z^2 + 1) * q^54 + (z^3 + 2z^2 - 5z + 5) * q^55 + (6z^3 - 6z^2 - 3) * q^56 + (3z^2 - 4z + 3) * q^57 + (6z^3 + 2z - 2) * q^58 + (-7z^3 + 6z^2 - 7z) q^59 + (z^3 - 3z^2 + 3z - 1) * q^60 + (3z^3 - 6z^2 + 6z - 3) q^61 + (-2z^3 - 3z^2 - 2z) q^62 - 3z^3 q^63 + (-2z^2 - z - 2) q^64 + (-z^3 + z^2 - 3) * q^65 + (3z^3 - 2z^2 - z - 3) * q^66 + (-9z^3 + 9z^2 + 5) * q^67 + (-z^2 + 2z - 1) q^68 + (3z^3 + 4z - 4) * q^69 + (-3z^3 + 3z^2 - 3z) q^70 + (9z^3 - 9) q^71 + (-z^3 - z^2 + z + 1) * q^72 + (2z^3 - 2z^2 + 2z) q^73 + (5z^3 + 3z - 3) * q^74 + (3z^2 + 3) q^75 + (-4z^3 + 4z^2 - 3) * q^76 + (6z^3 + 6z^2 - 3z) q^77 + (-5z^3 + 5z^2 + 7) * q^78 + (4z^2 - 7z + 4) * q^79 + (-3z + 3) q^80 + z^2 * q^81 + (9z^3 - 8z^2 + 8z - 9) q^82 + (-9z^3 + 3z^2 - 3z + 9) q^83 + (3z^3 - 3z^2 + 3z) q^84 + (z^3 - 2z + 2) q^85 + (-3z^2 + 2z - 3) * q^86 + (4z^3 - 4z^2 - 2) * q^87 + (2z^3 + 3z^2 - 6) * q^88 + (-4z^3 + 4z^2 + 7) * q^89 + (z^2 - z + 1) * q^90 + (-15z^3 - 6z + 6) * q^91 + (z^3 - 5z^2 + z) q^92 + (-2z^3 - z^2 + z + 2) q^93 + (-z^3 + z^2 - z + 1) * q^94 + (7z^3 - 11z^2 + 7z) q^95 + (4z^3 - z + 1) q^96 + (-13z^2 + 6z - 13) * q^97 + (-2z^3 + 2z^2 + 2) * q^98 + (-2z^2 - 2z + 1) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{2} - q^{3} + 3 q^{4} - q^{5} - 3 q^{6} - 3 q^{7} - 5 q^{8} - q^{9}+O(q^{10}) \) 4 * q - 3 * q^2 - q^3 + 3 * q^4 - q^5 - 3 * q^6 - 3 * q^7 - 5 * q^8 - q^9	\( 4 q - 3 q^{2} - q^{3} + 3 q^{4} - q^{5} - 3 q^{6} - 3 q^{7} - 5 q^{8} - q^{9} + 2 q^{10} + 9 q^{11} - 2 q^{12} - 9 q^{13} + 6 q^{14} + 4 q^{15} + 9 q^{16} + 2 q^{17} + 2 q^{18} - 10 q^{19} + 3 q^{20} + 12 q^{21} - 8 q^{22} - 4 q^{23} + 5 q^{24} - 6 q^{25} + 8 q^{26} - q^{27} - 6 q^{28} - 10 q^{29} - 3 q^{30} + 8 q^{31} - 18 q^{32} - q^{33} - 4 q^{34} - 3 q^{35} + 3 q^{36} - 3 q^{37} - 9 q^{39} + 10 q^{40} + 23 q^{41} - 9 q^{42} + 16 q^{43} - 2 q^{44} - 6 q^{45} + 13 q^{46} - 3 q^{47} - 6 q^{48} - 2 q^{49} + 12 q^{50} - 3 q^{51} + 7 q^{52} + 6 q^{53} + 2 q^{54} + 14 q^{55} + 5 q^{57} - 20 q^{59} + 3 q^{60} + 3 q^{61} - q^{62} - 3 q^{63} - 7 q^{64} - 14 q^{65} - 8 q^{66} + 2 q^{67} - q^{68} - 9 q^{69} - 9 q^{70} - 27 q^{71} + 5 q^{72} + 6 q^{73} - 4 q^{74} + 9 q^{75} - 20 q^{76} - 3 q^{77} + 18 q^{78} + 5 q^{79} + 9 q^{80} - q^{81} - 11 q^{82} + 21 q^{83} + 9 q^{84} + 7 q^{85} - 7 q^{86} - 25 q^{88} + 20 q^{89} + 2 q^{90} + 3 q^{91} + 7 q^{92} + 8 q^{93} + q^{94} + 25 q^{95} + 7 q^{96} - 33 q^{97} + 4 q^{98} + 4 q^{99}+O(q^{100}) \) 4 * q - 3 * q^2 - q^3 + 3 * q^4 - q^5 - 3 * q^6 - 3 * q^7 - 5 * q^8 - q^9 + 2 * q^10 + 9 * q^11 - 2 * q^12 - 9 * q^13 + 6 * q^14 + 4 * q^15 + 9 * q^16 + 2 * q^17 + 2 * q^18 - 10 * q^19 + 3 * q^20 + 12 * q^21 - 8 * q^22 - 4 * q^23 + 5 * q^24 - 6 * q^25 + 8 * q^26 - q^27 - 6 * q^28 - 10 * q^29 - 3 * q^30 + 8 * q^31 - 18 * q^32 - q^33 - 4 * q^34 - 3 * q^35 + 3 * q^36 - 3 * q^37 - 9 * q^39 + 10 * q^40 + 23 * q^41 - 9 * q^42 + 16 * q^43 - 2 * q^44 - 6 * q^45 + 13 * q^46 - 3 * q^47 - 6 * q^48 - 2 * q^49 + 12 * q^50 - 3 * q^51 + 7 * q^52 + 6 * q^53 + 2 * q^54 + 14 * q^55 + 5 * q^57 - 20 * q^59 + 3 * q^60 + 3 * q^61 - q^62 - 3 * q^63 - 7 * q^64 - 14 * q^65 - 8 * q^66 + 2 * q^67 - q^68 - 9 * q^69 - 9 * q^70 - 27 * q^71 + 5 * q^72 + 6 * q^73 - 4 * q^74 + 9 * q^75 - 20 * q^76 - 3 * q^77 + 18 * q^78 + 5 * q^79 + 9 * q^80 - q^81 - 11 * q^82 + 21 * q^83 + 9 * q^84 + 7 * q^85 - 7 * q^86 - 25 * q^88 + 20 * q^89 + 2 * q^90 + 3 * q^91 + 7 * q^92 + 8 * q^93 + q^94 + 25 * q^95 + 7 * q^96 - 33 * q^97 + 4 * q^98 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

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	33.2.e.a

Level	$33$
Weight	$2$
Character orbit	33.e
Analytic conductor	$0.264$
Analytic rank	$0$
Dimension	$4$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(0.263506326670\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\zeta_{10})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{4} - x^{3} + x^{2} - x + 1 \) x^4 - x^3 + x^2 - x + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

$p$	$F_p(T)$
$2$	\( T^{4} + 3 T^{3} + 4 T^{2} + 2 T + 1 \) T^4 + 3T^3 + 4T^2 + 2*T + 1
$3$	\( T^{4} + T^{3} + T^{2} + T + 1 \) T^4 + T^3 + T^2 + T + 1
$5$	\( T^{4} + T^{3} + 6 T^{2} - 4 T + 1 \) T^4 + T^3 + 6T^2 - 4T + 1
$7$	\( T^{4} + 3 T^{3} + 9 T^{2} + 27 T + 81 \) T^4 + 3T^3 + 9T^2 + 27*T + 81
$11$	\( T^{4} - 9 T^{3} + 41 T^{2} - 99 T + 121 \) T^4 - 9T^3 + 41T^2 - 99*T + 121
$13$	\( T^{4} + 9 T^{3} + 31 T^{2} - 11 T + 121 \) T^4 + 9T^3 + 31T^2 - 11*T + 121
$17$	\( T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1 \) T^4 - 2T^3 + 4T^2 - 3*T + 1
$19$	\( T^{4} + 10 T^{3} + 40 T^{2} + 25 T + 25 \) T^4 + 10T^3 + 40T^2 + 25*T + 25
$23$	\( (T^{2} + 2 T - 19)^{2} \) (T^2 + 2*T - 19)^2
$29$	\( T^{4} + 10 T^{3} + 60 T^{2} + \cdots + 400 \) T^4 + 10T^3 + 60T^2 + 200*T + 400
$31$	\( T^{4} - 8 T^{3} + 34 T^{2} - 77 T + 121 \) T^4 - 8T^3 + 34T^2 - 77*T + 121
$37$	\( T^{4} + 3 T^{3} + 19 T^{2} + 7 T + 1 \) T^4 + 3T^3 + 19T^2 + 7*T + 1
$41$	\( T^{4} - 23 T^{3} + 249 T^{2} + \cdots + 5041 \) T^4 - 23T^3 + 249T^2 - 1207*T + 5041
$43$	\( (T^{2} - 8 T + 11)^{2} \) (T^2 - 8*T + 11)^2
$47$	\( T^{4} + 3 T^{3} + 4 T^{2} + 2 T + 1 \) T^4 + 3T^3 + 4T^2 + 2*T + 1
$53$	\( T^{4} - 6 T^{3} + 76 T^{2} + \cdots + 5041 \) T^4 - 6T^3 + 76T^2 - 781*T + 5041
$59$	\( T^{4} + 20 T^{3} + 190 T^{2} + \cdots + 3025 \) T^4 + 20T^3 + 190T^2 + 825*T + 3025
$61$	\( T^{4} - 3 T^{3} + 54 T^{2} + 108 T + 81 \) T^4 - 3T^3 + 54T^2 + 108*T + 81
$67$	\( (T^{2} - T - 101)^{2} \) (T^2 - T - 101)^2
$71$	\( T^{4} + 27 T^{3} + 324 T^{2} + \cdots + 6561 \) T^4 + 27T^3 + 324T^2 + 1458*T + 6561
$73$	\( T^{4} - 6 T^{3} + 16 T^{2} - 16 T + 16 \) T^4 - 6T^3 + 16T^2 - 16*T + 16
$79$	\( T^{4} - 5 T^{3} + 85 T^{2} + 75 T + 25 \) T^4 - 5T^3 + 85T^2 + 75*T + 25
$83$	\( T^{4} - 21 T^{3} + 171 T^{2} + \cdots + 81 \) T^4 - 21T^3 + 171T^2 - 81*T + 81
$89$	\( (T^{2} - 10 T + 5)^{2} \) (T^2 - 10*T + 5)^2
$97$	\( T^{4} + 33 T^{3} + 634 T^{2} + \cdots + 44521 \) T^4 + 33T^3 + 634T^2 + 6752*T + 44521
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\(n\)	\(13\)	\(23\)
\(\chi(n)\)	\(\zeta_{10}^{2}\)	\(1\)

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