LMFDB - Newform orbit 3332.1.cc.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3332,1,Mod(135,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, base_ring=CyclotomicField(42))

chi = DirichletCharacter(H, H._module([21, 32, 21]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("3332.135");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3332 = 2^{2} \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3332.cc (of order $42$, degree $12$, 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:	$1.66288462209$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$2$ over $\Q(\zeta_{42})$
Coefficient field:	$\Q(\zeta_{84})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{24} + x^{22} - x^{18} - x^{16} + x^{12} - x^{8} - x^{6} + x^{2} + 1 $ x^24 + x^22 - x^18 - x^16 + x^12 - x^8 - x^6 + x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{42}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{42} - \cdots)$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times$.

$n$	$785$	$885$	$1667$
$\chi(n)$	$-1$	$\zeta_{84}^{16}$	$-1$

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} $

135.1

−0.294755	+	0.955573i
0.294755	−	0.955573i
−0.294755	−	0.955573i
0.294755	+	0.955573i
0.680173	−	0.733052i
−0.680173	+	0.733052i
−0.997204	+	0.0747301i
0.997204	−	0.0747301i
0.930874	−	0.365341i
−0.930874	+	0.365341i
−0.149042	+	0.988831i
0.149042	−	0.988831i
−0.997204	−	0.0747301i
0.997204	+	0.0747301i
0.930874	+	0.365341i
−0.930874	−	0.365341i
−0.563320	+	0.826239i
0.563320	−	0.826239i
−0.563320	−	0.826239i
0.563320	+	0.826239i

−0.826239

0.563320i

−0.997204

0.925270i

0.365341

−

0.930874i

0.302705

−

1.32624i

0.974928

−

0.222521i

0.222521

0.974928i

0.0635609

−

0.848162i

135.2

−0.826239

0.563320i

0.997204

−

0.925270i

0.365341

−

0.930874i

−0.302705

1.32624i

−0.974928

0.222521i

0.222521

0.974928i

0.0635609

−

0.848162i

543.1

−0.826239

−

0.563320i

−0.997204

−

0.925270i

0.365341

0.930874i

0.302705

1.32624i

0.974928

0.222521i

0.222521

−

0.974928i

0.0635609

0.848162i

543.2

−0.826239

−

0.563320i

0.997204

0.925270i

0.365341

0.930874i

−0.302705

−

1.32624i

−0.974928

−

0.222521i

0.222521

−

0.974928i

0.0635609

0.848162i

611.1

−0.0747301

0.997204i

−0.563320

0.173761i

−0.988831

−

0.149042i

−0.131178

−

0.574730i

−0.974928

−

0.222521i

0.222521

−

0.974928i

−0.539102

0.367554i

611.2

−0.0747301

0.997204i

0.563320

−

0.173761i

−0.988831

−

0.149042i

0.131178

0.574730i

0.974928

0.222521i

0.222521

−

0.974928i

−0.539102

0.367554i

1019.1

0.988831

0.149042i

−0.930874

0.634659i

0.955573

0.294755i

−1.01507

0.488831i

0.433884

−

0.900969i

0.900969

0.433884i

0.0983929

−

0.250701i

1019.2

0.988831

0.149042i

0.930874

−

0.634659i

0.955573

0.294755i

1.01507

−

0.488831i

−0.433884

0.900969i

0.900969

0.433884i

0.0983929

−

0.250701i

1087.1

0.733052

0.680173i

−0.294755

−

0.0444272i

0.0747301

0.997204i

−0.185853

−

0.233052i

−0.781831

−

0.623490i

−0.623490

0.781831i

−0.870666

−

0.268565i

1087.2

0.733052

0.680173i

0.294755

0.0444272i

0.0747301

0.997204i

0.185853

0.233052i

0.781831

0.623490i

−0.623490

0.781831i

−0.870666

−

0.268565i

1495.1

−0.955573

0.294755i

−0.680173

−

1.73305i

0.826239

−

0.563320i

1.16078

1.45557i

−0.781831

−

0.623490i

−0.623490

0.781831i

−1.80778

1.67738i

1495.2

−0.955573

0.294755i

0.680173

1.73305i

0.826239

−

0.563320i

−1.16078

−

1.45557i

0.781831

0.623490i

−0.623490

0.781831i

−1.80778

1.67738i

1563.1

0.988831

−

0.149042i

−0.930874

−

0.634659i

0.955573

−

0.294755i

−1.01507

−

0.488831i

0.433884

0.900969i

0.900969

−

0.433884i

0.0983929

0.250701i

1563.2

0.988831

−

0.149042i

0.930874

0.634659i

0.955573

−

0.294755i

1.01507

0.488831i

−0.433884

−

0.900969i

0.900969

−

0.433884i

0.0983929

0.250701i

1971.1

0.733052

−

0.680173i

−0.294755

0.0444272i

0.0747301

−

0.997204i

−0.185853

0.233052i

−0.781831

0.623490i

−0.623490

−

0.781831i

−0.870666

0.268565i

1971.2

0.733052

−

0.680173i

0.294755

−

0.0444272i

0.0747301

−

0.997204i

0.185853

−

0.233052i

0.781831

−

0.623490i

−0.623490

−

0.781831i

−0.870666

0.268565i

2447.1

−0.365341

0.930874i

−0.149042

1.98883i

−0.733052

−

0.680173i

−1.79690

−

0.865341i

−0.433884

−

0.900969i

0.900969

−

0.433884i

−2.94440

−

0.443797i

2447.2

−0.365341

0.930874i

0.149042

−

1.98883i

−0.733052

−

0.680173i

1.79690

0.865341i

0.433884

0.900969i

0.900969

−

0.433884i

−2.94440

−

0.443797i

2515.1

−0.365341

−

0.930874i

−0.149042

−

1.98883i

−0.733052

0.680173i

−1.79690

0.865341i

−0.433884

0.900969i

0.900969

0.433884i

−2.94440

0.443797i

2515.2

−0.365341

−

0.930874i

0.149042

1.98883i

−0.733052

0.680173i

1.79690

−

0.865341i

0.433884

−

0.900969i

0.900969

0.433884i

−2.94440

0.443797i

See all 24 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
68.d	odd	2	1	CM by $\Q(\sqrt{-17}) $
4.b	odd	2	1	inner
17.b	even	2	1	inner
49.g	even	21	1	inner
196.o	odd	42	1	inner
833.z	even	42	1	inner
3332.cc	odd	42	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3332.1.cc.c	✓	24
4.b	odd	2	1	inner	3332.1.cc.c	✓	24
17.b	even	2	1	inner	3332.1.cc.c	✓	24
49.g	even	21	1	inner	3332.1.cc.c	✓	24
68.d	odd	2	1	CM	3332.1.cc.c	✓	24
196.o	odd	42	1	inner	3332.1.cc.c	✓	24
833.z	even	42	1	inner	3332.1.cc.c	✓	24
3332.cc	odd	42	1	inner	3332.1.cc.c	✓	24

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3332.1.cc.c	✓	24	1.a	even	1	1	trivial
3332.1.cc.c	✓	24	4.b	odd	2	1	inner
3332.1.cc.c	✓	24	17.b	even	2	1	inner
3332.1.cc.c	✓	24	49.g	even	21	1	inner
3332.1.cc.c	✓	24	68.d	odd	2	1	CM
3332.1.cc.c	✓	24	196.o	odd	42	1	inner
3332.1.cc.c	✓	24	833.z	even	42	1	inner
3332.1.cc.c	✓	24	3332.cc	odd	42	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{24} + 11 T_{3}^{22} + 49 T_{3}^{20} + 118 T_{3}^{18} + 192 T_{3}^{16} + 336 T_{3}^{14} + 421 T_{3}^{12} - 168 T_{3}^{10} + 999 T_{3}^{8} - 736 T_{3}^{6} + 231 T_{3}^{4} - 26 T_{3}^{2} + 1 $ T3^24 + 11*T3^22 + 49*T3^20 + 118*T3^18 + 192*T3^16 + 336*T3^14 + 421*T3^12 - 168*T3^10 + 999*T3^8 - 736*T3^6 + 231*T3^4 - 26*T3^2 + 1 acting on $S_{1}^{\mathrm{new}}(3332, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{12} + T^{11} - T^{9} - T^{8} + T^{6} - T^{4} - T^{3} + \cdots + 1)^{2} $ (T^12 + T^11 - T^9 - T^8 + T^6 - T^4 - T^3 + T + 1)^2
$3$	$ T^{24} + 11 T^{22} + 49 T^{20} + 118 T^{18} + \cdots + 1 $ T^24 + 11T^22 + 49T^20 + 118T^18 + 192T^16 + 336T^14 + 421T^12 - 168T^10 + 999T^8 - 736T^6 + 231T^4 - 26*T^2 + 1
$5$	$ T^{24} $ T^24
$7$	$ (T^{12} - T^{10} + T^{8} - T^{6} + T^{4} - T^{2} + 1)^{2} $ (T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1)^2
$11$	$ T^{24} - 10 T^{22} + 49 T^{20} - 134 T^{18} + \cdots + 1 $ T^24 - 10T^22 + 49T^20 - 134T^18 + 108T^16 + 378T^14 - 713T^12 - 84T^10 + 957T^8 + 62T^6 + 126T^4 + 16*T^2 + 1
$13$	$ (T^{12} - 5 T^{11} + 17 T^{10} - 38 T^{9} + \cdots + 1)^{2} $ (T^12 - 5T^11 + 17T^10 - 38T^9 + 68T^8 - 92T^7 + 105T^6 - 92T^5 + 61T^4 - 10T^3 + 45T^2 - 12*T + 1)^2
$17$	$ (T^{12} - T^{11} + T^{9} - T^{8} + T^{6} - T^{4} + T^{3} + \cdots + 1)^{2} $ (T^12 - T^11 + T^9 - T^8 + T^6 - T^4 + T^3 - T + 1)^2
$19$	$ T^{24} $ T^24
$23$	$ T^{24} + 70 T^{18} - 98 T^{16} + \cdots + 2401 $ T^24 + 70T^18 - 98T^16 - 49T^14 + 1323T^12 - 3430T^10 + 7889T^8 - 1372T^6 - 2401T^4 - 2401*T^2 + 2401
$29$	$ T^{24} $ T^24
$31$	$ (T^{12} + 7 T^{10} + 35 T^{8} + 84 T^{6} + \cdots + 49)^{2} $ (T^12 + 7T^10 + 35T^8 + 84T^6 + 147T^4 + 98*T^2 + 49)^2
$37$	$ T^{24} $ T^24
$41$	$ T^{24} $ T^24
$43$	$ T^{24} $ T^24
$47$	$ T^{24} $ T^24
$53$	$ (T^{12} + T^{11} - T^{9} + 6 T^{8} - 21 T^{7} + \cdots + 1)^{2} $ (T^12 + T^11 - T^9 + 6T^8 - 21T^7 - 20T^6 + 69T^4 - 29T^3 + 49T^2 - 13*T + 1)^2
$59$	$ T^{24} $ T^24
$61$	$ T^{24} $ T^24
$67$	$ T^{24} $ T^24
$71$	$ T^{24} + 6 T^{22} + 13 T^{20} + 38 T^{18} + \cdots + 1 $ T^24 + 6T^22 + 13T^20 + 38T^18 + 258T^16 + 716T^14 + 1519T^12 + 1978T^10 + 1284T^8 + 234T^6 + 96T^4 - 11*T^2 + 1
$73$	$ T^{24} $ T^24
$79$	$ T^{24} + 11 T^{22} + 77 T^{20} + 328 T^{18} + \cdots + 1 $ T^24 + 11T^22 + 77T^20 + 328T^18 + 1018T^16 + 2128T^14 + 3270T^12 + 3283T^10 + 2308T^8 + 804T^6 + 196T^4 + 16*T^2 + 1
$83$	$ T^{24} $ T^24
$89$	$ (T^{12} - T^{11} - 6 T^{9} + 6 T^{8} - 7 T^{7} + \cdots + 1)^{2} $ (T^12 - T^11 - 6T^9 + 6T^8 - 7T^7 + 43T^6 - 35T^5 + 90T^4 - 104T^3 + 70T^2 - 15*T + 1)^2
$97$	$ T^{24} $ T^24
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Level:	\( N \)	\(=\)	\( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3332.cc (of order \(42\), degree \(12\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{84}^{40} q^{2} + ( - \zeta_{84}^{37} + \zeta_{84}^{21}) q^{3} - \zeta_{84}^{38} q^{4} + ( - \zeta_{84}^{35} + \zeta_{84}^{19}) q^{6} - \zeta_{84}^{15} q^{7} - \zeta_{84}^{36} q^{8} + ( - \zeta_{84}^{32} + \zeta_{84}^{16} - 1) q^{9} +O(q^{10}) \) q - z^40 * q^2 + (-z^37 + z^21) * q^3 - z^38 * q^4 + (-z^35 + z^19) * q^6 - z^15 * q^7 - z^36 * q^8 + (-z^32 + z^16 - 1) * q^9	\( q - \zeta_{84}^{40} q^{2} + ( - \zeta_{84}^{37} + \zeta_{84}^{21}) q^{3} - \zeta_{84}^{38} q^{4} + ( - \zeta_{84}^{35} + \zeta_{84}^{19}) q^{6} - \zeta_{84}^{15} q^{7} - \zeta_{84}^{36} q^{8} + ( - \zeta_{84}^{32} + \zeta_{84}^{16} - 1) q^{9} + ( - \zeta_{84}^{7} + \zeta_{84}^{3}) q^{11} + ( - \zeta_{84}^{33} + \zeta_{84}^{17}) q^{12} + (\zeta_{84}^{14} + \zeta_{84}^{10}) q^{13} - \zeta_{84}^{13} q^{14} - \zeta_{84}^{34} q^{16} + \zeta_{84}^{32} q^{17} + (\zeta_{84}^{40} - \zeta_{84}^{30} + \zeta_{84}^{14}) q^{18} + ( - \zeta_{84}^{36} - \zeta_{84}^{10}) q^{21} + ( - \zeta_{84}^{5} + \zeta_{84}) q^{22} + ( - \zeta_{84}^{19} - \zeta_{84}) q^{23} + ( - \zeta_{84}^{31} + \zeta_{84}^{15}) q^{24} - \zeta_{84}^{2} q^{25} + (\zeta_{84}^{12} + \zeta_{84}^{8}) q^{26} + (\zeta_{84}^{37} - \zeta_{84}^{27} - \zeta_{84}^{21} - \zeta_{84}^{11}) q^{27} - \zeta_{84}^{11} q^{28} + ( - \zeta_{84}^{41} + \zeta_{84}^{29}) q^{31} - \zeta_{84}^{32} q^{32} + ( - \zeta_{84}^{40} - \zeta_{84}^{28} + \zeta_{84}^{24} - \zeta_{84}^{2}) q^{33} + \zeta_{84}^{30} q^{34} + (\zeta_{84}^{38} - \zeta_{84}^{28} + \zeta_{84}^{12}) q^{36} + (\zeta_{84}^{35} + \zeta_{84}^{31} + \zeta_{84}^{9} + \zeta_{84}^{5}) q^{39} + ( - \zeta_{84}^{34} - \zeta_{84}^{8}) q^{42} + ( - \zeta_{84}^{41} - \zeta_{84}^{3}) q^{44} + (\zeta_{84}^{41} - \zeta_{84}^{17}) q^{46} + ( - \zeta_{84}^{29} + \zeta_{84}^{13}) q^{48} + \zeta_{84}^{30} q^{49} - q^{50} + (\zeta_{84}^{27} - \zeta_{84}^{11}) q^{51} + (\zeta_{84}^{10} + \zeta_{84}^{6}) q^{52} + ( - \zeta_{84}^{18} + \zeta_{84}^{16}) q^{53} + (\zeta_{84}^{35} - \zeta_{84}^{25} - \zeta_{84}^{19} + \zeta_{84}^{9}) q^{54} - \zeta_{84}^{9} q^{56} + ( - \zeta_{84}^{39} + \zeta_{84}^{27}) q^{62} + ( - \zeta_{84}^{31} + \zeta_{84}^{15} - \zeta_{84}^{5}) q^{63} - \zeta_{84}^{30} q^{64} + ( - \zeta_{84}^{38} - \zeta_{84}^{26} + \zeta_{84}^{22} - 1) q^{66} + \zeta_{84}^{28} q^{68} + ( - \zeta_{84}^{40} + \zeta_{84}^{38} - \zeta_{84}^{22} - \zeta_{84}^{14}) q^{69} + (\zeta_{84}^{25} + \zeta_{84}^{23}) q^{71} + (\zeta_{84}^{36} - \zeta_{84}^{26} + \zeta_{84}^{10}) q^{72} + (\zeta_{84}^{39} - \zeta_{84}^{23}) q^{75} + (\zeta_{84}^{22} - \zeta_{84}^{18}) q^{77} + (\zeta_{84}^{33} + \zeta_{84}^{29} + \zeta_{84}^{7} + \zeta_{84}^{3}) q^{78} + (\zeta_{84}^{33} + \zeta_{84}^{23}) q^{79} + (\zeta_{84}^{32} - \zeta_{84}^{22} - \zeta_{84}^{16} - \zeta_{84}^{6} + 1) q^{81} + ( - \zeta_{84}^{32} - \zeta_{84}^{6}) q^{84} + ( - \zeta_{84}^{39} - \zeta_{84}) q^{88} + ( - \zeta_{84}^{24} - \zeta_{84}^{8}) q^{89} + ( - \zeta_{84}^{29} - \zeta_{84}^{25}) q^{91} + (\zeta_{84}^{39} - \zeta_{84}^{15}) q^{92} + ( - \zeta_{84}^{36} + \zeta_{84}^{24} + \zeta_{84}^{20} - \zeta_{84}^{8}) q^{93} + ( - \zeta_{84}^{27} + \zeta_{84}^{11}) q^{96} + \zeta_{84}^{28} q^{98} + (\zeta_{84}^{39} - \zeta_{84}^{35} - \zeta_{84}^{23} + \zeta_{84}^{19} + \zeta_{84}^{7} - \zeta_{84}^{3}) q^{99} +O(q^{100}) \) q - z^40 * q^2 + (-z^37 + z^21) * q^3 - z^38 * q^4 + (-z^35 + z^19) * q^6 - z^15 * q^7 - z^36 * q^8 + (-z^32 + z^16 - 1) * q^9 + (-z^7 + z^3) * q^11 + (-z^33 + z^17) * q^12 + (z^14 + z^10) * q^13 - z^13 * q^14 - z^34 * q^16 + z^32 * q^17 + (z^40 - z^30 + z^14) * q^18 + (-z^36 - z^10) * q^21 + (-z^5 + z) * q^22 + (-z^19 - z) * q^23 + (-z^31 + z^15) * q^24 - z^2 * q^25 + (z^12 + z^8) * q^26 + (z^37 - z^27 - z^21 - z^11) * q^27 - z^11 * q^28 + (-z^41 + z^29) * q^31 - z^32 * q^32 + (-z^40 - z^28 + z^24 - z^2) * q^33 + z^30 * q^34 + (z^38 - z^28 + z^12) * q^36 + (z^35 + z^31 + z^9 + z^5) * q^39 + (-z^34 - z^8) * q^42 + (-z^41 - z^3) * q^44 + (z^41 - z^17) * q^46 + (-z^29 + z^13) * q^48 + z^30 * q^49 - q^50 + (z^27 - z^11) * q^51 + (z^10 + z^6) * q^52 + (-z^18 + z^16) * q^53 + (z^35 - z^25 - z^19 + z^9) * q^54 - z^9 * q^56 + (-z^39 + z^27) * q^62 + (-z^31 + z^15 - z^5) * q^63 - z^30 * q^64 + (-z^38 - z^26 + z^22 - 1) * q^66 + z^28 * q^68 + (-z^40 + z^38 - z^22 - z^14) * q^69 + (z^25 + z^23) * q^71 + (z^36 - z^26 + z^10) * q^72 + (z^39 - z^23) * q^75 + (z^22 - z^18) * q^77 + (z^33 + z^29 + z^7 + z^3) * q^78 + (z^33 + z^23) * q^79 + (z^32 - z^22 - z^16 - z^6 + 1) * q^81 + (-z^32 - z^6) * q^84 + (-z^39 - z) * q^88 + (-z^24 - z^8) * q^89 + (-z^29 - z^25) * q^91 + (z^39 - z^15) * q^92 + (-z^36 + z^24 + z^20 - z^8) * q^93 + (-z^27 + z^11) * q^96 + z^28 * q^98 + (z^39 - z^35 - z^23 + z^19 + z^7 - z^3) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 2 q^{2} + 2 q^{4} + 4 q^{8} - 24 q^{9}+O(q^{10}) \) 24 * q - 2 * q^2 + 2 * q^4 + 4 * q^8 - 24 * q^9	\( 24 q - 2 q^{2} + 2 q^{4} + 4 q^{8} - 24 q^{9} + 10 q^{13} + 2 q^{16} + 2 q^{17} + 10 q^{18} + 6 q^{21} + 2 q^{25} - 2 q^{26} - 2 q^{32} + 8 q^{33} + 4 q^{34} + 6 q^{36} + 4 q^{49} - 24 q^{50} + 2 q^{52} - 2 q^{53} - 4 q^{64} - 22 q^{66} - 12 q^{68} - 14 q^{69} - 4 q^{72} - 6 q^{77} + 30 q^{81} - 6 q^{84} + 2 q^{89} - 12 q^{98}+O(q^{100}) \) 24 * q - 2 * q^2 + 2 * q^4 + 4 * q^8 - 24 * q^9 + 10 * q^13 + 2 * q^16 + 2 * q^17 + 10 * q^18 + 6 * q^21 + 2 * q^25 - 2 * q^26 - 2 * q^32 + 8 * q^33 + 4 * q^34 + 6 * q^36 + 4 * q^49 - 24 * q^50 + 2 * q^52 - 2 * q^53 - 4 * q^64 - 22 * q^66 - 12 * q^68 - 14 * q^69 - 4 * q^72 - 6 * q^77 + 30 * q^81 - 6 * q^84 + 2 * q^89 - 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	3332.1.cc.c

Level	$3332$
Weight	$1$
Character orbit	3332.cc
Analytic conductor	$1.663$
Analytic rank	$0$
Dimension	$24$
Projective image	$D_{42}$
CM discriminant	-68
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(1.66288462209\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(2\) over \(\Q(\zeta_{42})\)
Coefficient field:	\(\Q(\zeta_{84})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{24} + x^{22} - x^{18} - x^{16} + x^{12} - x^{8} - x^{6} + x^{2} + 1 \) x^24 + x^22 - x^18 - x^16 + x^12 - x^8 - x^6 + x^2 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{42}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{42} - \cdots)\)

\(n\)	\(785\)	\(885\)	\(1667\)
\(\chi(n)\)	\(-1\)	\(\zeta_{84}^{16}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 135.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{12} + T^{11} - T^{9} - T^{8} + T^{6} - T^{4} - T^{3} + \cdots + 1)^{2} \) (T^12 + T^11 - T^9 - T^8 + T^6 - T^4 - T^3 + T + 1)^2
$3$	\( T^{24} + 11 T^{22} + 49 T^{20} + 118 T^{18} + \cdots + 1 \) T^24 + 11T^22 + 49T^20 + 118T^18 + 192T^16 + 336T^14 + 421T^12 - 168T^10 + 999T^8 - 736T^6 + 231T^4 - 26*T^2 + 1
$5$	\( T^{24} \) T^24
$7$	\( (T^{12} - T^{10} + T^{8} - T^{6} + T^{4} - T^{2} + 1)^{2} \) (T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1)^2
$11$	\( T^{24} - 10 T^{22} + 49 T^{20} - 134 T^{18} + \cdots + 1 \) T^24 - 10T^22 + 49T^20 - 134T^18 + 108T^16 + 378T^14 - 713T^12 - 84T^10 + 957T^8 + 62T^6 + 126T^4 + 16*T^2 + 1
$13$	\( (T^{12} - 5 T^{11} + 17 T^{10} - 38 T^{9} + \cdots + 1)^{2} \) (T^12 - 5T^11 + 17T^10 - 38T^9 + 68T^8 - 92T^7 + 105T^6 - 92T^5 + 61T^4 - 10T^3 + 45T^2 - 12*T + 1)^2
$17$	\( (T^{12} - T^{11} + T^{9} - T^{8} + T^{6} - T^{4} + T^{3} + \cdots + 1)^{2} \) (T^12 - T^11 + T^9 - T^8 + T^6 - T^4 + T^3 - T + 1)^2
$19$	\( T^{24} \) T^24
$23$	\( T^{24} + 70 T^{18} - 98 T^{16} + \cdots + 2401 \) T^24 + 70T^18 - 98T^16 - 49T^14 + 1323T^12 - 3430T^10 + 7889T^8 - 1372T^6 - 2401T^4 - 2401*T^2 + 2401
$29$	\( T^{24} \) T^24
$31$	\( (T^{12} + 7 T^{10} + 35 T^{8} + 84 T^{6} + \cdots + 49)^{2} \) (T^12 + 7T^10 + 35T^8 + 84T^6 + 147T^4 + 98*T^2 + 49)^2
$37$	\( T^{24} \) T^24
$41$	\( T^{24} \) T^24
$43$	\( T^{24} \) T^24
$47$	\( T^{24} \) T^24
$53$	\( (T^{12} + T^{11} - T^{9} + 6 T^{8} - 21 T^{7} + \cdots + 1)^{2} \) (T^12 + T^11 - T^9 + 6T^8 - 21T^7 - 20T^6 + 69T^4 - 29T^3 + 49T^2 - 13*T + 1)^2
$59$	\( T^{24} \) T^24
$61$	\( T^{24} \) T^24
$67$	\( T^{24} \) T^24
$71$	\( T^{24} + 6 T^{22} + 13 T^{20} + 38 T^{18} + \cdots + 1 \) T^24 + 6T^22 + 13T^20 + 38T^18 + 258T^16 + 716T^14 + 1519T^12 + 1978T^10 + 1284T^8 + 234T^6 + 96T^4 - 11*T^2 + 1
$73$	\( T^{24} \) T^24
$79$	\( T^{24} + 11 T^{22} + 77 T^{20} + 328 T^{18} + \cdots + 1 \) T^24 + 11T^22 + 77T^20 + 328T^18 + 1018T^16 + 2128T^14 + 3270T^12 + 3283T^10 + 2308T^8 + 804T^6 + 196T^4 + 16*T^2 + 1
$83$	\( T^{24} \) T^24
$89$	\( (T^{12} - T^{11} - 6 T^{9} + 6 T^{8} - 7 T^{7} + \cdots + 1)^{2} \) (T^12 - T^11 - 6T^9 + 6T^8 - 7T^7 + 43T^6 - 35T^5 + 90T^4 - 104T^3 + 70T^2 - 15*T + 1)^2
$97$	\( T^{24} \) T^24
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