LMFDB - Newform orbit 78.3.l.a

Newspace parameters

Level:	$ N $	$=$	$ 78 = 2 \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	78.l (of order $12$, degree $4$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$2.12534606201$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/78\mathbb{Z}\right)^\times$.

$n$	$53$	$67$
$\chi(n)$	$1$	$\zeta_{12}$

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.

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

7.1

0.866025	−	0.500000i
−0.866025	+	0.500000i
−0.866025	−	0.500000i
0.866025	+	0.500000i

−1.36603

0.366025i

0.866025

−

1.50000i

1.73205

−

1.00000i

−4.73205

−

4.73205i

−0.633975

2.36603i

−2.50000

−

0.669873i

−2.00000

2.00000i

−1.50000

−

2.59808i

8.19615

4.73205i

19.1

0.366025

1.36603i

−0.866025

1.50000i

−1.73205

1.00000i

−1.26795

1.26795i

−2.36603

−

0.633975i

−2.50000

9.33013i

−2.00000

−

2.00000i

−1.50000

−

2.59808i

−2.19615

−

1.26795i

37.1

0.366025

−

1.36603i

−0.866025

−

1.50000i

−1.73205

−

1.00000i

−1.26795

−

1.26795i

−2.36603

0.633975i

−2.50000

−

9.33013i

−2.00000

2.00000i

−1.50000

2.59808i

−2.19615

1.26795i

67.1

−1.36603

−

0.366025i

0.866025

1.50000i

1.73205

1.00000i

−4.73205

4.73205i

−0.633975

−

2.36603i

−2.50000

0.669873i

−2.00000

−

2.00000i

−1.50000

2.59808i

8.19615

−

4.73205i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.f	odd	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	78.3.l.a	✓	4
3.b	odd	2	1		234.3.bb.c		4
13.c	even	3	1		1014.3.f.e		4
13.e	even	6	1		1014.3.f.d		4
13.f	odd	12	1	inner	78.3.l.a	✓	4
13.f	odd	12	1		1014.3.f.d		4
13.f	odd	12	1		1014.3.f.e		4
39.k	even	12	1		234.3.bb.c		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
78.3.l.a	✓	4	1.a	even	1	1	trivial
78.3.l.a	✓	4	13.f	odd	12	1	inner
234.3.bb.c		4	3.b	odd	2	1
234.3.bb.c		4	39.k	even	12	1
1014.3.f.d		4	13.e	even	6	1
1014.3.f.d		4	13.f	odd	12	1
1014.3.f.e		4	13.c	even	3	1
1014.3.f.e		4	13.f	odd	12	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{4} + 12T_{5}^{3} + 72T_{5}^{2} + 144T_{5} + 144 $ T5^4 + 12*T5^3 + 72*T5^2 + 144*T5 + 144 acting on $S_{3}^{\mathrm{new}}(78, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4 $ T^4 + 2T^3 + 2T^2 + 4*T + 4
$3$	$ T^{4} + 3T^{2} + 9 $ T^4 + 3*T^2 + 9
$5$	$ T^{4} + 12 T^{3} + 72 T^{2} + \cdots + 144 $ T^4 + 12T^3 + 72T^2 + 144*T + 144
$7$	$ T^{4} + 10 T^{3} + 125 T^{2} + \cdots + 625 $ T^4 + 10T^3 + 125T^2 + 500*T + 625
$11$	$ T^{4} - 12 T^{3} + 180 T^{2} + \cdots + 24336 $ T^4 - 12T^3 + 180T^2 - 3744*T + 24336
$13$	$ T^{4} - 169 T^{2} + 28561 $ T^4 - 169*T^2 + 28561
$17$	$ T^{4} - 24 T^{3} + 96 T^{2} + \cdots + 9216 $ T^4 - 24T^3 + 96T^2 + 2304*T + 9216
$19$	$ T^{4} - 62 T^{3} + 986 T^{2} + \cdots + 14884 $ T^4 - 62T^3 + 986T^2 + 1220*T + 14884
$23$	$ T^{4} - 48 T^{3} + 636 T^{2} + \cdots + 17424 $ T^4 - 48T^3 + 636T^2 + 6336*T + 17424
$29$	$ T^{4} + 300 T^{2} + 90000 $ T^4 + 300*T^2 + 90000
$31$	$ T^{4} - 106 T^{3} + 5618 T^{2} + \cdots + 1868689 $ T^4 - 106T^3 + 5618T^2 - 144902*T + 1868689
$37$	$ T^{4} + 98 T^{3} + 5882 T^{2} + \cdots + 3587236 $ T^4 + 98T^3 + 5882T^2 + 223492*T + 3587236
$41$	$ T^{4} + 96 T^{3} + 3600 T^{2} + \cdots + 1218816 $ T^4 + 96T^3 + 3600T^2 + 79488*T + 1218816
$43$	$ T^{4} - 30 T^{3} - 201 T^{2} + \cdots + 251001 $ T^4 - 30T^3 - 201T^2 + 15030*T + 251001
$47$	$ T^{4} - 132 T^{3} + 8712 T^{2} + \cdots + 685584 $ T^4 - 132T^3 + 8712T^2 - 109296*T + 685584
$53$	$ (T^{2} - 36 T - 6024)^{2} $ (T^2 - 36*T - 6024)^2
$59$	$ T^{4} - 84 T^{3} + 4068 T^{2} + \cdots + 16353936 $ T^4 - 84T^3 + 4068T^2 - 388224*T + 16353936
$61$	$ T^{4} + 72 T^{3} + 5763 T^{2} + \cdots + 335241 $ T^4 + 72T^3 + 5763T^2 - 41688*T + 335241
$67$	$ T^{4} + 148 T^{3} + 8501 T^{2} + \cdots + 6723649 $ T^4 + 148T^3 + 8501T^2 + 285230*T + 6723649
$71$	$ T^{4} - 180 T^{3} + \cdots + 33315984 $ T^4 - 180T^3 + 8676T^2 - 277056*T + 33315984
$73$	$ T^{4} + 190 T^{3} + \cdots + 13830961 $ T^4 + 190T^3 + 18050T^2 + 706610*T + 13830961
$79$	$ (T^{2} - 48 T + 429)^{2} $ (T^2 - 48*T + 429)^2
$83$	$ T^{4} - 264 T^{3} + \cdots + 21678336 $ T^4 - 264T^3 + 34848T^2 - 1229184*T + 21678336
$89$	$ T^{4} - 288 T^{3} + \cdots + 137170944 $ T^4 - 288T^3 + 21312T^2 - 562176*T + 137170944
$97$	$ T^{4} - 310 T^{3} + 27389 T^{2} + \cdots + 8162449 $ T^4 - 310T^3 + 27389T^2 - 331412*T + 8162449
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Overview	Random
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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:	\( N \)	\(=\)	\( 78 = 2 \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	78.l (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{2} + (\zeta_{12}^{3} + \zeta_{12}) q^{3} + 2 \zeta_{12} q^{4} + (4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 4) q^{5} + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} - 1) q^{6} + (5 \zeta_{12}^{3} - 5 \zeta_{12}^{2}) q^{7} + ( - 2 \zeta_{12}^{3} - 2) q^{8} + (3 \zeta_{12}^{2} - 3) q^{9}+O(q^{10}) \) q + (z^3 - z^2 - z) * q^2 + (z^3 + z) * q^3 + 2z q^4 + (4z^3 + 2z^2 - 2z - 4) q^5 + (-2z^3 - z^2 + z - 1) q^6 + (5z^3 - 5z^2) * q^7 + (-2z^3 - 2) q^8 + (3z^2 - 3) q^9	\( q + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{2} + (\zeta_{12}^{3} + \zeta_{12}) q^{3} + 2 \zeta_{12} q^{4} + (4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 4) q^{5} + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} - 1) q^{6} + (5 \zeta_{12}^{3} - 5 \zeta_{12}^{2}) q^{7} + ( - 2 \zeta_{12}^{3} - 2) q^{8} + (3 \zeta_{12}^{2} - 3) q^{9} + ( - 6 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 6 \zeta_{12} + 4) q^{10} + (10 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 8 \zeta_{12} - 2) q^{11} + (4 \zeta_{12}^{2} - 2) q^{12} + 13 \zeta_{12} q^{13} + ( - 5 \zeta_{12}^{3} + 10 \zeta_{12} - 5) q^{14} + ( - 6 \zeta_{12} - 6) q^{15} + 4 \zeta_{12}^{2} q^{16} + (4 \zeta_{12}^{2} - 12 \zeta_{12} + 4) q^{17} + ( - 3 \zeta_{12}^{3} + 3) q^{18} + (12 \zeta_{12}^{3} - 7 \zeta_{12}^{2} - 19 \zeta_{12} + 19) q^{19} + (4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 8 \zeta_{12} - 8) q^{20} + ( - 10 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 5 \zeta_{12} - 10) q^{21} + ( - 4 \zeta_{12}^{3} - 18 \zeta_{12}^{2} + 2 \zeta_{12} + 18) q^{22} + ( - 18 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 18 \zeta_{12} + 16) q^{23} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 4) q^{24} + (\zeta_{12}^{3} - 24 \zeta_{12}^{2} + 12) q^{25} + ( - 13 \zeta_{12}^{3} - 13) q^{26} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{27} + ( - 10 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 10) q^{28} + ( - 10 \zeta_{12}^{3} - 10 \zeta_{12}) q^{29} + (6 \zeta_{12}^{2} + 6 \zeta_{12} + 6) q^{30} + ( - 29 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 5 \zeta_{12} + 29) q^{31} + ( - 4 \zeta_{12}^{2} - 4 \zeta_{12} + 4) q^{32} + (18 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 12 \zeta_{12} - 12) q^{33} + (16 \zeta_{12}^{3} - 8 \zeta_{12}^{2} - 8 \zeta_{12} + 16) q^{34} + ( - 20 \zeta_{12}^{3} + 10 \zeta_{12}) q^{35} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{36} + (23 \zeta_{12}^{3} + 23 \zeta_{12}^{2} + 13 \zeta_{12} - 36) q^{37} + (26 \zeta_{12}^{3} - 24 \zeta_{12}^{2} + 12) q^{38} + (26 \zeta_{12}^{2} - 13) q^{39} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12} + 12) q^{40} + ( - 8 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 20 \zeta_{12} - 28) q^{41} + ( - 5 \zeta_{12}^{3} + 15 \zeta_{12}^{2} - 5 \zeta_{12}) q^{42} + (5 \zeta_{12}^{2} + 24 \zeta_{12} + 5) q^{43} + (20 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 4 \zeta_{12} - 20) q^{44} + ( - 6 \zeta_{12}^{3} - 12 \zeta_{12}^{2} - 6 \zeta_{12} + 6) q^{45} + (16 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 26 \zeta_{12} - 26) q^{46} + (18 \zeta_{12}^{3} + 30 \zeta_{12}^{2} + 30 \zeta_{12} + 18) q^{47} + (8 \zeta_{12}^{3} - 4 \zeta_{12}) q^{48} + ( - \zeta_{12}^{3} + 25 \zeta_{12}^{2} + \zeta_{12} - 50) q^{49} + (11 \zeta_{12}^{3} + 11 \zeta_{12}^{2} + 13 \zeta_{12} - 24) q^{50} + (12 \zeta_{12}^{3} - 24 \zeta_{12}^{2} + 12) q^{51} + 26 \zeta_{12}^{2} q^{52} + ( - 46 \zeta_{12}^{3} + 92 \zeta_{12} + 18) q^{53} + (3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3 \zeta_{12} + 6) q^{54} + ( - 24 \zeta_{12}^{3} - 60 \zeta_{12}^{2} - 24 \zeta_{12}) q^{55} + (10 \zeta_{12}^{2} - 10 \zeta_{12} + 10) q^{56} + (5 \zeta_{12}^{3} - 26 \zeta_{12}^{2} + 26 \zeta_{12} - 5) q^{57} + (20 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 10 \zeta_{12} + 10) q^{58} + (2 \zeta_{12}^{3} - 46 \zeta_{12}^{2} + 44 \zeta_{12} + 44) q^{59} + ( - 12 \zeta_{12}^{2} - 12 \zeta_{12}) q^{60} + (50 \zeta_{12}^{3} + 36 \zeta_{12}^{2} - 25 \zeta_{12} - 36) q^{61} + (53 \zeta_{12}^{3} + 5 \zeta_{12}^{2} - 53 \zeta_{12} - 10) q^{62} + ( - 15 \zeta_{12} + 15) q^{63} + 8 \zeta_{12}^{3} q^{64} + (26 \zeta_{12}^{3} + 26 \zeta_{12}^{2} - 52 \zeta_{12} - 52) q^{65} + ( - 18 \zeta_{12}^{3} + 36 \zeta_{12} + 6) q^{66} + ( - 12 \zeta_{12}^{3} + 12 \zeta_{12}^{2} - 31 \zeta_{12} - 43) q^{67} + (8 \zeta_{12}^{3} - 24 \zeta_{12}^{2} + 8 \zeta_{12}) q^{68} + (18 \zeta_{12}^{2} + 24 \zeta_{12} + 18) q^{69} + (10 \zeta_{12}^{3} + 20 \zeta_{12}^{2} - 20 \zeta_{12} - 10) q^{70} + (22 \zeta_{12}^{3} - 46 \zeta_{12}^{2} - 68 \zeta_{12} + 68) q^{71} + ( - 6 \zeta_{12}^{2} + 6 \zeta_{12} + 6) q^{72} + ( - 59 \zeta_{12}^{3} + 23 \zeta_{12}^{2} + 23 \zeta_{12} - 59) q^{73} + ( - 72 \zeta_{12}^{3} - 10 \zeta_{12}^{2} + 36 \zeta_{12} + 10) q^{74} + ( - 36 \zeta_{12}^{3} + \zeta_{12}^{2} + 36 \zeta_{12} - 2) q^{75} + ( - 14 \zeta_{12}^{3} - 14 \zeta_{12}^{2} + 38 \zeta_{12} - 24) q^{76} + (30 \zeta_{12}^{3} - 80 \zeta_{12}^{2} + 40) q^{77} + ( - 13 \zeta_{12}^{3} - 13 \zeta_{12}^{2} - 13 \zeta_{12} + 26) q^{78} + ( - 7 \zeta_{12}^{3} + 14 \zeta_{12} + 24) q^{79} + (8 \zeta_{12}^{3} - 8 \zeta_{12}^{2} - 16 \zeta_{12} - 8) q^{80} - 9 \zeta_{12}^{2} q^{81} + (28 \zeta_{12}^{2} + 12 \zeta_{12} + 28) q^{82} + ( - 40 \zeta_{12}^{3} + 52 \zeta_{12}^{2} - 52 \zeta_{12} + 40) q^{83} + (10 \zeta_{12}^{3} - 10 \zeta_{12}^{2} - 20 \zeta_{12} + 20) q^{84} + ( - 24 \zeta_{12}^{2} + 24 \zeta_{12} + 24) q^{85} + ( - 19 \zeta_{12}^{3} - 10 \zeta_{12}^{2} - 10 \zeta_{12} - 19) q^{86} + ( - 30 \zeta_{12}^{2} + 30) q^{87} + ( - 36 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 36 \zeta_{12} + 8) q^{88} + (64 \zeta_{12}^{3} + 64 \zeta_{12}^{2} - 104 \zeta_{12} + 40) q^{89} + (18 \zeta_{12}^{3} + 12 \zeta_{12}^{2} - 6) q^{90} + ( - 65 \zeta_{12}^{3} + 65 \zeta_{12}^{2} - 65) q^{91} + ( - 16 \zeta_{12}^{3} + 32 \zeta_{12} + 36) q^{92} + (19 \zeta_{12}^{3} - 19 \zeta_{12}^{2} + 34 \zeta_{12} + 53) q^{93} + ( - 30 \zeta_{12}^{3} - 66 \zeta_{12}^{2} - 30 \zeta_{12}) q^{94} + ( - 10 \zeta_{12}^{2} + 42 \zeta_{12} - 10) q^{95} + ( - 4 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 8 \zeta_{12} + 4) q^{96} + (71 \zeta_{12}^{3} - 13 \zeta_{12}^{2} - 84 \zeta_{12} + 84) q^{97} + ( - 50 \zeta_{12}^{3} + 26 \zeta_{12}^{2} + 24 \zeta_{12} + 24) q^{98} + ( - 24 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 6 \zeta_{12} - 24) q^{99} +O(q^{100}) \) q + (z^3 - z^2 - z) * q^2 + (z^3 + z) * q^3 + 2z q^4 + (4z^3 + 2z^2 - 2z - 4) q^5 + (-2z^3 - z^2 + z - 1) q^6 + (5z^3 - 5z^2) * q^7 + (-2z^3 - 2) q^8 + (3z^2 - 3) q^9 + (-6z^3 - 2z^2 + 6z + 4) q^10 + (10z^3 + 10z^2 - 8z - 2) q^11 + (4z^2 - 2) q^12 + 13z q^13 + (-5z^3 + 10z - 5) * q^14 + (-6z - 6) q^15 + 4z^2 q^16 + (4z^2 - 12z + 4) * q^17 + (-3z^3 + 3) q^18 + (12z^3 - 7z^2 - 19z + 19) q^19 + (4z^3 + 4z^2 - 8z - 8) q^20 + (-10z^3 + 5z^2 + 5z - 10) q^21 + (-4z^3 - 18z^2 + 2z + 18) q^22 + (-18z^3 - 8z^2 + 18z + 16) q^23 + (-2z^3 - 2z^2 - 2z + 4) q^24 + (z^3 - 24z^2 + 12) q^25 + (-13z^3 - 13) q^26 + (3z^3 - 6z) * q^27 + (-10z^3 + 10z^2 - 10) * q^28 + (-10z^3 - 10z) * q^29 + (6z^2 + 6z + 6) * q^30 + (-29z^3 - 5z^2 + 5z + 29) q^31 + (-4z^2 - 4z + 4) * q^32 + (18z^3 - 6z^2 - 12z - 12) q^33 + (16z^3 - 8z^2 - 8z + 16) q^34 + (-20z^3 + 10z) * q^35 + (6z^3 - 6z) * q^36 + (23z^3 + 23z^2 + 13z - 36) q^37 + (26z^3 - 24z^2 + 12) * q^38 + (26z^2 - 13) q^39 + (-4z^3 + 8z + 12) * q^40 + (-8z^3 + 8z^2 - 20z - 28) q^41 + (-5z^3 + 15z^2 - 5z) q^42 + (5z^2 + 24z + 5) * q^43 + (20z^3 + 4z^2 - 4z - 20) q^44 + (-6z^3 - 12z^2 - 6z + 6) q^45 + (16z^3 + 10z^2 - 26z - 26) q^46 + (18z^3 + 30z^2 + 30z + 18) q^47 + (8z^3 - 4z) * q^48 + (-z^3 + 25z^2 + z - 50) q^49 + (11z^3 + 11z^2 + 13z - 24) q^50 + (12z^3 - 24z^2 + 12) * q^51 + 26z^2 q^52 + (-46z^3 + 92z + 18) * q^53 + (3z^3 - 3z^2 + 3z + 6) q^54 + (-24z^3 - 60z^2 - 24z) q^55 + (10z^2 - 10z + 10) * q^56 + (5z^3 - 26z^2 + 26z - 5) q^57 + (20z^3 + 10z^2 - 10z + 10) q^58 + (2z^3 - 46z^2 + 44z + 44) q^59 + (-12z^2 - 12z) * q^60 + (50z^3 + 36z^2 - 25z - 36) q^61 + (53z^3 + 5z^2 - 53z - 10) q^62 + (-15z + 15) q^63 + 8z^3 q^64 + (26z^3 + 26z^2 - 52z - 52) q^65 + (-18z^3 + 36z + 6) * q^66 + (-12z^3 + 12z^2 - 31z - 43) q^67 + (8z^3 - 24z^2 + 8z) q^68 + (18z^2 + 24z + 18) * q^69 + (10z^3 + 20z^2 - 20z - 10) q^70 + (22z^3 - 46z^2 - 68z + 68) q^71 + (-6z^2 + 6z + 6) * q^72 + (-59z^3 + 23z^2 + 23z - 59) q^73 + (-72z^3 - 10z^2 + 36z + 10) q^74 + (-36z^3 + z^2 + 36z - 2) * q^75 + (-14z^3 - 14z^2 + 38z - 24) q^76 + (30z^3 - 80z^2 + 40) * q^77 + (-13z^3 - 13z^2 - 13z + 26) q^78 + (-7z^3 + 14z + 24) * q^79 + (8z^3 - 8z^2 - 16z - 8) q^80 - 9z^2 q^81 + (28z^2 + 12z + 28) * q^82 + (-40z^3 + 52z^2 - 52z + 40) q^83 + (10z^3 - 10z^2 - 20z + 20) q^84 + (-24z^2 + 24z + 24) * q^85 + (-19z^3 - 10z^2 - 10z - 19) q^86 + (-30z^2 + 30) q^87 + (-36z^3 - 4z^2 + 36z + 8) q^88 + (64z^3 + 64z^2 - 104z + 40) q^89 + (18z^3 + 12z^2 - 6) * q^90 + (-65z^3 + 65z^2 - 65) * q^91 + (-16z^3 + 32z + 36) * q^92 + (19z^3 - 19z^2 + 34z + 53) q^93 + (-30z^3 - 66z^2 - 30z) q^94 + (-10z^2 + 42z - 10) * q^95 + (-4z^3 - 8z^2 + 8z + 4) q^96 + (71z^3 - 13z^2 - 84z + 84) q^97 + (-50z^3 + 26z^2 + 24z + 24) q^98 + (-24z^3 - 6z^2 - 6z - 24) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - 12 q^{5} - 6 q^{6} - 10 q^{7} - 8 q^{8} - 6 q^{9}+O(q^{10}) \) 4 * q - 2 * q^2 - 12 * q^5 - 6 * q^6 - 10 * q^7 - 8 * q^8 - 6 * q^9	\( 4 q - 2 q^{2} - 12 q^{5} - 6 q^{6} - 10 q^{7} - 8 q^{8} - 6 q^{9} + 12 q^{10} + 12 q^{11} - 20 q^{14} - 24 q^{15} + 8 q^{16} + 24 q^{17} + 12 q^{18} + 62 q^{19} - 24 q^{20} - 30 q^{21} + 36 q^{22} + 48 q^{23} + 12 q^{24} - 52 q^{26} - 20 q^{28} + 36 q^{30} + 106 q^{31} + 8 q^{32} - 60 q^{33} + 48 q^{34} - 98 q^{37} + 48 q^{40} - 96 q^{41} + 30 q^{42} + 30 q^{43} - 72 q^{44} - 84 q^{46} + 132 q^{47} - 150 q^{49} - 74 q^{50} + 52 q^{52} + 72 q^{53} + 18 q^{54} - 120 q^{55} + 60 q^{56} - 72 q^{57} + 60 q^{58} + 84 q^{59} - 24 q^{60} - 72 q^{61} - 30 q^{62} + 60 q^{63} - 156 q^{65} + 24 q^{66} - 148 q^{67} - 48 q^{68} + 108 q^{69} + 180 q^{71} + 12 q^{72} - 190 q^{73} + 20 q^{74} - 6 q^{75} - 124 q^{76} + 78 q^{78} + 96 q^{79} - 48 q^{80} - 18 q^{81} + 168 q^{82} + 264 q^{83} + 60 q^{84} + 48 q^{85} - 96 q^{86} + 60 q^{87} + 24 q^{88} + 288 q^{89} - 130 q^{91} + 144 q^{92} + 174 q^{93} - 132 q^{94} - 60 q^{95} + 310 q^{97} + 148 q^{98} - 108 q^{99}+O(q^{100}) \) 4 * q - 2 * q^2 - 12 * q^5 - 6 * q^6 - 10 * q^7 - 8 * q^8 - 6 * q^9 + 12 * q^10 + 12 * q^11 - 20 * q^14 - 24 * q^15 + 8 * q^16 + 24 * q^17 + 12 * q^18 + 62 * q^19 - 24 * q^20 - 30 * q^21 + 36 * q^22 + 48 * q^23 + 12 * q^24 - 52 * q^26 - 20 * q^28 + 36 * q^30 + 106 * q^31 + 8 * q^32 - 60 * q^33 + 48 * q^34 - 98 * q^37 + 48 * q^40 - 96 * q^41 + 30 * q^42 + 30 * q^43 - 72 * q^44 - 84 * q^46 + 132 * q^47 - 150 * q^49 - 74 * q^50 + 52 * q^52 + 72 * q^53 + 18 * q^54 - 120 * q^55 + 60 * q^56 - 72 * q^57 + 60 * q^58 + 84 * q^59 - 24 * q^60 - 72 * q^61 - 30 * q^62 + 60 * q^63 - 156 * q^65 + 24 * q^66 - 148 * q^67 - 48 * q^68 + 108 * q^69 + 180 * q^71 + 12 * q^72 - 190 * q^73 + 20 * q^74 - 6 * q^75 - 124 * q^76 + 78 * q^78 + 96 * q^79 - 48 * q^80 - 18 * q^81 + 168 * q^82 + 264 * q^83 + 60 * q^84 + 48 * q^85 - 96 * q^86 + 60 * q^87 + 24 * q^88 + 288 * q^89 - 130 * q^91 + 144 * q^92 + 174 * q^93 - 132 * q^94 - 60 * q^95 + 310 * q^97 + 148 * q^98 - 108 * q^99

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	78.3.l.a

Level	$78$
Weight	$3$
Character orbit	78.l
Analytic conductor	$2.125$
Analytic rank	$0$
Dimension	$4$
CM	no
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(2.12534606201\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\zeta_{12})\)
Defining polynomial:	\( x^{4} - x^{2} + 1 \) x^4 - x^2 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$p$	$F_p(T)$
$2$	\( T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4 \) T^4 + 2T^3 + 2T^2 + 4*T + 4
$3$	\( T^{4} + 3T^{2} + 9 \) T^4 + 3*T^2 + 9
$5$	\( T^{4} + 12 T^{3} + 72 T^{2} + \cdots + 144 \) T^4 + 12T^3 + 72T^2 + 144*T + 144
$7$	\( T^{4} + 10 T^{3} + 125 T^{2} + \cdots + 625 \) T^4 + 10T^3 + 125T^2 + 500*T + 625
$11$	\( T^{4} - 12 T^{3} + 180 T^{2} + \cdots + 24336 \) T^4 - 12T^3 + 180T^2 - 3744*T + 24336
$13$	\( T^{4} - 169 T^{2} + 28561 \) T^4 - 169*T^2 + 28561
$17$	\( T^{4} - 24 T^{3} + 96 T^{2} + \cdots + 9216 \) T^4 - 24T^3 + 96T^2 + 2304*T + 9216
$19$	\( T^{4} - 62 T^{3} + 986 T^{2} + \cdots + 14884 \) T^4 - 62T^3 + 986T^2 + 1220*T + 14884
$23$	\( T^{4} - 48 T^{3} + 636 T^{2} + \cdots + 17424 \) T^4 - 48T^3 + 636T^2 + 6336*T + 17424
$29$	\( T^{4} + 300 T^{2} + 90000 \) T^4 + 300*T^2 + 90000
$31$	\( T^{4} - 106 T^{3} + 5618 T^{2} + \cdots + 1868689 \) T^4 - 106T^3 + 5618T^2 - 144902*T + 1868689
$37$	\( T^{4} + 98 T^{3} + 5882 T^{2} + \cdots + 3587236 \) T^4 + 98T^3 + 5882T^2 + 223492*T + 3587236
$41$	\( T^{4} + 96 T^{3} + 3600 T^{2} + \cdots + 1218816 \) T^4 + 96T^3 + 3600T^2 + 79488*T + 1218816
$43$	\( T^{4} - 30 T^{3} - 201 T^{2} + \cdots + 251001 \) T^4 - 30T^3 - 201T^2 + 15030*T + 251001
$47$	\( T^{4} - 132 T^{3} + 8712 T^{2} + \cdots + 685584 \) T^4 - 132T^3 + 8712T^2 - 109296*T + 685584
$53$	\( (T^{2} - 36 T - 6024)^{2} \) (T^2 - 36*T - 6024)^2
$59$	\( T^{4} - 84 T^{3} + 4068 T^{2} + \cdots + 16353936 \) T^4 - 84T^3 + 4068T^2 - 388224*T + 16353936
$61$	\( T^{4} + 72 T^{3} + 5763 T^{2} + \cdots + 335241 \) T^4 + 72T^3 + 5763T^2 - 41688*T + 335241
$67$	\( T^{4} + 148 T^{3} + 8501 T^{2} + \cdots + 6723649 \) T^4 + 148T^3 + 8501T^2 + 285230*T + 6723649
$71$	\( T^{4} - 180 T^{3} + \cdots + 33315984 \) T^4 - 180T^3 + 8676T^2 - 277056*T + 33315984
$73$	\( T^{4} + 190 T^{3} + \cdots + 13830961 \) T^4 + 190T^3 + 18050T^2 + 706610*T + 13830961
$79$	\( (T^{2} - 48 T + 429)^{2} \) (T^2 - 48*T + 429)^2
$83$	\( T^{4} - 264 T^{3} + \cdots + 21678336 \) T^4 - 264T^3 + 34848T^2 - 1229184*T + 21678336
$89$	\( T^{4} - 288 T^{3} + \cdots + 137170944 \) T^4 - 288T^3 + 21312T^2 - 562176*T + 137170944
$97$	\( T^{4} - 310 T^{3} + 27389 T^{2} + \cdots + 8162449 \) T^4 - 310T^3 + 27389T^2 - 331412*T + 8162449
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\(n\)	\(53\)	\(67\)
\(\chi(n)\)	\(1\)	\(\zeta_{12}\)

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