Newform orbit 312.3.b.b

• Label: 312.3.b.b
• Level: $312$
• Weight: $3$
• Character orbit: 312.b
• Self dual: yes
• Analytic conductor: $8.501$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -312
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$312 = 2^{3} \cdot 3 \cdot 13$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	312.b (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	$8.50138424804$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

$f(q)$

$=$

q - 2 * q^2 + 3 * q^3 + 4 * q^4 - 6 * q^6 - 8 * q^8 + 9 * q^9 + 12 * q^12 + 13 * q^13 + 16 * q^16 - 18 * q^18 + 14 * q^19 - 24 * q^24 + 25 * q^25 - 26 * q^26 + 27 * q^27 - 46 * q^29 - 32 * q^32 + 36 * q^36 - 22 * q^37 - 28 * q^38 + 39 * q^39 + 74 * q^41 + 62 * q^47 + 48 * q^48 + 49 * q^49 - 50 * q^50 + 52 * q^52 + 2 * q^53 - 54 * q^54 + 42 * q^57 + 92 * q^58 + 64 * q^64 - 82 * q^67 + 14 * q^71 - 72 * q^72 + 44 * q^74 + 75 * q^75 + 56 * q^76 - 78 * q^78 - 154 * q^79 + 81 * q^81 - 148 * q^82 - 138 * q^87 - 22 * q^89 - 124 * q^94 - 96 * q^96 - 98 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/312\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$157$	$209$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

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.

Label  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

77.1

0

−2.00000

3.00000

4.00000

0

−6.00000

0

−8.00000

9.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
312.b	odd	2	1	CM by $\Q(\sqrt{-78})$

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	312.3.b.b	yes	1
3.b	odd	2	1		312.3.b.c	yes	1
8.b	even	2	1		312.3.b.a	✓	1
13.b	even	2	1		312.3.b.d	yes	1
24.h	odd	2	1		312.3.b.d	yes	1
39.d	odd	2	1		312.3.b.a	✓	1
104.e	even	2	1		312.3.b.c	yes	1
312.b	odd	2	1	CM	312.3.b.b	yes	1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
312.3.b.a	✓	1	8.b	even	2	1
312.3.b.a	✓	1	39.d	odd	2	1
312.3.b.b	yes	1	1.a	even	1	1	trivial
312.3.b.b	yes	1	312.b	odd	2	1	CM
312.3.b.c	yes	1	3.b	odd	2	1
312.3.b.c	yes	1	104.e	even	2	1
312.3.b.d	yes	1	13.b	even	2	1
312.3.b.d	yes	1	24.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{3}^{\mathrm{new}}(312, [\chi])$:

$T_{5}$

$T_{19} - 14$

$T_{29} + 46$

$T_{41} - 74$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T + 2$
$3$	$T - 3$
$5$	$T$
$7$	$T$
$11$	$T$