Newform orbit 8.19.d.a

• Label: 8.19.d.a
• Level: $8$
• Weight: $19$
• Character orbit: 8.d
• Self dual: yes
• Analytic conductor: $16.431$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -8
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$8 = 2^{3}$
Weight:	$k$	$=$	$19$
Character orbit:	$[\chi]$	$=$	8.d (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	$16.4308910168$
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 - 512 * q^2 - 3266 * q^3 + 262144 * q^4 + 1672192 * q^6 - 134217728 * q^8 - 376753733 * q^9 - 354349618 * q^11 - 856162304 * q^12 + 68719476736 * q^16 + 119842447106 * q^17 + 192897911296 * q^18 + 335013705758 * q^19 + 181427004416 * q^22 + 438355099648 * q^24 + 3814697265625 * q^25 + 2495793009052 * q^27 - 35184372088832 * q^32 + 1157305852388 * q^33 - 61359332918272 * q^34 - 98763730583552 * q^36 - 171527017348096 * q^38 + 522162604887122 * q^41 + 1000250360894414 * q^43 - 92890626260992 * q^44 - 224437811019776 * q^48 + 1628413597910449 * q^49 - 1953125000000000 * q^50 - 391405432248196 * q^51 - 1277846020634624 * q^54 - 1094154763005628 * q^57 - 10228968070290322 * q^59 + 18014398509481984 * q^64 - 592540596422656 * q^66 - 50467064407716994 * q^67 + 31415978454155264 * q^68 + 50567030058778624 * q^72 + 60615173082969074 * q^73 - 12458801269531250 * q^75 + 87821832882225152 * q^76 + 137810855503871605 * q^81 - 267347253702206464 * q^82 - 352960460737558306 * q^83 - 512128184777939968 * q^86 + 47560000645627904 * q^88 + 487058048464217618 * q^89 + 114912159242125312 * q^96 + 1501067319528053666 * q^97 - 833747762130149888 * q^98 + 133502541368623994 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/8\mathbb{Z}\right)^\times$.

$n$	$5$	$7$
$\chi(n)$	$-1$	$-1$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

3.1

0

−512.000

−3266.00

262144.

0

1.67219e6

0

−1.34218e8

−3.76754e8

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	8.19.d.a	✓	1
3.b	odd	2	1		72.19.b.a		1
4.b	odd	2	1		32.19.d.a		1
8.b	even	2	1		32.19.d.a		1
8.d	odd	2	1	CM	8.19.d.a	✓	1
24.f	even	2	1		72.19.b.a		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
8.19.d.a	✓	1	1.a	even	1	1	trivial
8.19.d.a	✓	1	8.d	odd	2	1	CM
32.19.d.a		1	4.b	odd	2	1
32.19.d.a		1	8.b	even	2	1
72.19.b.a		1	3.b	odd	2	1
72.19.b.a		1	24.f	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3} + 3266$ acting on $S_{19}^{\mathrm{new}}(8, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T + 512$
$3$	$T + 3266$
$5$	$T$
$7$	$T$
$11$	$T + 354349618$