Newform orbit 40.15.e.b

• Label: 40.15.e.b
• Level: $40$
• Weight: $15$
• Character orbit: 40.e
• Self dual: yes
• Analytic conductor: $49.732$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -40
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$40 = 2^{3} \cdot 5$
Weight:	$k$	$=$	$15$
Character orbit:	$[\chi]$	$=$	40.e (of order $2$, degree $1$, minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	$49.7315872608$
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 + 128 * q^2 + 16384 * q^4 - 78125 * q^5 + 67866 * q^7 + 2097152 * q^8 + 4782969 * q^9 - 10000000 * q^10 + 16052382 * q^11 - 125276586 * q^13 + 8686848 * q^14 + 268435456 * q^16 + 612220032 * q^18 + 644127118 * q^19 - 1280000000 * q^20 + 2054704896 * q^22 + 5951338874 * q^23 + 6103515625 * q^25 - 16035403008 * q^26 + 1111916544 * q^28 + 34359738368 * q^32 - 5302031250 * q^35 + 78364164096 * q^36 + 100482630246 * q^37 + 82448271104 * q^38 - 163840000000 * q^40 + 227781371922 * q^41 + 263002226688 * q^44 - 373669453125 * q^45 + 761771375872 * q^46 + 985929843946 * q^47 - 673617278893 * q^49 + 781250000000 * q^50 - 2052531585024 * q^52 - 1801585778906 * q^53 - 1254092343750 * q^55 + 142325317632 * q^56 + 4687670223678 * q^59 + 324600974154 * q^63 + 4398046511104 * q^64 + 9787233281250 * q^65 - 678660000000 * q^70 + 10030613004288 * q^72 + 12861776671488 * q^74 + 10553378701312 * q^76 + 1089410956812 * q^77 - 20971520000000 * q^80 + 22876792454961 * q^81 + 29156015606016 * q^82 + 33664285016064 * q^88 - 57601022824782 * q^89 - 47829690000000 * q^90 - 8502020785476 * q^91 + 97506736111616 * q^92 + 126199020025088 * q^94 - 50322431093750 * q^95 - 86223011698304 * q^98 + 76778045482158 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/40\mathbb{Z}\right)^\times$.

$n$	$17$	$21$	$31$
$\chi(n)$	$-1$	$-1$	$-1$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

19.1

0

128.000

0

16384.0

−78125.0

0

67866.0

2.09715e6

4.78297e6

−1.00000e7

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
40.e	odd	2	1	CM by $\Q(\sqrt{-10})$

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	40.15.e.b	yes	1
4.b	odd	2	1		160.15.e.a		1
5.b	even	2	1		40.15.e.a	✓	1
8.b	even	2	1		160.15.e.b		1
8.d	odd	2	1		40.15.e.a	✓	1
20.d	odd	2	1		160.15.e.b		1
40.e	odd	2	1	CM	40.15.e.b	yes	1
40.f	even	2	1		160.15.e.a		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
40.15.e.a	✓	1	5.b	even	2	1
40.15.e.a	✓	1	8.d	odd	2	1
40.15.e.b	yes	1	1.a	even	1	1	trivial
40.15.e.b	yes	1	40.e	odd	2	1	CM
160.15.e.a		1	4.b	odd	2	1
160.15.e.a		1	40.f	even	2	1
160.15.e.b		1	8.b	even	2	1
160.15.e.b		1	20.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{15}^{\mathrm{new}}(40, [\chi])$:

$T_{3}$

$T_{7} - 67866$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T - 128$
$3$	$T$
$5$	$T + 78125$
$7$	$T - 67866$
$11$	$T - 16052382$