Newform orbit 416.7.h.a

• Label: 416.7.h.a
• Level: $416$
• Weight: $7$
• Character orbit: 416.h
• Self dual: yes
• Analytic conductor: $95.702$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -104
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$416 = 2^{5} \cdot 13$
Weight:	$k$	$=$	$7$
Character orbit:	$[\chi]$	$=$	416.h (of order $2$, degree $1$, not minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	$95.7024987859$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

$f(q)$

$=$

q + 50 * q^3 - 218 * q^5 + 614 * q^7 + 1771 * q^9 - 2197 * q^13 - 10900 * q^15 + 3170 * q^17 + 30700 * q^21 + 31899 * q^25 + 52100 * q^27 + 27830 * q^31 - 133852 * q^35 + 13894 * q^37 - 109850 * q^39 + 111490 * q^43 - 386078 * q^45 - 128554 * q^47 + 259347 * q^49 + 158500 * q^51 + 1087394 * q^63 + 478946 * q^65 + 317990 * q^71 + 1594950 * q^75 + 1313941 * q^81 - 691060 * q^85 - 1348958 * q^91 + 1391500 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$.

$n$	$261$	$287$	$353$
$\chi(n)$	$-1$	$-1$	$-1$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

207.1

0

0

50.0000

0

−218.000

0

614.000

0

1771.00

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
104.h	odd	2	1	CM by $\Q(\sqrt{-26})$

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	416.7.h.a		1
4.b	odd	2	1		104.7.h.b	yes	1
8.b	even	2	1		104.7.h.a	✓	1
8.d	odd	2	1		416.7.h.b		1
13.b	even	2	1		416.7.h.b		1
52.b	odd	2	1		104.7.h.a	✓	1
104.e	even	2	1		104.7.h.b	yes	1
104.h	odd	2	1	CM	416.7.h.a		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
104.7.h.a	✓	1	8.b	even	2	1
104.7.h.a	✓	1	52.b	odd	2	1
104.7.h.b	yes	1	4.b	odd	2	1
104.7.h.b	yes	1	104.e	even	2	1
416.7.h.a		1	1.a	even	1	1	trivial
416.7.h.a		1	104.h	odd	2	1	CM
416.7.h.b		1	8.d	odd	2	1
416.7.h.b		1	13.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{7}^{\mathrm{new}}(416, [\chi])$:

$T_{3} - 50$

$T_{5} + 218$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$
$3$	$T - 50$
$5$	$T + 218$
$7$	$T - 614$
$11$	$T$