Newform orbit 320.4.n.d

• Label: 320.4.n.d
• Level: $320$
• Weight: $4$
• Character orbit: 320.n
• Analytic conductor: $18.881$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$320 = 2^{6} \cdot 5$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	320.n (of order $4$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$18.8806112018$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-1})$
Defining polynomial:	$x^{2} + 1$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $i = \sqrt{-1}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + (2*i + 2) * q^3 + (-5*i - 10) * q^5 + (18*i - 18) * q^7 - 19*i * q^9 - 16*i * q^11 + (-33*i + 33) * q^13 + (-30*i - 10) * q^15 + (67*i + 67) * q^17 + 116 * q^19 - 72 * q^21 + (110*i + 110) * q^23 + (100*i + 75) * q^25 + (-92*i + 92) * q^27 - 132*i * q^29 + 68*i * q^31 + (-32*i + 32) * q^33 + (-90*i + 270) * q^35 + (-65*i - 65) * q^37 + 132 * q^39 - 304 * q^41 + (154*i + 154) * q^43 + (190*i - 95) * q^45 + (-306*i + 306) * q^47 - 305*i * q^49 + 268*i * q^51 + (217*i - 217) * q^53 + (160*i - 80) * q^55 + (232*i + 232) * q^57 - 204 * q^59 + 748 * q^61 + (342*i + 342) * q^63 + (165*i - 495) * q^65 + (-166*i + 166) * q^67 + 440*i * q^69 - 524*i * q^71 + (-277*i + 277) * q^73 + (350*i - 50) * q^75 + (288*i + 288) * q^77 + 1232 * q^79 - 145 * q^81 + (-290*i - 290) * q^83 + (-1005*i - 335) * q^85 + (-264*i + 264) * q^87 - 800*i * q^89 + 1188*i * q^91 + (136*i - 136) * q^93 + (-580*i - 1160) * q^95 + (-651*i - 651) * q^97 - 304 * q^99
$\operatorname{Tr}(f)(q)$	$=$	2 * q + 4 * q^3 - 20 * q^5 - 36 * q^7 + 66 * q^13 - 20 * q^15 + 134 * q^17 + 232 * q^19 - 144 * q^21 + 220 * q^23 + 150 * q^25 + 184 * q^27 + 64 * q^33 + 540 * q^35 - 130 * q^37 + 264 * q^39 - 608 * q^41 + 308 * q^43 - 190 * q^45 + 612 * q^47 - 434 * q^53 - 160 * q^55 + 464 * q^57 - 408 * q^59 + 1496 * q^61 + 684 * q^63 - 990 * q^65 + 332 * q^67 + 554 * q^73 - 100 * q^75 + 576 * q^77 + 2464 * q^79 - 290 * q^81 - 580 * q^83 - 670 * q^85 + 528 * q^87 - 272 * q^93 - 2320 * q^95 - 1302 * q^97 - 608 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$-1$	$i$	$1$

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

63.1

	−	1.00000i
		1.00000i

0

2.00000

−

2.00000i

0

−10.0000

+

5.00000i

0

−18.0000

−

18.0000i

0

19.0000i

0

127.1

0

2.00000

+

2.00000i

0

−10.0000

−

5.00000i

0

−18.0000

+

18.0000i

0

−

19.0000i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
20.e	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	320.4.n.d		2
4.b	odd	2	1		320.4.n.a		2
5.c	odd	4	1		320.4.n.a		2
8.b	even	2	1		160.4.n.a	✓	2
8.d	odd	2	1		160.4.n.b	yes	2
20.e	even	4	1	inner	320.4.n.d		2
40.i	odd	4	1		160.4.n.b	yes	2
40.k	even	4	1		160.4.n.a	✓	2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
160.4.n.a	✓	2	8.b	even	2	1
160.4.n.a	✓	2	40.k	even	4	1
160.4.n.b	yes	2	8.d	odd	2	1
160.4.n.b	yes	2	40.i	odd	4	1
320.4.n.a		2	4.b	odd	2	1
320.4.n.a		2	5.c	odd	4	1
320.4.n.d		2	1.a	even	1	1	trivial
320.4.n.d		2	20.e	even	4	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(320, [\chi])$:

$T_{3}^{2} - 4T_{3} + 8$

$T_{13}^{2} - 66T_{13} + 2178$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2} - 4T + 8$
$5$	$T^{2} + 20T + 125$
$7$	$T^{2} + 36T + 648$
$11$	$T^{2} + 256$