Newform orbit 10.12.a.a

• Label: 10.12.a.a
• Level: $10$
• Weight: $12$
• Character orbit: 10.a
• Self dual: yes
• Analytic conductor: $7.683$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$10 = 2 \cdot 5$
Weight:	$k$	$=$	$12$
Character orbit:	$[\chi]$	$=$	10.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$7.68343180560$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

$f(q)$

$=$

q - 32 * q^2 - 12 * q^3 + 1024 * q^4 + 3125 * q^5 + 384 * q^6 - 14176 * q^7 - 32768 * q^8 - 177003 * q^9 - 100000 * q^10 - 756348 * q^11 - 12288 * q^12 - 905482 * q^13 + 453632 * q^14 - 37500 * q^15 + 1048576 * q^16 + 2803794 * q^17 + 5664096 * q^18 - 5428660 * q^19 + 3200000 * q^20 + 170112 * q^21 + 24203136 * q^22 - 10236672 * q^23 + 393216 * q^24 + 9765625 * q^25 + 28975424 * q^26 + 4249800 * q^27 - 14516224 * q^28 - 197498010 * q^29 + 1200000 * q^30 - 44362288 * q^31 - 33554432 * q^32 + 9076176 * q^33 - 89721408 * q^34 - 44300000 * q^35 - 181251072 * q^36 + 576737054 * q^37 + 173717120 * q^38 + 10865784 * q^39 - 102400000 * q^40 + 930058362 * q^41 - 5443584 * q^42 + 1605598988 * q^43 - 774500352 * q^44 - 553134375 * q^45 + 327573504 * q^46 - 1803684456 * q^47 - 12582912 * q^48 - 1776367767 * q^49 - 312500000 * q^50 - 33645528 * q^51 - 927213568 * q^52 + 1558674798 * q^53 - 135993600 * q^54 - 2363587500 * q^55 + 464519168 * q^56 + 65143920 * q^57 + 6319936320 * q^58 - 9501997020 * q^59 - 38400000 * q^60 + 6736320422 * q^61 + 1419593216 * q^62 + 2509194528 * q^63 + 1073741824 * q^64 - 2829631250 * q^65 - 290437632 * q^66 + 8402906564 * q^67 + 2871085056 * q^68 + 122840064 * q^69 + 1417600000 * q^70 - 4806306168 * q^71 + 5800034304 * q^72 + 7462713338 * q^73 - 18455585728 * q^74 - 117187500 * q^75 - 5558947840 * q^76 + 10721989248 * q^77 - 347705088 * q^78 - 20644540720 * q^79 + 3276800000 * q^80 + 31304552841 * q^81 - 29761867584 * q^82 - 68013349212 * q^83 + 174194688 * q^84 + 8761856250 * q^85 - 51379167616 * q^86 + 2369976120 * q^87 + 24784011264 * q^88 + 69871323210 * q^89 + 17700300000 * q^90 + 12836112832 * q^91 - 10482352128 * q^92 + 532347456 * q^93 + 57717902592 * q^94 - 16964562500 * q^95 + 402653184 * q^96 + 39960952514 * q^97 + 56843768544 * q^98 + 133875865044 * q^99

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  

$\iota_m(\nu)$

$a_{2}$

$a_{3}$

$a_{4}$

$a_{5}$

$a_{6}$

$a_{7}$

$a_{8}$

$a_{9}$

$a_{10}$

1.1

0

−32.0000

−12.0000

1024.00

3125.00

384.000

−14176.0

−32768.0

−177003.

−100000.

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$5$	$-1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	10.12.a.a	✓	1
3.b	odd	2	1		90.12.a.g		1
4.b	odd	2	1		80.12.a.d		1
5.b	even	2	1		50.12.a.d		1
5.c	odd	4	2		50.12.b.c		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
10.12.a.a	✓	1	1.a	even	1	1	trivial
50.12.a.d		1	5.b	even	2	1
50.12.b.c		2	5.c	odd	4	2
80.12.a.d		1	4.b	odd	2	1
90.12.a.g		1	3.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3} + 12$ acting on $S_{12}^{\mathrm{new}}(\Gamma_0(10))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T + 32$
$3$	$T + 12$
$5$	$T - 3125$
$7$	$T + 14176$
$11$	$T + 756348$