Newform orbit 10.16.a.d

• Label: 10.16.a.d
• Level: $10$
• Weight: $16$
• Character orbit: 10.a
• Self dual: yes
• Analytic conductor: $14.269$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$10 = 2 \cdot 5$
Weight:	$k$	$=$	$16$
Character orbit:	$[\chi]$	$=$	10.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$14.2693505100$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{239569})$
Defining polynomial:	$x^{2} - x - 59892$
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$2^{2}\cdot 5$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = 10\sqrt{239569}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + 128 * q^2 + (-b - 922) * q^3 + 16384 * q^4 + 78125 * q^5 + (-128*b - 118016) * q^6 + (-423*b - 492466) * q^7 + 2097152 * q^8 + (1844*b + 10458077) * q^9 + 10000000 * q^10 + (12006*b + 55776072) * q^11 + (-16384*b - 15106048) * q^12 + (28548*b + 144656798) * q^13 + (-54144*b - 63035648) * q^14 + (-78125*b - 72031250) * q^15 + 268435456 * q^16 + (13428*b + 710869674) * q^17 + (236032*b + 1338633856) * q^18 + (134964*b + 3079703060) * q^19 + 1280000000 * q^20 + (882472*b + 10587822352) * q^21 + (1536768*b + 7139337216) * q^22 + (-5204241*b - 2165082942) * q^23 + (-2097152*b - 1933574144) * q^24 + 6103515625 * q^25 + (3654144*b + 18516070144) * q^26 + (2190662*b - 40589178340) * q^27 + (-6930432*b - 8068562944) * q^28 + (20046888*b - 82147970970) * q^29 + (-10000000*b - 9220000000) * q^30 + (29450862*b - 141355482508) * q^31 + 34359738368 * q^32 + (-66845604*b - 339052079784) * q^33 + (1718784*b + 90991318272) * q^34 + (-33046875*b - 38473906250) * q^35 + (30212096*b + 171345133568) * q^36 + (59426568*b + 395052579614) * q^37 + (17275392*b + 394201991680) * q^38 + (-170978054*b - 817295148956) * q^39 + 163840000000 * q^40 + (224037468*b - 187358632638) * q^41 + (112956416*b + 1355241261056) * q^42 + (10226907*b - 461912216602) * q^43 + (196706304*b + 913835163648) * q^44 + (144062500*b + 817037265625) * q^45 + (-666142848*b - 277130616576) * q^46 + (-577835703*b + 2398358606214) * q^47 + (-268435456*b - 247497490432) * q^48 + (416626236*b - 218454588687) * q^49 + 781250000000 * q^50 + (-723250290*b - 977115092628) * q^51 + (467730432*b + 2370056978432) * q^52 + (2557045548*b - 1384460646042) * q^53 + (280404736*b - 5195414827520) * q^54 + (937968750*b + 4357505625000) * q^55 + (-887095296*b - 1032776056832) * q^56 + (-3204139868*b - 6072805272920) * q^57 + (2566001664*b - 10514940284160) * q^58 + (-3981059784*b - 10368616994940) * q^59 + (-1280000000*b - 1180160000000) * q^60 + (3129160464*b + 288943862582) * q^61 + (3769710336*b - 18093501761024) * q^62 + (-5331873875*b - 23836916830682) * q^63 + 4398046511104 * q^64 + (2230312500*b + 11301312343750) * q^65 + (-8556237312*b - 43398666212352) * q^66 + (4385860641*b + 43276629038834) * q^67 + (220004352*b + 11646888738816) * q^68 + (6963393144*b + 126673687685424) * q^69 + (-4230000000*b - 4924660000000) * q^70 + (20725115814*b + 36919453344732) * q^71 + (3867148288*b + 21932177096704) * q^72 + (-15861716892*b + 19986863510738) * q^73 + (7606600704*b + 50566730190592) * q^74 + (-6103515625*b - 5627441406250) * q^75 + (2211250176*b + 50457854935040) * q^76 + (-29505825252*b - 149133846085752) * q^77 + (-21885190912*b - 104613779066368) * q^78 + (-5107258116*b + 210832652437400) * q^79 + 20971520000000 * q^80 + (12110003468*b - 165120222310159) * q^81 + (28676795904*b - 23981904977664) * q^82 + (10814315895*b - 360573330019482) * q^83 + (14458421248*b + 173470881415168) * q^84 + (1049062500*b + 55536693281250) * q^85 + (1309044096*b - 59124763725056) * q^86 + (63664740234*b - 404520861892860) * q^87 + (25178406912*b + 116970900946944) * q^88 + (-74280619080*b + 181856418311610) * q^89 + (18440000000*b + 104580770000000) * q^90 + (-75248744922*b - 360537383531468) * q^91 + (-85266284544*b - 35472718921728) * q^92 + (114201787744*b - 575221600975424) * q^93 + (-73962969984*b + 306989901595392) * q^94 + (10544062500*b + 240601801562500) * q^95 + (-34359738368*b - 31679678775296) * q^96 + (-66757883220*b + 144515049698474) * q^97 + (53328158208*b - 27962187351936) * q^98 + (228410749230*b + 1113693798075144) * q^99
$\operatorname{Tr}(f)(q)$	$=$	2 * q + 256 * q^2 - 1844 * q^3 + 32768 * q^4 + 156250 * q^5 - 236032 * q^6 - 984932 * q^7 + 4194304 * q^8 + 20916154 * q^9 + 20000000 * q^10 + 111552144 * q^11 - 30212096 * q^12 + 289313596 * q^13 - 126071296 * q^14 - 144062500 * q^15 + 536870912 * q^16 + 1421739348 * q^17 + 2677267712 * q^18 + 6159406120 * q^19 + 2560000000 * q^20 + 21175644704 * q^21 + 14278674432 * q^22 - 4330165884 * q^23 - 3867148288 * q^24 + 12207031250 * q^25 + 37032140288 * q^26 - 81178356680 * q^27 - 16137125888 * q^28 - 164295941940 * q^29 - 18440000000 * q^30 - 282710965016 * q^31 + 68719476736 * q^32 - 678104159568 * q^33 + 181982636544 * q^34 - 76947812500 * q^35 + 342690267136 * q^36 + 790105159228 * q^37 + 788403983360 * q^38 - 1634590297912 * q^39 + 327680000000 * q^40 - 374717265276 * q^41 + 2710482522112 * q^42 - 923824433204 * q^43 + 1827670327296 * q^44 + 1634074531250 * q^45 - 554261233152 * q^46 + 4796717212428 * q^47 - 494994980864 * q^48 - 436909177374 * q^49 + 1562500000000 * q^50 - 1954230185256 * q^51 + 4740113956864 * q^52 - 2768921292084 * q^53 - 10390829655040 * q^54 + 8715011250000 * q^55 - 2065552113664 * q^56 - 12145610545840 * q^57 - 21029880568320 * q^58 - 20737233989880 * q^59 - 2360320000000 * q^60 + 577887725164 * q^61 - 36187003522048 * q^62 - 47673833661364 * q^63 + 8796093022208 * q^64 + 22602624687500 * q^65 - 86797332424704 * q^66 + 86553258077668 * q^67 + 23293777477632 * q^68 + 253347375370848 * q^69 - 9849320000000 * q^70 + 73838906689464 * q^71 + 43864354193408 * q^72 + 39973727021476 * q^73 + 101133460381184 * q^74 - 11254882812500 * q^75 + 100915709870080 * q^76 - 298267692171504 * q^77 - 209227558132736 * q^78 + 421665304874800 * q^79 + 41943040000000 * q^80 - 330240444620318 * q^81 - 47963809955328 * q^82 - 721146660038964 * q^83 + 346941762830336 * q^84 + 111073386562500 * q^85 - 118249527450112 * q^86 - 809041723785720 * q^87 + 233941801893888 * q^88 + 363712836623220 * q^89 + 209161540000000 * q^90 - 721074767062936 * q^91 - 70945437843456 * q^92 - 1150443201950848 * q^93 + 613979803190784 * q^94 + 481203603125000 * q^95 - 63359357550592 * q^96 + 289030099396948 * q^97 - 55924374703872 * q^98 + 2227387596150288 * 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

245.229
−244.229

128.000

−5816.58

16384.0

78125.0

−744522.

−2.56287e6

2.09715e6

1.94837e7

1.00000e7

1.2

128.000

3972.58

16384.0

78125.0

508490.

1.57794e6

2.09715e6

1.43247e6

1.00000e7

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	10.16.a.d	✓	2
3.b	odd	2	1		90.16.a.j		2
4.b	odd	2	1		80.16.a.f		2
5.b	even	2	1		50.16.a.f		2
5.c	odd	4	2		50.16.b.e		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
10.16.a.d	✓	2	1.a	even	1	1	trivial
50.16.a.f		2	5.b	even	2	1
50.16.b.e		4	5.c	odd	4	2
80.16.a.f		2	4.b	odd	2	1
90.16.a.j		2	3.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{2} + 1844T_{3} - 23106816$ acting on $S_{16}^{\mathrm{new}}(\Gamma_0(10))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T - 128)^{2}$
$3$	$T^{2} + 1844 T - 23106816$
$5$	$(T - 78125)^{2}$
$7$	T^2 + 984932*T - 4044061398944
$11$	T^2 - 111552144*T - 342274048299216