Properties

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

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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: \(\Q(\sqrt{239569}) \)
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.

\(f(q)\) \(=\) \( q + 128 q^{2} + ( -922 - \beta ) q^{3} + 16384 q^{4} + 78125 q^{5} + ( -118016 - 128 \beta ) q^{6} + ( -492466 - 423 \beta ) q^{7} + 2097152 q^{8} + ( 10458077 + 1844 \beta ) q^{9} +O(q^{10})\) \( q + 128 q^{2} + ( -922 - \beta ) q^{3} + 16384 q^{4} + 78125 q^{5} + ( -118016 - 128 \beta ) q^{6} + ( -492466 - 423 \beta ) q^{7} + 2097152 q^{8} + ( 10458077 + 1844 \beta ) q^{9} + 10000000 q^{10} + ( 55776072 + 12006 \beta ) q^{11} + ( -15106048 - 16384 \beta ) q^{12} + ( 144656798 + 28548 \beta ) q^{13} + ( -63035648 - 54144 \beta ) q^{14} + ( -72031250 - 78125 \beta ) q^{15} + 268435456 q^{16} + ( 710869674 + 13428 \beta ) q^{17} + ( 1338633856 + 236032 \beta ) q^{18} + ( 3079703060 + 134964 \beta ) q^{19} + 1280000000 q^{20} + ( 10587822352 + 882472 \beta ) q^{21} + ( 7139337216 + 1536768 \beta ) q^{22} + ( -2165082942 - 5204241 \beta ) q^{23} + ( -1933574144 - 2097152 \beta ) q^{24} + 6103515625 q^{25} + ( 18516070144 + 3654144 \beta ) q^{26} + ( -40589178340 + 2190662 \beta ) q^{27} + ( -8068562944 - 6930432 \beta ) q^{28} + ( -82147970970 + 20046888 \beta ) q^{29} + ( -9220000000 - 10000000 \beta ) q^{30} + ( -141355482508 + 29450862 \beta ) q^{31} + 34359738368 q^{32} + ( -339052079784 - 66845604 \beta ) q^{33} + ( 90991318272 + 1718784 \beta ) q^{34} + ( -38473906250 - 33046875 \beta ) q^{35} + ( 171345133568 + 30212096 \beta ) q^{36} + ( 395052579614 + 59426568 \beta ) q^{37} + ( 394201991680 + 17275392 \beta ) q^{38} + ( -817295148956 - 170978054 \beta ) q^{39} + 163840000000 q^{40} + ( -187358632638 + 224037468 \beta ) q^{41} + ( 1355241261056 + 112956416 \beta ) q^{42} + ( -461912216602 + 10226907 \beta ) q^{43} + ( 913835163648 + 196706304 \beta ) q^{44} + ( 817037265625 + 144062500 \beta ) q^{45} + ( -277130616576 - 666142848 \beta ) q^{46} + ( 2398358606214 - 577835703 \beta ) q^{47} + ( -247497490432 - 268435456 \beta ) q^{48} + ( -218454588687 + 416626236 \beta ) q^{49} + 781250000000 q^{50} + ( -977115092628 - 723250290 \beta ) q^{51} + ( 2370056978432 + 467730432 \beta ) q^{52} + ( -1384460646042 + 2557045548 \beta ) q^{53} + ( -5195414827520 + 280404736 \beta ) q^{54} + ( 4357505625000 + 937968750 \beta ) q^{55} + ( -1032776056832 - 887095296 \beta ) q^{56} + ( -6072805272920 - 3204139868 \beta ) q^{57} + ( -10514940284160 + 2566001664 \beta ) q^{58} + ( -10368616994940 - 3981059784 \beta ) q^{59} + ( -1180160000000 - 1280000000 \beta ) q^{60} + ( 288943862582 + 3129160464 \beta ) q^{61} + ( -18093501761024 + 3769710336 \beta ) q^{62} + ( -23836916830682 - 5331873875 \beta ) q^{63} + 4398046511104 q^{64} + ( 11301312343750 + 2230312500 \beta ) q^{65} + ( -43398666212352 - 8556237312 \beta ) q^{66} + ( 43276629038834 + 4385860641 \beta ) q^{67} + ( 11646888738816 + 220004352 \beta ) q^{68} + ( 126673687685424 + 6963393144 \beta ) q^{69} + ( -4924660000000 - 4230000000 \beta ) q^{70} + ( 36919453344732 + 20725115814 \beta ) q^{71} + ( 21932177096704 + 3867148288 \beta ) q^{72} + ( 19986863510738 - 15861716892 \beta ) q^{73} + ( 50566730190592 + 7606600704 \beta ) q^{74} + ( -5627441406250 - 6103515625 \beta ) q^{75} + ( 50457854935040 + 2211250176 \beta ) q^{76} + ( -149133846085752 - 29505825252 \beta ) q^{77} + ( -104613779066368 - 21885190912 \beta ) q^{78} + ( 210832652437400 - 5107258116 \beta ) q^{79} + 20971520000000 q^{80} + ( -165120222310159 + 12110003468 \beta ) q^{81} + ( -23981904977664 + 28676795904 \beta ) q^{82} + ( -360573330019482 + 10814315895 \beta ) q^{83} + ( 173470881415168 + 14458421248 \beta ) q^{84} + ( 55536693281250 + 1049062500 \beta ) q^{85} + ( -59124763725056 + 1309044096 \beta ) q^{86} + ( -404520861892860 + 63664740234 \beta ) q^{87} + ( 116970900946944 + 25178406912 \beta ) q^{88} + ( 181856418311610 - 74280619080 \beta ) q^{89} + ( 104580770000000 + 18440000000 \beta ) q^{90} + ( -360537383531468 - 75248744922 \beta ) q^{91} + ( -35472718921728 - 85266284544 \beta ) q^{92} + ( -575221600975424 + 114201787744 \beta ) q^{93} + ( 306989901595392 - 73962969984 \beta ) q^{94} + ( 240601801562500 + 10544062500 \beta ) q^{95} + ( -31679678775296 - 34359738368 \beta ) q^{96} + ( 144515049698474 - 66757883220 \beta ) q^{97} + ( -27962187351936 + 53328158208 \beta ) q^{98} + ( 1113693798075144 + 228410749230 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 256q^{2} - 1844q^{3} + 32768q^{4} + 156250q^{5} - 236032q^{6} - 984932q^{7} + 4194304q^{8} + 20916154q^{9} + O(q^{10}) \) \( 2q + 256q^{2} - 1844q^{3} + 32768q^{4} + 156250q^{5} - 236032q^{6} - 984932q^{7} + 4194304q^{8} + 20916154q^{9} + 20000000q^{10} + 111552144q^{11} - 30212096q^{12} + 289313596q^{13} - 126071296q^{14} - 144062500q^{15} + 536870912q^{16} + 1421739348q^{17} + 2677267712q^{18} + 6159406120q^{19} + 2560000000q^{20} + 21175644704q^{21} + 14278674432q^{22} - 4330165884q^{23} - 3867148288q^{24} + 12207031250q^{25} + 37032140288q^{26} - 81178356680q^{27} - 16137125888q^{28} - 164295941940q^{29} - 18440000000q^{30} - 282710965016q^{31} + 68719476736q^{32} - 678104159568q^{33} + 181982636544q^{34} - 76947812500q^{35} + 342690267136q^{36} + 790105159228q^{37} + 788403983360q^{38} - 1634590297912q^{39} + 327680000000q^{40} - 374717265276q^{41} + 2710482522112q^{42} - 923824433204q^{43} + 1827670327296q^{44} + 1634074531250q^{45} - 554261233152q^{46} + 4796717212428q^{47} - 494994980864q^{48} - 436909177374q^{49} + 1562500000000q^{50} - 1954230185256q^{51} + 4740113956864q^{52} - 2768921292084q^{53} - 10390829655040q^{54} + 8715011250000q^{55} - 2065552113664q^{56} - 12145610545840q^{57} - 21029880568320q^{58} - 20737233989880q^{59} - 2360320000000q^{60} + 577887725164q^{61} - 36187003522048q^{62} - 47673833661364q^{63} + 8796093022208q^{64} + 22602624687500q^{65} - 86797332424704q^{66} + 86553258077668q^{67} + 23293777477632q^{68} + 253347375370848q^{69} - 9849320000000q^{70} + 73838906689464q^{71} + 43864354193408q^{72} + 39973727021476q^{73} + 101133460381184q^{74} - 11254882812500q^{75} + 100915709870080q^{76} - 298267692171504q^{77} - 209227558132736q^{78} + 421665304874800q^{79} + 41943040000000q^{80} - 330240444620318q^{81} - 47963809955328q^{82} - 721146660038964q^{83} + 346941762830336q^{84} + 111073386562500q^{85} - 118249527450112q^{86} - 809041723785720q^{87} + 233941801893888q^{88} + 363712836623220q^{89} + 209161540000000q^{90} - 721074767062936q^{91} - 70945437843456q^{92} - 1150443201950848q^{93} + 613979803190784q^{94} + 481203603125000q^{95} - 63359357550592q^{96} + 289030099396948q^{97} - 55924374703872q^{98} + 2227387596150288q^{99} + O(q^{100}) \)

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
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 128 T )^{2} \)
$3$ \( 1 + 1844 T + 5590998 T^{2} + 26459384508 T^{3} + 205891132094649 T^{4} \)
$5$ \( ( 1 - 78125 T )^{2} \)
$7$ \( 1 + 984932 T + 5451061620942 T^{2} + 4676025253111178876 T^{3} + \)\(22\!\cdots\!49\)\( T^{4} \)
$11$ \( 1 - 111552144 T + 8012222290532086 T^{2} - \)\(46\!\cdots\!44\)\( T^{3} + \)\(17\!\cdots\!01\)\( T^{4} \)
$13$ \( 1 - 289313596 T + 103772781935696718 T^{2} - \)\(14\!\cdots\!72\)\( T^{3} + \)\(26\!\cdots\!49\)\( T^{4} \)
$17$ \( 1 - 1421739348 T + 6225862099428528262 T^{2} - \)\(40\!\cdots\!64\)\( T^{3} + \)\(81\!\cdots\!49\)\( T^{4} \)
$19$ \( 1 - 6159406120 T + 39410443325042817798 T^{2} - \)\(93\!\cdots\!80\)\( T^{3} + \)\(23\!\cdots\!01\)\( T^{4} \)
$23$ \( 1 + 4330165884 T - \)\(11\!\cdots\!22\)\( T^{2} + \)\(11\!\cdots\!88\)\( T^{3} + \)\(71\!\cdots\!49\)\( T^{4} \)
$29$ \( 1 + 164295941940 T + \)\(14\!\cdots\!98\)\( T^{2} + \)\(14\!\cdots\!60\)\( T^{3} + \)\(74\!\cdots\!01\)\( T^{4} \)
$31$ \( 1 + 282710965016 T + \)\(46\!\cdots\!66\)\( T^{2} + \)\(66\!\cdots\!16\)\( T^{3} + \)\(55\!\cdots\!01\)\( T^{4} \)
$37$ \( 1 - 790105159228 T + \)\(73\!\cdots\!82\)\( T^{2} - \)\(26\!\cdots\!04\)\( T^{3} + \)\(11\!\cdots\!49\)\( T^{4} \)
$41$ \( 1 + 374717265276 T + \)\(19\!\cdots\!46\)\( T^{2} + \)\(58\!\cdots\!76\)\( T^{3} + \)\(24\!\cdots\!01\)\( T^{4} \)
$43$ \( 1 + 923824433204 T + \)\(65\!\cdots\!18\)\( T^{2} + \)\(29\!\cdots\!28\)\( T^{3} + \)\(10\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 - 4796717212428 T + \)\(21\!\cdots\!82\)\( T^{2} - \)\(57\!\cdots\!04\)\( T^{3} + \)\(14\!\cdots\!49\)\( T^{4} \)
$53$ \( 1 + 2768921292084 T - \)\(84\!\cdots\!22\)\( T^{2} + \)\(20\!\cdots\!88\)\( T^{3} + \)\(53\!\cdots\!49\)\( T^{4} \)
$59$ \( 1 + 20737233989880 T + \)\(45\!\cdots\!98\)\( T^{2} + \)\(75\!\cdots\!20\)\( T^{3} + \)\(13\!\cdots\!01\)\( T^{4} \)
$61$ \( 1 - 577887725164 T + \)\(97\!\cdots\!26\)\( T^{2} - \)\(34\!\cdots\!64\)\( T^{3} + \)\(36\!\cdots\!01\)\( T^{4} \)
$67$ \( 1 - 86553258077668 T + \)\(63\!\cdots\!42\)\( T^{2} - \)\(21\!\cdots\!24\)\( T^{3} + \)\(60\!\cdots\!49\)\( T^{4} \)
$71$ \( 1 - 73838906689464 T + \)\(28\!\cdots\!26\)\( T^{2} - \)\(43\!\cdots\!64\)\( T^{3} + \)\(34\!\cdots\!01\)\( T^{4} \)
$73$ \( 1 - 39973727021476 T + \)\(12\!\cdots\!58\)\( T^{2} - \)\(35\!\cdots\!32\)\( T^{3} + \)\(79\!\cdots\!49\)\( T^{4} \)
$79$ \( 1 - 421665304874800 T + \)\(10\!\cdots\!98\)\( T^{2} - \)\(12\!\cdots\!00\)\( T^{3} + \)\(84\!\cdots\!01\)\( T^{4} \)
$83$ \( 1 + 721146660038964 T + \)\(24\!\cdots\!38\)\( T^{2} + \)\(44\!\cdots\!48\)\( T^{3} + \)\(37\!\cdots\!49\)\( T^{4} \)
$89$ \( 1 - 363712836623220 T + \)\(24\!\cdots\!98\)\( T^{2} - \)\(63\!\cdots\!80\)\( T^{3} + \)\(30\!\cdots\!01\)\( T^{4} \)
$97$ \( 1 - 289030099396948 T + \)\(11\!\cdots\!62\)\( T^{2} - \)\(18\!\cdots\!64\)\( T^{3} + \)\(40\!\cdots\!49\)\( T^{4} \)
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