Properties

Label 10.18.a.d
Level 10
Weight 18
Character orbit 10.a
Self dual yes
Analytic conductor 18.322
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 10 = 2 \cdot 5 \)
Weight: \( k \) = \( 18 \)
Character orbit: \([\chi]\) = 10.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.3222087345\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2941}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{5}\cdot 3\cdot 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 240\sqrt{2941}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 256 q^{2} + ( 8814 - \beta ) q^{3} + 65536 q^{4} -390625 q^{5} + ( 2256384 - 256 \beta ) q^{6} + ( 13842098 - 117 \beta ) q^{7} + 16777216 q^{8} + ( 117948033 - 17628 \beta ) q^{9} +O(q^{10})\) \( q + 256 q^{2} + ( 8814 - \beta ) q^{3} + 65536 q^{4} -390625 q^{5} + ( 2256384 - 256 \beta ) q^{6} + ( 13842098 - 117 \beta ) q^{7} + 16777216 q^{8} + ( 117948033 - 17628 \beta ) q^{9} -100000000 q^{10} + ( 32257632 + 90882 \beta ) q^{11} + ( 577634304 - 65536 \beta ) q^{12} + ( 1447919234 + 74988 \beta ) q^{13} + ( 3543577088 - 29952 \beta ) q^{14} + ( -3442968750 + 390625 \beta ) q^{15} + 4294967296 q^{16} + ( 790106298 - 1394748 \beta ) q^{17} + ( 30194696448 - 4512768 \beta ) q^{18} + ( 22106856380 + 7161912 \beta ) q^{19} -25600000000 q^{20} + ( 141824238972 - 14873336 \beta ) q^{21} + ( 8257953792 + 23265792 \beta ) q^{22} + ( 243774891414 + 23366889 \beta ) q^{23} + ( 147874381824 - 16777216 \beta ) q^{24} + 152587890625 q^{25} + ( 370667323904 + 19196928 \beta ) q^{26} + ( 2887563970980 - 144181062 \beta ) q^{27} + ( 907155734528 - 7667712 \beta ) q^{28} + ( 1993657431750 + 2589624 \beta ) q^{29} + ( -881400000000 + 100000000 \beta ) q^{30} + ( -2746130669668 - 48426966 \beta ) q^{31} + 1099511627776 q^{32} + ( -15111237442752 + 768776316 \beta ) q^{33} + ( 202267212288 - 357055488 \beta ) q^{34} + ( -5407069531250 + 45703125 \beta ) q^{35} + ( 7729842290688 - 1155268608 \beta ) q^{36} + ( -31357643818942 + 503271432 \beta ) q^{37} + ( 5659355233280 + 1833449472 \beta ) q^{38} + ( 58872947676 - 786975002 \beta ) q^{39} -6553600000000 q^{40} + ( 11705738638662 - 5025537684 \beta ) q^{41} + ( 36307005176832 - 3807574016 \beta ) q^{42} + ( -62428461595546 + 2880330687 \beta ) q^{43} + ( 2114036170752 + 5956042752 \beta ) q^{44} + ( -46073450390625 + 6885937500 \beta ) q^{45} + ( 62406372201984 + 5981923584 \beta ) q^{46} + ( -92973306061782 - 4923046557 \beta ) q^{47} + ( 37855841746944 - 4294967296 \beta ) q^{48} + ( -38707898443203 - 3239050932 \beta ) q^{49} + 39062500000000 q^{50} + ( 243236539707372 - 13083415170 \beta ) q^{51} + ( 94890834919424 + 4914413568 \beta ) q^{52} + ( 179669890323834 + 25138319508 \beta ) q^{53} + ( 739216376570880 - 36910351872 \beta ) q^{54} + ( -12600637500000 - 35500781250 \beta ) q^{55} + ( 232231868039168 - 1962934272 \beta ) q^{56} + ( -1018389519725880 + 41018235988 \beta ) q^{57} + ( 510376302528000 + 662943744 \beta ) q^{58} + ( 451089585180300 + 65066242428 \beta ) q^{59} + ( -225638400000000 + 25600000000 \beta ) q^{60} + ( -782211459483838 + 93596587488 \beta ) q^{61} + ( -703009451435008 - 12397303296 \beta ) q^{62} + ( 1982034966054834 - 257808423405 \beta ) q^{63} + 281474976710656 q^{64} + ( -565593450781250 - 29292187500 \beta ) q^{65} + ( -3868476785344512 + 196806736896 \beta ) q^{66} + ( -2919869465527342 + 31577428989 \beta ) q^{67} + ( 51780406345728 - 91406204928 \beta ) q^{68} + ( -1809756490699404 - 37819131768 \beta ) q^{69} + ( -1384209800000000 + 11700000000 \beta ) q^{70} + ( -33794280217068 - 232972538742 \beta ) q^{71} + ( 1978839626416128 - 295748763648 \beta ) q^{72} + ( 1766695349792834 + 214621625988 \beta ) q^{73} + ( -8027556817649152 + 128837486592 \beta ) q^{74} + ( 1344909667968750 - 152587890625 \beta ) q^{75} + ( 1448794939719680 + 469363064832 \beta ) q^{76} + ( -1354766773318464 + 1254223407492 \beta ) q^{77} + ( 15071474605056 - 201465600512 \beta ) q^{78} + ( 9501328198276040 + 349601077692 \beta ) q^{79} -1677721600000000 q^{80} + ( 34643663225567541 - 1881893058084 \beta ) q^{81} + ( 2996669091497472 - 1286537647104 \beta ) q^{82} + ( 10837543987559094 + 1106830681155 \beta ) q^{83} + ( 9294593325268992 - 974738948096 \beta ) q^{84} + ( -308635272656250 + 544823437500 \beta ) q^{85} + ( -15981686168459776 + 737364655872 \beta ) q^{86} + ( 17133410154446100 - 1970832485814 \beta ) q^{87} + ( 541193259712512 + 1524746944512 \beta ) q^{88} + ( -48749761096111110 - 1388020676040 \beta ) q^{89} + ( -11794803300000000 + 1762800000000 \beta ) q^{90} + ( 18555978732959332 + 868584694446 \beta ) q^{91} + ( 15976031283707904 + 1531372437504 \beta ) q^{92} + ( -16000790198908152 + 2319295391344 \beta ) q^{93} + ( -23801166351816192 - 1260299918592 \beta ) q^{94} + ( -8635490773437500 - 2797621875000 \beta ) q^{95} + ( 9691095487217664 - 1099511627776 \beta ) q^{96} + ( 49944968927693378 + 958545051660 \beta ) q^{97} + ( -9909222001459968 - 829197038592 \beta ) q^{98} + ( -267588140647395744 + 10150715598210 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 512q^{2} + 17628q^{3} + 131072q^{4} - 781250q^{5} + 4512768q^{6} + 27684196q^{7} + 33554432q^{8} + 235896066q^{9} + O(q^{10}) \) \( 2q + 512q^{2} + 17628q^{3} + 131072q^{4} - 781250q^{5} + 4512768q^{6} + 27684196q^{7} + 33554432q^{8} + 235896066q^{9} - 200000000q^{10} + 64515264q^{11} + 1155268608q^{12} + 2895838468q^{13} + 7087154176q^{14} - 6885937500q^{15} + 8589934592q^{16} + 1580212596q^{17} + 60389392896q^{18} + 44213712760q^{19} - 51200000000q^{20} + 283648477944q^{21} + 16515907584q^{22} + 487549782828q^{23} + 295748763648q^{24} + 305175781250q^{25} + 741334647808q^{26} + 5775127941960q^{27} + 1814311469056q^{28} + 3987314863500q^{29} - 1762800000000q^{30} - 5492261339336q^{31} + 2199023255552q^{32} - 30222474885504q^{33} + 404534424576q^{34} - 10814139062500q^{35} + 15459684581376q^{36} - 62715287637884q^{37} + 11318710466560q^{38} + 117745895352q^{39} - 13107200000000q^{40} + 23411477277324q^{41} + 72614010353664q^{42} - 124856923191092q^{43} + 4228072341504q^{44} - 92146900781250q^{45} + 124812744403968q^{46} - 185946612123564q^{47} + 75711683493888q^{48} - 77415796886406q^{49} + 78125000000000q^{50} + 486473079414744q^{51} + 189781669838848q^{52} + 359339780647668q^{53} + 1478432753141760q^{54} - 25201275000000q^{55} + 464463736078336q^{56} - 2036779039451760q^{57} + 1020752605056000q^{58} + 902179170360600q^{59} - 451276800000000q^{60} - 1564422918967676q^{61} - 1406018902870016q^{62} + 3964069932109668q^{63} + 562949953421312q^{64} - 1131186901562500q^{65} - 7736953570689024q^{66} - 5839738931054684q^{67} + 103560812691456q^{68} - 3619512981398808q^{69} - 2768419600000000q^{70} - 67588560434136q^{71} + 3957679252832256q^{72} + 3533390699585668q^{73} - 16055113635298304q^{74} + 2689819335937500q^{75} + 2897589879439360q^{76} - 2709533546636928q^{77} + 30142949210112q^{78} + 19002656396552080q^{79} - 3355443200000000q^{80} + 69287326451135082q^{81} + 5993338182994944q^{82} + 21675087975118188q^{83} + 18589186650537984q^{84} - 617270545312500q^{85} - 31963372336919552q^{86} + 34266820308892200q^{87} + 1082386519425024q^{88} - 97499522192222220q^{89} - 23589606600000000q^{90} + 37111957465918664q^{91} + 31952062567415808q^{92} - 32001580397816304q^{93} - 47602332703632384q^{94} - 17270981546875000q^{95} + 19382190974435328q^{96} + 99889937855386756q^{97} - 19818444002919936q^{98} - 535176281294791488q^{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
27.6155
−26.6155
256.000 −4201.44 65536.0 −390625. −1.07557e6 1.23193e7 1.67772e7 −1.11488e8 −1.00000e8
1.2 256.000 21829.4 65536.0 −390625. 5.58834e6 1.53649e7 1.67772e7 3.47384e8 −1.00000e8
\(n\): 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 10.18.a.d 2
3.b odd 2 1 90.18.a.j 2
4.b odd 2 1 80.18.a.b 2
5.b even 2 1 50.18.a.c 2
5.c odd 4 2 50.18.b.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.18.a.d 2 1.a even 1 1 trivial
50.18.a.c 2 5.b even 2 1
50.18.b.f 4 5.c odd 4 2
80.18.a.b 2 4.b odd 2 1
90.18.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 \( T_{3}^{2} - 17628 T_{3} - 91715004 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(10))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 256 T )^{2} \)
$3$ \( 1 - 17628 T + 166565322 T^{2} - 2276482793364 T^{3} + 16677181699666569 T^{4} \)
$5$ \( ( 1 + 390625 T )^{2} \)
$7$ \( 1 - 27684196 T + 654545766513618 T^{2} - \)\(64\!\cdots\!72\)\( T^{3} + \)\(54\!\cdots\!49\)\( T^{4} \)
$11$ \( 1 - 64515264 T - 387244327765443434 T^{2} - \)\(32\!\cdots\!44\)\( T^{3} + \)\(25\!\cdots\!41\)\( T^{4} \)
$13$ \( 1 - 2895838468 T + 18444722845435992222 T^{2} - \)\(25\!\cdots\!44\)\( T^{3} + \)\(74\!\cdots\!89\)\( T^{4} \)
$17$ \( 1 - 1580212596 T + \)\(13\!\cdots\!58\)\( T^{2} - \)\(13\!\cdots\!92\)\( T^{3} + \)\(68\!\cdots\!29\)\( T^{4} \)
$19$ \( 1 - 44213712760 T + \)\(27\!\cdots\!78\)\( T^{2} - \)\(24\!\cdots\!40\)\( T^{3} + \)\(30\!\cdots\!21\)\( T^{4} \)
$23$ \( 1 - 487549782828 T + \)\(24\!\cdots\!02\)\( T^{2} - \)\(68\!\cdots\!84\)\( T^{3} + \)\(19\!\cdots\!09\)\( T^{4} \)
$29$ \( 1 - 3987314863500 T + \)\(18\!\cdots\!18\)\( T^{2} - \)\(28\!\cdots\!00\)\( T^{3} + \)\(52\!\cdots\!81\)\( T^{4} \)
$31$ \( 1 + 5492261339336 T + \)\(52\!\cdots\!46\)\( T^{2} + \)\(12\!\cdots\!96\)\( T^{3} + \)\(50\!\cdots\!21\)\( T^{4} \)
$37$ \( 1 + 62715287637884 T + \)\(18\!\cdots\!98\)\( T^{2} + \)\(28\!\cdots\!28\)\( T^{3} + \)\(20\!\cdots\!89\)\( T^{4} \)
$41$ \( 1 - 23411477277324 T + \)\(10\!\cdots\!06\)\( T^{2} - \)\(61\!\cdots\!44\)\( T^{3} + \)\(68\!\cdots\!61\)\( T^{4} \)
$43$ \( 1 + 124856923191092 T + \)\(14\!\cdots\!02\)\( T^{2} + \)\(73\!\cdots\!56\)\( T^{3} + \)\(34\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 + 185946612123564 T + \)\(57\!\cdots\!98\)\( T^{2} + \)\(49\!\cdots\!68\)\( T^{3} + \)\(71\!\cdots\!69\)\( T^{4} \)
$53$ \( 1 - 359339780647668 T + \)\(33\!\cdots\!82\)\( T^{2} - \)\(73\!\cdots\!84\)\( T^{3} + \)\(42\!\cdots\!69\)\( T^{4} \)
$59$ \( 1 - 902179170360600 T + \)\(20\!\cdots\!38\)\( T^{2} - \)\(11\!\cdots\!00\)\( T^{3} + \)\(16\!\cdots\!61\)\( T^{4} \)
$61$ \( 1 + 1564422918967676 T + \)\(36\!\cdots\!86\)\( T^{2} + \)\(35\!\cdots\!96\)\( T^{3} + \)\(50\!\cdots\!41\)\( T^{4} \)
$67$ \( 1 + 5839738931054684 T + \)\(30\!\cdots\!18\)\( T^{2} + \)\(64\!\cdots\!68\)\( T^{3} + \)\(12\!\cdots\!29\)\( T^{4} \)
$71$ \( 1 + 67588560434136 T + \)\(50\!\cdots\!06\)\( T^{2} + \)\(20\!\cdots\!76\)\( T^{3} + \)\(87\!\cdots\!81\)\( T^{4} \)
$73$ \( 1 - 3533390699585668 T + \)\(90\!\cdots\!62\)\( T^{2} - \)\(16\!\cdots\!04\)\( T^{3} + \)\(22\!\cdots\!09\)\( T^{4} \)
$79$ \( 1 - 19002656396552080 T + \)\(43\!\cdots\!18\)\( T^{2} - \)\(34\!\cdots\!20\)\( T^{3} + \)\(33\!\cdots\!81\)\( T^{4} \)
$83$ \( 1 - 21675087975118188 T + \)\(75\!\cdots\!82\)\( T^{2} - \)\(91\!\cdots\!24\)\( T^{3} + \)\(17\!\cdots\!29\)\( T^{4} \)
$89$ \( 1 + 97499522192222220 T + \)\(48\!\cdots\!58\)\( T^{2} + \)\(13\!\cdots\!80\)\( T^{3} + \)\(19\!\cdots\!41\)\( T^{4} \)
$97$ \( 1 - 99889937855386756 T + \)\(14\!\cdots\!58\)\( T^{2} - \)\(59\!\cdots\!72\)\( T^{3} + \)\(35\!\cdots\!69\)\( T^{4} \)
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