Properties

Label 10.18.a.b
Level 10
Weight 18
Character orbit 10.a
Self dual yes
Analytic conductor 18.322
Analytic rank 1
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{36061}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{5}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 80\sqrt{36061}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -256 q^{2} + ( -3154 - \beta ) q^{3} + 65536 q^{4} -390625 q^{5} + ( 807424 + 256 \beta ) q^{6} + ( 3271922 + 843 \beta ) q^{7} -16777216 q^{8} + ( 111597953 + 6308 \beta ) q^{9} +O(q^{10})\) \( q -256 q^{2} + ( -3154 - \beta ) q^{3} + 65536 q^{4} -390625 q^{5} + ( 807424 + 256 \beta ) q^{6} + ( 3271922 + 843 \beta ) q^{7} -16777216 q^{8} + ( 111597953 + 6308 \beta ) q^{9} + 100000000 q^{10} + ( 594704352 + 12738 \beta ) q^{11} + ( -206700544 - 65536 \beta ) q^{12} + ( -1008959614 - 122772 \beta ) q^{13} + ( -837612032 - 215808 \beta ) q^{14} + ( 1232031250 + 390625 \beta ) q^{15} + 4294967296 q^{16} + ( -9377819718 + 1944132 \beta ) q^{17} + ( -28569075968 - 1614848 \beta ) q^{18} + ( -68352415300 + 2994168 \beta ) q^{19} -25600000000 q^{20} + ( -204875949188 - 5930744 \beta ) q^{21} + ( -152244314112 - 3260928 \beta ) q^{22} + ( -324617085354 - 22452951 \beta ) q^{23} + ( 52915339264 + 16777216 \beta ) q^{24} + 152587890625 q^{25} + ( 258293661184 + 31429632 \beta ) q^{26} + ( -1400497712860 - 2353222 \beta ) q^{27} + ( 214428680192 + 55246848 \beta ) q^{28} + ( -2348271710010 - 4841544 \beta ) q^{29} + ( -315400000000 - 100000000 \beta ) q^{30} + ( 3560433689372 + 342178986 \beta ) q^{31} -1099511627776 q^{32} + ( -4815505641408 - 634880004 \beta ) q^{33} + ( 2400721847808 - 497697792 \beta ) q^{34} + ( -1278094531250 - 329296875 \beta ) q^{35} + ( 7313683447808 + 413401088 \beta ) q^{36} + ( 4966557318722 + 1485224712 \beta ) q^{37} + ( 17498218316800 - 766507008 \beta ) q^{38} + ( 31516857611356 + 1396182502 \beta ) q^{39} + 6553600000000 q^{40} + ( 4811355940422 + 2262047724 \beta ) q^{41} + ( 52448242992128 + 1518270464 \beta ) q^{42} + ( 3854218005926 - 8809717953 \beta ) q^{43} + ( 38974544412672 + 834797568 \beta ) q^{44} + ( -43592950390625 - 2464062500 \beta ) q^{45} + ( 83101973850624 + 5747955456 \beta ) q^{46} + ( -233036287918998 - 4448723037 \beta ) q^{47} + ( -13546326851584 - 4294967296 \beta ) q^{48} + ( -57914073443523 + 5516460492 \beta ) q^{49} -39062500000000 q^{50} + ( -419109358542228 + 3246027390 \beta ) q^{51} + ( -66123177263104 - 8045985792 \beta ) q^{52} + ( -311666560142214 + 27224743188 \beta ) q^{53} + ( 358527414492160 + 602424832 \beta ) q^{54} + ( -232306387500000 - 4975781250 \beta ) q^{55} + ( -54893742129152 - 14143193088 \beta ) q^{56} + ( -475441712531000 + 58908809428 \beta ) q^{57} + ( 601157557762560 + 1239435264 \beta ) q^{58} + ( 327308035997580 - 85208012868 \beta ) q^{59} + ( 80742400000000 + 25600000000 \beta ) q^{60} + ( 664520192581442 - 118790012448 \beta ) q^{61} + ( -911471024479232 - 87597820416 \beta ) q^{62} + ( 1592400983393266 + 114716358355 \beta ) q^{63} + 281474976710656 q^{64} + ( 394124849218750 + 47957812500 \beta ) q^{65} + ( 1232769444200448 + 162529281024 \beta ) q^{66} + ( -2124508205938798 - 126467311491 \beta ) q^{67} + ( -614584793038848 + 127410634752 \beta ) q^{68} + ( 6205767829676916 + 395433692808 \beta ) q^{69} + ( 327192200000000 + 84300000000 \beta ) q^{70} + ( 1017565974673812 - 551192664438 \beta ) q^{71} + ( -1872302962638848 - 105830678528 \beta ) q^{72} + ( 1124296730673986 - 27932935932 \beta ) q^{73} + ( -1271438673592832 - 380217526272 \beta ) q^{74} + ( -481262207031250 - 152587890625 \beta ) q^{75} + ( -4479543889100800 + 196225794048 \beta ) q^{76} + ( 4424084493918144 + 543013511172 \beta ) q^{77} + ( -8068315548507136 - 357422720512 \beta ) q^{78} + ( 9594213839247560 + 84548615868 \beta ) q^{79} -1677721600000000 q^{80} + ( -9451507007857099 + 593303626844 \beta ) q^{81} + ( -1231707120748032 - 579084217344 \beta ) q^{82} + ( -9237324543305994 - 203422699965 \beta ) q^{83} + ( -13426750205984768 - 388677238784 \beta ) q^{84} + ( 3663210827343750 - 759426562500 \beta ) q^{85} + ( -986679809517056 + 2255287795968 \beta ) q^{86} + ( 8523830849749140 + 2363541939786 \beta ) q^{87} + ( -9977483369644032 - 213708177408 \beta ) q^{88} + ( -8729456458197510 - 882301557960 \beta ) q^{89} + ( 11159795300000000 + 630800000000 \beta ) q^{90} + ( -27187304105716508 - 1252253362386 \beta ) q^{91} + ( -21274105305759744 - 1471476596736 \beta ) q^{92} + ( -90201232906813688 - 4639666211216 \beta ) q^{93} + ( 59657289707263488 + 1138873097472 \beta ) q^{94} + ( 26700162226562500 - 1169596875000 \beta ) q^{95} + ( 3467859674005504 + 1099511627776 \beta ) q^{96} + ( 97967162911245122 - 252225032820 \beta ) q^{97} + ( 14826002801541888 - 1412213885952 \beta ) q^{98} + ( 84912097914073056 + 5172929777730 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 512q^{2} - 6308q^{3} + 131072q^{4} - 781250q^{5} + 1614848q^{6} + 6543844q^{7} - 33554432q^{8} + 223195906q^{9} + O(q^{10}) \) \( 2q - 512q^{2} - 6308q^{3} + 131072q^{4} - 781250q^{5} + 1614848q^{6} + 6543844q^{7} - 33554432q^{8} + 223195906q^{9} + 200000000q^{10} + 1189408704q^{11} - 413401088q^{12} - 2017919228q^{13} - 1675224064q^{14} + 2464062500q^{15} + 8589934592q^{16} - 18755639436q^{17} - 57138151936q^{18} - 136704830600q^{19} - 51200000000q^{20} - 409751898376q^{21} - 304488628224q^{22} - 649234170708q^{23} + 105830678528q^{24} + 305175781250q^{25} + 516587322368q^{26} - 2800995425720q^{27} + 428857360384q^{28} - 4696543420020q^{29} - 630800000000q^{30} + 7120867378744q^{31} - 2199023255552q^{32} - 9631011282816q^{33} + 4801443695616q^{34} - 2556189062500q^{35} + 14627366895616q^{36} + 9933114637444q^{37} + 34996436633600q^{38} + 63033715222712q^{39} + 13107200000000q^{40} + 9622711880844q^{41} + 104896485984256q^{42} + 7708436011852q^{43} + 77949088825344q^{44} - 87185900781250q^{45} + 166203947701248q^{46} - 466072575837996q^{47} - 27092653703168q^{48} - 115828146887046q^{49} - 78125000000000q^{50} - 838218717084456q^{51} - 132246354526208q^{52} - 623333120284428q^{53} + 717054828984320q^{54} - 464612775000000q^{55} - 109787484258304q^{56} - 950883425062000q^{57} + 1202315115525120q^{58} + 654616071995160q^{59} + 161484800000000q^{60} + 1329040385162884q^{61} - 1822942048958464q^{62} + 3184801966786532q^{63} + 562949953421312q^{64} + 788249698437500q^{65} + 2465538888400896q^{66} - 4249016411877596q^{67} - 1229169586077696q^{68} + 12411535659353832q^{69} + 654384400000000q^{70} + 2035131949347624q^{71} - 3744605925277696q^{72} + 2248593461347972q^{73} - 2542877347185664q^{74} - 962524414062500q^{75} - 8959087778201600q^{76} + 8848168987836288q^{77} - 16136631097014272q^{78} + 19188427678495120q^{79} - 3355443200000000q^{80} - 18903014015714198q^{81} - 2463414241496064q^{82} - 18474649086611988q^{83} - 26853500411969536q^{84} + 7326421654687500q^{85} - 1973359619034112q^{86} + 17047661699498280q^{87} - 19954966739288064q^{88} - 17458912916395020q^{89} + 22319590600000000q^{90} - 54374608211433016q^{91} - 42548210611519488q^{92} - 180402465813627376q^{93} + 119314579414526976q^{94} + 53400324453125000q^{95} + 6935719348011008q^{96} + 195934325822490244q^{97} + 29652005603083776q^{98} + 169824195828146112q^{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
95.4487
−94.4487
−256.000 −18345.8 65536.0 −390625. 4.69652e6 1.60786e7 −1.67772e7 2.07428e8 1.00000e8
1.2 −256.000 12037.8 65536.0 −390625. −3.08167e6 −9.53475e6 −1.67772e7 1.57682e7 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.b 2
3.b odd 2 1 90.18.a.n 2
4.b odd 2 1 80.18.a.e 2
5.b even 2 1 50.18.a.g 2
5.c odd 4 2 50.18.b.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.18.a.b 2 1.a even 1 1 trivial
50.18.a.g 2 5.b even 2 1
50.18.b.e 4 5.c odd 4 2
80.18.a.e 2 4.b odd 2 1
90.18.a.n 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 \( T_{3}^{2} + 6308 T_{3} - 220842684 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(10))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 256 T )^{2} \)
$3$ \( 1 + 6308 T + 37437642 T^{2} + 814616148204 T^{3} + 16677181699666569 T^{4} \)
$5$ \( ( 1 + 390625 T )^{2} \)
$7$ \( 1 - 6543844 T + 311955534578898 T^{2} - \)\(15\!\cdots\!08\)\( T^{3} + \)\(54\!\cdots\!49\)\( T^{4} \)
$11$ \( 1 - 1189408704 T + 1327120047514909846 T^{2} - \)\(60\!\cdots\!84\)\( T^{3} + \)\(25\!\cdots\!41\)\( T^{4} \)
$13$ \( 1 + 2017919228 T + 14840135954392751262 T^{2} + \)\(17\!\cdots\!24\)\( T^{3} + \)\(74\!\cdots\!89\)\( T^{4} \)
$17$ \( 1 + 18755639436 T + \)\(87\!\cdots\!78\)\( T^{2} + \)\(15\!\cdots\!72\)\( T^{3} + \)\(68\!\cdots\!29\)\( T^{4} \)
$19$ \( 1 + 136704830600 T + \)\(13\!\cdots\!78\)\( T^{2} + \)\(74\!\cdots\!00\)\( T^{3} + \)\(30\!\cdots\!21\)\( T^{4} \)
$23$ \( 1 + 649234170708 T + \)\(27\!\cdots\!22\)\( T^{2} + \)\(91\!\cdots\!24\)\( T^{3} + \)\(19\!\cdots\!09\)\( T^{4} \)
$29$ \( 1 + 4696543420020 T + \)\(20\!\cdots\!18\)\( T^{2} + \)\(34\!\cdots\!80\)\( T^{3} + \)\(52\!\cdots\!81\)\( T^{4} \)
$31$ \( 1 - 7120867378744 T + \)\(30\!\cdots\!06\)\( T^{2} - \)\(16\!\cdots\!84\)\( T^{3} + \)\(50\!\cdots\!21\)\( T^{4} \)
$37$ \( 1 - 9933114637444 T + \)\(42\!\cdots\!18\)\( T^{2} - \)\(45\!\cdots\!48\)\( T^{3} + \)\(20\!\cdots\!89\)\( T^{4} \)
$41$ \( 1 - 9622711880844 T + \)\(40\!\cdots\!46\)\( T^{2} - \)\(25\!\cdots\!64\)\( T^{3} + \)\(68\!\cdots\!61\)\( T^{4} \)
$43$ \( 1 - 7708436011852 T - \)\(61\!\cdots\!38\)\( T^{2} - \)\(45\!\cdots\!36\)\( T^{3} + \)\(34\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 + 466072575837996 T + \)\(10\!\cdots\!78\)\( T^{2} + \)\(12\!\cdots\!52\)\( T^{3} + \)\(71\!\cdots\!69\)\( T^{4} \)
$53$ \( 1 + 623333120284428 T + \)\(33\!\cdots\!22\)\( T^{2} + \)\(12\!\cdots\!64\)\( T^{3} + \)\(42\!\cdots\!69\)\( T^{4} \)
$59$ \( 1 - 654616071995160 T + \)\(97\!\cdots\!38\)\( T^{2} - \)\(83\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!61\)\( T^{4} \)
$61$ \( 1 - 1329040385162884 T + \)\(16\!\cdots\!06\)\( T^{2} - \)\(29\!\cdots\!64\)\( T^{3} + \)\(50\!\cdots\!41\)\( T^{4} \)
$67$ \( 1 + 4249016411877596 T + \)\(22\!\cdots\!58\)\( T^{2} + \)\(46\!\cdots\!92\)\( T^{3} + \)\(12\!\cdots\!29\)\( T^{4} \)
$71$ \( 1 - 2035131949347624 T - \)\(98\!\cdots\!74\)\( T^{2} - \)\(60\!\cdots\!84\)\( T^{3} + \)\(87\!\cdots\!81\)\( T^{4} \)
$73$ \( 1 - 2248593461347972 T + \)\(96\!\cdots\!02\)\( T^{2} - \)\(10\!\cdots\!16\)\( T^{3} + \)\(22\!\cdots\!09\)\( T^{4} \)
$79$ \( 1 - 19188427678495120 T + \)\(45\!\cdots\!18\)\( T^{2} - \)\(34\!\cdots\!80\)\( T^{3} + \)\(33\!\cdots\!81\)\( T^{4} \)
$83$ \( 1 + 18474649086611988 T + \)\(91\!\cdots\!82\)\( T^{2} + \)\(77\!\cdots\!24\)\( T^{3} + \)\(17\!\cdots\!29\)\( T^{4} \)
$89$ \( 1 + 17458912916395020 T + \)\(26\!\cdots\!58\)\( T^{2} + \)\(24\!\cdots\!80\)\( T^{3} + \)\(19\!\cdots\!41\)\( T^{4} \)
$97$ \( 1 - 195934325822490244 T + \)\(21\!\cdots\!58\)\( T^{2} - \)\(11\!\cdots\!28\)\( T^{3} + \)\(35\!\cdots\!69\)\( T^{4} \)
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