Properties

Label 5.10.a.b
Level 5
Weight 10
Character orbit 5.a
Self dual yes
Analytic conductor 2.575
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(2.57517918082\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1009}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{1009}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -5 - \beta ) q^{2} + ( 130 + 2 \beta ) q^{3} + ( 522 + 10 \beta ) q^{4} + 625 q^{5} + ( -2668 - 140 \beta ) q^{6} + ( 850 + 214 \beta ) q^{7} + ( -10140 - 60 \beta ) q^{8} + ( 1253 + 520 \beta ) q^{9} +O(q^{10})\) \( q + ( -5 - \beta ) q^{2} + ( 130 + 2 \beta ) q^{3} + ( 522 + 10 \beta ) q^{4} + 625 q^{5} + ( -2668 - 140 \beta ) q^{6} + ( 850 + 214 \beta ) q^{7} + ( -10140 - 60 \beta ) q^{8} + ( 1253 + 520 \beta ) q^{9} + ( -3125 - 625 \beta ) q^{10} + ( 11992 - 1900 \beta ) q^{11} + ( 88040 + 2344 \beta ) q^{12} + ( 57510 + 1352 \beta ) q^{13} + ( -220176 - 1920 \beta ) q^{14} + ( 81250 + 1250 \beta ) q^{15} + ( -156024 + 5320 \beta ) q^{16} + ( 206410 - 12856 \beta ) q^{17} + ( -530945 - 3853 \beta ) q^{18} + ( -148260 + 2840 \beta ) q^{19} + ( 326250 + 6250 \beta ) q^{20} + ( 542352 + 29520 \beta ) q^{21} + ( 1857140 - 2492 \beta ) q^{22} + ( -524610 - 19398 \beta ) q^{23} + ( -1439280 - 28080 \beta ) q^{24} + 390625 q^{25} + ( -1651718 - 64270 \beta ) q^{26} + ( -1346540 + 30740 \beta ) q^{27} + ( 2602960 + 120208 \beta ) q^{28} + ( -1833490 + 106960 \beta ) q^{29} + ( -1667500 - 87500 \beta ) q^{30} + ( 806572 - 154700 \beta ) q^{31} + ( 603920 + 160144 \beta ) q^{32} + ( -2275240 - 223016 \beta ) q^{33} + ( 11939654 - 142130 \beta ) q^{34} + ( 531250 + 133750 \beta ) q^{35} + ( 5900866 + 283970 \beta ) q^{36} + ( -10560970 - 205296 \beta ) q^{37} + ( -2124260 + 134060 \beta ) q^{38} + ( 10204636 + 290780 \beta ) q^{39} + ( -6337500 - 37500 \beta ) q^{40} + ( -13478638 + 155800 \beta ) q^{41} + ( -32497440 - 689952 \beta ) q^{42} + ( 26444850 - 25798 \beta ) q^{43} + ( -12911176 - 871880 \beta ) q^{44} + ( 783125 + 325000 \beta ) q^{45} + ( 22195632 + 621600 \beta ) q^{46} + ( 29206090 + 523334 \beta ) q^{47} + ( -9547360 + 379552 \beta ) q^{48} + ( 6577057 + 363800 \beta ) q^{49} + ( -1953125 - 390625 \beta ) q^{50} + ( 889892 - 1258460 \beta ) q^{51} + ( 43661900 + 1280844 \beta ) q^{52} + ( -19517570 - 1137448 \beta ) q^{53} + ( -24283960 + 1192840 \beta ) q^{54} + ( 7495000 - 1187500 \beta ) q^{55} + ( -21574560 - 2220960 \beta ) q^{56} + ( -13542680 + 72680 \beta ) q^{57} + ( -98755190 + 1298690 \beta ) q^{58} + ( -27497780 + 1544720 \beta ) q^{59} + ( 55025000 + 1465000 \beta ) q^{60} + ( -137289858 + 692000 \beta ) q^{61} + ( 152059440 - 33072 \beta ) q^{62} + ( 113346570 + 710142 \beta ) q^{63} + ( -84720608 - 4128480 \beta ) q^{64} + ( 35943750 + 845000 \beta ) q^{65} + ( 236399344 + 3390320 \beta ) q^{66} + ( -159290 - 2416706 \beta ) q^{67} + ( -21971020 - 4646732 \beta ) q^{68} + ( -107344464 - 3570960 \beta ) q^{69} + ( -137610000 - 1200000 \beta ) q^{70} + ( -3565468 + 6278500 \beta ) q^{71} + ( -44186220 - 5347980 \beta ) q^{72} + ( 60429090 + 8830952 \beta ) q^{73} + ( 259948514 + 11587450 \beta ) q^{74} + ( 50781250 + 781250 \beta ) q^{75} + ( -48736120 - 120 \beta ) q^{76} + ( -400066200 + 951288 \beta ) q^{77} + ( -344420200 - 11658536 \beta ) q^{78} + ( 3438760 - 18775640 \beta ) q^{79} + ( -97515000 + 3325000 \beta ) q^{80} + ( -137679679 - 8932040 \beta ) q^{81} + ( -89809010 + 12699638 \beta ) q^{82} + ( 701174370 + 2748402 \beta ) q^{83} + ( 580964544 + 20832960 \beta ) q^{84} + ( 129006250 - 8035000 \beta ) q^{85} + ( -106194068 - 26315860 \beta ) q^{86} + ( -22508420 + 10237820 \beta ) q^{87} + ( -6572880 + 18546480 \beta ) q^{88} + ( 415044330 + 13381680 \beta ) q^{89} + ( -331840625 - 2408125 \beta ) q^{90} + ( 340815452 + 13456340 \beta ) q^{91} + ( -469572240 - 15371856 \beta ) q^{92} + ( -207330240 - 18497856 \beta ) q^{93} + ( -674074456 - 31822760 \beta ) q^{94} + ( -92662500 + 1775000 \beta ) q^{95} + ( 401680192 + 22026560 \beta ) q^{96} + ( 319197290 - 2622216 \beta ) q^{97} + ( -399959485 - 8396057 \beta ) q^{98} + ( -981866024 + 3855140 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 10q^{2} + 260q^{3} + 1044q^{4} + 1250q^{5} - 5336q^{6} + 1700q^{7} - 20280q^{8} + 2506q^{9} + O(q^{10}) \) \( 2q - 10q^{2} + 260q^{3} + 1044q^{4} + 1250q^{5} - 5336q^{6} + 1700q^{7} - 20280q^{8} + 2506q^{9} - 6250q^{10} + 23984q^{11} + 176080q^{12} + 115020q^{13} - 440352q^{14} + 162500q^{15} - 312048q^{16} + 412820q^{17} - 1061890q^{18} - 296520q^{19} + 652500q^{20} + 1084704q^{21} + 3714280q^{22} - 1049220q^{23} - 2878560q^{24} + 781250q^{25} - 3303436q^{26} - 2693080q^{27} + 5205920q^{28} - 3666980q^{29} - 3335000q^{30} + 1613144q^{31} + 1207840q^{32} - 4550480q^{33} + 23879308q^{34} + 1062500q^{35} + 11801732q^{36} - 21121940q^{37} - 4248520q^{38} + 20409272q^{39} - 12675000q^{40} - 26957276q^{41} - 64994880q^{42} + 52889700q^{43} - 25822352q^{44} + 1566250q^{45} + 44391264q^{46} + 58412180q^{47} - 19094720q^{48} + 13154114q^{49} - 3906250q^{50} + 1779784q^{51} + 87323800q^{52} - 39035140q^{53} - 48567920q^{54} + 14990000q^{55} - 43149120q^{56} - 27085360q^{57} - 197510380q^{58} - 54995560q^{59} + 110050000q^{60} - 274579716q^{61} + 304118880q^{62} + 226693140q^{63} - 169441216q^{64} + 71887500q^{65} + 472798688q^{66} - 318580q^{67} - 43942040q^{68} - 214688928q^{69} - 275220000q^{70} - 7130936q^{71} - 88372440q^{72} + 120858180q^{73} + 519897028q^{74} + 101562500q^{75} - 97472240q^{76} - 800132400q^{77} - 688840400q^{78} + 6877520q^{79} - 195030000q^{80} - 275359358q^{81} - 179618020q^{82} + 1402348740q^{83} + 1161929088q^{84} + 258012500q^{85} - 212388136q^{86} - 45016840q^{87} - 13145760q^{88} + 830088660q^{89} - 663681250q^{90} + 681630904q^{91} - 939144480q^{92} - 414660480q^{93} - 1348148912q^{94} - 185325000q^{95} + 803360384q^{96} + 638394580q^{97} - 799918970q^{98} - 1963732048q^{99} + O(q^{100}) \)

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
16.3824
−15.3824
−36.7648 193.530 839.648 625.000 −7115.07 7647.66 −12045.9 17770.7 −22978.0
1.2 26.7648 66.4705 204.352 625.000 1779.07 −5947.66 −8234.11 −15264.7 16728.0
\(n\): 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 5.10.a.b 2
3.b odd 2 1 45.10.a.f 2
4.b odd 2 1 80.10.a.f 2
5.b even 2 1 25.10.a.b 2
5.c odd 4 2 25.10.b.b 4
7.b odd 2 1 245.10.a.d 2
8.b even 2 1 320.10.a.k 2
8.d odd 2 1 320.10.a.s 2
15.d odd 2 1 225.10.a.h 2
15.e even 4 2 225.10.b.h 4
20.d odd 2 1 400.10.a.t 2
20.e even 4 2 400.10.c.p 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.10.a.b 2 1.a even 1 1 trivial
25.10.a.b 2 5.b even 2 1
25.10.b.b 4 5.c odd 4 2
45.10.a.f 2 3.b odd 2 1
80.10.a.f 2 4.b odd 2 1
225.10.a.h 2 15.d odd 2 1
225.10.b.h 4 15.e even 4 2
245.10.a.d 2 7.b odd 2 1
320.10.a.k 2 8.b even 2 1
320.10.a.s 2 8.d odd 2 1
400.10.a.t 2 20.d odd 2 1
400.10.c.p 4 20.e even 4 2

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 10 T_{2} - 984 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(5))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 10 T + 40 T^{2} + 5120 T^{3} + 262144 T^{4} \)
$3$ \( 1 - 260 T + 52230 T^{2} - 5117580 T^{3} + 387420489 T^{4} \)
$5$ \( ( 1 - 625 T )^{2} \)
$7$ \( 1 - 1700 T + 35221550 T^{2} - 68601131900 T^{3} + 1628413597910449 T^{4} \)
$11$ \( 1 - 23984 T + 1217213446 T^{2} - 56553017420944 T^{3} + 5559917313492231481 T^{4} \)
$13$ \( 1 - 115020 T + 22672043710 T^{2} - 1219729517882460 T^{3} + \)\(11\!\cdots\!29\)\( T^{4} \)
$17$ \( 1 - 412820 T + 113016614470 T^{2} - 48955447175491540 T^{3} + \)\(14\!\cdots\!09\)\( T^{4} \)
$19$ \( 1 + 296520 T + 659218232758 T^{2} + 95683356145429080 T^{3} + \)\(10\!\cdots\!41\)\( T^{4} \)
$23$ \( 1 + 1049220 T + 3497852029390 T^{2} + 1889805395460208860 T^{3} + \)\(32\!\cdots\!69\)\( T^{4} \)
$29$ \( 1 + 3666980 T + 20832571957438 T^{2} + 53197414150592105620 T^{3} + \)\(21\!\cdots\!61\)\( T^{4} \)
$31$ \( 1 - 1613144 T + 29382323902526 T^{2} - 42650917850753459624 T^{3} + \)\(69\!\cdots\!41\)\( T^{4} \)
$37$ \( 1 + 21121940 T + 328931801286510 T^{2} + \)\(27\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!29\)\( T^{4} \)
$41$ \( 1 + 26957276 T + 811945448362966 T^{2} + \)\(88\!\cdots\!36\)\( T^{3} + \)\(10\!\cdots\!21\)\( T^{4} \)
$43$ \( 1 - 52889700 T + 1703843788760950 T^{2} - \)\(26\!\cdots\!00\)\( T^{3} + \)\(25\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 - 58412180 T + 2814913257457630 T^{2} - \)\(65\!\cdots\!60\)\( T^{3} + \)\(12\!\cdots\!89\)\( T^{4} \)
$53$ \( 1 + 39035140 T + 5675030678030830 T^{2} + \)\(12\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!89\)\( T^{4} \)
$59$ \( 1 + 54995560 T + 15674484224932678 T^{2} + \)\(47\!\cdots\!40\)\( T^{3} + \)\(75\!\cdots\!21\)\( T^{4} \)
$61$ \( 1 + 274579716 T + 41753623519328446 T^{2} + \)\(32\!\cdots\!56\)\( T^{3} + \)\(13\!\cdots\!81\)\( T^{4} \)
$67$ \( 1 + 318580 T + 48520062064444070 T^{2} + \)\(86\!\cdots\!60\)\( T^{3} + \)\(74\!\cdots\!09\)\( T^{4} \)
$71$ \( 1 + 7130936 T + 51935375688707086 T^{2} + \)\(32\!\cdots\!16\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} \)
$73$ \( 1 - 120858180 T + 42707263689423190 T^{2} - \)\(71\!\cdots\!40\)\( T^{3} + \)\(34\!\cdots\!69\)\( T^{4} \)
$79$ \( 1 - 6877520 T - 115982362290712162 T^{2} - \)\(82\!\cdots\!80\)\( T^{3} + \)\(14\!\cdots\!61\)\( T^{4} \)
$83$ \( 1 - 1402348740 T + 857904310704391270 T^{2} - \)\(26\!\cdots\!20\)\( T^{3} + \)\(34\!\cdots\!09\)\( T^{4} \)
$89$ \( 1 - 830088660 T + 692293619421117718 T^{2} - \)\(29\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!81\)\( T^{4} \)
$97$ \( 1 - 638394580 T + 1615411126351062630 T^{2} - \)\(48\!\cdots\!60\)\( T^{3} + \)\(57\!\cdots\!89\)\( T^{4} \)
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