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: \( 2 \)
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.

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

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.

Atkin-Lehner signs

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

Hecke kernels

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