Properties

Label 5.8.a.b
Level 5
Weight 8
Character orbit 5.a
Self dual Yes
Analytic conductor 1.562
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 5 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 5.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(1.56192512742\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{19}) \)
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 \(\beta = 2\sqrt{19}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 10 + \beta ) q^{2} \) \( + ( 10 - 8 \beta ) q^{3} \) \( + ( 48 + 20 \beta ) q^{4} \) \( -125 q^{5} \) \( + ( -508 - 70 \beta ) q^{6} \) \( + ( -50 + 56 \beta ) q^{7} \) \( + ( 720 + 120 \beta ) q^{8} \) \( + ( 2777 - 160 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 10 + \beta ) q^{2} \) \( + ( 10 - 8 \beta ) q^{3} \) \( + ( 48 + 20 \beta ) q^{4} \) \( -125 q^{5} \) \( + ( -508 - 70 \beta ) q^{6} \) \( + ( -50 + 56 \beta ) q^{7} \) \( + ( 720 + 120 \beta ) q^{8} \) \( + ( 2777 - 160 \beta ) q^{9} \) \( + ( -1250 - 125 \beta ) q^{10} \) \( + ( 2272 + 400 \beta ) q^{11} \) \( + ( -11680 - 184 \beta ) q^{12} \) \( + ( 1770 - 608 \beta ) q^{13} \) \( + ( 3756 + 510 \beta ) q^{14} \) \( + ( -1250 + 1000 \beta ) q^{15} \) \( + ( 10176 - 640 \beta ) q^{16} \) \( + ( -13670 - 1184 \beta ) q^{17} \) \( + ( 15610 + 1177 \beta ) q^{18} \) \( + ( 19380 - 320 \beta ) q^{19} \) \( + ( -6000 - 2500 \beta ) q^{20} \) \( + ( -34548 + 960 \beta ) q^{21} \) \( + ( 53120 + 6272 \beta ) q^{22} \) \( + ( -62070 - 408 \beta ) q^{23} \) \( + ( -65760 - 4560 \beta ) q^{24} \) \( + 15625 q^{25} \) \( + ( -28508 - 4310 \beta ) q^{26} \) \( + ( 103180 - 6320 \beta ) q^{27} \) \( + ( 82720 + 1688 \beta ) q^{28} \) \( + ( -36130 + 19520 \beta ) q^{29} \) \( + ( 63500 + 8750 \beta ) q^{30} \) \( + ( 153412 - 2800 \beta ) q^{31} \) \( + ( -39040 - 11584 \beta ) q^{32} \) \( + ( -220480 - 14176 \beta ) q^{33} \) \( + ( -226684 - 25510 \beta ) q^{34} \) \( + ( 6250 - 7000 \beta ) q^{35} \) \( + ( -109904 + 47860 \beta ) q^{36} \) \( + ( -61510 + 25536 \beta ) q^{37} \) \( + ( 169480 + 16180 \beta ) q^{38} \) \( + ( 387364 - 20240 \beta ) q^{39} \) \( + ( -90000 - 15000 \beta ) q^{40} \) \( + ( 132182 - 56800 \beta ) q^{41} \) \( + ( -272520 - 24948 \beta ) q^{42} \) \( + ( 211650 + 43192 \beta ) q^{43} \) \( + ( 717056 + 64640 \beta ) q^{44} \) \( + ( -347125 + 20000 \beta ) q^{45} \) \( + ( -651708 - 66150 \beta ) q^{46} \) \( + ( -52730 + 45496 \beta ) q^{47} \) \( + ( 490880 - 87808 \beta ) q^{48} \) \( + ( -582707 - 5600 \beta ) q^{49} \) \( + ( 156250 + 15625 \beta ) q^{50} \) \( + ( 583172 + 97520 \beta ) q^{51} \) \( + ( -839200 + 6216 \beta ) q^{52} \) \( + ( -1195790 - 53408 \beta ) q^{53} \) \( + ( 551480 + 39980 \beta ) q^{54} \) \( + ( -284000 - 50000 \beta ) q^{55} \) \( + ( 474720 + 34320 \beta ) q^{56} \) \( + ( 388360 - 158240 \beta ) q^{57} \) \( + ( 1122220 + 159070 \beta ) q^{58} \) \( + ( -560060 - 227360 \beta ) q^{59} \) \( + ( 1460000 + 23000 \beta ) q^{60} \) \( + ( 1128522 + 160000 \beta ) q^{61} \) \( + ( 1321320 + 125412 \beta ) q^{62} \) \( + ( -819810 + 163512 \beta ) q^{63} \) \( + ( -2573312 - 72960 \beta ) q^{64} \) \( + ( -221250 + 76000 \beta ) q^{65} \) \( + ( -3282176 - 362240 \beta ) q^{66} \) \( + ( 2258230 - 79384 \beta ) q^{67} \) \( + ( -2455840 - 330232 \beta ) q^{68} \) \( + ( -372636 + 492480 \beta ) q^{69} \) \( + ( -469500 - 63750 \beta ) q^{70} \) \( + ( 310892 - 70000 \beta ) q^{71} \) \( + ( 540240 + 218040 \beta ) q^{72} \) \( + ( 2284530 - 226208 \beta ) q^{73} \) \( + ( 1325636 + 193850 \beta ) q^{74} \) \( + ( 156250 - 125000 \beta ) q^{75} \) \( + ( 443840 + 372240 \beta ) q^{76} \) \( + ( 1588800 + 107232 \beta ) q^{77} \) \( + ( 2335400 + 184964 \beta ) q^{78} \) \( + ( 2166520 - 472480 \beta ) q^{79} \) \( + ( -1272000 + 80000 \beta ) q^{80} \) \( + ( -1198939 - 538720 \beta ) q^{81} \) \( + ( -2994980 - 435818 \beta ) q^{82} \) \( + ( -4896510 + 490392 \beta ) q^{83} \) \( + ( -199104 - 644880 \beta ) q^{84} \) \( + ( 1708750 + 148000 \beta ) q^{85} \) \( + ( 5399092 + 643570 \beta ) q^{86} \) \( + ( -12229460 + 484240 \beta ) q^{87} \) \( + ( 5283840 + 560640 \beta ) q^{88} \) \( + ( 3012810 + 317760 \beta ) q^{89} \) \( + ( -1951250 - 147125 \beta ) q^{90} \) \( + ( -2676148 + 129520 \beta ) q^{91} \) \( + ( -3599520 - 1260984 \beta ) q^{92} \) \( + ( 3236520 - 1255296 \beta ) q^{93} \) \( + ( 2930396 + 402230 \beta ) q^{94} \) \( + ( -2422500 + 40000 \beta ) q^{95} \) \( + ( 6652672 + 196480 \beta ) q^{96} \) \( + ( 2304770 + 561696 \beta ) q^{97} \) \( + ( -6252670 - 638707 \beta ) q^{98} \) \( + ( 1445344 + 747280 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 20q^{2} \) \(\mathstrut +\mathstrut 20q^{3} \) \(\mathstrut +\mathstrut 96q^{4} \) \(\mathstrut -\mathstrut 250q^{5} \) \(\mathstrut -\mathstrut 1016q^{6} \) \(\mathstrut -\mathstrut 100q^{7} \) \(\mathstrut +\mathstrut 1440q^{8} \) \(\mathstrut +\mathstrut 5554q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 20q^{2} \) \(\mathstrut +\mathstrut 20q^{3} \) \(\mathstrut +\mathstrut 96q^{4} \) \(\mathstrut -\mathstrut 250q^{5} \) \(\mathstrut -\mathstrut 1016q^{6} \) \(\mathstrut -\mathstrut 100q^{7} \) \(\mathstrut +\mathstrut 1440q^{8} \) \(\mathstrut +\mathstrut 5554q^{9} \) \(\mathstrut -\mathstrut 2500q^{10} \) \(\mathstrut +\mathstrut 4544q^{11} \) \(\mathstrut -\mathstrut 23360q^{12} \) \(\mathstrut +\mathstrut 3540q^{13} \) \(\mathstrut +\mathstrut 7512q^{14} \) \(\mathstrut -\mathstrut 2500q^{15} \) \(\mathstrut +\mathstrut 20352q^{16} \) \(\mathstrut -\mathstrut 27340q^{17} \) \(\mathstrut +\mathstrut 31220q^{18} \) \(\mathstrut +\mathstrut 38760q^{19} \) \(\mathstrut -\mathstrut 12000q^{20} \) \(\mathstrut -\mathstrut 69096q^{21} \) \(\mathstrut +\mathstrut 106240q^{22} \) \(\mathstrut -\mathstrut 124140q^{23} \) \(\mathstrut -\mathstrut 131520q^{24} \) \(\mathstrut +\mathstrut 31250q^{25} \) \(\mathstrut -\mathstrut 57016q^{26} \) \(\mathstrut +\mathstrut 206360q^{27} \) \(\mathstrut +\mathstrut 165440q^{28} \) \(\mathstrut -\mathstrut 72260q^{29} \) \(\mathstrut +\mathstrut 127000q^{30} \) \(\mathstrut +\mathstrut 306824q^{31} \) \(\mathstrut -\mathstrut 78080q^{32} \) \(\mathstrut -\mathstrut 440960q^{33} \) \(\mathstrut -\mathstrut 453368q^{34} \) \(\mathstrut +\mathstrut 12500q^{35} \) \(\mathstrut -\mathstrut 219808q^{36} \) \(\mathstrut -\mathstrut 123020q^{37} \) \(\mathstrut +\mathstrut 338960q^{38} \) \(\mathstrut +\mathstrut 774728q^{39} \) \(\mathstrut -\mathstrut 180000q^{40} \) \(\mathstrut +\mathstrut 264364q^{41} \) \(\mathstrut -\mathstrut 545040q^{42} \) \(\mathstrut +\mathstrut 423300q^{43} \) \(\mathstrut +\mathstrut 1434112q^{44} \) \(\mathstrut -\mathstrut 694250q^{45} \) \(\mathstrut -\mathstrut 1303416q^{46} \) \(\mathstrut -\mathstrut 105460q^{47} \) \(\mathstrut +\mathstrut 981760q^{48} \) \(\mathstrut -\mathstrut 1165414q^{49} \) \(\mathstrut +\mathstrut 312500q^{50} \) \(\mathstrut +\mathstrut 1166344q^{51} \) \(\mathstrut -\mathstrut 1678400q^{52} \) \(\mathstrut -\mathstrut 2391580q^{53} \) \(\mathstrut +\mathstrut 1102960q^{54} \) \(\mathstrut -\mathstrut 568000q^{55} \) \(\mathstrut +\mathstrut 949440q^{56} \) \(\mathstrut +\mathstrut 776720q^{57} \) \(\mathstrut +\mathstrut 2244440q^{58} \) \(\mathstrut -\mathstrut 1120120q^{59} \) \(\mathstrut +\mathstrut 2920000q^{60} \) \(\mathstrut +\mathstrut 2257044q^{61} \) \(\mathstrut +\mathstrut 2642640q^{62} \) \(\mathstrut -\mathstrut 1639620q^{63} \) \(\mathstrut -\mathstrut 5146624q^{64} \) \(\mathstrut -\mathstrut 442500q^{65} \) \(\mathstrut -\mathstrut 6564352q^{66} \) \(\mathstrut +\mathstrut 4516460q^{67} \) \(\mathstrut -\mathstrut 4911680q^{68} \) \(\mathstrut -\mathstrut 745272q^{69} \) \(\mathstrut -\mathstrut 939000q^{70} \) \(\mathstrut +\mathstrut 621784q^{71} \) \(\mathstrut +\mathstrut 1080480q^{72} \) \(\mathstrut +\mathstrut 4569060q^{73} \) \(\mathstrut +\mathstrut 2651272q^{74} \) \(\mathstrut +\mathstrut 312500q^{75} \) \(\mathstrut +\mathstrut 887680q^{76} \) \(\mathstrut +\mathstrut 3177600q^{77} \) \(\mathstrut +\mathstrut 4670800q^{78} \) \(\mathstrut +\mathstrut 4333040q^{79} \) \(\mathstrut -\mathstrut 2544000q^{80} \) \(\mathstrut -\mathstrut 2397878q^{81} \) \(\mathstrut -\mathstrut 5989960q^{82} \) \(\mathstrut -\mathstrut 9793020q^{83} \) \(\mathstrut -\mathstrut 398208q^{84} \) \(\mathstrut +\mathstrut 3417500q^{85} \) \(\mathstrut +\mathstrut 10798184q^{86} \) \(\mathstrut -\mathstrut 24458920q^{87} \) \(\mathstrut +\mathstrut 10567680q^{88} \) \(\mathstrut +\mathstrut 6025620q^{89} \) \(\mathstrut -\mathstrut 3902500q^{90} \) \(\mathstrut -\mathstrut 5352296q^{91} \) \(\mathstrut -\mathstrut 7199040q^{92} \) \(\mathstrut +\mathstrut 6473040q^{93} \) \(\mathstrut +\mathstrut 5860792q^{94} \) \(\mathstrut -\mathstrut 4845000q^{95} \) \(\mathstrut +\mathstrut 13305344q^{96} \) \(\mathstrut +\mathstrut 4609540q^{97} \) \(\mathstrut -\mathstrut 12505340q^{98} \) \(\mathstrut +\mathstrut 2890688q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−4.35890
4.35890
1.28220 79.7424 −126.356 −125.000 102.246 −538.197 −326.136 4171.85 −160.275
1.2 18.7178 −59.7424 222.356 −125.000 −1118.25 438.197 1766.14 1382.15 −2339.72
\(n\): 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 \(T_{2}^{2} \) \(\mathstrut -\mathstrut 20 T_{2} \) \(\mathstrut +\mathstrut 24 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(5))\).