Properties

Label 20.10.a.b
Level 20
Weight 10
Character orbit 20.a
Self dual Yes
Analytic conductor 10.301
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) = \( 10 \)
Character orbit: \([\chi]\) = 20.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(10.3007167233\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{79}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 16\sqrt{79}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -130 + \beta ) q^{3} \) \( -625 q^{5} \) \( + ( -190 + 69 \beta ) q^{7} \) \( + ( 17441 - 260 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -130 + \beta ) q^{3} \) \( -625 q^{5} \) \( + ( -190 + 69 \beta ) q^{7} \) \( + ( 17441 - 260 \beta ) q^{9} \) \( + ( 51360 + 30 \beta ) q^{11} \) \( + ( 89570 + 564 \beta ) q^{13} \) \( + ( 81250 - 625 \beta ) q^{15} \) \( + ( 158010 + 348 \beta ) q^{17} \) \( + ( 68636 + 5160 \beta ) q^{19} \) \( + ( 1420156 - 9160 \beta ) q^{21} \) \( + ( -332730 - 6393 \beta ) q^{23} \) \( + 390625 q^{25} \) \( + ( -4966780 + 31558 \beta ) q^{27} \) \( + ( -3446874 - 5880 \beta ) q^{29} \) \( + ( 145916 - 26250 \beta ) q^{31} \) \( + ( -6070080 + 47460 \beta ) q^{33} \) \( + ( 118750 - 43125 \beta ) q^{35} \) \( + ( 5630690 - 59976 \beta ) q^{37} \) \( + ( -237764 + 16250 \beta ) q^{39} \) \( + ( 14886726 + 72180 \beta ) q^{41} \) \( + ( -5854090 - 99615 \beta ) q^{43} \) \( + ( -10900625 + 162500 \beta ) q^{45} \) \( + ( 31246650 + 42573 \beta ) q^{47} \) \( + ( 55968957 - 26220 \beta ) q^{49} \) \( + ( -13503348 + 112770 \beta ) q^{51} \) \( + ( 4708890 - 212532 \beta ) q^{53} \) \( + ( -32100000 - 18750 \beta ) q^{55} \) \( + ( 95433160 - 602164 \beta ) q^{57} \) \( + ( -46465428 - 373980 \beta ) q^{59} \) \( + ( 97836962 + 721440 \beta ) q^{61} \) \( + ( -366132350 + 1252829 \beta ) q^{63} \) \( + ( -55981250 - 352500 \beta ) q^{65} \) \( + ( -109883710 + 203139 \beta ) q^{67} \) \( + ( -86037132 + 498360 \beta ) q^{69} \) \( + ( 155603508 - 1397610 \beta ) q^{71} \) \( + ( -49612030 - 2047716 \beta ) q^{73} \) \( + ( -50781250 + 390625 \beta ) q^{75} \) \( + ( 32105280 + 3538140 \beta ) q^{77} \) \( + ( 271130888 + 70020 \beta ) q^{79} \) \( + ( 940619189 - 3951740 \beta ) q^{81} \) \( + ( -628457850 + 192957 \beta ) q^{83} \) \( + ( -98756250 - 217500 \beta ) q^{85} \) \( + ( 329176500 - 2682474 \beta ) q^{87} \) \( + ( -231145926 + 5421960 \beta ) q^{89} \) \( + ( 770018884 + 6073170 \beta ) q^{91} \) \( + ( -549849080 + 3558416 \beta ) q^{93} \) \( + ( -42897500 - 3225000 \beta ) q^{95} \) \( + ( 835858370 - 2902956 \beta ) q^{97} \) \( + ( 738022560 - 12830370 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 260q^{3} \) \(\mathstrut -\mathstrut 1250q^{5} \) \(\mathstrut -\mathstrut 380q^{7} \) \(\mathstrut +\mathstrut 34882q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 260q^{3} \) \(\mathstrut -\mathstrut 1250q^{5} \) \(\mathstrut -\mathstrut 380q^{7} \) \(\mathstrut +\mathstrut 34882q^{9} \) \(\mathstrut +\mathstrut 102720q^{11} \) \(\mathstrut +\mathstrut 179140q^{13} \) \(\mathstrut +\mathstrut 162500q^{15} \) \(\mathstrut +\mathstrut 316020q^{17} \) \(\mathstrut +\mathstrut 137272q^{19} \) \(\mathstrut +\mathstrut 2840312q^{21} \) \(\mathstrut -\mathstrut 665460q^{23} \) \(\mathstrut +\mathstrut 781250q^{25} \) \(\mathstrut -\mathstrut 9933560q^{27} \) \(\mathstrut -\mathstrut 6893748q^{29} \) \(\mathstrut +\mathstrut 291832q^{31} \) \(\mathstrut -\mathstrut 12140160q^{33} \) \(\mathstrut +\mathstrut 237500q^{35} \) \(\mathstrut +\mathstrut 11261380q^{37} \) \(\mathstrut -\mathstrut 475528q^{39} \) \(\mathstrut +\mathstrut 29773452q^{41} \) \(\mathstrut -\mathstrut 11708180q^{43} \) \(\mathstrut -\mathstrut 21801250q^{45} \) \(\mathstrut +\mathstrut 62493300q^{47} \) \(\mathstrut +\mathstrut 111937914q^{49} \) \(\mathstrut -\mathstrut 27006696q^{51} \) \(\mathstrut +\mathstrut 9417780q^{53} \) \(\mathstrut -\mathstrut 64200000q^{55} \) \(\mathstrut +\mathstrut 190866320q^{57} \) \(\mathstrut -\mathstrut 92930856q^{59} \) \(\mathstrut +\mathstrut 195673924q^{61} \) \(\mathstrut -\mathstrut 732264700q^{63} \) \(\mathstrut -\mathstrut 111962500q^{65} \) \(\mathstrut -\mathstrut 219767420q^{67} \) \(\mathstrut -\mathstrut 172074264q^{69} \) \(\mathstrut +\mathstrut 311207016q^{71} \) \(\mathstrut -\mathstrut 99224060q^{73} \) \(\mathstrut -\mathstrut 101562500q^{75} \) \(\mathstrut +\mathstrut 64210560q^{77} \) \(\mathstrut +\mathstrut 542261776q^{79} \) \(\mathstrut +\mathstrut 1881238378q^{81} \) \(\mathstrut -\mathstrut 1256915700q^{83} \) \(\mathstrut -\mathstrut 197512500q^{85} \) \(\mathstrut +\mathstrut 658353000q^{87} \) \(\mathstrut -\mathstrut 462291852q^{89} \) \(\mathstrut +\mathstrut 1540037768q^{91} \) \(\mathstrut -\mathstrut 1099698160q^{93} \) \(\mathstrut -\mathstrut 85795000q^{95} \) \(\mathstrut +\mathstrut 1671716740q^{97} \) \(\mathstrut +\mathstrut 1476045120q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−8.88819
8.88819
0 −272.211 0 −625.000 0 −10002.6 0 54415.9 0
1.2 0 12.2111 0 −625.000 0 9622.57 0 −19533.9 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
\(2\) \(-1\)
\(5\) \(1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{2} \) \(\mathstrut +\mathstrut 260 T_{3} \) \(\mathstrut -\mathstrut 3324 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(20))\).