Properties

Label 20.8.a.b
Level 20
Weight 8
Character orbit 20.a
Self dual Yes
Analytic conductor 6.248
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 20.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(6.24770050968\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1129}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{1129}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -10 - \beta ) q^{3} \) \( + 125 q^{5} \) \( + ( 830 + 9 \beta ) q^{7} \) \( + ( 2429 + 20 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -10 - \beta ) q^{3} \) \( + 125 q^{5} \) \( + ( 830 + 9 \beta ) q^{7} \) \( + ( 2429 + 20 \beta ) q^{9} \) \( + ( 1800 - 90 \beta ) q^{11} \) \( + ( 6590 + 36 \beta ) q^{13} \) \( + ( -1250 - 125 \beta ) q^{15} \) \( + ( 2730 + 468 \beta ) q^{17} \) \( + ( -20236 + 180 \beta ) q^{19} \) \( + ( -48944 - 920 \beta ) q^{21} \) \( + ( -20910 + 63 \beta ) q^{23} \) \( + 15625 q^{25} \) \( + ( -92740 - 442 \beta ) q^{27} \) \( + ( 59334 + 360 \beta ) q^{29} \) \( + ( -57964 + 3150 \beta ) q^{31} \) \( + ( 388440 - 900 \beta ) q^{33} \) \( + ( 103750 + 1125 \beta ) q^{35} \) \( + ( 153470 - 4536 \beta ) q^{37} \) \( + ( -228476 - 6950 \beta ) q^{39} \) \( + ( -176574 + 4860 \beta ) q^{41} \) \( + ( 607670 + 3195 \beta ) q^{43} \) \( + ( 303625 + 2500 \beta ) q^{45} \) \( + ( -1034250 + 153 \beta ) q^{47} \) \( + ( 231153 + 14940 \beta ) q^{49} \) \( + ( -2140788 - 7410 \beta ) q^{51} \) \( + ( 700230 - 13428 \beta ) q^{53} \) \( + ( 225000 - 11250 \beta ) q^{55} \) \( + ( -610520 + 18436 \beta ) q^{57} \) \( + ( -996252 - 10440 \beta ) q^{59} \) \( + ( -839338 - 36720 \beta ) q^{61} \) \( + ( 2828950 + 38461 \beta ) q^{63} \) \( + ( 823750 + 4500 \beta ) q^{65} \) \( + ( 1831970 + 25569 \beta ) q^{67} \) \( + ( -75408 + 20280 \beta ) q^{69} \) \( + ( 897468 + 18630 \beta ) q^{71} \) \( + ( 2531090 - 44604 \beta ) q^{73} \) \( + ( -156250 - 15625 \beta ) q^{75} \) \( + ( -2163960 - 58500 \beta ) q^{77} \) \( + ( 5089112 - 38340 \beta ) q^{79} \) \( + ( -2388751 + 53420 \beta ) q^{81} \) \( + ( -3607050 + 1143 \beta ) q^{83} \) \( + ( 341250 + 58500 \beta ) q^{85} \) \( + ( -2219100 - 62934 \beta ) q^{87} \) \( + ( -7665414 + 44280 \beta ) q^{89} \) \( + ( 6932884 + 89190 \beta ) q^{91} \) \( + ( -13645760 + 26464 \beta ) q^{93} \) \( + ( -2529500 + 22500 \beta ) q^{95} \) \( + ( 7012010 + 2124 \beta ) q^{97} \) \( + ( -3756600 - 182610 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 20q^{3} \) \(\mathstrut +\mathstrut 250q^{5} \) \(\mathstrut +\mathstrut 1660q^{7} \) \(\mathstrut +\mathstrut 4858q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 20q^{3} \) \(\mathstrut +\mathstrut 250q^{5} \) \(\mathstrut +\mathstrut 1660q^{7} \) \(\mathstrut +\mathstrut 4858q^{9} \) \(\mathstrut +\mathstrut 3600q^{11} \) \(\mathstrut +\mathstrut 13180q^{13} \) \(\mathstrut -\mathstrut 2500q^{15} \) \(\mathstrut +\mathstrut 5460q^{17} \) \(\mathstrut -\mathstrut 40472q^{19} \) \(\mathstrut -\mathstrut 97888q^{21} \) \(\mathstrut -\mathstrut 41820q^{23} \) \(\mathstrut +\mathstrut 31250q^{25} \) \(\mathstrut -\mathstrut 185480q^{27} \) \(\mathstrut +\mathstrut 118668q^{29} \) \(\mathstrut -\mathstrut 115928q^{31} \) \(\mathstrut +\mathstrut 776880q^{33} \) \(\mathstrut +\mathstrut 207500q^{35} \) \(\mathstrut +\mathstrut 306940q^{37} \) \(\mathstrut -\mathstrut 456952q^{39} \) \(\mathstrut -\mathstrut 353148q^{41} \) \(\mathstrut +\mathstrut 1215340q^{43} \) \(\mathstrut +\mathstrut 607250q^{45} \) \(\mathstrut -\mathstrut 2068500q^{47} \) \(\mathstrut +\mathstrut 462306q^{49} \) \(\mathstrut -\mathstrut 4281576q^{51} \) \(\mathstrut +\mathstrut 1400460q^{53} \) \(\mathstrut +\mathstrut 450000q^{55} \) \(\mathstrut -\mathstrut 1221040q^{57} \) \(\mathstrut -\mathstrut 1992504q^{59} \) \(\mathstrut -\mathstrut 1678676q^{61} \) \(\mathstrut +\mathstrut 5657900q^{63} \) \(\mathstrut +\mathstrut 1647500q^{65} \) \(\mathstrut +\mathstrut 3663940q^{67} \) \(\mathstrut -\mathstrut 150816q^{69} \) \(\mathstrut +\mathstrut 1794936q^{71} \) \(\mathstrut +\mathstrut 5062180q^{73} \) \(\mathstrut -\mathstrut 312500q^{75} \) \(\mathstrut -\mathstrut 4327920q^{77} \) \(\mathstrut +\mathstrut 10178224q^{79} \) \(\mathstrut -\mathstrut 4777502q^{81} \) \(\mathstrut -\mathstrut 7214100q^{83} \) \(\mathstrut +\mathstrut 682500q^{85} \) \(\mathstrut -\mathstrut 4438200q^{87} \) \(\mathstrut -\mathstrut 15330828q^{89} \) \(\mathstrut +\mathstrut 13865768q^{91} \) \(\mathstrut -\mathstrut 27291520q^{93} \) \(\mathstrut -\mathstrut 5059000q^{95} \) \(\mathstrut +\mathstrut 14024020q^{97} \) \(\mathstrut -\mathstrut 7513200q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
17.3003
−16.3003
0 −77.2012 0 125.000 0 1434.81 0 3773.02 0
1.2 0 57.2012 0 125.000 0 225.189 0 1084.98 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 \(T_{3}^{2} \) \(\mathstrut +\mathstrut 20 T_{3} \) \(\mathstrut -\mathstrut 4416 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(20))\).