Properties

Label 80.10.a.c
Level $80$
Weight $10$
Character orbit 80.a
Self dual yes
Analytic conductor $41.203$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 80.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(41.2028668931\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 20)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 48q^{3} + 625q^{5} + 532q^{7} - 17379q^{9} + O(q^{10}) \) \( q + 48q^{3} + 625q^{5} + 532q^{7} - 17379q^{9} + 33180q^{11} - 99682q^{13} + 30000q^{15} - 443454q^{17} + 357244q^{19} + 25536q^{21} + 142956q^{23} + 390625q^{25} - 1778976q^{27} + 1527966q^{29} - 7323416q^{31} + 1592640q^{33} + 332500q^{35} - 2666842q^{37} - 4784736q^{39} - 7939014q^{41} + 21174520q^{43} - 10861875q^{45} - 16059636q^{47} - 40070583q^{49} - 21285792q^{51} - 87822234q^{53} + 20737500q^{55} + 17147712q^{57} - 120625212q^{59} + 93576542q^{61} - 9245628q^{63} - 62301250q^{65} - 193621688q^{67} + 6861888q^{69} - 417763488q^{71} - 450372742q^{73} + 18750000q^{75} + 17651760q^{77} + 91425472q^{79} + 256680009q^{81} + 652637376q^{83} - 277158750q^{85} + 73342368q^{87} - 170059206q^{89} - 53030824q^{91} - 351523968q^{93} + 223277500q^{95} - 10947022q^{97} - 576635220q^{99} + O(q^{100}) \)

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
0
0 48.0000 0 625.000 0 532.000 0 −17379.0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)

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 80.10.a.c 1
4.b odd 2 1 20.10.a.a 1
5.b even 2 1 400.10.a.e 1
5.c odd 4 2 400.10.c.h 2
8.b even 2 1 320.10.a.d 1
8.d odd 2 1 320.10.a.g 1
12.b even 2 1 180.10.a.b 1
20.d odd 2 1 100.10.a.b 1
20.e even 4 2 100.10.c.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.10.a.a 1 4.b odd 2 1
80.10.a.c 1 1.a even 1 1 trivial
100.10.a.b 1 20.d odd 2 1
100.10.c.b 2 20.e even 4 2
180.10.a.b 1 12.b even 2 1
320.10.a.d 1 8.b even 2 1
320.10.a.g 1 8.d odd 2 1
400.10.a.e 1 5.b even 2 1
400.10.c.h 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 48 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(80))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 48 T + 19683 T^{2} \)
$5$ \( 1 - 625 T \)
$7$ \( 1 - 532 T + 40353607 T^{2} \)
$11$ \( 1 - 33180 T + 2357947691 T^{2} \)
$13$ \( 1 + 99682 T + 10604499373 T^{2} \)
$17$ \( 1 + 443454 T + 118587876497 T^{2} \)
$19$ \( 1 - 357244 T + 322687697779 T^{2} \)
$23$ \( 1 - 142956 T + 1801152661463 T^{2} \)
$29$ \( 1 - 1527966 T + 14507145975869 T^{2} \)
$31$ \( 1 + 7323416 T + 26439622160671 T^{2} \)
$37$ \( 1 + 2666842 T + 129961739795077 T^{2} \)
$41$ \( 1 + 7939014 T + 327381934393961 T^{2} \)
$43$ \( 1 - 21174520 T + 502592611936843 T^{2} \)
$47$ \( 1 + 16059636 T + 1119130473102767 T^{2} \)
$53$ \( 1 + 87822234 T + 3299763591802133 T^{2} \)
$59$ \( 1 + 120625212 T + 8662995818654939 T^{2} \)
$61$ \( 1 - 93576542 T + 11694146092834141 T^{2} \)
$67$ \( 1 + 193621688 T + 27206534396294947 T^{2} \)
$71$ \( 1 + 417763488 T + 45848500718449031 T^{2} \)
$73$ \( 1 + 450372742 T + 58871586708267913 T^{2} \)
$79$ \( 1 - 91425472 T + 119851595982618319 T^{2} \)
$83$ \( 1 - 652637376 T + 186940255267540403 T^{2} \)
$89$ \( 1 + 170059206 T + 350356403707485209 T^{2} \)
$97$ \( 1 + 10947022 T + 760231058654565217 T^{2} \)
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