Properties

Label 20.10.e.a
Level 20
Weight 10
Character orbit 20.e
Analytic conductor 10.301
Analytic rank 0
Dimension 2
CM discriminant -4
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 20.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.3007167233\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -16 - 16 i ) q^{2} + 512 i q^{4} + ( 718 - 1199 i ) q^{5} + ( 8192 - 8192 i ) q^{8} -19683 i q^{9} +O(q^{10})\) \( q + ( -16 - 16 i ) q^{2} + 512 i q^{4} + ( 718 - 1199 i ) q^{5} + ( 8192 - 8192 i ) q^{8} -19683 i q^{9} + ( -30672 + 7696 i ) q^{10} + ( -142561 + 142561 i ) q^{13} -262144 q^{16} + ( -481437 - 481437 i ) q^{17} + ( -314928 + 314928 i ) q^{18} + ( 613888 + 367616 i ) q^{20} + ( -922077 - 1721764 i ) q^{25} + 4561952 q^{26} -2126876 i q^{29} + ( 4194304 + 4194304 i ) q^{32} + 15405984 i q^{34} + 10077696 q^{36} + ( -12321127 - 12321127 i ) q^{37} + ( -3940352 - 15704064 i ) q^{40} + 7561912 q^{41} + ( -23599917 - 14132394 i ) q^{45} + 40353607 i q^{49} + ( -12794992 + 42301456 i ) q^{50} + ( -72991232 - 72991232 i ) q^{52} + ( -12019671 + 12019671 i ) q^{53} + ( -34030016 + 34030016 i ) q^{58} + 216178092 q^{61} -134217728 i q^{64} + ( 68571841 + 273289437 i ) q^{65} + ( 246495744 - 246495744 i ) q^{68} + ( -161243136 - 161243136 i ) q^{72} + ( 262875349 - 262875349 i ) q^{73} + 394276064 i q^{74} + ( -188219392 + 314310656 i ) q^{80} -387420489 q^{81} + ( -120990592 - 120990592 i ) q^{82} + ( -922914729 + 231571197 i ) q^{85} -366771856 i q^{89} + ( 151480368 + 603716976 i ) q^{90} + ( -1216731347 - 1216731347 i ) q^{97} + ( 645657712 - 645657712 i ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 32q^{2} + 1436q^{5} + 16384q^{8} + O(q^{10}) \) \( 2q - 32q^{2} + 1436q^{5} + 16384q^{8} - 61344q^{10} - 285122q^{13} - 524288q^{16} - 962874q^{17} - 629856q^{18} + 1227776q^{20} - 1844154q^{25} + 9123904q^{26} + 8388608q^{32} + 20155392q^{36} - 24642254q^{37} - 7880704q^{40} + 15123824q^{41} - 47199834q^{45} - 25589984q^{50} - 145982464q^{52} - 24039342q^{53} - 68060032q^{58} + 432356184q^{61} + 137143682q^{65} + 492991488q^{68} - 322486272q^{72} + 525750698q^{73} - 376438784q^{80} - 774840978q^{81} - 241981184q^{82} - 1845829458q^{85} + 302960736q^{90} - 2433462694q^{97} + 1291315424q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(17\)
\(\chi(n)\) \(-1\) \(i\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
1.00000i
1.00000i
−16.0000 + 16.0000i 0 512.000i 718.000 + 1199.00i 0 0 8192.00 + 8192.00i 19683.0i −30672.0 7696.00i
7.1 −16.0000 16.0000i 0 512.000i 718.000 1199.00i 0 0 8192.00 8192.00i 19683.0i −30672.0 + 7696.00i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 20.10.e.a 2
4.b odd 2 1 CM 20.10.e.a 2
5.b even 2 1 100.10.e.c 2
5.c odd 4 1 inner 20.10.e.a 2
5.c odd 4 1 100.10.e.c 2
20.d odd 2 1 100.10.e.c 2
20.e even 4 1 inner 20.10.e.a 2
20.e even 4 1 100.10.e.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.10.e.a 2 1.a even 1 1 trivial
20.10.e.a 2 4.b odd 2 1 CM
20.10.e.a 2 5.c odd 4 1 inner
20.10.e.a 2 20.e even 4 1 inner
100.10.e.c 2 5.b even 2 1
100.10.e.c 2 5.c odd 4 1
100.10.e.c 2 20.d odd 2 1
100.10.e.c 2 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{10}^{\mathrm{new}}(20, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 32 T + 512 T^{2} \)
$3$ \( 1 + 387420489 T^{4} \)
$5$ \( 1 - 1436 T + 1953125 T^{2} \)
$7$ \( 1 + 1628413597910449 T^{4} \)
$11$ \( ( 1 - 2357947691 T^{2} )^{2} \)
$13$ \( ( 1 + 112806 T + 10604499373 T^{2} )( 1 + 172316 T + 10604499373 T^{2} ) \)
$17$ \( ( 1 + 407992 T + 118587876497 T^{2} )( 1 + 554882 T + 118587876497 T^{2} ) \)
$19$ \( ( 1 + 322687697779 T^{2} )^{2} \)
$23$ \( 1 + \)\(32\!\cdots\!69\)\( T^{4} \)
$29$ \( ( 1 - 7314710 T + 14507145975869 T^{2} )( 1 + 7314710 T + 14507145975869 T^{2} ) \)
$31$ \( ( 1 - 26439622160671 T^{2} )^{2} \)
$37$ \( ( 1 + 1923372 T + 129961739795077 T^{2} )( 1 + 22718882 T + 129961739795077 T^{2} ) \)
$41$ \( ( 1 - 7561912 T + 327381934393961 T^{2} )^{2} \)
$43$ \( 1 + \)\(25\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 + \)\(12\!\cdots\!89\)\( T^{4} \)
$53$ \( ( 1 - 68323684 T + 3299763591802133 T^{2} )( 1 + 92363026 T + 3299763591802133 T^{2} ) \)
$59$ \( ( 1 + 8662995818654939 T^{2} )^{2} \)
$61$ \( ( 1 - 216178092 T + 11694146092834141 T^{2} )^{2} \)
$67$ \( 1 + \)\(74\!\cdots\!09\)\( T^{4} \)
$71$ \( ( 1 - 45848500718449031 T^{2} )^{2} \)
$73$ \( ( 1 - 483419504 T + 58871586708267913 T^{2} )( 1 - 42331194 T + 58871586708267913 T^{2} ) \)
$79$ \( ( 1 + 119851595982618319 T^{2} )^{2} \)
$83$ \( 1 + \)\(34\!\cdots\!09\)\( T^{4} \)
$89$ \( ( 1 - 1125568310 T + 350356403707485209 T^{2} )( 1 + 1125568310 T + 350356403707485209 T^{2} ) \)
$97$ \( ( 1 + 1016663992 T + 760231058654565217 T^{2} )( 1 + 1416798702 T + 760231058654565217 T^{2} ) \)
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