Properties

Label 20.10.e.a
Level 20
Weight 10
Character orbit 20.e
Analytic conductor 10.301
Analytic rank 0
Dimension 2
CM disc. -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 \(4\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(10.3007167233\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
4.b Odd 1 CM by \(\Q(\sqrt{-1}) \) yes
5.c Odd 1 yes
20.e Even 1 yes

Hecke kernels

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