Properties

Label 20.7.d.c
Level 20
Weight 7
Character orbit 20.d
Analytic conductor 4.601
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: \( k \) = \( 7 \)
Character orbit: \([\chi]\) = 20.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(4.6010816724\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q\) \( + 2 \beta q^{2} \) \( -64 q^{4} \) \( + ( -117 + 11 \beta ) q^{5} \) \( -128 \beta q^{8} \) \( -729 q^{9} \) \(+O(q^{10})\) \( q\) \( + 2 \beta q^{2} \) \( -64 q^{4} \) \( + ( -117 + 11 \beta ) q^{5} \) \( -128 \beta q^{8} \) \( -729 q^{9} \) \( + ( -352 - 234 \beta ) q^{10} \) \( + 414 \beta q^{13} \) \( + 4096 q^{16} \) \( + 2444 \beta q^{17} \) \( -1458 \beta q^{18} \) \( + ( 7488 - 704 \beta ) q^{20} \) \( + ( 11753 - 2574 \beta ) q^{25} \) \( -13248 q^{26} \) \( -31878 q^{29} \) \( + 8192 \beta q^{32} \) \( -78208 q^{34} \) \( + 46656 q^{36} \) \( -21186 \beta q^{37} \) \( + ( 22528 + 14976 \beta ) q^{40} \) \( + 84942 q^{41} \) \( + ( 85293 - 8019 \beta ) q^{45} \) \( -117649 q^{49} \) \( + ( 82368 + 23506 \beta ) q^{50} \) \( -26496 \beta q^{52} \) \( + 74074 \beta q^{53} \) \( -63756 \beta q^{58} \) \( -234938 q^{61} \) \( -262144 q^{64} \) \( + ( -72864 - 48438 \beta ) q^{65} \) \( -156416 \beta q^{68} \) \( + 93312 \beta q^{72} \) \( + 162504 \beta q^{73} \) \( + 677952 q^{74} \) \( + ( -479232 + 45056 \beta ) q^{80} \) \( + 531441 q^{81} \) \( + 169884 \beta q^{82} \) \( + ( -430144 - 285948 \beta ) q^{85} \) \( + 1378962 q^{89} \) \( + ( 256608 + 170586 \beta ) q^{90} \) \( -269676 \beta q^{97} \) \( -235298 \beta q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 128q^{4} \) \(\mathstrut -\mathstrut 234q^{5} \) \(\mathstrut -\mathstrut 1458q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 128q^{4} \) \(\mathstrut -\mathstrut 234q^{5} \) \(\mathstrut -\mathstrut 1458q^{9} \) \(\mathstrut -\mathstrut 704q^{10} \) \(\mathstrut +\mathstrut 8192q^{16} \) \(\mathstrut +\mathstrut 14976q^{20} \) \(\mathstrut +\mathstrut 23506q^{25} \) \(\mathstrut -\mathstrut 26496q^{26} \) \(\mathstrut -\mathstrut 63756q^{29} \) \(\mathstrut -\mathstrut 156416q^{34} \) \(\mathstrut +\mathstrut 93312q^{36} \) \(\mathstrut +\mathstrut 45056q^{40} \) \(\mathstrut +\mathstrut 169884q^{41} \) \(\mathstrut +\mathstrut 170586q^{45} \) \(\mathstrut -\mathstrut 235298q^{49} \) \(\mathstrut +\mathstrut 164736q^{50} \) \(\mathstrut -\mathstrut 469876q^{61} \) \(\mathstrut -\mathstrut 524288q^{64} \) \(\mathstrut -\mathstrut 145728q^{65} \) \(\mathstrut +\mathstrut 1355904q^{74} \) \(\mathstrut -\mathstrut 958464q^{80} \) \(\mathstrut +\mathstrut 1062882q^{81} \) \(\mathstrut -\mathstrut 860288q^{85} \) \(\mathstrut +\mathstrut 2757924q^{89} \) \(\mathstrut +\mathstrut 513216q^{90} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
1.00000i
1.00000i
8.00000i 0 −64.0000 −117.000 44.0000i 0 0 512.000i −729.000 −352.000 + 936.000i
19.2 8.00000i 0 −64.0000 −117.000 + 44.0000i 0 0 512.000i −729.000 −352.000 936.000i
\(n\): 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.b Even 1 yes
20.d Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3} \) acting on \(S_{7}^{\mathrm{new}}(20, [\chi])\).