Properties

Label 20.11.d.c
Level 20
Weight 11
Character orbit 20.d
Analytic conductor 12.707
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: \( k \) = \( 11 \)
Character orbit: \([\chi]\) = 20.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(12.7071450535\)
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.

\(f(q)\) \(=\) \( q\) \( + 8 \beta q^{2} \) \( -1024 q^{4} \) \( + ( -237 + 779 \beta ) q^{5} \) \( -8192 \beta q^{8} \) \( -59049 q^{9} \) \(+O(q^{10})\) \( q\) \( + 8 \beta q^{2} \) \( -1024 q^{4} \) \( + ( -237 + 779 \beta ) q^{5} \) \( -8192 \beta q^{8} \) \( -59049 q^{9} \) \( + ( -99712 - 1896 \beta ) q^{10} \) \( -72834 \beta q^{13} \) \( + 1048576 q^{16} \) \( -452884 \beta q^{17} \) \( -472392 \beta q^{18} \) \( + ( 242688 - 797696 \beta ) q^{20} \) \( + ( -9653287 - 369246 \beta ) q^{25} \) \( + 9322752 q^{26} \) \( -32319798 q^{29} \) \( + 8388608 \beta q^{32} \) \( + 57969152 q^{34} \) \( + 60466176 q^{36} \) \( + 34559166 \beta q^{37} \) \( + ( 102105088 + 1941504 \beta ) q^{40} \) \( -207190098 q^{41} \) \( + ( 13994613 - 45999171 \beta ) q^{45} \) \( -282475249 q^{49} \) \( + ( 47263488 - 77226296 \beta ) q^{50} \) \( + 74582016 \beta q^{52} \) \( -73384934 \beta q^{53} \) \( -258558384 \beta q^{58} \) \( + 1330090102 q^{61} \) \( -1073741824 q^{64} \) \( + ( 907802976 + 17261658 \beta ) q^{65} \) \( + 463753216 \beta q^{68} \) \( + 483729408 \beta q^{72} \) \( + 447227016 \beta q^{73} \) \( -4423573248 q^{74} \) \( + ( -248512512 + 816840704 \beta ) q^{80} \) \( + 3486784401 q^{81} \) \( -1657520784 \beta q^{82} \) \( + ( 5644746176 + 107333508 \beta ) q^{85} \) \( -8562016398 q^{89} \) \( + ( 5887893888 + 111956904 \beta ) q^{90} \) \( -3704292684 \beta q^{97} \) \( -2259801992 \beta q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 2048q^{4} \) \(\mathstrut -\mathstrut 474q^{5} \) \(\mathstrut -\mathstrut 118098q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 2048q^{4} \) \(\mathstrut -\mathstrut 474q^{5} \) \(\mathstrut -\mathstrut 118098q^{9} \) \(\mathstrut -\mathstrut 199424q^{10} \) \(\mathstrut +\mathstrut 2097152q^{16} \) \(\mathstrut +\mathstrut 485376q^{20} \) \(\mathstrut -\mathstrut 19306574q^{25} \) \(\mathstrut +\mathstrut 18645504q^{26} \) \(\mathstrut -\mathstrut 64639596q^{29} \) \(\mathstrut +\mathstrut 115938304q^{34} \) \(\mathstrut +\mathstrut 120932352q^{36} \) \(\mathstrut +\mathstrut 204210176q^{40} \) \(\mathstrut -\mathstrut 414380196q^{41} \) \(\mathstrut +\mathstrut 27989226q^{45} \) \(\mathstrut -\mathstrut 564950498q^{49} \) \(\mathstrut +\mathstrut 94526976q^{50} \) \(\mathstrut +\mathstrut 2660180204q^{61} \) \(\mathstrut -\mathstrut 2147483648q^{64} \) \(\mathstrut +\mathstrut 1815605952q^{65} \) \(\mathstrut -\mathstrut 8847146496q^{74} \) \(\mathstrut -\mathstrut 497025024q^{80} \) \(\mathstrut +\mathstrut 6973568802q^{81} \) \(\mathstrut +\mathstrut 11289492352q^{85} \) \(\mathstrut -\mathstrut 17124032796q^{89} \) \(\mathstrut +\mathstrut 11775787776q^{90} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
32.0000i 0 −1024.00 −237.000 3116.00i 0 0 32768.0i −59049.0 −99712.0 + 7584.00i
19.2 32.0000i 0 −1024.00 −237.000 + 3116.00i 0 0 32768.0i −59049.0 −99712.0 7584.00i
\(n\): 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.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 \(S_{11}^{\mathrm{new}}(20, [\chi])\).