Properties

Label 20.5.f.a
Level 20
Weight 5
Character orbit 20.f
Analytic conductor 2.067
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(2.06739926168\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{241})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -2 + 2 \beta_{1} + \beta_{3} ) q^{3} \) \( + ( -6 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{5} \) \( + ( 26 + 29 \beta_{1} + 3 \beta_{2} ) q^{7} \) \( + ( -52 \beta_{1} - 5 \beta_{2} - 5 \beta_{3} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -2 + 2 \beta_{1} + \beta_{3} ) q^{3} \) \( + ( -6 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{5} \) \( + ( 26 + 29 \beta_{1} + 3 \beta_{2} ) q^{7} \) \( + ( -52 \beta_{1} - 5 \beta_{2} - 5 \beta_{3} ) q^{9} \) \( + ( -80 + 5 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} ) q^{11} \) \( + ( -93 + 93 \beta_{1} - 6 \beta_{3} ) q^{13} \) \( + ( 130 - 228 \beta_{1} + 2 \beta_{2} - 9 \beta_{3} ) q^{15} \) \( + ( 247 + 233 \beta_{1} - 14 \beta_{2} ) q^{17} \) \( + ( -62 \beta_{1} + 30 \beta_{2} + 30 \beta_{3} ) q^{19} \) \( + ( -464 - 35 \beta_{1} - 35 \beta_{2} + 35 \beta_{3} ) q^{21} \) \( + ( -194 + 194 \beta_{1} + 17 \beta_{3} ) q^{23} \) \( + ( 455 - 339 \beta_{1} + 21 \beta_{2} + 3 \beta_{3} ) q^{25} \) \( + ( 532 + 528 \beta_{1} - 4 \beta_{2} ) q^{27} \) \( + ( -552 \beta_{1} - 40 \beta_{2} - 40 \beta_{3} ) q^{29} \) \( + ( -284 + 75 \beta_{1} + 75 \beta_{2} - 75 \beta_{3} ) q^{31} \) \( + ( -440 + 440 \beta_{1} - 50 \beta_{3} ) q^{33} \) \( + ( -590 - 577 \beta_{1} - 87 \beta_{2} + 14 \beta_{3} ) q^{35} \) \( + ( -219 - 111 \beta_{1} + 108 \beta_{2} ) q^{37} \) \( + ( 273 \beta_{1} - 75 \beta_{2} - 75 \beta_{3} ) q^{39} \) \( + ( 736 + 5 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} ) q^{41} \) \( + ( 870 - 870 \beta_{1} + 105 \beta_{3} ) q^{43} \) \( + ( 365 + 1889 \beta_{1} + 89 \beta_{2} + 82 \beta_{3} ) q^{45} \) \( + ( -498 - 627 \beta_{1} - 129 \beta_{2} ) q^{47} \) \( + ( 196 \beta_{1} + 165 \beta_{2} + 165 \beta_{3} ) q^{49} \) \( + ( 692 - 205 \beta_{1} - 205 \beta_{2} + 205 \beta_{3} ) q^{51} \) \( + ( 509 - 509 \beta_{1} + 28 \beta_{3} ) q^{53} \) \( + ( -1800 + 1045 \beta_{1} + 45 \beta_{2} - 165 \beta_{3} ) q^{55} \) \( + ( -3416 - 3504 \beta_{1} - 88 \beta_{2} ) q^{57} \) \( + ( 4626 \beta_{1} + 110 \beta_{2} + 110 \beta_{3} ) q^{59} \) \( + ( 3792 + 165 \beta_{1} + 165 \beta_{2} - 165 \beta_{3} ) q^{61} \) \( + ( 3022 - 3022 \beta_{1} - 431 \beta_{3} ) q^{63} \) \( + ( -255 + 1788 \beta_{1} - 117 \beta_{2} - 261 \beta_{3} ) q^{65} \) \( + ( -1234 - 1171 \beta_{1} + 63 \beta_{2} ) q^{67} \) \( + ( -3061 \beta_{1} - 245 \beta_{2} - 245 \beta_{3} ) q^{69} \) \( + ( -1612 + 115 \beta_{1} + 115 \beta_{2} - 115 \beta_{3} ) q^{71} \) \( + ( -3383 + 3383 \beta_{1} + 264 \beta_{3} ) q^{73} \) \( + ( -2710 + 1558 \beta_{1} + 288 \beta_{2} + 509 \beta_{3} ) q^{75} \) \( + ( -280 - 230 \beta_{1} + 50 \beta_{2} ) q^{77} \) \( + ( -336 \beta_{1} - 240 \beta_{2} - 240 \beta_{3} ) q^{79} \) \( + ( 2159 - 115 \beta_{1} - 115 \beta_{2} + 115 \beta_{3} ) q^{81} \) \( + ( 5226 - 5226 \beta_{1} + 477 \beta_{3} ) q^{83} \) \( + ( 4595 - 254 \beta_{1} - 699 \beta_{2} + 303 \beta_{3} ) q^{85} \) \( + ( 5824 + 6576 \beta_{1} + 752 \beta_{2} ) q^{87} \) \( + ( 72 \beta_{1} + 280 \beta_{2} + 280 \beta_{3} ) q^{89} \) \( + ( -2676 - 105 \beta_{1} - 105 \beta_{2} + 105 \beta_{3} ) q^{91} \) \( + ( -8432 + 8432 \beta_{1} + 166 \beta_{3} ) q^{93} \) \( + ( -4060 - 10586 \beta_{1} + 214 \beta_{2} - 118 \beta_{3} ) q^{95} \) \( + ( -279 - 1311 \beta_{1} - 1032 \beta_{2} ) q^{97} \) \( + ( -2125 \beta_{1} + 115 \beta_{2} + 115 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 10q^{3} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut +\mathstrut 110q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 10q^{3} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut +\mathstrut 110q^{7} \) \(\mathstrut -\mathstrut 300q^{11} \) \(\mathstrut -\mathstrut 360q^{13} \) \(\mathstrut +\mathstrut 542q^{15} \) \(\mathstrut +\mathstrut 960q^{17} \) \(\mathstrut -\mathstrut 1996q^{21} \) \(\mathstrut -\mathstrut 810q^{23} \) \(\mathstrut +\mathstrut 1856q^{25} \) \(\mathstrut +\mathstrut 2120q^{27} \) \(\mathstrut -\mathstrut 836q^{31} \) \(\mathstrut -\mathstrut 1660q^{33} \) \(\mathstrut -\mathstrut 2562q^{35} \) \(\mathstrut -\mathstrut 660q^{37} \) \(\mathstrut +\mathstrut 2964q^{41} \) \(\mathstrut +\mathstrut 3270q^{43} \) \(\mathstrut +\mathstrut 1474q^{45} \) \(\mathstrut -\mathstrut 2250q^{47} \) \(\mathstrut +\mathstrut 1948q^{51} \) \(\mathstrut +\mathstrut 1980q^{53} \) \(\mathstrut -\mathstrut 6780q^{55} \) \(\mathstrut -\mathstrut 13840q^{57} \) \(\mathstrut +\mathstrut 15828q^{61} \) \(\mathstrut +\mathstrut 12950q^{63} \) \(\mathstrut -\mathstrut 732q^{65} \) \(\mathstrut -\mathstrut 4810q^{67} \) \(\mathstrut -\mathstrut 5988q^{71} \) \(\mathstrut -\mathstrut 14060q^{73} \) \(\mathstrut -\mathstrut 11282q^{75} \) \(\mathstrut -\mathstrut 1020q^{77} \) \(\mathstrut +\mathstrut 8176q^{81} \) \(\mathstrut +\mathstrut 19950q^{83} \) \(\mathstrut +\mathstrut 16376q^{85} \) \(\mathstrut +\mathstrut 24800q^{87} \) \(\mathstrut -\mathstrut 11124q^{91} \) \(\mathstrut -\mathstrut 34060q^{93} \) \(\mathstrut -\mathstrut 15576q^{95} \) \(\mathstrut -\mathstrut 3180q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut +\mathstrut \) \(121\) \(x^{2}\mathstrut +\mathstrut \) \(3600\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 61 \nu \)\()/60\)
\(\beta_{2}\)\(=\)\( \nu^{2} + \nu + 61 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} - 60 \nu^{2} + 121 \nu - 3660 \)\()/60\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut -\mathstrut \) \(122\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(61\) \(\beta_{3}\mathstrut -\mathstrut \) \(61\) \(\beta_{2}\mathstrut +\mathstrut \) \(181\) \(\beta_{1}\)\()/2\)

Character Values

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

\(n\) \(11\) \(17\)
\(\chi(n)\) \(1\) \(\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
7.26209i
8.26209i
7.26209i
8.26209i
0 −10.2621 10.2621i 0 −24.7863 3.26209i 0 50.7863 50.7863i 0 129.621i 0
13.2 0 5.26209 + 5.26209i 0 21.7863 + 12.2621i 0 4.21374 4.21374i 0 25.6209i 0
17.1 0 −10.2621 + 10.2621i 0 −24.7863 + 3.26209i 0 50.7863 + 50.7863i 0 129.621i 0
17.2 0 5.26209 5.26209i 0 21.7863 12.2621i 0 4.21374 + 4.21374i 0 25.6209i 0
\(n\): 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
5.c Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{5}^{\mathrm{new}}(20, [\chi])\).