Properties

Label 20.8.c.a
Level 20
Weight 8
Character orbit 20.c
Analytic conductor 6.248
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 20.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(6.24770050968\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7}\cdot 5 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{3} \) \( + ( -39 - \beta_{1} + \beta_{2} ) q^{5} \) \( + ( -5 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{7} \) \( + ( -509 - 4 \beta_{1} + 8 \beta_{2} + \beta_{3} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{3} \) \( + ( -39 - \beta_{1} + \beta_{2} ) q^{5} \) \( + ( -5 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{7} \) \( + ( -509 - 4 \beta_{1} + 8 \beta_{2} + \beta_{3} ) q^{9} \) \( + ( -660 - 8 \beta_{1} + 16 \beta_{2} + 2 \beta_{3} ) q^{11} \) \( + ( 222 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} ) q^{13} \) \( + ( 1388 - 253 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} ) q^{15} \) \( + ( -192 \beta_{1} + 20 \beta_{2} - 10 \beta_{3} ) q^{17} \) \( + ( -9084 + 24 \beta_{1} - 48 \beta_{2} - 6 \beta_{3} ) q^{19} \) \( + ( 15776 + 76 \beta_{1} - 152 \beta_{2} - 19 \beta_{3} ) q^{21} \) \( + ( 843 \beta_{1} + 58 \beta_{2} - 29 \beta_{3} ) q^{23} \) \( + ( 40509 - 274 \beta_{1} - 46 \beta_{2} - 55 \beta_{3} ) q^{25} \) \( + ( -430 \beta_{1} - 84 \beta_{2} + 42 \beta_{3} ) q^{27} \) \( + ( -120474 + 48 \beta_{1} - 96 \beta_{2} - 12 \beta_{3} ) q^{29} \) \( + ( 40336 - 240 \beta_{1} + 480 \beta_{2} + 60 \beta_{3} ) q^{31} \) \( + ( -4876 \beta_{1} - 168 \beta_{2} + 84 \beta_{3} ) q^{33} \) \( + ( 172812 + 4153 \beta_{1} + 252 \beta_{2} + 220 \beta_{3} ) q^{35} \) \( + ( 5746 \beta_{1} + 226 \beta_{2} - 113 \beta_{3} ) q^{37} \) \( + ( -591624 - 720 \beta_{1} + 1440 \beta_{2} + 180 \beta_{3} ) q^{39} \) \( + ( 282366 + 172 \beta_{1} - 344 \beta_{2} - 43 \beta_{3} ) q^{41} \) \( + ( -3163 \beta_{1} - 124 \beta_{2} + 62 \beta_{3} ) q^{43} \) \( + ( 597811 - 1251 \beta_{1} - 349 \beta_{2} - 275 \beta_{3} ) q^{45} \) \( + ( -8889 \beta_{1} - 506 \beta_{2} + 253 \beta_{3} ) q^{47} \) \( + ( -1078113 + 1932 \beta_{1} - 3864 \beta_{2} - 483 \beta_{3} ) q^{49} \) \( + ( 494672 + 208 \beta_{1} - 416 \beta_{2} - 52 \beta_{3} ) q^{51} \) \( + ( 10194 \beta_{1} + 1638 \beta_{2} - 819 \beta_{3} ) q^{53} \) \( + ( 1181660 - 2860 \beta_{1} - 340 \beta_{2} - 550 \beta_{3} ) q^{55} \) \( + ( 3564 \beta_{1} + 504 \beta_{2} - 252 \beta_{3} ) q^{57} \) \( + ( -1936788 - 632 \beta_{1} + 1264 \beta_{2} + 158 \beta_{3} ) q^{59} \) \( + ( 1076282 + 1356 \beta_{1} - 2712 \beta_{2} - 339 \beta_{3} ) q^{61} \) \( + ( 44893 \beta_{1} - 2778 \beta_{2} + 1389 \beta_{3} ) q^{63} \) \( + ( 847392 - 47502 \beta_{1} + 282 \beta_{2} + 1845 \beta_{3} ) q^{65} \) \( + ( -33319 \beta_{1} + 1916 \beta_{2} - 958 \beta_{3} ) q^{67} \) \( + ( -2339312 - 4996 \beta_{1} + 9992 \beta_{2} + 1249 \beta_{3} ) q^{69} \) \( + ( -301512 - 4224 \beta_{1} + 8448 \beta_{2} + 1056 \beta_{3} ) q^{71} \) \( + ( -33748 \beta_{1} - 736 \beta_{2} + 368 \beta_{3} ) q^{73} \) \( + ( 816472 + 72533 \beta_{1} + 3032 \beta_{2} - 440 \beta_{3} ) q^{75} \) \( + ( 87996 \beta_{1} - 6272 \beta_{2} + 3136 \beta_{3} ) q^{77} \) \( + ( 1429248 + 6768 \beta_{1} - 13536 \beta_{2} - 1692 \beta_{3} ) q^{79} \) \( + ( 142529 - 4676 \beta_{1} + 9352 \beta_{2} + 1169 \beta_{3} ) q^{81} \) \( + ( -74511 \beta_{1} + 6076 \beta_{2} - 3038 \beta_{3} ) q^{83} \) \( + ( -2064016 + 19696 \beta_{1} - 2036 \beta_{2} - 3410 \beta_{3} ) q^{85} \) \( + ( -95178 \beta_{1} + 1008 \beta_{2} - 504 \beta_{3} ) q^{87} \) \( + ( 3410394 - 2416 \beta_{1} + 4832 \beta_{2} + 604 \beta_{3} ) q^{89} \) \( + ( -1966056 + 23808 \beta_{1} - 47616 \beta_{2} - 5952 \beta_{3} ) q^{91} \) \( + ( -86144 \beta_{1} - 5040 \beta_{2} + 2520 \beta_{3} ) q^{93} \) \( + ( -3113484 + 19644 \beta_{1} - 10044 \beta_{2} + 1650 \beta_{3} ) q^{95} \) \( + ( 117032 \beta_{1} + 14444 \beta_{2} - 7222 \beta_{3} ) q^{97} \) \( + ( 11895140 + 6712 \beta_{1} - 13424 \beta_{2} - 1678 \beta_{3} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 156q^{5} \) \(\mathstrut -\mathstrut 2036q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 156q^{5} \) \(\mathstrut -\mathstrut 2036q^{9} \) \(\mathstrut -\mathstrut 2640q^{11} \) \(\mathstrut +\mathstrut 5552q^{15} \) \(\mathstrut -\mathstrut 36336q^{19} \) \(\mathstrut +\mathstrut 63104q^{21} \) \(\mathstrut +\mathstrut 162036q^{25} \) \(\mathstrut -\mathstrut 481896q^{29} \) \(\mathstrut +\mathstrut 161344q^{31} \) \(\mathstrut +\mathstrut 691248q^{35} \) \(\mathstrut -\mathstrut 2366496q^{39} \) \(\mathstrut +\mathstrut 1129464q^{41} \) \(\mathstrut +\mathstrut 2391244q^{45} \) \(\mathstrut -\mathstrut 4312452q^{49} \) \(\mathstrut +\mathstrut 1978688q^{51} \) \(\mathstrut +\mathstrut 4726640q^{55} \) \(\mathstrut -\mathstrut 7747152q^{59} \) \(\mathstrut +\mathstrut 4305128q^{61} \) \(\mathstrut +\mathstrut 3389568q^{65} \) \(\mathstrut -\mathstrut 9357248q^{69} \) \(\mathstrut -\mathstrut 1206048q^{71} \) \(\mathstrut +\mathstrut 3265888q^{75} \) \(\mathstrut +\mathstrut 5716992q^{79} \) \(\mathstrut +\mathstrut 570116q^{81} \) \(\mathstrut -\mathstrut 8256064q^{85} \) \(\mathstrut +\mathstrut 13641576q^{89} \) \(\mathstrut -\mathstrut 7864224q^{91} \) \(\mathstrut -\mathstrut 12453936q^{95} \) \(\mathstrut +\mathstrut 47580560q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut +\mathstrut \) \(1348\) \(x^{2}\mathstrut +\mathstrut \) \(93051\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\( -2 \nu^{3} + 42 \nu^{2} - 2318 \nu + 28308 \)\()/105\)
\(\beta_{3}\)\(=\)\((\)\( 16 \nu^{3} + 84 \nu^{2} + 19384 \nu + 56616 \)\()/105\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(8\) \(\beta_{2}\mathstrut -\mathstrut \) \(4\) \(\beta_{1}\mathstrut -\mathstrut \) \(2696\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(21\) \(\beta_{3}\mathstrut -\mathstrut \) \(42\) \(\beta_{2}\mathstrut -\mathstrut \) \(2402\) \(\beta_{1}\)\()/4\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
35.7074i
8.54284i
8.54284i
35.7074i
0 71.4148i 0 −279.408 7.49513i 0 860.515i 0 −2913.08 0
9.2 0 17.0857i 0 201.408 + 193.804i 0 1750.09i 0 1895.08 0
9.3 0 17.0857i 0 201.408 193.804i 0 1750.09i 0 1895.08 0
9.4 0 71.4148i 0 −279.408 + 7.49513i 0 860.515i 0 −2913.08 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.b Even 1 yes

Hecke kernels

There are no other newforms in \(S_{8}^{\mathrm{new}}(20, [\chi])\).