Properties

Label 80.4.c.b
Level $80$
Weight $4$
Character orbit 80.c
Analytic conductor $4.720$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 80.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.72015280046\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-19}) \)
Defining polynomial: \( x^{2} - x + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-19}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{3} + ( - \beta + 7) q^{5} + \beta q^{7} - 49 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{3} + ( - \beta + 7) q^{5} + \beta q^{7} - 49 q^{9} - 20 q^{11} - 6 \beta q^{13} + ( - 7 \beta - 76) q^{15} + 8 \beta q^{17} + 84 q^{19} + 76 q^{21} - 7 \beta q^{23} + ( - 14 \beta - 27) q^{25} + 22 \beta q^{27} + 6 q^{29} + 224 q^{31} + 20 \beta q^{33} + (7 \beta + 76) q^{35} + 14 \beta q^{37} - 456 q^{39} + 266 q^{41} + 35 \beta q^{43} + (49 \beta - 343) q^{45} - 43 \beta q^{47} + 267 q^{49} + 608 q^{51} - 42 \beta q^{53} + (20 \beta - 140) q^{55} - 84 \beta q^{57} + 28 q^{59} + 182 q^{61} - 49 \beta q^{63} + ( - 42 \beta - 456) q^{65} - 49 \beta q^{67} - 532 q^{69} - 408 q^{71} + 124 \beta q^{73} + (27 \beta - 1064) q^{75} - 20 \beta q^{77} - 48 q^{79} + 349 q^{81} + 23 \beta q^{83} + (56 \beta + 608) q^{85} - 6 \beta q^{87} - 1526 q^{89} + 456 q^{91} - 224 \beta q^{93} + ( - 84 \beta + 588) q^{95} + 64 \beta q^{97} + 980 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 14 q^{5} - 98 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 14 q^{5} - 98 q^{9} - 40 q^{11} - 152 q^{15} + 168 q^{19} + 152 q^{21} - 54 q^{25} + 12 q^{29} + 448 q^{31} + 152 q^{35} - 912 q^{39} + 532 q^{41} - 686 q^{45} + 534 q^{49} + 1216 q^{51} - 280 q^{55} + 56 q^{59} + 364 q^{61} - 912 q^{65} - 1064 q^{69} - 816 q^{71} - 2128 q^{75} - 96 q^{79} + 698 q^{81} + 1216 q^{85} - 3052 q^{89} + 912 q^{91} + 1176 q^{95} + 1960 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
0.500000 + 2.17945i
0.500000 2.17945i
0 8.71780i 0 7.00000 8.71780i 0 8.71780i 0 −49.0000 0
49.2 0 8.71780i 0 7.00000 + 8.71780i 0 8.71780i 0 −49.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.4.c.b 2
3.b odd 2 1 720.4.f.a 2
4.b odd 2 1 20.4.c.a 2
5.b even 2 1 inner 80.4.c.b 2
5.c odd 4 2 400.4.a.w 2
8.b even 2 1 320.4.c.b 2
8.d odd 2 1 320.4.c.a 2
12.b even 2 1 180.4.d.a 2
15.d odd 2 1 720.4.f.a 2
20.d odd 2 1 20.4.c.a 2
20.e even 4 2 100.4.a.d 2
28.d even 2 1 980.4.e.a 2
40.e odd 2 1 320.4.c.a 2
40.f even 2 1 320.4.c.b 2
40.i odd 4 2 1600.4.a.ck 2
40.k even 4 2 1600.4.a.cj 2
60.h even 2 1 180.4.d.a 2
60.l odd 4 2 900.4.a.s 2
140.c even 2 1 980.4.e.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.4.c.a 2 4.b odd 2 1
20.4.c.a 2 20.d odd 2 1
80.4.c.b 2 1.a even 1 1 trivial
80.4.c.b 2 5.b even 2 1 inner
100.4.a.d 2 20.e even 4 2
180.4.d.a 2 12.b even 2 1
180.4.d.a 2 60.h even 2 1
320.4.c.a 2 8.d odd 2 1
320.4.c.a 2 40.e odd 2 1
320.4.c.b 2 8.b even 2 1
320.4.c.b 2 40.f even 2 1
400.4.a.w 2 5.c odd 4 2
720.4.f.a 2 3.b odd 2 1
720.4.f.a 2 15.d odd 2 1
900.4.a.s 2 60.l odd 4 2
980.4.e.a 2 28.d even 2 1
980.4.e.a 2 140.c even 2 1
1600.4.a.cj 2 40.k even 4 2
1600.4.a.ck 2 40.i odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 76 \) acting on \(S_{4}^{\mathrm{new}}(80, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 76 \) Copy content Toggle raw display
$5$ \( T^{2} - 14T + 125 \) Copy content Toggle raw display
$7$ \( T^{2} + 76 \) Copy content Toggle raw display
$11$ \( (T + 20)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2736 \) Copy content Toggle raw display
$17$ \( T^{2} + 4864 \) Copy content Toggle raw display
$19$ \( (T - 84)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 3724 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( (T - 224)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 14896 \) Copy content Toggle raw display
$41$ \( (T - 266)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 93100 \) Copy content Toggle raw display
$47$ \( T^{2} + 140524 \) Copy content Toggle raw display
$53$ \( T^{2} + 134064 \) Copy content Toggle raw display
$59$ \( (T - 28)^{2} \) Copy content Toggle raw display
$61$ \( (T - 182)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 182476 \) Copy content Toggle raw display
$71$ \( (T + 408)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 1168576 \) Copy content Toggle raw display
$79$ \( (T + 48)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 40204 \) Copy content Toggle raw display
$89$ \( (T + 1526)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 311296 \) Copy content Toggle raw display
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