Properties

Label 80.6.n.c
Level 80
Weight 6
Character orbit 80.n
Analytic conductor 12.831
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 80.n (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.8307055850\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{155})\)
Defining polynomial: \(x^{4} - 77 x^{2} + 1521\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} + ( 55 + 10 \beta_{1} ) q^{5} -11 \beta_{3} q^{7} + 67 \beta_{1} q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} + ( 55 + 10 \beta_{1} ) q^{5} -11 \beta_{3} q^{7} + 67 \beta_{1} q^{9} + ( 5 \beta_{2} - 5 \beta_{3} ) q^{11} + ( 165 - 165 \beta_{1} ) q^{13} + ( -55 \beta_{2} + 10 \beta_{3} ) q^{15} + ( 1265 + 1265 \beta_{1} ) q^{17} + ( 110 \beta_{2} + 110 \beta_{3} ) q^{19} + 3410 q^{21} + 43 \beta_{2} q^{23} + ( 2925 + 1100 \beta_{1} ) q^{25} -176 \beta_{3} q^{27} + 2596 \beta_{1} q^{29} + ( -285 \beta_{2} + 285 \beta_{3} ) q^{31} + ( 1550 - 1550 \beta_{1} ) q^{33} + ( -110 \beta_{2} - 605 \beta_{3} ) q^{35} + ( 1385 + 1385 \beta_{1} ) q^{37} + ( -165 \beta_{2} - 165 \beta_{3} ) q^{39} + 2178 q^{41} + 1111 \beta_{2} q^{43} + ( -670 + 3685 \beta_{1} ) q^{45} + 397 \beta_{3} q^{47} -20703 \beta_{1} q^{49} + ( -1265 \beta_{2} + 1265 \beta_{3} ) q^{51} + ( -23915 + 23915 \beta_{1} ) q^{53} + ( 225 \beta_{2} - 325 \beta_{3} ) q^{55} + ( -34100 - 34100 \beta_{1} ) q^{57} + ( 1090 \beta_{2} + 1090 \beta_{3} ) q^{59} -35882 q^{61} -737 \beta_{2} q^{63} + ( 10725 - 7425 \beta_{1} ) q^{65} -2155 \beta_{3} q^{67} -13330 \beta_{1} q^{69} + ( 2695 \beta_{2} - 2695 \beta_{3} ) q^{71} + ( -21615 + 21615 \beta_{1} ) q^{73} + ( -2925 \beta_{2} + 1100 \beta_{3} ) q^{75} + ( -17050 - 17050 \beta_{1} ) q^{77} + ( -880 \beta_{2} - 880 \beta_{3} ) q^{79} + 70841 q^{81} -3113 \beta_{2} q^{83} + ( 56925 + 82225 \beta_{1} ) q^{85} + 2596 \beta_{3} q^{87} -114424 \beta_{1} q^{89} + ( 1815 \beta_{2} - 1815 \beta_{3} ) q^{91} + ( -88350 + 88350 \beta_{1} ) q^{93} + ( 7150 \beta_{2} + 4950 \beta_{3} ) q^{95} + ( -615 - 615 \beta_{1} ) q^{97} + ( -335 \beta_{2} - 335 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 220q^{5} + O(q^{10}) \) \( 4q + 220q^{5} + 660q^{13} + 5060q^{17} + 13640q^{21} + 11700q^{25} + 6200q^{33} + 5540q^{37} + 8712q^{41} - 2680q^{45} - 95660q^{53} - 136400q^{57} - 143528q^{61} + 42900q^{65} - 86460q^{73} - 68200q^{77} + 283364q^{81} + 227700q^{85} - 353400q^{93} - 2460q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 77 x^{2} + 1521\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - 38 \nu \)\()/39\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 78 \nu^{2} + 116 \nu - 3003 \)\()/39\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} - 78 \nu^{2} + 116 \nu + 3003 \)\()/39\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} + 2 \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + \beta_{2} + 154\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(19 \beta_{3} + 19 \beta_{2} + 116 \beta_{1}\)\()/2\)

Character values

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

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(\beta_{1}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
6.22495 + 0.500000i
−6.22495 + 0.500000i
6.22495 0.500000i
−6.22495 0.500000i
0 −12.4499 12.4499i 0 55.0000 + 10.0000i 0 −136.949 + 136.949i 0 67.0000i 0
47.2 0 12.4499 + 12.4499i 0 55.0000 + 10.0000i 0 136.949 136.949i 0 67.0000i 0
63.1 0 −12.4499 + 12.4499i 0 55.0000 10.0000i 0 −136.949 136.949i 0 67.0000i 0
63.2 0 12.4499 12.4499i 0 55.0000 10.0000i 0 136.949 + 136.949i 0 67.0000i 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
4.b odd 2 1 inner
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.6.n.c 4
4.b odd 2 1 inner 80.6.n.c 4
5.b even 2 1 400.6.n.b 4
5.c odd 4 1 inner 80.6.n.c 4
5.c odd 4 1 400.6.n.b 4
20.d odd 2 1 400.6.n.b 4
20.e even 4 1 inner 80.6.n.c 4
20.e even 4 1 400.6.n.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.6.n.c 4 1.a even 1 1 trivial
80.6.n.c 4 4.b odd 2 1 inner
80.6.n.c 4 5.c odd 4 1 inner
80.6.n.c 4 20.e even 4 1 inner
400.6.n.b 4 5.b even 2 1
400.6.n.b 4 5.c odd 4 1
400.6.n.b 4 20.d odd 2 1
400.6.n.b 4 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 96100 \) acting on \(S_{6}^{\mathrm{new}}(80, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 87122 T^{4} + 3486784401 T^{8} \)
$5$ \( ( 1 - 110 T + 3125 T^{2} )^{2} \)
$7$ \( 1 - 549771682 T^{4} + 79792266297612001 T^{8} \)
$11$ \( ( 1 - 306602 T^{2} + 25937424601 T^{4} )^{2} \)
$13$ \( ( 1 - 330 T + 54450 T^{2} - 122526690 T^{3} + 137858491849 T^{4} )^{2} \)
$17$ \( ( 1 - 2530 T + 3200450 T^{2} - 3592238210 T^{3} + 2015993900449 T^{4} )^{2} \)
$19$ \( ( 1 - 2549802 T^{2} + 6131066257801 T^{4} )^{2} \)
$23$ \( 1 + 68424579426718 T^{4} + \)\(17\!\cdots\!01\)\( T^{8} \)
$29$ \( ( 1 - 34283082 T^{2} + 420707233300201 T^{4} )^{2} \)
$31$ \( ( 1 - 6898802 T^{2} + 819628286980801 T^{4} )^{2} \)
$37$ \( ( 1 - 2770 T + 3836450 T^{2} - 192082760890 T^{3} + 4808584372417849 T^{4} )^{2} \)
$41$ \( ( 1 - 2178 T + 115856201 T^{2} )^{4} \)
$43$ \( 1 - 35368995141923122 T^{4} + \)\(46\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 + 62763367693678078 T^{4} + \)\(27\!\cdots\!01\)\( T^{8} \)
$53$ \( ( 1 + 47830 T + 1143854450 T^{2} + 20002290430190 T^{3} + 174887470365513049 T^{4} )^{2} \)
$59$ \( ( 1 + 693226598 T^{2} + 511116753300641401 T^{4} )^{2} \)
$61$ \( ( 1 + 35882 T + 844596301 T^{2} )^{4} \)
$67$ \( 1 - 2056557036860651602 T^{4} + \)\(33\!\cdots\!01\)\( T^{8} \)
$71$ \( ( 1 + 894616798 T^{2} + 3255243551009881201 T^{4} )^{2} \)
$73$ \( ( 1 + 43230 T + 934416450 T^{2} + 89618884965390 T^{3} + 4297625829703557649 T^{4} )^{2} \)
$79$ \( ( 1 + 5673984798 T^{2} + 9468276082626847201 T^{4} )^{2} \)
$83$ \( 1 - 7276763020942840082 T^{4} + \)\(24\!\cdots\!01\)\( T^{8} \)
$89$ \( ( 1 + 1924732878 T^{2} + 31181719929966183601 T^{4} )^{2} \)
$97$ \( ( 1 + 1230 T + 756450 T^{2} + 10562428516110 T^{3} + 73742412689492826049 T^{4} )^{2} \)
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