Properties

Label 80.6.n.b
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: \(\Q(i, \sqrt{195})\)
Defining polynomial: \(x^{4} - 97 x^{2} + 2401\)
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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} + ( -25 - 50 \beta_{1} ) q^{5} + 9 \beta_{3} q^{7} + 147 \beta_{1} q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} + ( -25 - 50 \beta_{1} ) q^{5} + 9 \beta_{3} q^{7} + 147 \beta_{1} q^{9} + ( -15 \beta_{2} + 15 \beta_{3} ) q^{11} + ( -695 + 695 \beta_{1} ) q^{13} + ( 25 \beta_{2} - 50 \beta_{3} ) q^{15} + ( -95 - 95 \beta_{1} ) q^{17} + ( -30 \beta_{2} - 30 \beta_{3} ) q^{19} -3510 q^{21} + 183 \beta_{2} q^{23} + ( -1875 + 2500 \beta_{1} ) q^{25} -96 \beta_{3} q^{27} + 2876 \beta_{1} q^{29} + ( -345 \beta_{2} + 345 \beta_{3} ) q^{31} + ( -5850 + 5850 \beta_{1} ) q^{33} + ( -450 \beta_{2} - 225 \beta_{3} ) q^{35} + ( -5155 - 5155 \beta_{1} ) q^{37} + ( 695 \beta_{2} + 695 \beta_{3} ) q^{39} + 9218 q^{41} + 111 \beta_{2} q^{43} + ( 7350 - 3675 \beta_{1} ) q^{45} -903 \beta_{3} q^{47} -14783 \beta_{1} q^{49} + ( 95 \beta_{2} - 95 \beta_{3} ) q^{51} + ( 11345 - 11345 \beta_{1} ) q^{53} + ( -375 \beta_{2} - 1125 \beta_{3} ) q^{55} + ( 11700 + 11700 \beta_{1} ) q^{57} + ( 30 \beta_{2} + 30 \beta_{3} ) q^{59} + 18678 q^{61} + 1323 \beta_{2} q^{63} + ( 52125 + 17375 \beta_{1} ) q^{65} + 1245 \beta_{3} q^{67} -71370 \beta_{1} q^{69} + ( -2085 \beta_{2} + 2085 \beta_{3} ) q^{71} + ( -24055 + 24055 \beta_{1} ) q^{73} + ( 1875 \beta_{2} + 2500 \beta_{3} ) q^{75} + ( -52650 - 52650 \beta_{1} ) q^{77} + ( -3360 \beta_{2} - 3360 \beta_{3} ) q^{79} + 73161 q^{81} -2553 \beta_{2} q^{83} + ( -2375 + 7125 \beta_{1} ) q^{85} + 2876 \beta_{3} q^{87} + 74296 \beta_{1} q^{89} + ( 6255 \beta_{2} - 6255 \beta_{3} ) q^{91} + ( -134550 + 134550 \beta_{1} ) q^{93} + ( 2250 \beta_{2} - 750 \beta_{3} ) q^{95} + ( -49255 - 49255 \beta_{1} ) q^{97} + ( 2205 \beta_{2} + 2205 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 100q^{5} + O(q^{10}) \) \( 4q - 100q^{5} - 2780q^{13} - 380q^{17} - 14040q^{21} - 7500q^{25} - 23400q^{33} - 20620q^{37} + 36872q^{41} + 29400q^{45} + 45380q^{53} + 46800q^{57} + 74712q^{61} + 208500q^{65} - 96220q^{73} - 210600q^{77} + 292644q^{81} - 9500q^{85} - 538200q^{93} - 197020q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 97 x^{2} + 2401\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - 48 \nu \)\()/49\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 98 \nu^{2} + 146 \nu - 4753 \)\()/49\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} - 98 \nu^{2} + 146 \nu + 4753 \)\()/49\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} + 2 \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + \beta_{2} + 194\)\()/4\)
\(\nu^{3}\)\(=\)\(12 \beta_{3} + 12 \beta_{2} + 73 \beta_{1}\)

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
6.98212 + 0.500000i
−6.98212 + 0.500000i
6.98212 0.500000i
−6.98212 0.500000i
0 −13.9642 13.9642i 0 −25.0000 50.0000i 0 125.678 125.678i 0 147.000i 0
47.2 0 13.9642 + 13.9642i 0 −25.0000 50.0000i 0 −125.678 + 125.678i 0 147.000i 0
63.1 0 −13.9642 + 13.9642i 0 −25.0000 + 50.0000i 0 125.678 + 125.678i 0 147.000i 0
63.2 0 13.9642 13.9642i 0 −25.0000 + 50.0000i 0 −125.678 125.678i 0 147.000i 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.b 4
4.b odd 2 1 inner 80.6.n.b 4
5.b even 2 1 400.6.n.d 4
5.c odd 4 1 inner 80.6.n.b 4
5.c odd 4 1 400.6.n.d 4
20.d odd 2 1 400.6.n.d 4
20.e even 4 1 inner 80.6.n.b 4
20.e even 4 1 400.6.n.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.6.n.b 4 1.a even 1 1 trivial
80.6.n.b 4 4.b odd 2 1 inner
80.6.n.b 4 5.c odd 4 1 inner
80.6.n.b 4 20.e even 4 1 inner
400.6.n.d 4 5.b even 2 1
400.6.n.d 4 5.c odd 4 1
400.6.n.d 4 20.d odd 2 1
400.6.n.d 4 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 152100 \) acting on \(S_{6}^{\mathrm{new}}(80, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 108882 T^{4} + 3486784401 T^{8} \)
$5$ \( ( 1 + 50 T + 3125 T^{2} )^{2} \)
$7$ \( 1 - 560853922 T^{4} + 79792266297612001 T^{8} \)
$11$ \( ( 1 - 146602 T^{2} + 25937424601 T^{4} )^{2} \)
$13$ \( ( 1 + 1390 T + 966050 T^{2} + 516097270 T^{3} + 137858491849 T^{4} )^{2} \)
$17$ \( ( 1 + 190 T + 18050 T^{2} + 269772830 T^{3} + 2015993900449 T^{4} )^{2} \)
$19$ \( ( 1 + 4250198 T^{2} + 6131066257801 T^{4} )^{2} \)
$23$ \( 1 - 82817669402722 T^{4} + \)\(17\!\cdots\!01\)\( T^{8} \)
$29$ \( ( 1 - 32750922 T^{2} + 420707233300201 T^{4} )^{2} \)
$31$ \( ( 1 + 35581198 T^{2} + 819628286980801 T^{4} )^{2} \)
$37$ \( ( 1 + 10310 T + 53148050 T^{2} + 714936196670 T^{3} + 4808584372417849 T^{4} )^{2} \)
$41$ \( ( 1 - 9218 T + 115856201 T^{2} )^{4} \)
$43$ \( 1 + 40420440476627918 T^{4} + \)\(46\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 - 85407260265966082 T^{4} + \)\(27\!\cdots\!01\)\( T^{8} \)
$53$ \( ( 1 - 22690 T + 257418050 T^{2} - 9488855736170 T^{3} + 174887470365513049 T^{4} )^{2} \)
$59$ \( ( 1 + 1429146598 T^{2} + 511116753300641401 T^{4} )^{2} \)
$61$ \( ( 1 - 18678 T + 844596301 T^{2} )^{4} \)
$67$ \( 1 + 746452483343412398 T^{4} + \)\(33\!\cdots\!01\)\( T^{8} \)
$71$ \( ( 1 - 217623202 T^{2} + 3255243551009881201 T^{4} )^{2} \)
$73$ \( ( 1 + 48110 T + 1157286050 T^{2} + 99735474339230 T^{3} + 4297625829703557649 T^{4} )^{2} \)
$79$ \( ( 1 - 2651775202 T^{2} + 9468276082626847201 T^{4} )^{2} \)
$83$ \( 1 - 2557737354504584722 T^{4} + \)\(24\!\cdots\!01\)\( T^{8} \)
$89$ \( ( 1 - 5648223282 T^{2} + 31181719929966183601 T^{4} )^{2} \)
$97$ \( ( 1 + 98510 T + 4852110050 T^{2} + 845938888717070 T^{3} + 73742412689492826049 T^{4} )^{2} \)
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