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Properties

Label 80.22.a.c.1.2
Level $80$
Weight $22$
Character 80.1
Self dual yes
Analytic conductor $223.582$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [80,22,Mod(1,80)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("80.1"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 80.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-30972] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(223.581875430\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1179649}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 294912 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4}\cdot 3\cdot 5 \)
Twist minimal: no (minimal twist has level 10)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-542.558\) of defining polynomial
Character \(\chi\) \(=\) 80.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+114848. q^{3} -9.76562e6 q^{5} +2.03949e8 q^{7} +2.72970e9 q^{9} +4.40105e10 q^{11} -1.80893e11 q^{13} -1.12156e12 q^{15} +1.29311e13 q^{17} -3.50261e13 q^{19} +2.34231e13 q^{21} +1.00465e14 q^{23} +9.53674e13 q^{25} -8.87849e14 q^{27} +2.65416e15 q^{29} -1.12512e15 q^{31} +5.05452e15 q^{33} -1.99169e15 q^{35} +2.62794e16 q^{37} -2.07752e16 q^{39} -2.80940e16 q^{41} -1.48308e17 q^{43} -2.66573e16 q^{45} +9.51528e16 q^{47} -5.16951e17 q^{49} +1.48511e18 q^{51} +1.03378e18 q^{53} -4.29790e17 q^{55} -4.02268e18 q^{57} +1.03132e18 q^{59} -8.22371e18 q^{61} +5.56719e17 q^{63} +1.76653e18 q^{65} +1.74709e19 q^{67} +1.15382e19 q^{69} -2.67322e19 q^{71} +5.60167e19 q^{73} +1.09528e19 q^{75} +8.97588e18 q^{77} +4.33055e19 q^{79} -1.30521e20 q^{81} +2.59402e20 q^{83} -1.26280e20 q^{85} +3.04825e20 q^{87} +5.09045e20 q^{89} -3.68929e19 q^{91} -1.29218e20 q^{93} +3.42052e20 q^{95} +6.58070e20 q^{97} +1.20136e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 30972 q^{3} - 19531250 q^{5} + 439959356 q^{7} + 13532817186 q^{9} - 105191777184 q^{11} + 308456648932 q^{13} + 302460937500 q^{15} + 18393996100404 q^{17} - 23821216715800 q^{19} - 10991999204616 q^{21}+ \cdots - 14\!\cdots\!12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 114848. 1.12292 0.561462 − 0.827503i \(-0.310239\pi\)
0.561462 + 0.827503i \(0.310239\pi\)
\(4\) 0 0
\(5\) −9.76562e6 −0.447214
\(6\) 0 0
\(7\) 2.03949e8 0.272892 0.136446 − 0.990647i \(-0.456432\pi\)
0.136446 + 0.990647i \(0.456432\pi\)
\(8\) 0 0
\(9\) 2.72970e9 0.260957
\(10\) 0 0
\(11\) 4.40105e10 0.511603 0.255802 − 0.966729i \(-0.417661\pi\)
0.255802 + 0.966729i \(0.417661\pi\)
\(12\) 0 0
\(13\) −1.80893e11 −0.363929 −0.181965 − 0.983305i \(-0.558246\pi\)
−0.181965 + 0.983305i \(0.558246\pi\)
\(14\) 0 0
\(15\) −1.12156e12 −0.502187
\(16\) 0 0
\(17\) 1.29311e13 1.55569 0.777844 − 0.628458i \(-0.216314\pi\)
0.777844 + 0.628458i \(0.216314\pi\)
\(18\) 0 0
\(19\) −3.50261e13 −1.31063 −0.655313 − 0.755357i \(-0.727463\pi\)
−0.655313 + 0.755357i \(0.727463\pi\)
\(20\) 0 0
\(21\) 2.34231e13 0.306437
\(22\) 0 0
\(23\) 1.00465e14 0.505677 0.252838 − 0.967509i \(-0.418636\pi\)
0.252838 + 0.967509i \(0.418636\pi\)
\(24\) 0 0
\(25\) 9.53674e13 0.200000
\(26\) 0 0
\(27\) −8.87849e14 −0.829889
\(28\) 0 0
\(29\) 2.65416e15 1.17152 0.585758 − 0.810486i \(-0.300797\pi\)
0.585758 + 0.810486i \(0.300797\pi\)
\(30\) 0 0
\(31\) −1.12512e15 −0.246548 −0.123274 − 0.992373i \(-0.539339\pi\)
−0.123274 + 0.992373i \(0.539339\pi\)
\(32\) 0 0
\(33\) 5.05452e15 0.574491
\(34\) 0 0
\(35\) −1.99169e15 −0.122041
\(36\) 0 0
\(37\) 2.62794e16 0.898458 0.449229 − 0.893417i \(-0.351699\pi\)
0.449229 + 0.893417i \(0.351699\pi\)
\(38\) 0 0
\(39\) −2.07752e16 −0.408665
\(40\) 0 0
\(41\) −2.80940e16 −0.326876 −0.163438 − 0.986554i \(-0.552258\pi\)
−0.163438 + 0.986554i \(0.552258\pi\)
\(42\) 0 0
\(43\) −1.48308e17 −1.04651 −0.523257 − 0.852175i \(-0.675283\pi\)
−0.523257 + 0.852175i \(0.675283\pi\)
\(44\) 0 0
\(45\) −2.66573e16 −0.116704
\(46\) 0 0
\(47\) 9.51528e16 0.263872 0.131936 − 0.991258i \(-0.457881\pi\)
0.131936 + 0.991258i \(0.457881\pi\)
\(48\) 0 0
\(49\) −5.16951e17 −0.925530
\(50\) 0 0
\(51\) 1.48511e18 1.74692
\(52\) 0 0
\(53\) 1.03378e18 0.811955 0.405978 − 0.913883i \(-0.366931\pi\)
0.405978 + 0.913883i \(0.366931\pi\)
\(54\) 0 0
\(55\) −4.29790e17 −0.228796
\(56\) 0 0
\(57\) −4.02268e18 −1.47173
\(58\) 0 0
\(59\) 1.03132e18 0.262692 0.131346 − 0.991337i \(-0.458070\pi\)
0.131346 + 0.991337i \(0.458070\pi\)
\(60\) 0 0
\(61\) −8.22371e18 −1.47606 −0.738031 − 0.674767i \(-0.764244\pi\)
−0.738031 + 0.674767i \(0.764244\pi\)
\(62\) 0 0
\(63\) 5.56719e17 0.0712132
\(64\) 0 0
\(65\) 1.76653e18 0.162754
\(66\) 0 0
\(67\) 1.74709e19 1.17092 0.585462 − 0.810700i \(-0.300913\pi\)
0.585462 + 0.810700i \(0.300913\pi\)
\(68\) 0 0
\(69\) 1.15382e19 0.567836
\(70\) 0 0
\(71\) −2.67322e19 −0.974591 −0.487296 − 0.873237i \(-0.662017\pi\)
−0.487296 + 0.873237i \(0.662017\pi\)
\(72\) 0 0
\(73\) 5.60167e19 1.52555 0.762777 − 0.646662i \(-0.223836\pi\)
0.762777 + 0.646662i \(0.223836\pi\)
\(74\) 0 0
\(75\) 1.09528e19 0.224585
\(76\) 0 0
\(77\) 8.97588e18 0.139613
\(78\) 0 0
\(79\) 4.33055e19 0.514587 0.257293 − 0.966333i \(-0.417169\pi\)
0.257293 + 0.966333i \(0.417169\pi\)
\(80\) 0 0
\(81\) −1.30521e20 −1.19286
\(82\) 0 0
\(83\) 2.59402e20 1.83507 0.917535 − 0.397655i \(-0.130176\pi\)
0.917535 + 0.397655i \(0.130176\pi\)
\(84\) 0 0
\(85\) −1.26280e20 −0.695724
\(86\) 0 0
\(87\) 3.04825e20 1.31552
\(88\) 0 0
\(89\) 5.09045e20 1.73046 0.865230 − 0.501376i \(-0.167173\pi\)
0.865230 + 0.501376i \(0.167173\pi\)
\(90\) 0 0
\(91\) −3.68929e19 −0.0993135
\(92\) 0 0
\(93\) −1.29218e20 −0.276854
\(94\) 0 0
\(95\) 3.42052e20 0.586130
\(96\) 0 0
\(97\) 6.58070e20 0.906085 0.453042 − 0.891489i \(-0.350339\pi\)
0.453042 + 0.891489i \(0.350339\pi\)
\(98\) 0 0
\(99\) 1.20136e20 0.133507
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 80.22.a.c.1.2 2
4.3 odd 2 10.22.a.d.1.1 ✓ 2
20.3 even 4 50.22.b.e.49.1 4
20.7 even 4 50.22.b.e.49.4 4
20.19 odd 2 50.22.a.d.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.22.a.d.1.1 ✓ 2 4.3 odd 2
50.22.a.d.1.2 2 20.19 odd 2
50.22.b.e.49.1 4 20.3 even 4
50.22.b.e.49.4 4 20.7 even 4
80.22.a.c.1.2 2 1.1 even 1 trivial