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Properties

Label 1860.3.k.a.61.8
Level $1860$
Weight $3$
Character 1860.61
Analytic conductor $50.681$
Analytic rank $0$
Dimension $44$
Inner twists $2$

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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] = [1860,3,Mod(61,1860)] 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("1860.61"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1860, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1860.k (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(50.6813291710\)
Analytic rank: \(0\)
Dimension: \(44\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 61.8
Character \(\chi\) \(=\) 1860.61
Dual form 1860.3.k.a.61.7

$q$-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)
 
\(f(q)\) \(=\) \(q+1.73205i q^{3} +2.23607 q^{5} -13.0283 q^{7} -3.00000 q^{9} -0.272475i q^{11} +9.95599i q^{13} +3.87298i q^{15} -1.62154i q^{17} -9.88133 q^{19} -22.5657i q^{21} -19.0412i q^{23} +5.00000 q^{25} -5.19615i q^{27} -16.1921i q^{29} +(-14.9856 + 27.1373i) q^{31} +0.471941 q^{33} -29.1322 q^{35} +15.7679i q^{37} -17.2443 q^{39} -0.376305 q^{41} -3.91590i q^{43} -6.70820 q^{45} -9.67636 q^{47} +120.737 q^{49} +2.80859 q^{51} -87.7153i q^{53} -0.609273i q^{55} -17.1150i q^{57} +73.0238 q^{59} +55.6746i q^{61} +39.0849 q^{63} +22.2623i q^{65} +60.1578 q^{67} +32.9804 q^{69} +12.7313 q^{71} -61.1408i q^{73} +8.66025i q^{75} +3.54989i q^{77} -39.5275i q^{79} +9.00000 q^{81} -38.8270i q^{83} -3.62587i q^{85} +28.0455 q^{87} -160.631i q^{89} -129.710i q^{91} +(-47.0031 - 25.9559i) q^{93} -22.0953 q^{95} -25.9980 q^{97} +0.817426i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 44 q - 16 q^{7} - 132 q^{9} - 32 q^{19} + 220 q^{25} - 60 q^{31} - 24 q^{33} - 40 q^{35} - 24 q^{39} - 80 q^{41} + 96 q^{47} + 252 q^{49} - 168 q^{51} - 248 q^{59} + 48 q^{63} + 208 q^{67} + 120 q^{71}+ \cdots + 96 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1860\mathbb{Z}\right)^\times\).

\(n\) \(931\) \(1117\) \(1241\) \(1801\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 1.73205i 0.577350i
\(4\) 0 0
\(5\) 2.23607 0.447214
\(6\) 0 0
\(7\) −13.0283 −1.86119 −0.930593 − 0.366056i \(-0.880708\pi\)
−0.930593 + 0.366056i \(0.880708\pi\)
\(8\) 0 0
\(9\) −3.00000 −0.333333
\(10\) 0 0
\(11\) − 0.272475i − 0.0247705i −0.999923 − 0.0123852i \(-0.996058\pi\)
0.999923 − 0.0123852i \(-0.00394244\pi\)
\(12\) 0 0
\(13\) 9.95599i 0.765845i 0.923780 + 0.382923i \(0.125082\pi\)
−0.923780 + 0.382923i \(0.874918\pi\)
\(14\) 0 0
\(15\) 3.87298i 0.258199i
\(16\) 0 0
\(17\) − 1.62154i − 0.0953846i −0.998862 − 0.0476923i \(-0.984813\pi\)
0.998862 − 0.0476923i \(-0.0151867\pi\)
\(18\) 0 0
\(19\) −9.88133 −0.520070 −0.260035 − 0.965599i \(-0.583734\pi\)
−0.260035 + 0.965599i \(0.583734\pi\)
\(20\) 0 0
\(21\) − 22.5657i − 1.07456i
\(22\) 0 0
\(23\) − 19.0412i − 0.827880i −0.910304 − 0.413940i \(-0.864152\pi\)
0.910304 − 0.413940i \(-0.135848\pi\)
\(24\) 0 0
\(25\) 5.00000 0.200000
\(26\) 0 0
\(27\) − 5.19615i − 0.192450i
\(28\) 0 0
\(29\) − 16.1921i − 0.558347i −0.960241 − 0.279174i \(-0.909939\pi\)
0.960241 − 0.279174i \(-0.0900605\pi\)
\(30\) 0 0
\(31\) −14.9856 + 27.1373i −0.483408 + 0.875395i
\(32\) 0 0
\(33\) 0.471941 0.0143012
\(34\) 0 0
\(35\) −29.1322 −0.832348
\(36\) 0 0
\(37\) 15.7679i 0.426158i 0.977035 + 0.213079i \(0.0683493\pi\)
−0.977035 + 0.213079i \(0.931651\pi\)
\(38\) 0 0
\(39\) −17.2443 −0.442161
\(40\) 0 0
\(41\) −0.376305 −0.00917817 −0.00458909 − 0.999989i \(-0.501461\pi\)
−0.00458909 + 0.999989i \(0.501461\pi\)
\(42\) 0 0
\(43\) − 3.91590i − 0.0910675i −0.998963 − 0.0455337i \(-0.985501\pi\)
0.998963 − 0.0455337i \(-0.0144988\pi\)
\(44\) 0 0
\(45\) −6.70820 −0.149071
\(46\) 0 0
\(47\) −9.67636 −0.205880 −0.102940 − 0.994688i \(-0.532825\pi\)
−0.102940 + 0.994688i \(0.532825\pi\)
\(48\) 0 0
\(49\) 120.737 2.46401
\(50\) 0 0
\(51\) 2.80859 0.0550703
\(52\) 0 0
\(53\) − 87.7153i − 1.65501i −0.561462 − 0.827503i \(-0.689761\pi\)
0.561462 − 0.827503i \(-0.310239\pi\)
\(54\) 0 0
\(55\) − 0.609273i − 0.0110777i
\(56\) 0 0
\(57\) − 17.1150i − 0.300262i
\(58\) 0 0
\(59\) 73.0238 1.23769 0.618846 − 0.785513i \(-0.287601\pi\)
0.618846 + 0.785513i \(0.287601\pi\)
\(60\) 0 0
\(61\) 55.6746i 0.912699i 0.889801 + 0.456349i \(0.150843\pi\)
−0.889801 + 0.456349i \(0.849157\pi\)
\(62\) 0 0
\(63\) 39.0849 0.620395
\(64\) 0 0
\(65\) 22.2623i 0.342496i
\(66\) 0 0
\(67\) 60.1578 0.897877 0.448939 − 0.893563i \(-0.351802\pi\)
0.448939 + 0.893563i \(0.351802\pi\)
\(68\) 0 0
\(69\) 32.9804 0.477977
\(70\) 0 0
\(71\) 12.7313 0.179313 0.0896567 − 0.995973i \(-0.471423\pi\)
0.0896567 + 0.995973i \(0.471423\pi\)
\(72\) 0 0
\(73\) − 61.1408i − 0.837545i −0.908091 − 0.418772i \(-0.862461\pi\)
0.908091 − 0.418772i \(-0.137539\pi\)
\(74\) 0 0
\(75\) 8.66025i 0.115470i
\(76\) 0 0
\(77\) 3.54989i 0.0461025i
\(78\) 0 0
\(79\) − 39.5275i − 0.500348i −0.968201 − 0.250174i \(-0.919512\pi\)
0.968201 − 0.250174i \(-0.0804879\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) − 38.8270i − 0.467796i −0.972261 − 0.233898i \(-0.924852\pi\)
0.972261 − 0.233898i \(-0.0751481\pi\)
\(84\) 0 0
\(85\) − 3.62587i − 0.0426573i
\(86\) 0 0
\(87\) 28.0455 0.322362
\(88\) 0 0
\(89\) − 160.631i − 1.80484i −0.430854 − 0.902422i \(-0.641788\pi\)
0.430854 − 0.902422i \(-0.358212\pi\)
\(90\) 0 0
\(91\) − 129.710i − 1.42538i
\(92\) 0 0
\(93\) −47.0031 − 25.9559i −0.505410 − 0.279096i
\(94\) 0 0
\(95\) −22.0953 −0.232582
\(96\) 0 0
\(97\) −25.9980 −0.268021 −0.134010 − 0.990980i \(-0.542786\pi\)
−0.134010 + 0.990980i \(0.542786\pi\)
\(98\) 0 0
\(99\) 0.817426i 0.00825683i
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 1860.3.k.a.61.8 yes 44
31.30 odd 2 inner 1860.3.k.a.61.7 ✓ 44
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1860.3.k.a.61.7 ✓ 44 31.30 odd 2 inner
1860.3.k.a.61.8 yes 44 1.1 even 1 trivial