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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1860,2,Mod(1141,1860)] 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("1860.1141"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1860, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1860.q (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-1,0,-1,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.8521747760\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1741.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1860.1741
Dual form 1860.2.q.b.1141.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{5} +(-0.500000 - 0.866025i) q^{9} +(2.00000 + 3.46410i) q^{11} +(1.50000 + 2.59808i) q^{13} +1.00000 q^{15} +(2.00000 - 3.46410i) q^{17} +(2.50000 - 4.33013i) q^{19} -2.00000 q^{23} +(-0.500000 + 0.866025i) q^{25} +1.00000 q^{27} -6.00000 q^{29} +(2.00000 - 5.19615i) q^{31} -4.00000 q^{33} +(-3.50000 + 6.06218i) q^{37} -3.00000 q^{39} +(4.00000 + 6.92820i) q^{41} +(0.500000 - 0.866025i) q^{43} +(-0.500000 + 0.866025i) q^{45} +6.00000 q^{47} +(3.50000 + 6.06218i) q^{49} +(2.00000 + 3.46410i) q^{51} +(2.00000 + 3.46410i) q^{53} +(2.00000 - 3.46410i) q^{55} +(2.50000 + 4.33013i) q^{57} +10.0000 q^{61} +(1.50000 - 2.59808i) q^{65} +(2.00000 + 3.46410i) q^{67} +(1.00000 - 1.73205i) q^{69} +(3.00000 + 5.19615i) q^{71} +(3.50000 + 6.06218i) q^{73} +(-0.500000 - 0.866025i) q^{75} +(6.00000 - 10.3923i) q^{79} +(-0.500000 + 0.866025i) q^{81} +(1.00000 + 1.73205i) q^{83} -4.00000 q^{85} +(3.00000 - 5.19615i) q^{87} +4.00000 q^{89} +(3.50000 + 4.33013i) q^{93} -5.00000 q^{95} +9.00000 q^{97} +(2.00000 - 3.46410i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - q^{5} - q^{9} + 4 q^{11} + 3 q^{13} + 2 q^{15} + 4 q^{17} + 5 q^{19} - 4 q^{23} - q^{25} + 2 q^{27} - 12 q^{29} + 4 q^{31} - 8 q^{33} - 7 q^{37} - 6 q^{39} + 8 q^{41} + q^{43} - q^{45}+ \cdots + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(931\) \(1117\) \(1241\) \(1801\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) −0.500000 + 0.866025i −0.288675 + 0.500000i
\(4\) 0 0
\(5\) −0.500000 − 0.866025i −0.223607 − 0.387298i
\(6\) 0 0
\(7\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(8\) 0 0
\(9\) −0.500000 − 0.866025i −0.166667 − 0.288675i
\(10\) 0 0
\(11\) 2.00000 + 3.46410i 0.603023 + 1.04447i 0.992361 + 0.123371i \(0.0393705\pi\)
−0.389338 + 0.921095i \(0.627296\pi\)
\(12\) 0 0
\(13\) 1.50000 + 2.59808i 0.416025 + 0.720577i 0.995535 − 0.0943882i \(-0.0300895\pi\)
−0.579510 + 0.814965i \(0.696756\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 2.00000 − 3.46410i 0.485071 − 0.840168i −0.514782 − 0.857321i \(-0.672127\pi\)
0.999853 + 0.0171533i \(0.00546033\pi\)
\(18\) 0 0
\(19\) 2.50000 − 4.33013i 0.573539 − 0.993399i −0.422659 − 0.906289i \(-0.638903\pi\)
0.996199 − 0.0871106i \(-0.0277634\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.00000 −0.417029 −0.208514 − 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 2.00000 − 5.19615i 0.359211 − 0.933257i
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −3.50000 + 6.06218i −0.575396 + 0.996616i 0.420602 + 0.907245i \(0.361819\pi\)
−0.995998 + 0.0893706i \(0.971514\pi\)
\(38\) 0 0
\(39\) −3.00000 −0.480384
\(40\) 0 0
\(41\) 4.00000 + 6.92820i 0.624695 + 1.08200i 0.988600 + 0.150567i \(0.0481100\pi\)
−0.363905 + 0.931436i \(0.618557\pi\)
\(42\) 0 0
\(43\) 0.500000 − 0.866025i 0.0762493 − 0.132068i −0.825380 − 0.564578i \(-0.809039\pi\)
0.901629 + 0.432511i \(0.142372\pi\)
\(44\) 0 0
\(45\) −0.500000 + 0.866025i −0.0745356 + 0.129099i
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) 3.50000 + 6.06218i 0.500000 + 0.866025i
\(50\) 0 0
\(51\) 2.00000 + 3.46410i 0.280056 + 0.485071i
\(52\) 0 0
\(53\) 2.00000 + 3.46410i 0.274721 + 0.475831i 0.970065 − 0.242846i \(-0.0780811\pi\)
−0.695344 + 0.718677i \(0.744748\pi\)
\(54\) 0 0
\(55\) 2.00000 − 3.46410i 0.269680 − 0.467099i
\(56\) 0 0
\(57\) 2.50000 + 4.33013i 0.331133 + 0.573539i
\(58\) 0 0
\(59\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.50000 − 2.59808i 0.186052 − 0.322252i
\(66\) 0 0
\(67\) 2.00000 + 3.46410i 0.244339 + 0.423207i 0.961946 − 0.273241i \(-0.0880957\pi\)
−0.717607 + 0.696449i \(0.754762\pi\)
\(68\) 0 0
\(69\) 1.00000 − 1.73205i 0.120386 − 0.208514i
\(70\) 0 0
\(71\) 3.00000 + 5.19615i 0.356034 + 0.616670i 0.987294 − 0.158901i \(-0.0507952\pi\)
−0.631260 + 0.775571i \(0.717462\pi\)
\(72\) 0 0
\(73\) 3.50000 + 6.06218i 0.409644 + 0.709524i 0.994850 − 0.101361i \(-0.0323196\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) −0.500000 − 0.866025i −0.0577350 − 0.100000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.00000 − 10.3923i 0.675053 − 1.16923i −0.301401 − 0.953498i \(-0.597454\pi\)
0.976453 − 0.215728i \(-0.0692125\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 1.00000 + 1.73205i 0.109764 + 0.190117i 0.915675 − 0.401920i \(-0.131657\pi\)
−0.805910 + 0.592037i \(0.798324\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) 0 0
\(87\) 3.00000 − 5.19615i 0.321634 − 0.557086i
\(88\) 0 0
\(89\) 4.00000 0.423999 0.212000 − 0.977270i \(-0.432002\pi\)
0.212000 + 0.977270i \(0.432002\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 3.50000 + 4.33013i 0.362933 + 0.449013i
\(94\) 0 0
\(95\) −5.00000 −0.512989
\(96\) 0 0
\(97\) 9.00000 0.913812 0.456906 − 0.889515i \(-0.348958\pi\)
0.456906 + 0.889515i \(0.348958\pi\)
\(98\) 0 0
\(99\) 2.00000 − 3.46410i 0.201008 − 0.348155i
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 1860.2.q.b.1741.1 yes 2
31.25 even 3 inner 1860.2.q.b.1141.1 ✓ 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1860.2.q.b.1141.1 ✓ 2 31.25 even 3 inner
1860.2.q.b.1741.1 yes 2 1.1 even 1 trivial