Show commands: Magma / Pari/GP / SageMath

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 = [6,0,3,0,3,0,-4] 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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.64616643.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 19x^{4} + 106x^{2} + 147 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1741.1
Root \(-3.05165i\) of defining polynomial
Character \(\chi\) \(=\) 1860.1741
Dual form 1860.2.q.g.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)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{3} +(0.500000 + 0.866025i) q^{5} +(-2.15628 + 3.73479i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(-1.50000 - 2.59808i) q^{11} +(-1.64281 - 2.84542i) q^{13} +1.00000 q^{15} +(-2.79909 + 4.84816i) q^{17} +(2.64281 - 4.57747i) q^{19} +(2.15628 + 3.73479i) q^{21} +3.31256 q^{23} +(-0.500000 + 0.866025i) q^{25} -1.00000 q^{27} -0.687440 q^{29} +(-5.29909 + 1.70871i) q^{31} -3.00000 q^{33} -4.31256 q^{35} +(-5.15628 + 8.93094i) q^{37} -3.28561 q^{39} +(-4.28561 - 7.42290i) q^{41} +(-0.0134745 + 0.0233386i) q^{43} +(0.500000 - 0.866025i) q^{45} -9.91073 q^{47} +(-5.79909 - 10.0443i) q^{49} +(2.79909 + 4.84816i) q^{51} +(1.66975 + 2.89210i) q^{53} +(1.50000 - 2.59808i) q^{55} +(-2.64281 - 4.57747i) q^{57} +(-1.66975 + 2.89210i) q^{59} -13.8838 q^{61} +4.31256 q^{63} +(1.64281 - 2.84542i) q^{65} +(-2.14281 - 3.71145i) q^{67} +(1.65628 - 2.86876i) q^{69} +(1.65628 + 2.86876i) q^{71} +(-6.81256 - 11.7997i) q^{73} +(0.500000 + 0.866025i) q^{75} +12.9377 q^{77} +(4.64281 - 8.04157i) q^{79} +(-0.500000 + 0.866025i) q^{81} +(0.357195 + 0.618679i) q^{83} -5.59817 q^{85} +(-0.343720 + 0.595341i) q^{87} -2.71439 q^{89} +14.1694 q^{91} +(-1.16975 + 5.44350i) q^{93} +5.28561 q^{95} -5.57122 q^{97} +(-1.50000 + 2.59808i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} + 3 q^{5} - 4 q^{7} - 3 q^{9} - 9 q^{11} + 3 q^{13} + 6 q^{15} + 5 q^{17} + 3 q^{19} + 4 q^{21} + 2 q^{23} - 3 q^{25} - 6 q^{27} - 22 q^{29} - 10 q^{31} - 18 q^{33} - 8 q^{35} - 22 q^{37}+ \cdots - 9 q^{99}+O(q^{100}) \) 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
\(7\) −2.15628 + 3.73479i −0.814997 + 1.41162i 0.0943328 + 0.995541i \(0.469928\pi\)
−0.909330 + 0.416076i \(0.863405\pi\)
\(8\) 0 0
\(9\) −0.500000 − 0.866025i −0.166667 − 0.288675i
\(10\) 0 0
\(11\) −1.50000 − 2.59808i −0.452267 − 0.783349i 0.546259 − 0.837616i \(-0.316051\pi\)
−0.998526 + 0.0542666i \(0.982718\pi\)
\(12\) 0 0
\(13\) −1.64281 − 2.84542i −0.455632 − 0.789178i 0.543092 − 0.839673i \(-0.317253\pi\)
−0.998724 + 0.0504950i \(0.983920\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) −2.79909 + 4.84816i −0.678878 + 1.17585i 0.296441 + 0.955051i \(0.404200\pi\)
−0.975319 + 0.220800i \(0.929133\pi\)
\(18\) 0 0
\(19\) 2.64281 − 4.57747i 0.606301 − 1.05014i −0.385543 − 0.922690i \(-0.625986\pi\)
0.991844 − 0.127455i \(-0.0406807\pi\)
\(20\) 0 0
\(21\) 2.15628 + 3.73479i 0.470539 + 0.814997i
\(22\) 0 0
\(23\) 3.31256 0.690716 0.345358 − 0.938471i \(-0.387757\pi\)
0.345358 + 0.938471i \(0.387757\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −0.687440 −0.127654 −0.0638272 − 0.997961i \(-0.520331\pi\)
−0.0638272 + 0.997961i \(0.520331\pi\)
\(30\) 0 0
\(31\) −5.29909 + 1.70871i −0.951744 + 0.306894i
\(32\) 0 0
\(33\) −3.00000 −0.522233
\(34\) 0 0
\(35\) −4.31256 −0.728956
\(36\) 0 0
\(37\) −5.15628 + 8.93094i −0.847687 + 1.46824i 0.0355798 + 0.999367i \(0.488672\pi\)
−0.883267 + 0.468870i \(0.844661\pi\)
\(38\) 0 0
\(39\) −3.28561 −0.526119
\(40\) 0 0
\(41\) −4.28561 − 7.42290i −0.669300 − 1.15926i −0.978100 − 0.208135i \(-0.933261\pi\)
0.308800 − 0.951127i \(-0.400073\pi\)
\(42\) 0 0
\(43\) −0.0134745 + 0.0233386i −0.00205485 + 0.00355910i −0.867051 − 0.498219i \(-0.833987\pi\)
0.864996 + 0.501778i \(0.167321\pi\)
\(44\) 0 0
\(45\) 0.500000 − 0.866025i 0.0745356 − 0.129099i
\(46\) 0 0
\(47\) −9.91073 −1.44563 −0.722814 − 0.691042i \(-0.757152\pi\)
−0.722814 + 0.691042i \(0.757152\pi\)
\(48\) 0 0
\(49\) −5.79909 − 10.0443i −0.828441 − 1.43490i
\(50\) 0 0
\(51\) 2.79909 + 4.84816i 0.391950 + 0.678878i
\(52\) 0 0
\(53\) 1.66975 + 2.89210i 0.229358 + 0.397260i 0.957618 − 0.288041i \(-0.0930039\pi\)
−0.728260 + 0.685301i \(0.759671\pi\)
\(54\) 0 0
\(55\) 1.50000 − 2.59808i 0.202260 − 0.350325i
\(56\) 0 0
\(57\) −2.64281 − 4.57747i −0.350048 − 0.606301i
\(58\) 0 0
\(59\) −1.66975 + 2.89210i −0.217383 + 0.376519i −0.954007 − 0.299784i \(-0.903086\pi\)
0.736624 + 0.676303i \(0.236419\pi\)
\(60\) 0 0
\(61\) −13.8838 −1.77764 −0.888818 − 0.458260i \(-0.848473\pi\)
−0.888818 + 0.458260i \(0.848473\pi\)
\(62\) 0 0
\(63\) 4.31256 0.543331
\(64\) 0 0
\(65\) 1.64281 − 2.84542i 0.203765 − 0.352931i
\(66\) 0 0
\(67\) −2.14281 − 3.71145i −0.261785 − 0.453425i 0.704931 − 0.709276i \(-0.250978\pi\)
−0.966716 + 0.255850i \(0.917645\pi\)
\(68\) 0 0
\(69\) 1.65628 − 2.86876i 0.199393 − 0.345358i
\(70\) 0 0
\(71\) 1.65628 + 2.86876i 0.196564 + 0.340459i 0.947412 − 0.320016i \(-0.103688\pi\)
−0.750848 + 0.660475i \(0.770355\pi\)
\(72\) 0 0
\(73\) −6.81256 − 11.7997i −0.797350 − 1.38105i −0.921336 − 0.388766i \(-0.872901\pi\)
0.123987 − 0.992284i \(-0.460432\pi\)
\(74\) 0 0
\(75\) 0.500000 + 0.866025i 0.0577350 + 0.100000i
\(76\) 0 0
\(77\) 12.9377 1.47439
\(78\) 0 0
\(79\) 4.64281 − 8.04157i 0.522356 − 0.904748i −0.477305 − 0.878738i \(-0.658386\pi\)
0.999662 − 0.0260103i \(-0.00828026\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 0.357195 + 0.618679i 0.0392072 + 0.0679089i 0.884963 − 0.465661i \(-0.154183\pi\)
−0.845756 + 0.533570i \(0.820850\pi\)
\(84\) 0 0
\(85\) −5.59817 −0.607207
\(86\) 0 0
\(87\) −0.343720 + 0.595341i −0.0368507 + 0.0638272i
\(88\) 0 0
\(89\) −2.71439 −0.287725 −0.143862 − 0.989598i \(-0.545952\pi\)
−0.143862 + 0.989598i \(0.545952\pi\)
\(90\) 0 0
\(91\) 14.1694 1.48536
\(92\) 0 0
\(93\) −1.16975 + 5.44350i −0.121298 + 0.564464i
\(94\) 0 0
\(95\) 5.28561 0.542292
\(96\) 0 0
\(97\) −5.57122 −0.565672 −0.282836 − 0.959168i \(-0.591275\pi\)
−0.282836 + 0.959168i \(0.591275\pi\)
\(98\) 0 0
\(99\) −1.50000 + 2.59808i −0.150756 + 0.261116i
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.g.1741.1 yes 6
31.25 even 3 inner 1860.2.q.g.1141.1 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1860.2.q.g.1141.1 ✓ 6 31.25 even 3 inner
1860.2.q.g.1741.1 yes 6 1.1 even 1 trivial