Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [390,2,Mod(61,390)] 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("390.61"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(390, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,1,1,-1,-2,-1,-3] 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: \(3.11416567883\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 61.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 390.61
Dual form 390.2.i.e.211.1

$q$-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^{2} +(0.500000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{4} -1.00000 q^{5} +(-0.500000 - 0.866025i) q^{6} +(-1.50000 - 2.59808i) q^{7} -1.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} +(-0.500000 + 0.866025i) q^{10} +(1.50000 - 2.59808i) q^{11} -1.00000 q^{12} +(-2.50000 + 2.59808i) q^{13} -3.00000 q^{14} +(-0.500000 + 0.866025i) q^{15} +(-0.500000 + 0.866025i) q^{16} -1.00000 q^{18} +(-1.50000 - 2.59808i) q^{19} +(0.500000 + 0.866025i) q^{20} -3.00000 q^{21} +(-1.50000 - 2.59808i) q^{22} +(2.00000 - 3.46410i) q^{23} +(-0.500000 + 0.866025i) q^{24} +1.00000 q^{25} +(1.00000 + 3.46410i) q^{26} -1.00000 q^{27} +(-1.50000 + 2.59808i) q^{28} +(2.00000 - 3.46410i) q^{29} +(0.500000 + 0.866025i) q^{30} +6.00000 q^{31} +(0.500000 + 0.866025i) q^{32} +(-1.50000 - 2.59808i) q^{33} +(1.50000 + 2.59808i) q^{35} +(-0.500000 + 0.866025i) q^{36} +(-4.50000 + 7.79423i) q^{37} -3.00000 q^{38} +(1.00000 + 3.46410i) q^{39} +1.00000 q^{40} +(5.00000 - 8.66025i) q^{41} +(-1.50000 + 2.59808i) q^{42} +(5.00000 + 8.66025i) q^{43} -3.00000 q^{44} +(0.500000 + 0.866025i) q^{45} +(-2.00000 - 3.46410i) q^{46} -3.00000 q^{47} +(0.500000 + 0.866025i) q^{48} +(-1.00000 + 1.73205i) q^{49} +(0.500000 - 0.866025i) q^{50} +(3.50000 + 0.866025i) q^{52} +9.00000 q^{53} +(-0.500000 + 0.866025i) q^{54} +(-1.50000 + 2.59808i) q^{55} +(1.50000 + 2.59808i) q^{56} -3.00000 q^{57} +(-2.00000 - 3.46410i) q^{58} +(-6.00000 - 10.3923i) q^{59} +1.00000 q^{60} +(3.00000 + 5.19615i) q^{61} +(3.00000 - 5.19615i) q^{62} +(-1.50000 + 2.59808i) q^{63} +1.00000 q^{64} +(2.50000 - 2.59808i) q^{65} -3.00000 q^{66} +(4.00000 - 6.92820i) q^{67} +(-2.00000 - 3.46410i) q^{69} +3.00000 q^{70} +(7.00000 + 12.1244i) q^{71} +(0.500000 + 0.866025i) q^{72} -8.00000 q^{73} +(4.50000 + 7.79423i) q^{74} +(0.500000 - 0.866025i) q^{75} +(-1.50000 + 2.59808i) q^{76} -9.00000 q^{77} +(3.50000 + 0.866025i) q^{78} +6.00000 q^{79} +(0.500000 - 0.866025i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(-5.00000 - 8.66025i) q^{82} +16.0000 q^{83} +(1.50000 + 2.59808i) q^{84} +10.0000 q^{86} +(-2.00000 - 3.46410i) q^{87} +(-1.50000 + 2.59808i) q^{88} +(1.50000 - 2.59808i) q^{89} +1.00000 q^{90} +(10.5000 + 2.59808i) q^{91} -4.00000 q^{92} +(3.00000 - 5.19615i) q^{93} +(-1.50000 + 2.59808i) q^{94} +(1.50000 + 2.59808i) q^{95} +1.00000 q^{96} +(-4.00000 - 6.92820i) q^{97} +(1.00000 + 1.73205i) q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + q^{3} - q^{4} - 2 q^{5} - q^{6} - 3 q^{7} - 2 q^{8} - q^{9} - q^{10} + 3 q^{11} - 2 q^{12} - 5 q^{13} - 6 q^{14} - q^{15} - q^{16} - 2 q^{18} - 3 q^{19} + q^{20} - 6 q^{21} - 3 q^{22}+ \cdots - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/390\mathbb{Z}\right)^\times\).

\(n\) \(131\) \(157\) \(301\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 − 0.866025i 0.353553 − 0.612372i
\(3\) 0.500000 − 0.866025i 0.288675 − 0.500000i
\(4\) −0.500000 − 0.866025i −0.250000 − 0.433013i
\(5\) −1.00000 −0.447214
\(6\) −0.500000 − 0.866025i −0.204124 − 0.353553i
\(7\) −1.50000 − 2.59808i −0.566947 − 0.981981i −0.996866 − 0.0791130i \(-0.974791\pi\)
0.429919 − 0.902867i \(-0.358542\pi\)
\(8\) −1.00000 −0.353553
\(9\) −0.500000 − 0.866025i −0.166667 − 0.288675i
\(10\) −0.500000 + 0.866025i −0.158114 + 0.273861i
\(11\) 1.50000 − 2.59808i 0.452267 − 0.783349i −0.546259 − 0.837616i \(-0.683949\pi\)
0.998526 + 0.0542666i \(0.0172821\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.50000 + 2.59808i −0.693375 + 0.720577i
\(14\) −3.00000 −0.801784
\(15\) −0.500000 + 0.866025i −0.129099 + 0.223607i
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.50000 − 2.59808i −0.344124 − 0.596040i 0.641071 − 0.767482i \(-0.278491\pi\)
−0.985194 + 0.171442i \(0.945157\pi\)
\(20\) 0.500000 + 0.866025i 0.111803 + 0.193649i
\(21\) −3.00000 −0.654654
\(22\) −1.50000 − 2.59808i −0.319801 − 0.553912i
\(23\) 2.00000 − 3.46410i 0.417029 − 0.722315i −0.578610 − 0.815604i \(-0.696405\pi\)
0.995639 + 0.0932891i \(0.0297381\pi\)
\(24\) −0.500000 + 0.866025i −0.102062 + 0.176777i
\(25\) 1.00000 0.200000
\(26\) 1.00000 + 3.46410i 0.196116 + 0.679366i
\(27\) −1.00000 −0.192450
\(28\) −1.50000 + 2.59808i −0.283473 + 0.490990i
\(29\) 2.00000 − 3.46410i 0.371391 − 0.643268i −0.618389 − 0.785872i \(-0.712214\pi\)
0.989780 + 0.142605i \(0.0455477\pi\)
\(30\) 0.500000 + 0.866025i 0.0912871 + 0.158114i
\(31\) 6.00000 1.07763 0.538816 − 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 0.500000 + 0.866025i 0.0883883 + 0.153093i
\(33\) −1.50000 − 2.59808i −0.261116 − 0.452267i
\(34\) 0 0
\(35\) 1.50000 + 2.59808i 0.253546 + 0.439155i
\(36\) −0.500000 + 0.866025i −0.0833333 + 0.144338i
\(37\) −4.50000 + 7.79423i −0.739795 + 1.28136i 0.212792 + 0.977098i \(0.431744\pi\)
−0.952587 + 0.304266i \(0.901589\pi\)
\(38\) −3.00000 −0.486664
\(39\) 1.00000 + 3.46410i 0.160128 + 0.554700i
\(40\) 1.00000 0.158114
\(41\) 5.00000 − 8.66025i 0.780869 − 1.35250i −0.150567 − 0.988600i \(-0.548110\pi\)
0.931436 − 0.363905i \(-0.118557\pi\)
\(42\) −1.50000 + 2.59808i −0.231455 + 0.400892i
\(43\) 5.00000 + 8.66025i 0.762493 + 1.32068i 0.941562 + 0.336840i \(0.109358\pi\)
−0.179069 + 0.983836i \(0.557309\pi\)
\(44\) −3.00000 −0.452267
\(45\) 0.500000 + 0.866025i 0.0745356 + 0.129099i
\(46\) −2.00000 − 3.46410i −0.294884 − 0.510754i
\(47\) −3.00000 −0.437595 −0.218797 − 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 0.500000 + 0.866025i 0.0721688 + 0.125000i
\(49\) −1.00000 + 1.73205i −0.142857 + 0.247436i
\(50\) 0.500000 − 0.866025i 0.0707107 − 0.122474i
\(51\) 0 0
\(52\) 3.50000 + 0.866025i 0.485363 + 0.120096i
\(53\) 9.00000 1.23625 0.618123 − 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) −0.500000 + 0.866025i −0.0680414 + 0.117851i
\(55\) −1.50000 + 2.59808i −0.202260 + 0.350325i
\(56\) 1.50000 + 2.59808i 0.200446 + 0.347183i
\(57\) −3.00000 −0.397360
\(58\) −2.00000 − 3.46410i −0.262613 − 0.454859i
\(59\) −6.00000 − 10.3923i −0.781133 − 1.35296i −0.931282 − 0.364299i \(-0.881308\pi\)
0.150148 − 0.988663i \(-0.452025\pi\)
\(60\) 1.00000 0.129099
\(61\) 3.00000 + 5.19615i 0.384111 + 0.665299i 0.991645 − 0.128994i \(-0.0411748\pi\)
−0.607535 + 0.794293i \(0.707841\pi\)
\(62\) 3.00000 − 5.19615i 0.381000 − 0.659912i
\(63\) −1.50000 + 2.59808i −0.188982 + 0.327327i
\(64\) 1.00000 0.125000
\(65\) 2.50000 − 2.59808i 0.310087 − 0.322252i
\(66\) −3.00000 −0.369274
\(67\) 4.00000 − 6.92820i 0.488678 − 0.846415i −0.511237 − 0.859440i \(-0.670813\pi\)
0.999915 + 0.0130248i \(0.00414604\pi\)
\(68\) 0 0
\(69\) −2.00000 − 3.46410i −0.240772 − 0.417029i
\(70\) 3.00000 0.358569
\(71\) 7.00000 + 12.1244i 0.830747 + 1.43890i 0.897447 + 0.441123i \(0.145420\pi\)
−0.0666994 + 0.997773i \(0.521247\pi\)
\(72\) 0.500000 + 0.866025i 0.0589256 + 0.102062i
\(73\) −8.00000 −0.936329 −0.468165 − 0.883641i \(-0.655085\pi\)
−0.468165 + 0.883641i \(0.655085\pi\)
\(74\) 4.50000 + 7.79423i 0.523114 + 0.906061i
\(75\) 0.500000 − 0.866025i 0.0577350 − 0.100000i
\(76\) −1.50000 + 2.59808i −0.172062 + 0.298020i
\(77\) −9.00000 −1.02565
\(78\) 3.50000 + 0.866025i 0.396297 + 0.0980581i
\(79\) 6.00000 0.675053 0.337526 − 0.941316i \(-0.390410\pi\)
0.337526 + 0.941316i \(0.390410\pi\)
\(80\) 0.500000 − 0.866025i 0.0559017 − 0.0968246i
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) −5.00000 − 8.66025i −0.552158 − 0.956365i
\(83\) 16.0000 1.75623 0.878114 − 0.478451i \(-0.158802\pi\)
0.878114 + 0.478451i \(0.158802\pi\)
\(84\) 1.50000 + 2.59808i 0.163663 + 0.283473i
\(85\) 0 0
\(86\) 10.0000 1.07833
\(87\) −2.00000 − 3.46410i −0.214423 − 0.371391i
\(88\) −1.50000 + 2.59808i −0.159901 + 0.276956i
\(89\) 1.50000 − 2.59808i 0.159000 − 0.275396i −0.775509 − 0.631337i \(-0.782506\pi\)
0.934508 + 0.355942i \(0.115840\pi\)
\(90\) 1.00000 0.105409
\(91\) 10.5000 + 2.59808i 1.10070 + 0.272352i
\(92\) −4.00000 −0.417029
\(93\) 3.00000 − 5.19615i 0.311086 − 0.538816i
\(94\) −1.50000 + 2.59808i −0.154713 + 0.267971i
\(95\) 1.50000 + 2.59808i 0.153897 + 0.266557i
\(96\) 1.00000 0.102062
\(97\) −4.00000 − 6.92820i −0.406138 − 0.703452i 0.588315 − 0.808632i \(-0.299792\pi\)
−0.994453 + 0.105180i \(0.966458\pi\)
\(98\) 1.00000 + 1.73205i 0.101015 + 0.174964i
\(99\) −3.00000 −0.301511
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 390.2.i.e.61.1 ✓ 2
3.2 odd 2 1170.2.i.c.451.1 2
5.2 odd 4 1950.2.z.d.1699.2 4
5.3 odd 4 1950.2.z.d.1699.1 4
5.4 even 2 1950.2.i.f.451.1 2
13.3 even 3 inner 390.2.i.e.211.1 yes 2
13.4 even 6 5070.2.a.r.1.1 1
13.6 odd 12 5070.2.b.b.1351.2 2
13.7 odd 12 5070.2.b.b.1351.1 2
13.9 even 3 5070.2.a.b.1.1 1
39.29 odd 6 1170.2.i.c.991.1 2
65.3 odd 12 1950.2.z.d.1849.2 4
65.29 even 6 1950.2.i.f.601.1 2
65.42 odd 12 1950.2.z.d.1849.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.i.e.61.1 ✓ 2 1.1 even 1 trivial
390.2.i.e.211.1 yes 2 13.3 even 3 inner
1170.2.i.c.451.1 2 3.2 odd 2
1170.2.i.c.991.1 2 39.29 odd 6
1950.2.i.f.451.1 2 5.4 even 2
1950.2.i.f.601.1 2 65.29 even 6
1950.2.z.d.1699.1 4 5.3 odd 4
1950.2.z.d.1699.2 4 5.2 odd 4
1950.2.z.d.1849.1 4 65.42 odd 12
1950.2.z.d.1849.2 4 65.3 odd 12
5070.2.a.b.1.1 1 13.9 even 3
5070.2.a.r.1.1 1 13.4 even 6
5070.2.b.b.1351.1 2 13.7 odd 12
5070.2.b.b.1351.2 2 13.6 odd 12