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Properties

Label 1014.2.e.i.991.1
Level $1014$
Weight $2$
Character 1014.991
Analytic conductor $8.097$
Analytic rank $0$
Dimension $4$
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] = [1014,2,Mod(529,1014)] 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("1014.529"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1014, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1014.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,2,-2,-2,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.09683076496\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 991.1
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1014.991
Dual form 1014.2.e.i.529.1

$q$-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} -3.73205 q^{5} +(0.500000 - 0.866025i) q^{6} +(-1.36603 + 2.36603i) q^{7} -1.00000 q^{8} +(-0.500000 + 0.866025i) q^{9} +(-1.86603 - 3.23205i) q^{10} +(0.633975 + 1.09808i) q^{11} +1.00000 q^{12} -2.73205 q^{14} +(1.86603 + 3.23205i) q^{15} +(-0.500000 - 0.866025i) q^{16} +(2.86603 - 4.96410i) q^{17} -1.00000 q^{18} +(2.36603 - 4.09808i) q^{19} +(1.86603 - 3.23205i) q^{20} +2.73205 q^{21} +(-0.633975 + 1.09808i) q^{22} +(-2.09808 - 3.63397i) q^{23} +(0.500000 + 0.866025i) q^{24} +8.92820 q^{25} +1.00000 q^{27} +(-1.36603 - 2.36603i) q^{28} +(2.23205 + 3.86603i) q^{29} +(-1.86603 + 3.23205i) q^{30} -1.46410 q^{31} +(0.500000 - 0.866025i) q^{32} +(0.633975 - 1.09808i) q^{33} +5.73205 q^{34} +(5.09808 - 8.83013i) q^{35} +(-0.500000 - 0.866025i) q^{36} +(1.76795 + 3.06218i) q^{37} +4.73205 q^{38} +3.73205 q^{40} +(-4.69615 - 8.13397i) q^{41} +(1.36603 + 2.36603i) q^{42} +(4.83013 - 8.36603i) q^{43} -1.26795 q^{44} +(1.86603 - 3.23205i) q^{45} +(2.09808 - 3.63397i) q^{46} -2.19615 q^{47} +(-0.500000 + 0.866025i) q^{48} +(-0.232051 - 0.401924i) q^{49} +(4.46410 + 7.73205i) q^{50} -5.73205 q^{51} -6.46410 q^{53} +(0.500000 + 0.866025i) q^{54} +(-2.36603 - 4.09808i) q^{55} +(1.36603 - 2.36603i) q^{56} -4.73205 q^{57} +(-2.23205 + 3.86603i) q^{58} +(4.00000 - 6.92820i) q^{59} -3.73205 q^{60} +(4.59808 - 7.96410i) q^{61} +(-0.732051 - 1.26795i) q^{62} +(-1.36603 - 2.36603i) q^{63} +1.00000 q^{64} +1.26795 q^{66} +(6.56218 + 11.3660i) q^{67} +(2.86603 + 4.96410i) q^{68} +(-2.09808 + 3.63397i) q^{69} +10.1962 q^{70} +(2.36603 - 4.09808i) q^{71} +(0.500000 - 0.866025i) q^{72} +6.26795 q^{73} +(-1.76795 + 3.06218i) q^{74} +(-4.46410 - 7.73205i) q^{75} +(2.36603 + 4.09808i) q^{76} -3.46410 q^{77} -2.53590 q^{79} +(1.86603 + 3.23205i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(4.69615 - 8.13397i) q^{82} +0.196152 q^{83} +(-1.36603 + 2.36603i) q^{84} +(-10.6962 + 18.5263i) q^{85} +9.66025 q^{86} +(2.23205 - 3.86603i) q^{87} +(-0.633975 - 1.09808i) q^{88} +(-4.73205 - 8.19615i) q^{89} +3.73205 q^{90} +4.19615 q^{92} +(0.732051 + 1.26795i) q^{93} +(-1.09808 - 1.90192i) q^{94} +(-8.83013 + 15.2942i) q^{95} -1.00000 q^{96} +(-3.00000 + 5.19615i) q^{97} +(0.232051 - 0.401924i) q^{98} -1.26795 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{3} - 2 q^{4} - 8 q^{5} + 2 q^{6} - 2 q^{7} - 4 q^{8} - 2 q^{9} - 4 q^{10} + 6 q^{11} + 4 q^{12} - 4 q^{14} + 4 q^{15} - 2 q^{16} + 8 q^{17} - 4 q^{18} + 6 q^{19} + 4 q^{20} + 4 q^{21}+ \cdots - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(5\) −3.73205 −1.66902 −0.834512 − 0.550990i \(-0.814250\pi\)
−0.834512 + 0.550990i \(0.814250\pi\)
\(6\) 0.500000 − 0.866025i 0.204124 − 0.353553i
\(7\) −1.36603 + 2.36603i −0.516309 + 0.894274i 0.483512 + 0.875338i \(0.339361\pi\)
−0.999821 + 0.0189356i \(0.993972\pi\)
\(8\) −1.00000 −0.353553
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) −1.86603 − 3.23205i −0.590089 − 1.02206i
\(11\) 0.633975 + 1.09808i 0.191151 + 0.331082i 0.945632 − 0.325239i \(-0.105445\pi\)
−0.754481 + 0.656322i \(0.772111\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −2.73205 −0.730171
\(15\) 1.86603 + 3.23205i 0.481806 + 0.834512i
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 2.86603 − 4.96410i 0.695113 − 1.20397i −0.275029 − 0.961436i \(-0.588688\pi\)
0.970143 − 0.242536i \(-0.0779791\pi\)
\(18\) −1.00000 −0.235702
\(19\) 2.36603 − 4.09808i 0.542803 − 0.940163i −0.455938 − 0.890011i \(-0.650696\pi\)
0.998742 − 0.0501517i \(-0.0159705\pi\)
\(20\) 1.86603 − 3.23205i 0.417256 − 0.722709i
\(21\) 2.73205 0.596182
\(22\) −0.633975 + 1.09808i −0.135164 + 0.234111i
\(23\) −2.09808 − 3.63397i −0.437479 − 0.757736i 0.560015 − 0.828482i \(-0.310795\pi\)
−0.997494 + 0.0707462i \(0.977462\pi\)
\(24\) 0.500000 + 0.866025i 0.102062 + 0.176777i
\(25\) 8.92820 1.78564
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −1.36603 − 2.36603i −0.258155 − 0.447137i
\(29\) 2.23205 + 3.86603i 0.414481 + 0.717903i 0.995374 − 0.0960774i \(-0.0306296\pi\)
−0.580892 + 0.813980i \(0.697296\pi\)
\(30\) −1.86603 + 3.23205i −0.340688 + 0.590089i
\(31\) −1.46410 −0.262960 −0.131480 − 0.991319i \(-0.541973\pi\)
−0.131480 + 0.991319i \(0.541973\pi\)
\(32\) 0.500000 − 0.866025i 0.0883883 − 0.153093i
\(33\) 0.633975 − 1.09808i 0.110361 − 0.191151i
\(34\) 5.73205 0.983039
\(35\) 5.09808 − 8.83013i 0.861732 − 1.49256i
\(36\) −0.500000 − 0.866025i −0.0833333 − 0.144338i
\(37\) 1.76795 + 3.06218i 0.290649 + 0.503419i 0.973963 − 0.226705i \(-0.0727955\pi\)
−0.683314 + 0.730124i \(0.739462\pi\)
\(38\) 4.73205 0.767640
\(39\) 0 0
\(40\) 3.73205 0.590089
\(41\) −4.69615 − 8.13397i −0.733416 − 1.27031i −0.955415 − 0.295267i \(-0.904592\pi\)
0.221999 − 0.975047i \(-0.428742\pi\)
\(42\) 1.36603 + 2.36603i 0.210782 + 0.365086i
\(43\) 4.83013 − 8.36603i 0.736587 − 1.27581i −0.217436 − 0.976075i \(-0.569769\pi\)
0.954023 − 0.299732i \(-0.0968974\pi\)
\(44\) −1.26795 −0.191151
\(45\) 1.86603 − 3.23205i 0.278171 − 0.481806i
\(46\) 2.09808 − 3.63397i 0.309344 − 0.535800i
\(47\) −2.19615 −0.320342 −0.160171 − 0.987089i \(-0.551205\pi\)
−0.160171 + 0.987089i \(0.551205\pi\)
\(48\) −0.500000 + 0.866025i −0.0721688 + 0.125000i
\(49\) −0.232051 − 0.401924i −0.0331501 − 0.0574177i
\(50\) 4.46410 + 7.73205i 0.631319 + 1.09348i
\(51\) −5.73205 −0.802648
\(52\) 0 0
\(53\) −6.46410 −0.887913 −0.443956 − 0.896048i \(-0.646425\pi\)
−0.443956 + 0.896048i \(0.646425\pi\)
\(54\) 0.500000 + 0.866025i 0.0680414 + 0.117851i
\(55\) −2.36603 − 4.09808i −0.319035 − 0.552584i
\(56\) 1.36603 − 2.36603i 0.182543 − 0.316173i
\(57\) −4.73205 −0.626775
\(58\) −2.23205 + 3.86603i −0.293083 + 0.507634i
\(59\) 4.00000 − 6.92820i 0.520756 − 0.901975i −0.478953 − 0.877841i \(-0.658984\pi\)
0.999709 − 0.0241347i \(-0.00768307\pi\)
\(60\) −3.73205 −0.481806
\(61\) 4.59808 − 7.96410i 0.588723 − 1.01970i −0.405677 − 0.914017i \(-0.632964\pi\)
0.994400 − 0.105682i \(-0.0337026\pi\)
\(62\) −0.732051 − 1.26795i −0.0929705 − 0.161030i
\(63\) −1.36603 − 2.36603i −0.172103 − 0.298091i
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.26795 0.156074
\(67\) 6.56218 + 11.3660i 0.801698 + 1.38858i 0.918498 + 0.395426i \(0.129403\pi\)
−0.116800 + 0.993155i \(0.537264\pi\)
\(68\) 2.86603 + 4.96410i 0.347557 + 0.601986i
\(69\) −2.09808 + 3.63397i −0.252579 + 0.437479i
\(70\) 10.1962 1.21867
\(71\) 2.36603 − 4.09808i 0.280796 − 0.486352i −0.690785 − 0.723060i \(-0.742735\pi\)
0.971581 + 0.236708i \(0.0760684\pi\)
\(72\) 0.500000 − 0.866025i 0.0589256 − 0.102062i
\(73\) 6.26795 0.733608 0.366804 − 0.930298i \(-0.380452\pi\)
0.366804 + 0.930298i \(0.380452\pi\)
\(74\) −1.76795 + 3.06218i −0.205520 + 0.355971i
\(75\) −4.46410 − 7.73205i −0.515470 − 0.892820i
\(76\) 2.36603 + 4.09808i 0.271402 + 0.470082i
\(77\) −3.46410 −0.394771
\(78\) 0 0
\(79\) −2.53590 −0.285311 −0.142655 − 0.989772i \(-0.545564\pi\)
−0.142655 + 0.989772i \(0.545564\pi\)
\(80\) 1.86603 + 3.23205i 0.208628 + 0.361354i
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 4.69615 − 8.13397i 0.518603 − 0.898247i
\(83\) 0.196152 0.0215305 0.0107653 − 0.999942i \(-0.496573\pi\)
0.0107653 + 0.999942i \(0.496573\pi\)
\(84\) −1.36603 + 2.36603i −0.149046 + 0.258155i
\(85\) −10.6962 + 18.5263i −1.16016 + 2.00946i
\(86\) 9.66025 1.04169
\(87\) 2.23205 − 3.86603i 0.239301 − 0.414481i
\(88\) −0.633975 − 1.09808i −0.0675819 − 0.117055i
\(89\) −4.73205 − 8.19615i −0.501596 − 0.868790i −0.999998 − 0.00184433i \(-0.999413\pi\)
0.498402 − 0.866946i \(-0.333920\pi\)
\(90\) 3.73205 0.393393
\(91\) 0 0
\(92\) 4.19615 0.437479
\(93\) 0.732051 + 1.26795i 0.0759101 + 0.131480i
\(94\) −1.09808 − 1.90192i −0.113258 − 0.196168i
\(95\) −8.83013 + 15.2942i −0.905952 + 1.56915i
\(96\) −1.00000 −0.102062
\(97\) −3.00000 + 5.19615i −0.304604 + 0.527589i −0.977173 − 0.212445i \(-0.931857\pi\)
0.672569 + 0.740034i \(0.265191\pi\)
\(98\) 0.232051 − 0.401924i 0.0234407 − 0.0406004i
\(99\) −1.26795 −0.127434
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 1014.2.e.i.991.1 4
13.2 odd 12 1014.2.b.e.337.4 4
13.3 even 3 1014.2.a.i.1.1 2
13.4 even 6 1014.2.e.g.529.2 4
13.5 odd 4 1014.2.i.a.361.2 4
13.6 odd 12 78.2.i.a.43.1 ✓ 4
13.7 odd 12 1014.2.i.a.823.2 4
13.8 odd 4 78.2.i.a.49.1 yes 4
13.9 even 3 inner 1014.2.e.i.529.1 4
13.10 even 6 1014.2.a.k.1.2 2
13.11 odd 12 1014.2.b.e.337.1 4
13.12 even 2 1014.2.e.g.991.2 4
39.2 even 12 3042.2.b.i.1351.1 4
39.8 even 4 234.2.l.c.127.2 4
39.11 even 12 3042.2.b.i.1351.4 4
39.23 odd 6 3042.2.a.p.1.1 2
39.29 odd 6 3042.2.a.y.1.2 2
39.32 even 12 234.2.l.c.199.2 4
52.3 odd 6 8112.2.a.bj.1.1 2
52.19 even 12 624.2.bv.e.433.2 4
52.23 odd 6 8112.2.a.bp.1.2 2
52.47 even 4 624.2.bv.e.49.1 4
65.8 even 4 1950.2.y.g.49.2 4
65.19 odd 12 1950.2.bc.d.901.2 4
65.32 even 12 1950.2.y.g.199.2 4
65.34 odd 4 1950.2.bc.d.751.2 4
65.47 even 4 1950.2.y.b.49.1 4
65.58 even 12 1950.2.y.b.199.1 4
156.47 odd 4 1872.2.by.h.1297.2 4
156.71 odd 12 1872.2.by.h.433.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.i.a.43.1 ✓ 4 13.6 odd 12
78.2.i.a.49.1 yes 4 13.8 odd 4
234.2.l.c.127.2 4 39.8 even 4
234.2.l.c.199.2 4 39.32 even 12
624.2.bv.e.49.1 4 52.47 even 4
624.2.bv.e.433.2 4 52.19 even 12
1014.2.a.i.1.1 2 13.3 even 3
1014.2.a.k.1.2 2 13.10 even 6
1014.2.b.e.337.1 4 13.11 odd 12
1014.2.b.e.337.4 4 13.2 odd 12
1014.2.e.g.529.2 4 13.4 even 6
1014.2.e.g.991.2 4 13.12 even 2
1014.2.e.i.529.1 4 13.9 even 3 inner
1014.2.e.i.991.1 4 1.1 even 1 trivial
1014.2.i.a.361.2 4 13.5 odd 4
1014.2.i.a.823.2 4 13.7 odd 12
1872.2.by.h.433.1 4 156.71 odd 12
1872.2.by.h.1297.2 4 156.47 odd 4
1950.2.y.b.49.1 4 65.47 even 4
1950.2.y.b.199.1 4 65.58 even 12
1950.2.y.g.49.2 4 65.8 even 4
1950.2.y.g.199.2 4 65.32 even 12
1950.2.bc.d.751.2 4 65.34 odd 4
1950.2.bc.d.901.2 4 65.19 odd 12
3042.2.a.p.1.1 2 39.23 odd 6
3042.2.a.y.1.2 2 39.29 odd 6
3042.2.b.i.1351.1 4 39.2 even 12
3042.2.b.i.1351.4 4 39.11 even 12
8112.2.a.bj.1.1 2 52.3 odd 6
8112.2.a.bp.1.2 2 52.23 odd 6