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Properties

Label 1014.2.i.a.361.2
Level $1014$
Weight $2$
Character 1014.361
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(361,1014)] 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("1014.361"); 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, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1014.i (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-2,2,0,0,-6,0,-2,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == 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 \(\Q(\zeta_{6})\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 361.2
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1014.361
Dual form 1014.2.i.a.823.2

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

Character values

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

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.866025 − 0.500000i 0.612372 − 0.353553i
\(3\) −0.500000 − 0.866025i −0.288675 − 0.500000i
\(4\) 0.500000 − 0.866025i 0.250000 − 0.433013i
\(5\) 3.73205i 1.66902i 0.550990 + 0.834512i \(0.314250\pi\)
−0.550990 + 0.834512i \(0.685750\pi\)
\(6\) −0.866025 − 0.500000i −0.353553 − 0.204124i
\(7\) −2.36603 − 1.36603i −0.894274 − 0.516309i −0.0189356 − 0.999821i \(-0.506028\pi\)
−0.875338 + 0.483512i \(0.839361\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 1.86603 + 3.23205i 0.590089 + 1.02206i
\(11\) −1.09808 + 0.633975i −0.331082 + 0.191151i −0.656322 − 0.754481i \(-0.727889\pi\)
0.325239 + 0.945632i \(0.394555\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −2.73205 −0.730171
\(15\) 3.23205 − 1.86603i 0.834512 − 0.481806i
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −2.86603 + 4.96410i −0.695113 + 1.20397i 0.275029 + 0.961436i \(0.411312\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 1.00000i 0.235702i
\(19\) −4.09808 − 2.36603i −0.940163 − 0.542803i −0.0501517 − 0.998742i \(-0.515970\pi\)
−0.890011 + 0.455938i \(0.849304\pi\)
\(20\) 3.23205 + 1.86603i 0.722709 + 0.417256i
\(21\) 2.73205i 0.596182i
\(22\) −0.633975 + 1.09808i −0.135164 + 0.234111i
\(23\) 2.09808 + 3.63397i 0.437479 + 0.757736i 0.997494 − 0.0707462i \(-0.0225381\pi\)
−0.560015 + 0.828482i \(0.689205\pi\)
\(24\) −0.866025 + 0.500000i −0.176777 + 0.102062i
\(25\) −8.92820 −1.78564
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −2.36603 + 1.36603i −0.447137 + 0.258155i
\(29\) 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
\(31\) 1.46410i 0.262960i 0.991319 + 0.131480i \(0.0419730\pi\)
−0.991319 + 0.131480i \(0.958027\pi\)
\(32\) −0.866025 − 0.500000i −0.153093 − 0.0883883i
\(33\) 1.09808 + 0.633975i 0.191151 + 0.110361i
\(34\) 5.73205i 0.983039i
\(35\) 5.09808 − 8.83013i 0.861732 − 1.49256i
\(36\) 0.500000 + 0.866025i 0.0833333 + 0.144338i
\(37\) −3.06218 + 1.76795i −0.503419 + 0.290649i −0.730124 − 0.683314i \(-0.760538\pi\)
0.226705 + 0.973963i \(0.427205\pi\)
\(38\) −4.73205 −0.767640
\(39\) 0 0
\(40\) 3.73205 0.590089
\(41\) −8.13397 + 4.69615i −1.27031 + 0.733416i −0.975047 − 0.221999i \(-0.928742\pi\)
−0.295267 + 0.955415i \(0.595408\pi\)
\(42\) 1.36603 + 2.36603i 0.210782 + 0.365086i
\(43\) −4.83013 + 8.36603i −0.736587 + 1.27581i 0.217436 + 0.976075i \(0.430231\pi\)
−0.954023 + 0.299732i \(0.903103\pi\)
\(44\) 1.26795i 0.191151i
\(45\) −3.23205 − 1.86603i −0.481806 − 0.278171i
\(46\) 3.63397 + 2.09808i 0.535800 + 0.309344i
\(47\) − 2.19615i − 0.320342i −0.987089 − 0.160171i \(-0.948795\pi\)
0.987089 − 0.160171i \(-0.0512045\pi\)
\(48\) −0.500000 + 0.866025i −0.0721688 + 0.125000i
\(49\) 0.232051 + 0.401924i 0.0331501 + 0.0574177i
\(50\) −7.73205 + 4.46410i −1.09348 + 0.631319i
\(51\) 5.73205 0.802648
\(52\) 0 0
\(53\) −6.46410 −0.887913 −0.443956 − 0.896048i \(-0.646425\pi\)
−0.443956 + 0.896048i \(0.646425\pi\)
\(54\) 0.866025 − 0.500000i 0.117851 − 0.0680414i
\(55\) −2.36603 − 4.09808i −0.319035 − 0.552584i
\(56\) −1.36603 + 2.36603i −0.182543 + 0.316173i
\(57\) 4.73205i 0.626775i
\(58\) 3.86603 + 2.23205i 0.507634 + 0.293083i
\(59\) 6.92820 + 4.00000i 0.901975 + 0.520756i 0.877841 − 0.478953i \(-0.158984\pi\)
0.0241347 + 0.999709i \(0.492317\pi\)
\(60\) − 3.73205i − 0.481806i
\(61\) 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
\(63\) 2.36603 − 1.36603i 0.298091 − 0.172103i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 1.26795 0.156074
\(67\) 11.3660 − 6.56218i 1.38858 − 0.801698i 0.395426 − 0.918498i \(-0.370597\pi\)
0.993155 + 0.116800i \(0.0372638\pi\)
\(68\) 2.86603 + 4.96410i 0.347557 + 0.601986i
\(69\) 2.09808 − 3.63397i 0.252579 − 0.437479i
\(70\) − 10.1962i − 1.21867i
\(71\) −4.09808 − 2.36603i −0.486352 − 0.280796i 0.236708 − 0.971581i \(-0.423932\pi\)
−0.723060 + 0.690785i \(0.757265\pi\)
\(72\) 0.866025 + 0.500000i 0.102062 + 0.0589256i
\(73\) 6.26795i 0.733608i 0.930298 + 0.366804i \(0.119548\pi\)
−0.930298 + 0.366804i \(0.880452\pi\)
\(74\) −1.76795 + 3.06218i −0.205520 + 0.355971i
\(75\) 4.46410 + 7.73205i 0.515470 + 0.892820i
\(76\) −4.09808 + 2.36603i −0.470082 + 0.271402i
\(77\) 3.46410 0.394771
\(78\) 0 0
\(79\) −2.53590 −0.285311 −0.142655 − 0.989772i \(-0.545564\pi\)
−0.142655 + 0.989772i \(0.545564\pi\)
\(80\) 3.23205 − 1.86603i 0.361354 − 0.208628i
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) −4.69615 + 8.13397i −0.518603 + 0.898247i
\(83\) − 0.196152i − 0.0215305i −0.999942 − 0.0107653i \(-0.996573\pi\)
0.999942 − 0.0107653i \(-0.00342676\pi\)
\(84\) 2.36603 + 1.36603i 0.258155 + 0.149046i
\(85\) −18.5263 − 10.6962i −2.00946 − 1.16016i
\(86\) 9.66025i 1.04169i
\(87\) 2.23205 − 3.86603i 0.239301 − 0.414481i
\(88\) 0.633975 + 1.09808i 0.0675819 + 0.117055i
\(89\) 8.19615 − 4.73205i 0.868790 − 0.501596i 0.00184433 − 0.999998i \(-0.499413\pi\)
0.866946 + 0.498402i \(0.166080\pi\)
\(90\) −3.73205 −0.393393
\(91\) 0 0
\(92\) 4.19615 0.437479
\(93\) 1.26795 − 0.732051i 0.131480 − 0.0759101i
\(94\) −1.09808 − 1.90192i −0.113258 − 0.196168i
\(95\) 8.83013 − 15.2942i 0.905952 − 1.56915i
\(96\) 1.00000i 0.102062i
\(97\) 5.19615 + 3.00000i 0.527589 + 0.304604i 0.740034 − 0.672569i \(-0.234809\pi\)
−0.212445 + 0.977173i \(0.568143\pi\)
\(98\) 0.401924 + 0.232051i 0.0406004 + 0.0234407i
\(99\) − 1.26795i − 0.127434i
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.i.a.361.2 4
13.2 odd 12 1014.2.a.k.1.2 2
13.3 even 3 1014.2.b.e.337.4 4
13.4 even 6 inner 1014.2.i.a.823.2 4
13.5 odd 4 1014.2.e.g.991.2 4
13.6 odd 12 1014.2.e.g.529.2 4
13.7 odd 12 1014.2.e.i.529.1 4
13.8 odd 4 1014.2.e.i.991.1 4
13.9 even 3 78.2.i.a.43.1 ✓ 4
13.10 even 6 1014.2.b.e.337.1 4
13.11 odd 12 1014.2.a.i.1.1 2
13.12 even 2 78.2.i.a.49.1 yes 4
39.2 even 12 3042.2.a.p.1.1 2
39.11 even 12 3042.2.a.y.1.2 2
39.23 odd 6 3042.2.b.i.1351.4 4
39.29 odd 6 3042.2.b.i.1351.1 4
39.35 odd 6 234.2.l.c.199.2 4
39.38 odd 2 234.2.l.c.127.2 4
52.11 even 12 8112.2.a.bj.1.1 2
52.15 even 12 8112.2.a.bp.1.2 2
52.35 odd 6 624.2.bv.e.433.2 4
52.51 odd 2 624.2.bv.e.49.1 4
65.9 even 6 1950.2.bc.d.901.2 4
65.12 odd 4 1950.2.y.b.49.1 4
65.22 odd 12 1950.2.y.g.199.2 4
65.38 odd 4 1950.2.y.g.49.2 4
65.48 odd 12 1950.2.y.b.199.1 4
65.64 even 2 1950.2.bc.d.751.2 4
156.35 even 6 1872.2.by.h.433.1 4
156.155 even 2 1872.2.by.h.1297.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.i.a.43.1 ✓ 4 13.9 even 3
78.2.i.a.49.1 yes 4 13.12 even 2
234.2.l.c.127.2 4 39.38 odd 2
234.2.l.c.199.2 4 39.35 odd 6
624.2.bv.e.49.1 4 52.51 odd 2
624.2.bv.e.433.2 4 52.35 odd 6
1014.2.a.i.1.1 2 13.11 odd 12
1014.2.a.k.1.2 2 13.2 odd 12
1014.2.b.e.337.1 4 13.10 even 6
1014.2.b.e.337.4 4 13.3 even 3
1014.2.e.g.529.2 4 13.6 odd 12
1014.2.e.g.991.2 4 13.5 odd 4
1014.2.e.i.529.1 4 13.7 odd 12
1014.2.e.i.991.1 4 13.8 odd 4
1014.2.i.a.361.2 4 1.1 even 1 trivial
1014.2.i.a.823.2 4 13.4 even 6 inner
1872.2.by.h.433.1 4 156.35 even 6
1872.2.by.h.1297.2 4 156.155 even 2
1950.2.y.b.49.1 4 65.12 odd 4
1950.2.y.b.199.1 4 65.48 odd 12
1950.2.y.g.49.2 4 65.38 odd 4
1950.2.y.g.199.2 4 65.22 odd 12
1950.2.bc.d.751.2 4 65.64 even 2
1950.2.bc.d.901.2 4 65.9 even 6
3042.2.a.p.1.1 2 39.2 even 12
3042.2.a.y.1.2 2 39.11 even 12
3042.2.b.i.1351.1 4 39.29 odd 6
3042.2.b.i.1351.4 4 39.23 odd 6
8112.2.a.bj.1.1 2 52.11 even 12
8112.2.a.bp.1.2 2 52.15 even 12