Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1134,2,Mod(865,1134)] 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("1134.865"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1134, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1134.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,0,2,0,0,-1] 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: \(9.05503558921\)
Analytic rank: \(1\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 378)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 865.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1134.865
Dual form 1134.2.e.c.919.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +(-0.500000 - 2.59808i) q^{7} -1.00000 q^{8} +(-3.00000 - 5.19615i) q^{11} +(-2.50000 - 4.33013i) q^{13} +(0.500000 + 2.59808i) q^{14} +1.00000 q^{16} +(-3.00000 + 5.19615i) q^{17} +(2.00000 + 3.46410i) q^{19} +(3.00000 + 5.19615i) q^{22} +(-3.00000 + 5.19615i) q^{23} +(2.50000 + 4.33013i) q^{25} +(2.50000 + 4.33013i) q^{26} +(-0.500000 - 2.59808i) q^{28} +(-3.00000 + 5.19615i) q^{29} -1.00000 q^{31} -1.00000 q^{32} +(3.00000 - 5.19615i) q^{34} +(0.500000 + 0.866025i) q^{37} +(-2.00000 - 3.46410i) q^{38} +(3.00000 + 5.19615i) q^{41} +(0.500000 - 0.866025i) q^{43} +(-3.00000 - 5.19615i) q^{44} +(3.00000 - 5.19615i) q^{46} -6.00000 q^{47} +(-6.50000 + 2.59808i) q^{49} +(-2.50000 - 4.33013i) q^{50} +(-2.50000 - 4.33013i) q^{52} +(3.00000 - 5.19615i) q^{53} +(0.500000 + 2.59808i) q^{56} +(3.00000 - 5.19615i) q^{58} -6.00000 q^{59} -1.00000 q^{61} +1.00000 q^{62} +1.00000 q^{64} -1.00000 q^{67} +(-3.00000 + 5.19615i) q^{68} +12.0000 q^{71} +(-1.00000 + 1.73205i) q^{73} +(-0.500000 - 0.866025i) q^{74} +(2.00000 + 3.46410i) q^{76} +(-12.0000 + 10.3923i) q^{77} -1.00000 q^{79} +(-3.00000 - 5.19615i) q^{82} +(-3.00000 + 5.19615i) q^{83} +(-0.500000 + 0.866025i) q^{86} +(3.00000 + 5.19615i) q^{88} +(-10.0000 + 8.66025i) q^{91} +(-3.00000 + 5.19615i) q^{92} +6.00000 q^{94} +(-8.50000 + 14.7224i) q^{97} +(6.50000 - 2.59808i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - q^{7} - 2 q^{8} - 6 q^{11} - 5 q^{13} + q^{14} + 2 q^{16} - 6 q^{17} + 4 q^{19} + 6 q^{22} - 6 q^{23} + 5 q^{25} + 5 q^{26} - q^{28} - 6 q^{29} - 2 q^{31} - 2 q^{32} + 6 q^{34}+ \cdots + 13 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 0 0
\(7\) −0.500000 − 2.59808i −0.188982 − 0.981981i
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −3.00000 − 5.19615i −0.904534 − 1.56670i −0.821541 − 0.570149i \(-0.806886\pi\)
−0.0829925 − 0.996550i \(-0.526448\pi\)
\(12\) 0 0
\(13\) −2.50000 − 4.33013i −0.693375 − 1.20096i −0.970725 − 0.240192i \(-0.922790\pi\)
0.277350 − 0.960769i \(-0.410544\pi\)
\(14\) 0.500000 + 2.59808i 0.133631 + 0.694365i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.00000 + 5.19615i −0.727607 + 1.26025i 0.230285 + 0.973123i \(0.426034\pi\)
−0.957892 + 0.287129i \(0.907299\pi\)
\(18\) 0 0
\(19\) 2.00000 + 3.46410i 0.458831 + 0.794719i 0.998899 − 0.0469020i \(-0.0149348\pi\)
−0.540068 + 0.841621i \(0.681602\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.00000 + 5.19615i 0.639602 + 1.10782i
\(23\) −3.00000 + 5.19615i −0.625543 + 1.08347i 0.362892 + 0.931831i \(0.381789\pi\)
−0.988436 + 0.151642i \(0.951544\pi\)
\(24\) 0 0
\(25\) 2.50000 + 4.33013i 0.500000 + 0.866025i
\(26\) 2.50000 + 4.33013i 0.490290 + 0.849208i
\(27\) 0 0
\(28\) −0.500000 − 2.59808i −0.0944911 − 0.490990i
\(29\) −3.00000 + 5.19615i −0.557086 + 0.964901i 0.440652 + 0.897678i \(0.354747\pi\)
−0.997738 + 0.0672232i \(0.978586\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605 −0.0898027 − 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 3.00000 − 5.19615i 0.514496 − 0.891133i
\(35\) 0 0
\(36\) 0 0
\(37\) 0.500000 + 0.866025i 0.0821995 + 0.142374i 0.904194 − 0.427121i \(-0.140472\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −2.00000 − 3.46410i −0.324443 − 0.561951i
\(39\) 0 0
\(40\) 0 0
\(41\) 3.00000 + 5.19615i 0.468521 + 0.811503i 0.999353 − 0.0359748i \(-0.0114536\pi\)
−0.530831 + 0.847477i \(0.678120\pi\)
\(42\) 0 0
\(43\) 0.500000 − 0.866025i 0.0762493 − 0.132068i −0.825380 − 0.564578i \(-0.809039\pi\)
0.901629 + 0.432511i \(0.142372\pi\)
\(44\) −3.00000 − 5.19615i −0.452267 − 0.783349i
\(45\) 0 0
\(46\) 3.00000 − 5.19615i 0.442326 − 0.766131i
\(47\) −6.00000 −0.875190 −0.437595 − 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) −6.50000 + 2.59808i −0.928571 + 0.371154i
\(50\) −2.50000 − 4.33013i −0.353553 − 0.612372i
\(51\) 0 0
\(52\) −2.50000 − 4.33013i −0.346688 − 0.600481i
\(53\) 3.00000 − 5.19615i 0.412082 − 0.713746i −0.583036 − 0.812447i \(-0.698135\pi\)
0.995117 + 0.0987002i \(0.0314685\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0.500000 + 2.59808i 0.0668153 + 0.347183i
\(57\) 0 0
\(58\) 3.00000 − 5.19615i 0.393919 − 0.682288i
\(59\) −6.00000 −0.781133 −0.390567 − 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 − 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 1.00000 0.127000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −1.00000 −0.122169 −0.0610847 − 0.998133i \(-0.519456\pi\)
−0.0610847 + 0.998133i \(0.519456\pi\)
\(68\) −3.00000 + 5.19615i −0.363803 + 0.630126i
\(69\) 0 0
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) −1.00000 + 1.73205i −0.117041 + 0.202721i −0.918594 − 0.395203i \(-0.870674\pi\)
0.801553 + 0.597924i \(0.204008\pi\)
\(74\) −0.500000 − 0.866025i −0.0581238 − 0.100673i
\(75\) 0 0
\(76\) 2.00000 + 3.46410i 0.229416 + 0.397360i
\(77\) −12.0000 + 10.3923i −1.36753 + 1.18431i
\(78\) 0 0
\(79\) −1.00000 −0.112509 −0.0562544 − 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −3.00000 − 5.19615i −0.331295 − 0.573819i
\(83\) −3.00000 + 5.19615i −0.329293 + 0.570352i −0.982372 − 0.186938i \(-0.940144\pi\)
0.653079 + 0.757290i \(0.273477\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −0.500000 + 0.866025i −0.0539164 + 0.0933859i
\(87\) 0 0
\(88\) 3.00000 + 5.19615i 0.319801 + 0.553912i
\(89\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 0 0
\(91\) −10.0000 + 8.66025i −1.04828 + 0.907841i
\(92\) −3.00000 + 5.19615i −0.312772 + 0.541736i
\(93\) 0 0
\(94\) 6.00000 0.618853
\(95\) 0 0
\(96\) 0 0
\(97\) −8.50000 + 14.7224i −0.863044 + 1.49484i 0.00593185 + 0.999982i \(0.498112\pi\)
−0.868976 + 0.494854i \(0.835222\pi\)
\(98\) 6.50000 − 2.59808i 0.656599 − 0.262445i
\(99\) 0 0
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 1134.2.e.c.865.1 2
3.2 odd 2 1134.2.e.m.865.1 2
7.2 even 3 1134.2.h.n.541.1 2
9.2 odd 6 378.2.g.b.109.1 ✓ 2
9.4 even 3 1134.2.h.n.109.1 2
9.5 odd 6 1134.2.h.d.109.1 2
9.7 even 3 378.2.g.e.109.1 yes 2
21.2 odd 6 1134.2.h.d.541.1 2
63.2 odd 6 378.2.g.b.163.1 yes 2
63.11 odd 6 2646.2.a.x.1.1 1
63.16 even 3 378.2.g.e.163.1 yes 2
63.23 odd 6 1134.2.e.m.919.1 2
63.25 even 3 2646.2.a.h.1.1 1
63.38 even 6 2646.2.a.w.1.1 1
63.52 odd 6 2646.2.a.g.1.1 1
63.58 even 3 inner 1134.2.e.c.919.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.2.g.b.109.1 ✓ 2 9.2 odd 6
378.2.g.b.163.1 yes 2 63.2 odd 6
378.2.g.e.109.1 yes 2 9.7 even 3
378.2.g.e.163.1 yes 2 63.16 even 3
1134.2.e.c.865.1 2 1.1 even 1 trivial
1134.2.e.c.919.1 2 63.58 even 3 inner
1134.2.e.m.865.1 2 3.2 odd 2
1134.2.e.m.919.1 2 63.23 odd 6
1134.2.h.d.109.1 2 9.5 odd 6
1134.2.h.d.541.1 2 21.2 odd 6
1134.2.h.n.109.1 2 9.4 even 3
1134.2.h.n.541.1 2 7.2 even 3
2646.2.a.g.1.1 1 63.52 odd 6
2646.2.a.h.1.1 1 63.25 even 3
2646.2.a.w.1.1 1 63.38 even 6
2646.2.a.x.1.1 1 63.11 odd 6