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(379,1134)] 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("1134.379"); 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, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1134.f (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,2,0,-2,-4,0,2,-4,0,-8,-2,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.05503558921\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 379.1
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1134.379
Dual form 1134.2.f.s.757.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.86603 - 3.23205i) q^{5} +(0.500000 - 0.866025i) q^{7} -1.00000 q^{8} -3.73205 q^{10} +(2.09808 - 3.63397i) q^{11} +(-0.232051 - 0.401924i) q^{13} +(-0.500000 - 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} +7.00000 q^{17} -2.73205 q^{19} +(-1.86603 + 3.23205i) q^{20} +(-2.09808 - 3.63397i) q^{22} +(-3.09808 - 5.36603i) q^{23} +(-4.46410 + 7.73205i) q^{25} -0.464102 q^{26} -1.00000 q^{28} +(-4.23205 + 7.33013i) q^{29} +(1.09808 + 1.90192i) q^{31} +(0.500000 + 0.866025i) q^{32} +(3.50000 - 6.06218i) q^{34} -3.73205 q^{35} -6.66025 q^{37} +(-1.36603 + 2.36603i) q^{38} +(1.86603 + 3.23205i) q^{40} +(-4.73205 - 8.19615i) q^{41} +(-2.73205 + 4.73205i) q^{43} -4.19615 q^{44} -6.19615 q^{46} +(-0.633975 + 1.09808i) q^{47} +(-0.500000 - 0.866025i) q^{49} +(4.46410 + 7.73205i) q^{50} +(-0.232051 + 0.401924i) q^{52} -2.53590 q^{53} -15.6603 q^{55} +(-0.500000 + 0.866025i) q^{56} +(4.23205 + 7.33013i) q^{58} +(3.09808 + 5.36603i) q^{59} +(4.96410 - 8.59808i) q^{61} +2.19615 q^{62} +1.00000 q^{64} +(-0.866025 + 1.50000i) q^{65} +(1.63397 + 2.83013i) q^{67} +(-3.50000 - 6.06218i) q^{68} +(-1.86603 + 3.23205i) q^{70} +13.4641 q^{71} +11.7321 q^{73} +(-3.33013 + 5.76795i) q^{74} +(1.36603 + 2.36603i) q^{76} +(-2.09808 - 3.63397i) q^{77} +(-7.56218 + 13.0981i) q^{79} +3.73205 q^{80} -9.46410 q^{82} +(7.29423 - 12.6340i) q^{83} +(-13.0622 - 22.6244i) q^{85} +(2.73205 + 4.73205i) q^{86} +(-2.09808 + 3.63397i) q^{88} -3.92820 q^{89} -0.464102 q^{91} +(-3.09808 + 5.36603i) q^{92} +(0.633975 + 1.09808i) q^{94} +(5.09808 + 8.83013i) q^{95} +(1.46410 - 2.53590i) q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{4} - 4 q^{5} + 2 q^{7} - 4 q^{8} - 8 q^{10} - 2 q^{11} + 6 q^{13} - 2 q^{14} - 2 q^{16} + 28 q^{17} - 4 q^{19} - 4 q^{20} + 2 q^{22} - 2 q^{23} - 4 q^{25} + 12 q^{26} - 4 q^{28} - 10 q^{29}+ \cdots - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(407\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\)

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 0
\(4\) −0.500000 − 0.866025i −0.250000 − 0.433013i
\(5\) −1.86603 − 3.23205i −0.834512 − 1.44542i −0.894427 − 0.447214i \(-0.852416\pi\)
0.0599153 − 0.998203i \(-0.480917\pi\)
\(6\) 0 0
\(7\) 0.500000 − 0.866025i 0.188982 − 0.327327i
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −3.73205 −1.18018
\(11\) 2.09808 − 3.63397i 0.632594 − 1.09568i −0.354426 − 0.935084i \(-0.615324\pi\)
0.987020 − 0.160600i \(-0.0513430\pi\)
\(12\) 0 0
\(13\) −0.232051 − 0.401924i −0.0643593 − 0.111474i 0.832050 − 0.554700i \(-0.187167\pi\)
−0.896410 + 0.443227i \(0.853834\pi\)
\(14\) −0.500000 − 0.866025i −0.133631 − 0.231455i
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) 7.00000 1.69775 0.848875 − 0.528594i \(-0.177281\pi\)
0.848875 + 0.528594i \(0.177281\pi\)
\(18\) 0 0
\(19\) −2.73205 −0.626775 −0.313388 − 0.949625i \(-0.601464\pi\)
−0.313388 + 0.949625i \(0.601464\pi\)
\(20\) −1.86603 + 3.23205i −0.417256 + 0.722709i
\(21\) 0 0
\(22\) −2.09808 − 3.63397i −0.447311 − 0.774766i
\(23\) −3.09808 − 5.36603i −0.645994 − 1.11889i −0.984071 − 0.177775i \(-0.943110\pi\)
0.338078 − 0.941118i \(-0.390223\pi\)
\(24\) 0 0
\(25\) −4.46410 + 7.73205i −0.892820 + 1.54641i
\(26\) −0.464102 −0.0910178
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −4.23205 + 7.33013i −0.785872 + 1.36117i 0.142605 + 0.989780i \(0.454452\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) 1.09808 + 1.90192i 0.197220 + 0.341596i 0.947626 − 0.319382i \(-0.103475\pi\)
−0.750406 + 0.660977i \(0.770142\pi\)
\(32\) 0.500000 + 0.866025i 0.0883883 + 0.153093i
\(33\) 0 0
\(34\) 3.50000 − 6.06218i 0.600245 − 1.03965i
\(35\) −3.73205 −0.630832
\(36\) 0 0
\(37\) −6.66025 −1.09494 −0.547470 − 0.836826i \(-0.684409\pi\)
−0.547470 + 0.836826i \(0.684409\pi\)
\(38\) −1.36603 + 2.36603i −0.221599 + 0.383820i
\(39\) 0 0
\(40\) 1.86603 + 3.23205i 0.295045 + 0.511032i
\(41\) −4.73205 − 8.19615i −0.739022 − 1.28002i −0.952936 − 0.303171i \(-0.901955\pi\)
0.213914 − 0.976853i \(-0.431379\pi\)
\(42\) 0 0
\(43\) −2.73205 + 4.73205i −0.416634 + 0.721631i −0.995598 − 0.0937217i \(-0.970124\pi\)
0.578965 + 0.815353i \(0.303457\pi\)
\(44\) −4.19615 −0.632594
\(45\) 0 0
\(46\) −6.19615 −0.913573
\(47\) −0.633975 + 1.09808i −0.0924747 + 0.160171i −0.908552 − 0.417772i \(-0.862811\pi\)
0.816077 + 0.577943i \(0.196144\pi\)
\(48\) 0 0
\(49\) −0.500000 − 0.866025i −0.0714286 − 0.123718i
\(50\) 4.46410 + 7.73205i 0.631319 + 1.09348i
\(51\) 0 0
\(52\) −0.232051 + 0.401924i −0.0321797 + 0.0557368i
\(53\) −2.53590 −0.348332 −0.174166 − 0.984716i \(-0.555723\pi\)
−0.174166 + 0.984716i \(0.555723\pi\)
\(54\) 0 0
\(55\) −15.6603 −2.11163
\(56\) −0.500000 + 0.866025i −0.0668153 + 0.115728i
\(57\) 0 0
\(58\) 4.23205 + 7.33013i 0.555695 + 0.962493i
\(59\) 3.09808 + 5.36603i 0.403335 + 0.698597i 0.994126 − 0.108228i \(-0.0345175\pi\)
−0.590791 + 0.806825i \(0.701184\pi\)
\(60\) 0 0
\(61\) 4.96410 − 8.59808i 0.635588 − 1.10087i −0.350802 − 0.936450i \(-0.614091\pi\)
0.986390 − 0.164421i \(-0.0525756\pi\)
\(62\) 2.19615 0.278912
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −0.866025 + 1.50000i −0.107417 + 0.186052i
\(66\) 0 0
\(67\) 1.63397 + 2.83013i 0.199622 + 0.345755i 0.948406 − 0.317059i \(-0.102695\pi\)
−0.748784 + 0.662814i \(0.769362\pi\)
\(68\) −3.50000 − 6.06218i −0.424437 − 0.735147i
\(69\) 0 0
\(70\) −1.86603 + 3.23205i −0.223033 + 0.386304i
\(71\) 13.4641 1.59789 0.798947 − 0.601401i \(-0.205391\pi\)
0.798947 + 0.601401i \(0.205391\pi\)
\(72\) 0 0
\(73\) 11.7321 1.37313 0.686566 − 0.727067i \(-0.259117\pi\)
0.686566 + 0.727067i \(0.259117\pi\)
\(74\) −3.33013 + 5.76795i −0.387119 + 0.670510i
\(75\) 0 0
\(76\) 1.36603 + 2.36603i 0.156694 + 0.271402i
\(77\) −2.09808 − 3.63397i −0.239098 − 0.414130i
\(78\) 0 0
\(79\) −7.56218 + 13.0981i −0.850811 + 1.47365i 0.0296655 + 0.999560i \(0.490556\pi\)
−0.880477 + 0.474089i \(0.842778\pi\)
\(80\) 3.73205 0.417256
\(81\) 0 0
\(82\) −9.46410 −1.04514
\(83\) 7.29423 − 12.6340i 0.800646 − 1.38676i −0.118546 − 0.992949i \(-0.537823\pi\)
0.919192 − 0.393810i \(-0.128843\pi\)
\(84\) 0 0
\(85\) −13.0622 − 22.6244i −1.41679 − 2.45396i
\(86\) 2.73205 + 4.73205i 0.294605 + 0.510270i
\(87\) 0 0
\(88\) −2.09808 + 3.63397i −0.223656 + 0.387383i
\(89\) −3.92820 −0.416389 −0.208194 − 0.978087i \(-0.566759\pi\)
−0.208194 + 0.978087i \(0.566759\pi\)
\(90\) 0 0
\(91\) −0.464102 −0.0486511
\(92\) −3.09808 + 5.36603i −0.322997 + 0.559447i
\(93\) 0 0
\(94\) 0.633975 + 1.09808i 0.0653895 + 0.113258i
\(95\) 5.09808 + 8.83013i 0.523052 + 0.905952i
\(96\) 0 0
\(97\) 1.46410 − 2.53590i 0.148657 − 0.257481i −0.782074 − 0.623185i \(-0.785838\pi\)
0.930731 + 0.365704i \(0.119172\pi\)
\(98\) −1.00000 −0.101015
\(99\) 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.f.s.379.1 4
3.2 odd 2 1134.2.f.r.379.2 4
9.2 odd 6 1134.2.a.m.1.1 yes 2
9.4 even 3 inner 1134.2.f.s.757.1 4
9.5 odd 6 1134.2.f.r.757.2 4
9.7 even 3 1134.2.a.l.1.2 ✓ 2
36.7 odd 6 9072.2.a.bp.1.2 2
36.11 even 6 9072.2.a.y.1.1 2
63.20 even 6 7938.2.a.bt.1.2 2
63.34 odd 6 7938.2.a.bg.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1134.2.a.l.1.2 ✓ 2 9.7 even 3
1134.2.a.m.1.1 yes 2 9.2 odd 6
1134.2.f.r.379.2 4 3.2 odd 2
1134.2.f.r.757.2 4 9.5 odd 6
1134.2.f.s.379.1 4 1.1 even 1 trivial
1134.2.f.s.757.1 4 9.4 even 3 inner
7938.2.a.bg.1.1 2 63.34 odd 6
7938.2.a.bt.1.2 2 63.20 even 6
9072.2.a.y.1.1 2 36.11 even 6
9072.2.a.bp.1.2 2 36.7 odd 6