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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [513,2,Mod(235,513)] 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("513.235"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(513, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 513 = 3^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 513.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,2,0,-2,-1] 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: \(4.09632562369\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 171)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 334.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 513.334
Dual form 513.2.h.a.235.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{4} +(-0.500000 - 0.866025i) q^{5} +(-1.50000 - 2.59808i) q^{7} -3.00000 q^{8} +(-0.500000 - 0.866025i) q^{10} +(1.50000 + 2.59808i) q^{11} -6.00000 q^{13} +(-1.50000 - 2.59808i) q^{14} -1.00000 q^{16} +(1.50000 - 2.59808i) q^{17} +(-4.00000 - 1.73205i) q^{19} +(0.500000 + 0.866025i) q^{20} +(1.50000 + 2.59808i) q^{22} -8.00000 q^{23} +(2.00000 - 3.46410i) q^{25} -6.00000 q^{26} +(1.50000 + 2.59808i) q^{28} +(-2.50000 + 4.33013i) q^{29} +(3.50000 - 6.06218i) q^{31} +5.00000 q^{32} +(1.50000 - 2.59808i) q^{34} +(-1.50000 + 2.59808i) q^{35} +2.00000 q^{37} +(-4.00000 - 1.73205i) q^{38} +(1.50000 + 2.59808i) q^{40} +(-0.500000 - 0.866025i) q^{41} +8.00000 q^{43} +(-1.50000 - 2.59808i) q^{44} -8.00000 q^{46} +(4.50000 - 7.79423i) q^{47} +(-1.00000 + 1.73205i) q^{49} +(2.00000 - 3.46410i) q^{50} +6.00000 q^{52} +(1.50000 + 2.59808i) q^{53} +(1.50000 - 2.59808i) q^{55} +(4.50000 + 7.79423i) q^{56} +(-2.50000 + 4.33013i) q^{58} +(1.50000 + 2.59808i) q^{59} +(-3.50000 + 6.06218i) q^{61} +(3.50000 - 6.06218i) q^{62} +7.00000 q^{64} +(3.00000 + 5.19615i) q^{65} -4.00000 q^{67} +(-1.50000 + 2.59808i) q^{68} +(-1.50000 + 2.59808i) q^{70} +(-7.50000 + 12.9904i) q^{71} +(2.50000 - 4.33013i) q^{73} +2.00000 q^{74} +(4.00000 + 1.73205i) q^{76} +(4.50000 - 7.79423i) q^{77} -12.0000 q^{79} +(0.500000 + 0.866025i) q^{80} +(-0.500000 - 0.866025i) q^{82} +(-0.500000 - 0.866025i) q^{83} -3.00000 q^{85} +8.00000 q^{86} +(-4.50000 - 7.79423i) q^{88} +(-0.500000 - 0.866025i) q^{89} +(9.00000 + 15.5885i) q^{91} +8.00000 q^{92} +(4.50000 - 7.79423i) q^{94} +(0.500000 + 4.33013i) q^{95} -2.00000 q^{97} +(-1.00000 + 1.73205i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{4} - q^{5} - 3 q^{7} - 6 q^{8} - q^{10} + 3 q^{11} - 12 q^{13} - 3 q^{14} - 2 q^{16} + 3 q^{17} - 8 q^{19} + q^{20} + 3 q^{22} - 16 q^{23} + 4 q^{25} - 12 q^{26} + 3 q^{28} - 5 q^{29}+ \cdots - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/513\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(325\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\)

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 0.353553 − 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) −0.500000 − 0.866025i −0.223607 − 0.387298i 0.732294 − 0.680989i \(-0.238450\pi\)
−0.955901 + 0.293691i \(0.905116\pi\)
\(6\) 0 0
\(7\) −1.50000 − 2.59808i −0.566947 − 0.981981i −0.996866 − 0.0791130i \(-0.974791\pi\)
0.429919 − 0.902867i \(-0.358542\pi\)
\(8\) −3.00000 −1.06066
\(9\) 0 0
\(10\) −0.500000 − 0.866025i −0.158114 − 0.273861i
\(11\) 1.50000 + 2.59808i 0.452267 + 0.783349i 0.998526 − 0.0542666i \(-0.0172821\pi\)
−0.546259 + 0.837616i \(0.683949\pi\)
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 − 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) −1.50000 − 2.59808i −0.400892 − 0.694365i
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 1.50000 − 2.59808i 0.363803 − 0.630126i −0.624780 − 0.780801i \(-0.714811\pi\)
0.988583 + 0.150675i \(0.0481447\pi\)
\(18\) 0 0
\(19\) −4.00000 − 1.73205i −0.917663 − 0.397360i
\(20\) 0.500000 + 0.866025i 0.111803 + 0.193649i
\(21\) 0 0
\(22\) 1.50000 + 2.59808i 0.319801 + 0.553912i
\(23\) −8.00000 −1.66812 −0.834058 − 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) 2.00000 − 3.46410i 0.400000 − 0.692820i
\(26\) −6.00000 −1.17670
\(27\) 0 0
\(28\) 1.50000 + 2.59808i 0.283473 + 0.490990i
\(29\) −2.50000 + 4.33013i −0.464238 + 0.804084i −0.999167 − 0.0408130i \(-0.987005\pi\)
0.534928 + 0.844897i \(0.320339\pi\)
\(30\) 0 0
\(31\) 3.50000 − 6.06218i 0.628619 − 1.08880i −0.359211 − 0.933257i \(-0.616954\pi\)
0.987829 − 0.155543i \(-0.0497126\pi\)
\(32\) 5.00000 0.883883
\(33\) 0 0
\(34\) 1.50000 − 2.59808i 0.257248 − 0.445566i
\(35\) −1.50000 + 2.59808i −0.253546 + 0.439155i
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −4.00000 − 1.73205i −0.648886 − 0.280976i
\(39\) 0 0
\(40\) 1.50000 + 2.59808i 0.237171 + 0.410792i
\(41\) −0.500000 − 0.866025i −0.0780869 − 0.135250i 0.824338 − 0.566099i \(-0.191548\pi\)
−0.902424 + 0.430848i \(0.858214\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) −1.50000 − 2.59808i −0.226134 − 0.391675i
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) 4.50000 − 7.79423i 0.656392 − 1.13691i −0.325150 − 0.945662i \(-0.605415\pi\)
0.981543 − 0.191243i \(-0.0612518\pi\)
\(48\) 0 0
\(49\) −1.00000 + 1.73205i −0.142857 + 0.247436i
\(50\) 2.00000 − 3.46410i 0.282843 − 0.489898i
\(51\) 0 0
\(52\) 6.00000 0.832050
\(53\) 1.50000 + 2.59808i 0.206041 + 0.356873i 0.950464 − 0.310835i \(-0.100609\pi\)
−0.744423 + 0.667708i \(0.767275\pi\)
\(54\) 0 0
\(55\) 1.50000 − 2.59808i 0.202260 − 0.350325i
\(56\) 4.50000 + 7.79423i 0.601338 + 1.04155i
\(57\) 0 0
\(58\) −2.50000 + 4.33013i −0.328266 + 0.568574i
\(59\) 1.50000 + 2.59808i 0.195283 + 0.338241i 0.946993 − 0.321253i \(-0.104104\pi\)
−0.751710 + 0.659494i \(0.770771\pi\)
\(60\) 0 0
\(61\) −3.50000 + 6.06218i −0.448129 + 0.776182i −0.998264 − 0.0588933i \(-0.981243\pi\)
0.550135 + 0.835076i \(0.314576\pi\)
\(62\) 3.50000 − 6.06218i 0.444500 − 0.769897i
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 3.00000 + 5.19615i 0.372104 + 0.644503i
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −1.50000 + 2.59808i −0.181902 + 0.315063i
\(69\) 0 0
\(70\) −1.50000 + 2.59808i −0.179284 + 0.310530i
\(71\) −7.50000 + 12.9904i −0.890086 + 1.54167i −0.0503155 + 0.998733i \(0.516023\pi\)
−0.839771 + 0.542941i \(0.817311\pi\)
\(72\) 0 0
\(73\) 2.50000 − 4.33013i 0.292603 − 0.506803i −0.681822 − 0.731519i \(-0.738812\pi\)
0.974424 + 0.224716i \(0.0721453\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) 4.00000 + 1.73205i 0.458831 + 0.198680i
\(77\) 4.50000 − 7.79423i 0.512823 − 0.888235i
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 − 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 0.500000 + 0.866025i 0.0559017 + 0.0968246i
\(81\) 0 0
\(82\) −0.500000 − 0.866025i −0.0552158 − 0.0956365i
\(83\) −0.500000 − 0.866025i −0.0548821 − 0.0950586i 0.837279 − 0.546776i \(-0.184145\pi\)
−0.892161 + 0.451717i \(0.850812\pi\)
\(84\) 0 0
\(85\) −3.00000 −0.325396
\(86\) 8.00000 0.862662
\(87\) 0 0
\(88\) −4.50000 − 7.79423i −0.479702 − 0.830868i
\(89\) −0.500000 − 0.866025i −0.0529999 − 0.0917985i 0.838308 − 0.545197i \(-0.183545\pi\)
−0.891308 + 0.453398i \(0.850212\pi\)
\(90\) 0 0
\(91\) 9.00000 + 15.5885i 0.943456 + 1.63411i
\(92\) 8.00000 0.834058
\(93\) 0 0
\(94\) 4.50000 − 7.79423i 0.464140 − 0.803913i
\(95\) 0.500000 + 4.33013i 0.0512989 + 0.444262i
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −1.00000 + 1.73205i −0.101015 + 0.174964i
\(99\) 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 513.2.h.a.334.1 2
3.2 odd 2 171.2.h.b.49.1 yes 2
9.2 odd 6 171.2.g.b.106.1 ✓ 2
9.7 even 3 513.2.g.b.505.1 2
19.7 even 3 513.2.g.b.64.1 2
57.26 odd 6 171.2.g.b.121.1 yes 2
171.7 even 3 inner 513.2.h.a.235.1 2
171.83 odd 6 171.2.h.b.7.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.g.b.106.1 ✓ 2 9.2 odd 6
171.2.g.b.121.1 yes 2 57.26 odd 6
171.2.h.b.7.1 yes 2 171.83 odd 6
171.2.h.b.49.1 yes 2 3.2 odd 2
513.2.g.b.64.1 2 19.7 even 3
513.2.g.b.505.1 2 9.7 even 3
513.2.h.a.235.1 2 171.7 even 3 inner
513.2.h.a.334.1 2 1.1 even 1 trivial