Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [513,2,Mod(64,513)] 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("513.64"); 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([2, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 513 = 3^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 513.g (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-1,0,1,2] 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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 171)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 505.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 513.505
Dual form 513.2.g.b.64.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{2} +(0.500000 - 0.866025i) q^{4} +1.00000 q^{5} +(-1.50000 + 2.59808i) q^{7} -3.00000 q^{8} +(-0.500000 - 0.866025i) q^{10} +(1.50000 - 2.59808i) q^{11} +(3.00000 - 5.19615i) q^{13} +3.00000 q^{14} +(0.500000 + 0.866025i) q^{16} +(1.50000 - 2.59808i) q^{17} +(-4.00000 - 1.73205i) q^{19} +(0.500000 - 0.866025i) q^{20} -3.00000 q^{22} +(4.00000 - 6.92820i) q^{23} -4.00000 q^{25} -6.00000 q^{26} +(1.50000 + 2.59808i) q^{28} +5.00000 q^{29} +(3.50000 + 6.06218i) q^{31} +(-2.50000 + 4.33013i) q^{32} -3.00000 q^{34} +(-1.50000 + 2.59808i) q^{35} +2.00000 q^{37} +(0.500000 + 4.33013i) q^{38} -3.00000 q^{40} +1.00000 q^{41} +(-4.00000 - 6.92820i) q^{43} +(-1.50000 - 2.59808i) q^{44} -8.00000 q^{46} -9.00000 q^{47} +(-1.00000 - 1.73205i) q^{49} +(2.00000 + 3.46410i) q^{50} +(-3.00000 - 5.19615i) q^{52} +(1.50000 + 2.59808i) q^{53} +(1.50000 - 2.59808i) q^{55} +(4.50000 - 7.79423i) q^{56} +(-2.50000 - 4.33013i) q^{58} -3.00000 q^{59} +7.00000 q^{61} +(3.50000 - 6.06218i) q^{62} +7.00000 q^{64} +(3.00000 - 5.19615i) q^{65} +(2.00000 - 3.46410i) q^{67} +(-1.50000 - 2.59808i) q^{68} +3.00000 q^{70} +(-7.50000 + 12.9904i) q^{71} +(2.50000 - 4.33013i) q^{73} +(-1.00000 - 1.73205i) q^{74} +(-3.50000 + 2.59808i) q^{76} +(4.50000 + 7.79423i) q^{77} +(6.00000 + 10.3923i) q^{79} +(0.500000 + 0.866025i) q^{80} +(-0.500000 - 0.866025i) q^{82} +(-0.500000 + 0.866025i) q^{83} +(1.50000 - 2.59808i) q^{85} +(-4.00000 + 6.92820i) q^{86} +(-4.50000 + 7.79423i) q^{88} +(-0.500000 - 0.866025i) q^{89} +(9.00000 + 15.5885i) q^{91} +(-4.00000 - 6.92820i) q^{92} +(4.50000 + 7.79423i) q^{94} +(-4.00000 - 1.73205i) q^{95} +(1.00000 + 1.73205i) q^{97} +(-1.00000 + 1.73205i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + q^{4} + 2 q^{5} - 3 q^{7} - 6 q^{8} - q^{10} + 3 q^{11} + 6 q^{13} + 6 q^{14} + q^{16} + 3 q^{17} - 8 q^{19} + q^{20} - 6 q^{22} + 8 q^{23} - 8 q^{25} - 12 q^{26} + 3 q^{28} + 10 q^{29}+ \cdots - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 − 0.866025i −0.353553 − 0.612372i 0.633316 − 0.773893i \(-0.281693\pi\)
−0.986869 + 0.161521i \(0.948360\pi\)
\(3\) 0 0
\(4\) 0.500000 − 0.866025i 0.250000 − 0.433013i
\(5\) 1.00000 0.447214 0.223607 − 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) −1.50000 + 2.59808i −0.566947 + 0.981981i 0.429919 + 0.902867i \(0.358542\pi\)
−0.996866 + 0.0791130i \(0.974791\pi\)
\(8\) −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 −0.546259 − 0.837616i \(-0.683949\pi\)
0.998526 + 0.0542666i \(0.0172821\pi\)
\(12\) 0 0
\(13\) 3.00000 − 5.19615i 0.832050 − 1.44115i −0.0643593 − 0.997927i \(-0.520500\pi\)
0.896410 − 0.443227i \(-0.146166\pi\)
\(14\) 3.00000 0.801784
\(15\) 0 0
\(16\) 0.500000 + 0.866025i 0.125000 + 0.216506i
\(17\) 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
\(22\) −3.00000 −0.639602
\(23\) 4.00000 − 6.92820i 0.834058 − 1.44463i −0.0607377 − 0.998154i \(-0.519345\pi\)
0.894795 − 0.446476i \(-0.147321\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) −6.00000 −1.17670
\(27\) 0 0
\(28\) 1.50000 + 2.59808i 0.283473 + 0.490990i
\(29\) 5.00000 0.928477 0.464238 − 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) 3.50000 + 6.06218i 0.628619 + 1.08880i 0.987829 + 0.155543i \(0.0497126\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −2.50000 + 4.33013i −0.441942 + 0.765466i
\(33\) 0 0
\(34\) −3.00000 −0.514496
\(35\) −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 \(0.447432\pi\)
\(38\) 0.500000 + 4.33013i 0.0811107 + 0.702439i
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) 1.00000 0.156174 0.0780869 − 0.996947i \(-0.475119\pi\)
0.0780869 + 0.996947i \(0.475119\pi\)
\(42\) 0 0
\(43\) −4.00000 − 6.92820i −0.609994 − 1.05654i −0.991241 − 0.132068i \(-0.957838\pi\)
0.381246 − 0.924473i \(-0.375495\pi\)
\(44\) −1.50000 − 2.59808i −0.226134 − 0.391675i
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) −9.00000 −1.31278 −0.656392 − 0.754420i \(-0.727918\pi\)
−0.656392 + 0.754420i \(0.727918\pi\)
\(48\) 0 0
\(49\) −1.00000 − 1.73205i −0.142857 − 0.247436i
\(50\) 2.00000 + 3.46410i 0.282843 + 0.489898i
\(51\) 0 0
\(52\) −3.00000 − 5.19615i −0.416025 − 0.720577i
\(53\) 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
\(59\) −3.00000 −0.390567 −0.195283 − 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 0 0
\(61\) 7.00000 0.896258 0.448129 − 0.893969i \(-0.352090\pi\)
0.448129 + 0.893969i \(0.352090\pi\)
\(62\) 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
\(67\) 2.00000 − 3.46410i 0.244339 − 0.423207i −0.717607 − 0.696449i \(-0.754762\pi\)
0.961946 + 0.273241i \(0.0880957\pi\)
\(68\) −1.50000 − 2.59808i −0.181902 − 0.315063i
\(69\) 0 0
\(70\) 3.00000 0.358569
\(71\) −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 \(0.0721453\pi\)
\(74\) −1.00000 − 1.73205i −0.116248 − 0.201347i
\(75\) 0 0
\(76\) −3.50000 + 2.59808i −0.401478 + 0.298020i
\(77\) 4.50000 + 7.79423i 0.512823 + 0.888235i
\(78\) 0 0
\(79\) 6.00000 + 10.3923i 0.675053 + 1.16923i 0.976453 + 0.215728i \(0.0692125\pi\)
−0.301401 + 0.953498i \(0.597454\pi\)
\(80\) 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 −0.892161 − 0.451717i \(-0.850812\pi\)
0.837279 + 0.546776i \(0.184145\pi\)
\(84\) 0 0
\(85\) 1.50000 − 2.59808i 0.162698 − 0.281801i
\(86\) −4.00000 + 6.92820i −0.431331 + 0.747087i
\(87\) 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
\(92\) −4.00000 − 6.92820i −0.417029 − 0.722315i
\(93\) 0 0
\(94\) 4.50000 + 7.79423i 0.464140 + 0.803913i
\(95\) −4.00000 − 1.73205i −0.410391 − 0.177705i
\(96\) 0 0
\(97\) 1.00000 + 1.73205i 0.101535 + 0.175863i 0.912317 − 0.409484i \(-0.134291\pi\)
−0.810782 + 0.585348i \(0.800958\pi\)
\(98\) −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.g.b.505.1 2
3.2 odd 2 171.2.g.b.106.1 ✓ 2
9.4 even 3 513.2.h.a.334.1 2
9.5 odd 6 171.2.h.b.49.1 yes 2
19.7 even 3 513.2.h.a.235.1 2
57.26 odd 6 171.2.h.b.7.1 yes 2
171.121 even 3 inner 513.2.g.b.64.1 2
171.140 odd 6 171.2.g.b.121.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.g.b.106.1 ✓ 2 3.2 odd 2
171.2.g.b.121.1 yes 2 171.140 odd 6
171.2.h.b.7.1 yes 2 57.26 odd 6
171.2.h.b.49.1 yes 2 9.5 odd 6
513.2.g.b.64.1 2 171.121 even 3 inner
513.2.g.b.505.1 2 1.1 even 1 trivial
513.2.h.a.235.1 2 19.7 even 3
513.2.h.a.334.1 2 9.4 even 3