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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [912,2,Mod(49,912)] 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("912.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(912, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,3,0,-2,0,2,0,-3,0,0,0,1] 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: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.954288.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 57)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 577.1
Root \(0.403374 - 1.68443i\) of defining polynomial
Character \(\chi\) \(=\) 912.577
Dual form 912.2.q.l.49.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{3} +(-1.66044 - 2.87597i) q^{5} -2.32088 q^{7} +(-0.500000 + 0.866025i) q^{9} +1.70739 q^{11} +(-2.01414 + 3.48859i) q^{13} +(1.66044 - 2.87597i) q^{15} +(-0.193252 + 4.35461i) q^{19} +(-1.16044 - 2.00994i) q^{21} +(-1.17458 + 2.03443i) q^{23} +(-3.01414 + 5.22064i) q^{25} -1.00000 q^{27} +(-3.32088 + 5.75194i) q^{29} -6.70739 q^{31} +(0.853695 + 1.47864i) q^{33} +(3.85369 + 6.67479i) q^{35} -1.00000 q^{37} -4.02827 q^{39} +(3.32088 + 5.75194i) q^{41} +(0.353695 + 0.612617i) q^{43} +3.32088 q^{45} +(-3.00000 + 5.19615i) q^{47} -1.61350 q^{49} +(4.98133 - 8.62791i) q^{53} +(-2.83502 - 4.91040i) q^{55} +(-3.86783 + 2.00994i) q^{57} +(0.853695 + 1.47864i) q^{59} +(-1.69325 + 2.93280i) q^{61} +(1.16044 - 2.00994i) q^{63} +13.3774 q^{65} +(-4.18872 + 7.25507i) q^{67} -2.34916 q^{69} +(-4.70739 - 8.15344i) q^{71} +(-5.82088 - 10.0821i) q^{73} -6.02827 q^{75} -3.96265 q^{77} +(-1.67458 - 2.90046i) q^{79} +(-0.500000 - 0.866025i) q^{81} +10.0565 q^{83} -6.64177 q^{87} +(1.33956 - 2.32018i) q^{89} +(4.67458 - 8.09661i) q^{91} +(-3.35369 - 5.80877i) q^{93} +(12.8446 - 6.67479i) q^{95} +(-8.86330 - 15.3517i) q^{97} +(-0.853695 + 1.47864i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} - 2 q^{5} + 2 q^{7} - 3 q^{9} + q^{13} + 2 q^{15} - 4 q^{19} + q^{21} + 14 q^{23} - 5 q^{25} - 6 q^{27} - 4 q^{29} - 30 q^{31} + 18 q^{35} - 6 q^{37} + 2 q^{39} + 4 q^{41} - 3 q^{43} + 4 q^{45}+ \cdots + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(1\)

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 0
\(3\) 0.500000 + 0.866025i 0.288675 + 0.500000i
\(4\) 0 0
\(5\) −1.66044 − 2.87597i −0.742572 − 1.28617i −0.951320 − 0.308204i \(-0.900272\pi\)
0.208748 − 0.977969i \(-0.433061\pi\)
\(6\) 0 0
\(7\) −2.32088 −0.877212 −0.438606 − 0.898679i \(-0.644528\pi\)
−0.438606 + 0.898679i \(0.644528\pi\)
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 1.70739 0.514797 0.257399 − 0.966305i \(-0.417135\pi\)
0.257399 + 0.966305i \(0.417135\pi\)
\(12\) 0 0
\(13\) −2.01414 + 3.48859i −0.558621 + 0.967560i 0.438991 + 0.898492i \(0.355336\pi\)
−0.997612 + 0.0690685i \(0.977997\pi\)
\(14\) 0 0
\(15\) 1.66044 − 2.87597i 0.428724 − 0.742572i
\(16\) 0 0
\(17\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) −0.193252 + 4.35461i −0.0443351 + 0.999017i
\(20\) 0 0
\(21\) −1.16044 − 2.00994i −0.253229 − 0.438606i
\(22\) 0 0
\(23\) −1.17458 + 2.03443i −0.244917 + 0.424208i −0.962108 − 0.272668i \(-0.912094\pi\)
0.717191 + 0.696876i \(0.245427\pi\)
\(24\) 0 0
\(25\) −3.01414 + 5.22064i −0.602827 + 1.04413i
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −3.32088 + 5.75194i −0.616673 + 1.06811i 0.373416 + 0.927664i \(0.378186\pi\)
−0.990089 + 0.140444i \(0.955147\pi\)
\(30\) 0 0
\(31\) −6.70739 −1.20468 −0.602341 − 0.798239i \(-0.705765\pi\)
−0.602341 + 0.798239i \(0.705765\pi\)
\(32\) 0 0
\(33\) 0.853695 + 1.47864i 0.148609 + 0.257399i
\(34\) 0 0
\(35\) 3.85369 + 6.67479i 0.651393 + 1.12825i
\(36\) 0 0
\(37\) −1.00000 −0.164399 −0.0821995 − 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) 0 0
\(39\) −4.02827 −0.645040
\(40\) 0 0
\(41\) 3.32088 + 5.75194i 0.518635 + 0.898302i 0.999766 + 0.0216532i \(0.00689298\pi\)
−0.481131 + 0.876649i \(0.659774\pi\)
\(42\) 0 0
\(43\) 0.353695 + 0.612617i 0.0539379 + 0.0934232i 0.891734 − 0.452561i \(-0.149489\pi\)
−0.837796 + 0.545984i \(0.816156\pi\)
\(44\) 0 0
\(45\) 3.32088 0.495048
\(46\) 0 0
\(47\) −3.00000 + 5.19615i −0.437595 + 0.757937i −0.997503 − 0.0706177i \(-0.977503\pi\)
0.559908 + 0.828554i \(0.310836\pi\)
\(48\) 0 0
\(49\) −1.61350 −0.230499
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.98133 − 8.62791i 0.684238 − 1.18513i −0.289438 − 0.957197i \(-0.593468\pi\)
0.973676 − 0.227938i \(-0.0731983\pi\)
\(54\) 0 0
\(55\) −2.83502 − 4.91040i −0.382274 − 0.662118i
\(56\) 0 0
\(57\) −3.86783 + 2.00994i −0.512307 + 0.266224i
\(58\) 0 0
\(59\) 0.853695 + 1.47864i 0.111142 + 0.192503i 0.916231 − 0.400651i \(-0.131216\pi\)
−0.805089 + 0.593154i \(0.797883\pi\)
\(60\) 0 0
\(61\) −1.69325 + 2.93280i −0.216799 + 0.375506i −0.953828 − 0.300355i \(-0.902895\pi\)
0.737029 + 0.675861i \(0.236228\pi\)
\(62\) 0 0
\(63\) 1.16044 − 2.00994i 0.146202 − 0.253229i
\(64\) 0 0
\(65\) 13.3774 1.65927
\(66\) 0 0
\(67\) −4.18872 + 7.25507i −0.511733 + 0.886348i 0.488174 + 0.872746i \(0.337663\pi\)
−0.999907 + 0.0136016i \(0.995670\pi\)
\(68\) 0 0
\(69\) −2.34916 −0.282805
\(70\) 0 0
\(71\) −4.70739 − 8.15344i −0.558664 − 0.967635i −0.997608 − 0.0691206i \(-0.977981\pi\)
0.438944 − 0.898514i \(-0.355353\pi\)
\(72\) 0 0
\(73\) −5.82088 − 10.0821i −0.681283 − 1.18002i −0.974590 − 0.223998i \(-0.928089\pi\)
0.293307 − 0.956018i \(-0.405244\pi\)
\(74\) 0 0
\(75\) −6.02827 −0.696085
\(76\) 0 0
\(77\) −3.96265 −0.451586
\(78\) 0 0
\(79\) −1.67458 − 2.90046i −0.188405 − 0.326327i 0.756314 − 0.654209i \(-0.226998\pi\)
−0.944719 + 0.327882i \(0.893665\pi\)
\(80\) 0 0
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) 10.0565 1.10385 0.551925 − 0.833894i \(-0.313894\pi\)
0.551925 + 0.833894i \(0.313894\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −6.64177 −0.712072
\(88\) 0 0
\(89\) 1.33956 − 2.32018i 0.141993 − 0.245939i −0.786254 − 0.617903i \(-0.787982\pi\)
0.928247 + 0.371964i \(0.121316\pi\)
\(90\) 0 0
\(91\) 4.67458 − 8.09661i 0.490029 − 0.848755i
\(92\) 0 0
\(93\) −3.35369 − 5.80877i −0.347762 − 0.602341i
\(94\) 0 0
\(95\) 12.8446 − 6.67479i 1.31783 − 0.684820i
\(96\) 0 0
\(97\) −8.86330 − 15.3517i −0.899931 − 1.55873i −0.827580 − 0.561347i \(-0.810283\pi\)
−0.0723511 − 0.997379i \(-0.523050\pi\)
\(98\) 0 0
\(99\) −0.853695 + 1.47864i −0.0857995 + 0.148609i
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 912.2.q.l.577.1 6
3.2 odd 2 2736.2.s.z.577.3 6
4.3 odd 2 57.2.e.b.7.3 ✓ 6
12.11 even 2 171.2.f.b.64.1 6
19.11 even 3 inner 912.2.q.l.49.1 6
57.11 odd 6 2736.2.s.z.1873.3 6
76.7 odd 6 1083.2.a.l.1.1 3
76.11 odd 6 57.2.e.b.49.3 yes 6
76.31 even 6 1083.2.a.o.1.3 3
228.11 even 6 171.2.f.b.163.1 6
228.83 even 6 3249.2.a.y.1.3 3
228.107 odd 6 3249.2.a.t.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.e.b.7.3 ✓ 6 4.3 odd 2
57.2.e.b.49.3 yes 6 76.11 odd 6
171.2.f.b.64.1 6 12.11 even 2
171.2.f.b.163.1 6 228.11 even 6
912.2.q.l.49.1 6 19.11 even 3 inner
912.2.q.l.577.1 6 1.1 even 1 trivial
1083.2.a.l.1.1 3 76.7 odd 6
1083.2.a.o.1.3 3 76.31 even 6
2736.2.s.z.577.3 6 3.2 odd 2
2736.2.s.z.1873.3 6 57.11 odd 6
3249.2.a.t.1.1 3 228.107 odd 6
3249.2.a.y.1.3 3 228.83 even 6