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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [936,2,Mod(217,936)] 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("936.217"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(936, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 936.t (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,-4,0,-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: \(7.47399762919\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.27870912.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 7x^{4} + 2x^{3} + 38x^{2} - 12x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 289.3
Root \(1.42789 - 2.47317i\) of defining polynomial
Character \(\chi\) \(=\) 936.289
Dual form 936.2.t.g.217.3

$q$-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)
 
\(f(q)\) \(=\) \(q+1.85577 q^{5} +(-1.42789 + 2.47317i) q^{7} +(0.649832 + 1.12554i) q^{11} +(3.00560 - 1.99157i) q^{13} +(0.278054 - 0.481604i) q^{17} +(-0.649832 + 1.12554i) q^{19} +(4.50560 + 7.80394i) q^{23} -1.55611 q^{25} +(2.22755 + 3.85823i) q^{29} -0.443892 q^{31} +(-2.64983 + 4.58964i) q^{35} +(-3.78366 - 6.55349i) q^{37} +(5.43349 + 9.41108i) q^{41} +(-0.871778 + 1.50996i) q^{43} +5.01121 q^{47} +(-0.577718 - 1.00064i) q^{49} -2.14423 q^{53} +(1.20594 + 2.08875i) q^{55} +(4.41188 - 7.64160i) q^{59} +(4.86138 - 8.42015i) q^{61} +(5.57772 - 3.69590i) q^{65} +(5.43910 + 9.42079i) q^{67} +(-0.505605 + 0.875733i) q^{71} -0.112217 q^{73} -3.71155 q^{77} -13.5785 q^{79} +4.72275 q^{83} +(0.516005 - 0.893747i) q^{85} +(-2.00000 - 3.46410i) q^{89} +(0.633827 + 10.2771i) q^{91} +(-1.20594 + 2.08875i) q^{95} +(1.07772 - 1.86666i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{5} - q^{7} - q^{13} - 2 q^{17} + 8 q^{23} - 2 q^{25} - 2 q^{29} - 10 q^{31} - 12 q^{35} + 6 q^{41} - 5 q^{43} - 8 q^{47} + 8 q^{49} - 28 q^{53} - 4 q^{55} + 4 q^{59} - 5 q^{61} + 22 q^{65}+ \cdots - 5 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/936\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(469\) \(703\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(1\)

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 0
\(4\) 0 0
\(5\) 1.85577 0.829927 0.414963 − 0.909838i \(-0.363794\pi\)
0.414963 + 0.909838i \(0.363794\pi\)
\(6\) 0 0
\(7\) −1.42789 + 2.47317i −0.539690 + 0.934771i 0.459230 + 0.888317i \(0.348125\pi\)
−0.998920 + 0.0464537i \(0.985208\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.649832 + 1.12554i 0.195932 + 0.339364i 0.947206 − 0.320627i \(-0.103894\pi\)
−0.751274 + 0.659991i \(0.770560\pi\)
\(12\) 0 0
\(13\) 3.00560 − 1.99157i 0.833605 − 0.552361i
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.278054 − 0.481604i 0.0674380 − 0.116806i −0.830335 − 0.557265i \(-0.811851\pi\)
0.897773 + 0.440459i \(0.145184\pi\)
\(18\) 0 0
\(19\) −0.649832 + 1.12554i −0.149082 + 0.258217i −0.930888 − 0.365304i \(-0.880965\pi\)
0.781807 + 0.623521i \(0.214298\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.50560 + 7.80394i 0.939483 + 1.62723i 0.766437 + 0.642319i \(0.222028\pi\)
0.173046 + 0.984914i \(0.444639\pi\)
\(24\) 0 0
\(25\) −1.55611 −0.311222
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.22755 + 3.85823i 0.413646 + 0.716455i 0.995285 − 0.0969913i \(-0.0309219\pi\)
−0.581640 + 0.813447i \(0.697589\pi\)
\(30\) 0 0
\(31\) −0.443892 −0.0797253 −0.0398626 − 0.999205i \(-0.512692\pi\)
−0.0398626 + 0.999205i \(0.512692\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.64983 + 4.58964i −0.447903 + 0.775791i
\(36\) 0 0
\(37\) −3.78366 − 6.55349i −0.622030 − 1.07739i −0.989107 − 0.147197i \(-0.952975\pi\)
0.367078 − 0.930190i \(-0.380358\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.43349 + 9.41108i 0.848569 + 1.46976i 0.882486 + 0.470339i \(0.155869\pi\)
−0.0339169 + 0.999425i \(0.510798\pi\)
\(42\) 0 0
\(43\) −0.871778 + 1.50996i −0.132945 + 0.230267i −0.924811 − 0.380428i \(-0.875777\pi\)
0.791866 + 0.610695i \(0.209110\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.01121 0.730960 0.365480 − 0.930819i \(-0.380905\pi\)
0.365480 + 0.930819i \(0.380905\pi\)
\(48\) 0 0
\(49\) −0.577718 − 1.00064i −0.0825312 − 0.142948i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.14423 −0.294532 −0.147266 − 0.989097i \(-0.547047\pi\)
−0.147266 + 0.989097i \(0.547047\pi\)
\(54\) 0 0
\(55\) 1.20594 + 2.08875i 0.162609 + 0.281647i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.41188 − 7.64160i 0.574378 − 0.994852i −0.421731 − 0.906721i \(-0.638577\pi\)
0.996109 − 0.0881308i \(-0.0280893\pi\)
\(60\) 0 0
\(61\) 4.86138 − 8.42015i 0.622436 − 1.07809i −0.366595 − 0.930381i \(-0.619477\pi\)
0.989031 − 0.147709i \(-0.0471901\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5.57772 − 3.69590i 0.691831 − 0.458420i
\(66\) 0 0
\(67\) 5.43910 + 9.42079i 0.664491 + 1.15093i 0.979423 + 0.201818i \(0.0646850\pi\)
−0.314932 + 0.949114i \(0.601982\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −0.505605 + 0.875733i −0.0600042 + 0.103930i −0.894467 − 0.447134i \(-0.852445\pi\)
0.834463 + 0.551064i \(0.185778\pi\)
\(72\) 0 0
\(73\) −0.112217 −0.0131340 −0.00656699 − 0.999978i \(-0.502090\pi\)
−0.00656699 + 0.999978i \(0.502090\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.71155 −0.422970
\(78\) 0 0
\(79\) −13.5785 −1.52770 −0.763852 − 0.645392i \(-0.776694\pi\)
−0.763852 + 0.645392i \(0.776694\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.72275 0.518390 0.259195 − 0.965825i \(-0.416543\pi\)
0.259195 + 0.965825i \(0.416543\pi\)
\(84\) 0 0
\(85\) 0.516005 − 0.893747i 0.0559686 − 0.0969405i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.00000 − 3.46410i −0.212000 − 0.367194i 0.740341 − 0.672232i \(-0.234664\pi\)
−0.952340 + 0.305038i \(0.901331\pi\)
\(90\) 0 0
\(91\) 0.633827 + 10.2771i 0.0664431 + 1.07733i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.20594 + 2.08875i −0.123727 + 0.214301i
\(96\) 0 0
\(97\) 1.07772 − 1.86666i 0.109426 − 0.189531i −0.806112 − 0.591763i \(-0.798432\pi\)
0.915538 + 0.402232i \(0.131765\pi\)
\(98\) 0 0
\(99\) 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 936.2.t.g.289.3 yes 6
3.2 odd 2 936.2.t.i.289.1 yes 6
4.3 odd 2 1872.2.t.t.289.3 6
12.11 even 2 1872.2.t.v.289.1 6
13.9 even 3 inner 936.2.t.g.217.3 ✓ 6
39.35 odd 6 936.2.t.i.217.1 yes 6
52.35 odd 6 1872.2.t.t.1153.3 6
156.35 even 6 1872.2.t.v.1153.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
936.2.t.g.217.3 ✓ 6 13.9 even 3 inner
936.2.t.g.289.3 yes 6 1.1 even 1 trivial
936.2.t.i.217.1 yes 6 39.35 odd 6
936.2.t.i.289.1 yes 6 3.2 odd 2
1872.2.t.t.289.3 6 4.3 odd 2
1872.2.t.t.1153.3 6 52.35 odd 6
1872.2.t.v.289.1 6 12.11 even 2
1872.2.t.v.1153.1 6 156.35 even 6