Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [784,2,Mod(177,784)] 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("784.177"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-2,0,0,0,0,0,-1,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.26027151847\)
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: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 177.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 784.177
Dual form 784.2.i.c.753.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.00000 - 1.73205i) q^{3} +(-0.500000 + 0.866025i) q^{9} -4.00000 q^{13} +(-3.00000 - 5.19615i) q^{17} +(1.00000 - 1.73205i) q^{19} +(2.50000 + 4.33013i) q^{25} -4.00000 q^{27} -6.00000 q^{29} +(-2.00000 - 3.46410i) q^{31} +(-1.00000 + 1.73205i) q^{37} +(4.00000 + 6.92820i) q^{39} +6.00000 q^{41} -8.00000 q^{43} +(-6.00000 + 10.3923i) q^{47} +(-6.00000 + 10.3923i) q^{51} +(-3.00000 - 5.19615i) q^{53} -4.00000 q^{57} +(-3.00000 - 5.19615i) q^{59} +(-4.00000 + 6.92820i) q^{61} +(-2.00000 - 3.46410i) q^{67} +(-1.00000 - 1.73205i) q^{73} +(5.00000 - 8.66025i) q^{75} +(4.00000 - 6.92820i) q^{79} +(5.50000 + 9.52628i) q^{81} +6.00000 q^{83} +(6.00000 + 10.3923i) q^{87} +(3.00000 - 5.19615i) q^{89} +(-4.00000 + 6.92820i) q^{93} -10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - q^{9} - 8 q^{13} - 6 q^{17} + 2 q^{19} + 5 q^{25} - 8 q^{27} - 12 q^{29} - 4 q^{31} - 2 q^{37} + 8 q^{39} + 12 q^{41} - 16 q^{43} - 12 q^{47} - 12 q^{51} - 6 q^{53} - 8 q^{57} - 6 q^{59}+ \cdots - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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 0
\(3\) −1.00000 − 1.73205i −0.577350 − 1.00000i −0.995782 − 0.0917517i \(-0.970753\pi\)
0.418432 − 0.908248i \(-0.362580\pi\)
\(4\) 0 0
\(5\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.00000 − 5.19615i −0.727607 − 1.26025i −0.957892 − 0.287129i \(-0.907299\pi\)
0.230285 − 0.973123i \(-0.426034\pi\)
\(18\) 0 0
\(19\) 1.00000 − 1.73205i 0.229416 − 0.397360i −0.728219 − 0.685344i \(-0.759652\pi\)
0.957635 + 0.287984i \(0.0929851\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 0 0
\(25\) 2.50000 + 4.33013i 0.500000 + 0.866025i
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −2.00000 − 3.46410i −0.359211 − 0.622171i 0.628619 − 0.777714i \(-0.283621\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 + 1.73205i −0.164399 + 0.284747i −0.936442 − 0.350823i \(-0.885902\pi\)
0.772043 + 0.635571i \(0.219235\pi\)
\(38\) 0 0
\(39\) 4.00000 + 6.92820i 0.640513 + 1.10940i
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.00000 + 10.3923i −0.875190 + 1.51587i −0.0186297 + 0.999826i \(0.505930\pi\)
−0.856560 + 0.516047i \(0.827403\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −6.00000 + 10.3923i −0.840168 + 1.45521i
\(52\) 0 0
\(53\) −3.00000 − 5.19615i −0.412082 − 0.713746i 0.583036 − 0.812447i \(-0.301865\pi\)
−0.995117 + 0.0987002i \(0.968532\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) −3.00000 − 5.19615i −0.390567 − 0.676481i 0.601958 − 0.798528i \(-0.294388\pi\)
−0.992524 + 0.122047i \(0.961054\pi\)
\(60\) 0 0
\(61\) −4.00000 + 6.92820i −0.512148 + 0.887066i 0.487753 + 0.872982i \(0.337817\pi\)
−0.999901 + 0.0140840i \(0.995517\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −2.00000 − 3.46410i −0.244339 − 0.423207i 0.717607 − 0.696449i \(-0.245238\pi\)
−0.961946 + 0.273241i \(0.911904\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −1.00000 − 1.73205i −0.117041 − 0.202721i 0.801553 − 0.597924i \(-0.204008\pi\)
−0.918594 + 0.395203i \(0.870674\pi\)
\(74\) 0 0
\(75\) 5.00000 − 8.66025i 0.577350 − 1.00000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 4.00000 − 6.92820i 0.450035 − 0.779484i −0.548352 − 0.836247i \(-0.684745\pi\)
0.998388 + 0.0567635i \(0.0180781\pi\)
\(80\) 0 0
\(81\) 5.50000 + 9.52628i 0.611111 + 1.05848i
\(82\) 0 0
\(83\) 6.00000 0.658586 0.329293 − 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6.00000 + 10.3923i 0.643268 + 1.11417i
\(88\) 0 0
\(89\) 3.00000 − 5.19615i 0.317999 − 0.550791i −0.662071 − 0.749441i \(-0.730322\pi\)
0.980071 + 0.198650i \(0.0636557\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −4.00000 + 6.92820i −0.414781 + 0.718421i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 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 784.2.i.c.177.1 2
4.3 odd 2 98.2.c.b.79.1 2
7.2 even 3 112.2.a.c.1.1 1
7.3 odd 6 784.2.i.i.753.1 2
7.4 even 3 inner 784.2.i.c.753.1 2
7.5 odd 6 784.2.a.b.1.1 1
7.6 odd 2 784.2.i.i.177.1 2
12.11 even 2 882.2.g.c.667.1 2
21.2 odd 6 1008.2.a.h.1.1 1
21.5 even 6 7056.2.a.bd.1.1 1
28.3 even 6 98.2.c.a.67.1 2
28.11 odd 6 98.2.c.b.67.1 2
28.19 even 6 98.2.a.a.1.1 1
28.23 odd 6 14.2.a.a.1.1 ✓ 1
28.27 even 2 98.2.c.a.79.1 2
35.2 odd 12 2800.2.g.h.449.1 2
35.9 even 6 2800.2.a.g.1.1 1
35.23 odd 12 2800.2.g.h.449.2 2
56.5 odd 6 3136.2.a.z.1.1 1
56.19 even 6 3136.2.a.e.1.1 1
56.37 even 6 448.2.a.a.1.1 1
56.51 odd 6 448.2.a.g.1.1 1
84.11 even 6 882.2.g.c.361.1 2
84.23 even 6 126.2.a.b.1.1 1
84.47 odd 6 882.2.a.i.1.1 1
84.59 odd 6 882.2.g.d.361.1 2
84.83 odd 2 882.2.g.d.667.1 2
112.37 even 12 1792.2.b.g.897.1 2
112.51 odd 12 1792.2.b.c.897.1 2
112.93 even 12 1792.2.b.g.897.2 2
112.107 odd 12 1792.2.b.c.897.2 2
140.19 even 6 2450.2.a.t.1.1 1
140.23 even 12 350.2.c.d.99.2 2
140.47 odd 12 2450.2.c.c.99.1 2
140.79 odd 6 350.2.a.f.1.1 1
140.103 odd 12 2450.2.c.c.99.2 2
140.107 even 12 350.2.c.d.99.1 2
168.107 even 6 4032.2.a.w.1.1 1
168.149 odd 6 4032.2.a.r.1.1 1
252.23 even 6 1134.2.f.f.379.1 2
252.79 odd 6 1134.2.f.l.757.1 2
252.191 even 6 1134.2.f.f.757.1 2
252.247 odd 6 1134.2.f.l.379.1 2
308.219 even 6 1694.2.a.e.1.1 1
364.51 odd 6 2366.2.a.j.1.1 1
364.135 even 12 2366.2.d.b.337.2 2
364.359 even 12 2366.2.d.b.337.1 2
420.23 odd 12 3150.2.g.j.2899.1 2
420.107 odd 12 3150.2.g.j.2899.2 2
420.359 even 6 3150.2.a.i.1.1 1
476.135 odd 6 4046.2.a.f.1.1 1
532.303 even 6 5054.2.a.c.1.1 1
644.275 even 6 7406.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.2.a.a.1.1 ✓ 1 28.23 odd 6
98.2.a.a.1.1 1 28.19 even 6
98.2.c.a.67.1 2 28.3 even 6
98.2.c.a.79.1 2 28.27 even 2
98.2.c.b.67.1 2 28.11 odd 6
98.2.c.b.79.1 2 4.3 odd 2
112.2.a.c.1.1 1 7.2 even 3
126.2.a.b.1.1 1 84.23 even 6
350.2.a.f.1.1 1 140.79 odd 6
350.2.c.d.99.1 2 140.107 even 12
350.2.c.d.99.2 2 140.23 even 12
448.2.a.a.1.1 1 56.37 even 6
448.2.a.g.1.1 1 56.51 odd 6
784.2.a.b.1.1 1 7.5 odd 6
784.2.i.c.177.1 2 1.1 even 1 trivial
784.2.i.c.753.1 2 7.4 even 3 inner
784.2.i.i.177.1 2 7.6 odd 2
784.2.i.i.753.1 2 7.3 odd 6
882.2.a.i.1.1 1 84.47 odd 6
882.2.g.c.361.1 2 84.11 even 6
882.2.g.c.667.1 2 12.11 even 2
882.2.g.d.361.1 2 84.59 odd 6
882.2.g.d.667.1 2 84.83 odd 2
1008.2.a.h.1.1 1 21.2 odd 6
1134.2.f.f.379.1 2 252.23 even 6
1134.2.f.f.757.1 2 252.191 even 6
1134.2.f.l.379.1 2 252.247 odd 6
1134.2.f.l.757.1 2 252.79 odd 6
1694.2.a.e.1.1 1 308.219 even 6
1792.2.b.c.897.1 2 112.51 odd 12
1792.2.b.c.897.2 2 112.107 odd 12
1792.2.b.g.897.1 2 112.37 even 12
1792.2.b.g.897.2 2 112.93 even 12
2366.2.a.j.1.1 1 364.51 odd 6
2366.2.d.b.337.1 2 364.359 even 12
2366.2.d.b.337.2 2 364.135 even 12
2450.2.a.t.1.1 1 140.19 even 6
2450.2.c.c.99.1 2 140.47 odd 12
2450.2.c.c.99.2 2 140.103 odd 12
2800.2.a.g.1.1 1 35.9 even 6
2800.2.g.h.449.1 2 35.2 odd 12
2800.2.g.h.449.2 2 35.23 odd 12
3136.2.a.e.1.1 1 56.19 even 6
3136.2.a.z.1.1 1 56.5 odd 6
3150.2.a.i.1.1 1 420.359 even 6
3150.2.g.j.2899.1 2 420.23 odd 12
3150.2.g.j.2899.2 2 420.107 odd 12
4032.2.a.r.1.1 1 168.149 odd 6
4032.2.a.w.1.1 1 168.107 even 6
4046.2.a.f.1.1 1 476.135 odd 6
5054.2.a.c.1.1 1 532.303 even 6
7056.2.a.bd.1.1 1 21.5 even 6
7406.2.a.a.1.1 1 644.275 even 6