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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 = [4,0,0,0,0,0,0,0,-10,0,-8] 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 392)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 753.1
Root \(-0.707107 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 784.753
Dual form 784.2.i.k.177.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.41421 + 2.44949i) q^{3} +(1.41421 + 2.44949i) q^{5} +(-2.50000 - 4.33013i) q^{9} +(-2.00000 + 3.46410i) q^{11} +2.82843 q^{13} -8.00000 q^{15} +(-2.82843 + 4.89898i) q^{17} +(1.41421 + 2.44949i) q^{19} +(-1.50000 + 2.59808i) q^{25} +5.65685 q^{27} +2.00000 q^{29} +(2.82843 - 4.89898i) q^{31} +(-5.65685 - 9.79796i) q^{33} +(-5.00000 - 8.66025i) q^{37} +(-4.00000 + 6.92820i) q^{39} -5.65685 q^{41} +4.00000 q^{43} +(7.07107 - 12.2474i) q^{45} +(-2.82843 - 4.89898i) q^{47} +(-8.00000 - 13.8564i) q^{51} +(-3.00000 + 5.19615i) q^{53} -11.3137 q^{55} -8.00000 q^{57} +(1.41421 - 2.44949i) q^{59} +(7.07107 + 12.2474i) q^{61} +(4.00000 + 6.92820i) q^{65} +(6.00000 - 10.3923i) q^{67} +(-4.24264 - 7.34847i) q^{75} +(4.00000 + 6.92820i) q^{79} +(-0.500000 + 0.866025i) q^{81} -14.1421 q^{83} -16.0000 q^{85} +(-2.82843 + 4.89898i) q^{87} +(8.00000 + 13.8564i) q^{93} +(-4.00000 + 6.92820i) q^{95} -5.65685 q^{97} +20.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{9} - 8 q^{11} - 32 q^{15} - 6 q^{25} + 8 q^{29} - 20 q^{37} - 16 q^{39} + 16 q^{43} - 32 q^{51} - 12 q^{53} - 32 q^{57} + 16 q^{65} + 24 q^{67} + 16 q^{79} - 2 q^{81} - 64 q^{85} + 32 q^{93}+ \cdots + 80 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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 0
\(3\) −1.41421 + 2.44949i −0.816497 + 1.41421i 0.0917517 + 0.995782i \(0.470753\pi\)
−0.908248 + 0.418432i \(0.862580\pi\)
\(4\) 0 0
\(5\) 1.41421 + 2.44949i 0.632456 + 1.09545i 0.987048 + 0.160424i \(0.0512862\pi\)
−0.354593 + 0.935021i \(0.615380\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −2.50000 − 4.33013i −0.833333 − 1.44338i
\(10\) 0 0
\(11\) −2.00000 + 3.46410i −0.603023 + 1.04447i 0.389338 + 0.921095i \(0.372704\pi\)
−0.992361 + 0.123371i \(0.960630\pi\)
\(12\) 0 0
\(13\) 2.82843 0.784465 0.392232 − 0.919866i \(-0.371703\pi\)
0.392232 + 0.919866i \(0.371703\pi\)
\(14\) 0 0
\(15\) −8.00000 −2.06559
\(16\) 0 0
\(17\) −2.82843 + 4.89898i −0.685994 + 1.18818i 0.287129 + 0.957892i \(0.407299\pi\)
−0.973123 + 0.230285i \(0.926034\pi\)
\(18\) 0 0
\(19\) 1.41421 + 2.44949i 0.324443 + 0.561951i 0.981399 − 0.191977i \(-0.0614899\pi\)
−0.656957 + 0.753928i \(0.728157\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) 0 0
\(25\) −1.50000 + 2.59808i −0.300000 + 0.519615i
\(26\) 0 0
\(27\) 5.65685 1.08866
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 2.82843 − 4.89898i 0.508001 − 0.879883i −0.491957 − 0.870620i \(-0.663718\pi\)
0.999957 − 0.00926296i \(-0.00294853\pi\)
\(32\) 0 0
\(33\) −5.65685 − 9.79796i −0.984732 − 1.70561i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.00000 − 8.66025i −0.821995 − 1.42374i −0.904194 − 0.427121i \(-0.859528\pi\)
0.0821995 − 0.996616i \(-0.473806\pi\)
\(38\) 0 0
\(39\) −4.00000 + 6.92820i −0.640513 + 1.10940i
\(40\) 0 0
\(41\) −5.65685 −0.883452 −0.441726 − 0.897150i \(-0.645634\pi\)
−0.441726 + 0.897150i \(0.645634\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 7.07107 − 12.2474i 1.05409 − 1.82574i
\(46\) 0 0
\(47\) −2.82843 − 4.89898i −0.412568 − 0.714590i 0.582601 − 0.812758i \(-0.302035\pi\)
−0.995170 + 0.0981685i \(0.968702\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −8.00000 − 13.8564i −1.12022 − 1.94029i
\(52\) 0 0
\(53\) −3.00000 + 5.19615i −0.412082 + 0.713746i −0.995117 − 0.0987002i \(-0.968532\pi\)
0.583036 + 0.812447i \(0.301865\pi\)
\(54\) 0 0
\(55\) −11.3137 −1.52554
\(56\) 0 0
\(57\) −8.00000 −1.05963
\(58\) 0 0
\(59\) 1.41421 − 2.44949i 0.184115 − 0.318896i −0.759163 − 0.650901i \(-0.774391\pi\)
0.943278 + 0.332004i \(0.107725\pi\)
\(60\) 0 0
\(61\) 7.07107 + 12.2474i 0.905357 + 1.56813i 0.820437 + 0.571737i \(0.193730\pi\)
0.0849208 + 0.996388i \(0.472936\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.00000 + 6.92820i 0.496139 + 0.859338i
\(66\) 0 0
\(67\) 6.00000 − 10.3923i 0.733017 − 1.26962i −0.222571 − 0.974916i \(-0.571445\pi\)
0.955588 − 0.294706i \(-0.0952216\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(74\) 0 0
\(75\) −4.24264 − 7.34847i −0.489898 − 0.848528i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 4.00000 + 6.92820i 0.450035 + 0.779484i 0.998388 − 0.0567635i \(-0.0180781\pi\)
−0.548352 + 0.836247i \(0.684745\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) −14.1421 −1.55230 −0.776151 − 0.630548i \(-0.782830\pi\)
−0.776151 + 0.630548i \(0.782830\pi\)
\(84\) 0 0
\(85\) −16.0000 −1.73544
\(86\) 0 0
\(87\) −2.82843 + 4.89898i −0.303239 + 0.525226i
\(88\) 0 0
\(89\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 8.00000 + 13.8564i 0.829561 + 1.43684i
\(94\) 0 0
\(95\) −4.00000 + 6.92820i −0.410391 + 0.710819i
\(96\) 0 0
\(97\) −5.65685 −0.574367 −0.287183 − 0.957876i \(-0.592719\pi\)
−0.287183 + 0.957876i \(0.592719\pi\)
\(98\) 0 0
\(99\) 20.0000 2.01008
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.k.753.1 4
4.3 odd 2 392.2.i.g.361.2 4
7.2 even 3 inner 784.2.i.k.177.1 4
7.3 odd 6 784.2.a.n.1.1 2
7.4 even 3 784.2.a.n.1.2 2
7.5 odd 6 inner 784.2.i.k.177.2 4
7.6 odd 2 inner 784.2.i.k.753.2 4
12.11 even 2 3528.2.s.be.361.1 4
21.11 odd 6 7056.2.a.cj.1.2 2
21.17 even 6 7056.2.a.cj.1.1 2
28.3 even 6 392.2.a.h.1.2 yes 2
28.11 odd 6 392.2.a.h.1.1 ✓ 2
28.19 even 6 392.2.i.g.177.1 4
28.23 odd 6 392.2.i.g.177.2 4
28.27 even 2 392.2.i.g.361.1 4
56.3 even 6 3136.2.a.bt.1.1 2
56.11 odd 6 3136.2.a.bt.1.2 2
56.45 odd 6 3136.2.a.bq.1.2 2
56.53 even 6 3136.2.a.bq.1.1 2
84.11 even 6 3528.2.a.bj.1.2 2
84.23 even 6 3528.2.s.be.3313.1 4
84.47 odd 6 3528.2.s.be.3313.2 4
84.59 odd 6 3528.2.a.bj.1.1 2
84.83 odd 2 3528.2.s.be.361.2 4
140.39 odd 6 9800.2.a.bw.1.2 2
140.59 even 6 9800.2.a.bw.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
392.2.a.h.1.1 ✓ 2 28.11 odd 6
392.2.a.h.1.2 yes 2 28.3 even 6
392.2.i.g.177.1 4 28.19 even 6
392.2.i.g.177.2 4 28.23 odd 6
392.2.i.g.361.1 4 28.27 even 2
392.2.i.g.361.2 4 4.3 odd 2
784.2.a.n.1.1 2 7.3 odd 6
784.2.a.n.1.2 2 7.4 even 3
784.2.i.k.177.1 4 7.2 even 3 inner
784.2.i.k.177.2 4 7.5 odd 6 inner
784.2.i.k.753.1 4 1.1 even 1 trivial
784.2.i.k.753.2 4 7.6 odd 2 inner
3136.2.a.bq.1.1 2 56.53 even 6
3136.2.a.bq.1.2 2 56.45 odd 6
3136.2.a.bt.1.1 2 56.3 even 6
3136.2.a.bt.1.2 2 56.11 odd 6
3528.2.a.bj.1.1 2 84.59 odd 6
3528.2.a.bj.1.2 2 84.11 even 6
3528.2.s.be.361.1 4 12.11 even 2
3528.2.s.be.361.2 4 84.83 odd 2
3528.2.s.be.3313.1 4 84.23 even 6
3528.2.s.be.3313.2 4 84.47 odd 6
7056.2.a.cj.1.1 2 21.17 even 6
7056.2.a.cj.1.2 2 21.11 odd 6
9800.2.a.bw.1.1 2 140.59 even 6
9800.2.a.bw.1.2 2 140.39 odd 6