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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [784,5,Mod(687,784)] 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("784.687"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 784.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,36,0,0,0,80,0,0,0,236,0,0,0,-24,0,0,0,0,0,0,0,624] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(81.0420510577\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.207528535809.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 21x^{6} - 2x^{5} + 265x^{4} - 66x^{3} + 1344x^{2} + 1080x + 3600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{22}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 687.7
Root \(-1.60961 + 2.78792i\) of defining polynomial
Character \(\chi\) \(=\) 784.687
Dual form 784.5.d.j.687.2

$q$-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)
 
\(f(q)\) \(=\) \(q+6.40550i q^{3} -38.0135 q^{5} +39.9695 q^{9} +70.7571i q^{11} +59.7612 q^{13} -243.496i q^{15} -359.030 q^{17} -24.0021i q^{19} -985.841i q^{23} +820.029 q^{25} +774.871i q^{27} +1112.52 q^{29} -656.478i q^{31} -453.235 q^{33} -1922.60 q^{37} +382.800i q^{39} +304.913 q^{41} -1214.41i q^{43} -1519.38 q^{45} -3315.46i q^{47} -2299.77i q^{51} +57.6259 q^{53} -2689.73i q^{55} +153.746 q^{57} +4473.09i q^{59} +4637.64 q^{61} -2271.73 q^{65} +2605.64i q^{67} +6314.81 q^{69} +4949.48i q^{71} +7893.99 q^{73} +5252.70i q^{75} +9566.11i q^{79} -1725.90 q^{81} -5924.74i q^{83} +13648.0 q^{85} +7126.26i q^{87} +4470.34 q^{89} +4205.07 q^{93} +912.406i q^{95} -15831.0 q^{97} +2828.13i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 36 q^{5} + 80 q^{9} + 236 q^{13} - 24 q^{17} + 624 q^{25} + 3864 q^{29} + 4160 q^{33} - 2408 q^{37} + 1464 q^{41} + 116 q^{45} + 6960 q^{53} + 23480 q^{57} + 10836 q^{61} - 15240 q^{65} + 8336 q^{69}+ \cdots - 19576 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 6.40550i 0.711722i 0.934539 + 0.355861i \(0.115812\pi\)
−0.934539 + 0.355861i \(0.884188\pi\)
\(4\) 0 0
\(5\) −38.0135 −1.52054 −0.760271 − 0.649606i \(-0.774934\pi\)
−0.760271 + 0.649606i \(0.774934\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 39.9695 0.493451
\(10\) 0 0
\(11\) 70.7571i 0.584770i 0.956301 + 0.292385i \(0.0944488\pi\)
−0.956301 + 0.292385i \(0.905551\pi\)
\(12\) 0 0
\(13\) 59.7612 0.353616 0.176808 − 0.984245i \(-0.443423\pi\)
0.176808 + 0.984245i \(0.443423\pi\)
\(14\) 0 0
\(15\) − 243.496i − 1.08220i
\(16\) 0 0
\(17\) −359.030 −1.24232 −0.621159 − 0.783685i \(-0.713338\pi\)
−0.621159 + 0.783685i \(0.713338\pi\)
\(18\) 0 0
\(19\) − 24.0021i − 0.0664879i −0.999447 − 0.0332439i \(-0.989416\pi\)
0.999447 − 0.0332439i \(-0.0105838\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 985.841i − 1.86359i −0.362980 − 0.931797i \(-0.618241\pi\)
0.362980 − 0.931797i \(-0.381759\pi\)
\(24\) 0 0
\(25\) 820.029 1.31205
\(26\) 0 0
\(27\) 774.871i 1.06292i
\(28\) 0 0
\(29\) 1112.52 1.32286 0.661428 − 0.750009i \(-0.269951\pi\)
0.661428 + 0.750009i \(0.269951\pi\)
\(30\) 0 0
\(31\) − 656.478i − 0.683120i −0.939860 − 0.341560i \(-0.889045\pi\)
0.939860 − 0.341560i \(-0.110955\pi\)
\(32\) 0 0
\(33\) −453.235 −0.416194
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1922.60 −1.40438 −0.702190 − 0.711990i \(-0.747794\pi\)
−0.702190 + 0.711990i \(0.747794\pi\)
\(38\) 0 0
\(39\) 382.800i 0.251677i
\(40\) 0 0
\(41\) 304.913 0.181388 0.0906941 − 0.995879i \(-0.471091\pi\)
0.0906941 + 0.995879i \(0.471091\pi\)
\(42\) 0 0
\(43\) − 1214.41i − 0.656794i −0.944540 − 0.328397i \(-0.893492\pi\)
0.944540 − 0.328397i \(-0.106508\pi\)
\(44\) 0 0
\(45\) −1519.38 −0.750313
\(46\) 0 0
\(47\) − 3315.46i − 1.50089i −0.660935 − 0.750443i \(-0.729840\pi\)
0.660935 − 0.750443i \(-0.270160\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) − 2299.77i − 0.884186i
\(52\) 0 0
\(53\) 57.6259 0.0205147 0.0102574 − 0.999947i \(-0.496735\pi\)
0.0102574 + 0.999947i \(0.496735\pi\)
\(54\) 0 0
\(55\) − 2689.73i − 0.889167i
\(56\) 0 0
\(57\) 153.746 0.0473209
\(58\) 0 0
\(59\) 4473.09i 1.28500i 0.766286 + 0.642500i \(0.222103\pi\)
−0.766286 + 0.642500i \(0.777897\pi\)
\(60\) 0 0
\(61\) 4637.64 1.24634 0.623171 − 0.782086i \(-0.285844\pi\)
0.623171 + 0.782086i \(0.285844\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2271.73 −0.537689
\(66\) 0 0
\(67\) 2605.64i 0.580450i 0.956958 + 0.290225i \(0.0937301\pi\)
−0.956958 + 0.290225i \(0.906270\pi\)
\(68\) 0 0
\(69\) 6314.81 1.32636
\(70\) 0 0
\(71\) 4949.48i 0.981844i 0.871203 + 0.490922i \(0.163340\pi\)
−0.871203 + 0.490922i \(0.836660\pi\)
\(72\) 0 0
\(73\) 7893.99 1.48133 0.740663 − 0.671877i \(-0.234512\pi\)
0.740663 + 0.671877i \(0.234512\pi\)
\(74\) 0 0
\(75\) 5252.70i 0.933813i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 9566.11i 1.53278i 0.642373 + 0.766392i \(0.277950\pi\)
−0.642373 + 0.766392i \(0.722050\pi\)
\(80\) 0 0
\(81\) −1725.90 −0.263055
\(82\) 0 0
\(83\) − 5924.74i − 0.860029i −0.902822 − 0.430015i \(-0.858508\pi\)
0.902822 − 0.430015i \(-0.141492\pi\)
\(84\) 0 0
\(85\) 13648.0 1.88900
\(86\) 0 0
\(87\) 7126.26i 0.941506i
\(88\) 0 0
\(89\) 4470.34 0.564366 0.282183 − 0.959361i \(-0.408941\pi\)
0.282183 + 0.959361i \(0.408941\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 4205.07 0.486192
\(94\) 0 0
\(95\) 912.406i 0.101098i
\(96\) 0 0
\(97\) −15831.0 −1.68254 −0.841270 − 0.540616i \(-0.818191\pi\)
−0.841270 + 0.540616i \(0.818191\pi\)
\(98\) 0 0
\(99\) 2828.13i 0.288555i
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 784.5.d.j.687.7 8
4.3 odd 2 inner 784.5.d.j.687.2 8
7.6 odd 2 112.5.d.b.15.2 ✓ 8
21.20 even 2 1008.5.m.e.127.2 8
28.27 even 2 112.5.d.b.15.7 yes 8
56.13 odd 2 448.5.d.d.127.7 8
56.27 even 2 448.5.d.d.127.2 8
84.83 odd 2 1008.5.m.e.127.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.d.b.15.2 ✓ 8 7.6 odd 2
112.5.d.b.15.7 yes 8 28.27 even 2
448.5.d.d.127.2 8 56.27 even 2
448.5.d.d.127.7 8 56.13 odd 2
784.5.d.j.687.2 8 4.3 odd 2 inner
784.5.d.j.687.7 8 1.1 even 1 trivial
1008.5.m.e.127.1 8 84.83 odd 2
1008.5.m.e.127.2 8 21.20 even 2