Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1008,5,Mod(127,1008)] 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("1008.127"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 1008.m (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,36,0,0,0,0,0,0,0,-236] 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: \(104.196922789\)
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: \( 2^{18}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.1
Root \(-1.60961 + 2.78792i\) of defining polynomial
Character \(\chi\) \(=\) 1008.127
Dual form 1008.5.m.e.127.2

$q$-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)
 
\(f(q)\) \(=\) \(q-38.0135 q^{5} -18.5203i q^{7} +70.7571i q^{11} -59.7612 q^{13} -359.030 q^{17} -24.0021i q^{19} -985.841i q^{23} +820.029 q^{25} -1112.52 q^{29} -656.478i q^{31} +704.021i q^{35} -1922.60 q^{37} +304.913 q^{41} +1214.41i q^{43} +3315.46i q^{47} -343.000 q^{49} -57.6259 q^{53} -2689.73i q^{55} -4473.09i q^{59} -4637.64 q^{61} +2271.73 q^{65} -2605.64i q^{67} +4949.48i q^{71} -7893.99 q^{73} +1310.44 q^{77} -9566.11i q^{79} +5924.74i q^{83} +13648.0 q^{85} +4470.34 q^{89} +1106.79i q^{91} +912.406i q^{95} +15831.0 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 36 q^{5} - 236 q^{13} - 24 q^{17} + 624 q^{25} - 3864 q^{29} - 2408 q^{37} + 1464 q^{41} - 2744 q^{49} - 6960 q^{53} - 10836 q^{61} + 15240 q^{65} - 12352 q^{73} - 5880 q^{77} + 27544 q^{85} + 16272 q^{89}+ \cdots + 19576 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(5\) −38.0135 −1.52054 −0.760271 − 0.649606i \(-0.774934\pi\)
−0.760271 + 0.649606i \(0.774934\pi\)
\(6\) 0 0
\(7\) − 18.5203i − 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 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 \(-0.556577\pi\)
−0.176808 + 0.984245i \(0.556577\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 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
\(27\) 0 0
\(28\) 0 0
\(29\) −1112.52 −1.32286 −0.661428 − 0.750009i \(-0.730049\pi\)
−0.661428 + 0.750009i \(0.730049\pi\)
\(30\) 0 0
\(31\) − 656.478i − 0.683120i −0.939860 − 0.341560i \(-0.889045\pi\)
0.939860 − 0.341560i \(-0.110955\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 704.021i 0.574711i
\(36\) 0 0
\(37\) −1922.60 −1.40438 −0.702190 − 0.711990i \(-0.747794\pi\)
−0.702190 + 0.711990i \(0.747794\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 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 \(0.106508\pi\)
−0.944540 + 0.328397i \(0.893492\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3315.46i 1.50089i 0.660935 + 0.750443i \(0.270160\pi\)
−0.660935 + 0.750443i \(0.729840\pi\)
\(48\) 0 0
\(49\) −343.000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −57.6259 −0.0205147 −0.0102574 − 0.999947i \(-0.503265\pi\)
−0.0102574 + 0.999947i \(0.503265\pi\)
\(54\) 0 0
\(55\) − 2689.73i − 0.889167i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 4473.09i − 1.28500i −0.766286 − 0.642500i \(-0.777897\pi\)
0.766286 − 0.642500i \(-0.222103\pi\)
\(60\) 0 0
\(61\) −4637.64 −1.24634 −0.623171 − 0.782086i \(-0.714156\pi\)
−0.623171 + 0.782086i \(0.714156\pi\)
\(62\) 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 \(-0.906270\pi\)
0.956958 − 0.290225i \(-0.0937301\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 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 \(-0.765488\pi\)
−0.740663 + 0.671877i \(0.765488\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1310.44 0.221022
\(78\) 0 0
\(79\) − 9566.11i − 1.53278i −0.642373 − 0.766392i \(-0.722050\pi\)
0.642373 − 0.766392i \(-0.277950\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5924.74i 0.860029i 0.902822 + 0.430015i \(0.141492\pi\)
−0.902822 + 0.430015i \(0.858508\pi\)
\(84\) 0 0
\(85\) 13648.0 1.88900
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4470.34 0.564366 0.282183 − 0.959361i \(-0.408941\pi\)
0.282183 + 0.959361i \(0.408941\pi\)
\(90\) 0 0
\(91\) 1106.79i 0.133654i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 912.406i 0.101098i
\(96\) 0 0
\(97\) 15831.0 1.68254 0.841270 − 0.540616i \(-0.181809\pi\)
0.841270 + 0.540616i \(0.181809\pi\)
\(98\) 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 1008.5.m.e.127.1 8
3.2 odd 2 112.5.d.b.15.7 yes 8
4.3 odd 2 inner 1008.5.m.e.127.2 8
12.11 even 2 112.5.d.b.15.2 ✓ 8
21.20 even 2 784.5.d.j.687.2 8
24.5 odd 2 448.5.d.d.127.2 8
24.11 even 2 448.5.d.d.127.7 8
84.83 odd 2 784.5.d.j.687.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.d.b.15.2 ✓ 8 12.11 even 2
112.5.d.b.15.7 yes 8 3.2 odd 2
448.5.d.d.127.2 8 24.5 odd 2
448.5.d.d.127.7 8 24.11 even 2
784.5.d.j.687.2 8 21.20 even 2
784.5.d.j.687.7 8 84.83 odd 2
1008.5.m.e.127.1 8 1.1 even 1 trivial
1008.5.m.e.127.2 8 4.3 odd 2 inner