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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1008,5,Mod(449,1008)] 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("1008.449"); 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([0, 0, 1, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 1008.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,472] 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: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{7})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 8x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 3\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.2
Root \(-1.16372i\) of defining polynomial
Character \(\chi\) \(=\) 1008.449
Dual form 1008.5.d.b.449.3

$q$-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)
 
\(f(q)\) \(=\) \(q-7.80683i q^{5} -18.5203 q^{7} +53.7974i q^{11} +192.081 q^{13} +46.1625i q^{17} -545.608 q^{19} -260.158i q^{23} +564.053 q^{25} +469.920i q^{29} +3.04052 q^{31} +144.584i q^{35} +63.2156 q^{37} -98.0690i q^{41} -2173.76 q^{43} -851.567i q^{47} +343.000 q^{49} -4633.53i q^{53} +419.987 q^{55} +496.598i q^{59} +1983.93 q^{61} -1499.54i q^{65} +1893.54 q^{67} +8581.96i q^{71} -8414.36 q^{73} -996.342i q^{77} -1985.35 q^{79} +12089.9i q^{83} +360.383 q^{85} +3823.51i q^{89} -3557.39 q^{91} +4259.46i q^{95} -7837.31 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 472 q^{13} + 40 q^{19} - 1596 q^{25} - 136 q^{31} - 4192 q^{37} - 9584 q^{43} + 1372 q^{49} + 5384 q^{55} + 4528 q^{61} + 1944 q^{67} - 21360 q^{73} - 10312 q^{79} + 29296 q^{85} - 5488 q^{91} - 44832 q^{97}+O(q^{100}) \) 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
\(5\) − 7.80683i − 0.312273i −0.987735 − 0.156137i \(-0.950096\pi\)
0.987735 − 0.156137i \(-0.0499040\pi\)
\(6\) 0 0
\(7\) −18.5203 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 53.7974i 0.444607i 0.974978 + 0.222303i \(0.0713575\pi\)
−0.974978 + 0.222303i \(0.928642\pi\)
\(12\) 0 0
\(13\) 192.081 1.13657 0.568287 − 0.822830i \(-0.307606\pi\)
0.568287 + 0.822830i \(0.307606\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 46.1625i 0.159732i 0.996806 + 0.0798659i \(0.0254492\pi\)
−0.996806 + 0.0798659i \(0.974551\pi\)
\(18\) 0 0
\(19\) −545.608 −1.51138 −0.755689 − 0.654930i \(-0.772698\pi\)
−0.755689 + 0.654930i \(0.772698\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 260.158i − 0.491792i −0.969296 − 0.245896i \(-0.920918\pi\)
0.969296 − 0.245896i \(-0.0790822\pi\)
\(24\) 0 0
\(25\) 564.053 0.902486
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 469.920i 0.558763i 0.960180 + 0.279382i \(0.0901295\pi\)
−0.960180 + 0.279382i \(0.909871\pi\)
\(30\) 0 0
\(31\) 3.04052 0.00316391 0.00158196 − 0.999999i \(-0.499496\pi\)
0.00158196 + 0.999999i \(0.499496\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 144.584i 0.118028i
\(36\) 0 0
\(37\) 63.2156 0.0461764 0.0230882 − 0.999733i \(-0.492650\pi\)
0.0230882 + 0.999733i \(0.492650\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 98.0690i − 0.0583397i −0.999574 − 0.0291698i \(-0.990714\pi\)
0.999574 − 0.0291698i \(-0.00928636\pi\)
\(42\) 0 0
\(43\) −2173.76 −1.17564 −0.587820 − 0.808992i \(-0.700014\pi\)
−0.587820 + 0.808992i \(0.700014\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 851.567i − 0.385499i −0.981248 − 0.192750i \(-0.938260\pi\)
0.981248 − 0.192750i \(-0.0617405\pi\)
\(48\) 0 0
\(49\) 343.000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 4633.53i − 1.64953i −0.565474 − 0.824766i \(-0.691307\pi\)
0.565474 − 0.824766i \(-0.308693\pi\)
\(54\) 0 0
\(55\) 419.987 0.138839
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 496.598i 0.142659i 0.997453 + 0.0713297i \(0.0227243\pi\)
−0.997453 + 0.0713297i \(0.977276\pi\)
\(60\) 0 0
\(61\) 1983.93 0.533172 0.266586 − 0.963811i \(-0.414104\pi\)
0.266586 + 0.963811i \(0.414104\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 1499.54i − 0.354921i
\(66\) 0 0
\(67\) 1893.54 0.421818 0.210909 − 0.977506i \(-0.432358\pi\)
0.210909 + 0.977506i \(0.432358\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8581.96i 1.70243i 0.524816 + 0.851216i \(0.324134\pi\)
−0.524816 + 0.851216i \(0.675866\pi\)
\(72\) 0 0
\(73\) −8414.36 −1.57898 −0.789488 − 0.613766i \(-0.789654\pi\)
−0.789488 + 0.613766i \(0.789654\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 996.342i − 0.168046i
\(78\) 0 0
\(79\) −1985.35 −0.318114 −0.159057 − 0.987269i \(-0.550845\pi\)
−0.159057 + 0.987269i \(0.550845\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12089.9i 1.75496i 0.479611 + 0.877481i \(0.340778\pi\)
−0.479611 + 0.877481i \(0.659222\pi\)
\(84\) 0 0
\(85\) 360.383 0.0498799
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3823.51i 0.482705i 0.970438 + 0.241352i \(0.0775910\pi\)
−0.970438 + 0.241352i \(0.922409\pi\)
\(90\) 0 0
\(91\) −3557.39 −0.429585
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4259.46i 0.471963i
\(96\) 0 0
\(97\) −7837.31 −0.832959 −0.416480 − 0.909145i \(-0.636736\pi\)
−0.416480 + 0.909145i \(0.636736\pi\)
\(98\) 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.d.b.449.2 4
3.2 odd 2 inner 1008.5.d.b.449.3 4
4.3 odd 2 126.5.b.a.71.3 yes 4
12.11 even 2 126.5.b.a.71.2 ✓ 4
28.27 even 2 882.5.b.d.197.4 4
84.83 odd 2 882.5.b.d.197.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.5.b.a.71.2 ✓ 4 12.11 even 2
126.5.b.a.71.3 yes 4 4.3 odd 2
882.5.b.d.197.1 4 84.83 odd 2
882.5.b.d.197.4 4 28.27 even 2
1008.5.d.b.449.2 4 1.1 even 1 trivial
1008.5.d.b.449.3 4 3.2 odd 2 inner