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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [448,5,Mod(15,448)] 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("448.15"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 448.k (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(46.3097434616\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(48\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 239.4
Character \(\chi\) \(=\) 448.239
Dual form 448.5.k.a.15.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(-10.8491 + 10.8491i) q^{3} +(-31.7296 + 31.7296i) q^{5} -18.5203 q^{7} -154.406i q^{9} +(126.465 + 126.465i) q^{11} +(-44.9589 - 44.9589i) q^{13} -688.475i q^{15} -110.269 q^{17} +(125.611 - 125.611i) q^{19} +(200.928 - 200.928i) q^{21} +915.524 q^{23} -1388.54i q^{25} +(796.390 + 796.390i) q^{27} +(-459.446 - 459.446i) q^{29} -1611.67i q^{31} -2744.06 q^{33} +(587.640 - 587.640i) q^{35} +(964.893 - 964.893i) q^{37} +975.527 q^{39} +936.914i q^{41} +(895.948 + 895.948i) q^{43} +(4899.24 + 4899.24i) q^{45} +1582.55i q^{47} +343.000 q^{49} +(1196.32 - 1196.32i) q^{51} +(-909.872 + 909.872i) q^{53} -8025.34 q^{55} +2725.53i q^{57} +(950.591 + 950.591i) q^{59} +(-658.183 - 658.183i) q^{61} +2859.64i q^{63} +2853.05 q^{65} +(2974.42 - 2974.42i) q^{67} +(-9932.62 + 9932.62i) q^{69} -7205.11 q^{71} -2350.55i q^{73} +(15064.4 + 15064.4i) q^{75} +(-2342.16 - 2342.16i) q^{77} +341.141i q^{79} -4773.34 q^{81} +(-6304.16 + 6304.16i) q^{83} +(3498.80 - 3498.80i) q^{85} +9969.15 q^{87} -4326.70i q^{89} +(832.650 + 832.650i) q^{91} +(17485.2 + 17485.2i) q^{93} +7971.18i q^{95} +3632.54 q^{97} +(19526.9 - 19526.9i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 96 q^{11} - 1408 q^{19} + 1152 q^{23} - 3648 q^{27} - 864 q^{29} + 1824 q^{37} + 5376 q^{39} + 6176 q^{43} + 32928 q^{49} + 8064 q^{51} + 480 q^{53} - 23552 q^{55} + 7488 q^{59} + 7552 q^{61} + 4032 q^{65}+ \cdots + 59552 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{3}{4}\right)\)

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
\(3\) −10.8491 + 10.8491i −1.20546 + 1.20546i −0.232973 + 0.972483i \(0.574845\pi\)
−0.972483 + 0.232973i \(0.925155\pi\)
\(4\) 0 0
\(5\) −31.7296 + 31.7296i −1.26918 + 1.26918i −0.322674 + 0.946510i \(0.604582\pi\)
−0.946510 + 0.322674i \(0.895418\pi\)
\(6\) 0 0
\(7\) −18.5203 −0.377964
\(8\) 0 0
\(9\) − 154.406i − 1.90625i
\(10\) 0 0
\(11\) 126.465 + 126.465i 1.04516 + 1.04516i 0.998931 + 0.0462313i \(0.0147211\pi\)
0.0462313 + 0.998931i \(0.485279\pi\)
\(12\) 0 0
\(13\) −44.9589 − 44.9589i −0.266029 − 0.266029i 0.561469 − 0.827498i \(-0.310236\pi\)
−0.827498 + 0.561469i \(0.810236\pi\)
\(14\) 0 0
\(15\) − 688.475i − 3.05989i
\(16\) 0 0
\(17\) −110.269 −0.381555 −0.190777 − 0.981633i \(-0.561101\pi\)
−0.190777 + 0.981633i \(0.561101\pi\)
\(18\) 0 0
\(19\) 125.611 − 125.611i 0.347953 − 0.347953i −0.511394 − 0.859347i \(-0.670871\pi\)
0.859347 + 0.511394i \(0.170871\pi\)
\(20\) 0 0
\(21\) 200.928 − 200.928i 0.455619 − 0.455619i
\(22\) 0 0
\(23\) 915.524 1.73067 0.865335 − 0.501194i \(-0.167106\pi\)
0.865335 + 0.501194i \(0.167106\pi\)
\(24\) 0 0
\(25\) − 1388.54i − 2.22166i
\(26\) 0 0
\(27\) 796.390 + 796.390i 1.09244 + 1.09244i
\(28\) 0 0
\(29\) −459.446 − 459.446i −0.546309 − 0.546309i 0.379062 − 0.925371i \(-0.376247\pi\)
−0.925371 + 0.379062i \(0.876247\pi\)
\(30\) 0 0
\(31\) − 1611.67i − 1.67707i −0.544844 − 0.838537i \(-0.683411\pi\)
0.544844 − 0.838537i \(-0.316589\pi\)
\(32\) 0 0
\(33\) −2744.06 −2.51979
\(34\) 0 0
\(35\) 587.640 − 587.640i 0.479706 − 0.479706i
\(36\) 0 0
\(37\) 964.893 − 964.893i 0.704816 − 0.704816i −0.260624 − 0.965440i \(-0.583928\pi\)
0.965440 + 0.260624i \(0.0839283\pi\)
\(38\) 0 0
\(39\) 975.527 0.641372
\(40\) 0 0
\(41\) 936.914i 0.557355i 0.960385 + 0.278678i \(0.0898961\pi\)
−0.960385 + 0.278678i \(0.910104\pi\)
\(42\) 0 0
\(43\) 895.948 + 895.948i 0.484558 + 0.484558i 0.906584 − 0.422026i \(-0.138681\pi\)
−0.422026 + 0.906584i \(0.638681\pi\)
\(44\) 0 0
\(45\) 4899.24 + 4899.24i 2.41938 + 2.41938i
\(46\) 0 0
\(47\) 1582.55i 0.716409i 0.933643 + 0.358204i \(0.116611\pi\)
−0.933643 + 0.358204i \(0.883389\pi\)
\(48\) 0 0
\(49\) 343.000 0.142857
\(50\) 0 0
\(51\) 1196.32 − 1196.32i 0.459947 − 0.459947i
\(52\) 0 0
\(53\) −909.872 + 909.872i −0.323913 + 0.323913i −0.850266 − 0.526353i \(-0.823559\pi\)
0.526353 + 0.850266i \(0.323559\pi\)
\(54\) 0 0
\(55\) −8025.34 −2.65301
\(56\) 0 0
\(57\) 2725.53i 0.838884i
\(58\) 0 0
\(59\) 950.591 + 950.591i 0.273080 + 0.273080i 0.830339 − 0.557259i \(-0.188147\pi\)
−0.557259 + 0.830339i \(0.688147\pi\)
\(60\) 0 0
\(61\) −658.183 − 658.183i −0.176883 − 0.176883i 0.613112 − 0.789996i \(-0.289917\pi\)
−0.789996 + 0.613112i \(0.789917\pi\)
\(62\) 0 0
\(63\) 2859.64i 0.720494i
\(64\) 0 0
\(65\) 2853.05 0.675279
\(66\) 0 0
\(67\) 2974.42 − 2974.42i 0.662603 − 0.662603i −0.293390 − 0.955993i \(-0.594783\pi\)
0.955993 + 0.293390i \(0.0947835\pi\)
\(68\) 0 0
\(69\) −9932.62 + 9932.62i −2.08625 + 2.08625i
\(70\) 0 0
\(71\) −7205.11 −1.42930 −0.714651 − 0.699481i \(-0.753414\pi\)
−0.714651 + 0.699481i \(0.753414\pi\)
\(72\) 0 0
\(73\) − 2350.55i − 0.441087i −0.975377 − 0.220543i \(-0.929217\pi\)
0.975377 − 0.220543i \(-0.0707830\pi\)
\(74\) 0 0
\(75\) 15064.4 + 15064.4i 2.67811 + 2.67811i
\(76\) 0 0
\(77\) −2342.16 − 2342.16i −0.395034 − 0.395034i
\(78\) 0 0
\(79\) 341.141i 0.0546613i 0.999626 + 0.0273306i \(0.00870069\pi\)
−0.999626 + 0.0273306i \(0.991299\pi\)
\(80\) 0 0
\(81\) −4773.34 −0.727533
\(82\) 0 0
\(83\) −6304.16 + 6304.16i −0.915105 + 0.915105i −0.996668 − 0.0815627i \(-0.974009\pi\)
0.0815627 + 0.996668i \(0.474009\pi\)
\(84\) 0 0
\(85\) 3498.80 − 3498.80i 0.484263 − 0.484263i
\(86\) 0 0
\(87\) 9969.15 1.31710
\(88\) 0 0
\(89\) − 4326.70i − 0.546231i −0.961981 − 0.273115i \(-0.911946\pi\)
0.961981 − 0.273115i \(-0.0880541\pi\)
\(90\) 0 0
\(91\) 832.650 + 832.650i 0.100549 + 0.100549i
\(92\) 0 0
\(93\) 17485.2 + 17485.2i 2.02164 + 2.02164i
\(94\) 0 0
\(95\) 7971.18i 0.883233i
\(96\) 0 0
\(97\) 3632.54 0.386070 0.193035 − 0.981192i \(-0.438167\pi\)
0.193035 + 0.981192i \(0.438167\pi\)
\(98\) 0 0
\(99\) 19526.9 − 19526.9i 1.99234 − 1.99234i
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 448.5.k.a.239.4 96
4.3 odd 2 112.5.k.a.99.6 yes 96
16.5 even 4 112.5.k.a.43.6 ✓ 96
16.11 odd 4 inner 448.5.k.a.15.4 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.k.a.43.6 ✓ 96 16.5 even 4
112.5.k.a.99.6 yes 96 4.3 odd 2
448.5.k.a.15.4 96 16.11 odd 4 inner
448.5.k.a.239.4 96 1.1 even 1 trivial