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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 15.20
Character \(\chi\) \(=\) 448.15
Dual form 448.5.k.a.239.20

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.91051 - 1.91051i) q^{3} +(33.8528 + 33.8528i) q^{5} -18.5203 q^{7} -73.6999i q^{9} +(-100.769 + 100.769i) q^{11} +(28.6391 - 28.6391i) q^{13} -129.353i q^{15} +549.495 q^{17} +(-58.0004 - 58.0004i) q^{19} +(35.3832 + 35.3832i) q^{21} +874.669 q^{23} +1667.03i q^{25} +(-295.556 + 295.556i) q^{27} +(-307.151 + 307.151i) q^{29} -763.779i q^{31} +385.040 q^{33} +(-626.963 - 626.963i) q^{35} +(-292.192 - 292.192i) q^{37} -109.431 q^{39} +1393.66i q^{41} +(-757.492 + 757.492i) q^{43} +(2494.95 - 2494.95i) q^{45} -494.734i q^{47} +343.000 q^{49} +(-1049.82 - 1049.82i) q^{51} +(203.756 + 203.756i) q^{53} -6822.61 q^{55} +221.621i q^{57} +(-3355.95 + 3355.95i) q^{59} +(-4499.31 + 4499.31i) q^{61} +1364.94i q^{63} +1939.03 q^{65} +(5209.57 + 5209.57i) q^{67} +(-1671.07 - 1671.07i) q^{69} +6529.16 q^{71} -1866.03i q^{73} +(3184.88 - 3184.88i) q^{75} +(1866.26 - 1866.26i) q^{77} +7145.38i q^{79} -4840.36 q^{81} +(1759.48 + 1759.48i) q^{83} +(18601.9 + 18601.9i) q^{85} +1173.63 q^{87} -1374.55i q^{89} +(-530.403 + 530.403i) q^{91} +(-1459.21 + 1459.21i) q^{93} -3926.95i q^{95} +5954.34 q^{97} +(7426.64 + 7426.64i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 96 q^{11} - 1408 q^{19} + 1152 q^{23} - 3648 q^{27} - 864 q^{29} + 1824 q^{37} + 5376 q^{39} + 6176 q^{43} + 32928 q^{49} + 8064 q^{51} + 480 q^{53} - 23552 q^{55} + 7488 q^{59} + 7552 q^{61} + 4032 q^{65}+ \cdots + 59552 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(3\) −1.91051 − 1.91051i −0.212279 − 0.212279i 0.592956 − 0.805235i \(-0.297961\pi\)
−0.805235 + 0.592956i \(0.797961\pi\)
\(4\) 0 0
\(5\) 33.8528 + 33.8528i 1.35411 + 1.35411i 0.881004 + 0.473109i \(0.156868\pi\)
0.473109 + 0.881004i \(0.343132\pi\)
\(6\) 0 0
\(7\) −18.5203 −0.377964
\(8\) 0 0
\(9\) − 73.6999i − 0.909875i
\(10\) 0 0
\(11\) −100.769 + 100.769i −0.832799 + 0.832799i −0.987899 − 0.155100i \(-0.950430\pi\)
0.155100 + 0.987899i \(0.450430\pi\)
\(12\) 0 0
\(13\) 28.6391 − 28.6391i 0.169462 − 0.169462i −0.617281 − 0.786743i \(-0.711766\pi\)
0.786743 + 0.617281i \(0.211766\pi\)
\(14\) 0 0
\(15\) − 129.353i − 0.574900i
\(16\) 0 0
\(17\) 549.495 1.90137 0.950683 − 0.310164i \(-0.100384\pi\)
0.950683 + 0.310164i \(0.100384\pi\)
\(18\) 0 0
\(19\) −58.0004 − 58.0004i −0.160666 − 0.160666i 0.622196 − 0.782862i \(-0.286241\pi\)
−0.782862 + 0.622196i \(0.786241\pi\)
\(20\) 0 0
\(21\) 35.3832 + 35.3832i 0.0802340 + 0.0802340i
\(22\) 0 0
\(23\) 874.669 1.65344 0.826719 − 0.562615i \(-0.190205\pi\)
0.826719 + 0.562615i \(0.190205\pi\)
\(24\) 0 0
\(25\) 1667.03i 2.66724i
\(26\) 0 0
\(27\) −295.556 + 295.556i −0.405427 + 0.405427i
\(28\) 0 0
\(29\) −307.151 + 307.151i −0.365221 + 0.365221i −0.865731 − 0.500510i \(-0.833146\pi\)
0.500510 + 0.865731i \(0.333146\pi\)
\(30\) 0 0
\(31\) − 763.779i − 0.794775i −0.917651 − 0.397387i \(-0.869917\pi\)
0.917651 − 0.397387i \(-0.130083\pi\)
\(32\) 0 0
\(33\) 385.040 0.353572
\(34\) 0 0
\(35\) −626.963 − 626.963i −0.511807 − 0.511807i
\(36\) 0 0
\(37\) −292.192 − 292.192i −0.213435 − 0.213435i 0.592290 − 0.805725i \(-0.298224\pi\)
−0.805725 + 0.592290i \(0.798224\pi\)
\(38\) 0 0
\(39\) −109.431 −0.0719465
\(40\) 0 0
\(41\) 1393.66i 0.829063i 0.910035 + 0.414532i \(0.136055\pi\)
−0.910035 + 0.414532i \(0.863945\pi\)
\(42\) 0 0
\(43\) −757.492 + 757.492i −0.409677 + 0.409677i −0.881626 − 0.471949i \(-0.843551\pi\)
0.471949 + 0.881626i \(0.343551\pi\)
\(44\) 0 0
\(45\) 2494.95 − 2494.95i 1.23207 − 1.23207i
\(46\) 0 0
\(47\) − 494.734i − 0.223963i −0.993710 − 0.111981i \(-0.964280\pi\)
0.993710 − 0.111981i \(-0.0357197\pi\)
\(48\) 0 0
\(49\) 343.000 0.142857
\(50\) 0 0
\(51\) −1049.82 − 1049.82i −0.403621 − 0.403621i
\(52\) 0 0
\(53\) 203.756 + 203.756i 0.0725369 + 0.0725369i 0.742445 − 0.669908i \(-0.233666\pi\)
−0.669908 + 0.742445i \(0.733666\pi\)
\(54\) 0 0
\(55\) −6822.61 −2.25541
\(56\) 0 0
\(57\) 221.621i 0.0682121i
\(58\) 0 0
\(59\) −3355.95 + 3355.95i −0.964077 + 0.964077i −0.999377 − 0.0352997i \(-0.988761\pi\)
0.0352997 + 0.999377i \(0.488761\pi\)
\(60\) 0 0
\(61\) −4499.31 + 4499.31i −1.20917 + 1.20917i −0.237871 + 0.971297i \(0.576450\pi\)
−0.971297 + 0.237871i \(0.923550\pi\)
\(62\) 0 0
\(63\) 1364.94i 0.343900i
\(64\) 0 0
\(65\) 1939.03 0.458941
\(66\) 0 0
\(67\) 5209.57 + 5209.57i 1.16052 + 1.16052i 0.984362 + 0.176157i \(0.0563668\pi\)
0.176157 + 0.984362i \(0.443633\pi\)
\(68\) 0 0
\(69\) −1671.07 − 1671.07i −0.350991 − 0.350991i
\(70\) 0 0
\(71\) 6529.16 1.29521 0.647606 − 0.761975i \(-0.275770\pi\)
0.647606 + 0.761975i \(0.275770\pi\)
\(72\) 0 0
\(73\) − 1866.03i − 0.350165i −0.984554 − 0.175082i \(-0.943981\pi\)
0.984554 − 0.175082i \(-0.0560192\pi\)
\(74\) 0 0
\(75\) 3184.88 − 3184.88i 0.566200 − 0.566200i
\(76\) 0 0
\(77\) 1866.26 − 1866.26i 0.314768 − 0.314768i
\(78\) 0 0
\(79\) 7145.38i 1.14491i 0.819937 + 0.572454i \(0.194009\pi\)
−0.819937 + 0.572454i \(0.805991\pi\)
\(80\) 0 0
\(81\) −4840.36 −0.737748
\(82\) 0 0
\(83\) 1759.48 + 1759.48i 0.255404 + 0.255404i 0.823182 − 0.567778i \(-0.192197\pi\)
−0.567778 + 0.823182i \(0.692197\pi\)
\(84\) 0 0
\(85\) 18601.9 + 18601.9i 2.57466 + 2.57466i
\(86\) 0 0
\(87\) 1173.63 0.155058
\(88\) 0 0
\(89\) − 1374.55i − 0.173532i −0.996229 − 0.0867662i \(-0.972347\pi\)
0.996229 − 0.0867662i \(-0.0276533\pi\)
\(90\) 0 0
\(91\) −530.403 + 530.403i −0.0640506 + 0.0640506i
\(92\) 0 0
\(93\) −1459.21 + 1459.21i −0.168714 + 0.168714i
\(94\) 0 0
\(95\) − 3926.95i − 0.435120i
\(96\) 0 0
\(97\) 5954.34 0.632835 0.316417 − 0.948620i \(-0.397520\pi\)
0.316417 + 0.948620i \(0.397520\pi\)
\(98\) 0 0
\(99\) 7426.64 + 7426.64i 0.757743 + 0.757743i
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.15.20 96
4.3 odd 2 112.5.k.a.43.13 ✓ 96
16.3 odd 4 inner 448.5.k.a.239.20 96
16.13 even 4 112.5.k.a.99.13 yes 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.k.a.43.13 ✓ 96 4.3 odd 2
112.5.k.a.99.13 yes 96 16.13 even 4
448.5.k.a.15.20 96 1.1 even 1 trivial
448.5.k.a.239.20 96 16.3 odd 4 inner