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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 239.20
Character \(\chi\) \(=\) 448.239
Dual form 448.5.k.a.15.20

$q$-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 \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

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

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
\(3\) −1.91051 + 1.91051i −0.212279 + 0.212279i −0.805235 − 0.592956i \(-0.797961\pi\)
0.592956 + 0.805235i \(0.297961\pi\)
\(4\) 0 0
\(5\) 33.8528 − 33.8528i 1.35411 − 1.35411i 0.473109 − 0.881004i \(-0.343132\pi\)
0.881004 − 0.473109i \(-0.156868\pi\)
\(6\) 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 0.155100 − 0.987899i \(-0.450430\pi\)
−0.987899 + 0.155100i \(0.950430\pi\)
\(12\) 0 0
\(13\) 28.6391 + 28.6391i 0.169462 + 0.169462i 0.786743 − 0.617281i \(-0.211766\pi\)
−0.617281 + 0.786743i \(0.711766\pi\)
\(14\) 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 −0.782862 − 0.622196i \(-0.786241\pi\)
0.622196 + 0.782862i \(0.286241\pi\)
\(20\) 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 0.500510 − 0.865731i \(-0.333146\pi\)
−0.865731 + 0.500510i \(0.833146\pi\)
\(30\) 0 0
\(31\) 763.779i 0.794775i 0.917651 + 0.397387i \(0.130083\pi\)
−0.917651 + 0.397387i \(0.869917\pi\)
\(32\) 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 −0.805725 − 0.592290i \(-0.798224\pi\)
0.592290 + 0.805725i \(0.298224\pi\)
\(38\) 0 0
\(39\) −109.431 −0.0719465
\(40\) 0 0
\(41\) − 1393.66i − 0.829063i −0.910035 − 0.414532i \(-0.863945\pi\)
0.910035 − 0.414532i \(-0.136055\pi\)
\(42\) 0 0
\(43\) −757.492 − 757.492i −0.409677 − 0.409677i 0.471949 − 0.881626i \(-0.343551\pi\)
−0.881626 + 0.471949i \(0.843551\pi\)
\(44\) 0 0
\(45\) 2494.95 + 2494.95i 1.23207 + 1.23207i
\(46\) 0 0
\(47\) 494.734i 0.223963i 0.993710 + 0.111981i \(0.0357197\pi\)
−0.993710 + 0.111981i \(0.964280\pi\)
\(48\) 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 −0.669908 − 0.742445i \(-0.733666\pi\)
0.742445 + 0.669908i \(0.233666\pi\)
\(54\) 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 0.0352997 − 0.999377i \(-0.488761\pi\)
−0.999377 + 0.0352997i \(0.988761\pi\)
\(60\) 0 0
\(61\) −4499.31 − 4499.31i −1.20917 − 1.20917i −0.971297 − 0.237871i \(-0.923550\pi\)
−0.237871 − 0.971297i \(-0.576450\pi\)
\(62\) 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 0.176157 − 0.984362i \(-0.443633\pi\)
0.984362 − 0.176157i \(-0.0563668\pi\)
\(68\) 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 \(0.0560192\pi\)
−0.984554 + 0.175082i \(0.943981\pi\)
\(74\) 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 \(-0.805991\pi\)
0.819937 − 0.572454i \(-0.194009\pi\)
\(80\) 0 0
\(81\) −4840.36 −0.737748
\(82\) 0 0
\(83\) 1759.48 − 1759.48i 0.255404 − 0.255404i −0.567778 − 0.823182i \(-0.692197\pi\)
0.823182 + 0.567778i \(0.192197\pi\)
\(84\) 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 \(0.0276533\pi\)
−0.996229 + 0.0867662i \(0.972347\pi\)
\(90\) 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.239.20 96
4.3 odd 2 112.5.k.a.99.13 yes 96
16.5 even 4 112.5.k.a.43.13 ✓ 96
16.11 odd 4 inner 448.5.k.a.15.20 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.k.a.43.13 ✓ 96 16.5 even 4
112.5.k.a.99.13 yes 96 4.3 odd 2
448.5.k.a.15.20 96 16.11 odd 4 inner
448.5.k.a.239.20 96 1.1 even 1 trivial