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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 239.2
Character \(\chi\) \(=\) 448.239
Dual form 448.5.k.a.15.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-11.4158 + 11.4158i) q^{3} +(8.99601 - 8.99601i) q^{5} +18.5203 q^{7} -179.643i q^{9} +(-16.8258 - 16.8258i) q^{11} +(-139.940 - 139.940i) q^{13} +205.394i q^{15} +327.145 q^{17} +(-134.031 + 134.031i) q^{19} +(-211.424 + 211.424i) q^{21} +352.136 q^{23} +463.143i q^{25} +(1126.09 + 1126.09i) q^{27} +(-533.411 - 533.411i) q^{29} +1492.89i q^{31} +384.161 q^{33} +(166.609 - 166.609i) q^{35} +(431.104 - 431.104i) q^{37} +3195.08 q^{39} +2824.85i q^{41} +(341.119 + 341.119i) q^{43} +(-1616.07 - 1616.07i) q^{45} -3999.13i q^{47} +343.000 q^{49} +(-3734.64 + 3734.64i) q^{51} +(524.825 - 524.825i) q^{53} -302.730 q^{55} -3060.16i q^{57} +(-25.1602 - 25.1602i) q^{59} +(202.929 + 202.929i) q^{61} -3327.03i q^{63} -2517.81 q^{65} +(-5554.26 + 5554.26i) q^{67} +(-4019.93 + 4019.93i) q^{69} -4830.28 q^{71} +4640.36i q^{73} +(-5287.17 - 5287.17i) q^{75} +(-311.618 - 311.618i) q^{77} +2215.33i q^{79} -11159.5 q^{81} +(-8545.70 + 8545.70i) q^{83} +(2943.00 - 2943.00i) q^{85} +12178.7 q^{87} -8352.39i q^{89} +(-2591.73 - 2591.73i) q^{91} +(-17042.6 - 17042.6i) q^{93} +2411.50i q^{95} +12536.7 q^{97} +(-3022.63 + 3022.63i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 96 q^{11} - 1408 q^{19} + 1152 q^{23} - 3648 q^{27} - 864 q^{29} + 1824 q^{37} + 5376 q^{39} + 6176 q^{43} + 32928 q^{49} + 8064 q^{51} + 480 q^{53} - 23552 q^{55} + 7488 q^{59} + 7552 q^{61} + 4032 q^{65}+ \cdots + 59552 q^{99}+O(q^{100}) \) 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
\(3\) −11.4158 + 11.4158i −1.26843 + 1.26843i −0.321527 + 0.946901i \(0.604196\pi\)
−0.946901 + 0.321527i \(0.895804\pi\)
\(4\) 0 0
\(5\) 8.99601 − 8.99601i 0.359841 − 0.359841i −0.503914 − 0.863754i \(-0.668107\pi\)
0.863754 + 0.503914i \(0.168107\pi\)
\(6\) 0 0
\(7\) 18.5203 0.377964
\(8\) 0 0
\(9\) − 179.643i − 2.21781i
\(10\) 0 0
\(11\) −16.8258 − 16.8258i −0.139056 − 0.139056i 0.634152 − 0.773208i \(-0.281349\pi\)
−0.773208 + 0.634152i \(0.781349\pi\)
\(12\) 0 0
\(13\) −139.940 − 139.940i −0.828050 − 0.828050i 0.159197 − 0.987247i \(-0.449109\pi\)
−0.987247 + 0.159197i \(0.949109\pi\)
\(14\) 0 0
\(15\) 205.394i 0.912863i
\(16\) 0 0
\(17\) 327.145 1.13199 0.565995 − 0.824409i \(-0.308492\pi\)
0.565995 + 0.824409i \(0.308492\pi\)
\(18\) 0 0
\(19\) −134.031 + 134.031i −0.371278 + 0.371278i −0.867943 − 0.496665i \(-0.834558\pi\)
0.496665 + 0.867943i \(0.334558\pi\)
\(20\) 0 0
\(21\) −211.424 + 211.424i −0.479420 + 0.479420i
\(22\) 0 0
\(23\) 352.136 0.665663 0.332832 − 0.942986i \(-0.391996\pi\)
0.332832 + 0.942986i \(0.391996\pi\)
\(24\) 0 0
\(25\) 463.143i 0.741030i
\(26\) 0 0
\(27\) 1126.09 + 1126.09i 1.54471 + 1.54471i
\(28\) 0 0
\(29\) −533.411 − 533.411i −0.634259 − 0.634259i 0.314875 − 0.949133i \(-0.398038\pi\)
−0.949133 + 0.314875i \(0.898038\pi\)
\(30\) 0 0
\(31\) 1492.89i 1.55348i 0.629823 + 0.776739i \(0.283128\pi\)
−0.629823 + 0.776739i \(0.716872\pi\)
\(32\) 0 0
\(33\) 384.161 0.352765
\(34\) 0 0
\(35\) 166.609 − 166.609i 0.136007 − 0.136007i
\(36\) 0 0
\(37\) 431.104 − 431.104i 0.314904 − 0.314904i −0.531902 − 0.846806i \(-0.678522\pi\)
0.846806 + 0.531902i \(0.178522\pi\)
\(38\) 0 0
\(39\) 3195.08 2.10064
\(40\) 0 0
\(41\) 2824.85i 1.68046i 0.542233 + 0.840228i \(0.317579\pi\)
−0.542233 + 0.840228i \(0.682421\pi\)
\(42\) 0 0
\(43\) 341.119 + 341.119i 0.184488 + 0.184488i 0.793308 − 0.608820i \(-0.208357\pi\)
−0.608820 + 0.793308i \(0.708357\pi\)
\(44\) 0 0
\(45\) −1616.07 − 1616.07i −0.798060 − 0.798060i
\(46\) 0 0
\(47\) − 3999.13i − 1.81038i −0.425009 − 0.905189i \(-0.639729\pi\)
0.425009 − 0.905189i \(-0.360271\pi\)
\(48\) 0 0
\(49\) 343.000 0.142857
\(50\) 0 0
\(51\) −3734.64 + 3734.64i −1.43585 + 1.43585i
\(52\) 0 0
\(53\) 524.825 − 524.825i 0.186837 − 0.186837i −0.607490 − 0.794327i \(-0.707824\pi\)
0.794327 + 0.607490i \(0.207824\pi\)
\(54\) 0 0
\(55\) −302.730 −0.100076
\(56\) 0 0
\(57\) − 3060.16i − 0.941878i
\(58\) 0 0
\(59\) −25.1602 − 25.1602i −0.00722788 − 0.00722788i 0.703484 − 0.710711i \(-0.251627\pi\)
−0.710711 + 0.703484i \(0.751627\pi\)
\(60\) 0 0
\(61\) 202.929 + 202.929i 0.0545361 + 0.0545361i 0.733849 − 0.679313i \(-0.237722\pi\)
−0.679313 + 0.733849i \(0.737722\pi\)
\(62\) 0 0
\(63\) − 3327.03i − 0.838255i
\(64\) 0 0
\(65\) −2517.81 −0.595932
\(66\) 0 0
\(67\) −5554.26 + 5554.26i −1.23731 + 1.23731i −0.276207 + 0.961098i \(0.589078\pi\)
−0.961098 + 0.276207i \(0.910922\pi\)
\(68\) 0 0
\(69\) −4019.93 + 4019.93i −0.844345 + 0.844345i
\(70\) 0 0
\(71\) −4830.28 −0.958198 −0.479099 − 0.877761i \(-0.659037\pi\)
−0.479099 + 0.877761i \(0.659037\pi\)
\(72\) 0 0
\(73\) 4640.36i 0.870775i 0.900243 + 0.435387i \(0.143389\pi\)
−0.900243 + 0.435387i \(0.856611\pi\)
\(74\) 0 0
\(75\) −5287.17 − 5287.17i −0.939942 − 0.939942i
\(76\) 0 0
\(77\) −311.618 − 311.618i −0.0525583 − 0.0525583i
\(78\) 0 0
\(79\) 2215.33i 0.354963i 0.984124 + 0.177482i \(0.0567950\pi\)
−0.984124 + 0.177482i \(0.943205\pi\)
\(80\) 0 0
\(81\) −11159.5 −1.70089
\(82\) 0 0
\(83\) −8545.70 + 8545.70i −1.24048 + 1.24048i −0.280684 + 0.959800i \(0.590561\pi\)
−0.959800 + 0.280684i \(0.909439\pi\)
\(84\) 0 0
\(85\) 2943.00 − 2943.00i 0.407336 − 0.407336i
\(86\) 0 0
\(87\) 12178.7 1.60902
\(88\) 0 0
\(89\) − 8352.39i − 1.05446i −0.849722 − 0.527231i \(-0.823230\pi\)
0.849722 − 0.527231i \(-0.176770\pi\)
\(90\) 0 0
\(91\) −2591.73 − 2591.73i −0.312973 − 0.312973i
\(92\) 0 0
\(93\) −17042.6 − 17042.6i −1.97047 − 1.97047i
\(94\) 0 0
\(95\) 2411.50i 0.267202i
\(96\) 0 0
\(97\) 12536.7 1.33241 0.666207 − 0.745767i \(-0.267917\pi\)
0.666207 + 0.745767i \(0.267917\pi\)
\(98\) 0 0
\(99\) −3022.63 + 3022.63i −0.308401 + 0.308401i
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.239.2 96
4.3 odd 2 112.5.k.a.99.5 yes 96
16.5 even 4 112.5.k.a.43.5 ✓ 96
16.11 odd 4 inner 448.5.k.a.15.2 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.k.a.43.5 ✓ 96 16.5 even 4
112.5.k.a.99.5 yes 96 4.3 odd 2
448.5.k.a.15.2 96 16.11 odd 4 inner
448.5.k.a.239.2 96 1.1 even 1 trivial