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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [272,4,Mod(33,272)] 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("272.33"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(272, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 272 = 2^{4} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 272.b (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,-40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(16.0485195216\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-37 +3 \sqrt{33}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 74x^{2} + 1072 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 17)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 33.2
Root \(-4.44593i\) of defining polynomial
Character \(\chi\) \(=\) 272.33
Dual form 272.4.b.d.33.3

$q$-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)
 
\(f(q)\) \(=\) \(q-4.44593i q^{3} -19.4389i q^{5} -14.9929i q^{7} +7.23369 q^{9} -31.1215i q^{11} -5.21194 q^{13} -86.4239 q^{15} +(68.2119 - 16.1286i) q^{17} +28.0000 q^{19} -66.6576 q^{21} +167.194i q^{23} -252.870 q^{25} -152.201i q^{27} +136.072i q^{29} +50.5604i q^{31} -138.364 q^{33} -291.446 q^{35} -260.558i q^{37} +23.1719i q^{39} +183.225i q^{41} +348.293 q^{43} -140.615i q^{45} -318.424 q^{47} +118.212 q^{49} +(-71.7066 - 303.266i) q^{51} -408.250 q^{53} -604.967 q^{55} -124.486i q^{57} -108.119 q^{59} -123.677i q^{61} -108.454i q^{63} +101.314i q^{65} +243.826 q^{67} +743.331 q^{69} -42.7075i q^{71} +875.172i q^{73} +1124.24i q^{75} -466.603 q^{77} -750.553i q^{79} -481.364 q^{81} +472.728 q^{83} +(-313.522 - 1325.96i) q^{85} +604.967 q^{87} +376.364 q^{89} +78.1422i q^{91} +224.788 q^{93} -544.288i q^{95} -303.169i q^{97} -225.123i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 40 q^{9} + 140 q^{13} - 24 q^{15} + 112 q^{17} + 112 q^{19} + 124 q^{21} - 460 q^{25} - 1036 q^{33} - 936 q^{35} + 520 q^{43} - 952 q^{47} + 312 q^{49} - 1160 q^{51} - 576 q^{53} - 168 q^{55} + 1176 q^{59}+ \cdots + 1060 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/272\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(239\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) − 4.44593i − 0.855620i −0.903869 − 0.427810i \(-0.859285\pi\)
0.903869 − 0.427810i \(-0.140715\pi\)
\(4\) 0 0
\(5\) − 19.4389i − 1.73867i −0.494228 − 0.869333i \(-0.664549\pi\)
0.494228 − 0.869333i \(-0.335451\pi\)
\(6\) 0 0
\(7\) − 14.9929i − 0.809542i −0.914418 − 0.404771i \(-0.867351\pi\)
0.914418 − 0.404771i \(-0.132649\pi\)
\(8\) 0 0
\(9\) 7.23369 0.267914
\(10\) 0 0
\(11\) − 31.1215i − 0.853045i −0.904477 − 0.426522i \(-0.859739\pi\)
0.904477 − 0.426522i \(-0.140261\pi\)
\(12\) 0 0
\(13\) −5.21194 −0.111195 −0.0555974 − 0.998453i \(-0.517706\pi\)
−0.0555974 + 0.998453i \(0.517706\pi\)
\(14\) 0 0
\(15\) −86.4239 −1.48764
\(16\) 0 0
\(17\) 68.2119 − 16.1286i 0.973166 − 0.230103i
\(18\) 0 0
\(19\) 28.0000 0.338086 0.169043 − 0.985609i \(-0.445932\pi\)
0.169043 + 0.985609i \(0.445932\pi\)
\(20\) 0 0
\(21\) −66.6576 −0.692661
\(22\) 0 0
\(23\) 167.194i 1.51575i 0.652399 + 0.757875i \(0.273763\pi\)
−0.652399 + 0.757875i \(0.726237\pi\)
\(24\) 0 0
\(25\) −252.870 −2.02296
\(26\) 0 0
\(27\) − 152.201i − 1.08485i
\(28\) 0 0
\(29\) 136.072i 0.871309i 0.900114 + 0.435654i \(0.143483\pi\)
−0.900114 + 0.435654i \(0.856517\pi\)
\(30\) 0 0
\(31\) 50.5604i 0.292933i 0.989216 + 0.146466i \(0.0467900\pi\)
−0.989216 + 0.146466i \(0.953210\pi\)
\(32\) 0 0
\(33\) −138.364 −0.729882
\(34\) 0 0
\(35\) −291.446 −1.40752
\(36\) 0 0
\(37\) − 260.558i − 1.15772i −0.815428 − 0.578858i \(-0.803499\pi\)
0.815428 − 0.578858i \(-0.196501\pi\)
\(38\) 0 0
\(39\) 23.1719i 0.0951404i
\(40\) 0 0
\(41\) 183.225i 0.697927i 0.937136 + 0.348964i \(0.113466\pi\)
−0.937136 + 0.348964i \(0.886534\pi\)
\(42\) 0 0
\(43\) 348.293 1.23521 0.617607 − 0.786486i \(-0.288102\pi\)
0.617607 + 0.786486i \(0.288102\pi\)
\(44\) 0 0
\(45\) − 140.615i − 0.465813i
\(46\) 0 0
\(47\) −318.424 −0.988232 −0.494116 − 0.869396i \(-0.664508\pi\)
−0.494116 + 0.869396i \(0.664508\pi\)
\(48\) 0 0
\(49\) 118.212 0.344641
\(50\) 0 0
\(51\) −71.7066 − 303.266i −0.196881 − 0.832660i
\(52\) 0 0
\(53\) −408.250 −1.05806 −0.529032 − 0.848602i \(-0.677445\pi\)
−0.529032 + 0.848602i \(0.677445\pi\)
\(54\) 0 0
\(55\) −604.967 −1.48316
\(56\) 0 0
\(57\) − 124.486i − 0.289273i
\(58\) 0 0
\(59\) −108.119 −0.238575 −0.119288 − 0.992860i \(-0.538061\pi\)
−0.119288 + 0.992860i \(0.538061\pi\)
\(60\) 0 0
\(61\) − 123.677i − 0.259593i −0.991541 − 0.129796i \(-0.958568\pi\)
0.991541 − 0.129796i \(-0.0414324\pi\)
\(62\) 0 0
\(63\) − 108.454i − 0.216888i
\(64\) 0 0
\(65\) 101.314i 0.193330i
\(66\) 0 0
\(67\) 243.826 0.444598 0.222299 − 0.974979i \(-0.428644\pi\)
0.222299 + 0.974979i \(0.428644\pi\)
\(68\) 0 0
\(69\) 743.331 1.29691
\(70\) 0 0
\(71\) − 42.7075i − 0.0713866i −0.999363 − 0.0356933i \(-0.988636\pi\)
0.999363 − 0.0356933i \(-0.0113639\pi\)
\(72\) 0 0
\(73\) 875.172i 1.40317i 0.712588 + 0.701583i \(0.247523\pi\)
−0.712588 + 0.701583i \(0.752477\pi\)
\(74\) 0 0
\(75\) 1124.24i 1.73088i
\(76\) 0 0
\(77\) −466.603 −0.690576
\(78\) 0 0
\(79\) − 750.553i − 1.06891i −0.845197 − 0.534454i \(-0.820517\pi\)
0.845197 − 0.534454i \(-0.179483\pi\)
\(80\) 0 0
\(81\) −481.364 −0.660308
\(82\) 0 0
\(83\) 472.728 0.625165 0.312582 − 0.949891i \(-0.398806\pi\)
0.312582 + 0.949891i \(0.398806\pi\)
\(84\) 0 0
\(85\) −313.522 − 1325.96i −0.400073 − 1.69201i
\(86\) 0 0
\(87\) 604.967 0.745509
\(88\) 0 0
\(89\) 376.364 0.448253 0.224127 − 0.974560i \(-0.428047\pi\)
0.224127 + 0.974560i \(0.428047\pi\)
\(90\) 0 0
\(91\) 78.1422i 0.0900169i
\(92\) 0 0
\(93\) 224.788 0.250639
\(94\) 0 0
\(95\) − 544.288i − 0.587819i
\(96\) 0 0
\(97\) − 303.169i − 0.317342i −0.987332 − 0.158671i \(-0.949279\pi\)
0.987332 − 0.158671i \(-0.0507209\pi\)
\(98\) 0 0
\(99\) − 225.123i − 0.228543i
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 272.4.b.d.33.2 4
4.3 odd 2 17.4.b.a.16.4 yes 4
12.11 even 2 153.4.d.b.118.2 4
17.16 even 2 inner 272.4.b.d.33.3 4
20.3 even 4 425.4.c.c.424.3 8
20.7 even 4 425.4.c.c.424.6 8
20.19 odd 2 425.4.d.c.101.1 4
68.47 odd 4 289.4.a.e.1.1 4
68.55 odd 4 289.4.a.e.1.2 4
68.67 odd 2 17.4.b.a.16.3 ✓ 4
204.203 even 2 153.4.d.b.118.1 4
340.67 even 4 425.4.c.c.424.5 8
340.203 even 4 425.4.c.c.424.4 8
340.339 odd 2 425.4.d.c.101.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.b.a.16.3 ✓ 4 68.67 odd 2
17.4.b.a.16.4 yes 4 4.3 odd 2
153.4.d.b.118.1 4 204.203 even 2
153.4.d.b.118.2 4 12.11 even 2
272.4.b.d.33.2 4 1.1 even 1 trivial
272.4.b.d.33.3 4 17.16 even 2 inner
289.4.a.e.1.1 4 68.47 odd 4
289.4.a.e.1.2 4 68.55 odd 4
425.4.c.c.424.3 8 20.3 even 4
425.4.c.c.424.4 8 340.203 even 4
425.4.c.c.424.5 8 340.67 even 4
425.4.c.c.424.6 8 20.7 even 4
425.4.d.c.101.1 4 20.19 odd 2
425.4.d.c.101.2 4 340.339 odd 2