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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [168,2,Mod(125,168)] 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("168.125"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(168, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 168.i (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,4,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.34148675396\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 125.3
Root \(1.22474 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 168.125
Dual form 168.2.i.c.125.4

$q$-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+(1.22474 - 0.707107i) q^{2} +(1.00000 + 1.41421i) q^{3} +(1.00000 - 1.73205i) q^{4} +1.41421i q^{5} +(2.22474 + 1.02494i) q^{6} +(-2.00000 - 1.73205i) q^{7} -2.82843i q^{8} +(-1.00000 + 2.82843i) q^{9} +(1.00000 + 1.73205i) q^{10} +2.44949 q^{11} +(3.44949 - 0.317837i) q^{12} -2.00000 q^{13} +(-3.67423 - 0.707107i) q^{14} +(-2.00000 + 1.41421i) q^{15} +(-2.00000 - 3.46410i) q^{16} -7.34847 q^{17} +(0.775255 + 4.17121i) q^{18} +4.00000 q^{19} +(2.44949 + 1.41421i) q^{20} +(0.449490 - 4.56048i) q^{21} +(3.00000 - 1.73205i) q^{22} -1.41421i q^{23} +(4.00000 - 2.82843i) q^{24} +3.00000 q^{25} +(-2.44949 + 1.41421i) q^{26} +(-5.00000 + 1.41421i) q^{27} +(-5.00000 + 1.73205i) q^{28} -4.89898 q^{29} +(-1.44949 + 3.14626i) q^{30} -6.92820i q^{31} +(-4.89898 - 2.82843i) q^{32} +(2.44949 + 3.46410i) q^{33} +(-9.00000 + 5.19615i) q^{34} +(2.44949 - 2.82843i) q^{35} +(3.89898 + 4.56048i) q^{36} +10.3923i q^{37} +(4.89898 - 2.82843i) q^{38} +(-2.00000 - 2.82843i) q^{39} +4.00000 q^{40} +2.44949 q^{41} +(-2.67423 - 5.90326i) q^{42} +3.46410i q^{43} +(2.44949 - 4.24264i) q^{44} +(-4.00000 - 1.41421i) q^{45} +(-1.00000 - 1.73205i) q^{46} +4.89898 q^{47} +(2.89898 - 6.29253i) q^{48} +(1.00000 + 6.92820i) q^{49} +(3.67423 - 2.12132i) q^{50} +(-7.34847 - 10.3923i) q^{51} +(-2.00000 + 3.46410i) q^{52} +(-5.12372 + 5.26758i) q^{54} +3.46410i q^{55} +(-4.89898 + 5.65685i) q^{56} +(4.00000 + 5.65685i) q^{57} +(-6.00000 + 3.46410i) q^{58} +5.65685i q^{59} +(0.449490 + 4.87832i) q^{60} +10.0000 q^{61} +(-4.89898 - 8.48528i) q^{62} +(6.89898 - 3.92480i) q^{63} -8.00000 q^{64} -2.82843i q^{65} +(5.44949 + 2.51059i) q^{66} -3.46410i q^{67} +(-7.34847 + 12.7279i) q^{68} +(2.00000 - 1.41421i) q^{69} +(1.00000 - 5.19615i) q^{70} -1.41421i q^{71} +(8.00000 + 2.82843i) q^{72} +(7.34847 + 12.7279i) q^{74} +(3.00000 + 4.24264i) q^{75} +(4.00000 - 6.92820i) q^{76} +(-4.89898 - 4.24264i) q^{77} +(-4.44949 - 2.04989i) q^{78} +8.00000 q^{79} +(4.89898 - 2.82843i) q^{80} +(-7.00000 - 5.65685i) q^{81} +(3.00000 - 1.73205i) q^{82} -11.3137i q^{83} +(-7.44949 - 5.33902i) q^{84} -10.3923i q^{85} +(2.44949 + 4.24264i) q^{86} +(-4.89898 - 6.92820i) q^{87} -6.92820i q^{88} +7.34847 q^{89} +(-5.89898 + 1.09638i) q^{90} +(4.00000 + 3.46410i) q^{91} +(-2.44949 - 1.41421i) q^{92} +(9.79796 - 6.92820i) q^{93} +(6.00000 - 3.46410i) q^{94} +5.65685i q^{95} +(-0.898979 - 9.75663i) q^{96} -13.8564i q^{97} +(6.12372 + 7.77817i) q^{98} +(-2.44949 + 6.92820i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 4 q^{4} + 4 q^{6} - 8 q^{7} - 4 q^{9} + 4 q^{10} + 4 q^{12} - 8 q^{13} - 8 q^{15} - 8 q^{16} + 8 q^{18} + 16 q^{19} - 8 q^{21} + 12 q^{22} + 16 q^{24} + 12 q^{25} - 20 q^{27} - 20 q^{28}+ \cdots + 16 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(85\) \(113\) \(127\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.22474 − 0.707107i 0.866025 − 0.500000i
\(3\) 1.00000 + 1.41421i 0.577350 + 0.816497i
\(4\) 1.00000 − 1.73205i 0.500000 − 0.866025i
\(5\) 1.41421i 0.632456i 0.948683 + 0.316228i \(0.102416\pi\)
−0.948683 + 0.316228i \(0.897584\pi\)
\(6\) 2.22474 + 1.02494i 0.908248 + 0.418432i
\(7\) −2.00000 − 1.73205i −0.755929 − 0.654654i
\(8\) − 2.82843i − 1.00000i
\(9\) −1.00000 + 2.82843i −0.333333 + 0.942809i
\(10\) 1.00000 + 1.73205i 0.316228 + 0.547723i
\(11\) 2.44949 0.738549 0.369274 − 0.929320i \(-0.379606\pi\)
0.369274 + 0.929320i \(0.379606\pi\)
\(12\) 3.44949 − 0.317837i 0.995782 − 0.0917517i
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −3.67423 − 0.707107i −0.981981 − 0.188982i
\(15\) −2.00000 + 1.41421i −0.516398 + 0.365148i
\(16\) −2.00000 − 3.46410i −0.500000 − 0.866025i
\(17\) −7.34847 −1.78227 −0.891133 − 0.453743i \(-0.850089\pi\)
−0.891133 + 0.453743i \(0.850089\pi\)
\(18\) 0.775255 + 4.17121i 0.182729 + 0.983163i
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 2.44949 + 1.41421i 0.547723 + 0.316228i
\(21\) 0.449490 − 4.56048i 0.0980867 − 0.995178i
\(22\) 3.00000 − 1.73205i 0.639602 − 0.369274i
\(23\) − 1.41421i − 0.294884i −0.989071 − 0.147442i \(-0.952896\pi\)
0.989071 − 0.147442i \(-0.0471040\pi\)
\(24\) 4.00000 − 2.82843i 0.816497 − 0.577350i
\(25\) 3.00000 0.600000
\(26\) −2.44949 + 1.41421i −0.480384 + 0.277350i
\(27\) −5.00000 + 1.41421i −0.962250 + 0.272166i
\(28\) −5.00000 + 1.73205i −0.944911 + 0.327327i
\(29\) −4.89898 −0.909718 −0.454859 − 0.890564i \(-0.650310\pi\)
−0.454859 + 0.890564i \(0.650310\pi\)
\(30\) −1.44949 + 3.14626i −0.264639 + 0.574427i
\(31\) − 6.92820i − 1.24434i −0.782881 − 0.622171i \(-0.786251\pi\)
0.782881 − 0.622171i \(-0.213749\pi\)
\(32\) −4.89898 − 2.82843i −0.866025 − 0.500000i
\(33\) 2.44949 + 3.46410i 0.426401 + 0.603023i
\(34\) −9.00000 + 5.19615i −1.54349 + 0.891133i
\(35\) 2.44949 − 2.82843i 0.414039 − 0.478091i
\(36\) 3.89898 + 4.56048i 0.649830 + 0.760080i
\(37\) 10.3923i 1.70848i 0.519875 + 0.854242i \(0.325978\pi\)
−0.519875 + 0.854242i \(0.674022\pi\)
\(38\) 4.89898 − 2.82843i 0.794719 − 0.458831i
\(39\) −2.00000 − 2.82843i −0.320256 − 0.452911i
\(40\) 4.00000 0.632456
\(41\) 2.44949 0.382546 0.191273 − 0.981537i \(-0.438738\pi\)
0.191273 + 0.981537i \(0.438738\pi\)
\(42\) −2.67423 − 5.90326i −0.412643 − 0.910893i
\(43\) 3.46410i 0.528271i 0.964486 + 0.264135i \(0.0850865\pi\)
−0.964486 + 0.264135i \(0.914913\pi\)
\(44\) 2.44949 − 4.24264i 0.369274 − 0.639602i
\(45\) −4.00000 − 1.41421i −0.596285 − 0.210819i
\(46\) −1.00000 − 1.73205i −0.147442 − 0.255377i
\(47\) 4.89898 0.714590 0.357295 − 0.933992i \(-0.383699\pi\)
0.357295 + 0.933992i \(0.383699\pi\)
\(48\) 2.89898 − 6.29253i 0.418432 − 0.908248i
\(49\) 1.00000 + 6.92820i 0.142857 + 0.989743i
\(50\) 3.67423 − 2.12132i 0.519615 − 0.300000i
\(51\) −7.34847 − 10.3923i −1.02899 − 1.45521i
\(52\) −2.00000 + 3.46410i −0.277350 + 0.480384i
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) −5.12372 + 5.26758i −0.697251 + 0.716827i
\(55\) 3.46410i 0.467099i
\(56\) −4.89898 + 5.65685i −0.654654 + 0.755929i
\(57\) 4.00000 + 5.65685i 0.529813 + 0.749269i
\(58\) −6.00000 + 3.46410i −0.787839 + 0.454859i
\(59\) 5.65685i 0.736460i 0.929735 + 0.368230i \(0.120036\pi\)
−0.929735 + 0.368230i \(0.879964\pi\)
\(60\) 0.449490 + 4.87832i 0.0580289 + 0.629788i
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) −4.89898 − 8.48528i −0.622171 − 1.07763i
\(63\) 6.89898 − 3.92480i 0.869190 − 0.494479i
\(64\) −8.00000 −1.00000
\(65\) − 2.82843i − 0.350823i
\(66\) 5.44949 + 2.51059i 0.670786 + 0.309032i
\(67\) − 3.46410i − 0.423207i −0.977356 − 0.211604i \(-0.932131\pi\)
0.977356 − 0.211604i \(-0.0678686\pi\)
\(68\) −7.34847 + 12.7279i −0.891133 + 1.54349i
\(69\) 2.00000 − 1.41421i 0.240772 − 0.170251i
\(70\) 1.00000 − 5.19615i 0.119523 − 0.621059i
\(71\) − 1.41421i − 0.167836i −0.996473 − 0.0839181i \(-0.973257\pi\)
0.996473 − 0.0839181i \(-0.0267434\pi\)
\(72\) 8.00000 + 2.82843i 0.942809 + 0.333333i
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 7.34847 + 12.7279i 0.854242 + 1.47959i
\(75\) 3.00000 + 4.24264i 0.346410 + 0.489898i
\(76\) 4.00000 − 6.92820i 0.458831 − 0.794719i
\(77\) −4.89898 − 4.24264i −0.558291 − 0.483494i
\(78\) −4.44949 − 2.04989i −0.503806 − 0.232104i
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 4.89898 − 2.82843i 0.547723 − 0.316228i
\(81\) −7.00000 − 5.65685i −0.777778 − 0.628539i
\(82\) 3.00000 − 1.73205i 0.331295 − 0.191273i
\(83\) − 11.3137i − 1.24184i −0.783874 − 0.620920i \(-0.786759\pi\)
0.783874 − 0.620920i \(-0.213241\pi\)
\(84\) −7.44949 − 5.33902i −0.812806 − 0.582535i
\(85\) − 10.3923i − 1.12720i
\(86\) 2.44949 + 4.24264i 0.264135 + 0.457496i
\(87\) −4.89898 − 6.92820i −0.525226 − 0.742781i
\(88\) − 6.92820i − 0.738549i
\(89\) 7.34847 0.778936 0.389468 − 0.921040i \(-0.372659\pi\)
0.389468 + 0.921040i \(0.372659\pi\)
\(90\) −5.89898 + 1.09638i −0.621807 + 0.115568i
\(91\) 4.00000 + 3.46410i 0.419314 + 0.363137i
\(92\) −2.44949 − 1.41421i −0.255377 − 0.147442i
\(93\) 9.79796 − 6.92820i 1.01600 − 0.718421i
\(94\) 6.00000 − 3.46410i 0.618853 − 0.357295i
\(95\) 5.65685i 0.580381i
\(96\) −0.898979 − 9.75663i −0.0917517 − 0.995782i
\(97\) − 13.8564i − 1.40690i −0.710742 − 0.703452i \(-0.751641\pi\)
0.710742 − 0.703452i \(-0.248359\pi\)
\(98\) 6.12372 + 7.77817i 0.618590 + 0.785714i
\(99\) −2.44949 + 6.92820i −0.246183 + 0.696311i
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 168.2.i.c.125.3 yes 4
3.2 odd 2 inner 168.2.i.c.125.2 yes 4
4.3 odd 2 672.2.i.a.209.2 4
7.6 odd 2 168.2.i.a.125.3 yes 4
8.3 odd 2 672.2.i.c.209.4 4
8.5 even 2 168.2.i.a.125.1 ✓ 4
12.11 even 2 672.2.i.a.209.4 4
21.20 even 2 168.2.i.a.125.2 yes 4
24.5 odd 2 168.2.i.a.125.4 yes 4
24.11 even 2 672.2.i.c.209.2 4
28.27 even 2 672.2.i.c.209.3 4
56.13 odd 2 inner 168.2.i.c.125.1 yes 4
56.27 even 2 672.2.i.a.209.1 4
84.83 odd 2 672.2.i.c.209.1 4
168.83 odd 2 672.2.i.a.209.3 4
168.125 even 2 inner 168.2.i.c.125.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.i.a.125.1 ✓ 4 8.5 even 2
168.2.i.a.125.2 yes 4 21.20 even 2
168.2.i.a.125.3 yes 4 7.6 odd 2
168.2.i.a.125.4 yes 4 24.5 odd 2
168.2.i.c.125.1 yes 4 56.13 odd 2 inner
168.2.i.c.125.2 yes 4 3.2 odd 2 inner
168.2.i.c.125.3 yes 4 1.1 even 1 trivial
168.2.i.c.125.4 yes 4 168.125 even 2 inner
672.2.i.a.209.1 4 56.27 even 2
672.2.i.a.209.2 4 4.3 odd 2
672.2.i.a.209.3 4 168.83 odd 2
672.2.i.a.209.4 4 12.11 even 2
672.2.i.c.209.1 4 84.83 odd 2
672.2.i.c.209.2 4 24.11 even 2
672.2.i.c.209.3 4 28.27 even 2
672.2.i.c.209.4 4 8.3 odd 2