Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [637,2,Mod(165,637)] 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("637.165"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(637, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-2,1,10,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.59066497296.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 7x^{6} + 38x^{4} - 16x^{3} + 15x^{2} + 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 165.3
Root \(-0.115680 - 0.200364i\) of defining polynomial
Character \(\chi\) \(=\) 637.165
Dual form 637.2.h.i.471.3

$q$-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+0.231361 q^{2} +(-1.66113 - 2.87716i) q^{3} -1.94647 q^{4} +(-1.11568 - 1.93242i) q^{5} +(-0.384320 - 0.665661i) q^{6} -0.913059 q^{8} +(-4.01868 + 6.96056i) q^{9} +(-0.258125 - 0.447085i) q^{10} +(-1.66113 - 2.87716i) q^{11} +(3.23334 + 5.60030i) q^{12} +(-3.40300 - 1.19146i) q^{13} +(-3.70657 + 6.41997i) q^{15} +3.68170 q^{16} +1.37578 q^{17} +(-0.929766 + 1.61040i) q^{18} +(1.61766 - 2.80186i) q^{19} +(2.17164 + 3.76139i) q^{20} +(-0.384320 - 0.665661i) q^{22} +0.838502 q^{23} +(1.51671 + 2.62701i) q^{24} +(0.0105144 - 0.0182115i) q^{25} +(-0.787321 - 0.275657i) q^{26} +16.7354 q^{27} +(0.303571 - 0.525800i) q^{29} +(-0.857556 + 1.48533i) q^{30} +(0.857556 - 1.48533i) q^{31} +2.67792 q^{32} +(-5.51868 + 9.55864i) q^{33} +0.318302 q^{34} +(7.82225 - 13.5485i) q^{36} +1.55361 q^{37} +(0.374262 - 0.648241i) q^{38} +(2.22480 + 11.7701i) q^{39} +(1.01868 + 1.76441i) q^{40} +(-4.58892 + 7.94824i) q^{41} +(-0.615680 - 1.06639i) q^{43} +(3.23334 + 5.60030i) q^{44} +17.9343 q^{45} +0.193997 q^{46} +(-0.814085 - 1.41004i) q^{47} +(-6.11577 - 10.5928i) q^{48} +(0.00243263 - 0.00421343i) q^{50} +(-2.28535 - 3.95833i) q^{51} +(6.62385 + 2.31915i) q^{52} +(-4.19803 + 7.27121i) q^{53} +3.87192 q^{54} +(-3.70657 + 6.41997i) q^{55} -10.7485 q^{57} +(0.0702344 - 0.121650i) q^{58} -8.82234 q^{59} +(7.21474 - 12.4963i) q^{60} +(2.73334 - 4.73428i) q^{61} +(0.198405 - 0.343647i) q^{62} -6.74383 q^{64} +(1.49426 + 7.90530i) q^{65} +(-1.27681 + 2.21149i) q^{66} +(5.09287 + 8.82111i) q^{67} -2.67792 q^{68} +(-1.39286 - 2.41250i) q^{69} +(2.60714 + 4.51570i) q^{71} +(3.66929 - 6.35540i) q^{72} +(-1.98177 + 3.43253i) q^{73} +0.359445 q^{74} -0.0698632 q^{75} +(-3.14872 + 5.45375i) q^{76} +(0.514731 + 2.72315i) q^{78} +(-3.22525 - 5.58630i) q^{79} +(-4.10760 - 7.11457i) q^{80} +(-15.7436 - 27.2687i) q^{81} +(-1.06170 + 1.83891i) q^{82} +4.64055 q^{83} +(-1.53493 - 2.65858i) q^{85} +(-0.142444 - 0.246721i) q^{86} -2.01708 q^{87} +(1.51671 + 2.62701i) q^{88} -9.12826 q^{89} +4.14929 q^{90} -1.63212 q^{92} -5.69803 q^{93} +(-0.188347 - 0.326227i) q^{94} -7.21915 q^{95} +(-4.44836 - 7.70479i) q^{96} +(-7.67944 - 13.3012i) q^{97} +26.7022 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} + q^{3} + 10 q^{4} - 7 q^{5} - 5 q^{6} - 12 q^{8} - 7 q^{9} - 11 q^{10} + q^{11} + 12 q^{12} - 4 q^{13} - 3 q^{15} + 38 q^{16} + 8 q^{17} + 3 q^{18} + q^{19} - 2 q^{20} - 5 q^{22} - 4 q^{23}+ \cdots + 46 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/637\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(248\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.231361 0.163597 0.0817984 − 0.996649i \(-0.473934\pi\)
0.0817984 + 0.996649i \(0.473934\pi\)
\(3\) −1.66113 − 2.87716i −0.959052 − 1.66113i −0.724811 − 0.688948i \(-0.758073\pi\)
−0.234241 − 0.972179i \(-0.575260\pi\)
\(4\) −1.94647 −0.973236
\(5\) −1.11568 − 1.93242i −0.498947 − 0.864202i 0.501052 − 0.865417i \(-0.332947\pi\)
−0.999999 + 0.00121496i \(0.999613\pi\)
\(6\) −0.384320 − 0.665661i −0.156898 − 0.271755i
\(7\) 0 0
\(8\) −0.913059 −0.322815
\(9\) −4.01868 + 6.96056i −1.33956 + 2.32019i
\(10\) −0.258125 − 0.447085i −0.0816262 − 0.141381i
\(11\) −1.66113 − 2.87716i −0.500848 − 0.867495i −1.00000 0.000980003i \(-0.999688\pi\)
0.499151 − 0.866515i \(-0.333645\pi\)
\(12\) 3.23334 + 5.60030i 0.933384 + 1.61667i
\(13\) −3.40300 − 1.19146i −0.943823 − 0.330452i
\(14\) 0 0
\(15\) −3.70657 + 6.41997i −0.957033 + 1.65763i
\(16\) 3.68170 0.920425
\(17\) 1.37578 0.333676 0.166838 − 0.985984i \(-0.446644\pi\)
0.166838 + 0.985984i \(0.446644\pi\)
\(18\) −0.929766 + 1.61040i −0.219148 + 0.379575i
\(19\) 1.61766 − 2.80186i 0.371116 − 0.642791i −0.618622 − 0.785689i \(-0.712309\pi\)
0.989737 + 0.142898i \(0.0456420\pi\)
\(20\) 2.17164 + 3.76139i 0.485594 + 0.841073i
\(21\) 0 0
\(22\) −0.384320 − 0.665661i −0.0819372 − 0.141919i
\(23\) 0.838502 0.174840 0.0874199 − 0.996172i \(-0.472138\pi\)
0.0874199 + 0.996172i \(0.472138\pi\)
\(24\) 1.51671 + 2.62701i 0.309596 + 0.536237i
\(25\) 0.0105144 − 0.0182115i 0.00210289 − 0.00364231i
\(26\) −0.787321 − 0.275657i −0.154406 − 0.0540609i
\(27\) 16.7354 3.22073
\(28\) 0 0
\(29\) 0.303571 − 0.525800i 0.0563717 − 0.0976386i −0.836462 − 0.548024i \(-0.815380\pi\)
0.892834 + 0.450386i \(0.148713\pi\)
\(30\) −0.857556 + 1.48533i −0.156568 + 0.271183i
\(31\) 0.857556 − 1.48533i 0.154022 − 0.266773i −0.778681 − 0.627420i \(-0.784111\pi\)
0.932702 + 0.360647i \(0.117444\pi\)
\(32\) 2.67792 0.473394
\(33\) −5.51868 + 9.55864i −0.960679 + 1.66395i
\(34\) 0.318302 0.0545883
\(35\) 0 0
\(36\) 7.82225 − 13.5485i 1.30371 − 2.25809i
\(37\) 1.55361 0.255413 0.127706 − 0.991812i \(-0.459239\pi\)
0.127706 + 0.991812i \(0.459239\pi\)
\(38\) 0.374262 − 0.648241i 0.0607133 − 0.105159i
\(39\) 2.22480 + 11.7701i 0.356253 + 1.88473i
\(40\) 1.01868 + 1.76441i 0.161068 + 0.278978i
\(41\) −4.58892 + 7.94824i −0.716668 + 1.24131i 0.245644 + 0.969360i \(0.421001\pi\)
−0.962313 + 0.271946i \(0.912333\pi\)
\(42\) 0 0
\(43\) −0.615680 − 1.06639i −0.0938904 − 0.162623i 0.815255 − 0.579103i \(-0.196597\pi\)
−0.909145 + 0.416480i \(0.863264\pi\)
\(44\) 3.23334 + 5.60030i 0.487444 + 0.844277i
\(45\) 17.9343 2.67348
\(46\) 0.193997 0.0286032
\(47\) −0.814085 − 1.41004i −0.118747 − 0.205675i 0.800525 − 0.599300i \(-0.204554\pi\)
−0.919271 + 0.393625i \(0.871221\pi\)
\(48\) −6.11577 − 10.5928i −0.882735 − 1.52894i
\(49\) 0 0
\(50\) 0.00243263 − 0.00421343i 0.000344025 − 0.000595870i
\(51\) −2.28535 − 3.95833i −0.320012 − 0.554278i
\(52\) 6.62385 + 2.31915i 0.918562 + 0.321608i
\(53\) −4.19803 + 7.27121i −0.576644 + 0.998777i 0.419217 + 0.907886i \(0.362305\pi\)
−0.995861 + 0.0908909i \(0.971029\pi\)
\(54\) 3.87192 0.526901
\(55\) −3.70657 + 6.41997i −0.499794 + 0.865669i
\(56\) 0 0
\(57\) −10.7485 −1.42368
\(58\) 0.0702344 − 0.121650i 0.00922223 − 0.0159734i
\(59\) −8.82234 −1.14857 −0.574285 − 0.818655i \(-0.694720\pi\)
−0.574285 + 0.818655i \(0.694720\pi\)
\(60\) 7.21474 − 12.4963i 0.931419 − 1.61326i
\(61\) 2.73334 − 4.73428i 0.349968 − 0.606162i −0.636276 − 0.771462i \(-0.719526\pi\)
0.986243 + 0.165300i \(0.0528592\pi\)
\(62\) 0.198405 − 0.343647i 0.0251974 − 0.0436432i
\(63\) 0 0
\(64\) −6.74383 −0.842979
\(65\) 1.49426 + 7.90530i 0.185341 + 0.980532i
\(66\) −1.27681 + 2.21149i −0.157164 + 0.272216i
\(67\) 5.09287 + 8.82111i 0.622193 + 1.07767i 0.989077 + 0.147403i \(0.0470913\pi\)
−0.366884 + 0.930267i \(0.619575\pi\)
\(68\) −2.67792 −0.324745
\(69\) −1.39286 − 2.41250i −0.167680 − 0.290431i
\(70\) 0 0
\(71\) 2.60714 + 4.51570i 0.309411 + 0.535915i 0.978234 − 0.207507i \(-0.0665349\pi\)
−0.668823 + 0.743422i \(0.733202\pi\)
\(72\) 3.66929 − 6.35540i 0.432430 − 0.748992i
\(73\) −1.98177 + 3.43253i −0.231949 + 0.401748i −0.958382 − 0.285490i \(-0.907844\pi\)
0.726432 + 0.687238i \(0.241177\pi\)
\(74\) 0.359445 0.0417847
\(75\) −0.0698632 −0.00806711
\(76\) −3.14872 + 5.45375i −0.361183 + 0.625588i
\(77\) 0 0
\(78\) 0.514731 + 2.72315i 0.0582818 + 0.308336i
\(79\) −3.22525 − 5.58630i −0.362869 − 0.628508i 0.625562 − 0.780174i \(-0.284870\pi\)
−0.988432 + 0.151666i \(0.951536\pi\)
\(80\) −4.10760 − 7.11457i −0.459243 − 0.795433i
\(81\) −15.7436 − 27.2687i −1.74929 − 3.02985i
\(82\) −1.06170 + 1.83891i −0.117245 + 0.203074i
\(83\) 4.64055 0.509367 0.254684 − 0.967024i \(-0.418029\pi\)
0.254684 + 0.967024i \(0.418029\pi\)
\(84\) 0 0
\(85\) −1.53493 − 2.65858i −0.166487 − 0.288363i
\(86\) −0.142444 − 0.246721i −0.0153602 − 0.0266046i
\(87\) −2.01708 −0.216253
\(88\) 1.51671 + 2.62701i 0.161681 + 0.280041i
\(89\) −9.12826 −0.967593 −0.483797 − 0.875180i \(-0.660743\pi\)
−0.483797 + 0.875180i \(0.660743\pi\)
\(90\) 4.14929 0.437373
\(91\) 0 0
\(92\) −1.63212 −0.170160
\(93\) −5.69803 −0.590859
\(94\) −0.188347 − 0.326227i −0.0194266 − 0.0336478i
\(95\) −7.21915 −0.740669
\(96\) −4.44836 − 7.70479i −0.454009 − 0.786367i
\(97\) −7.67944 − 13.3012i −0.779729 − 1.35053i −0.932098 − 0.362206i \(-0.882024\pi\)
0.152369 − 0.988324i \(-0.451310\pi\)
\(98\) 0 0
\(99\) 26.7022 2.68367
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 637.2.h.i.165.3 8
7.2 even 3 637.2.g.j.373.2 8
7.3 odd 6 91.2.f.c.22.2 ✓ 8
7.4 even 3 637.2.f.i.295.2 8
7.5 odd 6 637.2.g.k.373.2 8
7.6 odd 2 637.2.h.h.165.3 8
13.3 even 3 637.2.g.j.263.2 8
21.17 even 6 819.2.o.h.568.3 8
28.3 even 6 1456.2.s.q.113.1 8
91.3 odd 6 91.2.f.c.29.2 yes 8
91.4 even 6 8281.2.a.bt.1.2 4
91.16 even 3 inner 637.2.h.i.471.3 8
91.17 odd 6 1183.2.a.l.1.2 4
91.45 even 12 1183.2.c.g.337.4 8
91.55 odd 6 637.2.g.k.263.2 8
91.59 even 12 1183.2.c.g.337.5 8
91.68 odd 6 637.2.h.h.471.3 8
91.74 even 3 8281.2.a.bp.1.3 4
91.81 even 3 637.2.f.i.393.2 8
91.87 odd 6 1183.2.a.k.1.3 4
273.185 even 6 819.2.o.h.757.3 8
364.3 even 6 1456.2.s.q.1121.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.f.c.22.2 ✓ 8 7.3 odd 6
91.2.f.c.29.2 yes 8 91.3 odd 6
637.2.f.i.295.2 8 7.4 even 3
637.2.f.i.393.2 8 91.81 even 3
637.2.g.j.263.2 8 13.3 even 3
637.2.g.j.373.2 8 7.2 even 3
637.2.g.k.263.2 8 91.55 odd 6
637.2.g.k.373.2 8 7.5 odd 6
637.2.h.h.165.3 8 7.6 odd 2
637.2.h.h.471.3 8 91.68 odd 6
637.2.h.i.165.3 8 1.1 even 1 trivial
637.2.h.i.471.3 8 91.16 even 3 inner
819.2.o.h.568.3 8 21.17 even 6
819.2.o.h.757.3 8 273.185 even 6
1183.2.a.k.1.3 4 91.87 odd 6
1183.2.a.l.1.2 4 91.17 odd 6
1183.2.c.g.337.4 8 91.45 even 12
1183.2.c.g.337.5 8 91.59 even 12
1456.2.s.q.113.1 8 28.3 even 6
1456.2.s.q.1121.1 8 364.3 even 6
8281.2.a.bp.1.3 4 91.74 even 3
8281.2.a.bt.1.2 4 91.4 even 6