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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [725,2,Mod(349,725)] 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("725.349"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(725, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 725.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-4,0,8,0,0,-4] 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: \(5.78915414654\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 145)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 349.3
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 725.349
Dual form 725.2.b.c.349.2

$q$-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)
 
\(f(q)\) \(=\) \(q+0.414214i q^{2} +2.00000i q^{3} +1.82843 q^{4} -0.828427 q^{6} -4.82843i q^{7} +1.58579i q^{8} -1.00000 q^{9} +0.828427 q^{11} +3.65685i q^{12} +2.00000i q^{13} +2.00000 q^{14} +3.00000 q^{16} +2.82843i q^{17} -0.414214i q^{18} +4.82843 q^{19} +9.65685 q^{21} +0.343146i q^{22} +3.17157i q^{23} -3.17157 q^{24} -0.828427 q^{26} +4.00000i q^{27} -8.82843i q^{28} -1.00000 q^{29} +6.48528 q^{31} +4.41421i q^{32} +1.65685i q^{33} -1.17157 q^{34} -1.82843 q^{36} -8.48528i q^{37} +2.00000i q^{38} -4.00000 q^{39} -6.00000 q^{41} +4.00000i q^{42} +6.00000i q^{43} +1.51472 q^{44} -1.31371 q^{46} -11.6569i q^{47} +6.00000i q^{48} -16.3137 q^{49} -5.65685 q^{51} +3.65685i q^{52} +3.65685i q^{53} -1.65685 q^{54} +7.65685 q^{56} +9.65685i q^{57} -0.414214i q^{58} -3.65685 q^{61} +2.68629i q^{62} +4.82843i q^{63} +4.17157 q^{64} -0.686292 q^{66} +6.48528i q^{67} +5.17157i q^{68} -6.34315 q^{69} -15.3137 q^{71} -1.58579i q^{72} -8.48528i q^{73} +3.51472 q^{74} +8.82843 q^{76} -4.00000i q^{77} -1.65685i q^{78} +2.48528 q^{79} -11.0000 q^{81} -2.48528i q^{82} -7.17157i q^{83} +17.6569 q^{84} -2.48528 q^{86} -2.00000i q^{87} +1.31371i q^{88} +7.65685 q^{89} +9.65685 q^{91} +5.79899i q^{92} +12.9706i q^{93} +4.82843 q^{94} -8.82843 q^{96} -12.4853i q^{97} -6.75736i q^{98} -0.828427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 8 q^{6} - 4 q^{9} - 8 q^{11} + 8 q^{14} + 12 q^{16} + 8 q^{19} + 16 q^{21} - 24 q^{24} + 8 q^{26} - 4 q^{29} - 8 q^{31} - 16 q^{34} + 4 q^{36} - 16 q^{39} - 24 q^{41} + 40 q^{44} + 40 q^{46}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/725\mathbb{Z}\right)^\times\).

\(n\) \(176\) \(552\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.414214i 0.292893i 0.989219 + 0.146447i \(0.0467837\pi\)
−0.989219 + 0.146447i \(0.953216\pi\)
\(3\) 2.00000i 1.15470i 0.816497 + 0.577350i \(0.195913\pi\)
−0.816497 + 0.577350i \(0.804087\pi\)
\(4\) 1.82843 0.914214
\(5\) 0 0
\(6\) −0.828427 −0.338204
\(7\) − 4.82843i − 1.82497i −0.409106 − 0.912487i \(-0.634159\pi\)
0.409106 − 0.912487i \(-0.365841\pi\)
\(8\) 1.58579i 0.560660i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0.828427 0.249780 0.124890 − 0.992171i \(-0.460142\pi\)
0.124890 + 0.992171i \(0.460142\pi\)
\(12\) 3.65685i 1.05564i
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) 2.82843i 0.685994i 0.939336 + 0.342997i \(0.111442\pi\)
−0.939336 + 0.342997i \(0.888558\pi\)
\(18\) − 0.414214i − 0.0976311i
\(19\) 4.82843 1.10772 0.553859 − 0.832611i \(-0.313155\pi\)
0.553859 + 0.832611i \(0.313155\pi\)
\(20\) 0 0
\(21\) 9.65685 2.10730
\(22\) 0.343146i 0.0731589i
\(23\) 3.17157i 0.661319i 0.943750 + 0.330659i \(0.107271\pi\)
−0.943750 + 0.330659i \(0.892729\pi\)
\(24\) −3.17157 −0.647395
\(25\) 0 0
\(26\) −0.828427 −0.162468
\(27\) 4.00000i 0.769800i
\(28\) − 8.82843i − 1.66842i
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 6.48528 1.16479 0.582395 − 0.812906i \(-0.302116\pi\)
0.582395 + 0.812906i \(0.302116\pi\)
\(32\) 4.41421i 0.780330i
\(33\) 1.65685i 0.288421i
\(34\) −1.17157 −0.200923
\(35\) 0 0
\(36\) −1.82843 −0.304738
\(37\) − 8.48528i − 1.39497i −0.716599 − 0.697486i \(-0.754302\pi\)
0.716599 − 0.697486i \(-0.245698\pi\)
\(38\) 2.00000i 0.324443i
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 4.00000i 0.617213i
\(43\) 6.00000i 0.914991i 0.889212 + 0.457496i \(0.151253\pi\)
−0.889212 + 0.457496i \(0.848747\pi\)
\(44\) 1.51472 0.228352
\(45\) 0 0
\(46\) −1.31371 −0.193696
\(47\) − 11.6569i − 1.70033i −0.526519 − 0.850163i \(-0.676503\pi\)
0.526519 − 0.850163i \(-0.323497\pi\)
\(48\) 6.00000i 0.866025i
\(49\) −16.3137 −2.33053
\(50\) 0 0
\(51\) −5.65685 −0.792118
\(52\) 3.65685i 0.507114i
\(53\) 3.65685i 0.502308i 0.967947 + 0.251154i \(0.0808100\pi\)
−0.967947 + 0.251154i \(0.919190\pi\)
\(54\) −1.65685 −0.225469
\(55\) 0 0
\(56\) 7.65685 1.02319
\(57\) 9.65685i 1.27908i
\(58\) − 0.414214i − 0.0543889i
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −3.65685 −0.468212 −0.234106 − 0.972211i \(-0.575216\pi\)
−0.234106 + 0.972211i \(0.575216\pi\)
\(62\) 2.68629i 0.341159i
\(63\) 4.82843i 0.608325i
\(64\) 4.17157 0.521447
\(65\) 0 0
\(66\) −0.686292 −0.0844766
\(67\) 6.48528i 0.792303i 0.918185 + 0.396152i \(0.129655\pi\)
−0.918185 + 0.396152i \(0.870345\pi\)
\(68\) 5.17157i 0.627145i
\(69\) −6.34315 −0.763625
\(70\) 0 0
\(71\) −15.3137 −1.81740 −0.908701 − 0.417447i \(-0.862925\pi\)
−0.908701 + 0.417447i \(0.862925\pi\)
\(72\) − 1.58579i − 0.186887i
\(73\) − 8.48528i − 0.993127i −0.868000 − 0.496564i \(-0.834595\pi\)
0.868000 − 0.496564i \(-0.165405\pi\)
\(74\) 3.51472 0.408578
\(75\) 0 0
\(76\) 8.82843 1.01269
\(77\) − 4.00000i − 0.455842i
\(78\) − 1.65685i − 0.187602i
\(79\) 2.48528 0.279616 0.139808 − 0.990179i \(-0.455351\pi\)
0.139808 + 0.990179i \(0.455351\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) − 2.48528i − 0.274453i
\(83\) − 7.17157i − 0.787182i −0.919286 − 0.393591i \(-0.871233\pi\)
0.919286 − 0.393591i \(-0.128767\pi\)
\(84\) 17.6569 1.92652
\(85\) 0 0
\(86\) −2.48528 −0.267995
\(87\) − 2.00000i − 0.214423i
\(88\) 1.31371i 0.140042i
\(89\) 7.65685 0.811625 0.405812 − 0.913956i \(-0.366989\pi\)
0.405812 + 0.913956i \(0.366989\pi\)
\(90\) 0 0
\(91\) 9.65685 1.01231
\(92\) 5.79899i 0.604586i
\(93\) 12.9706i 1.34498i
\(94\) 4.82843 0.498014
\(95\) 0 0
\(96\) −8.82843 −0.901048
\(97\) − 12.4853i − 1.26769i −0.773461 − 0.633844i \(-0.781476\pi\)
0.773461 − 0.633844i \(-0.218524\pi\)
\(98\) − 6.75736i − 0.682596i
\(99\) −0.828427 −0.0832601
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 725.2.b.c.349.3 4
5.2 odd 4 725.2.a.c.1.1 2
5.3 odd 4 145.2.a.b.1.2 ✓ 2
5.4 even 2 inner 725.2.b.c.349.2 4
15.2 even 4 6525.2.a.p.1.2 2
15.8 even 4 1305.2.a.n.1.1 2
20.3 even 4 2320.2.a.k.1.2 2
35.13 even 4 7105.2.a.e.1.2 2
40.3 even 4 9280.2.a.w.1.2 2
40.13 odd 4 9280.2.a.be.1.1 2
145.28 odd 4 4205.2.a.d.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.b.1.2 ✓ 2 5.3 odd 4
725.2.a.c.1.1 2 5.2 odd 4
725.2.b.c.349.2 4 5.4 even 2 inner
725.2.b.c.349.3 4 1.1 even 1 trivial
1305.2.a.n.1.1 2 15.8 even 4
2320.2.a.k.1.2 2 20.3 even 4
4205.2.a.d.1.1 2 145.28 odd 4
6525.2.a.p.1.2 2 15.2 even 4
7105.2.a.e.1.2 2 35.13 even 4
9280.2.a.w.1.2 2 40.3 even 4
9280.2.a.be.1.1 2 40.13 odd 4