Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [475,2,Mod(1,475)] 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("475.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(475, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.66064384.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 9x^{4} + 13x^{2} - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-2.68667\) of defining polynomial
Character \(\chi\) \(=\) 475.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.82254 q^{2} +2.31446 q^{3} +1.32164 q^{4} +4.21819 q^{6} +1.45033 q^{7} -1.23634 q^{8} +2.35673 q^{9} -3.89655 q^{11} +3.05888 q^{12} +3.05888 q^{13} +2.64327 q^{14} -4.89655 q^{16} +3.92301 q^{17} +4.29522 q^{18} -1.00000 q^{19} +3.35673 q^{21} -7.10160 q^{22} -5.37334 q^{23} -2.86146 q^{24} +5.57491 q^{26} -1.48883 q^{27} +1.91681 q^{28} +6.00000 q^{29} -8.43637 q^{31} -6.45146 q^{32} -9.01841 q^{33} +7.14982 q^{34} +3.11474 q^{36} -5.95953 q^{37} -1.82254 q^{38} +7.07965 q^{39} +10.4364 q^{41} +6.11775 q^{42} -1.45033 q^{43} -5.14982 q^{44} -9.79310 q^{46} +4.90686 q^{47} -11.3329 q^{48} -4.89655 q^{49} +9.07965 q^{51} +4.04272 q^{52} -4.23127 q^{53} -2.71345 q^{54} -1.79310 q^{56} -2.31446 q^{57} +10.9352 q^{58} +3.35673 q^{59} +10.3329 q^{61} -15.3756 q^{62} +3.41802 q^{63} -1.96491 q^{64} -16.4364 q^{66} -9.84404 q^{67} +5.18479 q^{68} -12.4364 q^{69} +8.64327 q^{71} -2.91372 q^{72} -2.43418 q^{73} -10.8615 q^{74} -1.32164 q^{76} -5.65127 q^{77} +12.9029 q^{78} -12.4364 q^{79} -10.5160 q^{81} +19.0207 q^{82} +12.6635 q^{83} +4.43637 q^{84} -2.64327 q^{86} +13.8868 q^{87} +4.81746 q^{88} +12.3662 q^{89} +4.43637 q^{91} -7.10160 q^{92} -19.5256 q^{93} +8.94292 q^{94} -14.9316 q^{96} +3.05888 q^{97} -8.92414 q^{98} -9.18310 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 8 q^{4} + 14 q^{9} + 2 q^{11} + 16 q^{14} - 4 q^{16} - 6 q^{19} + 20 q^{21} + 8 q^{24} + 8 q^{26} + 36 q^{29} - 8 q^{34} - 32 q^{36} - 8 q^{39} + 12 q^{41} + 20 q^{44} - 8 q^{46} - 4 q^{49} + 4 q^{51}+ \cdots - 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.82254 1.28873 0.644364 − 0.764719i \(-0.277122\pi\)
0.644364 + 0.764719i \(0.277122\pi\)
\(3\) 2.31446 1.33625 0.668127 − 0.744047i \(-0.267096\pi\)
0.668127 + 0.744047i \(0.267096\pi\)
\(4\) 1.32164 0.660819
\(5\) 0 0
\(6\) 4.21819 1.72207
\(7\) 1.45033 0.548172 0.274086 − 0.961705i \(-0.411625\pi\)
0.274086 + 0.961705i \(0.411625\pi\)
\(8\) −1.23634 −0.437112
\(9\) 2.35673 0.785575
\(10\) 0 0
\(11\) −3.89655 −1.17485 −0.587427 − 0.809277i \(-0.699859\pi\)
−0.587427 + 0.809277i \(0.699859\pi\)
\(12\) 3.05888 0.883022
\(13\) 3.05888 0.848380 0.424190 − 0.905573i \(-0.360559\pi\)
0.424190 + 0.905573i \(0.360559\pi\)
\(14\) 2.64327 0.706445
\(15\) 0 0
\(16\) −4.89655 −1.22414
\(17\) 3.92301 0.951469 0.475735 − 0.879589i \(-0.342182\pi\)
0.475735 + 0.879589i \(0.342182\pi\)
\(18\) 4.29522 1.01239
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 3.35673 0.732498
\(22\) −7.10160 −1.51407
\(23\) −5.37334 −1.12042 −0.560209 − 0.828351i \(-0.689279\pi\)
−0.560209 + 0.828351i \(0.689279\pi\)
\(24\) −2.86146 −0.584093
\(25\) 0 0
\(26\) 5.57491 1.09333
\(27\) −1.48883 −0.286526
\(28\) 1.91681 0.362242
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −8.43637 −1.51522 −0.757609 − 0.652709i \(-0.773632\pi\)
−0.757609 + 0.652709i \(0.773632\pi\)
\(32\) −6.45146 −1.14047
\(33\) −9.01841 −1.56990
\(34\) 7.14982 1.22618
\(35\) 0 0
\(36\) 3.11474 0.519123
\(37\) −5.95953 −0.979741 −0.489871 − 0.871795i \(-0.662956\pi\)
−0.489871 + 0.871795i \(0.662956\pi\)
\(38\) −1.82254 −0.295654
\(39\) 7.07965 1.13365
\(40\) 0 0
\(41\) 10.4364 1.62989 0.814944 − 0.579540i \(-0.196768\pi\)
0.814944 + 0.579540i \(0.196768\pi\)
\(42\) 6.11775 0.943990
\(43\) −1.45033 −0.221173 −0.110586 − 0.993867i \(-0.535273\pi\)
−0.110586 + 0.993867i \(0.535273\pi\)
\(44\) −5.14982 −0.776365
\(45\) 0 0
\(46\) −9.79310 −1.44391
\(47\) 4.90686 0.715739 0.357869 − 0.933772i \(-0.383503\pi\)
0.357869 + 0.933772i \(0.383503\pi\)
\(48\) −11.3329 −1.63576
\(49\) −4.89655 −0.699507
\(50\) 0 0
\(51\) 9.07965 1.27140
\(52\) 4.04272 0.560625
\(53\) −4.23127 −0.581209 −0.290605 − 0.956843i \(-0.593856\pi\)
−0.290605 + 0.956843i \(0.593856\pi\)
\(54\) −2.71345 −0.369254
\(55\) 0 0
\(56\) −1.79310 −0.239613
\(57\) −2.31446 −0.306558
\(58\) 10.9352 1.43586
\(59\) 3.35673 0.437008 0.218504 − 0.975836i \(-0.429882\pi\)
0.218504 + 0.975836i \(0.429882\pi\)
\(60\) 0 0
\(61\) 10.3329 1.32300 0.661498 − 0.749947i \(-0.269921\pi\)
0.661498 + 0.749947i \(0.269921\pi\)
\(62\) −15.3756 −1.95270
\(63\) 3.41802 0.430631
\(64\) −1.96491 −0.245614
\(65\) 0 0
\(66\) −16.4364 −2.02318
\(67\) −9.84404 −1.20264 −0.601320 − 0.799008i \(-0.705358\pi\)
−0.601320 + 0.799008i \(0.705358\pi\)
\(68\) 5.18479 0.628749
\(69\) −12.4364 −1.49716
\(70\) 0 0
\(71\) 8.64327 1.02577 0.512884 − 0.858458i \(-0.328577\pi\)
0.512884 + 0.858458i \(0.328577\pi\)
\(72\) −2.91372 −0.343385
\(73\) −2.43418 −0.284899 −0.142449 − 0.989802i \(-0.545498\pi\)
−0.142449 + 0.989802i \(0.545498\pi\)
\(74\) −10.8615 −1.26262
\(75\) 0 0
\(76\) −1.32164 −0.151602
\(77\) −5.65127 −0.644022
\(78\) 12.9029 1.46097
\(79\) −12.4364 −1.39920 −0.699601 − 0.714534i \(-0.746639\pi\)
−0.699601 + 0.714534i \(0.746639\pi\)
\(80\) 0 0
\(81\) −10.5160 −1.16845
\(82\) 19.0207 2.10048
\(83\) 12.6635 1.39000 0.694999 − 0.719011i \(-0.255405\pi\)
0.694999 + 0.719011i \(0.255405\pi\)
\(84\) 4.43637 0.484048
\(85\) 0 0
\(86\) −2.64327 −0.285032
\(87\) 13.8868 1.48882
\(88\) 4.81746 0.513543
\(89\) 12.3662 1.31081 0.655407 − 0.755276i \(-0.272497\pi\)
0.655407 + 0.755276i \(0.272497\pi\)
\(90\) 0 0
\(91\) 4.43637 0.465058
\(92\) −7.10160 −0.740393
\(93\) −19.5256 −2.02472
\(94\) 8.94292 0.922392
\(95\) 0 0
\(96\) −14.9316 −1.52395
\(97\) 3.05888 0.310582 0.155291 − 0.987869i \(-0.450369\pi\)
0.155291 + 0.987869i \(0.450369\pi\)
\(98\) −8.92414 −0.901474
\(99\) −9.18310 −0.922936
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 475.2.a.j.1.5 6
3.2 odd 2 4275.2.a.br.1.2 6
4.3 odd 2 7600.2.a.ck.1.2 6
5.2 odd 4 95.2.b.b.39.5 yes 6
5.3 odd 4 95.2.b.b.39.2 ✓ 6
5.4 even 2 inner 475.2.a.j.1.2 6
15.2 even 4 855.2.c.d.514.2 6
15.8 even 4 855.2.c.d.514.5 6
15.14 odd 2 4275.2.a.br.1.5 6
19.18 odd 2 9025.2.a.bx.1.2 6
20.3 even 4 1520.2.d.h.609.2 6
20.7 even 4 1520.2.d.h.609.5 6
20.19 odd 2 7600.2.a.ck.1.5 6
95.18 even 4 1805.2.b.e.1084.5 6
95.37 even 4 1805.2.b.e.1084.2 6
95.94 odd 2 9025.2.a.bx.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.b.b.39.2 ✓ 6 5.3 odd 4
95.2.b.b.39.5 yes 6 5.2 odd 4
475.2.a.j.1.2 6 5.4 even 2 inner
475.2.a.j.1.5 6 1.1 even 1 trivial
855.2.c.d.514.2 6 15.2 even 4
855.2.c.d.514.5 6 15.8 even 4
1520.2.d.h.609.2 6 20.3 even 4
1520.2.d.h.609.5 6 20.7 even 4
1805.2.b.e.1084.2 6 95.37 even 4
1805.2.b.e.1084.5 6 95.18 even 4
4275.2.a.br.1.2 6 3.2 odd 2
4275.2.a.br.1.5 6 15.14 odd 2
7600.2.a.ck.1.2 6 4.3 odd 2
7600.2.a.ck.1.5 6 20.19 odd 2
9025.2.a.bx.1.2 6 19.18 odd 2
9025.2.a.bx.1.5 6 95.94 odd 2