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 1.2
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 \(-0.722878\pi\)
−0.644364 + 0.764719i \(0.722878\pi\)
\(3\) −2.31446 −1.33625 −0.668127 − 0.744047i \(-0.732904\pi\)
−0.668127 + 0.744047i \(0.732904\pi\)
\(4\) 1.32164 0.660819
\(5\) 0 0
\(6\) 4.21819 1.72207
\(7\) −1.45033 −0.548172 −0.274086 − 0.961705i \(-0.588375\pi\)
−0.274086 + 0.961705i \(0.588375\pi\)
\(8\) 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 \(-0.639441\pi\)
−0.424190 + 0.905573i \(0.639441\pi\)
\(14\) 2.64327 0.706445
\(15\) 0 0
\(16\) −4.89655 −1.22414
\(17\) −3.92301 −0.951469 −0.475735 − 0.879589i \(-0.657818\pi\)
−0.475735 + 0.879589i \(0.657818\pi\)
\(18\) −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 \(-0.310721\pi\)
0.560209 + 0.828351i \(0.310721\pi\)
\(24\) −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 \(-0.337044\pi\)
0.489871 + 0.871795i \(0.337044\pi\)
\(38\) 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 \(-0.464727\pi\)
0.110586 + 0.993867i \(0.464727\pi\)
\(44\) −5.14982 −0.776365
\(45\) 0 0
\(46\) −9.79310 −1.44391
\(47\) −4.90686 −0.715739 −0.357869 − 0.933772i \(-0.616497\pi\)
−0.357869 + 0.933772i \(0.616497\pi\)
\(48\) 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 \(-0.406144\pi\)
0.290605 + 0.956843i \(0.406144\pi\)
\(54\) −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 \(-0.294642\pi\)
0.601320 + 0.799008i \(0.294642\pi\)
\(68\) −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 \(-0.454502\pi\)
0.142449 + 0.989802i \(0.454502\pi\)
\(74\) −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 \(-0.744595\pi\)
−0.694999 + 0.719011i \(0.744595\pi\)
\(84\) 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 \(-0.549631\pi\)
−0.155291 + 0.987869i \(0.549631\pi\)
\(98\) 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.1.2 6
3.2 odd 2 4275.2.a.br.1.5 6
4.3 odd 2 7600.2.a.ck.1.5 6
5.2 odd 4 95.2.b.b.39.2 ✓ 6
5.3 odd 4 95.2.b.b.39.5 yes 6
5.4 even 2 inner 475.2.a.j.1.5 6
15.2 even 4 855.2.c.d.514.5 6
15.8 even 4 855.2.c.d.514.2 6
15.14 odd 2 4275.2.a.br.1.2 6
19.18 odd 2 9025.2.a.bx.1.5 6
20.3 even 4 1520.2.d.h.609.5 6
20.7 even 4 1520.2.d.h.609.2 6
20.19 odd 2 7600.2.a.ck.1.2 6
95.18 even 4 1805.2.b.e.1084.2 6
95.37 even 4 1805.2.b.e.1084.5 6
95.94 odd 2 9025.2.a.bx.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.b.b.39.2 ✓ 6 5.2 odd 4
95.2.b.b.39.5 yes 6 5.3 odd 4
475.2.a.j.1.2 6 1.1 even 1 trivial
475.2.a.j.1.5 6 5.4 even 2 inner
855.2.c.d.514.2 6 15.8 even 4
855.2.c.d.514.5 6 15.2 even 4
1520.2.d.h.609.2 6 20.7 even 4
1520.2.d.h.609.5 6 20.3 even 4
1805.2.b.e.1084.2 6 95.18 even 4
1805.2.b.e.1084.5 6 95.37 even 4
4275.2.a.br.1.2 6 15.14 odd 2
4275.2.a.br.1.5 6 3.2 odd 2
7600.2.a.ck.1.2 6 20.19 odd 2
7600.2.a.ck.1.5 6 4.3 odd 2
9025.2.a.bx.1.2 6 95.94 odd 2
9025.2.a.bx.1.5 6 19.18 odd 2