Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [275,6,Mod(199,275)] 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("275.199"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 275.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,-82] 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: \(44.1055504486\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.30597006400.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 61x^{4} + 980x^{2} + 1600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.6
Root \(5.25849i\) of defining polynomial
Character \(\chi\) \(=\) 275.199
Dual form 275.6.b.c.199.1

$q$-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)
 
\(f(q)\) \(=\) \(q+8.45512i q^{2} -26.7755i q^{3} -39.4891 q^{4} +226.390 q^{6} -22.2927i q^{7} -63.3212i q^{8} -473.926 q^{9} +121.000 q^{11} +1057.34i q^{12} -225.547i q^{13} +188.488 q^{14} -728.262 q^{16} +1059.59i q^{17} -4007.11i q^{18} -2525.94 q^{19} -596.899 q^{21} +1023.07i q^{22} -337.423i q^{23} -1695.46 q^{24} +1907.03 q^{26} +6183.16i q^{27} +880.320i q^{28} +7644.60 q^{29} +6754.80 q^{31} -8183.83i q^{32} -3239.83i q^{33} -8958.95 q^{34} +18714.9 q^{36} +5663.43i q^{37} -21357.2i q^{38} -6039.13 q^{39} -13317.4 q^{41} -5046.85i q^{42} +9007.03i q^{43} -4778.18 q^{44} +2852.96 q^{46} +16644.2i q^{47} +19499.6i q^{48} +16310.0 q^{49} +28371.0 q^{51} +8906.65i q^{52} +21261.8i q^{53} -52279.4 q^{54} -1411.60 q^{56} +67633.4i q^{57} +64636.1i q^{58} +44329.0 q^{59} -36591.1 q^{61} +57112.7i q^{62} +10565.1i q^{63} +45890.9 q^{64} +27393.2 q^{66} +45840.6i q^{67} -41842.2i q^{68} -9034.68 q^{69} -31877.6 q^{71} +30009.6i q^{72} +49936.7i q^{73} -47885.0 q^{74} +99747.3 q^{76} -2697.42i q^{77} -51061.6i q^{78} -48257.1 q^{79} +50393.1 q^{81} -112600. i q^{82} +66052.0i q^{83} +23571.0 q^{84} -76155.6 q^{86} -204688. i q^{87} -7661.87i q^{88} +124047. q^{89} -5028.06 q^{91} +13324.5i q^{92} -180863. i q^{93} -140729. q^{94} -219126. q^{96} -73854.3i q^{97} +137903. i q^{98} -57345.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 82 q^{4} + 202 q^{6} - 498 q^{9} + 726 q^{11} + 1926 q^{14} - 5374 q^{16} + 20 q^{19} + 2472 q^{21} - 10110 q^{24} - 5788 q^{26} - 3940 q^{29} + 13192 q^{31} - 33774 q^{34} + 40276 q^{36} - 39056 q^{39}+ \cdots - 60258 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.45512i 1.49467i 0.664448 + 0.747334i \(0.268667\pi\)
−0.664448 + 0.747334i \(0.731333\pi\)
\(3\) − 26.7755i − 1.71765i −0.512271 − 0.858824i \(-0.671196\pi\)
0.512271 − 0.858824i \(-0.328804\pi\)
\(4\) −39.4891 −1.23403
\(5\) 0 0
\(6\) 226.390 2.56731
\(7\) − 22.2927i − 0.171956i −0.996297 − 0.0859782i \(-0.972598\pi\)
0.996297 − 0.0859782i \(-0.0274015\pi\)
\(8\) − 63.3212i − 0.349804i
\(9\) −473.926 −1.95031
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) 1057.34i 2.11964i
\(13\) − 225.547i − 0.370151i −0.982724 − 0.185075i \(-0.940747\pi\)
0.982724 − 0.185075i \(-0.0592530\pi\)
\(14\) 188.488 0.257018
\(15\) 0 0
\(16\) −728.262 −0.711194
\(17\) 1059.59i 0.889232i 0.895721 + 0.444616i \(0.146660\pi\)
−0.895721 + 0.444616i \(0.853340\pi\)
\(18\) − 4007.11i − 2.91507i
\(19\) −2525.94 −1.60524 −0.802620 − 0.596491i \(-0.796561\pi\)
−0.802620 + 0.596491i \(0.796561\pi\)
\(20\) 0 0
\(21\) −596.899 −0.295360
\(22\) 1023.07i 0.450660i
\(23\) − 337.423i − 0.133001i −0.997786 − 0.0665006i \(-0.978817\pi\)
0.997786 − 0.0665006i \(-0.0211834\pi\)
\(24\) −1695.46 −0.600839
\(25\) 0 0
\(26\) 1907.03 0.553253
\(27\) 6183.16i 1.63230i
\(28\) 880.320i 0.212200i
\(29\) 7644.60 1.68795 0.843976 − 0.536381i \(-0.180209\pi\)
0.843976 + 0.536381i \(0.180209\pi\)
\(30\) 0 0
\(31\) 6754.80 1.26243 0.631217 − 0.775607i \(-0.282556\pi\)
0.631217 + 0.775607i \(0.282556\pi\)
\(32\) − 8183.83i − 1.41280i
\(33\) − 3239.83i − 0.517890i
\(34\) −8958.95 −1.32911
\(35\) 0 0
\(36\) 18714.9 2.40675
\(37\) 5663.43i 0.680104i 0.940407 + 0.340052i \(0.110445\pi\)
−0.940407 + 0.340052i \(0.889555\pi\)
\(38\) − 21357.2i − 2.39930i
\(39\) −6039.13 −0.635789
\(40\) 0 0
\(41\) −13317.4 −1.23725 −0.618627 − 0.785685i \(-0.712311\pi\)
−0.618627 + 0.785685i \(0.712311\pi\)
\(42\) − 5046.85i − 0.441466i
\(43\) 9007.03i 0.742866i 0.928460 + 0.371433i \(0.121133\pi\)
−0.928460 + 0.371433i \(0.878867\pi\)
\(44\) −4778.18 −0.372075
\(45\) 0 0
\(46\) 2852.96 0.198793
\(47\) 16644.2i 1.09905i 0.835477 + 0.549526i \(0.185192\pi\)
−0.835477 + 0.549526i \(0.814808\pi\)
\(48\) 19499.6i 1.22158i
\(49\) 16310.0 0.970431
\(50\) 0 0
\(51\) 28371.0 1.52739
\(52\) 8906.65i 0.456779i
\(53\) 21261.8i 1.03971i 0.854256 + 0.519853i \(0.174013\pi\)
−0.854256 + 0.519853i \(0.825987\pi\)
\(54\) −52279.4 −2.43975
\(55\) 0 0
\(56\) −1411.60 −0.0601509
\(57\) 67633.4i 2.75724i
\(58\) 64636.1i 2.52293i
\(59\) 44329.0 1.65790 0.828948 − 0.559325i \(-0.188940\pi\)
0.828948 + 0.559325i \(0.188940\pi\)
\(60\) 0 0
\(61\) −36591.1 −1.25907 −0.629537 − 0.776970i \(-0.716755\pi\)
−0.629537 + 0.776970i \(0.716755\pi\)
\(62\) 57112.7i 1.88692i
\(63\) 10565.1i 0.335369i
\(64\) 45890.9 1.40048
\(65\) 0 0
\(66\) 27393.2 0.774074
\(67\) 45840.6i 1.24757i 0.781598 + 0.623783i \(0.214405\pi\)
−0.781598 + 0.623783i \(0.785595\pi\)
\(68\) − 41842.2i − 1.09734i
\(69\) −9034.68 −0.228449
\(70\) 0 0
\(71\) −31877.6 −0.750481 −0.375240 − 0.926928i \(-0.622440\pi\)
−0.375240 + 0.926928i \(0.622440\pi\)
\(72\) 30009.6i 0.682227i
\(73\) 49936.7i 1.09676i 0.836228 + 0.548382i \(0.184756\pi\)
−0.836228 + 0.548382i \(0.815244\pi\)
\(74\) −47885.0 −1.01653
\(75\) 0 0
\(76\) 99747.3 1.98092
\(77\) − 2697.42i − 0.0518468i
\(78\) − 51061.6i − 0.950294i
\(79\) −48257.1 −0.869949 −0.434974 − 0.900443i \(-0.643243\pi\)
−0.434974 + 0.900443i \(0.643243\pi\)
\(80\) 0 0
\(81\) 50393.1 0.853411
\(82\) − 112600.i − 1.84928i
\(83\) 66052.0i 1.05242i 0.850353 + 0.526212i \(0.176388\pi\)
−0.850353 + 0.526212i \(0.823612\pi\)
\(84\) 23571.0 0.364485
\(85\) 0 0
\(86\) −76155.6 −1.11034
\(87\) − 204688.i − 2.89931i
\(88\) − 7661.87i − 0.105470i
\(89\) 124047. 1.66002 0.830009 − 0.557750i \(-0.188335\pi\)
0.830009 + 0.557750i \(0.188335\pi\)
\(90\) 0 0
\(91\) −5028.06 −0.0636498
\(92\) 13324.5i 0.164128i
\(93\) − 180863.i − 2.16842i
\(94\) −140729. −1.64272
\(95\) 0 0
\(96\) −219126. −2.42670
\(97\) − 73854.3i − 0.796978i −0.917173 − 0.398489i \(-0.869535\pi\)
0.917173 − 0.398489i \(-0.130465\pi\)
\(98\) 137903.i 1.45047i
\(99\) −57345.1 −0.588042
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 275.6.b.c.199.6 6
5.2 odd 4 55.6.a.a.1.1 ✓ 3
5.3 odd 4 275.6.a.c.1.3 3
5.4 even 2 inner 275.6.b.c.199.1 6
15.2 even 4 495.6.a.f.1.3 3
20.7 even 4 880.6.a.l.1.3 3
55.32 even 4 605.6.a.b.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.6.a.a.1.1 ✓ 3 5.2 odd 4
275.6.a.c.1.3 3 5.3 odd 4
275.6.b.c.199.1 6 5.4 even 2 inner
275.6.b.c.199.6 6 1.1 even 1 trivial
495.6.a.f.1.3 3 15.2 even 4
605.6.a.b.1.3 3 55.32 even 4
880.6.a.l.1.3 3 20.7 even 4