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 = [8,0,0,-122] 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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 70x^{6} + 1541x^{4} + 10660x^{2} + 5476 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.3
Root \(-3.50110i\) of defining polynomial
Character \(\chi\) \(=\) 275.199
Dual form 275.6.b.d.199.6

$q$-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)
 
\(f(q)\) \(=\) \(q-6.96278i q^{2} -22.6701i q^{3} -16.4803 q^{4} -157.847 q^{6} -169.118i q^{7} -108.060i q^{8} -270.932 q^{9} -121.000 q^{11} +373.610i q^{12} -25.3182i q^{13} -1177.53 q^{14} -1279.77 q^{16} -2016.26i q^{17} +1886.44i q^{18} +773.486 q^{19} -3833.91 q^{21} +842.497i q^{22} +541.643i q^{23} -2449.73 q^{24} -176.285 q^{26} +633.231i q^{27} +2787.11i q^{28} +5882.28 q^{29} -915.584 q^{31} +5452.83i q^{32} +2743.08i q^{33} -14038.8 q^{34} +4465.06 q^{36} +11360.3i q^{37} -5385.62i q^{38} -573.964 q^{39} -15477.7 q^{41} +26694.7i q^{42} -6097.77i q^{43} +1994.12 q^{44} +3771.34 q^{46} +15131.8i q^{47} +29012.5i q^{48} -11793.8 q^{49} -45708.8 q^{51} +417.252i q^{52} -10443.0i q^{53} +4409.05 q^{54} -18274.9 q^{56} -17535.0i q^{57} -40957.0i q^{58} +50295.3 q^{59} +45523.0 q^{61} +6375.01i q^{62} +45819.4i q^{63} -2985.73 q^{64} +19099.5 q^{66} +11285.0i q^{67} +33228.7i q^{68} +12279.1 q^{69} +64741.3 q^{71} +29277.0i q^{72} -77769.4i q^{73} +79099.6 q^{74} -12747.3 q^{76} +20463.2i q^{77} +3996.39i q^{78} +87890.2 q^{79} -51481.2 q^{81} +107768. i q^{82} -18403.3i q^{83} +63184.1 q^{84} -42457.4 q^{86} -133352. i q^{87} +13075.3i q^{88} -52660.1 q^{89} -4281.74 q^{91} -8926.45i q^{92} +20756.4i q^{93} +105359. q^{94} +123616. q^{96} -38745.0i q^{97} +82117.3i q^{98} +32782.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 122 q^{4} - 314 q^{6} - 44 q^{9} - 968 q^{11} + 3374 q^{14} - 5342 q^{16} + 6788 q^{19} - 9416 q^{21} - 5442 q^{24} - 7300 q^{26} + 10496 q^{29} + 9464 q^{31} - 19518 q^{34} + 14300 q^{36} + 42160 q^{39}+ \cdots + 5324 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 6.96278i − 1.23086i −0.788192 − 0.615429i \(-0.788983\pi\)
0.788192 − 0.615429i \(-0.211017\pi\)
\(3\) − 22.6701i − 1.45429i −0.686486 − 0.727143i \(-0.740848\pi\)
0.686486 − 0.727143i \(-0.259152\pi\)
\(4\) −16.4803 −0.515010
\(5\) 0 0
\(6\) −157.847 −1.79002
\(7\) − 169.118i − 1.30450i −0.758004 − 0.652249i \(-0.773826\pi\)
0.758004 − 0.652249i \(-0.226174\pi\)
\(8\) − 108.060i − 0.596953i
\(9\) −270.932 −1.11495
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) 373.610i 0.748973i
\(13\) − 25.3182i − 0.0415502i −0.999784 − 0.0207751i \(-0.993387\pi\)
0.999784 − 0.0207751i \(-0.00661340\pi\)
\(14\) −1177.53 −1.60565
\(15\) 0 0
\(16\) −1279.77 −1.24977
\(17\) − 2016.26i − 1.69209i −0.533108 − 0.846047i \(-0.678976\pi\)
0.533108 − 0.846047i \(-0.321024\pi\)
\(18\) 1886.44i 1.37234i
\(19\) 773.486 0.491551 0.245775 − 0.969327i \(-0.420957\pi\)
0.245775 + 0.969327i \(0.420957\pi\)
\(20\) 0 0
\(21\) −3833.91 −1.89711
\(22\) 842.497i 0.371118i
\(23\) 541.643i 0.213498i 0.994286 + 0.106749i \(0.0340441\pi\)
−0.994286 + 0.106749i \(0.965956\pi\)
\(24\) −2449.73 −0.868141
\(25\) 0 0
\(26\) −176.285 −0.0511424
\(27\) 633.231i 0.167168i
\(28\) 2787.11i 0.671830i
\(29\) 5882.28 1.29882 0.649412 − 0.760437i \(-0.275015\pi\)
0.649412 + 0.760437i \(0.275015\pi\)
\(30\) 0 0
\(31\) −915.584 −0.171117 −0.0855587 − 0.996333i \(-0.527268\pi\)
−0.0855587 + 0.996333i \(0.527268\pi\)
\(32\) 5452.83i 0.941342i
\(33\) 2743.08i 0.438484i
\(34\) −14038.8 −2.08273
\(35\) 0 0
\(36\) 4465.06 0.574210
\(37\) 11360.3i 1.36423i 0.731245 + 0.682115i \(0.238939\pi\)
−0.731245 + 0.682115i \(0.761061\pi\)
\(38\) − 5385.62i − 0.605029i
\(39\) −573.964 −0.0604259
\(40\) 0 0
\(41\) −15477.7 −1.43796 −0.718979 − 0.695031i \(-0.755390\pi\)
−0.718979 + 0.695031i \(0.755390\pi\)
\(42\) 26694.7i 2.33508i
\(43\) − 6097.77i − 0.502921i −0.967868 − 0.251460i \(-0.919089\pi\)
0.967868 − 0.251460i \(-0.0809108\pi\)
\(44\) 1994.12 0.155281
\(45\) 0 0
\(46\) 3771.34 0.262785
\(47\) 15131.8i 0.999185i 0.866260 + 0.499593i \(0.166517\pi\)
−0.866260 + 0.499593i \(0.833483\pi\)
\(48\) 29012.5i 1.81753i
\(49\) −11793.8 −0.701717
\(50\) 0 0
\(51\) −45708.8 −2.46079
\(52\) 417.252i 0.0213988i
\(53\) − 10443.0i − 0.510666i −0.966853 − 0.255333i \(-0.917815\pi\)
0.966853 − 0.255333i \(-0.0821851\pi\)
\(54\) 4409.05 0.205760
\(55\) 0 0
\(56\) −18274.9 −0.778724
\(57\) − 17535.0i − 0.714856i
\(58\) − 40957.0i − 1.59867i
\(59\) 50295.3 1.88104 0.940519 − 0.339741i \(-0.110339\pi\)
0.940519 + 0.339741i \(0.110339\pi\)
\(60\) 0 0
\(61\) 45523.0 1.56641 0.783206 − 0.621762i \(-0.213583\pi\)
0.783206 + 0.621762i \(0.213583\pi\)
\(62\) 6375.01i 0.210621i
\(63\) 45819.4i 1.45445i
\(64\) −2985.73 −0.0911172
\(65\) 0 0
\(66\) 19099.5 0.539711
\(67\) 11285.0i 0.307124i 0.988139 + 0.153562i \(0.0490745\pi\)
−0.988139 + 0.153562i \(0.950926\pi\)
\(68\) 33228.7i 0.871446i
\(69\) 12279.1 0.310487
\(70\) 0 0
\(71\) 64741.3 1.52418 0.762089 − 0.647473i \(-0.224174\pi\)
0.762089 + 0.647473i \(0.224174\pi\)
\(72\) 29277.0i 0.665572i
\(73\) − 77769.4i − 1.70805i −0.520229 − 0.854027i \(-0.674153\pi\)
0.520229 − 0.854027i \(-0.325847\pi\)
\(74\) 79099.6 1.67917
\(75\) 0 0
\(76\) −12747.3 −0.253154
\(77\) 20463.2i 0.393321i
\(78\) 3996.39i 0.0743757i
\(79\) 87890.2 1.58443 0.792214 − 0.610243i \(-0.208928\pi\)
0.792214 + 0.610243i \(0.208928\pi\)
\(80\) 0 0
\(81\) −51481.2 −0.871838
\(82\) 107768.i 1.76992i
\(83\) − 18403.3i − 0.293225i −0.989194 − 0.146613i \(-0.953163\pi\)
0.989194 − 0.146613i \(-0.0468371\pi\)
\(84\) 63184.1 0.977034
\(85\) 0 0
\(86\) −42457.4 −0.619024
\(87\) − 133352.i − 1.88886i
\(88\) 13075.3i 0.179988i
\(89\) −52660.1 −0.704704 −0.352352 − 0.935868i \(-0.614618\pi\)
−0.352352 + 0.935868i \(0.614618\pi\)
\(90\) 0 0
\(91\) −4281.74 −0.0542022
\(92\) − 8926.45i − 0.109954i
\(93\) 20756.4i 0.248854i
\(94\) 105359. 1.22985
\(95\) 0 0
\(96\) 123616. 1.36898
\(97\) − 38745.0i − 0.418106i −0.977904 − 0.209053i \(-0.932962\pi\)
0.977904 − 0.209053i \(-0.0670382\pi\)
\(98\) 82117.3i 0.863714i
\(99\) 32782.8 0.336170
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.d.199.3 8
5.2 odd 4 275.6.a.d.1.3 4
5.3 odd 4 55.6.a.b.1.2 ✓ 4
5.4 even 2 inner 275.6.b.d.199.6 8
15.8 even 4 495.6.a.g.1.3 4
20.3 even 4 880.6.a.n.1.1 4
55.43 even 4 605.6.a.c.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.6.a.b.1.2 ✓ 4 5.3 odd 4
275.6.a.d.1.3 4 5.2 odd 4
275.6.b.d.199.3 8 1.1 even 1 trivial
275.6.b.d.199.6 8 5.4 even 2 inner
495.6.a.g.1.3 4 15.8 even 4
605.6.a.c.1.3 4 55.43 even 4
880.6.a.n.1.1 4 20.3 even 4