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

$q$-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 \(0.211017\pi\)
−0.788192 + 0.615429i \(0.788983\pi\)
\(3\) 22.6701i 1.45429i 0.686486 + 0.727143i \(0.259152\pi\)
−0.686486 + 0.727143i \(0.740848\pi\)
\(4\) −16.4803 −0.515010
\(5\) 0 0
\(6\) −157.847 −1.79002
\(7\) 169.118i 1.30450i 0.758004 + 0.652249i \(0.226174\pi\)
−0.758004 + 0.652249i \(0.773826\pi\)
\(8\) 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 \(0.00661340\pi\)
−0.999784 + 0.0207751i \(0.993387\pi\)
\(14\) −1177.53 −1.60565
\(15\) 0 0
\(16\) −1279.77 −1.24977
\(17\) 2016.26i 1.69209i 0.533108 + 0.846047i \(0.321024\pi\)
−0.533108 + 0.846047i \(0.678976\pi\)
\(18\) − 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 \(-0.965956\pi\)
0.994286 − 0.106749i \(-0.0340441\pi\)
\(24\) −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 \(-0.761061\pi\)
0.731245 − 0.682115i \(-0.238939\pi\)
\(38\) 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 \(0.0809108\pi\)
−0.967868 + 0.251460i \(0.919089\pi\)
\(44\) 1994.12 0.155281
\(45\) 0 0
\(46\) 3771.34 0.262785
\(47\) − 15131.8i − 0.999185i −0.866260 − 0.499593i \(-0.833483\pi\)
0.866260 − 0.499593i \(-0.166517\pi\)
\(48\) − 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 \(0.0821851\pi\)
−0.966853 + 0.255333i \(0.917815\pi\)
\(54\) 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 \(-0.950926\pi\)
0.988139 − 0.153562i \(-0.0490745\pi\)
\(68\) − 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 \(0.325847\pi\)
−0.520229 + 0.854027i \(0.674153\pi\)
\(74\) 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 \(0.0468371\pi\)
−0.989194 + 0.146613i \(0.953163\pi\)
\(84\) 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 \(0.0670382\pi\)
−0.977904 + 0.209053i \(0.932962\pi\)
\(98\) − 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.199.6 8
5.2 odd 4 55.6.a.b.1.2 ✓ 4
5.3 odd 4 275.6.a.d.1.3 4
5.4 even 2 inner 275.6.b.d.199.3 8
15.2 even 4 495.6.a.g.1.3 4
20.7 even 4 880.6.a.n.1.1 4
55.32 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 5.2 odd 4
275.6.a.d.1.3 4 5.3 odd 4
275.6.b.d.199.3 8 5.4 even 2 inner
275.6.b.d.199.6 8 1.1 even 1 trivial
495.6.a.g.1.3 4 15.2 even 4
605.6.a.c.1.3 4 55.32 even 4
880.6.a.n.1.1 4 20.7 even 4