Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [400,2,Mod(107,400)] 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("400.107"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 400.s (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.19401608085\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 2 x^{16} - 4 x^{15} - 5 x^{14} - 14 x^{13} - 10 x^{12} + 6 x^{11} + 37 x^{10} + 70 x^{9} + \cdots + 512 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 243.4
Root \(1.41303 + 0.0578659i\) of defining polynomial
Character \(\chi\) \(=\) 400.243
Dual form 400.2.s.d.107.4

$q$-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+(-0.567819 + 1.29521i) q^{2} -1.96251 q^{3} +(-1.35516 - 1.47090i) q^{4} +(1.11435 - 2.54187i) q^{6} +(1.60205 - 1.60205i) q^{7} +(2.67461 - 0.920026i) q^{8} +0.851447 q^{9} +(0.754587 + 0.754587i) q^{11} +(2.65952 + 2.88665i) q^{12} +5.94580i q^{13} +(1.16532 + 2.98467i) q^{14} +(-0.327065 + 3.98661i) q^{16} +(-1.95574 + 1.95574i) q^{17} +(-0.483468 + 1.10281i) q^{18} +(0.780680 + 0.780680i) q^{19} +(-3.14404 + 3.14404i) q^{21} +(-1.40582 + 0.548884i) q^{22} +(-4.93121 - 4.93121i) q^{23} +(-5.24896 + 1.80556i) q^{24} +(-7.70109 - 3.37614i) q^{26} +4.21656 q^{27} +(-4.52748 - 0.185408i) q^{28} +(-1.44802 + 1.44802i) q^{29} +3.60859i q^{31} +(-4.97780 - 2.68729i) q^{32} +(-1.48089 - 1.48089i) q^{33} +(-1.42260 - 3.64361i) q^{34} +(-1.15385 - 1.25239i) q^{36} +10.2364i q^{37} +(-1.45443 + 0.567864i) q^{38} -11.6687i q^{39} +6.93334i q^{41} +(-2.28696 - 5.85745i) q^{42} +9.91344i q^{43} +(0.0873298 - 2.13251i) q^{44} +(9.18700 - 3.58694i) q^{46} +(-0.104270 - 0.104270i) q^{47} +(0.641868 - 7.82376i) q^{48} +1.86688i q^{49} +(3.83816 - 3.83816i) q^{51} +(8.74565 - 8.05753i) q^{52} +4.03213 q^{53} +(-2.39424 + 5.46135i) q^{54} +(2.81093 - 5.75878i) q^{56} +(-1.53209 - 1.53209i) q^{57} +(-1.05328 - 2.69771i) q^{58} +(-3.46736 + 3.46736i) q^{59} +(0.680578 + 0.680578i) q^{61} +(-4.67390 - 2.04902i) q^{62} +(1.36406 - 1.36406i) q^{63} +(6.30711 - 4.92142i) q^{64} +(2.75894 - 1.07719i) q^{66} -9.04721i q^{67} +(5.52703 + 0.226341i) q^{68} +(9.67754 + 9.67754i) q^{69} -3.64007 q^{71} +(2.27729 - 0.783353i) q^{72} +(2.94030 - 2.94030i) q^{73} +(-13.2583 - 5.81242i) q^{74} +(0.0903496 - 2.20625i) q^{76} +2.41777 q^{77} +(15.1135 + 6.62570i) q^{78} +10.7140 q^{79} -10.8294 q^{81} +(-8.98016 - 3.93688i) q^{82} +4.23845 q^{83} +(8.88523 + 0.363865i) q^{84} +(-12.8400 - 5.62904i) q^{86} +(2.84176 - 2.84176i) q^{87} +(2.71247 + 1.32399i) q^{88} +0.0426256 q^{89} +(9.52546 + 9.52546i) q^{91} +(-0.570698 + 13.9359i) q^{92} -7.08189i q^{93} +(0.194258 - 0.0758455i) q^{94} +(9.76898 + 5.27383i) q^{96} +(1.91173 - 1.91173i) q^{97} +(-2.41802 - 1.06005i) q^{98} +(0.642491 + 0.642491i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 4 q^{4} - 8 q^{6} - 2 q^{7} + 12 q^{8} + 10 q^{9} - 2 q^{11} - 12 q^{14} + 6 q^{17} + 24 q^{18} - 2 q^{19} - 16 q^{21} - 12 q^{22} + 2 q^{23} - 4 q^{24} - 16 q^{26} + 24 q^{27} - 40 q^{28} + 14 q^{29}+ \cdots + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.567819 + 1.29521i −0.401509 + 0.915855i
\(3\) −1.96251 −1.13306 −0.566528 − 0.824043i \(-0.691714\pi\)
−0.566528 + 0.824043i \(0.691714\pi\)
\(4\) −1.35516 − 1.47090i −0.677582 − 0.735448i
\(5\) 0 0
\(6\) 1.11435 − 2.54187i 0.454932 − 1.03772i
\(7\) 1.60205 − 1.60205i 0.605517 − 0.605517i −0.336254 − 0.941771i \(-0.609160\pi\)
0.941771 + 0.336254i \(0.109160\pi\)
\(8\) 2.67461 − 0.920026i 0.945618 − 0.325278i
\(9\) 0.851447 0.283816
\(10\) 0 0
\(11\) 0.754587 + 0.754587i 0.227517 + 0.227517i 0.811654 − 0.584138i \(-0.198567\pi\)
−0.584138 + 0.811654i \(0.698567\pi\)
\(12\) 2.65952 + 2.88665i 0.767738 + 0.833303i
\(13\) 5.94580i 1.64907i 0.565812 + 0.824534i \(0.308563\pi\)
−0.565812 + 0.824534i \(0.691437\pi\)
\(14\) 1.16532 + 2.98467i 0.311446 + 0.797687i
\(15\) 0 0
\(16\) −0.327065 + 3.98661i −0.0817662 + 0.996652i
\(17\) −1.95574 + 1.95574i −0.474336 + 0.474336i −0.903315 − 0.428978i \(-0.858874\pi\)
0.428978 + 0.903315i \(0.358874\pi\)
\(18\) −0.483468 + 1.10281i −0.113954 + 0.259934i
\(19\) 0.780680 + 0.780680i 0.179100 + 0.179100i 0.790964 − 0.611863i \(-0.209580\pi\)
−0.611863 + 0.790964i \(0.709580\pi\)
\(20\) 0 0
\(21\) −3.14404 + 3.14404i −0.686085 + 0.686085i
\(22\) −1.40582 + 0.548884i −0.299722 + 0.117022i
\(23\) −4.93121 − 4.93121i −1.02823 − 1.02823i −0.999590 − 0.0286378i \(-0.990883\pi\)
−0.0286378 − 0.999590i \(-0.509117\pi\)
\(24\) −5.24896 + 1.80556i −1.07144 + 0.368558i
\(25\) 0 0
\(26\) −7.70109 − 3.37614i −1.51031 − 0.662115i
\(27\) 4.21656 0.811477
\(28\) −4.52748 − 0.185408i −0.855614 − 0.0350388i
\(29\) −1.44802 + 1.44802i −0.268891 + 0.268891i −0.828653 − 0.559762i \(-0.810892\pi\)
0.559762 + 0.828653i \(0.310892\pi\)
\(30\) 0 0
\(31\) 3.60859i 0.648121i 0.946036 + 0.324061i \(0.105048\pi\)
−0.946036 + 0.324061i \(0.894952\pi\)
\(32\) −4.97780 − 2.68729i −0.879959 − 0.475050i
\(33\) −1.48089 − 1.48089i −0.257789 − 0.257789i
\(34\) −1.42260 − 3.64361i −0.243973 − 0.624874i
\(35\) 0 0
\(36\) −1.15385 − 1.25239i −0.192308 − 0.208732i
\(37\) 10.2364i 1.68285i 0.540371 + 0.841427i \(0.318284\pi\)
−0.540371 + 0.841427i \(0.681716\pi\)
\(38\) −1.45443 + 0.567864i −0.235940 + 0.0921197i
\(39\) − 11.6687i − 1.86849i
\(40\) 0 0
\(41\) 6.93334i 1.08281i 0.840763 + 0.541403i \(0.182107\pi\)
−0.840763 + 0.541403i \(0.817893\pi\)
\(42\) −2.28696 − 5.85745i −0.352885 − 0.903823i
\(43\) 9.91344i 1.51179i 0.654695 + 0.755893i \(0.272797\pi\)
−0.654695 + 0.755893i \(0.727203\pi\)
\(44\) 0.0873298 − 2.13251i 0.0131655 − 0.321488i
\(45\) 0 0
\(46\) 9.18700 − 3.58694i 1.35455 − 0.528865i
\(47\) −0.104270 − 0.104270i −0.0152093 − 0.0152093i 0.699461 − 0.714671i \(-0.253423\pi\)
−0.714671 + 0.699461i \(0.753423\pi\)
\(48\) 0.641868 − 7.82376i 0.0926457 − 1.12926i
\(49\) 1.86688i 0.266698i
\(50\) 0 0
\(51\) 3.83816 − 3.83816i 0.537450 − 0.537450i
\(52\) 8.74565 − 8.05753i 1.21280 − 1.11738i
\(53\) 4.03213 0.553856 0.276928 − 0.960891i \(-0.410684\pi\)
0.276928 + 0.960891i \(0.410684\pi\)
\(54\) −2.39424 + 5.46135i −0.325815 + 0.743195i
\(55\) 0 0
\(56\) 2.81093 − 5.75878i 0.375627 − 0.769550i
\(57\) −1.53209 − 1.53209i −0.202931 − 0.202931i
\(58\) −1.05328 − 2.69771i −0.138303 − 0.354227i
\(59\) −3.46736 + 3.46736i −0.451412 + 0.451412i −0.895823 − 0.444411i \(-0.853413\pi\)
0.444411 + 0.895823i \(0.353413\pi\)
\(60\) 0 0
\(61\) 0.680578 + 0.680578i 0.0871391 + 0.0871391i 0.749333 − 0.662194i \(-0.230374\pi\)
−0.662194 + 0.749333i \(0.730374\pi\)
\(62\) −4.67390 − 2.04902i −0.593585 − 0.260226i
\(63\) 1.36406 − 1.36406i 0.171855 − 0.171855i
\(64\) 6.30711 − 4.92142i 0.788388 − 0.615178i
\(65\) 0 0
\(66\) 2.75894 − 1.07719i 0.339602 − 0.132593i
\(67\) − 9.04721i − 1.10529i −0.833416 − 0.552646i \(-0.813618\pi\)
0.833416 − 0.552646i \(-0.186382\pi\)
\(68\) 5.52703 + 0.226341i 0.670251 + 0.0274479i
\(69\) 9.67754 + 9.67754i 1.16504 + 1.16504i
\(70\) 0 0
\(71\) −3.64007 −0.431997 −0.215998 − 0.976394i \(-0.569301\pi\)
−0.215998 + 0.976394i \(0.569301\pi\)
\(72\) 2.27729 − 0.783353i 0.268381 − 0.0923191i
\(73\) 2.94030 − 2.94030i 0.344136 − 0.344136i −0.513784 − 0.857920i \(-0.671757\pi\)
0.857920 + 0.513784i \(0.171757\pi\)
\(74\) −13.2583 − 5.81242i −1.54125 − 0.675681i
\(75\) 0 0
\(76\) 0.0903496 − 2.20625i 0.0103638 − 0.253074i
\(77\) 2.41777 0.275530
\(78\) 15.1135 + 6.62570i 1.71126 + 0.750213i
\(79\) 10.7140 1.20542 0.602711 − 0.797960i \(-0.294087\pi\)
0.602711 + 0.797960i \(0.294087\pi\)
\(80\) 0 0
\(81\) −10.8294 −1.20326
\(82\) −8.98016 − 3.93688i −0.991693 − 0.434756i
\(83\) 4.23845 0.465230 0.232615 − 0.972569i \(-0.425272\pi\)
0.232615 + 0.972569i \(0.425272\pi\)
\(84\) 8.88523 + 0.363865i 0.969458 + 0.0397009i
\(85\) 0 0
\(86\) −12.8400 − 5.62904i −1.38458 − 0.606995i
\(87\) 2.84176 − 2.84176i 0.304668 − 0.304668i
\(88\) 2.71247 + 1.32399i 0.289150 + 0.141138i
\(89\) 0.0426256 0.00451831 0.00225915 − 0.999997i \(-0.499281\pi\)
0.00225915 + 0.999997i \(0.499281\pi\)
\(90\) 0 0
\(91\) 9.52546 + 9.52546i 0.998539 + 0.998539i
\(92\) −0.570698 + 13.9359i −0.0594993 + 1.45292i
\(93\) − 7.08189i − 0.734358i
\(94\) 0.194258 − 0.0758455i 0.0200362 − 0.00782287i
\(95\) 0 0
\(96\) 9.76898 + 5.27383i 0.997042 + 0.538258i
\(97\) 1.91173 − 1.91173i 0.194106 − 0.194106i −0.603362 − 0.797468i \(-0.706172\pi\)
0.797468 + 0.603362i \(0.206172\pi\)
\(98\) −2.41802 − 1.06005i −0.244257 − 0.107081i
\(99\) 0.642491 + 0.642491i 0.0645728 + 0.0645728i
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 400.2.s.d.243.4 18
4.3 odd 2 1600.2.s.d.943.8 18
5.2 odd 4 400.2.j.d.307.1 18
5.3 odd 4 80.2.j.b.67.9 yes 18
5.4 even 2 80.2.s.b.3.6 yes 18
15.8 even 4 720.2.bd.g.307.1 18
15.14 odd 2 720.2.z.g.163.4 18
16.5 even 4 1600.2.j.d.143.8 18
16.11 odd 4 400.2.j.d.43.1 18
20.3 even 4 320.2.j.b.47.8 18
20.7 even 4 1600.2.j.d.1007.2 18
20.19 odd 2 320.2.s.b.303.2 18
40.3 even 4 640.2.j.c.607.2 18
40.13 odd 4 640.2.j.d.607.8 18
40.19 odd 2 640.2.s.c.223.8 18
40.29 even 2 640.2.s.d.223.2 18
80.3 even 4 640.2.s.d.287.2 18
80.13 odd 4 640.2.s.c.287.8 18
80.19 odd 4 640.2.j.d.543.2 18
80.27 even 4 inner 400.2.s.d.107.4 18
80.29 even 4 640.2.j.c.543.8 18
80.37 odd 4 1600.2.s.d.207.8 18
80.43 even 4 80.2.s.b.27.6 yes 18
80.53 odd 4 320.2.s.b.207.2 18
80.59 odd 4 80.2.j.b.43.9 ✓ 18
80.69 even 4 320.2.j.b.143.2 18
240.59 even 4 720.2.bd.g.523.1 18
240.203 odd 4 720.2.z.g.667.4 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.j.b.43.9 ✓ 18 80.59 odd 4
80.2.j.b.67.9 yes 18 5.3 odd 4
80.2.s.b.3.6 yes 18 5.4 even 2
80.2.s.b.27.6 yes 18 80.43 even 4
320.2.j.b.47.8 18 20.3 even 4
320.2.j.b.143.2 18 80.69 even 4
320.2.s.b.207.2 18 80.53 odd 4
320.2.s.b.303.2 18 20.19 odd 2
400.2.j.d.43.1 18 16.11 odd 4
400.2.j.d.307.1 18 5.2 odd 4
400.2.s.d.107.4 18 80.27 even 4 inner
400.2.s.d.243.4 18 1.1 even 1 trivial
640.2.j.c.543.8 18 80.29 even 4
640.2.j.c.607.2 18 40.3 even 4
640.2.j.d.543.2 18 80.19 odd 4
640.2.j.d.607.8 18 40.13 odd 4
640.2.s.c.223.8 18 40.19 odd 2
640.2.s.c.287.8 18 80.13 odd 4
640.2.s.d.223.2 18 40.29 even 2
640.2.s.d.287.2 18 80.3 even 4
720.2.z.g.163.4 18 15.14 odd 2
720.2.z.g.667.4 18 240.203 odd 4
720.2.bd.g.307.1 18 15.8 even 4
720.2.bd.g.523.1 18 240.59 even 4
1600.2.j.d.143.8 18 16.5 even 4
1600.2.j.d.1007.2 18 20.7 even 4
1600.2.s.d.207.8 18 80.37 odd 4
1600.2.s.d.943.8 18 4.3 odd 2