Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2775,2,Mod(1,2775)] 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("2775.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2775, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2775 = 3 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2775.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-4,4,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(22.1584865609\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.6224.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} - 2x + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 111)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.44579\) of defining polynomial
Character \(\chi\) \(=\) 2775.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.44579 q^{2} -1.00000 q^{3} +3.98187 q^{4} -2.44579 q^{6} -0.269264 q^{7} +4.84724 q^{8} +1.00000 q^{9} +5.96375 q^{11} -3.98187 q^{12} -0.658562 q^{13} -0.658562 q^{14} +3.89157 q^{16} -5.29303 q^{17} +2.44579 q^{18} +3.07217 q^{19} +0.269264 q^{21} +14.5861 q^{22} +9.02377 q^{23} -4.84724 q^{24} -1.61070 q^{26} -1.00000 q^{27} -1.07217 q^{28} -4.49012 q^{29} +1.73074 q^{31} -0.176523 q^{32} -5.96375 q^{33} -12.9456 q^{34} +3.98187 q^{36} +1.00000 q^{37} +7.51388 q^{38} +0.658562 q^{39} +7.16084 q^{41} +0.658562 q^{42} +8.05241 q^{43} +23.7469 q^{44} +22.0702 q^{46} +8.62231 q^{47} -3.89157 q^{48} -6.92750 q^{49} +5.29303 q^{51} -2.62231 q^{52} -5.81940 q^{53} -2.44579 q^{54} -1.30519 q^{56} -3.07217 q^{57} -10.9819 q^{58} -12.0646 q^{59} +7.78315 q^{61} +4.23301 q^{62} -0.269264 q^{63} -8.21489 q^{64} -14.5861 q^{66} +0.269264 q^{67} -21.0762 q^{68} -9.02377 q^{69} +3.30519 q^{71} +4.84724 q^{72} +3.69448 q^{73} +2.44579 q^{74} +12.2330 q^{76} -1.60582 q^{77} +1.61070 q^{78} +8.85532 q^{79} +1.00000 q^{81} +17.5139 q^{82} +5.34144 q^{83} +1.07217 q^{84} +19.6945 q^{86} +4.49012 q^{87} +28.9077 q^{88} +5.20925 q^{89} +0.177327 q^{91} +35.9315 q^{92} -1.73074 q^{93} +21.0883 q^{94} +0.176523 q^{96} -12.0475 q^{97} -16.9432 q^{98} +5.96375 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 4 q^{4} - 4 q^{7} + 6 q^{8} + 4 q^{9} - 4 q^{12} - 4 q^{13} - 4 q^{14} - 4 q^{16} + 2 q^{17} + 8 q^{19} + 4 q^{21} + 12 q^{22} + 10 q^{23} - 6 q^{24} - 8 q^{26} - 4 q^{27} - 2 q^{29} + 4 q^{31}+ \cdots - 24 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.44579 1.72943 0.864716 − 0.502261i \(-0.167498\pi\)
0.864716 + 0.502261i \(0.167498\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.98187 1.99094
\(5\) 0 0
\(6\) −2.44579 −0.998488
\(7\) −0.269264 −0.101772 −0.0508861 − 0.998704i \(-0.516205\pi\)
−0.0508861 + 0.998704i \(0.516205\pi\)
\(8\) 4.84724 1.71376
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.96375 1.79814 0.899069 − 0.437807i \(-0.144245\pi\)
0.899069 + 0.437807i \(0.144245\pi\)
\(12\) −3.98187 −1.14947
\(13\) −0.658562 −0.182652 −0.0913261 − 0.995821i \(-0.529111\pi\)
−0.0913261 + 0.995821i \(0.529111\pi\)
\(14\) −0.658562 −0.176008
\(15\) 0 0
\(16\) 3.89157 0.972894
\(17\) −5.29303 −1.28375 −0.641874 − 0.766810i \(-0.721843\pi\)
−0.641874 + 0.766810i \(0.721843\pi\)
\(18\) 2.44579 0.576478
\(19\) 3.07217 0.704805 0.352403 − 0.935849i \(-0.385365\pi\)
0.352403 + 0.935849i \(0.385365\pi\)
\(20\) 0 0
\(21\) 0.269264 0.0587582
\(22\) 14.5861 3.10976
\(23\) 9.02377 1.88159 0.940793 − 0.338983i \(-0.110083\pi\)
0.940793 + 0.338983i \(0.110083\pi\)
\(24\) −4.84724 −0.989439
\(25\) 0 0
\(26\) −1.61070 −0.315885
\(27\) −1.00000 −0.192450
\(28\) −1.07217 −0.202622
\(29\) −4.49012 −0.833794 −0.416897 − 0.908954i \(-0.636882\pi\)
−0.416897 + 0.908954i \(0.636882\pi\)
\(30\) 0 0
\(31\) 1.73074 0.310849 0.155425 − 0.987848i \(-0.450325\pi\)
0.155425 + 0.987848i \(0.450325\pi\)
\(32\) −0.176523 −0.0312052
\(33\) −5.96375 −1.03816
\(34\) −12.9456 −2.22016
\(35\) 0 0
\(36\) 3.98187 0.663646
\(37\) 1.00000 0.164399
\(38\) 7.51388 1.21891
\(39\) 0.658562 0.105454
\(40\) 0 0
\(41\) 7.16084 1.11833 0.559167 − 0.829055i \(-0.311121\pi\)
0.559167 + 0.829055i \(0.311121\pi\)
\(42\) 0.658562 0.101618
\(43\) 8.05241 1.22798 0.613991 − 0.789313i \(-0.289563\pi\)
0.613991 + 0.789313i \(0.289563\pi\)
\(44\) 23.7469 3.57998
\(45\) 0 0
\(46\) 22.0702 3.25407
\(47\) 8.62231 1.25769 0.628847 − 0.777529i \(-0.283527\pi\)
0.628847 + 0.777529i \(0.283527\pi\)
\(48\) −3.89157 −0.561700
\(49\) −6.92750 −0.989642
\(50\) 0 0
\(51\) 5.29303 0.741172
\(52\) −2.62231 −0.363649
\(53\) −5.81940 −0.799356 −0.399678 − 0.916656i \(-0.630878\pi\)
−0.399678 + 0.916656i \(0.630878\pi\)
\(54\) −2.44579 −0.332829
\(55\) 0 0
\(56\) −1.30519 −0.174413
\(57\) −3.07217 −0.406919
\(58\) −10.9819 −1.44199
\(59\) −12.0646 −1.57067 −0.785337 − 0.619069i \(-0.787510\pi\)
−0.785337 + 0.619069i \(0.787510\pi\)
\(60\) 0 0
\(61\) 7.78315 0.996530 0.498265 − 0.867025i \(-0.333971\pi\)
0.498265 + 0.867025i \(0.333971\pi\)
\(62\) 4.23301 0.537593
\(63\) −0.269264 −0.0339240
\(64\) −8.21489 −1.02686
\(65\) 0 0
\(66\) −14.5861 −1.79542
\(67\) 0.269264 0.0328958 0.0164479 − 0.999865i \(-0.494764\pi\)
0.0164479 + 0.999865i \(0.494764\pi\)
\(68\) −21.0762 −2.55586
\(69\) −9.02377 −1.08633
\(70\) 0 0
\(71\) 3.30519 0.392253 0.196127 − 0.980579i \(-0.437164\pi\)
0.196127 + 0.980579i \(0.437164\pi\)
\(72\) 4.84724 0.571253
\(73\) 3.69448 0.432407 0.216203 − 0.976348i \(-0.430633\pi\)
0.216203 + 0.976348i \(0.430633\pi\)
\(74\) 2.44579 0.284317
\(75\) 0 0
\(76\) 12.2330 1.40322
\(77\) −1.60582 −0.183000
\(78\) 1.61070 0.182376
\(79\) 8.85532 0.996302 0.498151 − 0.867090i \(-0.334013\pi\)
0.498151 + 0.867090i \(0.334013\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 17.5139 1.93408
\(83\) 5.34144 0.586299 0.293150 − 0.956067i \(-0.405297\pi\)
0.293150 + 0.956067i \(0.405297\pi\)
\(84\) 1.07217 0.116984
\(85\) 0 0
\(86\) 19.6945 2.12371
\(87\) 4.49012 0.481391
\(88\) 28.9077 3.08157
\(89\) 5.20925 0.552179 0.276090 − 0.961132i \(-0.410961\pi\)
0.276090 + 0.961132i \(0.410961\pi\)
\(90\) 0 0
\(91\) 0.177327 0.0185889
\(92\) 35.9315 3.74612
\(93\) −1.73074 −0.179469
\(94\) 21.0883 2.17510
\(95\) 0 0
\(96\) 0.176523 0.0180163
\(97\) −12.0475 −1.22324 −0.611621 − 0.791151i \(-0.709482\pi\)
−0.611621 + 0.791151i \(0.709482\pi\)
\(98\) −16.9432 −1.71152
\(99\) 5.96375 0.599379
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 2775.2.a.x.1.4 4
3.2 odd 2 8325.2.a.bv.1.1 4
5.4 even 2 111.2.a.b.1.1 ✓ 4
15.14 odd 2 333.2.a.g.1.4 4
20.19 odd 2 1776.2.a.u.1.2 4
35.34 odd 2 5439.2.a.u.1.1 4
40.19 odd 2 7104.2.a.cf.1.3 4
40.29 even 2 7104.2.a.cc.1.3 4
60.59 even 2 5328.2.a.bs.1.3 4
185.184 even 2 4107.2.a.i.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
111.2.a.b.1.1 ✓ 4 5.4 even 2
333.2.a.g.1.4 4 15.14 odd 2
1776.2.a.u.1.2 4 20.19 odd 2
2775.2.a.x.1.4 4 1.1 even 1 trivial
4107.2.a.i.1.4 4 185.184 even 2
5328.2.a.bs.1.3 4 60.59 even 2
5439.2.a.u.1.1 4 35.34 odd 2
7104.2.a.cc.1.3 4 40.29 even 2
7104.2.a.cf.1.3 4 40.19 odd 2
8325.2.a.bv.1.1 4 3.2 odd 2