Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1881,2,Mod(1,1881)] 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("1881.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1881, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1881 = 3^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1881.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [7,1,0,15,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(15.0198606202\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 14x^{5} + 10x^{4} + 59x^{3} - 27x^{2} - 66x + 30 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 209)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(1.19313\) of defining polynomial
Character \(\chi\) \(=\) 1881.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.19313 q^{2} -0.576442 q^{4} -1.08235 q^{5} -1.30958 q^{7} -3.07403 q^{8} -1.29138 q^{10} +1.00000 q^{11} +2.53997 q^{13} -1.56249 q^{14} -2.51483 q^{16} +5.43892 q^{17} +1.00000 q^{19} +0.623909 q^{20} +1.19313 q^{22} -3.87095 q^{23} -3.82853 q^{25} +3.03052 q^{26} +0.754894 q^{28} +2.41412 q^{29} +3.03647 q^{31} +3.14754 q^{32} +6.48934 q^{34} +1.41741 q^{35} +6.85067 q^{37} +1.19313 q^{38} +3.32716 q^{40} +6.11344 q^{41} -2.95329 q^{43} -0.576442 q^{44} -4.61854 q^{46} +12.0923 q^{47} -5.28501 q^{49} -4.56793 q^{50} -1.46415 q^{52} +0.992927 q^{53} -1.08235 q^{55} +4.02567 q^{56} +2.88036 q^{58} +14.2251 q^{59} -5.82518 q^{61} +3.62290 q^{62} +8.78508 q^{64} -2.74913 q^{65} +8.79718 q^{67} -3.13522 q^{68} +1.69116 q^{70} +2.44975 q^{71} +5.84063 q^{73} +8.17374 q^{74} -0.576442 q^{76} -1.30958 q^{77} +17.0397 q^{79} +2.72192 q^{80} +7.29413 q^{82} +7.01303 q^{83} -5.88679 q^{85} -3.52366 q^{86} -3.07403 q^{88} +9.13103 q^{89} -3.32629 q^{91} +2.23138 q^{92} +14.4277 q^{94} -1.08235 q^{95} -14.7950 q^{97} -6.30570 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + q^{2} + 15 q^{4} - 2 q^{5} + 10 q^{7} + 9 q^{8} - 6 q^{10} + 7 q^{11} - 4 q^{13} - 6 q^{14} + 27 q^{16} - 2 q^{17} + 7 q^{19} + 4 q^{20} + q^{22} - 10 q^{23} + 9 q^{25} + 8 q^{26} + 26 q^{28} + 18 q^{29}+ \cdots - 19 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.19313 0.843670 0.421835 − 0.906673i \(-0.361386\pi\)
0.421835 + 0.906673i \(0.361386\pi\)
\(3\) 0 0
\(4\) −0.576442 −0.288221
\(5\) −1.08235 −0.484040 −0.242020 − 0.970271i \(-0.577810\pi\)
−0.242020 + 0.970271i \(0.577810\pi\)
\(6\) 0 0
\(7\) −1.30958 −0.494973 −0.247487 − 0.968891i \(-0.579605\pi\)
−0.247487 + 0.968891i \(0.579605\pi\)
\(8\) −3.07403 −1.08683
\(9\) 0 0
\(10\) −1.29138 −0.408370
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 2.53997 0.704462 0.352231 − 0.935913i \(-0.385423\pi\)
0.352231 + 0.935913i \(0.385423\pi\)
\(14\) −1.56249 −0.417594
\(15\) 0 0
\(16\) −2.51483 −0.628708
\(17\) 5.43892 1.31913 0.659566 − 0.751646i \(-0.270740\pi\)
0.659566 + 0.751646i \(0.270740\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0.623909 0.139510
\(21\) 0 0
\(22\) 1.19313 0.254376
\(23\) −3.87095 −0.807148 −0.403574 − 0.914947i \(-0.632232\pi\)
−0.403574 + 0.914947i \(0.632232\pi\)
\(24\) 0 0
\(25\) −3.82853 −0.765706
\(26\) 3.03052 0.594334
\(27\) 0 0
\(28\) 0.754894 0.142662
\(29\) 2.41412 0.448290 0.224145 − 0.974556i \(-0.428041\pi\)
0.224145 + 0.974556i \(0.428041\pi\)
\(30\) 0 0
\(31\) 3.03647 0.545366 0.272683 − 0.962104i \(-0.412089\pi\)
0.272683 + 0.962104i \(0.412089\pi\)
\(32\) 3.14754 0.556412
\(33\) 0 0
\(34\) 6.48934 1.11291
\(35\) 1.41741 0.239587
\(36\) 0 0
\(37\) 6.85067 1.12624 0.563122 − 0.826374i \(-0.309600\pi\)
0.563122 + 0.826374i \(0.309600\pi\)
\(38\) 1.19313 0.193551
\(39\) 0 0
\(40\) 3.32716 0.526070
\(41\) 6.11344 0.954759 0.477380 − 0.878697i \(-0.341587\pi\)
0.477380 + 0.878697i \(0.341587\pi\)
\(42\) 0 0
\(43\) −2.95329 −0.450373 −0.225186 − 0.974316i \(-0.572299\pi\)
−0.225186 + 0.974316i \(0.572299\pi\)
\(44\) −0.576442 −0.0869019
\(45\) 0 0
\(46\) −4.61854 −0.680966
\(47\) 12.0923 1.76385 0.881925 − 0.471391i \(-0.156248\pi\)
0.881925 + 0.471391i \(0.156248\pi\)
\(48\) 0 0
\(49\) −5.28501 −0.755002
\(50\) −4.56793 −0.646003
\(51\) 0 0
\(52\) −1.46415 −0.203041
\(53\) 0.992927 0.136389 0.0681945 − 0.997672i \(-0.478276\pi\)
0.0681945 + 0.997672i \(0.478276\pi\)
\(54\) 0 0
\(55\) −1.08235 −0.145943
\(56\) 4.02567 0.537953
\(57\) 0 0
\(58\) 2.88036 0.378209
\(59\) 14.2251 1.85195 0.925977 − 0.377580i \(-0.123244\pi\)
0.925977 + 0.377580i \(0.123244\pi\)
\(60\) 0 0
\(61\) −5.82518 −0.745838 −0.372919 − 0.927864i \(-0.621643\pi\)
−0.372919 + 0.927864i \(0.621643\pi\)
\(62\) 3.62290 0.460109
\(63\) 0 0
\(64\) 8.78508 1.09814
\(65\) −2.74913 −0.340988
\(66\) 0 0
\(67\) 8.79718 1.07475 0.537373 − 0.843345i \(-0.319417\pi\)
0.537373 + 0.843345i \(0.319417\pi\)
\(68\) −3.13522 −0.380202
\(69\) 0 0
\(70\) 1.69116 0.202132
\(71\) 2.44975 0.290732 0.145366 − 0.989378i \(-0.453564\pi\)
0.145366 + 0.989378i \(0.453564\pi\)
\(72\) 0 0
\(73\) 5.84063 0.683594 0.341797 − 0.939774i \(-0.388965\pi\)
0.341797 + 0.939774i \(0.388965\pi\)
\(74\) 8.17374 0.950178
\(75\) 0 0
\(76\) −0.576442 −0.0661224
\(77\) −1.30958 −0.149240
\(78\) 0 0
\(79\) 17.0397 1.91711 0.958557 − 0.284902i \(-0.0919610\pi\)
0.958557 + 0.284902i \(0.0919610\pi\)
\(80\) 2.72192 0.304319
\(81\) 0 0
\(82\) 7.29413 0.805502
\(83\) 7.01303 0.769780 0.384890 − 0.922963i \(-0.374239\pi\)
0.384890 + 0.922963i \(0.374239\pi\)
\(84\) 0 0
\(85\) −5.88679 −0.638512
\(86\) −3.52366 −0.379966
\(87\) 0 0
\(88\) −3.07403 −0.327693
\(89\) 9.13103 0.967887 0.483944 − 0.875099i \(-0.339204\pi\)
0.483944 + 0.875099i \(0.339204\pi\)
\(90\) 0 0
\(91\) −3.32629 −0.348690
\(92\) 2.23138 0.232637
\(93\) 0 0
\(94\) 14.4277 1.48811
\(95\) −1.08235 −0.111046
\(96\) 0 0
\(97\) −14.7950 −1.50220 −0.751100 − 0.660188i \(-0.770476\pi\)
−0.751100 + 0.660188i \(0.770476\pi\)
\(98\) −6.30570 −0.636972
\(99\) 0 0
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 1881.2.a.p.1.5 7
3.2 odd 2 209.2.a.d.1.3 ✓ 7
12.11 even 2 3344.2.a.ba.1.1 7
15.14 odd 2 5225.2.a.n.1.5 7
33.32 even 2 2299.2.a.q.1.5 7
57.56 even 2 3971.2.a.i.1.5 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.d.1.3 ✓ 7 3.2 odd 2
1881.2.a.p.1.5 7 1.1 even 1 trivial
2299.2.a.q.1.5 7 33.32 even 2
3344.2.a.ba.1.1 7 12.11 even 2
3971.2.a.i.1.5 7 57.56 even 2
5225.2.a.n.1.5 7 15.14 odd 2