Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1089,6,Mod(1,1089)] 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("1089.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1089, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1089.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,12,38,0,0] 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: \(174.657979776\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{38}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 38 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 121)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-6.16441\) of defining polynomial
Character \(\chi\) \(=\) 1089.1

$q$-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)
 
\(f(q)\) \(=\) \(q-6.16441 q^{2} +6.00000 q^{4} +19.0000 q^{5} -49.3153 q^{7} +160.275 q^{8} -117.124 q^{10} -739.730 q^{13} +304.000 q^{14} -1180.00 q^{16} -1824.67 q^{17} -2120.56 q^{19} +114.000 q^{20} -3071.00 q^{23} -2764.00 q^{25} +4560.00 q^{26} -295.892 q^{28} -493.153 q^{29} -1581.00 q^{31} +2145.22 q^{32} +11248.0 q^{34} -936.991 q^{35} -9145.00 q^{37} +13072.0 q^{38} +3045.22 q^{40} -17408.3 q^{41} +14104.2 q^{43} +18930.9 q^{46} +16636.0 q^{47} -14375.0 q^{49} +17038.4 q^{50} -4438.38 q^{52} +16266.0 q^{53} -7904.00 q^{56} +3040.00 q^{58} -14505.0 q^{59} +7791.82 q^{61} +9745.94 q^{62} +24536.0 q^{64} -14054.9 q^{65} -10635.0 q^{67} -10948.0 q^{68} +5776.00 q^{70} -31045.0 q^{71} -33682.4 q^{73} +56373.6 q^{74} -12723.4 q^{76} -83737.4 q^{79} -22420.0 q^{80} +107312. q^{82} -84822.3 q^{83} -34668.7 q^{85} -86944.0 q^{86} +109481. q^{89} +36480.0 q^{91} -18426.0 q^{92} -102551. q^{94} -40290.6 q^{95} -13615.0 q^{97} +88613.5 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 12 q^{4} + 38 q^{5} + 608 q^{14} - 2360 q^{16} + 228 q^{20} - 6142 q^{23} - 5528 q^{25} + 9120 q^{26} - 3162 q^{31} + 22496 q^{34} - 18290 q^{37} + 26144 q^{38} + 33272 q^{47} - 28750 q^{49} + 32532 q^{53}+ \cdots - 27230 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −6.16441 −1.08972 −0.544862 − 0.838525i \(-0.683418\pi\)
−0.544862 + 0.838525i \(0.683418\pi\)
\(3\) 0 0
\(4\) 6.00000 0.187500
\(5\) 19.0000 0.339882 0.169941 − 0.985454i \(-0.445642\pi\)
0.169941 + 0.985454i \(0.445642\pi\)
\(6\) 0 0
\(7\) −49.3153 −0.380397 −0.190198 − 0.981746i \(-0.560913\pi\)
−0.190198 + 0.981746i \(0.560913\pi\)
\(8\) 160.275 0.885401
\(9\) 0 0
\(10\) −117.124 −0.370378
\(11\) 0 0
\(12\) 0 0
\(13\) −739.730 −1.21399 −0.606994 − 0.794706i \(-0.707625\pi\)
−0.606994 + 0.794706i \(0.707625\pi\)
\(14\) 304.000 0.414528
\(15\) 0 0
\(16\) −1180.00 −1.15234
\(17\) −1824.67 −1.53130 −0.765652 − 0.643255i \(-0.777583\pi\)
−0.765652 + 0.643255i \(0.777583\pi\)
\(18\) 0 0
\(19\) −2120.56 −1.34762 −0.673808 − 0.738906i \(-0.735343\pi\)
−0.673808 + 0.738906i \(0.735343\pi\)
\(20\) 114.000 0.0637279
\(21\) 0 0
\(22\) 0 0
\(23\) −3071.00 −1.21049 −0.605244 − 0.796040i \(-0.706924\pi\)
−0.605244 + 0.796040i \(0.706924\pi\)
\(24\) 0 0
\(25\) −2764.00 −0.884480
\(26\) 4560.00 1.32291
\(27\) 0 0
\(28\) −295.892 −0.0713244
\(29\) −493.153 −0.108890 −0.0544448 − 0.998517i \(-0.517339\pi\)
−0.0544448 + 0.998517i \(0.517339\pi\)
\(30\) 0 0
\(31\) −1581.00 −0.295480 −0.147740 − 0.989026i \(-0.547200\pi\)
−0.147740 + 0.989026i \(0.547200\pi\)
\(32\) 2145.22 0.370336
\(33\) 0 0
\(34\) 11248.0 1.66870
\(35\) −936.991 −0.129290
\(36\) 0 0
\(37\) −9145.00 −1.09819 −0.549097 − 0.835758i \(-0.685028\pi\)
−0.549097 + 0.835758i \(0.685028\pi\)
\(38\) 13072.0 1.46853
\(39\) 0 0
\(40\) 3045.22 0.300932
\(41\) −17408.3 −1.61732 −0.808662 − 0.588274i \(-0.799808\pi\)
−0.808662 + 0.588274i \(0.799808\pi\)
\(42\) 0 0
\(43\) 14104.2 1.16326 0.581630 − 0.813454i \(-0.302415\pi\)
0.581630 + 0.813454i \(0.302415\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 18930.9 1.31910
\(47\) 16636.0 1.09851 0.549255 − 0.835655i \(-0.314911\pi\)
0.549255 + 0.835655i \(0.314911\pi\)
\(48\) 0 0
\(49\) −14375.0 −0.855298
\(50\) 17038.4 0.963840
\(51\) 0 0
\(52\) −4438.38 −0.227623
\(53\) 16266.0 0.795410 0.397705 − 0.917513i \(-0.369807\pi\)
0.397705 + 0.917513i \(0.369807\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −7904.00 −0.336804
\(57\) 0 0
\(58\) 3040.00 0.118660
\(59\) −14505.0 −0.542485 −0.271242 − 0.962511i \(-0.587435\pi\)
−0.271242 + 0.962511i \(0.587435\pi\)
\(60\) 0 0
\(61\) 7791.82 0.268111 0.134055 − 0.990974i \(-0.457200\pi\)
0.134055 + 0.990974i \(0.457200\pi\)
\(62\) 9745.94 0.321992
\(63\) 0 0
\(64\) 24536.0 0.748779
\(65\) −14054.9 −0.412613
\(66\) 0 0
\(67\) −10635.0 −0.289435 −0.144717 − 0.989473i \(-0.546227\pi\)
−0.144717 + 0.989473i \(0.546227\pi\)
\(68\) −10948.0 −0.287119
\(69\) 0 0
\(70\) 5776.00 0.140891
\(71\) −31045.0 −0.730880 −0.365440 − 0.930835i \(-0.619081\pi\)
−0.365440 + 0.930835i \(0.619081\pi\)
\(72\) 0 0
\(73\) −33682.4 −0.739768 −0.369884 − 0.929078i \(-0.620603\pi\)
−0.369884 + 0.929078i \(0.620603\pi\)
\(74\) 56373.6 1.19673
\(75\) 0 0
\(76\) −12723.4 −0.252678
\(77\) 0 0
\(78\) 0 0
\(79\) −83737.4 −1.50956 −0.754782 − 0.655975i \(-0.772258\pi\)
−0.754782 + 0.655975i \(0.772258\pi\)
\(80\) −22420.0 −0.391661
\(81\) 0 0
\(82\) 107312. 1.76244
\(83\) −84822.3 −1.35150 −0.675748 − 0.737132i \(-0.736179\pi\)
−0.675748 + 0.737132i \(0.736179\pi\)
\(84\) 0 0
\(85\) −34668.7 −0.520463
\(86\) −86944.0 −1.26763
\(87\) 0 0
\(88\) 0 0
\(89\) 109481. 1.46509 0.732544 − 0.680720i \(-0.238333\pi\)
0.732544 + 0.680720i \(0.238333\pi\)
\(90\) 0 0
\(91\) 36480.0 0.461797
\(92\) −18426.0 −0.226966
\(93\) 0 0
\(94\) −102551. −1.19707
\(95\) −40290.6 −0.458031
\(96\) 0 0
\(97\) −13615.0 −0.146923 −0.0734613 − 0.997298i \(-0.523405\pi\)
−0.0734613 + 0.997298i \(0.523405\pi\)
\(98\) 88613.5 0.932040
\(99\) 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 1089.6.a.m.1.1 2
3.2 odd 2 121.6.a.c.1.2 yes 2
11.10 odd 2 inner 1089.6.a.m.1.2 2
33.32 even 2 121.6.a.c.1.1 ✓ 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.6.a.c.1.1 ✓ 2 33.32 even 2
121.6.a.c.1.2 yes 2 3.2 odd 2
1089.6.a.m.1.1 2 1.1 even 1 trivial
1089.6.a.m.1.2 2 11.10 odd 2 inner