Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [121,6,Mod(1,121)] 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("121.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(121, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 121.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(19.4064421974\)
Analytic rank: \(1\)
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: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(6.16441\) of defining polynomial
Character \(\chi\) \(=\) 121.1

$q$-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)
 
\(f(q)\) \(=\) \(q+6.16441 q^{2} +7.00000 q^{3} +6.00000 q^{4} -19.0000 q^{5} +43.1509 q^{6} -49.3153 q^{7} -160.275 q^{8} -194.000 q^{9} -117.124 q^{10} +42.0000 q^{12} -739.730 q^{13} -304.000 q^{14} -133.000 q^{15} -1180.00 q^{16} +1824.67 q^{17} -1195.90 q^{18} -2120.56 q^{19} -114.000 q^{20} -345.207 q^{21} +3071.00 q^{23} -1121.92 q^{24} -2764.00 q^{25} -4560.00 q^{26} -3059.00 q^{27} -295.892 q^{28} +493.153 q^{29} -819.867 q^{30} -1581.00 q^{31} -2145.22 q^{32} +11248.0 q^{34} +936.991 q^{35} -1164.00 q^{36} -9145.00 q^{37} -13072.0 q^{38} -5178.11 q^{39} +3045.22 q^{40} +17408.3 q^{41} -2128.00 q^{42} +14104.2 q^{43} +3686.00 q^{45} +18930.9 q^{46} -16636.0 q^{47} -8260.00 q^{48} -14375.0 q^{49} -17038.4 q^{50} +12772.7 q^{51} -4438.38 q^{52} -16266.0 q^{53} -18856.9 q^{54} +7904.00 q^{56} -14843.9 q^{57} +3040.00 q^{58} +14505.0 q^{59} -798.000 q^{60} +7791.82 q^{61} -9745.94 q^{62} +9567.17 q^{63} +24536.0 q^{64} +14054.9 q^{65} -10635.0 q^{67} +10948.0 q^{68} +21497.0 q^{69} +5776.00 q^{70} +31045.0 q^{71} +31093.3 q^{72} -33682.4 q^{73} -56373.6 q^{74} -19348.0 q^{75} -12723.4 q^{76} -31920.0 q^{78} -83737.4 q^{79} +22420.0 q^{80} +25729.0 q^{81} +107312. q^{82} +84822.3 q^{83} -2071.24 q^{84} -34668.7 q^{85} +86944.0 q^{86} +3452.07 q^{87} -109481. q^{89} +22722.0 q^{90} +36480.0 q^{91} +18426.0 q^{92} -11067.0 q^{93} -102551. q^{94} +40290.6 q^{95} -15016.5 q^{96} -13615.0 q^{97} -88613.5 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 14 q^{3} + 12 q^{4} - 38 q^{5} - 388 q^{9} + 84 q^{12} - 608 q^{14} - 266 q^{15} - 2360 q^{16} - 228 q^{20} + 6142 q^{23} - 5528 q^{25} - 9120 q^{26} - 6118 q^{27} - 3162 q^{31} + 22496 q^{34} - 2328 q^{36}+ \cdots - 27230 q^{97}+O(q^{100}) \) 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 \(-0.316582\pi\)
0.544862 + 0.838525i \(0.316582\pi\)
\(3\) 7.00000 0.449050 0.224525 − 0.974468i \(-0.427917\pi\)
0.224525 + 0.974468i \(0.427917\pi\)
\(4\) 6.00000 0.187500
\(5\) −19.0000 −0.339882 −0.169941 − 0.985454i \(-0.554358\pi\)
−0.169941 + 0.985454i \(0.554358\pi\)
\(6\) 43.1509 0.489341
\(7\) −49.3153 −0.380397 −0.190198 − 0.981746i \(-0.560913\pi\)
−0.190198 + 0.981746i \(0.560913\pi\)
\(8\) −160.275 −0.885401
\(9\) −194.000 −0.798354
\(10\) −117.124 −0.370378
\(11\) 0 0
\(12\) 42.0000 0.0841969
\(13\) −739.730 −1.21399 −0.606994 − 0.794706i \(-0.707625\pi\)
−0.606994 + 0.794706i \(0.707625\pi\)
\(14\) −304.000 −0.414528
\(15\) −133.000 −0.152624
\(16\) −1180.00 −1.15234
\(17\) 1824.67 1.53130 0.765652 − 0.643255i \(-0.222417\pi\)
0.765652 + 0.643255i \(0.222417\pi\)
\(18\) −1195.90 −0.869986
\(19\) −2120.56 −1.34762 −0.673808 − 0.738906i \(-0.735343\pi\)
−0.673808 + 0.738906i \(0.735343\pi\)
\(20\) −114.000 −0.0637279
\(21\) −345.207 −0.170817
\(22\) 0 0
\(23\) 3071.00 1.21049 0.605244 − 0.796040i \(-0.293076\pi\)
0.605244 + 0.796040i \(0.293076\pi\)
\(24\) −1121.92 −0.397590
\(25\) −2764.00 −0.884480
\(26\) −4560.00 −1.32291
\(27\) −3059.00 −0.807551
\(28\) −295.892 −0.0713244
\(29\) 493.153 0.108890 0.0544448 − 0.998517i \(-0.482661\pi\)
0.0544448 + 0.998517i \(0.482661\pi\)
\(30\) −819.867 −0.166318
\(31\) −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
\(36\) −1164.00 −0.149691
\(37\) −9145.00 −1.09819 −0.549097 − 0.835758i \(-0.685028\pi\)
−0.549097 + 0.835758i \(0.685028\pi\)
\(38\) −13072.0 −1.46853
\(39\) −5178.11 −0.545142
\(40\) 3045.22 0.300932
\(41\) 17408.3 1.61732 0.808662 − 0.588274i \(-0.200192\pi\)
0.808662 + 0.588274i \(0.200192\pi\)
\(42\) −2128.00 −0.186144
\(43\) 14104.2 1.16326 0.581630 − 0.813454i \(-0.302415\pi\)
0.581630 + 0.813454i \(0.302415\pi\)
\(44\) 0 0
\(45\) 3686.00 0.271346
\(46\) 18930.9 1.31910
\(47\) −16636.0 −1.09851 −0.549255 − 0.835655i \(-0.685089\pi\)
−0.549255 + 0.835655i \(0.685089\pi\)
\(48\) −8260.00 −0.517460
\(49\) −14375.0 −0.855298
\(50\) −17038.4 −0.963840
\(51\) 12772.7 0.687632
\(52\) −4438.38 −0.227623
\(53\) −16266.0 −0.795410 −0.397705 − 0.917513i \(-0.630193\pi\)
−0.397705 + 0.917513i \(0.630193\pi\)
\(54\) −18856.9 −0.880009
\(55\) 0 0
\(56\) 7904.00 0.336804
\(57\) −14843.9 −0.605147
\(58\) 3040.00 0.118660
\(59\) 14505.0 0.542485 0.271242 − 0.962511i \(-0.412565\pi\)
0.271242 + 0.962511i \(0.412565\pi\)
\(60\) −798.000 −0.0286170
\(61\) 7791.82 0.268111 0.134055 − 0.990974i \(-0.457200\pi\)
0.134055 + 0.990974i \(0.457200\pi\)
\(62\) −9745.94 −0.321992
\(63\) 9567.17 0.303691
\(64\) 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
\(69\) 21497.0 0.543570
\(70\) 5776.00 0.140891
\(71\) 31045.0 0.730880 0.365440 − 0.930835i \(-0.380919\pi\)
0.365440 + 0.930835i \(0.380919\pi\)
\(72\) 31093.3 0.706864
\(73\) −33682.4 −0.739768 −0.369884 − 0.929078i \(-0.620603\pi\)
−0.369884 + 0.929078i \(0.620603\pi\)
\(74\) −56373.6 −1.19673
\(75\) −19348.0 −0.397176
\(76\) −12723.4 −0.252678
\(77\) 0 0
\(78\) −31920.0 −0.594055
\(79\) −83737.4 −1.50956 −0.754782 − 0.655975i \(-0.772258\pi\)
−0.754782 + 0.655975i \(0.772258\pi\)
\(80\) 22420.0 0.391661
\(81\) 25729.0 0.435723
\(82\) 107312. 1.76244
\(83\) 84822.3 1.35150 0.675748 − 0.737132i \(-0.263821\pi\)
0.675748 + 0.737132i \(0.263821\pi\)
\(84\) −2071.24 −0.0320282
\(85\) −34668.7 −0.520463
\(86\) 86944.0 1.26763
\(87\) 3452.07 0.0488969
\(88\) 0 0
\(89\) −109481. −1.46509 −0.732544 − 0.680720i \(-0.761667\pi\)
−0.732544 + 0.680720i \(0.761667\pi\)
\(90\) 22722.0 0.295693
\(91\) 36480.0 0.461797
\(92\) 18426.0 0.226966
\(93\) −11067.0 −0.132685
\(94\) −102551. −1.19707
\(95\) 40290.6 0.458031
\(96\) −15016.5 −0.166300
\(97\) −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 121.6.a.c.1.2 yes 2
3.2 odd 2 1089.6.a.m.1.1 2
11.10 odd 2 inner 121.6.a.c.1.1 ✓ 2
33.32 even 2 1089.6.a.m.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.6.a.c.1.1 ✓ 2 11.10 odd 2 inner
121.6.a.c.1.2 yes 2 1.1 even 1 trivial
1089.6.a.m.1.1 2 3.2 odd 2
1089.6.a.m.1.2 2 33.32 even 2