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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [121,3,Mod(120,121)] 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("121.120"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(121, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 121.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] 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: \(3.29701119876\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 120.2
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 121.120
Dual form 121.3.b.a.120.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.41421i q^{2} -1.00000 q^{3} +2.00000 q^{4} -7.00000 q^{5} -1.41421i q^{6} +7.07107i q^{7} +8.48528i q^{8} -8.00000 q^{9} -9.89949i q^{10} -2.00000 q^{12} +16.9706i q^{13} -10.0000 q^{14} +7.00000 q^{15} -4.00000 q^{16} -4.24264i q^{17} -11.3137i q^{18} -16.9706i q^{19} -14.0000 q^{20} -7.07107i q^{21} -9.00000 q^{23} -8.48528i q^{24} +24.0000 q^{25} -24.0000 q^{26} +17.0000 q^{27} +14.1421i q^{28} +22.6274i q^{29} +9.89949i q^{30} +49.0000 q^{31} +28.2843i q^{32} +6.00000 q^{34} -49.4975i q^{35} -16.0000 q^{36} +17.0000 q^{37} +24.0000 q^{38} -16.9706i q^{39} -59.3970i q^{40} -16.9706i q^{41} +10.0000 q^{42} +46.6690i q^{43} +56.0000 q^{45} -12.7279i q^{46} +32.0000 q^{47} +4.00000 q^{48} -1.00000 q^{49} +33.9411i q^{50} +4.24264i q^{51} +33.9411i q^{52} +16.0000 q^{53} +24.0416i q^{54} -60.0000 q^{56} +16.9706i q^{57} -32.0000 q^{58} -71.0000 q^{59} +14.0000 q^{60} +11.3137i q^{61} +69.2965i q^{62} -56.5685i q^{63} -56.0000 q^{64} -118.794i q^{65} -31.0000 q^{67} -8.48528i q^{68} +9.00000 q^{69} +70.0000 q^{70} -73.0000 q^{71} -67.8823i q^{72} -39.5980i q^{73} +24.0416i q^{74} -24.0000 q^{75} -33.9411i q^{76} +24.0000 q^{78} +156.978i q^{79} +28.0000 q^{80} +55.0000 q^{81} +24.0000 q^{82} -35.3553i q^{83} -14.1421i q^{84} +29.6985i q^{85} -66.0000 q^{86} -22.6274i q^{87} -9.00000 q^{89} +79.1960i q^{90} -120.000 q^{91} -18.0000 q^{92} -49.0000 q^{93} +45.2548i q^{94} +118.794i q^{95} -28.2843i q^{96} -17.0000 q^{97} -1.41421i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 4 q^{4} - 14 q^{5} - 16 q^{9} - 4 q^{12} - 20 q^{14} + 14 q^{15} - 8 q^{16} - 28 q^{20} - 18 q^{23} + 48 q^{25} - 48 q^{26} + 34 q^{27} + 98 q^{31} + 12 q^{34} - 32 q^{36} + 34 q^{37} + 48 q^{38}+ \cdots - 34 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 0.707107i 0.935414 + 0.353553i \(0.115027\pi\)
−0.935414 + 0.353553i \(0.884973\pi\)
\(3\) −1.00000 −0.333333 −0.166667 − 0.986013i \(-0.553300\pi\)
−0.166667 + 0.986013i \(0.553300\pi\)
\(4\) 2.00000 0.500000
\(5\) −7.00000 −1.40000 −0.700000 − 0.714143i \(-0.746817\pi\)
−0.700000 + 0.714143i \(0.746817\pi\)
\(6\) − 1.41421i − 0.235702i
\(7\) 7.07107i 1.01015i 0.863075 + 0.505076i \(0.168536\pi\)
−0.863075 + 0.505076i \(0.831464\pi\)
\(8\) 8.48528i 1.06066i
\(9\) −8.00000 −0.888889
\(10\) − 9.89949i − 0.989949i
\(11\) 0 0
\(12\) −2.00000 −0.166667
\(13\) 16.9706i 1.30543i 0.757604 + 0.652714i \(0.226370\pi\)
−0.757604 + 0.652714i \(0.773630\pi\)
\(14\) −10.0000 −0.714286
\(15\) 7.00000 0.466667
\(16\) −4.00000 −0.250000
\(17\) − 4.24264i − 0.249567i −0.992184 − 0.124784i \(-0.960176\pi\)
0.992184 − 0.124784i \(-0.0398236\pi\)
\(18\) − 11.3137i − 0.628539i
\(19\) − 16.9706i − 0.893188i −0.894737 − 0.446594i \(-0.852637\pi\)
0.894737 − 0.446594i \(-0.147363\pi\)
\(20\) −14.0000 −0.700000
\(21\) − 7.07107i − 0.336718i
\(22\) 0 0
\(23\) −9.00000 −0.391304 −0.195652 − 0.980673i \(-0.562682\pi\)
−0.195652 + 0.980673i \(0.562682\pi\)
\(24\) − 8.48528i − 0.353553i
\(25\) 24.0000 0.960000
\(26\) −24.0000 −0.923077
\(27\) 17.0000 0.629630
\(28\) 14.1421i 0.505076i
\(29\) 22.6274i 0.780256i 0.920761 + 0.390128i \(0.127569\pi\)
−0.920761 + 0.390128i \(0.872431\pi\)
\(30\) 9.89949i 0.329983i
\(31\) 49.0000 1.58065 0.790323 − 0.612691i \(-0.209913\pi\)
0.790323 + 0.612691i \(0.209913\pi\)
\(32\) 28.2843i 0.883883i
\(33\) 0 0
\(34\) 6.00000 0.176471
\(35\) − 49.4975i − 1.41421i
\(36\) −16.0000 −0.444444
\(37\) 17.0000 0.459459 0.229730 − 0.973254i \(-0.426216\pi\)
0.229730 + 0.973254i \(0.426216\pi\)
\(38\) 24.0000 0.631579
\(39\) − 16.9706i − 0.435143i
\(40\) − 59.3970i − 1.48492i
\(41\) − 16.9706i − 0.413916i −0.978350 − 0.206958i \(-0.933644\pi\)
0.978350 − 0.206958i \(-0.0663564\pi\)
\(42\) 10.0000 0.238095
\(43\) 46.6690i 1.08533i 0.839950 + 0.542663i \(0.182584\pi\)
−0.839950 + 0.542663i \(0.817416\pi\)
\(44\) 0 0
\(45\) 56.0000 1.24444
\(46\) − 12.7279i − 0.276694i
\(47\) 32.0000 0.680851 0.340426 − 0.940271i \(-0.389429\pi\)
0.340426 + 0.940271i \(0.389429\pi\)
\(48\) 4.00000 0.0833333
\(49\) −1.00000 −0.0204082
\(50\) 33.9411i 0.678823i
\(51\) 4.24264i 0.0831890i
\(52\) 33.9411i 0.652714i
\(53\) 16.0000 0.301887 0.150943 − 0.988542i \(-0.451769\pi\)
0.150943 + 0.988542i \(0.451769\pi\)
\(54\) 24.0416i 0.445215i
\(55\) 0 0
\(56\) −60.0000 −1.07143
\(57\) 16.9706i 0.297729i
\(58\) −32.0000 −0.551724
\(59\) −71.0000 −1.20339 −0.601695 − 0.798726i \(-0.705508\pi\)
−0.601695 + 0.798726i \(0.705508\pi\)
\(60\) 14.0000 0.233333
\(61\) 11.3137i 0.185471i 0.995691 + 0.0927353i \(0.0295610\pi\)
−0.995691 + 0.0927353i \(0.970439\pi\)
\(62\) 69.2965i 1.11768i
\(63\) − 56.5685i − 0.897913i
\(64\) −56.0000 −0.875000
\(65\) − 118.794i − 1.82760i
\(66\) 0 0
\(67\) −31.0000 −0.462687 −0.231343 − 0.972872i \(-0.574312\pi\)
−0.231343 + 0.972872i \(0.574312\pi\)
\(68\) − 8.48528i − 0.124784i
\(69\) 9.00000 0.130435
\(70\) 70.0000 1.00000
\(71\) −73.0000 −1.02817 −0.514085 − 0.857740i \(-0.671868\pi\)
−0.514085 + 0.857740i \(0.671868\pi\)
\(72\) − 67.8823i − 0.942809i
\(73\) − 39.5980i − 0.542438i −0.962518 − 0.271219i \(-0.912573\pi\)
0.962518 − 0.271219i \(-0.0874268\pi\)
\(74\) 24.0416i 0.324887i
\(75\) −24.0000 −0.320000
\(76\) − 33.9411i − 0.446594i
\(77\) 0 0
\(78\) 24.0000 0.307692
\(79\) 156.978i 1.98706i 0.113572 + 0.993530i \(0.463771\pi\)
−0.113572 + 0.993530i \(0.536229\pi\)
\(80\) 28.0000 0.350000
\(81\) 55.0000 0.679012
\(82\) 24.0000 0.292683
\(83\) − 35.3553i − 0.425968i −0.977056 − 0.212984i \(-0.931682\pi\)
0.977056 − 0.212984i \(-0.0683182\pi\)
\(84\) − 14.1421i − 0.168359i
\(85\) 29.6985i 0.349394i
\(86\) −66.0000 −0.767442
\(87\) − 22.6274i − 0.260085i
\(88\) 0 0
\(89\) −9.00000 −0.101124 −0.0505618 − 0.998721i \(-0.516101\pi\)
−0.0505618 + 0.998721i \(0.516101\pi\)
\(90\) 79.1960i 0.879955i
\(91\) −120.000 −1.31868
\(92\) −18.0000 −0.195652
\(93\) −49.0000 −0.526882
\(94\) 45.2548i 0.481434i
\(95\) 118.794i 1.25046i
\(96\) − 28.2843i − 0.294628i
\(97\) −17.0000 −0.175258 −0.0876289 − 0.996153i \(-0.527929\pi\)
−0.0876289 + 0.996153i \(0.527929\pi\)
\(98\) − 1.41421i − 0.0144308i
\(99\) 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 121.3.b.a.120.2 yes 2
3.2 odd 2 1089.3.c.a.604.1 2
11.2 odd 10 121.3.d.e.40.1 8
11.3 even 5 121.3.d.e.112.2 8
11.4 even 5 121.3.d.e.94.1 8
11.5 even 5 121.3.d.e.118.1 8
11.6 odd 10 121.3.d.e.118.2 8
11.7 odd 10 121.3.d.e.94.2 8
11.8 odd 10 121.3.d.e.112.1 8
11.9 even 5 121.3.d.e.40.2 8
11.10 odd 2 inner 121.3.b.a.120.1 ✓ 2
33.32 even 2 1089.3.c.a.604.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.3.b.a.120.1 ✓ 2 11.10 odd 2 inner
121.3.b.a.120.2 yes 2 1.1 even 1 trivial
121.3.d.e.40.1 8 11.2 odd 10
121.3.d.e.40.2 8 11.9 even 5
121.3.d.e.94.1 8 11.4 even 5
121.3.d.e.94.2 8 11.7 odd 10
121.3.d.e.112.1 8 11.8 odd 10
121.3.d.e.112.2 8 11.3 even 5
121.3.d.e.118.1 8 11.5 even 5
121.3.d.e.118.2 8 11.6 odd 10
1089.3.c.a.604.1 2 3.2 odd 2
1089.3.c.a.604.2 2 33.32 even 2