Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [121,2,Mod(3,121)] 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("121.3"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(121, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([8])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 121.c (of order \(5\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,1] 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: no
Analytic conductor: \(0.966189864457\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 81.1
Root \(0.809017 - 0.587785i\) of defining polynomial
Character \(\chi\) \(=\) 121.81
Dual form 121.2.c.d.3.1

$q$-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+(0.809017 + 0.587785i) q^{2} +(0.618034 - 1.90211i) q^{3} +(-0.309017 - 0.951057i) q^{4} +(-0.809017 + 0.587785i) q^{5} +(1.61803 - 1.17557i) q^{6} +(0.618034 + 1.90211i) q^{7} +(0.927051 - 2.85317i) q^{8} +(-0.809017 - 0.587785i) q^{9} -1.00000 q^{10} -2.00000 q^{12} +(0.809017 + 0.587785i) q^{13} +(-0.618034 + 1.90211i) q^{14} +(0.618034 + 1.90211i) q^{15} +(0.809017 - 0.587785i) q^{16} +(-4.04508 + 2.93893i) q^{17} +(-0.309017 - 0.951057i) q^{18} +(-1.85410 + 5.70634i) q^{19} +(0.809017 + 0.587785i) q^{20} +4.00000 q^{21} +2.00000 q^{23} +(-4.85410 - 3.52671i) q^{24} +(-1.23607 + 3.80423i) q^{25} +(0.309017 + 0.951057i) q^{26} +(3.23607 - 2.35114i) q^{27} +(1.61803 - 1.17557i) q^{28} +(-2.78115 - 8.55951i) q^{29} +(-0.618034 + 1.90211i) q^{30} +(1.61803 + 1.17557i) q^{31} -5.00000 q^{32} -5.00000 q^{34} +(-1.61803 - 1.17557i) q^{35} +(-0.309017 + 0.951057i) q^{36} +(-0.927051 - 2.85317i) q^{37} +(-4.85410 + 3.52671i) q^{38} +(1.61803 - 1.17557i) q^{39} +(0.927051 + 2.85317i) q^{40} +(1.54508 - 4.75528i) q^{41} +(3.23607 + 2.35114i) q^{42} +1.00000 q^{45} +(1.61803 + 1.17557i) q^{46} +(0.618034 - 1.90211i) q^{47} +(-0.618034 - 1.90211i) q^{48} +(2.42705 - 1.76336i) q^{49} +(-3.23607 + 2.35114i) q^{50} +(3.09017 + 9.51057i) q^{51} +(0.309017 - 0.951057i) q^{52} +(-7.28115 - 5.29007i) q^{53} +4.00000 q^{54} +6.00000 q^{56} +(9.70820 + 7.05342i) q^{57} +(2.78115 - 8.55951i) q^{58} +(2.47214 + 7.60845i) q^{59} +(1.61803 - 1.17557i) q^{60} +(4.85410 - 3.52671i) q^{61} +(0.618034 + 1.90211i) q^{62} +(0.618034 - 1.90211i) q^{63} +(-5.66312 - 4.11450i) q^{64} -1.00000 q^{65} +2.00000 q^{67} +(4.04508 + 2.93893i) q^{68} +(1.23607 - 3.80423i) q^{69} +(-0.618034 - 1.90211i) q^{70} +(-9.70820 + 7.05342i) q^{71} +(-2.42705 + 1.76336i) q^{72} +(0.618034 + 1.90211i) q^{73} +(0.927051 - 2.85317i) q^{74} +(6.47214 + 4.70228i) q^{75} +6.00000 q^{76} +2.00000 q^{78} +(-8.09017 - 5.87785i) q^{79} +(-0.309017 + 0.951057i) q^{80} +(-3.39919 - 10.4616i) q^{81} +(4.04508 - 2.93893i) q^{82} +(4.85410 - 3.52671i) q^{83} +(-1.23607 - 3.80423i) q^{84} +(1.54508 - 4.75528i) q^{85} -18.0000 q^{87} -9.00000 q^{89} +(0.809017 + 0.587785i) q^{90} +(-0.618034 + 1.90211i) q^{91} +(-0.618034 - 1.90211i) q^{92} +(3.23607 - 2.35114i) q^{93} +(1.61803 - 1.17557i) q^{94} +(-1.85410 - 5.70634i) q^{95} +(-3.09017 + 9.51057i) q^{96} +(10.5172 + 7.64121i) q^{97} +3.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - 2 q^{3} + q^{4} - q^{5} + 2 q^{6} - 2 q^{7} - 3 q^{8} - q^{9} - 4 q^{10} - 8 q^{12} + q^{13} + 2 q^{14} - 2 q^{15} + q^{16} - 5 q^{17} + q^{18} + 6 q^{19} + q^{20} + 16 q^{21} + 8 q^{23}+ \cdots + 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.809017 + 0.587785i 0.572061 + 0.415627i 0.835853 − 0.548953i \(-0.184973\pi\)
−0.263792 + 0.964580i \(0.584973\pi\)
\(3\) 0.618034 − 1.90211i 0.356822 − 1.09819i −0.598123 − 0.801404i \(-0.704087\pi\)
0.954945 − 0.296781i \(-0.0959133\pi\)
\(4\) −0.309017 − 0.951057i −0.154508 − 0.475528i
\(5\) −0.809017 + 0.587785i −0.361803 + 0.262866i −0.753804 − 0.657099i \(-0.771783\pi\)
0.392000 + 0.919965i \(0.371783\pi\)
\(6\) 1.61803 − 1.17557i 0.660560 − 0.479925i
\(7\) 0.618034 + 1.90211i 0.233595 + 0.718931i 0.997305 + 0.0733714i \(0.0233759\pi\)
−0.763710 + 0.645560i \(0.776624\pi\)
\(8\) 0.927051 − 2.85317i 0.327762 − 1.00875i
\(9\) −0.809017 − 0.587785i −0.269672 − 0.195928i
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) −2.00000 −0.577350
\(13\) 0.809017 + 0.587785i 0.224381 + 0.163022i 0.694297 − 0.719689i \(-0.255716\pi\)
−0.469916 + 0.882711i \(0.655716\pi\)
\(14\) −0.618034 + 1.90211i −0.165177 + 0.508361i
\(15\) 0.618034 + 1.90211i 0.159576 + 0.491123i
\(16\) 0.809017 − 0.587785i 0.202254 − 0.146946i
\(17\) −4.04508 + 2.93893i −0.981077 + 0.712794i −0.957949 − 0.286938i \(-0.907363\pi\)
−0.0231281 + 0.999733i \(0.507363\pi\)
\(18\) −0.309017 − 0.951057i −0.0728360 − 0.224166i
\(19\) −1.85410 + 5.70634i −0.425360 + 1.30912i 0.477289 + 0.878746i \(0.341620\pi\)
−0.902649 + 0.430377i \(0.858380\pi\)
\(20\) 0.809017 + 0.587785i 0.180902 + 0.131433i
\(21\) 4.00000 0.872872
\(22\) 0 0
\(23\) 2.00000 0.417029 0.208514 − 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) −4.85410 − 3.52671i −0.990839 − 0.719887i
\(25\) −1.23607 + 3.80423i −0.247214 + 0.760845i
\(26\) 0.309017 + 0.951057i 0.0606032 + 0.186518i
\(27\) 3.23607 − 2.35114i 0.622782 − 0.452477i
\(28\) 1.61803 − 1.17557i 0.305780 − 0.222162i
\(29\) −2.78115 − 8.55951i −0.516447 − 1.58946i −0.780633 − 0.624989i \(-0.785103\pi\)
0.264186 − 0.964472i \(-0.414897\pi\)
\(30\) −0.618034 + 1.90211i −0.112837 + 0.347277i
\(31\) 1.61803 + 1.17557i 0.290607 + 0.211139i 0.723531 − 0.690292i \(-0.242518\pi\)
−0.432923 + 0.901431i \(0.642518\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) −5.00000 −0.857493
\(35\) −1.61803 − 1.17557i −0.273498 − 0.198708i
\(36\) −0.309017 + 0.951057i −0.0515028 + 0.158509i
\(37\) −0.927051 − 2.85317i −0.152406 − 0.469058i 0.845483 − 0.534003i \(-0.179313\pi\)
−0.997889 + 0.0649448i \(0.979313\pi\)
\(38\) −4.85410 + 3.52671i −0.787439 + 0.572108i
\(39\) 1.61803 − 1.17557i 0.259093 − 0.188242i
\(40\) 0.927051 + 2.85317i 0.146580 + 0.451126i
\(41\) 1.54508 − 4.75528i 0.241302 − 0.742650i −0.754921 − 0.655816i \(-0.772325\pi\)
0.996223 − 0.0868346i \(-0.0276752\pi\)
\(42\) 3.23607 + 2.35114i 0.499336 + 0.362789i
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 1.61803 + 1.17557i 0.238566 + 0.173328i
\(47\) 0.618034 − 1.90211i 0.0901495 − 0.277452i −0.895810 − 0.444438i \(-0.853403\pi\)
0.985959 + 0.166986i \(0.0534035\pi\)
\(48\) −0.618034 − 1.90211i −0.0892055 − 0.274546i
\(49\) 2.42705 − 1.76336i 0.346722 − 0.251908i
\(50\) −3.23607 + 2.35114i −0.457649 + 0.332502i
\(51\) 3.09017 + 9.51057i 0.432710 + 1.33175i
\(52\) 0.309017 − 0.951057i 0.0428529 − 0.131888i
\(53\) −7.28115 − 5.29007i −1.00014 − 0.726647i −0.0380244 − 0.999277i \(-0.512106\pi\)
−0.962119 + 0.272630i \(0.912106\pi\)
\(54\) 4.00000 0.544331
\(55\) 0 0
\(56\) 6.00000 0.801784
\(57\) 9.70820 + 7.05342i 1.28588 + 0.934249i
\(58\) 2.78115 − 8.55951i 0.365183 − 1.12392i
\(59\) 2.47214 + 7.60845i 0.321845 + 0.990536i 0.972845 + 0.231458i \(0.0743497\pi\)
−0.651000 + 0.759078i \(0.725650\pi\)
\(60\) 1.61803 − 1.17557i 0.208887 − 0.151765i
\(61\) 4.85410 − 3.52671i 0.621504 − 0.451549i −0.231942 − 0.972730i \(-0.574508\pi\)
0.853447 + 0.521180i \(0.174508\pi\)
\(62\) 0.618034 + 1.90211i 0.0784904 + 0.241569i
\(63\) 0.618034 − 1.90211i 0.0778650 − 0.239644i
\(64\) −5.66312 − 4.11450i −0.707890 − 0.514312i
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 − 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 4.04508 + 2.93893i 0.490539 + 0.356397i
\(69\) 1.23607 − 3.80423i 0.148805 − 0.457975i
\(70\) −0.618034 − 1.90211i −0.0738692 − 0.227346i
\(71\) −9.70820 + 7.05342i −1.15215 + 0.837087i −0.988766 − 0.149475i \(-0.952242\pi\)
−0.163386 + 0.986562i \(0.552242\pi\)
\(72\) −2.42705 + 1.76336i −0.286031 + 0.207813i
\(73\) 0.618034 + 1.90211i 0.0723354 + 0.222625i 0.980688 − 0.195580i \(-0.0626591\pi\)
−0.908352 + 0.418206i \(0.862659\pi\)
\(74\) 0.927051 − 2.85317i 0.107767 − 0.331674i
\(75\) 6.47214 + 4.70228i 0.747338 + 0.542973i
\(76\) 6.00000 0.688247
\(77\) 0 0
\(78\) 2.00000 0.226455
\(79\) −8.09017 − 5.87785i −0.910215 − 0.661310i 0.0308541 − 0.999524i \(-0.490177\pi\)
−0.941069 + 0.338214i \(0.890177\pi\)
\(80\) −0.309017 + 0.951057i −0.0345492 + 0.106331i
\(81\) −3.39919 − 10.4616i −0.377687 − 1.16240i
\(82\) 4.04508 − 2.93893i 0.446705 − 0.324550i
\(83\) 4.85410 − 3.52671i 0.532807 − 0.387107i −0.288600 − 0.957450i \(-0.593190\pi\)
0.821407 + 0.570343i \(0.193190\pi\)
\(84\) −1.23607 − 3.80423i −0.134866 − 0.415075i
\(85\) 1.54508 − 4.75528i 0.167588 − 0.515783i
\(86\) 0 0
\(87\) −18.0000 −1.92980
\(88\) 0 0
\(89\) −9.00000 −0.953998 −0.476999 − 0.878904i \(-0.658275\pi\)
−0.476999 + 0.878904i \(0.658275\pi\)
\(90\) 0.809017 + 0.587785i 0.0852779 + 0.0619580i
\(91\) −0.618034 + 1.90211i −0.0647876 + 0.199396i
\(92\) −0.618034 − 1.90211i −0.0644345 − 0.198309i
\(93\) 3.23607 − 2.35114i 0.335565 − 0.243802i
\(94\) 1.61803 − 1.17557i 0.166887 − 0.121251i
\(95\) −1.85410 − 5.70634i −0.190227 − 0.585458i
\(96\) −3.09017 + 9.51057i −0.315389 + 0.970668i
\(97\) 10.5172 + 7.64121i 1.06786 + 0.775847i 0.975527 − 0.219881i \(-0.0705669\pi\)
0.0923353 + 0.995728i \(0.470567\pi\)
\(98\) 3.00000 0.303046
\(99\) 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 121.2.c.d.81.1 4
11.2 odd 10 121.2.c.b.27.1 4
11.3 even 5 inner 121.2.c.d.3.1 4
11.4 even 5 inner 121.2.c.d.9.1 4
11.5 even 5 121.2.a.a.1.1 ✓ 1
11.6 odd 10 121.2.a.c.1.1 yes 1
11.7 odd 10 121.2.c.b.9.1 4
11.8 odd 10 121.2.c.b.3.1 4
11.9 even 5 inner 121.2.c.d.27.1 4
11.10 odd 2 121.2.c.b.81.1 4
33.5 odd 10 1089.2.a.i.1.1 1
33.17 even 10 1089.2.a.c.1.1 1
44.27 odd 10 1936.2.a.a.1.1 1
44.39 even 10 1936.2.a.b.1.1 1
55.39 odd 10 3025.2.a.b.1.1 1
55.49 even 10 3025.2.a.e.1.1 1
77.6 even 10 5929.2.a.g.1.1 1
77.27 odd 10 5929.2.a.a.1.1 1
88.5 even 10 7744.2.a.f.1.1 1
88.27 odd 10 7744.2.a.be.1.1 1
88.61 odd 10 7744.2.a.c.1.1 1
88.83 even 10 7744.2.a.bf.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.2.a.a.1.1 ✓ 1 11.5 even 5
121.2.a.c.1.1 yes 1 11.6 odd 10
121.2.c.b.3.1 4 11.8 odd 10
121.2.c.b.9.1 4 11.7 odd 10
121.2.c.b.27.1 4 11.2 odd 10
121.2.c.b.81.1 4 11.10 odd 2
121.2.c.d.3.1 4 11.3 even 5 inner
121.2.c.d.9.1 4 11.4 even 5 inner
121.2.c.d.27.1 4 11.9 even 5 inner
121.2.c.d.81.1 4 1.1 even 1 trivial
1089.2.a.c.1.1 1 33.17 even 10
1089.2.a.i.1.1 1 33.5 odd 10
1936.2.a.a.1.1 1 44.27 odd 10
1936.2.a.b.1.1 1 44.39 even 10
3025.2.a.b.1.1 1 55.39 odd 10
3025.2.a.e.1.1 1 55.49 even 10
5929.2.a.a.1.1 1 77.27 odd 10
5929.2.a.g.1.1 1 77.6 even 10
7744.2.a.c.1.1 1 88.61 odd 10
7744.2.a.f.1.1 1 88.5 even 10
7744.2.a.be.1.1 1 88.27 odd 10
7744.2.a.bf.1.1 1 88.83 even 10