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 3.1
Root \(0.809017 + 0.587785i\) of defining polynomial
Character \(\chi\) \(=\) 121.3
Dual form 121.2.c.b.81.1

$q$-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 \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{4}{5}\right)\)

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 \(-0.815027\pi\)
0.263792 + 0.964580i \(0.415027\pi\)
\(3\) 0.618034 + 1.90211i 0.356822 + 1.09819i 0.954945 + 0.296781i \(0.0959133\pi\)
−0.598123 + 0.801404i \(0.704087\pi\)
\(4\) −0.309017 + 0.951057i −0.154508 + 0.475528i
\(5\) −0.809017 − 0.587785i −0.361803 − 0.262866i 0.392000 − 0.919965i \(-0.371783\pi\)
−0.753804 + 0.657099i \(0.771783\pi\)
\(6\) −1.61803 − 1.17557i −0.660560 − 0.479925i
\(7\) −0.618034 + 1.90211i −0.233595 + 0.718931i 0.763710 + 0.645560i \(0.223376\pi\)
−0.997305 + 0.0733714i \(0.976624\pi\)
\(8\) −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 \(-0.744284\pi\)
0.469916 + 0.882711i \(0.344284\pi\)
\(14\) −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 \(-0.0926374\pi\)
0.0231281 + 0.999733i \(0.492637\pi\)
\(18\) 0.309017 − 0.951057i 0.0728360 − 0.224166i
\(19\) 1.85410 + 5.70634i 0.425360 + 1.30912i 0.902649 + 0.430377i \(0.141620\pi\)
−0.477289 + 0.878746i \(0.658380\pi\)
\(20\) 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 −0.264186 − 0.964472i \(-0.585103\pi\)
0.780633 − 0.624989i \(-0.214897\pi\)
\(30\) 0.618034 + 1.90211i 0.112837 + 0.347277i
\(31\) 1.61803 − 1.17557i 0.290607 − 0.211139i −0.432923 − 0.901431i \(-0.642518\pi\)
0.723531 + 0.690292i \(0.242518\pi\)
\(32\) 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 −0.997889 − 0.0649448i \(-0.979313\pi\)
0.845483 + 0.534003i \(0.179313\pi\)
\(38\) −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 −0.996223 − 0.0868346i \(-0.972325\pi\)
0.754921 − 0.655816i \(-0.227675\pi\)
\(42\) 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 0.985959 − 0.166986i \(-0.0534035\pi\)
−0.895810 + 0.444438i \(0.853403\pi\)
\(48\) −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 −0.962119 − 0.272630i \(-0.912106\pi\)
−0.0380244 + 0.999277i \(0.512106\pi\)
\(54\) −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 −0.651000 − 0.759078i \(-0.725650\pi\)
0.972845 − 0.231458i \(-0.0743497\pi\)
\(60\) 1.61803 + 1.17557i 0.208887 + 0.151765i
\(61\) −4.85410 − 3.52671i −0.621504 − 0.451549i 0.231942 − 0.972730i \(-0.425492\pi\)
−0.853447 + 0.521180i \(0.825492\pi\)
\(62\) −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 −0.163386 − 0.986562i \(-0.552242\pi\)
−0.988766 + 0.149475i \(0.952242\pi\)
\(72\) 2.42705 + 1.76336i 0.286031 + 0.207813i
\(73\) −0.618034 + 1.90211i −0.0723354 + 0.222625i −0.980688 − 0.195580i \(-0.937341\pi\)
0.908352 + 0.418206i \(0.137341\pi\)
\(74\) −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 \(-0.509823\pi\)
0.941069 + 0.338214i \(0.109823\pi\)
\(80\) −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 \(-0.406810\pi\)
−0.821407 + 0.570343i \(0.806810\pi\)
\(84\) 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 0.0923353 − 0.995728i \(-0.470567\pi\)
0.975527 + 0.219881i \(0.0705669\pi\)
\(98\) −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.b.3.1 4
11.2 odd 10 121.2.a.a.1.1 ✓ 1
11.3 even 5 inner 121.2.c.b.27.1 4
11.4 even 5 inner 121.2.c.b.81.1 4
11.5 even 5 inner 121.2.c.b.9.1 4
11.6 odd 10 121.2.c.d.9.1 4
11.7 odd 10 121.2.c.d.81.1 4
11.8 odd 10 121.2.c.d.27.1 4
11.9 even 5 121.2.a.c.1.1 yes 1
11.10 odd 2 121.2.c.d.3.1 4
33.2 even 10 1089.2.a.i.1.1 1
33.20 odd 10 1089.2.a.c.1.1 1
44.31 odd 10 1936.2.a.b.1.1 1
44.35 even 10 1936.2.a.a.1.1 1
55.9 even 10 3025.2.a.b.1.1 1
55.24 odd 10 3025.2.a.e.1.1 1
77.13 even 10 5929.2.a.a.1.1 1
77.20 odd 10 5929.2.a.g.1.1 1
88.13 odd 10 7744.2.a.f.1.1 1
88.35 even 10 7744.2.a.be.1.1 1
88.53 even 10 7744.2.a.c.1.1 1
88.75 odd 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 11.2 odd 10
121.2.a.c.1.1 yes 1 11.9 even 5
121.2.c.b.3.1 4 1.1 even 1 trivial
121.2.c.b.9.1 4 11.5 even 5 inner
121.2.c.b.27.1 4 11.3 even 5 inner
121.2.c.b.81.1 4 11.4 even 5 inner
121.2.c.d.3.1 4 11.10 odd 2
121.2.c.d.9.1 4 11.6 odd 10
121.2.c.d.27.1 4 11.8 odd 10
121.2.c.d.81.1 4 11.7 odd 10
1089.2.a.c.1.1 1 33.20 odd 10
1089.2.a.i.1.1 1 33.2 even 10
1936.2.a.a.1.1 1 44.35 even 10
1936.2.a.b.1.1 1 44.31 odd 10
3025.2.a.b.1.1 1 55.9 even 10
3025.2.a.e.1.1 1 55.24 odd 10
5929.2.a.a.1.1 1 77.13 even 10
5929.2.a.g.1.1 1 77.20 odd 10
7744.2.a.c.1.1 1 88.53 even 10
7744.2.a.f.1.1 1 88.13 odd 10
7744.2.a.be.1.1 1 88.35 even 10
7744.2.a.bf.1.1 1 88.75 odd 10