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 27.1
Root \(-0.309017 + 0.951057i\) of defining polynomial
Character \(\chi\) \(=\) 121.27
Dual form 121.2.c.d.9.1

$q$-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+(-0.309017 - 0.951057i) q^{2} +(-1.61803 - 1.17557i) q^{3} +(0.809017 - 0.587785i) q^{4} +(0.309017 - 0.951057i) q^{5} +(-0.618034 + 1.90211i) q^{6} +(-1.61803 + 1.17557i) q^{7} +(-2.42705 - 1.76336i) q^{8} +(0.309017 + 0.951057i) q^{9} -1.00000 q^{10} -2.00000 q^{12} +(-0.309017 - 0.951057i) q^{13} +(1.61803 + 1.17557i) q^{14} +(-1.61803 + 1.17557i) q^{15} +(-0.309017 + 0.951057i) q^{16} +(1.54508 - 4.75528i) q^{17} +(0.809017 - 0.587785i) q^{18} +(4.85410 + 3.52671i) q^{19} +(-0.309017 - 0.951057i) q^{20} +4.00000 q^{21} +2.00000 q^{23} +(1.85410 + 5.70634i) q^{24} +(3.23607 + 2.35114i) q^{25} +(-0.809017 + 0.587785i) q^{26} +(-1.23607 + 3.80423i) q^{27} +(-0.618034 + 1.90211i) q^{28} +(7.28115 - 5.29007i) q^{29} +(1.61803 + 1.17557i) q^{30} +(-0.618034 - 1.90211i) q^{31} -5.00000 q^{32} -5.00000 q^{34} +(0.618034 + 1.90211i) q^{35} +(0.809017 + 0.587785i) q^{36} +(2.42705 - 1.76336i) q^{37} +(1.85410 - 5.70634i) q^{38} +(-0.618034 + 1.90211i) q^{39} +(-2.42705 + 1.76336i) q^{40} +(-4.04508 - 2.93893i) q^{41} +(-1.23607 - 3.80423i) q^{42} +1.00000 q^{45} +(-0.618034 - 1.90211i) q^{46} +(-1.61803 - 1.17557i) q^{47} +(1.61803 - 1.17557i) q^{48} +(-0.927051 + 2.85317i) q^{49} +(1.23607 - 3.80423i) q^{50} +(-8.09017 + 5.87785i) q^{51} +(-0.809017 - 0.587785i) q^{52} +(2.78115 + 8.55951i) q^{53} +4.00000 q^{54} +6.00000 q^{56} +(-3.70820 - 11.4127i) q^{57} +(-7.28115 - 5.29007i) q^{58} +(-6.47214 + 4.70228i) q^{59} +(-0.618034 + 1.90211i) q^{60} +(-1.85410 + 5.70634i) q^{61} +(-1.61803 + 1.17557i) q^{62} +(-1.61803 - 1.17557i) q^{63} +(2.16312 + 6.65740i) q^{64} -1.00000 q^{65} +2.00000 q^{67} +(-1.54508 - 4.75528i) q^{68} +(-3.23607 - 2.35114i) q^{69} +(1.61803 - 1.17557i) q^{70} +(3.70820 - 11.4127i) q^{71} +(0.927051 - 2.85317i) q^{72} +(-1.61803 + 1.17557i) q^{73} +(-2.42705 - 1.76336i) q^{74} +(-2.47214 - 7.60845i) q^{75} +6.00000 q^{76} +2.00000 q^{78} +(3.09017 + 9.51057i) q^{79} +(0.809017 + 0.587785i) q^{80} +(8.89919 - 6.46564i) q^{81} +(-1.54508 + 4.75528i) q^{82} +(-1.85410 + 5.70634i) q^{83} +(3.23607 - 2.35114i) q^{84} +(-4.04508 - 2.93893i) q^{85} -18.0000 q^{87} -9.00000 q^{89} +(-0.309017 - 0.951057i) q^{90} +(1.61803 + 1.17557i) q^{91} +(1.61803 - 1.17557i) q^{92} +(-1.23607 + 3.80423i) q^{93} +(-0.618034 + 1.90211i) q^{94} +(4.85410 - 3.52671i) q^{95} +(8.09017 + 5.87785i) q^{96} +(-4.01722 - 12.3637i) 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 \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.309017 − 0.951057i −0.218508 − 0.672499i −0.998886 − 0.0471903i \(-0.984973\pi\)
0.780378 − 0.625308i \(-0.215027\pi\)
\(3\) −1.61803 − 1.17557i −0.934172 − 0.678716i 0.0128385 − 0.999918i \(-0.495913\pi\)
−0.947011 + 0.321202i \(0.895913\pi\)
\(4\) 0.809017 − 0.587785i 0.404508 − 0.293893i
\(5\) 0.309017 − 0.951057i 0.138197 − 0.425325i −0.857877 − 0.513855i \(-0.828217\pi\)
0.996074 + 0.0885298i \(0.0282169\pi\)
\(6\) −0.618034 + 1.90211i −0.252311 + 0.776534i
\(7\) −1.61803 + 1.17557i −0.611559 + 0.444324i −0.849963 − 0.526842i \(-0.823376\pi\)
0.238404 + 0.971166i \(0.423376\pi\)
\(8\) −2.42705 − 1.76336i −0.858092 − 0.623440i
\(9\) 0.309017 + 0.951057i 0.103006 + 0.317019i
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) −2.00000 −0.577350
\(13\) −0.309017 − 0.951057i −0.0857059 − 0.263776i 0.899014 − 0.437919i \(-0.144284\pi\)
−0.984720 + 0.174143i \(0.944284\pi\)
\(14\) 1.61803 + 1.17557i 0.432438 + 0.314184i
\(15\) −1.61803 + 1.17557i −0.417775 + 0.303531i
\(16\) −0.309017 + 0.951057i −0.0772542 + 0.237764i
\(17\) 1.54508 − 4.75528i 0.374738 − 1.15333i −0.568917 − 0.822395i \(-0.692637\pi\)
0.943655 − 0.330930i \(-0.107363\pi\)
\(18\) 0.809017 − 0.587785i 0.190687 − 0.138542i
\(19\) 4.85410 + 3.52671i 1.11361 + 0.809083i 0.983228 − 0.182381i \(-0.0583804\pi\)
0.130379 + 0.991464i \(0.458380\pi\)
\(20\) −0.309017 − 0.951057i −0.0690983 − 0.212663i
\(21\) 4.00000 0.872872
\(22\) 0 0
\(23\) 2.00000 0.417029 0.208514 − 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 1.85410 + 5.70634i 0.378467 + 1.16480i
\(25\) 3.23607 + 2.35114i 0.647214 + 0.470228i
\(26\) −0.809017 + 0.587785i −0.158661 + 0.115274i
\(27\) −1.23607 + 3.80423i −0.237881 + 0.732124i
\(28\) −0.618034 + 1.90211i −0.116797 + 0.359466i
\(29\) 7.28115 − 5.29007i 1.35208 − 0.982341i 0.353171 − 0.935559i \(-0.385103\pi\)
0.998905 − 0.0467821i \(-0.0148966\pi\)
\(30\) 1.61803 + 1.17557i 0.295411 + 0.214629i
\(31\) −0.618034 − 1.90211i −0.111002 − 0.341630i 0.880090 − 0.474807i \(-0.157482\pi\)
−0.991092 + 0.133177i \(0.957482\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) −5.00000 −0.857493
\(35\) 0.618034 + 1.90211i 0.104467 + 0.321516i
\(36\) 0.809017 + 0.587785i 0.134836 + 0.0979642i
\(37\) 2.42705 − 1.76336i 0.399005 − 0.289894i −0.370131 − 0.928980i \(-0.620687\pi\)
0.769135 + 0.639086i \(0.220687\pi\)
\(38\) 1.85410 − 5.70634i 0.300775 − 0.925690i
\(39\) −0.618034 + 1.90211i −0.0989646 + 0.304582i
\(40\) −2.42705 + 1.76336i −0.383750 + 0.278811i
\(41\) −4.04508 − 2.93893i −0.631736 − 0.458983i 0.225265 − 0.974298i \(-0.427675\pi\)
−0.857001 + 0.515314i \(0.827675\pi\)
\(42\) −1.23607 − 3.80423i −0.190729 − 0.587005i
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) −0.618034 − 1.90211i −0.0911241 − 0.280451i
\(47\) −1.61803 − 1.17557i −0.236015 − 0.171475i 0.463491 − 0.886101i \(-0.346597\pi\)
−0.699506 + 0.714627i \(0.746597\pi\)
\(48\) 1.61803 − 1.17557i 0.233543 − 0.169679i
\(49\) −0.927051 + 2.85317i −0.132436 + 0.407596i
\(50\) 1.23607 − 3.80423i 0.174806 − 0.537999i
\(51\) −8.09017 + 5.87785i −1.13285 + 0.823064i
\(52\) −0.809017 − 0.587785i −0.112190 − 0.0815111i
\(53\) 2.78115 + 8.55951i 0.382021 + 1.17574i 0.938619 + 0.344957i \(0.112106\pi\)
−0.556598 + 0.830782i \(0.687894\pi\)
\(54\) 4.00000 0.544331
\(55\) 0 0
\(56\) 6.00000 0.801784
\(57\) −3.70820 − 11.4127i −0.491164 − 1.51165i
\(58\) −7.28115 − 5.29007i −0.956062 − 0.694620i
\(59\) −6.47214 + 4.70228i −0.842600 + 0.612185i −0.923096 − 0.384570i \(-0.874350\pi\)
0.0804955 + 0.996755i \(0.474350\pi\)
\(60\) −0.618034 + 1.90211i −0.0797878 + 0.245562i
\(61\) −1.85410 + 5.70634i −0.237393 + 0.730622i 0.759401 + 0.650622i \(0.225492\pi\)
−0.996795 + 0.0799995i \(0.974508\pi\)
\(62\) −1.61803 + 1.17557i −0.205491 + 0.149298i
\(63\) −1.61803 − 1.17557i −0.203853 − 0.148108i
\(64\) 2.16312 + 6.65740i 0.270390 + 0.832174i
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 − 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) −1.54508 − 4.75528i −0.187369 − 0.576663i
\(69\) −3.23607 − 2.35114i −0.389577 − 0.283044i
\(70\) 1.61803 − 1.17557i 0.193392 − 0.140508i
\(71\) 3.70820 − 11.4127i 0.440083 − 1.35444i −0.447704 − 0.894182i \(-0.647758\pi\)
0.887787 − 0.460254i \(-0.152242\pi\)
\(72\) 0.927051 − 2.85317i 0.109254 − 0.336249i
\(73\) −1.61803 + 1.17557i −0.189377 + 0.137590i −0.678434 − 0.734662i \(-0.737341\pi\)
0.489057 + 0.872252i \(0.337341\pi\)
\(74\) −2.42705 − 1.76336i −0.282139 − 0.204986i
\(75\) −2.47214 − 7.60845i −0.285458 − 0.878548i
\(76\) 6.00000 0.688247
\(77\) 0 0
\(78\) 2.00000 0.226455
\(79\) 3.09017 + 9.51057i 0.347671 + 1.07002i 0.960138 + 0.279526i \(0.0901773\pi\)
−0.612467 + 0.790496i \(0.709823\pi\)
\(80\) 0.809017 + 0.587785i 0.0904508 + 0.0657164i
\(81\) 8.89919 − 6.46564i 0.988799 − 0.718404i
\(82\) −1.54508 + 4.75528i −0.170626 + 0.525133i
\(83\) −1.85410 + 5.70634i −0.203514 + 0.626352i 0.796257 + 0.604959i \(0.206810\pi\)
−0.999771 + 0.0213936i \(0.993190\pi\)
\(84\) 3.23607 − 2.35114i 0.353084 − 0.256531i
\(85\) −4.04508 − 2.93893i −0.438751 − 0.318771i
\(86\) 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 \(0.658275\pi\)
\(90\) −0.309017 − 0.951057i −0.0325733 − 0.100250i
\(91\) 1.61803 + 1.17557i 0.169616 + 0.123233i
\(92\) 1.61803 − 1.17557i 0.168692 − 0.122562i
\(93\) −1.23607 + 3.80423i −0.128174 + 0.394480i
\(94\) −0.618034 + 1.90211i −0.0637453 + 0.196188i
\(95\) 4.85410 − 3.52671i 0.498020 − 0.361833i
\(96\) 8.09017 + 5.87785i 0.825700 + 0.599906i
\(97\) −4.01722 − 12.3637i −0.407887 − 1.25535i −0.918460 − 0.395513i \(-0.870567\pi\)
0.510573 − 0.859834i \(-0.329433\pi\)
\(98\) 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.27.1 4
11.2 odd 10 121.2.c.b.9.1 4
11.3 even 5 121.2.a.a.1.1 ✓ 1
11.4 even 5 inner 121.2.c.d.3.1 4
11.5 even 5 inner 121.2.c.d.81.1 4
11.6 odd 10 121.2.c.b.81.1 4
11.7 odd 10 121.2.c.b.3.1 4
11.8 odd 10 121.2.a.c.1.1 yes 1
11.9 even 5 inner 121.2.c.d.9.1 4
11.10 odd 2 121.2.c.b.27.1 4
33.8 even 10 1089.2.a.c.1.1 1
33.14 odd 10 1089.2.a.i.1.1 1
44.3 odd 10 1936.2.a.a.1.1 1
44.19 even 10 1936.2.a.b.1.1 1
55.14 even 10 3025.2.a.e.1.1 1
55.19 odd 10 3025.2.a.b.1.1 1
77.41 even 10 5929.2.a.g.1.1 1
77.69 odd 10 5929.2.a.a.1.1 1
88.3 odd 10 7744.2.a.be.1.1 1
88.19 even 10 7744.2.a.bf.1.1 1
88.69 even 10 7744.2.a.f.1.1 1
88.85 odd 10 7744.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.2.a.a.1.1 ✓ 1 11.3 even 5
121.2.a.c.1.1 yes 1 11.8 odd 10
121.2.c.b.3.1 4 11.7 odd 10
121.2.c.b.9.1 4 11.2 odd 10
121.2.c.b.27.1 4 11.10 odd 2
121.2.c.b.81.1 4 11.6 odd 10
121.2.c.d.3.1 4 11.4 even 5 inner
121.2.c.d.9.1 4 11.9 even 5 inner
121.2.c.d.27.1 4 1.1 even 1 trivial
121.2.c.d.81.1 4 11.5 even 5 inner
1089.2.a.c.1.1 1 33.8 even 10
1089.2.a.i.1.1 1 33.14 odd 10
1936.2.a.a.1.1 1 44.3 odd 10
1936.2.a.b.1.1 1 44.19 even 10
3025.2.a.b.1.1 1 55.19 odd 10
3025.2.a.e.1.1 1 55.14 even 10
5929.2.a.a.1.1 1 77.69 odd 10
5929.2.a.g.1.1 1 77.41 even 10
7744.2.a.c.1.1 1 88.85 odd 10
7744.2.a.f.1.1 1 88.69 even 10
7744.2.a.be.1.1 1 88.3 odd 10
7744.2.a.bf.1.1 1 88.19 even 10