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Properties

Label 22.2.c.a.15.1
Level $22$
Weight $2$
Character 22.15
Analytic conductor $0.176$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [22,2,Mod(3,22)] 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("22.3"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(22, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([8])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 22 = 2 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 22.c (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.175670884447\)
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 15.1
Root \(0.809017 + 0.587785i\) of defining polynomial
Character \(\chi\) \(=\) 22.15
Dual form 22.2.c.a.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} +(0.118034 - 0.363271i) q^{3} +(0.309017 + 0.951057i) q^{4} +(-2.61803 + 1.90211i) q^{5} +(-0.309017 + 0.224514i) q^{6} +(-0.618034 - 1.90211i) q^{7} +(0.309017 - 0.951057i) q^{8} +(2.30902 + 1.67760i) q^{9} +3.23607 q^{10} +(0.309017 - 3.30220i) q^{11} +0.381966 q^{12} +(-1.00000 - 0.726543i) q^{13} +(-0.618034 + 1.90211i) q^{14} +(0.381966 + 1.17557i) q^{15} +(-0.809017 + 0.587785i) q^{16} +(0.500000 - 0.363271i) q^{17} +(-0.881966 - 2.71441i) q^{18} +(-1.80902 + 5.56758i) q^{19} +(-2.61803 - 1.90211i) q^{20} -0.763932 q^{21} +(-2.19098 + 2.48990i) q^{22} +1.23607 q^{23} +(-0.309017 - 0.224514i) q^{24} +(1.69098 - 5.20431i) q^{25} +(0.381966 + 1.17557i) q^{26} +(1.80902 - 1.31433i) q^{27} +(1.61803 - 1.17557i) q^{28} +(1.38197 + 4.25325i) q^{29} +(0.381966 - 1.17557i) q^{30} +(-1.61803 - 1.17557i) q^{31} +1.00000 q^{32} +(-1.16312 - 0.502029i) q^{33} -0.618034 q^{34} +(5.23607 + 3.80423i) q^{35} +(-0.881966 + 2.71441i) q^{36} +(-1.14590 - 3.52671i) q^{37} +(4.73607 - 3.44095i) q^{38} +(-0.381966 + 0.277515i) q^{39} +(1.00000 + 3.07768i) q^{40} +(1.73607 - 5.34307i) q^{41} +(0.618034 + 0.449028i) q^{42} -8.56231 q^{43} +(3.23607 - 0.726543i) q^{44} -9.23607 q^{45} +(-1.00000 - 0.726543i) q^{46} +(-2.00000 + 6.15537i) q^{47} +(0.118034 + 0.363271i) q^{48} +(2.42705 - 1.76336i) q^{49} +(-4.42705 + 3.21644i) q^{50} +(-0.0729490 - 0.224514i) q^{51} +(0.381966 - 1.17557i) q^{52} +(1.23607 + 0.898056i) q^{53} -2.23607 q^{54} +(5.47214 + 9.23305i) q^{55} -2.00000 q^{56} +(1.80902 + 1.31433i) q^{57} +(1.38197 - 4.25325i) q^{58} +(-2.66312 - 8.19624i) q^{59} +(-1.00000 + 0.726543i) q^{60} +(2.00000 - 1.45309i) q^{61} +(0.618034 + 1.90211i) q^{62} +(1.76393 - 5.42882i) q^{63} +(-0.809017 - 0.587785i) q^{64} +4.00000 q^{65} +(0.645898 + 1.08981i) q^{66} +11.0902 q^{67} +(0.500000 + 0.363271i) q^{68} +(0.145898 - 0.449028i) q^{69} +(-2.00000 - 6.15537i) q^{70} +(4.23607 - 3.07768i) q^{71} +(2.30902 - 1.67760i) q^{72} +(3.20820 + 9.87384i) q^{73} +(-1.14590 + 3.52671i) q^{74} +(-1.69098 - 1.22857i) q^{75} -5.85410 q^{76} +(-6.47214 + 1.45309i) q^{77} +0.472136 q^{78} +(-10.8541 - 7.88597i) q^{79} +(1.00000 - 3.07768i) q^{80} +(2.38197 + 7.33094i) q^{81} +(-4.54508 + 3.30220i) q^{82} +(-7.54508 + 5.48183i) q^{83} +(-0.236068 - 0.726543i) q^{84} +(-0.618034 + 1.90211i) q^{85} +(6.92705 + 5.03280i) q^{86} +1.70820 q^{87} +(-3.04508 - 1.31433i) q^{88} -8.09017 q^{89} +(7.47214 + 5.42882i) q^{90} +(-0.763932 + 2.35114i) q^{91} +(0.381966 + 1.17557i) q^{92} +(-0.618034 + 0.449028i) q^{93} +(5.23607 - 3.80423i) q^{94} +(-5.85410 - 18.0171i) q^{95} +(0.118034 - 0.363271i) q^{96} +(-5.78115 - 4.20025i) q^{97} -3.00000 q^{98} +(6.25329 - 7.10642i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} - 4 q^{3} - q^{4} - 6 q^{5} + q^{6} + 2 q^{7} - q^{8} + 7 q^{9} + 4 q^{10} - q^{11} + 6 q^{12} - 4 q^{13} + 2 q^{14} + 6 q^{15} - q^{16} + 2 q^{17} - 8 q^{18} - 5 q^{19} - 6 q^{20} - 12 q^{21}+ \cdots - 13 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/22\mathbb{Z}\right)^\times\).

\(n\) \(13\)
\(\chi(n)\) \(e\left(\frac{1}{5}\right)\)

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
\(3\) 0.118034 − 0.363271i 0.0681470 − 0.209735i −0.911184 − 0.412000i \(-0.864830\pi\)
0.979331 + 0.202265i \(0.0648303\pi\)
\(4\) 0.309017 + 0.951057i 0.154508 + 0.475528i
\(5\) −2.61803 + 1.90211i −1.17082 + 0.850651i −0.991107 − 0.133068i \(-0.957517\pi\)
−0.179714 + 0.983719i \(0.557517\pi\)
\(6\) −0.309017 + 0.224514i −0.126156 + 0.0916575i
\(7\) −0.618034 − 1.90211i −0.233595 − 0.718931i −0.997305 − 0.0733714i \(-0.976624\pi\)
0.763710 − 0.645560i \(-0.223376\pi\)
\(8\) 0.309017 − 0.951057i 0.109254 − 0.336249i
\(9\) 2.30902 + 1.67760i 0.769672 + 0.559200i
\(10\) 3.23607 1.02333
\(11\) 0.309017 − 3.30220i 0.0931721 − 0.995650i
\(12\) 0.381966 0.110264
\(13\) −1.00000 − 0.726543i −0.277350 − 0.201507i 0.440411 − 0.897796i \(-0.354833\pi\)
−0.717761 + 0.696290i \(0.754833\pi\)
\(14\) −0.618034 + 1.90211i −0.165177 + 0.508361i
\(15\) 0.381966 + 1.17557i 0.0986232 + 0.303531i
\(16\) −0.809017 + 0.587785i −0.202254 + 0.146946i
\(17\) 0.500000 − 0.363271i 0.121268 − 0.0881062i −0.525498 − 0.850795i \(-0.676121\pi\)
0.646766 + 0.762688i \(0.276121\pi\)
\(18\) −0.881966 − 2.71441i −0.207881 − 0.639793i
\(19\) −1.80902 + 5.56758i −0.415017 + 1.27729i 0.497219 + 0.867625i \(0.334355\pi\)
−0.912236 + 0.409666i \(0.865645\pi\)
\(20\) −2.61803 − 1.90211i −0.585410 − 0.425325i
\(21\) −0.763932 −0.166704
\(22\) −2.19098 + 2.48990i −0.467119 + 0.530848i
\(23\) 1.23607 0.257738 0.128869 − 0.991662i \(-0.458865\pi\)
0.128869 + 0.991662i \(0.458865\pi\)
\(24\) −0.309017 − 0.224514i −0.0630778 − 0.0458287i
\(25\) 1.69098 − 5.20431i 0.338197 − 1.04086i
\(26\) 0.381966 + 1.17557i 0.0749097 + 0.230548i
\(27\) 1.80902 − 1.31433i 0.348145 − 0.252942i
\(28\) 1.61803 − 1.17557i 0.305780 − 0.222162i
\(29\) 1.38197 + 4.25325i 0.256625 + 0.789809i 0.993505 + 0.113787i \(0.0362980\pi\)
−0.736881 + 0.676023i \(0.763702\pi\)
\(30\) 0.381966 − 1.17557i 0.0697371 − 0.214629i
\(31\) −1.61803 − 1.17557i −0.290607 − 0.211139i 0.432923 − 0.901431i \(-0.357482\pi\)
−0.723531 + 0.690292i \(0.757482\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.16312 − 0.502029i −0.202473 − 0.0873920i
\(34\) −0.618034 −0.105992
\(35\) 5.23607 + 3.80423i 0.885057 + 0.643032i
\(36\) −0.881966 + 2.71441i −0.146994 + 0.452402i
\(37\) −1.14590 − 3.52671i −0.188384 − 0.579788i 0.811606 − 0.584206i \(-0.198594\pi\)
−0.999990 + 0.00441771i \(0.998594\pi\)
\(38\) 4.73607 − 3.44095i 0.768292 − 0.558197i
\(39\) −0.381966 + 0.277515i −0.0611635 + 0.0444379i
\(40\) 1.00000 + 3.07768i 0.158114 + 0.486624i
\(41\) 1.73607 − 5.34307i 0.271128 − 0.834447i −0.719090 − 0.694917i \(-0.755441\pi\)
0.990218 − 0.139530i \(-0.0445591\pi\)
\(42\) 0.618034 + 0.449028i 0.0953647 + 0.0692865i
\(43\) −8.56231 −1.30574 −0.652870 − 0.757470i \(-0.726435\pi\)
−0.652870 + 0.757470i \(0.726435\pi\)
\(44\) 3.23607 − 0.726543i 0.487856 − 0.109530i
\(45\) −9.23607 −1.37683
\(46\) −1.00000 − 0.726543i −0.147442 − 0.107123i
\(47\) −2.00000 + 6.15537i −0.291730 + 0.897853i 0.692570 + 0.721350i \(0.256478\pi\)
−0.984300 + 0.176502i \(0.943522\pi\)
\(48\) 0.118034 + 0.363271i 0.0170367 + 0.0524337i
\(49\) 2.42705 − 1.76336i 0.346722 − 0.251908i
\(50\) −4.42705 + 3.21644i −0.626080 + 0.454873i
\(51\) −0.0729490 − 0.224514i −0.0102149 − 0.0314382i
\(52\) 0.381966 − 1.17557i 0.0529692 − 0.163022i
\(53\) 1.23607 + 0.898056i 0.169787 + 0.123357i 0.669434 − 0.742872i \(-0.266537\pi\)
−0.499647 + 0.866229i \(0.666537\pi\)
\(54\) −2.23607 −0.304290
\(55\) 5.47214 + 9.23305i 0.737863 + 1.24498i
\(56\) −2.00000 −0.267261
\(57\) 1.80902 + 1.31433i 0.239610 + 0.174087i
\(58\) 1.38197 − 4.25325i 0.181461 − 0.558480i
\(59\) −2.66312 − 8.19624i −0.346709 − 1.06706i −0.960663 − 0.277718i \(-0.910422\pi\)
0.613954 − 0.789342i \(-0.289578\pi\)
\(60\) −1.00000 + 0.726543i −0.129099 + 0.0937962i
\(61\) 2.00000 − 1.45309i 0.256074 − 0.186048i −0.452341 − 0.891845i \(-0.649411\pi\)
0.708414 + 0.705797i \(0.249411\pi\)
\(62\) 0.618034 + 1.90211i 0.0784904 + 0.241569i
\(63\) 1.76393 − 5.42882i 0.222235 − 0.683968i
\(64\) −0.809017 − 0.587785i −0.101127 − 0.0734732i
\(65\) 4.00000 0.496139
\(66\) 0.645898 + 1.08981i 0.0795046 + 0.134147i
\(67\) 11.0902 1.35488 0.677440 − 0.735578i \(-0.263089\pi\)
0.677440 + 0.735578i \(0.263089\pi\)
\(68\) 0.500000 + 0.363271i 0.0606339 + 0.0440531i
\(69\) 0.145898 − 0.449028i 0.0175641 − 0.0540566i
\(70\) −2.00000 − 6.15537i −0.239046 − 0.735707i
\(71\) 4.23607 − 3.07768i 0.502729 − 0.365254i −0.307330 − 0.951603i \(-0.599435\pi\)
0.810058 + 0.586349i \(0.199435\pi\)
\(72\) 2.30902 − 1.67760i 0.272120 − 0.197707i
\(73\) 3.20820 + 9.87384i 0.375492 + 1.15565i 0.943146 + 0.332378i \(0.107851\pi\)
−0.567654 + 0.823267i \(0.692149\pi\)
\(74\) −1.14590 + 3.52671i −0.133208 + 0.409972i
\(75\) −1.69098 − 1.22857i −0.195258 − 0.141863i
\(76\) −5.85410 −0.671512
\(77\) −6.47214 + 1.45309i −0.737568 + 0.165594i
\(78\) 0.472136 0.0534589
\(79\) −10.8541 − 7.88597i −1.22118 − 0.887241i −0.224984 − 0.974362i \(-0.572233\pi\)
−0.996198 + 0.0871218i \(0.972233\pi\)
\(80\) 1.00000 − 3.07768i 0.111803 − 0.344095i
\(81\) 2.38197 + 7.33094i 0.264663 + 0.814549i
\(82\) −4.54508 + 3.30220i −0.501921 + 0.364667i
\(83\) −7.54508 + 5.48183i −0.828181 + 0.601708i −0.919044 − 0.394155i \(-0.871037\pi\)
0.0908634 + 0.995863i \(0.471037\pi\)
\(84\) −0.236068 − 0.726543i −0.0257571 − 0.0792723i
\(85\) −0.618034 + 1.90211i −0.0670352 + 0.206313i
\(86\) 6.92705 + 5.03280i 0.746963 + 0.542700i
\(87\) 1.70820 0.183139
\(88\) −3.04508 − 1.31433i −0.324607 − 0.140108i
\(89\) −8.09017 −0.857556 −0.428778 − 0.903410i \(-0.641056\pi\)
−0.428778 + 0.903410i \(0.641056\pi\)
\(90\) 7.47214 + 5.42882i 0.787632 + 0.572248i
\(91\) −0.763932 + 2.35114i −0.0800818 + 0.246467i
\(92\) 0.381966 + 1.17557i 0.0398227 + 0.122562i
\(93\) −0.618034 + 0.449028i −0.0640871 + 0.0465620i
\(94\) 5.23607 − 3.80423i 0.540059 − 0.392376i
\(95\) −5.85410 − 18.0171i −0.600618 − 1.84851i
\(96\) 0.118034 − 0.363271i 0.0120468 − 0.0370762i
\(97\) −5.78115 − 4.20025i −0.586987 − 0.426471i 0.254249 − 0.967139i \(-0.418172\pi\)
−0.841236 + 0.540668i \(0.818172\pi\)
\(98\) −3.00000 −0.303046
\(99\) 6.25329 − 7.10642i 0.628479 − 0.714222i
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 22.2.c.a.15.1 yes 4
3.2 odd 2 198.2.f.e.37.1 4
4.3 odd 2 176.2.m.c.81.1 4
5.2 odd 4 550.2.ba.c.499.2 8
5.3 odd 4 550.2.ba.c.499.1 8
5.4 even 2 550.2.h.h.301.1 4
8.3 odd 2 704.2.m.a.257.1 4
8.5 even 2 704.2.m.h.257.1 4
11.2 odd 10 242.2.c.d.27.1 4
11.3 even 5 inner 22.2.c.a.3.1 ✓ 4
11.4 even 5 242.2.c.a.9.1 4
11.5 even 5 242.2.a.f.1.1 2
11.6 odd 10 242.2.a.d.1.1 2
11.7 odd 10 242.2.c.d.9.1 4
11.8 odd 10 242.2.c.c.3.1 4
11.9 even 5 242.2.c.a.27.1 4
11.10 odd 2 242.2.c.c.81.1 4
33.5 odd 10 2178.2.a.p.1.1 2
33.14 odd 10 198.2.f.e.91.1 4
33.17 even 10 2178.2.a.x.1.1 2
44.3 odd 10 176.2.m.c.113.1 4
44.27 odd 10 1936.2.a.o.1.2 2
44.39 even 10 1936.2.a.n.1.2 2
55.3 odd 20 550.2.ba.c.399.2 8
55.14 even 10 550.2.h.h.201.1 4
55.39 odd 10 6050.2.a.ci.1.2 2
55.47 odd 20 550.2.ba.c.399.1 8
55.49 even 10 6050.2.a.bs.1.2 2
88.3 odd 10 704.2.m.a.641.1 4
88.5 even 10 7744.2.a.bm.1.2 2
88.27 odd 10 7744.2.a.cz.1.1 2
88.61 odd 10 7744.2.a.bn.1.2 2
88.69 even 10 704.2.m.h.641.1 4
88.83 even 10 7744.2.a.cy.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
22.2.c.a.3.1 ✓ 4 11.3 even 5 inner
22.2.c.a.15.1 yes 4 1.1 even 1 trivial
176.2.m.c.81.1 4 4.3 odd 2
176.2.m.c.113.1 4 44.3 odd 10
198.2.f.e.37.1 4 3.2 odd 2
198.2.f.e.91.1 4 33.14 odd 10
242.2.a.d.1.1 2 11.6 odd 10
242.2.a.f.1.1 2 11.5 even 5
242.2.c.a.9.1 4 11.4 even 5
242.2.c.a.27.1 4 11.9 even 5
242.2.c.c.3.1 4 11.8 odd 10
242.2.c.c.81.1 4 11.10 odd 2
242.2.c.d.9.1 4 11.7 odd 10
242.2.c.d.27.1 4 11.2 odd 10
550.2.h.h.201.1 4 55.14 even 10
550.2.h.h.301.1 4 5.4 even 2
550.2.ba.c.399.1 8 55.47 odd 20
550.2.ba.c.399.2 8 55.3 odd 20
550.2.ba.c.499.1 8 5.3 odd 4
550.2.ba.c.499.2 8 5.2 odd 4
704.2.m.a.257.1 4 8.3 odd 2
704.2.m.a.641.1 4 88.3 odd 10
704.2.m.h.257.1 4 8.5 even 2
704.2.m.h.641.1 4 88.69 even 10
1936.2.a.n.1.2 2 44.39 even 10
1936.2.a.o.1.2 2 44.27 odd 10
2178.2.a.p.1.1 2 33.5 odd 10
2178.2.a.x.1.1 2 33.17 even 10
6050.2.a.bs.1.2 2 55.49 even 10
6050.2.a.ci.1.2 2 55.39 odd 10
7744.2.a.bm.1.2 2 88.5 even 10
7744.2.a.bn.1.2 2 88.61 odd 10
7744.2.a.cy.1.1 2 88.83 even 10
7744.2.a.cz.1.1 2 88.27 odd 10