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Properties

Label 550.2.ba.c.399.1
Level $550$
Weight $2$
Character 550.399
Analytic conductor $4.392$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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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] = [550,2,Mod(49,550)] 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("550.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(550, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 550.ba (of order \(10\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,2,0,2,0,0,-14,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.39177211117\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 22)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label 399.1
Root \(-0.587785 - 0.809017i\) of defining polynomial
Character \(\chi\) \(=\) 550.399
Dual form 550.2.ba.c.499.1

$q$-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+(-0.587785 - 0.809017i) q^{2} +(0.363271 - 0.118034i) q^{3} +(-0.309017 + 0.951057i) q^{4} +(-0.309017 - 0.224514i) q^{6} +(-1.90211 - 0.618034i) q^{7} +(0.951057 - 0.309017i) q^{8} +(-2.30902 + 1.67760i) q^{9} +(0.309017 + 3.30220i) q^{11} +0.381966i q^{12} +(0.726543 + 1.00000i) q^{13} +(0.618034 + 1.90211i) q^{14} +(-0.809017 - 0.587785i) q^{16} +(-0.363271 + 0.500000i) q^{17} +(2.71441 + 0.881966i) q^{18} +(1.80902 + 5.56758i) q^{19} -0.763932 q^{21} +(2.48990 - 2.19098i) q^{22} -1.23607i q^{23} +(0.309017 - 0.224514i) q^{24} +(0.381966 - 1.17557i) q^{26} +(-1.31433 + 1.80902i) q^{27} +(1.17557 - 1.61803i) q^{28} +(-1.38197 + 4.25325i) q^{29} +(-1.61803 + 1.17557i) q^{31} +1.00000i q^{32} +(0.502029 + 1.16312i) q^{33} +0.618034 q^{34} +(-0.881966 - 2.71441i) q^{36} +(-3.52671 - 1.14590i) q^{37} +(3.44095 - 4.73607i) q^{38} +(0.381966 + 0.277515i) q^{39} +(1.73607 + 5.34307i) q^{41} +(0.449028 + 0.618034i) q^{42} +8.56231i q^{43} +(-3.23607 - 0.726543i) q^{44} +(-1.00000 + 0.726543i) q^{46} +(6.15537 - 2.00000i) q^{47} +(-0.363271 - 0.118034i) q^{48} +(-2.42705 - 1.76336i) q^{49} +(-0.0729490 + 0.224514i) q^{51} +(-1.17557 + 0.381966i) q^{52} +(-0.898056 - 1.23607i) q^{53} +2.23607 q^{54} -2.00000 q^{56} +(1.31433 + 1.80902i) q^{57} +(4.25325 - 1.38197i) q^{58} +(2.66312 - 8.19624i) q^{59} +(2.00000 + 1.45309i) q^{61} +(1.90211 + 0.618034i) q^{62} +(5.42882 - 1.76393i) q^{63} +(0.809017 - 0.587785i) q^{64} +(0.645898 - 1.08981i) q^{66} +11.0902i q^{67} +(-0.363271 - 0.500000i) q^{68} +(-0.145898 - 0.449028i) q^{69} +(4.23607 + 3.07768i) q^{71} +(-1.67760 + 2.30902i) q^{72} +(-9.87384 - 3.20820i) q^{73} +(1.14590 + 3.52671i) q^{74} -5.85410 q^{76} +(1.45309 - 6.47214i) q^{77} -0.472136i q^{78} +(10.8541 - 7.88597i) q^{79} +(2.38197 - 7.33094i) q^{81} +(3.30220 - 4.54508i) q^{82} +(-5.48183 + 7.54508i) q^{83} +(0.236068 - 0.726543i) q^{84} +(6.92705 - 5.03280i) q^{86} +1.70820i q^{87} +(1.31433 + 3.04508i) q^{88} +8.09017 q^{89} +(-0.763932 - 2.35114i) q^{91} +(1.17557 + 0.381966i) q^{92} +(-0.449028 + 0.618034i) q^{93} +(-5.23607 - 3.80423i) q^{94} +(0.118034 + 0.363271i) q^{96} +(-4.20025 - 5.78115i) q^{97} +3.00000i q^{98} +(-6.25329 - 7.10642i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{4} + 2 q^{6} - 14 q^{9} - 2 q^{11} - 4 q^{14} - 2 q^{16} + 10 q^{19} - 24 q^{21} - 2 q^{24} + 12 q^{26} - 20 q^{29} - 4 q^{31} - 4 q^{34} - 16 q^{36} + 12 q^{39} - 4 q^{41} - 8 q^{44} - 8 q^{46}+ \cdots + 26 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/550\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(e\left(\frac{4}{5}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.587785 − 0.809017i −0.415627 − 0.572061i
\(3\) 0.363271 − 0.118034i 0.209735 − 0.0681470i −0.202265 − 0.979331i \(-0.564830\pi\)
0.412000 + 0.911184i \(0.364830\pi\)
\(4\) −0.309017 + 0.951057i −0.154508 + 0.475528i
\(5\) 0 0
\(6\) −0.309017 − 0.224514i −0.126156 − 0.0916575i
\(7\) −1.90211 − 0.618034i −0.718931 − 0.233595i −0.0733714 − 0.997305i \(-0.523376\pi\)
−0.645560 + 0.763710i \(0.723376\pi\)
\(8\) 0.951057 − 0.309017i 0.336249 − 0.109254i
\(9\) −2.30902 + 1.67760i −0.769672 + 0.559200i
\(10\) 0 0
\(11\) 0.309017 + 3.30220i 0.0931721 + 0.995650i
\(12\) 0.381966i 0.110264i
\(13\) 0.726543 + 1.00000i 0.201507 + 0.277350i 0.897796 − 0.440411i \(-0.145167\pi\)
−0.696290 + 0.717761i \(0.745167\pi\)
\(14\) 0.618034 + 1.90211i 0.165177 + 0.508361i
\(15\) 0 0
\(16\) −0.809017 − 0.587785i −0.202254 − 0.146946i
\(17\) −0.363271 + 0.500000i −0.0881062 + 0.121268i −0.850795 − 0.525498i \(-0.823879\pi\)
0.762688 + 0.646766i \(0.223879\pi\)
\(18\) 2.71441 + 0.881966i 0.639793 + 0.207881i
\(19\) 1.80902 + 5.56758i 0.415017 + 1.27729i 0.912236 + 0.409666i \(0.134355\pi\)
−0.497219 + 0.867625i \(0.665645\pi\)
\(20\) 0 0
\(21\) −0.763932 −0.166704
\(22\) 2.48990 − 2.19098i 0.530848 − 0.467119i
\(23\) − 1.23607i − 0.257738i −0.991662 − 0.128869i \(-0.958865\pi\)
0.991662 − 0.128869i \(-0.0411347\pi\)
\(24\) 0.309017 − 0.224514i 0.0630778 − 0.0458287i
\(25\) 0 0
\(26\) 0.381966 − 1.17557i 0.0749097 − 0.230548i
\(27\) −1.31433 + 1.80902i −0.252942 + 0.348145i
\(28\) 1.17557 − 1.61803i 0.222162 − 0.305780i
\(29\) −1.38197 + 4.25325i −0.256625 + 0.789809i 0.736881 + 0.676023i \(0.236298\pi\)
−0.993505 + 0.113787i \(0.963702\pi\)
\(30\) 0 0
\(31\) −1.61803 + 1.17557i −0.290607 + 0.211139i −0.723531 − 0.690292i \(-0.757482\pi\)
0.432923 + 0.901431i \(0.357482\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0.502029 + 1.16312i 0.0873920 + 0.202473i
\(34\) 0.618034 0.105992
\(35\) 0 0
\(36\) −0.881966 − 2.71441i −0.146994 − 0.452402i
\(37\) −3.52671 − 1.14590i −0.579788 − 0.188384i 0.00441771 − 0.999990i \(-0.498594\pi\)
−0.584206 + 0.811606i \(0.698594\pi\)
\(38\) 3.44095 − 4.73607i 0.558197 − 0.768292i
\(39\) 0.381966 + 0.277515i 0.0611635 + 0.0444379i
\(40\) 0 0
\(41\) 1.73607 + 5.34307i 0.271128 + 0.834447i 0.990218 + 0.139530i \(0.0445591\pi\)
−0.719090 + 0.694917i \(0.755441\pi\)
\(42\) 0.449028 + 0.618034i 0.0692865 + 0.0953647i
\(43\) 8.56231i 1.30574i 0.757470 + 0.652870i \(0.226435\pi\)
−0.757470 + 0.652870i \(0.773565\pi\)
\(44\) −3.23607 − 0.726543i −0.487856 − 0.109530i
\(45\) 0 0
\(46\) −1.00000 + 0.726543i −0.147442 + 0.107123i
\(47\) 6.15537 − 2.00000i 0.897853 − 0.291730i 0.176502 − 0.984300i \(-0.443522\pi\)
0.721350 + 0.692570i \(0.243522\pi\)
\(48\) −0.363271 − 0.118034i −0.0524337 − 0.0170367i
\(49\) −2.42705 − 1.76336i −0.346722 − 0.251908i
\(50\) 0 0
\(51\) −0.0729490 + 0.224514i −0.0102149 + 0.0314382i
\(52\) −1.17557 + 0.381966i −0.163022 + 0.0529692i
\(53\) −0.898056 − 1.23607i −0.123357 − 0.169787i 0.742872 − 0.669434i \(-0.233463\pi\)
−0.866229 + 0.499647i \(0.833463\pi\)
\(54\) 2.23607 0.304290
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) 1.31433 + 1.80902i 0.174087 + 0.239610i
\(58\) 4.25325 − 1.38197i 0.558480 − 0.181461i
\(59\) 2.66312 − 8.19624i 0.346709 − 1.06706i −0.613954 − 0.789342i \(-0.710422\pi\)
0.960663 − 0.277718i \(-0.0895779\pi\)
\(60\) 0 0
\(61\) 2.00000 + 1.45309i 0.256074 + 0.186048i 0.708414 − 0.705797i \(-0.249411\pi\)
−0.452341 + 0.891845i \(0.649411\pi\)
\(62\) 1.90211 + 0.618034i 0.241569 + 0.0784904i
\(63\) 5.42882 − 1.76393i 0.683968 − 0.222235i
\(64\) 0.809017 − 0.587785i 0.101127 − 0.0734732i
\(65\) 0 0
\(66\) 0.645898 − 1.08981i 0.0795046 − 0.134147i
\(67\) 11.0902i 1.35488i 0.735578 + 0.677440i \(0.236911\pi\)
−0.735578 + 0.677440i \(0.763089\pi\)
\(68\) −0.363271 − 0.500000i −0.0440531 − 0.0606339i
\(69\) −0.145898 − 0.449028i −0.0175641 − 0.0540566i
\(70\) 0 0
\(71\) 4.23607 + 3.07768i 0.502729 + 0.365254i 0.810058 − 0.586349i \(-0.199435\pi\)
−0.307330 + 0.951603i \(0.599435\pi\)
\(72\) −1.67760 + 2.30902i −0.197707 + 0.272120i
\(73\) −9.87384 − 3.20820i −1.15565 − 0.375492i −0.332378 − 0.943146i \(-0.607851\pi\)
−0.823267 + 0.567654i \(0.807851\pi\)
\(74\) 1.14590 + 3.52671i 0.133208 + 0.409972i
\(75\) 0 0
\(76\) −5.85410 −0.671512
\(77\) 1.45309 − 6.47214i 0.165594 − 0.737568i
\(78\) − 0.472136i − 0.0534589i
\(79\) 10.8541 − 7.88597i 1.22118 − 0.887241i 0.224984 − 0.974362i \(-0.427767\pi\)
0.996198 + 0.0871218i \(0.0277669\pi\)
\(80\) 0 0
\(81\) 2.38197 − 7.33094i 0.264663 − 0.814549i
\(82\) 3.30220 − 4.54508i 0.364667 − 0.501921i
\(83\) −5.48183 + 7.54508i −0.601708 + 0.828181i −0.995863 − 0.0908634i \(-0.971037\pi\)
0.394155 + 0.919044i \(0.371037\pi\)
\(84\) 0.236068 − 0.726543i 0.0257571 − 0.0792723i
\(85\) 0 0
\(86\) 6.92705 − 5.03280i 0.746963 − 0.542700i
\(87\) 1.70820i 0.183139i
\(88\) 1.31433 + 3.04508i 0.140108 + 0.324607i
\(89\) 8.09017 0.857556 0.428778 − 0.903410i \(-0.358944\pi\)
0.428778 + 0.903410i \(0.358944\pi\)
\(90\) 0 0
\(91\) −0.763932 − 2.35114i −0.0800818 − 0.246467i
\(92\) 1.17557 + 0.381966i 0.122562 + 0.0398227i
\(93\) −0.449028 + 0.618034i −0.0465620 + 0.0640871i
\(94\) −5.23607 − 3.80423i −0.540059 − 0.392376i
\(95\) 0 0
\(96\) 0.118034 + 0.363271i 0.0120468 + 0.0370762i
\(97\) −4.20025 − 5.78115i −0.426471 − 0.586987i 0.540668 − 0.841236i \(-0.318172\pi\)
−0.967139 + 0.254249i \(0.918172\pi\)
\(98\) 3.00000i 0.303046i
\(99\) −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 550.2.ba.c.399.1 8
5.2 odd 4 550.2.h.h.201.1 4
5.3 odd 4 22.2.c.a.3.1 ✓ 4
5.4 even 2 inner 550.2.ba.c.399.2 8
11.4 even 5 inner 550.2.ba.c.499.2 8
15.8 even 4 198.2.f.e.91.1 4
20.3 even 4 176.2.m.c.113.1 4
40.3 even 4 704.2.m.a.641.1 4
40.13 odd 4 704.2.m.h.641.1 4
55.2 even 20 6050.2.a.ci.1.2 2
55.3 odd 20 242.2.c.a.27.1 4
55.4 even 10 inner 550.2.ba.c.499.1 8
55.8 even 20 242.2.c.d.27.1 4
55.13 even 20 242.2.a.d.1.1 2
55.18 even 20 242.2.c.c.81.1 4
55.28 even 20 242.2.c.d.9.1 4
55.37 odd 20 550.2.h.h.301.1 4
55.38 odd 20 242.2.c.a.9.1 4
55.42 odd 20 6050.2.a.bs.1.2 2
55.43 even 4 242.2.c.c.3.1 4
55.48 odd 20 22.2.c.a.15.1 yes 4
55.53 odd 20 242.2.a.f.1.1 2
165.53 even 20 2178.2.a.p.1.1 2
165.68 odd 20 2178.2.a.x.1.1 2
165.158 even 20 198.2.f.e.37.1 4
220.103 even 20 176.2.m.c.81.1 4
220.123 odd 20 1936.2.a.n.1.2 2
220.163 even 20 1936.2.a.o.1.2 2
440.13 even 20 7744.2.a.bn.1.2 2
440.53 odd 20 7744.2.a.bm.1.2 2
440.123 odd 20 7744.2.a.cy.1.1 2
440.163 even 20 7744.2.a.cz.1.1 2
440.213 odd 20 704.2.m.h.257.1 4
440.323 even 20 704.2.m.a.257.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
22.2.c.a.3.1 ✓ 4 5.3 odd 4
22.2.c.a.15.1 yes 4 55.48 odd 20
176.2.m.c.81.1 4 220.103 even 20
176.2.m.c.113.1 4 20.3 even 4
198.2.f.e.37.1 4 165.158 even 20
198.2.f.e.91.1 4 15.8 even 4
242.2.a.d.1.1 2 55.13 even 20
242.2.a.f.1.1 2 55.53 odd 20
242.2.c.a.9.1 4 55.38 odd 20
242.2.c.a.27.1 4 55.3 odd 20
242.2.c.c.3.1 4 55.43 even 4
242.2.c.c.81.1 4 55.18 even 20
242.2.c.d.9.1 4 55.28 even 20
242.2.c.d.27.1 4 55.8 even 20
550.2.h.h.201.1 4 5.2 odd 4
550.2.h.h.301.1 4 55.37 odd 20
550.2.ba.c.399.1 8 1.1 even 1 trivial
550.2.ba.c.399.2 8 5.4 even 2 inner
550.2.ba.c.499.1 8 55.4 even 10 inner
550.2.ba.c.499.2 8 11.4 even 5 inner
704.2.m.a.257.1 4 440.323 even 20
704.2.m.a.641.1 4 40.3 even 4
704.2.m.h.257.1 4 440.213 odd 20
704.2.m.h.641.1 4 40.13 odd 4
1936.2.a.n.1.2 2 220.123 odd 20
1936.2.a.o.1.2 2 220.163 even 20
2178.2.a.p.1.1 2 165.53 even 20
2178.2.a.x.1.1 2 165.68 odd 20
6050.2.a.bs.1.2 2 55.42 odd 20
6050.2.a.ci.1.2 2 55.2 even 20
7744.2.a.bm.1.2 2 440.53 odd 20
7744.2.a.bn.1.2 2 440.13 even 20
7744.2.a.cy.1.1 2 440.123 odd 20
7744.2.a.cz.1.1 2 440.163 even 20