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Properties

Label 726.2.h.i.215.1
Level $726$
Weight $2$
Character 726.215
Analytic conductor $5.797$
Analytic rank $0$
Dimension $8$
Inner twists $8$

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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] = [726,2,Mod(161,726)] 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("726.161"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(726, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 726 = 2 \cdot 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 726.h (of order \(10\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,2,2,-2,0,-2,0,2,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.79713918674\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.64000000.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 66)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label 215.1
Root \(-1.34500 - 0.437016i\) of defining polynomial
Character \(\chi\) \(=\) 726.215
Dual form 726.2.h.i.233.1

$q$-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} +(-0.0222369 + 1.73191i) q^{3} +(-0.809017 + 0.587785i) q^{4} +(-1.34500 - 0.437016i) q^{5} +(1.65401 - 0.514040i) q^{6} +(2.49376 + 3.43237i) q^{7} +(0.809017 + 0.587785i) q^{8} +(-2.99901 - 0.0770245i) q^{9} +1.41421i q^{10} +(-1.00000 - 1.41421i) q^{12} +(-4.03499 + 1.31105i) q^{13} +(2.49376 - 3.43237i) q^{14} +(0.786780 - 2.31969i) q^{15} +(0.309017 - 0.951057i) q^{16} +(0.853491 + 2.87603i) q^{18} +(1.34500 - 0.437016i) q^{20} +(-6.00000 + 4.24264i) q^{21} -1.41421i q^{23} +(-1.03598 + 1.38807i) q^{24} +(-2.42705 - 1.76336i) q^{25} +(2.49376 + 3.43237i) q^{26} +(0.200088 - 5.19230i) q^{27} +(-4.03499 - 1.31105i) q^{28} +(-4.85410 + 3.52671i) q^{29} +(-2.44929 - 0.0314477i) q^{30} +(-1.23607 - 3.80423i) q^{31} -1.00000 q^{32} +(-1.85410 - 5.70634i) q^{35} +(2.47152 - 1.70046i) q^{36} +(-1.61803 + 1.17557i) q^{37} +(-2.18089 - 7.01739i) q^{39} +(-0.831254 - 1.14412i) q^{40} +(4.85410 + 3.52671i) q^{41} +(5.88909 + 4.39529i) q^{42} +8.48528i q^{43} +(4.00000 + 1.41421i) q^{45} +(-1.34500 + 0.437016i) q^{46} +(-5.81878 + 8.00886i) q^{47} +(1.64027 + 0.556338i) q^{48} +(-3.39919 + 10.4616i) q^{49} +(-0.927051 + 2.85317i) q^{50} +(2.49376 - 3.43237i) q^{52} +(-6.72499 + 2.18508i) q^{53} +(-5.00000 + 1.41421i) q^{54} +4.24264i q^{56} +(4.85410 + 3.52671i) q^{58} +(-6.65003 - 9.15298i) q^{59} +(0.726963 + 2.33913i) q^{60} +(-4.03499 - 1.31105i) q^{61} +(-3.23607 + 2.35114i) q^{62} +(-7.21444 - 10.4858i) q^{63} +(0.309017 + 0.951057i) q^{64} +6.00000 q^{65} -4.00000 q^{67} +(2.44929 + 0.0314477i) q^{69} +(-4.85410 + 3.52671i) q^{70} +(6.72499 + 2.18508i) q^{71} +(-2.38098 - 1.82509i) q^{72} +(1.61803 + 1.17557i) q^{74} +(3.10794 - 4.16422i) q^{75} +(-6.00000 + 4.24264i) q^{78} +(4.03499 - 1.31105i) q^{79} +(-0.831254 + 1.14412i) q^{80} +(8.98813 + 0.461994i) q^{81} +(1.85410 - 5.70634i) q^{82} +(-3.70820 + 11.4127i) q^{83} +(2.36034 - 6.95908i) q^{84} +(8.06998 - 2.62210i) q^{86} +(-6.00000 - 8.48528i) q^{87} -5.65685i q^{89} +(0.108929 - 4.24124i) q^{90} +(-14.5623 - 10.5801i) q^{91} +(0.831254 + 1.14412i) q^{92} +(6.61606 - 2.05616i) q^{93} +(9.41498 + 3.05911i) q^{94} +(0.0222369 - 1.73191i) q^{96} +(2.47214 + 7.60845i) q^{97} +11.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} + 2 q^{3} - 2 q^{4} - 2 q^{6} + 2 q^{8} + 2 q^{9} - 8 q^{12} + 4 q^{15} - 2 q^{16} - 2 q^{18} - 48 q^{21} - 2 q^{24} - 6 q^{25} - 10 q^{27} - 12 q^{29} - 4 q^{30} + 8 q^{31} - 8 q^{32} + 12 q^{35}+ \cdots + 88 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/726\mathbb{Z}\right)^\times\).

\(n\) \(485\) \(607\)
\(\chi(n)\) \(-1\) \(e\left(\frac{9}{10}\right)\)

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
\(3\) −0.0222369 + 1.73191i −0.0128385 + 0.999918i
\(4\) −0.809017 + 0.587785i −0.404508 + 0.293893i
\(5\) −1.34500 − 0.437016i −0.601501 − 0.195440i −0.00759122 − 0.999971i \(-0.502416\pi\)
−0.593910 + 0.804532i \(0.702416\pi\)
\(6\) 1.65401 − 0.514040i 0.675248 − 0.209856i
\(7\) 2.49376 + 3.43237i 0.942553 + 1.29731i 0.954757 + 0.297388i \(0.0961154\pi\)
−0.0122035 + 0.999926i \(0.503885\pi\)
\(8\) 0.809017 + 0.587785i 0.286031 + 0.207813i
\(9\) −2.99901 − 0.0770245i −0.999670 − 0.0256748i
\(10\) 1.41421i 0.447214i
\(11\) 0 0
\(12\) −1.00000 − 1.41421i −0.288675 − 0.408248i
\(13\) −4.03499 + 1.31105i −1.11911 + 0.363619i −0.809426 − 0.587222i \(-0.800222\pi\)
−0.309679 + 0.950841i \(0.600222\pi\)
\(14\) 2.49376 − 3.43237i 0.666486 − 0.917339i
\(15\) 0.786780 − 2.31969i 0.203146 − 0.598942i
\(16\) 0.309017 − 0.951057i 0.0772542 − 0.237764i
\(17\) 0 0 −0.951057 − 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(18\) 0.853491 + 2.87603i 0.201170 + 0.677887i
\(19\) 0 0 −0.809017 − 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(20\) 1.34500 − 0.437016i 0.300750 − 0.0977198i
\(21\) −6.00000 + 4.24264i −1.30931 + 0.925820i
\(22\) 0 0
\(23\) − 1.41421i − 0.294884i −0.989071 − 0.147442i \(-0.952896\pi\)
0.989071 − 0.147442i \(-0.0471040\pi\)
\(24\) −1.03598 + 1.38807i −0.211469 + 0.283339i
\(25\) −2.42705 − 1.76336i −0.485410 − 0.352671i
\(26\) 2.49376 + 3.43237i 0.489067 + 0.673143i
\(27\) 0.200088 − 5.19230i 0.0385069 − 0.999258i
\(28\) −4.03499 − 1.31105i −0.762542 − 0.247765i
\(29\) −4.85410 + 3.52671i −0.901384 + 0.654894i −0.938821 − 0.344405i \(-0.888081\pi\)
0.0374370 + 0.999299i \(0.488081\pi\)
\(30\) −2.44929 − 0.0314477i −0.447177 − 0.00574154i
\(31\) −1.23607 − 3.80423i −0.222004 − 0.683259i −0.998582 − 0.0532375i \(-0.983046\pi\)
0.776578 − 0.630022i \(-0.216954\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) −1.85410 − 5.70634i −0.313400 − 0.964547i
\(36\) 2.47152 − 1.70046i 0.411921 − 0.283410i
\(37\) −1.61803 + 1.17557i −0.266003 + 0.193263i −0.712789 − 0.701378i \(-0.752568\pi\)
0.446786 + 0.894641i \(0.352568\pi\)
\(38\) 0 0
\(39\) −2.18089 − 7.01739i −0.349222 − 1.12368i
\(40\) −0.831254 − 1.14412i −0.131433 − 0.180902i
\(41\) 4.85410 + 3.52671i 0.758083 + 0.550780i 0.898322 − 0.439338i \(-0.144787\pi\)
−0.140238 + 0.990118i \(0.544787\pi\)
\(42\) 5.88909 + 4.39529i 0.908707 + 0.678208i
\(43\) 8.48528i 1.29399i 0.762493 + 0.646997i \(0.223975\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) 4.00000 + 1.41421i 0.596285 + 0.210819i
\(46\) −1.34500 + 0.437016i −0.198309 + 0.0644345i
\(47\) −5.81878 + 8.00886i −0.848756 + 1.16821i 0.135380 + 0.990794i \(0.456775\pi\)
−0.984136 + 0.177418i \(0.943225\pi\)
\(48\) 1.64027 + 0.556338i 0.236753 + 0.0803004i
\(49\) −3.39919 + 10.4616i −0.485598 + 1.49452i
\(50\) −0.927051 + 2.85317i −0.131105 + 0.403499i
\(51\) 0 0
\(52\) 2.49376 − 3.43237i 0.345823 − 0.475984i
\(53\) −6.72499 + 2.18508i −0.923748 + 0.300144i −0.732003 − 0.681301i \(-0.761414\pi\)
−0.191744 + 0.981445i \(0.561414\pi\)
\(54\) −5.00000 + 1.41421i −0.680414 + 0.192450i
\(55\) 0 0
\(56\) 4.24264i 0.566947i
\(57\) 0 0
\(58\) 4.85410 + 3.52671i 0.637375 + 0.463080i
\(59\) −6.65003 − 9.15298i −0.865760 − 1.19162i −0.980165 − 0.198183i \(-0.936496\pi\)
0.114405 − 0.993434i \(-0.463504\pi\)
\(60\) 0.726963 + 2.33913i 0.0938505 + 0.301980i
\(61\) −4.03499 − 1.31105i −0.516628 − 0.167863i 0.0390866 − 0.999236i \(-0.487555\pi\)
−0.555714 + 0.831373i \(0.687555\pi\)
\(62\) −3.23607 + 2.35114i −0.410981 + 0.298595i
\(63\) −7.21444 − 10.4858i −0.908934 − 1.32109i
\(64\) 0.309017 + 0.951057i 0.0386271 + 0.118882i
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 2.44929 + 0.0314477i 0.294860 + 0.00378586i
\(70\) −4.85410 + 3.52671i −0.580176 + 0.421523i
\(71\) 6.72499 + 2.18508i 0.798109 + 0.259321i 0.679553 − 0.733626i \(-0.262174\pi\)
0.118556 + 0.992947i \(0.462174\pi\)
\(72\) −2.38098 − 1.82509i −0.280601 − 0.215089i
\(73\) 0 0 0.809017 − 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(74\) 1.61803 + 1.17557i 0.188093 + 0.136657i
\(75\) 3.10794 − 4.16422i 0.358874 − 0.480842i
\(76\) 0 0
\(77\) 0 0
\(78\) −6.00000 + 4.24264i −0.679366 + 0.480384i
\(79\) 4.03499 − 1.31105i 0.453972 − 0.147504i −0.0731009 − 0.997325i \(-0.523290\pi\)
0.527073 + 0.849820i \(0.323290\pi\)
\(80\) −0.831254 + 1.14412i −0.0929370 + 0.127917i
\(81\) 8.98813 + 0.461994i 0.998682 + 0.0513327i
\(82\) 1.85410 − 5.70634i 0.204751 − 0.630160i
\(83\) −3.70820 + 11.4127i −0.407028 + 1.25270i 0.512161 + 0.858889i \(0.328845\pi\)
−0.919190 + 0.393815i \(0.871155\pi\)
\(84\) 2.36034 − 6.95908i 0.257534 − 0.759298i
\(85\) 0 0
\(86\) 8.06998 − 2.62210i 0.870209 − 0.282748i
\(87\) −6.00000 − 8.48528i −0.643268 − 0.909718i
\(88\) 0 0
\(89\) − 5.65685i − 0.599625i −0.953998 − 0.299813i \(-0.903076\pi\)
0.953998 − 0.299813i \(-0.0969242\pi\)
\(90\) 0.108929 − 4.24124i 0.0114821 − 0.447066i
\(91\) −14.5623 − 10.5801i −1.52654 − 1.10910i
\(92\) 0.831254 + 1.14412i 0.0866642 + 0.119283i
\(93\) 6.61606 − 2.05616i 0.686053 − 0.213214i
\(94\) 9.41498 + 3.05911i 0.971081 + 0.315523i
\(95\) 0 0
\(96\) 0.0222369 − 1.73191i 0.00226954 − 0.176762i
\(97\) 2.47214 + 7.60845i 0.251007 + 0.772521i 0.994590 + 0.103877i \(0.0331249\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(98\) 11.0000 1.11117
\(99\) 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 726.2.h.i.215.1 8
3.2 odd 2 726.2.h.e.215.2 8
11.2 odd 10 726.2.h.e.233.2 8
11.3 even 5 66.2.b.a.65.1 ✓ 2
11.4 even 5 inner 726.2.h.i.239.2 8
11.5 even 5 inner 726.2.h.i.161.1 8
11.6 odd 10 726.2.h.e.161.1 8
11.7 odd 10 726.2.h.e.239.2 8
11.8 odd 10 66.2.b.b.65.1 yes 2
11.9 even 5 inner 726.2.h.i.233.2 8
11.10 odd 2 726.2.h.e.215.1 8
33.2 even 10 inner 726.2.h.i.233.1 8
33.5 odd 10 726.2.h.e.161.2 8
33.8 even 10 66.2.b.a.65.2 yes 2
33.14 odd 10 66.2.b.b.65.2 yes 2
33.17 even 10 inner 726.2.h.i.161.2 8
33.20 odd 10 726.2.h.e.233.1 8
33.26 odd 10 726.2.h.e.239.1 8
33.29 even 10 inner 726.2.h.i.239.1 8
33.32 even 2 inner 726.2.h.i.215.2 8
44.3 odd 10 528.2.b.b.65.2 2
44.19 even 10 528.2.b.c.65.2 2
55.3 odd 20 1650.2.f.b.1649.4 4
55.8 even 20 1650.2.f.a.1649.2 4
55.14 even 10 1650.2.d.b.1451.2 2
55.19 odd 10 1650.2.d.a.1451.2 2
55.47 odd 20 1650.2.f.b.1649.1 4
55.52 even 20 1650.2.f.a.1649.3 4
88.3 odd 10 2112.2.b.d.65.1 2
88.19 even 10 2112.2.b.b.65.1 2
88.69 even 10 2112.2.b.g.65.2 2
88.85 odd 10 2112.2.b.i.65.2 2
99.14 odd 30 1782.2.i.c.593.2 4
99.25 even 15 1782.2.i.f.1187.2 4
99.41 even 30 1782.2.i.f.593.2 4
99.47 odd 30 1782.2.i.c.1187.1 4
99.52 odd 30 1782.2.i.c.1187.2 4
99.58 even 15 1782.2.i.f.593.1 4
99.74 even 30 1782.2.i.f.1187.1 4
99.85 odd 30 1782.2.i.c.593.1 4
132.47 even 10 528.2.b.c.65.1 2
132.107 odd 10 528.2.b.b.65.1 2
165.8 odd 20 1650.2.f.b.1649.3 4
165.14 odd 10 1650.2.d.a.1451.1 2
165.47 even 20 1650.2.f.a.1649.4 4
165.74 even 10 1650.2.d.b.1451.1 2
165.107 odd 20 1650.2.f.b.1649.2 4
165.113 even 20 1650.2.f.a.1649.1 4
264.107 odd 10 2112.2.b.d.65.2 2
264.173 even 10 2112.2.b.g.65.1 2
264.179 even 10 2112.2.b.b.65.2 2
264.245 odd 10 2112.2.b.i.65.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
66.2.b.a.65.1 ✓ 2 11.3 even 5
66.2.b.a.65.2 yes 2 33.8 even 10
66.2.b.b.65.1 yes 2 11.8 odd 10
66.2.b.b.65.2 yes 2 33.14 odd 10
528.2.b.b.65.1 2 132.107 odd 10
528.2.b.b.65.2 2 44.3 odd 10
528.2.b.c.65.1 2 132.47 even 10
528.2.b.c.65.2 2 44.19 even 10
726.2.h.e.161.1 8 11.6 odd 10
726.2.h.e.161.2 8 33.5 odd 10
726.2.h.e.215.1 8 11.10 odd 2
726.2.h.e.215.2 8 3.2 odd 2
726.2.h.e.233.1 8 33.20 odd 10
726.2.h.e.233.2 8 11.2 odd 10
726.2.h.e.239.1 8 33.26 odd 10
726.2.h.e.239.2 8 11.7 odd 10
726.2.h.i.161.1 8 11.5 even 5 inner
726.2.h.i.161.2 8 33.17 even 10 inner
726.2.h.i.215.1 8 1.1 even 1 trivial
726.2.h.i.215.2 8 33.32 even 2 inner
726.2.h.i.233.1 8 33.2 even 10 inner
726.2.h.i.233.2 8 11.9 even 5 inner
726.2.h.i.239.1 8 33.29 even 10 inner
726.2.h.i.239.2 8 11.4 even 5 inner
1650.2.d.a.1451.1 2 165.14 odd 10
1650.2.d.a.1451.2 2 55.19 odd 10
1650.2.d.b.1451.1 2 165.74 even 10
1650.2.d.b.1451.2 2 55.14 even 10
1650.2.f.a.1649.1 4 165.113 even 20
1650.2.f.a.1649.2 4 55.8 even 20
1650.2.f.a.1649.3 4 55.52 even 20
1650.2.f.a.1649.4 4 165.47 even 20
1650.2.f.b.1649.1 4 55.47 odd 20
1650.2.f.b.1649.2 4 165.107 odd 20
1650.2.f.b.1649.3 4 165.8 odd 20
1650.2.f.b.1649.4 4 55.3 odd 20
1782.2.i.c.593.1 4 99.85 odd 30
1782.2.i.c.593.2 4 99.14 odd 30
1782.2.i.c.1187.1 4 99.47 odd 30
1782.2.i.c.1187.2 4 99.52 odd 30
1782.2.i.f.593.1 4 99.58 even 15
1782.2.i.f.593.2 4 99.41 even 30
1782.2.i.f.1187.1 4 99.74 even 30
1782.2.i.f.1187.2 4 99.25 even 15
2112.2.b.b.65.1 2 88.19 even 10
2112.2.b.b.65.2 2 264.179 even 10
2112.2.b.d.65.1 2 88.3 odd 10
2112.2.b.d.65.2 2 264.107 odd 10
2112.2.b.g.65.1 2 264.173 even 10
2112.2.b.g.65.2 2 88.69 even 10
2112.2.b.i.65.1 2 264.245 odd 10
2112.2.b.i.65.2 2 88.85 odd 10