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Properties

Label 726.2.h.e.161.2
Level $726$
Weight $2$
Character 726.161
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 161.2
Root \(-0.831254 + 1.14412i\) of defining polynomial
Character \(\chi\) \(=\) 726.161
Dual form 726.2.h.e.239.2

$q$-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} +(1.03598 + 1.38807i) q^{3} +(0.309017 + 0.951057i) q^{4} +(-0.831254 - 1.14412i) q^{5} +(-0.0222369 - 1.73191i) q^{6} +(4.03499 - 1.31105i) q^{7} +(0.309017 - 0.951057i) q^{8} +(-0.853491 + 2.87603i) q^{9} +1.41421i q^{10} +(-1.00000 + 1.41421i) q^{12} +(2.49376 - 3.43237i) q^{13} +(-4.03499 - 1.31105i) q^{14} +(0.726963 - 2.33913i) q^{15} +(-0.809017 + 0.587785i) q^{16} +(2.38098 - 1.82509i) q^{18} +(0.831254 - 1.14412i) q^{20} +(6.00000 + 4.24264i) q^{21} +1.41421i q^{23} +(1.64027 - 0.556338i) q^{24} +(0.927051 - 2.85317i) q^{25} +(-4.03499 + 1.31105i) q^{26} +(-4.87634 + 1.79480i) q^{27} +(2.49376 + 3.43237i) q^{28} +(-1.85410 - 5.70634i) q^{29} +(-1.96303 + 1.46510i) q^{30} +(3.23607 + 2.35114i) q^{31} +1.00000 q^{32} +(-4.85410 - 3.52671i) q^{35} +(-2.99901 + 0.0770245i) q^{36} +(0.618034 + 1.90211i) q^{37} +(7.34786 - 0.0943431i) q^{39} +(-1.34500 + 0.437016i) q^{40} +(1.85410 - 5.70634i) q^{41} +(-2.36034 - 6.95908i) q^{42} +8.48528i q^{43} +(4.00000 - 1.41421i) q^{45} +(0.831254 - 1.14412i) q^{46} +(9.41498 + 3.05911i) q^{47} +(-1.65401 - 0.514040i) q^{48} +(8.89919 - 6.46564i) q^{49} +(-2.42705 + 1.76336i) q^{50} +(4.03499 + 1.31105i) q^{52} +(-4.15627 + 5.72061i) q^{53} +(5.00000 + 1.41421i) q^{54} -4.24264i q^{56} +(-1.85410 + 5.70634i) q^{58} +(10.7600 - 3.49613i) q^{59} +(2.44929 - 0.0314477i) q^{60} +(2.49376 + 3.43237i) q^{61} +(-1.23607 - 3.80423i) q^{62} +(0.326787 + 12.7237i) q^{63} +(-0.809017 - 0.587785i) q^{64} -6.00000 q^{65} -4.00000 q^{67} +(-1.96303 + 1.46510i) q^{69} +(1.85410 + 5.70634i) q^{70} +(4.15627 + 5.72061i) q^{71} +(2.47152 + 1.70046i) q^{72} +(0.618034 - 1.90211i) q^{74} +(4.92081 - 1.66901i) q^{75} +(-6.00000 - 4.24264i) q^{78} +(-2.49376 + 3.43237i) q^{79} +(1.34500 + 0.437016i) q^{80} +(-7.54311 - 4.90933i) q^{81} +(-4.85410 + 3.52671i) q^{82} +(-9.70820 + 7.05342i) q^{83} +(-2.18089 + 7.01739i) q^{84} +(4.98752 - 6.86474i) q^{86} +(6.00000 - 8.48528i) q^{87} +5.65685i q^{89} +(-4.06732 - 1.20702i) q^{90} +(5.56231 - 17.1190i) q^{91} +(-1.34500 + 0.437016i) q^{92} +(0.0889475 + 6.92763i) q^{93} +(-5.81878 - 8.00886i) q^{94} +(1.03598 + 1.38807i) q^{96} +(-6.47214 - 4.70228i) 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 \(\left(\mathbb{Z}/726\mathbb{Z}\right)^\times\).

\(n\) \(485\) \(607\)
\(\chi(n)\) \(-1\) \(e\left(\frac{7}{10}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.809017 − 0.587785i −0.572061 − 0.415627i
\(3\) 1.03598 + 1.38807i 0.598123 + 0.801404i
\(4\) 0.309017 + 0.951057i 0.154508 + 0.475528i
\(5\) −0.831254 − 1.14412i −0.371748 − 0.511667i 0.581627 − 0.813456i \(-0.302416\pi\)
−0.953375 + 0.301788i \(0.902416\pi\)
\(6\) −0.0222369 − 1.73191i −0.00907817 − 0.707049i
\(7\) 4.03499 − 1.31105i 1.52508 − 0.495530i 0.577869 − 0.816130i \(-0.303885\pi\)
0.947215 + 0.320600i \(0.103885\pi\)
\(8\) 0.309017 − 0.951057i 0.109254 − 0.336249i
\(9\) −0.853491 + 2.87603i −0.284497 + 0.958677i
\(10\) 1.41421i 0.447214i
\(11\) 0 0
\(12\) −1.00000 + 1.41421i −0.288675 + 0.408248i
\(13\) 2.49376 − 3.43237i 0.691645 − 0.951968i −0.308355 − 0.951271i \(-0.599778\pi\)
1.00000 0.000696272i \(-0.000221630\pi\)
\(14\) −4.03499 − 1.31105i −1.07840 − 0.350392i
\(15\) 0.726963 − 2.33913i 0.187701 − 0.603961i
\(16\) −0.809017 + 0.587785i −0.202254 + 0.146946i
\(17\) 0 0 −0.587785 − 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(18\) 2.38098 − 1.82509i 0.561202 − 0.430178i
\(19\) 0 0 0.309017 − 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(20\) 0.831254 − 1.14412i 0.185874 − 0.255834i
\(21\) 6.00000 + 4.24264i 1.30931 + 0.925820i
\(22\) 0 0
\(23\) 1.41421i 0.294884i 0.989071 + 0.147442i \(0.0471040\pi\)
−0.989071 + 0.147442i \(0.952896\pi\)
\(24\) 1.64027 − 0.556338i 0.334819 − 0.113562i
\(25\) 0.927051 − 2.85317i 0.185410 − 0.570634i
\(26\) −4.03499 + 1.31105i −0.791327 + 0.257118i
\(27\) −4.87634 + 1.79480i −0.938452 + 0.345410i
\(28\) 2.49376 + 3.43237i 0.471277 + 0.648657i
\(29\) −1.85410 − 5.70634i −0.344298 − 1.05964i −0.961958 − 0.273196i \(-0.911919\pi\)
0.617660 − 0.786445i \(-0.288081\pi\)
\(30\) −1.96303 + 1.46510i −0.358399 + 0.267489i
\(31\) 3.23607 + 2.35114i 0.581215 + 0.422277i 0.839162 − 0.543882i \(-0.183046\pi\)
−0.257947 + 0.966159i \(0.583046\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) −4.85410 − 3.52671i −0.820493 − 0.596123i
\(36\) −2.99901 + 0.0770245i −0.499835 + 0.0128374i
\(37\) 0.618034 + 1.90211i 0.101604 + 0.312705i 0.988918 − 0.148460i \(-0.0474315\pi\)
−0.887314 + 0.461165i \(0.847432\pi\)
\(38\) 0 0
\(39\) 7.34786 − 0.0943431i 1.17660 − 0.0151070i
\(40\) −1.34500 + 0.437016i −0.212663 + 0.0690983i
\(41\) 1.85410 − 5.70634i 0.289562 − 0.891180i −0.695432 − 0.718592i \(-0.744787\pi\)
0.984994 − 0.172588i \(-0.0552131\pi\)
\(42\) −2.36034 − 6.95908i −0.364208 − 1.07381i
\(43\) 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
\(46\) 0.831254 − 1.14412i 0.122562 − 0.168692i
\(47\) 9.41498 + 3.05911i 1.37332 + 0.446217i 0.900466 − 0.434926i \(-0.143225\pi\)
0.472850 + 0.881143i \(0.343225\pi\)
\(48\) −1.65401 − 0.514040i −0.238736 − 0.0741954i
\(49\) 8.89919 − 6.46564i 1.27131 − 0.923663i
\(50\) −2.42705 + 1.76336i −0.343237 + 0.249376i
\(51\) 0 0
\(52\) 4.03499 + 1.31105i 0.559553 + 0.181810i
\(53\) −4.15627 + 5.72061i −0.570908 + 0.785787i −0.992662 − 0.120923i \(-0.961414\pi\)
0.421754 + 0.906710i \(0.361414\pi\)
\(54\) 5.00000 + 1.41421i 0.680414 + 0.192450i
\(55\) 0 0
\(56\) − 4.24264i − 0.566947i
\(57\) 0 0
\(58\) −1.85410 + 5.70634i −0.243456 + 0.749279i
\(59\) 10.7600 − 3.49613i 1.40083 − 0.455157i 0.491371 − 0.870951i \(-0.336496\pi\)
0.909459 + 0.415794i \(0.136496\pi\)
\(60\) 2.44929 − 0.0314477i 0.316202 − 0.00405988i
\(61\) 2.49376 + 3.43237i 0.319293 + 0.439470i 0.938251 − 0.345954i \(-0.112445\pi\)
−0.618958 + 0.785424i \(0.712445\pi\)
\(62\) −1.23607 − 3.80423i −0.156981 − 0.483137i
\(63\) 0.326787 + 12.7237i 0.0411713 + 1.60304i
\(64\) −0.809017 − 0.587785i −0.101127 − 0.0734732i
\(65\) −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
\(69\) −1.96303 + 1.46510i −0.236321 + 0.176377i
\(70\) 1.85410 + 5.70634i 0.221608 + 0.682038i
\(71\) 4.15627 + 5.72061i 0.493258 + 0.678912i 0.980985 − 0.194085i \(-0.0621736\pi\)
−0.487726 + 0.872997i \(0.662174\pi\)
\(72\) 2.47152 + 1.70046i 0.291272 + 0.200401i
\(73\) 0 0 −0.309017 − 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(74\) 0.618034 − 1.90211i 0.0718450 − 0.221116i
\(75\) 4.92081 − 1.66901i 0.568206 − 0.192721i
\(76\) 0 0
\(77\) 0 0
\(78\) −6.00000 − 4.24264i −0.679366 − 0.480384i
\(79\) −2.49376 + 3.43237i −0.280570 + 0.386172i −0.925923 − 0.377713i \(-0.876710\pi\)
0.645353 + 0.763885i \(0.276710\pi\)
\(80\) 1.34500 + 0.437016i 0.150375 + 0.0488599i
\(81\) −7.54311 − 4.90933i −0.838123 − 0.545481i
\(82\) −4.85410 + 3.52671i −0.536046 + 0.389460i
\(83\) −9.70820 + 7.05342i −1.06561 + 0.774214i −0.975119 − 0.221683i \(-0.928845\pi\)
−0.0904951 + 0.995897i \(0.528845\pi\)
\(84\) −2.18089 + 7.01739i −0.237955 + 0.765660i
\(85\) 0 0
\(86\) 4.98752 − 6.86474i 0.537818 − 0.740244i
\(87\) 6.00000 − 8.48528i 0.643268 − 0.909718i
\(88\) 0 0
\(89\) 5.65685i 0.599625i 0.953998 + 0.299813i \(0.0969242\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) −4.06732 − 1.20702i −0.428733 − 0.127231i
\(91\) 5.56231 − 17.1190i 0.583088 − 1.79456i
\(92\) −1.34500 + 0.437016i −0.140226 + 0.0455621i
\(93\) 0.0889475 + 6.92763i 0.00922343 + 0.718362i
\(94\) −5.81878 − 8.00886i −0.600161 − 0.826051i
\(95\) 0 0
\(96\) 1.03598 + 1.38807i 0.105734 + 0.141670i
\(97\) −6.47214 − 4.70228i −0.657146 − 0.477444i 0.208552 − 0.978011i \(-0.433125\pi\)
−0.865698 + 0.500567i \(0.833125\pi\)
\(98\) −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.e.161.2 8
3.2 odd 2 726.2.h.i.161.1 8
11.2 odd 10 726.2.h.i.215.2 8
11.3 even 5 inner 726.2.h.e.239.1 8
11.4 even 5 inner 726.2.h.e.233.1 8
11.5 even 5 66.2.b.b.65.2 yes 2
11.6 odd 10 66.2.b.a.65.2 yes 2
11.7 odd 10 726.2.h.i.233.1 8
11.8 odd 10 726.2.h.i.239.1 8
11.9 even 5 inner 726.2.h.e.215.2 8
11.10 odd 2 726.2.h.i.161.2 8
33.2 even 10 inner 726.2.h.e.215.1 8
33.5 odd 10 66.2.b.a.65.1 ✓ 2
33.8 even 10 inner 726.2.h.e.239.2 8
33.14 odd 10 726.2.h.i.239.2 8
33.17 even 10 66.2.b.b.65.1 yes 2
33.20 odd 10 726.2.h.i.215.1 8
33.26 odd 10 726.2.h.i.233.2 8
33.29 even 10 inner 726.2.h.e.233.2 8
33.32 even 2 inner 726.2.h.e.161.1 8
44.27 odd 10 528.2.b.c.65.1 2
44.39 even 10 528.2.b.b.65.1 2
55.17 even 20 1650.2.f.b.1649.2 4
55.27 odd 20 1650.2.f.a.1649.4 4
55.28 even 20 1650.2.f.b.1649.3 4
55.38 odd 20 1650.2.f.a.1649.1 4
55.39 odd 10 1650.2.d.b.1451.1 2
55.49 even 10 1650.2.d.a.1451.1 2
88.5 even 10 2112.2.b.i.65.1 2
88.27 odd 10 2112.2.b.b.65.2 2
88.61 odd 10 2112.2.b.g.65.1 2
88.83 even 10 2112.2.b.d.65.2 2
99.5 odd 30 1782.2.i.f.593.1 4
99.16 even 15 1782.2.i.c.1187.1 4
99.38 odd 30 1782.2.i.f.1187.2 4
99.49 even 15 1782.2.i.c.593.2 4
99.50 even 30 1782.2.i.c.593.1 4
99.61 odd 30 1782.2.i.f.1187.1 4
99.83 even 30 1782.2.i.c.1187.2 4
99.94 odd 30 1782.2.i.f.593.2 4
132.71 even 10 528.2.b.b.65.2 2
132.83 odd 10 528.2.b.c.65.2 2
165.17 odd 20 1650.2.f.a.1649.3 4
165.38 even 20 1650.2.f.b.1649.4 4
165.83 odd 20 1650.2.f.a.1649.2 4
165.104 odd 10 1650.2.d.b.1451.2 2
165.137 even 20 1650.2.f.b.1649.1 4
165.149 even 10 1650.2.d.a.1451.2 2
264.5 odd 10 2112.2.b.g.65.2 2
264.83 odd 10 2112.2.b.b.65.1 2
264.149 even 10 2112.2.b.i.65.2 2
264.203 even 10 2112.2.b.d.65.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
66.2.b.a.65.1 ✓ 2 33.5 odd 10
66.2.b.a.65.2 yes 2 11.6 odd 10
66.2.b.b.65.1 yes 2 33.17 even 10
66.2.b.b.65.2 yes 2 11.5 even 5
528.2.b.b.65.1 2 44.39 even 10
528.2.b.b.65.2 2 132.71 even 10
528.2.b.c.65.1 2 44.27 odd 10
528.2.b.c.65.2 2 132.83 odd 10
726.2.h.e.161.1 8 33.32 even 2 inner
726.2.h.e.161.2 8 1.1 even 1 trivial
726.2.h.e.215.1 8 33.2 even 10 inner
726.2.h.e.215.2 8 11.9 even 5 inner
726.2.h.e.233.1 8 11.4 even 5 inner
726.2.h.e.233.2 8 33.29 even 10 inner
726.2.h.e.239.1 8 11.3 even 5 inner
726.2.h.e.239.2 8 33.8 even 10 inner
726.2.h.i.161.1 8 3.2 odd 2
726.2.h.i.161.2 8 11.10 odd 2
726.2.h.i.215.1 8 33.20 odd 10
726.2.h.i.215.2 8 11.2 odd 10
726.2.h.i.233.1 8 11.7 odd 10
726.2.h.i.233.2 8 33.26 odd 10
726.2.h.i.239.1 8 11.8 odd 10
726.2.h.i.239.2 8 33.14 odd 10
1650.2.d.a.1451.1 2 55.49 even 10
1650.2.d.a.1451.2 2 165.149 even 10
1650.2.d.b.1451.1 2 55.39 odd 10
1650.2.d.b.1451.2 2 165.104 odd 10
1650.2.f.a.1649.1 4 55.38 odd 20
1650.2.f.a.1649.2 4 165.83 odd 20
1650.2.f.a.1649.3 4 165.17 odd 20
1650.2.f.a.1649.4 4 55.27 odd 20
1650.2.f.b.1649.1 4 165.137 even 20
1650.2.f.b.1649.2 4 55.17 even 20
1650.2.f.b.1649.3 4 55.28 even 20
1650.2.f.b.1649.4 4 165.38 even 20
1782.2.i.c.593.1 4 99.50 even 30
1782.2.i.c.593.2 4 99.49 even 15
1782.2.i.c.1187.1 4 99.16 even 15
1782.2.i.c.1187.2 4 99.83 even 30
1782.2.i.f.593.1 4 99.5 odd 30
1782.2.i.f.593.2 4 99.94 odd 30
1782.2.i.f.1187.1 4 99.61 odd 30
1782.2.i.f.1187.2 4 99.38 odd 30
2112.2.b.b.65.1 2 264.83 odd 10
2112.2.b.b.65.2 2 88.27 odd 10
2112.2.b.d.65.1 2 264.203 even 10
2112.2.b.d.65.2 2 88.83 even 10
2112.2.b.g.65.1 2 88.61 odd 10
2112.2.b.g.65.2 2 264.5 odd 10
2112.2.b.i.65.1 2 88.5 even 10
2112.2.b.i.65.2 2 264.149 even 10