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Properties

Label 605.2.g.d.81.1
Level $605$
Weight $2$
Character 605.81
Analytic conductor $4.831$
Analytic rank $0$
Dimension $4$
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] = [605,2,Mod(81,605)] 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("605.81"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(605, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.g (of order \(5\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,1,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.83094932229\)
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 81.1
Root \(0.809017 - 0.587785i\) of defining polynomial
Character \(\chi\) \(=\) 605.81
Dual form 605.2.g.d.366.1

$q$-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} +(-0.927051 + 2.85317i) q^{3} +(-0.309017 - 0.951057i) q^{4} +(-0.809017 + 0.587785i) q^{5} +(-2.42705 + 1.76336i) q^{6} +(-0.927051 - 2.85317i) q^{7} +(0.927051 - 2.85317i) q^{8} +(-4.85410 - 3.52671i) q^{9} -1.00000 q^{10} +3.00000 q^{12} +(-3.23607 - 2.35114i) q^{13} +(0.927051 - 2.85317i) q^{14} +(-0.927051 - 2.85317i) q^{15} +(0.809017 - 0.587785i) q^{16} +(-1.85410 - 5.70634i) q^{18} +(1.23607 - 3.80423i) q^{19} +(0.809017 + 0.587785i) q^{20} +9.00000 q^{21} -8.00000 q^{23} +(7.28115 + 5.29007i) q^{24} +(0.309017 - 0.951057i) q^{25} +(-1.23607 - 3.80423i) q^{26} +(7.28115 - 5.29007i) q^{27} +(-2.42705 + 1.76336i) q^{28} +(1.85410 + 5.70634i) q^{29} +(0.927051 - 2.85317i) q^{30} +(1.61803 + 1.17557i) q^{31} -5.00000 q^{32} +(2.42705 + 1.76336i) q^{35} +(-1.85410 + 5.70634i) q^{36} +(-2.47214 - 7.60845i) q^{37} +(3.23607 - 2.35114i) q^{38} +(9.70820 - 7.05342i) q^{39} +(0.927051 + 2.85317i) q^{40} +(-1.54508 + 4.75528i) q^{41} +(7.28115 + 5.29007i) q^{42} +5.00000 q^{43} +6.00000 q^{45} +(-6.47214 - 4.70228i) q^{46} +(-0.927051 + 2.85317i) q^{47} +(0.927051 + 2.85317i) q^{48} +(-1.61803 + 1.17557i) q^{49} +(0.809017 - 0.587785i) q^{50} +(-1.23607 + 3.80423i) q^{52} +(-3.23607 - 2.35114i) q^{53} +9.00000 q^{54} -9.00000 q^{56} +(9.70820 + 7.05342i) q^{57} +(-1.85410 + 5.70634i) q^{58} +(-0.618034 - 1.90211i) q^{59} +(-2.42705 + 1.76336i) q^{60} +(8.89919 - 6.46564i) q^{61} +(0.618034 + 1.90211i) q^{62} +(-5.56231 + 17.1190i) q^{63} +(-5.66312 - 4.11450i) q^{64} +4.00000 q^{65} -13.0000 q^{67} +(7.41641 - 22.8254i) q^{69} +(0.927051 + 2.85317i) q^{70} +(-1.61803 + 1.17557i) q^{71} +(-14.5623 + 10.5801i) q^{72} +(-2.47214 - 7.60845i) q^{73} +(2.47214 - 7.60845i) q^{74} +(2.42705 + 1.76336i) q^{75} -4.00000 q^{76} +12.0000 q^{78} +(-8.09017 - 5.87785i) q^{79} +(-0.309017 + 0.951057i) q^{80} +(2.78115 + 8.55951i) q^{81} +(-4.04508 + 2.93893i) q^{82} +(-3.23607 + 2.35114i) q^{83} +(-2.78115 - 8.55951i) q^{84} +(4.04508 + 2.93893i) q^{86} -18.0000 q^{87} +1.00000 q^{89} +(4.85410 + 3.52671i) q^{90} +(-3.70820 + 11.4127i) q^{91} +(2.47214 + 7.60845i) q^{92} +(-4.85410 + 3.52671i) q^{93} +(-2.42705 + 1.76336i) q^{94} +(1.23607 + 3.80423i) q^{95} +(4.63525 - 14.2658i) q^{96} +(6.47214 + 4.70228i) q^{97} -2.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + 3 q^{3} + q^{4} - q^{5} - 3 q^{6} + 3 q^{7} - 3 q^{8} - 6 q^{9} - 4 q^{10} + 12 q^{12} - 4 q^{13} - 3 q^{14} + 3 q^{15} + q^{16} + 6 q^{18} - 4 q^{19} + q^{20} + 36 q^{21} - 32 q^{23}+ \cdots - 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(486\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{5}\right)\)

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 0.835853 − 0.548953i \(-0.184973\pi\)
−0.263792 + 0.964580i \(0.584973\pi\)
\(3\) −0.927051 + 2.85317i −0.535233 + 1.64728i 0.207912 + 0.978148i \(0.433333\pi\)
−0.743145 + 0.669131i \(0.766667\pi\)
\(4\) −0.309017 − 0.951057i −0.154508 − 0.475528i
\(5\) −0.809017 + 0.587785i −0.361803 + 0.262866i
\(6\) −2.42705 + 1.76336i −0.990839 + 0.719887i
\(7\) −0.927051 − 2.85317i −0.350392 − 1.07840i −0.958633 − 0.284644i \(-0.908125\pi\)
0.608241 − 0.793752i \(-0.291875\pi\)
\(8\) 0.927051 − 2.85317i 0.327762 − 1.00875i
\(9\) −4.85410 − 3.52671i −1.61803 − 1.17557i
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 3.00000 0.866025
\(13\) −3.23607 − 2.35114i −0.897524 − 0.652089i 0.0403050 − 0.999187i \(-0.487167\pi\)
−0.937829 + 0.347098i \(0.887167\pi\)
\(14\) 0.927051 − 2.85317i 0.247765 − 0.762542i
\(15\) −0.927051 − 2.85317i −0.239364 − 0.736685i
\(16\) 0.809017 − 0.587785i 0.202254 − 0.146946i
\(17\) 0 0 −0.587785 − 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(18\) −1.85410 − 5.70634i −0.437016 − 1.34500i
\(19\) 1.23607 − 3.80423i 0.283573 − 0.872749i −0.703249 − 0.710943i \(-0.748268\pi\)
0.986823 − 0.161806i \(-0.0517318\pi\)
\(20\) 0.809017 + 0.587785i 0.180902 + 0.131433i
\(21\) 9.00000 1.96396
\(22\) 0 0
\(23\) −8.00000 −1.66812 −0.834058 − 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 7.28115 + 5.29007i 1.48626 + 1.07983i
\(25\) 0.309017 − 0.951057i 0.0618034 − 0.190211i
\(26\) −1.23607 − 3.80423i −0.242413 − 0.746070i
\(27\) 7.28115 − 5.29007i 1.40126 − 1.01807i
\(28\) −2.42705 + 1.76336i −0.458670 + 0.333243i
\(29\) 1.85410 + 5.70634i 0.344298 + 1.05964i 0.961958 + 0.273196i \(0.0880806\pi\)
−0.617660 + 0.786445i \(0.711919\pi\)
\(30\) 0.927051 − 2.85317i 0.169256 − 0.520915i
\(31\) 1.61803 + 1.17557i 0.290607 + 0.211139i 0.723531 − 0.690292i \(-0.242518\pi\)
−0.432923 + 0.901431i \(0.642518\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) 0 0
\(35\) 2.42705 + 1.76336i 0.410246 + 0.298062i
\(36\) −1.85410 + 5.70634i −0.309017 + 0.951057i
\(37\) −2.47214 − 7.60845i −0.406417 − 1.25082i −0.919707 − 0.392607i \(-0.871573\pi\)
0.513290 − 0.858215i \(-0.328427\pi\)
\(38\) 3.23607 − 2.35114i 0.524960 − 0.381405i
\(39\) 9.70820 − 7.05342i 1.55456 − 1.12945i
\(40\) 0.927051 + 2.85317i 0.146580 + 0.451126i
\(41\) −1.54508 + 4.75528i −0.241302 + 0.742650i 0.754921 + 0.655816i \(0.227675\pi\)
−0.996223 + 0.0868346i \(0.972325\pi\)
\(42\) 7.28115 + 5.29007i 1.12351 + 0.816275i
\(43\) 5.00000 0.762493 0.381246 − 0.924473i \(-0.375495\pi\)
0.381246 + 0.924473i \(0.375495\pi\)
\(44\) 0 0
\(45\) 6.00000 0.894427
\(46\) −6.47214 − 4.70228i −0.954264 − 0.693314i
\(47\) −0.927051 + 2.85317i −0.135224 + 0.416178i −0.995625 − 0.0934408i \(-0.970213\pi\)
0.860401 + 0.509618i \(0.170213\pi\)
\(48\) 0.927051 + 2.85317i 0.133808 + 0.411820i
\(49\) −1.61803 + 1.17557i −0.231148 + 0.167939i
\(50\) 0.809017 − 0.587785i 0.114412 − 0.0831254i
\(51\) 0 0
\(52\) −1.23607 + 3.80423i −0.171412 + 0.527551i
\(53\) −3.23607 − 2.35114i −0.444508 − 0.322954i 0.342916 − 0.939366i \(-0.388586\pi\)
−0.787424 + 0.616412i \(0.788586\pi\)
\(54\) 9.00000 1.22474
\(55\) 0 0
\(56\) −9.00000 −1.20268
\(57\) 9.70820 + 7.05342i 1.28588 + 0.934249i
\(58\) −1.85410 + 5.70634i −0.243456 + 0.749279i
\(59\) −0.618034 − 1.90211i −0.0804612 − 0.247634i 0.902732 − 0.430204i \(-0.141558\pi\)
−0.983193 + 0.182570i \(0.941558\pi\)
\(60\) −2.42705 + 1.76336i −0.313331 + 0.227648i
\(61\) 8.89919 − 6.46564i 1.13942 − 0.827840i 0.152385 − 0.988321i \(-0.451305\pi\)
0.987039 + 0.160481i \(0.0513046\pi\)
\(62\) 0.618034 + 1.90211i 0.0784904 + 0.241569i
\(63\) −5.56231 + 17.1190i −0.700785 + 2.15679i
\(64\) −5.66312 − 4.11450i −0.707890 − 0.514312i
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) −13.0000 −1.58820 −0.794101 − 0.607785i \(-0.792058\pi\)
−0.794101 + 0.607785i \(0.792058\pi\)
\(68\) 0 0
\(69\) 7.41641 − 22.8254i 0.892831 − 2.74785i
\(70\) 0.927051 + 2.85317i 0.110804 + 0.341019i
\(71\) −1.61803 + 1.17557i −0.192025 + 0.139515i −0.679644 − 0.733542i \(-0.737866\pi\)
0.487619 + 0.873057i \(0.337866\pi\)
\(72\) −14.5623 + 10.5801i −1.71618 + 1.24688i
\(73\) −2.47214 − 7.60845i −0.289342 − 0.890502i −0.985064 − 0.172191i \(-0.944915\pi\)
0.695722 − 0.718311i \(-0.255085\pi\)
\(74\) 2.47214 − 7.60845i 0.287380 − 0.884465i
\(75\) 2.42705 + 1.76336i 0.280252 + 0.203615i
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) 12.0000 1.35873
\(79\) −8.09017 − 5.87785i −0.910215 − 0.661310i 0.0308541 − 0.999524i \(-0.490177\pi\)
−0.941069 + 0.338214i \(0.890177\pi\)
\(80\) −0.309017 + 0.951057i −0.0345492 + 0.106331i
\(81\) 2.78115 + 8.55951i 0.309017 + 0.951057i
\(82\) −4.04508 + 2.93893i −0.446705 + 0.324550i
\(83\) −3.23607 + 2.35114i −0.355205 + 0.258071i −0.751049 − 0.660246i \(-0.770452\pi\)
0.395845 + 0.918318i \(0.370452\pi\)
\(84\) −2.78115 − 8.55951i −0.303449 − 0.933919i
\(85\) 0 0
\(86\) 4.04508 + 2.93893i 0.436193 + 0.316913i
\(87\) −18.0000 −1.92980
\(88\) 0 0
\(89\) 1.00000 0.106000 0.0529999 − 0.998595i \(-0.483122\pi\)
0.0529999 + 0.998595i \(0.483122\pi\)
\(90\) 4.85410 + 3.52671i 0.511667 + 0.371748i
\(91\) −3.70820 + 11.4127i −0.388725 + 1.19637i
\(92\) 2.47214 + 7.60845i 0.257738 + 0.793236i
\(93\) −4.85410 + 3.52671i −0.503347 + 0.365703i
\(94\) −2.42705 + 1.76336i −0.250331 + 0.181876i
\(95\) 1.23607 + 3.80423i 0.126818 + 0.390305i
\(96\) 4.63525 − 14.2658i 0.473084 − 1.45600i
\(97\) 6.47214 + 4.70228i 0.657146 + 0.477444i 0.865698 − 0.500567i \(-0.166875\pi\)
−0.208552 + 0.978011i \(0.566875\pi\)
\(98\) −2.00000 −0.202031
\(99\) 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 605.2.g.d.81.1 4
11.2 odd 10 605.2.g.b.511.1 4
11.3 even 5 inner 605.2.g.d.366.1 4
11.4 even 5 inner 605.2.g.d.251.1 4
11.5 even 5 605.2.a.a.1.1 ✓ 1
11.6 odd 10 605.2.a.c.1.1 yes 1
11.7 odd 10 605.2.g.b.251.1 4
11.8 odd 10 605.2.g.b.366.1 4
11.9 even 5 inner 605.2.g.d.511.1 4
11.10 odd 2 605.2.g.b.81.1 4
33.5 odd 10 5445.2.a.h.1.1 1
33.17 even 10 5445.2.a.d.1.1 1
44.27 odd 10 9680.2.a.bf.1.1 1
44.39 even 10 9680.2.a.be.1.1 1
55.39 odd 10 3025.2.a.c.1.1 1
55.49 even 10 3025.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
605.2.a.a.1.1 ✓ 1 11.5 even 5
605.2.a.c.1.1 yes 1 11.6 odd 10
605.2.g.b.81.1 4 11.10 odd 2
605.2.g.b.251.1 4 11.7 odd 10
605.2.g.b.366.1 4 11.8 odd 10
605.2.g.b.511.1 4 11.2 odd 10
605.2.g.d.81.1 4 1.1 even 1 trivial
605.2.g.d.251.1 4 11.4 even 5 inner
605.2.g.d.366.1 4 11.3 even 5 inner
605.2.g.d.511.1 4 11.9 even 5 inner
3025.2.a.c.1.1 1 55.39 odd 10
3025.2.a.g.1.1 1 55.49 even 10
5445.2.a.d.1.1 1 33.17 even 10
5445.2.a.h.1.1 1 33.5 odd 10
9680.2.a.be.1.1 1 44.39 even 10
9680.2.a.bf.1.1 1 44.27 odd 10