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Properties

Label 605.2.g.b.366.1
Level $605$
Weight $2$
Character 605.366
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 366.1
Root \(0.809017 + 0.587785i\) of defining polynomial
Character \(\chi\) \(=\) 605.366
Dual form 605.2.g.b.81.1

$q$-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 \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(486\)
\(\chi(n)\) \(1\) \(e\left(\frac{4}{5}\right)\)

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 \(-0.815027\pi\)
0.263792 + 0.964580i \(0.415027\pi\)
\(3\) −0.927051 − 2.85317i −0.535233 − 1.64728i −0.743145 − 0.669131i \(-0.766667\pi\)
0.207912 − 0.978148i \(-0.433333\pi\)
\(4\) −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 −0.608241 − 0.793752i \(-0.708125\pi\)
0.958633 − 0.284644i \(-0.0918755\pi\)
\(8\) −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 \(-0.512833\pi\)
0.937829 + 0.347098i \(0.112833\pi\)
\(14\) 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 \(-0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(18\) 1.85410 − 5.70634i 0.437016 − 1.34500i
\(19\) −1.23607 − 3.80423i −0.283573 − 0.872749i −0.986823 − 0.161806i \(-0.948268\pi\)
0.703249 − 0.710943i \(-0.251732\pi\)
\(20\) 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 0.617660 + 0.786445i \(0.288081\pi\)
−0.961958 + 0.273196i \(0.911919\pi\)
\(30\) −0.927051 − 2.85317i −0.169256 − 0.520915i
\(31\) 1.61803 − 1.17557i 0.290607 − 0.211139i −0.432923 − 0.901431i \(-0.642518\pi\)
0.723531 + 0.690292i \(0.242518\pi\)
\(32\) 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 0.513290 + 0.858215i \(0.328427\pi\)
−0.919707 + 0.392607i \(0.871573\pi\)
\(38\) 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 0.996223 + 0.0868346i \(0.0276752\pi\)
−0.754921 + 0.655816i \(0.772325\pi\)
\(42\) 7.28115 − 5.29007i 1.12351 − 0.816275i
\(43\) −5.00000 −0.762493 −0.381246 − 0.924473i \(-0.624505\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 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 0.860401 − 0.509618i \(-0.170213\pi\)
−0.995625 + 0.0934408i \(0.970213\pi\)
\(48\) 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 −0.787424 − 0.616412i \(-0.788586\pi\)
0.342916 + 0.939366i \(0.388586\pi\)
\(54\) −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 −0.983193 − 0.182570i \(-0.941558\pi\)
0.902732 + 0.430204i \(0.141558\pi\)
\(60\) −2.42705 − 1.76336i −0.313331 − 0.227648i
\(61\) −8.89919 − 6.46564i −1.13942 − 0.827840i −0.152385 − 0.988321i \(-0.548695\pi\)
−0.987039 + 0.160481i \(0.948695\pi\)
\(62\) −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 0.487619 − 0.873057i \(-0.337866\pi\)
−0.679644 + 0.733542i \(0.737866\pi\)
\(72\) 14.5623 + 10.5801i 1.71618 + 1.24688i
\(73\) 2.47214 − 7.60845i 0.289342 − 0.890502i −0.695722 − 0.718311i \(-0.744915\pi\)
0.985064 − 0.172191i \(-0.0550846\pi\)
\(74\) −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 \(-0.509823\pi\)
0.941069 + 0.338214i \(0.109823\pi\)
\(80\) −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 \(-0.229548\pi\)
−0.395845 + 0.918318i \(0.629548\pi\)
\(84\) 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 −0.208552 − 0.978011i \(-0.566875\pi\)
0.865698 + 0.500567i \(0.166875\pi\)
\(98\) 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.b.366.1 4
11.2 odd 10 605.2.a.a.1.1 ✓ 1
11.3 even 5 inner 605.2.g.b.511.1 4
11.4 even 5 inner 605.2.g.b.81.1 4
11.5 even 5 inner 605.2.g.b.251.1 4
11.6 odd 10 605.2.g.d.251.1 4
11.7 odd 10 605.2.g.d.81.1 4
11.8 odd 10 605.2.g.d.511.1 4
11.9 even 5 605.2.a.c.1.1 yes 1
11.10 odd 2 605.2.g.d.366.1 4
33.2 even 10 5445.2.a.h.1.1 1
33.20 odd 10 5445.2.a.d.1.1 1
44.31 odd 10 9680.2.a.be.1.1 1
44.35 even 10 9680.2.a.bf.1.1 1
55.9 even 10 3025.2.a.c.1.1 1
55.24 odd 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 11.2 odd 10
605.2.a.c.1.1 yes 1 11.9 even 5
605.2.g.b.81.1 4 11.4 even 5 inner
605.2.g.b.251.1 4 11.5 even 5 inner
605.2.g.b.366.1 4 1.1 even 1 trivial
605.2.g.b.511.1 4 11.3 even 5 inner
605.2.g.d.81.1 4 11.7 odd 10
605.2.g.d.251.1 4 11.6 odd 10
605.2.g.d.366.1 4 11.10 odd 2
605.2.g.d.511.1 4 11.8 odd 10
3025.2.a.c.1.1 1 55.9 even 10
3025.2.a.g.1.1 1 55.24 odd 10
5445.2.a.d.1.1 1 33.20 odd 10
5445.2.a.h.1.1 1 33.2 even 10
9680.2.a.be.1.1 1 44.31 odd 10
9680.2.a.bf.1.1 1 44.35 even 10