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Properties

Label 605.2.g.b.251.1
Level $605$
Weight $2$
Character 605.251
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 251.1
Root \(-0.309017 - 0.951057i\) of defining polynomial
Character \(\chi\) \(=\) 605.251
Dual form 605.2.g.b.511.1

$q$-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} +(2.42705 - 1.76336i) q^{3} +(0.809017 + 0.587785i) q^{4} +(0.309017 + 0.951057i) q^{5} +(-0.927051 - 2.85317i) q^{6} +(-2.42705 - 1.76336i) q^{7} +(2.42705 - 1.76336i) q^{8} +(1.85410 - 5.70634i) q^{9} +1.00000 q^{10} +3.00000 q^{12} +(-1.23607 + 3.80423i) q^{13} +(-2.42705 + 1.76336i) q^{14} +(2.42705 + 1.76336i) q^{15} +(-0.309017 - 0.951057i) q^{16} +(-4.85410 - 3.52671i) q^{18} +(3.23607 - 2.35114i) q^{19} +(-0.309017 + 0.951057i) q^{20} -9.00000 q^{21} -8.00000 q^{23} +(2.78115 - 8.55951i) q^{24} +(-0.809017 + 0.587785i) q^{25} +(3.23607 + 2.35114i) q^{26} +(-2.78115 - 8.55951i) q^{27} +(-0.927051 - 2.85317i) q^{28} +(4.85410 + 3.52671i) q^{29} +(2.42705 - 1.76336i) q^{30} +(-0.618034 + 1.90211i) q^{31} +5.00000 q^{32} +(0.927051 - 2.85317i) q^{35} +(4.85410 - 3.52671i) q^{36} +(6.47214 + 4.70228i) q^{37} +(-1.23607 - 3.80423i) q^{38} +(3.70820 + 11.4127i) q^{39} +(2.42705 + 1.76336i) q^{40} +(-4.04508 + 2.93893i) q^{41} +(-2.78115 + 8.55951i) q^{42} -5.00000 q^{43} +6.00000 q^{45} +(-2.47214 + 7.60845i) q^{46} +(2.42705 - 1.76336i) q^{47} +(-2.42705 - 1.76336i) q^{48} +(0.618034 + 1.90211i) q^{49} +(0.309017 + 0.951057i) q^{50} +(-3.23607 + 2.35114i) q^{52} +(1.23607 - 3.80423i) q^{53} -9.00000 q^{54} -9.00000 q^{56} +(3.70820 - 11.4127i) q^{57} +(4.85410 - 3.52671i) q^{58} +(1.61803 + 1.17557i) q^{59} +(0.927051 + 2.85317i) q^{60} +(3.39919 + 10.4616i) q^{61} +(1.61803 + 1.17557i) q^{62} +(-14.5623 + 10.5801i) q^{63} +(2.16312 - 6.65740i) q^{64} -4.00000 q^{65} -13.0000 q^{67} +(-19.4164 + 14.1068i) q^{69} +(-2.42705 - 1.76336i) q^{70} +(0.618034 + 1.90211i) q^{71} +(-5.56231 - 17.1190i) q^{72} +(-6.47214 - 4.70228i) q^{73} +(6.47214 - 4.70228i) q^{74} +(-0.927051 + 2.85317i) q^{75} +4.00000 q^{76} +12.0000 q^{78} +(-3.09017 + 9.51057i) q^{79} +(0.809017 - 0.587785i) q^{80} +(-7.28115 - 5.29007i) q^{81} +(1.54508 + 4.75528i) q^{82} +(-1.23607 - 3.80423i) q^{83} +(-7.28115 - 5.29007i) q^{84} +(-1.54508 + 4.75528i) q^{86} +18.0000 q^{87} +1.00000 q^{89} +(1.85410 - 5.70634i) q^{90} +(9.70820 - 7.05342i) q^{91} +(-6.47214 - 4.70228i) q^{92} +(1.85410 + 5.70634i) q^{93} +(-0.927051 - 2.85317i) q^{94} +(3.23607 + 2.35114i) q^{95} +(12.1353 - 8.81678i) q^{96} +(-2.47214 + 7.60845i) 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 \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(486\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.309017 − 0.951057i 0.218508 − 0.672499i −0.780378 − 0.625308i \(-0.784973\pi\)
0.998886 − 0.0471903i \(-0.0150267\pi\)
\(3\) 2.42705 − 1.76336i 1.40126 − 1.01807i 0.406737 − 0.913545i \(-0.366667\pi\)
0.994522 − 0.104528i \(-0.0333333\pi\)
\(4\) 0.809017 + 0.587785i 0.404508 + 0.293893i
\(5\) 0.309017 + 0.951057i 0.138197 + 0.425325i
\(6\) −0.927051 − 2.85317i −0.378467 − 1.16480i
\(7\) −2.42705 − 1.76336i −0.917339 − 0.666486i 0.0255212 − 0.999674i \(-0.491875\pi\)
−0.942860 + 0.333188i \(0.891875\pi\)
\(8\) 2.42705 − 1.76336i 0.858092 − 0.623440i
\(9\) 1.85410 − 5.70634i 0.618034 − 1.90211i
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) 3.00000 0.866025
\(13\) −1.23607 + 3.80423i −0.342824 + 1.05510i 0.619915 + 0.784669i \(0.287167\pi\)
−0.962739 + 0.270434i \(0.912833\pi\)
\(14\) −2.42705 + 1.76336i −0.648657 + 0.471277i
\(15\) 2.42705 + 1.76336i 0.626662 + 0.455296i
\(16\) −0.309017 − 0.951057i −0.0772542 − 0.237764i
\(17\) 0 0 0.951057 − 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(18\) −4.85410 − 3.52671i −1.14412 − 0.831254i
\(19\) 3.23607 − 2.35114i 0.742405 − 0.539389i −0.151058 − 0.988525i \(-0.548268\pi\)
0.893463 + 0.449136i \(0.148268\pi\)
\(20\) −0.309017 + 0.951057i −0.0690983 + 0.212663i
\(21\) −9.00000 −1.96396
\(22\) 0 0
\(23\) −8.00000 −1.66812 −0.834058 − 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 2.78115 − 8.55951i 0.567700 − 1.74720i
\(25\) −0.809017 + 0.587785i −0.161803 + 0.117557i
\(26\) 3.23607 + 2.35114i 0.634645 + 0.461097i
\(27\) −2.78115 − 8.55951i −0.535233 − 1.64728i
\(28\) −0.927051 − 2.85317i −0.175196 − 0.539198i
\(29\) 4.85410 + 3.52671i 0.901384 + 0.654894i 0.938821 − 0.344405i \(-0.111919\pi\)
−0.0374370 + 0.999299i \(0.511919\pi\)
\(30\) 2.42705 − 1.76336i 0.443117 − 0.321943i
\(31\) −0.618034 + 1.90211i −0.111002 + 0.341630i −0.991092 − 0.133177i \(-0.957482\pi\)
0.880090 + 0.474807i \(0.157482\pi\)
\(32\) 5.00000 0.883883
\(33\) 0 0
\(34\) 0 0
\(35\) 0.927051 − 2.85317i 0.156700 − 0.482274i
\(36\) 4.85410 − 3.52671i 0.809017 − 0.587785i
\(37\) 6.47214 + 4.70228i 1.06401 + 0.773050i 0.974827 − 0.222965i \(-0.0715734\pi\)
0.0891861 + 0.996015i \(0.471573\pi\)
\(38\) −1.23607 − 3.80423i −0.200517 − 0.617127i
\(39\) 3.70820 + 11.4127i 0.593788 + 1.82749i
\(40\) 2.42705 + 1.76336i 0.383750 + 0.278811i
\(41\) −4.04508 + 2.93893i −0.631736 + 0.458983i −0.857001 − 0.515314i \(-0.827675\pi\)
0.225265 + 0.974298i \(0.427675\pi\)
\(42\) −2.78115 + 8.55951i −0.429141 + 1.32076i
\(43\) −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
\(46\) −2.47214 + 7.60845i −0.364497 + 1.12181i
\(47\) 2.42705 − 1.76336i 0.354022 − 0.257212i −0.396533 − 0.918021i \(-0.629787\pi\)
0.750554 + 0.660809i \(0.229787\pi\)
\(48\) −2.42705 − 1.76336i −0.350315 − 0.254518i
\(49\) 0.618034 + 1.90211i 0.0882906 + 0.271730i
\(50\) 0.309017 + 0.951057i 0.0437016 + 0.134500i
\(51\) 0 0
\(52\) −3.23607 + 2.35114i −0.448762 + 0.326045i
\(53\) 1.23607 − 3.80423i 0.169787 − 0.522551i −0.829570 − 0.558403i \(-0.811414\pi\)
0.999357 + 0.0358519i \(0.0114145\pi\)
\(54\) −9.00000 −1.22474
\(55\) 0 0
\(56\) −9.00000 −1.20268
\(57\) 3.70820 − 11.4127i 0.491164 − 1.51165i
\(58\) 4.85410 − 3.52671i 0.637375 − 0.463080i
\(59\) 1.61803 + 1.17557i 0.210650 + 0.153046i 0.688108 − 0.725608i \(-0.258442\pi\)
−0.477458 + 0.878655i \(0.658442\pi\)
\(60\) 0.927051 + 2.85317i 0.119682 + 0.368343i
\(61\) 3.39919 + 10.4616i 0.435221 + 1.33947i 0.892860 + 0.450335i \(0.148695\pi\)
−0.457638 + 0.889138i \(0.651305\pi\)
\(62\) 1.61803 + 1.17557i 0.205491 + 0.149298i
\(63\) −14.5623 + 10.5801i −1.83468 + 1.33297i
\(64\) 2.16312 − 6.65740i 0.270390 − 0.832174i
\(65\) −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
\(69\) −19.4164 + 14.1068i −2.33746 + 1.69826i
\(70\) −2.42705 − 1.76336i −0.290088 − 0.210761i
\(71\) 0.618034 + 1.90211i 0.0733471 + 0.225739i 0.981009 − 0.193963i \(-0.0621343\pi\)
−0.907662 + 0.419703i \(0.862134\pi\)
\(72\) −5.56231 − 17.1190i −0.655524 − 2.01750i
\(73\) −6.47214 − 4.70228i −0.757506 − 0.550360i 0.140638 − 0.990061i \(-0.455085\pi\)
−0.898144 + 0.439701i \(0.855085\pi\)
\(74\) 6.47214 − 4.70228i 0.752371 − 0.546629i
\(75\) −0.927051 + 2.85317i −0.107047 + 0.329456i
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 12.0000 1.35873
\(79\) −3.09017 + 9.51057i −0.347671 + 1.07002i 0.612467 + 0.790496i \(0.290177\pi\)
−0.960138 + 0.279526i \(0.909823\pi\)
\(80\) 0.809017 − 0.587785i 0.0904508 − 0.0657164i
\(81\) −7.28115 − 5.29007i −0.809017 − 0.587785i
\(82\) 1.54508 + 4.75528i 0.170626 + 0.525133i
\(83\) −1.23607 − 3.80423i −0.135676 − 0.417568i 0.860018 − 0.510263i \(-0.170452\pi\)
−0.995695 + 0.0926948i \(0.970452\pi\)
\(84\) −7.28115 − 5.29007i −0.794439 − 0.577194i
\(85\) 0 0
\(86\) −1.54508 + 4.75528i −0.166611 + 0.512775i
\(87\) 18.0000 1.92980
\(88\) 0 0
\(89\) 1.00000 0.106000 0.0529999 − 0.998595i \(-0.483122\pi\)
0.0529999 + 0.998595i \(0.483122\pi\)
\(90\) 1.85410 − 5.70634i 0.195440 − 0.601501i
\(91\) 9.70820 − 7.05342i 1.01770 − 0.739400i
\(92\) −6.47214 − 4.70228i −0.674767 − 0.490247i
\(93\) 1.85410 + 5.70634i 0.192261 + 0.591720i
\(94\) −0.927051 − 2.85317i −0.0956180 − 0.294282i
\(95\) 3.23607 + 2.35114i 0.332014 + 0.241222i
\(96\) 12.1353 − 8.81678i 1.23855 − 0.899859i
\(97\) −2.47214 + 7.60845i −0.251007 + 0.772521i 0.743583 + 0.668644i \(0.233125\pi\)
−0.994590 + 0.103877i \(0.966875\pi\)
\(98\) 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.251.1 4
11.2 odd 10 605.2.g.d.366.1 4
11.3 even 5 inner 605.2.g.b.81.1 4
11.4 even 5 605.2.a.c.1.1 yes 1
11.5 even 5 inner 605.2.g.b.511.1 4
11.6 odd 10 605.2.g.d.511.1 4
11.7 odd 10 605.2.a.a.1.1 ✓ 1
11.8 odd 10 605.2.g.d.81.1 4
11.9 even 5 inner 605.2.g.b.366.1 4
11.10 odd 2 605.2.g.d.251.1 4
33.26 odd 10 5445.2.a.d.1.1 1
33.29 even 10 5445.2.a.h.1.1 1
44.7 even 10 9680.2.a.bf.1.1 1
44.15 odd 10 9680.2.a.be.1.1 1
55.4 even 10 3025.2.a.c.1.1 1
55.29 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 11.7 odd 10
605.2.a.c.1.1 yes 1 11.4 even 5
605.2.g.b.81.1 4 11.3 even 5 inner
605.2.g.b.251.1 4 1.1 even 1 trivial
605.2.g.b.366.1 4 11.9 even 5 inner
605.2.g.b.511.1 4 11.5 even 5 inner
605.2.g.d.81.1 4 11.8 odd 10
605.2.g.d.251.1 4 11.10 odd 2
605.2.g.d.366.1 4 11.2 odd 10
605.2.g.d.511.1 4 11.6 odd 10
3025.2.a.c.1.1 1 55.4 even 10
3025.2.a.g.1.1 1 55.29 odd 10
5445.2.a.d.1.1 1 33.26 odd 10
5445.2.a.h.1.1 1 33.29 even 10
9680.2.a.be.1.1 1 44.15 odd 10
9680.2.a.bf.1.1 1 44.7 even 10