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Properties

Label 90.2.i.b.49.3
Level $90$
Weight $2$
Character 90.49
Analytic conductor $0.719$
Analytic rank $0$
Dimension $8$
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] = [90,2,Mod(49,90)] 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("90.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 90.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.718653618192\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 49.3
Root \(-0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 90.49
Dual form 90.2.i.b.79.3

$q$-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+(0.866025 - 0.500000i) q^{2} +(0.158919 - 1.72474i) q^{3} +(0.500000 - 0.866025i) q^{4} +(-0.917738 + 2.03906i) q^{5} +(-0.724745 - 1.57313i) q^{6} +(0.389270 - 0.224745i) q^{7} -1.00000i q^{8} +(-2.94949 - 0.548188i) q^{9} +(0.224745 + 2.22474i) q^{10} +(1.72474 + 2.98735i) q^{11} +(-1.41421 - 1.00000i) q^{12} +(-2.12132 - 1.22474i) q^{13} +(0.224745 - 0.389270i) q^{14} +(3.37101 + 1.90691i) q^{15} +(-0.500000 - 0.866025i) q^{16} +5.89898i q^{17} +(-2.82843 + 1.00000i) q^{18} +5.44949 q^{19} +(1.30701 + 1.81431i) q^{20} +(-0.325765 - 0.707107i) q^{21} +(2.98735 + 1.72474i) q^{22} +(-5.97469 - 3.44949i) q^{23} +(-1.72474 - 0.158919i) q^{24} +(-3.31552 - 3.74264i) q^{25} -2.44949 q^{26} +(-1.41421 + 5.00000i) q^{27} -0.449490i q^{28} +(-3.00000 - 5.19615i) q^{29} +(3.87283 - 0.0340742i) q^{30} +(-0.775255 + 1.34278i) q^{31} +(-0.866025 - 0.500000i) q^{32} +(5.42650 - 2.50000i) q^{33} +(2.94949 + 5.10867i) q^{34} +(0.101021 + 1.00000i) q^{35} +(-1.94949 + 2.28024i) q^{36} -8.00000i q^{37} +(4.71940 - 2.72474i) q^{38} +(-2.44949 + 3.46410i) q^{39} +(2.03906 + 0.917738i) q^{40} +(0.500000 - 0.866025i) q^{41} +(-0.635674 - 0.449490i) q^{42} +(2.20881 - 1.27526i) q^{43} +3.44949 q^{44} +(3.82465 - 5.51109i) q^{45} -6.89898 q^{46} +(3.85337 - 2.22474i) q^{47} +(-1.57313 + 0.724745i) q^{48} +(-3.39898 + 5.88721i) q^{49} +(-4.74264 - 1.58346i) q^{50} +(10.1742 + 0.937458i) q^{51} +(-2.12132 + 1.22474i) q^{52} +3.55051i q^{53} +(1.27526 + 5.03723i) q^{54} +(-7.67423 + 0.775255i) q^{55} +(-0.224745 - 0.389270i) q^{56} +(0.866025 - 9.39898i) q^{57} +(-5.19615 - 3.00000i) q^{58} +(6.62372 - 11.4726i) q^{59} +(3.33694 - 1.96593i) q^{60} +(-2.22474 - 3.85337i) q^{61} +1.55051i q^{62} +(-1.27135 + 0.449490i) q^{63} -1.00000 q^{64} +(4.44414 - 3.20150i) q^{65} +(3.44949 - 4.87832i) q^{66} +(3.94086 + 2.27526i) q^{67} +(5.10867 + 2.94949i) q^{68} +(-6.89898 + 9.75663i) q^{69} +(0.587486 + 0.815515i) q^{70} -2.44949 q^{71} +(-0.548188 + 2.94949i) q^{72} +14.7980i q^{73} +(-4.00000 - 6.92820i) q^{74} +(-6.98200 + 5.12364i) q^{75} +(2.72474 - 4.71940i) q^{76} +(1.34278 + 0.775255i) q^{77} +(-0.389270 + 4.22474i) q^{78} +(3.67423 + 6.36396i) q^{79} +(2.22474 - 0.224745i) q^{80} +(8.39898 + 3.23375i) q^{81} -1.00000i q^{82} +(-3.46410 + 2.00000i) q^{83} +(-0.775255 - 0.0714323i) q^{84} +(-12.0284 - 5.41372i) q^{85} +(1.27526 - 2.20881i) q^{86} +(-9.43879 + 4.34847i) q^{87} +(2.98735 - 1.72474i) q^{88} -3.10102 q^{89} +(0.556696 - 6.68506i) q^{90} -1.10102 q^{91} +(-5.97469 + 3.44949i) q^{92} +(2.19275 + 1.55051i) q^{93} +(2.22474 - 3.85337i) q^{94} +(-5.00120 + 11.1118i) q^{95} +(-1.00000 + 1.41421i) q^{96} +(-11.2583 + 6.50000i) q^{97} +6.79796i q^{98} +(-3.44949 - 9.75663i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} - 4 q^{5} + 4 q^{6} - 4 q^{9} - 8 q^{10} + 4 q^{11} - 8 q^{14} - 8 q^{15} - 4 q^{16} + 24 q^{19} + 4 q^{20} - 32 q^{21} - 4 q^{24} - 24 q^{29} + 16 q^{30} - 16 q^{31} + 4 q^{34} + 40 q^{35}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.866025 − 0.500000i 0.612372 − 0.353553i
\(3\) 0.158919 − 1.72474i 0.0917517 − 0.995782i
\(4\) 0.500000 − 0.866025i 0.250000 − 0.433013i
\(5\) −0.917738 + 2.03906i −0.410425 + 0.911894i
\(6\) −0.724745 − 1.57313i −0.295876 − 0.642229i
\(7\) 0.389270 − 0.224745i 0.147130 − 0.0849456i −0.424628 − 0.905368i \(-0.639595\pi\)
0.571758 + 0.820422i \(0.306262\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −2.94949 − 0.548188i −0.983163 − 0.182729i
\(10\) 0.224745 + 2.22474i 0.0710706 + 0.703526i
\(11\) 1.72474 + 2.98735i 0.520030 + 0.900719i 0.999729 + 0.0232854i \(0.00741263\pi\)
−0.479699 + 0.877433i \(0.659254\pi\)
\(12\) −1.41421 − 1.00000i −0.408248 − 0.288675i
\(13\) −2.12132 − 1.22474i −0.588348 − 0.339683i 0.176096 − 0.984373i \(-0.443653\pi\)
−0.764444 + 0.644690i \(0.776986\pi\)
\(14\) 0.224745 − 0.389270i 0.0600656 − 0.104037i
\(15\) 3.37101 + 1.90691i 0.870391 + 0.492361i
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 5.89898i 1.43071i 0.698760 + 0.715356i \(0.253736\pi\)
−0.698760 + 0.715356i \(0.746264\pi\)
\(18\) −2.82843 + 1.00000i −0.666667 + 0.235702i
\(19\) 5.44949 1.25020 0.625099 − 0.780545i \(-0.285058\pi\)
0.625099 + 0.780545i \(0.285058\pi\)
\(20\) 1.30701 + 1.81431i 0.292256 + 0.405693i
\(21\) −0.325765 − 0.707107i −0.0710878 − 0.154303i
\(22\) 2.98735 + 1.72474i 0.636904 + 0.367717i
\(23\) −5.97469 − 3.44949i −1.24581 − 0.719268i −0.275538 − 0.961290i \(-0.588856\pi\)
−0.970271 + 0.242022i \(0.922189\pi\)
\(24\) −1.72474 − 0.158919i −0.352062 − 0.0324391i
\(25\) −3.31552 − 3.74264i −0.663103 − 0.748528i
\(26\) −2.44949 −0.480384
\(27\) −1.41421 + 5.00000i −0.272166 + 0.962250i
\(28\) − 0.449490i − 0.0849456i
\(29\) −3.00000 − 5.19615i −0.557086 − 0.964901i −0.997738 − 0.0672232i \(-0.978586\pi\)
0.440652 − 0.897678i \(-0.354747\pi\)
\(30\) 3.87283 − 0.0340742i 0.707079 − 0.00622106i
\(31\) −0.775255 + 1.34278i −0.139240 + 0.241171i −0.927209 − 0.374544i \(-0.877799\pi\)
0.787969 + 0.615715i \(0.211133\pi\)
\(32\) −0.866025 − 0.500000i −0.153093 − 0.0883883i
\(33\) 5.42650 − 2.50000i 0.944633 − 0.435194i
\(34\) 2.94949 + 5.10867i 0.505833 + 0.876129i
\(35\) 0.101021 + 1.00000i 0.0170756 + 0.169031i
\(36\) −1.94949 + 2.28024i −0.324915 + 0.380040i
\(37\) − 8.00000i − 1.31519i −0.753371 − 0.657596i \(-0.771573\pi\)
0.753371 − 0.657596i \(-0.228427\pi\)
\(38\) 4.71940 − 2.72474i 0.765587 − 0.442012i
\(39\) −2.44949 + 3.46410i −0.392232 + 0.554700i
\(40\) 2.03906 + 0.917738i 0.322403 + 0.145107i
\(41\) 0.500000 − 0.866025i 0.0780869 − 0.135250i −0.824338 − 0.566099i \(-0.808452\pi\)
0.902424 + 0.430848i \(0.141786\pi\)
\(42\) −0.635674 − 0.449490i −0.0980867 − 0.0693578i
\(43\) 2.20881 − 1.27526i 0.336840 − 0.194475i −0.322034 − 0.946728i \(-0.604366\pi\)
0.658874 + 0.752254i \(0.271033\pi\)
\(44\) 3.44949 0.520030
\(45\) 3.82465 − 5.51109i 0.570144 − 0.821544i
\(46\) −6.89898 −1.01720
\(47\) 3.85337 − 2.22474i 0.562072 − 0.324512i −0.191905 − 0.981414i \(-0.561466\pi\)
0.753977 + 0.656901i \(0.228133\pi\)
\(48\) −1.57313 + 0.724745i −0.227062 + 0.104608i
\(49\) −3.39898 + 5.88721i −0.485568 + 0.841029i
\(50\) −4.74264 − 1.58346i −0.670711 − 0.223936i
\(51\) 10.1742 + 0.937458i 1.42468 + 0.131270i
\(52\) −2.12132 + 1.22474i −0.294174 + 0.169842i
\(53\) 3.55051i 0.487700i 0.969813 + 0.243850i \(0.0784105\pi\)
−0.969813 + 0.243850i \(0.921590\pi\)
\(54\) 1.27526 + 5.03723i 0.173540 + 0.685481i
\(55\) −7.67423 + 0.775255i −1.03479 + 0.104535i
\(56\) −0.224745 − 0.389270i −0.0300328 − 0.0520183i
\(57\) 0.866025 − 9.39898i 0.114708 − 1.24493i
\(58\) −5.19615 − 3.00000i −0.682288 − 0.393919i
\(59\) 6.62372 − 11.4726i 0.862335 − 1.49361i −0.00733331 − 0.999973i \(-0.502334\pi\)
0.869669 − 0.493636i \(-0.164332\pi\)
\(60\) 3.33694 − 1.96593i 0.430796 − 0.253800i
\(61\) −2.22474 − 3.85337i −0.284849 − 0.493374i 0.687723 − 0.725973i \(-0.258610\pi\)
−0.972573 + 0.232599i \(0.925277\pi\)
\(62\) 1.55051i 0.196915i
\(63\) −1.27135 + 0.449490i −0.160175 + 0.0566304i
\(64\) −1.00000 −0.125000
\(65\) 4.44414 − 3.20150i 0.551228 − 0.397097i
\(66\) 3.44949 − 4.87832i 0.424603 − 0.600479i
\(67\) 3.94086 + 2.27526i 0.481452 + 0.277967i 0.721022 − 0.692913i \(-0.243673\pi\)
−0.239569 + 0.970879i \(0.577006\pi\)
\(68\) 5.10867 + 2.94949i 0.619517 + 0.357678i
\(69\) −6.89898 + 9.75663i −0.830540 + 1.17456i
\(70\) 0.587486 + 0.815515i 0.0702180 + 0.0974727i
\(71\) −2.44949 −0.290701 −0.145350 − 0.989380i \(-0.546431\pi\)
−0.145350 + 0.989380i \(0.546431\pi\)
\(72\) −0.548188 + 2.94949i −0.0646046 + 0.347601i
\(73\) 14.7980i 1.73197i 0.500070 + 0.865985i \(0.333308\pi\)
−0.500070 + 0.865985i \(0.666692\pi\)
\(74\) −4.00000 − 6.92820i −0.464991 − 0.805387i
\(75\) −6.98200 + 5.12364i −0.806212 + 0.591627i
\(76\) 2.72474 − 4.71940i 0.312550 − 0.541352i
\(77\) 1.34278 + 0.775255i 0.153024 + 0.0883485i
\(78\) −0.389270 + 4.22474i −0.0440761 + 0.478358i
\(79\) 3.67423 + 6.36396i 0.413384 + 0.716002i 0.995257 − 0.0972777i \(-0.0310135\pi\)
−0.581874 + 0.813279i \(0.697680\pi\)
\(80\) 2.22474 − 0.224745i 0.248734 − 0.0251272i
\(81\) 8.39898 + 3.23375i 0.933220 + 0.359306i
\(82\) − 1.00000i − 0.110432i
\(83\) −3.46410 + 2.00000i −0.380235 + 0.219529i −0.677920 − 0.735135i \(-0.737119\pi\)
0.297686 + 0.954664i \(0.403785\pi\)
\(84\) −0.775255 − 0.0714323i −0.0845873 − 0.00779390i
\(85\) −12.0284 − 5.41372i −1.30466 − 0.587200i
\(86\) 1.27526 − 2.20881i 0.137514 − 0.238182i
\(87\) −9.43879 + 4.34847i −1.01194 + 0.466205i
\(88\) 2.98735 − 1.72474i 0.318452 − 0.183858i
\(89\) −3.10102 −0.328708 −0.164354 − 0.986401i \(-0.552554\pi\)
−0.164354 + 0.986401i \(0.552554\pi\)
\(90\) 0.556696 − 6.68506i 0.0586809 − 0.704668i
\(91\) −1.10102 −0.115418
\(92\) −5.97469 + 3.44949i −0.622905 + 0.359634i
\(93\) 2.19275 + 1.55051i 0.227378 + 0.160780i
\(94\) 2.22474 − 3.85337i 0.229465 − 0.397445i
\(95\) −5.00120 + 11.1118i −0.513112 + 1.14005i
\(96\) −1.00000 + 1.41421i −0.102062 + 0.144338i
\(97\) −11.2583 + 6.50000i −1.14311 + 0.659975i −0.947199 − 0.320647i \(-0.896100\pi\)
−0.195911 + 0.980622i \(0.562766\pi\)
\(98\) 6.79796i 0.686698i
\(99\) −3.44949 − 9.75663i −0.346687 − 0.980578i
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 90.2.i.b.49.3 yes 8
3.2 odd 2 270.2.i.b.199.1 8
4.3 odd 2 720.2.by.c.49.2 8
5.2 odd 4 450.2.e.n.301.1 4
5.3 odd 4 450.2.e.k.301.2 4
5.4 even 2 inner 90.2.i.b.49.2 ✓ 8
9.2 odd 6 270.2.i.b.19.4 8
9.4 even 3 810.2.c.f.649.4 4
9.5 odd 6 810.2.c.e.649.1 4
9.7 even 3 inner 90.2.i.b.79.2 yes 8
12.11 even 2 2160.2.by.d.1009.2 8
15.2 even 4 1350.2.e.j.901.2 4
15.8 even 4 1350.2.e.m.901.1 4
15.14 odd 2 270.2.i.b.199.4 8
20.19 odd 2 720.2.by.c.49.3 8
36.7 odd 6 720.2.by.c.529.3 8
36.11 even 6 2160.2.by.d.289.3 8
45.2 even 12 1350.2.e.j.451.2 4
45.4 even 6 810.2.c.f.649.2 4
45.7 odd 12 450.2.e.n.151.1 4
45.13 odd 12 4050.2.a.bs.1.2 2
45.14 odd 6 810.2.c.e.649.3 4
45.22 odd 12 4050.2.a.bq.1.1 2
45.23 even 12 4050.2.a.bm.1.2 2
45.29 odd 6 270.2.i.b.19.1 8
45.32 even 12 4050.2.a.bz.1.1 2
45.34 even 6 inner 90.2.i.b.79.3 yes 8
45.38 even 12 1350.2.e.m.451.1 4
45.43 odd 12 450.2.e.k.151.2 4
60.59 even 2 2160.2.by.d.1009.3 8
180.79 odd 6 720.2.by.c.529.2 8
180.119 even 6 2160.2.by.d.289.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.2.i.b.49.2 ✓ 8 5.4 even 2 inner
90.2.i.b.49.3 yes 8 1.1 even 1 trivial
90.2.i.b.79.2 yes 8 9.7 even 3 inner
90.2.i.b.79.3 yes 8 45.34 even 6 inner
270.2.i.b.19.1 8 45.29 odd 6
270.2.i.b.19.4 8 9.2 odd 6
270.2.i.b.199.1 8 3.2 odd 2
270.2.i.b.199.4 8 15.14 odd 2
450.2.e.k.151.2 4 45.43 odd 12
450.2.e.k.301.2 4 5.3 odd 4
450.2.e.n.151.1 4 45.7 odd 12
450.2.e.n.301.1 4 5.2 odd 4
720.2.by.c.49.2 8 4.3 odd 2
720.2.by.c.49.3 8 20.19 odd 2
720.2.by.c.529.2 8 180.79 odd 6
720.2.by.c.529.3 8 36.7 odd 6
810.2.c.e.649.1 4 9.5 odd 6
810.2.c.e.649.3 4 45.14 odd 6
810.2.c.f.649.2 4 45.4 even 6
810.2.c.f.649.4 4 9.4 even 3
1350.2.e.j.451.2 4 45.2 even 12
1350.2.e.j.901.2 4 15.2 even 4
1350.2.e.m.451.1 4 45.38 even 12
1350.2.e.m.901.1 4 15.8 even 4
2160.2.by.d.289.2 8 180.119 even 6
2160.2.by.d.289.3 8 36.11 even 6
2160.2.by.d.1009.2 8 12.11 even 2
2160.2.by.d.1009.3 8 60.59 even 2
4050.2.a.bm.1.2 2 45.23 even 12
4050.2.a.bq.1.1 2 45.22 odd 12
4050.2.a.bs.1.2 2 45.13 odd 12
4050.2.a.bz.1.1 2 45.32 even 12