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Properties

Label 90.2.e.c.31.1
Level $90$
Weight $2$
Character 90.31
Analytic conductor $0.719$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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(31,90)] 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("90.31"); 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, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 90.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.718653618192\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 31.1
Root \(1.68614 + 0.396143i\) of defining polynomial
Character \(\chi\) \(=\) 90.31
Dual form 90.2.e.c.61.2

$q$-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+(0.500000 + 0.866025i) q^{2} +(0.500000 - 1.65831i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(0.500000 - 0.866025i) q^{5} +(1.68614 - 0.396143i) q^{6} +(1.18614 + 2.05446i) q^{7} -1.00000 q^{8} +(-2.50000 - 1.65831i) q^{9} +1.00000 q^{10} +(-0.686141 - 1.18843i) q^{11} +(1.18614 + 1.26217i) q^{12} +(-2.37228 + 4.10891i) q^{13} +(-1.18614 + 2.05446i) q^{14} +(-1.18614 - 1.26217i) q^{15} +(-0.500000 - 0.866025i) q^{16} -7.37228 q^{17} +(0.186141 - 2.99422i) q^{18} +3.37228 q^{19} +(0.500000 + 0.866025i) q^{20} +(4.00000 - 0.939764i) q^{21} +(0.686141 - 1.18843i) q^{22} +(2.18614 - 3.78651i) q^{23} +(-0.500000 + 1.65831i) q^{24} +(-0.500000 - 0.866025i) q^{25} -4.74456 q^{26} +(-4.00000 + 3.31662i) q^{27} -2.37228 q^{28} +(2.18614 + 3.78651i) q^{29} +(0.500000 - 1.65831i) q^{30} +(3.37228 - 5.84096i) q^{31} +(0.500000 - 0.866025i) q^{32} +(-2.31386 + 0.543620i) q^{33} +(-3.68614 - 6.38458i) q^{34} +2.37228 q^{35} +(2.68614 - 1.33591i) q^{36} -4.00000 q^{37} +(1.68614 + 2.92048i) q^{38} +(5.62772 + 5.98844i) q^{39} +(-0.500000 + 0.866025i) q^{40} +(1.50000 - 2.59808i) q^{41} +(2.81386 + 2.99422i) q^{42} +(5.68614 + 9.84868i) q^{43} +1.37228 q^{44} +(-2.68614 + 1.33591i) q^{45} +4.37228 q^{46} +(0.813859 + 1.40965i) q^{47} +(-1.68614 + 0.396143i) q^{48} +(0.686141 - 1.18843i) q^{49} +(0.500000 - 0.866025i) q^{50} +(-3.68614 + 12.2255i) q^{51} +(-2.37228 - 4.10891i) q^{52} +11.4891 q^{53} +(-4.87228 - 1.80579i) q^{54} -1.37228 q^{55} +(-1.18614 - 2.05446i) q^{56} +(1.68614 - 5.59230i) q^{57} +(-2.18614 + 3.78651i) q^{58} +(0.686141 - 1.18843i) q^{59} +(1.68614 - 0.396143i) q^{60} +(-4.55842 - 7.89542i) q^{61} +6.74456 q^{62} +(0.441578 - 7.10313i) q^{63} +1.00000 q^{64} +(2.37228 + 4.10891i) q^{65} +(-1.62772 - 1.73205i) q^{66} +(3.50000 - 6.06218i) q^{67} +(3.68614 - 6.38458i) q^{68} +(-5.18614 - 5.51856i) q^{69} +(1.18614 + 2.05446i) q^{70} -6.00000 q^{71} +(2.50000 + 1.65831i) q^{72} -14.1168 q^{73} +(-2.00000 - 3.46410i) q^{74} +(-1.68614 + 0.396143i) q^{75} +(-1.68614 + 2.92048i) q^{76} +(1.62772 - 2.81929i) q^{77} +(-2.37228 + 7.86797i) q^{78} +(-1.00000 - 1.73205i) q^{79} -1.00000 q^{80} +(3.50000 + 8.29156i) q^{81} +3.00000 q^{82} +(-0.813859 - 1.40965i) q^{83} +(-1.18614 + 3.93398i) q^{84} +(-3.68614 + 6.38458i) q^{85} +(-5.68614 + 9.84868i) q^{86} +(7.37228 - 1.73205i) q^{87} +(0.686141 + 1.18843i) q^{88} -1.11684 q^{89} +(-2.50000 - 1.65831i) q^{90} -11.2554 q^{91} +(2.18614 + 3.78651i) q^{92} +(-8.00000 - 8.51278i) q^{93} +(-0.813859 + 1.40965i) q^{94} +(1.68614 - 2.92048i) q^{95} +(-1.18614 - 1.26217i) q^{96} +(1.31386 + 2.27567i) q^{97} +1.37228 q^{98} +(-0.255437 + 4.10891i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 2 q^{3} - 2 q^{4} + 2 q^{5} + q^{6} - q^{7} - 4 q^{8} - 10 q^{9} + 4 q^{10} + 3 q^{11} - q^{12} + 2 q^{13} + q^{14} + q^{15} - 2 q^{16} - 18 q^{17} - 5 q^{18} + 2 q^{19} + 2 q^{20}+ \cdots - 24 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 + 0.866025i 0.353553 + 0.612372i
\(3\) 0.500000 − 1.65831i 0.288675 − 0.957427i
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) 0.500000 − 0.866025i 0.223607 − 0.387298i
\(6\) 1.68614 − 0.396143i 0.688364 − 0.161725i
\(7\) 1.18614 + 2.05446i 0.448319 + 0.776511i 0.998277 − 0.0586811i \(-0.0186895\pi\)
−0.549958 + 0.835192i \(0.685356\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.50000 − 1.65831i −0.833333 − 0.552771i
\(10\) 1.00000 0.316228
\(11\) −0.686141 − 1.18843i −0.206879 − 0.358325i 0.743851 − 0.668346i \(-0.232997\pi\)
−0.950730 + 0.310021i \(0.899664\pi\)
\(12\) 1.18614 + 1.26217i 0.342409 + 0.364357i
\(13\) −2.37228 + 4.10891i −0.657952 + 1.13961i 0.323192 + 0.946333i \(0.395244\pi\)
−0.981145 + 0.193274i \(0.938089\pi\)
\(14\) −1.18614 + 2.05446i −0.317009 + 0.549076i
\(15\) −1.18614 − 1.26217i −0.306260 − 0.325891i
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −7.37228 −1.78804 −0.894020 − 0.448026i \(-0.852127\pi\)
−0.894020 + 0.448026i \(0.852127\pi\)
\(18\) 0.186141 − 2.99422i 0.0438738 − 0.705744i
\(19\) 3.37228 0.773654 0.386827 − 0.922152i \(-0.373571\pi\)
0.386827 + 0.922152i \(0.373571\pi\)
\(20\) 0.500000 + 0.866025i 0.111803 + 0.193649i
\(21\) 4.00000 − 0.939764i 0.872872 − 0.205073i
\(22\) 0.686141 − 1.18843i 0.146286 − 0.253374i
\(23\) 2.18614 − 3.78651i 0.455842 − 0.789541i −0.542894 − 0.839801i \(-0.682672\pi\)
0.998736 + 0.0502598i \(0.0160049\pi\)
\(24\) −0.500000 + 1.65831i −0.102062 + 0.338502i
\(25\) −0.500000 − 0.866025i −0.100000 − 0.173205i
\(26\) −4.74456 −0.930485
\(27\) −4.00000 + 3.31662i −0.769800 + 0.638285i
\(28\) −2.37228 −0.448319
\(29\) 2.18614 + 3.78651i 0.405956 + 0.703137i 0.994432 − 0.105378i \(-0.0336052\pi\)
−0.588476 + 0.808515i \(0.700272\pi\)
\(30\) 0.500000 − 1.65831i 0.0912871 − 0.302765i
\(31\) 3.37228 − 5.84096i 0.605680 − 1.04907i −0.386264 − 0.922388i \(-0.626235\pi\)
0.991944 − 0.126680i \(-0.0404320\pi\)
\(32\) 0.500000 − 0.866025i 0.0883883 − 0.153093i
\(33\) −2.31386 + 0.543620i −0.402791 + 0.0946322i
\(34\) −3.68614 − 6.38458i −0.632168 − 1.09495i
\(35\) 2.37228 0.400989
\(36\) 2.68614 − 1.33591i 0.447690 − 0.222651i
\(37\) −4.00000 −0.657596 −0.328798 − 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 1.68614 + 2.92048i 0.273528 + 0.473765i
\(39\) 5.62772 + 5.98844i 0.901156 + 0.958918i
\(40\) −0.500000 + 0.866025i −0.0790569 + 0.136931i
\(41\) 1.50000 − 2.59808i 0.234261 − 0.405751i −0.724797 − 0.688963i \(-0.758066\pi\)
0.959058 + 0.283211i \(0.0913998\pi\)
\(42\) 2.81386 + 2.99422i 0.434188 + 0.462018i
\(43\) 5.68614 + 9.84868i 0.867128 + 1.50191i 0.864918 + 0.501913i \(0.167370\pi\)
0.00221007 + 0.999998i \(0.499297\pi\)
\(44\) 1.37228 0.206879
\(45\) −2.68614 + 1.33591i −0.400426 + 0.199145i
\(46\) 4.37228 0.644658
\(47\) 0.813859 + 1.40965i 0.118714 + 0.205618i 0.919258 − 0.393655i \(-0.128790\pi\)
−0.800545 + 0.599273i \(0.795456\pi\)
\(48\) −1.68614 + 0.396143i −0.243373 + 0.0571784i
\(49\) 0.686141 − 1.18843i 0.0980201 − 0.169776i
\(50\) 0.500000 − 0.866025i 0.0707107 − 0.122474i
\(51\) −3.68614 + 12.2255i −0.516163 + 1.71192i
\(52\) −2.37228 − 4.10891i −0.328976 − 0.569804i
\(53\) 11.4891 1.57815 0.789076 − 0.614295i \(-0.210560\pi\)
0.789076 + 0.614295i \(0.210560\pi\)
\(54\) −4.87228 − 1.80579i −0.663034 − 0.245737i
\(55\) −1.37228 −0.185038
\(56\) −1.18614 − 2.05446i −0.158505 − 0.274538i
\(57\) 1.68614 − 5.59230i 0.223335 − 0.740718i
\(58\) −2.18614 + 3.78651i −0.287054 + 0.497193i
\(59\) 0.686141 − 1.18843i 0.0893279 − 0.154720i −0.817899 − 0.575361i \(-0.804861\pi\)
0.907227 + 0.420641i \(0.138195\pi\)
\(60\) 1.68614 − 0.396143i 0.217680 − 0.0511419i
\(61\) −4.55842 − 7.89542i −0.583646 − 1.01090i −0.995043 − 0.0994483i \(-0.968292\pi\)
0.411397 − 0.911456i \(-0.365041\pi\)
\(62\) 6.74456 0.856560
\(63\) 0.441578 − 7.10313i 0.0556336 − 0.894910i
\(64\) 1.00000 0.125000
\(65\) 2.37228 + 4.10891i 0.294245 + 0.509648i
\(66\) −1.62772 − 1.73205i −0.200358 − 0.213201i
\(67\) 3.50000 − 6.06218i 0.427593 − 0.740613i −0.569066 − 0.822292i \(-0.692695\pi\)
0.996659 + 0.0816792i \(0.0260283\pi\)
\(68\) 3.68614 − 6.38458i 0.447010 − 0.774244i
\(69\) −5.18614 − 5.51856i −0.624338 − 0.664356i
\(70\) 1.18614 + 2.05446i 0.141771 + 0.245554i
\(71\) −6.00000 −0.712069 −0.356034 − 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 2.50000 + 1.65831i 0.294628 + 0.195434i
\(73\) −14.1168 −1.65225 −0.826126 − 0.563486i \(-0.809460\pi\)
−0.826126 + 0.563486i \(0.809460\pi\)
\(74\) −2.00000 − 3.46410i −0.232495 − 0.402694i
\(75\) −1.68614 + 0.396143i −0.194699 + 0.0457427i
\(76\) −1.68614 + 2.92048i −0.193414 + 0.335002i
\(77\) 1.62772 − 2.81929i 0.185496 − 0.321288i
\(78\) −2.37228 + 7.86797i −0.268608 + 0.890872i
\(79\) −1.00000 − 1.73205i −0.112509 − 0.194871i 0.804272 − 0.594261i \(-0.202555\pi\)
−0.916781 + 0.399390i \(0.869222\pi\)
\(80\) −1.00000 −0.111803
\(81\) 3.50000 + 8.29156i 0.388889 + 0.921285i
\(82\) 3.00000 0.331295
\(83\) −0.813859 − 1.40965i −0.0893327 − 0.154729i 0.817897 − 0.575365i \(-0.195140\pi\)
−0.907229 + 0.420637i \(0.861807\pi\)
\(84\) −1.18614 + 3.93398i −0.129419 + 0.429233i
\(85\) −3.68614 + 6.38458i −0.399818 + 0.692505i
\(86\) −5.68614 + 9.84868i −0.613152 + 1.06201i
\(87\) 7.37228 − 1.73205i 0.790392 − 0.185695i
\(88\) 0.686141 + 1.18843i 0.0731428 + 0.126687i
\(89\) −1.11684 −0.118385 −0.0591926 − 0.998247i \(-0.518853\pi\)
−0.0591926 + 0.998247i \(0.518853\pi\)
\(90\) −2.50000 − 1.65831i −0.263523 − 0.174801i
\(91\) −11.2554 −1.17989
\(92\) 2.18614 + 3.78651i 0.227921 + 0.394771i
\(93\) −8.00000 − 8.51278i −0.829561 − 0.882734i
\(94\) −0.813859 + 1.40965i −0.0839432 + 0.145394i
\(95\) 1.68614 − 2.92048i 0.172994 − 0.299635i
\(96\) −1.18614 − 1.26217i −0.121060 − 0.128820i
\(97\) 1.31386 + 2.27567i 0.133402 + 0.231059i 0.924986 − 0.380001i \(-0.124076\pi\)
−0.791584 + 0.611061i \(0.790743\pi\)
\(98\) 1.37228 0.138621
\(99\) −0.255437 + 4.10891i −0.0256724 + 0.412961i
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.e.c.31.1 ✓ 4
3.2 odd 2 270.2.e.c.91.2 4
4.3 odd 2 720.2.q.f.481.2 4
5.2 odd 4 450.2.j.g.49.1 8
5.3 odd 4 450.2.j.g.49.4 8
5.4 even 2 450.2.e.j.301.2 4
9.2 odd 6 270.2.e.c.181.2 4
9.4 even 3 810.2.a.i.1.1 2
9.5 odd 6 810.2.a.k.1.1 2
9.7 even 3 inner 90.2.e.c.61.2 yes 4
12.11 even 2 2160.2.q.f.1441.1 4
15.2 even 4 1350.2.j.f.199.3 8
15.8 even 4 1350.2.j.f.199.2 8
15.14 odd 2 1350.2.e.l.901.1 4
36.7 odd 6 720.2.q.f.241.1 4
36.11 even 6 2160.2.q.f.721.1 4
36.23 even 6 6480.2.a.bn.1.2 2
36.31 odd 6 6480.2.a.be.1.2 2
45.2 even 12 1350.2.j.f.1099.2 8
45.4 even 6 4050.2.a.bw.1.2 2
45.7 odd 12 450.2.j.g.349.4 8
45.13 odd 12 4050.2.c.v.649.4 4
45.14 odd 6 4050.2.a.bo.1.2 2
45.22 odd 12 4050.2.c.v.649.1 4
45.23 even 12 4050.2.c.ba.649.2 4
45.29 odd 6 1350.2.e.l.451.1 4
45.32 even 12 4050.2.c.ba.649.3 4
45.34 even 6 450.2.e.j.151.1 4
45.38 even 12 1350.2.j.f.1099.3 8
45.43 odd 12 450.2.j.g.349.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.2.e.c.31.1 ✓ 4 1.1 even 1 trivial
90.2.e.c.61.2 yes 4 9.7 even 3 inner
270.2.e.c.91.2 4 3.2 odd 2
270.2.e.c.181.2 4 9.2 odd 6
450.2.e.j.151.1 4 45.34 even 6
450.2.e.j.301.2 4 5.4 even 2
450.2.j.g.49.1 8 5.2 odd 4
450.2.j.g.49.4 8 5.3 odd 4
450.2.j.g.349.1 8 45.43 odd 12
450.2.j.g.349.4 8 45.7 odd 12
720.2.q.f.241.1 4 36.7 odd 6
720.2.q.f.481.2 4 4.3 odd 2
810.2.a.i.1.1 2 9.4 even 3
810.2.a.k.1.1 2 9.5 odd 6
1350.2.e.l.451.1 4 45.29 odd 6
1350.2.e.l.901.1 4 15.14 odd 2
1350.2.j.f.199.2 8 15.8 even 4
1350.2.j.f.199.3 8 15.2 even 4
1350.2.j.f.1099.2 8 45.2 even 12
1350.2.j.f.1099.3 8 45.38 even 12
2160.2.q.f.721.1 4 36.11 even 6
2160.2.q.f.1441.1 4 12.11 even 2
4050.2.a.bo.1.2 2 45.14 odd 6
4050.2.a.bw.1.2 2 45.4 even 6
4050.2.c.v.649.1 4 45.22 odd 12
4050.2.c.v.649.4 4 45.13 odd 12
4050.2.c.ba.649.2 4 45.23 even 12
4050.2.c.ba.649.3 4 45.32 even 12
6480.2.a.be.1.2 2 36.31 odd 6
6480.2.a.bn.1.2 2 36.23 even 6