↑

Properties

Label 90.2.l.b.47.1
Level $90$
Weight $2$
Character 90.47
Analytic conductor $0.719$
Analytic rank $0$
Dimension $16$
Inner twists $4$

Related objects

Downloads

Learn more

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(23,90)] 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("90.23"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([10, 9])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 90.l (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16] 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: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: 16.0.9349208943630483456.9
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + \cdots + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 47.1
Root \(0.500000 + 0.410882i\) of defining polynomial
Character \(\chi\) \(=\) 90.47
Dual form 90.2.l.b.23.1

$q$-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+(-0.258819 - 0.965926i) q^{2} +(-0.0795432 + 1.73022i) q^{3} +(-0.866025 + 0.500000i) q^{4} +(0.661570 + 2.13596i) q^{5} +(1.69185 - 0.370982i) q^{6} +(3.75574 - 1.00635i) q^{7} +(0.707107 + 0.707107i) q^{8} +(-2.98735 - 0.275255i) q^{9} +(1.89195 - 1.19185i) q^{10} +(-3.44125 - 1.98681i) q^{11} +(-0.796225 - 1.53819i) q^{12} +(0.956351 + 0.256253i) q^{13} +(-1.94411 - 3.36730i) q^{14} +(-3.74831 + 0.974763i) q^{15} +(0.500000 - 0.866025i) q^{16} +(-0.120239 + 0.120239i) q^{17} +(0.507306 + 2.95680i) q^{18} -1.88492i q^{19} +(-1.64092 - 1.51901i) q^{20} +(1.44246 + 6.57832i) q^{21} +(-1.02845 + 3.83821i) q^{22} +(1.36362 - 5.08911i) q^{23} +(-1.27970 + 1.16721i) q^{24} +(-4.12465 + 2.82617i) q^{25} -0.990087i q^{26} +(0.713876 - 5.14688i) q^{27} +(-2.74939 + 2.74939i) q^{28} +(-2.15618 + 3.73461i) q^{29} +(1.91168 + 3.36830i) q^{30} +(-4.70172 - 8.14362i) q^{31} +(-0.965926 - 0.258819i) q^{32} +(3.71134 - 5.79609i) q^{33} +(0.147262 + 0.0850217i) q^{34} +(4.63420 + 7.35634i) q^{35} +(2.72474 - 1.25529i) q^{36} +(3.26863 + 3.26863i) q^{37} +(-1.82070 + 0.487854i) q^{38} +(-0.519447 + 1.63432i) q^{39} +(-1.04255 + 1.97815i) q^{40} +(7.15775 - 4.13253i) q^{41} +(5.98083 - 3.09591i) q^{42} +(0.533983 + 1.99285i) q^{43} +3.97361 q^{44} +(-1.38840 - 6.56295i) q^{45} -5.26863 q^{46} +(0.897060 + 3.34787i) q^{47} +(1.45865 + 0.933998i) q^{48} +(7.03067 - 4.05916i) q^{49} +(3.79741 + 3.25264i) q^{50} +(-0.198476 - 0.217604i) q^{51} +(-0.956351 + 0.256253i) q^{52} +(-3.66571 - 3.66571i) q^{53} +(-5.15627 + 0.642559i) q^{54} +(1.96711 - 8.66478i) q^{55} +(3.36730 + 1.94411i) q^{56} +(3.26134 + 0.149933i) q^{57} +(4.16541 + 1.11612i) q^{58} +(2.72877 + 4.72637i) q^{59} +(2.75875 - 2.71832i) q^{60} +(-4.35623 + 7.54520i) q^{61} +(-6.64923 + 6.64923i) q^{62} +(-11.4967 + 1.97252i) q^{63} +1.00000i q^{64} +(0.0853460 + 2.21226i) q^{65} +(-6.55916 - 2.08475i) q^{66} +(-2.10759 + 7.86563i) q^{67} +(0.0440105 - 0.164249i) q^{68} +(8.69683 + 2.76418i) q^{69} +(5.90626 - 6.38026i) q^{70} +6.94911i q^{71} +(-1.91774 - 2.30701i) q^{72} +(-8.27728 + 8.27728i) q^{73} +(2.31127 - 4.00324i) q^{74} +(-4.56182 - 7.36137i) q^{75} +(0.942462 + 1.63239i) q^{76} +(-14.9238 - 3.99883i) q^{77} +(1.71307 + 0.0787547i) q^{78} +(-11.7529 - 6.78553i) q^{79} +(2.18058 + 0.495044i) q^{80} +(8.84847 + 1.64456i) q^{81} +(-5.84428 - 5.84428i) q^{82} +(6.75913 - 1.81110i) q^{83} +(-4.53837 - 4.97576i) q^{84} +(-0.336372 - 0.177279i) q^{85} +(1.78674 - 1.03157i) q^{86} +(-6.29020 - 4.02773i) q^{87} +(-1.02845 - 3.83821i) q^{88} +4.87832 q^{89} +(-5.97998 + 3.03971i) q^{90} +3.84968 q^{91} +(1.36362 + 5.08911i) q^{92} +(14.4643 - 7.48725i) q^{93} +(3.00162 - 1.73299i) q^{94} +(4.02612 - 1.24701i) q^{95} +(0.524648 - 1.65068i) q^{96} +(1.44518 - 0.387234i) q^{97} +(-5.74052 - 5.74052i) q^{98} +(9.73332 + 6.88249i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 12 q^{5} + 8 q^{7} - 8 q^{10} - 24 q^{15} + 8 q^{16} - 12 q^{20} + 24 q^{21} + 8 q^{22} - 24 q^{23} - 16 q^{25} - 16 q^{28} - 12 q^{30} - 8 q^{31} + 24 q^{36} + 24 q^{38} - 4 q^{40} + 24 q^{41} + 24 q^{42}+ \cdots - 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(11\) \(37\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.258819 − 0.965926i −0.183013 − 0.683013i
\(3\) −0.0795432 + 1.73022i −0.0459243 + 0.998945i
\(4\) −0.866025 + 0.500000i −0.433013 + 0.250000i
\(5\) 0.661570 + 2.13596i 0.295863 + 0.955230i
\(6\) 1.69185 − 0.370982i 0.690697 − 0.151453i
\(7\) 3.75574 − 1.00635i 1.41954 − 0.380364i 0.534217 − 0.845347i \(-0.320606\pi\)
0.885319 + 0.464984i \(0.153940\pi\)
\(8\) 0.707107 + 0.707107i 0.250000 + 0.250000i
\(9\) −2.98735 − 0.275255i −0.995782 − 0.0917517i
\(10\) 1.89195 − 1.19185i 0.598288 − 0.376898i
\(11\) −3.44125 − 1.98681i −1.03758 − 0.599044i −0.118430 − 0.992962i \(-0.537786\pi\)
−0.919145 + 0.393918i \(0.871119\pi\)
\(12\) −0.796225 − 1.53819i −0.229850 − 0.444037i
\(13\) 0.956351 + 0.256253i 0.265244 + 0.0710719i 0.388990 − 0.921242i \(-0.372824\pi\)
−0.123746 + 0.992314i \(0.539491\pi\)
\(14\) −1.94411 − 3.36730i −0.519586 − 0.899950i
\(15\) −3.74831 + 0.974763i −0.967810 + 0.251683i
\(16\) 0.500000 − 0.866025i 0.125000 − 0.216506i
\(17\) −0.120239 + 0.120239i −0.0291622 + 0.0291622i −0.721538 − 0.692375i \(-0.756564\pi\)
0.692375 + 0.721538i \(0.256564\pi\)
\(18\) 0.507306 + 2.95680i 0.119573 + 0.696923i
\(19\) − 1.88492i − 0.432431i −0.976346 − 0.216216i \(-0.930629\pi\)
0.976346 − 0.216216i \(-0.0693714\pi\)
\(20\) −1.64092 − 1.51901i −0.366920 − 0.339661i
\(21\) 1.44246 + 6.57832i 0.314771 + 1.43551i
\(22\) −1.02845 + 3.83821i −0.219265 + 0.818310i
\(23\) 1.36362 − 5.08911i 0.284335 − 1.06115i −0.664989 − 0.746853i \(-0.731564\pi\)
0.949324 − 0.314299i \(-0.101770\pi\)
\(24\) −1.27970 + 1.16721i −0.261217 + 0.238255i
\(25\) −4.12465 + 2.82617i −0.824930 + 0.565235i
\(26\) − 0.990087i − 0.194172i
\(27\) 0.713876 − 5.14688i 0.137386 − 0.990518i
\(28\) −2.74939 + 2.74939i −0.519586 + 0.519586i
\(29\) −2.15618 + 3.73461i −0.400392 + 0.693499i −0.993773 − 0.111422i \(-0.964459\pi\)
0.593381 + 0.804922i \(0.297793\pi\)
\(30\) 1.91168 + 3.36830i 0.349024 + 0.614965i
\(31\) −4.70172 − 8.14362i −0.844454 − 1.46264i −0.886095 − 0.463504i \(-0.846592\pi\)
0.0416413 − 0.999133i \(-0.486741\pi\)
\(32\) −0.965926 − 0.258819i −0.170753 − 0.0457532i
\(33\) 3.71134 − 5.79609i 0.646062 − 1.00897i
\(34\) 0.147262 + 0.0850217i 0.0252552 + 0.0145811i
\(35\) 4.63420 + 7.35634i 0.783323 + 1.24345i
\(36\) 2.72474 − 1.25529i 0.454124 − 0.209216i
\(37\) 3.26863 + 3.26863i 0.537360 + 0.537360i 0.922753 − 0.385393i \(-0.125934\pi\)
−0.385393 + 0.922753i \(0.625934\pi\)
\(38\) −1.82070 + 0.487854i −0.295356 + 0.0791404i
\(39\) −0.519447 + 1.63432i −0.0831781 + 0.261700i
\(40\) −1.04255 + 1.97815i −0.164842 + 0.312773i
\(41\) 7.15775 − 4.13253i 1.11785 − 0.645393i 0.177001 − 0.984211i \(-0.443360\pi\)
0.940852 + 0.338818i \(0.110027\pi\)
\(42\) 5.98083 − 3.09591i 0.922862 − 0.477709i
\(43\) 0.533983 + 1.99285i 0.0814316 + 0.303907i 0.994615 − 0.103643i \(-0.0330500\pi\)
−0.913183 + 0.407550i \(0.866383\pi\)
\(44\) 3.97361 0.599044
\(45\) −1.38840 − 6.56295i −0.206971 − 0.978347i
\(46\) −5.26863 −0.776818
\(47\) 0.897060 + 3.34787i 0.130850 + 0.488338i 0.999981 − 0.00624459i \(-0.00198773\pi\)
−0.869131 + 0.494582i \(0.835321\pi\)
\(48\) 1.45865 + 0.933998i 0.210537 + 0.134811i
\(49\) 7.03067 − 4.05916i 1.00438 − 0.579880i
\(50\) 3.79741 + 3.25264i 0.537035 + 0.459993i
\(51\) −0.198476 − 0.217604i −0.0277922 − 0.0304707i
\(52\) −0.956351 + 0.256253i −0.132622 + 0.0355359i
\(53\) −3.66571 − 3.66571i −0.503524 − 0.503524i 0.409007 − 0.912531i \(-0.365875\pi\)
−0.912531 + 0.409007i \(0.865875\pi\)
\(54\) −5.15627 + 0.642559i −0.701679 + 0.0874413i
\(55\) 1.96711 − 8.66478i 0.265245 − 1.16836i
\(56\) 3.36730 + 1.94411i 0.449975 + 0.259793i
\(57\) 3.26134 + 0.149933i 0.431975 + 0.0198591i
\(58\) 4.16541 + 1.11612i 0.546946 + 0.146554i
\(59\) 2.72877 + 4.72637i 0.355255 + 0.615320i 0.987162 − 0.159724i \(-0.0510606\pi\)
−0.631906 + 0.775045i \(0.717727\pi\)
\(60\) 2.75875 − 2.71832i 0.356153 − 0.350934i
\(61\) −4.35623 + 7.54520i −0.557758 + 0.966064i 0.439926 + 0.898034i \(0.355005\pi\)
−0.997683 + 0.0680302i \(0.978329\pi\)
\(62\) −6.64923 + 6.64923i −0.844454 + 0.844454i
\(63\) −11.4967 + 1.97252i −1.44845 + 0.248514i
\(64\) 1.00000i 0.125000i
\(65\) 0.0853460 + 2.21226i 0.0105859 + 0.274397i
\(66\) −6.55916 − 2.08475i −0.807377 − 0.256614i
\(67\) −2.10759 + 7.86563i −0.257483 + 0.960940i 0.709209 + 0.704998i \(0.249052\pi\)
−0.966692 + 0.255942i \(0.917615\pi\)
\(68\) 0.0440105 − 0.164249i 0.00533705 − 0.0199182i
\(69\) 8.69683 + 2.76418i 1.04698 + 0.332768i
\(70\) 5.90626 − 6.38026i 0.705933 − 0.762587i
\(71\) 6.94911i 0.824708i 0.911024 + 0.412354i \(0.135293\pi\)
−0.911024 + 0.412354i \(0.864707\pi\)
\(72\) −1.91774 − 2.30701i −0.226008 − 0.271883i
\(73\) −8.27728 + 8.27728i −0.968783 + 0.968783i −0.999527 − 0.0307446i \(-0.990212\pi\)
0.0307446 + 0.999527i \(0.490212\pi\)
\(74\) 2.31127 − 4.00324i 0.268680 − 0.465368i
\(75\) −4.56182 − 7.36137i −0.526754 − 0.850018i
\(76\) 0.942462 + 1.63239i 0.108108 + 0.187248i
\(77\) −14.9238 − 3.99883i −1.70073 − 0.455709i
\(78\) 1.71307 + 0.0787547i 0.193967 + 0.00891722i
\(79\) −11.7529 − 6.78553i −1.32230 − 0.763431i −0.338206 − 0.941072i \(-0.609820\pi\)
−0.984095 + 0.177641i \(0.943153\pi\)
\(80\) 2.18058 + 0.495044i 0.243796 + 0.0553475i
\(81\) 8.84847 + 1.64456i 0.983163 + 0.182729i
\(82\) −5.84428 − 5.84428i −0.645393 − 0.645393i
\(83\) 6.75913 − 1.81110i 0.741911 − 0.198795i 0.131984 − 0.991252i \(-0.457865\pi\)
0.609927 + 0.792457i \(0.291199\pi\)
\(84\) −4.53837 − 4.97576i −0.495176 − 0.542900i
\(85\) −0.336372 − 0.177279i −0.0364847 − 0.0192286i
\(86\) 1.78674 − 1.03157i 0.192669 − 0.111238i
\(87\) −6.29020 − 4.02773i −0.674380 − 0.431818i
\(88\) −1.02845 − 3.83821i −0.109633 − 0.409155i
\(89\) 4.87832 0.517100 0.258550 − 0.965998i \(-0.416755\pi\)
0.258550 + 0.965998i \(0.416755\pi\)
\(90\) −5.97998 + 3.03971i −0.630345 + 0.320414i
\(91\) 3.84968 0.403557
\(92\) 1.36362 + 5.08911i 0.142168 + 0.530576i
\(93\) 14.4643 − 7.48725i 1.49987 − 0.776392i
\(94\) 3.00162 − 1.73299i 0.309594 − 0.178744i
\(95\) 4.02612 − 1.24701i 0.413072 − 0.127940i
\(96\) 0.524648 − 1.65068i 0.0535466 − 0.168472i
\(97\) 1.44518 − 0.387234i 0.146736 − 0.0393177i −0.184704 − 0.982794i \(-0.559132\pi\)
0.331439 + 0.943477i \(0.392466\pi\)
\(98\) −5.74052 − 5.74052i −0.579880 − 0.579880i
\(99\) 9.73332 + 6.88249i 0.978235 + 0.691717i
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.l.b.47.1 yes 16
3.2 odd 2 270.2.m.b.197.3 16
4.3 odd 2 720.2.cu.b.497.3 16
5.2 odd 4 450.2.p.h.443.4 16
5.3 odd 4 inner 90.2.l.b.83.1 yes 16
5.4 even 2 450.2.p.h.407.4 16
9.2 odd 6 810.2.f.c.647.8 16
9.4 even 3 270.2.m.b.17.4 16
9.5 odd 6 inner 90.2.l.b.77.1 yes 16
9.7 even 3 810.2.f.c.647.1 16
15.2 even 4 1350.2.q.h.143.2 16
15.8 even 4 270.2.m.b.143.4 16
15.14 odd 2 1350.2.q.h.1007.1 16
20.3 even 4 720.2.cu.b.353.4 16
36.23 even 6 720.2.cu.b.257.4 16
45.4 even 6 1350.2.q.h.557.2 16
45.13 odd 12 270.2.m.b.233.3 16
45.14 odd 6 450.2.p.h.257.4 16
45.22 odd 12 1350.2.q.h.1043.1 16
45.23 even 12 inner 90.2.l.b.23.1 ✓ 16
45.32 even 12 450.2.p.h.293.4 16
45.38 even 12 810.2.f.c.323.1 16
45.43 odd 12 810.2.f.c.323.8 16
180.23 odd 12 720.2.cu.b.113.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.2.l.b.23.1 ✓ 16 45.23 even 12 inner
90.2.l.b.47.1 yes 16 1.1 even 1 trivial
90.2.l.b.77.1 yes 16 9.5 odd 6 inner
90.2.l.b.83.1 yes 16 5.3 odd 4 inner
270.2.m.b.17.4 16 9.4 even 3
270.2.m.b.143.4 16 15.8 even 4
270.2.m.b.197.3 16 3.2 odd 2
270.2.m.b.233.3 16 45.13 odd 12
450.2.p.h.257.4 16 45.14 odd 6
450.2.p.h.293.4 16 45.32 even 12
450.2.p.h.407.4 16 5.4 even 2
450.2.p.h.443.4 16 5.2 odd 4
720.2.cu.b.113.3 16 180.23 odd 12
720.2.cu.b.257.4 16 36.23 even 6
720.2.cu.b.353.4 16 20.3 even 4
720.2.cu.b.497.3 16 4.3 odd 2
810.2.f.c.323.1 16 45.38 even 12
810.2.f.c.323.8 16 45.43 odd 12
810.2.f.c.647.1 16 9.7 even 3
810.2.f.c.647.8 16 9.2 odd 6
1350.2.q.h.143.2 16 15.2 even 4
1350.2.q.h.557.2 16 45.4 even 6
1350.2.q.h.1007.1 16 15.14 odd 2
1350.2.q.h.1043.1 16 45.22 odd 12