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Properties

Label 90.2.l.b.77.1
Level $90$
Weight $2$
Character 90.77
Analytic conductor $0.719$
Analytic rank $0$
Dimension $16$
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(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 77.1
Root \(0.500000 - 1.74530i\) of defining polynomial
Character \(\chi\) \(=\) 90.77
Dual form 90.2.l.b.83.1

$q$-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+(-0.965926 - 0.258819i) q^{2} +(-1.73022 + 0.0795432i) q^{3} +(0.866025 + 0.500000i) q^{4} +(-1.51901 + 1.64092i) q^{5} +(1.69185 + 0.370982i) q^{6} +(-1.00635 + 3.75574i) 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} +(-1.53819 - 0.796225i) q^{12} +(-0.256253 - 0.956351i) q^{13} +(1.94411 - 3.36730i) q^{14} +(2.49770 - 2.95998i) q^{15} +(0.500000 + 0.866025i) q^{16} +(0.120239 - 0.120239i) q^{17} +(-2.95680 - 0.507306i) q^{18} -1.88492i q^{19} +(-2.13596 + 0.661570i) q^{20} +(1.44246 - 6.57832i) q^{21} +(3.83821 - 1.02845i) q^{22} +(5.08911 - 1.36362i) q^{23} +(1.27970 + 1.16721i) q^{24} +(-0.385214 - 4.98514i) q^{25} +0.990087i q^{26} +(-5.14688 + 0.713876i) q^{27} +(-2.74939 + 2.74939i) q^{28} +(2.15618 + 3.73461i) q^{29} +(-3.17870 + 2.21267i) q^{30} +(-4.70172 + 8.14362i) q^{31} +(-0.258819 - 0.965926i) q^{32} +(5.79609 - 3.71134i) 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} +(-0.487854 + 1.82070i) q^{38} +(0.519447 + 1.63432i) q^{39} +(2.23441 - 0.0862005i) q^{40} +(7.15775 + 4.13253i) q^{41} +(-3.09591 + 5.98083i) q^{42} +(-1.99285 - 0.533983i) q^{43} -3.97361 q^{44} +(-4.08614 + 5.32010i) q^{45} -5.26863 q^{46} +(3.34787 + 0.897060i) q^{47} +(-0.933998 - 1.45865i) q^{48} +(-7.03067 - 4.05916i) q^{49} +(-0.918161 + 4.91498i) q^{50} +(-0.198476 + 0.217604i) q^{51} +(0.256253 - 0.956351i) 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} +(0.149933 + 3.26134i) q^{57} +(-1.11612 - 4.16541i) q^{58} +(-2.72877 + 4.72637i) q^{59} +(3.64306 - 1.31457i) q^{60} +(-4.35623 - 7.54520i) q^{61} +(6.64923 - 6.64923i) q^{62} +(-1.97252 + 11.4967i) q^{63} +1.00000i q^{64} +(1.95854 + 1.03222i) q^{65} +(-6.55916 + 2.08475i) q^{66} +(7.86563 - 2.10759i) q^{67} +(0.164249 - 0.0440105i) q^{68} +(-8.69683 + 2.76418i) q^{69} +(2.57234 + 8.30510i) q^{70} -6.94911i q^{71} +(-2.30701 - 1.91774i) q^{72} +(-8.27728 + 8.27728i) q^{73} +(-2.31127 - 4.00324i) q^{74} +(1.06304 + 8.59476i) q^{75} +(0.942462 - 1.63239i) q^{76} +(-3.99883 - 14.9238i) q^{77} +(-0.0787547 - 1.71307i) 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} +(1.81110 - 6.75913i) q^{83} +(4.53837 - 4.97576i) q^{84} +(0.0146578 + 0.379946i) q^{85} +(1.78674 + 1.03157i) q^{86} +(-4.02773 - 6.29020i) q^{87} +(3.83821 + 1.02845i) q^{88} -4.87832 q^{89} +(5.32385 - 4.08125i) q^{90} +3.84968 q^{91} +(5.08911 + 1.36362i) q^{92} +(7.48725 - 14.4643i) q^{93} +(-3.00162 - 1.73299i) q^{94} +(3.09300 + 2.86322i) q^{95} +(0.524648 + 1.65068i) q^{96} +(-0.387234 + 1.44518i) 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 \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.965926 − 0.258819i −0.683013 − 0.183013i
\(3\) −1.73022 + 0.0795432i −0.998945 + 0.0459243i
\(4\) 0.866025 + 0.500000i 0.433013 + 0.250000i
\(5\) −1.51901 + 1.64092i −0.679322 + 0.733840i
\(6\) 1.69185 + 0.370982i 0.690697 + 0.151453i
\(7\) −1.00635 + 3.75574i −0.380364 + 1.41954i 0.464984 + 0.885319i \(0.346060\pi\)
−0.845347 + 0.534217i \(0.820606\pi\)
\(8\) −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 −0.919145 − 0.393918i \(-0.871119\pi\)
−0.118430 + 0.992962i \(0.537786\pi\)
\(12\) −1.53819 − 0.796225i −0.444037 − 0.229850i
\(13\) −0.256253 − 0.956351i −0.0710719 − 0.265244i 0.921242 − 0.388990i \(-0.127176\pi\)
−0.992314 + 0.123746i \(0.960509\pi\)
\(14\) 1.94411 − 3.36730i 0.519586 − 0.899950i
\(15\) 2.49770 − 2.95998i 0.644904 − 0.764263i
\(16\) 0.500000 + 0.866025i 0.125000 + 0.216506i
\(17\) 0.120239 − 0.120239i 0.0291622 − 0.0291622i −0.692375 − 0.721538i \(-0.743436\pi\)
0.721538 + 0.692375i \(0.243436\pi\)
\(18\) −2.95680 − 0.507306i −0.696923 − 0.119573i
\(19\) − 1.88492i − 0.432431i −0.976346 − 0.216216i \(-0.930629\pi\)
0.976346 − 0.216216i \(-0.0693714\pi\)
\(20\) −2.13596 + 0.661570i −0.477615 + 0.147932i
\(21\) 1.44246 − 6.57832i 0.314771 − 1.43551i
\(22\) 3.83821 − 1.02845i 0.818310 − 0.219265i
\(23\) 5.08911 − 1.36362i 1.06115 − 0.284335i 0.314299 − 0.949324i \(-0.398230\pi\)
0.746853 + 0.664989i \(0.231564\pi\)
\(24\) 1.27970 + 1.16721i 0.261217 + 0.238255i
\(25\) −0.385214 − 4.98514i −0.0770427 − 0.997028i
\(26\) 0.990087i 0.194172i
\(27\) −5.14688 + 0.713876i −0.990518 + 0.137386i
\(28\) −2.74939 + 2.74939i −0.519586 + 0.519586i
\(29\) 2.15618 + 3.73461i 0.400392 + 0.693499i 0.993773 − 0.111422i \(-0.0355406\pi\)
−0.593381 + 0.804922i \(0.702207\pi\)
\(30\) −3.17870 + 2.21267i −0.580348 + 0.403976i
\(31\) −4.70172 + 8.14362i −0.844454 + 1.46264i 0.0416413 + 0.999133i \(0.486741\pi\)
−0.886095 + 0.463504i \(0.846592\pi\)
\(32\) −0.258819 − 0.965926i −0.0457532 − 0.170753i
\(33\) 5.79609 − 3.71134i 1.00897 − 0.646062i
\(34\) −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 \(0.625934\pi\)
\(38\) −0.487854 + 1.82070i −0.0791404 + 0.295356i
\(39\) 0.519447 + 1.63432i 0.0831781 + 0.261700i
\(40\) 2.23441 − 0.0862005i 0.353291 − 0.0136295i
\(41\) 7.15775 + 4.13253i 1.11785 + 0.645393i 0.940852 − 0.338818i \(-0.110027\pi\)
0.177001 + 0.984211i \(0.443360\pi\)
\(42\) −3.09591 + 5.98083i −0.477709 + 0.922862i
\(43\) −1.99285 − 0.533983i −0.303907 − 0.0814316i 0.103643 − 0.994615i \(-0.466950\pi\)
−0.407550 + 0.913183i \(0.633617\pi\)
\(44\) −3.97361 −0.599044
\(45\) −4.08614 + 5.32010i −0.609126 + 0.793074i
\(46\) −5.26863 −0.776818
\(47\) 3.34787 + 0.897060i 0.488338 + 0.130850i 0.494582 − 0.869131i \(-0.335321\pi\)
−0.00624459 + 0.999981i \(0.501988\pi\)
\(48\) −0.933998 − 1.45865i −0.134811 − 0.210537i
\(49\) −7.03067 − 4.05916i −1.00438 − 0.579880i
\(50\) −0.918161 + 4.91498i −0.129848 + 0.695082i
\(51\) −0.198476 + 0.217604i −0.0277922 + 0.0304707i
\(52\) 0.256253 − 0.956351i 0.0355359 − 0.132622i
\(53\) 3.66571 + 3.66571i 0.503524 + 0.503524i 0.912531 − 0.409007i \(-0.134125\pi\)
−0.409007 + 0.912531i \(0.634125\pi\)
\(54\) 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
\(57\) 0.149933 + 3.26134i 0.0198591 + 0.431975i
\(58\) −1.11612 − 4.16541i −0.146554 − 0.546946i
\(59\) −2.72877 + 4.72637i −0.355255 + 0.615320i −0.987162 − 0.159724i \(-0.948939\pi\)
0.631906 + 0.775045i \(0.282273\pi\)
\(60\) 3.64306 − 1.31457i 0.470318 − 0.169710i
\(61\) −4.35623 − 7.54520i −0.557758 − 0.966064i −0.997683 − 0.0680302i \(-0.978329\pi\)
0.439926 − 0.898034i \(-0.355005\pi\)
\(62\) 6.64923 − 6.64923i 0.844454 − 0.844454i
\(63\) −1.97252 + 11.4967i −0.248514 + 1.44845i
\(64\) 1.00000i 0.125000i
\(65\) 1.95854 + 1.03222i 0.242927 + 0.128031i
\(66\) −6.55916 + 2.08475i −0.807377 + 0.256614i
\(67\) 7.86563 − 2.10759i 0.960940 − 0.257483i 0.255942 − 0.966692i \(-0.417615\pi\)
0.704998 + 0.709209i \(0.250948\pi\)
\(68\) 0.164249 − 0.0440105i 0.0199182 − 0.00533705i
\(69\) −8.69683 + 2.76418i −1.04698 + 0.332768i
\(70\) 2.57234 + 8.30510i 0.307453 + 0.992649i
\(71\) − 6.94911i − 0.824708i −0.911024 − 0.412354i \(-0.864707\pi\)
0.911024 − 0.412354i \(-0.135293\pi\)
\(72\) −2.30701 − 1.91774i −0.271883 − 0.226008i
\(73\) −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
\(75\) 1.06304 + 8.59476i 0.122749 + 0.992438i
\(76\) 0.942462 − 1.63239i 0.108108 − 0.187248i
\(77\) −3.99883 − 14.9238i −0.455709 − 1.70073i
\(78\) −0.0787547 − 1.71307i −0.00891722 − 0.193967i
\(79\) 11.7529 − 6.78553i 1.32230 − 0.763431i 0.338206 − 0.941072i \(-0.390180\pi\)
0.984095 + 0.177641i \(0.0568465\pi\)
\(80\) −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
\(83\) 1.81110 − 6.75913i 0.198795 − 0.741911i −0.792457 − 0.609927i \(-0.791199\pi\)
0.991252 − 0.131984i \(-0.0421347\pi\)
\(84\) 4.53837 − 4.97576i 0.495176 − 0.542900i
\(85\) 0.0146578 + 0.379946i 0.00158986 + 0.0412109i
\(86\) 1.78674 + 1.03157i 0.192669 + 0.111238i
\(87\) −4.02773 − 6.29020i −0.431818 − 0.674380i
\(88\) 3.83821 + 1.02845i 0.409155 + 0.109633i
\(89\) −4.87832 −0.517100 −0.258550 − 0.965998i \(-0.583245\pi\)
−0.258550 + 0.965998i \(0.583245\pi\)
\(90\) 5.32385 − 4.08125i 0.561183 − 0.430202i
\(91\) 3.84968 0.403557
\(92\) 5.08911 + 1.36362i 0.530576 + 0.142168i
\(93\) 7.48725 − 14.4643i 0.776392 − 1.49987i
\(94\) −3.00162 − 1.73299i −0.309594 − 0.178744i
\(95\) 3.09300 + 2.86322i 0.317335 + 0.293760i
\(96\) 0.524648 + 1.65068i 0.0535466 + 0.168472i
\(97\) −0.387234 + 1.44518i −0.0393177 + 0.146736i −0.982794 − 0.184704i \(-0.940868\pi\)
0.943477 + 0.331439i \(0.107534\pi\)
\(98\) 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.77.1 yes 16
3.2 odd 2 270.2.m.b.17.4 16
4.3 odd 2 720.2.cu.b.257.4 16
5.2 odd 4 450.2.p.h.293.4 16
5.3 odd 4 inner 90.2.l.b.23.1 ✓ 16
5.4 even 2 450.2.p.h.257.4 16
9.2 odd 6 inner 90.2.l.b.47.1 yes 16
9.4 even 3 810.2.f.c.647.8 16
9.5 odd 6 810.2.f.c.647.1 16
9.7 even 3 270.2.m.b.197.3 16
15.2 even 4 1350.2.q.h.1043.1 16
15.8 even 4 270.2.m.b.233.3 16
15.14 odd 2 1350.2.q.h.557.2 16
20.3 even 4 720.2.cu.b.113.3 16
36.11 even 6 720.2.cu.b.497.3 16
45.2 even 12 450.2.p.h.443.4 16
45.7 odd 12 1350.2.q.h.143.2 16
45.13 odd 12 810.2.f.c.323.1 16
45.23 even 12 810.2.f.c.323.8 16
45.29 odd 6 450.2.p.h.407.4 16
45.34 even 6 1350.2.q.h.1007.1 16
45.38 even 12 inner 90.2.l.b.83.1 yes 16
45.43 odd 12 270.2.m.b.143.4 16
180.83 odd 12 720.2.cu.b.353.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.2.l.b.23.1 ✓ 16 5.3 odd 4 inner
90.2.l.b.47.1 yes 16 9.2 odd 6 inner
90.2.l.b.77.1 yes 16 1.1 even 1 trivial
90.2.l.b.83.1 yes 16 45.38 even 12 inner
270.2.m.b.17.4 16 3.2 odd 2
270.2.m.b.143.4 16 45.43 odd 12
270.2.m.b.197.3 16 9.7 even 3
270.2.m.b.233.3 16 15.8 even 4
450.2.p.h.257.4 16 5.4 even 2
450.2.p.h.293.4 16 5.2 odd 4
450.2.p.h.407.4 16 45.29 odd 6
450.2.p.h.443.4 16 45.2 even 12
720.2.cu.b.113.3 16 20.3 even 4
720.2.cu.b.257.4 16 4.3 odd 2
720.2.cu.b.353.4 16 180.83 odd 12
720.2.cu.b.497.3 16 36.11 even 6
810.2.f.c.323.1 16 45.13 odd 12
810.2.f.c.323.8 16 45.23 even 12
810.2.f.c.647.1 16 9.5 odd 6
810.2.f.c.647.8 16 9.4 even 3
1350.2.q.h.143.2 16 45.7 odd 12
1350.2.q.h.557.2 16 15.14 odd 2
1350.2.q.h.1007.1 16 45.34 even 6
1350.2.q.h.1043.1 16 15.2 even 4