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Properties

Label 90.2.e.a.61.1
Level $90$
Weight $2$
Character 90.61
Analytic conductor $0.719$
Analytic rank $0$
Dimension $2$
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 = [2,-1] 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: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 61.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 90.61
Dual form 90.2.e.a.31.1

$q$-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+(-0.500000 + 0.866025i) q^{2} +(1.50000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(0.500000 + 0.866025i) q^{5} +1.73205i q^{6} +(0.500000 - 0.866025i) q^{7} +1.00000 q^{8} +(1.50000 - 2.59808i) q^{9} -1.00000 q^{10} +(-3.00000 + 5.19615i) q^{11} +(-1.50000 - 0.866025i) q^{12} +(-1.00000 - 1.73205i) q^{13} +(0.500000 + 0.866025i) q^{14} +(1.50000 + 0.866025i) q^{15} +(-0.500000 + 0.866025i) q^{16} +(1.50000 + 2.59808i) q^{18} -4.00000 q^{19} +(0.500000 - 0.866025i) q^{20} -1.73205i q^{21} +(-3.00000 - 5.19615i) q^{22} +(-4.50000 - 7.79423i) q^{23} +(1.50000 - 0.866025i) q^{24} +(-0.500000 + 0.866025i) q^{25} +2.00000 q^{26} -5.19615i q^{27} -1.00000 q^{28} +(-1.50000 + 2.59808i) q^{29} +(-1.50000 + 0.866025i) q^{30} +(2.00000 + 3.46410i) q^{31} +(-0.500000 - 0.866025i) q^{32} +10.3923i q^{33} +1.00000 q^{35} -3.00000 q^{36} +8.00000 q^{37} +(2.00000 - 3.46410i) q^{38} +(-3.00000 - 1.73205i) q^{39} +(0.500000 + 0.866025i) q^{40} +(1.50000 + 2.59808i) q^{41} +(1.50000 + 0.866025i) q^{42} +(-4.00000 + 6.92820i) q^{43} +6.00000 q^{44} +3.00000 q^{45} +9.00000 q^{46} +(1.50000 - 2.59808i) q^{47} +1.73205i q^{48} +(3.00000 + 5.19615i) q^{49} +(-0.500000 - 0.866025i) q^{50} +(-1.00000 + 1.73205i) q^{52} +6.00000 q^{53} +(4.50000 + 2.59808i) q^{54} -6.00000 q^{55} +(0.500000 - 0.866025i) q^{56} +(-6.00000 + 3.46410i) q^{57} +(-1.50000 - 2.59808i) q^{58} +(-3.00000 - 5.19615i) q^{59} -1.73205i q^{60} +(6.50000 - 11.2583i) q^{61} -4.00000 q^{62} +(-1.50000 - 2.59808i) q^{63} +1.00000 q^{64} +(1.00000 - 1.73205i) q^{65} +(-9.00000 - 5.19615i) q^{66} +(6.50000 + 11.2583i) q^{67} +(-13.5000 - 7.79423i) q^{69} +(-0.500000 + 0.866025i) q^{70} -6.00000 q^{71} +(1.50000 - 2.59808i) q^{72} -4.00000 q^{73} +(-4.00000 + 6.92820i) q^{74} +1.73205i q^{75} +(2.00000 + 3.46410i) q^{76} +(3.00000 + 5.19615i) q^{77} +(3.00000 - 1.73205i) q^{78} +(5.00000 - 8.66025i) q^{79} -1.00000 q^{80} +(-4.50000 - 7.79423i) q^{81} -3.00000 q^{82} +(4.50000 - 7.79423i) q^{83} +(-1.50000 + 0.866025i) q^{84} +(-4.00000 - 6.92820i) q^{86} +5.19615i q^{87} +(-3.00000 + 5.19615i) q^{88} +9.00000 q^{89} +(-1.50000 + 2.59808i) q^{90} -2.00000 q^{91} +(-4.50000 + 7.79423i) q^{92} +(6.00000 + 3.46410i) q^{93} +(1.50000 + 2.59808i) q^{94} +(-2.00000 - 3.46410i) q^{95} +(-1.50000 - 0.866025i) q^{96} +(-1.00000 + 1.73205i) q^{97} -6.00000 q^{98} +(9.00000 + 15.5885i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 3 q^{3} - q^{4} + q^{5} + q^{7} + 2 q^{8} + 3 q^{9} - 2 q^{10} - 6 q^{11} - 3 q^{12} - 2 q^{13} + q^{14} + 3 q^{15} - q^{16} + 3 q^{18} - 8 q^{19} + q^{20} - 6 q^{22} - 9 q^{23} + 3 q^{24}+ \cdots + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(11\) \(37\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(3\) 1.50000 − 0.866025i 0.866025 − 0.500000i
\(4\) −0.500000 − 0.866025i −0.250000 − 0.433013i
\(5\) 0.500000 + 0.866025i 0.223607 + 0.387298i
\(6\) 1.73205i 0.707107i
\(7\) 0.500000 − 0.866025i 0.188982 − 0.327327i −0.755929 − 0.654654i \(-0.772814\pi\)
0.944911 + 0.327327i \(0.106148\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.50000 − 2.59808i 0.500000 − 0.866025i
\(10\) −1.00000 −0.316228
\(11\) −3.00000 + 5.19615i −0.904534 + 1.56670i −0.0829925 + 0.996550i \(0.526448\pi\)
−0.821541 + 0.570149i \(0.806886\pi\)
\(12\) −1.50000 − 0.866025i −0.433013 − 0.250000i
\(13\) −1.00000 − 1.73205i −0.277350 − 0.480384i 0.693375 − 0.720577i \(-0.256123\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) 0.500000 + 0.866025i 0.133631 + 0.231455i
\(15\) 1.50000 + 0.866025i 0.387298 + 0.223607i
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 1.50000 + 2.59808i 0.353553 + 0.612372i
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0.500000 − 0.866025i 0.111803 − 0.193649i
\(21\) − 1.73205i − 0.377964i
\(22\) −3.00000 − 5.19615i −0.639602 − 1.10782i
\(23\) −4.50000 − 7.79423i −0.938315 − 1.62521i −0.768613 − 0.639713i \(-0.779053\pi\)
−0.169701 − 0.985496i \(-0.554280\pi\)
\(24\) 1.50000 − 0.866025i 0.306186 − 0.176777i
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) 2.00000 0.392232
\(27\) − 5.19615i − 1.00000i
\(28\) −1.00000 −0.188982
\(29\) −1.50000 + 2.59808i −0.278543 + 0.482451i −0.971023 − 0.238987i \(-0.923185\pi\)
0.692480 + 0.721437i \(0.256518\pi\)
\(30\) −1.50000 + 0.866025i −0.273861 + 0.158114i
\(31\) 2.00000 + 3.46410i 0.359211 + 0.622171i 0.987829 − 0.155543i \(-0.0497126\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) −0.500000 − 0.866025i −0.0883883 − 0.153093i
\(33\) 10.3923i 1.80907i
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) −3.00000 −0.500000
\(37\) 8.00000 1.31519 0.657596 − 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 2.00000 − 3.46410i 0.324443 − 0.561951i
\(39\) −3.00000 − 1.73205i −0.480384 − 0.277350i
\(40\) 0.500000 + 0.866025i 0.0790569 + 0.136931i
\(41\) 1.50000 + 2.59808i 0.234261 + 0.405751i 0.959058 − 0.283211i \(-0.0913998\pi\)
−0.724797 + 0.688963i \(0.758066\pi\)
\(42\) 1.50000 + 0.866025i 0.231455 + 0.133631i
\(43\) −4.00000 + 6.92820i −0.609994 + 1.05654i 0.381246 + 0.924473i \(0.375495\pi\)
−0.991241 + 0.132068i \(0.957838\pi\)
\(44\) 6.00000 0.904534
\(45\) 3.00000 0.447214
\(46\) 9.00000 1.32698
\(47\) 1.50000 − 2.59808i 0.218797 − 0.378968i −0.735643 − 0.677369i \(-0.763120\pi\)
0.954441 + 0.298401i \(0.0964533\pi\)
\(48\) 1.73205i 0.250000i
\(49\) 3.00000 + 5.19615i 0.428571 + 0.742307i
\(50\) −0.500000 − 0.866025i −0.0707107 − 0.122474i
\(51\) 0 0
\(52\) −1.00000 + 1.73205i −0.138675 + 0.240192i
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 4.50000 + 2.59808i 0.612372 + 0.353553i
\(55\) −6.00000 −0.809040
\(56\) 0.500000 − 0.866025i 0.0668153 − 0.115728i
\(57\) −6.00000 + 3.46410i −0.794719 + 0.458831i
\(58\) −1.50000 − 2.59808i −0.196960 − 0.341144i
\(59\) −3.00000 − 5.19615i −0.390567 − 0.676481i 0.601958 − 0.798528i \(-0.294388\pi\)
−0.992524 + 0.122047i \(0.961054\pi\)
\(60\) − 1.73205i − 0.223607i
\(61\) 6.50000 − 11.2583i 0.832240 − 1.44148i −0.0640184 − 0.997949i \(-0.520392\pi\)
0.896258 − 0.443533i \(-0.146275\pi\)
\(62\) −4.00000 −0.508001
\(63\) −1.50000 − 2.59808i −0.188982 − 0.327327i
\(64\) 1.00000 0.125000
\(65\) 1.00000 − 1.73205i 0.124035 − 0.214834i
\(66\) −9.00000 − 5.19615i −1.10782 − 0.639602i
\(67\) 6.50000 + 11.2583i 0.794101 + 1.37542i 0.923408 + 0.383819i \(0.125391\pi\)
−0.129307 + 0.991605i \(0.541275\pi\)
\(68\) 0 0
\(69\) −13.5000 − 7.79423i −1.62521 − 0.938315i
\(70\) −0.500000 + 0.866025i −0.0597614 + 0.103510i
\(71\) −6.00000 −0.712069 −0.356034 − 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 1.50000 − 2.59808i 0.176777 − 0.306186i
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) −4.00000 + 6.92820i −0.464991 + 0.805387i
\(75\) 1.73205i 0.200000i
\(76\) 2.00000 + 3.46410i 0.229416 + 0.397360i
\(77\) 3.00000 + 5.19615i 0.341882 + 0.592157i
\(78\) 3.00000 − 1.73205i 0.339683 − 0.196116i
\(79\) 5.00000 − 8.66025i 0.562544 − 0.974355i −0.434730 − 0.900561i \(-0.643156\pi\)
0.997274 − 0.0737937i \(-0.0235106\pi\)
\(80\) −1.00000 −0.111803
\(81\) −4.50000 − 7.79423i −0.500000 − 0.866025i
\(82\) −3.00000 −0.331295
\(83\) 4.50000 − 7.79423i 0.493939 − 0.855528i −0.506036 − 0.862512i \(-0.668890\pi\)
0.999976 + 0.00698436i \(0.00222321\pi\)
\(84\) −1.50000 + 0.866025i −0.163663 + 0.0944911i
\(85\) 0 0
\(86\) −4.00000 − 6.92820i −0.431331 − 0.747087i
\(87\) 5.19615i 0.557086i
\(88\) −3.00000 + 5.19615i −0.319801 + 0.553912i
\(89\) 9.00000 0.953998 0.476999 − 0.878904i \(-0.341725\pi\)
0.476999 + 0.878904i \(0.341725\pi\)
\(90\) −1.50000 + 2.59808i −0.158114 + 0.273861i
\(91\) −2.00000 −0.209657
\(92\) −4.50000 + 7.79423i −0.469157 + 0.812605i
\(93\) 6.00000 + 3.46410i 0.622171 + 0.359211i
\(94\) 1.50000 + 2.59808i 0.154713 + 0.267971i
\(95\) −2.00000 − 3.46410i −0.205196 − 0.355409i
\(96\) −1.50000 − 0.866025i −0.153093 − 0.0883883i
\(97\) −1.00000 + 1.73205i −0.101535 + 0.175863i −0.912317 − 0.409484i \(-0.865709\pi\)
0.810782 + 0.585348i \(0.199042\pi\)
\(98\) −6.00000 −0.606092
\(99\) 9.00000 + 15.5885i 0.904534 + 1.56670i
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.a.61.1 yes 2
3.2 odd 2 270.2.e.b.181.1 2
4.3 odd 2 720.2.q.b.241.1 2
5.2 odd 4 450.2.j.c.349.1 4
5.3 odd 4 450.2.j.c.349.2 4
5.4 even 2 450.2.e.e.151.1 2
9.2 odd 6 810.2.a.b.1.1 1
9.4 even 3 inner 90.2.e.a.31.1 ✓ 2
9.5 odd 6 270.2.e.b.91.1 2
9.7 even 3 810.2.a.g.1.1 1
12.11 even 2 2160.2.q.b.721.1 2
15.2 even 4 1350.2.j.e.1099.2 4
15.8 even 4 1350.2.j.e.1099.1 4
15.14 odd 2 1350.2.e.b.451.1 2
36.7 odd 6 6480.2.a.g.1.1 1
36.11 even 6 6480.2.a.v.1.1 1
36.23 even 6 2160.2.q.b.1441.1 2
36.31 odd 6 720.2.q.b.481.1 2
45.2 even 12 4050.2.c.a.649.1 2
45.4 even 6 450.2.e.e.301.1 2
45.7 odd 12 4050.2.c.t.649.2 2
45.13 odd 12 450.2.j.c.49.1 4
45.14 odd 6 1350.2.e.b.901.1 2
45.22 odd 12 450.2.j.c.49.2 4
45.23 even 12 1350.2.j.e.199.2 4
45.29 odd 6 4050.2.a.ba.1.1 1
45.32 even 12 1350.2.j.e.199.1 4
45.34 even 6 4050.2.a.n.1.1 1
45.38 even 12 4050.2.c.a.649.2 2
45.43 odd 12 4050.2.c.t.649.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.2.e.a.31.1 ✓ 2 9.4 even 3 inner
90.2.e.a.61.1 yes 2 1.1 even 1 trivial
270.2.e.b.91.1 2 9.5 odd 6
270.2.e.b.181.1 2 3.2 odd 2
450.2.e.e.151.1 2 5.4 even 2
450.2.e.e.301.1 2 45.4 even 6
450.2.j.c.49.1 4 45.13 odd 12
450.2.j.c.49.2 4 45.22 odd 12
450.2.j.c.349.1 4 5.2 odd 4
450.2.j.c.349.2 4 5.3 odd 4
720.2.q.b.241.1 2 4.3 odd 2
720.2.q.b.481.1 2 36.31 odd 6
810.2.a.b.1.1 1 9.2 odd 6
810.2.a.g.1.1 1 9.7 even 3
1350.2.e.b.451.1 2 15.14 odd 2
1350.2.e.b.901.1 2 45.14 odd 6
1350.2.j.e.199.1 4 45.32 even 12
1350.2.j.e.199.2 4 45.23 even 12
1350.2.j.e.1099.1 4 15.8 even 4
1350.2.j.e.1099.2 4 15.2 even 4
2160.2.q.b.721.1 2 12.11 even 2
2160.2.q.b.1441.1 2 36.23 even 6
4050.2.a.n.1.1 1 45.34 even 6
4050.2.a.ba.1.1 1 45.29 odd 6
4050.2.c.a.649.1 2 45.2 even 12
4050.2.c.a.649.2 2 45.38 even 12
4050.2.c.t.649.1 2 45.43 odd 12
4050.2.c.t.649.2 2 45.7 odd 12
6480.2.a.g.1.1 1 36.7 odd 6
6480.2.a.v.1.1 1 36.11 even 6