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Properties

Label 90.11.k.b.7.5
Level $90$
Weight $11$
Character 90.7
Analytic conductor $57.182$
Analytic rank $0$
Dimension $120$
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] = [90,11,Mod(7,90)] 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("90.7"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([8, 3])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 90.k (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [120,960] 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: \(57.1821527406\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(30\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 7.5
Character \(\chi\) \(=\) 90.7
Dual form 90.11.k.b.13.5

$q$-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+(21.8564 - 5.85641i) q^{2} +(-215.021 - 113.204i) q^{3} +(443.405 - 256.000i) q^{4} +(-962.611 - 2973.05i) q^{5} +(-5362.55 - 1214.99i) q^{6} +(23832.8 - 6385.97i) q^{7} +(8192.00 - 8192.00i) q^{8} +(33418.6 + 48682.4i) q^{9} +(-38450.6 - 59342.7i) q^{10} +(-92386.7 + 160018. i) q^{11} +(-124321. + 4849.96i) q^{12} +(526999. + 141209. i) q^{13} +(483500. - 279149. i) q^{14} +(-129580. + 748238. i) q^{15} +(131072. - 227023. i) q^{16} +(385625. + 385625. i) q^{17} +(1.01552e6 + 868310. i) q^{18} +3.48282e6i q^{19} +(-1.18793e6 - 1.07184e6i) q^{20} +(-5.84745e6 - 1.32485e6i) q^{21} +(-1.08211e6 + 4.03848e6i) q^{22} +(3.10066e6 + 830819. i) q^{23} +(-2.68882e6 + 834079. i) q^{24} +(-7.91238e6 + 5.72378e6i) q^{25} +1.23453e7 q^{26} +(-1.67464e6 - 1.42508e7i) q^{27} +(8.93275e6 - 8.93275e6i) q^{28} +(6.96840e6 + 4.02320e6i) q^{29} +(1.54983e6 + 1.71127e7i) q^{30} +(2.04072e6 + 3.53463e6i) q^{31} +(1.53522e6 - 5.72953e6i) q^{32} +(3.79798e7 - 2.39487e7i) q^{33} +(1.06868e7 + 6.17000e6i) q^{34} +(-4.19275e7 - 6.47087e7i) q^{35} +(2.72807e7 + 1.30309e7i) q^{36} +(6.72296e7 + 6.72296e7i) q^{37} +(2.03968e7 + 7.61218e7i) q^{38} +(-9.73301e7 - 9.00213e7i) q^{39} +(-3.22409e7 - 1.64695e7i) q^{40} +(-3.62849e7 - 6.28473e7i) q^{41} +(-1.35563e8 + 5.28851e6i) q^{42} +(5.41787e7 + 2.02198e8i) q^{43} +9.46040e7i q^{44} +(1.12566e8 - 1.46217e8i) q^{45} +7.26349e7 q^{46} +(2.29685e8 - 6.15440e7i) q^{47} +(-5.38832e7 + 3.39768e7i) q^{48} +(2.82589e8 - 1.63153e8i) q^{49} +(-1.39416e8 + 1.71439e8i) q^{50} +(-3.92629e7 - 1.26572e8i) q^{51} +(2.69823e8 - 7.22989e7i) q^{52} +(5.29139e8 - 5.29139e8i) q^{53} +(-1.20060e8 - 3.01665e8i) q^{54} +(5.64675e8 + 1.20634e8i) q^{55} +(1.42924e8 - 2.47552e8i) q^{56} +(3.94269e8 - 7.48877e8i) q^{57} +(1.75866e8 + 4.71230e7i) q^{58} +(-6.45902e8 + 3.72912e8i) q^{59} +(1.34092e8 + 3.64945e8i) q^{60} +(-2.05733e8 + 3.56340e8i) q^{61} +(6.53030e7 + 6.53030e7i) q^{62} +(1.10734e9 + 9.46826e8i) q^{63} -1.34218e8i q^{64} +(-8.74743e7 - 1.70272e9i) q^{65} +(6.89849e8 - 7.45858e8i) q^{66} +(2.60771e8 - 9.73211e8i) q^{67} +(2.69708e8 + 7.22681e7i) q^{68} +(-5.72653e8 - 5.29651e8i) q^{69} +(-1.29534e9 - 1.16875e9i) q^{70} +1.85185e8 q^{71} +(6.72572e8 + 1.25041e8i) q^{72} +(-4.61514e8 + 4.61514e8i) q^{73} +(1.86312e9 + 1.07567e9i) q^{74} +(2.34928e9 - 3.35014e8i) q^{75} +(8.91601e8 + 1.54430e9i) q^{76} +(-1.17996e9 + 4.40366e9i) q^{77} +(-2.65449e9 - 1.39754e9i) q^{78} +(-3.78509e9 - 2.18532e9i) q^{79} +(-8.01122e8 - 1.71148e8i) q^{80} +(-1.25317e9 + 3.25380e9i) q^{81} +(-1.16112e9 - 1.16112e9i) q^{82} +(-8.08859e8 - 3.01870e9i) q^{83} +(-2.93195e9 + 9.09500e8i) q^{84} +(7.75274e8 - 1.51769e9i) q^{85} +(2.36830e9 + 4.10202e9i) q^{86} +(-1.04290e9 - 1.65392e9i) q^{87} +(5.54040e8 + 2.06770e9i) q^{88} +4.73492e9i q^{89} +(1.60398e9 - 3.85502e9i) q^{90} +1.34616e10 q^{91} +(1.58754e9 - 4.25380e8i) q^{92} +(-3.86617e7 - 9.91035e8i) q^{93} +(4.65967e9 - 2.69026e9i) q^{94} +(1.03546e10 - 3.35260e9i) q^{95} +(-9.78710e8 + 1.05817e9i) q^{96} +(3.97451e9 - 1.06497e9i) q^{97} +(5.22089e9 - 5.22089e9i) q^{98} +(-1.08775e10 + 8.49990e8i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 960 q^{2} - 128 q^{3} - 8192 q^{6} + 1830 q^{7} + 983040 q^{8} + 105792 q^{10} + 217740 q^{11} - 65536 q^{12} - 3385658 q^{15} + 15728640 q^{16} + 2438244 q^{17} + 2534464 q^{18} + 1692672 q^{20}+ \cdots + 76966514304 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(11\) \(37\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 21.8564 − 5.85641i 0.683013 − 0.183013i
\(3\) −215.021 − 113.204i −0.884858 − 0.465861i
\(4\) 443.405 − 256.000i 0.433013 − 0.250000i
\(5\) −962.611 − 2973.05i −0.308036 − 0.951375i
\(6\) −5362.55 − 1214.99i −0.689628 − 0.156249i
\(7\) 23832.8 − 6385.97i 1.41803 − 0.379959i 0.533243 − 0.845962i \(-0.320973\pi\)
0.884783 + 0.466004i \(0.154307\pi\)
\(8\) 8192.00 − 8192.00i 0.250000 − 0.250000i
\(9\) 33418.6 + 48682.4i 0.565948 + 0.824441i
\(10\) −38450.6 − 59342.7i −0.384506 − 0.593427i
\(11\) −92386.7 + 160018.i −0.573649 + 0.993589i 0.422538 + 0.906345i \(0.361139\pi\)
−0.996187 + 0.0872438i \(0.972194\pi\)
\(12\) −124321. + 4849.96i −0.499620 + 0.0194909i
\(13\) 526999. + 141209.i 1.41936 + 0.380317i 0.885257 − 0.465102i \(-0.153982\pi\)
0.534104 + 0.845419i \(0.320649\pi\)
\(14\) 483500. − 279149.i 0.898992 − 0.519033i
\(15\) −129580. + 748238.i −0.170640 + 0.985333i
\(16\) 131072. − 227023.i 0.125000 − 0.216506i
\(17\) 385625. + 385625.i 0.271594 + 0.271594i 0.829742 − 0.558147i \(-0.188488\pi\)
−0.558147 + 0.829742i \(0.688488\pi\)
\(18\) 1.01552e6 + 868310.i 0.537433 + 0.459528i
\(19\) 3.48282e6i 1.40657i 0.710906 + 0.703287i \(0.248285\pi\)
−0.710906 + 0.703287i \(0.751715\pi\)
\(20\) −1.18793e6 − 1.07184e6i −0.371227 − 0.334948i
\(21\) −5.84745e6 − 1.32485e6i −1.43176 − 0.324393i
\(22\) −1.08211e6 + 4.03848e6i −0.209970 + 0.783619i
\(23\) 3.10066e6 + 830819.i 0.481743 + 0.129083i 0.491514 − 0.870870i \(-0.336444\pi\)
−0.00977135 + 0.999952i \(0.503110\pi\)
\(24\) −2.68882e6 + 834079.i −0.337680 + 0.104749i
\(25\) −7.91238e6 + 5.72378e6i −0.810228 + 0.586115i
\(26\) 1.23453e7 1.03904
\(27\) −1.67464e6 − 1.42508e7i −0.116708 − 0.993166i
\(28\) 8.93275e6 − 8.93275e6i 0.519033 − 0.519033i
\(29\) 6.96840e6 + 4.02320e6i 0.339737 + 0.196147i 0.660156 − 0.751129i \(-0.270490\pi\)
−0.320419 + 0.947276i \(0.603824\pi\)
\(30\) 1.54983e6 + 1.71127e7i 0.0637790 + 0.704225i
\(31\) 2.04072e6 + 3.53463e6i 0.0712811 + 0.123463i 0.899463 − 0.436997i \(-0.143958\pi\)
−0.828182 + 0.560459i \(0.810625\pi\)
\(32\) 1.53522e6 − 5.72953e6i 0.0457532 − 0.170753i
\(33\) 3.79798e7 − 2.39487e7i 0.970472 − 0.611945i
\(34\) 1.06868e7 + 6.17000e6i 0.235208 + 0.135797i
\(35\) −4.19275e7 − 6.47087e7i −0.798286 − 1.23203i
\(36\) 2.72807e7 + 1.30309e7i 0.451173 + 0.215507i
\(37\) 6.72296e7 + 6.72296e7i 0.969509 + 0.969509i 0.999549 − 0.0300402i \(-0.00956352\pi\)
−0.0300402 + 0.999549i \(0.509564\pi\)
\(38\) 2.03968e7 + 7.61218e7i 0.257421 + 0.960708i
\(39\) −9.73301e7 − 9.00213e7i −1.07876 − 0.997751i
\(40\) −3.22409e7 − 1.64695e7i −0.314853 − 0.160835i
\(41\) −3.62849e7 − 6.28473e7i −0.313189 − 0.542460i 0.665862 − 0.746075i \(-0.268064\pi\)
−0.979051 + 0.203615i \(0.934731\pi\)
\(42\) −1.35563e8 + 5.28851e6i −1.03728 + 0.0404657i
\(43\) 5.41787e7 + 2.02198e8i 0.368542 + 1.37542i 0.862556 + 0.505962i \(0.168862\pi\)
−0.494014 + 0.869454i \(0.664471\pi\)
\(44\) 9.46040e7i 0.573649i
\(45\) 1.12566e8 − 1.46217e8i 0.610021 − 0.792385i
\(46\) 7.26349e7 0.352660
\(47\) 2.29685e8 − 6.15440e7i 1.00148 − 0.268347i 0.279417 − 0.960170i \(-0.409859\pi\)
0.722067 + 0.691823i \(0.243192\pi\)
\(48\) −5.38832e7 + 3.39768e7i −0.211469 + 0.133345i
\(49\) 2.82589e8 − 1.63153e8i 1.00040 − 0.577582i
\(50\) −1.39416e8 + 1.71439e8i −0.446130 + 0.548606i
\(51\) −3.92629e7 − 1.26572e8i −0.113797 − 0.366847i
\(52\) 2.69823e8 − 7.22989e7i 0.709680 − 0.190158i
\(53\) 5.29139e8 − 5.29139e8i 1.26529 − 1.26529i 0.316797 − 0.948493i \(-0.397393\pi\)
0.948493 − 0.316797i \(-0.102607\pi\)
\(54\) −1.20060e8 − 3.01665e8i −0.261475 − 0.656986i
\(55\) 5.64675e8 + 1.20634e8i 1.12198 + 0.239694i
\(56\) 1.42924e8 − 2.47552e8i 0.259517 − 0.449496i
\(57\) 3.94269e8 − 7.48877e8i 0.655267 − 1.24462i
\(58\) 1.75866e8 + 4.71230e7i 0.267942 + 0.0717949i
\(59\) −6.45902e8 + 3.72912e8i −0.903455 + 0.521610i −0.878320 − 0.478074i \(-0.841335\pi\)
−0.0251357 + 0.999684i \(0.508002\pi\)
\(60\) 1.34092e8 + 3.64945e8i 0.172444 + 0.469322i
\(61\) −2.05733e8 + 3.56340e8i −0.243587 + 0.421906i −0.961733 − 0.273987i \(-0.911658\pi\)
0.718146 + 0.695892i \(0.244991\pi\)
\(62\) 6.53030e7 + 6.53030e7i 0.0712811 + 0.0712811i
\(63\) 1.10734e9 + 9.46826e8i 1.11578 + 0.954042i
\(64\) − 1.34218e8i − 0.125000i
\(65\) −8.74743e7 − 1.70272e9i −0.0753900 − 1.46750i
\(66\) 6.89849e8 − 7.45858e8i 0.550851 − 0.595575i
\(67\) 2.60771e8 − 9.73211e8i 0.193146 − 0.720830i −0.799593 − 0.600542i \(-0.794952\pi\)
0.992739 − 0.120288i \(-0.0383818\pi\)
\(68\) 2.69708e8 + 7.22681e7i 0.185502 + 0.0497052i
\(69\) −5.72653e8 − 5.29651e8i −0.366139 − 0.338645i
\(70\) −1.29534e9 − 1.16875e9i −0.770717 − 0.695398i
\(71\) 1.85185e8 0.102639 0.0513196 − 0.998682i \(-0.483657\pi\)
0.0513196 + 0.998682i \(0.483657\pi\)
\(72\) 6.72572e8 + 1.25041e8i 0.347597 + 0.0646234i
\(73\) −4.61514e8 + 4.61514e8i −0.222623 + 0.222623i −0.809602 − 0.586979i \(-0.800317\pi\)
0.586979 + 0.809602i \(0.300317\pi\)
\(74\) 1.86312e9 + 1.07567e9i 0.839619 + 0.484754i
\(75\) 2.34928e9 − 3.35014e8i 0.989985 − 0.141175i
\(76\) 8.91601e8 + 1.54430e9i 0.351643 + 0.609064i
\(77\) −1.17996e9 + 4.40366e9i −0.435926 + 1.62690i
\(78\) −2.65449e9 − 1.39754e9i −0.919407 − 0.484050i
\(79\) −3.78509e9 − 2.18532e9i −1.23010 − 0.710200i −0.263051 − 0.964782i \(-0.584729\pi\)
−0.967051 + 0.254582i \(0.918062\pi\)
\(80\) −8.01122e8 − 1.71148e8i −0.244483 − 0.0522302i
\(81\) −1.25317e9 + 3.25380e9i −0.359407 + 0.933181i
\(82\) −1.16112e9 − 1.16112e9i −0.313189 − 0.313189i
\(83\) −8.08859e8 − 3.01870e9i −0.205344 − 0.766355i −0.989344 − 0.145595i \(-0.953491\pi\)
0.784000 − 0.620761i \(-0.213176\pi\)
\(84\) −2.93195e9 + 9.09500e8i −0.701068 + 0.217474i
\(85\) 7.75274e8 − 1.51769e9i 0.174727 − 0.342049i
\(86\) 2.36830e9 + 4.10202e9i 0.503437 + 0.871979i
\(87\) −1.04290e9 − 1.65392e9i −0.209242 − 0.331833i
\(88\) 5.54040e8 + 2.06770e9i 0.104985 + 0.391809i
\(89\) 4.73492e9i 0.847935i 0.905677 + 0.423968i \(0.139363\pi\)
−0.905677 + 0.423968i \(0.860637\pi\)
\(90\) 1.60398e9 − 3.85502e9i 0.271635 − 0.652851i
\(91\) 1.34616e10 2.15719
\(92\) 1.58754e9 − 4.25380e8i 0.240871 − 0.0645413i
\(93\) −3.86617e7 − 9.91035e8i −0.00555733 − 0.142454i
\(94\) 4.65967e9 − 2.69026e9i 0.634915 − 0.366568i
\(95\) 1.03546e10 − 3.35260e9i 1.33818 − 0.433275i
\(96\) −9.78710e8 + 1.05817e9i −0.120032 + 0.129778i
\(97\) 3.97451e9 − 1.06497e9i 0.462834 − 0.124016i −0.0198643 − 0.999803i \(-0.506323\pi\)
0.482698 + 0.875787i \(0.339657\pi\)
\(98\) 5.22089e9 − 5.22089e9i 0.577582 − 0.577582i
\(99\) −1.08775e10 + 8.49990e8i −1.14381 + 0.0893794i
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 90.11.k.b.7.5 ✓ 120
5.3 odd 4 inner 90.11.k.b.43.20 yes 120
9.4 even 3 inner 90.11.k.b.67.20 yes 120
45.13 odd 12 inner 90.11.k.b.13.5 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.k.b.7.5 ✓ 120 1.1 even 1 trivial
90.11.k.b.13.5 yes 120 45.13 odd 12 inner
90.11.k.b.43.20 yes 120 5.3 odd 4 inner
90.11.k.b.67.20 yes 120 9.4 even 3 inner