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 7.19
Character \(\chi\) \(=\) 90.7
Dual form 90.11.k.b.13.19

$q$-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} +(81.4250 + 228.952i) q^{3} +(443.405 - 256.000i) q^{4} +(-1692.09 - 2627.25i) q^{5} +(3120.49 + 4527.21i) q^{6} +(7782.62 - 2085.35i) q^{7} +(8192.00 - 8192.00i) q^{8} +(-45788.9 + 37284.8i) q^{9} +(-52369.2 - 47512.8i) q^{10} +(-12954.7 + 22438.2i) q^{11} +(94716.0 + 80673.6i) q^{12} +(29157.1 + 7812.62i) q^{13} +(157887. - 91156.4i) q^{14} +(463736. - 601331. i) q^{15} +(131072. - 227023. i) q^{16} +(-716377. - 716377. i) q^{17} +(-782426. + 1.08307e6i) q^{18} -2.85484e6i q^{19} +(-1.42286e6 - 731763. i) q^{20} +(1.11114e6 + 1.61205e6i) q^{21} +(-151736. + 566286. i) q^{22} +(2.47212e6 + 662403. i) q^{23} +(2.54261e6 + 1.20854e6i) q^{24} +(-4.03930e6 + 8.89109e6i) q^{25} +683023. q^{26} +(-1.22648e7 - 7.44754e6i) q^{27} +(2.91700e6 - 2.91700e6i) q^{28} +(-1.63937e7 - 9.46493e6i) q^{29} +(6.61397e6 - 1.58588e7i) q^{30} +(-1.37025e7 - 2.37334e7i) q^{31} +(1.53522e6 - 5.72953e6i) q^{32} +(-6.19210e6 - 1.13897e6i) q^{33} +(-1.98528e7 - 1.14620e7i) q^{34} +(-1.86476e7 - 1.69183e7i) q^{35} +(-1.07581e7 + 2.82542e7i) q^{36} +(-4.03379e7 - 4.03379e7i) q^{37} +(-1.67191e7 - 6.23965e7i) q^{38} +(585404. + 7.31172e6i) q^{39} +(-3.53840e7 - 7.66088e6i) q^{40} +(3.92265e7 + 6.79422e7i) q^{41} +(3.37264e7 + 2.87262e7i) q^{42} +(-5.47389e7 - 2.04289e8i) q^{43} +1.32656e7i q^{44} +(1.75436e8 + 5.72099e7i) q^{45} +5.79110e7 q^{46} +(2.13150e8 - 5.71135e7i) q^{47} +(6.26500e7 + 1.15238e7i) q^{48} +(-1.88410e8 + 1.08779e8i) q^{49} +(-3.62148e7 + 2.17983e8i) q^{50} +(1.05685e8 - 2.22347e8i) q^{51} +(1.49284e7 - 4.00006e6i) q^{52} +(4.55030e8 - 4.55030e8i) q^{53} +(-3.11680e8 - 9.09489e7i) q^{54} +(8.08713e7 - 3.93211e6i) q^{55} +(4.66721e7 - 8.08384e7i) q^{56} +(6.53620e8 - 2.32455e8i) q^{57} +(-4.13738e8 - 1.10861e8i) q^{58} +(9.77030e7 - 5.64089e7i) q^{59} +(5.16823e7 - 3.85350e8i) q^{60} +(4.79515e8 - 8.30544e8i) q^{61} +(-4.38480e8 - 4.38480e8i) q^{62} +(-2.78606e8 + 3.85659e8i) q^{63} -1.34218e8i q^{64} +(-2.88106e7 - 8.98228e7i) q^{65} +(-1.42007e8 + 1.13696e7i) q^{66} +(5.04489e8 - 1.88278e9i) q^{67} +(-5.01038e8 - 1.34253e8i) q^{68} +(4.96342e7 + 6.19933e8i) q^{69} +(-5.06650e8 - 2.60566e8i) q^{70} -6.72955e8 q^{71} +(-6.96656e7 + 6.80540e8i) q^{72} +(-1.98303e9 + 1.98303e9i) q^{73} +(-1.11788e9 - 6.45407e8i) q^{74} +(-2.36453e9 - 2.00849e8i) q^{75} +(-7.30838e8 - 1.26585e9i) q^{76} +(-5.40300e7 + 2.01643e8i) q^{77} +(5.56152e7 + 1.56379e8i) q^{78} +(-7.30526e7 - 4.21769e7i) q^{79} +(-8.18233e8 + 3.97840e7i) q^{80} +(7.06467e8 - 3.41446e9i) q^{81} +(1.25525e9 + 1.25525e9i) q^{82} +(-6.46814e7 - 2.41394e8i) q^{83} +(9.05371e8 + 4.30336e8i) q^{84} +(-6.69932e8 + 3.09428e9i) q^{85} +(-2.39279e9 - 4.14444e9i) q^{86} +(8.32152e8 - 4.52406e9i) q^{87} +(7.76888e7 + 2.89938e8i) q^{88} -2.29353e8i q^{89} +(4.16944e9 + 2.22981e8i) q^{90} +2.43211e8 q^{91} +(1.26573e9 - 3.39150e8i) q^{92} +(4.31808e9 - 5.06970e9i) q^{93} +(4.32422e9 - 2.49659e9i) q^{94} +(-7.50038e9 + 4.83063e9i) q^{95} +(1.43679e9 - 1.15035e8i) q^{96} +(-3.89960e9 + 1.04489e9i) q^{97} +(-3.48092e9 + 3.48092e9i) q^{98} +(-2.43422e8 - 1.51043e9i) 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
\(3\) 81.4250 + 228.952i 0.335082 + 0.942189i
\(4\) 443.405 − 256.000i 0.433013 − 0.250000i
\(5\) −1692.09 − 2627.25i −0.541468 − 0.840721i
\(6\) 3120.49 + 4527.21i 0.401298 + 0.582203i
\(7\) 7782.62 − 2085.35i 0.463058 − 0.124076i −0.0197448 − 0.999805i \(-0.506285\pi\)
0.482803 + 0.875729i \(0.339619\pi\)
\(8\) 8192.00 − 8192.00i 0.250000 − 0.250000i
\(9\) −45788.9 + 37284.8i −0.775439 + 0.631422i
\(10\) −52369.2 − 47512.8i −0.523692 − 0.475128i
\(11\) −12954.7 + 22438.2i −0.0804384 + 0.139323i −0.903438 − 0.428718i \(-0.858965\pi\)
0.823000 + 0.568042i \(0.192299\pi\)
\(12\) 94716.0 + 80673.6i 0.380642 + 0.324209i
\(13\) 29157.1 + 7812.62i 0.0785286 + 0.0210417i 0.297869 − 0.954607i \(-0.403724\pi\)
−0.219341 + 0.975648i \(0.570391\pi\)
\(14\) 157887. − 91156.4i 0.293567 − 0.169491i
\(15\) 463736. − 601331.i 0.610682 − 0.791876i
\(16\) 131072. − 227023.i 0.125000 − 0.216506i
\(17\) −716377. − 716377.i −0.504542 − 0.504542i 0.408304 − 0.912846i \(-0.366120\pi\)
−0.912846 + 0.408304i \(0.866120\pi\)
\(18\) −782426. + 1.08307e6i −0.414077 + 0.573184i
\(19\) − 2.85484e6i − 1.15296i −0.817112 − 0.576479i \(-0.804426\pi\)
0.817112 − 0.576479i \(-0.195574\pi\)
\(20\) −1.42286e6 − 731763.i −0.444643 − 0.228676i
\(21\) 1.11114e6 + 1.61205e6i 0.272066 + 0.394713i
\(22\) −151736. + 566286.i −0.0294425 + 0.109881i
\(23\) 2.47212e6 + 662403.i 0.384088 + 0.102916i 0.445696 − 0.895184i \(-0.352956\pi\)
−0.0616080 + 0.998100i \(0.519623\pi\)
\(24\) 2.54261e6 + 1.20854e6i 0.319318 + 0.151777i
\(25\) −4.03930e6 + 8.89109e6i −0.413625 + 0.910448i
\(26\) 683023. 0.0574869
\(27\) −1.22648e7 − 7.44754e6i −0.854755 − 0.519032i
\(28\) 2.91700e6 − 2.91700e6i 0.169491 − 0.169491i
\(29\) −1.63937e7 − 9.46493e6i −0.799260 − 0.461453i 0.0439526 − 0.999034i \(-0.486005\pi\)
−0.843212 + 0.537581i \(0.819338\pi\)
\(30\) 6.61397e6 − 1.58588e7i 0.272180 − 0.652624i
\(31\) −1.37025e7 − 2.37334e7i −0.478620 − 0.828995i 0.521079 − 0.853508i \(-0.325530\pi\)
−0.999699 + 0.0245138i \(0.992196\pi\)
\(32\) 1.53522e6 − 5.72953e6i 0.0457532 − 0.170753i
\(33\) −6.19210e6 − 1.13897e6i −0.158223 − 0.0291034i
\(34\) −1.98528e7 − 1.14620e7i −0.436946 − 0.252271i
\(35\) −1.86476e7 − 1.69183e7i −0.355045 − 0.322120i
\(36\) −1.07581e7 + 2.82542e7i −0.177920 + 0.467274i
\(37\) −4.03379e7 − 4.03379e7i −0.581708 − 0.581708i 0.353664 − 0.935372i \(-0.384936\pi\)
−0.935372 + 0.353664i \(0.884936\pi\)
\(38\) −1.67191e7 − 6.23965e7i −0.211006 − 0.787485i
\(39\) 585404. + 7.31172e6i 0.00648832 + 0.0810394i
\(40\) −3.53840e7 − 7.66088e6i −0.345547 − 0.0748133i
\(41\) 3.92265e7 + 6.79422e7i 0.338579 + 0.586436i 0.984166 − 0.177251i \(-0.0567205\pi\)
−0.645587 + 0.763687i \(0.723387\pi\)
\(42\) 3.37264e7 + 2.87262e7i 0.258062 + 0.219802i
\(43\) −5.47389e7 − 2.04289e8i −0.372352 − 1.38964i −0.857175 − 0.515025i \(-0.827782\pi\)
0.484822 − 0.874613i \(-0.338884\pi\)
\(44\) 1.32656e7i 0.0804384i
\(45\) 1.75436e8 + 5.72099e7i 0.950726 + 0.310034i
\(46\) 5.79110e7 0.281172
\(47\) 2.13150e8 − 5.71135e7i 0.929388 − 0.249029i 0.237794 − 0.971316i \(-0.423576\pi\)
0.691594 + 0.722287i \(0.256909\pi\)
\(48\) 6.26500e7 + 1.15238e7i 0.245875 + 0.0452261i
\(49\) −1.88410e8 + 1.08779e8i −0.666997 + 0.385091i
\(50\) −3.62148e7 + 2.17983e8i −0.115887 + 0.697546i
\(51\) 1.05685e8 − 2.22347e8i 0.306310 − 0.644437i
\(52\) 1.49284e7 − 4.00006e6i 0.0392643 − 0.0105208i
\(53\) 4.55030e8 − 4.55030e8i 1.08808 − 1.08808i 0.0923532 − 0.995726i \(-0.470561\pi\)
0.995726 − 0.0923532i \(-0.0294389\pi\)
\(54\) −3.11680e8 − 9.09489e7i −0.678798 − 0.198074i
\(55\) 8.08713e7 − 3.93211e6i 0.160687 − 0.00781290i
\(56\) 4.66721e7 − 8.08384e7i 0.0847455 − 0.146784i
\(57\) 6.53620e8 − 2.32455e8i 1.08630 − 0.386336i
\(58\) −4.13738e8 − 1.10861e8i −0.630356 − 0.168903i
\(59\) 9.77030e7 − 5.64089e7i 0.136662 − 0.0789019i −0.430110 − 0.902776i \(-0.641525\pi\)
0.566772 + 0.823875i \(0.308192\pi\)
\(60\) 5.16823e7 − 3.85350e8i 0.0664639 − 0.495563i
\(61\) 4.79515e8 − 8.30544e8i 0.567745 − 0.983363i −0.429044 − 0.903284i \(-0.641149\pi\)
0.996789 − 0.0800790i \(-0.0255173\pi\)
\(62\) −4.38480e8 − 4.38480e8i −0.478620 − 0.478620i
\(63\) −2.78606e8 + 3.85659e8i −0.280729 + 0.388599i
\(64\) − 1.34218e8i − 0.125000i
\(65\) −2.88106e7 − 8.98228e7i −0.0248305 − 0.0774140i
\(66\) −1.42007e8 + 1.13696e7i −0.113394 + 0.00907877i
\(67\) 5.04489e8 − 1.88278e9i 0.373661 − 1.39452i −0.481631 − 0.876374i \(-0.659955\pi\)
0.855292 − 0.518147i \(-0.173378\pi\)
\(68\) −5.01038e8 − 1.34253e8i −0.344608 − 0.0923375i
\(69\) 4.96342e7 + 6.19933e8i 0.0317348 + 0.396369i
\(70\) −5.06650e8 − 2.60566e8i −0.301452 − 0.155034i
\(71\) −6.72955e8 −0.372987 −0.186494 − 0.982456i \(-0.559712\pi\)
−0.186494 + 0.982456i \(0.559712\pi\)
\(72\) −6.96656e7 + 6.80540e8i −0.0360044 + 0.351715i
\(73\) −1.98303e9 + 1.98303e9i −0.956568 + 0.956568i −0.999095 − 0.0425274i \(-0.986459\pi\)
0.0425274 + 0.999095i \(0.486459\pi\)
\(74\) −1.11788e9 − 6.45407e8i −0.503774 − 0.290854i
\(75\) −2.36453e9 − 2.00849e8i −0.996412 − 0.0846375i
\(76\) −7.30838e8 − 1.26585e9i −0.288239 − 0.499245i
\(77\) −5.40300e7 + 2.01643e8i −0.0199610 + 0.0744954i
\(78\) 5.56152e7 + 1.56379e8i 0.0192628 + 0.0541635i
\(79\) −7.30526e7 − 4.21769e7i −0.0237411 − 0.0137069i 0.488082 − 0.872797i \(-0.337697\pi\)
−0.511824 + 0.859091i \(0.671030\pi\)
\(80\) −8.18233e8 + 3.97840e7i −0.249705 + 0.0121411i
\(81\) 7.06467e8 − 3.41446e9i 0.202613 − 0.979259i
\(82\) 1.25525e9 + 1.25525e9i 0.338579 + 0.338579i
\(83\) −6.46814e7 − 2.41394e8i −0.0164206 − 0.0612825i 0.957229 − 0.289330i \(-0.0934325\pi\)
−0.973650 + 0.228048i \(0.926766\pi\)
\(84\) 9.05371e8 + 4.30336e8i 0.216486 + 0.102899i
\(85\) −6.69932e8 + 3.09428e9i −0.150986 + 0.697372i
\(86\) −2.39279e9 − 4.14444e9i −0.508643 − 0.880995i
\(87\) 8.32152e8 − 4.52406e9i 0.166958 − 0.907678i
\(88\) 7.76888e7 + 2.89938e8i 0.0147213 + 0.0549405i
\(89\) − 2.29353e8i − 0.0410728i −0.999789 − 0.0205364i \(-0.993463\pi\)
0.999789 − 0.0205364i \(-0.00653741\pi\)
\(90\) 4.16944e9 + 2.22981e8i 0.706098 + 0.0377621i
\(91\) 2.43211e8 0.0389741
\(92\) 1.26573e9 − 3.39150e8i 0.192044 − 0.0514580i
\(93\) 4.31808e9 − 5.06970e9i 0.620692 − 0.728732i
\(94\) 4.32422e9 − 2.49659e9i 0.589208 − 0.340180i
\(95\) −7.50038e9 + 4.83063e9i −0.969316 + 0.624290i
\(96\) 1.43679e9 − 1.15035e8i 0.176213 − 0.0141083i
\(97\) −3.89960e9 + 1.04489e9i −0.454110 + 0.121678i −0.478622 − 0.878021i \(-0.658864\pi\)
0.0245118 + 0.999700i \(0.492197\pi\)
\(98\) −3.48092e9 + 3.48092e9i −0.385091 + 0.385091i
\(99\) −2.43422e8 − 1.51043e9i −0.0255967 − 0.158828i
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.7.19 ✓ 120
5.3 odd 4 inner 90.11.k.b.43.3 yes 120
9.4 even 3 inner 90.11.k.b.67.3 yes 120
45.13 odd 12 inner 90.11.k.b.13.19 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.k.b.7.19 ✓ 120 1.1 even 1 trivial
90.11.k.b.13.19 yes 120 45.13 odd 12 inner
90.11.k.b.43.3 yes 120 5.3 odd 4 inner
90.11.k.b.67.3 yes 120 9.4 even 3 inner