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 7.11
Character \(\chi\) \(=\) 90.7
Dual form 90.11.k.b.13.11

$q$-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} +(-105.656 + 218.828i) q^{3} +(443.405 - 256.000i) q^{4} +(351.923 + 3105.12i) q^{5} +(-1027.72 + 5401.56i) q^{6} +(-9471.76 + 2537.95i) q^{7} +(8192.00 - 8192.00i) q^{8} +(-36722.5 - 46241.1i) q^{9} +(25876.6 + 65805.8i) q^{10} +(112176. - 194295. i) q^{11} +(9171.48 + 124077. i) q^{12} +(-270752. - 72547.9i) q^{13} +(-192155. + 110941. i) q^{14} +(-716671. - 251065. i) q^{15} +(131072. - 227023. i) q^{16} +(-713217. - 713217. i) q^{17} +(-1.07343e6 - 795603. i) q^{18} -2.28737e6i q^{19} +(950955. + 1.28673e6i) q^{20} +(445376. - 2.34084e6i) q^{21} +(1.31390e6 - 4.90355e6i) q^{22} +(8.18835e6 + 2.19406e6i) q^{23} +(927104. + 2.65818e6i) q^{24} +(-9.51793e6 + 2.18553e6i) q^{25} -6.34254e6 q^{26} +(1.39988e7 - 3.15025e6i) q^{27} +(-3.55011e6 + 3.55011e6i) q^{28} +(1.24233e7 + 7.17261e6i) q^{29} +(-1.71342e7 - 1.29026e6i) q^{30} +(-8.17510e6 - 1.41597e7i) q^{31} +(1.53522e6 - 5.72953e6i) q^{32} +(3.06651e7 + 4.50759e7i) q^{33} +(-1.97652e7 - 1.14115e7i) q^{34} +(-1.12140e7 - 2.85178e7i) q^{35} +(-2.81207e7 - 1.11026e7i) q^{36} +(2.61279e7 + 2.61279e7i) q^{37} +(-1.33958e7 - 4.99936e7i) q^{38} +(4.44822e7 - 5.15831e7i) q^{39} +(2.83201e7 + 2.25542e7i) q^{40} +(-1.20416e7 - 2.08566e7i) q^{41} +(-3.97458e6 - 5.37706e7i) q^{42} +(1.44889e7 + 5.40733e7i) q^{43} -1.14869e8i q^{44} +(1.30661e8 - 1.30301e8i) q^{45} +1.91817e8 q^{46} +(-1.16398e7 + 3.11889e6i) q^{47} +(3.58305e7 + 5.26687e7i) q^{48} +(-1.61358e8 + 9.31599e7i) q^{49} +(-1.95228e8 + 1.03509e8i) q^{50} +(2.31428e8 - 8.07161e7i) q^{51} +(-1.38625e8 + 3.71445e7i) q^{52} +(3.63350e8 - 3.63350e8i) q^{53} +(2.87515e8 - 1.50836e8i) q^{54} +(6.42788e8 + 2.79944e8i) q^{55} +(-5.68018e7 + 9.83836e7i) q^{56} +(5.00540e8 + 2.41675e8i) q^{57} +(3.13535e8 + 8.40114e7i) q^{58} +(1.02700e9 - 5.92936e8i) q^{59} +(-3.82048e8 + 7.21443e7i) q^{60} +(5.40453e7 - 9.36092e7i) q^{61} +(-2.61603e8 - 2.61603e8i) q^{62} +(4.65184e8 + 3.44785e8i) q^{63} -1.34218e8i q^{64} +(1.29986e8 - 8.66250e8i) q^{65} +(9.34212e8 + 8.05609e8i) q^{66} +(1.40255e8 - 5.23439e8i) q^{67} +(-4.98827e8 - 1.33660e8i) q^{68} +(-1.34527e9 + 1.56002e9i) q^{69} +(-4.12109e8 - 5.57623e8i) q^{70} -1.48814e9 q^{71} +(-6.79638e8 - 7.79766e7i) q^{72} +(1.76878e9 - 1.76878e9i) q^{73} +(7.24077e8 + 4.18046e8i) q^{74} +(5.27374e8 - 2.31370e9i) q^{75} +(-5.85566e8 - 1.01423e9i) q^{76} +(-5.69397e8 + 2.12502e9i) q^{77} +(6.70130e8 - 1.38793e9i) q^{78} +(-1.07248e9 - 6.19195e8i) q^{79} +(7.51062e8 + 3.27100e8i) q^{80} +(-7.89700e8 + 3.39618e9i) q^{81} +(-3.85330e8 - 3.85330e8i) q^{82} +(-1.40444e9 - 5.24143e9i) q^{83} +(-4.01773e8 - 1.15196e9i) q^{84} +(1.96363e9 - 2.46562e9i) q^{85} +(6.33350e8 + 1.09699e9i) q^{86} +(-2.88217e9 + 1.96074e9i) q^{87} +(-6.72718e8 - 2.51062e9i) q^{88} -3.87540e9i q^{89} +(2.09268e9 - 3.61312e9i) q^{90} +2.74862e9 q^{91} +(4.19243e9 - 1.12336e9i) q^{92} +(3.96229e9 - 2.92882e8i) q^{93} +(-2.36140e8 + 1.36335e8i) q^{94} +(7.10255e9 - 8.04978e8i) q^{95} +(1.09158e9 + 9.41310e8i) q^{96} +(-1.08409e10 + 2.90482e9i) q^{97} +(-2.98112e9 + 2.98112e9i) q^{98} +(-1.31038e10 + 1.94784e9i) 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
\(3\) −105.656 + 218.828i −0.434800 + 0.900527i
\(4\) 443.405 − 256.000i 0.433013 − 0.250000i
\(5\) 351.923 + 3105.12i 0.112615 + 0.993639i
\(6\) −1027.72 + 5401.56i −0.132166 + 0.694645i
\(7\) −9471.76 + 2537.95i −0.563560 + 0.151006i −0.529342 − 0.848409i \(-0.677561\pi\)
−0.0342189 + 0.999414i \(0.510894\pi\)
\(8\) 8192.00 − 8192.00i 0.250000 − 0.250000i
\(9\) −36722.5 − 46241.1i −0.621899 − 0.783098i
\(10\) 25876.6 + 65805.8i 0.258766 + 0.658058i
\(11\) 112176. − 194295.i 0.696528 − 1.20642i −0.273135 − 0.961976i \(-0.588061\pi\)
0.969663 − 0.244446i \(-0.0786059\pi\)
\(12\) 9171.48 + 124077.i 0.0368581 + 0.498640i
\(13\) −270752. − 72547.9i −0.729215 − 0.195393i −0.124935 − 0.992165i \(-0.539872\pi\)
−0.604280 + 0.796772i \(0.706539\pi\)
\(14\) −192155. + 110941.i −0.357283 + 0.206277i
\(15\) −716671. − 251065.i −0.943764 − 0.330620i
\(16\) 131072. − 227023.i 0.125000 − 0.216506i
\(17\) −713217. − 713217.i −0.502316 − 0.502316i 0.409841 − 0.912157i \(-0.365584\pi\)
−0.912157 + 0.409841i \(0.865584\pi\)
\(18\) −1.07343e6 − 795603.i −0.568082 − 0.421050i
\(19\) − 2.28737e6i − 0.923779i −0.886938 − 0.461889i \(-0.847172\pi\)
0.886938 − 0.461889i \(-0.152828\pi\)
\(20\) 950955. + 1.28673e6i 0.297174 + 0.402104i
\(21\) 445376. − 2.34084e6i 0.109051 − 0.573159i
\(22\) 1.31390e6 − 4.90355e6i 0.254947 − 0.951474i
\(23\) 8.18835e6 + 2.19406e6i 1.27221 + 0.340886i 0.830875 − 0.556459i \(-0.187840\pi\)
0.441330 + 0.897345i \(0.354507\pi\)
\(24\) 927104. + 2.65818e6i 0.116432 + 0.333832i
\(25\) −9.51793e6 + 2.18553e6i −0.974636 + 0.223798i
\(26\) −6.34254e6 −0.533822
\(27\) 1.39988e7 − 3.15025e6i 0.975602 − 0.219546i
\(28\) −3.55011e6 + 3.55011e6i −0.206277 + 0.206277i
\(29\) 1.24233e7 + 7.17261e6i 0.605686 + 0.349693i 0.771275 − 0.636502i \(-0.219619\pi\)
−0.165589 + 0.986195i \(0.552953\pi\)
\(30\) −1.71342e7 − 1.29026e6i −0.705110 − 0.0530971i
\(31\) −8.17510e6 − 1.41597e7i −0.285552 − 0.494590i 0.687191 − 0.726477i \(-0.258843\pi\)
−0.972743 + 0.231887i \(0.925510\pi\)
\(32\) 1.53522e6 − 5.72953e6i 0.0457532 − 0.170753i
\(33\) 3.06651e7 + 4.50759e7i 0.783565 + 1.15179i
\(34\) −1.97652e7 − 1.14115e7i −0.435018 − 0.251158i
\(35\) −1.12140e7 − 2.85178e7i −0.213511 − 0.542970i
\(36\) −2.81207e7 − 1.11026e7i −0.465064 − 0.183617i
\(37\) 2.61279e7 + 2.61279e7i 0.376787 + 0.376787i 0.869941 − 0.493155i \(-0.164156\pi\)
−0.493155 + 0.869941i \(0.664156\pi\)
\(38\) −1.33958e7 − 4.99936e7i −0.169063 − 0.630953i
\(39\) 4.44822e7 − 5.15831e7i 0.493019 − 0.571721i
\(40\) 2.83201e7 + 2.25542e7i 0.276564 + 0.220256i
\(41\) −1.20416e7 − 2.08566e7i −0.103935 − 0.180022i 0.809367 − 0.587303i \(-0.199810\pi\)
−0.913303 + 0.407281i \(0.866477\pi\)
\(42\) −3.97458e6 − 5.37706e7i −0.0304120 − 0.411432i
\(43\) 1.44889e7 + 5.40733e7i 0.0985582 + 0.367824i 0.997535 − 0.0701722i \(-0.0223549\pi\)
−0.898977 + 0.437997i \(0.855688\pi\)
\(44\) − 1.14869e8i − 0.696528i
\(45\) 1.30661e8 − 1.30301e8i 0.708081 − 0.706132i
\(46\) 1.91817e8 0.931319
\(47\) −1.16398e7 + 3.11889e6i −0.0507525 + 0.0135991i −0.284106 − 0.958793i \(-0.591697\pi\)
0.233353 + 0.972392i \(0.425030\pi\)
\(48\) 3.58305e7 + 5.26687e7i 0.140620 + 0.206703i
\(49\) −1.61358e8 + 9.31599e7i −0.571228 + 0.329798i
\(50\) −1.95228e8 + 1.03509e8i −0.624731 + 0.331228i
\(51\) 2.31428e8 − 8.07161e7i 0.670756 − 0.233942i
\(52\) −1.38625e8 + 3.71445e7i −0.364607 + 0.0976963i
\(53\) 3.63350e8 − 3.63350e8i 0.868853 − 0.868853i −0.123492 − 0.992346i \(-0.539409\pi\)
0.992346 + 0.123492i \(0.0394094\pi\)
\(54\) 2.87515e8 − 1.50836e8i 0.626169 − 0.328501i
\(55\) 6.42788e8 + 2.79944e8i 1.27719 + 0.556235i
\(56\) −5.68018e7 + 9.83836e7i −0.103139 + 0.178642i
\(57\) 5.00540e8 + 2.41675e8i 0.831888 + 0.401659i
\(58\) 3.13535e8 + 8.40114e7i 0.477690 + 0.127997i
\(59\) 1.02700e9 − 5.92936e8i 1.43651 − 0.829369i 0.438904 − 0.898534i \(-0.355367\pi\)
0.997605 + 0.0691649i \(0.0220335\pi\)
\(60\) −3.82048e8 + 7.21443e7i −0.491317 + 0.0927782i
\(61\) 5.40453e7 − 9.36092e7i 0.0639895 − 0.110833i −0.832256 − 0.554392i \(-0.812951\pi\)
0.896245 + 0.443559i \(0.146284\pi\)
\(62\) −2.61603e8 − 2.61603e8i −0.285552 − 0.285552i
\(63\) 4.65184e8 + 3.44785e8i 0.468730 + 0.347413i
\(64\) − 1.34218e8i − 0.125000i
\(65\) 1.29986e8 − 8.66250e8i 0.112029 − 0.746580i
\(66\) 9.34212e8 + 8.05609e8i 0.745978 + 0.643287i
\(67\) 1.40255e8 − 5.23439e8i 0.103883 − 0.387697i −0.894333 − 0.447402i \(-0.852349\pi\)
0.998216 + 0.0597050i \(0.0190160\pi\)
\(68\) −4.98827e8 − 1.33660e8i −0.343088 − 0.0919302i
\(69\) −1.34527e9 + 1.56002e9i −0.860132 + 0.997438i
\(70\) −4.12109e8 − 5.57623e8i −0.245201 − 0.331780i
\(71\) −1.48814e9 −0.824805 −0.412402 − 0.911002i \(-0.635310\pi\)
−0.412402 + 0.911002i \(0.635310\pi\)
\(72\) −6.79638e8 − 7.79766e7i −0.351249 − 0.0402997i
\(73\) 1.76878e9 − 1.76878e9i 0.853215 − 0.853215i −0.137313 − 0.990528i \(-0.543846\pi\)
0.990528 + 0.137313i \(0.0438465\pi\)
\(74\) 7.24077e8 + 4.18046e8i 0.326307 + 0.188393i
\(75\) 5.27374e8 − 2.31370e9i 0.222235 − 0.974993i
\(76\) −5.85566e8 − 1.01423e9i −0.230945 − 0.400008i
\(77\) −5.69397e8 + 2.12502e9i −0.210359 + 0.785071i
\(78\) 6.70130e8 − 1.38793e9i 0.232106 − 0.480722i
\(79\) −1.07248e9 − 6.19195e8i −0.348540 − 0.201230i 0.315502 − 0.948925i \(-0.397827\pi\)
−0.664042 + 0.747695i \(0.731160\pi\)
\(80\) 7.51062e8 + 3.27100e8i 0.229206 + 0.0998229i
\(81\) −7.89700e8 + 3.39618e9i −0.226484 + 0.974015i
\(82\) −3.85330e8 − 3.85330e8i −0.103935 − 0.103935i
\(83\) −1.40444e9 − 5.24143e9i −0.356543 − 1.33064i −0.878532 − 0.477684i \(-0.841476\pi\)
0.521989 − 0.852952i \(-0.325190\pi\)
\(84\) −4.01773e8 − 1.15196e9i −0.0960691 − 0.275448i
\(85\) 1.96363e9 − 2.46562e9i 0.442552 − 0.555689i
\(86\) 6.33350e8 + 1.09699e9i 0.134633 + 0.233191i
\(87\) −2.88217e9 + 1.96074e9i −0.578260 + 0.393390i
\(88\) −6.72718e8 − 2.51062e9i −0.127473 − 0.475737i
\(89\) − 3.87540e9i − 0.694010i −0.937863 − 0.347005i \(-0.887199\pi\)
0.937863 − 0.347005i \(-0.112801\pi\)
\(90\) 2.09268e9 − 3.61312e9i 0.354397 − 0.611885i
\(91\) 2.74862e9 0.440462
\(92\) 4.19243e9 − 1.12336e9i 0.636103 − 0.170443i
\(93\) 3.96229e9 − 2.92882e8i 0.569550 − 0.0420996i
\(94\) −2.36140e8 + 1.36335e8i −0.0321758 + 0.0185767i
\(95\) 7.10255e9 − 8.04978e8i 0.917902 − 0.104032i
\(96\) 1.09158e9 + 9.41310e8i 0.133874 + 0.115445i
\(97\) −1.08409e10 + 2.90482e9i −1.26243 + 0.338268i −0.827127 − 0.562014i \(-0.810027\pi\)
−0.435306 + 0.900282i \(0.643360\pi\)
\(98\) −2.98112e9 + 2.98112e9i −0.329798 + 0.329798i
\(99\) −1.31038e10 + 1.94784e9i −1.37792 + 0.204823i
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.7.11 ✓ 120
5.3 odd 4 inner 90.11.k.b.43.4 yes 120
9.4 even 3 inner 90.11.k.b.67.4 yes 120
45.13 odd 12 inner 90.11.k.b.13.11 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.k.b.7.11 ✓ 120 1.1 even 1 trivial
90.11.k.b.13.11 yes 120 45.13 odd 12 inner
90.11.k.b.43.4 yes 120 5.3 odd 4 inner
90.11.k.b.67.4 yes 120 9.4 even 3 inner