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 67.13
Character \(\chi\) \(=\) 90.67
Dual form 90.11.k.b.43.13

$q$-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+(-5.85641 + 21.8564i) q^{2} +(-68.3665 + 233.185i) q^{3} +(-443.405 - 256.000i) q^{4} +(-2400.71 - 2000.55i) q^{5} +(-4696.19 - 2859.87i) q^{6} +(7179.49 - 26794.2i) q^{7} +(8192.00 - 8192.00i) q^{8} +(-49701.0 - 31884.0i) q^{9} +(57784.4 - 40754.9i) q^{10} +(-38347.8 - 66420.4i) q^{11} +(90009.3 - 85893.3i) q^{12} +(-109174. - 407442. i) q^{13} +(543580. + 313836. i) q^{14} +(630626. - 423038. i) q^{15} +(131072. + 227023. i) q^{16} +(-1.71814e6 - 1.71814e6i) q^{17} +(987940. - 899560. i) q^{18} +2.67094e6i q^{19} +(552347. + 1.50164e6i) q^{20} +(5.75716e6 + 3.50598e6i) q^{21} +(1.67629e6 - 449161. i) q^{22} +(-719472. - 2.68510e6i) q^{23} +(1.35019e6 + 2.47031e6i) q^{24} +(1.76121e6 + 9.60550e6i) q^{25} +9.54459e6 q^{26} +(1.08328e7 - 9.40971e6i) q^{27} +(-1.00427e7 + 1.00427e7i) q^{28} +(-1.72346e7 + 9.95039e6i) q^{29} +(5.55289e6 + 1.62607e7i) q^{30} +(-2.23240e7 + 3.86663e7i) q^{31} +(-5.72953e6 + 1.53522e6i) q^{32} +(1.81099e7 - 4.40119e6i) q^{33} +(4.76145e7 - 2.74903e7i) q^{34} +(-7.08391e7 + 4.99623e7i) q^{35} +(1.38754e7 + 2.68610e7i) q^{36} +(1.72527e6 + 1.72527e6i) q^{37} +(-5.83773e7 - 1.56421e7i) q^{38} +(1.02473e8 + 2.39777e6i) q^{39} +(-3.60552e7 + 3.27812e6i) q^{40} +(9.46032e7 - 1.63857e8i) q^{41} +(-1.10344e8 + 1.05298e8i) q^{42} +(1.02282e8 + 2.74063e7i) q^{43} +3.92682e7i q^{44} +(5.55322e7 + 1.75974e8i) q^{45} +6.29003e7 q^{46} +(-8.56435e7 + 3.19626e8i) q^{47} +(-6.18993e7 + 1.50432e7i) q^{48} +(-4.21755e8 - 2.43500e8i) q^{49} +(-2.20256e8 - 1.77599e7i) q^{50} +(5.18108e8 - 2.83181e8i) q^{51} +(-5.58970e7 + 2.08611e8i) q^{52} +(5.00384e8 - 5.00384e8i) q^{53} +(1.42221e8 + 2.91872e8i) q^{54} +(-4.08153e7 + 2.36173e8i) q^{55} +(-1.60684e8 - 2.78313e8i) q^{56} +(-6.22823e8 - 1.82603e8i) q^{57} +(-1.16547e8 - 4.34959e8i) q^{58} +(-4.42465e8 - 2.55458e8i) q^{59} +(-3.87920e8 + 2.61370e7i) q^{60} +(7.31433e7 + 1.26688e8i) q^{61} +(-7.14369e8 - 7.14369e8i) q^{62} +(-1.21114e9 + 1.10279e9i) q^{63} -1.34218e8i q^{64} +(-5.53014e8 + 1.19656e9i) q^{65} +(-9.86486e6 + 4.21593e8i) q^{66} +(-1.10111e9 + 2.95043e8i) q^{67} +(3.21988e8 + 1.20168e9i) q^{68} +(6.75313e8 + 1.58017e7i) q^{69} +(-6.77133e8 - 1.84089e9i) q^{70} +1.24826e9 q^{71} +(-6.68345e8 + 1.45957e8i) q^{72} +(2.63992e9 - 2.63992e9i) q^{73} +(-4.78122e7 + 2.76044e7i) q^{74} +(-2.36026e9 - 2.46007e8i) q^{75} +(6.83762e8 - 1.18431e9i) q^{76} +(-2.05500e9 + 5.50636e8i) q^{77} +(-6.52531e8 + 2.22565e9i) q^{78} +(-3.60137e9 + 2.07925e9i) q^{79} +(1.39506e8 - 8.07234e8i) q^{80} +(1.45360e9 + 3.16934e9i) q^{81} +(3.02730e9 + 3.02730e9i) q^{82} +(-1.27987e9 - 3.42940e8i) q^{83} +(-1.65523e9 - 3.02840e9i) q^{84} +(6.87533e8 + 7.56200e9i) q^{85} +(-1.19801e9 + 2.07501e9i) q^{86} +(-1.14201e9 - 4.69911e9i) q^{87} +(-8.58261e8 - 2.29970e8i) q^{88} -4.73776e8i q^{89} +(-4.17138e9 + 1.83160e8i) q^{90} -1.17009e10 q^{91} +(-3.68369e8 + 1.37477e9i) q^{92} +(-7.49018e9 - 7.84910e9i) q^{93} +(-6.48431e9 - 3.74372e9i) q^{94} +(5.34336e9 - 6.41217e9i) q^{95} +(3.37179e7 - 1.44099e9i) q^{96} +(-3.39674e9 + 1.26768e10i) q^{97} +(7.79201e9 - 7.79201e9i) q^{98} +(-2.11823e8 + 4.52385e9i) 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 \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.85641 + 21.8564i −0.183013 + 0.683013i
\(3\) −68.3665 + 233.185i −0.281344 + 0.959607i
\(4\) −443.405 − 256.000i −0.433013 − 0.250000i
\(5\) −2400.71 − 2000.55i −0.768228 − 0.640176i
\(6\) −4696.19 − 2859.87i −0.603934 − 0.367782i
\(7\) 7179.49 − 26794.2i 0.427173 − 1.59423i −0.331959 − 0.943294i \(-0.607710\pi\)
0.759132 − 0.650937i \(-0.225624\pi\)
\(8\) 8192.00 − 8192.00i 0.250000 − 0.250000i
\(9\) −49701.0 − 31884.0i −0.841691 − 0.539959i
\(10\) 57784.4 − 40754.9i 0.577844 − 0.407549i
\(11\) −38347.8 − 66420.4i −0.238110 − 0.412418i 0.722062 − 0.691828i \(-0.243194\pi\)
−0.960172 + 0.279410i \(0.909861\pi\)
\(12\) 90009.3 − 85893.3i 0.361727 − 0.345186i
\(13\) −109174. − 407442.i −0.294037 − 1.09736i −0.941979 − 0.335671i \(-0.891037\pi\)
0.647942 − 0.761689i \(-0.275630\pi\)
\(14\) 543580. + 313836.i 1.01070 + 0.583529i
\(15\) 630626. − 423038.i 0.830454 − 0.557087i
\(16\) 131072. + 227023.i 0.125000 + 0.216506i
\(17\) −1.71814e6 − 1.71814e6i −1.21008 − 1.21008i −0.971000 − 0.239081i \(-0.923154\pi\)
−0.239081 − 0.971000i \(-0.576846\pi\)
\(18\) 987940. − 899560.i 0.522839 − 0.476067i
\(19\) 2.67094e6i 1.07869i 0.842085 + 0.539345i \(0.181328\pi\)
−0.842085 + 0.539345i \(0.818672\pi\)
\(20\) 552347. + 1.50164e6i 0.172608 + 0.469262i
\(21\) 5.75716e6 + 3.50598e6i 1.40965 + 0.858445i
\(22\) 1.67629e6 − 449161.i 0.325264 − 0.0871543i
\(23\) −719472. − 2.68510e6i −0.111783 − 0.417179i 0.887243 − 0.461301i \(-0.152617\pi\)
−0.999026 + 0.0441229i \(0.985951\pi\)
\(24\) 1.35019e6 + 2.47031e6i 0.169566 + 0.310238i
\(25\) 1.76121e6 + 9.60550e6i 0.180348 + 0.983603i
\(26\) 9.54459e6 0.803324
\(27\) 1.08328e7 − 9.40971e6i 0.754953 − 0.655779i
\(28\) −1.00427e7 + 1.00427e7i −0.583529 + 0.583529i
\(29\) −1.72346e7 + 9.95039e6i −0.840254 + 0.485121i −0.857351 − 0.514733i \(-0.827891\pi\)
0.0170965 + 0.999854i \(0.494558\pi\)
\(30\) 5.55289e6 + 1.62607e7i 0.228514 + 0.669165i
\(31\) −2.23240e7 + 3.86663e7i −0.779766 + 1.35059i 0.152311 + 0.988333i \(0.451328\pi\)
−0.932077 + 0.362261i \(0.882005\pi\)
\(32\) −5.72953e6 + 1.53522e6i −0.170753 + 0.0457532i
\(33\) 1.81099e7 − 4.40119e6i 0.462750 − 0.112461i
\(34\) 4.76145e7 − 2.74903e7i 1.04796 − 0.605040i
\(35\) −7.08391e7 + 4.99623e7i −1.34875 + 0.951266i
\(36\) 1.38754e7 + 2.68610e7i 0.229473 + 0.444232i
\(37\) 1.72527e6 + 1.72527e6i 0.0248799 + 0.0248799i 0.719437 − 0.694557i \(-0.244400\pi\)
−0.694557 + 0.719437i \(0.744400\pi\)
\(38\) −5.83773e7 − 1.56421e7i −0.736759 − 0.197414i
\(39\) 1.02473e8 + 2.39777e6i 1.13576 + 0.0265757i
\(40\) −3.60552e7 + 3.27812e6i −0.352101 + 0.0320128i
\(41\) 9.46032e7 − 1.63857e8i 0.816557 − 1.41432i −0.0916480 − 0.995791i \(-0.529213\pi\)
0.908205 − 0.418526i \(-0.137453\pi\)
\(42\) −1.10344e8 + 1.05298e8i −0.844313 + 0.805704i
\(43\) 1.02282e8 + 2.74063e7i 0.695755 + 0.186427i 0.589328 − 0.807894i \(-0.299392\pi\)
0.106427 + 0.994321i \(0.466059\pi\)
\(44\) 3.92682e7i 0.238110i
\(45\) 5.55322e7 + 1.75974e8i 0.300942 + 0.953643i
\(46\) 6.29003e7 0.305396
\(47\) −8.56435e7 + 3.19626e8i −0.373426 + 1.39365i 0.482204 + 0.876059i \(0.339836\pi\)
−0.855630 + 0.517587i \(0.826830\pi\)
\(48\) −6.18993e7 + 1.50432e7i −0.242929 + 0.0590382i
\(49\) −4.21755e8 − 2.43500e8i −1.49307 − 0.862024i
\(50\) −2.20256e8 − 1.77599e7i −0.704819 − 0.0568318i
\(51\) 5.18108e8 − 2.83181e8i 1.50165 − 0.820754i
\(52\) −5.58970e7 + 2.08611e8i −0.147018 + 0.548680i
\(53\) 5.00384e8 − 5.00384e8i 1.19653 − 1.19653i 0.221334 − 0.975198i \(-0.428959\pi\)
0.975198 − 0.221334i \(-0.0710411\pi\)
\(54\) 1.42221e8 + 2.91872e8i 0.309739 + 0.635658i
\(55\) −4.08153e7 + 2.36173e8i −0.0810979 + 0.469264i
\(56\) −1.60684e8 − 2.78313e8i −0.291764 − 0.505351i
\(57\) −6.22823e8 − 1.82603e8i −1.03512 − 0.303483i
\(58\) −1.16547e8 − 4.34959e8i −0.177567 − 0.662687i
\(59\) −4.42465e8 − 2.55458e8i −0.618898 − 0.357321i 0.157542 − 0.987512i \(-0.449643\pi\)
−0.776440 + 0.630191i \(0.782976\pi\)
\(60\) −3.87920e8 + 2.61370e7i −0.498869 + 0.0336123i
\(61\) 7.31433e7 + 1.26688e8i 0.0866015 + 0.149998i 0.906073 − 0.423122i \(-0.139066\pi\)
−0.819471 + 0.573121i \(0.805733\pi\)
\(62\) −7.14369e8 − 7.14369e8i −0.779766 − 0.779766i
\(63\) −1.21114e9 + 1.10279e9i −1.22037 + 1.11119i
\(64\) − 1.34218e8i − 0.125000i
\(65\) −5.53014e8 + 1.19656e9i −0.476617 + 1.03126i
\(66\) −9.86486e6 + 4.21593e8i −0.00787719 + 0.336646i
\(67\) −1.10111e9 + 2.95043e8i −0.815565 + 0.218530i −0.642407 − 0.766364i \(-0.722064\pi\)
−0.173158 + 0.984894i \(0.555397\pi\)
\(68\) 3.21988e8 + 1.20168e9i 0.221460 + 0.826501i
\(69\) 6.75313e8 + 1.58017e7i 0.431777 + 0.0101032i
\(70\) −6.77133e8 − 1.84089e9i −0.402888 − 1.09531i
\(71\) 1.24826e9 0.691854 0.345927 − 0.938261i \(-0.387564\pi\)
0.345927 + 0.938261i \(0.387564\pi\)
\(72\) −6.68345e8 + 1.45957e8i −0.345413 + 0.0754331i
\(73\) 2.63992e9 − 2.63992e9i 1.27343 − 1.27343i 0.329159 − 0.944274i \(-0.393235\pi\)
0.944274 − 0.329159i \(-0.106765\pi\)
\(74\) −4.78122e7 + 2.76044e7i −0.0215467 + 0.0124400i
\(75\) −2.36026e9 − 2.46007e8i −0.994612 − 0.103667i
\(76\) 6.83762e8 − 1.18431e9i 0.269673 − 0.467087i
\(77\) −2.05500e9 + 5.50636e8i −0.759204 + 0.203428i
\(78\) −6.52531e8 + 2.22565e9i −0.226010 + 0.770875i
\(79\) −3.60137e9 + 2.07925e9i −1.17039 + 0.675727i −0.953773 − 0.300528i \(-0.902837\pi\)
−0.216621 + 0.976256i \(0.569504\pi\)
\(80\) 1.39506e8 − 8.07234e8i 0.0425738 − 0.246348i
\(81\) 1.45360e9 + 3.16934e9i 0.416889 + 0.908958i
\(82\) 3.02730e9 + 3.02730e9i 0.816557 + 0.816557i
\(83\) −1.27987e9 − 3.42940e8i −0.324919 − 0.0870617i 0.0926727 − 0.995697i \(-0.470459\pi\)
−0.417591 + 0.908635i \(0.637126\pi\)
\(84\) −1.65523e9 − 3.02840e9i −0.395786 − 0.724131i
\(85\) 6.87533e8 + 7.56200e9i 0.154953 + 1.70428i
\(86\) −1.19801e9 + 2.07501e9i −0.254664 + 0.441091i
\(87\) −1.14201e9 − 4.69911e9i −0.229125 − 0.942799i
\(88\) −8.58261e8 − 2.29970e8i −0.162632 − 0.0435771i
\(89\) − 4.73776e8i − 0.0848444i −0.999100 − 0.0424222i \(-0.986493\pi\)
0.999100 − 0.0424222i \(-0.0135075\pi\)
\(90\) −4.17138e9 + 1.83160e8i −0.706426 + 0.0310183i
\(91\) −1.17009e10 −1.87505
\(92\) −3.68369e8 + 1.37477e9i −0.0558913 + 0.208589i
\(93\) −7.49018e9 − 7.84910e9i −1.07666 − 1.12825i
\(94\) −6.48431e9 − 3.74372e9i −0.883536 − 0.510110i
\(95\) 5.34336e9 − 6.41217e9i 0.690552 − 0.828680i
\(96\) 3.37179e7 − 1.44099e9i 0.00413527 − 0.176728i
\(97\) −3.39674e9 + 1.26768e10i −0.395552 + 1.47622i 0.425285 + 0.905059i \(0.360174\pi\)
−0.820837 + 0.571162i \(0.806493\pi\)
\(98\) 7.79201e9 − 7.79201e9i 0.862024 − 0.862024i
\(99\) −2.11823e8 + 4.52385e9i −0.0222740 + 0.475699i
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.67.13 yes 120
5.3 odd 4 inner 90.11.k.b.13.4 yes 120
9.7 even 3 inner 90.11.k.b.7.4 ✓ 120
45.43 odd 12 inner 90.11.k.b.43.13 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.k.b.7.4 ✓ 120 9.7 even 3 inner
90.11.k.b.13.4 yes 120 5.3 odd 4 inner
90.11.k.b.43.13 yes 120 45.43 odd 12 inner
90.11.k.b.67.13 yes 120 1.1 even 1 trivial