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 7.4
Character \(\chi\) \(=\) 90.7
Dual form 90.11.k.b.13.4

$q$-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} +(-233.185 + 68.3665i) q^{3} +(443.405 - 256.000i) q^{4} +(2932.88 - 1078.80i) q^{5} +(-4696.19 + 2859.87i) q^{6} +(-26794.2 + 7179.49i) 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} +(-85893.3 + 90009.3i) q^{12} +(407442. + 109174. i) q^{13} +(-543580. + 313836. i) q^{14} +(-610149. + 452071. i) q^{15} +(131072. - 227023. i) q^{16} +(-1.71814e6 - 1.71814e6i) q^{17} +(899560. - 987940. i) q^{18} +2.67094e6i q^{19} +(1.02428e6 - 1.22916e6i) q^{20} +(5.75716e6 - 3.50598e6i) q^{21} +(-449161. + 1.67629e6i) q^{22} +(2.68510e6 + 719472. i) q^{23} +(-1.35019e6 + 2.47031e6i) q^{24} +(7.43800e6 - 6.32800e6i) q^{25} +9.54459e6 q^{26} +(-9.40971e6 + 1.08328e7i) q^{27} +(-1.00427e7 + 1.00427e7i) q^{28} +(1.72346e7 + 9.95039e6i) q^{29} +(-1.06882e7 + 1.34539e7i) q^{30} +(-2.23240e7 - 3.86663e7i) q^{31} +(1.53522e6 - 5.72953e6i) q^{32} +(4.40119e6 - 1.81099e7i) 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} +(1.56421e7 + 5.83773e7i) q^{38} +(-1.02473e8 + 2.39777e6i) q^{39} +(1.51886e7 - 3.28637e7i) q^{40} +(9.46032e7 + 1.63857e8i) q^{41} +(1.05298e8 - 1.10344e8i) q^{42} +(-2.74063e7 - 1.02282e8i) q^{43} +3.92682e7i q^{44} +(1.11371e8 - 1.47130e8i) q^{45} +6.29003e7 q^{46} +(3.19626e8 - 8.56435e7i) q^{47} +(-1.50432e7 + 6.18993e7i) q^{48} +(4.21755e8 - 2.43500e8i) q^{49} +(1.25509e8 - 1.81867e8i) q^{50} +(5.18108e8 + 2.83181e8i) q^{51} +(2.08611e8 - 5.58970e7i) 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} +(-1.82603e8 - 6.22823e8i) q^{57} +(4.34959e8 + 1.16547e8i) q^{58} +(4.42465e8 - 2.55458e8i) q^{59} +(-1.54813e8 + 3.56649e8i) q^{60} +(7.31433e7 - 1.26688e8i) q^{61} +(-7.14369e8 - 7.14369e8i) q^{62} +(-1.10279e9 + 1.21114e9i) q^{63} -1.34218e8i q^{64} +(1.31276e9 - 1.19355e8i) q^{65} +(-9.86486e6 - 4.21593e8i) q^{66} +(2.95043e8 - 1.10111e9i) q^{67} +(-1.20168e9 - 3.21988e8i) q^{68} +(-6.75313e8 + 1.58017e7i) q^{69} +(-1.25569e9 + 1.50686e9i) q^{70} +1.24826e9 q^{71} +(1.45957e8 - 6.68345e8i) q^{72} +(2.63992e9 - 2.63992e9i) q^{73} +(4.78122e7 + 2.76044e7i) q^{74} +(-1.30180e9 + 1.98410e9i) q^{75} +(6.83762e8 + 1.18431e9i) q^{76} +(5.50636e8 - 2.05500e9i) q^{77} +(-2.22565e9 + 6.52531e8i) 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} +(3.42940e8 + 1.27987e9i) q^{83} +(1.65523e9 - 3.02840e9i) q^{84} +(-6.89265e9 - 3.18558e9i) q^{85} +(-1.19801e9 - 2.07501e9i) q^{86} +(-4.69911e9 - 1.14201e9i) q^{87} +(2.29970e8 + 8.58261e8i) q^{88} -4.73776e8i q^{89} +(1.57251e9 - 3.86796e9i) q^{90} -1.17009e10 q^{91} +(1.37477e9 - 3.68369e8i) q^{92} +(7.84910e9 + 7.49018e9i) q^{93} +(6.48431e9 - 3.74372e9i) q^{94} +(2.88142e9 + 7.83357e9i) q^{95} +(3.37179e7 + 1.44099e9i) q^{96} +(1.26768e10 - 3.39674e9i) 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 \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).

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

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
\(3\) −233.185 + 68.3665i −0.959607 + 0.281344i
\(4\) 443.405 − 256.000i 0.433013 − 0.250000i
\(5\) 2932.88 − 1078.80i 0.938523 − 0.345217i
\(6\) −4696.19 + 2859.87i −0.603934 + 0.367782i
\(7\) −26794.2 + 7179.49i −1.59423 + 0.427173i −0.943294 − 0.331959i \(-0.892290\pi\)
−0.650937 + 0.759132i \(0.725624\pi\)
\(8\) 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 −0.960172 − 0.279410i \(-0.909861\pi\)
0.722062 + 0.691828i \(0.243194\pi\)
\(12\) −85893.3 + 90009.3i −0.345186 + 0.361727i
\(13\) 407442. + 109174.i 1.09736 + 0.294037i 0.761689 − 0.647942i \(-0.224370\pi\)
0.335671 + 0.941979i \(0.391037\pi\)
\(14\) −543580. + 313836.i −1.01070 + 0.583529i
\(15\) −610149. + 452071.i −0.803489 + 0.595320i
\(16\) 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 \(-0.576846\pi\)
\(18\) 899560. − 987940.i 0.476067 − 0.522839i
\(19\) 2.67094e6i 1.07869i 0.842085 + 0.539345i \(0.181328\pi\)
−0.842085 + 0.539345i \(0.818672\pi\)
\(20\) 1.02428e6 − 1.22916e6i 0.320088 − 0.384114i
\(21\) 5.75716e6 − 3.50598e6i 1.40965 − 0.858445i
\(22\) −449161. + 1.67629e6i −0.0871543 + 0.325264i
\(23\) 2.68510e6 + 719472.i 0.417179 + 0.111783i 0.461301 − 0.887243i \(-0.347383\pi\)
−0.0441229 + 0.999026i \(0.514049\pi\)
\(24\) −1.35019e6 + 2.47031e6i −0.169566 + 0.310238i
\(25\) 7.43800e6 − 6.32800e6i 0.761651 − 0.647987i
\(26\) 9.54459e6 0.803324
\(27\) −9.40971e6 + 1.08328e7i −0.655779 + 0.754953i
\(28\) −1.00427e7 + 1.00427e7i −0.583529 + 0.583529i
\(29\) 1.72346e7 + 9.95039e6i 0.840254 + 0.485121i 0.857351 − 0.514733i \(-0.172109\pi\)
−0.0170965 + 0.999854i \(0.505442\pi\)
\(30\) −1.06882e7 + 1.34539e7i −0.439842 + 0.553660i
\(31\) −2.23240e7 − 3.86663e7i −0.779766 − 1.35059i −0.932077 − 0.362261i \(-0.882005\pi\)
0.152311 − 0.988333i \(-0.451328\pi\)
\(32\) 1.53522e6 − 5.72953e6i 0.0457532 − 0.170753i
\(33\) 4.40119e6 − 1.81099e7i 0.112461 − 0.462750i
\(34\) −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 \(0.744400\pi\)
\(38\) 1.56421e7 + 5.83773e7i 0.197414 + 0.736759i
\(39\) −1.02473e8 + 2.39777e6i −1.13576 + 0.0265757i
\(40\) 1.51886e7 − 3.28637e7i 0.148327 − 0.320935i
\(41\) 9.46032e7 + 1.63857e8i 0.816557 + 1.41432i 0.908205 + 0.418526i \(0.137453\pi\)
−0.0916480 + 0.995791i \(0.529213\pi\)
\(42\) 1.05298e8 − 1.10344e8i 0.805704 − 0.844313i
\(43\) −2.74063e7 − 1.02282e8i −0.186427 − 0.695755i −0.994321 − 0.106427i \(-0.966059\pi\)
0.807894 − 0.589328i \(-0.200608\pi\)
\(44\) 3.92682e7i 0.238110i
\(45\) 1.11371e8 − 1.47130e8i 0.603544 − 0.797330i
\(46\) 6.29003e7 0.305396
\(47\) 3.19626e8 − 8.56435e7i 1.39365 − 0.373426i 0.517587 − 0.855630i \(-0.326830\pi\)
0.876059 + 0.482204i \(0.160164\pi\)
\(48\) −1.50432e7 + 6.18993e7i −0.0590382 + 0.242929i
\(49\) 4.21755e8 − 2.43500e8i 1.49307 − 0.862024i
\(50\) 1.25509e8 − 1.81867e8i 0.401627 − 0.581975i
\(51\) 5.18108e8 + 2.83181e8i 1.50165 + 0.820754i
\(52\) 2.08611e8 − 5.58970e7i 0.548680 − 0.147018i
\(53\) 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
\(57\) −1.82603e8 − 6.22823e8i −0.303483 − 1.03512i
\(58\) 4.34959e8 + 1.16547e8i 0.662687 + 0.177567i
\(59\) 4.42465e8 − 2.55458e8i 0.618898 − 0.357321i −0.157542 − 0.987512i \(-0.550357\pi\)
0.776440 + 0.630191i \(0.217024\pi\)
\(60\) −1.54813e8 + 3.56649e8i −0.199091 + 0.458653i
\(61\) 7.31433e7 − 1.26688e8i 0.0866015 − 0.149998i −0.819471 − 0.573121i \(-0.805733\pi\)
0.906073 + 0.423122i \(0.139066\pi\)
\(62\) −7.14369e8 − 7.14369e8i −0.779766 − 0.779766i
\(63\) −1.10279e9 + 1.21114e9i −1.11119 + 1.22037i
\(64\) − 1.34218e8i − 0.125000i
\(65\) 1.31276e9 − 1.19355e8i 1.13140 − 0.102867i
\(66\) −9.86486e6 − 4.21593e8i −0.00787719 − 0.336646i
\(67\) 2.95043e8 − 1.10111e9i 0.218530 − 0.815565i −0.766364 − 0.642407i \(-0.777936\pi\)
0.984894 − 0.173158i \(-0.0553972\pi\)
\(68\) −1.20168e9 − 3.21988e8i −0.826501 − 0.221460i
\(69\) −6.75313e8 + 1.58017e7i −0.431777 + 0.0101032i
\(70\) −1.25569e9 + 1.50686e9i −0.747123 + 0.896566i
\(71\) 1.24826e9 0.691854 0.345927 − 0.938261i \(-0.387564\pi\)
0.345927 + 0.938261i \(0.387564\pi\)
\(72\) 1.45957e8 − 6.68345e8i 0.0754331 − 0.345413i
\(73\) 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
\(75\) −1.30180e9 + 1.98410e9i −0.548578 + 0.836099i
\(76\) 6.83762e8 + 1.18431e9i 0.269673 + 0.467087i
\(77\) 5.50636e8 − 2.05500e9i 0.203428 − 0.759204i
\(78\) −2.22565e9 + 6.52531e8i −0.770875 + 0.226010i
\(79\) 3.60137e9 + 2.07925e9i 1.17039 + 0.675727i 0.953773 − 0.300528i \(-0.0971630\pi\)
0.216621 + 0.976256i \(0.430496\pi\)
\(80\) 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
\(83\) 3.42940e8 + 1.27987e9i 0.0870617 + 0.324919i 0.995697 − 0.0926727i \(-0.0295410\pi\)
−0.908635 + 0.417591i \(0.862874\pi\)
\(84\) 1.65523e9 − 3.02840e9i 0.395786 − 0.724131i
\(85\) −6.89265e9 − 3.18558e9i −1.55343 − 0.717949i
\(86\) −1.19801e9 − 2.07501e9i −0.254664 − 0.441091i
\(87\) −4.69911e9 − 1.14201e9i −0.942799 − 0.229125i
\(88\) 2.29970e8 + 8.58261e8i 0.0435771 + 0.162632i
\(89\) − 4.73776e8i − 0.0848444i −0.999100 − 0.0424222i \(-0.986493\pi\)
0.999100 − 0.0424222i \(-0.0135075\pi\)
\(90\) 1.57251e9 − 3.86796e9i 0.266307 − 0.655043i
\(91\) −1.17009e10 −1.87505
\(92\) 1.37477e9 − 3.68369e8i 0.208589 − 0.0558913i
\(93\) 7.84910e9 + 7.49018e9i 1.12825 + 1.07666i
\(94\) 6.48431e9 − 3.74372e9i 0.883536 − 0.510110i
\(95\) 2.88142e9 + 7.83357e9i 0.372382 + 1.01238i
\(96\) 3.37179e7 + 1.44099e9i 0.00413527 + 0.176728i
\(97\) 1.26768e10 − 3.39674e9i 1.47622 − 0.395552i 0.571162 − 0.820837i \(-0.306493\pi\)
0.905059 + 0.425285i \(0.139826\pi\)
\(98\) 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.7.4 ✓ 120
5.3 odd 4 inner 90.11.k.b.43.13 yes 120
9.4 even 3 inner 90.11.k.b.67.13 yes 120
45.13 odd 12 inner 90.11.k.b.13.4 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.k.b.7.4 ✓ 120 1.1 even 1 trivial
90.11.k.b.13.4 yes 120 45.13 odd 12 inner
90.11.k.b.43.13 yes 120 5.3 odd 4 inner
90.11.k.b.67.13 yes 120 9.4 even 3 inner