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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [90,11,Mod(11,90)] 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("90.11"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 90.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(57.1821527406\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 41.18
Character \(\chi\) \(=\) 90.41
Dual form 90.11.h.a.11.18

$q$-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+(-19.5959 + 11.3137i) q^{2} +(216.480 + 110.387i) q^{3} +(256.000 - 443.405i) q^{4} +(-1210.31 - 698.771i) q^{5} +(-5491.02 + 286.069i) q^{6} +(7828.11 + 13558.7i) q^{7} +11585.2i q^{8} +(34678.6 + 47793.1i) q^{9} +31622.8 q^{10} +(194331. - 112197. i) q^{11} +(104365. - 67729.5i) q^{12} +(-86403.2 + 149655. i) q^{13} +(-306798. - 177130. i) q^{14} +(-184873. - 284872. i) q^{15} +(-131072. - 227023. i) q^{16} -684997. i q^{17} +(-1.22028e6 - 544207. i) q^{18} +1.00179e6 q^{19} +(-619677. + 357771. i) q^{20} +(197935. + 3.79931e6i) q^{21} +(-2.53873e6 + 4.39721e6i) q^{22} +(1.70978e6 + 987144. i) q^{23} +(-1.27886e6 + 2.50798e6i) q^{24} +(976562. + 1.69146e6i) q^{25} -3.91016e6i q^{26} +(2.23151e6 + 1.41743e7i) q^{27} +8.01598e6 q^{28} +(1.06041e7 - 6.12231e6i) q^{29} +(6.84571e6 + 3.49073e6i) q^{30} +(8.29198e6 - 1.43621e7i) q^{31} +(5.13695e6 + 2.96582e6i) q^{32} +(5.44539e7 - 2.83692e6i) q^{33} +(7.74985e6 + 1.34231e7i) q^{34} -2.18802e7i q^{35} +(3.00694e7 - 3.14161e6i) q^{36} -7.30867e7 q^{37} +(-1.96310e7 + 1.13340e7i) q^{38} +(-3.52245e7 + 2.28596e7i) q^{39} +(8.09543e6 - 1.40217e7i) q^{40} +(1.57101e8 + 9.07025e7i) q^{41} +(-4.68630e7 - 7.22116e7i) q^{42} +(-7.19385e7 - 1.24601e8i) q^{43} -1.14890e8i q^{44} +(-8.57527e6 - 8.20767e7i) q^{45} -4.46731e7 q^{46} +(6.39804e7 - 3.69391e7i) q^{47} +(-3.31417e6 - 6.36147e7i) q^{48} +(1.86791e7 - 3.23531e7i) q^{49} +(-3.82733e7 - 2.20971e7i) q^{50} +(7.56145e7 - 1.48288e8i) q^{51} +(4.42384e7 + 7.66232e7i) q^{52} +1.54551e8i q^{53} +(-2.04093e8 - 2.52512e8i) q^{54} -3.13600e8 q^{55} +(-1.57081e8 + 9.06905e7i) q^{56} +(2.16868e8 + 1.10584e8i) q^{57} +(-1.38532e8 + 2.39944e8i) q^{58} +(3.56937e8 + 2.06078e8i) q^{59} +(-1.73641e8 + 9.04628e6i) q^{60} +(5.82779e8 + 1.00940e9i) q^{61} +3.75252e8i q^{62} +(-3.76544e8 + 8.44325e8i) q^{63} -1.34218e8 q^{64} +(2.09149e8 - 1.20752e8i) q^{65} +(-1.03498e9 + 6.71667e8i) q^{66} +(9.22012e8 - 1.59697e9i) q^{67} +(-3.03731e8 - 1.75359e8i) q^{68} +(2.61167e8 + 4.02435e8i) q^{69} +(2.47547e8 + 4.28763e8i) q^{70} +3.30240e9i q^{71} +(-5.53695e8 + 4.01759e8i) q^{72} +6.65904e8 q^{73} +(1.43220e9 - 8.26881e8i) q^{74} +(2.46925e7 + 4.73967e8i) q^{75} +(2.56459e8 - 4.44199e8i) q^{76} +(3.04249e9 + 1.75658e9i) q^{77} +(4.31630e8 - 8.46473e8i) q^{78} +(2.00595e9 + 3.47440e9i) q^{79} +3.66357e8i q^{80} +(-1.08158e9 + 3.31479e9i) q^{81} -4.10473e9 q^{82} +(-4.94795e9 + 2.85670e9i) q^{83} +(1.73530e9 + 8.84858e8i) q^{84} +(-4.78656e8 + 8.29056e8i) q^{85} +(2.81940e9 + 1.62778e9i) q^{86} +(2.97141e9 - 1.54803e8i) q^{87} +(1.29983e9 + 2.25137e9i) q^{88} +9.92038e9i q^{89} +(1.09663e9 + 1.51135e9i) q^{90} -2.70549e9 q^{91} +(8.75410e8 - 5.05418e8i) q^{92} +(3.38044e9 - 2.19380e9i) q^{93} +(-8.35836e8 + 1.44771e9i) q^{94} +(-1.21248e9 - 7.00023e8i) q^{95} +(7.84663e8 + 1.20909e9i) q^{96} +(-3.64519e9 - 6.31365e9i) q^{97} +8.45318e8i q^{98} +(1.21014e10 + 5.39685e9i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q + 44 q^{3} + 20480 q^{4} - 13568 q^{6} - 24476 q^{7} - 103396 q^{9} + 1944 q^{11} + 45056 q^{12} + 561100 q^{13} - 175680 q^{14} - 687500 q^{15} - 10485760 q^{16} - 3888896 q^{18} + 5932480 q^{19}+ \cdots - 78067269744 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(11\) \(37\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −19.5959 + 11.3137i −0.612372 + 0.353553i
\(3\) 216.480 + 110.387i 0.890866 + 0.454266i
\(4\) 256.000 − 443.405i 0.250000 − 0.433013i
\(5\) −1210.31 − 698.771i −0.387298 − 0.223607i
\(6\) −5491.02 + 286.069i −0.706149 + 0.0367887i
\(7\) 7828.11 + 13558.7i 0.465765 + 0.806728i 0.999236 − 0.0390902i \(-0.0124460\pi\)
−0.533471 + 0.845818i \(0.679113\pi\)
\(8\) 11585.2i 0.353553i
\(9\) 34678.6 + 47793.1i 0.587285 + 0.809381i
\(10\) 31622.8 0.316228
\(11\) 194331. − 112197.i 1.20664 − 0.696655i 0.244618 − 0.969620i \(-0.421338\pi\)
0.962024 + 0.272965i \(0.0880042\pi\)
\(12\) 104365. − 67729.5i 0.419420 − 0.272190i
\(13\) −86403.2 + 149655.i −0.232709 + 0.403064i −0.958604 − 0.284741i \(-0.908092\pi\)
0.725895 + 0.687805i \(0.241426\pi\)
\(14\) −306798. − 177130.i −0.570443 − 0.329345i
\(15\) −184873. − 284872.i −0.243454 − 0.375140i
\(16\) −131072. − 227023.i −0.125000 − 0.216506i
\(17\) − 684997.i − 0.482441i −0.970470 − 0.241220i \(-0.922452\pi\)
0.970470 − 0.241220i \(-0.0775476\pi\)
\(18\) −1.22028e6 − 544207.i −0.645796 − 0.288006i
\(19\) 1.00179e6 0.404585 0.202292 − 0.979325i \(-0.435161\pi\)
0.202292 + 0.979325i \(0.435161\pi\)
\(20\) −619677. + 357771.i −0.193649 + 0.111803i
\(21\) 197935. + 3.79931e6i 0.0484647 + 0.930268i
\(22\) −2.53873e6 + 4.39721e6i −0.492609 + 0.853225i
\(23\) 1.70978e6 + 987144.i 0.265645 + 0.153370i 0.626907 − 0.779094i \(-0.284321\pi\)
−0.361262 + 0.932464i \(0.617654\pi\)
\(24\) −1.27886e6 + 2.50798e6i −0.160607 + 0.314969i
\(25\) 976562. + 1.69146e6i 0.100000 + 0.173205i
\(26\) − 3.91016e6i − 0.329100i
\(27\) 2.23151e6 + 1.41743e7i 0.155518 + 0.987833i
\(28\) 8.01598e6 0.465765
\(29\) 1.06041e7 − 6.12231e6i 0.516994 − 0.298487i −0.218710 − 0.975790i \(-0.570185\pi\)
0.735704 + 0.677303i \(0.236851\pi\)
\(30\) 6.84571e6 + 3.49073e6i 0.281717 + 0.143652i
\(31\) 8.29198e6 − 1.43621e7i 0.289634 − 0.501661i −0.684088 − 0.729399i \(-0.739800\pi\)
0.973722 + 0.227738i \(0.0731330\pi\)
\(32\) 5.13695e6 + 2.96582e6i 0.153093 + 0.0883883i
\(33\) 5.44539e7 − 2.83692e6i 1.39142 − 0.0724898i
\(34\) 7.74985e6 + 1.34231e7i 0.170568 + 0.295433i
\(35\) − 2.18802e7i − 0.416593i
\(36\) 3.00694e7 − 3.14161e6i 0.497293 − 0.0519566i
\(37\) −7.30867e7 −1.05397 −0.526987 − 0.849874i \(-0.676678\pi\)
−0.526987 + 0.849874i \(0.676678\pi\)
\(38\) −1.96310e7 + 1.13340e7i −0.247756 + 0.143042i
\(39\) −3.52245e7 + 2.28596e7i −0.390411 + 0.253364i
\(40\) 8.09543e6 − 1.40217e7i 0.0790569 − 0.136931i
\(41\) 1.57101e8 + 9.07025e7i 1.35600 + 0.782889i 0.989082 − 0.147364i \(-0.0470787\pi\)
0.366921 + 0.930252i \(0.380412\pi\)
\(42\) −4.68630e7 − 7.22116e7i −0.358578 − 0.552536i
\(43\) −7.19385e7 − 1.24601e8i −0.489349 − 0.847578i 0.510576 − 0.859833i \(-0.329432\pi\)
−0.999925 + 0.0122550i \(0.996099\pi\)
\(44\) − 1.14890e8i − 0.696655i
\(45\) −8.57527e6 − 8.20767e7i −0.0464714 − 0.444793i
\(46\) −4.46731e7 −0.216898
\(47\) 6.39804e7 − 3.69391e7i 0.278970 − 0.161063i −0.353987 − 0.935250i \(-0.615174\pi\)
0.632957 + 0.774187i \(0.281841\pi\)
\(48\) −3.31417e6 − 6.36147e7i −0.0130068 − 0.249661i
\(49\) 1.86791e7 − 3.23531e7i 0.0661264 − 0.114534i
\(50\) −3.82733e7 − 2.20971e7i −0.122474 − 0.0707107i
\(51\) 7.56145e7 − 1.48288e8i 0.219156 − 0.429790i
\(52\) 4.42384e7 + 7.66232e7i 0.116354 + 0.201532i
\(53\) 1.54551e8i 0.369567i 0.982779 + 0.184784i \(0.0591584\pi\)
−0.982779 + 0.184784i \(0.940842\pi\)
\(54\) −2.04093e8 − 2.52512e8i −0.444487 − 0.549938i
\(55\) −3.13600e8 −0.623107
\(56\) −1.57081e8 + 9.06905e7i −0.285221 + 0.164673i
\(57\) 2.16868e8 + 1.10584e8i 0.360431 + 0.183789i
\(58\) −1.38532e8 + 2.39944e8i −0.211062 + 0.365570i
\(59\) 3.56937e8 + 2.06078e8i 0.499265 + 0.288251i 0.728410 − 0.685141i \(-0.240260\pi\)
−0.229145 + 0.973392i \(0.573593\pi\)
\(60\) −1.73641e8 + 9.04628e6i −0.223304 + 0.0116336i
\(61\) 5.82779e8 + 1.00940e9i 0.690009 + 1.19513i 0.971834 + 0.235665i \(0.0757268\pi\)
−0.281825 + 0.959466i \(0.590940\pi\)
\(62\) 3.75252e8i 0.409605i
\(63\) −3.76544e8 + 8.44325e8i −0.379414 + 0.850760i
\(64\) −1.34218e8 −0.125000
\(65\) 2.09149e8 − 1.20752e8i 0.180256 − 0.104071i
\(66\) −1.03498e9 + 6.71667e8i −0.826440 + 0.536333i
\(67\) 9.22012e8 − 1.59697e9i 0.682909 − 1.18283i −0.291180 − 0.956668i \(-0.594048\pi\)
0.974089 − 0.226165i \(-0.0726187\pi\)
\(68\) −3.03731e8 − 1.75359e8i −0.208903 − 0.120610i
\(69\) 2.61167e8 + 4.02435e8i 0.166983 + 0.257306i
\(70\) 2.47547e8 + 4.28763e8i 0.147288 + 0.255110i
\(71\) 3.30240e9i 1.83037i 0.403037 + 0.915184i \(0.367955\pi\)
−0.403037 + 0.915184i \(0.632045\pi\)
\(72\) −5.53695e8 + 4.01759e8i −0.286159 + 0.207636i
\(73\) 6.65904e8 0.321216 0.160608 − 0.987018i \(-0.448655\pi\)
0.160608 + 0.987018i \(0.448655\pi\)
\(74\) 1.43220e9 − 8.26881e8i 0.645424 − 0.372636i
\(75\) 2.46925e7 + 4.73967e8i 0.0104054 + 0.199729i
\(76\) 2.56459e8 − 4.44199e8i 0.101146 − 0.175190i
\(77\) 3.04249e9 + 1.75658e9i 1.12402 + 0.648955i
\(78\) 4.31630e8 − 8.46473e8i 0.149499 − 0.293184i
\(79\) 2.00595e9 + 3.47440e9i 0.651905 + 1.12913i 0.982660 + 0.185416i \(0.0593632\pi\)
−0.330755 + 0.943717i \(0.607303\pi\)
\(80\) 3.66357e8i 0.111803i
\(81\) −1.08158e9 + 3.31479e9i −0.310194 + 0.950673i
\(82\) −4.10473e9 −1.10717
\(83\) −4.94795e9 + 2.85670e9i −1.25613 + 0.725227i −0.972320 − 0.233653i \(-0.924932\pi\)
−0.283810 + 0.958880i \(0.591599\pi\)
\(84\) 1.73530e9 + 8.84858e8i 0.414934 + 0.211581i
\(85\) −4.78656e8 + 8.29056e8i −0.107877 + 0.186848i
\(86\) 2.81940e9 + 1.62778e9i 0.599328 + 0.346022i
\(87\) 2.97141e9 − 1.54803e8i 0.596165 − 0.0310588i
\(88\) 1.29983e9 + 2.25137e9i 0.246305 + 0.426612i
\(89\) 9.92038e9i 1.77655i 0.459308 + 0.888277i \(0.348097\pi\)
−0.459308 + 0.888277i \(0.651903\pi\)
\(90\) 1.09663e9 + 1.51135e9i 0.185716 + 0.255949i
\(91\) −2.70549e9 −0.433550
\(92\) 8.75410e8 − 5.05418e8i 0.132823 − 0.0766852i
\(93\) 3.38044e9 − 2.19380e9i 0.485913 − 0.315342i
\(94\) −8.35836e8 + 1.44771e9i −0.113889 + 0.197262i
\(95\) −1.21248e9 − 7.00023e8i −0.156695 − 0.0904679i
\(96\) 7.84663e8 + 1.20909e9i 0.0962336 + 0.148287i
\(97\) −3.64519e9 − 6.31365e9i −0.424484 − 0.735228i 0.571888 − 0.820332i \(-0.306211\pi\)
−0.996372 + 0.0851036i \(0.972878\pi\)
\(98\) 8.45318e8i 0.0935168i
\(99\) 1.21014e10 + 5.39685e9i 1.27250 + 0.567498i
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.h.a.41.18 yes 80
3.2 odd 2 270.11.h.a.71.33 80
9.2 odd 6 inner 90.11.h.a.11.18 ✓ 80
9.7 even 3 270.11.h.a.251.33 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.18 ✓ 80 9.2 odd 6 inner
90.11.h.a.41.18 yes 80 1.1 even 1 trivial
270.11.h.a.71.33 80 3.2 odd 2
270.11.h.a.251.33 80 9.7 even 3