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Properties

Label 90.11.h.a.11.18
Level $90$
Weight $11$
Character 90.11
Analytic conductor $57.182$
Analytic rank $0$
Dimension $80$
Inner twists $2$

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

$q$-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 \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).

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

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 −0.533471 − 0.845818i \(-0.679113\pi\)
0.999236 + 0.0390902i \(0.0124460\pi\)
\(8\) − 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 0.962024 − 0.272965i \(-0.0880042\pi\)
0.244618 + 0.969620i \(0.421338\pi\)
\(12\) 104365. + 67729.5i 0.419420 + 0.272190i
\(13\) −86403.2 − 149655.i −0.232709 − 0.403064i 0.725895 − 0.687805i \(-0.241426\pi\)
−0.958604 + 0.284741i \(0.908092\pi\)
\(14\) −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 \(0.0775476\pi\)
−0.970470 + 0.241220i \(0.922452\pi\)
\(18\) −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 −0.361262 − 0.932464i \(-0.617654\pi\)
0.626907 + 0.779094i \(0.284321\pi\)
\(24\) −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 0.735704 − 0.677303i \(-0.236851\pi\)
−0.218710 + 0.975790i \(0.570185\pi\)
\(30\) 6.84571e6 − 3.49073e6i 0.281717 − 0.143652i
\(31\) 8.29198e6 + 1.43621e7i 0.289634 + 0.501661i 0.973722 − 0.227738i \(-0.0731330\pi\)
−0.684088 + 0.729399i \(0.739800\pi\)
\(32\) 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 0.366921 − 0.930252i \(-0.380412\pi\)
0.989082 + 0.147364i \(0.0470787\pi\)
\(42\) −4.68630e7 + 7.22116e7i −0.358578 + 0.552536i
\(43\) −7.19385e7 + 1.24601e8i −0.489349 + 0.847578i −0.999925 − 0.0122550i \(-0.996099\pi\)
0.510576 + 0.859833i \(0.329432\pi\)
\(44\) 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 0.632957 − 0.774187i \(-0.281841\pi\)
−0.353987 + 0.935250i \(0.615174\pi\)
\(48\) −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 \(-0.940842\pi\)
0.982779 − 0.184784i \(-0.0591584\pi\)
\(54\) −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 −0.229145 − 0.973392i \(-0.573593\pi\)
0.728410 + 0.685141i \(0.240260\pi\)
\(60\) −1.73641e8 − 9.04628e6i −0.223304 − 0.0116336i
\(61\) 5.82779e8 − 1.00940e9i 0.690009 − 1.19513i −0.281825 − 0.959466i \(-0.590940\pi\)
0.971834 − 0.235665i \(-0.0757268\pi\)
\(62\) − 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 0.974089 + 0.226165i \(0.0726187\pi\)
−0.291180 + 0.956668i \(0.594048\pi\)
\(68\) −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 \(-0.632045\pi\)
0.403037 − 0.915184i \(-0.367955\pi\)
\(72\) −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 −0.330755 − 0.943717i \(-0.607303\pi\)
0.982660 − 0.185416i \(-0.0593632\pi\)
\(80\) − 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 −0.283810 − 0.958880i \(-0.591599\pi\)
−0.972320 + 0.233653i \(0.924932\pi\)
\(84\) 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 \(-0.651903\pi\)
0.459308 − 0.888277i \(-0.348097\pi\)
\(90\) 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 −0.996372 − 0.0851036i \(-0.972878\pi\)
0.571888 + 0.820332i \(0.306211\pi\)
\(98\) − 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.11.18 ✓ 80
3.2 odd 2 270.11.h.a.251.33 80
9.4 even 3 270.11.h.a.71.33 80
9.5 odd 6 inner 90.11.h.a.41.18 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.18 ✓ 80 1.1 even 1 trivial
90.11.h.a.41.18 yes 80 9.5 odd 6 inner
270.11.h.a.71.33 80 9.4 even 3
270.11.h.a.251.33 80 3.2 odd 2