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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [270,11,Mod(71,270)] 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("270.71"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(270, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5, 0])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 270.h (of order \(6\), degree \(2\), not 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: \(171.546458222\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 71.33
Character \(\chi\) \(=\) 270.71
Dual form 270.11.h.a.251.33

$q$-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)
 
\(f(q)\) \(=\) \(q+(19.5959 - 11.3137i) q^{2} +(256.000 - 443.405i) q^{4} +(1210.31 + 698.771i) q^{5} +(7828.11 + 13558.7i) q^{7} -11585.2i q^{8} +31622.8 q^{10} +(-194331. + 112197. i) q^{11} +(-86403.2 + 149655. i) q^{13} +(306798. + 177130. i) q^{14} +(-131072. - 227023. i) q^{16} +684997. i q^{17} +1.00179e6 q^{19} +(619677. - 357771. i) q^{20} +(-2.53873e6 + 4.39721e6i) q^{22} +(-1.70978e6 - 987144. i) q^{23} +(976562. + 1.69146e6i) q^{25} +3.91016e6i q^{26} +8.01598e6 q^{28} +(-1.06041e7 + 6.12231e6i) q^{29} +(8.29198e6 - 1.43621e7i) q^{31} +(-5.13695e6 - 2.96582e6i) q^{32} +(7.74985e6 + 1.34231e7i) q^{34} +2.18802e7i q^{35} -7.30867e7 q^{37} +(1.96310e7 - 1.13340e7i) q^{38} +(8.09543e6 - 1.40217e7i) q^{40} +(-1.57101e8 - 9.07025e7i) q^{41} +(-7.19385e7 - 1.24601e8i) q^{43} +1.14890e8i q^{44} -4.46731e7 q^{46} +(-6.39804e7 + 3.69391e7i) q^{47} +(1.86791e7 - 3.23531e7i) q^{49} +(3.82733e7 + 2.20971e7i) q^{50} +(4.42384e7 + 7.66232e7i) q^{52} -1.54551e8i q^{53} -3.13600e8 q^{55} +(1.57081e8 - 9.06905e7i) q^{56} +(-1.38532e8 + 2.39944e8i) q^{58} +(-3.56937e8 - 2.06078e8i) q^{59} +(5.82779e8 + 1.00940e9i) q^{61} -3.75252e8i q^{62} -1.34218e8 q^{64} +(-2.09149e8 + 1.20752e8i) q^{65} +(9.22012e8 - 1.59697e9i) q^{67} +(3.03731e8 + 1.75359e8i) q^{68} +(2.47547e8 + 4.28763e8i) q^{70} -3.30240e9i q^{71} +6.65904e8 q^{73} +(-1.43220e9 + 8.26881e8i) q^{74} +(2.56459e8 - 4.44199e8i) q^{76} +(-3.04249e9 - 1.75658e9i) q^{77} +(2.00595e9 + 3.47440e9i) q^{79} -3.66357e8i q^{80} -4.10473e9 q^{82} +(4.94795e9 - 2.85670e9i) q^{83} +(-4.78656e8 + 8.29056e8i) q^{85} +(-2.81940e9 - 1.62778e9i) q^{86} +(1.29983e9 + 2.25137e9i) q^{88} -9.92038e9i q^{89} -2.70549e9 q^{91} +(-8.75410e8 + 5.05418e8i) q^{92} +(-8.35836e8 + 1.44771e9i) q^{94} +(1.21248e9 + 7.00023e8i) q^{95} +(-3.64519e9 - 6.31365e9i) q^{97} -8.45318e8i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q + 20480 q^{4} - 24476 q^{7} - 1944 q^{11} + 561100 q^{13} + 175680 q^{14} - 10485760 q^{16} + 5932480 q^{19} - 6946944 q^{22} + 2008908 q^{23} + 78125000 q^{25} - 25063424 q^{28} + 54816192 q^{29}+ \cdots - 10980388424 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/270\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(217\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\)

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
\(3\) 0 0
\(4\) 256.000 − 443.405i 0.250000 − 0.433013i
\(5\) 1210.31 + 698.771i 0.387298 + 0.223607i
\(6\) 0 0
\(7\) 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
\(9\) 0 0
\(10\) 31622.8 0.316228
\(11\) −194331. + 112197.i −1.20664 + 0.696655i −0.962024 − 0.272965i \(-0.911996\pi\)
−0.244618 + 0.969620i \(0.578662\pi\)
\(12\) 0 0
\(13\) −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
\(15\) 0 0
\(16\) −131072. − 227023.i −0.125000 − 0.216506i
\(17\) 684997.i 0.482441i 0.970470 + 0.241220i \(0.0775476\pi\)
−0.970470 + 0.241220i \(0.922452\pi\)
\(18\) 0 0
\(19\) 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
\(21\) 0 0
\(22\) −2.53873e6 + 4.39721e6i −0.492609 + 0.853225i
\(23\) −1.70978e6 − 987144.i −0.265645 − 0.153370i 0.361262 − 0.932464i \(-0.382346\pi\)
−0.626907 + 0.779094i \(0.715679\pi\)
\(24\) 0 0
\(25\) 976562. + 1.69146e6i 0.100000 + 0.173205i
\(26\) 3.91016e6i 0.329100i
\(27\) 0 0
\(28\) 8.01598e6 0.465765
\(29\) −1.06041e7 + 6.12231e6i −0.516994 + 0.298487i −0.735704 − 0.677303i \(-0.763149\pi\)
0.218710 + 0.975790i \(0.429815\pi\)
\(30\) 0 0
\(31\) 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
\(33\) 0 0
\(34\) 7.74985e6 + 1.34231e7i 0.170568 + 0.295433i
\(35\) 2.18802e7i 0.416593i
\(36\) 0 0
\(37\) −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
\(39\) 0 0
\(40\) 8.09543e6 − 1.40217e7i 0.0790569 − 0.136931i
\(41\) −1.57101e8 − 9.07025e7i −1.35600 − 0.782889i −0.366921 − 0.930252i \(-0.619588\pi\)
−0.989082 + 0.147364i \(0.952921\pi\)
\(42\) 0 0
\(43\) −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
\(45\) 0 0
\(46\) −4.46731e7 −0.216898
\(47\) −6.39804e7 + 3.69391e7i −0.278970 + 0.161063i −0.632957 − 0.774187i \(-0.718159\pi\)
0.353987 + 0.935250i \(0.384826\pi\)
\(48\) 0 0
\(49\) 1.86791e7 − 3.23531e7i 0.0661264 − 0.114534i
\(50\) 3.82733e7 + 2.20971e7i 0.122474 + 0.0707107i
\(51\) 0 0
\(52\) 4.42384e7 + 7.66232e7i 0.116354 + 0.201532i
\(53\) − 1.54551e8i − 0.369567i −0.982779 − 0.184784i \(-0.940842\pi\)
0.982779 − 0.184784i \(-0.0591584\pi\)
\(54\) 0 0
\(55\) −3.13600e8 −0.623107
\(56\) 1.57081e8 − 9.06905e7i 0.285221 − 0.164673i
\(57\) 0 0
\(58\) −1.38532e8 + 2.39944e8i −0.211062 + 0.365570i
\(59\) −3.56937e8 − 2.06078e8i −0.499265 − 0.288251i 0.229145 − 0.973392i \(-0.426407\pi\)
−0.728410 + 0.685141i \(0.759740\pi\)
\(60\) 0 0
\(61\) 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
\(63\) 0 0
\(64\) −1.34218e8 −0.125000
\(65\) −2.09149e8 + 1.20752e8i −0.180256 + 0.104071i
\(66\) 0 0
\(67\) 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
\(69\) 0 0
\(70\) 2.47547e8 + 4.28763e8i 0.147288 + 0.255110i
\(71\) − 3.30240e9i − 1.83037i −0.403037 − 0.915184i \(-0.632045\pi\)
0.403037 − 0.915184i \(-0.367955\pi\)
\(72\) 0 0
\(73\) 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
\(75\) 0 0
\(76\) 2.56459e8 − 4.44199e8i 0.101146 − 0.175190i
\(77\) −3.04249e9 − 1.75658e9i −1.12402 − 0.648955i
\(78\) 0 0
\(79\) 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
\(81\) 0 0
\(82\) −4.10473e9 −1.10717
\(83\) 4.94795e9 − 2.85670e9i 1.25613 − 0.725227i 0.283810 − 0.958880i \(-0.408401\pi\)
0.972320 + 0.233653i \(0.0750680\pi\)
\(84\) 0 0
\(85\) −4.78656e8 + 8.29056e8i −0.107877 + 0.186848i
\(86\) −2.81940e9 − 1.62778e9i −0.599328 − 0.346022i
\(87\) 0 0
\(88\) 1.29983e9 + 2.25137e9i 0.246305 + 0.426612i
\(89\) − 9.92038e9i − 1.77655i −0.459308 − 0.888277i \(-0.651903\pi\)
0.459308 − 0.888277i \(-0.348097\pi\)
\(90\) 0 0
\(91\) −2.70549e9 −0.433550
\(92\) −8.75410e8 + 5.05418e8i −0.132823 + 0.0766852i
\(93\) 0 0
\(94\) −8.35836e8 + 1.44771e9i −0.113889 + 0.197262i
\(95\) 1.21248e9 + 7.00023e8i 0.156695 + 0.0904679i
\(96\) 0 0
\(97\) −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
\(99\) 0 0
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 270.11.h.a.71.33 80
3.2 odd 2 90.11.h.a.41.18 yes 80
9.2 odd 6 inner 270.11.h.a.251.33 80
9.7 even 3 90.11.h.a.11.18 ✓ 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.18 ✓ 80 9.7 even 3
90.11.h.a.41.18 yes 80 3.2 odd 2
270.11.h.a.71.33 80 1.1 even 1 trivial
270.11.h.a.251.33 80 9.2 odd 6 inner