Show commands: Magma / Pari/GP / SageMath

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 251.33
Character \(\chi\) \(=\) 270.251
Dual form 270.11.h.a.71.33

$q$-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 \(\left(\mathbb{Z}/270\mathbb{Z}\right)^\times\).

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

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
\(3\) 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 −0.533471 − 0.845818i \(-0.679113\pi\)
0.999236 + 0.0390902i \(0.0124460\pi\)
\(8\) 11585.2i 0.353553i
\(9\) 0 0
\(10\) 31622.8 0.316228
\(11\) −194331. − 112197.i −1.20664 − 0.696655i −0.244618 − 0.969620i \(-0.578662\pi\)
−0.962024 + 0.272965i \(0.911996\pi\)
\(12\) 0 0
\(13\) −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
\(15\) 0 0
\(16\) −131072. + 227023.i −0.125000 + 0.216506i
\(17\) − 684997.i − 0.482441i −0.970470 − 0.241220i \(-0.922452\pi\)
0.970470 − 0.241220i \(-0.0775476\pi\)
\(18\) 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 −0.626907 − 0.779094i \(-0.715679\pi\)
0.361262 + 0.932464i \(0.382346\pi\)
\(24\) 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 0.218710 − 0.975790i \(-0.429815\pi\)
−0.735704 + 0.677303i \(0.763149\pi\)
\(30\) 0 0
\(31\) 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
\(33\) 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 −0.989082 − 0.147364i \(-0.952921\pi\)
−0.366921 + 0.930252i \(0.619588\pi\)
\(42\) 0 0
\(43\) −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
\(45\) 0 0
\(46\) −4.46731e7 −0.216898
\(47\) −6.39804e7 − 3.69391e7i −0.278970 − 0.161063i 0.353987 − 0.935250i \(-0.384826\pi\)
−0.632957 + 0.774187i \(0.718159\pi\)
\(48\) 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 \(0.0591584\pi\)
−0.982779 + 0.184784i \(0.940842\pi\)
\(54\) 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 −0.728410 − 0.685141i \(-0.759740\pi\)
0.229145 + 0.973392i \(0.426407\pi\)
\(60\) 0 0
\(61\) 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
\(63\) 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 0.974089 + 0.226165i \(0.0726187\pi\)
−0.291180 + 0.956668i \(0.594048\pi\)
\(68\) 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 \(0.367955\pi\)
−0.403037 + 0.915184i \(0.632045\pi\)
\(72\) 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 −0.330755 − 0.943717i \(-0.607303\pi\)
0.982660 − 0.185416i \(-0.0593632\pi\)
\(80\) 3.66357e8i 0.111803i
\(81\) 0 0
\(82\) −4.10473e9 −1.10717
\(83\) 4.94795e9 + 2.85670e9i 1.25613 + 0.725227i 0.972320 − 0.233653i \(-0.0750680\pi\)
0.283810 + 0.958880i \(0.408401\pi\)
\(84\) 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 \(0.348097\pi\)
−0.459308 + 0.888277i \(0.651903\pi\)
\(90\) 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 −0.996372 − 0.0851036i \(-0.972878\pi\)
0.571888 + 0.820332i \(0.306211\pi\)
\(98\) 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.251.33 80
3.2 odd 2 90.11.h.a.11.18 ✓ 80
9.4 even 3 90.11.h.a.41.18 yes 80
9.5 odd 6 inner 270.11.h.a.71.33 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.18 ✓ 80 3.2 odd 2
90.11.h.a.41.18 yes 80 9.4 even 3
270.11.h.a.71.33 80 9.5 odd 6 inner
270.11.h.a.251.33 80 1.1 even 1 trivial