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 251.30
Character \(\chi\) \(=\) 270.251
Dual form 270.11.h.a.71.30

$q$-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)
 
\(f(q)\) \(=\) \(q+(19.5959 + 11.3137i) q^{2} +(256.000 + 443.405i) q^{4} +(-1210.31 + 698.771i) q^{5} +(-5111.91 + 8854.09i) q^{7} +11585.2i q^{8} -31622.8 q^{10} +(83878.2 + 48427.1i) q^{11} +(-57466.9 - 99535.5i) q^{13} +(-200345. + 115669. i) q^{14} +(-131072. + 227023. i) q^{16} -1.26845e6i q^{17} +1.73688e6 q^{19} +(-619677. - 357771. i) q^{20} +(1.09578e6 + 1.89795e6i) q^{22} +(-6.63805e6 + 3.83248e6i) q^{23} +(976562. - 1.69146e6i) q^{25} -2.60065e6i q^{26} -5.23460e6 q^{28} +(-2.45691e7 - 1.41850e7i) q^{29} +(1.05043e6 + 1.81940e6i) q^{31} +(-5.13695e6 + 2.96582e6i) q^{32} +(1.43508e7 - 2.48564e7i) q^{34} -1.42882e7i q^{35} -1.00798e8 q^{37} +(3.40357e7 + 1.96505e7i) q^{38} +(-8.09543e6 - 1.40217e7i) q^{40} +(1.09096e8 - 6.29867e7i) q^{41} +(-4.70164e7 + 8.14347e7i) q^{43} +4.95894e7i q^{44} -1.73438e8 q^{46} +(-1.27799e8 - 7.37849e7i) q^{47} +(8.89743e7 + 1.54108e8i) q^{49} +(3.82733e7 - 2.20971e7i) q^{50} +(2.94230e7 - 5.09622e7i) q^{52} +5.19853e6i q^{53} -1.35358e8 q^{55} +(-1.02577e8 - 5.92227e7i) q^{56} +(-3.20969e8 - 5.55935e8i) q^{58} +(-3.10574e8 + 1.79310e8i) q^{59} +(3.95553e8 - 6.85118e8i) q^{61} +4.75372e7i q^{62} -1.34218e8 q^{64} +(1.39105e8 + 8.03124e7i) q^{65} +(5.84667e8 + 1.01267e9i) q^{67} +(5.62435e8 - 3.24722e8i) q^{68} +(1.61653e8 - 2.79991e8i) q^{70} -2.00363e9i q^{71} +3.30750e9 q^{73} +(-1.97523e9 - 1.14040e9i) q^{74} +(4.44640e8 + 7.70140e8i) q^{76} +(-8.57557e8 + 4.95111e8i) q^{77} +(-1.40038e8 + 2.42553e8i) q^{79} -3.66357e8i q^{80} +2.85045e9 q^{82} +(-3.27351e9 - 1.88996e9i) q^{83} +(8.86353e8 + 1.53521e9i) q^{85} +(-1.84266e9 + 1.06386e9i) q^{86} +(-5.61040e8 + 9.71749e8i) q^{88} -6.97588e9i q^{89} +1.17506e9 q^{91} +(-3.39868e9 - 1.96223e9i) q^{92} +(-1.66956e9 - 2.89177e9i) q^{94} +(-2.10215e9 + 1.21368e9i) q^{95} +(5.07737e9 - 8.79427e9i) q^{97} +4.02652e9i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q + 20480 q^{4} - 24476 q^{7} - 1944 q^{11} + 561100 q^{13} + 175680 q^{14} - 10485760 q^{16} + 5932480 q^{19} - 6946944 q^{22} + 2008908 q^{23} + 78125000 q^{25} - 25063424 q^{28} + 54816192 q^{29}+ \cdots - 10980388424 q^{97}+O(q^{100}) \) 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
\(7\) −5111.91 + 8854.09i −0.304154 + 0.526810i −0.977073 − 0.212906i \(-0.931707\pi\)
0.672919 + 0.739716i \(0.265040\pi\)
\(8\) 11585.2i 0.353553i
\(9\) 0 0
\(10\) −31622.8 −0.316228
\(11\) 83878.2 + 48427.1i 0.520818 + 0.300694i 0.737269 − 0.675599i \(-0.236115\pi\)
−0.216451 + 0.976293i \(0.569448\pi\)
\(12\) 0 0
\(13\) −57466.9 − 99535.5i −0.154775 − 0.268078i 0.778202 − 0.628014i \(-0.216132\pi\)
−0.932977 + 0.359936i \(0.882799\pi\)
\(14\) −200345. + 115669.i −0.372511 + 0.215069i
\(15\) 0 0
\(16\) −131072. + 227023.i −0.125000 + 0.216506i
\(17\) − 1.26845e6i − 0.893362i −0.894693 − 0.446681i \(-0.852606\pi\)
0.894693 − 0.446681i \(-0.147394\pi\)
\(18\) 0 0
\(19\) 1.73688e6 0.701457 0.350728 − 0.936477i \(-0.385934\pi\)
0.350728 + 0.936477i \(0.385934\pi\)
\(20\) −619677. − 357771.i −0.193649 − 0.111803i
\(21\) 0 0
\(22\) 1.09578e6 + 1.89795e6i 0.212623 + 0.368274i
\(23\) −6.63805e6 + 3.83248e6i −1.03134 + 0.595444i −0.917368 − 0.398040i \(-0.869690\pi\)
−0.113971 + 0.993484i \(0.536357\pi\)
\(24\) 0 0
\(25\) 976562. − 1.69146e6i 0.100000 − 0.173205i
\(26\) − 2.60065e6i − 0.218885i
\(27\) 0 0
\(28\) −5.23460e6 −0.304154
\(29\) −2.45691e7 − 1.41850e7i −1.19784 − 0.691574i −0.237768 − 0.971322i \(-0.576416\pi\)
−0.960074 + 0.279748i \(0.909749\pi\)
\(30\) 0 0
\(31\) 1.05043e6 + 1.81940e6i 0.0366910 + 0.0635507i 0.883788 − 0.467888i \(-0.154985\pi\)
−0.847097 + 0.531439i \(0.821652\pi\)
\(32\) −5.13695e6 + 2.96582e6i −0.153093 + 0.0883883i
\(33\) 0 0
\(34\) 1.43508e7 − 2.48564e7i 0.315851 − 0.547070i
\(35\) − 1.42882e7i − 0.272043i
\(36\) 0 0
\(37\) −1.00798e8 −1.45359 −0.726796 − 0.686853i \(-0.758991\pi\)
−0.726796 + 0.686853i \(0.758991\pi\)
\(38\) 3.40357e7 + 1.96505e7i 0.429553 + 0.248002i
\(39\) 0 0
\(40\) −8.09543e6 − 1.40217e7i −0.0790569 − 0.136931i
\(41\) 1.09096e8 − 6.29867e7i 0.941651 − 0.543662i 0.0511734 − 0.998690i \(-0.483704\pi\)
0.890477 + 0.455027i \(0.150371\pi\)
\(42\) 0 0
\(43\) −4.70164e7 + 8.14347e7i −0.319821 + 0.553946i −0.980450 − 0.196766i \(-0.936956\pi\)
0.660630 + 0.750712i \(0.270289\pi\)
\(44\) 4.95894e7i 0.300694i
\(45\) 0 0
\(46\) −1.73438e8 −0.842085
\(47\) −1.27799e8 − 7.37849e7i −0.557236 − 0.321720i 0.194800 − 0.980843i \(-0.437594\pi\)
−0.752035 + 0.659123i \(0.770928\pi\)
\(48\) 0 0
\(49\) 8.89743e7 + 1.54108e8i 0.314981 + 0.545563i
\(50\) 3.82733e7 − 2.20971e7i 0.122474 − 0.0707107i
\(51\) 0 0
\(52\) 2.94230e7 − 5.09622e7i 0.0773875 − 0.134039i
\(53\) 5.19853e6i 0.0124309i 0.999981 + 0.00621544i \(0.00197845\pi\)
−0.999981 + 0.00621544i \(0.998022\pi\)
\(54\) 0 0
\(55\) −1.35358e8 −0.268949
\(56\) −1.02577e8 − 5.92227e7i −0.186255 − 0.107535i
\(57\) 0 0
\(58\) −3.20969e8 − 5.55935e8i −0.489017 − 0.847002i
\(59\) −3.10574e8 + 1.79310e8i −0.434415 + 0.250810i −0.701226 − 0.712939i \(-0.747364\pi\)
0.266811 + 0.963749i \(0.414030\pi\)
\(60\) 0 0
\(61\) 3.95553e8 − 6.85118e8i 0.468334 − 0.811178i −0.531011 − 0.847365i \(-0.678188\pi\)
0.999345 + 0.0361868i \(0.0115211\pi\)
\(62\) 4.75372e7i 0.0518889i
\(63\) 0 0
\(64\) −1.34218e8 −0.125000
\(65\) 1.39105e8 + 8.03124e7i 0.119888 + 0.0692175i
\(66\) 0 0
\(67\) 5.84667e8 + 1.01267e9i 0.433046 + 0.750058i 0.997134 − 0.0756568i \(-0.0241053\pi\)
−0.564088 + 0.825715i \(0.690772\pi\)
\(68\) 5.62435e8 − 3.24722e8i 0.386837 − 0.223340i
\(69\) 0 0
\(70\) 1.61653e8 − 2.79991e8i 0.0961819 − 0.166592i
\(71\) − 2.00363e9i − 1.11052i −0.831677 − 0.555260i \(-0.812619\pi\)
0.831677 − 0.555260i \(-0.187381\pi\)
\(72\) 0 0
\(73\) 3.30750e9 1.59546 0.797728 − 0.603017i \(-0.206035\pi\)
0.797728 + 0.603017i \(0.206035\pi\)
\(74\) −1.97523e9 − 1.14040e9i −0.890140 − 0.513922i
\(75\) 0 0
\(76\) 4.44640e8 + 7.70140e8i 0.175364 + 0.303740i
\(77\) −8.57557e8 + 4.95111e8i −0.316818 + 0.182915i
\(78\) 0 0
\(79\) −1.40038e8 + 2.42553e8i −0.0455103 + 0.0788262i −0.887883 − 0.460069i \(-0.847825\pi\)
0.842373 + 0.538895i \(0.181158\pi\)
\(80\) − 3.66357e8i − 0.111803i
\(81\) 0 0
\(82\) 2.85045e9 0.768855
\(83\) −3.27351e9 − 1.88996e9i −0.831042 − 0.479802i 0.0231674 − 0.999732i \(-0.492625\pi\)
−0.854209 + 0.519929i \(0.825958\pi\)
\(84\) 0 0
\(85\) 8.86353e8 + 1.53521e9i 0.199762 + 0.345997i
\(86\) −1.84266e9 + 1.06386e9i −0.391699 + 0.226148i
\(87\) 0 0
\(88\) −5.61040e8 + 9.71749e8i −0.106312 + 0.184137i
\(89\) − 6.97588e9i − 1.24925i −0.780925 − 0.624625i \(-0.785252\pi\)
0.780925 − 0.624625i \(-0.214748\pi\)
\(90\) 0 0
\(91\) 1.17506e9 0.188302
\(92\) −3.39868e9 − 1.96223e9i −0.515670 − 0.297722i
\(93\) 0 0
\(94\) −1.66956e9 − 2.89177e9i −0.227491 − 0.394025i
\(95\) −2.10215e9 + 1.21368e9i −0.271673 + 0.156851i
\(96\) 0 0
\(97\) 5.07737e9 − 8.79427e9i 0.591263 − 1.02410i −0.402800 − 0.915288i \(-0.631963\pi\)
0.994063 − 0.108809i \(-0.0347037\pi\)
\(98\) 4.02652e9i 0.445450i
\(99\) 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.251.30 80
3.2 odd 2 90.11.h.a.11.20 ✓ 80
9.4 even 3 90.11.h.a.41.20 yes 80
9.5 odd 6 inner 270.11.h.a.71.30 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.20 ✓ 80 3.2 odd 2
90.11.h.a.41.20 yes 80 9.4 even 3
270.11.h.a.71.30 80 9.5 odd 6 inner
270.11.h.a.251.30 80 1.1 even 1 trivial