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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 71.38
Character \(\chi\) \(=\) 270.71
Dual form 270.11.h.a.251.38

$q$-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)
 
\(f(q)\) \(=\) \(q+(19.5959 - 11.3137i) q^{2} +(256.000 - 443.405i) q^{4} +(1210.31 + 698.771i) q^{5} +(-12296.7 - 21298.5i) q^{7} -11585.2i q^{8} +31622.8 q^{10} +(-87076.5 + 50273.7i) q^{11} +(248770. - 430883. i) q^{13} +(-481931. - 278243. i) q^{14} +(-131072. - 227023. i) q^{16} +1.31855e6i q^{17} -3.30546e6 q^{19} +(619677. - 357771. i) q^{20} +(-1.13756e6 + 1.97032e6i) q^{22} +(-741187. - 427924. i) q^{23} +(976562. + 1.69146e6i) q^{25} -1.12581e7i q^{26} -1.25918e7 q^{28} +(-2.44124e7 + 1.40945e7i) q^{29} +(-3.92422e6 + 6.79695e6i) q^{31} +(-5.13695e6 - 2.96582e6i) q^{32} +(1.49177e7 + 2.58381e7i) q^{34} -3.43704e7i q^{35} +8.06236e7 q^{37} +(-6.47735e7 + 3.73970e7i) q^{38} +(8.09543e6 - 1.40217e7i) q^{40} +(-4.04458e7 - 2.33514e7i) q^{41} +(-9.45742e7 - 1.63807e8i) q^{43} +5.14802e7i q^{44} -1.93656e7 q^{46} +(-1.68457e7 + 9.72589e6i) q^{47} +(-1.61181e8 + 2.79173e8i) q^{49} +(3.82733e7 + 2.20971e7i) q^{50} +(-1.27370e8 - 2.20612e8i) q^{52} +3.26059e8i q^{53} -1.40519e8 q^{55} +(-2.46749e8 + 1.42460e8i) q^{56} +(-3.18923e8 + 5.52391e8i) q^{58} +(8.59385e8 + 4.96166e8i) q^{59} +(2.34338e8 + 4.05885e8i) q^{61} +1.77590e8i q^{62} -1.34218e8 q^{64} +(6.02177e8 - 3.47667e8i) q^{65} +(1.23126e9 - 2.13261e9i) q^{67} +(5.84650e8 + 3.37548e8i) q^{68} +(-3.88856e8 - 6.73519e8i) q^{70} +1.82010e9i q^{71} -1.52969e9 q^{73} +(1.57989e9 - 9.12152e8i) q^{74} +(-8.46197e8 + 1.46566e9i) q^{76} +(2.14151e9 + 1.23640e9i) q^{77} +(2.30062e9 + 3.98479e9i) q^{79} -3.66357e8i q^{80} -1.05676e9 q^{82} +(1.55643e9 - 8.98608e8i) q^{83} +(-9.21363e8 + 1.59585e9i) q^{85} +(-3.70654e9 - 2.13997e9i) q^{86} +(5.82432e8 + 1.00880e9i) q^{88} +4.51468e9i q^{89} -1.22362e10 q^{91} +(-3.79488e8 + 2.19097e8i) q^{92} +(-2.20072e8 + 3.81175e8i) q^{94} +(-4.00062e9 - 2.30976e9i) q^{95} +(-6.08289e9 - 1.05359e10i) q^{97} +7.29421e9i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q + 20480 q^{4} - 24476 q^{7} - 1944 q^{11} + 561100 q^{13} + 175680 q^{14} - 10485760 q^{16} + 5932480 q^{19} - 6946944 q^{22} + 2008908 q^{23} + 78125000 q^{25} - 25063424 q^{28} + 54816192 q^{29}+ \cdots - 10980388424 q^{97}+O(q^{100}) \) 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
\(7\) −12296.7 − 21298.5i −0.731643 − 1.26724i −0.956181 − 0.292777i \(-0.905421\pi\)
0.224538 − 0.974465i \(-0.427913\pi\)
\(8\) − 11585.2i − 0.353553i
\(9\) 0 0
\(10\) 31622.8 0.316228
\(11\) −87076.5 + 50273.7i −0.540677 + 0.312160i −0.745353 − 0.666670i \(-0.767719\pi\)
0.204676 + 0.978830i \(0.434386\pi\)
\(12\) 0 0
\(13\) 248770. − 430883.i 0.670011 − 1.16049i −0.307890 − 0.951422i \(-0.599623\pi\)
0.977901 − 0.209071i \(-0.0670438\pi\)
\(14\) −481931. − 278243.i −0.896075 − 0.517349i
\(15\) 0 0
\(16\) −131072. − 227023.i −0.125000 − 0.216506i
\(17\) 1.31855e6i 0.928648i 0.885666 + 0.464324i \(0.153703\pi\)
−0.885666 + 0.464324i \(0.846297\pi\)
\(18\) 0 0
\(19\) −3.30546e6 −1.33495 −0.667473 − 0.744634i \(-0.732624\pi\)
−0.667473 + 0.744634i \(0.732624\pi\)
\(20\) 619677. − 357771.i 0.193649 − 0.111803i
\(21\) 0 0
\(22\) −1.13756e6 + 1.97032e6i −0.220730 + 0.382316i
\(23\) −741187. − 427924.i −0.115156 − 0.0664856i 0.441315 − 0.897352i \(-0.354512\pi\)
−0.556472 + 0.830866i \(0.687845\pi\)
\(24\) 0 0
\(25\) 976562. + 1.69146e6i 0.100000 + 0.173205i
\(26\) − 1.12581e7i − 0.947538i
\(27\) 0 0
\(28\) −1.25918e7 −0.731643
\(29\) −2.44124e7 + 1.40945e7i −1.19020 + 0.687165i −0.958353 − 0.285587i \(-0.907812\pi\)
−0.231851 + 0.972751i \(0.574478\pi\)
\(30\) 0 0
\(31\) −3.92422e6 + 6.79695e6i −0.137071 + 0.237413i −0.926387 − 0.376574i \(-0.877102\pi\)
0.789316 + 0.613987i \(0.210435\pi\)
\(32\) −5.13695e6 − 2.96582e6i −0.153093 − 0.0883883i
\(33\) 0 0
\(34\) 1.49177e7 + 2.58381e7i 0.328326 + 0.568678i
\(35\) − 3.43704e7i − 0.654401i
\(36\) 0 0
\(37\) 8.06236e7 1.16266 0.581331 − 0.813667i \(-0.302532\pi\)
0.581331 + 0.813667i \(0.302532\pi\)
\(38\) −6.47735e7 + 3.73970e7i −0.817484 + 0.471975i
\(39\) 0 0
\(40\) 8.09543e6 − 1.40217e7i 0.0790569 − 0.136931i
\(41\) −4.04458e7 − 2.33514e7i −0.349104 − 0.201555i 0.315187 − 0.949030i \(-0.397933\pi\)
−0.664290 + 0.747475i \(0.731266\pi\)
\(42\) 0 0
\(43\) −9.45742e7 − 1.63807e8i −0.643325 − 1.11427i −0.984686 − 0.174339i \(-0.944221\pi\)
0.341361 − 0.939932i \(-0.389112\pi\)
\(44\) 5.14802e7i 0.312160i
\(45\) 0 0
\(46\) −1.93656e7 −0.0940249
\(47\) −1.68457e7 + 9.72589e6i −0.0734515 + 0.0424072i −0.536276 − 0.844043i \(-0.680169\pi\)
0.462824 + 0.886450i \(0.346836\pi\)
\(48\) 0 0
\(49\) −1.61181e8 + 2.79173e8i −0.570602 + 0.988311i
\(50\) 3.82733e7 + 2.20971e7i 0.122474 + 0.0707107i
\(51\) 0 0
\(52\) −1.27370e8 − 2.20612e8i −0.335005 − 0.580246i
\(53\) 3.26059e8i 0.779681i 0.920882 + 0.389840i \(0.127470\pi\)
−0.920882 + 0.389840i \(0.872530\pi\)
\(54\) 0 0
\(55\) −1.40519e8 −0.279204
\(56\) −2.46749e8 + 1.42460e8i −0.448038 + 0.258675i
\(57\) 0 0
\(58\) −3.18923e8 + 5.52391e8i −0.485899 + 0.841601i
\(59\) 8.59385e8 + 4.96166e8i 1.20206 + 0.694012i 0.961014 − 0.276501i \(-0.0891749\pi\)
0.241050 + 0.970513i \(0.422508\pi\)
\(60\) 0 0
\(61\) 2.34338e8 + 4.05885e8i 0.277456 + 0.480567i 0.970752 − 0.240085i \(-0.0771755\pi\)
−0.693296 + 0.720653i \(0.743842\pi\)
\(62\) 1.77590e8i 0.193847i
\(63\) 0 0
\(64\) −1.34218e8 −0.125000
\(65\) 6.02177e8 − 3.47667e8i 0.518988 − 0.299638i
\(66\) 0 0
\(67\) 1.23126e9 − 2.13261e9i 0.911962 − 1.57957i 0.100674 − 0.994920i \(-0.467900\pi\)
0.811289 − 0.584646i \(-0.198766\pi\)
\(68\) 5.84650e8 + 3.37548e8i 0.402116 + 0.232162i
\(69\) 0 0
\(70\) −3.88856e8 − 6.73519e8i −0.231366 − 0.400737i
\(71\) 1.82010e9i 1.00880i 0.863471 + 0.504398i \(0.168286\pi\)
−0.863471 + 0.504398i \(0.831714\pi\)
\(72\) 0 0
\(73\) −1.52969e9 −0.737884 −0.368942 − 0.929452i \(-0.620280\pi\)
−0.368942 + 0.929452i \(0.620280\pi\)
\(74\) 1.57989e9 − 9.12152e8i 0.711982 − 0.411063i
\(75\) 0 0
\(76\) −8.46197e8 + 1.46566e9i −0.333736 + 0.578049i
\(77\) 2.14151e9 + 1.23640e9i 0.791164 + 0.456779i
\(78\) 0 0
\(79\) 2.30062e9 + 3.98479e9i 0.747668 + 1.29500i 0.948938 + 0.315463i \(0.102160\pi\)
−0.201270 + 0.979536i \(0.564507\pi\)
\(80\) − 3.66357e8i − 0.111803i
\(81\) 0 0
\(82\) −1.05676e9 −0.285042
\(83\) 1.55643e9 − 8.98608e8i 0.395130 − 0.228129i −0.289250 − 0.957253i \(-0.593406\pi\)
0.684381 + 0.729125i \(0.260073\pi\)
\(84\) 0 0
\(85\) −9.21363e8 + 1.59585e9i −0.207652 + 0.359664i
\(86\) −3.70654e9 − 2.13997e9i −0.787909 − 0.454899i
\(87\) 0 0
\(88\) 5.82432e8 + 1.00880e9i 0.110365 + 0.191158i
\(89\) 4.51468e9i 0.808495i 0.914650 + 0.404247i \(0.132466\pi\)
−0.914650 + 0.404247i \(0.867534\pi\)
\(90\) 0 0
\(91\) −1.22362e10 −1.96083
\(92\) −3.79488e8 + 2.19097e8i −0.0575782 + 0.0332428i
\(93\) 0 0
\(94\) −2.20072e8 + 3.81175e8i −0.0299864 + 0.0519380i
\(95\) −4.00062e9 − 2.30976e9i −0.517022 − 0.298503i
\(96\) 0 0
\(97\) −6.08289e9 − 1.05359e10i −0.708356 − 1.22691i −0.965467 − 0.260527i \(-0.916104\pi\)
0.257111 − 0.966382i \(-0.417230\pi\)
\(98\) 7.29421e9i 0.806953i
\(99\) 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.71.38 80
3.2 odd 2 90.11.h.a.41.4 yes 80
9.2 odd 6 inner 270.11.h.a.251.38 80
9.7 even 3 90.11.h.a.11.4 ✓ 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.h.a.11.4 ✓ 80 9.7 even 3
90.11.h.a.41.4 yes 80 3.2 odd 2
270.11.h.a.71.38 80 1.1 even 1 trivial
270.11.h.a.251.38 80 9.2 odd 6 inner