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Properties

Label 27.8.c.a.10.1
Level $27$
Weight $8$
Character 27.10
Analytic conductor $8.434$
Analytic rank $0$
Dimension $12$
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] = [27,8,Mod(10,27)] 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("27.10"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 27.c (of order \(3\), 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: \(8.43439568807\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 375 x^{10} - 1820 x^{9} + 50808 x^{8} - 192378 x^{7} + 3002887 x^{6} + \cdots + 754412211 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{21} \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 10.1
Root \(0.500000 + 9.80854i\) of defining polynomial
Character \(\chi\) \(=\) 27.10
Dual form 27.8.c.a.19.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-7.74445 - 13.4138i) q^{2} +(-55.9529 + 96.9133i) q^{4} +(-52.7641 + 91.3900i) q^{5} +(761.419 + 1318.82i) q^{7} -249.280 q^{8} +1634.51 q^{10} +(1045.08 + 1810.14i) q^{11} +(4667.91 - 8085.06i) q^{13} +(11793.5 - 20427.0i) q^{14} +(9092.51 + 15748.7i) q^{16} +20641.6 q^{17} -9369.08 q^{19} +(-5904.61 - 10227.1i) q^{20} +(16187.2 - 28037.1i) q^{22} +(-38695.2 + 67022.1i) q^{23} +(33494.4 + 58014.0i) q^{25} -144602. q^{26} -170415. q^{28} +(81478.1 + 141124. i) q^{29} +(-9829.95 + 17026.0i) q^{31} +(124879. - 216297. i) q^{32} +(-159857. - 276881. i) q^{34} -160702. q^{35} +105494. q^{37} +(72558.3 + 125675. i) q^{38} +(13153.0 - 22781.7i) q^{40} +(68590.9 - 118803. i) q^{41} +(-6130.74 - 10618.8i) q^{43} -233902. q^{44} +1.19869e6 q^{46} +(270053. + 467745. i) q^{47} +(-747747. + 1.29514e6i) q^{49} +(518791. - 898573. i) q^{50} +(522367. + 904766. i) q^{52} -17622.4 q^{53} -220572. q^{55} +(-189807. - 328755. i) q^{56} +(1.26201e6 - 2.18586e6i) q^{58} +(194345. - 336615. i) q^{59} +(-1.32379e6 - 2.29287e6i) q^{61} +304510. q^{62} -1.54079e6 q^{64} +(492596. + 853201. i) q^{65} +(-300341. + 520206. i) q^{67} +(-1.15496e6 + 2.00044e6i) q^{68} +(1.24455e6 + 2.15562e6i) q^{70} -1.63676e6 q^{71} -3.75812e6 q^{73} +(-816991. - 1.41507e6i) q^{74} +(524227. - 907988. i) q^{76} +(-1.59149e6 + 2.75655e6i) q^{77} +(1.30240e6 + 2.25582e6i) q^{79} -1.91903e6 q^{80} -2.12479e6 q^{82} +(192939. + 334181. i) q^{83} +(-1.08913e6 + 1.88643e6i) q^{85} +(-94958.4 + 164473. i) q^{86} +(-260519. - 451232. i) q^{88} +9.20167e6 q^{89} +1.42170e7 q^{91} +(-4.33022e6 - 7.50017e6i) q^{92} +(4.18282e6 - 7.24486e6i) q^{94} +(494350. - 856240. i) q^{95} +(-1.90585e6 - 3.30104e6i) q^{97} +2.31636e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 9 q^{2} - 321 q^{4} + 180 q^{5} - 84 q^{7} - 5922 q^{8} + 252 q^{10} + 8460 q^{11} - 1848 q^{13} + 16272 q^{14} - 12417 q^{16} - 30564 q^{17} + 24432 q^{19} + 40788 q^{20} - 35001 q^{22} + 51588 q^{23}+ \cdots + 95833314 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.74445 − 13.4138i −0.684519 − 1.18562i −0.973588 − 0.228313i \(-0.926679\pi\)
0.289069 − 0.957308i \(-0.406654\pi\)
\(3\) 0 0
\(4\) −55.9529 + 96.9133i −0.437132 + 0.757135i
\(5\) −52.7641 + 91.3900i −0.188774 + 0.326967i −0.944842 − 0.327527i \(-0.893785\pi\)
0.756067 + 0.654494i \(0.227118\pi\)
\(6\) 0 0
\(7\) 761.419 + 1318.82i 0.839036 + 1.45325i 0.890701 + 0.454589i \(0.150214\pi\)
−0.0516649 + 0.998664i \(0.516453\pi\)
\(8\) −249.280 −0.172136
\(9\) 0 0
\(10\) 1634.51 0.516879
\(11\) 1045.08 + 1810.14i 0.236743 + 0.410051i 0.959778 − 0.280761i \(-0.0905868\pi\)
−0.723035 + 0.690811i \(0.757253\pi\)
\(12\) 0 0
\(13\) 4667.91 − 8085.06i 0.589279 − 1.02066i −0.405048 − 0.914295i \(-0.632745\pi\)
0.994327 − 0.106366i \(-0.0339214\pi\)
\(14\) 11793.5 − 20427.0i 1.14867 − 1.98956i
\(15\) 0 0
\(16\) 9092.51 + 15748.7i 0.554963 + 0.961224i
\(17\) 20641.6 1.01899 0.509497 − 0.860473i \(-0.329832\pi\)
0.509497 + 0.860473i \(0.329832\pi\)
\(18\) 0 0
\(19\) −9369.08 −0.313371 −0.156686 − 0.987649i \(-0.550081\pi\)
−0.156686 + 0.987649i \(0.550081\pi\)
\(20\) −5904.61 − 10227.1i −0.165039 − 0.285856i
\(21\) 0 0
\(22\) 16187.2 − 28037.1i 0.324110 − 0.561375i
\(23\) −38695.2 + 67022.1i −0.663147 + 1.14860i 0.316637 + 0.948547i \(0.397446\pi\)
−0.979784 + 0.200058i \(0.935887\pi\)
\(24\) 0 0
\(25\) 33494.4 + 58014.0i 0.428728 + 0.742579i
\(26\) −144602. −1.61349
\(27\) 0 0
\(28\) −170415. −1.46708
\(29\) 81478.1 + 141124.i 0.620366 + 1.07450i 0.989418 + 0.145096i \(0.0463491\pi\)
−0.369052 + 0.929409i \(0.620318\pi\)
\(30\) 0 0
\(31\) −9829.95 + 17026.0i −0.0592632 + 0.102647i −0.894135 − 0.447798i \(-0.852208\pi\)
0.834872 + 0.550445i \(0.185542\pi\)
\(32\) 124879. − 216297.i 0.673697 − 1.16688i
\(33\) 0 0
\(34\) −159857. − 276881.i −0.697520 − 1.20814i
\(35\) −160702. −0.633554
\(36\) 0 0
\(37\) 105494. 0.342390 0.171195 − 0.985237i \(-0.445237\pi\)
0.171195 + 0.985237i \(0.445237\pi\)
\(38\) 72558.3 + 125675.i 0.214509 + 0.371540i
\(39\) 0 0
\(40\) 13153.0 − 22781.7i 0.0324949 − 0.0562829i
\(41\) 68590.9 − 118803.i 0.155426 − 0.269205i −0.777788 − 0.628526i \(-0.783658\pi\)
0.933214 + 0.359321i \(0.116992\pi\)
\(42\) 0 0
\(43\) −6130.74 − 10618.8i −0.0117591 − 0.0203673i 0.860086 − 0.510149i \(-0.170410\pi\)
−0.871845 + 0.489782i \(0.837076\pi\)
\(44\) −233902. −0.413952
\(45\) 0 0
\(46\) 1.19869e6 1.81575
\(47\) 270053. + 467745.i 0.379408 + 0.657154i 0.990976 − 0.134038i \(-0.0427944\pi\)
−0.611568 + 0.791192i \(0.709461\pi\)
\(48\) 0 0
\(49\) −747747. + 1.29514e6i −0.907964 + 1.57264i
\(50\) 518791. − 898573.i 0.586945 − 1.01662i
\(51\) 0 0
\(52\) 522367. + 904766.i 0.515186 + 0.892328i
\(53\) −17622.4 −0.0162592 −0.00812962 − 0.999967i \(-0.502588\pi\)
−0.00812962 + 0.999967i \(0.502588\pi\)
\(54\) 0 0
\(55\) −220572. −0.178764
\(56\) −189807. − 328755.i −0.144429 − 0.250158i
\(57\) 0 0
\(58\) 1.26201e6 − 2.18586e6i 0.849304 − 1.47104i
\(59\) 194345. − 336615.i 0.123194 − 0.213379i −0.797831 − 0.602881i \(-0.794019\pi\)
0.921026 + 0.389502i \(0.127353\pi\)
\(60\) 0 0
\(61\) −1.32379e6 − 2.29287e6i −0.746731 − 1.29338i −0.949381 − 0.314126i \(-0.898289\pi\)
0.202650 − 0.979251i \(-0.435045\pi\)
\(62\) 304510. 0.162267
\(63\) 0 0
\(64\) −1.54079e6 −0.734708
\(65\) 492596. + 853201.i 0.222482 + 0.385349i
\(66\) 0 0
\(67\) −300341. + 520206.i −0.121998 + 0.211307i −0.920556 − 0.390612i \(-0.872264\pi\)
0.798557 + 0.601919i \(0.205597\pi\)
\(68\) −1.15496e6 + 2.00044e6i −0.445435 + 0.771516i
\(69\) 0 0
\(70\) 1.24455e6 + 2.15562e6i 0.433680 + 0.751156i
\(71\) −1.63676e6 −0.542727 −0.271364 − 0.962477i \(-0.587475\pi\)
−0.271364 + 0.962477i \(0.587475\pi\)
\(72\) 0 0
\(73\) −3.75812e6 −1.13068 −0.565341 − 0.824857i \(-0.691255\pi\)
−0.565341 + 0.824857i \(0.691255\pi\)
\(74\) −816991. − 1.41507e6i −0.234372 − 0.405945i
\(75\) 0 0
\(76\) 524227. − 907988.i 0.136985 − 0.237264i
\(77\) −1.59149e6 + 2.75655e6i −0.397272 + 0.688095i
\(78\) 0 0
\(79\) 1.30240e6 + 2.25582e6i 0.297200 + 0.514765i 0.975494 − 0.220025i \(-0.0706139\pi\)
−0.678294 + 0.734790i \(0.737281\pi\)
\(80\) −1.91903e6 −0.419051
\(81\) 0 0
\(82\) −2.12479e6 −0.425567
\(83\) 192939. + 334181.i 0.0370380 + 0.0641517i 0.883950 − 0.467581i \(-0.154874\pi\)
−0.846912 + 0.531733i \(0.821541\pi\)
\(84\) 0 0
\(85\) −1.08913e6 + 1.88643e6i −0.192360 + 0.333177i
\(86\) −94958.4 + 164473.i −0.0160986 + 0.0278836i
\(87\) 0 0
\(88\) −260519. − 451232.i −0.0407521 − 0.0705846i
\(89\) 9.20167e6 1.38357 0.691786 − 0.722102i \(-0.256824\pi\)
0.691786 + 0.722102i \(0.256824\pi\)
\(90\) 0 0
\(91\) 1.42170e7 1.97771
\(92\) −4.33022e6 − 7.50017e6i −0.579766 − 1.00418i
\(93\) 0 0
\(94\) 4.18282e6 − 7.24486e6i 0.519424 − 0.899668i
\(95\) 494350. − 856240.i 0.0591565 − 0.102462i
\(96\) 0 0
\(97\) −1.90585e6 − 3.30104e6i −0.212026 − 0.367239i 0.740323 − 0.672252i \(-0.234673\pi\)
−0.952348 + 0.305012i \(0.901339\pi\)
\(98\) 2.31636e7 2.48607
\(99\) 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 27.8.c.a.10.1 12
3.2 odd 2 9.8.c.a.4.6 ✓ 12
4.3 odd 2 432.8.i.c.145.2 12
9.2 odd 6 9.8.c.a.7.6 yes 12
9.4 even 3 81.8.a.c.1.6 6
9.5 odd 6 81.8.a.e.1.1 6
9.7 even 3 inner 27.8.c.a.19.1 12
12.11 even 2 144.8.i.c.49.6 12
36.7 odd 6 432.8.i.c.289.2 12
36.11 even 6 144.8.i.c.97.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.8.c.a.4.6 ✓ 12 3.2 odd 2
9.8.c.a.7.6 yes 12 9.2 odd 6
27.8.c.a.10.1 12 1.1 even 1 trivial
27.8.c.a.19.1 12 9.7 even 3 inner
81.8.a.c.1.6 6 9.4 even 3
81.8.a.e.1.1 6 9.5 odd 6
144.8.i.c.49.6 12 12.11 even 2
144.8.i.c.97.6 12 36.11 even 6
432.8.i.c.145.2 12 4.3 odd 2
432.8.i.c.289.2 12 36.7 odd 6