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Properties

Label 27.5.b.c.26.1
Level $27$
Weight $5$
Character 27.26
Analytic conductor $2.791$
Analytic rank $0$
Dimension $2$
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] = [27,5,Mod(26,27)] 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("27.26"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 27.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.79098900326\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 26.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 27.26
Dual form 27.5.b.c.26.2

$q$-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)
 
\(f(q)\) \(=\) \(q-3.00000i q^{2} +7.00000 q^{4} -33.0000i q^{5} -19.0000 q^{7} -69.0000i q^{8} -99.0000 q^{10} +123.000i q^{11} +302.000 q^{13} +57.0000i q^{14} -95.0000 q^{16} +414.000i q^{17} -304.000 q^{19} -231.000i q^{20} +369.000 q^{22} +300.000i q^{23} -464.000 q^{25} -906.000i q^{26} -133.000 q^{28} -678.000i q^{29} +239.000 q^{31} -819.000i q^{32} +1242.00 q^{34} +627.000i q^{35} +740.000 q^{37} +912.000i q^{38} -2277.00 q^{40} +228.000i q^{41} -982.000 q^{43} +861.000i q^{44} +900.000 q^{46} +2166.00i q^{47} -2040.00 q^{49} +1392.00i q^{50} +2114.00 q^{52} -1593.00i q^{53} +4059.00 q^{55} +1311.00i q^{56} -2034.00 q^{58} -2922.00i q^{59} -316.000 q^{61} -717.000i q^{62} -3977.00 q^{64} -9966.00i q^{65} +4622.00 q^{67} +2898.00i q^{68} +1881.00 q^{70} +1818.00i q^{71} -3031.00 q^{73} -2220.00i q^{74} -2128.00 q^{76} -2337.00i q^{77} -10450.0 q^{79} +3135.00i q^{80} +684.000 q^{82} +12633.0i q^{83} +13662.0 q^{85} +2946.00i q^{86} +8487.00 q^{88} -7002.00i q^{89} -5738.00 q^{91} +2100.00i q^{92} +6498.00 q^{94} +10032.0i q^{95} -6517.00 q^{97} +6120.00i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 14 q^{4} - 38 q^{7} - 198 q^{10} + 604 q^{13} - 190 q^{16} - 608 q^{19} + 738 q^{22} - 928 q^{25} - 266 q^{28} + 478 q^{31} + 2484 q^{34} + 1480 q^{37} - 4554 q^{40} - 1964 q^{43} + 1800 q^{46} - 4080 q^{49}+ \cdots - 13034 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.00000i − 0.750000i −0.927025 − 0.375000i \(-0.877643\pi\)
0.927025 − 0.375000i \(-0.122357\pi\)
\(3\) 0 0
\(4\) 7.00000 0.437500
\(5\) − 33.0000i − 1.32000i −0.751266 − 0.660000i \(-0.770556\pi\)
0.751266 − 0.660000i \(-0.229444\pi\)
\(6\) 0 0
\(7\) −19.0000 −0.387755 −0.193878 − 0.981026i \(-0.562106\pi\)
−0.193878 + 0.981026i \(0.562106\pi\)
\(8\) − 69.0000i − 1.07812i
\(9\) 0 0
\(10\) −99.0000 −0.990000
\(11\) 123.000i 1.01653i 0.861201 + 0.508264i \(0.169713\pi\)
−0.861201 + 0.508264i \(0.830287\pi\)
\(12\) 0 0
\(13\) 302.000 1.78698 0.893491 − 0.449081i \(-0.148248\pi\)
0.893491 + 0.449081i \(0.148248\pi\)
\(14\) 57.0000i 0.290816i
\(15\) 0 0
\(16\) −95.0000 −0.371094
\(17\) 414.000i 1.43253i 0.697830 + 0.716263i \(0.254149\pi\)
−0.697830 + 0.716263i \(0.745851\pi\)
\(18\) 0 0
\(19\) −304.000 −0.842105 −0.421053 − 0.907036i \(-0.638339\pi\)
−0.421053 + 0.907036i \(0.638339\pi\)
\(20\) − 231.000i − 0.577500i
\(21\) 0 0
\(22\) 369.000 0.762397
\(23\) 300.000i 0.567108i 0.958956 + 0.283554i \(0.0915135\pi\)
−0.958956 + 0.283554i \(0.908487\pi\)
\(24\) 0 0
\(25\) −464.000 −0.742400
\(26\) − 906.000i − 1.34024i
\(27\) 0 0
\(28\) −133.000 −0.169643
\(29\) − 678.000i − 0.806183i −0.915160 − 0.403092i \(-0.867936\pi\)
0.915160 − 0.403092i \(-0.132064\pi\)
\(30\) 0 0
\(31\) 239.000 0.248699 0.124350 − 0.992238i \(-0.460316\pi\)
0.124350 + 0.992238i \(0.460316\pi\)
\(32\) − 819.000i − 0.799805i
\(33\) 0 0
\(34\) 1242.00 1.07439
\(35\) 627.000i 0.511837i
\(36\) 0 0
\(37\) 740.000 0.540541 0.270270 − 0.962784i \(-0.412887\pi\)
0.270270 + 0.962784i \(0.412887\pi\)
\(38\) 912.000i 0.631579i
\(39\) 0 0
\(40\) −2277.00 −1.42312
\(41\) 228.000i 0.135634i 0.997698 + 0.0678168i \(0.0216033\pi\)
−0.997698 + 0.0678168i \(0.978397\pi\)
\(42\) 0 0
\(43\) −982.000 −0.531098 −0.265549 − 0.964097i \(-0.585553\pi\)
−0.265549 + 0.964097i \(0.585553\pi\)
\(44\) 861.000i 0.444731i
\(45\) 0 0
\(46\) 900.000 0.425331
\(47\) 2166.00i 0.980534i 0.871572 + 0.490267i \(0.163101\pi\)
−0.871572 + 0.490267i \(0.836899\pi\)
\(48\) 0 0
\(49\) −2040.00 −0.849646
\(50\) 1392.00i 0.556800i
\(51\) 0 0
\(52\) 2114.00 0.781805
\(53\) − 1593.00i − 0.567106i −0.958957 − 0.283553i \(-0.908487\pi\)
0.958957 − 0.283553i \(-0.0915131\pi\)
\(54\) 0 0
\(55\) 4059.00 1.34182
\(56\) 1311.00i 0.418048i
\(57\) 0 0
\(58\) −2034.00 −0.604637
\(59\) − 2922.00i − 0.839414i −0.907660 − 0.419707i \(-0.862133\pi\)
0.907660 − 0.419707i \(-0.137867\pi\)
\(60\) 0 0
\(61\) −316.000 −0.0849234 −0.0424617 − 0.999098i \(-0.513520\pi\)
−0.0424617 + 0.999098i \(0.513520\pi\)
\(62\) − 717.000i − 0.186524i
\(63\) 0 0
\(64\) −3977.00 −0.970947
\(65\) − 9966.00i − 2.35882i
\(66\) 0 0
\(67\) 4622.00 1.02963 0.514814 − 0.857302i \(-0.327861\pi\)
0.514814 + 0.857302i \(0.327861\pi\)
\(68\) 2898.00i 0.626730i
\(69\) 0 0
\(70\) 1881.00 0.383878
\(71\) 1818.00i 0.360643i 0.983608 + 0.180321i \(0.0577138\pi\)
−0.983608 + 0.180321i \(0.942286\pi\)
\(72\) 0 0
\(73\) −3031.00 −0.568775 −0.284387 − 0.958709i \(-0.591790\pi\)
−0.284387 + 0.958709i \(0.591790\pi\)
\(74\) − 2220.00i − 0.405405i
\(75\) 0 0
\(76\) −2128.00 −0.368421
\(77\) − 2337.00i − 0.394164i
\(78\) 0 0
\(79\) −10450.0 −1.67441 −0.837206 − 0.546888i \(-0.815812\pi\)
−0.837206 + 0.546888i \(0.815812\pi\)
\(80\) 3135.00i 0.489844i
\(81\) 0 0
\(82\) 684.000 0.101725
\(83\) 12633.0i 1.83379i 0.399125 + 0.916897i \(0.369314\pi\)
−0.399125 + 0.916897i \(0.630686\pi\)
\(84\) 0 0
\(85\) 13662.0 1.89093
\(86\) 2946.00i 0.398323i
\(87\) 0 0
\(88\) 8487.00 1.09595
\(89\) − 7002.00i − 0.883979i −0.897020 − 0.441990i \(-0.854273\pi\)
0.897020 − 0.441990i \(-0.145727\pi\)
\(90\) 0 0
\(91\) −5738.00 −0.692911
\(92\) 2100.00i 0.248110i
\(93\) 0 0
\(94\) 6498.00 0.735401
\(95\) 10032.0i 1.11158i
\(96\) 0 0
\(97\) −6517.00 −0.692635 −0.346317 − 0.938117i \(-0.612568\pi\)
−0.346317 + 0.938117i \(0.612568\pi\)
\(98\) 6120.00i 0.637234i
\(99\) 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 27.5.b.c.26.1 ✓ 2
3.2 odd 2 inner 27.5.b.c.26.2 yes 2
4.3 odd 2 432.5.e.e.161.1 2
5.2 odd 4 675.5.d.d.674.1 2
5.3 odd 4 675.5.d.a.674.2 2
5.4 even 2 675.5.c.h.26.2 2
9.2 odd 6 81.5.d.b.53.2 4
9.4 even 3 81.5.d.b.26.2 4
9.5 odd 6 81.5.d.b.26.1 4
9.7 even 3 81.5.d.b.53.1 4
12.11 even 2 432.5.e.e.161.2 2
15.2 even 4 675.5.d.a.674.1 2
15.8 even 4 675.5.d.d.674.2 2
15.14 odd 2 675.5.c.h.26.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.5.b.c.26.1 ✓ 2 1.1 even 1 trivial
27.5.b.c.26.2 yes 2 3.2 odd 2 inner
81.5.d.b.26.1 4 9.5 odd 6
81.5.d.b.26.2 4 9.4 even 3
81.5.d.b.53.1 4 9.7 even 3
81.5.d.b.53.2 4 9.2 odd 6
432.5.e.e.161.1 2 4.3 odd 2
432.5.e.e.161.2 2 12.11 even 2
675.5.c.h.26.1 2 15.14 odd 2
675.5.c.h.26.2 2 5.4 even 2
675.5.d.a.674.1 2 15.2 even 4
675.5.d.a.674.2 2 5.3 odd 4
675.5.d.d.674.1 2 5.2 odd 4
675.5.d.d.674.2 2 15.8 even 4