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Properties

Label 135.4.b.a.109.3
Level $135$
Weight $4$
Character 135.109
Analytic conductor $7.965$
Analytic rank $0$
Dimension $4$
CM discriminant -15
Inner twists $4$

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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] = [135,4,Mod(109,135)] 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("135.109"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(135, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 135.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.96525785077\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 109.3
Root \(-1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 135.109
Dual form 135.4.b.a.109.2

$q$-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)
 
\(f(q)\) \(=\) \(q+3.38197i q^{2} -3.43769 q^{4} +11.1803i q^{5} +15.4296i q^{8} -37.8115 q^{10} -79.6838 q^{16} +87.3181i q^{17} -102.125 q^{19} -38.4346i q^{20} -121.807i q^{23} -125.000 q^{25} +337.371 q^{31} -146.051i q^{32} -295.307 q^{34} -345.382i q^{38} -172.508 q^{40} +411.945 q^{46} +545.601i q^{47} +343.000 q^{49} -422.746i q^{50} +706.813i q^{53} +943.735 q^{61} +1140.98i q^{62} -143.530 q^{64} -300.173i q^{68} +351.073 q^{76} +1339.35 q^{79} -890.892i q^{80} -1346.04i q^{83} -976.246 q^{85} +418.734i q^{92} -1845.20 q^{94} -1141.79i q^{95} +1160.01i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 54 q^{4} + 50 q^{10} + 446 q^{16} - 328 q^{19} - 500 q^{25} + 464 q^{31} - 14 q^{34} - 2300 q^{40} + 3298 q^{46} + 1372 q^{49} + 716 q^{61} - 6692 q^{64} + 3618 q^{76} + 608 q^{79} - 3100 q^{85} + 2440 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).

\(n\) \(56\) \(82\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.38197i 1.19571i 0.801606 + 0.597853i \(0.203979\pi\)
−0.801606 + 0.597853i \(0.796021\pi\)
\(3\) 0 0
\(4\) −3.43769 −0.429712
\(5\) 11.1803i 1.00000i
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 15.4296i 0.681897i
\(9\) 0 0
\(10\) −37.8115 −1.19571
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −79.6838 −1.24506
\(17\) 87.3181i 1.24575i 0.782321 + 0.622875i \(0.214036\pi\)
−0.782321 + 0.622875i \(0.785964\pi\)
\(18\) 0 0
\(19\) −102.125 −1.23310 −0.616552 − 0.787314i \(-0.711471\pi\)
−0.616552 + 0.787314i \(0.711471\pi\)
\(20\) − 38.4346i − 0.429712i
\(21\) 0 0
\(22\) 0 0
\(23\) − 121.807i − 1.10428i −0.833752 − 0.552139i \(-0.813812\pi\)
0.833752 − 0.552139i \(-0.186188\pi\)
\(24\) 0 0
\(25\) −125.000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 337.371 1.95463 0.977316 − 0.211788i \(-0.0679286\pi\)
0.977316 + 0.211788i \(0.0679286\pi\)
\(32\) − 146.051i − 0.806828i
\(33\) 0 0
\(34\) −295.307 −1.48955
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) − 345.382i − 1.47443i
\(39\) 0 0
\(40\) −172.508 −0.681897
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 411.945 1.32039
\(47\) 545.601i 1.69328i 0.532168 + 0.846639i \(0.321377\pi\)
−0.532168 + 0.846639i \(0.678623\pi\)
\(48\) 0 0
\(49\) 343.000 1.00000
\(50\) − 422.746i − 1.19571i
\(51\) 0 0
\(52\) 0 0
\(53\) 706.813i 1.83185i 0.401344 + 0.915927i \(0.368543\pi\)
−0.401344 + 0.915927i \(0.631457\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 943.735 1.98087 0.990434 − 0.137989i \(-0.0440639\pi\)
0.990434 + 0.137989i \(0.0440639\pi\)
\(62\) 1140.98i 2.33716i
\(63\) 0 0
\(64\) −143.530 −0.280331
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) − 300.173i − 0.535313i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 351.073 0.529880
\(77\) 0 0
\(78\) 0 0
\(79\) 1339.35 1.90745 0.953727 − 0.300674i \(-0.0972115\pi\)
0.953727 + 0.300674i \(0.0972115\pi\)
\(80\) − 890.892i − 1.24506i
\(81\) 0 0
\(82\) 0 0
\(83\) − 1346.04i − 1.78009i −0.455873 − 0.890045i \(-0.650673\pi\)
0.455873 − 0.890045i \(-0.349327\pi\)
\(84\) 0 0
\(85\) −976.246 −1.24575
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 418.734i 0.474522i
\(93\) 0 0
\(94\) −1845.20 −2.02466
\(95\) − 1141.79i − 1.23310i
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 1160.01i 1.19571i
\(99\) 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 135.4.b.a.109.3 yes 4
3.2 odd 2 inner 135.4.b.a.109.2 ✓ 4
5.2 odd 4 675.4.a.k.1.2 2
5.3 odd 4 675.4.a.o.1.1 2
5.4 even 2 inner 135.4.b.a.109.2 ✓ 4
15.2 even 4 675.4.a.o.1.1 2
15.8 even 4 675.4.a.k.1.2 2
15.14 odd 2 CM 135.4.b.a.109.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.4.b.a.109.2 ✓ 4 3.2 odd 2 inner
135.4.b.a.109.2 ✓ 4 5.4 even 2 inner
135.4.b.a.109.3 yes 4 1.1 even 1 trivial
135.4.b.a.109.3 yes 4 15.14 odd 2 CM
675.4.a.k.1.2 2 5.2 odd 4
675.4.a.k.1.2 2 15.8 even 4
675.4.a.o.1.1 2 5.3 odd 4
675.4.a.o.1.1 2 15.2 even 4