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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2916,3,Mod(1457,2916)] 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("2916.1457"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2916, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 2916 = 2^{2} \cdot 3^{6} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2916.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [36,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-180,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,252,0,0,0,0,0,0,0,0,0,0,0,18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(61)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(79.4552450875\)
Analytic rank: \(0\)
Dimension: \(36\)
Twist minimal: no (minimal twist has level 108)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1457.2
Character \(\chi\) \(=\) 2916.1457
Dual form 2916.3.c.b.1457.35

$q$-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)
 
\(f(q)\) \(=\) \(q-8.78835i q^{5} -4.31308 q^{7} +19.6964i q^{11} -7.57484 q^{13} +3.65486i q^{17} +33.6666 q^{19} -3.07366i q^{23} -52.2351 q^{25} -24.3137i q^{29} +28.8664 q^{31} +37.9048i q^{35} +29.8009 q^{37} -23.0041i q^{41} -8.94519 q^{43} +10.5400i q^{47} -30.3974 q^{49} -70.8948i q^{53} +173.099 q^{55} +68.2811i q^{59} +106.200 q^{61} +66.5703i q^{65} -24.6294 q^{67} -34.5416i q^{71} -41.8381 q^{73} -84.9519i q^{77} +50.5876 q^{79} +31.3979i q^{83} +32.1202 q^{85} +46.2146i q^{89} +32.6709 q^{91} -295.874i q^{95} +110.716 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q - 180 q^{25} + 252 q^{49} + 18 q^{61} - 90 q^{67} + 126 q^{73} - 198 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2916\mathbb{Z}\right)^\times\).

\(n\) \(1459\) \(2189\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) − 8.78835i − 1.75767i −0.477126 − 0.878835i \(-0.658322\pi\)
0.477126 − 0.878835i \(-0.341678\pi\)
\(6\) 0 0
\(7\) −4.31308 −0.616154 −0.308077 − 0.951361i \(-0.599685\pi\)
−0.308077 + 0.951361i \(0.599685\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 19.6964i 1.79058i 0.445485 + 0.895289i \(0.353031\pi\)
−0.445485 + 0.895289i \(0.646969\pi\)
\(12\) 0 0
\(13\) −7.57484 −0.582680 −0.291340 − 0.956620i \(-0.594101\pi\)
−0.291340 + 0.956620i \(0.594101\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.65486i 0.214992i 0.994206 + 0.107496i \(0.0342832\pi\)
−0.994206 + 0.107496i \(0.965717\pi\)
\(18\) 0 0
\(19\) 33.6666 1.77193 0.885964 − 0.463754i \(-0.153498\pi\)
0.885964 + 0.463754i \(0.153498\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 3.07366i − 0.133638i −0.997765 − 0.0668188i \(-0.978715\pi\)
0.997765 − 0.0668188i \(-0.0212849\pi\)
\(24\) 0 0
\(25\) −52.2351 −2.08940
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 24.3137i − 0.838404i −0.907893 − 0.419202i \(-0.862310\pi\)
0.907893 − 0.419202i \(-0.137690\pi\)
\(30\) 0 0
\(31\) 28.8664 0.931173 0.465587 − 0.885002i \(-0.345843\pi\)
0.465587 + 0.885002i \(0.345843\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 37.9048i 1.08299i
\(36\) 0 0
\(37\) 29.8009 0.805429 0.402714 − 0.915326i \(-0.368067\pi\)
0.402714 + 0.915326i \(0.368067\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 23.0041i − 0.561076i −0.959843 − 0.280538i \(-0.909487\pi\)
0.959843 − 0.280538i \(-0.0905129\pi\)
\(42\) 0 0
\(43\) −8.94519 −0.208028 −0.104014 − 0.994576i \(-0.533169\pi\)
−0.104014 + 0.994576i \(0.533169\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 10.5400i 0.224255i 0.993694 + 0.112127i \(0.0357665\pi\)
−0.993694 + 0.112127i \(0.964233\pi\)
\(48\) 0 0
\(49\) −30.3974 −0.620355
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 70.8948i − 1.33764i −0.743426 − 0.668818i \(-0.766800\pi\)
0.743426 − 0.668818i \(-0.233200\pi\)
\(54\) 0 0
\(55\) 173.099 3.14725
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 68.2811i 1.15731i 0.815574 + 0.578653i \(0.196422\pi\)
−0.815574 + 0.578653i \(0.803578\pi\)
\(60\) 0 0
\(61\) 106.200 1.74098 0.870489 − 0.492189i \(-0.163803\pi\)
0.870489 + 0.492189i \(0.163803\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 66.5703i 1.02416i
\(66\) 0 0
\(67\) −24.6294 −0.367603 −0.183802 − 0.982963i \(-0.558840\pi\)
−0.183802 + 0.982963i \(0.558840\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 34.5416i − 0.486502i −0.969963 − 0.243251i \(-0.921786\pi\)
0.969963 − 0.243251i \(-0.0782139\pi\)
\(72\) 0 0
\(73\) −41.8381 −0.573124 −0.286562 − 0.958062i \(-0.592512\pi\)
−0.286562 + 0.958062i \(0.592512\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 84.9519i − 1.10327i
\(78\) 0 0
\(79\) 50.5876 0.640350 0.320175 − 0.947358i \(-0.396258\pi\)
0.320175 + 0.947358i \(0.396258\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 31.3979i 0.378287i 0.981949 + 0.189144i \(0.0605712\pi\)
−0.981949 + 0.189144i \(0.939429\pi\)
\(84\) 0 0
\(85\) 32.1202 0.377884
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 46.2146i 0.519266i 0.965707 + 0.259633i \(0.0836015\pi\)
−0.965707 + 0.259633i \(0.916398\pi\)
\(90\) 0 0
\(91\) 32.6709 0.359020
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 295.874i − 3.11446i
\(96\) 0 0
\(97\) 110.716 1.14140 0.570700 − 0.821159i \(-0.306672\pi\)
0.570700 + 0.821159i \(0.306672\pi\)
\(98\) 0 0
\(99\) 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 2916.3.c.b.1457.2 36
3.2 odd 2 inner 2916.3.c.b.1457.35 36
27.5 odd 18 108.3.k.a.29.4 ✓ 36
27.11 odd 18 324.3.k.a.233.1 36
27.16 even 9 108.3.k.a.41.4 yes 36
27.22 even 9 324.3.k.a.89.1 36
108.43 odd 18 432.3.bc.b.257.3 36
108.59 even 18 432.3.bc.b.353.3 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
108.3.k.a.29.4 ✓ 36 27.5 odd 18
108.3.k.a.41.4 yes 36 27.16 even 9
324.3.k.a.89.1 36 27.22 even 9
324.3.k.a.233.1 36 27.11 odd 18
432.3.bc.b.257.3 36 108.43 odd 18
432.3.bc.b.353.3 36 108.59 even 18
2916.3.c.b.1457.2 36 1.1 even 1 trivial
2916.3.c.b.1457.35 36 3.2 odd 2 inner