Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [288,4,Mod(145,288)] 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("288.145"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(288, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 288.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,-40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(16.9925500817\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-10}, \sqrt{22})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 145.2
Root \(-2.34521 - 1.58114i\) of defining polynomial
Character \(\chi\) \(=\) 288.145
Dual form 288.4.d.c.145.3

$q$-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)
 
\(f(q)\) \(=\) \(q-6.32456i q^{5} -10.0000 q^{7} +37.9473i q^{11} +59.3296i q^{13} +75.0467 q^{17} -118.659i q^{19} +150.093 q^{23} +85.0000 q^{25} +246.658i q^{29} -62.0000 q^{31} +63.2456i q^{35} +59.3296i q^{37} +375.233 q^{41} +118.659i q^{43} +450.280 q^{47} -243.000 q^{49} +132.816i q^{53} +240.000 q^{55} +733.648i q^{59} -533.966i q^{61} +375.233 q^{65} +711.955i q^{67} +30.0000 q^{73} -379.473i q^{77} -94.0000 q^{79} -670.403i q^{83} -474.637i q^{85} -750.467 q^{89} -593.296i q^{91} -750.467 q^{95} +130.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 40 q^{7} + 340 q^{25} - 248 q^{31} - 972 q^{49} + 960 q^{55} + 120 q^{73} - 376 q^{79} + 520 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(5\) − 6.32456i − 0.565685i −0.959166 − 0.282843i \(-0.908723\pi\)
0.959166 − 0.282843i \(-0.0912774\pi\)
\(6\) 0 0
\(7\) −10.0000 −0.539949 −0.269975 − 0.962867i \(-0.587015\pi\)
−0.269975 + 0.962867i \(0.587015\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 37.9473i 1.04014i 0.854123 + 0.520071i \(0.174094\pi\)
−0.854123 + 0.520071i \(0.825906\pi\)
\(12\) 0 0
\(13\) 59.3296i 1.26577i 0.774244 + 0.632887i \(0.218130\pi\)
−0.774244 + 0.632887i \(0.781870\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 75.0467 1.07068 0.535338 − 0.844638i \(-0.320184\pi\)
0.535338 + 0.844638i \(0.320184\pi\)
\(18\) 0 0
\(19\) − 118.659i − 1.43275i −0.697715 − 0.716376i \(-0.745800\pi\)
0.697715 − 0.716376i \(-0.254200\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 150.093 1.36072 0.680361 − 0.732877i \(-0.261823\pi\)
0.680361 + 0.732877i \(0.261823\pi\)
\(24\) 0 0
\(25\) 85.0000 0.680000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 246.658i 1.57942i 0.613480 + 0.789710i \(0.289769\pi\)
−0.613480 + 0.789710i \(0.710231\pi\)
\(30\) 0 0
\(31\) −62.0000 −0.359211 −0.179605 − 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 63.2456i 0.305441i
\(36\) 0 0
\(37\) 59.3296i 0.263614i 0.991275 + 0.131807i \(0.0420779\pi\)
−0.991275 + 0.131807i \(0.957922\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 375.233 1.42931 0.714654 − 0.699479i \(-0.246584\pi\)
0.714654 + 0.699479i \(0.246584\pi\)
\(42\) 0 0
\(43\) 118.659i 0.420822i 0.977613 + 0.210411i \(0.0674802\pi\)
−0.977613 + 0.210411i \(0.932520\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 450.280 1.39745 0.698724 − 0.715391i \(-0.253751\pi\)
0.698724 + 0.715391i \(0.253751\pi\)
\(48\) 0 0
\(49\) −243.000 −0.708455
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 132.816i 0.344220i 0.985078 + 0.172110i \(0.0550584\pi\)
−0.985078 + 0.172110i \(0.944942\pi\)
\(54\) 0 0
\(55\) 240.000 0.588393
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 733.648i 1.61886i 0.587215 + 0.809431i \(0.300224\pi\)
−0.587215 + 0.809431i \(0.699776\pi\)
\(60\) 0 0
\(61\) − 533.966i − 1.12078i −0.828230 − 0.560388i \(-0.810652\pi\)
0.828230 − 0.560388i \(-0.189348\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 375.233 0.716030
\(66\) 0 0
\(67\) 711.955i 1.29820i 0.760705 + 0.649098i \(0.224854\pi\)
−0.760705 + 0.649098i \(0.775146\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 30.0000 0.0480991 0.0240496 − 0.999711i \(-0.492344\pi\)
0.0240496 + 0.999711i \(0.492344\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 379.473i − 0.561623i
\(78\) 0 0
\(79\) −94.0000 −0.133871 −0.0669356 − 0.997757i \(-0.521322\pi\)
−0.0669356 + 0.997757i \(0.521322\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 670.403i − 0.886582i −0.896378 − 0.443291i \(-0.853811\pi\)
0.896378 − 0.443291i \(-0.146189\pi\)
\(84\) 0 0
\(85\) − 474.637i − 0.605666i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −750.467 −0.893812 −0.446906 − 0.894581i \(-0.647474\pi\)
−0.446906 + 0.894581i \(0.647474\pi\)
\(90\) 0 0
\(91\) − 593.296i − 0.683454i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −750.467 −0.810487
\(96\) 0 0
\(97\) 130.000 0.136077 0.0680387 − 0.997683i \(-0.478326\pi\)
0.0680387 + 0.997683i \(0.478326\pi\)
\(98\) 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 288.4.d.c.145.2 4
3.2 odd 2 inner 288.4.d.c.145.4 4
4.3 odd 2 72.4.d.c.37.3 yes 4
8.3 odd 2 72.4.d.c.37.4 yes 4
8.5 even 2 inner 288.4.d.c.145.3 4
12.11 even 2 72.4.d.c.37.2 yes 4
16.3 odd 4 2304.4.a.bx.1.1 4
16.5 even 4 2304.4.a.cc.1.4 4
16.11 odd 4 2304.4.a.bx.1.4 4
16.13 even 4 2304.4.a.cc.1.1 4
24.5 odd 2 inner 288.4.d.c.145.1 4
24.11 even 2 72.4.d.c.37.1 ✓ 4
48.5 odd 4 2304.4.a.cc.1.2 4
48.11 even 4 2304.4.a.bx.1.2 4
48.29 odd 4 2304.4.a.cc.1.3 4
48.35 even 4 2304.4.a.bx.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.4.d.c.37.1 ✓ 4 24.11 even 2
72.4.d.c.37.2 yes 4 12.11 even 2
72.4.d.c.37.3 yes 4 4.3 odd 2
72.4.d.c.37.4 yes 4 8.3 odd 2
288.4.d.c.145.1 4 24.5 odd 2 inner
288.4.d.c.145.2 4 1.1 even 1 trivial
288.4.d.c.145.3 4 8.5 even 2 inner
288.4.d.c.145.4 4 3.2 odd 2 inner
2304.4.a.bx.1.1 4 16.3 odd 4
2304.4.a.bx.1.2 4 48.11 even 4
2304.4.a.bx.1.3 4 48.35 even 4
2304.4.a.bx.1.4 4 16.11 odd 4
2304.4.a.cc.1.1 4 16.13 even 4
2304.4.a.cc.1.2 4 48.5 odd 4
2304.4.a.cc.1.3 4 48.29 odd 4
2304.4.a.cc.1.4 4 16.5 even 4