↑

Properties

Label 1440.4.a.v.1.1
Level $1440$
Weight $4$
Character 1440.1
Self dual yes
Analytic conductor $84.963$
Analytic rank $1$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1440,4,Mod(1,1440)] 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("1440.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1440.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-10,0,0,0,0,0,0,0,76] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(84.9627504083\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{10}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 160)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.16228\) of defining polynomial
Character \(\chi\) \(=\) 1440.1

$q$-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)
 
\(f(q)\) \(=\) \(q-5.00000 q^{5} -18.9737 q^{7} -12.6491 q^{11} +38.0000 q^{13} -34.0000 q^{17} +101.193 q^{19} +82.2192 q^{23} +25.0000 q^{25} -270.000 q^{29} +341.526 q^{31} +94.8683 q^{35} +206.000 q^{37} +270.000 q^{41} -537.587 q^{43} -132.816 q^{47} +17.0000 q^{49} +258.000 q^{53} +63.2456 q^{55} -75.8947 q^{59} -250.000 q^{61} -190.000 q^{65} +815.868 q^{67} -645.105 q^{71} -1078.00 q^{73} +240.000 q^{77} +278.280 q^{79} -1106.80 q^{83} +170.000 q^{85} -890.000 q^{89} -720.999 q^{91} -505.964 q^{95} -254.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 10 q^{5} + 76 q^{13} - 68 q^{17} + 50 q^{25} - 540 q^{29} + 412 q^{37} + 540 q^{41} + 34 q^{49} + 516 q^{53} - 500 q^{61} - 380 q^{65} - 2156 q^{73} + 480 q^{77} + 340 q^{85} - 1780 q^{89} - 508 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) −18.9737 −1.02448 −0.512241 − 0.858842i \(-0.671184\pi\)
−0.512241 + 0.858842i \(0.671184\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −12.6491 −0.346714 −0.173357 − 0.984859i \(-0.555461\pi\)
−0.173357 + 0.984859i \(0.555461\pi\)
\(12\) 0 0
\(13\) 38.0000 0.810716 0.405358 − 0.914158i \(-0.367147\pi\)
0.405358 + 0.914158i \(0.367147\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −34.0000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 101.193 1.22185 0.610927 − 0.791687i \(-0.290797\pi\)
0.610927 + 0.791687i \(0.290797\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 82.2192 0.745387 0.372693 − 0.927955i \(-0.378434\pi\)
0.372693 + 0.927955i \(0.378434\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −270.000 −1.72889 −0.864444 − 0.502729i \(-0.832329\pi\)
−0.864444 + 0.502729i \(0.832329\pi\)
\(30\) 0 0
\(31\) 341.526 1.97871 0.989353 − 0.145537i \(-0.0464908\pi\)
0.989353 + 0.145537i \(0.0464908\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 94.8683 0.458162
\(36\) 0 0
\(37\) 206.000 0.915302 0.457651 − 0.889132i \(-0.348691\pi\)
0.457651 + 0.889132i \(0.348691\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 270.000 1.02846 0.514231 − 0.857652i \(-0.328078\pi\)
0.514231 + 0.857652i \(0.328078\pi\)
\(42\) 0 0
\(43\) −537.587 −1.90654 −0.953271 − 0.302117i \(-0.902307\pi\)
−0.953271 + 0.302117i \(0.902307\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −132.816 −0.412195 −0.206097 − 0.978531i \(-0.566076\pi\)
−0.206097 + 0.978531i \(0.566076\pi\)
\(48\) 0 0
\(49\) 17.0000 0.0495627
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 258.000 0.668661 0.334330 − 0.942456i \(-0.391490\pi\)
0.334330 + 0.942456i \(0.391490\pi\)
\(54\) 0 0
\(55\) 63.2456 0.155055
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −75.8947 −0.167469 −0.0837343 − 0.996488i \(-0.526685\pi\)
−0.0837343 + 0.996488i \(0.526685\pi\)
\(60\) 0 0
\(61\) −250.000 −0.524741 −0.262371 − 0.964967i \(-0.584504\pi\)
−0.262371 + 0.964967i \(0.584504\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −190.000 −0.362563
\(66\) 0 0
\(67\) 815.868 1.48767 0.743837 − 0.668362i \(-0.233004\pi\)
0.743837 + 0.668362i \(0.233004\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −645.105 −1.07831 −0.539154 − 0.842207i \(-0.681256\pi\)
−0.539154 + 0.842207i \(0.681256\pi\)
\(72\) 0 0
\(73\) −1078.00 −1.72836 −0.864181 − 0.503182i \(-0.832163\pi\)
−0.864181 + 0.503182i \(0.832163\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 240.000 0.355202
\(78\) 0 0
\(79\) 278.280 0.396316 0.198158 − 0.980170i \(-0.436504\pi\)
0.198158 + 0.980170i \(0.436504\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1106.80 −1.46370 −0.731848 − 0.681468i \(-0.761342\pi\)
−0.731848 + 0.681468i \(0.761342\pi\)
\(84\) 0 0
\(85\) 170.000 0.216930
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −890.000 −1.06000 −0.529999 − 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −720.999 −0.830563
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −505.964 −0.546430
\(96\) 0 0
\(97\) −254.000 −0.265874 −0.132937 − 0.991124i \(-0.542441\pi\)
−0.132937 + 0.991124i \(0.542441\pi\)
\(98\) 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 1440.4.a.v.1.1 2
3.2 odd 2 160.4.a.f.1.1 ✓ 2
4.3 odd 2 inner 1440.4.a.v.1.2 2
12.11 even 2 160.4.a.f.1.2 yes 2
15.2 even 4 800.4.c.j.449.4 4
15.8 even 4 800.4.c.j.449.2 4
15.14 odd 2 800.4.a.p.1.2 2
24.5 odd 2 320.4.a.p.1.2 2
24.11 even 2 320.4.a.p.1.1 2
48.5 odd 4 1280.4.d.u.641.4 4
48.11 even 4 1280.4.d.u.641.2 4
48.29 odd 4 1280.4.d.u.641.1 4
48.35 even 4 1280.4.d.u.641.3 4
60.23 odd 4 800.4.c.j.449.3 4
60.47 odd 4 800.4.c.j.449.1 4
60.59 even 2 800.4.a.p.1.1 2
120.29 odd 2 1600.4.a.ch.1.1 2
120.59 even 2 1600.4.a.ch.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.4.a.f.1.1 ✓ 2 3.2 odd 2
160.4.a.f.1.2 yes 2 12.11 even 2
320.4.a.p.1.1 2 24.11 even 2
320.4.a.p.1.2 2 24.5 odd 2
800.4.a.p.1.1 2 60.59 even 2
800.4.a.p.1.2 2 15.14 odd 2
800.4.c.j.449.1 4 60.47 odd 4
800.4.c.j.449.2 4 15.8 even 4
800.4.c.j.449.3 4 60.23 odd 4
800.4.c.j.449.4 4 15.2 even 4
1280.4.d.u.641.1 4 48.29 odd 4
1280.4.d.u.641.2 4 48.11 even 4
1280.4.d.u.641.3 4 48.35 even 4
1280.4.d.u.641.4 4 48.5 odd 4
1440.4.a.v.1.1 2 1.1 even 1 trivial
1440.4.a.v.1.2 2 4.3 odd 2 inner
1600.4.a.ch.1.1 2 120.29 odd 2
1600.4.a.ch.1.2 2 120.59 even 2