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Properties

Label 800.4.c.f.449.2
Level $800$
Weight $4$
Character 800.449
Analytic conductor $47.202$
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] = [800,4,Mod(449,800)] 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("800.449"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 800.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,46,0,120] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(47.2015280046\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 800.449
Dual form 800.4.c.f.449.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000i q^{3} +6.00000i q^{7} +23.0000 q^{9} +60.0000 q^{11} -50.0000i q^{13} -30.0000i q^{17} -40.0000 q^{19} -12.0000 q^{21} -178.000i q^{23} +100.000i q^{27} -166.000 q^{29} +20.0000 q^{31} +120.000i q^{33} +10.0000i q^{37} +100.000 q^{39} -250.000 q^{41} -142.000i q^{43} +214.000i q^{47} +307.000 q^{49} +60.0000 q^{51} -490.000i q^{53} -80.0000i q^{57} +800.000 q^{59} +250.000 q^{61} +138.000i q^{63} -774.000i q^{67} +356.000 q^{69} +100.000 q^{71} +230.000i q^{73} +360.000i q^{77} +1320.00 q^{79} +421.000 q^{81} -982.000i q^{83} -332.000i q^{87} -874.000 q^{89} +300.000 q^{91} +40.0000i q^{93} -310.000i q^{97} +1380.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 46 q^{9} + 120 q^{11} - 80 q^{19} - 24 q^{21} - 332 q^{29} + 40 q^{31} + 200 q^{39} - 500 q^{41} + 614 q^{49} + 120 q^{51} + 1600 q^{59} + 500 q^{61} + 712 q^{69} + 200 q^{71} + 2640 q^{79} + 842 q^{81}+ \cdots + 2760 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 2.00000i 0.384900i 0.981307 + 0.192450i \(0.0616434\pi\)
−0.981307 + 0.192450i \(0.938357\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 6.00000i 0.323970i 0.986793 + 0.161985i \(0.0517895\pi\)
−0.986793 + 0.161985i \(0.948210\pi\)
\(8\) 0 0
\(9\) 23.0000 0.851852
\(10\) 0 0
\(11\) 60.0000 1.64461 0.822304 − 0.569049i \(-0.192689\pi\)
0.822304 + 0.569049i \(0.192689\pi\)
\(12\) 0 0
\(13\) − 50.0000i − 1.06673i −0.845885 − 0.533366i \(-0.820927\pi\)
0.845885 − 0.533366i \(-0.179073\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 30.0000i − 0.428004i −0.976833 − 0.214002i \(-0.931350\pi\)
0.976833 − 0.214002i \(-0.0686499\pi\)
\(18\) 0 0
\(19\) −40.0000 −0.482980 −0.241490 − 0.970403i \(-0.577636\pi\)
−0.241490 + 0.970403i \(0.577636\pi\)
\(20\) 0 0
\(21\) −12.0000 −0.124696
\(22\) 0 0
\(23\) − 178.000i − 1.61372i −0.590743 − 0.806860i \(-0.701165\pi\)
0.590743 − 0.806860i \(-0.298835\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 100.000i 0.712778i
\(28\) 0 0
\(29\) −166.000 −1.06295 −0.531473 − 0.847075i \(-0.678361\pi\)
−0.531473 + 0.847075i \(0.678361\pi\)
\(30\) 0 0
\(31\) 20.0000 0.115874 0.0579372 − 0.998320i \(-0.481548\pi\)
0.0579372 + 0.998320i \(0.481548\pi\)
\(32\) 0 0
\(33\) 120.000i 0.633010i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 10.0000i 0.0444322i 0.999753 + 0.0222161i \(0.00707218\pi\)
−0.999753 + 0.0222161i \(0.992928\pi\)
\(38\) 0 0
\(39\) 100.000 0.410585
\(40\) 0 0
\(41\) −250.000 −0.952279 −0.476140 − 0.879370i \(-0.657964\pi\)
−0.476140 + 0.879370i \(0.657964\pi\)
\(42\) 0 0
\(43\) − 142.000i − 0.503600i −0.967779 − 0.251800i \(-0.918977\pi\)
0.967779 − 0.251800i \(-0.0810225\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 214.000i 0.664151i 0.943253 + 0.332076i \(0.107749\pi\)
−0.943253 + 0.332076i \(0.892251\pi\)
\(48\) 0 0
\(49\) 307.000 0.895044
\(50\) 0 0
\(51\) 60.0000 0.164739
\(52\) 0 0
\(53\) − 490.000i − 1.26994i −0.772538 − 0.634969i \(-0.781013\pi\)
0.772538 − 0.634969i \(-0.218987\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 80.0000i − 0.185899i
\(58\) 0 0
\(59\) 800.000 1.76527 0.882637 − 0.470056i \(-0.155766\pi\)
0.882637 + 0.470056i \(0.155766\pi\)
\(60\) 0 0
\(61\) 250.000 0.524741 0.262371 − 0.964967i \(-0.415496\pi\)
0.262371 + 0.964967i \(0.415496\pi\)
\(62\) 0 0
\(63\) 138.000i 0.275974i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 774.000i − 1.41133i −0.708545 − 0.705665i \(-0.750648\pi\)
0.708545 − 0.705665i \(-0.249352\pi\)
\(68\) 0 0
\(69\) 356.000 0.621121
\(70\) 0 0
\(71\) 100.000 0.167152 0.0835762 − 0.996501i \(-0.473366\pi\)
0.0835762 + 0.996501i \(0.473366\pi\)
\(72\) 0 0
\(73\) 230.000i 0.368760i 0.982855 + 0.184380i \(0.0590277\pi\)
−0.982855 + 0.184380i \(0.940972\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 360.000i 0.532803i
\(78\) 0 0
\(79\) 1320.00 1.87989 0.939947 − 0.341321i \(-0.110874\pi\)
0.939947 + 0.341321i \(0.110874\pi\)
\(80\) 0 0
\(81\) 421.000 0.577503
\(82\) 0 0
\(83\) − 982.000i − 1.29866i −0.760508 − 0.649328i \(-0.775050\pi\)
0.760508 − 0.649328i \(-0.224950\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 332.000i − 0.409128i
\(88\) 0 0
\(89\) −874.000 −1.04094 −0.520471 − 0.853879i \(-0.674244\pi\)
−0.520471 + 0.853879i \(0.674244\pi\)
\(90\) 0 0
\(91\) 300.000 0.345588
\(92\) 0 0
\(93\) 40.0000i 0.0446001i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 310.000i − 0.324492i −0.986750 − 0.162246i \(-0.948126\pi\)
0.986750 − 0.162246i \(-0.0518738\pi\)
\(98\) 0 0
\(99\) 1380.00 1.40096
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 800.4.c.f.449.2 2
4.3 odd 2 800.4.c.e.449.1 2
5.2 odd 4 800.4.a.h.1.1 1
5.3 odd 4 160.4.a.a.1.1 ✓ 1
5.4 even 2 inner 800.4.c.f.449.1 2
15.8 even 4 1440.4.a.o.1.1 1
20.3 even 4 160.4.a.b.1.1 yes 1
20.7 even 4 800.4.a.d.1.1 1
20.19 odd 2 800.4.c.e.449.2 2
40.3 even 4 320.4.a.f.1.1 1
40.13 odd 4 320.4.a.i.1.1 1
40.27 even 4 1600.4.a.bj.1.1 1
40.37 odd 4 1600.4.a.r.1.1 1
60.23 odd 4 1440.4.a.n.1.1 1
80.3 even 4 1280.4.d.k.641.2 2
80.13 odd 4 1280.4.d.f.641.1 2
80.43 even 4 1280.4.d.k.641.1 2
80.53 odd 4 1280.4.d.f.641.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.4.a.a.1.1 ✓ 1 5.3 odd 4
160.4.a.b.1.1 yes 1 20.3 even 4
320.4.a.f.1.1 1 40.3 even 4
320.4.a.i.1.1 1 40.13 odd 4
800.4.a.d.1.1 1 20.7 even 4
800.4.a.h.1.1 1 5.2 odd 4
800.4.c.e.449.1 2 4.3 odd 2
800.4.c.e.449.2 2 20.19 odd 2
800.4.c.f.449.1 2 5.4 even 2 inner
800.4.c.f.449.2 2 1.1 even 1 trivial
1280.4.d.f.641.1 2 80.13 odd 4
1280.4.d.f.641.2 2 80.53 odd 4
1280.4.d.k.641.1 2 80.43 even 4
1280.4.d.k.641.2 2 80.3 even 4
1440.4.a.n.1.1 1 60.23 odd 4
1440.4.a.o.1.1 1 15.8 even 4
1600.4.a.r.1.1 1 40.37 odd 4
1600.4.a.bj.1.1 1 40.27 even 4