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Properties

Label 400.6.c.j.49.2
Level $400$
Weight $6$
Character 400.49
Analytic conductor $64.154$
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] = [400,6,Mod(49,400)] 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("400.49"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 400.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,454] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(64.1535279252\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 5)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 400.49
Dual form 400.6.c.j.49.1

$q$-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)
 
\(f(q)\) \(=\) \(q+4.00000i q^{3} +192.000i q^{7} +227.000 q^{9} +148.000 q^{11} +286.000i q^{13} +1678.00i q^{17} +1060.00 q^{19} -768.000 q^{21} -2976.00i q^{23} +1880.00i q^{27} +3410.00 q^{29} +2448.00 q^{31} +592.000i q^{33} -182.000i q^{37} -1144.00 q^{39} -9398.00 q^{41} +1244.00i q^{43} -12088.0i q^{47} -20057.0 q^{49} -6712.00 q^{51} +23846.0i q^{53} +4240.00i q^{57} -20020.0 q^{59} +32302.0 q^{61} +43584.0i q^{63} +60972.0i q^{67} +11904.0 q^{69} +32648.0 q^{71} -38774.0i q^{73} +28416.0i q^{77} -33360.0 q^{79} +47641.0 q^{81} -16716.0i q^{83} +13640.0i q^{87} -101370. q^{89} -54912.0 q^{91} +9792.00i q^{93} +119038. i q^{97} +33596.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 454 q^{9} + 296 q^{11} + 2120 q^{19} - 1536 q^{21} + 6820 q^{29} + 4896 q^{31} - 2288 q^{39} - 18796 q^{41} - 40114 q^{49} - 13424 q^{51} - 40040 q^{59} + 64604 q^{61} + 23808 q^{69} + 65296 q^{71}+ \cdots + 67192 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 4.00000i 0.256600i 0.991735 + 0.128300i \(0.0409521\pi\)
−0.991735 + 0.128300i \(0.959048\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 192.000i 1.48100i 0.672054 + 0.740502i \(0.265412\pi\)
−0.672054 + 0.740502i \(0.734588\pi\)
\(8\) 0 0
\(9\) 227.000 0.934156
\(10\) 0 0
\(11\) 148.000 0.368791 0.184395 − 0.982852i \(-0.440967\pi\)
0.184395 + 0.982852i \(0.440967\pi\)
\(12\) 0 0
\(13\) 286.000i 0.469362i 0.972072 + 0.234681i \(0.0754045\pi\)
−0.972072 + 0.234681i \(0.924595\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1678.00i 1.40822i 0.710092 + 0.704109i \(0.248653\pi\)
−0.710092 + 0.704109i \(0.751347\pi\)
\(18\) 0 0
\(19\) 1060.00 0.673631 0.336815 − 0.941571i \(-0.390650\pi\)
0.336815 + 0.941571i \(0.390650\pi\)
\(20\) 0 0
\(21\) −768.000 −0.380026
\(22\) 0 0
\(23\) − 2976.00i − 1.17304i −0.809934 − 0.586521i \(-0.800497\pi\)
0.809934 − 0.586521i \(-0.199503\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1880.00i 0.496305i
\(28\) 0 0
\(29\) 3410.00 0.752938 0.376469 − 0.926429i \(-0.377138\pi\)
0.376469 + 0.926429i \(0.377138\pi\)
\(30\) 0 0
\(31\) 2448.00 0.457517 0.228758 − 0.973483i \(-0.426533\pi\)
0.228758 + 0.973483i \(0.426533\pi\)
\(32\) 0 0
\(33\) 592.000i 0.0946317i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 182.000i − 0.0218558i −0.999940 − 0.0109279i \(-0.996521\pi\)
0.999940 − 0.0109279i \(-0.00347853\pi\)
\(38\) 0 0
\(39\) −1144.00 −0.120438
\(40\) 0 0
\(41\) −9398.00 −0.873124 −0.436562 − 0.899674i \(-0.643804\pi\)
−0.436562 + 0.899674i \(0.643804\pi\)
\(42\) 0 0
\(43\) 1244.00i 0.102600i 0.998683 + 0.0513002i \(0.0163365\pi\)
−0.998683 + 0.0513002i \(0.983663\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 12088.0i − 0.798196i −0.916908 − 0.399098i \(-0.869323\pi\)
0.916908 − 0.399098i \(-0.130677\pi\)
\(48\) 0 0
\(49\) −20057.0 −1.19337
\(50\) 0 0
\(51\) −6712.00 −0.361349
\(52\) 0 0
\(53\) 23846.0i 1.16607i 0.812446 + 0.583037i \(0.198136\pi\)
−0.812446 + 0.583037i \(0.801864\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4240.00i 0.172854i
\(58\) 0 0
\(59\) −20020.0 −0.748745 −0.374373 − 0.927278i \(-0.622142\pi\)
−0.374373 + 0.927278i \(0.622142\pi\)
\(60\) 0 0
\(61\) 32302.0 1.11149 0.555744 − 0.831353i \(-0.312433\pi\)
0.555744 + 0.831353i \(0.312433\pi\)
\(62\) 0 0
\(63\) 43584.0i 1.38349i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 60972.0i 1.65937i 0.558231 + 0.829685i \(0.311480\pi\)
−0.558231 + 0.829685i \(0.688520\pi\)
\(68\) 0 0
\(69\) 11904.0 0.301003
\(70\) 0 0
\(71\) 32648.0 0.768618 0.384309 − 0.923204i \(-0.374440\pi\)
0.384309 + 0.923204i \(0.374440\pi\)
\(72\) 0 0
\(73\) − 38774.0i − 0.851596i −0.904818 − 0.425798i \(-0.859993\pi\)
0.904818 − 0.425798i \(-0.140007\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 28416.0i 0.546180i
\(78\) 0 0
\(79\) −33360.0 −0.601393 −0.300696 − 0.953720i \(-0.597219\pi\)
−0.300696 + 0.953720i \(0.597219\pi\)
\(80\) 0 0
\(81\) 47641.0 0.806805
\(82\) 0 0
\(83\) − 16716.0i − 0.266340i −0.991093 − 0.133170i \(-0.957484\pi\)
0.991093 − 0.133170i \(-0.0425157\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 13640.0i 0.193204i
\(88\) 0 0
\(89\) −101370. −1.35655 −0.678273 − 0.734810i \(-0.737271\pi\)
−0.678273 + 0.734810i \(0.737271\pi\)
\(90\) 0 0
\(91\) −54912.0 −0.695126
\(92\) 0 0
\(93\) 9792.00i 0.117399i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 119038.i 1.28457i 0.766468 + 0.642283i \(0.222013\pi\)
−0.766468 + 0.642283i \(0.777987\pi\)
\(98\) 0 0
\(99\) 33596.0 0.344508
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 400.6.c.j.49.2 2
4.3 odd 2 25.6.b.a.24.1 2
5.2 odd 4 80.6.a.e.1.1 1
5.3 odd 4 400.6.a.g.1.1 1
5.4 even 2 inner 400.6.c.j.49.1 2
12.11 even 2 225.6.b.e.199.2 2
15.2 even 4 720.6.a.a.1.1 1
20.3 even 4 25.6.a.a.1.1 1
20.7 even 4 5.6.a.a.1.1 ✓ 1
20.19 odd 2 25.6.b.a.24.2 2
40.27 even 4 320.6.a.j.1.1 1
40.37 odd 4 320.6.a.g.1.1 1
60.23 odd 4 225.6.a.f.1.1 1
60.47 odd 4 45.6.a.b.1.1 1
60.59 even 2 225.6.b.e.199.1 2
140.27 odd 4 245.6.a.b.1.1 1
220.87 odd 4 605.6.a.a.1.1 1
260.207 even 4 845.6.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.6.a.a.1.1 ✓ 1 20.7 even 4
25.6.a.a.1.1 1 20.3 even 4
25.6.b.a.24.1 2 4.3 odd 2
25.6.b.a.24.2 2 20.19 odd 2
45.6.a.b.1.1 1 60.47 odd 4
80.6.a.e.1.1 1 5.2 odd 4
225.6.a.f.1.1 1 60.23 odd 4
225.6.b.e.199.1 2 60.59 even 2
225.6.b.e.199.2 2 12.11 even 2
245.6.a.b.1.1 1 140.27 odd 4
320.6.a.g.1.1 1 40.37 odd 4
320.6.a.j.1.1 1 40.27 even 4
400.6.a.g.1.1 1 5.3 odd 4
400.6.c.j.49.1 2 5.4 even 2 inner
400.6.c.j.49.2 2 1.1 even 1 trivial
605.6.a.a.1.1 1 220.87 odd 4
720.6.a.a.1.1 1 15.2 even 4
845.6.a.b.1.1 1 260.207 even 4