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Properties

Label 400.3.bg.f.337.8
Level $400$
Weight $3$
Character 400.337
Analytic conductor $10.899$
Analytic rank $0$
Dimension $64$
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] = [400,3,Mod(17,400)] 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("400.17"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([0, 0, 13])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 400.bg (of order \(20\), degree \(8\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [64] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.8992105744\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(8\) over \(\Q(\zeta_{20})\)
Twist minimal: no (minimal twist has level 200)
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label 337.8
Character \(\chi\) \(=\) 400.337
Dual form 400.3.bg.f.273.8

$q$-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)
 
\(f(q)\) \(=\) \(q+(4.33681 + 2.20972i) q^{3} +(0.913560 + 4.91583i) q^{5} +(4.23023 + 4.23023i) q^{7} +(8.63502 + 11.8851i) q^{9} +(-5.29425 - 3.84650i) q^{11} +(-0.370366 + 2.33840i) q^{13} +(-6.90065 + 23.3377i) q^{15} +(22.0546 - 11.2374i) q^{17} +(-25.7489 - 8.36632i) q^{19} +(8.99811 + 27.6933i) q^{21} +(-15.6106 + 2.47248i) q^{23} +(-23.3308 + 8.98182i) q^{25} +(4.33305 + 27.3578i) q^{27} +(38.4796 - 12.5028i) q^{29} +(1.53159 - 4.71375i) q^{31} +(-14.4605 - 28.3803i) q^{33} +(-16.9305 + 24.6597i) q^{35} +(32.3795 + 5.12841i) q^{37} +(-6.77341 + 9.32279i) q^{39} +(-65.8117 + 47.8150i) q^{41} +(43.9478 - 43.9478i) q^{43} +(-50.5365 + 53.3061i) q^{45} +(1.81688 - 3.56582i) q^{47} -13.2103i q^{49} +120.478 q^{51} +(-7.55410 - 3.84901i) q^{53} +(14.0721 - 29.5397i) q^{55} +(-93.1809 - 93.1809i) q^{57} +(-53.8981 - 74.1844i) q^{59} +(74.4356 + 54.0806i) q^{61} +(-13.7485 + 86.8048i) q^{63} +(-11.8335 + 0.315612i) q^{65} +(21.9187 - 11.1681i) q^{67} +(-73.1639 - 23.7724i) q^{69} +(37.5374 + 115.528i) q^{71} +(115.118 - 18.2329i) q^{73} +(-121.029 - 12.6020i) q^{75} +(-6.12433 - 38.6675i) q^{77} +(-70.2432 + 22.8234i) q^{79} +(-0.804034 + 2.47456i) q^{81} +(50.1266 + 98.3789i) q^{83} +(75.3893 + 98.1507i) q^{85} +(194.506 + 30.8068i) q^{87} +(-9.28274 + 12.7766i) q^{89} +(-11.4587 + 8.32524i) q^{91} +(17.0583 - 17.0583i) q^{93} +(17.6043 - 134.220i) q^{95} +(-17.2696 + 33.8934i) q^{97} -96.1373i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 64 q + 6 q^{5} + 4 q^{7} - 40 q^{9} - 16 q^{11} + 24 q^{13} - 82 q^{15} - 8 q^{17} + 50 q^{19} - 100 q^{21} + 48 q^{23} - 200 q^{25} - 90 q^{27} - 108 q^{31} + 260 q^{33} - 2 q^{35} - 94 q^{37} - 320 q^{39}+ \cdots - 544 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(e\left(\frac{9}{20}\right)\) \(1\)

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
\(3\) 4.33681 + 2.20972i 1.44560 + 0.736572i 0.988269 − 0.152723i \(-0.0488041\pi\)
0.457335 + 0.889295i \(0.348804\pi\)
\(4\) 0 0
\(5\) 0.913560 + 4.91583i 0.182712 + 0.983166i
\(6\) 0 0
\(7\) 4.23023 + 4.23023i 0.604319 + 0.604319i 0.941456 − 0.337137i \(-0.109458\pi\)
−0.337137 + 0.941456i \(0.609458\pi\)
\(8\) 0 0
\(9\) 8.63502 + 11.8851i 0.959447 + 1.32057i
\(10\) 0 0
\(11\) −5.29425 − 3.84650i −0.481296 − 0.349682i 0.320531 − 0.947238i \(-0.396139\pi\)
−0.801827 + 0.597556i \(0.796139\pi\)
\(12\) 0 0
\(13\) −0.370366 + 2.33840i −0.0284897 + 0.179877i −0.997830 − 0.0658461i \(-0.979025\pi\)
0.969340 + 0.245723i \(0.0790254\pi\)
\(14\) 0 0
\(15\) −6.90065 + 23.3377i −0.460044 + 1.55585i
\(16\) 0 0
\(17\) 22.0546 − 11.2374i 1.29733 − 0.661022i 0.337426 − 0.941352i \(-0.390444\pi\)
0.959904 + 0.280330i \(0.0904438\pi\)
\(18\) 0 0
\(19\) −25.7489 − 8.36632i −1.35521 − 0.440333i −0.460766 − 0.887522i \(-0.652425\pi\)
−0.894439 + 0.447189i \(0.852425\pi\)
\(20\) 0 0
\(21\) 8.99811 + 27.6933i 0.428481 + 1.31873i
\(22\) 0 0
\(23\) −15.6106 + 2.47248i −0.678724 + 0.107499i −0.486272 − 0.873807i \(-0.661644\pi\)
−0.192451 + 0.981307i \(0.561644\pi\)
\(24\) 0 0
\(25\) −23.3308 + 8.98182i −0.933233 + 0.359273i
\(26\) 0 0
\(27\) 4.33305 + 27.3578i 0.160483 + 1.01325i
\(28\) 0 0
\(29\) 38.4796 − 12.5028i 1.32688 − 0.431130i 0.442029 − 0.897001i \(-0.354259\pi\)
0.884853 + 0.465871i \(0.154259\pi\)
\(30\) 0 0
\(31\) 1.53159 − 4.71375i 0.0494062 − 0.152057i −0.923310 − 0.384056i \(-0.874527\pi\)
0.972716 + 0.232000i \(0.0745269\pi\)
\(32\) 0 0
\(33\) −14.4605 − 28.3803i −0.438197 − 0.860010i
\(34\) 0 0
\(35\) −16.9305 + 24.6597i −0.483730 + 0.704563i
\(36\) 0 0
\(37\) 32.3795 + 5.12841i 0.875122 + 0.138606i 0.577807 − 0.816174i \(-0.303909\pi\)
0.297315 + 0.954779i \(0.403909\pi\)
\(38\) 0 0
\(39\) −6.77341 + 9.32279i −0.173677 + 0.239046i
\(40\) 0 0
\(41\) −65.8117 + 47.8150i −1.60516 + 1.16622i −0.728592 + 0.684948i \(0.759825\pi\)
−0.876571 + 0.481272i \(0.840175\pi\)
\(42\) 0 0
\(43\) 43.9478 − 43.9478i 1.02204 − 1.02204i 0.0222914 − 0.999752i \(-0.492904\pi\)
0.999752 − 0.0222914i \(-0.00709615\pi\)
\(44\) 0 0
\(45\) −50.5365 + 53.3061i −1.12303 + 1.18458i
\(46\) 0 0
\(47\) 1.81688 − 3.56582i 0.0386570 − 0.0758686i −0.870870 − 0.491513i \(-0.836444\pi\)
0.909527 + 0.415645i \(0.136444\pi\)
\(48\) 0 0
\(49\) − 13.2103i − 0.269597i
\(50\) 0 0
\(51\) 120.478 2.36231
\(52\) 0 0
\(53\) −7.55410 − 3.84901i −0.142530 − 0.0726228i 0.381271 − 0.924463i \(-0.375487\pi\)
−0.523801 + 0.851841i \(0.675487\pi\)
\(54\) 0 0
\(55\) 14.0721 − 29.5397i 0.255857 − 0.537085i
\(56\) 0 0
\(57\) −93.1809 − 93.1809i −1.63475 − 1.63475i
\(58\) 0 0
\(59\) −53.8981 − 74.1844i −0.913527 − 1.25736i −0.965948 − 0.258737i \(-0.916694\pi\)
0.0524204 − 0.998625i \(-0.483306\pi\)
\(60\) 0 0
\(61\) 74.4356 + 54.0806i 1.22026 + 0.886567i 0.996121 − 0.0879911i \(-0.0280447\pi\)
0.224134 + 0.974558i \(0.428045\pi\)
\(62\) 0 0
\(63\) −13.7485 + 86.8048i −0.218231 + 1.37785i
\(64\) 0 0
\(65\) −11.8335 + 0.315612i −0.182054 + 0.00485556i
\(66\) 0 0
\(67\) 21.9187 − 11.1681i 0.327145 − 0.166689i −0.282702 − 0.959208i \(-0.591231\pi\)
0.609847 + 0.792519i \(0.291231\pi\)
\(68\) 0 0
\(69\) −73.1639 − 23.7724i −1.06035 − 0.344527i
\(70\) 0 0
\(71\) 37.5374 + 115.528i 0.528695 + 1.62716i 0.756891 + 0.653542i \(0.226718\pi\)
−0.228195 + 0.973615i \(0.573282\pi\)
\(72\) 0 0
\(73\) 115.118 − 18.2329i 1.57696 − 0.249765i 0.694265 − 0.719720i \(-0.255730\pi\)
0.882691 + 0.469954i \(0.155730\pi\)
\(74\) 0 0
\(75\) −121.029 − 12.6020i −1.61371 − 0.168027i
\(76\) 0 0
\(77\) −6.12433 − 38.6675i −0.0795368 − 0.502176i
\(78\) 0 0
\(79\) −70.2432 + 22.8234i −0.889154 + 0.288904i −0.717753 − 0.696298i \(-0.754829\pi\)
−0.171401 + 0.985201i \(0.554829\pi\)
\(80\) 0 0
\(81\) −0.804034 + 2.47456i −0.00992635 + 0.0305502i
\(82\) 0 0
\(83\) 50.1266 + 98.3789i 0.603934 + 1.18529i 0.967298 + 0.253641i \(0.0816281\pi\)
−0.363364 + 0.931647i \(0.618372\pi\)
\(84\) 0 0
\(85\) 75.3893 + 98.1507i 0.886933 + 1.15471i
\(86\) 0 0
\(87\) 194.506 + 30.8068i 2.23570 + 0.354101i
\(88\) 0 0
\(89\) −9.28274 + 12.7766i −0.104300 + 0.143557i −0.857977 − 0.513689i \(-0.828279\pi\)
0.753676 + 0.657246i \(0.228279\pi\)
\(90\) 0 0
\(91\) −11.4587 + 8.32524i −0.125920 + 0.0914861i
\(92\) 0 0
\(93\) 17.0583 − 17.0583i 0.183422 − 0.183422i
\(94\) 0 0
\(95\) 17.6043 − 134.220i 0.185308 − 1.41285i
\(96\) 0 0
\(97\) −17.2696 + 33.8934i −0.178037 + 0.349417i −0.962729 − 0.270469i \(-0.912821\pi\)
0.784692 + 0.619886i \(0.212821\pi\)
\(98\) 0 0
\(99\) − 96.1373i − 0.971084i
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 400.3.bg.f.337.8 64
4.3 odd 2 200.3.u.b.137.1 yes 64
25.23 odd 20 inner 400.3.bg.f.273.8 64
100.23 even 20 200.3.u.b.73.1 ✓ 64
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.3.u.b.73.1 ✓ 64 100.23 even 20
200.3.u.b.137.1 yes 64 4.3 odd 2
400.3.bg.f.273.8 64 25.23 odd 20 inner
400.3.bg.f.337.8 64 1.1 even 1 trivial