Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [338,3,Mod(99,338)] 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("338.99"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(338, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 338.d (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,-12,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.20983293538\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.4739148267126784.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 21x^{8} + 98x^{4} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 13^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 99.2
Root \(-1.10568 + 1.10568i\) of defining polynomial
Character \(\chi\) \(=\) 338.99
Dual form 338.3.d.h.239.2

$q$-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+(-1.00000 - 1.00000i) q^{2} -3.23112 q^{3} +2.00000i q^{4} +(0.725610 + 0.725610i) q^{5} +(3.23112 + 3.23112i) q^{6} +(6.96533 - 6.96533i) q^{7} +(2.00000 - 2.00000i) q^{8} +1.44017 q^{9} -1.45122i q^{10} +(-15.2411 + 15.2411i) q^{11} -6.46225i q^{12} -13.9307 q^{14} +(-2.34454 - 2.34454i) q^{15} -4.00000 q^{16} +2.44995i q^{17} +(-1.44017 - 1.44017i) q^{18} +(16.3652 + 16.3652i) q^{19} +(-1.45122 + 1.45122i) q^{20} +(-22.5059 + 22.5059i) q^{21} +30.4823 q^{22} -24.3104i q^{23} +(-6.46225 + 6.46225i) q^{24} -23.9470i q^{25} +24.4268 q^{27} +(13.9307 + 13.9307i) q^{28} +28.8330 q^{29} +4.68907i q^{30} +(30.8436 + 30.8436i) q^{31} +(4.00000 + 4.00000i) q^{32} +(49.2460 - 49.2460i) q^{33} +(2.44995 - 2.44995i) q^{34} +10.1082 q^{35} +2.88033i q^{36} +(34.2593 - 34.2593i) q^{37} -32.7303i q^{38} +2.90244 q^{40} +(18.3596 + 18.3596i) q^{41} +45.0117 q^{42} +6.69265i q^{43} +(-30.4823 - 30.4823i) q^{44} +(1.04500 + 1.04500i) q^{45} +(-24.3104 + 24.3104i) q^{46} +(-9.85769 + 9.85769i) q^{47} +12.9245 q^{48} -48.0317i q^{49} +(-23.9470 + 23.9470i) q^{50} -7.91608i q^{51} +36.3764 q^{53} +(-24.4268 - 24.4268i) q^{54} -22.1182 q^{55} -27.8613i q^{56} +(-52.8778 - 52.8778i) q^{57} +(-28.8330 - 28.8330i) q^{58} +(-0.197517 + 0.197517i) q^{59} +(4.68907 - 4.68907i) q^{60} +13.5360 q^{61} -61.6873i q^{62} +(10.0312 - 10.0312i) q^{63} -8.00000i q^{64} -98.4919 q^{66} +(31.3877 + 31.3877i) q^{67} -4.89989 q^{68} +78.5498i q^{69} +(-10.1082 - 10.1082i) q^{70} +(35.3224 + 35.3224i) q^{71} +(2.88033 - 2.88033i) q^{72} +(62.1910 - 62.1910i) q^{73} -68.5186 q^{74} +77.3757i q^{75} +(-32.7303 + 32.7303i) q^{76} +212.319i q^{77} +30.7872 q^{79} +(-2.90244 - 2.90244i) q^{80} -91.8874 q^{81} -36.7193i q^{82} +(84.7223 + 84.7223i) q^{83} +(-45.0117 - 45.0117i) q^{84} +(-1.77771 + 1.77771i) q^{85} +(6.69265 - 6.69265i) q^{86} -93.1631 q^{87} +60.9645i q^{88} +(49.7700 - 49.7700i) q^{89} -2.09000i q^{90} +48.6207 q^{92} +(-99.6596 - 99.6596i) q^{93} +19.7154 q^{94} +23.7494i q^{95} +(-12.9245 - 12.9245i) q^{96} +(-50.5647 - 50.5647i) q^{97} +(-48.0317 + 48.0317i) q^{98} +(-21.9497 + 21.9497i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} - 8 q^{3} + 8 q^{5} + 8 q^{6} + 24 q^{7} + 24 q^{8} + 28 q^{9} + 16 q^{11} - 48 q^{14} + 116 q^{15} - 48 q^{16} - 28 q^{18} + 92 q^{19} - 16 q^{20} - 128 q^{21} - 32 q^{22} - 16 q^{24} - 68 q^{27}+ \cdots + 112 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/338\mathbb{Z}\right)^\times\).

\(n\) \(171\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 − 1.00000i −0.500000 − 0.500000i
\(3\) −3.23112 −1.07704 −0.538521 − 0.842612i \(-0.681017\pi\)
−0.538521 + 0.842612i \(0.681017\pi\)
\(4\) 2.00000i 0.500000i
\(5\) 0.725610 + 0.725610i 0.145122 + 0.145122i 0.775935 − 0.630813i \(-0.217278\pi\)
−0.630813 + 0.775935i \(0.717278\pi\)
\(6\) 3.23112 + 3.23112i 0.538521 + 0.538521i
\(7\) 6.96533 − 6.96533i 0.995047 − 0.995047i −0.00494055 − 0.999988i \(-0.501573\pi\)
0.999988 + 0.00494055i \(0.00157263\pi\)
\(8\) 2.00000 − 2.00000i 0.250000 − 0.250000i
\(9\) 1.44017 0.160018
\(10\) − 1.45122i − 0.145122i
\(11\) −15.2411 + 15.2411i −1.38556 + 1.38556i −0.551152 + 0.834405i \(0.685812\pi\)
−0.834405 + 0.551152i \(0.814188\pi\)
\(12\) − 6.46225i − 0.538521i
\(13\) 0 0
\(14\) −13.9307 −0.995047
\(15\) −2.34454 − 2.34454i −0.156302 − 0.156302i
\(16\) −4.00000 −0.250000
\(17\) 2.44995i 0.144115i 0.997400 + 0.0720573i \(0.0229564\pi\)
−0.997400 + 0.0720573i \(0.977044\pi\)
\(18\) −1.44017 − 1.44017i −0.0800092 − 0.0800092i
\(19\) 16.3652 + 16.3652i 0.861324 + 0.861324i 0.991492 − 0.130168i \(-0.0415517\pi\)
−0.130168 + 0.991492i \(0.541552\pi\)
\(20\) −1.45122 + 1.45122i −0.0725610 + 0.0725610i
\(21\) −22.5059 + 22.5059i −1.07171 + 1.07171i
\(22\) 30.4823 1.38556
\(23\) − 24.3104i − 1.05697i −0.848942 − 0.528486i \(-0.822760\pi\)
0.848942 − 0.528486i \(-0.177240\pi\)
\(24\) −6.46225 + 6.46225i −0.269260 + 0.269260i
\(25\) − 23.9470i − 0.957879i
\(26\) 0 0
\(27\) 24.4268 0.904695
\(28\) 13.9307 + 13.9307i 0.497524 + 0.497524i
\(29\) 28.8330 0.994242 0.497121 − 0.867681i \(-0.334390\pi\)
0.497121 + 0.867681i \(0.334390\pi\)
\(30\) 4.68907i 0.156302i
\(31\) 30.8436 + 30.8436i 0.994956 + 0.994956i 0.999987 − 0.00503120i \(-0.00160149\pi\)
−0.00503120 + 0.999987i \(0.501601\pi\)
\(32\) 4.00000 + 4.00000i 0.125000 + 0.125000i
\(33\) 49.2460 − 49.2460i 1.49230 − 1.49230i
\(34\) 2.44995 − 2.44995i 0.0720573 − 0.0720573i
\(35\) 10.1082 0.288806
\(36\) 2.88033i 0.0800092i
\(37\) 34.2593 − 34.2593i 0.925927 − 0.925927i −0.0715125 − 0.997440i \(-0.522783\pi\)
0.997440 + 0.0715125i \(0.0227826\pi\)
\(38\) − 32.7303i − 0.861324i
\(39\) 0 0
\(40\) 2.90244 0.0725610
\(41\) 18.3596 + 18.3596i 0.447796 + 0.447796i 0.894621 − 0.446825i \(-0.147445\pi\)
−0.446825 + 0.894621i \(0.647445\pi\)
\(42\) 45.0117 1.07171
\(43\) 6.69265i 0.155643i 0.996967 + 0.0778215i \(0.0247964\pi\)
−0.996967 + 0.0778215i \(0.975204\pi\)
\(44\) −30.4823 − 30.4823i −0.692778 − 0.692778i
\(45\) 1.04500 + 1.04500i 0.0232222 + 0.0232222i
\(46\) −24.3104 + 24.3104i −0.528486 + 0.528486i
\(47\) −9.85769 + 9.85769i −0.209738 + 0.209738i −0.804156 − 0.594418i \(-0.797382\pi\)
0.594418 + 0.804156i \(0.297382\pi\)
\(48\) 12.9245 0.269260
\(49\) − 48.0317i − 0.980238i
\(50\) −23.9470 + 23.9470i −0.478940 + 0.478940i
\(51\) − 7.91608i − 0.155217i
\(52\) 0 0
\(53\) 36.3764 0.686348 0.343174 − 0.939272i \(-0.388498\pi\)
0.343174 + 0.939272i \(0.388498\pi\)
\(54\) −24.4268 − 24.4268i −0.452348 − 0.452348i
\(55\) −22.1182 −0.402150
\(56\) − 27.8613i − 0.497524i
\(57\) −52.8778 − 52.8778i −0.927681 − 0.927681i
\(58\) −28.8330 − 28.8330i −0.497121 − 0.497121i
\(59\) −0.197517 + 0.197517i −0.00334774 + 0.00334774i −0.708779 − 0.705431i \(-0.750754\pi\)
0.705431 + 0.708779i \(0.250754\pi\)
\(60\) 4.68907 − 4.68907i 0.0781512 − 0.0781512i
\(61\) 13.5360 0.221901 0.110951 − 0.993826i \(-0.464610\pi\)
0.110951 + 0.993826i \(0.464610\pi\)
\(62\) − 61.6873i − 0.994956i
\(63\) 10.0312 − 10.0312i 0.159226 − 0.159226i
\(64\) − 8.00000i − 0.125000i
\(65\) 0 0
\(66\) −98.4919 −1.49230
\(67\) 31.3877 + 31.3877i 0.468473 + 0.468473i 0.901420 − 0.432946i \(-0.142526\pi\)
−0.432946 + 0.901420i \(0.642526\pi\)
\(68\) −4.89989 −0.0720573
\(69\) 78.5498i 1.13840i
\(70\) −10.1082 − 10.1082i −0.144403 − 0.144403i
\(71\) 35.3224 + 35.3224i 0.497498 + 0.497498i 0.910658 − 0.413160i \(-0.135575\pi\)
−0.413160 + 0.910658i \(0.635575\pi\)
\(72\) 2.88033 − 2.88033i 0.0400046 − 0.0400046i
\(73\) 62.1910 − 62.1910i 0.851931 − 0.851931i −0.138439 − 0.990371i \(-0.544209\pi\)
0.990371 + 0.138439i \(0.0442086\pi\)
\(74\) −68.5186 −0.925927
\(75\) 77.3757i 1.03168i
\(76\) −32.7303 + 32.7303i −0.430662 + 0.430662i
\(77\) 212.319i 2.75739i
\(78\) 0 0
\(79\) 30.7872 0.389712 0.194856 − 0.980832i \(-0.437576\pi\)
0.194856 + 0.980832i \(0.437576\pi\)
\(80\) −2.90244 − 2.90244i −0.0362805 − 0.0362805i
\(81\) −91.8874 −1.13441
\(82\) − 36.7193i − 0.447796i
\(83\) 84.7223 + 84.7223i 1.02075 + 1.02075i 0.999780 + 0.0209708i \(0.00667571\pi\)
0.0209708 + 0.999780i \(0.493324\pi\)
\(84\) −45.0117 − 45.0117i −0.535854 − 0.535854i
\(85\) −1.77771 + 1.77771i −0.0209142 + 0.0209142i
\(86\) 6.69265 − 6.69265i 0.0778215 − 0.0778215i
\(87\) −93.1631 −1.07084
\(88\) 60.9645i 0.692778i
\(89\) 49.7700 − 49.7700i 0.559214 − 0.559214i −0.369870 − 0.929084i \(-0.620598\pi\)
0.929084 + 0.369870i \(0.120598\pi\)
\(90\) − 2.09000i − 0.0232222i
\(91\) 0 0
\(92\) 48.6207 0.528486
\(93\) −99.6596 − 99.6596i −1.07161 − 1.07161i
\(94\) 19.7154 0.209738
\(95\) 23.7494i 0.249994i
\(96\) −12.9245 − 12.9245i −0.134630 − 0.134630i
\(97\) −50.5647 − 50.5647i −0.521285 − 0.521285i 0.396674 − 0.917960i \(-0.370164\pi\)
−0.917960 + 0.396674i \(0.870164\pi\)
\(98\) −48.0317 + 48.0317i −0.490119 + 0.490119i
\(99\) −21.9497 + 21.9497i −0.221715 + 0.221715i
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 338.3.d.h.99.2 ✓ 12
13.2 odd 12 338.3.f.l.19.5 24
13.3 even 3 338.3.f.l.319.5 24
13.4 even 6 338.3.f.k.89.5 24
13.5 odd 4 inner 338.3.d.h.239.2 yes 12
13.6 odd 12 338.3.f.l.249.5 24
13.7 odd 12 338.3.f.k.249.5 24
13.8 odd 4 338.3.d.i.239.2 yes 12
13.9 even 3 338.3.f.l.89.5 24
13.10 even 6 338.3.f.k.319.5 24
13.11 odd 12 338.3.f.k.19.5 24
13.12 even 2 338.3.d.i.99.2 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
338.3.d.h.99.2 ✓ 12 1.1 even 1 trivial
338.3.d.h.239.2 yes 12 13.5 odd 4 inner
338.3.d.i.99.2 yes 12 13.12 even 2
338.3.d.i.239.2 yes 12 13.8 odd 4
338.3.f.k.19.5 24 13.11 odd 12
338.3.f.k.89.5 24 13.4 even 6
338.3.f.k.249.5 24 13.7 odd 12
338.3.f.k.319.5 24 13.10 even 6
338.3.f.l.19.5 24 13.2 odd 12
338.3.f.l.89.5 24 13.9 even 3
338.3.f.l.249.5 24 13.6 odd 12
338.3.f.l.319.5 24 13.3 even 3