Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [550,2,Mod(201,550)] 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("550.201"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(550, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 8])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 550.h (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-2,4,-2,0,-1,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.39177211117\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.159390625.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 6x^{6} - 11x^{5} + 21x^{4} - 5x^{3} + 10x^{2} + 25x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 401.1
Root \(-0.628998 + 0.456994i\) of defining polynomial
Character \(\chi\) \(=\) 550.401
Dual form 550.2.h.k.251.1

$q$-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+(0.309017 + 0.951057i) q^{2} +(0.420275 + 0.305348i) q^{3} +(-0.809017 + 0.587785i) q^{4} +(-0.160531 + 0.494063i) q^{6} +(3.65351 - 2.65443i) q^{7} +(-0.809017 - 0.587785i) q^{8} +(-0.843657 - 2.59651i) q^{9} +(2.88699 - 1.63256i) q^{11} -0.519488 q^{12} +(-1.77748 - 5.47053i) q^{13} +(3.65351 + 2.65443i) q^{14} +(0.309017 - 0.951057i) q^{16} +(-0.493224 + 1.51799i) q^{17} +(2.20872 - 1.60473i) q^{18} +(4.63577 + 3.36809i) q^{19} +2.34600 q^{21} +(2.44479 + 2.24120i) q^{22} -7.30703 q^{23} +(-0.160531 - 0.494063i) q^{24} +(4.65351 - 3.38098i) q^{26} +(0.919862 - 2.83104i) q^{27} +(-1.39552 + 4.29496i) q^{28} +(-3.49406 + 2.53859i) q^{29} +(3.33461 + 10.2629i) q^{31} +1.00000 q^{32} +(1.71183 + 0.195411i) q^{33} -1.59611 q^{34} +(2.20872 + 1.60473i) q^{36} +(0.840550 - 0.610695i) q^{37} +(-1.77071 + 5.44968i) q^{38} +(0.923382 - 2.84188i) q^{39} +(2.17827 + 1.58261i) q^{41} +(0.724955 + 2.23118i) q^{42} +4.57418 q^{43} +(-1.37603 + 3.01770i) q^{44} +(-2.25800 - 6.94940i) q^{46} +(-3.91151 - 2.84188i) q^{47} +(0.420275 - 0.305348i) q^{48} +(4.13903 - 12.7386i) q^{49} +(-0.670803 + 0.487367i) q^{51} +(4.65351 + 3.38098i) q^{52} +(2.97458 + 9.15480i) q^{53} +2.97674 q^{54} -4.51599 q^{56} +(0.919862 + 2.83104i) q^{57} +(-3.49406 - 2.53859i) q^{58} +(-1.66906 + 1.21264i) q^{59} +(-0.639962 + 1.96960i) q^{61} +(-8.73013 + 6.34281i) q^{62} +(-9.97458 - 7.24695i) q^{63} +(0.309017 + 0.951057i) q^{64} +(0.343138 + 1.68843i) q^{66} -3.39901 q^{67} +(-0.493224 - 1.51799i) q^{68} +(-3.07096 - 2.23118i) q^{69} +(1.22252 - 3.76252i) q^{71} +(-0.843657 + 2.59651i) q^{72} +(-1.50283 + 1.09187i) q^{73} +(0.840550 + 0.610695i) q^{74} -5.73013 q^{76} +(6.21414 - 13.6279i) q^{77} +2.98813 q^{78} +(1.67894 + 5.16724i) q^{79} +(-5.37512 + 3.90525i) q^{81} +(-0.832025 + 2.56071i) q^{82} +(0.747032 - 2.29913i) q^{83} +(-1.89796 + 1.37895i) q^{84} +(1.41350 + 4.35030i) q^{86} -2.24362 q^{87} +(-3.29522 - 0.376160i) q^{88} +1.72146 q^{89} +(-21.0152 - 15.2685i) q^{91} +(5.91151 - 4.29496i) q^{92} +(-1.73229 + 5.33145i) q^{93} +(1.49406 - 4.59826i) q^{94} +(0.420275 + 0.305348i) q^{96} +(0.663119 + 2.04087i) q^{97} +13.3942 q^{98} +(-6.67460 - 6.11878i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} + 4 q^{3} - 2 q^{4} - q^{6} - 2 q^{7} - 2 q^{8} - 24 q^{9} + 5 q^{11} - 6 q^{12} - 4 q^{13} - 2 q^{14} - 2 q^{16} - 18 q^{17} + 11 q^{18} + 17 q^{19} - 40 q^{21} - 5 q^{22} + 4 q^{23} - q^{24}+ \cdots - 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/550\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(e\left(\frac{2}{5}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.309017 + 0.951057i 0.218508 + 0.672499i
\(3\) 0.420275 + 0.305348i 0.242646 + 0.176292i 0.702461 − 0.711722i \(-0.252084\pi\)
−0.459815 + 0.888015i \(0.652084\pi\)
\(4\) −0.809017 + 0.587785i −0.404508 + 0.293893i
\(5\) 0 0
\(6\) −0.160531 + 0.494063i −0.0655364 + 0.201700i
\(7\) 3.65351 − 2.65443i 1.38090 − 1.00328i 0.384103 − 0.923290i \(-0.374511\pi\)
0.996796 − 0.0799913i \(-0.0254892\pi\)
\(8\) −0.809017 − 0.587785i −0.286031 − 0.207813i
\(9\) −0.843657 − 2.59651i −0.281219 − 0.865503i
\(10\) 0 0
\(11\) 2.88699 − 1.63256i 0.870461 − 0.492237i
\(12\) −0.519488 −0.149963
\(13\) −1.77748 − 5.47053i −0.492985 − 1.51725i −0.820073 − 0.572259i \(-0.806067\pi\)
0.327087 − 0.944994i \(-0.393933\pi\)
\(14\) 3.65351 + 2.65443i 0.976443 + 0.709427i
\(15\) 0 0
\(16\) 0.309017 − 0.951057i 0.0772542 − 0.237764i
\(17\) −0.493224 + 1.51799i −0.119624 + 0.368166i −0.992883 − 0.119090i \(-0.962002\pi\)
0.873259 + 0.487256i \(0.162002\pi\)
\(18\) 2.20872 − 1.60473i 0.520601 − 0.378239i
\(19\) 4.63577 + 3.36809i 1.06352 + 0.772692i 0.974736 − 0.223359i \(-0.0717022\pi\)
0.0887832 + 0.996051i \(0.471702\pi\)
\(20\) 0 0
\(21\) 2.34600 0.511940
\(22\) 2.44479 + 2.24120i 0.521231 + 0.477826i
\(23\) −7.30703 −1.52362 −0.761810 − 0.647800i \(-0.775689\pi\)
−0.761810 + 0.647800i \(0.775689\pi\)
\(24\) −0.160531 − 0.494063i −0.0327682 − 0.100850i
\(25\) 0 0
\(26\) 4.65351 − 3.38098i 0.912629 − 0.663064i
\(27\) 0.919862 − 2.83104i 0.177028 − 0.544835i
\(28\) −1.39552 + 4.29496i −0.263728 + 0.811672i
\(29\) −3.49406 + 2.53859i −0.648831 + 0.471404i −0.862873 − 0.505421i \(-0.831337\pi\)
0.214042 + 0.976825i \(0.431337\pi\)
\(30\) 0 0
\(31\) 3.33461 + 10.2629i 0.598914 + 1.84327i 0.534188 + 0.845366i \(0.320617\pi\)
0.0647261 + 0.997903i \(0.479383\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.71183 + 0.195411i 0.297991 + 0.0340167i
\(34\) −1.59611 −0.273730
\(35\) 0 0
\(36\) 2.20872 + 1.60473i 0.368121 + 0.267455i
\(37\) 0.840550 − 0.610695i 0.138186 − 0.100398i −0.516545 − 0.856260i \(-0.672782\pi\)
0.654731 + 0.755862i \(0.272782\pi\)
\(38\) −1.77071 + 5.44968i −0.287247 + 0.884055i
\(39\) 0.923382 − 2.84188i 0.147860 − 0.455065i
\(40\) 0 0
\(41\) 2.17827 + 1.58261i 0.340189 + 0.247162i 0.744741 − 0.667353i \(-0.232573\pi\)
−0.404553 + 0.914515i \(0.632573\pi\)
\(42\) 0.724955 + 2.23118i 0.111863 + 0.344279i
\(43\) 4.57418 0.697556 0.348778 − 0.937205i \(-0.386597\pi\)
0.348778 + 0.937205i \(0.386597\pi\)
\(44\) −1.37603 + 3.01770i −0.207444 + 0.454936i
\(45\) 0 0
\(46\) −2.25800 − 6.94940i −0.332923 − 1.02463i
\(47\) −3.91151 − 2.84188i −0.570552 − 0.414531i 0.264753 − 0.964316i \(-0.414709\pi\)
−0.835306 + 0.549786i \(0.814709\pi\)
\(48\) 0.420275 − 0.305348i 0.0606615 − 0.0440731i
\(49\) 4.13903 − 12.7386i 0.591290 − 1.81980i
\(50\) 0 0
\(51\) −0.670803 + 0.487367i −0.0939312 + 0.0682450i
\(52\) 4.65351 + 3.38098i 0.645326 + 0.468857i
\(53\) 2.97458 + 9.15480i 0.408589 + 1.25751i 0.917861 + 0.396903i \(0.129915\pi\)
−0.509271 + 0.860606i \(0.670085\pi\)
\(54\) 2.97674 0.405083
\(55\) 0 0
\(56\) −4.51599 −0.603475
\(57\) 0.919862 + 2.83104i 0.121839 + 0.374981i
\(58\) −3.49406 − 2.53859i −0.458793 − 0.333333i
\(59\) −1.66906 + 1.21264i −0.217292 + 0.157872i −0.691107 − 0.722752i \(-0.742877\pi\)
0.473815 + 0.880625i \(0.342877\pi\)
\(60\) 0 0
\(61\) −0.639962 + 1.96960i −0.0819387 + 0.252181i −0.983630 − 0.180198i \(-0.942326\pi\)
0.901692 + 0.432380i \(0.142326\pi\)
\(62\) −8.73013 + 6.34281i −1.10873 + 0.805538i
\(63\) −9.97458 − 7.24695i −1.25668 − 0.913030i
\(64\) 0.309017 + 0.951057i 0.0386271 + 0.118882i
\(65\) 0 0
\(66\) 0.343138 + 1.68843i 0.0422374 + 0.207832i
\(67\) −3.39901 −0.415256 −0.207628 − 0.978208i \(-0.566574\pi\)
−0.207628 + 0.978208i \(0.566574\pi\)
\(68\) −0.493224 − 1.51799i −0.0598122 − 0.184083i
\(69\) −3.07096 − 2.23118i −0.369700 − 0.268603i
\(70\) 0 0
\(71\) 1.22252 − 3.76252i 0.145086 − 0.446529i −0.851936 − 0.523646i \(-0.824572\pi\)
0.997022 + 0.0771172i \(0.0245716\pi\)
\(72\) −0.843657 + 2.59651i −0.0994259 + 0.306002i
\(73\) −1.50283 + 1.09187i −0.175893 + 0.127794i −0.672248 − 0.740326i \(-0.734671\pi\)
0.496355 + 0.868119i \(0.334671\pi\)
\(74\) 0.840550 + 0.610695i 0.0977119 + 0.0709919i
\(75\) 0 0
\(76\) −5.73013 −0.657291
\(77\) 6.21414 − 13.6279i 0.708167 − 1.55305i
\(78\) 2.98813 0.338339
\(79\) 1.67894 + 5.16724i 0.188895 + 0.581360i 0.999994 − 0.00354969i \(-0.00112990\pi\)
−0.811098 + 0.584910i \(0.801130\pi\)
\(80\) 0 0
\(81\) −5.37512 + 3.90525i −0.597236 + 0.433917i
\(82\) −0.832025 + 2.56071i −0.0918818 + 0.282783i
\(83\) 0.747032 − 2.29913i 0.0819974 − 0.252362i −0.901650 − 0.432466i \(-0.857643\pi\)
0.983648 + 0.180104i \(0.0576435\pi\)
\(84\) −1.89796 + 1.37895i −0.207084 + 0.150455i
\(85\) 0 0
\(86\) 1.41350 + 4.35030i 0.152421 + 0.469105i
\(87\) −2.24362 −0.240541
\(88\) −3.29522 − 0.376160i −0.351272 − 0.0400988i
\(89\) 1.72146 0.182474 0.0912371 − 0.995829i \(-0.470918\pi\)
0.0912371 + 0.995829i \(0.470918\pi\)
\(90\) 0 0
\(91\) −21.0152 − 15.2685i −2.20299 − 1.60057i
\(92\) 5.91151 − 4.29496i 0.616318 − 0.447781i
\(93\) −1.73229 + 5.33145i −0.179630 + 0.552846i
\(94\) 1.49406 − 4.59826i 0.154101 − 0.474274i
\(95\) 0 0
\(96\) 0.420275 + 0.305348i 0.0428941 + 0.0311644i
\(97\) 0.663119 + 2.04087i 0.0673295 + 0.207219i 0.979061 − 0.203568i \(-0.0652539\pi\)
−0.911731 + 0.410787i \(0.865254\pi\)
\(98\) 13.3942 1.35302
\(99\) −6.67460 − 6.11878i −0.670823 − 0.614961i
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 550.2.h.k.401.1 yes 8
5.2 odd 4 550.2.ba.g.49.1 16
5.3 odd 4 550.2.ba.g.49.4 16
5.4 even 2 550.2.h.m.401.2 yes 8
11.3 even 5 6050.2.a.dg.1.3 4
11.8 odd 10 6050.2.a.cz.1.3 4
11.9 even 5 inner 550.2.h.k.251.1 ✓ 8
55.9 even 10 550.2.h.m.251.2 yes 8
55.14 even 10 6050.2.a.df.1.2 4
55.19 odd 10 6050.2.a.dn.1.2 4
55.42 odd 20 550.2.ba.g.449.4 16
55.53 odd 20 550.2.ba.g.449.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
550.2.h.k.251.1 ✓ 8 11.9 even 5 inner
550.2.h.k.401.1 yes 8 1.1 even 1 trivial
550.2.h.m.251.2 yes 8 55.9 even 10
550.2.h.m.401.2 yes 8 5.4 even 2
550.2.ba.g.49.1 16 5.2 odd 4
550.2.ba.g.49.4 16 5.3 odd 4
550.2.ba.g.449.1 16 55.53 odd 20
550.2.ba.g.449.4 16 55.42 odd 20
6050.2.a.cz.1.3 4 11.8 odd 10
6050.2.a.df.1.2 4 55.14 even 10
6050.2.a.dg.1.3 4 11.3 even 5
6050.2.a.dn.1.2 4 55.19 odd 10