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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [888,2,Mod(473,888)] 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("888.473"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 0, 6, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.br (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(152\)
Relative dimension: \(38\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 569.14
Character \(\chi\) \(=\) 888.569
Dual form 888.2.br.a.785.14

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.709178 + 1.58021i) q^{3} +(-0.440252 + 1.64304i) q^{5} +(-2.41379 - 4.18081i) q^{7} +(-1.99413 - 2.24130i) q^{9} +5.79998 q^{11} +(0.132470 + 0.0354952i) q^{13} +(-2.28414 - 1.86090i) q^{15} +(1.95366 - 0.523480i) q^{17} +(2.26131 + 0.605916i) q^{19} +(8.31836 - 0.849361i) q^{21} +(-2.57074 - 2.57074i) q^{23} +(1.82436 + 1.05329i) q^{25} +(4.95592 - 1.56167i) q^{27} +(1.61116 - 1.61116i) q^{29} +(2.85614 + 2.85614i) q^{31} +(-4.11322 + 9.16519i) q^{33} +(7.93193 - 2.12535i) q^{35} +(-4.69397 + 3.86868i) q^{37} +(-0.150034 + 0.184158i) q^{39} +(2.81750 + 4.88006i) q^{41} +(7.06828 - 7.06828i) q^{43} +(4.56048 - 2.28971i) q^{45} -7.92849i q^{47} +(-8.15277 + 14.1210i) q^{49} +(-0.558280 + 3.45843i) q^{51} +(4.24708 + 2.45205i) q^{53} +(-2.55345 + 9.52962i) q^{55} +(-2.56114 + 3.14364i) q^{57} +(4.36135 - 1.16862i) q^{59} +(-3.81348 + 14.2321i) q^{61} +(-4.55703 + 13.7471i) q^{63} +(-0.116640 + 0.202027i) q^{65} +(-1.58274 + 0.913795i) q^{67} +(5.88542 - 2.23920i) q^{69} +(10.1420 - 5.85550i) q^{71} -13.5662i q^{73} +(-2.95822 + 2.13590i) q^{75} +(-13.9999 - 24.2486i) q^{77} +(-6.87939 - 1.84333i) q^{79} +(-1.04687 + 8.93891i) q^{81} +(3.79517 + 2.19114i) q^{83} +3.44040i q^{85} +(1.40337 + 3.68856i) q^{87} +(1.55789 + 5.81411i) q^{89} +(-0.171356 - 0.639508i) q^{91} +(-6.53880 + 2.48779i) q^{93} +(-1.99109 + 3.44867i) q^{95} +(12.1216 - 12.1216i) q^{97} +(-11.5659 - 12.9995i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 152 q + 4 q^{13} - 12 q^{15} + 4 q^{19} - 44 q^{31} - 12 q^{39} + 28 q^{43} + 20 q^{45} - 80 q^{49} - 12 q^{51} - 8 q^{55} - 40 q^{57} - 28 q^{61} + 48 q^{63} + 56 q^{69} + 64 q^{75} + 20 q^{79} + 16 q^{81}+ \cdots - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/888\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(1\) \(e\left(\frac{11}{12}\right)\) \(1\) \(-1\)

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
\(3\) −0.709178 + 1.58021i −0.409444 + 0.912335i
\(4\) 0 0
\(5\) −0.440252 + 1.64304i −0.196887 + 0.734791i 0.794884 + 0.606762i \(0.207532\pi\)
−0.991770 + 0.128029i \(0.959135\pi\)
\(6\) 0 0
\(7\) −2.41379 − 4.18081i −0.912327 − 1.58020i −0.810768 − 0.585367i \(-0.800950\pi\)
−0.101559 − 0.994830i \(-0.532383\pi\)
\(8\) 0 0
\(9\) −1.99413 − 2.24130i −0.664711 − 0.747101i
\(10\) 0 0
\(11\) 5.79998 1.74876 0.874380 − 0.485242i \(-0.161269\pi\)
0.874380 + 0.485242i \(0.161269\pi\)
\(12\) 0 0
\(13\) 0.132470 + 0.0354952i 0.0367405 + 0.00984459i 0.277142 − 0.960829i \(-0.410613\pi\)
−0.240402 + 0.970673i \(0.577279\pi\)
\(14\) 0 0
\(15\) −2.28414 − 1.86090i −0.589762 − 0.480483i
\(16\) 0 0
\(17\) 1.95366 − 0.523480i 0.473831 − 0.126963i −0.0139976 − 0.999902i \(-0.504456\pi\)
0.487829 + 0.872939i \(0.337789\pi\)
\(18\) 0 0
\(19\) 2.26131 + 0.605916i 0.518780 + 0.139007i 0.508702 − 0.860943i \(-0.330125\pi\)
0.0100776 + 0.999949i \(0.496792\pi\)
\(20\) 0 0
\(21\) 8.31836 − 0.849361i 1.81522 − 0.185346i
\(22\) 0 0
\(23\) −2.57074 − 2.57074i −0.536036 − 0.536036i 0.386326 − 0.922362i \(-0.373744\pi\)
−0.922362 + 0.386326i \(0.873744\pi\)
\(24\) 0 0
\(25\) 1.82436 + 1.05329i 0.364871 + 0.210659i
\(26\) 0 0
\(27\) 4.95592 − 1.56167i 0.953768 − 0.300543i
\(28\) 0 0
\(29\) 1.61116 − 1.61116i 0.299184 − 0.299184i −0.541510 − 0.840694i \(-0.682147\pi\)
0.840694 + 0.541510i \(0.182147\pi\)
\(30\) 0 0
\(31\) 2.85614 + 2.85614i 0.512977 + 0.512977i 0.915437 − 0.402460i \(-0.131845\pi\)
−0.402460 + 0.915437i \(0.631845\pi\)
\(32\) 0 0
\(33\) −4.11322 + 9.16519i −0.716019 + 1.59545i
\(34\) 0 0
\(35\) 7.93193 − 2.12535i 1.34074 − 0.359250i
\(36\) 0 0
\(37\) −4.69397 + 3.86868i −0.771684 + 0.636007i
\(38\) 0 0
\(39\) −0.150034 + 0.184158i −0.0240247 + 0.0294888i
\(40\) 0 0
\(41\) 2.81750 + 4.88006i 0.440020 + 0.762137i 0.997690 − 0.0679253i \(-0.0216380\pi\)
−0.557670 + 0.830063i \(0.688305\pi\)
\(42\) 0 0
\(43\) 7.06828 − 7.06828i 1.07790 − 1.07790i 0.0812059 − 0.996697i \(-0.474123\pi\)
0.996697 − 0.0812059i \(-0.0258771\pi\)
\(44\) 0 0
\(45\) 4.56048 − 2.28971i 0.679836 − 0.341330i
\(46\) 0 0
\(47\) − 7.92849i − 1.15649i −0.815864 − 0.578244i \(-0.803738\pi\)
0.815864 − 0.578244i \(-0.196262\pi\)
\(48\) 0 0
\(49\) −8.15277 + 14.1210i −1.16468 + 2.01729i
\(50\) 0 0
\(51\) −0.558280 + 3.45843i −0.0781749 + 0.484277i
\(52\) 0 0
\(53\) 4.24708 + 2.45205i 0.583382 + 0.336816i 0.762476 − 0.647016i \(-0.223983\pi\)
−0.179094 + 0.983832i \(0.557317\pi\)
\(54\) 0 0
\(55\) −2.55345 + 9.52962i −0.344308 + 1.28497i
\(56\) 0 0
\(57\) −2.56114 + 3.14364i −0.339232 + 0.416386i
\(58\) 0 0
\(59\) 4.36135 − 1.16862i 0.567799 − 0.152141i 0.0365121 − 0.999333i \(-0.488375\pi\)
0.531287 + 0.847192i \(0.321709\pi\)
\(60\) 0 0
\(61\) −3.81348 + 14.2321i −0.488267 + 1.82224i 0.0766114 + 0.997061i \(0.475590\pi\)
−0.564878 + 0.825174i \(0.691077\pi\)
\(62\) 0 0
\(63\) −4.55703 + 13.7471i −0.574132 + 1.73197i
\(64\) 0 0
\(65\) −0.116640 + 0.202027i −0.0144674 + 0.0250583i
\(66\) 0 0
\(67\) −1.58274 + 0.913795i −0.193362 + 0.111638i −0.593556 − 0.804793i \(-0.702276\pi\)
0.400193 + 0.916431i \(0.368943\pi\)
\(68\) 0 0
\(69\) 5.88542 − 2.23920i 0.708522 − 0.269568i
\(70\) 0 0
\(71\) 10.1420 − 5.85550i 1.20364 − 0.694920i 0.242274 − 0.970208i \(-0.422107\pi\)
0.961362 + 0.275288i \(0.0887732\pi\)
\(72\) 0 0
\(73\) − 13.5662i − 1.58780i −0.608048 − 0.793900i \(-0.708047\pi\)
0.608048 − 0.793900i \(-0.291953\pi\)
\(74\) 0 0
\(75\) −2.95822 + 2.13590i −0.341586 + 0.246632i
\(76\) 0 0
\(77\) −13.9999 − 24.2486i −1.59544 − 2.76338i
\(78\) 0 0
\(79\) −6.87939 − 1.84333i −0.773992 − 0.207391i −0.149858 − 0.988708i \(-0.547882\pi\)
−0.624135 + 0.781317i \(0.714548\pi\)
\(80\) 0 0
\(81\) −1.04687 + 8.93891i −0.116319 + 0.993212i
\(82\) 0 0
\(83\) 3.79517 + 2.19114i 0.416574 + 0.240509i 0.693611 − 0.720350i \(-0.256019\pi\)
−0.277036 + 0.960859i \(0.589352\pi\)
\(84\) 0 0
\(85\) 3.44040i 0.373164i
\(86\) 0 0
\(87\) 1.40337 + 3.68856i 0.150457 + 0.395455i
\(88\) 0 0
\(89\) 1.55789 + 5.81411i 0.165136 + 0.616295i 0.998023 + 0.0628518i \(0.0200196\pi\)
−0.832887 + 0.553443i \(0.813314\pi\)
\(90\) 0 0
\(91\) −0.171356 − 0.639508i −0.0179630 − 0.0670387i
\(92\) 0 0
\(93\) −6.53880 + 2.48779i −0.678042 + 0.257972i
\(94\) 0 0
\(95\) −1.99109 + 3.44867i −0.204282 + 0.353826i
\(96\) 0 0
\(97\) 12.1216 − 12.1216i 1.23076 − 1.23076i 0.267094 − 0.963670i \(-0.413936\pi\)
0.963670 − 0.267094i \(-0.0860635\pi\)
\(98\) 0 0
\(99\) −11.5659 − 12.9995i −1.16242 − 1.30650i
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 888.2.br.a.569.14 yes 152
3.2 odd 2 inner 888.2.br.a.569.1 ✓ 152
37.8 odd 12 inner 888.2.br.a.785.1 yes 152
111.8 even 12 inner 888.2.br.a.785.14 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.br.a.569.1 ✓ 152 3.2 odd 2 inner
888.2.br.a.569.14 yes 152 1.1 even 1 trivial
888.2.br.a.785.1 yes 152 37.8 odd 12 inner
888.2.br.a.785.14 yes 152 111.8 even 12 inner