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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 569.12
Character \(\chi\) \(=\) 888.569
Dual form 888.2.br.a.785.12

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.898050 - 1.48105i) q^{3} +(0.719227 - 2.68419i) q^{5} +(2.26018 + 3.91475i) q^{7} +(-1.38701 + 2.66011i) q^{9} -4.17083 q^{11} +(-6.14914 - 1.64766i) q^{13} +(-4.62132 + 1.34533i) q^{15} +(-5.34826 + 1.43306i) q^{17} +(1.15795 + 0.310272i) q^{19} +(3.76818 - 6.86308i) q^{21} +(0.554289 + 0.554289i) q^{23} +(-2.35748 - 1.36109i) q^{25} +(5.18536 - 0.334684i) q^{27} +(-4.45960 + 4.45960i) q^{29} +(-5.75692 - 5.75692i) q^{31} +(3.74562 + 6.17721i) q^{33} +(12.1335 - 3.25117i) q^{35} +(-1.51344 - 5.89148i) q^{37} +(3.08197 + 10.5868i) q^{39} +(-4.00989 - 6.94533i) q^{41} +(2.15787 - 2.15787i) q^{43} +(6.14268 + 5.63623i) q^{45} +6.01642i q^{47} +(-6.71685 + 11.6339i) q^{49} +(6.92545 + 6.63408i) q^{51} +(-4.08709 - 2.35968i) q^{53} +(-2.99978 + 11.1953i) q^{55} +(-0.580370 - 1.99362i) q^{57} +(0.458872 - 0.122954i) q^{59} +(-1.89001 + 7.05361i) q^{61} +(-13.5486 + 0.582537i) q^{63} +(-8.84525 + 15.3204i) q^{65} +(0.496664 - 0.286749i) q^{67} +(0.323149 - 1.31871i) q^{69} +(1.25616 - 0.725245i) q^{71} +11.0829i q^{73} +(0.101292 + 4.71387i) q^{75} +(-9.42684 - 16.3278i) q^{77} +(6.35025 + 1.70154i) q^{79} +(-5.15240 - 7.37921i) q^{81} +(5.56039 + 3.21029i) q^{83} +15.3865i q^{85} +(10.6098 + 2.59994i) q^{87} +(-0.508881 - 1.89917i) q^{89} +(-7.44801 - 27.7963i) q^{91} +(-3.35628 + 13.6963i) q^{93} +(1.66566 - 2.88500i) q^{95} +(11.4369 - 11.4369i) q^{97} +(5.78499 - 11.0949i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 152 q + 4 q^{13} - 12 q^{15} + 4 q^{19} - 44 q^{31} - 12 q^{39} + 28 q^{43} + 20 q^{45} - 80 q^{49} - 12 q^{51} - 8 q^{55} - 40 q^{57} - 28 q^{61} + 48 q^{63} + 56 q^{69} + 64 q^{75} + 20 q^{79} + 16 q^{81}+ \cdots - 20 q^{97}+O(q^{100}) \) 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
\(3\) −0.898050 − 1.48105i −0.518490 − 0.855084i
\(4\) 0 0
\(5\) 0.719227 − 2.68419i 0.321648 − 1.20041i −0.595990 − 0.802992i \(-0.703240\pi\)
0.917639 − 0.397416i \(-0.130093\pi\)
\(6\) 0 0
\(7\) 2.26018 + 3.91475i 0.854269 + 1.47964i 0.877322 + 0.479902i \(0.159328\pi\)
−0.0230532 + 0.999734i \(0.507339\pi\)
\(8\) 0 0
\(9\) −1.38701 + 2.66011i −0.462337 + 0.886704i
\(10\) 0 0
\(11\) −4.17083 −1.25755 −0.628777 − 0.777586i \(-0.716444\pi\)
−0.628777 + 0.777586i \(0.716444\pi\)
\(12\) 0 0
\(13\) −6.14914 − 1.64766i −1.70546 − 0.456978i −0.731158 − 0.682208i \(-0.761020\pi\)
−0.974305 + 0.225231i \(0.927686\pi\)
\(14\) 0 0
\(15\) −4.62132 + 1.34533i −1.19322 + 0.347363i
\(16\) 0 0
\(17\) −5.34826 + 1.43306i −1.29714 + 0.347569i −0.840370 − 0.542014i \(-0.817662\pi\)
−0.456775 + 0.889582i \(0.650995\pi\)
\(18\) 0 0
\(19\) 1.15795 + 0.310272i 0.265652 + 0.0711812i 0.389186 − 0.921159i \(-0.372756\pi\)
−0.123534 + 0.992340i \(0.539423\pi\)
\(20\) 0 0
\(21\) 3.76818 − 6.86308i 0.822284 − 1.49765i
\(22\) 0 0
\(23\) 0.554289 + 0.554289i 0.115577 + 0.115577i 0.762530 − 0.646953i \(-0.223957\pi\)
−0.646953 + 0.762530i \(0.723957\pi\)
\(24\) 0 0
\(25\) −2.35748 − 1.36109i −0.471496 − 0.272218i
\(26\) 0 0
\(27\) 5.18536 − 0.334684i 0.997924 − 0.0644100i
\(28\) 0 0
\(29\) −4.45960 + 4.45960i −0.828126 + 0.828126i −0.987257 − 0.159131i \(-0.949131\pi\)
0.159131 + 0.987257i \(0.449131\pi\)
\(30\) 0 0
\(31\) −5.75692 − 5.75692i −1.03397 − 1.03397i −0.999402 − 0.0345718i \(-0.988993\pi\)
−0.0345718 − 0.999402i \(-0.511007\pi\)
\(32\) 0 0
\(33\) 3.74562 + 6.17721i 0.652028 + 1.07531i
\(34\) 0 0
\(35\) 12.1335 − 3.25117i 2.05094 − 0.549548i
\(36\) 0 0
\(37\) −1.51344 − 5.89148i −0.248808 − 0.968553i
\(38\) 0 0
\(39\) 3.08197 + 10.5868i 0.493511 + 1.69525i
\(40\) 0 0
\(41\) −4.00989 − 6.94533i −0.626240 − 1.08468i −0.988300 − 0.152524i \(-0.951260\pi\)
0.362060 − 0.932155i \(-0.382073\pi\)
\(42\) 0 0
\(43\) 2.15787 − 2.15787i 0.329072 − 0.329072i −0.523162 − 0.852233i \(-0.675248\pi\)
0.852233 + 0.523162i \(0.175248\pi\)
\(44\) 0 0
\(45\) 6.14268 + 5.63623i 0.915697 + 0.840200i
\(46\) 0 0
\(47\) 6.01642i 0.877584i 0.898589 + 0.438792i \(0.144594\pi\)
−0.898589 + 0.438792i \(0.855406\pi\)
\(48\) 0 0
\(49\) −6.71685 + 11.6339i −0.959550 + 1.66199i
\(50\) 0 0
\(51\) 6.92545 + 6.63408i 0.969756 + 0.928957i
\(52\) 0 0
\(53\) −4.08709 − 2.35968i −0.561405 − 0.324127i 0.192304 − 0.981335i \(-0.438404\pi\)
−0.753709 + 0.657208i \(0.771737\pi\)
\(54\) 0 0
\(55\) −2.99978 + 11.1953i −0.404490 + 1.50958i
\(56\) 0 0
\(57\) −0.580370 − 1.99362i −0.0768718 − 0.264061i
\(58\) 0 0
\(59\) 0.458872 − 0.122954i 0.0597401 − 0.0160073i −0.228825 − 0.973468i \(-0.573488\pi\)
0.288565 + 0.957460i \(0.406822\pi\)
\(60\) 0 0
\(61\) −1.89001 + 7.05361i −0.241991 + 0.903122i 0.732881 + 0.680356i \(0.238175\pi\)
−0.974872 + 0.222765i \(0.928492\pi\)
\(62\) 0 0
\(63\) −13.5486 + 0.582537i −1.70696 + 0.0733928i
\(64\) 0 0
\(65\) −8.84525 + 15.3204i −1.09712 + 1.90027i
\(66\) 0 0
\(67\) 0.496664 − 0.286749i 0.0606772 − 0.0350320i −0.469354 − 0.883010i \(-0.655513\pi\)
0.530032 + 0.847978i \(0.322180\pi\)
\(68\) 0 0
\(69\) 0.323149 − 1.31871i 0.0389026 − 0.158754i
\(70\) 0 0
\(71\) 1.25616 − 0.725245i 0.149079 − 0.0860707i −0.423605 − 0.905847i \(-0.639236\pi\)
0.572684 + 0.819776i \(0.305902\pi\)
\(72\) 0 0
\(73\) 11.0829i 1.29715i 0.761149 + 0.648577i \(0.224636\pi\)
−0.761149 + 0.648577i \(0.775364\pi\)
\(74\) 0 0
\(75\) 0.101292 + 4.71387i 0.0116962 + 0.544311i
\(76\) 0 0
\(77\) −9.42684 − 16.3278i −1.07429 − 1.86072i
\(78\) 0 0
\(79\) 6.35025 + 1.70154i 0.714458 + 0.191439i 0.597698 − 0.801722i \(-0.296082\pi\)
0.116761 + 0.993160i \(0.462749\pi\)
\(80\) 0 0
\(81\) −5.15240 − 7.37921i −0.572489 − 0.819912i
\(82\) 0 0
\(83\) 5.56039 + 3.21029i 0.610332 + 0.352375i 0.773095 − 0.634290i \(-0.218707\pi\)
−0.162763 + 0.986665i \(0.552041\pi\)
\(84\) 0 0
\(85\) 15.3865i 1.66890i
\(86\) 0 0
\(87\) 10.6098 + 2.59994i 1.13749 + 0.278743i
\(88\) 0 0
\(89\) −0.508881 − 1.89917i −0.0539413 − 0.201312i 0.933696 − 0.358066i \(-0.116564\pi\)
−0.987638 + 0.156754i \(0.949897\pi\)
\(90\) 0 0
\(91\) −7.44801 − 27.7963i −0.780763 − 2.91385i
\(92\) 0 0
\(93\) −3.35628 + 13.6963i −0.348030 + 1.42024i
\(94\) 0 0
\(95\) 1.66566 − 2.88500i 0.170893 − 0.295995i
\(96\) 0 0
\(97\) 11.4369 − 11.4369i 1.16124 − 1.16124i 0.177040 − 0.984204i \(-0.443348\pi\)
0.984204 − 0.177040i \(-0.0566521\pi\)
\(98\) 0 0
\(99\) 5.78499 − 11.0949i 0.581413 − 1.11508i
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.569.12 ✓ 152
3.2 odd 2 inner 888.2.br.a.569.25 yes 152
37.8 odd 12 inner 888.2.br.a.785.25 yes 152
111.8 even 12 inner 888.2.br.a.785.12 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.br.a.569.12 ✓ 152 1.1 even 1 trivial
888.2.br.a.569.25 yes 152 3.2 odd 2 inner
888.2.br.a.785.12 yes 152 111.8 even 12 inner
888.2.br.a.785.25 yes 152 37.8 odd 12 inner