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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 569.27
Character \(\chi\) \(=\) 888.569
Dual form 888.2.br.a.785.27

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.996973 + 1.41635i) q^{3} +(-0.601400 + 2.24446i) q^{5} +(-0.786614 - 1.36246i) q^{7} +(-1.01209 + 2.82412i) q^{9} -0.738125 q^{11} +(-5.22905 - 1.40112i) q^{13} +(-3.77851 + 1.38587i) q^{15} +(-6.88427 + 1.84463i) q^{17} +(-2.79810 - 0.749750i) q^{19} +(1.14548 - 2.47245i) q^{21} +(5.90122 + 5.90122i) q^{23} +(-0.345771 - 0.199631i) q^{25} +(-5.00897 + 1.38210i) q^{27} +(5.65819 - 5.65819i) q^{29} +(-1.19809 - 1.19809i) q^{31} +(-0.735890 - 1.04544i) q^{33} +(3.53104 - 0.946139i) q^{35} +(4.55273 + 4.03393i) q^{37} +(-3.22875 - 8.80304i) q^{39} +(-4.27854 - 7.41065i) q^{41} +(-7.64204 + 7.64204i) q^{43} +(-5.72995 - 3.97002i) q^{45} +10.6754i q^{47} +(2.26248 - 3.91872i) q^{49} +(-9.47607 - 7.91148i) q^{51} +(4.00020 + 2.30952i) q^{53} +(0.443908 - 1.65669i) q^{55} +(-1.72773 - 4.71057i) q^{57} +(-2.81778 + 0.755023i) q^{59} +(-2.69213 + 10.0472i) q^{61} +(4.64387 - 0.842568i) q^{63} +(6.28950 - 10.8937i) q^{65} +(-5.21175 + 3.00900i) q^{67} +(-2.47483 + 14.2415i) q^{69} +(2.51092 - 1.44968i) q^{71} -14.9577i q^{73} +(-0.0619772 - 0.688760i) q^{75} +(0.580619 + 1.00566i) q^{77} +(-1.59192 - 0.426553i) q^{79} +(-6.95135 - 5.71653i) q^{81} +(12.2447 + 7.06950i) q^{83} -16.5608i q^{85} +(13.6550 + 2.37291i) q^{87} +(3.62290 + 13.5208i) q^{89} +(2.20428 + 8.22649i) q^{91} +(0.502450 - 2.89138i) q^{93} +(3.36556 - 5.82932i) q^{95} +(0.824155 - 0.824155i) q^{97} +(0.747049 - 2.08456i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 152 q + 4 q^{13} - 12 q^{15} + 4 q^{19} - 44 q^{31} - 12 q^{39} + 28 q^{43} + 20 q^{45} - 80 q^{49} - 12 q^{51} - 8 q^{55} - 40 q^{57} - 28 q^{61} + 48 q^{63} + 56 q^{69} + 64 q^{75} + 20 q^{79} + 16 q^{81}+ \cdots - 20 q^{97}+O(q^{100}) \) 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
\(3\) 0.996973 + 1.41635i 0.575603 + 0.817730i
\(4\) 0 0
\(5\) −0.601400 + 2.24446i −0.268954 + 1.00375i 0.690831 + 0.723016i \(0.257245\pi\)
−0.959785 + 0.280735i \(0.909422\pi\)
\(6\) 0 0
\(7\) −0.786614 − 1.36246i −0.297312 − 0.514960i 0.678208 − 0.734870i \(-0.262757\pi\)
−0.975520 + 0.219910i \(0.929424\pi\)
\(8\) 0 0
\(9\) −1.01209 + 2.82412i −0.337363 + 0.941375i
\(10\) 0 0
\(11\) −0.738125 −0.222553 −0.111276 − 0.993789i \(-0.535494\pi\)
−0.111276 + 0.993789i \(0.535494\pi\)
\(12\) 0 0
\(13\) −5.22905 − 1.40112i −1.45028 − 0.388601i −0.554158 − 0.832412i \(-0.686960\pi\)
−0.896120 + 0.443811i \(0.853626\pi\)
\(14\) 0 0
\(15\) −3.77851 + 1.38587i −0.975608 + 0.357830i
\(16\) 0 0
\(17\) −6.88427 + 1.84463i −1.66968 + 0.447389i −0.965024 − 0.262160i \(-0.915565\pi\)
−0.704656 + 0.709549i \(0.748899\pi\)
\(18\) 0 0
\(19\) −2.79810 − 0.749750i −0.641929 − 0.172004i −0.0768521 − 0.997043i \(-0.524487\pi\)
−0.565077 + 0.825038i \(0.691154\pi\)
\(20\) 0 0
\(21\) 1.14548 − 2.47245i 0.249964 − 0.539533i
\(22\) 0 0
\(23\) 5.90122 + 5.90122i 1.23049 + 1.23049i 0.963776 + 0.266713i \(0.0859376\pi\)
0.266713 + 0.963776i \(0.414062\pi\)
\(24\) 0 0
\(25\) −0.345771 − 0.199631i −0.0691542 − 0.0399262i
\(26\) 0 0
\(27\) −5.00897 + 1.38210i −0.963977 + 0.265986i
\(28\) 0 0
\(29\) 5.65819 − 5.65819i 1.05070 − 1.05070i 0.0520557 − 0.998644i \(-0.483423\pi\)
0.998644 − 0.0520557i \(-0.0165773\pi\)
\(30\) 0 0
\(31\) −1.19809 − 1.19809i −0.215183 − 0.215183i 0.591282 − 0.806465i \(-0.298622\pi\)
−0.806465 + 0.591282i \(0.798622\pi\)
\(32\) 0 0
\(33\) −0.735890 − 1.04544i −0.128102 − 0.181988i
\(34\) 0 0
\(35\) 3.53104 − 0.946139i 0.596855 − 0.159927i
\(36\) 0 0
\(37\) 4.55273 + 4.03393i 0.748465 + 0.663175i
\(38\) 0 0
\(39\) −3.22875 − 8.80304i −0.517013 − 1.40962i
\(40\) 0 0
\(41\) −4.27854 − 7.41065i −0.668196 − 1.15735i −0.978408 − 0.206682i \(-0.933733\pi\)
0.310212 − 0.950667i \(-0.399600\pi\)
\(42\) 0 0
\(43\) −7.64204 + 7.64204i −1.16540 + 1.16540i −0.182125 + 0.983275i \(0.558298\pi\)
−0.983275 + 0.182125i \(0.941702\pi\)
\(44\) 0 0
\(45\) −5.72995 − 3.97002i −0.854170 − 0.591815i
\(46\) 0 0
\(47\) 10.6754i 1.55717i 0.627540 + 0.778584i \(0.284062\pi\)
−0.627540 + 0.778584i \(0.715938\pi\)
\(48\) 0 0
\(49\) 2.26248 − 3.91872i 0.323211 − 0.559818i
\(50\) 0 0
\(51\) −9.47607 − 7.91148i −1.32692 − 1.10783i
\(52\) 0 0
\(53\) 4.00020 + 2.30952i 0.549470 + 0.317237i 0.748908 − 0.662674i \(-0.230578\pi\)
−0.199438 + 0.979910i \(0.563912\pi\)
\(54\) 0 0
\(55\) 0.443908 − 1.65669i 0.0598566 − 0.223388i
\(56\) 0 0
\(57\) −1.72773 − 4.71057i −0.228843 − 0.623931i
\(58\) 0 0
\(59\) −2.81778 + 0.755023i −0.366844 + 0.0982956i −0.437532 − 0.899203i \(-0.644147\pi\)
0.0706878 + 0.997498i \(0.477481\pi\)
\(60\) 0 0
\(61\) −2.69213 + 10.0472i −0.344692 + 1.28641i 0.548280 + 0.836295i \(0.315283\pi\)
−0.892972 + 0.450112i \(0.851384\pi\)
\(62\) 0 0
\(63\) 4.64387 − 0.842568i 0.585072 − 0.106154i
\(64\) 0 0
\(65\) 6.28950 − 10.8937i 0.780117 − 1.35120i
\(66\) 0 0
\(67\) −5.21175 + 3.00900i −0.636716 + 0.367608i −0.783348 − 0.621583i \(-0.786490\pi\)
0.146632 + 0.989191i \(0.453157\pi\)
\(68\) 0 0
\(69\) −2.47483 + 14.2415i −0.297935 + 1.71448i
\(70\) 0 0
\(71\) 2.51092 − 1.44968i 0.297991 − 0.172045i −0.343549 − 0.939135i \(-0.611629\pi\)
0.641540 + 0.767089i \(0.278296\pi\)
\(72\) 0 0
\(73\) − 14.9577i − 1.75067i −0.483517 − 0.875335i \(-0.660641\pi\)
0.483517 − 0.875335i \(-0.339359\pi\)
\(74\) 0 0
\(75\) −0.0619772 − 0.688760i −0.00715651 − 0.0795311i
\(76\) 0 0
\(77\) 0.580619 + 1.00566i 0.0661677 + 0.114606i
\(78\) 0 0
\(79\) −1.59192 − 0.426553i −0.179105 − 0.0479910i 0.168152 − 0.985761i \(-0.446220\pi\)
−0.347257 + 0.937770i \(0.612887\pi\)
\(80\) 0 0
\(81\) −6.95135 − 5.71653i −0.772372 − 0.635170i
\(82\) 0 0
\(83\) 12.2447 + 7.06950i 1.34403 + 0.775979i 0.987397 − 0.158264i \(-0.0505896\pi\)
0.356638 + 0.934243i \(0.383923\pi\)
\(84\) 0 0
\(85\) − 16.5608i − 1.79627i
\(86\) 0 0
\(87\) 13.6550 + 2.37291i 1.46397 + 0.254403i
\(88\) 0 0
\(89\) 3.62290 + 13.5208i 0.384027 + 1.43321i 0.839695 + 0.543058i \(0.182733\pi\)
−0.455669 + 0.890149i \(0.650600\pi\)
\(90\) 0 0
\(91\) 2.20428 + 8.22649i 0.231072 + 0.862371i
\(92\) 0 0
\(93\) 0.502450 − 2.89138i 0.0521017 − 0.299822i
\(94\) 0 0
\(95\) 3.36556 − 5.82932i 0.345299 − 0.598076i
\(96\) 0 0
\(97\) 0.824155 − 0.824155i 0.0836803 − 0.0836803i −0.664028 − 0.747708i \(-0.731154\pi\)
0.747708 + 0.664028i \(0.231154\pi\)
\(98\) 0 0
\(99\) 0.747049 − 2.08456i 0.0750812 − 0.209506i
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.569.27 yes 152
3.2 odd 2 inner 888.2.br.a.569.13 ✓ 152
37.8 odd 12 inner 888.2.br.a.785.13 yes 152
111.8 even 12 inner 888.2.br.a.785.27 yes 152
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.br.a.569.13 ✓ 152 3.2 odd 2 inner
888.2.br.a.569.27 yes 152 1.1 even 1 trivial
888.2.br.a.785.13 yes 152 37.8 odd 12 inner
888.2.br.a.785.27 yes 152 111.8 even 12 inner