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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [855,2,Mod(406,855)] 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("855.406"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(855, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 855.k (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-1,0,1,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.82720937282\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 285)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 406.1
Root \(0.809017 + 1.40126i\) of defining polynomial
Character \(\chi\) \(=\) 855.406
Dual form 855.2.k.e.676.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.809017 - 1.40126i) q^{2} +(-0.309017 + 0.535233i) q^{4} +(-0.500000 - 0.866025i) q^{5} +0.763932 q^{7} -2.23607 q^{8} +(-0.809017 + 1.40126i) q^{10} -2.00000 q^{11} +(0.618034 - 1.07047i) q^{13} +(-0.618034 - 1.07047i) q^{14} +(2.42705 + 4.20378i) q^{16} +(-1.73607 - 3.00696i) q^{17} +(-4.35410 + 0.204441i) q^{19} +0.618034 q^{20} +(1.61803 + 2.80252i) q^{22} +(-1.23607 + 2.14093i) q^{23} +(-0.500000 + 0.866025i) q^{25} -2.00000 q^{26} +(-0.236068 + 0.408882i) q^{28} +(-0.381966 + 0.661585i) q^{29} -8.23607 q^{31} +(1.69098 - 2.92887i) q^{32} +(-2.80902 + 4.86536i) q^{34} +(-0.381966 - 0.661585i) q^{35} +2.47214 q^{37} +(3.80902 + 5.93583i) q^{38} +(1.11803 + 1.93649i) q^{40} +(3.85410 + 6.67550i) q^{41} +(-5.85410 - 10.1396i) q^{43} +(0.618034 - 1.07047i) q^{44} +4.00000 q^{46} +(0.881966 - 1.52761i) q^{47} -6.41641 q^{49} +1.61803 q^{50} +(0.381966 + 0.661585i) q^{52} +(-0.500000 + 0.866025i) q^{53} +(1.00000 + 1.73205i) q^{55} -1.70820 q^{56} +1.23607 q^{58} +(-1.00000 - 1.73205i) q^{59} +(-6.23607 + 10.8012i) q^{61} +(6.66312 + 11.5409i) q^{62} +4.23607 q^{64} -1.23607 q^{65} +(1.61803 - 2.80252i) q^{67} +2.14590 q^{68} +(-0.618034 + 1.07047i) q^{70} +(4.61803 + 7.99867i) q^{71} +(1.76393 + 3.05522i) q^{73} +(-2.00000 - 3.46410i) q^{74} +(1.23607 - 2.39364i) q^{76} -1.52786 q^{77} +(2.42705 - 4.20378i) q^{80} +(6.23607 - 10.8012i) q^{82} -15.1803 q^{83} +(-1.73607 + 3.00696i) q^{85} +(-9.47214 + 16.4062i) q^{86} +4.47214 q^{88} +(1.76393 - 3.05522i) q^{89} +(0.472136 - 0.817763i) q^{91} +(-0.763932 - 1.32317i) q^{92} -2.85410 q^{94} +(2.35410 + 3.66854i) q^{95} +(1.00000 + 1.73205i) q^{97} +(5.19098 + 8.99105i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} + q^{4} - 2 q^{5} + 12 q^{7} - q^{10} - 8 q^{11} - 2 q^{13} + 2 q^{14} + 3 q^{16} + 2 q^{17} - 4 q^{19} - 2 q^{20} + 2 q^{22} + 4 q^{23} - 2 q^{25} - 8 q^{26} + 8 q^{28} - 6 q^{29} - 24 q^{31}+ \cdots + 23 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/855\mathbb{Z}\right)^\times\).

\(n\) \(172\) \(191\) \(496\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.809017 − 1.40126i −0.572061 − 0.990839i −0.996354 − 0.0853143i \(-0.972811\pi\)
0.424293 − 0.905525i \(-0.360523\pi\)
\(3\) 0 0
\(4\) −0.309017 + 0.535233i −0.154508 + 0.267617i
\(5\) −0.500000 − 0.866025i −0.223607 − 0.387298i
\(6\) 0 0
\(7\) 0.763932 0.288739 0.144370 − 0.989524i \(-0.453885\pi\)
0.144370 + 0.989524i \(0.453885\pi\)
\(8\) −2.23607 −0.790569
\(9\) 0 0
\(10\) −0.809017 + 1.40126i −0.255834 + 0.443117i
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 0.618034 − 1.07047i 0.171412 − 0.296894i −0.767502 − 0.641047i \(-0.778500\pi\)
0.938914 + 0.344153i \(0.111834\pi\)
\(14\) −0.618034 − 1.07047i −0.165177 − 0.286094i
\(15\) 0 0
\(16\) 2.42705 + 4.20378i 0.606763 + 1.05094i
\(17\) −1.73607 − 3.00696i −0.421058 − 0.729294i 0.574985 − 0.818164i \(-0.305008\pi\)
−0.996043 + 0.0888696i \(0.971675\pi\)
\(18\) 0 0
\(19\) −4.35410 + 0.204441i −0.998899 + 0.0469020i
\(20\) 0.618034 0.138197
\(21\) 0 0
\(22\) 1.61803 + 2.80252i 0.344966 + 0.597499i
\(23\) −1.23607 + 2.14093i −0.257738 + 0.446415i −0.965636 − 0.259900i \(-0.916310\pi\)
0.707898 + 0.706315i \(0.249644\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) −0.236068 + 0.408882i −0.0446127 + 0.0772714i
\(29\) −0.381966 + 0.661585i −0.0709293 + 0.122853i −0.899309 − 0.437314i \(-0.855930\pi\)
0.828380 + 0.560167i \(0.189263\pi\)
\(30\) 0 0
\(31\) −8.23607 −1.47924 −0.739621 − 0.673024i \(-0.764995\pi\)
−0.739621 + 0.673024i \(0.764995\pi\)
\(32\) 1.69098 − 2.92887i 0.298926 − 0.517756i
\(33\) 0 0
\(34\) −2.80902 + 4.86536i −0.481742 + 0.834402i
\(35\) −0.381966 − 0.661585i −0.0645640 − 0.111828i
\(36\) 0 0
\(37\) 2.47214 0.406417 0.203208 − 0.979136i \(-0.434863\pi\)
0.203208 + 0.979136i \(0.434863\pi\)
\(38\) 3.80902 + 5.93583i 0.617904 + 0.962918i
\(39\) 0 0
\(40\) 1.11803 + 1.93649i 0.176777 + 0.306186i
\(41\) 3.85410 + 6.67550i 0.601910 + 1.04254i 0.992532 + 0.121987i \(0.0389266\pi\)
−0.390622 + 0.920551i \(0.627740\pi\)
\(42\) 0 0
\(43\) −5.85410 − 10.1396i −0.892742 − 1.54627i −0.836574 − 0.547853i \(-0.815445\pi\)
−0.0561679 − 0.998421i \(-0.517888\pi\)
\(44\) 0.618034 − 1.07047i 0.0931721 − 0.161379i
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 0.881966 − 1.52761i 0.128648 − 0.222825i −0.794505 − 0.607258i \(-0.792270\pi\)
0.923153 + 0.384433i \(0.125603\pi\)
\(48\) 0 0
\(49\) −6.41641 −0.916630
\(50\) 1.61803 0.228825
\(51\) 0 0
\(52\) 0.381966 + 0.661585i 0.0529692 + 0.0917453i
\(53\) −0.500000 + 0.866025i −0.0686803 + 0.118958i −0.898321 − 0.439340i \(-0.855212\pi\)
0.829640 + 0.558298i \(0.188546\pi\)
\(54\) 0 0
\(55\) 1.00000 + 1.73205i 0.134840 + 0.233550i
\(56\) −1.70820 −0.228268
\(57\) 0 0
\(58\) 1.23607 0.162304
\(59\) −1.00000 − 1.73205i −0.130189 − 0.225494i 0.793560 − 0.608492i \(-0.208225\pi\)
−0.923749 + 0.382998i \(0.874892\pi\)
\(60\) 0 0
\(61\) −6.23607 + 10.8012i −0.798447 + 1.38295i 0.122181 + 0.992508i \(0.461011\pi\)
−0.920627 + 0.390442i \(0.872322\pi\)
\(62\) 6.66312 + 11.5409i 0.846217 + 1.46569i
\(63\) 0 0
\(64\) 4.23607 0.529508
\(65\) −1.23607 −0.153315
\(66\) 0 0
\(67\) 1.61803 − 2.80252i 0.197674 − 0.342382i −0.750100 − 0.661325i \(-0.769994\pi\)
0.947774 + 0.318943i \(0.103328\pi\)
\(68\) 2.14590 0.260228
\(69\) 0 0
\(70\) −0.618034 + 1.07047i −0.0738692 + 0.127945i
\(71\) 4.61803 + 7.99867i 0.548060 + 0.949267i 0.998407 + 0.0564141i \(0.0179667\pi\)
−0.450348 + 0.892853i \(0.648700\pi\)
\(72\) 0 0
\(73\) 1.76393 + 3.05522i 0.206453 + 0.357586i 0.950595 − 0.310435i \(-0.100475\pi\)
−0.744142 + 0.668022i \(0.767141\pi\)
\(74\) −2.00000 − 3.46410i −0.232495 − 0.402694i
\(75\) 0 0
\(76\) 1.23607 − 2.39364i 0.141787 − 0.274569i
\(77\) −1.52786 −0.174116
\(78\) 0 0
\(79\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(80\) 2.42705 − 4.20378i 0.271353 − 0.469996i
\(81\) 0 0
\(82\) 6.23607 − 10.8012i 0.688659 − 1.19279i
\(83\) −15.1803 −1.66626 −0.833129 − 0.553078i \(-0.813453\pi\)
−0.833129 + 0.553078i \(0.813453\pi\)
\(84\) 0 0
\(85\) −1.73607 + 3.00696i −0.188303 + 0.326150i
\(86\) −9.47214 + 16.4062i −1.02141 + 1.76913i
\(87\) 0 0
\(88\) 4.47214 0.476731
\(89\) 1.76393 − 3.05522i 0.186976 − 0.323853i −0.757264 − 0.653109i \(-0.773465\pi\)
0.944241 + 0.329256i \(0.106798\pi\)
\(90\) 0 0
\(91\) 0.472136 − 0.817763i 0.0494933 − 0.0857249i
\(92\) −0.763932 − 1.32317i −0.0796454 − 0.137950i
\(93\) 0 0
\(94\) −2.85410 −0.294378
\(95\) 2.35410 + 3.66854i 0.241526 + 0.376385i
\(96\) 0 0
\(97\) 1.00000 + 1.73205i 0.101535 + 0.175863i 0.912317 − 0.409484i \(-0.134291\pi\)
−0.810782 + 0.585348i \(0.800958\pi\)
\(98\) 5.19098 + 8.99105i 0.524368 + 0.908233i
\(99\) 0 0
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 855.2.k.e.406.1 4
3.2 odd 2 285.2.i.e.121.2 yes 4
19.11 even 3 inner 855.2.k.e.676.1 4
57.11 odd 6 285.2.i.e.106.2 ✓ 4
57.26 odd 6 5415.2.a.q.1.1 2
57.50 even 6 5415.2.a.t.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.i.e.106.2 ✓ 4 57.11 odd 6
285.2.i.e.121.2 yes 4 3.2 odd 2
855.2.k.e.406.1 4 1.1 even 1 trivial
855.2.k.e.676.1 4 19.11 even 3 inner
5415.2.a.q.1.1 2 57.26 odd 6
5415.2.a.t.1.2 2 57.50 even 6