Show commands: Magma / Pari/GP / SageMath

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 406.2
Root \(-0.309017 - 0.535233i\) of defining polynomial
Character \(\chi\) \(=\) 855.406
Dual form 855.2.k.e.676.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.309017 + 0.535233i) q^{2} +(0.809017 - 1.40126i) q^{4} +(-0.500000 - 0.866025i) q^{5} +5.23607 q^{7} +2.23607 q^{8} +(0.309017 - 0.535233i) q^{10} -2.00000 q^{11} +(-1.61803 + 2.80252i) q^{13} +(1.61803 + 2.80252i) q^{14} +(-0.927051 - 1.60570i) q^{16} +(2.73607 + 4.73901i) q^{17} +(2.35410 - 3.66854i) q^{19} -1.61803 q^{20} +(-0.618034 - 1.07047i) q^{22} +(3.23607 - 5.60503i) q^{23} +(-0.500000 + 0.866025i) q^{25} -2.00000 q^{26} +(4.23607 - 7.33708i) q^{28} +(-2.61803 + 4.53457i) q^{29} -3.76393 q^{31} +(2.80902 - 4.86536i) q^{32} +(-1.69098 + 2.92887i) q^{34} +(-2.61803 - 4.53457i) q^{35} -6.47214 q^{37} +(2.69098 + 0.126351i) q^{38} +(-1.11803 - 1.93649i) q^{40} +(-2.85410 - 4.94345i) q^{41} +(0.854102 + 1.47935i) q^{43} +(-1.61803 + 2.80252i) q^{44} +4.00000 q^{46} +(3.11803 - 5.40059i) q^{47} +20.4164 q^{49} -0.618034 q^{50} +(2.61803 + 4.53457i) q^{52} +(-0.500000 + 0.866025i) q^{53} +(1.00000 + 1.73205i) q^{55} +11.7082 q^{56} -3.23607 q^{58} +(-1.00000 - 1.73205i) q^{59} +(-1.76393 + 3.05522i) q^{61} +(-1.16312 - 2.01458i) q^{62} -0.236068 q^{64} +3.23607 q^{65} +(-0.618034 + 1.07047i) q^{67} +8.85410 q^{68} +(1.61803 - 2.80252i) q^{70} +(2.38197 + 4.12569i) q^{71} +(6.23607 + 10.8012i) q^{73} +(-2.00000 - 3.46410i) q^{74} +(-3.23607 - 6.26662i) q^{76} -10.4721 q^{77} +(-0.927051 + 1.60570i) q^{80} +(1.76393 - 3.05522i) q^{82} +7.18034 q^{83} +(2.73607 - 4.73901i) q^{85} +(-0.527864 + 0.914287i) q^{86} -4.47214 q^{88} +(6.23607 - 10.8012i) q^{89} +(-8.47214 + 14.6742i) q^{91} +(-5.23607 - 9.06914i) q^{92} +3.85410 q^{94} +(-4.35410 - 0.204441i) q^{95} +(1.00000 + 1.73205i) q^{97} +(6.30902 + 10.9275i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} + q^{4} - 2 q^{5} + 12 q^{7} - q^{10} - 8 q^{11} - 2 q^{13} + 2 q^{14} + 3 q^{16} + 2 q^{17} - 4 q^{19} - 2 q^{20} + 2 q^{22} + 4 q^{23} - 2 q^{25} - 8 q^{26} + 8 q^{28} - 6 q^{29} - 24 q^{31}+ \cdots + 23 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.309017 + 0.535233i 0.218508 + 0.378467i 0.954352 − 0.298684i \(-0.0965477\pi\)
−0.735844 + 0.677151i \(0.763214\pi\)
\(3\) 0 0
\(4\) 0.809017 − 1.40126i 0.404508 − 0.700629i
\(5\) −0.500000 − 0.866025i −0.223607 − 0.387298i
\(6\) 0 0
\(7\) 5.23607 1.97905 0.989524 − 0.144370i \(-0.0461154\pi\)
0.989524 + 0.144370i \(0.0461154\pi\)
\(8\) 2.23607 0.790569
\(9\) 0 0
\(10\) 0.309017 − 0.535233i 0.0977198 − 0.169256i
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) −1.61803 + 2.80252i −0.448762 + 0.777278i −0.998306 − 0.0581865i \(-0.981468\pi\)
0.549544 + 0.835465i \(0.314802\pi\)
\(14\) 1.61803 + 2.80252i 0.432438 + 0.749004i
\(15\) 0 0
\(16\) −0.927051 − 1.60570i −0.231763 − 0.401425i
\(17\) 2.73607 + 4.73901i 0.663594 + 1.14938i 0.979664 + 0.200643i \(0.0643030\pi\)
−0.316071 + 0.948736i \(0.602364\pi\)
\(18\) 0 0
\(19\) 2.35410 − 3.66854i 0.540068 − 0.841621i
\(20\) −1.61803 −0.361803
\(21\) 0 0
\(22\) −0.618034 − 1.07047i −0.131765 − 0.228224i
\(23\) 3.23607 − 5.60503i 0.674767 − 1.16873i −0.301770 − 0.953381i \(-0.597578\pi\)
0.976537 − 0.215350i \(-0.0690891\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 4.23607 − 7.33708i 0.800542 − 1.38658i
\(29\) −2.61803 + 4.53457i −0.486157 + 0.842048i −0.999873 − 0.0159118i \(-0.994935\pi\)
0.513717 + 0.857960i \(0.328268\pi\)
\(30\) 0 0
\(31\) −3.76393 −0.676022 −0.338011 − 0.941142i \(-0.609754\pi\)
−0.338011 + 0.941142i \(0.609754\pi\)
\(32\) 2.80902 − 4.86536i 0.496569 − 0.860082i
\(33\) 0 0
\(34\) −1.69098 + 2.92887i −0.290001 + 0.502297i
\(35\) −2.61803 − 4.53457i −0.442529 − 0.766482i
\(36\) 0 0
\(37\) −6.47214 −1.06401 −0.532006 − 0.846740i \(-0.678562\pi\)
−0.532006 + 0.846740i \(0.678562\pi\)
\(38\) 2.69098 + 0.126351i 0.436535 + 0.0204969i
\(39\) 0 0
\(40\) −1.11803 − 1.93649i −0.176777 − 0.306186i
\(41\) −2.85410 − 4.94345i −0.445736 − 0.772037i 0.552367 − 0.833601i \(-0.313725\pi\)
−0.998103 + 0.0615637i \(0.980391\pi\)
\(42\) 0 0
\(43\) 0.854102 + 1.47935i 0.130249 + 0.225598i 0.923773 − 0.382941i \(-0.125089\pi\)
−0.793523 + 0.608540i \(0.791756\pi\)
\(44\) −1.61803 + 2.80252i −0.243928 + 0.422495i
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 3.11803 − 5.40059i 0.454812 − 0.787757i −0.543865 − 0.839172i \(-0.683040\pi\)
0.998677 + 0.0514150i \(0.0163731\pi\)
\(48\) 0 0
\(49\) 20.4164 2.91663
\(50\) −0.618034 −0.0874032
\(51\) 0 0
\(52\) 2.61803 + 4.53457i 0.363056 + 0.628831i
\(53\) −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
\(56\) 11.7082 1.56457
\(57\) 0 0
\(58\) −3.23607 −0.424917
\(59\) −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
\(61\) −1.76393 + 3.05522i −0.225848 + 0.391181i −0.956574 − 0.291491i \(-0.905849\pi\)
0.730725 + 0.682672i \(0.239182\pi\)
\(62\) −1.16312 − 2.01458i −0.147716 − 0.255852i
\(63\) 0 0
\(64\) −0.236068 −0.0295085
\(65\) 3.23607 0.401385
\(66\) 0 0
\(67\) −0.618034 + 1.07047i −0.0755049 + 0.130778i −0.901306 − 0.433184i \(-0.857390\pi\)
0.825801 + 0.563962i \(0.190724\pi\)
\(68\) 8.85410 1.07372
\(69\) 0 0
\(70\) 1.61803 − 2.80252i 0.193392 − 0.334965i
\(71\) 2.38197 + 4.12569i 0.282687 + 0.489629i 0.972046 − 0.234792i \(-0.0754408\pi\)
−0.689358 + 0.724420i \(0.742107\pi\)
\(72\) 0 0
\(73\) 6.23607 + 10.8012i 0.729877 + 1.26418i 0.956935 + 0.290302i \(0.0937558\pi\)
−0.227058 + 0.973881i \(0.572911\pi\)
\(74\) −2.00000 − 3.46410i −0.232495 − 0.402694i
\(75\) 0 0
\(76\) −3.23607 − 6.26662i −0.371202 − 0.718830i
\(77\) −10.4721 −1.19341
\(78\) 0 0
\(79\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(80\) −0.927051 + 1.60570i −0.103647 + 0.179523i
\(81\) 0 0
\(82\) 1.76393 − 3.05522i 0.194794 − 0.337393i
\(83\) 7.18034 0.788145 0.394072 − 0.919079i \(-0.371066\pi\)
0.394072 + 0.919079i \(0.371066\pi\)
\(84\) 0 0
\(85\) 2.73607 − 4.73901i 0.296768 − 0.514018i
\(86\) −0.527864 + 0.914287i −0.0569210 + 0.0985901i
\(87\) 0 0
\(88\) −4.47214 −0.476731
\(89\) 6.23607 − 10.8012i 0.661022 − 1.14492i −0.319326 − 0.947645i \(-0.603456\pi\)
0.980347 − 0.197278i \(-0.0632103\pi\)
\(90\) 0 0
\(91\) −8.47214 + 14.6742i −0.888121 + 1.53827i
\(92\) −5.23607 − 9.06914i −0.545898 − 0.945523i
\(93\) 0 0
\(94\) 3.85410 0.397520
\(95\) −4.35410 − 0.204441i −0.446721 − 0.0209752i
\(96\) 0 0
\(97\) 1.00000 + 1.73205i 0.101535 + 0.175863i 0.912317 − 0.409484i \(-0.134291\pi\)
−0.810782 + 0.585348i \(0.800958\pi\)
\(98\) 6.30902 + 10.9275i 0.637307 + 1.10385i
\(99\) 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.406.2 4
3.2 odd 2 285.2.i.e.121.1 yes 4
19.11 even 3 inner 855.2.k.e.676.2 4
57.11 odd 6 285.2.i.e.106.1 ✓ 4
57.26 odd 6 5415.2.a.q.1.2 2
57.50 even 6 5415.2.a.t.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.i.e.106.1 ✓ 4 57.11 odd 6
285.2.i.e.121.1 yes 4 3.2 odd 2
855.2.k.e.406.2 4 1.1 even 1 trivial
855.2.k.e.676.2 4 19.11 even 3 inner
5415.2.a.q.1.2 2 57.26 odd 6
5415.2.a.t.1.1 2 57.50 even 6