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Properties

Label 340.2.e.a.69.4
Level $340$
Weight $2$
Character 340.69
Analytic conductor $2.715$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [340,2,Mod(69,340)] 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("340.69"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(340, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 340 = 2^{2} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 340.e (of order \(2\), degree \(1\), 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: \(2.71491366872\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.2058981376.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 18x^{4} - 34x^{3} + 32x^{2} - 8x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 69.4
Root \(1.52153 + 1.52153i\) of defining polynomial
Character \(\chi\) \(=\) 340.69
Dual form 340.2.e.a.69.5

$q$-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)
 
\(f(q)\) \(=\) \(q-0.287336i q^{3} +(-2.19291 - 0.437190i) q^{5} +4.67316i q^{7} +2.91744 q^{9} -3.33039 q^{11} +5.04306i q^{13} +(-0.125620 + 0.630102i) q^{15} -1.00000i q^{17} +5.83488 q^{19} +1.34277 q^{21} +7.54754i q^{23} +(4.61773 + 1.91744i) q^{25} -1.70029i q^{27} -5.83488 q^{29} -2.45601 q^{31} +0.956942i q^{33} +(2.04306 - 10.2478i) q^{35} -9.83488i q^{37} +1.44905 q^{39} +2.63707 q^{41} +7.04306i q^{43} +(-6.39769 - 1.27547i) q^{45} -0.217146i q^{47} -14.8384 q^{49} -0.287336 q^{51} -4.52041i q^{53} +(7.30326 + 1.45601i) q^{55} -1.67657i q^{57} -2.43429 q^{59} -1.55991 q^{61} +13.6337i q^{63} +(2.20477 - 11.0590i) q^{65} +3.61773i q^{67} +2.16868 q^{69} +13.4165 q^{71} -8.30682i q^{73} +(0.550949 - 1.32684i) q^{75} -15.5635i q^{77} +1.37886 q^{79} +8.26376 q^{81} -11.9008i q^{83} +(-0.437190 + 2.19291i) q^{85} +1.67657i q^{87} +4.27954 q^{89} -23.5670 q^{91} +0.705701i q^{93} +(-12.7954 - 2.55095i) q^{95} -14.9120i q^{97} -9.71622 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{5} - 4 q^{15} + 4 q^{25} + 4 q^{31} - 4 q^{35} - 4 q^{39} - 4 q^{41} - 6 q^{45} - 8 q^{49} + 4 q^{51} + 4 q^{55} + 8 q^{59} + 12 q^{61} - 12 q^{65} + 40 q^{71} + 20 q^{75} - 12 q^{79} - 32 q^{81}+ \cdots - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/340\mathbb{Z}\right)^\times\).

\(n\) \(137\) \(171\) \(241\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) − 0.287336i − 0.165893i −0.996554 − 0.0829467i \(-0.973567\pi\)
0.996554 − 0.0829467i \(-0.0264331\pi\)
\(4\) 0 0
\(5\) −2.19291 − 0.437190i −0.980700 − 0.195517i
\(6\) 0 0
\(7\) 4.67316i 1.76629i 0.469101 + 0.883144i \(0.344578\pi\)
−0.469101 + 0.883144i \(0.655422\pi\)
\(8\) 0 0
\(9\) 2.91744 0.972479
\(10\) 0 0
\(11\) −3.33039 −1.00415 −0.502076 − 0.864824i \(-0.667430\pi\)
−0.502076 + 0.864824i \(0.667430\pi\)
\(12\) 0 0
\(13\) 5.04306i 1.39869i 0.714783 + 0.699346i \(0.246525\pi\)
−0.714783 + 0.699346i \(0.753475\pi\)
\(14\) 0 0
\(15\) −0.125620 + 0.630102i −0.0324350 + 0.162692i
\(16\) 0 0
\(17\) − 1.00000i − 0.242536i
\(18\) 0 0
\(19\) 5.83488 1.33861 0.669306 − 0.742987i \(-0.266591\pi\)
0.669306 + 0.742987i \(0.266591\pi\)
\(20\) 0 0
\(21\) 1.34277 0.293016
\(22\) 0 0
\(23\) 7.54754i 1.57377i 0.617099 + 0.786885i \(0.288308\pi\)
−0.617099 + 0.786885i \(0.711692\pi\)
\(24\) 0 0
\(25\) 4.61773 + 1.91744i 0.923546 + 0.383488i
\(26\) 0 0
\(27\) − 1.70029i − 0.327221i
\(28\) 0 0
\(29\) −5.83488 −1.08351 −0.541755 − 0.840537i \(-0.682240\pi\)
−0.541755 + 0.840537i \(0.682240\pi\)
\(30\) 0 0
\(31\) −2.45601 −0.441113 −0.220557 − 0.975374i \(-0.570787\pi\)
−0.220557 + 0.975374i \(0.570787\pi\)
\(32\) 0 0
\(33\) 0.956942i 0.166582i
\(34\) 0 0
\(35\) 2.04306 − 10.2478i 0.345340 − 1.73220i
\(36\) 0 0
\(37\) − 9.83488i − 1.61684i −0.588604 − 0.808422i \(-0.700322\pi\)
0.588604 − 0.808422i \(-0.299678\pi\)
\(38\) 0 0
\(39\) 1.44905 0.232034
\(40\) 0 0
\(41\) 2.63707 0.411840 0.205920 − 0.978569i \(-0.433981\pi\)
0.205920 + 0.978569i \(0.433981\pi\)
\(42\) 0 0
\(43\) 7.04306i 1.07406i 0.843564 + 0.537028i \(0.180453\pi\)
−0.843564 + 0.537028i \(0.819547\pi\)
\(44\) 0 0
\(45\) −6.39769 − 1.27547i −0.953711 − 0.190136i
\(46\) 0 0
\(47\) − 0.217146i − 0.0316740i −0.999875 − 0.0158370i \(-0.994959\pi\)
0.999875 − 0.0158370i \(-0.00504129\pi\)
\(48\) 0 0
\(49\) −14.8384 −2.11978
\(50\) 0 0
\(51\) −0.287336 −0.0402351
\(52\) 0 0
\(53\) − 4.52041i − 0.620926i −0.950585 − 0.310463i \(-0.899516\pi\)
0.950585 − 0.310463i \(-0.100484\pi\)
\(54\) 0 0
\(55\) 7.30326 + 1.45601i 0.984772 + 0.196329i
\(56\) 0 0
\(57\) − 1.67657i − 0.222067i
\(58\) 0 0
\(59\) −2.43429 −0.316918 −0.158459 − 0.987366i \(-0.550653\pi\)
−0.158459 + 0.987366i \(0.550653\pi\)
\(60\) 0 0
\(61\) −1.55991 −0.199726 −0.0998632 − 0.995001i \(-0.531841\pi\)
−0.0998632 + 0.995001i \(0.531841\pi\)
\(62\) 0 0
\(63\) 13.6337i 1.71768i
\(64\) 0 0
\(65\) 2.20477 − 11.0590i 0.273469 − 1.37170i
\(66\) 0 0
\(67\) 3.61773i 0.441976i 0.975277 + 0.220988i \(0.0709282\pi\)
−0.975277 + 0.220988i \(0.929072\pi\)
\(68\) 0 0
\(69\) 2.16868 0.261078
\(70\) 0 0
\(71\) 13.4165 1.59225 0.796123 − 0.605134i \(-0.206881\pi\)
0.796123 + 0.605134i \(0.206881\pi\)
\(72\) 0 0
\(73\) − 8.30682i − 0.972240i −0.873892 − 0.486120i \(-0.838412\pi\)
0.873892 − 0.486120i \(-0.161588\pi\)
\(74\) 0 0
\(75\) 0.550949 − 1.32684i 0.0636181 − 0.153210i
\(76\) 0 0
\(77\) − 15.5635i − 1.77362i
\(78\) 0 0
\(79\) 1.37886 0.155134 0.0775670 − 0.996987i \(-0.475285\pi\)
0.0775670 + 0.996987i \(0.475285\pi\)
\(80\) 0 0
\(81\) 8.26376 0.918196
\(82\) 0 0
\(83\) − 11.9008i − 1.30629i −0.757235 − 0.653143i \(-0.773450\pi\)
0.757235 − 0.653143i \(-0.226550\pi\)
\(84\) 0 0
\(85\) −0.437190 + 2.19291i −0.0474199 + 0.237855i
\(86\) 0 0
\(87\) 1.67657i 0.179747i
\(88\) 0 0
\(89\) 4.27954 0.453630 0.226815 − 0.973938i \(-0.427169\pi\)
0.226815 + 0.973938i \(0.427169\pi\)
\(90\) 0 0
\(91\) −23.5670 −2.47050
\(92\) 0 0
\(93\) 0.705701i 0.0731778i
\(94\) 0 0
\(95\) −12.7954 − 2.55095i −1.31278 − 0.261722i
\(96\) 0 0
\(97\) − 14.9120i − 1.51409i −0.653364 − 0.757044i \(-0.726643\pi\)
0.653364 − 0.757044i \(-0.273357\pi\)
\(98\) 0 0
\(99\) −9.71622 −0.976517
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 340.2.e.a.69.4 ✓ 8
3.2 odd 2 3060.2.g.f.2449.8 8
4.3 odd 2 1360.2.e.e.1089.5 8
5.2 odd 4 1700.2.a.g.1.2 4
5.3 odd 4 1700.2.a.f.1.3 4
5.4 even 2 inner 340.2.e.a.69.5 yes 8
15.14 odd 2 3060.2.g.f.2449.7 8
20.3 even 4 6800.2.a.bv.1.2 4
20.7 even 4 6800.2.a.bu.1.3 4
20.19 odd 2 1360.2.e.e.1089.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
340.2.e.a.69.4 ✓ 8 1.1 even 1 trivial
340.2.e.a.69.5 yes 8 5.4 even 2 inner
1360.2.e.e.1089.4 8 20.19 odd 2
1360.2.e.e.1089.5 8 4.3 odd 2
1700.2.a.f.1.3 4 5.3 odd 4
1700.2.a.g.1.2 4 5.2 odd 4
3060.2.g.f.2449.7 8 15.14 odd 2
3060.2.g.f.2449.8 8 3.2 odd 2
6800.2.a.bu.1.3 4 20.7 even 4
6800.2.a.bv.1.2 4 20.3 even 4