Properties

Label 55.6.a.b.1.2
Level $55$
Weight $6$
Character 55.1
Self dual yes
Analytic conductor $8.821$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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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] = [55,6,Mod(1,55)] 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("55.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(55, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 55 = 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 55.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(8.82111008971\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 33x^{2} - 8x + 116 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.50110\) of defining polynomial
Character \(\chi\) \(=\) 55.1

$q$-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)
 
\(f(q)\) \(=\) \(q-6.96278 q^{2} +22.6701 q^{3} +16.4803 q^{4} -25.0000 q^{5} -157.847 q^{6} -169.118 q^{7} +108.060 q^{8} +270.932 q^{9} +174.070 q^{10} -121.000 q^{11} +373.610 q^{12} +25.3182 q^{13} +1177.53 q^{14} -566.752 q^{15} -1279.77 q^{16} -2016.26 q^{17} -1886.44 q^{18} -773.486 q^{19} -412.008 q^{20} -3833.91 q^{21} +842.497 q^{22} -541.643 q^{23} +2449.73 q^{24} +625.000 q^{25} -176.285 q^{26} +633.231 q^{27} -2787.11 q^{28} -5882.28 q^{29} +3946.17 q^{30} -915.584 q^{31} +5452.83 q^{32} -2743.08 q^{33} +14038.8 q^{34} +4227.94 q^{35} +4465.06 q^{36} +11360.3 q^{37} +5385.62 q^{38} +573.964 q^{39} -2701.50 q^{40} -15477.7 q^{41} +26694.7 q^{42} +6097.77 q^{43} -1994.12 q^{44} -6773.31 q^{45} +3771.34 q^{46} +15131.8 q^{47} -29012.5 q^{48} +11793.8 q^{49} -4351.74 q^{50} -45708.8 q^{51} +417.252 q^{52} +10443.0 q^{53} -4409.05 q^{54} +3025.00 q^{55} -18274.9 q^{56} -17535.0 q^{57} +40957.0 q^{58} -50295.3 q^{59} -9340.26 q^{60} +45523.0 q^{61} +6375.01 q^{62} -45819.4 q^{63} +2985.73 q^{64} -632.954 q^{65} +19099.5 q^{66} +11285.0 q^{67} -33228.7 q^{68} -12279.1 q^{69} -29438.2 q^{70} +64741.3 q^{71} +29277.0 q^{72} +77769.4 q^{73} -79099.6 q^{74} +14168.8 q^{75} -12747.3 q^{76} +20463.2 q^{77} -3996.39 q^{78} -87890.2 q^{79} +31994.2 q^{80} -51481.2 q^{81} +107768. q^{82} +18403.3 q^{83} -63184.1 q^{84} +50406.5 q^{85} -42457.4 q^{86} -133352. q^{87} -13075.3 q^{88} +52660.1 q^{89} +47161.1 q^{90} -4281.74 q^{91} -8926.45 q^{92} -20756.4 q^{93} -105359. q^{94} +19337.2 q^{95} +123616. q^{96} -38745.0 q^{97} -82117.3 q^{98} -32782.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 5 q^{2} + 61 q^{4} - 100 q^{5} - 157 q^{6} - 90 q^{7} - 135 q^{8} + 22 q^{9} + 125 q^{10} - 484 q^{11} - 795 q^{12} + 820 q^{13} - 1687 q^{14} - 2671 q^{16} - 3800 q^{17} - 1610 q^{18} - 3394 q^{19}+ \cdots - 2662 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −6.96278 −1.23086 −0.615429 − 0.788192i \(-0.711017\pi\)
−0.615429 + 0.788192i \(0.711017\pi\)
\(3\) 22.6701 1.45429 0.727143 − 0.686486i \(-0.240848\pi\)
0.727143 + 0.686486i \(0.240848\pi\)
\(4\) 16.4803 0.515010
\(5\) −25.0000 −0.447214
\(6\) −157.847 −1.79002
\(7\) −169.118 −1.30450 −0.652249 − 0.758004i \(-0.726174\pi\)
−0.652249 + 0.758004i \(0.726174\pi\)
\(8\) 108.060 0.596953
\(9\) 270.932 1.11495
\(10\) 174.070 0.550456
\(11\) −121.000 −0.301511
\(12\) 373.610 0.748973
\(13\) 25.3182 0.0415502 0.0207751 − 0.999784i \(-0.493387\pi\)
0.0207751 + 0.999784i \(0.493387\pi\)
\(14\) 1177.53 1.60565
\(15\) −566.752 −0.650377
\(16\) −1279.77 −1.24977
\(17\) −2016.26 −1.69209 −0.846047 − 0.533108i \(-0.821024\pi\)
−0.846047 + 0.533108i \(0.821024\pi\)
\(18\) −1886.44 −1.37234
\(19\) −773.486 −0.491551 −0.245775 − 0.969327i \(-0.579043\pi\)
−0.245775 + 0.969327i \(0.579043\pi\)
\(20\) −412.008 −0.230320
\(21\) −3833.91 −1.89711
\(22\) 842.497 0.371118
\(23\) −541.643 −0.213498 −0.106749 − 0.994286i \(-0.534044\pi\)
−0.106749 + 0.994286i \(0.534044\pi\)
\(24\) 2449.73 0.868141
\(25\) 625.000 0.200000
\(26\) −176.285 −0.0511424
\(27\) 633.231 0.167168
\(28\) −2787.11 −0.671830
\(29\) −5882.28 −1.29882 −0.649412 − 0.760437i \(-0.724985\pi\)
−0.649412 + 0.760437i \(0.724985\pi\)
\(30\) 3946.17 0.800521
\(31\) −915.584 −0.171117 −0.0855587 − 0.996333i \(-0.527268\pi\)
−0.0855587 + 0.996333i \(0.527268\pi\)
\(32\) 5452.83 0.941342
\(33\) −2743.08 −0.438484
\(34\) 14038.8 2.08273
\(35\) 4227.94 0.583390
\(36\) 4465.06 0.574210
\(37\) 11360.3 1.36423 0.682115 − 0.731245i \(-0.261061\pi\)
0.682115 + 0.731245i \(0.261061\pi\)
\(38\) 5385.62 0.605029
\(39\) 573.964 0.0604259
\(40\) −2701.50 −0.266966
\(41\) −15477.7 −1.43796 −0.718979 − 0.695031i \(-0.755390\pi\)
−0.718979 + 0.695031i \(0.755390\pi\)
\(42\) 26694.7 2.33508
\(43\) 6097.77 0.502921 0.251460 − 0.967868i \(-0.419089\pi\)
0.251460 + 0.967868i \(0.419089\pi\)
\(44\) −1994.12 −0.155281
\(45\) −6773.31 −0.498620
\(46\) 3771.34 0.262785
\(47\) 15131.8 0.999185 0.499593 − 0.866260i \(-0.333483\pi\)
0.499593 + 0.866260i \(0.333483\pi\)
\(48\) −29012.5 −1.81753
\(49\) 11793.8 0.701717
\(50\) −4351.74 −0.246172
\(51\) −45708.8 −2.46079
\(52\) 417.252 0.0213988
\(53\) 10443.0 0.510666 0.255333 − 0.966853i \(-0.417815\pi\)
0.255333 + 0.966853i \(0.417815\pi\)
\(54\) −4409.05 −0.205760
\(55\) 3025.00 0.134840
\(56\) −18274.9 −0.778724
\(57\) −17535.0 −0.714856
\(58\) 40957.0 1.59867
\(59\) −50295.3 −1.88104 −0.940519 − 0.339741i \(-0.889661\pi\)
−0.940519 + 0.339741i \(0.889661\pi\)
\(60\) −9340.26 −0.334951
\(61\) 45523.0 1.56641 0.783206 − 0.621762i \(-0.213583\pi\)
0.783206 + 0.621762i \(0.213583\pi\)
\(62\) 6375.01 0.210621
\(63\) −45819.4 −1.45445
\(64\) 2985.73 0.0911172
\(65\) −632.954 −0.0185818
\(66\) 19099.5 0.539711
\(67\) 11285.0 0.307124 0.153562 − 0.988139i \(-0.450926\pi\)
0.153562 + 0.988139i \(0.450926\pi\)
\(68\) −33228.7 −0.871446
\(69\) −12279.1 −0.310487
\(70\) −29438.2 −0.718069
\(71\) 64741.3 1.52418 0.762089 − 0.647473i \(-0.224174\pi\)
0.762089 + 0.647473i \(0.224174\pi\)
\(72\) 29277.0 0.665572
\(73\) 77769.4 1.70805 0.854027 − 0.520229i \(-0.174153\pi\)
0.854027 + 0.520229i \(0.174153\pi\)
\(74\) −79099.6 −1.67917
\(75\) 14168.8 0.290857
\(76\) −12747.3 −0.253154
\(77\) 20463.2 0.393321
\(78\) −3996.39 −0.0743757
\(79\) −87890.2 −1.58443 −0.792214 − 0.610243i \(-0.791072\pi\)
−0.792214 + 0.610243i \(0.791072\pi\)
\(80\) 31994.2 0.558916
\(81\) −51481.2 −0.871838
\(82\) 107768. 1.76992
\(83\) 18403.3 0.293225 0.146613 − 0.989194i \(-0.453163\pi\)
0.146613 + 0.989194i \(0.453163\pi\)
\(84\) −63184.1 −0.977034
\(85\) 50406.5 0.756728
\(86\) −42457.4 −0.619024
\(87\) −133352. −1.88886
\(88\) −13075.3 −0.179988
\(89\) 52660.1 0.704704 0.352352 − 0.935868i \(-0.385382\pi\)
0.352352 + 0.935868i \(0.385382\pi\)
\(90\) 47161.1 0.613730
\(91\) −4281.74 −0.0542022
\(92\) −8926.45 −0.109954
\(93\) −20756.4 −0.248854
\(94\) −105359. −1.22985
\(95\) 19337.2 0.219828
\(96\) 123616. 1.36898
\(97\) −38745.0 −0.418106 −0.209053 − 0.977904i \(-0.567038\pi\)
−0.209053 + 0.977904i \(0.567038\pi\)
\(98\) −82117.3 −0.863714
\(99\) −32782.8 −0.336170
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 55.6.a.b.1.2 ✓ 4
3.2 odd 2 495.6.a.g.1.3 4
4.3 odd 2 880.6.a.n.1.1 4
5.2 odd 4 275.6.b.d.199.3 8
5.3 odd 4 275.6.b.d.199.6 8
5.4 even 2 275.6.a.d.1.3 4
11.10 odd 2 605.6.a.c.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.6.a.b.1.2 ✓ 4 1.1 even 1 trivial
275.6.a.d.1.3 4 5.4 even 2
275.6.b.d.199.3 8 5.2 odd 4
275.6.b.d.199.6 8 5.3 odd 4
495.6.a.g.1.3 4 3.2 odd 2
605.6.a.c.1.3 4 11.10 odd 2
880.6.a.n.1.1 4 4.3 odd 2