Properties

Label 55.6.a.b.1.3
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 1.3
Root \(1.74666\) 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)
 
\(f(q)\) \(=\) \(q+2.64192 q^{2} +6.66534 q^{3} -25.0202 q^{4} -25.0000 q^{5} +17.6093 q^{6} -12.8802 q^{7} -150.643 q^{8} -198.573 q^{9} -66.0481 q^{10} -121.000 q^{11} -166.769 q^{12} -485.167 q^{13} -34.0284 q^{14} -166.634 q^{15} +402.660 q^{16} +266.661 q^{17} -524.615 q^{18} -149.702 q^{19} +625.506 q^{20} -85.8507 q^{21} -319.673 q^{22} -3213.11 q^{23} -1004.09 q^{24} +625.000 q^{25} -1281.77 q^{26} -2943.24 q^{27} +322.265 q^{28} +2948.81 q^{29} -440.233 q^{30} +2145.87 q^{31} +5884.38 q^{32} -806.507 q^{33} +704.498 q^{34} +322.004 q^{35} +4968.35 q^{36} -808.357 q^{37} -395.500 q^{38} -3233.80 q^{39} +3766.08 q^{40} +10105.2 q^{41} -226.811 q^{42} +2763.15 q^{43} +3027.45 q^{44} +4964.33 q^{45} -8488.79 q^{46} +9973.36 q^{47} +2683.87 q^{48} -16641.1 q^{49} +1651.20 q^{50} +1777.39 q^{51} +12139.0 q^{52} +7126.92 q^{53} -7775.81 q^{54} +3025.00 q^{55} +1940.31 q^{56} -997.814 q^{57} +7790.52 q^{58} -33337.2 q^{59} +4169.21 q^{60} -11871.1 q^{61} +5669.22 q^{62} +2557.66 q^{63} +2660.94 q^{64} +12129.2 q^{65} -2130.73 q^{66} +4500.58 q^{67} -6671.93 q^{68} -21416.5 q^{69} +850.710 q^{70} -45977.8 q^{71} +29913.7 q^{72} -62039.1 q^{73} -2135.62 q^{74} +4165.84 q^{75} +3745.57 q^{76} +1558.50 q^{77} -8543.46 q^{78} -57486.6 q^{79} -10066.5 q^{80} +28635.6 q^{81} +26697.2 q^{82} -90511.7 q^{83} +2148.01 q^{84} -6666.53 q^{85} +7300.03 q^{86} +19654.8 q^{87} +18227.8 q^{88} -127861. q^{89} +13115.4 q^{90} +6249.03 q^{91} +80392.8 q^{92} +14303.0 q^{93} +26348.9 q^{94} +3742.54 q^{95} +39221.4 q^{96} +132338. q^{97} -43964.5 q^{98} +24027.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 5 q^{2} + 61 q^{4} - 100 q^{5} - 157 q^{6} - 90 q^{7} - 135 q^{8} + 22 q^{9} + 125 q^{10} - 484 q^{11} - 795 q^{12} + 820 q^{13} - 1687 q^{14} - 2671 q^{16} - 3800 q^{17} - 1610 q^{18} - 3394 q^{19}+ \cdots - 2662 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.64192 0.467030 0.233515 − 0.972353i \(-0.424977\pi\)
0.233515 + 0.972353i \(0.424977\pi\)
\(3\) 6.66534 0.427582 0.213791 − 0.976879i \(-0.431419\pi\)
0.213791 + 0.976879i \(0.431419\pi\)
\(4\) −25.0202 −0.781883
\(5\) −25.0000 −0.447214
\(6\) 17.6093 0.199694
\(7\) −12.8802 −0.0993520 −0.0496760 − 0.998765i \(-0.515819\pi\)
−0.0496760 + 0.998765i \(0.515819\pi\)
\(8\) −150.643 −0.832193
\(9\) −198.573 −0.817174
\(10\) −66.0481 −0.208862
\(11\) −121.000 −0.301511
\(12\) −166.769 −0.334319
\(13\) −485.167 −0.796219 −0.398110 − 0.917338i \(-0.630334\pi\)
−0.398110 + 0.917338i \(0.630334\pi\)
\(14\) −34.0284 −0.0464004
\(15\) −166.634 −0.191220
\(16\) 402.660 0.393223
\(17\) 266.661 0.223788 0.111894 − 0.993720i \(-0.464308\pi\)
0.111894 + 0.993720i \(0.464308\pi\)
\(18\) −524.615 −0.381645
\(19\) −149.702 −0.0951356 −0.0475678 − 0.998868i \(-0.515147\pi\)
−0.0475678 + 0.998868i \(0.515147\pi\)
\(20\) 625.506 0.349669
\(21\) −85.8507 −0.0424811
\(22\) −319.673 −0.140815
\(23\) −3213.11 −1.26650 −0.633251 − 0.773946i \(-0.718280\pi\)
−0.633251 + 0.773946i \(0.718280\pi\)
\(24\) −1004.09 −0.355831
\(25\) 625.000 0.200000
\(26\) −1281.77 −0.371859
\(27\) −2943.24 −0.776991
\(28\) 322.265 0.0776816
\(29\) 2948.81 0.651106 0.325553 − 0.945524i \(-0.394450\pi\)
0.325553 + 0.945524i \(0.394450\pi\)
\(30\) −440.233 −0.0893058
\(31\) 2145.87 0.401051 0.200525 − 0.979689i \(-0.435735\pi\)
0.200525 + 0.979689i \(0.435735\pi\)
\(32\) 5884.38 1.01584
\(33\) −806.507 −0.128921
\(34\) 704.498 0.104516
\(35\) 322.004 0.0444315
\(36\) 4968.35 0.638934
\(37\) −808.357 −0.0970731 −0.0485366 − 0.998821i \(-0.515456\pi\)
−0.0485366 + 0.998821i \(0.515456\pi\)
\(38\) −395.500 −0.0444312
\(39\) −3233.80 −0.340449
\(40\) 3766.08 0.372168
\(41\) 10105.2 0.938829 0.469414 − 0.882978i \(-0.344465\pi\)
0.469414 + 0.882978i \(0.344465\pi\)
\(42\) −226.811 −0.0198400
\(43\) 2763.15 0.227894 0.113947 − 0.993487i \(-0.463651\pi\)
0.113947 + 0.993487i \(0.463651\pi\)
\(44\) 3027.45 0.235746
\(45\) 4964.33 0.365451
\(46\) −8488.79 −0.591495
\(47\) 9973.36 0.658562 0.329281 − 0.944232i \(-0.393194\pi\)
0.329281 + 0.944232i \(0.393194\pi\)
\(48\) 2683.87 0.168135
\(49\) −16641.1 −0.990129
\(50\) 1651.20 0.0934061
\(51\) 1777.39 0.0956878
\(52\) 12139.0 0.622550
\(53\) 7126.92 0.348508 0.174254 − 0.984701i \(-0.444249\pi\)
0.174254 + 0.984701i \(0.444249\pi\)
\(54\) −7775.81 −0.362878
\(55\) 3025.00 0.134840
\(56\) 1940.31 0.0826800
\(57\) −997.814 −0.0406783
\(58\) 7790.52 0.304086
\(59\) −33337.2 −1.24681 −0.623403 − 0.781901i \(-0.714250\pi\)
−0.623403 + 0.781901i \(0.714250\pi\)
\(60\) 4169.21 0.149512
\(61\) −11871.1 −0.408476 −0.204238 − 0.978921i \(-0.565472\pi\)
−0.204238 + 0.978921i \(0.565472\pi\)
\(62\) 5669.22 0.187303
\(63\) 2557.66 0.0811878
\(64\) 2660.94 0.0812053
\(65\) 12129.2 0.356080
\(66\) −2130.73 −0.0602099
\(67\) 4500.58 0.122485 0.0612423 − 0.998123i \(-0.480494\pi\)
0.0612423 + 0.998123i \(0.480494\pi\)
\(68\) −6671.93 −0.174976
\(69\) −21416.5 −0.541533
\(70\) 850.710 0.0207509
\(71\) −45977.8 −1.08244 −0.541218 − 0.840882i \(-0.682037\pi\)
−0.541218 + 0.840882i \(0.682037\pi\)
\(72\) 29913.7 0.680046
\(73\) −62039.1 −1.36257 −0.681284 − 0.732019i \(-0.738578\pi\)
−0.681284 + 0.732019i \(0.738578\pi\)
\(74\) −2135.62 −0.0453361
\(75\) 4165.84 0.0855164
\(76\) 3745.57 0.0743848
\(77\) 1558.50 0.0299557
\(78\) −8543.46 −0.159000
\(79\) −57486.6 −1.03633 −0.518166 − 0.855280i \(-0.673385\pi\)
−0.518166 + 0.855280i \(0.673385\pi\)
\(80\) −10066.5 −0.175855
\(81\) 28635.6 0.484946
\(82\) 26697.2 0.438461
\(83\) −90511.7 −1.44215 −0.721074 − 0.692858i \(-0.756351\pi\)
−0.721074 + 0.692858i \(0.756351\pi\)
\(84\) 2148.01 0.0332152
\(85\) −6666.53 −0.100081
\(86\) 7300.03 0.106434
\(87\) 19654.8 0.278401
\(88\) 18227.8 0.250916
\(89\) −127861. −1.71105 −0.855524 − 0.517764i \(-0.826765\pi\)
−0.855524 + 0.517764i \(0.826765\pi\)
\(90\) 13115.4 0.170677
\(91\) 6249.03 0.0791059
\(92\) 80392.8 0.990256
\(93\) 14303.0 0.171482
\(94\) 26348.9 0.307569
\(95\) 3742.54 0.0425459
\(96\) 39221.4 0.434355
\(97\) 132338. 1.42809 0.714046 − 0.700099i \(-0.246861\pi\)
0.714046 + 0.700099i \(0.246861\pi\)
\(98\) −43964.5 −0.462420
\(99\) 24027.4 0.246387
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.1.3 ✓ 4
3.2 odd 2 495.6.a.g.1.2 4
4.3 odd 2 880.6.a.n.1.2 4
5.2 odd 4 275.6.b.d.199.5 8
5.3 odd 4 275.6.b.d.199.4 8
5.4 even 2 275.6.a.d.1.2 4
11.10 odd 2 605.6.a.c.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.6.a.b.1.3 ✓ 4 1.1 even 1 trivial
275.6.a.d.1.2 4 5.4 even 2
275.6.b.d.199.4 8 5.3 odd 4
275.6.b.d.199.5 8 5.2 odd 4
495.6.a.g.1.2 4 3.2 odd 2
605.6.a.c.1.2 4 11.10 odd 2
880.6.a.n.1.2 4 4.3 odd 2