Properties

Label 55.6.a.b.1.4
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 1.4
Root \(-3.95665\) 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)
 
\(f(q)\) \(=\) \(q+7.82466 q^{2} -16.3044 q^{3} +29.2253 q^{4} -25.0000 q^{5} -127.576 q^{6} -125.436 q^{7} -21.7111 q^{8} +22.8321 q^{9} -195.616 q^{10} -121.000 q^{11} -476.500 q^{12} +532.300 q^{13} -981.491 q^{14} +407.609 q^{15} -1105.09 q^{16} -1373.09 q^{17} +178.654 q^{18} -554.639 q^{19} -730.632 q^{20} +2045.15 q^{21} -946.784 q^{22} +4250.72 q^{23} +353.986 q^{24} +625.000 q^{25} +4165.06 q^{26} +3589.70 q^{27} -3665.89 q^{28} -6973.40 q^{29} +3189.40 q^{30} +3130.03 q^{31} -7952.21 q^{32} +1972.83 q^{33} -10744.0 q^{34} +3135.89 q^{35} +667.276 q^{36} +1384.70 q^{37} -4339.86 q^{38} -8678.81 q^{39} +542.778 q^{40} +679.385 q^{41} +16002.6 q^{42} -1721.06 q^{43} -3536.26 q^{44} -570.803 q^{45} +33260.4 q^{46} -15143.3 q^{47} +18017.8 q^{48} -1072.91 q^{49} +4890.41 q^{50} +22387.4 q^{51} +15556.6 q^{52} -9544.44 q^{53} +28088.1 q^{54} +3025.00 q^{55} +2723.35 q^{56} +9043.04 q^{57} -54564.5 q^{58} +27582.7 q^{59} +11912.5 q^{60} -40527.5 q^{61} +24491.5 q^{62} -2863.96 q^{63} -26860.4 q^{64} -13307.5 q^{65} +15436.7 q^{66} -58726.5 q^{67} -40129.0 q^{68} -69305.2 q^{69} +24537.3 q^{70} -42527.8 q^{71} -495.712 q^{72} +23753.0 q^{73} +10834.8 q^{74} -10190.2 q^{75} -16209.5 q^{76} +15177.7 q^{77} -67908.7 q^{78} -78690.1 q^{79} +27627.3 q^{80} -64075.9 q^{81} +5315.96 q^{82} +52252.1 q^{83} +59770.0 q^{84} +34327.3 q^{85} -13466.7 q^{86} +113697. q^{87} +2627.05 q^{88} +8156.09 q^{89} -4466.34 q^{90} -66769.3 q^{91} +124228. q^{92} -51033.2 q^{93} -118491. q^{94} +13866.0 q^{95} +129656. q^{96} +79010.1 q^{97} -8395.17 q^{98} -2762.69 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 5 q^{2} + 61 q^{4} - 100 q^{5} - 157 q^{6} - 90 q^{7} - 135 q^{8} + 22 q^{9} + 125 q^{10} - 484 q^{11} - 795 q^{12} + 820 q^{13} - 1687 q^{14} - 2671 q^{16} - 3800 q^{17} - 1610 q^{18} - 3394 q^{19}+ \cdots - 2662 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 7.82466 1.38322 0.691609 − 0.722272i \(-0.256902\pi\)
0.691609 + 0.722272i \(0.256902\pi\)
\(3\) −16.3044 −1.04593 −0.522963 − 0.852356i \(-0.675173\pi\)
−0.522963 + 0.852356i \(0.675173\pi\)
\(4\) 29.2253 0.913290
\(5\) −25.0000 −0.447214
\(6\) −127.576 −1.44674
\(7\) −125.436 −0.967555 −0.483778 − 0.875191i \(-0.660736\pi\)
−0.483778 + 0.875191i \(0.660736\pi\)
\(8\) −21.7111 −0.119938
\(9\) 22.8321 0.0939594
\(10\) −195.616 −0.618594
\(11\) −121.000 −0.301511
\(12\) −476.500 −0.955233
\(13\) 532.300 0.873570 0.436785 − 0.899566i \(-0.356117\pi\)
0.436785 + 0.899566i \(0.356117\pi\)
\(14\) −981.491 −1.33834
\(15\) 407.609 0.467752
\(16\) −1105.09 −1.07919
\(17\) −1373.09 −1.15233 −0.576166 − 0.817333i \(-0.695452\pi\)
−0.576166 + 0.817333i \(0.695452\pi\)
\(18\) 178.654 0.129966
\(19\) −554.639 −0.352474 −0.176237 − 0.984348i \(-0.556392\pi\)
−0.176237 + 0.984348i \(0.556392\pi\)
\(20\) −730.632 −0.408436
\(21\) 2045.15 1.01199
\(22\) −946.784 −0.417056
\(23\) 4250.72 1.67549 0.837746 − 0.546060i \(-0.183873\pi\)
0.837746 + 0.546060i \(0.183873\pi\)
\(24\) 353.986 0.125446
\(25\) 625.000 0.200000
\(26\) 4165.06 1.20834
\(27\) 3589.70 0.947651
\(28\) −3665.89 −0.883659
\(29\) −6973.40 −1.53975 −0.769874 − 0.638196i \(-0.779681\pi\)
−0.769874 + 0.638196i \(0.779681\pi\)
\(30\) 3189.40 0.647003
\(31\) 3130.03 0.584985 0.292493 − 0.956268i \(-0.405515\pi\)
0.292493 + 0.956268i \(0.405515\pi\)
\(32\) −7952.21 −1.37282
\(33\) 1972.83 0.315358
\(34\) −10744.0 −1.59392
\(35\) 3135.89 0.432704
\(36\) 667.276 0.0858122
\(37\) 1384.70 0.166284 0.0831422 − 0.996538i \(-0.473504\pi\)
0.0831422 + 0.996538i \(0.473504\pi\)
\(38\) −4339.86 −0.487548
\(39\) −8678.81 −0.913689
\(40\) 542.778 0.0536380
\(41\) 679.385 0.0631185 0.0315592 − 0.999502i \(-0.489953\pi\)
0.0315592 + 0.999502i \(0.489953\pi\)
\(42\) 16002.6 1.39980
\(43\) −1721.06 −0.141946 −0.0709732 − 0.997478i \(-0.522610\pi\)
−0.0709732 + 0.997478i \(0.522610\pi\)
\(44\) −3536.26 −0.275367
\(45\) −570.803 −0.0420199
\(46\) 33260.4 2.31757
\(47\) −15143.3 −0.999945 −0.499972 − 0.866041i \(-0.666656\pi\)
−0.499972 + 0.866041i \(0.666656\pi\)
\(48\) 18017.8 1.12875
\(49\) −1072.91 −0.0638372
\(50\) 4890.41 0.276643
\(51\) 22387.4 1.20525
\(52\) 15556.6 0.797823
\(53\) −9544.44 −0.466724 −0.233362 − 0.972390i \(-0.574973\pi\)
−0.233362 + 0.972390i \(0.574973\pi\)
\(54\) 28088.1 1.31081
\(55\) 3025.00 0.134840
\(56\) 2723.35 0.116047
\(57\) 9043.04 0.368661
\(58\) −54564.5 −2.12981
\(59\) 27582.7 1.03159 0.515794 − 0.856713i \(-0.327497\pi\)
0.515794 + 0.856713i \(0.327497\pi\)
\(60\) 11912.5 0.427193
\(61\) −40527.5 −1.39452 −0.697261 − 0.716817i \(-0.745598\pi\)
−0.697261 + 0.716817i \(0.745598\pi\)
\(62\) 24491.5 0.809162
\(63\) −2863.96 −0.0909109
\(64\) −26860.4 −0.819714
\(65\) −13307.5 −0.390672
\(66\) 15436.7 0.436209
\(67\) −58726.5 −1.59826 −0.799130 − 0.601159i \(-0.794706\pi\)
−0.799130 + 0.601159i \(0.794706\pi\)
\(68\) −40129.0 −1.05241
\(69\) −69305.2 −1.75244
\(70\) 24537.3 0.598523
\(71\) −42527.8 −1.00121 −0.500607 − 0.865675i \(-0.666890\pi\)
−0.500607 + 0.865675i \(0.666890\pi\)
\(72\) −495.712 −0.0112693
\(73\) 23753.0 0.521690 0.260845 − 0.965381i \(-0.415999\pi\)
0.260845 + 0.965381i \(0.415999\pi\)
\(74\) 10834.8 0.230007
\(75\) −10190.2 −0.209185
\(76\) −16209.5 −0.321911
\(77\) 15177.7 0.291729
\(78\) −67908.7 −1.26383
\(79\) −78690.1 −1.41857 −0.709287 − 0.704919i \(-0.750983\pi\)
−0.709287 + 0.704919i \(0.750983\pi\)
\(80\) 27627.3 0.482629
\(81\) −64075.9 −1.08513
\(82\) 5315.96 0.0873066
\(83\) 52252.1 0.832546 0.416273 − 0.909240i \(-0.363336\pi\)
0.416273 + 0.909240i \(0.363336\pi\)
\(84\) 59770.0 0.924241
\(85\) 34327.3 0.515338
\(86\) −13466.7 −0.196343
\(87\) 113697. 1.61046
\(88\) 2627.05 0.0361627
\(89\) 8156.09 0.109146 0.0545729 − 0.998510i \(-0.482620\pi\)
0.0545729 + 0.998510i \(0.482620\pi\)
\(90\) −4466.34 −0.0581227
\(91\) −66769.3 −0.845227
\(92\) 124228. 1.53021
\(93\) −51033.2 −0.611851
\(94\) −118491. −1.38314
\(95\) 13866.0 0.157631
\(96\) 129656. 1.43586
\(97\) 79010.1 0.852615 0.426308 − 0.904578i \(-0.359814\pi\)
0.426308 + 0.904578i \(0.359814\pi\)
\(98\) −8395.17 −0.0883007
\(99\) −2762.69 −0.0283298
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.1.4 ✓ 4
3.2 odd 2 495.6.a.g.1.1 4
4.3 odd 2 880.6.a.n.1.4 4
5.2 odd 4 275.6.b.d.199.7 8
5.3 odd 4 275.6.b.d.199.2 8
5.4 even 2 275.6.a.d.1.1 4
11.10 odd 2 605.6.a.c.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.6.a.b.1.4 ✓ 4 1.1 even 1 trivial
275.6.a.d.1.1 4 5.4 even 2
275.6.b.d.199.2 8 5.3 odd 4
275.6.b.d.199.7 8 5.2 odd 4
495.6.a.g.1.1 4 3.2 odd 2
605.6.a.c.1.1 4 11.10 odd 2
880.6.a.n.1.4 4 4.3 odd 2