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Properties

Label 2366.2.a.be.1.3
Level $2366$
Weight $2$
Character 2366.1
Self dual yes
Analytic conductor $18.893$
Analytic rank $1$
Dimension $6$
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] = [2366,2,Mod(1,2366)] 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("2366.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2366, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2366.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-6,1,6,-4] 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: yes
Analytic conductor: \(18.8926051182\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.6052921.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 11x^{4} + 7x^{3} + 33x^{2} - 9x - 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.05140\) of defining polynomial
Character \(\chi\) \(=\) 2366.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.05140 q^{3} +1.00000 q^{4} -2.26985 q^{5} +1.05140 q^{6} -1.00000 q^{7} -1.00000 q^{8} -1.89456 q^{9} +2.26985 q^{10} -0.977125 q^{11} -1.05140 q^{12} +1.00000 q^{14} +2.38653 q^{15} +1.00000 q^{16} +0.963254 q^{17} +1.89456 q^{18} +5.10479 q^{19} -2.26985 q^{20} +1.05140 q^{21} +0.977125 q^{22} +8.11859 q^{23} +1.05140 q^{24} +0.152241 q^{25} +5.14614 q^{27} -1.00000 q^{28} +9.77952 q^{29} -2.38653 q^{30} -5.15994 q^{31} -1.00000 q^{32} +1.02735 q^{33} -0.963254 q^{34} +2.26985 q^{35} -1.89456 q^{36} -0.794311 q^{37} -5.10479 q^{38} +2.26985 q^{40} +1.49967 q^{41} -1.05140 q^{42} -7.93286 q^{43} -0.977125 q^{44} +4.30037 q^{45} -8.11859 q^{46} -5.04879 q^{47} -1.05140 q^{48} +1.00000 q^{49} -0.152241 q^{50} -1.01277 q^{51} +3.68079 q^{53} -5.14614 q^{54} +2.21793 q^{55} +1.00000 q^{56} -5.36718 q^{57} -9.77952 q^{58} -12.8260 q^{59} +2.38653 q^{60} -3.93268 q^{61} +5.15994 q^{62} +1.89456 q^{63} +1.00000 q^{64} -1.02735 q^{66} -0.0496112 q^{67} +0.963254 q^{68} -8.53589 q^{69} -2.26985 q^{70} +3.71940 q^{71} +1.89456 q^{72} +8.87873 q^{73} +0.794311 q^{74} -0.160066 q^{75} +5.10479 q^{76} +0.977125 q^{77} -6.98214 q^{79} -2.26985 q^{80} +0.273022 q^{81} -1.49967 q^{82} -6.24779 q^{83} +1.05140 q^{84} -2.18645 q^{85} +7.93286 q^{86} -10.2822 q^{87} +0.977125 q^{88} +5.92385 q^{89} -4.30037 q^{90} +8.11859 q^{92} +5.42516 q^{93} +5.04879 q^{94} -11.5871 q^{95} +1.05140 q^{96} -7.29137 q^{97} -1.00000 q^{98} +1.85122 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + q^{3} + 6 q^{4} - 4 q^{5} - q^{6} - 6 q^{7} - 6 q^{8} + 5 q^{9} + 4 q^{10} - 6 q^{11} + q^{12} + 6 q^{14} + 5 q^{15} + 6 q^{16} - 9 q^{17} - 5 q^{18} - 10 q^{19} - 4 q^{20} - q^{21}+ \cdots + 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.05140 −0.607026 −0.303513 − 0.952827i \(-0.598160\pi\)
−0.303513 + 0.952827i \(0.598160\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.26985 −1.01511 −0.507555 − 0.861619i \(-0.669451\pi\)
−0.507555 + 0.861619i \(0.669451\pi\)
\(6\) 1.05140 0.429232
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) −1.89456 −0.631519
\(10\) 2.26985 0.717791
\(11\) −0.977125 −0.294614 −0.147307 − 0.989091i \(-0.547061\pi\)
−0.147307 + 0.989091i \(0.547061\pi\)
\(12\) −1.05140 −0.303513
\(13\) 0 0
\(14\) 1.00000 0.267261
\(15\) 2.38653 0.616198
\(16\) 1.00000 0.250000
\(17\) 0.963254 0.233623 0.116812 − 0.993154i \(-0.462733\pi\)
0.116812 + 0.993154i \(0.462733\pi\)
\(18\) 1.89456 0.446552
\(19\) 5.10479 1.17112 0.585560 − 0.810629i \(-0.300875\pi\)
0.585560 + 0.810629i \(0.300875\pi\)
\(20\) −2.26985 −0.507555
\(21\) 1.05140 0.229434
\(22\) 0.977125 0.208324
\(23\) 8.11859 1.69284 0.846422 − 0.532513i \(-0.178752\pi\)
0.846422 + 0.532513i \(0.178752\pi\)
\(24\) 1.05140 0.214616
\(25\) 0.152241 0.0304482
\(26\) 0 0
\(27\) 5.14614 0.990375
\(28\) −1.00000 −0.188982
\(29\) 9.77952 1.81601 0.908005 − 0.418959i \(-0.137605\pi\)
0.908005 + 0.418959i \(0.137605\pi\)
\(30\) −2.38653 −0.435718
\(31\) −5.15994 −0.926752 −0.463376 − 0.886162i \(-0.653362\pi\)
−0.463376 + 0.886162i \(0.653362\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.02735 0.178839
\(34\) −0.963254 −0.165197
\(35\) 2.26985 0.383675
\(36\) −1.89456 −0.315760
\(37\) −0.794311 −0.130584 −0.0652920 − 0.997866i \(-0.520798\pi\)
−0.0652920 + 0.997866i \(0.520798\pi\)
\(38\) −5.10479 −0.828107
\(39\) 0 0
\(40\) 2.26985 0.358896
\(41\) 1.49967 0.234208 0.117104 − 0.993120i \(-0.462639\pi\)
0.117104 + 0.993120i \(0.462639\pi\)
\(42\) −1.05140 −0.162235
\(43\) −7.93286 −1.20975 −0.604875 − 0.796321i \(-0.706777\pi\)
−0.604875 + 0.796321i \(0.706777\pi\)
\(44\) −0.977125 −0.147307
\(45\) 4.30037 0.641061
\(46\) −8.11859 −1.19702
\(47\) −5.04879 −0.736442 −0.368221 − 0.929738i \(-0.620033\pi\)
−0.368221 + 0.929738i \(0.620033\pi\)
\(48\) −1.05140 −0.151757
\(49\) 1.00000 0.142857
\(50\) −0.152241 −0.0215301
\(51\) −1.01277 −0.141816
\(52\) 0 0
\(53\) 3.68079 0.505596 0.252798 − 0.967519i \(-0.418649\pi\)
0.252798 + 0.967519i \(0.418649\pi\)
\(54\) −5.14614 −0.700301
\(55\) 2.21793 0.299066
\(56\) 1.00000 0.133631
\(57\) −5.36718 −0.710900
\(58\) −9.77952 −1.28411
\(59\) −12.8260 −1.66980 −0.834898 − 0.550404i \(-0.814474\pi\)
−0.834898 + 0.550404i \(0.814474\pi\)
\(60\) 2.38653 0.308099
\(61\) −3.93268 −0.503528 −0.251764 − 0.967789i \(-0.581011\pi\)
−0.251764 + 0.967789i \(0.581011\pi\)
\(62\) 5.15994 0.655313
\(63\) 1.89456 0.238692
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.02735 −0.126458
\(67\) −0.0496112 −0.00606097 −0.00303049 − 0.999995i \(-0.500965\pi\)
−0.00303049 + 0.999995i \(0.500965\pi\)
\(68\) 0.963254 0.116812
\(69\) −8.53589 −1.02760
\(70\) −2.26985 −0.271300
\(71\) 3.71940 0.441412 0.220706 − 0.975340i \(-0.429164\pi\)
0.220706 + 0.975340i \(0.429164\pi\)
\(72\) 1.89456 0.223276
\(73\) 8.87873 1.03918 0.519588 − 0.854417i \(-0.326085\pi\)
0.519588 + 0.854417i \(0.326085\pi\)
\(74\) 0.794311 0.0923368
\(75\) −0.160066 −0.0184829
\(76\) 5.10479 0.585560
\(77\) 0.977125 0.111354
\(78\) 0 0
\(79\) −6.98214 −0.785552 −0.392776 − 0.919634i \(-0.628485\pi\)
−0.392776 + 0.919634i \(0.628485\pi\)
\(80\) −2.26985 −0.253777
\(81\) 0.273022 0.0303357
\(82\) −1.49967 −0.165610
\(83\) −6.24779 −0.685785 −0.342892 − 0.939375i \(-0.611407\pi\)
−0.342892 + 0.939375i \(0.611407\pi\)
\(84\) 1.05140 0.114717
\(85\) −2.18645 −0.237153
\(86\) 7.93286 0.855422
\(87\) −10.2822 −1.10237
\(88\) 0.977125 0.104162
\(89\) 5.92385 0.627927 0.313964 − 0.949435i \(-0.398343\pi\)
0.313964 + 0.949435i \(0.398343\pi\)
\(90\) −4.30037 −0.453299
\(91\) 0 0
\(92\) 8.11859 0.846422
\(93\) 5.42516 0.562563
\(94\) 5.04879 0.520743
\(95\) −11.5871 −1.18882
\(96\) 1.05140 0.107308
\(97\) −7.29137 −0.740326 −0.370163 − 0.928967i \(-0.620698\pi\)
−0.370163 + 0.928967i \(0.620698\pi\)
\(98\) −1.00000 −0.101015
\(99\) 1.85122 0.186055
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 2366.2.a.be.1.3 ✓ 6
13.5 odd 4 2366.2.d.q.337.9 12
13.8 odd 4 2366.2.d.q.337.3 12
13.12 even 2 2366.2.a.bg.1.3 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2366.2.a.be.1.3 ✓ 6 1.1 even 1 trivial
2366.2.a.bg.1.3 yes 6 13.12 even 2
2366.2.d.q.337.3 12 13.8 odd 4
2366.2.d.q.337.9 12 13.5 odd 4