Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7406,2,Mod(1,7406)] 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("7406.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7406, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7406 = 2 \cdot 7 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7406.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,3,-1,3,4,-1,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(59.1372077370\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1509.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 7x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.551929\) of defining polynomial
Character \(\chi\) \(=\) 7406.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -0.551929 q^{3} +1.00000 q^{4} +1.55193 q^{5} -0.551929 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.69537 q^{9} +1.55193 q^{10} +3.69537 q^{11} -0.551929 q^{12} -4.69537 q^{13} -1.00000 q^{14} -0.856556 q^{15} +1.00000 q^{16} +2.24730 q^{17} -2.69537 q^{18} +3.14344 q^{19} +1.55193 q^{20} +0.551929 q^{21} +3.69537 q^{22} -0.551929 q^{24} -2.59152 q^{25} -4.69537 q^{26} +3.14344 q^{27} -1.00000 q^{28} -8.14344 q^{29} -0.856556 q^{30} -3.14344 q^{31} +1.00000 q^{32} -2.03959 q^{33} +2.24730 q^{34} -1.55193 q^{35} -2.69537 q^{36} -10.2473 q^{37} +3.14344 q^{38} +2.59152 q^{39} +1.55193 q^{40} +3.79923 q^{41} +0.551929 q^{42} -7.14344 q^{43} +3.69537 q^{44} -4.18303 q^{45} -1.65579 q^{47} -0.551929 q^{48} +1.00000 q^{49} -2.59152 q^{50} -1.24035 q^{51} -4.69537 q^{52} -8.35116 q^{53} +3.14344 q^{54} +5.73496 q^{55} -1.00000 q^{56} -1.73496 q^{57} -8.14344 q^{58} +7.39075 q^{59} -0.856556 q^{60} -10.1434 q^{61} -3.14344 q^{62} +2.69537 q^{63} +1.00000 q^{64} -7.28689 q^{65} -2.03959 q^{66} -2.20772 q^{67} +2.24730 q^{68} -1.55193 q^{70} +12.7992 q^{71} -2.69537 q^{72} +6.39075 q^{73} -10.2473 q^{74} +1.43033 q^{75} +3.14344 q^{76} -3.69537 q^{77} +2.59152 q^{78} +7.63805 q^{79} +1.55193 q^{80} +6.35116 q^{81} +3.79923 q^{82} -0.799233 q^{83} +0.551929 q^{84} +3.48766 q^{85} -7.14344 q^{86} +4.49461 q^{87} +3.69537 q^{88} -15.0000 q^{89} -4.18303 q^{90} +4.69537 q^{91} +1.73496 q^{93} -1.65579 q^{94} +4.87840 q^{95} -0.551929 q^{96} +11.7023 q^{97} +1.00000 q^{98} -9.96041 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - q^{3} + 3 q^{4} + 4 q^{5} - q^{6} - 3 q^{7} + 3 q^{8} + 6 q^{9} + 4 q^{10} - 3 q^{11} - q^{12} - 3 q^{14} - 16 q^{15} + 3 q^{16} - 8 q^{17} + 6 q^{18} - 4 q^{19} + 4 q^{20} + q^{21}+ \cdots - 42 q^{99}+O(q^{100}) \) 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
\(3\) −0.551929 −0.318657 −0.159328 − 0.987226i \(-0.550933\pi\)
−0.159328 + 0.987226i \(0.550933\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.55193 0.694044 0.347022 − 0.937857i \(-0.387193\pi\)
0.347022 + 0.937857i \(0.387193\pi\)
\(6\) −0.551929 −0.225324
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) −2.69537 −0.898458
\(10\) 1.55193 0.490763
\(11\) 3.69537 1.11420 0.557099 − 0.830446i \(-0.311915\pi\)
0.557099 + 0.830446i \(0.311915\pi\)
\(12\) −0.551929 −0.159328
\(13\) −4.69537 −1.30226 −0.651131 − 0.758965i \(-0.725705\pi\)
−0.651131 + 0.758965i \(0.725705\pi\)
\(14\) −1.00000 −0.267261
\(15\) −0.856556 −0.221162
\(16\) 1.00000 0.250000
\(17\) 2.24730 0.545051 0.272526 − 0.962149i \(-0.412141\pi\)
0.272526 + 0.962149i \(0.412141\pi\)
\(18\) −2.69537 −0.635306
\(19\) 3.14344 0.721156 0.360578 − 0.932729i \(-0.382580\pi\)
0.360578 + 0.932729i \(0.382580\pi\)
\(20\) 1.55193 0.347022
\(21\) 0.551929 0.120441
\(22\) 3.69537 0.787856
\(23\) 0 0
\(24\) −0.551929 −0.112662
\(25\) −2.59152 −0.518303
\(26\) −4.69537 −0.920839
\(27\) 3.14344 0.604956
\(28\) −1.00000 −0.188982
\(29\) −8.14344 −1.51220 −0.756100 − 0.654456i \(-0.772898\pi\)
−0.756100 + 0.654456i \(0.772898\pi\)
\(30\) −0.856556 −0.156385
\(31\) −3.14344 −0.564579 −0.282290 − 0.959329i \(-0.591094\pi\)
−0.282290 + 0.959329i \(0.591094\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.03959 −0.355046
\(34\) 2.24730 0.385409
\(35\) −1.55193 −0.262324
\(36\) −2.69537 −0.449229
\(37\) −10.2473 −1.68465 −0.842323 − 0.538973i \(-0.818813\pi\)
−0.842323 + 0.538973i \(0.818813\pi\)
\(38\) 3.14344 0.509934
\(39\) 2.59152 0.414975
\(40\) 1.55193 0.245382
\(41\) 3.79923 0.593340 0.296670 − 0.954980i \(-0.404124\pi\)
0.296670 + 0.954980i \(0.404124\pi\)
\(42\) 0.551929 0.0851646
\(43\) −7.14344 −1.08937 −0.544683 − 0.838642i \(-0.683350\pi\)
−0.544683 + 0.838642i \(0.683350\pi\)
\(44\) 3.69537 0.557099
\(45\) −4.18303 −0.623569
\(46\) 0 0
\(47\) −1.65579 −0.241522 −0.120761 − 0.992682i \(-0.538533\pi\)
−0.120761 + 0.992682i \(0.538533\pi\)
\(48\) −0.551929 −0.0796642
\(49\) 1.00000 0.142857
\(50\) −2.59152 −0.366496
\(51\) −1.24035 −0.173684
\(52\) −4.69537 −0.651131
\(53\) −8.35116 −1.14712 −0.573560 − 0.819163i \(-0.694438\pi\)
−0.573560 + 0.819163i \(0.694438\pi\)
\(54\) 3.14344 0.427769
\(55\) 5.73496 0.773302
\(56\) −1.00000 −0.133631
\(57\) −1.73496 −0.229801
\(58\) −8.14344 −1.06929
\(59\) 7.39075 0.962193 0.481097 − 0.876668i \(-0.340239\pi\)
0.481097 + 0.876668i \(0.340239\pi\)
\(60\) −0.856556 −0.110581
\(61\) −10.1434 −1.29873 −0.649367 − 0.760475i \(-0.724966\pi\)
−0.649367 + 0.760475i \(0.724966\pi\)
\(62\) −3.14344 −0.399218
\(63\) 2.69537 0.339585
\(64\) 1.00000 0.125000
\(65\) −7.28689 −0.903827
\(66\) −2.03959 −0.251056
\(67\) −2.20772 −0.269716 −0.134858 − 0.990865i \(-0.543058\pi\)
−0.134858 + 0.990865i \(0.543058\pi\)
\(68\) 2.24730 0.272526
\(69\) 0 0
\(70\) −1.55193 −0.185491
\(71\) 12.7992 1.51899 0.759495 − 0.650513i \(-0.225446\pi\)
0.759495 + 0.650513i \(0.225446\pi\)
\(72\) −2.69537 −0.317653
\(73\) 6.39075 0.747980 0.373990 − 0.927433i \(-0.377989\pi\)
0.373990 + 0.927433i \(0.377989\pi\)
\(74\) −10.2473 −1.19122
\(75\) 1.43033 0.165161
\(76\) 3.14344 0.360578
\(77\) −3.69537 −0.421127
\(78\) 2.59152 0.293431
\(79\) 7.63805 0.859348 0.429674 − 0.902984i \(-0.358629\pi\)
0.429674 + 0.902984i \(0.358629\pi\)
\(80\) 1.55193 0.173511
\(81\) 6.35116 0.705685
\(82\) 3.79923 0.419555
\(83\) −0.799233 −0.0877272 −0.0438636 − 0.999038i \(-0.513967\pi\)
−0.0438636 + 0.999038i \(0.513967\pi\)
\(84\) 0.551929 0.0602204
\(85\) 3.48766 0.378289
\(86\) −7.14344 −0.770297
\(87\) 4.49461 0.481872
\(88\) 3.69537 0.393928
\(89\) −15.0000 −1.59000 −0.794998 − 0.606612i \(-0.792528\pi\)
−0.794998 + 0.606612i \(0.792528\pi\)
\(90\) −4.18303 −0.440930
\(91\) 4.69537 0.492209
\(92\) 0 0
\(93\) 1.73496 0.179907
\(94\) −1.65579 −0.170782
\(95\) 4.87840 0.500514
\(96\) −0.551929 −0.0563311
\(97\) 11.7023 1.18819 0.594096 − 0.804395i \(-0.297510\pi\)
0.594096 + 0.804395i \(0.297510\pi\)
\(98\) 1.00000 0.101015
\(99\) −9.96041 −1.00106
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 7406.2.a.w.1.2 yes 3
23.22 odd 2 7406.2.a.v.1.2 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7406.2.a.v.1.2 ✓ 3 23.22 odd 2
7406.2.a.w.1.2 yes 3 1.1 even 1 trivial