Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2738,2,Mod(1,2738)] 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("2738.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2738, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2738 = 2 \cdot 37^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2738.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,-3,0,3,-1,0,0] 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: \(21.8630400734\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.404.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 74)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.65544\) of defining polynomial
Character \(\chi\) \(=\) 2738.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -3.05137 q^{3} +1.00000 q^{4} -1.25951 q^{5} +3.05137 q^{6} +3.05137 q^{7} -1.00000 q^{8} +6.31088 q^{9} +1.25951 q^{10} +5.31088 q^{11} -3.05137 q^{12} -2.00000 q^{13} -3.05137 q^{14} +3.84324 q^{15} +1.00000 q^{16} -5.10275 q^{17} -6.31088 q^{18} -4.79186 q^{19} -1.25951 q^{20} -9.31088 q^{21} -5.31088 q^{22} -1.74049 q^{23} +3.05137 q^{24} -3.41363 q^{25} +2.00000 q^{26} -10.1027 q^{27} +3.05137 q^{28} +4.05137 q^{29} -3.84324 q^{30} +0.791864 q^{31} -1.00000 q^{32} -16.2055 q^{33} +5.10275 q^{34} -3.84324 q^{35} +6.31088 q^{36} +4.79186 q^{38} +6.10275 q^{39} +1.25951 q^{40} -0.208136 q^{41} +9.31088 q^{42} +4.36226 q^{43} +5.31088 q^{44} -7.94863 q^{45} +1.74049 q^{46} -3.20814 q^{47} -3.05137 q^{48} +2.31088 q^{49} +3.41363 q^{50} +15.5704 q^{51} -2.00000 q^{52} +11.3109 q^{53} +10.1027 q^{54} -6.68912 q^{55} -3.05137 q^{56} +14.6218 q^{57} -4.05137 q^{58} +12.7245 q^{59} +3.84324 q^{60} -6.05137 q^{61} -0.791864 q^{62} +19.2569 q^{63} +1.00000 q^{64} +2.51902 q^{65} +16.2055 q^{66} -9.31088 q^{67} -5.10275 q^{68} +5.31088 q^{69} +3.84324 q^{70} -7.05137 q^{71} -6.31088 q^{72} -3.31088 q^{73} +10.4163 q^{75} -4.79186 q^{76} +16.2055 q^{77} -6.10275 q^{78} +11.4136 q^{79} -1.25951 q^{80} +11.8946 q^{81} +0.208136 q^{82} -3.57040 q^{83} -9.31088 q^{84} +6.42697 q^{85} -4.36226 q^{86} -12.3623 q^{87} -5.31088 q^{88} -3.27284 q^{89} +7.94863 q^{90} -6.10275 q^{91} -1.74049 q^{92} -2.41627 q^{93} +3.20814 q^{94} +6.03540 q^{95} +3.05137 q^{96} -2.20814 q^{97} -2.31088 q^{98} +33.5164 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} - q^{5} - 3 q^{8} + 7 q^{9} + q^{10} + 4 q^{11} - 6 q^{13} - 4 q^{15} + 3 q^{16} + 3 q^{17} - 7 q^{18} - 8 q^{19} - q^{20} - 16 q^{21} - 4 q^{22} - 8 q^{23} + 20 q^{25} + 6 q^{26}+ \cdots + 52 q^{99}+O(q^{100}) \) 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
\(3\) −3.05137 −1.76171 −0.880856 − 0.473385i \(-0.843032\pi\)
−0.880856 + 0.473385i \(0.843032\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.25951 −0.563270 −0.281635 − 0.959522i \(-0.590877\pi\)
−0.281635 + 0.959522i \(0.590877\pi\)
\(6\) 3.05137 1.24572
\(7\) 3.05137 1.15331 0.576656 − 0.816987i \(-0.304357\pi\)
0.576656 + 0.816987i \(0.304357\pi\)
\(8\) −1.00000 −0.353553
\(9\) 6.31088 2.10363
\(10\) 1.25951 0.398292
\(11\) 5.31088 1.60129 0.800646 − 0.599138i \(-0.204490\pi\)
0.800646 + 0.599138i \(0.204490\pi\)
\(12\) −3.05137 −0.880856
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −3.05137 −0.815514
\(15\) 3.84324 0.992320
\(16\) 1.00000 0.250000
\(17\) −5.10275 −1.23760 −0.618799 − 0.785549i \(-0.712381\pi\)
−0.618799 + 0.785549i \(0.712381\pi\)
\(18\) −6.31088 −1.48749
\(19\) −4.79186 −1.09933 −0.549664 − 0.835386i \(-0.685244\pi\)
−0.549664 + 0.835386i \(0.685244\pi\)
\(20\) −1.25951 −0.281635
\(21\) −9.31088 −2.03180
\(22\) −5.31088 −1.13228
\(23\) −1.74049 −0.362917 −0.181459 − 0.983399i \(-0.558082\pi\)
−0.181459 + 0.983399i \(0.558082\pi\)
\(24\) 3.05137 0.622859
\(25\) −3.41363 −0.682727
\(26\) 2.00000 0.392232
\(27\) −10.1027 −1.94427
\(28\) 3.05137 0.576656
\(29\) 4.05137 0.752321 0.376161 − 0.926554i \(-0.377244\pi\)
0.376161 + 0.926554i \(0.377244\pi\)
\(30\) −3.84324 −0.701676
\(31\) 0.791864 0.142223 0.0711115 − 0.997468i \(-0.477345\pi\)
0.0711115 + 0.997468i \(0.477345\pi\)
\(32\) −1.00000 −0.176777
\(33\) −16.2055 −2.82101
\(34\) 5.10275 0.875114
\(35\) −3.84324 −0.649626
\(36\) 6.31088 1.05181
\(37\) 0 0
\(38\) 4.79186 0.777343
\(39\) 6.10275 0.977222
\(40\) 1.25951 0.199146
\(41\) −0.208136 −0.0325054 −0.0162527 − 0.999868i \(-0.505174\pi\)
−0.0162527 + 0.999868i \(0.505174\pi\)
\(42\) 9.31088 1.43670
\(43\) 4.36226 0.665238 0.332619 − 0.943061i \(-0.392068\pi\)
0.332619 + 0.943061i \(0.392068\pi\)
\(44\) 5.31088 0.800646
\(45\) −7.94863 −1.18491
\(46\) 1.74049 0.256621
\(47\) −3.20814 −0.467955 −0.233977 − 0.972242i \(-0.575174\pi\)
−0.233977 + 0.972242i \(0.575174\pi\)
\(48\) −3.05137 −0.440428
\(49\) 2.31088 0.330126
\(50\) 3.41363 0.482761
\(51\) 15.5704 2.18029
\(52\) −2.00000 −0.277350
\(53\) 11.3109 1.55367 0.776835 − 0.629705i \(-0.216824\pi\)
0.776835 + 0.629705i \(0.216824\pi\)
\(54\) 10.1027 1.37481
\(55\) −6.68912 −0.901960
\(56\) −3.05137 −0.407757
\(57\) 14.6218 1.93670
\(58\) −4.05137 −0.531971
\(59\) 12.7245 1.65659 0.828296 − 0.560291i \(-0.189311\pi\)
0.828296 + 0.560291i \(0.189311\pi\)
\(60\) 3.84324 0.496160
\(61\) −6.05137 −0.774799 −0.387400 − 0.921912i \(-0.626627\pi\)
−0.387400 + 0.921912i \(0.626627\pi\)
\(62\) −0.791864 −0.100567
\(63\) 19.2569 2.42614
\(64\) 1.00000 0.125000
\(65\) 2.51902 0.312446
\(66\) 16.2055 1.99476
\(67\) −9.31088 −1.13751 −0.568753 − 0.822509i \(-0.692574\pi\)
−0.568753 + 0.822509i \(0.692574\pi\)
\(68\) −5.10275 −0.618799
\(69\) 5.31088 0.639355
\(70\) 3.84324 0.459355
\(71\) −7.05137 −0.836844 −0.418422 − 0.908253i \(-0.637417\pi\)
−0.418422 + 0.908253i \(0.637417\pi\)
\(72\) −6.31088 −0.743745
\(73\) −3.31088 −0.387510 −0.193755 − 0.981050i \(-0.562067\pi\)
−0.193755 + 0.981050i \(0.562067\pi\)
\(74\) 0 0
\(75\) 10.4163 1.20277
\(76\) −4.79186 −0.549664
\(77\) 16.2055 1.84679
\(78\) −6.10275 −0.691000
\(79\) 11.4136 1.28413 0.642067 − 0.766649i \(-0.278077\pi\)
0.642067 + 0.766649i \(0.278077\pi\)
\(80\) −1.25951 −0.140818
\(81\) 11.8946 1.32162
\(82\) 0.208136 0.0229848
\(83\) −3.57040 −0.391902 −0.195951 − 0.980614i \(-0.562779\pi\)
−0.195951 + 0.980614i \(0.562779\pi\)
\(84\) −9.31088 −1.01590
\(85\) 6.42697 0.697102
\(86\) −4.36226 −0.470394
\(87\) −12.3623 −1.32537
\(88\) −5.31088 −0.566142
\(89\) −3.27284 −0.346921 −0.173460 − 0.984841i \(-0.555495\pi\)
−0.173460 + 0.984841i \(0.555495\pi\)
\(90\) 7.94863 0.837859
\(91\) −6.10275 −0.639742
\(92\) −1.74049 −0.181459
\(93\) −2.41627 −0.250556
\(94\) 3.20814 0.330894
\(95\) 6.03540 0.619219
\(96\) 3.05137 0.311430
\(97\) −2.20814 −0.224202 −0.112101 − 0.993697i \(-0.535758\pi\)
−0.112101 + 0.993697i \(0.535758\pi\)
\(98\) −2.31088 −0.233435
\(99\) 33.5164 3.36852
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 2738.2.a.n.1.1 3
37.11 even 6 74.2.c.c.47.3 ✓ 6
37.27 even 6 74.2.c.c.63.3 yes 6
37.36 even 2 2738.2.a.o.1.1 3
111.11 odd 6 666.2.f.j.343.2 6
111.101 odd 6 666.2.f.j.433.2 6
148.11 odd 6 592.2.i.e.417.1 6
148.27 odd 6 592.2.i.e.433.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.c.c.47.3 ✓ 6 37.11 even 6
74.2.c.c.63.3 yes 6 37.27 even 6
592.2.i.e.417.1 6 148.11 odd 6
592.2.i.e.433.1 6 148.27 odd 6
666.2.f.j.343.2 6 111.11 odd 6
666.2.f.j.433.2 6 111.101 odd 6
2738.2.a.n.1.1 3 1.1 even 1 trivial
2738.2.a.o.1.1 3 37.36 even 2