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 = [6,-6,0,6,6,0,0] 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: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{36})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 6x^{4} + 9x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.28558\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.87939 q^{3} +1.00000 q^{4} +4.26414 q^{5} +1.87939 q^{6} +3.61144 q^{7} -1.00000 q^{8} +0.532089 q^{9} -4.26414 q^{10} +2.11681 q^{11} -1.87939 q^{12} +2.88400 q^{13} -3.61144 q^{14} -8.01396 q^{15} +1.00000 q^{16} -0.0267197 q^{17} -0.532089 q^{18} +3.22847 q^{19} +4.26414 q^{20} -6.78728 q^{21} -2.11681 q^{22} -2.97244 q^{23} +1.87939 q^{24} +13.1829 q^{25} -2.88400 q^{26} +4.63816 q^{27} +3.61144 q^{28} -5.73300 q^{29} +8.01396 q^{30} +6.76932 q^{31} -1.00000 q^{32} -3.97829 q^{33} +0.0267197 q^{34} +15.3997 q^{35} +0.532089 q^{36} -3.22847 q^{38} -5.42015 q^{39} -4.26414 q^{40} -1.49152 q^{41} +6.78728 q^{42} +5.53737 q^{43} +2.11681 q^{44} +2.26890 q^{45} +2.97244 q^{46} +2.61309 q^{47} -1.87939 q^{48} +6.04247 q^{49} -13.1829 q^{50} +0.0502166 q^{51} +2.88400 q^{52} +1.90902 q^{53} -4.63816 q^{54} +9.02635 q^{55} -3.61144 q^{56} -6.06754 q^{57} +5.73300 q^{58} +5.62482 q^{59} -8.01396 q^{60} +5.74087 q^{61} -6.76932 q^{62} +1.92160 q^{63} +1.00000 q^{64} +12.2978 q^{65} +3.97829 q^{66} -6.92148 q^{67} -0.0267197 q^{68} +5.58637 q^{69} -15.3997 q^{70} -9.27923 q^{71} -0.532089 q^{72} -16.2707 q^{73} -24.7757 q^{75} +3.22847 q^{76} +7.64471 q^{77} +5.42015 q^{78} +1.41514 q^{79} +4.26414 q^{80} -10.3131 q^{81} +1.49152 q^{82} -1.69642 q^{83} -6.78728 q^{84} -0.113936 q^{85} -5.53737 q^{86} +10.7745 q^{87} -2.11681 q^{88} -8.27093 q^{89} -2.26890 q^{90} +10.4154 q^{91} -2.97244 q^{92} -12.7222 q^{93} -2.61309 q^{94} +13.7667 q^{95} +1.87939 q^{96} -6.52527 q^{97} -6.04247 q^{98} +1.12633 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + 6 q^{4} + 6 q^{5} - 6 q^{8} - 6 q^{9} - 6 q^{10} - 6 q^{11} + 12 q^{13} - 6 q^{15} + 6 q^{16} + 12 q^{17} + 6 q^{18} + 12 q^{19} + 6 q^{20} - 12 q^{21} + 6 q^{22} - 6 q^{23} + 6 q^{25} - 12 q^{26}+ \cdots + 12 q^{99}+O(q^{100}) \) 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
\(3\) −1.87939 −1.08506 −0.542532 − 0.840035i \(-0.682534\pi\)
−0.542532 + 0.840035i \(0.682534\pi\)
\(4\) 1.00000 0.500000
\(5\) 4.26414 1.90698 0.953491 − 0.301423i \(-0.0974615\pi\)
0.953491 + 0.301423i \(0.0974615\pi\)
\(6\) 1.87939 0.767256
\(7\) 3.61144 1.36499 0.682497 − 0.730888i \(-0.260894\pi\)
0.682497 + 0.730888i \(0.260894\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0.532089 0.177363
\(10\) −4.26414 −1.34844
\(11\) 2.11681 0.638241 0.319120 − 0.947714i \(-0.396613\pi\)
0.319120 + 0.947714i \(0.396613\pi\)
\(12\) −1.87939 −0.542532
\(13\) 2.88400 0.799878 0.399939 − 0.916542i \(-0.369031\pi\)
0.399939 + 0.916542i \(0.369031\pi\)
\(14\) −3.61144 −0.965197
\(15\) −8.01396 −2.06920
\(16\) 1.00000 0.250000
\(17\) −0.0267197 −0.00648047 −0.00324024 − 0.999995i \(-0.501031\pi\)
−0.00324024 + 0.999995i \(0.501031\pi\)
\(18\) −0.532089 −0.125415
\(19\) 3.22847 0.740662 0.370331 − 0.928900i \(-0.379244\pi\)
0.370331 + 0.928900i \(0.379244\pi\)
\(20\) 4.26414 0.953491
\(21\) −6.78728 −1.48111
\(22\) −2.11681 −0.451304
\(23\) −2.97244 −0.619797 −0.309899 − 0.950770i \(-0.600295\pi\)
−0.309899 + 0.950770i \(0.600295\pi\)
\(24\) 1.87939 0.383628
\(25\) 13.1829 2.63658
\(26\) −2.88400 −0.565599
\(27\) 4.63816 0.892613
\(28\) 3.61144 0.682497
\(29\) −5.73300 −1.06459 −0.532296 − 0.846558i \(-0.678671\pi\)
−0.532296 + 0.846558i \(0.678671\pi\)
\(30\) 8.01396 1.46314
\(31\) 6.76932 1.21581 0.607903 − 0.794011i \(-0.292011\pi\)
0.607903 + 0.794011i \(0.292011\pi\)
\(32\) −1.00000 −0.176777
\(33\) −3.97829 −0.692532
\(34\) 0.0267197 0.00458239
\(35\) 15.3997 2.60302
\(36\) 0.532089 0.0886815
\(37\) 0 0
\(38\) −3.22847 −0.523727
\(39\) −5.42015 −0.867919
\(40\) −4.26414 −0.674220
\(41\) −1.49152 −0.232937 −0.116468 − 0.993194i \(-0.537157\pi\)
−0.116468 + 0.993194i \(0.537157\pi\)
\(42\) 6.78728 1.04730
\(43\) 5.53737 0.844441 0.422221 − 0.906493i \(-0.361251\pi\)
0.422221 + 0.906493i \(0.361251\pi\)
\(44\) 2.11681 0.319120
\(45\) 2.26890 0.338228
\(46\) 2.97244 0.438263
\(47\) 2.61309 0.381158 0.190579 − 0.981672i \(-0.438964\pi\)
0.190579 + 0.981672i \(0.438964\pi\)
\(48\) −1.87939 −0.271266
\(49\) 6.04247 0.863210
\(50\) −13.1829 −1.86434
\(51\) 0.0502166 0.00703173
\(52\) 2.88400 0.399939
\(53\) 1.90902 0.262224 0.131112 − 0.991368i \(-0.458145\pi\)
0.131112 + 0.991368i \(0.458145\pi\)
\(54\) −4.63816 −0.631173
\(55\) 9.02635 1.21711
\(56\) −3.61144 −0.482598
\(57\) −6.06754 −0.803665
\(58\) 5.73300 0.752780
\(59\) 5.62482 0.732289 0.366144 − 0.930558i \(-0.380678\pi\)
0.366144 + 0.930558i \(0.380678\pi\)
\(60\) −8.01396 −1.03460
\(61\) 5.74087 0.735043 0.367522 − 0.930015i \(-0.380206\pi\)
0.367522 + 0.930015i \(0.380206\pi\)
\(62\) −6.76932 −0.859705
\(63\) 1.92160 0.242099
\(64\) 1.00000 0.125000
\(65\) 12.2978 1.52535
\(66\) 3.97829 0.489694
\(67\) −6.92148 −0.845593 −0.422796 − 0.906225i \(-0.638951\pi\)
−0.422796 + 0.906225i \(0.638951\pi\)
\(68\) −0.0267197 −0.00324024
\(69\) 5.58637 0.672519
\(70\) −15.3997 −1.84061
\(71\) −9.27923 −1.10124 −0.550621 − 0.834755i \(-0.685609\pi\)
−0.550621 + 0.834755i \(0.685609\pi\)
\(72\) −0.532089 −0.0627073
\(73\) −16.2707 −1.90434 −0.952169 − 0.305571i \(-0.901153\pi\)
−0.952169 + 0.305571i \(0.901153\pi\)
\(74\) 0 0
\(75\) −24.7757 −2.86085
\(76\) 3.22847 0.370331
\(77\) 7.64471 0.871195
\(78\) 5.42015 0.613711
\(79\) 1.41514 0.159216 0.0796078 − 0.996826i \(-0.474633\pi\)
0.0796078 + 0.996826i \(0.474633\pi\)
\(80\) 4.26414 0.476745
\(81\) −10.3131 −1.14591
\(82\) 1.49152 0.164711
\(83\) −1.69642 −0.186206 −0.0931030 − 0.995656i \(-0.529679\pi\)
−0.0931030 + 0.995656i \(0.529679\pi\)
\(84\) −6.78728 −0.740553
\(85\) −0.113936 −0.0123581
\(86\) −5.53737 −0.597110
\(87\) 10.7745 1.15515
\(88\) −2.11681 −0.225652
\(89\) −8.27093 −0.876717 −0.438358 − 0.898800i \(-0.644440\pi\)
−0.438358 + 0.898800i \(0.644440\pi\)
\(90\) −2.26890 −0.239163
\(91\) 10.4154 1.09183
\(92\) −2.97244 −0.309899
\(93\) −12.7222 −1.31923
\(94\) −2.61309 −0.269519
\(95\) 13.7667 1.41243
\(96\) 1.87939 0.191814
\(97\) −6.52527 −0.662541 −0.331270 − 0.943536i \(-0.607477\pi\)
−0.331270 + 0.943536i \(0.607477\pi\)
\(98\) −6.04247 −0.610382
\(99\) 1.12633 0.113200
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.r.1.2 6
37.5 odd 36 74.2.h.a.25.2 yes 12
37.15 odd 36 74.2.h.a.3.2 ✓ 12
37.36 even 2 2738.2.a.s.1.1 6
111.5 even 36 666.2.bj.c.469.1 12
111.89 even 36 666.2.bj.c.595.1 12
148.15 even 36 592.2.bq.b.225.1 12
148.79 even 36 592.2.bq.b.321.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.h.a.3.2 ✓ 12 37.15 odd 36
74.2.h.a.25.2 yes 12 37.5 odd 36
592.2.bq.b.225.1 12 148.15 even 36
592.2.bq.b.321.1 12 148.79 even 36
666.2.bj.c.469.1 12 111.5 even 36
666.2.bj.c.595.1 12 111.89 even 36
2738.2.a.r.1.2 6 1.1 even 1 trivial
2738.2.a.s.1.1 6 37.36 even 2