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 1.3
Root \(0.684040\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +0.347296 q^{3} +1.00000 q^{4} -2.61144 q^{5} -0.347296 q^{6} -2.07935 q^{7} -1.00000 q^{8} -2.87939 q^{9} +2.61144 q^{10} -5.99619 q^{11} +0.347296 q^{12} +4.11695 q^{13} +2.07935 q^{14} -0.906942 q^{15} +1.00000 q^{16} +0.962542 q^{17} +2.87939 q^{18} +1.56408 q^{19} -2.61144 q^{20} -0.722150 q^{21} +5.99619 q^{22} -6.37054 q^{23} -0.347296 q^{24} +1.81960 q^{25} -4.11695 q^{26} -2.04189 q^{27} -2.07935 q^{28} -4.05523 q^{29} +0.906942 q^{30} +3.39997 q^{31} -1.00000 q^{32} -2.08246 q^{33} -0.962542 q^{34} +5.43008 q^{35} -2.87939 q^{36} -1.56408 q^{38} +1.42980 q^{39} +2.61144 q^{40} -10.3715 q^{41} +0.722150 q^{42} -3.76932 q^{43} -5.99619 q^{44} +7.51933 q^{45} +6.37054 q^{46} -6.17501 q^{47} +0.347296 q^{48} -2.67632 q^{49} -1.81960 q^{50} +0.334287 q^{51} +4.11695 q^{52} -8.02781 q^{53} +2.04189 q^{54} +15.6587 q^{55} +2.07935 q^{56} +0.543198 q^{57} +4.05523 q^{58} +5.10018 q^{59} -0.906942 q^{60} +9.73135 q^{61} -3.39997 q^{62} +5.98724 q^{63} +1.00000 q^{64} -10.7512 q^{65} +2.08246 q^{66} +10.5537 q^{67} +0.962542 q^{68} -2.21246 q^{69} -5.43008 q^{70} +10.8018 q^{71} +2.87939 q^{72} -3.55293 q^{73} +0.631940 q^{75} +1.56408 q^{76} +12.4682 q^{77} -1.42980 q^{78} -2.54971 q^{79} -2.61144 q^{80} +7.92902 q^{81} +10.3715 q^{82} +6.91602 q^{83} -0.722150 q^{84} -2.51362 q^{85} +3.76932 q^{86} -1.40837 q^{87} +5.99619 q^{88} +16.2636 q^{89} -7.51933 q^{90} -8.56057 q^{91} -6.37054 q^{92} +1.18080 q^{93} +6.17501 q^{94} -4.08449 q^{95} -0.347296 q^{96} -16.3015 q^{97} +2.67632 q^{98} +17.2653 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + 6 q^{4} + 6 q^{5} - 6 q^{8} - 6 q^{9} - 6 q^{10} - 6 q^{11} + 12 q^{13} - 6 q^{15} + 6 q^{16} + 12 q^{17} + 6 q^{18} + 12 q^{19} + 6 q^{20} - 12 q^{21} + 6 q^{22} - 6 q^{23} + 6 q^{25} - 12 q^{26}+ \cdots + 12 q^{99}+O(q^{100}) \) 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
\(3\) 0.347296 0.200512 0.100256 − 0.994962i \(-0.468034\pi\)
0.100256 + 0.994962i \(0.468034\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.61144 −1.16787 −0.583935 − 0.811801i \(-0.698488\pi\)
−0.583935 + 0.811801i \(0.698488\pi\)
\(6\) −0.347296 −0.141783
\(7\) −2.07935 −0.785919 −0.392960 − 0.919556i \(-0.628549\pi\)
−0.392960 + 0.919556i \(0.628549\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.87939 −0.959795
\(10\) 2.61144 0.825809
\(11\) −5.99619 −1.80792 −0.903960 − 0.427618i \(-0.859353\pi\)
−0.903960 + 0.427618i \(0.859353\pi\)
\(12\) 0.347296 0.100256
\(13\) 4.11695 1.14184 0.570918 − 0.821007i \(-0.306587\pi\)
0.570918 + 0.821007i \(0.306587\pi\)
\(14\) 2.07935 0.555729
\(15\) −0.906942 −0.234171
\(16\) 1.00000 0.250000
\(17\) 0.962542 0.233451 0.116725 − 0.993164i \(-0.462760\pi\)
0.116725 + 0.993164i \(0.462760\pi\)
\(18\) 2.87939 0.678678
\(19\) 1.56408 0.358824 0.179412 − 0.983774i \(-0.442581\pi\)
0.179412 + 0.983774i \(0.442581\pi\)
\(20\) −2.61144 −0.583935
\(21\) −0.722150 −0.157586
\(22\) 5.99619 1.27839
\(23\) −6.37054 −1.32835 −0.664174 − 0.747578i \(-0.731217\pi\)
−0.664174 + 0.747578i \(0.731217\pi\)
\(24\) −0.347296 −0.0708916
\(25\) 1.81960 0.363920
\(26\) −4.11695 −0.807400
\(27\) −2.04189 −0.392962
\(28\) −2.07935 −0.392960
\(29\) −4.05523 −0.753037 −0.376519 − 0.926409i \(-0.622879\pi\)
−0.376519 + 0.926409i \(0.622879\pi\)
\(30\) 0.906942 0.165584
\(31\) 3.39997 0.610653 0.305326 − 0.952248i \(-0.401234\pi\)
0.305326 + 0.952248i \(0.401234\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.08246 −0.362509
\(34\) −0.962542 −0.165075
\(35\) 5.43008 0.917851
\(36\) −2.87939 −0.479898
\(37\) 0 0
\(38\) −1.56408 −0.253727
\(39\) 1.42980 0.228952
\(40\) 2.61144 0.412904
\(41\) −10.3715 −1.61975 −0.809877 − 0.586599i \(-0.800466\pi\)
−0.809877 + 0.586599i \(0.800466\pi\)
\(42\) 0.722150 0.111430
\(43\) −3.76932 −0.574816 −0.287408 − 0.957808i \(-0.592794\pi\)
−0.287408 + 0.957808i \(0.592794\pi\)
\(44\) −5.99619 −0.903960
\(45\) 7.51933 1.12092
\(46\) 6.37054 0.939284
\(47\) −6.17501 −0.900717 −0.450359 − 0.892848i \(-0.648704\pi\)
−0.450359 + 0.892848i \(0.648704\pi\)
\(48\) 0.347296 0.0501279
\(49\) −2.67632 −0.382331
\(50\) −1.81960 −0.257330
\(51\) 0.334287 0.0468096
\(52\) 4.11695 0.570918
\(53\) −8.02781 −1.10270 −0.551352 − 0.834272i \(-0.685888\pi\)
−0.551352 + 0.834272i \(0.685888\pi\)
\(54\) 2.04189 0.277866
\(55\) 15.6587 2.11141
\(56\) 2.07935 0.277864
\(57\) 0.543198 0.0719484
\(58\) 4.05523 0.532478
\(59\) 5.10018 0.663987 0.331993 − 0.943282i \(-0.392279\pi\)
0.331993 + 0.943282i \(0.392279\pi\)
\(60\) −0.906942 −0.117086
\(61\) 9.73135 1.24597 0.622986 − 0.782233i \(-0.285919\pi\)
0.622986 + 0.782233i \(0.285919\pi\)
\(62\) −3.39997 −0.431797
\(63\) 5.98724 0.754322
\(64\) 1.00000 0.125000
\(65\) −10.7512 −1.33352
\(66\) 2.08246 0.256333
\(67\) 10.5537 1.28934 0.644671 − 0.764460i \(-0.276994\pi\)
0.644671 + 0.764460i \(0.276994\pi\)
\(68\) 0.962542 0.116725
\(69\) −2.21246 −0.266349
\(70\) −5.43008 −0.649019
\(71\) 10.8018 1.28193 0.640966 − 0.767569i \(-0.278534\pi\)
0.640966 + 0.767569i \(0.278534\pi\)
\(72\) 2.87939 0.339339
\(73\) −3.55293 −0.415839 −0.207920 − 0.978146i \(-0.566669\pi\)
−0.207920 + 0.978146i \(0.566669\pi\)
\(74\) 0 0
\(75\) 0.631940 0.0729701
\(76\) 1.56408 0.179412
\(77\) 12.4682 1.42088
\(78\) −1.42980 −0.161893
\(79\) −2.54971 −0.286865 −0.143433 − 0.989660i \(-0.545814\pi\)
−0.143433 + 0.989660i \(0.545814\pi\)
\(80\) −2.61144 −0.291967
\(81\) 7.92902 0.881002
\(82\) 10.3715 1.14534
\(83\) 6.91602 0.759132 0.379566 − 0.925165i \(-0.376073\pi\)
0.379566 + 0.925165i \(0.376073\pi\)
\(84\) −0.722150 −0.0787930
\(85\) −2.51362 −0.272640
\(86\) 3.76932 0.406457
\(87\) −1.40837 −0.150993
\(88\) 5.99619 0.639196
\(89\) 16.2636 1.72393 0.861967 − 0.506965i \(-0.169233\pi\)
0.861967 + 0.506965i \(0.169233\pi\)
\(90\) −7.51933 −0.792607
\(91\) −8.56057 −0.897391
\(92\) −6.37054 −0.664174
\(93\) 1.18080 0.122443
\(94\) 6.17501 0.636903
\(95\) −4.08449 −0.419059
\(96\) −0.347296 −0.0354458
\(97\) −16.3015 −1.65517 −0.827583 − 0.561344i \(-0.810284\pi\)
−0.827583 + 0.561344i \(0.810284\pi\)
\(98\) 2.67632 0.270349
\(99\) 17.2653 1.73523
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.1.3 6
37.17 odd 36 74.2.h.a.67.1 yes 12
37.24 odd 36 74.2.h.a.21.1 ✓ 12
37.36 even 2 2738.2.a.s.1.4 6
111.17 even 36 666.2.bj.c.289.2 12
111.98 even 36 666.2.bj.c.613.2 12
148.91 even 36 592.2.bq.b.289.2 12
148.135 even 36 592.2.bq.b.465.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.h.a.21.1 ✓ 12 37.24 odd 36
74.2.h.a.67.1 yes 12 37.17 odd 36
592.2.bq.b.289.2 12 148.91 even 36
592.2.bq.b.465.2 12 148.135 even 36
666.2.bj.c.289.2 12 111.17 even 36
666.2.bj.c.613.2 12 111.98 even 36
2738.2.a.r.1.3 6 1.1 even 1 trivial
2738.2.a.s.1.4 6 37.36 even 2