Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2178,4,Mod(1,2178)] 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("2178.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2178, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2178.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,6,0,12,-8,0,-11] 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: \(128.506159993\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.2117020.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 200x - 630 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(15.9813\) of defining polynomial
Character \(\chi\) \(=\) 2178.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +4.00000 q^{4} +12.9813 q^{5} -2.51728 q^{7} +8.00000 q^{8} +25.9626 q^{10} +65.4425 q^{13} -5.03457 q^{14} +16.0000 q^{16} +83.9411 q^{17} +23.4453 q^{19} +51.9253 q^{20} +120.816 q^{23} +43.5145 q^{25} +130.885 q^{26} -10.0691 q^{28} -156.970 q^{29} +195.221 q^{31} +32.0000 q^{32} +167.882 q^{34} -32.6777 q^{35} -48.8823 q^{37} +46.8907 q^{38} +103.851 q^{40} +93.8832 q^{41} -461.589 q^{43} +241.632 q^{46} +3.72345 q^{47} -336.663 q^{49} +87.0290 q^{50} +261.770 q^{52} -315.157 q^{53} -20.1383 q^{56} -313.940 q^{58} -98.4478 q^{59} +676.377 q^{61} +390.442 q^{62} +64.0000 q^{64} +849.530 q^{65} +656.459 q^{67} +335.765 q^{68} -65.3553 q^{70} -480.569 q^{71} -50.6519 q^{73} -97.7645 q^{74} +93.7814 q^{76} +737.353 q^{79} +207.701 q^{80} +187.766 q^{82} -631.178 q^{83} +1089.67 q^{85} -923.178 q^{86} -554.852 q^{89} -164.737 q^{91} +483.264 q^{92} +7.44690 q^{94} +304.351 q^{95} +1154.24 q^{97} -673.327 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 6 q^{2} + 12 q^{4} - 8 q^{5} - 11 q^{7} + 24 q^{8} - 16 q^{10} + 12 q^{13} - 22 q^{14} + 48 q^{16} + 24 q^{17} - 27 q^{19} - 32 q^{20} - 20 q^{23} + 47 q^{25} + 24 q^{26} - 44 q^{28} - 76 q^{29} - 75 q^{31}+ \cdots + 1944 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) 12.9813 1.16108 0.580542 − 0.814230i \(-0.302841\pi\)
0.580542 + 0.814230i \(0.302841\pi\)
\(6\) 0 0
\(7\) −2.51728 −0.135921 −0.0679603 − 0.997688i \(-0.521649\pi\)
−0.0679603 + 0.997688i \(0.521649\pi\)
\(8\) 8.00000 0.353553
\(9\) 0 0
\(10\) 25.9626 0.821010
\(11\) 0 0
\(12\) 0 0
\(13\) 65.4425 1.39619 0.698096 − 0.716004i \(-0.254031\pi\)
0.698096 + 0.716004i \(0.254031\pi\)
\(14\) −5.03457 −0.0961104
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 83.9411 1.19757 0.598786 − 0.800909i \(-0.295650\pi\)
0.598786 + 0.800909i \(0.295650\pi\)
\(18\) 0 0
\(19\) 23.4453 0.283091 0.141546 − 0.989932i \(-0.454793\pi\)
0.141546 + 0.989932i \(0.454793\pi\)
\(20\) 51.9253 0.580542
\(21\) 0 0
\(22\) 0 0
\(23\) 120.816 1.09530 0.547649 − 0.836708i \(-0.315523\pi\)
0.547649 + 0.836708i \(0.315523\pi\)
\(24\) 0 0
\(25\) 43.5145 0.348116
\(26\) 130.885 0.987257
\(27\) 0 0
\(28\) −10.0691 −0.0679603
\(29\) −156.970 −1.00512 −0.502562 − 0.864541i \(-0.667609\pi\)
−0.502562 + 0.864541i \(0.667609\pi\)
\(30\) 0 0
\(31\) 195.221 1.13106 0.565528 − 0.824729i \(-0.308672\pi\)
0.565528 + 0.824729i \(0.308672\pi\)
\(32\) 32.0000 0.176777
\(33\) 0 0
\(34\) 167.882 0.846811
\(35\) −32.6777 −0.157815
\(36\) 0 0
\(37\) −48.8823 −0.217194 −0.108597 − 0.994086i \(-0.534636\pi\)
−0.108597 + 0.994086i \(0.534636\pi\)
\(38\) 46.8907 0.200176
\(39\) 0 0
\(40\) 103.851 0.410505
\(41\) 93.8832 0.357612 0.178806 − 0.983884i \(-0.442777\pi\)
0.178806 + 0.983884i \(0.442777\pi\)
\(42\) 0 0
\(43\) −461.589 −1.63701 −0.818507 − 0.574496i \(-0.805198\pi\)
−0.818507 + 0.574496i \(0.805198\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 241.632 0.774493
\(47\) 3.72345 0.0115558 0.00577788 − 0.999983i \(-0.498161\pi\)
0.00577788 + 0.999983i \(0.498161\pi\)
\(48\) 0 0
\(49\) −336.663 −0.981526
\(50\) 87.0290 0.246155
\(51\) 0 0
\(52\) 261.770 0.698096
\(53\) −315.157 −0.816795 −0.408398 − 0.912804i \(-0.633912\pi\)
−0.408398 + 0.912804i \(0.633912\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −20.1383 −0.0480552
\(57\) 0 0
\(58\) −313.940 −0.710730
\(59\) −98.4478 −0.217234 −0.108617 − 0.994084i \(-0.534642\pi\)
−0.108617 + 0.994084i \(0.534642\pi\)
\(60\) 0 0
\(61\) 676.377 1.41969 0.709846 − 0.704357i \(-0.248765\pi\)
0.709846 + 0.704357i \(0.248765\pi\)
\(62\) 390.442 0.799778
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 849.530 1.62110
\(66\) 0 0
\(67\) 656.459 1.19700 0.598501 − 0.801122i \(-0.295763\pi\)
0.598501 + 0.801122i \(0.295763\pi\)
\(68\) 335.765 0.598786
\(69\) 0 0
\(70\) −65.3553 −0.111592
\(71\) −480.569 −0.803283 −0.401641 − 0.915797i \(-0.631560\pi\)
−0.401641 + 0.915797i \(0.631560\pi\)
\(72\) 0 0
\(73\) −50.6519 −0.0812103 −0.0406051 − 0.999175i \(-0.512929\pi\)
−0.0406051 + 0.999175i \(0.512929\pi\)
\(74\) −97.7645 −0.153580
\(75\) 0 0
\(76\) 93.7814 0.141546
\(77\) 0 0
\(78\) 0 0
\(79\) 737.353 1.05011 0.525055 − 0.851068i \(-0.324045\pi\)
0.525055 + 0.851068i \(0.324045\pi\)
\(80\) 207.701 0.290271
\(81\) 0 0
\(82\) 187.766 0.252870
\(83\) −631.178 −0.834708 −0.417354 − 0.908744i \(-0.637042\pi\)
−0.417354 + 0.908744i \(0.637042\pi\)
\(84\) 0 0
\(85\) 1089.67 1.39048
\(86\) −923.178 −1.15754
\(87\) 0 0
\(88\) 0 0
\(89\) −554.852 −0.660833 −0.330417 − 0.943835i \(-0.607189\pi\)
−0.330417 + 0.943835i \(0.607189\pi\)
\(90\) 0 0
\(91\) −164.737 −0.189771
\(92\) 483.264 0.547649
\(93\) 0 0
\(94\) 7.44690 0.00817116
\(95\) 304.351 0.328692
\(96\) 0 0
\(97\) 1154.24 1.20819 0.604097 − 0.796911i \(-0.293534\pi\)
0.604097 + 0.796911i \(0.293534\pi\)
\(98\) −673.327 −0.694043
\(99\) 0 0
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 2178.4.a.br.1.3 yes 3
3.2 odd 2 2178.4.a.bq.1.1 yes 3
11.10 odd 2 2178.4.a.bp.1.3 ✓ 3
33.32 even 2 2178.4.a.bs.1.1 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2178.4.a.bp.1.3 ✓ 3 11.10 odd 2
2178.4.a.bq.1.1 yes 3 3.2 odd 2
2178.4.a.br.1.3 yes 3 1.1 even 1 trivial
2178.4.a.bs.1.1 yes 3 33.32 even 2