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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3552,2,Mod(2737,3552)] 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("3552.2737"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3552, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3552 = 2^{5} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3552.o (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(28.3628627980\)
Analytic rank: \(0\)
Dimension: \(76\)
Twist minimal: no (minimal twist has level 888)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2737.55
Character \(\chi\) \(=\) 3552.2737
Dual form 3552.2.o.a.2737.56

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} -3.72444 q^{5} +1.69680 q^{7} -1.00000 q^{9} +4.68385i q^{11} -2.13131 q^{13} +3.72444i q^{15} -2.33582i q^{17} +4.49539 q^{19} -1.69680i q^{21} +4.43424i q^{23} +8.87145 q^{25} +1.00000i q^{27} -2.87264 q^{29} -5.38804i q^{31} +4.68385 q^{33} -6.31964 q^{35} +(-1.40098 - 5.91923i) q^{37} +2.13131i q^{39} +5.38413 q^{41} +4.73817 q^{43} +3.72444 q^{45} +4.50697 q^{47} -4.12086 q^{49} -2.33582 q^{51} +3.54067i q^{53} -17.4447i q^{55} -4.49539i q^{57} -12.2870 q^{59} -13.5409 q^{61} -1.69680 q^{63} +7.93794 q^{65} -2.62610i q^{67} +4.43424 q^{69} -5.83268 q^{71} +4.35697 q^{73} -8.87145i q^{75} +7.94757i q^{77} -5.01900i q^{79} +1.00000 q^{81} -14.1557i q^{83} +8.69964i q^{85} +2.87264i q^{87} -16.4080i q^{89} -3.61642 q^{91} -5.38804 q^{93} -16.7428 q^{95} +0.171488i q^{97} -4.68385i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 76 q - 8 q^{7} - 76 q^{9} + 84 q^{25} - 8 q^{41} + 60 q^{49} + 8 q^{63} + 48 q^{65} + 16 q^{71} + 24 q^{73} + 76 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3552\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(2369\) \(3073\) \(3109\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 1.00000i − 0.577350i
\(4\) 0 0
\(5\) −3.72444 −1.66562 −0.832810 − 0.553559i \(-0.813269\pi\)
−0.832810 + 0.553559i \(0.813269\pi\)
\(6\) 0 0
\(7\) 1.69680 0.641332 0.320666 − 0.947192i \(-0.396093\pi\)
0.320666 + 0.947192i \(0.396093\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.68385i 1.41223i 0.708095 + 0.706117i \(0.249555\pi\)
−0.708095 + 0.706117i \(0.750445\pi\)
\(12\) 0 0
\(13\) −2.13131 −0.591119 −0.295560 − 0.955324i \(-0.595506\pi\)
−0.295560 + 0.955324i \(0.595506\pi\)
\(14\) 0 0
\(15\) 3.72444i 0.961646i
\(16\) 0 0
\(17\) − 2.33582i − 0.566521i −0.959043 − 0.283260i \(-0.908584\pi\)
0.959043 − 0.283260i \(-0.0914160\pi\)
\(18\) 0 0
\(19\) 4.49539 1.03131 0.515657 − 0.856795i \(-0.327548\pi\)
0.515657 + 0.856795i \(0.327548\pi\)
\(20\) 0 0
\(21\) − 1.69680i − 0.370273i
\(22\) 0 0
\(23\) 4.43424i 0.924602i 0.886723 + 0.462301i \(0.152976\pi\)
−0.886723 + 0.462301i \(0.847024\pi\)
\(24\) 0 0
\(25\) 8.87145 1.77429
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −2.87264 −0.533436 −0.266718 − 0.963775i \(-0.585939\pi\)
−0.266718 + 0.963775i \(0.585939\pi\)
\(30\) 0 0
\(31\) − 5.38804i − 0.967721i −0.875145 − 0.483861i \(-0.839234\pi\)
0.875145 − 0.483861i \(-0.160766\pi\)
\(32\) 0 0
\(33\) 4.68385 0.815353
\(34\) 0 0
\(35\) −6.31964 −1.06821
\(36\) 0 0
\(37\) −1.40098 − 5.91923i −0.230319 − 0.973115i
\(38\) 0 0
\(39\) 2.13131i 0.341283i
\(40\) 0 0
\(41\) 5.38413 0.840859 0.420430 − 0.907325i \(-0.361879\pi\)
0.420430 + 0.907325i \(0.361879\pi\)
\(42\) 0 0
\(43\) 4.73817 0.722565 0.361282 − 0.932456i \(-0.382339\pi\)
0.361282 + 0.932456i \(0.382339\pi\)
\(44\) 0 0
\(45\) 3.72444 0.555207
\(46\) 0 0
\(47\) 4.50697 0.657409 0.328705 − 0.944433i \(-0.393388\pi\)
0.328705 + 0.944433i \(0.393388\pi\)
\(48\) 0 0
\(49\) −4.12086 −0.588694
\(50\) 0 0
\(51\) −2.33582 −0.327081
\(52\) 0 0
\(53\) 3.54067i 0.486349i 0.969983 + 0.243174i \(0.0781888\pi\)
−0.969983 + 0.243174i \(0.921811\pi\)
\(54\) 0 0
\(55\) − 17.4447i − 2.35224i
\(56\) 0 0
\(57\) − 4.49539i − 0.595430i
\(58\) 0 0
\(59\) −12.2870 −1.59964 −0.799818 − 0.600243i \(-0.795070\pi\)
−0.799818 + 0.600243i \(0.795070\pi\)
\(60\) 0 0
\(61\) −13.5409 −1.73373 −0.866865 − 0.498544i \(-0.833868\pi\)
−0.866865 + 0.498544i \(0.833868\pi\)
\(62\) 0 0
\(63\) −1.69680 −0.213777
\(64\) 0 0
\(65\) 7.93794 0.984580
\(66\) 0 0
\(67\) − 2.62610i − 0.320829i −0.987050 − 0.160415i \(-0.948717\pi\)
0.987050 − 0.160415i \(-0.0512831\pi\)
\(68\) 0 0
\(69\) 4.43424 0.533819
\(70\) 0 0
\(71\) −5.83268 −0.692212 −0.346106 − 0.938195i \(-0.612496\pi\)
−0.346106 + 0.938195i \(0.612496\pi\)
\(72\) 0 0
\(73\) 4.35697 0.509945 0.254973 − 0.966948i \(-0.417934\pi\)
0.254973 + 0.966948i \(0.417934\pi\)
\(74\) 0 0
\(75\) − 8.87145i − 1.02439i
\(76\) 0 0
\(77\) 7.94757i 0.905710i
\(78\) 0 0
\(79\) − 5.01900i − 0.564682i −0.959314 − 0.282341i \(-0.908889\pi\)
0.959314 − 0.282341i \(-0.0911109\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 14.1557i − 1.55379i −0.629631 − 0.776894i \(-0.716794\pi\)
0.629631 − 0.776894i \(-0.283206\pi\)
\(84\) 0 0
\(85\) 8.69964i 0.943608i
\(86\) 0 0
\(87\) 2.87264i 0.307979i
\(88\) 0 0
\(89\) − 16.4080i − 1.73925i −0.493717 − 0.869623i \(-0.664362\pi\)
0.493717 − 0.869623i \(-0.335638\pi\)
\(90\) 0 0
\(91\) −3.61642 −0.379104
\(92\) 0 0
\(93\) −5.38804 −0.558714
\(94\) 0 0
\(95\) −16.7428 −1.71778
\(96\) 0 0
\(97\) 0.171488i 0.0174119i 0.999962 + 0.00870597i \(0.00277123\pi\)
−0.999962 + 0.00870597i \(0.997229\pi\)
\(98\) 0 0
\(99\) − 4.68385i − 0.470745i
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 3552.2.o.a.2737.55 76
4.3 odd 2 888.2.o.a.517.69 yes 76
8.3 odd 2 888.2.o.a.517.7 ✓ 76
8.5 even 2 inner 3552.2.o.a.2737.74 76
37.36 even 2 inner 3552.2.o.a.2737.73 76
148.147 odd 2 888.2.o.a.517.8 yes 76
296.147 odd 2 888.2.o.a.517.70 yes 76
296.221 even 2 inner 3552.2.o.a.2737.56 76
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.o.a.517.7 ✓ 76 8.3 odd 2
888.2.o.a.517.8 yes 76 148.147 odd 2
888.2.o.a.517.69 yes 76 4.3 odd 2
888.2.o.a.517.70 yes 76 296.147 odd 2
3552.2.o.a.2737.55 76 1.1 even 1 trivial
3552.2.o.a.2737.56 76 296.221 even 2 inner
3552.2.o.a.2737.73 76 37.36 even 2 inner
3552.2.o.a.2737.74 76 8.5 even 2 inner