↑

Properties

Label 2535.2.a.ba.1.3
Level $2535$
Weight $2$
Character 2535.1
Self dual yes
Analytic conductor $20.242$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2535,2,Mod(1,2535)] 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("2535.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2535, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2535 = 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2535.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-3,6,-3,0,-3] 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: \(20.2420769124\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.756.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.26180\) of defining polynomial
Character \(\chi\) \(=\) 2535.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.26180 q^{2} -1.00000 q^{3} +3.11575 q^{4} -1.00000 q^{5} -2.26180 q^{6} +1.26180 q^{7} +2.52360 q^{8} +1.00000 q^{9} -2.26180 q^{10} -4.52360 q^{11} -3.11575 q^{12} +2.85395 q^{14} +1.00000 q^{15} -0.523604 q^{16} -4.49330 q^{17} +2.26180 q^{18} -5.11575 q^{19} -3.11575 q^{20} -1.26180 q^{21} -10.2315 q^{22} +2.23150 q^{23} -2.52360 q^{24} +1.00000 q^{25} -1.00000 q^{27} +3.93146 q^{28} -1.37755 q^{29} +2.26180 q^{30} -8.87085 q^{31} -6.23150 q^{32} +4.52360 q^{33} -10.1630 q^{34} -1.26180 q^{35} +3.11575 q^{36} +0.231499 q^{37} -11.5708 q^{38} -2.52360 q^{40} -1.14605 q^{41} -2.85395 q^{42} +6.37755 q^{43} -14.0944 q^{44} -1.00000 q^{45} +5.04721 q^{46} +10.7854 q^{47} +0.523604 q^{48} -5.40786 q^{49} +2.26180 q^{50} +4.49330 q^{51} +4.52360 q^{53} -2.26180 q^{54} +4.52360 q^{55} +3.18429 q^{56} +5.11575 q^{57} -3.11575 q^{58} +0.853947 q^{59} +3.11575 q^{60} +4.63935 q^{61} -20.0641 q^{62} +1.26180 q^{63} -13.0472 q^{64} +10.2315 q^{66} -13.1327 q^{67} -14.0000 q^{68} -2.23150 q^{69} -2.85395 q^{70} +9.60905 q^{71} +2.52360 q^{72} -13.7854 q^{73} +0.523604 q^{74} -1.00000 q^{75} -15.9394 q^{76} -5.70789 q^{77} -8.87085 q^{79} +0.523604 q^{80} +1.00000 q^{81} -2.59214 q^{82} +8.23150 q^{83} -3.93146 q^{84} +4.49330 q^{85} +14.4248 q^{86} +1.37755 q^{87} -11.4158 q^{88} -6.62245 q^{89} -2.26180 q^{90} +6.95279 q^{92} +8.87085 q^{93} +24.3945 q^{94} +5.11575 q^{95} +6.23150 q^{96} -10.6697 q^{97} -12.2315 q^{98} -4.52360 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 6 q^{4} - 3 q^{5} - 3 q^{7} - 6 q^{8} + 3 q^{9} - 6 q^{12} + 12 q^{14} + 3 q^{15} + 12 q^{16} - 12 q^{19} - 6 q^{20} + 3 q^{21} - 24 q^{22} + 6 q^{24} + 3 q^{25} - 3 q^{27} - 12 q^{28} + 6 q^{29}+ \cdots - 30 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.26180 1.59934 0.799668 − 0.600443i \(-0.205009\pi\)
0.799668 + 0.600443i \(0.205009\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.11575 1.55787
\(5\) −1.00000 −0.447214
\(6\) −2.26180 −0.923377
\(7\) 1.26180 0.476916 0.238458 − 0.971153i \(-0.423358\pi\)
0.238458 + 0.971153i \(0.423358\pi\)
\(8\) 2.52360 0.892229
\(9\) 1.00000 0.333333
\(10\) −2.26180 −0.715245
\(11\) −4.52360 −1.36392 −0.681959 − 0.731390i \(-0.738872\pi\)
−0.681959 + 0.731390i \(0.738872\pi\)
\(12\) −3.11575 −0.899439
\(13\) 0 0
\(14\) 2.85395 0.762749
\(15\) 1.00000 0.258199
\(16\) −0.523604 −0.130901
\(17\) −4.49330 −1.08979 −0.544893 − 0.838506i \(-0.683430\pi\)
−0.544893 + 0.838506i \(0.683430\pi\)
\(18\) 2.26180 0.533112
\(19\) −5.11575 −1.17363 −0.586817 − 0.809720i \(-0.699619\pi\)
−0.586817 + 0.809720i \(0.699619\pi\)
\(20\) −3.11575 −0.696703
\(21\) −1.26180 −0.275348
\(22\) −10.2315 −2.18136
\(23\) 2.23150 0.465300 0.232650 − 0.972561i \(-0.425260\pi\)
0.232650 + 0.972561i \(0.425260\pi\)
\(24\) −2.52360 −0.515129
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 3.93146 0.742976
\(29\) −1.37755 −0.255805 −0.127902 − 0.991787i \(-0.540824\pi\)
−0.127902 + 0.991787i \(0.540824\pi\)
\(30\) 2.26180 0.412947
\(31\) −8.87085 −1.59325 −0.796626 − 0.604472i \(-0.793384\pi\)
−0.796626 + 0.604472i \(0.793384\pi\)
\(32\) −6.23150 −1.10158
\(33\) 4.52360 0.787458
\(34\) −10.1630 −1.74293
\(35\) −1.26180 −0.213284
\(36\) 3.11575 0.519292
\(37\) 0.231499 0.0380582 0.0190291 − 0.999819i \(-0.493942\pi\)
0.0190291 + 0.999819i \(0.493942\pi\)
\(38\) −11.5708 −1.87703
\(39\) 0 0
\(40\) −2.52360 −0.399017
\(41\) −1.14605 −0.178983 −0.0894917 − 0.995988i \(-0.528524\pi\)
−0.0894917 + 0.995988i \(0.528524\pi\)
\(42\) −2.85395 −0.440374
\(43\) 6.37755 0.972568 0.486284 − 0.873801i \(-0.338352\pi\)
0.486284 + 0.873801i \(0.338352\pi\)
\(44\) −14.0944 −2.12481
\(45\) −1.00000 −0.149071
\(46\) 5.04721 0.744170
\(47\) 10.7854 1.57321 0.786607 − 0.617454i \(-0.211836\pi\)
0.786607 + 0.617454i \(0.211836\pi\)
\(48\) 0.523604 0.0755758
\(49\) −5.40786 −0.772551
\(50\) 2.26180 0.319867
\(51\) 4.49330 0.629188
\(52\) 0 0
\(53\) 4.52360 0.621365 0.310682 − 0.950514i \(-0.399442\pi\)
0.310682 + 0.950514i \(0.399442\pi\)
\(54\) −2.26180 −0.307792
\(55\) 4.52360 0.609963
\(56\) 3.18429 0.425519
\(57\) 5.11575 0.677598
\(58\) −3.11575 −0.409118
\(59\) 0.853947 0.111174 0.0555872 − 0.998454i \(-0.482297\pi\)
0.0555872 + 0.998454i \(0.482297\pi\)
\(60\) 3.11575 0.402242
\(61\) 4.63935 0.594008 0.297004 − 0.954876i \(-0.404012\pi\)
0.297004 + 0.954876i \(0.404012\pi\)
\(62\) −20.0641 −2.54815
\(63\) 1.26180 0.158972
\(64\) −13.0472 −1.63090
\(65\) 0 0
\(66\) 10.2315 1.25941
\(67\) −13.1327 −1.60441 −0.802205 − 0.597049i \(-0.796340\pi\)
−0.802205 + 0.597049i \(0.796340\pi\)
\(68\) −14.0000 −1.69775
\(69\) −2.23150 −0.268641
\(70\) −2.85395 −0.341112
\(71\) 9.60905 1.14038 0.570192 − 0.821511i \(-0.306869\pi\)
0.570192 + 0.821511i \(0.306869\pi\)
\(72\) 2.52360 0.297410
\(73\) −13.7854 −1.61346 −0.806730 − 0.590920i \(-0.798765\pi\)
−0.806730 + 0.590920i \(0.798765\pi\)
\(74\) 0.523604 0.0608678
\(75\) −1.00000 −0.115470
\(76\) −15.9394 −1.82837
\(77\) −5.70789 −0.650475
\(78\) 0 0
\(79\) −8.87085 −0.998049 −0.499024 − 0.866588i \(-0.666308\pi\)
−0.499024 + 0.866588i \(0.666308\pi\)
\(80\) 0.523604 0.0585408
\(81\) 1.00000 0.111111
\(82\) −2.59214 −0.286255
\(83\) 8.23150 0.903524 0.451762 − 0.892138i \(-0.350796\pi\)
0.451762 + 0.892138i \(0.350796\pi\)
\(84\) −3.93146 −0.428957
\(85\) 4.49330 0.487367
\(86\) 14.4248 1.55546
\(87\) 1.37755 0.147689
\(88\) −11.4158 −1.21693
\(89\) −6.62245 −0.701978 −0.350989 − 0.936380i \(-0.614155\pi\)
−0.350989 + 0.936380i \(0.614155\pi\)
\(90\) −2.26180 −0.238415
\(91\) 0 0
\(92\) 6.95279 0.724879
\(93\) 8.87085 0.919865
\(94\) 24.3945 2.51610
\(95\) 5.11575 0.524865
\(96\) 6.23150 0.636000
\(97\) −10.6697 −1.08334 −0.541670 − 0.840591i \(-0.682208\pi\)
−0.541670 + 0.840591i \(0.682208\pi\)
\(98\) −12.2315 −1.23557
\(99\) −4.52360 −0.454639
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 2535.2.a.ba.1.3 3
3.2 odd 2 7605.2.a.bw.1.1 3
13.4 even 6 195.2.i.d.16.3 ✓ 6
13.10 even 6 195.2.i.d.61.3 yes 6
13.12 even 2 2535.2.a.bb.1.1 3
39.17 odd 6 585.2.j.f.406.1 6
39.23 odd 6 585.2.j.f.451.1 6
39.38 odd 2 7605.2.a.bv.1.3 3
65.4 even 6 975.2.i.l.601.1 6
65.17 odd 12 975.2.bb.k.874.2 12
65.23 odd 12 975.2.bb.k.724.2 12
65.43 odd 12 975.2.bb.k.874.5 12
65.49 even 6 975.2.i.l.451.1 6
65.62 odd 12 975.2.bb.k.724.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.i.d.16.3 ✓ 6 13.4 even 6
195.2.i.d.61.3 yes 6 13.10 even 6
585.2.j.f.406.1 6 39.17 odd 6
585.2.j.f.451.1 6 39.23 odd 6
975.2.i.l.451.1 6 65.49 even 6
975.2.i.l.601.1 6 65.4 even 6
975.2.bb.k.724.2 12 65.23 odd 12
975.2.bb.k.724.5 12 65.62 odd 12
975.2.bb.k.874.2 12 65.17 odd 12
975.2.bb.k.874.5 12 65.43 odd 12
2535.2.a.ba.1.3 3 1.1 even 1 trivial
2535.2.a.bb.1.1 3 13.12 even 2
7605.2.a.bv.1.3 3 39.38 odd 2
7605.2.a.bw.1.1 3 3.2 odd 2