↑

Properties

Label 2535.2.a.s.1.1
Level $2535$
Weight $2$
Character 2535.1
Self dual yes
Analytic conductor $20.242$
Analytic rank $0$
Dimension $2$
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 = [2,2,2,4,2,2,-2] 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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) 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 1.1
Root \(-1.73205\) 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)
 
\(f(q)\) \(=\) \(q-0.732051 q^{2} +1.00000 q^{3} -1.46410 q^{4} +1.00000 q^{5} -0.732051 q^{6} -4.46410 q^{7} +2.53590 q^{8} +1.00000 q^{9} -0.732051 q^{10} +3.46410 q^{11} -1.46410 q^{12} +3.26795 q^{14} +1.00000 q^{15} +1.07180 q^{16} -6.73205 q^{17} -0.732051 q^{18} +5.46410 q^{19} -1.46410 q^{20} -4.46410 q^{21} -2.53590 q^{22} -0.535898 q^{23} +2.53590 q^{24} +1.00000 q^{25} +1.00000 q^{27} +6.53590 q^{28} -2.73205 q^{29} -0.732051 q^{30} +3.19615 q^{31} -5.85641 q^{32} +3.46410 q^{33} +4.92820 q^{34} -4.46410 q^{35} -1.46410 q^{36} -4.00000 q^{37} -4.00000 q^{38} +2.53590 q^{40} +5.26795 q^{41} +3.26795 q^{42} -0.267949 q^{43} -5.07180 q^{44} +1.00000 q^{45} +0.392305 q^{46} +0.196152 q^{47} +1.07180 q^{48} +12.9282 q^{49} -0.732051 q^{50} -6.73205 q^{51} -6.92820 q^{53} -0.732051 q^{54} +3.46410 q^{55} -11.3205 q^{56} +5.46410 q^{57} +2.00000 q^{58} +7.26795 q^{59} -1.46410 q^{60} +4.46410 q^{61} -2.33975 q^{62} -4.46410 q^{63} +2.14359 q^{64} -2.53590 q^{66} -12.4641 q^{67} +9.85641 q^{68} -0.535898 q^{69} +3.26795 q^{70} +12.7321 q^{71} +2.53590 q^{72} +15.3923 q^{73} +2.92820 q^{74} +1.00000 q^{75} -8.00000 q^{76} -15.4641 q^{77} +1.92820 q^{79} +1.07180 q^{80} +1.00000 q^{81} -3.85641 q^{82} +2.53590 q^{83} +6.53590 q^{84} -6.73205 q^{85} +0.196152 q^{86} -2.73205 q^{87} +8.78461 q^{88} +1.26795 q^{89} -0.732051 q^{90} +0.784610 q^{92} +3.19615 q^{93} -0.143594 q^{94} +5.46410 q^{95} -5.85641 q^{96} +16.4641 q^{97} -9.46410 q^{98} +3.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 4 q^{4} + 2 q^{5} + 2 q^{6} - 2 q^{7} + 12 q^{8} + 2 q^{9} + 2 q^{10} + 4 q^{12} + 10 q^{14} + 2 q^{15} + 16 q^{16} - 10 q^{17} + 2 q^{18} + 4 q^{19} + 4 q^{20} - 2 q^{21} - 12 q^{22}+ \cdots - 12 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.732051 −0.517638 −0.258819 − 0.965926i \(-0.583333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.46410 −0.732051
\(5\) 1.00000 0.447214
\(6\) −0.732051 −0.298858
\(7\) −4.46410 −1.68727 −0.843636 − 0.536916i \(-0.819589\pi\)
−0.843636 + 0.536916i \(0.819589\pi\)
\(8\) 2.53590 0.896575
\(9\) 1.00000 0.333333
\(10\) −0.732051 −0.231495
\(11\) 3.46410 1.04447 0.522233 − 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(12\) −1.46410 −0.422650
\(13\) 0 0
\(14\) 3.26795 0.873396
\(15\) 1.00000 0.258199
\(16\) 1.07180 0.267949
\(17\) −6.73205 −1.63276 −0.816381 − 0.577514i \(-0.804023\pi\)
−0.816381 + 0.577514i \(0.804023\pi\)
\(18\) −0.732051 −0.172546
\(19\) 5.46410 1.25355 0.626775 − 0.779200i \(-0.284374\pi\)
0.626775 + 0.779200i \(0.284374\pi\)
\(20\) −1.46410 −0.327383
\(21\) −4.46410 −0.974147
\(22\) −2.53590 −0.540655
\(23\) −0.535898 −0.111743 −0.0558713 − 0.998438i \(-0.517794\pi\)
−0.0558713 + 0.998438i \(0.517794\pi\)
\(24\) 2.53590 0.517638
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 6.53590 1.23517
\(29\) −2.73205 −0.507329 −0.253665 − 0.967292i \(-0.581636\pi\)
−0.253665 + 0.967292i \(0.581636\pi\)
\(30\) −0.732051 −0.133654
\(31\) 3.19615 0.574046 0.287023 − 0.957924i \(-0.407334\pi\)
0.287023 + 0.957924i \(0.407334\pi\)
\(32\) −5.85641 −1.03528
\(33\) 3.46410 0.603023
\(34\) 4.92820 0.845180
\(35\) −4.46410 −0.754571
\(36\) −1.46410 −0.244017
\(37\) −4.00000 −0.657596 −0.328798 − 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) −4.00000 −0.648886
\(39\) 0 0
\(40\) 2.53590 0.400961
\(41\) 5.26795 0.822715 0.411358 − 0.911474i \(-0.365055\pi\)
0.411358 + 0.911474i \(0.365055\pi\)
\(42\) 3.26795 0.504256
\(43\) −0.267949 −0.0408619 −0.0204309 − 0.999791i \(-0.506504\pi\)
−0.0204309 + 0.999791i \(0.506504\pi\)
\(44\) −5.07180 −0.764602
\(45\) 1.00000 0.149071
\(46\) 0.392305 0.0578422
\(47\) 0.196152 0.0286118 0.0143059 − 0.999898i \(-0.495446\pi\)
0.0143059 + 0.999898i \(0.495446\pi\)
\(48\) 1.07180 0.154701
\(49\) 12.9282 1.84689
\(50\) −0.732051 −0.103528
\(51\) −6.73205 −0.942676
\(52\) 0 0
\(53\) −6.92820 −0.951662 −0.475831 − 0.879537i \(-0.657853\pi\)
−0.475831 + 0.879537i \(0.657853\pi\)
\(54\) −0.732051 −0.0996195
\(55\) 3.46410 0.467099
\(56\) −11.3205 −1.51277
\(57\) 5.46410 0.723738
\(58\) 2.00000 0.262613
\(59\) 7.26795 0.946206 0.473103 − 0.881007i \(-0.343134\pi\)
0.473103 + 0.881007i \(0.343134\pi\)
\(60\) −1.46410 −0.189015
\(61\) 4.46410 0.571570 0.285785 − 0.958294i \(-0.407746\pi\)
0.285785 + 0.958294i \(0.407746\pi\)
\(62\) −2.33975 −0.297148
\(63\) −4.46410 −0.562424
\(64\) 2.14359 0.267949
\(65\) 0 0
\(66\) −2.53590 −0.312148
\(67\) −12.4641 −1.52273 −0.761366 − 0.648322i \(-0.775471\pi\)
−0.761366 + 0.648322i \(0.775471\pi\)
\(68\) 9.85641 1.19526
\(69\) −0.535898 −0.0645146
\(70\) 3.26795 0.390595
\(71\) 12.7321 1.51102 0.755508 − 0.655139i \(-0.227390\pi\)
0.755508 + 0.655139i \(0.227390\pi\)
\(72\) 2.53590 0.298858
\(73\) 15.3923 1.80153 0.900767 − 0.434304i \(-0.143006\pi\)
0.900767 + 0.434304i \(0.143006\pi\)
\(74\) 2.92820 0.340397
\(75\) 1.00000 0.115470
\(76\) −8.00000 −0.917663
\(77\) −15.4641 −1.76230
\(78\) 0 0
\(79\) 1.92820 0.216940 0.108470 − 0.994100i \(-0.465405\pi\)
0.108470 + 0.994100i \(0.465405\pi\)
\(80\) 1.07180 0.119831
\(81\) 1.00000 0.111111
\(82\) −3.85641 −0.425869
\(83\) 2.53590 0.278351 0.139176 − 0.990268i \(-0.455555\pi\)
0.139176 + 0.990268i \(0.455555\pi\)
\(84\) 6.53590 0.713125
\(85\) −6.73205 −0.730193
\(86\) 0.196152 0.0211517
\(87\) −2.73205 −0.292907
\(88\) 8.78461 0.936443
\(89\) 1.26795 0.134402 0.0672012 − 0.997739i \(-0.478593\pi\)
0.0672012 + 0.997739i \(0.478593\pi\)
\(90\) −0.732051 −0.0771649
\(91\) 0 0
\(92\) 0.784610 0.0818012
\(93\) 3.19615 0.331426
\(94\) −0.143594 −0.0148105
\(95\) 5.46410 0.560605
\(96\) −5.85641 −0.597717
\(97\) 16.4641 1.67168 0.835838 − 0.548976i \(-0.184982\pi\)
0.835838 + 0.548976i \(0.184982\pi\)
\(98\) −9.46410 −0.956019
\(99\) 3.46410 0.348155
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.s.1.1 2
3.2 odd 2 7605.2.a.y.1.2 2
13.2 odd 12 195.2.bb.a.121.2 ✓ 4
13.7 odd 12 195.2.bb.a.166.2 yes 4
13.12 even 2 2535.2.a.n.1.2 2
39.2 even 12 585.2.bu.a.316.1 4
39.20 even 12 585.2.bu.a.361.1 4
39.38 odd 2 7605.2.a.bk.1.1 2
65.2 even 12 975.2.w.a.199.2 4
65.7 even 12 975.2.w.f.49.1 4
65.28 even 12 975.2.w.f.199.1 4
65.33 even 12 975.2.w.a.49.2 4
65.54 odd 12 975.2.bc.h.901.1 4
65.59 odd 12 975.2.bc.h.751.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.bb.a.121.2 ✓ 4 13.2 odd 12
195.2.bb.a.166.2 yes 4 13.7 odd 12
585.2.bu.a.316.1 4 39.2 even 12
585.2.bu.a.361.1 4 39.20 even 12
975.2.w.a.49.2 4 65.33 even 12
975.2.w.a.199.2 4 65.2 even 12
975.2.w.f.49.1 4 65.7 even 12
975.2.w.f.199.1 4 65.28 even 12
975.2.bc.h.751.1 4 65.59 odd 12
975.2.bc.h.901.1 4 65.54 odd 12
2535.2.a.n.1.2 2 13.12 even 2
2535.2.a.s.1.1 2 1.1 even 1 trivial
7605.2.a.y.1.2 2 3.2 odd 2
7605.2.a.bk.1.1 2 39.38 odd 2