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Properties

Label 2535.2.a.ba.1.2
Level $2535$
Weight $2$
Character 2535.1
Self dual yes
Analytic conductor $20.242$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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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 1.2
Root \(-0.339877\) 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)
 
\(f(q)\) \(=\) \(q+0.339877 q^{2} -1.00000 q^{3} -1.88448 q^{4} -1.00000 q^{5} -0.339877 q^{6} -0.660123 q^{7} -1.32025 q^{8} +1.00000 q^{9} -0.339877 q^{10} -0.679754 q^{11} +1.88448 q^{12} -0.224361 q^{14} +1.00000 q^{15} +3.32025 q^{16} +7.42909 q^{17} +0.339877 q^{18} -0.115516 q^{19} +1.88448 q^{20} +0.660123 q^{21} -0.231033 q^{22} -7.76897 q^{23} +1.32025 q^{24} +1.00000 q^{25} -1.00000 q^{27} +1.24399 q^{28} +5.54461 q^{29} +0.339877 q^{30} +9.97370 q^{31} +3.76897 q^{32} +0.679754 q^{33} +2.52498 q^{34} +0.660123 q^{35} -1.88448 q^{36} -9.76897 q^{37} -0.0392613 q^{38} +1.32025 q^{40} -4.22436 q^{41} +0.224361 q^{42} -0.544607 q^{43} +1.28098 q^{44} -1.00000 q^{45} -2.64049 q^{46} +5.01963 q^{47} -3.32025 q^{48} -6.56424 q^{49} +0.339877 q^{50} -7.42909 q^{51} +0.679754 q^{53} -0.339877 q^{54} +0.679754 q^{55} +0.871525 q^{56} +0.115516 q^{57} +1.88448 q^{58} -2.22436 q^{59} -1.88448 q^{60} -4.20473 q^{61} +3.38983 q^{62} -0.660123 q^{63} -5.35951 q^{64} +0.231033 q^{66} +7.63382 q^{67} -14.0000 q^{68} +7.76897 q^{69} +0.224361 q^{70} -7.31357 q^{71} -1.32025 q^{72} -8.01963 q^{73} -3.32025 q^{74} -1.00000 q^{75} +0.217689 q^{76} +0.448721 q^{77} +9.97370 q^{79} -3.32025 q^{80} +1.00000 q^{81} -1.43576 q^{82} -1.76897 q^{83} -1.24399 q^{84} -7.42909 q^{85} -0.185099 q^{86} -5.54461 q^{87} +0.897442 q^{88} -13.5446 q^{89} -0.339877 q^{90} +14.6405 q^{92} -9.97370 q^{93} +1.70606 q^{94} +0.115516 q^{95} -3.76897 q^{96} -9.90411 q^{97} -2.23103 q^{98} -0.679754 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 6 q^{4} - 3 q^{5} - 3 q^{7} - 6 q^{8} + 3 q^{9} - 6 q^{12} + 12 q^{14} + 3 q^{15} + 12 q^{16} - 12 q^{19} - 6 q^{20} + 3 q^{21} - 24 q^{22} + 6 q^{24} + 3 q^{25} - 3 q^{27} - 12 q^{28} + 6 q^{29}+ \cdots - 30 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.339877 0.240329 0.120165 − 0.992754i \(-0.461658\pi\)
0.120165 + 0.992754i \(0.461658\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.88448 −0.942242
\(5\) −1.00000 −0.447214
\(6\) −0.339877 −0.138754
\(7\) −0.660123 −0.249503 −0.124752 − 0.992188i \(-0.539813\pi\)
−0.124752 + 0.992188i \(0.539813\pi\)
\(8\) −1.32025 −0.466778
\(9\) 1.00000 0.333333
\(10\) −0.339877 −0.107479
\(11\) −0.679754 −0.204953 −0.102477 − 0.994735i \(-0.532677\pi\)
−0.102477 + 0.994735i \(0.532677\pi\)
\(12\) 1.88448 0.544004
\(13\) 0 0
\(14\) −0.224361 −0.0599629
\(15\) 1.00000 0.258199
\(16\) 3.32025 0.830062
\(17\) 7.42909 1.80182 0.900910 − 0.434007i \(-0.142901\pi\)
0.900910 + 0.434007i \(0.142901\pi\)
\(18\) 0.339877 0.0801098
\(19\) −0.115516 −0.0265013 −0.0132506 − 0.999912i \(-0.504218\pi\)
−0.0132506 + 0.999912i \(0.504218\pi\)
\(20\) 1.88448 0.421383
\(21\) 0.660123 0.144051
\(22\) −0.231033 −0.0492563
\(23\) −7.76897 −1.61994 −0.809971 − 0.586470i \(-0.800517\pi\)
−0.809971 + 0.586470i \(0.800517\pi\)
\(24\) 1.32025 0.269494
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 1.24399 0.235092
\(29\) 5.54461 1.02961 0.514804 − 0.857308i \(-0.327865\pi\)
0.514804 + 0.857308i \(0.327865\pi\)
\(30\) 0.339877 0.0620527
\(31\) 9.97370 1.79133 0.895664 − 0.444730i \(-0.146701\pi\)
0.895664 + 0.444730i \(0.146701\pi\)
\(32\) 3.76897 0.666266
\(33\) 0.679754 0.118330
\(34\) 2.52498 0.433030
\(35\) 0.660123 0.111581
\(36\) −1.88448 −0.314081
\(37\) −9.76897 −1.60601 −0.803004 − 0.595973i \(-0.796766\pi\)
−0.803004 + 0.595973i \(0.796766\pi\)
\(38\) −0.0392613 −0.00636903
\(39\) 0 0
\(40\) 1.32025 0.208749
\(41\) −4.22436 −0.659734 −0.329867 − 0.944027i \(-0.607004\pi\)
−0.329867 + 0.944027i \(0.607004\pi\)
\(42\) 0.224361 0.0346196
\(43\) −0.544607 −0.0830518 −0.0415259 − 0.999137i \(-0.513222\pi\)
−0.0415259 + 0.999137i \(0.513222\pi\)
\(44\) 1.28098 0.193116
\(45\) −1.00000 −0.149071
\(46\) −2.64049 −0.389319
\(47\) 5.01963 0.732188 0.366094 − 0.930578i \(-0.380695\pi\)
0.366094 + 0.930578i \(0.380695\pi\)
\(48\) −3.32025 −0.479236
\(49\) −6.56424 −0.937748
\(50\) 0.339877 0.0480659
\(51\) −7.42909 −1.04028
\(52\) 0 0
\(53\) 0.679754 0.0933714 0.0466857 − 0.998910i \(-0.485134\pi\)
0.0466857 + 0.998910i \(0.485134\pi\)
\(54\) −0.339877 −0.0462514
\(55\) 0.679754 0.0916580
\(56\) 0.871525 0.116462
\(57\) 0.115516 0.0153005
\(58\) 1.88448 0.247445
\(59\) −2.22436 −0.289587 −0.144794 − 0.989462i \(-0.546252\pi\)
−0.144794 + 0.989462i \(0.546252\pi\)
\(60\) −1.88448 −0.243286
\(61\) −4.20473 −0.538361 −0.269180 − 0.963090i \(-0.586753\pi\)
−0.269180 + 0.963090i \(0.586753\pi\)
\(62\) 3.38983 0.430509
\(63\) −0.660123 −0.0831677
\(64\) −5.35951 −0.669938
\(65\) 0 0
\(66\) 0.231033 0.0284381
\(67\) 7.63382 0.932620 0.466310 − 0.884621i \(-0.345583\pi\)
0.466310 + 0.884621i \(0.345583\pi\)
\(68\) −14.0000 −1.69775
\(69\) 7.76897 0.935274
\(70\) 0.224361 0.0268162
\(71\) −7.31357 −0.867962 −0.433981 − 0.900922i \(-0.642891\pi\)
−0.433981 + 0.900922i \(0.642891\pi\)
\(72\) −1.32025 −0.155593
\(73\) −8.01963 −0.938627 −0.469313 − 0.883032i \(-0.655499\pi\)
−0.469313 + 0.883032i \(0.655499\pi\)
\(74\) −3.32025 −0.385971
\(75\) −1.00000 −0.115470
\(76\) 0.217689 0.0249706
\(77\) 0.448721 0.0511365
\(78\) 0 0
\(79\) 9.97370 1.12213 0.561064 − 0.827772i \(-0.310392\pi\)
0.561064 + 0.827772i \(0.310392\pi\)
\(80\) −3.32025 −0.371215
\(81\) 1.00000 0.111111
\(82\) −1.43576 −0.158553
\(83\) −1.76897 −0.194169 −0.0970847 − 0.995276i \(-0.530952\pi\)
−0.0970847 + 0.995276i \(0.530952\pi\)
\(84\) −1.24399 −0.135731
\(85\) −7.42909 −0.805798
\(86\) −0.185099 −0.0199598
\(87\) −5.54461 −0.594444
\(88\) 0.897442 0.0956677
\(89\) −13.5446 −1.43573 −0.717863 − 0.696185i \(-0.754879\pi\)
−0.717863 + 0.696185i \(0.754879\pi\)
\(90\) −0.339877 −0.0358262
\(91\) 0 0
\(92\) 14.6405 1.52638
\(93\) −9.97370 −1.03422
\(94\) 1.70606 0.175966
\(95\) 0.115516 0.0118517
\(96\) −3.76897 −0.384669
\(97\) −9.90411 −1.00561 −0.502805 − 0.864400i \(-0.667699\pi\)
−0.502805 + 0.864400i \(0.667699\pi\)
\(98\) −2.23103 −0.225368
\(99\) −0.679754 −0.0683178
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.1.2 3
3.2 odd 2 7605.2.a.bw.1.2 3
13.4 even 6 195.2.i.d.16.2 ✓ 6
13.10 even 6 195.2.i.d.61.2 yes 6
13.12 even 2 2535.2.a.bb.1.2 3
39.17 odd 6 585.2.j.f.406.2 6
39.23 odd 6 585.2.j.f.451.2 6
39.38 odd 2 7605.2.a.bv.1.2 3
65.4 even 6 975.2.i.l.601.2 6
65.17 odd 12 975.2.bb.k.874.3 12
65.23 odd 12 975.2.bb.k.724.3 12
65.43 odd 12 975.2.bb.k.874.4 12
65.49 even 6 975.2.i.l.451.2 6
65.62 odd 12 975.2.bb.k.724.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.i.d.16.2 ✓ 6 13.4 even 6
195.2.i.d.61.2 yes 6 13.10 even 6
585.2.j.f.406.2 6 39.17 odd 6
585.2.j.f.451.2 6 39.23 odd 6
975.2.i.l.451.2 6 65.49 even 6
975.2.i.l.601.2 6 65.4 even 6
975.2.bb.k.724.3 12 65.23 odd 12
975.2.bb.k.724.4 12 65.62 odd 12
975.2.bb.k.874.3 12 65.17 odd 12
975.2.bb.k.874.4 12 65.43 odd 12
2535.2.a.ba.1.2 3 1.1 even 1 trivial
2535.2.a.bb.1.2 3 13.12 even 2
7605.2.a.bv.1.2 3 39.38 odd 2
7605.2.a.bw.1.2 3 3.2 odd 2