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Properties

Label 1521.2.a.o.1.3
Level $1521$
Weight $2$
Character 1521.1
Self dual yes
Analytic conductor $12.145$
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] = [1521,2,Mod(1,1521)] 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("1521.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1521, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,-2,0,0,-4,0,3,-3,0,5,-8,0,0,5,0,2,2,0,4,0,0,3,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(12.1452461474\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 169)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.445042\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.801938 q^{2} -1.35690 q^{4} +0.246980 q^{5} +2.35690 q^{7} -2.69202 q^{8} +0.198062 q^{10} -4.24698 q^{11} +1.89008 q^{14} +0.554958 q^{16} -2.15883 q^{17} +0.0881460 q^{19} -0.335126 q^{20} -3.40581 q^{22} -1.49396 q^{23} -4.93900 q^{25} -3.19806 q^{28} -4.63102 q^{29} +6.63102 q^{31} +5.82908 q^{32} -1.73125 q^{34} +0.582105 q^{35} -5.69202 q^{37} +0.0706876 q^{38} -0.664874 q^{40} -11.5918 q^{41} -0.295897 q^{43} +5.76271 q^{44} -1.19806 q^{46} -7.35690 q^{47} -1.44504 q^{49} -3.96077 q^{50} +10.3937 q^{53} -1.04892 q^{55} -6.34481 q^{56} -3.71379 q^{58} -6.78017 q^{59} +3.47219 q^{61} +5.31767 q^{62} +3.56465 q^{64} -7.67994 q^{67} +2.92931 q^{68} +0.466812 q^{70} -8.66487 q^{71} -6.73556 q^{73} -4.56465 q^{74} -0.119605 q^{76} -10.0097 q^{77} +9.97046 q^{79} +0.137063 q^{80} -9.29590 q^{82} +1.60925 q^{83} -0.533188 q^{85} -0.237291 q^{86} +11.4330 q^{88} -2.88471 q^{89} +2.02715 q^{92} -5.89977 q^{94} +0.0217703 q^{95} +8.05861 q^{97} -1.15883 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} - 4 q^{5} + 3 q^{7} - 3 q^{8} + 5 q^{10} - 8 q^{11} + 5 q^{14} + 2 q^{16} + 2 q^{17} + 4 q^{19} + 3 q^{22} + 5 q^{23} - 5 q^{25} - 14 q^{28} + q^{29} + 5 q^{31} + 7 q^{32} - 13 q^{34} - 4 q^{35}+ \cdots + 5 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.801938 0.567056 0.283528 − 0.958964i \(-0.408495\pi\)
0.283528 + 0.958964i \(0.408495\pi\)
\(3\) 0 0
\(4\) −1.35690 −0.678448
\(5\) 0.246980 0.110453 0.0552263 − 0.998474i \(-0.482412\pi\)
0.0552263 + 0.998474i \(0.482412\pi\)
\(6\) 0 0
\(7\) 2.35690 0.890823 0.445411 − 0.895326i \(-0.353057\pi\)
0.445411 + 0.895326i \(0.353057\pi\)
\(8\) −2.69202 −0.951773
\(9\) 0 0
\(10\) 0.198062 0.0626328
\(11\) −4.24698 −1.28051 −0.640256 − 0.768161i \(-0.721172\pi\)
−0.640256 + 0.768161i \(0.721172\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 1.89008 0.505146
\(15\) 0 0
\(16\) 0.554958 0.138740
\(17\) −2.15883 −0.523594 −0.261797 − 0.965123i \(-0.584315\pi\)
−0.261797 + 0.965123i \(0.584315\pi\)
\(18\) 0 0
\(19\) 0.0881460 0.0202221 0.0101110 − 0.999949i \(-0.496782\pi\)
0.0101110 + 0.999949i \(0.496782\pi\)
\(20\) −0.335126 −0.0749364
\(21\) 0 0
\(22\) −3.40581 −0.726122
\(23\) −1.49396 −0.311512 −0.155756 − 0.987796i \(-0.549781\pi\)
−0.155756 + 0.987796i \(0.549781\pi\)
\(24\) 0 0
\(25\) −4.93900 −0.987800
\(26\) 0 0
\(27\) 0 0
\(28\) −3.19806 −0.604377
\(29\) −4.63102 −0.859959 −0.429980 − 0.902839i \(-0.641479\pi\)
−0.429980 + 0.902839i \(0.641479\pi\)
\(30\) 0 0
\(31\) 6.63102 1.19097 0.595483 − 0.803368i \(-0.296961\pi\)
0.595483 + 0.803368i \(0.296961\pi\)
\(32\) 5.82908 1.03045
\(33\) 0 0
\(34\) −1.73125 −0.296907
\(35\) 0.582105 0.0983937
\(36\) 0 0
\(37\) −5.69202 −0.935763 −0.467881 − 0.883791i \(-0.654983\pi\)
−0.467881 + 0.883791i \(0.654983\pi\)
\(38\) 0.0706876 0.0114670
\(39\) 0 0
\(40\) −0.664874 −0.105126
\(41\) −11.5918 −1.81033 −0.905167 − 0.425056i \(-0.860254\pi\)
−0.905167 + 0.425056i \(0.860254\pi\)
\(42\) 0 0
\(43\) −0.295897 −0.0451239 −0.0225619 − 0.999745i \(-0.507182\pi\)
−0.0225619 + 0.999745i \(0.507182\pi\)
\(44\) 5.76271 0.868761
\(45\) 0 0
\(46\) −1.19806 −0.176645
\(47\) −7.35690 −1.07311 −0.536557 − 0.843864i \(-0.680275\pi\)
−0.536557 + 0.843864i \(0.680275\pi\)
\(48\) 0 0
\(49\) −1.44504 −0.206435
\(50\) −3.96077 −0.560138
\(51\) 0 0
\(52\) 0 0
\(53\) 10.3937 1.42769 0.713844 − 0.700304i \(-0.246952\pi\)
0.713844 + 0.700304i \(0.246952\pi\)
\(54\) 0 0
\(55\) −1.04892 −0.141436
\(56\) −6.34481 −0.847861
\(57\) 0 0
\(58\) −3.71379 −0.487645
\(59\) −6.78017 −0.882703 −0.441351 − 0.897334i \(-0.645501\pi\)
−0.441351 + 0.897334i \(0.645501\pi\)
\(60\) 0 0
\(61\) 3.47219 0.444568 0.222284 − 0.974982i \(-0.428649\pi\)
0.222284 + 0.974982i \(0.428649\pi\)
\(62\) 5.31767 0.675344
\(63\) 0 0
\(64\) 3.56465 0.445581
\(65\) 0 0
\(66\) 0 0
\(67\) −7.67994 −0.938254 −0.469127 − 0.883131i \(-0.655431\pi\)
−0.469127 + 0.883131i \(0.655431\pi\)
\(68\) 2.92931 0.355231
\(69\) 0 0
\(70\) 0.466812 0.0557947
\(71\) −8.66487 −1.02833 −0.514166 − 0.857691i \(-0.671898\pi\)
−0.514166 + 0.857691i \(0.671898\pi\)
\(72\) 0 0
\(73\) −6.73556 −0.788338 −0.394169 − 0.919038i \(-0.628968\pi\)
−0.394169 + 0.919038i \(0.628968\pi\)
\(74\) −4.56465 −0.530629
\(75\) 0 0
\(76\) −0.119605 −0.0137196
\(77\) −10.0097 −1.14071
\(78\) 0 0
\(79\) 9.97046 1.12176 0.560882 − 0.827896i \(-0.310462\pi\)
0.560882 + 0.827896i \(0.310462\pi\)
\(80\) 0.137063 0.0153241
\(81\) 0 0
\(82\) −9.29590 −1.02656
\(83\) 1.60925 0.176638 0.0883192 − 0.996092i \(-0.471850\pi\)
0.0883192 + 0.996092i \(0.471850\pi\)
\(84\) 0 0
\(85\) −0.533188 −0.0578323
\(86\) −0.237291 −0.0255877
\(87\) 0 0
\(88\) 11.4330 1.21876
\(89\) −2.88471 −0.305778 −0.152889 − 0.988243i \(-0.548858\pi\)
−0.152889 + 0.988243i \(0.548858\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.02715 0.211345
\(93\) 0 0
\(94\) −5.89977 −0.608515
\(95\) 0.0217703 0.00223358
\(96\) 0 0
\(97\) 8.05861 0.818227 0.409114 − 0.912483i \(-0.365838\pi\)
0.409114 + 0.912483i \(0.365838\pi\)
\(98\) −1.15883 −0.117060
\(99\) 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 1521.2.a.o.1.3 3
3.2 odd 2 169.2.a.c.1.1 yes 3
12.11 even 2 2704.2.a.ba.1.3 3
13.5 odd 4 1521.2.b.l.1351.2 6
13.8 odd 4 1521.2.b.l.1351.5 6
13.12 even 2 1521.2.a.r.1.1 3
15.14 odd 2 4225.2.a.bb.1.3 3
21.20 even 2 8281.2.a.bj.1.1 3
39.2 even 12 169.2.e.b.147.2 12
39.5 even 4 169.2.b.b.168.5 6
39.8 even 4 169.2.b.b.168.2 6
39.11 even 12 169.2.e.b.147.5 12
39.17 odd 6 169.2.c.c.146.1 6
39.20 even 12 169.2.e.b.23.2 12
39.23 odd 6 169.2.c.c.22.1 6
39.29 odd 6 169.2.c.b.22.3 6
39.32 even 12 169.2.e.b.23.5 12
39.35 odd 6 169.2.c.b.146.3 6
39.38 odd 2 169.2.a.b.1.3 ✓ 3
156.47 odd 4 2704.2.f.o.337.5 6
156.83 odd 4 2704.2.f.o.337.6 6
156.155 even 2 2704.2.a.z.1.3 3
195.194 odd 2 4225.2.a.bg.1.1 3
273.272 even 2 8281.2.a.bf.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.3 ✓ 3 39.38 odd 2
169.2.a.c.1.1 yes 3 3.2 odd 2
169.2.b.b.168.2 6 39.8 even 4
169.2.b.b.168.5 6 39.5 even 4
169.2.c.b.22.3 6 39.29 odd 6
169.2.c.b.146.3 6 39.35 odd 6
169.2.c.c.22.1 6 39.23 odd 6
169.2.c.c.146.1 6 39.17 odd 6
169.2.e.b.23.2 12 39.20 even 12
169.2.e.b.23.5 12 39.32 even 12
169.2.e.b.147.2 12 39.2 even 12
169.2.e.b.147.5 12 39.11 even 12
1521.2.a.o.1.3 3 1.1 even 1 trivial
1521.2.a.r.1.1 3 13.12 even 2
1521.2.b.l.1351.2 6 13.5 odd 4
1521.2.b.l.1351.5 6 13.8 odd 4
2704.2.a.z.1.3 3 156.155 even 2
2704.2.a.ba.1.3 3 12.11 even 2
2704.2.f.o.337.5 6 156.47 odd 4
2704.2.f.o.337.6 6 156.83 odd 4
4225.2.a.bb.1.3 3 15.14 odd 2
4225.2.a.bg.1.1 3 195.194 odd 2
8281.2.a.bf.1.3 3 273.272 even 2
8281.2.a.bj.1.1 3 21.20 even 2