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Properties

Label 1521.4.a.bk.1.5
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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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] = [1521,4,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, 4); 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, 4, names="a")
 
Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,60,0,0,0,0,0,80,0,0,0,60,0,500,-210] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(89.7419051187\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 70x^{8} + 1645x^{6} - 14700x^{4} + 44100x^{2} - 27648 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-0.917374\) 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)
 
\(f(q)\) \(=\) \(q-0.917374 q^{2} -7.15843 q^{4} +15.4704 q^{5} -20.5833 q^{7} +13.9059 q^{8} -14.1922 q^{10} +65.8420 q^{11} +18.8826 q^{14} +44.5105 q^{16} -44.2956 q^{17} +147.053 q^{19} -110.744 q^{20} -60.4017 q^{22} -53.1586 q^{23} +114.334 q^{25} +147.344 q^{28} +38.6257 q^{29} +88.3894 q^{31} -152.080 q^{32} +40.6357 q^{34} -318.433 q^{35} +78.9587 q^{37} -134.903 q^{38} +215.131 q^{40} +354.966 q^{41} -407.846 q^{43} -471.325 q^{44} +48.7663 q^{46} +67.9674 q^{47} +80.6738 q^{49} -104.887 q^{50} -226.572 q^{53} +1018.60 q^{55} -286.231 q^{56} -35.4342 q^{58} -142.031 q^{59} +266.831 q^{61} -81.0862 q^{62} -216.569 q^{64} -411.187 q^{67} +317.087 q^{68} +292.122 q^{70} +91.5052 q^{71} +63.1328 q^{73} -72.4347 q^{74} -1052.67 q^{76} -1355.25 q^{77} -287.115 q^{79} +688.595 q^{80} -325.637 q^{82} +373.812 q^{83} -685.272 q^{85} +374.147 q^{86} +915.595 q^{88} -119.403 q^{89} +380.532 q^{92} -62.3515 q^{94} +2274.97 q^{95} +554.650 q^{97} -74.0080 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 60 q^{4} + 80 q^{10} + 60 q^{14} + 500 q^{16} - 210 q^{17} + 580 q^{22} + 120 q^{23} + 960 q^{25} - 990 q^{29} + 120 q^{35} - 1380 q^{38} + 2000 q^{40} - 740 q^{43} + 1550 q^{49} - 330 q^{53} + 520 q^{55}+ \cdots + 2760 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.917374 −0.324341 −0.162170 − 0.986763i \(-0.551849\pi\)
−0.162170 + 0.986763i \(0.551849\pi\)
\(3\) 0 0
\(4\) −7.15843 −0.894803
\(5\) 15.4704 1.38372 0.691858 − 0.722034i \(-0.256792\pi\)
0.691858 + 0.722034i \(0.256792\pi\)
\(6\) 0 0
\(7\) −20.5833 −1.11140 −0.555698 − 0.831384i \(-0.687549\pi\)
−0.555698 + 0.831384i \(0.687549\pi\)
\(8\) 13.9059 0.614562
\(9\) 0 0
\(10\) −14.1922 −0.448795
\(11\) 65.8420 1.80474 0.902369 − 0.430964i \(-0.141827\pi\)
0.902369 + 0.430964i \(0.141827\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 18.8826 0.360471
\(15\) 0 0
\(16\) 44.5105 0.695476
\(17\) −44.2956 −0.631957 −0.315979 − 0.948766i \(-0.602333\pi\)
−0.315979 + 0.948766i \(0.602333\pi\)
\(18\) 0 0
\(19\) 147.053 1.77560 0.887798 − 0.460234i \(-0.152234\pi\)
0.887798 + 0.460234i \(0.152234\pi\)
\(20\) −110.744 −1.23815
\(21\) 0 0
\(22\) −60.4017 −0.585350
\(23\) −53.1586 −0.481928 −0.240964 − 0.970534i \(-0.577464\pi\)
−0.240964 + 0.970534i \(0.577464\pi\)
\(24\) 0 0
\(25\) 114.334 0.914669
\(26\) 0 0
\(27\) 0 0
\(28\) 147.344 0.994480
\(29\) 38.6257 0.247331 0.123666 − 0.992324i \(-0.460535\pi\)
0.123666 + 0.992324i \(0.460535\pi\)
\(30\) 0 0
\(31\) 88.3894 0.512104 0.256052 − 0.966663i \(-0.417578\pi\)
0.256052 + 0.966663i \(0.417578\pi\)
\(32\) −152.080 −0.840133
\(33\) 0 0
\(34\) 40.6357 0.204969
\(35\) −318.433 −1.53786
\(36\) 0 0
\(37\) 78.9587 0.350831 0.175415 − 0.984495i \(-0.443873\pi\)
0.175415 + 0.984495i \(0.443873\pi\)
\(38\) −134.903 −0.575898
\(39\) 0 0
\(40\) 215.131 0.850379
\(41\) 354.966 1.35211 0.676054 − 0.736852i \(-0.263689\pi\)
0.676054 + 0.736852i \(0.263689\pi\)
\(42\) 0 0
\(43\) −407.846 −1.44642 −0.723208 − 0.690630i \(-0.757333\pi\)
−0.723208 + 0.690630i \(0.757333\pi\)
\(44\) −471.325 −1.61489
\(45\) 0 0
\(46\) 48.7663 0.156309
\(47\) 67.9674 0.210938 0.105469 − 0.994423i \(-0.466366\pi\)
0.105469 + 0.994423i \(0.466366\pi\)
\(48\) 0 0
\(49\) 80.6738 0.235201
\(50\) −104.887 −0.296664
\(51\) 0 0
\(52\) 0 0
\(53\) −226.572 −0.587209 −0.293604 − 0.955927i \(-0.594855\pi\)
−0.293604 + 0.955927i \(0.594855\pi\)
\(54\) 0 0
\(55\) 1018.60 2.49724
\(56\) −286.231 −0.683021
\(57\) 0 0
\(58\) −35.4342 −0.0802196
\(59\) −142.031 −0.313404 −0.156702 − 0.987646i \(-0.550086\pi\)
−0.156702 + 0.987646i \(0.550086\pi\)
\(60\) 0 0
\(61\) 266.831 0.560069 0.280035 − 0.959990i \(-0.409654\pi\)
0.280035 + 0.959990i \(0.409654\pi\)
\(62\) −81.0862 −0.166096
\(63\) 0 0
\(64\) −216.569 −0.422987
\(65\) 0 0
\(66\) 0 0
\(67\) −411.187 −0.749768 −0.374884 − 0.927072i \(-0.622317\pi\)
−0.374884 + 0.927072i \(0.622317\pi\)
\(68\) 317.087 0.565477
\(69\) 0 0
\(70\) 292.122 0.498789
\(71\) 91.5052 0.152953 0.0764765 − 0.997071i \(-0.475633\pi\)
0.0764765 + 0.997071i \(0.475633\pi\)
\(72\) 0 0
\(73\) 63.1328 0.101221 0.0506105 − 0.998718i \(-0.483883\pi\)
0.0506105 + 0.998718i \(0.483883\pi\)
\(74\) −72.4347 −0.113789
\(75\) 0 0
\(76\) −1052.67 −1.58881
\(77\) −1355.25 −2.00578
\(78\) 0 0
\(79\) −287.115 −0.408899 −0.204449 − 0.978877i \(-0.565540\pi\)
−0.204449 + 0.978877i \(0.565540\pi\)
\(80\) 688.595 0.962341
\(81\) 0 0
\(82\) −325.637 −0.438543
\(83\) 373.812 0.494352 0.247176 − 0.968971i \(-0.420497\pi\)
0.247176 + 0.968971i \(0.420497\pi\)
\(84\) 0 0
\(85\) −685.272 −0.874449
\(86\) 374.147 0.469132
\(87\) 0 0
\(88\) 915.595 1.10912
\(89\) −119.403 −0.142209 −0.0711047 − 0.997469i \(-0.522652\pi\)
−0.0711047 + 0.997469i \(0.522652\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 380.532 0.431231
\(93\) 0 0
\(94\) −62.3515 −0.0684157
\(95\) 2274.97 2.45692
\(96\) 0 0
\(97\) 554.650 0.580579 0.290290 − 0.956939i \(-0.406248\pi\)
0.290290 + 0.956939i \(0.406248\pi\)
\(98\) −74.0080 −0.0762851
\(99\) 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 1521.4.a.bk.1.5 10
3.2 odd 2 507.4.a.r.1.6 10
13.6 odd 12 117.4.q.e.10.3 10
13.11 odd 12 117.4.q.e.82.3 10
13.12 even 2 inner 1521.4.a.bk.1.6 10
39.5 even 4 507.4.b.i.337.5 10
39.8 even 4 507.4.b.i.337.6 10
39.11 even 12 39.4.j.c.4.3 ✓ 10
39.32 even 12 39.4.j.c.10.3 yes 10
39.38 odd 2 507.4.a.r.1.5 10
156.11 odd 12 624.4.bv.h.433.2 10
156.71 odd 12 624.4.bv.h.49.4 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.c.4.3 ✓ 10 39.11 even 12
39.4.j.c.10.3 yes 10 39.32 even 12
117.4.q.e.10.3 10 13.6 odd 12
117.4.q.e.82.3 10 13.11 odd 12
507.4.a.r.1.5 10 39.38 odd 2
507.4.a.r.1.6 10 3.2 odd 2
507.4.b.i.337.5 10 39.5 even 4
507.4.b.i.337.6 10 39.8 even 4
624.4.bv.h.49.4 10 156.71 odd 12
624.4.bv.h.433.2 10 156.11 odd 12
1521.4.a.bk.1.5 10 1.1 even 1 trivial
1521.4.a.bk.1.6 10 13.12 even 2 inner