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Properties

Label 1521.4.a.bk.1.1
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 1.1
Root \(-5.36472\) 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)
 
\(f(q)\) \(=\) \(q-5.36472 q^{2} +20.7803 q^{4} +2.69631 q^{5} -15.2025 q^{7} -68.5626 q^{8} -14.4650 q^{10} -66.8848 q^{11} +81.5570 q^{14} +201.577 q^{16} -4.16354 q^{17} -26.0850 q^{19} +56.0301 q^{20} +358.819 q^{22} -47.3242 q^{23} -117.730 q^{25} -315.911 q^{28} -257.007 q^{29} -206.242 q^{31} -532.906 q^{32} +22.3362 q^{34} -40.9906 q^{35} -175.686 q^{37} +139.939 q^{38} -184.866 q^{40} +156.463 q^{41} +51.9845 q^{43} -1389.88 q^{44} +253.881 q^{46} -354.222 q^{47} -111.885 q^{49} +631.588 q^{50} +10.4723 q^{53} -180.342 q^{55} +1042.32 q^{56} +1378.77 q^{58} +445.114 q^{59} +119.696 q^{61} +1106.43 q^{62} +1246.28 q^{64} +22.4078 q^{67} -86.5195 q^{68} +219.903 q^{70} +285.207 q^{71} -740.989 q^{73} +942.507 q^{74} -542.053 q^{76} +1016.81 q^{77} -547.679 q^{79} +543.516 q^{80} -839.378 q^{82} +603.056 q^{83} -11.2262 q^{85} -278.882 q^{86} +4585.80 q^{88} +215.668 q^{89} -983.409 q^{92} +1900.31 q^{94} -70.3333 q^{95} -1447.50 q^{97} +600.233 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 60 q^{4} + 80 q^{10} + 60 q^{14} + 500 q^{16} - 210 q^{17} + 580 q^{22} + 120 q^{23} + 960 q^{25} - 990 q^{29} + 120 q^{35} - 1380 q^{38} + 2000 q^{40} - 740 q^{43} + 1550 q^{49} - 330 q^{53} + 520 q^{55}+ \cdots + 2760 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.36472 −1.89672 −0.948358 − 0.317201i \(-0.897257\pi\)
−0.948358 + 0.317201i \(0.897257\pi\)
\(3\) 0 0
\(4\) 20.7803 2.59753
\(5\) 2.69631 0.241165 0.120583 − 0.992703i \(-0.461524\pi\)
0.120583 + 0.992703i \(0.461524\pi\)
\(6\) 0 0
\(7\) −15.2025 −0.820856 −0.410428 − 0.911893i \(-0.634621\pi\)
−0.410428 + 0.911893i \(0.634621\pi\)
\(8\) −68.5626 −3.03007
\(9\) 0 0
\(10\) −14.4650 −0.457423
\(11\) −66.8848 −1.83332 −0.916661 − 0.399666i \(-0.869126\pi\)
−0.916661 + 0.399666i \(0.869126\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 81.5570 1.55693
\(15\) 0 0
\(16\) 201.577 3.14965
\(17\) −4.16354 −0.0594004 −0.0297002 − 0.999559i \(-0.509455\pi\)
−0.0297002 + 0.999559i \(0.509455\pi\)
\(18\) 0 0
\(19\) −26.0850 −0.314963 −0.157482 − 0.987522i \(-0.550338\pi\)
−0.157482 + 0.987522i \(0.550338\pi\)
\(20\) 56.0301 0.626436
\(21\) 0 0
\(22\) 358.819 3.47729
\(23\) −47.3242 −0.429034 −0.214517 − 0.976720i \(-0.568818\pi\)
−0.214517 + 0.976720i \(0.568818\pi\)
\(24\) 0 0
\(25\) −117.730 −0.941839
\(26\) 0 0
\(27\) 0 0
\(28\) −315.911 −2.13220
\(29\) −257.007 −1.64569 −0.822845 − 0.568266i \(-0.807614\pi\)
−0.822845 + 0.568266i \(0.807614\pi\)
\(30\) 0 0
\(31\) −206.242 −1.19491 −0.597455 − 0.801903i \(-0.703821\pi\)
−0.597455 + 0.801903i \(0.703821\pi\)
\(32\) −532.906 −2.94392
\(33\) 0 0
\(34\) 22.3362 0.112666
\(35\) −40.9906 −0.197962
\(36\) 0 0
\(37\) −175.686 −0.780611 −0.390305 − 0.920685i \(-0.627631\pi\)
−0.390305 + 0.920685i \(0.627631\pi\)
\(38\) 139.939 0.597396
\(39\) 0 0
\(40\) −184.866 −0.730748
\(41\) 156.463 0.595984 0.297992 − 0.954568i \(-0.403683\pi\)
0.297992 + 0.954568i \(0.403683\pi\)
\(42\) 0 0
\(43\) 51.9845 0.184362 0.0921809 − 0.995742i \(-0.470616\pi\)
0.0921809 + 0.995742i \(0.470616\pi\)
\(44\) −1389.88 −4.76211
\(45\) 0 0
\(46\) 253.881 0.813755
\(47\) −354.222 −1.09933 −0.549666 − 0.835384i \(-0.685245\pi\)
−0.549666 + 0.835384i \(0.685245\pi\)
\(48\) 0 0
\(49\) −111.885 −0.326196
\(50\) 631.588 1.78640
\(51\) 0 0
\(52\) 0 0
\(53\) 10.4723 0.0271412 0.0135706 − 0.999908i \(-0.495680\pi\)
0.0135706 + 0.999908i \(0.495680\pi\)
\(54\) 0 0
\(55\) −180.342 −0.442134
\(56\) 1042.32 2.48725
\(57\) 0 0
\(58\) 1378.77 3.12141
\(59\) 445.114 0.982185 0.491092 − 0.871107i \(-0.336598\pi\)
0.491092 + 0.871107i \(0.336598\pi\)
\(60\) 0 0
\(61\) 119.696 0.251238 0.125619 − 0.992079i \(-0.459908\pi\)
0.125619 + 0.992079i \(0.459908\pi\)
\(62\) 1106.43 2.26640
\(63\) 0 0
\(64\) 1246.28 2.43413
\(65\) 0 0
\(66\) 0 0
\(67\) 22.4078 0.0408589 0.0204294 − 0.999791i \(-0.493497\pi\)
0.0204294 + 0.999791i \(0.493497\pi\)
\(68\) −86.5195 −0.154295
\(69\) 0 0
\(70\) 219.903 0.375478
\(71\) 285.207 0.476731 0.238365 − 0.971176i \(-0.423388\pi\)
0.238365 + 0.971176i \(0.423388\pi\)
\(72\) 0 0
\(73\) −740.989 −1.18803 −0.594015 − 0.804454i \(-0.702458\pi\)
−0.594015 + 0.804454i \(0.702458\pi\)
\(74\) 942.507 1.48060
\(75\) 0 0
\(76\) −542.053 −0.818128
\(77\) 1016.81 1.50489
\(78\) 0 0
\(79\) −547.679 −0.779983 −0.389992 − 0.920818i \(-0.627522\pi\)
−0.389992 + 0.920818i \(0.627522\pi\)
\(80\) 543.516 0.759586
\(81\) 0 0
\(82\) −839.378 −1.13041
\(83\) 603.056 0.797518 0.398759 − 0.917056i \(-0.369441\pi\)
0.398759 + 0.917056i \(0.369441\pi\)
\(84\) 0 0
\(85\) −11.2262 −0.0143253
\(86\) −278.882 −0.349682
\(87\) 0 0
\(88\) 4585.80 5.55509
\(89\) 215.668 0.256863 0.128431 − 0.991718i \(-0.459006\pi\)
0.128431 + 0.991718i \(0.459006\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −983.409 −1.11443
\(93\) 0 0
\(94\) 1900.31 2.08512
\(95\) −70.3333 −0.0759583
\(96\) 0 0
\(97\) −1447.50 −1.51517 −0.757586 − 0.652735i \(-0.773622\pi\)
−0.757586 + 0.652735i \(0.773622\pi\)
\(98\) 600.233 0.618700
\(99\) 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.1.1 10
3.2 odd 2 507.4.a.r.1.10 10
13.6 odd 12 117.4.q.e.10.5 10
13.11 odd 12 117.4.q.e.82.5 10
13.12 even 2 inner 1521.4.a.bk.1.10 10
39.5 even 4 507.4.b.i.337.1 10
39.8 even 4 507.4.b.i.337.10 10
39.11 even 12 39.4.j.c.4.1 ✓ 10
39.32 even 12 39.4.j.c.10.1 yes 10
39.38 odd 2 507.4.a.r.1.1 10
156.11 odd 12 624.4.bv.h.433.3 10
156.71 odd 12 624.4.bv.h.49.3 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.j.c.4.1 ✓ 10 39.11 even 12
39.4.j.c.10.1 yes 10 39.32 even 12
117.4.q.e.10.5 10 13.6 odd 12
117.4.q.e.82.5 10 13.11 odd 12
507.4.a.r.1.1 10 39.38 odd 2
507.4.a.r.1.10 10 3.2 odd 2
507.4.b.i.337.1 10 39.5 even 4
507.4.b.i.337.10 10 39.8 even 4
624.4.bv.h.49.3 10 156.71 odd 12
624.4.bv.h.433.3 10 156.11 odd 12
1521.4.a.bk.1.1 10 1.1 even 1 trivial
1521.4.a.bk.1.10 10 13.12 even 2 inner